Question

The Midland Tool Shop has four heavy presses it uses to stamp out prefabricated metal covers and housings for electric consumer products. All four presses operate differently and are of different sizes. Currently the firm has a contract to produce three products. The contract calls for 400 units of product A, 570 units of product B, and 320 units of product C. The time (in minutes) required for each product to be produced on each machine is as follows:

Machine

Product | 1 | 2 | 3 | 4 |

A | 35 | 41 | 34 | 39 |

B | 40 | 36 | 32 | 43 |

C | 38 | 37 | 33 | 40 |

Machine 1 is available for 150 hours, machine 2 for 240 hours, machine 3 for 200 hours, and machine 4 for 250 hours. The products also result in different profits, according to the machine they are produced on, because of time, waste, and operating cost. The profit per unit per machine for each product is summarized as follows:

Machine

Product | 1 | 2 | 3 | 4 |

A | $6.68 | $7.80 | $8.20 | $7.90 |

B | $6.70 | $8.90 | $8.20 | $6.30 |

C | $8.40 | $6.10 | $9.00 | $5.80 |

The company wants to know how many units of each product to produce on each machine to maximize profit.

a. Formulate this problem as a linear programming model.

Step 1

Given values:

Product "A" having 400 units

Product"B" having 570 units

Product "C" having 320 units

Then the time required for given machines is

Product | 1 | 2 | 3 | 4 |

A | 35 | 41 | 34 | 39 |

B | 40 | 36 | 32 | 43 |

C | 38 | 37 | 33 | 40 |

Time available for machine 1 is 150hours

For machine 2 is 240hours

Machine 3 is 200hours

Machine 4 is 250hours

Product | 1 | 2 | 3 | 4 |

A | $6.68 | $7.80 | $8.20 | $7.90 |

B | $6.70 | $8.90 | $8.20 | $6.30 |

C | $8.40 | $6.10 | $9.00 | $5.80 |

Step 2

Formulate the linear programming Model

Considering

Product | 1 | 2 | 3 | 4 |

1 | 35 | 41 | 34 | 39 |

2 | 40 | 36 | 32 | 43 |

3 | 38 | 37 | 33 | 40 |

Product avilability | 9000 | 14400 | 12000 | 15000 |

Then the maximization function is

maximize Z=

Then these are subjected to

...

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