Multi-objective Nonlinear Programming Model Research Articles

Finding Efficient Solutions in Interval Multi-Objective Linear Programming Models by Uncertainty Theory

Interval multi-objective linear programming (IMOLP) ímodels are one of the methods to tackle uncertainties. In this paper, we propose two methods to determine the efficient solutions in the IMOLP models through the expected value, variance and entropy operators which have good properties. One of the most important properties of these methods is to obtain different efficient solutions set according to decision makers’ preferences as available information. We first develop the concept of the expected value, variance and entropy operators on the set of intervals and study some properties of the expected value, variance and entropy operators. Then, we present an IMOLP model with uncertain parameters in the objective functions. In the first method, we use the expected value and variance operators in the IMOLP models and then we apply the weighted sum method to convert an IMOLP model into a multi-objective non-linear programming (MONLP) model. In the second method, the IMOLP model using the expected value, variance and entropy operators can be converted into a multi-objective linear programming (MOLP) model. The proposed methods are applicable for large scale models. Finally, to illustrate the efficiency of the proposed methods, numerical examples and two real-world models are solved.

Multi-Objective Constrained Optimization Model and Molten Iron Allocation Application Based on Hybrid Archimedes Optimization Algorithm

The challenge of distributing molten iron involves the optimal allocation of blast furnace output to various steelmaking furnaces, considering the blast furnace’s production capacity and the steelmaking converter’s consumption capacity. The primary objective is to prioritize the distribution from the blast furnace to achieve a balance between iron and steel production while ensuring that the volume of hot metal within the system remains within a safe range. To address this, a constrained multi-objective nonlinear programming model is abstracted. A linear weighting method combines multiple objectives into a single objective function, while the Lagrange multiplier method addresses constraints. The proposed hybrid Archimedes optimization algorithm effectively solves this problem, demonstrating significant improvements in time efficiency and precision compared to existing methods.

A fairness-based multi-objective distribution and restoration model for enhanced resilience of supply chain transportation networks

Disruptive events play a significant role in the performance of supply chain networks (SCNs) due to their high dependency on transportation networks. Therefore, it is critical to develop coordinated strategies that consider (i) efficient restoration plans for the transportation network (TN) and (ii) distribution plans for downstream SCN, which consider critical aspects such as road congestion, travel time, and required service levels. Thus, we propose a novel multi-objective non-linear mixed-integer programming (NMIP) model to deal with disruptions in transportation networks while maximizing SCN performance. We developed a model for optimal restoration that reduces the effects of disruptions that disrupt the performance of the road TN and SCNs while considering a fair distribution of commodities. The model provides optimal restoration strategies for the TN after unpredicted events, not only based on a cost-effectiveness approach but also on the fair distribution of commodities among demand nodes on a complex network structure with a multi-supplier, multi-demand, and multi-commodity framework. To illustrate the model's capabilities, we study the transportation of commodities in Colombia and how disruptions impact their SCN performance. The results show that a fairness-based distribution helps to obtain satisfaction rates faster when compared to restoration plans based on cost-effectiveness only.

A Multi-Objective Mathematical Programming Model for Transit Network Design and Frequency Setting Problem

In this study, we propose a novel multi-objective nonlinear mixed-integer mathematical programming model for the transit network design and frequency setting problem that aims at designing the routes and determining the frequencies of the routes to satisfy passenger demand in a transit network. The proposed model incorporates the features of real-life transit network systems and reflects the views of both passengers and the transit agency by considering the in-vehicle travel time, transfers, waiting times at the boarding and transfer stops, overcrowding and under-utilization of vehicles, and vehicle fleet size. Unlike previous studies that simplify several aspects of the transit network design and frequency setting problem, the proposed model is the first to determine routes and their frequencies simultaneously from scratch, i.e., without using line and frequency pools while considering the aforementioned issues, such as transfers and waiting. We solve the proposed model using Gurobi. We provide the results of what-if analyses conducted using a real-world public bus transport network in the city of Kayseri in Türkiye. We also present the results of computational tests implemented to validate and verify the model using Mandl benchmark instances from the literature. The results indicate that the model produces better solutions than the state-of-the-art algorithms in the literature and that the model can be used by public transit planners as a decision aid.

FUZZY MULTI-OBJECTIVE NONLINEAR PROGRAMMING PROBLEMS UNDER VARİOUS MEMBERSHIP FUNCTIONS: A COMPARATİVE ANALYSIS

Fuzzy sets have been applied to various decision-making problems when there is uncertainty in real-life problems. In decision-making problems, objective functions and constraints sometimes cannot be expressed linearly. In such cases, the problems discussed are expressed by nonlinear programming models. Fuzzy multi-objective programming models are problems containing multiple objective functions, where objective functions and/or constraints include fuzzy parameters. Membership functions are crucial to obtain optimal solution of fuzzy multi-objective programming model. In this study, a green supply chain network model with fuzzy parameters is proposed. Proposed model with nonlinear constraints is a fuzzy multi-objective nonlinear programming model that minimizes both transportation costs and emissions generated by two vehicle types during transportation. The model is used in Zimmermann's Min-Max approach by considering triangular, hyperbolic and exponential membership functions and optimal solutions are obtained. When optimal solutions are compared, it is seen that optimal solution obtained using the hyperbolic membership function is better than the optimal solutions obtained from triangular and exponential ones. Maximum common satisfaction level calculated using hyperbolic membership function for proposed model is λ=0.97. Sensitivity analysis is also carried out by taking into account distances between suppliers, manufacturers, distribution centers and customers, as well as customer demands.

Dynamic and harmonious allocation of irrigation water resources under climate change: A SWAT-based multi-objective nonlinear framework

Efficient allocation of water resources in irrigation districts can alleviate regional water shortages and promote sustainable irrigated agriculture development. However, existing research on water resource allocation in irrigation districts does not address the lack of coordination within the “diversion-delivery-irrigation” chain for multiple water sources and users in a changing environment. Hence, poor water supply and demand matching, low efficiency and poor climate change responses pose challenges for efficient water resource allocation in irrigation districts. Therefore, this study couples the SWAT runoff simulation model with a multiobjective nonlinear programming model and proposes a weather-driven dynamic and optimal allocation model for multiple water sources. This model accounts for fluctuations in water supply and fine-tunes the allocation of water resources to different water sources, different channels and different crop fertility periods in the irrigation area. The model is designed to achieve synergistic improvements in water supply and demand, economic efficiency, equity in water distribution and efficiency in water use. The model was applied to the Qindeli Irrigation District in Heilongjiang Province. The results show that an increase in water supply at the head of the channel promotes a synergistic increase in economic efficiency and water supply and demand matching. This model can improve water use efficiency under water scarcity by reasonably optimizing the water use structure of the irrigation district. Compared with the traditional irrigation method, the optimized model saves 4 % of water and increases yield by 399 kg/ha, economic efficiency by 0.2 yuan per cubic meter of water, water use efficiency by 9 %, and water supply and demand matching by >80 % at all stages of fertility. The model ensures that water resources are allocated in an equitable manner at all levels.

A multi-objective nonlinear integer programming model for mixed runway operations within the TMAs

AbstractGlobal air traffic demand has shown rapid growth for the last three decades. This growth led to more delays and congestion within terminal manoeuvring areas (TMAs) around major airports. The efficient use of airport capacities through the careful planning of air traffic flows is imperative to overcome these problems. In this study, a mixed-integer nonlinear programming (MINLP) model with a multi-objective approach was developed to solve the aircraft sequencing and scheduling problem for mixed runway operations within the TMAs. The model contains fuel cost functions based on airspeed, altitude, bank angle, and the aerodynamic characteristics of the aircraft. The optimisation problem was solved by using the $\varepsilon$ -constraint method where total delay and total fuel functions were simultaneously optimised. We tested the model with different scenarios generated based on the real traffic data of Istanbul Sabiha Gökçen Airport. The results revealed that the average total delay and average total fuel were reduced by 26.4% and 6.7%, respectively.

Multiobjective Collaborative Optimization Method for the Urban Rail Multirouting Train Operation Plan

The Train Operation Plan (TOP) of urban rail transit (URT) is a comprehensive plan for the operation of trains, the use of facilities and equipment, and the organization of other operational tasks. The TOP should not only be formulated in terms of time-varying passenger flow periods, but it should also be arranged to consider the substitutability of trains between multiple routes combined with the passenger choice. Based on the principle of “operating by the flow” and the requirement for precise allocation of transport capacity for multiple routes, this article constructs a multiobjective nonlinear integer programming model by taking the minimized generalized travel cost of passengers, total running mileage of trains, fluctuation of trains for each route (as optimization targets), and the combination of requirements of both headways and fully loaded rates as constraints. A multiobjective genetic-based algorithm is designed to simultaneously optimize the TOP and the two-way train stopping time in each period. Finally, the proposed model and algorithm are validated with the real data from the Guangzhou Metro Line 2. The results show that the Pareto optimal TOP and dynamic train stopping time are significantly improved compared to the original values.

Multi-objective optimization for AGV energy efficient scheduling problem with customer satisfaction

<abstract><p>In recent years, it has been gradually recognized that efficient scheduling of automated guided vehicles (AGVs) can help companies find the balance between energy consumption and workstation satisfaction. Therefore, the energy consumption of AGVs for the manufacturing environment and the AGV energy efficient scheduling problem with customer satisfaction (AGVEESC) in a flexible manufacturing system are investigated. A new multi-objective non-linear programming model is developed to minimize energy consumption while maximizing workstation satisfaction by optimizing the pick-up and delivery processes of the AGV for material handling. Through the introduction of auxiliary variables, the model is linearized. Then, an interactive fuzzy programming approach is developed to obtain a compromise solution by constructing a membership function for two conflicting objectives. The experimental results show that a good level of energy consumption and workstation satisfaction can be achieved through the proposed model and algorithm.</p></abstract>

A water-energy-food-carbon nexus optimization model for sustainable agricultural development in the Yellow River Basin under uncertainty

Water scarcity, limited land resources and global warming are the most challenging issues facing sustainable agricultural development, and modern agriculture requires efficient and environmentally friendly agricultural water and land management practices. This paper proposes a water-energy-food-carbon nexus system optimization model, which aims to formulate a scientific and rational water and land resource allocation strategy for sustainable agricultural development to increase irrigation water productivity, reduce carbon emissions and enhance regional agriculture competitiveness. First, the carbon footprint life cycle assessment method was used to measure carbon sinks and carbon emissions across the ecosystem. Second, a multiobjective nonlinear programming model with the goal of maximizing irrigation water productivity, minimizing carbon emissions, and maximizing the competitiveness of low-carbon agriculture is established. Finally, an empirical study is performed in the Yellow River Basin, and an elitist nondominated sorting genetic algorithm is used to solve the model. The research results show that the water-energy-food-carbon nexus system optimization model based on uncertain conditions established in this paper, can maximize resource utilization efficiency and effectively measure the impact of regional agricultural production on the environment. It has reference significance for the realization of carbon neutrality in the region.

Investigating the Effect of Priority in Customer Service in Optimizing the Hierarchical Location-Allocation of Crowded Facilities in the Framework of Queuing Systems

In this research, the problem of location-hierarchical allocation of crowded facilities has been investigated by considering customers’ priority in serving within the framework of queuing systems. The objective functions of the problem are focused on minimizing the total waiting time of customers and minimizing the maximum unemployment of each facility. The problem model is a multi-objective nonlinear programming model, and to evaluate the efficiency of the model, examples in different dimensions have been solved by a multi-objective genetic algorithm (NSGA-II) based on minimal sorting. Since the performance of meta-heuristic algorithms is highly dependent on their parameters, the parameters of this algorithm are adjusted using the Taguchi design. The results show that the establishment of the priority system has reduced the average waiting time of all customers compared to the shift system, so it can be concluded that if in designing hierarchical facilities, the goal is to reduce the waiting time of a particular class of customers, they should Prioritize.

Optimizing the Multi-Level Location-Assignment Problem in Queue Networks Using a Multi-Objective Optimization Approach

Abstract Using hubs in distribution networks is an efficient approach. In this paper, a model for the location-allocation problem is designed within the framework of the queuing network in which services have several levels, and customers must go through these levels to complete the service. The purpose of the model is to locate an appropriate number of facilities among potential locations and allocate customers. The model is presented as a multi-objective nonlinear mixed-integer programming model. The objective functions include the summation of the customer and the waiting time in the system and the waiting time in the system and minimizing the maximum possibility of unemployment in the facility. To solve the model, the technique of accurate solution of the epsilon constraint method is used for multi-objective optimization, and Pareto solutions of the problem will be calculated. Moreover, the sensitivity analysis of the problem is performed, and the results demonstrate sensitivity to customer demand rate. Based on the results obtained, it can be concluded that the proposed model is able to greatly summate the customer and the waiting time in the system and reduce the maximum probability of unemployment at several levels of all facilities. The model can also be further developed by choosing vehicles for each customer.

OSWMI: An objective-subjective weighted method for minimizing inconsistency in multi-criteria decision making

In Multi-Criteria Decision Making (MCDM), alternatives are evaluated by considering different criteria. In MCDM, there is a requirement to integrate the objective and subjective weights, since the objective weighting methods ignore the decision-maker’s (DM’s) experiences and the subjective weighting methods ignore the performance ratings of the alternatives with respect to different criteria. To integrate the two types of weights and evaluate the best alternative, three well-established methods, namely “CRiteria Importance Through Intercriteria Correlation (CRITIC)”, “Best Worst Method (BWM)”, and “LINear programming techniques for Multidimensional Analysis of Preferences (LINMAP)” are considered in our study. Based on these methods, we have proposed a new method, namely “Objective-Subjective Weighted method for Minimizing Inconsistency (OSWMI)” which considers both pairwise comparisons of the criteria and alternatives along with their corresponding performance ratings. We have first improved both the methods, CRITIC (named as improved CRITIC) and LINMAP (named as LINMAP II). Finally, the proposed OSWMI method is developed by integrating the improved CRITIC method, BWM, and LINMAP II using a multi-objective non-linear programming (MONLP) model. The OSWMI method may reduce the problem of strategic weight manipulation, since the integrated weights and the two ideal solutions are priori unknown and obtained simultaneously for selecting the best alternative. A case study of the web service selection is used to demonstrate the implementation of the OSWMI method. From the analysis, the proposed OSWMI method reveals a promising result. Further, sensitivity of the OSWMI method is checked by using the standard regression coefficients obtained by multiple linear regression.

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 New Approach for Solving Bi-Level Multi-Objective NonLinear Programming Model under Neutrosophic Environment

A New Approach for Solving Bi-Level Multi-Objective NonLinear Programming Model under Neutrosophic Environment

Agricultural water and land resources allocation considering carbon sink/source and water scarcity/degradation footprint

There is an urgent need for scientific management of agricultural water and land resources to cope with global warming and water shortages. Therefore, a stochastic multi-objective non-linear programming model was established under the society-economy-ecology framework in this study, which is capable of (1) considering the carbon sink function of farmland vegetation and the carbon emissions produced from the input of production materials; (2) dealing with the potential impact of agricultural water use on ecology by the indexes of water scarcity and degradation footprint (3) weighing the conflicts and contradictions among different developing targets with economic benefits, and water productivity as well as ecological considerations; (4) obtaining the planting structure, irrigation water allocation schemes and irrigation schedules. In order to verify the applicability and effectiveness of the model, it was applied to Huangyang Irrigation District in Shiyang River Basin, northwest China. After optimization, the net carbon sink increased by 2.55 × 104 t of which fertilizer contributed nearly 50% of carbon emission, while the water footprint reduced by 0.48 × 108 m3. To analyze the impact of the different allocation schemes of water and land resources on the ecological, economic and social subsystems as well as their interaction, coupling coordination degree (CCD) models were introduced to evaluate status quo and the optimization results of different models. The results showed that, compared with the single-objective models and the status quo, the proposed model has improved the value of CCD from 0.428 to 0.674. The proposed model can promote the harmonious and sustainable development of agricultural production and is equally applicable to agricultural management systems in the regions with similar conditions.

A method to improve the resilience of multimodal transport network: Location selection strategy of emergency rescue facilities

This paper studies the location selection strategy of emergency rescue facilities in the multimodal transport network to improve the resilience of the network. Based on the overall perspective, we propose a cooperative coverage model. On one hand, it helps network operators improve the recovery speed of the entire multimodal transport network in the event of a disaster. On the other hand, it maximizes the effect of the rescue cost of government managers. But we must note that in most transport networks, failure data is difficult to be collected or is insufficient. Therefore, we propose the premise that the network failure may occur at any point. The rescue demand is quantified by the links/nodes importance, which is calculated by the multi-attribute decision-making method based on hesitant intuitionistic fuzzy set-technique for order preference by similarity to an ideal solution (TOPSIS). Then, a multi-objective non-linear programming model is established, and fast non-dominated sorting genetic algorithm-cellular automata (NSGA-II-CA) is designed to solve the model. Taking the multimodal transport network in the Sichuan-Tibet region of China as an example, calculation and analysis are performed to verify the effectiveness and advancement of the proposed model and algorithm. The innovation of this paper is to propose a method of emergency rescue facility location from the perspective of global optimization.

A conceptual framework for a multi-criteria decision support tool to select technologies for resource recovery from urban wastewater

In the context of circular economy, wastewater can be used to address some of the 21st century's challenges regarding the transition to renewable resources for water, energy, and nutrients. Despite all the research, development, and experience with resource recovery from urban wastewater, its implementation is still limited. The transition from treatment to resource recovery is complex due to the difficulty of selecting unit processes from a large number of candidate processes considering the operational limitations of each process, and sustainability objectives. Presently, a multi-criteria decision support tool that deals with the difficulty of unit process selection for resource recovery from wastewater has not been developed. Therefore, this paper presents the conceptual framework of a decision support tool to find the optimum treatment train consisting of compatible unit processes which can recover water, energy and/or nutrients from a specified influent composition. The framework presents the relationship between the user input, the knowledge library of technologies and a weighted multi-objective nonlinear programming model to aid process selection. The model presented here shows, not only how the processes are selected, but also the four-dimensional sustainability impact of the generated treatment train while considering the weight provided by the user. Thus, this study presents a reproducible framework which can support private and public decision-makers in transparent evidence-based decision making and eventually the systematic implementation of resource recovery from urban wastewater.

Thermo-economic and environmental optimization of a solar-driven zero-liquid discharge system for shale gas wastewater desalination

Wastewater management is one of the main hurdles encountered by the shale gas industry for boosting overall process cost-effectiveness while reducing environmental impacts. In this light, this paper introduces a new multi-objective model for the thermo-economic and environmental optimization of solar-based zero-liquid discharge (ZLD) desalination systems. The solar-driven ZLD system is especially developed for desalinating high-salinity wastewaters from shale gas process. A decentralized system is proposed, encompassing a solar thermal system, a Rankine power cycle, and a multiple-effect evaporator combined with mechanical vapor recompression. The environment-friendly ZLD operation is ensured by specifying the salt concentration of brine discharges close to saturation conditions. The mathematical modelling approach is centered on a multi-objective non-linear programming (MoNLP) formulation, which is aimed at simultaneously minimizing thermo-economic and environmental objective functions. The latter objective function is quantified by the ReCiPe methodology based on life cycle assessment. The MoNLP model is implemented in GAMS software, and solved through the epsilon-constraint method. A set of trade-off Pareto-optimal solutions is presented to support decision-makers towards implementing more sustainable and cost-efficient solar-driven ZLD desalination systems. The comprehensive energy, economic and environmental analysis reveals that the innovative system significantly decreases costs and environmental impacts in shale gas wastewater operations.

Multi-Objective Nonlinear Programming Model for Reducing Octane Number Loss in Gasoline Refining Process Based on Data Mining Technology

To simultaneously reduce automobile exhaust pollution to the environment and satisfy the demand for high-quality gasoline, the treatment of fluid catalytic cracking (FCC) gasoline is urgently needed to minimize octane number (RON) loss. We presented a new systematic method for determining an optimal operation scheme for minimising RON loss and operational risks. Firstly, many data were collected and preprocessed. Then, grey correlative degree analysis and Pearson correlation analysis were used to reduce the dimensionality, and the major variables with representativeness and independence were selected from the 367 variables. Then, the RON and sulfur (S) content were predicted by multiple nonlinear regression. A multi-objective nonlinear optimization model was established with the maximum reduction in RON loss and minimum operational risk as the objective function. Finally, the optimal operation scheme of the operating variable corresponding to the sample with a RON loss reduction greater than 30% in 325 samples was solved in Python.