Network Flow Problems – The Assignment Problem

1. Network Flow Problems –The Assignment Problem Consider the problem of assigning n assignees to n tasks. Only one task can be assigned to an assignee, and each task must be assigned. There is also a cost associated with assigning an assignee i to task j, cij. The objective is to assign all tasks such that the total cost is minimized.

2. Network Flow Problems –The Assignment Problem • Examples: • Assign people to project assignments • Assign jobs to machines • Assign products to plants • Assign tasks to time slots

3. Network Flow Problems –The Assignment Problem • To fit the assignment problem definition, the following assumptions must be satisfied: • The number of assignees and the number of tasks are the same (denoted by n). • Each assignee is to be assigned to exactly one task. • Each task is to be assigned to exactly one assignee. • There is a cost cij associated with assignee i performing task j. • The objective is to determine how all n assignments should be made to minimize the total cost.

4. Assignment Problem – Flow Diagram a – assignee t – tasks a1 c11 1 1 t1 c12 a2 2 2 t2 a3 3 3 t3 an n n t4 cnn assignees tasks

5. Assignment Problem –Cost Matrix Let the following represent the standard assignment problem cost matrix, c:

6. Assignment Problem –Conversion to Standard Cost Matrix Consider following cost matrix, how do you convert to satisfy the standard definition of the assignment problem? Add “big M” to avoid incompatible assignments, and add a dummy assignee (or task) to have equal assignees and tasks.

7. Assignment Problem – Math Formulation Minimize s.t. Total Cost i j Does this formulation look familiar? Is this a Linear Program?

8. Assignment Problem –Hungarian Method Consider following cost matrix. Reduce by Row Minimum.

9. Assignment Problem –Hungarian Method Reduced by Column Minimum: Reduce by Minimum of uncovered cells (1):

10. Assignment Problem –Hungarian Method Solution is now optimal since minimum number of lines to cover all 0 is 4 (equal to n). A3 -> T3, A1 -> T2, A2 -> T4, A4 -> T1 Z = 3 + 5 + 5 + 2 = 15

11. Assignment Problem –Hungarian Method Summary of Hungarian Method: Step 1 – Find the minimum element in each row. Construct a new matrix by subtracting from each cost the minimum cost in its row. For this new matrix, find the minimum cost in each column. Construct a new matrix by subtracting from each cost the minimum cost in its column. Step 2 – Draw the minimum number of lines (horizontal or vertical) that are needed to cover all the zeros in the reduced cost matrix. If n lines are required, an optimal solutions is available among the covered zeros in the matrix. If fewer than n lines are needed, proceed to step 3. Step 3 – Find the smallest nonzero element (call its value k) in the reduced cost matrix that is uncovered by the lines drawn in Step 2. Now subtract k from each uncovered element of the reduced cost matrix and add k to each element that is covered by two lines. Return to step 2.