Question

A cargo plane can carry a maximum weight of 100 tons and a maximum volume of 60 cubic meters. There are three materials to be transported, and the cargo company may choose to carry any amount of each, up to the maximum available limits given below.

• Material 1 has density $\mathbf{2}\mathbf{}\mathit{t}\mathit{o}\mathit{n}\mathit{s}\mathbf{/}\mathit{c}\mathit{u}\mathit{b}\mathit{i}\mathit{c}$meters, maximum available amount 40 cubic meters, and revenue \(1,000 per cubic meter.

• Material 2 has density $\mathbf{1}\mathbf{}\mathit{t}\mathit{o}\mathit{n}\mathbf{/}\mathit{c}\mathit{u}\mathit{b}\mathit{i}\mathit{c}$meters,maximum available amount 30 cubic meters, and revenue \)1,200 per cubic meter.

• Material 3 has density $\mathbf{3}\mathbf{}\mathit{t}\mathit{o}\mathit{n}\mathit{s}\mathbf{/}\mathit{c}\mathit{u}\mathit{b}\mathit{i}\mathit{c}$meters, maximum available amount 20 cubic meters, and revenue $12,000 per cubic meter.

Write a linear program that optimizes revenue within the constraints.

Short answer

The aim of a linear programming is to optimize the operations according to the constraints. In our question, we need to optimize the revenue.

Step-by-step solution

Defining the variables of constraints

Let the first material given be: x₁ and corresponding revenue=1000$/m³. So total revenue for $x_1$will be=1000$x_1$.

Let the second material given be: x₂ and corresponding revenue=1200$/m³. So total revenue for $x_2$will be=1200$x_2$.

Let the third material given be: x₃ and corresponding revenue=12000$/m³. So total revenue for $x_3$will be=12000$x_3$.

Now in the question, we have given ‘A cargo plane can carry a maximum weight of 100 tons’. So the constraint according it can be represented as:

$2x_1+x_2+3x_3\le 100$.

Also it is mention that, ‘: A cargo plane can carry and a maximum volume of 60 cubic meters’. This can be represented as:

$x_1+x_2+x_3\le 60$.

Linear Programming

According to the given constraints, we have to maximize(optimize) our revenue:

$$\begin{gathered} \mathrm{Maximun}:1000x_1+1200x_2+12000x_3 \\ 2x_1+x_2+3x_3\le 100 \\ {x}_1\le 40 \\ {x}_2\le 30 \\ {x}_3\le 20 \\ {x}_1,x_2,x_3\ge 0 \end{gathered}$$

This is our required linear program.