*1.2.1 Optimization model*

Optimization means 'the action of finding the best solution'. Optimization modeling is also known as Mathematical Programming. Mathematical programming is the use of mathematical models, particularly optimizing models, to assist in making decisions. It is a branch of operation research which has wide applications in various areas of human activity. Optimization can help solve problems where there are two situations as (1) many ways of doing something or (2) limited resource available.

### **1.3 Classification of optimization problem**

Any real-world optimization problem may be characterized by five qualities. The problem function may all be linear or be nonlinear. The functional relationships may be known i.e. deterministic, or there may be uncertainty about them i.e. probabilistic. The optimization may take place at a fixed point in time (static) or it may be an optimization over time (dynamic). The variables may be continuous or discrete. And lastly, the problem functions may all be continuously differentiable (smooth) or may have points where the functions are non-differentiable (non-smooth).

#### **1.4 Linear programming (LP)**

Linear programming is an optimization technique of a linear objective function, subject to linear equality and linear inequality constraints. It is a mathematical method that is used to determine the best possible outcome or solution from a given set of parameters or a list of requirements, which are represented in the form of linear relationships. It is most often used in computer modeling or simulation in order to find the best solution in allocating finite resources such as money, energy, manpower, machine resources, time, space and many other variables. In most cases, the "best outcome" needed from linear programming is maximum profit or lowest cost. It was first developed by Soviet mathematician and economist Leonid Kantorvich in 1937 during the second world-war.

#### **1.5 Standard form of LP**

Here we present the standard form of linear programming. A linear programming problem may be defined as the problem of maximizing or minimizing.

The standard linear programming problem can be expressed in a compact form as: Maximize (or Minimize)

$$\mathbf{z} = \sum\_{i=1}^{n} c\_i \mathbf{x}\_i \tag{1}$$

$$\text{subject to } \sum\_{j=1}^{m} a\_{ij} \mathbf{x}\_j \{ \le \text{, } = \text{, } \ge \} b\_i, \quad i = 1, 2, \dots, n \tag{2}$$

$$\mathbf{x}\_j \ge \mathbf{0} \qquad j = 1, 2, \dots, m$$

**1.2 Preliminaries**

*1.2.1 Optimization model*

**1.3 Classification of optimization problem**

the chapter.

available.

(non-smooth).

**1.4 Linear programming (LP)**

**1.5 Standard form of LP**

**180**

Maximize (or Minimize)

In the current section, we discuss some preliminaries of the terms we mention in

*Concepts, Applications and Emerging Opportunities in Industrial Engineering*

Optimization means 'the action of finding the best solution'. Optimization modeling is also known as Mathematical Programming. Mathematical programming is the use of mathematical models, particularly optimizing models, to assist in making decisions. It is a branch of operation research which has wide applications in various areas of human activity. Optimization can help solve problems where there are two situations as (1) many ways of doing something or (2) limited resource

Any real-world optimization problem may be characterized by five qualities. The problem function may all be linear or be nonlinear. The functional relationships may be known i.e. deterministic, or there may be uncertainty about them i.e. probabilistic. The optimization may take place at a fixed point in time (static) or it may be an optimization over time (dynamic). The variables may be continuous or discrete. And lastly, the problem functions may all be continuously differentiable

Linear programming is an optimization technique of a linear objective function,

subject to linear equality and linear inequality constraints. It is a mathematical method that is used to determine the best possible outcome or solution from a given set of parameters or a list of requirements, which are represented in the form of linear relationships. It is most often used in computer modeling or simulation in order to find the best solution in allocating finite resources such as money, energy, manpower, machine resources, time, space and many other variables. In most cases, the "best outcome" needed from linear programming is maximum profit or lowest

cost. It was first developed by Soviet mathematician and economist Leonid

ming problem may be defined as the problem of maximizing or minimizing.

*<sup>z</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*¼1

*x <sup>j</sup>* ≥0 *j* ¼ 1, 2, … , *m*

Here we present the standard form of linear programming. A linear program-

The standard linear programming problem can be expressed in a compact form as:

*cixi* (1)

*aijx <sup>j</sup>*f g ≤, ¼ , ≥ *bi*, *i* ¼ 1, 2, … , *n* (2)

Kantorvich in 1937 during the second world-war.

subject to <sup>X</sup>*<sup>m</sup>*

*j*¼1

(smooth) or may have points where the functions are non-differentiable

The basic components of linear programming are as follows:

