ZZ'Y
sets and systems
ELSEVIER
Fuzzy Sets and Systems 98 (1998) 291-298
Fuzzy integer transportation problem
Stefan Chanas*, Dorota Kuchta
Institute of Industrial Engineering and Management, Technical University of Wroctaw, ul. Smoluchowskiego 25, 50372 Wroetaw,Poland
Received January 1996; revised November 1996
Abstract
An algorithm has been proposed which solves the transportation problem with fuzzy supply and demand values and
the integrality condition imposed on the solution. This algorithm is exact and computationally effective,although the
problem is formulated in the general way, i.e. its fuzzy supply and demand values can differ from each other and be fuzzy
numbers of any type. © 1998 Elsevier Science B.V. All rights reserved.
Keywords: Fuzzy programming; Integer programming; Transportation problem
I. Introduction
The transportation model has wide practical
applications, not only in transportation problems
per se, but also in such problems as production
planning. The parameters of each transportation
problem are unit costs (profits) and demand and
supply (production, storage capacity) values. In
practice, these parameters are not always exactly
known and stable. This paper deals with the case
when the unit costs (profits) are known exactly, but
the estimate of the supply and demand (capacities)
values are only imprecise. This imprecision may
follow from the lack of exact information but
may also be the consequence of a certain flexibility
the given enterprise has in planning its capacities.
A frequently used means to express the imprecision
are fuzzy numbers and is the approach we have
adopted.
*Corresponding author.
0165-0114/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved
Pll S01 6 5 - 0 1 1 4 ( 9 6 ) 0 0 3 8 0 - 6
In the classical transportation problem with integer demands and supply values there is always an
integer solution. This solution can be determined
with the transportation simplex method, one of the
most popular solution methods of the transportation problem. This property (the possibility of finding an integer solution) is not preserved in the fuzzy
transportation problem with fuzzy demands and
supplies, even if the characteristics of the fuzzy
numbers occurring in the problem are integer. In
order to obtain an integer optimal solution (which
might be necessary for reasons of feasibility),
a special algorithm has to be used.
Such an algorithm has been presented in [4].
However, it requires solving a parametric transportation problem with a parameter in the demand
and supply values. In this paper we propose an
algorithm determining the optimal integer solution
of a more general fuzzy transportation problem
than the one considered in [-4] making use only of
the classical (i.e. non-parametric) transportation
problem.
292
S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 (1998) 291-298
where At and Bj are fuzzy numbers of the following
form:
2. Formulation of the problem and notation
All fuzzy numbers we will consider in this paper
are fuzzy numbers of the L-R type. A fuzzy number
A of the L-R type is denoted as A = ( a , d ,
c~A,flA)L-R and has the following membership function [6]:
Ai = (if.z, -ai, o~A,, flA,)L,- R,,
Bj = (b j, -~j, ~B~, flB~)~- T~,
i = 1,...,m,
for t~<a
\
#A(t) =
~A
/t
for t E ~ , d ]
1
(t e R),
for t>~d
where a, d, CA, fla are real and non-negative parameters and L, R are shape functions.
F is a shape function if it fulfills the following conditions: F is a continuous non-increasing
function on the half-line [0, ~), F(0) = 1 and F is
strictly decreasing on this part of the domain on
which it is positive.
The following functions are examples of shape
functions:
• Linear: F(y) = max{0, 1 - y}, y e R + w { O } .
• Exponential: F(y) = e -pr, p ~> 1, y e R + w { O } .
• Power: F(y) = max{0, 1 - yP}, p >~ 1, y e R + w
{0}.
• Rational: F(y) = 1/(1 + yP), p/> 1, y ~ R + u { 0 } .
Some special cases are possible, in which the type
of one or both of the functions L and R may have
no significance:
• _a=-oo:#A(t)=l
for t~<~.
• -a= + o O : # A ( t ) = 1 f o r t ~ > a .
• C~A = O: lla(t) = 0 for t ~< a.
j=l,...,n.
The symbol x is a solution matrix. Its elements
are corresponding decision variables, i.e. x =
[Xu]m× .. The unit transportation costs c~j,
i = 1 , . . . , m , j = 1 , . . . , n , are assumed to be crisp
numbers.
The above formulation of the objective means
that the goal is also expressed by a fuzzy number.
This fuzzy number is denoted by G and takes the
following form:
G = ( - ~ , Co, 0,/~G)Lo- Ro.
In this place we can say more precisely in which
sense the integer fuzzy transportation problem considered in this paper is more general than the one
considered in [4]: here the couples (Li, Ri)
(i = 1,..., m), (S j, Tj) ( j = 1 .... , n) can all be different, whereas in [4] they have to be identical (linear).
In the case when the shape functions are not linear
but the same for all parameters, i.e. (L, R~)=
(S j, T j) = (F, F), the problem may be easily transformed to the linear parametric transportation
problem (see [2, 3]) and solved again by a method
proposed in [4].
The following definition makes it clear how the
satisfaction of the fuzzy constraints and of the fuzzy
goal is understood in problem (1).
• flA=O:#A(t)=lfort~>~.
The fuzzy transportation problem considered in
this paper is formulated as follows:
c(x)= ~
~ cijxij--*n~n,
#c(x) = m i n { # A , ~ = l x u )
i = 1 j=l
~ xij~-Ai,
(i = l ' ' ' ' ' m ) '
i = 1, ... ,m ,
j=l
~ xij'~Bj,
Definition 1. Let x be an arbitrary solution.
(a) The value
(1)
j = 1, ... ,n
i=1
xij >~ 0 and integer, i = 1, ..., m, j = 1, ... , n ,
is called the degree of satisfaction of the constraints
of problem (1) through x.
S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 H998) 291-298
(b) The value
293
A is not less than 2, i.e.
A ~ = {t~gl~A(t)
~G(x) = ~ d c ( x ) ) = ~
c~x~j
i
j=l
is called the degree of satisfaction of the goal of
problem (1) through x.
According to Bellmann-Zadeh approach (see
[1]) the optimal solution of problem (1) (called the
maximizing solution) is such a solution of the problem which simultaneously fulfills the constraints
and the goal to a maximal degree.
t> ~}.
It is easy to notice that under the assumptions on
A~ and Bj accepted in this paper h-cuts A~ and
B} are intervals of the following form:
A~ = [a__i--Li-l(,~)~A,,ai + Ri-l(2)fla,],
i = 1 ..... m,
B~ = [bi - Sfl(2)a~j, bj + Tfl()OflBj],
j = 1, ... ,n.
(3)
The 2-cut for the fuzzy goal G is the set
G ~ = ( - oc, Co + Rc, l(~)flG].
Definition 2. The maximizing solution of problem
(1) is such x for which the function # o ( x ) =
rain {~c(X), #G(x)} attains the maximal value. If this
maximal value is zero, we say that problem (1) is
infeasible.
(4)
Then we can rewrite problem (2) in the following
way:
~ max,
c(x) e G ~ ,
~ xijeA~,
3. Solution of the problem
i= l,...,m
j=l
According to Definition 2, solving problem (1) is
equivalent to solving the following integer mathematical programming problem:
xijeB~.,
j = 1,... ,n
(5)
i=1
2>0,
min{#c(X), #dx)} --. max,
xij>lO and integer, i = l .... ,m, j = l , . . . , n .
xij >~ 0 and integer, i-- 1,...,m, j = 1 , . . . , n .
2>0,
The above problem is not a transportation problem - because of its objective function and the first
constraint. For this reason we could not make use
of any transportation algorithm to solve it. But we
can associate with it an interval transportation
problem. This auxiliary problem can be solved by
means of any algorithm solving the classical transportation problem (the transition from the interval
transportation problem to the classical one will be
described in Section 4). Moreover, the solution of
this auxiliary problem will permit us to find the
solution of problem (5) and thus (1).
This auxiliary problem will have the following
form (it is defined for any fixed 2 > 0):
x~j ~> 0 and integer, i = 1 . . . . . m, j = 1 . . . . . n.
c(x) --, min,
Solving the above problem is in turn equivalent
to solving the following one:
2 --, max,
~dC(X)) >. ;~ ,
J2A,
Xij
>~ 2,
i= l,...,m,
",.j= 1
#sj
xi~ ) 2 ,
j = 1 . . . . . n,
(2)
i
Let us recall the following well-known definition:
Definition 3. Let A be any fuzzy number. The 2-cut
of A, denoted by A z, is the set of those real numbers,
for which the value of the membership function of
~ xijeA~,
i = l . . . . . m,
~ xijeB},
j = 1 ..... n,
j=l
i=1
xlj/> 0 and integer, i = 1, ..., m, j = 1 . . . . . n.
(6)
S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 (1998) 291-298
294
Solving problem (6) for any fixed 2 > 0 allows us
to find out whether problem (5) is feasible for this
value of 2. It is enough to check whether the objective function value of problem (6) fulfills the first
constraint of problem (5). What we need is the
maximal value of 2 for which problem (5) is feasible
and the corresponding solution of problem (5).
The ends of the intervals from the constraints of
problem (6) may be non-integer. This would mean
that after the transition to the classical transportation problem described in section (6) we might get
a classical transportation problem with non-integer
supply and/or demand values and the classical
algorithm would not guarantee that an integer
solution can be obtained. However, we can replace
problem (6) with problem (7), having already
integer ends in the supply and demand intervals,
without changing the set of feasible and optimal
solutions. In the definition of problem (7) we use the
following notation.
Definition 4. Let A be an arbitrary interval. The
symbol [A] denotes the widest interval having integer ends and contained in A, i.e. [A] = [a, b],
where
a = min{t It ~ A, t integer},
b = max{tit e A, t integer}.
The problem with which we can replace problem
(6) has the following form:
c(x) --, min,
~ xije[A~],
i = l ..... m,
j=l
~'. xi~[B}],
j = 1,...,n,
(7)
i=1
xij >~ 0 and integer i = 1,...,m, j = 1,...,n.
The sets of feasible and optimal solutions of
problems (6) and (7) are identical - because of the
integrality condition imposed on x.
For any fixed 2 > 0 we can solve problem (5)
converting it into a classical transportation problem with integer supply and demand values
(according to Section 4) and applying, for example,
the simplex transportation algorithm.
If we were able to solve problem (7) (or the
corresponding classical transportation problem) as
a parametric problem with the parameter 2, we
would be able to solve the original problem (1), too.
However, the coefficients of the problem depend on
parameter 2 in a non-linear way, which makes the
matter rather difficult. In order to avoid the necessity of solving a parametric transportation problem,
we propose the following algorithm, which only
requires solving several classical transportation
problems.
This algorithm starts from the utmost values of
the parameter 2, i.e. 2 = 0 and 2 = 1. We check for
which of these parameter values problem (5) is
feasible. If it is infeasible for 2 = 0, problem (1) is
infeasible. If problem (5) is feasible for 2 = 1, then
2 = 1 is the optimal objective function value in
problem (5) and the corresponding solution of
problem (5) is the solution of problem (1). If problem (5) turns out to be feasible for 2 = 0 and infeasible for 2 = 1 (which will be the most frequent
case), we consider the value 2 = 1 and then the
interval (0, ½] (if problem (5) is infeasible for 2 = ½)
or [½, 1] (in the other case). 1 Proceeding in this way,
we will approach from both sides the optimal value
of the objective function of problem (5).
Thus, in each step of the algorithm we will consider an interval [21, 22], such that problem (5) is
feasible for 2 = 21 and infeasible for 2 = 21. It
would not make any sense to divide this interval
further, when problem (7) for 2 = 21 is a minimal
extension of problem (7) for 2 = 22, as defined in
Definition 5.
Definition 5. Problem (7) for 2 = 21 is a minimal
extension of problem (7) for 2 = 22 when problem
(7) for 2 = 21 is identical to problem (7) for 2 = 2",
where
2* = m a x {
max
[~A,(t), max
~Bj(t)}
where t is an integer.
From the above paragraph it follows that the
corresponding algorithm, in which the consecutive
1In fact, these intervals may be slightly modified. The details
are given in the description of the algorithm.
S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 (1998) 291-298
intervals of the values of 2 are divided into two,
terminates after a finite number of steps, if we check
in each step whether problem (7) for 2 = 21 is
a minimal extension of problem (7) for 2 = 22. In
fact, it is not necessary to check it from the very
beginning. The user can determine the length of the
interval [-21, 22], beginning with which this condition should be verified.
The algorithm works as follows:
Step 1: Set 2(1):= 0, 2(2):= 1.
Step 2: Solve problem (7) for 2 = 2(1). If the
problem is feasible and c(x()~(1)))• Gm), then go to
step 3. Otherwise S T O P - problem (1) is infeasible
(#o(x) = 0 for all x).
Step 3: Solve problem (7) for 2 = 2(2). If the
problem is feasible and c(x(2(2)))•G ~2), then
S T O P - x(2(2)) is the optimal solution of problem
(1) and #o(x(2(2))) = 1. Otherwise go to step 4.
Step 4: Set 2(ha/f):= (2(1)+ 2(2))/2 and go to
step 5.
Step 5: Solve problem (7) for 2 = 2(half). If the
problem is infeasible, then set 2(2):= 2(halJ) and go
to step 6. Otherwise three cases are possible.
(i) #o(x(2(ha/f)))= ~(x(2(ha/f))); then x(2(ha/f))
is an optimal solution of problem (1). STOP.
(ii) /ao(x(2(ha/f))) > ~(x(2(ha/f))); then set 2(1):=
#c(X()~(half))) and go to step 6.
(iii) #o(x(2(half))) < #c(X(2(half))); then set
2(2):= #c(X(2(half))) or, if 2(2) = #c(X(2(half))),
then 2(2):= 2(ha/f). Go to step 6.
Step 6: If 4 ( 2 ) - 4 ( 1 ) > e , then go to step 4.
Otherwise check whether problem (7) for 2 = 4(1) is
a minimal extension of problem (7) for 2 = 2(2). If
not, then go to step 4. Otherwise S T O P - one of the
solutions, x(2(1)) or x(2(2)), is the optimal solution
of problem (1). If problem (5) was infeasible for
2 = 2(2), then x(2(1)) is an optimal solution of
problem (1).
Number e is given by the user. It seems that it
should not be greater than 0.1 and not smaller than
0.05.
4. Transformation of an interval transportation
problem into a classical one
Let us consider the following transportation
problem with interval supply and demand values:
clxl =
295
i c,jx, min,
i=1 j = l
~, xij e [a~,a{],
i = 1 . . . . . m,
j=l
xij • [bJ, b2], j = 1,...,n,
i=1
xii~>0andinteger, i = l , . . - , m ,
j=l,...,n.
(8)
Problem (8) can be transformed into a classical
transportation problem, i.e. with real supply and
demand values, by adding origins and destinations
with suitable supply and demand levels [4, 3]. The
corresponding classical transportation problem is
defined in the following way.
It will have 2m + 1 origins with the following
supply values al, i = 1, ... ,2m + 1:
ai=a I
fori=l
ai = a~-m - al-,n
. . . . . m,
fori=m+l,...,2m,
azm+l = ~ ( b E - b)).
j=l
It will have 2n + 1 destinations with the following
demand values:
bj=bJ
forj=l
bj=b2_,-bJ_,
i=1
. . . . . n,
forj=n+l,...,2n,
j=l
The cost coefficients d~j will be defined in the following way:
dij = Cij for i = 1 . . . . . m, j = 1 .... , n,
d~j = Ci-z,j for i = m + l , . . . ,2m, j = l . . . . . n,
dij=ci,j-, fori=l,...,m,j=n+l
. . . . . 2n,
dgj=C~_m,j_, f o r i = m + l
.... ,2m,
j=n+
1, ... ,2n,
d~,2, + 1 is a big number, as the corresponding route
is prohibited, for i -- 1 . . . . . m,
di,zn+l = 0 for i = m + 1, ... ,2m + 1,
d2m+ 1,j is a big number, as the corresponding route
is prohibited, for j = 1 .... , n,
d2,,+lj=0
forj=n+
1,...,2n.
[Xij](2ra+l)x(2n+l)
Once we have a solution 2
of the auxiliary, classical problem defined above,
=
296
S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 (1998) 291-298
we can obtain solution of p r o b l e m (8) in the following way:
F o r i:= 1 to m do
F o r j : = 1 to 2n do xij:= ~ij + ~,,+i,j
F o r i:= 1 to m do
F o r j : = 1 to n do xi3:= Rij + Ri.,+j
Step 1: 2(1):= 0, 2(2):= 1.
Step 2: P r o b l e m (7) is feasible for 2 = 0 and
c(x(O)) = 150 ~ G O = [0, 800].
Step 3: P r o b l e m (7) is infeasible for 2 = 1.
Step 4: 2(ha/f):= (0 + 1)/2 = 0.5.
Step 5: P r o b l e m (7) is feasible for 2 = 0.5
and #~(x(0.5)) = 0.74 > pc(X(0.5)) = 0.5. Therefore
5. Computational e x a m p l e
The algorithm will be illustrated at the following
example:
10Xl~ + 20x12 + 30x13 +
20x2~ + 50x22
+ 60X23 --* man,
x t l + x12 + x13 ~ (10, 10,5, 5)L-E,
XZl + X22 + X23 -- (16, 16, 5, 5)E-L,
X l l + X21 ~_ (10, 10, 5, 5)v-E,
(9, 9, 4, 4)L-g,
~--- (1, 1, 1, 1)p_R,
X12 "t- X22 ~
X13 + X23
xij>lO and integer
i=1,2,
j=1,2,3.
T h e fuzzy goal is determined by the following
fuzzy number: G -- (0, 300, 0, 500)L-L. L stands for
the linear shape function, E for the exponential
with the p a r a m e t e r p = 1, P for the p o w e r shape
function with the p a r a m e t e r p = 2 and R for the
rational one with p - 1.
T h e 2-cuts of the fuzzy supply and d e m a n d
values and of the fuzzy goal, with the shape functions a d o p t e d in the example, are as follows (see (3)
and (4)):
A~ = [ 1 0 - - 5(1 -- 2), 1 0 - - 51n(2)],
AZz
=
[16 + 51n(2), 16 + 5(1 - 2)],
B~ : [10 - 5 ~ / O - 2), 10 - 5 ln(2)l,
B~ = [9 -- 4(1 -- 2), 9 - 41n(2),l,
[1
~
l
= [ 0 , 3 0 0 + (1 - , ~ ) 5 0 0 ] .
T h e steps of the algorithm for the example are as
follows:
).(1):=
Step
Step
Step
0.5.
6: 2(2) - 2(1) = 1 - 0.5 > 0.05.
4: 2(ha/f):= (0.5 + 1)/2 = 0.75.
5: P r o b l e m (7) is infeasible for 2 = 0.75.
Therefore 2(2):= 0.75.
Step 6: 2(2) - 2(1) = 0.75 - 0.5 > 0.05.
Step 4: 2(ha/f):= (0.5 + 0.75)/2 = 0.625.
Step 5: P r o b l e m (7) is feasible for 2 = 0.625 and
#~(x(0.625)) = 0.54 < #c(X(0.625)) = 0.64. Therefore
2(2):= 0.64.
Step 6: 2(2) - 2(1) = 0.64 - 0.5 > 0.05.
Step 4: 2(ha/f):= (0.5 + 0.64)/2 = 0.57.
Step 5: P r o b l e m (7) is feasible for ,t = 0.57 and
pG(x(0.57)) = 0.58 < #c(X(0.57)) = 0.6. Therefore
2(2) := 0.6.
Step 6: 2(2) - 2(1) = 0.6 - 0.5 > 0.05.
Step 4: 2(ha/f):= (0.5 + 0.6)/2 = 0.55.
Step 5: P r o b l e m (7) is feasible for 2 = 0.55 and
#~(x(0.55)) = 0.58 < #c(X(0.55)) = 0.6. As 2(2) =
#c(X(0.55)) = 0.6 therefore 2(2):= 0.55.
Step 6: 2(2) - 2(1) = 0.55 - 0.5 ~< 0.05. But problem (7) for 2 = 2(1) = 0.5 is not a minimal extension
of p r o b l e m (7) for 2 = 2(2):= 0.55 and step 4 is
performed once more.
Step 4: 2(ha/f):= (0.5 + 0.55)/2 = 0.525.
Step 5: P r o b l e m (7) is feasible for 2 = 0.525 and
#6(x(0.525)) = 0.7 >/~c(X(0.525)) = 0.5488. Therefore
2(1) := 0.5488.
Step 6: 2(2) - 2(1) = 0.55 - 0.5488 ~< 0.05. Therefore it is checked whether problem (7) for 2 = 2(1):=
0.5488 is a minimal extension of p r o b l e m (7) for
2 = 2(1):= 0.55. Since the answer is positive, it is
the end of the algorithm. O n e of solutions x(0.5488)
and x(0.55) is the optimal solution of p r o b l e m (1).
As /~D(x(0.55)) = 0.58 > #o(x(0.5488)) = 0.5488,
the second solution, whose full form is given in
Table 1, is better.
Total cost = 510.
#c(X) = 0.6; #~(x) = 0.58; #o(x) = 0.58.
S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 (1998) 291-298
Table 1
Receivers
Suppliers
1
2
1
2
3
12
8
1
1
In order to illustrate fully step 5 of the algorithm,
we will show how problem (5) is built up and solved
for a chosen value of the parameter. Let 2 = 0.55.
In this case,
A °55 = [7.55, 12.99],
A °55 = [13.01, 18.251,
B °55 = [6.46, 12.991,
B °Ss = [7.2, 11.391,
[A °55] = [14, 18],
[B°'551 = [7, 12],
[B°'551 = [8, 11],
a transportation problem with prohibited routes,
presented in Table 2.
The optimal solution of the above transportation
problem is as given in Table 3.
The optimal solution of this problem is reduced,
after the application of the formulae at the end of
Section 4, to the solution of the initial problem (7).
This solution is given in Table 1.
During the last realization of step 6, while checking the stop condition, 2", defined in Definition 5, is
calculated from the following expression:
2" = max
#A,(7), #A1(13), /tA2(13), #a2(19),#Bl(6), } = 0.5488
#8,(13),#n2(7 ), ~B2(12), #8~(0), #B3(2)
B °'55 = [0.33, 1.82],
[A °55] = [8, 121,
297
Problem (7) for 2 = 2* is obviously the same one as
for ~,(1), which means a stop of the algorithm.
6. Conclusions
[B°S' 1 -- [1, 1].
The transportation problem (5) with interval
supply and demand values can be reduced, according to the procedure presented in section 4, to
In this paper we have proposed an algorithm
solving the integer fuzzy transportation problem
(with fuzzy supply and demand values as well as
Table 2
1
2
3
4
5
Demand
1
2
3
4
5
6
7
Supply
10
20
10
20
20
50
20
50
30
60
30
60
8
•
•
8
1
30
60
30
60
0
0
•
0
0
0
14
14
4
4
8
7
20
50
20
50
0
3
•
•
10
20
10
20
0
5
3
4
5
6
7
Supply
1
5
Table 3
1
1
2
3
4
5
Demand
2
7
7
7
1
8
1
1
5
2
3
8
0
4
4
6
14
14
4
4
8
298
S. Chanas, D. Kuchta / Fuzzy Sets and Systems 98 (1998) 291-298
with the fuzzy goal) in the sense of maximizing
the joint satisfaction of the goal and of the constraints. This algorithm requires, apart from some
easy transformations, only solving the classical
transportation problem (in particular, it is not
necessary to solve any parametric problems). Fuzzy
numbers defining the problem do not have to be
trapezoidal. They can differ from each other and be
of any type.
References
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environment, Management Sci. B17(1970) 203-218.
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Interface between Artificial Intelligence and Operations
Research in Fuzzy Environment (Verlag TOV Rheinland,
Krln, 1989) 105-116.
I3] S. Chanas, M. Delgado, J.L Verdegay and M.A. Vila, Interval and fuzzy extensions of classical transportation
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