By Stephen P. Bradley
Ebook by way of Bradley, Stephen P., Hax, Arnoldo C., Magnanti, Thomas L.
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1 SIMPLEX METHOD—A PREVIEW Optimal Solutions Consider the following linear program: Maximize z = 0x1 + 0x2 − 3x3 − x4 + 20, (Objective 1) subject to: x1 − 3x3 + 3x4 = 6, x2 − 8x3 + 4x4 = 4, xj ≥ 0 ( j = 1, 2, 3, 4). Note that as stated the problem has a very special form. It satisfies the following: 1. All decision variables are constrained to be nonnegative. 2. All constraints, except for the nonnegativity of decision variables, are stated as equalities. 1 Simplex Method—A Preview 39 3. The righthand-side coefficients are all nonnegative.
We should emphasize that at each stage the form of the given problem is exactly equivalent to the original problem. 3 SIMPLEX METHOD—A FULL EXAMPLE The simplex method for solving linear programs is but one of a number of methods, or algorithms, for solving optimization problems. By an algorithm, we mean a systematic procedure, usually iterative, for solving a class of problems. The simplex method, for example, is an algorithm for solving the class of linear-programming problems. Any finite optimization algorithm should terminate in one, and only one, of the following possible situations: 1.
Before presenting a formal description of the algorithm, we consider some examples. Though elementary, these examples illustrate the essential algebraic and geometric features of the method and motivate the general procedure. 1 SIMPLEX METHOD—A PREVIEW Optimal Solutions Consider the following linear program: Maximize z = 0x1 + 0x2 − 3x3 − x4 + 20, (Objective 1) subject to: x1 − 3x3 + 3x4 = 6, x2 − 8x3 + 4x4 = 4, xj ≥ 0 ( j = 1, 2, 3, 4). Note that as stated the problem has a very special form. It satisfies the following: 1.