n 2. Consider the following linear optimisation problem x1 - x2 +2x 4; 2x1x2 4; I1,

n

Transcribed Image Text:

2. Consider the following linear optimisation problem x1 - ¹x2 ₁+2x₂ ≤ 4; 2x1x2 4; I1, I2 ≥ 0, maximise subject to for some free parameter c + 0. (a) Plot the set of constraints in a two-dimensional graph, with values of ₁ on the horizontal axis and values of 2 on the vertical axis. Mark the area in the graph that corresponds to the feasibility set. [3 marks] (b) Plot the objective function on the graph from part (a). What is the slope of the line containing all points that yield the same value of the objective? [3 marks] (c) Using your graph, determine all the solutions to the above linear program, depending on the value of the parameter c. [6 marks] (d) Reformulate the linear program into its slack form. Specify the basic and non- basic variables. [3 marks] (e) Let c = 1. Find a solution to the optimisation problem using the simplex method. Describe each step in your solution. What is the maximised value of the objective function? Specify which constraints are binding. [5 marks] 2. Consider the following linear optimisation problem x1 - ¹x2 ₁+2x₂ ≤ 4; 2x1x2 4; I1, I2 ≥ 0, maximise subject to for some free parameter c + 0. (a) Plot the set of constraints in a two-dimensional graph, with values of ₁ on the horizontal axis and values of 2 on the vertical axis. Mark the area in the graph that corresponds to the feasibility set. [3 marks] (b) Plot the objective function on the graph from part (a). What is the slope of the line containing all points that yield the same value of the objective? [3 marks] (c) Using your graph, determine all the solutions to the above linear program, depending on the value of the parameter c. [6 marks] (d) Reformulate the linear program into its slack form. Specify the basic and non- basic variables. [3 marks] (e) Let c = 1. Find a solution to the optimisation problem using the simplex method. Describe each step in your solution. What is the maximised value of the objective function? Specify which constraints are binding. [5 marks]