4 The "Cuddly Friends Company" produces soft toys. For one day's production run it has available \(11 \mathrm {~m} ^ { 2 }\) of furry material, \(24 \mathrm {~m} ^ { 2 }\) of woolly material and 30 glass eyes. It has three soft toys which it can produce:
The "Cuddly Aardvark", each of which requires \(0.5 \mathrm {~m} ^ { 2 }\) of furry material, \(2 \mathrm {~m} ^ { 2 }\) of woolly material and two eyes. Each sells at a profit of \(\pounds 3\).
The "Cuddly Bear", each of which requires \(1 \mathrm {~m} ^ { 2 }\) of furry material, \(1.5 \mathrm {~m} ^ { 2 }\) of woolly material and two eyes. Each sells at a profit of \(\pounds 5\).
The "Cuddly Cat", each of which requires \(1 \mathrm {~m} ^ { 2 }\) of furry material, \(1 \mathrm {~m} ^ { 2 }\) of woolly material and two eyes. Each sells at a profit of \(\pounds 2\).
An analyst formulates the following LP to find the production plan which maximises profit.
$$\begin{array} { l l }
\text { Maximise } & 3 a + 5 b + 2 c
\text { subject to } & 0.5 a + b + c \leqslant 11 ,
& 2 a + 1.5 b + c \leqslant 24 ,
& 2 a + 2 b + 2 c \leqslant 30 .
\end{array}$$
- Explain how this formulation models the problem, and say why the analyst has not simplified the last inequality to \(a + b + c \leqslant 15\).
- The final constraint is different from the others in that the resource is integer valued. Explain why that does not impose an additional difficulty for this problem.
- Solve this problem using the simplex algorithm.
Interpret your solution and say what resources are left over.
On a particular day an order is received for two Cuddly Cats, and the extra constraint \(c \geqslant 2\) is added to the formulation.
- Set up an initial simplex tableau to deal with the modified problem using either the big-M approach or two-phase simplex. Do not perform any iterations on your tableau.
- Show that the solution given by \(a = 8 , b = 2\) and \(c = 5\) uses all of the resources, but that \(a = 6 , b = 6\) and \(c = 2\) gives more profit.
What resources are left over from the latter solution?