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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