In this paper a network comprising alternative branching nodes with probabilistic outcomes is considered. In other words, network nodes are probabilistic with exclusive-or receiver and exclusive-or emitter. First, an analytical approach is proposed to simplify the structure of network. Then, it is assumed that the duration of activities is positive trapezoidal fuzzy number (TFN). This paper combines the randomness and fuzziness and shows that the fuzzy completion time of alternative stochastic network is a fuzzy-valued random variable. Then, the probability function of network fuzzy completion time and its expected value is defined. Finally, the applications and computations are illustrated in a numerical example.

Probabilistic or stochastic programming is a framework for modeling optimization problems that involve uncertainty. In this paper, we focus on multi-objective linear programming problems in which the coefficients of constraints and the right hand side vector are fuzzy random variables. There are several methods in the literature that convert this problem to a stochastic or fuzzy problem. By using a special type of fuzzy inequality, we transform the problem into a convenient stochastic problem. Then some known methods are applied to obtain the optimal solution. Finally, the equivalent multi-objective problem is solved by an interactive approach. A numerical example is provided to illustrate the procedure.

In this paper, we consider an anticipating stochastic differential equation in which the integrands are not adapted to the filtration generated by a Wiener process in the stochastic integrals. By leveraging the correspondence between the Skorohod integral and the Itô,-Skorohod integral, we propose solving these equations using standard iterative techniques. Subsequently, we discuss the existence and uniqueness of strong solutions to these equations. The incorporation of non-adapted, fuzzy, and random processes in such equations makes them applicable in financial models.

Due to the vital role resources play in the project's success or failure, in the last 60 years, much research has been done in the field of resource-leveling. The first studies considered the conditions of the project to be definite, but the following researches led to the uncertainty of the project conditions. Some of these uncertain studies assumed that the project conditions were only fuzzy, and some assumed that they were only stochastic. After introducing fuzzy-stochastic theory, project management research considered the conditions for a project to be fuzzy-stochastic. Due to the gap of this approach in resource leveling, this quantitative and developing research developed a multi-objective fuzzy-random resource-leveling model. In this research, the project execution time is considered as a fuzzy-random variable. Finally, the proposed model, which is among the NP-hard models, was solved by an NSGA-II algorithm in Matlab software. Two other algorithms developed this algorithm, namely control algorithm and variable decision preparation algorithm, and a control-memory, to solve the problem of project's diversity. The innovation of this research is noteworthy in two cases. The first is that the multi-objective resource-leveling model was presented in a fuzzy-random manner, and the second is that the NSGA-II algorithm was developed to solve it. Finally, the proposed algorithm's reproducibility, convergence, efficiency, and validity were discussed and approved.

In this paper we developed an inventory model in mixed imprecise and uncertain environment. Presented model is developed form of (r, Q) and is a multi-items model with two objectives as minimizing costs (holding & shortage) and risk level under constraints including available budgetary, the least service level, storage spaces & allowable quantities of shortage. Demand distribution functions are assumed to be exponential and extra demands are supposed in two situations as lost sales and backlogging. At first we develop crisp model then fuzzy stochastic model with fuzzy budgetary, allowable quantities of shortage and shortage spaces (i.e. stochastic with normal distribution function) parameter. All of fuzzy numbers are triangular type. In methodology of solution we change model to a crisp multi objective by using difuzzification of fuzzy constraints and fuzzy chance-constrained programming methods, and then solve it by fuzzy logic method. Finally an illustrated example is taken and solved using LINGO package.

In this paper, a new type of equation namely fuzzy fractional stochastic Pantograph delay differential system (FSPDDS) is proposed. In our previous work, a first extension of fuzzy stochastic differential system into fuzzy fractional stochastic differential system by using Granular differentiability has been established. Here we study the existence and uniqueness results for the fuzzy FSPDDS which are obtained by using generalized Granular differentiability and contraction principle with weaker conditions. This kind of equation is used in many real world problems. Finally, we provide two numerical examples for the effectiveness of the theoretical results.

fuzzy fractional pantograph stochastic differential equations $($FFPSDEs$)$ is investigated here. The initial objective is to show the existence and uniqueness of solutions using Banach fixed point theorem. The second objective is discussing averaging principle of FFPSDEs, precisely, we will prove that the solutions of FFPSDEs can be approximated in the sense of mean square by the solutions of averaged fuzzy fractional stochastic system.

One of the best techniques for evaluating the performance of organizations is data envelopment analysis. Data Envelopment Analysis (DEA) is a non-parametric method for evaluating the performance of Decision-Making Units (DMUs) that recognizes the relative performance of DMUs based on mathematical programming. The classic DEA model were initially formulated for optimal inputs and outputs, but in real-world problems the values observed from input and output data are often ambiguous and random. In fact, decision makers may be faced with a specific hybrid environment where there is fuzziness and randomness in the problem. To overcome this problem, data envelopment analysis models in random fuzzy environment have been proposed. Although the DEA has many advantages, one of the disadvantages of this method is that the classic DEA does not actually give us a definitive conclusion and does not allow random changes in input and output. In this research data envelopment analysis models in fuzzy random environments is reviewed.

In this paper, the numerical solution of the improper fuzzy integral ~I=ò+¥~a ~¦(~x)~dx is proposed where, ~¦(~x) is a bounded and closed fuzzy valued function defined on the closed fuzzy real number system and a is ~a fuzzy number with triangular or bell-shape membership function. For this purpose, the a -level sets of the fuzzy number ~I, -Ia and -Ia, 0£a£1, are evaluated whose end points are the crisp improper integrals. Then, a reliable method is introduced to estimate -Ia and -Ia by using the stochastic arithmetic. In this case, an algorithm is presented which evaluates these integrals by using the Simpson rule. By using the CESTAC method, we find the optimal natural numbers -ma and -ma such that these improper integrals are estimated by definite integrals. Also, for a given r, we evaluate the approximate value of the membership function m~I (r) and determine the accuracy of the results. At last, two fuzzy integrals are computed using the given algorithm to show the results of the research.