Edexcel A Level Further Maths Paper 4D: Decision Mathematics 2 June 2019 question paper 9FM0/4D

Table 1 shows the cost, in pounds, of transporting one unit of stock from each of four supply points, A, B, C and D, to each of four demand points, P, Q, R and S. It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required.

Table 2 shows an initial solution given by the north-west corner method.

Taking DQ as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear.

Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by stating the

entering cell and exiting cell.

Determine whether the solution obtained from this second iteration is optimal, giving a reason for your answer.

State the cost of the solution found in (b).

Four workers, Ted (T), Harold (H), James (J) and Margaret (M), are to be assigned to four tasks, 1, 2, 3 and 4. Each worker must be assigned to just one task and each task must be done by just one worker.

The profit, in pounds, resulting from allocating each worker to each task, is shown in the table below. The profit is to be maximised.

Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total profit. You must make your method clear and show the table after each stage.

Determine the resulting total profit.

In Figure 1 the weight of arc SB is denoted by x where x 0

Explain why Dijkstra’s algorithm cannot be used on the directed network in Figure 1.

It is given that the minimum weight route from S to T passes through B.

Use dynamic programming to find

the range of possible values of x

the minimum weight route from S to T.

A two person zero-sum game is represented by the pay-off matrix for player A shown above.

Verify that there is no stable solution to this game.

Player A intends to make a random choice between options P, Q and R, choosing option P with probability p₁, option Q with probability p₂ and option R with probability p₃

Player A wants to find the optimal values of p₁, p₂ and p₃ using the Simplex algorithm. Player A formulates the following linear programming problem for the game, writing the constraints as inequalities.

Maximise P = V

Correct the errors made by player A in the linear programming formulation of the game, giving reasons for your answer.

Write down an initial Simplex tableau for the corrected linear programming problem.

The Simplex algorithm is used to solve the corrected linear programming problem.

The optimal values are p₁ = 0.6, p₂ = 0 and p₃= 0.4

Calculate the value of the game to player A.

Determine the optimal strategy for player B, making your reasoning clear.

An increasing sequence {u_n} for n is such that the difference between the nth term of {u_n} and the mean of the previous two terms of {u_n} is always 6

Show that, for n 3

Given that u₁ = 2 and u₂ = 8

find the solution of this second order recurrence relation to obtain an expression for u_n in terms of n.

Show that as n ’, u_n kn where k is a constant to be determined. You must give reasons for your answer.

Two cuts C₁ and C₂ are shown.

Find the capacity of

cut C₁

cut C₂

Explain why the arcs AE and CE cannot be at their upper capacities simultaneously.

Explain why a flow of 31 litres per second through the system is not possible.

Hence determine a minimum feasible flow and a maximum feasible flow through the system. You must draw these feasible flows on the diagrams in the answer book and give reasons to justify your answer. You should not apply the labelling procedure to find these flows.

Aisha is deciding whether or not to play a game.

The game involves rolling three fair six-sided dice, which have faces numbered from 1 to 6

If the total score on the three dice is 16 or more then she wins a prize. If the total score is 15 or less then she loses and will have to pay the person running the game £3

Given that the prize is £15

draw a decision tree to model Aisha’s possible decisions and the possible outcomes

determine Aisha’s optimal EMV and state the optimal strategy indicated by the decision tree.

The utility function of the game to Aisha is u(m) = 1 – e where m is the amount of money that Aisha has available. Given that Aisha has exactly £3 and that the prize is now £ x

find the expected utility to Aisha of playing the game in the form

Aisha decides to use the expected utilities to determine whether she should play the game or not.

Find the minimum prize for which Aisha would consider playing the game.