Multi-objective Programming Model Research Articles

Multi-objective mixed integer programming modelling for closed-loop supply chain network design: an enhanced Benders decomposition algorithm

Considering a closed-loop supply chain (CLSC) network with uncertain demand and recycling rates, this article innovatively designs a multi-objective mixed-integer programming model that incorporates corporate social responsibility (CSR), a facility retrofit strategy (FRS), a flexible supply strategy (FSS) and a vehicle selection strategy (VSS). Then, robust optimization methods are applied to construct robust models under three situations of uncertainty. For the computational complexity of large-scale problems, an enhanced Benders decomposition algorithm (EBDA) is designed. A numerical case analysis is conducted using five different scale instances. First, compared to other algorithms, EBDA accelerates the solution efficiency while ensuring convergence. Secondly, the sensitivity of the objective weights and the trade-offs of multiple objectives are analysed. Finally, the impact is analysed of the uncertain environment, CSR, an FRS, an FSS and a VSS on the CLSC network. Decision makers need to balance three objectives to manage CLSC and use these strategies appropriately to address the negative impact of an uncertain environment.

A Multi-Objective Model for Designing a Sustainable Closed-Loop Supply Chain Logistics Network

Background: The growing concern for environmental and social issues has led to a focus on designing sustainable supply chains and increasing industrial responsibility towards society. In this paper, a multi-objective mixed-integer programming model is presented for designing a sustainable closed-loop supply chain. The model is aimed at the minimization of the total cost with the total used facilities, the negative environmental impacts, and the maximization of the positive social impacts. Methods: The epsilon-constraint method is utilized for solving the model and further extracting the Pareto solutions. Results: The result of the research clearly shows an optimal trade-off between the conflicting objectives, where, by paying more attention to the social and environmental aspects of sustainability, the total costs are increased or by optimizing the number of facilities, a better balance between the dynamics associated with the short-term and long-term goals is reached. The results of the sensitivity analysis also show that increasing the demand of the supply chain has the greatest impact on the supply chain costs compared to other objectives. Conclusions: Consequently, investigating such comprehensive sustainable objectives provides better insights into the impact of design variables on the expectations of stakeholders.

Estimation of travel flux between urban blocks by combining spatio-temporal and purpose correlation

Understanding the travel flux between urban blocks is fundamental for traffic demand prediction, urban area planning and urban traffic management. However, the uncertainty of human mobility patterns and the complexity of urban transportation systems usually yield challenges in accurately estimating the travel flux within a city. Thus, we propose a novel travel flux estimation method that integrates traffic flow characteristics (traffic volume and travel time), spatio-temporal autocorrelation, and travel purpose correlation. First, the geographically weighted method was used to model and verify the spatio-temporal autocorrelation of origin–destination flows, whereas the purpose correlation of origin–destination flows was expressed through the function feature vectors of urban blocks. Then, a multi-objective bi-level programming model, according to the generalized least squares method, was constructed to estimate the travel flux between blocks. This was used to solve the problem of accurate estimation of travel flux by combining microscopic traffic flow characteristics with macroscopic spatio-temporal and purpose characteristics. Finally, an empirical analysis of the Hankou district, Wuhan City, demonstrated that in contrast to the existing method, the accuracy of the proposed method for predicting the human travel flux improved by approximately 20%. The estimated results were consistent with the spatial distribution pattern of human travel. Moreover, these results can provide targeted decision support for planning urban spaces, allocating urban resources, and guiding vehicular traffic.

A Multiscale Approach for Free-Float Bike-Sharing Electronic Fence Location Planning: A Case Study of Shenzhen City

As an emerging technological means for managing free-float bike-sharing parking, electronic fences have attracted increasing attention in major cities as a solution to the challenges posed by disorderly parking of free-float bikes. Existing research has predominantly focused on employing clustering methods from the perspectives of free-float bike-sharing companies and users to plan and deploy electronic fences. However, the results often deviate significantly from the actual phenomenon. Therefore, scientific location selection is particularly important to fully harness the effectiveness of electronic fences. This paper proposes a multiscale clustering method based on free-float bike-sharing parking features to determine the optimal locations for electronic fences. A multiobjective mixed-integer programming model is established to address the location planning problem of electronic fences, determining the planning positions, quantities, and areas of electronic fences. A case study is conducted using a local area free-float bike-sharing dataset from Shenzhen city to validate the effectiveness of the proposed method. Comparative results with traditional approaches solely relying on K-means or DBSCAN methods demonstrate that the proposed approach achieves efficient location selection, through multiscale fusion site selection in the study area of 1.5∗1 km, and only 25 electronic fences need to be planned and deployed, covering a total area of 1691.88 square meters, which can provide rational placement solutions and better utilize the effectiveness of electronic fences. This method can thus offer decision-making support for the planning and location selection of electronic fences in free-float bike-sharing systems.

An ordered multi-objective fuzzy stochastic approach to sustainable water resources management: a case study from Taiyuan City, China

Abstract The uncertainty of socioeconomic development and climate change poses challenges to the sustainable management of water resources in Taiyuan, China. The study proposes a type-2 fuzzy chance-constrained ordered multi-objective fractional programming (T2FCC-ORMOFP) model to address the allocation of water under uncertainties. The model incorporates type-2 fuzzy programming and chance-constrained programming into an overall framework to handle multiple uncertainties under multi-level and multi-objective conflicts, while using fractional programming to reflect the marginal benefits of the system. It prioritizes the principles of optimal, fair, and stable distribution by the upper-level governing authorities, while considering the social and economic environmental benefits of lower-level decision-making. The optimal water allocation scenarios were obtained under different hydrological guarantee rates, violation levels of water supply constraints, and net economic benefits of type-2 fuzzy numbers for 2030. Additionally, groundwater is replaced by reclaimed water in varying proportions for flexible water supply. The results show that the water demand in Taiyuan cannot be ignored, and the risk of water shortage is more sensitive for the agricultural and industrial sectors. The recycled water substitution strategy can optimize the water supply structure and improve social and economic benefits (9–12%). Also T2FCC-ORMOFP can improve overall efficiency (20%) compared with a single-level model.

A novel approach for multi-objective linear programming model under spherical fuzzy environment and its application

Every decision-making process particularly those involving real-life issues is disproportionately plagued by uncertainty. It is also unavoidable and obvious. Since its conception are several ways for representing uncertainty have been proposed by numerous academics to cope with uncertainty. Fuzzy sets and hierarchical such as picture fuzzy sets stand out among them as excellent representation techniques for modeling uncertainty. However, there are several significant drawbacks to the current uncertainty modeling techniques. Due to its vast versatility and benefits we here embrace the idea of the spherical fuzzy set, an extension of the picture fuzzy set. On the other hand amid uncertainty in real life the multi-objective plays a critical role. In this research paper determining a Multi-Objective Linear Programming Problem of Spherical fuzzy sets serves to stimulate nous. The score function corresponding to the degree positive, negative and neutral is the foundation upon which the suggested approach is developed. Additionally we apply the suggested strategy to the solution of the multi-objective linear programming problem to demonstrate its superiority through certain numerical examples. Maximization or Minimizing of the cost is the primary goal of the multi-objective linear programming problem. Using an explicitly defined score function the suggested solution transformed the Spherical Fuzzy Multi-Objective Linear Programming Problem into a Crisp Multi-Objective Linear Programming Problem (CMOLPP). We establish some theorems to show that the efficient solution of CMOLPP is likewise an efficient solution of SFMOLPP. The CMOLPP is then further simplified into a single-objective Linear Programming Problem (LPP) thus we revamp the modified Zimmermann’s approach in the environment of a nonlinear membership function with the aid of the suggested technique. It is possible to simply solve this single-objective LPP using any software or standard LPP algorithm. The suggested approach achieves the fuzzy optimum result without altering the nature of the issue. An application of the suggested approach has been used to illustrate it and its results have been distinguished from those of other preexisting methods found in the literature. To determine the importance of the suggested technique which adjudicate thorough theorem and result analysis is conducted.

A multi-objective two-stage stochastic programming approach for wind farms and demand side provider decisions using the weighted metric method

The rapid expansion of wind-based renewable energy sources highlights the critical need to enhance reserve capacity within power systems. This research aims to develop an effective model for harnessing the combined potential of flexible demand-side resources, acting as demand-side providers (DSPs), to address the inherent unpredictability of wind power. The DSP framework comprises three demand response aggregators: the capacity market program (CAP), energy storage systems (ESSs), and plug-in electric vehicle parking lots (PEVs PLs). To achieve this, a multi-objective two-stage stochastic programming model is introduced. It not only minimizes operational costs for system operators but also maximizes returns for demand-side resource owners supervised by DSPs in the energy and reserve market environment. The problem is tackled using a mixed-integer linear programming (MILP) approach employing a weighted metric methodology. Within this framework, the Weibull probability distribution function (WPDF) is utilized to model wind power uncertainty, while the truncated Gaussian distribution function (GDF) encapsulates the uncertainty regarding electric vehicle owner behavior. The results demonstrate that employing this method for simultaneous programming of energy and reserve not only reduces system operator costs but also enhances profits for resource owners.

Optimization model growth and equalization of farmer’s income in the rotiklot dam service area on the border area of Indonesia-Timor Leste

The study aims to analyze optimization models of growth and income distribution. The study used a survey method in Fatuketi Village, Kakuluk Mesak District as the Rotiklot Dam service area, with a sample of 300 respondents. Data analysis uses a multi-objective linear programming model for growth and equal distribution of agricultural income. The results of the analysis show that an income of IDR 11,576,192.00 per year is an income that optimizes growth and income distribution. Optimizing growth and equal distribution shows that the average income of farmers in the first planting season was IDR. 4,132,108 and IDR. 7,444,084 in planting season II. The marginal land used is 300 ha with the volume of water needed to meet all plant needs as much as 1,986,565.16 m3. Agricultural income can still be increased for the third planting season because there is still excess water of 948,434.84 m3.

Power Generation Mix Optimization under Auction Mechanism for Carbon Emission Rights

As the international community attaches importance to environmental and climate issues, carbon dioxide emissions in various countries have been subject to constraints and limits. The carbon trading market, as a market tool to reduce greenhouse gas emissions, has gone through a development process from a pilot carbon market to a national carbon market in China. At present, the industries included in the national carbon market are mainly the electric power industry, and the carbon emissions of the electric power industry account for about 40% of the national carbon emissions. According to the construction history of foreign carbon markets, China’s future carbon quota allocation will gradually transition from free allocation to auction allocation, and the auction mechanism will bring a heavy economic burden to the electric power industry, especially the thermal power generation industry. Therefore, this study takes Guangdong Province as an example to optimize the power generation mix with the objective of minimizing the total economic cost after the innovative introduction of the carbon quota auction mechanism, constructs an optimization model of the power generation mix based on the auction ratio by comprehensively applying the system dynamics model and the multi-objective linear programming model, systematically researches the power generation structure under different auction ratios with the time scale of months, and quantitatively evaluates the economic inputs needed to reduce the greenhouse gas emissions. The results of the study show that after comprehensively comparing the total economic cost, renewable energy development, and carbon emissions, it is the most scientific and reasonable to set the auction ratio of carbon allowances at 20%, which achieves the best level of economic and environmental benefits.

Research on the Optimization of Multibeam Bathymetry Survey Line Layout Based on Simulated Annealing Algorithm

Multibeam bathymetry systems are widely used in water depth measurement, and their principle is based on the propagation characteristics of sound waves in water. The system has multiple advantages, such as high resolution, high accuracy, wide measurement range, automated mapping, and high efficiency. This study is based on the propagation characteristics of sound waves in water and utilizes a multibeam bathymetry system for water depth measurement, focusing on key issues such as coverage width, survey path, and seawater depth. Based on a two-dimensional plane model, the direction of the emitted beams from the measurement vessel and the relationship with seafloor slope angles are considered. The key parameter calculation model of the multibeam bathymetry system is established using the sine theorem and cosine theorem. To enhance the analysis accuracy, the calculation model is extended to three dimensions, incorporating more factors that have a significant impact on the measurement, such as the slope angle along the survey lines and changes in water depth. In practical applications, the survey lines are set to run in a north-south direction, and a greedy algorithm is used to ensure complete coverage of the measurements. Taking into account the complex seafloor topography, the fitted lines are used to represent the seafloor slopes. The multi-objective programming model is transformed into an unconstrained single-objective programming problem for further optimization. By solving the problem using simulated annealing algorithm, the optimized solution provides specific values for the spacing between adjacent survey lines, total track length, missed coverage rate of the surveyed area, and duplicate measurement coverage, enabling the optimal utilization of multibeam bathymetry systems in oceanic surveys.

Research on Pricing and Replenishment Strategies for Vegetable Commodities Based on an Enhanced Nonlinear Objective Programming Model

With the increasing consumption of fresh food ingredients in the Chinese market and the perishable nature of vegetable commodities, which represent the largest proportion, optimizing procurement strategies plays a crucial role in enhancing the profitability of supermarkets. This has garnered significant attention from scholars and practitioners. Drawing on principles from microeconomics, this study employs multiple linear regression to describe the relationship between pricing and sales volume. Additionally, time series forecasting methods are utilized. By considering both supermarket profitability and social welfare maximization as optimization objectives, single-objective optimization and multi-objective nonlinear programming models are established. Various decision strategies for supermarkets under different scenarios are discussed, and a satisfactory decision is obtained by solving the models using differential evolution algorithm. Through research on the expected conditions and data provided by real supermarkets, this paper provides guidance for intelligent supermarket procurement strategies and supply chain management in practical settings.

Research on the Profit Maximization Model for Automatic Pricing and Restocking of Vegetable Commodities

Due to the perishable nature of fresh agricultural products and the uncertainty of the types and prices of individual items before daily procurement in supermarkets, these establishments typically need to determine daily replenishment and pricing strategies based on the historical sales and demand for various goods. This paper starts from the perspective of maximizing supermarket revenue, predicting the sales of various product categories through seasonal indices. Under the constraints of the sales space for goods, the paper further uses ARIMA time series forecasting for the sales volume and procurement prices of individual items during the corresponding time period. Cost markup equivalent coefficients, daily replenishment quantities, and discount coefficients are considered as decision variables. Based on the demand curve model in economics, the paper takes into account the counteraction of cost markup pricing coefficients on sales volume and the counteraction of discount coefficients on the discounted portion of sales. The objective functions include maximizing supermarket revenue and satisfying the market demand for various categories of vegetable products. The paper adopts a novel sales model of offering discounts on predicted perishable vegetables and establishes a multi-objective programming model, providing the optimal strategies for replenishment quantities and pricing for each item on July 1st.

Research On LSTM-Based Pricing Strategy for Vegetable Category Wholesale Price Prediction and Multi-Objective Planning

Fresh vegetable products constitute an indispensable facet of both daily life and the developmental trajectory of the food industry. Owing to the susceptibility to damage and spoilage, vegetable commodities frequently engender substantial financial losses for food retail establishments. This paper addresses the replenishment and pricing decision problem for vegetable commodities and proposes a solution based on goal programming and LSTM time series forecasting models. Firstly, historical data analysis reveals a linear or logarithmic correlation between vegetable sales volume and cost-plus pricing. Subsequently, an LSTM neural network model is employed to forecast wholesale vegetable prices for the upcoming week. Building upon the forecasted results, a multi-objective programming model is established with the objective of maximizing revenue, utilizing daily replenishment total and pricing as decision variables. The model takes into account the constrained relationship between sales volume and pricing, as well as the lower limit constraint on wholesale prices. Additionally, the paper solves the programming model using the least squares method to determine the optimal daily replenishment quantity and pricing for each vegetable category. Finally, the computational results validate the effectiveness of the proposed method in guiding replenishment and pricing decisions for supermarkets, achieving revenue maximization. This study provides a feasible quantitative decision-making solution for supermarkets facing complex replenishment and pricing issues.

A Multi-Objective Optimization Method for Maritime Search and Rescue Resource Allocation: An Application to the South China Sea

To effectively address the increase in maritime accidents and the challenges posed by the trend toward larger ships for maritime safety, it is crucial to rationally allocate the limited maritime search and rescue (MSAR) resources and enhance accident response capabilities. We present a comprehensive method for allocating MSAR resources, aiming to improve the overall efficiency of MSAR operations. First, we use long short-term memory to predict the number of future accidents and employ the K-medoids algorithm to identify the accident black spots in the studied area. Next, we analyze the multi-constraint conditions in the MSAR resource allocation process. A multi-objective integer programming model is constructed to minimize the response time and allocation cost. Finally, we use the non-dominated sorting genetic algorithm II (DNSGA-II) with Deb’s rules to solve the model, and we propose a multi-attribute decision optimization-based method for MSAR resource allocation. We found that the DNSGA-II exhibits better convergence and generates higher-quality solutions compared to the NSGA-II, particle swarm optimization (PSO), and enhanced particle swarm optimization (EPSO) algorithms. Compared with the existing MSAR resource emergency response system, the optimized scheme reduces the response time and allocation cost by 11.32% and 6.15%, respectively. The proposed method can offer decision makers new insights when formulating MSAR resource allocation plans.

Job rotations based on physical and psychological workloads: A proposal for the footwear industry

IntroductionThe aim of this study is to propose a model of job rotations in shoes (sandals and tennis shoes) production cells, aiming at reducing the risk exposure to work-related musculoskeletal disorders considering physical and psychological workloads. MethodsThe physical load was assessed by the Occupational Repetitive Actions method, and the psychological load by the Job Stress Scale. A total of 259 workers in the footwear industry participated in this study. These loads were incorporated into a multi-objective mathematical programming model, where the objective functions aimed to minimize the maximum work-related musculoskeletal disorders and the maximum total work demand, and to maximize the minimum work control and social support to which a worker might be exposed during a daily journey. To the best of our knowledge, this is the first time in the literature that physical and psychological aspects are considered when solving job rotation problems. To tackle the problem, we propose a matheuristic based on an epsilon-constraint framework. This method allows us to generate different solutions with respect to the specific objective functions. To assess the effectiveness of the solutions, we calculated the variance of the scores of the physical and psychological loads in the conditions without and with rotations. ResultsFive rotation solutions were reached for the sandal production cells and twelve for the tennis shoe production cells. There was a reduction in the variation of physical and psychological loads in all solutions for both cells when compared to the no rotation condition. However, for the sandal production cells, the reduction in variation between the physical loads of the conditions with and without rotations was not significant due to the similarity of the exposure indices. ConclusionsThe model reduces the risk of work-related musculoskeletal disorders, promoting variability in muscle recruitment and enrichment of work content, being applicable in other segments.

A Multi-Objective Mathematical Programming Model for Project-Scheduling Optimization Considering Customer Satisfaction in Construction Projects

The aim of this study was to develop a multi-objective mathematical programming model for the trade-off of time, cost, and quality in the project-scheduling problem (PSP) by taking priorities and resource constraints as well as activity preemption into account. First, a small-sized problem instance that was a sub-project of an oil and gas construction project was used for te validation of the proposed model and algorithm. Subsequently, considering the sensitivity, complexity, and importance of oil and gas projects, the proposed model was implemented in a large-sized oil and gas construction project. Considering the NP-hardness of this problem, the NSGA-II metaheuristic algorithm was used to deal with the time, cost, and quality trade-off problem. Finally, a sensitivity analysis was implemented on the three main parameters of time, cost, and quality to investigate the effects of changes on the results. The findings show that the proposed model is more sensitive to cost changes, so an increase in project costs leads to a drastic change in the values of other objective functions.

A novel and dynamic land use/cover change research framework based on an improved PLUS model and a fuzzy multiobjective programming model

Spatial reconstruction and scenario simulation of historical processes and future trends of land use/cover change (LUCC) can help to reveal the historical background of land conversion and the spatial distribution of future land. Moreover, there is a close relationship between the spatiotemporal dynamics of land use/cover and changes in different ecosystem services (ESs). Using this relationship to simulate future land use scenarios is important. In this study, an LUCC dynamic analysis framework (LSTM-PLUS-FMOP) was constructed based on a deep learning time series forecasting model (LSTM), a parallelized urban land use simulation (PLUS) model and a fuzzy multiobjective programming (FMOP) model. The PLUS model was used to analyze the driving mechanism of land expansion and explore the land conversion pattern. In addition, three land conversion scenarios were established: natural land expansion (NLE), economic development priority (EDP) and regional sustainable development (RSD). The FMOP model and the relationship between LUCC and ES were used to perform a spatial simulation of land conversion. The uncertainty parameters in the model were treated by intuitionistic fuzzy numbers (IFSs). This study applied the constructed framework to the Yellow River Basin of Shaanxi Province (YRB-SX). The results showed that (1) from 2000 to 2020, the cropland area of the YRB-SX continuously decreased by 12.67 × 104 ha, while the built-up area continuously increased by 28.25 × 104 ha. The net reduction in woodland and grassland area was 13.90 × 104 ha. (2) The relative error range of land prediction using the LSTM model was 0.0003– 0.0042. This model had better accuracy than the Markov chain prediction model. (3) The cropland area decreased by 0.26% (NLE), 0.85% (EDP) and 1.68% (RSD) under the three scenarios. The built-up area increased by 25.01%, 32.76% and 14.72%, respectively. The RSD scenario followed the principles of ecological protection and spatial constraints, which mitigated the degradation of the ecosystem to some extent. This coupled simulation framework will help to obtain land allocation schemes that meet the requirements of ecological protection and provide solutions for rational land management.

Multi-product disassembly line balancing optimization method for high disassembly profit and low energy consumption with noise pollution constraints

Remanufacturing attains much attention from both industrial and academic sectors due to its beneficial roles in energy conservation and environment protection, where disassembly is a crucial part. To reach the comprehensive sustainability and global optimization of disassembling multiple end-of-life products, this paper suggests a multi-objective multi-product disassembly line balancing problem with considering disassembly profit, energy consumption and noise pollution simultaneously. According to the natures of the problem under consideration, a multi-objective integer programming model is constructed. Its goals are to reach maximum disassembly profit and realize minimum energy consumption while observing noise pollution requirements and resource constraints. Accordingly, a multi-objective group teaching optimization algorithm is specially devised. In it, rank and crowding distance methods are employed to partition the population into two groups. Moreover, precedence preserving crossover and mutation methods are severally used on the two groups to realize the teacher phase. Furthermore, to achieve the student phase, an adaptive local search method is applied to refine solutions in an external archive, and thus its exploitation ability is enhanced. By executing contrast experiments between the devised approach and its powerful competitors on a set of real-world test instances, the experimental results validate that it has highly-adaptive and well-superior performance in tackling the problem of concern.

Solution of Z-number-based multi-objective linear programming models with different membership functions

In a universe full of fuzziness and uncertainty, it is an absolute blessing if some information is reliable to some degree. Considering the amount of uncertainty, Zadeh proposed the idea of Z-number, which carries both uncertainty and the reliability of the information. Uncertain information can be reliably conveyed using Z-numbers. On the other hand, one of the important and widely used decision-making techniques is linear programming. Currently, linear programming has been extensively explored using fuzzy information. Decisions made using linear programming with uncertain data are highly uncertain due to the lack of reliability of the information. An effective decision-making method is multi-objective linear programming (MOLP), in which the decision-maker attempts to draw conclusions from information containing conflicting attributes. In the MOLP problem, there may be different solution vectors corresponding to each objective function. One needs to find a compromise solution vector that satisfies each objective function to a certain extent based on the decision maker's suggested preferences. In this study, we use Z-number and MOLP to model the Z-number MOLP problem (ZMOLPP). Furthermore, we propose a strategy for solving ZMOLPP by transforming it into a crisp MOLP problem, and finally converting this crisp MOLP problem into a single-objective linear programming problem (LPP) using different shape functions. Additionally, we compare the results with existing MOLP problems in fuzzy settings to validate the proposed strategy.

Optimization of spatial utilization efficiency of captive Hu sheep based on multi-objective programming model

Optimization of spatial utilization efficiency of captive Hu sheep based on multi-objective programming model