jhshsh - Chapter 4 - Sensitivity Analysis & The...

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Unformatted text preview: Chapter 4 - Sensitivity Analysis & The Simplex Method : S-2 ------------------------------------------------------------------------------------------ 4. See file: Prb4_3.xls Microsoft Excel Sensitivity Report Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $C$4 Value: X1 5 0 4 1E+30 2.8 $D$4 Value: X2 0 -4.666666667 2 4.666666667 1E+30 Constraints Cell Name $E$8 Used: $E$9 Used: a. b. c. d. e. Final Value 10 15 Constraint R.H. Side 0 20 1.333333333 15 Shadow Price Allowable Allowable Increase Decrease 1E+30 10 15 15 f. The objective function coefficient for X1 can decrease by 2.8 or increase by any amount without changing the optimal solution. The optimal solution is unique. None of the allowable increase or decrease values for the objective coefficients are zero. 4.67 If X2 were forced to equal 1, the optimal objective function value would be approximately 20 - 4.67 = 15.33. An increase of 10 in the RHS value of the second constraint is within the allowable range of increase for the shadow price of this constraint. Therefore, if the RHS for the second constraint increases from 15 to 25 the new objective function value would be approximately 20 + 1.33 10=33.33. The new reduced cost for X2 would be 2 - (4 0 + 1 1.333) = 0.67. Therefore, it would be profitable to increase the value of X2 and the current solution would no longer be optimal. MIN: 260X13 + 220X14 + 290X15 + 230X23 + 240X24 + 310X25 S.T.: X13 + X14 + X15 d 20 X23 + X24 + X25 d 20 X13 + X23 t 10 X14 + X24 t 15 X15 + X25 t 10 Xij t 0 See file: Prb4_6.xls See below 5. a. b. c. Chapter 4 - Sensitivity Analysis & The Simplex Method : S-3 ------------------------------------------------------------------------------------------ Microsoft Excel Sensitivity Report Adjustable Cells Cell $C$10 $D$10 $E$10 $C$11 $D$11 $E$11 Final Reduced Objective Allowable Allowable Name Value Cost Coefficient Increase Decrease Eustis Miami 0 50 260 1E+30 50 Eustis Orlando 10 0 220 20 0 Eustis Tallahassee 10 0 290 0 310 Clermont Miami 10 0 230 50 230 Clermont Orlando 5 0 240 0 20 Clermont Tallahassee 0 0 310 1E+30 0 Constraints Cell $C$12 $D$12 $E$12 $F$10 $F$11 d. e. No. No. Miami 0 10 10 10 $8,600 Orlando 15 0 15 15 Tallahassee 5 5 10 10 Capacity Used Available 20 20 15 20 Name Shipped Miami Shipped Orlando Shipped Tallahassee Eustis Used Clermont Used Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease 10 230 10 5 10 15 240 15 5 5 10 310 10 5 5 20 -20 20 5 5 15 0 20 1E+30 5 Eustis Clermont Shipped Demand Total Cost f. g. h. The solution would not change. The current solution uses only 15 of the 20 tons of capacity available at Clermont. Reducing the capacity in Eustis would increase costs by $20 per unit decrease yielding an objective function value of $8,600+205 = $8,700 Every additional ton of concentrate unit shipped from Eustis to Miami would increase costs by $50. 19. See file: Prb4_10.xls Microsoft Excel Sensitivity Report Adjustable Cells Final Cell $C$5 $D$5 Constraints Final Cell $E$9 $E$10 $E$11 Name Cutting Used Sanding Used Finishing Used Value 40 40 50 Shadow Price 350 300 0 Constraint R.H. Side 40 40 60 Allowable Increase 40 6.666667 1E+30 Allowable Decrease 13.33333 20 10 Name Qty Doors Qty Windows Value 20 40 Reduced Cost 0 0 Objective Coefficient 500 400 Allowable Increase 300 350 Allowable Decrease 233.3333 150 a. b. c. d. No, the objective coefficient on doors could increase to $800 without changing the optimal solution. Yes, the objective coefficient on windows can only decrease to $250 without changing the optimal solution. The shadow price (marginal value) is $0 because there is a surplus of this resource. $350*20=$7000. 26. See file: Prb4_9.xls Microsoft Excel Sensitivity Report Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$19 Acres to Plant Watermelons 60 0 256 28.5 66.33333333 $C$19 Acres to Plant Cantaloupes 40 0 284.5 99.5 28.5 Constraints Cell Name $F$11 Water Used per hour) $F$19 Acres Planted per hour) a. b. c. d. Final Shadow Constraint Allowable Value Price R.H. Side Increase 6,000 1 6000 1500 100 199 100 20 Allowable Decrease 1000 20 The profit per acre for watermelons can drop by $66.33. The profit per acre of cantaloupes would have to increase by $99.50. 60/66.33 + 50/99.5 = 1.407 > 100%. Therefore we cannot guarantee that the solution is still optimal. The farmer should lease all 20 acres and be willing to pay up to $199 per acre (assuming he can use his own water on the additional 20 acres). ...
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This note was uploaded on 05/29/2010 for the course NGD 254 taught by Professor Kajue during the Spring '05 term at École Normale Supérieure.

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