Linear Programming III- The Dual ProblemMaumita Roy

Duality TheoremFor every maximisation (or minimisation) problem in linear programming, there is a unique similar problem of minimisation (or maximisation) involving the same data which describes the original problem.

The given problem is called primal or direct , the corresponding intimately related problem is called its dual problem.

In fact, either can be called original as both are derived from the same data.

Standard form of the Primal Problem

Standard form of the Dual Problem

Rules for constructing the dual from primalNumber of variables in the dual problem is equal to the number of constraints in the original problem and vice versa.Coefficients of the objective function in the dual problem come from the right hand side of the original problem.If the original problem is a max. model, the dual is a min. model.The coefficients for the first constraint function for the dual problem are the coefficients of the first variable in the constraints for the original problem, and similarly for the other constraints. The right hand side of the dual constraints come from the objective function coefficients in the original problem.The sense of the ith dual constraint is = if and only if the ith variable in the original problem is unconstrained in sign.The ith variable in the dual problem is unconstrained in sign if and only if the ith constraint in the original problem is an equality.

There is a small company in Mumbai which has recently become engaged in the production of office furniture. The company manufactures tables, desks and chairs. The production of a table requires 8 kgs of wood and 5 kgs of metal and is sold for Rs.800; a desk uses 6 kgs of wood and 4 kgs of metal and is sold for Rs.600; and a chair requires 4 kgs of both metal and wood and is sold for Rs.500. We would like to determine the revenue maximizing strategy for this company, given that their resources are limited to 100 kgs of wood and 60 kgs of metal.

Objective function:

Max Z = 800x1 + 600x2 + 500x3

Subject to : 8x1 + 6x2 + 4x3 100 Wood constraint 5x1 + 4x2 + 4x3 60 Metal constraint

x1; x2; x3 0 non negativity constraintExample 1:

Now consider that there is a much bigger company in Mumbai 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.

Objective function:

Min Z = 100 y1 + 60 y2

Subject to : 8y1 + 5y2 800 6y1 + 4y2 600 4y1 + 4y2 500

y1; y2 0 non negativity constraint

An individual has a choice of two types of food to eat, meat and potatoes, each offering varying degrees of nutritional benefit. He has been warned by his doctor that he must receive at least 400 units of protein, 200 units of carbohydrates and 100 units of fat from his daily diet. Given that a kg of steak costs Rs.100 and provides 80 units of protein, 20 units of carbohydrates and 30 units of fat, and that a kg of potatoes costs Rs.20 and provides 40 units of protein, 50 units of carbohydrates and 20 units of fat, he would like to find the minimum cost diet which satisfies his nutritional requirements Example 2 :Objective function: Min Z = 100x1 +20x2Subject to:

80x1 + 40x2 400 protein constraint 20x1 + 50x2 200 carbohydrates constraint 30x1 + 20x2 100 fat constraint

x1; x2 0 non negativity constraint

Now consider a chemical company which hopes to attract this individual away from his present diet by offering him synthetic nutrients in the form of pills. This company would like determine prices per unit for their synthetic nutrients which will bring them the highest possible revenue while still providing an acceptable dietary alternative to the individual.

Objective function:

Max Z = 400y1 +200y2 + 100y3Subject to: 80y1 +20y2 + 30y3 100 40y1 + 50y2 + 20y3 20

y1; y2; y3 0 non negativity constraint

Example:

Find the dual of the following primal LP problem:Objective function: Max Z = 20x1 +15x2 + 18x3 + 10x4

Subject to:4x1 - 3x2 + 10x3 + 4x4 60 x1 + x2 + x3 = 27 - x2 + 4x3 + 7x4 35 x1; x2; x3 0; x4 unrestricted in sign

Objective function: Min Z = 2x1 + 3x2 + 4x3

Subject to: 2x1 + 3x2 + 5x3 2 3x1 + x2 + 7x3 = 3 x1 + 4x2 + 6x3 5 x1; x2 0; x3 unrestricted in sign

Objective function: Min Z = x1 - 3x2 - 2x3

Subject to: 3x1 - x2 + 2x3 7 2x1 - 4x2 12 -4x1 + 3x2 + 8x3 = 10 x1; x2 0; x3 unrestricted in sign