Assignment Problems

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Assignment Problems

A production supervisor is considering how he should assign the four jobs that are to be performed, to the four of the workers. He wants to assign the jobs to the workers such that aggregate time to perform the jobs is the least. Based on the previous experience, he has the information on the time taken by the four workers in performing these jobs, as given in the table:

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There are four method of solving assigning problems:

(a) Complete enumeration method

(b) Transportation method

(c) Simplex method

(d) Hungarian Assignment Method

(a) Complete enumeration method

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(c) Simplex Method:

(d) Hungarian Assignment Method

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Hungarian Assignment Method

Step 1: Locate the smallest cost element in each row of the cost table. Now subtract this smallest element from each element in that row. As a result, there shall be at least one zero in each row of this new table called the reduced cost table.

Step 2: In the reduced cost table obtained, consider each column and locate the smallest element in it. Subtract the smallest value from every other entry in the column. As a consequence of this action, there would be at least one zero in each of the rows and the columns in the second reduced cost table.

Step 3: Draw minimum number of horizontal lines and vertical lines (not the diagonal one) that are required to cover the “Zero” elements. If the number of lines drawn is equal to n (the number of rows/columns) the solution is optimal, and proceed to the step 6. If the number of lines drawn is smaller than n, go to step 4.

Step 4: Select the smallest uncovered (by the lines) cost element. Subtract this element from all uncovered elements including it and add this element to each value located at the intersection of any two lines. The cost elements through which only one line passes remain unaltered.

Step 5: Repeat steps 3 and 4 until an optimal solution is obtained.

Step 6: Given the optimal solution, make the job assignments as indicated by the zero elements. This is done as follows.

a. Locate a row which contains only one zero element. Assign the job corresponding to this element to its corresponding person. Cross out the zeros, if any, in the column corresponding to the element, which is indicative of the fact that the particular job and the person is no more available.

b. Repeat (a) for each of such rows which contain only one zero. Similarly, perform the same operation in respect of each column containing only one zero element, crossing out the zero(s), if any, in the row in which the element lies.

c. If there is no row or column with only a single zero element left, then select a row/column arbitrarily and choose one of the jobs (persons) and make the assignment. Now cross the remaining zeros in the column and row in respect of which the assignment is made.

d. Repeat steps (a) through (c) until all assignments are made.

e. Determine the total cost with reference to the original cost table.

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Unbalanced and Assignment

You are given the information about the cost of performing different jobs by different persons. The job-person marking X indicates that the individual involved cannot perform the particular job. Using this information, state (i) optimal assignment of jobs (ii) the cost of such assignments.

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Unique Vs Multiple Solutions

Solve the following assignment problem and obtain the minimum cost at which all the jobs can be optimized.

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Maximization Case

A company plans to assign 5 salesmen to 5 districts in which it operates. Estimates of sales revenue in thousands of rupees for each salesman in different districts are given in the following table. In your opinion, what should be the placement of the salesmen if the objective is to maximize the expected sales revenue?

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Covert the given maximization problem into minimization problem by subtracting all the elements from the highest value of the element of the matrix i.e. 41. Then the minimization matrix will be

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REFERENCES:

1. Operations Research: Theory and Applications - Sharma J. K, 4/e , Macmilan, 2010

2. Operations Research - Vohra N. D, 4/e, TMH, 2010.

3. Operations Research – Kalavathy S, 3/e, Vikas Publishing House.

4. Operations Research – Anand Sharma, HPH.

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