The capacitated hub covering location-routing problem for simultaneous pickup and delivery systems

Accepted Manuscript The capacitated hub covering location-routing problem for simultaneous pickup and delivery systems Hossein Karimi PII: DOI: Reference:

S0360-8352(17)30587-9 https://doi.org/10.1016/j.cie.2017.12.020 CAIE 5024

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

3 June 2017 27 October 2017 19 December 2017

Please cite this article as: Karimi, H., The capacitated hub covering location-routing problem for simultaneous pickup and delivery systems, Computers & Industrial Engineering (2017), doi: https://doi.org/10.1016/j.cie. 2017.12.020

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The capacitated hub covering location-routing problem for simultaneous pickup and delivery systems Hossein Karimi Corresponding author: Assistant professor, Department of Industrial Engineering, University of Bojnord, Email: [email protected] , Postal Code: 94531-55111, Tel.: +98 584 2284611; fax: +98 584 2410700.

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The capacitated hub covering location-routing problem for simultaneous pickup and delivery systems Abstract In this study, a specific type of hub network topology, called hub location-routing, is presented in which the routes between the nodes assigned to a hub form a tour in this topology. The model minimizes the total cost of hub location and vehicle routing, subject to predefined travel time, hub capacity, vehicle capacity, and simultaneous pickups and deliveries. A polynomial-size mixed integer programming formulation is introduced for the single allocation type of the problems. In this paper, a set of valid inequalities is proposed for the formulation. In addition, a tabu-search based heuristic is suggested which determines the hub location and vehicle routes simultaneously. Series of computational tests are then executed to evaluate the performance of valid inequalities and tabu-search based heuristic. The results show that using all valid inequalities improves the solution time of the pure proposed model. Meanwhile, the proposed heuristic works efficiently in finding good-quality solutions for the proposed hub location-routing model. Keywords: Hub network; vehicle routing problem; tabu search; covering; valid inequality

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1. Introduction Flows of goods, passengers, and information between origins and destinations (O-Ds) need a complex connected network. Hub network is designed for servicing these flows between multiple O-Ds as a kind of transportation networks. Hubs allow completely interconnected links to be replaced with fewer indirect links. Lower network construction costs, consolidation of flow handling, and sorting are the merits of these configurations. Moreover, hubs allow vehicles to benefit economies of scale by the unification of flows. In a general form, a hub-and-spoke problem involves (1) finding the optimal number of hubs, (2) deciding on their optimal locations, (3) and allocating non-hub O-Ds to the hubs (single or multiple allocation). Hub-and-spoke networks are pertinent to many different kinds of the transportation problem. Real-world examples of these networks include airline passenger travel (Shaw, 1993), package delivery systems (Chen, 2008), postal systems (Ernst and Krishnamoorthy, 1996), transportation system (Gelareh and Nickel, 2010), cargo delivery services (Tan and Kara, 2007), and telecommunication networks (Carello et al., 2004). At present, even small and medium companies must be conscious that the future profit, market share, and other plausible factors with this nature may be contingent on the location and distribution of decisions (for an illustrative example, see (Wasner and Zapfel, 2004; Yang, 2009)). Location-routing models handle these issues when customers with the lessthan-truckload demands are serviced from the routes making multiple stops (Belenguer et al., 2011). Moreover, the delivery/pickup cost to/from the customer for these systems is related to the routing of vehicles (Tuzun and Burke, 1999). Furthermore, the location-routing problem (LRP) integrates depot location and vehicle routing decisions. In LRPs, there is no flow between the nodes assigned to different facilities, like hub networks (i.e. depot to depot, customer to customer, depot to customer, and vice versa). In fact, in the hub-and-spoke network, hub and non-hub nodes play as depot and customer, respectively. Nagy and Salhi (1998) presented a hub network, which considered some links between nonhubs based on vehicle capacity, and was cheaper than hub-and spoke network. Considering vehicle capacity makes some tours in a network for each hub. The tour was a network topology that operated in a circular type in which the goods traveled around the tour in a specified direction. A hub location and routing formulation for less-than-truckload flows under time coverage bound and capacity of hub nodes were presented and called capacitated hub covering LRP (CHCLRP). 3

Wasner and Zapfel (2004) described the reasons and necessity of combining both the hub location as a strategic decision and the vehicle routing problem as a tactical/operational decision. In CHCLRP, pickup and delivery should be discussed, since commodities are picked up from non-hubs and sent to a hub; also, after sorting and some other plausible procedures, they are delivered to their destination. It means that, if the origin is before the destination in the same tour, the flows must go to the hub, since we suppose that flows should be sorted and scheduled at hub nodes. In this paper, it is supposed that the flows of the origin picked up in the previous process of gathering flows in the tour (e.g. one day before) are delivered to the destination, and the flows picked up in the current operation of the vehicle will be delivered in the next process of the vehicle (e.g. next-day). In fact, the model is dedicated to a warmed up transportation system, which is similar to the problem known as simultaneous pickup and delivery vehicle routing that has been greatly studied in recent years (as a clear-cut example, see Karaoglan et al., 2012). An example of a CHCLRP network topology is depicted in Figure 1. In this figure, hubs and non-hubs are shown by rectangles and cycles, respectively. {Insert Figure (1) here} Parcel and postal services are the real-world application of CHCLRP, where their services employ the hub location-routing networks (Çetiner et al., 2010; Kuby and Gray, 1993). Commonly, flow switching offices are scattered in a geographic area. Local pickup tours are used to collect the parcels or mails, instead of directly connecting them, which is very expensive. The parcels or mails are sorted, consolidated according to the destinations in a place. Then, they are sent to other places, where they are redistributed in order to be delivered to the destinations by delivery tours. The pickup and delivery tours can be happened at the same time by the same vehicle. The aforementioned description is the characterization of a specialized version of a CHCLRP network topology. In fact, this topology can be effective in reducing their operational and setup costs. In the described application, all the customers are delighted to use the service time up to a certain service time. Routing of vehicles from the viewpoint of the capacity of hub centers plays a significant role in the deliver with high speed in the promised time. In this study, after reviewing the related works in Section 2, first, a polynomial-size mixed integer programming (MIP) formulation is proposed in Section 3. Then, in Section 4, a group of polynomial valid inequalities is presented for strengthening the formulations as well as 4

improving the model size. In the next section, a heuristic approach based on tabu search (TS) is used, in which locating and routing phases are decided at the same time. The performance of the proposed formulations and the TS approach is evaluated in small- and medium-sized test problems taken from the literature in Section 6. Two phases are performed for the experimental study. The first phase evaluates the efficiency of valid inequalities in the CHCLRP formulation. This phase is executed on small-size problems. The second phase is dedicated to investigating the performance of the TS algorithm in small- and medium-sized test problems. The computational results point out that the proposed solution is highly competent

in obtaining good-quality solutions in reasonable computational time.

Additionally, valid inequalities perform efficiently. Finally, Section 7 concludes this research.

2. Literature review In this section, related works are briefly reviewerd after presenting some features of CHCLRP. Travel cost, which represents a significant fraction of expenses plays a momentous role in this network topology (Wasner and Zapfel, 2004). Beside travel costs of the routes, there are many factors related to the construction cost of hub networks, one of which is the location of hubs and another is the number of them. These two items act the special roles of collecting and distributing cost in a network (O’Kelly, 1986). In hub network literature, when this number is predetermined, the model is called p-hub. If the number of hubs is not predefined, the model's solution space is larger than a p-hub; so, it is hard to be solved. In this research, the number of hubs is unknown in the formulation, like Wasner and Zapfel (2004) and Nagy and Salhi (1998). The proposed model, in this study, decides about the number of hubs in addition to their location. Travel time is another important subject in hub network systems. For time sensitive flow, such as drug, meat, and perishable goods, travel time must be considered. Furthermore, it is applied to passenger flows if the dissatisfaction of passengers with a long trip is supposed to be reduced (Kara and Tansel, 2003). This issue called "hub covering" includes three coverage types (Campbell, 1994): time coverage (1) between O-Ds, (2) between origin and hub, hub and hub, as well as hub and destination distinctly (Karimi and Bashiri, 2001), and finally (3) just for hub to non-hub link. The first type is usually studied in the literature such as Ghodratnama et al., (2013) and employed in this research as well. Nonetheless, the second and third types are considered for special applications (as a living example, in the metro network) (Karimi and Bashiri, 2001). 5

Hubs can be (1) fully interconnected (Campbell, 1994), (2) constructed in a tree shape (Contreras et al., 2010), and (3) incompletely interconnected (Setak et al., 2013; Karimi and Setak, 2014). In the present research, hubs are fully interconnected. Moreover, travel time from origins to destinations (Tan and Kara, 2007), vehicle capacity (Çetiner et al., 2010), and capacity of hub nodes (Aykin, 1994) are the other factors in a hub network. CHCLRP is composed of two sub problems: the hub location problem and the vehicle routing problem with simultaneous pickup and delivery and covering constraints. Both of these problems have been investigated widely in the literature. Interested readers can study the review papers on hub location problems such as O’Kelly and Miller (1994), Alumur and Kara (2008) and Farahani et al., (2013) and simultaneously pickup and delivery vehicle routing problem such as Berbeglia et al., (2007). First, the literature related to solution approaches is presented in this section. To tackle the hub location-routing network, three heuristic approaches mentioned in the LRP literature can be used, which include (1) successive methods (Jacobson and Madsen, 1980), (2) iterative methods (Çetiner et al., 2010), and (3) integrated or simultaneous approaches (Wasner and Zapfel, 2004). Successive methods first determine the locations of hubs and then use the resulted data to determine a good solution for the remained routing problem. In this method, there are two strategies: locate-route and route-locate. Nevertheless, there is no feedback in this approach as well as iterative method. It is obvious that this method cannot always find the optimum solution, since location and routing problems are linked together. Iterative methods are repeated between a pure location problem and routing problem. In this approach, the result of a problem works as the input data for the other problem. Here, like successive methods, there is difference between locate-route and route-locate methods. The first approaches begin with location problems. The following problem is a routing problem, and the route-locate methods are vice versa. Despite the fact that the mentioned methods clearly differentiate between a location aspect and a routing aspect (i.e. each location or routing problem is entirely solved in an independent way). In the integrated or simultaneous methods, the choice of routes is the substantial ingredient of the location aspect (i.e. location and routing problems are nested). In CHCLRP, a simultaneous method is utilized which means the formulation integrates hub

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location phase and routing aspect into one polynomial-size MIP model and the solution algorithm combines these two phases together. Another approach is exact solution method to handle LRP models. A number of these methods for LRP have been discussed in the literature (Laporte and Norbert, 1981; Gourdin et al., 2000). Moreover, in this method, some valid inequality is used. A set of valid inequalities for CHCLRP is proposed to tighten the formulation. In the literature, some studies have more similar characteristics to CHCLRP. First, Nagy and Salhi (1998) proposed many to many locations-routing problem in which several customers can send commodities to others and flows between depots are permitted. The authors presented an MIP formulation for the problem. Additionally, a locate-route successive method was proposed to solve an instance with 249 customers in order to represent its performance. As the second related work, a case study concentrated on a medium-size Austrian parcel distribution company, which was closely related to the many to many locations routing problem (Wasner and Zapfel, 2004). This case study considered two levels of facilities and constructed the travel cost relying on the predefined number of vehicles; also, all the interhub flows were transshipped just by a central hub. The application points of view, more than the theoretical contributions, have been highlighted. Moreover, a nonlinear model was provided which seems to be specific to the studied organization. In the end, the authors reported 14.7% expense saving over the present condition by solving the problem by the proposed simultaneous heuristic. In this paper, almost like their research, CHCLRP mainly combines hub location problem, involved in strategic decisions, with a vehicle routing problem, involved in the tactical decisions. Çetiner et al. (2010) considered the integrated hub location and routing problem in postal delivery systems. They did not propose any formulation of the problem and just developed an iterative solution procedure. In the first step, postal hub nodes were determined and the other offices allocated to them were multiplied. The second step routed the offices which were related to hub regions (i.e. locate-route). In this paper, the Turkish postal delivery system data were used for a case study. As another example, Karaoglan et al. (2012) proposed a model seeking to minimize the total cost of location and routing by simultaneously locating multiple depots and planning the vehicle routes subject to pickup and delivery demand of each customer concurrently. Two MIP formulations of the problem were proposed (node- and arc7

based). Moreover, to strengthen the formulations, some valid inequalities were presented. To handle the model, the authors suggested a heuristic successive method that was a kind of locate-route strategy. The computational results for 360 test instances showed that using valid inequalities in the formulations not only made better lower bounds, but also increased the velocity of the solver. In a study which set out to formulated hub LRP, de Camargo et al. (2013) suggested a model for many-to-many LRP as a type of hub LRP. A Benders algorithm for tackling the classical Australian post data set was suggested, which was able to be decomposed into two smaller subproblems. Also, Rodríguez-Martín et al. (2014) studied a kind of many-to-many location routing problem. Each location in the network is a potential hub, and exactly predefined number of hubs should be installed. There are no fixed costs for using a location as a hub, and fully inter-connected hub network is considered. Moreover, At most a predefined number of customers can be assigned to a hub, and at each hub, there is exactly one vehicle for visiting the allocated customers on a route. Recently, Lopes et al. (2016) formulated and solve a variant of the hub LRP, which consists in allocating each tour to one hub and one hub to each tour. Moreover, a tour was used to interconnect all hubs. A variable neighborhood descent was used as a local search procedure. {Insert Table (1) here} There remain several aspects of hub LRP about which relatively little is known. In order to highlight the contributions of CHCLRP, Table 1 gives a synopsis of the contents of the mentioned more related articles and shows their similarities and differences to/from CHCLRP. It can be observed that this paper improves a unified approach to the CHCLRP. One contribution of this work is the consideration of service time constraint for any O-D in the location and routing decisions. This constraint may be either offered as delivery and pickup options or imposed by nodes. Furthermore, CHCLRP does not consider the scheduled time at hubs as an assumption. Also, a polynomial-size MIP is proposed for CHCLRP. Moreover, for practical problems, an integrated, heuristic method based on TS is presented, since integrated or simultaneous methods promise the highest quality of solutions (Wasner and Zapfel, 2004). In the next section, a modeling formulation develops the above-discussed ideas. This modeling framework is used to develop a mathematical model for CHCLRP. Moreover, some valid inequality is proposed to strengthen the formulation.

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3. Model formulation The CHCLRP can be defined as follows: Let G  ( N , A) be a complete directed network where N is a set of nodes and

A  (i, j ) : i, j  N  is the set of arcs. Each arc has

nonnegative distance, cost, and weight. In addition, triangular inequality is considered. An unrestricted fleet of homogeneous vehicles is available to serve the nodes. The model selects the optimum number of vehicles. The problem is to determine the locations of hubs, allocation of non-hubs to just one hub, and the corresponding vehicle routes with minimum total cost under the following requirements: (1) Each node is served by one vehicle; (2) Each vehicle is used at most by one tour; (3) The total vehicle load at any point of the tour is not more than the vehicle capacity; (4) Total pickup and total delivery load of the nodes allocated to a hub do not exceed the capacity of the hub; (5) The cost time from i to j via k and m does not exceed the time-bound. To formulate the CHCLRP, the following notations are considered: Decision variables: 1  x ik  0 

1 z ijk  0 i ykm

u ik v ik

gik

If the node i is allocated to hub at node k . xkk  1 indicates a hub is at node k Otherwise If a vehicle for the hub at node k moves directly from node i to node j Otherwise the flow transported between hubs at node k and m generated from noed i . The total amount of flow delivered on the route related to hub at node k just before having serviced node i The total amount of flow picked up on the route related to hub at node k that includes up to node i Total travel time remained after departing node i on the route related to the hub at the node Total travel time passed just before arriving at node i on the route related to the hub at node k k

qik

Let t ij , wij , and cij be travel time, flow, and cost between two nodes i and j, respectively.

n  N ; 0    1 is a discount factor,   1 is a delay factor for consolidation/distribution at hubs, Oi   wij is the quantity originated by node i, Di   w ji is the flow destined to jN

jN

node i, and  ,  , as well as T are the maximum capacity of a hub, vehicle, and maximum 9

time-bound of each O-D. Fk is the fixed cost of making a hub at node k and F  is the fixed cost of employing a vehicle. Moreover, M    tij acts as a tight big number. iN jN

i min    cij zijk    ckm ykm  iN jN kN

  F z iN kN

x

kN

kik

kN mN

(1)

  Fk xkk kN

1

ik

xik  xkk

y

mN

x iN

z

z

jN

(2)

i  N,k  N

(3)

i  N, k  N

(4)

kN

(5)

ijk

 xik

i  N , k  N , (i  k )

(6)

ijk

 x jk

j  N, k  N,( j  k)

(7)

jN , j  i

iN ,i  j

mN

 xkk

ik

x x

i   y mk  Oi xik   wij x jk

i km

iN

ik

Oi  xkk

kN

(8)

ik

Di  xkk

kN

(9)

uik  u jk  Di    zijk  (  D j  Di ) z jik

i  N , j  N , k  N , (i  j ), (i  k )

(10)

v jk  vik  O j    zijk  (  O j  Oi ) z jik

i  N , j  N , k  N , (i  j ), (i  k )

(11)

g ik  g jk  t ji  M  Mzijk  (M  t ji  tij ) z jik

i  N , j  N , k  N , (i  j ), (i  k )

(12)

q jk  qik  t ij  M  Mz ijk  (M  t ji  t ij ) z jik

i  N , j  N , k  N , (i  j ), (i  k )

(13)

u ik    ( Di  ) z ikk vik    (Oi  ) z kik g ik  M  (t ik  M ) z ikk ) qik  M  (t ki  M ) z kik u ik  Di xik

i  N , k  N , (i  k )

(14)

i  N , k  N , (i  k )

(15)

i  N , k  N , (i  k )

(16)

i  N , k  N , (i  k )

(17)

i  N , k  N , (i  k )

(18)

vik  Oi xik g ik  t ik z ikk qik  t ki zkik uik  vik  Di  xik

i  N , k  N , (i  k )

(19)

i  N , k  N , (i  k )

(20)

i  N , k  N , (i  k )

(21)

i  N , k  N , (i  k )

(22)

i  N , k  N , m  N , j  N , (i  k ), (i  j ), (m  j )

(23)

i  N, k  N

(24)

i  N, j  N, k  N

(25)

iN

iN

g ik  t km ykm  q jm  T

xik  0,1 z ijk  0,1 10

y km  0,1 u ik  0 vik  0 g ik  0 qik  0

k  N, m  N

(26)

i  N, k  N

(27)

i  N,k  N

(28)

i  N,k  N

(29)

i  N,k  N

(30)

where (i, j, k , m)  N . In this formulation, objective function (1) minimizes the network system cost, including distance dependent shipping cost, inter-hub transportation costs and fixed costs of vehicles and hubs. Constraints (2) ensure that each node should be supported by one hub. Constraints (3) declare that a node can be assigned to an established hub. Constraints (4) cause the model to create the hub link between two hub nodes. Constraints (5) forbid the model to make a node as a hub without any allocated node to it. Constraints (6) and (7) ensure that each node must be visited exactly once if it is allocated to a hub. Constraints (8) and (9) guarantee that the total input and output flows on any hub do not surpass its capacity. Constraints (10) and (11) are flow divergences for delivery and pickup shipment, respectively. Moreover, constraints (12) and (3) are similar to constraints (10) and (11); but, they are travel time inequalities. Constraints (10), (11), (12), and (13) remove subtours and are kinds of Miller-Tucker-Zemlin (MTZ) subtour elimination constraints (Miller et al., 1960). They are flow inequalities for delivery and pickup flows, respectively. Constraints (14)–(21) limit the

u ik , v ik , and gik variables. Note that the lower bounds of these variables

are delivery, pickup, and travel time, while their upper bounds for uik and vik are capacity and time-bound for

gik . Constraints (22) ensure that the whole load on any link does not exceed

the vehicle capacity. Constraints (23) are satisfying the time-bound constraint. Constraints (24), (25), and (26) indicate that the variables xik , z ijk , and y km are binary variables, while constraints (27), (28), (29), and (30) imply that the others are non-negative.

4. Valid inequalities In this section, four valid inequalities are introduced in order to strengthen the linear relaxation of our formulation. These inequalities remove some fractional solutions from the solution space. A stronger lower bound can be obtained for the problem using valid inequalities.

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x iN

ik



z

iN jN ,i  j

kN

ijk

(31)

Proposition 1. Constraints (31) are valid inequalities for CHCLRP. Proof. In any feasible solution, since at least a tour should be constructed, the minimum number of arcs (i.e. the right hand side of constraints (31)) to construct this tour for a hub node equals the number of the assigned nodes (i.e. the left hand side of constraints (31)). The second and third valid inequalities which bound the minimum number of tours in the total system is given in constraints (32) and (33). The validity of these inequalities for our formulation is shown in proposition 2. Karaoglan et al. (2012) employed similar valid inequalities for LRPs with simultaneous pickup and delivery. Proposition 2. Constraint (32) is a valid inequality for CHCLRP.

z iN kN



ikk

w iN jN

ij

k    y km kN mN

(32)



Proof. The number of tours (i.e. vehicles) must distinctly satisfy the minimum requirement of delivery and pickup flow. Since the hub network is interconnected, and flow which its origin and destination is hub must be removed from total flow of the network. These issues are shown in constraint (32). Another valid inequality is a special case of the classical subtour elimination of the travelling salesman problem (Dantzig et al., 1954). The constraints (33) do not allow any subtour with just two different nodes.

 z

ijk k N , k  i , k  j

 z jik   1

i  N , j  N , (i  j )

(33)

One of the properties of a CHCLRP is the minimum number of hub nodes (MNH). It is obvious that the number of established hubs should be greater than MNH. This property can be calculated according to the capacity and flow of the nodes. First, the capacity is sorted in a descending order, called "sortcap". The following procedure finds MNH. This property is applied to define the final inequality (34); this valid inequality performs excellently when

MNH  1 . MNH  1;

w iN jN

While{( MNH

ij

 sortcap

)  1;

MNH  MNH  1; }

k

12

x

k N

kk

 MNH

(34)

To be able to answer the CHCLRP for a real network size, TS as a heuristic approach is introduced in the next section.

5. Tabu-search based huristic In this section, for solving the CHCLRP model, a simultaneous heuristic solution approach is presented. This method is specially appropriate for solving realistic problems. TS is a method for solving combinatorial problems (Glover, 1989). TS explores the solution space beyond a local optimum. Adaptive memory is one of the main components of the TS. Also, it makes a more flexible search conduct and tries to prevent falling back to the previous local optimum. In reality, TS can discover high-quality solutions for the hub network (Skorin-Kapov and Skorin-Kapov, 1994). Moreover, TS has been used in the hub network studies such as Klincewicz (1992), Setak and Karimi (2014) and Abyazi-Sani and Ghanbari (2016). {Insert Figure (2) here} In addition, in TS, two components, called diversification and intensification, are carried out to equip the algorithm with a long-term memory in many cases. The diversification periodically redirects the search process toward optimum areas. The intensification, on the contrary, searches for more special promising areas of the solution space. To represent a solution describes the variables, a permutation-based representation is proposed. This permutation, after generating a feasible solution, can be an input of the objective function. The generation pseudo-code is explained in Figure 2. In real-world problems, fixed cost of establishing a hub is far more expensive than the travel cost between two nodes (see AP dataset from Ernst and Krishnamoorthy, 1996) and vehicle usage cost (http://prodhonc.free.fr/homepage). Based on these realities, in the presented pseudo-code for the generation of a solution from a permutation, the first number in a permutation is determined as the first hub. Then, the remainder makes tours for this hub until the capacity of the hub is overflowed. When capacity constraint of the hub is not satisfied, the current number in permutation constructs another hub. The other considered constraints (i.e. time-bound and vehicle capacity) affect the number of the required tours. An example of the presentation and the generated solution by the mentioned pseudo-code is depicted (see Figure 3). The generated solution in Figure 3 is shown in Figure 4. Components of Figure 4 are similar to those in Figure 1. 13

{Insert Figure (3) here} {Insert Figure (4) here} The pseudo-code of TS, developed for this research, is described in Figure 5. The TS parameters are set at the beginning of the algorithm. The while loop carries out the whole TS. The search begins with a random solution in which generation solution procedures are employed. In the sequential iterations, the permutation changes under a special interchange (i.e. swap and insertion). From a given permutation, the solution can be generated by proposing a generation pseudo-code. The network must be feasible by allocating at least one tour to hubs. {Insert Figure (5) here} Clearly, in practice, the search has to be stopped at some point. The most regularly used stopping criteria in TS are: after a fixed number of iterations (or a fixed amount of computational time); after some numbers of iterations without any upgrading in the objective function value (this criterion is utilized in most implementations); and when the objective arrives at a predefined threshold value. In this paper, the TS stops after a fixed number of iterations which is set at 20n (n is the number nodes) or 1000-second search. Swap and insertion are two of the most useful neighborhood strategies. In a swap, two places in a permutation are selected and the values of these places are interchanged together. However, in the insertion strategy, the value of the second place is put into the position before the first place. Using heuristics to solve the real-world problems affects the quality of the obtained solutions. In the absence of optimal solutions to be used as benchmarks, upper or lower bounds are needed. These lower bounds are required to be tight for the solution evaluation in roder to be accurate in our problem. The lower bounds in this research are gained by CPLEX solver's cut and the proposed valid inequalities.

6. Computational study and discussion The purpose of this section is to appraise the performance of valid inequalities and TS solution approach.

6.1. Test instances In the computational experiments, two datasets of benchmark instances usually employed in the literature, called Australian post (AP) and Turkish network (TN), are used. AP is downloaded from the renowned OR-Library (http://people.brunel.ac.uk/~mastjjb/jeb/info.html). AP is derived 14

from a real application to a postal delivery network, which describe postcode districts, the direction of their coordinates, and flow volumes (mail flow); in addition,   0.75 . The values of the collection and distribution coefficients are set at 1. A feature of this data is that the flow matrix is not symmetrical (i.e. Wij  W ji and Wii  0 ). Moreover, the AP dataset contains capacities and fixed costs at the nodes. Two types of fixed costs and capacities: tight (T) and loose (L), are considered. Hence, for each benchmark, four types of problems (i.e. LL, LT, TL, and TT) are used. Tan and Kara (2007) introduced TN as the benchmark for hub location problems. It included the fixed cost of opening hub node; but, the capacity of the nodes was not embedded in it. Loose and tight capacities of the hubs are set according to

i 

n(Oi  Di ) n(Oi  Di ) and i  for TN. We also consider   0.75 like AP. The n node 2 4

subsets are generated by taking the top n×n sub-matrices. Moreover, the flow matrix is divided by 10000 in TN. The instances from this dataset are known by their node number and capacity type of hubs. Since the AP and TN datasets do not consider time-bound, vehicle fixed cost, and vehicle capacity, T  max (  t ij ) , F   i

jN

F iN

i

n2

and   unifrnd (max (  wij ),   wij ) (where i

jN

iN jN

unifrnd (,) generates a uniform random between and) is taken into account.  is a set at

1.25 for the instances. CPLEX solver (version 12) is applied in this study and run by a machine equipped with Intel(R) Core(TM)2 with 2.53 GHz as the processor and 3 GB of RAM.

6.2. Valid inequality performance In the second part of the computational study after introducing the datasets, comparisons among valid inequalities are made. To this end, the AP and TN datasets with n  10 and n  5,7,10 are used, respectively. Six forms of CHCLRP are implemented with respect to the valid inequalities. The first form is relied just on pure CHCLRP, called pure. Valid inequalities (31), (32), (33), and (34) are added to CHCLRP and run in the 2nd, 3rd, 4th, and 5th forms, respectively. Finally, CHCLRP with valid inequalities (31)-(34) is set as the last form. The results are given in Tables 2-4 for each instance. The performances of the model (without using valid inequalities) and the set of valid inequalities are evaluated with the gap, time, and nodes used in the branch and bound tree. Gap is the difference between the optimal value found and the value of linear relaxation at the root node of the CPLEX solver, divided by the optimum 15

value. The CPLEX maximum running time is limited to one hour for each instance. When the computational time exceeds the time limit, it is around 3600 s in the result tables. A more systematic and itemized view of the results is described in Tables 2-4. Table 2 represents the performances of the implementations regarding the computational time. Table 3 gives the performances of the implementations in terms of gap. Finally,Table 4 shows the performances of the implementations in terms of the number of nodes explored in the tree search. {Insert Table (2) here} {Insert Table (3) here} {Insert Table (4) here} An inclusive overview of the benchmark results running on some of the AP and TN smallsize instances are provided in Figure 6, in which the horizontal axis plots the implementations, while the vertical axis plots the average gap, time, and number of nodes used in the branch and bound tree, respectively from up to down. Using ANOVA test for the gap results described in Table 2 shows that all the implementations, except using valid inequality (31), have meant significantly different from using all valid inequalities with the 95% confidence level.

The detailed results of the

ANOVA test for the gap performance is provided in Table 5. The ANOVA test for solution time and the explored nodes in the branch and bound tree show that there is no significant difference between all the implementation strategies with the 95% confidence level. However, as demonstrated in Figure 1, the average solution time and explored node in branch and bound tree for using valid inequalities (32) and all valid inequalities are less than the other implementation strategies. Therefore, using all valid inequalities (i.e. a column called (31)-(34)) almost gives a tighter and faster formulation than any other implementations). {Insert Table (5) here} To portray a good picture of the solution network in all the instances, it is provided for the small-sized problems in Figures 7, 8, 9, and 10. These figure definitions are the same as those of Figure 1. The results show that the tight capacity needs more hub and hub link than the loose one. Furthermore, it can be concluded from Table 2 that its objective function value is much more than the loose capacity problem. Moreover, tightening the fixed cost does not influence the result of a loose fixed cost problem in AP results. However, due to the value of fixed cost in the tighter problem, the objective function values are much more than those of 16

the loose problems.

6.3. Evaluating the TS performance In this subsection, TS solution approaches are executed 10 different times for small- and medium-sized instances and the average values are reported. The optimal solution is achieved for the small size (i.e. for AP and TN n  10 , respectively) in reasonable computational time, since we compare TS with the optimal solution. The small-sized problems were defined by Gelareh (2008), while, for the medium-size problem mentioned former, a different strategy is used. Table 6 shows the detailed results. {Insert Figure (6) here} {Insert Table (6) here} The AP and TN datasets with n  20,25 and n  15,20,25,30 are employed as some instances for medium-sized problems, respectively. The size of medium problems was defined by Gelareh (2008) Table 7 summarizes the computational results of the TS and CPLEX on medium-sized instances. For these instances, the machine spends much more time on obtaining the exact solution. Hence, first, TS is run; then, the obtained solution time is set for CPLEX. Now, we can compare TS and CPLEX, since an equal state of their solution time is reported in the column titled CPU time in Table 7. {Insert Table (7) here} The gap in Table 6 is the difference between the optimal value found and the value of TS, divided by the optimal value. Moreover, the gap in Table 7 is the difference of CPLEX lower bound (LB) and CPLEX/TS, divided by the lower bound. The lower bounds are achieved using all valid inequalities in the set solution time with CPLEX. Biter is the iteration where the TS finds a good solution and Miter is the maximum iteration of TS. {Insert Figure (7) here} {Insert Figure (8) here} {Insert Figure (9) here} Below, more discussions about the results shown in Tables 6 and 7 are presented. As seen in Table 6, the gap for total small-sized instances is zero. It means that the TS is able to solve all the employed small-sized instances optimally. However, the average for CPU time of TS is 17

11.495 s greater than that of the CPLEX. These results also show that TS yields the optimum solutions to small-sized problems, but its computational time is not at the level of the CPLEX. {Insert Figure (10) here} Moreover, as can be seen from Table 6, averagely, 16% of iterations in TS explore the solution space with improving the solution and find the better solution. It means that the TS converges almost quickly to the best solution for small-sized problems. It is apparent from Table 6 that, although the average gap for the CPLEX is 63.78%, for TS, it is reduced to 24.13%. The improvement produced by the TS is 39.65% in the mid-sized instances, while the CPLEX and TS employ the same computational time. In addition, according to Table 7, averagely 40.43% iterations are improved in seeking a good solution. In other words, in comparison with small-sized instances, TS in mid-sized problems needs more time to converge. In addition, for mid-sized instances, CHCLRP is hard to solve to optimality. Using ANOVA test for the gap results described in Table 7 shows that the means of CPLEX and TS are significantly different with the 95% confidence level (see Figure 11 for detail result). It is concluded from Tables 6 that TS solution is the same as CPLEX with the same quality. In addition, from Table 7, it is concluded that the proposed TS for mid-sized problems can find better quality solutions than CPLEX in the same CPU time for solving the CHCLRP. {Insert Figure (11) here}

7. Conclusion and future research In this study, CHCLRP was proposed over a complete hub network. In contrast to other hub location problems, CHCLRP integrated hub location and simultaneous pickup and delivery vehicle routing problem. A polynomial-sized MIP formulation of the problem was also presented. In order to tighten the model, the set of valid inequalities was suggested. Furthermore, a TS was introduced. To evaluate the performance of the valid inequalities and the TS algorithm, the sets of the experiments were performed, which were extracted from the literature. The results revealed that the incorporation of valid inequalities into the CHCLRP not only provided better lower bounds, but also reduced computational time. Outcomes of the 18

computational study also showed that, for all the test instances, the proposed TS almost performed efficiently in terms of solve the CHCLRP. In fact, in the small-sized problems, the TS yielded the optimal solution; in the mid-sized instances, the TS worked better than CPLEX solver in the same computational time. The future works may concentrate on the uncertainty environment for the CHCLRP. Moreover, in public transportation, the hub location routing model should be differently formulated and discussed. Furthermore, for some applications in commodity transportation, more than two levels in the hub location-routing network should be considered. Also, the future research may handle the congestion of hubs and the flow between them. Acknowledgement The author appreciate the anonymous referees for their constructive criticism that helped considerably to improve this article. References Abyazi-Sani, R., & Ghanbari, R. (2016). An efficient tabu search for solving the uncapacitated single allocation hub location problem. Computers & Industrial Engineering, 93, 99-109. Alumur, S., & Kara B. Y. (2008) Network hub location problems: The state of the art. European Journal of Operational Research, 190, 1-12. Aykin, T. (1994) Lagrangian relaxation based approaches to capacitated hub-and-spoke network design problem. European Journal of Operational Research, 79, 501-523. Belenguer, J. M., Benavent, E., Prins, C., Prodhon, C. & Calvo, R.W. (2011) A branch-and-cut method for the capacitated location-routing problem. Computers & Operation Research, 38, 931-941. Berbeglia, G., Cordeau, J. F., Gribkovskaia, I. & Laporte, G. (2007) Static pickup and delivery problems: a classification scheme and survey. TOP, 15, 1-31. Campbell, J. F. (1994) Integer programming formulations of discrete hub location problems. European Journal of Operational Research, 72, 387-405. Carello, G., Della Croce, F., Ghirardi, M. & Tadei, R. (2004) Solving the hub location problem in telecommunication network design: A local search approach. Networks, 44, 94-105. Çetiner, S., Sepil, C. & Süral, H. (2010) Hubbing and routing in postal delivery systems. Annals of Operations Research, 181, 109-124. Chen, H., Campbell, A. M. & Thomas, B. W. (2008) Network design for time-constrained delivery. Naval Research Logistics, 55, 493-511. 19

Contreras, I., Fernandez, E. & Marin, A. (2010) The tree of hubs location problem. European Journal of Operational Research, 202, 390-400. Dantzig, G. B., Fulkerson, D. R. & Johnson, S. M. (1954) Solution of a large scale traveling salesman problem, Operations Research, 2, 393-410. De Camargo, R. S., De Miranda, G. & Løkketangen, A. (2013) A new formulation and an exact approach for the many-to-many hub location-routing problem. Applied Mathematical Modelling, 37, 7465-7480. Ernst, A. T. & Krishnamoorthy, M. (1996) Efficient algorithms for the uncapacitated single allocation p-hub median problem. Location Science, 4, 139-154. Farahani, R. Z., Hekmatfar, M., Arabani, A. B. & Nikbakhsh, E. (2013) Survey: Hub location problems: A review of models, classification, solution techniques, and applications. Computers & Industrial Engineering, 64, 1096-1109. Gelareh, S. (2008) Hub location models in public transport planning, PhD Dissertation. University of Technology. Kaiserslautern. Gelareh, S. & Nickel, S. (2011) Hub location problems in transportation networks. Transportation Research Part E Logistics and Transportation Review, 47, 1092-1111. Ghodratnama, A., Tavakkoli-Moghaddam, R. & Azaron, A. (2013) A fuzzy possibilistic biobjective hub covering problem considering production facilities, time horizons and transporter vehicles, The International Journal of Advanced Manufacturing Technology, 66, 187-206. Glover, F. (1989) Tabu search – Part I. ORSA J Comput, 1, 190-206. Gourdin, E., Labbe, M. & Laporte, G. (2000) The uncapacitated facility location problem with client matching. Operations Research, 48, 671-685. Jacobson, S. K. & Madsen, O. B. G. (1980) A comparative study of heuristics for a two-level routing–location problem. European Journal of Operational Research, 5, 378-387. Kara, B. & Tansel, B. (2003) The single-assignment hub covering problem: Models and linearizations. Journal of the Operational Research Society, 54, 59-64. Karaoglan, I., Altiparmak, F., Kara, I. & Dengiz, B. (2012) The location-routing problem with simultaneous pickup and delivery: Formulations and a heuristic approach. Omega, 40, 465-477. Karimi, H. & Bashiri, M. (2011) Hub covering location problems with different coverage types. Scientia Iranica, 18, 1571–1578. Karimi, H. & Setak, M. (2014) Proprietor and customer costs in the incomplete hub locationrouting network topology. Applied Mathematical Modelling, 38,1011-1023. Klincewicz. J. G. (1992) Avoiding local optima in the p-hub location problem using tabu search and grasp. Annals of Operations Research, 40, 121-132. Kuby, M. J. & Gray, R. G. (1993) The hub network design problem with stopovers and feeders: The case of Federal Express. Transportation Research Part A: Policy and Practice, 27, 1-12. 20

Laporte, G. & Norbert, Y. (1981) An exact algorithm for minimizing routing and operating costs in depot location. European Journal of Operational Research, 6, 224-226. Lopes, M. C., de Andrade, C. E., de Queiroz, T. A., Resende, M. G., & Miyazawa, F. K. (2016). Heuristics for a hub location‐ routing problem. Networks, 68(1), 5490. Miller, C. E., Tucker, A. W. & Zemlin, R. A. (1960) Integer programming formulation of traveling salesman problems. Journal of the ACM, 7, 326-329. Nagy, G. & Salhi, S. (1998) The many-to-many location-routing problem. Top, 6, 261-275. O’Kelly, M. E. & Miller, H. J. (1994) The hub network design problem-a review and synthesis. Journal of Transport Geography, 2, 31-40. O’Kelly, M. E. (1986) The location of interacting hub facilities. Transportation Science, 20, 92-106. Perl, J. & Daskin, M. S. (1993) A warehouse location–routing problem. Transportation Research Part B: Methodological, 19, 381-396. Rodríguez-Martín, I., Salazar-González, J. J., & Yaman, H. (2014). A branch-and-cut algorithm for the hub location and routing problem. Computers & Operations Research, 50, 161-174. Setak, M., Karimi, H. & Rastani, S. (2013) Designing incomplete hub location-routing network in urban transportation problem. International Journal of Engineering, 26, 997-1006. Setak, M. & Karimi, H. (2014) Hub Covering Location Problem under Gradual Decay Function. Journal of Scientific and Industrial Research, 73, 145-148. Shaw, S. L. (1993) Hub structures of major US passenger airlines. Journal of Transport Geography, 1, 47-58. Skorin-Kapov, D. & Skorin-Kapov, J. (1994) On tabu search for the location of interacting hub facilities. European Journal of Operational Research, 73, 502-509. Tan, P. Z. & Kara, B. Y. (2007) A hub covering model for cargo delivery systems. Networks, 49, 28-39. Tuzun, D. & Burke, L. I. (1999) A two-phase tabu search approach to the location routing problem. European Journal of Operational Research, 116, 87-99. Wasner, M. & Zapfel, G. (2004) An integrated multi-depot hub-location vehicle routing model for network planning of parcel service. International Journal of Production Economics. 90, 403-419. Yang, T. H. (2009) Stochastic air freight hub location and flight routes planning. Applied Mathematical Modelling, 33, 4424-4430.

21

Figure 1: An example of a CHCLRP network topology

22

make the first number in permutation as the first hub; current tour=[](null); for each i ( number in permutation except the first number) do { calculate the required capacity for the current tour, and called if

"rccr" ;

rccr   { add permutation(i) to the current tour; calculate the required travel time for the current network, and called if

rttcn  T { remove the permutation(i) from the current tour; generate a new tour for the current hub; add the permutation(i) to the current tour;

} calculate rccr ; if

rccr   {

remove the permutation(i) from the current tour; generate a new tour for the current hub; add the permutation(i) to the current tour; } }

Figure 2: Feasibility calculate the required capacity for the current hub, procedure and called "rcch" ; if

rcch   { remove the permutation(i) from the current tour; add the permutation(i) as a new hub;

} i=i+1; } report the generated solution;

23

"rttcn" ;

2

5

1

A 3 6 9

4

8

7

 2 6

B 5 1 2 9 4 8

3 7

Figure 3: Representation of the solution.

24

Figure 4: Generated solution of the example.

25

Tabu Search { max_frequency 20n, frequency 0, tabu list (TL)zero square matrix of order

n  (n  1) , max_time1000(s) 2

current_solution  generate randomized solution; best_solution  current_solution; newbest_solution current_solution; while (frequency <= max_frequency or CPU time
Figure 5: Pseudo-code for TS.

} }

current_solution newbest _solution if (current_solution
26

Figure 6: Average performances of the implementations.

27

9 10 7

8

5

6 4

3

9 10 7

8

2 1

5

6

10LL 4

3

9 10 7

8 2

5

1

6

10LT, 10TT

4

3

2 1

10TL Figure 7: Optimal solution for AP with 10 nodes.

28

5

4 3

2 1

5L 5

4 3

2 1

5T Figure 8: Optimal solution for TN with 5 nodes.

29

5 6 4 3

2 7

1

7L, 7T Figure 9: Optimal solution for TN with 7 nodes.

30

8 5 10

6 4 3

9 2 7

1

10L, 10T

Figure 10: Optimal solution for TN with 10 nodes.

31

Figure 11: Statistical result of GAP performance for comparison of TS versus CPLEX.

32

Table 1: Literature of more related works to CHCLRP Research Nagy and Salhi (1998)

Wasner and Zapfel (2004)





Çetiner et al. (2010)

Karaoglan et al. (2012)

De Camargo et al. (2013)

RodríguezMartín et al. (2014)

Lopes et al. (2016)

Current research

component Hub network



-









Allocation

SA

MA

MA

SA

SA

SA

SA

SA

Hub connection

FI

-

PI

-

FI

FI

PI

FI

Formulation

NPS MIP

MINLP

Not presented

PS MIP

PS MIP

PS MIP

NPS MIP

PS MIP

Number of hubs/depots

N

D (just one hub)

-

N

N

D

D

N

Predefined maximum number of vehicles





-

-







-

Fixed cost of vehicle

-

-

-



-

-

-



Fixed cost of hubs



-

-





-

-



Capacity of hubs/depots





-



-

-

-



Time constraint for O-Ds

-



-

-

-

-

-



Heuristic method

Su

I

Si

Si (SA)

-

Si

Si(VND)

Valid inequality

-

-

-



-



-

Si (TS) 

SA-single allocation; MA-multiple allocation; FI- full interconnection; PI- partial interconnection; NPS MIP- non polynomial-sized mixed integer programming; MINLP- mixed integer nonlinear programming; PS MIPpolynomial-sized mixed integer programming; N- not predetermined; D- predetermined; Su-successive; I- iterative; Si- Simultaneous; SA-simulated annealing; VND

33

Table 2: GAP performances of the CHCLRP implementations Instance

Name

AP

10LL 10LT 10TL 10TT 5L 5T 7L 7T 10L 10T

TN

 2799.299 2799.299 2799.299 2799.299 17.694 17.694 46.725 46.725 71.948 71.948 Average

Optimal value 29182.862 61002.313 32212.041 67756.837 4438.121 7149.853 4089.430 4089.430 5593.388 5593.388

Pure

(31)

(32)

(33)

(34)

28.495 29.797 25.812 30.371 42.640 34.880 29.022 31.198 39.056 28.495 31.977

14.288 17.708 12.944 19.496 21.805 20.039 14.024 13.437 34.635 14.288 18.266

9.165 19.770 8.553 21.090 33.987 29.796 17.208 21.926 31.257 9.165 20.192

28.495 29.781 25.812 30.366 40.266 33.226 25.246 15.234 29.898 28.495 28.682

28.495 15.597 25.812 14.474 42.640 28.901 29.022 31.198 39.056 28.495 28.369

34

(31)(34) 9.165 0.051 8.553 0.048 21.805 9.939 9.591 4.866 23.660 9.165 9.684

Table 3: Computational time performances of the CHCLRP implementations Instance

Name



Pure

(31)

(32)

(33)

(34)

(31)-(34)

AP

10LL 10LT 10TL 10TT 5L 5T 7L 7T 10L 10T Average

2799.299 2799.299 2799.299 2799.299 17.694 17.694 46.725 46.725 71.948 71.948

3600 1789.887 3600 1160.011 0.524 0.257 1.641 0.475 201.634 53.549 1040.798

3600 4.353 3600 5.161 0.488 0.255 1.200 0.504 176.231 21.585 740.978

1.669 5.67 1.837 5.365 1.279 0.164 1.695 0.398 33.151 12.342 6.357

3600 1711.535 3600 962.577 0.646 0.249 1.374 0.431 239.191 67.643 1018.365

3600 386.441 3600 557.89 0.679 0.178 1.507 0.445 190.145 87.816 842.510

1.0633 2.443 1.854 2.325 0.397 0.144 0.699 0.409 17.829 13.019 4.018

TN

35

Table 4: Node explored performances in tree search for the CHCLRP implementations Instance AP

TN

Name 10LL 10LT 10TL 10TT 5L 5T 7L 7T 10L 10T Average

 2799.299 2799.299 2799.299 2799.299 17.694 17.694 46.725 46.725 71.948 71.948

Pure 600164 55692 939566 142820 135 0 30 6 4149 1545 174410.7

(31) 1010010 87 1164261 88 97 0 23 22 3748 1044 217938

36

(32) 176 116 246 56 110 0 0 0 769 382 185.5

(33) 597147 55022 919828 126235 130 0 35 23 4475 1694 170458.9

(34) 600004 65754 928823 127575 118 0 37 6 3765 3668 172975

(31)-(34) 105 108 261 141 92 0 0 0 534 356 159.7

Table 5: Results of ANOVA analysis to compare GAP performances of the CHCLRP implementations. Pairwise comparison Pure-(31) Pure-(32) Pure-(33) Pure, (34) Pure-“(31)-(34)” (31)- (32) (31)-(33) (31)-(34) (31)-“(31)-(34)” (32)-(33) (32)-(34) (32)-“(31)-(34)” (33)-(34) (33)-“(31)-(34)” (34)-“(31)-(34)”

Lower limit for 95% confidence interval 3.800 1.874 -6.616 -6.303 12.382 -11.836 -20.326 -20.013 -1.328 -18.401 -18.088 0.597 -9.598 9.087 8.774

Mean difference 13.710 11.785 3.295 3.608 22.292 -1.925 -10.416 -10.103 8.582 -8.490 -8.177 10.507 0.313 18.998 18.685

37

Upper limit for 95% confidence interval 23.621 21.695 13.205 13.518 32.203 7.985 -0.505 -0.192 18.493 1.420 1.733 20.418 10.223 28.908 28.595

P-value

Significant

0.002 0.011 0.922 0.889 0.000 0.992 0.034 0.043 0.126 0.133 0.162 0.032 1.000 0.000 0.000

Yes Yes No No Yes No Yes Yes No No No Yes No Yes Yes

Table 6: Computational results of TS on small-sized problems Instance

Name



AP

10LL

2799.299

CPLEX Opt CPU time (s) 29182.862 1.063

10LT 10TL 10TT 5L 5T 7L 7T 10L 10T

2799.299 2799.299 2799.299 17.694 17.694 46.725 46.725 71.948 71.948

61002.313 32212.041 67756.837 4438.121 7149.853 4089.430 4089.430 5593.388 5593.388

TN

2.443 1.854 2.325 0.397 0.144 0.699 0.409 17.829 13.019

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Gap (%) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

TS Biter Miter 24 200 36 42 25 17 13 23 19 29 39

200 200 200 100 100 140 140 200 200

CPU time(s) 22.345 21.423 22.153 20.983 3.421 2.912 8.682 9.021 21.732 22.462

Table 7: Computational results of TS on medium-sized problems Instance

Name



AP

20LL

1033.532

CPLEX LB Gap (%) 226547.565 34.51

TN

20LT 20TL 20TT 25LL 25LT 25TL 25TT 15L 15T 20L 20T 25L 25T 30L 30T

1033.532 1033.532 1033.532 3259.192 3259.192 3259.192 3259.192 99.169 99.169 101.818 101.818 122.645 122.645 135.414 135.414

251842.880 244466.358 285455.438 25232.403 66000.747 39549.885 97303.027 5256.019 5372.632 5644.393 5682.752 6552.395 6566.372 7423.777 7448.530

46.81 19.27 41.77 0.03 0.20 0.01 41.25 6.42 1.06 41.83 75.92 181.85 131.63 202.54 195.42

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Gap (%) 20.37

TS Biter 23

Miter 359

CPU time (s) 1000

28.28 15.84 20.26 0.03 0.20 0.01 10.12 4.53 1.06 34.86 32.65 54.47 57.31 54.68 51.49

88 78 216 135 116 102 127 124 206 73 28 71 138 81 105

400 357 400 154 282 147 279 300 300 363 362 145 140 125 119

822.844 1000 861.031 1000 1000 1000 1000 237.154 231.656 1000 1000 1000 1000 1000 1000

Highlights o A new hub network called as capacitated hub covering location-routing problem is studied o We solve the model with a tabu-search based heuristic. o A set of valid inequalities is introduced to tighten the formulation o Computational results demonstrate the high efficiency of valid inequalities and tabu-search based heuristic.

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