LINEAR PROGRAMMING ASSIGNMENT Case Solution
Problem 1
Let Xijbe the amount of the space in square feet which the company leases in month ifor jmonths.
Then, the total variables which we would be having would be 6 as follows:
= x1,1; x1,2; x1,3; x2,1; x2,2; x3,1.
Then, the LP model would be as follows:
Minimize 280(x1,1 + x2,1 + x3,1) + 450(x1,2 + x2,2) + 600(x1,3)
Subject to:
x1,1 + x1,2 + x1,3 >= 25000
x1,2 + x1,3 + x2,1 + x2,2 >=10000
x1,3 + x2,2 + x3,1 >=20000
x1,1; x1,2; x1,3; x2,1; x2,2; x3,1 >=0
The objective function shown above would minimize the total cost of leasing space. The first three constraints make sure that the required amount of space is leased for months 1, 2 & 3 respectively. The fourth constraint is for the non-negativity for all the variables.
Problem 2
The assignment for this problem would be as follows:
Row Reductions | 1 | 2 | 3 | 4 | 5 |
1 | 4 | 2 | 1 | 3 | 0 |
2 | 4 | 3 | 1 | - | 0 |
3 | 4 | 5 | 2 | 3 | 0 |
Column Reductions | 1 | 2 | 3 | 4 | 5 |
1 | 0 | 0 | 0 | 0 | 0 |
2 | 0 | 1 | 0 | - | 0 |
3 | 0 | 3 | 1 | 0 | 0 |
Assignment | Total Cost | ||
Contractor 1 | Job 1 + Job 2 | 14 | |
Contractor 2 | Job 3 + Job 5 | 5 | |
Contractor 3 | Job 4 | 6 | |
Optimal Total Cost | 25 |
Problem 3
The LP model is as follows:
Min = 4x1− 5x2− 3x3
- t. x1 + x2 + x3 + x4 = 10
x3 ≥ 1 x1 + 2x3 ≤ 4 x1 ≥ 0,x2 ≤ 0
This could be transformed into standard form as follows:
Min Z = 4x1− 5x2− 3x3
Subject to:
x1 + x2 + x3 + x4 = 10
x3 – s1= 1 x1 + 2x3 + s2= 4 x1 – s3= 0,x2 +s4= 0
x1, x2, x3, x4, s1, s2, s3, s4 >= 0
We can see that the constraint system has m=4 and n = 4 variables, therefore we need to get 4-4=0 variables equal to zero to solve for phase 1 problem. For example if we substitute x2, x2, x3 and x4 to 0 then:
S1: 0
S2: 0
S3: 0
S4: 0
This is clearly feasible and also called as the basic feasible solution.................
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