6 P72094A 4. A linear programming problem in x, y and z is to be solved using the big‑M method. The initial tableau is shown below. b.v. x y z s1 s2 s3 a1 a2 Value s1 2 3 4 1 0 0 0 0 13 a1 1 –2 2 0 –1 0 1 0 8 a2 3 0 –4 0 0 –1 0 1 12 P 2 – 4M –3 + 2M –1 + 2M 0 M M 0 0 –20M (a) Using the information in the above tableau, formulate the linear programming problem. You should • list each of the constraints as an inequality • state the two possible objectives (4) (b) Obtain the most efficient pivot for a first iteration of the big‑M method. You must give reasons for your answer. (2) (Total for Question 4 is 6 marks) 7 Turn over P72094A Please turn over for Question 5 8 P72094A 5. 4 D(x) A(4) C(7) E(4) G(3) H(5) J(7) K(6) B(5) 14 22 11 0 7 16 F(x) Figure 2 The network in Figure 2 shows the activities that need to be completed for a project. Each activity is represented by an arc and the duration of the activity, in days, is shown in brackets. The early event times are shown in Figure 2. (a) Complete Table 1 in the answer book to show the immediately preceding activities for each activity. (2) It is given that  4 < x  m (b) State the largest possible integer value of m. (1) (c) (i) Complete Diagram 1 in the answer book to show the late event times. (ii) State the activities that must be critical. (3) (d) Calculate the total float for activity G. (1) 9 Turn over P72094A The resource histogram in Figure 3 shows the number of workers required when each activity starts at its earliest possible time. The histogram also shows which activities happen at each time. K 2 4 6 8 10 12 14 16 18 20 22 time number of workers 8 7 6 5 4 3 2 1 J A K J H H G B C C D D E D F F F B C Figure 3 (e) Complete Table 2 in the answer book to show the number of workers required for each activity of the project. (2) (f) Draw a Gantt chart on Grid 1 in the answer book to represent the activity network. (5) (Total for Question 5 is 14 marks)