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Assignment Meeting 15 Course: D0744 - Deterministic Optimization Year: 2009

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Introduction 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, c ij. The objective is to assign all tasks such that the total cost is minimized. Bina Nusantara University 3

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Assignment in General LP Bina Nusantara University 4 In general the LP formulation is given as Minimize Each supply is 1 Each demand is 1

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Example Assign people to project assignments Assign jobs to machines Assign products to plants Assign tasks to time slots Bina Nusantara University 5

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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 c ij associated with assignee i performing task j. The objective is to determine how all n assignments should be made to minimize the total cost. Bina Nusantara University 6

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Flow Diagram Bina Nusantara University 7 1 1 3 3 2 2 1 1 3 3 2 2 n n assignees tasks a – assignee t – tasks a1a1 a2a2 a3a3 t1t1 t2t2 t3t3 t4t4 c 11 c 12 n n c nn anan

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Cost Matrix Let the following represent the standard assignment problem cost matrix, c: Bina Nusantara University 8

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Conversion to Standard Cost Matrix Consider following cost matrix, how do you convert to satisfy the standard definition of the assignment problem? Bina Nusantara University 9

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Cont’d Add “big M” to avoid incompatible assignments, and add a dummy assignee (or task) to have equal assignees and tasks. Bina Nusantara University 10

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Math Formulation Bina Nusantara University 11 Minimize s.t. Total Cost i j Does this formulation look familiar? Is this a Linear Program?

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Hungarian Method Consider following cost matrix Bina Nusantara University 12

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Cont’d Reduce by Row Minimum Bina Nusantara University 13

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Cont’d Reduced by Column Minimum Bina Nusantara University 14

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Cont’d Reduce by Minimum of uncovered cells (1): Bina Nusantara University 15

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Cont’d Solution is now optimal since minimum number of lines to cover all 0 is 4 (equal to n). Bina Nusantara University 16 A3 -> T3, A1 -> T2, A2 -> T4, A4 -> T1 Z = 3 + 5 + 5 + 2 = 15

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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. Bina Nusantara University 17

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Allocation problems - The Hungarian Algorithm The Hungarian algorithm Step 1Reduce the array by both row and column subtractions Step 2Cover the zero elements.

Allocation problems - The Hungarian Algorithm The Hungarian algorithm Step 1Reduce the array by both row and column subtractions Step 2Cover the zero elements.

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