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	- 7	0	0	0	0
   0	2	1.5	2.5	1	0	0	10000
   0	1	0.5	1.5	0	1	0	7500
   0	300	200	400	0	0	1	2000000

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.