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Chapter 10 The primal linear program solution answers the tactical question when it tells us how much to produce. But the dual can have far greater impact because it addresses strategical issues regarding the structure of the business itself. 1 Sensitivity Analysis and Duality in Linear Programming Shadow Prices and Opportunity Costs The Redwood furniture problem involves XT and XC denote the number of tables and chairs to be made. Maximize P = 6XT + 8XC Subject to: 30XT + 20XC < 300 (wood) 5XT + 10XC < 110 (labor) where XT, XC > 0 What is the effect of one additional unit of the respective resources? It will allow more product to be made, increasing profit. That is the resource’s marginal value or shadow price. The marginal value of a resource can be found by changing the available level and solving the revised problem. There is a faster way to find the marginal value. 2 Knowing levels for marginal values enhances planning capabilities. They can be used to “price” out a new product. Consider the wood used in making furniture. Raising the available level from 300 to 301 board feet will change the feasible solution region, as shown on the following slide. It will increase profit by $.10. Shadow Prices and Opportunity Costs 3 The Dual Linear Program The original problem is here referred to as the primal linear program. The dual linear program expresses the problem with a resource orientation. Let UW and UL denote the marginal values of wood and labor. The objective is to Minimize C = 300UW + 110UL Subject to: 30UW + 5UL > 6 (table) 20UW + 10UL > 8 (chair) where UW, UL > 0 The total value of available resources is minimized. The constraint left-hand sides express the opportunity cost of making one unit of the respective product. It is an opportunity cost because making the product may divert resources from their optimal utilization, and theUs measure their value. Each product is required to have an opportunity cost > its unit profit. 4 The dual has a column (variable) for each primal row (constraint). Primal right-hand sides are dual objective coefficients and vice versa. Redwood Furniture Problem: Graphical Solution of Dual 5 Meaning of Dual Solution The values UW = .10 and UL = .60 tell us that additional wood (acquired at present costs) would raise profit by $.10. Similarly, an additional hour of labor raises P by $.60. Redwood would benefit by more resources. They would even pay a small premium to get it (not more than the respective U.) The Us are shadow prices for resource in pricing out a new product, such a desk. Making a desk would divert resources from tables and chairs, and fewer would be made. 6 Evaluating New Products Using the Dual 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. 7 Sensitivity Limits for Right-Hand Sides QuickQuant provides the following report that lists for each constraint the optimal level of the applicable dual variable. wood labor 8 Included are the sensitivity limits for the right-hand sides. The dual values will be the same for any available quantity of wood and labor falling within those limits. Sensitivity Limits for Objective Coefficients QuickQuant also gives sensitivity limits for objective coefficients (unit profits, costs). The present primal optimal quantities will be exactly the same for any level of profit falling with those limits. 9 For example, if the table and chair profits are both $10, only P changes, to $10(4) + 10(9) = $130. Post-Optimality Analysis Perhaps the most important role of the dual is helping the decision maker improve the business itself. That can be done with a post-optimality analysis in which a series of changes are made to the problem structure ( and hence business environment). Find economic bottlenecks, get more resources: Authorize overtime or train new employees. Install faster machines or expand plant. Target marketing, expand profitable products. Adjust pricing. Solve New LPs after each possible change. New Us will suggest further possible improvements. 10 Solver’s Sensitivity Report Solver’s Sensitivity Report yields the values of the: dual variables allowable increases and decreases for the right-hand sides (upper and lower limits). 11 allowable increases and decreases for the coefficients of the objective function (upper and lower limits). Solver’s Sensitivity Report To get Solver’s Sensitivity Report, highlight Sensitivity Report in the Report box of the Solver Results dialog box before clicking the OK button. 12 Sensitivity Report for Swatville Sluggers (Figure 10-3 ) 3. Allowable increases and decreases for right-hand sides. 2. Dual slack or surplus variables (in the Reduced Cost column). Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $D$17 X30 80.46 0.000 12.05 3.68 0.857 $E$17 X32 0 0.000 11.75 0.50 2.537 $F$17 X34 80.46 0.000 13.19 3.57 0.516 $G$17 X36 0 -1.269 12.64 1.27 1E+30 $H$17 X38 0.000 2.01 1.94 4. Allowable increases and decreases80.46 for coefficients of the 15.48 objective function. $I$17 X40 0 -1.202 15.18 1.20 1E+30 Constraints Final Shadow Constraint Allowable Allowable 1. Shadow Cell Name Value Price R.H. Side Increase Decrease prices. $N$6 Wood 8206.90 0.000 12000 1E+30 3793.10 $N$7 Lathe 2655.17 0.000 6000 1E+30 3344.83 $N$8 Finishing 7000.00 0.468 7000 3235.29 7000.00 $N$9 Boxes 241.38 0.000 1000 1E+30 758.62 $N$10 Stain 563.22 0.000 1500 1E+30 936.78 $N$11 Varnish 1689.66 0.000 3000 1E+30 1310.34 $N$12 Production NB: Multiple 0.00 0.252solutions 0exist to the 0dual. 164.71 optimal $N$13 < 0.00solution 0.635 0 is different 0 250.00 This dual found by Excel $N$14 Restrictions than the 0.00 0.601given in the 0 primal. 0 114.75 dual solution 13 Sensitivity Report for Redwood Furniture (Figure 10-6) 2. Dual slack or surplus variables (in the Reduced Cost column). 3. Allowable increases and decreases for righthand sides. These give lower and upper limits. Adjustable Cells 1. Shadow prices. Cell Name $B$9 XT $C$9 XC Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease 4 0 6 6 2 9 0 8 4 4 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $G$5 Wood used 300 0.1 300 360 80 $G$6 Labor used 110 0.6 110 40 60 14 4. Allowable increases and decreases for the coefficients of the objective function. These give the lower and upper limits.