Bi-objective Mathematical Model Research Articles

A hybrid learning-based meta-heuristic algorithm for scheduling of an additive manufacturing system consisting of parallel SLM machines

Additive manufacturing (AM) has been recognised as a promising technology under the context of Industry 4.0, which is reshaping manufacturing paradigms. A prominent type of AM machine is the selective laser melting (SLM) machine, in which several parts may form a job and be produced concurrently. This paper aims to investigate a scheduling problem in an AM system with non-identical parallel SLM machines. Since, in this system, there might be differences in the material types of parts, the required setup time between two consecutive jobs on the relevant machine is dependent on their material types. Accordingly, a bi-objective mathematical model is extended for the problem, considering the makespan and the total tardiness penalty as two objective functions. Due to the high complexity of the problem, an efficient hybrid meta-heuristic algorithm is developed by combining the non-dominated sorting genetic algorithm (NSGA-II) with a novel learning-based local search founded on the k-means clustering algorithm and a regression neural network. The local search enhances the exploitation ability of the NSGA-II while intelligently being taught during the solving procedure. Finally, the superiority of the proposed hybrid algorithm is demonstrated through a computational experiment.

Ecotourism supply chain during the COVID-19 pandemic: A real case study

The coronavirus (COVID-19) disease has caused serious and irreversible damage to the ecotourism industry, posing serious challenges to all parts of the ecotourism supply chain. The ecotourism supply chain is made up of various components, the most important of which are ecotourism centers. During these pandemic times, the primary concerns of these centers are to improve their deplorable economic conditions and retain customers for the post-coronavirus era. As a result, an investigation should be conducted to address these concerns and provide appropriate solutions to help them overcome the challenges that have emerged. To achieve the research goal, a bi-objective mathematical model for the ecotourism supply chain in an uncertain environment is developed, accounting for the effects of COVID-19. The first objective function minimizes the total cost of the supply chain, while the second maximizes customer satisfaction. The proposed mathematical model is solved using a fuzzy goal programming (FGP) method. A sensitivity analysis study is also carried out to examine the performance of some basic parameters. Furthermore, the model is tested in a real case study to determine its efficacy. Finally, some effective managerial insights are proposed to improve the situation of the centers during the pandemic.

Multigraph Modeling for Urban Distribution of Emergency Commodities with Semisoft Time Windows under Uncertainty

In this paper, we studied a stochastic bi-objective mathematical model for effective and reliable rescue operations in multigraph network. The problem is addressed by a two-stage stochastic nonlinear mixed-integer program where the reliability of routes is explicitly traded-off with total weighted completion time. The underlying transportation network is able to keep a group of multiattribute parallel arcs between every pair of nodes. By this, the proposed model should consider the routing decision in logistic planning along with the path selection in an uncertain condition. The first stage of the model concerns with the vehicle routing decisions which is not involved with random parameters; besides, the second stage of the model involves with the departure time at each demand node and path finding decisions after observation of random vectors in the first stage considering a finite number of scenarios. To efficiently solve the presented model, an enhanced nondominated sorting genetic algorithm II (NSGA-II) is proposed. The effectiveness of the introduced method is then evaluated by conducting several numerical examples. The results implied the high performance of our method in comparison to the standard NSGA-II. In further analyses, we investigated the beneficiary of using multigraph setting and showed the applicability of the proposed model using a real transportation case.

Designing an uncertain bi-objective green leagile capacitated lot sizing problem considering FM/M/C queue system

PurposeThis paper aims to present a novel bi-objective mathematical model for a production-inventory system under uncertainty.Design/methodology/approachThis paper addresses agile and lean manufacturing concepts alongside with green production methods to design an integrated capacitated lot sizing problem (CLSP). From a methodological perspective, the problem is solved in three phases. In the first step, an FM/M/C queuing system is used to minimize the number of customers waited to receive their orders. In the second step, an effective approach is applied to deal with the fuzzy bi-objective model and finally, a hybrid metaheuristic algorithm is used to solve the problem.FindingsSome numerical test problems and sensitivity analyzes are conducted to measure the efficiency of the proposed model and the solution method. The results validate the model and the performance of the solution method compared to Gams results in small size test problems and prove the superiority of the hybrid algorithm in comparison with the other well-known metaheuristic algorithms in large size test problems.Originality/valueThis paper presents a novel bi-objective mathematical model for a CLSP under uncertainty. The proposed model is conducted on a practical case and several sensitivity analysis are conducted to assess the behavior of the model. Using a queue system, this problem aims to reduce the items waited in the queue to receive service. Two objective functions are considered to maximize the profit and minimize the negative environmental effects. In this regard, the second objective function aims to reduce the amount of emitted carbon.

Green dynamic multimodal logistics network design problem considering financing decisions: a case study of cement logistics.

Logistics network is one of the most important parts of supply chains with significant share in achieving sustainability across them. In this paper, we investigate a new multi-objective mixed integer linear programming model for the design of multimodal logistics network. A bi-objective mathematical model is introduced and two conflicting objectives including the minimization of total cost and the total environmental impact are taken into account. Effective environmental life cycle assessment-based method is incorporated in the model to estimate the relevant environmental impacts. Due to budget constraints, financing decisions for facility construction are considered in the proposed model. To cope with the model objective functions, the augmented ε-constraint method is applied. Computational analysis is also provided by using a cement multimodal rail-road logistics network case study to present the significance of the proposed model. Results show that utilizing the proposed multi-period optimization model influences the location of multimodal terminals and their construction time. Also, the results show that the use of the proposed model enhances the efficiency of terminals. On the other hand, computational results indicate that preferences of decision-makers and the importance of environmental objective have significant impacts on the topology of transportation network.

Push/Pull Parts Production Policy Optimization in the ATO Environment

Within the Assembly to Order (ATO) production strategy, the common approach is to produce the parts to assemble with a Push-Make to Stock policy.In recent decades, the effects of the modern Just in Time (JIT) moved to a Pull-Make to Order policy. Assembled parts characterized by wide variety and huge storage space utilization are critical, and a proper Push/Pull production policy definition is required. An appropriate balance of storage space utilization and setup times leads to the optimization of the production policy. The aim of this paper is to define a bi-objective mathematical optimization model to assign the most suitable production policy to the parts within the production mix in an ATO industrial context. A numerical simulation and an operative case study showcases the model application, proving the industrial relevance of this research.

Optimal cross-trained worker assignment for a hybrid seru production system to minimize makespan and workload imbalance

As a worker-centred assembly mode developed in the electronics industry in Japan, seru is receiving growing attention due to improved flexibility and responsiveness. In seru implementation, cross-trained worker assignment is a vital problem. Most previous studies focused on assigning cross-trained workers into pure divisional or rotating seru separately, but overlooked the problem for a hybrid seru production system that includes both seru types. This research fills this gap by minimizing the makespan and balancing the workers’ workload of each seru in a bi-objective mathematical model. For medium-scale instances, the exact solutions are obtained. For large-scale instances, we propose an NSGA-II-based memetic algorithm that uses two-level encoding and incorporates the bat algorithm as a local search and two K-means-based NSGA-II algorithms. The experimental results illustrate that the K-means-based NSGA-II not only outperforms other algorithms with respect to common proximity and diversity metrics but also runs an order of magnitude faster (in seconds versus minutes required by others on the same computer). Some management insights are obtained based on many numerical experiments.

Bi-objective grey wolf optimization algorithm combined Levy flight mechanism for the FMC green scheduling problem

As more and more enterprises have laid great emphasis on eco-friendly manufacturing processes while enhancing agility, this paper focuses on the green scheduling problem of the Flexible Manufacturing Cell with material handling robots (FMC-R). In the FMC, each job is characterized by several operations, processing machines, and machining time. Robots at one straight track transfer jobs to the related machines. Distinguished from most of the FMC scheduling problems, this paper considers not only the operation sequencing of jobs but also the robots’ transportation processes. A bi-objective mathematical model is established aiming to minimize the total makespan and the total energy consumption of the FMC system simultaneously. Due to the NP-hard nature of the problem, a Levy Flight and Weighted Distance-updated Multi-objective Grey Wolf Algorithm (LWMOGWO) is proposed. It combines the Levy flight with Multi-objective Grey Wolf Algorithm (MOGWO). A weighted distance updating mechanism is integrated to calculate the positions of individuals. A local neighbourhood search strategy is adopted, which makes it possible to take full advantage of the three leading wolves in population. The performance of the LWMOGWO algorithm is evaluated through groups of experiments by comparing with multiple hybrid meta-heuristic algorithms, hybrid particle swarm optimization (NGPSO) algorithm, an improved grey wolf optimization (IGWO) algorithm, MOGWO, NSGA-II (Non-dominated sorting genetic algorithm), and PLMEAPS. Results reveal the proposed LWMOGWO algorithm has a higher quality of solutions in solving the scheduling of the FMC-R problem.

Metaheuristic approach proposal for the solution of the bi-objective course scheduling problem

Timetabling problems are among the commonly encountered problems in real life, from education institutions to airline companies. It is generally difficult to obtain optimal solutions for the timetabling problems that vary in terms of structures of constraints and objective functions, and these problems are considered being in NP-hard category, which cannot be solved in polynomial time in real life. In this study, a bi-objective mathematical model is proposed for a course scheduling problem in Kutahya Dumlupinar University Department of Industrial Engineering. While it is aimed in the first objective function to maximize the sum of the preferences of instructors determined by using the Analytic Hierarchy Process Method, it is aimed to minimize the students’ course overlap in the other. Conic scalarization method is used to combine the objective functions. Due to NP-hard nature of the problem, the Tabu Search Algorithm, one of metaheuristic approaches is used to solve it. Using the obtained data, the Tabu Search Algorithm by considering the proposed bi-objective mathematical model is designed for the problem and a software is developed in Excel Visual Basic program. The experimental results are evaluated with Analysis of Variance by using Minitab Program, comparing the results, satisfactory solutions are obtained.

Determining the optimal portfolio for healthcare processes management using a hybrid decision-making approach

Due to the increasing progress in various industries, paying attention to the internal processes of the organizations is more visible to stay on the competitive scene. Therefore, many organizations attempt to simplify and evaluate their internal processes using re-engineering. By reviewing the conducted studies, it can be stated that one of the existing problems in the implementation of re-engineering projects is the selection of the optimal portfolio of processes. Hence, this study aims to provide a bi-objective mathematical model for selecting processes in the re-engineering project by considering two key assumptions include improvement in achieving organizational goals and staff resistance. To this end, first, the impact of processes on organizational goals is specified by experts and then the goals’ weights are obtained using a fuzzy Best Worst Method. Finally, the proposed model is solved by an augmented e-constraint method and the optimal portfolio of processes is selected. Also, a public Hospital of Sari as a real-world case study is employed to set the values of model parameters. Finally, the obtained results are reported and using a sensitivity analysis, several directions are provided. The results show that changes in the staff resistance directly affects the second objective function, while changes in the improvement created by each process affect the first objective function. Also, changes in costs have little effect on either objective functions.

A heuristic optimization approach for multi-vehicle and one-cargo green transportation scheduling in shipbuilding

To actively respond to the call for green shipbuilding, block cooperative transportation has been particularly concerned in reducing carbon emission in the shipyard, and hence a “multi-vehicle and one-cargo” (MVOC) green transportation scheduling problem emerges. Aiming to solve this problem effectively and improve transportation efficiency and reduce energy consumption, a bi-objective mathematical model combined routing model with synchronization constraints is proposed to simultaneously minimize non-value-added transportation time cost and total CO2 emission. A Pareto-based multi-objective Tabu Search (MOTS) algorithm is then designed to solve the model, in which local improvements are developed to generate promising neighboring individuals. Experimental results show that the proposed MOTS algorithm can effectively solve the problem even on a large scale and outperform the classic algorithm of nondominated sorting genetic algorithm-II (NSGA-Ⅱ). It is hoped that this work enables an operation mode with high efficiency and low energy consumption and provides useful insights for flatcar transportation scheduling operators in the shipyard.

A bi-objective mathematical model for integrated planning of sugarcane harvesting and transport operations

Motivated by a real situation arising in the Brazilian sugarcane industry, this paper addresses the integrated planning of harvesting and transport operations over a multi-period planning horizon. The aim is to develop a tactical plan for the deployment of harvesting and transport equipment that sets the periods for the execution of the harvesting operations on the sugarcane fields and the type of harvesting machines and road haulage vehicles to be operated. These decisions are made subject to multiple constraints related to the projected crop yield, resource availability, demand for sugarcane at the mills and further technical requirements specific to the harvesting operations. The tactical plan to be determined minimizes the total cost incurred by the equipment used and the total time required to harvest all the fields. We propose a bi-objective mixed-integer non-linear programming model for this new problem. A computational study is conducted for test instances capturing the characteristics of a Brazilian milling company. Pareto-optimal solutions are identified by the Progressive Bounded Constraint method that is extended to the problem at hand. A comparative analysis highlights the trade-offs between economic performance and harvesting efficiency, thereby supporting the decision maker in making a more informed choice of the preferred tactical plan. Useful managerial insights are also provided into the profile of the harvesting and transport resources that should be used under different weather conditions and work schedules.

An Integrated Bi-Objective Mathematical Model for Minimizing Take-off Delay and Passenger Dissatisfaction

As air transportation has increased in recent years, it is necessary for airport planners to optimally manage aircraft ground traffic on stands, taxiways and runways in order to minimize flight delay and passenger dissatisfaction. A closer look at the literature in this area indicates that most studies have merely focused on one of these resources which in a macroscopic level may result in aircrafts’ collision and ground traffic at the airport. In this paper, a new bi-objective Mixed-Integer Linear Programming (MILP) model is developed to help airport management to integrate Gate Assignment Problem (GAP) and Runway Scheduling Problem (RSP) considering taxiing operation for departing flights. The proposed model aims to help airport planners to 1) minimize any deviation from preferred schedule and 2) minimize transit passengers’ walking distance. Due to the complexity of the research problem, a Normalized Weighted Sum Method (NWSM) is applied to solve small-sized problems and two meta-heuristics, namely NSGA-II and MOGWO, are used for large-scale instances to generate Pareto optimal solutions. The performance of these algorithms is assessed by well-known coverage and convergence measures. Based on the most criteria, the results indicate that MOGWO outperforms NSGA-II.

Multi-Depot Pickup and Delivery Problem with Resource Sharing

Resource sharing (RS) integrated into the optimization of multi-depot pickup and delivery problem (MDPDP) can greatly reduce the logistics operating cost and required transportation resources by reconfiguring the logistics network. This study formulates and solves an MDPDP with RS (MDPDPRS). First, a bi-objective mathematical programming model that minimizes the logistics cost and the number of vehicles is constructed, in which vehicles are allowed to be used multiple times by one or multiple logistics facilities. Second, a two-stage hybrid algorithm composed of a k-means clustering algorithm, a Clark-Wright (CW) algorithm, and a nondominated sorting genetic algorithm II (NSGA-II) is designed. The k-means algorithm is adopted in the first stage to reallocate customers to logistics facilities according to the Manhattan distance between them, by which the computational complexity of solving the MDPDPRS is reduced. In the second stage, CW and NSGA-II are adopted jointly to optimize the vehicle routes and find the Pareto optimal solutions. CW algorithm is used to select the initial solution, which can increase the speed of finding the optimal solution during NSGA-II. Fast nondominated sorting operator and elite strategy selection operator are utilized to maintain the diversity of solutions in NSGA-II. Third, benchmark tests are conducted to verify the performance and effectiveness of the proposed two-stage hybrid algorithm, and numerical results prove that the proposed methodology outperforms the standard NSGA-II and multi-objective particle swarm optimization algorithm. Finally, optimization results of a real-world logistics network from Chongqing confirm the applicability of the mathematical model and the designed solution algorithm. Solving the MDPDPRS provides a management tool for logistics enterprises to improve resource configuration and optimize logistics operation efficiency.

Two-echelon multi-period location routing problem with shared transportation resource

Solving a two-echelon multi-period location routing problem (2E-MPLRP) involves facility location selection and two-echelon vehicle routing optimization. Based on the periodic time characteristics of logistics facilities and customers, the optimal solutions provide periodic location decisions and vehicle routing schemes simultaneously in each service period of the planning horizon. Transportation resource configuration is tweaked by enabling resource sharing across multiple service periods to maximize resource utilization in the context of growing emphasis on sustainable development. A bi-objective mathematical model is developed to formulate the 2E-MPLRP to obtain the minimum total operating cost and number of vehicles. A two-stage hybrid algorithm including three-dimensional (3D) k-means clustering and multi-objective improved particle swarm optimization (MOIPSO) algorithm is proposed to solve the 2E-MPLRP. The 3D k-means clustering algorithm is adapted to assign customers to distribution centers (DCs) to receive service in multiple service periods, and the MOIPSO algorithm is then designed to optimize the vehicle routes and find the Pareto optimal solutions. With an external repository strategy and a rapidly decreasing mutation strategy incorporated in the iterative process, the proposed hybrid algorithm performs well in expanding the particles’ searching region and achieving robust optimal results. An algorithm comparison demonstrates the superiority of the proposed hybrid algorithm over other existing algorithms. A real-world case study of 2E-MPLRP in Chongqing, China is conducted, and results show that the proposed model and algorithm are of practical significance in minimizing operating cost, improving transportation efficiency, and contributing to sustainable two-echelon logistics network operations.

A multi-objective robust possibilistic programming approach to coordinating procurement operations in the disaster supply chain using a multi-attribute reverse auction mechanism

This paper presents a multi-attribute reverse auction framework to facilitate the purchase operations between suppliers and a relief organization. The proposed auction framework consists of announcement, construction, and evaluation phases. In the announcement phase, the relief organization provides suppliers with information about demand and required relief items. In the construction phase, a fuzzy bi-objective mathematical model is developed from a supplier’s perspective; by solving the model, the suppliers offer their bids, each of which stipulating price, quantity, and delivery time for each item. The evaluation phase is formulated from the relief organization’s perspective; suppliers are first ranked based on their proposed bids and other qualitative criteria by the fuzzy PROMETHEE method. Then, a fuzzy tri-objective mathematical model is solved to determine winner(s) and allocate orders. For the first time, this paper develops a multi-objective robust possibilistic programming (MORPP) approach to cope with the uncertainty of parameters in the relief chain. Furthermore, multi-objective models are solved by an augmented ∊-constraint method. Finally, applicability and validity of the proposed auctioning framework are tested through a real case study in Iran and the managerial insights are provided.

Multidepot Recycling Vehicle Routing Problem with Resource Sharing and Time Window Assignment

This study aims to provide tactical and operational decisions in multidepot recycling logistics networks with consideration of resource sharing (RS) and time window assignment (TWA) strategies. The RS strategy contributes to efficient resource allocation and utilization among recycling centers (RCs). The TWA strategy involves assigning time windows to customers to enhance the operational efficiency of logistics networks. A biobjective mathematical model is established to minimize the total operating cost and number of vehicles for solving the multidepot recycling vehicle routing problem with RS and TWA (MRVRPRSTWA). A hybrid heuristic algorithm including 3D k-means clustering algorithm and nondominated sorting genetic algorithm- (NSGA-) II (NSGA-II) is designed. The 3D k-means clustering algorithm groups customers into clusters on the basis of their spatial and temporal distances to reduce the computational complexity in optimizing the multidepot logistics networks. In comparison with NSGA algorithm, the NSGA-II algorithm incorporates an elitist strategy, which can improve the computational speed and robustness. In this study, the performance of the NSGA-II algorithm is compared with the other two algorithms. Results show that the proposed algorithm is superior in solving MRVRPRSTWA. The proposed model and algorithm are applied to an empirical case study in Chongqing City, China, to test their applicability in real logistics operations. Four different scenarios regarding whether the RS and TWA strategies are included or not are developed to test the efficacy of the proposed methods. The results indicate that the RS and TWA strategies can optimize the recycling services and resource allocation and utilization and enhance the operational efficiency, thus promoting the sustainable development of the logistics industry.

Modelling and an improved NSGA-II algorithm for sustainable manufacturing systems with energy conservation under environmental uncertainties: a case study

ABSTRACT This study addresses a flexible job shop scheduling problem considering the energy tax regulations. We introduce two strategies as reactive scheduling to deal with two different kinds of uncertainty. The first uncertainty is about the start and processing times of the jobs with a known probability distribution. The second kind is related to machine breakdowns, modification or cancellation of the orders, and receive new orders without any known probability distribution. Two conflict objective functions are considered as minimising tax cost on surplus energy consumption and minimising total cost of jobs lateness based on soft time-windows. A bi-objective mathematical model is developed to formulate the problem. Since the problem is well-known strongly NP-hard, a new solution approach is introduced based on the NSGA-II algorithm to solve the problem in a reasonable computational time. Some test problems based on a real case study are used to evaluate the performance of the proposed solution approach. The result analysis confirms the effectiveness of the proposed algorithm and its superiority comparing with another algorithm based on the classic NSGA-II. Moreover, sensitivity analysis is done on the main parameters to provide proper managerial suggestions based on the obtained Pareto front solutions.

Designing an integrated municipal solid waste management system using a fuzzy chance-constrained programming model considering economic and environmental aspects under uncertainty

In recent decades, municipal solid waste (MSW) management has become one of the major concerns of human societies. Therefore, an efficient MSW system of collecting, recycling, and disposing is required to be designed, maintained and improved continuously. However, designing a proper waste collection management system by controlling the spread of diseases and environmental pollutants is very costly. To provide an efficient MSW management system, this study proposes an integrated location-allocation problem to plan the operations such as collection, recycling, disposal, and transportation under uncertainty. To deal with the problem uncertainty, a triangular fuzzy number approach is employed and finally, a fuzzy chance-constrained programming model is developed. Moreover, the proposed bi-objective mathematical programming model aims to minimize the total costs and the total volume of pollutant emissions simultaneously. To solve and validate the model, a weighted goal programming (WGP) method is employed and solved by CPLEX solver of GAMS software. Finally, a real case study problem was implemented in Qazvin, Iran. The results showed that, in most cases, increasing the confidence level of the case study problem leads to the improvement of objective functions. Managers can take into account the output of this research provided along with a sensitivity analysis under different conditions.

A Trial-and-Error Toll Design Method for Traffic Congestion Mitigation on Large River-Crossing Channels in a Megacity

In this study, we addresse traffic congestion on river-crossing channels in a megacity which is divided into several subareas by trunk rivers. With the development of urbanization, cross-river travel demand is continuously increasing. To deal with the increasing challenge, the urban transport authority may build more river-crossing channels and provide more high-volume public transport services to alleviate traffic congestion. However, it is widely accepted that even though these strategies can mitigate traffic congestion to a certain level, they are not essential approaches to address traffic congestion. In this study, we consider a channel toll scheme for addressing this issue. Additional fares are applied to private vehicles, that an appropriate number of private vehicle drivers are motivated to take public transport or switch to neighboring uncongested river-crossing channels. To minimize the toll surcharge on both neighboring channels, while alleviating the traffic flow to a certain level, in this study, we provide a bi-objective mathematical model. Some properties of this model are discussed, including the existence and uniqueness of the Pareto optimal solution. To address this problem, a trial-and-error method is applied. Numerical experiments are provided to validate the proposed solution method.