Characteristics of the dual problem
Duality in linear programming has the following major characteristics:
1. Dual of the dual is primal.
2. If either the primal of dual problem has a solution, then the other also has a solution and their optimum values are equal.
3. If any of the two problems has only an infeasible solution then the value of the objective function of the other is unbounded.
4. The value of the objective function for any feasible solution of the primal is less than value of the objective function for any feasible solution of the dual.
5. If either the primal or the dual problem has an unbounded solution then the solution to the other problem is infeasible.
6. If the primal has a feasible solution but the dual does not have, then the primal will not have a finite optimum solution and vice-versa.
Advantages of Duality
1. It yields a number of powerful theorems.
2. Computational procedure can be considerably reduced by converting it into dual if the primal problem contains a large number of rows (constraints) and a smaller number of columns (variables).
3. Solution of the dual cheeks the accuracy of the primal solution for computational enors.
4. Gives additional information as to how the optimum solution changes as a result of the changes in the coefficients and the formulation of the problem (this is termed as postopimality or sensitivity analysis)
5. Indicates that fairly close relationships exits between linear programming duality.
6. Economic interpretation of the dual helps the management m making future decisions.
Illustration. The concept of duality is more effectively demonstrated in the following illustration:
PRIMAL DUAL
Minimize Z = 6,000x1+ 4,000x2 Maximize Z* = 12y1 + 18 y3 + 40y4
Subject to the constraints subject to the constraints
4x1 +x2 > 12 4y1+9y2+7y3+10y4 < 6000
9x1 + x2 > 20 y1 + y2 +3y3 + 40y4 < 4000
7x1 + 3x2 > 18 y1> 0, Y2 > 0, y3 > 0, y4 > 0
10x1 + 40x2 > 40
X1 > 0,x2 > 0
PRIMAL
Variables x1 x2 Relation Constraints
Y1 4 1 > 12
Y2 9 1 > 20
Dual Y3 7 3 > 18
Y4 10 40 > 40
Relation < < Max Z
Constraints 6000 4000 Min. Z
It may be noted that:
1. Primal here involves minimization Dual involves maximization
2. In primal we write objective In dual we write objective function
Function as Z as Z*
3. In primal the variables are Dual has a new set of variables, i.e.,
X1 and x2 y1,y2,y3 and y4
4. Primal has four constraints The dual therefore has two constraints
X1 and x2
5. In primal has four constraints The dual therefore has four variables,
Variables viz y1 y2 y3 and y4
6. In primal's objective function In dual, 6000 and 4000 become
6000 and 4000 are the coefficients Constants of constraints on the
right hand side.
7. In primal the coefficients of in dual each column takes the position
Constraints columnwise are:
4 1 4 9 7 10
9 1 1 1 3 40
7 3
10 40
8. In primal the signs of constraints In dual the signs of the constraints
Traints are greater than or equal are just the reverse, i.e., less than
To. Or equal to.
9. The non-negativity constraints The non-negativity constraints
Are as many as the variables as many as the variables in the dual,
In the primal, i.e. 2 i.e., 4
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