15.053 February 24, 2005Goals of today’s lecture on Sensitivity AnalysisOn Varying the RHSShadow price (assume maximization).Quick exercisesWhy are shadow prices valid in intervals?The Excel Sensitivity ReportShadow prices of inequality constraintsShadow prices and managerial interpretations.Excel Sensitivity ReportSome Managerial QuestionsBrief SummaryVarying two RHS coefficients at onceNext: Varying the Cost CoefficientsExcel Spreadsheets: Varying the Cost coefficientsAllowable increases and decreasesTreating shadow prices as real prices:pricing outThe impact of producing 100 cases of Champ 2Reduced Costs (maximization)Suppose that the number of production hours of Champ2 were 8.On pricing outExcel Spreadsheets: The reduced costsReduced costs for basic variables are always 0. Pricing Out of x1.Pricing Out of x2Exercise Price Out x3Managerial InterpretationsShadow Prices for “” constraints are non-negative.Shadow Prices for “” constraints are nonpositive.Requiring at least u units of champagne glasses: the shadow price as a function of uSummary slide115.053 February 24, 2005 z Sensitivity Analysis and shadow pricesSuggestion:Please try to complete at least 2/3 of the homework set by next Thursday2Goals of today’s lecture on Sensitivity Analysisz Changes in RHS– motivation from 2 dimensions– shadow prices, and their intervalsz Changes in the cost coefficientsz Treating shadow prices as real prices– pricing out and reduced costs3The DTC ProblemK = number of slingshot kits manufactured (in 10s)S = number of stone shields manufactured (in 10s)Gathering time:Smoothing time:Delivery time:Shield demand:Slingshot demand:K,S ≥ 0Non-negativity:2 K + 3 S ≤ 10K + 2 S ≤6K + S ≤ 5S ≤3K ≤4z = 3 K + 5 S (in 10s)Maximize Profit4KS1234123Isoprofitgathering time5KS123IsoprofitlineK + 2S = 712346On Varying the RHSz Suppose we vary the amount of smoothing time from 0 to 10, and keep all other coefficients the same.z How will the profit change?z Let G(t) be the profit for DTC if smoothing time is t and all other data stays the same.7Profit as a function of Smoothing Time0246810121416180246810Smoothing TimeOptimum ProfitSlope = 2.5Slope = 0Slope = 1Slope = 35 1/34 6.508Shadow price (assume maximization).z The shadow price of the i-th constraint is the increase in the optimum objective value per unit increase of the RHS of the i-th constraint.z The shadow price is the derivative! It is defined based on the math formulation of the LP.z The shadow price of smoothing time is G’(6) = 1.z The shadow price is valid when the RHS changes within an interval: in this case it is valid for smoothing time from 5 1/3 to 6.5, if all other data remains the same.9Quick exercisesz Using just the shadow price information and the range information, answer the following.– What is the optimum profit if smoothing time changes from 6 to 6.3?– What is the optimum profit if smoothing time changes from 6 to 5.7?– What is the optimum profit if smoothing time changes from 6 to 5?10Why are shadow prices valid in intervals?z The shadow price is valid in the interval in which “structure of the optimal solution” stays the same.z In 2 dimensions, this means that the shadow price is valid when the RHS is in an interval in which the same “corner point” solution remain z In multiple dimensions, it means that the shadow price is valid when the RHS changes so long as the optimal basic variables remain optimal.11KS1234123Isoprofitgathering timeThe shadow price of the “constraint K ≤4” is 0.12The Excel Sensitivity ReportGathering 10 1 1 1Smoothing 6 1 0.5 0.67Delivery 5 0 1E+30 1Slingshot 4 0 1E+30 2Shield 3 0 1E+30 1Name RHS Price Increase DecreaseShadow Allowable Allowable13Shadow prices of inequality constraintsz If an inequality constraint is not binding for the optimal solution, then its shadow price is 0.14Shadow prices and managerial interpretations.z Shadow prices are derivatives.z They have a mathematical interpretation.– If the RHS of smoothing goes from 6 to 6 + ∆, then the objective value increases by ∆, for-2/3 <= ∆ <= .5z And the shadow prices are useful to managers in sensitivity analysis.z Rule: Get the mathematical interpretation correct first. Then interpret it managerially.15Glass Example (from AMP)A manufacturer of glasses needs to determine the optimal mix of 6-oz juice glasses, 10 oz juice glasses, and champagne glasses.The constraints are as follows:z production capacity is limited to 60 hoursz storage space is limited to 15,000 sq. ft.z The total demand for 6-oz juice glasses is at most 800 cases.Using data obtained by his assistant, the manufacturer formulated the linear program on the next slide.16Glass Examplex1= # of cases of 6-oz juice glasses (in 100s)x2 = # of cases of 10-oz cocktail glasses (in 100s)x3= # of cases of champagne glasses (in 100s)max 5 x1+ 4.5 x2+ 6 x3($100s)s.t 6 x1+ 5 x2+ 8 x3≤ 60 (prod. cap. in hrs)10 x1+ 20 x2+ 10 x3≤ 150 (wareh. cap. in 100s of ft2)x1≤ 8 (6-0z. glass dem. in 100s of cases)x1≥ 0, x2≥ 0, x3≥ 017Excel Sensitivity ReportConstraint Shadow Allowable AllowableName R.H. Side Price Increase DecreaseProd. Cap.60 11/14 5 1/2 22 1/2W-house Cap.150 1/35 90 22Juice Cap.8 0 1E+30 1 4/7optimal objective value is z = 51 3/7 ($100s)optimal solution: x1= 6 3/7 x2= 4 2/7 x3= 018Some Managerial QuestionsQ. How much should you be willing to pay for two extra hours of production?A. The optimal objective value will increase by 2 ×11/14 = 11/7. This is $1,100/7 = $157 1/7Q. How much would it cost you if production capacity decreased by 20 hours? By 25 hours?A. Q. How much would you be willing to pay for 50 extra square feet of warehouse space?A.19Brief Summaryz The shadow price is the unit change in the optimal objective value per unit change in the RHS.z Shadow prices usually (but not always) have economic interpretations that are managerially useful.z Shadow prices are valid in an interval, which is provided by the Excel Sensitivity Report. This assumes that one RHS coefficient changes, but all others stay the same.z Excel provides shadow prices for our LPs**20Varying two RHS coefficients at onceQ. Suppose that you are offered a package deal of 2 hours of labor plus 350 square feet of warehouse space. What would this deal be worth to you?A. Determine the value of 2