Model 22: Lumber Company

A lumber company has three sources of wood and five markets to be supplied. The annual availability of wood at sources 1, 2, and 3 are 10, 20, and 15 million board feet, respectively. The amount that can be sold annually at markets 1, 12, 3, 4, and 5 are 7, 11, 9, 10, and 8 million board feet, respectively.

In the past the company has shipped the wood by train. However, because shipping costs have been increasing, the alternative of using ships to make some of the deliveries is being investigated. This alternative would require the company to invest in some ships. Except for these investment costs, the shipping costs in thousands of dollars per million board feet by rail and by water (when feasible) would be the following for each route:

Source Unit cost by rail
Unit cost by ship
1 2 3 4 5 1 2 3 4 5
1 6172455566313824- 35
2 69786049563643282431
3 5966636147- 33363226

The capital investment (in thousands of dollars) in ships required for each million board feet to be transported annually by ship along each route is given as follows:

Source Investment for ships
1 2 3 4 5

Considering the expected useful life of the ships and the time value of money, the equivalent uniform annual cost of these investments is one-tenth the amount given in the table. The company is able to raise only $6,750,000 to invest in ships. The object is to determine the overall shipping plan that minimizes the total equivalent uniform annual cost while meeting this investment budget and the sales demand at the markets. Formulate the linear programming model for this problem.