Chapter 4, Problem 1FCCE

What is the standard maximization form of a linear programming problem?

To determine

The standard maximization form of a linear programming problem.

Answer to Problem 1FCCE

Solution:

The objective function is written as:

Maximize: $z=ax+by+\cdots$

Where, $a,b,c\cdots$ denotes the profit per unit product $\text{type }A\text{, type }B\text{, type }C\cdots$ and so on respectively.

And, $x,y,z\cdots$ denote the number of products of $\text{type }A\text{, type }B\text{, type }C\cdots$ and so on respectively

Also, $x\ge 0,y\ge 0,z\ge 0\cdots$

The constraints are joined by the inequality sign $(\le )$ and it is written as:

$\left[\text{linear equations}\right]\le \left[\text{non-negative constants}\right]$

Explanation of Solution

In maximization problem,

Let, $x,y,z\cdots$ denote the number of products of $\text{type }A\text{, type }B\text{, type }C\cdots$ and so on respectively.

The objective function is written as:

Maximize: $z=ax+by+\cdots$

Where, $a,b,c\cdots$ denotes the profit per unit selling of product $\text{type }A\text{, type }B\text{, type }C\cdots$ and so on respectively.

And, constraints are joined by the inequality $(\le )$ and written as:

$\{\begin{array}{l}{a}_{1}x+b_{1}y+\cdots \le c_1\\ a_{2}x+b_{2}y+\cdots \le c_2\\ a_{2}x+b_{2}y+\cdots \le c_3\\ ⋮\\ ⋮\end{array}$

Or:

$\left[\text{linear equations}\right]\le \left[\text{non-negative constants}\right]$

Here, the right side of inequalities always remain positive or $c_1,c_2,c_3\cdots \ge 0$.

And, the number of products $x,y,z\cdots$ are also always positive.

So,

$x\ge 0,y\ge 0,z\ge 0\cdots$