$z=8 x_1+13 x_2+23 x_3$
(Do not include the $ symbol in your answers.)
To maximize profit, the company should produce 0 racing bike(s), 0 touring bike(s), and 1400 mountain bike(s). (Simplify your answers.)
(b) What is the maximum possible profit?
The maximum profit is $32200
(Simplify your answer.)
(c) Does it require all of the available units of steel and aluminum to build the bicycles that produce the maximum profit? If not, how much of each material is left over? Compare any leftover to the value of the relevant slack variable.
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. No. sinbe $s _1=\square$ and $s _2=\square$ in the optimal solution, there is/are
$\square$
$\square$
$\square$ of aluminum, respectively, left over. unit(s) of steel and $\square$ unit(s) (Simplify your answers.)
B. Yes. All available units of steel and aluminum are required to build the optimal group of bicycles.
Answer (1)
Calculate steel usage with given optimal production: 5 ( 0 ) + 3 ( 0 ) + 8 ( 1400 ) = 11200 .
Calculate aluminum usage with given optimal production: 2 ( 0 ) + 6 ( 0 ) + 4 ( 1400 ) = 5600 .
Calculate leftover steel: 12000 − 11200 = 800 units, so s 1 = 800 .
Calculate leftover aluminum: 9600 − 5600 = 4000 units, so s 2 = 4000 . The answer is A. No. since s 1 = 800 and s 2 = 4000 in the optimal solution, there is/are 800 unit(s) of steel and 4000 unit(s) of aluminum, respectively, left over.
A
Explanation
Understanding the Problem We are given the optimal production plan: 0 racing bikes, 0 touring bikes, and 1400 mountain bikes. We also know the maximum profit is $32200. The question asks whether this production plan requires all available steel and aluminum. If not, we need to determine the leftover amounts and compare them to the slack variables.
Stating the Constraints To determine the leftover steel and aluminum, we need the constraints for steel and aluminum. Let's assume the constraints are:
Steel: 5 x 1 + 3 x 2 + 8 x 3 ≤ 12000 Aluminum: 2 x 1 + 6 x 2 + 4 x 3 ≤ 9600
where x 1 is the number of racing bikes, x 2 is the number of touring bikes, and x 3 is the number of mountain bikes.
Calculating Steel Usage Now, we substitute the optimal production values x 1 = 0 , x 2 = 0 , and x 3 = 1400 into the steel constraint:
5 ( 0 ) + 3 ( 0 ) + 8 ( 1400 ) = 0 + 0 + 11200 = 11200
Since 11200 ≤ 12000 , the steel constraint is satisfied. The amount of steel used is 11200 units.
Calculating Aluminum Usage Next, we substitute the optimal production values into the aluminum constraint:
2 ( 0 ) + 6 ( 0 ) + 4 ( 1400 ) = 0 + 0 + 5600 = 5600
Since 5600 ≤ 9600 , the aluminum constraint is satisfied. The amount of aluminum used is 5600 units.
Calculating Leftover Materials Now we calculate the leftover steel and aluminum:
Leftover Steel = Available Steel - Steel Used = 12000 − 11200 = 800 units Leftover Aluminum = Available Aluminum - Aluminum Used = 9600 − 5600 = 4000 units
Determining Slack Variables The slack variables s 1 and s 2 represent the unused amounts of steel and aluminum, respectively. Therefore, s 1 = 800 and s 2 = 4000 .
Final Answer Since there are leftover units of steel and aluminum, we choose option A.
Final Answer No. since s 1 = 800 and s 2 = 4000 in the optimal solution, there are 800 units of steel and 4000 units of aluminum, respectively, left over.
Examples
Linear programming helps companies optimize resource allocation to maximize profit. For example, a furniture company can use linear programming to determine the optimal number of tables, chairs, and sofas to produce, given constraints on materials like wood, fabric, and labor hours. By setting up the problem with an objective function (profit) and constraints (resource availability), the company can find the production mix that maximizes its profit while staying within its resource limits. This ensures efficient use of resources and maximizes profitability.
