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.
(a) How many of each type of bicycle should be made in order to maximize profit if the company makes $8 per racing bike, $13 per touring bike, and $23 per mountain bike?
Let [tex]$x_1$[/tex] be the number of racing bikes, let [tex]$x_2$[/tex] be the number touring bikes, and let [tex]$x_3$[/tex] be the number of mountain bikes. What is the objective function?
[tex]$z=8 x_1+13 x_2+23 x_3$[/tex]
(Do not include the $ symbol in your answers.)
To maximize profit, the company should produce (Simplify your answers.)
$\square$ racing bike(s), $\square$ touring bike(s), and $\square$ mountain bike(s).
Answer (1)
Define the objective function: Maximize z = 8 x 1 + 13 x 2 + 23 x 3 .
Establish the constraints: 11 x 1 + 14 x 2 + 22 x 3 ≤ 30800 (steel), 15 x 1 + 24 x 2 + 18 x 3 ≤ 27000 (aluminum), and x 1 , x 2 , x 3 ≥ 0 .
Solve the linear programming problem to find the optimal values for x 1 , x 2 , and x 3 .
The solution is to produce 0 racing bikes, 0 touring bikes, and 1400 mountain bikes.
Explanation
Problem Setup We are given a linear programming problem where we want to maximize the profit from selling racing bikes ($8 each), touring bikes ($13 each), and mountain bikes ( 23 e a c h ) , s u bj ec tt oco n s t r ain t so n t h e a v ai l ab l es t ee l ( 30 , 800 u ni t s ) an d a l u min u m ( 27 , 000 u ni t s ) . L e t x_1 , x_2 , an d x_3$ represent the number of racing, touring, and mountain bikes, respectively.
Objective Function The objective function to maximize is the total profit, which is given by: z = 8 x 1 + 13 x 2 + 23 x 3
Constraints The constraints are given by the amount of steel and aluminum available: Steel constraint: 11 x 1 + 14 x 2 + 22 x 3 ≤ 30800 Aluminum constraint: 15 x 1 + 24 x 2 + 18 x 3 ≤ 27000 Also, we have non-negativity constraints: x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0
Solving the problem Using linear programming, the optimal solution is found to be: x 1 = 0 x 2 = 0 x 3 = 1400
Interpretation of the solution This means the company should produce 0 racing bikes, 0 touring bikes, and 1400 mountain bikes to maximize profit.
Examples
Linear programming is used in various real-world scenarios, such as optimizing production plans, managing supply chains, and determining the most efficient allocation of resources. For example, a delivery company can use linear programming to determine the optimal routes for its trucks to minimize fuel consumption and delivery time. Similarly, a farmer can use it to decide how much of each crop to plant to maximize profit, given constraints on land, water, and fertilizer.
