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Math Help - Generating a Linear Programming Model for Weenies and Buns

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    Post Generating a Linear Programming Model for Weenies and Buns

    1. Weenies and Buns is a food processing plant which manufacturer’s hot dogs and hot dog buns. They grind their own flour for the hot dog buns at a maximum rate of 200 pounds per week. Each hot dog bun requires 0.1 pound of flour. They currently have a contract with Pig land, Inc. which specifies that a delivery of 800 pounds of pork product is delivered every Monday. Each hot dog requires ¼ pound of pork product. All the other ingredients in the hot dogs and hot dog buns are in plentiful supply. Finally the labour force at Weenies and Buns consists of 5 employees working full time (40 hours per week each). Each hot dog requires 3 minutes of labour, and each hot dog bun requires 2 minutes of labour. Each hot dog yields a profit of $0.20, and each bun yields a profit of $0.10.
    a) Use the graphical solution to determine the number of hot dogs and hot dog buns to produce each week that would maximize profit.
    b) Determine the optimality range of the ratio of hot dog to hot dog buns that will keep the solution unchanged.
    c) Determine the worth per unit change in the availability of the pork product and its range of applicability.
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    Quote Originally Posted by kellyplatinum View Post
    1. Weenies and Buns is a food processing plant which manufacturer’s hot dogs and hot dog buns. They grind their own flour for the hot dog buns at a maximum rate of 200 pounds per week. Each hot dog bun requires 0.1 pound of flour. They currently have a contract with Pig land, Inc. which specifies that a delivery of 800 pounds of pork product is delivered every Monday. Each hot dog requires ¼ pound of pork product. All the other ingredients in the hot dogs and hot dog buns are in plentiful supply. Finally the labour force at Weenies and Buns consists of 5 employees working full time (40 hours per week each). Each hot dog requires 3 minutes of labour, and each hot dog bun requires 2 minutes of labour. Each hot dog yields a profit of $0.20, and each bun yields a profit of $0.10.
    a) Use the graphical solution to determine the number of hot dogs and hot dog buns to produce each week that would maximize profit.
    b) Determine the optimality range of the ratio of hot dog to hot dog buns that will keep the solution unchanged.
    c) Determine the worth per unit change in the availability of the pork product and its range of applicability.
    hey, another Jamaican!
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    hi do u know the answer to the question, i'm not sure about some of the constraints
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by kellyplatinum View Post
    hi do u know the answer to the question, i'm not sure about some of the constraints
    Nah, i've never done Linear Programming, at least not in this capacity. you're better off waiting for CaptainBlack or someone else who's good at this
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    ok thanks
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    Quote Originally Posted by kellyplatinum View Post
    1. Weenies and Buns is a food processing plant which manufacturer’s hot dogs and hot dog buns. They grind their own flour for the hot dog buns at a maximum rate of 200 pounds per week. Each hot dog bun requires 0.1 pound of flour. They currently have a contract with Pig land, Inc. which specifies that a delivery of 800 pounds of pork product is delivered every Monday. Each hot dog requires ¼ pound of pork product. All the other ingredients in the hot dogs and hot dog buns are in plentiful supply. Finally the labour force at Weenies and Buns consists of 5 employees working full time (40 hours per week each). Each hot dog requires 3 minutes of labour, and each hot dog bun requires 2 minutes of labour. Each hot dog yields a profit of $0.20, and each bun yields a profit of $0.10.
    a) Use the graphical solution to determine the number of hot dogs and hot dog buns to produce each week that would maximize profit.
    b) Determine the optimality range of the ratio of hot dog to hot dog buns that will keep the solution unchanged.
    c) Determine the worth per unit change in the availability of the pork product and its range of applicability.
    Decision variables:
    B ----number of buns to be produced, per week.
    D ----number of hot dogs to be produced, per week.

    The constraints:

    They grind their own flour for the hot dog buns at a maximum rate of 200 pounds per week. Each hot dog bun requires 0.1 pound of flour.
    0.1B <= 200
    B <= 2000 -------------(1)

    They currently have a contract with Pig land, Inc. which specifies that a delivery of 800 pounds of pork product is delivered every Monday. Each hot dog requires ¼ pound of pork product.
    (1/4)W <= 800
    D <= 3200 ----------------(2)

    Finally the labour force at Weenies and Buns consists of 5 employees working full time (40 hours per week each). Each hot dog requires 3 minutes of labour, and each hot dog bun requires 2 minutes of labour.

    5 employees at 40 each = 200 hrs available = 200*60 = 12,000 minutes avalable time for labor each week.

    2B +3D <= 12,000 ----------(3)

    And the two non-negative constraits,
    B >= 0 -------------------(4)
    D >= 0 -----------------(5)

    Using, say, a rectangular axes, where B is horizontal and D is vertical, so the ordered pairs or points are in (B,D):
    Graph those 5 inequalities, determine their intersections, determine the feasible region.

    Feasible region is a pentagon whose vertices are:
    (0,0) -----------intersection of (4) and (5).
    (0,3200) -------intersection of (5) and (2).
    (1200,3200) ----intersection of (2) and (3).
    (2000,2666) ----intersection of (3) and (1).
    (2000,0) ----intersection of (1) and (4).

    ------------------------------------------------------

    a) Use the graphicalp solution to maximize profit.

    Each hot dog yields a profit of $0.20, and each bun yields a profit of $0.10.

    Profit, P = 0.1B +0.2D -------------***
    Test that on the 5 vertices.
    Max. P is at vertex (1200,3200), therefore, for maximum profit, 1200 buns and 3200 hot dogs must be produced. --------answer.

    b) Determine the optimality range of the ratio of hot dog to hot dog buns that will keep the solution unchanged.

    If I understand what that means,
    The optimum range is that side of the feasible region defined by vertices (1200,3200) and (2000,2666).
    At (1200,3200), the ratio of D over B is 3200/1200 = 8/3.
    At (2000,2666), it is 2666/2000 = 1.333 = 4/3
    So, the optimality range of the ratio of hot dogs to buns is from 4/3 up to 8/3.

    c) Determine the worth per unit change in the availability of the pork product and its range of applicability.

    If I understand that again,

    At (1200,3200), the pork product needed is (1/4)(3200) = 800 lbs, which is the weekly delivery.
    At (2000,2666), the pork product needed is (1/4)(2666) = 666.5 lbs only, so there is 800 -666.5 = 133.5 lbs not being used per week.

    Now what?
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