## Formulating a Problem for the Simplex Algorithm

The simplex algorithm is used to find optimal combinations of certain quantities, subject to constraints on those quantities. Before the algorithm can be used the problem must be suitably expressed, which involves defining the constraint conditions and the objective function which is to be maximised. Each constrain, and the objective function, is a linear combination of the quantities

Suppose a factory produces three types of car A, B and C. The time, money and labour needed to produce each car is shown in the table below.

Car | Time | Money | Labour |

A | 3 | 3,000 | 6 |

B | 5 | 2,500 | 9 |

C | 4 | 2,000 | 10 |

Total Available | 100 | 90,000 | 150 |

Each car of type A produces £200 profit, each car of type B £350 and each car of type C £150

The total time needed to produce a cars of type A, b cars of type B, c cars of type C isso

The total money needed to produce a cars of type A, b cars of type B, c cars of type C is so

The total labour needed to produce a cars of type A, b cars of type B, c cars of type C is so

Also, of course,and

The produced oncars of type A,cars of type B,cars of type C isThe simplex algorithm finds valuesandthat maximises the profit subject to time, money and labour constraints.