The Transportation and Assignment Problems

1. Chapter 9 The Transportation and Assignment Problems

2. Introduction • Transportation problem • Many applications involve deciding how to optimally transport goods (or schedule production) • Assignment problem • Deals with assigning people to tasks • Transportation and assignment problems • Special cases of minimum cost flow problem • Presented in Chapter 10

3. 9.1 The Transportation Problem • Prototype example • P&T Company produces products including canned peas • Production occurs at three canneries • Four distribution warehouses are spread across the U.S. • Management initiates a study to reduce shipping expenses

4. The Transportation Problem

5. The Transportation Problem • Arrows represent possible truck routes • Number on arrow: shipping cost per truckload • Bracketed number: truckloads out

6. The Transportation Problem • Let Z represent total shipping cost • xij represents number of truckloads shipped from cannery i to warehouse j • Problem: choose values of the 12 decision variables xijto minimize Z

7. The Transportation Problem

8. The Transportation Problem

9. The Transportation Problem • The transportation problem model • Concern: distributing any commodity from sources to destinations • The requirements assumption • Each source has a fixed supply • Entire supply must be distributed to the destinations • Each destination has a fixed demand • Entire demand must be received from the sources

10. The Transportation Problem • The feasible solutions property • A transportation problem will have feasible solutions if and only if: • The cost assumption • Cost is directly proportional to number of units distributed

11. The Transportation Problem • The transportation problem type: any linear programming problem that fits the structure in Table 9.6 • Is

12. The Transportation Problem • Solving the P&T Co. example using a spreadsheet • See Figure 9.4 in the text • Solver uses the general simplex method • Rather than the streamlined version specifically designed for the transportation problem

13. 9.2 A Streamlined Simplex Method for the Transportation Problem • Transportation simplex method • No artificial variables needed • Current row zero can be obtained without using any other row • Leave basic variable identified in a simple way • New BF solution can be identified immediately • Without algebraic manipulation on simplex tableau • Almost the entire simplex tableau can be eliminated

14. A Streamlined Simplex Method for the Transportation Problem • Values needed to apply the transportation simplex method • Current BF solution • Current values of ui and vj • Resulting values of cij − ui − vj for nonbasic variables xij • Transportation simplex tableau • Used to record values for each iteration

16. A Streamlined Simplex Method for the Transportation Problem • Transportation simplex method is more efficient • Especially for large problems • For transportation problems with m sources and n destinations: • Number of basic variables is equal to m+n-1

17. A Streamlined Simplex Method for the Transportation Problem • General procedure for constructing an initial BF solution • To begin, all source rows and columns of the transportation simplex tableau are under consideration for providing a basic variable • From the rows and columns still under consideration, select the next basic variable according to some criterion • Make that allocation large enough to exactly use up the smaller of the remaining supply in its row or the remaining demand in its column

18. A Streamlined Simplex Method for the Transportation Problem • General procedure (cont’d.) • Eliminate that row or column from further consideration • If both row and column are the same, arbitrarily choose the row to eliminate • If only one row or column remains under consideration, complete the procedure by selecting every remaining variable associated with that row or column to be basic with the only feasible allocation • Otherwise, return to step 1

19. A Streamlined Simplex Method for the Transportation Problem • Alternative criteria for step one • Northwest corner rule • Select the northwest corner, move one column to the right and then one row down • Vogel’s approximation method • Based on the arithmetic difference between the smallest and next-to-smallest unit cost cij still remaining in that row or column

20. A Streamlined Simplex Method for the Transportation Problem • Alternative criteria for step one (cont’d.) • Russel’s approximation method • For each row still under consideration, determine largest unit cost i still remaining in the row • For each column still under consideration, determine largest unit cost still remaining in the row • For each variable xij not previously selected in these rows and columns, calculate Δij =cij - i - • Select the largest (absolute) negative value of Δij

21. A Streamlined Simplex Method for the Transportation Problem • Next step • Check whether the initial solution is optimal by applying the optimality test • Optimality test • A BF solution is optimal if and only if for every (i,j) such that xij is nonbasic • If the current solution is not optimal, go to an iteration

22. A Streamlined Simplex Method for the Transportation Problem • An iteration • Find the entering basic variable • See Page 343 in the text • Find the leaving basic variable • See Pages 343-344 in the text • Find the new BF solution • See Pages 344-345 in the text

23. 9.3 The Assignment Problem • Special type of linear programming problem • Assignees are being asked to perform tasks • Assignees could be people, machines, plants, or time slots • Requirements to fit assignment problem definition • The number of assignees and tasks are the same • Designated by n

24. The Assignment Problem • Requirements to fit assignment problem definition (cont’d.) • Each assignee is assigned to exactly one task • Each task is to be performed by exactly one assignee • Cost cij is associated with each assignee i performing task j • Objective: determine how n assignments should be made to minimize the total cost

25. The Assignment Problem • If problem does not fit requirement 1 or 2 • Dummy assignees and dummy tasks may be constructed • Prototype example • The Job Shop Co. problem • Assign new machines to locations to minimize total cost of materials handling

26. The Assignment Problem

27. The Assignment Problem • xij can have only values zero or one • One if assignee i performs task j • Zero if not

28. The Assignment Problem

29. The Assignment Problem • Can use simplex method or transportation simplex method to solve • Recommendation: use specialized solution procedures for the assignment problem • Will be more efficient for large problems • Example: Pages 353-356 of the text

30. 9.4 A Special Algorithm for the Assignment Problem • Summary of the Hungarian algorithm • Subtract the smallest number in each row from every number in the row. Enter the results in a new table. • Subtract the smallest number in each column of the new table from every number in the column. Enter the results in another table.

31. A Special Algorithm for the Assignment Problem • Summary of the Hungarian algorithm (cont’d.) • Test whether an optimal set of assignments can be made. To do this, determine the minimum number of lines needed to cross out all zeros • If the minimum number of lines equals the number of rows, an optimal set of assignments is possible. Proceed with step 6. • If not, proceed with step 4.

32. A Special Algorithm for the Assignment Problem • Summary of the Hungarian algorithm (cont’d.) • If the number of lines is less than the number of rows, modify the table as follows: • Subtract the smallest uncovered number from every uncovered number in the table • Add the smallest uncovered number to the numbers at intersections of covering lines • Numbers crossed out but not at intersections of cross-out lines carry over unchanged to the next table

33. A Special Algorithm for the Assignment Problem • Summary of the Hungarian algorithm (cont’d.) • Repeat steps 3 and 4 until an optimum set of assignments is possible • Make assignments one at a time in positions that have zero elements. Begin with rows or columns with only one zero. Cross out both row and column after each assignment is made. Move on, with preference given to any row with only one zero not crossed out.

34. A Special Algorithm for the Assignment Problem • Summary of the Hungarian algorithm (cont’d.) • (cont’d.) Continue until every row and column has exactly one assignment and so has been crossed out. This will be an optimal solution for the problem

35. 9.5 Conclusions • General simplex method is a powerful algorithm • Simplified approaches are available when problem fits the special structure • Transportation problem • Assignment problem • Both problem types studied in this chapter: • Have a number of common applications