Using possibilistic linear programming for fuzzy
transportation planning decisions
Tien-Fu Liang, Cheng-Shing Chiu, Hung-Wen Cheng
Abstract
In real-world transportation planning decision (TPD) problems, input data or related
parameters are often imprecise/fuzzy owing to incomplete or unobtainable information.
This study develops a novel possibilistic linear programming (PLP) approach for solving
TPD problems with imprecise goal and constraints. The proposed approach attempts to
minimize the total transportation costs with reference to imprecise cost coefficients,
available supply and forecast demand. An industrial case is used to demonstrate the
feasibility of applying the proposed approach to real TPD problems. Consequently, the
proposed approach yields an efficient compromise solution and overall degree of
decision maker satisfaction with the determined goal values. Particularly, several
significant findings and features of the proposed approach that distinguish it from the
ordinary linear programming and other fuzzy TPD models are presented. Overall, the
proposed PLP approach is practically applicable for solving TPD problems in uncertain
environments, and can generate better decisions than other models.
Keywords: possibilistic linear programming, fuzzy transportation planning decisions,
fuzzy set theory.
Tien-Fu Liang: Associate Professor, Department of Industrial Management, HIT.
Cheng-Shing Chiu: Instructor, Department of Industrial Management, HIT.
Hung-Wen Cheng: Instructor, Department of Industrial Management, HIT.
以可能性線性規劃求解模糊運輸規劃決策問題
梁添富、邱振興、鄭鴻文
摘要
自產銷實務來看，企業如何建構及尋求供應鏈體系的最適運輸規劃決策(TPD)
模式，藉以有效調配產銷資源及降低物流成本，實為攸關整體營運績效及競爭力
之重要課題。本文目的在於發展一適於模糊環境下之可能性線性規劃(PLP)方法，
用以求解模糊 TPD 問題，內容結合模糊決策概念及模糊規劃方法，建構一符合企
業需求的決策模式，企圖達成總運輸成本的極小化。同時，本文亦進一步發展模
式之系統化求解程序，提供決策者實際執行及修正模式的適當步驟，有效尋求模
糊 TPD 問題之最適妥協解。此外，本文特舉一產業個案進行模式測試與比較，結
果發現本文 PLP 方法除可求得一組有效的妥協解與較高整體決策滿意度外，並具
有彈性修正程序、提供多元決策資訊及較高的模式建構與計算效率等正面特色，
將可解決傳統 TPD 模式在實際應用上程度不足的問題。
關鍵詞：可能性線性規劃、模糊運輸規劃決策、模糊集理論。
梁添富：修平技術學院工業管理系副教授
邱振興：修平技術學院工業管理系講師
鄭鴻文：修平技術學院工業管理系講師
method and algorithms cannot solve all
1. Introduction
fuzzy TPD problems. Fuzzy set theory
The transportation planning decision
was presented by Zadeh [21] and has
(TPD) problem involves identifying the
been applied extensively in various fields.
lowest shipping cost plan for distributing
In
goods
(e.g.
introduced fuzzy set theory into an
factories) to multiple destinations (e.g.
ordinary LP problem with fuzzy goal and
warehouses). Basically, the TPD is a
constraints.
linear programming (LP) problem that
decision-making method proposed by
can be solved using the ordinary simplex
Bellman and Zadeh [1], that study
method. The transportation model for
confirmed the existence of an equivalent
solving TPD problems can be extended to
LP problem. Subsequently, fuzzy linear
encompass various important applications,
programming (FLP) has developed into
including the assignment problem, the
several fuzzy optimization methods for
transshipment problem, the periodical
solving TPD problems. Chanas et al. [5]
aggregate production planning problem,
presented a FLP approach for solving the
and the problem of locating new facilities.
TPD problem with crisp cost coefficients
However, when any of the LP models or
and fuzzy supply and demand volumes.
the existing effective algorithms (e.g. the
Moreover,
stepping stone method and the modified
proposed the concept of the optimal
distribution method) are used to solve the
solution of the TPD problem with fuzzy
TPD problems, the goal and model inputs
coefficients expressed as L-L fuzzy
are
numbers, and developed an algorithm for
from
multiple
generally
sources
assumed
to
be
deterministic/crisp [19].
1976,
Zimmermann
Following
Chanas
and
[23]
the
Kuchta
first
fuzzy
[6]
obtaining this solution. Additionally,
In real-world TPD problems, input
Chanas and Kuchta [7] designed an
data or related parameters, such as unit
algorithm for solving the integer fuzzy
cost coefficients, available supply and
transportation problem with fuzzy supply
demand volumes, are often imprecise/
and demand volumes in the sense of
fuzzy due to incomplete or unobtainable
maximizing the joint satisfaction of the
information. Obviously, traditional LP
fuzzy goal and constraints. Related works
on fuzzy TPD programming problems
Tang et al. [18] designed two types of
included Bit et al. [2], Li and Lai [16],
PLP with general possibilistic distribution,
and El-Wahed [8].
including LP problems with general
In 1978, Zadeh [22] presented the
possibilistic
resources
and
general
theory of possibility, which is related to
possibilistic objective coefficients. Wang
the theory of fuzzy sets. A possibility
and Liang [20] presented an interactive
distribution
fuzzy
PLP approach for solving aggregate
restriction, which elastically constraints
production planning problems involving
the values that can be assigned to a
imprecise forecast demand, operating
variable. That study demonstrated that the
costs and capacity. That approach can
importance of the theory of possibility
yield an efficient compromise solution
stems from the fact that much of the
and the overall levels of satisfaction of
information on which human decisions is
the decision maker (DM) with the
based
than
determined goal values. Related studies
probabilistic in nature. Buckley [3-4]
on PLP problems included Inuiguchi and
formulated a mathematical programming
Sakawa [12], Tanaka et al. [17], Hsu and
problem in which all parameters may be
Wang [9], and Jensen and Maturana [13].
fuzzy
on
is
defined
possibilistic
variables
as
a
rather
specified
by
their
On the whole, the PLP method not
possibility distribution, and illustrated
only
this problem using the possibilistic linear
efficiency and flexible doctrines than the
programming (PLP) approach. Lai and
FLP
Hwang [14] developed an auxiliary
techniques, but also facilities possibilistic
multi-objective
decision-making
linear
programming
provides
and
more
stochastic
in
computational
programming
uncertain
(MOLP) model for solving a PLP
environments [22, 15, 18]. This work
problem with imprecise objective and/or
develops
constraint
[10]
approach for solving the TPD problems
developed a parametric programming
with imprecise unit cost coefficients,
approach to deal with the complete
available supply and forecast demand.
solutions of multiple objective TPD
The rest is organized as follows. Section
problems with possibilistic coefficients.
2 describes the problem and formulates
coefficients.
Hussein
a
novel
interactive
PLP
the problem. Section 3 then develops the
at a cost of aij per unit, and destination j
interactive PLP approach and algorithm
( j  1, 2, , n ) has a demand of Dj units
for solving TPD problems. Subsequently,
of the commodity to be received from the
Section 4 presents an industrial case
sources. The TPD proposed here attempts
designed to implement the feasibility of
to determine the right volumes to be
applying the proposed PLP approach to
transported from each source to each
real TPD problems. Next, Section 5
destination
discusses the findings for the practical
transportation
application of the proposed approach.
demonstrates the TPD problem as a
Conclusions finally are drawn in Section
network. Each node represents a source
6.
or a destination. The arc joining a source
to
minimize
costs.
total
Figure
1
and a destination represents the route
2. Problem formulation
through which the commodity is shipped.
Generally, the available capacities
2.1. Problem description and notation
The TPD problem examined here
for each source, the forecast demand for
can be described as follows. Assume that
each destination, and the unit cost
a distribution center seeks to determine
coefficients are imprecise over the
the transportation plan of a homogeneous
planning horizon. Therefore, assigning a
commodity
n
set of crisp values for the environmental
destinations. Source i ( i  1, 2, , m ) has
coefficients and related parameters is
a supply of Si units of the commodity
inappropriate for dealing with such
available to distribute to each destination
ambiguous TPD problems. Alternatively,
from
m
sources
a~11 : Q11
~
S1
Imprecise available
supply
to
1
~
S2
2
1
~
D1
2
~
D2
⋮
⋮
~
Sm
Destinations
n
m
a~mn : Qmn
Figure 1. The TPD problem as a network
~
Dn
Imprecise forecast
demand
possibility
distribution
an
the planning horizon. Correspondingly,
effectual method for proceeding with
the imprecise objective function of the
inherent
proposed PLP model is as follows.
ambiguous
provides
phenomena
in
determining environmental coefficients
and related parameters [22, 14, 12]. The
m
n
Min ~
z   a~ij Qij
(1)
i 1 j 1
TPD problems proposed in this study
where a~ij denote imprecise unit cost
focuses on developing an interactive PLP
coefficients with triangular possibility
approach
distributions.
for
transportation
optimizing
plan
in
the
imprecise
The
proposed
model
assumes that the transportation costs on a
environments.
given route are directly proportional to
2.2. Possibilistic linear programming
the number of units transported.
(PLP) model
2.2.2. Constraints
2.2.1. Objective function
 Constraints on total supply available
Most practical decisions for solving
TPD
problems
consider
total
transportation costs. The original PLP
model
proposed
here
selects
total
transportation costs as the objective
function, after reviewing the literature
and considering practical situations [5-6,
8,
16].
Letting
transportation
(
i  1, 2,  , m
z
denotes
total
and
Qij
costs
;
j  1, 2, , n
)
for each source i
n
 Qij  Si
~
i  1, 2, , m
(2)
j 1
 Constraints on total demand for each
destination j
m
~
 Qij  D j
j  1, 2, , n
(3)
i 1
 Non-negativity constraints on decision
variables
Qij  0
i  1, 2, , m
j  1, 2, , n
(4)
units
Equations (2) and (3) are normally
transported from source i to destination j.
imprecise constraints. In practice, the
The related unit cost coefficients in the
total supply available from each source in
objective
frequently
Eq. (2) and the total demand required by
imprecise because of some information
each destination in Equation (3) are never
being incomplete or unobtainable over
obtained
represents
the
number
function
are
of
precisely
in
a
dynamic
environment,
owing
to
uncertainty
regarding the available capacities, worker
skills, forecast demand, public policy, and
other factors over the planning horizon.
has a feasible solution only if
n
 Si   D j
~
i 1
~
if normalized).
2. The most optimistic value ( aijo ) that
has a very low likelihood of belonging
Notably, the model described above
m
available values (possibility degree  1
to
the
set
of
available
values
(possibility degree  0 if normalized).
(5)
j 1
Equation (5) is the necessary and
sufficient conditions for the existence of a
feasible solution to the above original
3. The most pessimistic value ( aijp ) that
has a very low likelihood of belonging
to
the
set
of
available
values
(possibility degree  0 if normalized).
Figure 2 presents the triangular
PLP model.
possibility
distribution
of
imprecise
number a~ij  (aijm , aijo , aijp ) .
3. Model development
3.1. Model the imprecise data with
triangular possibility distribution
 aij
1
The possibility distribution can be
stated as the degree of occurrence of an
a ijo
a ijm
a~ij
p
ij
event with imprecise data. This work
0
assumes the DM to have already adopted
Figure 2. The triangular possibility
distribution of a~
the pattern of triangular possibility
a
ij
distribution for all imprecise numbers. In
practice,
triangular
a
DM
can
possibility
construct
the
distribution
of
a~
imprecise unit cost coefficients
ij
based on the three prominent data, as
follows.
1. The most possible value ( aijm ) that
definitely belongs to the set of
Similarly, the related imprecise data
for the original PLP model thus can be
modeled
using
triangular
possibility
distributions, as follows.
a~ij  (aijm , aijo , aijp ) i  1, 2, , m j  1, 2, , n
~
Si  (Sim , Sio , Sip )
i  1, 2, , m
~
D j  ( D mj , D oj , D jp )
j  1, 2, , n
3.2. Developing
an
auxiliary
multi-
maximizing the possibility of obtaining
(zm  zo ) ,
objective linear programming (MOLP)
lower
model
minimizing the risk of obtaining higher
3.2.1. Strategy for solving the imprecise
goal value, ( z p  z m ) . The last two
objective function
the original PLP model in the previous
has
a
distribution.
value,
and
objectives actually are relative measures
The imprecise objective function of
section
goal
triangular
possibility
Geometrically,
this
imprecise objective is fully defined by
three prominent points ( z o , 0) , ( z m , 1)
and ( z p , 0) . The imprecise objective can
be minimized by pushing the three
prominent points towards the leftwards.
Because of the vertical coordinates of the
prominent points being fixed at either 1
from z m , the most possible value of the
imprecise total transportation cost. Figure
3 presents the strategy for minimizing the
imprecise objective function.
As indicated in Figure 3, possibility
~
distribution B is preferred to possibility
~
distribution A . The following lists the
results for the three new crisp objective
functions of total transportation cost in
Equation (1).
z
or 0, the three horizontal coordinates are
the only considerations. Consequently,
1
solving the imprecise objective requires
minimizing
zo
zm
,
and
zp
simultaneously. Using Lai and Hwang’s
approach [14], the approach developed
here
minimizes
(zm  zo )
zm
,
maximizes
and minimizes
(z p  zm ) ,
rather than simultaneously minimizing
z o , z m and z p . Restated, the proposed
approach
involves
~
A
~
B
simultaneously
minimizing the most possible goal value
of the imprecise objective function, z m ,
(Ⅰ) (Ⅱ)
0
z
o
z
m
z
p
~
z
Figure 3. The strategy to minimize the
imprecise objective function
m
n
Min z1  z m   a ijm Qij
(6)
i 1 j 1
m
n
Max z 2  ( z m  z o )   (aijm  aijo )Qij
(7)
i 1 j 1
m
n
Min z3  ( z p  z m )   (aijp  aijm )Qij
i 1 j 1
(8)
Equations (6) to (8) are equivalent to
simultaneously minimizing the most
possible value of total transportation
costs, maximizing the possibility of
crisp
equality
constraints
can
be
presented as follows.
n
 Qij  w1Sim,   w2 Sio,   w3Sip, 
j 1
i  1, 2, , m
obtaining lower total transportation costs
(9)
(regionⅠof the possibility distribution in
where w1  w2  w3  1 , w1 , w2 and w3 repre
Figure 3), and minimizing the risk of
sent the weights of the most pessimistic,
obtaining higher total transportation costs
most possible and most optimistic values
(regionⅡof the possibility distribution in
of the imprecise number, respectively.
Figure 3).
Similarly,
3.2.2. Strategy for solving the imprecise
constraints
the
minimum
acceptable possibility,  , is given, the
auxiliary crisp equality constraints of
Recalling Equation (2) from the
previous
if
PLP
model,
consider
Equation (3) can be presented as follows.
the
m
situations in which the available supply
~
for each source (the right-hand side), Si ,
are
imprecise
and
have
triangular
 Qij  w1Dim,   w2 Dio,   w3 Dip, 
i 1
j  1, 2, , n
3.3. Solving the auxiliary MOLP problem
possibility distribution with the most and
least possible values. In real-world TPD
problems, the DM can estimate a possible
interval for imprecise demand based on
experience and knowledge. The main
problem
is
obtaining
a
crisp
representative number for the imprecise
supply. This study applies the weighted
average method proposed by Lai and
~
Hwang [14] to convert Si into a crisp
(10)
The
auxiliary
MOLP
problem
developed above can be converted into an
equivalent single-goal LP problem using
the
linear
membership
Zimmermann
[22]
to
function
represent
of
the
imprecise goal of the DM, together with
the fuzzy decision-making of Bellman
and Zadeh [1]. First, the positive ideal
solutions
(PIS)
and
negative
ideal
solutions (NIS) of the three objective
number. If the minimum acceptable
functions of the auxiliary MOLP problem
possibility,  , is given, the auxiliary
can be specified as follows [11, 14],
respectively.
f1 ( z1 ) ( f 3 ( z 3 ))
z1PIS  Min z m ,
z1NIS  Max z m
(11a)
1
z 2PIS  Max ( z m  z o ) , z 2NIS  Min ( z m  z o )
(11b)
z3PIS  Min ( z p  z m ) , z3NIS  Max ( z p  z m )
z 1PIS
( z 3PIS )
0
(11c)
The
corresponding
linear
of z1 and z3
function is defined by
f 2 (z2 )
z1  z1PIS
z1PIS  z1  z1NIS
z2  z2PIS
z2 
z2NIS
1

 z3NIS  z3
f 3 ( z3 )   NIS
PIS
 z3  z3

0
z3 
z3PIS
z2NIS  z2  z2PIS
z 2NIS
0
z 2PIS
Figure 5. Linear membership function of
z2
and Zadeh [1], the complete equivalent
single-goal LP model for solving TPD
z3PIS  z3  z3NIS (14)
problems can be formulated as follows.
z3  z3NIS
Max L
functions
z2
(13)
Figures 4 and 5 show the graphs of
membership
1
(12)
z1  z1NIS
1

 z2  z2NIS
f 2 ( z2 )   PIS
NIS
 z2  z2

0
linear
z1 ( z 3 )
Figure 4. Linear membership functions
membership function for each objective
1
 NIS
 z1  z1
f1 ( z1 )   NIS
PIS
 z1  z1

0
z 1NIS
( z 3NIS )
for
s.t.
L  f g ( zg )
m
i 1 j 1
m
n
z2    (aijm  aijo )Qij
i 1 j 1
determined by asking the DM to specify
m
the imprecise objective value interval
n
z1    aijmQij
Equations (12) to (14). In practice, each
linear membership function can be
g  1, 2, 3
n
z3    (aijp  aijm )Qij
i 1 j 1
(PIS and NIS).
n
Finally, using the minimum operator
of the fuzzy decision-making of Bellman
 Qij  w1Sim,   w2 Sio,   w3Sip, 
j 1
i  1, 2, , m
m
 Qij  w1Dim,   w2 Dio,   w3 Dip, 
i 1
be modified until a satisfactory solution
j  1, 2, , n
Qij  0
i  1, 2, , m
with the initial solution, the model must
j  1, 2, , n
is identified.
where the auxiliary variable L represents
the overall degree of DM satisfaction
4. Model implementation
with determined goal values.
4.1 Case description
Dali Company was used as a case
3.4. Solution procedure
The interactive solution procedure of
study to demonstrate the practicality of
the proposed PLP approach for solving
the
proposed
methodology.
TPD problems is as follows.
Company is the leading producer of soft
Step 1. Formulate the original PLP model
drinks and low-temperature foods in
for the TPD problems.
Taiwan. Currently, Dali plans to develop
Step 2. Model the imprecise coefficients
the South-East Asian market and broaden
and right-hand sides using the triangular
the visibility of Dali products in the
possibility distributions.
Chinese market. Notably, following the
Step 3. Develop the three new crisp
entry of Taiwan to the World Trade
objective functions of the auxiliary
Organization, Dali plans to seek strategic
MOLP problem for the imprecise goal.
alliance with prominent international
Step 4. Given the minimum acceptable
companies, and introduced international
possibility,  , convert the imprecise
bread to lighten the embedded future
constraints into crisp ones using the
impact. In the domestic soft drinks
weighted average method.
market, Dali produces tea beverages to
Step 5. Specify the linear membership
meet demand from four distribution
functions for all of the objective functions,
centers in Taichung, Chiayi, Kaohsiung,
and then convert the auxiliary MOLP
and Taipei, with production being based
problem into an equivalent single-goal LP
at three plants in Changhua, Touliu, and
model using the minimum operator.
Hsinchu. According to the preliminary
Step 6. Solve and modify the model
environmental
interactively. If the DM is dissatisfied
summarizes the potential supply available
information,
Dali
Table
1
from these three plants, the forecast
focuses on developing an interactive PLP
demand from the four distribution centers,
approach
and the unit transportation costs for each
transportation costs.
route used by Dali for the upcoming
4.2. Solving procedures
to
minimize
the
total
season. The environmental coefficients
The interactive solution procedure
and related parameters generally are
using the proposed PLP approach for the
imprecise
Dali case is described as follows.
numbers
with
triangular
possibility distributions over the planning
Step 1. Formulate the original PLP model
horizon
for the TPD problem according to
due
to
incomplete
or
unobtainable information. For example,
Equations (1) to (4).
the available supply of the Changhua
Step 2. Model all imprecise data with
plant is (7.2, 8, 8.8) thousand dozen
triangular possibility distributions, as
bottles, the forecast demand of the
listed in Table 1.
Taichung Distribution center is (6.2, 7,
Step 3. Develop three new crisp objective
7.8) thousand dozen bottles, and the
functions of the auxiliary MOLP problem
transportation cost per dozen bottles from
according to Equations (6) to (8), and are
Changhua to Taichung is ($8, $10,
presented as follows.
$10.8).
Min z1  10Q11  22Q12  10Q13  20Q14 15Q21
20Q22  12Q23  8Q24
20Q31  12Q32 10Q33  15Q34 (15)
Due to transportation costs being a
major expense, the management of Dali
is initiating a study to reduce these costs
as much as possible. This TPD problem
for the industrial case presented here
Max z 2  ( z m  z o )  2Q11  1.6Q12  2Q13
1.2Q14  1Q21  1.8Q22  2Q23
2Q24  1.6Q31 2.4Q32  2.2Q33
(16)
1Q34
Table 1. Summarized data in the Dali case (in U.S. dollar)
Source
Changhua
Touliu
Hsinchu
Demand (000
dozen
bottles)
Destination
Supply (000
dozen bottles)
Taichung
Chiayi
Kaohsiung
Taipei
($8, $10, $10.8) ($20.4, $22, $24) ($8, $10, $10.6) ($18.8, $20, $22) (7.2, 8, 8.8)
($14, $15, $16) ($18.2, $20, $22) ($10, $12, $13)
($6, $8, $8.8)
(12, 14, 16)
($18.4, $20, $21) ($9.6, $12, $13) ($7.8, $10, $10.8) ($14, $15, $16) (10.2, 12, 13.8)
(6.2, 7, 7.8)
(8.6, 10, 11.4)
(6.5, 8, 9.5)
(7.8, 9, 10.2)
Min z3  ( z p  z m )  0.8Q11  2Q12  0.6Q13
2Q14  1Q21  2Q22  1Q23 0.8Q24
(17)
1Q31 1Q32  0.8Q33 1Q34
Step 4. Specify the minimum acceptable
1

 50 ,000  z 3
f 3 ( z3 )  
 40 ,000
0
z 3  10 ,000
10 ,000  z 3  50 ,000
z 3  50 ,000
(20)
possibility level,   0.5 . The auxiliary
crisp constraints then can be formulated
using Equations (9) and (10). This work
applies the concept of the most likely
values proposed by the approach of Lai
and Hwang [14], setting w2  4/6 and
Furthermore,
the
complete
equivalent single-goal LP model for
solving the TPD problem for the Dali
case can be formulated based on the LP
model as presented in the previous
section.
w1  w3  1/6. The reason the most likely
Step 6. Solve and modify the model
values are used here is that the most
interactively. LINDO computer software
possible values generally are the most
is used to run this ordinary LP model and
important ones, and thus should be
obtains the following results. The initial
assigned greater weights.
total transportation cost is imprecise and
Step 5. Define the corresponding linear
has a triangular possibility distribution of
membership functions for the three new
($382,000, $346,800, $399,000), and the
objective functions of the auxiliary
overall degree of DM satisfaction with
MOLP problem according to Equations
determined goal values is 0.5040.
(12) to (14), and are shown as follows.
Moreover, the DM may attempt to
z1  300 ,000
1

 900 ,000  z1
f1 ( z1 )  
300 ,000  z1  900 ,000
 600 ,000
0
z1  900 ,000
(18)
1

 z  10 ,000
f 2 (z2 )   2
 50 ,000
0
modify the results interactively by
a d j u st i n g t h e l i n e ar m e m b e r sh i p
functions and related model parameters to
obtain a satisfactory solution.
Consequentl y, t he i mproved total
z 2  60 ,000
10 ,000  z 2  60 ,000
z 2  10 ,000
transportation cost is imprecise and has a
triangular possibility distribution of
($352,000, $315,800, $367,000), and
(19)
overall degree of DM satisfaction is
Table 2. Initial and improved TPD plan for the Dali case
Initial compromise solutions
Improved compromise solutions
Taichung Chiayi Kaohsiung Taipei Taichung Chiayi Kaohsiung Taipei
7
0
1
0
7
0
1
0
Allocation Changhua
(000
Touliu
0
5
0
9
0
0
5
9
dozen)
Hsinchu
0
5
7
0
0
10
2
0
Objective values
L  0.5040
L  0.8733
z1  $382,000, z2  $35,200,
z1  $352,000, z2  $36,200,
z3  $17,000
z3  $15,000
~
~
z  ($382,000, $346,800, $399,000) z  ($352,000, $315,800, $367,000)
PIS
($300,000, $60,000, $10,000)
($320,000, $40,000, $12,000)
NIS
($900,000, $10,000, $50,000)
($900,000, $10,000, $60,000)
Note: The total transportation cost has a triangular possibility distribution of ~
z  ( z1, z1  z2 , z1  z3 ) .
Item
Table 3. Comparison of the crisp LP and the proposed PLP solutions
Item
The crisp ordinary LP model
Objective function
Min z
L (DM’s overall degree of satisfaction)
100%
z (Total transportation cost)
0.8733.
Table
2
lists
$352,000
the
initial
The proposed PLP approach
Max L
87.33%
($352,000, $315,800,
$367,000)
assumed that the DM specified the most
andimproved TPD plans for the Dali case
possible
values
of
the
possibility
based on the present information.
distribution of each imprecise data as the
precise numbers. From Table 3, applying
LP to minimize the total transportation
5. Findings
cost, the optimal value was $35,200. In
5.1. Results analysis
Several
significant
findings
regarding the practical TPD application
of the proposed PLP approach are as
follows. First, the proposed PLP approach
yields an efficient compromise solution.
Alternatively, the TPD problem of the
Dali case presented above was solved
with the crisp ordinary LP model. Table 3
compares the results using the LP model
with the proposed PLP approach. Herein,
contrast with the proposed PLP approach,
the improved compromise solution was
($352,000, $315,800, $367,000). These
figures indicate that the PLP solution are
an efficient solution, compared to the
optimal goal values obtained by the
ordinary
LP
model.
Thereafter,
an
improved TPD plan is obtained by the
proposed
PLP
approach
under
an
acceptable degree of DM satisfaction in
imprecise environments. Notably, the
up to 0.8733. Figure 6 illustrates the
analytical results show that replacing the
change in the triangular possibilistic
imprecise data with the most possible
distributions
values or average of the possibility
transportation cost for the Dali case.
distributions
as
the
classical
TPD
techniques is unreasonable.
of
the
imprecise
total
Third, comparing the initial and
improved solutions reveals that the
Second, the proposed PLP approach
changes in the PIS and NIS of the three
determines the overall degree of DM
objective functions of the MOLP problem,
satisfaction under the proposed strategy
influences the objective and L values.
of minimizing the most possible values
From Table 2, the L values rapidly
and the risk of obtaining higher values,
increased from 0.5040 to 0.8733 when
and
of
the PIS changed from ($300,000, $60,000,
obtaining lower values for the imprecise
$10,000) to ($320,000, $40,000, $12,000)
objective function. If the solution is L  1 ,
and NIS changed from ($900,000,
then each goal is fully satisfied; if
$10,000, $50,000) to ($900,000, $10,000,
0  L  1 , then all of the goals are satisfied
$60,000). Conversely, total transportation
at the level of L, and if L  0 , then none
cost sharply decreases from ($382,000,
of the goals are satisfied. For example,
$346,800, $399,000) to ($352,000,
the overall degree of DM satisfaction
$315,800, $367,000). This finding
with the imprecise goal values ($382,000,
suggests that the DM must specify an
maximizing
$346,800,
the
$399,000)
possibility
initially
z
was
generated as 0.5040. Moreover, if the DM
Improved solution
1
Initial solution
did not accept the initial overall degree of
this satisfaction values, then the L value
was adjusted to seek a better compromise
solution. Consequently, the improved
total transportation cost is imprecise and
has a triangular possibility distribution of
~
z
0
315,800
352,000 367,000
346,800
382,000 399,000
Figure 6. The possibilistic distribution of
($352,000, $315,800, $367,000), and the
the improved total
overall degree of DM satisfaction can be
transportation cost
appropriate set of PIS and NIS of the
thus generally is distinguished from FLP
objective functions to make a right TPD,
and
some
FLP
and
inapplicable
for
to
effectively seek
the
linear
methods
PLP.
may
The
be
major
membership function of each imprecise
limitation in applying the FLP approach
objective. In practice, the LP solutions
is the lack of computational efficiency.
frequently served as a starting point of
Alternatively, the PLP approach not only
the PIS and NIS, and both intervals must
provides more computational efficiency
cover the LP solutions. For example,
and more flexible doctrines, but also
three crisp objective functions of the
facilities possibilistic decision-making in
auxiliary MOLP problem presented in the
imprecise environments [15, 18, 22].
Dali case were respectively solved using
5.2. Model Comparisons
the crisp single-goal LP model, and the
corresponding PIS and NIS of the initial
The PLP approach proposed in this study
solutions are listed in Table 4.
offers a practical method of solving TPD
Finally, the meaning of the PLP
problems,
and
can
generate
better
be
decisions than other models. Table 5
distinguished from the current FLP
compares the proposed PLP approach
approaches. The FLP is based on the
presented here with the ordinary LP and
subjective
FLP [6] models.
approach
studied
here
preferred
must
concept
for
establishing membership functions with
Several characteristics distinguish
fuzzy data, while the PLP is based on the
the proposed PLP approach from other
objective degree of event occurrence
TPD models. The proposed approach
required
involves an imprecise objective function
to
obtain
possibilistic
distributions with imprecise data. PLP
and constraints. In practice, the available
Table 4. The PIS and NIS of the initial solutions
Item
LP-1
LP-2
LP-3
( zkPIS , zkNIS ) (k  1, 2, 3)
Objective function
Min z1
Max z2
Min z3

z1
$352,000


($300,000, $900,000)
z2

$36,200

($60,000, $10,000)
z3


$15,000
($10,000, $50,000)
Table 5. Comparisons of the major TPD models
Factor
The ordinary LP
FLP [6]
The proposed PLP approach
Objective function
Objective value
Constraints property
DM overall satisfaction
Available capacities
Forecast demand
Unit costs coefficients
Revised flexibility
Single/Deterministic
Deterministic
Deterministic
Considered (100%)
Deterministic
Deterministic
Deterministic
Low
Single/Fuzzy
Deterministic
Deterministic
Not considered
Deterministic
Deterministic
Fuzzy
Medium
Single/ Imprecise
Imprecise/fuzzy
Imprecise/fuzzy
Considered
Imprecise/fuzzy
Imprecise/fuzzy
Imprecise/fuzzy
High
capacities for each source, the forecast
information than other TPD methods.
demand for each destination, and the
Notably, the goal values using the
related
are
proposed approach should be imprecise
imprecise over the planning horizon.
in nature because the related cost
Assigning a set of crisp values for these
coefficients are always imprecise in
imprecise
certain environments.
unit
cost
coefficients
parameters
is
not
an
inappropriate means of dealing with such
ambiguous TPD problems. The proposed
6. Conclusions
PLP approach provides an effective
method for proceeding in cases involving
inherent
ambiguities.
Moreover,
the
proposed PLP approach can solve most
real-world TPD problems with imprecise
goal
and
constraints
through
an
interactive decision-making process. The
proposed
approach
constitutes
a
systematic framework that facilitates the
decision-making process, enabling the
DM interactively to modify the imprecise
data and related model parameters until a
satisfactory
solution
is
found.
Additionally, the PLP approach presented
here also outputs more diverse decision
The TPD problem involves the
distribution of goods and services from a
set of sources to a set of destinations. In
real-wor1d TPD problems, related input
data or parameters, such as the unit cost
coefficients, available supply and demand
volumes, are frequently imprecise/fuzzy
due
to
incomplete
or
unobtainable
information. This study develops a novel
interactive PLP approach for solving
TPD problems with imprecise goal and
constraints. The proposed PLP approach
attempts
to
minimize
the
total
transportation costs in terms of imprecise
cost coefficients, available supply and
forecast demand. The proposed strategy
proposed provides a practical method of
involves simultaneously minimizing the
solving TPD problems, and can generate
most possible values, maximizing the
better decisions than other models.
possibility of obtaining lower values, and
minimizing the risk of obtaining higher
values
for
the
imprecise
objective
function. The linear membership function
is employed to represent the imprecise
goals, and the original problem can be
converted into an equivalent single-goal
linear programming model using the
minimum aggregate operator.
feasibility of applying the proposed
to
real
TPD
Consequently,
the
proposed
PLP
efficient
TPD
approach
yields
an
problems.
compromise solution and overall degree
of DM satisfaction with determined goal
values.
Furthermore,
approach
framework
provides
that
the
a
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