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LINEAR PROGRAMMINGLINEAR PROGRAMMINGIntroduction to Introduction to

Sensitivity AnalysisSensitivity Analysis

Professor Ahmadi

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Chapter 3 Chapter 3 Linear Programming: Linear Programming:

Sensitivity Analysis and Interpretation of Sensitivity Analysis and Interpretation of SolutionSolution

Introduction to Sensitivity AnalysisIntroduction to Sensitivity Analysis Graphical Sensitivity AnalysisGraphical Sensitivity Analysis Spreadsheet Solution & Sensitivity Spreadsheet Solution & Sensitivity

AnalysisAnalysis

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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 Function Range of Optimality: The Objective Function CoefficientsCoefficients

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 constant. keeping all other data of the problem constant. (The value of the objective function may (The value of the objective function may change in this range.)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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ExampleExample

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 dual Price (Shadow Price) for the assembly stage.stage.

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

stage.stage.

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Example 2: Your TurnExample 2: Your Turn

Refer to The Olympic Bike Co. (From Chapter 2)Refer to The Olympic Bike Co. (From Chapter 2)• Solve this Problem and find the optimum solution.Solve this Problem and find the optimum solution.

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 Available)(Steel Available)

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

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

Range of OptimalityRange of Optimality

Question:Question:

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?

Answer:Answer:

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

Range of OptimalityRange of Optimality

Question:Question:

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?

Answer:Answer:

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Range of FeasibilityRange of Feasibility

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

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

Range of Feasibility and Relevant CostsRange of Feasibility and Relevant Costs

Question:Question:

If aluminum were a relevant cost, what is the If aluminum were a relevant cost, what is the maximum amount the company should pay for maximum amount the company should pay for 50 extra pounds of aluminum? 50 extra pounds of aluminum?

Answer:Answer:

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Example 3Example 3

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

Min Z = 6Min Z = 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

Use Excel to solve this problem.Use Excel to solve this problem.

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Example 3Example 3

Optimal SolutionOptimal Solution

According to the output:According to the output:

xx11 = 1.5 = 1.5

xx22 = 2.0 = 2.0

Z (the objective function value) = 27.00.Z (the objective function value) = 27.00.

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Example 3Example 3

Range of OptimalityRange of Optimality

Question:Question:

Suppose the unit cost of Suppose the unit cost of xx11 is decreased to $4. Is is decreased to $4. Is the current solution still optimal? What is the the current solution still optimal? What is the value of the objective function when this unit value of the objective function when this unit cost is decreased to $4?cost is decreased to $4?

Answer:Answer:

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Example 3Example 3

Range of OptimalityRange of Optimality

Question:Question:

How much can the unit cost of How much can the unit cost of xx22 be decreased be decreased without concern for the optimal solution without concern for the optimal solution changing?changing?

Answer:Answer:

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Example 3Example 3

Range of FeasibilityRange of Feasibility

Question:Question:

If the right-hand side of constraint 3 is increased If the right-hand side of constraint 3 is increased by 1, what will be the effect on the optimal by 1, what will be the effect on the optimal solution?solution?

Answer:Answer:

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A Note on Sunk Cost and Relevant CostA Note on Sunk Cost and Relevant Cost

A resource cost is a A resource cost is a relevant costrelevant cost if the amount if the amount paid for it is dependent upon the amount of the paid for it is dependent upon the amount of the resource used by the decision variables. resource used by the decision variables.

Relevant costs Relevant costs areare reflected in the objective reflected in the objective function coefficients. function coefficients.

A resource cost is a A resource cost is a sunk costsunk cost if it must be paid if it must be paid regardless of the amount of the resource regardless of the amount of the resource actually used by the decision variables. actually used by the decision variables.

Sunk resource costs Sunk resource costs are not are not reflected in the reflected in the objective function coefficients.objective function coefficients.

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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.