Multi-objective Programming Model Research Articles

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 minimising 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.

Dual-layer optimization planning for offshore wind farms considering power system security

The vigorous development of offshore wind farms (OWFs) has further promoted the utilization of wind energy. Owing to the demand for low-carbon environmental protection, the increase in the scale of OWF constructions has also had a significant impact on the power system, which cannot be ignored. Specifically, the uncertainty and volatility of wind-power generation should be considered in power system optimization planning schemes. Therefore, this study proposes a probabilistic power-flow analysis index construction method that employs the static-security region theory to optimize the OWFs–grid-connection topology, including the number, location, and capacity of the public connection points connected to the onshore power grid. This method can be used to calculate the probability indicators of power grid insecurity for OWFs–grid-connected systems with different network topologies. Furthermore, a dual-layer multi-objective programming model that considers the economics and stability for integrated planning of electrical collector and multiterminal direct current transmission systems is established. The proposed model comprises two inner-layer models and one outer-layer model. The inner-layer models are used to construct the collection and transmission topologies of the OWFs–grid-connected system that minimize the probability of grid insecurity and the planning cost of the outer-layer model. Finally, the effectiveness of the proposed OWFs planning model was verified using an improved IEEE 24-bus system that can be used for topology planning of electrical collector and transmission systems while ensuring grid stability.

Optimization of Pickup Vehicle Scheduling for Steel Logistics Park with Mixed Storage

Pickup vehicle scheduling in steel logistics parks is an important problem for determining the outbound efficiency of steel products. In a steel logistics park, each yard contains different types of steel products, which provides flexible yard selection for each pickup operation. In this case, the yard allocation and the loading sequence for each vehicle must be considered simultaneously in pickup vehicle scheduling, which greatly increases the scheduling complexity. To overcome this challenge, in this paper, we propose a pickup vehicle scheduling problem with mixed steel storage (PVSP-MSS) to optimize the makespan of pickup vehicles and the makespan of steel logistics parks simultaneously. The optimization problem is formulated as a multi-objective mixed-integer linear programming model, and an enhanced algorithm based on SPEA2 (ESPEA) is proposed to solve the problem with a high efficiency. In the ESPEA, a cooperative initialization strategy is firstly proposed to initialize the vehicle pickup sequence for each yard. Then, an insertion decoding method is designed to improve the scheduling efficiency, utilizing the idle time of a yard. Furthermore, local search technology based on critical paths is proposed for the ESPEA to improve the solution quality. Experiments are executed based on data collected from a real steel logistics park. The results confirm that the ESPEA can significantly reduce both the makespan of each pickup vehicle and the makespan of the steel logistics park.

Designing a reliable-sustainable supply chain network: adaptive m-objective ε-constraint method

AbstractIn the current era emphasizing sustainability and circularity, supply chain network design is a critical challenge for making reliable decisions. The optimization of facility location-allocation inventory problems (FLAIPs) holds the key to achieving dependable product delivery with reduced costs and carbon emissions. Despite the importance of these challenges, a substantial research gap exists regarding economic, reliability, and sustainability criteria for FLAIPs. This paper aims to fill this gap by introducing a multi-objective mixed-integer linear programming model, focusing on configuring a reliable sustainable supply chain network. The model addresses three key objectives: minimizing costs, minimizing emissions, and maximizing reliability. A notable contribution of this research lies in elaborating on five levels of a supply chain network catering to the delivery of multiple products across various periods. Another novelty is the simultaneous incorporation of economic, environmental, and reliability objectives in the network design—a facet rarely addressed in prior research. Results highlight that varying demand levels for each facility lead to altered trade-offs between objectives, empowering practitioners to make diverse decisions in facility location allocation. The proposed mathematical model undergoes validation through numerical examples and sensitivity analysis of parameters. The paper concludes by presenting theoretical and managerial implications, contributing valuable insights to the field of sustainable supply chains.

A Sustainable Multi-Objective Model for Capacitated-Electric-Vehicle-Routing-Problem Considering Hard and Soft Time Windows as Well as Partial Recharging.

Due to the high pollution of the transportation sector, nowadays the role of electric vehicles has been noticed more and more by governments, organizations, and environmentally friendly people. On the other hand, the problem of electric vehicle routing (EVRP) has been widely studied in recent years. This paper deals with an extended version of EVRP, in which electric vehicles (EVs) deliver goods to customers. The limited battery capacity of EVs causes their operational domains to be less than those of gasoline vehicles. For this purpose, several charging stations are considered in this study for EVs. In addition, depending on the operational domain, a full charge may not be needed, which reduces the operation time. Therefore, partial recharging is also taken into account in the present research. This problem is formulated as a multi-objective integer linear programming model, whose objective functions include economic, environmental, and social aspects. Then, the preemptive fuzzy goal programming method (PFGP) is exploited as an exact method to solve small-sized problems. Also, two hybrid meta-heuristic algorithms inspired by nature, including MOSA, MOGWO, MOPSO, and NSGAII_TLBO, are utilized to solve large-sized problems. The results obtained from solving the numerous test problems demonstrate that the hybrid meta-heuristic algorithm can provide efficient solutions in terms of quality and non-dominated solutions in all test problems. In addition, the performance of the algorithms was compared in terms of four indexes: time, MID, MOCV, and HV. Moreover, statistical analysis is performed to investigate whether there is a significant difference between the performance of the algorithms. The results indicate that the MOSA algorithm performs better in terms of the time index. On the other hand, the NSGA-II-TLBO algorithm outperforms in terms of the MID, MOCV, and HV indexes.

Social transfer policies and child poverty in European countries: Evaluating policy alternatives through a multiobjective programming model

Child poverty remains a significant concern across Europe and citizens regard policy interventions that effectively address this issue to be of utmost importance. In this work, we analyse the potential of social transfer policy alternatives to alleviate child and overall poverty simultaneously in European countries. Using a logistic regression to define the objective functions and constraints, we rely on multiobjective programming to assess key features of national social transfer systems that can minimize child and overall poverty and tend to equal both levels downwards. Taking into account the starting levels of each country, the results underline the importance of finding an adequate combination of pro-child and pro-poor targeting of social transfer policies. For most European countries, an increase in the level of cash benefits for children is advisable – particularly universal benefits – except in Croatia, the Netherlands, and Norway. Nonetheless, in France, Greece, Italy, Ireland, the Netherlands, and Spain, the recommended proportion of GDP allocated to universal cash benefits is lower than the proportion allocated to means-tested benefits. In this regard, our research provides an innovative approach for policymakers to optimize poverty reduction strategies by contributing valuable insights for the design and implementation of effective social transfer policies to combat child and overall poverty in European countries.

Automated pricing and replenishment decisions for vegetable products based on evaluation optimization models

Based on the commodity information of the supermarket in the annex, the detailed data of historical sales flow, the wholesale price of vegetable commodities and the recent loss rate of vegetable commodities, and through the data analysis of each category and each single product, the automatic pricing and replenishment decision-making model of commodities is established. Use the optimization evaluation algorithm to formulate the total daily replenishment and pricing strategy of each category and each single product. In order to solve the first problem, firstly, the outliers in the original data of Annexes 2 and 3 are cleaned, normalized, feature selected and dimensionally reduced. Secondly, a quarter is taken as a sales cycle of supermarkets, so as to find the proportion of sales volume of a certain category in the same quarter of three years to the total sales volume, and give the distribution law of sales volume of different categories, the results are shown. Considering different periods again, the daily sales volume distribution law is calculated by taking one day as a sales cycle, and the results are shown. Finally, the Pearson grade correlation coefficient is used to judge the relationship between the processing indicators, and the matrix heat map is obtained. According to the two results, it was concluded that there was a significant positive correlation between the sales volume of mosaic and cauliflower vegetables, and a significant negative correlation between the sales volume of nightshade and aquatic root vegetables. In view of the second problem, firstly, considering the functional relationship between the total sales volume and the cost pricing, the correlation analysis and linear fitting were carried out to obtain the linear relationship between the sales price of each category and the maximum value of the sales volume of each category in July of the previous year can be described as Through further nonlinear fitting and optimization problem solving, the total daily replenishment volume and pricing strategy of each vegetable category in the coming week (July 1-7, 2023) are shown in Table 1 and Table 2, which makes the supermarket have the largest revenue In response to the third question, based on the known data, we can analyze the data requirements for each data: we need to know the sales volume of various vegetables during this period, we need to determine the purchase cost of each vegetable, we need to understand the past pricing strategy and response, and we need to know the inventory of various vegetables on June 30. On this basis, a multi-objective dynamic programming model is established, and the total number of saleable items is 30 by using the greedy algorithm to obtain the replenishment quantity of single items on July 1, and the pricing strategy is further solved by using the linear equation fitted in problem 2. In response to the fourth problem , on the basis of the existing sales, wholesale price and loss rate data, in order to better formulate the replenishment and pricing decisions of vegetable products, supermarkets also need to consider and collect the following 12 aspects of relevant data to assist in planning the pricing and replenishment decisions of vegetable products, such as: customer preference and satisfaction survey, seasonality and availability of vegetables, competitor information, inventory costs and storage conditions, historical sales data and trend analysis, customer flow and purchase period, nutritional value and health benefits of vegetables, Socio-economic factors, external environmental factors, policy and regulatory factors, technological and innovation factors, and supply chain and logistics information to ensure more comprehensive and accurate decision-making. Among them, the analysis of historical sales data and trends is mainly carried out.

Supporting tactical harvest planning decisions of major fruits through a multi-objective modeling approach by using exact methods

An important stage of the agri-food supply chain is production (agricultural practices and harvesting), where relevant decisions are made in a short period of intensive activity. Regarding the major fruit supply chain, the complexity of these decisions increases because most fruit export companies manage several orchards with different fruit species and varieties. Moreover, the amount of labor and resources must be estimated at least two months in advance of the harvest so that they can be relied upon during the harvest season. In this study, a multi-objective modeling approach for supporting tactical harvest planning decisions of major fruits is developed, aiming to evidence its adequacy and to identify different harvest plans according to the prioritized objectives. Therefore, a multi-objective linear programming (MOLP) model, which considers three conflicting objectives, is proposed. The three objectives correspond to minimizing harvest costs, fruit loss and harvest days. For solving the MOLP model, five a posteriori multi-objective exact methods are compared using a synthetic instance based on a real case. The Pareto frontiers obtained by these methods are analyzed using three metrics related to accuracy, cardinality, and diversity criteria, so that the most suitable method can be selected. Thus, the selected method is applied to an actual case study of a Chilean fruit export company. In addition, the ε-constraint method is used in this case study to compare the obtention of non-dominated solutions in the convex and non-convex regions of the Pareto frontier. The results show that harvest plans vary significantly according to the prioritization of the objective functions. Thus, the percentage difference between each objective function’s worst and best values is around 497% when the harvest costs are minimized, around 1,448% when the fruit loss is minimized, and around 18% when the number of harvest days is minimized. In addition, the adequacy of the multi-objective approach rather than the mono-objective approach to support tactical harvest planning decisions is demonstrated. Furthermore, a scenario analysis for considering changes in the harvest periods is carried out by considering three possible scenarios. This analysis is required because these periods can change every year due to different factors, such as weather conditions and agricultural practices, among others. The results show that the harvest costs and fruit loss increase, on average, 87% and 100%, respectively, when the harvest period is short, that is, one day. Finally, managerial insights about the information obtained by the MOLP model are discussed.

A Socio-Economic and Quality-Oriented Optimal Fruit Supply Chain Network Design in a Multi-Market Multi-Product Environment: A Real Case Study

This paper focuses on developing an optimal sustainable fruit Closed-Loop Supply Chain (CLSC). A novel multi-objective Mixed-Integer Linear Programming (MILP) model is suggested to formulate a multi-product, multi-market, multi-period, multi-level, multi-mode selling problem which aims to minimize cost and emissions and maximize the whole social segments’ responsiveness to fruit, vinegar, and compost. The model is solved by the LP-metric method and implemented in a real case in Iran. The findings indicate that achieving the highest level of responsiveness comes with a substantial cost for the network. However, this does not apply to environmental considerations. For decision-makers prioritizing sustainability, the proposed decision support system strongly advises against buying fruit from fruit cars. Due to the absence of positive impacts on the level of responsiveness, buying from fruit cars results in increased costs and emissions. For a cost- and environmentally-sensitive decision-maker, it is recommended to provide higher-quality fruits. Because the lower the quality of the fruit, leads to increased waste rates and emissions. Meeting increased vinegar demand, with its elevated costs and emissions, becomes justifiable only when the policymaker is actively seeking to achieve maximum responsiveness.

Optimal sizing of behind-the-meter battery energy storage systems under optimal battery operation: A case study in Ireland

The global shift toward a sustainable future has led to new policies to promote the use of renewable energy resources (RESs) in residential buildings. In Ireland, the number of installed roof-top photovoltaics (PVs) and home batteries, known as behind-the-meter (BTM) resources, has significantly increased in recent years, accelerated by grants administered by the Sustainable Energy Authority of Ireland (SEAI). In this work, an optimization tool that uses a combination of a Genetic Algorithm (GA) and a multi-objective mixed-integer linear programming (MOMILP) model is developed. This tool integrates an optimal battery control scheme from the MOMILP model aiming to minimize electricity bills and battery degradation to optimize the sizing of BTM batteries with minimum installation costs. Various scenarios have been simulated to assess the potential of the proposed model under different metering and billing systems, and optimization constraints. Compared to a case with a rule-based optimization model, results show that our optimization model can save 5.3 % more on annual bill costs while keeping battery degradation to a minimum of 2.1 % per year.

Evaluation of disassembly line layouts using an integrated fermatean fuzzy decision-making methodology: An application for refrigerator disassembly line

The layout focusing on the physical arrangement of workstations is an important consideration when designing a disassembly line. The layout of a line has a significant impact on its efficiency. The new problem addressed within this study is evaluating the main disassembly line layout types with respect to a thorough hierarchy of evaluation criteria classifying quantitative and qualitative. The addressed problem is modeled as multi-criteria decision-making (MCDM). To this end, a novel methodology that combines Stepwise Weight Assessment Ratio Analysis (SWARA) and Weighted Aggregated Sum Product Assessment (WASPAS) methods under the Fermatean fuzzy environment is proposed for sustainable waste management. Fermatean fuzzy SWARA (FF-SWARA) is used to find out the weight of criteria consisting of disassembly line balancing characteristics, reconfigurability, working conditions, and end-of-life product characteristics. Fermatean fuzzy WASPAS (FF-WASPAS) is used to rank the alternative disassembly line layouts consisting of straight, U-shaped, two-sided, and parallel lines considering the weight of criteria. The multi-objective mixed integer linear programming (MILP) model for the disassembly line balancing problem (DLBP) of each line layout type is integrated into the proposed methodology. This integration reflects the effect of balancing performance measurements such as the number of stations, removing hazardous parts early, removing high demand parts early, and line lead time. Reconfigurability principles, working conditions, and product characteristics that rely on expert opinions are considered under the qualitative criteria. The developed methodology was applied to a refrigerator disassembly line and verified by sensitivity analysis and comparative studies. The results show that the two-sided disassembly line layout is the best option.

Fuzzy-random robust flexible programming on sustainable closed-loop renewable energy supply chain

Photovoltaic (PV) systems have become a widely accepted method of harnessing solar energy in recent decades. In addition to the growing use of renewable energies are being used due to the depletion of fossil fuel supplies and the corresponding pollution generated by these non-renewable fuels. Since PV systems possess a limited lifespan, and are predicted to eventually become outdated, sustainability and recycling that should be incorporated into the structure of the solar system supply chain. The management of high and low moisture waste (H&LMW) is crucial due to its harmful environmental impacts, including sludge waste, carbon dioxide (CO2), and acid compounds. This study explores the integration of research on H&LMW and PV systems, focusing on energy generation from biogas, incineration power plants, and PV systems. In this respect, we propose a multi-objective mixed-integer programming model to construct a sustainable closed-loop renewable energy supply chain by considering forward and reverse flow through solar, biogas, and incineration power plants. Reducing CO2 with power-to-gas technology is thought to be an environmentally friendly approach. A novel fuzzy-random robust flexible programming approach based on Me measure is proposed, which overcomes the limitations of addressing the uncertainty of the parameters. Next, in order to solve the suggested multi-objective model, a novel strategy known as utility function based multi-choice conic goal programming is presented. A case study explores the sustainable challenges of a closed-loop renewable energy supply network in India’s renewable energy sector. The experimental results indicate that 9 candidate locations are selected for solar wafer production facilities, while 10, 9, and 15 locations are chosen for solar cell production, module manufacturing, and solar power plants, respectively. A sensitivity analysis reveals that the proportion of air pollution limitations possessively rises from 100% to 80% corresponding to the aggregate cost increases from INR 8021452 to INR 8056542.

A Stochastic Goal-Based Economic Order Quantity Model for Dairy Products

Resilient efforts are still sought to optimize the inventory of dairy products under demand uncertainty. Several industries operate under uncertainties that severely affect company performance. In this paper, a multi-objective goal programming model is proposed to optimize the economic order quantity of dairy products under stochastic demand. Adopting a Markov chain approach, the model initially defines the demand transition matrix, inventory cost matrix, objective function, priorities, and goal constraints of dairy products. The model seeks to minimize the deviation variables from the targeted economic order quantity and total inventory costs of dairy products over a finite period planning horizon. The sum of deviations is minimized so that the actual economic order quantity and inventory costs meet the targeted levels of dairy products. Using the simplex method, an analytical solution is obtained and a numerical example is presented for illustration; demonstrating the overachievement or underachievement of the targeted economic order quantity and inventory costs of dairy products. Results indicate how priority-based goal programming solution is sensitive to the highest priority to be achieved. The proposed model can be effective where the relevant economic order quantity and inventory costs can be prioritized if necessary.

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.