Linear ProgrammingSensitivity of the Right Hand Side Coefficients

Sensitivity of RHS CoefficientsRHS coefficients usually give some maximum limit for a resource or some minimum requirement that must be met.Changes to the RHS can happen when extra units of the resource become available or when some of the original resource becomes unavailable.Or the minimum requirement is loosened (made less) or strengthened (made greater).Extra units may be available for a price. The question becomes how much would an extra unit add to the value of the objective function, that is, what is the most we would be willing to pay for extra units of the resource?

Finding the Optimal Point - ReviewX1, X2 0

Optimal Point WithOne Extra Unit of PlasticX1, X2 0Max 8X1 + 5X2s.t.Still determined by Plastic and Time constraintsShadow Price (for Plastic)

4363.40 4360 =(new profit) - (old profit)$3.40

Shadow PricesThe shadow price for a constraint is the amount the objective function value will change given:1 additional unit on the RHS of the constraintNo other changesThis shadow price is valid as long as the same constraints (including the one whose RHS is changing) determine the optimal point.In this case plastic and production timeIt can be shown that if the RHS for plastic were 1002 the profit would increase another $3.40 to $4366.80.It can also be shown that if the RHS for plastic were 999 the profit would decrease by $3.40 to $4356.60.

Allowable Increase andAllowable Decreaseof a RHS ValueThe shadow prices remain valid as long as the same constraints (called the binding constraints) determine the optimal point.When the RHS of the constraint is increased or decreased to the point that another constraint replaces one of the binding constraints to determine the optimal point a new shadow price becomes valid for the constraint.The amount the RHS can increase or decrease before another constraint becomes one of the binding constraints is what Excel calls the Allowable Increase and the Allowable Decrease respectively.

Increasing the Right Handside for PlasticX1, X2 0Max 8X1 + 5X2s.t.

Further Increasingthe Right Hand Side for PlasticX1, X2 0Max 8X1 + 5X2s.t.The shadow prices will now CHANGE

Decreasing the RHS for PlasticX1, X2 0Max 8X1 + 5X2s.t.2X1 + 1X2 (Plastic)Optimal solution determined by Plastic and Time Constraintsand by X2 axis!

Further Decreasingthe RHS for PlasticX1, X2 0Max 8X1 + 5X2s.t.2X1 + 1X2 (Plastic)The shadow prices will now CHANGE

Comparison With ExcelHere is the printout out of the sensitivity analysis dealing with the objective RHS coefficients for the original Galaxy Industries problem.Range of Feasibility is the range of values that an RHS coefficient can assume without changing the shadow prices as long as no other changes are made.

Exact Meaning of Shadow PricesA shadow price always means the amount the objective function will change given a one unit increase in the RHS value of a constraint.But does this mean that this is the value (the most you would be willing to pay) for an extra unit? The answer depends on how the objective function coefficients were calculated.If the objective function coefficients did not take the value of the resource into consideration, these are sunk costs.Shadow price = the value of an extra unit of the resource.If the objective function coefficients did take the value of the resource into consideration, these are included costs.Shadow price = a premium above the current price of the item that one would be willing to pay for an extra unit.

EXAMPLESuppose the $8 objective function coefficient for dozens of Space Rays and the $5 objective function coefficient for dozens of Zappers were calculated as follows:

DOZ. DOZ.SPACE RAYS ZAPPERSSelling Price $24 $26Costs Plastic ($3/lb) $ 6 (2 lbs.) $ 3 (1 lb.) Other Variable Costs $10 $18 =========== ==========Total Profit Per Dozen $ 8 $ 5The $3.40 shadow price for plasticmeans we would be willing to payup to $3.40 more than the currentprice of $3 per pound (that is up to$6.40/ lb.) for extra plastic.Production time is a sunk costIt is not included in the objectivefunction coefficient calculation. The $0.40 shadow price is thevalue of an extra minute of production time.

Complementary SlacknessComplementary slackness also holds for RHS values. This property for RHS values states:

Again, it can happen, that both are 0.Complementary SlacknessFor RHS Coefficients

For each constraint, either the slack(difference between RHS LHS) is 0 or its shadow price will be 0.Plastic: Shadow Price 0; Slack = 1000-1000 = 0Time: Shadow Price 0; Slack = 2400-2400 = 0Prod. Limit: Slack = 700-680 0; Shadow Price = 0Prod. Mix: Slack = 350-(-40) 0; Shadow Price = 0

ReviewShadow priceFound by subtracting the original objective function value from the objective function value with one more unit of the resource on the RHSMeaningIncluded CostSunk CostRange of FeasibilityRange of RHS value in which shadow price does not changeThe same constraints determine the optimal solution in the range of feasibilityComplementary SlacknessEither the slack is 0 or the shadow price is 0