# Transportation, Transshipment, and Assignment Problems

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Transportation, Transshipment, and Assignment Problems
Part 2 Deterministic Decision Models Chapter 6 Transportation, Transshipment, and Assignment Problems

Learning Objectives After completing this chapter, you should be able to: Describe the nature of transportation, transshipment, and assignment problems. Formulate a transportation problem as a linear programming model. Use the transportation method to solve problems with Excel. Solve maximization transportation problems, unbalanced problems, and problems with prohibited routes. Solve aggregate planning problems using the transportation model. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Learning Objectives (cont’d)
After completing this chapter, you should be able to: Formulate a transshipment problem as a linear programming model. Solve transshipment problems with Excel. Formulate an assignment problem as a linear programming model. Use the assignment method to solve problems with Excel. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Transportation Problems
A distribution-type problem in which supplies of goods that are held at various locations are to be distributed to other receiving locations. The solution of a transportation problem will indicate to a manager the quantities and costs of various routes and the resulting minimum cost. Used to compare location alternatives in deciding where to locate factories and warehouses to achieve the minimum cost distribution configuration. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Formulating the Model A transportation problem
Typically involves a set of sending locations, which are referred to as origins, and a set of receiving locations, which are referred to as destinations. To develop a model of a transportation problem, it is necessary to have the following information: Supply quantity (capacity) of each origin. Demand quantity of each destination. Unit transportation cost for each origin-destination route. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Transshipment Problems
A transportation problem in which some locations are used as intermediate shipping points, thereby serving both as origins and as destinations. Involve the distribution of goods from intermediate nodes in addition to multiple sources and multiple destinations. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Assignment Problems The Assignment-type Problems
Involve the matching or pairing of two sets of items such as jobs and machines, secretaries and reports, lawyers and cases, and so forth. Have different cost or time requirements for different pairings. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 6–1 Schematic of a Transportation Problem

Table 6–1 Transportation Table for Harley’s Sand and Gravel

Special Cases of Transportation Problems
Maximization Transportation-type problems that concern profits or revenues rather than costs with the objective to maximize profits rather than to minimize costs. Unacceptable Routes Certain origin-destination combinations may be unacceptable due to weather factors, equipment breakdowns, labor problems, or skill requirements that either prohibit, or make undesirable, certain combinations (routes). Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Special Cases of Transportation Problems (cont’d)
Unequal Supply and Demand Situations in which supply and demand are not equal such that it is necessary to modify the original problem so that supply and demand are equalized. Quantities in dummy routes in the optimal solution are not shipped and serve to indicate which supplier will hold the excess supply, and how much, or which destination will not receive its total demand, and how much it will be short. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 6-1 Input and Output Worksheet for the Transportation (topsoil) Problem

Exhibit 6-2 Parameter Specification Screen for the Topsoil Transportation Problem

Exhibit 6–3 Solver Options Screen

Exhibit 6–4 Solver Results

Exhibit 6–5 Answer Report for the Topsoil Transportation Problem

Exhibit 6–6 Sensitivity Report for the Topsoil Transportation Problem

Exhibit 6–7 Input and Output Sheet for the Revised Transportation (topsoil) Problem When the Shipping Route between Farm B and Project 1 Is Prohibited Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 6–2 A Network Diagram of a Transshipment Problem

Table 6–2 Cost of Shipping One Unit from the Farms to Warehouses
Example 6-2 Transshipment Problem The manager of Harley’s Sand and Gravel Pit has decided to utilize two intermediate nodes as transshipment points for temporary storage of topsoil. The revised diagram of the transshipment problem is given in Figure 6-3. Table 6–2 Cost of Shipping One Unit from the Farms to Warehouses Table 6–2 Cost of Shipping One Unit from the Warehouses to Projects Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 6–3 A Network Diagram of Harley’s Sand and Gravel Pit Transshipment Example

Exhibit 6–8 Excel Input and Output Screen for the Transshipment Problem

Exhibit 6–9 Parameter Specifications Screen for the Transshipment Problem

Using the Transportation Problem to Solve Aggregate Planning Problems
Involves creating a long-term production plan for achieving a demand-supply balance. Aggregate planners usually avoid in terms of thinking of individual products. Planners are concerned about the quantity and timing of production to meet the expected demand. Aggregate planners attempt to minimize the production cost over the planning horizon. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 6–4 Transportation Table for Aggregate Planning Purposes

Table 6–5 Transportation Table for the Aggregate Planning Problem of Example 6-3

Using the Transportation Problem to Solve Location Planning Problems
Location Analysis Comparing transportation costs for alternative locations for new facilities to minimize total cost. Provides planners an opportunity to assess the impact of each warehouse location on the total distribution costs for the system. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 6–6 System with Chicago Warehouse

Example 6-4 A manager has prepared a table that shows the cost of performing each of five jobs by each of five employees (see Table 6-8). According to this table, job I will cost \$15 if done by Al. \$20 if it is done by Bill, and so on. The manager has stated that his goal is to develop a set of job assignments that will minimize the total cost of getting all four jobs done. It is further required that the jobs be performed simultaneously, thus requiring one job being assigned to each employee. In the past, to find the minimum-cost set of assignments, the manager has resorted to listing all of the different possible assignments (i.e., complete enumeration) for small problems such as this one. But for larger problems, the manager simply guesses because there are too many possibilities to try to list them. For example, with a 5X5 table, there are 5! = 120 different possibilities; but with, say, a 7X7 table, there are 7! = 5,040 possibilities. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 6–8 Numerical Example for the Assignment Problem

Exhibit 6–10 Excel Input and Output Worksheet for the Assignment Problem

Exhibit 6–11 Parameter Specifications Screen for the Assignment Problem

Exhibit 6–12 Excel Worksheet for the Transportation Problem in Solved Problem 1

Exhibit 6–13 Parameter Specification Screen for Solved Problem 1

Exhibit 6–14 Excel Worksheet for the Assignment Problem in Solved Problem 2

Exhibit 6–15 Parameter Specification Screen for Solved Problem 2

Exhibit 6–16 Excel Worksheet for the Transportation Problem in Solved Problem 3