Project management decisions using fuzzy linear programming

This work presents an interactive fuzzy linear programming (FLP) approach for solving project management (PM) decision problems in a fuzzy environment. The proposed approach attempts to minimize total costs with reference to direct, indirect and penalty costs, durations of activities, specified project completion time and total allocated budget. A numerical example demonstrates the feasibility of applying the proposed FLP approach to actual PM decision problems. Accordingly, the proposed approach yields an efficient solution and determines the overall degree of decision maker (DM) satisfaction. Moreover, the proposed approach offers a systematic framework that facilitates the decision-making process, enabling a DM to interactively modify the range of the results when the environment data are vague until a satisfactory solution is obtained. In particular, several significant characteristics of the proposed FLP approach are elucidated in contrast to those of the main PM decision methods.     

Dependent-chance programming: A class of stochastic optimization

This paper provides a theoretical framework of dependent-chance programming, as well as dependent-chance multiobjective programming and dependent-chance goal programming which are new types of stochastic optimization. A stochastic simulation based genetic algorithm is also designed for solving dependent-chance programming models.     

Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method

The aim of this paper is to develop an interactive two-phase method that can help the Project Manager (PM) with solving the fuzzy multi-objective decision problems. Therefore, in this paper, we first revisit the related papers and focus on how to develop an interactive two-phase method. Next, we establish to consider the imprecise nature of the data by fulfilling the possibilistic programming model, and we also assume that each objective work has a fuzzy goal. Finally, for reaching our objective, the detailed numerical example is presented to illustrate the feasibility of applying the proposed approach to PM decision problems at the end of this paper. Results show that our model can be applied as an effective tool. Furthermore, we believe that this approach can be applied to solve other multi-objective decision making problems.     

Interactive fuzzy programming for fuzzy random two-level linear programming problems through probability maximization with possibility

This paper considers two-level linear programming problems involving fuzzy random variables under cooperative behavior of the decision makers. Through the introduction of fuzzy goals together with possibility measures, the formulated fuzzy random two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. By adopting probability maximization, the transformed stochastic two-level programming problem can be reduced to a deterministic one. Interactive fuzzy programming to derive a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. An illustrative numerical example demonstrates the feasibility and efficiency of the proposed method.     

Duality theory in fuzzy mathematical programming problems with fuzzy coefficients

In this paper, the notions of subgradient, subdifferential, and differential with respect to convex fuzzy mappings are investigated, which provides the basis for the fuzzy extremum problem theory. We consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and/or sufficient optimality conditions. Furthermore, the concept of saddle-points and minimax theorems under fuzzy environment is discussed. The results obtained are used to formulate the Lagrangian dual of fuzzy programming. Under certain fuzzy convexity assumptions, KKT conditions for fuzzy programming are derived, and the “perturbed” convex fuzzy programming is considered. Finally, these results are applied to fuzzy linear programming and fuzzy quadratic programming.     

Qualitative programming for selection decisions

Many selection decision problems involve qualitative information, the exclusion of which from optimization processes may lead to suboptimal (and possibly, disastrous) solutions. This paper presents an approach to selection decisions where some or all of the available information may be qualitative as well as incomplete. Appropriate mappings which accomplish this task are discussed in detail, and an example of the real-world application of this approach to software selection decisions is presented.     

Approximate dynamic programming with a fuzzy parameterization

Dynamic programming (DP) is a powerful paradigm for general, nonlinear optimal control. Computing exact DP solutions is in general only possible when the process states and the control actions take values in a small discrete set. In practice, it is necessary to approximate the solutions. Therefore, we propose an algorithm for approximate DP that relies on a fuzzy partition of the state space, and on a discretization of the action space. This fuzzy Q-iteration algorithm works for deterministic processes, under the discounted return criterion. We prove that fuzzy Q-iteration asymptotically converges to a solution that lies within a bound of the optimal solution. A bound on the suboptimality of the solution obtained in a finite number of iterations is also derived. Under continuity assumptions on the dynamics and on the reward function, we show that fuzzy Q-iteration is consistent, i.e., that it asymptotically obtains the optimal solution as the approximation accuracy increases. These properties hold both when the parameters of the approximator are updated in a synchronous fashion, and when they are updated asynchronously. The asynchronous algorithm is proven to converge at least as fast as the synchronous one. The performance of fuzzy Q-iteration is illustrated in a two-link manipulator control problem.     

Linear programming with fuzzy coefficients in constraints

This paper presents a new method for solving linear programming problems with fuzzy coefficients in constraints. It is shown that such problems can be reduced to a linear semi-infinite programming problem. The relations between optimal solutions and extreme points of the linear semi-infinite program are established. A cutting plane algorithm is introduced with a convergence proof, and a numerical example is included to illustrate the solution procedure.     

Neural fuzzy logic programming

A foundational development of propositional fuzzy logic programs is presented. Fuzzy logic programs are structured knowledge bases including uncertainties in rules and facts. The precise specifications of uncertainties have a great influence on the performance of the knowledge base. It is shown how fuzzy logic programs can be transformed to neural networks, where adaptations of uncertainties in the knowledge base increase the reliability of the program and are carried out automatically.     

Exact fuzzy optimal solution of fully fuzzy linear programming problems with unrestricted fuzzy variables

Kumar et al. (Appl. Math. Model. 35:817?C823, 2011) pointed out that there is no method in literature to find the exact fuzzy optimal solution of fully fuzzy linear programming (FFLP) problems and proposed a new method to find the fuzzy optimal solution of FFLP problems with equality constraints having non-negative fuzzy variables and unrestricted fuzzy coefficients. There may exist several FFLP problems with equality constraints in which no restriction can be applied on all or some of the fuzzy variables but due to the limitation of the existing method these types of problems can not be solved by using the existing method. In this paper a new method is proposed to find the exact fuzzy optimal solution of FFLP problems with equality constraints having non-negative fuzzy coefficients and unrestricted fuzzy variables. The proposed method can also be used to solve the FFLP problems with equality constraints having non-negative fuzzy variables and unrestricted fuzzy coefficients. To show the advantage of the proposed method over existing method the results of some FFLP problems with equality constraints, obtained by using the existing and proposed method, are compared. Also, to show the application of proposed method a real life problem is solved by using the proposed method.     

Interactive fuzzy programming for random fuzzy two-level programming problems through possibility-based fractile model

This paper focuses on interactive decision making methods for random fuzzy two-level linear programming problems. Considering the probabilities that the decision makers’ objective function values are smaller than or equal to target variables, fuzzy goals of the decision makers are introduced. Using the fractile model to optimize the target variables under the condition that the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, the original random fuzzy two-level programming problems are reduced to deterministic ones. Interactive fuzzy nonlinear programming to obtain a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. An illustrative numerical example demonstrates the feasibility and efficiency of the proposed method.     

系数为对称梯形模糊数的模糊线性规划

针对模糊系数的线性规划, 提出了一种系数为对称梯形模糊数的线性规划的方法, 同时得出一些定理、命题以及相应的算法, 并通过实例验证了该方法的有效性. 该方法与常规方法的不同之处在于无须将模糊线性规划转化为经典线性规划就能得到满意的模糊优化解, 因此提出的方法所取得的规划结果更加满足决策者的需要.     

Linear programming approach for solving fuzzy critical path problems with fuzzy parameters

To the best of our knowledge, there is no method in the literature to find the fuzzy optimal solution of fully fuzzy critical path (FFCP) problems i.e., critical path problems in which all the parameters are represented by LR flat fuzzy numbers. In this paper, a new method is proposed for the same. Also, it is shown that it is better to use JMD representation of LR flat fuzzy numbers in the proposed method as compared to the other representation of LR flat fuzzy numbers.     

含有模糊和随机参数的混合机会约束规划模型

提出一类混合机会约束规划模型，该模型同时含有模糊和随机参数，运用随机模拟与模糊模拟相结合的技术，给出了求解该机会约束规划模型的遗传算法，通过对生产过程最优化决策的典型问题进行分析建模和数值求解，说明了该模型和算法的合理性和有效性。     

Statistical and fuzzy optimal decisions: II. Fuzzy decisions

In the paper, methods for optimal possibility-theoretical decisions are considered on examples of fuzzy identification and fuzzy estimation. The quality of the decision is determined in terms of the risk of losses accompanying the decision and is characterized by the values of possibility and/or inevitability of losses. Yurii P. Pyt’ev. Graduated from the Faculty of Physics, Moscow State University. Professor, Head of the Department of Computer Methods in Physics, Faculty of Physics, Moscow State University. Scientific interests: mathematical methods of data analysis and interpretation, mathematical modeling of measuring-computing systems of superhigh resolution, and fuzzy and uncertain fuzzy mathematics. Author of more than 300 publications on mathematics, mathematical physics, and information science, including several monographs.     

Statistical and fuzzy optimal decisions: I. Statistical decisions

In the proposed series of two articles, the methods of optimal statistical and fuzzy (possibility-theoretical) decisions are described from a unified point of view. In the second article [22], we consider the possibility-theoretical methods for optimal decisions (to be more precise, optimal identification methods or, in other words, methods for optimal choice of one of a number of alternative hypotheses concerning the affiliation of the object of investigation to one of a finite number of classes), as well as the methods for optimal estimation of fuzzy elements. The quality of the decision is determined in terms of the risk of losses accompanying the decision and is characterized by the values of possibility and/or inevitability of losses. The presentation follows the scheme adopted in the theory of statistical decisions [16, 17]. In the present work, which is the first in the series, elements of statistical decision theory, whose analogs were obtained in [22] and which make it possible to reconstruct the possibility empirically, are presented for the convenience of the readers who want to trace the analogy between the probability-theoretical methods and the possibility-theoretical methods for decision optimization [22]. Yurii P. Pyt’ev. Graduated from the Faculty of Physics, Moscow State University. Professor, Head of the Department of Computer Methods in Physics, Faculty of Physics, Moscow State University. Scientific Interests: mathematical methods of data analysis and interpretation, mathematical modeling of measuring-computing systems of superhigh resolution, and fuzzy and uncertain fuzzy mathematics. Author of more than 300 publications on mathematics, mathematical physics, and information science, including several monographs.     

The quadratic programming problem with fuzzy relation inequality constraints

The quadratic programming has been widely applied to solve real world problems. The quadratic functions are often applied in the inventory management, portfolio selection, engineering design, molecular study, and economics, etc. Fuzzy relation inequalities (FRI) are important elements of fuzzy mathematics, and they have recently been widely applied in the fuzzy comprehensive evaluation and cybernetics. In view of the importance of quadratic functions and FRI, we present a fuzzy relation quadratic programming model with a quadratic objective function subject to the max-product fuzzy relation inequality constraints. Some sufficient conditions are presented to determine its optimal solution in terms of the maximum solution or the minimal solutions of its feasible domain. Also, some simplification operations have been given to accelerate the resolution of the problem by removing the components having no effect on the solution process. The simplified problem can be converted into a traditional quadratic programming problem. An algorithm is also proposed to solve it. Finally, some numerical examples are given to illustrate the steps of the algorithm.     

Knowledge-based function optimization using fuzzy culturalalgorithms with evolutionary programming

In this paper, the advantages of a fuzzy representation in problem solving and search is investigated using the framework of Cultural algorithms (CAs). Since all natural languages contain a fuzzy component, the natural question is "Does this fuzzy representation facilitate the problem-solving process, within these systems". In order to investigate this question we use the CA framework of Reynolds (1996), CAs are a computational model of cultural evolution derived from and used to express basic anthropological models of culture and its development. A mathematical model of a full fuzzy CA is developed there. In it, the problem solving knowledge is represented using a fuzzy framework. Several theoretical results concerning its properties are presented. The model is then applied to the solution of a set of 12 difficult, benchmark problems in nonlinear real-valued function optimization. The performance of the full fuzzy model is compared with 8 other fuzzy and crisp architectures. The results suggest that a fuzzy approach can produce a statistically significant improvement in search efficiency over nonfuzzy versions for the entire set of functions, the then investigate the class of performance functions for which the full fuzzy system exhibits the greatest improvements over nonfuzzy systems. In general, these are functions which require some preliminary investigation in order to embark on an effective search.     

Optimality conditions for linear programming problems with fuzzy coefficients

The optimality conditions for linear programming problems with fuzzy coefficients are derived in this paper. Two solution concepts are proposed by considering the orderings on the set of all fuzzy numbers. The solution concepts proposed in this paper will follow from the similar solution concept, called the nondominated solution, in the multiobjective programming problem. Under these settings, the optimality conditions will be naturally elicited.