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Chapter 3 Introduction to Sensitivity Analysis Sensitivity Analysis: Computer Solution

Sensitivity Analysis (LP Section 3). Chapter 3 Introduction to Sensitivity Analysis Sensitivity Analysis: Computer Solution. Sensitivity Analysis.

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Chapter 3 Introduction to Sensitivity Analysis Sensitivity Analysis: Computer Solution

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  1. Sensitivity Analysis (LP Section 3) Chapter 3 • Introduction to Sensitivity Analysis • Sensitivity Analysis: Computer Solution Dr. C. Lightner Fayetteville State University

  2. Sensitivity Analysis • Sensitivity analysis (or post-optimality analysis) is used to determine how the optimal solution is affected by changes, within specified ranges, in: • the objective function coefficients • the right-hand side (RHS) values • Sensitivity analysis allows you to answer questions about making such changes without actually resolving the problem. • Sensitivity analysis is important to the manager who must operate in a dynamic environment with imprecise estimates of the coefficients. • Sensitivity analysis allows managers to ask certain what-ifquestions about the problem. Dr. C. Lightner Fayetteville State University

  3. Standard Computer Output Software packages such as LINDO provide the following LP information: • Information about the objective function: • its optimal value • coefficient ranges (ranges of optimality) • Information about the decision variables: • their optimal values • their reduced costs • Information about the constraints: • the amount of slack or surplus • the dual prices • right-hand side ranges (ranges of feasibility) Dr. C. Lightner Fayetteville State University

  4. Standard Computer Output • In LP Section 2(Chapter 2) we discussed: • objective function value • values of the decision variables • reduced costs • slack/surplus • In this section we will address the following questions: • If a single objective function coefficient changes will the current solution still be optimal? If yes, what is the new objective function value? 2. If more than one objective function coefficient changes will the current solution still be optimal? If yes, what is the new objective function value? (Simultaneous changes) • How much would you be will to pay for one more unit of the right hand side of a constraint? (dual price) • By how much does my objective function value have to change in order for it to be profitable to produce the corresponding variable? (reduced cost from LP Section 2) Dr. C. Lightner Fayetteville State University

  5. Example 1 • LP Formulation Max 5x1 + 7x2 s.t. x1< 6 2x1 + 3x2< 19 x1 + x2< 8 x1, x2> 0 Dr. C. Lightner Fayetteville State University

  6. Example 1 • Graphical Solution x2 x1 + x2< 8 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 Max 5x1 + 7x2 x1< 6 Optimal: x1 = 5, x2 = 3, z = 46 2x1 + 3x2< 19 x1 Dr. C. Lightner Fayetteville State University

  7. Objective Function Coefficients • Let us consider how changes in the objective function coefficients might affect the optimal solution. • The range of optimality for each coefficient provides the range of values over which the current solution will remain optimal. • Managers should focus on those objective coefficients that have a narrow range of optimality and coefficients near the endpoints of the range. Dr. C. Lightner Fayetteville State University

  8. Example 1 • Changing Slope of Objective Function 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 By changing coefficients in the objective function you are altering its slope. You can verify this by simply changing a coefficient and graphing the new objective function. x2 5 Feasible Region 4 3 1 2 x1 Dr. C. Lightner Fayetteville State University

  9. Range of Optimality • Graphically, the limits of a range of optimality are found by changing the slope of the objective function line within the limits of the slopes of the binding constraint lines. Binding constraints are the constraints that intersect to form the optimal point. • We will focus on using the computer output to analyze the range of optimality for objective function coefficients. Dr. C. Lightner Fayetteville State University

  10. Example 1: Question 1 (Using LINDO) • Question 1 (In general) If a single objective function coefficient changes, will the current solution still be optimal? If yes, what is the new objective function value? • To answer the first part of this question, you must locate the section of the spreadsheet that discusses objective coefficient ranges. This section is highlighted on the following output slides. • The output shows that the objective function coefficient of x1 (currently 5) can increase by as much as 2 and decrease by as much as 0.333333 and the optimal solution will remain optimal. Thus provided the coefficient of x1 is between 4.666667 and 7, the optimal solution will remain x*1 =5, x*2 =3. • The objective function coefficient of x2 can vary between 5 and 7.5 and the current solution will remain optimal. Dr. C. Lightner Fayetteville State University

  11. Example 1: Question 1 • Range of Optimality for c1 and c2 OBJECTIVE FUNCTION VALUE 1) 46.00000 VARIABLE VALUE REDUCED COST X1 5.000000 0.000000 X2 3.000000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 1.000000 0.000000 3) 0.000000 2.000000 4) 0.000000 1.000000 5) 5.000000 0.000000 6) 3.000000 0.000000 Dr. C. Lightner Fayetteville State University

  12. Example 1: Question 1 OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X1 5.000000 2.000000 0.333333 X2 7.000000 0.500000 2.000000 RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 6.000000 INFINITY 1.000000 3 19.000000 5.000000 1.000000 4 8.000000 0.333333 1.666667 5 0.000000 5.000000 INFINITY 6 0.000000 3.000000 INFINITY Dr. C. Lightner Fayetteville State University

  13. Example 1: Question 2 • If more than one objective function coefficient changes will the current solution still be optimal? If yes, what is the new objective function value? (Simultaneous changes) Dr. C. Lightner Fayetteville State University

  14. The 100% Rule The 100% rule states that simultaneous changes in objective function coefficients will not change the optimal solution as long as the sum of the percentages of the change for the coefficients does not exceed 100%. % of change for a coefficient = Dr. C. Lightner Fayetteville State University

  15. Example 1: Question 2 • Suppose that simultaneously the coefficient of x1 changed to 6 (currently 5) and the coefficient of x2 changed to 6 (currently 7), simultaneously. • The coefficient of x1 (c1) is increased by 1 unit. The allowable increase is 2. Thus the % change is ½ * 100= 50% • The coefficient of x2 (c2) is decreased by 1 unit. The allowable decrease is also 2. Thus the % change is ½ * 100= 50% • The sum of the % of change is 50% + 50% =100%. This does not exceed 100% Thus the optimal solution is still x*1 =5, x*2 =3. The new objective function value is 6 (5) + 6 (3)= 48. OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X1 5.000000 2.0000000.333333 X2 7.000000 0.500000 2.000000 Dr. C. Lightner Fayetteville State University

  16. Example 1: Question 3 How much would you be will to pay for one more unit of the right hand side of a constraint? (dual price) The dual price tells us how much each unit of the right hand side (rhs) of a constraint affects the objective function value. This is particularly useful if you are considering purchasing more rhs units, or if units are downsized for some reason. Dr. C. Lightner Fayetteville State University

  17. Right-Hand Sides: Question 3 • Let us consider how a change in the right-hand side for a constraint might affect the feasible region and perhaps cause a change in the optimal solution. • The improvement in the value of the objective function value per unit increase in the right-hand side is called the dual price. • The range of feasibility is the range over which the dual price is applicable. • As the RHS increases, other constraints will become binding and limit the change in the value of the objective function. Dr. C. Lightner Fayetteville State University

  18. Dual Price • The dual price for a nonbinding constraint is 0. • RHS changes will affect the objective function value (Provided the change is within the allowable range) as follows: Let z= the objective function value • For a minimization problem: znew= zold - (change in rhs) * (the dual price for constraint with changing rhs) For a maximization problem: znew= zold + (change in rhs) * (the dual price for constraint with changing rhs) NOTE: If the rhs value decreases by 5 then the change is -5. Dr. C. Lightner Fayetteville State University

  19. Example 1: Question 3 OBJECTIVE FUNCTION VALUE 1) 46.00000 VARIABLE VALUE REDUCED COST X1 5.000000 0.000000 X2 3.000000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 1.000000 0.000000 3) 0.000000 2.000000 4) 0.000000 1.000000 5) 5.000000 0.000000 6) 3.000000 0.000000 Dr. C. Lightner Fayetteville State University

  20. Example 1: Question 3 • The dual price of Constraint 2 reveals that if its right hand side increases by one unit, the objective function will increase by 2. Thus one extra unit of constraint 2 would be worth up to $2. • The dual price of Constraint 3 reveals that if its right hand side increases by one unit, the objective function will increase by 1. Thus one extra unit of constraint 3 would be worth up to $1. Dr. C. Lightner Fayetteville State University

  21. Floataway Tours (LP Section 1) • Refer to the Floataway Tours problem formulation in LP Section 1. Answer the following questions: 1. If Sleekboats redesigns their Speekhawk boats (x1) resulting in a increased expected daily profit of $90 (currently $70), would the company still only purchase 28 (the current optimal value of x1) Speedhawk boats? 2. If the expected daily profit of Sleekboats (x1) and Classys (x4) simultaneously decreases to $69 and $100, respectively, would the current solution (x*1=28 and x*4=28) remain optimal? LINDO output on following slides. Dr. C. Lightner Fayetteville State University

  22. Floataway Tours LINDO Solution OBJECTIVE FUNCTION VALUE 1) 5040.000 VARIABLE VALUE REDUCED COST X1 28.000000 0.000000 X2 0.000000 2.000000 X3 0.000000 12.000000 X4 28.000000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 0.012000 3) 6.000000 0.000000 4) 0.000000 -2.000000 5) 52.000000 0.000000 6) 28.000000 0.000000 7) 0.000000 0.000000 8) 0.000000 0.000000 9) 28.000000 0.000000 Dr. C. Lightner Fayetteville State University

  23. Floataway Tours LINDO Solution OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X1 70.000000 44.999996 1.875001 X2 80.000000 2.000001 INFINITY X3 50.000000 12.000001 INFINITY X4 110.000000 INFINITY 16.363636 RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 420000.000000 INFINITY 45000.000000 3 50.000000 6.000000 INFINITY 4 0.000000 70.000000 30.000000 5 200.000000 52.000000 INFINITY 6 0.000000 28.000000 INFINITY 7 0.000000 0.000000 INFINITY 8 0.000000 0.000000 INFINITY 9 0.000000 28.000000 INFINITY Dr. C. Lightner Fayetteville State University

  24. Floataway Tours: Sensitivity Analysis • Question 1: The allowable increase for the objective function coefficient of x1 is 44.999996. Since increasing to $90 is only a $20 increase, this is an allowable increase. Thus the current solution is still optimal. • Question 2: The coefficient of x1 is decreased by 1. The allowable decrease is 1.875001. The % change is 53.33%. The coefficient of x2 is decreased by 10. The allowable decrease is 16.363636. The % change is 61.11%. The sum of the % changes is 114.444%. This exceeds 100%, therefore the optimal solution would change. Dr. C. Lightner Fayetteville State University

  25. U.S. Navy Example (LP Section 1) Refer to the U.S. Navy Example problem formulation in LP Section 1. Answer the following questions: 1. If the cost to ship to Pensacola via railroad is increased to 11 (currently 9) would the current solution still be optimal? 2. Suppose truck costs increased to the following: San Diego Norfolk Pensacola $13 (currently 12) $ 6.5 (currently 6) $ 5.5 (currently 5) Will the optimal solution change? 3. If San Diego changes its requirements to 3750 (currently 3700) can you determine how much will this cost the US Navy? 4. Currently the cost to ship to San Diego via railroad is $20. The optimal solution states that no cargo should be shipped to San Diego via railroad (x21*=0). How much would the cost have to be in order for us to use this mode of shipment to this destination? LINDO output on following slides. Dr. C. Lightner Fayetteville State University

  26. U.S. Navy Problem LINDO Solution OBJECTIVE FUNCTION VALUE 1) 115900.0 VARIABLE VALUE REDUCED COST X11 1700.000000 0.000000 X12 2500.000000 0.000000 X13 500.000000 0.000000 X21 0.000000 4.000000 X22 0.000000 1.000000 X23 2000.000000 0.000000 X31 2000.000000 0.000000 X32 0.000000 2.000000 X33 0.000000 5.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 -12.000000 3) 0.000000 -6.000000 4) 0.000000 -5.000000 5) 2700.000000 0.000000 6) 0.000000 -18.000000 7) 0.000000 -4.000000 8) 100.000000 0.000000 9) 1700.000000 0.000000 10) 2500.000000 0.000000 11) 500.000000 0.000000 12) 0.000000 0.000000 13) 0.000000 0.000000 14) 2000.000000 0.000000 15) 2000.000000 0.000000 16) 0.000000 0.000000 17) 0.000000 0.000000 Dr. C. Lightner Fayetteville State University

  27. U.S. Navy Problem LINDO Solution OBJ COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE X11 12.000000 4.000000 2.000000 X12 6.000000 1.000000 6.000000 X13 5.000000 4.000000 1.000000 X21 20.000000 INFINITY 4.000000 X22 11.000000 INFINITY 1.000000 X23 9.000000 1.000000 4.000000 X31 30.000000 2.000000 18.000000 X32 26.000000 INFINITY 2.000000 X33 28.000000 INFINITY 5.000000 RIGHTHAND SIDE RANGES ROW CURRENT ALLOWABLE ALLOWABLE RHS INCREASE DECREASE 2 3700.000000 100.000000 1700.000000 3 2500.000000 100.000000 2500.000000 4 2500.000000 100.000000 500.000000 5 2000.000000 2700.000000 INFINITY 6 2000.000000 1700.000000 2000.000000 7 2000.000000 500.000000 2000.000000 8 8800.000000 INFINITY 100.000000 9 0.000000 1700.000000 INFINITY 10 0.000000 2500.000000 INFINITY 11 0.000000 500.000000 INFINITY 12 0.000000 0.000000 INFINITY 13 0.000000 0.000000 INFINITY 14 0.000000 2000.000000 INFINITY 15 0.000000 2000.000000 INFINITY 16 0.000000 0.000000 INFINITY 17 0.000000 0.000000 INFINITY Dr. C. Lightner Fayetteville State University

  28. U.S. Navy Problem: Questions 1 & 2 • Question 1: The variable for shipping to Pensacola via railroad is x23. The current objective function coefficient for this variable is 9, the allowable increase is 1. By increasing to 11, this is an increase of 2 which exceeds the allowable increase. Thus the optimal solution would change. • Question 2: Shipping via truck to San Diego (x21) has increased costs by 1. The allowable increase is 4. The % change is ¼*100=25 %. Shipping via truck to Norfolk (x22) has increased costs by 0.5. The allowable increase is 1. The % change is 0.5/1*100=50 %. Shipping via truck to Pensacola (x23) has increased costs by 0.5. The allowable increase is 4. The % change is 0.5/4*100=12.5 %. The sum of the % changes is 25% + 50% + 12.5% =82.5%. Since this does not exceed 100% the current solution remains optimal. Dr. C. Lightner Fayetteville State University

  29. U.S. Navy Problem: Questions 3 & 4 • Question 3: Constraint 1 (row 2 in LINDO printout) deals with meeting San Diego’s requirements. The current rhs of this constraint is 3700, the allowable increase is 100. Thus increasing to 3750 is allowable. Since this rhs increase is allowable the dual price is meaningful. The dual price for this constraint is -12. This is a minimization problem and the current optimal cost (optimal objective function value) is 115900. znew= zold - (change in rhs) * (the dual price for constraint with changing rhs) Or znew=115900 – (+50) *(-12)= 116500. Thus our cost will increase by $600 (116500 – 115900) • Question 4: The reduced cost for the variable x21 is 4. The current cost for this variable (the objective function coefficient) is 20. Thus this cost must be improved by 4 in order for its optimal value to be nonzero. For minimization problems improved means decreased. Thus if the cost reduced to 16, the optimal value for x21 would be nonzero, i.e. Some cargo would be shipped to San Diego via railroad. Dr. C. Lightner Fayetteville State University

  30. THE END See your textbook for more examples and detailed explanations of all topics discussed in these notes. Dr. C. Lightner Fayetteville State University

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