Question

KPNG Consulting has five projects to consider. Each will require time in the next four quarters according to the table below

Project	Time in 1st quarter	Time in 2nd quarter	Time in 3rd quarter	Time in 4th quarter	Revenue $k
A	6	9	3	1	240
B	4	13	6	4	120
C	8	0	8	6	150
D	3	3	1	7	60
E	12	3	4	9	250

Revenue from each project is also shown. Develop a model whose solution would maximize revenue, meet the time budget of: 28 in the 1st quarter, 20 in the 2nd quarter, 18 in the 3rd quarter and 16 in the 4th quarter; at least three projects; and not do both projects C and D. If project B is chosen, project D must be chosen.

Answer 1

Answer: We will formulate the required model as mentioned below:

Decision Variable:

Let Decision Variables:

A = 1 If Project A is Selected, 0 Otherwise,

B = 1 If Project B is Selected, 0 Otherwise,

C = 1 If Project C is Selected, 0 Otherwise,

D = 1 If Project D is Selected, 0 Otherwise, and

E = 1 If Project E is Selected, 0 Otherwise

Objective Function:

Here, the objective is to maximize total revenue. Hence, the objective function:

MaxZ = 240000 A + 120000 B + 150000 C + 60000 D + 250000 E

(Note: Revenue is converted in full amount by multiplying with 1000 (k), as mentioned in the question)

Subject to Constraint:

C1 = 6A + 4B + 8C + 3D + 12E < 28 (TIme Budget for 1st Quarter)

C2 = 9A + 13B + 0C + 3D + 3E < 20 (TIme Budget for 2nd Quarter)

C3 = 3A + 6B + 8C + 1D + 4E < 18 (TIme Budget for 3rd Quarter)

C4 = 1A + 4B + 6C + 7D + 9E < 16 (TIme Budget for 4th Quarter)

C5 = A + B + C + D + E > 3 (At least three projects to be selected)

C6 = C + D = 1 (Either Project C or Project D, but not both Projects)

C7 = B < D (If Project B is chosen, Project D must be chosen) (Alternatively, B - D < 0)

Where A, B, C, D, E ∈ {0, 1}