Ó Indian Academy of Sciences
Sādhanā Vol. 41, No. 5, May 2016, pp. 531–539
DOI 10.1007/s12046-016-0491-x
Uncertain multi-objective multi-product solid transportation problems
DEEPIKA RANI* and T R GULATI
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247 667, India
e-mail: [email protected]
MS received 17 July 2014; revised 28 April 2015; accepted 18 December 2015
Abstract. The solid transportation problem is an important generalization of the classical transportation
problem as it also considers the conveyance constraints along with the source and destination constraints. The
problem can be made more effective by incorporating some other factors, which make it useful in real life
situations. In this paper, we consider a fully fuzzy multi-objective multi-item solid transportation problem and
present a method to find its fuzzy optimal-compromise solution using the fuzzy programming technique. To take
into account the imprecision in finding the exact values of parameters, all the parameters are taken as trapezoidal
fuzzy numbers. A numerical example is solved to illustrate the methodology.
Keywords. Solid transportation problem; fuzzy optimal-compromise solution; fuzzy programming technique;
trapezoidal fuzzy number; multi-product (item).
1. Introduction
The need of generalization of traditional transportation problem to solid transportation problem arises when different kinds
of conveyances are available for the transportation of goods to
save time as well as money. It was first stated by Schell [1] and
later on Haley [2] described its solution procedure. The
multi-objective multi-item solid transportation problem
(MOMISTP) is further generalization of the solid transportation problem. It deals with optimizing multiple objectives and
using different types of conveyances to transport heterogeneous products from the warehouses to the consumer points.
This type of transportation problem is very beneficial in many
industries, where more than one kind of products are shipped.
In multiple objective problems, the objectives are generally
conflicting in nature, so the concept of optimal solution is
replaced by optimal compromise solution also called efficient
solution or pareto optimal solution or non-dominated solution.
In the conventional solid transportation problem, it is
assumed that all the parameters are known exactly and
many algorithms have been developed to solve these
problems. But in real world situations, it is not always so.
Due to uncontrollable factors like lack of information and
uncertainty in judgement, the values of the transportation
parameters, i.e., unit cost of transportation, availability and
demand are not exact always. This impreciseness in the
values of the parameters can be represented by using the
fuzzy set theory given by Zadeh [3]. A systematic study of
fuzzy mathematical programming has also been given by
Bector and Chandra [4]. Many authors have used the fuzzy
*For correspondence
numbers to represent the uncertainty in transportation
parameters and proposed methods to solve them. The
MOMISTPs in which all the parameters are represented by
fuzzy numbers are called fully fuzzy multi-objective multiitem solid transportation problems (FFMOMISTPs).
Fuzzy programming technique for the multi-objective
transportation problems was given by Zimmermann [5]. Bit
et al [6] applied the fuzzy programming technique to solve
MOSTP. Li et al [7] solved the multi-objective solid
transportation problem (MOSTP) using the genetic algorithm in which only objective function coefficients are taken
as fuzzy numbers. Liu and Liu [8] presented the expected
value model in fuzzy programming. Islam and Roy [9]
studied the geometric programming approach for the multiobjective transportation problems. Ojha et al [10] proposed
methods to solve fuzzy MOSTP, where all the parameters
except decision variables are taken as fuzzy numbers. Gupta
et al [11] proposed a method, called Mehar’s method, to find
the exact fuzzy optimal solution of unbalanced fully fuzzy
multi-objective transportation problems.
Uncertainty theory based expected constrained programming for the solid transportation problems in uncertain
environment is studied by Cui and Sheng [12]. Recently,
Kundu et al [13] have proposed a method to find the crisp
optimal compromise solution of the fuzzy MOMISTP,
using the fuzzy programming technique and global criterion
method. Solid transportation problems are also studied by
Baidya et al [14] and Kundu et al [15]. Ebrahimnejad [16]
studied the transportation problem with generalized trapezoidal fuzzy numbers. To the best of our knowledge, no
method has been proposed in the literature to find the fuzzy
optimal compromise solution of the FFMOMISTP.
531
532
D Rani and T R Gulati
In this paper, a method is proposed to find the fuzzy
optimal compromise solution of FFMOMISTP. The application of the proposed method is shown by obtaining the
fuzzy optimal compromise solution of the numerical
example for which Kundu et al [13] found the crisp optimal
compromise solution. Since the proposed problem is the
generalization of the traditional solid transportation problem, it is also applicable to solve both single and multiobjective solid transportation problems, single or multiobjective transportation problems (both for single and
multi-item).
e 2 ¼ ða1 þ a2 ; b1 þ b2 ; c1 þ c2 ; d1 þ d2 Þ
e1 A
(i) A
e 2 ¼ ða1 d2 ; b1 c2 ; c1 b2 ; d1 a2 Þ
e
(ii) A 1 A
e 1 ¼ ðka1 ; kb1 ; kc1 ; kd1 Þ; k 0
(iii) k A
ðkd1 ; kc1 ; kb1 ; ka1 Þ; k 0
e
e
(iv) A 1 A 2 ¼ ða; b; c; dÞ,
where
a ¼ minða1 a2 ; a1 d2 ; d1 a2 ; d1 d2 Þ,
b ¼ minðb1 b2 ; b1 c2 ; c1 b2 ; c1 c2 Þ,
c ¼ maxðb1 b2 ; b1 c2 ; c1 b2 ; c1 c2 Þ,
d ¼ maxða1 a2 ; a1 d2 ; d1 a2 ; d1 d2 Þ:
2. Preliminaries
3. Mathematical model
In this section, some basic definitions and ranking approach
for trapezoidal fuzzy numbers are presented.
A FFMOMISTP with parameters as trapezoidal fuzzy
numbers can be stated as a transportation problem with R
objectives in which l different items are to be transported
from m sources (Si ; 1 i m) to n destinations
(Dj ; 1 j n) via K different conveyances. Let e
a pi denote
p
the fuzzy availability of item p at source Si ; e
b j denote the
fuzzy demand of item p at destination Dj ; e
e k be the total
fuzzy capacity of kth conveyance, e
c rp
ijk be the fuzzy penalty
for transporting one unit of item p from Si to Dj via kth
x pijk be the fuzzy
conveyance for rth objective Zr and e
quantity of item p to be transported from Si to Dj using kth
conveyance in order to minimize R objective functions. The
problem is mathematically modeled as follows:
2.1 Basic definitions [17, 18]
e defined on the universal
Definition 1 A fuzzy number A
e ¼ ða; b; c; dÞ, is said to
set of real numbers R, denoted as A
be a trapezoidal fuzzy number if its membership function
le ðxÞ is given by
A
8
ðx aÞ
>
>
>
>
>
ðb
aÞ
>
>
<
1
le ðxÞ ¼
A
ðx
dÞ
>
>
>
>
>
ðc
dÞ
>
>
:
0
a x\b
bxc
c\x d
otherwise.
e ¼ ða; b; c; dÞ
Definition 2 A trapezoidal fuzzy number A
is said to be zero trapezoidal fuzzy number if and only if
a ¼ 0; b ¼ 0; c ¼ 0 and d ¼ 0.
e ¼ ða; b; c; dÞ
Definition 3 A trapezoidal fuzzy number A
is said to be non-negative trapezoidal fuzzy number if and
only if a 0.
e1 ¼
Definition 4 Two trapezoidal fuzzy numbers A
e
ða1 ; b1 ; c1 ; d1 Þ and A 2 ¼ ða2 ; b2 ; c2 ; d2 Þ are said to be equal
if and only if a1 ¼ a2 ; b1 ¼ b2 ; c1 ¼ c2 and d1 ¼ d2 ; and is
e 2.
e1 ¼ A
denoted by A
Remark 1 If for a trapezoidal fuzzy number
e ¼ ða; b; c; dÞ; b ¼ c, then it is called a triangular fuzzy
A
number and is denoted by (a, b, b, d) or (a, b, d) or
(a, c, d).
2.2 Arithmetic operations [17]
e 1 ¼ ða1 ; b1 ; c1 ; d1 Þ and A
e 2 ¼ ða2 ; b2 ; c2 ; d2 Þ be two
Let A
trapezoidal fuzzy numbers. Then
ðPÞ
e 1; Z
e 2 ; . . .; Z
e RÞ
Minimize ð Z
subject to
n X
K
X
e
x pijk e
a pi ; 1 i m; 1 p l;
j¼1 k¼1
m X
K
X
p
e
x pijk e
b j ; 1 j n; 1 p l;
i¼1 k¼1
l X
m X
n
X
e
x pijk e
e k ; 1 k K;
p¼1 i¼1 j¼1
where ex pijk is a non-negative trapezoidal fuzzy number,
P P Pn PK
c rp for all i; j; k; p and Zer ¼ lp¼1 m
i¼1
j¼1
k¼1 ðe
ijk
e
x pijk Þ; 1 r R:
For the above problem to be balanced it should satisfy:
Pm p Pn
p
(i)
a i ¼ j¼1 e
b j ; 1 p l, i.e., for an item, its
i¼1 e
total availability at all sources should be equal to its
demand at all the destinations.
Pl Pm p Pl Pn
P
p
(ii)
a i ¼ p¼1 j¼1 e
ek,
b j ¼ Kk¼1 e
i.e.,
p¼1
i¼1 e
overall availability/demand of all the items at all
the sources/destinations and total conveyance capacity should be equal.
Uncertain multi-objective multi-product solid transportation
Definition 5 A fuzzy feasible solution e
x ¼ fex pijk g of (P) is
said to be a fuzzy optimal compromise (efficient) solution if
there exists no other feasible solution e
y ¼ fe
y pijk g such that,
l X
m X
n X
K
X
Rðe
c rp
y pijk Þ
ijk e
l X
m X
n X
K
X
Rðe
c rp
x pijk Þ
ijk e
for all r
p¼1 i¼1 j¼1 k¼1
and
l X
m X
n X
K
X
Rðe
c rp
y pijk Þ
ijk e
p¼1 i¼1 j¼1 k¼1
\
x0p Þg þ maxf0; ðzp yp Þ ðz0p y0p Þg þ maxf0;
ðy0p ðwp zp Þ ðw0p z0p ÞgÞ. Now go to Step 2.
Ps
a pi ¼
Step 2: After Step 1, we get
i¼1 e
Pt
p
e
j¼1 b j for all p; 1 p l, where s ¼ m or m þ 1 and
P Pt
Pl Ps
p
e
e
t ¼ n or n þ 1, i.e.,
ap ¼ l
b ¼
p¼1
p¼1 i¼1 j¼1 k¼1
l X
m X
n X
K
X
Rðe
c rp
x pijk Þ for atleast one r:
ijk e
p¼1 i¼1 j¼1 k¼1
4. Proposed method
In this section a method has been proposed to find the fuzzy
optimal compromise solution of Problem (P).
The proposed method consists of the following steps:
Step 1: Verify whether the problem under consideration is
balanced.
Pn
Pm p
ep
For all p; 1 p l, find
a i and
i¼1 e
j¼1 b j . Let
P
Pm p
p
a i ¼ ðxp ; yp ; zp ; wp Þ and nj¼1 e
b j ¼ ðx0p ; y0p ; z0p ; w0p Þ.
i¼1 e
P
P
p
a pi ¼ nj¼1 e
b j for all p; 1 p l. Go to Step 2.
Case 1: m
i¼1 e
P
P
p
b j for all or any p, 1 p l. To
Case 2: m
a pi 6¼ nj¼1 e
i¼1 e
Pm p Pn
p
a i ¼ j¼1 e
b j , proceed according to the
make
i¼1 e
following subcases. One or all may apply.
Subcase 2a: If xp x0p ; yp xp y0p x0p ; zp yp z0p y0p
and wp zp w0p z0p (for one or more values of p), then
introduce a dummy source with fuzzy availability of the
pth item(s) equal to ðx0p xp ; y0p yp ; z0p zp ; w0p wp Þ.
Subcase 2b: If xp x0p ; yp xp y0p x0p ; zp yp z0p y0p
and wp zp w0p z0p (for one or more values of p), then
introduce a dummy destination having fuzzy demand of
the pth item(s) equal to ðxp x0p ; yp y0p ; zp z0p ; wp w0p Þ.
Subcase 2c: None of the Subcases 2a or 2b is satisfied. In
such a situation add a dummy source with availability of
the pth item(s) equal to ðmaxf0; ðx0p xp Þg; maxf0; ðx0p xp Þg þ maxf0; ðy0p x0p Þ ðyp xp Þg; maxf0; ðx0p xp Þg
þ
maxf0; ðy0p x0p Þ ðyp xp Þg þ maxf0; ðz0p y0p Þ ðzp yp Þg; maxf0; ðx0p xp Þg þ maxf0; ðy0p x0p Þ ðyp xp Þg þ maxf0; ðz0p y0p Þ ðzp yp Þg
þ maxf0; ðw0p z0p Þ ðwp zp ÞgÞ and also a dummy destination having
demand of these item(s) as ðmaxf0; ðxp x0p Þg; maxf0;
ðxp x0p Þg þ maxf0; ðyp xp Þ ðy0p x0p Þg; maxf0; ðxp x0p Þg
þ maxf0; ðyp xp Þ ðy0p x0p Þg þ maxf0; ðzp 0
yp Þ ðzp y0p Þg; maxf0; ðxp x0p Þg þ maxf0; ðyp xp Þ 533
i¼1
i
p¼1
j¼1
j
ðu; v; w; aÞ (say).
P P
P P
p
Now, check whether lp¼1 si¼1 e
a pi ¼ lp¼1 tj¼1 e
bj ¼
P
PK
e k . Let Kk¼1 e
e k ¼ ðu0 ; v0 ; w0 ; a0 Þ.
k¼1 e
Pl Ps
P P
P
p
Case 1: If
a pi ¼ lp¼1 tj¼1 e
ek,
b j ¼ Kk¼1 e
p¼1
i¼1 e
i.e., the total availability of all the items, their demand
and the total conveyance capacity are equal, then go to
Step 3.
Pl Ps
P P
P
p
Case 2: If
a pi ¼ lp¼1 tj¼1 e
ek,
b j 6¼ Kk¼1 e
p¼1
i¼1 e
then proceed according to the following subcases:
Subcase 2a: If u u0 ; v u v0 u0 ; w v w0 v0 and
a w a0 w0 , then check whether in Step 1 a dummy
source and/or a dummy destination have been added and
proceed as below:
Case (i): If both a dummy source and a dummy
destination have been introduced, then increase their
total availability and demand by the fuzzy quantity ðu0 u; v0 v; w0 w; a0 aÞ (The demand of only those
items is to be increased whose availability has been
increased at the dummy source).
Case (ii): If only a dummy source (destination) has been
introduced, then increase its total availability (demand)
by the fuzzy number ðu0 u; v0 v; w0 w; a0 aÞ and
also introduce a dummy destination (source) having the
demand (availability) of these added items as
ðu0 u; v0 v; w0 w; a0 aÞ.
Case (iii): If neither a dummy source nor a dummy
destination has been added, then introduce a dummy
source with the total availability equal to the fuzzy
number ðu0 u; v0 v; w0 w; a0 aÞ and a dummy
destination with demand of these added items equal to
ðu0 u; v0 v; w0 w; a0 aÞ.
Subcase 2b: If u u0 ; v u v0 u0 ; w v w0 v0
and a w a0 w0 , then introduce a dummy conveyance having capacity ðu u0 ; v v0 ; w w0 ; a a0 Þ.
Subcase 2c: If neither Subcase 2a nor Subcase 2b
applies, then check whether in Step 1 a dummy source or
a dummy destination or both have been added and
proceed according to the following cases. Let ð
u; v;
aÞ ¼ ðmaxf0; u0 ug; maxf0; u0 ug þ maxf0; ðv0 w;
u0 Þ ðv uÞg; maxf0; u0 ug þ maxf0; ðv0 u0 Þ ðv uÞg
þ maxf0; ðw0 v0 Þ ðw vÞg; maxf0; u0 ug þ
maxf0; ðv0 u0 Þ ðv uÞg þ maxf0; ðw0 v0 Þ ðw 0 ; a0 Þ ¼
u0 ; v0 ; w
vÞg þ maxf0; ða0 w0 Þ ða wÞgÞ and ð
0
0
ðmaxf0; u u g; maxf0; u u g
þ maxf0; ðv uÞ ðv0 u0 Þg; maxf0; u u0 g þ maxf0; ðv uÞ ðv0 u0 Þg
þ maxf0; ðw vÞ ðw0 v0 Þg; maxf0; u u0 gþ maxf0;
534
D Rani and T R Gulati
ðv uÞ ðv0 u0 Þg
þ maxf0; ðw vÞ ðw0 v0 Þgþ
0
maxf0; ða wÞ ða w0 ÞgÞ.
Case (i): If both a dummy source and a dummy
destination have been introduced, then increase their
total availability and demand by the fuzzy quantity
aÞ (The demand of only those items is to be
ð
u; v; w;
increased whose availability has been increased at the
dummy source). Also, introduce a dummy conveyance
0 ; a0 Þ.
with capacity ð
u0 ; v0 ; w
Case (ii): If only a dummy source (destination) has been
introduced, then increase its total availability (demand)
aÞ and introduce a dummy
by the fuzzy number ð
u; v; w;
destination (source) having the demand (availability) of
aÞ. Also, introduce a
these added items as ð
u; v; w;
0 ; a0 Þ.
dummy conveyance with capacity ð
u0 ; v0 ; w
Case (iii): If neither a dummy source nor a dummy
destination has been added, then introduce both a dummy
source as well as a dummy destination with the
aÞ.
availability and demand of added items as ð
u; v; w;
Also, introduce a dummy conveyance with capacity
0 ; a0 Þ.
ð
u0 ; v0 ; w
Assume the unit transportation costs required due to the
dummy source/destination/conveyance to be zero trapezoidal fuzzy number.
Now, the problem is balanced and takes the form:
ðP01 Þ
Minimize
t X
K
X
l X
s X
t X
K
X
p
arp
ijk xijk
fR Þ
ðP0 Þ Minimize ðf
Z1 ; f
Z2 ; . . .; Z
subject to
t X
K
X
j¼1 k¼1
1pl
s X
K
X
0p 0p 0p
ðxpijk ; ypijk ; zpijk ; wpijk Þ ¼ ða0p
j ; bj ; cj ; dj Þ;
i¼1 k¼1
1 j t; 1 p l
l X
s X
t
X
1 k K;
where Zer ¼
Xl
X
Step 3: Corresponding to each objective, convert the
problem ðP0 Þ into following four crisp problems
ðP01 Þ ðP04 Þ:
ðP02 Þ
Minimize
l X
s X
t X
K
X
p¼1 i¼1 j¼1 k¼1
p¼1 i¼1 j¼1 k¼1
subject to
t X
K
X
xpijk ¼ api ; 1 i s; 1 p l
p
brp
ijk yijk
ypijk ¼ bpi ; 1 i s; 1 p l
j¼1 k¼1
s X
K
X
xpijk ¼ a0p
j ; 1 j t; 1 p l
ypijk ¼ b0p
j ; 1 j t; 1 p l
i¼1 k¼1
l X
m X
n
X
xpijk ¼ a00k ; 1 k K
ypijk xp
ijk
xpijk 0
Minimize
ypijk ¼ b00k ; 1 k K
p¼1 i¼1 j¼1
p¼1 i¼1 j¼1
l X
s X
t X
K
X
p
crp
ijk zijk
ðP04 Þ
Minimize
p¼1 i¼1 j¼1 k¼1
zpijk ¼ cpi ; 1 i s; 1 p l
j¼1 k¼1
l X
s X
t X
K
X
rp p
dijk
wijk
p¼1 i¼1 j¼1 k¼1
subject to
s X
K
X
X
subject to
l X
m X
n
X
t X
K
X
X
s
t
K
rp
rp
rp
ððarp
ijk ; bijk ; cijk ; dijk Þ
p¼1
i¼1
j¼1
k¼1
ðxpijk ; ypijk ; zpijk ; wpijk ÞÞ; 1 r R; ðxpijk ; ypijk ; zpijk ; wpijk Þ is a nonrp
rp
negative trapezoidal fuzzy number and e
c rp
ijk ¼ ðaijk ; bijk ;
p
rp
0p 0p 0p
a pi ¼ ðapi ; bpi ; cpi ; dip Þ; e
b j ¼ ða0p
ek ¼
crp
j ; bj ; cj ; dj Þ; e
ijk ; dijk Þ; e
ða00k ; b00k ; c00k ; dk00 Þ.
i¼1 k¼1
ðP03 Þ
ðxpijk ; ypijk ; zpijk ; wpijk Þ ¼ ða00k ; b00k ; c00k ; dk00 Þ;
p¼1 i¼1 j¼1
j¼1 k¼1
s X
K
X
ðxpijk ; ypijk ; zpijk ; wpijk Þ ¼ ðapi ; bpi ; cpi ; dip Þ; 1 i s;
subject to
t X
K
X
wpijk ¼ dip ; 1 i s; 1 p l
j¼1 k¼1
zpijk ¼ c0p
j ; 1 j t; 1 p l
i¼1 k¼1
s X
K
X
wpijk ¼ dj0p ; 1 j t; 1 p l
i¼1 k¼1
l X
m X
n
X
zpijk ¼ c00k ; 1 k K
p¼1 i¼1 j¼1
zpijk yp
ijk
l X
m X
n
X
ypijk ¼ dk00 ; 1 k K
p¼1 i¼1 j¼1
wpijk zp
ijk
Uncertain multi-objective multi-product solid transportation
Solve the four problems sequentially using the optimal
p
solution xp
ijk of ðP1 Þ in ðP2 Þ; optimal solution yijk of ðP2 Þ in
p
ðP3 Þ and optimal solution zp
ijk of ðP3 Þ in ðP4 Þ. Let wijk be the
optimal solution of ðP4 Þ. This leads to the fuzzy optimal
p p
p
0
solution e
x pijk ¼ ðxp
ijk ; yijk ; zijk ; wijk Þ of ðP Þ.
Step 4: Apply the fuzzy programming technique to obtain
the optimal-compromise solution and calculate the value of
each objective function.
535
Table 3. Unit transportation penalties for item 1 in the second
objective.
Destinations !
Sources #
Conveyance k ¼ 1
S1
S2
Conveyance k ¼ 2
S1
S2
D1
D2
D3
(4,5,7,8)
(6,8,9,11)
(3,5,6,8)
(5,6,7,8)
(7,9,10,12)
(6,7,9,10)
(6,7,8,9)
(4,6,7,9)
(5,7,9,11)
(4,6,8,10) (7,9,11,13) (9,10,11,12)
5. Numerical example
Consider the multi-objective multi-item solid transportation
problem solved by Kundu et al [13]. In this problem, the
number of destinations is three, while that of sources, items,
conveyances and objectives is two each. The authors have
proposed a method to find the crisp optimal compromise
solution. Since the fuzzy solution has more information than
the crisp one, we solve the same problem using the method
proposed by us to find the fuzzy optimal compromise solution. The data of the problem is as follows (tables 1–4):
From table 5, we find that for the first item, total availP
a 1i ¼ ð21; 24; 26; 28Þ ð28; 32; 35; 37Þ ¼ ð49;
ability 2i¼1 e
P
1
b ¼ ð14; 16; 19; 22Þ 56; 61; 65Þ and total demand 3 e
j¼1
j
ð17; 20; 22; 25Þ ð12; 15; 18; 21Þ ¼ ð43; 51; 59; 68Þ: SimP2
ilarly for the second item, total availability
a 2i ¼
i¼1 e
P3
2
b j = (51,58,63,71).
ð57; 62; 67; 72Þ and total demand j¼1 e
Table 1. Unit transportation penalties for item 1 in the first
objective.
Destinations !
Sources #
Conveyance k ¼ 1
S1
S2
Conveyance k ¼ 2
S1
S2
D1
D2
D3
(5,8,9,11)
(8,10,13,15)
(4,6,9,11)
(6,7,8,9)
(10,12,14,16)
(11,13,15,17)
(9,11,13,15) (6,8,10,12) (7,9,12,14)
(10,11,13,15) (6,8,10,12) (14,16,18,20)
Table 2. Unit transportation penalties for item 2 in the first
objective.
Destinations !
Sources #
Conveyance k ¼ 1
S1
S2
Conveyance k ¼ 2
S1
S2
D1
D2
D3
(9,10,12,13)
(11,13,14,16)
(5,8,10,12)
(7,9,12,14)
(10,11,12,13)
(12,14,16,18)
(11,13,14,15) (6,7,9,11)
(8,10,11,13)
(14,16,18,20) (9,11,13,14) (13,14,15,16)
Table 4. Unit transportation penalties for item 2 in the second
objective.
Destinations !
Sources #
Conveyance k ¼ 1
S1
S2
Conveyance k ¼ 2
S1
S2
D1
D2
D3
(5,7,9,10)
(10,11,13,14)
(4,6,7,9)
(6,7,8,9)
(9,11,12,13)
(7,9,11,12)
(7,8,9,10)
(6,8,10,12)
(4,5,7,8) (8,10,11,12)
(5,7,9,11) (9,10,12,14)
Table 5. Availability and demand data.
Fuzzy availability
Fuzzy demand
Conveyance
capacity
1
e 1 ¼ ð46; 49; 51; 53Þ
e
a 11 ¼ ð21; 24; 26; 28Þ e
b 1 ¼ ð14; 16; 19; 22Þ e
1
1
e 2 ¼ ð51; 53; 56; 59Þ
e
a 2 ¼ ð28; 32; 35; 37Þ e
b 2 ¼ ð17; 20; 22; 25Þ e
2
1
e
a 1 ¼ ð32; 34; 37; 39Þ e
b ¼ ð12; 15; 18; 21Þ
3
2
e
a 22 ¼ ð25; 28; 30; 33Þ e
b 1 ¼ ð20; 23; 25; 28Þ
2
e
b 2 ¼ ð16; 18; 19; 22Þ
2
e
b ¼ ð15; 17; 19; 21Þ
3
P2
P
P2
P
1
2
Since
a 1i 6¼ 3j¼1 e
a 2i 6¼ 3j¼1 e
b j and
b j , the
i¼1 e
i¼1 e
problem is unbalanced. Now, first step is to balance the
problem.
We find that neither Subcase 2a nor Subcase 2b of Step 1
holds for any of the items. So, according to Subcase 2c, we
introduce a dummy source (S3 ) having availabilities of the
first and second items as e
a 13 ¼ ð0; 1; 4; 9Þ and
e
a 23 ¼ ð0; 2; 2; 5Þ, respectively. Also, we introduce a dummy
destination (D4 ) with demand of the first and second items
1
2
as e
b4 ¼ e
b 4 ¼ ð6; 6; 6; 6Þ so that the total availability and
P
total demand of both the items become equal, i.e., 2p¼1
P3
P P
p
a pi ¼ 2p¼1 4j¼1 e
b j ¼ ð106; 121; 134; 151Þ:
i¼1 e
P2
e k ¼ ð97;
Since, the total conveyance capacity
k¼1 e
P2 P3
P2 P4
p
102; 107; 112Þ. Clearly,
a i ¼ p¼1 j¼1
p¼1
i¼1 e
536
D Rani and T R Gulati
P
p
e
b j 6¼ 2k¼1 e
e k . For (106, 121, 134, 151) = ðu; v; w; aÞ and
(97, 102, 107, 112) = ðu0 ; v0 ; w0 ; a0 Þ, the condition u u0 ; v u v0 u0 ; w v w0 v0 and a w a0 w0 is met and
so according to Subcase 2b of Step 2, we introduce a
dummy conveyance having capacity ee 3 = (9,19,27,39).
Thus the problem becomes balanced.
Since, a dummy source (S3 ), a dummy destination (D4 )
and a dummy conveyance are introduced so we assume
e
e
c rp
c rp
c rp
ij3 ¼ ð0; 0; 0; 0Þ for all r ¼ 1; 2; p ¼ 1; 2;
3jk ¼ e
i4k ¼
i ¼ 1; 2; 3; j ¼ 1; 2; 3; 4 and k ¼ 1; 2; 3. The obtained balanced problem can be written as
e 1 ¼ ð5; 8; 9; 11Þ e
Minimize Z
x 1111 ð9; 11; 13; 15Þ
e
x 1112 ð4; 6; 9; 11Þ e
x 1121 ð6; 8; 10; 12Þ e
x 1122 ð10; 12;
1
1
14; 16Þ ex 131 ð7; 9; 12; 14Þ e
x 132 ð8; 10; 13; 15Þ
1
1
e
x 211 ð10; 11; 13; 15Þ e
x 212 ð6; 7; 8; 9Þ e
x 1221 ð6; 8;
1
1
10; 12Þ ex 222 ð11; 13; 15; 17Þ e
x 231 ð14; 16; 18; 20Þ 1
2
e
x 111 ð11; 13; 14; 15Þ e
x 2112 ð5;
x 232 ð9; 10; 12; 13Þ e
2
2
x 122 ð10; 11; 12; 13Þ 8; 10; 12Þ e
x 121 ð6; 7; 9; 11Þ e
2
2
e
x 132 ð11; 13; 14; 16Þ e
x 2211 x 131 ð8; 10; 11; 13Þ e
2
2
ð14; 16; 18; 20Þ ex 212 ð7; 9; 12; 14Þ ex 221 ð9; 11; 3; 14Þ
ex 2222 ð12; 14; 16; 18Þ e
x 2231 ð13; 14; 15; 16Þ ex 2232 1
1
x 123 e
x 1133 e
x 1141 e
x 1142 e
x 1143 e
x 1213
ð0; 0; 0; 0Þ ðe
x 113 e
ex 1223 e
x 1233 e
x 1241 e
x 1242 e
x 1243 ex 1311 e
x 1312 e
x 1313 1
1
1
1
1
1
1
e
x3 2 2 e
x 3 2 3 ex 3 3 1 e
x3 3 2 e
x3 3 3 e
x3 4 1 e
x 13 4 2 x3 2 1 e
e
x 2113 e
x 2123 ex 2133 e
x 2141 e
x 2142 e
x 2143 e
x 2213 ex 2223
x 1343 e
2
2
2
2
2
2
2
ex 233 e
x 241 e
x 242 e
x 243 e
x 311 e
x 312 e
x 313 e
x 2321 e
x 2323 ex 2331 e
x 2332 e
x 2333 e
x 2341 ex 2342 ex 2343 Þ:
x 2322 e
1
e 2 ¼ ð4; 5; 7; 8Þ e
Minimize Z
x 111 ð6; 7; 8; 9Þ e
x 1112 x 1122 ð7; 9; 10; 12Þ e
x 1131
ð3; 5; 6; 8Þ e
x 1121 ð4; 6; 7; 9Þ e
1
1
ð5; 7; 9; 11Þ e
x 132 ð6; 8; 9; 11Þ e
x 211 (4,6,8,10)
1
1
e
x 212 ð5; 6; 7; 8Þ e
x 221 ð7; 9; 11; 13Þ e
x 1222 ð6; 7; 9; 10Þ
1
1
ex 231 ð9; 10; 11; 12Þ e
x 232 ð5; 7; 9; 10Þ ex 2111 ð7; 8;
x 21 2 1 ð4; 5; 7; 8Þ e
x 21 2 2 9; 1 0Þ e
x 21 1 2 ð4; 6; 7; 9Þ e
2
2
ð9; 11; 12; 13Þ e
x 131 ð8; 10; 11; 12Þ e
x 132 (10,11,13,14)
2
2
x 2221 ð5; 7; 9;
ex 211 ð6; 8; 10; 12Þ ex 212 ð6; 7; 8; 9Þ e
2
2
x 2232
11Þ ex 222 ð7; 9; 11; 12Þ ex 231 ð9; 10; 12; 14Þ e
ð0; 0; 0; 0Þ ðe
x 1113 e
x 1123 e
x 1133 ex 1141 e
x 1142 e
x 1143 1
1
1
1
1
1
1
e
x 241 e
x 242 e
x 243 e
x 311 e
x 1312 x 213 ex 223 ex 233 e
e
ex 1322 e
x 1313 ex 1321 x 1323 e
x 1331 e
x 1332 e
x 1333 e
x 1341 1
1
2
2
2
2
2
e
e
x 342 ex 343 ex 113 e
x 123 e
x 133 e
x 141 x 142 e
x 2143 e
e
x 2213 ex 2223 ex 2233 e
x 2241 e
x 2242 x 2243 e
x 2311 e
x 2312 2
2
2
2
2
2
2
e
x 313 e
x 321 ex 322 e
x 323 e
x 331 e
x 332 e
x 333 e
x 2341 e
x 2342 ex 2343 Þ
subject to
4 X
3
X
ðx11jk ; y11jk ; z11jk ; w11jk Þ ¼ ð21; 24; 26; 28Þ;
j¼1 k¼1
4 X
3
X
ðx21jk ; y21jk ; z21jk ; w21jk Þ ¼ ð32; 34; 37; 39Þ
j¼1 k¼1
4 X
3
X
ðx12jk ; y12jk ; z12jk ; w12jk Þ ¼ ð28; 32; 35; 37Þ;
j¼1 k¼1
4 X
3
X
ðx22jk ; y22jk ; z22jk ; w22jk Þ ¼ ð25; 28; 30; 33Þ
j¼1 k¼1
4 X
3
X
ðx13jk ; y13jk ; z13jk ; w13jk Þ ¼ ð0; 1; 4; 9Þ;
j¼1 k¼1
4 X
3
X
ðx23jk ; y23jk ; z23jk ; w23jk Þ ¼ ð0; 2; 2; 5Þ
j¼1 k¼1
3 X
3
X
ðx1i1k ; y1i1k ; z1i1k ; w1i1k Þ ¼ ð14; 16; 19; 22Þ;
i¼1 k¼1
3 X
3
X
ðx2i1k ; y2i1k ; z2i1k ; w2i1k Þ ¼ ð20; 23; 25; 28Þ
i¼1 k¼1
3 X
3
X
ðx1i2k ; y1i2k ; z1i2k ; w1i2k Þ ¼ ð17; 20; 22; 25Þ;
i¼1 k¼1
3 X
3
X
ðx2i2k ; y2i2k ; z2i2k ; w2i2k Þ ¼ ð16; 18; 19; 22Þ
i¼1 k¼1
3 X
3
X
ðx1i3k ; y1i3k ; z1i3k ; w1i3k Þ ¼ ð12; 15; 18; 21Þ;
i¼1 k¼1
3 X
3
X
ðx2i3k ; y2i3k ; z2i3k ; w2i3k Þ ¼ ð15; 17; 19; 21Þ
i¼1 k¼1
3 X
3
X
ðx1i4k ; y1i4k ; z1i4k ; w1i4k Þ ¼ ð6; 6; 6; 6Þ;
i¼1 k¼1
3 X
3
X
ðx2i4k ; y2i4k ; z2i4k ; w2i4k Þ ¼ ð6; 6; 6; 6Þ
i¼1 k¼1
2 X
3 X
4
X
ðxpij1 ; ypij1 ; zpij1 ; wpij1 Þ ¼ ð46; 49; 51; 53Þ;
p¼1 i¼1 j¼1
2 X
3 X
4
X
ðxpij2 ; ypij2 ; zpij2 ; wpij2 Þ ¼ ð51; 53; 56; 59Þ
p¼1 i¼1 j¼1
2 X
3 X
4
X
ðxpij3 ; ypij3 ; zpij3 ; wpij3 Þ ¼ ð9; 19; 27; 39Þ
p¼1 i¼1 j¼1
ðxpijk ; ypijk ; zpijk ; wpijk Þ for all i, j, k, p is a non-negative trapezoidal fuzzy number.
e 1 by solving the following four problems:
We minimize Z
1
Minimize Z1 ¼ 5x1111 þ 9x1112 þ 4x1121 þ 6x1122 þ 10x1131 þ
1
7x132 þ 8x1211 þ 10x1212 þ 6x1221 þ 6x1222 þ 11x1231 þ 14x1232 þ
9x2111 þ 11x2112 þ 5x2121 þ 6x2122 þ 10x2131 þ 8x2132 þ 11x2211 þ
14x2212 þ 7x2221 þ 9x2222 þ 12x2231 þ 13x2232
Uncertain multi-objective multi-product solid transportation
subject to
4 X
3
X
subject to
x11jk ¼ 21;
4 X
3
X
j¼1 k¼1
j¼1 k¼1
4 X
3
X
4 X
3
X
x22jk ¼ 25;
j¼1 k¼1
4 X
3
X
3 X
3
X
x23jk ¼ 0;
x1i2k ¼ 17;
x13jk ¼ 0;
3 X
3
X
x1i1k ¼ 14;
x2i1k ¼ 20;
i¼1 k¼1
3 X
3
X
x1i3k ¼ 12;
3 X
3
X
i¼1 k¼1
3 X
3
X
x2i3k ¼ 15;
2 X
3 X
4
X
xpij1 ¼ 46;
x1i4k ¼ 6;
y11jk ¼ 24;
y22jk ¼ 28;
4 X
3
X
y21jk ¼ 34;
xpij2 ¼ 51;
y1i1k ¼ 16;
4 X
3
X
y13jk ¼ 1;
y2i2k ¼ 18;
i¼1 k¼1
3 X
3
X
3 X
3
X
y1i4k ¼ 6;
2 X
3 X
4
X
4 X
3
X
y12jk ¼ 32;
y23jk ¼ 2;
j¼1 k¼1
y2i1k ¼ 23;
y1i3k ¼ 15;
3 X
3
X
3 X
3
X
y1i2k ¼ 20;
i¼1 k¼1
i¼1 k¼1
p¼1 i¼1 j¼1
z13jk ¼ 4;
3 X
3
X
z1i1k ¼ 19;
z2i2k ¼ 19;
i¼1 k¼1
3 X
3
X
z2i3k ¼ 19;
p¼1 i¼1 j¼1
3 X
3
X
z1i3k ¼ 18;
i¼1 k¼1
3 X
3
X
z1i4k ¼ 6;
i¼1 k¼1
zpij1 ¼ 51;
z2i1k ¼ 25;
i¼1 k¼1
3 X
3
X
z1i2k ¼ 22;
z2i4k ¼ 6;
i¼1 k¼1
2 X
3 X
4
X
zpij2 ¼ 56;
p¼1 i¼1 j¼1
2 X
3 X
4
X
4 X
3
X
i¼1 k¼1
2 X
3 X
4
X
z12jk ¼ 35;
j¼1 k¼1
i¼1 k¼1
i¼1 k¼1
j¼1 k¼1
j¼1 k¼1
i¼1 k¼1
3 X
3
X
z23jk ¼ 2;
2 X
3 X
4
X
p¼1 i¼1 j¼1
j¼1 k¼1
i¼1 k¼1
3 X
3
X
3 X
3
X
4 X
3
X
zpij3 ¼ 27;
zpijk yp
ijk
8 i; j; k; p:
p¼1 i¼1 j¼1
j¼1 k¼1
3 X
3
X
4 X
3
X
z21jk ¼ 37;
j¼1 k¼1
2 X
3 X
4
X
xpij3 ¼ 9; xpijk 0 8 i; j; k; p:
j¼1 k¼1
3 X
3
X
z22jk ¼ 30;
i¼1 k¼1
Minimize Z12 ¼ 8y1111 þ 11y1112 þ 6y1121 þ 8y1122 þ 12y1131 þ
9y1132 þ 10y1211 þ 11y1212 þ 7y1221 þ 8y1222 þ 13y1231 þ 16y1232 þ
10y2111 þ 13y2112 þ 8y2121 þ 7y2122 þ 11y2131 þ 10y2132 þ 13y2211 þ
16y2212 þ 9y2221 þ 11y2222 þ 14y2231 þ 14y2232
subject to
4 X
3
X
j¼1 k¼1
4 X
3
X
i¼1 k¼1
p¼1 i¼1 j¼1
4 X
3
X
j¼1 k¼1
j¼1 k¼1
x2i2k ¼ 16;
p¼1 i¼1 j¼1
2 X
3 X
4
X
4 X
3
X
z11jk ¼ 26;
4 X
3
X
3 X
3
X
i¼1 k¼1
x2i4k ¼ 6;
4 X
3
X
j¼1 k¼1
i¼1 k¼1
i¼1 k¼1
3 X
3
X
x12jk ¼ 28;
j¼1 k¼1
i¼1 k¼1
i¼1 k¼1
3 X
3
X
4 X
3
X
x21jk ¼ 32;
j¼1 k¼1
j¼1 k¼1
3 X
3
X
537
3 X
3
X
y2i3k ¼ 17;
i¼1 k¼1
y2i4k ¼ 6;
i¼1 k¼1
ypij1 ¼ 49;
2 X
3 X
4
X
ypij2 ¼ 53;
p¼1 i¼1 j¼1
ypij3 ¼ 19; ypijk xp
ijk 8 i; j; k; p:
p¼1 i¼1 j¼1
Minimize
Z13 ¼ 9z1111 þ 13z1112 þ 9z1121 þ 10z1122 þ 14z1131 þ
1
1
12z132 þ 13z211 þ 13z1212 þ 8z1221 þ 10z1222 þ 15z1231 þ 18z1232 þ
12z2111 þ 14z2112 þ 10z2121 þ 9z2122 þ 12z2131 þ 11z2132 þ 14z2211 þ
18z2212 þ 12z2221 þ 13z2222 þ 16z2231 þ 15z2232
Minimize
Z14 ¼ 11w1111 þ 15w1112 þ 11w1121 þ 12w1122 þ
1
1
16w131 þ 14w132 þ 15w1211 þ 15w1212 þ 9w1221 þ 12w1222 þ
17w1231 þ 20w1232 þ 13w2111 þ 15w2112 þ 12w2121 þ 11w2122 þ
13w2131 þ 13w2132 þ 16w2211 þ 20w2212 þ 14w2221 þ 14w2222 þ
18w2231 þ 16w2232
subject to
4 X
3
X
4 X
3
X
w11jk ¼ 28;
j¼1 k¼1
j¼1 k¼1
4 X
3
X
4 X
3
X
w22jk ¼ 33;
j¼1 k¼1
j¼1 k¼1
3 X
3
X
3 X
3
X
w1i1k ¼ 22;
i¼1 k¼1
3 X
3
X
w2i2k ¼ 22;
3 X
3
X
w13jk ¼ 9;
i¼1 k¼1
2 X
3 X
4
X
3 X
3
X
w2i1k ¼ 28;
3 X
3
X
w2i3k ¼ 21;
i¼1 k¼1
w2i4k ¼ 6;
i¼1 k¼1
p¼1 i¼1 j¼1
wpijk zp
ijk 8 i; j; k; p:
w1i2k ¼ 25;
i¼1 k¼1
w1i3k ¼ 21;
3 X
3
X
wpij2 ¼ 59;
w23jk ¼ 5;
j¼1 k¼1
i¼1 k¼1
w1i4k ¼ 6;
w12jk ¼ 37;
j¼1 k¼1
4 X
3
X
i¼1 k¼1
i¼1 k¼1
3 X
3
X
4 X
3
X
w21jk ¼ 39;
2 X
3 X
4
X
2 X
3 X
4
X
wpij1 ¼ 53;
p¼1 i¼1 j¼1
wpij3 ¼ 39;
p¼1 i¼1 j¼1
p
p
xp
ijk ; yijk and zijk are the optimal solutions of the previous
problems. On solving these problems sequentially, the
obtained values of xpijk ; ypijk ; zpijk , and wpijk ; for p ¼ 1; 2; i ¼
1; 2; 3; j ¼ 1; 2; 3; 4 and k ¼ 1; 2; 3 are
e
x 1111 ¼ ð9; 9; 9; 9Þ;
538
D Rani and T R Gulati
e
x 1113 ¼ ð0; 0; 2; 2Þ; e
x 1121 ¼ ð0; 2; 2; 2Þ; e
x 1132 ¼ ð12; 12; 12; 12Þ;
1
1
e
x 211 ¼ ð5; 5; 5; 5Þ; e
x 1213 ¼ ð0; 1; 2; 2Þ;
x 133 ¼ ð0; 1; 1; 3Þ; e
e
x 1223 ¼ ð0; 1; 1; 3Þ;
x 1221 ¼ ð5; 5; 7; 7Þ; ex 1222 ¼ ð12; 12; 12; 12Þ; e
1
1
e
x 242 ¼ ð6; 6; 6; 6Þ; ex 1312 ¼ ð0; 1; 1; 4Þ;
x 233 ¼ ð0; 2; 2; 2Þ; e
e
x 1332 ¼ ð0; 0; 3; 3Þ; e
x 1333 ¼ ð0; 0; 0; 1Þ;
x 1321 ¼ ð0; 0; 0; 1Þ; e
2
2
e
e
x 111 ¼ ð1; 1; 1; 1Þ; e
x 113 ¼ ð0; 2; 2; 4Þ;
x 2121 ¼ ð16; 16;
x 2132 ¼ ð15; 15; 15; 15Þ; e
x 2133 ¼
16; 16Þ; e
x 2123 ¼ ð0; 0; 1; 1Þ; e
2
2
ð0; 0; 2; 2Þ; e
x 211 ¼ ð10; 10; 10; 10Þ; ex 213 ¼ ð9; 9; 11; 11Þ;
2
e
x 223 ¼ ð0; 1; 1; 4Þ; e
x 2233 ¼ ð0; 2; 2; 2Þ; e
x 2242 ¼ ð6; 6; 6; 6Þ;
2
2
2
e
x 311 ¼ ð0; 1; 1; 2Þ; e
x 322 ¼ ð0; 1; 1; 1Þ; e
x 333 ¼ ð0; 0; 0; 2Þ: The
remaining variables are zero trapezoidal fuzzy numbers and
e 1 ¼ ð590; 791; 961; 1131Þ:
Z
e 2 in a similar way and then applying the
Minimizing Z
fuzzy programming technique, the obtained optimal-compromise solution is e
x 1111 ¼ ð9; 9; 9; 9Þ; e
x 1121 ¼ ð0; 2; 2; 2Þ;
1
1
e
x 123 ¼ ð0; 1; 1; 3Þ; e
x 132 ¼ ð12; 12; 12; 12Þ; ex 1133 ¼ ð0; 0; 2;
x 1213 ¼ ð0; 1; 1; 1Þ; e
x 1221 ¼ ð17; 17; 19;
2Þ; ex 1212 ¼ ð5; 5; 5; 5Þ; e
1
1
x 242 ¼ ð6; 6; 6; 6Þ; e
x 1311 ¼ ð0; 1; 1;
19Þ; e
x 223 ¼ ð0; 3; 4; 6Þ; e
¼ ð0; 0; 3; 6Þ; e
x 1323 ¼ ð0; 0; 0; 1Þ; e
x 1333
¼ ð0; 0;
1Þ; ex 1312
2
2
x 113 ¼ ð0; 0; 2; 2Þ; e
x 2122 ¼ ð11;
0; 1Þ; ex 111 ¼ ð15; 15; 15; 16Þ; e
x 2132 ¼ ð6; 6; 6; 6Þ; ex 2212 ¼
11; 11; 11Þ; e
x 2123 ¼ ð0; 2; 3; 4Þ; e
2
x 2221 ¼ ð5; 5; 5; 5Þ; ex 2233 ¼
ð5; 5; 5; 5Þ; e
x 213 ¼ ð0; 1; 1; 2Þ; e
x 2311 ¼ ð0; 0; 0; 1Þ; e
x 2312 ¼
ð9; 11; 13; 15Þ; ex 2242 ¼ ð6; 6; 6; 6Þ; e
2
ð0; 2; 2; 2Þ; ex 323 ¼ ð0; 0; 0; 2Þ: All other variables are zero
trapezoidal fuzzy numbers.
e 2 are found to be
e 1 and Z
The values of Z
(635,778,955,1100) and (428,566,724,847), respectively.
6. Interpretation of results
In this section, the results of the numerical example
obtained by using the proposed method are interpreted
graphically.
The graph of membership functions of the obtained
e 2 are in figures 1 and 2.
e 1 and Z
optimal values of Z
From figure 1, the following information about the
minimum value of the objective function Z1 can be
interpreted:
(i) 635 MinZ1 1100.
e 1.
Figure 1. Optimal value of Z
e 2.
Figure 2. Optimal value of Z
(ii) The chances that the minimum value of Z1 will lie in
the range 778–955 units are maximum.
(iii) The overall level of satisfaction for other values of
Z1 (say y) is le ðyÞ %, where
Z1
8
ðy 635Þ
>
>
>
>
143
>
>
<
1
le ðyÞ ¼
Z1
>
ð1100
yÞ
>
>
>
>
145
>
:
0
635 y\778
778 y 955
955\y 1100
otherwise
e 2 can be
The obtained results of the objective function Z
interpreted in a similar manner.
7. Advantages of the proposed method
(i) In the method proposed by Kundu et al [13], the
multi-objective multi-item solid transportation problem with transportation parameters as trapezoidal
fuzzy numbers is first converted to the equivalent
crisp problem. The obtained results are thus real
numbers, while the method proposed in this paper
provides the fuzzy optimal compromise solution.
(ii) Kumar and Kaur [19] pointed out the limitations of
existing methods [10, 20–23, 25] and the shortcomings of the method proposed by Liu [24] to solve the
single and multi-objective solid transportation problems. To overcome these limitations and resolve the
shortcomings, they have proposed a method to obtain
the fuzzy optimal solution of the fuzzy solid transportation problem. They have also solved two
existing fuzzy solid transportation problems by their
method and showed that the problems which could be
solved by the existing methods can also be solved by
their method.
However, none of the method proposed in the above
cited papers can be applied to a FFMOMISTP, for
which a method has been proposed in the present
paper. This method is also applicable to the problems
considered in Gen et al [25], [10, 20–23]. We have
also solved the examples in Kumar and Kaur [19] by
the method proposed by us. The results obtained are
same as shown in table 6.
Uncertain multi-objective multi-product solid transportation
539
Table 6. Results obtained by using the existing as well as proposed method.
Example
Example 3.1 ([19])
Example 5.1 ([19])
Numerical example in section 5
Existing method
Proposed method
(1800,1900,1900,2800)
(226,540,750,879)
Not applicable
(1800,1900,1900,2800)
(226,540,750,879)
e 2 ¼ ð428; 566; 724; 847Þ
e 1 ¼ ð635; 778; 955; 1100Þ Z
Z
8. Conclusions
In this work, the fuzzy optimal compromise solution is
obtained for the multi-objective multi-item solid transportation problem, where all the parameters are represented
by trapezoidal fuzzy numbers. As in the real world applications, the transportation parameters are not always precise and the fuzzy numbers handle more information than
the crisp ones, the obtained results are more beneficial for
the decision maker.
Since the proposed method is for the FFMOMISTP,
same is also applicable to the single and multi-objective
solid transportation problems as well as to the single and
multi-objective solid transportation problems with fuzzy
parameters.
Acknowledgments
The authors are thankful to the reviewers for their valuable
comments and suggestions, which improved the presentation of the paper. The first author is also thankful to CSIR,
Government of India, for providing financial support.
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