Tribhuvan University BSc Fourth Year Linear Programming Exam Question 2082

Tribhuvan University · Linear Programming (BSc Fourth Year) · 2082 BS · Annual Examination — 5 questions — a free old question paper on Goodone.club’s exam-paper archive.

  1. (a) Define a hyperplane and a convex set. Show that the set of all feasible solutions of a linear programming problem is a convex set. [1+1+3] (b) If P and Q are two convex sets in Rⁿ, then for any scalars α and β, prove that αP + βQ is a convex set. Using this, show that P − Q is also a convex set. [4+1] [10 marks]
  2. (a) Rewrite the following linear programming problem in standard form and convert it into canonical form. Also convert it into a minimization problem. [2+2+1] Maximize Z = 12x₁ + 5x₂ Subject to 6x₁ + 3x₂ ≥ 5; 7x₁ + 2x₂ ≤ 14; x₁, x₂ ≥ 0. (b) Show that the three forms of a linear programming problem are equivalent. [5] [10 marks]
  3. (a) What do you mean by a basic feasible solution? When does the solution become degenerate or non-degenerate? [1+1+2] (b) Describe the steps of the simplex method for computing an optimal solution in a linear programming problem. [5] OR Solve the following linear programming problem using the two-phase method: Maximize Z = 10x₁ + 8x₂, subject to x₁ + x₂ ≤ 6; 2x₁ + x₂ ≥ 8; x₁, x₂ ≥ 0. [5] [10 marks]
  4. (a) Convert the following linear programming problem into its dual. [5] Minimize Z = 2x₂ + 5x₃ Subject to x₁ + x₂ ≥ 2; 2x₁ + x₂ + 6x₃ ≤ 6; x₁ − x₂ + 3x₃ = 4; x₁, x₂, x₃ ≥ 0. (b) If the i-th constraint of the primal is an equality constraint, prove that the i-th dual variable is unrestricted in sign. [5] [10 marks]
  5. (a) Formulate the linear programming model for a transportation problem, including a description of the essential variables, constraints and objective function. Additionally, provide an example of a real-life application of this problem. [4+1] (b) Formulate the linear programming model for a diet problem, including a description of the essential variables, constraints and objective function. Additionally, provide an example of a real-life application of this problem. [4+1] OR Write a general linear programming model for the maximum-flow problem and justify that the model is linear. Briefly des... [10 marks]