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Edexcel D1 2008 January Q5
7 marks Moderate -0.3
5. (a) Draw the activity network described in this precedence table, using activity on arc and exactly two dummies.
(4)
ActivityImmediately preceding activities
A-
B-
CA
DB
EB, C
FB, C
(b) Explain why each of the two dummies is necessary.
(3)
Edexcel D1 2008 January Q6
13 marks Moderate -0.8
6. (a) Define the term 'cut' as it applies to a directed network.
(2) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7396d930-0143-4876-b019-a4d73e09b172-7_659_1367_376_349} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} Figure 6 shows a capacitated, directed network. The number on each arc represents the capacity of that arc. The numbers in circles represent an initial flow.
(b) Complete the labelling procedure on Diagram 2 in your answer book by entering values along CE, EG, HT and GT.
(c) Find the maximum flow through the network. You must list each flow-augmenting route you use together with its flow.
(d) Show a maximal flow pattern on Diagram 3 in your answer book.
(e) State the value of the maximum flow through the network.
(f) Prove that your flow is maximal.
Edexcel D1 2008 January Q7
16 marks Moderate -0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7396d930-0143-4876-b019-a4d73e09b172-8_2158_1803_239_137} \captionsetup{labelformat=empty} \caption{Figure 7}
\end{figure}
  1. Phil sells boxed lunches to travellers at railway stations. Customers can select either the vegetarian box or the non-vegetarian box.
Phil decides to use graphical linear programming to help him optimise the numbers of each type of box he should produce each day. Each day Phil produces \(x\) vegetarian boxes and \(y\) non-vegetarian boxes.
One of the constraints limiting the number of boxes is $$x + y \geqslant 70$$ This, together with \(x \geqslant 0 , y \geqslant 0\) and a fourth constraint, has been represented in Figure 7. The rejected region has been shaded.
  1. Write down the inequality represented by the fourth constraint. Two further constraints are: $$\begin{aligned} & x + 2 y \leqslant 160 \\ & \text { and } y > 60 \end{aligned}$$
  2. Add two lines and shading to Diagram 4 in your answer book to represent these inequalities.
  3. Hence determine and label the feasible region, R .
  4. Use your graph to determine the minimum total number of boxes he needs to prepare each day. Make your method clear. Phil makes a profit of \(\pounds 1.20\) on each vegetarian box and \(\pounds 1.40\) on each non-vegetarian box. He wishes to maximise his profit.
  5. Write down the objective function.
  6. Use your graph to obtain the optimal number of vegetarian and non-vegetarian boxes he should produce each day. You must make your method clear.
  7. Find Phil's maximum daily profit.
Edexcel D1 2009 January Q1
9 marks Easy -1.8
1. Max Lauren John Hannah Kieran Tara Richard Imogen
  1. Use a quick sort to produce a list of these names in ascending alphabetical order. You must make your pivots clear.
  2. Use the binary search algorithm on your list from part (a) to try to locate the name 'Hugo'.
Edexcel D1 2009 January Q2
8 marks Easy -1.3
2.
\(\mathbf { A }\)\(\mathbf { B }\)\(\mathbf { C }\)\(\mathbf { D }\)\(\mathbf { E }\)\(\mathbf { F }\)
\(\mathbf { A }\)-24--2322
\(\mathbf { B }\)24-18191720
\(\mathbf { C }\)-18-1114-
\(\mathbf { D }\)-1911-13-
\(\mathbf { E }\)23171413-21
\(\mathbf { F }\)2220--21-
The table shows the distances, in metres, between six vertices, \(\mathbf { A } , \mathbf { B } , \mathbf { C } , \mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\), in a network.
  1. Draw the weighted network using the vertices given in Diagram 1 in the answer booklet.
  2. Use Kruskal's algorithm to find a minimum spanning tree. You should list the edges in the order that you consider them and state whether you are adding them to your minimum spanning tree.
  3. Draw your tree on Diagram 2 in the answer booklet and find its total weight.
Edexcel D1 2009 January Q3
7 marks Moderate -0.8
3. (a) Draw the activity network described in this precedence table, using activity on arc and exactly two dummies.
(5)
ActivityImmediately preceding activities
A-
B-
C-
DB
EB, C
FB, C
GF
HF
IG, H
JI
(b) Explain why each of the two dummies is necessary.
Edexcel D1 2009 January Q4
8 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef029462-ffed-4cdf-87bc-56c8a13d671f-4_540_626_244_316} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef029462-ffed-4cdf-87bc-56c8a13d671f-4_531_613_246_1128} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the possible allocations of six people, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F , to six tasks, 1, 2, 3, 4, 5 and 6.
Figure 2 shows an initial matching.
  1. Starting from this initial matching, use the maximum matching algorithm to find an improved matching. You must list the alternating path used, and your improved matching.
  2. Explain why it is not possible to find a complete matching. D now has task 2 added to their possible allocation.
  3. Using the improved matching found in part (a) as the new initial matching, use the maximum matching algorithm to find a complete matching. You must list the alternating path used and your complete matching.
    (3)
Edexcel D1 2009 January Q5
8 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef029462-ffed-4cdf-87bc-56c8a13d671f-5_806_1211_264_427} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} (The total weight of the network in Figure 3 is 543 km .)
Figure 3 models a network of railway tracks that have to be inspected. The number on each arc is the length, in km , of that section of railway track.
Each track must be traversed at least once and the length of the inspection route must be minimised.
The inspection route must start and finish at the same vertex.
  1. Use an appropriate algorithm to find the length of the shortest inspection route. You should make your method and working clear. It is now permitted to start and finish the inspection at two distinct vertices.
  2. State which two vertices should be chosen to minimise the length of the new route. Give a reason for your answer.
Edexcel D1 2009 January Q6
7 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef029462-ffed-4cdf-87bc-56c8a13d671f-6_609_1283_260_392} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a network of roads through eight villages, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }\) and H . The number on each arc is the length of that road in km .
  1. Use Dijkstra's algorithm to find the shortest route from A to H. State your shortest route and its length.
    (5) There is a fair in village C and you cannot drive through the village. A shortest route from A to H which avoids C needs to be found.
  2. State this new minimal route and its length.
    (2)
Edexcel D1 2009 January Q7
12 marks Moderate -0.3
7. A linear programming problem is modelled by the following constraints $$\begin{aligned} 8 x + 3 y & \leqslant 480 \\ 8 x + 7 y & \geqslant 560 \\ y & \geqslant 4 x \\ x , y & \geqslant 0 \end{aligned}$$
  1. Use the grid provided in your answer book to represent these inequalities graphically. Hence determine the feasible region and label it R . The objective function, \(F\), is given by $$F = 3 x + y$$
  2. Making your method clear, determine
    1. the minimum value of the function \(F\) and the coordinates of the optimal point,
    2. the maximum value of the function \(F\) and the coordinates of the optimal point.
Edexcel D1 2009 January Q8
16 marks Moderate -0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef029462-ffed-4cdf-87bc-56c8a13d671f-8_574_1362_242_349} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} The network in Figure 5 shows the activities involved in a process. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, taken to complete the activity.
  1. Calculate the early time and the late time for each event, showing them on the diagram in the answer book.
  2. Determine the critical activities and the length of the critical path.
  3. Calculate the total float on activities F and G . You must make the numbers you used in your calculation clear.
  4. On the grid in the answer book, draw a cascade (Gantt) chart for the process. Given that each task requires just one worker,
  5. use your cascade chart to determine the minimum number of workers required to complete the process in the minimum time. Explain your reasoning clearly.
Edexcel D1 2010 January Q1
6 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{17bc9fb2-13bf-4ffa-93ac-bef170467570-2_611_629_360_717} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the possible allocation of six people, Alice (A), Brian (B), Christine (C), David (D), Elizabeth (E) and Freddy (F), to six tasks, 1, 2, 3, 4, 5 and 6. An initial matching is Alice to task 1, Christine to task 3, David to task 4 and Elizabeth to task 5.
  1. Show this initial matching on Diagram 1 in the answer book.
    (1)
  2. Starting from this initial matching, use the maximum matching algorithm to find a complete matching. List clearly the alternating paths that you use, and give your final matching.
    (5)
Edexcel D1 2010 January Q2
9 marks Easy -1.8
2. Prim's algorithm finds a minimum spanning tree for a connected graph.
  1. Explain the terms
    1. connected graph,
    2. tree,
    3. spanning tree.
  2. Name an alternative algorithm for finding a minimum spanning tree. \begin{table}[h]
    CambridgeLondonNorwichOxfordPortsmouthSalisburyYork
    Cambridge (C)-606281132139156
    London (L)60-116567488211
    Norwich (N)62116-144204201181
    Oxford (O)8156144-8463184
    Portsmouth (P)1327420484-43269
    Salisbury (S)139882016343-248
    York (Y)156211181184269248-
    \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{table} Figure 2 shows the distances by road, in miles, between seven cities.
    1. Use Prim's algorithm, starting at London, to find the minimum spanning tree for these cities. You must clearly state the order in which you selected the edges of your tree, and the weight of the final tree.
    2. Draw your tree using the vertices given in Diagram 2 in the answer book.
      (5)
Edexcel D1 2010 January Q3
10 marks Moderate -0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{17bc9fb2-13bf-4ffa-93ac-bef170467570-4_579_1273_239_397} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} [The total weight of the network is 167]
Figure 3 represents a network of paths. The number on each arc gives the time, in minutes, to travel along that path.
  1. Use Dijkstra's algorithm to find the quickest route from A to H. State your quickest route and the time taken.
    (5) Kevin must walk along each path at least once and return to his starting point.
  2. Use an appropriate algorithm to find the time of Kevin's quickest possible route, starting and finishing at A. You should make your method and working clear.
Edexcel D1 2010 January Q4
11 marks Easy -1.2
4. A builder is asked to replace the guttering on a house. The lengths needed, in metres, are $$0.6,4.0,2.5,3.2,0.5,2.6,0.4,0.3,4.0 \text { and } 1.0$$ Guttering is sold in 4 m lengths.
  1. Carry out a quick sort to produce a list of the lengths needed in descending order. You should show the result of each pass and identify your pivots clearly.
  2. Apply the first-fit decreasing bin-packing algorithm to your ordered list to determine the total number of 4 m lengths needed.
  3. Does the answer to part (b) use the minimum number of 4 m lengths? You must justify your answer.
Edexcel D1 2010 January Q5
7 marks Easy -1.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{17bc9fb2-13bf-4ffa-93ac-bef170467570-6_2228_613_269_861} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} An algorithm is described by the flowchart shown in Figure 4.
  1. Given that \(\mathrm { S } = 25000\), complete the table in the answer book to show the results obtained at each step when the algorithm is applied. This algorithm is designed to model a possible system of income tax, T, on an annual salary, £S.
  2. Write down the amount of income tax paid by a person with an annual salary of \(\pounds 25000\).
  3. Find the maximum annual salary of a person who pays no tax.
Edexcel D1 2010 January Q6
15 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{17bc9fb2-13bf-4ffa-93ac-bef170467570-7_614_1315_1027_374} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 is the activity network relating to a building project. The number in brackets on each arc gives the time taken, in days, to complete the activity.
  1. Explain the significance of the dotted line from event (2) to event (3).
  2. Complete the precedence table in the answer booklet.
  3. Calculate the early time and the late time for each event, showing them on the diagram in the answer booklet.
  4. Determine the critical activities and the length of the critical path.
  5. On the grid in the answer booklet, draw a cascade (Gantt) chart for the project.
Edexcel D1 2010 January Q7
17 marks Easy -1.2
7. You are in charge of buying new cupboards for a school laboratory. The cupboards are available in two different sizes, standard and large.
The maximum budget available is \(\pounds 1800\). Standard cupboards cost \(\pounds 150\) and large cupboards cost \(\pounds 300\).
Let \(x\) be the number of standard cupboards and \(y\) be the number of large cupboards.
  1. Write down an inequality, in terms of \(x\) and \(y\), to model this constraint.
    (2) The cupboards will be fitted along a wall 9 m long. Standard cupboards are 90 cm long and large cupboards are 120 cm long.
  2. Show that this constraint can be modelled by $$3 x + 4 y \leqslant 30$$ You must make your reasoning clear. Given also that \(y \geqslant 2\),
  3. explain what this constraint means in the context of the question. The capacity of a large cupboard is \(40 \%\) greater than the capacity of a standard cupboard. You wish to maximise the total capacity.
  4. Show that your objective can be expressed as $$\text { maximise } 5 x + 7 y$$
  5. Represent your inequalities graphically, on the axes in your answer booklet, indicating clearly the feasible region, R.
  6. Find the number of standard cupboards and large cupboards that need to be purchased. Make your method clear.
Edexcel D1 2011 January Q1
8 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-2_750_1285_388_390} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a network of roads between eight villages, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }\) and H . The number on each arc gives the length, in miles, of the corresponding road.
  1. Use Dijkstra's algorithm to find the shortest distance from A to H .
    (5)
  2. State your shortest route.
    (1)
  3. Write down the shortest route from H to C and state its length.
    (2)
Edexcel D1 2011 January Q2
12 marks Moderate -0.8
2. $$\begin{array} { l l l l l l l l } 23 & 29 & 11 & 34 & 10 & 14 & 35 & 17 \end{array}$$ The numbers represent the sizes, in megabytes (MB), of eight files.
The files are to be stored on 50 MB discs.
  1. Calculate a lower bound for the number of discs needed to store all eight files.
  2. Use the first-fit bin packing algorithm to fit the files onto the discs.
  3. Perform a bubble sort on the numbers in the list to sort them into descending order. You need only write down the final result of each pass.
  4. Use the first-fit decreasing bin packing algorithm to fit the files onto the discs.
Edexcel D1 2011 January Q3
10 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-4_986_1255_269_402} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure}
  1. Use Kruskal's algorithm to find a minimum spanning tree for the network shown in Figure 2. You should list the arcs in the order in which you consider them. In each case, state whether you are adding the arc to your minimum spanning tree.
    (3)
  2. Starting at A, use Prim's algorithm to find a minimum spanning tree for the network in Figure 2. You must clearly state the order in which you include the arcs in your tree.
    (3)
  3. Draw a minimum spanning tree for the network in Figure 2 using the vertices given in Diagram 1 of the answer book. State the weight of the minimum spanning tree.
    (2) A new spanning tree is required which includes the arcs DI and HG, and which has the lowest possible total weight.
  4. Explain which algorithm you would choose to complete the tree, and how the algorithm should be adapted. (You do not need to find the tree.)
Edexcel D1 2011 January Q4
7 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-5_736_602_276_301} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-5_730_588_278_1171} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Six workers, Anthony, Beth, David, Jacob, Kantola and Miri, are to be allocated to six tasks, 1, 2, \(3,4,5\) and 6 . Figure 3 shows the possible allocations of the workers, and an initial matching is shown in Figure 4.
  1. Write down the technical name given to the type of diagram shown in Figure 3.
  2. Use the maximum matching algorithm once to find an improved matching. You must state the alternating path you use and your improved matching. Anthony now agrees to add task 6 to his possible allocations.
  3. Starting with your improved matching, use the maximum matching algorithm to obtain a complete matching. You must state the alternating path you use and your complete matching.
Edexcel D1 2011 January Q5
11 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-6_867_1381_260_342} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} [The total weight of the network is 31.6 km ]
Figure 5 models a network of roads. The road markings on these roads are to be renewed. The number on each arc represents the length, in km , of that road. In order to renew the road markings, each road must be traversed at least once.
  1. Use the route inspection algorithm, starting and finishing at A , to find a suitable route, which should be stated. You must make your method and working clear.
  2. State the roads that must be traversed twice and the length of the route.
    (3) The machine that will be used to renew the road markings can only be delivered to D . It will start at D, but it may finish at any vertex.
    Each road must still be traversed at least once.
  3. Given that the route is to be minimised, determine where the machine should finish. Give reasons to justify your answer.
    (3)
Edexcel D1 2011 January Q6
11 marks Moderate -0.5
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-7_1214_1581_251_242} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} The graph in Figure 6 is being used to solve a linear programming problem. Two of the constraints have been drawn on the graph and the rejected regions shaded out.
  1. Write down the constraints shown on the graph. Two further constraints are $$\begin{aligned} x + y & \geqslant 30 \\ \text { and } \quad 5 x + 8 y & \leqslant 400 \end{aligned}$$
  2. Add two lines and shading to Graph 1 in your answer book to represent these constraints. Hence determine the feasible region and label it R . The objective is to $$\text { minimise } 15 x + 10 y$$
  3. Draw a profit line on Graph 1 and use it to find the optimal solution. You must label your profit line clearly.
    (3)
Edexcel D1 2011 January Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0360f78d-e18c-4c47-a2ec-ddd705a4175f-8_888_1701_198_180} \captionsetup{labelformat=empty} \caption{Figure 7}
\end{figure} The network in Figure 7 shows the activities that need to be undertaken to complete a maintenance project. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the activity. The numbers in circles are the events. Each activity requires one worker. The project is to be completed in the shortest possible time.
  1. Complete the precedence table for this network in the answer book.
  2. Explain why each of the following is necessary.
    1. The dummy from event 6 to event 7 .
    2. The dummy from event 8 to event 9 .
  3. Complete Diagram 2 in the answer book to show the early and the late event times.
  4. State the critical activities.
  5. Calculate the total float on activity K . You must make the numbers used in your calculation clear.
  6. Calculate a lower bound for the number of workers needed to complete the project in the minimum time.