Use the two steps for solving a linear programming problem,...

Use the two steps for solving a linear programming problem, given in the box on page $888,$ to solve the problems. In 1978 , a ruling by the Civil Aeronautics Board allowed Federal Express to purchase larger aircraft. Federal Express's options included 20 Boeing 727 s that United Airlines was retiring and/or the French-built Dassault Fanjet Falcon $20 .$ To aid in their decision, executives at Federal Express analyzed the following data: $$\begin{array}{ll} {} & {\text { Boeing } 727 \quad \text { Falcon } 20} \\ {\text { Direct Operating cost }} & {\$ 1400 \text { per hour } \$ 500 \text { per hour }} \\ {\text { Payload }} & {42,000 \text { pounds } \quad 6000 \text { pounds }} \end{array}$$ Federal Express was faced with the following constraints: $\cdot$ Hourly operating cost was limited to 35,000. $\cdot$ Total payload had to be at least 672,000 pounds. $\cdot$ Only twenty 727 s were available. Given the constraints, how many of each kind of aircraft should Federal Express have purchased to maximize the number of aircraft?

Use the two steps for solving a linear programming problem, given in the box on page $888,$ to solve the problems.
In 1978 , a ruling by the Civil Aeronautics Board allowed Federal Express to purchase larger aircraft. Federal Express's options included 20 Boeing 727 s that United Airlines was retiring and/or the French-built Dassault Fanjet Falcon $20 .$ To aid in their decision, executives at Federal Express analyzed the following data:
$$\begin{array}{ll}
{} & {\text { Boeing } 727 \quad \text { Falcon } 20} \\
{\text { Direct Operating cost }} & {\$ 1400 \text { per hour } \$ 500 \text { per hour }} \\
{\text { Payload }} & {42,000 \text { pounds } \quad 6000 \text { pounds }}
\end{array}$$
Federal Express was faced with the following constraints:
$\cdot$ Hourly operating cost was limited to 35,000.
$\cdot$ Total payload had to be at least 672,000 pounds.
$\cdot$ Only twenty 727 s were available.
Given the constraints, how many of each kind of aircraft should Federal Express have purchased to maximize the number of aircraft?

Key Concepts

Corner Point Principle

The corner point principle is a key aspect of linear programming which states that if an optimal solution exists, it will be found at one of the vertices (or corner points) of the feasible region. This principle simplifies the search for the optimal solution by allowing evaluation of the objective function only at these key points.

Feasible Region

The feasible region is the collection of all possible points that satisfy all the constraints in a linear programming problem. It is typically a polygonal area when graphed, and every point within this region represents a potential solution to the problem.

Linear Programming

Linear programming is a mathematical method used for optimizing a linear objective function, subject to a set of linear inequalities or equations known as constraints. It is commonly applied in resource allocation, production scheduling, and many other decision-making problems where the goal is to maximize or minimize a particular quantity.

Objective Function

The objective function in linear programming is a linear equation that represents the quantity to be optimized—either maximized or minimized. This function encapsulates the goal of the problem, such as maximizing profit or minimizing cost, and is evaluated over the set of feasible solutions defined by the constraints.

Constraints

Constraints are the linear inequalities or equations that define the restrictions or limitations in a linear programming problem. They represent the conditions and resource limitations that must be adhered to, ensuring that any potential solution falls within a realistic and acceptable range.

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