1. Irwin/McGraw-Hill © The McGraw-Hill Companies, Inc., 2003 6.1 Chapter 6 Transportation and Assignment Problems

2. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.2 Transportation Problem The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply s i ) to n destinations (each with a demand d j ), when the unit shipping cost from an origin, i, to a destination, j, is c ij.

3. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.3 Network Representation of Transportation Problem

4. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.4

5. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.5 P&T Company Distribution Problem

6. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.6 Shipping Data CanneryOutputWarehouseAllocation Bellingham75 truckloadsSacramento80 truckloads Eugene125 truckloadsSalt Lake City65 truckloads Albert Lea100 truckloadsRapid City70 truckloads Total300 truckloadsAlbuquerque85 truckloads Total300 truckloads

7. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.7 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.

8. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.8 The Transportation Model Any problem (whether involving transportation or not) fits the model for a transportation problem if 1.It can be described completely in terms of a table like Table 6.5 that identifies all the sources, destinations, supplies, demands, and unit costs, and 2.satisfies both the requirements assumption and the cost assumption. The objective is to minimize the total cost of distributing the units.

9. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.9 Network Representation

10. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.10 The Transportation Problem is an LP Let x ij = the number of truckloads to ship from cannery i to warehouse j (i = 1, 2, 3; j = 1, 2, 3, 4) Minimize Cost = $464x 11 + $513x 12 + $654x 13 + $867x 14 + $352x 21 + $416x 22 + $690x 23 + $791x 24 + $995x 31 + $682x 32 + $388x 33 + $685x 34 subject to Cannery 1:x 11 + x 12 + x 13 + x 14 = 75 Cannery 2:x 21 + x 22 + x 23 + x 24 = 125 Cannery 3:x 31 + x 32 + x 33 + x 34 = 100 Warehouse 1:x 11 + x 21 + x 31 = 80 Warehouse 2:x 12 + x 22 + x 32 = 65 Warehouse 3:x 13 + x 23 + x 33 = 70 Warehouse 4:x 14 + x 24 + x 34 = 85 and x ij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4)

11. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.11 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.

12. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.12 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 1996. Saved $200 million per year.

13. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.13 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:1234 Capacity Available Plant 1$41$27$28$2475 24029—2375 33730272145 Required production2030 40 Question: Which plants should produce which products?

14. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.14 Transportation Problem Formulation Unit Cost Destination (Product):1234Supply Source(Plant) 1$41$27$28$2475 24029—2375 33730272145 Demand2030 40

15. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.15 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?

16. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.16 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 BerdooLos DevilsSan GoHollyglassAvailable Colombo River$160$130$220$1705 Sacron River1401301901506 Calorie River190200230—5 Needed2541.5 (million acre feet) Question: How much water should Metro take from each river, and how much should they send from each river to each city?

17. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.17 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 TimeOvertime Regular TimeOvertime 11020101.081.1015 2 30151.111.1215 325 101.101.1115 4205101.131.15 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?

18. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.18 Optimal Production at Northern Airplane Month 1 (RT) 2 (RT) 3 (RT) 3 (OT) 4 (RT) Production 20 10 25 10 5 Installations 10 15 25 0 20 Stored 10 5 10 0

19. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.19 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?

20. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.20 Data for the Middletown School District Distance (Miles) to School Tract123 Number of High School Students 12.21.92.5500 21.41.31.7400 30.51.81.1450 41.20.32.0400 50.90.71.0500 61.11.60.6450 72.70.71.5450 81.81.20.8400 91.51.70.7500 Minimum enrollment1,2001,1001,000 Maximum enrollment1,8001,7001,500

21. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.21 Assignment Problem An assignment problem seeks to minimize the total cost assignment of m workers to m jobs, given that the cost of worker i performing job j is c ij. It assumes all workers are assigned and each job is performed. An assignment problem is a special case of a transportation problem in which all supplies and all demands are equal to 1; hence assignment problems may be solved as linear programs. The network representation of an assignment problem with three workers and three jobs is shown on the next slide.

22. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.22 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: 1.The number of assignees and the number of tasks are the same. 2.Each assignee is to be assigned to exactly one task. 3.Each task is to be performed by exactly one assignee. 4.There is a cost associated with each combination of an assignee performing a task. 5.The objective is to determine how all the assignments should be made to minimize the total cost.

23. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.23 Network Representation of Assignment Problem

24. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.24

25. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.25 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?

26. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.26 Data for the Sellmore Problem Required Time per Task (Hours) Temporary Employee Word ProcessingGraphicsPacketsRegistrations Hourly Wage Ann35412740$14 Ian4745325112 Joan3956364313 Sean3251254615

27. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.27 The Network Representation

28. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.28 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?

29. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.29 Materials-Handling Cost Data Cost per Hour Location:12345 Machine 1$13$16$12$14$15 215—132016 3471067

30. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.30 Assignment Problem Example The coach of a swim team needs to assign swimmers to a 200-yard medley relay team (four swimmers, each swims 50 yards of one of the four strokes). Since most of the best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and their best times (in seconds) they have achieved in each of the strokes (for 50 yards) are shown below. BackstrokeBreaststrokeButterflyFreestyle Carl37.743.433.329.2 Chris32.933.128.526.4 David33.842.238.929.6 Tony37.034.730.428.5 Ken35.441.833.631.1 Question: How should the swimmers be assigned to make the fastest relay team?

31. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.31 Spreadsheet Formulation

32. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.32 Bidding for Classes In the MBA program at a prestigious university in the Pacific Northwest, students bid for electives in the second year of their program. Each of the 10 students has 100 points to bid (total) and must take two electives. There are four electives available: –Quantitative Methods –Finance –Operations Management –Accounting Each class is limited to 5 students. Question: How should students be assigned to the classes?

33. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.33 Points Bid for Electives Electives Student Quantitative MethodsFinance Operations ManagementAccounting George6010 20 Fred20 4020 Ann45 55 Eric5020525 Susan30 10 Liz50 00 Ed7020100 David25 3515 Tony35153515 Jennifer6010 20

34. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.34 Spreadsheet Solution (Maximizing Total Points)

35. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.35 Spreadsheet Solution (Maximizing the Minimum Student Point Total)