Computational aspects of linear programming Simplex method D.T. Nguyen * , Y. Bai, J. Qin, B. Han, Y. Hu CEE Department, Old Dominion University, 135 Kauf, Norfolk, VA 23529, USA Abstract In this paper, the Simplex method is re-examined from the computational view points. Efficient numerical implementation for the Simplex procedure is suggested. Special features of artificial variables, and variables with unrestriction in signs are exploited to reduce the com- putational efforts, and computer memory requirement. The developed Simplex code has been tested on several examples, and its performance has been compared with existing Simplex codes.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Simplex method; Basic variables; Non-basic variables; Linear programming 1. Introduction Any linear programming (LP) problems can [1,2] be expressed in the following standard form. Find the design variable vector ~ x such that Minimize F  ~ c T ~ x 1 Subject To A ~ x  ~ b 2 and ~ x  ~ o 3 As an example, consider the following simple problem: Find the vector ~ x  x 1 x 2 ( ) ; such that Minimize F  -x 1 - 3x 2 4 Subject to x 1 + x 2  2 5 2x 1 - x 2  1 6 x 1 ; x 2  0 7 The inequality constraints shown in Eqs. (5) and (6) can be put in the standard (equality) form by introducing slack and/or surplus variables x 3 and x 4 as following: x 1 + x 2 + x 3  2 8 2x 1 - x 2 + x 4  1 9 Eqs. (8) and (9) represent system of two equations and four unknowns. Therefore, there exists infinite number of possible solutions. One possible set of solution is to set x 1  0  x 2 10 and therefore, one can easily find x 3  2  right-hand-side; or rhs; of Eq: 8 x 4  1  rhs of Eq: 9: In the Simplex method, Eqs. (8) and (9) are said to be in the Canonical form (the columns associated with variables x 3 and x 4 only contain 0’s and 1’s). Furthermore, the variables with zero values are defined as “Non-Basic” variables, whereas the variables with rhs values are referred to as “Basic” variables. Thus, for the LP problem defined in Eqs. (4)–(7), x 3 and x 4 are the basic variables, and x 1 and x 2 are non-basic variables. 2. Simplex method Observing Eqs. (8) and (9) carefully, one can see that the Advances in Engineering Software 31 (2000) 539–545 0965-9978/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S0965-9978(00)00022-3 www.elsevier.com/locate/advengsoft * Corresponding author. Tel.: + 1-757-683-3761; fax: + 1-757-683- 5354. E-mail address: nguyen@cee.odu.edu (D.T. Nguyen).