Large Size Test Problems Research Articles

The cumulative vehicle routing problem with arc time windows

In this paper, we introduce a new variant of the cumulative Vehicle Routing Problem (cum-VRP), where certain arcs of the undirected complete graph can only be utilized within specified time constraints. The problem is referred to as the Cumulative Vehicle Routing Problem with Arc Time Windows (Cum-VRPATW) and emerges in routing situations with flow disruptions across road segments, which generally emerges from military and civilian transportation. We devise a mixed integer programming (MIP) formulation to model the problem and solve it using CPLEX. To effectively address this problem, an effective algorithm is designed. It is a hybrid algorithm that combines principles of the Greedy Randomized Adaptive Search Procedure (GRASP) and decomposition strategies reducing computational complexity. Also, we propose two indexes, the RatioIn and the RatioTWL index that can identify difficult problems. Furthermore, we compare the CPLEX results vis-a-vis the new hybrid heuristic we have developed. Computational tests on converted Solomon (1987) data sets show that the new hybrid heuristic provides extremely good performance results for the cum-VRPATW large size test problems, justifying its effectiveness and robustness.

Convexification of Queueing Formulas by Mixed-Integer Second-Order Cone Programming: An Application to a Discrete Location Problem with Congestion

Mixed-integer second-order cone programs (MISOCPs) form a novel class of mixed-integer convex programs, which can be solved very efficiently as a result of the recent advances in optimization solvers. This paper shows how various performance metrics of M/G/1 queues can be modeled by different MISOCPs. To motivate the reformulation method, it is first applied to a challenging stochastic location problem with congestion, which is broadly used to design socially optimal service systems. Three different MISOCPs are developed and compared on different sets of benchmark test problems. The new formulations efficiently solve very large-size test problems that cannot be solved by the two existing methods developed based on linear programming within reasonable time. The superiority of the conic reformulation method is next shown over a state-space decomposition method recently used to solve an assignment problem in queueing systems. Finally, the general applicability of the method is shown for similar optimization problems that use queue-theoretic performance measures to address customer satisfaction and service quality.

Designing a bi-objective decision support model for the disaster management

This paper addresses the allocation and scheduling of the relief teams as one of the main issues in the response phase of the disaster management. In this study, a bi-objective mixed-integer programming (BOMIP) model is proposed to assign and schedule the relief teams in the disasters. The first objective function aims to minimize the sum of weighted completion times of the incidents. The second objective function also minimizes the sum of weighted tardiness of the relief operations. In order to be more similar to the real world, time windows for the incidents and damaged routes are considered in this research. Furthermore, the actual relief time of an incident by the relief team is calculated according to the position of the corresponding relief team and the fatigue effect. Due to NP-hardness of the considered problem, the proposed model cannot present the Pareto solution in a reasonable time. Thus, NSGA-II and PSO algorithms are applied to solve the problem. Furthermore, the obtained results of the proposed algorithms are compared with respect to different performance metrics in large-size test problems. Finally, the sensitivity analysis and the managerial suggestions are provided to investigate the impact of some parameters on the Pareto frontier.

The capacitated minimum spanning tree problem with arc time windows

We consider a new variant of the minimum spanning tree problem, where time windows are associated with the arcs of the underlying graph and capacities relate to the maximum number of vertices that subtrees may incorporate. The problem is referred to as the Capacitated Minimum Spanning Tree with Arc Time Windows (CMSTP_ATW) and emerges in routing situations with flow disruptions across road segments. We devise a Mixed Integer Programming (MIP) formulation to model the problem, which can be solved using CPLEX. To examine the quality of the solutions obtained, we convert the data sets of Solomon (1987) to appropriately capture CMSTP_ATW instances and provide results for the problems with 25, 50 and 100 vertices. Furthermore, we compare the CPLEX built-in heuristic that determines the initial integer solution for the CMSPT_ATW, vis-a-vis a greedy heuristic we have developed that offers high quality solutions in short computational times for the large size test problems. Experimental results show that there is a strong negative correlation between the GAP of CPLEX and the total number of iterations in one-hour performance time, against the no or positive correlation between the GAP of CPLEX and the initial iterations. Finally, we modify the MIP by adding parts of the solution derived, using the greedy heuristic, in the set of problem constraints, and observe that the CPLEX results for the CMSTP_ATW are in general improved, offering evidence that it is a promising solution approach.

Robotic disassembly line balancing problem: A mathematical model and ant colony optimization approach

The use of robots is significantly increasing day by day in manufacturing systems, and especially improving the efficiency of the lines. Robots can be used to complete disassembly tasks, and each of the robots can need different operation times to perform the tasks. In this paper, the balancing of the robotic disassembly line problem has been studied to develop efficient solution techniques. Firstly, a mixed-integer linear mathematical model is proposed to determine and solve the problem optimally. A case study from literature is addressed to assess and show the efficiency and effectiveness of the model to minimize cycle time. Secondly, a heuristic algorithm based on ant colony optimization is also proposed to discover a solution for especially the large-size test problems due to the complexity of the problem. The performance of the proposed heuristic algorithm is verified and compared with the different heuristic on data sets. The computational results indicate that the proposed mathematical model and the algorithms are promising for the small and large-size test problems, respectively. Finally, it should be stated that robots have great potential to use in the area of disassembly line and useful solutions provide according to test results.

Flexible flow shop scheduling problem to minimize makespan with renewable resources

This paper deals with a flexible flow shop (FFS) scheduling problem with unrelated parallel machines and renewable resource shared among the stages. The FFS scheduling problem is one of the most common manufacturing environment in which there is more than a machine in at least one production stage. In such a system, to decrease the processing times, additional renewable resources are assigned to the jobs or machines, and it can lead to decrease the total completion time. For this purpose, a mixed integer linear programming (MILP) model is proposed to minimize the maximum completion time (makespan) in an FFS environment. The proposed model is computationally intractable. Therefore, a particle swarm optimization (PSO) algorithm as well as a hybrid PSO and simulated annealing (SA) algorithm named SA-PSO, are developed to solve the model. Through numerical experiments on randomly generated test problems, the authors demonstrate that the hybrid SA-PSO algorithm outperforms the PSO, especially for large size test problems.

A robust fuzzy approach for constrained multi-product economic production quantity with imperfect items and rework process

ABSTRACTThis paper aims to develop a robust multi-item EPQ formulation considering rework process and imperfect items. The mathematical model consists of two objective functions targeting minimization of total inventory costs and the total required warehouse space, respectively. Since, in real-world applications, parameters of the mathematical formulation are due to uncertainty, in this paper, Basic Chance Constraint Programming and Robust Fuzzy Chance Constraint Programming models are developed to handle uncertainties in both objective function and constraints. By solving examples, the superiority of the RFCCP over the BCCP model is showed and its ability to provide risk-averse solutions is ensured. Due to the nonlinearity of the RFCCP model and to provide the decision maker with the Pareto front, two algorithms, namely, Multi-Objective Grey Wolf Optimizer and Multi-Objective Water Cycle Algorithm, are used to optimize the decision variables. The performance of the algorithms is evaluated within different large-size test problems, using diversity, Spacing, number of non-dominated solutions and CPU-time. In the end, to investigate significant differences and to determine the best algorithm ANOVA test is implemented. In the end, we also offer new research directions, e.g. in terms of further ways of uncertainty modelling and risk management.

Solving a new cost-oriented assembly line balancing problem by classical and hybrid meta-heuristic algorithms

In this study, a new cost-oriented assembly line balancing problem is proposed and formulated. A single objective function consisting of minimizing the cost associated with equipment, labor wage, and station establishment is considered for the problem. This problem is more complicated comparing to the literature as worker qualification is considered for determining his/her wage. As this problem is of NP-hard optimization problems, some meta-heuristic solution approaches, e.g., simulated annealing, variable neighborhood search, genetic algorithm, tabu search, population-based simulated annealing, and their hybrid versions are proposed. In the proposed algorithms, a novel encoding–decoding scheme is applied. This scheme uses the Hungarian method to assign the workers to the station to reduce the total wage of the workers. To study the performance of the proposed meta-heuristic algorithms, ten test problems are generated randomly, and using one of them the parameters of the algorithms are tuned by the Taguchi method. The final experiments on the proposed algorithms and the test problems show that in the most of the experiments, among the proposed algorithms, the single-solution-based algorithms, except TS, perform better than the population-based algorithms, especially for the case of large size test problems.

Optimizing a multi-item economic order quantity problem with imperfect items, inspection errors, and backorders

This paper proposes a multi-item economic order quantity model with imperfect items in supply deliveries. The inspection process to classify the items is not perfect and involves two types of error: Type-I and Type-II. To cope with the uncertainty involved in real-world applications and to bring the problem closer to reality, operational constraints are assumed stochastic. The aim is to determine the optimal order and back order sizes of the items in order to achieve maximum total profit. As the proposed mathematical model is a constrained nonlinear programming, three different solution methods including an exact method named the interior-point and two novel meta-heuristics named grey wolf optimizer (GWO) and moth-flame optimization (MFO) algorithms are utilized to solve the problem. In order to demonstrate the most efficient solution method, the performance of the three solution methods is evaluated when they solve some test problems of different sizes. Various comparison measures including percentage relative error, relative percentage deviation, and computation time are used to compare the solution methods. Based on the results, MFO performs better in small and medium instances in terms of percentage relative error; meanwhile, GWO shows a better performance in terms of relative percentage deviation in large-size test problems. In the end, sensitivity analyses are carried out to investigate how any parameter change affects the objective function value of the mathematical model in order to determine the most critical parameter.

Integration of efficient multi-objective ant-colony and a heuristic method to solve a novel multi-objective mixed load school bus routing model

In this paper, a novel mixed-load school bus routing problem (MLSBRP) is introduced. MLSBRP assumes that several students from different schools can simultaneously take a ride on the same bus. A bi-objective mixed integer linear programming (BO-MILP) formulation is proposed to model MLSBRP. The objectives are: a) minimizing the number of the buses; and b) minimizing the average riding time of the students. A hybrid multi-objective ant colony optimization (h-MOACO) algorithm, incorporating a novel routing heuristic algorithm, is developed to solve the associated BO-MILP. Performance of the proposed h-MOACO is compared with those achieved by commercial operation research software called CPLEX, and a customized NSGAII algorithm through multi-objective diversity and accuracy metrics over several small-size and large-size test problems, respectively. Sensitivity analysis are conducted on the main parameters of the MLSBRP. The computational results indicate the capability of the new proposed MLSBRP and suitability of the proposed h-MOACO algorithm.

Minimizing Piecewise-Concave Functions Over Polyhedra

We introduce an iterative method for solving linearly constrained optimization problems, whose nonsmooth nonconvex objective function is defined as the pointwise maximum of finitely many concave functions. Such problems often arise from reformulations of certain constraint structures (e.g., binary constraints, finite max-min constraints) in diverse areas of optimization. We state a local optimization strategy, which exploits piecewise-concavity of the objective function, giving rise to a linearized model corrected by a proximity term. In addition we introduce an approximate line-search strategy, based on a curvilinear model, which, similarly to bundle methods, can return either a satisfactory descent or a null-step declaration. Termination at a point satisfying an approximate stationarity condition is proved. We embed the local minimization algorithm into a Variable Neighborhood Search scheme and a Coordinate Direction Search heuristic, whose aim is to improve the current estimate of the global minimizer by exploring different parts of the feasible region. We finally provide computational results obtained on two sets of small- to large-size test problems.

A fuzzy multi-objective optimization model for sustainable reverse logistics network design

Decreasing the environmental impact, increasing the degree of social responsibility, and considering the economic motivations of organizations are three significant features in designing a reverse logistics network under sustainability respects. Developing a model, which can simultaneously consider these environmental, social, and economic aspects and their indicators, is an important problem for both researchers and practitioners. In this paper, we try to address this comprehensive approach by using indicators for measurement of aforementioned aspects and by applying fuzzy mathematical programming to design a multi-echelon multi-period multi-objective model for a sustainable reverse logistics network. To reflect all aspects of sustainability, we try to minimize the present value of costs, as well as environmental impacts, and optimize the social responsibility as objective functions of the model. In order to deal with uncertain parameters, fuzzy mathematical programming is used, and to obtain solutions on Pareto front, a customized multi-objective particle swarm optimization (MOPSO) algorithm is applied. The validity of the proposed solution procedure has been analyzed in small and large size test problems based on four comparison metrics and computational time using analysis of variance. Finally, in order to indicate the applicability of the suggested model and the practicality of the proposed solution procedure, the model has been implemented in a medical syringe recycling system. The results reveal that the suggested MOPSO algorithm overtakes epsilon-constraint method from the aspects of quality of the solutions as well as computational time. Proper use of the proposed process could help managers efficiently manage the flow of recycled products with regard to environmental and social considerations, and the process offers a sustainable competitive advantage to corporations.

Applying soft-computing techniques in solving dynamic multi-objective layout problems in cellular manufacturing system

The formation of machine cells to process subsequent part families is not the only goal of the designing of an efficient cellular manufacturing system (CMS). A competent layout of the newly acquired cells is also essential to restrict the total inter-cell material handling cost which is primarily significant with large production volume. Furthermore, in realistic industrial scenario, the uncertainty of product demand can influence the layout configuration to be altered from period to period. Albeit there are numerous articles exist in the domain of CMS research considering cell formation problems, layout issues have not been addressed significantly. Therefore, the aim of our paper is to portray a reformed mathematical model of the inter-cell layout design problem in dynamic production situation considering material handling cost and a modified proximity relationship of manufacturing cells. The proposed Quadratic Assignment Programming (QAP) model is combinatorial in nature and is difficult to solve using traditional exact solution methods. The state-of-the-art soft-computing techniques are extremely advantageous for such QAP paradigms. Thus, we developed an improved genetic algorithm (IGA) and a simulated annealing heuristic (SAH) to sort out the abovementioned problem. Due to the inadequacy of datasets, we formed small to large size test problems (6 × 6 × 2 to 24 × 24 × 10) in logical way to cater the purpose. The proposed algorithms are successfully employed to attain near-optimal solutions to the test problems. Computational results demonstrate the proficiency of the IGA over SAH for all the test problems in terms of solution quality and computational time. In addition, we conducted a statistical data analysis to validate the test results.

Design of high-performing hybrid meta-heuristics for unrelated parallel machine scheduling with machine eligibility and precedence constraints

This study involves an unrelated parallel machine scheduling problem in which sequence-dependent set-up times, different release dates, machine eligibility and precedence constraints are considered to minimize total late works. A new mixed-integer programming model is presented and two efficient hybrid meta-heuristics, genetic algorithm and ant colony optimization, combined with the acceptance strategy of the simulated annealing algorithm (Metropolis acceptance rule), are proposed to solve this problem. Manifestly, the precedence constraints greatly increase the complexity of the scheduling problem to generate feasible solutions, especially in a parallel machine environment. In this research, a new corrective algorithm is proposed to obtain the feasibility in all stages of the algorithms. The performance of the proposed algorithms is evaluated in numerical examples. The results indicate that the suggested hybrid ant colony optimization statistically outperformed the proposed hybrid genetic algorithm in solving large-size test problems.

Robust optimization approach to production system with failure in rework and breakdown under uncertainty: evolutionary methods

Purpose – This paper aims to consider a multi-site production planning problem with failure in rework and breakdown subject to demand uncertainty. Design/methodology/approach – In this new mathematical model, at first, a feasible range for production time is found, and then the model is rewritten considering the demand uncertainty and robust optimization techniques. Here, three evolutionary methods are presented: robust particle swarm optimization, robust genetic algorithm (RGA) and robust simulated annealing with the ability of handling uncertainties. Firstly, the proposed mathematical model is validated by solving a problem in the LINGO environment. Afterwards, to compare and find the efficiency of the proposed evolutionary methods, some large-size test problems are solved. Findings – The results show that the proposed models can prepare a promising approach to fulfill an efficient production planning in multi-site production planning. Results obtained by comparing the three proposed algorithms demonstrate that the presented RGA has better and more efficient solutions. Originality/value – Considering the robust optimization approach to production system with failure in rework and breakdown under uncertainty.

Dynamic Vehicle Routing Problem with Multiple Depots

The Vehicle Routing Problems (VRPs) has been extensively studied and applied in many fields. Variations of VRPs have been proposed and appeared in research for many decades. Dynamic Vehicle Routing Problem with Multiple Depots (D-MDVRP) extends the variation of VRPs to dynamism of customers by knowing the information of customers (both locations and due dates) at diverse times. An application of this problem can be found in food delivery services which have many service stores. The customer delivery orders are fulfilled by a group of scattered service stores which can be analogous to depots in D-MDVRP. In this example the information of all customer orders are not known at the same time depending on arrivals of customers. Thus the objective of this operation is to determine vehicle routing from service stores as well as dispatching time. This paper aims to develop a heuristic approach for D-MDVRP. The proposed heuristic method comprises of two phases: route construction and vehicle dispatch. Routes are constructed by applying the Nearest Neighbor Procedure (NNP) to cluster customers and select a proper depot, Sweeping and Reordering Procedures (SRP) to generate initial feasible routes, and Insertion Procedure (IP) to improve routing. Then the determination of dispatch is followed in the next phase. In order to deal with the dynamism, the dispatch time of each vehicle is determined by maximizing the waiting time to provide the opportunity to add more arriving customers in the future. An iterative process between two phases is adopted when a new customer enters the problem, and the vehicles are dispatched when the time becomes critical. From the computational study, the heuristic method performs well on small sized test problems in a shorter CPU time compared to the optimal solutions from CPLEX, and provides an overall average of 8.36 % Gap. For large size test problems, the heuristic method is compared with static problems, and provides an overall average of 3.48 % Gap.

The center location-dependent relocation problem with a probabilistic line barrier

This paper presents a new mixed-integer nonlinear programming (MINLP) for a multi-period rectilinear distance center location-dependent relocation problem in the presence of a probabilistic line-shaped barrier that uniformly occurs on a given horizontal route. In this problem, the demand and location of the existing facilities have a dynamic nature and the relocation is dependent to the location of new facilities in previous period. The objective function of the presented model is to minimize the maximum expected weighted barrier distance between the new facility and the existing facilities during the planning horizon. The optimum solution of small-sized test problems is obtained by the optimization software. For large-size test problems which the optimization software is unable to find the optimum solution in the runtime limitation, two meta-heuristics based on the genetic algorithm (GA) and imperialist competitive algorithm (ICA) are applied. To validate the meta-heuristics, a lower bound problem based on the forbidden region instead of the line barrier is generated. Related results of numerical experiments are illustrated and are then compared.

Determining the parameters of MSG algorithm for multi period layout problem

PurposeThe facility layout problem aims to assign machines/departments to locations and modeled as a quadratic assignment problem (QAP). Multi period facility layout is a special case of this problem where the sum of material handling and re‐layout costs are minimized. Since the problem is proved to be NP‐hard, several exact and heuristic methods are proposed in the literature. The purpose of this paper is to solve the multi period layout problem by using the modified sub‐gradient (MSG) algorithm for the first time and to determine its parameters.Design/methodology/approachThe MSG algorithm can solve a large‐scale of optimization problems that also includes multi period facility layout. Since the performance of the algorithm depends on parameters, a design of experiment is made to determine the appropriate parameter values.FindingsThe proposed method evaluates the parameters of the MSG algorithm and most suitable general algebraic modeling solvers. It is observed that the parameter α value and solver type have main effects for small and large size test problems. Further, the results stated that solver type has more influence on large size test problem.Research limitations/implicationsThis study is limited with the determination of the MSG algorithm parameters and solver types on the well known small and large size test problems. Further studies may include other test problem results obtained from the presented MSG algorithm parameters and compare them with best known results in the literature.Originality/valueThe paper determines the parameters of the MSG algorithm that is used to solve the multi period layout problem, for the first time in the literature.

An artificial immune system based algorithm to solve unequal area facility layout problem

This study introduces an artificial immune system (AIS) based algorithm to solve the unequal area facility layout problem (FLP) with flexible bay structure (FBS). The proposed clonal selection algorithm (CSA) has a new encoding and a novel procedure to cope with dummy departments that are introduced to fill the empty space in the facility area. The algorithm showed consistent performance for the 25 test problem cases studied. The problems with 100 and 125 were studied with FBS first time in the literature. CSA provided four new best FBS solutions and reached to sixteen best-so-far FBS solutions. Further, the two very large size test problems were solved first time using FBS representation, and results significantly improved the previous best known solutions. The overall results state that CSA with FBS representation was successful in 95.65% of the test problems when compared with the best-so-far FBS results and 90.90% compared with the best known solutions that have not used FBS representation.

A Heuristic Algorithm for Resource Allocation/Reallocation Problem

This paper presents a 1-opt heuristic approach to solve resource allocation/reallocation problem which is known as 0/1 multichoice multidimensional knapsack problem (MMKP). The intercept matrix of the constraints is employed to find optimal or near‐optimal solution of the MMKP. This heuristic approach is tested for 33 benchmark problems taken from OR library of sizes upto 7000, and the results have been compared with optimum solutions. Computational complexity is proved to be O(klmn2) of solving heuristically MMKP using this approach. The performance of our heuristic is compared with the best state‐of‐art heuristic algorithms with respect to the quality of the solutions found. The encouraging results especially for relatively large‐size test problems indicate that this heuristic approach can successfully be used for finding good solutions for highly constrained NP‐hard problems.