Question

The Valley Research Inc. is contemplating a research and development (R&D) program encompassing eight potential research projects. All the projects are expected to go on simultaneously. The company is constrained from embarking on all projects by the number of scientists it has and the budget available for R&D projects. The company has 25 scientists and 300 thousand dollars budgeted for R&D projects. Furthermore, since projects 2 and 5 require the involvement of the same scientists, both of them would not be selected, but either may or may not be selected. The Table below shows the expenses in thousands of dollars, the number of scientists required, and the estimated profit in millions of dollars associated with each project. (a). Formulate the integer programming model for this problem that will maximize profit. That is, you will define the decision variables, state the objective function, and thirdly, state the constraints. (Hint: This is a 0-1 integer programming problem). (b). Use the computer software (QM for Windows or its equivalent) to determine the integer optimal solution to this problem by determining; (i) The optimal projects to be selected for execution. (ii) The optimal profit.

Project	Expense ($1,000s)	Scientists Required	Estimated Profit ($1,000,000s)
1	$?50	7	$0.35
2	90	15	0.80
3	40	5	0.30
4	47	6	0.16
5	80	8	0.55

Answer 1

(a) ILP model is as follows

Decision variables: Let Xi be the binary variable such that Xi = 1 if project i is undertaken, otherwise not.

Objective function: Max 0.35X1+0.8X2+0.3X3+0.16X4+0.55X5

Constraints:

7X1+15X2+5X3+6X4+8X5 <= 25

50X1+90X2+40X3+47X4+80X5 <= 300

X2+X5 <= 1

(b) Solution using LINGO software follows

(i) Optimal solution is following

X1=1, X3=1, X5=1

Therefore, projects 1, 3 and 5 should be undertaken

(ii) Optimal profit = $ 1.2 m