1 the role of sensitivity analysis of the optimal solution is the optimal solution sensitive to...

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The Role of Sensitivity Analysis The Role of Sensitivity Analysis of the Optimal Solutionof the Optimal Solution

• Is the optimal solution sensitive to changes in input parameters?

• Possible reasons for asking this question:– Parameter values used were only best estimates.– Dynamic environment may cause changes.– “What-if” analysis may provide economical and

operational information.

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Max 8X1 + 5X2 (Weekly profit)subject to

2X1 + 1X2 1000 (Plastic)

3X1 + 4X2 2400 (Production Time)

X1 + X2 700 (Total production)

X1 - X2 350 (Mix)

Xj> = 0, j = 1,2 (Nonnegativity)

The Galaxy Linear Programming ModelThe Galaxy Linear Programming Model

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• Range of Optimality– The optimal solution will remain unchanged as long as

• An objective function coefficient lies within its range of

optimality • There are no changes in any other input parameters.

– The value of the objective function will change if the

coefficient multiplies a variable whose value is nonzero.

Sensitivity Analysis of Sensitivity Analysis of Objective Function Coefficients.Objective Function Coefficients.

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500

1000

500 800

X2

X1Max 8X

1 + 5X2

Max 4X1 + 5X

2

Max 3.75X1 + 5X

2

Max 2X1 + 5X

2

Sensitivity Analysis of Sensitivity Analysis of Objective Function Coefficients.Objective Function Coefficients.

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500

1000

400 600 800

X2

X1

Max8X1 + 5X

2

Max 3.75X1 + 5X

2

Max 10 X

1 + 5X2

Range of optimality: [3.75, 10](Coefficient of X1)

Sensitivity Analysis of Sensitivity Analysis of Objective Function Coefficients.Objective Function Coefficients.

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• Reduced costAssuming there are no other changes to the input parameters, the reduced cost for a variable Xj that has a value of “0” at the optimal solution is:

– The negative of the objective coefficient increase of the variable Xj (-Cj) necessary for the variable to be positive in the optimal solution

– Alternatively, it is the change in the objective value per unit increase of Xj.

• Complementary slackness At the optimal solution, either the value of a variable is zero, or its reduced cost is 0.

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• In sensitivity analysis of right-hand sides of constraints we are interested in the following questions:– Keeping all other factors the same, how much would the

optimal value of the objective function (for example, the profit) change if the right-hand side of a constraint changed by one unit?

– For how many additional or fewer units will this per unit change be valid?

Sensitivity Analysis of Sensitivity Analysis of Right-Hand Side ValuesRight-Hand Side Values

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• Any change to the right hand side of a binding constraint will change the optimal solution.

• Any change to the right-hand side of a non-binding constraint that is less than its slack or surplus, will cause no change in the optimal solution.

Sensitivity Analysis of Sensitivity Analysis of Right-Hand Side ValuesRight-Hand Side Values

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Shadow PricesShadow Prices

• Assuming there are no other changes to the input parameters, the change to the objective function value per unit increase to a right hand side of a constraint is called the “Shadow Price”

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1000

500

X2

X1

500

2X1 + 1x

2 <=1000

When more plastic becomes available (the plastic constraint is relaxed), the right hand side of the plastic constraint increases.

Production timeconstraint

Maximum profit = $4360

2X1 + 1x

2 <=1001 Maximum profit = $4363.4

Shadow price = 4363.40 – 4360.00 = 3.40

Shadow Price – graphical demonstrationShadow Price – graphical demonstrationThe Plastic constraint

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Range of FeasibilityRange of Feasibility

• Assuming there are no other changes to the input parameters, the range of feasibility is– The range of values for a right hand side of a constraint, in

which the shadow prices for the constraints remain unchanged.

– In the range of feasibility the objective function value changes as follows:Change in objective value = [Shadow price][Change in the right hand side value]

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Range of FeasibilityRange of Feasibility

1000

500

X2

X1

500

2X1 + 1x

2 <=1000

Increasing the amount of plastic is only effective until a new constraint becomes active.

The Plastic constraint

This is an infeasible solutionProduction timeconstraint

Production mix constraintX1 + X2 700

A new activeconstraint

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Range of FeasibilityRange of Feasibility

1000

500

X2

X1

500

The Plastic constraint

Production timeconstraint

Note how the profit increases as the amount of plastic increases.

2X1 + 1x

2 1000

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Range of FeasibilityRange of Feasibility

1000

500

X2

X1

5002X1 + 1X2 1100

Less plastic becomes available (the plastic constraint is more restrictive).

The profit decreases

A new activeconstraint

Infeasiblesolution

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Other Post - Optimality Changes Other Post - Optimality Changes

• Addition of a constraint.

• Deletion of a constraint.

• Addition of a variable.

• Deletion of a variable.

• Changes in the left - hand side coefficients.

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Using Excel Solver to Find an Optimal Using Excel Solver to Find an Optimal Solution and Analyze ResultsSolution and Analyze Results

• To see the input screen in Excel click Galaxy.xls• Click Solver to obtain the following dialog box.

Equal To:By Changing cells

These cells containthe decision variables

$B$4:$C$4

To enter constraints click…

Set Target cell $D$6This cell contains the value of the objective function

$D$7:$D$10 $F$7:$F$10

All the constraintshave the same direction, thus are included in one “Excel constraint”.

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Using Excel SolverUsing Excel Solver

• To see the input screen in Excel click Galaxy.xls• Click Solver to obtain the following dialog box.

Equal To:

$D$7:$D$10<=$F$7:$F$10

By Changing cellsThese cells containthe decision variables

$B$4:$C$4

Set Target cell $D$6This cell contains the value of the objective function

Click on ‘Options’and check ‘Linear Programming’ and‘Non-negative’.

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• To see the input screen in Excel click Galaxy.xls• Click Solver to obtain the following dialog box.

Equal To:

$D$7:$D$10<=$F$7:$F$10

By Changing cells$B$4:$C$4

Set Target cell $D$6

Using Excel SolverUsing Excel Solver

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Space Rays ZappersDozens 320 360

Total LimitProfit 8 5 4360

Plastic 2 1 1000 <= 1000Prod. Time 3 4 2400 <= 2400

Total 1 1 680 <= 700Mix 1 -1 -40 <= 350

GALAXY INDUSTRIES

Using Excel Solver – Optimal SolutionUsing Excel Solver – Optimal Solution

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Space Rays ZappersDozens 320 360

Total LimitProfit 8 5 4360

Plastic 2 1 1000 <= 1000Prod. Time 3 4 2400 <= 2400

Total 1 1 680 <= 700Mix 1 -1 -40 <= 350

GALAXY INDUSTRIES

Using Excel Solver – Optimal SolutionUsing Excel Solver – Optimal Solution

Solver is ready to providereports to analyze theoptimal solution.

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Using Excel Solver –Answer ReportUsing Excel Solver –Answer ReportMicrosoft Excel 9.0 Answer ReportWorksheet: [Galaxy.xls]GalaxyReport Created: 11/12/2001 8:02:06 PM

Target Cell (Max)Cell Name Original Value Final Value

$D$6 Profit Total 4360 4360

Adjustable CellsCell Name Original Value Final Value

$B$4 Dozens Space Rays 320 320$C$4 Dozens Zappers 360 360

ConstraintsCell Name Cell Value Formula Status Slack

$D$7 Plastic Total 1000 $D$7<=$F$7 Binding 0$D$8 Prod. Time Total 2400 $D$8<=$F$8 Binding 0$D$9 Total Total 680 $D$9<=$F$9 Not Binding 20$D$10 Mix Total -40 $D$10<=$F$10 Not Binding 390

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Using Excel Solver –Sensitivity Using Excel Solver –Sensitivity ReportReport

Microsoft Excel Sensitivity ReportWorksheet: [Galaxy.xls]Sheet1Report Created:

Adjustable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$B$4 Dozens Space Rays 320 0 8 2 4.25$C$4 Dozens Zappers 360 0 5 5.666666667 1

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$D$7 Plastic Total 1000 3.4 1000 100 400$D$8 Prod. Time Total 2400 0.4 2400 100 650$D$9 Total Total 680 0 700 1E+30 20$D$10 Mix Total -40 0 350 1E+30 390

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Another Example:Another Example: Cost Minimization Diet Problem Cost Minimization Diet Problem

• Mix two sea ration products: Texfoods, Calration.• Minimize the total cost of the mix. • Meet the minimum requirements of Vitamin A, Vitamin D, and Iron.

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• Decision variables– X1 (X2) -- The number of two-ounce portions of

Texfoods (Calration) product used in a serving.

• The ModelMinimize 0.60X1 + 0.50X2Subject to

20X1 + 50X2 100 Vitamin A 25X1 + 25X2 100 Vitamin D 50X1 + 10X2 100 Iron X1, X2 0

Cost per 2 oz.

% Vitamin Aprovided per 2 oz.

% required

Cost Minimization Diet Problem Cost Minimization Diet Problem

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10

2 44 5

Feasible RegionFeasible Region

Vitamin “D” constraint

Vitamin “A” constraint

The Iron constraint

The Diet Problem - Graphical solutionThe Diet Problem - Graphical solution

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• Summary of the optimal solution

– Texfood product = 1.5 portions (= 3 ounces)Calration product = 2.5 portions (= 5 ounces)

– Cost =$ 2.15 per serving. – The minimum requirement for Vitamin D and iron are met with

no surplus. – The mixture provides 155% of the requirement for Vitamin A.

Cost Minimization Diet Problem Cost Minimization Diet Problem