Individual Assignment: Task $1$

BNM$860$ DECISION MODELS

Problem Description

A company has two factories, one each at Durham and Norwich. The factories produce paints which are sold to four wholesalers. The wholesalers are either supplied directly from the factories or through one of the company warehouses, the transportation costs being paid by the company. The company has three warehouses, one each in Reading, Sheffield, and Edinburgh. Table $1$ shows the transportation costs per ton for deliveries from the factories to the warehouses or wholesalers and also from the warehouses to the wholesalers, omitting entries when delivery from a certain supplier or warehouse is impossible for some destination.

$\backslash$table$[[$Supplier$,$Warehouse,Wholesaler$],[$Reading$,$Sheffield,Edinburgh,$1,2,3,4],[$Durham$,-,57,48,121,101,\backslash$table$[[-]],137],[$Norwich$,54,58,-,\backslash$table$[[-],[-]],98,105,150],[$Reading$,-,-,-,59,-,41,67],[$Sheffield$,-,-,-,52,62,67,72],[$Edinburgh$,-,-,-,72,67,48,59]]$

Table $1$: Transportation costs in $f^{\text{'}}/$ton delivered.

The two factories at Durham and Norwich can produce up to $83,000$ and $67,000$ tons per week respectively. No more than $40,000,38,000$ and $36,000$ tons can be moved each week through the warehouse in Reading, Sheffield, and Edinburgh, respectively. Wholesalers $1,2,3,$ and $4$ require $48,000,38,000,21,000,$ and $36,000$ tons per week respectively.

Answer the following three parts of the problem. Parts A and B are worth $30\%$ each while part C is worth $40\%$ of the overall mark for this problem.

A$.$ Formulate a linear programming model to determine the minimum cost transportation schedule. Clearly explain the variables you use and the constraints you construct. What is the minimum cost transportation schedule and what is the minimum transportation cost?

B$.$ Discuss the effect on the minimum transportation cost when capacity at each factory or warehouse is altered by adding or subtracting one ton. What are the minimum capacity changes $($in both directions$)$ at Sheffield that will alter the optimum set of routes and what will those alterations be$?$ Explain how you arrive at each one of your answers.

C$.$ The management of the company is considering the possibility of closing down one of the warehouses as this is expected to result in substantial labour and maintenance savings. Further, the manager of the Edinburgh warehouse is considering sub$-$letting some of the capacity of this warehouse. Such sub$-$lets woulc have to be in exact multiples of $1000$ tons. It is estimated that each $1000$ tons of capacity could be let for $28,000$ per week. Formulate a mixed integer linear programming model $-$ or$,$ if necessary, different model variants $-$ to examine and evaluate the alternative courses of action. What would you recommend the company to do$,$ and why? Discuss the alternatives, also considering the solution from part A and explain which additional information you might need $($if any$)$ to give the company more specific advice.

Presentation requirements: Your report should have a main body and associated appendices. Parts A and B should take no more than one page each within the main body. Part $C$ can take up to two pages. You can use single spaced pages if you wish. Font size is $12.$ The appendices for each of parts A and $C$ can have up to two pages in addition to any LP software output. Any appendix for B need not exceed one page in total, in additi to any LP software output. LP software to use for this assignment must be LINDO.