1
2
3
4
5
6
7
8
9
A B C D E F
Activity 1 Activity 2
Unit Profit $1 $2
Used Available
Resource 1 1 3 8 <= 8
Resource 2 1 1 4 <= 4
Activity 1 Activity 2 Total Profit
Solution 2 2 $6.00
Resource Usage
1
2
3
4
5
6
7
8
9
A B C D E F
Activity 1 Activity 2
Unit Profit $1 $2
Used Available
Resource 1 1 3 9 <= 9
Resource 2 1 1 4 <= 4
Activity 1 Activity 2 Total Profit
Solution 1.5 2.5 $6.50
Resource Usage
1
2
3
4
5
6
7
8
9
A B C D E F
Activity 1 Activity 2
Unit Profit $1 $2
Used Available
Resource 1 1 3 8 <= 8
Resource 2 1 1 5 <= 5
Activity 1 Activity 2 Total Profit
Solution 3.5 1.5 $6.50
Resource Usage
1
2
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
36
37
38
39
40
41
Solution
A B C D E F
Activity 1 Activity 2
Unit Profit $1 $2
Used Available
Resource 1 1 3 8 <= 8
Resource 2 1 1 4 <= 4
Activity 1 Activity 2 Total Profit
Solution 2 2 $6.00
Available Total Incremental
Resource 1 Activity 1 Activity 2 Profit Profit
2 2 $6.00
4 4 0 $4.00
53.5 0.5 $4.50 $0.50
6 3 1 $5.00 $0.50
72.5 1.5 $5.50 $0.50
8 2 2 $6.00 $0.50
91.5 2.5 $6.50 $0.50
10 1 3 $7.00 $0.50
11 0.5 3.5 $7.50 $0.50
Resource Usage
Solution
1
2
3
4
5
6
7
8
9
10
A B C D E F
Activity 1 Activity 2
Unit Profit $3 $2
Used Available
Resource 1 1 0 3 <= 4
Resource 2 1 3 15 <= 15
Resource 3 2 1 10 <= 10
Activity 1 Activity 2 Total Profit
Solution 3 4 $17.00
Resource Usage
1
2
3
4
5
6
7
8
9
10
A B C D E F
Activity 1 Activity 2
Unit Profit $3 $2
Used Available
Resource 1 1 0 3 <= 5
Resource 2 1 3 15 <= 15
Resource 3 2 1 10 <= 10
Activity 1 Activity 2 Total Profit
Solution 3 4 $17.00
Resource Usage
1
2
3
4
5
6
7
8
9
10
A B C D E F
Activity 1 Activity 2
Unit Profit $3 $2
Used Available
Resource 1 1 0 2.8 <= 4
Resource 2 1 3 16 <= 16
Resource 3 2 1 10 <= 10
Activity 1 Activity 2 Total Profit
Solution 2.8 4.4 $17.20
Resource Usage
1
2
3
4
5
6
7
8
9
10
A B C D E F
Activity 1 Activity 2
Unit Profit $3 $2
Used Available
Resource 1 1 0 3.6 <= 4
Resource 2 1 3 15 <= 15
Resource 3 2 1 11 <= 11
Activity 1 Activity 2 Total Profit
Solution 3.6 3.8 $18.40
Resource Usage
1
2
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
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
A B C D E F
Activity 1 Activity 2
Unit Profit $3 $2
Used Available
Resource 1 1 0 3 <= 4
Resource 2 1 3 15 <= 15
Resource 3 2 1 10 <= 10
Activity 1 Activity 2 Total Profit
Solution 3 4 $17.00
Resource Usage
53
54
55
56
57
58
A B C D E F
11 3.6 3.8 $18.40 $1.40
12 43.667 $19.33 $0.93
13 43.667 $19.33 $0.00
14 43.667 $19.33 $0.00
15 43.667 $19.33 $0.00
16 43.667 $19.33 $0.00
Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost
Coefficient
Increase Decrease
$B$10 Solution Activity 1 3 0 3 1 2.333
$C$10 Solution Activity 2 4 0 2 7 0.5
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$D$5 Resource 1 Used 3 0 4 1E+30 1
$D$6 Resource 2 Used 15 0.2 15 15 5
$D$7 Resource 3 Used 10 1.4 10 1.667 5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A B C D E F G H I J K
Super Grain Corp. Advertising-Mix Problem
TV Spots Magazine Ads SS Ads Range Name Cells
Exposures per Ad 1,300 600 500 BudgetAvailable H8:H9
(thousands) BudgetSpent F8:F9
Budget Budget CostPerAd C8:E9
Cost per Ad ($thousands) Spent Available ExposuresPerAd C4:E4
Ad Budget 300 150 100 4,000 <= 4,000 MaxTVSpots C15
Planning Budget 90 30 40 1,000 <= 1,000 NumberOfAds C13:E13
TotalExposures H13
Total Exposures TVSpots C13
TV Spots Magazine Ads SS Ads (thousands)
Number of Ads 0 20 10 17,000
<=
Max TV Spots 5
Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost
Coefficient
Increase Decrease
$C$13 TVSpots 0 -50 1300 50 1E+30
$D$13 Number of Ads Magazine Ads 20 0600 150 50
$E$13 Number of Ads SS Ads 10 0500 300 33.333
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$F$8 Ad Budget Spent 4,000 34000 1000 1500
$F$9 Planning Budget Spent 1,000 51000 600 200
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
A B C D E F G H I J K
Super Grain Corp. Advertising-Mix Problem
TV Spots Magazine Ads SS Ads Range Name Cells
Exposures per Ad 1,300 600 500 BudgetAvailable H7:H8
(thousands) BudgetSpent F7:F8
Cost per Ad ($thousands) Budget Spent Budget Available CostPerAd C7:E8
Ad Budget 300 150 100 3,775 <= 4,000 CouponRedemptionPerAd C15:E15
Planning Budget 90 30 40 1,000 <= 1,000 ExposuresPerAd C4:E4
MaxTVSpots C21
Number Reached per Ad (millions) Total Reached Minimum Acceptable MinimumAcceptable H11:H12
Young Children 1.2 0.1 0 5 >= 5 NumberOfAds C19:E19
Parents of Young Children 0.5 0.2 0.2 5.85 >= 5 NumberReachedPerAd C11:E12
RequiredAmount H15
TV Spots Magazine Ads SS Ads Total Redeemed Required Amount TotalExposures H19
Coupon Redemption per Ad 0 40 120 1,490 = 1,490 TotalReached F11:F12
($thousands) TotalRedeemed F15
Total Exposures TVSpots C19
TV Spots Magazine Ads SS Ads (thousands)
Number of Ads 3 14 7.75 16,175
<=
Maximum TV Spots 5
Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost
Coefficient
Increase Decrease
$C$19 TVSpots 3 0 1300 1040 1E+30
$D$19 Number of Ads Magazine Ads 14 0600 1E+30 192.59
$E$19 Number of Ads SS Ads 7.75 0500 577.78 1E+30
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$F$7 Ad Budget Budget Spent 3,775 04000 1E+30 225
$F$8 Planning Budget Budget Spent 1,000 35 1000 22.5 85
$F$15 TotalRedeemed 1,490 -8 1490 385 90
$F$11 Young Children Total Reached 5 -1575.76 5 1.32 0.45
$F$12 Parents of Young Children Total Reached 5.85 0 5 0.85 1E+30
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 $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
Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost
Coefficient
Increase Decrease
$C$21 Number Working Shift 48 0170 1E+30 10
$D$21 Number Working Shift 31 0160 10 160
$E$21 Number Working Shift 39 0175 5175
$F$21 Number Working Shift 43 0180 1E+30 5
$G$21 Number Working Shift 15 0195 1E+30 195
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$H$8 6am-8am Working 48 10 48 648
$H$9 8am-10am Working 79 160 79 1E+30 6
$H$10 10am– 12pm Working 79 065 14 1E+30
$H$11 12pm-2pm Working 118 087 31 1E+30
$H$12 2pm-4pm Working 70 064 6 1E+30
$H$13 4pm-6pm Working 82 073 9 1E+30
$H$14 6pm-8pm Working 82 175 82 1E+30 6
$H$15 8pm-10pm Working 43 543 6 6
$H$16 10pm-12am Working 58 052 6 1E+30
$H$17 12am-6am Working 15 195 15 1E+30 6