Solving Linear
Programming Problems
The Primal-Dual
Relationship
Shubhagata Roy
Concept of Duality
One part of a Linear Programming Problem (LPP) is called
the Primal and the other part is called the Dual. In other
words, each maximization problem in LP has its
corresponding problem, called the dual, which is a
minimization problem. Similarly, each minimization problem
has its corresponding dual, a maximization problem. For
example, if the primal is concerned with maximizing the
contribution from the three products A, B, and C and from
the three departments X, Y, and Z, then the dual will be
concerned with minimizing the costs associated with the
time used in the three departments to produce those three
products. An optimal solution from the primal and the dual
problem would be same as they both originate from the
same set of data.
Rules for Constructing the Dual from Primal
1. A dual variable is defined for each constraint in the primal
problem,i.e, the no. of variables in the dual problem is equal
to no. of constraints in the primal problem and vice-versa. If
there are m constraints and n variables in the primal
problem then there would be m variables and n constraints
in the dual problem.
2. The RHS of primal,i.e, b1,b2,……,bm, become the
coefficients of dual variables(Y1,Y2,……,Ym) in the dual
objective function(ZY). Also the coefficients of primal
variables(X1,X2,……,Xn),i.e, c1,c2,……,cn, become RHS of
the dual constraints.
3. For a maximization primal problem(with all < or =
constraints), there exists a minimization dual problem(with
all > or = constraints) and vice-versa.
4.The matrix of coefficients of variables in dual problem is
the transpose of matrix of coefficients in the primal
problem and vice-versa.
5. If any of the primal constraint (say ith) is an equality then
the corresponding dual variable is unrestricted in sign
and vice-versa.
Relation Constraints
…….
…….
b1
b2
…….
< or =
< or =
…….
Xn
a1n
a2n
…….
Ym
Relation
Constraint
X1
a11
a21
Primal Variables
…….
X2
a12
…….
…….
a22
…….
Y1
Y2
…….
Dual Variables
The Primal-Dual Relationship
am1
> or =
c1
am2
> or =
c2
…….
…….
…….
amn
> or =
cn
< or =
bm
Min ZY
Max ZX
Example:
Maximize ZX = 6000X1+4000X2
s.t. 4X1 + X2 <or= 12
9X1 + X2 <or= 20
7X1 + 3X2 <or= 18
10X1 +40X2 <or= 40
X1,X2 >or= 0
Dual
Variables
Primal Dual Relationship
Y1
Y2
Y3
Y4
Relation
Constraint
Primal Variables
X1
X2
4
1
9
1
7
3
10
40
> or =
> or =
6000
4000
Relation
Constraints
< or =
< or =
12
20
18
40
Min ZY
< or =
< or =
Max ZX
Hence the Dual Problem looks like,
Minimize ZY = 12Y1+20Y2+18Y3+40Y4
subject to the constraints,
4Y1+9Y2+7Y3+10Y4 >or= 6000
Y1+ Y2+3Y3+40Y4 >or= 4000
Y1,Y2 >or= 0