Download Duality Theory for Interval Linear Programming Problems G. Ramesh and K. Ganesan

IOSR Journal of Mathematics (IOSRJM)
ISSN: 2278-5728 Volume 4, Issue 4 (Nov-Dec, 2012), 39-47
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Duality Theory for Interval Linear Programming Problems
1
G. Ramesh and 2K. Ganesan
1,2
Department of Mathematics, Faculty of Engineering and Technology, SRM University, Kattankulathur,
Chennai - 603203, India.
Abstract : We define the primal and dual linear programming problems involving interval numbers as the way
of traditional linear programming problems. We discuss the solution concepts of primal and dual linear
programming problems involving interval numbers without converting them to classical linear programming
problems. By introducing new arithmetic operations between interval numbers, we prove the weak and strong
duality theorems. Complementary slackness theorem is also proved. A numerical example is provided to
illustrate the theory developed in this paper.
Keywords: Interval Numbers, Interval Arithmetic, Linear Programming, Weak Duality, Strong Duality,
Complementary Slackness.
I.
INTRODUCTION
Linear programming is a most widely and successfully used decision tool in the quantitative analysis of
practical problems where rational decisions have to be made. In order to solve a Linear Programming Problem,
the decision parameters of the model must be fixed at crisp values. But to model real-life problems and perform
computations we must deal with uncertainty and inexactness. These uncertainty and inexactness are due to
measurement inaccuracy, simplification of physical models, variations of the parameters of the system,
computational errors etc. Interval analysis is an efficient and reliable tool that allows us to handle such problems
effectively.
Linear programming problems with interval coefficients have been studied by several authors, such as
Atanu Sengupta et al. [2, 3], Bitran [5], Chanas and Kuchta [6], Nakahara et al. [20], Steuer [26] and Tong
Shaocheng [31]. Numerous methods for comparison of interval numbers can be found as in Atanu Sengupta and
Tapan Kumar Pal [2, 3], Ganesan and Veeramani [8, 9] etc.
By taking maximum value range and minimum value range inequalities as constraint conditions, Tong
Shaocheng [31] reduced the interval linear programming problem in to two classical linear programming
problems and obtained an optimal interval solution to it. Ramesh and Ganesan [24] proposed a method for
solving interval number linear programming problems without converting them to classical linear programming
problems.
The duality theory for inexact linear programming problems was proposed by Soyster [27–29] and
Thuente [30]. Falk [7] provided some properties on this problem. However, Pomerol [23] pointed out some
drawbacks of Soyster‟s results and provided some mild conditions to improve them. Masahiro Inuiguchia [14]
et al has studied the duality of interval number linear programming problems through fuzzy linear programming
problems. Bector and Chandra [4] introduced a pair of linear primal-dual problems under fuzzy environment
and established the duality relationship between them. Hsien-Chung Wu [12,13] introduced the concept of
scalar product for closed intervals in the objective and inequality constraints of the primal and dual linear
programming problems with interval numbers. He introduced a solution concept that is essentially similar to the
notion of nondominated solution in multiobjective programming problems by imposing a partial ordering on the
set of all closed intervals. He then proved the weak and strong duality theorems for linear programming
problems with interval numbers. Rohn [25] also discussed the duality in a interval linear programming problem
with real-valued objective function. In this paper, we attempt to develop the duality theory for interval linear
programming problems without converting them to classical linear programming problems.
The rest of this paper is organised as follows: In section 2, we recall the definitions of interval number
linear programming, interval numbers and some related results of interval arithmetic on them. In section 3, we
define the interval number primal and dual linear programming problems as the way of traditional linear
programming problems. We then prove the weak and strong duality theorems. Complementary Slackness
theorem is also proved. In section 4, a numerical example is provided to illustrate the theory developed in this
paper.
II. PRELIMINARIES
The aim of this section is to present some notations, notions and results which are of useful in our further
consideration.
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Duality theory for interval linear programming problems
Let
IR = {a = [a1, a 2 ] : a1  a 2 and a1, a 2  R}
be
the
set
of
all
proper
intervals
and
IR = {a = [a1 , a 2 ] : a1 > a 2 and a1 , a 2  R} be the set of all improper intervals on the real line R. If a = a1 = a 2 = a,
then a = [a, a] = a is a real number (or a degenerate interval). We shall use the terms “interval” and “interval
number” interchangeably. The mid-point and width (or half-width) of an interval number are defined as
a + a2 
 =  1
The mid-point and width (or half-width) of an interval number a  [a1 , a 2 ] are defined as m(a)
 and
 2 
a -a
 =  2 1  . The interval number a can also be expressed in terms of its midpoint and width as
w(a)
 2 
a  [a1 ,a 2 ]  m(a),
 w(a)
 .
2.1. A New Interval Arithmetic
Ming Ma et al.[18] have proposed a new fuzzy arithmetic based upon both location index and
fuzziness index function . The location index number is taken in the ordinary arithmetic, whereas the
fuzziness index functions are considered to follow the lattice rule which are the least upper bound and
greatest lower bound in the lattice L. That is for a, b  L we define a  b = max{a,b} and a  b = min{a,b}.
For any two intervals a = [a , a ], b = [b , b ]  IR and for * +, -, ·, ÷ , the arithmetic operations on a and b
1
are defined as:
a * b = [a1 , a 2 ]*[b1, b2 ] =
2
1
2

 w(b)
  m(a)
 max w(a),
 .
 w(a)
  m(b),
  m(b),
 w(b)
m(a),
In particular

 w(b)
  m(a)
 max w(a),
 .
 w(a)
  m(b),
  m(b),
 w(b)
(i). Addition : a + b = m(a),

 w(b)
  m(a)
 max w(a),
 .
 w(a)
  m(b),
  m(b),
 w(b)
(ii). Subtraction : a - b = m(a),

 w(b)
  m(a)
 max w(a),
 .
 w(a)
  m(b),
  m(b),
 w(b)
(iv). Multiplication : a  b = m(a),
 w(b)
  m(a)
 max{w(a),
 , provided m(b)
  0.

 w(a)
  m(b),
  m(b),
 w(b)}
(v). Division : a  b  m(a),
III.
Main Results
Now we are in a position to prove interval analogue of some important relationships between the
primal and dual linear programming problems. We consider the primal and dual linear programming problems
involving interval numbers as follows:
Consider the following linear programming problem involving interval numbers
n
(P)
maxz   c j x j
j=1
n
subject to  a ij x j  b i , i = 1, 2, 3…m
j=1
and x j  0 for all j  1, 2,3, ., n,
where a ij , c j , x j , b i  IR, i = 1, 2,3,…….m and j = 1, 2,3,……., n.
(3.1)
We call the above problem (P) as the primal interval linear programming problem, and it can be
rewritten as
 x  b and x  0 ,
  subject to A
(P)
max z  cx
3.2
Let


 x are (m  n) , (m 1), (1 n),(n 1) matrices involving interval numbers.
where A, b, c,
 x  b,
 x  0} be the feasible region of problem (3.1). We say that x  is a
X = {x = (x 1, x 2, x 3, ......, x n ) : A


feasible solution to the primal problem (3.1), if x  X . A feasible solution x  X is said to be
  *  cx

cx
an optimum solution to the primal problem (3.1),
for all x  X ,
where
n
c x =  c j x j = (c1x 1 + c 2 x 2 + c 3 x 3 +...... + c n x n ) and c = (c1,c 2,c 3, ......., c n ).
j1
Now we consider the following linear programming problem involving interval numbers:
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Duality theory for interval linear programming problems
m
   b i y i
min w
(D)
i=1
m
subject to  a ij y i  c j , j = 1, 2, 3…n
i=1
and y i  0 for all i  1, 2,3, ., m,
where a ij , c j , x j , b i  IR, i = 1, 2,3,…….m and j = 1, 2,3,……., n.
(3.3)
We call the above problem (D) or (3.3) as the dual interval linear programming problem of the primal
problem (P), and it can be rewritten as
 y  c and y  0 ,
  subject to A
  by
(D)
min w
 3.4


 y are (m  n), (1 × m), (n × 1), (m × 1) matrices involving interval numbers.
where A, b, c,
 y  c, y  0 be the feasible region of problem (3.3). We say that y  is a
Let Y  y   y 1 , y 2 , y 3 , · · · , ym  : A
feasible solution to the dual problem (3.3), if y   Y. A feasible solution y   Y is said to be an optimum
m
    by
  for all y  Y , where by
  =  b y = b y  b y + b y + · · · + b y and
solution to the dual (3.3) if by
i i
1 1
2 2
3 3
m m
i 1
b = (b 1, b 2, b 3, ......., b n ).
Standard Form:
For the general study, we convert the given interval number linear programming problem into its standard form
 x  b and x  0,
 where A,
 b,
 c,
  subject to A
 x are (m n) , (m1), (1n), (n1) matrices
as max z  cx
consisting of interval numbers.
 x  b . If all x  [-α ,α ] for some α  0 , then x is said to be
Definition 3.1. Let x = (x 1, x 2 ,x 3 ,...,x n) solves A
j
j
j
j
a
basic
solution.
If
x j [-α j ,α j ]
some α j  0,
for
then x
has
some
non-zero
components,
 x  b can be written as:
say x 1, x 2, x 3, ......, x k , 1  k  n. Then A
a 1x 1 + a 2 x 2 + a 3 x 3 +...+ a k x k + a k+1[-βk+1 ,βk+1 ] + a k+2[-βk+2 ,βk+2 ] +...+ a n [-βn ,βn ]  b
If the columns a 1 ,a 2 ,a 3 ,...,a k corresponding to these non-zero components x 1 , x 2 , x 3 ,....., x k are linearly
independent, then x is said to be a basic solution.
Remark 3.1. Given a system of m simultaneous linear equations involving interval numbers in n unknowns
 is an (m n) interval matrix and rank of A
 is m. Let B
 x  b , b  IR m , where A
 be any (m m)
(m  n) A
.
interval matrix formed by m linearly independent columns of A
-1 
T
-1 
 
 is
Let x = B b = (x , x , x ..., x ) or simply x = B b = (x , x , x ..., x ) and x = (x , x , x ,..., x ,0,0,...,0)
B
1
2
3
m
B
1
2
3
m
1
2
3
m
a basic solution. In this case, we also say that x B is a basic solution.
 x  b . where A
 x  b .
 = (a ) , a  IR. Then x = B-1b is a solution of A
Theorem 3.1. Consider A
ij m n
ij
B
Theorem 3. 2 (Weak duality theorem) If x = (x 1 , x 2 , x 3 ,..., x n ) is any feasible solution to the primal interval
linear programming problem (3.1) and y   y 1 , y 2 , y 3 , · · · , y m  is any feasible solution to the dual interval
linear programming problem (3.3), then
n
m
j1
i 1
  or  c x   b y .
   by
cx
j j
i i
Proof. Since x is a feasible solution to the primal interval linear programming problem (3.1), we have
n
 a x  b , x  0, i  1, 2, 3, · · · , m.
j i
n
ij
j
i
j
   ,
 m  a ij  , w  a ij  m  x j  , w  x j   m b i , w b i
j i
x j  0,i  1, 2, 3, · · · , m.
Multiplying the ith (i  1, 2, 3, · · · , m) primal constraint by y i  m  y i  , w  y i  and adding we have
    m  y  , w  y  , i  1, 2, 3, · · · , m
  m  b  , w  b  m  y  , w  y 
 m  a ij  , w  a ij  m  x j  , w  x j  m  y i  , w  y i   m b i , w b i
n
j i
m
n
i 1
j i
  m  a ij  , w  a ij  m  x j  , w  x j  m  y i  , w  y i 
i 1
i
 .
  m  a ij  , w  a ij  m  x j  , w  x j  m  y i  , w  y i    b i y i  by
m
n
i 1 ji
i
i
m
m
i 1
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i
i
i
(3.5)
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Duality theory for interval linear programming problems
Multiplying the jth (j = 1, 2, 3,…,n) dual constraint by x j  m  x j  , w  x j  and adding we have
m
 a ij y i x j  c j x j , j = 1, 2, 3,…, n
i=1
m
 m  a ij  , w  a ij  m  y i  , w  y i  m  x j  , w  x j   m  c j  , w  c j  m  x j  , w  x j  , j = 1, 2, 3,…, n
i=1
n
m
n
j1
m
i=1
n
j=1
n
  m  a ij  , w  a ij  m  y i  , w  y i  m  x j  , w  x j    m  c j  , w  c j  m  x j  , w  x j 
  m  a ij  , w  a ij  m  y i  , w  y i  m  x j  , w  x j    m  c j  , w  c j  m  x j  , w  x j 
i=1 j1
j=1
 .
  m  a ij  , w  a ij  m  y i  , w  y i  m  x j  , w  x j    c j x j  cx
m
n
n
i=1 j1
(3.6)
j=1
From equations (3.5) and (3.6), we have
 m  c j  , w  c j  m  x j  , w  x j     m  a ij  , w  a ij  m  y i  , w  y i  m  x j  , w  x j 
n
m
n
i=1 j1
m
j=1
    m  y  , w  y 
m  c  , w  c  m  x  , w  x    m  b  , w  b  m  y  , w  y 
  m b i , w b i
i
i 1
n
(3.7)
i
m
 
j
j=1
j
n
m
j=1
i 1
j
j
i
i 1
i
i
i
 .
   by
cx
 
    c j x j   b i y i  by
cx
Proposition 3.1. Suppose that x  = (x 1 , x 2 , x 3 ,..., x n ) ) and y    y 1 , y 2 , y 3 , · · · , y m  are feasible solutions to
   , then x  and y  are optimal solutions to
    by
the primal (3.1) and the dual (3.3) respectively, such that cx
the primal and dual problems respectively.
   . From the weak duality theorem, we have cx
    cx
    by
    by
  for all x  X ⇒ cx
    cx
  for
Proof. Let cx
    cx
    by
  for all y   Y.
   for all y    y , y , y , · · · , y   Y ⇒ by
all x  X . Similarly, by
1
Proposition
 ,
   by
If cx
3.2.
then
there
2
exist
3
m

x  X and
y   Y such that
    cx
  for
cx
all






x  X and by  by for all y  Y. That is x is an optimum solution to the primal problem (3.1) and y an
optimum solution to the dual problem (3. 3).
   0 , there exist x  X and y   Y such that cx
   . Then the results follow
   by
    by
Proof. Since cx
immediately from proposition (3.1).
Theorem 3. 3 (Strong duality theorem)
  ( x1, x2 , x3 ,...xn ) is an optimal solution to the primal problem (3.1), then there exit a feasible
If x
 .
   by
  ( y1, y 2 , y3 ,... y m ) to the dual problem (3.3) such that cx
solution y
Proof: We convert the primal problem (3.1) to its standard form by adding slack variables as follows:
n
n
(3.8)
max z   c x subject to  a x  x  b , i  1,2,...,m and x  0
j1
j
j
j1
ij
j
n i
i
j
for all j  1,2,...,n, n  1,...,n  m, where x n i are slack variables.
 x  b and x  0 ,
  subject to A
That is max z  cx
 b,
 c,
 x are
where A,
 m   n  m , (m 1), (1 (n  m)), ((n  m) 1)
(3.9)
matrices consisting of interval
 b is an optimal basic feasible solution to (3.8), where B is the corresponding basis
numbers. Let x B  B
 B
 1a .
matrix, c B is the cost vector corresponding to x B . We know that z j  c B y j so that y jB
j
,
-1


c B B a j , for j = 1, 2,3,..., n.
Also (z j - c j ) = (c B y j - c j ) = 

 -1

(c B B e j - 0), for j = n +1, n + 2, n + 3,..., n + m
1
 1b is an optimal basic feasible solution to (3.8), we have (z  c )  0 for all j. So that
Since x B  B
j
j
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Duality theory for interval linear programming problems
 0 .
 1a  c )  0 and (c B
 1e  0)
(z j  c j )  0  (c B B
j
j
B
j
-1
-1
 - c )  0 and c B
 a - c )  0 and c B
  0  (c B
 -1A
 -1  0 .
 (c B B
j
j
B
B
j
B
-1
-1
  c and c B
 A
  0 .
 c B B
B
  c  y A
  c  A
 y  c  y is a feasible solution to the dual.
 1A
 1 , then y  0 and c B
Suppose that y  c B B
B
   yb
 1b  c B
 

   by.
   c BB
 is an optimal solution to the primal (3.8). Hence cx
Also by
B  cx , when ever x
Theorem 3.4. (Complementary Slackness theorem)
If x  = (x 1 , x 2 , x 3 ,..., x n ) is a feasible solution to the primal (3.1) and y    y 1 , y 2 , y 3 , · · · , y m  is a
feasible solution to the dual (3.3), then they must satisfy the so-called complementary slackness conditions:
n
n
 then  a x   b . (ii).  a x   b , then y   0.

(i). If y   0,
i
ij
j=1
j
i
j=1
ij
j
m
m
i=1
i=1
i
i
 then  a y   c . (iv). If  a y   c , then x   0.

(iii). If x j  0,
ij i
j
ij i
j
j
Proof. If x  is a feasible solution to the primal (3.1) and y  is a feasible solution to the dual (3.3), then from
the strong duality theorem x  is an optimal solution to the primal (3.1) and y  is an optimal solution to the dual
(3.3) such that from equation (3.7), we have
n
m
n
m

 

 c x    a x y   b y .
j
j1
n
j
i 1
j i
i. e.  m(c j ), w(c j )
j1
ij
j
i
i
i 1
i
m
n
i 1
m
j i
m(x j ), w(x j )    m(a ij ), w(a ij )
  m(b i ), w(b i )
m(x j ), w(x j )
m(y i ), w(y i )
m(y i ), w(y i ) .
i 1
Now
(3.10).
from equation (3.10), we have
m n
m
   m(a ij ), w(a ij ) m(x j ), w(x j ) m(y i ), w(y i )   m(b i ), w(b i ) m(y i ), w(y i ) .
i 1 j i
m n
i 1
m
   m(a ij ), w(a ij ) m(x ), w(x ) m(y ), w(y )   m(b i ), w(b i ) m(y i ), w(y i )  0.
i 1 j i
m
n

j

j

i


i
i 1



    m(a ij ), w(a ij ) m(x j ), w(x j )  m(b i ), w(b i )  m(y i ), w(y i )  0.
i 1  j i

n









   m(a ij ), w(a ij ) m(x j ), w(x j )  m(bi ), w(b i )  m(y i ), w(y i )  0.
 j i

n
  m(a ij ), w(a ij ) m(x j ), w(x j )  m(b i ), w(b i )  0 (or) m(y i ), w(y i )  0.

j i
n



  m(a ij ), w(a ij ) m(x ), w(x )  m(b i ), w(b i )

j
j i
n
  a ij x j  b i
j i
(or)

j
(or)
m(y i ), w(y i )  0.
y i  0
n
n
 then  a x   b and (ii) if  a x   b , then y   0.

So, (i) if y i  0,
ij j
i
ij j
i
i
ji
ji
m
n
n
i 1
ji
j1
Similarly from equation (3.10), we have   a ij x j y i   c jx j
m
n
n
That is   m(a ij ), w(a ij ) m(x j ), w(x j ) m(y i ), w(y i )   m(c j ), w(c j )
i 1 ji
m
j1
n
n
   m(a ij ), w(a ij ) m(x j ), w(x j ) m(y i ), w(y i )   m(c j ), w(c j )
i 1 ji
j1
m
    a ij y i  c j  x j  0
i1
ji 

n
m
m(x j ), w(x j )
m(x j ), w(x j )  0
m
   a ij y i  c j  x j  0
i1

m
  a ij y *i - c j  0 (or) x *j  0   a ij y *i  c j (or) x *j  0
i=1
i=1
m
m
i=1
i=1
 then  a y   c and (iv). If  a y   c , then x   0.

So (iii). If x  0,
ij i
j
ij i
j
j

j
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Duality theory for interval linear programming problems
IV.
Numerical Examples
Example 4.1. Consider the following interval number linear programming problem:
(P)
Max z = [29,31]x 1 +[22, 24]x 2 +[28,30]x 3
subject to constraints 6x 1 + 5x 2 + 3x 3  [25, 27]
4x 1 + 2x 2 + 5x 3  [6,8]

and x , x , x  0.
1
2
3
(4.1)
We
call
the above problem as the
primal problem. Then the corresponding dual problem is given by
  [25, 27]y 1 +[6,8]y 2
Min w
subject to constraints 6y 1 + 4y 2  [29,31]
5y 1 + 2y 2  [22, 24]
3y 1 + 5y 2  [28,30]

and y 1 , y 2  0.
(D)
(4.2)
(i). Optimal solution to the primal interval number linear programming problem:
Let us apply the interval version of simplex algorithm and the new interval arithmetic to solve the primal
problem. The standard form of the given primal interval number linear programming problem based upon both
location index (mid point) and fuzziness index function (width) as:
Max z  30,1 x 1 + 23,1 x 2 + 29,1 x 3 + 0s1 + 0s 2
subject to constraints 6x 1 + 5x 2 + 3x 3 + s1 + 0s 2  26,1
4x 1 + 2x 2 + 5x 4 + 0s1 + s 2  7,1

and
x 1 , x 2 , x 3 ,s1 ,s 2  0.
Initial iteration: Initial basic feasible solution is given by s1 = 26,1 , s 2 = 7,1 .
c j
x B
30,1
x 1
23,1
x 2
29,1
x 3
0
s1
0
s 2
c B
0
y B
s1
26,1
6
5
3
1
0
0
s 2
7,1
 4
2
5
0
1
z j

(z j - c j )
0
-30,1
0
-23,1
0
-29,1
0
0
0
0
θ
4.33,1
1.75,1
 for some j, the current basic feasible solution is not Optimal.
Since (z j  c j )  0,
First iteration: Here s 2 leaves the basis and x 1 enters in to the basis
c B
0
30,1
y B
s1
x 1
c j
x B
15.5,1
1.75,1
z j
(z j - c j )
30,1
x 1
23,1
x 2
29,1
x 3
0
1
30,1
0, 0
2
0.5
15,1
-8,1
-4.5
1.25
37.5,1
8.5,1
0
s1
1
0
0, 0
0, 0
0
s 2
-1.5
0.25
7.5,1
7.5,1
θ
7.75,1
3.5,1
 for some j, the current basic feasible solution is not Optimal.
Since (z j  c j )  0,
The improved basic feasible solution is given by s1 = 15.5,1 , x 1 = 1.75,1 .
Second iteration: Here x 1 leaves the basis and x 2 enters in to the basis
c B
0
23,1
y B
s1
x 2
c j
x B
30,1
x 1
23,1
x 2
29,1
x 3
0
s1
0
s 2
8.5,1
3.5,1
z j
(z j - c j )
-4
2
46,1
16,1
0
1
23,1
0,0
-9.5
2.5
57,1
28.5,1
1
0
0,0
0,0
-2.5
0.5
11.5,1
11.5,1
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θ
44 | Page
Duality theory for interval linear programming problems
Since (z j - c j )  0
for all j, the current basic feasible solution is optimal. The optimal solution
is s1 = 8.5,1 , x 2 = 3.5,1 and max z = 80.5,1 .
Hence the optimal solution for the given primal interval number linear programming problem is
x 1 = [0,0], x 2 = [2.5,4.5] and
maxz = [79.5,81.5].
(ii). Optimal solution to the dual interval number linear programming problem:
The standard form of the given dual interval number linear programming problem based upon both location
index (mid point) and fuzziness index function (width) as:
  26,1 y 1 + 7,1 y 2 + 0s1 + 0s 2 + 0s3 + MR 1 + MR 2 + MR 3
min w
(D)
subject to 6y 1 + 4y 2 - s1 + 0s 2 + 0s3 + R1 + 0R 2 + 0R 3  30,1
5y 1 + 2y 2 + 0s1 - s 2 + 0s3 + 0R1 + R 2 + 0R 3  23,1
3y 1 + 5y 2 + 0s1 + 0s 2 - s3 + 0R1 + 0R 2 + R 3  29,1
and y 1 , y 2 ,s1 ,s 2 s3 0
 = 30,1 , R
 = 23,1 and R
 = 29,1 .
Initial iteration: Initial basic feasible solution is given by R
1
2
3

C
B
M
M
M

Y
B
R 1
R 2
R 3
b j

X
B
30,1
23,1
29,1
j
w
 j  b j )
(w
26,1
7,1
0
0
0
s3
0
0
-1
-M
M
R
1
M
R
2
M
R
y 1
6
(5)
3
14M
y 2
4
2
5
11M
s1
-1
0
0
-M
s 2
0
-1
0
-M
1
0
0
M
0
1
0
M
0
0
1
M
14M - 26,1
11M - 7,1
-M
-M
-M
0
0
0
3
θ
5,1
4.6,1
9.6,1
 for some j, the current basic feasible solution is not Optimal.
 j  b j )  0,
Since (w
First iteration: Here R 2 leaves the basis and y 1 enters in to the basis
b j
26,1
7,1
0
0
0 M M
c B
y B
x B
y 1
y 2
s1
s 2
s3 R 1 R 2
M
R 1
2.4,1
0
1.0
-1
1.2
0 1 0
26,1 y 1
4.6,1
1
0.4
0
0
0 0 0
M
R 3
15.2,1
0
3.8
0
0
-1 0 1
w j
26,1
5.4M +10.4,1 -M
1.8M - 5.2,1 -M M M

(w j  b j ) 0,0
5.4 + 3.4,1
-M 1.8M - 5.2,1 -M 0 0
θ
1.5,1
11.5,1
4,1
 for some j, the current basic feasible solution is not Optimal.
 j  b j )  0,
Since (w
  2.4,1 , y  4.6,1 , R
  15.2,1 .
The improved basic feasible solution is given by R
1
1
3
Second iteration: Here R 1 leaves the basis and y 2 enters in to the basis
c B
7,1
26,1
M
y B
y 2
y 1
R
3
b j
x B
1.5,1
26,1
y 1
0
7,1
y 2
1
4,1
9.5,1
j
w
 j  b j )
(w
1
0
26,1
0
0
7,1
0.25
2.37
2.37M + 2.12,1
0, 0
0, 0
2.37M + 2.12,1
0
s1
-0.62
0
s 2
0.75
-0.5
-2.25
-2.25M - 7.75,1
-2.25M - 7.75,1
0
s3
0
M
R 3
0
0
-1
-M
0
1
M
-M
0
θ
.....
16,1
4,1
 for some j, the current basic feasible solution is not Optimal.
 j  b j )  0,
Since (w
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Duality theory for interval linear programming problems
The improved basic feasible solution is given by y 2 = 1.5,1 , y 1 = 4,1 , R 3 = 9.5,1 .
Third iteration: Here R 3 leaves the basis and s1 enters in to the basis
b j
26,1
7,1
0
0
cB
s1
s 2
y B
x B
y1
y 2
7,1
y 2
4,1
0
1
0
0.16
26,1
y 1
3,1
1
0
0
-0.27
0,0
s1
4,1
0
0
1
-0.94

wj
26,1
7,1
0,0
-5.9,1

 -b )
(w
0,0
0,0
0,0
-5.9,1
j
j
0
s3
-0.26
0.10
-0.42
0.88,1
θ
...
28.57,1
...
0.88,1
 for some j, the current basic feasible solution is not Optimal.
 j  b j )  0,
Since (w
The improved basic feasible solution is given by y 2  4,1 , y 1  3,1 ,s1  4,1 .
Fourth iteration: Here y 1 leaves the basis and s 3 enters in to the basis
b j
26,1
7,1
0
c B
y B
x B
y 1
y 2
s1
7,1
0, 0
0, 0
y 2
s3
s1
0
s 2
0
s3
11.51,1
2.50
1
0
-0.5
0
28.57,1
16.02,1
j
w
  b )
(w
9.52
4
17.5,1
0
0
7,1
0
1
0, 0
-2.57
-2.02
-3.5,1
1
0
0, 0
-8.5,1
0, 0
0, 0
-3.5,1
0, 0
j
j
θ
...
...
...
 for all j, the current basic feasible solution is Optimal. The optimal solution is
 j  b j )  0,
Since (w
 = 80.5, 1 .
y 2 = 11.51,1 , s3 = 28.57,1 , s1 = 16.02,1 and min w
Hence the optimal solution for the dual interval number linear programming problem is
 = [79.5, 81.5].
y 1 = [0 ,0], y 2 = [10.5 ,12.5], y 3 = [0 ,0] and min w
From the optimal solutions for the primal and dual interval number linear programming problem, we see that
 = [79.5, 81.5] and Dual : minw = [79.5, 81.5].
Primal : maxz

Hence Primal : maxz = [79.5, 81.5] = Dual: min w
We see that both primal and dual problems have optimal solutions and the two optimal values are equal. Also
both optimal solutions obey the strong duality theorem.
III.
CONCLUSION
We introduced the notation of primal and dual linear programming problems involving interval
numbers as the way of traditional linear programming problems. We discuss the solution concepts of primal and
dual linear programming problems involving interval numbers without converting them to classical linear
programming problems. Under new arithmetic operations between interval numbers, we have proved the weak
and strong duality theorems. Complementary slackness theorem is also proved. These results will be useful for
post optimality analysis. A numerical example is provided to show that both primal and dual problems have
optimal solutions and the two optimal values are equal.
Acknowledgements
The authors are grateful to the anonymous referees and the editors for their constructive comments and
suggestions.
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