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Computers & Industrial Engineering 62 (2012) 408–420

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Facility location dynamics: An overview of classifications and applications Alireza Boloori Arabani a, Reza Zanjirani Farahani b,⇑ a b

Department of Industrial and Systems Engineering, Wayne State University, Detroit, MI 48202, USA Department of Informatics and Operations Management, Kingston Business School, Kingston University, Kingston Hill, Kingston Upon Thames, Surrey KT2 7LB, UK

a r t i c l e

i n f o

Article history: Received 18 January 2011 Received in revised form 6 July 2011 Accepted 30 September 2011 Available online 8 October 2011 Keywords: Static facility location Dynamic facility location Location–relocation Multi-period Time-dependent Stochastic

a b s t r a c t In order to modify the current facility or develop a new facility, the dynamics of facility location problems (FLPs) ought to be taken into account so as to efficiently deal with changing parameters such as market demand, internal and external factors, and populations. Since FLPs have a strategic or long-term essence, the inherited uncertainty of future parameters must be incorporated in relevant models, so these models can be considered applicable and ready to implement. Furthermore, due to largely capital outlaid, location or relocation of facilities is basically considered as a long-term planning. Hence, regarding the way in which relevant criteria will change over time, decision makers not only are concerned about the operability and profitability of facilities for an extended period, but also seek to robust locations fitting well with variable demands. Concerning this fact, a trade-off should be set between benefits brought by facility location changes and costs incurred by possible modifications. This review reports on literature pointing out some aspects and characteristics of the dynamics of FLPs. In fact, this paper aims not only to review most variants of these problems, but also to provide a broad overview of their mathematical formulations as well as case studies that have been studied by the literature. Finally, based on classified research works and available gaps in the literature, some possible research trends will be pointed out. Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved.

1. Introduction A common application of FLPs includes a distributer choosing where to site a distribution center in a supply chain, a manufacturer selecting a right location for placing its warehouse, or even urban planners setting where to position a recreational facility; however, newer applications of FLPs contain location of bank accounts, database location in computer networks, vendor selection, etc. In each of these cases and also many other instances of FLPs, there will be no difficulty in dealing with system’s requirements as long as factors and parameters are fixed and consistent with the planning time horizon. Such condition usually happens in static facility location problems (STFLPs). However, with the development of FLPs, STFLPs might not fulfill the requirements of a system because not only main parameters are prone to change during the corresponding time horizon, but also is a considerable amount of investment and capital required for developing or obtaining a new facility. Since facility location decisions are particularly costly and time-sensitive, they are expected to perform in the most beneficial way in a long-term period. Therefore, in order to effectively handle probable fluctuations in future as well as changing parameters, a dynamic model seems be indispensable. ⇑ Corresponding author. Tel.: +44 (0)20 8417 5098; fax: +44 (0)20 8417 7026. E-mail addresses: [email protected], [email protected] (R.Z. Farahani).

From a general viewpoint, FLPs are defined in terms of two elements: space and time. By space, a planning area where facilities are located is referred, and by time, the time of location (establishing a new facility or modifying the existing facility) is indicated. According to their essence, both of space and time can be analyzed by discrete and continuous aspects. For example, if a discrete space is considered, the location of a facility can just be sited in specific points while in a continuous space, the facility is allowed to be sited anywhere in the planning area. Furthermore, discrete time means that the establishment of a new facility or the modification of the existing facility is permitted in predetermined points of time, whereas such restriction does not exist for continuous time. Such classification forms the main part of this review paper in that continuous-space, discrete-space, and network-space location problems are addressed under the category of STFLPs. On the other hand, time spans constitute the main parts of the dynamics of FLPs which forms the most important part of this paper (as a complementary research addressing other aspects of facility location dynamics, one can refer to Farahani, Abedian, & Sharahi, 2009). Even though this paper addresses STFLPs, it puts the emphasis on dynamic facility location problems (DFLPs) and their variants. For example, one of these areas is stochastic facility location problems (SFLPs) dealing with the intrinsic uncertainty of models’ parameters. In fact, such problems try to identify appropriate locations in which any configuration of random parameters in a model can be implemented. As other relevant areas, multi-period and

0360-8352/$ - see front matter Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.09.018

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single-period facility location problems (MPFLPs and SPFLPs) can be considered as discrete time span and continuous time span, respectively. In fact, in MPFLPs, a decision maker deals with changing parameters in each of several discrete-time planning horizons while in single-period models there is just one such period. Furthermore, another relevant area is the time-dependent facility location problem (TDFLP) (sometimes called demand-dependent problem), for which companies face with inconsistent demands fluctuating all over a year. Regarding these models, one of the primary instances are companies selling seasonal products. As another area, the relocation of facilities in a specific period, after they have been located, can be taken into account; meanwhile, this is called the facility location/relocation problem (FLRP). As a matter of fact, it is quite common for a firm to consider relocating its facilities over the time horizon without any potential disruption of activities in the firm. This review paper is categorized as follows. First, STFLPs are addressed in Section 2 consisted of three parts: continuous facility location problems (CFLPs), discrete facility location problems (DIFLPs), and network facility location problems (NFLPs). Then, Section 3 throws light on DFLPs. Regarding this section, the following parts of dynamic models will be discussed: (1) dynamic deterministic facility location problems (DDFLPs), (2) FLRPs, (3) MPFLPs, (4) TDFLPs, (5) SFLPs which are relatively similar to probabilistic facility location problems (PRFLPs), and (6) fuzzy facility location problems (FFLPs). In Section 4, available literature will be discussed and classified based on their performance measures (the type of objective functions in corresponding problems). Section 5 addresses the implementation of dynamic models in terms of case studies and various applications. Moreover, Section 6 reveals some possible trends for future research, and finally, Section 7 presents the conclusion of the whole review paper.

2.1.1. Single facility location problem In the SIFLP, a new facility should be located in a way such that its distances with other facilities are minimized as much as possible; meanwhile, this distance can be defined in many terms such as Euclidean distance and Manhattan distance. As one of the primary models in SIFLPs, the generalized Weber problem can be pointed out by which the site of a new facility is selected from a set of existing facilities (Wesolowsky, 1973):

2. Static facility location problems

As can be easily observed, Eq. (1) has been extended to Eq. (2), in which D is the set of new facilities; wij denotes the weight between the existing facility i and new facility j; Xj and Pi respectively denote the positions of new facility j, identified by the problem, and the existing facility i; thus, d(Xj, Pi) represents the distance between these two positions. This model considers no weight between new facilities; however, another type of the MUFLP takes these weights into account (Daneshzand & Shoeleh, 2009):

As mentioned before, the space issue is as much important as the time issue in the analysis of FLPs. Due to this fact, the space issue is essentially taken into account in STFLPs while the time issue is generally discussed in DFLPs; therefore, it has been preferred to discuss static problems in this section and address dynamic problems in the subsequent section. Now, STFLPs are thrown light on in terms of CFLPs, DIFLPs, and NFLPs. However, before going through details, it should be noted that if the space issue is considered for a FLP, two things should be identified: (1) customers with predetermined locations and (2) facilities with locations to be specified based on concerned objective function(s) (ReVelle, Eiselt, & Daskin, 2008). Therefore, the shape or topography of potential facilities will be the primary factor affecting models in continuous (plane) and discrete problems (Klose & Drexl, 2005). 2.1. Continuous facility location problems In CFLPs, facilities are generally supposed to be located anywhere in a planning area. As a matter of fact, the performance of such models is affected by two primary factors: (1) the continuous solution space in which facilities are allowed to be sited on every point in the plane and (2) distance between facilities and customers is measured by means of corresponding distance criteria (Ballou, 1968). As applications of continuous models, the location of video cameras or pollution censors to monitor certain environments can be mentioned (ReVelle et al., 2008). From a general point of view, CFLPs can be divided into three categories: single-facility location problems (SIFLPs), multiplefacility location problems (MUFLPs), and facility location–allocation problems (FLAPs) which will be explained as follows.

Minimize Z ¼

X

wi dðX; Pi Þ

ð1Þ

i2F

According to this model, the total incurred costs are minimized, for which F is the set of existing facilities; wi denotes a weight transforming distances into costs for the existing facility i; X and Pi respectively denote the positions of a new facility, identified by the problem, and the existing facility i; hence, d(X, Pi) represents the distance between these two positions. Even though the Weber problem is the primary type of CFLPs, for more advanced modifications of continuous problems (the location of single line and multiple lines in the plane, half-line facilities, hyperplanes and spheres, and polygonal curves), one can refer to Diaz-Banez, Mesa, and Schobel (2004). 2.1.2. Multi-facility location problem The MUFLP is quite similar to the SIFLP; however, instead of a new facility, several new facilities must recognize their optimal locations. Concerning the interaction of SIFLPs and MUFLPs, it should be noted that each SIFLP can be transformed to its multifacility equivalent. For example, the foregoing Weber problem can be considered with several facilities (Akyuz, Oncan, & Altnel, 2009):

Minimize Z ¼

XX i2F

Minimize Z ¼

X

wij dðX j ; Pi Þ

ð2Þ

j2D

v ik dðX i ; X k Þ þ

XX i2F

i;k2D i < 1 if candidate node j yij ¼ gives service to demand node i; > : 0 otherwise

8i 2 D;

ð8Þ

8j 2 F ð9Þ

This problem minimizes the total costs incurred by the allocation of clients to facilities. Here, F is the set of candidate locations where facilities may be sited, and D is the set of demand nodes; C(i, j) denotes the cost of assigning customer i to facility j; furthermore, Eq. (5) reveals that p facilities are exactly opened. Moreover, Eq. (6) certifies that each demand node is assigned, and constraint (7) indicates that open facilities are only allowed to be assigned to demand nodes. Finally, constraints (8) and (9) set binary conditions for the variables of this model. In addition to the preceding model, there are other types of location–allocation problems such as capacitated problems, in which the allocation of facilities to customers may be constrained due to the capacity limitation of facility, cost, time, etc. Furthermore, multi-commodity location–allocation problem can be taken into account, in which several product items are handled instead of a single product. For further discussions regarding FLAPs, one can refer to Cooper (1963), Sule (2001), Nickel and Puetro (2005), Azarmand and Neishabouri Jami (2009), and Akyuz et al. (2009). 2.2. Discrete facility location problems In DIFLPs there are two discrete sets of demands as well as candidate locations. The reason for which demand or candidate locations are considered as discrete parameters is that demands are usually expected to be occurred at specific geographical points. Quadratic assignment problem (QAP) and plant location problem (PLP) are the most important elements of DIFLPs which will be explained as follows. 2.2.1. Quadratic assignment problem From a general point of view, the QAP is one of the branches of assignment problem, in which a set of people must be assigned to a set of jobs. However, in the area of FLPs, this problem is interpreted as the assignment of some facilities to some customers/clients. Following this concept, Koopmans and Beckmann (1957) proposed the integer QAP for the first time:

Minimize Z ¼

D X D X D X D X i¼1

Cði; jÞ dðk; lÞ xik xjl

Subject to : m X xij ¼ 1 j ¼ 1; . . . ; D i¼1 m X

ð10Þ

j¼1 k¼1 l¼1

xij ¼ 1 i ¼ 1; . . . ; D

ð11Þ ð12Þ

j¼1

8 > < 1 if facility i is assigned xij ¼ to customer j; i; j ¼ 1; . . . ; D > : 0 otherwise

ð13Þ

This model minimizes the total cost incurred by the assignment of facilities to customers. Here, there are D facilities and D customers (their number should be equal); C(i, j) denotes that cost of assigning facility i to customer j; d(k, l) identifies the distance between facility k and customer l; xij represents an integer variable revealing

whether facility i is assigned to customer j or not (Eq. 13). Furthermore, Eqs. (11) and (12) are the main constraints of assignment problems identifying that each facility can serve just one customer, and every customer can be served by just one facility. While this problem was formulated as an integer QAP, there are other types of QAPs addressed by the literature such as Trace formulation and other linearized versions of this problem. (For more details, see Cela, 1998; Commander, 2003; Bayat & Sedghi, 2009.) 2.2.2. Plant location problem Without loss of generality, a plant is referred to a set of facilities, each of which can be a production, manufacturing, warehouse, or distribution facility. Therefore, PLPs have been applied for a wide range of areas. Regarding this fact, the following instances can be mentioned: plant location with procurement planning by Lim and Kim (2001), a multi-commodity PLP with technology selection by Dasci and Verter (2001), uncapacitated PLP by Fernandez and Puerto (2003), capacitated PLP with distribution by Berman and Drezner (2006), and PLP with service areas by Drezner, Drezner, and Salhi (2006). However, the main framework of PLP, sometimes considered as the uncapacitated facility location problem, has the following formulation (ReVelle et al., 2008):

Minimize Z ¼

X

C j xj þ a

j2F

XX j2F

wj dði; jÞ yij

ð14Þ

i2D

Similar to the foregoing model, this model minimizes the total incurred costs. Moreover, candidate facility j incurs cost Cj per time unit; wj denotes a weight transforming distances into costs for the existing facility j; d(i, j) is the distance from demand node i to candidate facility j; in addition, a converts the demand-weighted total distance to cost units; furthermore, the constraints of this model are similar to constraints (6)–(9). 2.3. Network facility location problems If a location problem is configured in terms of a network, consisted of nodes and links, demands typically arise on nodes; however, demands can also occur on links and nodes simultaneously. Based on this definition, NFLPs can be classified as five main categories: median problems, center problems, covering problems, hub location problems (HULPs), and hierarchical location problems (HILPs); meanwhile, it should be noted that the first problem (median) particularly optimizes weighted distance between demand points of candidate locations. In fact, this model is capable for dealing with cost minimization or profit maximization objectives. However, such optimization approach is not of interest in center and covering problems, in which completely different criteria are being applied. In addition to these three problems, HULPs and HILPs may be of interest in the analysis of NFLPs. 2.3.1. Median problem As the most important part of median models, the p-median extends the 1-median problem to find the most appropriate sites for p facilities and to identify facilities serving each demand node such that the sum of weighted distances from each demand node to its nearest facility is minimized. The formulation of p-median problem can be mentioned as follows (ReVelle et al., 2008):

Minimize Z ¼

XX i2D

wi dði; jÞ yij

ð15Þ

j2F

This model optimizes the weighted total distance, for which the constraints of this model are similar to constraints (5)–(9). Furthermore, wj and d(i, j) are alike those explained for the PLP in Section 2.2.2.

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Similar to PLPs, the median problem can be defined in terms of capacitated and uncapacitated problems. Regarding this point and other details of median problems, one can refer to Nickel and Puetro (2005) and Jamshidi (2009). 2.3.2. Covering problem From a general point of view, in the covering problem, each customer can be served by each facility provided that the customer and the facility are located in a specific distance called the coverage distance. Regarding this condition, the total covering problem was initially proposed by Toregas, Swain, ReVelle, and Bergman (1971):

Minimize Z ¼

X

xj

X

dði; jÞ yij  Z 6 0 8i 2 D

This condition represents the maximum distance measured by assignment variables. Concerning more details about center problems and relevant explanations, one can refer to Daskin (1995), Nickel and Puetro (2005), and Biazaran and SeyediNezhad (2009). 2.3.4. Hub location problem As one of the primary types of this problem, the p-hub problem was initially proposed by O’Kelly (1987):

ð16Þ Minimize Z ¼

j2F

Subject to : X xj P 1 8 i 2 D

ð23Þ

j2F

XX

Cði; kÞ yik

X

i2D k2D

ð17Þ

þ

j2F C

j2F

XX

Cðk; iÞ yik

k2D i2D

Regarding the objective function (Eq. 16) and in order to fulfill customers’ orders, the number of required facilities is expected to be minimized. By constraint (17), the number of candidate facility locations covering a demand node (from set Fc) cannot be lower than 1; meanwhile, xj is set according to constraint (8). In an attempt to revise this condition, Church and ReVelle (1974) proposed the maximum covering problem as follows:

Maximize Z ¼

X

wi zi

ð18Þ

Subject to : X xj 6 0 8 i 2 D zi 

ð19Þ

i2D

j2F C

 zi ¼

1 if demand node i is covered; 8i 2 D 0

otherwise

ð20Þ

The number of covered demands should be maximized by the model (Eq. 18). Furthermore, constraint (19) clarifies that if there is no facility at one of the candidate locations serving the node, demand node i cannot be covered. Moreover, other constraints of this model are set according to constraints (5) and (8). Concerning more details about covering problems and their variants (set covering, partial covering, etc.), one can refer to Francis, McGinnis, and White (1992), Nickel and Puetro (2005), Fallah, NaimiSadigh, and Aslanzadeh (2009). 2.3.3. Center problem In comparison with the covering problem, the center problem searches for the sites of locations to fulfill all demands; meanwhile, facilities should have the minimum distance with corresponding demand points (Daskin, 1995). As one of the most significant types of center problems, the vertex p-center problem is now pointed out, in which locations are only allowed to be considered at the nodes of a network (ReVelle et al., 2008):

Minimize Z Subject : X dði; jÞ yij P 0 8j 2 F Z

ð21Þ

þB

XXXX i2D

! f ði; jÞ

j2F

X

! f ði; jÞ

j2F

f ði; jÞ Cðk; lÞ yik yjl

ð24Þ

k2D l2D

The objective function of this problem minimizes the total cost. Furthermore, C(i, j) denotes the movement cost between nodes i and j; f(i, j) denotes the flow between nodes i and j; B is a discount factor. In addition, other constraints of this model are taken into account as constraints (5)–(9). For other varieties of HULPs as well as their vast applications, one can refer to Daskin (1995) and Hekmatfar and Pishvaee (2009). 2.3.5. Hierarchical location problem Consider a distribution system in which there are several hierarchical facilities. In such a system, facilities on a higher level are able to select their locations independently from facilities in a lower layer. However, an important criterion must be met in the foregoing distribution network: ‘‘higher level nodes have a sufficiently high capacity, and handling costs as well as transshipment costs, associated with these nodes, are proportional to the amount of items reloaded and shipped, respectively. Then, the transshipment cost from the source to the destination is measured with regard to the cost of allotted demand. Otherwise, transshipments covering several stages of the distribution system have to be considered explicitly’’ (Klose & Drexl, 2005). According to Sahin and Sural (2007), one of the primitive types of HILPs is the problem with a single-flow two-level system:

Minimize Z ¼

XX j

Subject to : X f1 ðj; kÞ ¼ dk j2F 1

X

k

XX i

f 2 ði; jÞ Cði; jÞ

ð25Þ

j

8k 2 D

ð26Þ

f1 ðj; kÞ 8j 2 F 1

ð27Þ

f1 ðj; kÞ 6 M j yj

8j 2 F 1

ð28Þ

f2 ði; jÞ 6 M i xi

8i 2 F 2

ð29Þ

f2 ði; jÞ ¼

i2F 2

X

X

f 1 ðj; kÞ Cðj; kÞ þ

k2D

k2D

X j2F 1

ð22Þ

j2F

In this model, the maximal facility customer distance is minimized via the objective function as well as constraint (22). In addition, other constraints of this model are considered as constraints (5)–(9); furthermore, d(i, j) is alike that explained for the PLP in Section 2.2.2. Regarding this model, it should be added that Hakimi (1964) presented a modified version of this model, called p-center model, in which constraint (22) is replaced by the following constraint:

X

yj ¼ p1

ð30Þ

xi ¼ p2

ð31Þ

j2F 1

X i2F 2

xi ¼ yj ¼



1 if level 2 facility is located at node i; 0



1 if level 1 facility is located at node j; 0

8i 2 F 2

otherwise

otherwise

8j 2 F 1

ð32Þ

ð33Þ

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Regarding this problem, two levels are considered for facilities (level 1 and level 2) since the problem is hierarchical. f1(j, k) and f2(i, j) respectively stand for the flow from facility node j and demand node k (in level 1) and the flow from facility node i (at level 2) and facility node j (at level 1); in addition, the unit flow between different levels of facilities incurs cost C(j, k) while C(i, j) is considered for flows between facilities at level 1 and demand nodes; furthermore, F1 and F2 denote the set of candidate locations where facilities may be sited for level 1 and level 2, respectively; p1 and p2 are the number of facilities that are expected to be located/ opened at levels 1 and 2; Mj and Mi are the capacities of facilities at levels 1 and 2, and dk is the demand placed by demand node k. Concerning the model, the total demand weighted distance is minimized (Eq. 25). Eq. (26) certifies that the demand of a demand node is fulfilled. By constraint (27), the total demand of a level 1 facility is equal to the demand transferred from that facility to a facility at other level. Furthermore, constraints (28) and (29) demonstrate the capacity of facilities at two levels, and constraints (30) and (31) are alike Eq. (5). As other instances of HILPs and their applications, one can refer to Moore and ReVelle (1982), Sahin and Sural (2007), and Bastani and Kazemzadeh (2009). Finally, an important point should be taken into account. Even though a wide range of STFLPs were described in this section, they are not limited to the abovementioned models, and others like ‘‘obnoxious facility location’’ and ‘‘competitive facility location’’ can be added which are more sophisticated. Since this paper focuses on dynamics of FLP rather than describing the models of this area, it has been preferred not to mention the other models; meanwhile, interested reader can refer to Daskin (1995), Drezner (1995), Drezner and Hamacher (2002), Farahani and Hekmatfar (2009), Francis et al. (1992), Handler and Mirchandani (1979), Love, Morris, and Wesolowsky (1988), Mirchandani and Francis (1990), and Nickel and Puetro (2005) to see many other models.

models or time-dependent models. Therefore, this section throws light on main elements of facility location dynamics consisted of DDFLPs, FLRPs, MPFLPs, TDFLPs, SFLPs, PFLPs, and FFLPs. 3.1. Dynamic deterministic facility location problem As indicated in Section 2, the generalized Weber problem is one of the primary static models, in which a facility’s location should be chosen from a set of m candidate points (costs measured based on Eq. (1)). However, there might be some circumstances in which effective parameters (e.g. population, market trends, distribution costs, demand patterns, environmental factors, etc.) are being changed or modified over time. In fact, in the majority of real-world problems, these changes are usually occurred, for which an action must be taken to revise, relocate, and adapt facilities. Therefore, rather than static models, the emergence of dynamic models has been necessary for such problems. In fact, the mentioned static model can be modified to a dynamic deterministic model in which p time periods are considered instead of a single period while in each of these periods, an optimal location is identified (Wesolowsky, 1973):

Minimize Z ¼

mp p X X l¼1

fli ðxl ; yl Þ þ

i¼1

p X

c l zl

ð34Þ

ðfor l ¼ 1; . . . ; pÞ

ð35Þ

l¼2

Subject to : zl ¼



0

if dl1;l ¼ 0;

1 else if dl1;l > 0

Regarding Eq. (34), there are ml candidate destinations in period l; fli(xl, yl) is the shipping cost between a facility located at fli(xl, yl) and destination i; cl stands for the moving cost in period l; and dl1,l is the distance by which the facility is transited in period l. Correspondingly, Eq. (35) is affected by this distance. 3.2. Facility location–relocation problem

3. Dynamic facility location problems In the previous section, STFLPs were thoroughly addressed; however, the main focus of this review paper is on DFLPs. Therefore, in the case of the interaction of STFLPs and DFLPs, an important point should not be overlooked. All types of STFLPs can be transformed to an equivalent problem to be discussed for DFLPs. For example, the SIFLP can be transformed to the DDFLP to be explained later. Furthermore, it should be noted that all other types of STFLPs (e.g. FLAP, QAP, PLP, etc.) can be remodeled as their dynamic counterparts. In DFLPs, there are two main criteria affecting the decision to identify the right location for a facility: (1) cost for which a trade-off must be set between expenditures incurred by developing a new facility or revising the current facility and profits supposed to be acquired as a result of such development. For such facilities, several instances can be pointed out such as warehouses, distribution centers, hospitals, recreation centers, schools, and depots planned for being operable for decades and (2) time for which the opening and closing of facilities are considered over the planning horizon. Furthermore, from another point of view, dynamic models can be divided into two sub-categories: (1) explicitly dynamic models in which facilities are opened/closed at pre-specified times and locations and (2) facilities are supposed to be opened and remained open throughout the planning horizon at the beginning of the horizon (implicitly dynamic models). Although dynamic models might cover a wide variety of conditions, looking at different areas of FLPs, one might arrive at different notions of what dynamic means. For example, as described before, dynamic models might have interaction with multi-period

According to what explained about changing parameters and uncertain decision variables in a FLP, it is common for a firm to consider relocating its facilities (modifying the current facility) over the planning time horizon because the main parameters of models may change as a consequence of many factors such as the interaction of customers and suppliers, distribution networks, business climate, and government legislation. From a general point of view, the relocation of a facility is primarily concerned with the time of relocation, the number of relocation, and the cost of relocation. In the case of relocation time, facilities’ relocation can be made at discrete or continuous time. For the former (discrete time), the relocation of one facility or several facilities is allowed in definite, predetermined, and discrete points of time; however, for the latter (continuous time), predetermined time points for relocation do not exist, and these relocations are permissible in any time during the planning horizon. Moreover, in the case of relocation number, there can be a single relocation called server, or multiple relocations (Farahani, Drezner, & Asgari, 2009). Finally, for relocation cost, it is obvious that this cost depends particularly on the current and future locations of that facility. Additionally, Min and Melachrinoudis (1999) presumed that the relocation decision of a facility is affected by many factors from which the most significant ones are as follows: (1) cost of land acquisition, zoning permits, building construction, moving equipment and personnel, etc., (2) accessibility and quick delivery to customers, (3) reachability to suppliers, (4) easy access to transportation networks, (5) tax incentives, (6) quality of labor, and (7) labor-management relations. Based on these factors, a decision-maker might come up with the following concerns in the relocation problem: in order to minimize cost, when and where

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to relocate? How to gradually phase-out the existing facility without violating firm’s activities? In a two-echelon entity, how to fulfill the company’s targets for short time periods? What is the time schedule for relocating capacity? And, how cost minimization can be compensated by reducing transit times and how to take advantage of local incentives? Finally, as one of the most important derivations of FLRPs, it should be noted that Emergency Medical Services (EMSs) applies this concept to ensure satisfactory response time to incidents. As research papers in this area, Alanis, Ingolfsson, and Kolfal (2010) can be pointed out analyzing an EMS system by a two-dimensional Markov chain model. In addition, Harewood (2002) worked on a case study of an ambulance deployment problem with multiple objective functions (cost minimization and customer’s coverage maximization), and Maxwell, Restrepo, Henderson, and Topaloglu (2010) proposed a decision making problem of ambulance redeployment solved by a dynamic programming method. 3.3. Multi-period (discrete time) vs. single-period (continuous time) facility location problems As already emphasized, decision makers should think about robust FLPs to be operable for a considerable time horizon which can be defined in terms of multi-period time horizons. As a matter of fact, by considering such a planning horizon, three achievements can be obtained: (1) the appropriate timing of location decision, (2) clarifying the best location(s), and (3) allowing a firm to better anticipate any favorable/unfavorable fluctuations in market demand in the corresponding time horizon, whereas single-period models (continuous time horizon) do not show such characteristics (Miller, Friesz, Tobin, & Kwon, 2007). Another advantage of multiperiod models over single-period models are its correspondence with dynamic models because in each subordinate planning horizon, a decision maker can deal with changing parameters more effectively in comparison with single-period models in which the decision maker is hardly able to cope with the uncertain essence of changing parameters (Hale & Moberg, 2003; Klose & Drexl, 2005; Melo, Nickel, & Saldanha-da-Gama, 2009; ReVelle & Eiselt, 2005; Sahin & Sural, 2007; Snyder, 2006). Based on Hormozi and Khumawala (1996), the opening/closing of facilities can incur no cost (theoretically) or bring cost in addition to fixed cost; meanwhile, the second status is more applicable. Regarding these two conditions, Albareda-Sambola, Fernández, Hinojosa, and Puerto (2009) addressed another interpretation to decide upon opening/closing of new/existing facilities. Therefore, based on the multi-period nature of a model, some new decision variables are added to the related problem such as transportation plan and time-staged establishment of facilities. Since multi-period models and location/relocation models, described in the previous section, have many common elements, many references considered them as a single model. For example, Wesolowsky and Truscott (1975) proposed a model in which both of the mentioned models were incorporated:

Minimize Z ¼

K X n X m X k¼1 j¼1

Subject to : m X xijk ¼ 1 8j; k i¼1 n X

xijk 6 nxiik

8i; k

i¼1

Aijk xijk þ

K X m X 

c0ik y0ik þ c00ik y00ik



ð36Þ

k¼2 i¼1

ð37Þ ð38Þ

j¼1 m X i¼1

xiik ¼ G 8k

ð39Þ

m X

y0ik 6 mk ;

for k ¼ 2; . . . ; K

413

ð40Þ

i¼1

xiik  xii;k1 þ y0ik  y00ik ¼ 0;

xijk

for k ¼ 2; . . . ; K

8 > < 1 if location node j gives service to demand ¼ node i in time span k; 8i; j; k > : 0 otherwise

ð41Þ

ð42Þ

8 > < 1 if a facility is closed at node i y0ik ¼ in time span k; 8i; k > : 0 otherwise

ð43Þ

8 > < 1 if a facility is erected at node i y00ik ¼ in time span k; 8i; k > : 0 otherwise

ð44Þ

In this model, there are m demand nodes (i = 1, . . . , m), n candidate locations (j = 1, . . . , n), and K periods (k = 1, . . . , K). In addition, the assignment of a facility at node j to demand at node i in period k incurs cost whose present value is Aijk; the removal and establishment of a facility from/at node i in period k incurs cost whose present value are c0ik and c00ik , respectively. Furthermore, the facility location can be changed at most mk times. Regarding the description of the model, Eq. (36) sets the objective function minimizing costs; meanwhile, it consists of three types of cost: (1) cost of assigning all facilities to their corresponding demand nodes, (2) the removal cost of facilities from their current nodes, and (3) the establishment cost of new facilities to their respective demand nodes. In addition, Eq. (37) clarifies that each facility at node j cannot serve more than one demand node. Constraint (38) identifies that each facility at node j is allotted to demand point i providing that point i is self-assigned. By Eq. (39), G self-assignments are made from a set of m demand nodes. Moreover, the number of sites quitted is constrained by inequality 40. Furthermore, Eq. (41) measures the cost of relocation based on the second part of Eq. (36) (all these formulations are considered in each period). 3.4. Time-dependent facility location problems In comparison with classical STFLPs, DFLPs have many benefits from which the existence of changing demands over time can be mentioned. In fact, based on the type of a company’s activities, demands may vary throughout the whole planning horizon. Regarding this condition, there are some commodities with daily or seasonal demand, for which the existence of dynamic models is quite necessary. As an instance, for an area with a dynamically increasing population, dynamic problems ought to be considered for different sections of that area such as hospitals and fire stations. As another example, in a tourist area where demand patterns have distinguishing fluctuations, if there are many different facilities such as stores and fire stations, two or more locations are supposed to be operable at any specific time (according to the availability of locations and the corresponding demand). As one of the landmark research subjected to the case of timedependent models, Drezner and Wesolowsky (1991) can be referred who proposed a modified version of the Weber problem in which not only demand points have a predictable manner (configured by demand weights), but also is the location of a facility likely to change one or more times during the planning horizon. Meanwhile, in comparison with primary approaches, the main contribution of their work is that their model conforms better to real applications such as seasonal demand. Generally, they put their emphasis on identifying two things: the time of changing a

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facility’s location and the new place for locating the facility in each time interval. According to what Drezner and Wesolowsky (1991) and Farahani, Drezner et al. (2009) discussed, the TDFLP can be represented as follows:

Minimize Z ¼ Fðx; yÞ ¼

Z

T

( m X

0

¼

m X i¼1

dðX j ; Pi Þ

)

Z

wi ðtÞ dðX j ; Pi Þ dt

ð52Þ

Z  Rk þ mk ð1  zk Þ P 0; 

8k

ð53Þ

1 if candidate node j includes a facility; 8j 0 otherwise

ð54Þ

i¼1

Z

T

wi ðtÞ dt

ð45Þ

0

yijk

T

wi ðtÞ dt;

qk zk P a

k¼1

xj ¼

In which F(x, y) is the cost in the finite period [0, T]; Moreover, the distance and weight parameters are alike those applied for the Weber problem. Furthermore, to find the solution of this objective function, it is required to incorporate constant weights (wi) into the model which is obtained as follows:

Wi ¼

K X

i ¼ 1; . . . ; m

ð46Þ

8 > < 1 if facility node j gives service to demand node i ¼ ðby scenario kÞ; 8i; j; k > : 0 otherwise ð55Þ

8 > < 1 if the maximum regret is minimized zk ¼ under a set including scenario k; > : 0 otherwise

8k

ð56Þ

0

As a result of this approach, the resulting model will be the Weber problem. 3.5. Stochastic, probabilistic, and fuzzy facility location problems As indicated before, dynamic models mainly address the timing of locating facilities in the planning time horizon; however, determining robust facility locations which conform to any customization of models’ parameters is one of the critical objectives of decision makers because in many facilities functioning over years or decades, the main parameters such as cost, demand, and delivery time are likely to be uncertain during the planning horizon. Therefore, models considering uncertainty would be of interest in such conditions. According to what Owen and Daskin (1998) proposed, this uncertainty might be come up by two reasons as either future conditions incur planning uncertainty or absence of knowledge for input parameters incurs the related uncertainty. To cope with these reasons, two approaches are exploited as follows: (1) Probabilistic(stochastic) approach in which variables and parameters are taking probability distributions. Rosenhead, Elton, and Gupta (1972) compared this condition to a risk situation in which the values of uncertain parameters, having probability distributions, are primarily known by decision makers. (2) Scenario planning(robust) approach in which a set of possible future values are taken into account for each parameter/ variable. Meanwhile, Rosenhead et al. (1972) categorized this condition as an uncertainty situation in which parameters are uncertain, and information about probabilities is even unavailable. Based on these two interpretations, it can be stated that FLPs dealing with uncertainties are classified as stochastic and robust FLPs (Snyder, 2006). As one of the most fundamental types of stochastic models, one can refer to the following stochastic problem applying a scenario planning approach (Chen, Daskin, Shen, & Uryasev, 2006; Owen & Daskin, 1998):

Minimize Z

ð47Þ

Subject to : n X xj ¼ F

ð48Þ

j¼1 n X

yijk ¼ 1;

8i; k

ð49Þ

j¼1

xj P yijk ; Rk 

8i; j; k

m X

n X

i¼1

j¼1

bk hik dijk yijk  V

ð50Þ

! ¼ 0;

8k

ð51Þ

In this model, there are m demand nodes (i = 1, . . . , m), n candidate locations (j = 1, . . . , n), and K possible scenarios (index k = 1, . . . , K means that the corresponding parameter has been considered under scenario k). Moreover, hik is the demand of node i; dijk is the distance from node i to candidate site j; there are F availb k is the minimum weighted distance acquired by able facilities; V scenario k (the best p-median value); qk is the probability of the scenario k; Rk is the regret obtained by scenario k, for which there is a large constant mk such that mk P Rk. Concerning the foregoing description, the a-reliable Minimax regret is minimized by Eq. (47). By Eq. (48), F facilities are exactly supposed to be located. By Eq. (49), any demand node cannot be linked to more than one facility (under each scenario). Constraint (50) determines that if a facility is not located at node j, it cannot satisfy the demand of node i under scenario k. Furthermore, the regret attributed to scenario k is defined by Eq. (51). Inequality (52) determines that the least possible probability of selecting scenarios must be a. Moreover, the maximum regrets are identified by inequality (53). Furthermore, as other model dedicated to the case of SFLPs, the work of Rosenthal, White, and Young (1978) can be mentioned in which the location and relocation of facilities were considered through stochastic processes:

Minimize Z ¼ E

" # a X fFðX t1 ; X t Þ þ GðX t ; At ÞgBt1

ð57Þ

t¼1

This objective tries to obtain the minimum expected present value of costs. Moreover, the location of a server and the location of a customer (at time t) are identified by Xt and At, respectively. N = {1, . . . , n} is the set of possible locations; Fand G are n  n matrices respectively denoting the server relocation cost and the service cost. (B stands for discount.) Moreover, demand nodes have a transition matrix denoted by P. Furthermore, this probabilistic process evolves as follows: (1) (Xt, At1) is observed, and then,Xt is chosen; (2) the relocations cost is measured in terms of f(Xt, At1); (3) the chance probability At is observed; and (4) the service cost g(Xt, At) is incurred. Finally, the application of fuzzy approaches in FLPs has been primarily divided into two categories: (1) the selection of facility location (a decision making problem) and (2) the location–allocation problem (an optimization problem). For the first category, the following strategies are generally applied (for more mathematical details, see Kahraman, Cebi, & Tuysuz, 2010): Fuzzy Analytic Hierarchy Process (AHP), Fuzzy TOPSIS (the technique for order preference by similarity to ideal solution) method, and Fuzzy information axiom.

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Furthermore, in the case of application of fuzzy logic in location–allocation problems, one can refer to the following case (Wen & Iwamura, 2008a):

Cðx; yÞ ¼ min z2Z

n X m X i¼1

zij

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi  aj Þ2 þ ðyi  bj Þ2

Subject to : n X zij ¼ nj ; 8j i¼1 m X

zij 6 ui ;

ð58Þ

j¼1

ð59Þ

8i

ð60Þ

j¼1

zij P 0;

8i; j

ð61Þ

In this model, there are n facilities and m customers; furthermore, (xi, yi) denotes the location of facility i(1 6 i 6 n); (aj, bj) stands for the location of customer j(1 6 j 6 m); facility i, whose capacity is ui, supplies zij units for customer j; and nj is the random fuzzy demand of customer j. Concerning this model, the objective function (Eq. 58) minimizes transportation costs. Eq. (59) identifies that the whole amount supplied by all facilities to a specific customer must satisfy his/her demand. Moreover, constraint (60) reveals that the whole amount supplied by a facility cannot be greater than its capacity. Now, after addressing the foregoing types of DFLPs, Table 1 summarizes a group of papers and publications applied for each of these problems. Regarding this classification, it should be noted that the corresponding papers have been sorted based on the year of their publication. 4. The classification of optimization methods and methodologies The purpose of this section is not only to give an overview of the types of applied objective functions measuring the performance of dynamic models and other relevant areas, but also to give an

insight of the types of methodologies that have been applied for solving these models. In the case of performance measures, the majority of papers exploit a cost minimization objective consisting of operational costs, transportation costs, material handling costs, etc. On the other hand, profit maximization is another objective that has received less attention in the literature. Furthermore, the last category belongs to several (and possibly incompatible) objectives that are considered as either multi-criteria or multiobjective optimization problems. Table 2 classifies the literature by their applied objective function/performance measure. Furthermore, in the case of solution methodologies, two general classes are offered: general applied approaches and specially configured approaches, within each of which two further parts are considered. For example, a mathematical programming approach solving a problem to optimality or near optimality is categorized as ‘‘general method, exact approach’’. Otherwise, if any solution approach does not find a feasible solution within 5% of the optimal solution, that will be considered as ‘‘general method, heuristic approach’’. This procedure is the same for the second general class for which two categories are also considered: ‘‘specific algorithm, exact approach’’ and ‘‘specific algorithm, heuristic approach’’. Table 3 categorizes the literature according to their applied methodologies.

5. Applications and case studies In this section, some applications of DFLPs are addressed through Table 4. According to this table, two measures are considered to classify these applications: the type of context and the type of industry. In the case of context, two sub-classes are further studied: (1) case studies referring to real-world applications of dynamic models in industries and (2) industrial context standing for a study being theoretically applied for a specific industry. In the case of the type of industry, it should be observed that an industry for which the corresponding application comes from is studied. Even though the categorization of various applications is presented in this section, some potential areas of dynamic models

Table 1 A classification of literature in DFLPs. Type of static problems

Article

DDFLP

Erlenkotter (1981) Shulman (1991) Daskin et al. (1992) Saldanha-da-Gama and Captivo (1998)

Puerto and Rodriguez-Chia (2006) Miller et al. (2007) Behmardi and Lee (2008) Thanh et al. (2008)

Abravaya and Berend (2009) Gebennini et al. (2009) Bozkaya et al. (2010)

FLRP

Frantzeskakis and Watson-Gandy (1989) Emamizadeh and Farahani (1997) Min and Melachrinoudis (1999) Melachrinoudis and Min (2000) Gendreau et al. (2001)

Brotcorne et al. (2003) Wang et al. (2003) Melo et al. (2006) Dias et al. (2006) Dias et al. (2007a, 2007b)

Dias et al. (2008) Hinojosa et al. (2008) Farahani, Drezner et al. (2009) Koontz et al. (2009) Naraharisetti and Karimi (2010)

MPFLP

Sherali (1990) Melachrinoudis et al. (1995) Chardaire et al. (1996) Antunes and Peeters (2000) Hinojosa et al. (2000) Antunes and Peeters (2001)

Canel et al. (2001) Gue (2003) Romeijn and Morales (2004) Gen and Syarif (2005) Manzini and Gebennini (2008) Rajagopalan et al. (2008)

Acar et al. (2009) Albareda-Sambola et al. (2009) Mahar et al. (2009) Alanis et al. (2010)

TDFLP

Batta (1989) Drezner and Wesolowsky (1991)

Drezner and Wesolowsky (1996) Averbakh et al. (1998)

Averbakh et al. (2007)

SFLP/PRFLP

Chan et al. (2001) Killmer et al. (2001) Alonso-Ayuso et al. (2003) Averbakh (2003) Aghezzaf (2005)

Romauch and Hartl (2005) Gabor and Van Ommeren (2006) Snyder et al. (2007) Baron et al. (2008) Schütz et al. (2008)

Lee and Jeong (2009) Lin (2009) Yao et al. (2010) Albareda-Sambola et al. (2011) Wang et al. (2011)

FFLP

Yang et al. (2007) Chou et al. (2008) Liu (2008) Wen and Iwamura (2008b)

Bashiri and Hosseininezhad (2009) Esnaf and Küçükdeniz (2009) Liu and Tian (2009) Vahidnia et al. (2009)

Lau et al. (2010) Moheb-Alizadeh et al. (2011) Mokhtarian (2011) Wen and Kang (2011)

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Table 2 A classification of literature by their performance measures. Type of objective function

Article

Minimization (cost, time, distance, risk)

Erlenkotter (1981) Batta (1989) Frantzeskakis and Watson-Gandy (1989) Drezner and Wesolowsky (1991) Shulman (1991) Chardaire et al. (1996) Drezner and Wesolowsky (1996) Emamizadeh and Farahani (1997) Averbakh et al. (1998) Saldanha-da-Gama and Captivo (1998) Antunes and Peeters (2000) Hinojosa et al. (2000) Canel et al. (2001) Chan et al. (2001) Killmer et al. (2001) Gue (2003) Romeijn and Morales (2004)

Averbakh et al. (2007) Dias et al. (2007a, 2007b) Snyder et al. (2007) Baron et al. (2008) Chou et al. (2008) Hinojosa et al. (2008) Manzini and Gebennini (2008) Schütz et al. (2008) Thanh et al. (2008) Wen and Iwamura (2008b)

Maximization (profit, availability of services)

Gendreau et al. (2001) Alonso-Ayuso et al. (2003) Brotcorne et al. (2003)

Miller et al. (2007) Behmardi and Lee (2008) Bozkaya et al. (2010)

Naraharisetti and Karimi (2010)

Multi-criteria/multi-objective

Sherali (1990) Melachrinoudis et al. (1995) Min and Melachrinoudis (1999) Melachrinoudis and Min (2000)

Yang et al. (2007) Dias et al. (2008) Bashiri and Hosseininezhad (2009) Vahidnia et al. (2009)

Moheb-Alizadeh et al. (2011) Mokhtarian (2011) Wang et al. (2011)

have not received much attention. Based on what Melo, Nickel, and Saldanha-da-Gama (2006) proposed, there could be some reasons for the lack of more applications: (1) decision makers are not possibly permitted to reveal the available data of a company; (2) decision support systems have difficulty in dealing with quantitative

Aghezzaf (2005) Gen and Syarif (2005) Romauch and Hartl (2005) Dias et al. (2006) Gabor and Van Ommeren (2006) Melo et al. (2006)

Abravaya and Berend (2009) Acar et al. (2009) Albareda-Sambola et al. (2009) Esnaf and Küçükdeniz (2009) Farahani, Drezner et al. (2009) Gebennini et al. (2009) Lee and Jeong (2009) Lin (2009) Liu and Tian (2009) Mahar et al. (2009) Alanis et al. (2010) Lau et al. (2010) Yao et al. (2010) Albareda-Sambola et al. (2011) Wen and Kang (2011)

models for which there is no specific method for implementing these models in strategic planning; (3) based on the type of company, collecting data is relatively hard, or even no data is available; and (4) preparation and aggregation of obtained data are difficult and time-consuming based on the type of an industry.

Table 3 A classification of literature by their methodologies. Type of methodology

Article

General method/solver Exact Batta (1989) Sherali (1990) Shulman (1991) Melachrinoudis et al. (1995) Emamizadeh and Farahani (1997) Averbakh et al. (1998) Antunes and Peeters (2000) Hinojosa et al. (2000) Melachrinoudis and Min (2000) Killmer et al. (2001)

Alonso-Ayuso et al. (2003) Brotcorne et al. (2003) Gue (2003) Aghezzaf (2005) Melo et al. (2006) Miller et al. (2007) Baron et al. (2008) Hinojosa et al. (2008) Manzini and Gebennini (2008) Schütz et al. (2008)

Thanh et al. (2008) Acar et al. (2009) Bashiri and Hosseininezhad (2009) Farahani, Drezner et al. (2009) Gebennini et al. (2009) Lee and Jeong (2009) Mahar et al. (2009) Naraharisetti and Karimi (2010) Moheb-Alizadeh et al. (2011)

Frantzeskakis and Watson-Gandy (1989) Min and Melachrinoudis (1999) Behmardi and Lee (2008)

Albareda-Sambola et al. (2009) Vahidnia et al. (2009) Alanis et al. (2010)

Albareda-Sambola et al. (2011) Mokhtarian (2011)

Drezner and Wesolowsky (1991) Chardaire et al. (1996) Drezner and Wesolowsky (1996) Saldanha-da-Gama and Captivo (1998) Canel et al. (2001) Chan et al. (2001)

Gendreau et al. (2001) Gen and Syarif (2005) Averbakh et al. (2007) Snyder et al. (2007) Yang et al. (2007) Wen and Iwamura (2008b)

Abravaya and Berend (2009) Esnaf and Küçükdeniz (2009) Lau et al. (2010) Wen and Kang (2011)

Erlenkotter (1981) Antunes and Peeters (2001) Romeijn and Morales (2004) Romauch and Hartl (2005) Dias et al. (2006)

Gabor and Van Ommeren (2006)

Lin (2009) Liu and Tian (2009) Bozkaya et al. (2010) Wang et al. (2011) Yao et al. (2010)

Heuristic

Specific algorithm Exact

Heuristic

Dias et al. (2007a, 2007b) Chou et al. (2008) Dias et al. (2008)

A. Boloori Arabani, R.Z. Farahani / Computers & Industrial Engineering 62 (2012) 408–420 Table 4 Applications of DFLPs. Article

Killmer et al. (2001) Yao et al. (2010) Antunes and Peeters (2000, 2001) Gendreau et al. (2001), Harewood (2002), Brotcorne et al. (2003), Alanis et al. (2010) Manzini and Gebennini (2008) Mamada et al. (2005)

Type of context Case study p

Gebennini et al. (2009) Melachrinoudis et al. (1995) Gue (2003) Min and Melachrinoudis (1999), Melachrinoudis and Min (2000) Schütz et al. (2008) Mokhtarian (2011) Koontz et al. (2009) Chardaire et al. (1996), Taheri and Zomaya (2007) Esnaf and Küçükdeniz (2009) Bozkaya et al. (2010)

Industrial context

p p p

p p p p

Yang et al. (2007) Vahidnia et al. (2009) p p

p p

p p p p p p

Area

Albany, New York transportation Chemistry industries Portuguese schools Emergency medical services

Electronic industries Evacuation Management Fire station systems Healthcare (Hospital) systems Italian electronics industries Landfill industries Military North-American manufacturing/ distribution system Norwegian meat industries Oil industries Public library policies Telecommunication Turkeys road industries Turkeys transportation

6. Future trends and prospects In order to conduct further studies in the case of dynamics of FLPs and based on available gaps in the literature, this section delivers some research trends. One of the areas requiring more attention is how to combine continuous models with dynamic problems. At the first glance, it might be perceived that the essence of dynamic problems is based on discrete models while the interaction of continuous models with dynamic problems can facilitate favorable results as discussed by Fleischer and Tardos (1998), Klose and Drexl (2005), and Suzuki and Drezner (2009). Furthermore, dynamic location problems can be studied in supply chain management (SCM) (Melo et al., 2009) from four aspects:  Since SCM generally deals with collaborating domestic and international companies, the location problem in SCM has been considered as one of the most influential issues. As a consequence, the dynamic types of FLPs (e.g. FLRP, SFLP, MPFLP, etc.) can follow this trend as well; meanwhile, it should be noted that the relocation decision (as one type of these dynamics) of a facility should be in accordance with the relocation decision of other related, yet differently owned facilities.  As one of the main elements of SCM, supply chain networks can be mentioned. Since these networks are highly applicable in real-world problems, the implementation of these dynamics needs to be further studied by NFLPs. In fact, NFLPs not only can be integrated with SFLPs, but also might be analyzed from time/demand point of view, for which TDFLPs can be inserted in network models.

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 In order to effectively handle fluctuating demands received from miscellaneous customers, industries and companies are rapidly moving towards handling of several facilities instead of a single facility. Even if some papers have dedicated their subject to dynamic multi-facility location problems (Bashiri & Hosseininezhad, 2009; Chan, Carter, & Burnes, 2001; Current, Ratick, & ReVelle, 1997; Ghiani, Guerriero, & Musmanno, 2002; Pirkul & Jayaraman, 1998), more efforts should be made to incorporate multiple facilities in such problems.  Since over the recent decade, many destructive happenings have been occurred in the world, disaster management is another area that in the case of applications and case studies could be addressed well by the literature. In fact, this category can be divided into two sections: natural disaster (e.g. earthquake, hurricane, Tsunami, etc.) or man-made disaster (e.g. war, terrorist attack, etc.), for which one the most applicable problems is the FLRP. Regarding this problem, a state-of-theart approach can be to implement the effect of disaster on the related problem. In this case, Snyder, Scaparra, Daskin, and Church (2006) presented an application of STFLPs in disaster and supply chain disruption; therefore, it can be generalized to dynamic circumstances. Finally, in the case of performance measures (applied methodologies) and proposed solution approaches, three points should be thrown light on:  Concerning performance measures, it seems be necessary to apply new objective functions rather than ordinary measures such as cost/distance minimization and profit maximization. In fact, the vast majority of research works have put their emphasis on these two measures; meanwhile, other performance measures have not been considerably applied. As a matter of fact, practical models, dealing with real-world problems, have captured more attention in recent years because their application can bring more benefits for companies. Hence, the existence of more multi-criteria and multi-objective problems, with more conflicting objectives, might be required. For example, as a novel implementation of non-traditional objective functions, Chen (2007) developed a multi-objective optimization problem for a dynamic model in which two objective functions were taken into account: the minimization of the distance-based objective and the maximization of the adjacency-based objective which are quite unique and has been rarely addressed in the literature. From another point of view, as the main parameters of dynamic problems change over the planning time horizon, they are supposed to fulfill robustness and reliability criteria; therefore, one of the primary performance measures can be a risk percentage incurred on the concerned facility by the related model. For more novel objective functions in a multi-objective facility location optimization problem, one can refer to Farahani, SteadieSeifi, and Asgari (2010).  Traditionally, economic aspects of DFLPs are just considered by the literature (as described above). However, sustainability is another emerging concept comprising social and environmental criteria other than economic ones. Therefore, sustainable facility location and consequently, sustainable dynamic facility location can be taken into account for having sustainable locations. Regarding this fact, a trade-off must be set between handling changing parameters of the problem and maintaining the sustainability of the location decision.  In the case of proposed solution approaches, it should be noted that those approaches finding exact solutions, either general solvers or specific algorithms, have not been considerably modified with the passage of time. Since the inherent complexity of

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FLPs (specifically dynamic problems) has been troublesome, existing methods are no longer effective in dealing with these problems (particularly large-scale problems). Hence, methods bringing heuristic or metaheuristics solutions have been initiated and established. For this category of methodologies, one can refer to approaches proposed by the literature such as genetic algorithm (GA), simulated annealing (SA), ant colony optimization (ACO), and particle swarm optimization (PSO). However, they may not be sufficient in dealing with dynamic problems, and hence, more state-of-the-art hybrid techniques can be applied in upcoming research works. 7. Conclusions In this review paper, it has been attempted to prepare a passage of literature of DFLPs and other pertinent concepts particularly the most recent literature on these cases. For this purpose, all reviewed papers have been classified according to their field of study. In addition, objective functions (interpreted as performance measures) and optimization models and techniques have been analyzed and categorized based on their exact or heuristic essence. Furthermore, some applications have been discussed based on corresponding case studies and industrial contexts. Moreover, according to what have been gathered in this review paper and also available gaps in the literature, some possible trends for future works have been presented, for which practitioners and researches can come up with novel and contributing achievements. Finally, based on the whole contents of this review paper, a primary conclusion can be made. Dynamic models have a leading role in strategic planning and decision making for FLPs, and they are gradually becoming more important due to their vast applicability. Hence, in order to fulfill many necessities of real-world problems concurrently, these models are supposed to be provided with more comprehensive and flexible approaches. In other words, in comparison with what have been addressed in this review paper, there are still possible opportunities by which new models (as well as solution techniques) can be advanced and developed. Acknowledgement The authors are grateful for the three anonymous referees for their constructive comments which improved the paper at all. References Abravaya, S., & Berend, D. (2009). Multi-dimensional dynamic facility location and fast computation at query points. Information Processing Letters, 109(8), 386–390. Acar, Y., Kadipasaoglu, S. N., & Day, J. M. (2009). Incorporating uncertainty in optimal decision making: Integrating mixed integer programming and simulation to solve combinatorial problems. Computers and Industrial Engineering, 56(1), 106–112. Aghezzaf, E. (2005). Capacity planning and warehouse location in supply chains with uncertain demands. Journal of the Operational Research Society, 56, 453–462. Akyuz, M. H., Oncan, T., & Altnel, I. K. (2009). The multi-commodity capacitated multi-facility Weber problem: Heuristics and confidence intervals. In Proceedings of the international multiconference of engineers and computer scientists (Vol. 2). Alanis, R., Ingolfsson, A., & Kolfal, B. (2010). A Markov chain model for an EMS system with repositioning. . Albareda-Sambola, M., Fernández, E., Hinojosa, Y., & Puerto, J. (2009). The multiperiod incremental service facility location problem. Computers and Operations Research, 36(5), 1356–1375. Albareda-Sambola, M., Fernández, E., & Saldanha-da-Gama, F. (2011). The facility location problem with Bernoulli demands. Omega, 39(3), 335–345. Alonso-Ayuso, A., Escudero, L. F., Garin, A., Ortuno, M. T., & Perez, G. (2003). An approach for strategic supply chain planning under uncertainty based on stochastic 0-1 programming. Journal of Global Optimization, 26(1), 97–124. Antunes, A., & Peeters, D. (2000). A dynamic optimization model for school network planning. Socio-Economic Planning Sciences, 34(2), 101–120.

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