635 words - 3 pages

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.

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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

The objective function is:

Ζ =15Χ1+11Χ2+10.5Χ3

Subject to the constraints:

Χ1+1.5Χ2 + 1.8Χ3 400 (1)

2Χ1 + 1.7Χ2 + 1.6Χ3 200 (2)

3Χ1 + 1.4Χ2 + 2Χ3 250 (3)

Χ1, Χ2, Χ3 0

Problem 3

A company produces three types of tools (A, B, C). The resources used to produce each type of tool are shown in the following table.

| Tools |

| A | B | C |

Iron (kg) | 9 | 8 | 7.5 |

Machine time (minutes) | 50 | 110 | 90 |

Labor time (minutes) | 60 | 40 | 55 |

Profit (€) | 12 | 15 | 13.5 |

The company works on a weekly schedule of 5 days, with 2 shifts of 7.5 hours each. It has 4 machines available for production and 15 employees on each shift.

The marketing department requires that at least 300 tools of all types be produced each week.

The company’s capacity is 3 tons of iron.

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

The problem can be formulated as the following linear program.

Χ1 = the number of tools type A to be produced

Χ2 = the number of tools type B to be produced

Χ3 = the number of tools type C to be produced

The objective function is:

Ζ=12Χ1+15Χ2+113.5Χ3

Subject to the constraints:

9Χ1 + 8Χ2 + 7.5Χ3 3000 (1)

50Χ1 + 110Χ2 + 90Χ3 18000 (2)

60Χ1 + 40Χ2 + 55Χ3 135000 (3)

Χ1 + Χ2 + Χ3 300 (4)

Χ1, Χ2, Χ3 0

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