Simplex Method. Introduction:


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1 Introduction: In the previous chapter, we discussed about the graphical method for solving linear programming problems. Although the graphical method is an invaluable aid to understand the properties of linear programming models, it provides very little help in handling practical problems. In this chapter, we concentrate on the simplex method for solving linear programming problems with a larger number of variables. Many different methods have been proposed to solve linear programming problems, but simplex method has proved to be the most effective. This method is applicable to any problem that can be formulated in terms of linear obective function, subect to a set of linear constraints. Basic Terminology Slack variable It is a variable that is added to the lefthand side of a less than or equal to type constraint to convert the constraint into an equality. In economic terms, slack variables represent leftover or unused capacity. Specifically: a i x + a i x + a i3 x a in x n b i can be written as a i x + a i x + a i3 x a in x n + s i = b i Where i =,,..., m Surplus variable It is a variable subtracted from the lefthand side of a greater than or equal to type constraint to convert the constraint into equality. It is also known as negative slack variable. In economic terms, surplus variables represent over fulfillment of the requirement. Specifically: a i x + a i x + a i3 x a in x n b i can be written as a i x + a i x + a i3 x a in x n  s i = b i Where i =,,..., m Artificial variable It is a non negative variable introduced to facilitate the computation of an initial basic feasible solution. In other words, a variable added to the lefthand side of a greater than or equal to type constraint to convert the constraint into an equality is called an artificial variable. Basic Variables and NonBasic Variables: The variables attached to the independent column vectors of the basis matrix are known as basic variables and the remaining (nm) variables whose values are assumed to be zero are known as nonbasic variables. Basic Solution: Given a system of m simultaneous linear equations containing n variables (n>m) and the set of equations be AX=b, R(A)=m. If any m*n, non singular matrix be arbitrating selected from A and if we assume all (nm) variables zero which are not associated with column matrix, the solution so obtained is basic solution. Nondegenerate basic solution: If all components of a solution set corresponding to a basic variable are zero quantities then the basic solution is known as nondegenerate basic solution.
2 Degenerate basic solution: If some components of the solution set corresponding to the basic variables are zero the basic solution is known as degenerate solution. A Basic feasible solution is said to be degenerate solution if one or more than one basic variable are zero. Non Degenerate Solution: A Non Degenerate Solution feasible solution is the basic feasible solution which has exactly m positive xi (i=,,,m) i.e. None of the basic variables are zero. At what condition Simplex method has an unbounded solution? a) If corresponding to any negative Z C, all elements of X column are negative or zero ( 0). Then the solution is unbounded. b) If all artificial vectors are driven out from the basis but some non basis victors are less than zero. At what condition Simplex method has an no feasible solution? If all Z C 0 but some artificial variables are present at the positive level in the optimal solution. At what condition Simplex method has an alternative solution? If all Z C 0, the alternative solution exist if any nonbasic Z C is also zero. Consider the general linear programming problem Maximize z = c x + c x + c 3 x c n x n Subect to a x + a x + a 3 x a n x n b a x + a x + a 3 x a n x n b... a m x + a m x + a m3 x a mn x n x, x,..., x n 0 b m Where: c ( =,,..., n) in the obective function are called the cost or profit coefficients. b i (i =,,..., m) are called resources. a i (i =,,..., m; =,,..., n) are called technological coefficients or inputoutput coefficients. Converting inequalities to equalities Introducing slack variables to convert inequalities to equalities a x + a x + a 3 x a n x n + s = b a x + a x + a 3 x a n x n + s = b...
3 a m x + a m x + a m3 x a mn x n + s m = b m x, x,..., x n 0 s, s,..., s m 0 3 An initial basic feasible solution is obtained by setting x = x =... = x n = 0 s = b s = b... s m = b m The initial simplex table is formed by writing out the coefficients and constraints of a LPP in a systematic tabular form. The following table shows the structure of a simplex table. Example0: Using simplex method to solve the following LPP: Max Z=3x + x Subect to x + x 4 x  x x, x 0. Solution: Introducing slack variables x 3 and x 4 in st and nd constraints respectively. The reformulated LPP becomes Max Z=3x + x +0x 3 +0x 4 Subect to x + x + x 3 =4 x x + x 4 =, x, x, x 3, x 4 0. Using simplex method, we have Table C C B X B b y y y 3 y 4 Min Ratio 0 x x 40 Z 03* C 4/=4 /=* 0 x 3* 0  *
4 3 x Z C 6 05* 0 3 x 3 x 3 0 / / 0 / / Z 0 0 5/ / C 4 Since all Z C 0.Hence the solution is optimal. The optimal solution is Max Z= at x =3, x =. Example0: Using simplex method to solve the following LPP: Max Z=60x + 50x Subect to x + x 40 3x +x 60 x, x 0. Solution: Introducing slack variables x 3 and x 4 in st and nd constraints respectively. The reformulated LPP becomes Max Z=60x + 50x +0x 3 +0x 4 Subect to x + x + x 3 =40 3x +x + x 4 =60 x, x, x 3, x 4 0. Using simplex method, we have Table C C B X B b y y y 3 y 4 Min Ratio 0 x /=40 0 x 4 C Z x 3* 0 0 4/3 /3 /3 30
5 60 x 0 /3 0 /3 Z C x 5 0 4/3 / x 0 0 / / Z /4 70/4 C Since all Z C 0.Hence the solution is optimal and the optimal solution is Max Z=350 at x =0, x =5.
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