. State which of the following statements are true and which are false. Give reasons for your answer with a short proof or a counter example.
a) The intersection of finite number of convex sets is not convex.
b) If value of the matrix game
is 4, then
.
c) If 10 is added to each of the entries of the cost matrix of a assignment problem, then the total cost of an optimal assignment for the changed cost matrix will increase by 10.
d) For maximization LP model, the simplex method is terminated when all values .
e) The dummy source or destination in a transportation problem is added to prevent solution from becoming degenerate.
a) False. The intersection of a finite number of convex sets is always convex. Reason: Let (C_1, C_2, \dots, C_n) be convex sets. For any (x, y \in \bigcap_{i=1}^n C_i), since each (C_i) is convex, (\lambda x + (1-\lambda)y \in C_i) for all (\lambda \in __________ _________ __________ _____ __________ _____ _________ ____ _________.
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___ ___ ________ ___ ___.
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Solve the following LPP graphically:
Maximize:
$
subject to the constraints:$
$
$
b) Find all values of k for which the vectors:$
are linearly independent.
A department has five employees with five jobs to be performed. The time (in hours) each employee will take to perform each job is given in the following matrix. How should the jobs be allocated, one per employee, so as to minimize the total man hours?
b) For the following pay-off matrix, transform the zero-sum game into an equivalent linear programming problem:
Write the LPP formulation of the following transportation problem:
| Destination | Supply | |||
|---|---|---|---|---|
| $D_1$ | $D_2$ | $D_3$ | ||
| Source $O_1$ | 10 | 18 | 12 | 200 |
| Source $O_2$ | 15 | 17 | 9 | 300 |
| Source $O_3$ | 13 | 15 | 7 | 500 |
| Requirement | 400 | 200 | 400 |
A company makes two kinds of leather belts. Belts A is high quality belt and belt B is of lower quality. The respective profits on A and B are ₹ 4 and ₹ 3 per belt. The production of each type A requires twice as much time as a belt. The production of each type of type B, and if all belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (both A
and B combined). Belt A require a fancy buckle and only 400 buckles per day are available. There are only 700 buckles a day available for belt B. What should be the daily production of each type of belt? Formulate this problem as an LP model and solve it by the graphical method.
b) Find the initial basic feasible solution of the following transportation problem using North-West Corner method.
) निम्नलिखित दो खिलाड़ी शून्य योग खेल को प्रमुखता सिद्धांत द्वारा खेल में समानीत कीजिए और इस प्रकार खेल को हल कीजिए।
Write the LPP formulation of the following assignment problem:
b) Solve the following game graphically:
Reduce the following two person zero sum game to game using principle of dominance. And hence solve the game.
Player B
b) Obtain the dual of the following primal LP problem:
Maximize
Subject to $
$