Multi-objective Mixed-integer Nonlinear Programming Model Research Articles

Multi-objective location-routing model and algorithm study of material reserve bases for large-scale railway engineering construction projects

Material supply is an important support for engineering construction and an important factor affecting the progress and quality of engineering construction. A multi-objective mixed-integer nonlinear programming model is proposed for the location routing problem of the material reserve base of large-scale railway engineering construction projects, considering realistic factors like the construction period, the cycle of the material reserve, the variety of materials, the heterogeneous fleet, and demand splitting. An improved Non-dominated Sorting Genetic Algorithm-II (NSGA- II) is developed for solving this model. This paper optimises obtaining the initial population based on the linear feature of railway engineering, improves the fitness function, and designs a two-stage crossover operator. In addition, the algorithm is compared with the augmented ϵ-constraint method with the help of a practical example. Three metrics are used to assess the performance of the algorithm's outputs, and two metrics are used to assess the similarity of the Pareto fronts obtained by the two algorithms. The results show that the proposed algorithm performs better in terms of ‘quantity’ and ‘spacing’, that the Pareto solutions obtained are accurate, and that the speed of solution is almost 10 times that of the augmented ϵ-constraint (AUGMECON) method.

Uncertain programming model for designing multi-objective reverse logistics networks

Given the contradiction between the rapid growth of products and the modest recovery rate of end-of-life products, there is a pressing need to understand the societal significance of establishing a reverse logistics network for end-of-life products. This research constructs an open-loop five-tier reverse logistic network model encompassing customers, centres for collection, disassembly and inspection, remanufacturing, and disposal. A multi-objective mixed-integer nonlinear programming model under uncertainty has been developed. Unlike previous research, this model accounts for surrounding residents' disutility of facilities while simultaneously minimizing economic costs and environmental impact. Besides, uncertainty theory is introduced in solving the proposed model. More specifically, the formulated model converts all uncertain variables into uncertain distributions by implementing the uncertain multi-objective programming method. Furthermore, a customised non-dominated sorting genetic algorithm III (NSGA-III) is proposed and is employed for the first time to address facility selection and recycling volume distribution within the network. The model is then validated using a real-life case study focusing on end-of-life vehicles in Changchun, China. This research could assist decision-makers in both governmental and private sectors in achieving a balanced approach to the interests of the economy, environment, and local communities comprehensively when designing reverse supply chains.

A location-inventory model for the sustainable supply chain of perishable products based on pricing and replenishment decisions: A case study.

In recent decades, changes in people's lifestyles and occupations have led to new food consumption patterns around the world, with a notable growth in the demand for ready meals and meat products. Accordingly, the food industry has tried to transform the global food culture by promoting its more profitable products to become able to set a higher price for meat. Considering the short lifetime of products in perishable food supply chains, inventory decisions are considered crucial. In addition, the demand for perishable food products is greatly affected by their freshness. In this paper, we develop a multi-objective mixed-integer non-linear programming model for a four-level sustainable supply chain (SC) of a perishable product with price-dependent demand and deterioration rates. The SC consists of suppliers, a production center (PC), distribution centers (DCs), and retailers. We aim to ascertain the optimal pricing policy and cycle length to maximize profit, achieve specific social objectives, and minimize the total cost and adverse environmental impacts. The proposed model determines the deterioration rate of each product according to its expiration date, the optimal location of supply-side facilities, and the flow rate between the facilities based on the selected mode of delivery. The products are delivered to the retailers by a mechanized transportation system either directly from the PC or indirectly through the DCs. Finally, an actual case study is provided to demonstrate the applicability of the model and our theoretical results under real-world conditions. We solve the case study by a preemptive fuzzy goal programming method and perform several sensitivity analyses on the results. By performing simultaneous sensitivity analyses on the demand and expiration dates, the optimal values of the problem's parameters are determined. The optimal values help decision-makers make optimal decisions regarding the selling price of products and replenishment times. The model is applicable to supply chains of any perishable items with expiration dates.

Multi-objective energy-efficient hybrid flow shop scheduling using Q-learning and GVNS driven NSGA-II

The urgent mission for carbon peak and carbon neutrality is demanding greater industrial sustainability. Energy-efficient hybrid flow shop scheduling problem (EEHFSP) has been raising increasing attention in recent years. This paper studies a new EEHFSP with uniform machines to minimize total tardiness, total energy cost, and carbon trading cost. Time-of-use tariffs and power down strategies are simultaneously adopted. A novel multi-objective mixed-integer nonlinear programming model for the problem is proposed. To solve the model, we propose a Q-learning and general variable neighborhood search (GVNS) driven non-dominated sorting genetic algorithm II (QVNS-NSGA-II). The novelty of the algorithm is that we incorporate Q-learning into GVNS to guide premium adaptive operator selection throughout the shaking and local search processes. A distinguishing feature is that the states and actions of Q-learning are set as neighborhood structures and local search operators. The Q-learning-driven GVNS is embedded into NSGA-II to promote the exploration and exploitation capability. Experimental results show that the proposed QVNS-NSGA-II outperforms NSGA-II, improved Jaya, and modified MOEA/D in terms of the quantity, quality of Pareto solutions, and computational efficiency. Sensitivity analysis also derives several managerial implications. The proposed approach can be applied to improve sustainability and productivity for hybrid flow shop manufacturers.

Novel empty train return strategy and passenger control strategy to satisfy asymmetric passenger demand: A joint optimization with train timetabling

Big cities’ bi-directional metro lines often have unbalanced passenger demand, i.e., up-directional passenger demand is oversaturated but down-directional passenger demand is unsaturated. In recent years, the method to relieve congestion in oversaturated direction has attracted much attention, but many problems in the unsaturated direction have not gotten as much attention, e.g., the waste of transportation resources caused by low loading rate, and the insufficient utilization of platform space resources caused by less waiting passengers. To address these problems, this paper introduces a new collaborative optimization model for the train timetable, passenger control and rolling stock schedule on a bi-directional metro line with unbalanced passenger demand. With consider the real-time state of bi-directional platform passengers, a platform passenger control strategy is designed to promote the utilization of platform resources, and an empty train return strategy is proposed to transfer transportation resources to oversaturated direction. To characterize the problem mathematically, a multi-objective mixed-integer nonlinear programming model (MINLP) is formulated to minimize the passengers’ waiting time, the difference of bi-directional platform passenger density and the use cost of trains. An effective heuristic algorithm based on the non-dominated sorting genetic algorithm (NSGA-II) and the adaptive large neighborhood search (ALNS) is designed to find high-quality solutions for the proposed model. Finally, five sets of numerical examples based on the Beijing metro Batong line, are conducted to validate the performance of the proposed method in this paper. The experimental results demonstrate the method can fully utilize the transportation and platform resources of the unsaturated direction, to improve transport efficiency and relieve platform congestion of oversaturated direction.

An inexact multi-objective mixed-integer nonlinear programming approach for water-soil-fertilizer management under uncertainty considering “footprint family-planetary boundary” assessment

Water, energy, and food (WEF) are interlinked and create a dynamic system that impacts both human well-being and ecology. Given the importance of ecology within the WEF system, water resources serve as the core issue. The allocation of water resources in irrigation districts is a challenging problem for the coordinated development of agricultural production, water resources, and the ecological environment. The integration of stochastic multi-objective programming, fuzzy credibility-constrained programming, and mixed integer programming offers a solution to this issue, with the construction of a fuzzy credibility-constrained stochastic multi-objective mixed-integer nonlinear programming model. The applicability and validity of this model were verified by applying it to the Kaikong Irrigation District (KID) of the Tarim River Basin in northwest China, with notable findings indicating that the optimized system reduces agricultural costs by 5.82%, increases irrigation water use efficiency by 1.80%, and reduces global warming potential by 6.45%. This study investigates the effects of diverse allocation strategies of water and land resources on the social, economic, and ecological subsystems and their interactions by downscaling four subprocesses PBs to the KID scale. The optimization model reveals that only the nitrogen footprint of Kuerle City surpasses the nitrogen boundary of the KID. The proposed solutions based on the model can encourage the green and ecologically-friendly development of agricultural production and can be applied to agricultural systems in arid regions with similar conditions.

Integrated optimization of demand-driven timetable, train formation plan and rolling stock circulation with variable running times and dwell times

Timetable, train formation plan and rolling stock circulation are the key operational issues for efficient railway operation. The three issues are optimized sequentially based on the order of the operational plan. Sequential optimization manner cannot balance the costs of the three stages effectively and could create an infeasible solution for a later stage when the resources are limited. In addition, because of the computational complexity, multiple variable elements, such as the variable running times, variable dwell times and the coupling/decoupling operations, of a transportation system are ignored. This paper focuses on optimizing timetable, train formation plan and rolling stock circulation simultaneously to minimize costs and meet passenger demand. The key to solving the problem is to determine operation times (i.e., arrival times, departure times, running times and dwell times), the formation type of each train service and rolling stock connections (including the turnaround operations and the coupling/decoupling operations) between these train services. Considering the passenger costs and operator costs, a multi-objective mixed-integer nonlinear programming (MINLP) model is proposed to minimize the total passenger waiting time (TWT), the number of rolling stocks (NR), the number of formations (NF) and the number of coupling/decoupling operations (NC) based on a time–space network. The multi-objective MINLP model is further reformulated into a single-objective mixed-integer linear programming (MILP) model by logical linearizing, piecewise linearizing and fuzzy programming. A numerical example is used in which the exact solution of MINLP obtained by the BARON solver is compared with the approximate solution of MILP obtained by the CPLEX solver to verify the effectiveness of the piecewise linearization. Finally, the model is tested on a real-world case study of the Jinshan Line in Shanghai. The CPLEX solver can efficiently produce the approximate solution within a given computation time in acceptable gaps. The results demonstrate that the integrated model can reduce the number of rolling stocks and improve the utilization rate of rolling stocks. Furthermore, the integrated model can effectively resolve the shortage of rolling stocks when only one depot has rolling stocks or the number of available rolling stocks is limited. In addition, considering the multiple variable elements have a significant effect on the improvement of all the objectives and reduce both the passenger costs and the operator costs.

Multi-objective steel plate cutting optimization problem based on real number coding genetic algorithm

The rectangular packing problem is an NP-complete combinatorial optimization problem. This problem occurs widely in social production scenarios, with steel plate cutting being one example. The cutting scheme for the rectangular packing problem needs to be improved because, without the globally optimal solution, there are many unnecessary edges in the steel cutting process. Based on a practical roll-fed disc shearing steel plate optimization problem, this paper explores a generalized packing method for rectangles of special dimensions and abstractly condenses complex quantitative relationships to establish a multi-objective mixed-integer nonlinear programming model. An innovative algorithm design based on a genetic algorithm is established to plan the cutting scheme in a high-speed and efficient way. The outcome is a utilization rate of up to 92.73% for raw materials and a significant reduction in labor, providing a guide for practical production and processing tasks. The advantages and disadvantages of the model and algorithm are discussed, and it is concluded that this rectangular packing method has strong universality and generalization ability, allowing rectangular packing tasks with large data volumes to be completed within a short time.

A methodology for propellant composition optimization in aerosol consumer products, considering economic, safety, and environmental objectives

In this work we present for the first time a methodology to simultaneously optimize the composition of aerosol propellant mixes for consumer products, considering economical, safety, and environmental objectives. Selection of the suitable components for the propellant mix follows a decomposition approach that takes into account product and package compatibility, the regulatory framework, and the type of propellant. An inherent safety approach is considered for the safety objective. The core of the mathematical model is the multicomponent vapor–liquid equilibrium using the Peng–Robinson–Stryjek–Vera cubic equation of state. The optimization problem is defined as a multi-objective mixed-integer nonlinear programming model solved using the ϵ-constraint method. The application of the methodology is demonstrated with a case study for an underarm deodorant formulation. One propellant mix is obtained from the Pareto front as the best compromise of the three objective functions. The rationale to choose a different solution, based on stakeholder’s preferences, is discussed.

Robust possibilistic programming to design a closed-loop blood supply chain network considering service-level maximization and lateral resupply.

Reconfiguring the structure of the supply chain network is one of the most strategic and vital decisions in designing a supply chain network. In this study, a Closed-Loop Blood Supply Chain Network (CLBSCN) considering blood group compatibility, ABO-Rh(D), and blood product shelf life has been studied to determine the best strategic and tactical decisions simultaneously considering lateral resupply/transshipment and service-level maximization. Several vital parameters, including supply and demand, are considered fuzzy numbers to approximate reality due to the nature of the world. Furthermore, two crucial factors include ABO-Rh(D) and blood product shelf life considered, while the concept of lateral resupply governs the interconnections of hospitals’ excess blood units. We propose a fuzzy multi-objective Mixed-Integer Non-Linear Programming (MINLP) model to consider two critical objective functions: minimizing the total costs of the network and maximizing the minimum service level to the patients at each Hospital. The fuzzy multi-objective MINLP model is converted to a deterministic multi-objective model using the equivalent auxiliary crisp model to deal with uncertainty. Then, by utilizing two interactive fuzzy solution approaches, the results have been compared based on a real case study to suggest the best solution for the proposed model. Also, we conduct sensitivity analysis on essential parameters such as demand, supply, and capacity to understand how these parameter variations impact two proposed objective functions. Then, the proposed model is tested on a real case study for model validation. The results confirmed that considering the lateral resupply could significantly save the costs of the designed network by a total of $343,000. Interestingly, maximizing the minimum service level at hospitals increased the service level from 58% to 68%.

A congested capacitated location problem with continuous network demand

This paper presents a multi-objective mixed-integer non-linear programming model for a congested multiple-server discrete facility location problem with uniformly distributed demands along the network edges. Regarding the capacity of each facility and the maximum waiting time threshold, the developed model aims to determine the number and locations of established facilities along with their corresponding number of assigned servers such that the traveling distance, the waiting time, the total cost, and the number of lost sales (uncovered customers) are minimized simultaneously. Also, this paper proposes modified versions of some of the existing heuristics and metaheuristic algorithms currently used to solve NP-hard location problems. Here, the memetic algorithm along with its modified version called the stochastic memetic algorithm, as well as the modified add and modified drop heuristics are used as the solution methods. Computational results and comparisons demonstrate that although the results obtained from the developed stochastic memetic algorithm are slightly better, the applied memetic algorithm could be considered as the most efficient approach in finding reasonable solutions with less required CPU times.

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 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.

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.

Multiobjective Scheduling Method for Short-Term Peak Shaving Operation of Cascade Hydro Plants

With structural changes in power consumption, peak load demands and peak-valley differences are growing, which motivates peak shaving operations of hydropower systems. A multiobjective mixed-integer nonlinear programming model is presented for short-term hydropower generation scheduling that aims to regulate peak loads and enhance power efficiency. The combined feasible operation zones and the aggregate generation function of each plant are embedded in this model to deal with the operation constraints of units. A set of Pareto solutions of this model is generated by an improved normalized constraint method that narrows the optimization range to ensure the practicality and compact distribution of solutions, each of which represents an aggregated scheduling of cascaded hydropower plants. Then, a mixed integer nonlinear programming formulation is developed to solve the unit commitment problem for each solution, and a fuzzy-based membership value assignment method is employed to extract a best compromise solution for final implementation. A case study of the hydropower plants of the Beipanjiang cascade in China is used to test the proposed approach, verifying its success in generating a practical operating schedule while providing abundant information to decision makers for decision support.

Passenger demand oriented train scheduling and rolling stock circulation planning for an urban rail transit line

We study the integration of train scheduling and rolling stock circulation planning under time-varying passenger demand for an urban rail transit line, where the practical train operation constraints, e.g., the capacity of trains, the number of available rolling stocks, and the entering/exiting depot operations, are considered. Three solution approaches are proposed to solve the resulting multi-objective mixed-integer nonlinear programming (MINLP) problem to deliver both an irregular train schedule (i.e., departure and arrival times of all train services) and a rolling stock circulation plan (including entering/exiting depot operations of rolling stocks and connections between train services) simultaneously. We first present an iterative nonlinear programming (INP) approach, where the solutions of the original MINLP problem are obtained by solving a nonlinear programming problem and a mixed integer linear programming (MILP) problem iteratively. Moreover, an equivalent MILP formulation of the original MINLP model is developed and an approximated MILP approach is proposed to reduce the number of constraints introduced by passenger demand. A case study is conducted based on the practical data of the Beijing Yizhuang line, where the three proposed approaches are compared with a state-of-the-art approach and a practical method used by the traffic planners. This comparison shows the effectiveness and efficiency of the three proposed approaches.

Joint decision model of supplier selection and order allocation for the mass customization of logistics services

This paper formulates a multi-objective mixed-integer nonlinear programming model to simultaneously determine the optimal selection strategy of functional logistics service providers and the order allocation strategy under the mass customization of logistics services. An improved genetic algorithm based on multi-layer coding is designed to solve the model. Numerical experiments show that the supplier selection strategy and the order allocation strategy are mutually influential, therefore, decision makers need to consider both simultaneously to optimize the overall performance of the supply chain. The influence of some important parameters on the model results was observed through a sensitivity analysis.

A novel multi-objective programming model of relief distribution for sustainable disaster supply chain in large-scale natural disasters

To save lives and reduce suffering of victims, the focus of this paper is to design the strategies of relief distribution regarding beneficiary perspective on sustainability. This problem is formulated as a multi-objective mixed-integer nonlinear programming model to maximize the lowest victims' perceived satisfaction, and minimize respectively the largest deviation on victims’ perceived satisfaction for all demand points and sub-phases. Then, genetic algorithm is proposed to solve this mathematical model. To validate the proposed methodologies, a case study from Wenchuan earthquake is illustrated. Computational results demonstrate genetic algorithm here can achieve the trade-off between solution quality and computation time for relief distribution with the concern of sustainability. Furthermore, it indicates that the methodology provides the tools for decision-makers to optimize the structure of relief distribution network and inventory, as well as alleviate the suffering of victims. Increasingly, this paper expects to not only validate the proposed model and method, but also highlight the importance and urge of considering beneficiary perspective on sustainability into relief distribution problem.

Involving Environmental Assessment in the Optimal Design of Domestic Cogeneration Systems

This paper presents a multi-objective optimization formulation for designing domestic cogeneration systems to satisfy thermal and electric energy demands. The proposed model is formulated as a multi-objective mixed-integer nonlinear programming model, which incorporates as economic objective function the minimization of the total annual cost, which includes the capital costs for the new units and the thermal storage system as well as the operating and maintenance costs. The model also incorporates an environmental objective function, which accounts for the life cycle assessment through the Eco-indicator 99 method to account for the damages to the resources, human health, and ecosystem quality. The proposed model allows considering the interaction with the external users and the external electric grid. The optimization formulation is solved to generate design alternatives to establish tradeoffs among the considered objectives.

A Multi-Objective Model for Location-Allocation Problem in a Supply Chain

The fast changing and dynamic global business environment require companies to plan their entire supply chain from the raw material supplier to the end customer. In this paper, we design an integrated supply chain including multiple suppliers, multiple factories, multiple distributors, multiple customers, multiple products, and multiple transportation alternatives. A new multi-objective mixed-integer nonlinear programming model is proposed to deal with this facility location-allocation problem. It considers two conflicting objectives simultaneously, and then the problem is transformed into a multi-objective linear one. The first objective function aims to minimize total losses of the supply chain including raw material purchasing costs, transportation costs and establishment costs of factories and distributions. The second objective function is to minimize the sum deterioration rate of end products and raw materials incurred by transportation alternatives. Finally, the proposed model is solved as a single-objective, mixed-integer, programming model applying the Global Criteria Method. We test their model with numerical example and the results indicate that the proposed model can provide a promising approach to fulfill customer demand and design an efficient supply chain.