Fuzzy Chance-constrained Programming Research Articles

Development of a fuel management model for a multi-source district heating system under multi-uncertainty and multi-dimensional constraints

In this paper, an interval single-sided fuzzy chance-constrained mixed-integer programming model is developed for the fossil fuel management of a multi-source district heating system under multiple uncertainties, where the heat-supply-capacity expansion planning can also be reflected. The non-dimensional comprehensive equations method is simultaneously improved to quantify the ambiguous heat provisions, utilized as a type of boundary condition inputs to the proposed model. A real-world case study of a heating system located in northeastern China is undertaken to show the feasibility and applicability of the proposed methods. To obtain the reasonable fossil-fuel management schemes, multi-dimensional constraints are incorporated into the model based on a comprehensive consideration in terms of fuel supply and demand, quality and quantity, economic cost and environment protection, as well as their interactions. Results obtained from the case study indicate that the solutions for both continuous and binary variables have been generated, which are useful for identifying suitable fuel-supply patterns and heat-source operational modes for a heating system under different system reliabilities and heating-load distribution states. In addition, the results also reveal that the fossil-fuel management and heating-capacity-expansion pattern, as well as the economic cost and pollutant emission performances are sensitive to the thermal coefficient and system reliability level, which may provide in-depth analyses of tradeoffs for further supporting robust fossil-fuel management under uncertainty.

A multi-fuel management model for a community-level district heating system under multiple uncertainties

In this study, an interval two-stage double-stochastic single-sided fuzzy chance-constrained programming model is developed for supporting fuel management of a community-level district heating system (DHS) fed with both traditional fossil fuels and renewable biofuels under multiple uncertainties. The proposed model is based on the integration of interval parameter programming and single-sided fuzzy chance-constrained programming within an improved stochastic programming framework to tackle the uncertainties expressed as crisp intervals, fuzzy relationship, and probability distributions. Through transforming and solving the model, the related fuzzy and stochastic information can be effectively reflected in the generated solutions. A real fuel management case of a DHS located in Junpu New District of Dalian is utilized to demonstrate the model applicability. The obtained solutions provides an effective linkage in terms of both ‘‘quality’’ and ‘‘quantity’’ aspects for fuel management under various scenarios associated with multiple factors, and thus can help the decision makers to identify desired fuel allotment patterns. Moreover, this study is also useful for decision makers to address the other challenges (e.g. the imbalance between fuel supply and demand, the contradiction between air-pollution emission and environmental protection, as well as the tradeoff between the total heating cost and system satisfaction degree) generated in the fuel management processes.

A type-2 fuzzy chance-constrained programming method for planning Shanghai’s energy system

In this study, a type-2 fuzzy chance-constrained programming (TFCP) method is developed for supporting energy systems planning (ESP) under uncertainty. TFCP incorporates chance-constrained programming within a type-2 fuzzy programming framework to handle uncertainties (i.e. type-2 fuzzy sets and probabilistic distributions) in the objective and constraints, as well as to examine the reliability of satisfying (or risk of violating) system constraints. A TFCP-based energy system (TFCP-ES) model is then formulated for energy systems planning of Shanghai, where multi-energy resources, multi-processing technologies, multi-conversion technologies and multi-end users are considered. Solutions of energy supply, electricity generation, capacity expansion, and air-pollutant emission associated with different constraint-violation risks are obtained. Results reveal that (i) natural gas is one of the major energy-supply sources for the city in the future (increasing by 13.0%); (ii) the city’s electricity-generation structure tends to the transition from coal-dominated into clean energy-dominated (e.g., natural gas, onshore wind, offshore wind, and photovoltaic power); (iii) the city’s energy-supply security is enhanced by provoking the utilization of renewable energies (increasing by 1.0%). The results are helpful for managers to adjust the city’s current energy structure, enhance the energy supply security, as well as make tradeoff between system cost and constraint-violation risk.

Fuzzy capacitated location-routing problem with simultaneous pickup and delivery demands

ABSTRACTIn this work, the fuzzy capacitated location-routing problem with simultaneous pickup and delivery demands (FCLRP-SPD) is considered. This problem is a more realistic case of the capacitated location-routing problem and belongs to the reverse logistics. In FCLRP-SPD, a set of customers with simultaneous pickup and delivery demands should be supplied by a fleet of vehicles that start and end their tours at a single depot. In addition, both pickup and delivery demands of customers are fuzzy variables. The objective function is minimizing the routing costs, fixed costs of establishing the depot(s), and employing the vehicles. To model the problem, a fuzzy chance-constrained programming model is designed based upon the fuzzy credibility theory. To solve the FCLRP-SPD, a greedy clustering method (GCM) is developed, which includes four iterative phases. To achieve the best value of two types of parameters of the model, named ‘vehicle indexes’ and ‘depots indexes’, and to analyze their influences on the final solution, numerical experiments are carried out. Moreover, the efficiency of the GCM is demonstrated using a standard benchmark set of test problems.

Mathematical Programming for Modelling Green Supply Chains Under Randomness and Fuzziness

Nowadays by growing concerns about environmental problems, businesses and industries are under pressure to decrease their negative impact on environment, consequently firms and industries have to reconsider about their activities and make their business compatible with environment. So industries should green their supply chains to optimize economic and environmental concerns, but because of uncertainty in the real world like inconsistency of world economy, the process of greening supply chains can be more complex. To optimize total costs and the unfavourable sides of supply chains simultaneously in an uncertain situation, this paper presents a multi-objective mixed integer programming with fuzzy random variables (FRVs) and by using fuzzy theory and fuzzy random chance-constrained programming (FRCCP), the proposed model is converted to deterministic model. This paper can be also suitable for decision making with optimistic, pessimistic and realistic notion. Finally, a numerical example is presented to illustrate the model.

A Multiobjective Fuzzy Chance Constrained Programming Model for Land Allocation in Agricultural Sector: A case study

In this article a fuzzy multiobjective chance constrained programming model is used for modeling and solving land allocation problems efficiently with the help of fuzzy goal programming. Optimal production of seasonal crops and related expenditures are considered from the viewpoint of proper utilization of total cultivating land and different farming resources. Some resource parameters associated with the probabilistic constraints are taken as normally distributed fuzzy random variables. The potential use of this methodology is illustrated by a case example.

A copula-based fuzzy chance-constrained programming model and its application to electric power generation systems planning

This study developed a copula-based fuzzy chance-constrained programming (CFCCP) model and applied it to electric power generation systems planning under multiple uncertainties. The CFCCP model was formulated by incorporating existing joint-probabilistic constrained programming and generalized fuzzy linear programming techniques within a general mixed-integer linear programming framework. The CFCCP model can not only effectively reflect uncertain interactions among random variables even when the random variables follow different probability distributions and have previously unknown correlations, but can also provide information about the membership grades for the decision variables and objective-function values. Thus, it would have a wider application scope than existing optimization models for power generation systems planning. Its applicability has been demonstrated through a case study of electric power generation planning within a region of North China. As a result, fuzzy interval solutions related to power generation and capacity expansion patterns of electricity-generation facilities, and primary energy supply structures were generated within six scenarios of constraint-violation levels under different α-cut levels. The results are helpful to investigate dynamic features of the regional power generation system, identify desired decision alternatives, and analyze the influences of interactions among multiple uncertainties on system outputs.

A stochastic fuzzy chance-constrained programming model for energy–environment system planning and management in the City of Beijing

ABSTRACTA stochastic and fuzzy chance-constrained programming (SFCCP) model was developed in this study for supporting energy–environment management in the city of Beijing, China. SFCCP was capable of tackling the variables in constraints as fuzzy random variables, which were integration of randomness and vagueness. SFCCP was applied to an energy–environment management system in the city of Beijing. The study results indicated that SFCCP was useful in helping decision makers gain in-depth insights into proposed management system and establish environment-friendly energy allocation alternatives. The application of SFCCP is expected to provide a good demonstration to energy–environment management problems under complex uncertainties.

A fuzzy chance-constrained programming model with type 1 and type 2 fuzzy sets for solid waste management under uncertainty

ABSTRACTEffective management of municipal solid waste (MSW) is critical for urban planning and development. This study aims to develop an integrated type 1 and type 2 fuzzy sets chance-constrained programming (ITFCCP) model for tackling regional MSW management problem under a fuzzy environment, where waste generation amounts are supposed to be type 2 fuzzy variables and treated capacities of facilities are assumed to be type 1 fuzzy variables. The evaluation and expression of uncertainty overcome the drawbacks in describing fuzzy possibility distributions as oversimplified forms. The fuzzy constraints are converted to their crisp equivalents through chance-constrained programming under the same or different confidence levels. Regional waste management of the City of Dalian, China, was used as a case study for demonstration. The solutions under various confidence levels reflect the trade-off between system economy and reliability. It is concluded that the ITFCCP model is capable of helping decision makers to generate reasonable waste-allocation alternatives under uncertainties.

Conjunctive Water Use Optimization for Watershed-Lake Water Distribution System under Uncertainty: a Case Study

In this research, a large-scale inexact optimization method was developed for the conjunctive use management of a watershed-lake water distribution system. The modeling framework has the advantages in taking into account the water balance of multi-reservoirs, satisfying the municipal industrial and agricultural water demands in the multi-period context, reflecting the relationship among multi-reservoirs, multiple water related projects, and maintaining the operational rules of certain lake levels. Moreover, such a method can also handle the uncertainty expressed as fuzzy membership functions through integrating the fuzzy credibility chance-constrained programming. The developed method was applied to the conjunctive use management of water resources in the lake Dianchi watershed, China. Cost-effective water allocation schemes for the groundwater project and the water transfer project, and optimal operational rules for the lake and multiple reservoirs were successfully obtained. Also, the annual water balance of the watershed-lake system and the system cost of the conjunctive water use were investigated and analyzed under multiple credibility levels of meeting the water demands for municipal, industrial and agricultural users.

Bi-level programming modeling and hierarchical hybrid algorithm for antimissile dynamic firepower allocation problem with uncertain environment

Aiming at the shortcomings of antimissile dynamic firepower allocation (ADFA) researches under uncertain environment, the fuzzy chance-constrained bi-level programming model with complex constraints is proposed by introducing the uncertain programming theory. Firstly, maximization cost-effectiveness ratio and earliest interception time as the upper and the lower objective functions of the model, respectively, are used. In order to close to the battlefield environment, the model constraint includes interception time window, effective damage lower bound and intercept strategy, etc. Secondly, a particle coding scheme and repairing scheme are given with hierarchical structure for multi-constrained bi-level ADFA problem. Furthermore, the improved variable neighborhood PSO algorithm with convergence criterions and the PSO algorithm with doubt and repulsion factor (PSO-DR) are effectively combined. On these bases, the hierarchical hybrid fuzzy particle swarm optimization algorithm is presented with fuzzy simulation technique. Finally, the results of comparison show the proposed algorithm has stronger global searching ability and faster convergence speed, which can effectively solve large-scale ADFA problem and adapt to the requirements of real-time decision.

Fuzzy multi-objective chance-constrained programming model for hazardous materials transportation

Hazardous materials transportation is an important and hot issue of public safety. Based on the shortest path model, this paper presents a fuzzy multi-objective programming model that minimizes the transportation risk to life, travel time and fuel consumption. First, we present the risk model, travel time model and fuel consumption model. Furthermore, we formulate a chance-constrained programming model within the framework of credibility theory, in which the lengths of arcs in the transportation network are assumed to be fuzzy variables. A hybrid intelligent algorithm integrating fuzzy simulation and genetic algorithm is designed for finding a satisfactory solution. Finally, some numerical examples are given to demonstrate the efficiency of the proposed model and algorithm.

On Solving the Fuzzy Customer Information Problem in Multicommodity Multimodal Routing with Schedule-Based Services

In this study, we combine the fuzzy customer information problem with the multicommodity multimodal routing with schedule-based services which was explored in our previous study [1]. The fuzzy characteristics of the customer information are embodied in the demanded volumes of the multiple commodities and the time windows of their due dates. When the schedule-based services are considered in the routing, schedule constraints emerge because the operations of block container trains should follow their predetermined schedules. This will restrict the routes selection from space-time feasibility. To solve this combinatorial optimization problem, we first build a fuzzy chance-constrained nonlinear programming model based on fuzzy possibility theory. We then use a crisp equivalent method and a linearization method to transform the proposed model into the classical linear programming model that can be effectively solved by the standard mathematical programming software. Finally, a numerical case is presented to demonstrate the feasibility of the proposed method. The sensitivity of the best solution with respect to the values of the confidence levels is also examined.

A Comparative Study of Fuzzy Techniques to Handle Uncertainty: An Industrial Grinding Process

AbstractFuzzy‐based approaches like fuzzy chance constrained programming (FCCP) and fuzzy expected value model (FEVM) have been applied to a multi‐objective optimization problem of the industrial grinding process to carry out the uncertainty analysis. Results are compared with respect to the power of risk averseness adopted in the approaches used. The extent of constraint satisfaction due to the presence of uncertain parameters can be accommodated assuming credibility of constraint satisfaction under the FCCP framework whereas the robust set of parameters in the FEVM approach is determined by considering the expectation terms for objectives and constraints. Nonlinear relation of uncertain parameters has been handled by adopting simulation‐based approaches while computing the credibility. These approaches are very generic and can be applied for the study of parametric sensitivity for any process model in a novel manner.

A Comparative Study of Fuzzy Techniques to Handle Uncertainty: An Industrial Grinding Process

AbstractThe deterministic optimization models for chemical processes assume perfect information, i.e., the system parameters involved in the models should have fixed values. However, the knowledge about these parameters is often either not easy to obtain or, if available, subjected to uncertainty involved in determining them (e.g. experiments, regression etc.). The optimization results obtained using these deterministic models are, therefore, not robust to uncertainties in the parameters of the system. In this work, fuzzy based approaches, e.g. fuzzy chance constrained programming (FCCP) and fuzzy expected value model (FEVM), have been applied to a multi‐objective optimization problem of the industrial grinding process to carry out the uncertainty analysis and the results are compared with respect to the power of risk averseness adopted in the approaches used. These methods assume the uncertain parameters as fuzzy numbers and membership function is used to represent the degree of uncertainty in them. The extent of constraint satisfaction due to the presence of uncertain parameters can be accommodated assuming credibility of constraint satisfaction under FCCP framework whereas the robust set of parameters in the FEVM approach is determined by considering the expectation terms for objectives and constraints. Moreover, the issue of nonlinear relation of uncertain parameters has been handled by adopting simulation based approaches while computing the credibility. The results obtained by FCCP technique show how the presence of uncertainty leads to an operating zone of varied risk appetite of a decision maker by using different uncertain measures such as credibility, possibility and necessity. These approaches are very generic and can be adopted for the study of parametric sensitivity for any process model in a novel manner.

Reliability Evaluation of a Microgrid Considering Its Operating Condition

Microgrids offer several reliability benefits, such as the improvement of load-point reliability and the opportunity for reliability-differentiated services. The primary goal of this work is to investigate the impacts of operating condition on the reliability index for microgrid system. It relies on a component failure rate model which quantifies the relationship between component failure rate and state variables. Some parameters involved are characterized by subjective uncertainty. Thus, fuzzy numbers are introduced to represent such parameters, and an optimization model based on Fuzzy Chance Constrained Programming (FCCP) is established for reliability index calculation. In addition, we present a hybrid algorithm which combines scenario enumeration and fuzzy simulation as a solution tool. The simulations in a microgrid test system show that reliability indices without considering operating condition can often prove to be optimistic. We also investigate two groups of situations, which include the different penetration levels of microsource and different confidence levels. The results support the necessity of considering operating condition for achieving accurate reliability evaluation.

3Es System Optimization under Uncertainty Using Hybrid Intelligent Algorithm: A Fuzzy Chance-Constrained Programming Model

Harmonious development of 3Es (economy-energy-environment) system is the key to realize regional sustainable development. The structure and components of 3Es system are analyzed. Based on the analysis of causality diagram, GDP and industrial structure are selected as the target parameters of economy subsystem, energy consumption intensity is selected as the target parameter of energy subsystem, and the emissions of COD, ammonia nitrogen, SO2, andNOXand CO2emission intensity are selected as the target parameters of environment system. Fixed assets investment of three industries, total energy consumption, and investment in environmental pollution control are selected as the decision variables. By regarding the parameters of 3Es system optimization as fuzzy numbers, a fuzzy chance-constrained goal programming (FCCGP) model is constructed, and a hybrid intelligent algorithm including fuzzy simulation and genetic algorithm is proposed for solving it. The results of empirical analysis on Shandong province of China show that the FCCGP model can reflect the inherent relationship and evolution law of 3Es system and provide the effective decision-making support for 3Es system optimization.

Uncertain and multi-objective programming models for crop planting structure optimization

Crop planting structure optimization is a significant way to increase agricultural economic benefits and improve agricultural water management. The complexities of fluctuating stream conditions, varying economic profits, and uncertainties and errors in estimated modeling parameters, as well as the complexities among economic, social, natural resources and environmental aspects, have led to the necessity of developing optimization models for crop planting structure which consider uncertainty and multi-objectives elements. In this study, three single-objective programming models under uncertainty for crop planting structure optimization were developed, including an interval linear programming model, an inexact fuzzy chance-constrained programming (IFCCP) model and an inexact fuzzy linear programming (IFLP) model. Each of the three models takes grayness into account. Moreover, the IFCCP model considers fuzzy uncertainty of parameters/variables and stochastic characteristics of constraints, while the IFLP model takes into account the fuzzy uncertainty of both constraints and objective functions. To satisfy the sustainable development of crop planting structure planning, a fuzzy-optimizationtheory-based fuzzy linear multi-objective programming model was developed, which is capable of reflecting both uncertainties and multi-objective. In addition, a multiobjective fractional programming model for crop structure optimization was also developed to quantitatively express the multi-objective in one optimization model with the numerator representing maximum economic benefits and the denominator representing minimum crop planting area allocation. These models better reflect actual situations, considering the uncertainties and multi-objectives of crop planting structure optimization systems. The five models developed were then applied to a real case study in Minqin County, north-west China. The advantages, the applicable conditions and the solution methods of each model are expounded. Detailed analysis of results of each model and their comparisons demonstrate the feasibility and applicability of the models developed, therefore decision makers can choose the appropriate model when making decisions.

Fuzzy chance-constrained geometric programming: the possibility, necessity and credibility approaches

Geometric programming (GP) is a powerful tool for solving a variety of optimization problems. Most GP problems involve precise parameters. However, the observed values of the parameters in real-life GP problems are often imprecise or vague and, consequently, the optimization process and the related decisions take place in the face of uncertainty. The uncertainty associated with the coefficients of GP problems can be formalized using fuzzy variables. In this paper, we propose chance-constrained GP to deal with the impreciseness and the ambiguity inherent to real-life GP problems. Given a fuzzy GP model, we formulate three variants of chance-constrained GP based on the possibility, necessity and credibility approaches and show how they can be transformed into equivalent deterministic GP problems to be solved via the duality algorithm. We demonstrate the applicability of the proposed models and the efficacy of the introduced procedures with two numerical examples.

Fuzzy chance-constrained goal programming model for multi-attribute financial portfolio selection

The aim of this paper is to propose a fuzzy chance constrained goal programming model for solving a multi-attribute financial portfolio selection problem under two types of uncertainty namely randomness and fuzziness. The chance-constrained goals are considered as random variables. The obtained portfolio through this model is the portfolio of the best compromise where the financial decision-maker was asked to make tradeoffs among conflicting and incommensurable attributes such as the expected return, risk and the earning price ratio. The proposed model has been applied to the Tunisian stock exchange market for the period July 2003 to December 2007.