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Math Help - Figure out how to maximize profit by solving linear programming problems graphically

  1. #1
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    Figure out how to maximize profit by solving linear programming problems graphically

    I'm new to this site so I'm hoping this is the right area to post this. I'm dealing with a finite mathematics problem. Okay so my professor made up a bunch of problems only thing is they are very different from the ones in our book so I'm not as sure how to go about them. The problem reads:

    • A small country grows only 2 crops for export:coffee and cocoa. The country has 500,000 hectares of land available for the crops. Long term contracts require that at least 100,000 hectares of land be devoted to coffee and at least 200,000 hectares to cocoa. Cocoa must be processed locally and production conditions limit cocoa to 270,000 hectares. Coffee requires 2 workers per hectare and cocoa requires 5 workers per hectare. No more than 1,750,000 people are available for working these crops. Coffee produces a profit of $220 per hectare and cocoa a profit of $550 per hectare. How many hectares should the country devote to each crop in order to maximize profit?
    I need to be able to:
    1. clearly define the two variables
    2. determine the system of constraints that models the situation.
    3. determine the objective function for the situation
    4. Graph and shade the feasible solutions region (which i think i can handle if i had the right system of constraints)
    5. Evaluate the objective function for each of the corner points. (have serious problems with)
    6. Determine the values of the variables that will optimize the objective function.
    Can anyone help me figure this out without just giving me the answer , kinda like help me out step by step?Thanks.
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  2. #2
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    Quote Originally Posted by ChristinaC20 View Post
    I'm new to this site so I'm hoping this is the right area to post this. I'm dealing with a finite mathematics problem. Okay so my professor made up a bunch of problems only thing is they are very different from the ones in our book so I'm not as sure how to go about them. The problem reads:

    • A small country grows only 2 crops for export:coffee and cocoa. The country has 500,000 hectares of land available for the crops. Long term contracts require that at least 100,000 hectares of land be devoted to coffee and at least 200,000 hectares to cocoa. Cocoa must be processed locally and production conditions limit cocoa to 270,000 hectares. Coffee requires 2 workers per hectare and cocoa requires 5 workers per hectare. No more than 1,750,000 people are available for working these crops. Coffee produces a profit of $220 per hectare and cocoa a profit of $550 per hectare. How many hectares should the country devote to each crop in order to maximize profit?
    I need to be able to:
    1. clearly define the two variables
    1. It's hard to give a "hint" for this- just do what it says. Since the question asked was "How many hectares should the country devote to each crop in order to maximize profit?", the two variables are x= "number of hectares devoted to coffee", y= "number of hectares devoted to cocoa".

    2. determine the system of constraints that models the situation.
    Write things like "The country has 500,000 hectares of land available for the crops. Long term contracts require that at least 100,000 hectares of land be devoted to coffee and at least 200,000 hectares to cocoa. Cocoa must be processed locally and production conditions limit cocoa to 270,000 hectares." as inequalities. For example, "The country has 500,000 hectares of land available for the crops" means that, if x is the amount used for coffee and y the amount used for cocoa, then x+ y is the total hectares of land used for both: x+ y\le 500,000. The boundary of that is the straight line x+ y= 500,000. Since x= 0 y= 0 certainly satisfies x+ y< 500,000, the region satisfying x+ y< 500,000 is the side of that line containing (0,0).

  3. determine the objective function for the situation
Since the problem is to maximize profit, the "object function" is the profit. If x hectares are devoted to coffee and y hectares are devoted to cocoa, how much profit is made? Of course, you get that from "Coffee produces a profit of $220 per hectare and cocoa a profit of $550 per hectare."

  • Graph and shade the feasible solutions region (which i think i can handle if i had the right system of constraints)
  • Yes, each constraint will graph as one side of a straight line. You might do that by graphing the straight line, then shading the approriate side. If you use different "shading" (vertical lines, horizontal lines, diagonal lines, etc.) the "feasible region" will be the region with all the different shadings.

  • Evaluate the objective function for each of the corner points. (have serious problems with)
  • Finding the corner points involves solving the two equations representing the lines that intersect at that corner.

    [quote]
  • Determine the values of the variables that will optimize the objective function.
  • [quote]
    Once you have done the previous part, see which corner point gives the largest value. The (x, y) values for that point are the values of the variables you want.

    Can anyone help me figure this out without just giving me the answer , kinda like help me out step by step?Thanks.
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  • #3
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    problem with objective function i think

    thanks. I think my problem is still in the constraints and objective function. When I set it up I use A=Coffee, B=cocoa:

    Contstraints
    A+B=500,000
    A greater then or equal to 100,000
    B greater then or equal to 200,000
    B less then or equal to 270,000
    2A+5B less then or equal to 1,750,000
    A greater than or equal to 0
    B greater than or equal to 0

    Objective function
    P=220A+550B
    (but to me this part doesnt sound right because i added in another variable P=profit)

    Any clue as to where I'm going wrong?

    Thanks
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  • #4
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    Quote Originally Posted by ChristinaC20 View Post
    I'm new to this site so I'm hoping this is the right area to post this. I'm dealing with a finite mathematics problem. Okay so my professor made up a bunch of problems only thing is they are very different from the ones in our book so I'm not as sure how to go about them. The problem reads:

    • A small country grows only 2 crops for export:coffee and cocoa. The country has 500,000 hectares of land available for the crops. Long term contracts require that at least 100,000 hectares of land be devoted to coffee and at least 200,000 hectares to cocoa. Cocoa must be processed locally and production conditions limit cocoa to 270,000 hectares. Coffee requires 2 workers per hectare and cocoa requires 5 workers per hectare. No more than 1,750,000 people are available for working these crops. Coffee produces a profit of $220 per hectare and cocoa a profit of $550 per hectare. How many hectares should the country devote to each crop in order to maximize profit?
    I need to be able to:
    1. clearly define the two variables
    2. determine the system of constraints that models the situation.
    3. determine the objective function for the situation
    4. Graph and shade the feasible solutions region (which i think i can handle if i had the right system of constraints)
    5. Evaluate the objective function for each of the corner points. (have serious problems with)
    6. Determine the values of the variables that will optimize the objective function.
    Can anyone help me figure this out without just giving me the answer , kinda like help me out step by step?Thanks.
    1. Let x denote the number of hectares of coffee;
    let y denote the number of hectares of cocoa

    then you know:

    x\geq 100,000
    y\geq200,000~\wedge~y\leq270,000
    x+y\leq500,000~\implies~y\leq -x+500,000
    2x+5y\leq1,750,000~\implies~y \leq -\dfrac25 x+350,000

    2. Draw a sketch of these 5 lines and shade the region of the feasible solutions.

    3. The profit P is calculated by:

    P=220x + 550y~\implies~y=-\dfrac25 x + \underbrace{\dfrac P{550}}_{y-intercept}

    4. Determine the coordinates of the 5 vertices. The objective function passes through the vertices (or any point of the feasible region). Choose the vertex such that the y-intercept of the objective function is the largest and consequently that P is the largest.

    5. Use the maximum y-intercept to calculate the maximum of P.

    EDIT: Obviously I'm a little bit late with my answer. I'll post it nevertheless to show you how I have done the last part of the question.
    Attached Thumbnails Attached Thumbnails Figure out how to maximize profit by solving linear programming problems graphically-kaffkakao.png  
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  • #5
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    Quote Originally Posted by earboth View Post

    EDIT: Obviously I'm a little bit late with my answer. I'll post it nevertheless to show you how I have done the last part of the question.
    Thanks a bunch....my graph was to small so i was having a debate wether or not there were 5 corner points or not because the 1 was so close and your graph proved me right so thanks alot
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