Assignment - Options for Engineering Simulation

Please consider to choose other option only if you think your background is more appropriate to show your ability in simulation and report writing.

Option - Transport Modelling and Assignment Optimisation

Note: This is an optional problem and please only consider it if the first main option (Option 1) and other options do not fit into your area of expertise or if you think that transport and assignment problems are most relevant to your interest and applications.

Problem Statement

The main objective to manage all the resources subject to various supply and demand constraints so as to complete the tasks with the minimum costs (or the maximum profits).

Please choose a case study either from your own real-world application or from a hypothetical scenario about a product (such as food, electronic device or clothing).

The product is being produced by n>5 suppliers and each supplier can produce S_i. The product is distributed to m>5 retailers/buyers and each retailer demands R_j of the product. The rest of the product can be stored in a warehouse (W). The cost of transporting one unit of this product from Supplier Si to Retail Rj is C_ij>0. The total quantity of the product is P and thus

S₁ + S₂ + ? S_n = P.

Similarly, all the product must be distributed to retailers and the warehouse, which means that

R₁ + R₂ + ? R_m + W = P.

For example, in the case of n=3, m=4 and P=1000, we have

and its cost matrix is simply

Let x_ij be the integer quantity to be distributed from Supplier (S_i) to Retailer (R_j). The objective is to distribute the product such that

Minimize ∑ⁿ_i=1 ∑^m_j=1 c_ij x_ij,

Subject to various constraints.

Your main tasks are as follows:

1) Formulate this resource allocation/assignment problem in a clear mathematical manner. Write down the mathematical equations including the objective function and all the constraints. Discuss your formulation and its relevance to your real-world application.

2) For a simple case of n=5, m=5, and P=1000. Generate any non-negative meaningful numbers of supply quantities and demand quantities as well as the transport costs. List all these in tables and explain what they mean. Consider a case where there is no product stored in the warehouse (thus all the products have been distributed to retailers).

3) Use any method (such as the north-west corner method) to find an initial feasible solution to this problem and calculate its overall transport cost.

4) Use Excel and its Solver or any programming language to implement the above scenario and solve it so as to find the minimize cost. Explain what it means and test if all constraints are satisfied.

5) Extend your computer code to solve any case of n>5 and m>10 (say, n=10, m=25, P=2000), and discuss your results and the solution method used. Can you extend the above case to the case with k>2 warehouses? What has to be done in order to minimize the costs and also minimize the storage of the warehouses?

Write a brief report (about 3000 to 5000 words) to summarize your formulation, the main solution procedure and key findings.