This presentation is based on PRIMAL & DUAL realed to operation research.

1. PRIMAL & DUAL PROBLEMS
OPERATION RESEARCH
Submitted by :
Khambhayata Mayur (130010119042)
Khant Vijaykumar (130010119045)
Lad Yashkumar (130010119047)
Submitted to :

2. Content
 Duality in Linear Programming Problems
 Different useful aspects
 Points to remember while converting into dual
 Conversion of primal to its dual
 Important modification
 Relationship between solutions of Primal and Dual
 Example

3. 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 programs are very closely related and optimal solution of dual
gives complete information about optimal solution of primal and vice
versa.

4. Different useful aspects
 If primal has large number of constraints and small number of variables,
computation can be considerably reduced by converting problem to Dual
and then solving it.
 Economic interpretation of the dual helps the management in making future
decision.
 Calculation of the dual checks the accuracy of the primal solution.
 Duality is used to solve LPP in which the initial solution is infeasible.

5. Points to remember while converting into dual
 If the primal contains n variables and m constraints, the dual will contain
m variables and n constraints.
 The maximization problem in the primal becomes the minimization
problem in the dual and vice versa.
 The maximization problem has (≤) constraints while the minimization
problem has (≥) constraints.
 The constants c1, c2, …, cn in the objective function of the primal appear
in the constraints of the dual.
 The constants b1, b2, …, bm in the constraints of the primal appear in the
objective function of the dual.
 The variables in both problems are non-negative.

6. Conversion of primal to its dual
The general L.P.P. or primal in
canonical form is:
Maximize z = c1x1+c2x2+…+cnxn
Subject to a11x1+a12x2+…+a1nxn ≤ b1
a21x1+a22x2+…+a2nxn≤b2
…………..……………………………………
…………………………………………………
am1x1+am2x2+…+amnxn ≤ bm
where, x1, x2, ...,xn, all ≥ 0.
Now its dual is:
Minimize w = b1y1+b2y2+…+bmym
Subject to a11y1+a21y2+…+am1ym ≤ c1
a12y1+a22y2+…+am2ym ≤ c2
…………..…………………………………………..
………………………………………………………...
a1ny1+a2ny2+…+amnym ≤ cn
where, the dual variables y1, y2,…, ym,
all ≥ 0.

7. Important modification
 Objective function :
If the objective function is maximum in Primal then it will change to
minimize in Dual LPP and vice versa.
 Co-efficient of objective function :
Co-efficient of objective function in Primal LPP will be the RHS of
constraints in Dual LPP.
 Constraints :
RHS of constraints in Primal LPP will function as co-efficient of
objective function in Dual LPP.

8.  Co-efficient of decision variables in each constraints :
Transverse matrix of Primal LPP is applicable to the co-efficient of
decision variables in each constraint for Dual LPP.
 Constraint type :
Primal LPP Dual LPP
Constraint type ≤ type Constraint type ≥ type
Constraint type ≥ type Constraint type ≤ type
Primal constraint “ i ” is = type Dual variable “ Yj ” unrestricted in sign
Primal variable “ Xi ” unrestricted in sign Dual constraint “ j ” is = type

9.  Relationship between solutions of Primal and Dual
 Values for the non-basic variables of the Primal are given by the base
raw of the Dual solution, under the slack variables (if any) neglecting the
–ve sign if any and under the artificial variables neglecting –ve sign as
well as deleting M.
 Values of sack variables of the Primal are given by the base row under
the non-basic variables of the Dual solution neglecting the –ve sign if
any.
 The values of the objective function is same for Primal and Dual
problem.

10. Example
Primal :
Dual :