Question

The Valley Research Inc. is contemplating a research and development (R&D) program encompassing five potential research projects named 1, 2, 3, 4, and 5. 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 23 scientists and $300,000 (three hundred 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. From your optimal solution determine the following.

         (i)     List the optimal projects to be selected for execution.

         

         (ii)    The optimal total profit from all the selected projects.

Project	Expense ($)	Scientists Required	Estimated Profit ($Millions) ($1,000,000)
1	60,000	7	3.0
2	90,000	10	7.0
3	45,000	5	2.5
4	47,000	6	1.6
5	80,000	4	5.5

When using QM for Windows to solve the problem, select the “integer & mixed integer programming” option from the module listing. While entering your data, also select the 0/1 option in the last row.

Answer 1

Anshler : Let's define he dedston Varables as ano Xa each of the Pechs. The Values a,?11 bei ft. the meet ts se ered and O MX3 Re the Pnect 8 rot selected. e can thus Pamulahe 가& Problem as Subiect to Lets use uProos excel Solver to obha?n a Solution. The model s souun beleco

1Prcject Expense Scientist Profit Selection SCC00 10 4SCOD 47000 80000 2.5 1.60 5.5 8 Constraint 1o 9 Constraint 10 Constaint 0 23 300000 12 Obj func 1-1 125

The formulas used is shown below:

1 Proje Expense Scientist Profit Selection 90000 45000 47000 80000 2.5 1.6 5.5 8 Constraint 1 9 Constraint 2 10 Constraint 3 SUMPRODUCT(E2:E6,C2:C6)< =SUMPRODUCT(E2:E6,B2 :B6) <- =E2+EG 23 2Obj func SUMPRODUCT E2:E6,02 D6)

The solver parameters are shown below:

1 Praject Expense Scientist Profit Selection 6CC00 CCoo 5COD 47000 80C00 10 2.5 1.6 5.5 8 Constraint 1 9 Constraint 20 10 Constraint 30 23 30D000 12 Obj func 1.1 14 15

The result is shown below:

1 Praject Expense Scientist Profit Selection 6CC00 CCoo 5COD 47000 80C00 10 2.5 1.6 5.5 8 Constraint 1 9 Constraint 20 10 Constraint 30 23 30D000 12 Obj func 1.1 14 15

The op inal Preuts to seleut are ,3,y and 5 Ps 1.6mln The OpHo
​​​​​​​