11.      Write dual of the following linear programming problems:
Maximize                  Z = 10y1 – 8y2 – 6y3                  Subject to     3y1 + y2 – 2y3 ≤ 10
-2y1 + 3y2 – y3 >/Â 12
y1 , y2 , y3 ≥ 0
12.      Write dual of the following linear programming problems:
Minimize                   Z = -4x1 + 3x2                               Subject to     x1 – 2x2 ≥ -4
2x1 + 3x2 ≥ 13
-x1 + x2 ≤ -4
x1, x2 ≥ 0
13.      Write dual of the following linear programming problems:
Maximize                  Z = 5x + 7y                            Subject to     x + y ≤ 4
3x + 8y ≤ 26
10x + 7y ≤ 35
and x, y ≥ 0
14.      Give the dual in complete mathematical form for the following primal of a linear programming problem:
Maximize                  Z = 3x1 – 2x2                         Subject to     x1 ≤ 4, x2 ≤ 6
x1 + x2 ≤ 5
-x2 ≤ -1
x1, x2 ≥ 0
15.      M/s. Raj and Bilimoria Associates produce three items ‘X’, ‘Y’ and ‘Z’ each of which have to be processed through three machines ‘P’, ‘Q’ and ‘R’. Each unit of the product ‘X’ requires 3, 4 and 2 hours on machines ‘P’, ‘Q’ and ‘R’ respectively. Similarly each unit of product ‘Y’ requires 5, 4 and 4 hours on machine P, Q and R respectively, where as for product Z, these requirements are 2, 4 and 5 hours on these three machines P, Q and R. Every day 60 hours are available on machine P, 72 hours on machine Q and 100 hours on machine R. the unit contribution of these products X, Y and Z are ` 5, ` 10 and ` 8 respectively.
- Formulate the linear programming problem and using simplex method find the optimal solution for the product mix, also find the unused capacity of machines if any.
- What should be the effect on the solution of each of the following:
(i)Â Â Â Â Â Â Obtaining an order of 12 units of X which has to be met.
(ii)Â Â Â Â Â An increase of 20% in the capacity of machine P
Additional questions:
(iii)Â Â Â Â Obtaining an order of 6 units of item X which has to be met.
If the optimal solution obtained does not require the production of some item explain as to why such item would not be produced. In this context indicate the quantity (quantities) of other item/s that would be foregone for producing such product.
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