Optimum Solution of Linear Programming Problem by Simplex Method

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1 Optimum Solution of Linear Programming Problem by Simplex Method U S Hegde 1, S Uma 2, Aravind P N 3 1 Associate Professor & HOD, Department of Mathematics, Sir M V I T, Bangalore, India 2 Associate Professor, Department of Mathematics, Sir M V I T, Bangalore, India 3 Assistant Professor, Department of Mathematics, Sir M V I T, Bangalore, India Abstract In this paper we will learn how to prepare a linear programming problem in order to solve it by pivoting using a matrix method. The Simplex Method is matrix based method used for solving linear programming problems with any number of variables. The simplex algorithm can be used to solve linear programming problems that already are, or can be converted to, standard maximum-type problems. Keywords LPP, Constraints, Algorithm, NZV(Non Zero Variables),Optimization (maximization or minimization) I. STANDARD LINEAR PROGRAMMING PROBLEM A linear programming problem is mathematically formulated by identifying a set of variables x 1, x 2,... x n which are subject to certain constraints in the form : a 11 x 1 + a 12 x a 1n x n ( ) b 1 a 21 x 1 + a 22 x a 2n x n ( ) b 2... a m1 x 1 + a m2 x a mn x n ( ) b n where the coefficients a ij, b i (1 i m, 1 j n ) are constraints and x 1 0, x x n 0 The objective function involving the variables x 1, x 2,... x n along with the given constants c 1, c 2,... c n will be a linear function of the form Z = c 1 x 1 +c 2 x c n x n II. FORMULATION OF THE PROBLEM Statement of the example : Two spare parts X and Y are to be produced in a batch. Each one has to go through two processes A and B. The time required in hours per unit and the total time available are given in the table. Profits per unit of X and Y are Rs. 5 & 6.The number of spare parts X & Yare to be produced in this batch to maximize the profit. X Y Total hours available Process A Process B The mathematical formulation of the LPP is to maximize is Z = 5X + 6Y subject to the constraints 3X + 4Y 48, 9X + 4Y 72 Page 197

2 III. SIMPLEX METHOD Consider the LPP with objective function Z = ax + by for maximization subject to the constraints a 1 x + b 1 y k 1, a 2 x + b 2 y k 2, x, y 0 Step 1 : Convert the linear inequalities into equations by introducing slack variables s 1, s 2 a 1 x + b 1 y + s 1 = k 1 a 2 x + b 2 y + s 2 = k 2 Step 2 : Initial Simplex Tableau s 1 a 1 b k 1 s 2 a 2 b k 2 λ -a -b Step 3 : (1) Identify least negative indicator (2) Suppose a is the least indicator first column is pivotal column and examine ratios k 1 /a 1, k 2 / a 2 identify least positive ratio (3) Suppose k 1 /a 1 is least then first row is called pivotal row and a 1 is called pivot Further replace s 1 by x Step 4 : First Simplex Tableau x 1 b 1 /a 1 1/a 1 0 k 1 /a 1 s 2 a 2 b k 2 λ -a -b Step 5 : Second Simplex Tableau x 1 b 1 /a 1 1/a 1 0 k 1 /a 1 s 2 0 b 2 -a 2 /a 1 1 k 2 λ 0 -b c 0 d IF THERE ARE NO NEGATIVE INDICATORS THE PROCESS IS COMPLETED THEN Z IS MAXIMUM AT x = k 1 /a 1, y = 0 Page 198

3 IV. SIMPLEX METHOD ALGORITHM Insert slack variables and find slack equations V. Rewrite the objective function and put it below the slack equations Write the initial simplex tableau Find the pivot element by finding the most negative indicator in last row and using the smallest quotient rule. Perform the pivot operation. yes Are there any VI. more negative indicators in the last row? no The maximum has been reached. VII. Page 199

4 VIII. ILLUSTRATIVE EXAMPLES Example 1 : Maximize P = 4x 1 2x 2 - x 3 subject to constraints x 1 + x 2 + x 3 3 2x 1 + 2x 2 + x 3 4 x 1 - x 2 0 x 1 + x 2 + x 3 + s 1 = 3 2x 1 +2 x 2 + x 3 + s 2 = 4 x 1 - x 2 + s 3 = 0 4x 1 2x 2 - x 3 = P is the objective function Table 1 : s x 1 =s 3 s s λ Table 2 : s R 1 = -R 3 +R 1 s R 2 = -2R 3 +R 2 x λ R 4 = 4R 3 +R 4 Table 3 : s x 2 =s 2 s x λ Table 4 : s R 1 =-2R 2 +R 1 x x R 3 =R 2 +R 3 λ R 4 =2R 2 +R 4 Table 5 : s x x λ No Negative indicators Thus the maximum value of P = 2 at x 1 = 1, x 2 = 1, x 3 = 0 Page 200

5 IX. CONCLUSIONS This paper presents a insight into understanding, analyzing a linear programming problem and obtaining a optimum solution (maximization or minimization) by simplex method. The procedure and algorithm of simplex method with examples are discussed in detail to understand the LPP more precisely and effectively. REFERENCES [1] Dr. B S Grewal, Higher Engineering Mathematics, Khanna Publishers [2] Dr. K S Chandrashekar, Engineering Mathematics, Sudha Publications [3] Prof G K Ranganath, Introduction to Linear Programming, S Chand & Company [4] [5] Page 201

1. Introduce slack variables for each inequaility to make them equations and rewrite the objective function in the form ax by cz... + P = 0.

1. Introduce slack variables for each inequaility to make them equations and rewrite the objective function in the form ax by cz... + P = 0. 3.4 Simplex Method If a linear programming problem has more than 2 variables, solving graphically is not the way to go. Instead, we ll use a more methodical, numeric process called the Simplex Method.