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Operations Management Coursework 1

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Operations Management Coursework 1 Atiqah Erlina binti Abdullah K1455621 King’s College London Department of Informatics

Operations Management Coursework 1

Question 1 Model this problem as a linear program with integer constraints. What do your

variables represent? Provide a tabular representation of the above information and explain how the constraints in your program are derived from it. Constraints identified: Plant A: 40 golf carts and 3 snowmobiles 5 Plant B: 65 golf carts Plant C: 53

snowmobiles

Hence;

Minimize z = 210000x1 + 190000x2 + 182000x3 Subject to; 40x1 + 65x2 ≥ 1500 35x1 + 53x3 ≥ 1100

x1 ≤ 30

x2 ≤ 30

x3 ≤ 30

x1, x2, x3 ≥ 0

Initial Tableau

x1 40 35 1 0 0 210000 x2 65 0 53 1 0 190000 x3 0 53

0 1 182000 z 0 0 0 0 0 1

Therefore; Plant A=x1 Plant B=x2 Plant C=x3

ans 1500 1100 30 30 30 0

Question 2 Convert the p roblem into c anonical form, identifying any slack and surplus variables

that are needed. State the problem in this form.

In canonical form:

Maximize z = -­‐210000x1 -­‐190000x2 -­‐182000x3

Subject to; 40x1 + 65x2 -­‐ x4 = 1500 35x1 + 53x3 -­‐ x5 = 1100

x1+x6 = 30

x2+x7 = 30

x3+x8 = 30

x1, x2, x3, x4, x5, x6, x7, x8 ≥ 0

The problem found in this canonical form is that the s lack variable x4 & x5 have a

negative value. But x4 & x5 is constrained to be non-­‐negative!

Hence solving this problem from the all-­‐slack point becomes infeasible.

Question 4

Write down the pseudo-­‐objective function for this problem, showing all your working. We first introduce artificial variables for each constraint in the target problem, so the

3 new variables are x9, x10, x11, x12 and x13.

Maximize z' = -­‐x9 -­‐x10 -­‐x11 -­‐ x12 -­‐ x13

Subject to; 40x1 + 65x2 -­‐ x4 + x9= 1500

35x1 + 53x3 -­‐ x5 + x10= 1100

x1 + x6 + x11 = 30

x2 + x7 + x12= 30

x3 + x8 + x13= 30

x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13 ≥ 0

Question 5

Construct the initial tableau for phase 1 and find the basic feasible point for phase 2. Our pseudo p roblem is:

Question 3

Explain how y ou know that the all-­‐slack point is not feasible?

Step 1: Initial T ableau

-­‐x9 -­‐x10 -­‐x11 -­‐ x12 -­‐ x13

Maximize z' =

Subject to; 40x1 + 65x2 -­‐ x4 + x9= 1500

35x1 + 53x3 -­‐ x5 + x10= 1100

x1 + x6 + x11 = 30

x2 + x7 + x12= 30

x3 + x8 + x13= 30

x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13 ≥ 0

The constraints are summed: 76x1 + 66x2 + 54x3 -­‐ x4 -­‐ x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 = 2690 = -­‐ x9 -­‐ x10 -­‐ x11 -­‐ x12 -­‐ x = 76x1 + 66x2 + 54x3 -­‐ x4 -­‐ x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 = 2690

z' 13 z' -­‐ 76x1 -­‐ 66x2 -­‐ 54x3 + x4 + x5 -­‐ x6 -­‐ x7 -­‐ x8 = -­‐2690

(continued in excel document along with Question 6)

Question 7 What is the solution to the integer problem? Why is the value of this solution different to

that obtained in 6?

LPSolve was r un to get an answer for this question and the answer obtained was as follows:

Hence;

The value of t his solution is different compared to Question 6 because the feasible

regions are very different. The feasible region for the integer problem is a set of discrete points. z x1 x2 x3 7432 30 5 1

Question 8 Could the solution to an integer problem ever be better than the solution found to its

relaxed version?

Optimal solution found through LPSolve is 7432.

This ultimately depends on w hich objective is better achieved. In our case, we want to minimize the

cost hence the relaxed version is more prefered.

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