Transportation Optimization with Fuzzy Trapezoidal Numbers Based on Possibility Theory Dayi He, Ran Li, Qi Huang, Ping Lei* School of Humanities & Economic Management, Lab of Resources & Environment Management, China University of Geosciences (Beijing), Beijing, P. R. China
Abstract In this paper, a parametric method is introduced to solve fuzzy transportation problem. Considering that parameters of transportation problem have uncertainties, this paper develops a generalized fuzzy transportation problem with fuzzy supply, demand and cost. For simplicity, these parameters are assumed to be fuzzy trapezoidal numbers. Based on possibility theory and consistent with decision-makers’ subjectiveness and practical requirements, the fuzzy transportation problem is transformed to a crisp linear transportation problem by defuzzifying fuzzy constraints and objectives with application of fractile and modality approach. Finally, a numerical example is provided to exemplify the application of fuzzy transportation programming and to verify the validity of the proposed methods. Citation: He D, Li R, Huang Q, Lei P (2014) Transportation Optimization with Fuzzy Trapezoidal Numbers Based on Possibility Theory. PLoS ONE 9(8): e105142. doi:10.1371/journal.pone.0105142 Editor: Catalin Buiu, Politehnica University of Bucharest, Romania Received April 11, 2014; Accepted July 19, 2014; Published August 19, 2014 Copyright: ß 2014 He et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and its Supporting Information files. Funding: This research is supported by the Fundamental Research Funds for the Central Universities (2-9-2012-86) and supported by Key Laboratory of Carrying Capacity Assessment for Resource and Environment, MLR (CCA2012.05). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * Email: [email protected]
Natarajan [6] proposed a method, namely fuzzy zero point methods, for finding a fuzzy optimal solution for a fuzzy transportation problem where all parameters are trapezoidal fuzzy numbers. Basirzadeh [7] introduced an approach for solving a wide range of fuzzy transportation problem by ranking fuzzy numbers. Narayanamoorthy et al. [8] proposed a new algorithm called Fuzzy Russell’s method for the initial basic feasible solution to a Fuzzy transportation problem. Chaudhuri, et al. [9] present the closed, bounded and nonempty feasible region of the transportation problem using fuzzy trapezoidal numbers. Most previous research on fuzzy transportation problem assumes that all parameters are fuzzy numbers where the representation is either normal or abnormal, triangular or trapezoidal. Previous reports in literatures typically focus on comparing two or more fuzzy numbers and ranking fuzzy numbers for the purpose of optimization. This approach often achieved a bounded range or nonempty feasible region of transportation quantities and total cost, which are instructive rather than a straightforward solution for decision-makers. Furthermore, in order to effectively solve fuzzy transportation problem, the key factor is the decision-maker’s attitude about fuzziness. The possibility theory provides a feasible and scientific tool to describe the decision-maker’s attitude about fuzziness. Therefore, we will try to apply possibility theory to solve fuzzy transportation problem. In this paper, we will also consider all parameters in a fuzzy transportation problem as fuzzy numbers. However, we will utilize possibility theory to formalize the fuzzy transportation problem by introducing fractile and modality approach, therefore the decision-maker’s attitude about uncertainty can be considered into the decision process, which is of great
Introduction The transportation problem is a special type of the linear programming problem which arises in many practical applications. And it is one of the earliest and most fruitful applications of linear programming technique. It has been widely studied in Logistics and Operations Management where distribution of goods and commodities from sources to destinations is an important issue. The problem was formalized by the French Mathematician Gaspard Monge in 1781. Tolstoi was one of the first to study the transportation problem mathematically. In transportation problem the optimal shipping patterns between origins or supply centers and destinations or demand centers should be determined. Suppose that m origins are to supply n destinations with a certain product. Let ai be the amount of the product available at origin i, and bj be the amount of the product required at destination j. Further, the cost of shipping a unit of product from origin i to destination j is assumed to be ci j , and let xi j be the amount shipped from origin i to destination j. If shipping cost are assumed to be proportional to the amount shipped from each origin to each destination so as to minimize total shipping cost turns out to be a linear programming problem. In practice, there are many situations where it is impossible to get precise data for the cost parameters, due to complexity and uncertainty of information. A fuzzy transportation problem (FTP) is a transportation problem in which transportation costs, supply and/or demand quantities are fuzzy quantities. Earlier methods for solving fuzzy transportation problems are proposed by [1–4]. More recently, Gani and Razak [5] obtained a fuzzy solution for a two stage cost minimizing fuzzy transportation problem in which supplies and demands are trapezoidal fuzzy numbers. Pandian and PLOS ONE | www.plosone.org
1
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
Figure 1. The membership function and a-cut of a trapezoidal fuzzy number. doi:10.1371/journal.pone.0105142.g001
importance when a decision-maker is confronted with uncertainty or fuzziness. This paper has five sections. Section 1 is an introduction. In section 2, some basic concepts and possibility theory are introduced. Section 3 is the main part of this paper, where we proposed fuzzy transportation problem and defuzzied it into four crisp linear programming problems by utilizing the fractile or modality approach. In section 4, a numerical example is provided to illustrate the methods developed in this paper. And section 5 concludes.
Preliminaries In this section the basic concepts of transportation problem, fuzzy number, trapezoidal fuzzy number, possibility theory and their properties are recalled.
2.1 The crisp transportation problem Mathematically a transportation problem can be stated as follows:
min z~
m X n X
cij xij
ð1Þ
i~1 j~1
Figure 2. Framework of defuzzifying FTP. doi:10.1371/journal.pone.0105142.g002
PLOS ONE | www.plosone.org
2
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
Figure 3. The treatment of constraints. doi:10.1371/journal.pone.0105142.g003 Figure 4. Treatment of objective: necessity-fractile approach. doi:10.1371/journal.pone.0105142.g004
8 Pn > < Pj~1 xi j ~ai , i~1,2, ,m (supply constraints) m subject to i~1 xi j ~bj , j~1,2, ,n (destination constraints) ð2Þ > : Vi and j xi j §0,
~ )~fx [ X Dm ~ (x)w0g is a closed and bounded iv) Supp(A A interval, i.e., ½a1 {b,a2 zc. Specifically an LR-type fuzzy number is obtained from a generalized fuzzy number f (x) and h(x) are if the shape functions
where ci j is the cost of transportation of a unit from the it h source to the j t h destination, and the quantity xi j is some non-negative number, which is to be transported from the it h origin to the j t h destination. An obvious necessary and sufficient condition for the linear programming problem given in (1) to have a solution is that m X
ai ~
i~1
n X
bj
approximated by L
ð3Þ
x{a2 c
respectively.
2.3 Trapezoidal fuzzy number
j~1
~ is a trapezoidal fuzzy number denoted by A fuzzy number T ~ ~(t1 ,t2 ; b,c) where t1 , t2 , a, b are real number and its T membership function mT (x) is given below. 8 0, > > > x{t1 > > > < b , 1, mT (x)~ > > > t2 {x , > > c > : 0,
2.2 Generalized fuzzy number and its a-cut A fuzzy number is an generalization of a regular, real number in the sense that it does not refer to one single value but rather to a connected set of possible values, where each possible value has its own weight between 0 and 1. This weight is called the membership function. A fuzzy number is thus a special case of a convex, normalized fuzzy set of the real line. ~ , conventionally Mathematically a generalized fuzzy number A ~ represented by A~(a1 ,a2 ; b,c), i.e., (left point, right point; left spread, right spread), is a normalized convex fuzzy subset on the real line R if i) a1 {bva1 ƒa2 va2 zc; ii) the membership function is of the following form:
xƒt1 {b t1 {bƒxƒt1 t1 ƒxƒt2 t2 ƒxƒt2 zc x§t2 zc
~ which can be denoted by The a-cut of the fuzzy number T Ta ~½Tal ,Tau , is shown in the Fig. 1. Tal ~t1 {(1{a)b,
Tau ~t2 z(1{a)c
2.4 Possibility theory Possibility theory is an uncertainty theory devoted to handle incomplete information. As such, it complements probability theory. It differs from the latter by using of a pair of dual setfunctions (possibility and necessity measures) instead of only one. This feature makes it easier to capture partial ignorance. Besides, it is not additive and makes sense on ordinal structures. The name Theory of Possibility was coined in Negoita and Zadeh [10], inspired by Gaines [11]. In Zadeh’s view, possibility distributions were meant to provide a graded semantics to natural language statements. However, possibility and necessity measures can also be the basis of a full-fledged representation of partial belief that
Vx[½a1 {b,a1 Vx[½a1 ,a2 Vx[½a2 ,a2 zc
where f (x) and h(x) are the monotonic increasing and decreasing functions in ½a1 {b,a1 and ½a2 ,a2 zc respectively; iii) mA~ (x) is an upper semi-continuous function;
PLOS ONE | www.plosone.org
and R
~ is an interval number denoted by ½A ~ ~½Al ,Ar , The a-cut of A a a a which is explicitly shown for a LR-type fuzzy number, i.e., ½Ala ,Ara ~½a1 {bL{1 (a),a2 zcR{1 (a) for all a [ ½0,1.
i.e., it is assumed that the total available is equal to the total required. If it is not true, a fictitious origin or destination can be added. It should be noted that the problem has a feasible solution if and only if condition (2) is satisfied. Now the problem is to determine xij , in such a way that the total transportation cost is minimum.
8 > < f (x), mA~ (x)~ 1, > : h(x),
a1 {x b
3
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
Figure 6. Treatment of objective: necessity-modality approach. doi:10.1371/journal.pone.0105142.g006 Figure 5. Treatment of objective: possibility-fractile approach. doi:10.1371/journal.pone.0105142.g005
Mathematical formulation of fuzzy transportation problem
parallels probability [12]. Then, it can be seen as a coarse, nonnumerical version of probability theory, or as a framework for reasoning with extreme probabilities, or yet as a simple approach to reasoning with imprecise probabilities [13]. ~ and B ~ , the ~ are two fuzzy sets and a fuzzy event v [ A If A ~ possibility and necessity of v [ B are defined as:
In this section, we first present the Fuzzy Transportation Problem (FTP). Then, based on possibility theory, treatment of constraints and objective are discussed in order to develop the four formulations of the FTP.
3.1 Fuzzy transportation problem Mathematically the FTP can be described as follows.
~ )~ sup minfm ~ (r),m ~ (r)g P A~ (B B A r
min ~z~
m X n X
~ci j xi j
ð4Þ
i~1 j~1
~ )~ inf maxf1{m ~ (r),m ~ (r)g NA~ (B B A r
8 Xn ~ ~ai , i~1,2, ,m (supply constraints) > < X j~1 xi j ~ (5) m ~bj , j~1,2, ,n (destination constraints) ð5Þ subject to x ~ ~ i~1 i j > : Vi and j xi j §0,
If ~a and ~b are two fuzzy numbers and their membership function are m~a (x) and m~b (x) respectively, the measure of possibility and necessity are defined as: Pos(~a ~b)~ supfmin (m~a (x),m~b (y)Dx,y [ R, x yg
In the constraints of FTP, the left side is a crisp number while the right side is a fuzzy number. Hence, the constraints produce a fuzzy set. To determine whether a crisp number is in a fuzzy set, the classical method is to use the membership function. If the membership function value of the crisp number is positive, then it can be sure that it belongs to the fuzzy set, and the larger is the value, the higher is the extent. As for the objective function, it is a linear non-negative combination of fuzzy variables. So the objective of FTP is a fuzzy variable too. To minimize the objective, a ranking method of fuzzy numbers should be introduced. Therefore, in order to solve FTP, we should introduce a measure to determine whether a crisp number belongs to a fuzzy set and a ranking index to rank fuzzy numbers. Generally ~ , and speaking, let the feasible fuzzy set of the FTP is denoted by X ~ . And there is a all possible objective values is in the fuzzy set Z measure system M to determine whether a crisp number belongs ~ )ƒ1. Then the FTP can be to a fuzzy set, where 0ƒM(a [ X described as follows conceptually.
Nes(~a ~b)~ inffmax (1{m~a (x),m~b (y))Dx,y [ R, x yg If ~b is a crisp number, then the we can achieved that Pos(~aƒb)~ supfm~a (x)Dx [ R,xƒbg
Nes(~aƒb)~ supf1{m~a (x)Dx [ R,xwbg Generally speaking, the possibility degree evaluates to what extent an event is consistent with the knowledge P, while the necessity degree evaluates to what extent an event is certainly implied by the knowledge.
max
PLOS ONE | www.plosone.org
4
~) M(z(x) [ Z
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
We think that how to solve FTP depends on the decisionmaker’s attitude to the fuzziness, so it is hard to define which model should be used. Decision-maker can choose any proposed method to solve FTP just according to his attitude to the fuzziness. We propose possibility theory to describe decision-maker’s attitude. As FTP consists of fuzzy constraints and objective, this paper deals with them individually as shown in Figure 2. With respect to fuzzy constraints, we utilize possibility theory to convert them into crisp and bounded intervals. As for fuzzy objective, by applying fractile and modality approach we convert it into four formations from the view of necessity and possibility respectively. Then by recombining the converted constraints and objective, four models are established.
3.2 Treatment of constraints
Figure 7. Treatment of objective: possibility-modality approach. doi:10.1371/journal.pone.0105142.g007
As constraints of FTP are equations, and according to the definition of possibility and necessity, we can get Pos(~ a~b)~m~a (b),
Nes(~ a~b)~0
~ )§z, 0ƒzƒ1 M(x [ X
s:t:
Hence, the treatment of constraints can be considered from the view of possibility exclusively. It is reasonable and frequently-used that when decision makers are confronted with uncertainty, they can only make sure that the th constraints are satisfied to some extent. For Pn the i origin Ai in FTP, the decision-maker expects that the j~1 xi j ~~ai should be satisfied as possible because the production is uncertain. According to the definition of possibility, we can set that the possibility of the constraint be true is larger than a given level ai , where 0ƒai ƒ1, which reflects the extent how the constraint is satisfied.
or
max ( s:t:
q
~ )§q, 0ƒqƒ1 M(z(x)[Z ~ M(x[X )§z, 0ƒzƒ1
~ and q is the degree of x where z is the degree of z(x) belongs to Z ~. belongs to X To be simple, in this paper we use trapezoidal numbers to represent the fuzziness of parameters in FTP, i.e., we assume that
Pos
n X
! xi j ~~ a §ai , i~1,2, ,m
ð6Þ
j~1
~cij ~(ci j 1 ,ci j2 ; Li j ,Ri j ),i~1,2, ,m; j~1,2, ,n
According to the definition of possibility, the equation (6) is depicted in the Figure 3, where P1 and P2 is the left and right bound of ~ ai ’s a-cut. To ensure the equation (6) can be satisfied, the point O should be above of P1 and P2 , which converts the constraint P n ai to j~1 xi j ~~
~ai ~(ai1 ,ai2 ; Li ,Ri ),i~1,2, ,m
~bj ~(bj1 ,bj2 ; Lj ,Rj ),j~1,2, ,n ai1 {(1{ai )Li ~aLiai ƒ
And the possibility and necessity measure are adopted as ranking index.
n X j~1
xi j ƒaR iai ~ai2 z(1{ai )Ri , i~1,2, ,m ð7Þ
Table 1. Data of the example.
B1
B2
B3
B4
B5
Sup.
A1
(8,9;3,2)
(2,3;1,5)
(2,5;1,6)
(10,16;4,8)
(7,12;5,5)
(22,28;20,30)
A2
(3,6;1,8)
(8,11;4,3)
(12,16;5,8)
(3,5;1,6)
(2,6;1,7)
(28,35;25,36)
A3
(13,16;5,8)
(3,8;1,10)
(10,18;4,9)
(4,10;1,1)
(12,20;5,2)
(16,24;12,28)
A4
(18,23;3,6)
(12,18;5,9)
(11,15;4,8)
(30,36;6,4)
(6,9;2,1)
(28,36;25,40)
Dem.
(20,30;12,30)
(22,32;15,30)
(32,40;25,40)
(12,28;6,18)
(28,36;18,40)
doi:10.1371/journal.pone.0105142.t001
PLOS ONE | www.plosone.org
5
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
C1 ~
m X n X
ci j1 xi j ,
C2 ~
i~1 j~1
L~
m X n X
ci j2 xi j
i~1 j~1
Li j xi j ,
R~
m X n X
i~1 j~1
Ri j xi j
i~1 j~1
There are two approaches to treat the objective: fractile and modality approach. 3.3.1 Fractile approach. A fractile approach corresponds to the Kataoka’s model of a stochastic programming problem [14,15]. Geoffrion [16] calls the Kataoka’s model the fractile criterion approach. By the definition in statistics, p-fractile is the value u which satisfies
Figure 8. The relationship between supply and demand. doi:10.1371/journal.pone.0105142.g008
Similarly, for the j t h destination Bj , the constraint should be satisfied with the possibility larger than bj (0ƒbj ƒ1), i.e., ! m X ~ Pos xi j ~b §bj , j~1,2, ,n
m X n X
Prob(X ƒu)~p where X is a random variable. In this definition, p-fractile does not generally exist for all p [ (0,1). That is why we define p-fractile as the smallest value up of u which satisfies
ð8Þ
i~1
And it can be converted to Prob(X ƒu)§p bj1 {(1{bj )Lj ~bLjbj ƒ
m X i~1
xij ƒbR bbj ~bj2 z(1{bj )Rj , j~1,2, ,n
ð9Þ
From the viewpoint of Dempster-Shafer theory of evidence [17], it is known that Pos(X ƒu) and Nes(X ƒu) can be regarded as the upper and lower bounds of Prob(X ƒu) (see [18]). In this sense, p-possibility fractile is defined as the smallest value of u which satisfies
In such ways, the constraints of FTP are transformed into crisp and bounded intervals.
3.3 Treatment of objective
Pos(X ƒu)§p
According to the operation algorithm of fuzzy numbers, because the unit cost of FTP is a linear non-negative combination of m|n trapezoidal numbers, the objective is a trapezoidal numbers too. ~ ~(C1 ,C2 ; L,R) be the objective value. When xi j §0, there Let C exists
and p-necessity fractile is defined as the smallest value of u which satisfies
Table 2. The crisp transportation problem originated from Model I.
c~0:95
B1
B2
B3
B4
B5
Sup.
A1
10.90
7.75
10.70
23.60
16.75
[19.0, 32.0]
A2
13.60
13.85
23.60
10.70
12.65
[25.5, 38.6]
A3
23.60
17.50
26.55
10.95
21.90
[13.6, 29.6]
A4
28.70
26.55
22.60
39.80
9.95
[23.0, 44.0]
Dem.
[17, 37.5]
[20.5, 35]
[30.75, 42]
[11.1, 30.7]
[24.4, 44]
c~0:05
B1
B2
B3
B4
B5
Sup.
A1
9.10
3.25
5.30
16.40
12.25
[19.0, 32.5]
A2
6.40
11.15
16.40
5.30
6.35
[25.5, 38.6]
A3
16.40
8.50
18.45
10.05
20.10
[13.6, 29.6]
A4
23.30
18.45
15.40
36.20
9.05
[23.0, 44.0]
Dem.
[17.0,37.5]
[20.5,35.0]
[30.75,42.0]
[11.1,30.7]
[24.4,44.0]
doi:10.1371/journal.pone.0105142.t002
PLOS ONE | www.plosone.org
6
August 2014 | Volume 9 | Issue 8 | e105142
0.80
As for the objective of FTP, from the point of necessity it can convert to
103.75
13.6
Nes(X ƒu)§p
0.83
1.00
1.00 32.5
33.82
Sup.
Pos
Fuzzy Transportation Optimization Based on Possibility Theory
0
0.57
0
23.83
24.40
0.80
0
11.1
0
23.83
11.1
0.85
B5 B4
min C
~ ƒC)§c s:t: Nes(C
where c is a parameter to reflect to what extent the decision-maker ~ ƒC. Eq.(10) is certain about the lower bound of the fuzzy event C expresses that the decision-maker minimizes the total cost at a given necessity level. As shown in Figure 4, we can get
~ ~(375:43,699:85; 143:03,575:10) C
0.95
30.75 20.5
0.90 0.75
17
0
0 13.6
0 0
0
30.75
0 5.15
1.75 0
17
B1
B2
B3
~ ƒC)~ Nes(C
c~0:05
C{C2 §c[C2 zcRƒC R Hence, Eq. (10) turns to
0.80
0.86
13.6
24.4
s:t:C2 zcRƒC
ð11Þ
It is easy to find that the optimal solution of (11) is C~C2 zcR. Therefore, the objective function of FTP can be substituted by 103.75
1.00
1.00
32.5
33.25
Pos
C{C2 R
then the constraint in (10) can be converted to
min C
Supply
ð10Þ
min
C2 zcR~
m X n X
(ci j2 zcRi j )xi j
ð12Þ
0.80
24.4
And with the constraints (7) and (9), the FTP turns to the following model, which is a crisp linear programming problem.
0.85
11.1
0
24.4 0
11.1
0
0 0
0
B4
B5
i~1 j~1
cost
PLOS ONE | www.plosone.org
D
m X n X
(ci j2 zcRi j )xi j
i~1 j~1
( s:t:
Eq: ð7Þ Eq: ð9Þ
From the point of the possibility and by the fractile approach, the objective of FTP means that
doi:10.1371/journal.pone.0105142.t003
~ ~(444:3,790:35; 176:9,438:5) C Total
30.75
0.95 0.90 0.75 Pos
20.5 17 Dem.
0
0 0 0 A4
2.5 0 A3
0
30.75 1.75
17 A2
16.25
0 A1
B3 B2 B1 c~0:95
Table 3. Solution from Model I (c~0:95).
Model I
min
min C
~ ƒC)§h s:t: Pos(C
ð13Þ
i.e., the decision-maker expects that the total cost should be minimized as the possibility that total cost is not larger than some given level C.
7
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
greater than a given level C0 , i.e., ~ ƒC0 ) Nes(C
max
ð16Þ
From Figure 6, we can get ~ ƒC0 )~ Nes(C
C0 {C2 R
hence, the objective of FTP can be converted to
min
Figure 9. Fuzzy total cost when c~0:95 and ª~0:05. doi:10.1371/journal.pone.0105142.g009
As shown in Figure 5, we can get ~ ƒC)~ Pos(C
C0 {C2 ~ R
Xm Xn i~1 j~1 X m Xn i~1
Xm Xn
C{C1 zL §h L
i~1 j~1 X m Xn
min
i~1
s:t:
C1 {(1{h)LƒC Hence, (13) can be converted to s:t: C1 {(1{h)LƒC
m X n X
1 y~ Xm Xn i~1
m X n X
(cij1 {(1{h)Lij )xij
i~1 j~1
( s:t:
Eq: ð7Þ
Ri j xi j
Eq: ð7Þ Eq: ð9Þ
j~1
Ri j xi j
, zi j ~xi j y,Vi, j
D
min
m X n X
cij2 zij {C0 y
i~1 j~1
s:t:
8Xn zij {½ai1 {(1{ai )Li y§0, > > > X j~1 > > > n > > j~1 zij {½ai2 z(1{ai )Ri yƒ0, > > > X > > m < i~1 zij {½bj1 {(1{bj )Lj y§0, X > m > > > i~1 zij {½bj2 z(1{bj )Rj yƒ0, > > > X X > m n > > > i~1 j~1 Rij zij ~1 > > : y§0,zij §0; i~1,2, ,m; j~1,2, ,n
i~1,2, ,m i~1,2, ,m j~1,2, ,n j~1,2, ,n
Eq: ð9Þ From the point of possibility, the decision-maker expects that the total cost should not be greater than a given level C0 , i.e.,
3.3.2 Modality approach. A modality optimization model corresponds to the minimum-risk approach to a stochastic programming problem [15]. The minimum-risk approach is also called the maximum probability approach [14] or the aspiration criterion approach [16]. A modality optimization approach is a dual approach to the fractile optimization one. In this approach, the decision-maker puts more importance on the certainty degree comparing to the fractile approach. From the point of necessity and modality approach, the decision-maker in FTP expects that the total cost should not be
PLOS ONE | www.plosone.org
j~1
ð15Þ
Model III
D
ci j2 xi j {C0
We obtain the following model. (ci j1 {(1{h)Li j )xi j
And with the constraints (7) and (9), the FTP turns to the following model.
Model II
ð17Þ
ð14Þ
i~1 j~1
min
Ri j xi j
This is a fractional programming problem which can be transformed to a linear programming problem by the substitution
It is easy to find that the optimal solution of (14) is C1 {(1{h)L, i.e.,
C1 {(1{h)L~
j~1
Such that FTP turns to
or
min C
ci j2 xi j {C0
max
8
~ ƒC0 ) Pos(C
ð18Þ
August 2014 | Volume 9 | Issue 8 | e105142
PLOS ONE | www.plosone.org
0
17
0.75
A4
Dem.
Pos
9
doi:10.1371/journal.pone.0105142.t005
~ ~(377:7,701:55; 143:60,571:70) C
0.75
cost
Pos
Total
0
17
0
A3
Dem.
17
A2
A4
0
A1
B1
Table 5. Solution from Model III (C0 ~1000).
doi:10.1371/journal.pone.0105142.t004
~ ~(346:35,681:9; 126:75,616:15) C
0
A3
cost
17
A2
Total
0
A1
B1
Table 4. Solution from Model II (h~0:95).
0.90
20.5
0
13.6
5.15
1.75
B2
0.90
20.5
0
18.75
0
1.75
B2
0.95
30.75
0
0
0
30.75
B3
0.95
30.75
0
0
0
30.75
B3
0.85
11.1
0
0
11.1
0
B4
0.85
11.1
0
0
11.1
0
B4
0.80
24.4
24.4
0
0
0
B5
0.80
24.4
23
0
1.4
0
B5
103.75
24.4
13.6
33.25
32.5
Sup.
103.75
23
18.75
29.5
32.5
Sup.
0.856
0.80
1.00
1.00
Pos
0.80
1.00
1.00
1.00
Pos
Fuzzy Transportation Optimization Based on Possibility Theory
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
From Figure 7, we can get C0 {C1 zL L
0.80
1.00
1.00
1.00
Pos
~ ƒC0 )~ Pos(C
Hence, the objective of FTP can be transformed to C0 {C1 zL [ min L
103.75
23
18.75
32.5
29.5
Sup.
max
Pm Pn
i~1 j~1 P m Pn i~1
cij1 xij {C0
j~1
Li j xi j
ð19Þ
Such that the FTP turns to Pm Pn C~
ci j1 xi j {C0 i~1 Pm j~1 Pn i~1 j~1 Li j xi j
0.80
24.4
23
0
0
1.4
B5
min
s:t:
Eq: ð7Þ Eq: ð9Þ
0.85
11.1
0
0
11.1
0
B4
Again, this is a fractional programming problem which can be transformed to a linear programming problem by the substitution 1 y~ Pm Pn i~1
j~1
Li j xi j
, zi j ~xi j y,Vi, j
0.95
30.75
0
0
0
30.75
B3
We obtain the following model.
0.90
D
m X n X
cij1 zij {C0 y
i~1 j~1
s:t:
8Xn zij {½ai1 {(1{ai )Li y§0 i~1,2, ,m > > > X j~1 > > > n > i~1,2, ,m > j~1 zij {½ai2 z(1{ai )Ri yƒ0 > > > X > > m < j~1,2, ,n i~1 zij {½bj1 {(1{bj )Lj y§0 X > m > > j~1,2, ,n > i~1 zij {½bj2 {(1{bj )Lj y§0 > > > X X > m n > > > i~1 j~1 Lij zij ~1 > > : y§0,zij §0; i~1,2, ,m; j~1,2, ,n
To sum up, from the view of possibility and necessity, and by fractile and modality approach, we proposed four ways to defuzzify FTP to four crisp linear programming problems. During the transformation process, some parameters are introduced to clarify the decision-maker’s subjectiveness about fuzziness, which makes the solutions more practical. As for constraints, we introduced ai and bj to reflect the decision-maker’s requirement on the extent how the constraint is satisfied in the view of possibility. With respect to the objective function, in the fractile approach we proposed c as the lower bound of necessity and h as the lower bound of possibility that the total cost is not larger than a given level set by the decision-maker. While in the modality approach, C0 was introduced as the upper bound of total cost that decision-maker expected. In brief, these parameters reflect the decision-maker’s attitude
doi:10.1371/journal.pone.0105142.t006
~ ~(346:35,681:9; 126:75,616:15) C
0.75
PLOS ONE | www.plosone.org
Total
cost Pos
20.5 17 Dem.
0 0 A4
18.75 0 A3
1.75
17 A2
0
0 A1
B1
Table 6. Solution from Model IV (C0 ~1000).
B2
Model IV
min
10
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
some given level C is 0.95, i.e., let h~0:95. The solution is in the Table 4. In model III, let C0 equals 1000, the solution is listed in Table 5. From Table 5, we can easily get that
to the fuzziness in FTP from the view of possibility or necessity.
Numerical example In this section, numerical examples of FTP will be presented to demonstrate and verify the proposed approaches. All established models are originated from the above models based on the different parameter settings. All models are solved by the Lingo software, so the solving process is omitted for simplification. The description of transportation problem is standard tables, where the central part is the cost cij , the column ‘‘Supply’’ are ai and the row ‘‘Demand’’ are bj . While in the solution tables, the central part is the quantities of transportation from Ai to Bj . Table 1 is a FTP with four origins Ai (i~1,2,3,4) and five destinations Bj (j~1,2,3,4,5) because the supply, demand and cost of unit are assumed to be fuzzy trapezoidal numbers. By adding up for origins’ supply and five destinations’ demand in Table 1, the total supply and demand are fuzzy trapezoidal numbers (94,123;82,134) and (114,166;76,158) depicted in the Figure 8. To solve this FTP, firstly we set the value of ai , bj , c and C0 as follows: ai :~(0:85,0:90,0:80,0:80);
c :~0:95,0:05;
~ ƒC0 ~1000)~1 Pos(C Hence, the possibility that the total cost is not larger than given C0 is satisfied. This is reasonable because C0 is set to be the upper bound of the minimal total cost which a decision-maker can afford. In model IV, similarly let C0 equals 1000, and solution is listed in Table 6. From Table 6, the following relationship exist. ~ ƒC0 ~1000)~1 Pos(C Similar results can be achieved as shown in Model III.
Conclusions In this paper, a simple but effective parametric method was introduced to solve fuzzy transportation problem. By using possibility theory in fractile and modality approach, the fuzzy transportation problem is transformed into four types of crisp linear programming problems. In the process of transformation, some parameters are introduced to reflect decision-maker’s attitude to the uncertainty or fuzziness. The methods proposed in this paper can be used for all kinds of fuzzy transportation problem, whether triangular and trapezoidal fuzzy numbers with normal or abnormal data.
bj :~(0:75,0:90,0:95,0:85,0:80)
h :~0:95;
C0 :~1000
Then according to (7) and (9), the supply and demand constraints can be converted to different bounded intervals as shown in Table 2 based on the given value of ai and bj . By using Model I, let c to be 0.95 and 0.05 respectively. And based on the Eq.(12), the cost of FTP can be transformed into a crisp value as shown in Table 2. Solve the two crisp transportation problems by linear programming, the optimal solution is in Table 3, and the fuzzy total cost is depicted in Figure 9 for comparative analysis. From Table 3 and Figure 9, it can be found that the distribution varies with the changing of c. However, the total transportation, supply from different origins and demand to different destinations are same. And, the possibility requirement of supply and demand constraints are almost satisfied. From the point of fuzzy total cost, it gets smaller when c is smaller. According to Model II, assume that the decision-maker expects that the lower bound of possibility that total cost is not larger than
Acknowledgments This research is supported by the Fundamental Research Funds for the Central Universities (2-9-2012-86) and supported by Key Laboratory of Carrying Capacity Assessment for Resource and Environment, MLR (CCA2012.05). The authors would like to heartily thank the Editor in Chief and anonymous reviewers for their careful reading of this paper and for their helpful comments and suggestions.
Author Contributions Analyzed the data: RL. Contributed to the writing of the manuscript: DH PL. Developed models: DH. Designed the numerical examples: QH.
References 1. Chanas S, Kołodziejczyk W, Machaj A (1984) A fuzzy approach to the transportation problem. Fuzzy Sets and Systems 13: 211–221. 2. Chanas S, Delgado M, Verdegay J, Vila M (1993) Interval and fuzzy extensions of classical transportation problems. Transportation Planning and Technology 17: 203–218. 3. Chanas S, Kuchta D (1996) A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets and Systems 82: 299–305. 4. Liu ST, Kao C (2004) Solving fuzzy transportation problems based on extension principle. European Journal of Operational Research 153: 661–674. 5. Gani A, Razak KA (2006) Two stage fuzzy transportation problem. Journal of Physical Sciences 10: 63–69. 6. Pandian P, Natarajan G (2010) A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Applied Mathematical Sciences 4: 79–90. 7. Basirzadeh H (2011) An approach for solving fuzzy transportation problem. Applied Mathematical Sciences 5: 1549–1566.
PLOS ONE | www.plosone.org
8. Narayanamoorthy S, Saranya S, Maheswari S (2013) A method for solving fuzzy transportation problem (ftp) using fuzzy russell’s method. International Journal of Intelligent Systems & Applications 5. 9. Chaudhuri A, De K, Chatterjee D, Mitra P (2013) Trapezoidal fuzzy numbers for the transportation problem. arXiv preprint arXiv:13071893. 10. Negoita C, Zadeh L, Zimmermann H (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy sets and systems 1: 3–28. 11. Gaines BR, Kohout TL (1975) Possible automata. 12. Dubois D, Prade H (1988) Default reasoning and possibility theory. Artificial Intelligence 35: 243–257. 13. Dubois D, Nguyen HT, Prade H (2000) Possibility theory, probability and fuzzy sets misunderstandings, bridges and gaps. In: Fundamentals of fuzzy sets, Springer. pp. 343–438. 14. Kataoka S (1963) A stochastic programming model. Econometrica: Journal of the Econometric Society: 181–196. 15. Stancu-Minasian I (1990) Overview of different approaches for solving stochastic programming problems with multiple objective functions. In: Stochastic versus
11
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
17. Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. The annals of mathematical statistics : 325–339. 18. Dubois D, Prade H (1987) The mean value of a fuzzy number. Fuzzy sets and systems 24: 279–300.
fuzzy approaches to multiobjective mathematical programming under uncertainty, Springer. pp. 71–101. 16. Geoffrion AM (1967) Stochastic programming with aspiration or fractile criteria. Management Science 13: 672–679.
PLOS ONE | www.plosone.org
12
August 2014 | Volume 9 | Issue 8 | e105142
Abstract In this paper, a parametric method is introduced to solve fuzzy transportation problem. Considering that parameters of transportation problem have uncertainties, this paper develops a generalized fuzzy transportation problem with fuzzy supply, demand and cost. For simplicity, these parameters are assumed to be fuzzy trapezoidal numbers. Based on possibility theory and consistent with decision-makers’ subjectiveness and practical requirements, the fuzzy transportation problem is transformed to a crisp linear transportation problem by defuzzifying fuzzy constraints and objectives with application of fractile and modality approach. Finally, a numerical example is provided to exemplify the application of fuzzy transportation programming and to verify the validity of the proposed methods. Citation: He D, Li R, Huang Q, Lei P (2014) Transportation Optimization with Fuzzy Trapezoidal Numbers Based on Possibility Theory. PLoS ONE 9(8): e105142. doi:10.1371/journal.pone.0105142 Editor: Catalin Buiu, Politehnica University of Bucharest, Romania Received April 11, 2014; Accepted July 19, 2014; Published August 19, 2014 Copyright: ß 2014 He et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and its Supporting Information files. Funding: This research is supported by the Fundamental Research Funds for the Central Universities (2-9-2012-86) and supported by Key Laboratory of Carrying Capacity Assessment for Resource and Environment, MLR (CCA2012.05). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * Email: [email protected]
Natarajan [6] proposed a method, namely fuzzy zero point methods, for finding a fuzzy optimal solution for a fuzzy transportation problem where all parameters are trapezoidal fuzzy numbers. Basirzadeh [7] introduced an approach for solving a wide range of fuzzy transportation problem by ranking fuzzy numbers. Narayanamoorthy et al. [8] proposed a new algorithm called Fuzzy Russell’s method for the initial basic feasible solution to a Fuzzy transportation problem. Chaudhuri, et al. [9] present the closed, bounded and nonempty feasible region of the transportation problem using fuzzy trapezoidal numbers. Most previous research on fuzzy transportation problem assumes that all parameters are fuzzy numbers where the representation is either normal or abnormal, triangular or trapezoidal. Previous reports in literatures typically focus on comparing two or more fuzzy numbers and ranking fuzzy numbers for the purpose of optimization. This approach often achieved a bounded range or nonempty feasible region of transportation quantities and total cost, which are instructive rather than a straightforward solution for decision-makers. Furthermore, in order to effectively solve fuzzy transportation problem, the key factor is the decision-maker’s attitude about fuzziness. The possibility theory provides a feasible and scientific tool to describe the decision-maker’s attitude about fuzziness. Therefore, we will try to apply possibility theory to solve fuzzy transportation problem. In this paper, we will also consider all parameters in a fuzzy transportation problem as fuzzy numbers. However, we will utilize possibility theory to formalize the fuzzy transportation problem by introducing fractile and modality approach, therefore the decision-maker’s attitude about uncertainty can be considered into the decision process, which is of great
Introduction The transportation problem is a special type of the linear programming problem which arises in many practical applications. And it is one of the earliest and most fruitful applications of linear programming technique. It has been widely studied in Logistics and Operations Management where distribution of goods and commodities from sources to destinations is an important issue. The problem was formalized by the French Mathematician Gaspard Monge in 1781. Tolstoi was one of the first to study the transportation problem mathematically. In transportation problem the optimal shipping patterns between origins or supply centers and destinations or demand centers should be determined. Suppose that m origins are to supply n destinations with a certain product. Let ai be the amount of the product available at origin i, and bj be the amount of the product required at destination j. Further, the cost of shipping a unit of product from origin i to destination j is assumed to be ci j , and let xi j be the amount shipped from origin i to destination j. If shipping cost are assumed to be proportional to the amount shipped from each origin to each destination so as to minimize total shipping cost turns out to be a linear programming problem. In practice, there are many situations where it is impossible to get precise data for the cost parameters, due to complexity and uncertainty of information. A fuzzy transportation problem (FTP) is a transportation problem in which transportation costs, supply and/or demand quantities are fuzzy quantities. Earlier methods for solving fuzzy transportation problems are proposed by [1–4]. More recently, Gani and Razak [5] obtained a fuzzy solution for a two stage cost minimizing fuzzy transportation problem in which supplies and demands are trapezoidal fuzzy numbers. Pandian and PLOS ONE | www.plosone.org
1
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
Figure 1. The membership function and a-cut of a trapezoidal fuzzy number. doi:10.1371/journal.pone.0105142.g001
importance when a decision-maker is confronted with uncertainty or fuzziness. This paper has five sections. Section 1 is an introduction. In section 2, some basic concepts and possibility theory are introduced. Section 3 is the main part of this paper, where we proposed fuzzy transportation problem and defuzzied it into four crisp linear programming problems by utilizing the fractile or modality approach. In section 4, a numerical example is provided to illustrate the methods developed in this paper. And section 5 concludes.
Preliminaries In this section the basic concepts of transportation problem, fuzzy number, trapezoidal fuzzy number, possibility theory and their properties are recalled.
2.1 The crisp transportation problem Mathematically a transportation problem can be stated as follows:
min z~
m X n X
cij xij
ð1Þ
i~1 j~1
Figure 2. Framework of defuzzifying FTP. doi:10.1371/journal.pone.0105142.g002
PLOS ONE | www.plosone.org
2
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
Figure 3. The treatment of constraints. doi:10.1371/journal.pone.0105142.g003 Figure 4. Treatment of objective: necessity-fractile approach. doi:10.1371/journal.pone.0105142.g004
8 Pn > < Pj~1 xi j ~ai , i~1,2, ,m (supply constraints) m subject to i~1 xi j ~bj , j~1,2, ,n (destination constraints) ð2Þ > : Vi and j xi j §0,
~ )~fx [ X Dm ~ (x)w0g is a closed and bounded iv) Supp(A A interval, i.e., ½a1 {b,a2 zc. Specifically an LR-type fuzzy number is obtained from a generalized fuzzy number f (x) and h(x) are if the shape functions
where ci j is the cost of transportation of a unit from the it h source to the j t h destination, and the quantity xi j is some non-negative number, which is to be transported from the it h origin to the j t h destination. An obvious necessary and sufficient condition for the linear programming problem given in (1) to have a solution is that m X
ai ~
i~1
n X
bj
approximated by L
ð3Þ
x{a2 c
respectively.
2.3 Trapezoidal fuzzy number
j~1
~ is a trapezoidal fuzzy number denoted by A fuzzy number T ~ ~(t1 ,t2 ; b,c) where t1 , t2 , a, b are real number and its T membership function mT (x) is given below. 8 0, > > > x{t1 > > > < b , 1, mT (x)~ > > > t2 {x , > > c > : 0,
2.2 Generalized fuzzy number and its a-cut A fuzzy number is an generalization of a regular, real number in the sense that it does not refer to one single value but rather to a connected set of possible values, where each possible value has its own weight between 0 and 1. This weight is called the membership function. A fuzzy number is thus a special case of a convex, normalized fuzzy set of the real line. ~ , conventionally Mathematically a generalized fuzzy number A ~ represented by A~(a1 ,a2 ; b,c), i.e., (left point, right point; left spread, right spread), is a normalized convex fuzzy subset on the real line R if i) a1 {bva1 ƒa2 va2 zc; ii) the membership function is of the following form:
xƒt1 {b t1 {bƒxƒt1 t1 ƒxƒt2 t2 ƒxƒt2 zc x§t2 zc
~ which can be denoted by The a-cut of the fuzzy number T Ta ~½Tal ,Tau , is shown in the Fig. 1. Tal ~t1 {(1{a)b,
Tau ~t2 z(1{a)c
2.4 Possibility theory Possibility theory is an uncertainty theory devoted to handle incomplete information. As such, it complements probability theory. It differs from the latter by using of a pair of dual setfunctions (possibility and necessity measures) instead of only one. This feature makes it easier to capture partial ignorance. Besides, it is not additive and makes sense on ordinal structures. The name Theory of Possibility was coined in Negoita and Zadeh [10], inspired by Gaines [11]. In Zadeh’s view, possibility distributions were meant to provide a graded semantics to natural language statements. However, possibility and necessity measures can also be the basis of a full-fledged representation of partial belief that
Vx[½a1 {b,a1 Vx[½a1 ,a2 Vx[½a2 ,a2 zc
where f (x) and h(x) are the monotonic increasing and decreasing functions in ½a1 {b,a1 and ½a2 ,a2 zc respectively; iii) mA~ (x) is an upper semi-continuous function;
PLOS ONE | www.plosone.org
and R
~ is an interval number denoted by ½A ~ ~½Al ,Ar , The a-cut of A a a a which is explicitly shown for a LR-type fuzzy number, i.e., ½Ala ,Ara ~½a1 {bL{1 (a),a2 zcR{1 (a) for all a [ ½0,1.
i.e., it is assumed that the total available is equal to the total required. If it is not true, a fictitious origin or destination can be added. It should be noted that the problem has a feasible solution if and only if condition (2) is satisfied. Now the problem is to determine xij , in such a way that the total transportation cost is minimum.
8 > < f (x), mA~ (x)~ 1, > : h(x),
a1 {x b
3
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
Figure 6. Treatment of objective: necessity-modality approach. doi:10.1371/journal.pone.0105142.g006 Figure 5. Treatment of objective: possibility-fractile approach. doi:10.1371/journal.pone.0105142.g005
Mathematical formulation of fuzzy transportation problem
parallels probability [12]. Then, it can be seen as a coarse, nonnumerical version of probability theory, or as a framework for reasoning with extreme probabilities, or yet as a simple approach to reasoning with imprecise probabilities [13]. ~ and B ~ , the ~ are two fuzzy sets and a fuzzy event v [ A If A ~ possibility and necessity of v [ B are defined as:
In this section, we first present the Fuzzy Transportation Problem (FTP). Then, based on possibility theory, treatment of constraints and objective are discussed in order to develop the four formulations of the FTP.
3.1 Fuzzy transportation problem Mathematically the FTP can be described as follows.
~ )~ sup minfm ~ (r),m ~ (r)g P A~ (B B A r
min ~z~
m X n X
~ci j xi j
ð4Þ
i~1 j~1
~ )~ inf maxf1{m ~ (r),m ~ (r)g NA~ (B B A r
8 Xn ~ ~ai , i~1,2, ,m (supply constraints) > < X j~1 xi j ~ (5) m ~bj , j~1,2, ,n (destination constraints) ð5Þ subject to x ~ ~ i~1 i j > : Vi and j xi j §0,
If ~a and ~b are two fuzzy numbers and their membership function are m~a (x) and m~b (x) respectively, the measure of possibility and necessity are defined as: Pos(~a ~b)~ supfmin (m~a (x),m~b (y)Dx,y [ R, x yg
In the constraints of FTP, the left side is a crisp number while the right side is a fuzzy number. Hence, the constraints produce a fuzzy set. To determine whether a crisp number is in a fuzzy set, the classical method is to use the membership function. If the membership function value of the crisp number is positive, then it can be sure that it belongs to the fuzzy set, and the larger is the value, the higher is the extent. As for the objective function, it is a linear non-negative combination of fuzzy variables. So the objective of FTP is a fuzzy variable too. To minimize the objective, a ranking method of fuzzy numbers should be introduced. Therefore, in order to solve FTP, we should introduce a measure to determine whether a crisp number belongs to a fuzzy set and a ranking index to rank fuzzy numbers. Generally ~ , and speaking, let the feasible fuzzy set of the FTP is denoted by X ~ . And there is a all possible objective values is in the fuzzy set Z measure system M to determine whether a crisp number belongs ~ )ƒ1. Then the FTP can be to a fuzzy set, where 0ƒM(a [ X described as follows conceptually.
Nes(~a ~b)~ inffmax (1{m~a (x),m~b (y))Dx,y [ R, x yg If ~b is a crisp number, then the we can achieved that Pos(~aƒb)~ supfm~a (x)Dx [ R,xƒbg
Nes(~aƒb)~ supf1{m~a (x)Dx [ R,xwbg Generally speaking, the possibility degree evaluates to what extent an event is consistent with the knowledge P, while the necessity degree evaluates to what extent an event is certainly implied by the knowledge.
max
PLOS ONE | www.plosone.org
4
~) M(z(x) [ Z
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
We think that how to solve FTP depends on the decisionmaker’s attitude to the fuzziness, so it is hard to define which model should be used. Decision-maker can choose any proposed method to solve FTP just according to his attitude to the fuzziness. We propose possibility theory to describe decision-maker’s attitude. As FTP consists of fuzzy constraints and objective, this paper deals with them individually as shown in Figure 2. With respect to fuzzy constraints, we utilize possibility theory to convert them into crisp and bounded intervals. As for fuzzy objective, by applying fractile and modality approach we convert it into four formations from the view of necessity and possibility respectively. Then by recombining the converted constraints and objective, four models are established.
3.2 Treatment of constraints
Figure 7. Treatment of objective: possibility-modality approach. doi:10.1371/journal.pone.0105142.g007
As constraints of FTP are equations, and according to the definition of possibility and necessity, we can get Pos(~ a~b)~m~a (b),
Nes(~ a~b)~0
~ )§z, 0ƒzƒ1 M(x [ X
s:t:
Hence, the treatment of constraints can be considered from the view of possibility exclusively. It is reasonable and frequently-used that when decision makers are confronted with uncertainty, they can only make sure that the th constraints are satisfied to some extent. For Pn the i origin Ai in FTP, the decision-maker expects that the j~1 xi j ~~ai should be satisfied as possible because the production is uncertain. According to the definition of possibility, we can set that the possibility of the constraint be true is larger than a given level ai , where 0ƒai ƒ1, which reflects the extent how the constraint is satisfied.
or
max ( s:t:
q
~ )§q, 0ƒqƒ1 M(z(x)[Z ~ M(x[X )§z, 0ƒzƒ1
~ and q is the degree of x where z is the degree of z(x) belongs to Z ~. belongs to X To be simple, in this paper we use trapezoidal numbers to represent the fuzziness of parameters in FTP, i.e., we assume that
Pos
n X
! xi j ~~ a §ai , i~1,2, ,m
ð6Þ
j~1
~cij ~(ci j 1 ,ci j2 ; Li j ,Ri j ),i~1,2, ,m; j~1,2, ,n
According to the definition of possibility, the equation (6) is depicted in the Figure 3, where P1 and P2 is the left and right bound of ~ ai ’s a-cut. To ensure the equation (6) can be satisfied, the point O should be above of P1 and P2 , which converts the constraint P n ai to j~1 xi j ~~
~ai ~(ai1 ,ai2 ; Li ,Ri ),i~1,2, ,m
~bj ~(bj1 ,bj2 ; Lj ,Rj ),j~1,2, ,n ai1 {(1{ai )Li ~aLiai ƒ
And the possibility and necessity measure are adopted as ranking index.
n X j~1
xi j ƒaR iai ~ai2 z(1{ai )Ri , i~1,2, ,m ð7Þ
Table 1. Data of the example.
B1
B2
B3
B4
B5
Sup.
A1
(8,9;3,2)
(2,3;1,5)
(2,5;1,6)
(10,16;4,8)
(7,12;5,5)
(22,28;20,30)
A2
(3,6;1,8)
(8,11;4,3)
(12,16;5,8)
(3,5;1,6)
(2,6;1,7)
(28,35;25,36)
A3
(13,16;5,8)
(3,8;1,10)
(10,18;4,9)
(4,10;1,1)
(12,20;5,2)
(16,24;12,28)
A4
(18,23;3,6)
(12,18;5,9)
(11,15;4,8)
(30,36;6,4)
(6,9;2,1)
(28,36;25,40)
Dem.
(20,30;12,30)
(22,32;15,30)
(32,40;25,40)
(12,28;6,18)
(28,36;18,40)
doi:10.1371/journal.pone.0105142.t001
PLOS ONE | www.plosone.org
5
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
C1 ~
m X n X
ci j1 xi j ,
C2 ~
i~1 j~1
L~
m X n X
ci j2 xi j
i~1 j~1
Li j xi j ,
R~
m X n X
i~1 j~1
Ri j xi j
i~1 j~1
There are two approaches to treat the objective: fractile and modality approach. 3.3.1 Fractile approach. A fractile approach corresponds to the Kataoka’s model of a stochastic programming problem [14,15]. Geoffrion [16] calls the Kataoka’s model the fractile criterion approach. By the definition in statistics, p-fractile is the value u which satisfies
Figure 8. The relationship between supply and demand. doi:10.1371/journal.pone.0105142.g008
Similarly, for the j t h destination Bj , the constraint should be satisfied with the possibility larger than bj (0ƒbj ƒ1), i.e., ! m X ~ Pos xi j ~b §bj , j~1,2, ,n
m X n X
Prob(X ƒu)~p where X is a random variable. In this definition, p-fractile does not generally exist for all p [ (0,1). That is why we define p-fractile as the smallest value up of u which satisfies
ð8Þ
i~1
And it can be converted to Prob(X ƒu)§p bj1 {(1{bj )Lj ~bLjbj ƒ
m X i~1
xij ƒbR bbj ~bj2 z(1{bj )Rj , j~1,2, ,n
ð9Þ
From the viewpoint of Dempster-Shafer theory of evidence [17], it is known that Pos(X ƒu) and Nes(X ƒu) can be regarded as the upper and lower bounds of Prob(X ƒu) (see [18]). In this sense, p-possibility fractile is defined as the smallest value of u which satisfies
In such ways, the constraints of FTP are transformed into crisp and bounded intervals.
3.3 Treatment of objective
Pos(X ƒu)§p
According to the operation algorithm of fuzzy numbers, because the unit cost of FTP is a linear non-negative combination of m|n trapezoidal numbers, the objective is a trapezoidal numbers too. ~ ~(C1 ,C2 ; L,R) be the objective value. When xi j §0, there Let C exists
and p-necessity fractile is defined as the smallest value of u which satisfies
Table 2. The crisp transportation problem originated from Model I.
c~0:95
B1
B2
B3
B4
B5
Sup.
A1
10.90
7.75
10.70
23.60
16.75
[19.0, 32.0]
A2
13.60
13.85
23.60
10.70
12.65
[25.5, 38.6]
A3
23.60
17.50
26.55
10.95
21.90
[13.6, 29.6]
A4
28.70
26.55
22.60
39.80
9.95
[23.0, 44.0]
Dem.
[17, 37.5]
[20.5, 35]
[30.75, 42]
[11.1, 30.7]
[24.4, 44]
c~0:05
B1
B2
B3
B4
B5
Sup.
A1
9.10
3.25
5.30
16.40
12.25
[19.0, 32.5]
A2
6.40
11.15
16.40
5.30
6.35
[25.5, 38.6]
A3
16.40
8.50
18.45
10.05
20.10
[13.6, 29.6]
A4
23.30
18.45
15.40
36.20
9.05
[23.0, 44.0]
Dem.
[17.0,37.5]
[20.5,35.0]
[30.75,42.0]
[11.1,30.7]
[24.4,44.0]
doi:10.1371/journal.pone.0105142.t002
PLOS ONE | www.plosone.org
6
August 2014 | Volume 9 | Issue 8 | e105142
0.80
As for the objective of FTP, from the point of necessity it can convert to
103.75
13.6
Nes(X ƒu)§p
0.83
1.00
1.00 32.5
33.82
Sup.
Pos
Fuzzy Transportation Optimization Based on Possibility Theory
0
0.57
0
23.83
24.40
0.80
0
11.1
0
23.83
11.1
0.85
B5 B4
min C
~ ƒC)§c s:t: Nes(C
where c is a parameter to reflect to what extent the decision-maker ~ ƒC. Eq.(10) is certain about the lower bound of the fuzzy event C expresses that the decision-maker minimizes the total cost at a given necessity level. As shown in Figure 4, we can get
~ ~(375:43,699:85; 143:03,575:10) C
0.95
30.75 20.5
0.90 0.75
17
0
0 13.6
0 0
0
30.75
0 5.15
1.75 0
17
B1
B2
B3
~ ƒC)~ Nes(C
c~0:05
C{C2 §c[C2 zcRƒC R Hence, Eq. (10) turns to
0.80
0.86
13.6
24.4
s:t:C2 zcRƒC
ð11Þ
It is easy to find that the optimal solution of (11) is C~C2 zcR. Therefore, the objective function of FTP can be substituted by 103.75
1.00
1.00
32.5
33.25
Pos
C{C2 R
then the constraint in (10) can be converted to
min C
Supply
ð10Þ
min
C2 zcR~
m X n X
(ci j2 zcRi j )xi j
ð12Þ
0.80
24.4
And with the constraints (7) and (9), the FTP turns to the following model, which is a crisp linear programming problem.
0.85
11.1
0
24.4 0
11.1
0
0 0
0
B4
B5
i~1 j~1
cost
PLOS ONE | www.plosone.org
D
m X n X
(ci j2 zcRi j )xi j
i~1 j~1
( s:t:
Eq: ð7Þ Eq: ð9Þ
From the point of the possibility and by the fractile approach, the objective of FTP means that
doi:10.1371/journal.pone.0105142.t003
~ ~(444:3,790:35; 176:9,438:5) C Total
30.75
0.95 0.90 0.75 Pos
20.5 17 Dem.
0
0 0 0 A4
2.5 0 A3
0
30.75 1.75
17 A2
16.25
0 A1
B3 B2 B1 c~0:95
Table 3. Solution from Model I (c~0:95).
Model I
min
min C
~ ƒC)§h s:t: Pos(C
ð13Þ
i.e., the decision-maker expects that the total cost should be minimized as the possibility that total cost is not larger than some given level C.
7
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
greater than a given level C0 , i.e., ~ ƒC0 ) Nes(C
max
ð16Þ
From Figure 6, we can get ~ ƒC0 )~ Nes(C
C0 {C2 R
hence, the objective of FTP can be converted to
min
Figure 9. Fuzzy total cost when c~0:95 and ª~0:05. doi:10.1371/journal.pone.0105142.g009
As shown in Figure 5, we can get ~ ƒC)~ Pos(C
C0 {C2 ~ R
Xm Xn i~1 j~1 X m Xn i~1
Xm Xn
C{C1 zL §h L
i~1 j~1 X m Xn
min
i~1
s:t:
C1 {(1{h)LƒC Hence, (13) can be converted to s:t: C1 {(1{h)LƒC
m X n X
1 y~ Xm Xn i~1
m X n X
(cij1 {(1{h)Lij )xij
i~1 j~1
( s:t:
Eq: ð7Þ
Ri j xi j
Eq: ð7Þ Eq: ð9Þ
j~1
Ri j xi j
, zi j ~xi j y,Vi, j
D
min
m X n X
cij2 zij {C0 y
i~1 j~1
s:t:
8Xn zij {½ai1 {(1{ai )Li y§0, > > > X j~1 > > > n > > j~1 zij {½ai2 z(1{ai )Ri yƒ0, > > > X > > m < i~1 zij {½bj1 {(1{bj )Lj y§0, X > m > > > i~1 zij {½bj2 z(1{bj )Rj yƒ0, > > > X X > m n > > > i~1 j~1 Rij zij ~1 > > : y§0,zij §0; i~1,2, ,m; j~1,2, ,n
i~1,2, ,m i~1,2, ,m j~1,2, ,n j~1,2, ,n
Eq: ð9Þ From the point of possibility, the decision-maker expects that the total cost should not be greater than a given level C0 , i.e.,
3.3.2 Modality approach. A modality optimization model corresponds to the minimum-risk approach to a stochastic programming problem [15]. The minimum-risk approach is also called the maximum probability approach [14] or the aspiration criterion approach [16]. A modality optimization approach is a dual approach to the fractile optimization one. In this approach, the decision-maker puts more importance on the certainty degree comparing to the fractile approach. From the point of necessity and modality approach, the decision-maker in FTP expects that the total cost should not be
PLOS ONE | www.plosone.org
j~1
ð15Þ
Model III
D
ci j2 xi j {C0
We obtain the following model. (ci j1 {(1{h)Li j )xi j
And with the constraints (7) and (9), the FTP turns to the following model.
Model II
ð17Þ
ð14Þ
i~1 j~1
min
Ri j xi j
This is a fractional programming problem which can be transformed to a linear programming problem by the substitution
It is easy to find that the optimal solution of (14) is C1 {(1{h)L, i.e.,
C1 {(1{h)L~
j~1
Such that FTP turns to
or
min C
ci j2 xi j {C0
max
8
~ ƒC0 ) Pos(C
ð18Þ
August 2014 | Volume 9 | Issue 8 | e105142
PLOS ONE | www.plosone.org
0
17
0.75
A4
Dem.
Pos
9
doi:10.1371/journal.pone.0105142.t005
~ ~(377:7,701:55; 143:60,571:70) C
0.75
cost
Pos
Total
0
17
0
A3
Dem.
17
A2
A4
0
A1
B1
Table 5. Solution from Model III (C0 ~1000).
doi:10.1371/journal.pone.0105142.t004
~ ~(346:35,681:9; 126:75,616:15) C
0
A3
cost
17
A2
Total
0
A1
B1
Table 4. Solution from Model II (h~0:95).
0.90
20.5
0
13.6
5.15
1.75
B2
0.90
20.5
0
18.75
0
1.75
B2
0.95
30.75
0
0
0
30.75
B3
0.95
30.75
0
0
0
30.75
B3
0.85
11.1
0
0
11.1
0
B4
0.85
11.1
0
0
11.1
0
B4
0.80
24.4
24.4
0
0
0
B5
0.80
24.4
23
0
1.4
0
B5
103.75
24.4
13.6
33.25
32.5
Sup.
103.75
23
18.75
29.5
32.5
Sup.
0.856
0.80
1.00
1.00
Pos
0.80
1.00
1.00
1.00
Pos
Fuzzy Transportation Optimization Based on Possibility Theory
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
From Figure 7, we can get C0 {C1 zL L
0.80
1.00
1.00
1.00
Pos
~ ƒC0 )~ Pos(C
Hence, the objective of FTP can be transformed to C0 {C1 zL [ min L
103.75
23
18.75
32.5
29.5
Sup.
max
Pm Pn
i~1 j~1 P m Pn i~1
cij1 xij {C0
j~1
Li j xi j
ð19Þ
Such that the FTP turns to Pm Pn C~
ci j1 xi j {C0 i~1 Pm j~1 Pn i~1 j~1 Li j xi j
0.80
24.4
23
0
0
1.4
B5
min
s:t:
Eq: ð7Þ Eq: ð9Þ
0.85
11.1
0
0
11.1
0
B4
Again, this is a fractional programming problem which can be transformed to a linear programming problem by the substitution 1 y~ Pm Pn i~1
j~1
Li j xi j
, zi j ~xi j y,Vi, j
0.95
30.75
0
0
0
30.75
B3
We obtain the following model.
0.90
D
m X n X
cij1 zij {C0 y
i~1 j~1
s:t:
8Xn zij {½ai1 {(1{ai )Li y§0 i~1,2, ,m > > > X j~1 > > > n > i~1,2, ,m > j~1 zij {½ai2 z(1{ai )Ri yƒ0 > > > X > > m < j~1,2, ,n i~1 zij {½bj1 {(1{bj )Lj y§0 X > m > > j~1,2, ,n > i~1 zij {½bj2 {(1{bj )Lj y§0 > > > X X > m n > > > i~1 j~1 Lij zij ~1 > > : y§0,zij §0; i~1,2, ,m; j~1,2, ,n
To sum up, from the view of possibility and necessity, and by fractile and modality approach, we proposed four ways to defuzzify FTP to four crisp linear programming problems. During the transformation process, some parameters are introduced to clarify the decision-maker’s subjectiveness about fuzziness, which makes the solutions more practical. As for constraints, we introduced ai and bj to reflect the decision-maker’s requirement on the extent how the constraint is satisfied in the view of possibility. With respect to the objective function, in the fractile approach we proposed c as the lower bound of necessity and h as the lower bound of possibility that the total cost is not larger than a given level set by the decision-maker. While in the modality approach, C0 was introduced as the upper bound of total cost that decision-maker expected. In brief, these parameters reflect the decision-maker’s attitude
doi:10.1371/journal.pone.0105142.t006
~ ~(346:35,681:9; 126:75,616:15) C
0.75
PLOS ONE | www.plosone.org
Total
cost Pos
20.5 17 Dem.
0 0 A4
18.75 0 A3
1.75
17 A2
0
0 A1
B1
Table 6. Solution from Model IV (C0 ~1000).
B2
Model IV
min
10
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
some given level C is 0.95, i.e., let h~0:95. The solution is in the Table 4. In model III, let C0 equals 1000, the solution is listed in Table 5. From Table 5, we can easily get that
to the fuzziness in FTP from the view of possibility or necessity.
Numerical example In this section, numerical examples of FTP will be presented to demonstrate and verify the proposed approaches. All established models are originated from the above models based on the different parameter settings. All models are solved by the Lingo software, so the solving process is omitted for simplification. The description of transportation problem is standard tables, where the central part is the cost cij , the column ‘‘Supply’’ are ai and the row ‘‘Demand’’ are bj . While in the solution tables, the central part is the quantities of transportation from Ai to Bj . Table 1 is a FTP with four origins Ai (i~1,2,3,4) and five destinations Bj (j~1,2,3,4,5) because the supply, demand and cost of unit are assumed to be fuzzy trapezoidal numbers. By adding up for origins’ supply and five destinations’ demand in Table 1, the total supply and demand are fuzzy trapezoidal numbers (94,123;82,134) and (114,166;76,158) depicted in the Figure 8. To solve this FTP, firstly we set the value of ai , bj , c and C0 as follows: ai :~(0:85,0:90,0:80,0:80);
c :~0:95,0:05;
~ ƒC0 ~1000)~1 Pos(C Hence, the possibility that the total cost is not larger than given C0 is satisfied. This is reasonable because C0 is set to be the upper bound of the minimal total cost which a decision-maker can afford. In model IV, similarly let C0 equals 1000, and solution is listed in Table 6. From Table 6, the following relationship exist. ~ ƒC0 ~1000)~1 Pos(C Similar results can be achieved as shown in Model III.
Conclusions In this paper, a simple but effective parametric method was introduced to solve fuzzy transportation problem. By using possibility theory in fractile and modality approach, the fuzzy transportation problem is transformed into four types of crisp linear programming problems. In the process of transformation, some parameters are introduced to reflect decision-maker’s attitude to the uncertainty or fuzziness. The methods proposed in this paper can be used for all kinds of fuzzy transportation problem, whether triangular and trapezoidal fuzzy numbers with normal or abnormal data.
bj :~(0:75,0:90,0:95,0:85,0:80)
h :~0:95;
C0 :~1000
Then according to (7) and (9), the supply and demand constraints can be converted to different bounded intervals as shown in Table 2 based on the given value of ai and bj . By using Model I, let c to be 0.95 and 0.05 respectively. And based on the Eq.(12), the cost of FTP can be transformed into a crisp value as shown in Table 2. Solve the two crisp transportation problems by linear programming, the optimal solution is in Table 3, and the fuzzy total cost is depicted in Figure 9 for comparative analysis. From Table 3 and Figure 9, it can be found that the distribution varies with the changing of c. However, the total transportation, supply from different origins and demand to different destinations are same. And, the possibility requirement of supply and demand constraints are almost satisfied. From the point of fuzzy total cost, it gets smaller when c is smaller. According to Model II, assume that the decision-maker expects that the lower bound of possibility that total cost is not larger than
Acknowledgments This research is supported by the Fundamental Research Funds for the Central Universities (2-9-2012-86) and supported by Key Laboratory of Carrying Capacity Assessment for Resource and Environment, MLR (CCA2012.05). The authors would like to heartily thank the Editor in Chief and anonymous reviewers for their careful reading of this paper and for their helpful comments and suggestions.
Author Contributions Analyzed the data: RL. Contributed to the writing of the manuscript: DH PL. Developed models: DH. Designed the numerical examples: QH.
References 1. Chanas S, Kołodziejczyk W, Machaj A (1984) A fuzzy approach to the transportation problem. Fuzzy Sets and Systems 13: 211–221. 2. Chanas S, Delgado M, Verdegay J, Vila M (1993) Interval and fuzzy extensions of classical transportation problems. Transportation Planning and Technology 17: 203–218. 3. Chanas S, Kuchta D (1996) A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets and Systems 82: 299–305. 4. Liu ST, Kao C (2004) Solving fuzzy transportation problems based on extension principle. European Journal of Operational Research 153: 661–674. 5. Gani A, Razak KA (2006) Two stage fuzzy transportation problem. Journal of Physical Sciences 10: 63–69. 6. Pandian P, Natarajan G (2010) A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Applied Mathematical Sciences 4: 79–90. 7. Basirzadeh H (2011) An approach for solving fuzzy transportation problem. Applied Mathematical Sciences 5: 1549–1566.
PLOS ONE | www.plosone.org
8. Narayanamoorthy S, Saranya S, Maheswari S (2013) A method for solving fuzzy transportation problem (ftp) using fuzzy russell’s method. International Journal of Intelligent Systems & Applications 5. 9. Chaudhuri A, De K, Chatterjee D, Mitra P (2013) Trapezoidal fuzzy numbers for the transportation problem. arXiv preprint arXiv:13071893. 10. Negoita C, Zadeh L, Zimmermann H (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy sets and systems 1: 3–28. 11. Gaines BR, Kohout TL (1975) Possible automata. 12. Dubois D, Prade H (1988) Default reasoning and possibility theory. Artificial Intelligence 35: 243–257. 13. Dubois D, Nguyen HT, Prade H (2000) Possibility theory, probability and fuzzy sets misunderstandings, bridges and gaps. In: Fundamentals of fuzzy sets, Springer. pp. 343–438. 14. Kataoka S (1963) A stochastic programming model. Econometrica: Journal of the Econometric Society: 181–196. 15. Stancu-Minasian I (1990) Overview of different approaches for solving stochastic programming problems with multiple objective functions. In: Stochastic versus
11
August 2014 | Volume 9 | Issue 8 | e105142
Fuzzy Transportation Optimization Based on Possibility Theory
17. Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. The annals of mathematical statistics : 325–339. 18. Dubois D, Prade H (1987) The mean value of a fuzzy number. Fuzzy sets and systems 24: 279–300.
fuzzy approaches to multiobjective mathematical programming under uncertainty, Springer. pp. 71–101. 16. Geoffrion AM (1967) Stochastic programming with aspiration or fractile criteria. Management Science 13: 672–679.
PLOS ONE | www.plosone.org
12
August 2014 | Volume 9 | Issue 8 | e105142
