Karush-Kuhn-Tucker Research Articles

Optimizing Distribution Transformer Ratings in the Presence of Electric Vehicle Charging and Renewable Energy Integration

Aims and Background: A re-evaluation of distribution transformer ratings is necessary to ensure efficient and reliable operation due to the significant impact of the growing number of electric vehicles (EVs) on load dynamics. The objective of this study is to optimize the rating of the distribution transformers to accommodate the year-round demand for electric vehicle charging while minimizing expenses and no-load losses. The performance and lifespan of transformers under dynamic hourly loads are evaluated by analysing real-world EV charging data, load profiles, and transformer parameters. Using state-of-the-art simulation, real-world data analysis, and optimization algorithms, this study optimizes the transformer distribution ratings under the integration of EV charging and renewable energy. Objectives and Methodology: Dynamic hourly load changes can be captured by analysing realworld electric vehicle charging data in conjunction with seasonal load profiles. In order to maximise transformer lifetime and minimise losses, convex optimisation is subjected to Karush-Kuhn- Tucker (KKT) conditions. Transformer performance is also assessed by using energy storage devices and solar energy simulations. To determine the impact of various electric vehicle charging scenarios on thermal stress and transformer ageing, sensitivity analyses are performed. Results and Discussion: Transformers with a 10% higher rating can manage maximum electric vehicle charging loads with a 15% slower loss of life acceleration, according to our models. Transformer performance is further optimised with the integration of solar energy and energy storage systems, which improves load control and reduces operational costs by up to 20%. Conclusion: Using a convex optimisation framework and the Karush-Kuhn-Tucker (KKT) criteria, the study achieves a 95% accuracy rate in predicting hourly load fluctuations.

AoI-Minimal Task Assignment and Trajectory Optimization in Multi-UAV-Assisted Wireless Powered IoT Networks

This paper investigates the energy transfer and data collection problem of multiple unmanned aerial vehicle (UAV)-assisted wireless-powered Internet of Things (IoT) networks. To ensure information freshness for IoT devices and reduce UAVs’ energy consumption, we minimize the average Age of Information (AoI) of IoT devices by jointly optimizing the energy harvesting (EH) and data collection time for IoT devices, the selection of data collection points (DCPs), DCP-IoT associations, and task assignment, flight speed, and trajectories of UAVs, subject to the limited endurance of UAVs. As this problem is nonconvex, we propose a novel DCP association and trajectory-planning scheme that seeks age-optimal solutions through an iterative three-step process. First, we calculate the EH and data collection time for IoT devices using Karush–Kuhn–Tucker (KKT) conditions. Then, we introduce an optimal hovering time allocation-based affinity propagation (OHTAP) clustering algorithm to determine optimal DCP locations and establish DCP-IoT associations. Finally, we develop two algorithms to optimize UAVs’ trajectories: an improved partheno-genetic algorithm with enhancement mechanisms (EIPGA) and a hybrid algorithm that combines improved MinMax k-means clustering with EIPGA. Numerical results confirm that our scheme consistently outperforms benchmark schemes in AoI performance and solution stability across diverse scenarios.

Optimal operation and offering strategy for profit‐seeking of a coal‐mine energy prosumer in a liberalised market

AbstractCoal mines consume electricity from grids while making use of various derived energy resources to generate heat and power. This allows coal mines to act as energy prosumers. This paper presents an integrated optimisation model of optimal operation and offers strategies for profit‐seeking of a coal‐mine energy prosumer in a liberalised market. The discussed problem is formulated as a tri‐stage bi‐level programming model. In the upper level, the coal mine‐integrated energy system (CMIES) operator decides the retail contract management, equipment operation plan, and retail electricity price offered to end‐users through three stages to maximise its expected profits at a predefined risk level. In the lower level, the end‐users react to the price bids offered by the CMIES operator under study and other market retailers to minimise their costs for energy procurement. Unlike previous approaches, the endogenous uncertainties associated with market subjects' strategic behaviours are explicitly considered. To solve the proposed problem efficiently, the Karush–Kuhn–Tucker conditions and multicut benders decomposition method are employed. The simulation results show that the collaborative optimisation of the energy cycle and production scheduling can reduce daily operating costs by nearly $3576 and carbon emissions by 6.72 tons, which verifies the effectiveness of the proposed method.

Day-Ahead Two-Stage Bidding Strategy for Multi-Photovoltaic Storage Charging Stations Based on Bidding Space

Against the backdrop of a “dual-carbon” strategy, the use of photovoltaic storage charging stations (PSCSs), as an effective way to aggregate and manage electric vehicles, new energy sources, and energy storage, will be an important primary component of the electricity market. The operational characteristics of the aggregated resources within a PSCS determine its bidding space, which has an important influence on its bidding strategy. In this paper, a novel bidding space model is constructed for PSCSs, which dynamically integrates electric vehicles, photovoltaic generation, and energy storage. A two-stage bidding strategy for multiple PSCSs is established, with stage I aiming at achieving the lowest cost for the power purchased by a PSCS to optimize the power generation and power plan and stage II aiming at achieving the lowest cost of the grid operator’s power purchase to optimize the system’s power balance. Thirdly, the two-stage model is transformed into a single-layer, mixed-integer linear programming problem using dyadic theory and Karush–Kuhn–Tucker (KKT) conditions, enabling the derivation of the optimal bidding strategy. Finally, the example analysis verifies that the proposed model can achieve a reduction in the PSCS’s day-ahead power purchase cost and flexibly dispatch each resource within the PSCS to maximize revenue, as well as reducing power consumption behavior during peak tariff hours, to enhance the market power of the PSCS in the electricity market.

A non-interior-point continuation method for the optimal control problem with equilibrium constraints

This study presents a numerical method for the optimal control problem with equilibrium constraints (OCPEC). It is extremely difficult to solve OCPEC owing to the absence of constraint regularity and strictly feasible interior points. To solve OCPEC efficiently, we first relax the discretized OCPEC to recover the constraint regularity and then map its Karush–Kuhn–Tucker (KKT) conditions into a parameterized system of equations. Subsequently, we solve the parameterized system using a novel two-stage solution method called the non-interior-point continuation method. In the first stage, a non-interior-point method is employed to find an initial solution, which solves the parameterized system using Newton’s method and globalizes convergence using a dedicated merit function. In the second stage, a predictor–corrector continuation method is utilized to track the solution trajectory as a function of the parameter, starting at the initial solution. The proposed method regularizes the KKT matrix and does not enforce iterates to remain in the feasible interior, which mitigates the numerical difficulties in solving OCPEC. Convergence properties are analyzed under certain assumptions. Numerical experiments demonstrate that the proposed method can solve OCPEC while demanding remarkably less computation time than the interior-point method.

Assessment of Stakeholder Benefits from Participating in Community-Shared Solar Photovoltaics Through Monthly Renting and Load Management in South Korea

Various studies have explored community-shared solar (CSS) initiatives to help lower energy costs and increase the use of renewable energy sources. Various forms of CSS have been developed worldwide, specifically adapted to meet local economic and environmental conditions as well as technological readiness. This study proposes a variant of CSS that incorporates monthly photovoltaic (PV) rental options and load management functions for households in South Korea, a country characterized by limited land availability, high population density, and extremely high land-use costs. This study evaluates the feasibility of the proposed CSS by assessing the economic benefits for all stakeholders involved, including households, the CSS business (or government), and the grid service provider. It utilizes a mathematical programming model for its formulation and employs an iterative algorithm based on Karush–Kuhn–Tucker conditions for solving it. Additionally, a numerical assessment is conducted with 400 customers classified into three different categories of energy usage. The findings indicate that participating households experienced a reduction in electricity costs ranging from 36.8% to 56.7%, depending on the season and specific scenarios. The CSS business also realized significant profits while the grid service provider benefited from reduced fluctuations in power supply, leading to improved efficiency in grid operations and maintenance.

Unveiling coopetition dynamics between shared mobility and public transport: A game-theoretic approach

The liberalization of the transport market and advancements in real-time information technologies have prospered various shared mobility services, such as ridesourcing and carsharing. The emergence of these services complicates the relationships between them and public transport, as they often compete and cooperate simultaneously. This study develops a game-theoretic model to unveil these interactions using a multi-leader single-follower framework. In this framework, operators set their service rates as leaders, while travelers are assigned to services based on a logit model, which influences the profitability of both operators. The public transport operator may also subsidize travelers who use shared mobility service to access first- or last-mile trips, referring to as the bundle services. We reformulate the resulting nonlinear, nonconvex problem into a standard convex bilevel model by using outer linear approximations and applying KKT conditions to replace the lower-level problem. An iterative algorithm is developed to solve the game-theoretical model, complemented by an optimization-based bound tightening technique to enhance solution efficiency and accuracy. Our findings show that smaller operators, limited by budget constraints, are more likely to cooperate in bundle services for longer distances but tend to compete for shorter distances. In contrast, larger operators strategically alternate between competition and cooperation based on market conditions. Furthermore, well-designed subsidies in the bundle services can incentivize cooperation between shared mobility and public transport, benefiting both operators and travelers.

Two-Stage, Three-Layer Stochastic Robust Model and Solution for Multi-Energy Access System Based on Hybrid Game Theory

This paper proposes a two-stage, three-layer stochastic robust model and its solution method for a multi-energy access system (MEAS) considering different weather scenarios which are described through scenario probabilities and output uncertainties. In the first stage, based on the principle of the master–slave game, the master–slave relationship between the grid dispatch department (GDD) and the MEAS is constructed and the master–slave game transaction mechanism is analyzed. The GDD establishes a stochastic pricing model that takes into account the uncertainty of wind power scenario probabilities. In the second stage, considering the impacts of wind power and photovoltaic scenario probability uncertainties and output uncertainties, a max–max–min three-layer structured stochastic robust model for the MEAS is established and its cooperation model is constructed based on the Nash bargaining principle. A variable alternating iteration algorithm combining Karush–Kuhn–Tucker conditions (KKT) is proposed to solve the stochastic robust model of the MEAS. The alternating direction method of multipliers (ADMM) is used to solve the cooperation model of the MEAS and a particle swarm algorithm (PSO) is employed to solve the non-convex two-stage model. Finally, the effectiveness of the proposed model and method is verified through case studies.

Optimal Operation of Generation Company’s Participating in Multiple Markets with Allocation and Exchange of Energy-Consuming Rights and Carbon Credits

The proposal of the energy-consuming right (ECR) market may lead to generation companies (GenCos) facing the risk of being overcharged due to the inaccurate calculation of carbon emission reduction, since it claims the same credit as the carbon market does. To estimate the carbon emission reduction accurately for the GenCos that participate in electricity, carbon, and ECR markets simultaneously, this paper proposes a market framework where a flexible exchange mechanism between the ECR and carbon markets is specially considered. To investigate the influence of the allocation and exchange of ECR and carbon credits on the behavior of GenCos that participate in multi-type markets, a bi-level model based on the leader–follower game theory is proposed. In the upper level of the proposed model, a decision problem for maximizing the profit of GenCos is developed, which is especially constrained to the primary allocation of ECR and carbon credits. While the multi-type market clearing model and an exchange mechanism between the ECR and carbon credits are proposed in the lower level of the model. The bi-level problem is converted into the mathematical program with equilibrium constraints (MPECs) through the Karush–Kuhn–Tucker (KKT) condition to solve. The results illustrate that the interaction between the ECR market and the carbon market can improve the energy efficiency and reduce the carbon emissions of GenCos.

A Tri-Level Transaction Method for Microgrid Clusters Considering Uncertainties and Dynamic Hydrogen Prices

The advancement of hydrogen technology and rising environmental concerns have shifted research toward renewable energy for green hydrogen production. This study introduces a novel tri-level transaction methodology for microgrid clusters, addressing uncertainties and price fluctuations in hydrogen. We establish a comprehensive microgrid topology with distributed power generation and hydrogen production facilities. A polygonal uncertainty set method quantifies wind and solar energy uncertainties, while an enhanced interval optimization technique refines the model. We integrate a sophisticated demand response model for hydrogen loading, capturing users’ behavior in response to price changes, thereby improving renewable energy utilization and supporting economically viable management practices. Additionally, we propose a tri-level game-theoretic framework for analyzing stakeholder interactions in microgrid clusters, incorporating supply–demand dynamics and a master–slave structure for microgrids and users. A distributed algorithm, “KKT & supply-demand ratio”, solves large-scale optimization problems by integrating Karush–Kuhn–Tucker conditions with a heuristic approach. Our simulations validate the methodology, demonstrating that accounting for uncertainties and dynamic hydrogen prices enhances renewable energy use and economic efficiency, optimizing social welfare for operators and economic benefits for microgrids and users.

Low-Carbon Optimization Scheduling of Integrated Energy Systems Based on Bilateral Demand Response and Two-Level Stackelberg Game

In the context of low-carbon energy transformation, fully utilizing the integrated demand response (IDR) resources on the load side can improve the operational flexibility and economy of the integrated energy system (IES). However, establishing a reasonable trading mechanism to enhance users’ participation in IDR has become a key issue that IES urgently needs to solve. To this end, this paper first establishes an IES model that includes electricity, heat, and gas. To reduce carbon emissions, a ladder-type carbon trading mechanism is introduced while adding low-carbon technologies such as carbon capture devices and power-to-gas conversion. Secondly, a bilateral IDR mechanism centered on the load aggregator (LA) is proposed, and a multi-agent operation model including IES, LA, and users is established. The IDR subsidy price is dynamically determined through a two-level Stackelberg game model involving IES, LA, and users. Then, through KKT conditions and the Big M method, the two-level game model is turned into an IES-LA game model, which is solved using a combination of the White Shark Optimization method and the Gurobi solver. The final simulation results show that the scheduling model can fully reflect the time value of IDR resources, and the IES cost is decreased by USD 152.22, while LA and user benefits are increased by USD 54.61 and USD 31.85. Meanwhile, the ladder-type carbon trading mechanism and low-carbon technology have effectively achieved low-carbon operation of the system.

Chance-constrained bi-level optimal scheduling model for distribution network with thermal controllable load aggregators

An increasing amount of distributed renewable energy is being integrated into distribution networks to achieve decarbonization. It is essential to exploit demand-side flexible resources further to enhance system flexibility in response to the intermittency and unpredictability of renewable energy sources. This paper introduces a polytope-based aggregation method for thermostatically controlled loads (TCLs), aggregating numerous individual TCLs into a unified virtual battery via the aggregator (AGG). This approach avoids the dimensionality curse faced by the distribution system operator when directly controlling each TCL, while efficiently utilizing TCL flexibility. Subsequently, a bi-level optimization model is established, where AGGs are treated as independent stakeholders participating in the distribution network scheduling optimization through the local energy market. This model incorporates chance constraints to address the uncertainty of renewable energy sources. Finally, the distributionally robust chance constraint (DRCC) method is used to convert chance constraints into a linear form, and strong duality theory and Karush–Kuhn–Tucker conditions are applied to transform the bi-level model into a single-level model with equilibrium solutions. Case studies on the IEEE 33-bus network demonstrate that the proposed polytope-based aggregation method substantially improves computational efficiency with minimal optimality loss. Additionally, the DRCC method offers superior economic performance compared to robust and deterministic optimization approaches, while maintaining robustness.

The Sharing Energy Storage Mechanism for Demand Side Energy Communities

Energy storage (ES) units are vital for the reliable and economical operation of the power system with a high penetration of renewable distributed generators (DGs). Due to ES’s high investment costs and long payback period, energy management with shared ESs becomes a suitable choice for the demand side. This work investigates the sharing mechanism of ES units for low-voltage (LV) energy prosumer (EP) communities, in which energy interactions of multiple styles among the EPs are enabled, and the aggregated ES dispatch center (AESDC) is established as a special energy service provider to facilitate the scheduling and marketing mechanism. A shared ES operation framework considering multiple EP communities is established, in which both the energy scheduling and cost allocation methods are studied. Then a shared ES model and energy marketing scheme for multiple communities based on the leader–follower game is proposed. The Karush–Kuhn–Tucker (KKT) condition is used to transform the double-layer model into a single-layer model, and then the large M method and PSO-HS algorithm are used to solve it, which improves convergence features in both speed and performance. On this basis, a cost allocation strategy based on the Owen value method is proposed to resolve the issues of benefit distribution fairness and user privacy under current situations. A case study simulation is carried out, and the results show that, with the ES scheduling strategy shared by multiple renewable communities in the leader–follower game, the energy cost is reduced significantly, and all communities acquire benefits from shared ES operators and aggregated ES dispatch centers, which verifies the advantageous and economical features of the proposed framework and strategy. With the cost allocation strategy based on the Owen value method, the distribution results are rational and equitable both for the groups and individuals among the multiple EP communities. Comparing it with other algorithms, the presented PSO-HS algorithm demonstrates better features in computing speed and convergence. Therefore, the proposed mechanism can be implemented in multiple scenarios on the demand side.

A Hybrid Fuzzy Mathematical Programming Approach for Manufacturing Inventory Models with Partial Trade Credit Policy and Reliability

This study introduces an inventory model for manufacturing that prioritizes product quality and cost efficiency. Utilizing fuzzy logic and mathematical programming, the model integrates fuzzy numbers to describe uncertainties associated with manufacturing costs and quality control parameters. The model extends beyond conventional inventory systems by incorporating a dynamic mechanism to halt production, employing fuzzy decision variables to optimize the economic order quantity and minimize total costs. Key innovations include the application of approaches related to graded mean integration for defuzzification and the use of Kuhn–Tucker conditions to ensure optimal solutions under complex constraints. These approaches facilitate the precise management of production rates, inventory levels, and cost factors, which are essential in achieving a balance between supply and demand. A computational analysis validates the model’s effectiveness, demonstrating cost reductions while maintaining optimal inventory levels. This underscores the potential of integrating fuzzy arithmetic with traditional optimization techniques to enhance decision making in inventory management. The model’s adaptability and accuracy indicate its broad applicability across various sectors facing similar challenges, offering a valuable tool for operational managers and decision makers to improve efficiency and reduce waste in production cycles.

An LQR‐Lagrange Algorithm Generated by Interdisciplinary Integration With Optimal Control and Optimization

ABSTRACTInterdisciplinary integration is a superior method to improve the optimization algorithm. In this paper, control theory and optimization are combined, and the optimization algorithm is regarded as a control process. Based on the premise of optimal control, the state equation corresponding to Lagrange Algorithm is established with the Karush–Kuhn–Tucker (KKT) conditions as the objective. As an optimal control method, linear quadratic regulator (LQR) is utilized to control the calculation process, and an innovative LQR‐Lagrange Algorithm is proposed. The Lyapunov stability criterion is applied to analyze the convergence, and it is proved that the proposed LQR‐Lagrange Algorithm is bound to converge as long as the parameter matrices and are positive definite. The analysis indicates that the influence of parameters in LQR‐Lagrange Algorithm on the calculation speed is monotonic, and the elements in and has no effect on the convergence. Therefore, the proposed algorithm has a monotonic and user‐friendly parameter tuning strategy. It perfectly tackles the game between parameter tuning strategy and calculation speed, and cracks the difficulties and dilemmas of conventional algorithms in this issue, realizing a win‐win situation.

Transactive energy management for CIES considering tri-level hybrid game-theoretic approaches with multiple weight allocation factors

In city integrated energy systems, to reduce the costs of energy production and usage, it is extremely important for different entities to design proper energy management approaches since they have diverse behaviours and benefits. The hierarchy of the system is naturally divided into three layers, and hybrid game theories are employed on the basis of the vertical master–slave and horizontal equal cooperation characteristics among energy providers, aggregators, and users. First, to reflect the interactions between different levels' entities, Stackelberg game theory is adopted vertically, and the model complexity is lessened by the KKT conditions. Then, for energy providers, the asymmetric bargaining game is improved by combining multiple weight allocation factors, which can help reduce renewable energy fluctuations and encourage the consumption of clean energy to reduce the carbon emissions of the system. Also, the results show that the proposed method can enhance the utility of renewable energy generation units by approximately 64 % to 69 %. Finally, the optimality and effectiveness of the proposed game theoretic approaches are verified in case studies and in strict mathematical proofs. In addition, the impact mechanisms on the demand response from the perspectives of the user load and price sensitivity are analysed. To explore pricing methods that are more suitable for user aggregators, two pricing strategies are designed for the user aggregator and compared in terms of computation time and revenue. The results show that a unified pricing strategy can increase the aggregator's revenue by 1.24 % and reduce the computation time by 24.47 %.

Supply chain network equilibrium with outsourcing for fresh agricultural products under stochastic demands

Abstract Accepted by: M. Zied Babai This paper examines a supply chain network focusing on the distribution of fresh agricultural products, incorporating outsourcing and stochastic demands considerations. Initially, we establish a generalized Nash equilibrium model with stochastic demands among fresh produce firms. Subsequently, we transform this model into a mixed complementarity system using the Karush–Kuhn–Tucker conditions. Utilizing the Fischer–Burmeister function, we further convert the mixed complementarity system into a set of nonlinear equations, amenable to solution via GAMS software. We conduct numerical experiments and perform sensitivity analysis on key parameters. Our findings suggest that higher subsidy rates incentivize firms towards production outsourcing, particularly benefiting low-income farmers, thereby potentially increasing profits. Moreover, market fluctuations play a pivotal role in ensuring the stability and profitability of firms, with moderate fluctuations presenting opportunities for fresh enterprises to adapt, innovate and capitalize on evolving market conditions. These results offer valuable insights for effective management of fresh produce supply chains.

Resilient market bidding strategy for Mobile energy storage system considering transfer uncertainty

The participation of Mobile Energy Storage Systems (MESS) in the electricity market can not only increase its own profit but also alleviate power transmission congestion and increase market clearing balance. However, relevant market trading strategies have yet to be explored. Accordingly, this paper proposes a resilient market bidding strategy for MESS considering the operation of transportation network. Firstly, this paper proposes a joint optimization framework of energy and transportation systems. In this framework, the upper layer is to make decisions on the space-time distribution and bidding strategy of MESS, and the lower layers is designed for the equilibrium of the transportation network and electricity market clearing. Subsequently, for the newly introduced transportation layer, this paper proposes a Logit-type Robust Stochastic User Equilibrium (LRSUE) to calculate the uncertain transfer time of MESS in the transportation network. Further, based on strong duality theory and Karush-Kuhn-Tucker Conditions (KKT) conditions, the two-layer model can achieve an efficient solution. Finally, two cases were provided to validate the proposed methods, demonstrating that MESS increase its revenue, alleviate power transmission congestion, and increase the electricity market clearing balance.

Electrification of transportation: A hybrid Benders/SDDP algorithm for optimal charging station trading

This paper examines the electrification of transportation as a response to environmental challenges caused by fossil fuels, exploring the potential of battery electric vehicles and hydrogen fuel cell vehicles as alternative solutions. However, a significant barrier to their widespread adoption is the limited availability of charging infrastructure. Therefore, this study proposes the development of comprehensive charging stations capable of accommodating both battery and hydrogen vehicles to address this challenge. The energy is purchased from the day-ahead and intraday auction-based electricity markets, where the electricity price is subject to uncertainty. Therefore, a two-stage stochastic programming model is formulated while the price scenarios are generated utilizing a k-means clustering algorithm. Given the complexity of the proposed model, an efficient solution approach is developed through the hybridization of the Benders decomposition algorithm and stochastic dual dynamic programming. In the Benders master problem, day-ahead bidding variables are determined, whereas the Benders sub-problem addresses intraday bidding and charging station scheduling variables, employing stochastic dual dynamic programming to tackle its intractability. Additionally, we transform the mixed integer linear program model of the second stage problem into a linear program, confirming its validity through KKT conditions. Our model provides practical insights for making informed decisions in electricity markets based on sequential auctions. While the bidding curves submitted to the day-ahead market remain unaffected by scenarios, those submitted to the intra-day market show dependence on fluctuations in day-ahead market prices.

MULTI-ENTITY STOCK DEPENDENT MODEL WITH CAPACITY AND MANUFACTURE COST RESTRAINT’S

Inventory has an impact on the manufacturing process as well as supply chain operations. The fundamental goal of this research paper is to optimize the cost associated with inventories and to provide flow less continuous production process in time. Normally, demand rate of any entity in inventory control model are treated as predictable and at the same time constant too, and that the cost associated to unit inventory must be independent and non-variable in nature. Nevertheless, in practical circumstances, the unit price and demand rate of an entity may be interconnected. When the asking for an article is enormous, an entity is manufactured in huge quantities and the static charges of manufacturing being diffused over a multiple component. Henceforth, per unit article cost decreases significantly. i.e., per unit article cot and the demand of an article are related under inverse variation. So, better to consider the demand rate of an article as a variable constraint than to fixed one. In this research article, a mathematical model for multiple articles through permitted and restricted shortage and per article cost based on demand accompanied by upper and lower limits viz restricted storage space and manufacturing expenses has been constructed. Overall, investigating the simultaneous effect of storage space and manufacturing expenses in an inventory model provides valuable insights that enable cost optimization, resource allocation, capacity planning, and risk mitigation. It helps companies make informed decisions and improve their overall operational efficiency and profitability. The Multi-Entity Stock Dependent Model with Capacity and Manufacture Cost Restraints can be customized and used in a variety of sectors that include managing inventory across numerous entities and complicated supply chain networks. Here are a few examples of industries that can benefit from such a model: manufacturing industry, the retail and distributor sector, e-commerce companies, pharmaceutical and healthcare industry, automotive industry and food and beverage industry. The article cost is explored at this juncture in a fuzzy atmosphere and solutions of the model being obtained through KKT condition. Finally, a conclusion is offered in the final portion.