A manufacturer of bicycles builds racing, touring, and mountain models. The bicycles are made of both steel and aluminum. The company has available 30,800 units of steel and 27,000 units of aluminum. The racing, touring, and mountain models need 11, 14, and 22 units of steel, and 15, 24, and 18 units of aluminum, respectively. Complete parts (a) through (d) below.
[tex]z=8 x_1+13 x_2+23 x_3[/tex]
To maximize profit, the company should produce 0 racing bike(s), 0 touring bike(s), and 1400 mountain bike(s).
(b) What is the maximum possible profit?
The maximum profit is $32200.
(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. Since [tex]s_1=[/tex] and [tex]s _2=[/tex] in the optimal solution, there is/are unit(s) of steel and unit(s) of aluminum, respectively, left over.
B. Yes. All available units of steel and aluminum are required to build the optimal group of bicycles.
Answer (2)
Calculate the amount of steel used: 22 × 1400 = 30800 .
Calculate the amount of aluminum used: 18 × 1400 = 25200 .
Calculate the leftover steel: 30800 − 30800 = 0 .
Calculate the leftover aluminum: 27000 − 25200 = 1800 . Therefore, the answer is A. No. Since s 1 = 0 and s 2 = 1800 in the optimal solution, there is/are 0 unit(s) of steel and 1800 unit(s) of aluminum, respectively, left over.
Explanation
Understanding the Problem We are given that the company produces 0 racing bikes, 0 touring bikes, and 1400 mountain bikes to maximize profit. We need to determine if this production plan uses all available steel and aluminum, and if not, how much of each material is left over.
Calculating Steel Usage First, let's calculate the amount of steel used in the production of 1400 mountain bikes. Each mountain bike requires 22 units of steel, so the total steel used is: 22 × 1400 = 30800 Thus, 30800 units of steel are used.
Calculating Aluminum Usage Next, let's calculate the amount of aluminum used. Each mountain bike requires 18 units of aluminum, so the total aluminum used is: 18 × 1400 = 25200 Thus, 25200 units of aluminum are used.
Calculating Leftover Steel Now, let's determine the amount of steel left over. The company has 30,800 units of steel available, and they used 30,800 units. Therefore, the amount of steel left over is: 30800 − 30800 = 0 So, there are 0 units of steel left over.
Calculating Leftover Aluminum Next, let's determine the amount of aluminum left over. The company has 27,000 units of aluminum available, and they used 25,200 units. Therefore, the amount of aluminum left over is: 27000 − 25200 = 1800 So, there are 1800 units of aluminum left over.
Determining Leftover Material and Slack Variables Since there are 0 units of steel and 1800 units of aluminum left over, not all available units of steel and aluminum are required to build the optimal group of bicycles. The slack variables s 1 and s 2 represent the unused amounts of steel and aluminum, respectively. Therefore, s 1 = 0 and s 2 = 1800 .
Final Answer Therefore, the correct choice is A. No. Since s 1 = 0 and s 2 = 1800 in the optimal solution, there is/are 0 unit(s) of steel and 1800 unit(s) of aluminum, respectively, left over.
Examples
Linear programming helps companies optimize their resource allocation to maximize profits. 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, labor, and demand. By solving the linear program, the company can identify the production plan that maximizes its profit while satisfying all constraints. This ensures efficient use of resources and maximizes profitability.
The company uses 30,800 units of steel and 25,200 units of aluminum to produce 1400 mountain bikes, leaving 0 units of steel and 1,800 units of aluminum leftover. Thus, the company does not require all of the available materials. The correct answer is A. No. Since s_1 = 0 and s_2 = 1800 in the optimal solution, there are 0 units of steel and 1,800 units of aluminum, respectively, left over.
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