Question

Your small consulting business has its annual budget planning meeting in two weeks. Every year the process of budgeting, deciding on the projects to accept for the coming year, makes life very difficult for everyone in the firm. The firm specializes in forensic analysis of financial data and the projects are made available in advance, usually by large industrial conglomerates. The expertise you have developed over years allows you to pick-and-choose the projects you would like to accept at the beginning of the planning year. You would like to select projects that maximize revenue and result in a feasible use of resources.

Projects are categorized by Type and information about the number of each Type is also available. The revenue for each project and the associated use of resources (A-D) is provided in the adjacent table. Resources are special skill activities that the firm possesses like financial modeling, report writing, project management, etc. The total number of units of each resource is also provided--e.g. there are 800 units of Res-A available in a year. Thus, for example, we can state that there are 25 projects of Type 1 available at a revenue of $45,000 for each project, and the use of resources for each project is 6 units of A, 12 units of B, 0 units of C, and 5 units of D.

The shown data permits us to create an LP that maximizes the revenue generated by selecting units of each project type to accept into our budget, while adhering to the constrained resource limits.

a) Create the LP for the data provided. Use X1, X2,... as the decision variables for the number of each Type of project selected

		Project Data
Project Type	# Prjts Available	Revenue	Res-A	Res-B	Res-C	Res-D
1	25	45000	6	12	0	5
2	30	63000	9	16	4	8
3	47	27500	4	10	4	5
4	53	19500	4	5	0	7
5	16	71000	7	10	8	4
6	19	56000	10	5	7	0
7	36	48500	6	7	10	3
		Total Resources available==>	800	900	700	375

Answer 1

A(i) represent number of project (i) for i = 1,2 .. 7

Z = 45000*A1 +63000*A2 + 27500*A3 +19500A*A4 + 71000*A5 +56000*A6 +48500*A7

Maximize Z

Constraints

6*A1 +9*A2 +4 *A3 + 4*A4 +7*A5 +10*A6 +6*A7 <= 800

12*A1 +16*A2 +10*A3 + 5*A4 +10*A5 +5*A6 +7*A7 <= 900

0*A1 +4*A2 +4 *A3 + 0*A4 +8*A5 +7*A6 +10*A7 <= 700

5*A1 +8*A2 +5*A3 + 7*A4 +4*A5 +0*A6 +3*A7 <= 375

Solving

A1	11.5942
A2	0
A3	0
A4	0
A5	60.86957
A6	30.43478
A7	0

Z = 6547826

We have to round to nearest integer to get exact Z