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Unformatted text preview: Hillier and Lieberman Chapter 4 The Essence of the Simplex It is an algebraic procedure Wyndor glass example Constraint boundaries, figure 4.1 Wyndor Glass Maximize Z = 3X1 + 5X2 Subject to: X1 4 2X2 12 3X1 + 2X2 18 Figure 4.1 0, 6 4,0 4,3 2, 6 Corners: (0,0) (4,0) (4.3) (2,6) and (0,6) Corner Point Feasible Solutions Corner point feasible solutions (CPF solutions) occur at the intersections of the constraint boundaries, which are in the feasible region. The problem has 2 decision variables, so each corner occurs at the intersection of two constraints. The dimension of the problem determines the number of constraints that intersect at each corner. If there are, for example, 5 decision variables the corner points occur at the intersection of 5 constraints. Adjacent feasible solutions For any linear programming model with n decision variables, two CPF solutions are adjacent to each other if they share n1 constraints. The two adjacent CPF solutions are connected by a line segment that lies on the same shared constraint boundaries. Such a constraint boundary is referred to as an edge of the feasible region. Figure 4.1 0, 6 4,0 4,3 2, 6 0,0 (0,0) and (0,6) adjacent (0,6) and (2,6) adjacent (2,6) and (4,3) adjacent (4,3) and (4,0) adjacent (4,0) and (0,0) adjacent There are 5 edges. From each CPF solution, 2 edges emanate. Each CPF solution has 2 adjacent solutions. Optimality test If a CPF solution has no adjacent CPF solution that is better (in terms of the value of the objective function) then it must be an optimal solution. Solving the example Initialization. Choose point (0,0) because it is convenient (no calculation needed to show it is feasible) Optimality test: Conclude that (0,0) is not optimal because adjacent CPF solutions are better Iteration 1: Move to a better CPF solution Simplex Iteration Consider the two edges that emanate from (0,0). Move up the X2 axis because X2 contributes more to the objective function of Z = 3X1 + 5X2. Stop at the next constraint boundary, 2X2 = 12 (or X2 = 6). This is point (0,6). You cant move further in this direction without leaving the feasible region. Find the other adjacent CPF solution for this point (2,6) from the intersection of constraints. Optimality test: (0,6) is not optimal because (2,6) is better. Repeat steps. (2,6) is optimal solution. Solution Concepts Simplex focuses only on CPF solutions, a finite set. It is an iterative procedure, meaning a fixed series of steps is repeated (an iteration) until a desired result is found. When possible, the initial CPF solution is the origin because it is convenient. (Possible when all variables are nonnegative.) If origin is infeasible, special procedures are needed, described in section 4.6 of this chapter. Solution Concepts, Continued At each iteration, we always choose an adjacent CPF solution. Steps take place along the edges of the feasible region....
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This note was uploaded on 11/15/2011 for the course AGEC 7100 taught by Professor Duffy,p during the Fall '08 term at Auburn University.
 Fall '08
 Duffy,P

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