Table of Contents Chapter 6 (Transportation and Assignment Problems)

1. The P&T Company Distribution Problem (Section 6.1) 6.2–6.5 Characteristics of Transportation Problems (Section 6.2) 6.6–6.14 Variants of Transportation Problems: Better Products (Section 6.3) 6.15–6.17 Variants of Transportation Problems: Nifty (Section 6.3) 6.18–6.20 Applications of Transportation Problems: Metro Water (Section 6.4) 6.21–6.22 Applications of Transportation Problems: Northern Airplane (Section 6.4) 6.23–6.25 Applications of Transportation Problems: Middletown (Section 6.4) 6.26–6.28 Applications of Transportation Problems: Energetic (Section 6.4) 6.29–6.31 A Case Study: Texago Corp. Site Selection Problem (Section 6.5) 6.32–6.46 Characteristics of Assignment Problems: Sellmore (Section 6.6) 6.47–6.51 Variants of Assignment Problems: Job Shop (Section 6.7) 6.52-6.54 Variants of Assignment Problems: Better Products (Section 6.7) 6.55 Variants of Assignment Problems: Revised Middletown (Section 6.7) 6.56 Transportation & Assignment Problems (UW Lecture) 6.57–6.75 These slides are based upon a lecture to second-year MBA students at the University of Washington that discusses transportation and assignment problems (as taught by one of the authors). Table of ContentsChapter 6 (Transportation and Assignment Problems) © The McGraw-Hill Companies, Inc., 2003

2. P&T Company Distribution Problem © The McGraw-Hill Companies, Inc., 2003

3. Shipping Data © The McGraw-Hill Companies, Inc., 2003

4. Current Shipping Plan © The McGraw-Hill Companies, Inc., 2003

5. Shipping Cost per Truckload Total shipping cost = 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685) = $165,595 © The McGraw-Hill Companies, Inc., 2003

6. Terminology for a Transportation Problem © The McGraw-Hill Companies, Inc., 2003

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. © The McGraw-Hill Companies, Inc., 2003

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 6.5 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. © The McGraw-Hill Companies, Inc., 2003

9. The P&T Co. Transportation Problem © The McGraw-Hill Companies, Inc., 2003

10. Spreadsheet Formulation © The McGraw-Hill Companies, Inc., 2003

11. Network Representation © The McGraw-Hill Companies, Inc., 2003

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 + $685x34subject 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 = 85andxij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4) © The McGraw-Hill Companies, Inc., 2003

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. © The McGraw-Hill Companies, Inc., 2003

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 1996. 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”, downloadable at www.mhhe.com/hillier2e/articles © The McGraw-Hill Companies, Inc., 2003

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. Question: Which plants should produce which products? © The McGraw-Hill Companies, Inc., 2003

16. Transportation Problem Formulation © The McGraw-Hill Companies, Inc., 2003

17. Spreadsheet Formulation © The McGraw-Hill Companies, Inc., 2003

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? © The McGraw-Hill Companies, Inc., 2003

19. Data for the Nifty Company Question: How many units should Nifty sell to each customer and how many units should they ship from each plant to each customer? © The McGraw-Hill Companies, Inc., 2003

20. Spreadsheet Formulation © The McGraw-Hill Companies, Inc., 2003

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. Question: How much water should Metro take from each river, and how much should they send from each river to each city? © The McGraw-Hill Companies, Inc., 2003

22. Spreadsheet Formulation © The McGraw-Hill Companies, Inc., 2003

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. 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? © The McGraw-Hill Companies, Inc., 2003

24. Spreadsheet Formulation © The McGraw-Hill Companies, Inc., 2003

25. Optimal Production at Northern Airplane © The McGraw-Hill Companies, Inc., 2003

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? © The McGraw-Hill Companies, Inc., 2003

27. Data for the Middletown School District © The McGraw-Hill Companies, Inc., 2003

28. Spreadsheet Formulation © The McGraw-Hill Companies, Inc., 2003

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? © The McGraw-Hill Companies, Inc., 2003

30. Cost Data for Energetic © The McGraw-Hill Companies, Inc., 2003

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

32. Location of Texago’s Facilities © The McGraw-Hill Companies, Inc., 2003

33. Potential Sites for Texago’s New Refinery © The McGraw-Hill Companies, Inc., 2003

34. Production Data for Texago © The McGraw-Hill Companies, Inc., 2003

35. Cost Data for Shipping to Refineries © The McGraw-Hill Companies, Inc., 2003

36. Cost Data for Shipping to Distribution Centers © The McGraw-Hill Companies, Inc., 2003

37. Estimated Operating Costs for Refineries © The McGraw-Hill Companies, Inc., 2003

38. Basic Spreadsheet for Shipping to Refineries © The McGraw-Hill Companies, Inc., 2003

39. Shipping to Refineries, Including Los Angeles © The McGraw-Hill Companies, Inc., 2003

40. Shipping to Refineries, Including Galveston © The McGraw-Hill Companies, Inc., 2003

41. Shipping to Refineries, Including St. Louis © The McGraw-Hill Companies, Inc., 2003

42. Basic Spreadsheet for Shipping to D.C.’s © The McGraw-Hill Companies, Inc., 2003

43. Shipping to D.C.’s When Choose Los Angeles © The McGraw-Hill Companies, Inc., 2003

44. Shipping to D.C.’s When Choose Galveston © The McGraw-Hill Companies, Inc., 2003

45. Shipping to D.C.’s When Choose St. Louis © The McGraw-Hill Companies, Inc., 2003

46. Annual Variable Costs © The McGraw-Hill Companies, Inc., 2003

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? © The McGraw-Hill Companies, Inc., 2003

48. Data for the Sellmore Problem © The McGraw-Hill Companies, Inc., 2003

49. Spreadsheet Formulation © The McGraw-Hill Companies, Inc., 2003

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. © The McGraw-Hill Companies, Inc., 2003