Modelling linear fractional transportation problem in belief degree—based uncertain environment

A linear fractional transportation problem in uncertain environment is studied in this paper where the uncertain parameters of the problem are of belief degree—based uncertainty. For the first time, this type of uncertainty is considered for the linear fractional transportation problem. Belief degree—based uncertainty is useful for the cases that no historical information of an uncertain event is available. Zigzag type uncertainty distribution is used to show the belief degree—based uncertainty of the parameters of the problem. As solution methodology, the uncertain linear fractional transportation problem is converted to a crisp form using three approaches of expected value model, expected value and chance-constrained model, and chance-constrained model, separately. An extensive computational study on a real illustrative example shows the efficiency of the proposed formulation and the conversion approaches. The sensitivity analysis over the example illustrates the high dependency of the objective function value to the changes of the confidence level values of the chance constraints in the expected value and chance-constrained programming approach and the chance-constrained programming approach.     

General Barnes-Godunova-Levin type inequalities for Sugeno integral

Integral inequalities play important roles in classical probability and measure theory. Non-additive measure is a generalization of additive probability measure. Sugeno’s integral is a useful tool in several theoretical and applied statistics which has been built on non-additive measure. For instance, in decision theory, the Sugeno integral is a median, which is indeed a qualitative counterpart to the averaging operation underlying expected utility. In this paper, Barnes-Godunova-Levin type inequalities for the Sugeno integral on abstract spaces are studied in a rather general form and, for this, we introduce some new technics for the treatment of concave functions in the Sugeno integration context. Also, several examples are given to illustrate the validity of this inequality. Moreover, a strengthened version of Barnes-Godunova-Levin type inequality for Sugeno integrals on a real interval based on a binary operation ★ is presented.     

Uncertain goal programming models for bicriteria solid transportation problem

This paper investigates an uncertain bicriteria solid transportation problem. The supplies, demands, conveyance capacities, transportation cost and transportation time are regarded as uncertain variables. According to two types of methods to rank the uncertain variables, expected value goal programming model and chance-constrained goal programming model for the bicriteria solid transportation problem are constructed. It is proved that the expected value goal programming model and chance-constrained goal programming model can be respectively transformed into the corresponding deterministic equivalents by taking advantage of some properties of uncertainty theory. Based on these equivalence relations, the optimal transportation plans of the uncertain goal programming models can be obtained. Several numerical experiments are presented to illustrate the applications of the models.     

The key theorem and the bounds on the rate of uniform convergence of learning theory on Sugeno measure space

In the 1970s, Vapnik[1―3] proposed the Statistical Learning Theory (SLT), which deals mainly with the statistical learning principles when samples are limited. SLT is an im- portant development and supplement of traditional statistics, whose kernel idea …     

Connectivity reliability in uncertain networks with stability analysis

This paper treats the fundamental problems of reliability and stability analysis in uncertain networks. Here, we consider a collapsed, post-disaster, traffic network that is composed of nodes (centers) and arcs (links), where the uncertain operationality or reliability of links is evaluated by domain experts. To ensure the arrival of relief materials and rescue vehicles to the disaster areas in time, uncertainty theory, which neither requires any probability distribution nor fuzzy membership function, is employed to originally propose the problem of choosing the most reliable path (MRP). We then introduce the new problems of α-most reliable path (α-MRP), which aims to minimize the pessimistic risk value of a path under a given confidence level α, and very most reliable path (VMRP), where the objective is to maximize the confidence level of a path under a given threshold of pessimistic risk. Then, exploiting these concepts, we give the uncertainty distribution of the MRP in an uncertain traffic network. The objective of both α-MRP and VMRP is to determine a path that comprises the least risky route for transportation from a designated source node to a designated sink node, but with different decision criteria. Furthermore, a methodology is proposed to tackle the stability analysis issue in the framework of uncertainty programming; specifically, we show how to compute the arcs’ tolerances. Finally, we provide illustrative examples that show how our approaches work in realistic situation.     

The multi-level warehouse layout problem with uncertain information: uncertainty theory method

This paper investigates the multi-level warehouse layout problem with indeterminate factors, in which the monthly demands and horizontal transportation distances are described by uncertain variables. We first consider the distribution function of the total cost for transportation. Second, two uncertain models, namely, the chance-constrained programming model and the chance-maximum programming model, are developed to lay out the multi-level warehouse under uncertainty. Some properties of the models are discussed to solve the models. The properties point out that the optimal solution to the chance-constrained programming model is equivalent to a corresponding deterministic model. Additionally, we also discuss the relation between the chance-constrained programming model and the chance-maximum programming model, which leads to an effective approach to search for the optimal solution to the chance-maximum programming model. Finally, a numerical experiment is illustrated to show the ideas of the proposed models.     

Degree-constrained minimum spanning tree problem with uncertain edge weights

Uncertainty theory has shown great advantages in solving many nondeterministic problems, one of which is the degree-constrained minimum spanning tree (DCMST) problem in uncertain networks. Based on different criteria for ranking uncertain variables, three types of DCMST models are proposed here: uncertain expected value DCMST model, uncertain α-DCMST model and uncertain most chance DCMST model. In this paper, we give their uncertainty distributions and fully characterize uncertain expected value DCMST and uncertain α-DCMST in uncertain networks. We also discover an equivalence relation between the uncertain α-DCMST of an uncertain network and the DCMST of the corresponding deterministic network. Finally, a related genetic algorithm is proposed here to solve the three models, and some numerical examples are provided to illustrate its effectiveness.     

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.     

资源受限的项目调度问题的求解算法

本文建立了不确定资源环境下的资源受限的项目调度模型,用不确定规划的方法将不确定问题转化为等价的确定性问题,并给出了一个解决该问题的二阶段算法及实例。     

Interactive fuzzy random cooperative two-level linear programming through level sets based probability maximization

In this paper, assuming cooperative behavior of the decision makers, two-level linear programming problems under fuzzy random environments are considered. To deal with the formulated fuzzy random two-level linear programming problems, α-level sets of fuzzy random variables are introduced and an α-stochastic two-level linear programming problem is defined for guaranteeing the degree of realization of the problem. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced and the α-stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through 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 is provided to demonstrate the feasibility and efficiency of the proposed method.     

Finding the least influenced set in uncertain databases

Due to the inherent existence of uncertainty in many real-world applications, in this paper, we investigate an important query in uncertain databases, namely probabilistic least influenced set (PLIS) query, which retrieves all the uncertain objects in an uncertain database such that they are the least affected by a given query object with high probabilities. Such a PLIS query is useful in applications such as business planning. We propose and tackle both monochromatic and bichromatic versions (i.e. M-PLIS and B-PLIS, respectively) of the PLIS query. In order to efficiently answer PLIS queries, we present three pruning methods, MINMAX, Regional, and Candidate pruning, which can effectively reduce the PLIS search space. The proposed pruning methods can be seamlessly integrated into efficient query procedures. Moreover, we also study important variants of PLIS query with uncertain query object (i.e. UQ-PLIS). Furthermore, we formulate and tackle the PLIS problem on uncertain moving objects (i.e. UMOD-PLIS). Extensive experiments have demonstrated the efficiency and effectiveness of our proposed approaches under various settings.     

Fuzzy fixed charge solid transportation problem and algorithm

This paper mainly investigates the fixed charge solid transportation problem under fuzzy environment, in which the direct costs, the fixed charges, the supplies, the demands and the conveyance capacities are supposed to be fuzzy variables. As a result, several new models, i.e., expected value model, chance-constrained programming model and dependent-chance programming model, are constructed on the basis of credibility theory. After that, the crisp equivalences are also discussed for different models. In order to solve the models, hybrid intelligent algorithm is designed based on the fuzzy simulation technique and tabu search algorithm. Finally, two application results are given to show the applications of the models and algorithm.     

Uncertain Shapley value of coalitional game with application to supply chain alliance

Uncertain coalitional game deals with situations in which the transferable payoffs are uncertain variables. The uncertain core has been proposed as the solution of uncertain coalitional game. This paper goes further by presenting two definitions of uncertain Shapley value: expected Shapley value and α-optimistic Shapley value. Meanwhile, some characterizations of the uncertain Shapley value are investigated. Finally, as an application, uncertain Shapley value is used to solve a profit allocation problem of supply chain alliance.     

Fuzzy random chance-constrained programming

By fuzzy random programming, we mean the optimization theory dealing with fuzzy random decision problems. This paper presents a new concept of chance of fuzzy random events, and constructs a general framework of fuzzy random chance-constrained programming. We also design a spectrum of fuzzy random simulations for computing uncertain functions arising in the area of fuzzy random programming. To speed up the process of handling uncertain functions, we train a neural network to approximate uncertain functions based on the training data generated by fuzzy random simulation. Finally, we integrate the fuzzy random simulation, neural network, and genetic algorithm to produce a more powerful and effective hybrid intelligent algorithm for solving fuzzy random programming models and illustrate its effectiveness by some numerical examples     

Related Fritz Carlson type inequalities for Sugeno integrals

In this paper, two general related inequalities to Carlson type inequality for the Sugeno integrals on an abstract fuzzy measure space $(X, \mathcal{F})$ are studied. Several examples are given to illustrate the validity of these inequalities.     

基于分布鲁棒优化的城市轨道交通网络末班车衔接研究

随着我国城市轨道交通网络规模快速扩张,线路间协调配合的高度复杂性给城市轨道交通的运营组织与管理带来极大挑战.针对客流需求及其分布双重不确定条件下的城市轨道交通网络末班车衔接优化问题,提出一种分布鲁棒机会约束规划模型,即在给定容忍度下最小化最坏条件下的换乘失败客流量.通过分析分布鲁棒优化模型与其对应鲁棒优化模型之间的联系,证明该模型为鲁棒优化模型的推广形式.基于有限的期望和方差信息构造高斯分布非精确集,采用对偶理论将原模型转化为可利用CPLEX求解的混合整数二阶锥规划形式,并通过数值实验验证所构建模型的有效性.算例结果表明:分布鲁棒模型对于小规模网络可利用CPLEX快速求得精确解;相比鲁棒模型可有效避免产生过于保守的优化结果;相比随机模型可有效降低极端情况下换乘失败客流量,具有较强的鲁棒性.     

基于随机模拟与PSO算法相结合的随机机会约束规划算法

随机机会约束规划作为一类重要的随机规划,广泛存在于许多领域中.为了寻找更有效的求解随机机会约束规划的算法,通过采用随机模拟来逼近随机函数,并在微粒群算法PSO(Particle Swarm Optimization)中利用随机模拟实现估计适应值和检验解的可行性,从而给出了求解随机机会约束规划的新算法,最后,测试其性能并与遗传算法进行了比较,实例结果表明该算法的正确性和有效性.     

求解随机机会约束规划的混合智能算法

随机机会约束规划是一类有着广泛应用背景的随机规划问题,采用随机仿真产生样本训练BP网络以逼近随机函数,然后在微粒群算法中利用神经网络计算适应值和实现检验解的可行性,从而提出了一种求解随机机会约束规划的混合智能算法。最后通过两个实例的仿真结果说明了算法的正确性和有效性。     

Fuzzy data envelopment analysis: A discrete approach

Data envelopment analysis (DEA) as introduced by Charnes, Cooper, and Rhodes (1978) is a linear programming technique that has widely been used to evaluate the relative efficiency of a set of homogenous decision making units (DMUs). In many real applications, the input-output variables cannot be precisely measured. This is particularly important in assessing efficiency of DMUs using DEA, since the efficiency score of inefficient DMUs are very sensitive to possible data errors. Hence, several approaches have been proposed to deal with imprecise data. Perhaps the most popular fuzzy DEA model is based on α-cut. One drawback of the α-cut approach is that it cannot include all information about uncertainty. This paper aims to introduce an alternative linear programming model that can include some uncertainty information from the intervals within the α-cut approach. We introduce the concept of “local α-level” to develop a multi-objective linear programming to measure the efficiency of DMUs under uncertainty. An example is given to illustrate the use of this method.     

An intelligent quality-based approach to fusing multi-source probabilistic information

Our objective here is to obtain quality-fused values from multiple sources of probabilistic distributions, where quality is related to the lack of uncertainty in the fused value and the use of credible sources. We first introduce a vector representation for a probability distribution. With the aid of the Gini formulation of entropy, we show how the norm of the vector provides a measure of the certainty, i.e., information, associated with a probability distribution. We look at two special cases of fusion for source inputs those that are maximally uncertain and certain. We provide a measure of credibility associated with subsets of sources. We look at the issue of finding the highest quality fused value from the weighted aggregations of source provided probability distributions.