graduate program in business information systems linear programming: sensitivity analysis and...
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Graduate Program in Business Information Systems
Linear Programming:Sensitivity Analysis and Duality
Aslı Sencer
BIS 517- Aslı Sencer 2
Shadow Prices and Opportunity Costs
LP solution answers the tactical question, i.e., how much to produce
Suppose the focus is on resources rather than the products, i.e., Each resource has a shadow price that
reflects the true impact of scarcity. To find these, we need a transformation
of the primal problem which is referred to as the dual problem.
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Ex:Redwood Furniture Product Mix problem (revisited)
: number of tables produced in a period
: number of chairs produced in a period
labor)(110105
wood)(3002030
ct
ct
XX
XX
cXMaximize 86XProfit t
)itynonnegativ(0, ct XX
tX
cX
Optimal Solution: Xt=4 tables, Xc=9 ChairsProfit*=$96
Optimal Solution: Xt=4, Xc=9Profit*=$96
Resource used
Resource Available
Resource Left
Wood 300 300 0
Labor 110 110 0
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Increasing the Available Resources
What happens if the available wood is increased by 1 ft?
Need to resolve LP with the new constraint
which yields XT*=4.05, XC*=8.975, P*=$96.10
wood)(3012030 ct XX
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Graphical Representation
Constraint 1
Constraint 2
11
15
10 22
(4,9)
Xt
Xc
NEW OPTIMAL SOLUTION
Xt=4.05,Xc=8.975P=$96.10
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Shadow Price and Opportunity Cost
Optimal profit in the new problem is $96.10-$96=$0.1 greater!
SHADOW PRICESHADOW PRICE
Shadow price is the marginal value of a resource. Shadow price is the opportunity cost of not
increasing the resource.
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Question?
Question 1:How much should the DM be willing to pay for a unit increase in wood resource?
Answer:Infact, the DM should not pay more than $0.1 for a unit increase in the current wood capacity of $300
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Question?
Question 2:If the wood resource is to be increased by 100ft (i.e., it will be 400 ft now), what will be the new optimal profit?
Answer:Can not tell directly! Shadow prices are valid only for certain ranges of change in the available resources.
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Question? Why do you think it is so?
Constraint 1
Constraint 2
11
15
10 22
(4,9)
Xt
Xc
NEW OPTIMUM
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The Dual Problem: Technical Approach
0,
)(81020
)(6530
110300
LW
Lw
Lw
Lw
UU
ChairUU
TableUU
UUCMin
0,
U110105
U3002030
86XP
L
W
t
ct
ct
ct
c
XX
XX
XX
XMax
PRIMAL PROBLEM DUAL PROBLEM
For any primal solution Xt, Xc (not necessarily optimal), there is a corresponding dual solution Uw, UL.
If the primal solution is not optimal, then dual solution is infeasible!
If they are both feasible then both solutions are optimal and P*=C*!
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Dual Problem: Economical Interpretation
Primal problem: Production Manager’s perspective:Optimize resource allocation to maximize Total Profit.
Dual problem:Economist’s perspective:Optimize resource allocation to minimize aggregate value of increasing any resource by one more unit.
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Dual formulation
product)each for ()(
subject to
)(
return marginalproduct
costy opportunit marginalproduct
profitunitUunitpertrequiremen
UquantityavailableCMin
resourceresourcesall
resourceresourcesall
If for any productMarginal opportunity cost > Marginal return Do not produce
Marginal opportunity cost < Marginal return Produce more Marginal opportunity cost = Marginal return Current production
level is optimal
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Sensitivity Analysis Using Excel Solver
Adjustable Cells
Cell NameFinalValue
ReducedCost
ObjectiveCoefficient
AllowableIncrease
AllowableDecrease
$C$9 Xt 4 0 6 6 2
$D$9 Xc 9 0 8 4 4
Constraints
Cell NameFinalValue
ShadowPrice
CurrentR.H. Side
AllowableIncrease
AllowableDecrease
$G$5 <= LHS 300 0,1 300 360 80
$G$6 <= LHS 110 0,6 110 40 60
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Questions?
If available wood is 310ft, what is the new optimal solution? 310-300=10ft increase is required From the sensitivity analysis allowable increase is
360, so shadow prices are valid! Pnew*=96+10*0.1=$97. The optimal solution is
found by solving
If the available wood is 700ft what is the new optimal solution? 700>300+360=660, so a new solution will exist. We
need to resolve it with new constraint!
labor)(110105
wood)(3102030
ct
ct
XX
XX
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Questions?
If the unit profit of a table is decreased to $5, new optimum? Current value is 6, thus $1 decrease is required. In the sensitivity table, allowable decrease is 2. So current solution is still optimal. Xt=4, Xc=9 and P=5(4)+8(9)=$92
Would you hire an extra labor for 10 hrs at a total cost of $5? In the sensitivity table, allowable increase is 40,
so dual prices are valid. increase in optimal profit=0.6(10)=$6.
Net saving=$6-$5=$1>0, so hire labor!
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Questions?
How is the optimal solution found in this case? The optimal solution is found by solving
labor)(120105
wood)(3002030
ct
ct
XX
XX
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Pricing new products using shadow prices
Making a deskdesk would divert resources from tables and chairs, and fewer would be made.
Redwood evaluates new products: Bench having profit of $7, needing 25 board feet of
wood and 7 hours of labor. Planter box having profit of $2, needing 10 board
feet of wood and 2 hours of labor. The opportunity costs for one of each are:
Bench: $.10(25) + .60(7) = $6.70 (< $7). Make it, because doing so increases P by $.30/unit.
Planter box: $.10(10) + .60(2) = $2.40 (> $2). Do not make. Resources are too valuable.