Question

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

                                                                                      Project

                                                                 1           2           3           4           5          

           __________________________________________________________

            Supervisors Required                 4            3           5           3           4          

          Workers Required                      15          24         35         20        24        

          NPV (in thousands of dollars)   200        280       320       220      280     

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

- Use no more than 15 supervisors

- Use no more than 80 workers

- At least three projects are to be done.

(a) Formulate a capital budgeting problem by defining the decision variables, the objective function, and all the constraints. Briefly describe what the objective function and each constraint represent.

(b) The optimal solution for the above problem is given below.

Variable values are X1 = 0, X2 = 1, X3 = 1, X4 = 1, X5 = 0

Objective function value is 820

Interpret the optimal solution to make a recommendation to the company.

Answer 1

a. The capital budgeting problem can be formulated by taking the decision variables as X1, X2, X3, X4, X5 where each variable represents if the project will be chosen or not.

The constraints are as follows,

Objective Function = Z ( max NPV) = 200X1 +280X2 + 320X3+220 X4 + 280X5

Constraints :-

0<=X1, X2, X3, X4, X5<=1 ( This means that if a variable is 1 then that project is chosen if it is 0 it is excluded)

S = 4X1 +3X2 + 5X3+3 X4 + 4X5 <= 15 -- ( No. of supervisors constraint)

W = 15X1 +24 X2 + 35 X3+ 20 X4 + 24 X5 <= 80 -- ( No. of workers constraint)

Also , X1 +X2 + X3+ X4 + X5 >= 3 -- ( Min no. of project constraints)

b.

Given:-

The optimal solution for the above problem is X1 = 0, X2 = 1, X3 = 1, X4 = 1, X5 = 0 , This means that the project 2,3 and 4 are chosen.

So, for those three choices the value of the objective function will be :-

Z ( NPV) = 200X1 +280X2 + 320X3+220 X4 + 280X5

= 200 * 0  +280*1 + 320 * 1+220 * 1 + 280* 0

= 280 + 320 +220

= 820