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Unformatted text preview: Chapter 3 SIMPLEX METHOD In this chapter, we put the theory developed in the last to practice. We develop the simplex method algorithm for LP problems given in feasible canonical form and standard form. We also discuss two methods, the MMethod and the TwoPhase Method, that deal with the situation that we have an infeasible starting basic solution. 3.1 Simplex Method for Problems in Feasible Canonical Form The Simplex method is a method that proceeds from one BFS or extreme point of the feasible region of an LP problem expressed in tableau form to another BFS, in such a way as to continually increase (or decrease) the value of the objective function until optimality is reached. The simplex method moves from one extreme point to one of its neighboring extreme point. Consider the following LP in feasible canonical form, i.e. its right hand side vector b ≥ 0: max x = c T x subject to ( A x ≤ b , x ≥ . Its initial tableau is x 1 x 2 ··· x s ··· x n x n +1 ··· x n + r ··· x n + m b x n +1 a 11 a 12 ··· a 1 s ··· a 1 n 1 ··· ··· b 1 x n +2 a 21 a 22 ··· a 2 s ··· a 2 n ··· ··· b 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x n + r a r 1 a r 2 ··· a rs ··· a rn ··· 1 ··· b r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x n + m a m 1 a m 2 ··· a ms ··· a mn ··· ··· 1 b r x c 1 c 2 ··· c s ··· c n ··· ··· Here x n + i , i = 1 ··· ,m are the slack variables. The original variables x i , i = 1 , ··· ,n are called the structural or decision variables . Since all b i ≥ 0, we can read off directly from the tableau a starting 1 2 Chapter 3. SIMPLEX METHOD BFS given by [0 , , ··· , ,b 1 ,b 2 , ··· ,b m ] T , i.e. all structural variables x j are set to zero. Note that this corresponds to the origin of the ndimensional subspace R n of R n + m . In matrix form, the original constraint A x ≤ b has be augmented to [ A I ] • x x s ‚ = A x + I x s = b . (3.1) Here x s is the vector of slack variables. Since the columns of the augmented matrix [ A . . . I ] that correspond to the slack variables { x n + i } m i =1 is an identity matrix which is clearly invertible, the slack variables { x n + i } m i =1 are basic. We denote by B the set of current basic variables, i.e. B = { x n + i } m i =1 . The set of nonbasic variables, i.e. { x i } n i =1 will be denoted by N . Consider now the process of replacing an x r ∈ B by an x s ∈ N . We say that x r is to leave the basis and x s is to enter the basis. Consequently after this operation, x r becomes nonbasic, i.e. x r ∈ N and x s becomes basic, i.e. x s ∈ B . This of course amounts to a different (selection of columns of matrix A to give a different) basis B . We shall achieve this change of basis by a pivot operation (or simply called a pivot ). This pivot operation is designed to maintain an identity matrix as the basis in the tableau at all time.as the basis in the tableau at all time....
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 Spring '10
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