J E Beasley

OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following conditions.

A full list of the topics available in OR-Notes can be found here.

Linear programming tutorial class question

A company has three plants 1, 2 and 3 producing product X. This product is a powder formed by a process of mixing raw material ingredients in carefully controlled proportions. After the mixing process the then constituted product is ready for packing into one kg cartons. The latter operation is normally carried out at the plant of origin but may, if necessary, be carried out at one of the other plants if "in house" capacity is insufficient to meet demand. The distribution of the product to customers is always made from the plant where the mixing takes place.

The raw materials for product X cost £300 per thousand kg of product X produced. The cost per thousand kg to ship the product made at one plant for packing into cartons at another plant (and to return it to the original plant) are as follows:

              Packing plant 
              1   2   3
From plant 1  -   20  23
           2  25  -   24
           3  30  27  -

The manufacturing costs in £'s per thousand kgs of the mixing and packing processes at each of the three plants are as follows:

                Plant cost
                1   2   3
Mixing process  100 120 150
Packing process 60  30  40

The capacities of the mixing and packing process at each of the three plants in the next month are:

                Plant capacity (1,000's of kgs)
                1   2   3
Mixing process  60  80  40
Packing process 70  50  50

To ensure that a plant is not given over entirely to packing product from other plants the maximum amount of packing of product from other plants that can take place at plants 1, 2 and 3 is 5, 10 and 15 (thousand kgs) respectively.

The company can currently sell as much of product X as it can make. Formulate the problem of determining the mixing and packing arrangements for the three plants as an LP so as to minimise costs.