Logistics distribution centers location problem and algorithm under fuzzy environment

Distribution centers location problem is concerned with how to select distribution centers from the potential set so that the total relevant cost is minimized. This paper mainly investigates this problem under fuzzy environment. Consequentially, chance-constrained programming model for the problem is designed and some properties of the model are investigated. Tabu search algorithm, genetic algorithm and fuzzy simulation algorithm are integrated to seek the approximate best solution of the model. A numerical example is also given to show the application of the algorithm.     

Portfolio selection based on fuzzy cross-entropy

In this paper, the Kapur cross-entropy minimization model for portfolio selection problem is discussed under fuzzy environment, which minimizes the divergence of the fuzzy investment return from a priori one. First, three mathematical models are proposed by defining divergence as cross-entropy, average return as expected value and risk as variance, semivariance and chance of bad outcome, respectively. In order to solve these models under fuzzy environment, a hybrid intelligent algorithm is designed by integrating numerical integration, fuzzy simulation and genetic algorithm. Finally, several numerical examples are given to illustrate the modeling idea and the effectiveness of the proposed algorithm.     

A note on chance constrained programming with fuzzy coefficients

This paper deals with nonlinear chance constrained programming as well as multiobjective case and goal programming with fuzzy coefficients occurring in not only constraints but also objectives. We also present a fuzzy simulation technique for handling fuzzy objective constraints and fuzzy goal constraints. Finally, a fuzzy simulation based genetic algorithm is employed to solve a numerical example.     

Fuzzy minimum weight edge covering problem

Minimum weight edge covering problem, known as a classic problem in graph theory, is employed in many scientific and engineering applications. In the applications, the weight may denote cost, time, or opponent’s payoff, which can be vague in practice. This paper considers the edge covering problem under fuzzy environment, and formulates three models which are expected minimum weight edge cover model, α-minimum weight edge cover model, and the most minimum weight edge cover model. As an extension for the models, we respectively introduce the crisp equivalent of each model in the case that the weights are independent trapezoidal fuzzy variables. Due to the complexity of the problem, a hybrid intelligent algorithm is employed to solve the models, which can deal with the problem with any type of fuzzy weights. At last, some numerical experiments are given to show the application of the models and the robustness of the algorithm.     

Extension of an algorithm for product mix problems with fuzzy conditions

In the real world of production, there are some uncertain parameters such as capacity, profit, and processing time. On the other hand, the fuzzy sets theory is applied as a suitable tool for managing production where dynamicity of the production environment prevents from determining target function, constraints and other parameters of the model. In this paper, a genetic algorithm embedded in fuzzy revised theory of constraints is presented for product mix problems. Results of implementing the proposed algorithm show efficiency and flexibility of this algorithm.     

Entropy maximization model for the trip distribution problem with fuzzy and random parameters

Many trip distribution problems can be modeled as entropy maximization models with quadratic cost constraints. In this paper, the travel costs per unit flow between different zones are assumed to be given fuzzy variables and the trip productions at origins and trip attractions at destinations are assumed to be given random variables. For this case, an entropy maximization model with chance constraint is proposed, and is proved to be convex. In order to solve this model, fuzzy simulation, stochastic simulation and a genetic algorithm are integrated to produce a hybrid intelligent algorithm. Finally, a numerical example is presented to demonstrate the application of the model and the algorithm.     

New models for shortest path problem with fuzzy arc lengths

This paper considers the shortest path problem with fuzzy arc lengths. According to different decision criteria, the concepts of expected shortest path, α-shortest path and the most shortest path in fuzzy environment are originally proposed, and three types of models are formulated. In order to solve these models, a hybrid intelligent algorithm integrating simulation and genetic algorithm is provided and some numerous examples are given to illustrate its effectiveness.     

Mean-variance-skewness model for portfolio selection with fuzzy returns

Numerous empirical studies show that portfolio returns are generally asymmetric, and investors would prefer a portfolio return with larger degree of asymmetry when the mean value and variance are same. In order to measure the asymmetry of fuzzy portfolio return, a concept of skewness is defined as the third central moment in this paper, and its mathematical properties are studied. As an extension of the fuzzy mean-variance model, a mean-variance-skewness model is presented and the corresponding variations are also considered. In order to solve the proposed models, a genetic algorithm integrating fuzzy simulation is designed. Finally, several numerical examples are given to illustrate the modelling idea and the effectiveness of the proposed algorithm.     

模糊反向物流网络设计模型的一种改进遗传算法

反向物流是物流研究中的一个重要分支,其相关问题是目前研究的热点问题。该研究在模糊环境中根据不同的决策标准,建立了关于反向物流问题中的回收问题的三种不同类型的模型:期望值模型,机会约束模型和相关机会模型,并设计了一个模糊模拟和遗传算法相结合的混合智能算法来解决提出的模型,最后给出了一个数值例子,结果证明了将此混合智能算法用于求解模糊反向物流网络设计模型问题的有效性。     

Fuzzy inventory model with two warehouses under possibility measure on fuzzy goal

Multi-item inventory model with stock-dependent demand and two-storage facilities is developed in fuzzy environment (purchase cost, investment amount and storehouse capacity are imprecise) under inflation and time value of money. Joint replenishment and simultaneous transfer of items from one warehouse to another is proposed using basic period (BP) policy. As some parameters are fuzzy in nature, objective (average profit) function as well as some constraints are imprecise in nature. Model is formulated as to optimize the possibility/necessity measure of the fuzzy goal of the objective function and constraints are satisfied with some pre-defined necessity. A genetic algorithm (GA) is developed with roulette wheel selection, binary crossover and mutation and is used to solve the model when the equivalent crisp form of the model is available. In other cases fuzzy simulation process is proposed to measure possibility/necessity of the fuzzy goal as well as to check the constraints of the problem and finally the model is solved using fuzzy simulation based genetic algorithm (FSGA). The models are illustrated with some numerical examples and some sensitivity analyses have been done.     

A hybrid intelligent algorithm for portfolio selection problem with fuzzy returns

Portfolio selection theory with fuzzy returns has been well developed and widely applied. Within the framework of credibility theory, several fuzzy portfolio selection models have been proposed such as mean–variance model, entropy optimization model, chance constrained programming model and so on. In order to solve these nonlinear optimization models, a hybrid intelligent algorithm is designed by integrating simulated annealing algorithm, neural network and fuzzy simulation techniques, where the neural network is used to approximate the expected value and variance for fuzzy returns and the fuzzy simulation is used to generate the training data for neural network. Since these models are used to be solved by genetic algorithm, some comparisons between the hybrid intelligent algorithm and genetic algorithm are given in terms of numerical examples, which imply that the hybrid intelligent algorithm is robust and more effective. In particular, it reduces the running time significantly for large size problems.     

A class of multiobjective linear programming model with fuzzy random coefficients

The aim of this paper is to deal with a multiobjective linear programming problem with fuzzy random coefficients. Some crisp equivalent models are presented and a traditional algorithm based on an interactive fuzzy satisfying method is proposed to obtain the decision maker’s satisfying solution. In addition, the technique of fuzzy random simulation is adopted to handle general fuzzy random objective functions and fuzzy random constraints which are usually hard to be converted into their crisp equivalents. Furthermore, combined with the techniques of fuzzy random simulation, a genetic algorithm using the compromise approach is designed for solving a fuzzy random multiobjective programming problem. Finally, illustrative examples are given in order to show the application of the proposed models and algorithms.     

A fuzzy multi-objective programming for optimization of fire station locations through genetic algorithms

Location of fire stations is an important factor in its fire protection capability. This paper aims to determine the optimal location of fire station facilities. The proposed method is the combination of a fuzzy multi-objective programming and a genetic algorithm. The original fuzzy multiple objectives are appropriately converted to a single unified ‘min–max’ goal, which makes it easy to apply a genetic algorithm for the problem solving. Compared with the existing methods of fire station location our approach has three distinguish features: (1) considering fuzzy nature of a decision maker (DM) in the location optimization model; (2) fully considering the demands for the facilities from the areas with various fire risk categories; (3) being more understandable and practical to DM. The case study was based on the data collected from the Derbyshire fire and rescue service and used to illustrate the application of the method for the optimization of fire station locations.     

Maximum cut in fuzzy nature: Models and algorithms

The maximum cut (Max-Cut) problem has extensive applications in various real-world fields, such as network design and statistical physics. In this paper, a more practical version, the Max-Cut problem with fuzzy coefficients, is discussed. Specifically, based on credibility theory, the Max-Cut problem with fuzzy coefficients is formulated as an expected value model, a chance-constrained programming model and a dependent-chance programming model respectively according to different decision criteria. When these fuzzy coefficients are represented by special fuzzy variables like triangular fuzzy numbers and trapezoidal fuzzy numbers, the crisp equivalents of the fuzzy Max-Cut problem can be obtained. Finally, a genetic algorithm combined with fuzzy simulation techniques is designed for the general fuzzy Max-Cut problem under these models and numerical experiment confirms the effectiveness of the designed genetic algorithm.     

Fuzzy project scheduling problem and its hybrid intelligent algorithm

Project scheduling problem is to determine the schedule of allocating resources so as to balance the total cost and the completion time. This paper considers a type of project scheduling problem with fuzzy activity duration times. According to some management goals, three types of fuzzy models are built to solve the project scheduling problem. Moreover, the technique of fuzzy simulation and genetic algorithm are integrated to design a hybrid intelligent algorithm to solve the fuzzy models. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm.     

Fuzzy facility location-allocation problem under the Hurwicz criterion

Facility location-allocation (FLA) problem has been widely studied by operational researchers due to its many practical applications. Many researchers have studied the FLA problem in a deterministic environment. However, the models they proposed cannot accommodate satisfactorily various customer demands in the real world. Thus, we consider the FLA problem with uncertainties. In this paper, a new model named α-cost model under the Hurwicz criterion is presented with fuzzy demands. In order to solve this model, the simplex algorithm, fuzzy simulations and a genetic algorithm are integrated to produce a hybrid intelligent algorithm. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed algorithm.     

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.     

Optimal stopping in a fuzzy environment

The problem under consideration is that of optimally controlling and stopping either a deterministic or a stochastic system in a fuzzy environment. The optimal decision is the sequence of controls that maximizes the membership function of the intersection of the fuzzy constraints and a fuzzy goal. The fuzzy goal is a fuzzy set in the cartesian product of the state space with the set of possible stopping times. Dynamic programming is applied to yield a numerical solution. This approach yields an algorithm that corrects a result of Kacprzyk.     

Optimisation of integrated reverse logistics networks with different product recovery routes

The awareness of importance of product recovery has grown swiftly in the past few decades. This paper focuses on a problem of inventory control and production planning optimisation of a generic type of an integrated Reverse Logistics (RL) network which consists of a traditional forward production route, two alternative recovery routes, including repair and remanufacturing and a disposal route. It is assumed that demand and return quantities are uncertain. A quality level is assigned to each of the returned products. Due to uncertainty in the return quantity, quantity of returned products of a certain quality level is uncertain too. The uncertainties are modelled using fuzzy trapezoidal numbers. Quality thresholds are used to segregate the returned products into repair, remanufacturing or disposal routes. A two phase fuzzy mixed integer optimisation algorithm is developed to provide a solution to the inventory control and production planning problem. In Phase 1, uncertainties in quantity of product returns and quality of returns are considered to calculate the quantities to be sent to different recovery routes. These outputs are inputs into Phase 2 which generates decisions on component procurement, production, repair and disassembly. Finally, numerical experiments and sensitivity analysis are carried out to better understand the effects of quality of returns and RL network parameters on the network performance. These parameters include quantity of returned products, unit repair costs, unit production cost, setup costs and unit disposal cost.     

Lot sizing for a single product recovery system with variable setup numbers

Inventory systems for joint remanufacturing and manufacturing have recently received considerable attention. In such systems, used products are collected from customers and are kept at the recoverable inventory warehouse for future remanufacturing. In this paper a production–remanufacturing inventory system is considered, where the demand can be satisfied by production and remanufacturing. The cost structure consists of the EOQ-type setup costs, holding costs and shortage costs. The model with no shortage case in serviceable inventory is first studied. The serviceable inventory shortage case is discussed next. Both models are considered for the case of variable setup numbers of equal sized batches for production and remanufacturing processes. For these two models sufficient conditions for the optimal type of policy, referring to the parameters of the models, are proposed.