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.

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