Linear Programming

Paint Transhipment Problem

A company has two factories, one each at Bristol and Leeds. The factories produce paints which are
sold to five 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 London, Birmingham and Glasgow. Table 1 shows the
transportation costs per ton for deliveries from the suppliers 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.

Warehouse Wholesaler
Supplier London Birmingham Glasgow 1 2 3 4 5
Bristol 25 23 – 80 – 90 100 86
Leeds 30 27 30 – 70 54 – 100
London – – – 37 31 – 40 44
Birmingham – – – 36 40 43 40 46
Glasgow – – – 45 42 30 – 36

TABLE 1: TRANSPORTATION COSTS IN £’S/TON DELIVERED
The two factories at Bristol and Leeds can produce up to 40,000 and 50,000 tons per week
respectively. No more than 20,000 15,000 and 12,000 tons can be moved each week through the
warehouse in London, Birmingham and Glasgow, respectively. Wholesalers 1, 2, 3, 4 and 5 require at
least 15,000, 20,000, 13,000, 14,000 and 16,000 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.
Explain clearly the variables you use and the constraints you construct. What is the minimum
cost transportation schedule and what are the corresponding costs?

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
at Glasgow 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 Birmingham warehouse is considering sub–letting some of the capacity of
this warehouse. Such sub–lets would have to be in exact multiples of 1000 tons. It is estimated
that each 1000 tons of capacity could be let for £21,000 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 taking into account the solution from part (a) and explain which
additional information you might need (if any) to give the company more specific advice