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21 December, 16:07

A furniture company produces three types of couches. The first type uses 1 foot of framing wood and 3 feet of cabinet wood. The second type uses 2 feet of framing wood and 2 feet of cabinet wood. The third type uses 2 feet of framing wood and 1 foot of cabinet wood. The profit of the three types of couches are $10, $8, and $5, respectively. The factory produces 500 couches each month of the first type, 300 of the second type, and 200 of the third type. However, this month there is a shortage of cabinet wood to only 600 feet, but the supply of framing wood is increased by 100 feet. How should the production of the three types of couches be adjusted to minimize the decrease in profit? Formulate this problem as a linear programming problem.

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  1. 21 December, 16:18
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    Objective Function:

    z = 10*x₁ + 8*x₂ + 5*x₃ To maximize

    Constraints:

    x₁ + 2x₂ + 2x₃ ≤ 1600

    3x₁ + 2x₂ + x₃ ≤ 1700

    x₁; x₂; x₃ ≥ 0

    Step-by-step explanation:

    The company produces three types of couches

    Type Framing wood (ft) Cabinet wood (ft) Profit $ Previous Pro

    1 (x₁) 1 3 10 500

    2 (x₂) 2 2 8 300

    3 (x₃) 2 1 5 200

    Objective Function:

    z = 10*x₁ + 8*x₂ + 5*x₃ To maximize

    First Constraint

    Framing wood: If factory could produce 1*500 + 2*300 + 2*200 = 1500 ft

    and this quantity is increased by 100 that means factory wil have 1600 ft of framing wood available, then

    (1) x₁ + 2x₂ + 2x₃ ≤ 1600

    Second constraint

    Cabinet wood:

    If factory produced 3*500 + 2*300 + 1*200 = 2300 ft and factory has a shortage 600 ft then only 1700 ft of cabinet wood will be available that means

    3x₁ + 2x₂ + x₃ ≤ 1700

    And x₁; x₂; x₃ ≥ 0
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