Multi-objective Mixed Integer Nonlinear Programming Research Articles

Utilizing energy transition to drive sustainability in cold supply chains: a case study in the frozen food industry

In alignment with the ever-growing interest in adopting sustainable practices, this paper devises a cold supply chain (CSC) planning model that integrates the three pillars of sustainability into the decision-making process while accounting for the shift towards clean energy sources. Interrelated decisions pertaining to production-distribution strategy, backorder and inventory levels, choice of truck type, and selection of third-party logistics (3PLs) providers are jointly optimized. For global CSCs in specific, such decisions are particularly sensitive to the energy sources of the refrigerated facilities and the accompanying levels of CO2 emissions generated. As such, a multi-objective mixed-integer non-linear programming (MINLP) model is developed and then solved via the weighted-sum method. In essence, the model seeks to operationalize sustainability goals by considering the rapidly evolving transition in energy sources across different regions when deciding on which 3PLs to engage in a contractual agreement with while adjusting the production and distribution strategy accordingly. The practical relevance of the model is illustrated using a case study drawn from the North American frozen food industry. The conducted trade-off analysis indicates the possibility of obtaining a drastic improvement of 86% in jobs’ stability levels (social measure) with a maximum cost increase of around 9% as compared to the economic measure. Furthermore, the analysis reveals that it is possible to reduce 71% of CO2 emissions while attaining 63% reduction in worker variations at the expense of only 4.47% cost increase once compared to solely optimizing the economic objective.

Reliability Enhancement by Strategic Placement of Remotely-Controlled Automation Devices: A Case Study on Tabriz Distribution Network

Achieving high distribution-reliability levels and concurrently minimizing operating costs are as the main issues in distribution system optimization. Determination of the optimal number and location of remote control switches (RCS) in the distribution system network is an essential issue from the reliability and economic points of view. To address these issues, this paper develops a novel multiobjective model from the distribution system viewpoint, wherein the primary objective, optimal RCS placement is implemented aiming at minimizing the operating costs, while in the second objective, the reliability index improvement is taken into account. So, a novel approach from a robust heuristic algorithm, modified non dominated sorting genetic algorithm (MNSGA-II), is developed and presented to solve this multiobjective mixed-integer non-linear programming problem. Simulation results received by the genetic algorithm have been compared with the other popular optimization algorithms, seperately. Results show that the proposed algorithm provides extensively the best performance, in terms of quality of the answer received and computational efficiency. The feasibility of the proposed algorithm was examined by application to two distribution feeders of the Tabriz city distribution network containing the fourth feeder of the TractorSazai substation.

Last-Mile Shuttle Planning for Improving Bus Commuters’ Travel Time Reliability: A Case Study of Beijing

This study proposes a multiobjective mixed integer nonlinear programming model for a last-mile shuttle service to improve bus commuters’ travel time reliability. The approach aims to assign the routes that pick up the transit passengers located at the different stops by shuttle service. A bilevel optimization model is established: the upper model of route design considers the tradeoff between time cost and fare cost when some of the passengers take the shuttles, and the lower model assigns the demand of transit passengers. The proposed model effectively captures the reliability of travel time because related parameters are estimated by a statistical fitting test with a large number of real-world bus geographic information system (GPS) data. Moreover, dynamic demand diverting from conventional transit to shuttle service and travel time reliability, including passenger in-vehicle time (IVT) and waiting time (WT), are fully considered in this model. Since the task is a nonlinear programming model, a two-stage algorithm combined with linearization processing is presented to find an optimal solution. Finally, from the case study of Zhongguancun Software Park zone in Beijing, it is indicated that when last-mile shuttle service is provided, bus passengers’ travel time reliability of last-mile trips can be improved by 14%. The study can be an important reference for improving the low reliability widely existing in the current transit commuters’ last-mile problem.

A data-driven approach to support voltage profiles & loss reduction in wind generator integrated active distribution network considering solid-state transformers with twofold reactive power compensation

ABSTRACT The paper proposes an optimal planning strategy for an active distribution network (ADN) with wind energy-based distributed generation (WDG) by considering the twofold reactive power compensation (TRPC) feature of a three-stage type solid-state transformer (SST). A model for wind speed and load demand estimation for each hour of the day is formulated by using the partition around medoids (PAM) technique over an annual database of load demand and wind speed. A multi-objective mixed-integer non-linear programming (MINLP) approach for optimally locating and sizing the SST installations in the radial distribution network (RDN) is presented to address the dual objectives of voltage profile improvement (VPI) and energy loss reduction (ELR). Along with the SST parameters, the locations and number of wind turbine (WT) units to be integrated into the system are treated as optimization variables. Furthermore, the distribution power flow accounts for the distribution transformer’s (DT) operating losses as well as the approximate losses of the SST. The presented approach is tested on a 33-bus RDN and the results are reported for multiple case studies. The programming was developed in the MATLAB R2020a environment and the paretosearch algorithm (PSA) is used to address the MINLP problem. Other standard multi-objective optimization algorithms, including multi-objective multi-verse optimization (MOMVO), non-dominated sorting genetic algorithm (NSGA-II), and multi-objective salp swarm algorithm (MSSA) are used to compare the performance of the proposed method. The outcomes of the peak hour evaluation, with a 20% over-rating, show the active and reactive power demand on the MV side of the substation to have decreased by 56.3% and 30.1% respectively. The active line loss has been lowered by 76.95% and the reactive line loss has decreased by 76.14%. The absolute minimum voltage has increased by 7.314%. Furthermore, according to the annual technical evaluation, there was a 29.42% reduction in active energy served by the DTs and SSTs.

Smart border patrol using drones and wireless charging system under budget limitation

Harsh environmental factors surrounding the national borders as well as their extensive length make the use of manned systems for monitoring every location on the border a risky, expensive, and hardly possible task. Smart border patrol using small-size drones may provide significant help in patrolling areas inaccessible to patrol agents, reduce agent response time, and increase the safety of patrol agents working in dangerous regions. However, the short flight duration associated with small-size drones due to battery limitation can be a serious drawback for such a system for seamless surveillance. Therefore, this paper proposes a drone-based continuous surveillance system for border patrol with a dynamic wireless battery charging system that is built onto an electrification line (E-line). To control the maximum time interval between two consecutive flights to a particular section of the border, a permitted revisiting gap is assigned to each location on the border based on its criticality. A multi-objective mixed-integer non-linear programming (MINLP) model is developed to minimize both the total length of the E-line system for the wireless charging system and the number of drones to satisfy the revisiting gap constraint for a secure border patrol. A solution algorithm is proposed to achieve the Pareto-optimal set. This method provides the decision-maker with a set of candidate solutions to choose from based on their priorities and budgetary constraints. We illustrate our method using a case study on a segment of the US-Mexico borderline.

Optimal Design of Battery Swapping-Based Electrified Public Bus Transit Systems

This article proposes a novel model to optimize the design of fully electrified public bus transit (PBT) systems that are operating using the concept of battery swapping. The proposed optimization model is formulated as a multiobjective mixed-integer nonlinear programming model that aims at minimizing the overall capital and operation expenditures of the electrified bus transit system. The formulated model determines the optimal configuration design parameters of the electrified bus transit system, including capacity of the onboard electric bus batteries, rated power of chargers, and the number of installed chargers and battery modules at the battery swapping station (BSS). Also, the model yields the optimal schedules of batteries swapping and charging scheme of batteries at the BSS. The model takes into consideration several physical and operation constraints, such as the limits of batteries state of charge, dynamic changes in the electricity prices, impacts of traffic conditions and heat, ventilation, and air conditioning (HVAC) operation on the battery electric bus (BEB) energy consumption, stiff and flexible schedule of bus assignments, and impacts of bus transit rush hour periods on batteries swapping. Several case studies are carried out on a real PBT system in the province of Ontario, Canada, to validate the effectiveness of the proposed model. The proposed model could be utilized as a decision-making tool to investigate the applicability of using the battery swapping concept to electrify bus transits based on the operation requirements and preferences of their operators.

A Decision Framework for Solar PV Panels Supply Chain in Context of Sustainable Supplier Selection and Order Allocation

Sustainable supplier selection and order allocation (SSSOA) is paramount to sustainable supply chain management. It is a complex multi-dimensional decision-making process augmented with the triple bottom line of sustainability. This research presents a multi-phase decision framework to address a SSSOA problem for the multi-echelon renewable energy equipment (Solar PV Panels) supply chain. The framework comprises of fuzzy Multi-Criteria Decision-Making techniques augmented with fuzzy multi-objective mixed-integer non-linear programming mathematical model. The various economic, environmental, and social objectives were optimized for a multi-period, multi-modal transportation network of the supply chain. The results show that among the various sustainable criteria selected in this study, product cost, environmental management system, and health and safety rights of employees are the most important for decision-makers. The results of the mathematical model highlighted the impact of multimodal transportation on overall cost, time, and environmental impact for all periods. An analysis of results revealed that transfer cost and customer clearance cost contribute significantly towards overall cost. Furthermore, defect rate was also observed to play a critical role in supplier selection and order allocation.

Economic and Environmental Sustainability of Waste Plastics Chemical Recycling from the Consequential Perspective

Consequential life cycle assessment (CLCA) enables evaluating the environmental consequences of market dynamics overlooked by attributional life cycle assessment (ALCA). A consequential life cycle optimization (CLCO) framework is developed in this work to determine the economically optimal and most environmentally sustainable chemical recycling pathway for high-density polyethylene waste. The framework includes CLCA and techno-economic analysis (TEA), and the CLCO problem is formulated as a multi-objective mixed-integer nonlinear fractional programming problem (MINFP) that is effectively solved by an optimization algorithm. This multi-objective optimization problem aims to minimize the unit life cycle environmental impacts with maximum unit net present value (NPV). Environmental assessment results show that the total greenhouse gas (GHG) emissions evaluated by the CLCA approach are 14.22 % lower than those assessed by the ALCA approach. By evaluating the ReCiPe end-point score, CLCA further reduces the total environmental impacts corresponding to particulate matter formation by 17.37 %.

A Multi-Objective Meta-Heuristic Approach to Improve the Bus Transit Network: A Case Study of Fargo-Moorhead Area

In this study, the Transit Network Design Problem (TNDP) is studied to determine the set of routes and frequency on each route for public transportation systems. To ensure the important concerns of planners like route length, route configuration, demand satisfaction, and attractiveness of the transit routes, the TNDP is solved to generate a set of routes by proposing an initial route set generation (IRSG) procedure embedded into the NSGA-II algorithm. The proposed IRSG algorithm aims to produce high-quality initial route set solutions to reach better optimization procedures. Moreover, the Multi-Objective Mixed-Integer Non-Linear Programming (MOMINLP) model is proposed to formulate the frequency setting problem on each route by minimizing the total travel time of passengers (user costs) and operator costs simultaneously, while maximizing the service coverage area near all the bus stops. The MOMINLP model is solved by applying the NSGA-II algorithm to produce a Pareto front between the first and the second objective functions. The model was applied to the Fargo-Moorhead Area (FMA), a small urban area. Results were compared with the existing transit network to measure the efficiency of the NSGA-II solution methodology. The proposed algorithm was found to considerably decrease the total travel time of passengers.

Closed-Loop Supply Chain Design with Sustainability Aspects and Network Resilience under Uncertainty: Modelling and Application

In this study, we present a multiobjective mixed-integer nonlinear programming (MINLP) model to design a closed-loop supply chain (CLSC) from production stage to distribution as well as recycling for reproduction. The given network includes production centers, potential points for establishing of distribution centers, retrieval centers, collecting and recycling centers, and the demand points. The presented model seeks to find optimal locations for distribution centers, second-hand product collection centers, and recycling centers under the uncertainty situation alongside the factory’s fixed points. The purpose of the presented model is to minimize overall network costs including processing, establishing, and transportation of products and return flows as well as environmental impacts while maximizing social scales and network flexibility according to the presence of uncertainty parameters in the problem. To solve the proposed model with fuzzy uncertainty, first, the improved epsilon (ε)-constraints approach is used to transform a multiobjective to a single-objective problem. Afterward, the Lagrangian relaxation approach is applied to effectively solve the problem. A real-world case study is used to evaluate the performance of the proposed model. Finally, sensitivity analysis is performed to study the effects of important parameters on the optimal solution.

Multi-Objective Optimization for Skip-Stop Strategy Based on Smartcard Data Considering Total Travel Time and Equity

With the rise of urban sprawl, urban railways extend out further to the city’s outer district, installing additional stations. Passengers who travel from the outer district to the center of the city therefore experience long travel times. Although skip-stop strategy helps save total travel time, deviation of travel time among all origin–destination pairs may be increased, leading to equity problems. This study aims to minimize the inequity and total travel time through train stop planning and train scheduling. A coefficient of variation is adopted as a measure of inequity. The problem is formulated as a multi-objective mixed integer nonlinear programming model. Origin–destination demand is extracted from smartcard data and a case study of four urban railway lines in Seoul is conducted. The results indicate that the number of transfer stations for equity-oriented skip-stop strategy is smaller than that for total-travel-time-oriented skip-stop strategy. We also discover that as the number of transfer stations rises, inequity increases and total travel time is reduced. For skip-stop strategy considering total travel time and equity simultaneously, average total travel time and the average deviation are reduced by up to 10.3% and 10.6%, respectively, compared with those of all-stop strategy. We analyze the gradient of Pareto optimal sets to find out which factors (equity or total travel time) are more significant. Skip-stop strategy on lines 5 and 9 can be designed based on equity, while line 4 can be planned based on total travel time.

Modularity-based quality assessment of a disruptive reconfigurable manufacturing system-A hybrid meta-heuristic approach

This study considers quality aspects in the process planning of a reconfigurable manufacturing system. The goal is to analyze how the variation in quality impacts the process planning, i.e., cost-based design and modular features. Besides this, the analysis helps in identifying the number of conforming and failed products delivered by a process plan. First, a multi-objective mixed integer non-linear programming model is proposed that contains the novel objectives of cost, quality decay, and modular efforts. Secondly, the model is implemented on an industrial case study by using an exact solution approach and a novel hybrid version of two popular meta-heuristics, namely non-sorting genetic algorithm and multi-objective particle swarm optimization. The hybrid heuristic helps strengthening the application of approaches by creating a balance in searching the solution space. The performance of different approaches is assessed by using two metrics and two termination criteria. The findings will help the decision-makers in assessing how quality-related issues impact the choice of a process plan and in understanding the trade-off among cost, quality, and modularity. Finally, conclusion and future research avenues are provided.

Speed Optimization for Container Ship Fleet Deployment Considering Fuel Consumption

In recent years, low energy consumption has become the common choice of economic development in the world. In order to control energy consumption, shipping line speed optimization has become strategically important. to reduce fuel consumption, this study optimizes the container ship fleet deployment problem by adopting the strategy of adjusting each leg of each route’s sailing speed. To calculate fuel consumption more accurately, both sailing speed and the ship’s payload are considered. A multi-objective mixed integer nonlinear programming model is established to optimize the allocation of liner routes with multiple ship types on multiple routes. A linear outer-approximation algorithm and an improved piecewise linear approximation algorithm are used for linearization. If segments of an interval increase, the results will be more accurate but will take more time to compute. As fuel prices increase, to make trade-offs among economic and environmental considerations, the shipping company is adopting the “adding ship and slow down its speed” strategy, which verifies the validity and applicability of the established model.

Efficient and sustainable closed-loop supply chain network design: A two-stage stochastic formulation with a hybrid solution methodology

In recent years, consumers and legislators have pushed companies to design their supply chain networks to consider environmental and social impacts as an important performance outcome. Due to the role of resource utilization as a key component of logistics network design, another primary goal of design is ensuring available scarce resources are used as efficiently as possible across all facilities. To address efficiency issues in a sustainable closed-loop supply chain network, a stochastic integrated multi-objective mixed integer nonlinear programming model is developed in this paper, in which sustainability outcomes as well as efficiency of facility resource utilization are considered in the design of a sustainable supply chain network. In doing so, efficiency is assessed using a bi-objective output-oriented data envelopment analysis model. A hybrid three-step solution methodology is presented that creates a linear form of the original mixed integer nonlinear programming problem using piecewise McCormick envelopes approach. In the second step, an aggregated single objective programming model is derived by exploiting the multi-choice goal programming. Finally, a Lagrangian relaxation algorithm is developed to effectively solve the latter stochastic single objective mixed integer linear programming problem. The application of the proposed approach is investigated with data drawn from a case study in the electronics industry. This case study illustrates how firms may balance sustainability and efficiency in the supply chain network design problem. Further, it demonstrates the integration of efficiency results in improving economic aspects of sustainability as well as social responsibility outcomes, but also highlights the trade-offs that exist between efficiency and environmental impacts.

A hybrid NSGA-II algorithm for the closed-loop supply chain network design in e-commerce

Designing the supply chain network is one of the significant areas in e-commerce business management. This concept plays a crucial role in e-commerce systems. For example, location-inventory-pricing-routing of an e-commerce supply chain is considered a crucial issue in this field. This field established many severe challenges in the modern world, like maintaining the supply chain for returned items, preserving customers’ trust and satisfaction, and developing an applicable supply chain with cost considerations. The research proposes a multi-objective mixed integer nonlinear programming model to design a closed-loop supply chain network based on the e-commerce context. The proposed model incorporates two objectives that optimize the business’s total profits and the customers’ satisfaction. Then, numerous numerical examples are generated and solved using the epsilon constraint method in GAMS optimization software. The validation of the given model has been tested for the large problems via a hybrid two-level non-dominated sort genetic algorithm. Finally, some sensitivity analysis has been performed to provide some managerial insights.

Economic-environmental evaluation and multi-objective optimization of supercritical CO2 based-central tower concentrated solar power system with thermal storage

We address the optimal design of central tower-concentrated solar power (CSP) system combined with supercritical CO2 Brayton cycle and thermal storage under economic and environmental objectives. The economic objective is measured by the levelized cost of electricity (LCOE), and the environmental objective by the power plant’s total environmental impact potential (TEIP) considering the system’s global warming, acidification, and eutrophication emission. A multi-objective mixed-integer nonlinear programming (MINLP) model is developed that takes into account the main characteristics of CSP plant, e.g. unit operations, working fluid thermodynamics, equipment sizing, thermal storage capacity. Life cycle assessment and economic evaluation of the manufacturing, construction, operation and decommission stages of the systems are also embedded in the model. The multi-objective MINLP problem is solved by a tailored algorithm, and the resulting Pareto solutions are analyzed to identify the tradeoffs between the economic and environmental performance. The proposed approach is illustrated through a case study of a 50 MWe CSP plant. Results show that the framework is able to obtain the system’s economically-environmentally optimal design in reasonable time. A minimum LCOE of 115.82 $/MWh can be achieved for the most cost-effective design, and a minimum TEIP of 320.54 × 103 mPE90 can also be achieved for the most environmentally friendly design. A ‘balanced’ solution is identified with an LCOE of 116.89 $/MWh and TEIP of 330.29 × 103 mPE90. Numerical studies also reveal that while the Brayton cycle only accounts for a small proportion of the total investment, spending more on enhancing the efficiency of its equipment is cost-effective to improve the overall economic and environmental performance of the system.

A branch-and-bound algorithm based on NSGAII for multi-objective mixed integer nonlinear optimization problems

Solving Multi-Objective Mixed Integer NonLinear Programming (MO-MINLP) problems is a point of interest for many researchers as they appear in several real-world applications, especially in the mechanical engineering field. Many researchers have proposed using hybrids of metaheuristics with mono-objective branch and bound. Others have suggested using heuristics with Multi-Criteria Branch and Bound (MCBB). A general hybrid approach is proposed based on MCBB and Non-dominated Sorting Genetic Algorithm 2 (NSGAII) to enhance the approximated Pareto front of MO-MINLP problems. A computational experiment based on statistical assessment is presented to compare the performance of the proposed algorithm (BnB-NSGAII) with NSGAII using well-known metrics from the literature. To evaluate the computational efficiency, a new metric, the Investment Ratio (IR), is proposed that relates the quality of solution to the consumed effort. Experimental results on five real-world mechanical engineering problems and two mathematical ones indicate that BnB-NSGAII could be a competitive alternative for solving MO-MINLP problems.

A multi-criteria decision framework for sustainable supplier selection and order allocation using multi-objective optimization and fuzzy approach

ABSTRACT With the growing social and environmental concerns, the integration of sustainability in supplier selection and order allocation is of paramount importance. This research presents a holistic multi-phase decision-support framework to solve the sustainable supplier selection and order allocation problem for the multi-echelon supply chain. The framework comprises a multi-objective mixed-integer nonlinear programming mathematical model augmented with fuzzy multi-criteria decision making techniques and forecast demand. The various economic, environmental and social objectives were optimized for a multi-modal transportation network of a multi-echelon supply chain. The results of the mathematical model highlighted the impact of multi-modal transportation on the total cost and total travel time. The results also demonstrated the relationship between the multi-modal transport network and the environmental impact of the supply chain. The proposed multi-phase holistic decision support framework can be used in the comprehensive sustainability-based analysis of supply chains.

Hybridization of an interactive fuzzy methodology with a lexicographic min-max approach for optimizing a multi-period multi-product multi-echelon sustainable closed-loop supply chain network

Here, a fuzzy multi-period multi-echelon multi-objective mixed-integer non-linear programming (MOMINLP) model is considered for a sustainable multi-product multi-site multi-distribution multi-customer supply chain in forward flow having multi-centers for collecting, checking, repairing and decomposing and multi-disposal centers in the reverse flow. Minimizing the total cost of the closed-loop supply chain (CLSC), elevating the customer satisfaction degrees, minimizing the total waiting time, minimizing the manufacturing site greenhouse gases and minimizing the CO2 emissions from vehicles are considered as the objective functions. Furthermore, integration of strategic decisions of flow allocations and vehicle routing with tactical and operational decisions such as production and workforce planning and improving upon customer satisfaction are considered. Also, we are concerned with the stability of the model and the accuracy of the obtained solution. We consider uncertainty and propose an appropriate method to develop a fair optimization of the distribution of raw materials and products to the supply chain participants. The presented solution method initially considers an adaptation of the lexicographic min–max fairness approach to finding a short delay for the delivery time of all the existing flows between every two consecutive echelons of the CLSC network. Then, a service quality measure is introduced to evaluate the uncertain delivery time considering the delay unpleasantness measure (DUM) index and measure the delay unpleasantness of all the existing delivery time delays. The model is converted to an auxiliary crisp MOMINLP problem by taking appropriate strategies, and a novel interactive fuzzy approach is proposed to find a compromised solution. The effectiveness of the algorithm is illustrated through a generated case study. The validity of the proposed method is confirmed by comparing the obtained results with the ones obtained by some other valid approaches, making use of distance and dispersion measure functions. Computational results show the proposed fuzzy method to be more efficient than other approaches. The encouraging results provide motivations for the use of our proposed fuzzy approach to solving other kinds of multi-objective mixed-integer models.

Voltage regulation methods for active distribution networks considering the reactive power optimization of substations

Due to the increasing penetration of photovoltaic (PV) power systems in active distribution networks (ADNs), PV power fluctuations may result in significant voltage variations of ADNs. Therefore, this paper proposes a voltage regulation method for ADNs to minimize the operational losses while keeping the nodal voltages within the limit with the reduced PV power curtailment and the reduced switching numbers of on-load tap changers (OLTCs) and capacitor banks (CBs). Meanwhile, the proposed voltage regulation method also aims to minimize the reactive power flowing through OLTCs, and to minimize the switching numbers of substation CBs. In this study, the centralized voltage regulation is performed based on the worst voltage variation scenarios of ADNs, where a multi-objective mixed integer nonlinear programming (MINP) model with time-varying decision variables is established. The MINP model is solved using the non-dominated sorting genetic algorithm II (NSGA-II), and a practical decision-making algorithm is developed to select the best solution from the Pareto optimal set. Moreover, the decentralized voltage regulation aims at mitigating real-time nodal voltage variations via adjusting the real-time active and reactive power of each PV plant. Several simulations and comparisons are carried out on a modified IEEE 33-node system to verify the effectiveness of the proposed methods, and to compare with some previous voltage regulation methods. Simulation results show that the proposed voltage regulation methods can not only effectively control voltage variations of ADNs but also improve the economics of ADNs, substations, and PV plants.