Assignment Problems

1. Assignment Problems Example: Machineco has four jobs to be completed. Each machine must be assigned to complete one job. The time required to setup each machine for completing each job is shown in the table below. Machineco wants to minimize the total setup time needed to complete the four jobs. Adapted from Sarper’s slides by Monique Guignard-Spielberg

2. Setup times (more generally called the cost matrix)

3. Let us define decision variables Xij as follows: Xij=1 if machine i is assigned to do job j Xij=0 if machine i is not assigned to do job j In general an assignment problem is a balanced transportation problem in which all supplies and demands are equal to 1, i.e., the number of jobs is equal to the number of machines (m, in general).

4. The Model According to the setup table Machineco’s problem can be formulated as follows (for i,j=1,2,3,4):

5. Solution Method • Although the transportation simplex (to be studied next) appears to be very efficient, there is a certain class of transportation problems, called assignment problems, for which the transportation simplex is often very inefficient. For that reason we will use another method called The Hungarian Method.

6. Basic Ideas • Suppose we ranked all feasible assignments in increasing order of cost. • The ranking does not change if one subtracts resp. adds) the same amount, say, D, from/to all costs in the same row, since all assignment costs are reduced (resp. increased) by D. • The ranking does not change if one subtracts (resp. adds) the same amount, say, D, from/to all costs in the same column, since all assignment costs are reduced (resp. increased) by D.

7. Basic Ideas (cont’d) • An assignment selects one entry in each row and one in each column. • As long as all costs are kept nonnegative, if the reduced cost matrix allows a zero cost assignment, that assignment is optimal. • If there is a zero cost in each row and a zero cost in each column of the reduced cost matrix, this assignment is optimal.

8. The steps of the Hungarian Method are as listed below: Step1. Find the minimum element in each row of the mxm cost matrix. 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 (reduced cost matrix) by subtracting from each cost the minimum cost in its column.

9. Step2. Draw the minimum number of lines (horizontal and/or vertical) that are needed to cover all zeros in the reduced cost matrix. If m lines are required, an optimal solution is available among the covered zeros in the matrix. If fewer than m lines are required, proceed to step3, we do not yet have a valid assignment. Step3. 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 (i.e., where two lines cross). Return to step2.

10. Setup times (more generally called the cost matrix)

11. Setup times (more generally called the cost matrix)

12. Setup times (also called the cost matrix)

13. Setup times (also called the cost matrix)

14. Setup times (Also called the cost matrix)

15. Setup times (Also called the cost matrix) only three lines needed: we do not have enough 0’s yet…

16. Setup times (Also called the cost matrix) Same as subtracting 1 from every uncovered row and adding 1 to every covered column: this adds at least one zero to the matrix.

17. Setup times (Also called the cost matrix)

18. Setup times (Also called the cost matrix) Now we need four lines for our 4x4 matrix => there is an assignment

19. Optimal Assignment For the Reduced Cost Matrix

20. Optimal Assignment For the Original Cost Matrix Cost = 15