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L15_10 - Assignment Problems Example Machineco has four...

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2/22 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 following table (also called the cost matrix ): Time (Hours) Job 1 Job 2 Job 3 Job 4 Machine 1 14 5 8 7 Machine 2 2 12 6 5 Machine 3 7 8 3 9 Machine 4 2 4 6 10 Machinco wants to minimize the total setup time needed to complete the four jobs.
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3/22 Assignment Problems In general, an assignment problem is a balanced transportation problem, in which all supplies and demands are equal to 1. This is a special subclass of transportation problems that can be solved more efficiently than the general transportation problems. Next, we will develop a linear programming model for our example. The decision variables: x ij = 1 , if machine i is assigned to meet the demands of job j 0 , otherwise .
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4/22 Assignment Problems Based on the cost matrix, 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 x ij = 0 or x ij = 1 , i , j = 1 , . . . , 4
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5/22 Assignment Problems Note that we have the integrality constraints: x ij = 0 or x ij = 1 , i , j = 1 , . . . , 4 However, if we remove the integrality constraints and solve the resulting transportation problem with the transportation simplex, we will obtain an integer solution. Indeed, all the input data are integer, and we use only additions and subtractions in the transportation simplex method, therefore the optimal solution has to be integer.
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