Systems approach to equipment replacement in wood
products manufacturing
Carino, Honorio F ; Lin, Wenjie; Muehlenfeld, Ken
Journal 45. 6 (Jun 1995): 61.
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; Li, Yuenian. Forest Products
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A new approach to equipment replacement analysis involving the use of system optimization
and the discounted cash flow method of incremental investment analysis is presented. This
new approach ensures that replacement alternatives are evaluated within the framework of
maximizing profit and that the selection of a replacement alternative will enhance the
productivity of the entire production system, not just a given machine center. A computer
program called REPLACE, which embodies this new approach to replacement analysis, was
developed to facilitate the evaluation process and ensure the accuracy of the results obtained.
The efficacy of REPLACE was demonstrated in a case study involving an equipment
replacement analysis for a southern pine dimension lumber mill.
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Replacing equipment in a manufacturing operation is a major management decision that
invariably involves allocating capital for enhancing the firm's economic efficiency and ability
to maintain its competitive edge. However, many industrial managers, including some in the
wood industry, do not seem to appreciate the strategic value of conducting the timely
evaluations needed to maintain good equipment replacement programs or schedules. This
observation is based on close contacts with numerous wood products manufacturers,
including on-site mill visitations made by the principal author working as a forest products
extension specialist in Alabama in the past 7 years. "If it ain't broke, don't fix it" seems to be
the prevalent philosophy, particularly among operators of small wood products mills.
These managers do not realize there is an opportunity cost associated with any delay in
replacing economically inefficient equipment with more efficient alternatives. And if they do
consider replacing equipment either due to physical impairment or obsolescence, heavy
reliance is placed on experience or observation alone, often without regard for system and
theory. Often the same type of equipment that has been observed to perform well at another
mill is adopted as a replacement by the mill in question with little or no consideration given
to differences in layout and operating conditions at each mill. In many cases, this approach
has led to an unnecessarily high capital outlay for a replacement with a capacity or technical
capability that was more than adequate for the intended service.
Conceptually, equipment must be replaced at the end of its economic life even though it may
appear to be functionally efficient. The economic life of an asset is the time interval that
minimizes the asset's total equivalent annual costs or maximizes its equivalent annual net
income (7,8). It is also referred to as the minimum-cost life or the optimum replacement
interval. Clearly, economic life is the service period that must be terminated when a new
piece of equipment (i.e., a possible replacement) has an equivalent annual cost (or uniform
annual cost) lower than the cost of keeping the old equipment 1 or more years longer. If the
replaceable equipment is the determinant of production (i.e., the "bottleneck" of the
production line), the replacement should have adequate capacity to meet current levels of
production and any planned expansion. Extra capacity may be viewed as a necessary reserve
against the uncertainties that make the best predictions go wrong. But nevertheless, investing
for excess capacity beyond what is required to maintain current production levels must be
justified in economic terms. The additional investment must yield a return no less than the
minimum required by the firm as a matter of policy. In other words, the replacement should
be able to generate additional income from increased production (in terms of quantity and/or
quality) to offset the additional costs associated with it. Opting for a higher capacity
replacement might result in the bottleneck being shifted to another machine center, which
then becomes the new determinant of production at a higher level. If the replaceable
equipment is not the bottleneck, a replacement alternative with a smaller capacity and
presumably lower cost may be considered.
Evidently, a traditional or conventional approach to equipment replacement analysis (1,5,8),
which assumes a fixed production output in comparing alternatives and considers individual
machines or machine centers independently of others in the system, is inadequate for the
continuous production line settings found in many modern wood products manufacturing
operations. In such operations, keeping high machine uptime and a well-balanced (i.e., in
terms of capacity) production line is imperative for achieving production targets and desired
levels of profitability. This means that profit realization is not determined by an individual
machine or machine center, but by the entire production system with its interdependent
elements. A systems approach seems more appropriate in this case. It ensures that
replacement alternatives are evaluated within the framework of maximizing profit and that
the selection of a replacement alternative enhances the productivity of the entire production
system, not just the productivity of a given machine center. A review of the literature,
including previous surveys (3,4,6), reveals this is a new approach to equipment replacement
analysis. Therefore, this paper will discuss the development and implementation of a systems
approach through a case analysis involving a southern pine dimension lumber manufacturing
system.
NEW SYSTEMS APPROACH
The new systems approach utilizes a system optimization procedure in combination with a
discounted cash flow method of incremental investment analysis. The system optimization
procedure is primarily used to determine the economic impact of a machine replacement
alternative on overall mill profitability. This means that the efficacy of each replacement
alternative is evaluated within the framework of maximizing profit. Specifically, the potential
increases in net revenue associated with each replacement alternative are first determined
through optimization techniques (e.g., linear programming analysis). By identifying these and
other economic data associated with each alternative, the economic desirability of the
incremental investment needed for the potential replacement is subsequently evaluated based
on discounted cash flow calculations.
SYSTEM OPTIMIZATION ANALYSIS
The first step in this new systems approach is to establish a baseline estimate of the maximum
profitability of the existing production system under current conditions or a given set of
conditions (planned or expected) and identify the machine(s) that may have to be replaced.
This involves developing and using a system optimization model of the production system
under consideration. The model should be formulated to provide information about the
maximum attainable net revenue, capacity utilization of equipment in the system, and
optimum operating policy. A linear programming (LP) model is one type of system
optimization model that can be used and was in fact used by the authors in the case analysis
that will be discussed later in this paper. However, other mathematical programming and
system simulation models may be considered as alternatives. With profit maximization as an
objective, an evaluation of the present mill setup (i.e., with the machine(s) suspected of
needing replacement) is initially undertaken to obtain baseline information for the
replacement analysis. The optimization analysis (e.g., LP analysis) yields a set of valuable
information regarding system performance including attainable net revenue and machine
capacity utilization at various machine centers. Capacity utilization data are of special interest
to us because they are good indicators of the bottleneck (i.e., with 100% capacity utilization
status) machine that is inhibiting production and, hence, mill profitability. Therefore, such a
machine is often considered to be a good candidate for replacement regardless of its physical
condition.
Evidently, the capacity utilization data obtained from the system optimization analysis can
help the analyst determine the appropriate direction of change in terms of capacity of the
equipment to be replaced. If the replaceable equipment is the bottleneck, the replacement
alternative(s) should have a capacity no less than that of the equipment to be replaced.
Otherwise, if the replaceable equipment is not the bottleneck, the replacement alternative(s)
should have a capacity no more than that of the equipment to be replaced. Such a
predetermination is necessary to keep the number of qualified alternatives needing further
analysis to a minimum, shortening the search for the best alternative.
The optimization analysis is used to calculate the maximum attainable net revenue associated
with all replacement alternatives considered, including the "do nothing" alternative of
maintaining the status quo, i.e., continuing with the existing mill setup. Each replacement
alternative is evaluated within the framework of maximizing profit and potential increases in
net revenue (using net revenue figures from the existing mill setup as baseline) associated
with each alternative are calculated. Such incremental net revenue data will later be used in
the discounted cash flow method of the incremental investment analysis that follows
DISCOUNTED CASH FLOW ANALYSIS.
Because the replacement situation in question is considered to be a mutually exclusive event
(i.e., the selection of one alternative precludes the selection of others), the selection of the
best alternative is based on an iterative process of comparing two alternatives ("defender" or
"current best" vs. "challenger") at a time until all alternatives in the pool are considered using
discounted cash flow techniques in conjunction with incremental analysis, i.e., examining the
differences between the cash flows of the defender and challenger. In this connection, all
possible alternatives should be listed or ranked in ascending order based on their initial
capital outlays. The base alternative from which other replacement alternatives are initially
compared should be identified. In most cases, the one with the lowest ranking (i.e., with
smallest capital outlay) is designated as the initial base or current best alternative and often it
is the do nothing alternative (i.e., continue with the status quo). The capital outlay includes all
the costs incident to buying, installing, and making the equipment fully ready to operate on
the intended service. It should be pointed out that the challenger is always the next higher
alternative in order of capital outlay that has not been involved previously in a comparison.
With the net revenue data from the optimization analysis and other economic data (more
information on these will follow) associated with each replacement alternative, after-tax cash
flows are then calculated for the required economic comparisons between defenders and
challengers. Only the cost and income elements that vary among the replacement alternatives
are considered for comparison purposes.
Cost and economic life estimation. -- A good estimation of the economic life of an asset is
required in replacement analysis. As stated earlier, the economic life of an asset is the time
interval that minimizes the asset's total equivalent annual cost (7). If a machine is kept until
the end of year n, its equivalent annual cost can be calculated using the following relationship
(7):
(Equation (1) omitted)
The economic life of an asset will be equal to the n in Equation [1] that minimizes EAC(n).
In this study, the maximum number of years to be considered in determining the economic
life of a replacement alternative is its expected useful life (for new equipment) or remaining
useful life (for existing equipment). In estimating economic life, it is assumed that the
discount rate is equal to the minimum attractive rate of return (MARR).
It is also assumed that mill revenue associated with each replacement alternative will not
change from year to year, therefore, it will not affect the determination of economic life.
After-tax cash flow is estimated by adding the tax liability adjustment to the before-tax cash
flow. The tax liability adjustment is determined by applying the capital gain tax rate to the
capital gain (loss) resulting from the sale of the equipment. If a machine is to be sold in year
n, then the amount of capital gain (loss) is equal to the resale value of this machine at the end
of the year minus its book value at the end of the year. Here, the book value at the end of year
n is the undepreciated balance of the equipment, which is obtained by subtracting the
depreciation allowance that year from the book value at the end of the previous year. The
depreciation allowance is calculated according to the depreciation schedule specified in the
Modified Accelerated Cost Recovery System (MACRS) specified by the Internal Revenue
Service (9).
Before-tax cash flow is the summation of capital cost, interest on capital, maintenance cost,
insurance cost, and property tax. It is assumed that energy and labor costs associated with
equipment or machines will not change from year to year. Hence, these are not included in
the calculation of before-tax cash flow. Capital cost is the difference between the resale
values of the asset at the end of the current year and that of the previous year. An
exponential-function-decay method (2) is used in determining the resale value. An applicable
functional relationship is:
V sub n = V sub 0 X e sup -an [2]
where:
V sub n = resale value at the end of year n
V sub 0 = current resale value
n = year of resale
a = exponential parameter; inputted by user or approximated using the following relationship:
a = 1n(V sub 0 /V sub n )/n
V sub 0 and V sub n are initially inputted by the user.
Investment interest is estimated by multiplying the resale value at the end of the previous
year by the MARR. Similarly, insurance cost and property tax are estimated by multiplying
the property value by the insurance rate and property tax rate, respectively. Maintenance cost
was estimated using the following relationship:
C sub n+1 = C sub n X (1+b) [3]
where:
C sub n+1 = maintenance cost in year n+l
C sub n = maintenance cost in year n
b = acceleration rate (inputted by user)
Incremental investment analysis. -- Because replacement alternatives are considered to be
mutually exclusive, they have to be compared using a discounted cash flow technique in
conjunction with incremental analysis. In this connection, the internal rate of return method
can be and was used in evaluating the economic desirability of the incremental investment for
a replacement machine. The incremental investment is considered desirable or acceptable if
the after-tax rate of return associated with it was greater than or equal to the MARR.
The after-tax rate of return on incremental investment (RORII) is the discount rate i at which
the equivalent annual worth of the after-tax cash flows associated with the defender and the
challenger are equal (7). In other words, it is the discount rate that makes the equivalent
annual worth of the incremental after-tax cash flow equal to zero. The equivalent annual
worth approach to internal rate of return calculations was adopted because it provides a fair
basis of comparing alternatives, particularly when they are of unequal lives (7). Also, the
calculations are a lot easier to do because cost and revenue figures are readily available in
annual form. Equivalent annual worth can be calculated using the following relationship (7):
(Equation (4) omitted)
The RORTI between the challenger and the defender is the discount rate i that satisfies the
following equation (7):
EAW sub C = EAW sub D [5]
or
PW sub C X (A/P,i,n sub C ) = PW sub D X (A/P,i,n sub D ) [6]
where:
EAW sub c and EAW sub D = the equivalent annual worths for the challenger and the
defender, respectively, which can be calculated using Equation [4]
n sub C and n sub D = the economic lives of the challenger and the defender, respectively
PW sub C and PW sub D = the present worth of the after-tax cash flows for n years
(economic life) of the challenger and defender, respectively
As stated before, the comparison of mutually exclusive alternatives is appropriately done
through incremental analysis, i.e., examining the differences between their respective cash
flows. The differences in initial capital investment, operating costs, and net revenues
associated with the challenger and the defender are considered major factors influencing
RORII. The initial capital investment for an existing machine includes the current resale
value plus any costs incurred in upgrading or extending its useful life. For alternative new
machines, it is the installed cost. The incremental investment to be evaluated is the difference
between the two. The incremental revenue is estimated by taking the difference between the
net revenues associated with the challenger and the defender. These net revenue figures are
included in the output of the system optimization analysis. It is assumed that revenue will not
change from year to year within the economic life of a given machine. Operating costs are
estimated only for the machine or machine center considered in the replacement analysis and
not for the whole mill. Only the operating cost elements that vary among alternatives are
considered and these include maintenance cost, insurance cost, property tax, energy cost, and
labor cost. The incremental operating cost is the difference between those operating costs
associated with the challenger and the defender.
COMPUTER MODEL AND PROGRAMMING
A computer program called REPLACE, which embodies the new systems approach to
equipment replacement, was developed to facilitate and simplify what normally could be a
complex and tedious analytical process. REPLACE can be considered user-friendly because
it allows interactive input/output of data and convenient execution of the necessary programs
for system optimization and discounted cash flow analyses. This was accomplished using
Windows programming with Visual Basic in conjunction with Lotus 1-2-3 and What'sBest!
The Lotus 1-2-3 environment was used in the system optimization model formulation (a
linear programming model in this case), input data entry, estimation of economic life, cash
flow analysis, and output report writing. What'sBest, an optimization software developed by
LINDO System, Inc., was used for solving the LP model in the case study to be discussed
later. The Visual Basic environment was used to create the graphical user interface that acts
as a navigator in running this program. The specification of software used should not be
construed as endorsement of these products. With the help of the graphical user interface, a
user only needs to click an appropriate button to invoke a procedure. For example, to invoke
the system optimization procedure, simply click (with a mouse) on the system optimization
button appearing on the main menu window. All replacement alternatives provided by a user
will be evaluated in one run. The final output of this program includes information on the
return rate of incremental investment, economic life of both defender and challenger, and
discounted pay-back periods of replacing the initial machine for each comparison.
It must be pointed out that the LP model is mill specific. This means that a new LP model has
to be formulated and developed when considering other mills with a layout that is different
from that of the study mill. All other aspects of REPLACE will remain the same.
CASE ANALYSIS
MILL CONFIGURATION
The systems approach to equipment replacement analysis was tested in a case study involving
a southern pine dimension lumber mill in Alabama. Figure 1 shows a simplified layout of the
mill, which is a two-sided sawmill. (Figure 1 omitted) On one side is a chip-n-saw (CNS) that
processes small-diameter (6 to 17 in. on the small end) logs. On the other side is a bandmill
(or bandsaw headrig) that processes logs with small-end diameters of at least 10 inches. Other
equipment in the sawmill includes a debarker, a cut-off saw, a linebar resaw, 2 three-saw
edgers, a chipping edger, a twin-band resaw, and a multi-saw trimmer. Input logs are bucked
or cut to the desired lengths by the cut-off saw and then debarked with the debarker. The
debarked logs then go to either the bandsaw or the CNS. From the headrig, side boards are
conveyed to a three-saw edger for edging. Doubles and cants can be directed to the twinband
resaw or to the linebar resaw depending on the volume at those stations. Cants from the CNS
are also directed to the twinband resaw. Boards from the CNS move up through another
three-saw edger for edging. A chipping edger handles further manufacturing on boards from
the twinband resaw. All breakdown stations can feed boards directly to the gang trimmer.
BASELINE EVALUATION OF SYSTEM PERFORMANCE
The first step is to establish a baseline estimate of the maximum profitability of the existing
production system. This was accomplished using a linear programming model as shown in
the Appendix. The objective function of this model is to maximize the net revenue (i.e.,
product revenues minus log costs) of the mill with log input, lumber output, and machine
capacity as constraining factors, and assuming that everything produced could be sold at
current market prices. In addition to the maximum mill revenue, the output of the LP model
also includes data on the capacity utilization of equipment in the system, the volume of
lumber produced by sizes, and the volume of logs processed by each side of the mill. In
discussing the results that follow, it is assumed that the reader is familiar with LP analysis, so
further details on LP are not presented.
LP analysis of the existing mill setup revealed a net revenue of approximately $3,025 per
hour of operation (Table 1). (Table 1 omitted) This involved producing 21.04 thousand board
feet (MBF) of lumber per hour from the 14.07 cunit of logs processed through the bandsaw
headrig and 16.57 cunit of logs processed through the CNS (Table 1). This is equivalent to a
lumber recovery factor of about 6.8 board feet (BF) of lumber per cubic foot of log input.
The LP results also show that the existing CNS, bandsaw headrig, and the twinband resaw
were being operated at full capacity under current conditions (Table 2). (Table 2 omitted)
However, the machine center that had attracted management's attention the most was the old
CNS, which recently had been breaking down quite often. On average, it had an uptime of
about 50 minutes per hour of operation and was generally considered as inhibiting higher
production. Also, the estimated cost of maintenance on this machine had sharply risen from
$4,000 (when it was new) to about $12,000 annually, which is excluding the cost of lost
production due to delays. For these reasons, management had been evaluating replacement
options.
It should be pointed out that the LP model formation is mill dependent. In other words, the
LP model needs to be modified if the mill layout is changed. As stated earlier, LP is not the
only system modeling tool that can be used in the systems approach to equipment
replacement problems. Other mathematical programming models and simulation models may
be used as alternatives to estimate the performance of the system under various replacement
options.
REPLACEMENT OPTIONS
Management could keep the status quo, i.e., continue operating with the existing CNS. Or, it
could modify the current CNS to increase throughput by reducing unnecessary delay. The
CNS setup time can be reduced with a better log input scanning system and better handling
equipment. Another alternative is to replace the CNS altogether with another log breakdown
system that can provide higher volume yield as well as higher throughput.
A headrig considered as a possible replacement for the existing CNS was a 6-foot twin
bandsaw/reducer head with a sharp chain (SC) log feeding system. A heavy roller impales the
log on a sharp chain conveyor, thereby holding it firm as it is led to the twinband saws for
conversion into boards and a cant. Two chipping heads initially convert the log into a cant
and the two bandsaws would reduce it into a smaller cant by cutting one board from each
side, in one forward pass. High production rates (up to 250 feet per minute or 8 to 12 logs per
minute) are possible with this system. Compared to the CNS, it provides higher lumber
recovery mainly because it offers greater flexibility in the sawing method to use for different
log types and sizes.
Another headrig considered as a possible replacement for the existing CNS was a twinband
saw with an end dogging log feeder (EDLF) system. The twinband saw with the EDLF
system provides higher lumber recovery mainly due to smaller target sizes and greater
flexibility in the sawing method used for different kinds of logs. The EDLF holds the log
more securely than the SC does, thus minimizing sawing variations and hence smaller target
sizes. It allows accurate log taper and offset sawing, which are not possible with the SC
system.
Table 3 shows the estimated present value of investments, useful lives, and salvage values for
the various replacement alternatives considered in the study. (Table 3 omitted) The existing
CNS, which had an estimated remaining life of 4 years, was designated as the do-nothing
option or replacement alternative No. 0. The alternatives considered were: No. 1 modify
and/or upgrade the existing CNS for an initial capital outlay of $400,000 so as to extend its
life to 8 years; No. 2 -- replace the existing CNS with a 6-foot twinband/SC system with a 12year life for an initial capital outlay of $1,200,000; and No. 3 -- replace the existing CNS with
a 6-foot twinband/EDLF system with a 12-year life for an initial capital outlay of $1,500,000.
SYSTEM PERFORMANCE WITH REPLACEMENT ALTERNATIVES
After the baseline estimation, LP analysis was again undertaken to reevaluate system
performance considering each replacement alternative. The resulting production data and
potential net revenues associated with each alternative are presented in Table 1, and the
machine uptime and capacity utilization rates are listed in Table 2.
With a modified CNS system, the hourly net revenue determined from LP analysis would
increase to about $3,035 (Table 1.) This increase in net revenue is due primarily to log cost
savings and not to increased yield or increased throughput. Although the modified CNS is
capable of processing, on average, about 6 to 8 logs per minute (i.e., about 2 logs per minute
more than the existing machine), it is prevented from operating consistently at that level by
the twinband resaw, which shows a 100 percent utilization of capacity (Table 2). In essence,
the twinband resaw is now inhibiting higher production even if the CNS is modified or
upgraded.
With the twinband/SC system, the hourly net revenue would increase to $3,367, $342 more
than under the existing mill setup (Table 1). The increase can be attributed both to improved
lumber recovery and increased throughput, particularly on side 1 (bandsaw headrig).
Evidently, the critical bottleneck would also transfer to the twinband resaw if the
twinband/SC headrig system were in place (Table 2).
With the twinband/EDLF system, the hourly net revenue would increase to $3,821, $796
more than under the existing mill setup (Table 1). The increase can be attributed both to
improved lumber recovery and increased throughput, particularly on side 2
(twinband/EDLF). Again, the increase in net revenue was not so much from increased lumber
value yield per log, but more from log cost savings due to the large number of smaller
diameter logs allocated for processing on side 2. As in the two other replacement situations,
the critical bottleneck would also transfer to the twinband resaw section if the
twinband/EDLF headrig system were selected (Table 2).
The replacement alternatives to the existing CNS would allow higher production rates but
could also inhibit performing at capacity levels by the twinband resaw. Certainly,
management also could look into the possible replacement of this machine in order to attain
higher production, but that is beyond the scope of this case study. The same analytical
procedure presented in this paper could be followed.
INCREMENTAL INVESTMENT ANALYSIS
Incremental analysis in conjunction with discounted cash flow analysis was used to determine
the economic desirability of the replacement alternatives. This analysis procedure is invoked
by clicking the financial analysis button appearing in the main menu window of REPLACE.
In addition to the initial capital cost, useful life, and salvage value, operating costs associated
with each replacement alternative including maintenance cost, insurance cost, property tax,
energy cost, and labor cost are required as inputs. The attainable net revenue associated with
each replacement alternative is automatically transferred (through macro programming) from
the output module of the system optimization procedure to the input module of the financial
analysis procedure for use in the investment analysis.
Initially, the existing CNS was compared to the modified CNS. The RORII of $200,000
turned out to be 12.3 percent, which is greater than the MARR of 10 percent (Table 4). (Table
4 omitted) Therefore, the incremental investment is considered economically desirable, i.e.,
the existing CNS should be modified. In the second comparison, the modified CNS became a
defender and was then challenged by the twinband/SC system, which involved an
incremental investment of $1,000,000. The RORII was only 6.2, so the incremental
investment on a twinband/SC system was not justified. The next comparison was therefore
between the modified CNS and the twinband/EDLF system, which involved an incremental
initial capital outlay of $1,300,000. In this case, the RORII was 12.4 percent, which is greater
than the MARR. Hence, the incremental investment on a twinband/EDLF system as a
replacement for the existing CNS was considered justifiable. At this point, it can be
concluded that the twinband with EDLF system is the most desirable replacement alternative
to the existing CNS.
LITERATURE CITED
1. Butler, D.A. and D.P. Dykstra. 1981. Logging replacement equipment: a quantitative
approach. Forest Sci. 27(1):2-12.
2. Collier, C.A. and D.E. Jacques. 1984. Optimum equipment life by minimum life cycle
costs. J. Constr. Eng. Mgmt. ASCE 110(2):248-265.
3. Forston, J.C. and R.C. Field. 1979. Capital budgeting techniques for forestry: a review.
South. J. of Appl. Forestry 3(4):141-143.
4. Mao, J.C.T. 1970. Survey of capital budgeting: theory and practice. J. of Finance
25(2):349-360.
5. Mills, W.L., Jr. and R.A. Tufts. 1985. Equipment replacement: a comparison of two
methods. Forest Sci. 31(3):661-670.
6. Schall, L.D., G.L. Syndem, and W.R. Geijsbeek, Jr. 1978. Survey and analysis of capital
budgeting methods. J. of Finance 33(1):281-287.
7. Thuesen, G.J. and W.J. Fabrycky. 1993. Engineering Economy (8th edition). Prentice-Hall.
Englewood Cliffs, N.J. 717 pp.
8. Tufts, R.A. and W.L. Mills, Jr. 1982. Financial analysis of equipment replacement. Forest
Prod. J. 32(10):45-52.
9. U.S. Department of Treasury -- Internal Revenue Service (USDT-IRS). 1993.
Depreciation. Internal Revenue Service Pub. No. 534.
APPENDIX
LP MODEL FORMULATION
Maximize
Z = Sigma sub t Sigma sub w Sigma sub h P1 sub twh X F sub twh + Sigma sub d Sigma sub
l (P2 X W X G sub dl + P3 X W X E sub dl - C1 sub dl ) X1 sub dl + Sigma sub d Sigma sub
l (P2 X W X G sub dl + P3 X W X E sub dl - C1 sub dl ) X2 sub dl + Sigma sub d Sigma sub
l (P2 X W - C1 sub dl ) X3 sub dl [1]
Subject to the following constraints:
Log input: Xl sub dl + X2 sub dl + X3 sub dl <= IT sub dl [2]
Lumber output:
F sub twh - Sigma sub d Sigma sub l CF sub dltwh X X1 sub dl - Sigma sub d Sigma sub l
CF sub dltwh X X2 sub dl <= 0 [3]
Sigma sub d Sigma sub l T1 sub dl X X1 sub dl + Sigma sub d Sigma sub l T1 sub dl X X2
sub dl + Sigma sub d Sigma sub l T1 sub dl X X3 sub dl <= M1 [4]
Sigma sub d Sigma sub l T2 sub dl <= M2 [5]
Sigma sub d Sigma sub l T3 sub dl X X2 sub dl <= M3 [6]
Sigma sub d Sigma sub l T4 sub dl X X1 sub dl <= M4 [7]
Sigma sub d Sigma sub l T5 sub dl X X1 sub dl + Sigma sub d Sigma sub l T5 sub dl X X2
sub dl <= M5 [8]
Sigma sub d Sigma sub l T6 sub dl X X1 sub dl <= M6 [9]
Sigma sub d Sigma sub l T7 sub dl X X2 sub dl <= M7 [10]
Sigma sub d Sigma sub l T8 sub dl X X1 sub dl + Sigma sub d Sigma sub l T8 sub dl X X2
sub dl <= M8 [11]
Non-negativity: Xl sub dl , X2 sub dl X3 sub dl F sub thw >= 0
where:
Z = net revenue (i.e., product revenues minus log costs) for an average or typical hour
operation, $/hour
Index sets
t = nominal lumber thickness class (in.) w = nominal lumber width class (in.) h = nominal
lumber length class (ft.) d = nominal log small-end diameter class (in.) l = nominal log length
class (ft.) n = index for processing machine or equipment 1 -- debarker; 2 -- headrig; 3 -chip-n-saw; 4 -- linebar resaw; 5 -- twin band resaw; 6 -- edger No. 1; 7 -- edger No. 2; 8 -chipping edger
Data
Cl sub dl = delivered cost of conventional sawlogs ($/cunit)
CF sub dltwh = lumber conversion factor by log diameter and length classes and lumber
thickness, width, and length classes (MBF/cunit)
E sub dl = percent of green log volume converted into sawdust by diameter and length classes
G sub dl = percent of green log volume converted into chips from green mill residues by
diameter and length classes
IT sub dl = quantity of available supply of logs by diameter and length classes (cunits/hr.)
Mn = machine uptime (min./hr.); n = 1, 2,....8
P1 sub twh = average price of lumber (i.e., weighted by grade) by thickness, width, and
length classes ($/MBF)
P2 = average price of green wood chips ($/ton)
P3 = average price of green sawdust ($/ton)
Tn = processing rate of machine n by log diameter and length classes (min./cunit); n = 1,
2,...8
Y = mill lumber output (MBF/hr.)
W = weight factor (tons per cunit of log input)
Decision variables
F sub twh = volume of lumber produced by thickness, width, and length classes (MBF/hr.)
X1 sub dl = volume of conventional sawlogs processed through the bandmill by diameter and
length classes (cunits/hr.)
X2 sub dl = volume of conventional sawlogs processed through the chip-n-saw by diameter
and length classes (cunits/hr.)
X3 sub dl = volume of logs chipped entirely by diameter and length classes (cunits/hr.)
Equation [1] is the linear objective function to be maximized, which in this case is the total
net revenue for an average or typical hour operation. The objective function indicates that net
revenue, expressed as the difference between product revenues and delivered costs of log
input, can be maximized considering various options:
1) Produce lumber and its byproducts (chips and sawdust) by routing through the bandmill
sawlogs with small-end diameters ranging from 10 to 30 inches and lengths of 8, 10, 12, 14,
16, 18, and 20 feet; and/or
2) Produce lumber and its byproducts (chips and sawdust) by routing through the chip-n-saw
sawlogs with small-end diameters ranging from 6 to 17 inches and lengths of 8, 10, 12, 14,
16, 18, and 20 feet; and/or
3) Convert into chips whole logs with small-end diameters ranging from 6 to 9 inches and
lengths of 8, 10, 12, 14, 16, 18, and 20 feet.
It was assumed that the costs of converting a unit volume of input sawlogs into lumber or
chips do not change with log size. The maximization of the objective function is subject to
constraints represented by Equations [2] through [11], as well as the non-negativity
requirement for the values of the decision variables. Equation [2] indicates that the total
volume of logs utilized for the options discussed should not exceed the amount of available
log supply. Equation [3] requires that the total volume of lumber produced should not exceed
the volume yield expected from the log input. Equations [4] to [11] limit the available
processing time at the various machine centers. It is assumed in the formulation that the mill
can sell at the prevailing market prices all the lumber, chips, and sawdust produced. It is also
assumed there are no capacity constraints for the cut-off saw, whole log chipper, trimmer, dry
kiln, and planer mill, since these have been observed to be utilized far below their capacities.
Copyright Forest Products Society Jun 1995
Word count: 5858
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Indexing (details)
Cite
Subject
Software;
Computer based modeling;
Wood products;
Capital expenditures;
Studies;
Manufacturing;
Benefit cost analysis
Classification
9130: Experimental/theoretical treatment, 5240: Software & systems, 8630: Lumber & wood
products industries, includes paper
Title
Systems approach to equipment replacement in wood products manufacturing
Author
Carino, Honorio; Lin, Wenjie; Muehlenfeld, Ken; Li, Yuenian
Publication title
Forest Products Journal
Volume
45
Issue
6
Pages
61
Number of pages
8
Publication year
1995
Publication date
Jun 1995
Year
1995
Publisher
Forest Products Society
Place of publication
Madison
Country of publication
United States
Journal subject
Forests And Forestry--Lumber And Wood, Building And Construction
ISSN
00157473
CODEN
FPJOAB
Source type
Scholarly Journals
Language of publication
English
Document type
PERIODICAL
Subfile
Wood products, Studies, Software, Manufacturing, Computer based modeling, Capital
expenditures, Benefit cost analysis
Accession number
01062551
ProQuest document ID
214633640
Document URL
http://search.proquest.com/docview/214633640?accountid=62692
Copyright
Copyright Forest Products Society Jun 1995
Last updated
2010-06-08
Database
ProQuest Agriculture Journals