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

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

7. 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

8. The Transportation Problem

9. The Transportation Problem

10. 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

11. The Transportation Problem • The feasible solutions property • A transportation problem will have feasible solutions if and only if: • If a problem does not quite satisfy this requirement, it is possible to reformulate the problem by introducing a dummy supply (if there’s more demand than supply) or a dummy destination (if there’s more supply than demand) to take up the extra. • The cost assumption • Cost is directly proportional to number of units distributed

12. The Transportation Problem

13. The Transportation Problem

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

15. The Transportation Problem • Solving the P&T Co. example using a spreadsheet • Solver uses the general simplex method • Rather than the streamlined version specifically designed for the transportation problem which is discussed in Section 9.2

16. The Transportation Problem An example with a dummy destination

17. The Transportation Problem Since it is impossible to produce engines in one month for installation in an earlier month, xij must be 0 if i > j. Therefore, the corresponding cij must set to be a big number M to force xij = 0 in the optimal solution (due to minimizing the total cost). Also, ci5 = 0 (i = 1, 2, 3, 4) since destination 5 is fictional and there is no real cost associate with it. The parameter table is therefore as follows. The problem is solved using Excel solver and see also a different formulation of the problem (when it’s not formulated as a transportation problem).

18. The Transportation Problem

19. 9.2 A Streamlined Simplex Method for the Transportation Problem • Transportation simplex method • No artificial variables needed (thus it’s simpler, easier and more efficient than the general simplex method, especially for large problems) • 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 (row operations) on simplex tableau • Almost the entire simplex tableau can be eliminated

20. A Streamlined Simplex Method for the Transportation Problem • Values needed to apply the transportation simplex method • Current BF solution • Number of basic variables = m + n – 1 where m = number of sources and n = number of destinations; Although number of functional constraints = m + n, one is redundant due to the requirement of total supply = total demand • Current values of ui and vj (they are the multipliers that are subtracted from row 0) • Resulting values of cij − ui − vj for nonbasic variables xij; for a basic variable xij :cij − ui − vj = 0 • Transportation simplex tableau • Used to record values for each iteration

22. 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;

23. 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 variables with the only feasible allocation • Otherwise, return to step 1

24. A Streamlined Simplex Method for the Transportation Problem • Alternative criteria for step one • Northwest corner rule (simplest but not consider cij’s) • Select the northwest corner (x11), if any supply remains, move one column to the right and then one row down

25. A Streamlined Simplex Method for the Transportation Problem • Vogel’s approximation method • Calculate the second smallest cij – smallest cij for each remaining row and column, and then select the row or column having the largest difference, assign the variable corresponding to the smallest cijin that row or column and let it equals the minimum of supply and demand and then eliminate the row containing the supply or column containing the demand. Eliminate row if there’s a tie.

26. A Streamlined Simplex Method for the Transportation Problem

27. 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 column • For each variable xij not previously selected in these rows and columns, calculate Δij =cij - i - • Select the largest (absolute) negative (i.e. most negative) value of Δij(breaking tie arbitrarily)

28. A Streamlined Simplex Method for the Transportation Problem

29. 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; need to be first solved from cij = ui + vj for all basic variable xij. There are (m + n) ui and vj from m + n – 1 equations, thus we can let one of them be 0. Choose the ui with most basic variables in the ith row (break tie arbitrarily). • If the current solution is optimal, then stop; otherwise, go to an iteration.

30. A Streamlined Simplex Method for the Transportation Problem

31. A Streamlined Simplex Method for the Transportation Problem

32. A Streamlined Simplex Method for the Transportation Problem • An iteration: - Determine the entering basic variable: select the one corresponding to the most negative • Determine the leaving basic variable: construct a chain reaction, starting from the entering basic variable (called recipient cell), then the donor cell in the same column, then the recipient cell in the row, until the donor cell in the row containing the entering basic variable. From the donor cells, select the one with smallest value. In general, there is always just one chain reaction (in either direction) that can be completed successfully to maintain feasibility.

33. A Streamlined Simplex Method for the Transportation Problem • Determine the new BF: add the value of leaving basic variable to all recipient cells and subtract it from all donor cells.

34. A Streamlined Simplex Method for the Transportation Problem

37. 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

38. 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

39. 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

40. The Assignment Problem C22 = M (a big number) to prevent this assignment, i.e., by forcing x22 = 0 in the optimal solution.

41. The Assignment Problem • xij can have only values zero or one • One if assignee i performs task j • Zero if not • Each assignee is assigned to one and only one task (first n constraints) • Each task is to be performed by one and only one assignee (the next n constraints) • Although xij should be binary, without adding this restriction in the model for the assignment problem, xij will still be binary due to the integer property of the model.

42. The Assignment Problem

43. The Assignment Problem • Can use simplex method or streamlined transportation simplex method to solve (note that the assignment problem is just a special type of transportation problem where # of sources (assignees) = # of destinations (tasks) and si = 1 and dj = 1 for every i and j). • Recommendation: use specialized solution procedures for the assignment problem (in Section 9.4) • Will be more efficient for large problems • See two ways to find the optimal solutions on the course website (job shop) • Example: Better Products Co. problem

44. The Assignment Problem

45. The Assignment Problem Option 1: Permit Product splitting Formulate the problem as a transportation problem. See the solution using Excel Solver on the course website.

46. The Assignment Problem

47. The Assignment Problem See the solution using Excel Solver on the course website.

48. 9.4 A Special Algorithm for the Assignment Problem • Using Hungarian algorithm to solve the assignment is much faster and easier than the simplex method • Operate directly on the cost table until all the assignments can be made to the zero element positions. • Summary of the Hungarian algorithm • Subtract the smallest number in each row from every number in the row (row reduction). Enter the results in a new table. • Subtract the smallest number in each column of the new table from every number in the column (column reduction). Enter the results in another table.

49. 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.

50. 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