Assignment: Design of an optimisation model for industrial problems

A bio-fuel refining supply chain in the UK consists of

i) five collection points of wheat straw as feedstock are located in Peterborough, King's Lynn, Norwich, Ipswich and Letchworth,

ii) three storages located in Thetford, Cambridge and Bury St. Edmunds, and

iii) two bio-refineries located in Cambridge and Colchester.

The cost of the wheat straw at the place of collection points is £20 per tonne, and availability at each of the collection points is given in Table 1.

Table 1 Feedstock availability

The maximum capacities of the storages are: Thetford 50000 tonnes, Cambridge 20000 tonnes and Bury St. Edmunds 40000 tonnes. All collected feedstock has to be transported to storages first and before it is distributed to bio-refineries.

The cost of storing the wheat straw (per tonne of feedstock) in all three storages is minimised cost which is a complex relationship of energy requirements x₁, operational cost associated with labour and equipment x₂ and repayment of capital investment x₃, defined as
Coststorage = x² + x² + x² - x₁ - x₂ - x₃ + 6.0

The available feedstock (the wheat straw) is transported between collection points and storages using smaller trucks and at a cost of £5 per mile per tonne, whereas between storages and bio- refineries larger trucks are used at a cost of £2.5 per mile per tonne.

The conversion rate of bio-refining is 30%; 1 tonne of wheat straw produces 0.3 tonne of bio-fuel. The cost of bio-refining is £95 per tonne of biofuel and biofuel is sold at a price of £1025 per tonne. Assume that all produced biofuel is sold at the bio-refinery location.

The maximum capacities of the bio-refineries are Cambridge 60000 and Colchester 40000 tonnes of straw and they are expected to operate at full capacities.

The two scenarios are to be considered:

i) Maximising the utilisation of bio-refinery capacities of 60000 tonnes for the bio-refinery located in Cambridge and 40000 tonnes for the bio-refinery located in Colchester;

ii) Maximising the utilisation of bio-mass if the capacities of both bio-refineries is increased to 90000 tonnes each;

Task

Solve the stated problem by addressing the following issues:

1. Formulate the optimisation problem which will account for all costs and revenues in the supply chain and optimise its operation, dominantly the logistic, such that the whole supply chain (bio-refineries) gets maximum profit in both scenarios.

2. Provide a graphical representation (block diagram) of the supply chain and in appropriate way mark all the actors of the supply chain (collection points, storages, bio-refineries, transportation) and proposed (optimisation) variables.

3. Formulate the supply chain optimisation problem mathematically for both scenarios. The complexity of the problem suggests using a matrix formulation, the approach which could be easier adaptable to other similar problems. Explain and justify the selection of variables and their meaning in terms of the problem;

4. Implement the optimisation problem(s) in GAMS and hence calculate the optimum for both scenarios and using appropriate solvers.

5. Provide a graphical representation (block diagram) of the solution with clearly indicate values of the variables. Explain the results in terms of the set problem. Explain how trustful results are, any additional activity you took to increase trust and why.

Report

Write up report with a full account of your design and solution, which will include:

1. Account of your design which will show supply chain (graphical) representation, analytical form of the optimisation model for the two scenarios, justification of your analytical form in relation to the given problem, selection of the parameters and variables, as well as constraints and solver for GAMS program. Specify all the costs and distances used in your model;

2. Electronic, non-compressed and fully commented version of the GAMS programme (text or MS word format of GAMS programmes will be disregarded) for both scenarios and storage cost calculation.

3. Description and explanation of the results which will include result analysis, graphical representation of results in terms of set problem and description (and verification) of accuracy and trustfulness into obtained results, all for both scenarios.