1. The Transportation and Assignment Problems
Chapter 9
The Transportation and Assignment Problems

2. Introduction Transportation problem Assignment 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 xij to 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 𝑢 i still remaining in the row
For each column still under consideration, determine largest unit cost 𝑣𝑗 still remaining in the row
For each variable xij not previously selected in these rows and columns, calculate Δij =cij - 𝑢 i - 𝑣𝑗
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
𝑐𝑖𝑗–𝑢𝑖–𝑣𝑗 ≥ 0
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 in the text
Find the new BF solution
See Pages 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 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