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SC Design

Facility Location under UncertaintyChapter 6

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A tree representation of uncertainty

One way to represent Uncertainty is a binomial tree Up by 1 down by -1 move with equal probability

),0( 2TNormal

T steps

1)5.0()1()5.0()1( 222

<Show Applet balldrop.htm>

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Decision tree

– One column of nodes for each time period– Each node corresponds to a future state

» What is in a state? Price, demand, inflation, exchange rate, your OPRE 6366 grade

– Each path corresponds to an evolution of the states into the future

– Transition from one node to another determined by probabilities

– Evaluate the cost of a path starting from period T and work backwards in time to period 0.

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Evaluating Facility Investments: AM Tires. Section 6.5 of Chopra.

Dedicated Plant Flexible Plant Plant Fixed Cost Variable Cost Fixed Cost Variable Cost

US 100,000

$1 M /year.

$15 /tire

$1.1 M /year

$15 /tire

Mexico 50,000

4 M pesos / year

110 pesos /tire

4.4 M pesos /year

110 pesos /tire

Now

U.S. Demand = 100,000; Mexico demand = 50,000. Demand is not to be met always. But selling more increases profit.

1US$ = 9 pesos.Sale price $30 in US and 240 pesos in Mexico.

FutureDemand goes up or down by 20 percent with probability 0.5 andExchange rate goes up or down by 25 per cent with probability 0.5.

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AM Tires

DU=100DM=50

E=9

Period 0 Period 1 Period 2

DU=120DM = 60E=11.25

DU=120DM = 60E=6.75

DU=120DM = 40E=11.25

DU=120DM = 40E=6.75

DU=80DM = 60E=11.25

DU=80DM = 60E=6.75

DU=80DM = 40E=11.25

DU=80DM = 40E=6.75

DU=144DM = 72E=14.06

DU=144DM = 72E=8.44

DU=144DM = 48E=14.06

DU=144DM = 48E=8.44

DU=96DM = 72E=14.06

DU=96DM = 72E=8.44

DU=96DM = 48E=14.06

DU=96DM = 48E=8.44

How many states in period 2? Consider US demand 4 or 3 states Consider the rest also 4x4x4 or 3x3x3

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AM Tires

Four possible capacity configurations:•Both dedicated•Both flexible•U.S. flexible, Mexico dedicated•U.S. dedicated, Mexico flexible

Consider the both flexible configurationFor each node solve the demand allocation model.

Plants Markets

U.S.

Mexico

U.S.

Mexico

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AM Tires in period 2: Demand Allocation for DUS = 144; DMex = 72, E = 14.06Source

i Destination

j Variable

cost Shipping

cost E Sale price Margin($)

mij

U.S. U.S. $15 0 14.06 $30 $15 U.S. Mexico $15 $1 14.06 240 pesos $1.1

Mexico U.S. 110 pesos $1 14.06 $30 $21.2 Mexico Mexico 110 pesos 0 14.06 240 pesos $9.2

0

such that

xmMax

2

1

2

1

2

1

2

1jijij

ij

ij

ij

ji

ij

i

x

Kx

Dx Compare this formulation to the Transportation problem.We maximize the profit now.

1.1=240/14.06-15-121.2=30-110/14.06-19.2=(240-110)/14.06

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AM Tires: Demand Allocation for DU = 144; DM = 72, E = 14.06; Cheap Peso

Plants Markets

U.S.

Mexico

U.S.

Mexico

100K; $15

44K; $21.2

6K; $9.2

Profit =Revenue-Cost

US Production’s contribution=100,000*15-1,100,000=$400,000Mex Production’s contribution=44,000*21.2+6000*9.2-4,400,000/14.06=$675,055Profit(DU = 144; DM = 72, E = 14.06; Period 2; Both flexible)=$1,075,055

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AM Tires: Demand Allocation for DU = 144; DM = 72, E = 8.44; Expensive Peso

Plants Markets

U.S.

Mexico

U.S.

Mexico

100K; $15

44K; $16

6K; $15.4

US Production’s contribution=100,000*15-1,100,000=$400,000Mex Production’s contribution=44,000*16+6000*15.4-4,400,000/8.44 =704000+92400-521327=$275,073Profit(DU = 144; DM = 72, E = 8.44; Period 2; Both flexible)=$675,073

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AM Tires: Demand Allocation for DU = 144; DM = 72, E = 5.06; Very Expensive Peso

Plants Markets

U.S.

Mexico

U.S.

Mexico

78K; $15

22K; $31.4

50K; $25.7

US Production’s contribution=78000*15+22000*31.4-1,100,000=$760,800Mex Production’s contribution=50000*25.7-4,400,000/5.06=$415,435Profit(DU = 144; DM = 72, E = 8.44; Period 2; Both flexible)=$1,176,235

Cheap Peso profit=$1,075K; Expensive Peso profit=$675K; Very Expensive Peso profit=$1,176K

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Facility Decision at AM Tires

Plant Configuration United States Mexico

NPV

Dedicated Dedicated $1,629,319 Flexible Dedicated $1,514,322

Dedicated Flexible $1,722,447 Flexible Flexible $1,529,758

Make profit computations for the first year nodes one by one:Compute the profit for a node and add to that(0.9)(1/8)(Sum of the profits of all 8 nodes

connected to the current one)

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Capacity Investment Strategies

Single sourcing is risky Hedging Strategy

– Risk management? » Too much capacity or too little capacity» E.g. 200 leading financial services companies are examined from 1997-

2002. Every other company struck at least once by a risky event. Source: Running with Risk. The McKinsey Quarterly. No.4. 2003.

» Managers unfamiliar with risk often focus on relatively simple accounting metrics as net income, earnings per share, return on investment, etc.

– Match revenue and cost exposure Flexible Strategy

– Excess total capacity in multiple plants– Flexible technologies

More will be said in aggregate planning chapter

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Summary

Decisions under uncertainty– Location– Flexibility

Decision trees