Q1. 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. (Solution length varies)
- b)) If value of the 2×2 matrix game [1 2; p 4] is 4, then p≥ 4. (Solution length varies)
- c)) If 10 is added to each of the entries of the cost matrix of a 3×3 assignment problem, then the total cost of an optimal assignment for the changed cost matrix will increase by 10. (Solution length varies)
- d)) For maximization LP model, the simplex method is terminated when all values C-Zj≥0. (Solution length varies)
- e)) The dummy source or destination in a transportation problem is added to prevent solution from becoming degenerate. (Solution length varies)
Key Points:
- Intersection of any number of convex sets is always convex.
- For 2x2 game [1 2; p 4] with value 4, p must be ≥ 4.
- Adding 'k' to n×n assignment matrix increases total cost by n*k.
- Maximization LP terminates when all Cj-Zj values are ≤ 0.
Answer: This response evaluates five statements related to Linear Programming, Game Theory, and Transportation/Assignment problems, identifying them as true or false and providing detailed reasoning with proofs or explanations, drawing upon fundamental concepts from the MTE-12 course material. Each sub-question is addressed individually, adhering to principles such as convex set properties, saddle point conditions in matrix games, cost calculation in assignment problems, optimality criteria in the simp...