Fuzzy Mathematical Programming Method Research Articles

Method of operating assessment of the state of network elements to ensure the quality of services in corporate high­speed multiservice communication networks

The article proposes intelligent methods and algorithms for operational assessment of the state of network elements to ensure indicators of the quality of communication services in corporate multi-service communication networks that function in a mode close to real time. A feature of corporate multi-service networks is the high dynamics of changes in their state. And the task of the automated network management system is to ensure the given quality of communication services. Thus, the relevance of this study is determined by the need to implement network management processes in a mode close to real time, with a given quality. The basis of the method of operational assessment of the state of network elements is the concept of using intelligent agents. In the proposed approach, intelligent agents are created as hierarchical fuzzy situational networks, in which management decisions, in contrast to known methods based on the application of reference situations, are produced on the basis of solving a hierarchical system of optimization tasks based on fuzzy mathematical programming methods. The main paradigm of their functioning is «situation – action». This makes it possible to significantly reduce the dimension of the task to be solved and obtain Pareto-optimal solutions for managing the quality of communication services. The intelligent agent of management solutions can be implemented automatically, provided that the network administrator delegates such capabilities to it. The method of estimating traffic parameters is based on the concept of conditional nonlinear Pareto-optimal filtering. Its essence is to apply a two-stage procedure; at the first stage, a forecast of the values of parameter estimates is made, but in the second, with the receipt of the following observations, their values are adjusted. The average relative error of traffic parameter estimation does not exceed 10%, which is sufficiently operational for managing the quality of communication services.

Optimal Water Resources Allocation in the Yinma River Basin in Jilin Province, China, Using Fuzzy Programming

Being one of the most important sources of water in the Jilin Province in China, the Yinma River Basin (YRB) is facing problems of water scarcity in low economic areas and low utilization in richer areas mainly caused by the irrational allocation of water, excessive pursuit of economic benefits, and neglect of environmental problems. Restricting watershed development involves potential decision-making risks. Some scholars have used the interval two-stage stochastic planning method to adjust water resource allocation in the Drinking Horse River Basin, but the method uses historical statistics for projection and does not take into account the ambiguity and uncertainty in real planning situations. Therefore, this study addresses the problems prevalent in the allocation of water resources in the YRB through optimization using stochastic programming methods, interval and two-stage, and introduces the fuzzy mathematical programming method, with the aim of coordinating the water balance of various water-consuming sectors in the YRB, so as to reconfigure the water allocation. The goal is to solve the existing problems of irrational water allocation, reduce system risks posed by excessive economic development, mitigate water shortages in the water-consuming sectors, and alleviate potential decision-making risks and vague uncertainties associated with the allocation of water resources. Additionally, optimization of the pollution-holding capacity improvement project was carried out. The interval fuzzy two-stage model simulation developed in this study shows that the distribution of water across the different administrative regions can be reduced by up to 30% compared with the original model, effectively reducing the problem of water wastage. Post-optimization, the impact of water shortage in the water resources allocation scheme is alleviated to a significant degree, and there is no water shortage in some areas. At the same time, the eco-environmental sector has gradually taken the leading role in the distribution of water reuse among the different water-consuming sectors. The pollution-holding capacity has been enhanced, and the discharge and river entry chemical oxygen demand (COD) and ammonia nitrogen, two typical pollutants, have been reduced. The membership interval in the interval fuzzy two-stage model reflects the relationship between the possible level of the target value and the risk level. This study provides a guideline for decision makers for balancing the relationship between benefits and risks and proposes a planning scheme that is more conducive to the development of the river basin.

Optimization of Ecological Water Replenishment Scheme Based on the Interval Fuzzy Two-Stage Stochastic Programming Method: Boluo Lake National Nature Reserve, Jilin Province, China.

In this paper, a fuzzy mathematical programming method is introduced on the basis of the interval two-stage stochastic programming (ITSP) optimisation model for the wetland ecological water replenishment scheme in Boluo Lake National Nature Reserve. The minimum ecological water supply is taken as the objective function, and the lake bubble water diversion capacity, lake bubble water supply capacity, water diversion sequence, ecological service value, and minimum capacity of the wetland water supply are taken as constraints. The ecological water replenishment schemes of five lakes in the Boluo Lake National Nature Reserve are optimised at the levels of low flow years, normal flow years, and high flow years, and an optimised model for the wetland ecological water replenishment scheme in Boluo Lake National Nature Reserve based on the interval fuzzy two-stage stochastic programming (IFTSP) method is constructed. The model fully considers the waste of water resources and the protection of migratory bird habitat and makes rational allocation of water resources to make full use of flood resources. The IFTSP model proposed herein fully considers the fuzzy and uncertain characteristics of the planning area in the lake bubble area of Boluo Lake National Nature Reserve and improves the decision-making efficiency of decision-makers by providing technical support for smooth implementation of the ecological water replenishment project in nature reserves. The model can also be used as a theoretical guide for ecological recharge projects in other regions of the world.

Fuzzy programming approach to Bi-level linear programming problems

In this study, we discussed a fuzzy programming approach to bi-level linear programming problems and their application. Bi-level linear programming is characterized as mathematical programming to solve decentralized problems with two decision-makers in the hierarchal organization. They become more important for the contemporary decentralized organization where each unit seeks to optimize its own objective. In addition to this, we have considered Bi-Level Linear Programming (BLPP) and applied the Fuzzy Mathematical Programming (FMP) approach to get the solution of the system. We have suggested the FMP method for the minimization of the objectives in terms of the linear membership functions. FMP is a supervised search procedure (supervised by the upper Decision Maker (DM)). The upper-level decision-maker provides the preferred values of decision variables under his control (to enable the lower level DM to search for his optimum in a wider feasible space) and the bounds of his objective function (to direct the lower level DM to search for his solutions in the right direction).

Intellectual method of operational evaluation of the network element state to ensure the quality of services in corporate multiservice communications networks

The work proposes and investigates an intelligent method and algorithms for on-line assessment of the state of network elements to ensure the required quality indicators of provided communication services in corporate high-speed multiservice communication networks. The developed method and algorithms operate in a mode close to real time. One of the features of corporate multiservice communication networks is the high dynamics of changes in their state. The main task of the automated control system, which is an integral part of the corporate multiservice communication network, is to ensure the specified quality of the provided communication services to the consumer. Thus, the relevance of the research presented in the work is due to the fact that most of the management processes in corporate high-speed multiservice communication networks must be implemented in a mode close to real time with a given quality. The basis of the method for operational assessment of the state of network elements is the concept of creating and using intelligent agents. In the proposed approach, intelligent agents are created as hierarchical fuzzy situational networks, in which control solutions, in contrast to known methods based on the use of reference situations, are applied based on solving a hierarchical set of optimization problems using fuzzy mathematical programming methods. The main paradigm of their functioning is “situation -action”.

A hesitant fuzzy mathematical programming method for hybrid multi-criteria group decision making with hesitant fuzzy truth degrees

This paper aims to develop a new hesitant fuzzy mathematical programming method for hybrid multi-criteria group decision making (MCGDM) with hesitant fuzzy truth degrees and incomplete criteria weight information. In this method, the types of assessment information on criteria are expressed by Atanassov intuitionistic fuzzy sets, hesitant fuzzy sets, trapezoidal fuzzy numbers, intervals and real numbers, respectively. Firstly, the distances of each alternative to positive ideal solution (PIS) and negative ideal solution (NIS) are calculated. Then the hesitant fuzzy positive ideal group consistency index (HFPGCI) and hesitant fuzzy positive ideal group inconsistency index (HFPGICI), the hesitant fuzzy negative ideal group consistency index (HFNGCI) and hesitant fuzzy negative ideal group inconsistency index (HFNGICI) are defined, respectively. To derive the PIS, NIS and the criteria weights simultaneously, a new four-objective hesitant fuzzy mathematical programming model is constructed by minimizing the HFPGICI and HFNGICI as well as maximizing the HFPGCI and HFNGCI. Using the geometric-mean score functions of hesitant fuzzy sets, the four-objective programming model is transformed to a single objective program to resolve. Subsequently, the relative closeness degrees of alternatives for each decision maker (DM) are obtained and applied to derive the individual ranking order of alternatives. To generate the collective ranking order of alternatives, a multi-objective assignment model is established and converted into a single objective programming model to resolve. Thus, a new hesitant fuzzy mathematical programming method is proposed to solve hybrid MCGDM. Finally, a real example is provided to demonstrate the applicability and validity of the proposed method.

Pythagorean fuzzy mathematical programming method for multi-attribute group decision making with Pythagorean fuzzy truth degrees

This paper develops a Pythagorean fuzzy (PF) mathematical programming method to solve multi-attribute group decision-making problems under PF environments. The main work is summarized as four aspects: (1) Considering the fuzziness and hesitancy in pairwise comparisons of alternatives, we firstly introduce PF sets to depict the fuzzy truth degrees of alternative comparisons. (2) According to the information entropy, individual subjective attribute weight vectors of decision makers (DMs) are calculated and integrated into a collective one by a cross-entropy optimization model. Then DMs’ weights are objectively derived from the collective subjective attribute weight vector. (3) PF group consistency and inconsistency indices are defined based on PF-positive ideal solution (PFPIS) and PF-negative ideal solution (PFNIS), respectively. To determine comprehensive attribute weights, a biobjective PF mathematical programming model is constructed through minimizing two inconsistency indices based on PFPIS and PFNIS simultaneously. A linear programming method is technically developed to solve this model. (4) Using the cross-entropy again, collective relative closeness degrees of alternatives are explicitly derived to rank the alternatives. Finally, an example of green supplier selection is analyzed to verify the effectiveness of the proposed method.

Fuzzy mathematical programming approach to heterogeneous multiattribute decision-making with interval-valued intuitionistic fuzzy truth degrees

Considering the hesitancy degrees on pair-wise comparisons of alternatives as interval-valued intuitionistic fuzzy (IVIF) sets (IVIFSs), we develop a new fuzzy mathematical programming method for solving heterogeneous multiattribute decision-making problems based on the Linear Programming Technique for Multidimensional Analysis of Preference. In this method, IVIFSs, intuitionistic fuzzy sets (IFSs), trapezoidal fuzzy numbers, linguistic variables, intervals and real numbers are used to represent multiple types of attribute values and the attribute weights are not completely known. The preference relations between alternatives given by decision maker are expressed with IVIFSs of ordered pairs of alternatives. The consistency and inconsistency indices are defined as IVIFSs on the basis of comparisons of alternatives with IVIF truth degrees. The attribute weights and fuzzy ideal solution (FIS) are estimated through constructing a fuzzy mathematical programming model, which is solved by the technically developed method of IVIF mathematical programming. Hereby the distances of alternatives to the FIS are computed to rank the alternatives. Some generalization and discussion on the constructed IVIF mathematical programming model are also presented. A green supplier selection example is provided to illustrate the effectiveness of the proposed model and method. The comparison analyses verify the superiorities of the proposed method.

Novel robust fuzzy mathematical programming methods

Robust optimization has attracted great attention in recent decade owing to its ability to cope with risk and uncertainty in real-life optimization problems. Recently, the scope of robust optimization is extended into fuzzy mathematical programming by proposing robust possibilistic programming approach. As a complementary for robust possibilistic programming approach, this paper proposes different novel robust flexible programming and robust mixed possibilistic-flexible programming models. The advantages and strengths of the proposed models are described and a real-life problem is employed to show the efficiency and practicability of the propounded models against the traditional fuzzy programming approach.

Interval-valued intuitionistic fuzzy mathematical programming method for hybrid multi-criteria group decision making with interval-valued intuitionistic fuzzy truth degrees

As an important component of group decision making, the hybrid multi-criteria group decision making (MCGDM) is very complex and interesting in real applications. The purpose of this paper is to develop a novel interval-valued intuitionistic fuzzy (IVIF) mathematical programming method for hybrid MCGDM considering alternative comparisons with hesitancy degrees. The subjective preference relations between alternatives given by each decision maker (DM) are formulated as an IVIF set (IVIFS). The IVIFSs, intuitionistic fuzzy sets (IFSs), trapezoidal fuzzy numbers (TrFNs), linguistic variables, intervals and real numbers are used to represent the multiple types of criteria values. The information of criteria weights is incomplete. The IVIFS-type consistency and inconsistency indices are defined through considering the fuzzy positive and negative ideal solutions simultaneously. To determine the criteria weights, we construct a novel bi-objective IVIF mathematical programming of minimizing the inconsistency index and meanwhile maximizing the consistency index, which is solved by the technically developed linear goal programming approach. The individual ranking order of alternatives furnished by each DM is subsequently obtained according to the comprehensive relative closeness degrees of alternatives to the fuzzy positive ideal solution. The collective ranking order of alternatives is derived through establishing a new multi-objective assignment model. A real example of critical infrastructure evaluation is provided to demonstrate the applicability and effectiveness of this method.

Superiority–inferiority modeling coupled minimax-regret analysis for energy management systems

In this study, a superiority–inferiority-based minimax-regret analysis (SI-MRA) model is developed for supporting the energy management systems (EMS) planning under uncertainty. In SI-MRA model, techniques of fuzzy mathematical programming (FMP) with the superiority and inferiority measures and minimax regret analysis (MMR) are incorporated within a general framework. The SI-MRA improves upon conventional FMP methods by directly reflecting the relationships among fuzzy coefficients in both the objective function and constraints with a high computational efficiency. It can not only address uncertainties expressed as fuzzy sets in both of the objective function and system constraints but also can adopt a list of scenarios to reflect the uncertainties of random variables without making assumptions on their possibilistic distributions. The developed SI-MRA model is applied to a case study of long-term EMS planning, where fuzziness and randomness exist in the costs for electricity generation and demand. A number of scenarios associated with various alternatives and outcomes under different electricity demand levels are examined. The results can help decision makers identify an optimal strategy of planning electricity generation and capacity expansion based on a minimax regret level under uncertainty.

Application of Fuzzy Sets to Aggregate Production Planning With Multiproducts and Multitime Periods

The objective of this study is to develop a fuzzy mathematical programming method to solve aggregate production planning (APP) decision problems that involve multiproducts and multitime periods in a fuzzy environment. The fuzzy APP model that is developed here attempts to minimize total cost with respect to inventory carrying levels, available labor levels, machine capacity and warehouse space, and the constraint of available budget. The proposed APP method evaluates monetary interest of related operating cost categories and provides greater computational efficiency and flexibility by adopting triangular fuzzy numbers and piecewise linear membership functions to represent both imprecise data and fuzzy goals. The actual performance of an industrial company was used to demonstrate the feasibility of applying the proposed method to real-world APP decisions. The proposed method yields an efficient solution and presents overall decision-maker satisfaction with the given goal values. This paper also presents several significant management implications that are related to the practical application of the proposed method.

Selection of Rational Trends in Development of an Automated System for Controlling the Operation of Electronic Facilities Under Electronic Conflict Conditions

In the interests of technical and economical substantiation of the prospects of development of an automated system for controlling the operation of electronic facilities under electronic conflict conditions, an algorithm for solving the problem of selecting rational trends for its development, based upon fuzzy mathematical programming methods, is proposed.

An interactive fuzzy satisficing method based on a probability maximization model for multiobjective linear programming problems involving random variable coefficients

AbstractAs an approach to the optimization of systems containing fuzziness and uncertainty, the probabilistic programming method including uncertainty based on probability theory and the fuzzy mathematical programming method representing fuzziness in terms of fuzzy theory are typical ones that have been developed in various forms. In the present research, we target multiobjective linear programming problems in which the coefficients included in the program are random variables. We develop a formulation based on the probabilistic maximization model in which the probability that several objective functions are below certain values is maximized under the stochastic constraint condition that the constraints need not be satisfied all the time but only above a certain probability. For the multiobjective probability maximization model, the fuzzy target of the decision maker is introduced. Also, an interactive algorithm based on the reference point method that derives a solution satisfactory to the decision maker by interaction with the decision maker is applied. The new decision making process is a combination of the probabilistic programming method and the fuzzy programming method. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 87(3): 67–76, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10080

Fuzzy programming approach to multi-level programming problems

Multi-level programming is characterized as mathematical programming to solve decentralized planning problems. We have considered a multi-level linear programming problem and applied fuzzy mathematical programming (FMP) approach to obtain the solution of the system. We have suggested FMP method for the minimization of the objectives using linear membership functions. FMP is a supervised search procedure (supervised by the higher level decision maker (DM)). The higher level DM provides the preferred values of decision variables under his control (to enable the lower level DM to search for his optimum in a wider feasible space) and the bounds of his objective function (to direct the lower level DM to search for his solutions in the right direction).

Fuzzy mathematical programming applied to multi-level programming problems

An alternative multi-level programming technique based on fuzzy mathematical programming (FMP) is developed which gives better solutions than that proposed by Shih et al. (Comput. Oper. Res. 23(1) (1996) 73). The method uses the objectives in a sequential order of the hierarchy interactively and takes into account the desire of the DM at each stage. The technique is equally applicable to linear/nonlinear multi-level (say P levels) programming problems (MLPPs). Scope and purpose In this paper, we propose an alternative FMP method for a linear/nonlinear MLPP of maximization- and minimization-type objectives which gives better optimal solutions.

Network Models and Optimization of Ecologo-Economic Systems

An ecologo-economic system is a kind of integrated large-scale systems involving both ecological and economic factors. In this paper, a new type of network flow model -- multiple attribute network model of ecologo-economic system is suggested for the analysis and optimization of regional development. Methods of describing the multiple attribute of energy, mass and financial fund flow are introduced. The expression of objective function and constraints are given, and algorithms of optimization are discussed. In order to overcome the vagueness of ecological and economic factors, the paper introduces fuzzy mathematical programming methods. Furthermore it studies the example of an ecologo-economic system of a county in Northeast China.