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.

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”

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

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

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

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

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

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.

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 15.5 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.

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

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

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)

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

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.

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”

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?

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

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

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?

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?

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

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 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 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?

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

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.

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

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?

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.

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

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

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

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?

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—

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

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?

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.