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McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.1 Transportation and Assignment Models.

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Presentation on theme: "McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 6.1 Transportation and Assignment Models."— Presentation transcript:

1 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Transportation and Assignment Models

2 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Outline Network models in general – Transportation, Assignment and Transshipment models belong to a special class of linear programming problems called network flow problems Characteristics of Transportation models Characteristics of Assignment models Variations on a theme

3 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Network Optimization Problems Many optimization problems can be represented by a graphical network representation. Some examples: – Distribution problems – Routing problems – Maximum flow problems – Designing computer / phone / road networks – Equipment replacement Arcs Nodes

4 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Transportation problem Frequently arises in planning for distribution of goods and services from several supply locations to several demand locations. – Examples? Characteristics (typical) – Quantity of goods available at each supply location is limited. – Quantity of goods needed at each of several demand locations is known. – Usual objective of a transportation problem is to minimize the cost of shipping goods from the origins to the destinations. Variations on the Transportation Problem theme – Total supply not equal to total demand. – Maximization objective function. – Route capacities or route minimums. – Unacceptable routes.

5 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Example: Supply Chains – the generic model Raw materials supplier Manufacturing plant Distribution center Customers/ Retailers upstream downstream

6 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Example: Forest industry supply chain Wagner, H.M. (1975). Principles of Operations Research 2 nd ed. Englewood Cliffs NJ: Prentice-Hall

7 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Assumptions 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 multiplied by the number of units distributed.

8 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The network for a transportation problem Variable costs - c ij Quantities - x ij Demands - D j Capacities - K i Region 1 Region 2 Region 4 Region 3 Plant 1 Plant 2 Plant 3 The Decision Variable

9 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., A basic transportation model – Foster Generators Transportation cost per unit Plants (origin nodes) Distribution Centers Chicago 4000 St. Louis 2000 Bedford 6000 Boston 6000 Distribution Routes (arcs) Lexington 1500 York 2500 Cleveland 5000 Supplies / Plant Capacities (units) Demands (units)

10 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Parameters c ij variable costs per unit transported from Plant i to Region j K i capacity for Plant i D j demand for Region j mnumber of regions nnumber of plants Decision variables x ij quantity transported from Plant i to Region j All demands are satisfied No capacities are exceeded Transportation Problem: Demand Allocation Model

11 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., A simple Transportation/Demand Allocation problem solved

12 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The Capacitated Plant Location Model (CPLM) Open/Closed Fixed costs Capacities Variable costs Quantities Demands Ware- house 1 Ware- house 2 Ware- house 3 Plant 1 Plant 2 Plant 3

13 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The Capacitated Plant Location Model (CPLM) Decision variables y i binary variable indicating whether Plant i should be open (1) or closed (0) x ij quantity transported from Plant i to Region j Parameters F i fixed costs for Plant i c ij variable costs per unit transported from Plant i to Region j K i capacity for Plant i D j demand for Region j mnumber of regions nnumber of potential plants

14 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The CPLM with single sourcing Ware- house 1 Ware- house 2 Ware- house 3 Plant 1 Plant 2 Plant 3 Variable costs Open/Closed Fixed costs Capacities Assigning plants to warehouses Demands

15 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The CPLM with single sourcing Parameters F i fixed costs for Plant i c ij variable costs per unit transported from Plant i to Region j K i capacity for Plant i D j demand for Region j mnumber of regions nnumber of potential plants Decision variables y i binary variable indicating whether Plant i should be open(1) or closed (0) x ij binary variable indicating whether Plant i should supply market in Region j

16 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Transshipment model with sourcing Market 1 Market 1 Market 2 Market 2 Market 3 Market 3 Plant 1 Supplier 1 Supplier 1 Supplier 2 Supplier 2 Plant 2 Plant 3 Ware- house 1 Ware- house 1 Ware- house 2 Ware- house 2 Variable costs Open/ Closed Quantities Capacities Fixed costs Demands Open/ Closed Quantities Variable costs Fixed costs Capacities (Combines plant location, warehouse location and sourcing)

17 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Transshipment model with sourcing subject to: for i = 1, 2, 3, …, n for e = 1, 2, 3, …, t for j = 1, 2, 3, …, m for h = 1, 2, 3, …, l Warehouse capacity constraint Warehouse flow balance Market satisfaction Plant flow balance Plant capacity constraint Source capacity constraint Warehouse fixed costs Supplier-Plant variable costs Plant-Warehouse variable costs Warehouse-Market variable costs Plant fixed costs

18 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Transshipment model with sourcing Market 1 D 1 Market 1 D 1 Market 2 D 2 Market 2 D 2 Market m D m Market m D m Plant 1 K 1, F 1, y i Plant 1 K 1, F 1, y i Supplier 1 S 1 Supplier 1 S 1 Supplier l S l Supplier l S l Plant 2 K 2, F 2, y i Plant 2 K 2, F 2, y i Plant n K n, F n, y i Plant n K n, F n, y i Ware- house 1 W 1, f 1, y e Ware- house 1 W 1, f 1, y e Ware- house t W t, f t, y e Ware- house t W t, f t, y e X 12, c 12 X tm, c tm X 21, c 21 X t2, c t2 X 11, c 11 X l2, c l2 X ln, c ln X 11, c 11 X 21, c 21 X 2t, c 2t X nt, c nt x ie, c ie x ej, c ej x hi, c hi

19 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Transshipment model with sourcing Parameters mnumber of markets or demand points nnumber of potential plant or factory locations lnumber of suppliers tnumber of potential warehouse locations D j annual demand from customer j K i potential capacity of plant at site i S h supply capacity at supplier h W e potential warehouse capacity at site e F i fixed cost of locating a plant at site i f e fixed cost of locating a warehouse at site e c hi cost of shipping one unit from supply source h to plant i c ie cost of producing and shipping one unit from plant i to warehouse e c ej cost of shipping one unit from warehouse e to customer j Decision variables y i 1 if plant is located at site i, 0 otherwise y e 1 if warehouse is located at site e, 0 otherwise x ei quantity shipped from warehouse e to market j x ie quantity shipped from plant at site i to warehouse e x hi quantity shipped from supplier h to plant at site i

20 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The Assignment Problem Arises in a variety of decision making situations: – Jobs to machines – Agents to tasks – Sales personnel to sales territories – Contracts to bidders – Spacecraft to planetary missions – Nuclear warheads to targets Distinguishing feature of the basic assignment problem – One agent is assigned to one and only one task We seek a set of assignments that optimizes a stated objective – Minimize costs – Minimize time – Maximize profit – Maximize observation time – Maximize damage – …etc.

21 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., An Assignment Problem – Fowle Marketing Completion time in days Project leaders (origin nodes) Clients (destination nodes) Client Client Kari - 1 Client Possible assignments (arcs) Gudmund - 1 Terry - 1 Supplies Demands

22 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The Assignment Problem Parameters c ij cost of assigning Agent i to Task j mnumber of Agents nnumber of Tasks Decision variables x ij assignment of Agent i to Task j, 0 if not assigned, 1 if assigned

23 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., A simple Assignment Problem solved

24 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Table of Contents Chapter 6 (Transportation and Assignment Problems) 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) 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).

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

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

27 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Current Shipping Plan Warehouse From \ To SacramentoSalt Lake CityRapid CityAlbuquerque Cannery Bellingham75000 Eugene Albert Lea001585

28 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Shipping Cost per Truckload Warehouse From \ To SacramentoSalt Lake CityRapid CityAlbuquerque Cannery Bellingham$464$513$654$867 Eugene Albert Lea Total shipping cost = 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685) = $165,595

29 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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

30 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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.

31 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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.

32 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The P&T Co. Transportation Problem data Unit Cost Destination (Warehouse): Sacrament oSalt Lake CityRapid CityAlbuquerqueSupply Source (Cannery) Bellingham$464$513$654$86775 Eugene Albert Lea Demand

33 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Network Representation

34 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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) From Cannery 2 to all destinations From Cannery 3 to all destinations From Cannery 1 to all destinations All canneries can supply all warehouses

35 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Formulation

36 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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.

37 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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”, downloadable at

38 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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$28$ — Required production Question: Which plants should produce which products?

39 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Transportation Problem Formulation Unit Cost Destination (Product):1234Supply Source(Plant) 1$41$27$28$ — Demand

40 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Formulation

41 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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?

42 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Data for the Nifty Company Unit Cost Product:1234 Capacity Available Plant 1$41$27$28$ — Required production Question: How many units should Nifty sell to each customer and how many units should they ship from each plant to each customer?

43 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Formulation

44 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Metro Water (Distributing Natural Resources) Metro Water District is an agency that administers water distribution in a large geographic 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 River Calorie River —5 Needed (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?

45 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Formulation

46 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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 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?

47 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Formulation

48 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Optimal Production at Northern Airplane Month 1 (RT) 2 (RT) 3 (RT) 3 (OT) 4 (RT) Production Installations Stored

49 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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?

50 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Data for the Middletown School District Distance (Miles) to School Tract123 Number of High School Students Minimum enrollment1,2001,1001,000 Maximum enrollment1,8001,7001,500

51 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Formulation

52 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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?

53 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Cost Data for Energetic Unit Cost Energy Need:ElectricityWater HeatingSpace Heating Source of Energy Electricity$400$500$600 Natural gas— Solar heater—300400

54 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Formulation

55 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Location of Texago’s Facilities Type of FacilityLocations Oil fields1. Several in Texas 2. Several in California 3. Several in Alaska Refineries1. Near New Orleans, Louisiana 2. Near Charleston, South Carolina 3. Near Seattle, Washington Distribution Centers1. Pittsburgh, Pennsylvania 2. Atlanta, Georgia 3. Kansas City, Missouri 4. San Francisco, California

56 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Potential Sites for Texago’s New Refinery Potential SiteMain Advantages Near Los Angeles, California1. Near California oil fields. 2. Ready access from Alaska oil fields. 3. Fairly near San Francisco distribution center. Near Galveston, Texas1. Near Texas oil fields. 2. Ready access from Middle East imports. 3. Near corporate headquarters. Near St. Louis, Missouri1. Low operating costs. 2. Centrally located for distribution centers. 3. Ready access to crude oil via the Mississippi River.

57 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Production Data for Texago Refinery Crude Oil Needed Annually (Million Barrels)Oil Fields Crude Oil Produced Annually (Million Barrels) New Orleans100Texas80 Charleston60California60 Seattle80Alaska100 New site120Total240 Total360Needed imports = 360 – 240 = 120

58 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Cost Data for Shipping to Refineries Cost per Unit Shipped to Refinery or Potential Refinery (Millions of Dollars per Million Barrels) New OrleansCharlestonSeattle Los AngelesGalvestonSt. Louis Source Texas California Alaska Middle East235434

59 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Cost Data for Shipping to Distribution Centers Cost per Unit Shipped to Distribution Center (Millions of Dollars) PittsburghAtlantaKansas CitySan Francisco Refinery New Orleans Charleston7547 Seattle7843 Potential Refinery Los Angeles8632 Galveston5436 St. Louis4315 Number of units needed

60 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Estimated Operating Costs for Refineries SiteAnnual Operating Cost (Millions of Dollars) Los Angeles Galveston St. Louis

61 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Basic Spreadsheet for Shipping to Refineries

62 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Shipping to Refineries, Including Los Angeles

63 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Shipping to Refineries, Including Galveston

64 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Shipping to Refineries, Including St. Louis

65 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Basic Spreadsheet for Shipping to D.C.’s

66 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Shipping to D.C.’s When Choose Los Angeles

67 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Shipping to D.C.’s When Choose Galveston

68 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Shipping to D.C.’s When Choose St. Louis

69 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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 Galveston920 million1.63 billion570 million3.12 billion St. Louis960 million1.43 billion530 million2.92 billion

70 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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?

71 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Data for the Sellmore Problem Required Time per Task (Hours) Temporary Employee Word ProcessingGraphicsPacketsRegistrations Hourly Wage Ann $14 Ian Joan Sean

72 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Formulation

73 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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.

74 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The Network Representation

75 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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?

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

77 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Formulation

78 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Better Products (No Product Splitting)

79 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Middletown School District (No Tract Splitting)

80 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The Transportation Problem A common problem in logistics is how to transport goods from a set of sources (e.g., plants, warehouses, etc.) to a set of destinations (e.g., warehouses, customers, etc.) at the minimum possible cost. Given – a set of sources, each with a given supply, – a set of destinations, each with a given demand, – a cost table (cost/unit to ship from each source to each destination) Goal – Choose shipping quantities from each source to each destination so as to minimize total shipping cost.

81 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The Network Representation

82 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Transportation Problem Example A company has two plants (in Seattle and Atlanta) producing a certain product that is to be shipped to three distribution centers (in Sacramento, St. Louis, and Pittsburgh). – The unit production costs are the same at the two plants, and the shipping costs per unit are shown in the table below. – Shipments are made once per week. – During each week, each plant produces at most 60 units and each distribution center needs at least 40 units. Unit Shipping CostDistribution Center SacramentoSt. Louis Pittsburg h Plant Seattle$2$6$8 Atlanta$7$5$3 Question: How many units should be shipped from each plant to each distribution center?

83 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Solution

84 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Shipping from D.C.’s to Customers The same company ships one of its products from its three distribution centers to four different customers – The shipping costs per unit are shown in the table below. – Shipments are made once per week. – During each week, each distribution center has received 40 units. – Customer demand is also shown in the table below. Unit Shipping CostCustomer 1234 Distributio n Center Sacramento$8$10$7$11 St. Louis$12$11$9$6 Pittsburgh$10$9$15$10 Customer Demand Question: How many units should be shipped from each distribution center to each customer?

85 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Solution

86 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Managing the Whole Supply Chain (Plant to D.C. to Customer)

87 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Site Selection The lease is up on their distribution center in St. Louis. They now must decide whether to sign a new lease in St. Louis, or move the distribution center to a new location. One possible new location is Omaha, Nebraska, which is offering a better deal on the lease. Question: Should they move their distribution center to Omaha?

88 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Solution to Site Selection

89 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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”, downloadable at

90 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The Assignment Problem The job of assigning people (or machines or whatever) to a set of tasks is called an assignment problem. Given – a set of assignees – a set of tasks – a cost table (cost associated with each assignee performing each task) Goal – Match assignees to tasks so as to perform all of the tasks at the minimum possible cost.

91 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Network Representation

92 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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 Carl Chris David Tony Ken Question: How should the swimmers be assigned to make the fastest relay team?

93 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Algebraic Formulation Letx ij = 1 if swimmer i swims stroke j; 0 otherwise t ij = best time of swimmer i in stroke j Minimize Time = ∑ i ∑ j t ij x ij subject to: each stroke swum:∑ i x ij = 1 for each stroke j each swimmer swims 1:∑ j x ij ≤ 1 for each swimmer i and x ij ≥ 0 for all i and j.

94 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Spreadsheet Formulation

95 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 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?

96 McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Points Bid for Electives Electives Student Quantitative MethodsFinance Operations ManagementAccounting George Fred Ann45 55 Eric Susan30 10 Liz50 00 Ed David Tony Jennifer

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

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


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