1
2
3
4
5
6
7
8
9
A B C D E F
Toys Subassemblies
Unit Profit $3.00 -$2.50
Used Available
Subassembly A 2 -1 3,000 <= 3,000
Subassembly B 1 -1 1,000 <= 1,000
Toys Subassemblies Total Profit
Production 2,000 1,000 $3,500
Resource Usage
1
2
3
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6
7
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10
11
12
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A B C D E F
Toys Subassemblies
Unit Profit $3.00 -$2.50
Used Available
Subassembly A 2 -1 3,000 <= 3,000
Subassembly B 1 -1 1,000 <= 1,000
Toys Subassemblies Total Profit
Production 2,000 1,000 $3,500
Unit Profit
for Toys Toys Subassemblies Total Profit
2,000 1,000 $3,500
$2.00 1000 0$2,000
$2.25 1000 0$2,250
$2.50 1000 0$2,500
$2.75 2000 1000 $3,000
$3.00 2000 1000 $3,500
$3.25 2000 1000 $4,000
$3.50 2000 1000 $4,500
$3.75 2000 1000 $5,000
$4.00 2000 1000 $5,500
Resource Usage
Production
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2
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7
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10
11
12
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21
22
A B C D E F
Toys Subassemblies
Unit Profit $3.00 -$2.50
Used Available
Subassembly A 2 -1 3,000 <= 3,000
Subassembly B 1 -1 1,000 <= 1,000
Toys Subassemblies Total Profit
Production 2,000 1,000 $3,500
Unit Profit
for Subassemblies Toys Subassemblies Total Profit
2,000 1,000 $3,500
-$3.50 1000 0$3,000
-$3.25 1000 0$3,000
-$3.00 1000 0$3,000
-$2.75 2000 1000 $3,250
-$2.50 2000 1000 $3,500
-$2.25 2000 1000 $3,750
-$2.00 2000 1000 $4,000
-$1.75 2000 1000 $4,250
-$1.50 2000 1000 $4,500
Resource Usage
Production
Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$B$9 Production Toys 2,000 0 3 2 0.5
$C$9 Production Subassemblies 1,000 0 -2.5 1 0.5
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$D$5 Subassembly A Used 3,000 13000 1E+30 1000
$D$6 Subassembly B Used 1,000 21000 500 1E+30
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A B C D E F G H I J K
Toys Subassemblies
Unit Profit $3.00 -$2.50
Used Available
Subassembly A 2 -1 3,000 <= 3,000
Subassembly B 1 -1 1,000 <= 1,000
Toys Subassemblies Total Profit
Production 2,000 1,000 $3,500
Total Profit Unit Profit for Subassemblies
$3,500 -$3.50 -$3.25 -$3.00 -$2.75 -$2.50 -$2.25 -$2.00 -$1.75 -$1.50
$2.00 $2,000 $2,000 $2,000 $2,000 $2,000 $2,000 $2,000 $2,250 $2,500
$2.25 $2,250 $2,250 $2,250 $2,250 $2,250 $2,250 $2,500 $2,750 $3,000
$2.50 $2,500 $2,500 $2,500 $2,500 $2,500 $2,750 $3,000 $3,250 $3,500
Unit Profit $2.75 $2,750 $2,750 $2,750 $2,750 $3,000 $3,250 $3,500 $3,750 $4,000
for Toys $3.00 $3,000 $3,000 $3,000 $3,250 $3,500 $3,750 $4,000 $4,250 $4,500
$3.25 $3,250 $3,250 $3,500 $3,750 $4,000 $4,250 $4,500 $4,750 $5,000
$3.50 $3,500 $3,750 $4,000 $4,250 $4,500 $4,750 $5,000 $5,250 $5,500
$3.75 $4,000 $4,250 $4,500 $4,750 $5,000 $5,250 $5,500 $5,750 $6,000
$4.00 $4,500 $4,750 $5,000 $5,250 $5,500 $5,750 $6,000 $6,250 $6,500
Resource Usage
1
2
3
4
5
6
7
8
9
A B C D E F
Activity 1 Activity 2
Unit Profit $2 $5
Used Available
Resource 1 1 2 10 <= 10
Resource 2 1 3 12 <= 12
Activity 1 Activity 2 Total Profit
Solution 6 2 $22
Resource Usage
Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$B$9 Solution Activity 1 6 0 2 0.5 0.33333
$C$9 Solution Activity 2 2 0 5 1 1
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$D$5 Resource 1 Used 10 110 2 2
$D$6 Resource 2 Used 12 112 3 2
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2
3
4
5
6
7
8
9
A B C D E F
Activity 1 Activity 2
Unit Profit $1 $5
Used Available
Resource 1 1 2 8 <= 10
Resource 2 1 3 12 <= 12
Activity 1 Activity 2 Total Profit
Solution 0 4 $20
Resource Usage
1
2
3
4
5
6
7
8
9
A B C D E F
Activity 1 Activity 2
Unit Profit $3 $5
Used Available
Resource 1 1 2 10 <= 10
Resource 2 1 3 10 <= 12
Activity 1 Activity 2 Total Profit
Solution 10 0$30
Resource Usage
1
2
3
4
5
6
7
8
9
A B C D E F
Activity 1 Activity 2
Unit Profit $2 $2.50
Used Available
Resource 1 1 2 10 <= 10
Resource 2 1 3 10 <= 12
Activity 1 Activity 2 Total Profit
Solution 10 0$20
Resource Usage
1
2
3
4
5
6
7
8
9
A B C D E F
Activity 1 Activity 2
Unit Profit $2 $7.50
Used Available
Resource 1 1 2 8 <= 10
Resource 2 1 3 12 <= 12
Activity 1 Activity 2 Total Profit
Solution 0 4 $30
Resource Usage
1
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40
Solution
A B C D E F
Activity 1 Activity 2
Unit Profit $2 $5
Used Available
Resource 1 1 2 8 <= 10
Resource 2 1 3 12 <= 12
Activity 1 Activity 2 Total Profit
Solution 0 4 $20
Unit Profit for Total
Activity 1 Activity 1 Activity 2 Profit
0 4 $20.00
$1.00 0 4 $20.00
$1.20 0 4 $20.00
$1.40 0 4 $20.00
$1.60 0 4 $20.00
$1.80 6 2 $20.80
$2.00 6 2 $22.00
$2.20 6 2 $23.20
$2.40 6 2 $24.40
$2.60 10 0$26.00
$2.80 10 0$28.00
Resource Usage
Solution
1
2
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35
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37
A B C D E F G H I J K L M
Activity 1 Activity 2
Unit Profit $2 $5
Used Available
Resource 1 1 2 10 <= 10
Resource 2 1 3 12 <= 12
Activity 1 Activity 2 Total Profit
Solution 6 2 $22
Total Profit Unit Profit for Activity 2
$22 $2.50 $3.00 $3.50 $4.00 $4.50 $5.00 $5.50 $6.00 $6.50 $7.00 $7.50
$1.00 $11.00 $12.00 $14.00 $16.00 $18.00 $20.00 $22.00 $24.00 $26.00 $28.00 $30.00
$1.20 $12.20 $13.20 $14.20 $16.00 $18.00 $20.00 $22.00 $24.00 $26.00 $28.00 $30.00
$1.40 $14.00 $14.40 $15.40 $16.40 $18.00 $20.00 $22.00 $24.00 $26.00 $28.00 $30.00
Unit Profit $1.60 $16.00 $16.00 $16.60 $17.60 $18.60 $20.00 $22.00 $24.00 $26.00 $28.00 $30.00
for $1.80 $18.00 $18.00 $18.00 $18.80 $19.80 $20.80 $22.00 $24.00 $26.00 $28.00 $30.00
Activity 1 $2.00 $20.00 $20.00 $20.00 $20.00 $21.00 $22.00 $23.00 $24.00 $26.00 $28.00 $30.00
$2.20 $22.00 $22.00 $22.00 $22.00 $22.20 $23.20 $24.20 $25.20 $26.20 $28.00 $30.00
$2.40 $24.00 $24.00 $24.00 $24.00 $24.00 $24.40 $25.40 $26.40 $27.40 $28.40 $30.00
$2.60 $26.00 $26.00 $26.00 $26.00 $26.00 $26.00 $26.60 $27.60 $28.60 $29.60 $30.60
$2.80 $28.00 $28.00 $28.00 $28.00 $28.00 $28.00 $28.00 $28.80 $29.80 $30.80 $31.80
$3.00 $30.00 $30.00 $30.00 $30.00 $30.00 $30.00 $30.00 $30.00 $31.00 $32.00 $33.00
Solution Unit Profit for Activity 2
(6,2) $2.50 $3.00 $3.50 $4.00 $4.50 $5.00 $5.50 $6.00 $6.50 $7.00 $7.50
$1.00 (6,2) (0,4) (0,4) (0,4) (0,4) (0,4) (0,4) (0,4) (0,4) (0,4) (0,4)
$1.20 (6,2) (6,2) (6,2) (0,4) (0,4) (0,4) (0,4) (0,4) (0,4) (0,4) (0,4)
$1.40 (10,0) (6,2) (6,2) (6,2) (0,4) (0,4) (0,4) (0,4) (0,4) (0,4) (0,4)
Unit Profit $1.60 (10,0) (10,0) (6,2) (6,2) (6,2) (0,4) (0,4) (0,4) (0,4) (0,4) (0,4)
for $1.80 (10,0) (10,0) (10,0) (6,2) (6,2) (6,2) (0,4) (0,4) (0,4) (0,4) (0,4)
Activity 1 $2.00 (10,0) (10,0) (10,0) (6,2) (6,2) (6,2) (6,2) (0,4) (0,4) (0,4) (0,4)
$2.20 (10,0) (10,0) (10,0) (10,0) (6,2) (6,2) (6,2) (6,2) (6,2) (0,4) (0,4)
$2.40 (10,0) (10,0) (10,0) (10,0) (10,0) (6,2) (6,2) (6,2) (6,2) (6,2) (0,4)
$2.60 (10,0) (10,0) (10,0) (10,0) (10,0) (10,0) (6,2) (6,2) (6,2) (6,2) (6,2)
$2.80 (10,0) (10,0) (10,0) (10,0) (10,0) (10,0) (10,0) (6,2) (6,2) (6,2) (6,2)
$3.00 (10,0) (10,0) (10,0) (10,0) (10,0) (10,0) (10,0) (6,2) (6,2) (6,2) (6,2)
Resource Usage
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2
3
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5
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9
10
11
12
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14
15
A B C D E F G H I J K
Big M Company Distribution Problem
Shipping Cost
(per Lathe) Customer 1 Customer 2 Customer 3 Range Name Cells
Factory 1 $700 $900 $800 OrderSize C15:E15
Factory 2 $800 $900 $700 Output H11:H12
ShippingCost C5:E6
Total TotalCost H15
Shipped TotalShippedOut F11:F12
Units Shipped Customer 1 Customer 2 Customer 3 Out Output TotalToCustomer C13:E13
Factory 1 10 2 0 12 =12 UnitsShipped C11:E12
Factory 2 0 6 9 15 =15
Total To Customer 10 8 9
= = = Total Cost
Order Size 10 8 9 $20,500
Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$C$11 Factory 1 Customer 1 10 0700 100 1E+30
$D$11 Factory 1 Customer 2 2 0 900 100 100
$E$11 Factory 1 Customer 3 0 100 800 1E+30 100
$C$12 Factory 2 Customer 1 0 100 800 1E+30 100
$D$12 Factory 2 Customer 2 6 0 900 100 100
$E$12 Factory 2 Customer 3 9 0 700 100 1E+30
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$F$11 Factory 1 Out 12 012 0 1E+30
$F$12 Factory 2 Out 15 015 2 0
$C$13 Total To Customer Customer 1 10 700 10 010
$D$13 Total To Customer Customer 2 8 900 8 0 2
$E$13 Total To Customer Customer 3 9 700 9 0 2
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21
A B C D E F G H I J K L M
Union Airways Personnel Scheduling Problem
6am-2pm 8am-4pm Noon-8pm
4pm-midnight
10pm-6am Range Name Cells
Shift Shift Shift Shift Shift CostPerShift C5:G5
Cost per Shift $170 $160 $175 $180 $195 MinimumNeeded J8:J17
Total Minimum NumberWorking C21:G21
Time Period Shift Works Time Period? (1=yes, 0=no) Working Needed ShiftWorksTimePeriod C8:G17
6am-8am 1 0 0 0 0 48 >= 48 TotalCost J21
8am-10am 1 1 0 0 0 79 >= 79 TotalWorking H8:H17
10am- 12pm 1 1 0 0 0 79 >= 65
12pm-2pm 1 1 1 0 0 118 >= 87
2pm-4pm 0 1 1 0 0 70 >= 64
4pm-6pm 0 0 1 1 0 82 >= 73
6pm-8pm 0 0 1 1 0 82 >= 82
8pm-10pm 0 0 0 1 0 43 >= 43
10pm-12am 0 0 0 1 1 58 >= 52
12am-6am 0 0 0 0 1 15 >= 15
6am-2pm 8am-4pm Noon-8pm
4pm-midnight
10pm-6am
Shift Shift Shift Shift Shift Total Cost
Number Working 48 31 39 43 15 $30,610
1
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A B C D E F G H I J K L M
Union Airways Personnel Scheduling Problem
6am-2pm 8am-4pm Noon-8pm
4pm-midnight
10pm-6am Range Name Cells
Shift Shift Shift Shift Shift CostPerShift C5:G5
Cost per Shift $170 $165 $175 $180 $195 MinimumNeeded J8:J17
Total Minimum NumberWorking C21:G21
Time Period Shift Works Time Period? (1=yes, 0=no) Working Needed ShiftWorksTimePeriod C8:G17
6am-8am 1 0 0 0 0 48 >= 48 TotalCost J21
8am-10am 1 1 0 0 0 79 >= 79 TotalWorking H8:H17
10am- 12pm 1 1 0 0 0 79 >= 65
12pm-2pm 1 1 1 0 0 118 >= 87
2pm-4pm 0 1 1 0 0 70 >= 64
4pm-6pm 0 0 1 1 0 82 >= 73
6pm-8pm 0 0 1 1 0 82 >= 82
8pm-10pm 0 0 0 1 0 43 >= 43
10pm-12am 0 0 0 1 1 58 >= 52
12am-6am 0 0 0 0 1 15 >= 15
6am-2pm 8am-4pm Noon-8pm
4pm-midnight
10pm-6am
Shift Shift Shift Shift Shift Total Cost
Number Working 48 31 39 43 15 $30,765
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
A B C D E F G H I J K L M
Union Airways Personnel Scheduling Problem
6am-2pm 8am-4pm Noon-8pm
4pm-midnight
10pm-6am Range Name Cells
Shift Shift Shift Shift Shift CostPerShift C5:G5
Cost per Shift $170 $160 $175 $170 $195 MinimumNeeded J8:J17
Total Minimum NumberWorking C21:G21
Time Period Shift Works Time Period? (1=yes, 0=no) Working Needed ShiftWorksTimePeriod C8:G17
6am-8am 1 0 0 0 0 48 >= 48 TotalCost J21
8am-10am 1 1 0 0 0 79 >= 79 TotalWorking H8:H17
10am- 12pm 1 1 0 0 0 79 >= 65
12pm-2pm 1 1 1 0 0 112 >= 87
2pm-4pm 0 1 1 0 0 64 >= 64
4pm-6pm 0 0 1 1 0 82 >= 73
6pm-8pm 0 0 1 1 0 82 >= 82
8pm-10pm 0 0 0 1 0 49 >= 43
10pm-12am 0 0 0 1 1 64 >= 52
12am-6am 0 0 0 0 1 15 >= 15
6am-2pm 8am-4pm Noon-8pm
4pm-midnight
10pm-6am
Shift Shift Shift Shift Shift Total Cost
Number Working 48 31 33 49 15 $30,150
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
A B C D E F G H I J K L M
Union Airways Personnel Scheduling Problem
6am-2pm 8am-4pm Noon-8pm
4pm-midnight
10pm-6am Range Name Cells
Shift Shift Shift Shift Shift CostPerShift C5:G5
Cost per Shift $170 $165 $175 $170 $195 MinimumNeeded J8:J17
Total Minimum NumberWorking C21:G21
Time Period Shift Works Time Period? (1=yes, 0=no) Working Needed ShiftWorksTimePeriod C8:G17
6am-8am 1 0 0 0 0 48 >= 48 TotalCost J21
8am-10am 1 1 0 0 0 79 >= 79 TotalWorking H8:H17
10am- 12pm 1 1 0 0 0 79 >= 65
12pm-2pm 1 1 1 0 0 112 >= 87
2pm-4pm 0 1 1 0 0 64 >= 64
4pm-6pm 0 0 1 1 0 82 >= 73
6pm-8pm 0 0 1 1 0 82 >= 82
8pm-10pm 0 0 0 1 0 49 >= 43
10pm-12am 0 0 0 1 1 64 >= 52
12am-6am 0 0 0 0 1 15 >= 15
6am-2pm 8am-4pm Noon-8pm
4pm-midnight
10pm-6am
Shift Shift Shift Shift Shift Total Cost
Number Working 48 31 33 49 15 $30,305
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
A B C D E F G H I J K L M
Union Airways Personnel Scheduling Problem
6am-2pm 8am-4pm Noon-8pm
4pm-midnight
10pm-6am Range Name Cells
Shift Shift Shift Shift Shift CostPerShift C5:G5
Cost per Shift $166 $164 $171 $184 $199 MinimumNeeded J8:J17
Total Minimum NumberWorking C21:G21
Time Period Shift Works Time Period? (1=yes, 0=no) Working Needed ShiftWorksTimePeriod C8:G17
6am-8am 1 0 0 0 0 48 >= 48 TotalCost J21
8am-10am 1 1 0 0 0 79 >= 79 TotalWorking H8:H17
10am- 12pm 1 1 0 0 0 79 >= 65
12pm-2pm 1 1 1 0 0 118 >= 87
2pm-4pm 0 1 1 0 0 70 >= 64
4pm-6pm 0 0 1 1 0 82 >= 73
6pm-8pm 0 0 1 1 0 82 >= 82
8pm-10pm 0 0 0 1 0 43 >= 43
10pm-12am 0 0 0 1 1 58 >= 52
12am-6am 0 0 0 0 1 15 >= 15
6am-2pm 8am-4pm Noon-8pm
4pm-midnight
10pm-6am
Shift Shift Shift Shift Shift Total Cost
Number Working 48 31 39 43 15 $30,618