Questions — Edexcel D2 (237 questions)

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Edexcel D2 2006 June Q5
Moderate -0.5
Victor owns some kiosks selling ice cream, hot dogs and soft drinks. The network below shows the choices of action and the profits, in thousands of pounds, they generate over the next four years. The negative numbers indicate losses due to the purchases of new kiosks. \includegraphics{figure_5} Use a suitable algorithm to determine the sequence of actions so that the profit over the four years is maximised and state this maximum profit. (Total 12 marks)
Edexcel D2 2006 June Q6
14 marks Moderate -0.5
  1. Explain briefly the circumstances under which a degenerate feasible solution may occur to a transportation problem. [2]
  2. Explain why a dummy location may be needed when solving a transportation problem. [1]
The table below shows the cost of transporting one unit of stock from each of three supply points \(A\), \(B\) and \(C\) to each of two demand points 1 and 2. It also shows the stock held at each supply point and the stock required at each demand point.
12Supply
\(A\)624715
\(B\)614812
\(C\)685817
Demand1611
  1. Complete the table below to show a possible initial feasible solution generated by the north-west corner method.
    123
    \(A\)
    \(B\)0
    \(C\)
    [1]
  2. Use the stepping-stone method to obtain an optimal solution and state its cost. You should make your method clear by stating shadow costs, improvement indices, stepping-stone route, and the entering and exiting squares at each stage. [10]
(Total 14 marks)
Edexcel D2 2006 June Q7
16 marks Standard +0.8
A two person zero-sum game is represented by the following pay-off matrix for player \(A\).
\(B\) plays 1\(B\) plays 2\(B\) plays 3
\(A\) plays 1572
\(A\) plays 2384
\(A\) plays 3649
  1. Formulate the game as a linear programming problem for player \(A\), writing the constraints as equalities and clearly defining your variables. [5]
  2. Explain why it is necessary to use the simplex algorithm to solve this game theory problem. [1]
  3. Write down an initial simplex tableau making your variables clear. [2]
  4. Perform two complete iterations of the simplex algorithm, indicating your pivots and stating the row operations that you use. [8]
(Total 16 marks)
Edexcel D2 2006 June Q8
16 marks Standard +0.3
The tableau below is the initial tableau for a maximising linear programming problem.
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)710101003600
\(s\)69120103600
\(t\)2340012400
\(P\)-35-55-600000
  1. Write down the four equations represented in the initial tableau above. [4]
  2. Taking the most negative number in the profit row to indicate the pivot column at each stage, solve this linear programming problem. State the row operations that you use. [9]
  3. State the values of the objective function and each variable. [3]
(Total 16 marks)
Edexcel D2 2006 June Q9
14 marks Moderate -0.3
\includegraphics{figure_9} The figure above shows a capacitated, directed network. The capacity of each arc is shown on each arc. The numbers in circles represent an initial flow from \(S\) to \(T\). Two cuts \(C_1\) and \(C_2\) are shown on the figure.
  1. Write down the capacity of each of the two cuts and the value of the initial flow. [3]
  2. Complete the initialisation of the labelling procedure on the diagram below by entering values along arcs \(AC\), \(CD\), \(DE\) and \(DT\). \includegraphics{figure_9b} [2]
  3. Hence use the labelling procedure to find a maximal flow through the network. You must list each flow-augmenting path you use, together with its flow. [5]
  4. Show your maximal flow pattern on the diagram below. \includegraphics{figure_9d} [2]
  5. Prove that your flow is maximal. [2]
(Total 14 marks)
Edexcel D2 Q1
6 marks Moderate -0.8
This question should be answered on the sheet provided. The table below shows the distances in miles between five villages. Jane lives in village A and is about to take her daughter's friends home to villages B, C, D and E. She will begin and end her journey at A and wishes to travel the minimum distance possible.
ABCDE
A\(-\)4782
B4\(-\)156
C71\(-\)27
D852\(-\)3
E2673\(-\)
  1. Obtain a minimum spanning tree for the network and hence find an upper bound for the length of Jane's journey. [4 marks]
  2. Using a shortcut, improve this upper bound to find an upper bound of less than 15 miles. [2 marks]
Edexcel D2 Q2
8 marks Standard +0.3
The payoff matrix for player A in a two-person zero-sum game with value V is shown below.
B
IIIIII
\multirow{3}{*}{A}I6\(-4\)\(-1\)
II\(-2\)53
III51\(-3\)
Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player B.
  1. Rewrite the matrix as necessary and state the new value of the game, v, in terms of V. [2 marks]
  2. Define your decision variables. [2 marks]
  3. Write down the objective function in terms of your decision variables. [2 marks]
  4. Write down the constraints. [2 marks]
Edexcel D2 Q3
9 marks Moderate -0.3
This question should be answered on the sheet provided. The table below gives distances, in miles, for a network relating to a travelling salesman problem.
ABCDEFG
A\(-\)83576810391120
B83\(-\)7863418252
C5778\(-\)37596374
D686337\(-\)605262
E103415960\(-\)4851
F9182635248\(-\)77
G1205274625177\(-\)
  1. Use the nearest neighbour algorithm, starting at A, to find an upper bound for the length of a tour beginning and ending at A and state the tour. [4 marks]
  2. By deleting A, obtain a lower bound for the length of a tour. [4 marks]
  3. Hence, write down an inequality which must by satisfied by d, the minimum distance travelled in miles. [1 mark]
Edexcel D2 Q4
10 marks Challenging +1.8
This question should be answered on the sheet provided. A rally consisting of four stages is being planned. The first stage will begin at A and the last stage will end at L. Various routes are being considered, with the end of one stage being the start of the next. The organisers want the shortest stage to be as long as possible. The table below shows the length, in miles, of each of the possible stages.
Finishing point
CDEFGHI
\multirow{3}{*}{Starting point}A14.513108114
B510.5
C96
D12715
E
F5
G8
H10
I
J
K
Finishing point
JKL
2
923
29
5
6
10
Use dynamic programming to find the route which satisfies the wish of the organisers. State the length of the shortest stage on this route. [10 marks]
Edexcel D2 Q5
11 marks Standard +0.3
Four athletes are put forward for selection for a mixed stage relay race at a local competition. They may each be selected for a maximum of one stage and only one athlete can be entered for each stage. The average time, in seconds, for each athlete to complete each stage is given below, based on past performances.
Stage
123
Alex1969168
Darren2264157
Leroy2072166
Suraj2366171
Use the Hungarian algorithm to find an optimal allocation which will minimise the team's total time. Your answer should show clearly how you have applied the algorithm. [11 marks]
Edexcel D2 Q6
13 marks Moderate -0.3
The payoff matrix for player X in a two-person zero-sum game is shown below.
Y
\(Y_1\)\(Y_2\)
\multirow{2}{*}{X}\(X_1\)\(-2\)4
\(X_2\)6\(-1\)
  1. Explain why the game does not have a saddle point. [3 marks]
  2. Find the optimal strategy for
    1. player X, [8 marks]
    2. player Y.
  3. Find the value of the game. [2 marks]
Edexcel D2 Q7
18 marks Standard +0.3
A transportation problem has costs, in pounds, and supply and demand, in appropriate units, as given in the transportation tableau below.
DEFSupply
A13111420
B1091215
C156825
Demand30525
  1. Find the initial solution given by the north-west corner rule and state why it is degenerate. [3 marks]
  2. Use the stepping-stone method to obtain an optimal solution minimising total cost. State the resulting transportation pattern and its total cost. [15 marks]