Managing facility disruption in hubandspoke networks: formulations and efficient solution methods
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Abstract
Hub disruption result in substantially higher transportation cost and customer dissatisfaction. In this study, first a mathematical model to design hubandspoke networks under hub failure is presented. For a fast and inexpensive recovery, the proposed model constructs networks in which every single demand point will have a backup hub to be served from in case of disruption. The problem is formulated as a mixed integer quadratic program in a way that could be linearized without significantly increasing the number of variables. To further ease the model’ computational burden, indicator constraints are employed in the linearized model. The resulting formulation produced optimal solutions for small and some medium size instances. To tackle large problems, three efficient particle swarm optimisationbased metaheuristics which incorporate efficient solution representation, shortterm memory and special crossover operator are proposed. We present the results for two scenarios relating to high and low probabilities of hub failures and provide managerial insight. The computational results, using problem instances with various sizes taken from CAB and TR datasets, confirm the effectiveness and efficiency of the proposed problem formulation and our new solution techniques.
Keywords
Hubandspoke Reliability Hub failure Particle swarm optimization Indicator constraints1 Introduction
The classical hub location problem aims to find suitable locations for hubs, allocate the existing demand to these facilities, and direct the flow between origindestination pairs at minimum cost. In this class of locational problems it is common to assume there is a link between every hub pair, there is no direct path between non hub nodes, and there is economies of scale for using the interhub connections (Alumur and Kara 2008). Hub location problems are categorised into two distinctive groups namely single and multiple allocation problems. In the former, all incoming and outgoing traffic to and from every node is transferred via a single hub, while in the latter each node is allowed to receive and send flow through more than one hub. The focus of this paper is on the Uncapacitated Single Allocation pHub Location problem where the number of hub facilities to open is predetermined. This problem is also known as the Single Allocation pHub Median problem.
Early research on hubandspoke systems focused on developing efficient mathematical formulations and solution techniques for this paradigm. O’Kelly (1986, 1987) presented the first mathematical model for the single allocation phub median problem. Another pioneer of the hub location research, Campbell (1994), developed a linear integer formulation for the problem. Based on the idea of multicommodity flow, Ernst and Krishnamoorthy (1996) proposed a new set of formulations for both single and multiple allocation cases. A widely used formulation in the literature for single and multiple allocation phub median problem is the work of SkorinKapov et al. (1996). The MIP models proposed by SkorinKapov et al. (1996) yield to tight linear relaxation.
Over the years, a number of solution techniques e.g., approximation and exact methods have been developed and successfully applied to solve instances of the hub locationallocation problem. Examples of such methods include Simulated Annealing (Ernst and Krishnamoorthy 1999), Genetic Algorithm (Kratica et al. 2007), Hybrid GA and Tabu Search (AbdinnourHelm 1998), Lagrangian Relaxation (Elhedhli and Wu 2010; Contreras et al. 2009), Benders Decomposition (Contreras et al. 2012), Branch and Bound (Ernst and Krishnamoorthy 1998), and Branch and Cut (Yaman and Carello 2005) among others.
More recent studies focus on the extension of the classical version and aim to develop more realistic hub locationallocation problems such as models with congestion (e.g., Elhedhli and Wu 2010; Azizi et al. 2017), flow dependent economies of scale (Camargo et al. 2009; Campbell et al. 2005), and competitive hub location models (Eiselt and Marianov 2009). For an overview of hub location problem the reader may refer to Campbell and O’Kelly (2012) and Contreras (2015).
In traditional approaches to network design, supply/facility disruption is often presumed to be a rare event. These approaches are mainly concerned with the location of the facilities and the allocation of demand to these facilities in such a way to minimise the total network cost. In practice, however, it is likely that over the course of time one or even more than one facility become disrupted. The cause of disruption may vary from severe weather condition and natural disasters to labour dispute and manmade sabotage. While a set of disruption causes (e.g., earth quake) may occur less frequently, a large number of other causes are likely to strike a supply chain network. For instance, to deal with frequent supply disruption WallMart uses an emergency operations centre. The centre is responsible to prepare for and mitigate the effects of manmade and/or natural disasters (Snyder et al. 2016).
Research have shown that even small disruption in supply may have devastating impacts. In 1998, strikes at two General Motors parts plants resulted in closure of 100 other parts plants and then 26 assembly plants. In the context of locational problems, facility disruption result in excessive transportation cost as customers initially served by these facilities now must be served by other operational facilities in the network (Snyder and Daskin 2005).
Azad et al. (2012) proposed a capacitated supply chain network design model under random disruptions. The formulation aims to determine the locations and types of distribution centres and allocate customers to each opened facility. Azad et al. (2012) considered partially disrupted facilities which can operate with a fraction of their initial capacities. The authors considered a “general supply chain network” which is different from the topology of the hubandspoke system. In a hubandspoke network the flow is processed at least in one facility, is bidirectional and there are interhub flow interactions.
Hub location problems with reliability consideration have received very limited attention in the literature (Kim and O’Kelly 2009). Kim and O’Kelly (2009) formulated two phub location problems in telecommunication network with reliability consideration. The first model, phub maximum reliability, maximises the expected network flow. The second model, phub mandatory dispersion, specifies the optimal hub locations to increase network reliability and prevents excessive concentration of interacting flows in particular hub facilities. Kim and O’Kelly (2009) investigated both single and multiple allocation cases without considering backup hubs and rerouting the flows.
An et al. (2015) proposed a mixed integer nonlinear formulation for reliable hubandspoke problem. The authors successfully solved the formulation for problem instances up to 25 nodes using branch and bound and Lagrangian methods. An et al. (2015) compared the number of passengers served in a classical and that in a reliable network. Using a numerical example, they show that a reliable network can transport more passengers by its regular routes than a network with classical configuration. They also showed that the configuration of classical and reliable networks may not be identical. An et al. (2015) assumed (a) the flow transported in the network is symmetrical and (b) in designing alternative routes a multiple allocation is used (regardless of single allocation or multiple allocation structure used to determine regular routes). Furthermore, in the proposed model by An et al. (2015) flow transported between a pair of nodes which are allocated to the same hub facility will be rerouted through a single backup hub i.e., restricting a pair of demand points initially assigned to the same hub to be rerouted via a single backup hub. Tran et al. (2017) proposed a mixed integer nonlinear model to find the optimal location of predetermined number of hubs. They linearize the model using a specialized flow network called a probability lattice and used a tabu search algorithm to solve problem instances up to 25 nodes with symmetrical flows. Tran et al. (2017) used a levelset approach to allocate spokes to hubs in a specific order. This method of allocation though effective may prevent the model to be extended by e.g., incorporating capacity constraints and/or adapting multiple allocation scheme. Azizi et al. (2016) also proposed a mixed integer nonlinear formulation to design hubandspoke network taking into account the probability of hub failure and rerouting cost. They solved a number of relatively small problem instances to optimality and proposed an evolutionary algorithm to solve large instances. Similar to An et al. (2015), the authors showed that configuration of networks with and without hub failure consideration may not be the same. Azizi et al. (2016) consider a backup facility for each hub and assumed that once a facility becomes unavailable all demand points initially served by the disrupted facility is served by one of the operating facilities in the network. Reallocating the affected nodes to only one of the operating facilities in the network though effective in cases where demand points need to communicate via a single hub, may not lead to optimal solution in other cases.
In this paper, we present models that simultaneously minimises the daytoday network operations cost and the expected transportation cost that incurs when one of the hub facilities experience short term disruption. In doing so, our proposed models seek to find the optimal locations for the hub facilities, allocate the demand to these hubs and find the least expensive backup facility for each demand point in the network. The resulting network(s) is expected to satisfy the demand with a minimum additional cost in the event of supply disruption. Our models intend to provide robust solutions (i.e., networks) that perform well even when part of the network fails. Handling instances with and without symmetrical flow, reallocating the affected demand without any restriction to all operating hubs and offering solutions under a range of hub failure probabilities are other important characteristics of the proposed formulation.
 (i)
An investigation into a reliable hubandspoke system incorporating heterogeneous probability of hub failure
 (ii)
A nonlinear model purposely designed to simultaneously construct networks with and without symmetrical flow and select backup facilities for every demand possibly affected by a disruption
 (iii)
An improved linear model with the use of indicator constraints and
 (iv)
Efficient PSObased metaheuristics for large instances.
2 Problem statement and formulation
In the single allocation phub median problem, each node is assigned to a particular hub and all incoming and outgoing flow are routed through that hub. The total number of nodes in the network is assumed to be N, each node is considered a potential site and the number of hubs to open is p. A path from origin spoke i to destination spoke j includes three parts: collection from spoke i to the first hub k, transfer between first hub k and the second hub m, and distribution from hub m to destination j. The cost per unit flow along this route, \(i\rightarrow k\rightarrow m\rightarrow j\), is calculated as \(\chi \times c_{ik}+\alpha \times c_{km}+\delta \times c_{mj}\) where \(\chi \) and \(\delta \) are coefficients of collection and distribution respectively and \(\alpha \) is the interhub discount factor. Let \(\lambda _{ij}\) be the amount of flow to be routed from origin i to destination j, the transportation cost from i to j routed via hubs k and m, \(C_{ikmj}\) is then calculated as \(C_{ikmj}=\lambda _{ij}\left( \chi \times c_{ik}+\alpha \times c_{km}+\delta \times c_{mj} \right) \).
2.1 Notation

\(z_{ik}=1\) if node i is assigned to hub k and \(=\) 0 otherwise;

\(x_{ikmj}=1\) if flow from i to j passes through hub k and \(m \text { and} = 0\text { otherwise;} \)

\(u_{ikn}=1\) if n is the backup for node i when hub k in the path from origin i to any destination j is disrupted and \(=\) 0 otherwise.
2.2 Model formulation
To formulate the problem, in the event of any hub disruption we recognise four different types of flows namely: (1) flow that is not affected by the disruption (2) flow that uses the disrupted facility as its first hub (3) flow that uses the disrupted facility as the second hub (4) flow that originated from or destined to a hub facility.
The proposed model minimizes the weighted objective function \(wF1+\left( 1w \right) F2\) where
The proposed formulation for RSApHLI is a Mixed Integer Quadratic Programing (MIQP) and includes \(n^{2}+4n^{3}+n^{4}\) variables. In the next section we will show that because of our novel modelling approach linearizing the model using conventional techniques will not significantly increase the number of variables. Our approach is similar to the technique proposed by Chaovalitwongse et al. (2004) to linearize multiquadratic 0–1 programming problems.
2.3 Linearization
The Number of variables and constraints after linearization
Problem  Number of variables  Number of constraints 

RSApHLI (MIQP)  \(n^{2}+4n^{3}+n^{4}\)  \(1+n+2n^{2}+7n^{3}\) 
RSApHLII (MIP)  \(n^{2}+7n^{3}+n^{4}+n^{5}\)  \(1+n+2n^{2}+16n^{3}+4n^{5}\) 
2.4 Linear model with indicator constraints
In MILP formulations, it is common to model constraints that either hold or are relaxed depending on the value of a binary variable. These constraints are often modelled using the so called BigM formulations to enforce the relationship between a new variable and its corresponding nonlinear term. These BigM formulations though useful are often blamed for unstable behaviour in MILP model. Indicator Constraints have the advantage of preventing the problems associated with BigM formulations. Similar to BigM formulations, indicator constraints use a binary variable to turn on or turn off the enforcement of a constraint but without using a big M. Further information about the indicator constraints could be found in the paper by Bonami et al. (2015).
Subject to:
2.5 Preliminary study
To evaluate and compare the performance of the proposed (linear) formulations with and without indicator constraints, we conducted a limited computational experiment as described in the following. A relatively small problem instance with 10 nodes and 3 hubs is selected from the U.S. Civil Aeronautics Board which is known as CAB dataset (O’Kelly 1987). We generated three instances of the original problem by setting the objective function weight w to 0.3, 0.5 and 0.7. The discount parameter \(\alpha \) is set to 0.2; the coefficients of collection \(\chi \) and distribution cost \(\delta \) are set to one per unit for all three instances. The problems are solved using CPLEX 12.6 with default options values. The algorithm is run for 5000 s on Intel Core i5 PC with 3.2 GHz processor with 4 GB of RAM. The computational results are presented in Table 2.
The results pertaining to RSApHLII model show that CPLEX is unable to produce the optimal solution for the problem instance with Open image in new window in 5000 s but manages to locate the optimal solution for the other two instances with Open image in new window and 0.7.
CPLEX results for three instances of RSApHLII and RSApHLIII
\(\hbox {n}^\mathrm{a}\)  \(\hbox {p}^\mathrm{b}\)  \(\upalpha ^\mathrm{c}\)  \(\hbox {w}^\mathrm{d}\)  RSApHLII  % GAP  RSApHLIII  

Lower bound  Best solution  Time (s)  Lower bound  Best solution  Time (s)  
10 (CAB)  3  0.2  0.3  250.05  408.48  5000  39  367.24  367.24\(^\mathrm{e}\)  824 
0.5  409.98  409.89\(^\mathrm{e}\)  1405  0  409.98  409.98\(^\mathrm{e}\)  356  
0.7  452.54  452.54\(^\mathrm{e}\)  936  0  452.54  452.54\(^\mathrm{e}\)  189 
This improved formulation will be used to provide when possible lower and upper bounds for benchmarking purposes. As will be shown in our computational results, the above improved model formulation could only be used to solve relatively small to medium size problem instances to optimality.
Due to the limitation in solving the above model formulation to optimality for realistically sized problem instances with CPLEX, one way forward is to design an efficient metaheuristic. In this study, we developed three metaheuristics based on the wellknown evolutionary algorithm of Particle Swarm Optimisation (PSO). These algorithms are discussed in the following section.
3 Particle swarm optimization
Particle swarm optimization (PSO) is one of the most efficient natureinspired optimization algorithms. It was proposed by Kennedy and Eberhart (1995) about two decades ago. The algorithm was developed based on the motion of a flock of birds searching for foods and aimed to mainly optimise continuous nonlinear optimisation problems.
In PSO based algorithms, at the beginning of the evolutionary process a set of particles which is called a swarm is generated randomly. Each member of a swarm is called a particle. During the search process, particles are flown through a hyperspace. A particle’s status on the space is characterized by two elements namely position and velocity. Particles may change their position in the search space similar to flying birds searching for food in the sky. While searching the hyperspace, a particle adjusts its (new) velocity according to its own best experience, the best experience of all particles in the swarm and its previous velocity. The particle’s new velocity and current position are then used to determine its new position.
A large portion of the published research works on PSO is dedicated to continuous optimization problems whereas there is a lack of research dealing with its discrete counterpart namely combinatorial optimization problems (Liu et al. 2008). To the best of our knowledge, our proposed algorithms is the first PSObased optimisation technique to be developed for solving this interesting though challenging class of hub location problem.
In this paper two variations of a basic PSO and a hybrid PSO algorithm that incorporate crossover and memory are proposed. The two versions which we refer to as PSOv1 and PSOv2, differ in terms of solutions representation and the mechanism by which a continuous particle position is transformed into one or more discrete solution(s). The hybrid PSO algorithm on the other hand, is an extension of PSOv2 which incorporates a shortterm memory and a crossover operator, both of which are in our knowledge, new ingredients that are successfully embedded for the first time into the PSO search methodology.
3.1 PSOv1 algorithm
Particle velocity
Updating particle velocity and position
Due to the continuous nature of the particle position in PSO, standard encoding schemes cannot be directly adopted for discrete location problems such as hub and spoke systems. One procedure to transform a continuous particle position into a discrete solution in PSOv1 is described via the following example. Consider the particle’ position presented in Fig. 5. As mentioned earlier, the number of columns/elements in a particle position array corresponds to the number of nodes in the problem (in this example the number of elements are 10). The integer part of the largest element of the array refers to the number of hubs which in the above example is 3.
All elements in the particle position that have the same value constitute a cluster. For example, the following three clusters \(\{3,7\},\, \{ 2,5,10\}\) and \(\{1,4,6,8,9\}\) are presented in Fig. 6a.
As each member of a cluster (i.e., a node) is a potential site for a hub facility, a discrete feasible solution can be generated by connecting the selected hubs. For example, a possible configuration is given in Fig. 6b where nodes 3, 2, and 4 corresponding to clusters 1, 2 and 3 are chosen. Considering the combination of potential hub facilities in each cluster node, a quite large number of feasible solutions could be constructed from one single particle position. In the example presented in Fig. 6a, a total of 30 \((3\times 2\times 5)\) solutions could be generated. In general, if \(n_{i}\) refers to the number of nodes in cluster i, the total number of configurations become \( \prod \nolimits _{i=1}^p n_{i} \).
Clearly, as the number of hubs and/nodes increases the number of solutions/possibilities could increase significantly. Examining all possible solutions produced by changing hub facilities in each cluster could be, particularly, for relatively large instances computationally expensive. Therefore, it is more practical to either randomly select one solution among all possible solutions or generate a fraction of potential solutions and select the best one with the lowest total cost.
We set up a limited number of experiments using instances with 10, 15, 20, and 25 nodes and concluded that examining a fraction of the potential solutions (e.g., \(k_{0})\) and selecting the best one among the subgroup could provide good overall best results.
Backup selection
PSOv1 encoding scheme transfers a continuous particle position to a discrete solution to a single allocation median hub location problem. Although the selection process of backup facilities could also be a part of the solution representation, to retain the simplicity of our solution representation we opted to treat this decision separately. Backup facilities for the existing demand points could be chosen (1) randomly (2) by examining all possibilities or (3) based on proximity to a potential backup facility. Examining all combinations (i.e., complete enumeration) will provide the best solution as it guarantees the selection of the most economically suitable backup but it is computationally expensive even for small problem instances. We therefore compared the two other options using a number of test problems and concluded that in large instances selecting nearby backups to demand nodes outperforms the random backup selection strategy. For the small to medium size problem instances, random backup selection may result in better quality solutions.
4 PSOv2 algorithm
In this variant PSOv2, a continuous particle position is directly encoded/mapped to only one discrete solution. The main difference(s) between this version and the earlier one are the solution representations and the mechanisms through which a continuous particle position is encoded into a solution.
Particle position and representation
Entities in the second row of the matrix denote the allocation of each node to one of the hubs specified in the first row.
Similar to PSOv1 position array, the second row of the matrix is filled with randomly generated continuous numbers in the range of [\(1,p+1\varepsilon \)] where \(0< \upvarepsilon < 1\).
Particle velocity
Similar to that in PSOv1, the initial values of particles’ velocities, \(v_{k0}\) , are generated using Eq. (52) with \(v_{max}\) and \(v_{min}\) defined as in PSOV1 (Eq. 53). Particles velocities and positions are updated in each iteration using Eqs. (54) and (44) respectively.
Transforming a continuous particle position into a single discrete solution
To transfer a continuous particle position into a discrete solution, first all entities in the first row of the particle position matrix are ranked. Then p elements of the first row are considered as potential hub locations. For instance, in the particle position presented in Fig. 8 the smallest number is the fifth element “0.7” and is ranked “1”; the third element “1.1” is ranked second (i.e., “2”); and the first element “1.6” is ranked third (i.e., “3”) etc. After ranking the numbers in the first row of the particle position matrix, the three potential hub facilities (p assume to be 3) are located as node 3 (first hub), 5(second hub) and 2(third hub).
For instance, in Fig. 9 the third and the seventh elements of the second row are identical and have the value of “1” which means nodes 3 and 7 are allocated to the first hub which is hub “3”.
Repair algorithm
The transfer of a continuous particle position into a discrete solution is presented in Fig. 11.
4.1 Hybrid PSO algorithm
Our preliminary results for relatively small problem instances show that both variants (i.e., PSOv1 and PSOv2) perform well in finding good quality solutions to the RSApHLIII. Nevertheless in the majority of cases the same solutions are obtained by PSOv2 requiring relatively shorter computational times.
Research have shown that PSObased optimization algorithms are highly randomised and, in general, are susceptible to early convergence (i.e., could be trapped into a local optima too early in the search). To overcome this drawback and hence improve the performance of the PSO2 algorithm, two new features are imbedded into the algorithm to make our hybrid PSO which we refer to HPSO for short. (a) The first feature is a memory which is setup by collecting a number of global best particle positions at the end of each iteration. The size of the memory is assumed to be the same as that of the PSO swarm size e.g., 20. (b) The second feature is a crossover operator aiming to improve the quality of the solutions in the PSO population as well as diversifying the search. Once a memory is completed, solutions stored in the memory are combined using a special type of crossover operator. To perform the crossover operation, two parent particles are selected each time randomly to produce an offspring particle. The fitness of the newly generated solution is then compared with those in the PSO population; the offspring replaces the first inferior particle found in the population. The offspring particle is ignored if it cannot replace any of the particles in the population. Upon completion of the new population the memory is reset before the next iteration of the algorithm begins. The crossover operator is described next.
The proposed crossover operator
The solution representation used in HPSO is the same as that in PSOv2 (2 dimensional matrix). Classical crossover operators are often designed for onedimensional arrays and could not be directly applied here. Therefore, we propose a special type of crossover that combines the classical “single point” with what we name a “constructive” crossover operator.
To produce a new offspring, a template 2dimentional matrix for the offspring solution is first constructed. Next, two parents are marked randomly from the current population. Then, one of the two parents is selected randomly and the first element on the “first row” of its matrix is transferred to the same place on the offspring particle template matrix. The second element of the offspring template is taken from the other parent. These steps are repeated until the first row of the offspring array is populated with elements transferred from the two parents.
5 Computational results and analysis
The computational time for all test problems
Nodes  10  15  20  25  55  

Hubs  3  5  3  5  3  5  3  5  3  5 
CPU (s)  10  10  20  20  20  30  40  60  90  120 
In this section we analyse two scenarios reflecting real life applications. In the first one we consider networks that are quite highly susceptible to hub disruptions. Whereas in the second, we attempt to address other networks in which hub disruption is less likely to occur. The former scenario includes a network in which hubs are assumed to be, for instance, distribution centres. In these types of networks the probabilities of hub failures are expected to be relatively high. In the second scenario, such as hubandspoke networks in airline industry, the chance of hub failure is believed to be relatively low though extremely costly if that happens.
For the case with high probability of hub failure, the probabilities, \(q_{i}\), for all nodes in the network \(i =1 \ldots n\) are generated randomly from U [0.1, 0.3]. For the other case, we use the low probabilities of hub failures as given in the paper by An et al. (2015).
5.1 Scenario I: the case of high hub failure probability
5.1.1 CAB dataset
In CAB dataset the flow is transported equally in both directions (i.e., symmetrical flow). In such networks, the size of the problem can be reduced by calculating the transportation cost in just one direction and then doubling the cost to present the total cost of transporting the flow. This setting has been adopted by a number of studies and, to some extent, reduces the computational time required to solve problems with symmetrical flow.
A summary of the computational results and comparisons with CPLEXCAB dataset
PSOv1  PSOV2  HPSO  CPLEX  Gap (%)  

\(\hbox {ATNC}^\mathrm{a}\)  \(\hbox {Time}^\mathrm{b}\)  ATNC  Time  ATNC  Time  \(\hbox {ALB}^\mathrm{c}\)  \(\hbox {AUB}^\mathrm{d}\)  Time  
10 Nodes  466.48  3.00  465.20  0.35  465.05  0.34  464.99  464.99  372.5  0.0 
15 Nodes  840.92  12.40  804.94  8.60  801.78  7.50  533.34  850.15  1809.24  36 
20 Nodes  759.50  12.51  726.01  13.16  708.83  12.85  –  –  –  – 
25 Noes  878.21  20.07  833.90  29.85  815.58  21.52  –  –  –  – 
CPLEX solved all 18 benchmark problems in a reasonable computational times with 17 within 1–11 min and the other requiring just 18 min with a large portion of this time being spent in closing the last 5% gap. The average computational time to solve all 18 problems is 372.5 s. Nevertheless these are still significantly high when compared against the time required by any of the PSObased algorithms.
The results in Table 4 further show that the three PSObased algorithms performed well though HPSO found more optimal solutions than the other two algorithms (i.e., PSOv1 and PSOv2). For test problems with 10 nodes, the optimal locations of hubs and allocations of nonhubs to the selected hub facilities as well as backup facilities for each node are presented in Table 5. For example, in this table the optimal value of the objective function for the problem with 3 hubs, discount factor of 0.2, and the objective weight of 0.3 is 367.24; the locations of the hubs are 4 (Chicago), 2 (Baltimore) and 7 (DallasFW); the backup facility for nodes 1, 5, 6, 9 is 2 (Baltimore) and for nodes 3, 8 and 10 is 4 (Chicago) (Fig. 13).
Detailed results for problems with 10 nodes
\(\hbox {n}^\mathrm{a}\)  \(\hbox {p}^\mathrm{b}\)  \(\upalpha ^\mathrm{c}\)  \(\hbox {w}^\mathrm{d}\)  Hub locations  Backup hubs  \(\hbox {TNC}^\mathrm{e}\)  Time (s)  Gap (%) 

10 (CAB)  3  0.2  0.3  4224447747  2040220424  367.24  183.8  0.0 
0.5  4224447747  2040220424  409.89  75.2  0.0  
0.7  4224447747  2040220424  452.54  57.5  0.0  
0.4  0.3  4224447747  2040220424  401.54  256.8  0.0  
0.5  4224447747  2040220424  457.13  138.8  0.0  
0.7  4224447747  2040220424  512.70  70.6  0.0  
0.8  0.3  4224447447  2040220724  469.07  286.4  0.0  
0.5  4224447447  2040220724  550.10  233.4  0.0  
0.7  4994497497  9440940704  620.35  118.2  0.0  
5  0.2  0.3  1224447847  0040120021  375.22  556.2  0.0  
0.5  1224447847  0040120021  368.98  571.6  0.0  
0.7  1224997797  0090440401  362.31  560.3  0.0  
0.4  0.3  1224447847  0040220021  432.53  638.1  0.0  
0.5  1224447847  0040220021  437.50  536.9  0.0  
0.7  1224447847  0040120021  442.46  585.5  0.0  
0.8  0.3  1224447847  0040220021  547.03  1063.4  0.0  
0.5  1224447847  0040120021  574.43  665.3  0.0  
0.7  1994997897  0140440001  588.85  107.3  0.0  
Average  464.99  372.5  0.0 
In test problems with 15 nodes, our proposed PSO based algorithms improved the upper bounds for the 18 instances with HPSO algorithm achieving improvement up to 20%.
CPLEX did not provide any solution for larger instances with 20 and 25 nodes due to the lack of sufficient memory. A summary of the computational results for these 36 large problems are also presented in Table 4. Here, the three PSO algorithms could be clearly ranked with HPSO on top in terms of both solutions quality and the computational times followed by PSOv2 and PSOv1.
Further observations and analysis
The network presented in Fig. 14b is based on our approach and realistically reflects the real world. The (unweighted) regular transportation cost of the network in Fig. 14b is 752.65 unit which is 4.7% higher than that of the network provided by the classical model. This finding suggests that using our proposed model that takes to account facility unavailability may not result in a significant increase in regular transportation cost while providing a more robust configuration in case of hub failure. Furthermore, as illustrated in Fig. 14 the topologies of the two networks with and without hub failure consideration are different. This observation suggests that solving a classical hub location problem to optimality and then deciding on the backup facilities is a simple but an illinformed strategy that will lead to an inferior solution. This is an important strategic decision that ought not to be taken lightly.
5.1.2 TR dataset
The proposed formulation and algorithms have been also tested on 54 benchmark problems derived from TR dataset with nonsymmetrical flow ranging from 10 to 55 nodes. The average costs and computational times over 18 problem instances with 10 nodes and 3 and 5 hubs are presented in the first row of Tables 6. Similar to the CAB dataset, CPLEX solved all benchmark problems with 10 nodes in reasonable computational times. It appears that, in general, CPLEX has less difficulty in solving instances from TR dataset as the average computational time required (280 sec) is about 25% lower than that used for CAB dataset (373 sec). This could be due to the dataset’s structure and properties.
Computational results and comparisons with CPLEXTR dataset
PSOv1  PSOV2  HPSO  CPLEX  Gap (%)  

\(\hbox {ATNC}^{\mathrm{a}}\)  \(\hbox {Time}^{\mathrm{b}}\)  ATNC  Time  ATNC  Time  \(\hbox {ALB}^\mathrm{c}\)  \(\hbox {AUB}^{\mathrm{d}}\)  Time  
10 Nodes  404.19  3.80  401.79  0.20  401.79  0.30  401.78  401.78  280.12  0.0 
25 Nodes  645.87  22.29  592.33  25.81  563.79  18.33  –  –  –  – 
55 Nodes  705.50  38.51  672.68  65.96  662.32  54.54  –  –  –  – 
5.2 Scenario II: the case of low hub failure probability
In some applications of the hubandspoke systems it is common to assume a low probability of hub failure. To examine how the proposed formulation behaves under these circumstances and to assess what impact this change might have on the final solutions, we tested the same problem instances from the CAB dataset but with low probability of hub failures instead. As mentioned earlier, in this study we use the low probabilities of hub failures as given in the paper by An et al. (2015).
To be consistent with the analysis conducted in the first scenario, all other parameters used in the formulation and the PSO algorithm are the same.
Furthermore, as HPSO proved to be more powerful than the other two proposed algorithms from now on we only discuss the results provided by this algorithm. We begin with the small instances with 10 nodes and 3 and 5 hubs from the CAB dataset.
A summary of the computational results and comparisons CAB dataset
High probability of hub failure  Low probability of hub failure  

HPSO  CPLEX  HPSO  CPLEX  
\(\hbox {ATNC}^{\mathrm{a}}\)  \(\hbox {Time}^{\mathrm{b}}\)  \(\hbox {AUB}^{\mathrm{c}}\)  Time  Gap (%)  ATNC  Time  AUB  Time  Gap (%)  
10 Nodes  465.05  0.34  464.99  372.5  0.0  303.02  2.59  303.02  126.59  0.0 
15 Nodes  801.78  7.50  883.25  915.60  37.1  487.58  11.11  485.40  882.56  7.18 
20 Nodes  708.83  12.85  –  –  –  483.42  15.33  –  –  – 
25 Noes  815.58  21.52  –  –  –  564.49  32.29  –  –  – 
Detailed results for problems with 20 and 25 nodes—CAB dataset
\(\hbox {n}^{\mathrm{a}}\)  \(\hbox {p}^{\mathrm{b}}\)  \(\upalpha ^\mathrm{c}\)  \(\hbox {w}^{\mathrm{d}}\)  \(\hbox {TNC}^{\mathrm{e}}\)  Time (s)  Hubs  Backups 

20  3  0.2  0.3  306.76  19.42  20, 7, 8  7, 7, 7, 7, 8, 7, 0, 0, 8, 8, 20, 7, 7, 7, 8, 8, 7, 7, 7, 0 
0.5  443.86  10.91  4, 18, 12  18, 4, 4, 0, 18, 18, 12, 12, 18, 18, 18, 0, 18, 4, 18, 18, 4, 0, 4, 18  
0.7  558.99  5.85  4, 17, 12  17, 4, 4, 0, 17, 17, 12, 12, 17, 12, 17, 0, 17, 4, 17, 17, 0, 4, 4, 4  
0.4  0.3  329.38  3.80  5, 18, 8  18, 5, 5, 18, 0, 18, 8, 0, 18, 8, 8, 5, 18, 18, 8, 8, 5, 0, 5, 18  
0.5  446.41  15.26  4, 18, 12  18, 4, 4, 0, 12, 18, 12, 18, 18, 12, 18, 0, 18, 4, 12, 12, 4, 0, 18, 4  
0.7  652.82  8.47  5, 17, 12  17, 5, 5, 17, 0, 17, 12, 12, 17, 12, 17, 0, 17, 17, 17, 17, 0, 5, 5, 17  
0.8  0.3  379.05  16.72  20, 7, 8  7, 7, 7, 7, 8, 7, 0, 0, 8, 20, 8, 7, 8, 7, 7, 20, 7, 7, 7, 0  
0.5  596.22  19.86  20, 4, 11  4, 4, 4, 0, 11, 11, 4, 4, 4, 4, 0, 4, 4, 11, 11, 20, 4, 11, 4, 0  
0.7  800.75  15.76  20, 4, 11  4, 4, 4, 0, 4, 4, 4, 4, 4, 4, 0, 4, 11, 4, 11, 11, 4, 4, 4, 0  
5  0.2  0.3  268.90  6.23  4, 2, 7, 19, 14  2, 0, 4, 0, 4, 4, 0, 19, 2, 4, 4, 7, 7, 0, 7, 14, 4, 4, 0, 4  
0.5  362.65  23.35  4, 18, 16, 12, 14  14, 4, 14, 0, 18, 16, 18, 16, 18, 16, 14, 0, 14, 0, 18, 0, 4, 0, 14, 18  
0.7  448.84  21.10  20, 4, 17, 7, 12  17, 17, 20, 0, 4, 4, 0, 12, 20, 17, 20, 0, 12, 4, 17, 20, 0, 20, 4, 0  
0.4  0.3  321.04  25.80  1, 20, 7, 8, 14  0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 7, 7, 7, 0, 8, 1, 1, 1, 7, 0  
0.5  465.64  21.29  20, 17, 4, 11, 17  14, 14, 20, 0, 17, 17, 17, 4, 20, 20, 0, 4, 20, 0, 4, 20, 0, 20, 4, 0  
0.7  561.16  9.33  5, 17, 4, 12, 17  4, 14, 5, 0, 0, 17, 4, 14, 4, 4, 4, 0, 12, 0, 4, 12, 0, 5, 5, 17  
0.8  0.3  393.51  7.00  21, 15, 8, 7, 20  2, 0, 2, 15, 2, 2, 0, 0, 2, 8, 15, 7, 20, 2, 0, 7, 2, 2, 7, 0  
0.5  593.36  26.58  1, 20, 7, 8, 15  0, 1, 1, 15, 7, 1, 0, 0, 15, 20, 1, 1, 7, 1, 0, 1, 1, 1, 7, 0  
0.7  772.22  19.16  20, 4, 7, 12, 15  15, 4, 15, 0, 15, 4, 0, 15, 4, 15, 20, 0, 15, 4, 0, 4, 4, 15, 12, 0  
25  3  0.2  0.3  348.35  11.59  13, 8, 18  18, 18, 13, 18, 18, 18, 8, 0, 18, 8, 8, 13, 0, 18, 8, 8, 13, 0, 13, 13, 8, 13, 13, 18, 13 
0.5  491.23  38.95  5, 18, 12  12, 18, 5, 18, 0, 18, 12, 18, 18, 12, 18, 0, 12, 18, 12, 5, 5, 0, 18, 12, 18, 5, 5, 18, 18  
0.7  635.83  15.30  5, 17, 22  17, 5, 5, 17, 0, 17, 17, 22, 17, 17, 17, 5, 17, 17, 17, 17, 0, 5, 22, 17, 17, 0, 5, 17, 17  
0.4  0.3  379.10  3.14  5, 18, 8  18, 5, 5, 18, 0, 18, 8, 0, 18, 8, 8, 5, 18, 18, 8, 8, 5, 0, 5, 18, 8, 5, 8, 5, 5  
0.5  579.34  20.78  4, 2, 12  4, 0, 4, 0, 4, 4, 2, 12, 2, 4, 4, 0, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4  
0.7  704.09  29.45  5, 18, 12  18, 5, 5, 18, 0, 18, 12, 12, 18, 18, 18, 0, 18, 18, 18, 18, 5, 0, 12, 18, 18, 5, 5, 18, 5  
0.8  0.3  429.57  34.18  20, 8, 15  15, 15, 15, 15, 15, 15, 8, 0, 15, 8, 15, 15, 15, 15, 0, 15, 15, 15, 15, 0, 15, 15, 8, 15, 15  
0.5  668.43  26.47  20, 8, 15  15, 15, 15, 15, 15, 15, 8, 0, 15, 8, 15, 15, 15, 15, 0, 15, 15, 15, 15, 0, 15, 15, 15, 15, 15  
0.7  921.65  29.03  5, 25, 12  25, 5, 5, 25, 0, 25, 12, 25, 12, 5, 25, 0, 25, 25, 25, 25, 5, 5, 25, 25, 25, 25, 5, 25, 0  
5  0.2  0.3  343.23  45.18  1, 5, 17, 11, 12  0, 1, 5, 11, 0, 1, 1, 12, 17, 17, 0, 0, 1, 11, 5, 11, 0, 5, 5, 1, 5, 11, 11, 17, 17  
0.5  459.70  50.06  9, 14, 18, 21, 12  21, 18, 9, 9, 9, 9, 9, 12, 0, 21, 9, 0, 9, 0, 9, 14, 9, 0, 12, 18, 0, 21, 12, 14, 9  
0.7  569.32  28.19  4, 17, 7, 12, 14  14, 17, 4, 0, 17, 17, 0, 7, 17, 7, 7, 0, 7, 0, 7, 7, 0, 17, 12, 17, 7, 7, 12, 14, 4  
0.4  0.3  392.07  47.76  20, 7, 12, 14, 18  14, 18, 18, 18, 18, 18, 0, 7, 18, 7, 7, 0, 7, 0, 7, 7, 20, 0, 7, 0, 7, 7, 12, 14, 18  
0.5  529.29  44.31  5, 17, 8, 12, 14  14, 5, 5, 17, 0, 17, 8, 0, 17, 8, 8, 0, 14, 0, 8, 14, 0, 17, 12, 17, 8, 8, 8, 14, 17  
0.7  655.40  41.14  4, 17, 12, 14, 25  25, 25, 25, 0, 25, 25, 14, 12, 25, 14, 25, 0, 25, 0, 25, 14, 0, 25, 12, 25, 25, 4, 4, 14, 0  
0.8  0.3  471.56  34.87  5, 2, 7, 8, 20  20, 0, 20, 20, 0, 20, 0, 0, 20, 7, 7, 8, 7, 2, 8, 7, 20, 2, 7, 0, 7, 7, 8, 2, 2  
0.5  693.74  42.53  20, 4, 8, 11, 19  4, 4, 4, 0, 4, 4, 11, 0, 4, 11, 0, 19, 11, 4, 4, 11, 4, 4, 0, 0, 11, 19, 19, 4, 4  
0.7  888.99  38.24  20, 4, 8, 12, 19  4, 4, 4, 0, 4, 4, 8, 0, 4, 8, 4, 0, 4, 4, 8, 4, 4, 4, 0, 0, 4, 19, 8, 4, 4  
Average  523.96  23.81 
A further examination of the results show that, in the majority of the cases in this scenario the locations of the hubs, allocations of demand to these hubs and the backup facilities differ from those in the previous scenario. To further explore this issue we repeated the experiment that resulted in two networks depicted in Fig. 14 but using low probabilities of failures, see Fig. 15. It is worth mentioning that the two network configurations provided by RSApHLIII with high (Fig. 14b) and low (Fig. 15b) probabilities of hub failure differ both in terms of the selected hubs and the allocations of the demand to these hubs. As expected the regular transportation cost of the network in Fig. 15a is slightly higher than that of the network of Fig. 15b with hub failure consideration (i.e., 716.98 vs. 767.84). However, the expected transportation cost of the network provided by our formulation (Fig. 15b) is 35% (52.5 vs. 80.4) lower than that provided by the classical model (Fig. 15a). This observation clearly shows that even with very low probabilities of hub failure a significant cost saving could be achieved by considering hub failure. This is an interesting finding as one may expect less saving in expected transportation cost as the probabilities of hub failure decreases.
Table 7 also summarises the average costs, computational times and Gaps for instances with 15, 20, and 25 nodes and 3 and 5 hubs. For test problems with 15 nodes, CPLEX is first run for a limited time of 20 min and the results are recorded. The computational times are then extended to 2.5 hours to see whether the quality of the final solutions can possibly be improved. Similar to what we observed in small sized problem instances, our proposed formulation with low probabilities of hub failures is solved with less computational effort. For the test problems with 15 nodes the average optimality gap based on 20 min computing times, is about 8.75% which is significantly lower than that recorded in previous scenario i.e., 36%. Extending the computational times to 2.5 hours, CPLEX found optimal solutions for two test problems with 3 hubs. For the other instances, longer computational times either improved the best upper bounds or in some cases significantly reduced the optimality gaps through tightening the lower bounds.
HPSO also solved two instances to optimality and improved the upper bounds for a number of those problem instances. The quality of the solutions provided by the HPSO are clearly comparable with those provided by CPLEX in extended computational times. This observation demonstrates the efficiency and stability of our proposed hybrid PSO algorithm.
A total of 36 larger instances with 20 and 25 nodes are also solved using our hybrid PSO (HPSO) algorithm. A summary of the results and detail description of each solution for these classes of benchmark problems are presented in Tables 7, 8 respectively. In Table 8 we report the total network cost, the hub locations, demand allocation and the selected backup facilities. For instance, for the 20 nodes instance with 3 hubs, discount factor 0.2, and the objective function weight of 0.3, the cost of the best solution found is 306.76 units; the selected hub locations are 20 (Pittsburgh), 7 (DallasFW), and 8 (Denver); the backup facility for nodes 1–4, 6, 12–14, and 17–19 is 7 (DallasFW); for nodes 5,9–10, 15 and 16 is hub 8 (Denver); and for node 11 it is hub 20 (Pittsburgh).
6 Conclusion
In this study, a mixed integer quadratic programming (MIQP) formulation is proposed for the reliable single allocation hub location problem with heterogeneous probability of hub failure. The model assigned a backup facility to every demand point in the network for a fast and a lowcost recovery after hub failure. The objective function of the model minimises the weighted transportation cost in regular situation and the expected cost resulted from hub failures.
The proposed formulation is developed in such way to maintain the size of the problem after linearization. To improve the computational efficiency of the resulting MILP model, all constraints initially formulated using Big Ms are substituted with logical constraints known as Indicator Constraints. Using the proposed formulation, we solved all small instances and some medium size problem instances to optimality and provided lower and/or upper bounds for other problems. To tackle large instances, three novel algorithms based on PSO namely PSOv1, PSOv2 and HPSO are proposed. In PSOv1 and PSOv2, we investigated the effect of two different solution representations and the mechanisms through which a continuous particle is transferred to a discrete solution. It was found that PSOv2 that maps a continuous particle to a single discrete solution performs better than PSOv1. The performance of the PSOv2 is then further improved by introducing two new interesting features namely a crossover operator and a memory to keep track of good solutions that are explored during the search. To our knowledge, this is the first time the hybridisation of PSO with those attributes is examined.
We presented and discussed the computational results for two scenarios. In the first scenario it is assumed that hub disruption is more likely to occur and therefore the selected probabilities of hub failures are set relatively high. Whereas in the second a hub disruption is assumed to be an event with a low probability of occurrence. Our numerical results show that even with low levels of probabilities of hub failures the regular transportation cost of the reliable network will not increase significantly and the amount of cost saving remains considerable. This finding, however, is based on experiments involving relatively small instances. Further research required to examine optimal solutions to large problem instances.
In this study the single allocation version of the hub location problem is considered, however the proposed model and our metaheuristic could be easily adopted to address the multiple allocation cases. Other venues of future research including related problems that incorporate capacity constraints and fixed costs are also worth examining.
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