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318
PROBLEM SUMMARY
1. Balanced transportation
2. Balanced transportation
3. Short answer, discussion
4. Unbalanced transportation
5. Balanced transportation
6. Unbalanced transportation
7. Unbalanced transportation
8. Unbalanced transportation
9. Unbalanced transportation, multiple optimal
10. Sensitivity analysis (B–9)
11. Unbalanced transportation, multiple optimal
12. Unbalanced transportation, degenerate
13. Unbalanced transportation, degenerate
14. Balanced transportation
15. Balanced transportation
16. Sensitivity analysis (B–15)
17. Unbalanced transportation, multiple optimal
18. Sensitivity analysis (B–17)
19. Shortage costs (B–17)
20. Unbalanced transportation
21. Unbalanced transportation, multiple optimal
22. Balanced transportation
23. Unbalanced transportation, multiple optimal
24. Sensitivity analysis (B–23)
25. Unbalanced transportation
26. Sensitivity analysis (B–25)
27. Sensitivity analysis (B–25)
28. Unbalanced transportation
29. Unbalanced transportation, degenerate
30. Unbalanced transportation
31. Unbalanced transportation, production scheduling(B–30)
32. Unbalanced transportation
33. Sensitivity analysis (B–32)
34. Shortage costs
35. Multiperiod scheduling
36. Unbalanced assignment, LP formulation
37. Assignment
38. Assignment
39. Assignment
40. Unbalanced assignment, multiple optimal
41. Assignment, multiple optimal
42. Assignment
43. Unbalanced assignment, multiple optimal
44. Balanced assignment
45. Balanced assignment (B–9)
46. Unbalanced assignment
47. Unbalanced assignment
48. Unbalanced assignment
49. Unbalanced assignment, prohibited assignment
50. Unbalanced assignment, multiple optimal
51. Assignment
52. Unbalanced assignment (maximization)
PROBLEM SOLUTIONS
1. Using the VAM initial solutions:
Module B: Transportation and Assignment Solution Methods
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$1290
b.) Solutions achieved using minimum cost cell method is optimal
Optimal (with multiple optimal solutions)Total travel time = 20,700 minutes
FromTo
A B C D
North
South
East
West
12 23 35 17
26 15 21 27
18 20 22 31
29 24 35 10
15 10 23 16
0 0 0 0
Central
Dummy
Demand
Supply
250 250
340300 40
200 200
150 310
210 210
160
100 190 290
400 400 400 400 1,600
FromTo
A B C D
North
South
East
West
12 23 35 17
26 15 21 27
18 20 22 31
29 24 35 10
15 10 23 16Central
Demand
Supply
250 250
340300
310
210 210
310
100 4010 140 290
350 350 350 350 1,400
–7
24.
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Demand 350 350 350 350 1,400
FromTo
A B C D
North
South
East
West
12 23 35 17
26 15 21 27
18 20 22 31
29 24 35 10
15 10 23 16Central
Supply
250 250
340300 40
310
210 210
310
100 50 140 290
–3
338
Allocate 40 students to cell (South, C):
Allocate 100 students to cell (East, A):
Optimal. Total travel time = 21,200 minutes.
The overall travel time increased by 500 minutes,which divided by all 1,400 students is only an increaseof .357 minutes per student. This does not seem to bea significantly large increase.
Demand 350 350 350 350 1,400
FromTo
A B C D
North
South
East
West
12 23 35 17
26 15 21 27
18 20 22 31
29 24 35 10
15 10 23 16Central
Supply
250 250
340200 140
100 310
210 210
210
150 140 290
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Demand
FromTo
1 2 3 4
A
B
C
D
9 8 11 12 7 8 0
10 10 8 6
8 6 6 5
4 6 9 5
12 10 8 9
9
7
8
6
7
4
10
7
0
0
0
0E
5 6
8 18 26
13 27 40
20 20
40
25 15 4 1 45
25 15 30 18 27 35 21 171
SupplyDummy
Demand
FromTo
1 2 3 4
A
B
C
D
9 8 11 12 7 8 0
10 10 8 6
8 6 6 5
4 6 9 5
12 10 8 9
9
7
8
6
7
4
10
7
0
0
0
0E
5 6
8 18 26
13 27 40
20 20
5 35 40
25 2 17 1 45
25 15 30 18 27 35 21 171
SupplyDummy
0
0
0
0
5 35
339
23. The initial solution is determined using VAM, as follows.
Optimal, Total Profit = $1,528 with multiple optimalsolutions at (B,2), (E,4) and (B, Dummy)
Alternate, allocate 13 cases to (B,2)
25.
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Alternate, allocate 17 cases to (E,4)
Demand
FromTo
1 2 3 4
A
B
C
D
9 8 11 12 7 8 0
10 10 8 6
8 6 6 5
4 6 9 5
12 10 8 9
9
7
8
6
7
4
10
7
0
0
0
0E
5 6
25 1 26
13 27 40
20 20
5 35 40
25 2 17 1 45
25 15 30 18 27 35 21 171
SupplyDummy
24. If Easy Time purchased all the baby food demanded at each store from the distributortotal profit would be $1,246, which is less thanbuying it from the other locations asdetermined in problem 25. This profit iscomputed by multiplying the profit at eachstore by the demand. In order to determine ifsome of the demand should be met by thedistributor a new source (F) must be added tothe transportation tableau from problem 25.This source represents the distributor and hasan available supply of 150 cases, the totaldemand from all the stores. The tableau andoptimal solution is shown as follows.
Demand
FromTo
1 2 3 4
A
B
C
D
9 8 11 12 7 8 0
10 10 8 6
8 6 6 5
4 6 9 5
9 8 9 9
9
7
8
7
7
4
10
8
7
0
0
0
0
12 10 8 9 6 0E
F
5 6
8 18 26
27 13 40
20 20
35 5 40
25 15 5 45
22 128 150
25 15 30 18 27 35 171 321
SupplyDummy
Optimal, Total profit = $1,545 with multipleoptimal solutions
26.
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This new solution results in a greater profit than thesolution in problem 25 ($1,545 > $1,528). Thus, someof the demand (specifically at store 3) should be metfrom the distributors.
25. Solve the model as a linear programming model to obtain the shadow prices. Among the5 purchase locations, the store at Albany hasthe highest shadow price of $3. The sensitivityrange for supply at Albany is 25 ≤ q1 ≤ 43.Thus, as much as 17 additional cases can bepurchased from Albany which would increaseprofit by $51 for a total of $1,579.
From ToFrom Denver St. Paul Louisville Akron Topeka Dummy Supply
3.7 4.6 4.9 5.5 4.3 0Sac. 13 5 18
3.4 5.1 4.4 5.9 5.2 0Bak. 8 2 5 15
3.3 4.1 3.7 2.9 2.6 0San. 10 10
1.9 4.2 2.7 5.4 3.9 0Mont. 12 12
6.1 5.1 3.8 2.5 4.1 0Jack. 15 5 20
6.6 4.8 3.5 3.6 4.5 0Ocala 15 15
Demand 20 15 15 15 20 5 90
Total cost, Z = $278,000
It is cheaper for National Foods to continue to operate its own trucking firm.
33. Increasing the supply at Sacramento, Jacksonville and Ocala to 25 tons would have little effect, reducing the over-all monthly shipping cost to $276,000, which is still higher than the $245,000 the company is currently spendingwith its own trucks.
Alternatively, increasing the supply at San Antonio and Montgomery to 25 tons per month reduces the monthlyshipping cost to $242,500 which is less than the company’s cost with their own trucks.
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From To
From Hong Kong Singapore Taipei Supply
300 210 340
L.A. 150 300 450
490 520 610
Savannah 400 200 600
360 320 500
Galveston 350 350
Order 800 920 1100
Shortages 200 200
Orders 600 500 500 1600
Z = $723,500
Penalty costs = 200 x $800 = $160,000
Period of Production
Period of Use 1 2 3 4 Capacity
0 2 4 6Beginning Inventory 300 300
20 22 24 26Regular 8,700 300 9,000
1 25 27 29 31Overtime 1000 1,000
27 29 31 33Subcontract 3,000
M 20 22 24Regular 10,000 10,000
2 M 25 27 29Overtime 700 800 1,500
M 27 29 31Subcontract 200 3,000
M M 20 22Regular 12,000 12,000
3 M M 25 27Overtime 2,000 2,000
M M 27 29Subcontract 1,000 2,000 3,000
M M M 20Regular 12,000 12,000
4 M M M 25Overtime 2,000 2,000
M M M 27Subcontract 3,000 3,000
Demand 9,000 12,000 16,000 19,000
Total cost; Z = $1,198,500; multiple optimal solutions exist
35.
34.
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36. 37.
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38.
39.
40.
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41.
42.
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43.
44. a) It actually can be solved by either method—transportation or assignment. However, if theassignment method is used there will be 9 desti-nations, with each city repeated 3 times.
b) The solution with the transportation method.
From ToFrom Athens Columbia Nashville Supply
165 90 1301 1 1
75 210 3202 1 1
180 170 1403 1 1
220 80 604 1 1
410 140 805 1 1
150 170 1906 1 1
170 110 1507 1 1
105 125 1608 1 1
240 200 1559 1 1
Demand 3 3 3 9
Total mileage, Z = 985; multiple optimal solutions exist.
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45. This changes the solution and increases the totalmileage. The new assignments are:
Officials 3, 6 and 7 – AthensOfficials 1, 2 and 8 – ColumbiaOfficials 4, 5 and 9 – Nashville