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Sensitivity Analysis of Transportation Problems
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Journal of Applied Sciences Research, 3(8): 668-675, 2007
© 2007, INSInet Publication
Sensitivity Analysis of Transportation Problems
N.M. Badra
Department of Physics and Eng. Math., Faculty of Engineering, Ain Shams University, Egypt.
Abstract: This paper presents a sensitivity analysis to multiobjective transportation problem. The proposed
approach yields to maximum tolerance percentages in multiobjective transportation problem. The weighted
sum approach is used to solve the multiobjective transportation problem. The proposed approach allows
changing in both the weights and the objective function coefficients and the right hand side constraints
simultaneously and independently from their specified values while remaining the same basis optimal.
Formulation of both perturbed solution and the corresponding perturbed objective values are presented.
An illustrative example is presented to clarify the idea of the proposed approach.
Key words: Sensitivity analysis; Parametric programming; transportation problem; Tolerance approach.
INTRODUCTION
Sensitivity analysis is to analyze the effect of the
changes of the objective function coefficients and the
effect of changes of the right hand side constraints on
the optimal value of the objective function as well as
the validity ranges of these effects. Studies on the
sensitivity analysis in linear programming and
multiobjective programming had been developed [1 -6 ].
The implementation of sensitivity analysis is based
primarily on basic optimal solutions. Degeneracy is a
complex phenomenon that may influence the
computation of optimal basic solutions and sensitivity
analysis. Therefore, it may be difficult to determine the
ranges of parameter values for which a given optimal
solution remains optimal for the transportation problem.
Thus, it has been pointed out that when degeneracy
occurs, the sensitivity analysis approach to determine
sensitivity ranges of parameters yields impractical
results. Apparently traditional sensitivity analysis is not
suitable for transportation problem. Saaty and Gass [7 ]
worked on perturbations in the objective function
coefficients and in the right hand side terms of the
constraints. They expressed these coefficients as linear
functions of parameters. Gal [8 ] had been expressed these
coefficients in matrix form. W endell [9 ] had presented
tolerance approach to sensitivity analysis in linear
programming. His approach had considered changing in
the objective function coefficients and in the right hand
terms of constraints simultaneously and independently.
The objective function coefficients and the right hand
terms of constraints were changing within the
maximum tolerance percentage of their specified values
simultaneously and independently while, maintaining
same optimal solution. B arnett [ 1 0 ] worked on
perturbations in the matrix coefficients. He described
an algorithm for finding a rectangular region within
which the components of a column vector may vary.
Shetty[1 1 ] had studied the case where a single element
of the coefficient matrix was perturbed. The usual
analysis of this case, yielded the largest interval in
which the same basis was optimal. In general, this
interval may not be closed. Saaty and Kim [1 2 ,1 3 ] had
considered perturbations in a column with a scalar
parameter. In fact, the analysis of a column
perturbation with a scalar parameter can be reduced to
the analysis of an element of the coefficient matrix
with a scalar parameter. In contrast, the tolerance
approach considers a vector perturbation of the column
which cannot be further reduced. Ravi and W endell[1 4 ]
had presented tolerance approach to sensitivity analysis
of matrix coefficients in linear programming. Their
approach had considered changing in the elements of
a column or a row of the matrix coefficients
simultaneously and independently in a standard linear
programming problem. Their approach yielded to a
maximum tolerance percentage within which the
elements of a column may all vary simultaneously and
independently from their estimated values while still
retaining the same set of basic variables in an optimal
solution. Hansen, Labbe and W endell [1 5 ] had presented
perturbations in multiple linear programming. Their
approach determined the maximum percentage by
which all weights can deviate simultaneously and
independently from their estimated values while
retaining the same optimal basis. They also shown how
a priori bounds on the ranges of the weights can be
exploited to yield a larger maximum tolerance
percentage. Marmol and Puerto [1 6 ] had presented special
cases of the tolerance approach in multiobjective linear
programming. They enlarged the range of meaningful
regions of weights that can be handled easily from the
tolerance approach point of view. They represented the
information that the decision maker is able to offer the
relationships between the importance of different
objectives, in terms of marginal substitution rates.
They also had presented an algorithm which
reduced the
computational effort, in order to
Corresponding author: N.M. Badra, Department of Physics and Eng. Math., Faculty of Engineering, Ain Shams University, Egypt.
668
J. Appl. Sci. Res., 3(8): 668-675, 2007
get the maximum tolerance percentage. Lin and W en [1 7 ]
had presented two other types of sensitivity range for
the assignment problem. The first type of sensitivity
range has used to determine the range in which the
current optimal assignment remains optimal. The
second type of sensitivity range was to determine those
values of assignment model parameters for which the
rate of change of optimal value function remains
constant.
In this paper, the tolerance approach to sensitivity
analysis in multiobjective transportation problems is
presented. The weighted sum approach is used to
convert the multiobjective transportation problem into
a single transportation problem. The proposed approach
allows changing in both the weights, the objective
function coefficients and in the right hand side
constraints terms simultaneously and independently
from their specified values while remaining the same
solution optimal. The proposed approach can be
considered as an extension of W endell[9 ] tolerance
approach in a special case of linear programming and
in the case of multiobjective transportation problem.
Also, the proposed approach can be considered as
another tolerance approach of both Hansen, Labbe and
W endell[1 5 ] tolerance approach in a special case of the
transportation problem that allows changing in both the
weights, the objective functions coefficients and in the
right hand side terms simultaneously and independently
from their specified values while remaining the same
solution optimal. The proposed approach differs from
their approach in allowing both weighting and objective
functions coefficients to change parametrically
according to maximum certain specified tolerance
intervals while remaining the same solution optimal.
The weights and coefficients of each objective function
are changed parametrically according to certain
specified intervals on the equivalent objective function
coefficients from the ordinary analysis only to
simultaneous and independent perturbations of the
coefficients of nonbasic variables. Similarly, the
intervals for the right hand side terms are applied only
to simultaneous and independent perturbations of the
terms in those constraints that having basic variables in
the optimal tableau. The condition of balanced
transportation problem is taken into consideration
when dealing with the right hand side constraints. The
proposed tolerance approach uses only the data of
initial and final simplex tableau of the equivalent single
linear problem. Formulation of both obtained perturbed
solution and the corresponding perturbed objective
values are presented. An illustrative example is
presented to clarify the idea of the proposed approach.
c i j ( k ) x i j, k=1,2,… ,K ]
P(MTP): Min [
(1)
subject to:
x i j = a i , i=1,2,… ,m
(2)
x i j = b j , j=1,2,… ,n
(3)
ai =
x
i j
$ 0,
where,
c i j (k)
xij
ai
bj
b j ( Balanced condition )
i=1,2,… ,m,
(4)
j=1,2,… ,n
is the cost of transporting a unit of objective
k from source i to destination j,
is the amount transported from source i to
destination j,
i=1,2,… ,m are the availability at i source and
j=12,… ,n are the requirement at j destination.
The m constraints (2) represent the supplies and
the n constraints (3) represent the destinations. The
balanced condition (4) will lead to reduce the total
number of constraints given by either (2) or (3) by one
i.e., total number of constraints = m + n - 1. This
means that the number of optimal basic variables in the
final optimal simples tableau is equal to m + n – 1.
The previous multiobjective transportation problem
P(MTP) given by (1-4) can be put in matrix form as
follows:
P(MTP): Min [ C
subject to:
A X = B, X $ 0
where,
C (k)
X
A
B
( k )
X, k=1,2,… ,K ]
(5)
(6)
is a matrix of m x n dimensional represents
the K objective coefficients,
is a column vector of n dimensional represents
the decision variables,
is an (m+n-1) x ( m n ) matrix representing
constraint matrix and
is a column vector of (m+n–1) dimensional
represents the right hand side constraints.
Note that for any matrix D of r x c dimensional,
the notation D R i , i=1,2,… ,r will be used to denote the
i th row, while, the notation D C j , j=1,2,… ,c will be
used to denote the j th column. This means that the
matrix D can be written in row form as:
Problem Formulation: It is well known that the
transportation problem TP is a special case of linear
programming (LP). Consider the multiobjective
transportation problem P(MTP) of K objectives, each
of m sources and n destinations as follows:
669
J. Appl. Sci. Res., 3(8): 668-675, 2007
D = [ D
R 1
D
R 2
… D
R i
…. D
R r
]
(7)X
*
= [ x 1 * x 2 * … .. x m
*
n
]t
(13)
Also, the matrix D can be written in column form as:
This optimum solution will be divided into two
sets of variables, the set of m + n – 1 basic variables
D = [ DC1 DC2 … DCj … DCc ]
(8)X * B that takes nonnegative values and the set of nm –
m – n + 1 non basic variables X *N B that takes the zero
values as follows:Also, suppose that the index set of basic variables
is denoted by I and the index set of nonbasic variables
X * B = [ x B 1 * x B 2 * … x B m + n -1 * ] t
(14)
is denoted by J as follows:
X
I = {1,2,… , m+n-1}, J = {1,2,..., m n – m – n + 1:
i
j for all i = 1,2,.., m+ n-1 }
(9)
Min [ ë
X ]
A X = B
X, ë $ 0, ë
(10)
(6)
= 1
(11)
where,
ë is a row vector of K–dimensional representing the
objective function weights,
is an K x m n matrix representing the objective
function coefficients and
is an identity column vector of K – dimensional.
For simplicity, let,
C = ë
P (ëTP): Min [ C X ]
subject to:
o
C
o
= [ C
B
X, ë $ 0, ë Î = 1
X*
B1
= [ C
NB
]t = 0
(15)
C
… C
B2
C
NB1
… C
NB2
]
B m + n -1
(16)
]
N B m n -m -n +1
(17)
Also, the initial matrix constraints coefficients
A o will be divided in column form into two sets,
the set of m + n - 1 basic A o B that takes the identity
value only at the corresponding basic variable while
the others are zeros and the set of nm – m – n + 1
non basic A o N B that takes any values.
A
o
A
o
= [ A
B
A
C B1
= [ A
NB
= [
B
NB
= [
B
= [
NB
C NB1
C B2
A
…… A
C NB2
C B m + n -1
…… A
]
(18)
C N B n m -m -n +1
]
(19)
= [
1
2
… ..
B1
B2
NB1
C B1
C NB1
]
m + n -1
……
NB2
B m + n -1
..…
C B2
(20)
(21)
N B m n -m -n +1
..…
C NB2
]
..…
]
C B m + n -1
(22)
]
C N B n m -m -n +1
(23)
]
It is clear that an optimum solution of problem
P (ëTP) is an efficient solution for problem P (MTP).
Also, note that if X * is an efficient solution of
(11) problem P(MTP), then, there exists ë = (ë ë … ë )
1
2
K
such that X * is an optimal solution of problem
of problem P (ëTP)
P (ëTP).
(6)
solution
C
= [
(12)
A X = B
*
m n -m -n +1
W hile, in the final optimum simplex tableau the
pervious matrices will be defined as follows:
Thus, the equivalent single transportation problem
P(ëTP) is given by:
The optimum
is given by: -
= [ x NB1* x NB2* … x NB
NB
Also, the initial objective function coefficients C o
will be divided into two sets of coefficients, the set of
m+n-1 basic coefficients of the final optimal tableau
C o B and the set of nm–m–n+1 non basic coefficients
of the final optimal tableau C o N B and can be expressed
as follows:
W eighting
Approach:
Several
approaches for
solving
P(MTP) whether direct, scalarization,
interactive or fuzzy approaches. The scalarization
approach is the most common approach and the
weighting approach is the most common one
among them. W eighting approach can be used for
transforming the multiobjective transportation problem
P(MTP) into single transportation problem P(ëTP) as
follows:
P (ëTP):
subject to:
*
670
(24)
J. Appl. Sci. Res., 3(8): 668-675, 2007
Perturbed Problem: Perturbations to the right hand
side constraints and to the objective function
coefficients to the problem P(ëTP) can be viewed as a
perturbed problem P(PëTP) as follows:
P(PëTP): Min [
( 1 + á
i j
) c
i j
= ( 1 + â i ) a i, i=1,2,… ,m
x
i j
= ( 1 + ã
j
i j
- ä
(25)
]
) b j, j=1,2,… ,n
- T
(1 + â i) a i =
(1 + ã
j
) b
(Balanced condition)
x
i j
$ 0,
j=1,2,… ,n
R
where,
á ij are the multiplicative parameters of c i j
representing the allowable tolerance percentage of
the objective function coefficients perturbations,
â I are the multiplicative parameters of á i representing
the allowable tolerance percentage of the supplies
and
ã j are the multiplicative parameters of b j representing
the allowable tolerance percentage of the
destinations.
A X = ( B ± Ä )
X, ë $ 0
i
# ä* ,
i 0 I
(33)
*
# á
i j
# T *,
( i, j ) 0 J
(34)
ä, á , ë, C
= { ä, á, ë, C: - ä * # ä i # ä *,
i 0 I
- T * # a i j # T *,
( i, j ) 0 J
(1 - ä i ) b i # b i # (1 + ä i ) b i , i 0 I
(1 - á i j ) c i j # ë
# (1 + á i j ) c i j,
(i, j ) 0 J
ë $ 0, ë
= 1 }
(35)
The tolerance approach aims at finding this critical
region of the parameters of both weights of the
objectives, objective functions coefficients and the right
hand side constraints at which any change inside their
ranges of this region does not affect the optimal basis,
while, any change outside their rages will affect the
optimal basis.
The multiplicative parameters á i j must not
exceed a non negative number ô. W hile, the both
multiplicative parameters â i and ã j must not exceed a
non negative number ñ.
P(PëTP): Min [ ( C ± Ù ) X ]
subject to:
# ä
Thus, the critical region of the parameters of the
objective functions coefficients and the right hand side
constraints perturbations is denoted by
R ä, á , ë, C and is defined as follows:
j
(28)
i=1,2,… ,m,
*
Also, suppose that the maximum allowable
tolerance for the objective function coefficients
perturbations T * having the property that the same
(26) basis is optimal in the perturbed problem P(PëP) as
long as the absolute value of each parameter T j
satisfies the following:
(27)
subject to:
x
i j x
property that the same basis is optimal in the perturbed
problem P(PëP) as long as the absolute value of each
parameter ä i satisfies the following:
Theorem: (W endell [9]): The value ä * that represents
the maximum allowable tolerance for the right hand
(29) side constraints perturbations can be determined as
follows:-
(30)
(36)
where,
Ù = [ á
Ä = [ ä
11
1
ä
á
2
12
… .. á
… .. ä
mn
m + n -1
]
]t
W hile, the value ô * that represents the maximum
allowable tolerance for the objective function
coefficients perturbations can be determined as
(32)
follows:-
(31)
Let,
T * is a multiplicative parameter of C that represents
the maximum allowable tolerance percentage of the
objective function coefficients perturbations and
ä * is a multiplicative parameter of B that represents
the maximum allowable tolerance percentage of the
right hand side constraints perturbations.
(37)
Note that all the information of ä * and T * is
readily available from the original and final optimal
simplex tableau. Moreover, the critical region of the
parameters of both weights of the objectives, objective
functions coefficients and the right hand side
Definition: (Critical Region of the Parameters):
Suppose that the maximum allowable tolerance for the
right hand side constraints perturbations ä * having the
671
J. Appl. Sci. Res., 3(8): 668-675, 2007
Table 1: D ata for the m ultiobjective
4
1
4
6
1
4
1
1.5
D em ands
60
60
constraints at which any change inside their ranges
of this region does not affect the optimal basis can
be determined from (35). The values of perturbed
optimal basis
perturbed objective
values
coefficients
and perturbed objective value
can be determined as follows:
=
+
=
=
Ä = [ ä
Ù
B
NB
+
1
ä
= { á
2
i j
Table 2: Initial sim plex tableau for the illustrative exam ple.
all
-----------------------------------------------------------------------------x 22 x 23 u 1 u 2
u4 B
(38)
Basic x 1 1 x 1 2 x 1 3 x 2 1
u3
1
u1
1
1
0
0
0
1
0
0
0
90
0
u (39)
0
0
1
1
1
0
1
0
0
90
2
1
u3
0
0
1
0
0
0
0
1
0
60
0
u4
1
0
0
1
0
0
0
0
1
60
(40)
- z
-4
-3
-2
-1
-3
-4
0
0
0
0
0
Ä
- Ù
+
. Ù
. Ä +
… .. ä
m + n -1
Ù
NB
B
]t
| ( i, j ) 0 I }, Ù
(41)
NB
= { á
i j
| ( i, j ) 0 J}
(42)
where,
Ä
is a column vector of m+n-1 – dimensional
represents the optimal basis variables in the final
optimal simplex tableau,
is a square matrix of m+n-1 dimensional
represents the coefficients of initial basis in the
final optimal simplex tableau,
is a column vector of m+n-1 dimensional
represents the allowable tolerance in the right
hand side constraints perturbations as defined in
the critical region,
is a column vector of m+n-1 dimensional
represents the allowable tolerance in objective
function coefficients perturbations of basis
variables,
is a column vector of mn-m-n+1 dimensional
represents the allowable tolerance in the
objective function coefficients perturbations of
non basic variables and
is an m+n-1 x mn-m-n+1 matrix represents the
column coefficients constraints matrix of non
basis variables in the final optimal simplex
tableau.
A X = B
X $ 0
( k )
X, k=1,2]
Table 3: Final optim al sim plex tableau for the illustrative exam ple.
all
-----------------------------------------------------------------------------x 22 x 23 u 1 u 2
Basic x 1 1 x 1 2 x 1 3 x 2 1
u3 u4 B
0
x 13
0
1
0
0
1
1
1
-1 -1
60
-1
x 22
0
0
0
1
1
0
1
-1 0
30
1
x 21
0
0
1
0
0
0
0
1
0
60
1
x 12
1
0
0
0
-1
0
-1
1
1
30
- z
3
0
0
0
0
2
-2
-2
1
-1
-360
X = [ x 11 x 12 x 13 x 21 x 22 x 23 ] t
(45)
The multiobjective transportation problem P(MTP1)
can be transformed into single transportation problem
P(ëTP1) by using the weighting sum approach as
follows:
P (ëTP1):
subject to:
Min [ C X ]
A X = B
(46)
X, ë $ 0,
ë = 1
where,
5. Illustrative Example: Consider the multiobjective
transportation problem P(MTP1) of two sources ( m=2)
and three destinations ( n=3 ) as follows:
P(MTP1): Min [C
subject to:
transportation problem .
3
90
0.5
2
90
7
60
Total = 180
ël = [ ë 1 ë 2 ], C = ë
(43)
(44)
,
= [1 1]
t
(47)
In this example, the weights of the objective
functions are chosen by [0.6 0.4] so that the resultant
objective function coefficients C is given by:
C = [ 4 3 2 1 3 4 ]
(48)
The results of the original and final optimal
simplex tableau for the example are given in Table 2
and Table 3.
From Tables 2 and 3, the required data for
perturbations are given as follows:
The data for the multiobjective transportation
problem is given in Table 1, where, the cell (i, j) in
the north-west corner gives the c i j ( 1 ) while, the cell
in the south-east corner gives the c i j ( 2 ). Each row
corresponds to a supply point and each column
corresponds to a demand point. The total supply equals
the total demand.
I = {x1 2 , x 13 , x 21 , x 22 }, J = {x1 1 , x 23 , u 1 , u 2 , u 3 , u 4 }
672
(49)
J. Appl. Sci. Res., 3(8): 668-675, 2007
For simplicity, the decision variables x 1 1 , x 1 2 , x 1 3 ,
x 1 4 , x 1 5 and x 1 6 will be denoted by numbers 1, 2, 3, 4,5
and 6 respectively. W hile, the decision variables u 1 , u 2 ,
u 3 and u 4 will be denoted by numbers 7, 8, 9 and 10
respectively. Thus,
R
(50)
= { ä, á, ë, C:
- T * # áj # T
(1 - ä i ) b i # b i #
(1 - á j ) c j # ë 1 c j (1 ) +
I = { 2, 3, 4, 5 }, J = { 1, 6, 7, 8, 9, 10 }
A
o
B
= [ A
R2
A
R3
A
R4
A
R5
The critical region of the parameters of the
objective functions coefficients and the right hand side
constraints perturbations R ä, á , ë, C of the problem
P(ëTP1) given by (36) and is defined as follows:
ä, á , ë, C
] =
(1 + ä i ) a i =
Ao
C
C
–1
B
o
B
o
Ao
NB
NB
- ä * # ä i # ä *, i0{2,3,4,5}
,
j 0 {1,6,7,8,9,10}
(1 + ä i ) b i, i 0 {2,3,4,5}
ë 2 c j (2 ) # (1 + á j) c j, j 0
{1,6,7,8,9,10}
*
(1 + ä j ) b
j
=[A o B -1 R 2 A o B -1 R 3 A o B -1 R 4 A o B -1 R 5 ]=
= [ c2 c3 c4 c5 ] = [ 3 2 1 3 ]
= [ c 1 c 6 c 7 c 8 c 9 c 10 ] = [ 4 4 0 0 0 0 ]
= [ A oC 1 A oC 6 A oC 7
AoC8 AoC9 AoC10 ] =
(51)
The maximum allowable tolerance for the right
hand side constraints perturbations ä * of the problem
P(ëTP1) given by (37) and can be calculated as
follows:
(52)
Note that, for the balanced condition, the following
perturbed condition must be satisfied:á 1 + á 2 = â
condition)
1
+ â
2
+ â
3
ä * =0.1429, T *=0.1111, ë 1 , ë 2 $0, ë 1 + ë 2 = 1}
(Balanced perturbed
(53)
(55)
The values of perturbed optimal basis
,
perturbed objective values coefficients
and
perturbed objective value
of the problem P(W P1)
are given by (22-24) and can be determined as follows:
The maximum allowable tolerance for the
objective function coefficients perturbations T * of the
problem P(ëTP1) given by (38) and can be calculated
as follows:
(56)
(54)
673
J. Appl. Sci. Res., 3(8): 668-675, 2007
(57)
= 360 + [2 2 -1 1]
+[60+ä 1 ä 2 -ä 3 -ä 4
30+ä 2 -ä 3 60+ä 3 30-ä 2 +ä 3 +ä 4 ]
(58)
where,
Ä = [ ä
1
ä
2
ä
3
ä
4
]t
Ù B = [á 2 á 3 á 4 á 5 ] t,
Ù N B = [á 1 á 6 á 7 á 8 á
C
o
B
= [ 3 2 1 3 ], C
9
á 10 ] t
o
NB
= [ 4 4 0 0 0 0 ]
simultaneously and independently from their specified
values while remaining the same solution optimal. The
proposed approach can be considered as an extension
of W endell[1 2 ] tolerance approach in a special case of
linear programming and in the case of multiobjective
transportation problem. Also, the proposed approach
can be considered as another tolerance approach of
both Hansen, Labbe and W endell[1 8 ] tolerance approach
in a special case of the transportation problem that
allows changing in both the weights, the objective
functions coefficients and in the right hand side terms
simultaneously and independently from their specified
values while remaining the same solution optimal. The
condition of balanced transportation problem is taken
into consideration when dealing with the right hand
side constraints. The proposed approach differs from
their approach in allowing both weighting and objective
functions coefficients to change parametrically
according to maximum certain specified tolerance
intervals while remaining the same solution optimal.
The proposed tolerance approach aims at finding the
balanced region of the parameters of both weights of
the objectives, objective functions coefficients and the
right hand side constraints at which any change inside
their ranges of this region does not affect the optimal
basis. The proposed tolerance approach uses only the
data of initial and final simplex tableau of the
equivalent single linear problem. Formulation of both
obtained perturbed solution and the corresponding
perturbed objective values are presented. An illustrative
example is presented to clarify the idea of the proposed
approach.
REFERENCES
Ao
NB
1.
=
2.
3.
(59)
Conclusion: This paper presents the tolerance approach
to sensitivity analysis in multiobjective transportation
problem. The weighted sum approach is used to
convert the multiobjective transportation problem into
single transportation problem. The proposed approach
allows changing in both the weights, objective
functions coefficients and in the right hand side terms
4.
5.
674
Gallagher, R. and O. Saleh, 1994. “Constructing
the Set of Efficient Objective Values in Linear
Multiple Objective Transportation Problems ”,
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