Question

6. The Charming City Construction Company is considering six projects. The projects, the number of supervisors and the number of workers required for each project, and the expected profits for each project are given below.

                                                                                      Project

                                                                 1           2           3           4           5           6

           ________________________________________________________________

            Supervisors Required                 5            4           6           3           4           2

          Workers Required                      15          24         35         22        26         32

          Profit (in thousands of dollars) 210        290       330       240      300      200

The objective is to maximize the company's total expected profit subject to the following constraints:

- Use no more than 20 supervisors

- Use no more than 108 workers

- If project 4 is done, then project 2 must be done and vice versa

- At least four projects are to be done.

Formulate a capital budgeting integer optimization problem by defining

(a) The decision variables.

(b) The objective function. What does it represent?

(c) All the constraints. What does each constraint represent?

How to solve each question

Answer 1

ANSWER :

We will formulate this problem as below:

(a) Decision variables

Xi= 1 if a project i is selected else 0 (binary variable)

X1= 1 if a project 1 is selected else 0

X2= 1 if a project 2 is selected else 0

X3= 1 if a project 3 is selected else 0

X4= 1 if a project 4 is selected else 0

X5= 1 if a project 5 is selected else 0

X6= 1 if a project 6 is selected else 0

(b) Objective function

The objective is to maximize total expected profit from the projects. Total profit will be sum of the product of expected profit from projects and variable to select the project.

MAX z= 210X1+290X2+330X3+240X4+300X5+200X6

(c)

Constraints:

5X1+4X1+6X3+3X4+4X5+2X6 <= 20

(this represents use no more than 20 supervisors)

15X1+24X2+35X3+22X4+26X5+32X6 <= 108

( this represents use no more than 108 workers)

X2=X4

(this represents if project 4 is done, then project 2 must be done and vice versa)

X1+X2+X3+X4+X5+X6 >= 4

(this represents at least four projects are to be done)

X1, X2, X3, X4, X5, X6= {0,1}; all Xi's are binary variable.