Chapter 6 Duality Theory 2015 Mc GrawHill Education

Chapter 6 Duality Theory © 2015 Mc. Graw-Hill Education. All rights reserved.

6. 1 The Essence of Duality Theory • Every linear programming problem has an associated problem called the dual – Original problem is known as the primal – These relationships prove useful in a variety of ways • Consider a maximization primal problem in standard form – Dual is a minimization problem – Dual uses same parameters in different locations © 2015 Mc. Graw-Hill Education. All rights reserved. 2

The Essence of Duality Theory © 2015 Mc. Graw-Hill Education. All rights reserved. 3

The Essence of Duality Theory • Coefficients in the objective function of the primal problem: – Are right-hand sides of the functional constraints in the dual problem • Right-hand sides of the functional constraints in the primal problem: – Are the coefficients in the objective function of the dual problem © 2015 Mc. Graw-Hill Education. All rights reserved. 4

The Essence of Duality Theory • Coefficients of a variable in the functional constraints of the primal problem: – Are the coefficients in a functional constraint of the dual problem © 2015 Mc. Graw-Hill Education. All rights reserved. 5

© 2015 Mc. Graw-Hill Education. All rights reserved. 6

The Essence of Duality Theory © 2015 Mc. Graw-Hill Education. All rights reserved. 7

The Essence of Duality Theory • General relationships between primal and dual problems – Parameters for a functional constraint in either problem are the coefficients of a variable in the other problem – Coefficients in the objective function of either problem are the right-hand sides for the other problem © 2015 Mc. Graw-Hill Education. All rights reserved. 8

The Essence of Duality Theory • Origin of the dual problem – Duality theory based on the fundamental insight presented in Chapter 5 • Summary of primary-dual relationships – Weak duality property – Strong duality property – Complementary solutions property – Complementary optimal solutions property – Symmetry property © 2015 Mc. Graw-Hill Education. All rights reserved. 9

The Essence of Duality Theory • Summary of primary-dual relationships (cont’d. ) – Duality theorem • Weak duality property – If x is a feasible solution for the primal problem and y is a feasible solution for the dual problem, then cx ≤ yb. © 2015 Mc. Graw-Hill Education. All rights reserved. 10

The Essence of Duality Theory • Strong duality property – If x* is an optimal solution for the primal problem and y* is an optimal solution for the dual problem, then cx* = y*b. • Complementary solutions property – At each iteration, the simplex method simultaneously identifies a CPF solution x for the primal problem and a complementary solution y for the dual problem • Where cx = yb © 2015 Mc. Graw-Hill Education. All rights reserved. 11

The Essence of Duality Theory • Complementary solutions property (cont’d. ) – If x is not optimal for the primal problem, then y is not feasible for the dual problem • Complementary optimal solutions property – The simplex method identifies (at its final iteration) an optimal solution x* for the primal problem and a complementary optimal solution y* for the dual problem • Where cx* = y*b © 2015 Mc. Graw-Hill Education. All rights reserved. 12

The Essence of Duality Theory • Symmetry property – For any primal problem and its dual problem • All relationships between them must be symmetric • Duality theorem – Identifies the only possible relationships between the primal and dual problems – If one problem has feasible solutions and a bounded objective function, then so does the other problem • Both weak and strong duality properties apply © 2015 Mc. Graw-Hill Education. All rights reserved. 13

The Essence of Duality Theory • Duality theorem (cont’d. ) – If one problem has feasible solutions and an unbounded objective function, then the other problem has no feasible solutions – If one problem has no feasible solutions, then the other problem either has no feasible solutions or an unbounded objective function © 2015 Mc. Graw-Hill Education. All rights reserved. 14

The Essence of Duality Theory • Applications – Dual problem can be solved directly by the simplex method to identify an optimal solution for the primal problem • Can be useful if one of the problems has fewer functional constraints – Evaluation of a proposed solution for the primal problem – Economic interpretation of the dual problem • Insights for the primal problem © 2015 Mc. Graw-Hill Education. All rights reserved. 15

6. 2 Economic Interpretation of Duality • The dual variable yi – Interpreted as the contribution to profit per unit of resource i when the current set of basic variables is used to obtain the primal solution © 2015 Mc. Graw-Hill Education. All rights reserved. 16

Economic Interpretation of Duality • The simplex method: – Examines all nonbasic variables in the current BF solution • To see which ones can provide a more profitable use of the resources by being increased • If none can, the current solution must be optimal • If one or more can, the simplex method increases this variable (entering basic variable) until marginal value of resources changes • Increase results in a new BF solution and process is repeated © 2015 Mc. Graw-Hill Education. All rights reserved. 17

6. 3 Primal-Dual Relationships • Augmented form of the dual problem – Obtained by subtracting the surplus from the left-hand side of each constraint j – n functional constraints and n + m variables – Each basic solution has n basic variables and m nonbasic variables © 2015 Mc. Graw-Hill Education. All rights reserved. 18

Primal-Dual Relationships • Basic solutions of primal and dual problems have direct correspondence – Key: row 0 of simplex tableau for the primal basic solution – Dual solution read from row 0 is also a basic solution • Complementary basic solutions property – Primal problem basic solution has a complementary basic solution in the dual problem © 2015 Mc. Graw-Hill Education. All rights reserved. 19

Primal-Dual Relationships • Complementary slackness property – Shows how to identify the basic and nonbasic solutions © 2015 Mc. Graw-Hill Education. All rights reserved. 20

Primal-Dual Relationships © 2015 Mc. Graw-Hill Education. All rights reserved. 21

Primal-Dual Relationships © 2015 Mc. Graw-Hill Education. All rights reserved. 22

6. 4 Adapting to Other Primal Forms • Option to convert any model to standard form – Construct dual problem in the usual way • Constructing the dual of the dual yields the primal problem • Symmetry property implies: – It doesn’t matter which problem is called the dual and which is the primal © 2015 Mc. Graw-Hill Education. All rights reserved. 23

Adapting to Other Primal Forms © 2015 Mc. Graw-Hill Education. All rights reserved. 24

Adapting to Other Primal Forms • Sensible-odd-bizarre (SOB) method for determining the form of constraints in the dual – Formulate the primal problem in either maximization or minimization form • Dual problem will be in other form – Label the different forms of the functional and variable constraints as being sensible, odd, or bizarre • See Table 6. 14 for guidance © 2015 Mc. Graw-Hill Education. All rights reserved. 25

Adapting to Other Primal Forms © 2015 Mc. Graw-Hill Education. All rights reserved. 26

Adapting to Other Primal Forms • SOB method (cont’d. ) – For each constraint on an individual variable in the dual problem, use the form with the same label as the corresponding functional constraint in the primal problem – For each functional constraint in the dual problem, use the form with the same label as the constraint on the corresponding individual variable in the primal problem © 2015 Mc. Graw-Hill Education. All rights reserved. 27

6. 5 The Role of Duality in Sensitivity Analysis • Sensitivity analysis involves examining the impact of parameter changes – You can choose which problem to use to investigate each change: primal or dual • Changes in coefficients of a nonbasic variable – Does not affect solution feasibility – May affect whether the solution is optimal © 2015 Mc. Graw-Hill Education. All rights reserved. 28

The Role of Duality in Sensitivity Analysis • Introduction of a new variable – Adds another activity into the model – Implies a new constraint in the dual problem • Question: Is the complementary basic solution for the dual problem still feasible? – To answer this: simply check against the new constraint for the dual problem © 2015 Mc. Graw-Hill Education. All rights reserved. 29

6. 6 Conclusions • Every linear programming problem has a corresponding dual problem • Important relationships exist between the two problems • The simplex method can be applied to either problem – Sometimes dealing with dual problem saves computational effort © 2015 Mc. Graw-Hill Education. All rights reserved. 30