Other Algorithms for Linear Programming Chapter 7

Other Algorithms for Linear Programming Chapter 7

Introduction The Simplex method is only a part of the arsenal of algorithms regularly used by LP practitioners. Variants of the simplex method the dual simplex method parametric LP The interiorpoint approach

7. The Dual Simplex Method Based on the duality theory( ch 6) Can be thought of as the mirror image of the simplex method. Useful in special types of situations: If you have to introduce many artificial variables to construct an initial BF solution artificially, it is easier to begin with a dual feasible solution..fewer iterations as well To estimate the optimal value of the objective Sensitivity analysis

7. Parametric LP Used for conducting sensitivity analysis systematically by gradually changing various model parameters simultaneously. Systematic changes in the c parameters Systematic changes in the b parameters

Systematic Changes in the c parameters The objective function of the ordinary linear programming model n Z = j= Is replaced by c j x j n Z ( q ) = ( c j a q ) x j= j Where a j are given input constants representing the relative rates at which the coefficients are to be changed. j

Summary of the Parametric LP procedure for Systematic Changes in the c Parameters. Solve the problem with q=0 by the Simplex method.. Use the sensitivity analysis procedure to introduce the Dc j =a j q changes into Eq. (0).. Increase q until one of the nonbasic variables has its coefficient in Eq. (0) go negative ( or until q has been increased as far as desired). 4. Use this variable as the entering basic variable for an iteration of the simplex method to find the new optimal solution. Return to step.

Example Wyndor Glass Co. a = and a = Z ( q ) = ( q ) x x (5 q )

Beginning with the final simplex tableau q=0 Z / x x5 4 = 6

Step Would first have these changes from the original coefficients added into it on the left hand side Z qx qx / x x = 4 5 6

X and x as basic variables both need to be eliminated from Eq (0) Z qx (0) q times q qx / x Eq.() = 6 4 times Eq.() Z (/ 7 / 6q ) x 4 x 5 ( / q ) x 5 = 6 q Optimality test: the current BF solution will remain Optimal as long as these coefficients of the nonbasic Variables remain nonnegative: / 7 / 6q 0 / q 0 for for 0 allq q 0 9 / 7

Table 7.

Systematic Changes in the b Parameters Max Z( q ) = n j= c j x j subject to n j= and x j a ij x 0 j b i for a q i i =,,... m j =,,..., n

Summary of the Parametric LP Procedure for Systematic Changes in the b Parameters. Solve the problem with q=0 by the Simplex method.. Use the sensitivity analysis procedure to introduce the Db=aq changes to the right side column.. Increase q until one of the basic variables has its value in the right side column go negative ( or until q has been increased as far as desired). 4. Use this variable as the leaving variable for an iteration of the dual simplex method to find the new optimal solution. Return to step.

Example: dual problem for the Wyndor Glass Co. Table 6.; a = and a = functional constraints q q q q 5 5 y y or y y y y or y y

Table 7.

[ ] œ œ œ ß ø Œ Œ Œ º Ø = œ ß ø Œ º Ø œ ß ø Œ º Ø = = = œ ß ø Œ º Ø = = 6 7 5 / / 0 / * * 6 5 6 * * q q q q q q q b S b b y Z The role of the sensitivity analysis

7.4 An InteriorPoint Algorithm Concept : Shoot through the interior of the feasible region toward an optimal solution. Concept : Move in a direction that improves the objective function value at the fastest possible rate. Concept : Transform the feasible region to place the current trial solution near its center, thereby enabling a large improvement when concept is implemented.

Example Max Z = x x subject to x x 8 and x 0 x 0

x (0,8) optimal Z=6=x x (,4) (,) x

The Relevance of the Gradient for Concepts and The algorithm begins with an initial trial solution that lies in the interior of the feasible region i.e. inside the boundary of the feasible region. Arbitrarily chosen (,) to be the initial trial solution.

Using the Projected Gradient to Implement Concepts and The augmented form: C = [,, 0], X = [x, x, x] T, A = [,, ] B = 8 Adding the gradient: (, 4, 4) = (,, 4) (,, 0)

Using the Projected Gradient to Implement Concepts and The constraint fails, the projection of the result to x x x =8 is used: (, 4, 4) > (,, ). OR The projected gradient C p is used: (,, ) = (,, 4) (0,, ), here C p = (0,, ) is the projection of the gradient (,, 0) to x x x =0 P = I A T (A A T ) A C p = P c

A Centering Scheme for Implementing Concept New rescaled variables are introduceed: x = x /, x = x /, x = x x = x = x = / 4 New variables are equidistant from the three constraint boundaries: x = 0, x = 0, x = The new problem: 0 Z = x 4x x x 4x = 8

Summary and Illustration of the Algorithm The algorithm is designed for dealing with big problems. The algorithm tends to require smaller number of iterations (although with more work per iteration)

Conclusions Duality theory and parametric linear programming are especially valuable for postoptimality analysis Interior point method provides a powerful approach for efficiently solving of very large problems