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Question

A firm produces three products. These products are processed on three different machines. The time required to manufacture one unit of each of the three products and the daily capacity of the three machines are given in the table below:

It is required to determine the daily number of units to be manufactured for each product. The profit per unit for product 1,2 and 3 is Rs.4, Rs.3 and Rs.6 respectively. It is assumed that all the amounts produced are consumed in the market. Formulate the mathematical(L.P) model that will maximize the daily profit.

Machine | Time per unit (in minutes) of product-1 | Time per unit (in minutes) of product-2 | Time per unit (in minutes) of product-3 | Machine capacity (minutes/day) |

M1 | 2 | 3 | 2 | 440 |

M2 | 4 | - | 3 | 470 |

M2 | 2 | 5 | - | 430 |

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Solution

From the

Assume symbols for variable quantities noticed in step-1.Let the number of units of products,1,2, and 3 manufactured daily be x1,x2 and x3.

where x1,x2,x3≥0.

since negative production has no meaning and is not feasible.

i.e.,maximize Z=4x1+3x2+6x3

Here,constraints are on the machine capacities and can be mathematically expressed as

2x1+3x2+2x3≤440

4x1+0.x2+3x3≤470

2x1+5x2+0.x3≤430

∴, the complete mathematical(L.P) model for the problem can be written as

Maximize Z=4x1+3x2+6x3

subject to constraints, 2x1+3x2+2x3≤440

4x1+3x3≤470

2x1+5x2≤430

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