6 P72118A 6. Bernie makes garden sheds. He can build up to four sheds each month. If he builds more than two sheds in any one month, he must hire an additional worker at a cost of £250 for that month. In any month in which sheds are made, the overhead costs are £35 for each shed made that month. A maximum of three sheds can be held in storage at the end of any one month, at a cost of £80 per shed per month. Sheds must be delivered at the end of the month. The order schedule for sheds is Month January February March April May Number ordered 1 3 3 5 2 There are no sheds in storage at the beginning of January and Bernie plans to have no sheds left in storage after the May delivery. Use dynamic programming to determine the production schedule that minimises the costs given above. Complete the working in the table provided in the answer book and state the minimum cost. (14) (Total for Question 6 is 14 marks) 7 Turn over P72118A 7. Player B Option W Option X Option Y Option Z Player A Option Q 4 3 –1 –2 Option R –3 5 –4 k Option S –1 6 3 –3 A two person zero‑sum game is represented by the pay‑off matrix for player A shown above. It is given that k is an integer. (a) Show that Q is the play‑safe option for player A regardless of the value of k. (2) Given that Z is the play‑safe option for player B, (b) determine the range of possible values of k. You must make your working clear. (2) (c) Explain why player B should never play option X. You must make your reasoning clear. (2) Player A intends to make a random choice between options Q, R and S, choosing option Q with probability p1, option R with probability p2 and option S with probability p3 Player A wants to find the optimal values of p1, p2 and p3 using the Simplex algorithm. Given that  k > –4, player A formulates the following objective function for the corresponding linear program. Maximise P = V, where V = the value of the original game + 4 (d) (i) Formulate the constraints of the linear programming problem for player A. You should write the constraints as equations. (ii) Write down an initial Simplex tableau, making your variables clear. (7) The Simplex algorithm is used to solve the linear programming problem. It is given that in the final Simplex tableau the optimal value of  p1 = 7 37 , the optimal value of  p2 = 17 37 and all the slack variables are zero. (e) Determine the value of k, making your method clear. (4) (Total for Question 7 is 17 marks)