LINEAR PROGRAMMING



Introduction

In Linear programming, we deal with the optimization (maximization or minimization) of a linear function of a number of variables subject to a number of restrictions (or constraints) on variables in the form of linear in-equations in the variables of the optimization function.

In Linear programming, we deal with the optimization (maximization or minimization) of a linear function of a number of variables subject to a number of restrictions (or constraints) on variables in the form of linear in-equations in the variables of the optimization function.

The Idea of linear programming was formulated by the Russian mathematician, L. V. Kantorovich and the American economist FL.Hitchcock in 1941. This was the well known transportation problem. In 1947, the American economist, G.B. Dantzig suggested an efficient method known as the simplex method. Linear programming is the method used in decision making in business for obtaining the maximum or minimum value of a linear expression, subject to satisfying certain given linear in-equations.

LINEAR PROGRAMMING PROBLEMS AND ITS MATHEMATICAL FORMULATION

In this section, we .shall discuss the, general form of a linear programming problem.

The general mathematical description of a linear programming (LPP) is given below optimize (Maximize or Minimize)

Z= ax + by + c



subjected to

ax + by ( ≤ = ≥) c





The definition of various terms related to LPP are as follows:

Objective Function : Linear function Z = ax + by + c, where a, b, c are constants, which has to be maximized or minimized is called linear objective function.





Constraints : The linear inequalities or equations or restrictions on the variables of a linear programming problem are called constraints. The conditions x ≥ 0, y ≥ 0 are called non-negative

constraints.





Optimization problem : A problem which seeks to maximize: or minimize a linear function (say of two variables x and y ) subject to certain constraints as determined by a set of linear inequalities is called an optimization problem.





Solution : A set values of variables x, y is called a solution of LPP, if it satisfies the constraints of LPP.





Feasible Solution : A set of values of the variables x, y is called a feasible solution of a LPP, if it satisfies the constraints and non-negativity restrictions of the problem.





Infeasible Solution : A solution of LPP is an infeasible solution, if the system of constraints has no point which satisfies all the constraints and non-negativity restrictions.





Feasible region : The common region determined . by the constraints including non-negative constraints x, y ≥ 0 of LPP is called the feasible region (or solution region) for the problem.

The region other than feasible region is called an infeasible region. Points within and on the boundary of the Feasible region represent feasible solution of the constraints. Any point outside the feasible region is called an infeasible solution.





Optimal Feasible Solution : Any point in the feasible region that gives the optimal value (maximum or minimum) of the objective function is called an optimal solution.





Convex set : A set is convex set, if every point on the segment joining any two points in it lies in it.

FORMULATION OF A LINEAR PROGRAMMING

In order to solve a linear programming problem, the following steps are followed. 1.Identify the variables in the given LPP and denote these by x and y. 2. Translate all constraints in the form of linear inequation. No constraint is left out. 3. Formulate the objective function in terms of x and y and decide the objective 'function is to be maximized or minimized. 4. Solve these in-equations simultaneously. Mark the common area by a shaded region. This is the feasible region.

5. Find the coordinates of all the vertices of the feasible region

6.Find the value of the objective function at each vertex of the feasible region.

7.Find the values of x and y which maximize or minimize the value of the objective function.

SOLVING LINEAR PROGRAMMING PROBLEMS

After the formulation of the linear programming problems it is required to solve it for the optimum values of the variable,

There are three methods of solving a LPP.

(i)Graphical method or Corner point method,

(ii) ISO profit or ISO cost line method.

(iii) Simplex method.

We use the following theorems which are fundamental in solving

linear programming problems.





If there is a solution of a linear programming problems then it will occur at a corner point or on a line segment between two corner points. The fundamental theorem of linear programming is a great help. Instead of testing all of the infinite number of points in the feasible region, you only have to test the corner point. Whichever corner point yields the largest value for the objective function is the maximum and whichever corner point yields the smallest value for the objective function is minimum.

Corner Point Method

The method comprises o f the following steps.

1. Find the feasible region of the linear programming problem and determine its corner point (vertices) either by inspection or by solving the two equations of the line intersecting at that point.

2. Evaluate the objective function Z=ax + by at each corner

point. Let M and m respectively denotes the largest and smallest

values at these point.

3. When the feasible region. is bounded, •M and m are the

maximum and minimum values of Z

4.In case, the feasible region is unbounded, we have

(a) M is the maximum value of Z, if the open half plane determined by ax + by > M has no point in common with the feasible region. Otherwise, Z has no maximum value.

(b) Similarly, m is the minimum value of Z if open half plane determined by ax + by< m has no point in common With the feasible region. Otherwise, Z has no minimum value.





Note : If two corner points of the feasible region are both optimal solution of the same type, i. e. both produce the same maximum or minimum, then any points is also an optimal solution of the same type.





Iso-Profit or Iso-Cost Method

This is an alternate and more general method for finding the optimal solution of an L.P.P.

In this method, we first give any suitable constant value, say Z1, to the objective function and draw the corresponding line of the objective function. This line is called Iso- profit or Iso-

cost line, since every point on this line will yield the same profit or cost Z. After that, we draw another line by giving another value say Z2 to the objective function. The two lines are parallel to each other. If Z1 < Z2 and objective is to maximize Z, then we move the line corresponding to Z1 to the line corresponding to Z2, parallel to itself as far as possible until the farthest point within the feasible region is touched by this line.





In order to find the minimum value of Z, we have to find the line nearest to the origin and having at least one point common with the shaded region.

DIFFERENT TYPES OF LINEAR PROGRAMMING

PROBLEMS





1. Diet Problems

In this type of problem, we have to find the amount of different kinds of constituents/nutrients which should be included in a diet so as to be minimize the cost of the desired diet such that it contains a certain Minimum amount of each constituents/nutrient.





2. Manufacturing Problems

In this type of problem, we have to determine the number of different products which should be produced and sold by a firm where each product requires man power, machine hours, labor

hours per unit Of the product, warehouse space per unit of output, etc. in order to make maximum profit.





3. Transportation Problems

In this type of problems, we have to determine the transportation

schedule for a commodity from different plants or factories

situated at different location to different markets at different

locations in such a way that the total cost of transportation is ,

minimum, subject to the Imitation (constraints) as regards the

demand of each market and supply, from each plant or factory.



