Linear Program Problem ARC Consultants needs to know how long it will take to co

Question

Linear Program Problem

ARC Consultants needs to know how long it will take to complete a particular job. The job consists of (n) different tasks. Each task (k) takes (tk) hours to complete (regardless of how many people work on it).

Additionally, certain tasks cannot be performed until other tasks have been completed. This information is represented in a (n)x(n) matrix P where the (i, j)th element of the matrix is set to 1 if task (j) must be performed before task (i) and a 0 otherwise. (Note that all elements on the diagonal must be zero as a task cannot be performed before itself.) There is no limit on the number of tasks that can be performed at once.

Question:

Set up a linear program that determines how soon ARC can complete the job. As always, clearly specify your decision variables, objective function and constraints.

Explanation / Answer

For product 1 applying exponential smoothing with a smoothing constant of 0.7 we get:

M1 = Y1 = 23
M2 = 0.7Y2 + 0.3M1 = 0.7(27) + 0.3(23) = 25.80
M3 = 0.7Y3 + 0.3M2 = 0.7(34) + 0.3(25.80) = 31.54
M4 = 0.7Y4 + 0.3M3 = 0.7(40) + 0.3(31.54) = 37.46

The forecast for week five is just the average for week 4 = M4 = 37.46 = 31 (as we cannot have fractional demand).

For product 2 applying exponential smoothing with a smoothing constant of 0.7 we get:

M1 = Y1 = 11
M2 = 0.7Y2 + 0.3M1 = 0.7(13) + 0.3(11) = 12.40
M3 = 0.7Y3 + 0.3M2 = 0.7(15) + 0.3(12.40) = 14.22
M4 = 0.7Y4 + 0.3M3 = 0.7(14) + 0.3(14.22) = 14.07

The forecast for week five is just the average for week 4 = M4 = 14.07 = 14 (as we cannot have fractional demand).

We can now formulate the LP for week 5 using the two demand figures (37 for product 1 and 14 for product 2) derived above.

Let

x1 be the number of units of product 1 produced

x2 be the number of units of product 2 produced

where x1, x2>=0

The constraints are:

15x1 + 7x2 <= 20(60) machine X

25x1 + 45x2 <= 15(60) machine Y

x1 <= 37 demand for product 1

x2 <= 14 demand for product 2

The objective is to maximise profit, i.e.

maximise 10x1 + 4x2 - 3(37- x1) - 1(14-x2)

i.e. maximise 13x1 + 5x2 - 125

The graph is shown below, from the graph we have that the solution occurs on the horizontal axis (x2=0) at x1=36 at which point the maximum profit is 13(36) + 5(0) - 125 =

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.