1. Table of Contents CD Chapter 15 (Transportation and Assignment Problems)
The P&T Company Distribution Problem (Section 15.1) 15.2–15.5
Characteristics of Transportation Problems (Section 15.2) 15.6–15.14
Variants of Transportation Problems: Better Products (Section 15.3) 15.15–15.17
Variants of Transportation Problems: Nifty (Section 15.3) 15.18–15.20
Applications of Transportation Problems: Metro Water (Section 15.4) 15.21–15.22
Applications of Transportation Problems: Northern Airplane (Section 15.4) 15.23–15.25
Applications of Transportation Problems: Middletown (Section 15.4) 15.26–15.28
Applications of Transportation Problems: Energetic (Section 15.4) 15.29–15.31
A Case Study: Texago Corp. Site Selection Problem (Section 15.5) 15.32–15.46
Characteristics of Assignment Problems: Sellmore (Section 15.6) 15.47–15.51
Variants of Assignment Problems: Job Shop (Section 15.7)
Variants of Assignment Problems: Better Products (Section 15.7) 15.55
Variants of Assignment Problems: Revised Middletown (Section 15.7) 15.56
McGraw-Hill/Irwin
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

2. P&T Company Distribution Problem
Figure Location of the canneries and warehouses for the P&T Company problem.
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3. Shipping Data Cannery Output Warehouse Allocation Bellingham
75 truckloads
Sacramento
80 truckloads
Eugene
125 truckloads
Salt Lake City
65 truckloads
Albert Lea
100 truckloads
Rapid City
70 truckloads
Total
300 truckloads
Albuquerque
85 truckloads
Table Shipping data for the P&T Company.
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4. Current Shipping Plan From \ To Warehouse Sacramento Salt Lake City
Rapid City
Albuquerque
Cannery
Bellingham 75
Eugene 5 65 55
Albert Lea 15 85
Table Current Shipping Plan for the P&T Company.
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5. Shipping Cost per Truckload
Warehouse
From \ To
Sacramento
Salt Lake City
Rapid City
Albuquerque
Cannery
Bellingham
$464
$513
$654
$867
Eugene
352
416
690
791
Albert Lea
995
682
388
685
Table Shipping Costs per Truckload for the P&T Company.
Total shipping cost = 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685) = $165,595
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6. Terminology for a Transportation Problem
P&T Company Problem
Truckloads of canned peas
Canneries
Warehouses
Output from a cannery
Allocation to a warehouse
Shipping cost per truckload from a cannery to a warehouse
General Model
Units of a commodity
Sources
Destinations
Supply from a source
Demand at a destination
Cost per unit distributed from a source to a destination
Table Terminology for a transportation problem
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7. Characteristics of Transportation Problems
The Requirements Assumption
Each source has a fixed supply of units, where this entire supply must be distributed to the destinations.
Each destination has a fixed demand for units, where this entire demand must be received from the sources.
The Feasible Solutions Property
A transportation problem will have feasible solutions if and only if the sum of its supplies equals the sum of its demands.
The Cost Assumption
The cost of distributing units from any particular source to any particular destination is directly proportional to the number of units distributed.
This cost is just the unit cost of distribution times the number of units distributed.
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8. The Transportation Model
Any problem (whether involving transportation or not) fits the model for a transportation problem if
It can be described completely in terms of a table like Table 15.5 that identifies all the sources, destinations, supplies, demands, and unit costs, and
satisfies both the requirements assumption and the cost assumption.
The objective is to minimize the total cost of distributing the units.
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9. The P&T Co. Transportation Problem
Unit Cost
Destination (Warehouse):
Sacramento
Salt Lake City
Rapid City
Albuquerque
Supply
Source (Cannery)
Bellingham
$464
$513
$654
$867 75
Eugene
352
416
690
791
125
Albert Lea
995
682
388
685
100
Demand 80 65 70 85
Table The data for the P&T Co. problem formulated as a transportation problem.
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10. Spreadsheet Formulation
Figure A spreadsheet formulation of the P&T Co. problem as a transportation problem, including the target cell Total Cost (J17), the changing cells Shipment Quantity (D12:G14), and the optimal shipping plan obtained by the Solver.
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11. Network Representation
Figure The network representation of the P&T Company transportation problem shows all the data in Table 15.5 graphically.
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12. The Transportation Problem is an LP
Let xij = the number of truckloads to ship from cannery i to warehouse j (i = 1, 2, 3; j = 1, 2, 3, 4) Minimize Cost = $464x11 + $513x12 + $654x13 + $867x14 + $352x21 + $416x22 + $690x23 + $791x24 + $995x31 + $682x32 + $388x33 + $685x34 subject to Cannery 1: x11 + x12 + x13 + x14 = 75 Cannery 2: x21 + x22 + x23 + x24 = 125 Cannery 3: x31 + x32 + x33 + x34 = 100 Warehouse 1: x11 + x21 + x31 = 80 Warehouse 2: x12 + x22 + x32 = 65 Warehouse 3: x13 + x23 + x33 = 70 Warehouse 4: x14 + x24 + x34 = 85 and xij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4)
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13. Integer Solutions Property
As long as all its supplies and demands have integer values, any transportation problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its decision variables. Therefore, it is not necessary to add constraints to the model that restrict these variables to only have integer values.
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14. Distribution System at Proctor and Gamble
Proctor and Gamble needed to consolidate and re-design their North American distribution system in the early 1990’s.
50 product categories
60 plants
15 distribution centers
1000 customer zones
Solved many transportation problems (one for each product category).
Goal: find best distribution plan, which plants to keep open, etc.
Closed many plants and distribution centers, and optimized their product sourcing and distribution location.
Implemented in Saved $200 million per year.
For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply Chain”
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15. Better Products (Assigning Plants to Products)
The Better Products Company has decided to initiate the product of four new products, using three plants that currently have excess capacity.
Unit Cost
Product: 1 2 3 4
Capacity Available
Plant
$41
$27
$28
$24 75 40 29 — 23 37 30 27 21 45
Required production 20
Table Data for the Better Products Co. problem.
Question: Which plants should produce which products?
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16. Transportation Problem Formulation
Unit Cost
Destination (Product): 1 2 3 4
Supply
Source(Plant)
$41
$27
$28
$24 75 40 29 — 23 37 30 27 21 45
Demand 20
Table Data for the Better Products Co. problem formulated as a variant of a transportation problem.
15-16

17. Spreadsheet Formulation
Figure A spreadsheet formulation of the Better Products Co. problem as a variant of a transportation problem, including the target cell Total Cost (I16), the changing cells Daily Production (C11:F13), and the optimal production plan obtained by the Solver.
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18. Nifty Co. (Choosing Customers)
The Nifty Company specializes in the production of a single product, which it produces in three plants.
Four customers would like to make major purchases. There will be enough to meet their minimum purchase requirements, but not all of their requested purchases.
Due largely to variations in shipping cost, the net profit per unit sold varies depending on which plant supplies which customer.
Question: How many units should Nifty sell to each customer and how many units should they ship from each plant to each customer?
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19. Data for the Nifty Company
Unit Cost
Product: 1 2 3 4
Capacity Available
Plant
$55
$42
$46
$53
8.000 37 18 32 48
5.000 29 59 51 35
7.000
Required production
9.000
6.000
Table Data for the Nifty Company problem.
Question: How many units should Nifty sell to each customer and how many units should they ship from each plant to each customer?
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20. Spreadsheet Formulation
Figure A spreadsheet formulation of the Nifty Company problem as a variant of a transportation problem, including the target cell Total Profit (I17), the changing cells Shipment (C11:F13), and the optimal shipping plan obtained by the Solver.
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21. Metro Water (Distributing Natural Resources)
Metro Water District is an agency that administers water distribution in a large goegraphic region. The region is arid, so water must be brought in from outside the region.
Sources of imported water: Colombo, Sacron, and Calorie rivers.
Main customers: Cities of Berdoo, Los Devils, San Go, and Hollyglass.
Cost per Acre Foot
Berdoo
Los Devils
San Go
Hollyglass
Available
Colombo River
$160
$130
$220
$170 5
Sacron River
140
130
190
150 6
Calorie River
200
230 —
Needed 2 4
1.5
(million acre feet)
Table Water resources data for Metro Water District.
Question: How much water should Metro take from each river, and how much should they send from each river to each city?
15-21

22. Spreadsheet Formulation
Figure A spreadsheet formulation of the Metro Water District problem as a variant of a transportation problem, including the target cell of Total Cost (I17), the changing cells Water Distribution (C11:F13), and the optimal solution obtained by the Solver.
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23. Northern Airplane (Production Scheduling)
Northern Airplane Company produces commercial airplanes. The last stage in production is to produce the jet engines and install them.
The company must meet the delivery deadline indicated in column 2.
Production and storage costs vary from month to month.
Maximum Production
Unit Cost of Production ($million)
Unit Cost of Storage ($thousand)
Month
Scheduled Installations
Regular Time
Overtime 1 10 20
1.08
1.10 15 2 30
1.11
1.12 3 25 4 5
1.13
1.15
Table Production scheduling data for the Northern Airplane Company problem.
Question: How many engines should be produced in each of the four months so that the total of the production and storage costs will be minimized?
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24. Spreadsheet Formulation
15-24

25. Optimal Production at Northern Airplane
Month
1 (RT)
2 (RT)
3 (RT)
3 (OT)
4 (RT)
Production 20 10 25 5
Installations 15
Stored
Table Optimal production schedule for the Northern Airplane Company.
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26. Middletown School District
Middletown School District is opening a third high school and thus needs to redraw the boundaries for the area of the city that will be assigned to the respective schools.
The city has been divided into 9 tracts with approximately equal populations.
Each school has a minimum and maximum number of students that should be assigned.
The school district management has decided that the appropriate objective is to minimize the average distance that students must travel to school.
Question: How many students from each tract should be assigned to each school?
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27. Data for the Middletown School District
Distance (Miles) to School
Tract 1 2 3
Number of High School Students
2.2
1.9
2.5
500
1.4
1.3
1.7
400
0.5
1.8
1.1
450 4
1.2
0.3
2.0 5
0.9
0.7
1.0 6
1.6
0.6 7
2.7
1.5 8
0.8 9
Minimum enrollment
1,200
1,100
1,000
Maximum enrollment
1,800
1,700
1,500
Table Data for the Middletown School District problem.
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28. Spreadsheet Formulation
Figure A spreadsheet formulation of the Middletown School District problem as a variant of a transportation problem, including the target cell Total Distance (H30), the changing cells Number of Students (C17:E25), and the optimal zoning plan obtained by the Solver.
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29. Energetic (Meeting Energy Needs)
The Energetic Company needs to make plans for the energy systems for a new building.
The energy needs fall into three categories:
electricity (20 units)
heating water (10 units)
heating space (30 units)
The three possible sources of energy are
electricity
natural gas
solar heating unit (limited to 30 units because of roof size)
Question: How should Energetic meet the energy needs for the new building?
15-29

30. Cost Data for Energetic
Unit Cost
Energy Need:
Electricity
Water Heating
Space Heating
Source of Energy
$400
$500
$600
Natural gas —
600
500
Solar heater
300
400
Table Cost data for the Energetic Co. problem.
15-30

31. Spreadsheet Formulation
Figure A spreadsheet formulation of the Energetic Co. problem as a variant of a transportation problem, including the target cell Total Cost (I18), the changing cells Daily Energy Use (D12:F14), and the optimal energy-sourcing plan obtained by the Solver.
15-31

32. Location of Texago’s Facilities
Type of Facility
Locations
Oil fields
1. Several in Texas 2. Several in California 3. Several in Alaska
Refineries
1. Near New Orleans, Lousiana 2. Near Charleston, South Carolina 3. Near Seattle, Washington
Distribution Centers
1. Pittsburgh, Pennsylvania 2. Atlanta, Georgia 3. Kansas City, Missouri 4. San Francisco, California
Table Location of Texago’s current facilities.
15-32

33. Potential Sites for Texago’s New Refinery
Main Advantages
Near Los Angeles, California
1. Near California oil fields. 2. Ready access from Alaska oil fields. 3. Fairly near San Francisco distribution center.
Near Galveston, Texas
1. Near Texas oil fields. 2. Ready access from Middle East imports. 3. Near corporate headquarters.
Near St. Louis, Missouri
1. Low operating costs. 2. Centrally located for distribution centers. 3. Ready access to crude oil via the Mississippi River.
Table Potential sites for Texago’s new refinery and their main advantages.
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34. Production Data for Texago
Refinery
Crude Oil Needed Annually (Million Barrels)
Oil Fields
Crude Oil Produced Annually (Million Barrels)
New Orleans
100
Texas 80
Charleston 60
California
Seattle
Alaska
New site
120
Total
240
360
Needed imports = 360 – 240 = 120
Table Production data for Texago Corp.
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35. Cost Data for Shipping to Refineries
Cost per Unit Shipped to Refinery or Potential Refinery (Millions of Dollars per Million Barrels)
New Orleans
Charleston
Seattle
Los Angeles
Galveston
St. Louis
Source
Texas 2 4 5 3 1
California
Alaska 7
Middle East
Table Cost data for shipping crude oil to a Texago refinery.
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36. Cost Data for Shipping to Distribution Centers
Cost per Unit Shipped to Distribution Center (Millions of Dollars)
Pittsburgh
Atlanta
Kansas City
San Francisco
Refinery
New Orleans
6.5
5.5 6 8
Charleston 7 5 4
Seattle 3
Potential Refinery
Los Angeles 2
Galveston
St. Louis 1
Number of units needed
100 80
Table Cost data for shipping finished product to a distribution center.
15-36

37. Estimated Operating Costs for Refineries
Site
Annual Operating Cost (Millions of Dollars)
Los Angeles
Galveston
St. Louis
620
570
530
Table Estimated operating costs for a Texago refinery at each potential site.
15-37

38. Basic Spreadsheet for Shipping to Refineries
Figure The basic spreadsheet formulation for the Texago transportation problem for shipping crude oil from the oil fields to the refineries, including the new refinery at a site still to be selected. The target cell is Total Cost (J20). Before entering the data for a new site and then clicking on the Solve button, a trial solution of 0 has been entered into each of the changing cells Shipment Quantity (D13:G16).
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39. Shipping to Refineries, Including Los Angeles
Figure The changing cells Shipment Quantity (D13:G16) give Texago management an optimal plan for shipping crude oil if Los Angeles is selected as the new site for the refinery in column G of Figure
15-39

40. Shipping to Refineries, Including Galveston
Figure The changing cells Shipment Quantity (D13:G16) give Texago management an optimal plan for shipping crude oil if Galveston is selected as the new site for the refinery in column G of Figure
15-40

41. Shipping to Refineries, Including St. Louis
Figure The changing cells Shipment Quantity (D13:G16) give Texago management an optimal plan for shipping crude oil if St. Louis is selected as the new site for the refinery in column G of Figure
15-41

42. Basic Spreadsheet for Shipping to D.C.’s
Figure The basic spreadsheet formulation for the Texago transportation problem for shipping finished product from the refineries (including the new one at a site still to be selected) to the distribution centers. The target cell is Total Cost (J20). Before entering the data for a new site and then clicking on the Solve button, a trial solution of 0 has been entered into each of the changing cells Shipment Quantity (D13:G16).
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43. Shipping to D.C.’s When Choose Los Angeles
Figure The changing cells Shipment Quantity (D13:G16) give Texago management an optimal plan for shipping finished product if Los Angeles is selected as the new site for a refinery in rows 8 and 16 of Figure
15-43

44. Shipping to D.C.’s When Choose Galveston
Figure The changing cells Shipment Quantity (D13:G16) give Texago management an optimal plan for shipping finished product if Galveston is selected as the new site for a refinery in rows 8 and 16 of Figure
15-44

45. Shipping to D.C.’s When Choose St. Louis
Figure The changing cells Shipment Quantity (D13:G16) give Texago management an optimal plan for shipping finished product if St. Louis is selected as the new site for a refinery in rows 8 and 16 of Figure
15-45

46. Annual Variable Costs Site Total Cost of Shipping Crude Oil
Total Cost of Shipping Finished Product
Operating Cost for New Refinery
Total Variable Cost
Los Angeles
$880 million
$1.57 billion
$620 million
$3.07 billion
Galveston
920 million
1.63 billion
570 million
3.12 billion
St. Louis
960 million
1.43 billion
530 million
2.92 billion
Table Annual variable costs resulting from the choice of each site for the new Texago refinery.
15-46

47. Sellmore Company Assignment Problem
The marketing manager of Sellmore Company will be holding the company’s annual sales conference soon.
He is hiring four temporary employees:
Ann
Ian
Joan
Sean
Each will handle one of the following four tasks:
Word processing of written presentations
Computer graphics for both oral and written presentations
Preparation of conference packets, including copying and organizing materials
Handling of advance and on-site registration for the conference
Question: Which person should be assigned to which task?
15-47

48. Data for the Sellmore Problem
Required Time per Task (Hours)
Temporary Employee
Word Processing
Graphics
Packets
Registrations
Hourly Wage
Ann 35 41 27 40
$14
Ian 47 45 32 51 12
Joan 39 56 36 43 13
Sean 25 46 15
Table Data for the Sellmore Company problem.
15-48

49. Spreadsheet Formulation
Figure A spreadsheet formulation of the Sellmore Co. problem as an assignment problem, including the target cell Total Cost (J30). The values of 1 in the changing cells Assignment (D24:G27) show the optimal plan obtained by the Solver for assigning the people to the tasks.
15-49

50. The Model for Assignment Problems
Given a set of tasks to be performed and a set of assignees who are available to perform these tasks, the problem is to determine which assignee should be assigned to each task.
To fit the model for an assignment problem, the following assumptions need to be satisfied:
The number of assignees and the number of tasks are the same.
Each assignee is to be assigned to exactly one task.
Each task is to be performed by exactly one assignee.
There is a cost associated with each combination of an assignee performing a task.
The objective is to determine how all the assignments should be made to minimize the total cost.
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51. The Network Representation
Figure The network representation of the Sellmore Co. assignment problem shows all the possible assignments and their costs graphically.
15-51

52. Job Shop (Assigning Machines to Locations)
The Job Shop Company has purchased three new machines of different types.
There are five available locations where the machine could be installed.
Some of these locations are more desirable for particular machines because of their proximity to work centers that will have a heavy work flow to these machines.
Question: How should the machines be assigned to locations?
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53. Materials-Handling Cost Data
Cost per Hour
Location: 1 2 3 4 5
Machine
$13
$16
$12
$14
$15 15 — 13 20 16 7 10 6
Table Materials-handling cost data for the Job Shop Co. problem.
15-53

54. Spreadsheet Formulation
Figure A spreadsheet formulation of the Job Shop Co. problem as a variant of an assignment problem, including the target cell Total Cost (J17). The values of 1 in the changing cells Assignment (C11:G13) show the optimal plan obtained by the Solver for assigning the machines to the locations.
15-54

55. Better Products (No Product Splitting)
Figure In contrast to Figure 15.4, product splitting is not allowed, so the Better Products Co. problem becomes a variant of an assignment problem. The target cell is Total Cost (I24). The values of 1 in the changing cells Assignment (C19:F21) display the optimal production plan obtained by the Solver.
15-55

56. Middletown School District (No Tract Splitting)
Figure In contrast to Figure 15.8, tract splitting is no longer allowed, so the Middletown School District problem becomes a variant of an assignment problem. The target cell is Total Distance (H30). The values of 1 in the changing cells Assignment (C18:E26) show the optimal zoning plan found by the Solver.
15-56