Fuzzy Goal Programming in the Case of Exponential Membership Functions with Quasiconcave Piecewise Linear Exponents

A fuzzy nonlinear goal programming approach is presented for solving multiobjective optimization problems involving vague and imprecise information. Several computational models, including simple additive, weighted additive, and preemptive priority models, are given for the numerical solution of the problem. The methodolo- gies are illustrated with the help of two structural optimization problems involving multiple goals. The solution of the first example is obtained using a graphical procedure whereas the second example is solved using nonlinear programming techniques. Linear membership functions are used in the numerical work for simplicity. The methodologies presented in this work aid in the preliminary design of structural systems involving imprecise and vague information about the goals and/or constraints. INEAR goal programming has been extensively used in solving decision-making problems involving linear equa- tions and multiple conflicting goals. The goals can be rank ordered depending on their importance to the decision maker. Goal programming attempts to achieve as man}7 of these goals as possible by minimizing deviational variables from the goal levels depending on their relative weights. Linear goal pro- gramming algorithms were developed by Charnes et al., 1 Ig- nizio,2 and Zanakis and Gupta.3 The extension of goal pro- gramming to nonlinear optimization problems has also been considered by several authors.3'4 A fuzzy programming ap- proach for linear programming problems with several objec- tives was suggested by Zimmerman. 5 Subsequently, several authors proposed different fuzzy goal programming ap- proaches for solving linear goal programming problems in- volving imprecise statements and information.610 Narasimhan6 suggested a method for solving fuzzy linear goal programming problems under the assumption of linear membership functions. His method involves solving a set of 2k linear programming problems, each containing 3k constraints where k denotes the number of goals in the original problem. Hannan7 indicated a procedure for formulating a fuzzy goal programming problem as an equivalent single linear program- ming problem with 2k goal-related constraints. The problems associated with the definition of fuzzy priorities were dis- cussed in Ref. 7. A brief review of the history and the state of the art in fuzzy multicriteria programming as of 1982 was presented by Ignizio.8 The distinction between fuzzy goal pro- gramming and fuzzy multicriteria formulations was given in Ref. 9. Models are presented by Hannan7 for the use of fuzzy goal programming with preemptive priorities, with Archime- dian weights, and with the maximization of the membership function corresponding to the minimum goal. A methodology based on the use of a nested hierarchy of priorities for each goal was presented by Rubin and Narasimhan.10 The impor- tance of multiple objectives in the design of practical engineer-

Optimizing sustainable and renewable energy portfolios using a fuzzy interval goal programming approach

Weighted-additive fuzzy multi-choice goal programming (WA-FMCGP) for supporting renewable energy site selection decisions

Multi-objective inventory models of deteriorating items with some constraints in a fuzzy environment

Fuzzy goal programming for health-care organization

In this paper, the fuzzy goal programming is investigated when both the coefficients and the aspiration levels are considered fuzzy numbers with either trapezoidal or triangular membership functions. The possibility programming approach has been utilised in the case of exceedance possibility and the case of strict exceedance possibility. In many situations, the decision-maker cannot set a precise priority structure for the possibility functions of the fuzzy goals. A membership function for the imprecise relation between different pairs of possibilities has been defined. This function reflects a scale for the degree of importance between any pair of goals. This scale starts from 'almost not important' to 'certainly more important'. Accordingly, there are two types of the membership functions. The first represents the possibility functions of all fuzzy goals, while the second is the membership functions of the imprecise importance relations. The weighted max-min approach is utilised for the two types. The suggested approach is illustrated by a numerical example.

In this paper, the fuzzy goal programming is investigated when both the coefficients and the aspiration levels are considered fuzzy numbers with either trapezoidal or triangular membership functions. The possibility programming approach has been utilised in the case of exceedance possibility and the case of strict exceedance possibility. In many situations, the decision-maker cannot set a precise priority structure for the possibility functions of the fuzzy goals. A membership function for the imprecise relation between different pairs of possibilities has been defined. This function reflects a scale for the degree of importance between any pair of goals. This scale starts from 'almost not important' to 'certainly more important'. Accordingly, there are two types of the membership functions. The first represents the possibility functions of all fuzzy goals, while the second is the membership functions of the imprecise importance relations. The weighted max-min approach is utilised for the two types. The suggested approach is illustrated by a numerical example.

It is sometimes difficult in real situations to estimate the coefficients of decision variables in multi-objective model. Even though mathematical analysis may contribute to determine these coefficients, historical data used may contain fuzzy and random properties and should be treated properly. Thus, this paper introduces a fuzzy random regression to approximate the coefficients; specifically the goal constraints of goal programming model. We propose a two phase-based approach for the solution model; first, we construct the goal constraints using fuzzy random regression model and, second, we solve the multi-objective problem with a fuzzy additive goal programming. A numerical example is presented to illustrate the model.

The objective of this article is to tie a knot between distance measure and fuzzy and intuitionistic fuzzy optimization through goal programming. Firstly, a distance measure for an intuitionistic fuzzy number is developed, and then it is implemented into an intuitionistic fuzzy nonlinear goal programming. Then using some conditions, the distance measure of intuitionistic fuzzy number is converted into distance measure of fuzzy number and a comparative study using a numerical example is shown for highest applicability of distance measure based intuitionistic fuzzy goal programming than distance measure based fuzzy goal programming.

Studies show that most actively managed mutual funds struggle to beat the market, driving an increase in the popularity of index investing. Index investing instruments, including index funds and Exchange-traded Funds, aim to track market performance. This study pursues both tracking error minimization and excess return maximization, two conflicting objectives, to construct an index portfolio. In the real-world financial environment, the desires and expectations of decision makers are generally imprecise. This study applies fuzzy theory to deal with imprecise objectives. This study represents minimizing tracking error and maximizing excess return as ‘fuzzy goals’ to improve traditional goal programming, which is suitable for handling multiple conflicting objectives, but subject to establishing crisp goals. Three fuzzy goal programming (FGP) models that track indexes are compared and discussed, and the results show that through certain membership functions and tracking models, an index tracking portfolio with a tracking error lower than the 0050 index fund, and a similar excess return to 0050 index fund can be constructed using additive type FGP. max-min type FGP underperforms the additive type FGP in index fund construction.

Assembly line balancing is attached to great significance in terms of improving the assembly efficiency and cutting down the assembly cost, Business tends to adopt a mixed assembly line to satisfy the customers' demands and reduce the costs nowadays, while in the actual productive process, productive activities are often influenced by some uncertain (ambiguous) factors, and end up in vague surroundings. Assembly line balancing generally requires a set of acceptable solutions to the several conflicting objectives. This paper addresses the existing problem in the balance of a mixed-model U-shaped assembly line and proposes a fuzzy goal (the number of workstations and cycle time goals) programming model. Moreover, the numerical example further tests and verifies the validity and feasibility of the proposed approach.

Activity assigning of fourth party logistics by particle swarm optimization-based preemptive fuzzy integer goal programming

It has become the leading world trend that all countries join hands to respond to climate change and promote green and low‐carbon development. In this background, more and more enterprises seek to control carbon emissions from the source and reduce indirect carbon emissions, consequently paying increased importance to the carbon footprint of their suppliers. Most of the earlier studies concerning supplier selection mainly focus on traditional factors, such as quality, service, and lead time, but minimal importance to the carbon emission of supplies. This study fills this gap by incorporating carbon emission criteria into supplier selection and presenting a method of combining fuzzy analytic hierarchy process (AHP) and fuzzy goal programming (GP) to address the problem of supplier selection and order quota allocation. Firstly, we use fuzzy AHP to evaluate the relevance and importance of supplier selection criteria according to the experts’ opinions. Second, each objective is given weights according to the fuzzy AHP results, and then the fuzzy GP method is used for supplier selection and order quota allocation. Finally, we implement an example study with a data set from a realistic situation, and the results confirm the effectiveness of the proposed method in an uncertain environment.

Fuzzy Waste Load Allocation Model (FWLAM), developed in an earlier study, derives the optimal fractional levels, for the base flow conditions, considering the goals of the Pollution Control Agency (PCA) and dischargers. The Modified Fuzzy Waste Load Allocation Model (MFWLAM) developed subsequently is a stochastic model and considers the moments (mean, variance and skewness) of water quality indicators, incorporating uncertainty due to randomness of input variables along with uncertainty due to imprecision. The risk of low water quality is reduced significantly by using this modified model, but inclusion of new constraints leads to a low value of acceptability level, λ, interpreted as the maximized minimum satisfaction in the system. To improve this value, a new model, which is a combination of FWLAM and MFWLAM, is presented, allowing for some violations in the constraints of MFWLAM. This combined model is a multiobjective optimization model having the objectives, maximization of acceptability level and minimization of violation of constraints. Fuzzy multiobjective programming, goal programming and fuzzy goal programming are used to find the solutions. For the optimization model, Probabilistic Global Search Lausanne (PGSL) is used as a nonlinear optimization tool. The methodology is applied to a case study of the Tunga–Bhadra river system in south India. The model results in a compromised solution of a higher value of acceptability level as compared to MFWLAM, with a satisfactory value of risk. Thus the goal of risk minimization is achieved with a comparatively better value of acceptability level.

Production planning is one of the most important issues of managers in industry. Production managers face up with several goals that sometimes conflict with each other. Operations research techniques with considering the constraints, optimize organizational goals. Objectives of all these techniques are raising productivity in the organization. In this paper, the researchers present a multi-objective linear fractional model of production planning in wood and metal company. One of the difficulties in solving the multiple objective fractional problems is computational problem that arises from of variability, in the example of Charnes and Cooper (1961) methods. In this research, fuzzy approach issued to solve multiple objective fractional mathematical problems of Khavar-E-Miane Wood and Metal Company. First, with some assumptions, the fuzzy linear fractional goal programming method of Pal has been used to solve the problem of production planning. Then, the fuzzy method of Dutta is utilized to solve the problem. Comparison of results showed that both methods have identical solution. Key words: Production planning, fuzzy goal programming, productivity, fuzzy fractional programming, wood and metal company.