lecture 7 : costs
this lecture is about how to handle costs in
hard-OR models
•
•
•
•
types of costs
costs that are relevant to decision making
an exam question
case example
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• cost is a decrease in wealth that results from
committing a resource to a particular use before the
benefits are calcuated
• costs can be classified in many ways:
–
–
–
–
–
–
–
–
explicit cost - implicit cost
tangible cost - intangible cost
accounting cost - economic (or opportunity) cost,
direct cost - indirect cost
fixed cost - variable cost
short run cost - long run cost
relevant cost - irrelevant cost
...
• these partitions necessarily overlap, like for example,
– a fixed, explicit accounting cost that is irrelevant
– an implicit, intangible opportunity cost that is relevant etc..
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explicit - implicit costs
1. explicit costs: out-of-pocket costs involving an
actual transfer of funds; examples:
• payrol costs
• purchasing costs
• ...
2. implicit costs: no funds change hand but cost is
incurred; examples:
• depreciation
• cost of reduced productivity and disruption caused by
rescheduling due to an emergency
• the social cost of unemployment
• cost of goodwill loss
• cost of late delivery
• ...
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economists’ fixed-variable / short-run-long-run costs
1. variable costs change in the short-run with the
output level
2. fixed costs do not change in the short-run with
the level of output, but they can change in the
long-run
accountants’ direct (prime) - indirect (overhead) costs
1. direct costs are directly associated with one
product or service
2. indirect costs are associated with all products and
services; hence these are joint costs
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• note that:
– capital, or capacity costs are fixed in the short-run but
variable in the long-run
– in the long-run all costs are variable, they change with
output level and with output capacity
– indirect costs remain fixed over a wide interval of output
level but will change when output exceeds that interval
– direct costs can be variable or fixed when for example a
machine is dedicated to a product so its fixed
depreciation is a cost directly related to the product
• relevant costs are costs that change as a result of
making decisions, irrelevant costs are costs that do not
change with the decision; this means that the relevant
cost of an activity would change when that activity
takes place
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• implicit costs that are shown in a balance sheet are
intangible costs from an accounting point of view; these
are often not relevant to OR; examples are:
– depreciation
– goodwill (when purchase cost exceeds the net value of assets)
– reserves for bad debts, a form of asset value reduction
it is relatively easy to put an exact figure on the amount of
these costs
• there are also implicit intangible costs relevant to OR
decisions; these are potential future costs or loss of
benefits:
− goodwill cost that might result from late deliveries is a
relevant intangible cost for OR; this is not the same goodwill
cost that accountants record in books
− depreciation is often not relevant for OR decisions
it is very difficult to put an exact figure on this type of
intangible costs
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accounting costs - economic costs
1. accounting costs are recorded in balance
sheets and income statements; they are
either explicit or implicit costs
2. economic costs or opportunity costs; the
implicit (or explicit) cost of a resource
calculated as forgone benefits from the best
alternative activity that the resource could
have been used for
•
•
an opportunity cost can be equal to the explicit
accounting cost
an opportunity cost can be intangible
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• opportunity cost of a resource is the answer to the
question: what is the highest net additional
income that not committing this resource can
generate?
• opportunity cost is measured in units of an objective
function (cost or profit), so it is a shadow price
• opportunity cost of an identically replaceable
resource is equal to the replacement cost, (which is
an explicit cost) eg. cost of cement or plain sugar
• opportunity cost of an identically irreplaceable
resource is equal to the cost of replacing the
resource with less desirable substitutes (eg. the
opportunity cost of assigning a uniquely talented engineer
from project B to project A would equal the income lost in B
plus the wages of replacing engineers)
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OBJECTIVE FUNCTION VALUE
Max 90 xA + 60 xB
subject to
2xA + xB < 8
xA < 3
xB < 4
1)
VARIABLE
XA
XB
420.0000
VALUE
2.000000
4.000000
REDUCED COST
0.000000
0.000000
ROW SLACK OR SURPLUS
DUAL PRICES
2)
0.000000
45.000000
3)
1.000000
0.000000
4)
0.000000
15.000000
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Max 90xA + 60xB
subject to
2xA + xB < 7
xA < 3
xB < 4
OBJECTIVE FUNCTION VALUE
1)
VARIABLE
COST
XA
XB
the shadow price of one
hour of labour, £45, is
an opportunity cost
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375.0000
VALUE
1.500000
4.000000
REDUCED
0.000000
0.000000
ROW SLACK OR SURPLUS DUAL
PRICES
2)
0.000000
45.000000
3)
1.500000
0.000000
4)
0.000000
15.000000
IE 398 Lecture 7
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relevant costs and benefits
• explicit costs are often relevant unless they are
− sunk costs, which are irrelevant (eg. suppose the LOD project has
costed $ 8 000 already and requires a further spending of $ 30 000 in order
to provide benefits of $ 35 000; the $ 30 000 should be payed since the
$ 8 000 is only a sunk cost and irrelevant for decision making)
sunk costs can be hidden, (eg. as depreciation entered for
accounting purposes rather than as a wear-and-tear cost;
identifying a sunk cost may not be always straightforward
• all relevant costs, explicit or implicit, should be
opportunity costs of resources used; in general they
are not the same as historical ie. accounting costs
(eg. the cost of warehousing space at LOD could be zero if
there were no alternative use for the warehouse, or it could
be $ 44/m2 for example, if it could be rented out or used for
parking)
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– replacement costs represent opportunity costs that are often
relevant, unlike historical purchase costs which are irrelevant
– future costs and benefits are relevant opportunity costs that
can be explicit such as a decommissioning cost or a salvage
value
– intangible costs are relevant opportunity costs in the form of
shortage costs, goodwill costs etc.; these are often important
but difficult to determine; extensive sensitivity analysis will be
needed to estimate intangibles
• conclusions:
• accounting records provide important historical cost and benefit data
• not all of these are relevant to managerial decision making
• all costs used for decision making, whether implicit or explicit or
intangible must be opportunity costs
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an exam question
A customer has offered to buy from LOD, a special type of hightechnology lubricant which is not among the lubricants that
LOD normally produces. Therefore the present offer represents
a one-time project and the customer has offered to pay
€200 000 for the required amount. The lubricant requires
mixing an oil A and a special agent B. The necessary amount of
A is available in the stocks of LOD, but B must be obtained from
an outside supplier and an order has already been placed when
the project was offered to LOD. The mixing process is quite
complicated and requires the use of two kinds of capital
equipment C and D; C is already owned by LOD but D must be
leased from outside. Mixing also requires that expert scientific
help be obtained from a nearby University. LOD accountants
calculate the following cost estimates for the project:
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cost item
material
type
accounting costs in €
A (in stock)
16 000
B (on order)
18 000
labour
70 000
capital equipment C (owned)
40 000
D (leased)
50 000
general overhead
30 000
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a) Based on accounting costs determine whether LOD should accept or
reject the project offer. Calculate the profit margin.
b) LOD manager has the following additional information about the cost
items:
 If oil A is not used in this project, it can either be distilled at a cost of
€5 000 and used in some other project to replace an input that would
cost €18 000; or it can be sold for €12 000.
 If agent B is not used in the project it has to be sold for €12 000.
However it is also possible to cancel the order before delivery, but a
penalty of €4 000 must then be payed to the supplier.
 Much of the labour cost is to be payed to the University except for the
wages of an LOD manager who must be transferred to the project. His
wage cost is €20 000; but if he is transferred to the project someone
else must be hired at a cost of €10 000 to replace him in his current
duties.
 Equipment C cost €240 000 when it was new. It is depreciated on a
straight-line basis and a depreciation cost of €40 000 has been
assigned to the project. If the project is not taken C will be sold at the
second-hand price of €60 000; if the project is taken C will be sold at
the second-hand price of €40 000 after the completion of the project.
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cost item
material
type
acc. costs opp. costs
A (in stock)
16 000
13 000
B (on order)
18 000
14 000
labour
70 000
60 000
capital equipment C (owned)
40 000
20 000
D (leased)
50 000
50 000
general overhead
30 000
0
total cost
224 000
157 000
profit
-24 000
43 000
opportunity cost is the real cost of an item, it is the answer to the question:
“what is the highest net additional income that I can generate if I don’t
commit this item to this project?”
net additional income = foregone income + expenses incurred, because the
resource is committed
cost of A:
max { 18 000 – 5 000 ; 12 000 } = 13 000
cost of B:
max { 18 000 – 4 000 ; 12 000 } = 14 000
labour cost: 50 000 + 10 000 = 60 000
cost of C: 60 000 – 40 000 = 20 000
Therefore the project would be rejected on the basis of accounting costs
whereas it should have been accepted as shown by the opportunity cost
analysis.
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case example: Champignons Galore,
a French mushroom grower
• a three stage process:
– put compost in trays
– mycelium growing – 26 to 28 days
– mushroom harvesting – in flushes with decreasing
yield and size at 6 to 8 day intervals (<5 times per
cycle)
• 66 sheds (400 trays in each shed) 65 of them
always in use except for 2 days cleaning
between cycles
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• three alternatives considered to increase production:
1. adopting two-layer sheds at the current site
2. acquiring a second site for new sheds
3. optimising the number of flushes without adding
new sheds
an OR study on costs and yields is initiated:
• only the third alternative is investigated based on the
objective to maximise profits making the following
boundary judgements:
– the production process remains unchanged, using 66 sheds and
400 trays/shed
– additional output can be sold at current prices
– additional inputs can be obtained at current prices
– the number of flushes per cycle can be 1, 2, 3, 4 or 5
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• the problem elements are:
– decision maker : Gerard Mousse
– objective : high annual profit
– performance measure :
profit (= revenue – total cost)
– decision criterion : maximize profit
– alternative courses of action : number
of flushes in each cycle
– problem context : defined by the
boundary judgements
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~~--
Section
Relevance Nature of effect
Input parameter
Compost/tray preparation:
raw materials
yes
prop.ltray
prep. cost/tray (1)
electricity
yes
variable/tray
prep. cost/tray (2)
diesel fuel
yes
prop.ltray
prep. cost/tray (3)
maintenance on vehicles
yes
variable/tray
prep. cost/tray (4)
vehicle depreciation
no
constant
fixed cost
yard/build. maintenance
no
constant
fixed cost
yard workers wages
yes
prop.ltray
salary J .Brownsey
no
constant
fixed cost
Shed loading and
maintenance:
supplies & diesel fuel
yes
prop.ltray
shed cost/tray
compost pack material
yes
prop.ltray
shed cost/tray
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prep. cost/tray (5)
21
tray repair & maint 'nce
yes
prop.ltray
shed cost/tray
vehicle maintenance
vehicle depreciation
shed workers' wages
salary ofM. MeTrae
yes
no
yes
no
variable/tray
constant
prop.ltray
constant
shed cost/tray
fixed cost
shed cost/tray
fixed cost
Climate control:
electricity
materials
lab. maintenance
equipment depreciation
new equipment
no
no
no
no
no
approx. constant
approx. constant
constant
constant
sunk cost
fixed cost
fixed cost
fixed cost
fixed cost
excluded
Picking, sorting, packaging:
packing materials
supplies & electricity
new equipment
maintenance equip.
equip. depreciation
diesel fuel trucks
maintenance trucks
depreciation trucks
wages sort/pack
wages for drivers
salary J. Fleurette
wages picking
vacation/sick pay
yes
yes
no
yes
no
yes
yes
no
yes
yes
no
yes
yes
prop.lkg
variable/kg
sunk cost
variable/kg
constant
prop.lkg
variable/kg
constant
prop.lkg
variable/kg
constant
variable/hour
variable/hour
sort/pack. cost/kg
sort/pack. cost/kg
excluded
sort/pack. cost/kg
fixed cost
sort/pack. cost/kg
sort/pack. cost/kg
fixed cost
sort/pack. cost/kg
sort/pack. cost/kg
fixed cost
pick pay/hr
pick pay/hr
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Local marketing:
salary J. Marchand
no
constant
fixed cost
travel J. Marchand
no
approx. constant
fixed cost
no
no
no
no
no
no
no
no
no
no
approx. constant
sunk cost
approx. constant
constant
approx. constant
constant
constant
constant
constant
constant
fixed cost
excluded
fixed cost
fixed cost
fixed cost
fixed cost
fixed cost
fixed cost
fixed cost
fixed cost
yes
prop.l€ paid
Other costs:
office supplies
new equipment
company cars running cost
depreciation company cars
office salaries
gardener
maintenance office
depreciation office bldg
insurance, mortgage interest
salaried employees accident
insurance, pension fund
worker accident
insurance &
pension fund
prep. cost/tray &
shed cost/tray &
sort/pack. cost/kg
& pick cost/hour
salary C. Mousse
travel C. Mousse
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no
no
IE 398 Lecture 7
constant
approx. constant
fixed cost
fixed cost
23
Mathematical model for annual profit
All expressions are defined for a decision choice of n consecutive
flushes in each cycle. where n can be 5, 4, 3, 2, or 1. The
expressions are labelled by the numbers shown in each circle or
output of the influence diagram.
1. Length of each cycle:
[cycle length] = [growing phase duration]+[average flush length] x n
= 28 + 7n
2.Number of cycles per year:
[cycles/year] = [days in year]/([cycle length] + [change-over time])
= 365/([ cycle length] + 2)
3. Number of trays prepared per year:
[trays/year] = [number of sheds] x [trays/shed] x [cycles/year]
= 65 x 400 x [cycles/year]
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5. Annual output of mushrooms:
[annual output] = [trays/year] x [cumulative yield/tray for n flushes]
6. Annual tray preparation cost:
[annual tray preparation cost] = [preparation cost/tray] x [trays/year]
7. Annual shed loading and unloading cost:
[annual shed cost] = [shed cost/tray] x [trays/year]
8. Mushroom picking time per tray: The picking time per kg increases
with each consecutive flush. As for the yield, Jennifer's figures allow
us to compute an average picking rate for each consecutive flush
9. Annual number of hours required for picking mushroom output:
[annual picking labour] = [trays/year] x [cumulative hours/tray]
10. Annual cost of picking labour:
[annual picking cost] = [annual picking labour] x [picking pay/hour]
11. Annual sorting and packing cost:
[annual sorting & packing cost] = [annual output] x [sort. & pack.
cost-kg]
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12. Annual compost revenue:
[annual compost revenue] = [spent compost/tray] x [trays/year] x
[net compost selling price/kg]
13. Annual mushroom revenue:
[annual mushroom revenue] = [annual output] x [mushroom
revenue/kg]
14. Annual net profit, total revenue less total costs:
[annual net profit] = [annual mushroom revenue]
+ [annual compost revenue]
- [annual tray preparation cost]
- [annual shed cost]
- [annual picking cost]
- [annual sorting & packing cost]
-[annualfixedcost]
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• Yield data : yield and picking data are collected by
observations over a 5-flush cycle in two sheds:
Flushes per cycle:
1
2
3
4
5
Cycle length in days
37
44
51
58
65
9.865
8.295
7.157
6.293
5.615
Yield/tray in kg
22.973
16.080
7.103
4.278
2.610
Cumul. yield/tray in kg
22.973
39.053
46.155
50.433
53.043
Picking hours/tray
2.126
1.670
1.024
0.841
0.625
Cumul. picking hr/tray
2.126
3.796
4.820
5.661
6.286
256,486
215,682
186,078
163,621
146,000
5,892,136
8,422,914
8,588,450
8,251,800
7,744,205
Number of cycles/year
Number of trays/year
Total output/year in kg
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relevant and irrelevant costs
1. C. Brun’s section (composting and preparation):
– € 5,493,557 for raw materials (straw, manure, peat, casing
sand, sterilization agent etc.). Most raw materials were
purchased regularly, some like straw almost daily, with raw
material stocks being small and remaining fairly constant over
the entire year.
– € 267,844 for electricity for peak heating, operation of tray
filling equipment, etc.
– € 27,911 diesel fuel for yard vehicles.
– € 25,787 maintenance and repair costs on yard vehicles.
– € 123,000 depreciation on yard vehicles and equipment.
– € 86,520 yard ground and building maintenance and repair
costs.
– € 428,211 for yard workers’ wages, including vacation pay.
– € 36,400 salary of C. Brun.
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2. Michel Boite’s section (shed loading/unloading):
–
–
–
–
–
–
–
–
€ 51,333 for supplies (sterilization and cleaning chemicals for sheds).
€ 219, 102 packing material for spent compost.
€ 24,451 for diesel fuel.
€ 12,211 for vehicle maintenance and repairs; € 35,600 depreciation
on vehicles.
€ 44,898 repairs and maintenance of sheds.
€ 281,346 for replacement, repairs, and maintenance oftrays.
€ 188,231 wages, including vacation pay.
€ 31,200 salary of M. Boite.
3. Karl Scharf’s section (climate control):
–
–
–
–
–
–
€
€
€
€
€
€
37,866
34,613
12,452
33,200
55,800
68,400
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electricity for climate control (approx. const.)
for materials (approx. const.)
laboratory maintenance, cleaning, etc.
depreciation on climate control and lab equipment.
new climate control and lab equipment (sunk)
salary of K. Scharf and Tina Paille.
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30
4. Jennifer Fleurette’s section (picking, sorting, and packaging):
–
–
–
–
–
–
–
–
–
–
–
–
€ 4,071,758 for packing materials.
€ 14,881 for supplies; €6,554 for electricity.
€ 24,600 for new sorting machine (sunk)
€ 4,212 for sorting and packing equipment maintenance.
€ 14,600 depreciation on sorting and packing equipment.
€ 5,602 maintenance and repair of sorting and packing shed.
€ 18,006 for diesel fuel for trucks.
€ 5,787 for truck maintenance and repairs; € 37,800 depreciation on trucks
€ 2,044,324 wages for sorting and packing staff, including vacation pay.
€ 52,120 wages for drivers.
€ 36,000 salary of J. Fleurette.
Pickers hourly wage: € 9.20; vacation and sick pay allowance: 8% on
wages.
5. local marketing:
– € 34,800 salary of J. Marchand.
– € 26,922 travel and daily allowances (approx. cons.)
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6. other costs:
•
•
•
•
•
•
•
•
•
•
•
•
•
•
€
€
€
€
€
€
€
€
€
€
€
€
€
€
48,766 for office supplies (approx. cons.)
32,688 for new office equipment and computers (sunk)
3,435 for office building electricity.
12,111 fuel and maintenance costs for two cars (approx. cons.)
6,600 depreciation on company cars.
288,420 salaries of office staff (approx. cons.) €22,600 gardener.
8,688 office building maintenance € I90,000 depreciation
I,360,000 mortgage interest; €124,005 fire and property insurance.
301,887 accident insurance on waged workers (2.5%).
8,517 accident insurance on salaried employees (1.5%).
483,019 pension fund contribution on waged workers (4%).
34,069 pension fund contribution on salaried employees (6%).
72,000 salary of G. Mousse.
35,210 travel cost for G. Mousse (approx. cons.)
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Revenue and cost computations
flushes / cycle
Mushroom revenue
Compost revenue
1
2
3
4
5
21,211,689
30,322,491
30,918,420
29,706,48
2
27 879 138
2,154,486
1,811,727
1,563,059
1,374,414
1,226,400
11,017,629
9,264,824
7,993,182
7,028,487
6,271,573
Shed cost
1,385,840
1,165,366
1,005,414
884,071
788,863
Picking cost
5,744,763
8,625,051
9,447,924
9,757,619
9,668,026
Sorting & packing
cost
4,834,338
6,910,773
7,046,591
6,770,379
6,353,911
Fixed cost
2,914,294
2,914,294
2,914,294
2,914,294
2,914,294
3,253,910
4,074,075
3,726,045
3,108,871
965,204
617,174
0
Tray preparation cost
Net profit
(2,530,689)
Difference
-578,182
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33
growing three flushes per cycle instead of five,
provides higher profits as well as a 10% increase
in output
• this is effectively equivalent to acquiring seven
additional sheds, so it seems that the other
options can be ignored for the time being
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