Chapter 11.8, Problem 32AYU

Explain in your own words what a linear programming problem is and how it can be solved.

To determine

What is linear programming problem and how it can be solved.

Answer to Problem 32AYU

Solution:

All LP problems have the following general form:

$MaximizeorMinimizeZ={\displaystyle \sum _{n=1}^n{c}_i{X}_i}$

$subjectto{\displaystyle \sum _{i=1}^n{a}_{ji}}{X}_i\le b_{j}j=1,2,\dots ,m$

Where $X_i=i^{th}$ decision variable.

$c_i=$ objective function coefficient corresponding to the $i^{th}$ variable.

$a_{ji}=$ the coefficient on $X_i$ in constraint $j$, and

$b_j=$ the right-hand-side coefficient on constraint $j$.

Step-1: Formulating the problem – Involves deducing the objective function and constraints equation.

Step-2: Solving the system of inequalities and obtaining the range of values (corner points) for decision variables. This can be done using graphical or computer programmes.

Step-3: Substituting the corner points in the objective functions and finding the values of maximum/minimum values of the objective function.

Explanation of Solution

Linear Programming:

Linear Programming (LP) is a mathematical technique for determining optimal solutions to problems that can be expressed using linear equations and inequalities. When real-world problems are represented accurately as a set of the linear equations, Linear Programming finds the best solution to the problems.

A linear program consists of a set of variables, a linear objective function indicating the contribution of each variable to the desired outcome, and a set of linear constraints describing the limits on the values of the variables. The solution to a linear program is a set of values for the problem variables that results in the best (either largest or smallest ) value of the objective function and yet satisfies all the constraints. Formulation is the process of translating a real-world problem into a linear program. Once a problem has been formulated as a linear program, it can be solved by analytical, graphical or using computer software.

Definitions in Linear Programming:

Linear Equations: All of the equations and inequalities in a linear program must, by definition, be linear. A linear function has the following form:

$a_0+a_1{x}_1+a_2{x}_2+a_3{x}_3\dots \dots +a_n{x}_n=0$

Where,

a. a’s are called the coefficients of the equation; they are also sometimes called parameters. The important thing to know about the coefficients is that they are fixed values, based on the nature of the problem being solved.

b. x’s are called the variables of the equation; they are allowed to take a range of values within the limits defined by the constraints.

Decision Variables: The variables in a linear program are a set of quantities that need to be determined in order to solve the problem; i.e., the problem is solved when the best values of the variables have been identified. The variables are sometimes called decision variables because the problem is to decide what value each variable should take. Typically, the variables represent the amount of a resource to use or the level of some activity.

Objective Function: The objective of a linear programming problem is to maximize or to minimize some numerical value. The objective function indicates how each variable contributes to the value to be optimized in solving the problem. The objective function takes the following general form:

$MaximizeorMinimizeZ={\displaystyle \sum _{i=1}^n{c}_i{X}_i}$

$where{c}_i=$ objective function coefficient corresponding to the $i^{th}$ variable.

$X_i=$ $i^{th}$ decision variable.

The coefficients of the objective function indicate the contribution to the value of the objective function of one unit of the corresponding variable.

Constraints: Constraints define the possible values that the variables of a linear programming problem may take. They typically represent resource constraints, or the minimum or maximum level of some activity or condition. They take the following general form:

$subjectto{\displaystyle \sum _{i=1}^n{a}_{ji}}{X}_i\le b_{j}j=1,2,\dots ,m$

Where $X_i=i^{th}$ decision variable.

$a_{ji}=$ the coefficient on $X_i$ in constraint $j$, and

$b_j=$ the right-hand-side coefficient on constraint $j$.

It should be noted that $j$ is an index that runs from 1 to m, and each value of $j$ corresponds to a constraint.

Solving General Linear Programming Problem

All LP problems have the following general form:

$MaximizeorMinimizeZ={\displaystyle \sum _{i=1}^n{c}_i{X}_i}$

$subjectto{\displaystyle \sum _{i=1}^n{a}_{ji}}{X}_i\le b_{j}j=1,2,\dots ,m$

Where $X_i=i^{th}$ decision variable.

$c_i=$ objective function coefficient corresponding to the $i^{th}$ variable.

$a_{ji}=$ the coefficient on $X_i$ in constraint $j$, and

$b_j=$ the right-hand-side coefficient on constraint $j$.

Step-1: Formulating the problem – Involves deducing the objective function and constraints equation.

Step-2: Solving the system of inequalities and obtaining the range of values (corner points) for decision variables. This can be done using graphical or computer programmes.

Step-3: Substituting the corner points in the objective functions and finding the values of maximum/minimum values of the objective function.