The effect of different initial solutions on the metaheuristic algorithms for the single allocation p-hub center and routing problem

Introduction

Heuristic algorithms are methods developed to solve problems more efficiently when traditional techniques are either too slow for finding an exact or approximate solution or unable to find any exact solution within a given search space. The goal of a heuristic algorithm is to generate a solution within a reasonable time frame that is sufficiently effective for addressing the problem at hand. Although this solution may not be the optimal one or might only approximate the exact solution, it remains valuable because it can be obtained without taking an excessive amount of time.

Generally, heuristics are classified into improvement and constructive heuristics, based on their respective strategies to solve the problems. Constructive heuristics begin their approach with an empty solution, subsequently employing an iterative process to progressively build the complete solution. Hence, constructive heuristics do not explore the entire solution space. Instead, they utilize a specific type of strategy to determine the approach for constructing the whole solution. As implied by their name, constructive heuristics generate solutions by composing their individual components, as opposed to improving whole solutions, which is accomplished by sequentially introducing one element at a time to an incomplete (partial) solution. Constructive heuristics commonly adjust and utilize greedy algorithms, where the best possible element is incorporated during each step. Nonetheless, due to using some greedy approach in building whole solutions, constructive heuristics might have the potential to obtain local optima.

Improvement heuristics require an initial solution to start the exploration of the solution space for the given problem, aiming to reach the best possible solution. This is done through a series of iterative steps, each deploying a range of diverse neighborhood operations. Improvement heuristics have the potential to yield good solutions for combinatorial optimization problems due to their comprehensive exploration of the solution space.

Improvement heuristics rely on an initial solution to start the exploration of the solution space for the given problem, aiming to reach the best possible solution. There are two main sources of the initial solutions which are: a random solution generation mechanism and constructive heuristics. The former generates solutions using random processes, the latter is an iterative method. Both can quickly generate/build complete solutions from scratch that can be integrated into complex constructive or iterative heuristics.

It should be noted that constructive heuristics are designed with the problem at hand in mind. For instance, the “Savings Algorithm” of Clarke & Wright (1964) can be an example of such constructive heuristic when the particular problem to be addressed is the vehicle routing problem (VRP). Throughout the years, several variations and specializations of the VRP have been introduced (see Elatar, Abouelmehdi & Riffi, 2023 for a comprehensive list of problem variants). Consequently, the Savings algorithm was reformulated to handle the time-window variant in Solomon (1987), and it may well be extended to other VRP variants in the future. However, it cannot be used to solve an arbitrary combinatorial optimization problem such as scheduling. Thus, the problem and the heuristics are tightly coupled. For this very reason, the existing studies investigating the impact of the initial solution, in which problems and heuristics to form initial solutions are closely intertwined, consider only a single problem type per publication at a time.

Metaheuristic algorithms, in contrast to heuristic algorithms which are designed to solve a wide range of optimization problem types, are adaptable and can be employed across various problem domains. They are generally classified into two groups: population-based or single trajectory. The former concentrate on maintaining and enhancing multiple candidate solutions (genetic algorithms, ant colony etc.), while the latter focus on modifying and improving a single initial solution (simulated annealing, tabu search etc.). The initial solution in a metaheuristic algorithm constitutes a critical element impacting its capacity to ascertain solutions of superior or acceptable quality. In the absence of any prior domain-specific information about the problem at hand, it is customary for the initial solution to be predominantly formed through entirely random procedures. When information about the problem is available, constructive heuristics can be utilized to generate an initial solution.

Metaheuristic algorithms typically comprise iterative optimization algorithms, often sharing a common step in their algorithmic framework, namely solution(s) initialization. Within the research community, it is widely acknowledged that this initialization step holds significant importance in the optimization process. This is because all subsequent solutions generated rely, to some extent, on their predecessors and ultimately on the initial solution or initial population of solutions. In the literature, the impact of initial solutions on metaheuristics has predominantly been examined within the context of population-based metaheuristics. As pointed out in a recent by review article by Sarhani, Voß & Jovanovic (2023), which clearly reveals the research gap on initialization of metaheuristics, including single trajectory-based metaheuristics.

The ‘initial solution effects’ line of research is summarized in the following paragraph. These problems extend from the quadratic assignment problem to energy resource planning, and from the production routing problem to transmission expansion planning. The effects of the initial solution on an algorithm are examined as stated above, and each article addresses a specific problem.

In Liu’s (2021) study, the impact of the initial solution was examined for a genetic algorithm developed for the quadratic assignment problem. The following initial solution strategies were proposed: random, random plus local search, random plus bad local search, a greedy randomized adaptive search procedure (GRASP), and a greedy randomized adaptive search procedure plus local search. When the genetic algorithm ran on top of different initial solution strategies, the authors observed that the quality of the final (best-found) solution was formed in reverse order to the above ranking. Antonio, Ramon & Ruben (2011) investigated how selecting good initial populations for genetic algorithms influences both the speed of convergence and the overall quality of the final solutions of transmission expansion planning problem. Sousa et al. (2016) proposed five different initial solutions for a simulated annealing algorithm to solve the energy resource scheduling in smart grids. These initial solutions are (a) random solution; (b) ant colony optimization; (c) naive-scheduling heuristic; (d) pre-scheduling heuristic; and (e) mixed-integer linear programming. The authors used the results of a relaxed version by removing certain constraints from the resource scheduling problem in the mixed-integer linear programming based initial solution. Ahmed, Hvattum & Agra (2023) investigated the impact of the initial solutions obtained by mathematical modeling relaxation of a two-commodity flow formulation, a three-index flow formulation, and a four-index flow formulation for the production routing problem on the performance of the proposed matheuristic. Oman & Cunningham (2001) investigated how seeding genetic algorithms with high-quality initial solutions-evaluated based on their objective values-impacted performance on the traveling salesman problem (TSP) and the job-shop scheduling problem. Burke, Newall & Weare (1998) applied various initialization strategies to the population of a genetic algorithm for the timetabling problem. In their study, Fischetti, Ljubic & Sinnl (2016) evaluated the performance of three distinct initial solutions for the prepack optimization problem using an ad-hoc heuristic. These initial solutions were constructed through three different approaches: random generation, selecting the most dissimilar options, and the most similar ones. Their effectiveness was further assessed by incorporating a refinement heuristic.

In line with the literature review provided in the above paragraph, this study also focuses on a single problem, namely the single allocation p-hub center and routing problem. The primary motivation of this study is to investigate the effects of three different constructive heuristics developed for the single allocation p-hub center and routing problem on a single trajectory metaheuristic, which is simulated annealing (SA). The reason for selecting SA is the need for a well-balanced interplay between diversification and intensification throughout the search process (Assad & Deep, 2018). Intensification, commonly referred to as exploitation, signifies the algorithm’s ability to exploit the search space surrounding the current good solution, while diversification, also known as exploration, entails exploring new regions of the search space to introduce novel information. We also note here that SA had been selected due to its recognized capability to strike a proper balance between these two conflicting characteristics in combinatorial optimization problems and especially in hub location and routing problems (Ernst & Krishnamoorthy, 1996; Kartal, Krishnamoorthy & Ernst, 2019; Ghaffarinasab, 2018).

Despite the existence of a wide variety of metaheuristics in the literature, several interesting metaheuristic algorithms have also been introduced in recent years. Interested readers may examine a novel genetic algorithm for solving the fuzzy dynamic ship routing and scheduling problem (Das et al., 2022), a novel discrete rat swarm optimization algorithm designed for the quadratic assignment problem (Mzili et al., 2023b), a discrete penguins search algorithm proposed for the multiple traveling salesman problem (Mzili, Mzili & Riffi, 2023), and hybrid genetic and penguin search optimization algorithms developed for the flow shop scheduling problem (Mzili et al., 2023a).

In the literature, the performance of metaheuristic algorithms is influenced by factors such as parameter optimization and the quality of the initial solution used. While considerable attention has been devoted to parameter tuning, the impact of a strong initial solution has often been overlooked. Yet, an effective initialization can play a role similar to that of parameter tuning by increasing the likelihood of reaching better final objective function values, especially when early-stage diversification is incorporated. Most studies that focus on initialization tend to address population-based algorithms (Li, Liu & Yang, 2020; Oman & Cunningham, 2001), leaving its effect on single trajectory-based metaheuristics less explored.

Research focusing on the impact of initial solutions in single trajectory-based algorithms remains scarce. For instance, Alfonzetti, Dilettos & Salerno (2006) demonstrated that in SA, the algorithm restarts with a new starting solution when no improvement is observed; a mechanism that often involves randomly selecting a new starting point or choosing the best candidate from a randomly generated pool to escape local optima. Similarly, Li, Liu & Yang (2020) showed that even if the initial solutions are not near optimal, the algorithm can still yield good outcomes provided that high diversity is maintained, and sufficient iterations are performed. This extensive search allows the algorithm to effectively explore the solution space, mitigating any drawbacks of an initial starting point.

Building on these insights, our study addresses this gap by investigating how the quality of the initial solution affects the final outcome in single trajectory-based metaheuristic algorithms. Specifically, we focus on the uncapacitated single allocation p-hub center and routing problem; a decomposable problem and evaluate three constructive heuristics, designed to exploit its structure within a simulated annealing framework.

The uncapacitated single allocation p-hub center and routing problem allocates demand centers to hubs and designs vehicle routes in such a way that the maximum distance is minimized between all origin-destination pairs (Kartal, Krishnamoorthy & Ernst, 2019). This problem is a combination of two well-known problems in the literature, namely the uncapacitated single allocation p-hub center (USApHCP) and the (multiple) traveling salesman problem (mTSP). USApHCP is NP-hard, which has been proven by Ernst et al. (2009). The NP-Complete class of the traveling salesman problem (TSP) has been proven by Karp (1972). Therefore, developing algorithms that are capable of obtaining optimal solutions for such problems within a reasonable time is not possible. As a result, a diverse range of heuristic algorithms can be employed to address both hub location and routing type problems. Although almost every hub location and routing problem in the literature takes into account different routing constraints and objective functions, in the following paragraph and Table 1 we provide an overview of both the metaheuristic and matheuristic algorithms developed for hub location and routing problems also the initial solutions strategies used strategies employed by these methods.

De Freitas et al. (2023) used a biased random-key genetic algorithm for the hub location routing problem with directed tours, incorporating a random initial solution. Aloullal, Saldanha-da-Gama & Todosijević (2023) introduced an input into matheuristic algorithms by relaxing integrality constraints formed by constraints involving only related to hub locations for the multi-period single-allocation hub location-routing problem, aiming to find a feasible solution. Wang, Liu & Yang (2023) developed a genetic algorithm for the robust hub location and routing problem with a third-party logistics strategy, generating initial solution populations based on random and greedy heuristics. For the multiple allocation hub location and routing problem, Wu, Qureshi & Yamada (2022) used a greedy approach as the initial solution for the adaptive large neighborhood search (ALNS) algorithm they developed. This approach was based on a two-stage construction phase. Bütün, Petrovic & Muyldermans (2021) proposed a tabu search algorithm for the capacitated directed cycle hub location and routing problem under congestion, suggesting a three-stage decomposition-based greedy heuristic for the initial solution. Yang, Bostel & Dejax (2019) developed a memetic algorithm for a hub location and routing problem with distinct collection and delivery tours, deriving an initial population consisting of random and greedy-based solutions. Danach, Gelareh & Neamatian Monemi (2019) solved the capacitated single-allocation p-hub location and routing problem using a feasible initial solution based on the Lagrangian relaxation technique. Clarke & Wright (1964) introduced three metaheuristic algorithms based on iterated local search, ant colony system, and discrete particle swarm optimization for the USApHCRP. The authors utilized random initial solutions for the iterated local search and discrete particle swarm optimization algorithm. Lopes et al. (2016) developed a biased random key genetic algorithm using both greedy-generated initial solutions and using random chromosomes for generating initial solutions for a hub location and routing problem. The authors also used a random solution for their variable neighborhood search-based heuristic. Ratli et al. (2022) addressed the problem studied by Lopes et al. (2016) and proposed a solution in the form of the General Variable Neighborhood Search algorithm. The initial solution was randomly generated. Pandiri & Singh (2021) also considered the same problem studied by Lopes et al. (2016). The authors developed two greedy heuristics for the initial solution to be used in the hyper heuristic. Kartal, Hasgul & Ernst (2017) utilized the initial solution obtained by the first iteration of an ant colony system algorithm in their study as the starting solution for the simulated annealing algorithm. Indeed, the final objective function results obtained by the simulated annealing algorithm are better than the final objective results obtained by the ant colony system. Rieck, Ehrenberg & Zimmermann (2014), worked on many-to-may location routing problem with inter-hub transport and multi commodity pickup-and-delivery and for the solution of the problem, the authors developed a fix-and-optimize heuristic and a genetic algorithm.

As motivation drawn from real-life applications, readers can refer to the works of Kuby & Gray (1993), Aykin (1995), Bruns, Klose & Stahly (2000), Grünert & Sebastian (2000), Wasner & Zapfel (2004), Cetiner, Sepil & Sural (2010), De Camargo, Miranda & Lokketangen (2013), and Rodríguez-Martín, Salazar-González & Yaman (2014) which incorporate routing decisions into the hub location problem for various versions of the problem. For interested readers, Wandelt, Wang & Sun (2025) provide a recent comprehensive review of hub location-routing problems (HLRPs), exploring the integration of hub location and vehicle routing to optimize logistics and transportation networks. The article examines various HLRP models, their mathematical formulations, and solution techniques, including exact and heuristic methods.

Recent algorithmic advancements in both metaheuristics and exact methods have led researchers to consider techniques from these two domains together. It has become common to integrate metaheuristic elements (typically local search) with exact algorithms devised for (mixed integer) linear programming. These techniques are called as matheuristics. Archetti & Speranza (2014) have categorized matheuristics into three distinct classes in routing related problems. The first of these is the decomposition approach, where the method subdivides the problem into smaller and less complex subproblems, each of which is addressed with a particular solution technique. The second class is improvement heuristics. In this type of matheuristics, the solution is improved by using mathematical programming models. The third class is branch-and-price/column generation-based approaches. These approaches are commonly used in solving routing problems. Such algorithms utilize a set partitioning formulation where a binary or integer variable is associated with each possible column. Given the exponential number of variables, the solution of the linear relaxation of the formulation is achieved through column generation. In the context of matheuristics based on branch-and-price or column generation, the exact method is adjusted to enhance convergence speed, albeit at the expense of losing optimality assurance. For example, the column generation phase may be terminated prematurely.

In the literature of the hub location and routing problem, there are few studies that include a matheuristic algorithm. Aloullal, Saldanha-da-Gama & Todosijević (2023) developed a four-phase matheuristic for the multi-period single-allocation hub location-routing problem, combining principles of relax-and-fix. Geçer (2020) worked on fixed p-hub center and routing problem where the locations of the hubs are predetermined. The author suggested a mathematical programming formulation for the problem. For finding an initial solution with a mathematical model, we modified Geçer’s (2020) mathematical programming formulation to form the routes with a min-max multiple travelling salesman problem. Unlike Geçer (2020), the locations of the hubs are not known in priori, and we find the hubs by using a modified radius-based p-hub center mathematical model and for creating the routes we use a modified mTSP formulation with min-max objective function to develop a matheuristic based strategy.

In this study, inspired by the concept of starting from a high-quality initial solution obtained through a constructive algorithm; an approach widely used in local search metaheuristics, we investigate the impact of various initial solution strategies on the final solution of the USApHCRP. To generate initial solutions, we propose four strategies: random, greedy, a hybrid approach combining random and greedy methods, and a decomposition-based matheuristic that formulates the problem in two hierarchical parts—hub location and node allocation, followed by routing part. These initial solutions serve as starting points for the proposed SA algorithm.

The second section presents the proposed initial solution strategies, detailing the random, greedy, hybrid, and decomposition-based matheuristic approaches, along with the modified formulations of the USApHCP and mTSP. The third section focuses on the specifics of the developed SA algorithm. The fourth section provides numerical results and their detailed analysis. The fifth section includes a discussion of the findings, and finally, the conclusion section summarizes the key insights and offers suggestions for future research.

Initial solution strategies

USApHCRP involves determining hubs, assigning non-hub nodes to hubs, and creating the associated routing structure. The mathematical model for this problem was initially presented in the literature by Geçer (2020) in three formulations: two-index vehicle-flow, three-index vehicle-flow, and finally, the four-index multi-commodity flow-based formulation. For the mathematical model of the problem, references can be made to Geçer (2020) and Kartal, Krishnamoorthy & Ernst (2019).

Before outlining the initial solution structures, a corresponding illustration of a hub location and routing example is presented in Fig. 1.

Figure 1 illustrates a hub location and routing structure. In this example, nodes 1 and 2 serve as hub nodes. Each route starts and ends at its respective hub node: the first hub follows the route 1 → 5 → 7 → 6 → 1, while the second hub follows the route 2 → 8 → 4 → 3 → 2. As observed, the routes involve multiple stopovers, with the primary objective being the minimization of the maximum distance or cost between any origin-destination pair. To accurately compute this maximum value, the model differentiates between collection and distribution routes for each vehicle. In many real-world scenarios, vehicles often return along the same path they took outbound, albeit in reverse order; thus, the outbound journey is considered the collection route, while the return journey is treated as the distribution route.

Simulated annealing

Simulated annealing (SA) a stochastic search method originally proposed by Kirkpatrick, Gelatt & Vecchi (1983), is employed to address optimization problems. It has been successfully applied to numerous large-scale real-world problems including TSP. Due to its ease of implementation, convergence characteristics, and the strategy it employs to avoid getting trapped in local optima. SA has become a widely used method for solving optimization problems in recent years. The name “Simulated Annealing” is derived from its analogy to the annealing process in solid materials. SA operates as a gradual refinement process. This progression involves the consideration not only of improved solutions but also the acceptance of poor solutions with a certain likelihood. This strategy leads the algorithm to escape from being trapped in local optima. This process constitutes one of the fundamental characteristics of the SA method. The acceptance probability is expressed as $e^{(-\mathrm{\Delta }E/T)}$ based on a conceptual temperature. Here, ΔE represents the change in objective functions between the current solution and the generated neighboring solution. In this context, T represents the control parameter, which corresponds to the temperature. For small values of $\mathrm{\Delta }$E, the likelihood of accepting a poor solution is higher than for larger values. Additionally, at higher temperature values, most new neighborhood solutions will be accepted. As the temperature approaches 0, the acceptance probability of new neighborhood solutions will decrease. Hence, in the SA algorithm, the initial temperature is typically set to a relatively high value to escape local optima.

Simulated annealing neighborhood moves

The hub location and routing problem representation for the SA is structured as follows: In this solution representation, we selected the problem structure from Fig. 1 for clarity and ease of understanding for the reader. Accordingly, the bold numbers 1 and 2 represent the hubs, while the nodes adjacent to the hubs are the non-hub nodes included in the route, listed in the order they are visited is shown in Fig. 2.

As seen in Fig. 2, the routes start and end at hub nodes 1 and 2. In the route of hub 1, the nodes 6, 7, and 5 are visited sequentially, departing from the hub and returning to it (1 → 6 → 7 → 5 → 1). Similarly, in the route of hub 2, the nodes 3, 4, and 8 are visited in order, before returning to the hub node 2. In Table 2, we provide all SA neighborhood moves, demonstrating the independent application of each to the initial state to derive the perturbed solution/state for better clarity.

Table 2:

Example of each move applied to the initial state independently to obtain the perturbated solution/state.

No	State/Move type	Route 1	Route 2
--	Initial state/solution	1-6-7-5-9-1;	2-3-4-8-2;
1.	Inter-node-insertion	1-7-5-6-9-1;	2-3-4-8-2;
2.	Intra-node-insertion	1-6-5-9-1;	2-3-4-7-8-2;
3.	Inter-node-swap	1-9-7-5-6-1;	2-3-4-8-2;
4.	Intra-node-swap	1-6-7-8-9-1;	2-3-4-5-2;
5.	Inter-hub-swap	7-6-1-5-9-7;	2-3-4-8-2;
6.	Inter-edge-opt	1-5-9-6-7-1;	2-3-4-8-2;
7.	Intra-edge-opt	1-6-4-8-9-1;	2-3-7-5-2;

DOI: 10.7717/peerj-cs.2840/table-2

Note:

The bold values in Table 2 indicate the hub locations. Italicized nodes indicate the ones that have undergone positional changes due to the corresponding neighborhood move operations.

Table 2 summarizes the solution representation for the initial state (the first row), while the subsequent rows present the perturbed solution representations obtained after applying each move type independently to the initial state. The bold values in Table 2 indicate the hub locations. Italicized nodes indicate the ones that have undergone positional changes due to the corresponding neighborhood move operations. In the following, we explain how the neighborhood moves operate and provide examples for each.

1. Inter-node-insertion: A randomly selected node from a randomly selected route is placed next to another randomly chosen city within the same route. In routes containing only one node, both the node to be moved and the node where the operation will take place are the same, resulting in no change. Therefore, intra-insertion cannot be performed within routes containing only one non-hub node. An example of an insertion within the same route is provided in the first row of Table 2, where node 6 has been inserted after node 5 within the same route (Route 1).

2. Intra-node-insertion: A randomly chosen node from one route is placed adjacent to a randomly selected node from another route (these two different routes may belong to the same hub), both of which are chosen randomly. If a non-hub node is chosen where there is only one non-hub node for relocation, the insertion is prohibited, it would result in only a hub node remaining in the route. An example of inter-route insertion is shown in the second row of Table 2, where node 7 has been relocated and positioned after the randomly chosen node 4 in a different vehicle’s route.

3. Inter-node-swap: A route containing at least two cities is randomly selected, and two cities are then selected randomly from this route to swap their positions. The third row of Table 2. illustrates an example of a swap within the same route, where node 9 and node 6 have exchanged positions in Route 1.

4. Intra-node-swap: First, two routes are randomly selected. Then, one city is randomly chosen from each of these routes, and their positions are swapped. The fourth row of Table 2, presents an example of inter-route swap, where node 5 from the Route 1 and node 8 from the Route 2.

5. Inter-hub-swap: A node is randomly chosen from within a randomly selected route, and this node is located as the new hub, while the node that was previously the hub is placed at the location where the new hub is chosen. An example of the inter-hub swap is provided in fifth row of Table 2, where the hub node (node 1) has swapped positions with node 7 within its route.

6. Inter-edge-opt: Exchange the positions of two consecutive non-hub nodes within a single route. An example is provided in sixth row of Table 2, where two edges from hub 2’s route have been swapped—specifically, edges 5–9 and 6–7 have been exchanged.

7. Intra-edge-opt: Swap the positions of two consecutive non-hub nodes (edges) between two routes that are randomly selected. An example of intra-edge-opt is provided in seventh row of Table 2, where the 4–8 edge from Route 2 has been swapped with the 7–5 edge from Route 1.

Computational results

Tan & Kara (2007) introduced Turkish Cargo Delivery (Turkiye network) into the literature. There are 81 demand centers in this dataset, flow amongst these demand centers, and distances in this network. The Turkish network dataset is also available through the OR library (Beasley, 1990).

We employed SA with four different initial solution strategies independently to address the USApHCRP, and Gurobi Optimizer version 9.0.2 was utilized to run mathematical models. We conducted the experimental tests on a server operating with the UBUNTU operating system, equipped with a 3.5 GHz Intel Xeon E5-2643 v2 processor and a total of 128.863 GB RAM, with 30 GB allocated to the heap. The SA algorithm was implemented in Java, while the Gurobi components were implemented in Python. The code is available at https://github.com/AbdKa/pHMLRP.

Discussion

In this section, we discuss the advantages and disadvantages of our proposed methodology, highlighting key trade-offs. This discussion not only revisits our primary research objectives but also contextualizes our findings within the broader literature.

Our study has several key strengths. First, we show that tailored initial solutions can greatly improve solution quality within a single-trajectory metaheuristic, specifically the SA algorithm.

Our initial solution strategies integrate problem-specific techniques from matheuristics and greedy heuristics, drawing on expertise knowledge. This approach takes advantage of the problem’s decomposable structure, resulting in good solutions.

Furthermore, our methods exhibit strong performance on large-scale problem instances. While NP-hard problems are inherently challenging, employing matheuristic-based and greedy initial solution strategies provide a competitive advantage over established methods, as demonstrated in our comparison with Kartal, Krishnamoorthy & Ernst (2019).

Notwithstanding the advantages, the matheuristic initial solution approach also has some limitations. The main drawback is the significant computational time required to solve mixed-integer programming models using a commercial solver to achieve optimal solutions, especially for large-scale problems. As the problem size increases, the computational time required on the solver grows, prompting us to impose a 2-h time limit on the solution process. While this constraint ensures practical computation times, it creates a trade-off between solution quality and efficiency.

Additionally, since SA is a single-trajectory-based metaheuristic that perturbs solutions through neighborhood moves and often accepts worse solutions, especially in the early stages, the impact of the initial solution may diminish as the algorithm progresses. This effect is particularly evident in the case of random initial solutions, where despite starting from poor-quality solutions, the final objective function values remain competitive. However, this may not completely eliminate the benefits of starting from a good initial solution, such as those generated by matheuristic or greedy approaches. On the contrary, starting from a well-structured solution is likely to have a positive impact, particularly in increasing the chances of outperforming existing methods in the literature.

Conclusions

This study addresses the uncapacitated single allocation p-hub center and routing problem by proposing an SA algorithm enhanced with four distinct initial solution strategies. Our findings suggest that the quality and structure of the initial solution can significantly impact the performance of an improvement-type metaheuristic, particularly across different problem structures, such as varying hub numbers and vehicle configurations. Our analysis indicates that random initialization is more effective for smaller problem instances, while greedy-based initialization performs better in cases with fewer hubs. This might be because the greedy approach selects hubs based on proximity to other nodes, often selecting more central locations. When the number of hubs is low, this approach might lead to a more balanced distribution of routes. The random-greedy strategy effectively balances exploration and exploitation, producing strong results across multiple instances. In contrast, the matheuristic-based strategy yields the most significant gains in larger problem instances. Specifically, the matheuristic approach employs the decomposition of the problem, allowing the underlying mathematical models to find either exact solutions or high-quality approximations within the given time constraints.

Our research contributes to the literature in two folds. First, it systematically examines how different initial solution generation strategies affect the performance of a single-trajectory metaheuristic, filling an existing gap where most studies have focused on population-based methods. Second, it introduces a novel matheuristic algorithm for the USApHCRP, which utilizes a modified p-hub center and an mTSP-based routing formulation, demonstrating the potential of hybrid methodologies in complex network optimization problems.

The initial solution strategies developed in this study are problem-specific, constructive-based algorithms that incorporate expert knowledge. This underscores the importance of domain expertise in designing effective constructive heuristics for decomposed problem structures. The integration of expert-driven insights into constructive solution generation could significantly influence the final outcomes of an optimization algorithm. Given its potential advantages, the methodologies developed in this study may also be applicable to similar problem structures where location-routing decisions play a crucial role.

The computational results on the Turkish Network validate the effectiveness of our approach, as our SA algorithm outperformed other metaheuristics in the literature. These promising results encourage further exploration into the use of constructive heuristics within metaheuristic frameworks.

Future work could extend our methodology to other variants of the hub location and routing problem, such as capacitated or multiple allocation versions, and explore additional hybrid strategies that integrate metaheuristic and exact optimization techniques. Expanding these results to other domains, it is evident that any problem with an underlying location-routing model could benefit from this approach. Additionally, future research may compare the performance of single-trajectory and population-based algorithms for this decomposable network design problem, broadening the applicability of our findings and contributing to the ongoing development of efficient solution methods for combinatorial optimization challenges.