1. 1 1 Slide © 2006 Thomson South-Western. All Rights Reserved. Slides prepared by JOHN LOUCKS St. Edward’s University

2. 2 2 Slide © 2006 Thomson South-Western. All Rights Reserved. Chapter 10 Transportation, Assignment, and Transshipment Problems n Transportation Problem Network Representation Network Representation General LP Formulation General LP Formulation

3. 3 3 Slide © 2006 Thomson South-Western. All Rights Reserved. Transportation, Assignment, and Transshipment Problems n A network model is one which can be represented by a set of nodes, a set of arcs, and functions (e.g. costs, supplies, demands, etc.) associated with the arcs and/or nodes. n Transportation, assignment, and transshipment problems of this chapter as well as the PERT/CPM problems (in another chapter) are all examples of network problems.

4. 4 4 Slide © 2006 Thomson South-Western. All Rights Reserved. Transportation, Assignment, and Transshipment Problems n Each of the three models of this chapter can be formulated as linear programs and solved by general purpose linear programming codes. n For each of the three models, if the right-hand side of the linear programming formulations are all integers, the optimal solution will be in terms of integer values for the decision variables. n However, there are many computer packages (including The Management Scientist ) that contain separate computer codes for these models which take advantage of their network structure.

5. 5 5 Slide © 2006 Thomson South-Western. All Rights Reserved. Transportation Problem n The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply s i ) to n destinations (each with a demand d j ), when the unit shipping cost from an origin, i, to a destination, j, is c ij. n The network representation for a transportation problem with two sources and three destinations is given on the next slide.

6. 6 6 Slide © 2006 Thomson South-Western. All Rights Reserved. Transportation Problem n Network Representation 2 2 c 11 c 12 c 13 c 21 c 22 c 23 d1d1d1d1 d2d2d2d2 d3d3d3d3 s1s1s1s1 s2s2 SourcesDestinations 3 3 2 2 1 1 1 1

7. 7 7 Slide © 2006 Thomson South-Western. All Rights Reserved. Transportation Problem n LP Formulation The LP formulation in terms of the amounts shipped from the origins to the destinations, x ij, can be written as: Min  c ij x ij Min  c ij x ij i j i j s.t.  x ij < s i for each origin i s.t.  x ij < s i for each origin i j  x ij = d j for each destination j  x ij = d j for each destination j i x ij > 0 for all i and j x ij > 0 for all i and j

8. 8 8 Slide © 2006 Thomson South-Western. All Rights Reserved. n LP Formulation Special Cases The following special-case modifications to the linear programming formulation can be made: Minimum shipping guarantee from i to j : Minimum shipping guarantee from i to j : x ij > L ij x ij > L ij Maximum route capacity from i to j : Maximum route capacity from i to j : x ij < L ij x ij < L ij Unacceptable route: Unacceptable route: Remove the corresponding decision variable. Remove the corresponding decision variable. Transportation Problem

9. 9 9 Slide © 2006 Thomson South-Western. All Rights Reserved. Example: Acme Block Co. Acme Block Company has orders for 80 tons of concrete blocks at three suburban locations as follows: Northwood -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons. Acme has two plants, each of which can produce 50 tons per week. Delivery cost per ton from each plant to each suburban location is shown on the next slide. How should end of week shipments be made to fill the above orders?

10. 10 Slide © 2006 Thomson South-Western. All Rights Reserved. n Delivery Cost Per Ton Northwood Westwood Eastwood Northwood Westwood Eastwood Plant 1 24 30 40 Plant 1 24 30 40 Plant 2 30 40 42 Plant 2 30 40 42 Example: Acme Block Co.

11. 11 Slide © 2006 Thomson South-Western. All Rights Reserved. n Partial Spreadsheet Showing Problem Data Example: Acme Block Co.

12. 12 Slide © 2006 Thomson South-Western. All Rights Reserved. n Partial Spreadsheet Showing Optimal Solution Example: Acme Block Co.

13. 13 Slide © 2006 Thomson South-Western. All Rights Reserved. n Optimal Solution From To Amount Cost From To Amount Cost Plant 1 Northwood 5 120 Plant 1 Westwood 45 1,350 Plant 1 Westwood 45 1,350 Plant 2 Northwood 20 600 Plant 2 Northwood 20 600 Plant 2 Eastwood 10 420 Plant 2 Eastwood 10 420 Total Cost = $2,490 Total Cost = $2,490 Example: Acme Block Co.

14. 14 Slide © 2006 Thomson South-Western. All Rights Reserved. n Partial Sensitivity Report (first half) Example: Acme Block Co.

15. 15 Slide © 2006 Thomson South-Western. All Rights Reserved. n Partial Sensitivity Report (second half) Example: Acme Block Co.

16. 16 Slide © 2006 Thomson South-Western. All Rights Reserved. End of Chapter 10