Reinforcing critical links for robust network logistics: A centrality measure for substitutability

Logistics networks are becoming more complex and interconnected. Guaranteeing the performance of the entire system when a part of the network is disrupted (e.g. due to excessive demands and extreme weather conditions) is one of the important issues. However, how much transportation resources should be allocated to which part of the network while maintaining efficiency is an open question. In this paper, we propose a novel metric, the substitutability centrality, which quantifies how much each transport link in the network contributes to the robustness of the system against disruptions. This metric is compelling in the following aspects: (1) it is intuitively interpretable; (2) it does not require simulation or optimization calculations; and (3) it takes into account changes in transportation routes of delivery due to disruptions. Furthermore, as a proof of concept, we demonstrate a simple case study, in which capacity allocation based on the proposed metric can maintain high performance of the system against various types of disruptions. We also found that this approach might not be effective for further increasing the robustness of networks that have many bypass routes.

are included in this category. For such systems, the occurrence of cascading failures and network disconnection has been mainly studied as important indicators of robustness performance (Schneider et al 2011, Wang and Rong 2011, Parandehgheibi and Modiano 2013, Chen et al 2018. For example, Schneider et al (2011) defined a performance measure by considering the largest component of connected nodes in the network during all possible attacks and analyzed the power supply system in the EU and Internet.
The second category includes the systems where the time of travel of what flows in the network is an important performance index. Various types of transportation networks, such as urban traffic networks (Tian et al 2016, Zhou andWang 2018), public transportation networks (Rodríguez-Núñez and García-Palomares 2014, Jenelius andCats 2015, Cats et al 2017), and logistics networks Chang 2021, He et al 2021), are included in this category. For such systems, average travel time, capacity reliability, and connectivity of the network have been widely used as key indicators. For example, He et al (2021) used a performance indicator based on the travel time of freight and analyzed the multimodal transportation networks including road, rail and waterway transportation.
Previous studies have shown that the robustness of the system depends on the network structure and the type of disruptions (Albert et al 2000, Holme et al 2002, Paul et al 2005, Wu and Holme 2011. For example, the contribution of loop structure to the robustness of supply network has been studied in detail (Katifori et al 2010, Kaiser et al 2020. Also, measures to quantify the robustness of a network have been proposed (Chen et al 2002, Sakakibara et al 2004, Al-Deek and Emam 2006, Scott et al 2006, Ellens and Kooij 2013, Qin et al 2013, Duan et al 2016, Darayi et al 2017, Liu et al 2017, Zhou and Wang 2018, Dong et al 2019. Furthermore, algorithms to enhance the robustness of the system by changing the network structure have been studied (Schneider et al 2011, Zeng and Liu 2012, Louzada et al 2013, Zhou and Liu 2014, Tang et al 2015, Chan and Akoglu 2016, Liu et al 2017, Sharkey et al 2021. The robustness of various systems has been successfully controlled by these studies; however, there are still gaps to effectively designing and operating robust network logistics for the following reasons. First, although many studies have been conducted, there are still very few on logistics networks. They might not be practically useful in the future logistics networks because they ignore the dynamical features of the delivery route (re) planning. For example, adaptive routing that takes into account current congestion and disruption patterns in the network can be effective in reducing travel time (Ezaki et al 2022). To take this mechanism into consideration, the metric should be based on the shortest paths when some links are not available. Second, although previous studies have elucidated methods to find important links to improve robustness, which can be used to find, protect or add important links, they do not provide a quantitative guide for how much capacity should be allocated to these links, given a level of acceptable risk of delivery delay and efficiency loss.
Here, we propose a new metric, which we call the substitutability centrality (SC), to quantify the contribution of each link to robustness. This metric quantifies how much traffic on a link would be increased if another link is blocked. It incorporates the network structure, demand pattern, risk factors, and cost of using links, while keeping the computation algorithm simple so that it can be easily understood as a measure of robustness. By allocating additional capacity resources proportional to SC, the robustness of the system can be effectively increased. As a proof of concept, we demonstrate that capacity allocation based on the SC effectively increases the robustness of the system using the previous model (Ezaki et al 2022). This metric does not require solving an optimization problem or simulating a model. Therefore, our metric has the advantage that it can be used to intuitively understand the importance of a link for robustness.
The remainder of this paper is organized as follows. In section 1, we describe the computation algorithm for the proposed metric and other baseline metrics. Additionally, we explain the simulation model (Ezaki et al 2022) that is used to demonstrate that our approach is useful for decreasing the travel time in the logistics network under disruptions. In section 2 we show how the proposed metric is computed on an example network and present the results of our experiments. Finally, in section 3, we discuss the results and implications of our approach.

Simulation model
To better contextualize the measures proposed in this paper, we first explain the simulation model (Ezaki et al 2022) we used to evaluate their performance. Since the simulations were performed for proof of concept, we used example parameter values.
The definition of the model is essentially the same as the one in Ezaki et al (2022), except for the network topology and allocation of traffic capacity. Here we give a brief description of the model. Consider a network of N distribution centers (DCs) that are connected by directed links. A link-ij (1 i, j N) represents that the DC-i at the beginning of the link can directly deliver goods to the DC-j at the end of the link. We denote the set of links by . The goods are represented by a set of packets. Cost and capacity values are defined for each link. The capacity value defines the maximum number of packets that can be transported on the link at one time.
In the network, m packets are generated with randomly selected origin and destination DCs. For simulations, we used a network with four inter-connected hub DCs, each of which was also connected to five leaf DCs. This mimics a common land logistics network. We set demand patterns for the origin and destination DCs, as shown in figures 1(a) and (b). Five leaf DCs of a hub DC had large demands of origin of D origin = 3, while the other leaf DCs had small demands of origin of D origin = 1. Another set of five leaf DCs had large demands of destination of D destination = 3, while the other leaf DCs had small demands of destination of D destination = 1. The four hub DCs had no demands of origin and destination.
The origin and destination DCs were selected with a probability that is proportional to each demand value, respectively. When a packet was generated, its delivery path was determined by the adaptive fastest path algorithm, which was proposed by Ezaki et al (2022). Briefly, the algorithm finds the path with the shortest travel time (i.e. the fastest path) that satisfies the capacity constraint, given the planned paths of other existing packets, by finding a bypath route that may not be the shortest path when the network is vacant, if necessary. If more than one fastest path exists, the one with the lowest cost value is selected. It also adaptively changes the path by recomputing the fastest path if the pattern of blocked links (which will be explained in section 1.4) in the network is changed. Once a packet had been delivered to its destination DC, it was removed from the system, and then a new packet with random origin and destination DCs was generated. This algorithm can find better paths than the traditional (static) shortest path algorithms, which do not consider the current congestion patterns.

Substitutability centrality (SC)
To define the importance of each link in the network for the allocation of traffic capacity, we consider three types of centrality measures, two of which were prepared for comparison.
The first centrality measure is based on the link betweenness centrality (BC) of the network (Freeman 1977). The BC of a link, ij BC c , is defined as where σ kl and σ kl (i, j) denote the number of shortest paths between DC k and l, and the number of shortest paths between DCs k and l that pass through link-ij, respectively. This measure quantifies the importance of links in a transportation network. In a previous study (Ezaki et al 2022), the authors showed that when the network was not disturbed, the capacity allocation that was proportional to the BC effectively reduced the travel time of delivery.
The second centrality measure is based on the demand-based betweenness centrality (DBC). The DBC of a link, ij DBC c , is defined as where P(k, l) is the probability that DCs k and l are selected as the origin and destination, respectively. This measure, which considers the occurrence frequency of paths, captures the effective usage demand of the link.
The third measure, which we propose as a measure of importance for robustness, is the SC. We denote the expected probability that link-ij becomes blocked by p ij . If there is no prior knowledge about this probability, one may assume a uniform probability distribution, p ij = 1/L, where L is the number of links in the network. In this study, we set p ij = 1/L. Note that, technically, this probability distribution minimizes the expected amount of blocked flow: where D ij is the demand on link-ij, under the assumption that p ij is proportional to the demand and inversely proportional to the resource R ij allocated to prevent disruption (i.e. p ij ∝ D ij /R ij ), with a given total resource available, In the optimal assignment, R ij ∝ D ij (see appendix A). We denote the DBC of link-ij of a network after removing link-kl by ij kl DBC, c -. This measure quantifies the usage demand for link-ij when link-kl is unavailable. If the difference between before and after the removal of link kl, when link-kl is unavailable. We define the demandbased substitutability centrality (DSC) measure as an expected increase in the demand as follows: If we include the negative contributions, the performance of this method deteriorates (appendix B, figure 5) because they tend to reduce the capacity of links with high betweenness centrality.

Capacity distribution
We define three types of the capacity of each link (i.e. the maximum number of packets that can be carried at once in the link), C ij BC , C ij DBC , and C ij DSC , using the three centrality measures, ij where X represents BC, DBC, or DSC, and α is an arbitrary constant that controls the total capacity. This distribution allocates αL of capacity resources on an existing network having a uniform capacity distribution with C ij = 1, in proportion to the selected centrality measure. This ensures that the network does not become disconnected by links with 0 ij X c = . The sum of the link capacity values is equal to (1 + α)L. Note that Ezaki et al (2022) used uniform capacity distributions (i.e. C ij = 1 and C ij = 2) and C ij BC with α = 1.
By comparing the performance of C ij BC , C ij DBC , and mixtures of C ij DBC and C ij DSC , we show that increasing the capacity of links considering the SC can reduce the travel time of delivery when the network is disturbed.

Simulation settings
As a demonstrative example, we used a hub-and-spoke network that has four major (i.e. hub) and twenty minor DCs (figure 1). The use of this network was motivated by a previous study (Ezaki et al 2022). We set the demand patterns for the origin and destination DCs as shown in figures 1(a) and (b), respectively. Similar to the previous study (Ezaki et al 2022), the cost values defined for each link were drawn uniformly at random from the interval [0.5, 1.5] (an example shown in figure 1(c)).
The number of packets in the system varied between 10 and 1000. To compute the capacity value, we set α = 20. For each run, we performed a simulation for T 1000 max = time steps. For each condition, we performed 10 runs and averaged the results.
We consider a single disruption-free baseline scenario and two types of scenarios with disruptions, which we call the random-disruption and targeted-attack scenarios. In the random-disruption scenario, M( = 2, 4, or 6) links were randomly selected from 12 links wired between hub DCs and were blocked for five time steps, after which another M links were reselected to be blocked. In the targeted-attack scenario, the link with the highest DBC, χ DBC , was permanently removed from the network. (figure 1(a)) and destination ( figure 1(b)) of deliveries. Unlike the BC ( figure 1(d)), the DBC (figure 1(e)) reflected the effective usage demand of the link. The DSC (figure 1(f)) showed different patterns from the DBC. The links with high DSC values will be used for bypath routing when the system is disrupted. Figures 2(a)-(c) show the capacity allocation based on these centrality measures (equation (5)). As an example, we selected a mixture of 70% C ij DBC and 30% C ij DSC , which was found to be optimal by scanning various mixture ratios (figure 3). The optimal mixing ratio depends on the situation and the level of risk assumed, which will be discussed later.

Simulation results
First, we compare the performance of the three types of capacity allocation. We focus on the average travel time, which is defined by the average number of time steps spent for each delivery. In the baseline scenario, the capacity allocation based on the BC resulted in a longer travel time than the other two types of capacity allocation ( figure 3(a)). The DBC allocation slightly outperformed the 70% DBC + 30% DSC allocation. The travel costs were similar among the three types of capacity allocations ( figure 3(b)). In contrast, in the random-disruption scenario (M = 6), the 70% DBC + 30% DSC allocation succeeded in reducing the travel time as compared to the DBC allocation (figure 3(c)) without increasing the travel cost ( figure 3(d)).
In both scenarios, the choice of the capacity distributions had no discernible impact on the travel cost (figures 3(b) and (d)), which is consistent with the results of the previous study (Ezaki et al 2022).
We also varied the mixture ratio of the DBC and DSC capacity distributions (figures 3(e)-(h)). In the baseline scenario, an increase in the fraction of the DSC increased the travel time (figure 3(e)) because capacity resources are allocated to the links that are not frequently used. In contrast, in the random-disruption scenario, the travel time was minimized when the fraction of DSC was around 30% ( figure 3(g)).
The results remained qualitatively similar when we reduced the number of blocked links (M = 2 and 4; figures 4(a)-(d)) except that the optimal fraction of DSC was smaller.
In the targeted attack scenario, the travel time was significantly reduced when the fraction of DSC was around 30% (figure 4(e)) without increasing the travel cost ( figure 4(f)).
Note that, given that the average cost for using a link is equal to 1, the variations in the travel cost were negligible in all the cases.

Conclusion
In this paper, we proposed a measure, the DSC, which quantifies the importance of links in improving the robustness of a logistics network. We also performed a case study assuming a simple model and network and showed that for some conditions, increasing the capacity of the links with large DSC values indeed reduced the travel time when the network was disturbed. We also studied how much proportion of the capacity distribution should be allocated based on the DBC and DSC values. If no risk of network disturbance is assumed, the capacity allocation based on the DBC is the most effective. However, if disruptions to the network are anticipated, increasing the capacity of the links with large DSC will effectively reduce the travel time of deliveries. If the capacity is determined only by the DBC, the system cannot cope with the baseline demand for each link because the DSC does not take into account how much the link is normally used. Thus, in considering the capacity, there is an optimal ratio of the DSC to DBC. In our example simulations, the optimal ratio ranged from ≈0.1 to ≈0.3. The optimal ratio should be affected by the type of disruption, number of packets, and network structure, and thus it is not straightforward to find it. Additionally, the level of risk assumed and how much cost can be paid to assure the robustness of the network are also important factors. Our contribution is to provide an efficient way to perform such analyses. Although simulations must be performed for a practical scenario, the effective capacity allocation can be searched with a single parameter (i.e. the fraction of DSC). Also, even without simulations, simply computing the DSC (figure 1(f)) provides useful insights into which link should be reinforced first.
We demonstrated the usefulness of the proposed measure (i.e. the DSC) by using a model we introduced in the previous study and three types of example scenarios. Although the performance of using the DSC may vary depending on the scenario and model assumed (or the practical situation considered), we believe that the DSC provides a useful baseline for finding important links for the robustness of the network because it simply quantifies how much each link is used as a bypass route when a link in the network is disrupted. In our results, the standard deviation of the travel time was substantially large even when the system had no disruption, which was due to the random nature of the model. Even under such randomness, the choice of capacity allocation strategy substantially impacted the average travel time, which reflects the total travel time (which is converted to cost in reality) and is the most important criterion in engineering logistics networks. Note that the effect of sampling errors in such averages are negligible because they were computed from very large samples.
The potential limitation of the DSC is that it does not consider the situations in which more than one link is disrupted. However, in our simulations, the capacity allocation using the DSC was effective even when more than one link was disrupted. Thus, we believe that this limitation has a minor impact on the performance in practical scenarios.
We assumed static demand patterns for the origin and destination DCs. While this is a reasonable assumption as the first step, its variability (Waller et al 2001, Clark andWatling 2005) should be incorporated in future work. Also, the capacity of each link may vary. It should be fruitful to consider strategies to dynamically allocate capacity resources, achieving both efficiency and robustness (Jin 2007, Ben-Akiva et al 2012. In this paper, motivated by Ezaki et al (2022), we focused on the hub-and-spoke network with relatively few bypass routes that mimics common traffic between DCs to provide insights into how to increase the robustness of such systems. While we confirmed that the DSC metric is useful for this type of lean networks, it may not be useful for redundant networks. In fact, for networks with many bypass routes, the capacity allocation based on the DSC did not outperform the one based on the DBC (appendix C, figure 6). This is because the effect of the DSC-based capacity allocation is small, since each packet can find bypass routes without significantly increasing travel time and cost.
The DSC requires computations of betweenness centrality for each of L links for L patterns of removing a link. This can become computationally expensive for large and dense networks. However, it may be affordable for many practical problems. For example, it took 20 min » to process a hub-and-spoke network with 520 nodes and 1380 links with a common single-core CPU (appendix D, figure 8). In cases where computational cost is prohibitive, several alternatives can be considered such as use of parallel computation or use of efficient approximate algorithms for computing betweenness centrality, and only taking into account the contributions from links with high betweenness centrality.
We proposed the SC as a robustness measure for logistics networks, but we believe that it is also useful for analyzing other types of transportation networks, such as urban traffic networks (Thonhofer et al 2018, Ding et al 2019 and public transportation networks (Mandl 1980, Schöbel 2007. Also, the SC of multimodal

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Appendix A. Derivation of optimal allocation of capacity resources
Here, we minimize equation (3) by finding a set of optimal resource allocation R ij under a constraint , using the method of Lagrange multipliers. We rewrite equation (3) and the resource constraint where Z is a normalization constant. Then, we find a stationary point for the Lagrangian function, F = f − λg. By solving   Note that the random network and BA network were generated at random in each run of simulations.

ORCID iDs
Takahiro Ezaki https:/ /orcid.org/0000-0003-4175-3028  We tested the hub-and-spoke network used in the main text (with 4 hub nodes and 5 × 4 leaf nodes), and two different hub-and-spoke networks with 8 hub nodes and 10 × 8 leaf nodes and 20 hub nodes and 25 × 20 leaf nodes, respectively. Hub nodes were completely connected to each other and leaf nodes were connected to one of the hub nodes. For each network, we randomly generated cost values and measured the execution time of computing the DSC, which was repeated 10 times. Each marker represents the computation time of a single run.