Edexcel FD1 AS (Further Decision 1 AS) 2021 June

2. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the corresponding activity. The exact duration, \(x\), of activity N is unknown, but it is given that \(5 < x < 10\) Each activity requires one worker. The project is to be completed in the shortest possible time.

1. Complete the precedence table in the answer book.
2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
3. List the critical activities. It is given that activity J can be delayed by up to 4 hours without affecting the shortest possible completion time of the project.
4. Determine the value of \(x\). You must make the numbers used in your calculation clear.
5. Draw a cascade chart for this project on Grid 1 in the answer book.

3. Donald plans to bake and sell cakes. The three types of cake that he can bake are brownies, flapjacks and muffins. Donald decides to bake 48 brownies and muffins in total.
Donald decides to bake at least 5 brownies for every 3 flapjacks.
At most \(40 \%\) of the cakes will be muffins.
Donald has enough ingredients to bake 60 brownies or 45 flapjacks or 35 muffins.
Donald plans to sell each brownie for \(\pounds 1.50\), each flapjack for \(\pounds 1\) and each muffin for \(\pounds 1.25\) He wants to maximise the total income from selling the cakes. Let \(x\) represent the number of brownies, let \(y\) represent the number of flapjacks and let \(z\) represent the number of muffins that Donald will bake. Formulate this as a linear programming problem in \(x\) and \(y\) only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem.

4. \begin{figure}[h] \end{figure} Dijkstra's algorithm has been applied to the network in Figure 2.
A working value has only been replaced at a node if the new working value is smaller.

1. State the length of the shortest path from A to G .
2. Complete the table in the answer book giving the weight of each arc listed. (Note that arc CE and arc EF are not in the table.)
3. State the shortest path from A to G. It is now given that
   • when Prim's algorithm, starting from A, is applied to the network, the order in which the arcs are added to the tree is \(\mathrm { AB } , \mathrm { BC } , \mathrm { CD } , \mathrm { CE } , \mathrm { EF }\) and FG
   • the weight of the corresponding minimum spanning tree is 80
   • the shortest path from A to F via E has weight 67
   • Determine the weight of arc CE and the weight of arc EF , making your reasoning clear.