(a) A small firm specialises in making three types of spare automobile parts. Each part is first cast from iron in the casting shop and then sent to the finishing shop where holes are drilled, surfaces are turned, and edges are ground. The required worker-hours (per 100 units) for each of the parts of the two shops are shown below: Part A B Casting (worker-hours) 2. 2 1 3 1 2 Finishing (worker-hours) The profits from the parts are $20, $15 and $30 (per 100 units), respectively. The capacities of the casting and finishing shops over the next month are 650 and 1,000 worker- hours, respectively. The owner of this firm wants to determine the quantities of each spare automobile part to be made next month so as to maximise profit. Formulate this optimisation problem as a linear programming problem in standard form, working through the following steps: (i) Define the decision variables. (ii) Write down the objective function. (iii) Formulate the LP problem in standard form (you do not need to solve it). (b) Given the following linear programming problem maximise subject to Z= -2x1 + x2 4x1 + 2x2 0 and x2 > 0 (i) Find an initial feasible corner point. (ii) Sketch the feasible region and plot the cost function Z=1 in the 21X2 -plane. Illustrate the optimal solution in your sketch, explaining your solution. (iii) Solve the LP problem using the Simplex Algorithm. Explain your steps; in particular, write down which variables enter/leave the basis at each step