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Thread: Applications of Linear Programming

  1. #1
    Member classicstrings's Avatar
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    Applications of Linear Programming

    I'm having trouble setting up the variables and constraints in these questions. Wondering if someone can help? Thanks!

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  2. #2
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    Hello, classicstrings!

    Here's the second one . . .


    A furniture maker produces cupboards and bookshelves.
    Both products use wooden planks, sawing time, sanding time, and assembly time.
    The cupboards use 20 m of wood, 40 minutes of sawing, 60 minutes of sanding,
    and 10 minute of assembly time for a profit of $280.

    The bookshelves use 10 m of wood, 30 minutes of sawing, 90 minutes of sanding,
    and 30 minutes of assembly time for a profit of $340.

    There are 220 m of wood, 480 minutes of sawing time, 1080 minutes of sanding time,
    and 330 minutes of assembly time available.

    How many of each item should be made for maximum profit?

    Let $\displaystyle x$ = number of cupboards, $\displaystyle y$ = number of bookshelves. .$\displaystyle x \geq 0,\;y \geq 0$ [1]
    A chart helps to organize the information . . .
    Code:
                    | wood + saw | sand | assem |
    - - - - - - - - + - - -+ - - + - - -+ - - - +
      cupboards (x) |  20x | 40x |  60x |  10x  |
    - - - - - - - - + - -  + - - + - - -+ - - - +
    bookshelves (y) |  10y | 30y |  90y |  30y  |
    - - - - - - - - + - -  + - - + - - -+ - - - +
      available     |  220 | 480 | 1080 |  330  |
    - - - - - - - - + - - -+ - - + - - -+ - - - +

    Wood: .$\displaystyle 20x + 10y \:\leq \:220\quad\Rightarrow\quad 2x + y \:\leq \:22$ [2]

    Sawing: .$\displaystyle 40x + 30y \:\leq \:480\quad\Rightarrow\quad 4x + 3y \:\leq \:48$ [3]

    Sanding: .$\displaystyle 60x + 90y \:\leq \:1080\quad\Rightarrow\quad 2x + 3y \:\leq \:36$ [4]

    Assembly: .$\displaystyle 10x + 30y \:\leq \:330\quad\Rightarrow\quad x + 3y\:\leq\:33$ [5]


    [1] places us in Quadrant 1.

    [2] Graph the line: $\displaystyle 2x + y \:=\:22$. .It has intercepts: $\displaystyle (11,0),\;(0,22)$. .
    . . .Shade the region below the line.

    [3] Graph the line: $\displaystyle 4x + 3y \:=\:48$. .It has intercepts: $\displaystyle (12,0),\;(0,16)$
    . . .Shade the region below the line.

    [4] Graph the line: $\displaystyle 2x + 3y \:=\:36$. .It has intercepts: $\displaystyle (18,0),\;(0,12)$
    . . .Shade the region below the line.

    [5] Graph the line: $\displaystyle x + 3y \:=\:33$. .It has intercepts: $\displaystyle (33,0),\;(0,11)$
    . . .Shade the region below the line.

    The final region is a hexagon. .Its vertices are (clockwise from the origin):
    . . $\displaystyle (0,0),\;(0,11),\;(3,10),\6,8),\;(9,4),\;(0,11)$

    Test them in the profit function: $\displaystyle P \:= \:280x + 340y$
    . . to see which one produces maximum profit.

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  3. #3
    Member classicstrings's Avatar
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    Hey Soroban! You have done the harder one for me, and I have gone through it a couple of times myself after, I have done the first one by looking @ how you did them. Cheers!
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