. 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 [0,1]). Therefore, (\lambda x + (1-\lambda)y \in \bigcap_{i=1}^n C_i). b) False. Consider _________ _________ ________ _______ ________ ________ _________.
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__________ ________ _______ _____.
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. 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.
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 |
निम्नलिखित आद्य रैखिक प्रोग्रामन (एलपी) समस्या की द्वैती प्राप्त कीजिए :
का अधिकतमीकरण कीजिए
जबकि
wo breakfast food manufacturers ABC and XYZ are competing for an increased market share. The pay-off matrix, shown in the following table, describes the increase in market share for ABC and decrease in market share of XYZ. Determine optimal strategies for both the manufacturers and the value of the game.
b) Find all the basic feasible solutions of the following system of linear equations:
$
$
$
Check if any of them is degenerate solution. Justify your answer.
) Using matrix – minima method, find the initial basic feasible solution of the following transportation problem:
Hence find the optimal solution.
b) Check whether the following set is convex:
$
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 $
$
) For what value of k are the following vectors linearly independent?$
b) Solve the following LP problem using simplex method:
Maximize
Subject to $
$
$