Best-known Solutions Research Articles

Constraint Programming and Local Search Heuristic: a Matheuristic Approach for Routing and Scheduling Feeder Vessels in Multi-terminal Ports

In the liner shipping business, shipping ports represent the main nodes in the maritime transportation network. These ports have a collection of terminals where container vessels can load and discharge containers. However, the logistics and planning of operations differ depending on the vessel size. Large container vessels visit a single terminal, whereas smaller container vessels, or feeder vessels, visit several terminals to transport all containers within the multiple terminals of the port. In this paper, we study the Port Scheduling Problem, the problem of scheduling the operations of feeder vessels in multi-terminal ports. The resulting problem can be identified as a version of the General Shop Scheduling Problem. We consider a Constraint Programming formulation of the problem, and we propose a matheuristic solution approach for solving large instances. The proposed matheuristic is a hybrid solution method that combines Constraint Programming with a local search heuristic. The solution approach benefits from the fast search capabilities of local search algorithms to explore the solution space using an Adaptive Large Neighbourhood Search heuristic. During the search, we further use the Constraint Programming model as an intensification technique, every time a new best-known solution is found. We conduct detailed computational experiments on the PortLib instances, showing that the incorporation of Constraint Programming within the heuristic search can result in significant benefits. The high instability in solution quality obtained by local search heuristics can be lowered by a simple combination of both methods.

A two-phase tabu search based evolutionary algorithm for the maximum diversity problem

In this paper, we study the maximum diversity problem (MDP) which is equivalent to the quadratic unconstrained binary optimization (QUBO) problem with cardinality constraint. The MDP aims to select a subset of elements with given cardinality such that the sum of pairwise distances between any two elements in the selected subset is maximized. For solving this computationally challenging problem, we propose a two-phase tabu search based evolutionary algorithm (TPTS/EA), which integrates several distinguishing features to ensure the diversity and the quality of the evolution, such as a two-phase tabu search algorithm which consists of a dynamic candidate list (DCL) strategy-based traditional tabu search in the first phase and a solution-based tabu search procedure to refine the search in the second phase, and two path-relinking based recombination operators to generate new offspring solutions. Tested on three sets of totally 140 public instances in the literature, the study demonstrates the efficacy of the proposed TPTS/EA algorithm in terms of both solution quality and computational efficiency. Specifically, our proposed TPTS/EA algorithm is able to improve the previous best known results for 2 instances, while matching the previous best-known solutions for 130 instances. We also provide experimental evidences to highlight the beneficial effect of several important components in our TPTS/EA algorithm.

Dynamic Gaussian bare-bones fruit fly optimizers with abandonment mechanism: method and analysis

The Fruit Fly Optimization Algorithm (FOA) is a recent algorithm inspired by the foraging behavior of fruit fly populations. However, the original FOA easily falls into the local optimum in the process of solving practical problems, and has a high probability of escaping from the optimal solution. In order to improve the global search capability and the quality of solutions, a dynamic step length mechanism, abandonment mechanism and Gaussian bare-bones mechanism are introduced into FOA, termed as BareFOA. Firstly, the random and ambiguous behavior of fruit flies during the olfactory phase is described using the abandonment mechanism. The search range of fruit fly populations is automatically adjusted using an update strategy with dynamic step length. As a result, the convergence speed and convergence accuracy of FOA have been greatly improved. Secondly, the Gaussian bare-bones mechanism that overcomes local optimal constraints is introduced, which greatly improves the global search capability of the FOA. Finally, 30 benchmark functions for CEC2017 and seven engineering optimization problems are experimented with and compared to the best-known solutions reported in the literature. The computational results show that the BareFOA not only significantly achieved the superior results on the benchmark problems than other competitive counterparts, but also can offer better results on the engineering optimization design problems.

A matrix-cube-based estimation of distribution algorithm for the distributed assembly permutation flow-shop scheduling problem

Abstract The distributed assembly permutation flow-shop scheduling problem (DAPFSP) is a typical NP-hard combinatorial optimization problem that has wide applications in advanced manufacturing systems and modern supply chains. In this work, an innovative three-dimensional matrix-cube-based estimation of distribution algorithm (MCEDA) is first proposed for the DAPFSP to minimize the maximum completion time. Firstly, a matrix cube is designed to learn the valuable information from elites. Secondly, a matrix-cube-based probabilistic model with an effective sampling mechanism is developed to estimate the probability distribution of superior solutions and to perform the global exploration for finding promising regions. Thirdly, a problem-dependent variable neighborhood descent method is proposed to perform the local exploitation around these promising regions, and several speedup strategies for evaluating neighboring solutions are utilized to enhance the computational efficiency. Furthermore, the influence of the parameters setting is analyzed by using design-of-experiment technique, and the suitable parameters are suggested for different scale problems. Finally, a comprehensive computational campaign against the state-of-the-art algorithms in the literature, together with statistical analyses, demonstrates that the proposed MCEDA produces better results than the existing algorithms by a significant margin. Moreover, the new best-known solutions for 214 instances are improved.

A variable neighbourhood search enhanced estimation of distribution algorithm for quadratic assignment problems

Quadratic Assignment Problem (QAP) is one of the most complex combinatorial optimization problems. Many real-world problems such as printed circuit board design, facility location problems, assigning gates to airplanes can be modelled as QAP. Problems of size greater than 35 is not able to solve optimally using conventional optimization methods. This warrants the use of evolutionary optimization methods for obtaining optimal or near optimal solutions for QAPs. This work proposes a hybridization on a univariate Estimation of Distribution Algorithm, namely the Population Based Incremental Learning Algorithm (PBILA), with Variable Neighbourhood Search (VNS) for solving QAPs. The proposed algorithm is employed to solve benchmark instances of QAP and the results are reported. The results of this work reveals that PBILA on its own is not efficient for solving the QAPs. However, when hybridised with VNS, the algorithm performs well providing best known solutions for 95 test instances out of the 101 instances considered. For most of the test instances, the percentage deviation is less than one percentage. The overall average percentage deviation of the obtained solutions from the best-known solutions is 0.037%, which is a significant improvement when compared with state-of-the-art algorithms.

A multistage optimisation algorithm for the large vehicle routing problem with time windows and synchronised visits

We propose a multistage algorithm for the vehicle routing problem with time windows and synchronised visits, which is capable of solving large problem instances arising in home health care applications. The algorithm is based on a constraint programming formulation of the daily home care scheduling and routing problem. It contains visits with hard time windows and pairwise synchronisation to be staffed by carers who have different skills and work custom shift patterns with contractual breaks. In a computational study, we first experiment with a benchmark set from the literature for the vehicle routing problem with time windows and synchronised visits. Our algorithm reproduced the majority of the best-known solutions, and strictly improved results for several other instances. Most importantly, we demonstrate that the algorithm can effectively solve real scheduling instances obtained from a UK home care provider. Their size significantly surpass similar scheduling problems considered in the literature. The multistage algorithm solved each of these instances in a matter of minutes, and outperformed human planners, scheduling more visits and significantly reducing total travel time.

Quantum-inspired algorithm for general minimum conical hull problems

A wide range of fundamental machine learning tasks that are addressed by the maximum a posteriori estimation can be reduced to a general minimum conical hull problem. The best-known solution to tackle general minimum conical hull problems is the divide-and-conquer anchoring learning scheme (DCA), whose runtime complexity is polynomial in size. However, big data is pushing these polynomial algorithms to their performance limits. In this paper, we propose a sublinear classical algorithm to tackle general minimum conical hull problems when the input has stored in a sample-based low-overhead data structure. The algorithm's runtime complexity is polynomial in the rank and polylogarithmic in size. The proposed algorithm achieves the exponential speedup over DCA and, therefore, provides advantages for high dimensional problems.

A two-phase local search with a discrete-event heuristic for the omnichannel vehicle routing problem

This work tackles a pick up and delivery vehicle routing problem that emerges from the integration of delivery problems from two neighbouring layers of a supply chain: (i) retailer replenishment; and (ii) online customers deliveries, where online customer orders have to be picked up from retailers on route. This integration is motivated by recent developments in marketing, e.g.: the move to the omnichannel marketing paradigm, in which companies manage multiple sales channels in a seamless, unified, and integrated way. This paper proposes a novel model and an original solving approach. In order to tackle the complex capacity constraints, previous models propose to solve a delivery vehicle assignment problem before solving the associated vehicle routing problem. In contrast, our model deals with the whole routing problem by considering only the capacity feasible routing decisions. Our solving approach is based on a two-phase algorithm. In the first phase, a discrete-event constructive heuristic is employed. In the second phase, the most ‘promising’ solutions obtained in the previous phase are refined using a sequence of local search neighbourhoods. A series of extensive computational experiments show that our algorithm is able to identify new best-known solutions for the vast majority of problem instances. Also, for a set of small instances we obtain improved lower bounds compared to a previously proposed lower bound formulation.

A parallelised large neighbourhood search heuristic for the asymmetric two-echelon vehicle routing problem with swap containers for cargo-bicycles

Cargo-bicycles are a promising alternative to conventional vans in city logistics in response to increasing urbanisation and environmental damage caused by city traffic. The delivery structure is modelled with the well studied Two-Echelon Capacitated Vehicle Routing Problem (2E-CVRP), which uses cross-docking from vans to cargo-bicycles at so-called satellites. To reduce the extra handling effort compared to single-tier systems, swap containers are used. Furthermore, for cargo-bicycles the consideration of asymmetric distance matrices is important. Therefore, we present the Asymmetric 2E-CVRP with Swap Containers and develop an efficient Parallelised Large Neighbourhood Search heuristic, that is further improved using a first-level heuristic. The heuristic is tested using the symmetric 2E-CVRP benchmark instances from the literature, outperforms previous heuristics for large instances and finds new best-known solutions. Subsequently, the heuristic is applied to a case study in Munich with 22 newly generated instances, each containing 200 customers and asymmetric distances. The results allow quantitative insights into the cost and CO2e emissions savings of the investigated cargo-bicycle set-up compared to conventional van delivery.

The multi-mode resource investment problem: a benchmark library and a computational study of lower and upper bounds

The multi-mode resource investment problem (MRIP) is the multi-mode extension of the resource investment problem, which is also known under the name resource availability cost problem. It is a project scheduling problem with a given due date as well as precedence and resource constraints. The goal is to find a precedence feasible schedule that minimises the resource costs of the allocated resources. To compute these costs, the maximum resource peak is considered regarding renewable resource types, whereas the sum of allocated nonrenewable resource units is used in the case of nonrenewable resources. Many practical and complex project scheduling settings can be modelled with this type of problem. Especially with the usage of different processing modes, time and cost compromises can be utilised by the project manager. In the literature, some procedures for the MRIP have been investigated; however, the computational experiments in these studies have not been carried out on common benchmark instances. This makes a fair comparison of methods difficult. Therefore, we generated novel instances specifically designed for this problem and published them on the website https://riplib.hsu-hh.de. On this website, the instances as well as best-known solution values are available and researchers can also contribute their findings. We investigate these novel instances by proposing and evaluating lower bounds for the MRIP. Additionally, we analyse the proposed instances by applying heuristic as well as exact methods. These experiments suggest that most of the instances are challenging and further research is needed. Finally, we show some computational complexity properties of the NP-hard MRIP.

MILP Models for Complex System Reliability Redundancy Allocation with Mixed Components

The redundancy allocation problem (RAP) aims to find an optimal allocation of redundant components subject to resource constraints. In this paper, mixed integer linear programming (MILP) models and MILP-based algorithms are proposed for complex system reliability redundancy allocation problem with mixed components, where the system have bridges or interconnecting subsystems and each subsystem can have mixed types of components. Unlike the other algorithms in the literature, the proposed MILP models view the problem from a different point of view and approximate the nonconvex nonlinear system reliability function of a complex system using random samples. The solution to the MILP converges to the optimal solution of the original problem as sample size increases. In addition, data aggregation-based algorithms are proposed to improve the solution time and quality based on the proposed MILP models. A computational experiment shows that the proposed models and algorithms converge to the optimal or best-known solution as sample size increases. The proposed algorithms outperform popular metaheuristic algorithms in the literature.

Metaheuristics applied for storage yards allocation in an Amazonian sustainable forest management area

In the sustainable management of Amazonian forests, it is essential to carry out the optimal planning of logging infrastructures to reduce costs and environmental impacts. However, there is a high degree of complexity due to the number of variables involved. Among these infrastructures, wood storage yards are of utmost importance as they directly influence the opening of forest roads and trails. The objective of this research was to evaluate the allocation of wood storage yards through exact solution and metaheuristics in a forest management area. The study area was a native forest under sustainable forest management regime located in the Brazilian Amazon. Three instances were formulated involving 5947 trees and 3172 wood storage yards facilities. We used a binary integer programming model solved by CPLEX and the metaheuristics Greedy Randomized Adaptive Search Procedure (GRASP), Tabu Search (TS), Variable Neighborhood Search (VNS) and Simulated Annealing (SA). GAP values increased as a function of instances. Although all metaheuristics obtained significant solutions with shorter processing times, only SA obtained feasible solutions in all executions for all three instances. In general, the metaheuristics were efficient in obtaining feasible solutions faster than CPLEX, which represents the feasibility of the planning of allocation storage large areas, and without significant losses of best-known solution. The SA presented the best performance in the three evaluated instances. Contribution of this study can be highlighted: evaluation of alternative computational methods for planning the allocation of wooden storage yards; evidence was obtained of effectiveness and efficiency of assessed metaheuristics and, the applicability of approximate methods in this problem was evaluated.

A hybrid adaptive large neighborhood search heuristic for the team orienteering problem

The Team Orienteering Problem (TOP) is a well-known NP-Hard vehicle routing problem in which one maximizes the collected profits for visiting some nodes. In this paper, we propose a Hybrid Adaptive Large Neighborhood Search (HALNS) to solve this problem. Our algorithm combines the exploration power of ALNS with local search procedures and an optimization stage using a Set Packing Problem to further improve the solutions. Extensive computational experiments demonstrate the high performance of our HALNS outperforming all the competing algorithms in the literature on a large set of benchmark instances in terms of solution quality and/or computational time. Our HALNS identifies all the 387 Best Known Solutions (BKS) from the literature on a first dataset including small-scale benchmark instances and all the 333 BKS for large-scale benchmark instances within very short computational times. Moreover, we improve one large-scale instance solution.

A hybrid method integrating an elite genetic algorithm with tabu search for the quadratic assignment problem

The Quadratic Assignment Problem (QAP) is one of the most studied classical combinatorial optimization problems. QAP has many practical applications. Designing enhanced meta-heuristic approaches for the QAP is an active research area. In this work, we propose a hybrid algorithm (EGATS) that combines an elite genetic algorithm and tabu search to solve the QAP. In the optimization process, EGATS employs two kinds of elite crossovers, repeated 2-exchange mutation, and tabu search to strike a balance between exploitation and exploration. We evaluated the performance of EGATS through computational experiments on 135 well-known benchmark instances from the quadratic assignment problem library, QAPLIB. EGATS obtained the best-known solution for 131 instances. Compared to other popular meta-heuristic algorithms in the literature, EGATS is a competitive method for the QAP.

The Time-dependent Electric Vehicle Routing Problem: Model and solution

We study a new problem named the Time-dependent Electric Vehicle Routing Problem (TDEVRP) which involves routing a fleet of electric vehicles to serve a set of customers and determining the vehicle’s speed and departure time at each arc of the routes with the purpose of minimizing a cost function. We propose an integer linear programming (ILP) model to formulate the TDEVRP and show that the state-of-the-art commercial optimizer (CPLEX) can only solve instances of very limited sizes (with no more than 15 customers). We thus propose an iterated variable neighbourhood search (IVNS) algorithm to find near-optimal solutions for larger instances. The key ingredients of IVNS include a fast evaluation method that allows local search moves to be evaluated in constant time O(1), a variable neighbourhood descent (VND) procedure to optimize the node sequences, and a departure time and speed optimization procedure(DSOP) to optimize the speed and departure time on each arc of the routes. The proposed algorithm demonstrates excellent performances on a set of newly created instances. In particular, it can achieve optimal or near-optimal solutions for all small-size instances (with no more than 15 customers) and is robust for large-size instances where the gap between the average and the best solution value is consistently lower than 2.38%. Additional experimental results on 40 benchmark instances of the closely related Time-Dependent Pollution Routing Problem indicate that the proposed IVNS algorithm also performs very well and even discovers 39 new best-known solutions (improved upper bounds).

The two-echelon production-routing problem

This paper introduces the Two-Echelon Production-Routing Problem. This problem is motivated from the petrochemical industry, enlarging the supply chain integration by taking into account production, inventory, and routing decisions in a two-echelon vendor-managed inventory system. We describe, model, and design a branch-and-cut (B&C) to solve the problem under different inventory policies. We also propose a novel exact algorithm, by employing parallel computing techniques, in order to combine local search procedures within a traditional B&C scheme. We evaluate the performance of our methods through extensive computational experiments, both by comparing the algorithms, the effectiveness of the different inventory policies, and the impact of these policies on the partial costs. We derive many managerial insights based on the results. We also validate our new exact algorithm by solving similar problems from the literature, such as the two-echelon multi-depot inventory-routing (2E-MDIRP) and the classical multi-vehicle production-routing problem (MV-PRP). Computational experiments show that our method is very competitive. Based on 512 experiments for the 2E-MDIRP, our algorithm was able to find 111 new best known solutions (BKS), besides proving 412 optimal solutions, against 298 from the literature. For 336 experiments over small and medium size MV-PRP instances, we proved 242 optimal solutions, 11 more than the exact methods from the literature, besides providing 95 new BKS. Moreover, we were the first to tackle large MV-PRP instances exactly, and in this case, our algorithm provides all BKS for instances up to 50 customers, 20 periods and 5 vehicles, outperforming all meta/matheuristics procedures from the literature.

Addressing Unequal Area Facility Layout Problems with the Coral Reef Optimization algorithm with Substrate Layers

The Unequal Area Facility Layout Problem (UA-FLP) is a relevant task in industrial manufacturing, in which the disposition of a number of facilities (or departments) in a manufacturing system must be obtained, under several optimization criteria and different constraints. The UA-FLP is a hard optimization problem, in which traditional optimization techniques do not obtain good results. Thus, it has been successfully tackled with different heuristics and meta-heuristics in the last years. In this work we address the UA-FLP with a multi-method ensemble approach, the Coral Reefs Optimization algorithm with Substrate Layers (CRO-SL). It is a novel multi-method evolutionary algorithm that encourages the evolution of several searching procedures at the same time over a single population. The CRO-SL has been previously applied to very difficult optimization problems, obtaining excellent performance. In this case, we adapt the CRO-SL to the UA-FLP, by means of increasing the diversity generation within the algorithm, which is helpful to improve the exploration of the searching space, avoiding to fall into local minima. Specifically, we propose to include several reproduction mechanisms (adapted to the UA-FLP) within each substrate of the algorithm, which will highly increase the diversity generation in the CRO-SL. An exhaustive experimental study of the CRO-SL performance in a large number of UA-FLP instances is carried out, including a comparison with the state-of-the-art algorithms for this problem. We will show the ability of the CRO-SL to reach or surpass the best-known solutions in most of the tested UA-FLP cases.

An iterated greedy algorithm for the obnoxious p-median problem

The obnoxious p-median problem (OpM) is one of the NP-hard combinatorial optimization problems, in which the goal is to find optimal places to facilities that are undesirable (e.g. noisy, dangerous, or pollutant) such that the sum of the minimum distances between each non-facility location and its nearest facility is maximized. In this paper, for the first time in the literature, Iterated Greedy (IG) metaheuristic has been applied at a higher level to solve this problem. A powerful composite local search method has also been developed by combining two fast and effective local search algorithms, namely RLS1 and RLS2, which were previously used to solve the OpM. Comprehensive experiments have been conducted to test the performance of the proposed algorithm using a common benchmark for the problem. The computational results show the effectiveness of the IG algorithm that it can find high-quality solutions in a short time. Based on the set of selected instances, the results also reveal that the developed IG algorithm outperforms most of the state-of-the-art algorithms and contributes to the literature with 5 new best-known solutions.

A Hybrid Grasshopper Optimization Algorithm Applied to the Open Vehicle Routing Problem

This paper presents a hybrid grasshopper optimization algorithm using a novel decoder and local search to solve instances of the open vehicle routing problem with capacity and distance constraints. The algorithm’s decoder first defines the number of vehicles to be used and then it partitions the clients, assigning them to the available routes. The algorithm performs a local search in three neighborhoods after decoding. When a new best solution is found, every route is locally optimized by solving a traveling salesman problem, considering the depot and clients in the route. Three sets containing a total of 30 benchmark problems from the literature were used to test the algorithm. The experiments considered two cases of the problem. In the first, the primary objective is to minimize the total number of vehicles and then the total distance to be traveled. In the second case, the total distance traveled by the vehicles is minimized. The obtained results showed the algorithm’s proficient performance. For the first case, the algorithm was able to improve or match the best-known solutions for 21 of the 30 benchmark problems. For the second case, the best-known solutions for 18 of the 30 benchmark problems were found or improved by the algorithm. Finally, a case study from a real-life problem is included.

Iterative Local-Search Heuristic for Weighted Vehicle Routing Problem

Solutions to the weighted vehicle routing problem (WVRP) find numerous applications, such as home-to-work bus service, toll-by-weight cargo transportation, perishable-food delivery, and hazardous waste collection. This paper presents mathematical WVRP models corresponding to both collection and delivery cases. This study demonstrates WVRPs to be NP-hard, and proposes an efficient heuristic method (RI-ILS) to solve WVRPs. RI-ILS is based on principles of regret-insertion and iterative local-search. Several computational experiments were performed in this study to demonstrate the utility of the RI-ILS approach. As observed, when solving traditional vehicle routing problems (VRPs), RI-ILS yields good results under all testing conditions. In particular, it provides new best-known solutions under 12 testing conditions. When solving WVRPs, the RI-ILS method outperforms two recently published state-of-the-art algorithms in terms of solution quality, computational time, and stability. Case studies have also been performed in this research based on real-world data obtained from two Chinese companies that provide home-to-work bus service to their employees.