Chapter05tw - Table of Contents Chapter 5(What-If Analysis...

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Unformatted text preview: Table of Contents Chapter 5 (What-If Analysis for Linear Programming) Continuing the Wyndor Case Study (Section 5.2) 5.2 Changes in One Objective Function Coefficient (Section 5.3) 5.3–5.9 Simultaneous Changes in Objective Function Coefficients (Section 5.4)5.10– 5.17 Single Changes in a Constraint (Section 5.5) 5.18–5.23 Simultaneous Changes in the Constraints (Section 5.6) 5.24–5.26 Sensitivity Analysis (UW Lecture) 5.27–5.43 These slides are based upon a lecture to first-year MBA students at the University of Washington that discusses sensitivity analysis for linear programming models (as taught by one of the authors). 1 Wyndor (Before What-If Analysis) B 3 4 5 6 7 8 9 10 11 12 Unit Profit C Doors $300 D Windows $500 E F G Plant 1 Plant 2 Plant 3 Hours Used Per Unit Produced 1 0 0 2 3 2 Doors 2 Windows 6 Hours Used 2 12 18 <= <= <= Hours Available 4 12 18 Total Profit $3,600 Units Produced 2 Using the Spreadsheet to do Sensitivity Analysis B 3 4 5 6 7 8 9 10 11 12 Unit Profit C Doors $200 D Windows $500 E F G Plant 1 Plant 2 Plant 3 Hours Used Per Unit Produced 1 0 0 2 3 2 Doors 2 Windows 6 Hours Used 2 12 18 <= <= <= Hours Available 4 12 18 Total Profit $3,400 Units Produced The profit per door has been revised from $300 to $200. No change occurs in the optimal solution (D*,W*) = (2,6). NOTE: The optimal value of the objective function DOES CHANGE 3 Using the Spreadsheet to do Sensitivity Analysis B 3 4 5 6 7 8 9 10 11 12 Unit Profit C Doors $500 D Windows $500 E F G Plant 1 Plant 2 Plant 3 Hours Used Per Unit Produced 1 0 0 2 3 2 Doors 2 Windows 6 Hours Used 2 12 18 <= <= <= Hours Available 4 12 18 Total Profit $4,000 Units Produced The profit per door has been revised from $300 to $500. No change occurs in the optimal solution. 4 Using the Spreadsheet to do Sensitivity Analysis B 3 4 5 6 7 8 9 10 11 12 Unit Profit C Doors $1,000 D Windows $500 E F G Plant 1 Plant 2 Plant 3 Hours Used Per Unit Produced 1 0 0 2 3 2 Doors 4 Windows 3 Hours Used 4 6 18 <= <= <= Hours Available 4 12 18 Total Profit $5,500 Units Produced The profit per door has been revised from $300 to $1,000. The optimal solution changes. 5 Using Solver Table to do Sensitivity Analysis B 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Unit Profit C Doors $300 D Windows $500 E F G Hours Used Per Unit Produced 1 0 0 2 3 2 Doors 2 Windows 6 Hours Used 2 12 18 Hours Available 4 12 18 Total Profit $3,600 Plant 1 Plant 2 Plant 3 <= <= <= Units Produced Unit Profit for Doors $100 $200 $300 $400 $500 $600 $700 $800 $900 $1,000 Optimal Units Produced Doors Windows 2 6 Total Profit $3,600 Select these cells (B18:E28), before choosing the Solver Table. C D Optimal Units Produced 16 17 Doors Windows 18 =DoorsProduced =WindowsProduced E Total Profit =TotalProfit 6 Using Solver Table to do Sensitivity Analysis 16 17 18 19 20 21 22 23 24 25 26 27 28 B Unit Profit for Doors $100 $200 $300 $400 $500 $600 $700 $800 $900 $1,000 C D Optimal Units Produced Doors Windows 2 6 2 6 2 6 2 6 2 6 2 6 2 6 2 6 4 3 4 3 4 3 E Total Profit $3,600 $3,200 $3,400 $3,600 $3,800 $4,000 $4,200 $4,400 $4,700 $5,100 $5,500 7 Using the Sensitivity Report to Find the Allowable Range Adjustable Cells Cell $C$12 $D$12 Name Units Produced Doors Units Produced Windows Final Value 2 6 Reduced Cost 0 0 Objective Coefficient 300 500 Allowable Increase 450 1E+30 Allowable Decrease 300 300 8 Graphical Insight into the Allowable Range W Production rate for windows 8 (2, 6) is optimal for 0 < PD < 750 6 Line B PD = 0 (Profit = 0 D + 500 W) Objective function: P = PD∙D + PW ∙W 4 Feasible region 2 PD = 750 (Profit = 750 D + 500 W) Line A 0 2 4 Production rate for doors 6 D Line C Slope = - PD/PW PD = 300 (Profit = 300 D + 500 W) The two dashed lines that pass through the solid constraint boundary lines are the objective function lines when PD (the unit profit for doors) is at an endpoint of its allowable range, 0 ≤ PD ≤ 750. 9 Using the Spreadsheet to do Sensitivity Analysis B 3 4 5 6 7 8 9 10 11 12 Unit Profit C Doors $450 D Windows $400 E F G Plant 1 Plant 2 Plant 3 Hours Used Per Unit Produced 1 0 0 2 3 2 Doors 2 Windows 6 Hours Used 2 12 18 <= <= <= Hours Available 4 12 18 Total Profit $3,300 Units Produced The profit per door has been revised from $300 to $450. The profit per window has been revised from $500 to $400. No change occurs in the optimal solution. 10 Using the Spreadsheet to do Sensitivity Analysis B 3 4 5 6 7 8 9 10 11 12 Unit Profit C Doors $600 D Windows $300 E F G Plant 1 Plant 2 Plant 3 Hours Used Per Unit Produced 1 0 0 2 3 2 Doors 4 Windows 3 Hours Used 4 6 18 <= <= <= Hours Available 4 12 18 Total Profit $3,300 Units Produced The profit per door has been revised from $300 to $600. The profit per window has been revised from $500 to $300. The optimal solution changes. 11 Using Solver Table to do Sensitivity Analysis B 3 4 Unit Profit 5 6 7 Plant 1 8 Plant 2 9 Plant 3 10 11 12 Units Produced 13 14 15 16 Total Profit 17 18 19 Unit Profit 20 for Doors 21 C Doors $300 D Windows $500 E F G H I Hours Used Per Unit 1 0 0 2 3 2 Doors 2 Windows 6 Hours Used 2 12 18 <= <= <= Hours Available 4 12 18 Total Profit $3,600 Select these cells (C17:H21), before choosing the Solver Table. $3,600 $300 $400 $500 $600 $100 Unit Profit for Windows $200 $300 $400 $500 C 17 =TotalProfit 12 Using Solver Table to do Sensitivity Analysis B 16 Total Profit 17 18 19 Unit Profit 20 for Doors 21 C $3,600 $300 $400 $500 $600 D $100 $1,500 $1,900 $2,300 $2,700 E F G Unit Profit for Windows $200 $300 $400 $1,800 $2,400 $3,000 $2,200 $2,600 $3,200 $2,600 $2,900 $3,400 $3,000 $3,300 $3,600 H $500 $3,600 $3,800 $4,000 $4,200 13 Using Solver Table to do Sensitivity Analysis 24 25 26 27 28 29 B C D Units Produced (Doors, Windows) (2, 6) $100 $300 (4, 3) Unit Profit $400 (4, 3) for Doors $500 (4, 3) $600 (4, 3) E F G Unit Profit for Windows $200 $300 $400 (4, 3) (2, 6) (2, 6) (4, 3) (2, 6) (2, 6) (4, 3) (4, 3) (2, 6) (4, 3) (4, 3) (4, 3) H $500 (2, 6) (2, 6) (2, 6) (2, 6) C 25 ="(" & DoorsProduced & ", " & WindowsProduced & ")" 14 The 100 Percent Rule The 100 Percent Rule for Simultaneous Changes in Objective Function Coefficients: If simultaneous changes are made in the coefficients of the objective function, calculate for each change the percentage of the allowable change (increase or decrease) for that coefficient to remain within its allowable range. If the sum of the percentage changes does not exceed 100 percent, the original optimal solution definitely will still be optimal. (If the sum does exceed 100 percent, then we cannot be sure.) 15 Graphical Insight into 100 Percent Rule W Production rate for windows 10 Objective function line now is Profit = $3150 = 525 D + 350 W since PD = $525, PW = $350. (2, 6) Entire line segment is optimal 4 Feasible region (4, 3) 8 6 2 The estimates of the unit profits for doors and windows change to PD = $525 and PW = $350, which lies at the edge of what is allowed by the 100 percent rule. 0 2 4 Production rate for doors 6 8 D 16 Graphical Insight into 100 Percent Rule Production rate W for windows 8 Optimal solution 6 (2, 6) Profit = $1800 = 150D + 250 W 4 Feasible region 2 0 2 4 Production rate for doors 6 8 D When the estimates of the unit profits for doors and windows change to PD = $150 and PW = $250 (half their original values), the graphical method shows that the optimal solution still is (D, W) = (2, 6) even though the 100 percent rule says that the optimal solution might change. 17 Using the Spreadsheet to do Sensitivity Analysis B 3 4 5 6 7 8 9 10 11 12 Unit Profit C Doors $300 D Windows $500 E F G Plant 1 Plant 2 Plant 3 Hours Used Per Unit Produced 1 0 0 2 3 2 Doors 1.667 Windows 6.5 Hours Used 1.667 13 18 <= <= <= Hours Available 4 13 18 Total Profit $3,750 Units Produced The hours available in plant 2 have been increased from 12 to 13. The total profit increases by $150 per week. 18 Using the Spreadsheet to do Sensitivity Analysis B 3 4 5 6 7 8 9 10 11 12 Unit Profit C Doors $300 D Windows $500 E F G Plant 1 Plant 2 Plant 3 Hours Used Per Unit Produced 1 0 0 2 3 2 Doors 0 Windows 9 Hours Used 0 18 18 <= <= <= Hours Available 4 18 18 Total Profit $4,500 Units Produced The hours available in plant 2 have been further increased from 13 to 18. The total profit increases by $750 per week ($150 per hour added in plant 2). 19 Using the Spreadsheet to do Sensitivity Analysis B 3 4 5 6 7 8 9 10 11 12 Unit Profit C Doors $300 D Windows $500 E F G Plant 1 Plant 2 Plant 3 Hours Used Per Unit Produced 1 0 0 2 3 2 Doors 0 Windows 9 Hours Used 0 18 18 <= <= <= Hours Available 4 20 18 Total Profit $4,500 Units Produced The hours available in plant 2 have been further increased from 18 to 20. The total profit does not increase any further. 20 Using Solver Table to do Sensitivity Analysis B 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Unit Profit C Doors $300 D Windows $500 E F G Plant 1 Plant 2 Plant 3 Hours Used Per Unit Produced 1 0 0 2 3 2 Doors 2 Windows 6 Hours Used 2 12 18 <= <= <= Hours Available 4 12 18 Total Profit $3,600 Units Produced Time Available in Plant 2 (hours) 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Optimal Units Produced Doors Windows 2 6 4 2 4 2.5 4 3 3.667 3.5 3.333 4 3 4.5 2.667 5 2.333 5.5 2 6 1.667 6.5 1.333 7 1 7.5 0.667 8 0.333 8.5 0 9 0 9 0 9 Total Profit $3,600 $2,200 $2,450 $2,700 $2,850 $3,000 $3,150 $3,300 $3,450 $3,600 $3,750 $3,900 $4,050 $4,200 $4,350 $4,500 $4,500 $4,500 Incremental Profit Select these cells (B18:E35), before choosing the Solver Table. $250 $250 $150 $150 $150 $150 $150 $150 $150 $150 $150 $150 $150 $150 $0 $0 21 Using the Sensitivity Report Adjustable Cells Cell $C$12 $D$12 Constraints Cell $E$7 $E$8 $E$9 Name Plant 1 Used Plant 2 Used Plant 3 Used Final Value 2 12 18 Shadow Price 0 150 100 Constraint R.H. Side 4 12 18 Allowable Increase 1E+30 6 6 Allowable Decrease 2 6 6 Name Units Produced Doors Units Produced Windows Final Value 2 6 Reduced Cost 0 0 Objective Coefficient 300 500 Allowable Increase 450 1E+30 Allowable Decrease 300 300 22 Graphical Interpretation of the Allowable Range W Production rate for windows 10 (0, 9) 8 Line B 2 W = 18 → Profit = 300 (0) + 500 (9) = $4,500 6 (2, 6) 2 W = 12 → Profit = 300 (2) + 500 (6) = $3,600 4 2 Feasible region for original problem (4, 3) 2 W = 6 → Profit = 300 (4) + 500 (3) = $2,700 Line C (3 D + 2 W = 18) Line A (D = 4) 0 2 4 Production rate for doors 6 D 23 Using the Spreadsheet to do Sensitivity Analysis B 3 4 5 6 7 8 9 10 11 12 Unit Profit C Doors $300 D Windows $500 E F G Plant 1 Plant 2 Plant 3 Hours Used Per Unit Produced 1 0 0 2 3 2 Doors 1.333 Windows 6.5 Hours Used 1.333 13 17 <= <= <= Hours Available 4 13 17 Total Profit $3,650 Units Produced One available hour in plant 3 has been shifted to plant 2. The total profit increases by $50 per week. 24 Using Solver Table to do Sensitivity Analysis B C D E 3 Doors Windows 4 Unit Profit $300 $500 5 Hours 6 Used Hours Used Per Unit Produced 7 Plant 1 1 0 2 8 Plant 2 0 2 12 9 Plant 3 3 2 18 10 11 Doors Windows 12 Units Produced 2.000 6 13 14 15 16 17 Time Available Time Available Optimal Units Produced 18 in Plant 2 (hours) in Plant 3 (hours) Doors Windows 19 2 6 20 12 18 2 6 21 13 17 1.333 6.5 22 14 16 0.667 7 23 15 15 0 7.5 24 16 14 0 7 25 17 13 0 6.5 26 18 12 0 6 F G H Hours Available 4 12 18 Total Profit $3,600 <= <= <= Total (Plants 2 & 3) 30 Total Profit $3,600 $3,600 $3,650 $3,700 $3,750 $3,500 $3,250 $3,000 Incremental Profit Select these cells (C19:F26), before choosing the Solver Table. $50 $50 $50 -$250 -$250 -$250 25 The 100 Percent Rule The 100 Percent Rule for Simultaneous Changes in Right-Hand Sides: The shadow prices remain valid for predicting the effect of simultaneously changing the right-hand sides of some of the functional constraints as long as the changes are not too large. To check whether the changes are small enough, calculate for each change the percentage of the allowable change (decrease or increase) for that right-hand side to remain within its allowable range. If the sum of the percentage changes does not exceed 100 percent, the shadow prices definitely will still be valid. (If the sum does exceed 100 percent, then we cannot be sure.) 26 A Production Problem Weekly supply of raw materials: 8 Small Bricks 6 Large Bricks Products: Table Profit = $20 / Table Chair Profit = $15 / Chair 27 Sensitivity Analysis Questions With the given weekly supply of raw materials and profit data, how many tables and chairs should be produced? What is the total weekly profit? What if one more large brick were available. How much would you be willing to pay for it? What if an additional two large bricks were available (to make a total of 9). How much would you be willing to pay for these two additional bricks? What if the profit per table were now $25. (Assume now there are only 6 large bricks again.) How many tables and chairs should now be produced? What if the profit per table were now $35. How many tables and chairs should now be produced? 28 Graphical Solution (Original Problem) T 4 3 2 1 Optimal Solution (2, 2). Profit = $70 2T + 2C < 8 small bricks Maximize Profit = ($20)T + ($15)C subject to 2T + C ≤ 6 large bricks 2T + 2C ≤ 8 small bricks and T ≥ 0, C ≥ 0. 2T + C < 6 large bricks 1 2 3 4 5 6 C Z = ($20)T + ($15)C = $70 29 7 Large Bricks T 4 3 2 1 2T + C < 6 large bricks 1 2 3 4 5 6 C 2T + 2C < 8 small bricks Maximize Profit = ($20)T + ($15)C subject to 2T + C ≤ 7 large bricks 2T + 2C ≤ 8 small bricks and T ≥ 0, C ≥ 0. New Optimal Solution (3, 1). Profit = $75 Old Optimal Solution (2, 2). Profit = $70 2T + C < 7 large bricks Z = ($20)T + ($15)C = $75 30 9 Large Bricks T 4 3 2 1 2T + 2C < 8 small bricks 1 2 3 4 2T + C < 6 large bricks 5 6 New Optimal Solution (4, 0). Profit = $80 Maximize Profit = ($20)T + ($15)C subject to 2T + C ≤ 9 large bricks 2T + 2C ≤ 8 small bricks and T ≥ 0, C ≥ 0. 2T + C < 9 large bricks Old Optimal Solution (2, 2). Profit = $70 C Z = ($20)T + ($15)C = $80 31 $25 Profit per Table T 4 3 2 1 Optimal Solution (2, 2). Profit = $80 2T + 2C < 8 small bricks Maximize Profit = ($25)T + ($15)C subject to 2T + C ≤ 6 large bricks 2T + 2C ≤ 8 small bricks and T ≥ 0, C ≥ 0. 2T + C < 6 large bricks 1 2 3 4 5 6 C Z = ($25)T + ($15)C = $80 32 $35 Profit per Table T 4 3 2 1 2T + 2C < 8 small bricks Maximize Profit = ($35)T + ($15)C subject to 2T + C ≤ 6 large bricks 2T + 2C ≤ 8 small bricks and T ≥ 0, C ≥ 0. New Optimal Solution (3, 0). Profit = $105 Old Optimal Solution (2, 2). Profit = $100 Z = ($35)T + ($15)C = $105 1 2 3 4 5 6 C 33 Generating the Sensitivity Report B 3 4 5 6 7 8 9 10 11 C D E F G Profit Large Bricks Small Bricks Tables $20.00 Chairs $15.00 Total Used 6 8 <= <= Available 6 8 Total Profit $70.00 Bill of Materials 2 1 2 2 Tables 2 Chairs 2 Production Quantity: After solving with Solver, choose “Sensitivity” under reports: 34 The Sensitivity Report B 3 4 5 6 7 8 9 10 11 C D E F G Profit Large Bricks Small Bricks Tables $20.00 Chairs $15.00 Total Used 6 8 <= <= Available 6 8 Total Profit $70.00 Bill of Materials 2 1 2 2 Tables 2 Chairs 2 Production Quantity: Adjustable Cells Cell Name $C$11 Production Quantity: Tables $D$11 Production Quantity: Chairs Constraints Cell $E$7 $E$8 Name Large Bricks Total Used Small Bricks Total Used Final Value 6 8 Shadow Price 5 5 Constraint R.H. Side 6 8 Allowable Increase 2 4 Allowable Decrease 2 2 Final Reduced Value Cost 2 0 2 0 Objective Coefficient 20 15 Allowable Increase 10 5 Allowable Decrease 5 5 35 The Sensitivity Report The solution Adjustable Cells Cell Name $C$11 Production Quantity: Tables $D$11 Production Quantity: Chairs Constraints Cell $E$7 $E$8 Name Large Bricks Total Used Small Bricks Total Used Final Value 6 8 Shadow Price 5 5 Constraint R.H. Side 6 8 Allowable Increase 2 4 Allowable Decrease 2 2 Final Reduced Value Cost 2 0 2 0 Objective Coefficient 20 15 Allowable Increase 10 5 Allowable Decrease 5 5 Allowable range (Solution stays the same) Usage of the resource (Left-hand-side of constraint) Allowable range (Shadow price is valid) Increase in objective function value per unit increase in right-hand-side (RHS) ∆Z = (shadow price)(∆RHS) 36 $35 Profit per Table B 3 4 5 6 7 8 9 10 11 C D E F G Profit Large Bricks Small Bricks Tables $35.00 Chairs $15.00 Total Used 6 6 <= <= Available 6 8 Total Profit $105.00 Bill of Materials 2 1 2 2 Tables 3 Chairs 0 Production Quantity: Adjustable Cells Final Cell Name Value $C$11 Production Quantity: Tables 3 $D$11 Production Quantity: Chairs 0 Constraints Cell Name $E$7 Large Bricks Total Used $E$8 Small Bricks Total Used Final Value 6 6 Shadow Price 17.5 0 Constraint R.H. Side 6 8 Allowable Increase 2 1E+30 Allowable Decrease 6 2 Reduced Cost 0 -2.5 Objective Coefficient 35 15 Allowable Increase 1E+30 2.5 Allowable Decrease 5 1E+30 37 7 Large Bricks B 3 4 5 6 7 8 9 10 11 C D E F G Profit Large Bricks Small Bricks Tables $20.00 Chairs $15.00 Total Used 7 8 <= <= Available 7 8 Total Profit $75.00 Bill of Materials 2 1 2 2 Tables 3 Chairs 1 Production Quantity: Adjustable Cells Cell Name $C$11 Production Quantity: Tables $D$11 Production Quantity: Chairs Constraints Cell Name $E$7 Large Bricks Total Used $E$8 Small Bricks Total Used Final Value 7 8 Shadow Price 5 5 Constraint R.H. Side 7 8 Allowable Increase 1 6 Allowable Decrease 3 1 Final Reduced Value Cost 3 0 1 0 Objective Coefficient 20 15 Allowable Increase 10 5 Allowable Decrease 5 5 38 9 Large Bricks B 3 4 5 6 7 8 9 10 11 C D E F G Profit Large Bricks Small Bricks Tables $20.00 Chairs $15.00 Total Used 8 8 <= <= Available 9 8 Total Profit $80.00 Bill of Materials 2 1 2 2 Tables 4 Chairs 0 Production Quantity: Adjustable Cells Cell Name $C$11 Production Quantity: Tables $D$11 Production Quantity: Chairs Constraints Cell Name $E$7 Large Bricks Total Used $E$8 Small Bricks Total Used Final Value 8 8 Shadow Price 0 10 Constraint R.H. Side 9 8 Allowable Increase 1E+30 1 Allowable Decrease 1 8 Final Reduced Value Cost 4 0 0 -5 Objective Coefficient 20 15 Allowable Increase 1E+30 5 Allowable Decrease 5 1E+30 39 100% Rule for Simultaneous Changes in the Objective Coefficients For simultaneous changes in the objective coefficients, if the sum of the percentage changes does not exceed 100%, the original solution will still be optimal. (If it does exceed 100%, we cannot be sure—it may or may not change.) Adjustable Cells Cell Name $C$11 Production Quantity: Tables $D$11 Production Quantity: Chairs Constraints Cell $E$7 $E$8 Name Large Bricks Total Used Small Bricks Total Used Final Value 6 8 Shadow Price 5 5 Constraint R.H. Side 6 8 Allowable Increase 2 4 Allowable Decrease 2 2 Final Reduced Value Cost 2 0 2 0 Objective Coefficient 20 15 Allowable Increase 10 5 Allowable Decrease 5 5 Examples: (Does solution stay the same?) Profit per Table = $24 Profit per Table = $25 Profit per Table = $28 & & & Profit per Chair = $13 Profit per Chair = $12 Profit per Chair = $18 40 100% Rule for Simultaneous Changes in the Right-Hand-Sides For simultaneous changes in the right-hand-sides, if the sum of the percentage changes does not exceed 100%, the shadow prices will still be valid. (If it does exceed 100%, we cannot be sure—they may or may not be valid.) Adjustable Cells Cell Name $C$11 Production Quantity: Tables $D$11 Production Quantity: Chairs Constraints Cell $E$7 $E$8 Name Large Bricks Total Used Small Bricks Total Used Final Value 6 8 Shadow Price 5 5 Constraint R.H. Side 6 8 Allowable Increase 2 4 Allowable Decrease 2 2 Final Reduced Value Cost 2 0 2 0 Objective Coefficient 20 15 Allowable Increase 10 5 Allowable Decrease 5 5 Examples: (Are the shadow prices valid? If so, what’s the new total profit?) (+1 Large Brick) & (+2 Small Bricks) (+1 Large Brick) & (–1 Small Brick) 41 Summary of Sensitivity Report for Changes in the Objective Function Coefficients Final Value The value of the decision variables (changing cells) in the optimal solution. Increase in the objective function value per unit increase in the value of a zero-valued variable (for small increases)—may be interpreted as the shadow price for the nonnegativity constraint. The current value of the objective coefficient. Defines the range of the coefficients in the objective function for which the current solution (value of the decision variables or changing cells in the optimal solution) will not change. Reduced Cost Objective Coefficient Allowable Increase/Decrease 42 Summary of Sensitivity Report for Changes in the Right-Hand-Sides Final Value The usage of the resource (or level of benefit achieved) in the optimal solution—the left-hand side of the constraint. The change in the value of the objective function per unit increase in the right-hand-side of the constraint (RHS): ∆Z = (Shadow Price)(∆RHS) (Note: only valid if change is within the allowable range—see below.) The current value of the right-hand-side of the constraint. Defines the range of values for the RHS for which the shadow price is valid and hence for which the new objective function value can be calculated. (NOT the range for which the current solution will not change.) Shadow Price Constraint R.H. Side Allowable Increase/Decrease 43 ...
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