Small-sized Instances Research Articles

A column-and-row generation approach for the flying sidekick travelling salesman problem

The flying sidekick traveling salesman problem (FS-TSP) is a parcel delivery problem arising in the last-mile logistics, where the distribution plan of a driver-operated truck assisted by a drone (unmanned aerial vehicle, UAV) has to be defined. The FS-TSP is a variant of the TSP where routing decisions are integrated with customer-to-drone and customer-to-truck assignment decisions and truck-and-drone synchronization constraints. The objective is the minimization of the time required to serve all the customers, taking into account drone payload capacity and battery power constraints. In this work we provide a new representation of the FS-TSP based on the definition of an extended graph. This representation allows to model the problem by a new and compact integer linear programming formulation, where the synchronization issue is tackled in a column generation fashion, thus avoiding the usage of big-M constraints, representing one of the main drawbacks of the models present in literature. The proposed formulation has been solved by an exact approach which combines a Branch-and-Cut algorithm and a column generation procedure, strengthened by variable fixing strategies and new valid inequalities specifically defined for the problem. The proposed method has been experienced on a large set of benchmark instances. Computational results show that the proposed approach either is competitive or outperforms the best exact approach present in literature for the FS-TSP. Indeed, it is able to provide the optimal solution for all small size instances with 10 customers and for several medium size instances with 20 customers, some of them never solved before.

Locust swarm optimisation for the permutation flow shop scheduling problem

The permutation flow shop scheduling problem (PFSP) is one of the most prominent types of combinatorial optimisation problems. The fact comes from its wide applications in manufactures and industries. In this paper, the locust swarm optimisation algorithm (LO) is proposed to find high quality solution for the PFSP, it simulates the behaviour of locust swarm. In this work, the LO algorithm is adapted to solve the PFSP with the objective of minimising the makespan where a heuristic rule called smallest position value (SPV) is triggered to transform the random numbers into sequences of jobs. Then the NEH heuristic of Nawaz-Enscore-Ham is applied to generate good quality initial population. Also, a simple and efficient iterated local search method is employed to explore the solution space efficiently. An experimental study is discussed in this paper to evaluate the performance of the introduced algorithm. Different well-known PFSP benchmarks of small, medium and large size instances are used for this purpose. The computational and statistical studies demonstrate that the proposed algorithm is evidently outperforming the recent algorithms in obtaining better solutions.

Allocation of flights to land at the terminals of an airport

In this paper, an algorithm for solving flight landing problem has been developed to reduce the chaos at the airport. This algorithm starts with an initial basic feasible solution and then the algorithm works on the principle of steepest descent direction search method from current solution. The algorithm is easy to comprehend. The performance of this algorithm has been checked using a numerical example with known results. The flight landing problem is an application of Quadratic Assignment Problem. Since Quadratic Assignment Problem is NP- hard, it is quite tedious to get an optimal solution even for small size instances. However, as far as optimality of the solution is concerned, this algorithm leads to very favourable results.

The vehicle routing problem with relaxed priority rules

The Vehicle Routing Problem (VRP) is one of the most studied topics in Operations Research. Among the numerous variants of the VRP, this research addresses the VRP with relaxed priority rules (VRP-RPR) in which customers are assigned to several priority groups and customers with the highest priorities typically need to be served before lower priority ones. Additional rules are used to control the trade-off between priority and cost efficiency. We propose a Mixed Integer Linear Programming (MILP) model to formulate the problem and to solve small-sized instances. A metaheuristic based on the Adaptive Large Neighborhood Search (ALNS) algorithm with problem-tailored components is then designed to handle the problem at larger scales. The experimental results demonstrate the performance of our proposed algorithm. Remarkably, it outperforms a metaheuristic recently proposed to solve the Clustered Traveling Saleman Problem with d-relaxed priority rule (CTSP-d), a special case of VRP-RPR, in both solution quality and computational time.

A hub location model in the sustainable supply chain considering customer segmentation

Purpose This paper aims to study the hub transportation system in supply chain networks which would contribute to reducing costs and environmental pollution, as well as to economic development and social responsibility. As not all customers tend to buy green products, several customer groups should be considered in terms of need type. Design/methodology/approach In this paper, a multi-objective hub location problem is developed for designing a sustainable supply chain network based on customer segmentation. It deals with the aspects of economic (cost reduction), environment (minimizing greenhouse gas emissions by the transport sector) and social responsibility (creating employment and community development). The epsilon-constraint method and augmented epsilon-constraint (AEC) method are used to solve the small-sized instances of this multi-objective problem. Due to the non-deterministic polynomial-time hardness of this problem, two non-dominated sorting genetic algorithm-II (NSGA-II) and multi-objective grey wolf optimizer (MOGWO) metaheuristic algorithms are also applied to tackle the large-sized instances of this problem. Findings As expected, the AEC method is able to provide better Pareto solutions according to the goals of the decision-makers. The Taguchi method was used for setting the parameters of the two metaheuristic algorithms. Considering the meaningful difference, the MOGWO algorithm outperforms the NSGA-II algorithm according to the rate of achievement to two objectives simultaneously and the spread of non-dominance solutions indexes. Regarding the other indexes, there was no meaningful difference between the performance of the two algorithms. Practical implications The model of this research provides a comprehensive solution for supply chain companies that want to achieve a rational balance between the three aspects of sustainability. Originality/value The importance of considering customer diversity on the one hand and saving on hub transportation costs, on the other hand, triggered us to propose a hub location model for designing a sustainable supply chain network based on customer segmentation.

A model and two heuristic methods for The Multi-Product Inventory-Location-Routing Problem with heterogeneous fleet

<p style='text-indent:20px;'>The Multi-Product Inventory-Location-Routing Problem with heterogeneous fleet considers a supply chain, which comprises multiple producers, potential distribution centers (DCs) with opening capacity levels and geographically scattered retailers each of which has deterministic demand over a discrete planning horizon. The goal is determining a set of DCs with their capacity levels to open, assigning retailers to the opened DCs, finding product quantities to be ordered by and distributed from opened DCs and determining the fleet and routes to satisfy the demands of retailers with minimum cost. A mixed-integer linear programming model is proposed to describe the problem, which is strengthened by two valid inequalities. Since the commercial solver can only solve the very small-sized instances within a reasonable time, two heuristic methods are developed. Results show that the proposed valid inequalities are effective and both methods provide important savings in acceptable run times compared to the commercial solver.</p>

Two-dimensional bandwidth minimization problem: Exact and heuristic approaches

Reducing the bandwidth in a graph is an optimization problem which has generated significant attention over time due to its practical application in Very Large Scale Integration (VLSI) layout designs, solving system of equations, or matrix decomposition, among others. The bandwidth problem is considered as a graph labeling problem where each vertex of the graph receives a unique label. The target consists in finding an embedding of the graph in a line, according to the labels assigned to each vertex, that minimizes the maximum distance between the labels of adjacent vertices. In this work, we are focused on a 2D variant where the graph has to be embedded in a two-dimensional grid instead. To solve it, we have designed two constructive and three local search methods which are integrated in a Basic Variable Neighborhood Search (BVNS) scheme. To assess their performance, we have designed three Constraint Satisfaction Problem (CSP) models. The experimental results show that our CSP models obtain remarkable results in small or medium size instances. On the other hand, BVNS is capable of reaching equal or similar results than the CSP models in a reduced run-time for small instances, and it can provide high quality solutions in those instances which are not optimally solvable with our CSP models.

Modeling and solving a real-world cutting stock problem in the marble industry via mathematical programming and stochastic diffusion search approaches

In this study, one-dimensional marble plane cutting problem is studied based on the cutting equipment productivity and effective use of marble blocks. Different types of marble planes should be cut from multiple stock sized marble blocks in parallel by using a stone block-cutting machine (gang saw). Marble blocks, which have different dimensions, and quality gradations are supplied by the marble processing factory’s own marble quarries. Different quality gradations of the marble blocks and cutting equipment productivity need to be taken into account in addition to minimize the total trim loss or cost in modelling the present cutting problem. The nature of the production process in the marble industry consists of large amount of waste. Minimizing the amount of waste can be controlled indirectly by establishing an appropriate cutting plan. Therefore, minimization of the total cost of transportation and overgrading (supplying higher quality material than specified by the customer) while determining a suitable cutting plan for the marble blocks is aimed. A mixed integer linear programming (MILP) model is developed for solving the present marble plane cutting problem for small-size instances. The salient feature of the developed model is not requiring a priori enumeration of all possible cutting patterns and to permit generation of retail to reuse. For larger size problem instances a Stochastic Diffusion Search (SDS) algorithm is developed. Extensive computational studies are performed by using test instances, which are collected from a marble processing company operating at İzmir. The computational results show that the developed models and SDS algorithm are very useful in providing effective cutting plans.

The integrated production-distribution scheduling in parallel machine environment by using improved genetic algorithms

ABSTRACT This study investigates the integrated production-distribution scheduling in a workshop which utilizes the non-identical parallel machines in production line. The objective is to schedule the jobs on machines and forming them into the batches so as to minimize the sum of tardiness and delivery costs. First, a mixed integer linear programming model is proposed which can solve the small-size instances in a reasonable time. To deal with large-size instances, a genetic algorithm (GA) and the improved genetic algorithm (IGA) equipped with optimal properties mechanism is developed. Areal life instance in bedroom furniture manufacturing is also presented that indicates the algorithm can reach the optimal solution for this problem. In order to evaluate the performance of the algorithms, 150 small size instances and more than 800 large size instances are generated which is conducted from benchmarks. The results indicate that IGA outperforms GA with respect to relative percentage deviations.

Iterated local search for single machine total weighted tardiness batch scheduling

This paper presents an iterated local search (ILS) algorithm for the single machine total weighted tardiness batch scheduling problem. To our knowledge, this is one of the first attempts to apply ILS to solve a batching scheduling problem. The proposed algorithm contains a local search procedure that explores five neighborhood structures, and we show how to efficiently implement them. Moreover, we compare the performance of our algorithm with dynamic programming-based implementations for the problem, including one from the literature and two other ones inspired in biased random-key genetic algorithms and ILS. We also demonstrate that finding the optimal batching for the problem given a fixed sequence of jobs is $$\mathcal {NP}$$ -hard, and provide an exact pseudo-polynomial time dynamic programming algorithm for solving such problem. Extensive computational experiments were conducted on newly proposed benchmark instances, and the results indicate that our algorithm yields highly competitive results when compared to other strategies. Finally, it was also observed that the methods that rely on dynamic programming tend to be time-consuming, even for small size instances.

Robust car sequencing for automotive assembly

Just-in-sequence material supply is the status quo in the automotive industry. In this process, the assembly sequence of vehicles is set several days prior to production, and communicated to the suppliers. The committed sequence is essential for efficient operations both at the original equipment manufacturer and its suppliers. In practice, however, sequence stability is insufficient. Short-term disruptions, such as quality problems and missing parts, put the sequence at risk. If a disruption occurs, the affected vehicle is removed from the sequence. The resulting gap is closed by bringing the succeeding vehicles forward. Such sequence alterations, however, cause workload changes and potentially work overloads at the assembly stations. As a remedial measure, additional sequence alterations are necessary, which further disturb material supply. Robustness against short-term sequence alterations is currently a key objective of automotive manufacturers.In this paper, we propose a sequencing approach that includes the vehicles’ failure probabilities in order to generate robust sequences. Robust sequences are sequences that can be operated without modifications, even when vehicles fail. We develop a branch-and-bound algorithm that optimally solves small-sized instances. For large-sized instances, we design a sampling-based adaptive large neighborhood search heuristic. The superiority of our approach is validated in a simulation study using real-world data from a major European manufacturer. We find reductions in the expected work overloads of 72% and 80%, compared to the industry solution, and compared to an approach taken from literature which does not take failures into account.

Shipment Path Planning for Rail Networks Considering Elastic Capacity

The rail shipment path (RSP) optimization problem is part of the operating plan. Unlike other transportation modes, the physical path of traffic flow in a rail system has its own unique characteristics, among which the most particular is that shipments gradually converge into one single traffic flow and then travel together through a tree-shaped path. More specifically, when two or more shipments arrive at a rail yard, those intended for the same destination are treated as one single freight flow and are shipped through the same path during the remainder of the trip, regardless of their origins. The RSP plan is intended for an intermediate planning horizon (one to two years), not for controlling day-to-day operations. Thus, storage capacities are required for links, resulting in available link capacities having an elastic feature. Based on this background, this article introduces the forming mechanism of the tree-shaped path from the perspective of rail yard operations and develops a nonlinear fuzzy 0-1 programming model for the RSP problem. Since the fuzzy capacity constraints and a notably large number of high-order terms are generated from the tree recursive expression, the model is hard to solve even for small-sized instances by typical methods of operational research. In this article, to solve RSP problems with the tree-shaped and the elastic capacity constraints in the rail network, we employ a simulated annealing algorithm. The numerical experiments based on the southwest rail network of China show that the solution efficiency of this approach is significant.

Optimization of Continuous Berth Scheduling by Taking into Account Double-Line Ship Mooring

“Double-Line Ship Mooring” (DLSM) mode has been applied as an initiative operation mode for solving berth allocation problems (BAP) in certain giant container terminals in China. In this study, a continuous berth scheduling problem with the DLSM model is illustrated and solved with exact and heuristic methods with an objective to minimize the total operation cost, including both the additional transportation cost for vessels not located at their minimum-cost berthing position and the penalties for vessels not being able to leave as planned. First of all, this problem is formulated as a mixed-integer programming model and solved by the CPLEX solver for small-size instances. Afterwards, a particle swarm optimization (PSO) algorithm is developed to obtain good quality solutions within reasonable execution time for large-scale problems. Experimental results show that DLSM mode can not only greatly reduce the total operation cost but also significantly improve the efficiency of berth scheduling in comparison with the widely used single-line ship mooring (SLSM) mode. The comparison made between the results obtained by the proposed PSO algorithm and that obtained by the CPLEX solver for both small-size and large-scale instances are also quite encouraging. To sum up, this study can not only validate the effectiveness of DLSM mode for heavy-loaded ports but also provide a powerful decision support tool for the port operators to make good quality berth schedules with the DLSM mode.

Quay Cranes and Yard Trucks Scheduling Problem at Container Terminals

A bi-objective mathematical model is developed to simultaneously consider the quay crane and yard truck scheduling problems at container terminals. Main real-world assumptions, such as quay cranes with non-crossing constraints, quay cranes’ safety margins and precedence constraints are considered in this model. This integrated approach leads to better efficiency and productivity at container terminals. Based on numerical experiments, the proposed mathematical model is effective for solving small-sized instances. Two versions of the simulated annealing algorithm are developed to heuristically solve the large-sized instances. Considering the allocation of trucks as a grouping problem, a grouping version of the simulated annealing algorithm is proposed. Effectiveness of the presented algorithms is compared to the optimal results of the mathematical model on small-sized problems. Moreover, the performances of the proposed algorithms on large-sized instances are compared with each other and the numerical results revealed that the grouping version of simulated annealing algorithm outperformed simulated annealing algorithm. Based on numerical investigations, there is a trade-off between the tasks’ completion time and the cost of utilizing more trucks. Moreover increasing the number of YTs leads to better outcomes than increasing the number of QCs. Besides two-cycle strategy and using dynamic assignment of yard truck to quay cranes leads to faster loading and unloading procedure.

An Efficient Local Search Heuristic for Earth Observation Satellite Integrated Scheduling

Earth observation satellites (EOSs) are taking a large number of pictures with increasing resolution which produce massive image data. Satellite data transmission becomes the bottleneck part in the process of EOS resource management. In this paper, we study the earth observation satellite integrated scheduling problem (EOSIS) where the imaging activities and download activities are considered integratively. We propose an integer linear programming model to formulate the problem. Due to the NP-hardness of the problem, we propose an efficient local search heuristic (ELSH) to solve problems of large size. ELSH uses a dedicated local search method to guarantee algorithm performance and efficient constraint handling mechanisms to guarantee algorithm efficiency. Numerical experimental results show that the algorithm demonstrates excellent performance on a set of benchmark instances. The ELSH achieves optimal results for all small-size instances (with 50 targets, two satellites, and three ground stations), and is very robust for large instances with up to 2000 targets. Moreover, the proposed ELSH easily dominates the state-of-the-art algorithm.

Stochastic programming approach for unidirectional quay crane scheduling problem with uncertainty

Quay crane scheduling is a key aspect of container terminal operation, which can be regarded as a decision-making process with uncertainty. Each task involves stochastic loading and unloading operation times owing to the existence of uncertainty. In this study, we investigate the unidirectional quay crane scheduling problem for a stochastic processing time, which requires that all the quay cranes move in the same direction either from bow to stern, or vice versa, throughout the planning horizon. The problem is formulated as a two-stage stochastic mixed-integer programming model, where the binary first-stage decision variables correspond to the assignment of tasks to quay cranes, and the mixed-integer second-stage decision variables are related to the generation of detailed schedules. To make the model solvable, we develop an alternative equivalent reformulation with a special structure that involves binary variables in the first stage and continuous variables in the second stage. To solve this reformulated model, an integer L-shaped method is presented for small-size instances, and a simulated annealing algorithm is presented for large-size instances to obtain near-optimal solutions. Numerical experiments show that the integer L-shaped method and simulated annealing algorithm could efficiently solve the unidirectional quay crane scheduling problem with uncertainty. The results also indicate that the stochastic model has distinct advantages in terms of shortening the completion time of vessels and improving the service level of container terminals compared with the expected value problem solutions.

The Multi-Period Workforce Scheduling and Routing Problem

Motivated by the problem faced by a company that provides installation and maintenance services of electrical infrastructures, we study the Multi-Period Workforce Scheduling and Routing Problem (MPWSRP). A set of jobs requiring a service is given. A given set of workers, each proficient in a number of skills, is available to serve the jobs. Each job requires a team of workers providing the required skills, and is associated with a given priority level. A service time, which might span over multiple time periods, is associated with each job, and depends on the number of workers that have been assigned. The MPWSRP calls for determining an optimal dispatch plan to serve the given set of jobs by routing a number of teams over a finite planning horizon. The MPWSRP is a highly complex combinatorial optimization problem where routing, scheduling, and assignment decisions have to be taken simultaneously. We cast the MPWSRP as a Mixed-Integer Linear Program (MILP). As an off-the-shelf solver can solve only small-size instances of the proposed MILP in reasonable computing times, we devise an Adaptive Large Neighborhood Search (ALNS) heuristic and a three-phase Decomposition Algorithm (DA) to solve large-size instances. Extensive computational experiments are conducted on a large set of instances comprising up to 100 jobs, 20 workers, and 20 time periods. The results show that ALNS is competitive with the solver on the small-size instances, producing solutions of similar average quality in considerably shorter computing times. In addition, they point out that ALNS consistently outperforms the DA in terms of solution quality. Further computational experiments are conducted applying our ALNS and DA on a set of large-size instances. Finally, a case study using real data sets is provided.

Radiotherapy treatment scheduling considering time window preferences

External-beam radiotherapy treatments are delivered by a linear accelerator (linac) in a series of high-energy radiation sessions over multiple days. With the increase in the incidence of cancer and the use of radiotherapy (RT), the problem of automatically scheduling RT sessions while satisfying patient preferences regarding the time of their appointments becomes increasingly relevant. While most literature focuses on timeliness of treatments, several Dutch RT centers have expressed their need to include patient preferences when scheduling appointments for irradiation sessions. In this study, we propose a mixed-integer linear programming (MILP) model that solves the problem of scheduling and sequencing RT sessions considering time window preferences given by patients. The MILP model alone is able to solve the problem to optimality, scheduling all sessions within the desired window, in reasonable time for small size instances up to 66 patients and 2 linacs per week. For larger centers, we propose a heuristic method that pre-assigns patients to linacs to decompose the problem in subproblems (clusters of linacs) before using the MILP model to solve the subproblems to optimality in a sequential manner. We test our methodology using real-world data from a large Dutch RT center (8 linacs). Results show that, combining the heuristic with the MILP model, the problem can be solved in reasonable computation time with as few as 2.8% of the sessions being scheduled outside the desired time window.

Problem on the integration between production and delivery with parallel batching machines of generic job sizes and processing times

In recent years the manufacturing industries are currently focused on adopting optimal hybrid mass/customized strategies. Working solely on customized production or mass production is not a desirable strategy for many manufacturers anymore. The rapid advances in technology, applications, and customers’ requirements are increasing the competitiveness and, therefore, productivity. Given this new scenario, the integration between production and delivery with parallel batching machines comes as an essential issue in logistics. Efficient integrated production and delivery operation is a critical factor for minimizing costs and increasing customer satisfaction. The particular case discussed in this paper consists of scheduling jobs of generic sizes and processing times on identical and parallel batching machines, aiming to maximize profits. The objective is to get a joint production and delivery schedule that maximizes total profits, focusing on attending consumers on time. We propose a mathematical formulation and show that the addition of a set of constraints improves the commercial solver performance. Additionally, we focus on the development and analys is of two fast polynomial methods. The first, RKP, aims to obtain relaxed solutions, while the last, CMFFDA, aims to obtain feasible solutions. Furthermore, computational experiments and sensitivity analysis with random instances of different sizes are conducted to evaluate the proposed methods and the integrated system. The results show tight gaps for the mathematical formulation and algorithms RKP and CMFFDA. The proposed algorithms present short computational times for small, moderate, and large size instances, an essential quality for daily operation.

Mixed Integer linear programming and constraint programming models for the online printing shop scheduling problem

In this work, the online printing shop scheduling problem is considered. This challenging real problem, that appears in the nowadays printing industry, can be seen as a flexible job shop scheduling problem with sequence flexibility in which precedence constraints among operations of a job are given by an arbitrary directed acyclic graph. In addition, several complicating particularities such as periods of unavailability of the machines, resumable operations, sequence-dependent setup times, partial overlapping among operations with precedence constraints, release times, and fixed operations are also present in the addressed problem. In the present work, mixed integer linear programming and constraint programming models for the minimization of the makespan are presented. Modeling the problem is twofold. On the one hand, the problem is precisely defined. On the other hand, the capabilities and limitations of a commercial software for solving the models are analyzed. Extensive numerical experiments with small-, medium-, and large-sized instances are presented. Numerical experiments show that the commercial solver is able to optimally solve only a fraction of the small-sized instances when considering the mixed integer linear programming model; while all small-sized and a fraction of the medium-sized instances are optimally solved when considering the constraint programming formulation of the problem. Moreover, the commercial solver is able to deliver feasible solutions for the large-sized instances that are of the size of the instances that appear in practice.