A goal programming approach for fuzzy multiobjective fractional programming problems

Fuzzy multiple objective fractional programming (FMOFP) is an important technique for solving many real-world problems involving the nature of vagueness, imprecision and/or random. Following the idea of binary behaviour of fuzzy programming (Chang 2007 Chang, C-T. 2007. Binary Behavior of Fuzzy Programming with Piecewise Membership Functions. IEEE Transactions on Fuzzy Systems, 15: 342–349.  [Google Scholar]), there may exist a situation where a decision-maker would like to make a decision on FMOFP involving the achievement of fuzzy goals, in which some of them may meet the behaviour of fuzzy programming (i.e. level achieved) or the behaviour of binary programming (i.e. completely not achieved). This is turned into a fuzzy multiple objective mixed binary fractional programming (FMOMBFP) problem. However, to the best of our knowledge, this problem is not well formulated by mathematical programming. Therefore, this article proposes a linearisation strategy to formulate the FMOMBFP problem in which extra binary variable is not required. In addition, achieving the highest membership value of each fuzzy goal defined for the fractional objective function, the proposed method can alleviate the computational difficulties when solving the FMOMBFP problem. To demonstrate the usefulness of the proposed method, a real-world case is also included.     

Application of fuzzy goal programming approach to multi-objective linear fractional inventory model

In this paper, we propose a model and solution approach for a multi-item inventory problem without shortages. The proposed model is formulated as a fractional multi-objective optimisation problem along with three constraints: budget constraint, space constraint and budgetary constraint on ordering cost of each item. The proposed inventory model becomes a multiple criteria decision-making (MCDM) problem in fuzzy environment. This model is solved by multi-objective fuzzy goal programming (MOFGP) approach. A numerical example is given to illustrate the proposed model.     

A two-phase approach for multi-objective programming problems with fuzzy coefficients

In this study, a two-phase procedure is introduced to solve multi-objective fuzzy linear programming problems. The procedure provides a practical solution approach, which is an integration of fuzzy parametric programming (FPP) and fuzzy linear programming (FLP), for solving real life multiple objective programming problems with all fuzzy coefficients. The interactive concept of the procedure is performed to reach simultaneous optimal solutions for all objective functions for different grades of precision according to the preferences of the decision-maker (DM). The procedure can be also performed to obtain lexicographic optimal and/or additive solutions if it is needed. In the first phase of the procedure, a family of vector optimization models is constructed by using FPP. Then in the second phase, each model is solved by FLP. The solutions are optimal and each one is an alternative decision plan for the DM.     

Interactive intuitionistic fuzzy methods for multilevel programming problems

Multilevel programming problems model a decision-making process with a hierarchy structure. Traditional solution methods including vertex enumeration algorithms and penalty function methods are not only inefficient to obtain the solution of the multilevel programming problems, but also lead to a paradox that the follower’s decision power dominates the leader’s. In this paper, both multilevel programming and intuitionistic fuzzy set are used to model problems in hierarchy expert and intelligent systems. We first present a score function to objectively depict the satisfactory degrees of decision makers by virtue of the intuitionistic fuzzy set for solving multilevel programming problems. Then we develop three optimization models and three interactive intuitionistic fuzzy methods to consider different satisfactory solutions for the requirements of expert decision makers. Furthermore, a new distance function is proposed to measure the merits of a satisfactory solution. Finally, a case study for cloud computing pricing problems and several numerical examples are given to verify the applicability and the effectiveness of the proposed models and methods.     

Genetic algorithm-based fuzzy goal programming for class of chance-constrained programming problems

This paper presents a procedure for solving a multiobjective chance-constrained programming problem. Random variables appearing on both sides of the chance constraint are considered as discrete random variables with a known probability distribution. The literature does not contain any deterministic equivalent for solving this type of problem. Therefore, classical multiobjective programming techniques are not directly applicable. In this paper, we use a stochastic simulation technique to handle randomness in chance constraints. A fuzzy goal programming formulation is developed by using a stochastic simulation-based genetic algorithm. The most satisfactory solution is obtained from the highest membership value of each of the membership goals. Two numerical examples demonstrate the feasibility of the proposed approach.     

An extended outranking approach for multi-criteria decision-making problems with linguistic intuitionistic fuzzy numbers

Linguistic intuitionistic fuzzy numbers (LIFNs), characterized by a linguistic membership degree, linguistic non-membership degree, and linguistic indeterminacy degree, represent a helpful tool for depicting uncertain information under complex environments. This paper focuses on developing an innovative method to address multi-criteria decision-making (MCDM) problems with LIFNs in which the weight information is completely unknown. First, the distance of LIFNs is defined with the aid of linguistic scale functions (LSFs). Second, some extended outranking relationships between each pair of LIFNs are proposed based on the elicitation of the classic relation models. Moreover, a ranking method is constructed to deal with MCDM problems according to the proposed outranking relationships of LIFNs. Finally, an illustrative example concerning coal mine safety evaluation is provided to demonstrate the proposed method, and its feasibility and validity are further verified by a sensitivity analysis and comparison with other existing methods.     

A hybrid fuzzy goal programming approach with different goal priorities to aggregate production planning

In this study a hybrid (including qualitative and quantitative objectives) fuzzy multi objective nonlinear programming (H-FMONLP) model with different goal priorities will be developed for aggregate production planning (APP) problem in a fuzzy environment. Using an interactive decision making process the proposed model tries to minimize total production costs, carrying and back ordering costs and costs of changes in workforce level (quantitative objectives) and maximize total customer satisfaction (qualitative objective) with regarding the inventory level, demand, labor level, machines capacity and warehouse space. A real-world industrial case study demonstrates applicability of proposed model to practical APP decision problems. GENOCOP III (Genetic Algorithm for Numerical Optimization of Constrained Problems) has been used to solve final crisp nonlinear programming problem.     

区间直觉模糊动态规划方法

基于直觉模糊集理论的思想与方法,探讨多阶段决策问题．运用分析技巧,构建直觉模糊集比较可能度公式及区间直觉模糊集比较可能度公式．在普通动态规划的基础上,提出区间直觉模糊动态规划及其最优解的概念,建立相应的数学模型及其最优解的算法,并指出直觉模糊动态规划是区间直觉模糊动态规划的特例．通过算例分析,说明了算法的合理性和可行性．为不确定动态规划和直觉模糊集理论的应用研究提供了新的思路．     

A goal programming approach to fuzzy linear regression with fuzzy input�Coutput data

Fuzzy linear regression is an active area of research. In the literature, fuzziness is considered in outputs and/or in inputs. This paper focuses on both fuzzy inputs and fuzzy outputs. First, some approximations for multiplication of two triangular fuzzy numbers are introduced. Then, to evaluate the fuzzy linear regression, the best approximation is selected to minimize a suitable function via goal programming. An important feature of the proposed model is that it takes into account the centers of fuzzy data as well as their spreads. Moreover, it is flexible to deal with both symmetric and non-symmetric data. Furthermore, it can handle the crisp inputs and trapezoidal fuzzy outputs easily. To show the efficiency of the proposed model, some numerical examples are solved and compared with some earlier methods.     

Development and optimization of unrestricted LR-type intuitionistic fuzzy mathematical programming problems

To the best of our knowledge, there is no method to find the product of unrestricted LR-type intuitionistic fuzzy numbers (IFNs) as well as the optimal solutions of LR-type intuitionistic fuzzy linear programming problems (IFLPPs) in which some or all the decision variables are unrestricted. Therefore it is necessary to pay attention to this issue and there is need to find the product of unrestricted LR-type IFNs as well as the optimal solutions of such programming problems. In this paper, the product of unrestricted LR-type IFNs based on the α-cut, β-cut and (α, β)-cut is proposed. Then with the help of proposed product, a new method is proposed to find the optimal solutions of unrestricted LR-type IFLPPs. Finally, an illustrative example is given to support the proposed method and investigated the applicability of existing approaches exist in the literature.     

Interactive balance space approach for solving multi-level multi-objective programming problems

This paper studies a multi-level multi-objective decision-making (ML-MODM) problems with linear or non-linear constraints. The objective functions at each level are non-linear functions, which are to be maximized or minimized.This paper presents a three-level multi-objective decision-making (TL-MODM) model and an interactive algorithm for solving such a model. The algorithm simplifies three-level multi-objective decision-making problems by transforming them into separate multi-objective decision making problems at each level, thereby avoiding the difficulty associated with non-convex mathematical programming. Our algorithm is an extension of the work of Shi and Xia [X. Shi, H. Xia, Interactive bi-level multi-objective decision making, Journal of the Operational Research Society 48 (1997) 943-949], which dealt with interactive bi-level multi-objective decision-making problems, with some modifications in assigning satisfactoriness to each objective function in all the levels of the TL-MODM problem. Also, we solve each separate multi-objective decision making problem of the TL-MODM problem by the balance space approach.A new formula is introduced to interconnect the satisfactoriness and the proportions of deviation needed to reflect the relative importance of each objective function. Thus, we have the proportions of deviation including satisfactoriness.In addition, we present new definitions for the satisfactoriness and the preferred solution in view of singular-level multi-objective decision making problems that corresponds to the η-optimal solution of the balance space approach. Also, new definitions for the feasible solution and the preferred solution (η-optimal point) of the TL-MODM problem are presented. An illustrative numerical example is given to demonstrate the algorithm.     

支付值为区间直觉模糊集的矩阵对策的线性规划求解方法

提出了支付值为区间直觉模糊集的矩阵对策定义及其解的概念,将求解局中人的极大-极小与极小-极大策略问题转化为求解一对辅助的非线性多目标规划,进而转化为一对易于求解的原始-对偶线性规划.数值实例表明了所提方法的有效性和实用性.所提出的区间直觉模糊集矩阵对策理论与方法既是对经典矩阵对策理论的发展,又可为解决其他带有区间直觉模糊信息的对策问题提供新的途径.     

An approach for multiple attribute group decision making problems with interval-valued intuitionistic trapezoidal fuzzy numbers

This article proposes an approach to resolve multiple attribute group decision making (MAGDM) problems with interval-valued intuitionistic trapezoidal fuzzy numbers (IVITFNs). We first introduce the cut set of IVITFNs and investigate the attitudinal score and accuracy expected functions for IVITFNs. Their novelty is that they allow the comparison of IVITFNs by taking into accounting of the experts’ risk attitude. Based on these expected functions, a ranking method for IVITFNs is proposed and a ranking sensitivity analysis method with respect to the risk attitude is developed. To aggregate the information with IVITFNs, we study the desirable properties of the interval-valued intuitionistic trapezoidal fuzzy weighted geometric (IVITFWG) operator, the interval-valued intuitionistic trapezoidal fuzzy ordered weighted geometric (IVITFOWG) operator, and the interval-valued intuitionistic trapezoidal fuzzy hybrid geometric (IVITFHG) operator. It is worth noting that the aggregated value by using these operators is also an interval-valued intuitionistic trapezoidal fuzzy value. Then, based on these expected functions and aggregating operators, an approach is proposed to solve MAGDM problems in which the attribute values take the form of interval-valued intuitionistic fuzzy numbers and the expert weights take the form of real numbers. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

An evaluation approach to logistics service using fuzzy theory,quality function development and goal programming

Logistics customer service is an important factor in the success of supply chain management. The aim of this study is to propose a novel approach for customer service management. For the improvement of logistics service operations, the proposed method integrates quality function development (QFD), fuzzy extended analytic hierarchy process (FEAHP), and multi-segment goal programming (MSGP). The advantage of the method includes the consideration of various logistics goals and the flexibility of setting multi-aspiration levels of evaluation criteria.     

Genetic based fuzzy goal programming for multiobjective chance constrained programming problems with continuous random variables

Solution procedure consisting of fuzzy goal programming and stochastic simulation-based genetic algorithm is presented, in this article, to solve multiobjective chance constrained programming problems with continuous random variables in the objective functions and in chance constraints. The fuzzy goal programming formulation of the problem is developed first using the stochastic simulation-based genetic algorithm. Without deriving the deterministic equivalent, chance constraints are used within the genetic process and their feasibilities are checked by the stochastic simulation technique. The problem is then reduced to an ordinary chance constrained programming problem. Again using the stochastic simulation-based genetic algorithm, the highest membership value of each of the membership goal is achieved and thereby the most satisfactory solution is obtained. The proposed procedure is illustrated by a numerical example.     

An efficient procedure for finding best compromise solutions to the multi-objective assignment problem

In this paper, we consider the problem of determining a best compromise solution for the multi-objective assignment problem. Such a solution minimizes a scalarizing function, such as the weighted Tchebychev norm or reference point achievement functions. To solve this problem, we resort to a ranking (or k-best) algorithm which enumerates feasible solutions according to an appropriate weighted sum until a condition, ensuring that an optimal solution has been found, is met. The ranking algorithm is based on a branch and bound scheme. We study how to implement efficiently this procedure by considering different algorithmic variants within the procedure: choice of the weighted sum, branching and bounding schemes. We present an experimental analysis that enables us to point out the best variants, and we provide experimental results showing the remarkable efficiency of the procedure, even for large size instances.     

Standard goal programming with fuzzy hierarchies: a sequential approach

This paper proposes a pragmatic model for multi-objective decision-making processes involving clusters of objectives which have a decisional meaning for the decision maker (DM). We provide the DMs with a comfortable tool that allows them to express their preferences both by comparing criteria of the same cluster and via the comparison between the different clusters. In standard goal programming the importance of the goals is modeled by the introduction of preferential weights or/and the incorporation of pre-emptive priorities. However, in many cases the DM is not able to establish a precise preference structure. Even in the case of precise weights the solution does not match necessarily the relative weights or, in the case of precise pre-emptive priority, the result could be very restrictive. In order to overcome these drawbacks, in this paper the normalized unwanted deviations are interpreted in terms of achievement degrees of the goals and fuzzy relations are used to model the relative importance of the goals. Thus, we show how several methodologies from the fuzzy goal programming literature can be tailored for solving standard GP problems. We apply this new modeling to problems where there is a “natural” clustering between goals of the same class. We address this situation by solving two phases; in the first one each class is handled separately taking into account the hierarchy of their goals and, in the second phase, we integrate the results of the first phase and the imprecise hierarchy of the different classes. We formulate a new goal programming model called as sequential goal programming with fuzzy hierarchy model. Because many real situations involve decision making in this environment, our proposal can be a useful tool of broad application. A numerical example illustrates the methodology.     

Using binary fuzzy goal programming and linear programming to resolve airport logistics center expansion plan problems

In recent years, the number of direct flights between Taiwan and mainland China has grown rapidly, as charter flights have been turned into regular flights. This important issue has prompted airport ground handling service (AGHS) companies in Taiwan to enhance convenient services for passengers and to invest in airport logistics center expansion plans (ALCEP) to broaden the AGHS market. Due to their budgetary restrictions, AGHS companies need to outsource many of their services to contractors to implement these plans. This study proposes an ALCEP solution procedure to guide AGHS companies in adjusting their priority goals and selecting the best contractor according to their needs. This proposed procedure successfully solves the ALCEP problem and facilitates the assignment of contractors by considering both qualitative and quantitative methods.     

A mathematical programming approach to multi-attribute decision making with interval-valued intuitionistic fuzzy assessment information

This article proposes an approach to handle multi-attribute decision making (MADM) problems under the interval-valued intuitionistic fuzzy environment, in which both assessments of alternatives on attributes (hereafter, referred to as attribute values) and attribute weights are provided as interval-valued intuitionistic fuzzy numbers (IVIFNs). The notion of relative closeness is extended to interval values to accommodate IVIFN decision data, and fractional programming models are developed based on the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method to determine a relative closeness interval where attribute weights are independently determined for each alternative. By employing a series of optimization models, a quadratic program is established for obtaining a unified attribute weight vector, whereby the individual IVIFN attribute values are aggregated into relative closeness intervals to the ideal solution for final ranking. An illustrative supplier selection problem is employed to demonstrate how to apply the proposed procedure.     

一类直觉模糊线性规划的求解及其应用

研究一类直觉模糊线性规划及其应用.首先,定义直觉模糊不等式,给出直觉模糊线性规划模型;然后,提出一种基于总精确函数的直觉模糊线性规划求解方法,并给出其求解步骤;最后,建立证劵投资组合的直觉模糊线性规划模型.数值算例表明了所提出理论是合理有效的.