BOWEN UNIVERSITY, IWO, OSUN STATE,NIGERIA.

PRINCIPLES OF DUALITY

STA322 (OPERATIONS RESEARCH I)A.C. OMOYAJOWOadefunke.omoyajowo@bowen.edu.ng

Learning outcome

At the end of the lesson, students should be able to:

▪ Understand the meaning of duality.

▪ Understand the principle behind duality

▪ Understand the rules guiding the principle of duality

▪ Write the dual of a primal problem

Take a look at this interesting illustration

There is a small company in Ogun state which has recently become engaged in the production of office furniture. The company manufactures tables, desks and chairs. The production of a table requires 8kgs of wood and 5kgs of metal and is sold for ₦80; a desk uses 6kgs of wood and 4kgs of metal and is sold for ₦60; and a chair requires 4kgs of both metal and wood and is sold for ₦50. We would like to determine the revenue maximizing strategy for this company, given that their resources are limited to 100kgs of wood and 60kgs of metal.

Illustration (cont’d)

Now consider that there is a much bigger company in Lagos which has been the lone producer of this type of furniture for many years. They don’t appreciate the competition from this new company; so they have decided to tender an offer to buy all of their competitor’s resources and therefore put them out of business.

Then the problem becomes a dual

Rules of duality

i. If the primal contains n variables and m constraints, the dual will contain a m variables and n constraints.

ii. The maximization problem in the primal becomes the minimization problem in the dual and vice versa

iii. The maximization problem has ≤ constraints while minimization problem has ≥ constraint

iv. constraints ≤ type in the primal becomes ≥ type in the dual and vice versa

v. The coefficient matrix of the constraints of the dual is the transpose of the primal

vi. A new set of variables appears in the dual.

vii. The constants C1, C2…. Cn in the objective function of the primal appear in the right hand side of the

constraints of the dual.

viii.The constants b1, b2…. bn in the right hand side of the constraints of the primal appear in the objective function

of the dual.

ix. The variables in both problems are Non-negative

Difference between Weak and Strong Duality.

Watch the lecture video for more explanation on how to write dual of a Linear programming Problem!