Expected Value Operator Research Articles

Mini-Batch Risk Forms

Risk forms are real functionals of two arguments: a bounded measurable function on a Polish space and a probability measure on that space. They are convenient mathematical structures adapting the coherent risk measures to the situation of a variable reference probability measure. We introduce a new class of risk forms called mini-batch forms. We construct them by using a random empirical probability measure as the second argument and by post-composition with the expected value operator. We prove that coherent and law invariant risk forms generate mini-batch risk forms which are well defined on the space of integrable random variables, and we derive their dual representation. We demonstrate how unbiased stochastic subgradients of such risk forms can be constructed. Then, we consider pre-compositions of mini-batch risk forms with nonsmooth and nonconvex functions, which are differentiable in a generalized way, and we derive generalized subgradients and unbiased stochastic subgradients of such compositions. Finally, we study the dependence of risk forms and mini-batch risk forms on perturbation of the probability measure and establish quantitative stability in terms of optimal transport metrics. We obtain finite-sample expected error estimates for mini-batch risk forms involving functions on a finite-dimensional space.

New approach to solve fuzzy multi-objective multi-item solid transportation problem

This paper explores the study of Multi-Objective Multi-item Solid Transportation Problem (MMSTP) under the fuzzy environment. Realizing the impact of real-life situations, here we consider MMSTP with parameters, e.g., transportation cost, supply, and demand, treat as trapezoidal fuzzy numbers. Trapezoidal fuzzy numbers are then converted into nearly approximation interval numbers by using (P. Grzegorzewski, Fuzzy Sets Syst. 130 (2002) 321–330.) conversation rule, and we derive a new rule to convert trapezoidal fuzzy numbers into nearly approximation rough interval numbers. We derive different models of MMSTP using interval and a rough interval number. Fuzzy programming and interval programming are then applied to solve converted MMSTP. The expected value operator is used to solve MMSTP in the rough interval. Thereafter, two numerical experiments are incorporated to show the application of the proposed method. Finally, conclusions are provided with the lines of future study of this manuscript.

Characterization of matrix transformation of complex uncertain sequences via expected value operator

The aim of this paper is to study the concept of matrix transformation between complex uncertain sequences in mean. The characterization of the matrix transformation has been made by applying the concept of convergence of complex uncertain series. Moreover, in this context, some well-known theorems of real sequence spaces have been established by considering complex uncertain sequence via expected value operator.

Solving the multi-modal transportation problem via the rough interval approach

This research studies a transportation problem to minimize total transportation cost under the rough interval approximation by considering the multi-modal transport framework, referred to here as the rough Multi-Modal Transportation Problem (MMTP). The parameters of MMTP are rough intervals, because the problem is chosen from a real-life scenario. To solve MMTP under a rough environment, we employ rough chance-constrained programming and the expected value operator for the rough interval and then outline the main advantages of our suggested method over those existing methods. Next, we propose an algorithm to optimally solve the problem and present a numerical example to examine the proposed technique. Finally, the solution is analyzed by the proposed method with rough-chance constrained programming and expected value operator.

Arithmetic Operations and Expected Values of Regular Interval Type-2 Fuzzy Variables

High computation complexity restricts the application prospects of the interval type-2 fuzzy variable (IT2-FV), despite its high degree of freedom in representing uncertainty. Thus, this paper studies the fuzzy operations for the regular symmetric triangular IT2-FVs (RSTIT2-FVs)—the simplest IT2-FVs having the greatest membership degrees of 1. Firstly, by defining the medium of an RSTIT2-FV, its membership function, credibility distribution, and inverse distribution are analytically and explicitly expressed. Secondly, an operational law for fuzzy arithmetic operations regarding mutually independent RSTIT2-FVs is proposed, which can simplify the calculations and directly output the inverse credibility of the functions. Afterwards, the operational law is applied to define the expected value operator of the IT2-FV and prove the linearity of the operator. Finally, some comparative examples are provided to verify the efficiency of the proposed operational law.

Rough approximation‐based inventory optimization with random prices for B2C companies during the promotion period

Most B2C e-commerce giants such as Amazon, Bestbuy, Alibaba, periodically or occasionally conduct promotional activities that put significant stress on their inventory systems. The sharp demand and price fluctuations caused by these promotional activities distress stocks over an inventory period if customer satisfaction is maintained. This article develops a chance-constrained multiobjective programming model to obtain optimal order quantities in an e-commerce inventory system with stochastic demand and prices. Traditional techniques for coping with the stochastic coefficients using an expected value operator or probability measures are usually inefficient in estimating quantities. With rational assumptions regarding market behavior, stochastic demand and price are characterized with a normal distribution based on historical data. A novel method called rough approximation is designed to flexibly formulate the feasible region, from which a crisp model is obtained that has no uncertain information. A proxy ideal point-based genetic algorithm (PIP-GA) is then developed to solve the model, allowing decision makers to fix the optimal order quantities for different promotional activity scales. Finally, the results are illustrated using a practical example from the Chinese B2C e-commerce giant “JD.COM.” Some special features are discussed to show the availability of the proposed model and the solution methodology.

Bifuzzy-Bilevel Programming Model: Solution and Application

Bi-level programming is widely used in processing various questions, but it cannot deal with the complex and fuzzy information contained in problems. In order to solve such problems better with intricate and vague information that can be efficiently handled by bifuzzy theory, a bifuzzy–bilevel programming model that sets the parameters to bifuzzy variables is proposed in this paper, which can process complex realistic data more accurately and improve the feasibility and validity of bi-level programming models. To ensure the solvability of the model, the equivalent form of the bifuzzy–bilevel programming model is obtained by utilizing the expected value operator. According to the linear and nonlinear characteristics of the model, the Karush–Kuhn–Tucker condition and particle swarm optimization algorithm are employed to handle the problem, respectively. Finally, by taking the distribution center location problem of the supplier as an example, the bifuzzy–bilevel programming model is applied in practice to balance highly intricate customer demands and corporate cost minimization, obtaining the feasible solution of functions at the upper and lower levels, and the bifuzzy information in the problem can also be processed well, which proves the effectiveness of the proposed methodology.

The Scalar Mean Chance and Expected Value of Regular Bifuzzy Variables

As a natural extension of the fuzzy variable, a bifuzzy variable is defined as a mapping from a credibility space to the collection of fuzzy variables, which is an appropriate tool to model the two-fold fuzzy phenomena. In order to enrich its theoretical foundation, this paper explores some important measures for regular bifuzzy variables, the most commonly used type of bifuzzy variables. Firstly, we introduce the regular bifuzzy variables’ mean chance measure and some properties, including self-duality and its calculation formulas. Furthermore, we also investigate the mean chance distribution for strictly monotone functions of regular bifuzzy variables based on the proposed operational law. Finally, we present the expected value operator as well as equivalent analytical formulas of the expected value of regular bifuzzy variables and their strictly monotone functions.

The two-sorted algebraic theory of states, and the universal states of MV-algebras

States of unital Abelian lattice-groups (normalised positive group homomorphisms to R ) provide an abstraction of expected-value operators. A well-known theorem due to Mundici asserts that the category of unital lattice-groups (with unit-preserving lattice-group homomorphisms as morphisms) is equivalent to the algebraic category of MV-algebras, and their homomorphisms. Through this equivalence, states of lattice-groups correspond to certain [ 0 , 1 ] -valued functionals on MV-algebras, which are also known as states. In this paper we allow states to take values in any unital lattice-group (or in any MV-algebra) rather than just in R (or just in [ 0 , 1 ] , respectively). We introduce a two-sorted algebraic theory whose models are precisely states of MV-algebras. We extend Mundici's equivalence to one between the category of MV-algebras with states as morphisms , and the category of unital Abelian lattice-groups with, again, states as morphisms. Thus, the models of our two-sorted theory may also be regarded as states between unital Abelian lattice-groups. As our first main result, we derive the existence of the universal state of any MV-algebra from the existence of free algebras in multi-sorted algebraic categories. The significance of the universal state of a given algebra is that it provides the most general expected-value operator on that algebra—a construct that is not available if one insists that states be real-valued. In the remaining part of the paper, we seek concrete representations of such universal states. We begin by clarifying the relationship of universal states with the theory of affine representations: the universal state A → B of the MV-algebra A coincides with a certain modification of Choquet's affine representation (of the lattice-group corresponding to A ) if, and only if, B is semisimple . Locally finite MV-algebras are semisimple, and Boolean algebras are instances of locally finite MV-algebras. Our second main result is then that the universal state of any locally finite MV-algebra has semisimple codomain , and can thus be described through our adaptation of Choquet's affine representation.

Fuzzy-rough multi-objective product blending fixed-charge transportation problem with truck load constraints through transfer station

In this contribution, for the first time, an efficient model of multi-objective product blending fixed-charge transportation problem with truck load constraints through transfer station is formulated. Transfer station inserts transfer cost and type-I fixed-charge. Our aim is to analyze an extra cost that treats as type-II fixed-charge and truck load constraints in the designed model that required when the amount of items exceeds the capacity of vehicle for fulfilling the shipment by more than one trip. Type-II fixed-charge is added with transportation cost and other cost from transfer station. We consider here an important issue of the multi-objective transportation problem as product blending constraints for transporting raw materials with different purity levels for customers’ satisfaction. In realistic point of view, the parameters of the model are imprecise in nature due to existing several unpredictable factors. These factors are apprehended by incorporating the fuzzy-rough environment on the parameters. Expected-value operator is utilized to derive the deterministic form of fuzzy-rough data, and the model is experienced with help of fuzzy programming, neutrosophic linear programming and global criteria method. Two numerical examples are illustrated to determine the applicability of the proposed model.

Characterization of statistical convergence of complex uncertain double sequence

The main aim of this paper is to introduce statistical convergence of a complex uncertain double sequence. Characterization of statistical convergence is given via uncertain measure, uncertain expected value operator, uncertain distribution function and also with respect to almost surely, uniformly almost surely. The concept of boundedness is introduced to establish some results. Finally, the notion of statistically complex uncertain double Cauchy sequence is proposed to obtain interrelationship with statistically convergent complex uncertain double sequence.

Intuitionistic fuzzy multi-stage multi-objective fixed-charge solid transportation problem in a green supply chain

This research mainly focuses on presenting an innovative study of a multi-stage multi-objective fixed-charge solid transportation problem (MMFSTP) with a green supply chain network system under an intuitionistic fuzzy environment. The most controversial issue in recent years is that greenhouse gas emissions such as carbon dioxide, methane, etc. induce air pollution and global warming, thus motivating us to formulate the proposed research. In real-world situations the parameters of MMFSTP via a green supply chain network system usually have unknown quantities, and thus we assume trapezoidal intuitionistic fuzzy numbers to accommodate them and then employ the expected value operator to convert intuitionistic fuzzy MMFSTP into deterministic MMFSTP. Next, the methodologies are constructed to solve the deterministic MMFSTP by weighted Tchebycheff metrics programming and min-max goal programming, which provide Pareto-optimal solutions. A comparison is then drawn between the Pareto-optimal solutions that are extracted from the programming, and thereafter a procedure is performed to analyze the sensitivity analysis of the target values in the min–max goal programming. Finally, we incorporate an application example connected with a real-life industrial problem to display the feasibility and potentiality of the proposed model. Conclusions about the findings and future study directions are also offered.

Fuzzy Bi-Objective Closed-Loop Supply Chain Network Design Problem with Multiple Recovery Options

A network design of a closed-loop supply chain (CLSC) with multiple recovery modes under fuzzy environments is studied in this article, in which all the cost coefficients (e.g., for facility establishment, transportation, manufacturing and recovery), customer demands, delivery time, recovery rates and some other factors that cannot be precisely estimated while designing are modeled as triangular fuzzy numbers. To handle these uncertain factors and achieve a compromise between the two conflicting objectives of maximizing company profit and improving customer satisfaction, a fuzzy bi-objective programming model and a corresponding two-stage fuzzy interactive solution method are presented. Applying the fuzzy expected value operator and fuzzy ranking method, the fuzzy model is transformed into a deterministic counterpart. Subsequently, Pareto optimal solutions are determined by employing the fuzzy interactive solution method to deal with the conflicting objectives. Numerical experiments address the efficiency of the proposed model and its solution approach. Furthermore, by comparing these results with the CLSC network design in deterministic environments, the benefits of modeling the CLSC network design problem with fuzzy information are highlighted.

The mean chance conditional value at risk under interval type-2 intuitionistic fuzzy random environment

The interval type-2 intuitionistic fuzzy random variable is an extension of the intuitionistic fuzzy random variable such that it can be a effective tool to determine some high-uncertainty phenomena. In this paper, the interval type-2 intuitionistic fuzzy random variable is introduced for the first time, and then, a scalar expected value operator of interval type-2 intuitionistic fuzzy random variable is proposed. Moreover, the new concepts of mean chance value at risk and mean chance conditional value at risk are discussed for the interval type-2 intuitionistic fuzzy random variables which have application in uncertain optimization, like fuzzy inverse location problems. Finally, it is proven that mean chance value at risk and mean chance conditional value at risk fulfill the convex risk metric properties.

Multi-objective fixed-charge transportation problem using rough programming

This paper analyses the multi-objective fixed-charge transportation problem (MOFCTP) using rough programming. Due to globalisation of market, the parameters of MOFCTP may not be defined precisely, so the parameters of the MOFCTP are treated as rough intervals. Expected value operator is used to convert rough MOFCTP to deterministic MOFCTP. Fuzzy programming method and linear weighted-sum method are used to obtain Pareto-optimal solution from deterministic MOFCTP. A comparative study is made between the obtained solutions extracted from the methods; and thereafter we perform a procedure to analyse the sensitive analysis of the parameters in MOFCTP. Finally, in order to show the applicability of our proposed study, an example on MOFCTP is included in this paper.

A bi-level programming model of rough stochastic MRCPSP in large-scale hydropower construction project

This paper focuses on developing a bi-level programming model for rough stochastic multi-mode resource constrained project scheduling problem (RS-MRCPSP/bi-level). The metal structure installation project (MSIP) in Wudongde Hydropower Station is considered as the prototype, and then it is extended to be a generalised bi-level MRCPSP. In the upper level, the construction contractor is responsible for the investment on the activity, while the outsourcing partner is in charge of construction and implementation in the lower level. In order to deal with the rough stochastic parameters, the rough stochastic expected value operator is employed. Subsequently, in order to obtain the optimal schedule, an interactive method-based passive congregation particle swarm optimisation (IM-based PCPSO) is designed. Finally, a practical application for MSIP is presented to highlight the practicability of proposed model and solution method.

A Stochastic Control Strategy for Safely Driving a Powered Wheelchair

In this paper we deal with model-based control design in the presence of uncertainties. We use Stochastic Dynamic Programming to solve two problems, called longitudinal and lateral vehicle control. The goal is to allow safe driving (navigation) of a moving vehicle in an environment with static obstacles. We show how to define the optimization problems given the stochastic driver behavior and environment. The vehicle dynamics model is deterministic (obeys physical laws) and is explicitly integrated into the optimization problem. In terms of results, the numerically computed control policies provide best-on-average performance (according to the expected value operator). In simulation, it is shown that the vehicle effectively avoids obstacles, thus ensuring a safe drive experience.

A bi-level programming model of rough stochastic MRCPSP in large-scale hydropower construction project

This paper focuses on developing a bi-level programming model for rough stochastic multi-mode resource constrained project scheduling problem (RS-MRCPSP/bi-level). The metal structure installation project (MSIP) in Wudongde Hydropower Station is considered as the prototype, and then it is extended to be a generalised bi-level MRCPSP. In the upper level, the construction contractor is responsible for the investment on the activity, while the outsourcing partner is in charge of construction and implementation in the lower level. In order to deal with the rough stochastic parameters, the rough stochastic expected value operator is employed. Subsequently, in order to obtain the optimal schedule, an interactive method-based passive congregation particle swarm optimisation (IM-based PCPSO) is designed. Finally, a practical application for MSIP is presented to highlight the practicability of proposed model and solution method.

Multi-objective multi-item fixed-charge solid transportation problem under twofold uncertainty

In this paper, we investigate a multi-objective multi-item fixed-charge solid transportation problem (MOMIFCSTP) with fuzzy-rough variables as coefficients of the objective functions and of the constraints. The main focus of the paper is to analyze MOMIFCSTP under a fuzzy-rough environment for a transporting system. In practical situations, the parameters of a MOMIFCSTP are imprecise in nature, due to several uncontrollable factors. For these reasons, we introduce the fuzzy-rough variables in MOMIFCSTP to tackle vague data which are different from fuzziness and roughness. Fuzzy-rough expected-value operator is employed to convert fuzzy-rough MOMIFCSTP into deterministic MOMIFCSTP. Thereafter, we develop a methodology to solve the deterministic MOMIFCSTP by technique for order preference by similarity to ideal solution (TOPSIS). Three distinct approaches, namely extended TOPSIS, weighted goal programming (WGP) and fuzzy programming, are used to derive Pareto-optimal solution from the suggested model. A comparison is drawn among the optimal solutions which are derived from different approaches. It is observed from the extracted results that TOPSIS provides a better optimal solution than WGP and fuzzy programming. TOPSIS also overcomes some difficulties which arise in WGP. Finally, a real-world (industrial) problem is incorporated to show the applicability and feasibility of the proposed problem.

A risk-averse multi-item inventory problem with uncertain demand

The observed values of demands in real-life inventory problems are sometimes imprecise due to the lack of information and historical data, thus a growing research is committed to study the properties of risk measures in fuzzy inventory optimization problems. In this paper, a risk-averse fuzzy optimization method is adopted for the multi-item inventory problem, in which the demands are described by common possibility distributions. Firstly, three classes of fuzzy inventory optimization models are built by combining the absolute semi-deviation with expected value operator and then model analysis is given for the min-max inventory models. To make the inventory problem tractable and computable, the equivalent forms of the proposed optimization models are discussed. Subsequently, several useful absolute semi-deviation formulas are presented under triangular, trapezoidal and Erlang possibility distributions. Finally, some numerical experiments are performed to highlight the modeling idea, and the computational results demonstrate the effectiveness of the solution method.