Binary fuzzy goal programming approach to single model straight and U-shaped assembly line balancing

Assembly line balancing generally requires a set of acceptable solutions to the several conflicting objectives. In this study, a binary fuzzy goal programming approach is applied to assembly line balancing. Models for balancing straight and U-shaped assembly lines with fuzzy goals (the number of workstations and cycle time goals) are proposed. The binary fuzzy goal programming models are solved using the methodology introduced by Chang [Chang, C.T., 2007. Binary fuzzy goal programming. European Journal of Operational Research 180 (1), 29–37]. An illustrative example is presented to demonstrate the validity of the proposed models and to compare the performance of straight and U-shaped line configurations.     

Fractional Goal Programming for Fuzzy Solid Transportation Problem with Interval Cost

In this paper, we study a solid transportation problem with interval cost using fractional goal programming approach (FGP). In real life applications of the FGP problem with multiple objectives, it is difficult for the decision-maker(s) to determine the goal value of each objective precisely as the goal values are imprecise, vague, or uncertain. Therefore, a fuzzy goal programming model is developed for this purpose. The proposed model presents an application of fuzzy goal programming to the solid transportation problem. Also, we use a special type of non-linear (hyperbolic) membership functions to solve multi-objective transportation problem. It gives an optimal compromise solution. The proposed model is illustrated by using an example.     

Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach

In this paper, two new algorithms are presented to solve multi-level multi-objective linear programming (ML-MOLP) problems through the fuzzy goal programming (FGP) approach. The membership functions for the defined fuzzy goals of all objective functions at all levels are developed in the model formulation of the problem; so also are the membership functions for vectors of fuzzy goals of the decision variables, controlled by decision makers at the top levels. Then the fuzzy goal programming approach is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtain the most satisfactory solution for all decision makers.     

A displayed inventory model with L–R fuzzy number

In this study, we formulate a multi-item displayed inventory model under shelf-space constraint in fuzzy environment. Here demand rate of an item is considered as a function of the displayed inventory level. The problem is formulated to maximize average profit. In real life situation, the goals and inventory parameters are may not precise. Such type of uncertainty may be characterized by fuzzy numbers. Here, the constraint goal and the inventory cost parameters are assumed to be triangular shaped fuzzy numbers with different types of left and right membership functions. The fuzzy numbers are then approximated to a nearest interval number. Using arithmetic of interval numbers, the problem is described as a multi-objective inventory problem. The problem is then solved by fuzzy geometric programming approach. Finally a numerical example is given to illustrate the problem.     

The Advantages of Fuzzy Optimization Models in Practical Use

Classical mathematical programming models require well-defined coefficients and right hand sides. In order to avoid a non satisfying modeling usually a broad information gathering and processing is necessary. In case of real problems some model parameters can be only roughly estimated. While in case of classical models the vague data is replaced by "average data", fuzzy models offer the opportunity to model subjective imaginations of the decision maker as precisely as a decision maker will be able to describe it. Thus the risk of applying a wrong model of the reality and selecting solutions which do not reflect the real problem can be clearly reduced. The modeling of real problems by means of deterministic and stochastic models requires extensive information processing. On the other hand we know that an optimum solution is finally defined only by few restrictions. Especially in case of larger systems we notice afterwards that most of the information is useless. The dilemma of data processing is due to the fact that first we have to calculate the solution in order to define, whether the information must be well-defined or whether vague data may be sufficient. Based on multicriteria programming problems it should be demonstrated that the dilemma of data processing in case of real programming problems can be handled adequately by modeling them as fuzzy system combined with an interactive problem-solving. Describing the real problem by means of a fuzzy system first of all only the available information or such information which can be achieved easily will be considered. Then we try to develop an optimum solution. With reference to the cost-benefit relation further information can be gathered in order to describe the solution more precisely. Furthermore it should be pointed out that some interactive fuzzy solution algorithms, e.g. FULPAL provide the opportunity to solve mixed integer multicriteria programming models as well.     

Comment to “interactive fuzzy goal programming approach for bilevel programming problem” by S.R. Arora and R. Gupta

The fuzzy bilevel programming problem is solved in the paper [1] using an approach of fuzzy goal programming. Using a simple example we show that this will not lead to a satisfactory solution. Especially the fuzzy constraint that the upper level variable is near some desired one has no influence on the computed solution in the example. This constraint is used to model the hierarchy in the approach in [1]. At the end of the paper we suggest one straightforward possibility for modeling the fuzzy bilevel programming problem and converting it into a crisp substitute.     

Revised multi-choice goal programming

Chang [C.-T. Chang, Multi-choice goal programming, Omega, The Inter. J. Manage. Sci. 35 (2007) 389–396] has recently proposed a new method namely multi-choice goal programming (MCGP) for multi-objective decision problems. The multi-choice goal programming allows the decision maker to set multi-choice aspiration levels for each goal to avoid underestimation of the decision. However, to express the multi-choice aspiration levels, multiplicative terms of binary variables are involved in their model. This leads to difficult implementation and it is not easily understood by industrial participants. In this paper, we propose an alternative method to formulate the multi-choice aspiration levels with two contributions: (1) the alternative approach does not involve multiplicative terms of binary variables, this leads to more efficient use of MCGP and is easily understood by industrial participants, and (2) the alternative approach represents a linear form of MCGP which can easily be solved by common linear programming packages, not requiring the use of integer programming packages. In addition, a new concept of constrained MCGP is introduced for constructing the relationships between goals in this paper. Finally, to demonstrate the usefulness of the proposed method, an illustrate example is included.     

Multiple goal non-cooperative conflicts under uncertainty: A fuzzy set approach

This paper investigates strategy selection for a participant in a two-party non-cooperative conflict which involves both uncertainty and multiple goals. Uncertainty arises from the players not knowing the utility functions. Multiple objectives appear as the result of the payoff being a vector of prizes and the players attempt to attain various goals for each prize separately. The main objective is to present a fuzzy set/fuzzy programming solution concept to the conflict situation. In doing so, we compare a Bayesian player to one that employs fuzzy set techniques. We point out some of the advantages of the fuzzy set method. The necessary computations in the fuzzy set method are explained in detail through an example.     

Fuzzy stochastic goal programming problems

In this paper, we present a model to measure attainment value of fuzzy stochastic goals. Then, the new measure is used to de-randomize and de-fuzzify the fuzzy stochastic goal programming problem and obtain a standard linear program (LP). A numerical example is provided to illustrate the proposed method.     

Tackling an MCDM-problem with the help of some results from fuzzy set theory

The problem to be addressed and tackled in this paper arose as a byproduct from some efforts at solving problems involving multiple goals by linking linear and goal programming models. The critical issue was that some forms for interdependence among the goals could not be handled in the programming models. Here we will deal with a set of goals — with realistic counterparts in a Finnish plywood industry — in which a subset of the goals are (i) conflicting, another subset (ii) unilaterally supporting and a third subset (iii) mutually supporting. It is furthermore observed that the elements of a studied set of goals may be partly independent and partly interdependent, which makes the context a fullfledged MCDM-problem. It is tackled with a technique which is based on the theory of fuzzy sets, the conceptual framework for fuzzy decisions and the algorithms developed for fuzzy mathematical programming. The resulting fuzzy multiobjective programming model is simplified and tested with the help of a fairly complex numerical example.     

Priority based fuzzy goal programming technique to fractional fuzzy goals using dynamic programming

This paper describes the use of preemptive priority based fuzzy goal programming method to fuzzy multiobjective fractional decision making problems under the framework of multistage dynamic programming. In the proposed approach, the membership functions for the defined objective goals with fuzzy aspiration levels are determined first without linearizing the fractional objectives which may have linear or nonlinear forms. Then the problem is solved recursively for achievement of the highest membership value (unity) by using priority based goal programming methodology at each decision stages and thereby identifying the optimal decision in the present decision making arena. A numerical example is solved to represent potentiality of the proposed approach.     

A fuzzy goal programming method with imprecise goal hierarchy

Two most widely used approaches to treating goals of different importance in goal programming (GP) are: (1) weighted GP, where importance of goals is modelled using weights, and (2) preemptive priority GP, where a goal hierarchy is specified implying infinite trade-offs among goals placed in different levels of importance. These approaches may be too restrictive in modelling of real life decision making problems. In this paper, a novel fuzzy goal programming method is proposed, where the hierarchical levels of the goals are imprecisely defined. The imprecise importance relations among the goals are modelled using fuzzy relations. An additive achievement function is defined, which takes into consideration both achievement degrees of the goals and degrees of satisfaction of the fuzzy importance relations. Examples are given to illustrate the proposed method.     

A new risk criterion in fuzzy environment and its application

Since the observed values of security returns in real-world problems are sometimes imprecise or vague, an increasing effort in research is devoted to study the properties of risk measures in fuzzy portfolio optimization problems. In this paper, a new risk measure is suggested to gauge the risk resulted from fuzzy uncertainty. For this purpose, the absolute deviation and absolute semi-deviation are first defined for fuzzy variable by nonlinear fuzzy integrals. To compute effectively the absolute semi-deviations of single fuzzy variable as well as its functions, this paper discusses the methods of computing the absolute semi-deviation by classical Lebesgue–Stieltjes (L–S) integral. After that, several useful absolute deviation and absolute semi-deviation formulas are established for common triangular, trapezoidal and normal fuzzy variables. Applying the absolute semi-deviation as a new risk measure in portfolio optimization, three classes of fuzzy portfolio optimization models are developed by combining the absolute semi-deviation with expected value operator and credibility measure. Based on the analytical representation of absolute semi-deviations, the established fuzzy portfolio selection models can be turned into their equivalent piecewise linear or fractional programming problems. Since the absolute semi-deviation is a piecewise fractional function and pseudo-convex on the feasible subregions of deterministic programming models, we take advantage of the structural characteristics to design a domain decomposition method to separate a deterministic programming problem into three convex subproblems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the effectiveness of the solution method.     

Bi-matrix Games with Fuzzy Goals and Fuzzy

A bi-matrix game with fuzzy goal is shown to be equivalent to a (crisp) non-linear programming problem in which the objective as well as all constraint functions are linear except two constraint functions, which are quadratic. This equivalence is further extended to bi-matrix games with fuzzy pay-offs, as well as to bi-matrix games with fuzzy goals and fuzzy payoffs, whose equilibrium strategies are conceptualized by employing a suitable ranking (defuzzification) function.     

Fuzzy goal programming with nested priorities

This paper proposes a new approach to formulating fuzzy priorities in a goal programming problem. The proposed methodology remedies certain shortcomings of the composite membership function approach discussed in previous works [7, 10]. The principal advantage of the proposed method is that it leads to a formulation in which tradeoffs between goals more closely reflect the decision maker's intentions than in other noninteractive approaches [8, 9, 10, 14], in some of which a fixed hierarchy of goals is assumed.     

Reliability and cost analysis of series system models using fuzzy parametric geometric programming

In this paper, we have discussed series system models with system reliability and cost. We have considered two types of the model; the former focuses on a problem of optimal reliability for series system with cost constraint and the latter is a center system cost model with reliability goal. It is necessary to improve the reliability of the system under limited available cost of system and also to minimize the systems cost subject to target goal of the reliability. Practically, cost of components has always been imprecise with vague in nature. So they are taken as fuzzy in nature and the reliability models are formulated as a fuzzy parametric geometric programming problem. Numerical examples are given to illustrate the model through fuzzy parametric geometric programming technique.     

A multi-objective joint replenishment inventory model of deteriorated items in a fuzzy environment

In this study, a fuzzy multi-objective joint replenishment inventory model of deteriorating items is developed. The model maximizes the profit and return on inventory investment (ROII) under fuzzy demand and shortage cost constraint. We propose a novel inverse weight fuzzy non-linear programming (IWFNLP) to formulate the fuzzy model. A soft computing, differential evolution (DE) with/without migration operation, is proposed to solve the problem. The performances of the proposed fuzzy method and the conventional fuzzy additive goal programming (FAGP) are compared. We show that the solution derived from the IWFNLP method satisfies the decision maker’s desirable achievement level of the profit objective, ROII objective and shortage cost constraint goal under the desirable possible level of fuzzy demand. It is an effective decision tool since it can really reflect the relative importance of each fuzzy component.     

Multi-product production planning: A goal programming approach

Cost minimization multi-product production problems with static production resource usage and internal product flow requirements have been solved by linear programming (LP) with input/output analysis. If the problem is complicated by interval resource estimates, interval linear programming (ILP) can be used. The solution of realistic problems by the above method is cumbersome. This paper suggests that linear goal programming (LGP) can be used to model a multi-product production system. LGP's unique modeling capabilities are used to solve a production planning problem with variable resource parameters. Input/output analysis is used to determine the technological coefficients for the goal constraints and is also used to derive an information sub-model that is used to reduce the number of variable resource goal constraints. Preliminary findings suggest that the LGP approach is more cost-efficient (in terms of CPU time) and in addition provides valuable information for aggregate planning.     

Enhanced interactive satisficing method via alternative tolerance for fuzzy goal programming with progressive preference

This paper proposes an enhanced interactive satisficing method via alternative tolerance for fuzzy goal programming with progressive preference. The alternative tolerances of the fuzzy objectives with three types of fuzzy relations are used to model progressive preference of decision maker. In order to improve the dissatisficing objectives, the relaxed satisficing objectives are sacrificed by modifying their tolerant limits. By means of attainable reference point, the auxiliary programming is designed to generate the tolerances of the dissatisficing objectives for ensuring feasibility. Correspondingly, the membership functions are updated or the objective constraints are added. The Max–Min goal programming model (or the revised one) and the test model of the M-Pareto optimality are solved lexicographically. By our method, the dissatisficing objectives are improved iteratively till the preferred result is acquired. Illustrative examples show its power.     

Pattern Classification by Linear Goal Programming and its Extensions

Pattern classification is one of the main themes in pattern recognition, and has been tackled by several methods such as the statistic one, artificial neural networks, mathematical programming and so on. Among them, the multi-surface method proposed by Mangasarian is very attractive, because it can provide an exact discrimination function even for highly nonlinear problems without any assumption on the data distribution. However, the method often causes many slits on the discrimination curve. In other words, the piecewise linear discrimination curve is sometimes too complex resulting in a poor generalization ability. In this paper, several trials in order to overcome the difficulties of the multi-surface method are suggested. One of them is the utilization of goal programming in which the auxiliary linear programming problem is formulated as a goal programming in order to get as simple discrimination curves as possible. Another one is to apply fuzzy programming by which we can get fuzzy discrimination curves with gray zones. In addition, it will be shown that using the suggested methods, the additional learning can be easily made. These features of the methods make the discrimination more realistic. The effectiveness of the methods is shown on the basis of some applications.