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Edexcel FD1 Specimen Q2
7 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{37435cc9-1e38-4c55-bd72-e2a1ec415ba7-03_570_663_175_701} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Define what is meant by a planar graph.
  2. Starting at A, find a Hamiltonian cycle for the graph in Figure 1. Arc AG is added to Figure 1 to create the graph shown in Figure 2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{37435cc9-1e38-4c55-bd72-e2a1ec415ba7-03_568_666_1226_701} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Taking ABCDEFGA as the Hamiltonian cycle,
  3. use the planarity algorithm to determine whether the graph shown in Figure 2 is planar. You must make your working clear and justify your answer.
Edexcel FD1 Specimen Q3
11 marks Standard +0.3
3.
  1. Explain clearly the difference between the classical travelling salesperson problem and the practical travelling salesperson problem.
    ABCDEFG
    A-172416211841
    B17-35253031\(x\)
    C2435-28203532
    D162528-291945
    E21302029-2235
    F1831351922-37
    G41\(x\)32453537-
    The table shows the least distances, in km, by road between seven towns, A, B, C, D, E, F and G . The least distance between B and G is \(x \mathrm {~km}\), where \(x > 25\) Preety needs to visit each town at least once, starting and finishing at A. She wishes to minimise the total distance she travels.
  2. Starting by deleting B and all of its arcs, find a lower bound for Preety's route. Preety found the nearest neighbour routes from each of A and C . Given that the sum of the lengths of these routes is 331 km ,
  3. find \(x\), making your method clear.
  4. Write down the smallest interval that you can be confident contains the optimal length of Preety's route. Give your answer as an inequality.
Edexcel FD1 Specimen Q4
14 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{37435cc9-1e38-4c55-bd72-e2a1ec415ba7-05_572_799_228_632} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The network in Figure 3 shows the roads linking a depot, D, and three collection points \(\mathrm { A } , \mathrm { B }\) and C . The number on each arc represents the length, in miles, of the corresponding road. The road from B to D is a one-way road, as indicated by the arrow.
  1. Explain clearly if Dijkstra's algorithm can be used to find a route from D to A . The initial distance and route tables for the network are given in the answer book.
  2. Use Floyd's algorithm to find a table of least distances. You should show both the distance table and the route table after each iteration.
  3. Explain how the final route table can be used to find the shortest route from D to B . State this route. There are items to collect at \(\mathrm { A } , \mathrm { B }\) and C . A van will leave D to make these collections in any order and then return to D. A minimum route is required. Using the final distance table and the Nearest Neighbour algorithm starting at D,
  4. find a minimum route and state its length. Floyd's algorithm and Dijkstra's algorithm are applied to a network. Each will find the shortest distance between vertices of the network.
  5. Describe how the results of these algorithms differ.
Edexcel FD1 Specimen Q5
15 marks Standard +0.3
5. A garden centre makes hanging baskets to sell to its customers. Three types of hanging basket are made, Sunshine, Drama and Peaceful. The plants used are categorised as Impact, Flowering or Trailing. Each Sunshine basket contains 2 Impact plants, 4 Flowering plants and 3 Trailing plants. Each Drama basket contains 3 Impact plants, 2 Flowering plants and 4 Trailing plants. Each Peaceful basket contains 1 Impact plant, 3 Flowering plants and 2 Trailing plants. The garden centre can use at most 80 Impact plants, at most 140 Flowering plants and at most 96 Trailing plants each day. The profit on Sunshine, Drama and Peaceful baskets are \(\pounds 12 , \pounds 20\) and \(\pounds 16\) respectively. The garden centre wishes to maximise its profit. Let \(x , y\) and \(z\) be the number of Sunshine, Drama and Peaceful baskets respectively, produced each day.
  1. Formulate this situation as a linear programming problem, giving your constraints as inequalities.
  2. State the further restriction that applies to the values of \(x , y\) and \(z\) in this context. The Simplex algorithm is used to solve this problem. After one iteration, the tableau is
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)\(- \frac { 1 } { 4 }\)0\(- \frac { 1 } { 2 }\)10\(- \frac { 3 } { 4 }\)8
    \(s\)\(\frac { 5 } { 2 }\)0201\(- \frac { 1 } { 2 }\)92
    \(y\)\(\frac { 3 } { 4 }\)1\(\frac { 1 } { 2 }\)00\(\frac { 1 } { 4 }\)24
    \(P\)30-6005480
  3. State the variable that was increased in the first iteration. Justify your answer.
  4. Determine how many plants in total are being used after only one iteration of the Simplex algorithm.
  5. Explain why for a second iteration of the Simplex algorithm the 2 in the \(z\) column is the pivot value. After a second iteration, the tableau is
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)\(\frac { 3 } { 8 }\)001\(\frac { 1 } { 4 }\)\(- \frac { 7 } { 8 }\)31
    \(s\)\(\frac { 5 } { 4 }\)010\(\frac { 1 } { 2 }\)\(- \frac { 1 } { 4 }\)46
    \(y\)\(\frac { 1 } { 8 }\)100\(- \frac { 1 } { 4 }\)\(\frac { 3 } { 8 }\)1
    \(P\)\(\frac { 21 } { 2 }\)0003\(\frac { 7 } { 2 }\)756
  6. Use algebra to explain why this tableau is optimal.
  7. State the optimal number of each type of basket that should be made. The manager of the garden centre is able to increase the number of Impact plants available each day from 80 to 100 . She wants to know if this would increase her profit.
  8. Use your final tableau to determine the effect of this increase. (You should not carry out any further calculations.)
Edexcel FD1 Specimen Q6
12 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{37435cc9-1e38-4c55-bd72-e2a1ec415ba7-08_1113_1319_169_374} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A project is modelled by the activity network shown in Figure 4. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete that activity. Each activity requires one worker. The project is to be completed in the shortest possible time.
  1. Calculate the early time and the late time for each event, using Diagram 1 in the answer book.
  2. On Grid 1 in the answer book, complete the cascade (Gantt) chart for this project.
  3. On Grid 2 in the answer book, draw a resource histogram to show the number of workers required each day when each activity begins at its earliest time. The supervisor of the project states that only three workers are required to complete the project in the minimum time.
  4. Use Grid 2 to determine if the project can be completed in the minimum time by only three workers. Give reasons for your answer.
Edexcel FD1 Specimen Q7
12 marks Standard +0.8
7. A linear programming problem in \(x , y\) and \(z\) is described as follows. Maximise \(P = 3 x + 2 y + 2 z\) subject to $$\begin{aligned} & 2 x + 2 y + z \leqslant 25 \\ & x + 4 y \leqslant 15 \\ & x \geqslant 3 \end{aligned}$$
  1. Explain why the Simplex algorithm cannot be used to solve this linear programming problem. The big-M method is to be used to solve this linear programming problem.
  2. Define, in this context, what M represents. You must use correct mathematical language in your answer. The initial tableau for a big-M solution to the problem is shown below.
    b.v.\(x\)\(y\)\(z\)\(s _ { 1 }\)\(s _ { 2 }\)\(s _ { 3 }\)\(t _ { 1 }\)Value
    \(s _ { 1 }\)221100025
    \(s _ { 2 }\)140010015
    \(t _ { 1 }\)10000-113
    \(P\)\(- ( 3 + M )\)-2-200M0\(- 3 M\)
  3. Explain clearly how the equation represented in the b.v. \(t _ { 1 }\) row was derived.
  4. Show how the equation represented in the b.v. \(P\) row was derived. The tableau obtained from the first iteration of the big-M method is shown below.
    b.v.\(x\)\(y\)\(z\)\(s _ { 1 }\)\(s _ { 2 }\)\(s _ { 3 }\)\(t _ { 1 }\)Value
    \(s _ { 1 }\)021102-219
    \(s _ { 2 }\)040011-112
    \(x\)10000-113
    P0-2-200-3\(3 +\) M9
  5. Solve the linear programming problem, starting from this second tableau. You must
Edexcel FD2 2019 June Q1
10 marks Challenging +1.2
  1. Table 1 shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to each of four demand points, \(\mathrm { P } , \mathrm { Q } , \mathrm { R }\) 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.
\begin{table}[h]
PQRSSupply
A1514171123
B109161242
C111381018
D1513161719
Demand25451220
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Table 2 shows an initial solution given by the north-west corner method. \begin{table}[h]
PQRS
A23
B240
C5121
D19
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
  1. Taking DQ as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear.
  2. Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by stating the
    • shadow costs
    • improvement indices
    • route
    • 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).
Edexcel FD2 2019 June Q2
7 marks Standard +0.8
  1. 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.
1234
T103977480
H201155145155
J111807792
M203188137184
  1. 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.
  2. Determine the resulting total profit.
Edexcel FD2 2019 June Q3
13 marks Challenging +1.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5274a614-7862-49f0-ad1d-b59b73aa51ad-04_1047_1691_210_187} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} In Figure 1 the weight of \(\operatorname { arc } \mathrm { SB }\) is denoted by \(x\) where \(x \geqslant 0\)
  1. Explain why Dijkstra's algorithm cannot be used on the directed network in Figure 1.
    (1) It is given that the minimum weight route from S to T passes through B .
  2. Use dynamic programming to find
    1. the range of possible values of \(x\)
    2. the minimum weight route from S to T .
      (12)
Edexcel FD2 2019 June Q4
14 marks Challenging +1.2
4.
\multirow{2}{*}{}Player B
Option XOption YOption Z
\multirow{3}{*}{Player A}Option P3-20
Option Q-44-2
Option R12-1
A two person zero-sum game is represented by the pay-off matrix for player A shown above.
  1. Verify that there is no stable solution to this game. Player A intends to make a random choice between options \(\mathrm { P } , \mathrm { Q }\) and R , choosing option P with probability \(p _ { 1 }\), option Q with probability \(p _ { 2 }\) and option R with probability \(p _ { 3 }\) Player A wants to find the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) using the Simplex algorithm. Player A formulates the following linear programming problem for the game, writing the constraints as inequalities. $$\begin{aligned} & \text { Maximise } P = V \\ & \text { subject to } V \geqslant 3 p _ { 1 } - 4 p _ { 2 } + p _ { 3 } \\ & \\ & V \geqslant - 2 p _ { 1 } + 4 p _ { 2 } + 2 p _ { 3 } \\ & V \geqslant - 2 p _ { 2 } - p _ { 3 } \\ & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 \\ & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , V \geqslant 0 \end{aligned}$$
  2. Correct the errors made by player A in the linear programming formulation of the game, giving reasons for your answer.
  3. 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 \(p _ { 1 } = 0.6 , p _ { 2 } = 0\) and \(p _ { 3 } = 0.4\)
  4. Calculate the value of the game to player A.
  5. Determine the optimal strategy for player B, making your reasoning clear.
Edexcel FD2 2019 June Q5
11 marks Challenging +1.2
5. An increasing sequence \(\left\{ u _ { n } \right\}\) for \(n \in \mathbb { N }\) is such that the difference between the \(n\)th term of \(\left\{ u _ { n } \right\}\) and the mean of the previous two terms of \(\left\{ u _ { n } \right\}\) is always 6
  1. Show that, for \(n \geqslant 3\) $$2 u _ { n } - u _ { n - 1 } - u _ { n - 2 } = 12$$ Given that \(u _ { 1 } = 2\) and \(u _ { 2 } = 8\)
  2. find the solution of this second order recurrence relation to obtain an expression for \(u _ { n }\) in terms of \(n\).
  3. Show that as \(n \rightarrow \infty , u _ { n } \rightarrow k n\) where \(k\) is a constant to be determined. You must give reasons for your answer.
Edexcel FD2 2019 June Q6
8 marks Challenging +1.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5274a614-7862-49f0-ad1d-b59b73aa51ad-07_983_1513_210_283} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a capacitated, directed network. The network represents a system of pipes through which fluid flows from a source, S , to a sink, T . The numbers ( \(l , u\) ) on each arc represent, in litres per second, the lower capacity, \(l\), and the upper capacity, \(u\), of the corresponding pipe. Two cuts \(C _ { 1 }\) and \(C _ { 2 }\) are shown.
  1. Find the capacity of
    1. \(\operatorname { cut } C _ { 1 }\)
    2. \(\operatorname { cut } C _ { 2 }\)
  2. Explain why the arcs AE and CE cannot be at their upper capacities simultaneously.
  3. Explain why a flow of 31 litres per second through the system is not possible.
  4. 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.
Edexcel FD2 2021 June Q1
6 marks Moderate -0.3
  1. Four workers, A, B, C and D, are to be assigned to three tasks, 1, 2 and 3 . Each task must be assigned to just one worker and each worker can do one task only.
Worker A cannot do task 2 and worker D cannot do task 3
The cost of assigning each worker to each task is shown in the table below.
The total cost is to be minimised.
123
A53-62
B485759
C556358
D6949-
Formulate the above situation as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.
(6) \section*{(Total for Question 1 is 6 marks)}
Edexcel FD2 2021 June Q2
7 marks Moderate -0.3
  1. Alka is considering paying \(\pounds 5\) to play a game. The game involves rolling two fair six-sided dice. If the sum of the numbers on the two dice is at least 8 , she receives \(\pounds 10\), otherwise she loses and receives nothing.
If Alka loses, she can pay a further \(\pounds 5\) to roll the dice again. If both dice show the same number then she receives \(\pounds 35\), otherwise she loses and receives nothing.
  1. Draw a decision tree to model Alka's possible decisions and the possible outcomes.
  2. Determine Alka's optimal EMV and state the optimal strategy indicated by the decision tree.
Edexcel FD2 2021 June Q3
11 marks Standard +0.8
3. The table below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to four sales points, \(\mathrm { P } , \mathrm { Q } , \mathrm { R }\) and S . It also shows the number of units held at each supply point and the number of units required at each sales point. A minimum cost solution is required.
PQRSSupply
A1819171328
B1615141943
C2117222329
D1620192136
Demand25414030
  1. Use the north-west corner method to obtain an initial solution.
  2. Taking AS as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear.
  3. Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by showing the route and stating the
Edexcel FD2 2021 June Q4
11 marks Challenging +1.2
  1. Sequences \(\left\{ x _ { n } \right\}\) and \(\left\{ y _ { n } \right\}\) for \(n \in \mathbb { N }\), are defined by
$$\begin{gathered} x _ { n + 1 } = 2 y _ { n } + 3 \quad \text { and } \quad y _ { n + 1 } = 3 x _ { n + 1 } - 4 x _ { n } \\ x _ { 1 } = 1 \quad \text { and } \quad y _ { 1 } = a \end{gathered}$$ where \(a\) is a constant.
  1. Show that \(x _ { n + 2 } - 6 x _ { n + 1 } + 8 x _ { n } = 3\)
  2. Solve the second-order recurrence relation given in (a) to obtain an expression for \(x _ { n }\) in terms of \(a\) and \(n\). Given that \(x _ { 7 } = 28225\)
  3. find the value of \(a\).
Edexcel FD2 2021 June Q5
16 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{262aa0e6-479f-447a-94db-aeb901b3c6fe-5_1095_1666_212_203} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a capacitated, directed network. The network represents a system of pipes through which fluid can flow. The weights on the arcs show the lower and upper capacities for the corresponding pipes, in litres per second.
  1. Calculate the capacity of
    1. cut \(C _ { 1 }\)
    2. cut \(C _ { 2 }\)
  2. Using only the capacities of cuts \(C _ { 1 }\) and \(C _ { 2 }\), state what can be deduced about the maximum flow through the system. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{262aa0e6-479f-447a-94db-aeb901b3c6fe-6_775_1516_169_278} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows an initial flow through the same network.
  3. State the value of the initial flow.
  4. By entering values along \(B C , C F\) and \(D T\), complete the labelling procedure on Diagram 1 in the answer book.
  5. Use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
  6. Use your answer to (e) to find a maximum flow pattern for this system of pipes and draw it on Diagram 2 in the answer book.
  7. Prove that the answer to (f) is optimal. A vertex restriction is now applied to \(B\) so that no more than 16 litres per second can flow through it.
    1. Complete Diagram 3 in the answer book to show this restriction.
    2. State the value of the maximum flow with this restriction.
Edexcel FD2 2021 June Q6
12 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{262aa0e6-479f-447a-94db-aeb901b3c6fe-7_782_1426_219_322} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The staged, directed network in Figure 3 represents a series of roads connecting 12 towns, \(S , A , B , C , D , E , F , G , H , I , J\) and \(T\). The number on each arc shows the distance between these towns, in miles. Bradley is planning a four-day cycle ride from \(S\) to \(T\).
He plans to leave his home at \(S\). On the first night he will stay at \(A , B\) or \(C\), on the second night he will stay at \(D , E , F\) or \(G\), on the third night he will stay at \(H , I\) or \(J\), and he will arrive at his friend's house at \(T\) on the fourth day. Bradley decides that the maximum distance he will cycle on any one day should be as small as possible.
  1. Write down the type of dynamic programming problem that Bradley needs to solve.
  2. Use dynamic programming to complete the table in the answer book.
  3. Hence write down the possible routes that Bradley could take.
Edexcel FD2 2021 June Q7
12 marks Challenging +1.2
7. Alexis and Becky are playing a zero-sum game. Alexis has two options, Q and R . Becky has three options, \(\mathrm { X } , \mathrm { Y }\) and Z .
Alexis intends to make a random choice between options Q and R , choosing option Q with probability \(p _ { 1 }\) and option R with probability \(p _ { 2 }\) Alexis wants to find the optimal values of \(p _ { 1 }\) and \(p _ { 2 }\) and formulates the following linear programme, writing the constraints as inequalities. $$\begin{aligned} & \text { Maximise } P = V \\ & \text { where } V = 3 + \text { the value of the gan } \\ & \text { subject to } V \leqslant 6 p _ { 1 } + p _ { 2 } \\ & \qquad \begin{aligned} & V \leqslant 8 p _ { 2 } \\ & V \leqslant 4 p _ { 1 } + 2 p _ { 2 } \\ & p _ { 1 } + p _ { 2 } \leqslant 1 \\ & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , V \geqslant 0 \end{aligned} \end{aligned}$$
  1. Complete the pay-off matrix for Alexis in the answer book.
  2. Use a graphical method to find the best strategy for Alexis.
  3. Calculate the value of the game to Alexis. Becky intends to make a random choice between options \(\mathrm { X } , \mathrm { Y }\) and Z , choosing option X with probability \(q _ { 1 }\), option Y with probability \(q _ { 2 }\) and option Z with probability \(q _ { 3 }\)
  4. Determine the best strategy for Becky, making your method and working clear.
Edexcel FD2 2023 June Q1
7 marks Moderate -0.5
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae354c58-6de8-4f94-b404-2f0feecb5bf3-02_953_1687_251_191} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a capacitated, directed network of pipes. The number on each arc represents the capacity of that pipe. The numbers in circles represent a feasible flow from S to T .
  1. State the value of the feasible flow.
    (1)
  2. Find the capacity of cut \(\mathrm { C } _ { 1 }\) and the capacity of cut \(\mathrm { C } _ { 2 }\) (2)
  3. By inspection, find a flow-augmenting route to increase the flow by two units. You must state your route.
    (1)
  4. Prove that, once the flow-augmenting route found in (c) has been applied, the flow is now maximal.
    (3)
Edexcel FD2 2023 June Q2
5 marks Moderate -0.8
2. An outdoor theatre is holding a summer gala performance. The theatre owner must decide whether to take out insurance against rain for this performance. The theatre owner estimates that
  • on a fine day, the total profit will be \(\pounds 15000\)
  • on a wet day, the total loss will be \(\pounds 20000\)
Insurance against rain costs \(\pounds 2000\). If the performance must be cancelled due to rain, then the theatre owner will receive \(\pounds 16000\) from the insurer. If the performance is not cancelled due to rain, then the theatre owner will receive nothing from the insurer. The probability of rain on the day of the gala performance is 0.2
Draw a decision tree and hence determine whether the theatre owner should take out the insurance against rain for this performance.
Edexcel FD2 2023 June Q3
9 marks Standard +0.3
3. The table below shows the stock held at each supply point and the stock required at each demand point in a standard transportation problem. The table also shows the cost, in pounds, of transporting the stock from each supply point to each demand point.
\cline { 2 - 5 } \multicolumn{1}{c|}{}QRSSupply
A23181245
B8101427
C11142134
D19151150
Demand753744
The problem is partially described by the linear programming formulation below.
Let \(x _ { i j }\) be the number of units transported from i to j $$\begin{aligned} & \text { where } \quad i \in \{ A , B , C , D \} \\ & \quad j \in \{ Q , R , S \} \text { and } x _ { i j } \geqslant 0 \\ & \text { Minimise } P = 23 x _ { A Q } + 18 x _ { A R } + 12 x _ { A S } + 8 x _ { B Q } + 10 x _ { B R } + 14 x _ { B S } \\ & \quad + 11 x _ { C Q } + 14 x _ { C R } + 21 x _ { C S } + 19 x _ { D Q } + 15 x _ { D R } + 11 x _ { D S } \end{aligned}$$
  1. Write down, as inequalities, the constraints of the linear program.
  2. Use the north-west corner method to obtain an initial solution to this transportation problem.
  3. Taking AS as the entering cell, use the stepping-stone method to find an improved solution. Make your route clear.
  4. Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by showing the route and the
Edexcel FD2 2023 June Q4
8 marks Standard +0.8
  1. Four students, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , are to be allocated to four rounds, \(1,2,3\) and 4 , in a competition. Each student is to take part in exactly one round and no two students may play in the same round.
Each student has been given an estimated score for each round. The estimated scores for each student are shown in the table below.
\cline { 2 - 5 } \multicolumn{1}{c|}{}1234
A34201815
B49311234
C48272326
D52454242
  1. Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total estimated score. You must make your method clear and show the table after each stage.
  2. Find this total estimated score.
Edexcel FD2 2023 June Q5
8 marks Challenging +1.2
5. A sequence \(\left\{ u _ { n } \right\}\), where \(\mathrm { n } \geqslant 0\), satisfies the second order recurrence relation $$u _ { n + 2 } = \frac { 1 } { 2 } \left( u _ { n + 1 } + u _ { n } \right) + 3 \text { where } u _ { 0 } = 15 \quad u _ { 1 } = 20$$
  1. By considering the sequence \(\left\{ v _ { n } \right\}\), where \(u _ { n } = v _ { n } + 2 n\) for \(\mathrm { n } \geqslant 0\), determine an expression for \(u _ { n }\) as a function of n .
  2. Describe the long-term behaviour of \(u _ { n }\)
Edexcel FD2 2023 June Q6
13 marks Standard +0.3
6. Polly is a motivational speaker who is planning her engagements for the next four weeks. Polly will
  • visit four different countries in these four weeks
  • visit just one country each week
  • leave from her home, S , and return there only after visiting the four countries
  • travel directly from one country to the next
Polly wishes to determine a schedule of four countries to visit.
Table 1 shows the countries Polly could visit each week. \begin{table}[h]
Week1234
Possible countries to visitA or BC, D or EF or GH, I or J
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Table 2 shows the speaker fee, in \(\pounds 100\) s, Polly would expect to earn in each country. \begin{table}[h]
CountryABCDEFGHIJ
Earnings in \(\boldsymbol { \pounds } \mathbf { 1 0 0 s }\)47454847494445474948
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table} Table 3 shows the cost, in \(\pounds 100\) s, of travelling between the countries. \begin{table}[h]
ABCDEFGHIJ
S52788
A345
B546
C75
D67
E76
F678
G786
\captionsetup{labelformat=empty} \caption{Table 3}
\end{table} Polly's expected income is the value of the speaker fee minus the cost of travel.
She wants to find a schedule that maximises her total expected income for the four weeks. Use dynamic programming to determine the optimal schedule. Complete the table provided in the answer book and state the maximum expected income.
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