Applied Mathematical Modelling 38 (2014) 3987–4005
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Optimizing fuzzy p-hub center problem with generalized value-at-risk criterion Kai Yang, Yankui Liu ⇑, Guoqing Yang Risk Management & Financial Engineering Lab, College of Mathematics & Computer Science, Hebei University, Baoding 071002, Hebei, China
a r t i c l e
i n f o
Article history: Received 10 October 2011 Received in revised form 30 December 2013 Accepted 21 January 2014 Available online 5 February 2014 Keywords: Parametric possibility distribution Uncertainty reduction Hub location problem Parametric programming Mixed-integer programming
a b s t r a c t In this study, we reduce the uncertainty embedded in secondary possibility distribution of a type-2 fuzzy variable by fuzzy integral, and apply the proposed reduction method to p-hub center problem, which is a nonlinear optimization problem due to the existence of integer decision variables. In order to optimize p-hub center problem, this paper develops a robust optimization method to describe travel times by employing parametric possibility distributions. We first derive the parametric possibility distributions of reduced fuzzy variables. After that, we apply the reduction methods to p-hub center problem and develop a new generalized value-at-risk (VaR) p-hub center problem, in which the travel times are characterized by parametric possibility distributions. Under mild assumptions, we turn the original fuzzy p-hub center problem into its equivalent parametric mixed-integer programming problems. So, we can solve the equivalent parametric mixed-integer programming problems by general-purpose optimization software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the efficiency of the proposed solution methods. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The study of p-hub center problem was firstly introduced by Campbell [1], in which a quadratic programming model was formulated. Kara and Tansel [2] provided several linear formulations for the single allocation p-hub center problem. Ernst et al. [3] proposed a mixed-integer linear programming for the multiple allocation p-hub center problem based on the concept radius of hubs. For comprehensive surveys on the deterministic p-hub center problems, the interested reader may refer to Alumur and Kara [4] and Campbell et al. [5]. In this paper, we concentrate on p-hub center problem under uncertainty, which is an active research area in the literature. The significance of uncertainty has motivated some researchers to address hub location problems with random hub nodes [6], random demands [7], random costs [8] and random travel times [9,10]. In the meantime, some new methods have also been developed to model hub location problems under possibilistic uncertainty. For example, Bashiri et al. [11] used fuzzy VIKOR to model a hub location problem, in which the location of hub facilities is determined by both qualitative and quantitative parameters. Taghipourian et al. [12] presented a fuzzy dynamic model, in which a group of facilities is considered as virtual hubs for backup in case the main hubs fail to operate. Yang et al. [13] investigated hub location problems based on the credibility criterion, and developed local search based hybrid algorithms to solve their optimization problem.
⇑ Corresponding author. Tel./fax: +86 312 5066629. E-mail addresses:
[email protected] (K. Yang),
[email protected] (Y. Liu),
[email protected] (G. Yang). http://dx.doi.org/10.1016/j.apm.2014.01.009 0307-904X/Ó 2014 Elsevier Inc. All rights reserved.
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However, for hub location problem, the exact possibility distributions of fuzzy travel times are sometimes difficult determined. It is necessary to find ways to evade the exact evaluation of possibility distributions of uncertain travel times. Type-2 fuzzy vector [14, ] is a natural extension of type-1 fuzzy vector, it provides an appropriate representation of uncertain parameters in certain applications [15,16]. During the applications of type-2 fuzzy theory in practice, several type-reduction methods have been developed in the literature [17–22]. Motivated by the above mentioned research, the purpose of this paper is to develop a robust parametric method for optimizing p-hub center problem, in which the uncertain travel times are characterized by type-2 fuzzy variables. We employ Choquet integrals [23] to reduce the uncertainty embedded in the secondary possibility distributions of uncertain travel times, and develop a novel fuzzy p-hub center problem. The Choquet integral was first introduced in [24], and its properties were also documented in [25,26]. When the travel times are mutually independent type-2 fuzzy parameters, we transform the established p-hub center problem to its equivalent parametric mixed-integer programming problems. In addition, the equivalent mixed-integer programming problems can be solved by general-purpose optimization softwares. At the end of this paper, we perform some numerical experiments to illustrate the new modeling ideas and the efficiency of the proposed solution methods. The rest of this paper is organized as follows. In Section 2, we deal with the reduction methods for type-2 trapezoidal fuzzy variables, and deduce the parametric possibility distributions of reduced fuzzy variables. In Section 3, we develop a new modeling approach to p-hub center problem, and discuss the equivalent parametric programming problems. Section 4 performs some numerical experiments to illustrate the new modeling idea. Section 5 gives our conclusions. 2. Parametric possibility distributions of uncertain parameters If a fuzzy variable n takes its values in the unit interval ½0; 1, then it is called a regular fuzzy variable [14]. Suppose ln ðtÞ is a generalized possibility distribution (not necessarily normalized) of regular fuzzy variable n. Then for any t 2 ½0; 1, the possibility, necessity and credibility of fuzzy event fn 6 tg are calculated by the following formulas:
Posfn 6 tg ¼ sup 06u6t
ln ðuÞ;
Necfn 6 tg ¼ sup 06u61
ð1Þ
ln ðuÞ sup ln ðuÞ
ð2Þ
t 2 > < l ln1 ðx; hl Þ ¼ 2h ; 2 > > > : ð2hl Þ 2
xr 1 r 2 r 1
; r1 6 x 6 r2 r2 6 x 6 r3
r 4 x r 4 r 3
; r3 6 x 6 r4 :
(ii) By the ER method, the reduced fuzzy variable n2 has the following parametric possibility distribution
8 ð4h h Þ r l > > 4 > < lÞ ln2 ðx; hl ; hr Þ ¼ ð4h ; 4 > > > : ð4hl hr Þ 4
xr 1 r 2 r 1
þ h4r ; r1 6 x 6 r2 r2 6 x 6 r3
r 4 x r 4 r 3
þ h4r ; r3 6 x 6 r 4 :
(iii) By the UER method, the reduced fuzzy variable n3 has the following parametric possibility distribution
8 ð2hr Þ > > < 2 ln3 ðx; hr Þ ¼ 1; > > : ð2hr Þ 2
xr1 r 2 r 1
þ h2r ; r 1 6 x 6 r 2 r2 6 x 6 r3
r 4 x r 4 r 3
hr 2
þ ; r3 6 x 6 r4 :
Theorem 1 shows that the reduced fuzzy variables n1 ; n2 and n3 have parametric possibility distributions with respect to parameter ~ h ¼ ðhl ; hr Þ. When the parameter ~ h varies in the unit interval ½0; 1, the parametric possibility distributions run over the entire footprints of type-2 fuzzy variables, which are plotted in Fig. 1. In particular, when hl ¼ 0 and hr ¼ 0, the paramet-
1 ξ3 0.25(4−θl) ξ2
μ(x)
0.5(2−θl) 0.5
ξ
1
0.5θr
0.25θr 0r
1
r2
x
r
3
r4
Fig. 1. The parametric possibility distributions of reduced fuzzy variables n1 ; n2 and n3 .
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ric possibility distributions are degenerated to the fixed possibility distribution of type-1 fuzzy variable. As a consequence, a reduced fuzzy variable with parametric possibility distribution is more flexible than a type-1 fuzzy variable with fixed possibility distribution. n be a type-2 trapezoidal fuzzy variable, and ln1 ðx; hl Þ; ln2 ðx; hl ; hr Þ and Corollary 1. Let ~ distributions of the reduced fuzzy variables n1 ; n2 and n3 , respectively. Then we have
ln3 ðx; hr Þ the parametric possibility
ln3 ðx; hr Þ P ln2 ðx; hl ; hr Þ P ln1 ðx; hl Þ: The reduced fuzzy variables obtained by expectation reduction methods are not always normalized, which is shown in Fig. 1. Therefore, it is necessary to generalize the term VaR of fuzzy variable, which is stated as follows. Suppose n is a reduced fuzzy variable with a parametric possibility distribution ln ðx; ~ hÞ. For any given a 2 ð0; 1Þ, the generalized a-VaR of the fuzzy variable n is defined as
e nVaR ðaÞ ¼ minfrj Crfn 6 rg P ag;
ð5Þ
e where the generalized credibility Crfn 6 rg is defined as
1 e Crfn 6 rg ¼ sup lðx; ~hÞ þ sup lðx; ~hÞ suplðx; ~hÞ ; 2 x2R x6r x>r
r 2 R:
ð6Þ
3. Hub location problem with uncertain travel times 3.1. Problem formulation The p-hub center problem is to locate p hubs and to allocate non-hub nodes to hub nodes such that the maximum travel time between any origin-destination pair is minimized. In some real applications, the travel times cannot be considered deterministic since their values may vary due to traffic condition, speed ambulances, time of day, climate conditions, and land and road type. In present paper, we address this issue via robust parametric optimization method. We will employ parametric possibility distributions to describe travel times, and the parametric possibility distributions are obtained by using three reduction methods developed in Section 2. When the parameters embedded in possibility distributions vary in the unit interval ½0; 1, the distribution functions run over the entire footprints of uncertain travel times, therefore the important information about travel times cannot be lost. In the following, we will adopt this modeling idea to construct robust phub center problems. The following assumptions are required to build our model:
The number of hubs to be located is predetermined. There are no capacities involved. Each origin/destination node is assigned to a single hub. Direct transportation between non-hub nodes is not allowed. In order to model the network system, we adopt the following notations:
N ¼ f1; 2; . . . ; ng: the set of nodes in the network. a 2 ð0; 1Þ: the predetermined generalized credibility level. e ij according to the LER, ER and UER methods. T ij : the reduction of travel time T d: the economies of scale parameter (the discount factor on links between hubs). p: the number of hubs to be selected. Decision variables: For each pair i; k 2 N, we define the following binary decision variables,
X ik ¼
1; if node i is assigned to hub k 0; otherwise:
When i ¼ k, the variable X kk represents the establishment or not establishment of a hub at node k. We define additional binary decision variables X ijkm that represent path in network from node i to node j through hub k first then hub m, i.e.,
X ijkm ¼
1; if exists a path from node i to j through hub k first then m 0;
otherwise:
K. Yang et al. / Applied Mathematical Modelling 38 (2014) 3987–4005
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Objective function: The objective function includes the total travel time on a path i ! k ! m ! j and is represented as
ðT ik þ dT km þ T mj ÞX ijkm ;
8i; j; k; m 2 N:
With a prescribed generalized credibility level a 2 ð0; 1Þ, the objective is to minimize the generalized VaR of the total travel time in the sense that
e minfuj CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a; 8i; j; k; m 2 Ng; e which is equivalent to following representation: minimize u such that CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a; 8i; j; k; m 2 N. Constrains: I: Constraint (7) ensures that path i ! k ! m ! j is a valid path in network if and only if nodes i and j are assigned to hubs k and m, respectively, i.e., X ik ¼ X jm ¼ 1,
X ijkm P X ik þ X jm 1;
ð7Þ
where X ijkm 2 f0; 1g. II: Constraint (8) imposes single assignment of nodes to hubs,
X X ik ¼ 1;
ð8Þ
k2N
where X ik 2 f0; 1g. III: Constraint (9) states that a non-hub node i can only be assigned to an open hub at node k,
X ik 6 X kk :
ð9Þ
VI: Constraint (10) requires that exactly p hubs are established in the network,
X X kk ¼ p:
ð10Þ
k2N
Based on the notations above, we formally build the fuzzy p-hub center problem as the following credibility-based parametric programming model:
8 min u > > > > e > s:t: : CrfðT > ik þ dT km þ T mj ÞX ijkm 6 ug P a; 8i; j; k; m 2 N > > > > > P X ik þ X jm 1; 8i; j; k; m 2 N X ijkm > > X > > > X ik ¼ 1; 8i 2 N > < k2N
> > > > > > > > > > > > > > > > > :
X ik 6 X kk ; 8i; k 2 N X X kk ¼ p
ð11Þ
k2N
X ik 2 f0; 1g; 8i; k 2 N X ijkm 2 f0; 1g; 8i; j; k; m 2 N:
Remark 1. The parameter d is the economies of scale factor. That is the travel times between the hub facilities k and m are smaller than the original times since hub facilities concentrate time, so 0 6 d < 1. This assumption is commonly imposed in the hub location literature [4,5]. Model (11) is a mixed-integer programming problem with generalized credibility constraints. The general solution methods require conversion of generalized credibility constraints to their respective deterministic equivalents. As we know, this conversion is usually hard to perform and only successfully for special cases. In the subsequent subsection, we will discuss the equivalent formulation of model (11) in the case when uncertain travel times are characterized by type-2 trapezoidal fuzzy variables. 3.2. Handing generalized credibility consraints To solve model (11), it is required to compute generalized a-VaR u, which involves the computation of generalized credibility in the constraints. In this section, we consider some special cases, where the travel times are characterized by type-2 trapezoidal fuzzy variables, and turn the constraints into their equivalent parametric forms. For this purpose, we assume that e ik , T e km and T e mj are mutually independent type-2 fuzzy variables. uncertain travel times T For the sake of presentation, all technical details and proofs in this section are provided in Appendix A. First, we discuss the equivalent parametric forms of generalized credibility constraints by using the ER method.
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1 2 3 4 1 2 3 e ik ¼ ðr 1 ; r 2 ; r3 ; r4 ; ~ e e ~ Theorem 2. Suppose uncertain travel times T ik ik ik ik hik Þ, T km ¼ ðr km ; r km ; r km ; r km ; hkm Þ and T mj ¼ ðr mj ; r mj ; r mj ; r4mj ; ~ hmj Þ are type-2 trapezoidal fuzzy variables, and T ik ; T km and T mj the corresponding reduced fuzzy variables by the ER method. For any i; j; k; m 2 N, the parametric inequalities hrik P hrkm P hrmj hold.
e (i) If hrmj =8 6 a 6 hrkm =8, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP
ð8a hrmj Þðr 2mj r1mj Þ 4 hlmj hrmj
!
þ
r 1ik
þ
1 dr km
þ
r1mj
X ijkm :
e (ii) If hrkm =8 < a 6 hrik =8, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP d
ð8a hrkm Þðr2km r 1km Þ 4 hlkm hrkm
þ
ð8a hrmj Þðr2mj r 1mj Þ 4 hlmj hrmj
!
þ
r1ik
þ
1 dr km
þ
r 1mj
X ijkm :
e (iii) If hrik =8 < a 6 ð4 hlik Þ=8, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP
ð8a hrik Þðr 2ik r 1ik Þ 4 hlik hrik
þd
ð8a hrkm Þðr 2km r 1km Þ 4 hlkm hrkm
þ
ð8a hrmj Þðr 2mj r 1mj Þ 4 hlmj hrmj
þ
r 1ik
þ
1 dr km
! þ
r 1mj
X ijkm :
e (iv) If ð4 hlik Þ=8 < a 6 ð8 2hlik hrik Þ=8, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP
ð8a þ 2hlik þ hrik 8Þðr4ik r3ik Þ 4 hlik hrik
þd
ð8a þ 2hlik þ hrkm 8Þðr4km r3km Þ 4 hlkm hrkm
þ
ð8a þ 2hlik þ hrmj 8Þðr4mj r3mj Þ 4 hlmj hrmj
! þ r4ik
4 þ drkm þ r4mj
X ijkm :
e (v) If ð8 2hlik hrik Þ=8 < a 6 ð8 2hlik hrkm Þ=8, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP d
ð8a þ 2hlik þ hrkm 8Þðr 4km r 3km Þ 4 hlkm hrkm
þ
ð8a þ 2hlik þ hrmj 8Þðr 4mj r 3mj Þ 4 hlmj hrmj
!
þ
r 4ik
þ
4 dr km
þ
r4mj
X ijkm :
e (vi) If ð8 2hlik hrkm Þ=8 < a 6 ð8 2hlik hrmj Þ=8, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP
ð8a þ 2hlik þ hrmj 8Þðr4mj r 3mj Þ 4 hlmj hrmj
!
þ
r4ik
þ
4 drkm
þ
r 4mj
X ijkm :
According to Theorem 2, we have obtained the structure of the equivalent constraints. First, we observe that the generalized credibility constraint can be represented equivalently as six different parametric forms according to the values of credibility level a. The credibility level reflects a decision maker’s attitude towards risk during decision making process. Applying the analytical expressions obtained in Theorem 2, the decision maker can specify a confidence level to optimize his objective function. Second, we observe that the equivalent parametric forms depend on the parameters hl and hr . The decision maker may adjust the values of hl and hr to make more flexible decisions according to his philosophy of modeling uncertainty. The next theorem deals with the equivalent parametric forms of generalized credibility constraints by using the LER method. 1 2 3 4 1 2 3 4 ~ e ik ¼ ðr1 ; r 2 ; r 3 ; r4 ; ~ e e ~ Theorem 3. Suppose the travel times T ik ik ik ik hik Þ, T km ¼ ðr km ; r km ; r km ; r km ; hkm Þ and T mj ¼ ðr mj ; r mj ; r mj ; r mj ; hmj Þ are type-2 trapezoidal fuzzy variables, and T ik ; T km and T mj the corresponding reduced fuzzy variables by the LER method. For any i; j; k; m 2 N, the parametric inequalities hlik P hlkm P hlmj hold.
e (i) If 0 < a 6 ð2 hlik Þ=4, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
u P 4a
r 2ik r 1ik 2 hlik
þd
r 2km r 1km 2 hlkm
þ
r 2mj r1mj 2 hlmj
!
!
þ
r 1ik
þ
1 dr km
þ
r1mj
X ijkm :
e (ii) If ð2 hlik Þ=4 < a 6 ð2 hlik Þ=2, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP
r4 r3 r 4km r3km r 4mj r 3mj ik ik 4a þ 2hlik 4 þ d þ 2 hlik 2 hlkm 2 hlmj
!
!
þ
r 4ik
þ
4 dr km
þ
r 4mj
X ijkm :
Finally, the next theorem gives the equivalent parametric forms of generalized credibility constraints by using the UER method. 1 2 3 4 1 2 3 4 ~ e ik ¼ ðr1 ; r 2 ; r 3 ; r4 ; ~ e e ~ Theorem 4. Suppose the travel times T ik ik ik ik hik Þ, T km ¼ ðr km ; r km ; r km ; r km ; hkm Þ and T mj ¼ ðr mj ; r mj ; r mj ; r mj ; hmj Þ are type-2 trapezoidal fuzzy variables, and T ik ; T km and T mj the corresponding reduced fuzzy variables by the UER method. For any i; j; k; m 2 N, the parametric inequalities hrik P hrkm P hrmj hold.
K. Yang et al. / Applied Mathematical Modelling 38 (2014) 3987–4005
3993
e (i) If hrmj =4 6 a 6 hrkm =4, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP
ð4a hrmj Þðr 2mj r 1mj Þ 2 hrmj
!
1
þ r 1ik þ dr km þ r1mj X ijkm :
e (ii) If hrkm =4 < a 6 hrik =4, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP d
! r ð4a hrkm Þðr2km r 1km Þ ð4a hmj Þðr2mj r 1mj Þ 1 1 1 þ þ r þ dr þ r km ik mj X ijkm : 2 hrmj 2 hrkm
e (iii) If hrik =4 < a 1=2, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP
! r ð4a hrik Þðr 2ik r 1ik Þ ð4a hrkm Þðr 2km r 1km Þ ð4a hmj Þðr2mj r 1mj Þ 1 1 1 þ d þ þ r þ dr þ r km ik mj X ijkm : 2 hrik 2 hrkm 2 hrmj
e (iv) If 1=2 < a 1 hrik =4, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP
! r ð4a þ hrik 4Þðr4ik r3ik Þ ð4a þ hrkm 4Þðr 4km r 3km Þ ð4a þ hmj 4Þðr 4mj r 3mj Þ 4 4 4 þ d þ þ r þ dr þ r km ik mj X ijkm : 2 hrik 2 hrkm 2 hrmj
e (v) If 1 hrik =4 < a 1 hrkm =4, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP
! r ð4a þ hrkm 4Þðr 4km r3km Þ ð4a þ hmj 4Þðr4mj r 3mj Þ 4 4 4 d þ þ rik þ dr km þ r mj X ijkm : 2 hrkm 2 hrmj
e (vi) If 1 hrkm =4 < a 1 hrmj =4, then CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a is equivalent to
uP
ð4a þ hrmj 4Þðr4mj r 3mj Þ 2 hrmj
!
þ
r4ik
þ
4 dr km
þ
r 4mj
X ijkm :
3.3. Solution methods In this subsection, we take advantage of the structural characteristics, and discuss the solution method for model (11). For e ik ; T e km and T e mj are mutually independent type-2 trapezoidal fuzzy varithis purpose, we assume the uncertain travel times T ables, and T ik , T km and T mj are their reduced fuzzy variables, respectively. In addition, we introduce the following index sets:
8 I1 ¼ fði; j; k; mÞjhrmj 6 a 6 hrkm =8; i; j; k; m 2 Ng; > > > > > > I2 ¼ fði; j; k; mÞjhrkm < a 6 hrik =8; i; j; k; m 2 Ng; > > > > < I3 ¼ fði; j; k; mÞjhr =8 < a 6 ð4 hl Þ=8; i; j; k; m 2 Ng; ik ik
> I4 ¼ fði; j; k; mÞjð4 hlik Þ=8 < a 6 ð8 2hlik hrik Þ=8; i; j; k; m 2 Ng; > > > > > I5 ¼ fði; j; k; mÞjð8 2hl hr Þ=8 < a 6 ð8 2hl hr Þ=8; i; j; k; m 2 Ng; > ik ik ik km > > > : I ¼ fði; j; k; mÞjð8 2hl hr Þ=8 < a 6 ð8 2hl hr Þ=8; i; j; k; m 2 Ng: 6 ik km ik mj Using the notations above, the feasible region of model (11) can be decomposed into six disjoint subregions according to the values of credibility level a, the solution process is implemented at most six times by solving six different equivalent parametric programming problems, which is described as follows. In the case of a 2 I1 , according to Theorem 2, we can transform model (11) to the following equivalent parametric programming problem
8 min > > > > > > > s:t: : > > > > > > > > > < > > > > > > > > > > > > > > > > :
u;
ð8ahrmj Þðr 2mj r 1mj Þ 4hlmj hrmj
1 þ r 1ik þ dr km þ r 1mj X ijkm 6 u; 8i; j; k; m 2 I1 ;
X ijkm P X ik þ X jm 1; 8i; j; k; m 2 N; P k2N X ik ¼ 1; 8i 2 N;
ð12Þ
X ik 6 X kk ; 8i; k 2 N; P k2N X kk ¼ p; X ik 2 f0; 1g; 8i; k 2 N; X ijkm 2 f0; 1g; 8i; j; k; m 2 N:
In the case of a 2 I2 , the equivalent parametric programming problem is obtained by replacing the first group constraints in problem (12) with the following constraints
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K. Yang et al. / Applied Mathematical Modelling 38 (2014) 3987–4005
d
ð8a hrkm Þðr 2km r 1km Þ 4 hlkm hrkm
þ
ð8a hrmj Þðr 2mj r 1mj Þ 4 hlmj hrmj
! 1
þ r 1ik þ dr km þ r 1mj X ijkm 6 u;
8i; j; k; m 2 I2 :
In the case of a 2 I3 , the equivalent parametric programming problem is obtained by replacing the first group constraints in problem (12) with the following constraints
ð8a hrik Þðr 2ik r1ik Þ 4 hlik hrik
þd
ð8a hrkm Þðr 2km r1km Þ 4 hlkm hrkm
þ
!
ð8a hrmj Þðr 2mj r1mj Þ
þ
4 hlmj hrmj
r 1ik
þ
1 dr km
þ
r1mj
X ijkm 6 u
for any i; j; k; m 2 I3 . In the case of a 2 I4 , the equivalent parametric programming problem is obtained by replacing the first group constraints in problem (12) with the following constraints
ð8a þ 2hlik þ hrik 8Þðr4ik r 3ik Þ 4 hlik hrik
þd
ð8a þ 2hlik þ hrkm 8Þðr 4km r 3km Þ 4 hlkm hrkm
þ
ð8a þ 2hlik þ hrmj 8Þðr 4mj r 3mj Þ 4 hlmj hrmj
! 4
þ r 4ik þ dr km þ r 4mj
X ijkm 6 u; 8i; j; k; m 2 I4 : In the case of a 2 I5 , the equivalent parametric programming problem is obtained by replacing the first group constraints in problem (12) with the following constraints
d
ð8a þ 2hlik þ hrkm 8Þðr 4km r3km Þ 4 hlkm hrkm
þ
ð8a þ 2hlik þ hrmj 8Þðr 4mj r 3mj Þ 4 hlmj hrmj
! 4
þ r 4ik þ drkm þ r 4mj X ijkm 6 u
for any i; j; k; m 2 I5 . In the case of a 2 I6 , the equivalent parametric programming problem is obtained by replacing the first group constraints in problem (12) with the following constraints
ð8a þ 2hlik þ hrmj 8Þðr 4mj r 3mj Þ 4 hlmj hrmj
! þ
r 4ik
þ
4 dr km
þ
r 4mj
X ijkm 6 u;
8i; j; k; m 2 I6 :
Finally, comparing the optimal solutions to six different equivalent parametric programming problems, we obtain the minimum as the global optimal solution to model (11). We refer to this solution procedure as the parameter-based domain decomposition method. Remark 2. When the type-2 travel times are reduced by LER method or UER method, we can get the optimal solutions to model (11) by the similar solution procedures described in the above. Problem (12) is a parametric mixed-integer programming problem with binary variables, one possible solution method is to use a standard branch-and-bound code. The size of the problem is, however, a serious disadvantage, which makes this way impossible for the solution of large problems, due to the memory requirements. The code LINGO (version 8.0), which is a state-of-the-art commercial general branch-and-bound IP-code [27], can be applied to deal with this problem. Furthermore, today’s IP codes have become increasingly complex with the incorporation of sophisticated algorithmic components, such as advanced search strategies, preprocessing and probing techniques, cutting plane algorithms, and primal heuristics. The structure of the constraints in problem (12) makes the use of modeling language particularly appropriate. This yields a rather efficient solution method for this kind of problem. In the next section, we perform some numerical experiments, in which we employ LINGO to solve the problems. 4. Numerical experiments 4.1. Data and implementation In this subsection, the data from 15 cities (N ¼ f1; 2; . . . ; 15g) in a service region are used to test our proposed method. The locations of the 15 cities are shown in Fig. 2. We summarize the possibility distributions of the type-2 trapezoidal fuzzy travel times in Table 1. Based on the data, the generalized credibility level a may be roughly separated to three levels, middle, high and low. Their corresponding generalized credibility levels are 0.55 (middle), 0.70 (high), and 0.30 (low), respectively. Because the discount factors are based on the 15 cities data set, they should represent a realistic application. In addition, we provide results with two levels of inter-hub transportation discounts d = 0.2 and 0.8. The small value of d ¼ 0:2 reflects a strong degree of consolidation and economies of scale that corresponds to the use of much faster transport on hub arcs than on non-hub arcs to generate large time savings. The big value d = 0.8 reflects a lesser degree of consolidation. For simplicity, we set hlij ¼ hl and hrij ¼ hr for i ¼ 1; 2; . . . ; 15; j ¼ 1; 2; . . . ; 15, and ~ h ¼ ðhl ; hr Þ is the parameter characterizing the degree of uncertainty. Several instances of model (11) are obtained by using different savings factor d in hub-to-hub transportation time, together with different values of the degree of uncertainty ~ h ¼ ðhl ; hr Þ and different service levels a. For any given parameters,
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model (11) becomes a mixed-integer programming problem (12). The mixed-integer programming problem is then implemented by LINGO solver. All of the numerical tests were executed on a personal computer (Lenovo with Intel Pentium (R) Dual-Core E5700 3.00 GHZ CPU and RAM 2.00 GB), using the Microsoft Windows 7 operating system. For each test problem, we report: Obj. value: for the optimal objective. Hubs: for the optimal hub nodes, and CPU (sec): the CPU time in seconds required for solving the formulation with LINGO. 4.2. Test results 4.2.1. Computational results with deterministic travel times Firstly, we study hub network design with deterministic travel times in model (11). For the sake of comparison, the deterministic travel times in Table 2 are the mean values of type-1 trapezoidal fuzzy travel times collected in Table 4. Using the deterministic p-hub center model formulated in ([2]), we can obtain the computational results shown in Table 3. The results in this case can serve as a reference for the alternative cases performed in the next numerical experiments. 4.2.2. Computational results with travel times having fixed possibility distributions Secondly, we study hub network design with travel times having fixed possibility distributions, which are collected in Table 4. As discussed in Section 4, in the case of hl ¼ 0 and hr ¼ 0, the reduced type-2 travel times T ij are degenerated to type-1 travel times tij for i ¼ 1; 2; . . . ; 15; j ¼ 1; 2; . . . ; 15. That is, when the travel times are characterized by type-1 trapezoidal fuzzy variables, model (11) is turned into its equivalent mixed-integer programming problem. Using this data, Table 5 summarizes the computational results with type-1 travel times. The computational results show that the objective value increases as the credibility level a increases. 4.2.3. Computational results with travel times having parametric possibility distributions Finally, we investigate hub network design with travel times having parametric possibility distributions. As discussed in Section 4, when the travel times are characterized by type-2 trapezoidal fuzzy variables, model (11) is turned into its equivalent parametric mixed-integer programming problem (12) by reduced methods. In this case, we report the optimal solutions to model (12) in the numerical experiments. Using the LER, UER and ER methods, the solution results for 15 city
Fig. 2. The location of 15 cities.
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Table 1 e ij from city i to city j. The data set about type-2 fuzzy travel time T 1
2
3
1 2
0 ð10; 13; 14; 15; ~ h21 Þ
0
3
ð16; 18; 20; 25; ~ h31 Þ ð40; 48; 49; 50; ~ h41 Þ
ð10; 17; 18; 20; ~ h32 Þ ð35; 37; 39; 40; ~ h42 Þ
ð25; 30; 33; 35; ~ h25 Þ ~61 Þ ð35; 40; 44; 45; h
ð19; 20; 21; 25; ~ h52 Þ ~62 Þ ð25; 30; 31; 35; h
ð25; 30; 35; 40; ~ h71 Þ ~81 Þ ð50; 55; 59; 60; h
ð18; 20; 22; 25; ~ h72 Þ ~82 Þ ð40; 45; 46; 50; h
ð45; 50; 51; 55; ~ h91 Þ ~101 Þ ð65; 68; 70; 75; h
ð30; 35; 38; 40; ~ h92 Þ ~102 Þ ð54; 55; 58; 60; h
ð30; 35; 37; 40; ~ h111 Þ ~121 Þ ð37; 40; 41; 45; h
ð25; 30; 36; 40; ~ h112 Þ ~122 Þ ð28; 30; 35; 40; h
14
ð45; 50; 57; 60; ~ h131 Þ ~141 Þ ð45; 49; 50; 55; h
ð48; 49; 50; 55; ~ h132 Þ ~142 Þ ð30; 35; 37; 40; h
15
ð55; 58; 59; 60; ~ h151 Þ 6
4 5 6 7 8 9 10 11 12 13
6 7
4
5
0 0
ð25; 30; 31; 35; ~ h43 Þ ~53 Þ ð17; 18; 20; 25; h
ð16; 17; 19; 20; ~ h53 Þ ~64 Þ ð20; 25; 30; 35; h
ð30; 35; 39; 40; ~ h63 Þ ð30; 35; 37; 40; ~ h73 Þ
ð35; 40; 42; 45; ~ h74 Þ
ð45; 50; 54; 55; ~ h83 Þ ð45; 48; 50; 55; ~ h93 Þ
ð35; 40; 41; 45; ~ h84 Þ ð40; 45; 47; 50; ~ h94 Þ
ð65; 68; 70; 75; ~ h103 Þ ð10; 15; 19; 20; ~ h113 Þ
ð55; 59; 60; 65; ~ h104 Þ ð20; 25; 32; 35; ~ h114 Þ
ð15; 18; 20; 25; ~ h123 Þ ð34; 35; 38; 40; ~ h133 Þ
ð15; 17; 18; 20; ~ h124 Þ ð10; 12; 13; 15; ~ h134 Þ
ð44; 45; 46; 50; ~ h152 Þ
ð30; 35; 38; 40; ~ h143 Þ ð45; 48; 51; 55; ~ h153 Þ
ð10; 15; 18; 20; ~ h144 Þ ð27; 30; 33; 35; ~ h154 Þ
7
8
9
0 ð14; 15; 19; 20; ~ h65 Þ ð15; 20; 25; 30; ~ h75 Þ ð25; 30; 34; 35; ~ h85 Þ ð28; 30; 34; 35; ~ h95 Þ ~105 Þ ð45; 50; 52; 55; h ð25; 30; 33; 35; ~ h115 Þ ~125 Þ ð15; 20; 23; 25; h ð29; 30; 31; 35; ~ h135 Þ ~145 Þ ð10; 15; 18; 20; h ð25; 28; 30; 35; ~ h155 Þ 10
0 ð10; 15; 16; 20; ~ h76 Þ ð10; 14; 15; 20; ~ h86 Þ ~96 Þ ð14; 15; 16; 20; h
8 9 10
ð20; 30; 32; 35; ~ h106 Þ ~116 Þ ð45; 50; 52; 55; h
11 12
ð38; 40; 41; 45; ~ h126 Þ ð38; 40; 44; 45; ~ h136 Þ
13 14
ð10; 13; 15; 20; ~ h146 Þ ð12; 14; 15; 20; ~ h156 Þ
15
11 11 12
0 ð20; 25; 27; 30; ~ h87 Þ
0
ð10; 13; 16; 20; ~ h97 Þ ð33; 35; 36; 40; ~ h107 Þ
ð10; 12; 14; 15; ~ h98 Þ ð14; 15; 19; 20; ~ h108 Þ ~118 Þ ð60; 65; 66; 70; h
0
ð35; 40; 55; 60; ~ h128 Þ ð45; 50; 53; 55; ~ h138 Þ
ð50; 55; 56; 60; ~ h129 Þ ð50; 55; 60; 65; ~ h139 Þ ~149 Þ ð25; 28; 30; 35; h
ð50; 52; 54; 55; ~ h117 Þ ~127 Þ ð30; 45; 47; 50; h ð45; 50; 55; 60; ~ h137 Þ ~147 Þ ð25; 30; 31; 35; h ð25; 30; 31; 35; ~ h157 Þ
ð20; 21; 24; 25; ~ h148 Þ ð10; 11; 12; 15; ~ h158 Þ
12
13
ð15; 18; 20; 25; ~ h109 Þ ~119 Þ ð60; 65; 66; 70; h
0 ð70; 80; 84; 85; ~ h1110 Þ ~1210 Þ ð68; 70; 73; 75; h ð65; 70; 73; 75; ~ h1310 Þ
ð18; 20; 23; 25; ~ h159 Þ
ð30; 40; 43; 45; ~ h1410 Þ ð24; 28; 29; 30; ~ h1510 Þ
14
15
0
14
ð10; 13; 15; 20; ~ h1211 Þ ð25; 30; 31; 35; ~ h1311 Þ ~1411 Þ ð40; 45; 46; 50; h
15
ð55; 60; 61; 65; ~ h1511 Þ
13
0 ð5; 7; 8; 10; ~ h1412 Þ
0
ð20; 30; 32; 35; ~ h1412 Þ ð40; 45; 48; 50; ~ h1512 Þ
ð25; 28; 30; 35; ~ h1413 Þ ð35; 40; 45; 50; ~ h1513 Þ
0 ð14; 15; 16; 20; ~ h1514 Þ
0
Table 2 The data set about deterministic travel time T ij from city i to city j.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
11 12 13 14 15
1
2
3
4
5
6
7
8
9
10
0 13 19.75 46.75 30.75 41 32.5 56 50.25 69.5 35.5 40.75 53 49.75 58
0 16.25 37.75 21.25 30.25 21.25 45.25 35.75 56.75 32.75 33.25 50.5 35.5 46.25
0 30.25 20 36 35.5 51 49.5 69.5 16 19.5 36.75 35.75 49.75
0 18 27.5 40.5 40.25 45.5 59.75 28 17.5 12.5 15.75 31.25
0 17 22.5 31 31.75 50.5 30.75 20.75 31.25 15.75 29.5
0 15.25 14.75 16.25 29.25 50.5 41 41.75 14.5 15.25
0 25.5 14.75 36 52.75 43 52.5 30.25 30.25
0 12.75 17 65.25 47.5 50.75 22.5 12
0 19.5 65.25 55.25 57.25 29.5 21.5
0 79.75 71.5 70.75 39.5 27.75
11
12
13
14
15
0 14.5 30.25 45.25 60.25
0 7.5 29.25 45.75
0 29.5 42.5
0 16.25
0
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K. Yang et al. / Applied Mathematical Modelling 38 (2014) 3987–4005 Table 3 Solution results with deterministic travel times. p
d
Optimal
CPU (s)
Hubs
Obj. value
2
0.2 0.8
5, 9 5, 10
62.5000 72.1500
444 507
3
0.2 0.8
3, 4, 9 5, 9, 10
51.1500 71.6500
362 400
4
0.2 0.8
2, 6, 10, 12 2, 5, 8, 10
42.7000 69.5000
606 599
nodes with various parameters p; ðhl ; hr Þ; d and a are reported in Tables 6–8, respectively. From the computational results, we observe that the optimal objective value varies while hl and hr change between 0 and 1.
4.3. Comparison of computational results In this subsection, we analyze the results with our robust model (11) with type-2 travel times about the economies of scale parameter d and the generalized credibility level a. For this purpose, we summarize the optimal network configurations in Figs. 3 and 4, respectively. Fig. 3 plots the optimal solutions obtained by the deterministic model and the robust type-2 model with different economies of scale parameter d. From Fig. 3, we find that the network structure changes greatly in the deterministic model. Specifically, when d ¼ 0:2, the optimal hub locations are nodes 3, 4 and 9. When d ¼ 0:8, the optimal hub located nodes 3 and 4 are changed to nodes 5 and 10. However, Fig. 3 shows that there are two hub nodes 8 and 12 unchanged in the results obtained by our robust type-2 model. As a consequence, the hub locations in the type-2 model are robust with respect to the economies of scale parameter.
Table 4 The data set about type-1 fuzzy travel time T ij from city i to city j. 1
2
3
4
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 ð10; 13; 14; 15Þ ð16; 18; 20; 25Þ ð40; 48; 49; 50Þ ð25; 30; 33; 35Þ ð35; 40; 44; 45Þ ð25; 30; 35; 40Þ ð50; 55; 59; 60Þ ð45; 50; 51; 55Þ ð65; 68; 70; 75Þ ð30; 35; 37; 40Þ ð37; 40; 41; 45Þ ð45; 50; 57; 60Þ ð45; 49; 50; 55Þ ð55; 58; 59; 60Þ
0 ð10; 17; 18; 20Þ ð35; 37; 39; 40Þ ð19; 20; 21; 25Þ ð25; 30; 31; 35Þ ð18; 20; 22; 25Þ ð40; 45; 46; 50Þ ð30; 35; 38; 40Þ ð54; 55; 58; 60Þ ð25; 30; 36; 40Þ ð28; 30; 35; 40Þ ð48; 49; 50; 55Þ ð30; 35; 37; 40Þ ð44; 45; 46; 50Þ
0 ð25; 30; 31; 35Þ ð17; 18; 20; 25Þ ð30; 35; 39; 40Þ ð30; 35; 37; 40Þ ð45; 50; 54; 55Þ ð45; 48; 50; 55Þ ð65; 68; 70; 75Þ ð10; 15; 19; 20Þ ð15; 18; 20; 25Þ ð34; 35; 38; 40Þ ð30; 35; 38; 40Þ ð45; 48; 51; 55Þ
0 ð16; 17; 19; 20Þ ð20; 25; 30; 35Þ ð35; 40; 42; 45Þ ð35; 40; 41; 45Þ ð40; 45; 47; 50Þ ð55; 59; 60; 65Þ ð20; 25; 32; 35Þ ð15; 17; 18; 20Þ ð10; 12; 13; 15Þ ð10; 15; 18; 20Þ ð27; 30; 33; 35Þ
0 ð14; 15; 19; 20Þ ð15; 20; 25; 30Þ ð25; 30; 34; 35Þ ð28; 30; 34; 35Þ ð45; 50; 52; 55Þ ð25; 30; 33; 35Þ ð15; 20; 23; 25Þ ð29; 30; 31; 35Þ ð10; 15; 18; 20Þ ð25; 28; 30; 35Þ
6
7
8
9
10
6 7 8 9 10 11 12 13 14 15
0 ð10; 15; 16; 20Þ ð10; 14; 15; 20Þ ð14; 15; 16; 20Þ ð20; 30; 32; 35Þ ð45; 50; 52; 55Þ ð38; 40; 41; 45Þ ð38; 40; 44; 45Þ ð10; 13; 15; 20Þ ð12; 14; 15; 20Þ
0 ð20; 25; 27; 30Þ ð10; 13; 16; 20Þ ð33; 35; 36; 40Þ ð50; 52; 54; 55Þ ð30; 45; 47; 50Þ ð45; 50; 55; 60Þ ð25; 30; 31; 35Þ ð25; 30; 31; 35Þ
0 ð10; 12; 14; 15Þ ð14; 15; 19; 20Þ ð60; 65; 66; 70Þ ð35; 40; 55; 60Þ ð45; 50; 53; 55Þ ð20; 21; 24; 25Þ ð10; 11; 12; 15Þ
0 ð15; 18; 20; 25Þ ð60; 65; 66; 70Þ ð50; 55; 56; 60Þ ð50; 55; 60; 65Þ ð25; 28; 30; 35Þ ð18; 20; 23; 25Þ
0 ð70; 80; 84; 85Þ ð68; 70; 73; 75Þ ð65; 70; 73; 75Þ ð30; 40; 43; 45Þ ð24; 28; 29; 30Þ
11
12
13
14
15
0 ð10; 13; 15; 20Þ ð25; 30; 31; 35Þ ð40; 45; 46; 50Þ ð55; 60; 61; 65Þ
0 ð5; 7; 8; 10Þ ð20; 30; 32; 35Þ ð40; 45; 48; 50Þ
0 ð25; 28; 30; 35Þ ð35; 40; 45; 50Þ
0 ð14; 15; 16; 20Þ
0
11 12 13 14 15
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Table 5 Solution results with type-1 travel times. p
a
d
2
0.2
0.8
3
0.2
0.8
4
0.2
0.8
Optimal
CPU (s)
Hubs
Obj. value
0.30 0.55 0.70 0.30 0.55 0.70
5, 6 5, 10 5, 9 5, 8 5, 10 5, 10
59.2000 66.4000 67.6000 66.6000 75.9400 76.9600
574 456 736 862 1071 658
0.30 0.55 0.70 0.30 0.55 0.70
3, 4, 9 3, 4, 9 2, 8, 12 5, 8, 13 5, 6, 10 5, 8, 10
45.7600 53.8000 56.2000 65.0000 75.0400 76.3600
387 376 418 532 555 601
0.30 0.55 0.70 0.30 0.55 0.70
2, 5,8,12 3, 4, 6, 10 3, 4, 6, 10 1, 5, 8,12 1, 5, 8, 10 5, 8, 10, 12
38.2000 44.9800 47.4800 57.0000 75.0400 76.3600
491 617 462 428 550 433
Fig. 4 plots the optimal solutions obtained by the type-1 model and the robust type-2 model with different generalized credibility level a and fixed d ¼ 0:2. From Fig. 3, we observe that the network structure changes greatly in the type-1 model with different values of parameter a. For example, when a ¼ 0:30, the optimal hub locations are nodes 3, 4 and 9. When a ¼ 0:70, the optimal hub located nodes 3, 4 and 9 are changed to nodes 2, 8 and 12. Thus, the network configurations are completely changed. However, Fig. 4 show the robust type-2 model use the same hub location nodes 3, 4 and 9, the difference is just the non-hub to hub assignments. Therefore, the hub locations in the type-2 model are robust with respect to the generalized credibility level. Based on the computational results, we next give several findings about our hub location problem. In deterministic model, the deterministic travel times are usually obtained by replacing all fuzzy parameters with their mean values. If the optimal hub locations are sensitive to environmental change, then this method is not acceptable. Therefore, it is necessary for the decision maker to obtain robust hub locations solutions. Table 6 Solution results by LER method with different parameters p; hl ; d and a. p
2
hl
0.3
d
0.2
0.8
3
0.5
0.2
0.8
0.8
0.2
0.8
4
1.0
0.2
0.8
a
Optimal
CPU (s)
Hubs
Obj. value
0.30 0.55 0.70 0.30 0.55 0.70
5, 8 5, 8 5, 9 5, 8 5, 10 5, 10
59.4118 67.1765 68.5882 67.2353 76.6000 77.8000
525 492 380 357 767 563
0.30 0.55 0.70 0.30 0.55 0.70 0.30 0.55 0.70 0.30 0.55 0.70
3, 4, 9 3, 4, 9 2, 8, 12 5, 8, 12 5, 6, 10 5, 9, 10 3, 4, 9 2, 8, 12 2, 8, 12 5, 8, 12 5, 9, 10 5, 8, 10
46.6800 56.7333 59.3600 67.0000 76.6533 78.4133 47.6000 59.2000 61.9333 69.0000 78.2667 81.3333
460 252 361 482 877 826 261 347 380 351 496 563
0.30 0.55 0.70 0.30 0.55 0.70
3, 6, 10, 12 2, 6, 10, 12 2, 6, 8, 12 1, 5, 8, 12 5, 6, 10, 13 4, 5, 6, 10
41.6000 51,1600 57.6000 61.6000 79.8800 82.5200
306 262 361 568 474 826
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K. Yang et al. / Applied Mathematical Modelling 38 (2014) 3987–4005 Table 7 Solution results by UER method with different parameters p; hr ; d and a. p
2
hr
0.2
a
d
0.2
0.8
3
0.5
0.2
0.8
0.7
0.2
0.8
4
1.0
0.2
0.8
Optimal
CPU (s)
Hubs
Obj. value
0.30 0.55 0.70 0.30 0.55 0.70
5, 9 5, 8 5, 9 5, 8 5, 10 5, 10
59.1111 66.4444 67.7778 66.3333 75.9778 77.1111
499 355 432 595 1311 578
0.30 0.55 0.70 0.30 0.55 0.70 0.30 0.55 0.70 0.30 0.55 0.70
3, 4, 9 3, 4, 9 3, 4, 9 5, 6, 12 5, 6, 10 5, 6, 10 3, 4, 9 3, 4, 9 3, 4, 9 4, 5, 6 5, 6, 10 5, 6, 10
45.1467 54.2687 57.2667 63.5733 75.1867 76.9467 44.7692 54.2308 57.9231 62.2769 75.2769 77.3077
339 266 368 513 491 874 343 278 401 427 543 638
0.30 0.55 0.70 0.30 0.55 0.70
2, 5, 8, 12 3, 4, 6, 10 3, 4, 6, 10 1, 5, 8, 12 5, 6, 10, 11 5, 8, 10, 13
35.6000 45.7600 51.1600 55.0000 75.4800 78.1200
301 298 432 318 799 903
Table 8 Solution results by ER method with different parameters p; ðhl ; hr Þ; d and a. p
2
ðhl ; hr Þ
ð0:3; 0:2Þ
d
0.2
0.8
3
ð0:5; 0:5Þ
0.2
0.8
ð0:8; 0:7Þ
0.2
0.8
4
ð1:0; 1:0Þ
0.2
0.8
a
Optimal
CPU (s)
Hubs
Obj. value
0.30 0.55 0.70 0.30 0.55 0.70
5, 6 5, 9 5, 8 5, 8 5, 10 5, 10
59.2571 66.8000 68.1714 66.7714 76.2800 77.4457
558 723 653 534 1291 634
0.30 0.55 0.70 0.30 0.55 0.70 0.30 0.55 0.70 0.30 0.55 0.70
3, 4, 9 3, 4, 9 2, 8, 12 5, 8, 12 5, 6, 10 5, 8, 10 3, 4, 9 3, 4, 9 3, 4, 9 5, 8, 12 5, 8, 10 5, 6, 10
45.9133 55.4000 58.6000 65.3330 75.9200 77.6800 46.1280 56.8400 59.8080 65.8000 76.7120 78.8240
269 286 639 490 1014 836 276 281 465 905 810 586
0.30 0.55 0.70 0.30 0.55 0.70
3, 4, 6, 10 3, 4, 6, 10 2, 6, 10, 12 1, 5, 8, 12 2, 5, 6, 10 4, 5, 6, 10
39.2000 48.0933 51.7733 58.4000 76.6533 78.4133
338 335 763 300 837 852
In type-1 model, the fuzzy travel times are assumed to have fixed possibility distributions, which usually require the knowledge and experience of decision makers in the related fields. In this case, the optimal hub locations depend on the fixed possibility distributions. If the fixed possibility distributions are unavailable, then we cannot obtain the optimal network configurations.
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Fig. 3. Robustness analysis about the economies of scale parameter d.
In our robust type-2 model, the fuzzy travel times are characterized by parametric possibility distributions. In the case that the fixed possibility distributions cannot be determined exactly in the designing network configuration process, the developed robust parametric optimization method provide an effective way for decision makers to design hub location networks. Therefore, with the proposed robust method, the decision maker can identify hub locations which are robust to both environmental change and allocation decisions.
5. Conclusions On the basis of fuzzy Chouqet integral, this paper reduced the uncertainty embedded in the secondary possibility distribution, and presented a new robust parametric optimization approach to p-hub center problem. The obtained major results of the paper are summarized as follows.
K. Yang et al. / Applied Mathematical Modelling 38 (2014) 3987–4005
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Fig. 4. Robustness analysis about the generalized credibility level a.
Firstly, applying the proposed reduction methods, we derived the parametric possibility distributions of reduced fuzzy variables for type-2 trapezoidal fuzzy variables. Secondly, a new generalized VaR p-hub center problem was built, where the uncertain travel times are characterized by parametric possibility distributions. When the travel times are type-2 trapezoidal fuzzy variables, we transform the original p-hub center problem to its equivalent parametric mixed-integer programming model, which is solved by general-purpose optimization software. Finally, we tested the proposed method with a data set for 15 city nodes. The computational results showed the credibility of the developed modeling idea and the efficiency of the proposed solution method.
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Acknowledgments The authors wish to thank the Editor and anonymous reviewers, whose valuable comments led to an improved version of this paper. This work was supported by National Natural Science Foundation of China (No. 61374184), and the Training Foundation of Hebei Province Talent Engineering. Appendix A
Proof of Theorem 1. We only prove assertion (iii), and the rest assertions can be proved similarly. ~ ~n ðxÞ of ~ Note that the secondary possibility distribution l n is the triangular RFV
x r1 x r1 x r1 x r1 r2 x hl ; ; þ hr r2 r1 r2 r1 r2 r1 r2 r1 r2 r1
for x 2 ½r 1 ; r 2 , the triangular RFV ð1 hl ; 1; 1Þ for x 2 ½r 2 ; r 3 , and the triangular RFV
r4 x r4 x r4 x r4 x x r3 hl ; ; þ hr r4 r3 r4 r3 r4 r3 r4 r3 r4 r3
n by the UER method. Then, according to Eq. (4), we have for x 2 ½r 3 ; r 4 . Let n3 be the reduction of ~
8 xr1 r2 x 8 2r r þhr r r ð4hr Þ > 2 1 2 1 > > ; r 6 x 6 r > > 1 2 < < 2 2 ln3 ðx; hr Þ ¼ 1; r 2 6 x 6 r 3 ¼ 1; > > r x xr > > > : ð2hr Þ : 2r33r2 þhr r3 r22 2 ; r 6 x 6 r 3 4 2
xr 1 r 2 r 1
þ h2r ; r 1 6 x 6 r 2 ;
r 4 x r 4 r 3
þ h2r ; r 3 6 x 6 r 4 ;
r2 6 x 6 r3 ;
which completes the proof of assertion (iii). h Proof of Theorem 2. We only prove assertion (iv), and the rest assertions can be proved similarly. e ik ; T e km and T e mj , respectively. Then, by the ER method, Let T ik ; T km and T mj be the reduced variables of type-2 travel times T the parametric possibility distributions of T ik ; T km and T mj are given as
8 ð4hl hr Þ ik ik > > 4 > > > < l lT ik ðx; hlik ; hrik Þ ¼ 4h4 ik ; > > > > l r > : ð4hik hik Þ 4
xr 1ik
r 2ik r 1ik
þ
hrik 4
; r 1ik 6 x 6 r2ik ; r 2ik 6 x 6 r3ik ;
r 4ik x r 4 r 3 ik ik
8 ð4hl hr Þ km km > > 2 > > > < l lT km ðx; hlkm ; hrkm Þ ¼ 4h4km ; > > > > ð4hl hr Þ > : km km 2
þ
hrik 4
; r 3ik 6 x 6 r4ik ;
xr 1km
r 2km r 1km
þ
hrkm 4
; r 1km 6 x 6 r 2km ; r 2km 6 x 6 r 3km ;
r 4km x
r 4km r 3km
þ
hrkm 4
; r 3km 6 x 6 r 4km
and
8 ð4hl hr Þ mj mj > > > 2 > > > < l lT mj ðx; hlmj ; hrmj Þ ¼ 4h4 mj ; > > > > > ð4hlmj hrmj Þ > : 2
xr 1mj r 2mj r 1mj
þ
hrmj 4
; r 1mj 6 x 6 r 2mj ; r 2mj 6 x 6 r 3mj ;
r4mj x r 4mj r 3mj
þ
hrmj 4
; r 3mj 6 x 6 r 4mj :
h i h i h i Denote the k-cuts of T ik ; T km and T mj by T ik;k ¼ T Lik;k ; T Rik;k , T km;k ¼ T Lkm;k ; T Rkm;k and T jm;k ¼ T Ljm;k ; T Rjm;k , where
8 < T Lik;k ¼ ð4k hrik Þðr 2ik r1ik Þ=ð4 hlik hrik Þ þ r 1ik ;
: T R ¼ ð4k hr Þðr 4 r 3 Þ=ð4 hl hr Þ þ r 4 ; ik ik ik ik;k ik ik ik (
T Lkm;k ¼ ð4k hrkm Þðr 2km r1km Þ=ð4 hlkm hrkm Þ þ r 1km ; T Rkm;k ¼ ð4k hrkm Þðr 4km r 3km Þ=ð4 hlkm hrkm Þ þ r 4km
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K. Yang et al. / Applied Mathematical Modelling 38 (2014) 3987–4005
and
(
T Lmj;k ¼ ð4k hrmj Þðr 2mj r 1mj Þ=ð4 hlmj hrmj Þ þ r1mj ; T Rmj;k ¼ ð4k hrmj Þðr4mj r 3mj Þ=ð4 hlmj hrmj Þ þ r 4mj :
For simplicity, we denote nijkm ¼ T ik þ dT kl þ T lj . By the supposition hlik P hlkm P hlmj and hrik P hrkm P hrmj , we obtain l hrmj 4 hlik 4 hlkm 4 hmj hrik hr 6 6 ; P km P : 4 4 4 4 4 4
Since T ik ; T km and T mj are mutually independent ([28]), we get the k-cut of nijkm as
h i h i L R nijkm;k ¼ nLijkm;k ; nRijkm;k ¼ T Lik;k þ dT km;k þ T Ljm;k ; T Rik;k þ dT km;k þ T Rjm;k :
For the sake of presentation, we adopt the following notations 1
r1 ¼ r 1ik þ dr km þ r1mj ; 1
r2 ¼ r 1ik þ dr km þ r1mj þ
ðhrkm hrmj Þðr 2mj r 1mj Þ 4 hlmj hrmj
1
r3 ¼ r 1ik þ dr km þ r1mj þ d
1
r4 ¼ r 2ik þ dr km þ r1mj þ d
4
r5 ¼ r 3ik þ dr km þ r4mj d
4
r6 ¼ r 4ik þ dr km þ r4mj d
4
r7 ¼ r 4ik þ dr km þ r4mj
;
ðhrik hrkm Þðr 2km r1km Þ 4 hlkm hrkm
þ
ðhrik hrmj Þðr 2mj r1mj Þ 4 hlmj hrmj
ð4 hlik hrkm Þðr 2km r 1km Þ 4
hlkm
hrkm
ð4 hlik hrkm Þðr 4km r 3km Þ 4
hlkm
hrkm
ðhrik hrkm Þðr 4km r3km Þ 4 hlkm hrkm
ðhrkm hrmj Þðr 4mj r 3mj Þ 4 hlmj hrmj
þ
;
ð4 hlik hrmj Þðr 2mj r1mj Þ 4 hlmj hrmj ð4 hlik hrmj Þðr 4mj r3mj Þ 4 hlmj hrmj
ðhrik hrmj Þðr 4mj r3mj Þ 4 hlmj hrmj
;
;
;
;
4
r8 ¼ r 4ik þ dr km þ r4mj : The parameter k is divided into three cases: (a) hrmj =4 6 k 6 hrkm =4; (b) hrkm =4 6 k 6 hrik =4 and (c) hrik =4 6 k 6 ð4 hlik Þ=4. For case (a), we deduce the parametric possibility distribution of fuzzy variable nijkm as follows:
_
L1 ðxÞ ¼
( 1
kjr 1ik þ dr km þ r 1mj þ
ð4k hrmj Þðr 2mj r1mj Þ 4 hlmj hrmj
hr hr mj 6k6 km 4 4
R3 ðxÞ ¼
_ hr hr mj 6k6 km 4 4
( kjr 4ik
þ
4 dr km
þ
r 4mj
)
ð4k hrmj Þðr4mj r 3mj Þ 4 hlmj hrmj
1
6x ¼
ð4 hlmj hrmj Þ x ðr 1ik þ dr km þ r1mj Þ hrmj þ ; r1 6 x 6 r2 ; 4 4 r 2mj r1mj
) Px
4
¼
ð4 hlmj hrmj Þ ðr 4ik þ dr km þ r4mj Þ x hrmj þ ; r7 6 x 4 4 r 4mj r 3mj
6 r8 : For case (b), we deduce the parametric possibility distribution of fuzzy variable nijkm as follows:
L2 ðxÞ ¼
R2 ðxÞ ¼
x ðr 1ik þ dðhlkm r1km þ hrkm r2km 4r 1km Þ=ð4 hlkm hrkm Þ þ ðhlmj r1mj þ hrmj r 2mj 4r1mj Þ=ð4 hlmj hrmj ÞÞ 4ðdðr2km r 1km Þ=ð4 hlkm hrkm Þ þ ðr 2mj r 1mj Þ=ð4 hlmj hrmj ÞÞ ðr 4ik þ dð4r 4km hrkm r 3km hlkm r4km Þ=ð4 hlkm hrkm Þ þ ð4r 4mj hrmj r3mj hlmj r 4mj Þ=ð4 hlmj hrmj ÞÞ x 4ðdðr 4km r3km Þ=ð4 hrkm hrkm Þ þ ðr4mj r 3mj Þ=ð4 hrmj hrmj ÞÞ
For case (c), we deduce the parametric possibility distribution of fuzzy variable nijkm as follows:
;
r2 6 x 6 r3 ;
;
r6 6 x 6 r7 :
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K. Yang et al. / Applied Mathematical Modelling 38 (2014) 3987–4005
L3 ðxÞ ¼
x ððhlik r 1ik þ hrik r 2ik 4r1ik Þ=ð4 hlik hrik Þ þ dðhlkm r 1km þ hrkm r 2km 4r1km Þ=ð4 hlkm hrkm ÞÞ 4ððr 2ik r1ik Þ=ð4 hrik hrik Þ þ dðr2km r 1km Þ=ð4 hlkm hrkm Þ þ ðr 2mj r 1mj Þ=ð4 hlmj hrmj ÞÞ
ðhlmj r 1mj þ hrmj r2mj 4r 1mj Þ=ð4 hlmj hrmj Þ 4ððr2ik r1ik Þ=ð4
hrik
hrik Þ
þ dðr 2km r 1km Þ=ð4 hlkm hrkm Þ þ ðr 2mj r1mj Þ=ð4 hlmj hrmj ÞÞ
;
r3 6 x 6 r4 ; R1 ðxÞ ¼
ðð4r 4ik hrik r 3ik hlik r 4ik Þ=ð4 hlik hrik Þ þ dð4r 4km hrkm r 3km hlkm r 4km Þ=ð4 hlkm hrkm ÞÞ x 4ððr 4ik r 3ik Þ=ð4 hlik hrik Þ þ dðr 4km r 3km Þ=ð4 hlkm hrkm Þ þ ðr 4mj r 3mj Þ=ð4 hlmj hrmj ÞÞ þ
ð4r4mj hrmj r 3mj hlmj r 4mj Þ=ð4 hlmj hrmj Þ 4ððr 4ik r 3ik Þ=ð4
hlik
hrik Þ
þ dðr4km r 3km Þ=ð4 hlkm hrkm Þ þ ðr 4mj r 3mj Þ=ð4 hlmj hrmj ÞÞ
;
r5 6 x 6 r6 : Using the notations above, we obtain the following piecewise fractional parametric possibility distribution
8 L1 ðxÞ; > > > > > L2 ðxÞ; > > > > > > L ðxÞ; > < 3l lnijkm ðxÞ ¼ 4h4 ik ; > > > > R1 ðxÞ; > > > > > > R2 ðxÞ; > > : R3 ðxÞ;
lnilkm of nijkm ,
r1 6 x 6 r2 r2 6 x 6 r3 r3 6 x 6 r4 r4 6 x 6 r5 r5 6 x 6 r6 r6 6 x 6 r7 r7 6 x 6 r8 :
By the parametric possibility distribution lnijkm ðxÞ and the computational method for generalized credibility, we have the following calculation results. If ð4 hlik Þ=8 < a 6 ð8 hlik hrik Þ=8, then the following inequality
ðð4r 4ik hrik r3ik hlik r 4ik Þ=ð4 hlik hrik Þ þ dð4r 4km hrkm r 3km hlkm r 4km Þ=ð4 hlkm hrkm Þ 4 hlik 4 8ððr4ik r 3ik Þ=ð4 hlik hrik Þ þ dðr 4km r 3km Þ=ð4 hlkm hrkm Þ þ ðr 4mj r3mj Þ=ð4 hlmj hrmj ÞÞ þ
ð4r 4mj hrmj r 3mj hlmj r4mj Þ=ð4 hlmj hrmj ÞÞ u 8ððr 4ik r3ik Þ=ð4 hlik hrik Þ þ dðr4km r 3km Þ=ð4 hlkm hrkm Þ þ ðr 4mj r 3mj Þ=ð4 hlmj hrmj ÞÞ
Pa is equivalent to
uP
ð8a þ 2hlik þ hrik 8Þðr 4ik r 3ik Þ 4
hlik
hrik
þd
ð8a þ 2hlik þ hrkm 8Þðr 4km r3km Þ 4
hlkm
hrkm
þ
ð8a þ 2hlik þ hrmj 8Þðr 4mj r 3mj Þ 4 hlmj hrmj
4
þ r 4ik þ dr km þ r4mj : e As a consequence, the generalized credibility constraint CrfðT ik þ dT km þ T mj ÞX ijkm 6 ug P a can be expressed as
uP
ð8a þ 2hlik þ hrik 8Þðr4ik r3ik Þ 4 hlik hrik
þd
ð8a þ 2hlik þ hrkm 8Þðr4km r3km Þ 4 hlkm hrkm
þ
ð8a þ 2hlik þ hrmj 8Þðr4mj r3mj Þ 4 hlmj hrmj
! 4
þ r4ik þ drkm þ r4mj X ijkm :
The proof of the assertion (iv) is complete. h Proof of Theorem 3. The proof is similar to that of Theorem 2. h Proof of Theorem 4. The proof is similar to that of Theorem 2. h References [1] J.F. Campbell, Integer programming formulations of discrete hub location problems, Eur. J. Oper. Res. 72 (1994) 387–405. [2] B.Y. Kara, B.C. Tansel, On the single-assignment p-hub center problem, Eur. J. Oper. Res. 125 (2000) 648–655. [3] A.T. Ernst, H.W. Hamacher, H. Jiang, M. Krishnamoorthy, G. Woeginger, Uncapacitated single and multiple allocation p-hub center problems, Comput. Oper. Res. 36 (2000) 2230–2241. [4] S. Alumur, B.Y. Kara, Network hub location problems: the state of the art, Eur. J. Oper. Res. 190 (2008) 1–21.
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