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Unit 4

Unit 4

Simplex Method

Structure: 4.1 Introduction Objectives 4.2 Standard Form of LPP Fundamental theorem of LPP 4.3 Solution of LPP – Simplex Method Initial basic feasible solution of an LPP To solve an LPP in canonical form by simplex method 4.4 The Simplex Algorithm Steps 4.5 Penalty Cost Method or Big M-method 4.6 Two Phase Method 4.7 Solved Problems on Minimisation 4.8 Summary 4.9 Glossary 4.10 Terminal Questions 4.11 Answers 4.12 Case Study

4.1 Introduction

In the previous unit we dealt with graphical analysis of linear programming problems. We discussed some basic definitions, graphical methods to solve LPP, some exceptional cases, advantages, limitations, and important geometric properties of LPP. In this unit, we will deal with the simplex method, which focuses on solving an LPP of any enormity involving two or more decision variables. The simplex algorithm is an iterative procedure for finding the optimal solution to a linear programming problem. The objective function controls the development and evaluation of each feasible solution to the problem. If a feasible solution exists, it is located at a corner point of the feasible region determined by the constraints of the system. The simplex method simply selects the optimal solution amongst the set of feasible solutions of the problem. The efficiency of this algorithm is because it considers only those feasible solutions which are provided by the corner points, and that too not all of them. Sikkim Manipal University Page No. 57

Operations Research

Unit 4

Objectives: After studying this unit, you should be able to: create a standard form of LPP from the given problem apply the simplex algorithm to the system of equations find solution using big M-technique explain the importance of the two phase method

4.2 Standard Form of LPP

The characteristics of the standard form of LPP are: All constraints are equations. The right-hand side element of each constraint equation is non-negative. All the variables are non-negative. The objective function is of maximisation type. The inequality constraints of equations are obtained by adding or subtracting the left-hand side of each such constraint by a non-negative variable. 1. Introduce slack variables (Si) for “” type of constraint. 2. Introduce surplus variables (Si) and artificial variables (Ai) for “” type of constraint. 3. Introduce only artificial variable for “=” type of constraint. 4. Cost (Cj) of slack and surplus variables will be zero and that of artificial variable will be “M”. To make the right-hand side of a constraint equation positive, multiply both sides of the resulting equation by -1. Use the elementary transformations introduced with the canonical form to achieve the remaining characteristics. Any standard form of the LPP is given by Maximise or Minimise z C i x i i 1 n

Subject to:

a ij x j S i b i ( b i 0 )i 1, 2.......... .m. j 1

n

and xj 0, j = 1, 2, --- n S i 0, i = 1, 2, --- m

Sikkim Manipal University Page No. 58

Operations Research

Unit 4

4.2.1 Fundamental theorem of LPP A set of m simultaneous linear equations in n unknowns/variables, where n m, AX = b, with r (A) = m. If there is a feasible solution X 0, then there exists a basic feasible solution. Self Assessment Questions 1. We add surplus variable for “≤”constraint. (True/False) 2. The right-hand side element of each constraint is non-negative in the standard form. (True/False)

4.3 Solution of the LPP – Simplex Method

Consider an LPP given in the standard form. To optimise z = c1 x1 + c2 x2 + ---+ cnxn Subject to a11 x1 + a12 x2 + -- + an x n S1 = b1 a21 x1 + a22 x2 + ----+ a2nxn S2 = b2 ………………………………………………………………….

am1 x1 + am2 x2 + -- + amnxnSm = bm x1, x2, --- xn, S1, S2 ---, Sm 0 To each of the constraint equations, add a new variable called an artificial variable on the left-hand side of every equation, which does not...