(Solved): 2. Consider the following linear optimisation problem \[ \begin{array}{l} \text { maximise } \quad ...

2. Consider the following linear optimisation problem \[ \begin{array}{l} \text { maximise } \quad x_{1}-\frac{1}{c} x_{2} \\ \text { subject to }-x_{1}+2 x_{2} \leq 4 \text {; } \\ 2 x_{1}-x_{2} \leq 4 \text {; } \\ x_{1}, x_{2} \geq 0 \text {, } \\ \end{array} \] for some free parameter \( c \neq 0 \). (a) Plot the set of constraints in a two-dimensional graph, with values of \( x_{1} \) on the horizontal axis and values of \( x_{2} \) 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 nonbasic 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]