B.1 Transportation Review Questions

Transcription

1 Lesson Topics Network Models are nodes, arcs, and functions (costs, supplies, demands, etc.) associated with the arcs and nodes, as in transportation, assignment, transshipment, and shortest-route problems. Transportation (1) Problems are Resource Allocation Problems when outputs are fixed, and when outputs and inputs occur at different locations, so goods must be transported from origins to destinations. Transportation Problems with Modes of Transport re-interpret some of the different origins in a basic transportation problem to include not only location but modes of transportation (truck, rail, ). Assignment (1) Problems are Transportation Problems when the goods are workers that are transported to jobs, and each worker either does all of a job or none of it, so the fraction completed is binary. 1

2 Transportation Question. Consider Forbelt Corporation has a oneyear contract to supply motors for all refrigerators produced by the Ice Age Corporation. Ice Age manufactures the refrigerators at four locations around the country: Boston, Dallas, Los Angeles, and St. Paul. Plans call for the following number (in thousands) of refrigerators to be produced at each location: Boston 50 Dallas 70 Los Angeles 60 St. Paul 80 Forbelt s three plants are capable of producing the motors. The plants and production capacities (in thousands) are Denver 100 Atlanta 100 Chicago 150 Because of varying production and transportation costs, the profit that Forbelt earns on each lot of 1000 units depends on which plant produced the lot and which destination it was shipped to. The following table gives the accounting department estimates of the profit per unit (shipments will be made in lots of 1000 units): Shipped to Produced at Boston Dallas Los Angeles St. Paul Denver Atlanta Chicago Consider the profit maximization problem of determining how many motors should be produced at each plant and how many motors should be shipped from each plant to each destination. a. Develop a network representation of the problem. b. Compute an optimum using a computer. What are optimal profits? 2

3 Answer to Question: a. 1 7 Boston Denver Atlanta Dallas Los Angeles Chicago St. Paul 80 b. There are alternative optimal solutions (you need only provide one). Solution #1 Solution # 2 Denver to St. Paul: 10 Denver to St. Paul: 10 Atlanta to Boston: 50 Atlanta to Boston: 50 Atlanta to Dallas: 50 Atlanta to Los Angeles: 50 Chicago to Dallas: 20 Chicago to Dallas: 70 Chicago to Los Angeles: 60 Chicago to Los Angeles: 10 Chicago to St. Paul: 70 Chicago to St. Paul: 70 Total Profit: $4240 If solution #1 is used, Forbelt should produce 10 motors at Denver, 100 motors at Atlanta, and 150 motors at Chicago. There will be idle capacity for 90 motors at Denver. If solution #2 is used, Forbelt should adopt the same production schedule but a modified shipping schedule. 3

4 Assignment Question. A professor has been contacted by four not-for-profit agencies that are willing to work with student consulting teams. The agencies need help with such things as budgeting, information systems, coordinating volunteers, and forecasting. Although each of the four student teams could work with any of the agencies, the professor feels that there is a difference in the amount of time it would take each group to solve each problem. The professor s estimate of the time, in days, is given in the table below. Formulate the problem of determining which team should work with which project. Compute an optimum using a computer. Projects Team Budgeting Information Volunteers Forecasting A B C D Tip: Your written answer should define the decision variables, formulate the objective and constraints, and solve for the optimum. --- You will not earn full credit if you just solve for the optimum; you must also define the decision variables, and formulate the objective and constraints. 4

5 Answer to Question: This is a standard assignment problem. The optimal assignment is: Team A works with the forecast, Team B works with volunteers, Team C works with budgeting, and Team D works with information. The total time is

6 Assignment Question. A market research film s three clients each requested that the firm conduct a sample survey. Four available statisticians can be assigned to these three projects; however, all four statisticians are busy, and therefore each can handle only one client. The following data show the number of hours required for each statistician to complete each job; the differences in time are based on experience and ability of the statisticians. Client Statistician A B C a. Formulate a linear programming model for this problem. Compute an optimum using a computer. b. Suppose that the time statistician 4 needs to complete the job for client A is increased from 160 to 165 hours. What effect will this change have on the solution? c. Suppose that the time statistician 4 needs to complete the job for client A is decreased to 140 hours. What effect will this change have on the solution? d. Suppose that the time statistician 3 needs to complete the job for client B increases to 250 hours. What effect will this change have on the solution? 6

7 Answer to Question: a. Min 150x x x x x x x x x x x x 43 s.t. x 11 + x 12 + x 13 1 x 21 + x 22 + x 23 1 x 31 + x 32 + x 33 1 x 41 + x 42 + x 43 1 x 11 + x 21 + x 31 + x 41 = 1 x 12 + x 22 + x 32 + x 42 = 1 x 13 +x 23 +x 33 +x 43 = 1 xij > for all i, j (The first four constraints are that the 4 statisticians each take at most one job; the last three constraints are that the 3 jobs are each filled by exactly one statistician.) Optimal Solution: x12 = 1, x23 = 1, x41 = 1 Total hours required: 590 Note: statistician 3 is not assigned. b. The solution will not change, but the total hours required will increase by 5. This is the extra time required for statistician 4 to complete the job for client A. c. The solution will not change, but the total time required will decrease by 20 hours. d. The solution will not change; statistician 3 will not be assigned. Note that this occurs because increasing the time for statistician 3 makes statistician 3 an even less attractive candidate for assignment. 7