Questions D2 (553 questions)

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OCR D2 Q4
10 marks Standard +0.3
A construction company has three teams of workers available, each of which is to be assigned to one of four jobs at a site. The following table shows the estimated cost, in tens of pounds, of each team doing each job: \begin{array}{c|c|c|c|c} & Windows & Conservatory & Doors & Greenhouse
\hline Team A & 27 & 80 & 8 & 81
Team B & 28 & 60 & 5 & 71
Team C & 30 & 90 & 7 & 73
\end{array} Use the Hungarian algorithm to find an allocation of jobs which will minimise the total cost. Show the state of the table after each stage in the algorithm and state the cost of the final assignment. [10 marks]
OCR D2 Q5
22 marks Standard +0.3
The following matrix gives the capacities of the pipes in a system. \begin{array}{c|c|c|c|c|c|c|c} To & & S & T & A & B & C & D
From & & & & & & &
\hline S & & -- & -- & 16 & 26 & -- & --
T & & -- & -- & -- & -- & -- & --
A & & -- & -- & -- & -- & 13 & 5
B & & -- & 16 & -- & -- & -- & 11
C & & -- & 11 & -- & -- & -- & --
D & & -- & 11 & -- & -- & -- & --
\end{array}
  1. Represent this information as a digraph. [3 marks]
  2. Find the minimum cut, expressing it in the form \(\{\) \(\}|\{\) \(\}\), and state its value. [2 marks]
  3. Starting from having no flow in the system, use the labelling procedure to find a maximal flow through the system. You should list each flow-augmenting route you use, together with its flow. [5 marks]
  4. Explain how you know that this flow is maximal. [1 mark]
[11 marks]
OCR D2 Q6
42 marks Challenging +1.2
The payoff matrix for player \(A\) in a two-person zero-sum game is shown below. \begin{array}{c|c|c|c|c} & & \multicolumn{3}{c}{B}
& & \text{I} & \text{II} & \text{III}
\hline \multirow{2}{*}{A} & \text{I} & -2 & 3 & -1
& \text{II} & 4 & -5 & 2
\end{array}
  1. Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player \(B\). [7 marks]
  2. By solving this linear programming problem, find the optimal strategy for player \(B\) and the value of the game. [14 marks]
[21 marks]