OCR MEI D2 2012 June — Question 4 20 marks

Exam BoardOCR MEI
ModuleD2 (Decision Mathematics 2)
Year2012
SessionJune
Marks20
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear Programming
TypeFormulation from word problem
DifficultyModerate -0.3 This is a standard linear programming formulation and simplex algorithm question typical of Decision Mathematics modules. Part (i) requires matching given tableau to constraints (routine verification), part (ii) is mechanical application of simplex algorithm, part (iii) involves shadow price interpretation, and part (iv) requires adding an artificial variable for two-stage simplex. While multi-part with several marks, all components are textbook exercises requiring no novel insight—slightly easier than average A-level due to being algorithmic rather than requiring mathematical creativity.
Spec7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations7.07c Interpret simplex: values of variables, slack, and objective
4 A publisher is considering producing three books over the next week: a mathematics book, a novel and a biography. The mathematics book will sell at \(\pounds 10\) and costs \(\pounds 4\) to produce. The novel will sell at \(\pounds 5\) and costs \(\pounds 2\) to produce. The biography will sell at \(\pounds 12\) and costs \(\pounds 5\) to produce. The publisher wants to maximise profit, and is confident that all books will be sold. There are constraints on production. Each copy of the mathematics book needs 2 minutes of printing time, 1 minute of packing time, and \(300 \mathrm {~cm} ^ { 3 }\) of temporary storage space. Each copy of the novel needs 1.5 minutes of printing time, 0.5 minutes of packing time, and \(200 \mathrm {~cm} ^ { 3 }\) of temporary storage space. Each copy of the biography needs 2.5 minutes of printing time, 1.5 minutes of packing time, and \(400 \mathrm {~cm} ^ { 3 }\) of temporary storage space. There are 10000 minutes of printing time available on several printing presses, 7500 minutes of packing time, and \(2 \mathrm {~m} ^ { 3 }\) of temporary storage space.
  1. Explain how the following initial feasible tableau models this problem.
    P\(x\)\(y\)\(z\)\(s 1\)\(s 2\)\(s 3\)RHS
    1- 6- 3- 70000
    021.52.510010000
    010.51.50107500
    03002004000012000000
  2. Use the simplex algorithm to solve your LP, and interpret your solution.
  3. The optimal solution involves producing just one of the three books. By how much would the price of each of the other books have to be increased to make them worth producing? There is a marketing requirement to provide at least 1000 copies of the novel.
  4. Show how to incorporate this constraint into the initial tableau ready for an application of the two-stage simplex method. Briefly describe how to use the modified tableau to solve the problem. You are NOT required to perform the iterations.
4 A publisher is considering producing three books over the next week: a mathematics book, a novel and a biography. The mathematics book will sell at $\pounds 10$ and costs $\pounds 4$ to produce. The novel will sell at $\pounds 5$ and costs $\pounds 2$ to produce. The biography will sell at $\pounds 12$ and costs $\pounds 5$ to produce. The publisher wants to maximise profit, and is confident that all books will be sold.

There are constraints on production. Each copy of the mathematics book needs 2 minutes of printing time, 1 minute of packing time, and $300 \mathrm {~cm} ^ { 3 }$ of temporary storage space.

Each copy of the novel needs 1.5 minutes of printing time, 0.5 minutes of packing time, and $200 \mathrm {~cm} ^ { 3 }$ of temporary storage space.

Each copy of the biography needs 2.5 minutes of printing time, 1.5 minutes of packing time, and $400 \mathrm {~cm} ^ { 3 }$ of temporary storage space.

There are 10000 minutes of printing time available on several printing presses, 7500 minutes of packing time, and $2 \mathrm {~m} ^ { 3 }$ of temporary storage space.\\
(i) Explain how the following initial feasible tableau models this problem.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | r | }
\hline
P & $x$ & $y$ & $z$ & $s 1$ & $s 2$ & $s 3$ & RHS \\
\hline
1 & - 6 & - 3 & - 7 & 0 & 0 & 0 & 0 \\
\hline
0 & 2 & 1.5 & 2.5 & 1 & 0 & 0 & 10000 \\
\hline
0 & 1 & 0.5 & 1.5 & 0 & 1 & 0 & 7500 \\
\hline
0 & 300 & 200 & 400 & 0 & 0 & 1 & 2000000 \\
\hline
\end{tabular}
\end{center}

(ii) Use the simplex algorithm to solve your LP, and interpret your solution.\\
(iii) The optimal solution involves producing just one of the three books. By how much would the price of each of the other books have to be increased to make them worth producing?

There is a marketing requirement to provide at least 1000 copies of the novel.\\
(iv) Show how to incorporate this constraint into the initial tableau ready for an application of the two-stage simplex method.

Briefly describe how to use the modified tableau to solve the problem. You are NOT required to perform the iterations.

\hfill \mbox{\textit{OCR MEI D2 2012 Q4 [20]}}
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Q1 16 Q2 16 Q3 20 Q4 20