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Unformatted text preview: 6 The simplex method We next formalize the method we developed in the previous section. We again consider a general linear program in standard equality form: maximize c T x subject to Ax = b x . As before, we introduce a new variable z to keep track of the objective value, so the system of equations defining the linear program is z- c T x = 0 Ax = b. (6.1) We break down each iteration of the solution algorithm, which is called the simplex method , into a sequence of steps. Step 0: Initialization. We (somehow) find an initial feasible basis B , form a list N consisting of its complement, and find a corresponding tableau: z- X j N c j x j = v x i + X j N a ij x j = b i ( i B ) . (6.2) This system is equivalent to the original constraints (6.1), and from it, we can read off the corresponding basic feasible solution: basic variables x i = b i ( i B ) nonbasic variables x j = 0 ( j N ) objective value z = v. Step 1: Check optimality. If each reduced cost satisfies c j 0 (for all indices j N ), then we stop: the current solution is optimal . To see this, notice that any feasible solution satisfies z = v + X j N c j x j v, 35 since c j 0 and x j 0 (for all j N ). Thus no feasible solution can have objective value larger than v . But the current solution attains this value, so must be optimal. Step 2: Choose entering index. If we dont stop in Step 1, we can find an entering index k N with corresponding reduced cost c k > 0. Consider now the effect of increasing the value of the corresponding variable x k (called the entering variable ) away from its current value zero to a new value t 0. At the next iteration of the simplex method, if the value of x k is nonzero, the index k must have entered the basis: hence the term entering index....
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- Fall '07