A firm manufactures two products; the net profit on product 1 is Rupees 3 per unit and Rupees 5 per unit on product 2. The manufacturing process is such that each product has to be processed in two departments D1 and D2. Each unit of product1 requires processing for 1 minute at D1 and 3 minutes at D2; each unit of product 2 requires processing for 2 minutes at D1 and 2 minutes at D2. Machine time available per day is 860 minutes at D1 and 1200 minutes at D2. How much of product 1 and 2 should be produced every day so that total profit is maximum. (solve with graphical method) Use the graphical method to solve the LPP
Maximize Z = 2X1 + 4X2
Subject to constraints:
X1 + 2X2 ≤ 5
X1 + X2 ≤ 4
And X1, X2 ≥ 0
Write the dual of following problems:
Maximize Z = 7X1 + 5X2
Subject to:
X1 + 2X2 ≤ 6
4X1 + 3X2 ≤ 12
X1, X2 ≥ 0
Maximize Z= 3X1 + 4X2
Subject to:
5X1 + 4X2 ≤ 200
3X1 + 5X2 ≤ 150
5X1 + 4X2 ≥ 100
8X1 + 4X2 ≥ 80
X1, X2 ≥ 0

Determine initial basic feasible solution to the following transportation problem using the north west corner rule.

D1
D2
D3
D4
Availability

O1
6
4
1
5
14

O2
8
9
2
7
16

O3
4
3
6
2
5

Requirement
6
10
15
4
35

Obtain an initial basic feasible solution to the following transportation problem using Vogel’s approximation method.

D1
D2
D3
D4

A
5
1
3
3
34

B
3
3
5
4
15

C
6
4
4
3
12

D
4
-1
4
2
19

21
25
17
17
80