AN INTERACTIVE POSSIBILISTIC LINEAR
PROGRAMMING APPROACH FOR
MULTIPLE OBJECTIVE TRANSPORTATION
PROBLEMS
Dr. Celal Hakan Kagnicioglu, Assistant
Anadolu University
ESYO Yunusemre Kampusu
26470 Eskisehir/TURKEY
Tel:00-90-222-3351775
e-mail: chkagnic@anadolu.edu.tr
• Transportation problem(TP) is an
important linear programming that includes
many applications such as job scheduling,
production inventory, production planning,
production distribution, allocation problems
and investment analysis.
• Efficient algorithms have been developed
for solving the transportation problem
when the cost coefficients and the supply
and demand quantities are known exactly.
• However, these parameters are not always
in a precise manner. The unit shipping
cost may vary in a time frame can be an
example of it. The supplies and demands
may be uncertain due to some
uncontrollable factors.
• Due to incompleteness and/or
unavailability of required data over the
mid-term decision horizon, critical
parameters ( such as transportation cost
and demand ) are assumed to be
imprecise in nature.
• To deal quantitatively with imprecise
information in making decisions, Bellman
and Zadeh(1970) and Zadeh(1978)
introduce the notion of fuzziness.
• Since the transportation problem is
essentially a linear program, fuzzy linear
programming techniques are applied to
the fuzzy transportation problem.
• This paper deals with solution of the multiple
objective transportation problems by an
interactive possibilistic linear programming
approach.
• In this transportation problem, there are two
objective functions which are
• Minimization of transportation cost and
• Minimization of transportation time.
• Transportation costs, demand and supply are
fuzzy coefficients with triangular possibility
distribution.
• Transportation times are assumed to be crisp in
the model.
Mathematical Model
m
n
Minimize Zk =   C X, k=1,2,…,K
subject to
X a ,
i =1,2,…,m,
,
j =1,2,…,n,
X b
k
ij
i 1
ij
j 1
n
ij
i
ij
j
j 1
m
i 1
X ij  0 ,
for all i and j.
The decision variable Xij represents the quantity to be transported from
origin Oi to destination Dj.
The sources can be production facilities and the destinations can be
warehouses, and at each origin Oi let ai be the amount of a
homogenous product transported to n destinations Dj to satisfy the
demand for bj units of the product there.
A penalty Cij is associated with transporting a unit of the product from
origin i to destination j.
Interactive possibilistic linear
programming model
Minimize Z1 =
Minimize TT =
m
n
i 1
m
j 1
n

~
C ij X
 
T ij X
i 1
j 1
n

X ij  a~ i
m
~
X ij  b j
j 1

i 1
X ij  0 ,
ij
ij
1
ΠC
Cp
Cm
Co
Figure1.The triangular possibility distribution of
•
•
•
C
The most pessimistic value (Cp) that has a very low likelihood of
belonging to the set of available values (possibility degree = 0 if
normalized)
The most possible value (Cm) that definitely belongs to the set of
available values (possibility degree = 1 if normalized)
The most optimistic value (Co) that has a very low likelihood of
belonging to the set of available values (possibility degree = 0 if
normalized)
Fuzzy objective with a triangular possibility distribution can
be converted into three crisp objectives. This objective
can be minimized by pushing the three points toward the
left. Solving the fuzzy objective becomes the process of
minimizing zp, zm and zo simultaneously. However,
there may exist a conflict in the simultaneous
optimization process.
By using Lai and Hwang’s approach, zm and (zo-zm) are
minimized and (zm-zp) is maximized, rather than
simultaneously minimization of zp, zm and zo. That is,
this approach simultaneously minimizes the most
possible value of the imprecise total costs, zm,
maximizes the possibility of obtaining lower total costs,
(zm-zp), and minimizes the risk of obtaining higher total
costs, (zo-zm).
•
•
•
•
m
n
C
X
Min Z1 = Zm =
Max Z2 = Zm - Zp =   (C  C ) X
Min Z3 = Zo - Zm =   (C  C ) X
Min Z4 = TT =   T X
i 1
m
ij
ij
j 1
m
n
m
ij
i 1
m
j 1
n
o
ij
i 1
m
n
i 1
j 1
ij
p
ij
j 1
ij
m
ij
ij
ij
.
1
Min Zm
ΠZ
Max (Zm-Zp) Min (Zo-Zm)
Zp
Zm
Zo
Figure 2.The strategy to minimize the total costs
This auxiliary multiobjective linear programming problem can be solved by
converting into an equivalent single goal linear programming problem by
using the Zimmermann’s (1978) fuzzy programming method. At first, the
positive ideal solutions (PIS) and negative ideal solutions (NIS) of the three
objective functions should be obtained:
Z1(Pis)= Min Zm,
Z2(Pis)= Max ( Zm- Zp ),
Z3(Pis)= Min ( Zo- Zm ),
Z4(Pis)= Min TT,
Z1(Nis)= Max Zm
Z2(Nis)= Min ( Zm - Zp )
Z3(Nis)= Max ( Zo- Zm )
Z4(Nis)= Max TT.
. The linear membership function of these
objective functions
f1(3)(Z1(3))=
1,......... .......... .......... Z 1 ( 3 )  Z 1PIS
,
(3)

NIS
 Z 1( 3 )  Z 1( 3 )
PIS
NIS
, Z 1( 3 )  Z 1( 3 )  Z 1( 3 )
 NIS
PIS
 Z 1( 3 )  Z 1( 3 )

NIS
 0 ,......... .......... ......... Z 1 ( 3 )  Z 1 ( 3 ) ,



,



f2(Z2)=
1,......... .......... ......... Z 2  Z 2PIS ,

NIS
 Z2  Z2
NIS
PIS
, Z2  Z2  Z2
 PIS
NIS
Z

Z
2
 2
 0 ,......... .......... ......... Z  Z NIS ,
2
2




,



. If the minimum acceptable possibility, β, is given, then the auxiliary crisp
equality constraints can be presented as follows:
n
X
ij
 w1 a i ,  w 2 a i ,  w 3 a i ,
ij
 w1 b j ,   w 2 b j ,   w 3 b j , 
p
m
o
İ=1,2,…m
j 1
m
X
i 1
p
m
o
j=1,2,….n
• Finally, after having the crisp constraints for the TP,
Zimmerman’s equivalent single objective linear
programming model is applied to obtain the overall
satisfaction compromise solution.
Max L
s. to
L ≤ f1(Z1)
L ≤ f2(Z2)
L ≤ f3(Z3)
L ≤ f4(Z4)
n

X ij  w 1 a i ,   w 2 a i ,   w 3 a i , 
p
m
o
j 1
m
X
 w1 b j ,   w 2 b j ,   w 3 b j , 
p
ij
m
o
i =1,2,…,m,
j =1,2,…,n.
i 1
0 ≤ L ≤ 1,
Xij ≥ 0 for all i and j,
where fi (Zi) is the satisfaction degree of the ith objective function.
Algorithm
•
•
•
•
•
The algorithm for solving the TP is as follows.
Model the imprecise coefficient and right-hand sides of the formulated
multiobjective TP model using triangular possibility distribution.
Three new crisp objective functions for the imprecise objective function is
developed. These three new objective functions minimize the most
possible total cost value, maximize the possibility of obtainin lower costs
and minimize the risk of obtaining higher costs, respectively.
Given the minimum acceptable possibility, β, imprecise constraints are
converted into crisp ones by the weighted average method.
After having PIS and NIS of the objective functions separately, linear
membership functions are specified for all the objective functions, and
then the auxiliary multiobjective TP is converted into an equivalent linear
programming model by the method of Bellman and Zadeh (1970) and
Zimmermann’s (1978) fuzzy programming method.
The model is solved and linear membership function is modified
interactively until the solution is satisfactory. When the solution is
accepted as satisfactory, it is stopped.
W1
W2
W3
W4 W5 W6 Supply
PF1
(20,25,28) (28,33.36) (19,23.25) ………..
(35,37,40)
PF2
(13,16.20) (9.12.16) (18,22,25) ……………… (62,64,67)
PF3
(36,38,39) (23,25.26) (14,16,19)………………. (38,40,41)
PF4
(64,66,70) (37,40,44) (45,47,50)………………. (37,39,40)
PF5
(44,46,50) (25,28,31) (46,49,53)…………….… (40,43,45)
PF6
(67,69,73) (23,26,28) (33,36,38)…………….… (22,24,28)
Demand(20,24,26) (55,60,63) (17,19,22)……………….
Table1. Transportation costs, supply and demand data
All values are imprecise numbers with triangular possibility
distributions.
Transportation time (minute) of a product from production
facilities to warehouses are precise values.
• Transportation problem is formulated according to the
multiobjective linear programming model described
above.
• The imprecise data is modeled by triangular possibility
distributions. There are two objective functions of the
model
• Three new crisp objective functions are developed for
the imprecise objective function. Therefore, the model
has four objective functions.
• Since right-hand side of the constraints (demand and
supply) are imprecise, auxiliary crisp constraints are
formulated by using weighted average method at β =
0.5.
• Four crisp objective functions of the auxiliary
multiobjective linear programming problem presented in
the example are solved using the crisp single-goal linear
programming model
• According to these solutions, the
corresponding PIS and NIS of the initial
solutions are specified.
• By using different set of PIS and NIS of
the objective functions, the corresponding
linear membership function of all the
objective functions can be defined.
• Two different set of PIS and NIS values
are used for the solutions
• Since transporatation cost (Z1)is imprecise and
has a triangular possibility distribution, it is
calculated by (Z1 – Z2, Z1, Z1 + Z3).
• Therefore, the initial transportation cost is
imprecise and has a triangular possibility
distribution of ( 5599.9, 6368.0, 6841.1 ), and
transportation time (Z4), is 8191.0 minutes.
• The improved solution is also imprecise and has
a triangular possibility distribution of ( 5338.5,
6147.9, 6749.5 ), and transportation time is
8288.8 minutes.
• Overall satisfaction degree of the objectives
increased from 0.44 to 0.70 by the improved
solution.
• Since TP is not balanced (supply =
demand) due to imprecise values, the
number of products transported in the
improved solution ( ∑xij = 228.7 ) is
greater than in the initial solution ( ∑xij
=220.6 ).
• Changes in the PIS and NIS of the
objective functions are very effective on
the value of the overall satisfaction degree
of the objectives (L).
CONCLUSION
This approach helps to solve most of the real life
transportation problems with multiobjective and
imprecise and precise parameters through an
interactive decision making process.
This work aims to present an interactive
possibilistic linear programming problem
approach for solving multiobjective
transportation problems with imprecise cost,
demand and supply.
By this approach, simultaneously the most
possible value of the imprecise total costs are
minimized, possibility of obtaining lower total
costs are maximized and, the risk of obtaining
higher total costs are minimized.