Edexcel D1 (Decision Mathematics 1) 2017 January

1. Use the binary search algorithm to try to locate the name Hilbert in the following alphabetical list. Clearly indicate how you chose your pivots and which part of the list is being rejected at each stage.

2. The table represents a network that shows the average journey time, in minutes, between eight towns, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }\) and H .

1. Use Prim's algorithm, starting at A , to find the minimum spanning tree for this network. You must clearly state the order in which you select the edges of your tree.
2. Draw the minimum spanning tree using the vertices given in Diagram 1 in the answer book.
3. State the weight of the minimum spanning tree.

3. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows the possible allocations of six workers, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F , to six tasks, \(1,2,3\), 4, 5 and 6. Each task must be assigned to only one worker and each worker must be assigned to exactly one task. Figure 2 shows an initial matching.

1. Starting from the given initial matching, use the maximum matching algorithm to find an alternating path from A to 4 . Hence find an improved matching. You should list the alternating path you use, and state your improved matching.
2. Explain why it is not possible to find a complete matching. After training, task 1 is added to worker A's possible allocations.
3. Starting from the improved matching found in (a), use the maximum matching algorithm to find a complete matching. You should list the alternating path you use, and state your complete matching.
   (3)

4. \(\begin{array} { l l l l l l l l l } 23 & 18 & 27 & 9 & 25 & 10 & 12 & 30 & 24 \end{array}\) The numbers in the list represent the weights, in kilograms, of nine suitcases. The suitcases are to be transported in containers that will each hold a maximum weight of 45 kilograms.

1. Calculate a lower bound for the number of containers that will be needed to transport the suitcases.
2. Use the first-fit bin packing algorithm to allocate the suitcases to the containers.
3. Using the list provided, carry out a bubble sort to produce a list of the weights in descending order. You need only give the state of the list after each complete pass.
4. Use the first-fit decreasing bin packing algorithm to allocate the suitcases to the containers.
5. Explain why it is not possible to transport the suitcases using fewer containers than the number used in (d).

5. \begin{figure}[h] \end{figure} [The total weight of the network is 106.7]
Figure 3 models a network of cycle tracks that have to be inspected. The number on each arc represents the length, in km , of the corresponding track. Angela needs to travel along each cycle track at least once and wishes to minimise the length of her inspection route. She must start and finish at A.

1. Use an appropriate algorithm to find the tracks that will need to be traversed twice. You should make your method and working clear.
2. Find a route of minimum length, starting and finishing at A . State the length of your route. A new cycle track, AC, is under construction. It will be 15 km long. Angela will have to include this new track in her inspection route.
3. State the effect this new track will have on the total length of her route. Justify your answer.

6. \begin{figure}[h] \end{figure} Figure 4 represents a network of roads. The number on each arc represents the time taken, in minutes, to drive along the corresponding road. Stieg wishes to minimise the time spent driving from his home at A , to his office at H . The amount of traffic on two of the roads leading into H varies each day, and so the length of time taken to drive along these roads is expressed in terms of \(x\), where \(x > 7\)

1. Use Dijkstra's algorithm to find the possible routes that minimise the driving time from A to H . State the length of each route, leaving your answer in terms of \(x\) where necessary.
   (7) On a particular day, the quickest route from A to H via G is 2 minutes quicker than the quickest route from A to H via E .
2. Calculate the value of \(x\). You must make your method and working clear.

7. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 5. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the corresponding activity. Each activity requires exactly one worker. The project is to be completed in the shortest possible time.

1. Complete Diagram 1 in the answer book to show the early event times and late event times.
2. Explain what is meant by a critical path.
3. List the critical path for this network.
4. For each of the situations below, state the effect that the delay would have on the project completion date.
   1. A 4-day delay during activity J.
   2. A 4-day delay during activity M . The delays mentioned in (d) do not occur.
5. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
6. Schedule the activities using the minimum number of workers so that the project is completed in the minimum time.

8. A shop sells three types of pen. These are ballpoint pens, rollerball pens and fountain pens. The shop manager knows that each week she should order

• at least 50 pens in total
• at least twice as many rollerball pens as fountain pens

In addition,

• at most \(60 \%\) of the pens she orders must be ballpoint pens
• at least a third of the pens she orders must be rollerball pens

Each ballpoint pen costs \(\pounds 2\), each rollerball pen costs \(\pounds 3\) and each fountain pen costs \(\pounds 5\)
The shop manager wants to minimise her costs.
Let \(x\) represent the number of ballpoint pens ordered, let \(y\) represent the number of rollerball pens ordered and let \(z\) represent the number of fountain pens ordered.

1. Formulate this information as a linear programming problem. State the objective and list the constraints as simplified inequalities with integer coefficients. The shop manager decides to order exactly 10 fountain pens. This reduces the problem to the following $$\begin{array} { l r } \text { Minimise } & P = 2 x + 3 y
   \text { subject to } & x + y \geqslant 40
   & 2 x - 3 y \leqslant 30
   - x + 2 y \geqslant 10
   & y \geqslant 20
   & x \geqslant 0 \end{array}$$
2. Represent these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region R .
3. Use the objective line method to find the optimal vertex, V, of the feasible region. You must make your objective line clear and label the optimal vertex V.
4. Write down the number of each type of pen that the shop manager should order. Calculate the cost of this order.
   (Total \(\mathbf { 1 6 }\) marks)