DUALITY IN LINEAR PROGRAMMING PROBLEMS

For every Linear programming Problem, there is a corresponding unique problem involving the same data and it also describes the original problem. The original problem is called primal programme and the corresponding unique problem is called Dual programme. The two programmes are very closely related and optimal solution of dual gives complete information about optimal solution of primal and vice versa.

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Key Features of Duality in Linear Programming Problems

This relationship between the primal and dual programmes offers several valuable insights and applications in optimization. Different useful aspects of this property are:

(a) If primal has a large number of constraints and a small number of variables, computation can be considerably reduced by converting the problem to Dual and then solving it.

(b) Students pursuing a B.Sc. or M.Sc. course in Jaipur or anywhere in India are taught that duality in linear programming has certain far reaching consequences of economic nature. This can help managers answer questions about alternative courses of action and their relative values.

(c) Calculation of the dual checks the accuracy of the primal solution.

(d) Duality in linear programming shows that each linear programme is equivalent to a two-person zero-sum game. This indicates that fairly close relationships exist between linear programming and the theory of games.

Forming Dual when Primal is in Canonical Form

Let us explore the dual property when primal is in canonical form.

From the above programmes, the following points are clear:

(i) The maximization problem in the primal becomes the minimization problem in the dual and vice versa.

(ii) The maximization problem has (<) constraints while the minimization problem has (>) constraints.

(iii) If the primal contains n variables and m constraints, the dual will contain m variables and n constraints.

(iv) The constants b1 b2, b3……… bm in the constraints of the primal appear in the objective function of the dual.

(v) The constants c1, c2, c3 cn in the objective function of the primal appear in the constraints of the dual.

(vi) The variables in both problems are non-negative.

The constraint relationships of the primal and dual are represented below.

Example 1:

Construct the dual to the primal problem

Solution:

Firstly the ≥ constraint is converted into ≤ constraint by multiplying both sides by -1.

Example 2:

Construct the dual to the primal problem

Solution:

Let Y1,Y2 , V3 and V4 be the corresponding dual variables, then the dual problem is given by

As the dual problem has fewer constraints than the primal (2 instead of 4), it requires less work and effort to solve it. This follows from the fact that the computational difficulty in the linear programming problem is mainly associated with the number of constraints rather than number of variables.

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