Assignment problems can be used and practically applied in business worlds. Take

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Assignment problems can be used and practically applied in business worlds.

Take the example of optimization problems. This problem can be applied in the real business world in case of manufacturing industries and also in logistic industries.

Many manufacturing companies are using optimization and linear programming to decide on how to allocate labor to do jobs optimally while minimizing its costs. The speed and accuracy of computers are used to find accurate solutions and take out guess works in any from the solution.

Similarly such assignment problems can be used by a company which has different manufacturing plants and different warehouses. Transportation cost is different for each route and each plant has a limited capacity and each warehouse has a demand that has to be met. Optimization problem will help to determine the quantity to be transported from each plant to each warehouse in such a way that all supply constraints (capacity of plants) and all demand constraints (requirements of warehouse) are met while minimizing total costs.

Suppose that the assignment problem is about minimizing costs in a maintenance shop. The shop has 3 group of employees – G1 (skilled), G2 (semi skilled), G3 (specialists).

Now consider there are 3 employees in G1, 4 in G2 and 2 in G3.

Now the maintenance shop has to perform 9 jobs in total – 6 of these jobs are easy while 3 of he jobs are complex.

Costs of workers, group wise and job wise type, is;

The task is to perform all the jobs at the lowest possible cost. This assignment can be solved using Linear programming. Excel's solver function can be used.

Solution:

Total workers = 9. Let each worker have two binary variables i.e. 1 and 0. It means that if worker 1 of category G1 has a binary variable of 1 for easy he will perform the easy repair and will not perform the complex repair (in which case the binary variable will be 0)

Thus there are a total of 9*2 = 18 variables (all of them binary)

The variables are defined below:

Objective function: 10EG11+24CG11+10EG12+24CG12+10EG13+24CG13+9EG21+28CG21+9EG22+28CG22+9EG23+28CG23+9EG24+28CG24+12EG31+20CG31+12EG32+20CG32

This has to be minimized.

Constraints are:

1. EG11+CG11 = 1

2. EG12+CG12 = 1

3. EG13+CG13 = 1

4. EG21+CG21 = 1

5. EG22+CG22 = 1

6. EG23+CG23 = 1

7. EG24+CG24 = 1

8. EG31+CG31 = 1

9. EG32+CG32 = 1

(The above constraints states that each worker can do only one type of job i.e. either easy or complex but not both)

10. EG11+CG11+EG12+CG12+EG13+CG13 = 3

11. EG21+CG21+EG22+CG22+EG23+CG23+EG24+CG24 = 4

12. EG31+CG31+EG32+CG32 = 2

(Constraints 10 to 12 state that 3 workers in G1, 4 workers in G2 and 2 workers in G3 are available)

13. EG11+EG12+EG13+EG21+EG22+EG23+EG24+EG31+EG32 = 6

14. CG11+CG12+CG13+CG21+CG22+CG23+CG24+CG31+CG32 = 3

(Constraints 13 and 14 state that there are 6 easy jobs and 3 complex jobs).

15. All variables are binary

Solving the above in excel's solver the following solution is obtained:

Thus minimized cost is $120 and 2 workers of G1 do easy task and 1 worker of G1 does complex task. All the 4 workers of G2 do easy tasks. All the 2 workers of G3 do complex task.

Such assignment problems can be applied in the real business world.

Easy job Complex job G1 $10.00 $24.00 G2 $9.00 $28.00 G3 $12.00 $20.00

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Provide one example of how an assignment problem could be used in the business v orld

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