Question

A department head has four managers, and three projects (A, B, C) to be executed. The managers differ in efficiency, and the projects differ in their intrinsic complexity. The estimate of the times each manager would take to complete each project is given in the matrix below. Formulate a linear programming (LP) model for determining how the projects should be assigned, to a manager, so as to minimize the total project completion time. No manager should be assigned more than one project.

                                                  

                                                     Manager

Project

	1	2	3	4
A	8	14	12	11
B	6	9	10	8
C	7	10	9	11

Answer 1

Decision variable: as shown in fig1 in green

the decision variables are binary in nature. which means that they can take only values of 0 or 1. 0 means manager not assigned to the project and 1 means manager assigned

Which manager assigned to each project:

Project A: Xi, (where i = 1,2,3,4 representing each manager)

Project B: Yi (where i = 1,2,3,4 representing each manager)

Project C: Zi (where i = 1,2,3,4 representing each manager)

Hence we have total 12 decision variables

Objective function: as shown in fig1 in yellow

minimize total time taken to complete each project.

X1*8 + X2*14 + ....+ Z3*9 + Z4*11

we multiply each decision variable with the corresponding time taken and sum all of them. this is the function we minimize.

Constraints:

1. all decision variable binary. as explained above.

2. 1 project assigned to 1 manger:

Project A,B,C: X1+X2+X3+X4; Y1+Y2+Y3+Y4; Z1+Z2+Z3+Z4 = 1, in excel: F7:F9 = 1

3. 1 manager to 1 project:

Manager1,2,3,4: X1+Y1+Z1, X2+Y2+Z2, X3+Y3+Z3, X4+Y4+Z4 1, in excel: B10:E10 1

solving the LP program:

fig1

fig2