Problems
Chapter Assignment 7, p. 393-400. 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. Machinco wants to minimize the total setup time needed to complete the four jobs.
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Setup times (Also called the cost matrix)
Time (Hours) Job1 Machine 1 Machine 2 Machine 3 Machine 4 14 2 7 2 Job2 5 12 8 4 Job3 8 6 3 6 Job4 7 5 9 10
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The Model
According to the setup table Machinco's problem can be formulated as follows (for i,j=1,2,3,4):
min Z = 14 X 11 + 5 X 12 + 8 X 13 + 7 X 14 + 2 X 21 + 12 X 22 + 6 X 23 + 5 X 24 +7 X 31 + 8 X 32 + 3 X 33 + 9 X 34 + 2 X 41 + X 42 + 6 X 43 + 10 X 44 s.t. X 11 + X 12 + X 13 + X 14 = 1 X 21 + X 22 + X 23 + X 24 = 1 X 31 + X 32 + X 33 + X 34 = 1 X 41 + X 42 + X 43 + X 44 = 1 X 11 + X 21 + X 31 + X 41 = 1 X 12 + X 22 + X 32 + X 42 = 1 X 13 + X 23 + X 33 + X 43 = 1 X 14 + X 24 + X 34 + X 44 = 1
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Xij = 0 or Xij = 1
For the model on the previous page note that: Xij=1 if machine i is assigned to meet the demands of job j Xij=0 if machine i is assigned to meet the demands of job j In general an assignment problem is a balanced transportation problem in which all supplies and demands are equal to 1.
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Solution Method
Although the transportation simplex 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 there is an other method called The Hungarian Method.
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Basic Ideas
Suppose we ranked all feasible assignments in increasing order of cost. The ranking does not change if one subtracts the same amount, say, D, from all costs in the same row, since all assignment costs are reduced by D. The ranking does not change if one subtracts the same amount, say, D, from all costs in the same column, since all assignment costs are reduced by D.
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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 zero a cost in each row and a zero cost in each column of the reduced cost matrix, this assignment is optimal.
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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.
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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 step 3. 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. Return to step2.
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Setup times (Also called the cost matrix)
Time (Hours) Job1 Machine 1 Machine 2 Machine 3 Machine 4 14-5 2-2 7-3 2-2 Job2 5-5 12-2 8-3 4-2 Job3 8-5 6-2 3-3 6-2 Job4 7-5 5-2 9-3 10-2
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Setup times (Also called the cost matrix)
Time (Hours) Job1 Machine 1 Machine 2 Mac...