Fuzzy Chance-constrained Programming Research Articles

Low-carbon economic operation strategy for multi-agent integrated energy system considering uncertainty of renewable energy power generation

The uncertainty of renewable energy output threatens the operation safety of multi-agent integrated energy system (MAIES), which makes it difficult to balance the low-carbon economic operation demands of various stakeholders. However, the existing research solely focuses on the operational strategy of multi-agent game involving integrated energy suppliers and users in deterministic scenarios, overlooking the complementary supporting role and game interaction of shared energy storage and wind farm as independent entities of interest under the instability of renewable energy power generation. Hence, this paper first establishes the optimal operation models for integrated energy system operator (IESO), user aggregator (UA), shared energy storage operator (SESO), and wind farm operator (WFO) considering the stepped carbon trading. Second, in the face of the actual situation of uncertainty of photovoltaic and wind power output, fuzzy chance-constrained programming is adopted for processing. Then, a bi-layer game equilibrium model with IESO as a leader and UA, SESO, and WFO as followers is proposed, and the existence and uniqueness of Stackelberg equilibrium solution are proved. Finally, simulation calculation is carried out based on the YALMIP toolbox in the Matlab R2023a software, and the improved particle swarm optimization algorithm and CPLEX solver are used to solve the model. The results demonstrate that the participation of SESO and WFO as independent stakeholders in the game interaction can improve the economic and environmental benefits of MAIES. The iterative optimization of demand response subsidy prices can effectively motivate users to participate in demand response, improve the ability of MAIES to cope with the uncertain risks of renewable energy generation and load, and reduce the power grid dispatch pressure.

Chance-Constrained Optimization for a Green Multimodal Routing Problem with Soft Time Window under Twofold Uncertainty

This study investigates a green multimodal routing problem with soft time window. The objective of routing is to minimize the total costs of accomplishing the multimodal transportation of a batch of goods. To improve the feasibility of optimization, this study formulates the routing problem in an uncertain environment where the capacities and carbon emission factors of the travel process and the transfer process in the multimodal network are considered fuzzy. Taking triangular fuzzy numbers to describe the uncertainty, this study proposes a fuzzy nonlinear programming model to deal with the specific routing problem. To make the problem solvable, this study adopts the fuzzy chance-constrained programming approach based on the possibility measure to remove the fuzziness of the proposed model. Furthermore, we use linear inequality constraints to reformulate the nonlinear equality constraints represented by the continuous piecewise linear functions and realize the linearization of the nonlinear programming model to improve the computational efficiency of problem solving. After model processing, we can utilize mathematical programming software to run exact solution algorithms to solve the specific routing problem. A numerical experiment is given to show the feasibility of the proposed model. The sensitivity analysis of the numerical experiment further clarifies how improving the confidence level of the chance constraints to enhance the possibility that the multimodal route planned in advance satisfies the real-time capacity constraint in the actual transportation, i.e., the reliability of the routing, increases both the total costs and carbon emissions of the route. The numerical experiment also finds that charging carbon emissions is not absolutely effective in emission reduction. In this condition, bi-objective analysis indicates the conflicting relationship between lowering transportation activity costs and reducing carbon emissions in routing optimization. The sensitivity of the Pareto solutions concerning the confidence level reveals that reliability, economy, and environmental sustainability are in conflict with each other. Based on the findings of this study, the customer and the multimodal transport operator can organize efficient multimodal transportation, balancing the above objectives using the proposed model.

A novel two-phase approach for a sustainable p-hub median problem with multiple service levels and carbon cap-and trade policy under uncertainty

ABSTRACT To reduce the carbon emissions for society's sustainable development and satisfy customers' personalized delivery service, this paper addresses the cap-and-trade policy and multiple service levels into a p-hub median problem. In the hub network designing, single-assignment patterns between hubs and demand nodes, the choice of transportation mode for each O-D pair and the uncertainties of carbon emission and price are taken into account. Based on a fuzzy chance-constrained programming approach, a fuzzy mathematical model under an expected cost-minimization context with some fuzzy constraints is first presented. Then a two-phase approach is proposed to solve the proposed model. Finally, in view of the comparative analysis with small- and large-scale Turkish instances, serval managerial insights are finally obtained for government and logistics companies in the reduction of carbon emission and the hub network's total designing cost for the sustainable development.

Type-2 fuzzy chance-constrained linear fractional programming model for a water resource management system: A case study of Taiyuan city, China

Abstract Uncertainties arising from extreme climate events and human activities pose a challenge to the efficient allocation of water resources. In this study, a type-2 fuzzy chance-constrained linear fractional programming (T2F-CCLFP) is developed to support the water resource management system under uncertainty by incorporating type-2 fuzzy sets, chance-constrained programming, and fractional programming into a comprehensive multi-objective optimization framework. The model enables the trade-off between economic, social, and environmental sustainability and provides water supply solutions associated with different levels of fuzzy uncertainty and risk of violating constraints. The T2F-CCLFP model is applied to Taiyuan, Shanxi Province, China, to support its water resource management. Results reveal that: (i) the industrial structure is transitioning toward diverse industries from energy and heavy industry dominance; (ii) external water transfer will be the major water-supply sources for the city in the future, accounting for 55 and 50% of the total water supply in 2025 and 2030, respectively; (iii) the water-supply security of the city is enhanced by provoking the utilization of reclaimed water (the annual growth rate is 13.9%). The results are helpful for managers in adjusting the current industry structure, enhancing water supply security, and contributing to the sustainable development of socio-economic and water systems.

Optimization Method of Photovoltaic Microgrid Energy Storage System Based on Price-based DR

Extreme events lead to an increasing number of power outages in the distribution network. It is currently the most effective method to restore power supply after distribution network failure to connect distributed photovoltaic to the distribution network in the form of microgrids. However, the randomness of distributed PV output and load is the biggest obstacle limiting its development. Therefore, an optimization method of photovoltaic microgrid energy storage system (ESS) based on price-based demand response (DR) is proposed in this paper. Firstly, based on the influence of the uncertainty of the time of use (TOU) and load on the price-based DR, a price-based DR model is built. Then, taking the minimum difference between photovoltaic output and load demand as the optimization objective, the optimal operation model of price-based DR based on the fuzzy chance constrained program (FCCP) is established, and an optimization model of photovoltaic microgrid ESS based on FCCP is obtained. Finally, the fuzzy chance optimization problem is converted into a deterministic optimization problem through constraint equivalent treatment. The proposed method is verified by numerical simulations.

Data-Driven State Transition Algorithm for Fuzzy Chance-Constrained Dynamic Optimization.

Many actual industrial production processes are dynamic and uncertain. When uncertain information are described by subjective experience and experts' knowledge based on scanty or vague information, fuzzy uncertainty exists. Fuzzy chance-constrained dynamic programming are applicable to industrial production modeling accompanied by fuzzy uncertainty and dynamics, where constraints need not or cannot be completely satisfied. In this article, a fuzzy chance-constrained dynamic optimization (FCCDO) formulation on the basis of credibility theory is established, in which, the credibility is used to measure the fuzzy uncertainty level of constraints. To solve the FCCDO problem (FCCDOP), an improved fuzzy simulation technique based on Hammersley sequence sampling is raised to transform fuzzy chance constraints to their deterministic equivalents, and then a data-driven state transition algorithm (DDSTA) using deep neural networks (DNNs) is put forward to achieve a stable, global and robust optimization performance. Finally, the successful applications of the FCCDO method to industrial studies demonstrate its advantages.

A chance-constrained programming method with credibility measure for solving the multi-skill multi-mode resource-constrained project scheduling problem

This paper addresses a multi-skilled extension of the multi-mode resource-constrained project scheduling problem (MRCPSP) with preemption of activities and generalized precedence relations under uncertainty. The problem is formulated mathematically as a single objective optimization model to minimize the project makespan. Concerning real-world circumstances, amount of non-renewable resources and availability are considered as interval type-2 fuzzy numbers. Therefore, a type-2 fuzzy chance-constrained programming (TFCP) method is developed to tackle the problem. Four unique characteristics are taken into account in this paper concurrently: (i) multi-skilled resources; (ii) renewable and non-renewable resources; (iii) preemptable activities; and (iv) interval type-2 fuzzy parameters. The existence of these practical contributions causes projects schedule, especially construction projects, to be closer to the real-world conditions. The proposed mathematical model is validated and tested on instances from PSPLIB (j12, j16 and j18) and MMLIB (MM50 and MM100) datasets. In this study, the computational results for the first time demonstrate that the number of preemption points depend heavily on fluctuations of non-renewable resources and number of skills. It is also shown that considering multi-skill renewable resources is an effective factor in project makespan. In J20, there were both a four-unit reduction and increase in the number of preemption points achieving by increasing the average amount of available non-renewable resources by 23 units and decreasing it by 22 units, respectively. Additionally, when seven skills of renewable resources were reduced, the number of preemption points increased by seven units. In this problem instance, project is completed within 28 days considering multi-skill renewable resources, whereas the project makespan is reported 33 days without considering them. Various sensitivity analyses are presented to evaluate the proposed mathematical model and the overall solution approach. Comparative analysis is conducted to compare the schedule in crisp and uncertain conditions. Eventually, the importance of considering multi-skill resources in project scheduling is illustrated compared to not considering them.

Fuzzy multi-objective truck scheduling in multi-modal rail–road Physical Internet hubs

The Physical Internet (PI) is an innovative concept that has the potential to significantly improve the efficiency, cost-effectiveness, and sustainability of the global supply chain industry, particularly in cross-docking operations. This paper addresses the truck-scheduling problem in rail–road PI-Hubs, taking into account simultaneously both uncertainty and multi-objective decision-making, which has not been fully explored in the literature, particularly for PI-structures. Our proposed approach defines a Multi-Objective Mixed-Integer Programming model (FMO-MIP) that incorporates fuzzy chance-constrained programming and ϵ-constraint to minimize both the total delay and the sum of PI-containers traveled distances, while considering the uncertainty on truck arrival times. This work takes into account the particularities of the Physical Internet and presents a novel decision-making solution to generate a robust Pareto front that aligns with decision-makers’ attitudes towards risk (optimistic/pessimistic) while balancing trade-offs between conflicting objectives.

Solving a new bi-objective mathematical model for a hybrid flow shop scheduling problem with robots and fuzzy maintenance time

This paper presents a bi-objective mathematical model of a hybrid flow shop scheduling problem (HFSSP) with robots and fuzzy maintenance time. The proposed model minimizes the mean completion time of jobs and the total cost of the processes based on maintenance costs and incurred transportation costs via robots. A fuzzy chance-constrained programming (FCCP) model is used to present its deterministic-equivalent problem. It is then solved by two multi-objective decision-making approaches, namely LP-metric and goal attainment (GA). A sensitivity analysis of the weights of the objective functions is conducted and the behavior of the model is studied for each instance. Also, a TOPSIS (the technique for order of preference by similarity to ideal solution) method is used to rank the efficiency of the proposed solution approaches. Future research areas are also suggested.

Demand response comprehensive incentive mechanism-based multi-time scale optimization scheduling for park integrated energy system

With the increasing uncertainty of energy supply side output, fully encouraging users to participate in demand response through different types of demand response incentive mechanisms has become one of the effective ways to deal with the uncertainty of integrated energy system operation and improve the overall energy efficiency. However, in existing studies, the coordination of uncertainty handling, optimization of demand response incentive strategies, and demand response measures at different time scales have not been adequately considered in the operation of integrated energy systems. Based on these considerations, this paper proposes a multi time-scale game optimization scheduling model for Park-level Integrated Energy System considering multiple types of demand response models. In the day-ahead stage, a Park-level Integrated Energy System optimization game scheduling model based on the demand response comprehensive incentive mechanism is established, and the uncertainty of the predicted value of distributed renewable energy and multi-type energy load was characterized based on the fuzzy chance-constrained programming method. In the intraday and real-time stages, a rolling optimization scheduling model is established with the minimum cost of Park-level Integrated Energy System operator scheduling. For the proposed model, an improved particle swarm optimization algorithm and an iterative solution strategy of CPLEX solver are introduced. Finally, the simulation results of an actual case show that the proposed model can effectively improve the Park-level Integrated Energy System operator and user economy while ensuring reliability.

Quartet trade-off for regional water resources allocation optimization with multiple water sources: A decentralized bi-level multi-objective model under hybrid uncertainty

Multiple uncertainties from water resource allocation have posed new challenges to balanced regional development. On the one hand, in the context of global water shortage crisis, groundwater, as an important water supply module, is facing severe risk of shortage. On the other hand, the regional authorities need to coordinate the contradictory water demand brought by the heterogeneous water use objectives from multiple water use levels, and these demands are also subject to complex uncertainties. How to mitigate the risks posed by uncertainties, ensure regional water security and balance participants, is attracting the keen attention of decision-makers around the world. Therefore, this study establishes a decentralized bi-level multi-objective regional water resources allocation (RWRA) optimization model under hybrid uncertainty, aiming at an optimal quartet trade-off of equity, sustainability, equilibrium, and efficiency. To deal with the hybrid uncertainty in the model, water demand is set as a random fuzzy number, and the groundwater constraint is combined with fuzzy chance-constrained programming (FCCP). Then, the decentralized bi-level multi-objective model is solved by the bi-level interactive method based on the satisfactory solution incorporated with the chaotic-particle swarm optimization (CPSO) algorithm. The proposed model and solution approach are then applied to the Dongjiang River Basin in Guangdong Province of China to verify the validity and utility of the model. The results show that (1) the model can effectively meet the different objectives requirements of regional authority and sub-areas; (2) groundwater constraint parameters has a significant impact on the model results, especially the Gini coefficient decreases by 13.6% when groundwater is used as a water source supplement; (3) from single-level model to decentralized bi-level model, the satisfactory balance of the two levels increases from 0.1457 to 0.793, which indicates that the decentralized bi-level model makes the two levels more balanced than the single-level model. Based on the results, some management recommendations for participants are finally provided.

Energy Storage Configuration of Distribution Networks Considering Uncertainties of Generalized Demand-Side Resources and Renewable Energies

With the growing proportion of advanced metering infrastructures and intelligent controllable equipment in power grids, demand response has been regarded as an effective and easily implemented approach to meet the demand–supply equilibrium. This paper innovatively proposes generalized demand-side resources combining the demand response with an energy storage system and constructs a configuration model to obtain scheduling plans. Firstly, this paper analyzes the characteristics of generalized demand-side resources and models the translational loads, reducible loads and energy storage system. Secondly, a deterministic energy storage configuration model aiming at achieving the lowest operation cost of distribution networks is established, from which the scheduling scheme of generalized demand-side resources can be obtained. Then, the fuzzy membership function and the probability density function are used to represent the uncertainty of the demand response, the prediction error of renewable energy output and the generalized demand-side resources that do not participate in the demand response. Therefore, this paper simulates daily operations to modify the capacity of energy storage. The problem is solved by using Monte Carlo simulation, fuzzy chance-constrained programming and mixed-integer programming. Finally, the effectiveness of this model is demonstrated with case studies in a 33-node distribution network. The results show that the uncertainty of this system is solved effectively. When only considering generalized demand-side resources, the total cost is reduced by 9.5%. After considering the uncertainty, the total cost is also decreased 0.3%. Simultaneously, the validity of the model is verified.

Designing an integrated humanitarian logistics network for the preparedness and response phases under uncertainty

In the current paper, a model of possibilistic location-allocation is proposed for designing a multi-period bi-objective humanitarian logistics network that pursues a reduction in the total cost and maximizes the total network coverage. The model considers the inventory management of perishable relief items and the flow of affected and wounded people in pre-disaster and post-disaster phases, simultaneously. Moreover, wounded people are categorized according to the injury severity. The uncertainty associated with key parameters is also addressed. To handle the uncertainty, a fuzzy chance-constrained programming method originated in the Me measure is used. The purpose of this measure is to prevent the extreme attitudes of the decision maker by considering the combination of possibility and necessity measures and using the optimistic-pessimistic parameter. This paper presents a solution procedure derived from a fuzzy interactive programming approach and two meta-heuristic algorithms, imperialist competitive algorithm and invasive weed optimization, so as to solve the study model. As for the validation of the proposed model and solution procedures, a number of test problems have been generated. A real-life case study is also implemented to instantiate whether the proposed model is applicable or not.

Agricultural Productive Carrying Capacity Improve and Water Optimal Allocation under Uncertainty Based on Remote Sensing Data in Lancang County, Southwest China

Through the reasonable calculation of water resources, evaluating the irrigation carrying capacity of farmland under the constraints of water resources is crucial for optimizing the spatial distribution of agricultural production and ecology and rationally adjusting the scale of agricultural production. This paper proposes an optimization framework based on Type 2 fuzzy chance-constrained programming (T2FCCP) to solve the problem of regional water resources optimal allocation and evaluation of farmland irrigation carrying capacity under uncertain conditions. To illustrate the applicability of the proposed framework, this paper conducts a case study on Lancang County, Puer City, Yunnan Province. Methods, such as watershed harmony evaluation method, remote sensing data, and shared socioeconomic pathways (SSPs), are applied and integrated into the proposed optimization framework to systematically deal with uncertainties in water resource systems and agricultural systems. The results include the costs and benefits of regional water and soil resources systems, water resources optimal allocation, and crop planting structure results under different SSPs in Lancang County, Puer City. The results also show that the total cost under T2FCCP is about 5% lower than that under fuzzy chance-constrained programming (FCCP) and about 17% lower than that under chance-constrained programming (CCP). By 2025, the water resources carrying capacity of different tributaries in Lancang County, Puer City will increase, and based on the evaluation results of agricultural production irrigation carrying capacity, suggestions are given to ensure agricultural production carrying capacity.

Fuzzy Multi-Objective Chance-Constrained Portfolio Optimization Under Uncertainty Considering Investment Return, Investment Risk, and Sustainability

Effective portfolio management can increase business value by maximizing investment return, reducing possible risk and, making the best use of limited resources. Achieving this goal is constrained by multiple conflicting objectives and uncertainty, which are inherent in all projects. The investment return, investment risk, and sustainability have been simultaneously evaluated in this study by fuzzy multi-objective chance-constrained portfolio optimization, aligned with an organization’s strategic direction. Through the fuzzy chance-constrained programming, the obtained portfolio under uncertainties can be optimized with a specified confidence level imposed by decision makers. With such an NP-hard problem, a Genetic Algorithm (GA) is embedded in the developed decision support tool for sustainable investment portfolio selection and optimization. The applicability of the proposed approach and the constructed tool are illustrated through an example.

Fuzzy Demand Vehicle Routing Problem with Soft Time Windows

Considering the vehicle routing problem with fuzzy demand and fuzzy time windows, a vehicle routing optimization method is proposed considering both soft time windows and uncertain customer demand. First, a fuzzy chance-constrained programming model is established based on credibility theory, minimizing the total logistics cost. At the same time, a random simulation algorithm is designed to calculate the penalty cost of delivery failures caused by demand that cannot be satisfied. In order to overcome the shortcomings of GA, which easily falls into the local optimum in the process of searching, and the slow convergence speed of SA when the population is too large, a hybrid simulated annealing–genetic algorithm is adopted to improve the solution quality and efficiency. Finally, the Solomon standard example is used to verify the effectiveness of the algorithm, and the influence of decision-makers’ subjective cost preference is analyzed.

A new fuzzy tri-objective model for a home health care problem with green ambulance routing and congestion under uncertainty

With the increasing number of older people and high hospital treatment costs, older people tend to receive cures at home and more demand for home health care centers (HHCCs). Therefore, HHCCs should use appropriate planning to carry out their activities to reduce costs while maintaining health care quality and patient satisfaction. According to previous studies, the present study shows that most studies have examined the problem in terms of logistical costs and have used lower multi-criteria decision-making (MCDM) methods. This has driven our efforts to simultaneously consider the financial, human resource, time and environmental pollutants, and problem-solving through MCDM methods. In this research, a mixed-integer linear programming (MILP) model with three objectives in both definite and indefinite states is presented to optimize the basic HHC processes. In the first objective function, an attempt is made to minimize costs. On the other hand, due to the high importance of environmental pollutants, the second objective function minimizes the greenhouse gas (GHG) emitted by ambulances. Furthermore, for the first time, a new objective function is presented to minimize the congestion of laboratory samples in HHCCs with the M/M/C queueing system. Then, a hybrid model based on the expected value (EV) and fuzzy chance-constrained programming (FCCP) is presented to bring the problem closer to reality. Finally, this mathematical model is solved using goal attainment (GA) and LP-metric methods ranked by a simple additive weighting (SAW) method.

Green closed-loop supply chain optimization strategy considering CER and incentive-compatibility theory under uncertainty.

Under the uncertain market demand and quality level, a total profit model of green closed-loop supply chain system (GCL-SCS) considering corporate environmental responsibility (CER) and government differential weight subsidy (GDWS) is constructed. Based on incentive-compatibility theory, the optimal subsidy allocation policy and green investment level were explored. Fuzzy chance-constrained programming (FCCP) is used to clarify the uncertainty factors of this model; while genetic algorithm (GA) and CPLEX are used to find and compare a calculating example's approximate optimal solution about this model. The main calculating results indicate that: (1) Enterprises can make optimal recycling, production and sales strategies according to different potential demand; (2) Without government subsidy, enterprises' higher green investment level will reduce their average gross profit, increase the quality level of recycled products and decrease the recycling rate, hence reduce their environmental protection willingness; (3) Based on incentive-compatibility theory, when government subsidy weight is set as 0.34~0.41 for consumers, enterprises' higher green investment level will enhance their average gross profit, reduce the quality level of recycled products and increase the recycling rate, which will improve their environmental protection willingness; (4) Under uncertain environment, the combination of reasonable government subsidy policy and enterprises green investment can make up for the defect of enterprises green investment alone, maximize utilities of government and enterprises, and optimize the green closed loop supply chain.

Multiobjective Lot Sizing and Scheduling of Multiproduct Switching Production in the Process Industry Considering Uncertain Market Information Under Mass Customization

With the gradual diversification of customer demand, to improve the rapid response ability of enterprises, this paper fully considers uncertain market information under the background of mass customization and establishes a new process industry multi-product switching production lot sizing and scheduling model with the goal of minimizing the maximum completion time and total switching cost. Fuzzy chance-constrained programming is used to explicitly incorporate market demand with uncertain quantities into the model. Starting from reality, this paper considers the switching cost of equipment when processing multiple varieties of products and skillfully integrates the conversion rate of materials during processing into the novel model, making the entire production system closer to the real state. It provides a new concept to consider cost reduction for actual workshop scheduling management. In addition, this paper proposes an improved multi-objective genetic particle swarm optimization (SMOPSO-IIs) algorithm to simulate and solve the model. The operation results show that the Pareto solution obtained by the SMOPSO-IIs algorithm is better overall. Finally, the model is solved by example simulation, and the operation results are analyzed along with a scheduling Gantt chart to verify its applicability and effectiveness. The model presented in this paper can be used to further shorten the gap between production theory and practical application and improve the current workshop scheduling management system of the process industry.

Green road\u2013rail intermodal routing problem with improved pickup and delivery services integrating truck departure time planning under uncertainty: an interactive fuzzy programming approach

This paper addresses the multi-objective optimization for the road–rail intermodal routing problem that aims to minimize the total costs and carbon dioxide emissions of the routes. To achieve high timeliness of the entire transportation process, pickup and delivery services are simultaneously improved based on the employment of fuzzy soft time windows to measure their service levels. The modeling of road–rail intermodal routing considers fixed schedules of rail and time flexibility of road to match the real-world transportation scenario, in which travel times and carbon dioxide emission factors of road services are considered to be time-varying. To improve the feasibility of the routing, uncertainty of travel times and carbon dioxide emission factors of road services and capacities of rail services are incorporated into the problem. By applying trapezoidal fuzzy numbers to formulate the uncertainty, we propose a fuzzy multi-objective nonlinear optimization model for the routing problem that integrates the truck departure time planning for road services. After processing the model with fuzzy chance-constrained programming and linearization, we obtain an auxiliary equivalent crisp linear model and solve it by designing an interactive fuzzy programming approach with the Bounded Objective Function method. Based on an empirical case study, we demonstrate the validity of the proposed approach and discuss the effects of improving the confidence levels and service levels on the optimization results. The case analysis reveals several managerial insights that help to realize an efficient transportation organization by making effective trade-offs among lowering costs, reducing emissions, improving service levels, and enhancing feasibility.