Your company’s refinery produces two types of automotive fuels: the production cost for the established (A) and the new (B) product is at £1,500 and £2,200 per tonne, respectively. Based on inventory analysis and outstanding orders, the refinery’s production manager notifies you that at least 30 tonnes of A and at least 10 tonnes of B must be produced during your next shift. To ensure the production target is met, you must produce a total of at least 80 tonnes of fuels (A+B) during the shift duration (8 hours). Also, an existing inventory of a perishable additive (C) which is essential for both fuels must be used as soon as possible.

Blending requires 5% of C for fuel A and 1% for fuel B (weight basis); only 5 tonnes of C are available. The blender requires 2.4 minutes per tonne of A and 1.8 minutes per tonne of B.

The current sale prices for these fuel products are £2,000/tonne (A) and £2,500/tonne (B).

(a) Express the profit generated from selling each of these two fuels (at given prices) as an explicit function of fuel quantities produced.

(b) Express the total income as a function of fuel quantities produced.

(c) Express the objective function for refinery profit maximisation. Derive it by means of defining all terms necessary, and obtain the simplest form explicitly.

(d) Express all constraints related to refinery operation and to physical limitations.

(e) Plot the feasible region for the problem formulation you derived.

(f) State the key LP theorem and compute the optimal operating point manually.

(g) Formulate your problem (use the linprog command in MATLABTM), provide your full code, present your results and discuss the significance of conclusions.

(h) Prices of A and B change: Solve the same problem for at least five more price combinations of A and B, for which you obtain at least one (1) optimal solution which is different from that in (g) (use the linprog command in MATLABTM).

(i) Find the limiting case (price combination) for which the optimal point switches.