Bi-objective Mixed-integer Programming Model Research Articles

A bi-objective optimization model for the medical supplies' simultaneous pickup and delivery with drones

In the COVID-19 pandemic, it is essential to transport medical supplies to specific locations accurately, safely, and promptly on time. The application of drones for medical supplies delivery can break ground traffic restrictions, shorten delivery time, and achieve the goal of contactless delivery to reduce the likelihood of contacting COVID-19 patients. However, the existing optimization model for drone delivery is cannot meet the requirements of medical supplies delivery in public health emergencies. Therefore, this paper proposes a bi-objective mixed integer programming model for the multi-trip drone location routing problem, which allows simultaneous pick-up and delivery, and shorten the time to deliver medical supplies in the right place. Then, a modified NSGA-II (Non-dominated Sorting Genetic Algorithm II) which includes double-layer coding, is designed to solve the model. This paper also conducts multiple sets of data experiments to verify the performance of modified NSGA-II. Comparing with separate pickup and delivery modes, this study demonstrates that the proposed optimization model with simultaneous pickup and delivery mode achieves a shorter time, is safer, and saves more resources. Finally, the sensitivity analysis is conducted by changing some parameters, and providing some reference suggestions for medical supplies delivery management via drones.

A new vocation queuing model to optimize truck appointments and yard handling-equipment use in dual transactions systems of container terminals

In a dual transaction at a marine terminal, an external truck drops off one container and picks up another on a single trip. To ensure a successful dual transaction operation and smooth external truck arrivals, a bi-objective mixed integer programming model is developed that optimizes the allocation of appointment quotas simultaneously with the deployment of (yard) cargo handling equipment. Our model addresses the challenges posed by the different types of truck movement in the terminal, i.e., delivery, pickup, and dual transaction. These require different handling equipment, various deadlines, and multiple priorities. To estimate the queuing length of external trucks in single or dual transactions (as well as that of internal trucks), a novel three-level vocation queuing model is set up. For the bi-objectives optimization, we propose a revised non-dominated genetic algorithm to obtain the approximate optimal solution. Experimental results prove the efficiency and effectiveness of the method. It shows that the newly built vocation queuing model can estimate the prioritized queuing process more effectively in three respects: a) the 3-level queuing; b) discrete truck arrivals in the queuing system; c) non-interruption of servers. Our quota optimization design improves the model’s applicability to real cases, especially in the case of dual transactions. It is demonstrated that the method proposed in the paper could help terminal operators allocate quotas and simultaneously match the capacity of yard handling, thus improving truck services and terminal profitability.

A goal programming model for the optimization of log logistics considering sorting decisions and social objective

Log logistics include sorting, processing, and transporting of logs from their place of harvest to demand locations. These activities account for a significant portion of the total log procurement costs; therefore, attempts were made in previous studies to optimize some aspects of log logistics. However, operational details, such as sorting decisions, truck compatibility requirements, and social objectives, are often disregarded in the optimization literature. Incorporating these details into the model makes the results more realistic and applicable. To address these gaps, a bi-objective mixed-integer programming model is developed in this paper to optimize log logistics. The first objective is to minimize total logistics costs, and the second objective is to provide a balanced workload for trucking contractors. The bi-objective model is solved using the goal programming approach. The model is applied to log logistics of a large Canadian forest company, where trucking contractors use heterogeneous fleet of trucks to carry various log sorts from cutblocks to sort yards for sorting. The planning horizon is 4 weeks with daily decisions. The goal programming model generates balanced workloads for the contractors with less than 0.4% increase in total costs compared to the single objective model where only the total cost is minimized.

Multi-Objective Vehicle Routing Problem for a Mixed Fleet of Electric and Conventional Vehicles with Time Windows and Recharging Stations

The vehicle routing problem as a challenging decision problem has been studied extensively. More specifically, solving it for a mixed fleet requires realistic calculation of the performance of electric and combustion vehicles. This study addresses a new variant of the vehicle routing problem for a mixed fleet of electric and combustion vehicles under the presence of time windows and charging stations. A bi-objective mixed-integer programming model is developed which aims at minimizing cost and pollution level concurrently. To accurately quantify travel quantities, such as fuel consumption, emission, and battery charge level, a set of realistic mathematical formulas are used. The model is first converted to a single-objective counterpart using the epsilon-constraint method and a simulated annealing algorithm is tailored to obtain Pareto optimal solutions. A discussion is also made on how the final solution can be selected from the Pareto frontier according to the design objectives. The presented framework can find a set of Pareto optimal solutions as a trade-off between cost and pollution objectives by considering different combinations of electric and combustion vehicles. It was shown that those solutions that involve more electric fleet than combustion fleet, lead to higher total costs and smaller emissions and vice versa.

Risk-Averse Co-Decision for Lower-Carbon Product Family Configuration and Resilient Supplier Selection

Due to global pandemics, political unrest and natural disasters, the stability of the supply chain is facing the challenge of more uncertain events. Although many scholars have conducted research on improving the resilience of the supply chain, the research on integrating product family configuration and supplier selection (PCSS) under disruption risks is limited. In this paper, the centralized supply chain network, which contains only one major manufacturer and several suppliers, is considered, and one resilience strategy (i.e., the fortified supplier) is used to enhance the resilience level of the selected supply base. Then, an improved stochastic bi-objective mixed integer programming model is proposed to support co-decision for PCSS under disruption risks. Furthermore, considering the above risk-neutral model as a benchmark, a risk-averse mixed integer program with Conditional Value-at-Risk (CVaR) is formulated to achieve maximum potential worst-case profit and minimum expected total greenhouse gases (GHG) emissions. Then, NSGA-II is applied to solve the proposed stochastic bi-objective mixed integer programming model. Taking the electronic dictionary as a case study, the risk-neutral solutions and risk-averse solutions that optimize, respectively, average and worst-case objectives of co-decision are also compared under two different ranges of disruption probability. The sensitivity analysis on the confidence level indicates that fortifying suppliers and controlling market share in co-decision for PCSS can effectively reduce the risk of low-profit/high-cost while minimizing the expected GHG emissions. Meanwhile, the effects of low-probability risk are more likely to be ignored in the risk-neutral solution, and it is necessary to adopt a risk-averse solution to reduce potential worst-case losses.

Synchronizing victim evacuation and debris removal: A data-driven robust prediction approach

This study introduces a new perspective in disaster management's response and post-disaster phases to synchronize multiple vehicles for victim evacuation and debris removal processes. A broad range of interrelated scheduling and routing operations and various synchronization aspects of heterogeneous vehicles are considered in this regard. A novel bi-objective mixed-integer programming model is presented, where the first objective function aims to minimize the total costs of the relief logistics network, and the second one minimizes the total operation times of vehicles. Moreover, due to extensive empirical and analytical errors, preliminary travel and service times are inexact and unreliable. Hence, a novel two-stage data-driven approach is rendered to predict reliable travel and service times. In the first stage, a new hybrid machine learning model is rendered to predict these times, and in the second stage, the distributionally robust optimization with φ-divergence is employed to surmount the unreliability of predicted times. A real case study is examined to illustrate the validity of the proposed model and solution approach. In addition, several simulation experiments are conducted to demonstrate the superiority of the proposed solution method in terms of robustness. Finally, the proposed framework can improve the planning by rendering meaningful insights concerning significant parameters' influence over the schedule and routing consequences.

Scheduling optimisation of multi-type special vehicles in an airport

An insufficient number of ground support vehicles and improper dispatch have a significant impact on the quality of airport services. The reasonable dispatch of airport special vehicles can greatly improve the efficiency of airport services. In this paper, a bi-objective mixed-integer programming model is proposed which takes the minimum number of vehicles required and minimum total extra time cost of special vehicles as objectives. Additionally, the service constraint relationship of various vehicles and the reusable nature of vehicles to make full use of existing special vehicle resources are considered. To easily solve the model, the comprehensive scheduling problem is reduced to several sub-problems with parallel services and timing constraints. A non-dominated sorting genetic algorithm with multiple chromosomes and an elite strategy are developed to solve them separately. Finally, using the actual flight data of a domestic airport, we demonstrate the effectiveness of the proposed model and offer useful managerial insights.

A flexible mathematical model for crew pairing optimization to generate n-day pairings considering the risk of COVID-19: a real case study

PurposeOne of the main advantages of the proposed model is that it is flexible to generate n-day pairings simultaneously. It means that, despite previous researches, one-day to n-day pairings can be generated in a single model. The flexibility in generating parings causes that the proposed model leads to better solutions compared to existing models. Another advantage of the model is minimizing the risk of COVID-19 by limitation of daily flights as well as elapsed time minimization. As airports are among high risk places in COVID-19 pandemic, minimization of infection risk is considered in this model for the first time. Genetic algorithm is used as the solution approach, and its efficiency is compared to GAMS in small and medium-size problems.Design/methodology/approachOne of the most complex issues in airlines is crew scheduling problem which is divided into two subproblems: crew pairing problem (CPP) and crew rostering problem (CRP). Generating crew pairings is a tremendous and exhausting task as millions of pairings may be generated for an airline. Moreover, crew cost has the largest share in total cost of airlines after fuel cost. As a result, crew scheduling with the aim of cost minimization is one of the most important issues in airlines. In this paper, a new bi-objective mixed integer programming model is proposed to generate pairings in such a way that deadhead cost, crew cost and the risk of COVID-19 are minimized.FindingsThe proposed model is applied for domestic flights of Iran Air airline. The results of the study indicate that genetic algorithm solutions have only 0.414 and 0.380 gap on average to optimum values of the first and the second objective functions, respectively. Due to the flexibility of the proposed model, it improves solutions resulted from existing models with fixed-duty pairings. Crew cost is decreased by 12.82, 24.72, 4.05 and 14.86% compared to one-duty to four-duty models. In detail, crew salary is improved by 12.85, 24.64, 4.07 and 14.91% and deadhead cost is decreased by 11.87, 26.98, 3.27, and 13.35% compared to one-duty to four-duty models, respectively.Originality/valueThe authors confirm that it is an original paper, has not been published elsewhere and is not currently under consideration of any other journal.

Pareto-based grouping meta-heuristic algorithm for humanitarian relief logistics with multistate network reliability

This article considers a biobjective location-routing problem to deliver relief resources to the victims affected by a disaster under uncertainty in demand, transportation infrastructure, and travel time. Since transportation networks are exposed to a considerable level of uncertainty, choosing the reliable path for relief goods to be transmitted to the affected areas ensures the arrival of these supplies. For the first time, route reliability is calculated based on the multistate theory, and the universal generating function technique is used for network reliability assessment. The problem is formulated as a multiperiod robust biobjective mixed-integer programming model. Two objective functions are considered: (a) decreasing the sum of arrival times of relief vehicles at the demand nodes for delivering aids to the affected areas, and (b) increasing the minimum route reliability for all the serving vehicles. A novel multiobjective grouping algorithm is proposed to obtain the Pareto-optimal solutions of the problem. Then, its performance is compared with two other multiobjective grouping algorithms. To evaluate the solution method, the algorithms are implemented on various test problems and compared statistically. A case study is presented to illustrate the potential applicability of our model. Additionally, to determine the effect of the changes in the main parameters of the problem on the value of objective functions, the sensitivity analyses are performed and the managerial insights are given.

Emergency logistics network design based on space–time resource configuration

The occurrence of natural disasters or accidents causes the obstruction or interruption of road traffic connectivity and affects the transportation of essential materials, especially for cross-regional delivery under emergency situations. Affected by COVID-19, government administrators establish cross-regional quarantine roadblocks to reduce the risk of virus transmission caused by cross-regional transportation. In this study, we propose an emergency logistics network design problem with resource sharing under collaborative alliances. We construct a state–space–time network-based bi-objective mixed integer programming model to optimize the vehicle routes in order to meet customer demands for essential materials with the lowest cost and highest emergency response speed under limited transportation resources. A two-stage hybrid heuristic algorithm is then proposed to find good-quality solutions for the problem. Clustering results are obtained using a 3D k-means clustering algorithm with the consideration of time and space indices. The optimization of the initial population generated by the improved Clarke and Wright saving method and improved nondominated sorting genetic algorithm-II with elite retention strategy provides stable and excellent performance for the searching of Pareto frontier. The cost difference of the entire emergency logistics network before and after collaboration, i.e., the profit, is fairly allocated to the participants (i.e., logistics service providers) through the Shapley value method. A real-world case in Chongqing City, China is used to validate the effectiveness of the proposed model and algorithm. This study contributes to smart transportation and logistics system in emergency planning and has particular implications for the optimal response of existing logistics system to the current COVID-19 pandemic.

Selection of consolidation center locations for China railway express to reduce greenhouse gas emission

To explore the environmental concerns created by China’s Belt and Road initiative, and to solve problems such as duplication of routes, inefficient use of resources, and excessive competition in the operation of the initiative’s China Railway Express component of the initiative, we studied optimization of the locations of China Railway Express consolidation centers. We selected 28 candidate nodes through a three-criterion screening mechanism. To account for the network’s operating costs and greenhouse gas emission, we developed a bi-objective mixed-integer programming model and applied the augmented ε-constraint method to solve it. Our analysis selected Qingdao, Harbin, Changchun, Taiyuan, Urumqi, and Tianjin as consolidation centers.

Bi-objective robust project scheduling with resource constraints and flexible activity execution lists

This paper analyses a type of extension of the resource-constrained project scheduling problem (RCPSP) whose activities have multiple execution lists. This sort of problem is titled resource-constrained project scheduling problem with flexible activity execution lists (abbreviated as RCPSP-FAL). The goal is to minimize makespan and maximize robustness simultaneously to improve the ability of the project to handle emergencies within the design completion time. Six robustness measures are proposed depending on the free slack determined by start time and renewable resource surplus per unit time. Moreover, a bi-objective mixed-integer programming model is developed to solve the considered problem. Besides, a simple data structure, namely selection-tree, is designed to detect available activity execution lists. Two algorithms are proposed, whose names are variant-genetic algorithm (VGA) and variant-simulated-annealing algorithm (VSA), to solve the developed model. Also, an aggregation function is established for assessing the fitness consisting of makespan and robustness. Several computation experiments are designed to evaluate the proposed algorithms and the six robustness measures. Results show the characteristics of different robust measures. Meanwhile, they also show the effectiveness of VSA and VGA, as well as the relations among robustness, makespan, and resource-usage.

Multi-objective optimization for a combined location-routing-inventory system considering carbon-capped differences

<p style='text-indent:20px;'>A combined location-routing-inventory system (CLRIS) in a three-echelon supply chain network is studied with environmental considerations. Specifically, a bi-objective mixed integer programming model is formulated for the CLRIS to deal with the trade-offs between the total cost and the carbon-capped difference (CCD). A multi-objective particle swarm optimization (MOPSO) heuristic solution procedure is developed and implemented to solve the bi-objective mixed integer programming problem. The bi-objective mixed integer programming model and the MOPSO heuristic procedure are applied to a real-life problem as an illustrative example. The approximate nondominated frontier formed by solutions not dominated by others can be used for the decision makers to make trade-offs between the total cost and the CCD. Sensitivity analyses are conducted, and the relationship among the carbon cap, CCD, the total cost and the carbon prices are examined, and relevant managerial insights are provided. Comparisons with other existing algorithms show that the MSPSO heuristic procedure has very good performance.</p>

A bi-objective model for scheduling of multiple projects under multi-skilled workforce for distributed load energy usage

Satisfying the clients' uncompromising priorities is a challenge for decision makers of organizations that face multiple projects. This paper considers an organization with a multi-skilled workforce working on several predetermined projects under time-of-use energy tariffs where distributed load energy usage is the primary concern of energy suppliers. This paper also considers different time-of-use energy tariffs, which are among the most common strategies to reach a more balanced energy utilization. The problem is stated as a bi-objective mixed-integer programming model containing two conflicting objectives; to minimize the total cost of the multi-skilled workforce and obtain a sustainable schedule with minimum deviation from the projects' priorities. The problem is formulated mathematically, and the GAMS solver is applied for justifying the conflict between the objectives and validating the proposed formulation. In order to tackle real-life instances of the problem, intelligent algorithms based on cuckoo search, particle swarm, and genetic algorithms are developed. In the proposed algorithms, the application of a well-designed encoding and decoding structure efficiently ensures the generated solutions' feasibility. The Taguchi method is used for calibrating the parameters of the proposed meta-heuristics. Performance of the solving methods is evaluated based on some experiments, where ELECTRE method is utilized as a decision-making technique to prioritize the developed algorithms. To this aim, some well-known multi-objective measures are applied for comparative analysis of the results, where the supremacy of FSCS in terms of all metrics is declared.

Resilience-Based Optimization of Postdisaster Restoration Strategy for Road Networks

This work proposes a framework for the optimization of postdisaster road network restoration strategies from a perspective of resilience. The network performance is evaluated by the total system travel time (TSTT). After the implementation of a postdisaster restoration schedule, the network flows in a certain period of days are on a disequilibrium state; thus, a link-based day-to-day traffic assignment model is employed to compute TSTT and simulate the traffic evolution. Two indicators are developed to assess the road network resilience, i.e., the resilience of performance loss and the resilience of recovery rapidity. The former is calculated based on TSTT, and the latter is computed according to the restoration makespan. Then, we formulate the restoration optimization problem as a resilience-based bi-objective mixed integer programming model aiming to maximize the network resilience. Due to the NP-hardness of the model, a genetic algorithm is developed to solve the model. Finally, a case study is conducted to demonstrate the effectiveness of the proposed method. The effects of key parameters including the number of work crews, travelers’ sensitivity to travel time, availability of budget, and decision makers’ preference on the values of the two objectives are investigated as well.

Half Open Multi-Depot Heterogeneous Vehicle Routing Problem for Hazardous Materials Transportation

How to reduce the accidents of hazardous materials has become an important and urgent research topic in the safety management of hazardous materials. In this study, we focus on the half open multi-depot heterogeneous vehicle routing problem for hazardous materials transportation. The goal is to determine the vehicle allocation and the optimal route with minimum risk and cost for hazardous materials transportation. A novel transportation risk model is presented considering the variation of vehicle loading, vehicle types, and hazardous materials category. In order to balance the transportation risk and the transportation cost, we propose a bi-objective mixed integer programming model. A hybrid intelligent algorithm is developed based on the ε-constraint method and genetic algorithm to obtain the Pareto optimal solutions. Numerical experiments are provided to demonstrate the effectiveness of the proposed model. Compared with the close multi-depot heterogeneous vehicle routing problem, the average risk and cost obtained by the proposed bi-objective mixed integer programming model can be reduced by 3.99% and 2.01%, respectively. In addition, compared with the half open multi-depot homogeneous vehicle routing problem, the cost is significantly reduced with the acceptable risk.

Optimizing configuration and scheduling of double 40 ft dual-trolley quay cranes and AGVs for improving container terminal services

Meeting the individual needs of customers will increase the competitiveness of container terminals. This paper optimizes the configuration and scheduling of double 40 ft dual-trolley quay cranes (QCs) and automated guided vehicles (AGVs) to improve the service of a container terminal. A two-stage biobjective mixed integer programming model with the objectives of maximizing customer satisfaction, minimizing the ineffective time of AGVs, and minimizing the delay time of QCs is proposed. For the proposed model, the piecewise function in the first stage is linearized by adding constraints of special ordered sets, and the GUROBI software is used to solve it. Then, an improved nondominated sorting genetic algorithm-Ⅲ is applied that aims to efficiently solve the second-stage model. Numerical experiments are presented to verify the effectiveness of models and algorithms. The results demonstrate the rationality and validity of the method, which selects the configuration and scheduling scheme based on customer satisfaction. • The satisfaction of shipowner and cargo owners are analyzed. • The handling equipment are double 40 ft dual-trolley quay cranes and AGV. • A two-stage bi-objective mixed integer programming model is proposed. • The capacity constraints of transport platform and buffer bracket are considered. • The improved non-dominated sorting genetic algorithmⅡ is used to solve the model.

Metaheuristics for the distance constrained generalized covering traveling salesman problem

In this work, we present an extension of the generalized covering salesman problem as distance constrained generalized m-covering salesman problem. Given a set of vertices including a depot, facilities, customer locations and demand associated with each customer location, the objective is to determine an optimal tour of each salesman such that the total covered demand is maximized and the tour length is minimized. A bi-objective mixed-integer programming model is formulated for the problem. The proposed model is solved using GUROBI 8.0 solver’s in-built hierarchical optimization approach. However, the computational complexity of the problem demands a specialized solution approach. To solve the problem efficiently, we propose two metaheuristics ant colony optimization (ACO) algorithm and greedy randomized adaptive search procedure (GRASP). Extensive computational experiments were performed using the benchmark instances from the literature. The results of computational experiments show the efficiency and effectiveness of the proposed metaheuristics algorithms. Although, the GRASP metaheuristics outperform the ACO algorithm in case of medium and large size instances.

The Biobjective Bike-Sharing Rebalancing Problem with Balance Intervals: A Multistart Multiobjective Particle Swarm Optimization Algorithm

The bike-sharing system (BSS), as a sustainable way to deal with the “last mile” problem of mass transit systems, is increasingly popular in recent years. Despite its success, the BSS tends to suffer from the mismatch of bike supply and user demand. BSS operators have to transfer bikes from surplus stations to deficit stations to redistribute them among stations by means of trucks. In this paper, we deal with the bike-sharing rebalancing problem with balance intervals (BRP-BIs), which is a variant of the static bike-sharing rebalancing problem. In this problem, the equilibrium of station is characterized by a balance interval instead of a balance point in the literature. We formulate the BRP-BI as a biobjective mixed-integer programming model with the aim of determining both the minimum cost route for a single capacitated vehicle and the maximum average rebalance utility, an index for the balanced degree of station. Then, a multistart multiobjective particle swarm optimization (MS-MOPSO) algorithm is proposed to solve the model such that the Pareto optimal solutions can be derived. The proposed algorithm is extended with crossover operator and variable neighbourhood search to enhance its exploratory capability. Compared with Hybrid NSGA-II and MOPSO, the computational experimental results demonstrate that our MS-MOPSO can obtain Pareto optimal solutions with higher quality.

A multi-objective stochastic model for a reverse logistics supply chain design with environmental considerations

Some electronic devices have a short lifetime, and variety-seeking and consumerism are increasingly growing in today’s societies. Moreover, electronic wastes contain precious substances such as gold, silver, copper, and aluminum. The proper disposal and processing of them by recycling offer considerable advantages to the environment, given the hazardous natures of such devices’ substances. The proposed reverse logistics with waste electrical and electronic equipment (WEEE) is an important task considered by researchers, the use of which offers economic benefits and reduces the environmental impacts of wastes. The present study models the electrical and electronic equipment (EEE) reverse logistics process as a bi-objective mixed-integer programming model under uncertainties. The mathematical model investigates two objectives: an economic objective and an environmental objective. The first is minimizing cost, while the second is maximizing the environmental score by reverse logistics processes in recovering and recycling. The parameters of demand and WEEE return rate which is obtained from the customer were considered as two uncertain parameters. A scenario-based stochastic programming (SSP) approach is applied to deal with the uncertainties. A case study of an electronic equipment manufacturer in Esfahan, Iran was included. The model was solved by a nominal approach and an SSP approach via the epsilon-constraint (EC) and augmented epsilon-constraint (AEC) methods to obtain optimal Pareto solutions and compare the methods. Finally, the optimal results of the two approaches were evaluated. The results indicated that the SSP approach using the AEC method had better outcomes.