Fuzzy nonlinear programming approach to the evaluation of manufacturing processes

A fuzzy robust nonlinear programming model is developed for the assessment of filter allocation and replacement strategies in hydraulic systems under uncertainty. It integrates the methods of fuzzy mathematic programming (FMP) and robust programming (RP) within the mixed integer nonlinear programming framework, and can facilitate dynamic analysis and optimization of filters allocation and replacement planning where the uncertainties are expressed as fuzzy membership functions. In modeling formulation, theory of contamination wear of typical hydraulic components is introduced to strengthen the presentation of relationship between system contamination and work performance. The fuzzy decision space is delimited into a more robust one by specifying the uncertainties through dimensional enlargement of the original fuzzy constraints. The piecewise linearization approach is employed to handle the nonlinearities of problem. The developed method has been applied to a case of planning filter allocation and replacement strategies under uncertainty and the generated optimal solution will help to reduce the total system cost and failure-risk level of the FPS.

Due to the complexity of the manufacturing environment, problems that can be solved by mathematical programming techniques are usually represented with non-linear programming (NP) models instead of linear programming (LP) models. When non-stochastic vagueness exists between the problem description and its corresponding NP model, fuzzy set theory can be applied to the mathematical model for the purpose of vividly representing the problem. This paper discusses the idea of the fuzzy NP model. Fuzzy set concepts are adapted to the NP objective function and constraints. An identical crisp NP model is derived from the fuzzy NP model for solving the problem numerically. Kuhn-Tucker conditions are addressed to determine the existence of a global optimal solution. A fuzzy machining economics model, which attempts to find optimal manufacturing parameters under vague elements of influence, is used to demonstrate the theory of the fuzzy NP model.

This paper presents an optimization method to solve a non-linear separable programming problem with coefficients and variables as generalized trapezoidal intuitionistic fuzzy numbers. Such optimization problems are known as fully intuitionistic fuzzy non-linear separable programming problems. The optimization method is based on the linear approximation of fully intuitionistic fuzzy non-linear separable functions. The concept of an intuitionistic fuzzy line segment between two intuitionistic fuzzy points is introduced to find the required linear approximation. In this way, a fully intuitionistic fuzzy non-linear programming problem is converted into an intuitionistic fuzzy linear programming problem. The defuzzification and component-wise comparison techniques are then used to convert the fully intuitionistic fuzzy linear programming problem to a linear programming problem with crisp coefficients which can then be solved by using traditional optimization techniques. The application of the proposed approach in an investment problem faced by a businessman has been presented.

It is sometimes necessary to consider more general nonlinear programs than the ones we have already studied: linear, quadratic, or conic programs. We give a brief introduction to this vast topic, and we discuss an application to estimating a volatility surface. Nonlinear Programming Consider a very general optimization problem of the form or the equivalent more concise form where. In the special case when all functions , hi are linear, problem (20.1) is a linear program as discussed in Chapter 2. When some of the functions , are nonlinear, problem (20.1) is a nonlinear program . Many practical problems are naturally formulated as nonlinear programs. We already saw quadratic programming and conic programming in earlier chapters. However, the family of problems that can be formulated as nonlinear programs is enormous, as many, if not most, imaginable kinds of constraints and objectives can be cast in terms of nonlinear functions. (For some noteworthy examples, see the exercises at the end of the chapter.) The immense modeling power of nonlinear programming comes at a cost: unlike linear, quadratic, and conic programming, which have a solid theory and are solvable via a few algorithmic templates, both the theory and methods to solve general nonlinear programs are far more complicated. Different types of nonlinear programs, determined by structural properties of the objective and constraint functions, are amenable to different types of algorithms. The subsequent sections sketch the main theory and most popular algorithmic ideas. A comprehensive treatment of this vast topic is beyond the scope of this textbook. We refer the reader to the excellent references Bertsekas (1999), Guler (2010), and Nocedal and Wright (2006) for more details. Numerical Nonlinear Programming Solvers There are numerous software packages for solving nonlinear programs. The following are some popular ones. We list them according to the class of algorithms (discussed in Section 20.4) that they are based on: (1) CONOPT, GRG2, Excel SOLVER. These solvers are based on the generalized reduced-gradient method . (2) MATLAB optimization toolbox, SNOPT, NLPQL. These solvers are based on sequential quadratic programming. (3) MINOS, LANCELOT. These solvers are based on an augmented Lagrangian approach. (4) MOSEK, LOQO, IPOPT. These solvers are based on interior-point methods.

This research proposesalpha ([Formula: see text]-cut Exponentially Weighted Moving Average (EWMA) and Cumulative Sum (CUSUM) control charts with fuzzy response observations in a manufacturing process under the existence of mean shift utilizing the fuzzy logic. In this research, the replicate’s observation is a fuzzy number represented by a triangular membership function, with the lower, average, and upper observation values. The fuzzy numbers are then normalized and assigned as input to the fuzzy logic, while a common output measure (COM) value is the output. Finally, the original values of the COM values are employed in developing the EWMA and CUSUM control charts with different [Formula: see text]-cut values. Three real case studies are adopted to illustrate the proposed EWMA and CUSUM control charts; including piston inside diameter, cap’s angel, and tablet weight. Results showed that the proposed EWMA and CUSUM control charts efficiently monitor fuzzy observations and detect the shift in process means. Moreover, the amount mean shift and [Formula: see text]-cut values affect the decision on process condition. In conclusion, the proposed approach is found effective in monitoring quality characteristic of fuzzy observations under mean shift which can be applied in a wide range of business applications.

Relaxed conditions for stability of nonlinear, continuous and discrete-time systems given by fuzzy models are presented. A theoretical analysis shows that the proposed methods provide better or at least the same results of the methods presented in the literature. Numerical results exemplify this fact. These results are also used for fuzzy regulators and observers designs. The nonlinear systems are represented by fuzzy models proposed by Takagi and Sugeno (1985). The stability analysis and the design of controllers are described by linear matrix inequalities, that can be solved efficiently using convex programming techniques. The specification of the decay rate, constrains on control input and output are also discussed.

This paper deals with the synthesis of fuzzy controller applied to the induction motor with a guaranteed model reference tracking performance. First, the Takagi-Sugeno (T-S) fuzzy model is used to approximate the nonlinear system in the synchronous d-q frame rotating with field-oriented control strategy. Then, a fuzzy state feedback controller is designed to reduce the tracking error by minimizing the disturbance level. The proposed controller is based on a T-S reference model in which the desired trajectory has been specified. The inaccessible rotor flux is estimated by a T-S fuzzy observer. The developed approach for the controller design is based on the synthesis of an augmented fuzzy model which regroups the model of induction machine, fuzzy observer, and reference model. The gains of the observer and controller are obtained by solving a set of linear matrix inequalities (LMIs). Finally, simulation and experimental results are given to show the performance of the observer-based tracking controller.

In this article, we would like to attempt finding a convergent near optimal solution of a fuzzy nonlinear programming problem (FNLPP) in which the objective as well as the constraints defining the relationship between variables are so imprecisely stated that it is practically impossible to represent by any existing crisp model. The primary aim of this research is to provide more flexibility to the decision-maker. Accordingly, to solve such a FNLPP, we use approximate reasoning which involves three stages: fuzzification, aggregation and defuzzification. The process is repeated till a convergent near optimal solution is attained. An algorithm is designed for the implementation of the proposal. Artificial examples have been considered to illustrate our approach and existing results are used for comparison. Finally, a case study is undertaken to demonstrate the effectiveness of our proposal.

A new method to design a fuzzy bilinear observer (FBO) with unknown inputs is developed for a class of nonlinear systems. The nonlinear system is modeled as a fuzzy bilinear model (FBM). This kind of T‐S fuzzy model is especially suitable for a nonlinear system with a bilinear term. The proposed fuzzy bilinear observer subject to unknown inputs is developed to ensure the asymptotic convergence of the error dynamic using the Lyapunov method. The proposed design conditions are given in linear matrix inequality (LMI) formulation. The paper studies also the problem of fault detection and isolation. An unknown input fuzzy bilinear fault diagnosis observer design is proposed. This work is given for both continuous and discrete cases of fuzzy bilinear models. Illustrative examples are chosen to provide the effectiveness of the given methodology.

: Several algorithms for solving problems in linear, quadratic, and nonlinear programming, network flows, and facilities location were developed and analyzed. Results include: (1) The analysis of the computational complexity of the problem of determining an optimally sparse representation of the null space of a matrix, and the development of worst-case bounds for the shadow-vertex simplex algorithm and several heuristics for distance constrained discrete facility location problems and conditional covering problems; (2) The development of efficient algorithms for dense and sparse assignment problems, strictly convex quadratic programming problems, and a nonlinear programming problem that arises when maximizing a correlation coefficient subject to linear constraints; and (3) The application of iterative methods to large sparse equality-constraind quadratic programs and the development of multiple constraint deletion strategies for active-set algorithms for linearly consrained nonlinear programming problems. (Author)

A nonlinear programming approach to kinematic shakedown analysis of frictional materials

Tensor product model transformation based fuzzy observer design for the unmanned helicopter is studied in this work. Different from the traditional Takagi-Sugeno (TS) fuzzy type observer design, tensor product model transformation based fuzzy observer is designed, and the quasi linear parameter varying polytopic model is involved in observer design, and the principle aims to implement the observer design is its numerical implementation with fuzzy model representation, the distributed gains of the observer can be calculated via higher order singular value and linear matrix inequalities(LMI). Finally, the tensor product model transformation based fuzzy observer is compared with TS fuzzy observer. The simulation result shows that the observation performance of tensor product distributed compensation (TPDC) observer is better than that of TS observer.

A multi-item cost minimising stochastic inventory model is developed in stochastic environment. Demand is considered to be function of income level of consumer and price of commodity. Budget, income and price are assumed to be random and their randomness is expressed by normal distribution. Deterioration rate is modelled using Weibull distribution. Stochastic inventory model is converted into equivalent crisp inventory model by using stochastic nonlinear programming (SNLP) technique. Crisp problem is solved by fuzzy nonlinear programming (FNLP) technique and intuitionistic fuzzy optimisation (IFO) technique. Sensitivity analysis of the optimum solution with respect to feasible changes in average and variance of income level and price is discussed and conclusions are noted.

The concept of duality plays an important role in optimization theory. This paper discusses some relations between primal and dual nonlinear programming problems in fuzzy environment. Here, fuzzy feasible region for a general fuzzy nonlinear programming is formed and the concept of fuzzy feasible solution is defined. First order dual relation for fuzzy nonlinear programming problem is studied.

Synthesis of a non dispersive solvent extraction plant for effluent treatment and metal recovery