7 Consider the linear programming problem: $$\begin{array} { l l } \text { maximise } & P = 4 y - x ,
\text { subject to } & x + 4 y \leqslant 22 ,
& x + y \leqslant 10 ,
& - x + 2 y \leqslant 8 ,
\text { and } & x \geqslant 0 , y \geqslant 0 . \end{array}$$

1. Represent the constraints graphically, shading out the regions where the inequalities are not satisfied. Calculate the value of \(x\) and the value of \(y\) at each of the vertices of the feasible region. Hence find the maximum value of \(P\), clearly indicating where it occurs.
2. By introducing slack variables, represent the problem as an initial Simplex tableau and use the Simplex algorithm to solve the problem.
3. Indicate on your diagram for part (i) the points that correspond to each stage of the Simplex algorithm carried out in part (ii). \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS
   4736
   Decision Mathematics 1
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4. STEP	A	\(B\)	C
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5. STEP	A	\(B\)	C
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6. 
   \includegraphics[max width=\textwidth, alt={}, center]{b1227633-913e-41a9-8bf8-1f064056963e-7_661_1004_285_557}
7. Upper bound = \(\_\_\_\_\) km Lower bound = \(\_\_\_\_\) km
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   Best upper bound = \(\_\_\_\_\) km 6
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   Least travel time = \(\_\_\_\_\) minutes Route: A- \(\_\_\_\_\) \(- D\)