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Math Help - Maximize Profit

  1. #1
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    Maximize Profit

    Product mix. Better Products, Inc. is a small manufacturer of three products it produces on two machines. In a typical week, 40 hours of time are available on each machine. Profit contribution and production time in hours per unit are given in the following table:

    Product
    1 2 3
    Profit/unit $30 $50 $20
    Machine 1 time/unit 0.5 2.0 0.75
    Machine 2 time/unit 1.0 1.0 0.5

    Two operators are required for machine 1. Thus, 2 hours of labor must be scheduled for each hour of machine 1 time. Only one operator is required for machine 2. A maximum of 100 labor-hours is available for assignment to the machines during the coming week. Other production requirements are that product 1 cannot account for more than 50 percent of the units produced and that product 3 must account for at least 20 percent of the units produced.

    a. How many units of each product should be produced to maximize the profit contribution?

    What is the projected weekly profit associated with your solution?
    b. How many hours of production time will be scheduled on each machine?
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  2. #2
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    Quote Originally Posted by totally-confused View Post
    Product mix. Better Products, Inc. is a small manufacturer of three products it produces on two machines. In a typical week, 40 hours of time are available on each machine. Profit contribution and production time in hours per unit are given in the following table:

    Product
    1 2 3
    Profit/unit $30 $50 $20
    Machine 1 time/unit 0.5 2.0 0.75
    Machine 2 time/unit 1.0 1.0 0.5

    Two operators are required for machine 1. Thus, 2 hours of labor must be scheduled for each hour of machine 1 time. Only one operator is required for machine 2. A maximum of 100 labor-hours is available for assignment to the machines during the coming week. Other production requirements are that product 1 cannot account for more than 50 percent of the units produced and that product 3 must account for at least 20 percent of the units produced.

    a. How many units of each product should be produced to maximize the profit contribution?

    What is the projected weekly profit associated with your solution?
    b. How many hours of production time will be scheduled on each machine?
    Hello,

    let x be the number of product #1
    let y be the number of product #2
    let z be the number of product #3

    then you get the following inequalities:
    x ≥ 0
    y ≥ 0
    z ≥ 0

    x ≤ y + z product 1 cannot account for more than 50 percent of the units produced

    2(1/2x+2y+3/4z) + (x+y+1/2z) ≤ 100
    2x + 5y + 2z ≤ 100 A maximum of 100 labor-hours is available

    x + y ≤ 4z product 3 must account for at least 20 percent of the units produced

    These 3 inequalities describe half spaces(?). The intersection of the planes which border thes half spaces is the point
    Code:
    ⎡     500         300       200 ⎤
    ⎢x = ⎯⎯⎯ ∧ y = ⎯⎯⎯ ∧ z = ⎯⎯⎯⎥
    ⎣      29          29        29 ⎦
    This point describes the maximum for all 3 variables

    The profit function is:

    P = 30x + 50y + 20z

    Plug in the coordinates of the common point of all 3 planes and evaluate:
    Code:
        500       300       200 
    30⎯⎯⎯ + 50⎯⎯⎯ + 20⎯⎯⎯
         29        29        29
    I've attached a diagram of the 3 conditional planes

    EB

    PS: It's the first time that I have done such a problem, so please check my calculations.
    Attached Thumbnails Attached Thumbnails Maximize Profit-lin_opt3dimens.gif  
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  3. #3
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    Hello,

    if I understand your problem correctly then the values of x, y and z have to be integers. Therefore:

    x = 500/29 --> x = 17
    y = 300/29 --> y = 10
    z = 200/29 --> z = 6

    Now you have a problem because the x-value is more than 50% of all products.
    So you should check which of the triples
    (16, 10, 6); (16, 11, 6); (16, 10, 7); (16, 11, 7) satisfies all conditions.

    EB
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