Fuzzy Chance-constrained Programming Research Articles

A location-routing problem model with multiple periods and fuzzy demands

This paper puts forward a dynamic capacitated location-routing problem with fuzzy demands (DCLRP-FD). It is given on input a set of identical vehicles (each having a capacity, a fixed cost and availability level), a set of depots with restricted capacities and opening costs, a set of customers with fuzzy demands, and a planning horizon with multiple periods. The problem consists of determining the depots to be opened only in the first period of the planning horizon, the customers and the vehicles to be assigned to each opened depot, and performing the routes that may be changed in each time period due to fuzzy demands. A fuzzy chance-constrained programming (FCCP) model has been designed using credibility theory and a hybrid heuristic algorithm with four phases is presented in order to solve the problem. To obtain the best value of the fuzzy parameters of the model and show the influence of the availability level of vehicles on final solution, some computational experiments are carried out. The validity of the model is then evaluated in contrast with CLRP-FD’s models in the literature. The results indicate that the model and the proposed algorithm are robust and could be used in real world problems.

Coupling fuzzy-chance constrained program with minimax regret analysis for water quality management

Mathematical modeling technique plays an important role for regionalization assessment of integrated economy and environment problems, resulting in provision of decision makers with break-through insights and risk-informed strategies. However, such a planning effort is complicated with a variety of uncertain and dynamic factors as well as their interactions. In this study, a fuzzy-chance constrained programming (FCP) method is firstly developed for addressing uncertainties characterized as fuzzy sets and random variables and, then, minimax regret (MMR) analysis technique is advanced to determine desired alternative that can reflect compromises between maximized system benefit and minimized system-failure risk. FCP coupled with MMR is applied to a real-case study of water quality management through optimizing chemical industry activities of the New Binhai District, an economically and industrially fast growing region in the center of northern China. Modeling formulation can analyze interactions among criteria of industry layout, economic benefit, pollution mitigation, and water quality security. Solutions for planning the water quality management have been generated, reflecting that there is trade-off among industrial structure, environmental protection, and economic development.

A solid waste management model with fuzzy random parameters

A novel solid waste management model with fuzzy random parameters (SWMMFRP) was proposed. The model improves the traditional stochastic mathematical programming (SMP) and fuzzy mathematical programming (FMP) methods through tackling dual uncertain parameters in the right-hand side of model constraints. The dual uncertainty is associated with both fuzzy imprecision and randomness. Based on stochastic and fuzzy chance-constrained programming (FCCP) techniques, SWMMFRP was capable of offering more-optimal solution scenarios under multiple uncertainties through incorporating various confidence information into a model framework, and evaluating tradeoffs between system economy and reliability. A long-term municipal waste management case was used to demonstrate the applicability of the proposed method. The results indicated that the model could help generate a spectrum of waste management patterns under different constraint-satisfaction and violation levels and provide support for decision makers to identify desired waste-management strategies. The study case also demonstrated that SWMMFRP is suitable in handling a municipal waste management system that allows somewhat compromise of environmental quality for achieving high economic benefits.

Fuzzy chance-constrained linear fractional programming approach for optimal water allocation

A fuzzy chance-constrained linear fractional programming method was developed for agricultural water resources management under multiple uncertainties. This approach improved upon the previous programming methods, and could reflect the ratio objective function and multiple uncertainties expressed as probability distributions, fuzzy sets, and their combinations. The proposed approach is applied to an agricultural water resources management system where many crops are considered under different precipitation years. Through the scenarios analyses, the multiple alternatives are presented. The solutions show that it is applicable to practical problems to address the crop water allocation under the precipitation variation and sustainable development with ratio objective function of the benefit and the irrigation amount. It also provides bases for identifying desired agriculture water resources management plans with reasonable benefit and irrigation schedules under crops.

Solving Water Quality Management Problem through Combined Genetic Algorithm and Fuzzy Simulation

A combined genetic algorithm and fuzzy simulation approach (GAFSA) was developed through integrating fuzzy chance-constrained programming (FCCP) and genetic algorithm (GA) into a general optimization framework. The major advantage of GAFSA is that it could tackle generally-shaped fuzzy membership functions on both sides of the model constraints, rather than handle single special forms like triangular or trapezoidal. An agricultural water quality management problem that has been investigated by a number of previous studies was used to demonstrate the applicability of the proposed method. The results showed that GAFSA allowed violation of system constraints at specified possibilistic confidence levels, leading to model solutions with higher system benefits. A conservative planning scheme could bring a more reliable system, but would be less economically attractive. Conversely, planning towards a higher system benefit would lead to a higher risk of system failure. The proposed model could help agricultural water managers analyse the trade-off between the overall system benefit and the failure risk of environmental compliance. A comparison of GAFSA to FCCP was given, and the potential limitations of the proposed method were also discussed.

Inexact Fuzzy Chance-Constrained Nonlinear Programming Approach for Crop Water Allocation under Precipitation Variation and Sustainable Development

AbstractIn this study, an inexact fuzzy chance-constrained nonlinear programming (IFCCNP) method was developed for agriculture water resources management under multiple uncertainties. It improved upon the previous stochastic programming methods, and could reflect uncertainties expressed as probability distributions, fuzzy sets, intervals, and their combinations. By using the integration of above programming to represent the multiple uncertain parameters, the IFCCNP can generate results that are more reliable reflecting practical problem than the previous methods. The developed model is then applied to a case study of agriculture water resources management system where many crops and their planting stages are considered under different precipitation years for demonstrating its applicability. The results indicate that comprehensive solutions have been obtained. Through analysis under different precipitation levels, it provides bases for identifying desired agriculture water resources management plans with r...

A dual-uncertainty-based chance-constrained model for municipal solid waste management

A double-sided dual-uncertainty-based chance-constrained programming (DDCCP) model was developed for supporting municipal solid waste management under uncertainty. The model was capable of tackling left-hand- and right-hand-side variables in constraints where those variables were affected by dual uncertainties (i.e. e.g. both fuzziness and randomness); and they were expressed as fuzzy random variables (FRVs). In this study, DDCCP model were formulated and solved based on stochastic and fuzzy chance-constrained programming techniques, leading to optimal solutions under different levels of constraints violation and satisfaction reliabilities. A long-term solid waste management problem was used to demonstrate the feasibility and applicability of DDCCP model. The obtained results indicated that DDCCP was effective in handling constraints with FRVs through satisfying them at a series of allowable levels, generating various solutions that facilitated evaluation of trade-offs between system economy and reliability. The proposed model could help decision makers establish cost-effective waste-flow allocation patterns under complex uncertainties, and gain in-depth insights into the municipal solid waste management system.

English

  In this paper we developed an inventory model in mixed imprecise and uncertain environment. Presented model is the developed form of r,Q and is a multi-itemsmodel with two objectives as minimizing costs (holding and shortage) and risk level under constraints including available budgetary, the least service level, storage spaces and allowable quantities of shortage. Demand distribution functions are assumed to be exponential and extra demands are supposed in two situations as lost sales and backlogging. At first we develop crisp model then fuzzy stochastic model with fuzzy budgetary, allowable quantities of shortage and shortage spaces (that is stochastic with normal distribution function) parameter. All of fuzzy numbers are triangular type. In methodology of solution we change model to a crisp multi-objective by using difuzzification of fuzzy constraints and fuzzy chance-constrained programming methods, and then solve it by fuzzy logic method. Finally an illustrated example is taken and solved using LINGO package.   Key words: r,Q model, lost sales, backlogging, multi objective programming, fuzzy-chance constrained programming.

Seeking optimal groundwater pumping strategies at Pinggu District in Beijing, China

A simulation-based fuzzy optimization method (SFOM) was proposed for regional groundwater pumping management in considering uncertainties. SFOM enhanced the traditional groundwater management models by incorporating a response matrix model (RMM) into a fuzzy chance-constrained programming (FCCP) framework. RMM was used to approximate the input–output relationship between pumping actions and subsurface hydrologic responses. Due to its explicit expression, RMM could be easily embedded into an optimization model to help seek cost-effective pumping solutions. A groundwater management case in Pinggu District of Beijing, China, was used to demonstrate the method's applicability. The study results showed that the obtained system cost and pumping rates would vary significantly under different confidence levels of constraints satisfaction. The decision-makers could identify the best groundwater pumping strategy through analyzing the tradeoff between the risk of violating the related water resources conservation target and the economic benefit. Compared with traditional approaches, SFOM was particularly advantageous in linking simulation and optimization models together, and tackling uncertainties using fuzzy chance constraints.

Optimization of the industrial structure facing sustainable development in resource-based city subjected to water resources under uncertainty

An inexact stochastic fuzzy programming (ISFP) approach has been developed for the optimization of the industrial structure in resource-based city subjected to water resources under uncertainty in present study. The ISFP method incorporates the techniques of inexact stochastic programming and inexact fuzzy chance-constrained programming, where the uncertainties are expressed as interval, fuzzy sets, and probability distribution, respectively. Moreover, it can also examine the risk of violating fuzzy tolerance constraints. The developed method is subsequently employed in a realistic case for industrial development in the Jinchang city, Gansu province, China. The result can help to analyze whether the water resources carrying capacity of Jinchang can meet the need of local economic development plan under uncertainty and help decision maker to optimize the industry structure under water resource constraints to meet the maximum economic efficiency.

Efficient management of air quality considering fuzzy confidences with varied reliabilities

Regional air quality management systems are complicated by uncertainties due to their interactive, dynamic, and multi-objective features. In this study, an inexact double-sided fuzzy chance-constrained programming (IDFCCP) model was developed and applied to a hypothetical case of regional air quality management. The results indicated that the proposed IDFCCP improved upon the existing ILP and DFCCP approaches; the fuzzy confidences at different levels could be analysed with varied reliability scenarios, making it possible to handle fuzzy uncertainties originating from both sides of the model constraints; other uncertain parameters could be expressed in terms of discrete intervals. The trade-off between system economy and reliability could be analysed by decision makers according to their preferences. The study results demonstrated that the proposed method could help decision makers identify desired policies under various environmental, economic, system-feasibility and system-reliability constraints.

Minimizing makespan in a single-machine scheduling problem with a learning effect and fuzzy processing times

In scheduling problems with learning effects, most research assumes that processing times are deterministic. This paper studies a single-machine scheduling problem with a position-based learning effect and fuzzy processing times where the objective is to minimize the makespan. The position-based learning effect of a job is assumed to be a function of its position. The processing times are considered to be triangular fuzzy numbers. Two different polynomial-time algorithms are developed for the problem. The first solution methodology is based on the fuzzy chance-constrained programming, whereas the second is based on a method to rank fuzzy numbers. Computational experiments are then conducted in order to evaluate the performance of the algorithms.

A hybrid waste-flow allocation model considering multiple stage and interval–fuzzy chance constraints

An inexact two-stage fuzzy chance-constrained programming (ITSFCCP) model was developed in this study for dealing with multiple uncertainties associated with solid waste management systems. The model was formulated by incorporating fuzzy chance-constrained programming and interval linear programming within a two-stage stochastic programming framework. The model could be used to facilitate analysis of the policy scenarios; meanwhile, the uncertainties associated with solid waste management systems could be expressed as probability distributions, possibility distribution and discrete intervals. A long-term waste planning problem was used to demonstrate the applicability of the ITSFCCP model. The study results indicated that ITSFCCP could provide a linkage to pre-defined policies and allowed violation of system constraints at predefined confidence levels. Moreover, it allowed uncertain information presented as discrete intervals to be communicated into the optimisation process. ITSFCCP could help waste managers to identify desired policies and gain insights into the tradeoffs between system economy and reliability.

Support Vector Machine Based on Type-2 Fuzzy Training Samples

In order to deal with the classification problems of type-2 fuzzy training samples on generalized credibility space. Firstly the type-2 fuzzy training samples are reduced to ordinary fuzzy samples by the mean reduction method. Secondly the definition of strong fuzzy linear separable data for type-2 fuzzy samples on generalized credibility space is introduced. Further, by utilizing fuzzy chance-constrained programming and classic support vector machine, a support vector machine based on type-2 fuzzy training samples and established on generalized credibility space is given. An example shows the efficiency of the support vector machine.

A Fuzzy Programming Model of Supplier Selection

An integrated supplier selection problem under fuzzy environment is studied in this paper. Firstly, the linear weight method is used to calculate the scores of suppliers according to their different attributes, such as: quality, service, warranty, delivery, reputation and position, which are assumed as fuzzy variables. Secondly, a fuzzy expected value programming model and a fuzzy chance-constrained programming model are proposed to select the best combination of the suppliers and determine the order quantities. A hybrid intelligent algorithm, based on fuzzy simulation, genetic algorithm and neural network, is used to solve the two models. Finally, a numerical example is given to illustrate the effectiveness of the proposed models.

A Production Inventory Model with Flexibility and Reliability under mλ-measure

In this paper, a production inventory model is developed with flexibility and reliability consideration of production process under mλ-measure which is linear combination of Possibility measure and Necessity measure. The model has incorporated selling price dependent demand, an imprecise cost parameters and time dependent partial backlogging. The critical values of fuzzy profit function with respect to mλ–measure are introduced and are employed to construct the fuzzy chance-constrained programming model. The objective function is optimized using a genetic algorithm and confidence limits of fuzzy profit function are obtained for numerical illustration with the sensitivity analysis

An uncertainty based formal approach towards parametric sensitivity: A case study on epoxy polymerization

AbstractKinetic constants in a proposed reaction scheme act as a set of parameters that are to be tuned with the experimental data and henceforth any model‐based optimization study related to the concerned reactor performance assumes the values of these kinetic parameters as fixed as obtained from the previous curve fitting exercise. However, it is known that these parameters are subjected to inherent source of uncertainties such as errors in measurement, change in concentrations for which they are not tuned for, etc. Assuming these kinetic parameters fixed for rest of the optimization is, therefore, not realistic and one should ideally check the sensitivity of these parameters on the final results. Fuzzy Chance Constrained Programming (FCCP) is one such approach that allows a decision maker to carry out such an analysis and this has been depicted here for determining the effect of uncertain kinetic parameters on the optimal performance of a semibatch epoxy polymerization reactor. Additionally, this study captures the tradeoff between solution quality and solution reliability that provides the angle of parametric sensitivity to this study. POLYM. ENG. SCI., 2011. © 2011 Society of Plastics Engineers

Municipal Solid Waste–Flow Allocation Planning with Trapezoidal-Shaped Fuzzy Parameters

A modified trapezoidal-shaped fuzzy chance-constrained mixed-integer programming (TFCMP) model was advanced for municipal solid waste (MSW) management. Compared with conventional methods, TFCMP was advantageous in handling fuzzy-type uncertainties in both the left- and right-hand sides of model constraints and could be used to reflect the possibility of constraints violation at predefined confidence levels. Mixed-integer programming (MIP) was embedded into the general framework of TFCMP for handling capacity-expansion issues. The solid waste management system in a typical Chinese city was used to demonstrate the applicability of TFCMP. Study results indicated that a variety of cost-effective MSW-flow allocation solutions could be obtained from TFCMP under various scenarios of system reliability. A trade-off between the total system cost and the reliability of satisfying model constraints can be analyzed to gain an in-depth insight into the characteristics of MSW systems. Generally, decision alternatives with lower system cost would be obtained if the environmental requirement was less strict; however, this may lead to a higher system failure risk. On the contrary, a more costly management scheme would ensure the environmental constraints be satisfied with a higher degree of reliability. The proposed model could assist decision makers in identifying effective waste allocation patterns and expansion options, with both cost and risk information being considered under complex uncertainties. TFCMP is also applicable to other management fields and its function could be extended through coupling with other inexact optimization methods.

A trust-based approach to selection of business services

With the increasing popularity of the service-oriented thinking in the business area, an overwhelming number of business services have arisen. These business services collaborate with each other and form a business network or a service-oriented business ecosystem (SOBE). It is a key issue to select appropriate business services to share and integrate them in the SOBE. Trust degrees play an important role in the selection of business services. However, existing researches mainly focus on quality of service (QoS)-based selection methods and few of them take trust degrees into account. In order to select services more accurately, a trust-based approach for selection of business services is proposed based on the formal definition of the SOBE. First, the method for acquisition and calculation of trust values is developed. The method also considers other parameters including QoS attributes, the physical distance between users and business services and the waiting time. Second, one node in a business flow may need to select two or more business services to achieve the user's requirements. Thus, the multi-service selection method for one node is presented. Third, a fuzzy chance-constrained programming model is proposed for the selection of business services by considering four kinds of factors: QoS attributes, trust relationship, physical distance and waiting time. Using the characteristic of fuzzy constraints, a function is developed to convert the fuzzy chance constraints into its crisp equivalents and then the equivalent crisp model is solved by the generic algorithm (GA)-based algorithm. For the case in which no satisfying solution can be found by the GA-based algorithm, a negotiation method is proposed to reach at a satisfying solution. Finally, a case study is conducted to demonstrate the feasibility and effectiveness of the proposed approach.

An Inexact Fuzzy-Chance-Constrained Air Quality Management Model

Regional air pollution is a major concern for almost every country because it not only directly relates to economic development, but also poses significant threats to environment and public health. In this study, an inexact fuzzy-chance-constrained air quality management model (IFAMM) was developed for regional air quality management under uncertainty. IFAMM was formulated through integrating interval linear programming (ILP) within a fuzzy-chance-constrained programming (FCCP) framework and could deal with uncertainties expressed as not only possibilistic distributions but also discrete intervals in air quality management systems. Moreover, the constraints with fuzzy variables could be satisfied at different confidence levels such that various solutions with different risk and cost considerations could be obtained. The developed model was applied to a hypothetical case of regional air quality management. Six abatement technologies and sulfur dioxide (SO2) emission trading under uncertainty were taken into consideration. The results demonstrated that IFAMM could help decision-makers generate cost-effective air quality management patterns, gain in-depth insights into effects of the uncertainties, and analyze tradeoffs between system economy and reliability. The results also implied that the trading scheme could achieve lower total abatement cost than a nontrading one.