This study tackles the multiple allocation -hub location problem (MApHLP) in the context of video-on-demand services, where digital content is partitioned into segments and stored exclusively at selected hub locations. Users, distributed across a wide geographical area, can connect to multiple hubs based on demand patterns, with all hubs assumed to be fully interconnected. The goal is to jointly determine hub locations, segment placement, user-to-hub assignments, and optimal routing paths to minimize total routing costs. To solve this complex problem, an enhanced adaptive large neighborhood search (EALNS) algorithm is proposed. The algorithm features an efficient objective evaluation strategy for rapid assessment of candidate solutions and employs adaptive operator selection that dynamically updates neighborhood strategies based on their past performance. The proposed EALNS algorithm is benchmarked against solutions obtained from CPLEX and a differential evolution-based metaheuristic. The reported results confirm its efficiency and competitiveness in addressing large-scale MApHLP instances.

This paper addresses the Bi-objective Maximal Covering Location Problem with Compactness (BOMCLP-C), a new variant of the classical maximal covering location problem that simultaneously maximizes total demand coverage and minimizes the spatial dispersion of selected facilities. The problem is NP-hard and exhibits a highly multimodal search space due to the combinatorial interaction between coverage and compactness objectives under cardinality constraints. To effectively approximate its Pareto front, two evolutionary multi-objective optimization paradigms are investigated: NSGA-II, representing dominance-based search, and MOEA/D, representing decomposition-based search. For each paradigm, two variants are implemented: a penalty-based formulation that relaxes the facility-count constraint through additive penalties, and a customized constraint-handling variant enhanced with local search (LS) that maintains feasibility and refines solutions in the neighborhood structure. Computational experiments on real-world instances drawn from the literature demonstrate that LS-based variants consistently achieve higher-quality Pareto fronts, attaining full feasibility and superior hypervolume values. A Wilcoxon signed-rank analysis confirms the significant performance difference between NSGA-II-LS and MOEA/D-LS. The study shows the effectiveness of integrating problem-specific constraint handling and local improvement in evolutionary multi-objective frameworks for large-scale discrete location optimization. It emphasizes the need for concise, feasible space representation when dealing with integer or combinatorial constraints.

In this article, one of the most commonly encountered problems in manufacturing systems, known as the Tool Indexing Problem (TIP), is considered. TIP involves allocating cutting tools to different slots in a tool magazine of a Computer Numerically Controlled (CNC) machine to reduce the processing time of jobs on the machine. In this article, three mixed-integer linear programming formulations of single-objective TIP without tool duplication and lifespan are presented. A comparative study of these three linear formulations of TIP is also included in this article. During the study, it was found that the exact solver CPLEX with these linear formulations struggles to find optimal solutions in a reasonable time for larger instances. Therefore, a Hybrid Large Neighborhood Search with Local Search (HLNS-LS) algorithm, which is a metaheuristic approach, is proposed for solving TIP. The LNS phase iteratively destroys and repairs solutions to explore different regions of the solution space, while the LS phase intensifies the search by applying multiple neighborhood operators, such as swap, insert, shift, and customized 2-Opt and 3-Opt, to refine solutions further. This hybrid approach balances diversification and intensification, aiming to find high-quality solutions that minimize the total turret rotation cost associated with tool indexing. The performance of the proposed HLNS-LS algorithm is evaluated against an improved Harmony Search algorithm, a Weighted Superposition Attraction-based algorithm, and a Constraint Programming model across 85 small-, medium-, and large-sized benchmark instances from the existing TIP literature.

This article introduces a geometric facility location problem that minimizes total Riesz interaction energy among p selected sites in the Euclidean plane. The objective generalizes the classical p-dispersion criterion by penalizing spatial proximity through inverse power distances, thereby promoting global dispersion in a smooth manner. The problem is shown to be NP-hard in general metric spaces. A Polynomial-Time Approximation Scheme (PTAS) is developed for the planar case for a truncated Riesz objective (for fixed p), with the same approximation guarantee extending to the original objective whenever the candidate set satisfies a natural minimum-separation condition. Furthermore, limiting analysis shows that the Riesz objective provides a smooth interpolation between strict dispersion and uniform selection, depending on the strength of the interaction exponent.

This article presents the Energy-Regularized Bi-Objective Maximal Covering Location Problem (ERBOMCLP), an extension of the classical Maximal Covering Location Problem (MCLP) that incorporates spatial dispersion among selected facilities. The classical MCLP maximizes covered demand under a fixed facility budget and service radius, but may place facilities in spatial clusters when high-demand areas are concentrated. ERBOMCLP addresses this limitation by augmenting coverage maximization with a Riesz s-energy objective that penalizes short inter-facility distances. To solve the model, we develop a memetic Non-dominated Sorting Genetic Algorithm II (NSGA-II) with binary fixed-cardinality encoding, problem-specific crossover and mutation operators, coverage-oriented local search, archive management, and a set-convergence-based termination criterion for detecting approximation-boundary stabilization. Experiments on ERBOMCLP instances derived from a real-world MCLP dataset show that the main memetic variant improves selected-solution coverage and often gives competitive or lower Riesz s-energy, while the spread of the Pareto front approximation remains instance dependent. A controlled hypervolume experiment with a Riesz-nonincreasing local-search rule further improves standard dominated hypervolume on the tested instances, at the cost of substantially more objective evaluations. The study connects potential-theoretic dispersion modeling with facility-location decision support for settings where high demand coverage is primary and dispersion is a complementary proxy for geographic balance.

This article considers a variant of the capacitated facility location problem, where each customer has a non-negative demand and each facility has a non-negative capacity limit and a service distance beyond which it cannot serve customers. A fixed number of facilities are opened from potential locations. The open facilities can only provide service to the customers incurring service costs. Due to the capacity limits and service distance of facilities, all the customers may not be serviced by the open facilities, and these customers can be outsourced incurring outsourcing costs. Each customer has its own non-negative outsourcing cost. The goal is to minimize the sum of service and outsourcing costs. This problem is termed the Capacitated Facility Location Problem with Service Distance and Outsourcing (CFLPSDO). A 0-1 integer linear programming formulation for CFLPSDO is presented in this article. For experimentation, 45 instances are generated from real-world maximal covering location problem instances. These are solved using the exact solver CPLEX to create benchmark solutions. Then, a customized Artificial Bee Colony (ABC) algorithm-based metaheuristic, with two operators for enhanced search space exploration and exploitation, is employed and found to produce optimal solutions efficiently.

In this article, an improved harmony search algorithm (IHSA) that utilizes opposition-based learning is presented for solving the maximal covering location problem (MCLP). The MCLP is a well-known facility location problem where a fixed number of facilities are opened at a given potential set of facility locations such that the sum of the demands of customers covered by the open facilities is maximized. Here, the performance of the harmony search algorithm (HSA) is improved by incorporating opposition-based learning that utilizes opposite, quasi-opposite and quasi-reflected numbers. Moreover, a local search heuristic is used to improve the performance of the HSA further. The proposed IHSA is employed to solve 83 real-world MCLP instances. The performance of the IHSA is compared with a Lagrangean/surrogate relaxation-based heuristic, a customized genetic algorithm with local refinement, and an improved chemical reaction optimization-based algorithm. The proposed IHSA is found to perform well in solving the MCLP instances.

We address the challenge of cost management for pre-disaster emergency funds, with ample warning time available for completing emergency operations. We formulate a mathematical model for a complex vehicle routing problem involving a single depot, a fleet of heterogeneous vehicles with limited capacities, and simultaneous delivery and pickup tasks. Each vehicle type is assigned a specific road network based on vehicle-road compatibility. We develop heuristic approaches to generate high-quality solutions for this problem and compare them with a state-of-the-art commercial solver. Our findings reveal that our heuristics perform exceptionally well for very large problem instances, while the commercial solver outperforms them for smaller instances. Moreover, our algorithms can handle scenarios where customers have either delivery or pickup demands, as well as cases where both operations are required. We evaluate the performance of our exact formulations extending existing data, as well as generate new data sets demonstrating the effectiveness of our bioinspired heuristic methods in achieving satisfactory outcomes.

The Maximal Covering Location Problem (MCLP) is concerned with the optimal placement of a fixed number of facilities to cover the maximum number of customers. This article considers a new variant of MCLP where both the coverage radii of facilities and the distance between customer and facility are fuzzy. Moreover, the finite capacity of each facility is considered. We call this problem the capacitated MCLP with fuzzy coverage area (FCMCLP), and it is formulated as a 0–1 linear programming problem. In this article, two classical metaheuristics: particle swarm optimization, differential evolution, and two new-generation metaheuristics: artificial bee colony algorithm, firefly algorithm, are proposed for solving FCMCLP. Each of the customized metaheuristics utilizes a greedy deterministic heuristic to generate their initial populations. They also incorporate a local neighborhood search to improve their convergence rates. New instances of FCMCLP are generated from the traditional MCLP instances available in the literature, and IBM’s CPLEX solver is used to generate benchmark solutions. An experimental comparative study among the four customized metaheuristics is described in this article. The performances of the proposed metaheuristics are also compared with the benchmark solutions obtained from CPLEX.

The maximal covering location problem (MCLP) is a well-known combinatorial optimization problem with several applications in emergency and military services as well as in public services. Traditionally, MCLP is a single objective problem where the objective is to maximize the sum of the demands of customers which are served by a fixed number of open facilities. In this article, a multi-objective MCLP is proposed where each customer has a preference for each facility. The multi-objective MCLP with customers' preferences (MOMCLPCP) deals with the opening of a fixed number of facilities from a given set of potential facility locations and then customers are assigned to these opened facilities such that both (i) the sum of the demands of customers and (ii) the sum of the preferences of the customers covered by these opened facilities are maximized. A Pareto-based multi-objective harmony search algorithm (MOHSA), which utilizes a harmony refinement strategy for faster convergence, is proposed to solve MOMCLPCP. The proposed MOHSA is terminated based on the stabilization of the density of non-dominated solutions. For experimental purposes, 82 new test instances of MOMCLPCP are generated from the existing single objective MCLP benchmark data sets. The performance of the proposed MOHSA is compared with the well-known non-dominated sorting genetic algorithm II (NSGA-II), and it has been observed that the proposed MOHSA always outperforms NSGA-II in terms of computation time. Moreover, statistical tests show that the objective values obtained from both algorithms are comparable.

• A new variant of the p-hub location problem is proposed with a backbone ring network. • Network flow-based and path-based MILP formulations of SApHLP are introduced. • A customized PSO-based method with factoradic-based solution encoding is proposed. • Problem instances are created from a real-world movie data set. • CPLEX optimizer with Benders decomposition is used to generate benchmark solutions. In this article, the single allocation p -hub location problem (SApHLP) with a ring backbone network for content placement in VoD services is proposed. In VoD services, a large volume of digital data is kept as data segments in spatially distributed hubs. In SApHLP, each user is restricted to be allocated only to a single hub, and here hubs form a ring backbone network. SApHLP jointly addresses (i) the locations of hubs, (ii) the placement of segments to hubs, (iii) the allocation of users to hubs as per their demands, and (iv) the optimal paths to route the demands from users to hubs. We have introduced network flow-based 3-subscripted and path-based 4-subscripted MILP formulations of SApHLP. This article presents a novel discrete particle swarm optimization (PSO)-based approach where factoradic numbers are used to encode solution. It also incorporates three problem-specific solution refinement methods for faster convergence. In this article, SApHLP instances are generated from a real-world database of video files obtained from a movie recommender system. The benchmark solutions are generated using IBM’s CPLEX optimizer with default settings and Benders decomposition strategy. The performance of the proposed PSO is compared with the benchmark results produced by CPLEX.

In video-on-demand (VoD) services, large volumes of digital data are kept at hubs which are spatially distributed over large geographic areas and users are connected to these hubs based on their demands. In this article, we consider a large database of video files, that are pre-partitioned to multiple segments based on the demand patterns of users. These segments are restricted to be located only in hubs. Here, users are allowed to be allocated to multiple hubs and all hubs are assumed to be connected with each other. We jointly decide the location of hubs, the placement of segments to these hubs and then the assignment of users to these hubs as per their demand patterns and finally, we find the optimal paths to route the demands of users for different segments having the objective of minimizing the total routing cost. In this article, a differential evolution (DE) based method is proposed to solve the problem. The proposed DE-based method utilizes an efficient function to evaluate the objective value of a candidate solution to the proposed problem. It also incorporates two problem-specific solution refinement techniques for faster convergence. Instances of the problem are generated from the real world movie database and the proposed method is applied to these instances and the performance is evaluated against the benchmark results obtained from CPLEX.

Motivated by the challenges faced by a logistics company, we present a new variant of the dynamic capacitated pickup and delivery problem with time windows (PDPTW) where excessive changes of unaffected routes are undesirable. In real-life scenarios, different dynamism sources such as canceled requests, change of demands, change of pickup, or delivery time windows often disrupt the existing planning of routes. The static PDPTW is solved with the current information about the problem well before executing the routes, such as the previous night. We present an algorithmic idea of a dynamic solver quickly addressing changes that occur due to the dynamism while avoiding excessive modifications to the previous solution. Since the company has not yet the dynamic data, new dynamic instances are generated from the existing static PDPTW instances in the literature. Preliminary results demonstrate that we can quickly incorporate the required changes. Future perspectives of this ongoing work are discussed in the end.

The uncapacitated facility location problem (UFLP) is a well-known combinatorial optimization problem having single-objective function. The objective of UFLP is to find a subset of facilities from a given set of potential facility locations such that the sum of the opening costs of the opened facilities and the service cost to serve all the customers is minimized. In traditional UFLP, customers are served by their nearest facilities. In this article, we have proposed a multi-objective UFLP where each customer has a preference for each facility. Hence, the objective of the multi-objective UFLP with customers’ preferences (MOUFLPCP) is to open a subset of facilities to serve all the customers such that the sum of the opening cost and service cost is minimized and the sum of the preferences is maximized. In this article, the elitist non-dominated sorting genetic algorithm II (NSGA-II), a popular Pareto-based GA, is employed to solve this problem. Moreover, a weighted sum genetic algorithm (WSGA)-based approach is proposed to solve MOUFLPCP where conflicting two objectives of the problem are aggregated to a single quality measure. For experimental purposes, new test instances of MOUFLPCP are created from the existing UFLP benchmark instances and the experimental results obtained using NSGA-II and WSGA-based approaches are demonstrated and compared for these newly created test instances.

A well-known combinatorial optimization problem, known as the uncapacitated facility location problem (UFLP) is considered in this article. A deterministic heuristic algorithm and a randomized heuristic algorithm are presented to solve UFLP. Though the proposed deterministic heuristic algorithm is very simple, it produces good solution for each instance of UFLP considered in this article. The main purpose of this article is to process all the data sets of UFLP available in the literature using a single algorithm. The proposed two algorithms are applied on these test instances of UFLP to determine their effectiveness. Here, the solution obtained from the proposed randomized algorithm is at least as good as the solution produced by the proposed deterministic algorithm. Hence, the proposed deterministic algorithm gives upper bound on the solution produced by the randomized algorithm. Although the proposed deterministic algorithm gives optimal results for most of the instances of UFLP, the randomized algorithm achieves optimal results for all the instances of UFLP considered in this article including those for which the deterministic algorithm fails to achieve the optimal solutions.

The Single-Row Facility Layout Problem (SRFLP) is a well-known combinatorial optimization problem. The objective of SRFLP is to find out the arrangement of facilities with given lengths on a line so that the weighted sum of the distances between all pairs of facilities is minimized. This problem is known to be NP-hard. Hence, a population-based improvement heuristic algorithm with local search is presented in this article to solve SRFLP.The proposed algorithm works well also for the Single-Row Equidistant Facility Layout Problem (SREFLP), where the length of each facility is equal. The computational efficiency of the proposed algorithm is checked with the instances of sizes ranging from 5 to 300 available in the literature for SRFLP and SREFLP. The obtained results are compared to those from different state-of-the-art algorithms.The proposed algorithm achieves best known solutions to date for every instance considered in this article in reasonable computational time.

The tool indexing problem (TIP) is the problem of allocating cutting tools to different slots in a tool magazine of Computer Numerically Controlled machine to reduce the processing time of jobs on the machine. This is one of the mostly encountered optimization problems in manufacturing systems. In TIP, the number of tools used by the machine is at most the number of slots available in the tool magazine. In this article, a customized harmony search (HS) algorithm, which utilizes a harmony refinement strategy for faster convergence, is presented to solve TIP. The harmony refinement method also helps to avoid getting stuck into local optima. The performance of the proposed method is tested on 27 instances taken from the literature and out of these it is found to improve the best known solutions for 16 instances. For the remaining instances, it gives the same results as found in the literature. Moreover, the performance of the proposed algorithm is tested on newly adapted 41 instances and for some of these instances the results obtained using the proposed algorithm are compared with that obtained using CPLEX.

For a fixed integer $$D (\ge 3)$$ and $$\lambda $$$$\in $$$${\mathbb {Z}}^+$$, a $$\lambda $$-L(D, 2, 1)-labeling of a graph $$G = (V, E)$$ is the problem of assigning non-negative integers (known as labels) from the set $$\{0, \ldots , \lambda \}$$ to the vertices of G such that if any two vertices in V are one, two and three distance apart from each other, then the assigned labels to these vertices must have a difference of at least D, 2 and 1, respectively. The vertices which are at least 4 distance apart can receive the same label. The minimum value among all the possible values of $$\lambda $$ for which there exists a $$\lambda $$-L(D, 2, 1)-labeling is known as the labeling number. In this paper, $$\lambda $$-L(D, 2, 1)-labeling of square grid is considered. The lower bound on the labeling number for square grid is presented, and a formula for $$\lambda $$-L(D, 2, 1)-labeling of square grid is proposed. The correctness proof of the proposed formula is given here. The upper bound of the labeling number obtained from the proposed labeling formula for square grid matches exactly with the lower bound of the labeling number.

The Uncapacitated Facility Location Problem (UFLP) is considered in this paper. Given a set of customers and a set of potential facility locations, the objective of UFLP is to open a subset of facilities to satisfy the demands of all the customers such that the sum of the opening cost for the opened facilities and the service cost is minimized. UFLP is a well-known combinatorial optimization problem which is also NP-hard. So, a metaheuristic algorithm for solving this problem is natural choice. In this paper, a relatively new swarm intelligence-based algorithm known as the Monkey Algorithm (MA) is applied to solve UFLP. To validate the efficiency of the proposed binary MA-based algorithm, experiments are carried out with various data instances of UFLP taken from the OR-Library and the results are compared with those of the Firefly Algorithm (FA) and the Artificial Bee Colony (ABC) algorithm.