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Outline

• Capacity• Economies and Diseconomies of Scale• Break Even Quantity• A Dynamic Capacity Expansion Policy

LESSON 3: CAPACITY GROWTH PLANNING

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Capacity

• Capacity – Capacity of a plant is the number of units that the plant

can produce in a given time. – The term capacity may refer to average capacity,

design capacity, and peak capacity.• Best operating level

– Best operating level is the design capacity– Design capacity provides minimum average cost

• Capacity utilization

Level Operating Best

UsedCapacity

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Economies and Diseconomies of Scale

• Economies and diseconomies of scale – A higher volume of production or production of multiple

products results in savings• Two or more products can be produced in the same

location.• Duplication of support functions can be eliminated.

These functions include information storage and retrieval systems and clerical and support staff.

– However, too many products or product lines produced at the same facility could cause an increase in costs because the various manufacturing operations may interfere with each other. This is diseconomies of scale.

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Economies and Diseconomies of Scale

250 roomhotel

Ave

rage

cos

t pe

r un

it

Best operating

level

500 roomhotel

1000 roomhotel

Best operating

levelBest

operating level

Economies of scale Diseconomies of scale

Number of Units Produced

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Cost of Capacity and Economies of Scale

Let f(y) = the cost of adding capacity y.

Economies of scale can be represented as:

Where, k is a constant and a<1.

akyyf )(

Economies/Diseconomies of Scale

0.0

50.0

100.0

150.0

200.0

250.0

300.0

0 5 10 15 20

y

f(y)

with

k=

10a=0.75a=1a=1.25

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Break-Even Quantity

If you invest in a new product, how high must sales be so that you break even? — i.e., when does your net profit 0?

Example: Suppose a company produces Q units of a product each year. The selling price is c1 per unit. Production of this product results in a fixed cost of K per year and a variable cost of c2 per unit. What is the breakeven quantity Qb per year?

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0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1 1.2

Production Quantity Q

Co

sts

an

d R

eve

nu

es

Profit

Loss

Qb

Break-Even Quantity

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A Dynamic Capacity Expansion Policy

• Frequency of capacity additions

– Too frequent: installation, training, premium for up to date technology, loss of production time

– Too infrequent: cost of capital tied to build and maintain the excess capacity

• Let

D = Annual increase in demand

x = Years between successive capacity addition

• Then, each capacity addition = xD

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Units

CapacityxD

Time

DemandD per year

Units

Time

Units

Time

Units

Incrementalexpansion

Time

Lead strategy Lag strategy

Average addition Incremental vs. one-step

One-step expansion

A Dynamic Capacity Expansion Policy

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• First, a review on the present worth analysis• See Appendix 1-A, p. 51 for details• Let

i = discount rate per period

F = future value

P = present value

n = number of periods

Then,

A Dynamic Capacity Expansion Policy

niF

P

1

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• If r is the annual discount rate, and the compounding is continuous,

• If r is the annual discount rate, the compounding is continuous, and the future value is discounted over t years,

A Dynamic Capacity Expansion Policy

rn

n

Fe

nr

FP

1lim

rtFeP

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• Let

r = Annual discount rate

f(y) = Cost of opening a plant of capacity y

• What is the present cost of adding a capacity of xD if the capacity is added x years from now and the compounding is annual?

• What is the present cost of adding a capacity of xD if the capacity is added x years from now and the compounding is semi-annual?

A Dynamic Capacity Expansion Policy

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• Let

r = Annual discount rate

f(y) = Cost of opening a plant of capacity y

• What is the present cost of adding a capacity of xD if the capacity is added x years from now and the compounding is continuous?

• What is the present cost of adding a capacity of xD if the capacity is added 2x years from now and the compounding is continuous?

A Dynamic Capacity Expansion Policy

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A Dynamic Capacity Expansion Policy

rx

a

rx

rxrx

rxrx

e

xDk

e

xDf

eexDf

xDfexdfexDf

xCxxD

1

)(

1

)(

]1)[(

)()()(

)(

2

2

years,every capacity adding of costs present The

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A Dynamic Capacity Expansion Policy

141- Figure from (ii)

error and trial (i)

by computed bemay known, is If

:following the satisfies if minimum is function cost The

rxae

rxa

xxC

rx 1

)(

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Figure 1-14

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 2 4 6

rx

Fu

nct

ion

= a

A Dynamic Capacity Expansion Policy

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Problem 1-37: Based on past experience, a chemicals firm estimates that the cost of new capacity additions obeys the law where y is measured in tons per year and f(y) in millions of dollars. Demand is growing at the rate of 3,000 tons per year and the accounting department recommends a rate of 12 percent per year for discounting future costs.

a. Determine the optimal timing of plant additions and the optimal size of each addition.

58.00205.0)( yyf

A Dynamic Capacity Expansion Policy

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b. What is the cost of each addition?

Size of each addition

Cost of each addition

A Dynamic Capacity Expansion Policy

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c. What is the present value of the cost of the next four additions? Assume an addition has just been made for the purposes of your calculation.

A Dynamic Capacity Expansion Policy

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Problem: From the past experience it is known that each doubling of the size results in an increase in cost of about 75 percent. Compute the value of a, if the cost function is of the form

akyyf )()(yf

A Dynamic Capacity Expansion Policy

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READING AND EXERCISES

Lesson 3

Reading:

Section 1.11, pp. 38-48 (4th Ed.), pp. 36-45 (5th Ed.)

Exercises:

34, 37, 38 pp. 47-48 (4th Ed.), pp. 44-45 (5th Ed.)