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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.

Sensitivity Analysis and Duality in

Linear Programming

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Shadow Prices andOpportunity 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.

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.

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Shadow Prices andOpportunity Costs

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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.

The dual has a column (variable) for each primal row (constraint). Primal right-hand sides are dual objective coefficients and vice versa.

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Redwood Furniture Problem:Graphical Solution of Dual

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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.

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Evaluating New ProductsUsing 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.

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Sensitivity Limits for Right-Hand Sides

wood wood

labor labor

QuickQuant provides the following report that lists for each constraint the optimal level of the applicable dual variable.

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.

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Sensitivity Limitsfor 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. For example, if the table and chair profits are

both $10, only P changes, to $10(4) + 10(9) = $130.

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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.

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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).

allowable increases and decreases for the coefficients of the objective function

(upper and lower limits).

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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.

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Sensitivity Report forSwatville Sluggers (Figure 10-3 )

Adjustable CellsFinal 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$17 X38 80.46 0.000 15.48 2.01 1.94$I$17 X40 0 -1.202 15.18 1.20 1E+30

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$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$12 Production 0.00 0.252 0 0 164.71$N$13 < 0.00 0.635 0 0 250.00$N$14 Restrictions 0.00 0.601 0 0 114.75

1. Shadow prices.

1. Shadow prices.

2. Dual slack or surplus variables (in the Reduced Cost column).

2. Dual slack or surplus variables (in the Reduced Cost column).

3. Allowable increases and decreases for right-hand sides.3. Allowable increases and decreases for right-hand sides.

NB: Multiple optimal solutions exist to the dual. This dual solution found by Excel is different than the dual solution given in the primal.

NB: Multiple optimal solutions exist to the dual. This dual solution found by Excel is different than the dual solution given in the primal.

4. Allowable increases and decreases for coefficients of the objective function.4. Allowable increases and decreases for coefficients of the objective function.

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Sensitivity Report forRedwood Furniture (Figure 10-6)

Adjustable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$B$9 XT 4 0 6 6 2$C$9 XC 9 0 8 4 4

ConstraintsFinal 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

1. Shadow prices.

1. Shadow prices.

2. Dual slack or surplus variables (in the Reduced Cost column).

2. Dual slack or surplus variables (in the Reduced Cost column). 3. Allowable increases

and decreases for right-hand sides. These give lower and upper limits.

3. Allowable increases and decreases for right-hand sides. These give lower and upper limits.

4. Allowable increases and decreases for the coefficients of the objective function. These give the lower and upper limits.

4. Allowable increases and decreases for the coefficients of the objective function. These give the lower and upper limits.