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Examination style paper 1 Answer all questions 1. Time allowed 90 minutes 13 A 100 m 64 385 12 65 B 400 m 390 63 14 C 1500 m 375 Downhill School has been invited to enter three pupils for an interschool athletic competition. The competition consists of three track events: 100 m, 400 m and 1500 m. A pupil can only enter one event and the winning team is the one whose total time is a minimum. Downhill School has held trials to find the times, in seconds, taken by three pupils, A, B and C, in each of the events. These times are given in the diagram above. The teacher's problem is to decide who shall run which event to obtain the best result in the competition. Formulate the teacher's problem as a linear programming problem. (7 marks) 2. An electrical shop has just received three new repair jobs: a television (1), a microwave (2) and a vacuum cleaner (3). Four men are available to do the repairs. The manager estimates what it will cost, in wages, to assign each of the workers to each of the jobs. These estimates, in £s, are given in the table below. Jobs Workers 1 2 3 Alf 14 16 11 Ben 13 15 12 Cyril 12 12 11 Dennis 16 18 16 D2 Examination style paper 1 Use the Hungarian algorithm to obtain the assignment of workers to jobs that results in the minimum overall cost. (9 marks) 3. D 15 18 16 A 30 S H E 16 20 24 19 14 B T 15 25 20 10 12 26 F I 14 C 13 18 G Members of a scout troup assemble at S and are told to make their way to T by any of the routes shown in the above figure. The weights on the edges give the time, in minutes, required to complete that leg. At least one person will take each of the possible routes from S to T. Use dynamic programming to determine the maximum time that any scout could take to travel from S to T. State the route used. Show your calculations in tabular form. (1 mark) 4. Allan and Barbara play a zero-sum game and each has three possible strategies. The payoff matrix is shown below. B A I II III I 3 5 4 II 2 3 3 III 6 3 8 (a) Explain why Allan will never choose strategy II. (2 marks) (b) Explain why Barbara will never choose strategy III. (2 marks) (c) Obtain the optimal strategies for both players and the value of the game. (10 marks) 121 122 Examination style paper 1 5. Destination Source A B C Available 1 25 20 26 30 2 15 20 20 100 3 30 18 12 80 Required 100 50 60 The above table shows the unit transportation costs, the source availabilities and the destination requirements for a transportation problem. Use the north-west corner rule to obtain an initial basic feasible solution and then use the stepping-stone method to obtain the optimal solution. State the transportation pattern and give its cost. (16 marks)