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AQA D1 2015 June Q4
8 marks Easy -1.2
4 The network opposite shows roads connecting 10 villages, \(A , B , \ldots , J\). The time taken to drive along a road is not proportional to the length of the road. The number on each edge shows the average time, in minutes, to drive along each road.
  1. A commuter wishes to drive from village \(A\) to the railway station at \(J\).
    1. Use Dijkstra's algorithm, on the diagram opposite, to find the shortest driving time from \(A\) to \(J\).
    2. State the corresponding route.
  2. A taxi driver is in village \(D\) at 10.30 am when she receives a radio call asking her to pick up a passenger at village \(A\) and take him to the station at \(J\). Assuming that it takes her 3 minutes to load the passenger and his luggage, at what time should she expect to arrive at the station?
    [0pt] [2 marks]
    \includegraphics[max width=\textwidth, alt={}]{f5890e58-38c3-413c-8762-6f80ce6dcec7-09_2484_1717_223_150}
AQA D1 2015 June Q5
7 marks Moderate -0.5
5 The network shows the paths mown through a wildflower meadow so that visitors can use these paths to enjoy the flowers. The lengths of the paths are shown in metres.
\includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-10_1097_1603_413_214} The total length of all the paths is 1400 m .
The mower is kept in a shed at \(A\). The groundskeeper must mow all the paths and return the mower to its shed.
  1. Find the length of an optimal Chinese postman route starting and finishing at \(A\).
  2. State the number of times that the mower, following the optimal route, will pass through:
    1. \(C\);
    2. \(D\).
AQA D1 2015 June Q6
12 marks Easy -1.2
6 The network shows the roads linking a warehouse at \(A\) and five shops, \(B , C , D , E\) and \(F\). The numbers on the edges show the lengths, in miles, of the roads. A delivery van leaves the warehouse, delivers to each of the shops and returns to the warehouse.
\includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-12_1241_1239_484_402}
  1. Complete the table, on the page opposite, showing the shortest distances between the vertices.
    1. Find the total distance travelled if the van follows the cycle \(A E F B C D A\).
    2. Explain why your answer to part (b)(i) provides an upper bound for the minimum journey length.
  3. Use the nearest neighbour algorithm starting from \(D\) to find a second upper bound.
  4. By deleting \(A\), find a lower bound for the minimum journey length.
  5. Given that the minimum journey length is \(T\), write down the best inequality for \(T\) that can be obtained from your answers to parts (b), (c) and (d).
    [0pt] [1 mark] \section*{Answer space for question 6} REFERENCE
  6. \(\boldsymbol { A }\)\(\boldsymbol { B }\)\(\boldsymbol { C }\)\(\boldsymbol { D }\)\(\boldsymbol { E }\)\(\boldsymbol { F }\)
    \(\boldsymbol { A }\)-7
    \(\boldsymbol { B }\)7-5
    \(\boldsymbol { C }\)5-4
    \(\boldsymbol { D }\)4-6
    \(\boldsymbol { E }\)6-10
    \(\boldsymbol { F }\)10-
AQA D1 2015 June Q7
6 marks Moderate -0.8
7
  1. A simple connected graph has 4 edges and \(m\) vertices. State the possible values of \(m\).
  2. A simple connected graph has \(n\) edges and 4 vertices. State the possible values of \(n\).
  3. A simple connected graph, \(G\), has 5 vertices and is Eulerian but not Hamiltonian. Draw a possible graph \(G\).
    [0pt] [2 marks]
AQA D1 2015 June Q8
6 marks Easy -1.2
8 A student is tracing the following algorithm.
\includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-18_1431_955_372_539}
  1. Trace the algorithm illustrated in the flowchart for the case where the input value of \(N\) is 5 .
  2. Explain the role of \(N\) in the algorithm.
AQA D1 2015 June Q9
17 marks Moderate -0.8
9 A company producing chicken food makes three products, Basic, Premium and Supreme, from wheat, maize and barley. A tonne \(( 1000 \mathrm {~kg} )\) of Basic uses 400 kg of wheat, 200 kg of maize and 400 kg of barley.
A tonne of Premium uses 400 kg of wheat, 500 kg of maize and 100 kg of barley.
A tonne of Supreme uses 600 kg of wheat, 200 kg of maize and 200 kg of barley.
The company has 130 tonnes of wheat, 70 tonnes of maize and 72 tonnes of barley available. The company must make at least 75 tonnes of Supreme.
The company makes \(\pounds 50\) profit per tonne of Basic, \(\pounds 100\) per tonne of Premium and \(\pounds 150\) per tonne of Supreme. They plan to make \(x\) tonnes of Basic, \(y\) tonnes of Premium and \(z\) tonnes of Supreme.
  1. Write down four inequalities representing the constraints (in addition to \(x , y \geqslant 0\) ).
    [0pt] [4 marks]
  2. The company want exactly half the production to be Supreme. Show that the constraints in part (a) become $$\begin{aligned} x + y & \leqslant 130 \\ 4 x + 7 y & \leqslant 700 \\ 2 x + y & \leqslant 240 \\ x + y & \geqslant 75 \\ x & \geqslant 0 \\ y & \geqslant 0 \end{aligned}$$
  3. On the grid opposite, illustrate all the constraints and label the feasible region.
  4. Write an expression for \(P\), the profit for the whole production, in terms of \(x\) and \(y\) only.
    [0pt] [2 marks]
    1. By drawing an objective line on your graph, or otherwise, find the values of \(x\) and \(y\) which give the maximum profit.
      [0pt] [2 marks]
    2. State the maximum profit and the amount of each product that must be made.
      [0pt] [2 marks] \section*{Answer space for question 9}
      \includegraphics[max width=\textwidth, alt={}]{f5890e58-38c3-413c-8762-6f80ce6dcec7-21_1349_1728_310_148}
      QUESTION
      PART
      REFERENCE
      \includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-24_2488_1728_219_141}
AQA D1 2016 June Q1
5 marks Easy -1.2
1 Alfred has bought six different chocolate bars. He wants to give a chocolate bar to each of his six friends. The table gives the names of the friends and indicates which of Alfred's six chocolate bars they like.
AQA D1 2016 June Q2
7 marks Easy -1.8
2
  1. Use a shuttle sort to rearrange into alphabetical order the following list of names:
    Rob, Eve, Meg, lan, Xavi
    Show the list at the end of each pass.
  2. A list of ten numbers is sorted into ascending order, using a shuttle sort.
    1. How many passes are needed?
    2. Give the maximum number of comparisons needed in the sixth pass.
    3. Given that the list is initially in descending order, find the total number of swaps needed.
      [0pt] [4 marks]
AQA D1 2016 June Q3
7 marks Moderate -0.3
3 The network below shows vertices \(A , B , C , D\) and \(E\). The number on each edge shows the distance between vertices.
\includegraphics[max width=\textwidth, alt={}, center]{fb95068f-f76d-492a-b385-bce17b26ae30-06_563_736_402_651}
    1. In the case where \(x = 8\), use Kruskal's algorithm to find a minimum spanning tree for the network. Write down the order in which you add edges to your minimum spanning tree.
    2. Draw your minimum spanning tree.
    3. Write down the length of your minimum spanning tree.
  2. Alice draws the same network but changes the value of \(x\). She correctly uses Kruskal's algorithm and edge \(C D\) is included in her minimum spanning tree.
    1. Explain why \(x\) cannot be equal to 7 .
    2. Write down an inequality for \(x\).
AQA D1 2016 June Q4
11 marks Moderate -0.3
4 Amal delivers free advertiser magazines to all the houses in his village. The network shows the roads in his village. The number on each road shows the time, in minutes, that Amal takes to walk along that road.
\includegraphics[max width=\textwidth, alt={}, center]{fb95068f-f76d-492a-b385-bce17b26ae30-08_846_1264_445_388}
  1. Amal starts his delivery round from his house, at vertex \(A\). He must walk along each road at least once.
    1. Find the length of an optimal Chinese postman route around the village, starting and finishing at Amal's house.
    2. State the number of times that Amal passes his friend Dipak's house, at vertex \(D\).
  2. Dipak offers to deliver the magazines while Amal is away on holiday. Dipak must walk along each road at least once. Assume that Dipak takes the same length of time as Amal to walk along each road.
    1. Dipak can start his journey at any vertex and finish his journey at any vertex. Find the length of time for an optimal route for Dipak.
    2. State the vertices at which Dipak could finish, in order to achieve his optimal route.
    1. Find the length of time for an optimal route for Dipak, if, instead, he wants to finish at his house, at vertex \(D\), and can start his journey at any other vertex.
    2. State the start vertex.
AQA D1 2016 June Q5
8 marks Standard +0.8
5 A fair comes to town one year and sets up its rides in two large fields that are separated by a river. The diagram shows the ten main rides, at \(A , B , C , \ldots , J\). The numbers on the edges represent the times, in minutes, it takes to walk between pairs of rides. A footbridge connects the rides at \(D\) and \(F\).
    1. Use Dijkstra's algorithm on the diagram below to find the minimum time to walk from \(A\) to each of the other main rides.
    2. Write down the route corresponding to the minimum time to walk from \(A\) to \(G\).
  2. The following year, the fair returns. In addition to the information shown on the diagram, another footbridge has been built to connect the rides at \(E\) and \(G\). This reduces the time taken to travel from \(A\) to \(G\), but the time taken to travel from \(A\) to \(J\) is not reduced. The time to walk across the footbridge from \(E\) to \(G\) is \(x\) minutes, where \(x\) is an integer. Find two inequalities for \(x\) and hence state the value of \(x\). \section*{Answer space for question 5}

    1. \includegraphics[max width=\textwidth, alt={}, center]{fb95068f-f76d-492a-b385-bce17b26ae30-12_659_1591_1692_223}
AQA D1 2016 June Q6
7 marks Moderate -0.5
6 A connected graph is semi-Eulerian if exactly two of its vertices are of odd degree.
  1. A graph is drawn with 4 vertices and 7 edges. What is the sum of the degrees of the vertices?
  2. Draw a simple semi-Eulerian graph with exactly 5 vertices and 5 edges, in which exactly one of the vertices has degree 4 .
  3. Draw a simple semi-Eulerian graph with exactly 5 vertices that is also a tree.
  4. A simple graph has 6 vertices. The graph has two vertices of degree 5 . Explain why the graph can have no vertex of degree 1.
AQA D1 2016 June Q7
17 marks Moderate -0.5
7 A company operates a steam railway between six stations. The minimum cost (in euros) of travelling between pairs of stations is shown in the table below.
  1. On Figure 1 below, use Prim's algorithm, starting from \(P\), to find a minimum spanning tree for the graph connecting \(P , Q , R , S , T\) and \(U\). State clearly the order in which you select the vertices and draw your minimum spanning tree. \section*{Question 7 continues on page 20} \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1}
    \(\boldsymbol { P }\)\(\boldsymbol { Q }\)\(\boldsymbol { R }\)\(\boldsymbol { S }\)\(\boldsymbol { T }\)\(\boldsymbol { U }\)
    \(\boldsymbol { P }\)-14711612
    \(\boldsymbol { Q }\)14-810910
    \(\boldsymbol { R }\)78-121315
    \(\boldsymbol { S }\)111012-511
    \(\boldsymbol { T }\)69135-10
    \(\boldsymbol { U }\)1210151110-
    \end{table}
  2. Another station, \(V\), is opened. The minimum costs (in euros) of travelling to and from \(V\) to each of the other stations are added to the table in part (a), as shown in Figure 2(i) below. Further copies of this table are shown in Figure 2(ii). Arjen is on holiday and he plans to visit each station. He intends to board a train at \(V\) and visit all the other stations, once only, before returning to \(V\).
    1. By first removing \(V\), obtain a lower bound for the minimum travelling cost of Arjen's tour. (You may use Figure 2(i) for your working.)
    2. Use the nearest neighbour algorithm twice, starting each time from \(V\), to find two different upper bounds for the minimum cost of Arjen's tour. State, with a reason, which of your two answers gives the better upper bound. (You may use Figure 2(ii) for your working.)
    3. Hence find an optimal tour of the seven stations. Explain how you know that it is optimal. Answer space for question 7(b) \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2(ii)}
      \(\boldsymbol { P }\)\(Q\)\(\boldsymbol { R }\)\(\boldsymbol { S }\)\(T\)\(\boldsymbol { U }\)\(V\)
      \(\boldsymbol { P }\)-1471161215
      \(Q\)14-81091018
      \(\boldsymbol { R }\)78-12131514
      \(\boldsymbol { S }\)111012-51114
      \(T\)69135-1017
      \(\boldsymbol { U }\)1210151110-12
      \(V\)151814141712-
      \end{table}
AQA D1 2016 June Q8
13 marks Easy -1.2
8 Nerys runs a cake shop. In November and December she sells Christmas hampers. She makes up the hampers herself, in two sizes: Luxury and Special. Each day, Nerys prepares \(x\) Luxury hampers and \(y\) Special hampers.
It takes Nerys 10 minutes to prepare a Luxury hamper and 15 minutes to prepare a Special hamper. She has 6 hours available, each day, for preparing hampers. From past experience, Nerys knows that each day:
  • she will need to prepare at least 5 hampers of each size
  • she will prepare at most a total of 32 hampers
  • she will prepare at least twice as many Luxury hampers as Special hampers.
Each Luxury hamper that Nerys prepares makes her a profit of \(\pounds 15\); each Special hamper makes a profit of \(\pounds 20\). Nerys wishes to maximise her daily profit, \(\pounds P\).
  1. Show that \(x\) and \(y\) must satisfy the inequality \(2 x + 3 y \leqslant 72\).
  2. In addition to \(x \geqslant 5\) and \(y \geqslant 5\), write down two more inequalities that model the constraints above.
  3. On the grid opposite draw a suitable diagram to enable this problem to be solved graphically. Indicate a feasible region and the direction of an objective line.
    1. Use your diagram to find the number of each type of hamper that Nerys should prepare each day to achieve a maximum profit.
    2. Calculate this profit.
      \includegraphics[max width=\textwidth, alt={}]{fb95068f-f76d-492a-b385-bce17b26ae30-27_2490_1719_217_150}
      \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
AQA D2 Q4
Standard +0.3
4 [Figures 3, 4 and 5, printed on the insert, are provided for use in this question.]
The network shows a system of pipes, with the lower and upper capacities for each pipe in litres per second.
\includegraphics[max width=\textwidth, alt={}, center]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-005_547_1214_555_404}
  1. Figure 3, on the insert, shows a partially completed diagram for a feasible flow of 10 litres per second from \(S\) to \(T\). Indicate, on Figure 3, the flows along the edges \(M N , P Q , N P\) and \(N T\).
    1. Taking your answer from part (a) as an initial flow, use flow augmentation on Figure 4 to find the maximum flow from \(S\) to \(T\).
    2. State the value of the maximum flow and illustrate this flow on Figure 5.
  3. Find a cut with capacity equal to that of the maximum flow.
AQA D2 Q7
Moderate -0.3
7 The network below shows a system of one-way roads. The number on each edge represents the number of bags for recycling that can be collected by driving along that road. A collector is to drive from \(A\) to \(I\).
\includegraphics[max width=\textwidth, alt={}, center]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-144_867_1644_552_191}
  1. Working backwards from \(\boldsymbol { I }\), use dynamic programming to find the maximum number of bags that can be collected when driving from \(A\) to \(I\). You must complete the table opposite as your solution.
  2. State the route that the collector should take in order to collect the maximum number of bags.
  3. StageStateFromValue
    1GI
    HI
    2
AQA D2 Q8
Moderate -0.3
8 The network below represents a system of pipes. The capacity of each pipe, in litres per second, is indicated on the corresponding edge.
\includegraphics[max width=\textwidth, alt={}, center]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-146_743_977_404_536}
  1. Find the maximum flow along each of the routes \(A B E H , A C F H\) and \(A D G H\) and enter their values in the table on Figure 4 opposite.
    1. Taking your answers to part (a) as the initial flow, use the labelling procedure on Figure 4 to find the maximum flow through the network. You should indicate any flow-augmenting routes in the table and modify the potential increases and decreases of the flow on the network.
    2. State the value of the maximum flow and, on Figure 5 opposite, illustrate a possible flow along each edge corresponding to this maximum flow.
  3. Confirm that you have a maximum flow by finding a cut of the same value. List the edges of your cut. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4}
    RouteFlow
    \(A B E H\)
    \(A C F H\)
    \(A D G H\)
    \end{table} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-147_746_972_397_845}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{c18db720-6fe8-4e6c-bd0c-dc51cc341b47-147_739_971_1311_539}
    \end{figure}
AQA D2 2006 January Q1
9 marks Moderate -0.5
1 Five trainers, Ali, Bo, Chas, Dee and Eve, held an initial training session with each of four teams over an assault course. The completion times in minutes are recorded below.
AliBoChasDeeEve
Team 11619182524
Team 22221202625
Team 32122232124
Team 42021212320
Each of the four teams is to be allocated a trainer and the overall time for the four teams is to be minimised. No trainer can train more than one team.
  1. Modify the table of values by adding an extra row of values so that the Hungarian algorithm can be applied.
  2. Use the Hungarian algorithm, reducing columns first then rows, to decide which four trainers should be allocated to which team. State the minimum total training time for the four teams using this matching.
AQA D2 2006 January Q2
9 marks Moderate -0.8
2 A manufacturing company is planning to build three new machines, \(A , B\) and \(C\), at the rate of one per month. The order in which they are built is a matter of choice, but the profits will vary according to the number of workers available and the suppliers' costs. The expected profits in thousands of pounds are given in the table.
\multirow[b]{2}{*}{Month}\multirow[b]{2}{*}{Already built}Profit (in units of £1000)
\(\boldsymbol { A }\)\(\boldsymbol { B }\)\(\boldsymbol { C }\)
1-524748
\multirow[t]{3}{*}{2}A-5854
B70-54
\(\boldsymbol { C }\)6863-
\multirow[t]{3}{*}{3}\(\boldsymbol { A }\) and \(\boldsymbol { B }\)--64
\(\boldsymbol { A }\) and \(\boldsymbol { C }\)-67-
\(\boldsymbol { B }\) and \(\boldsymbol { C }\)69--
  1. Draw a labelled network such that the most profitable order of manufacture corresponds to the longest path within that network.
  2. Use dynamic programming to determine the order of manufacture that maximises the total profit, and state this maximum profit.
AQA D2 2006 January Q3
18 marks Moderate -0.3
3 [Figures 1 and 2, printed on the insert, are provided for use in this question.] A building project is to be undertaken. The table shows the activities involved.
ActivityImmediate PredecessorsDuration (days)Number of Workers Required
A-23
BA42
CA61
D\(B , C\)83
EC32
FD22
GD, E42
HD, E61
I\(F , G , H\)23
  1. Complete the activity network for the project on Figure 1.
  2. Find the earliest start time for each activity.
  3. Find the latest finish time for each activity.
  4. Find the critical path and state the minimum time for completion.
  5. State the float time for each non-critical activity.
  6. Given that each activity starts as early as possible, draw a resource histogram for the project on Figure 2.
  7. There are only 3 workers available at any time. Use resource levelling to explain why the project will overrun and state the minimum extra time required.
AQA D2 2006 January Q4
14 marks Standard +0.3
4 [Figures 3, 4 and 5, printed on the insert, are provided for use in this question.]
The network shows a system of pipes, with the lower and upper capacities for each pipe in litres per second.
\includegraphics[max width=\textwidth, alt={}, center]{30a88efe-fe9e-4384-a3e3-da2a05326797-04_547_1214_555_404}
  1. Figure 3, on the insert, shows a partially completed diagram for a feasible flow of 10 litres per second from \(S\) to \(T\). Indicate, on Figure 3, the flows along the edges \(M N , P Q , N P\) and \(N T\).
    1. Taking your answer from part (a) as an initial flow, use flow augmentation on Figure 4 to find the maximum flow from \(S\) to \(T\).
    2. State the value of the maximum flow and illustrate this flow on Figure 5.
  3. Find a cut with capacity equal to that of the maximum flow.
AQA D2 2006 January Q5
13 marks Standard +0.3
5
  1. Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l c } \text { Maximise } & P = 3 x + 2 y + 4 z \\ \text { subject to } & x + 4 y + 2 z \leqslant 8 \\ & 2 x + 7 y + 3 z \leqslant 21 \\ & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
  2. Use the Simplex method to perform one iteration of your tableau for part (a), choosing a value in the \(z\)-column as pivot.
    1. Perform one further iteration.
    2. State whether or not this is the optimal solution, and give a reason for your answer.
AQA D2 2006 January Q6
11 marks Moderate -0.8
6 Sam is playing a computer game in which he is trying to drive a car in different road conditions. He chooses a car and the computer decides the road conditions. The points scored by Sam are shown in the table.
Road Conditions
\cline { 2 - 5 }\(\boldsymbol { C } _ { \mathbf { 1 } }\)\(\boldsymbol { C } _ { \mathbf { 2 } }\)\(\boldsymbol { C } _ { \mathbf { 3 } }\)
\cline { 2 - 5 }\(\boldsymbol { S } _ { \mathbf { 1 } }\)- 224
\cline { 2 - 5 } Sam's Car\(\boldsymbol { S } _ { \mathbf { 2 } }\)245
\cline { 2 - 5 }\(\boldsymbol { S } _ { \mathbf { 3 } }\)512
\cline { 2 - 5 }
\cline { 2 - 5 }
Sam is trying to maximise his total points and the computer is trying to stop him.
  1. Explain why Sam should never choose \(S _ { 1 }\) and why the computer should not choose \(C _ { 3 }\).
  2. Find the play-safe strategies for the reduced 2 by 2 game for Sam and the computer, and hence show that this game does not have a stable solution.
  3. Sam uses random numbers to choose \(S _ { 2 }\) with probability \(p\) and \(S _ { 3 }\) with probability \(1 - p\).
    1. Find expressions for the expected gain for Sam when the computer chooses each of its two remaining strategies.
    2. Calculate the value of \(p\) for Sam to maximise his total points.
    3. Hence find the expected points gain for Sam.
      SurnameOther Names
      Centre NumberCandidate Number
      Candidate Signature
      \section*{General Certificate of Education January 2006
      Advanced Level Examination} \section*{MATHEMATICS
      Unit Decision 2} MD02 \section*{Insert} Wednesday 18 January 20061.30 pm to 3.00 pm Insert for use in Questions 3 and 4.
      Fill in the boxes at the top of this page.
      Fasten this insert securely to your answer book.
AQA D2 2007 January Q1
11 marks Easy -1.2
1 [Figure 1, printed on the insert, is provided for use in this question.]
A building project is to be undertaken. The table shows the activities involved.
ActivityImmediate PredecessorsDuration (weeks)
A-2
B-1
CA3
DA, B2
EB4
FC1
G\(C , D , E\)3
HE5
I\(F , G\)2
J\(H , I\)3
  1. Complete an activity network for the project on Figure 1.
  2. Find the earliest start time for each activity.
  3. Find the latest finish time for each activity.
  4. State the minimum completion time for the building project and identify the critical paths.
AQA D2 2007 January Q2
13 marks Moderate -0.5
2 Five successful applicants received the following scores when matched against suitability criteria for five jobs in a company.
Job 1Job 2Job 3Job 4Job 5
Alex131191013
Bill1512121112
Cath121081414
Don1112131410
Ed1214141314
It is intended to allocate each applicant to a different job so as to maximise the total score of the five applicants.
  1. Explain why the Hungarian algorithm may be used if each number, \(x\), in the table is replaced by \(15 - x\).
  2. Form a new table by subtracting each number in the table from 15. Use the Hungarian algorithm to allocate the jobs to the applicants so that the total score is maximised.
  3. It is later discovered that Bill has already been allocated to Job 4. Decide how to alter the allocation of the other jobs so as to maximise the score now possible.