5-1 Wyndor (Before What-If Analysis) 3 4 5 6 7 8 9 10 11 12 B C D E F G Doors Windows Unit Profit $300 $500 Hours Hours Used Available Plant 1 1 0 2 <= 4 Plant 2 0 2 12 <= 12 Plant 3 3 2 18 <= 18 Doors Windows Total Prof Units Produced 2 6 $3,600 Hours Used Per Unit Produce
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5-1 Wyndor (Before What-If Analysis). 5-2 Using the Spreadsheet to do Sensitivity Analysis The profit per door has been revised from $300 to $200. No.
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Cell Name Value Cost Coefficient Increase Decrease$C$12 Units Produced Doors 2 0 300 450 300$D$12 Units Produced Windows 6 0 500 1E+30 300
5-8
Graphical Insight into the Allowable Range
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.
W
D
(2, 6) is optimal for 0 < PD < 750
PD = 0 (Profit = 0 D + 500 W)
PD = 300 (Profit = 300 D + 500 W)
PD = 750 (Profit = 750 D + 500 W)
Line A
Line C
Line B
0 2 4 6
2
4
6
8
Production rate for doors
Production ratefor windows
Feasibleregion
5-9
Using the Spreadsheet to do Sensitivity Analysis
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.
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.)
5-15
Graphical Insight into 100 Percent Rule
W
D0 2 4 6
2
4
6
8
Production rate for doors
Production rate
for windows
Feasible
region
10
Objective function line now is
Profit = $3150 = 525 D + 350 W
since PD = $525, PW = $350.
Entire line segment is optimal
(4, 3)
(2, 6)
8
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.
5-16
Graphical Insight into 100 Percent Rule
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.
0 2 4 6
2
4
6
8
(2, 6)
Feasible region
Optimal solution
Production rate for doors
Production rate for windows
Profit = $1800 = 150D + 250 W
8
W
D
5-17
Using the Spreadsheet to do Sensitivity Analysis
The hours available in plant 2 have been increased from 12 to 13.The total profit increases by $150 per week.
Select these cells (C19:F26), before choosing the Solver Table.
5-25
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.)
5-26
A Production Problem
Weekly supply of raw materials:
6 Large Bricks
Products:
TableProfit = $20 / Table
ChairProfit = $15 / Chair
8 Small Bricks
5-27
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?
5-28
Graphical Solution (Original Problem)
Maximize Profit = ($20)T + ($15)Csubject to
2T + C ≤ 6 large bricks2T + 2C ≤ 8 small bricks
andT ≥ 0, C ≥ 0.
1 2 3 4 5 6
1
2
3
4
T
C
2T + 2C < 8 small bricks
2T + C < 6 large bricks
Z = ($20)T + ($15)C = $70
Optimal Solution (2, 2). Profit = $70
5-29
7 Large Bricks
Maximize Profit = ($20)T + ($15)Csubject to
2T + C ≤ 7 large bricks2T + 2C ≤ 8 small bricks
andT ≥ 0, C ≥ 0.
1 2 3 4 5 6
1
2
3
4
T
C
2T + 2C < 8 small bricks
2T + C < 7 large bricks
Z = ($20)T + ($15)C = $75
Old Optimal Solution (2, 2). Profit = $70
New Optimal Solution (3, 1). Profit = $75
2T + C < 6 large bricks
5-30
9 Large Bricks
Maximize Profit = ($20)T + ($15)Csubject to
2T + C ≤ 9 large bricks2T + 2C ≤ 8 small bricks
andT ≥ 0, C ≥ 0.
1 2 3 4 5 6
1
2
3
4
T
C
2T + 2C < 8 small bricks
2T + C < 9 large bricks
Z = ($20)T + ($15)C = $80
Old Optimal Solution (2, 2). Profit = $70
New Optimal Solution (4, 0). Profit = $80
2T + C < 6 large bricks
5-31
$25 Profit per Table
Maximize Profit = ($25)T + ($15)Csubject to
2T + C ≤ 6 large bricks2T + 2C ≤ 8 small bricks
andT ≥ 0, C ≥ 0.
1 2 3 4 5 6
1
2
3
4
T
C
2T + 2C < 8 small bricks
2T + C < 6 large bricks
Z = ($25)T + ($15)C = $80
Optimal Solution (2, 2). Profit = $80
5-32
$35 Profit per Table
Maximize Profit = ($35)T + ($15)Csubject to
2T + C ≤ 6 large bricks2T + 2C ≤ 8 small bricks
andT ≥ 0, C ≥ 0.
1 2 3 4 5 6
1
2
3
4
T
C
2T + 2C < 8 small bricks
Z = ($35)T + ($15)C = $105
Old Optimal Solution (2, 2). Profit = $100
New Optimal Solution (3, 0). Profit = $105
5-33
Generating the Sensitivity Report
After solving with Solver, choose “Sensitivity” under reports:
34567891011
B C D E F GTables Chairs
Profit $20.00 $15.00
Total Used AvailableLarge Bricks 2 1 6 <= 6Small Bricks 2 2 8 <= 8
Tables Chairs Total ProfitProduction Quantity: 2 2 $70.00
Cell Name Value Price R.H. Side Increase Decrease$E$7 Large Bricks Total Used 8 0 9 1E+30 1$E$8 Small Bricks Total Used 8 10 8 1 8
5-39
100% Rule for Simultaneous Changesin the Objective Coefficients
For simultaneous changes in the objective coefficients, if the sum of the percentage changes does not exceed 100%, the original solution will still be optimal. (If it does exceed 100%, we cannot be sure—it may or may not change.)
Cell Name Value Price R.H. Side Increase Decrease$E$7 Large Bricks Total Used 6 5 6 2 2$E$8 Small Bricks Total Used 8 5 8 4 2
Examples: (Does solution stay the same?)Profit per Table = $24 & Profit per Chair = $13Profit per Table = $25 & Profit per Chair = $12Profit per Table = $28 & Profit per Chair = $18
5-40
100% Rule for Simultaneous Changesin 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.)