Linear Programming

Problem 1

A company produces two types of dolls (D1 and D2). The following table gives the selling price, the labor time, machine time and raw material required for the production of each type of doll, D1 and D2 respectively. Each week, up to 450 kg of raw material can be purchased. The company employs four workers, who regularly work 40 hours per week. Each week 400 hours of machine time are available.

| D1 | D2 |

Selling price | € 12 | € 8 |

Labor required | 0.7 hour | 0.5 hour |

Machine time required | 1.5 hours | 0.8 hour |

Raw Material required | 2 kg | 1 kg |

At most 50 dolls D1 and 60 toys D2 are expected to be demanded each week.

The production manager is interested in developing the optimal production plan that maximizes the total profit.

The problem can be formulated as the following linear program.

Let:

Χ1 = the number of dolls D1 to be produced

Χ2 = the number of dolls D2 to be produced

The objective function is:

Ζ=12Χ1+8Χ2

Subject to the constraints:

0.7Χ1 + 0.5Χ2 160 (1)

1.5Χ1 + 0.8Χ2 400 (2)

2Χ1 + Χ2 450 (3)

Χ1 50 (4)

Χ2 60 (5)

Χ1, Χ2 0

Problem 2

A firm produces three different types of toys: T1, T2, and T3. For the production three different kinds of raw materials are required (plastic, metal and material). The cost per unit and the available quantities of raw materials, the selling price for each type of toy are required in the next table.

Type ofToy | Required Content in | Price ($) |

| Plastic | Metal | Material | |

T1 | 1 | 2 | 3 | 20 |

T2 | 1.5 | 1.7 | 1.4 | 17 |

T3 | 1.8 | 1.6 | 2 | 15 |

Cost ($) | 5 | 6 | 4.5 | |

Available Quantities | 400 | 200 | 250 | |

The firm wishes to determine the optimum production level of each toy that maximizes its total profit.

The problem can be formulated as the following linear program.

Let:

Χ1 = the number of toys type T1 to be produced

Χ2 = the number of tools type T2 to be produced

Χ3 = the number of tools type T3 to be produced...