AN ABSTRACT OF THE THESIS OF
Karl Juhani Keipi
in
for the degree of Doctor of Philosophy
Forest Management
Title:
presented on
December 8, 1975
TRANSFER PRICING ALTERNATIVES FOR ALLOCATING LOGS IN A
FOREST PRODUCTS FIRM
Abstract approved:
(J
uuriit n.
The primary objective of this study is to test the hypothesis that
the transfer prices of the corporate resource, logs, should be their
competitive market prices.
This is what economic theory suggest.
The
study tests it in short term log allocation decision making of a
typical, decentralized Pacific Northwest forest products firm.
In a decentralized firm, the milling division and timber division
are profit centers.
The headquarters sets transfer prices and lets
the milling and timber divisions determine the allocation of logs.
The goal is to maximize the corporate profit which is the sum of the
divisional profits.
The transfer prices should be such that the
corporate profit is as close as possible to the globally maximum
corporate profit of a centralized firm under perfect knowledge.
Three
cases of decentralized organizations are studied:
1)
mill dominance where the milling division dictates the
quantity and quality of harvested logs to be delivered
internally and has the right to purchase logs from the
outside market; and the timber division harvests the required
logs and has the right to sell harvested logs to the outside
market;
timber dominance where the timber division dictates, the
quality and possibly the quantity of logs to be delivered
internally and has the right to buy logs from and sell harvest-'
ed logs to the outside market; and the milling division may
determine their total log demands accepting the quality of
supplied logs;
'mixed dominance where the milling division dictates the
quantity and quality of internally delivered logs; and the
timber division either harvests the' required logs or purchases
them from the outside market, and has the right to sell
harvested logs to the outside market.
The results of the case study show.that the hypothesis of economic
theory does not always hold.
ly the market price.
structure.
The best transfer price is not necessari-
It depends primarily on the organizational
Under timber dominance the firm's best transfer prices
'always follow the harvesting costs, not market prices.
Under mill and
mixed dominances the best transfer prices are based on log market
prices, with the transfer prices of individual log classes following
their relative market prices.
However, the best transfer price level
may be equal .to, above or below the market price depending on the
amount of logs available through harvest.
Under mill dominance the
best transfer price is below market price when the firm has a good or
fair amount (above 80 per cent of the mills' total log need) of logs
available from harvest.
When the harvestable logs are particularly
scarce, the best transfer price is above the market price.
Under mixed
dominance the best transfer price is market price when the firm has
good or fair amount of logs available from harvest.
a
When the harvest-
able logs are particularly scarce the best transfer price is above the
market price.
Mixed dominance is probably the most common of the three
organizations among forest products firms.
An average firm has good or
fair amount (above 80 per cent of the mills' total log need) of logs
available from harvest.
Therefore, for a typical Pacific Northwest
forest products firm the hypothesis does hold.
The above observations are valid when the transfer price decisions
are made under certainty: the prices, costs and production functions
are known to the decision makers.
But the results of the case study
linear programming calculations suggest that even under uncertainty
market price-based transfer prices bring the highest expected corporate
profit in a mixed or mill dominant firm and cost-based prices in a
timber dominant firm.
In a mixed dominance firm the expected market
price, in a timber dominant firm the expected harvesting cost is
probably the best transfer price level.
In a mill dominant firm the
best transfer price level may be above, equal to or less than the
expected market price.
TRANSFER PRICING ALTERNATIVES FOR ALLOCATING LOGS
IN A FOREST PRODUCTS FIRM
by
Kari Juhani Keipi
A THESIS
submitted to
Oregon State University
in partial fulfillrneht of
the requirements for the
degree of
Doctor of Philosophy
June
1976
AC KNOt4L EDG EM ENT
I wish to thank all who helped me in the formulation of the hazy
ideas into clear objectives of this study.
lam grateful to Dr. Norm
Johnson,.Dr. Bill McKillop, Tom Stoffleand Dennis Dykstra for the
many discussions to initiate the research.
I thank Dr. Beuter, my
major professor arid Dr. Halter for their good counsel in every step.
Many practical men helped me first in developing the project framework
and thenprovided data for the calculations.
Of them I should
especially mention Larry Chapman of Bohemia, Inc., Ted Nelson of Weyerhaeuser Co., Adam Ferrie of McMillan Bloedel
Crown Zellerbach Co.
Ltd., and Don Baack of
My stay at Oregon State University was made
financially possible by a generous scholarship of W.K. Kellogg
Foundation and a research assistantship granted by the School of
Forestry.
Special thanks belong to Gil-Won Sona, my faithful friend, who
encouraged and helped me in completing the theses.
Greg Grimes, my
collaborator of over half a year, developed the computer routines.
II thank the international friends at Oregon State University with whom
I have shared my life starting with wilderness experiences and extending
to every area of mind and spirit.
r
1
TABLE OF CONTENTS
page
INTRODUCTION
1.1.
1
BACKGROUND AND SCOPE
1
Decentralization
1
Divisional Dominance
2
Log Allocation Problem
5
Transfer Pricing Problem
7
1.2; OBJECTIVES
10
TRANSFER PRICING ALTERNATIVES
12
Cost Pricing
12
Market Pricing
13
Value Pricing
15
Negotiated Transfer Prices
15
Multiple Transfer Prices
16
THEORETICALLY OPTIMAL TRANSFER PRICE
18
3.1.
3.2.
OPTIMAL TRANSFER PRICE WHEN THERE IS NO OR A
NONCOMPETITIVE OUTSIDE LOG MARKET
18
Derived Internal Demand
18
Marginal Harvesting Cost
20
Negotiated Transfer Price Under Bilateral Monopoly
23
Optimal Transfer Price When There Is No Outside
Log Market
24
Optimal Transfer Price When There Is A NonCompetitive Outside Log Market
27
OPTIMAL TRANSFER PRICE WHEN THERE IS A COMPETITIVE
OUTSIDE LOG MARKET
30
Optimal Solution rn A Centralized Firm
30
11
Mill Dominance
32
Timber Dominance
35
Mixed Dominance
37
"Market Price Rule" of Transfer Pricing
37
Optimal Transfer Price For A Firm With Several Milling
Divisions
39
Optimal Transfer Price For A Firm With Several Timber
Divisions
3.3.
OPTIMAL TRANSFER PRICES FOR SEVERAL LOG CLASSES
43
Diagrams
44
Calculus
45
Mathematical Programming
46
Shadow Prices As Transfer Prices
46
Shadow Prices Plus Variable Costs As Transfer Prices
47
General Rule For Transfer Pricing?
4.
41
EXPERIMENT FOR TESTING THE HYPOTHESIS OF THE THEORY
48
50
4.1.
CASE FIRM
50
4.2.
PROBLEM FORMULATION
53
Centralization
53
Divisional Dominances
54
Solution Techniques
61
PROBLEM SOLVING
63
Objectives
63
Steps of Problem Solving
64
RESULTS
70
Harvesting Decision
71
Optimal Types of Transfer Pricing
74
4.3.
4.4.
111
Mill Dominance
74
Timber Dominance
77
Mixed Dominance
81
Best Levels of Transfer Prices
82.
Mill Dominance
82
Timber Dominance
87
Mixed Dominance
89
Best Divisional Dominance
91
Conclusion
94
OPTIMAL TRANSFER PRICE ANALYSIS UNDER PRICE UNCERTAINTY
95
5.1. OBJECTIVES OF THE ANALYSIS
95
5.2. STAGES OF THE ANALYSIS
98
Actions
99
Sawntimber Price States and Their Probabilities
99
Veneer and Log Market Price States and Their Probabilities
5.3. OUTCOMES
101
105
Corporate Profit Outcomes
105
Utility Outcomes
106
DISCUSSION
110
Practice
110
Theory
110
Experiment
111
Comparison Between Experiment, Theory and Practice
111
Uncertainty
113
Extensions
114
BIBLIOGRAPHY
117
iv
Appendix 1.
LINEAR PROGRAM FORMULATIONS
124
1.1.
Corporate Program
124
1.2.
Divisional Programs Under Dominance Organizations
131
Mill Dominance
131
Timber Dominance
135
Mixed Dominance
139
Appendix 2.
CASE FIRM DATA FOR THE LINEAR PROGRAMS
144
2.1.
Timber Data
144
2.2.
Milling Data
150
2.3.
User Cost Computations
159
Appendix 3.
LINEAR PROGRAMMING COMPUTATIONS FOR DOMINANCE
ALLOCATIONS
171
3.1.
Simplex Input Computer Flowcharts
171
3.2.
Examples of Computer Inputs and Outputs
177
Appendix 4.
ALTERNATIVES TO DIVISIONAL DOMINANCE APPROACH FOR
ALLOCATING LOGS IN A DECENTALIZED FIRM
188
4.1.
Shadow Price Allocation
190
4.2.
Decomposition Allocation
193
Equal Milling and Timber Division
193
Two Milling Divisions and a Dominated Timber
Division
4.3.
Price Adjustment Allocation
Appendix 5.
MULTIPERIOD LINEAR PROGRAM FORMULATIONS
Appendix 6.
JOINTLY OPTIMAL TRANSFER PRICES FOR DIVISIONAL
MANAGERS:
A SPECIAL CASE
198
202
206
214
V
LIST OF FIGURES
Figure
rage
1.1
Log allocation problem
3.l.a
Internal demand for logs
19
3,1 .b
Marginal harvesting cost
19
3.2.
Log allocation, upper and lower bounds of negotiated
transfer prices under bilateral monopoly
19
Log allocation and divisional profits when transfer
price equals to, is greater than or smaller than the
equilibrium price
25
Log allocation, internal and external log pricing
when there is a competitive outside log market, in
a price discrimination situation
25
3.3.a-c
3.4.
3.5.a-b
3.6.a-b
3.7.a-b
3.8.a-b
3 .9. a -d
3.lO.a-c
6
Log allocation and divisional profits under mill and
mixed dominances whentransferprice equals to
market price and is greater than or less than internal
equilibrium price
31
Log allocation and divisional profits under timber
dominance when transfer price equals to market price
and is greater than or less than internal equilibrium
price.
Timber division alone dictates the quantity
of logs to be used by the milling division
31
Examples of log allocation and divisional profits
under mill dominance when transfer price is greater
than or less than market price
33
Examples of log allocation and divisional profits
under timber dominance when transfer price is greater
thanor less than market price
33
Examples of log allocations and divisional profits
under mixed dominance when transfer price is greater
than or less than market price
33
Examples of log allocation and veneer, sawmill and
timber division profits under mixed dominance when
vi
transfer price is equal to, greater than or less
than market price
40
Log allocation and divisional profits of milling
divisions and two timber divisions under mixed
dominance when transfer price is market price
40
4.1
Case firm log allocation problem
51
4.2.a-d
Information and log flows in centralized organization,
under mill, timber and mixed dominance
55
Six divisional allocation problems and data needed for
solving them
59
Corporate profit losses from decentralization with
alternative transfer pricing types under Tnill, timber
and mixed dominance
80
3.11
4.3
4.4.a-c
4.5.a-c
Sensitivity of corporate profit losses from decentralization to changesin market-based transfer pricelevels
under mill dominance and mixed dominance, and costbased transfer price levels under timber dominance
when total harvest exceeds mills' total log need
83
4.6.a-c
Sensitivity of corporate profit losses from decentralization to changes in market-based transfer price
levels under mill dominance and mixed dominance, and
cost-based transfer price levels under timber dominance
when total harvest is much less than mills' total
log need
85
5.1
Examples of utility functions of a risk averse and
risk taking top management
107
Examples of two utility functions of a partially
risk averse, partially risk taking top management
107
A.2.i
User cost computations
160
A.2.2
Flow chart of subroutine SWITCH
165
A.3.l
Simplex input computer routines
172
A.3.2
Flow chart of
FORMAT PROGRAM for the milling division
174
A.3.3
Flow chart of
FORMAT PROGRAM for the timber division
175
5.2
vii
A.4.1.a-b
A.4.2
A.5.l
A.5.2.a-b
A.5.3
Information and log flows of Dantzig-Wolfe
decomposition algorithm organization of equal
milling and timber divisions, and of two
milling divisions and a dominated timber division
195
Information and log flows of price adjustment
organization
203
A boundary of milling and timber division profits
of several transfer prices, and a Jower milling
division profit acceptance level
215
A lexicographic utility function of the milling
division and corporate profits and the partitioning of several transfer prices to acceptable and
nonacceptable
215
Exponential milling division, timber division and
corporate utility functions for profits
215
viii
LIST Of TABLES
Table
4.1
4.2
Page
Results of the harvesting cost computations: Choosing
the nine tracts with the lowest harvesting cost to the
linear programs
72
Profits with alternative transfer pricing types under
mill dominance
75
4.3. Summary of log allocation toiiiills with alternative
transfer prices under mill dominance
4.4
4.5
4.6
5.1
5.2
5.3
76
Profits with alternative transfer pricing types under
timber dominance
78
Profits with alternative transfer pricing types under
mtxed dominance
79
Best range for transfer prices for the case firm as a
functIon of harvest level and divisional dominance,
and suggestion of economic theory for transfer prices
92
Derivation of a combined distribution from subjective
and objective veneer price probabilities
102.
Derivation of a combined distribution from subjective
and objective log market price probabilities
102
Profits of two transfer price alternatives under
price uncertainty in mill dominance organization
104
A.2.i
End product prices
152
A.2.2
End product recoveries
154
TRANSFER PRICING ALTERNATIVES FOR ALLOCATING LOGS IN A
FOREST PRODUCTS FIRM
INTRODUCTION
1.
1.1.
BACKGROUND AND SCOPE
Decentral ization
Transfer pricing is a requirement that arises from interactions
between decentralized sub-units.
In a decentralized finii, inter-
mediate products are transferred from the supplying division to the
user division.
Management literature abounds with reasoning why
the top management might want to choose decentralization (cf. Argyris,
p
239;
Arrow
(a) pp. 11-12; Arrow (b), p. 400; Baron, pp. 163-5;
Dopuch and Drake, p. 1; Hirschleifer (c), p. 29; Marschak, p. 400;
Martin, p. 94; Morris, pp. 18-22; Shillinglaw, p. 149; Whinston,
p. 418).
Horngren (p. 693) summarizes the benefits of decentraliza-
tion as improving division managers' incentives, innovations and
motivation in better daily decisions; facilitating fast allocation
adjustments to changes in market conditions; providing more time to
top management for strategic planning; training the division managers in independent decision making.
Decentralization facilitates
meaningful corporate resource allocation even when the headquarters
has at most sketchy knowledge about production technology.
Cor-
respondingly, the major disadvantage of decentralization is dysfunctional decision making due to lack of congruence between top man-
agement and divisional goals or due to lack of information.
Another
2
disadvantage is the increased costs of information gathering (Horngren, p. 694).
The divisions of a decentralized firm are profit centers-somewhat independent responsibility units.
two problems:
The top management faces
(1) how to divide responsibilities and (2) how to
coordinate the divisions.
The delegation of responsibilities appears
in the organizational structure of the firm.
In this study, transfer
prices serve for coordination of the allocation of the intermediate
products.
Total decentralization means minimum constraints and
maximum freedom for the divisional decision making.
Full centraliza-
tion means maximum constraints and minimum freedom.
Divisional Dominance
In a decentralized firm it is of crucial importance for
a
division manager to know what level of managerial freedom he has.
It affects immediately his divisional profit prospects and influences
also in the long run his motivation to manage the division well.
In
a forest products firm where the intermediate goods to be allocated
are logs the freedom of purchasing external logs may be delegated
either to the mills or to timber division.
If a division has this
right it has prospects for better profits than if it does not have
it.
Naturally a division is the better off the more it can decide
the firm's log allocation.
Some firms are market oriented (mills
prevalently determine the allocation) others resource oriented
(timber division leads the allocation).
In the two division forest
products firm of this study there are three alternative decentralized
organization patterns:
miii dominance where the milling division dictates the
quantity and quality of harvested logs to be delivered
internally and has the right to purchase logs from the
outside market; and the timber division harvests the re-
quired logs and has the right to sell harvested logs to the
outside market;
timber dominance where the timber division dictates the
quality and possibly the quantity of logs to be delivered
internally and has the right to buy logs from and sell
harvested logs to the outside market; and the milling divi-
sion may determine their total log demands accepting the
quality of supplied logs;
mixed dominance where the milling division dictates the
quantity and quality of internally delivered logs; and the
timber division either harvests the required logs or pur-
chases them from the outside market, and has the right to
sell harvested logs to the outside market.
In a centralized forest products firm the top management dictates all the activities.
It determines the quantities and qualities
of internal deliveries and external sales, and external log purchases.
There are no divisions but rather nonautonomous departments. that are
not profit centers.
4
Why have we chosen the dominance organization?
Why not use some
other structure of decentralization as transrer pricing
framework
where both divisions are true profit centers at the same time?
have proved to be very common in
divisional dominances because they
practice according to
Ferrie2,
We use
forest products firms executives (Baack,
Liimatta, Taylor4
;
personal interviews).
Purely
negotiated allocation after top management has announced transfer prices has the same disadvantage as negotiated transfer prices in Chaper 2:
it leads to bargaininci.
Bargaining behavior is difficult to quantify.
The best approach, game theory, has succeeded to produce workable
solution techniques only for two person zero-sum games under perfect
knowledge.
This is insufficient for most log allocation situations.
Other possible approaches are:
shadow price allocation
decomposition allocation
price adjustment allocation
Divisional dominance is superior to the three approaches.
Their
advantages and disadvantages are discussed in Appendix 4.
Don Baack, Crown Zellerbach Co., Portland, Ore.
Adam Ferrie, McMillan Bloedel Ltd., Vancouver, B.C., Canada
Into Liimatta, Hines Lumber Co., Hines, Ore.
4).
Sam Taylor, Boise-Cascade Co., Medford, Ore.
5
Log Allocation Problem
Figure 1.1 shows the log allocation problem framework faced by
the typical centralized Pacific Northwest forest products firm of
this study.
The sources of logs are timber tracts through harvest
and the competitive outside log market through purchase.
The tracts
are owned by the company or obtained from public timber sales.
In
a decentralized firm the purchaser can be either timber division
(timber and mixed dominance) or mill division (mill dominance).
The logs have three possible destinations:
they are used internally
by a veneer plant, sawmill or sold to the outside market.
always the timber division that sells logs externally.
process the logs and sell the end products.
It is
The mills
Log allocation comprises
the decisions of what tracts to harvest, where to take the harvested
logs, and what logs to purchase, where to use the purchased logs.
It does not in this study include bidding for future public timber
sales which is a long term decision.
The allocation decision is
made for three months--one quarter--but the mills consider also the
following quarter:
they have the option of producing veneer or
sawntimber for inventory.
Also in deciding about harvesting the
timber tracts the opportunities lost from cutting them at various
times, now and in the future, are considered.
The longer a tract
is available for harvest (in this study the maximum is five years--
the length of the firm's assumed cutting plan) the greater is the
opportunity cost of harvesting.
This is discussed in section 3.1
6
Centralized Firm:
Decentralized Firm:
No Divisions
Autonomous Divisions
Tlffiber Division
Sell
I Veneer
Mill. Division
/
(Buy)
/
I
3
Veneer
arvest
Plants
Logs at
Mi llyard
Use
Use
Harves
Saanills
ft
Buy
/ Sell
/
n
n
II
n
II
II
II
Other Firms
pOther Firms
I
p
_.-_j
Lumber
p
Other Firnis
L.
-J
Outside Log Market
(Possible Responsi-
bility Boundary)
/
Figure 1.1.
Log allocation problem.
(Possible Responsi-
bility Boundary)
\
7
and Appendix 2.3.
The short term character of the log allocation
problem of this study can be seen in the problem formulations of
Appendix 1.
Transfer Pricing Problem
In this study the transfer prices are the only means by which
the top management can control the divisional log allocations.
The
headquarters' goal is to choose the transfer price among a group of
alternatives that maximizes the top management utility.
In a major
part of this paper this is the same as maximizing the corporate
profit.
Log allocation through transfer pricing consists of two parts
(cf. Jennergren, p. 23):
finding the best transfer prices
implementing the best transfer prices
The best transfer prices can be found through information exchange
between headquarters and divisions and also between divisions themselves.
The information exchange would have the following stages:
the top management announces a set of tentative transfer
prices to both divisions
the dominant division solves its log allocation problem
given these prices and some guiding restraints from the
dominated division; these restraints prohibit technically
infeasible allocation to the dominated division
8
3)
the dominant division announces its log demand (milling
division) or supply (timber division) to the dominated
division
the dominated division solves its log allocation problem
given the transfer prices from the headquarters and allocation enforcement from the dominant division
both divisions report their profits to the top management
the top management starts a new information exchange
iteration by announcing a new set of transfer prices; the
iterations end when all alternative transfer prices have
been announced.
If the firm has mill or mixed dominance the information exchange
starts in stage 1 by top management asking the milling division man-
ager the following question:
"Assume you can get different classes of logs from the
timber division at these transfer prices and you process
them so that your divisional profits are maximized.
What would your profit be?"
The question to the timber division is:
"Assume the log demand schedule by the milling division.
What would your best profit be?"
Under timber dominance top management would ask the timber division
this question:
"Suppose you could deliver logs at these internal prices
and want to select the procurement program which maximizes
9
your divisional profit.
What would your profit be?"
The question to the milling division is:
"Assume the log supply schedule by the timber division.
What would your best profit be?"
The information exchange between the divisions themselves is
for their mutual coordination.
The dominated division delivers
information on its technically feasible log flows to the dominant
division.
Submitting this information guarantees that any of the
enforced allocations by the dominant division could be put into
action by the dominated division.
Thus any transfer price leads
to allocation feasible to both divisions.
Of the set of feasible
transfer prices the top management would choose and implement the
one that in its decision making situation maximizes its utility
(corporate profit).
10
1.2.
OBJECTIVES
The
p r i m a r y
o b j e c t
i
v e
of this study is to
test the hypothesis of economic theory that the log transfer prices
of a profit maximizing firm should be the competitive log market
prices.
The hypothesis is tested in a case study of a typical Pacif-
ic Northwest Douglas-fir region forest products firm.
The log allo-
cation problem is first solved given a centralized organization and
under perfect knowledge.
optimal log allocation.
The resulting allocation is the globally
This is the target of transfer pricing.
An experiment is set up to find the, best transfer. prices from
a great man,f candidates.
The experiment simulates the information
exchange actually taking place in decentrali'zed firms under mill,
timber or mixed ominance.
The transfer prices are based on log
market price, harvesting cost or log mill value which are the types
of transfer prices used in practice.
objectives
1)
The
s e c o n d a r y
are:
to find the optimal transfer pricing types under all three
divisional dominances;
'
to find the best levels of'transfer prices of the optimal
pricing types under all three divisional dominances;
3)
to rank the divisional dominances.
The criterion used for measuring the "optimality" or "goodness"
of transfer prices and the ranking of an organization is the corporate profit.
The profit is compared with the corporate' profit of
11
the globally optimal allocation and their difference is computed.
We wish to find the transfer price and the divisional
minimize the profit difference.
dominance that
The difference represents the
impact of dysfunctional decisions caused by the lack of goal congruence between the corporate and divisional managers on the corporate profit.
We do not try to measure the effect of different trans-
fer prices or organizations on other disadvantages or advantages of
decentralization.
Their impact on the efficiency of a firms
information system and divisional managers' motivation are beyond
the scope of this study.
The
tertiary objectives
are:
to discuss the transfer price analysis under uncertainty
when top management either maximizes the expected corporate
profit, or its expected utility when the utility is a nonlinear function of corporate profit;
to show the detailed calculations of the divisional log
allocation problems for corporate profit or utility maxi-
mizing firms that wish to find the best transfer prices in
their own, specific planning situations.
12
2.
TRANSFER PRICING ALTERNATIVES
There is a great variety of internal pricing procedures in forest
products firms today.
The major types are cost-based, market-based
value_based:andneqotiated
transfer prices.
Cost Pricing
There are no comprehensive surveys of actual practices of transfer
pricing in forest products firms.
However, the discussion
with
executives have shown that the use of cost-based transfer prices is
widespread in the Pacific Northwest forest industry, as in business
firms in general (Baack, Ferrie, Fisher5, Liimatta, Nelson6
personal interviews; cf. Horrigren, p. 740).
The most common of them
are the variable harvesting costs as transfer prices.
historical or standard variable harvesting costs.
They can be
Historical costs
are actual costs of previous periods recorded in accounting books.
Standard costs are target costs in the budget for the coming period.
Standard costs are preferable for planning purposes because they are
projections to the future (e.g.
Schattke etal., p. 281).
Historical
cost pricing fails to provide incentive to control costs.
Both historical and standard cost pricing result in very low
timber division profit.
R. Fisher,
Most or all of the firm's profit is transfer-
Bohemia Inc., Eugene, Ore.
Ted Nelson, Weyerhaeuser Co., North Bend, Ore.
13
ed to.the milling division.
This easily causes timber division manager
to feel as a "second class" executive in the firm.
incentives for good management.
decisions.
He may lose his
Cost pricing then can lead to poor
The situation is not much betterwhen the historical or
standard costs as transfer prices include an allowance for timber
division fixed cost and profit.
Now the higher transfer prices result
in greater timber division profit.
But this profit is a standard
planned profit by the top management.
The timber division manager
still has poor incentives for good management.
He feels that his
divisional freedom is restricted because he cannot plan his profits
independently.
rises:
profit?
When there is an allowance for profit the question
What is a reasonable standard as a target of the divisional
Is it an estimated "good" or "satisfactory"
profit of timber divisions in other firms?
or "average"
Should it be higher than,
the.same as or lower than thecompany's mills' profits?
Cost-based prices do not vary between different log classes
because harvesting costs vary little as a function of grade.
Thus,
cost-based transfer prices do not reflect the values of the logs at
market or at mills.
Market Pricing
When a firm operates in a competitive log market it is tempting
to use market prices to determine transfer prices.
Unlike cost-based
pricesthey naturally vary from log class to log class.
Under market
14
pricing, decisions have to be made whether short term or long term,
past or future estimated market prices should beapplied.
The length
of theallocation planning horizon naturally determines whether to use
short- or long-term market prices.
Si.nce the allocation decision is
made for the future, some would argue that future prices should be
used.
But whose estimate is the best?
Is it that of the analyst
working at headquarters or that of the divisional decision maker?
The
former's estimate is lacking the contribution of the "feel' at the
market that makes a subjective estimate often superior.
The divisional
prediction, on the other hand, may be misguiding, exaggerated or
prejudiced.
To avoid these problems in practice, historical log
market prices are used often as transfer prices (Chapman7; personal
interview).
Market prices establish a ceiling for transfer pricing when top
management wishes mills to buy predominantly internal logs.
A lower
price is justified when the timber division obtains economies of scale
from large sized internal deliveries (cf. Cook, p.90).
Transfer
prices greater than market prices are seldom used in practice (Baack,
personal interview),
though they could be justified by the improved
raw material availability the mills may obtain from the guaranteed
steady flow of internal log deliveries.
7)
Larry Chapman, Bohemia Inc., Eugene, Ore.
15
Value Pricing
As cost-based transferprices favor the mills by being low
compared to market prices, value-based prices favor timber division by
being high.
p. 69).
Value pricing is popular among oil companies (cf. Dean,
Due to the special structure of oil industry it helps to
minimize, their tax burden and maximize the corporate profit.
The
interviews with Pacific Northwest forest products firms showed that
the computatfonally difficult mill value-based prices are nonexistent
in forest industry.
They are computed by subtracting mills' direct
processing costs from the log values at end product level or also
subtracting allowances for mills' indirect fixed costs and profits.
Allotting the allowances to different log classes is arbitrary and
determining 'reasonable" profits for the mills difficult.
The value-
based transfer prices naturally vary from log class to log class.
They also vary from mill to mill.
Negotiated Transfer Prices
Negotiated transfer prices without top management as an umpire
and as the final decision maker have the disadvantages of bargaining.
Bargaining increases the chances of misunderstanding and conflicts
between the divisional managers.
It causes the managers to present
misguidedexaggerated and prejudiced pricing facts, and generates heat
and bad temper.
It rarely produces transfer prices that are not
dysfunctional for the firm as a whole (Cf. Dean, p. 72).
The procedure
16
also.requires a great amount of the executives' time (Cook, p. 93).
Discussions with top management as an umpireare, however, quite common
to find feasible internal prices and allocations for both divisions.
They help top management to narrow down the alternatives of cost-,
market- or value-based transfer prices for the execution.
Multiple Transfer Prices
A great majority of forest products firms prefer simple pricing
practices with the same prices for logsof the same type for timber
division as the supplier and all the mills as the users. Cost-based
transfer prices, might, however, vary between mills as a function of
the log transportation costs.
On value pricing the transfer prices
vary from mill toniill as a function of end product prices and
recoveries of each mill.
Top management might Want to apply different
transfer prices for the seller and buyer.
This could occur to encoura-
ge transactions of certain types of logs.
The reason might be risk
sharing between divisional managers.
A risk averse timber division
manager could receive a low cost-based transfer price plus a lump sum
for his log while a timber division manager as a risk taker pays the
log market price.
The harvesting costs vary little and thus cost-
based transfer prices are subject to little risk.
The coming period's
log market prices are more uncertain and allocation based on them as
transferprices is risky.
Practice has shownthat price discrimination is poor business
17
within a firm (Dean, p. 66).
It creats
bad feelings between managers
and friction in interdivisional relations.
This study focuses on
singular market-, cost- and value-based transfer prices set by top
management.
A price of the timber division as a supplier is the same
as a price of the veneer plant or sawmill as a user of the internal
logs.
18
I
3.
THEORETICALLY OPTIMAL TRANSFER PRICE
3.1.
OPTIMAL TRANSFER PRICE WHEN THERE IS NO OR A NON-COMPETITIVE
OUTSIDE LOG MARKET
According to economic theory a firm will use each input in such
quantity as to maximize its profits.
Profits will be maximized when
the amount added to the revenue of the firm by an additional unit of
the input equals the amount it adds to cost (Stigler, p. 239).
To
demonstrate this for logs as inputs we develop the concept of derived
internal demand and marginal harvesting cost of logs in Figure 3.1.
We discuss the intrafirm log market of a forest products company but
the analysis is applicable in any firm with an intermediate product.
Derived Internal Demand
We assume that the end products --veneer and lumber-- in a forest
products firm consist of two inputs:
includes all other production factors
logs, and processing which
such as labor and energy.
Whether the end product markets are competitive or not does not change
the analytical framework as long as they are independent (Hirsch
leifer
(a)
,
p.
176).
-
Here we assume competitive market.
employ processing inputs as a function of log quantity.
We
We assume
both technological and demand independence in the operations between
supplier and demander.
Technological independence means that the
19
Log end product
or mill vulue
or processing
cost, $/t113F
[V
lIP C
Log quantity, MBF
Figure 3.l.a.
Internal demand for logs.
Note: [V = niaxirnun value of logs after processing when the end product
market is competitive.
MPC
rnarginnl processing cost
derived internal demand
[V - MPG.
tiC
Ilarvestj ng cost,
$/ FIB F
i-User cost
Logging and transportation cost
Log quantity, by tract,
'IBE
Figure 3.Lb. Marginal harvesting cost (MC).
Log price,
value or
Cost, $/MBF
tog quantity, MBF
Figure 3.2.
Note:
IC
MEM
Log allocation, tipper (PT) and lower (PM bounds of negotiated
transfer prices under bilateral monopoly.
marginal harvesting Cost
marginal expense of logs as inputs of a nnonopsoniistic milling
division
pint
MRT
den ned internal deisund
= marginal revenue of a monopolistic timber division
20
operating costs of each
operations by the other.
division are independent of the level of
Demand independence means that an additional
external sale by either division does not reduce the external demand
for the products of the other (Hirschleifer (a), p. 172).
Internal
log demand depicts the locus of points of the highest prices the mills
can afford to pay for logs.
It is the maximum value of logs before
processing--the marginal log values after processing minus the marginal
(variable) processing costs of the mills.
The fixed processing costs
will be relevant in deciding whether a process should be instituted or
not, or whether it will be continued or not.
short term allocation.
They are irrelevant to
That is why they are not considered in Figure
3.l.a. (cf. Solomons, p. 213).
Marginal Harvesting Cost
In a forest products firm the marginal cost of the intermediate
product--log--is named the marginal harvesting cost.
It is somewhat
different from that normally found in economic text.
In a forest
products firm the technology requires that a tract is cut as a whole,
not partially.
The marginal harvesting cost function is thus discrete.
Its each step is the average (direct) harvesting cost of a tract.
The
average harvesting costs for all tracts available for cut, ordered
from
smallest to largest, form the nondecreasing marginal harvesting
cost (MC) function of the firm as Figure 3.l.b shows.
It is the locus
of points of the minimum prices the timber division would accept for
21
the logs.
The harvesting cost of a tract consists of two part: the
logging and transportation cost and the user cost which is the
opportunities lost if the tract is cut in the coming quarter instead
of in the future (Scott,
p
6;
Nautiyal, p. 338;
Johnson, p. 10;
Walker, p. 31)
The logging and transportation cost of a tract is a common concept
but the idea of user cost needs more explanation.
In our log alloca-
tion framework the forest products firm has tracts to be harvested in
three categories:
available tracts accessible for harvest
available tracts not accessible for harvest
tracts not available for harvest.
In this study of short, three month, time horizon we are only interested in category 1:
tracts with roads.
purchased public timber tracts and company's own
The second category concerns purchased tracts and
company tracts with no roads included in the company's cutting plan
for the next five years.
The third group includes public timber
sales to bid for within the coming five years and company timber to be
harvested within the following five year period.
We assume that decisions of bidding and the five year cutting
budget for company timber have been made by log range planning.
The
decisions regarding road building are intermediate time range decisions.
Short-term log allocation determines only whether the available and
accessible tracts should be harvested during the coming quarter when
22
thealternatjve is to harvest the tracts during some later quarter.
The public timber tracts are available for as many quarters as the
sale contract states.
The company timber is available for five years
(, = 20 three month periods).
In our log allocation framework of all the production decisions
the timber harvesting decision is of longest term.
action is irreversible:
Harvesting as an
once the trees of a tract have been felled
the possibility of harvesting them later is gone.
Timber deteriorates
after cutting but can be held standing long periods without deterioration.
In fact, standing timber may grow and increase in value.
By
harvesting just before the use the capital costs of logging and
transportation are reduced.
Thus we assume in this study that if logs
are used in the future, harvesting will occur in the future also.
Following Duerr (p. 203) we list the advantages of keeping tracts
unharvested:
the readiness to meet the mills
future log requirements,
the opportunity gain from future timber value increases,
the value increase of the stand due to volume growth.
The disadvantages of holding standing timber uncut are:
the costs of interest on the investment in wood stock, the
risk of fire and deterioration of wood quality,
the opportunity loss from future timber value decreases.
Guaranteeingthe mills' log demands for the future periods is a task
of intermediate or long range planning.
In this study we take advan-
23
tages (2) and C3)and
disadvantagesl) and (2) into consideration
through the user cost of eachtract.
For every accessible tract we
compute its discounted expectedvalue after stumpage for all quarters
the.tract is available.
To be ableto compute them, estimates are
needed for future prices of sawntimber, veneer and logs in the outside
market.
The costs of logging, transporting and processing are
estimated.
The tree growth is predicted.
From these we compute the
expected discounted value of each tract in the best uses of the logs
ineach period.
The user cost is computed by subtracting the estimat-
ed value of each tract in the best use of the logs in the coming
quarter from its discounted value in the best use of the logs in the
best
quarter as shown in Appendix 2.3.
The user cost is the temporal
counterpart of the opportunity cost (Walker, p. 31).
Negotiated Transfer Price Under Bilateral Monopoly
Bilateral monopoly exists in the intrafirm market when the timber
division is the only source of logs and the milling division their
only destination.
They negotiate the transfer prices and log
allocation among themselves without the headquarters acting as an
umpire
Economic theory is unable to derive any "optimal" or "best"
transfer price in this situation.
The transfer pricing and allocation
solution is based on the bargaining skills of the divisional managers.
Figure 3.2 shows the derived internal demand for logs faced by the
timber division.
As a monopolist the timber division's marginal
24
revenue isJ'1RT and it would like to sellq
mills.
logs at price
As a monopsonist the milling division's marginal expense of
logs is MEM and it would like to purchase
the timberdivision.
logs at price
M
from
The timber division can fulfill its desire only
in case the milling division acts as if
titive.
to the
T
the log market were compe-
Correspondingly the milling division can induce the transfer
price and log allocation as a nionopsonist only in case the timber
division acts as if the log market is competitive.
Since neither of the divisions can induce the transfer price and
log allocation the price may be any value between
allocation any value between
and
M
and
T
and the
(Ferguson, pp. 281-2).
Only by
accident the price and quantity would be one desired by the top
management.
Negotiated transfer prices under bilateral monopoly lead
easily serious dysfunctional log allocations.
Optimal Transfer Price When There Is No Outside Log Market
When there is no outside log market and the headquarters determines the transfer prices some of the dysfunctional log allocation of
the bilateral monopoly is avoided.
Top management considers the
derived internal demand as its marginal revenue curve.
cost curve is the marginal harvesting cost.
Its marginal
It maximizes its profit
by ch.00sing a transfer price such that in the resulting log allocation
marginal cost equals to marginal revenue.
Since the top management
sets the transfer price the' timber division has to accept Dt as its
25
Log value
Log value
or cost,
or cost,
$/MBF
MC
MC
A
A
TP
Bp
nt
C
q14q1.
Dint
C
Log quantity,
q1
Log quantity,
MUF
tIN F
Figure 3.3.a.
Figure 3.3.b.
Log value
or cost,
SI lB F
IC
A
a
Tp
E
D0t
C
M
Log quantity, MBF
Figure 3.3.c.
Figures 3.3 a-c.
Log allocation and divisional profits s,hen transfer once
(TP) equals to (Fig. a), is greaten than (Fig. b) or
smaller than (Fig. c) the equilibrium price (p ). Note:
MC = marginal harvesting cost;
= derived nterna1
demand.
Log price,
value or cost,
$/tIBF
p2
MC
Ip =
out
I
'I.
MR
lRtt
out
i
t.
q
Figure 3.4.
mt
Log quantity, MIlE
Log allocation, internal and external log pricing when there
is a noncoi'petitive outside log market, in a price discritsination situation (Johnson, p. 5a).
26
marginal revenue curve.
The'.milling division cannot exploit the
timber division by applying the marginal expense of logs-curve in the
log allocation, either.
Marginal cost equals to marginal revenue
at equilibrium point D in Figure 3.3.a.
The corporate profit maximiz-
ing transfer price (TP) is the ordinate of this point, pe__the equiliis the quantity
With this optimal transfer price q1 =
brium price.
of logs delivered from timber division to milling division.
Quantity
represents the amount of logs the milling division desires to
receive and cIT the quantity of logs thetimber division wishes to
Area ABD is the milling division profit and area BCD the
deliver.
timber division profit.
The sum of
transfer price TP is greatertharm equilibriumprice
occurs between the divisions:
e
A conflict
milling division wishes to acquire
logs and timber division supply
desire
In Figure 3.3.b,
these, ADC, is thecorporate profit.
T
logs.
M
If the milliny division's
comes through its profit will be BCDE and milling division's
profitABD
The corporate profit is ACED which is less than the
maximum profit by DEE. If the timber division's desire comes through
its profit will be BCG and milling division's profit ABD and profit
loss DGH.
The corporate net profit is ACF-EGH which is less than the
maximum corporate profit by FHG.
equilibrium price
maximum profit.
When transfer price TP is less than
the firm again incurs profits less than the
e
This is shown in Figure 3.3.c.
Assuming that the
timberdivision has to deliver the logs the millin
division wishes
27
to aquirethe corporate profitloss Is FGH.
Assuming that the.
milling division has toaccept the logs timber division wishes to
deliver the corporate loss corresponds to are DEF.
The corporate profit losses from poor transfer pricing may diminish
through divisional negotiations.
Quantities
closer to the optimal allocation.
and
may become
But they may not because the final
solution depends rather on the divisional managers' bargaining skills
than economic theory.
Analyses of divisional bargaining behavior are,
however, beyond the scope of this study.
Here we examine the optimal
transfer prices assuming log allocation according to wishes of only
milling division or only timber division.
According to the terminol-
ogy of Chapter 1 we call the first organizational structure mu
dominance and the second timber dominance
outside log market
.
In the absence of an
mill dominance is equivalent to mixed dominance.
The analyses indicate that under either divisional dominance the
optimal transfer price is the equilibrium price.
At the equilibrium
price the corporate marginal revenue (Dut) equals to the corporate
marginal cost (MC).
There are no corporate losses from decentralizing
the allocation if the equilibrium price prevails.
Optimal Transfer Price When There Is
A Noncompetitive Outside Log Market
The marginal cost-marginal revenue rule of transfer pricing and
log allocation holds also when there is a nOncompetitive outside log
market.
Figure 3.4 shows this.
There MC is the corporate marginal
28
cost curve.
But there are twocorporate marginal revenue curves, one
for the internal logs (D1t) and one for logs to be sold to the outside market (MRout) which is derived from the outside log demand
In economic theory this situation of several markets with
differently shaped demand and marginal revenue curves is called price
discrimination (Stigler, p. 209;
Leftwich, p. 216).
The total margin-
al revenue (MRt0t) is the sum of the two marginal revenue curves.
The optimum total log quantity procured is determined by equating the
total marginal revenue and the(total) marginal harvesting cost,
at point X.
The abscissa of this point is the total harvest q.
Econo-
mic theory says that the firm will maximize its profit by allocating
logs among the submarkets so as to equate its marginal revenue in
each submarket with total marginal revenue at point X.
Price in each
submarket is determined directly from the submarket demand curve,
given the submarket log allocation (Ferguson, pp. 280-281).
Thus the
optimal internal log delivery will be q1 and the external log sales
q2.
The optimal monopolistic outside market price is p2.
The optimal
transfer price, p1, is determined by the quantity allocated from
internal deliveries q1, and the internal demand curve,
which
happens to be also the marginal revenue of internal log deliveries.
The optimal transfer price is always the ordinate of point X.
Price p1
is the optimal transfer price under both mill, timber and
mixed dominance.
It produces the same corporate profit in all of them
as in a centralized organization.
Using it prevents dysfunctional
29
decisions and causes the profit.losses of decentralization to bezero.
We willshowthis in thenextsection, dealing with transfer pricing
when there is a competitive outside log market.
Whether the outside
log market is competitive or not, the basis for log allocation and
transfer pricing is the economic theory of price discrimination.
30
3.2.
OPTIMAL TRANSFER PRICE.WHENTHERE IS A COMPETITIVE OUTSIDE
LOG MARKET
Optimal Solution In A Centralized Firm
The marginal cost-marginal revenue rule of transfer pricing and
log allocation holds when there is a competitive outside log market.
In this section we will see that this rule causes market price to be
the optimal transfer price.
For showing it we apply again the
economic theory of price discrimination.
Market pricihg may result
in the allocation and maximum profit of a centralized organization.
There may be no corporate profit losses from decentralization.
First we wish to find the optimal allocation under centralization;
this allocation is the target of transfer pricing.
The log market
price can be either a marginal cost or a marginal revenue for the firm
as a.whole.
It is marginal cost when logs are purchased from, and
marginal revenue when logs are sold to the outside market.
When the
firm is a net seller of logs its total marginal revenue is the maximum of derived internal demand and market price:
Its total marginal, cost is MC.
max
C
Dint; MPJ.
If Figure 3.5.a depicted the profits
and log allocation of a centralized firm, the firm's total marginal
revenue curve of logs would be ADE -
MP, and marginal cost curve MC.
When the firm is a net buyer'of logs its total marginal revenue is
Dint and total marginal cost the minimum of the marginal cost and
market price:
'nun
[ MC; MP]
If Figure 3.5.b depicts the profits
31
Log value
Log value
or cost,
or cost,
S/MB F
S/MB F
MC
A
A
B
e
TP-MP
B
C
mt
Log quantity,
q1
Log quantity,
MBF
MBF
Figure 3.5.b.
Figure 3.5.a.
Log allocation and divisional profits under mill and mixed
Figures 3.5.a-b.
dcxninances when transfer price (TP) equals to market price
(NIP) and is greeter than (Fig. a) or less than (Fig. b)
internal equilibrium price
(gain for tiraber
division, loss for
Log price,
value or
mills)
cost, $/MB
lost profit
r
B
TP
Log price,
value or
cost, $/[3F
Lost profit
[P
-
0int
l
C
g
TP=I[P
I
t=x
C
It
(q1)
q
Log quantity,
(°q
II )
Log quantity
Mi[F
[[3 F
Figure 3.6.a
Figure 3.6.a-b.
log al locution end divisional profi t
Figure 3.6.b
under limber dominance
i:hcn transfer price (II') equals to until price (II) arid is
In) rntriinl equilibrium
greater then (Fig. a) or lccc than (I
the quantity of
price (H. Timber division alone di taleS
ii
logs to be used by the vi 111 ry di cr51
32
and log allocation of a centralized firm,the firm's total marginal
revenue curve would be Dt and the total marginal cost curve CD
NP.
The profit maximizing centralized firm's total wood procurement would
be the abscissa of the intersection of total marginal revenue curve
and total marginal cost curve.
3.5.a and b.
This intersection is point X in Figures
According to price discrimination theory the firm deter-
mines its optimal log allocation between the internal and external
markets by equating the marginal revenue in each market with the total
marginal revenue at point X. (cf. Ferguson, pp. 280-1).
result in a flow of
logs to the mills and
market in Figure 3.5.a.
In Figure 3.5.b
logs to the mills of which
logs to the outside
it would cause a flow of
are internal logs.
the optimal allocation of centralization.
firm's profits would be ADEC and AEDC.
This would
This would, then be
The corresponding maximum
These are the target alloca-
tion and profit to aim at in setting transfer prices.
Mill Dominance
When transfer
p r i c e
price
equals market
under mill dominance the log allocation and corporate pro-
fit are the same as those of the optimal solution in a centralized
firm.
There are no corporate profit losses from decentralization.
This is shown in Figure 3.5.a when themarket price (= transfer price)
is greater than the internal equilibrium price
The mills get
all their logs internally (q) and timberdivjsion sells part of the
33
Redistributed
109 price,
value or
tog price,
Cost,
cost
5/Mgi
profit
Loot profit
value or
,,/C
s/Mu
en-.,
P
A
l5'X
H
LIUfl
B
-
C
toil quan-
tity, tlBf
Figure
Figures
3. 7.a-b.
l 1TS Log quan-
Figure 3.7. b.
tity, 1191
[rumples of lnq allocation and divisional profits nuder
p
l'jnijranr chen transfer price is greater then (11g.
a
or less than (Fig. B) market price.
Loot
Log price,
value or
Redistributed
profit
cost
profIt
5/tigi
profit
iog price,
Lost
FtC
value or
cost,
redistributed profit
'IC
5/1113 F
A
'H
h
l-\us-:2
GYV
B
quantity, MBF
(q11)
Figure 3.8.a.
lIP
ii_IHj iIIil;rr.'_;-
Log quantity,
q
MBF
Figure 3.8.b.
Figures 3,B,u-b,
Exaniples of log allocation and divisional profits undnr
tiedor d.ju?ncC rheir transfer price it greater than
(fig, a) or less than (fig. b) market price.
Redistributed
Leg price,
vulvo or
Redistributed
profit
Log price,
or
Lost profit
tiC
profit
j
Lost profit
'I
5/tII1P
C
c
q1
c71.oij quautty,
It
F
guin
profit
1tust profit
$/MulF
A
H
B
C
'
\fl
Figures 3.9 .a-d.
lug price
value or
/llC
Redistributed
prof1it
5/1-1131
for mills
but loss
for tuber Loot profit
divisioo
MC
A
lp
IP
9int
1
Figure 3.9.c.
ttur
Figure 3.9k.
Redist rib rited
Log price,
valve or cost,
Log quastity,
p
p
tttup
Figure 3.9.a.
p1
1 og ibm-
tity, uhf
tog quantity, 11111
Figure 3.9.d.
[cantleS of log al locations and divisional trofits
under erju_desJiflgn rico transfer price is greater
(Iran (Figs, a, In) or less than (Figs. c, d) ni,nrl,ct
price.
34
harvest q) to the outside market(q_q).
Timber division profit
es.
area ABD.
There are no log purchas-
is area BCE and milling division profit
The corporate profit is their sum ADEC.
When the market
price ( = transfer price) is less than the internal equilibrium price
in Figure 3.5.b, the mills get only part of their logs from the
timber division harvest () and the rest is purchased from the outside market.
The timber division profit is area BCD and the mills'
profit area ABE.
The corporate profit is their sum AECD.
same as the maximum profit of a centralized firm.
price
(
This is the
When the market
transfer price) is the internal equilibrium price
the
allocation and profits are the same as for Figure 3.3.a.
Undermjll dominancewith
m a r k e t
p r i c e
transfer price above
the divisional and corporate profits are
the same as with transfer price equallingmarket price in Figure 3.5:
there are no corporate profit losses from decentralization. The log
allocations, however, differ:
milling division buys,all its logs from
the market (q) and timber division sells all harvested logs to the
market (q).
No internal log transactions take place.
in Figure 3.7.a.
b e 1
o w
losses.
But when
m a r k e t
t r a n s f e r
p r i c e
p r
i
This is shown
c e
i
s
the firm experiences profit
In Figure 3.7.b the corporate profit loss is DEG.
This is
the lost profit to timber division when milling division forces it to
deliver a total of q
logs internally at the low transfer price.
It
would be more economical for the timberdivision and for the firm to
35
sell part of their logs to the.outside market.
Area HBGE is the
redistributed profit from timber division to milling division caused
by the reduction of transfer price below market price.
Timber Dominance
When
t r a n s f e r
p r i c e
p r I c e
e q u a
1
s
m a r k e t
under timbr'r dominance a firm may or may not be facing
corporateprofit losses from decentralization.
Figure 3.6.a when market price
(
the internal equilibrium price
This is shown in
transfer price) is greater than
The timber division wants to
harvest q1 logs and the milling division wishes to acquire
logs.
The timber division is indifferent between delivering to the mills or
selling them to the outside market.
If the organization is flexible
enough to let the milling division inform the timber division about
its log acquisition desire the allocation and corporate profit is the
same as in the optimal solution of a centralized organization:
are no corporate profit losses from decentralization.
there
But if this
information flow between the divisions does not occur the lost corporate profit corresponds are EFG.
When market price
equilibrium price
profit loss.
(
= transfer price) is less than the internal
there again may or may not be a corporate
This is shown in Figure 3.7.b.
indifferent between delivering quantity
The timber division is
of logs to the mills,
If it does not know about the mills' desire to acquire these logs the
36
corporate.profit loss is are EFG.
price
When transfer
p r
i
c e
theprofit
is above market
losses from decentralization are substantial.
Figure 3.8.a shows that the timber division wishes to deliver
harvested logs to the mills.
But it would also like to deliver an
infinite amount of bought logs to them because for each unit delivered
it makesa constant (TP-MP) net revenue.
acquire only
The mills would like to
logs which would give them a profit of area ABD.
But
the dominant timber division forces themilling division and the whole
firm to incur an infinite profit loss which will cause the firm to
go
out of business.
Any transfer price below market
p r
i
3.8.b.
c e
is also disastrous for the firm as we can see in Figure
Now the timber division wants to sell all harvestable logs to
the outside market leaving the mills without raw material.
The mills
go out of business.
tAte can conclude that timber dominance without clear minimum and
maximum log requirements by the mills,enforced to the timber division,
does not work unless transfer price is market price.
Even if transfer
price is market price timber division has to know desired allocations
by the mills.
occur.
Otherwise
corporate profit losses from decentralization
37
Mixed Dominance
market
When transfer price equals
p r i c e
under mixed dominance the log allocation and corporate
profit. are the same as those of the optimal solution in a centralized
and mill dominant firm.
They are shown in Figure 3.5.a and b.
There
are nO corporate profit losses from decentralization.
Iith
p r i c e
tion.
t r a n s f e r
p r i c e
a b o v e
in a r k e t
the firm faces lost corporate profits from decentraliza-
Figure 3.9.a and b show two examples of this.
in both cases is marked by area DEG.
The profit loss
The.redistributed profit from
milling division to timber division caused by the increase of transfer
price above market price is BDEH.
The mills are going to use
and the timber division harvest amounts to
p r i c e
is b e
1
o w
m a r k e t
and d corporate profit losses incur.
When
p r i c e
logs
t r a n s f e r
as in Figures 3.9.c
In the figures their area is DEG.
"Market Price Rule" of Transfer Pricing
We have been using the price discrimination theory to find optimal
transfer prices both when the outside log market is competitive and
noncompetitive.
According to that theory the optimal transfer price
is the ordinate of the intersection of the total marginal revenue
function and total marginal cost function.
point X in figures 3.5 - 3.9.
This intersection is
In the figures it always lies in the
38
horizontal line depicting.the competitiveniarket price.
It has to lie
on themarket price line because that line is.at its whole length part
of either firm's total marginal revénüe curve or total marginal cost
curve.
The marginal revenues result from selling logs to the outside
market, theniarginal costs from buying them from the outside market.
But because the market price line is horizontal the ordinate of X is
always the market price.
price.
Thus market price is the optimal transfer
We have named the procedure of setting the optimal transfer
price as the ordinate of the intersection of total marginal cost and
total marginal revenue the "marginal cost - marginal revenue' rule of
transfer pricing.
calls it
Hirchleifer(a), p. 176; (b), p. 108; (c), p. 34)
'marginal cost" rule.
Inthis section we have seen that
when there is a competitive outside ba market, optimal transfer price
equals the market price according to these rules.
It is optimal
because it produces the same allocation and corporate profit as the
optimal solution of a centralized firm.
Through market pricing the
corporate profit losses of decentralization are avoided under any of
the three divisional dominance structures.
The "marginal cost" rule
then becomes "market price" rule.
The log market price reflects the opportunity cost of the intrafirm transaction to the divisions.
The timber division will not be
ready to sell logs to the mills at a price lower than the market price
(TP
NP).
Voluntary internal log deliveries occur only when TP
Voluntary internal log acquisitions occur when TP
MP.
MP.
The union of
39
thetwo sets of feasible solutions is TP = MP.
Transfer price should
then equal market price in order toavoiduneconomic allocations from
both. one of the divisions' and corporate point of view.
Optimal Transfer Price For A Firm
With Several Milling Divisions
The market price--marginal cost--rules of price discrimination
hold for the cases where we have several milling or timber divisions
(Ferguson, pp. 281, 272).
We restrictour attention only to the
organization of mixed dominance.
The reasonings we are going to
present apply also to the othertwo discussed organizations.
We
assume two independent internal users--veneer and sawmill divisions.
Areas ABC and DEF represent the milling divisions' profits in Figures
3.lO.a-c.
The summed mill profits are GHK.
The timber division
profit is HIJ in Figure 3.1O.a, HIJMK in Figure 3.lO.b and HIJ + LKM
inFigure 3.lO.c.
Symbols q
and q5 represent veneer and sawmill
divisions' internal demands for log, and
internal and external supplies of logs.
and q-1-5 timber division's
The corporate profit = summed
divisional profits are at their maximum when t r a n s f e r
i
S
in a r k e t
decentralization.
divisions.
p r i c é:
p r
i
There are no profit losses from
The analysis is similar for more than two milling
C
e
Log value
$/tBF
40
icy price,
Log value
value or cost,
$/FIBF
/MBF
Veneer Division
Sawriiil Division
DvtlRv
£
D
A
B
Log quantity, MBF
Figure 3.lO.a
Log value,
Log value,
log price,
$/FlriF
S/ ia F
value or cost
Veneer Division
Division
Dv
Lost profit
D
A
B
Redistributed profit
Saivini 11
D
///
lIni
//,
MC
1K
JI.
q5
1=il-
Figure 3.lO.b
Log value,
Log value,
s/liar
Log price,
value or cost,
$/tIBF
Veneer
,,
Redistributed Profit
Gain for nills , loss
for timber division
$/lBF
Sasosi 11
Division
Division
D =MR
DMR
v
v
I/tost profit
nt
(/1//i, ,,
'1/. /i/,,,',
F
MC
I
i-i P
fiTffl1J,
1l11Ii!lIIliIL F
TP
D
q5
qJ1q15
Figure 3.lO.c
Figures 3.lO.a-c.
Log quantity, PIIF
Examples of log allocation and veneer, snwsmill arid timutmcr
division profi Lu under vii ced mioni nancy wimes transfer price
(TP) is equal to (Fig. ), greater than (Fig. b) or less
than (Fig.c) market price (lIP).
Log price,
value or cost,
Log cost
Log cost
s/air
s/Fill
5/ MB
I4C
'.rc
rlc
A
-'P
A
B
5
q TA
Figure 3.11.
0int
q11-q1 1iq111
Log quantity, IBI
TB
Log allocation and divisional profits of milling divisions and two
titmer divisions (A, ii) under ni xed doaiimnnce when transfer price
(1P) is market price (rr)
41
Optimal TransferPrice For A Firm
With.Several Timber Divisions
When the company has'several log sources the analytical allocation
solution resembles a solution to a multiplant monopoly.
Again, the
firm maximizes profit by equating thetotal marginal cost to the total
marginal revenue.
This point is X in Figure 3.11.
The abscissa of
X is the total optimal procurement of logs by the timber divisions.
Optimal allotment of log procurement among the different timber
divisions requires each of them to procure that amount of logs for
which the divisional marginal cost is equal to the ordinate of point X
Figure 3.11 presents an example of log
(cf. Ferguson, p. 272).
allocation and profits of a mixed dominant firm with two timber
divisions when transfer price is log market price.
A and B deliver q
logs internally.
and q
Timber divisions
In the example the
mills' log demand (q) exceeds the total supply from harvest
and amount
-
is purchased from the log market.
TI
This allocation
produces a total profit of area GEIK for the mills and area HIJ = ABC
+ DEF for the timber divisions.
these area.
The corporate profit is the sum of
The allocation is such that each timber divisional margin-
al cost is equal to the ordinate of X and point X lies on the
competitive market price line.
But according to price discrimination
theory the ordinate of X is also the optimal transfer price.
rn a r k e t
p ri c e
i
s
the optimal
t r a n s f e r
Thus,
p r i c e.
It produces an.allocation equivalent toa firm with a centralized
42
organization.
ization.
There are no corporate profitlosses from decentral-
Theanalyses and results arethe same for amillor a
timber dominant firm.
43
3.3.
OPTIMAL TRANSFER PRICES FOR SEVERAL LOG CLASSES
According to economic theory the rule of equalling marginal
revenues and marginalcosts of logs holds also for profit maximizing
firms with several log classes.
Each class is employed to a point
where its marginal mill revenue equals to its marginal procurement
When the end product market is competitive the following
cost.
equalities must hold (cf. Friedman; p. 176):
MR1 = (EP
MLP
- MPC1) = MC1;
i
= l,...,I
where
MR
= marginal net revenue at mill of class
EP
= end product price, $/MBM
i
logs, $/MBF
MLP1= marginal productivity of class i logs, MBM/MBF
MPC= marginal processing cost of class i logs, $/MBF
MC
= marginal harvesting cost of class
i
logs, $/MBF
Following the same reasoning as for the one log class allocation
we would like to prove that if in a decentralized firm any class i
logs
are sold to the market or purchased from the market and delivered
internally then class i optimal transfer price must equal to its
market price.
Could this hold for all
i
= l,...,I log classes?
We
try to answer this question by discussing three means of finding
optimal transfer prices for several log classes: diagram (as in the
previous section), calculus and mathematical programming.
44
Diagrams
In order to use diagrams for finding the "optimal" or "best"
transfer prices we wish to use the same internal demand-marginal cost
framework as for the one log class situation.
If we can show that
the marginal log revenue MR1 for each log class is its derived internal
demand Dt and if the marginal harvesting cost MC1 for each log
class i is comparable with the MC in the one log class situation we
can solve the problem for each class graphically as we did in the one
log class situation (cf. Alchian
and Allen, p. 452; Ferguson,p.366;
Friedman, p. 177; Stigler, p. 240; etc.).
MR
But this requires that all
for i = l,...,I are independent and all r'iC1 for i = l,...,I are
also independent.
components
:
MR1 for i = l,...,I are independent only if their
marginal log productivities MLP1, and marginal log
processing costs MPC
for i = l,...,I, are all independent.
Product-
ivity--recovery--of each class is independent on the levels of other
log classes employed.
times.
Log processing costs are functions of processing
The time a log class spends in the process is independent of
the amounts of other log classes processed (Baack, Chapman, Nelson;
personal interviews).
In short run the total processing time of a
mill -- its capacity -- is limited.
After the maximum capacity has
been reached the marginal processing cost (of the next log unit) is
infinity.
The marginal cost of the log class of this unit is not
independent on the quantities of other log classes employed. We can
conclude that the marginal log revenue MR
of each log class is not
45
its derived internal demand.
The marginal harvesting costs of the log classes are not independent, either.
Each tract now comprises several log classes.
Every log
class or none is subject to harvest because in a tract the felling
of only one log class is not technically possible.
interdependencies MC
Due to these
does not represent the internal supply of log
class i by the timber division.
The solving of the log allocation
problems through internal pricing by using diagrams is impossible
because of the resulting multidimensionality.
Calculus
Finding the optimal allocation and transfer prices by solving
the simultaneous equations
MR1 = MCi;
is difficult.
I
= l,...,I
The solution becomes even more bothersome when there
are many mills with many end products as in our log allocation problem.
This is one of the main reasons why practical men in forest
productsfirms believe that economic theory has little relevance for
log allocation.
The theory's elementary acsumption is usually that
a firm has a single input and a single output (cf. Hadley, p. 481).
Where the theory proves the multi-input case in full generality,, the
analysis is frequently too complex or too empty for use (cf. Hirschleifer, (b), p. 96).
The statement, said about 20 years ago, is still
valid despite the advances of calculus for solving multi-input, multioutput production problems.
Unconstrained profit maximization is
46
unable to solve complex allocation problems.
Even constrained
optimization may be impossible through the traditional means--calculus.
A good example of the deficiency of calculus is its inability to
solve constrained or unconstrained log allocation problems with linear
or piecewise linear harvesting or processing functions (cf. Henderson
and Quandt, p. 334).
Mathematical Programming
Mathematical programming techniques have proved to be applicable
for solving complex constrained resource allocation problems.
Since
this discovery economists have been mapping the similarities and dis-
similarities of the traditional marginal, and mathematical programming
analyses in profit maximization.
Dorfman, etal.(pp. 183-4) have come
to the conclusion that the optimal resource allocation through linear
programming, for example, is equivalent to that through the marginal
economic analysis.
Shadow Prices As Transfer Prices
For every mathematical programming problem, which we call a
primal, there is a dual problem.
The dual variable values of basic
feasible primal solutions can be interpreted as "shadow prices". A
positive shadow price in a primal maximization problem indicates a
revenue to be gained through adding one unit of a scarce resource. A
linear or nonlinear log allocation problem for a forest products firm
with constrained log availabilities of log classes produces shadow
47
prices for the log classes.
The shadow prices indicate the marginal
values of the different log classes as limited inputs.
The shadow
prices of the log avilability constraints would then be announced as
transfer prices by the headquarters to the decentralized divisions
(Arrow, (a) pp. 9, 13; Arrow (b) p. 405; Dowdie, p. 93; Koopmans, p. 73;
etc.).
This idea has originated from the theory of resource allocation
of a decentralized socialist national economy.
The theory has produced
the following so-called Lange-Lerner rule:
"To attain maximum social welfare in a decentralized socialist
society, the state planning agency should solve the constrained
maximization problem and obtain the shadow prices of all inputs
and outputs.
Publish this price list and distribute it to all
members of the society.
Instruct all consumers and all plant
managers to behave as though they were satisfaction or profit
maximizers operating in perfectly competitive markets."
(Ferguson, p. 447; cf. Gordon, p. 20).
Shadow Prices Plus Variable Costs As Transfer Prices
Setting shadow prices as transfer prices has met criticism by
theorists who have actually been working with practical problems.
This criticism deals with how to use shadow prices as transfer prices.
Appendices 4.1 and 4.2 discuss more the difficulties of computing
shadow prices and alternative ways of using them as transfer prices.
Dopuch and Drake (p. 345) state that announcing shadow prices as
48
transfer prices would lead the divisional managers ". . .to select the
types of outputs" (logs by timber division; end-products by mills)
"which should be produced, but the exact levels of outputs must be
determined in alternative means."
Solomons (p. 187) does not recommend
the use of shadow prices in short term allocation.
But when used, the
transfer price should equal to the sum of shadow price and procurement
variable cost (Solomons, p. 191).
According to him a mathematical
programming dual solution tells that the intermediate products (logs)
are worth their shadow prices over their variable costs.
When shadow
prices are zero, transfer prices are equal to procurement costs.
General Rule For Transfer Pricing.?
The disagreement between the men of theory (Arrow, (b) p. 405;
Dowdle, p. 93; Koopmans, p. 73), and Solomons (p. 191) confirms the
nonexistence of a general all-pervasive rule for transfer pricing in
the great variety of conditions where the forest products firms work
today.
Dopuch and Drake (p. 341) stress that the transfer prices should
"reflect the opportunity costs of the intrafirm transactions'.
The
executives could consider the following general rule as the first step
in the allocation analysis:
The transfer prices should be equal to
additional variable loq procurement costs plus opportunity costs of
internal deliveries for the firm as a whole.(cf. Horngren, p. 733),.
The shadow price of Solomons obviously is equivalent to the opportunity
cost of internal deliveries.
The shadow prices of Arrow, Dowdle and
Koopmans have some other meaning.
49
When there is a competitive log market the opportunity cost of each
log class is its market price less its harvesting cost (cf. Anthony et
al., p. 274).
This is in accordance with the axiom of Solomons and the
general rule of Horngren.
Koopnians (p. 93) is the first economist to
elaborate the result that for one or several intermediate goods the
transfer price should always equal the market price if a competitive
market price exists.
Most decentralization theorists have agreed with
this result although there are few formal proofs (Anthony etal.,
p. 274; Arrow (b), p. 404; Cook, p. 88; Dopuch and Drake, p. 341;
Hirschleifer (b), p. 32; Solomons, p. 199).
Practical men do not always agree with the market price rule.
Many of the Pacific Northwest forest products firms applying marketbased transfer pricing use slightly lower than log market prices.
Most companies apply cost-based transfer pricing.
The theorists
usually assume few interdependencies between the divisions and a log
market with infinite demand and supply at a constant price.
In forest
products firms these interdependencies between timber division and
mills are great and neither in the short nor in the long run the
competitive log market can supply or receive infinite amounts of logs.
Market prices may in many cases be the most desirable transfer prices
when competitive log market exists.
In the following chapter we wish
to see whether this is true for a typical , hypothetical forest products
f i rm.
50
4.
4.1.
EXPERIMENT FOR TESTING THE HYPOTHESIS OF THE THEORY
CASE FIRM
The hypothesis of this study is that the economic theory of
transfer pricing holds for a typical, decentralized, corporate profit
maximizing forest products firm.
The theory suggests that the optimal
log transfer prices should equal market prices if a competitive log
market exists.
The experiment for testing the hypothesis consists of
three parts: constructing a case firm with its production and market
functions, formulating the log allocation problems and solving the
problems for alternative transfer prices.
The corporate profit maximi-
zing case firm has been constructed with Mr. Larry Chapman's cooperation using Bohemia Inc. 's veneer plant and sawmill in Coburg, Willa-
mette Valley, Oregon,as a basis.
The case firm's milling division
consists of these two plants with their capacities, approximate
recoveries, variable and fixed processing costs, and taxed.
The case
firm's timber division consists of the types of the firm's own and
public timber sale tracts which are available for the Coburg unit's
use.
Examples of the actual numbers and sources of physical tract
qualities, logging and transportation costs are listed in the appendices.
As Figure 4.1 shows we now have a more complex problem than
economic theory could easily handle.
It consists of five log classes:
four usable by the mills and one pulpwood class.
There are 21 public
51
(Respoiisibi 1 ity
botndary)
lIMBER DIVISION
MILLING DIVISIOII
Sources of logs
through harvesc
Market
1
(Unfil led
orders)
Veneer plant products:
7 veneer and
residue grades
Market 2
( Ant IC ipa ted
LLLI
5
orders)
5 log
I
classes
Saeinill products:
6
21
market)
1 uinbe r and
larket
(Unfi lied
orders)
1
sale tracts
9 own fores
tracts
ses
/
residue grades
public
timber
Slog
/ Farkd 3
/ (Future
/
I\
I
Market 2
(Ant ici-
\
pa ted
orders)
arket 3
(Future
market)
,"-
1T'
Outside /iog market:
4 log class purchase 'murket
S log class sales /
market
(possible responsibility
(possible responsibility
boundary)
boundry)
Figure 4.1.
Case
finn log allocation problem.
52
timber sale tracts and nine company-owned forest tracts ready for
harvest in the quarter.
The destinations of these logs are the
companys veneer plant, sawmill or outside log market.
There are four
veneer and three residue grades of the veneer plant products.
The
sawmill produces four sawntimber grades and two residue grades. There
are three veneer and lumber markets, as a function of time:
-- unfilled orders for which exact prices and quantities are known
-- anticipated orders with estimated prices, and known minimum
quantities
-- future market: producing for inventory with estimated prices, and
inventory capacity as maximum total quantity.
The log allocation decision is made for one quarter while for the
longer term harvesting decisions a maximum period of five years is
considered in order to estimate the possible opportunity losses of
harvesting tracts during the coming quarter instead of later in the
future.
The computations are carried out with historical 1967-71 pric-
es and costs instead of 1975-79 prices in order to avoid making price
predictions.
Predicting prices could be an objective for a separate,
comprehensive study.
1967 has been chosen as the basis year because
we want to avoid the market irregularities at 1972 government price
control s.
53
4.2.
PROBLEM FORMULATION
Centralization
The ideal transfer prices of logs result in the same allocation
and corporate profit as the globally optimal solution of the log
allocation problem for a centralized firm under perfect knowledge.
Thus we have to formulate and solve a corporate log allocation problem
given centralization.
This produces a target for allocation and
corporate profit through transfer pricing given divisional dominances.
In a centralized firm there is one log allocation problem for
the firm, formulated and solved by the top management to maximize the
corporate profit.
The top management's log allocation decisions are
executed by the nonautonomous timber and mill departments.
As Figure
4.2.a shows little internal information exchange is needed within the
firm because top management is the only decision maker.
Transfer
prices are not necessary because there are no internal log transactions
between profit centers.
The information "exchange" contains only the
one way flow of commands to the cost center departments dictating the
quantity and quality of logs to be allocated from different sources to
different destinations.
A centralized firm's log allocation problem
can be divided into three parts:
availability and employment of internal logs
availability and employment of external logs
availability and employment of other inputs; processing and
marketing of end products; selling of internal logs.
54
This partitioning emphasizes the separation of logs as a special
'corporate resource' from all other resources of the firm (cf. Jennergren, p. 21).
It is the approach used for developing the corporate
log allocation model for the case firm in Appendix 1.1.
Another
possible partitioning is:
intrafirm part:
external part:
internal allocation functions
log market conditions
The internal part deals with hiring and employing all inputs
except external logs for harvesting and milling activities.
The out-
come of the firm's processes--the sales of end products--belongs to
this part also.
The external part describes the prices of logs to be
sold to or purchased from the competitive outside log market and the
quantities of logs in the market available for purchase.
This type of
partitioning draws our attention to the firm's self sufficiency or
dependence as a decision unit on the external world in its log allocation decisions.
It is especially useful in formulating log allocation
problems for decentralized organizations.
Divisional Dominances
Log allocation in a decentralized firm requires information
exchange between the decision units.
An information exchange iteration
starts with the announcing of (1) the tentative transfer prices by the
top management.
At the same time the dominated division delivers (2)
information of the bounds of its technically feasible log transaction
solution set to the dominant division.
These bounds appear as
orofi t
corpo-ate
I
-
N
N
N.
N.
N
'trans far
'N .prices
-
profi ts".
..
.-"
profit
milling
division
divisional
l
-
.
I
I
ing
e
log
internal
(_..giirchases
demand
Figure 4.2.b
sales
internal deliveries
division
ivisio,ivi
ti:vbcr
supply
of mt.
es
log narket
'id"oduc t
markets)
log market
sales "-""
'
markets)
(eoduc
(sales)
deliaeries
- 1-- .profits lrestraints1enforc-
transfer'
-
log
irit.
Figure 4.2.a
departisent
L. timber
I
corporate profit
demand
-supoly
- - - -= -I
prices,.
headquarters '
headquarters
internal
enforcing
department
milling
-
I
N
Notation:
''
enforcing
N
"
supply
Figure 4.2.c
profit
division
divisional
timber
demand
I
V
log
internal
purchases
purchases
_
market)
-
log market
markets)
'1'
log market
N
---i information flow;
). log flow
lnformation'and lcp floms in ccntrnlized organization
(Fig. a), under mill (Fig. b), timber (Fig. c) and mixed
Figure 4.2.d
div.pfit
sales
internal deliveries
demand
division
1lT:nber
supply
ing
enforc-
p'c fit
estr.
of
div
iv jill sq
dlvi sion
p
I
dominance (Fig. d).
'N
\. "i
N
transfer
log
I nternal
lr)
(of iIit. internal deliveries
traiintsi
1res
rofits lint.
// /
V
"
'N
\prices
profits
L9iLJi"
Figures 4.2.a-d.
sm-P-
N
trans fey
prices
profits 'N
'N
N. .grices
N.transfer
'N
4N
--
headquarters L/
profi t
Corpora to
headquarters
1111111 ny
divis ion
7 divisional'
transfer,profit
prices -profits
1'
56
allocation guidelines or restrictions to the dominant division.
They
guarantee that every allocation decided by the dominant division is
feasible to the dominated division.
The dominant division solves its
log allocation problem given the transfer prices and the log guidelines
by the dominated division.
It announces (3) the enforcement of its
decisions to the dominated devision.
The dominated division solves its
log allocation problems given the transfer prices, and log enforcements
by the dominant division.
The iteration ends by the divisions report-
ing their (4) divisional profits to the headquarters.
The next
iteration starts by the announcement of another set of tentative transfer prices for logs.
allocation follow.
Steps (3) and (4), guidance and enforcement of
Step (2) is not affected by the transfer price
levels and does not need to be repeated.
The iterations continue until
the top management has announced all its alternative transfer price
sets.
At the implementation the top management chooses the set that
gives the highest sum of the divisional profits.
est corporate profit.
set to the division.
This sum is the high-
The top management enforces the transfer price
Finally, the divisions execute their allocation
optimal with respect to the transfer prices.
Figure 4.2.b-d show all four types of information flows under
mill, timber and mixed dominances.
It also shows the lo
flows at the
implementation of log allocation through transfer pricing.
The information flows of Figure 4.2.b under mill dominance in the
chronological order of their occurrence are:
57
transfer prices by the top management
restraints of internal log supply by the timber division to
the milling division
enforcement of internal log demand by the milling division to
the timber division
4)
divisional profits from the divisions to the headquarters
Correspondingly, the log flows under mill dominance are:
1)
internal deliveries of harvested logs by the timber division
to the milling division
2)
sales of harvested logs by the timber division to the outside
market
3)
purchases of external logs by the milling division
The information flows of Figure 4.2.c under timber dominance in the
chronological order of their occurrence are:
transfer prices by the top management to the divisions
restraints of internal log demand by the milling division to
the timber division
enforcement of internal log supply by the timber division to
the milling division
divisional profits from the divisions to the headquarters
Correspondingly, the log flows under timber dominance are:
internal deliveries of harvested logs by the timber division
to the milling division
sales of harvested logs by the timber division to the outside
58
market
3)
purchases of external logs by the timber division
The information flows of Figure 4.2.d, under mixed dominance are
the same as those under mill dominance; correspondingly, the log flows
under mixed dominance are the same as those under timber dominance.
(This is the reason for naming this organization "mixed dominance". )
Every divisional log allocation problem can be divided into four
parts according to the divisionts self sufficiency or dependence on
the outside world in its log allocation decisions.
Accordingly, there
are four sources of data to a divisional allocation problem:
intradivisional part:
internal allocation functions
interdivisional part:
--allocation guidance received by the dominant division from
the dominated division or
--allocation enforcement received by the dominated division
from the dominant division
external part:
control part:
log market conditions
transfer prices
Figure 4.3 shows the six divisional allocation problems of this
study and the data needed for them.
Purely divisional data to be used in the milling division allocation problem are end product market prices, milling costs, endproduct recoveries and processing times productivities of different
classes of logs and outstanding end product orders.
Purely timber
as
rcduct
r.g
Figure 4.3.
end :roduct
croers
prOcessing
times
reco.er:as
ccsts
ri 11
orices
v'arket
er.d
purely
iisicnal
pu rc na S
log
maxmum
/
cc
dric
under mid
solve allocation problem
under
(solve allocation problem
1'
dminvncn
)under nih
/
/ solve al boa- \
-tior problem
1.
D IV IS r ON
1ILLI:G
log
\Purchases
.._(
/raxirum
log class
harvests
n:a xi :v
minimum
(end maximum) total
harvest
I
\
t ran s'
price
log
/log class
\ harvests
log
I
Li
I
0
I
/solve allocatio
problem under
mixçd dominance
rnce
ocr timber
ahlocation problem un-
5.
\ocinance
under nih
ha rues t
minimum and
maximum total
able in
tracts
class avail-
maximum log
costs
1og selling
costs
harvesting
purely
divisional
data:
/
/
rarkt /
prices.
/ maximum
/ log
purchases
/
O
allocation enforceiunt from dominant
allocation guidance fro:,i dominated
control data:
divisions
transfer prices from hcadquarters to
external data of log market condition
interdivision-il data:
to dominated division
interdivisionel data:
to dominant division
intradivisional data for the internal part of the
allocation problen
divisional allocation problem under divisional dominance
Nototi on:
/soive ahboca-
4
DIV IS tO:
T IbiS ER
\. tics problem
//
(mills
log dnends
requi rem.
total log
mi 11
requ i rem.
\ class
/ lii
/mills
demands
(sills
St
/maximun
ha rv e
ruin) total
(and maxi-
Six divisional allocation problems and the data
needed for solving them.
-
,ark7L
60
division data are harvesting costs.and log class availabilities in
different tracts, smallest and largest wanted total harvest during
theperiod and log selling costs.
Guidelines to the mills under mill andmixed dominances are the
restrictions of iiii nimum and maximum total harvest and maximum log class
availabilities in tracts through harvest.
Under mixed dominance the
mills have also to know the approximate maximum log calss availabilities to timber division of logs in the outside market.
For the dominant
timber division guideline restraints consist of minimum log class
requirement to complete the unfilled edn product orders and mill total
log requirement to help mills running at desired capacity.
Without
these guidelines an allocation suggestion by the dominant division may
lead to infeasible solutions in the dominated division.
Any allocation
infeasible to a division is undesirable for the firm as a whole
because it, i.f implemented, forces the division out of business.
Enforcement restraints are mills' log demands under mill and mixed
dominances and timber division log supply under timber dominance which
the other party has to accept.
Market conditions appear in the
allocation problem formulation of the division
purchase logs.
that has the right to
They are the log market prices and maximum log class
availabilities of purchased logs
In a decentralized firm, given a divisional dominance, log alloc-
ation consists of two problems for the two divisions and the corporate
'mini" problem of adding divisional profits to find a corporate profit
corresponding to a transfer price.
Headquarters might choose only
61
one transfer price and the.corresponding allocation would be implemented.
Headquarters might announce several tentative transfer prices,
choose the one which produces the highest sum of the divisional
profits, and the allocation corresponding toit is implemented.
is the procedure in this study.
This
In order to find the optimal transfer
prices among those in use in today's forest products firms we compute
the corporate profit for each of them using the divisional log allocation problem formulations of Appendix 1.2.
Solution Techniques
Linear programming has been used to formulate and solve each of
the allocation problems.
The detailed formulations for, the firm as a
whole under centralization and for both divisions under decentralization appear in Appendices 1.1 and 1.2.
For our log allocation problem
the underlying assumptions of linear programming either hold or can be
worked around by special formulations.
linearity, additivity
The well-known assumptions of
divisibility and determinism are not restrictive
in our case firm (Hillier and Lieberman, pp. 136-138).
and harvesting the functional relationships are linear.
has particularly asked the practical men:
bach Co.,
Both in milling
The author
Don Baack of Crown-Zeller-
Adam Ferrie of McMillan Bloedel, Ltd., and Ted Nelson of
Weyerhaeuser Co,m whether there are serious nonlinearities in short
term log allocation problems as ours. They said they have not found any.
The additivity assumption seems according to them hold to a satisfact-
62
ory degree.
For example in milling all combinations of labor and logs
have constant marginal productivities: one unit of labor and logs
results in one unit of end products, hundred units of labor and logs
result in hundred units of end products, etc.
Divisibility assumption
causes a difficulty in harvesting decisions because from a tract no
amount of a certain type of logs can be cut without cutting a portion
of other log classes.
This problem is removed by adding a set of tract
log class structure constraints as in Appendix 1.
The deterministic
nature of linear programs can be overcome through repetitive "sensiti-
Studying log allocation under uncertainty in this
vity" computations.
way is expensive.
For every particular allocation problem we know,
however, that if formulated properly and if the other three assumptions
hold the resulting best basic feasible solution is optimal.
not necessarily the case in simulation approach.
This is
Attempts to create
an allocation model in FORTRAN showed that simulation model of this
type becomes complex.
Thus the first successful run might be very
expensive although successive runs have low additional costs.
Due to
the apparent advantages of linear programming it has been the major
operations research tool for solving log allocation problems (cf.
Bare, pp. 9, 28).
Linear programming is used also in this study to
solve both the corporate and divisional programs of Appendix 1
63
4.3.
PROBLEM SOLVING
Objectives
In this section we assert the amount and type of computations that
are necessary to reach the objectives set for the case study.
The main
objective of our experiment is to test the hypothesis whether transfer
prices should equal to market prices under certainty in a typical corporate profit maximizing forest products firm.
For the three division-
al dominances we first find the optimal transfer pricing type :market,
cost, value, or distorted market pricing.
We call the corporate profit
maximizing pricing type "optimal" because we exhaustively study all
the types of transfer pricing appearing in the literature and in the
practice in the forest products firms today.
We have had plenty of ob-
servations to find that, in varying situations, some pricing type
is
always the very best--optimal--for a divisional dominance.
After finding the optimal transfer pricing types we study the
sensitivity of the corporate profit to the changes in the transfer
price levels of the optimal pricing types under the three dominance
organization.
We call the corporate profit maximizing transfer price
level "best" because this level changes when log allocation conditions
vary and there is no "optimal" transfer price for all situations.
best transfer price level is especially sensitive
internal logs available through harvest.
The
tO the amount of
Comparing the corporate
profits of the best levels of the optimal transfer pricing types of
64
the three organization we find the ranking of the divisional dominances.
Observing whether the best transfer price level of the optimal
transfer pricing type within the highest ranking organization is
equivalent to log market price or not we finally conclude whether our
hypothesis holds or not.
Steps Of Problem Solving
The f i r s t
s t e p
in arriving at the objectives of this
study is to generate the data for the allocation problems. They are
the same for the corporate linear programming problem under centralization of Appendix 1.1 and the divisional linear programming problems
under divisional dominances of Appendix 1 .2.
several sources as Appendix 2 shows.
The data come from
The computationally most complex
of these data are the user costs of tracts.
When summed with the
logging and transportation costs they are used for deciding which
tracts to include in the linear programs as shown in Appendix 2.3 and
in the beginning of the next chapter.
The s e c o n d
s t e p
program under centralization.
is solving the corporate linear
It produces the target allocation and
corporate profit for the transfer pricing.
This target is called the
globlly optimal solution.
The t
h
I
r d
s t e p
is to name the transfer prices to be
used in the divisional log allocation problems under decentralization.
They are chosen so that for each divisional dominance the optimal
65
transfer pricing type can be found.
Since the study uses 1967 prices
and costs the 11 transfer price sets for the four log classes are 1967
values also.
These represent all major transfer pricing types currently used
by Pacific Northwest forest products companies (all types show unit
prices for four log classes)
A.
Log market price based transfer prices:
TP1 = log market prices ($100, 90, 68, 58)
TP2 = 10 per cent higher than market prices ($110, 99, 75, 64)
TP3
10 per cent lower than market prices ($90, 81 , 61, 52)
TP4 = $6 lower than market prices ($94, 84, 62, 52)
B.
Harvesting cost based transfer prices:
TP5 = logging and transportation cost($36.5,36.5,
36.5,36.5)8)
TP6 = logging and transportation cost plus allowance for timber division fixed costs and profits ($64,64,64,64)
C.
Combined harvesting cost and market price based transfer
prices:
TP7 = logging and transportation
cost plus allowance for
Standard cost for the period
Standard timber division fixed cost and taxes are $15,100 and
the minimum timber division profit allowance before stumpage
costs for the coming quarter is $265,400.
log deliveries to the mills are 10,200 MBF.
The acticipated
66
timber division fixed costs and profits in market price
ratios ($94, 85, 64,
D.
54)10)
Log mill value based transfer prices:l1)
TP8 = veneer plant log values ($114, 94, 76, 70)
sawmill log values ($111, 107, 84, 82)
TP9 = mill log values minus allowance for mills' fixed costs
and profits:
veneer plant log values ($96, 76, 58, 52)
sawmill log values ($91, 87, 64, 62)
E.
Transfer prices with distorted relative market prices:
TP10 = 10 per cent higher for the two best and 10 per cent
lower than market prices for the two poorest log
classes ($110, 99, 61, 52)
TP11 = 10 per cent lower for the two best and 10 per cent
higher than market prices for the two poorest log
Log class 3 price is used as a basis because in the past
its portion of the total harvest has been over 55 per cent.
Log mill values before fixed milling costs and taxes for
the last quarter of 1966.
Standard fixed costs and taxes
for the coming quarter are $25,200 for the veneer plant and
$37,800 for the sawmill.
$36,000 and $84,600.
and 6800 MBF.
The minimum profit allowances are
The estimated mill usages are 3400MBF
67
classes ($90, 81, 75, 64)
The transfer price alternatives have the following characteristics:
TP1
The listed four prices represent the log market prices
of the first quarter of 1967 for the four log classes
usable by the mills
TP2, TP3
A resource oriented top management may argue that
transfer prices somewhat higher (TP2) than market
prices should be used because milling division is gaining from having a secure, steady flow of logs, coming
from timber division.
Corresponding by a market orien-
ted top management may defend transfer prices somewhat
lower (TP3) than market prices because timber division
is gaining from having milling division as its permanent customer.
In both cases the top management consid-
ers it important to maintain the relative market prices
of the different classes of logs in the transfer prices.
TP4
A very common practice is setting transfer prices less
than log market prices by a constant.
TP5, TP6
The most cornon pricing practice of forest products
firms favoring mills in decision making is setting
transfer prices to equal to the average harvesting
cost.
Harvesting costs do not significantly vary from
log class to log class thus also their transfer prices
are equal
68
TP7
One pricing possibility is to combine cost and marketbased transfer prices.
The transfer price of the
average log class (class 3) would be the average
harvesting cost but for each log class the transfer
price would follow its relative log market price.
TP, TP9
The log mill value-based transfer prices, used by
firms that favor timber division for decision making
are difficult to determine.
Their accurate computation
requires advanced knowledge of end product prices,and
end product shipments, recoveries, and processing
costs.
That is why historical data, not future proje-
ctions, are usually used for computing them.
TP10, TP11
A "radical" top management may want to exaggerate the
market price difference between high and low quality
logs.
This is done in the transfer prices of the log
classes in TP10.
A "conservative' top management may
wish to even out the market price differences between
log classes, as is done in TP11.
The f o u r t h
s t e p
is to solve the divisional linear
programs for the 11 transfer prices.
Appendix 3.1 shows the procedures
for creating inputs for these programs.
Appendix 3.2 presents
examples of the linear program inputs and outputs.
Based on the
resulting corporate profits the optimal transfer pricing type is found
for each divisional dominance.
69
The f i f t h
s t e p
is to name a range of price levels of the
optimal transfer price types.
The divisional linear programs are
solved for them and the best transfer price level is found for each
divisional dominance.
From the results above we can decide which of
the three divisional dominance types ranks highest.
Finally, we conclude whether the hypothesis of economic theory
holds for the case firm or not:
price or not.
should the transfer price be market
This is done in the
price analysis under certainty.
s
i
x t h
s t e p
of our transfer
70
4.4.
RESULTS
The log allocation computations have been carried out in the order
mentioned in the previous section.
section follow the same order.
Therefore, the results of this
All the outcomes of the transfer pric-
ing analyses are recorded as the differences between the log allocation
corporate profits of the globally optimal solution and the summed
We
divisional profits of the allocations through transfer pricing.
recall that the globally optimal solution is the target produced by the
The
centralized firm's optimal allocation under perfect knowledge.
allocations through transfer pricing are the allocations produced by
the alternative transfer prices in the decentralized case firm under
mill, timber or mixed dominance.
The outcomes then depict the
corporate profit losses from decentralization.
More specifically, they
represent the impact of dysfunctional decisions made by the divisional
managers.
They are caused by the lack of goal congruence between the
divisional managers and top management.
Other advantages or disadvan-
tages of decentralization do not appear in the results of this study.
They belong to the sphere of psychology and are hard to measure.
task here is
The
Lo find the transfer prices that minimize the measurable
profit losses of decentralization under the three dominances.
A
summary of the globally optimal log allocation is presented in the
last column of Table 4.3 and the globally maximum profit in the last
row of Table 4.2.
71
Harvesting Decision
The first step in the log allocation computations is to estimate
the user cost of each tract.
It shows the opportunities lost in the
future periods if the tract is harvested in the coming quarter.
computational details are presented in Appendix 2.3.
Adding user cost
to the tract logging and transportation cost produces the
cost of a tract.
The
harvesting
Column 5 of table 4.1 shows the average harvesting
The tracts are ranked
cost of all 30 tracts under consideration.
according to rising average harvesting cost.
The harvesting costs are
used for two purposes:
for narrowing down the number of tracts to be considered for
harvest in the linear programs
as objective function cost coefficients of the harvestable
logs in each tract in the linear programs.
Pilot computations have indicated that in the linear programs the
harvesting of a tract depends greatly on its harvesting cost.
They
have shown that at any alternative log transfer or market price of this
study at most the nine tracts with the lowest average harvesting costs
of the 30 may be cut in the next quarter.
Thus in the linear programs
only these nine or fewer tracts are included to reduce the costs of
computations.
4.1.
A line separates the nine tracts from others in Table
The tract harvesting revenue consists of three components: log
class transfer prices, market prices and volumes.
The results of the
linear programming computations show that the harvesting schedule is,
72
Table 4.1.
Tract
Number
6
1
10
5
4
18
23
13*
11
17*
14
9
12
3
20
25
28
22*
19
2*
27*
24*
8
21*
30
29
16
26*
7
Note:
Results of the harvesting cost computations: choosing
the nine tracts with the lowest harvesting costs for
the linear programs.
Logging and
Transportation
Cost, $/MBF
28.00
29.00
22.00
31.00
28.00
34.00
26.00
19.00
21.00
28.00
31.00
25.00
23.00
36.00
39.00
24.00
30.00
35.00
34.00
33.00
31.00
34.00
33.00
38.00
38.00
38.00
38.00
36.00
24.00
36.00
User**
Cost,
$/MBF
Tract Rank
Quarter with
Harvest(i) Acc.
Highest Disc. ing Cost, to Incr.
Tract Value
Harv. Cost
$/MBF
7
28.00
29.00
31.17
33.69
36.33
36.68
37.47
42.56
43.88
44.26
44.55
44.84
7
45.31
5
46.54
47.33
48.55
48.70
49.63
53.78
55.96
56.30
58.49
60.85
62.62
64.04
65.03
0
1
0
1
9.17
2.69
8.33
2.68
11.47
23.56
22.88
16.26
13.55
19.84
22.31
10.54
8.33
24.55
18.70
14.63
19.78
22.96
25.30
24.49
27.85
24.62
26.04
27.03
10
8
10
27.91
7
42.90
56.23
80.47
9
5
3
5
3
6
8
7
19
6
5
8
8
6
7
8
10
10
8
9
9
65.91
78.90
80.23
116.47
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
*Conlpany owned tracts
**Ifighest discounted tract value over 20 quarters less the
highest value in quarter 1
73
indeed, veryinsensitive to changesin harvesting revenues: log transfer and market prices, and rather insensitive to changes in tract
log class composition.
The 11 transfer prices produce at most two
kinds of harvesting schedules for each divisional dominance.
The more
common schedule includes the tracts with the lowest harvesting costs.
The other schedule differs froni it by only one tract, by the marginal
tract which is accepted for harvest last, and may be cut only partially.
The form
of the timber division linear programs guarantees that
whole tracts are harvested except the marginal trat.
The tract log
class structure constraints (e.g. constrains (2) of Appendix l.2.(B))
secure that the log classes even of the marginal tract are felled in
proportion to their volume.
It is economically infeasible to cut only
one log class from a tract with several classes.
The company's own and public timber sales tracts are
treated
equally in the computations, except that the company tracts are avail-
able for harvest a total of 20 periods because the assumed company's
cutting plan is for five years, but as an average the public timber
is available for harvest only from three
to nine quarters.
The user
cost tends then to be higher for own timber tract than kublic tract
lowering the own tract's ranking for harvest.
tract among the ten
There is one company
highest ranking tracts for harvest of Table 4.1.
Tracts available only during the coming quarter and the partly cut
marginal tracts from the previous quarter would always be harvested.
Our case firm does not, however, have such tracts.
74
In the short term log allocation of the case firm the (sunk)
stumpage costs are excluded from the harvesting costs.
Due to this
exclusion the procurement costs of internal logs tend then to be lower
than those of external logs which are purchased at the market price.
This causes internal logs to be preferred to external logs.
The
maximum harvest constraint (e.g. (3a) of Appendix l.2.(C)) is for every
run of the experiment at the upper limit indicating that it is the
most efficient element in determining how many tracts are cut.
Optimal Types Of Transfer Pricing
Mill Dominance
Under mill dominance market-based transfer prices.minimize the
corporate profit losses from decentralization.
Table 4.2 and Figure 4.4.a on p. 73.
This we can see from
In the table, column 3 presents
the summary of the linear program outputs: the before fixed cost, tax
and stumpage corporate profit as a sum of the divisional profits.
The
transfer prices are ranked according to these profits in column 5.
Market-based transfer prices obtain ranks 1-5.
lowest ranks.
Cost pricing gets the
The ranking of the transfer prices according to profits
would not change from the above after fixed costs and taxes because
they are the same for each alternative.
It does not change after
stumpage, either, because including different stumpage prices cause
only minor deviations in the harvest pattern.
The computations have
shown that under mill dominance the corporate profit is quite sensi-
75
Table 4.2.
Profits with alternative transfer pricing types under
mill dominance, dollars.
Profits before stumpage,
fixed costs and taxes
Mill
Timber
division
Firm
division
Transfer
price
andtype
Market
pricing
Cost
pricing
Cost/market
pricing
Value
pricing
Distorted
market
pricing
3
Profit
loss from
decentralization
Rank
4
2
TP1
191 ,682
389,981
581 ,663
-2,625
4
TP2
156,341
424,169
580,510
-3,778
5
TP3
233,573
349,749.
583,322
-966
3
TP4
226,255
357,068
583,323
-965
2
TP5
501,333
53,820
555,153
-39,135
11
TP6
254,101
309,819
563,920
-20,368
10
TP
215,806
367,520
583,326
-962
1
TP8
132,270
444,947
577,217
-7,071
8
TP9
222,179
358,062
580,241
-4,047
6
TP
210,756
355,573
566,329
-17,959
9
200,915
378,027
578,942
-5,344
7
7
10
TP11
Centr.
CX)
alloc. =
global
opt.
*Note:
5
1
..
..
584,288
0
= difference between column (3) profit and the globally
optimal profit (X) of centralization.
MB F.
logs
Saw -
logs
tx
0
4411
x32
x42
0
4411
4411
2101
0
2101
2101
331
x22
3072
3072
3072
331
0
0
339
o
0
339
3
TP3
o
339
2
331
11
1
TP4
xl2
1x
mill
Veneer
k
TP7
TP2
Transfer
TP9
rice
TP11
TP8
0
4411
0
4553
0
4273
4273
2101
0
1614
2577
331
2577
3072
3066
670
0
0
339
7
0
0
3072
3072
0
334
6
0
0
0
339
5
0
0
339
4
0
4553
1542
2512
2801
1714
0
0
3072
0
0
339
9
TP10
670
2421
0
334
8
Rank according to corporate profit
TP1
1 780
2284
2181
0
4245
0
808
331
0
3073
2363
3072
4272
0
2578
0
0
339
Centr. alloc.
= global opt.
911
0
104
11
T P5
0
0
339
10
T P6
Summary of log allocation to mills, with alternative transfer prices, under mill dominance,
Allocation, X.
j = log ciassjk
Table 4.3.
77
tive to the availability of internal and external loqs.
figure concern a
log need.
The table and
situation where total harvest exceeds mills' total
The optimal transfer pricing type under mill dominance for
our case firm is, however, always market pricing.
Figure 4.4.a illust-
rates the profit losses from decentralizations of column 4 in the
table.
It shows that profit losses from decentralization can be
in-
significant by choosing market-based transfer prices but very substantial especially with cost pricing.
Table 4.3 summarizes the allocations of the four log classes to
the mills.
The last column is the target:
the globally optimal
allocation of a centralized organization under perfect knowledge.
The
other columns show the transfer prices according to their corporate
profit ranking in Table 4.2.
The farther away is the allocation from
the global optimum the poorer is the corporate profit.
This rule
generally holds for the table although it only shows sutiaiiaries of
allocation, not the exact sources and destinations of logs moved.
Timber Dominance
Under timber dominance cost-based transfer prices minimize the
corporate profit losses from decentralization.
4.4.b show this.
Table 4.3 and Figure
In the table, column 3 again presents the sunuary
of the linear program outputs:
the before fixed cost, tax and stumpage
corporate profit as a sum of the divisional profits.
Cost-based
transfer prices obtain the highest ranking according to these profits
78
Table 4.4.
Profits with alternative transfer pricing types under
timber dominance, dollars.
Profit before stumpage,
fixed costs and taxes
Mill
Timber
division
Firm
division
Transfer
price
andtype
Market
pricing
Cost
pricing
Cost/market
Value
pricing
Distorted
market
pricing
1
2
3
Profit
loss from
decentralization*
5
1P1
149,355
405,914
555,269
-29,019
6
TP2
40,492
485,795
526,287
-58,001
11
TP
224,571
331,728
556,299
-27,989
5
TP4
210,697
343,994
554,691
-29,597
7
TP5
486,883
83,359
570,242
-14,046
2
TP6
206,451
363,877
570,328
-13,960
1
TP7
195,154
361,501
556,655
-27,633
4
TP8
-23,848
587,937
564,089
-20,199
3
TP9
170,286
376,349
546,635
-37,653
8
TP10
121,103
435,230
541,993
-42,295
10
TP11
106,776
437,834
554,610
-39,678
9
)
Centr.
alloc. =
global
Rank
4
(x)
..
..
584,288
0
opt.
*Note: = difference between column (3) profit and the global
optimal profit (X) of centralization.
79
Table 4.5.
Profits with alternative transfer pricing types under
mixed dominance, dollars.
Profit before stumpage,
fixed costs and taxes
Mill
Timber
Firm
Division
Division
Transfer
price and
type
1
/
Market
pricing
Cost
pricing
2
3
Profi t
loss from
Decentralization*
TP1
191,612
392,611
584,223
-65
2
TP2
118,385
465,841
584,226
-62
1
TP3
262,263
321,958
584,221
-67
4
TP4
253,177
331,045
584,222
-66
3
TP5
629,263
-107,303
521,960
-62,328
TP6
335,893
186,067
521,960
-62,328
TP7
235,980
345,735
581,698
-2,573
5
TP8
12,280
538,297
540,577
-43,711
9
TP9
209,977
353,974
563,951
-20,337
7
TP10
211 ,229
364,236
575,465
-8,823
6
TP11
197,559
361,779
559,338
-24,950
8
10
Cost/market
Value
pricing
Distorted
market
pricing
Centr.
alloc. =
global
(X)
..
..
584,288
0
opt.
*Note:
Rank
5
4
= difference between column (3) profit and the globally
optimal profit (X) of centralization.
2.000
5.000
10 .000
15 .000
20.000
Profit
loss.
2.000
5.000
10.000
15,000
20,000
Profit
loss. $
Value
pricing
C/M'
Cost
pricing
Market pricing
Figure 4.4.b.
TV5 TV9
1P7
TV5 TV5
1P1 IF2 IF3 104
Figure 4.4.a.
I
pri cing
C/H" Value
Cost
pricing
TV
Market pricing
TV
TV
ii
0
1P1 IF2 TP3TP.
r
0
transfer price (S/MOP)
Transfer price (S/MOP)
Distorted
nkt. pricing
TP1ØTP1
Distorted
mkt. pricing
0 P1 0TP1
2,000
5,000
10.000
15.000
20.000
Profit
loss. $
C/11'
TP7
Transfer price (S/MOP)
Distorted
mkt. pricing
TV10 TV11
cost based average transfer price for all log classes following
the relative Oarket prices of the classes
shaws the best transfer prices.
C/H
Value
pricing
IF3 TV9
Corporate profit losses frcri decentralization with
alternative transfer pricing types under mill (Fig. a).
timber (Fig. b) and mixed (Fig. c) dominance.
Fipure 4.d.c.
Market pricing
Figures 4.4.a-c.
P5 TV6
Cost
pricing
TP1 IF2 TP3 OP4
81
in column 5.
poorer.
The other transfer pricing types perform considerably
These rankings hold for profits after fixed costs, taxes and
stumpage.
Figure 4.4.b illustrates the profit losses from decentral-
ization of column 4 in the table.
It shows that profit losses from
decentralization can be significant with cost-based transfer prices
and very significant with the other pricing types.
The table and figure
deal with a situation where total harvest exceedes mills' total log
need.
The computations have shown that cost pricing is, however,
always the optimal transfer pricing type for our case firm under
timber dominance.
Mixed Dominance
Under mixed dominance market-based transfer prices minimize the
corporate profit loss from decentralization.
show
this.
Table 4.4.and Figure 4.4.c
In the table, the corporate profits of the first four
transfer price allocations approximately reach the globally maximum
profit.
The cost-based transfer prices rank poorest but also value-
and distorted market price-based transfer prices perform very poorly.
The rankings hold for profits after fixed costs, taxes and stumpage.
Figure 4.4.c illustrates the profit losses from decentralization of
column 4 in the table.
The table and figure deal with a situation
where total harvest exceeds mills' total log need.
The computations
have shown, however, that market pricing is always the optimal pricing
type for our case firm under mixed dominance.
82
Best Levels Of Transfer Prices
Mill Dominance
Extensive computations have helped to analyze the sensitivity of
the corporate profit losses of decentralization to the transfer price
Under
levels of the optimal pricing types in the three organizations.
mill dominance usually transfer price slightly below market price
minimizes the corporate profit loss from decentralization.
4.5.a shows this.
Figure
In the figure, the profit loss is at its minimum
when transfer price is lowered below that best level.
When transfer
price is greater than market price only moderate profit losses occur
until it is raised to a level where the mills are forced to curtail
their production.
The transfer prices of the four log classes follow
the relative market prices in Figure 4.5.a. Computations have been
carried out also with transfer prices of all log classes differing
from market prices by a constant(for example TP4).
They result in high-
er corporate profit losses of decentralization than the procedure of
maintaining the relative market prices of the log classes (for example,
TP3).
Therefore, in all the sensitivity analyses the transfer prices
follow the relative market prices of the log classes.
If the log market price is known the top management would choose
a transfer price slightly less than market price.
But if there is
It
any market price uncertainty the decision becomes more difficult.
is evident that a conservative top management might choose a transfer
price equal to or even greater than the expected market price.
It
2,000
5,000
10,000
15,000
20,000
loss, $
.Profi t
2,000
5,000
10,000
15,000
20,000
loss, $
Prof i t
cm
0
cm
cm
cm
0
transp. Costs
50
C-
C
to
a
0'
'5
Cd
*
6
6
NIP+
64
en
Ci
to
cm
70
en
0'
0
to
timber dlv. cost
¶tTcrage
0
en
to
Ci
cm
Figure 4.5.b.
5
cm
Ci
Figure 4.5-u.
,to
18
MPI-
price = $68/iBF
lP+
12
-0
N. C- C-C-to
0' Cd
cc
Cd N. to cm to
Market
ton
3'
*
Lflto
U
tOto
tip-rip-
.0
to
12
'IP-
to
0
Cd0to-n
to
18
C-
cm
0
6.5 0
Logging are
to
0
to
cm
24
Cd
0
C-
_
cc
7,
en
Li
Ci
cm
80
-
to
IP+ MP+
33
24
NJ
N.
cm
cm
85
-
cc
O
90
Fcc
0'
0
100
24
classes, S/MBF
for all log
Uote:
8
to
to
to
cfl
to
Cd
6
12
Figure 4.5.c.
6
*
Market*
price = $68/MBF
*
18
MP+
riP+
33
I-iP+
24
priCe, $/liBF
Log class 3 transfer
changes in market-based transfer price levels under mill
dominance (Fig. a) and mixed dominance (Fig. c), and costbased
transfer price levels under timber dominance (Fig. b) when total
harvest exceeds nulls total log need.
*
lP-f+MP+
to
to
U
Sensitivity of corporate profit losses from decentralization to
*
12
to
cc
* shows the best transfer prices.
Figures 4.5.a-c.
Transfer price
price, $/MBF
Log class 3 transfer
2,000
5,000
10,000
15,000
20,000
Profit
loss, $
84
does so in order to avoid the risk of high profit losses resulting from
a possible underestimation of the market price level.
The next chapter
discusses this phenomenon more.
The results of the pilot sensitivity computations have shown that
except on the market price levels the transfer price decision may
depend on the amount of logs available for the mills.
It depends espec-
ially on the level of the maximum allowable harvest because internal
logs are preferred to a external timber in our firm with lower average
log harvesting costs than average log market prices.
That is why an
experiment was set up to study the best transfer prices at three levels
of harvest.
The maximum amount that can be spent for external logs is
At an average log market price of $68/MBF
$350,000, a constant.
external purchases correspond to 50 per cent of the mills' total log
need of 10,200 MBF.
Thus the purchase-need ratio (P/N) is .50.
The
three harvest levels, chosen for the experiment, are 15,000 MBF,
9,500 IvIBF and 7,500 MBF including the pulp logs, not usable by the
mills. Without the pulp logs (l6 per cent of total volume) the harvest
levels are about 12,600 MBF, 8,100 MBF and 6,300 MBF.
Thus the harvest
-need ratios of usable logs to the mills (H/N) are 1.25, .80 and .60
Figure 4.5.a shows the corporate profit losses of decentralization with
varying transfer prices at the highest harvest level , when total
harvest exceeds mills' total log harvest by 25 per cent.
the next harvest level
,
Results for
when the total harvest is 20 per cent less than
mills' total log need, are very similar to those of the first harvest
Profit
2,000
5,000
10,000
15,000
20,000
loss, $
Profit
2,000
5,000
10,000
15,000
20,000
loss, $
nd
transp. cost
Logging
t
36 5
0
to
18
M?-
to
45
55
7
lIP+
0
18
MP+
64
Figure 4.6.b.
tiniber div. cost
*
75
Figure 4.6.a.
= $68
price
Market
I
en
tO
Total average
7
MP-
en
"S
85
F..
to
100
03
03
log classes, $/MBF
Transfer price of all
price, $/MBF
Log class 3 transfer
Prof i t
18
lip-
tO
to
03
=
*
n
rip+
18
0
Figure 4.6.c.
68
price
Market
4.
717
Ip-
en
0
"SO.
.
tO
to
0'
price
transfer
Optimal
tO
to
price, $/MBF
Log class 3 transfer
(J'1
cc
Sensitivity of corporate profit losses from decentralization to
changes in rnarket'based transfer price levels cnder mill diinance
(Fig. a) and mixed dc.-snance (Fig. c), and costbased transfer
price levels under tirber dominance (Fig. b), when total harvest
is much less than mills total log need.
Note: * shows the best 6ransfer prices.
Figures 4.6.a-c.
2,000
5,000 -
10,000 -
15,000 -
20,000
loss, $
86
level: transfer price slightly less than market price is the best one.
But with H/N =.60, when mills' total log need is 40 per cent greater
than total harvest, the best transfer price level is different.
With
purchase-need ratio of only .50 there are only ten per cent (.60 + .50
- 1.00
.10) of total logs excessive to the mills' need.
The change
in the best transfer price level for our case firm then occurs for
P/N =
.50 when .60H/N
.80.
In this situation--when logs are
particularly scarce--transfer price slightly above market price
minimizes the corporate profit loss from decentralization.
4.6.a shows this.
Figure
In the figure, profit loss is at its minimum when
the transfer price is ten per cent higher than the log market price.
It becomes substantial when transfer price exceeds this best level.
When transfer price is equal to or somewhat less than market price
only moderate profit losses occur.
To summarize the best transfer prices of mill dominance at
different harvest levels (Table 4.6.): When harvest exceeds or is
somewhat less than mills' total log need transfer price should be
five to ten per cent below market price.
When harvest is much less
than mills' total log need, causing extreme scarcity of internal logs,
transfer price should be about ten per cent higher than market price.
For the two lowest harvest levels all the log purchasing possibilities
(about 50 per cent of the mills' total log need) are exhausted.
There-
fore the "best" transfer prices above may not hold for them with a
higher, for example, infinite purchasing possibility.
The computations
87
indicate that with it the best transfer prices may he greater than the
above recommendations.
Timber Dominance
Under timber dominance usually a wide range of cost-based transfer
price levels minimizes the corporate profit loss from decentralization.
Figure 4.5.b. shows this.
In the figure, the profit loss is at its
minimum when transfer price varies between the smallest average
harvesting cost ($28-30/MBF) of a tract and the highest transfer price
($75/MBF) the mills can afford to pay before starting to work at less
than capacity.
The latter transfer price is greater than the total
average timber division cost ($64) which is the highest possible
'harvesting cost" in the case firm.
Thus any cost-based transfer price
is the "best" transfer price under timber dominance.
The choice of
the cost-based transfer price is independent of the fluctuating log
market prices, and also changes in harvesting costs.
This insensitiv-
ity is a good property in a world of price and cost uncertainty.
The
choice of the transfer price level is somewhat sensitive to changes in
the amount of logs available, especially in the amount of
logs available through harvest.
internal
A similar experiment as under mill
dominance has been set up to find the best transfer price at three
levels of the maximum harvest.
Thus the harvest-need ratios(H/N) of
usable logs to the mills are 1.25, .80 and .60.
purchase-need ratio (P/N) is a constant .50.
At the same time the
Figure 4.5.b, discussed
88
above, shows the corporate profit losses from decentralization at Fl/N
.80 the result is the same: any harvesting cost level is the 'best'
transfer price.
But with H/N = .60, when total mills' log need is 40
per cent greater than total harvest, the result is different.
The ch-
ange from a range of "best" transfer prices to a single best price
then occurs for P/N = .50 when .60H/N
.80.
In this situation--when
logs are particularly scarce--transfer price just below the highest
price the mills can afford to pay for the logs before starting to oper-
ate at less than capacity minimizes the corporate profit loss from
decentralization.
Figure 4.6.b shows this.
In the figure, profit
loss decreases slowly with rising of the transfer price to its best
level.
It becomes substantial when transfer price exceeds this best
level
To summarize the best transfer prices of timber dominance at
different harvest levels (Table 4.6):
With harvest exceeding or being
somewhat less than mills' total log need the transfer price can be any
average harvest cost between the lowest harvesting cost of a tract and
the maximum transfer price the mills can afford to pay for the logs
before starting to work at less capacity(MAX).
This price ($75)
is
higher than the total average timber division cost ($64) which
includes the (variable) harvesting costs, fixed costs and an allowance
for timber division
profit.
With harvest being much less than mills'
total log need causing particular scarcity of internal logs transfer
price (MAX).
For the lowest harvest level all the log purchasing
89
possibilities (about 50 per cent of the mills' total log need) are
exhausted.
Therefore the "best" transfer price above for it may not
hold with a higher, for example, infinite purchasing possibility.
Mixed Dominance
Under mixed dominance usually a range of market-based transfer
prices minimize the corporate profit loss from decentralization.
Figure 4.5.c shows this.
In the figure, the profit loss is at its
minimum when transfer price varies from twenty per cent below to
twenty per cent above the market price.
It becomes very substantial
when the transfer price is lowered below the best range.
When transfer
price is greater than market price only moderate profit losses occur
until it is raised to a level where the mills are forced to shut down.
The transfer prices of the four log classes follow the relative
market prices in Figure 4.5.c.
Pilot computations have been carried
out also with transfer prices of all log classes differing from market
prices by a constant
.
They result in higher corporate profit losses
of decentralization than the procedure of maintaining the relative
market prices of the log classes. Therefore, all the sensitivity
analyses of transfer price levels are done following the relative
market prices of log classes.
Log market price uncertainty does not affect the choice of a
transfer price because of the insensitivity
price to the exact market price level.
of
the "best" transfer
However,if there is extreme
90
uncertainty about the market prices, a conservative top management
might choose transfer prices lightly above the expected market price.
It does so because a possibly too low (for example, market price minus
$18) transfer price, results in greater profit losses than a possibly
too high (for example, market price plus $18) transfer price.
The choice of the transfer price level is somewhat sensitive to
changes in the amount of logs available, especially of the amount of
internal logs available through harvest.
A similar experiment as under
mill and timber dominances has been set up to find the best transfer
price at three levels of the maximum harvest.
Thus the harvest-need
ratios (H/N) of usable logs to the mills are 1.25, .80 and .60.
At
the same time the purchase-need ratio (P/N) is a constant .50.
Figure 4.5.c, discussed above, shows the corporate profit losses
of decentralization at H/N = 1.25.
With H/N = .80 the result is simi-
lar: any transfer price within a range around the log market price is
the "best".
But with H/N = .60, when total mills' log need is 40 per
In this
cent greater than total harvest, the result is different.
situation-- when logs are particularly scarce--transfer price slightly
above market price minimizes the corporate profit loss from decentralization.
The change from a range of "best" transfer prices to a
single best price then occurs when .60
this.
H/N
.80.
Figure 4.6.c shows
In the figure, profit loss is at its minimum when the transfer
price is ten per cent higher than the log market price.
It becomes
substantial when the transfer price exceeds this best level.
When the
transfer price is equal to or somewhat less than the market price only
91
moderate profit losses occur.
To summarize the best transfer prices
of mixed dominance at different harvest levels (Table 4.6):
With
harvest exceeding the mills' total log need the transfer price, should
be within a 20 per cent range from the market price.
With harvest
being somewhat less than mills' total log need the range is ten per
With harvest being much less than mills' total log need, causing
cent.
extreme scarcity of internal logs, transfer price should be about ten
per cent higher than market price.
Best Divisional Dominance
The most common of the three divisional dominance structures in
practice is probably the mixed dominance.
It is also the best of them
according to our measurements of goodness.
The measurements are based
on (1) the corporate profit loss from decentralization of the best
transfer price level of the optimal pricing type and (2) the insensit-
ivity of this profit loss to changes in the transfer price level.
The criteria are then the height of the shortest bar and the
difference between its height and the height of the other bars surrounding it in Figures 4.5.a-c and 4.6.a-c.
According to both criteria
mixed dominance (with market pricing) is the most preferable organization.
It brings the lowest corporate profit losses from decentraliz-
ation.
Under conditions fo certainty it is better than the other two
organizations.
The low level of corporate profit losses is maintained
at a wide range of transfer prices.
It then performs well when the
92
Table 4.6.
Best range for transfer prices for the case firm as a
function of harvest level and divisional dominance, and
suggestion of economic theory for transfer prices.*
Harvest level
Organization and its optimal transfer pricing typ e
Mill dominance;
Timber dominance; Mixed dominance;
market pricing
cost pricing
market pricing
TP
Harvest exceeds .90MP
mills' need by
.95MP
25 per cent
Harvest is 20
per cent less
than mills'
need
.9OMP
Harvest is 40
per cent less
than mills'
need
TP
TP
HC
.
mm
**
.95. MP
l.lOMP **
HC
TP
MAX
.80MP
TP
MAX
.90MP
.
mm
.80MAX
TP
1 .20.MP
TP
1 .lOMP
TP
**
TP
l.lOMP
MAX
Suggestion of economic theory for all harvest levels and dominance:
TP = MP
Note:
*TP = transfer price
MP = market price (average of $68/MBF for four log classes)
HCmjn = lowest unit harvesting cost among all tracts
MAX = the highest cost based transfer price the mills can
afford to pay before starting to operate at less than
capacity (this was $75/MBF for our case firm which is
above the over-all average timber division cost of
$64/MBF that includes average variable cost over all
available tracts, fixed costs and profit allowance
for timber division)
**For these entries the log purchasing constraint (about 50 per cent
of the mills' total log need) is limiting; therefore these 'best"
transfer prices may not apply to a situation with a higher constraint, for example, with an infinite log purchasing possibility.
93
basis of pricing:
log market prices are known only with uncertainty.
According to the first criterion mill dominance (with market pricinc,)
performs better than timber dominance (with cost pricing).
the second best divisional dominance.
We rate it
Timber dominance offers low but
steady corporate profits at a very wide range of cost-based transfer
prices that can be determined accurately.
offer high corporate profit.
transfer price level
.
At best mill dominance can
But the profit is sensitive to the
The transfer price is based on market price
that can usually be estimated only imprecisely.
These results are valid for a firm with good or fair amount (above
80 per cent of the mills' total log need) of internal logs available
from harvest.
Many firms belong to this category.
When the internal
logs become more scarce, however, there is less and less difference
between the corporate profits of the three divisional dominances.
But
only at an extremely low level of self sufficiency of logs the top
management would be indifferent in its choice of a divisional dominance.
94
Conclusion
The results of the case study show that the hypothesis of economic
theory does not always hold.
The best log transfer price is not
necessarily the market price in a corporate profit maximizing forest
products firm.
It depends on the organizational structure, and on the
amount of harvestable logs available.
This is shown in Table 4.6 which
summarizes the results of the case firm transfer price analyzes.
Under timber dominance (col
.
2) the firm's best transfer prices
follow harvesting costs, not market prices. Under mill dominance (col.l)
the best transfer prices are based on log market prices, with the
transfer prices of individual log classes following their relative
market prices.
However, in our experiment the best transfer price
level is slightly below or above the market price.
ance (col. 3) the best transfer prices are based on
Under mixed dominlog market prices,
with the transfer prices of individual log classes following their
relative market prices.
When the internal logs are particularly scarce
the best transfer price is above market price.
The best transfer price is the market price when the firm applies
mixed dominance, the best of the three organizations studied here, and
the logs are not particularly scarce.
Mixed dominance is probably the
most common of the three organizations among forest products firms.
An average forest products firm has a good or fair amount (above 80
per cent of the mills' total log need) of internal logs available from
harvest.
does hold.
Therefore, for a typical forest products firm the hypothesis
95
5.
5.1.
OPTIMAL TRANSFER PRICE ANALYSIS UNDER PRICE UNCERTAINTY
OBJECTIVES OF THE ANALYSIS
In the previous chapters of this study we have assumed that all
the prices, costs and production functions are known to the decision
makers.
In practice a firm's management is facing uncertainty in log
allocation.
His knowledge of especially log and end product market
price is imperfect.
The purpose of this chapter is to show that the
top management's transfer price choice under price uncertainty is not
necessarily its choice under price certainty even when the expected
market prices are the actual "uncertaintyt' prices. We wish to show
that the corporate profit (R
)
of the expected market prices may not
be the expected profit (E(R)) of the market prices.
This phenomenon
is well known in the literature of decision making under uncertainty.
We can state this fact for our example as follows:
R = f (E(mp), E(sp), E(vp))
E(R) = E (f (mp, sp, vp))
where mp, sp and vp are log, sawntimber and veneer market prices;
E(mp), E(sp), E(vp) are their expected values.
The hypothesis that
R= E(R)
holds is quite common among the forest products firm's managers.
It
only hold if the corporate profit is very insensitive to changes in
96
market prices.
When it does not hold, choosing a transfer price becomes an
elaborate task.
The log allocation problem has to be solved and corp-
orate profit obtained for all alternative transfer prices and every
possible long and end product market price level that might occur.
Obviously log market price uncertainty will have a very significant
effect on corporate profits in organizations where transfer prices are
based on the log market prices:
in mill or mixed dominance firms.
Of the two, corporate profit is more sensitive to changes in the marketbased transfer prices under mill dominance.
Market price uncertainty
thus may affect most the transfer price choice in a mill dominant and
least in a timber dominant firm.
It clearly has effect also on the
choice of the organization.
We construct an example of two transfer price alternatives, IF1
(transfer price is log market price) and TP3(transfer price is 10 per
cent less than log market price), under mill dominance.
From Table 4.2
and Figure 4.4.a we see that under price certainty TP3 is preferred to
TP1 .
In the previous chapter we stated that under log market price
uncertainty a "conservative" top management might choose a transfer
price equal to or greater than the expected market price.
We came to
this suggestion because the possible profit losses are smaller for
transfer prices above the expected log market price than for transfer
prices below it.
In this chapter we show formally that the suggestion is justified
97
even for a top management indifferent to risk--not necessarily
conservative.
Moreover, even a top management, ready to take risks,
would prefer 1P1 to TP3 which is contrary to the transfer pricing
analysis under certainty of Table 4.2 and Figure 4.4.a.
98
STAGES OF THE ANALYSIS
5.2.
The stages of choosing transfer prices under market price
uncertainty for a profit maximizing top management are:
1
.
Choosing transfer price actions
Naming market price states
Deriving state probabilities
Computing the corporate profit outcomes for all state-action
combinations
Computing the expected corporate profit for each transfer
price action
Choosing the action with the highest expected corporate profit.
When the top management chooses transfer prices that maximize
the expected corporate profit it is indifferent to risk of achieving
profit.
But top management may not be indifferent to risk.
It may be
risk averse which means that the increase of satisfaction of an increase in profit is greater for small profit levels that
a similar
increase for high profit levels:
the top management's utility function
for corporate profit is concave.
Correspondingly, a risk taking top
management's utility function is convex.
The stages of choosing
transfer prices under market price uncertainty for a top management
whose utility to corporate profit is nonlinear are below.
Top
management now maximizes expected utility, not expected corporate
profit in
choosing a transfer price
as follows:
99
1-4. (as for a corporate profit maximizing top management)
Deriving the top management's utility function to corporate
profit
Computing utility of the corporate profit outcome of each
state-action combination
Computing the expected utility of each transfer price action
Choosing the action with the highest expected utility
Actions
Our example of optimal transfer price analysis under uncertainty
concerns two alternative transfer price actions, TP1 and TP3 in a mill
dominant organization.
According to the economic theory, TP1 (market
price) is the optimal price.
and Figure 4.5.a),
better than TP1.
According to our experiment (Table 4.2
TP3 (transfer price less than market price) is
We have chosen mill dominance as the organization
because under it the corporate profits are most sensitive to changes
especially in log market prices.
Sawntimber Price States And Their Probabilities
We assume uncertainty in sawntimber, veneer and log market prices.
Sawntimber price states and their probabilities result from the
sawntimber marketing manager's
subjective valuation.
Veneer and log
price states and their probabilities are derived from the veneer
marketing manager's and log the log sale manager's subjective market
100
assessments, and'bjective" predictions, carried out by the top management, conditional to the subjective sawntimber prices.
Sawntimber
prices are taken as the basis because they predict the forest industry
market situations best.
The price states are discrete because typical-
ly subjective price evaluations are discrete,
A continuous state
space would theoretically give more precise results but it would
increase the computational burden excessively.
Top management receives the subjective sawntimber price states
and their probabilities from the sawntimber marketing manaqer--the best
Top management does not believe
expert on these prices in the firm.
costly regression analyses would improve the price estimates
antly.
signific-
The sawntimber marketing manager is convinced that the relative
sawntimber grade prices (sp) remain stable.l2)
We assume that within
each state the prices are uniformly distributed.
Grade 3 subjective
sawntimber price states and their probabilities are:
Sawntimber price, $/MBM
-61.5
State name1
Subj. Probab.
C
B
Expected price = $72 =
B
A
A'
Sawntimher grade price ratios are: sp1
.98
.003
.232
.560
.183
.022
C'
61.6-68.5
68.6-75.5
75.6-82.5
82.6+
sp3, sp4 = .50
C = low price,
1 .000
1.84
sp3,
sp3
B = intermediate price,
A
high price.
101
Veneer And Log Market Price States And Their Probabilities
Top management has also acquired knowledge of subjective veneer
and log price states and probabilities from veneer marketing department
and timber division log sales manage.
The headquarters has carried out
regression analyses for quarterly veneer and log market prices with
sawntimber price as independent variable.
As an outcome from the
regression analyses we get objective veneer and log market price
probability distributions.
It is believed that the historically stable
relative veneer grade (vp.) and the log class (mp.) prices remain
unchanged for the coming period14.
The following linear regressions
give the best fits of veneer and log prices for quarters of 1950-66
and 1961-66, respectively:
vp3 (t) = .413 sp3 (t)
mp3(t)
R2=.98
R2=.97
.873 sp3 (t-l)
Corresponding to the five sawntimber price states we get the
following objective t-distributed predictions and standard deviations
of veneer and log market prices:
Sawntimber price
(sp3), $/MBM
C'
Veneer price prediction
(vp3), $/MBM
= 58
C=
= 72
= 79
A' = 86
B
A
Std. dev. of prediction
14)
24.0
26.9
29.8
32.7
35.5
3.72
Log market price
prediction (mp3),
$/MBM
50.6
56.7
62.8
68,9
75.1
7.19
Veneer grade price ratios are: vp1=l.77 vp3,vp2=l.14vp3,vp4=
.73vp
Log cTass price ratios are:mp1=l .47mp3,mp2=1 .33mp3,mp4=.85mp3.
Table 5.1. Derivation of a combined distribution from subjective and
102
objective veneer price probabilities.
Normalized
Veneer price
intervals,
/MBt1
State
Subjective
name*
prob. disn.
Objective
prob. disn.
Combined
combined
distrib. distribution
-19.5
0
.010
0
0
19.6-22.5
0
.029
0
0
22.6-25.5
0
.107
0
0
25.6-28.5
S'
.007
.216
.0015
.008
28.6-31.5
S
.106
.276
.0293
.164
31.6-34.5
M
.542
.216
.1169
.656
34.6-37.5
G
.285
.107
.0305
.171
.060
.029
.0002
.001
0
.010
37.6-40.5
40.6+
Total
Expected
.1.000
price
0
0
1.000
.1784
1.000
=33=M
*Note:
small , M = medium; G
S
Table 5.2.
great.
Derivation of a combined distribution from subjective
and
objective log market price probabilities.
Log Market
Normalized
price internals, State Subjective Objective Combined combined
nanie
prob. disn. prob. disn. distribu. distribution
$/t4BF
.
-43.5
0
.013
0
0
43.6-50.5
0
.068
0
0
50.6-57.5
0
.209
0
0
57.6-64.5
L
.086
.420
.6361
.366
64.6-71.5
1
.149
.209
.0313
.317
71.6-78.5
H
.380
.068
.0258
.262
78.6-85.5
0
.395
.013
.0054
.055
Total
Expected
1.000
1.000
.0986
1.000
price
68 = I
L = low, I = intermediate, H = high, 0 = very high.
103
Top management wishes to give weights of half and half to the
subjective and objective price probability distributions of the
regression analysis.
The combined subjective and objective veneer and
log prices probability distributions are derived in Tables 5.1 and 5.2.
Each objective probability has been computed from the t-distributed
veneer and log price prediction (cf. Eidman
J., p. 191)
From the tabulation of sawntimber price probability distribution
on p.100 we see that the expected sawntimber price is $72/MBM, corresponding to state B.
In Tables 5.1 and 5.2 expected veneer and log
market prices are $ 33/MBM, corresponding to state M, and $68/MBF,
corresponding to state I, respectively.
The joint state space includes
the five sawntimber market price, five veneer market price and four
log market price states with positive probabilities.
total of 5 x 5 x 4 = 100 states.
Thus we have a
A joint state probability is computed
by multiplying the probabilities of the individual sawntimber, veneer
and log market price states forming the joint state.
Some of the
joint states and their probabilities are listed in Table 5.3.
Due to
the symmetry of the state probability distributions the expected joint
log price is $68/MBF, sawntimber price $72/MBM and veneer price $33/MBF.
The"expected joint state" then is IBM.
439.3
221.1
183.1
153.4
105.8
l-tA1
H3G
HBM
HCM
1,i
503.0
516.5
329.4
329.4
329.4
329.4
133.8
289.4
251.2
231.5
211.8
173.6
Expectec profit
!3i
1CM
L4
LBS
L81
LSS
Lc.M.
Ncte:
lBS
580.5
560.9
541.2
518.8
523.8
71.1
- 6.5
70.9
51.3
31.6
109.1
14.2
52.4
72.1
91.7
129.9
35.5
93.1
112.8
150.7
105.0
299.0
299.0
299.0
299.0
299.0
349.7
349.7
_349.7
349.7
349.7
352.9
352.9
352.1
352.9
387.8
Expected profit*
204.0
261.9
242.2
281 .5
319.7
176.0
253.2
233.6
213.9
291 .5
168.3
246.9
227.4
286.6
225.4
irm
575.9
503.0
580.5
550.9
541.2
618.7
525.7
563.7 -
603.0
583.3
641.2
521.2
599.8
579.6
639.5
613.2
Profit before fixed cost,
taxes and stum ace
1V. limber dlv.
H=high
1= intermediate
price:
price:
Sawetirber
Stnp.age paid varies from L- an average it is $3LI0'3F for the about
.Ar.tf.cjzed fi>:eci uxosts ,rd tees are:
L1ow
Qvery high
Log market
$94,333
total
65.7
- 6.5
70.9
51.3
31.6
109.1
16.1
S1ow
Mintermediate
Grhigh
81.000
056
034
.134
025
044
048
00
.148
.037
73.7
54.0
038
.040
.025
.096
Q3]
.020
State
probability
93.4
131.6
11.6
90.2
70.7
129.9
103.5
after fixed
cost, taxes &
sturoace oaid4'
Firm profit
All states are
lOgs the mills use cuarterly.
$15,200
hnrrcrm
of
$15 ,.00
tici d0visic:n
10,230
$63,003
Veneer
price:
rr.iJitmg division
A=high
Brintemnediate
C=low
Cniy states with probability .02 or greater of all 100 states are presented here.
included in calculating the expected profits.
The states are:
329.4
390.0
390.0
390.0
390.0
601.3
581.7
562.O
390.0
L9.
211.3
191.7
172.0
639.5
545.1
622.4
602.7
650.3
614.6
249.6
439.3
439.3
439.3
459.9
154.7
0E1
State
Profit before fixed cost,
taxes and stumpaqe
Miil.div. Timber dlv.
Firm
Firm profit
after fixed
cost, taxes &
stumpane paid'
($90,81,61,52)
under price uncertainty in mifl dominance organization,
Transfer once
31C9 market price less l0
Profits of two transfer price alternatives
TP1log flarket price (Sl00,90,53,58)
b1e 5.3.
105
5.3.
OUTCOMES
Corporate Profit Outcomes
The divisional before fixed cost, tax and stumpage profits of
the 15 joint state-transfer price action combinations of Table 5.3
have been computed with the linear proqrmas of Appendix 1.2.
(A-B).
The rest--85 outcomes for the two transfer price actions and states
with a probability
15 outcomes.
of occurrence -- are interpolations of the
The corporate before fixed cost, tax and stumpage profits
are computed by summing the corresponding two divisional profits.
For intermediate and long term decisions the headquarters is accustomed
to work withafter fixed cost, tax and stumpage corporate profit.
These profits have significance in the utility analysis of the following section.
Therefore, they are calculated in Table 5.3 although
fixed cost and taxes are constant for the two transfer price alternatives and stumpage paid is a sunk cost.
The corporate profit outcomes of each action and joint state, and
each corresponding state probability are the ingredients in calculating
the expected corporate profits.
From the last row of Table 5.3 we
see that in our example an expected profit maximizing top manaqement
prefers TP1 to TP3.
This result is contrary to that of the expected
state IBM where top management would choose TP3 instead of TP1.
Thus
the headquarters should be careful in making transfer price decisions
based only on the expected price information.
It should rather analyze
1 06
the profit outcomes of all essential price states and calculate the
expected corporate profits for all transfer price actions.
It would
choose the transfer price alternative maximizing the expected corporate
profit.
This type of uncertainty analysis is important under mill
dominance but less important under mixed and timber dominance where
the corporate profit is rater insensitive to changes in market prices.
Utility Outcomes
Maximizing expected corporate profit means that the top management
is indifferent to risk.
(Halter and Dean, p. 45).
It has a linear utility function for profit
If the top management maximizes its utility
and the utility function for money is nonlinear it might or might not
choose TP1 as the transfer price.
Halter and Dean (pp. 32-53) and
Halter etal. (pp.54-61) describe well how to find utility functions of
decision makers through interviews.
Figure 5.1 shows examples of a
concave utility function (u1) of a risk averse and a convex utility
function (u2) to the after fixed cost, tax and stumpage corporate
profit of a risk searching top management.
Figure 5.2 presents examples
of two utility functions of combined risk aversion and risk preference
(u3, u4) to the after fixed cost, tax and stumpage corporate profit.
The utility outcomes are computed by inserting the corporate profit
outcomes (of Table 5.3) to the utility functions.
The expected
utilities of the alternative transfer price actions are calculated from
utility
outcomes and joint state probabilities (of Table 5.3).
The
107
Top
management
utility (u)
100
= 1.188 M - .0033
80
60
u2 = .19 M + .0028
40
20
-40
--
40
80
160 200
120
220
Corporate net profit after
stumpage paid (M), $1 000
-20
-40
Figure 5.1.
Examples of utility functions of a risk averse (u1) and risk
taking (u2) top management.
Top
management
uti1ty (u)
U.
-
.M
2.50
-
r
2
100
= 1.38 M - .019 M2 + .0009 N3
80
60
40
20
80
-40
40
U3
-20
80
120
160
200
220
Corporate net profit after
stumpage paid (N), $1,000
U4
-40
Figure 5.2.
Examples of two utility functions of a partially risk averse,
partially risk taking top management.
108
transfer price is chosen that maximizes the expected utility.
The
functions produce the following expected utilities for the two transfer
price actions:
Utility
function
Expected utility
for TP1
Expected utility
for TP2
U1
57.90
57.60
- choose TP1
U2
30.95
30.55
- choose TP1
66.56
66.46
- choose TP1
244.84
242.88
- choose TP1
In this case the form of the utility function does not affect the
choice of the transfer price: TP1 is always preferred to TP3. Several
other types of utility functions tested gave the same result.
It
does not change even when the state probability distribution is greatly
distorted.
Although the form of the utility function does not in this
example affect the transfer price choice it would be easy to costruct
situations where it might (cf. Halter and Dean, p. 47; Raiffa, p. 68).
From this chapter we can conclude that under price uncertainty the
headquarters should be careful in making transfer price decisions
based only on the expected price information.
It should rather analyze
the profit or utility outcomes of all essential price states and
calculate the expected corporate profit or utility for all transfer
price actions.
It would choose the transfer price alternative maximiz-
ing the expectedcorporate profit or utility.
The management may feel
109
that there are other sources of uncertainty in loq allocation: log
class distributions in the tracts, recoveries, harvesting or milling
costs, etc.
The states, their probabilities, and profit and utility
payoffs for each transfer price action should be determined in a similar
fashion as we have done under price uncertainty.
The expected corporate
profits or top management utilities are compared and the transfer price
action is chosen that gives the highest expected profit or utility.
110
6.
DISCUSSION
Practice
f'lost Pacific Northwest forest products firms use harvesting cost-
based transfer prices for their internal log transactions.
market-based transfer prices are quite common.
Also
The cost-based transfer
price level varies greatly from firm to firm. In some firms it can be
the average logging and transportation cost, in others the average
total timber division cost including an allowance for a 'normal"
timber division profit, etc.
The market-based transfer prices are
generally slightly below or at the market price level.
The discussion
with the executives have also indicated that of the three decentralized
organizations studied, mixed dominance is the most common. The log
allocation is hardly ever totally based on transfer prices.
the top
Usually
management does not set only the transfer prices but some
quantitative restrictions on internal log transactions as well .
These
firms call themselves decentralized and the divisions bear the name
"profit center."
In many cases they are not, indeed, truly decentral-
ized.
Theory
According to economic theory of a corporate profit maximizing
firm, log market price is always the optimal transfer price.
Applying
any other pricing types, as cost or value pricing, leads to corporate
profit losses.
Setting a market-based transfer price exactly at the
111
true market price level maximizes the corporate profit.
The theory
suggests that the mill and mixed dominances are equally acceptable
organizations.
Timber dominance is inferior to the two.
These
recommendations are based on many simplifying assumptions.
The theory
operates with a single input-single output firm or with a firm where
the inputs or the outputs are little or not at all interdependent.
The
decision makers are assumed to have perfect knowledge about the
allocation situation.
In practice the managers make decisions in
multi-input multi-output firms under conditions of uncertainty.
Experiment
The transfer pricing analyses for the corporate profit maximizing
case firm of this study show that market pricing is the optimal transfer pricing type.
The best transfer price level is equal to or slight-
ly lower than the log market price. The best organization is mixed
dominance.
analysis.
The case firm log allocation is a multi-input multi-output
The input and output dependencies of the linear programs
reflect the best knowledge of the author.
The log allocation decisions
are assumed to be made under certainty.
Comparison Between Experiment, Theory and Practice
The results of the experiment agree fairly well with the recommendations of the theory:
the optimal transfer pricing type is market
pricing, the best organization mixed dominance.
The greatest disagree-
112
ment concerns transfer prices under timber dominance: the experiment
proposes cost pricing,the theory market pricingin practice in
forest products firms the most common transfer pricing-organization
conibination is cost pricing under mixed dominance.
Closer examination in these firms has shown, however, that they
are not truly decentralized but the log allocation quantities are
somewhat enforced directly by the top management.
Obviously the top
management does not want to delegate decision making to the divisions
in fear of great profit losses from decentralization due to dysfunctional decisions.
The case study has shown that the fear is justified.
Cost-based transfer pricing under mixed dominance leads to great profit
losses.
15)
In these firms either the organization should be changed to
For an average one log class timber dominance firm the
experiment and the theory may not contradict.
This can be seen in
Figure 3.6.
There market price equal to marginal harvesting cost
at point X.
This cost is the average harvesting cost of the most
expensive tract to be harvested (cf. Figure 3.l.b).
highest average tract harvesting cost.
It is the
For an average profit
maximizing firm it would be somewhat below the
highest transfer
price the mills can afford to pay before starting to go out of
business ( MAX in Table 4.6).
For an average efficiency
one log
class firm the market price would then be within the recommended
cost-based transfer range of Table 4.6.
113
timber dominance while maintainingthe cost pricing of internal logs;
or preferably thecost pricing should be changed to the market pricing
while maintaining mixed dominance.
After these changes true decentral-
ization can be reinforced without great risk of dysfunctional log
allocation.
Uncertai nty
The results'of the case study of Chapter 4 apply in log allocation
through transfer pricing under certainty.
In practice the decisions
are always made under conditions of uncertainty.
it is, however,
evident from the results of the many case study sensitivity analyses
that even under uncertainty market pricing produces the highest corporate profit in a mixed or mill dominant firm, and cost pricinq in
a timber dominant firm.
A major advantage of using cost-based transfer prices and applying
timber dominance is a high level of accuracy in the corporate profit
estimation in changing price and cost conditions.
is a low expected (average) profit.
A major disadvantage
On the other hand, a great
advantage of using market-based transfer prices and applying mixed or
mill dominance is a high expected profit and a great disadvantage,
especially under mill dominance, a low level of accuracy in the
corporate profit estimation.
The transfer price-organization choice depends on the top manage-
ment's preference to the expected corporate profit and to the risk
114
involved in arriving at it.
According to interviews the managers rate
But a high
high the certainty connected with obtaining a profit.
expected profit is usually a more important requirement.
If the top
management is indifferent to risk--its utility is a linear function of
profit--it chooses the transfer price--organization combination that
maximizes the expected corporate profit.
If it does not appear to be
indifferent to risk--its utility is a nonlinear function of profit-first the exact form of the utility function must be found.
Then the
top management chooses the transfer price-organization combination that
maximizes its expected utility.
Whether the top management is indiff-
erent to risk or not, extensive sensitivity analyses become necessary
to aid decision making under uncertainty as the examples of Chapter 5
show.
Extensions
There might be situations where the case firm results of Chapter
4 may not apply.
This happens, for example, when the top management
wants to make long term transfer pricing decisions.
Then a multi-
period model as that in Appendix 5 should be formulated and solved.
Top management may be best conceived as a group of division managers.
Then group decision theory can be applied for choosing transfer prices.
Appendix 6 presents a special case of group decisions.
The decentralized organization of a company might be different
from any of the three structures of the case firm:
possibly top
115
management wishes the divisions to have equal rights in log allocation
There are three approaches
after the announcement of transfer prices.
to find optimal transfer prices inthat case: decompositions methods,
divisional trading, or price adjustment procedure.
in Appendix 4.
They are discussed
The current decomposition algorithms are more decentral-
ized information gathering than decentralized decision making tools
because in the execution corporate headquarters enforces the log
allocation.
We have not based the allocation on divisional trading to
avoid the disadvantages from bargaining.
Game theory which would cre-
ate the best trading model framework is not developed far enough today
to solve complex nonzero sum games.
Price adjustment procedure is
the mostpromising of the three approaches.
It has the disadvantage
of alternative optima which may increase the cost of the information
exchange within the firm.
optimal solution.
Besides that, it does not guarantee an
Its objective is rather to equate internal supply
and demand than to maximize the corporate profit.
Despite these short-
comings the divisional trading and price adjustment methods are
promising procedures which might apply to other kinds of log allocation
problem than those discussed in this study.
Price adjustment is an
especially attractive method to problems with nonlinear utility
functions of the decision makers.
Managers of decentralized forest products firms might not agree
with the results of the case firm in Chapter 4, for still some other
reasons: their fir.m is much larger or smaller in size, it contains
116
several other mills or only one mill, thetimber tract composition or
mill recoveries or end products are different, etc.
If.the differences
are substantial enough to justify the costs of the necessary computations, analyses of similar nature as in this study should be done.
117
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,
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124
Appendix 1.
LINEAR PROGRAM FORMULATIONS
Appendix 1.1.
Corporate Program
The classical mathematical log allocation problem for a centralized firm is of the following general form (cf. Jenneryren,
(A)
p.
1):
Maximize
K
(0)
tk (xk)
k=O
subject to
Xk
Xk,
fork
O,...,K
K
g
'
(xk)
a
k=l
where
Xk
= vector of the (j = 1,... ,J) classes of logs to be
allocated from the timber department (k=O) to the
milling departments (k = 1,... ,K)
tk(xk) = profit function of department k
Xk
= the set of feasible log allocations of department
k
gk(xk) = log requirement vector for classes j = l,...,i of
the (k = 1,... ,K) milling departments
a
= vector of the (j = l,...,J) log classes available
125
The corporate profit is the sum of the departmental net revenues
which can be linear or nonlinear functions of the logs allocated.
Constraints (1) indicate the feasible log allocations to the timber
department and mill departments.
Constraints (2) show the availabil-
ities of logs.
When the objective function and constraints are linear the
problem becomes (cf Dantzig, p. 448):
(B)
Maximize
K
(0)
Y
pkXk
k=O
subject to
Bkxk
kl
--
for k
bk
Akxk
0,... ,K
a
where
= vector of department k net revenues for the (j =
1,...,J) classes of logs xk
k = 0,...,K
= department k requirement coefficient matrix for
the departmental resources, k
bk
0,...
= vector of department k resources, k = 0,... ,K
= milling department k log requirement coefficient
matrix, k =
xk,a = as for problem (A)
126
Coefficients p1< for the mills (k = l,...,K) indicate the net
revenue per log unit from processing logs and selling end products.
For timber department (k=O) they denote the wood procurement costs
for internal deliveries and net revenues for purchasing logs.
The
first constraint set shows again the departmental resource (labor,
machine capacity, etc.) availabilities, and the second constraints
the availabilities of the scarce corporate resource, logs.
The corporate program for the centralized case firm log allocation is based on the general structure of formulation (B) but much
more detailed:
(C)
Maximize
end product sales revenues
M
K
N
ep
(o)
k=l
m=1
kmn
e
kmn
processing and acquisition costs of internal logs
'
-
K
'1
(Pcjk + hcjjk) X..k
il J='l
processing and acquisition costs of external logs
(pcJk + mp) Xojk
j'l
l
net revenues from external sales of harvested logs
I
J
(m
+
jl
- sc
- hc0)
x0
127
subject to
EU
end
m = 1,...,M; n = l,...,N
for
mn\
product
orders
(lb)
EL
ekmn
.
k
N
(2)
J
I
ekmn
ib
ri<
-
for
(3)
recoveries
Xjjk = 0
k = 1,... ,K; m = l,...,M
tjk
i0
for
Tk
Xijk
k
= l,...,K
processing
times
K
X.
(4)
for
i
1,... ,I;
3K
(5)
log
availability
on tracts
k=0
/
mp. X0.k
;
external log availabilities
jl k1
K
K
(6)
x.
ijk
for
I
- d
x
k=b
X1
i=l j=l k=0
:T
= 0
tract log class
structure
= 1,...,I; j = l,...,J-1
2.
Id
ij+lk
maxim urn
k
and
minimum
harvest
K
.
i=l j=l k=0
13
XL
1 28
where the variables are:
ekmn
= processing of end product m at mill k for market n,
MBM - k = l,...,K; m
l,...,M; n = l,...,N (K = no.
of mills)
Xjjk
= harvesting of class j logs in tract
i
Ci = l,...,I)
for mill k (k = l,...,K), or purchasing of class j
logs from outside market (i=O) for mill k (k = 1,...,
K), or harvesting of class j logs in tract i
,I) for outside market (k=O), MBF -
i
(i
= 1,
= 0,...
j= l,...,J; k = 0,...,K
where the objective function coefficients are:
epkmn
= net selling price of end product m at mill K for
market n, $/MBM
Pck
= log class j processing cost at mill k, $/MBF
hcJk
= log class j harvesting cost in tract i for destination
k, $/MBF
= log class j market price, $/MBF
sc.
= cost of selling class j logs to outside market, $/MBF
where the constraint coefficients are:
r
km
= recovery of end product m from class j logs at mill k,
MBM/MBF
= log class j processing time at mill k, shifts/MBF
k
d
= ratio of log class quantity to log class j+l quantity
j
in tract i
(= XiX1+i)
129
where the constraint right-hand-side constants are:
EUmn
= maximum demand of end product m in market n,
ELmn
= minimum demand of end product m in market n, MBM
Tk
= tOtal number of eight-hour shifts available during the
1BM
planning period at mill k
= amount of class j logs available in tract i, MBF
= total amount of money at which outside market logs can
F
be purchased, $
XU
= maximum desired total harvest, MBF.
XL
= minimum desired total harvest, MBF
The objective function in this formulation is more complex than
that of problem (B).
The firm's revenues result from end product
The
sales (first term) and log sales to other firms (fourth term).
possibility of log sales was missing from the classical resource
allocation formulations of (A) and (B).
The costs consist of har-
vesting and processing costs and costs of purchasing logs from the
outside market (third term).
The first, "divisional", constraint
set for the mills of problem (B) is represented by constraints
(3).
(la)-
It is hard to make a distinction between timber division and
"corporate" constraints.
Closest to "corporate" constraints of log
availability as we saw them in the first constraint set of problem
(B) are constraints (4), (5) and (7b).
Constraints (6) and (7a)
would then be wood procurement "divisional" constraints.
Constraints
(6) guarantee that all log classes of the tracts are cut in their
130
existing proportions.
cutting others.
None of the log classes can be cut without
In centralized log allocation problem (C) is solved
and the allotments are announced by the headquarters.
131
Appendix 1.2.
Divisional Programs Under Dominance Organizations
Mill Dominance
When the mills are dominant they determine their internal log
Timber division has to deliver the
demand given transfer prices.
logs demanded through harvesting of timber tracts.
from the outside market.
Mills buy logs
Under mill dominance, the milling division
has the following log allocation linear program:
(A)
(0)
sales revenues
Maximize
Th
-'
epknln ekmn
nl n
kl
processing and acquisition costs of internal logs
K
tJ
kl
-
(Pcik + tPik) xJk
processing and acquisition costs of external logs
K
(pc.k + mp.) X0
Y
-
k
j=l k=l
subject to
(la)
e
k=l
kmn
EU
for
m =
mn
end
product
orders
(lb)
kl
ekmn
ELmn
for
m = 1,... ,M;
132
(2)
rjkm (xljk + xojk) = 0
ekmn
recoveries
j'1
fork=l,...,K;m=l,.. ,M
J
(3)
t.
5
3k
(x
13k
+
processing times
03
j=l
for
k=l,...,K
K
J
mp.
(4)
j=l k=l
F
external log availability
'
K
x1
(5)
k=l
k
x
for
j = 1,... ,J
internal log
availability
J.K
XL
(6).
minimum harvest
j=l k=l
where the variables are:
ekmn
processing of end product m at mill k for market
n, MBM - k = l,...,K; m = l,...,M; n = l,...,N
Xljk
= class j logs delivered internally for mill k, MBF
- j = 1,... ,J; k = 1,... ,K
xOJk
= class j logs purchased from outside market for
mill k, MBF - j = 1,... ,J; k = 1,... ,K
where the constants are:
epkmn
PCjk
rjkm
tik as in the corporate program,
Appendix l.l.(C)
tPjk
= log class j transfer price for mill k, $/MBF
= log class j market price, $/MBF
133
Tk, F, X1 as in the corporate program,
EU, ELmn
Appendix l.l.(C)
X
= maximum amount of class j logs available through
harvest, MBF
In this formulation the objective function lacks the detailed
tract-by-tract log class variables of the corporate program.
Mill-
ing division revenues and costs result from sales of end products
and purchases of internal or external logs.
delivered logs are gathered under x1
All the internally
-variables.
ing these logs equals to transfer prices tp....
purchased logs are denoted by variables x0
The cost of buy-
The externally
and their prices by
mp. as in the corporate program of Appendix l.l.(C).
(la)-(3) are the same as milling constraints (la)-(3).
Constraints
Constraint
(4) here shows the availability of outside market logs as (5).
Restraints (5) and (6) are guidelines to wood acquisition which
guarantee feasible harvesting by the timber division.
Under mill dominance the timber division has the following log
allocation linear program:
(B)
Maximize
net revenues from internal deliveries of harvested logs
I
J
K
il
):
k'l
(0)
(tPjk - hck) Xijk
net revenues from external sales of harvested logs
(mp
+ i:l
jl
- sc
-
hc10)
x0
134
subject to
K
k0
for
X..
1J
1JI\
i
= l,...,I; j =
log
availability
in tracts
l,."'J
K
K
Xjj+lk = 0
Xjjk - d1
l,...,J-1
= l,...,I; j
i
tract log
class
structure
for
k'O
k=O
Id
K
k0 uk
-=1
j
maximum
XU
x..
)
and
minimum
harvest
K
XL
k
i=l j=l k=0
(4)
Xjjk
Xjk
for
j = l,...,J;
mills' demand
k = 1,... ,K
where the variables are:
Xijk = harvesting of class j logs in tract i for mill k
1,... ,K), or harvesting of class j logs in
(k
tract i for outside market (k=0), MBF
-
i
= 1,... ,I; j
l,...,J; k = 0,... ,K
where the constants are:
tPjk = log class j transfer price for mill k deliveries,
$/MBF
hcJk
sc
l.l.(C)
as in the corporate program, Appendix
135
Xi., XU, XL as in the corporate program, Appendix l.l.(C)
= log class j internal demand by mills [comes from
Xjk
the optimal solution of program (A)], MBF
Timber division revenues and costs result from internal deliveries and sales to the outside market of harvested logs.
Restraints
(l)-(3b) are wood procurement constraints (4), (6), (7a), (7b) of
the corporate program in Appendix l.l.(C).
Restraints (4) are the
internal log demands enforced by the mills.
Timber Dominance
When the timber division is dominant it determines the internal
log supply.
ket.
Its sources of logs are timber tracts and outside marThe timber division has
Mills have to use the logs supplied.
the following log allocation linear program:
(C)
Maximize
net revenues from internal deliveries of harvested logs
-----------------------.. ---------------------------------------------
jk - hcJk) Xjjk
(°)
net revenues from internal deliveries of purchased logs
--
(tpjk - mp) Xbjk
+
jl kl
net revenues from external sales of harvested logs
-I
+
J
3
il j'l
(mp. -
Sc. 3
hc.
.
130
)
x.
130
136
subject to
K
for
X..
(1)
"
k=O
i
= l,....,I;
log availability
in tracts
j =
K
(2)
Y
k=O
K
x.
ijk
- d.
x.
ij+lk
kO
tract log
=
class
structure
for
= l,...,I; j = l,...,J-1
i
13K
5"
'
il j=l k=O
I
3
maximum
XU
X
and
minimum
harvest
K
/
/
Xj.XL
i=l j=l k=O
3
(4)
external log availability
F
mp. XO.k
j=1 k=1
3
I.
j=l
i=O
=
for
Xk
k = l,...,K
mill
capacity
I
'
Xjjk_Xjk
for
mill minimum
j = l,...,J;
log
1=1
requirements
k = l,...,K
where the variables are:
xlk
= harvesting of class j logs in tract
i
(i = 1,...,
I) for mill k (k = 1,... ,K), or purchasing of
class j logs from outside market i
(i
for mill k (k = 1,... ,K), or harvesting of class
j logs in tract i
(I
= 1,... ,I) for outside market
137
k (k=O), MBF -
= 0,... ,I; j = 1,...,J; k = 0,
i
where the constants are
hciJk
tPjk
d. .
1]
5C
as in (B)
as in (B)
X, XU, XL, Xk as in (B)
Xjk = amount of class j logs needed by mill k to satisfy
the unfilled end product orders, MBF
F as in (A)
Xk
= amount of logs needed to fill mill k capacity, MBF
Timber division revenues and costs result from internal deliv-
eries of harvested or purchased logs, and outside market sales of
harvested logs.
Restraints (l)-(3b) are the same wood procurement
constraints as in program (B).
Constraint (4) shows the availabili-
ty of outside market logs as (4) in program (A).
Restraints (5) and
(6) are guideline constraints of milling which guarantee that enough
logs are available to fill the existing end product orders, and mi
mills' current capacity is fully utilized.
Under timber dominance the milling division has the following
log allocation linear program:
(D)
Maximize
processing and acquisition
costs of logs
sales revenues
-- -------- -
(0)
m1
L
epkmn ekmn
kl
-. -.- -- .----,
-,-
(PcJk + tPjk) Xjk
- jl kl
1 38
subject to
K
Y
(la)
4EU
e
for
m = l,...,M;
end product
orders
K
(lb)
'
k=l
for
EL
ekmn
m = l,...,M;
n=l,...,N
J
N
(2)
rjkmxjk=O
ekmn
n=l
recoveries
j=l
for k = l,...,K; m = l,...,M
3:Ji
for k = 1,... ,K
tik Xjk LTk
Xjk
for
X.k
j = l,...,J;
processing time
timber division
supply
k=
where the variables are:
= processing of end product m at mill k for market
ekmn
n, MBM --k = l,...,K; m =
= class j logs, all delivered by the timber division
Xjk
for mill k, MBF - j = 1,... ,J; k =
where the constants are:
epkmn
EU
,
mn
Xjk
tPjk
Pcjk
EL
,
mn
rkffl
tjk as in (A)
as in (A)
T
k
= log class j internal supply by the timber division
to mill k [comes from the optimal solution of
139
program (C)], MBF
Milling division revenues and costs result from end product
sales and internal log purchases.
Constraints (la)-(3) are the same
as in mill problem under mill dominance (A).
Constraints (4) are
the internal log supply enforcements by the timber division.
Mixed Dominance
Under mixed dominance the milling division determines the mills
The timber division has to fill mill needs by
total log needs.
procuring logs internally or externally.
buys logs.
Only the timber division
Under mixed dominance, the milling division has the
following log allocation linear program:
(E)
Maximize
processing and acquisition
costs of logs
sales revenues
(0)
k1
(pck + tPjk) Xjk
epkmn ekmn
-
jl k=l
subject to
K
(la)
e
k=l
kmn
EU
mn
end product
form = l,...,M
orders
I
n =
/
(lb)
e
EL
n=l,...,N
form = 1,... ,M;
140
recoveries
rjkm Xik = 0
-
l
for k = 1,... ,K; m = 1,...,M
J
Xj<
'
for
T<
k = l,...,K
processing
times
j =1
J
K
nip.
x
j
(4)
j=l k1
external log availability
F
ojk
K
x1 .
J
3
internal log
availability
K
Ji
x
for
minimum harvest
XL
uk
k='l
Xljk + x
(7)
j = 1,... ,J
for
X.
.
iJr
k=l
j
ojk
- x.
3k
one log source:
timber division
0
= l,...,J; k = 1,... ,K
where the variables are:
= processing of end product ni at mill k for market
ekmn
n, MBM --k = l,...,K; m = l,...,M; n =
= class j logs all delivered by timber division for
Xjk
mill k [= x
ajk
+ x
bjk
as shown in constraint (7)]
where the constants are:
epkmn
EU
,
nm
F, Xaj
tPik
PCjk
EL
,
mn
r.k
as in (A) and (D)
as in (A) and (D)
T
k
XL as in (A)
141
Milling
division revenues and costs result from end product sales
and internal log purchases as in (D).
Constraints (l)-(3) are the
same as those of programs (A) and (D).
Constraints (4), (5), (6)
and (7) are guideline constraints of wood procurement which guarantee that timber division will be able to deliver enough internal
(x1) and external (x0) logs to the mills.
Under mixed dominance timber division has the following log
allocation linear program:
(F)
Maximize
net revenues from internal deliveries of harvested logs
il 3:l
kl
(tPik - hcJk) Xiik
net revenues from internal deliveries of purchased logs
(tPik -
j1
+
m)
Xojk
k=l
net revenues from external sales of harvested logs
T
+
(mp
-
- hc0) x0
sc
subject t
Xjjk
= 1,... ,I;.
for
k=O
avilability
in tracts
j
=
K
x.
k0
for
- d.
x.
k=O
I
+lk
=
= l,...,I; j = 1,... ,J-1
tract log class
structure
142
K.
3
I
X.XU
maximum and
minimum harvest
i=1 j=l kO
13K
.-XL
j
.i'
j=1 k=O
i=
(4)
Xik
j1 k1
(5)
Xjjk
iO
Xjk
=
external log availability
F
for
j = 1,... ,J;
mills' demand
l,...,K
k
where the variables are:
Xijk
= harvesting of class j logs in tract i
(i = 1,... ,I)
for mill k (k = l,...,K), or purchasing of class j
logs from outside market (i=O) for mill k (k = 1,
,K), or harvesting of class j logs in tract i
= 1,...,I) for outside market (k=O), fiBF
(i
-
i
= U,...,I; j = 1,... ,d; k = 0,... ,K
where the constants are:
tPjk
mp hck sc
as in (B) and (C)
X., XU, XL as in (B) and (C)
Xjk as in (B)
F as in (A) and (C)
Timber division revenues and costs result from internal deliv-
eries of harvested or purchased logs, and outside market sales of
harvested logs as in (C).
Constraints show log availabilities
143
through harvest as those of programs (B) and (C).
Constraint (4)
shows the availability of outside market logs as (4) in programs (A)
and (C).
the mills.
Restraints (5) are the internal log demands enforced by
1 44
Appendix 2.
Appendix 2.1.
CASE FIRM DATA FOR THE LINEAR PROGRAMS
Timber Data
The timber price data come from the Oregon State University
Extension Service for 1967.
Timber cost and quantity data come from
Bohemia Company's Coburg unit's bookkeeping, the cost data
1967, and the quantity data for 1967 and later periods.
are for
The timber
division log allocation problems are in two parts in this study.
The main part consists of the timber division linear programs of this
section.
2.3.
The other part contains the user cost computations of Appendix
It serves for creating the harvesting cost data.
These are
used for reducing the number of alternative tracts for harvest in the
linear programs from a total of 30 to nine as shown in Table 4.1.
following is
The
a short discussion of the wood procurement data used
in the case firm linear programs.
The study concerns five
old-growth
Douglas-fir log classes which roughly correspond to the new USDA
Pacific Northwest Forest and Range Experiment Station gradinq rules
(Lane et al. (a), pp. 3-5).
We follow the notation of timber division
linear program formulation (C) of Appendix 1.2 for logs of class j,
x.
The symbol for logs from tract i
varies from 610 to 5200 MBF.
is x.
The size of the tracts
The log volumes (MBF) of the nine tracts
with the lowest harvesting cost, used in the linear programs are:
145
Tract
Logclass
1
2
3
4
6
5
7
8
9
X1
10
9
288
199
10
150
10
218
10
X2
56
54
480
527
10
216
348
437
761
X3
333
369 2304
527
267
138 2263 1529
571
X4
167
153
672
857
326
24
871 2186 2473
X5
44
315 1056
290
107
72
418
Total
610
900 4800 2400
720
830
495
600 3910 5200 4310
The log market prices of the first quarter of 1967 for Benton
County in Oregon, used in the linear programs, are (Oregon...):
Log Class
Log Grade
mP
$/MBF
Relative Price
x1
No. 2 Peeler
100
1.47
x2
No. 3 Peeler
90
1.33
x3
No. 2 Sawlog
68
1.00
x4
No. 3 Sawlog
58
.85
x5
Pulp Log
22
.33
To simplify the computations, log class 3 observed market price
is used as an "indicator" price and those for others are generated
by multiplying it by the relative prices.
The relative prices
above follow.wellthose of the whole area of Western Washington
146
and Northwestern Oregon (Adams, p. 5).
sc
We assume that the log selling price is not the same as its
purchasing price.
mill.
The above figures are FOB at the purchasing
The log selling costs are $3/MBF for all log classes.
Pulp logs (x5) cannot be used by the mills and are always sold to
the outside market.
hc
The harvesting costs for tract i consist of two components:
logging and transportation, and user costs.
Stumpage prices in
short term log allocation of our study are sunk costs.
For
public tiniber tracts they have been paid at the time of purchase.
For including the firm's own tracts in the five year cutting plan
from where they have been taken to the log allocation problem
their stumpage values have been already considered.
Douglas-fir stumpage paid for the tracts is $34/MBF.
The average
The first
quarter of 1967 average stumpage price is $52/MBF (Hamilton).
Logging and transportation costs are assumed to be $l9/MBF
to $39/MBF per tract as shown in Table 4.1
.
Tracts which cannot
be harvested in winter have infinite logging and transportation
costs and are not included in the harvestable tracts of this
study.
User cost computations are discussed in Appendix 2.3 and
their outcome is shown in Table 4.1.
The harvesting costs are
the same for all log classes in a tract.
One log class cannot be
147
harvested separately from the others.
XU, XL
The dependence of the best transfer prices on the harvest
level has been studied by setting the maximum harvest (XU) to
15,000 MBF, 9,500 MBF or 7,500 MBF.
Because the amount of pulp
logs is about 16 per cent of the timber volume on tracts the
maximum harvest of logs, usable to the mills is 12,600 MBF, 8,100
MBF or 6,300 MBF.
This means that the total harvest is 25 per
cent above, 20 per cent less or 40 per cent less than the mills'
total log need (about 10,200 MBF).
The minimumharvest is dictated by the need of keeping the
company's loggers employed.
It varies from 2,500 MBF to 8,500 MBF.
It does not have any significance in the allocation decisions,
however.
The harvesting cost is usually iuch lower than the log
market price.
This causes harvested, internal logs to be preferr-
ed to be bought, external logs.
In the linear program solutions
the harvest incurs always on its upper limit, XU, and XLconstraints are degenerate.
The upper limit of periodic external lo
purchases is $350,000.
It is the minimum of
the highest possible exchange
of logs for money in the
market at market prices or
the maximum amount of money allowed by the headquarters for
log purchases because of the firm's marketing strategy.
148
At the average log market price of $68/1BF the $350,000
corresponds 5.0 per cent of the mills1 approximate total log need
of 10,200 MBF.
The desire of keeping the mills running at capacity
Xk, Xjk
requires that they are provided a certain amount of logs.
The
Bohemia Inc.'s Coburg veneer plant is working in one shift and
sawmill in two shifts.
Given their average productivities the
maximum quarterly capacity of the veneer plant is about 3,420 MBF
and that of the sawmill 6,900 MBF.
our case firm.
These capacities are used for
The timber division faces these volumes under
timber dominance in the log delivery
r a
I
n t s.
g u
i
d i
n g
c o n
S
t -
It has to deliver those amounts of logs in order to
fill themills' capacities (Xk).
The timber division also has to deliver certain amounts of
certain classes of logs to the mills so that these can through
conversion satisfy the quarterly unfilled orders (ekml) of end
products.
g u
I
d
I
It faces these volumes under timber dominance in the
n g
c o n s t r a
i
n t s.
The log requirements
depend on the amount of unfilled orders and the log recoveries.
The approximate requirements (Xjk) are:
Log class
Veneer logs, MBF
Sawloqs, MBF
x1
300
400
x2
500
900
x3
300
700
x4
150
900
149
Under mill or mixed dominance the timber division must deliver
the logs demanded by the mills.
Symbol Xjk represents also these
demands in the right hand sides of the log
C
o n s t r a
i
n t s.
e n f o r c e m e n t
Table 4.3 consists of some Xjk values
which also represent the summaries of the log allocation under
mill dominance.
In this short term, one quarter log allocation analysis we
assume that the mills' log
kept constant.
starting and ending inventories, are
That is why the linear programs do not have any
symbols for log inventories.
Anticipated fixed costs and taxed of wood procurement do not appear
in the linear programs because they are the same for every organization
and transfer price alternative at the corporate level.
quarterly expected fixed cost and taxed are $15,100.
Timber division
The reason for
the low figure is the non-existence of the firm's own harvesting
machinery.
We assume that logging is contracted to other firms.
Appendix 3.2. (E) shows the use of the data presented in this
section for an example linear program.
output of that program.
Appendix 3..2.(G) presents the
150
Appendix 2.2.
Milling Data
- The end product prices and recoveries come from several studies
done recently in the Pacific Northwest forest products firms.
The
processing cost, processing time, capacity and end product demand
information comes primarily from Bohemia Inc.
The logs are committed
to four veneer grades (e16...,e41) on 3/8-inch basis and four sawntimber grades (e12,.
.
.
,e4).
Veneer quantity unit
and sawntimber unit thousand board feet.
is thousand square feet
In this study symbol MBM
(thousand board measure) is used for both of them. These have been
adopted from Forest Service studies (Lane et al.
Chapman of Bohemia Company (personal interview).
(a), p.
7) and Larry
Veneer plant produces
three types (e51,...,e71) and sawmill two types (e52,...,e62) of
residues.
End Product
Grade
End Product
A Veneer
e12
B Veneer
e22
11
Veneer
21
e31
C Veneer
C41
D Veneer
Sawn- /
Grade
Select: B,C and D
Select
Shop: Factory Select,
No.1 Shop, No. 2
Shop, No. 3 Shop
Moulding
t i mb e
e32
Dimension: Select
Structural
Construction, Standard
,
e51
Iindergrade
Veneer
42
Veneer '
Utility and Economy
Peeling Cores
residues
e71
Pulp Chips
Sawntimber.
residues
( e5
e62
Sawdust
Pulp Chips
151
The veneer and sawntimber grades are produced for three markets.
These markets are not functions of place but time in this study.
Mar-
ket 1 contains the unfilled orders (from last quarter of '66), market 2
anticipated orders (of first quarter of '67) and market 3 production
for inventory (for second quarter of '67).
The following is a short
discussion of the milling data used in the case firm linear programs.
epkmn
The sources of veneer (k1) and sawntimber (k2) prices are
the price indices of the Bureau of Labor Statistics for 1967.
The quarterly veneer prices (ep11
to ep14) have been computed as
.
functions of these indices from the regression equations of
Beuter's dissertation (pp. 137-8).
The sawntimber prices (ep21.
to ep24.) come from the same source.
Beuter, however, has thirt-
een regression equations that produce as amany sawntimber prices.
Our four sawntimber prices are weighted average of the prices of
the thirteen grades.
The weights are approximate shipment volumes.
They have been adapted from Beuter (p. 121).
Regression analyses
show that the relative veneer and sawtimber prices vary little
from quarter to quarter.
Therefore, for all three markets (m is
one to three) veneer grade three and sawtimber grade three prices
are chosen as "indicator end product prices'.
From market three
end product prices a storing cost, $5/MBM per period, is subtracted to get the net selling price.
The price for other grades are
computed by multiplying 'grade three prices for the first quarter
of '67 by the observed 1952-66 relative prices.
The veneer and
152
Table A.2.l.
End product
Veneer
grades
,
Market 1
dues
resi-
Relative
Market 3
price
47.8
58.4
55.5
1.77
e2
30.6
37.6
35.4
1.14
e31
27.0
33.0
30.9
1.00
e
19.7
24.1
22.2
.73
$.32/cu.ft.
20/cu. ft.
I
$.13/cu.ft.
len
Sawntimber
grades
Sawntimber
Price, $/MBM
Market 2
e11
[e51
Veneer
resi-
End product prices
L
e12
128.8
132.5
131.1
1.84
e22
68.6
70.6
67.9
.98
e32
70.0
72.0
69.4
1.00
e42
35.0
36.0
32.7
.50
$.09/cu.ft.
dues
$.13/cu.ft.
.
153
sawntimber relative prices come from regression analyses with very
high r-squared (over .95) values.
presented in Table A.2.l.
The end product prices are
The residue costs have been obtained
from several firms.
PC
We assume that the variable processing Costs are direct
jk
function of processing time and log class (Chapman, personal
interview).
By multiplying the processing time requirements (tjk)
of the log classes (j) at mills (k) by the following constants
we get the processing costs:
r.
j km
Veneer logs
Sawlogs
1540
t11 = $26/MBF
1460
t12
1315
t21 = $23/MBF
1370
t22 = $26/MBF
970
t31 = $l9/MBF
1235
t23 = $24/MBF
1020
t41 = $20/MBF
1285
t42
$28/MBF
$25/MBF
The recoveries have been recorded as functions of
and log top diameter.
log class
However, for this study only the recoveries
of the average diameter have been used for each log class.
They
are weighted averages of USDA Forest Service recovery studies for
old growth Douglas-fir in western Oregon and Washington, and
northwestern California (Lane
et al.
(b), (c)).
The weights are
the end product grade log volumes of these studies.
product recoveries appear in Table A.2.2.
The end
Sawntimber
residues
Total
Sawntimber
grades
resi dues
Veneer
Total
Veneer
grades
End Product
Table A.2.2.
10.0
.14
.19
11.3
44.78 cu.ft./MBF
43.40 cu.ft./MBF
e52
100.0
13.1
75.9
5.5
5.5
34.20 cu.ft./MBF
17.15 cu.ft./MBF
1.45
1.10
40.8
.08
46.41 cu.ft./MBF
15.00 cu.ft./MBF
.08
100.0
100.0
63.8
25.2
8.3
2.7
Per Cent
10.24 cu.ft./MBF
2.90
1.85
.73
.24
.08
MBM/MBF
14.1
14.42 cu.ft./MBF
1.42
.16
.58
.20
33.8
14.34 cu.ft./MBF
100.0
30.0
.42
1.40
17.9
.25
.48
44.95 cu.ft./MBF
10.88 cu.ft./MBF
2.56 cu.ft./MBF
100.0
28.2
30.5
22.5
18.8
Per Cent
e52
e42
e22
42.1
e12
.59
47.60 cu.ft./MBF
9.57 cu.ft./MBF
2.40 cu.ft./MBF
e71
e61
e51
2.98
.84
22.9
.72
e41
100.0
.91
30.0
.94
e31
3.13
.67
21.0
.56
.66
26.1
.82
MBM/MBF
Log Class
Log Class 2, X2
Log Class 3, X3
e21
e11
Per Cent
MBM/MBF
Log Class 1, X1
End product recoveries.
100.0
86.8
7.5
3.4
2.3
Per Cent
100.0
32.3
60.2
5.6
1.9
46.22 cu.ft./MBF
18.87 cu.ft./MBF
1.61
.52
.97
.09
.03
66.41 cu.ft./MBF
21.99 cu.ft./MBF
23.60 cu.ft./MBF
2.60
2.31
.20
.09
.06
MBM/MBF
Log Class 4, X4
155
Based on discussions with the supervisors of the Bohemia
t.
jk
Company Coburg mills, Ted Nelson of Weyerhaeuser Company and
Dobie's dissertation (pp. 60-69) the following processing time
functions have been developed:
Veneer logs
.154
= .00346
t.
\ID
+
- .0217
\1D
Sawlogs
.143
= .000156
t.
D.
+
D.
+
.0074121
'3
where t
=
log class j processin.g time, shifts/MBF
=
class j logts top diameter
Di
The period of time a log needs to be processed thus only depends
on the log diameter.
The average diameters and the resulting process-
ing time requirement (eight-hour shifts per MBF) of the four log
classes are:
Log class
Diameter, inches
Veneer log
processing time,
shifts/MBF
Sawlog
processing time,
shifts/1BF
x1
37
.01692
.01916
x2
31
.01748
.01817
x3
22
.01962
.01945
x4
22
.01962
.01945
Following the example of the veneer plant and sawmill of Bohemia
Inc.
in Coburg.the veneer plant and sawmill ofour case firm
work in one shift and two shifts, respectively.
In an average
156
In the veneer plant there are
there are 22 work days per jiionth.
then 66 and in the sawmill 132 shifts available in the cominq
qua rter.
EUmn
ELmn
We assume that the end products (m) may be sold to three
markets (n) which are a function of time.
and ELmn the lower limit of these sales.
EUmn depicts the upper
Market 1
(n=1)
comprises the unfilled orders of end products that must be promptly satisfied:
EUm1 = EL1
anticipated orders.
..
Some of them are from old customers and their
Market 3 (n=3) includes produc-
lower limit is known: ELm2
tion of inventory.
EUm3
Market 2 (n=2) includes the
The inventory capacity has an upper limit:
The following summarizes the end product sales limits:
Unfilled orders,
, MBM
= EL
EU
ml
ml
300
0
300
200
e31
200
100
e41
100
70
300
100
e22
700
200
e32
600
100
e42
500
400
e11
veneer
12
sawritimber )
Anticipated
orders, ELm2
MBM
Production for
inventory,EUm3,
MBM
800
}
1900
}
157
XL
The desire to keep the loggers employed requires certain minimum
These requirements are included in the
harvest activities.
g u
I
d
I
dominance.
MBF.
c o n s t r a
n g
i
n t s
to the mills under mill
In the calculations they vary from 2,500 MBF to 8,500
They did not have significance in calculations.
These
constraints seldom were effective.
X
Under mill or mixed dominance the milling division has to consider
the limited volumes of the different log classes available through
harvest before dictating its internal log demands to the timber
division.
These maximum volumes (X) are included in milling
division
g u
i
d
i
n g
c o n s t r a i n t s.
The internal
log availability to the mills depends directly on the log class
distribution of the tracts and the total maximum harvest by the
timber division.
The following volumes were used at the two
extreme maximum harvest levels as the X
values for the four log
classes usable to the mills:
Log class
Harvest = 15,000 MBF
Harvest = 7,500 MBF
x1
670
500
x2
1780
1100
x3
6340
2600
x4
3510
1400
These figures were found through pilot calculations.
They were
158
given as high values as possible when at the same time maintaining
feasibility in the timber divisions log allocations.
Under timber dominance the mills receive lou supply announcements
(X.) from the timber division.
c e m e n t
c o n s t r a
i
These are included in the
n t s
e n f 0 r -
to the milling division.
Their
function corresponds to that of the enforcement constraints (sums of
log allocation) to the timber division under mill dominance presented
in Table 4.3.
Anticipated fixed costs and taxes of milling do not appear in the
linear programs because they are at the corporate
every organization and transfer price alternative.
level the same for
The milling
division quarterly expected fixed costs and taxes are $63,000.
Appendix 3.2JF) shows the use of the data presented in this
section for an example linear program.
output of that linear program.
Appendix 3.2.(H) presents the
159
Appendix 2.3.
User Cost Computations
The opportunity cost of harvesting a tract--its user cost--is the
difference between the current estimated average tract value and the
future expected discounted value with the harvested logs to be delivered to their highest value use.
of five years (20 quarters
)
For company-owned tracts a total period
is considered, for public timber tracts a
period as long as the sales contract is valid before expiration.
The
sum of the average logging and transportation cost for a tract in a
quarter and its user cost is the tract harvesting cost in that quarter.
Figure A.2..l depicts the general user cost computation process.
For
the case study a more specific Tract Opportunity Cost Program (TOC)
has been written.
The program first arranges data
for computations:
log and end product market prices, log volumes, recoveries, growth
rates, logging and transportation, and milling costs.
These data
are used for computing discounted tract unit values for each quarter
which are the basis for determining the best harvest schedule.
The
tracts are scheduled for harvest in the quarter having the highest
discounted value while cutting
at most
the maximum allowable harvest.
The assumed allowable periodic maximum harvest is 14,000 MBF.
Since the average pulp log portion in the tracts is 16 per cent
this
means about 11,800 MBF of logs usable to the mills--l5 per cent above
the average mills' capacity of 10,200 MBF.
case firm is a net log seller.
We thus assume that the
The following is a short comment on
160
I
/MPtsct/
/ TC0
/
/
I pc.jk
Class j log
value at
market in
quarter t
II
km
j I
R.
jkm
t
Class j log
value at mill
k in quarter
/
TC.k
t
Check that
each quarter
some logs are
going to all
mills k
Find best destin, for class
logs in
j
t
quarter t
.°
1
Coniputo log
Compute tract
i disc, value
vol. for class
j and total
vol. for tract
i inquarter t
in quarter t
Check that
total harvesting constr. is
satisfied for
V.
/
/
Find bestquarter t* for tract
i
to be harvested
quarter 1)
each quarter t
Discounted
value of each
tract at its
best
Notation:
Notation:
t
= log transp.
cost from tract
i
to outside
Compute the
market inquarter diff. between
discounted valt, $/MBF
LC
= tract i logging
ues of quarters
for ea.
tract i = user,
cost of tract
cost inquartert, t* and
$/MBF
MP,t = class j log market price in
quarter t, $/MBF
= class j log selling cost to the
market in quarter
t, /MBF
Figure A.2.l.
1
1
Ur cost =
opportunity
cost of harvesting tract
User cost computations.
TCjk
= log transp. cost from tract
i
to mill k in quarter t,
R
= recovery of class j logs to
end product in at mill k in
$/ riB F
j km
quarter t, i1BM/MBF
Dt = discount factor
V. 0 = initial volume of class j
13
G.
13
log on tract i
= net volume growth percentage
of class j logs on tract i
in quarter t
= class j log process cost
in quarter t, $/MBF
EPkt = end product m price at mill
k in quarter t, $/MBF
PCik
161
the other assumptions about the inputs to the user cost computations
of Figure A.2.1:
TCiot, TCjkt, LCit
1
The logging and transportation costs for quarter
are the same for each tract and destination as those of Appendix
2.1.
Their yearly increase is assumed to be 5.8 per cent--an
estimate made in Bohemia Inc.
MPt
The log market price for the "indicator" log class 3 is the
same for quarter 1 as those of Appendix 2.1.
For the other 19
quarters they are the recorded prices of Benton County, Oregon,
with a minimum of $69/MBF in quarter 2 and a maximum of $lO3/MBF
in quarter 10 (Oregon...).
For quarters 2-9 the prices of other
classes follow those of quarter 1.
For quarters 10-20 they are
assumed to follow the relative prices:
Log class
Relative price
.xl
1.35
x2
1.28
x3
1.00
x4
.88
x5
.45
These relative prices mirror a slight decrease in the differences
between the recorded prices of the log classes from 1967 to 1971
162
SC
The log selling costs are supposed to remain constant $3 for
k
all log classes and all 20 quarters.
The log processing costs are assumed to increase 5.5 per cent
PC.
jk
per year from those of the first quarter 1967 in Appendix 2.1.
EPkmt
The relative end product prices
through the 20 quarters.
remain those of Appendix 2.1
The recorded veneer grade 3 prices vary
from a low $32/MBM in quarter 2 to a high $68/MBM in quarter 9.
The recorded sawntimber grade 3 prices vary from a low $74 in
quarter 2 to a
Rjkmt
high $117/MBM in quarter 9.
The recoveries are assumed to remain unchanged at the levels
of Appendix 2.1 through the 20 quarters.
The rate of interest in the discount factor is assumed to be
six per cent per year.
V.
13
The quarter 1 volumes of log classes for some tracts are
listed in Appendix 2.1.
Gt
13
In the high volume Douglas-fir.old growth tracts the percentage volume growth is small. The trees are also susceptible to rot.
The tracts are divided into six groups with respect to their
163
assumed yearly net growth percentages:
Tract group
Log class
l
2
1
.46
.70
2
.61
3
4
5
6
2.22
1.70
.31
.67
.50
1.75
1.59
.44
.51
.27
.38
.69
1 .40
.58
.49
4
.26
.31
.12
.96
.76
.62
5
.05
.09
.03
.10
.11
.16
3
Based on Oregon State University studies on tree growth it is
assumed that 70 per cent of the yearly growth incurs by the end of
the spring (2nd, 6th,..., 18th) quarter and 30 per cent by the end
of the summer (3rd, 7th,..., 19th) quarter (Bill Emmingham, per
-
sonal interview). The company tracts are available the whole five
year period (20 quarters).
t
The public timber sale tracts are available from three to
nine quarters, as an average seven quarters.
Table 4.1 shows
the periods when the tract values are at their highest.
After finding the optimal timing of the harvest of all tracts the
difference between the value of each tract in the best quarter and
quarter 1 is computed.
of quarter 1.
The difference is divided by the tract volume
This is the unit loss from cutting a tract the first
quarter instead of later in the future--it is the user cost.
of quarter 1
The sum
logging and transportation costs and user cost is its
164
harvesting cost, used in the log allocation linear programs.
The
tracts are ordered according to the harvesting costs from smallest to
greatest as Table 4.1 shows.
The rank of each tract indicates its
number (i) in the log allocation linear program.
Pilot timber division linear program computations have shown that
for the case firm, a tract with a lower harvesting cost is almost
always preferred to be cut for one with a higher cost.
Therefore, of
the 30 tracts available only the nine tracts with the lowest harvesting
costs have been chosen for the linear programming computations.
The
Tract Opportunity Cost program then serves for computing harvesting
costs for each tract and reducing the costs of the linear programming
calculations by decreasing the number of alternative tracts to be cut.
The program routines follow.
The most complex of these, subroutine
SWITCH is flow-charted in Figure a.2.2.
PROGRAM TOC
Computes discount factors, logging and transportation costs, and
processing costs.
Coordinates user cost computations and produce
output.
SUBROUTINE PRICESET
Reads prices of indicator log class (class 3) and indicator end
products (veneer and sawntimber grades 3), and generates price tables,
given the relative log class and end product market prices.
designed for uncertainty studies.
Originally
165
)
( Start
S.,
Generate a list of all tracts
scheduled for harvest on best
period (1), and compute nextbest period (NP) for each
tract.
4,
Compute differences between
the discounted values of each
tract's scheduled harvest at
period I and next-best
harvest at period NP.
Remove tract N from
the list.
1
Find the tract (N) in the
list having the smallest
differences.
.1
Remove tract N to its next
best harvest period (NP).
oe s
the period
tract N is moved to
NP) now harvest more
han maximum
cut?
Yes
Reschedule this
tract back to the
present period (I).
Does
the present
period (I) now harvest
less than maximum
No
cut?
Yes
Figure A.2.2.
Flow chart of subroutine SWITCH.
166
SUBROUTINE VOLUME
Sets up recovery tables.
SUBROUTINE TREES
Sets up log volume tables for 20 quarters (periods) simulating
tree growth from quarter 1 log volumes.
SUBROUTINE COMPUTE
Computes log values at mills and at outside market for each log
class and each quarter.
Selects the best log uses for all 20 quarters.
SUBROUTINE MAXCHECK
Computes discounted tract unit values for each quarter each tract
is available for harvesting.
Records the period with the highest
discounted value to be the quarter of harvest.
Checks that total
harvest is less than maximum cut (14,000 MBF) for each quarter and
computes the iser cost and harvesting cost of period 1.
SUBROUTINE SWITCH (I)
Determines tract(s) to be removed to next best harvest quarter to
reduce harvest quantity on quarter 1
below the maximum cut (14,000 MBF).
FUNCTION SUM(I)
Computes the total quantity of harvest on quarter (period) I.
FUNCTION XMIZE(K,I,NQ)
Determines the next best harvest quarter (period), NO, for tract
K, scheduled to be harvested on period I.
Computes the difference
(XMIZE) between the discounted values for quarters I and NQ.
Tract Opportunity Cost Program
053 FOPIRAN
0FIN
I
09/12/75
VFRSION T.t3
PFOI
167
2367
COIIMO'I
I
(,O'cMOM PV(7.2)).PS(6,20).'U(5,0).Ct4Vf5.2C).CMf5,Zo34
LT43C.20).RECVVIS.73.RECVS(5.'.).QISC12O3.0V1(30,20).
NUt 30) .UN (5, 3) .1330,5. 20) ,LY( 30.33 , Ut,) 30)
'V I 3 Zo
5)
PROGOA(4 TOC
t)CIUI)E COP4MOW
REII.
CO3'ION
iiI?
13
1.
),OS(6.20),PMIS.201
10149073 Pv(7,
C09'13U
'CIT (30, Z) .IF
.f.MV35,20) CIISt3,20).
VJ (5 .7) ,QECVS (5,',) ,OTSC(?0),
Dill 30. 20).
't1XT)ju).TTB(50).(l,I(5,3),It30.5.2O),IK(3C.S),OCl30I.
'
i)T1414S13N 3CV(5).SCS)5)
DiTiIS'JcZF,.,?T..tl. .?)..t000.),
Ic'CO. ?5..2'. ......1503.)
I)
flhIlUsION 01.17 (.10 I .HC(3C )
it.
I]1C'i.u1'.6739
OISC(iI'i.
15
OcT10 1.?,:?1
17
otcc(r)rr.(r-1)/'01C
10
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1)')
'CV) I)
C4V 31, 1)
20
2t
1) usc.' (1)
I
5 J?.2)
CHY (1, 3) rC.MV (1, J-1) 'DC
22
OcT
23
SCMSN.J)rC,rlS(T.J-i(DC
PtA(T(5,7) (I.t.T
6731
5 TI.3)
jr,,fl
U 1'.
1
'3
1)3
00
23
)(,t) c(j,3Q)
103(3 (OF?.?)
7
2'
3 CIT (1. 3) C1T( 1, 3-1) '0
33
T57[FSjr)
COLt.
31
CIII "iC1CFT
C.II IVOIIIM7
37
3t.
35
C3LI C0"PIJTE
CO'.I
3',
C
35
C
C
3'
XCkt.Cl<
ia
co'lpulE 0PP0'1U11ITY COST ((IC) AND WAQVEST!35 COST (OCIT)
00 57 )(5,13
3-)
'3
'I
'K
Ii's (K I
5 rc'lXT ('0)
oKivT(K,T)-I1VI(K.1)
((CIT (Y)'CIT(K,i)
17 r)Q
I-"IK):OCI'0! 'CLO('0,i)
'I.
10 1rI.29
11'I1J'It.
00
'3
((IC (3) IT H (K)) K' 3
I
9 COr4T1PUC
IF It. S
JrTBU 1
II GOlDS
507'3(T)70'3(K)
SIT'3(cc)'J
SOC 11) zOC (K)
501 3k)
.5'cXT (T('MXT(K)
(1)
'OS (1)
3,3.151'
5951(K) '3
T0CIT(t) TOCLT )T)'OCLT(K) 50CtT(X)rI
SHC('0)zT
SHC)I)'HC(K)
TzHC(I)
10 CO5)t'J
SO)
I
OC It) ad I ( I '4C ( ) ,95r( 1)
W'0 II U 10 2N (TO U (1
2tr'Onil(:1S03010 or-c H0RVESTtN(.00iT5//ITt..3V1?.2.2It2))
I
I
lOLL ((II
1735
OulTJt TRIISOM
6'
1,3
C0cl-O)1
'a."") l.'3'
(rICIUDS C09r1073
((SAL
0
n5(f,,Ø),p(6.J),rr-t;I5 ?0),I'MS(5,20),
ccIocc p(7,7
1.
.001 NO.20)
'CLI )3U,Zt) .PFCVV(S.7).RSCVS(5.5) ,cTtcC23I
I, K (3 Q 5) (t( 3 35)
4
(4
11(0) '5)
c 1) 33)
l((( (5
3) 5. 3 30,5 20)
V)i.Z0,5)
'CO'MO)')
eIcE)cc't0U 1T(3O).R1'T(6.5)
C
I.
READ PERIOD 1 LOG VOl.1)915 FOR EOCN TRACT
C
Si
1'
1
7:
C
75
C
1''
C
73
C
7',
C
N
PEA05.t)
((t.(I,J.i),J'I,5) .t1,30)
rosclATI5F7.0)
II
350W 11,015 F0
GROWTH RATS OASIS
1051 AVOILA(IE HA'VFST PFPTI)O roo TRACT
1113
10011 FOW AP'I'IUAL 1,PQWT(I PATS
7AflI5,flhl.!TF)
2 F,)Q$tiT (331 'I
PSAHS,3)RCT
3
)q t. i'l.S
7',
77
74
(3)
I,
(:i.j3
SI
57
rt?rIct. i3
F',
I I(<,J.l 3
57
I(rntftt.UE
IK.J.I'3('L('.i,I'2)
Ut
SI
'.
l'2.?0.'.
1.3K.
J.1-I)'((0.7)'i.I
I,IF )K.J,I'/)'L(".
J.!'i)
IT Sq. 10)C0T0'.
Si
AS
90
91
c059'3I(TINF PRTCET
oiLU'3r cori'i0I
'N
'CO"13,1
it.
PEAL I
()rI'3't
'CIT (33.2c)) 1(UVV(5.7),PCCVS(5.6) D!SC(23).DVT'35,21
.CC( .31),
't01 (30) .110(33) ,l(S.3) .L(30.5.201 .11(310.',)
0
'0)1.20.")
irlEOSIC)" scALs1nq.))
Tract Opportunity Cost Program (cont.)
C
05T() PNC5S Or I D(CU0
C
C
f)I5, I) (21) 3,J)
Eli!) PRO!)tJ(T5
)I,,2O)
168
P10(5.11 ((°V(!, 31 Jt ?) .0.7)
'fl
PEfl.i)(°(.T,J),J1,C,
1)1
P110(5,1) ()PS(j,J),J1,2Jl.I'5.f,l
121.
132
133
1).
C
C
PRICES Of'
PEII
C
125
105
127
LOC. CLASS
0IIOIC1TO
PEAol.1))pMc.J)
1 rO"'41T(2F5.0)
11
13
C
C)fIDUTE V.''iECP PRICFS FRUM INOICAIOP VENEER
C
ill
C
It!
113
fl,')
K)
lit.
2 J't.21
CO 2
T1.'.
115
Ii',
it?
lii
it)
121
1)2
('V I I
3) PV I I
2 Cr)&T I WUF
C
JI
I)
'SC AL AP
CO'(PUTE SIWNT!MPEP PRICES FRU'4-INUICA000 cowuTrpil(1
C
C
3 Jt.20
00
-123
00 3
12'.
It,'.
125
Pj,JIPS(3,,J)SCALARIK,2)
12'
I2
I CONTINUE
12)
C
112
C
111.
C
COMPUTE 'IARI(ET P°ICES FUQ'I INOTCAT()P LOS CLASC
!NOEXQ
t.
J1.2C
IJ.EQ.9)IN0EX'DiOFX4
13!
ifl
l3.
IY'
1",
137
131
01
Trl.5
00
t(I.i). 31'lTo.
i10TtiCEX.
1'.)
1*1.
'(#1
CONTInUE
'.
153
Pl(2,1)'17.
1'..
II.;
pp!i..11r55.
1'.
'S'JAROUTTNI (VOLUME
ROUTINE TO CDM('UTE RECOVERY VOL'IME lADLES
RECOVERY RATIOS.
VOM
-
PRV
PECOVtPY PATTO' F'JP VEILK
055
P:r.ovPv DATIn15 FO SAW'1T4'1F'1
IflCVV r PCOVE'1Y VOLUME -TARLE F
VENEER
RCCOVEPY V0t.U'IE TABlE FOR SAwuT TlEQ
PECV3
IIJCUJOE COMMON
PEAL I
COrIPIOS °vI?.!CI.PS.20 P'16.2O ,CP'V)5.20).CMSIS.20!
cLT(3o.2o).pECvV(s,7).pFv5(5.c),oIscI2o),tJvI(1o.?oI.
'NAT (21) .118(30) 1!.I5..).) .L(30.5.20) (_l(-) i0.S).00) 30).
'V(T.Zl.S)
CO''ON
01H1rjSIOti RAV(t.),P,RS('..1
RCin(s. t)RJ.4S
t FGNMAT('.Ft..
)-)EAD (S 2) RF.CVV,RFCVS
152
I '.1
2 cO?.1AT(''..2)
15'165
0')
0')
It.'.
5
3
J'I.'.
WECVV (I, J) PLC VV It, 2) PBV(i) .11
3 RF.CVS(I.J) RET,'IS(t,J)'RRS)I).Oi
16-.
I6F
FM!)
iF."
JfSOCIIYTIE CO,l.'UTE
I 6
17)
C
171
17)
11'
C
C
17'.
0
C
175
C
17..
C
177
171
17)
II)
ill
C
C
C
1P'.
1
-
pOUT1N COMPuTES LOS VALUE TABLES rop
CLASS I ON 2(0100 J
T'4I
(.055 O
1(t.J.Tl
V(2.J.1)
V)1.J.I)
LOT,
AT VFPIEEP PLANT
'lAL'J[
LOG VALUE AT SAWMILL
(.00 SELLIMT, °V,tCE
= COST CF )IILLI!H', VENEER
C'l'I(I, II
COIl Oc MILLING LUMBER
Cr1511.3)
OESI USE FITS LOG
ROUTINE MUST MAINTAIN NOT LF.SS TNAN ONE AN!) NOT
MORE TWiN TV!) LOS CLASSES r.OINC TO EACH MILL. Alto
lkft.J)
C
C
C
br, CLASS FIVE (11351 BE SOLO.
C
C
INCLUOE
s
PEAL L
COPIWON
CO' "AM
C0'0'I
CflM'1PI
COIIMIT'l
-
PV,7O),P5(E,2I).2M)5,I.CM'lI5.2O).t'5(f,.2hI
'CLTITO.201.Q1C'JV(5.7) PECv;S c.t1I';r.)281 011(39 701.
M(30I.TTR)3O).NP4(..L)3O,.20),IK)I0.).0C(3).
CP l'l)N
''1(1,20.51
COl'OP4
I
'11 tIE HG TON '1(20 51 LN (I)
I AT
00
Ii
VIi,J.t1.V(I,J.t)RECVV(I.K)PV(1.3)
Ir(,(.Tu.7,c.0T02
V(2.JI)rV(2.J.I)'RECVS(X.'PS((.J)
-
2
I 97
-
00 2
19!
11
1I.5
V I I J. 0) rV 12.J.II '0.
I 'It
j '17
101
1
O1 I J.1.2)
IA)
193
rQtJ-TI.I'JE
V(l,J.'T).V(l.J,I)'CPlV(T.Jl
-
V (2 J I) 'V (2 2.1) -CHS (I
V (3,2 I) PM)!. 2) -3.-
I CONTINUE
0')
'.0 J1.2)
LU(lI.TLNI2I'LN(3)'3
-
2)
-
Tract Opportunity Cost Program (cont.)
2t
I-I,'.
iBtc=V11,J,1I
:, n=t
00
211
2).
1.
Q 3 V:. 3
1(I(tC..VtK,J,t)I(O103
iK(J, 1)''<
205
20'.
)IIC,=v(J(,J.TP
1 co4TU4I
?C7
210
169
,.LV(J,1)
'. 1if)
LPflK)I
LF(LU(t1 .GE.tlC,0TOtO
9 I1.5
21)
211
212
21'
21'.
21.
Do
21'.
3, I
'(
I -VII 3
1)
IFt0.r,F.SMALI_,'orog
217
210
SIOALIrO
'1)
223
221
72?
225
'I
22.
775
22'.
1
27
11
221
21)
PU=T
LPfltIr1U(1)1
LIUI) -1t'I-1
1((J,N1)1
S(111t'lEIQO
(10 10 1=1,5
V(L'4(21.EQ.2)101fl18
0=Vl1.J, 1) -V(7,J,1)
23
rALL0
23.
71;
205
717
700
1,ID.Vti,J,I,-V3,JT)
2.7
2.1
7'.?
2'.(
2'.'.
2.5
2'.'.
I=1
19 CQ'ITIIIJF:
lrfo.G1.suALL,ofo1q
£'IIILrD
23'
LN(1)1N(1)-1
23
2'.'
2'.'
S11Lt.110)
03 23 t1.5
0=V(F,.J,II-I!(2,J.I)
751
2,2
25!
1F(D.G,SMALL)G0T079
25'.
255
255
U0=1
23 C0;T1NIJE
27
tN(ZILP4(2)'l
L'l Ufl
250
750
751
50T0'.Q
3) 17(LN(2).17.3)GOTO'.Q
37 S(1ALL1100
251
252
761
Dl) 35 t'1.5
26'.
IF (11< (3.0) .NF.2)G011139
I'(L(t.Eq.2IGoTo3
2'.c
D=V(2,J,t)-l1J,I)
IF)D.C,0.SMOLL)G01018
¶rlALLD
tirl
255
26'
260
253
773
271
27?
CrV(2.J,1I_V(3.J.J)
3
1)O.5E.SHALUGOT039
SMALL.0
273
N3
27.
77.5
27,
NII
39 C0JT1N1,E
LU(2)=LN(?)-t
277
770
273
703
701
252
203
20.
205
20,
207
700
COM'O'1
LiI'lI 1
154)'4)
J'(LN(2).f:Q.))c,01017
'.0 r.oT1iu
RETUPtO
ItO))
SUOtOOUTIN{ 'IATCHECK
I4CLUO( C0tl'ION
C
C011UTE D!C0U?ITE0 VALUES (flY!)
C
C
P101
C04l0'O
L
(0'10N PV(7,2Q) ,P5(5,7Q) ,PM(S,20) .C'IV(5,201 .(l5 (5,20).
CLO (1O.2Q),PfCOV(5,7),prCS(5,S) ,DTSC(20( .0'/1 (30,70)
4X70(,13(30I,Wu(5,3),L130,5,20)
L'((30,5),OC(301
V13,20,5)
co'1o'
0) 1 Krl.30
701
291
L4r!T5('
(115'-tIIOt)
203
SH=5O=0.
2_Ia
!t.LA
00 2 ii,S
21'
00 1
21.
215
295
297
290
201
331
331
3-)?
'NSN*I )K.J,t).VIN.I,J)
2 SOSO.LIK.J,1)
C
C
STLECT 01st P19100 10 HARVEST (MAT)
C
IFoOIG,G1.0VI(K,I)IGOTOI
373
30.
3-35
335
307
lEO
1
C
C
00 5 ('1.20
3-jO
31.)
311
312
313
COHTI';UF
CF1CCK ROR 'IAXT'I'J'i 0(11 bR EACW °IRIOt)
C
5
1)1'.000..LT.SUM(0I)CALL SWTTCW(!,
(0T1TINUC
RITU'II0
Tract Opportunity Cost Program (cont.)
314
315
C0MIOM
S'J(POUTIHL 'WITCHCT)
INCLUDE COIIMr)P4
'
REF,L
31'.
ST1AL11E103
MTO
31?
31'
3!.)
321
12?
323
1).
325
327
121
C
GEIICHATI (_TSI 01 Al.). TRACTS
INS '44°VESTLO CN ENIO)
11 r 0130011(3 VAlUE DIFFIRENCES BETWEEN PERIODS
Uk
REST 0111005
NP
NEXT (lEST P1PtO0S
PIT
SIIOSCRIPT FO LIST
C
fl
C
C
C
C
C
00 1 K1,3
MTHT41
323
33)
331
337
(fl
31'.
335
33,
337
333
131
3d.)
0SM(Ht)X(IZ1l(.t,'UP((lT))
C
C
3.'.
345
34'.
347
3'.'
343
C' A3 174 SUALLESI OICCO'J'l(Ffl vA'uE DIrFEWENCE
F"ACT WITH LEAST SIC,N!FTCAHT 5IIITCH
SElECT
N
C
14115 TRACT$S NEXT BEST PERIOD
NI
C
ARRAY INDEX FOR )ITH 1)1801
145
C
C
tr(o.00.SllAL1)GnTot
P4r1<
NIHP)T1T)
NI-MI
'41
3'.2
343
I.
C)HPIIPI Pv(7,?3).PSI7,,2) r'Ht5,?T) rMvlc,pD,,CHSIS,20.
CtT (30.20) ,RICVV)5,7).RECVS(5 7,).OIt(20) .041(39,20)
M01 30) 11013,?) ,T((5.3).LT0,,20) .LK(TO.51.0C(30)
1130) N30) ,N°l30)
I1H171SION
1. CONTINUE
C
S34!TCH N-TN TPACT AND CHECk TO SEF IF SWITCH FO(JLR.0
C
C
UP
C
C
C
C
7,
1.-Il N-TN 1'ACEiS UC1 1(151 0111)0(1
IF SO. TAkE NEXT REST (PACT TO SWTTCHJOF
IF HOT. CHCk TO SEE IF WERE OCUC
I' SO. I(IORE THIS TRACT AP13 TAKE WE1T.
P4)7T(74)r)4
35)
IF(5UM(NI).LT,I7,303.)&0T02
35'.
7411(N)zt
SHAIL-574(NS)rXHIZE(N.T,14p1N5)l
00 3 l('l,Mt
Tr)SHALL.1T.SM(K))GOTOJ.
351
35?
353
DVI
5
3-'S
11SIllKI
3',',
35'
354
35)
iSrk
I COIITINUI
1010.
36C
35?
C
P.3
0
31-S
C
C
C
3E.
'51,
37,7
I.NI)rT)VI(N,I)P1.
(F THE LAST SWITCH PUT THE I-TN FFRIOO
101 TOTAL U'lflER THE 1.310 UhF PXIHUM CUT. fF 50. PFT'jP.
CHECk TO SEE
IF 7401,
T9Y THE PIlOT TRACT WITH LEAST STF,NTFICANI SWITCH.
2 II (SUNlIt .LT.11.000.)RETUR9
SM I NA I' SM) 'IT I
37,3
117<7 1451
NK I 741)
373
371
NP'PIS) N((IT)
374
375
I 740
STIALL'1C100
IT 14 1
(,3T05
37!
375
377
371
371
t 4')
FUNCTION 5074(1)
C
C
C
C
CO'I'ON
333
3 4'.
355
345
00 5 Jr1,I.
ST1UFSHI1F,t(K,J.1)
345
I SR
E TURN
140
'39)
391
317
I'll
00'IMPII
CO''OM
3.11
431
41!
7,31
'2'.
'CS
t.
S CONTINUE
A CONTINUE
SUNSM8F
347
395
115
INCLUDE COMNON
WEAL
COON
3S1
33!
39.
H.RV1ST ON PELIOO I.
tQH'IoN PV) 1,73) ,S(o,2)) PM(5,?3) .014/(5.70) C1(S(5.20)
C11I!0.,?Q).RECVV(5.?I,RECVS(I.7,),OISC(20).OVIUO.20),
74xT1301.TTR(3I).NN(5,3),L(3O,5.20),LKI3Q.5),0C130),
(3,20 5)
SURF '0.
00 ( I(1,3Q
Ic(M1T(k) .01.1)10101.
C0'I'ON
C04'ON
39?
303
COIPIJIE SUN OF LOS OUA7ITXTIES FROP4 TRACTS SCHEDULED FOR
UHCTIO'l XIII2C(K.I.N0)
C
0
C
C
C
OITERTITNE 14FYT BEST PERIOD FOP k-TN TRACT
WARVEIT 0I P0flQ I.
11401001 COtIMON
RI:AL
SCI4EDUIFO FOR
TO NEXT 131ST FAPTOD
i
CO,1'4flr4 PV(7,?O).P5(c,,?9),fl..1(5,70),CNV(5 201.(:r.(1,2O(,
CLTI30,701 RICV'JS,7,PECVSS.cI.0I5C1231 031130,20),
.3! (30) 1T830 ,s.n .LU0.S,2QI ,LXtJO.SI .011301
IAIjtEjQQ
1 J'1,20
0'OVI I7.T)-OVTl(,i)
00
UII.F.O.J)GOTOO
(F(o.1T.0.,r,OTQI
IrID.CE.SMALL)GOTO1
S'IAIL'O
llqrJ
7,37
1 CONTINUE
'.13
PETLIRPI
1110
401
'.00
SET 719 113041
XHIZE EQUAL. 10 THE DIFFERINCE RETWIIP4 OIS00074TEQ
VALUES FOR PERIODS I 0)40 HO.
3)40
170
171
LINEAR PROGRAMMING COMPUTATIONS FOR DOMINANCE ALLOCATIONS
Appendix 3.
Appendix 3.1
.
Simplex Input Computer Flow Charts
The simplex program of the Oregon State University CDC 3300 computer, under direction of a control deck, receives the linear programs'
inputs, optimizes the problem and outputs the results.
Due to the
large size of the problems, simplex program inputs are created by
computer, as is shown in Figure A.3.l.
Parametric cost, price and
right-hand-side data are in the CONSTANTS deck or in function definitions associated with the FORMAT PROGRAM.
The actual parameter changes
are, however, made by the SIMPLEX CONTROL deck directing the SIMPLEX
PROGRAM how to process the INPUT.
Constraints may be added or deleted
by changing the Fortran FORMAT PROGRAMS.
In order to optimize the parametric linear programs as economically
as possible, two characteristics of the SIMPLEX PROGRAM are used.
First,
it has the ability to declare more than one objective function, and
chooses one of them for the optimization process at the time of
program execution.
This is usable as long as the constraint coeffici-
ents and right-hand-sides remain the same in iterations.
Second, names
are defined for constraint coefficients and right-hand-side constants.
These names are assigned numeric values which may be changed between
parametric runs.
In all milling division runs, the names are defined,
but we have not needed to reassign their values for this study.
172
F 0 RMAT
PROGRAM
CONSTANTS
SIMPLEX
CONTROL
INPUT
Figure A.3.l.
SIMPLEX
PROGRAM
(*REXX)
Simplex input computer routines.
173
Figure A.3.2. is a flow diagram of the FORMAT PROGRAM used in
developing simplex input for themillirig division problems.Procedure 1
generates the DEFINE statements for the end product recovery and
processing time constraint coefficients.
These could be redefined at
some additional points 9... as many times as required, but that has not
been necessary for this study.
Point 2 shows the guiding restrictions
by the timber division under mill or mixed dominance.
It shows
enforced allocations by the timber division under timber dominance.
Procedure 4 uses end product price defining functions and end product
demand requirements to produce coefficients for the end product column
vectors.
Procedure 6 uses transfer price and processing cost functions,
and recovery and processing time definitions from procedure 1 to compute coefficients for the log column vectors.
In mill dominance calcul-
ations the market prices of purchased logs are inserted using a text
editors.
Figure A.3.3. is a flow diagram of the FORMAT PROGRAM for generating timber division simplex input. First, data are read from the
CONSTANTS deck.
These are used in procedure 2 to generate DEFINE state-
ments for the linear programs' coefficients.
These are changed in
later DEFINE statements (procedure 7...) for the parametric runs.
Procedure 4 uses data for log availabilities on tracts from point 1.
It computes coefficients for constraints which insure that in harvested
tracts all log classes are cut in their existing volume proportions.
Points 3, 5, and 6 generate the actual simplex input that uses the
defined names from procedure 2.
174
(Begin
Generate
parametric
constraint
1-Recoveries
-Processing time
function
L
coeffi ci ent
dpfinit1innc
'I,
(Read timber
division
guiding or
enforcement
2
requi rements
/ Write
3
row name
and sign
declarations
'I,
Compute
end product
objective
function and
nonparametric
constraint
coefficients
/
/
/
4
'-End product
demand requi rements
-End product price
functions
f
Write
/ end product
5
column
vectors
/
/
-Processing cost function
-Transfer price function
-Recovery and processing
time definitions from
procedure 1
Compute
objective
function and
nonparametri c
constraint
6
coeffi ci ents
'I,
7
8
Write log
/ column
vectors
/
1
/ Write
constraint
I
/
rhs's
/
-Availability of logs in market (under
mill dominance only)
-Guiding or enforcement requirements from (2)
-Storage capacities of end products for
market 3
-Mill capacities (processing times)
Generate new
9
pararnetri c
constraint
10
( End
Figure A.3.2.
coeffi ci ent
definitions
Flow chart of FORMAT PROGRAM for the milling division.
175
( Begin
/'Read market prices,
transfer prices,
harvesting costs,
log availabilities
on tracts, guiding
or enforcement
requi rements
\I!
Generate parametric
constraint and
objective function
coefficient definitions
3 / Write row name declarations/
ompu e arv-s -in
proporti onal i ty
4
constraint
coefficients
Write column vectors
5
6
/ Write constraint rhs's
/
\J(
7
Figure A.3.3.
Generate new parametric
constraint and objective
function coefficient
definitions
Flow chart of FORMAT PROGRAM for the timber
division.
176
The simp'ex input for both milling and timber divisions are
created using FORMAT PROGRAMS.
The final adjustments to fit them to
the individual organizations are made using a text editor.
177
Appendix 3.2.
Examples of Computer Inputs And Outputs
This appendix presents an example of the linear programming
computations.
Of the numerous transfer price alternatives, of three
dominance organizations and of three harvesting levels it concerns
transfer price TP9 under mixed dominance when total harvest (15,000
MBF) exceeds mills' total log need (about 10,200 MBF).
For other
linear programming computations, only minor changes are needed.
Of the
thirty original timber tracts available for harvest, those nine with
the lowest harvesting cost are included (cf. Table 4.1).
For the first
seven of these, the tract log class structure constraints (2) of the
Appendix l.2.F formulation are missing.
For them, the log availability
constraints (2) are equalities instead of inequalities.
done in order to reduce the costs of computations.
This has been
It has been found
in pilot calculations that under mixed dominance, for all alternative
transfer prices, tracts 10 to 30 are harvested and tracts 1 to 7 are
always harvested when total harvest exceeds mills' total log need.
Milling Division FORMAT PROGRAM
A.
0S3 FuTRAN VESI0N 3.12
08/15/75 2241
I
2
P3O(,RAM MULP
LNILGL0 IJ,C
3
4
S
6
U11l.IlI )N
4)
S I 13
,lJ (2) ,C ('.,2,2) .0(2 C ,E (2) ,oPi10)
'. I .10 (2)
0814 (3?I1,.13,.Uj,.I3)
C
1tfYi: /531, 2)16)
(U: 1HV,lIflI ,
7
3 !IUIIV
2 HS
, 3(ICHS)
1, (05Cr').)
C
(C:Jd3,UJ,26j,tjU,O,23J,1u0,fu,i00,700,600,500.
8
406)
150,2:3, II
9
It
12
RESI 5), ORES (6)
DIII: N 1)1 EVIL)
178
(I) ill, ihO)
(:f.O1),8)jflQIJ1l0S), (ErSil
10
CALL. DEINL.R(0l
,5HPINF )
-
C
13
14
15
16
it
C
C
18
C
19
C
(,ULUINS OR ENIORCELIENE REQUIEMEN1S REAO
REAO(1,20)L.V,_3
20 FORli4l(Ll)15)
C
ROW NAME AND SIGN DECLARATIONS
W010E(10,1( (1,1:1,1))
26
1 FOI3T(OWSI'1J1/
1311))
2)) (3, 3:1 .4) .1 1,2) , COPES (JI , j, 5)
)3 1)
FOQ'IOIU.(t rVlEl(,'.Ct rSfl1)5U rA3)/0<FV
21
22
23
24
25
26
I 10
2
-LE11 r1L21 :L.L,I :LL41 rLL12
OUY
HQO/
'I'L1.cL2 cL,3 CLt.
LL22 rLL.32
FS
PTV
<PTS/
LL420/
COLUIINS')
27
28
C
2')
C
(NC PPOOIJCT COLUM
C
3)
DO
.5
31
32
00
3
VECTORS
1:1,2
00 3 Jr),
KI,4
00 5'. 11:1 13
1FU.EQ.2Iu1O'.
33
34
QPIII)rlP(K,J, 11)
36
3?
60105'.
30.'(Ii)rSP(K,J,El)
CONTINUE
5.
38
3WP11L.(10.5)IJU),K,J,O(J),C(K,J,i),E(JC,UU),K,U(I),K,J
3')
'.0
5 EUHA1(A1,21l, 1X,AtI, 1'.,IX,A5,A1 ,I1,
41
42
11 F0RMAT(l+,10(t00LL,F4.Z,1X)A4t1t)
4'.
00 12 I1,Z
45
46
47
48
65 12 3:1,4
DO 55 I('l,l)
OF)! .(Q. 2)6)1013
UPILI)rOFPIJ.1i)-VSS
6010',
4')
50
13 I')1I):FP(J,1(I-SSC
51
1.2
5, GOIIFINUE
53
12
14
5'.
55
50,
C
57
58
C
6!
68
5 1'1,2
00 6 J:t,'.
DO 1') 11:1,15
I)DEA11Lt0I)(Jl
(r1iI(IJ,1)
19 QPCI IC:- CIP(J, 1,01) POJ, 1,1))
P'ITL (1: .7) 13(L) ,J,r,u(1( .0,1, lUtE ),'<,J,(i(IC ,K,K=1 .4).
'1-3 (LI J. I (11.0)' (1 [C I 1:1, 10)
l-03M(.0 )OLIAC.1tt0 P11A1. 1o211.'.(IX.AL,11, ix, 1lL,Li,A1,I.)Ji.))LA1,
7
'201,11(0 0/11,51.?))
IF(1.EQ.2C60703
70
71.
1
73
77
70
7')
00
81
82
83
8'.
85
80,
87
88
09
90
91
WRITE)1J.-0)11H1_)..J,1,J,I,J,J,j,J
FORIIAT(ILOAI,Z1L! )..L!211! 1 DV L/11/IJV CO LOI1OCO dIV
L$I&CV LVO
j,)
11
60106
7'.
75
/5
b L:L,2
0))
0')
5')
72
1',CUC IC .3,0,1 11,0(1) ,J,U(I),J, (1I,QP( 11),! Cr1.10)
LOG COLUNC) VECTORS
10
51
62
53
65
t'11E CII
F0'OIAHCHF,lt 11,2NF,A1,JH 1 .81,11,0 -10/IHFAI,L1,
10(t 0011,13.2))
C
5.')
65
-j/A1,2I1
'ii)' O1l,F.Z))
,t111L(j0.l(1R(S(I),(1I,RES(1),I1r1,1U)',1R(1),11,5)
1,3
5
10
P-U
,J, I,J,J,J
(Eli), 10)10(1.1.3,1
LLO2I1O 150 LsI15O
FUF13T(lL0A1.211
Cl1I.ILUU:
PU (C 13,25)(11, J, i.J, In .4)
I
25 F1)RHST)CLLOCLL* LLO2ELI -10)
CHS LOIIOCS LStiO 10)
,Jnj,2)
P31(1(10.15)
15 FORMATIORHS)
DO 16 IL1,13
1:11
)FCi.EQ.10CI:3
PRITEC1O .17)1,1, CJ,LV(II,31 .3:1 .1.),!, CJ,LS(I1 ,J) ,J1,4)
IJ FOR311),HILLOEIO FV lOu 13 1)00 P10 56 P15 1320/
15
'1'-I!LL,11,',(. _0*1E,15)/MLLLOII,'.(0 LS1i. IS) I
lJR100C1o,5o)
18 FORMAT(OEOF)
CALL (XLI
1)10
Milling Division FORMAT PROGRAM (cont..)
92
93
SUBROUTINE OEFLPIER(IC)
C
C
C
C
C
C
9'.
95
9')
9/
98
99
DEFINE SrATLrnNr GERt,TOR
FUR ENU P000UCI RECOVERY VOLUMES
TAULE
160
C
huit
C
102
1, 00 2 1L,4,
109
U=OtA;1EIR(1)
110
101
00
113
114
105
116
3 CUNTINU
.
5 F090Iu. LO CASSi2 OIAMETER CLASS131 FOR DIAME.TERF8.2)
TrT;1 (0,1)
03 6 K1,7
111
118
1.
119
120
125
126
127
128
129
V(KI(At1LL(Kj3,J,t)
k11E(t ,1j(If, K,V(K,K1,'.)
10
FI?VA0('.(1,(((1tt,FlQ,5,jx,)
2 WR110(1o,lj)t,r t.V(5(,[,V(6)
11 FOYM0T(LH(j1tZ*11.5,2H Lt1S0AFjU.5,2H 1I1CSF10.5)
132
133
12
WRI((10.12)
FOR1AT(tEOF)
RETURN
ENO
13'.
FUNCTIoN YP(0,J,K)
C
TRANSFER PP(CE FUNCTION
C
C
C
C
I
C
X
C
10
C
C
MULTIPLIER P 118111181145 MA11KETPP ICE PRIPORTIO1IALITY
CLOSS
r LOG
MILL (1VEr4EE8, 25AWNTIMPEP)
J
C
On1JECTIVE FUnCTION DEFINTIOT) 110149CR
TRA?lFEP PRICE
OTREFISTOM
147
[)4
711
(prI
(4),L t',,2)
'.?05'(9 2:150,
1
.0(00)
32 3929611 8,1., 0. 8529411765)
00 TA (Ll4. 76,58 , 52 .91 , 8?
149
150
151
152
153
,6'. , 52)
OA(A(Dr1.,1?., 18.,2'.. 33.,45.,-j.,-j. ,0.,-1.)
TrL(I,J)
1PT
RETURN
15'.
ETIO
155
162
163
KrL,b
03 9
9
13,
131
lt.I.
,Krj ,t,)
3 Fo29AIt1HrL1tL:tj.5,2H L1iUV$F1u.5,2u LIIVCOYFIU.5,2H LIIOCV
Ft.5)
1T1(0,2)
12'.
154
160
V(K)TA0LE(K.8,J,t)
WRI TEl 10, 7)11 K ,V( (i
7
121
122
123
11.8
K-L,lL
IAULE(L,K,IUt3oTo'.
.1
IF(3.LI.
JK
112
156
157
1&,hi.i.),V(7)
13 FQRIAT(l')F3.7)
1113
146
TA'3..E(
FORiAT (15F5.2I
WRITE(15, I0)TAJLE
1.
006
1'..
11.5
I A0L. OEFO0 WRI TONG I)EFINE STATEMENTS
, WRITE OCFL4E STATEMENTS ONLY
0IMENSIO4
1.5
1.2
143
READ
,
IF'Itc.IIE.u)(,oro[o
R000(,,I(IAuj.E
107
141
TABLE
ACT.L,JN LNUTCATJR
0
0
103
035
136
137
138
13)
140
RECOVERY
IC
C
FUNCTiON VIP(J,c)
MARKET 3 VEUFEa PRISE FUNCTION
C
C
C
C
J
LOG CLASS
K
U3JLC(11E FJN3T loll DEFINITION lUNGER
VFP
C
VENEER FUTURE PRICE (MARKET 3)
C
OIMENS[3r1 0(.),D2(4),U3(z,)
16'.
'6b.5, (.4,31.
(005..8, 32. 7,28. 9,23.3)
7,21.0)
, (U
165
166
167
166
169
170
171
172
173
,
1
VFPO(J)
RETURN
2 VFP02(J)
RETURN
3 VFP03(J)
RETURN
ENO
17'.
FUNCTION SFP(J, C)
171
C
MARKET 3 sAwNrrIlo(R PRICE FUNCTION
J
LUG C,ASS
1?)
C
K
118
180
o9J1:r.TIv FUllTION 1(UFTNIT!OTI 13110CR
sSwTlIIiiAEp (InURE PRICE UIARKII 3)
SFP
C
C
1111
I)IMUTIIOFI
103
.111
[62
180
186
187
186
189
190
131
192
((AlA (Cl:
I) I, I, 01
I 4) .1)'.)'.)
1I.li.1.,7J.L,ij.
.(U4'121.b.66.11,G6.j,33.l'j
1
1.FP0(J(
PEIURN
2 SFPDt(J)
RLTURU
1 SFP'D4(J)
RETURN
END
179
Milling Division FORMAT PROGRAM (cont.)
FUHt'TION Vc'([.J,K)
A93
19'.
C
195
C
197
195
159
C
C
C
C
1)6
200
201
VE.HELR PRIEL FUICTIOPI
L0. CLASS
1
J
K
VP
C
MA8K1 (1 00 2)
O)JLCTIVE FUNCtION DEFINITION (IUHOER
VLNtLR PRICE
DIHENSIN D(t,,2),02(4) ,1J3(4)
,,.8,5d.b,2?.rJ,tq.7,
0616(0
203
0ATl(Dl:h3.(,'.k.J,36,26.3)
DAT A (OS53. 1, 3.. 2, 31. U. 21.9)
CTTO(1,L,1,1,t,t,t,1,1,t,2.3),K
lu's
205
206
207
1 VPO(I,J)
209
2 RETURN
V't)2(I)
RETURN
216
211
262
3 vP03(I)
RETURN
END
263
FUNCTION SP(T,J,K)
21'.
219
216
217
268
210
C
SAWUTIIIBER (R(CE FU'lTI0N
C
C
I
C
U
C
K
C
C
SP
220
221
222
223
224
225
226
227
228
220
UIMEP4SIOrI 0(4,2),D1t'.),04t',)
DATA (0 :125.5,68.S,T0.0,35.0,
OATA(D1ls9.4,77.5,?h0,39.'j)
D4Ti,(D',11Y.6,63.7,65.0,32.5)
IF (J.E1. t)GOTOL
COT:)(6,1.L,1,L,i,1,1,1,1,2,3),K
5P=)(J,J)
I.
?30
RETURN
SPrL)1(I)
ktIURFI
2
3 SPD', (I)
23'.
RETURN
235
END
FUNCTIO9 PC(I,i,T)
236
237
238
C
240
241
242
C
T
C
C
PC
PROCESSINI, C)51 FUNCT ION
I
LUG CLASS
C
C
C
243
24'.
246
2'.?
END
C
C
C
C
C
C
C
C
C
THIS Fu1CTtoIl WAS DESIGNED FOR PAROtIETRICALLY
CHANGIN; RE';ToE V VOL UHE, T'ROCESS INC TITlES 8)10
PROCISSINO COStS AS A FUNCTION OF LOG DIAMETER
DIMENSION (1(4)
(D37.,$j.,22,,22.)
END
C
280
251
LOG CLASS
UIRMIIER = LOG DIAMETER
RETURN
2&'O
282
I
DIAr1ETERO(j)
C
C
275
276
217
278
270
FUNCTION D.IA)IEIER(I)
LOG OIAIETER F(HCFIUN
DATA
26'.
27'.
PROCLSING C3S1
DIMENSION 0(..lI
RETURN
265
26)'
267
268
272
273
POC[SiIH& 1IH
OATA(rJr6536.b$,1315.79, 9,8.40,10t9.37,
l'.b1.35,1310.59,1203.93,j285.35)
250
270
211
MILL (1VENEER1 2SAWNTIHOER)
U
248
240
252
253
254
259
256
257
258
299
250
261
262
263
It 00 2)
O9JEUTIOE FUTJCTION DEFINITION DJHBER
SA(rITIH31R PRICE
132.9,7.6,72.0,"s.u)
231
232
233
2'1
L0(. (;LASS
MARKt.T
FUNCTION TN(O,J)
C
C
C
P0CESSIPlC TIME FUTISFION
0
U
IN
LOG DIAMEtER
MILL )1rVENER8 2SAWNTIMQER)
POOCESSIRI TIi1
DAtA (S10.U3012A35),(S2,.t1jO'.),(SJJ.jU7,A26)
DATA
(V1,..u2A'.'.3) .(V20.IXJJ) , (VJQ.U2L)'58)
6010(1.2) ,J
6 5SURT(J)
JVtslV2/SsVS
TH9 3. 'T / 39
RETURN
2 PS1'D.52/0s93
TH=9.1/(.
RETURN
END
180
181
Timber Division FORMAT PROGRAM
B.
0S3 FORTRAN
08/14/75
VFRSOH 3.12
212.
I
PROGPAII 101)'
4
COHOU AL)q,;i,tP('.,2) ,DM(4,2),HCI9) ,MP(5)
J1I1CN10N ID()
REAL
2
5
6
7
8
DATA
C
C
READ(6,1)AL
9
15
16
17
18
19
20
21
1 FORMAT(3F5.0)
C
THTFfl)E STAIEIITIIIS
FOR LOG AVAOLIBILIIY RHSAS
GENERATE
C
C
C
wRITE(10,2)((t,J,AL)I,J),J1,5),Ij,9)
2 FOR9AT(5(AXL211,F8.Q,IX))
CALL PR)FIT
CALL IPRICE
C
ROW NAII
C
WRITE(t0,3)(tt0(I),J,J=t,4),t=1,2),0(I,J,trj,4),Jj,2),
H (I.J,J I ,5),It,9)
23
2'.
25
26
30
31
32
33
AND SIGN DECLARATIONS
C
22
27
28
29
(ID.IHV,LHS)
INPUT LOG AVATLATIILITIES AND HARVESTING COSTS
C
10
11
12
13
14
(P
3 FOPMAI(AROMSY/tZFIPR3FIT/8( DA1,I1)/8(t
(5(1 <LI2IIH
WRITE(1O,6)(I.I,I 111,9)
'11flI
I. FO
.TTIArPYIIYA
rPI10t)
W1TE(t3,5)
5 FORMAT (1 <HO 'RANE
7000 <0UY/COLUiNSt)
rPItC
C
C
COLUMN VECTOR GENERATION
C
DO 16 IL,9
00 6 J1,4
00 7 K1,2
3.
35
36
37
30
8
WRITE(10,8)[,J,K,t,J,K,I,J,1O(K( J,J,K
FOQFIATII(IL3ItY PIPROFIT T311A L2I1 I
211t
7 CALL CUIS(t,J,K)
'.2
9 FOMMAI(IHL3IIY ((PROFIT YP211
43
45
1f,WRITE(1U,1f)
49
00 1.0 KrI,2
00 60 Jt,'.
60 WRIT(1U,F.t)),K,J,K,tO(K),i,J
17 FORIIAT(IH
1,8
50
91
EORMA((ULO211 MPR3FII PH2Il./ OtAt,Ij$ I OUT PIPYII)
WEITE()0,,0)(([,J,I,J,I'1,4),J1,2)
50 FORMATIa(tsY2Ilt S211
52
53
1.).
-lA/il)
5'.
C
RHS GE1IERATIQO
C
C
59
60
61
HR I TE (10 ,
Tilt
10 FO
II.
C
C
I
C
C
C
K
J
C
X1t211)()
15000 OUT 350 i100t/IEOF /1
= IPACT PIIJM)IER
= LOG CLASS
PtA). LU
CIIMMO1I I tJP'T.5)
7)
I_ 2(5)
0 (II. U1) 11 LII) .1
DAtA IL1=5,i,I,3,2),(L2r5,2,(.L.,5) )CO-'-I.)
DATA (L1HC,tH0,1IlA,tHO,1HD,1HA,11(A,tHC)
80
At
02
83
MLL(J)
NL2(J)
CA=LIJ)I,H)/L1J(I,N)
1r(J.E).1)CA-1.
8'.
85
8,
9.
1,1
= LOG USE (tVEMEER, 2=SAWNTIP-IDER, 3=MARKET)
CC(ISTRAINTS
CENTRATES PARTIAL LOG COLUMN VECTORS WITH
WIlICIP INSURE HA I IN PARTIALLY CUT TRACTS, ALL LOG
CLAGSCS ARE HIRVESTED IN THEIR F.XISIING VOLUME PROPORTIONS
C
C
93
I
HAPVEST1NG PROPORIIOUOLITY CONSTAINT
COEFFICIENT GENERATOR
C
78
92
J= I 5
SUBROUTINE CCIS (t,J,K)
C
C
C
77
09
90
91
J, 1, J
f(tLOGSA5(211 L2I1
[NO
71.
87
80
113
CALL EXIT
65
71.
75
I
Eli' IAT(?(tLO3SA',(2H O31,I1,1X,At/OIt)/l
7I.UGS
6'.
69
70
71
72
73
13)1(1 ,
WRITE (10, LII (110(1) ,J , 10(1), I,J"l,'.) ,Irl,2)
62
63
60
It)
CAll. C0iS(T,J,K)
47
67
((9
J5
WPTTE(10,q)T,J,K,I,J,1,J
46
66
L2IIt I
6 CALL COIIS(I.J,'<)
44
57
50
I DA1,I1t I St
k'3
40
41
Sb
HI)
(fl
39
95
S2Ii)/
GOIO)t,t.1,/,Z) ,J
IiHITE(lJ,3)1,J,(,I,L()) CO,I,L(J5),C8
3 FORMA.T(IHL3IIA PAII,A1,F3.0 PI1,A1,FJZ.8)
1
RETURN
2 WRITE(10,4)1,J,K,I,L(J) CA
I. FOMAT(1HL3I1 PtII,AL,F12.8)
P.EIURN
END
182
Timber Division FORMAT PROGRAN (cont.)
SUIORO)JTINE F9OFIT
OS
96
91
98
9')
100
C
READS LOG NA".K(T
C
REAL
PIP
DATA)tt0)
101.
REA)(2,2)MP
2 FORMAT(SFS.0)
11=1141
WRIEE(t0,3)II
3 F0PMAT( 'EP()FIT DEFEPIITTOPI13)
105
106
107
100
109
WRITE(13.t0)(t,MP(I ,Irl,14)
110
111
112
113
10 FORPIAT(14(tMPOII,F9.2,IX))
00 6 In,')
00 5 J=(,5
SF(J)MP(J)-HC(I)-3.
6 W9ITE(t0.iH1,J,F(J),J1,5)
11'4
115
116
117
118
7 FORMAT(5(tYP211,F8.2,1X))
RETURN
END
5U0901)TIUE TPRIr,E
C
P1105 1 (ANSIER PRI.CS APP) Cr PAlES .OLFTNF STATENEN1
FOR ILL COEFFICIENT
INVOLVING TRANSFER PRICES
C
C
C
°FAL
PIP'
125
Cr)M'PON
12?
DATA(It'))
REAP) (".1 )DM ER
'L 19,9), TP(l.,2),ON(14,2),IIC(9) ,MP(5)
OIP1FNI9i) 1)15)
125
12)
129
1 FSRiAI (sr5.)
131
132
133
9
110
I1
12
1140
2
1141
1142
1143
I.
11.'.
3
1146
1147
1148
1149
F ORPIA I (A
(PRICE DEFT)) I TIOPI 3)
0) 7 K=1,2
Jrj,l.
1
Lr)K-1)144J
1 F)J)nII(L)-MP(JJ
7WRITE)1A,12))J,K,F(J),Jnjl.)
150
139
1145
1141
WIIrr(1i ,q)It
00
1314
135
136
137
1,ENEP.AIFS DEFINE STO1EMENOS
DIMENSION F(S)
DATA(HC=26.,29.,31.t1,33.69,36.32,36.68,37.'il,'42.56,63.88)
103
12'.
API))
COMMON 4LM,5),TP(14,2),OM.,2),HC(9),MP(5)
101
102
119
120
121
122
123
pqicES
FOR ALL COEFFICIENTS INVUI.VI6G LOG MARKET PRICES
C
C
AT(,(/PHtHj,F15.7,jX))
FOP
R'T 11(13,2) (1, DPI (I) .11,5),( t,OM(t+t. ) , I:1, 5)
FORMAl P. ('ADA!) ,P9.0, IX)/',(ASOI1,F9.U, IX))
0)
3
01)
14
KnJ,O
F)P(=TP(J)-P5O(K)
W1tF.(tJ,fl(K,J,F(j),i=1,I,),(K J,F(J414),j41,l.)
5 FORMAl I. (?Tt2()A1AFA.2,1X)/14(I2jjt2AF0.2,jX))
W'1EC (11) ,J.j).
II. FORP)TL1.EOF)
RE 1URN
EN))
183
CONSTANTS
Milling Division Constants:
19
19
I,
'5 11'.
'32
C..
O C to'
000 '35
3,4
1,
0 CC .3
60
17
5'.
7?
97
16
71
(43
33)
150
'33
29
29
1')
19
39
37
Jot jo
00001,
10
37
65
7,1,
'I.
1,7
0,
0(4
2"
'$7
5!
70
57.
73
10'.
'31
110
[52
'.5'.
',90
'1'.
57
16
17
37
'0'3
25
13
19
62
33
21
68
60
59
62
38
20
29
22
113
559
61
62
02
16
23
16'.
'.7',
26
7
'34
'3'.
12',
11
1')
51.3
49
10
75
79
16
3"
'.9
62
56
1,6
7,7
131
116
30
5)'.
12 15'.
1'. 16',
1'. 155
'7
2'.
570
132
8
1?
8
161
3?5
358
12 118
213
30 , 0 7
13119
1fi 1')
5,_C 13
30001
011017
CCC I S
o0 1'.
130 19
31
3'.
ccc i
0&j 1(
00610
13113
00003
19
1,
9'.
69
10
93
'4',
75
87
14
53
'3)
.07
11
(41
(43
12
89
100
13
16
13,
7
9
17
29
41
52
9
08
71
'3
79
13
7'.
'31'
77
51
53
10',
33
85
",
'15
7'.
171
ii'.
o
150
30
11.9
73
5:
.395
47'.
46')
710
53
38
'.2
St
52
'0
13
1')
20
19
12 161
13,
II,'.
3')
59'
153
13 175
05
11
2'.
1',
113
1(5
I"
'49,4
lb
'.3
179
'+59
ICC .11.
£00.'?
03,3. 03
000 3'.
C 012,
17,
1')
o to 21,
13
;
1002?
011 2
CC , 21
.51
30001
3'.
001,31
01332
17
5
'-.1
.31
'43
3
5
oc
0003'.
'ISO'S
10
3
16
'3 C 335
131
.1
3.3
OOli 31
05050
00051
78
OOJ4S
.37
31
1'.
7,7
71
15
01,
'31
47
50
79
0
7
94
CCC
15
13
11
11
'8
5
3
19
3 I;
'3
5,
.32
3'.
"C,
'.3
'.1
'.9
75
1')
6
(4
7
3
7,
ll)
17
29
73
'.7
(40
5
112
09
29
25
36
68
7'.
79
82
6
21,
1S
3
111
;5
48
57
56
67
19'.
39
27
32
131
1'I,
63
733
'.9
195 106.
17') 111
1"l
053
.5510
121
1.38
16
117 l'12
110 11
103 107
1'')
'.',
2/5
312
700 23.
2.34 227
p3',
253
379
1',
13
26
ill,
21
'.65
1'.? 31')
15
67 617
7347 0(311.09 1(16 PAGE IS MOLP 0614 FPOFI 1(333
1,73)
00031
00000
1/00
OCt03
.3'.Q
COb'.
1.75
236
1°2
1'.'.
121
323
111
1'.
I.C6
"5
71'.
11
577
69'.
S
'.39
519
510
537
2?
17'')
2
(4
0
11
17
'."O
27'
25
25
3',
7,35
19?
69 1216
275
759
'53
2
1.
1.3
6735
17'.
2
(4
"C'
7,')
1'
'2
10
21
52
65
55
21
25
1
1
1
1
1
7
1.
92?
.3
06
5
13371
6
123
42/
611
7,
'3
25
1(9
331
.309
338
335
j7'4
3(44
.337
10.9
.
1
33 211
'.3 i'41
52 191
06 173
.33 106
7?
61
5',
55
33
1')
.23
308
'.2 205
9')
85
07
2)
536
1'
22 191
22 177
1') 2
1113
15
13
55
3') 135
(4
101,
6
.196
'.1"
Os's
')2
5?
'.31
347
39
13?
35'.
3.0
'.31,
1
TUIP
DATA 70034 (31301 2 F0
905A'l
(PREVIOUS P0(E LUM 5)
003.09
00006
00007
06308
1'lO
Timber Division
Constants:
90
08
56
22
5 FOP TOIP PTXOG'CAM
DATA 000,4 (37111
'0
31'.
5
10(41,
51,
70,.
St
58
3?
52
62
9 2927 3131
D.
21.
3.3
3
232
112
14
1'. 177
17 153
10 11
3
1!
2
tSr,
113
0
1'.
13
10
132
110
6'.
1
SIMPLEX COTROLS
Milling Division
Simplex Control:
00001
0 0 I) 0 7
0l) 303
0 3 33 )
0 0 00 5
11
3' 1 0 3'
0400f
00008
0 :j t
03010
00011
03512
0 3 31 3
'3 0 0 1 1.
0 0 3 1 ,
DEl 1330 , 0' 10
jt413IJ (.0 10 .R'tZO
13 511 III 11,1' 30
lit ,L='J,C'?O
TI! lF, .3.3 I MOle 02= (0)
ox it F, (3,! J 01
Out 0311,1" 20
0101.13.3.1.5 MILP TPMNL2
MOO tiIiF,013J'O2
13.370.1)1
1111130.3.1 MOLP TPMP-t8
3100 (11/F 0(1.3 '7) .3
(1
'30 1'IJI , 1= 2 3
T111133.3.1 IULP TP"HP-2',
MAO [ti1?E,OI.'J'O(4
000 3.5
13.131 Ph r , I. "20
00018
MAO I 'II IF, ORJ'O5
00517
o 301.0
T(IIF,3.1.1 MOLP 0031P_33
03)3 PlJ1,L=IC0
Timber Division
Simplex Control:
000)1
00002
(30003
TItLE.3.7.1 (IOLP rp=Mp-'.5
30009
COt 22
OUTPIJI .L"20
(AS (30131,1=20, S=3O
0ou0 7
O 002:.
EMO
O 3 3 2 '0
06621
00023
'lOX I'll! 20,OL'J"06
DEFINI I"tO
I'(PIJI f =10
009 Is!N, 1 =30
GIL ,C+2J1L2O
11111,3....?
0006
31'.x IMOZE
00008
00009
740
1012
0.310(31 ,120
07.31 S3U 1,1320, 333
IPSO)
03000
03302
OI004
0000',
00005
35404
00007
34008
00009
00311
00011
03012
03133
0)00
I,
00030
00006
OJOtO
03300
03009
10200
03001
('3520
03020
05025
00004
03037
£00233
03097610 Co355 0 (000
(.05 1100%
.46000 1003
.0203-0 1117
1101
,Gl0i
4.03210 tIC" (0,33.300 (VU
.02000 IV'.
.05(120 1)'.)
(5,03:100 .11100507030
311
.5331.1 (.132
3.
1.37
.01 033? 001 00', 035 09', 007 090 009 003
S2 '53 '5',
=01 =03 '03 =0', 'SI
'Cll'/'SJ
'03(0
'cs
oF 2r3.r'T0'l'1S
'11(1 1L71 =1130 '111,0 '1102 =1122 =1132 '1102
'LI '12 '13 '1'. '100 '3(3
=sv
0 020_C JUJ2ULS
VII '105010' lC3
'.7.50 02
VI) VI
03113
03)
000.00
12030
C3')S',
33037
313(1
2V40
3301,1
000,2
32153
31-3','.
03-063
035'',
1(31,2
03148
020',')
335")
03)03
03302
22,52
37fl9'
7.30.0
32o56
30057
10550
03;S'l
53260
02060
OIJ'll'
(111(1
37,00
.91,000 110'.
.;'?Or.O
41.000011
.11.00)
.0191, 1)1:3
1110(10013',
'100 CLI,0' fll01'l(.'0 CLII;.;
01.00
1,7000
(.05-0 (307
VS
.90000 (.211,
(5000
L 200
.010,0 .100 3.93003 120') 1(0.0000 03:0 40.10010
021
.40342 1,02
.?r,053 13:09.57000 1/05.10,003
1231
.00007 (.217 06.05010 LL3'1 30,97000
022
LOG CLVII 3 1112(111.0 GLOSS ', (00 000"E0Cl(
(2.00
.10207. 1222
.73303 13.'',
1371
.05010 130.1
1.05003
.0351,2 .001
030
13.00053 1 (CV I'(.20010 137.0 00.0000)
.001000 1150
1000
.0 000 1 353
0 .10000 12., I.
.1900)
132
.01955 3.20') 11,40000 17'S 30.10003
LOG CUSS I, 0)097.11.8 CLASS 5 107 00430.0(3
02.00
153)
.04,17 1/10.01033 ('03
.7)300 I'll. 0.3)303
11.1
.01400 LI/jO 39.00000 1'.CO 20.301150 107.0 81.201000
1031
. 030o0 1,02
.03000 L'.LS
.00000 1',S'.
.00000
('.3
.41905 LOS'S 19.10000 1400; 39.70000
1
III
LIII
112
00030
00031
000(2
184
Milling Division INPUT
E.
VI -1
47,83
021 "0021-Il' 253
VII
.
30.0.3 72
('1
((3
02 '8
30.00 (03
03 -1
03) 01
27.0') 02
27.93 03
0,1 '13312):')' 003
06 -1
V.3 01
l').72 02
19.70 03
71-37 01 -t
50.1,1 07
VI) 05
09.03 03
232 '-(O'J.93(S' 200 70(11 02 -1
027 01
31.00 132
37.03 03
',33i)33.l)' 200
1
10,) 007(1 03 1
037 (00
3.3.7.0 02
3.0.30 03
002 '000355' (I ('10(0 0'. -1
4.02
0'30'105'
07.00 0',
00.63 0'.
27.03 0',
07.0005
07.0004
01.8000
3,0.0340
30.00 010,
30.60 07
00.3.0 00
07.80 09
30,60 09
47.00 03
30.00 0')
27.110
27.0006
20.0007
I'O.70 04
19,70 00
27.0009
19.7009
27.03 00
09.70 00
50.43 015
17.0.00',
37.6000
09.70 07
55.',O 07
37.0007
77.00 07
19.70 08
50.1,0 U'.
133.7004
50.40 00,
33.000',
33.00 05
035
37.6006
33.0006
33.0007
1,0.00 4800.03033033.0 CO
37.60 08
33.00 08
20.10 00
120.33 07
60.10 08
70.00 03
35.00 Cl
04.10 07
24.10 00
0,2 III
10 00
24.13 03
24.0) 00
24.10 06
001 '140210' 333
51 -1
S;1 00 170.03 02 12V.60 00 120.0)04 170.7305 120.8006 023.8007
SOt '0700(6' 700
02-1
3,0.1,0 0',
0,0.63 05
003.0004
10.1.1 0?
000 01
63.60 03
('0.1,0 02
501 '1)O'3/13l' 1,53
SI -1
70.03 00
70.0) 03
70.33 04
Sit 0)
73,03 05
70.00 07
70.00 06
S'.l '(3777.0' 553
0', -I
35,22 02 .25.8) 03
05.0005
30.0005
30.0001
39.00 0'.
SI ('1
33l,''001('1)I5' 12') 7(3(1 01 -1
512 01 13.5:) 0'l 1.12.0) 03 132.50 04 132.01 (35 332.50 00, 132.00 07 137.50 CO
921 '"7215';' 27.-, 0002 03 -1
10.60 03
70.6004
70.3,:) IV
70.1,30'.
70.5006
70.60 17
70.63 00
7.20 00
5(2 '00111140' 113 '13(1 03
72,00 08
012 00
72,23-22
70.3000
72.0005
70,0000
72.00 07
72.05 03
0-2 '301131,0' '.00 51(31 50 -0
7.3
36.00 07
31,3150
34.03 06
0)
(6.00 02
36.03 01
300.000%
S.03 3'
,(1, co .3', 03 .05 s' .35 5% .10 05 ..os 37 .3'; 0)3 .30 03 .35 00 .35 1.'! -1
u-i
so
1
.25 (2
04 .22
.32
.20 00 .21 00 .23
.70 00 .1,)
30.633 C)
07.6019
30.1.000
32.00 C9
20.10 09
128.80 34
63.60-09
70.00 03
30.00 CO
2',.IO 03
170.00 CO
67.10 03
70.00 00
30.0000
25.00 CO
'.7,70
30.80
27.00
19.70
50.60
37.60
33.00
25.00
120.80
65.60
70.00
30.00
132.00 00 .132.50
70.6000
70.60
135.50 09
70.6009
72.0009
72.00 03
36.00 C)
36.00 09
72.03
30.03
30
-1
3016'. Cl 01 .1302 .1303 VS 3'. .1.1
1'', .1345.03117 .0.003 .03 01 .1)05.13 CIV -1
5.)
-i
01 .109 00 .1') (00 .00 0'. .01 05 .2500 .09 07 .0') (10 .0') 00 .0070 .0300
3070,3
03,65
3000.6
CII
320033
55).'l
J,'2
0173
10,0'.
J',J(l.
60306
-20,17
£0000
2,,'l/')
73354
00231
03102
5004.)
70034
00235
00006
0000)
01500
0004')
00530
01341
033'(3
00013
0009',
00094
033(4
00017
000'(8
03049
00100
00101
33102
04003
03103,
00103
00106
00107
00135
00009
03010
00111
00112
00113
0011'.
.00010
00118,
t011?
00110
00110
00)23
00071
00013
00(73
0537',
2)131
021(0
07130
-3St30
341,13
120(3,
00135
201.1'.
00137
34130
103
.30
I'S
0')
CO
.20 35.1
1
1.5.7
3
I
1
0
0
1
1
1
1
1
1
1
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1
0
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1142 ((II 1
1
1
50 135') COO L3CS 13 1
('.7 So 13.51 5,0 11.02 53 L4SI 37, 160'.
1.04
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111.2 00
1142 111.2 I 013 (,1,5') 1113% 1400 1'.
-59,00 00
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1
1(11 Oil III II (.321 '17 1t03 33 (003
0,, Liv'.
190101 -120.00 02 -110.03 03 -3118,00 C'. -107.00 05 -03.00 06 -81.00 07 -172.00 06 -127.00 09 -320.00 00 -022.00
1311 ((00 003 1)11
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10(20 III 121 VI .001 071700 13 L71'3 2'. 121'.
1970331
'93.00 (12 -171.00 01 -','..03 0', -09.00 05
1,021 UUO '10 (.1.30
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10001110 031 10 1901 VI' 1(0003 1.105 0'. 130'.
LUll 01 -77.01 00 -05.00 33 -0.9.37. 0', -53.0005
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'39.00033-103.00 00 -99.00
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1
11.130
1/14310 CO IOIIU 011(3 L"CV
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111,1 11117
937
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1
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-63.0006 -(0.00 01-013.40 VO -013.00 08-300.00.03 -I13.'.O
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47 00 07
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08 00
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80110 001 350000 (''0 '.503
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2 1I.S2 59
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-33.0001
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9'..CO 06
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1
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-77.0000 -1,',,Jl 33 -1,J,,30 0'. -6l.,)3 05
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01 -1
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60.50 CO
50.330
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50.5) 07
50.7.0 07
50,1(0 09
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50.53 00
50.5) 03
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30.4005
30.0006
30.307 30.40 00 30,1,009 32.4100 30.60
30.40 02
30.40 0)3
0.3 -1
6
23,910 04
25.93 00
20.00 03
20.90
25.90 06
25.90 00
75.90 09
123 01
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25.90 03
25.93 07
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87,300',
17,30 -02
17.30 07
1.7'. 01
17.3300
17.3306
07.30 08
17.3001
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030.10 13 124.10 03 154.10 04 126.1) 00 120.10 06 126.03 07 126,13 08 121.00 09 106.1) 00 120.13
1.32 1.5 1 52 -1
1,7,fl7 02
62.93 05
62.90 07
62.90 08
62.330 09
62.90 Cl
62.333
152 20
62.93 03
62.03 0'.
62.90 05
11,1 15
53 -1
61.43 41
1.4.40 07
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64.40
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66.40 0')
1.03 01
0.',.40 07
60.1,0 03
6',.',0 05
64.60 06
13', 15 1 ".'. -1
27.10 03
27,70 07
27.70 03
27.71 114
27.73 00
235.7306
27.70 09
27.70 00
07.7)
00.73 00
1.0'. Ct
705 1)1 VI 1101 07 1100 V.) 1013 0'. 110'.
0110 II')
1101 01 '102.04 '12 111.33 33 '(7.3.00 0', -102.00 05 -93.00 00, -01.00 07 -122.00 00 -120.00 09 -026.00 00 -Ilo.00
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1171 03
-59.00 07 -1,0.03 00 -'35.0) 0'. -89.0) 00 -00.03 06 -60.00 07 -1(9.00 00 -339.80 69-110.03 00 -'39.07
1121 1121
01 3.730 001200 CIII 171.0 10
P10 13) VI (lIt 00 1307 VI 1103 V'. 1300
1001 ''1
1031 01 -77,30 3' -(5,00 03 -79.05 3'. -63.30 05 -51,,03 06
42.00 01 -77.03 00 -77.00 09 -87.00 00 -77.03
U/ 1330 Co 137.0 COO 3.31.0 13
1031 1131
1001 ((03
('00 041 VI 1471 02 1,02 00 11,53 VI. 1401,
1141 Cl -72.00 0) -63.7.3 00 -00.00 0'. -54.03 05 -40.03 06 -33.00 01 -72.00 03
72.0O 09 '122.70 03 -352.00
11.1 11'.) 1 UI 141/ CS 150)) CII I'll 00
I'll (02 91 1001 0? 115? .53 1003 1'. ItS'.
1112 00
1013 01 -11(00 0' -015.30 35 -110.40 95 -104,33 04 -50,53 06 -03,00 07 -019.00 08 -119.00 00 -028.00 00 -(19.00
511 1)50 (II'S 1105 11 1
1112 1102
1127 335
703 323' 01 (.751 023.257 53 1753 51, 17335
102701 -113.025? -12.20 03 -94.30 0'. -92.13 05 -83.00 06 -71.0007-113,00 00-113.0009 -(16.00 00 -113.00
1122 1172
50 LI'S') 03(3 LOGS 17
1032 (II) 1 I' 00 132 01 (151 52 1(92 03 1953 5'. 135'.
1032 01 -VIVO III -33.30 03 -04.00 3'. -66.02 05 -59.00 06 -41.00 0? -80,00 05 -32.00 09 -90.20 00 -88.00
110 ('I
80
I'll
101 01
10
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5233.?
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-
Timber Division INPUT
F.
001)01
00)1
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0,3)
0001)3
708110.0 (hOtS 01,11 10 l)')'i. 700080)1
6), 01 13
335 XLI'.
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9 OLOI
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288 0132
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15)) 81.1,2
11
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06 01',?
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60007
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11)3.1)0 1)72
20.80
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717)
58.00 1023
01)116
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00.1) 1,
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00011)
00015
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01)1 1')
00020
00221
03023
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00025
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0O,.Jd
12231
20282
00,0 .5 3
00 a 30
00,31)
('0 23)'
0803 7
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1)01)1,2
0)1)1)1,3
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00,47
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1)
00 0
1
00252
1)0.53
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10057
01)060
0)059
00060
00161
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00 399
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00192
01193
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00197
00198
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00 20 0
00232
00200
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01)1)05
1)3220
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00200
00201
00213
00211
00212
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021)0,'.
00215
00216
02717
00210
00219
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01)210
01)231
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1151
03 Xl.!?
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60.25 0I'o2
6,.03 1)33
1,3.31 001,2
£a.69 yr,2
0'30
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0)1,6
21)1,3 XL?'.
111)3
571 019',
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710
1
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09.00 8'13
5.01
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63.31 11,3
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2,10 8"',)
61.03 Y)'74
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6',. '.1. X'13
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03.12 II'')?
'105000. 0tF01i11109 1
'110
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7'),)
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811
312
611
2007
69.10 1121
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02.60 1312
1,2,11 1,21
07.31 1,22
57.08 7520
50.00
60.53
55.03
'.2.11
63.31
33.01
(.1.06 102,.'
1,7.63 1211
62.11.! 3222
63.1)3 0131
211
212
311
312
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022
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'7.28
5,.),')
152'S,-.'.)
1)3,30 11,21
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0101
2,.01 1701
5,.0j 1172
53.6', 1831
2700
1712
11)11
60.,', 11110
50.10 51)20
1.7.11 192
11)12
1-111
1512
01)1'
027 010',
267 XLS',
30) 2173
011 iLl)
218 0182
0101
X'a3
210', -013',
50.30
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33,00
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32.12
03.12
1)3
130
1,20 /.1.00
68,00 1171,
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30.00 0)'26
33.91 9)3',
31.10 II'',',
20.50 1(5',
05.12 ['OIl,
27.53 1)70
22.01, 7(0',
21,12 01'')',
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73
130
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331
732
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332
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73.2
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30..)) 1151
32,0,, 5)1,2
31,3!)
35.1.0
121,0
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20.53 13,1
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30. 31 3062
21.1,) T"',l
27.19 1502
20.32 11,0)
27,31 1!.-.,!
2-).'..) 1761
1)1,,'..) 1/03
15.5', 11161
21.'.'. 1(156
0.02 1951
20.12 1952
16? 0115
253 0.2',
1,12 1)135
8',? 0.1'S
3.'), 210,,
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Oil 61,72
21110 0185
2013 X195
55.00
27.00 1015
20.00 1025
2.0..13 0135
70.31 11'65
111.1,8 Y1'5S
)'),31) 11'60
17.03 )('75
12.0'. 1P85
11.12 1P95
-5.00
0.60
301.6
0
20.00
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2.1.30
10.30
23.03
13.1)3
03.31
20.31
00.1,0
25.1,1)
02.12
22.32
1',.53
20.53
1.'.',
8.02
('1.''.
18.12
05.11)
151''!l)"l P
'001 1)112 ''303 '01', 0171 =0192 1(153 0105
'011 ':;21 =531 "8,1 =51? =822 =232 =502
'LII '112 '113 'Li'. '1.15
'[01 -'11? 'L13 '-1,'6 '173
'131 '0)1 '1,3 '1)', 099
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111.32 IL 13 '1'), 'L85
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01')I '('50 "'(0 -"I'll)
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11)01 '1)5 01 I'S,'. -1.01633393
01)1? P.'6Ji11 1912 L'01 1 III) 0 02) 1 S.12 1
09)? "SC
01
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1110
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1
1903
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1721 ''8 -1 71)6
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01)32 '90 -0 1'3C
1503 /188011 0093 L'13 1 I))) I
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L331 '90 -1 PSC
L8-.','/i'COI',I 13'.! 2'3' 1 IllS 0 01', i 1)01 1
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01)43 790. 1.5373731'.
0:111 Hi'')l80t 0811 1)80 1 COY M't
L-311 111'401 10 '(131 1)12 1 OUX M'2
183
11)30
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1,0.1 1091)0)) 1'l!',l 00'. 1 'lilY lIi.
1.
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0,33,! 1,1)113 7)130 01)3
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sot ?;10 -i
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10 23-0
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81.3'.
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00 23',
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00 2,16
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LII,.') 1180 (-('0 032 '2)12 1)01 303 3S'. 30'.
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1)1.1,',
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1056
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020
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1!.l,'1
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185
186
Milling Division OUTPUT
G.
00/truES
01
09.1
lOtOlilt
II)
3.C.1 .001.0 TP
8585
=
1500IF0l1
15.1111
ROWS SECT 0055
IN))
ROil
O1_
0,?
2
2
03
1
00.
2
07
05
05
00
Vt
REPORT
UPPER LIMIT
IRVIL
SLACO ACTIVITY
209972. 1121055
-233077.5 12)01,
501 0)5'
1111)00
P 15SF
P 15SF
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551 1SF
P1510
555 lIP
P 1580
200527. 550'.21
351355.015..)),
4005,53.7721,27
1.3055,7. 135130
174358.393755
2
209977.117,1.
2
7
7
167391 2'00309
209917. 112101,
2 0)977 , 11 20055
100470 105)37
-295501.. 150/0
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-516997. 1.50633
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55055'
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2J1917. 102505
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137.00000
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170.00006
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71.00030
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103.10000
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360 .00 3100
330 . 00 100 0
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188
Appendix 4.
ALTERNATIVES TO DIVISIONAL DOMINANCE APPROACH FOR
ALLOCATING LOGS IN A DECENTRALIZED FIRM
There are several alternatives for divisional dominance for
solving log allocation problems through transfer pricing in a decentralized forest products firm.
One of theiii is divisional negotiations
after top. management has announced some transfer prices.
same disadvantage as negotiated transfer prices:
ing.
It has the
it leads to bargain-
Finding solutions to bargaining problems is beyond the scope of
this study.
Other suggested approaches are:
shadow price allocation
decomposition allocation
price adjustment allocation
Their advantages and disadvantages are discussed in this appendix.
Shadow prices, meaningful as transfer prices, are hard to find.
When
implemented they easily lead disequilibrium of internal demand and
supply of logs.
This results in uneconomic decisions.
Decomposition
approach does not provide the divisional managers freedom to execute
the allocation.
It involves a great number of information exchange
iterations between headquarters and divisions which is time consuming
and expensive.
Also price adjustment method requires iterations.
It
does not necessarily lead to a feasible optimal allocation in a finite
number of steps.
Divisional dominance is the decentralized log allocation approach
used in this study.
It produces always an equality of internal demand
189
and supply.
It provides the dominant division with full freedom of
making log allocation decisions given some transfer prices.
Divisional
dominance procedure does not necessarily require information exchange
iterations.
The actual market or timber resource oriented organizations
of forest products firms today are close to the dominance formulations
presented in Appendix 1 .2.
1 90
Appendix 4.1.
Shadow Price Allocation
If the headquarters has perfect knowledge about the divisional
production function, prices and costs, it could solve the corporate
problem, as in Appendix 1.1., and find the optimal allocation.
If the
company is centralized headquarters announces the optimal allocation.
Transfer prices do not have a role in log allocation.
The information
and log flows are as those of Fig. 4.2.a.
Top management might, however, try to follow the resource allocation theory of a socialist decentralized economy, discussed in Chapter 3.
It would announce shadow prices as log transfer prices.
Or the sum of
the harvesting cost and shadow prices could be used as transfer prices
as suggested by Solomons (p. 191).
The divisions would choose their
log allocation schedules by maximizing their profits independently on
each other given these transfer prices.
The sum of their profits would
be the corporate profit.
These procedures have been developed for simple problems with only
a few sources of logs.
For every source there is a log availability
constraint and a shadow price.
The opportunity cost character of a
shadow price lets us quess that the shadow prices might vary from log
class to log class.
Ideally each log class would come from only one
source so that there would be as many shadow prices as there are log
classes.
It would be wrong to assign different prices for homogeneous
logs from different sources.
191
Shadow price procedure
Our
does not work for our, case firm.
allocation problem of Appendix 1.1
has, one one hand, log availability
constraints for homogeneous logs from different sources (constraints
These sources are the several tracts.
(4)).
For 1 tracts and 3 log
classes we have I x 3 constraints with I xJ shadow prices although we
are looking for only 3 shadow prices.
No theoretically correct way has
yet been presented in management science literature for taking
3
"average" or "representative" values for the I x 3 shadow prices.
There is also a shadow price connected with the overall harvesting
possibility constraint (7a).
price?
Could we use its shadow as a log transfer
It is not desirable for transfer price, either, because it
suggests only one price for all the log classes.
Even if the problem of identifying shadow prices as transfer
prices was solved their implementation would cause problems.
sion 'manager
A
divi-
might not agree about the outside market prices, costs,
capacities, etc., with the top management's "perfect knowledge".
He
would then allocate logs differently from what the optimal centralized
solution suggests.
An outward sign of the resulting allocational dis-
harmony between divisions is the inequality of internal log supply and
demand.
This disharmony causes the shadow price allocation to fail.
It can be avoided by negotiating and readjusting the prices or inducing
additional restrictions to the divisions.
Both the decomposition
allocation and price adjustment allocation approach are based on readjustments.
They are discussed below.
Charnes, Clower and Kortanek
192
(p. 310) call additional corporate restrictions preemptive goals.
Baron (p. 171) uses term guideline constraints.
The guiding and
enforcing constraints of log allocation under divisional dominance
belong to this category.
They have been discussed in Chapter 4.
193
Appendix 4.2.
Decomposition Allocation
Decomposition theory was redeveloped in 1960 when Dantzig and Wolfe
(pp. 767-778) created their linear program decomposition algorithm.
Since then it has been extended to nonlinear programs.
But perhaps a
more important extension of this purely computational method has been
its use in the economic theory of decentralization (Dantzig, pp. 455466).
Baumal and Fabian (pp. 1-32) have presented the exact structure
and numerical examples of Dantzig-Wolfe algorithm in decentralization.
Several algorithms have been developed which resemble it (e.g. Balas,
pp. 847-873
Hass, pp. B3l0-B331;
Whinston, pp. 405-447).
Before
this algorithm decomposition methods--also then suggested to be used
in decentralization--were based on gradient adjustment procedures.
The convergence of these procedures to optimum took infinite iterations
(i-lass, p. B3l2).
Jennergren has compared the
theoretical properties
of several mathematical decentralized resource allocation methods.
He
ranks Dantzig-Wolfe algorithm as the second best of them for linear
programs after divisional trading procedures (Jennergren, p. 84).
Equal Milling And Timber Division
We wish to see whether optimal corporate allocation can be found
while letting the divisions maximize their profits independently of
each other, without any external enforcing or guiding constraints for
a division to allocate logs.
This would seem to mean more divisional
194
freedom than a divisional dominance allocation allows.
Figure A.4.l.a.
shows the information and log flows of the procedure.
Dantzig-Wolfe algorithm is an iterative method.
It should converge
to an optimal solution after one feasible solution has been found
(Dantzig and Wolfe, p. 772;
Fabian, pp.5, 24-31).
Burton
etal., p. 310;
Baumol and
The algorithm has the following steps:
The headquarters announces aset of tentative transfer prices
or the log classes.
It asks the profit maximizing timber
division about its profits and supply schedule and the profit
maximizing milling division about its demand schedule given
these prices.
The divisions solve their divisional allocation problems
independently of each other, and report back to headquarters
their optimal allocations and profits.
Depending on whether
the milling or timber division buys logs the mills solve
problem of type (A), or (E) of Appendix 1.
The timber
division solves problem of type (B) without constraint (4), or
(C) of Appendix 1.2.
The headquarters uses divisional allocations and profits as
an input to a master program.
The master program finds the
weighted average of the divisional allocations which yields
the highest corporate profit.
For every acceptable solution
internal log demands must equal to supplies.
produces a shadow price for each log class.
The program
Nonnegative
195
I laflocS
trial transfer prices
mill.
division
I
(sale
(end product
markets)
divisional profits
Iand
div. orofits
headquarters
internal
enforcingJ._l9g_ j
log
deli veries
supplies = demands
at execution
transfer prices
sales -.--.-_-------
i
/
allocation proposals and
divisional profits
timber
division
> log market
pur- 'chases
div. profit
Figure A.4.l.a.
_tri a ltransferpriCes
- ---1
'allocation
Iproposals
and their
- - - -
-
1
sawmi 11
veneer
division
tend product
markets)
sales
division
1profits
dlv
div. profit
.rofit
't
'ii
headqua rters
I.
enforcing intj log supplyJ
corp. profit
i trial
tio.n
allocation and
profits
I
-
deliveries
w
timber
division
transfer prices
internal
log
= demands at execu-
sales
log market
1
div. profit
pu r-
chases
Figure A.4.l.b.
Figure A.4.l.a-b.
Notation:
Information and log flows of Dantzig-Wolfe decomposition
algorithm organization of equal milling and timber divisions
(Fig. a) and of two milling divisions and a dominated timber
division (Fig. b).
--+information flow
-+ log flow
196
shadow prices are announced as new log transfer prices.
Negative prices are
announced as transfer prices with zero
val ues.
The divisions solve the sub-problems and report their optimal
allocations and profits to the headquarters as in step 2.
The iterations continue until some master program solution
gives the same solution as the previous one.
From that point
shadow prices would not change in new iteration, either.
The
last iteration produces the optimal convex combination of the
allocations of all previous iterations.
At the execution step the headquarters enforces the optimal
log allocation.
This step reveals the most obvious disadvant-
age of Dantzig-Wolfe decomposition algorithm.
In execution...
division managers must be told what combination of their
proposals they must employ.
There is no automatic motivation
mechanism which will lead division managers to arrive at such
a combination of outputs of their own volition." (Baumol and
Fabian, p. 14).
The divisions, although enjoying decentral-
ization in the information exchange phase, have no freedom to
actually allocate logs.
in the execution.
The master program is:
(A) Maximize
Transfer prices do not play any role
197
I
K
t
t
R
(°)
k=O
Mk
t=l
subject to
T
K
M0t
(l)
tl
kl
'
k
K
I
k=i
t=i
XM = 0
-
forj =1,...,J
T
Mkl
(2)
fork=
t= 1
where the variables are:
Mkt = weights to be given to division k allocation at
iteration t
--k = 0
is timber division;
divisions;
k = l,...K are milling
t = l,...,T are information exchange
iterations
where the constants are:
Rkt = division k profit at iteration t, $
jk
= timber division supply of class j logs to mill
k at
iteration t, MBF
xjkt = mill
k demand for class j logs at iteration t, MBF
198
Constraints (1) equate the internal supply and demand of logs and
produce the J shadow prices for the next iteration.
Constraints (2)
guarantee that for each division, the sum of the convex combination
weights is
one.
Although an optimal allocation can be found the algorithm may
require thousands of iterations for an ordinary linear programming
problem (Charnes, Clower and Kortanek, p. 297).
Pilot computations for
milling division of Appendix l.2.(A), timber division of Appendix 1.2.
CC)
and the master program (A) above have shown that this is a serious
problem.
The shadow prices of the master program constraints (1) which
equate internal log supply and demand oscillated irregularly from small
negative to large positive numbers without seeming to converge.
Inclusion of guiding
constraints to the divisional problems does
improve the algorithm performance.
Dantzig-Wolfe decomposition
algorithm does not work in orqanizations where log supplier and log
user divisions can independently send their allocation suggestions in
response to transfer prices set by top management.
Two Milling Divisions And A Dominated Timber Division
Dantzig-Wolfe decomposition algorithm has originally been designed
for firms with a nonindependent raw materials procurement division
(timber division) and autonomous milling divisions (cf. Baumol and
Fabian, pp. 1-32).
Dantzig, p. 455).
This organization is shown in Figure A.4.l.b (cf.
The driving force of the decomposition algorithms
199
are -infeasible allocations for the timber division.
This has resulted
in one of their names: "infeasibility-pricing decomposition method,"
The infeasibilities produce nonzero
used by Balas (pp. 847-873).
shadow prices which are used as transfer prices in the iteration.
The
master program then includes only the milling divisions' allocations
and profits.
The objective is to find the optimal convex combination
tf milling divisions'internal log allocation suggestions given some
transfer prices.
The following formulation is for an organization
where the K milling divisions buy external logs:
K
(B)
Maximize
k=i tl
subject to
1i\
U)
K
T
t
I
x
ki tl
J
K
.
ljk
k
T
(3)
)
T
j
k=l t=l
Mt
forj=l,...,J
a.
3
mp. x
j=l
L-
t
NI
=
1
ojk
L
M
F
k
for k = l,..,K
t=l
where the variables are:
= weights to be given to milling division k allocation at
iteration t
-k=l,...,
=l,...,T
200
where the constants are:
= milling division k profit at iteration t,. $
Rk.
= milling division k internal log demand of class j logs at
Xljk
iteration t, MBF
= milling division k purchases of external class j logs at
Xojk
iteration t, MBF
= class j log market price, $/MBF
mp
= availability of internal class j locis, MBF
F = total amount of money at which external logs can be purchased,$
Constraints (1) show the log availabilities of internal logs.
They produce the J transfer prices which are used in the
iteration.
(J+])th
Constraints (2) show the availabilities of external logs.
Constraints (3) guarantee that for each milling division, the sum of
the convex combination weights is one.
The decomposition algorithm
did not produce very good results in an experiment with the case firm.
The corporate profits were about $40,000 less than those of the best
mill dominance
allocation which this decomposition approach resembles.
It took,however, less than ten iterations to arrive at this optimal
sol uti on.
Dantzig-Wolfe algorithm and other decomposition methods based on
it may be useful in solving some types of log allocation problems.
In
those problems, the timber division is a nonautonomous part of the
headquarters and the company has several milling divisions.
For reach-
201
ing the objectives of this study decomposition algorithms cannot be
used.
They have the most undesirable property that the divisions have
no freedom to execute the log allocation.
this right to itself.
The headquarters reserve
Transfer prices are used for collecting
info-
rmation to the headquarters but not for executing the top managenient's
decisions.
202
Appendix 4.3.
Price Adjustment Allocation
It is appealing to think a firm as a miniature economy.
nal
The inter-
log supplies and demands would be direct functions of their prices.
Headquarters could act as the famous "invisible hand" of Adam Smith in
quiding allocation proposals by divisions towards equilibrium.
This
guidance would occur through transfer prices only to minimize top
management's interference as is shown in Figure A.4.2.
Price adjust-
ment procedure is an iterative process based on these ideas.
It has
the following steps (cf. Jennergren, pp. 30-31):
1
.
The headquarters announces a set of tentative transfer prices
for the log classes.
It asks the timber division about its
supply schedule and the mills about their demand schedule
given these prices.
The divisions solve their allocation problems and report back
tothe headquarters their optimal allocation and profits.
Depending on whether the milling or timber division purchases
log mills solve problem of type (A) or (E) of Appendix 1.2.
The timber division solves a problem of type (B) without
constraints (4), or of type (C) of Appendix 1.
The headquarters compares supplies and demands.
If they do
not match, it computes and announces a new set of tentative
transfer prices:
203
I- -
trial transfer prices
I
milling
allocation
di vision
proposals
enforci n
(end product
market)
div. profit
11'
headquarters
sales)
/
transfer
internal
log
deliveries
prfes at execution
corp. jrofi t
1,
Ltri al transfer
L
prices
/
timber
division
ales
'
allocation proposals
Figure A.4.2.
Notation:
div. profit
log market
purchases
Information and log flows of price adjustment organization.
- --> information flow
> log flow
204
tp.
t+l
[0; tp
max
+
t)
Ft
X
(
k'l
where
tt+l
13k
X
-
k1
ojk
J
= trial transfer price of class .i logs at iteration t+l
$/MBF
tp
t
.
.
trial transfer price of class j logs at iteration t;
$/MBF
Xljkt = internal demand of class j logs of milling division k
at iteration t, MBF
Xojk
internal supply of class j logs to milling division k
at iteration t
Ft
adjustment factor expressing the size of the movement from
transfer price at iteration t to new price at iteration
t+l;
FtOfort=l,...,TandFt=0fortT+l.
- j = 1,...,J;
k
l,...,K;
t = l,...,T
The divisions solve the sub-problems and report back to the
headquarters their optimal allocation and profits, as in (2).
The iterations continue until the internal supplies and
demands match.
The last iteration produces the optimal trans-
fer prices, optimal allocation and maximum corporate profit
under price adjustment procedure.
At the execution step the headquarters announces the optimal
transfer prices.
The divisions allocate logs as they
indicated in the last iteration.
205
Price adjustment procedure is a simple method.
It is economically
more meaningful than decomposition allocation where divisions do not
have the true right to allocate.
It does not deny freedom from one of
the divisions as does dominance allocation (Jennergren, p. 40).
The
procedure may require a great many of information exchange iterations.
The most important drawback of the price adjustment procedure is
that it only works when divisional profit functions are strictly
concave (Jennergren, p. 40; cf. Kortanek, p. 74).
This results from
the phenomenon known as alternative optima (Hadley, p. 99).
It causes
distorted results in iterative linear programming problems.
Price
adjustment method does not then necessarily lead to a feasible optimal
solution in a finite number of steps.
used in this study.
This is the reason why it is not
206
Appendix 5.
MULTIPERIOD LINEAR PROGRAM FORMULATIONS
As examples of possible multiperiod (say, 5 years with 20 quarters) decentralized log allocation formulations we present the millThe
ing division and timber division programs under mill dominance.
mills determine their internal log demands each period given transfer
The timber division has to deliver the
prices by top management.
Mills buy logs from the
logs demanded by harvesting timber tracts.
The milling division program resembles that of
outside market.
Appendix l.2.(A):
(A)
Maximize
sales revenues
K
M
T
t
(0)
zpkm
t
t
Zkm
m=l k'l tl
costs of purchasing and processing internal logs
-
0
-
--------------------------------------------------------
-
K
-'
T
t
V' tPjk Xljk
tt
3
f
-
K
T
t
5
lJk
jk
tt
j=l k=I t=l
j=l k=1 t=l
costs of purchasing and processing external logs
0
c'
-'K
L
c'T
t
mpk X.k
OK
T
v
t
t
f
j=l k=l t=l
-
t
t
t
pC.k
?
j=l k=l t=l
subject to
K
(la)
kl
t
km
- Zk
= S
t-1
t
-
for m = 1,... ,M; t = 1,... ,T
desired end
product buffer
stocks
207
K
(ib)
ELt
Zkm
for
minimum end
product
order
m = 1,... ,M;
k= 1
t = 1,... ,T
3
km
jl
-
rJkffl
(qt
q.t)
recoveries
0
for m = 1,... ,M; k = 1,... ,K; t = 1,...
q.t) LT
(i
processing time
for k =
mpt
jl
Ft
Xik
external log
availability
fork=1,...,K
T
K
for
]
k=l t=1
(6)
3KI
x
j=1 k'1
Uk
t
t
(7)
+ XO.k
X]jk
1,... ,T
t
t
-
-
0Jk
internal log
availability
minimum harvest
desired log buffer
stock at mills
t-1
t
- Uk
tXLt fort
3
- Lik
for j = l,...,J; k = 1,...,K; t = 1,... ,T
where the variables are:
Zkm
= end product m sales from mill k in period t, MBM
Xljk
= class j internal logs received by mill k in
period t, MBF
Xojk
= class j external logs purchased from outside
208
market to mill k in period t, MBF
qt
= class j internal logs processed at mill k in
period t, MBF
qt
class j external logs processed at mill k in
period t, MBF
'km
= end product m processed at mill k in period t,
MBM --j = l,...,J; k =
m
l,...,M;
t=l,...,T
where the coefficients are
ZPkm
= net selling price of end product m at mill k in
period t, $/MBM
pcjkt
= log class j processing cost at mill k in period
t, $/MBF
mpjkt
= log class j market price at mill k in period t,
$ /M B F
tPjk
= transfer price of class j logs to mill k in
period t, $/MBF
= discount factor for period t
rjkm
= recovery of end product m from class j logs at
mill k, MBM/MBF
tik
= log class j processing time at mill k, shifts/MBF
where the constraint right-hand-side constants are:
Smt
= desired end product m buffer stock in the end of
period t, MBM
209
ELt
= minimum demand of end product m in period t, MBM
Tk
= total no. of eight-hour shifts available during
each period at mill k
Ft
= total amount of money at which outside market logs
can be purchased in period t, $
maximum amount of class j logs available internally in period t, MBF
XL
..
t
= minimum total harvest in period t, MBF
= desired mill k log class j inventory in the end
Ljkt
of period t
The milling division anticipated profits before fixed costs and
taxes are discounted end product sales net revenues minus costs of
using internal or external logs.
tion of internal
q0.
The mills are involved in acquisi(x0..t) logs, logs
(x1..t) and external
t) are processed to end products
are sold each period t.
t)
(
(q1t,
and end products (z.
t)
The mills maintain both end product and log
inventories which prevent stockouts in the changing periodic raw
material acquisition and end product demand conditions.
Constraints
(la) and (7) regulate procurement and production according to buffer
stock requirements.
Constraints (lb) show the unfilled or antici-
pated minimum end product orders.
milling processes.
in the mills.
Constraints (2) and (3) regulate
Constraint (4) depicts the log market situation
Constraints (5) and (6) are guidelines of wood
acquisition which guarantee feasible harvesting by timber division.
210
Under mill dominance the timber division has the following
multiperiod linear program which resembles the program of Appendix
1 .2. (B):
(B)
Maximize
discounted net revenues
from internal deliveries of harvested logs
J
I
T
K
ijk
i=l j=l k=l t=l
ijk
discounted net revenues
from external sales of harvested logs
J
I
+
\
T
L..
(
t
mp
i=1j=l t=l
t
sc.
j
-
ft
hc3)
3
-
subject to
I
K
t
-,
(1)
k=0 t=l
ijk
LX..
- 13
t
for i = l,...,I; j =
log availability
in tracts
,...
K
xk -
(2)
t
dijt
Xij+lk
= 0
k=0
tract log class
structure
for i = 1,...,I; j = l,...,J-l;
t = 1,... ,T
1
3
Y
V'
T
K
i=l j=l k=O t=l
3
I
T
(4)
maximum
and
minimum
harvest
x.k
xLt
l tl
IT? x.
il t=l
XUt
'
T
K
j1
'
kt
x.
kt = Xkt
13
for j = l,...,J
mills' log
demand
211
where the variables are:
Xijk
= harvesting of class j logs in tract i
for mill k
(k = l,...,K), or for outside market (k=O) in
period t, MBF ---i = 1,...,I; j = 1,...,J;
k = O,...,K; t
where the coefficients are:
hcjjkt = log class j logging and transportation cost in
tract i
sct
for mill k in period t, $/MBF
cost of selling class j logs to outside market
in period t, $/MBF
= log class 3 market price in period t, $/MBF
tPjk
= transfer price of class j. logs to mill k in
period t, $/MBF
= discount factor of period t
dijt
= ratio of log class j quantity to log class j + 1
quantity in tract i and period t (=
where the constraint right-hand-side constants are:
= amount of class j logs available in tract i and
period t, MBF
xut
= maximum desired harvest by timber division in
period t, MBF
XLt
= minimum desired harvest by timber division in
period t, MBF
Xjk
= log class j internal demand by mills in period t,
MB F
212
Timber division anticipated net revenues are the discounted net
income from internal and external log sales minus cost of harvesting
and selling external logs.
The harvesting cost includes only the
periodic logging and transportation costs.
Computation of user costs
as in Appendix 2.3 is not necessary in the multiperiod problem.
The
number of tracts (I) is not fixed for all periods but new public or
company timber tracts enter the computations in any period 1...20
when theybecome accessible for harvest through road building.
Although of longer term than the programs of Appendix 1 this
formulation does not try to optimize such intermediate and long time
range decisions as road building and bidding.
The idea behind pro-
gram Appendix 1.2.(C) was to solve a one period problem repeatedly
when the time comes to decide quarterly harvesting of accessible
tracts.
costs.
The myopia is partly overcome by the inclusion of user
The idea behind this program is to solve a, say 20 period,
problem repeatedly when the time comes to decide quarterly harvesting of accessible tracts.
The myopia is overcome by inclusion in
the computations of tracts which come accessible after period
The gains through increased precision of using a multiperiod
1.
in-
stead of a one period model may not be great, however, since there
still remains the task of finding expected values for the uncertain
future prices, costs, etc.
immensely.
The computational burden also increases
As a result of several computational experiments the
writer recommends that linear programming or other mathematical
213
optimization techniques
than two periods.
might notbe applied for problems with more
To reduce the costs of timber division calcula-
tions simple FORTRAN simulations resembling user cost computations
of Appendix 2.3 instead of mathematical programming could be used.
This recommendation holds especially for small firms with limited
research and development budgets.
C-,
214
Appendix 6.
JOINTLY OPTIMAL TRANSFER PRICES FOR DIVISIONAL MANAGERS:
A SPECIAL CASE
A major part of this study has dealt with a corporate profit
maximizing top management choosing transfer prices.
The level of top
management satisfaction is a linear function of the corporate profit
as a sum of the divisional profits:
Uc
= alco = a(Mmd +
where u0
Mco
top management utility
Mmd
Mtd = corporate, milling division and timber division
profit
a = constant
Figure A.5.l
.
shows the before fixed cost, tax and stumpage divisional
profits of the ten best transfer prices under mill dominance as an
example (column 3 of Table 4.1).
The optimal transfer price is TP7
with the highest sum of the abscissa (Mmd) and ordinate (Mtd)
Top management might, however, set a minimum limit for a division's
profits.
This could happen when the division manager has an influen-
tial position in the top management.
In Figure A.5.l., for example,
only transfer prices producing a milling division before fixed cost,
tax and stumpage profit over $230,000 could be acceptable.
the points above line b viould be feasible.
Then only
The requirement causes
TP3 to replace TP4 as the optimal transfer price.
The top management
preferences can be depicted by the lexicographic utility function of
Milling
division
profit,
$1000
215
_TP
260
p3
240
b
p4
p
p2
P
340
320
Figure A.5.l.
360
400
420 40 'G0
Timber division
profit, $1000
A boundary of milling and timber division profits of
several transfer prices, and a lower milling division
profit acceptance level (b).
Top myt.
utility
Hilling
division
profit, $1000
.80
TP6
260
I
0
.60
Ic'
240
I
Acceptable
transfer
prices
.40
TP
220
TP9
.20
200
TP
alP4
T7
Nonacceptable
trans fer
prices
F
550
560
Figures A.5.2.a. and b.
580
570
Corporate
profit,
$1000
550
560
570
580
Corporate
profit,
$1000
A lexicographic utility function of the milling division and
Corporate profits and the partitioning of several transfer
prices to acceptable and nonacceptable.
Utility
utd (c=200)
1.00
u0 (c=300)
.80
.60
.40
.20
200
Figure A.5.3.
400
600
800
Exponential milling division (Umd)
ProfIt, $1000
timber division
(utd) and Corporate (uco) utility functions (u =
1
- elk) for profits (M).
216
It is linear for milling division profits greater
Figure A.5.2.a.
than $230,000.. Top management utility for actions producing mill
profits less than or equal to that amount is negarive infinity.
Only
transfer prices TP3 and TP6 remain acceptable as we can see in Figure
A.5.2.b.
If top management consists of a group of profit maximizing divisional managers it might seek for a Pareto-optimal transfer price.
In our example it is found on the piecewise linear boundary of Figure
A.5.l.
Transfer prices TP9 and TP10 are dominated by others and lie to
the left of the boundary.
Every boundary point has a supporting
tangent line
or
h1M
+ h2Mtd = k
1td =
k
Mmd
-
where h1
1; h1,h2, k are constants
The Pareto-optimal point has a supporting tangent line with a
slope h1*/h2*.
Constants h1* and h2* cannot be determined "objectively"
but their values are found through bargaining between divisional
managers (cf. Raiffa, pp. 204-5).
If top management consists of utility maximizing division managers
who are not indifferent to risk the analysis of Figure A.5.l. is not
possible.
Replacing divisional profits by divisional utilities would
become necessary.
comparisons.
Dean, p. 52).
But this would lead to interpersonal utility
Interpersonal utility comparisons do not work (Halter and
Raiffa has presented, however, a special form of expo-
nential utility function of conservative group members.
He has
217
demonstrated that from the utility function of the individuals, a
group function can be constructed that produces a Pareto-optimal
solution (Raiffa, pp. 210-11).
The utility function of the n individ-
uals (1) and group (co) are of the following form:
Ui =
=
1
-
e1iki
n
M/
-
Uco
(M0) =
1
- e
i=l
Figure A.5.3. shows possible divisional and corporate utility
functions for the before fixed cost, tax and stumpage profits of
forest products firm.
a
The optimal top management transfer price is
found by maximizing the expected corporate utility of function Uco
as in Chapter 5 of this study.