ENGM 535 Optimization Assignment Problems

Prototype Problem K-Corp has 3 parts, each of which can be assigned to 1 of 3 machines. The problem is to assign 1 part to one machine only so as to minimize the total production costs. Relevant data are: Cost to Produce Part i on Machine j

LP Formulation Min Z total production t s Each machine to be used once 1 , machine i assigned to task j X = ij , else Min Z total production t s Each machine to be used once part assigned a = cost .

LP Formulation Min Z X s t = + 9 12 11 1 . machine 1 assigned to task j else ij = 1 , Min Z X s t = + 9 12 11 1 13 33 21 22 23 31 32 . machine 1 assigned machine 2 assigned machine 3 assigned part 1 assigned part 2 assigned part 3 assigned

LP Formulation Min Z c X s t j n i ij = å 1 2 . , X all i j ij ³ ,

Transportation Formulation Assignment special case of Transportation suppliers = destinations = n si = dj = 1

Excel Approach Excel doesn’t have special Assignment problem tool, but we can set up the problems like an general LP or Transportation problem. The books graphic for Excel use follow.

Modeled as Network Demand Supply 1 3 2 9 12 11 10 8 13 -1 1

Class Problem K-Corp has 4 projects underway and 3 possible EngManagers who could possibly be assigned to the projects. In order to determine who to assign to each project, the enterprise manager ranks each employee on background, experience, interpersonal skills, and organizational skills. A final score for each employee for each project is determined. Maximize the assignment score.

Class Problem

Class Problem This will not work, why not?

Prototype Problem Hungarian Algorithm K-Corp has 3 parts, each of which can be assigned to 1 of 3 machines. The problem is to assign 1 part to one machine only so as to minimize the total production costs. Relevant data are: Cost to Produce Part i on Machine j

Prototype Problem Hungarian Algorithm Step 1) Subtract the smallest number in each row from every number in the row. (This is called row reduction.) Enter the results in a new table. Cost to Produce Part i on Machine j

Prototype Problem Hungarian Algorithm Step 2) Subtract the smallest number in each column from every number in the column. (This is called column reduction.) Enter the results in a new table. Cost to Produce Part i on Machine j

Prototype Problem Hungarian Algorithm Step3. Test whether an optimal assignment can be made. You do this by determining the minimum number of lines needed to cover (i.e., cross out) all zeros. If the number of equals the number of rows, an optimal set of assignments is possible. In that case, go to step 6. Otherwise go on to step 4. Optimal, go to step 6.

Prototype Problem Hungarian Algorithm 6. Make the assignments one at a time in positions that have zero elements. Begin rows or columns that have only one zero. Since each row and each column nee receive exactly one assignment, cross out both the row and the column involved each assignment is made. Then move on to the rows and columns that are n crossed out to select the next assignment, with preference again given to any such or column that has only one zero that is not crossed out. Continue until every row every column has exactly one assignment, and has been crossed out. Machine 1 to Part 1 Machine 2 to Part 2 Machine 3 to Part 3

Hungarian Algorithm other 4. If the number of lines is less than the number of rows, modify the table in the following way: a. Subtract the smallest uncovered number from every uncovered number in table. b. Add the smallest uncovered number to the numbers at intersections of cover lines. c. Numbers crossed out but not at the intersections of cross-out lines carry over changed to the next table. 5. Repeat steps 3 and 4 until an optimal set of assignments is possible.