Piecewise Linear Membership Functions Research Articles

Application of Fuzzy Multi-Objective Linear Programming Model in a Multi-Period Multi- Product Production Planning Problem

The Multi-Period Multi-Product (MPMP) production planning, as a well known problem in literature, attempts to match production rates of individual products with fluctuated market demand over planning horizon. This study, demonstrates how the conventional MPMP Linear Programming (LP) model may fail to utilize available capacity of machines, and also a novel Multi Objective Linear Programming (MOLP) model is developed to simultaneously minimize net present value of production costs and maximize machine utilization. The proposed model consists of production constraints such as available labor, inventory, maximum subcontracting levels and also forecasted demands. The proposed MOLP model is further converted to a Fuzzy Multi Objective Linear Programming (FMOLP) model utilizing piecewise linear membership functions. The model, accommodates the Decision Maker (DM) with a more systematic decision making approach enabling the DM to adjust the search direction during the solving procedure to achieve the most satisf...

Application of Fuzzy Multi-Objective Linear Programming Model in a Multi-Period Multi-Product Production Planning Problem

The Multi-Period Multi-Product (MPMP) production planning, as a well known problem in literature, attempts to match production rates of individual products with fluctuated market demand over planning horizon. This study, demonstrates how the conventional MPMP Linear Programming (LP) model may fail to utilize available capacity of machines, and also a novel Multi Objective Linear Programming (MOLP) model is developed to simultaneously minimize net present value of production costs and maximize machine utilization. The proposed model consists of production constraints such as available labor, inventory, maximum subcontracting levels and also forecasted demands. The proposed MOLP model is further converted to a Fuzzy Multi Objective Linear Programming (FMOLP) model utilizing piecewise linear membership functions. The model, accommodates the Decision Maker (DM) with a more systematic decision making approach enabling the DM to adjust the search direction during the solving procedure to achieve the most satisf...

Fuzzy multi-objective production/distribution planning decisions with multi-product and multi-time period in a supply chain

This work develops a fuzzy multi-objective linear programming (FMOLP) model with piecewise linear membership function to solve integrated multi-product and multi-time period production/distribution planning decisions (PDPD) problems with fuzzy objectives. The original multi-objective linear programming designed in this work model attempts to simultaneously minimize total costs and total delivery time in relation to inventory levels, available machine capacity and labor levels at each source, and forecast demand and available warehouse space at each destination and total budget. The proposed FMOLP model provides a systematic framework that facilitates fuzzy decision-making process, enabling the decision maker (DM) to interactively adjust the search direction during the solution procedure to obtain a DM’s preferred satisfactory solution. Additionally, the DM computes the value in each cost category by considering the time value of money in the proposed model, which is appropriate for practical application to the PDPD problem in a supply chain. Finally, an industrial case demonstrates the feasibility of applying the proposed model to a real-world integrated PDPD problem in a supply chain under uncertain environments.

Binary Behavior of Fuzzy Programming With Piecewise Linear Membership Functions

The nature of vagueness, imprecision, and uncertainly is fuzzy rather than crisp and/or random, especially for a multiple objectives decision-making problem. A key component of fuzzy programming is the membership function that represents a mathematical expression of level function for the decision-maker's preference. In fact, a decision-making problem involves the achievement of fuzzy goals, some of which are met while others are not because these fuzzy goals are subject to real-world constraints. To represent this situation, the binary piecewise linear membership function is then employed. In order to solve the problem, we propose a new idea of how to formulate the binary piecewise linear membership function. The formulated problem can be easily solved using common integer programming packages. In addition, an illustrative example is included for demonstrating the usefulness of the proposed model. Finally, the analytical superiority of the proposed method in terms of the execution time can be seen, through a computation experiment conducted on a set of generated test examples.

Binary Behavior of Fuzzy Programming With Piecewise Linear Membership Functions

The nature of vagueness, imprecision, and uncertainly is fuzzy rather than crisp and/or random, especially for a multiple objectives decision-making problem. A key component of fuzzy programming is...

Distribution planning decisions using interactive fuzzy multi-objective linear programming

This work develops an interactive fuzzy multi-objective linear programming (i-FMOLP) method for solving the fuzzy multi-objective transportation problems with piecewise linear membership function. The proposed i-FMOLP method aims to simultaneously minimize the total distribution costs and the total delivery time with reference to fuzzy available supply and total budget at each source, and fuzzy forecast demand and maximum warehouse space at each destination. Additionally, the proposed method provides a systematic framework that facilitates the fuzzy decision-making process, enabling a decision maker (DM) to interactively modify the fuzzy data and related parameters until a set of satisfactory solutions is obtained. An industrial case is presented to demonstrate the feasibility of applying the proposed method to real transportation problems. Consequently, the proposed method yields an efficient solution and overall degree of DM satisfaction with the determined objective values. Especially, several significant management implications regarding the practical application of the proposed i-FMOLP method are presented.

A hybrid methodology for fuzzy optimization of electromagnetic devices

Typical formulations for fuzzy optimization utilize piecewise linear membership functions, which introduces regions of no differentiability and prevents the employment of deterministic methods. We propose the utilization of continuously differentiable membership functions, which permit the use of gradient-based methods. We present a general expression for the gradient of the decision degree function. Then, a stochastic method can be used to find a good starting point for the deterministic technique, in a hybrid approach. The formulation is applied to the optimization of an electrostatic micromotor. The average torque is maximized subject to a fuzzy constraint for the torque ripple. The results show the validity of the methodology in electromagnetic design.

The approximation of piecewise linear membership functions and Łukasiewicz operators

In this paper we propose an approximation of piecewise linear membership functions with the help of sigmoid functions and certain arithmetic operations. The gradient-based tuning of piecewise linear membership functions can be achieved with the proposed efficient approximation because it has simple continuous derivatives. With this construction we can even approximate the Łukasiewicz operator family which plays an important role in fuzzy logic, first of all from the theoretical point of view, although in practice in optimization and learning it is rarely used because the lack of good analytical properties, e.g. a continuous gradient. The proposed approximation enlarges the applicability of fuzzy methods to the operators and membership functions where the differentiability is desirable.

Inverse planning for low-dose-rate prostate brachytherapy by simulated annealing under fuzzy expert control

Simulated annealing (SA) is a multivariate combinatorial optimization process that searches the configuration space of possible solutions by a random walk, guided only by the goal of minimization of the objective function. The decision-making capabilities of a fuzzy inference system are applied to guide the SA search, to look for solutions which, in addition to optimizing a plan in dosimetric terms, also present some clinically desirable spatial features. No a priori constraints are placed on the number or position of needles or on the seed loading sequence of individual needles. These additional degrees of freedom are balanced by giving preference to consider plans with seed distributions that are balanced in the right/left and anterior/posterior halves in each axial slice, and with local seed density that is about uniform. Piecewise linear membership functions are constructed to represent these requirements. Before a step in the random search is subject to the SA test, the expert functions representing the spatial seed-distribution requirements are evaluated. Thus, the expert planner’s knowledge enters into the decision as to the “goodness” of a seed configuration regarding the spatial seed-distribution goals. When a step in the random walk yields a seed configuration that is found wanting, a specific number of additional steps in the local neighborhood is attempted until either improvement in the spatial requirements is achieved, or the allowed number of attempts is exhausted. In the latter case, the expert system desists and the unfavorable step is taken, moving on to the simulated annealing test. The number of attempts is determined by the fuzzy logic inference engine and depends on just how badly the expert requirement is not met. The program is interfaced with a commercial treatment planning system (TPS) to import optimized seed plans for isodose display and analysis. Execution in a 1.5 GHz computer is less than a minute, adequate for real-time planning. Results for phantom and real patients are dosimetrically comparable to generic methods such as modified peripheral loading, but in some cases, fewer needles and less radioactivity are required. Sensitivity to simulated random seed placement error is similar to manual plans.

Self‐tuning of fuzzy rules when learning data have a radically changing distribution

AbstractIn this paper, we propose a new type of membership functions (MSFs) and their efficient use to improve optimization of fuzzy reasoning using a steepest descent method. In self‐tuning of fuzzy rules using the steepest descent method, an algorithm to avoid suboptimal solutions by modifying learning coefficients has been proposed, where piecewise linear MSFs were introduced. In such an algorithm, when learning data have a radically changing distribution, it is impossible to avoid suboptimal solutions.To overcome this problem, we propose to apply double right triangular MSFs to the self‐tuning of fuzzy reasoning. By using these MSFs, radically changing grades can be represented easily. In addition, by a technique of simulated annealing (SA) we propose to move the peak positions of the MSFs according to the progress of learning so as to arrange the MSFs in certain positions where the learning data are changing radically. Compared with the algorithm using piecewise linear MSFs, this new algorithm makes it possible to avoid suboptimal solutions more effectively. The advantages of this new technique are demonstrated by numerical examples involving function approximations. © 2003 Wiley Periodicals, Inc. Electr Eng Jpn, 144(4): 63–74, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.10168

Generalization of Yang et al.'s method for fuzzy programming with piecewise linear membership functions

Li and Yu (Fuzzy Sets and Systems 101 (1999) 109) argued that the model of Yang et al. for fuzzy programming (Fuzzy Sets and Systems 41 (1991) 39) is correct only for a specific type of piecewise linear membership function and proposed their model for other type of membership functions. This study generalizes the model of Yang et al. A numerical example indicates that the model of Li and Yu is inapplicable to a fuzzy programming problem that involves more than one membership function, whereas the proposed model gives the optimal solution.

Information granulation and signal quantization

Shows that signal quantization can be conveniently captured and quantified in the language of information granules. Optimal codebooks exploited in any signal quantization (discretization) lend themselves to the underlying fundamental issues of information granulation. The paper elaborates on and contrasts between various forms of information granulation such as set theory, shadowed sets, and fuzzy sets. It is revealed that a set‐based codebook can be easily enhanced by the use of the shadowed sets. This also raises awareness about the performance of the quantization process and helps increase its quality by defining additional elements of the codebook and specifying their range of applicability. We show how different information granules contribute to the performance of signal quantification. The role of clustering techniques giving rise to information granules is also analyzed. Some pertinent theoretical results are derived. It is shown that fuzzy sets defined in terms of piecewise linear membership functions with 1/2 overlap between any two adjacent terms of the codebook give rise to the effect of lossless quantization. The study addresses both scalar and multivariable quantization. Numerical studies are included to help illustrate the quantization mechanisms carried out in the setting of granular computing.

Fuzzy Control Using Piecewise Linear Membership Functions Based on Knowledge of Tuning a PID Controller

Tuning control parameters of a fuzzy controller depends on trial-and-error. How this can be accomplished efficiently is an important subject in fuzzy control that should be investigated. We propose a method in which membership functions of a fuzzy controller can be simply set up using the knowledge of tuning parameters in a conventional PID controller. In this method, fuzzy control is realized as follows: Piecewise linear membership functions determined using the knowledge of tuning parameters in a PID controller are set up in antecedent parts of four fuzzy control rules having simple structures. Then the simplified inference method that enables high-speed inference is applied to fuzzy control rules.

Solving a system of infinitely many fuzzy inequalities with piecewise linear membership functions

A cutting plane type algorithm for solving a system of infinitely many fuzzy inequalities with piecewise linear membership functions is proposed. In each iteration, we solve a finite nonlinear optimization problem and add one or two more constraints. The iterative process ends when an optimal solution is identified. A convergence proof, under some mild conditions, is given. An efficient implementation based on the concepts of constraint surrogation and maximum entropy is included. Some computational results are also reported.

折れ線型メンバーシップ関数による調整簡便なファジィ制御

折れ線型メンバーシップ関数による調整簡便なファジィ制御

Solving fuzzy inequalities with piecewise linear membership functions

This paper deals with systems of fuzzy inequalities. It shows that a system of fuzzy inequalities with piecewise linear membership functions can be converted to a one-constraint nonlinear programming problem by employing the concepts of surrogate constraints and maximum entropy. An augmented Lagrangean algorithm is then applied to solve the resulting problem. Some computational results are included.

Fuzzy Control Using Piecewise Linear Membership Functions Based on Knowledge of Tuning PID Controller.

Tuning of the control parameters of a fuzzy controller depends on trial and error. How this can be accomplished efficiently is an important subject that should be investigated in fuzzy control. We propose a method in which the membership functions of a fuzzy controller can be simply set up by utilizing the knowledge of tuning parameters in a conventional PID controller. In this method, fuzzy control is realized as follows: the piecewise linear membership functions that are determined from the knowledge of tuning parameters in a PID controller are set up in the antecedent parts of the four fuzzy control rules of a simple structure. Then, the simplified inference method that enables high-speed inference is applied to the fuzzy control rules.

Effects of membership function parameters on the performance of a fuzzy signal detector

This paper describes a signal-detection algorithm based on fuzzy logic. The detector combines evidence provided by two waveform features and explicitly considers uncertainty in the detection decision. The detector classifies waveforms including a signal, not including a signal, or being uncertain, in which case no conclusion regarding presence or absence of a signal is drawn. Piecewise linear membership functions are used, and a method to describe the membership functions in terms of two parameters is developed. The performance of the detector is compared to a Bayesian maximum likelihood detector, using brainstem auditory evoked potential signals in simulated noise, and the effects of the steepness (slope) and overlap of the membership functions on detector performance are evaluated. By varying the membership function steepness and overlap, the fuzzy detector can almost completely eliminate classification errors at the cost of a large number of uncertain classifications or it can be made to perform similarly to the Bayesian detector.

DETECTION OF ROLLER BEARING DEFECTS USING EXPERT SYSTEM AND FUZZY LOGIC

A fuzzy expert system, Bearing_Expert, has been developed. This expert system is PC-based. It is menu driven and it is very user-friendly. This expert system offers specific knowledge of various aspects about bearing monitoring, such as diagnostic methods, defect frequencies, feature selection, fuzzy bearing classification etc. By responding to different questions successively, the user can obtain very specific knowledge via the shortest path. In order to demonstrate the effectiveness of Bearing_Expert, experiments have been conducted. Radial accelerations have been measured and analysed. Feature selection has been performed effectively by the integration of Bearing_Expert and a computer program. The average of the top five values of frequency response in the high frequency region (5–22 kHz) has been found to be the best feature for the detection of roller bearing defects. Fuzzy reasoning has also been used via Bearing_Expert. Piecewise linear membership functions have been developed. A 100% reliability has been achieved for the detection of roller bearing defects via fuzzy reasoning. Bearing_Expert is very beneficial for factory automation and preventive maintenance.

折れ線型メンバーシップ関数を用いたファジィルールの自動チューニング

折れ線型メンバーシップ関数を用いたファジィルールの自動チューニング