Question

The Akron Police Department has five detective squads available for assignment to five open crime cases. The chief of detectives, Paul Kuzdrall, wishes to assign the squads so that the total time to conclude the cases is minimized. The average number of days, based on past performance, for each squad to complete each case is as follows:

                                                    CASE

SQUAD	A	B	C	D	E
1	14	7	3	7	27
2	20	7	12	6	30
3	10	3	4	5	21
4	8	12	7	12	21
5	13	25	24	26	8

Write an LP formulation that could be used to solve this problem with the relevant LP software packages like excel "solver".

Answer 1

First we need to use a set of decision variables that represents each assignment. Let’s call them Xij. Here i ={A,B,C,D,E} and j ={1,2,3,4,5}. Also we need to set this decision variable as binary. That means

Xij = {0,1}

Let the time taken between each of the case and a squad be represented by Tij. Then the objective function will be

Minimize Z where Z = summation ij (Tij*Xij)

Subject to the constraints

Summation j (Xij) = 1

Summation I (Xij) = 1

The constraints ensure that every row and column will sum to 1.

The solver model is shown below

The formulas are shown below

The solver parameters are shown below

The result is shown below. Assignment should be squad 1 = C, squad 2 = D, squad 3 = B, squad 4 = A, squad 5 = E. Total time will be 28