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LINEAR PROGRAMMINGLINEAR PROGRAMMINGIntroduction to Introduction to

Sensitivity AnalysisSensitivity Analysis

Professor Ahmadi

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Learning ObjectivesLearning Objectives

Understand, using graphs, impact of changes in Understand, using graphs, impact of changes in objective function coefficients, right-hand-side objective function coefficients, right-hand-side values, and constraint coefficients on optimal values, and constraint coefficients on optimal solution of a linear programming problem.solution of a linear programming problem.

Generate answer and sensitivity reports using Generate answer and sensitivity reports using Excel's Solver.Excel's Solver.

Interpret all parameters of reports for Interpret all parameters of reports for maximization and minimization problems.maximization and minimization problems.

Analyze impact of simultaneous changes in Analyze impact of simultaneous changes in input data values using 100% rule.input data values using 100% rule.

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Sensitivity AnalysisSensitivity Analysis

Sensitivity analysisSensitivity analysis (or post-optimality (or post-optimality analysis) is used to determine how the optimal analysis) is used to determine how the optimal solution is affected by changes, within solution is affected by changes, within specified ranges, in:specified ranges, in:• the objective function coefficientsthe objective function coefficients• the right-hand side (RHS) valuesthe right-hand side (RHS) values

Sensitivity analysis is important to the Sensitivity analysis is important to the manager who must operate in a manager who must operate in a dynamic dynamic environmentenvironment with imprecise estimates of the with imprecise estimates of the coefficients. coefficients.

Sensitivity analysis allows the manager to ask Sensitivity analysis allows the manager to ask certain certain what-if questionswhat-if questions about the problem. about the problem.

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Range of Optimality: The Objective Range of Optimality: The Objective Function CoefficientsFunction Coefficients

A A range of optimalityrange of optimality of an objective function of an objective function coefficient is found by determining an interval coefficient is found by determining an interval for the coefficient in which the original for the coefficient in which the original optimal solution remains optimal while optimal solution remains optimal while keeping all other data of the problem keeping all other data of the problem constant. (The value of the objective function constant. (The value of the objective function may change in this range.)may change in this range.)

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The Right Hand Sides: Shadow Price (Dual The Right Hand Sides: Shadow Price (Dual Price)Price)

A A shadow priceshadow price for a right hand side value (or for a right hand side value (or resource limit) is the amount the objective function resource limit) is the amount the objective function will change per unit increase in the right hand side will change per unit increase in the right hand side value of a constraint. value of a constraint.

The The range of feasibilityrange of feasibility for a change in the right hand for a change in the right hand side value is the range of values for this coefficient in side value is the range of values for this coefficient in which the original shadow price remains constant.which the original shadow price remains constant.

Graphically, a shadow price is determined by adding Graphically, a shadow price is determined by adding +1 to the right hand side value in question and then +1 to the right hand side value in question and then resolving for the optimal solution in terms of the resolving for the optimal solution in terms of the same two binding constraints. same two binding constraints.

The shadow price is equal to the difference in the The shadow price is equal to the difference in the values of the objective functions between the new values of the objective functions between the new and original problems.and original problems.

The shadow price for a non-binding constraint is 0. The shadow price for a non-binding constraint is 0.

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Example 1Example 1

Refer to the “Woodworking” example of Chapter Refer to the “Woodworking” example of Chapter 2, where X1 = Tables and X2= Chairs. The 2, where X1 = Tables and X2= Chairs. The problem is shown below.problem is shown below.

  Max. Z=Max. Z= $100X1+60X2$100X1+60X2

  

s.t.s.t. 12X1+4X2 12X1+4X2 < < 60 (Assembly time in hours)60 (Assembly time in hours)

4X1+8X2 4X1+8X2 << 40 (Painting time in hours) 40 (Painting time in hours)

The optimum solution was X1=4, X2=3, and The optimum solution was X1=4, X2=3, and Z=$580. Answer the following questions Z=$580. Answer the following questions regarding this problem.regarding this problem.

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Answer the following Questions:Answer the following Questions:

1.1. Compute theCompute the range of optimalityrange of optimality for the contribution of X1 for the contribution of X1 (Tables)(Tables)

  

  

2.2. Compute theCompute the range of optimalityrange of optimality for the contribution of X2 for the contribution of X2 (Chairs)(Chairs)

  

  

3.3. Determine theDetermine the dual Price (Shadow Price) for the assembly stage.dual Price (Shadow Price) for the assembly stage.

  

  

4.4. Determine theDetermine the dual Price (Shadow Price) for the painting stage.dual Price (Shadow Price) for the painting stage.

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Standard Computer OutputStandard Computer Output Software packages such as Microsoft Excel and

LINDO provide the following LP information: Information about the objective function:Information about the objective function: its optimal valueits optimal value coefficient ranges (ranges of optimality)coefficient ranges (ranges of optimality) Information about the decision variables:Information about the decision variables: their optimal valuestheir optimal values their reduced coststheir reduced costs Information about the constraints:Information about the constraints: the amount of slack or surplusthe amount of slack or surplus the dual pricesthe dual prices right-hand side ranges (ranges of feasibility)right-hand side ranges (ranges of feasibility)

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Sensitivity ReportSensitivity Report

Sensitivity report has two distinct components. Sensitivity report has two distinct components. (1) Table titled Adjustable Cells (1) Table titled Adjustable Cells (2) Table titled Constraints. (2) Table titled Constraints. Tables permit one to answer several "what-if" Tables permit one to answer several "what-if"

questions regarding problem solution.questions regarding problem solution. Consider a change to only a single input data Consider a change to only a single input data

value. value. Sensitivity information does not always apply to Sensitivity information does not always apply to

simultaneous changes in several input data simultaneous changes in several input data values.values.

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Example 2: Olympic Bike Co.Example 2: Olympic Bike Co.

Model FormulationModel Formulation

Max 10Max 10xx11 + 15 + 15xx22 (Total Weekly (Total Weekly Profit) Profit)

s.t. 2s.t. 2xx11 + 4 + 4xx2 2 << 100 100 (Aluminum (Aluminum Available)Available)

33xx11 + 2 + 2xx2 2 << 80 80 (Steel (Steel Available)Available)

xx11, , xx2 2 >> 0 0 (Non-(Non-negativity)negativity)

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Example 2: Olympic Bike Co.Example 2: Olympic Bike Co.

Optimal SolutionOptimal Solution

According to the output: According to the output: xx11 (Deluxe frames) = (Deluxe frames) = 15, 15,

xx22 (Professional frames) = 17.5, and the (Professional frames) = 17.5, and the objective function value = $412.50.objective function value = $412.50.

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Example 2: Olympic Bike Co.Example 2: Olympic Bike Co.

Range of OptimalityRange of Optimality

QuestionQuestion

Suppose the profit on deluxe frames is increased Suppose the profit on deluxe frames is increased to $20. Is the above solution still optimal? What to $20. Is the above solution still optimal? What is the value of the objective function when this is the value of the objective function when this unit profit is increased to $20?unit profit is increased to $20?

AnswerAnswer

The output states that the solution remains The output states that the solution remains optimal as long as the objective function optimal as long as the objective function coefficient of coefficient of xx11 is between 7.5 and 22.5. Since is between 7.5 and 22.5. Since 20 is within this range, the optimal solution will 20 is within this range, the optimal solution will not change. The optimal profit will change: 20not change. The optimal profit will change: 20xx11 + 15+ 15xx22 = 20(15) + 15(17.5) = $562.50 = 20(15) + 15(17.5) = $562.50

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Example 2: Olympic Bike Co.Example 2: Olympic Bike Co.

Range of OptimalityRange of Optimality

QuestionQuestion

If the unit profit on deluxe frames were $6 If the unit profit on deluxe frames were $6 instead of $10 would the optimal solution instead of $10 would the optimal solution change?change?

AnswerAnswer

The output states that the solution remains The output states that the solution remains optimal as long as the objective function optimal as long as the objective function coefficient of coefficient of xx11 is between 7.5 and 22.5. Since 6 is between 7.5 and 22.5. Since 6 is outside this range, the optimal solution would is outside this range, the optimal solution would change.change.

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Example 2: Olympic Bike Co.Example 2: Olympic Bike Co.

Range of Feasibility: Range of Feasibility: The The range of feasibilityrange of feasibility for a for a change in a right-hand side value is the range of change in a right-hand side value is the range of values for this parameter in which the original values for this parameter in which the original shadow price remains constant.shadow price remains constant.

QuestionQuestion

What is the maximum amount the company should What is the maximum amount the company should pay for 50 extra pounds of aluminum? pay for 50 extra pounds of aluminum?

AnswerAnswer

The shadow price provides the value of extra The shadow price provides the value of extra aluminum. The shadow price for aluminum is $3.125 aluminum. The shadow price for aluminum is $3.125 per pound and the maximum allowable increase is per pound and the maximum allowable increase is 60 pounds. Since 50 is in this range, then the 60 pounds. Since 50 is in this range, then the $3.125 is valid. Thus, the value of 50 additional $3.125 is valid. Thus, the value of 50 additional pounds is: 50($3.125) = $156.25pounds is: 50($3.125) = $156.25

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Example 3Example 3

Consider the following linear program:Consider the following linear program:

Min 6Min 6xx11 + 9 + 9xx22 ($ cost) ($ cost)

s.t. s.t. xx11 + 2 + 2xx2 2 << 8 8

1010xx11 + 7.5 + 7.5xx2 2 >> 30 30

xx2 2 >> 2 2

xx11, , xx22 >> 0 0

Solve the above problem and perform Solve the above problem and perform sensitivity analysis.sensitivity analysis.

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Range of Optimality and 100% RuleRange of Optimality and 100% Rule

The 100% rule states that simultaneous changes The 100% rule states that simultaneous changes in objective function coefficients will not change in objective function coefficients will not change the optimal solution as long as the sum of the the optimal solution as long as the sum of the percentages of the change divided by the percentages of the change divided by the corresponding maximum allowable change in the corresponding maximum allowable change in the range of optimality for each coefficient does not range of optimality for each coefficient does not exceed 100%.exceed 100%.

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Range of FeasibilityRange of Feasibility

A dual price represents the improvement A dual price represents the improvement (increase or decrease) in the objective function (increase or decrease) in the objective function value per unit increase in the right-hand side. As value per unit increase in the right-hand side. As long as the right-hand side remain within the long as the right-hand side remain within the range of feasibility, there will be is no change in range of feasibility, there will be is no change in the shadow price. The the shadow price. The range of feasibilityrange of feasibility is the is the range over which the dual price is applicable.range over which the dual price is applicable.

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Range of Feasibility and 100% RuleRange of Feasibility and 100% Rule

The 100% rule states that simultaneous changes The 100% rule states that simultaneous changes in right-hand sides will not change the dual in right-hand sides will not change the dual prices as long as the sum of the percentages of prices as long as the sum of the percentages of the changes divided by the corresponding the changes divided by the corresponding maximum allowable change in the range of maximum allowable change in the range of feasibility for each right-hand side does not feasibility for each right-hand side does not exceed 100%.exceed 100%.

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Reduced CostReduced Cost

The The reduced costreduced cost for a decision variable whose for a decision variable whose value is 0 in the optimal solution is the amount value is 0 in the optimal solution is the amount the variable's objective function coefficient the variable's objective function coefficient would have to improve (increase for would have to improve (increase for maximization problems, decrease for maximization problems, decrease for minimization problems) before this variable minimization problems) before this variable could assume a positive value. could assume a positive value.

The reduced cost for a decision variable with a The reduced cost for a decision variable with a positive value is 0.positive value is 0.

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Example 4Example 4

In a product-mix-problem, X1, X2, X3, and X4 indicate the In a product-mix-problem, X1, X2, X3, and X4 indicate the units of products 1, 2, 3, and 4, respectively, and the units of products 1, 2, 3, and 4, respectively, and the linear programming model islinear programming model is

MAX Z = $5X1+$7X2+$8X3+$6X4MAX Z = $5X1+$7X2+$8X3+$6X4  S.T.S.T. 1) 3X1+2X2+4X3+3X41) 3X1+2X2+4X3+3X4600600 Machine A hoursMachine A hours

2) 4X1+1X2+2X3+6X42) 4X1+1X2+2X3+6X4700700 Machine B hoursMachine B hours3) 2X1+3X2+1X3+2X43) 2X1+3X2+1X3+2X4800800 Machine C hours Machine C hours 

Input the data in an Excel file and save the file. Using Input the data in an Excel file and save the file. Using SolverSolver, solve the problem and obtain sensitivity results., solve the problem and obtain sensitivity results.

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SummarySummary

Sensitivity analysis used by management to Sensitivity analysis used by management to answer series of “ what-if ” questions about LP answer series of “ what-if ” questions about LP model inputs. model inputs.

Tests sensitivity of optimal solution to changes: Tests sensitivity of optimal solution to changes: Profit or cost coefficients, and Profit or cost coefficients, and Constraint RHS values.Constraint RHS values. Explored sensitivity analysis graphically (with two Explored sensitivity analysis graphically (with two

decision variables). decision variables). Discussed interpretation of information:Discussed interpretation of information: In answer and sensitivity reports generated by In answer and sensitivity reports generated by

Solver. Solver. In reports used to analyze simultaneous changes In reports used to analyze simultaneous changes

in model parameter values.in model parameter values.