Presentation on theme: "Sensitivity of the Right Hand Side Coefficients"— Presentation transcript:
1 Sensitivity of the Right Hand Side Coefficients Linear ProgrammingSensitivity of the Right Hand Side Coefficients
2 Sensitivity of RHS Coefficients RHS 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?”
4 Optimal Point With One Extra Unit of Plastic 1000900800700600500400300200100X1Shadow Price (for Plastic)– 4360 =(new profit) (old profit)$3.40Max 8X1 + 5X2s.t.New OPTIMAL POINT (320.8,359.4)=10012X1 + 1X2 ≤ (Plastic)3X1 + 4X2 ≤ (Time)1X1 + 1X2 ≤(Limit)1X1 - 1X2 ≤ (Mix)X1, X2 ≥ 0Still determined by Plastic and Time constraints
5 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 $It can also be shown that if the RHS for plastic were 999 the profit would decrease by $3.40 to $
6 Allowable Increase and Allowable Decrease of a RHS Value The 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.
7 Increasing the Right Handside for Plastic X21000900800700600500400300200100X1Max 8X1 + 5X2s.t.1030106011002X1 + 1X2 ≤ (Plastic)3X1 + 4X2 ≤ (Time)1X1 + 1X2 ≤(Limit)1X1 - 1X2 ≤ (Mix)X1, X2 ≥ 0Plastic and Time constraints still determine the optimal solution to this point.
8 Further Increasing the Right Hand Side for Plastic X21000900800700600500400300200100X1Max 8X1 + 5X2s.t.10301060110011022X1 + 1X2 ≤ (Plastic)3X1 + 4X2 ≤ (Time)1X1 + 1X2 ≤(Limit)1X1 - 1X2 ≤ (Mix)X1, X2 ≥ 0Further increases to the RHS side of plastic have now made the plastic and the Limit constraints as the ones that determine the optimal point.The shadow prices will now CHANGE
9 Decreasing the RHS for Plastic X21000900800700600500400300200100X1Max 8X1 + 5X2s.t.60070085010002X1 + 1X2 ≤ (Plastic)3X1 + 4X2 ≤ (Time)1X1 + 1X2 ≤(Limit) Redundant1X1 - 1X2 ≤ (Mix)X1, X2 ≥ 0Optimal solution determined by Plastic and Time Constraintsand by X2 axis!
10 Further Decreasing the RHS for Plastic X21000900800700600500400300200100X1Max 8X1 + 5X2s.t.59060070085010002X1 + 1X2 ≤ (Plastic)3X1 + 4X2 ≤ (Time)1X1 + 1X2 ≤(Limit) Redundant1X1 - 1X2 ≤ (Mix)X1, X2 ≥ 0Optimal Point now determined by the plastic constraint and the X2-axisThe shadow prices will now CHANGE
11 Comparison With ExcelHere is the printout out of the sensitivity analysis dealing with the objective RHS coefficients for the original Galaxy Industries problem.Shadow Price for each constraintRange of Feasibility for RHS11000 – 400 Range of Feasibility for RHS2Range of Feasibility for RHS3700 – 20 ∞ ∞Range of Feasibility for RHS4350 – 390 ∞ ∞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.
12 Exact Meaning of Shadow Prices A 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.
13 Plastic is an included cost 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 $ $26CostsPlastic ($3/lb) $ 6 (2 lbs.) $ 3 (1 lb.)Other Variable Costs $ $18=========== ==========Total Profit Per Dozen $ $ 5Plastic is an included costProduction time is a sunk costThe $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.It is not included in the objectivefunction coefficient calculation.The $0.40 shadow price is thevalue of an extra minute ofproduction time.
14 Complementary Slackness Complementary slackness also holds for RHS values. This property for RHS values states:Again, it can happen, that both are 0.Complementary SlacknessFor RHS CoefficientsFor each constraint, either the slack(difference between RHS – LHS) is 0 or its shadow price will be 0.Plastic: Shadow Price ≠ 0; Slack = = 0Time: Shadow Price ≠ 0; Slack = = 0Prod. Limit: Slack = ≠ 0; Shadow Price = 0Prod. Mix: Slack = 350-(-40) ≠ 0; Shadow Price = 0
15 Review Shadow price Range of Feasibility Complementary Slackness Found 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