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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
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