Bi-objective Model Research Articles

Bi-objective Optimization for Multi-depot Green Vehicle Routing problem with Time Windows Considering Carbon Emission

The issue of carbon emission reduction has attracted extensive attention in academic circles, but there are few studies on the Pareto frontier analysis of the multi-depot green vehicle routing problem with multi-objective. In order to effectively balance economic and environmental effects, this paper constructs a bi-objective optimization model for the open multi-depot green vehicle routing problems with time windows, which takes operating cost and carbon emission as the objective functions. NSGA-III is used to obtain the Pareto solution set of the bi-objective model. The experimental results indicate that the proposed bi-objective optimization model is effective in controlling costs and reducing carbon emissions.

Capital-constrained maintenance scheduling for road networks considering traffic dynamics

Road network maintenance scheduling mainly considers budget limits in the previous studies and largely ignores the capital constraints. This study proposes a bi-objective mixed integer programming model, in which the net present value (NPV) is maximized and the increased total system travel time (ITSTT) due to maintenance activities under the capital constraints is minimized. Since the road network flows cannot reach an equilibrium state overnight due to the variation of network capacity, a link-based day-to-day dynamics model is developed to simulate the transient fluctuation in the traffic flows and calculate the total system travel time of each day. The bi-objective model is solved by the nondominated sorting genetic algorithm-II (NSGA-II) that generates a set of optimal Pareto solutions. The TOPSIS method is then adopted to determine the best compromise solution. Finally, a case study is conducted to demonstrate the effects of key parameters on the values of the two objectives.

Bi-level and multi-objective optimization of renewable energy sources and storage planning to support existing overloaded electricity grids

The paper proposes a bi-level multi-objective optimization model to optimally design and operate renewable energy sources and storage systems in an existing electrical grid with increasing demand. while respecting network constraints based. The bi-level optimization model combines particle swarm optimization (PSO) and dynamic linear AC-optimal power flow (DLOPF). The aim of the bi-objective model is to minimize costs while limiting carbon emissions. PSO sizes and identifies the placement of battery energy storage (BES) systems, and DLOPF locates and sizes RES in a spatial–temporal framework using the Levelized Cost of Energy and Storage. Different scenarios were applied to the IEEE-30 bus system to reveal the behavior of the dynamic network for the different cases, and the associated effect of integrating RES and BES. The model resulted in optimized scenarios in terms of placement and sizing of RES and BES at the lowest cost while considering cost minimization and carbon constraints. This shows how the complexity of the CO2 constraints requires more RES and BES installations and thus more funds. The maximum CO2 limit achieved is 30%, which has reached the storage limits and increased the overall cost by $15 m.

Efficiency improvement and balance in fixed cost allocation: A trade-off approach based on DEA

Fixed cost allocation (FCA) is a problem that decision-makers usually encounter in practice. The existing FCA studies based on data envelopment analysis (DEA) have not recognized the trade-off between improving efficiency and balancing FCA result and failed to provide an approach to generate an equilibrium solution between the two goals. In this study, under DEA common-weight evaluation mechanism, we propose an approach for centralized FCA using several principles featuring fairness in the situation where the fixed cost is regarded as a complement of one of the current inputs. It first considers the principle of making equal-proportional efficiency improvements for all the decision-making units (DMUs) in FCA. Then, we define a measure, called “balanced indicator”, to measure the balance of a FCA result. Finally, a bi-objective model is built that considers both the objectives of maximizing the proportional efficiency improvement of the DMUs and improving the balance of the FCA result. A bargaining-game-like model and a trade-off algorithm are provided to generate the final FCA result. Our research contributes by elucidating the trade-off between enhancing efficiency and achieving balance in FCA result. It also makes a distinct contribution by proposing a bargaining-game-like model and a trade-off algorithm to generate a FCA result that is an equilibrium solution between the two goals and ensures no DMU is allocated zero cost. Finally, we demonstrate the superiority of our approach by comparing it with a previous representative approach and applying it to a case study of China National Heavy Duty Truck Group.

Comprehensive quantity discount model for dynamic green supplier selection and order allocation

We model and solve a deterministic multi-period single-product green supplier selection and order allocation problem in which the considered suppliers’ availability, cost, and green performance change from one period to another in the planning horizon. Moreover, the available suppliers may offer an all-unit or an incremental quantity discount (QD) scheme, resulting in three problem configurations. In one configuration, all suppliers offer all-unit QD. In the second, all suppliers offer incremental QD. In the third, some suppliers offer all-unit QD, and others offer incremental QD. The problem is modeled using a bi-objective integer linear programming formulation that maximizes the total green value of the purchased items from all the suppliers and minimizes their total corresponding cost, including the fixed cost, variable cost, inventory holding cost, and shortage cost. The proposed bi-objective model is scalarized and solved using the branch-and-cut algorithm and a population-based heuristic. A numerical analysis is conducted, which allows first to validate the heuristic approach using small-size instances by comparing its results with those of the exact approach. Moreover, an extensive comparison between the exact and heuristic solution approaches is carried out. The results reveal different findings. First, the economic and environmental solutions of an instance are different, and the environmental solution is independent of the suppliers’ pricing schemes. Second, the maximum difference between the heuristic approach and the exact approach in terms of the bi-objective function value is 4.72%, which makes the proposed heuristic recommended for large-size instances due to its short computation time and good accuracy. Third, there is no difference in terms of the heuristic performance between the combined model and the models with a single type of discount. Fourth, the all-unit discount scheme seems to be generally better in terms of the trade-off between the green value of purchasing and cost.

Trade credit or credit insurance? A green supply chain finance design scheme with multi-objective programming

Classical supply chain finance (SCF) primarily focuses on the financial service among all upstream and downstream supply chain participants. Due to the continuously deteriorating of the ecological environment, an environmental-friendly SCF system is urgently needed. In this paper, we propose a novel SCF design scheme with environmental concerns, i.e., green supply chain finance (GSCF), consider the financing channels both from banks and from consumers, and design a bi-objective optimization model that depicts the trade-off between the benefit and the emission. Further, an improved normalized normal constraint (INNC) Pareto method is developed to address the optimal financing strategy of the bi-objective model. We then conduct a numerical case of a Taiwanese steel firm to verify the effectiveness and efficiency of our method. Results show that our model provides a portfolio of optimal solutions on Pareto frontier which can be applied as an effective decision support system when designing a GSCF. Furthermore, the sensitivity analysis also presents the impact of environmental investment cost, technological ratio of companies and the interest rate of trade credit on the optimal configuration of the GSCF.

A comparison among optimization software to solve bi-objective sectorization problem

In this study, we compare the performance of optimization software to solve the bi-objective sectorization problem. The used solution method is based on an approach that has not been used before in the literature on sectorization, in which, the bi-objective model is transformed into single-objective ones, whose results are regarded as ideal points for the objective functions in the bi-objective model. Anti-ideal points are also searched similarly. Then, using the ideal and anti-ideal points, the bi-objective model is redefined as a single-objective one and solved. The difficulties of solving the models, which are basically non-linear, are discussed. Furthermore, the models are linearized, in which case how the number of variables and constraints changes is discussed. Mathematical models are implemented in Python's Pulp library, Lingo, IBM ILOG CPLEX Optimization Studio, and GAMS software, and the obtained results are presented. Furthermore, metaheuristics available in Python's Pymoo library are utilized to solve the models' single- and bi-objective versions. In the experimental results section, benchmarks of different sizes are derived for the problem, and the results are presented. It is observed that the solvers do not perform satisfactorily in solving models; of all of them, GAMS achieves the best results. The utilized metaheuristics from the Pymoo library gain feasible results in reasonable times. In the conclusion section, suggestions are given for solving similar problems. Furthermore, this article summarizes the managerial applications of the sectorization problems.

Direct and indirect emissions: a bi-objective model for hybrid vehicle routing problem

Direct and indirect emissions: a bi-objective model for hybrid vehicle routing problem

A bi-objective model for scheduling green investments in two-stage supply chains

Investing in green technologies to increase sustainability in supply chains has become a common practice for two reasons: the first is directly related to the defense of the environment and people’s health to smooth the emissions of pollutants; the second is the increasing consumer awareness of green products. Despite the higher costs of producing with green technologies and processes, there is also a higher markup on the price of products which rewards the former costs. This study proposes a mathematical model for scheduling green investments over time in a two-stage supply chain to minimize the impact of production on the environment and the economic costs deriving from the investment. The resulting bi-objective model has nonlinear constraints and is solved using a commercial solver. Given its complexity, we propose an upper-bound heuristic and a lower-bound model to reduce the optimality gap attained at a given time limit. Tests on synthetic instances have been conducted, and an example demonstrates the applicability and efficacy of the proposed model.

Toward circular economy for pomegranate fruit supply chain under dynamic uncertainty: A case study

Food security is a fundamental prerequisite of human survival. This study focuses on providing a novel bi-objective model for the design of a green closed-loop pomegranate supply chain based on the value chain. To maximize the benefits of circular economy, we propose a thermochemical conversion process and develop a novel hybrid risk-neutral and risk-averse multi-stage stochastic programming (RNRAMSSP) approach to cope with supply uncertainty. We employ the expected value of perfect information (EVPI) and value of stochastic solution (VSS) to validate the proposed approach. For the VSS metric, we define a novel measure to compare the worst-case cost of the stochastic and deterministic model solutions. To demonstrate the applicability of the proposed model, we provide a real case study in Iran. Based on the VSS metric, the risk-neutral and risk-averse stochastic model solutions save up to 1% and 1.76% in the worst-case cost, respectively, compared to the deterministic model.

Resilience-oriented operation of microgrids in the presence of power-to-hydrogen systems

This study presents a novel framework for improving the resilience of microgrids based on the power-to-hydrogen concept and the ability of microgrids to operate independently (i.e., islanded mode). For this purpose, a model is being developed for the resilient operation of microgrids in which the compressed hydrogen produced by power-to-hydrogen systems can either be used to generate electricity through fuel cells or sold to other industries. The model is a bi-objective optimization problem, which minimizes the cost of operation and resilience by (i) reducing the active power exchange with the main grid, (ii) reducing the ohmic power losses, and (iii) increasing the amount of hydrogen stored in the tanks. A solution approach is also developed to deal with the complexity of the bi-objective model, combining a goal programming approach and Generalized Benders Decomposition, due to the mixed-integer nonlinear nature of the optimization problem. The results indicate that the resilience approach, although increasing the operation cost, does not lead to load shedding in the event of main grid failures. The study concludes that integrating distributed power-to-hydrogen systems results in significant benefits, including emission reductions of up to 20 % and cost savings of up to 30 %. Additionally, the integration of the decomposition method improves computational performance by 54 % compared to using commercial solvers within the GAMS software.

An integrated bi-objective model under warranty, technology level, and pricing by considering pre-sale and post-sale costs

An integrated bi-objective model under warranty, technology level, and pricing by considering pre-sale and post-sale costs

Reassigning Departure Slots with Preferences of the Airline and Passengers

The recovery of a departure flight is an important part of airline disruption management. The commonly used optimization objectives for assessing airport slot-scheduling efficiency are delay cost and delay time. However, the current literature does not consider the preferences of airlines and passengers in the recovery process simultaneously. The objective of this paper is to focus on the departure slot reassignment problem and develop a bi-objective optimization model. We introduce a metric for the price of fairness and formulate the airport slot scheduling problem as a bi-objective optimization model that considers the trade-off between the total airline delay cost and total passenger delay time. To quantify the trade-off between total airline delay costs and total passenger delay time, we apply the metric of the price of fairness in the bi-objective model, treating the total airline delay cost as an efficiency metric and the total passenger delay time as a fairness metric, calculating the price of fairness for each Pareto solution. An adaptive non-dominated sorting genetic algorithm-II based on dominant strengths (ANSGA2-DS) is developed to solve this problem. More precisely, three improved operations are presented: the improved fast dominant strength sorting method, new crowding distance improvement, and an adaptive elitist retention strategy. Three scenarios derived from a Chinese airline’s operation data are applied to the proposed bi-objective model and algorithm. The experimental findings demonstrate that the proposed model and method can effectively and efficiently address the problem. This may provide a basis for airline operation controllers to achieve a generally acceptable solution.

Dynamic Evacuation Path Planning for Multi-Exit Building Fire: Bi-Objective Model and Algorithm

Dynamic Evacuation Path Planning for Multi-Exit Building Fire: Bi-Objective Model and Algorithm

Real-time multi-factory scheduling in Industry 4.0 with virtual alliances

The ever-changing and dynamic market environment requires applying job shop systems based on real-time data. The establishment of physical-virtual systems in the production process has led to the emergence of intelligent factories. Compared with those employing traditional production methods, such factories manufacture products with higher quality, higher production speed, and other economic benefits. Regarding the virtual connections of factories, events such as the arrival of new jobs, and machine breakdowns, are identified by Radio Frequency Identification System between different production units, and related decisions are made quickly and carefully. In this intelligent job shop scheduling, it is assumed that independent factories create virtual production networks in which, each factory focuses on its own interests. Regarding the importance of this issue in today’s industry, this research explores the real-time scheduling problem in multi-agent production networks distributed in smart factories. Since this problem is a combination of static scheduling and real-time scheduling, a bi-objective model of mixed-integer linear programming is first developed. An approach is then proposed to solve the dynamic real-time scheduling problem. In addition, a learning-based memetic algorithm for solving large-size bi-objective instances is proposed due to the NP-hardness of the considered problem. Afterward, the results of the proposed algorithm are compared with the hybrid Pareto-based tabu search algorithm. The computational results show that in large-size instances, the proposed algorithm outperforms the competing algorithm.

Optimizing sustainable biomass–coal co-firing power plant location problem under ambiguous supply

Promoting biomass–coal co-firing power generation technology in existing coal-fired power plants is a practical and relatively low-cost short-run choice for the coal-fired power industry to reduce carbon emissions and develop sustainably. However, the uncertainty of biomass supply complicates the location of co-firing power plants among existing coal-fired plants. To address the challenge that only limited historical data of uncertain supply are available, this paper proposes a new ambiguity distribution set to characterize the uncertain supply capacity, supply price, and supply quality (i.e., energy conversion coefficient). Accordingly, a novel bi-objective distributionally robust fuzzy optimization model is developed for the sustainable co-firing power plant location (SCPPL) problem to balance conflicting objectives of minimizing total cost and maximizing carbon emission reduction. Subsequently, the computationally tractable counterpart of the proposed model is derived against the worst-case distribution by reformulating the distributionally robust credibility objective and constraints as their solvable equivalent forms. Furthermore, the Pareto optimal solutions of the proposed bi-objective model are generated using the augmented ϵ-constraint method. Finally, a realistic case study of Heilongjiang Province in China is conducted to verify the feasibility and effectiveness of the proposed optimization method. The experimental results demonstrate that the proposed model can hedge against the distributional ambiguity of uncertain parameters and provides a robust co-firing power plant location strategy. By comparison with the nominal distribution model and analysis of the sensitivity of selected parameters, management insights are summarized for decision-makers.

Joint Optimization of Platoon Control and Resource Scheduling in Cooperative Vehicle-Infrastructure System

Vehicle platooning technology is essential in achieving group consensus, on-road safety, and fuel-saving. Meanwhile, Vehicle-to-Infrastructure (V2I) communication significantly facilitates the development of connected vehicles. However, the coupled effects of the longitudinal vehicle's mobility, platoon control and V2I communication may result in a low reliable communication network between the platoon vehicle and the roadside unit, there is a tradeoff between the platoon control and communication reliability. In this paper, we investigate a bi-objective joint optimization problem where the first objective is to maximize the success probability of data transmission (communication reliability) and the second objective function is to minimize the traffic oscillation flow. The vehicle's mobility state of the platoon vehicle affects the channel capacity and transmission performance. In this context, we deeply explore the relationship between control signals and resource scheduling and theoretically deduce a closed-form expression of the optimal communication reliability objective. Through this closed expression, we transform the bi-objective model into a single objective MPC model by using <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -constraint method. We design an efficient algorithm for solving the joint optimization model and prove the convergence. To verify the effectiveness of the proposed method, we finally evaluate the spacing error, speed error, and resource scheduling of platooning vehicles through simulation experiments in two experimental scenarios. The results show that the proposed control-communication co-design can improve the platoon control performance while satisfying the high reliability of V2I communications.

Bi-objective model for multi-level supply chain by focusing on quality of agricultural products: a case study

PurposeDue to the importance of quality to customers, this study considers criteria of quality and profit and optimizes both in a multi-echelon cold chain of perishable agricultural products whose quality immediately begins to deteriorate after harvest. The two objectives of the proposed cold chain are to maximize profit and quality. Since postharvest quality loss in the supply chain depends on various decisions and factors, in addition to strategic decisions, the authors consider the temperature setting in refrigerated facilities and transportation vehicles due to the unfixed shelf life of the products which is related to the temperature found by Arrhenius formula.Design/methodology/approachThe authors use bi-objective mixed-integer nonlinear programming to design a four-echelon supply chain. The authors integrate the supply chain echelons to detect the sources and factors of quality loss. The four echelons include supply, processing, storage and customer. The decisions, including facility location, assigning nodes of each echelon to corresponding nodes from the adjacent echelon, allocation of vehicles to transport the products from farms to wholesalers, processing selection, and temperature setting in refrigerated facilities, are made in an integrated way. Model verification and validation in the case study are done based on three perishable herbal plants.FindingsThe model obtains a 29% profit against a total cost of 71 and 93% of original quality of the crops is maintained, indicating a 7% quality loss. The final quality of 93% is the result of making a US$6m investment in the supply chain, including the procurement of high-quality raw materials; facility establishment; high-speed, high-capacity vehicles; location assignment; processing selection and refrigeration equipment in the storage and transportation systems, helping to maximize both the final quality of the products and the total profit.Research limitations/implicationsThe proposed supply chain model should help managers with modeling decisions, especially when it comes to cold chains for agricultural products. The model yields these results – optimal location-allocation decisions for the facilities to minimize distances between the network nodes, which save time and maintain the majority of the products’ original quality; choosing the most appropriate processing method, which reduces the perishability rate; providing high-capacity, high-speed vehicles in the logistics system, which minimizes transportation costs and maximizes the quality; and setting the right temperature in the refrigerated facilities, which mitigates the postharvest decay reaction rate of the products.Practical implicationsComparison of the results of the present research with those of the traditional chain (obtained through experts) shows that since the designed chain increases the profit as well as the final quality, it has benefits for the main chain stakeholders, which are customers of agricultural products. This study model is expected to have a positive impact on the environment by placing strong emphasis on quality and preventing excessive waste generation and air pollution by imposing a financial penalty on extra demand production.Social implicationsSince profit and quality of the final product are two important factors in all cultures and communities, the proposed supply chain model can be used in any food industry around the world. Applying the proposed model induces growth in local industries and promotes the culture of prioritizing quality in societies.Originality/valueTo the best of the authors’ knowledge, this is the first research on a bi-objective four-echelon (supply, processing, storage and customer) postharvest supply chain for agricultural products including that integrates transportation logistics and considers the deterioration rate of products as a time-dependent variable at different levels of decision-making.

Sustainable hierarchical multi-modal hub network design problem: bi-objective formulations and solution algorithms

This paper presents a bi-objective model for the design and optimization of a sustainable hierarchical multi-modal hub network. The proposed model focuses on sustainability by considering economic, environmental, and social aspects of the decisions in a hierarchical network. A case of Turkish network for freight transportation is used to validate the proposed model. To solve the small-sized problems, the augmented epsilon constraint method version 2 (AUGMECON2) is applied. It can be inferred from the Pareto-optimal set obtained by AUGMECON2 that the effect of increasing the number of hubs after a threshold is marginal. The current contribution proposes two multi-objective genetic algorithms (NSGA-II and NRGA), which incorporate LP solving and Dijkstra algorithm. The results show the superiority of NRGA compared to NSGA-II in terms of solution time. Also, we present an alternative, more efficient formulation to the problem. Based on the alternative formulation, in addition to AUGMECON2, we use two exact methods, including Torabi and Hassini (TH) method and augmented weighted Tchebycheff procedure (AWTP), to find Pareto-optimal solutions for small, medium, and large-sized problems (including the case study). The performance of the proposed solution methods is measured using some multi-objective indicators. The results show the superiority of AUGMECON2.

Designing Airline Hub-and-Spoke Network and Fleet Size by a Biobjective Model Based on Passenger Preferences and Value of Time

This study presents a biobjective hub-and-spoke (HS) network design model for the global air passenger networks. The model explores the tradeoff between the total airline cost (airline preference) and lost time cost for passengers (user preference) as the model’s objectives. Most previous studies have focused on airline objectives and established HS networks based on the viewpoint of airlines, despite the importance of passenger objectives. Poor passenger service and inconvenience and dissatisfaction may lead to network breakdown. The major criteria for passenger dissatisfaction in HS networks are schedule and trip delays caused by nondirect flights. These delays (the lost time cost for passengers) are multiplied by the passengers’ value of time (VOT) and minimized as one of the model’s objectives. Another objective that is minimized is the transportation costs of the airline depending on the services provided (short-, medium-, and large-haul flights). The model is solved in a case study (Iranian Aeronautics Network) that is applied to the well-known yearbook of tourism statistics data. Pareto frontier was found for all candidate airports. Also, the number of aircraft required (short-, medium-, and large-haul), as well as the average load factor for different types of aircraft in various weights of the first objective (airline costs), was presented. The results of Pareto frontier indicated that Imam Khomeini International Airport should be selected as the global hub airport for Iran international flight network. Otherwise, Shiraz International Airport and Tabriz International Airport (as the first alternative), as well as Isfahan International Airport and Mashhad International Airport (as the second alternative), would be the best choices. The weight of the first objective (airline costs) seems to be between 0.7 to 1, a practical and logical weight that can reduce passenger costs (as the second objective) by 20% on average by adding only 15 long-haul, 40 medium-haul, and 37 short-haul aircraft to the airline’s fleet. Also, in this range, the average load factor for medium- and long-haul aircrafts is greater than 0.9, which seems to be ideal.