Expected Value Operator Research Articles

Rough set approach to non-cooperative continuous differential games

In this article, we analyze the non-cooperative continuous differential games under rough interval environment. The combination of rough set and continuous differential games expresses a new class defined as rough non-cooperative continuous differential games. We develop an effective and simple technique for solving the problem under study; in this methodology, the rough non-cooperative continuous differential games are transformed into two problems with interval environment corresponding to the upper and lower approximation of rough intervals. Furthermore, four crisp problems of non-cooperative continuous differential games are derived as follows: upper lower problem (ULP), lower lower problem (LLP), lower upper problem (LUP) and upper upper problem (UUP). Moreover, the trust measure and the expected value operator of rough interval are used to find the α-trust equilibrium strategies and the expected equilibrium strategies. Sufficient and necessary conditions for equilibrium strategies of rough non-cooperative continuous differential games are also derived. Finally, applicability and validity of the models and technique proposed in this article are illustrated with a practical example.

A study on two-person zero-sum rough interval continuous differential games

In this paper, we concentrate on solving the zero-sum two-person continuous differential games using rough programming approach. A new class defined as rough continuous differential games is resulted from the combination of rough programming and continuous differential games. An effective and simple technique is given for solving such problem. In addition, the trust measure and the expected value operator of rough interval are used to find the $$ \upalpha $$ -trust and expected equilibrium strategies for the rough zero-sum two-person continuous differential games. Moreover, sufficient and necessary conditions for an open loop saddle point solution of rough continuous differential games are also derived. Finally, a numerical example is given to confirm the theoretical results.

Developing Multiobjective Equilibrium Optimization Method for Sustainable Uncertain Supply Chain Planning Problems

This paper proposes a new multiobjective two-stage equilibrium optimization method for a supply chain planning problem with uncertain demand. To handle the ambiguity in the distribution of demand, the probability and possibility distributions are integrated to characterize the uncertain demand. As a result, the decision process in our equilibrium optimization problem is divided into two stages. In the first stage, decision variables should be taken before knowing the realizations of uncertain demand; while the second-stage decision variables must be taken after knowing the outcome of subjective uncertainty embedded in demand. On the basis of the proposed dynamic decision scheme, the objectives in the first-stage are constructed via credibilistic optimization methods. The objective and constraints in the second-stage are built via stochastic optimization methods. More specifically, three objectives in the first-stage are constructed based on the expected value operator and conditional value-at-risk of fuzzy variable, and the second-stage optimization model is built as a stochastic expected value model under a probabilistic constraint. When the random parameters follow normal distributions, the proposed equilibrium optimization model is equivalent to a triobjective two-stage credibilistic optimization model. To solve this model, we first employ a sequence of discrete possibility distributions to approximate continuous possibility distributions. Then, we design a new archive-guided multiobjective particle swarm optimization based on decomposition to solve the obtained approximate optimization model. Finally, numerical experiments via a light emitting diode industry problem are conducted to demonstrate the feasibility and effectiveness of the proposed optimization method and new heuristic algorithm.

Robust Optimization of PDEs with Random Coefficients Using a Multilevel Monte Carlo Method

This paper addresses optimization problems constrained by partial differential equations with uncertain coefficients. In particular, the robust control problem and the average control problem are considered for a tracking type cost functional with an additional penalty on the variance of the state. The expressions for the gradient and Hessian corresponding to either problem contain expected value operators. Due to the large number of uncertainties considered in our model, we suggest evaluating these expectations using a multilevel Monte Carlo (MLMC) method. Under mild assumptions, it is shown that this results in the gradient and Hessian corresponding to the MLMC estimator of the original cost functional. Furthermore, we show that the use of certain correlated samples yields a reduction in the total number of samples required. Two optimization methods are investigated: the nonlinear conjugate gradient method and the Newton method. For both, a specific algorithm is provided that dynamically decides which and how many samples should be taken in each iteration. The cost of the optimization up to some specified tolerance $\tau$ is shown to be proportional to the cost of a gradient evaluation with requested root mean square error $\tau$. The algorithms are tested on a model elliptic diffusion problem with lognormal diffusion coefficient. An additional nonlinear term is also considered.

Research on Fuzzy Multiple Attribute Route Evaluation Decision Method

The route evaluation is an important basis for airlines to make route decisions. In order to solve the fuzzy multiple attribute route decision problem, route attribute weight was completely unknown, and the route evaluation fuzzy language information was given, the fuzzy language and preference information were converted into trapezoidal fuzzy numbers. A programming model was constructed by minimizing the total deviation between trapezoidal fuzzy numbers to determining the weight of each attribute in the route decision, By weighting the average of the route attribute value and the weights value, compared the expected value of fuzzy comprehensive evaluation for each route, and ranking alternatives by using the expected value operator of fuzzy variable. Finally, a numerical example was used to illustrate the proposed method. The results show that the method can be used by air-lines to solve multiple attribute route decision problems with unknown weights, and the method is scientific and effective.

Multi-Objective Fixed-Charge Transportation Problem with Random Rough Variables

This paper considers a multi-objective fixed-charge transportation problem (MOFCTP) in which the parameters of the objective functions are random rough variables, while the supply and the demand parameters are rough variables. In real-life situations, the parameters of a multi-objective fixed-charge transportation problem may not be defined precisely, because of globalization of the market, uncontrollable factors, etc. As such, the multi-objective fixed-charge transportation problem is proposed under rough and random rough environments. To tackle uncertain (rough and random rough) parameters, the proposed model employs an expected value operator. Furthermore, a procedure is developed for converting the uncertain multi-objective fixed-charge transportation problem into a deterministic form and then solving the deterministic model. Three different methods, namely, the fuzzy programming, global criterion, and ϵ-constrained methods, are used to derive the optimal compromise solutions of the suggested model. To provide the preferable optimal solution of the formulated problem, a comparison is drawn among the optimal solutions that are extracted from different methods. Herein, the ϵ-constrained method derives a set of optimal solutions and generates an exact Paretofront. Finally, in order to show the applicability and feasibility of the proposed model, the paper includes a real-life example of a multi-objective fixed-charge transportation problem. The main contribution of the paper is that it deals with MOFCTP using two types of uncertainties, thus making the decision making process more flexible.

On solution of constraint matrix games under rough interval approach

The aim of this article is to propose an effective technique for solving constraint matrix games with rough interval payoffs, which are a class of non-cooperative two-person matrix games with rough interval payoffs and the strategies of the players are constrained. Since the payoffs of the rough constraint matrix games are rough intervals, then its game value is also a rough interval. In this technique, we derived four linear programming problems, which are used to obtain the upper–lower bound, lower–lower bound, lower–upper bound and upper–upper bound of the rough interval game values of the players in rough constraint matrix games. Moreover, the expected value operator and trust measure of rough interval have been used to obtain the α-trust equilibrium strategies and the expected equilibrium strategies of the problem under study. In addition, the different advantages of the proposed technique over those existing are discussed. Finally, a numerical experiment of market share game model is given and solved by the three mentioned methods to illustrate the effectiveness and practicality of the proposed method.

Forward and reverse flows pricing decisions for two competing supply chains with common collection centers in an intuitionistic fuzzy environment

This paper studies the selling and the acquisition price decisions for the products of two partially overlapping competing supply chains in an intuitionistic fuzzy environment. In the forward flows, the supply chains are assumed to produce substitutable products and sell them in a duopoly marketplace. The reverse flows are also concentrated on collecting and recycling the used products. While regularly it is assumed that the structures of the competing supply chains are completely separated, this paper assumes that the supply chains benefit from the services of the same collection centers. Considering the collection centers and the two supply chains to act as three integrated entities having enough bargaining powers to affect the decisions of each other, this paper concentrates on investigating the effects of different power structure scenarios on the optimal pricing decisions made by all of the parties involved in the problem. Another important feature of this study is that it employs the concept of intuitionistic fuzzy sets to integrate uncertainties and impressions inevitable in real-world situations into the decision-making process. To achieve this purpose, first, a new credibility-based definition for the concept of intuitionistic fuzzy variables is provided. This concept is then employed in the process of developing the intuitionistic fuzzy pricing models. A new expected value operator is also defined which is employed to transform the credibility-based intuitionistic fuzzy programming problems into their crisp equivalents. A numerical example is employed to discuss the properties of the introduced concepts, providing some interesting managerial implications regarding the defined problem.

A data-driven optimization model to collaborative manufacturing system considering geometric and physical performances for hypoid gear product

Abstract A data-driven optimization model to collaborative manufacturing system considering geometric and physical performances is proposed to improve competitiveness of hypoid gear product development facing with economic globalization. Firstly, to deal with the vagueness or impreciseness of the voice of customer (VOC), the fuzzy quality function deployment (fuzzy-QFD) using fuzzy weighted average method in the fuzzy expected value operator is introduced into hypoid gear manufacturing. It can convert them into the critical to qualities (CTQs), and the technical geometric and physical performance requirements. And then, the multi-objective optimization (MOO) modification of machine settings is proposed to establish a basic data-driven model for collaborative system. Different with the conventional modification only considering geometric performance, it provides an improved modification also considering the physical performances. Finally, a double-curved shell model of hypoid gear finite element is used to perform the numerical loaded tooth contact analysis (NLTCA), as well as to establish the data-driven relationships between machine settings and physical performance evaluations. Immediately, whole development is divided into three sub-problems: i) optimal operations of the noise factors by measurement and numerical control (NC) compensation, ii) identification of the prescribed ease-off topography by multi-objective optimization using iterative reference point approach and iii) modification considering geometric performance by a trust region algorithm with step strategy. The numerical instance in practical applications is given to verify the proposed methodology.

Stackelberg game optimization for integrated production-distribution-construction system in construction supply chain

Although construction supply chain management has attracted significant research attention, this field remains somewhat fragmented. This paper examines an integrated production-distribution-construction system consisting of the construction department and material suppliers under a fuzzy random environment with the aim of optimizing the global equilibrium. A novel bi-level multistage programming method with multiple objective optimization is developed to examine the inherent conflicts and complex interactions among decision makers in order to obtain the Stackelberg–Nash equilibrium solution, in which the construction department, as the leader, decides on the material allocations to construction sites, while the material supplier, as the follower, produces and transports the corresponding materials. For dealing with uncertainties, a hybrid crisp approach with an expected value operator is proposed to convert the fuzzy random parameters into definitive parameters. A hybrid algorithm combining an evolved genetic algorithm and particle swarm optimization is developed to solve this novel Stackelberg game model. The results from a practical example demonstrate the practicality and efficiency of the proposed optimization method, highlight the significance of quantitative analysis for the construction supply chain, and provide objective guidelines for its real-world application.

Fuzzy pay-off method for real options: The center of gravity approach with application in oilfield abandonment

This paper presents the CoG-FPOM, an adaptation of the Fuzzy Pay-Off Method (FPOM). The FPOM is a scenario-based real option valuation method that uses fuzzy numbers as possibility distributions. The paper shows that there are situations in which the original FPOM calculates the real option value to be negative, what is theoretically incorrect. The cause is related to the method used for obtaining a single representative value out of a fuzzy number. Instead of the possibilistic expected value operation used in the original FPOM, the CoG-FPOM uses the center of gravity (CoG) to accomplish the task – a proof of its general validity is presented. An application to calculate the abandonment real option for petroleum producing fields shows that the proposal can be easily used for project valuation under uncertainty.

Determining the importance ratings of customer requirements in quality function deployment based on interval linguistic information

Quality function deployment (QFD) is a customer-oriented tool and is widely applied to design and improve products and services. Determining the importance ratings (IRs) of customer requirements (CRs) is an essential step in QFD application and will affect the quality of product design and improvement. In this study, a group decision-making method is proposed to obtain realistic IRs. Low-carbon environment is considered in recognising CRs. Interval linguistic information (ILI) is used to express the vague evaluations in product improvement. In addition, an interval linguistic weighted arithmetic averaging operator, a normalised formula, and an expected value operator are integrated to deal with evaluation matrices expressed by ILI. A relationship matrix is used reversely to acquire accurate basic IRs (BIRs). The improved cosine method based on ILI is also employed to derive BIRs. The modified entropy method based on ILI is proposed to determine a competitive priority rating (CPR) of a CR. The final IR of every CR can be obtained by integrating its BIR and CPR. Finally, a practical product improvement of turbine engine is provided to illustrate the validity and feasibility of the proposed approach.

A Scalar Expected Value of Intuitionistic Fuzzy Random Individuals and Its Application to Risk Evaluation in Insurance Companies

Randomness and uncertainty always coexist in complex systems such as decision-making and risk evaluation systems in the real world. Intuitionistic fuzzy random variables, as a natural extension of fuzzy and random variables, may be a useful tool to characterize some high-uncertainty phenomena. This paper presents a scalar expected value operator of intuitionistic fuzzy random variables and then discusses some properties concerning the measurability of intuitionistic fuzzy random variables. In addition, a risk model based on intuitionistic fuzzy random individual claim amount in insurance companies is established, in which the claim number process is regarded as a Poisson process. The mean chance of the ultimate ruin is investigated in detail. In particular, the expressions of the mean chance of the ultimate ruin are presented in the cases of zero initial surplus and arbitrary initial surplus, respectively, if individual claim amount is an exponentially distributed intuitionistic fuzzy random variable. Finally, two illustrated examples are provided.

A Profit Maximizing Solid Transportation Model Under a Rough Interval Approach

This study attempts to establish an innovative solid transportation problem that intends to maximize profit under the rough interval approximation methodology. Two transportation problems were constructed in this regard with interval coefficients corresponding to the upper approximations and the lower approximations of the rough intervals under study. Furthermore, from the contingent solid transportation problems, four different classical solid transportation problems were derived, which were subsequently solved on the LINGO iteration platform. The concepts of completely satisfactory solution and rather satisfactory solution , surely optimal range , possibly optimal range , and rough optimal range have been discussed with a perspective to its relevance to real-world practical problems. The rough chance-constrained programming and the expected value operator for rough interval have been applied to solve the problem under study. The distinct advantages of the proposed method over those existing have been outlined. Numerical examples have also been provided to illustrate the solution procedure and the methodologies adopted.

Reliable Portfolio Selection Problem in Fuzzy Environment: An mλ Measure Based Approach

This paper investigates a fuzzy portfolio selection problem with guaranteed reliability, in which the fuzzy variables are used to capture the uncertain returns of different securities. To effectively handle the fuzziness in a mathematical way, a new expected value operator and variance of fuzzy variables are defined based on the m λ measure that is a linear combination of the possibility measure and necessity measure to balance the pessimism and optimism in the decision-making process. To formulate the reliable portfolio selection problem, we particularly adopt the expected total return and standard variance of the total return to evaluate the reliability of the investment strategies, producing three risk-guaranteed reliable portfolio selection models. To solve the proposed models, an effective genetic algorithm is designed to generate the approximate optimal solution to the considered problem. Finally, the numerical examples are given to show the performance of the proposed models and algorithm.

Asymptotic trends in time-varying oscillatory period for a dual-staged torsional system

The primary goal of this article is to propose a new analysis tool that estimates the asymptotic trends in the time-varying oscillatory period of a non-linear mechanical system. The scope is limited to the step-response of a torsional oscillator containing a dry friction element and dual-staged spring. Prior work on the stochastic linearization techniques is extended and modified for application in time domain. Subsequently, an instantaneous expected value operator and the concept of instantaneous effective stiffness are proposed. The non-linear system is approximated at some instant during the step-response by a linear time-invariant mechanical system that utilizes the instantaneous effective stiffness concept. The oscillatory period of the non-linear step-response at that instant is then approximated by the natural period of the corresponding linear system. The proposed method is rigorously illustrated via two computational example cases (a near backlash and near pre-load non-linearities), and the necessary digital signal processing parameters for time domain analysis are investigated. Finally, the feasibility and applicability of the proposed method is demonstrated by estimating the softening and hardening trends in the time-varying oscillatory period of the measured response for two laboratory experiments that contain clearance elements and multi-staged torsional springs.

Integrating geometric programming with rough set theory

Geometric programming has been applied in the problems of engineering design, economics and management science. The conventional deterministic geometric programming method requires precise single values for the coefficients and exponents of decision variables. However, there may exist uncertainty and impreciseness about the parameters as well as data in complex real-life problems. In such situations, the deterministic geometric programming method is inappropriate. In this paper, we integrate the deterministic geometric programming with rough set theory to propose a rough geometric programming method. Our proposed method has mainly three characteristics. Firstly, the coefficients in the objective function and constraints are rough variables. Secondly, the expected-value operator of rough variables is implemented. Thirdly, the method can determine both lower and upper bounds of the objective function at a specific trust level. Three illustrative examples are presented to demonstrate the efficacy of our novel method.

Correlation of Trapezoidal Fuzzy Variables Mistreatment Credibility Measure

The emphasis during this paper is principally on a replacement measure of correlation coefficient between trapezoidal fuzzy variables, called credibility correlation coefficient, is developed by creating use of the conception of credibility measure. This new measures are found to satisfy sure properties expected value operator and variance of fuzzy variables. The performance of credibility coefficient of correlation is compared with some existing measures of coefficient of correlation for fuzzy variety and therefore the results are mentioned. The man of science makes an attempt to clarify with the assistance of few examples.

A FUZZY BASED APPROACH FOR EARLY REQUIREMENT PRIORITIZATION

The importance of the prioritization in commercial software development has been analyzed by many researchers. The gathered requirements are required to be put into an order of some priority. In other words we can say that there is a need to prioritize the requirements. It is evident that most of the approaches and techniques proposed in recent research to prioritize the requirements have not been widely adopted. These approaches are too complex, time consuming, or inconsistent and difficult to implement In this paper we propose a fuzzy based approach for requirement prioritization in which requirement are prioritized in early phase of requirement engineering as post elicitation step. This category of prioritization is known as early requirement prioritization. The proposed fuzzy based approach considers the nature of requirements by modeling their attributes as fuzzy variables. As such, these variables are integrated into a fuzzy based inference system in which the requirements represented as input attributes and ranked via the expected value operator of a fuzzy variable.

A note on inequalities and critical values of fuzzy rough variables

A fuzzy rough variable is defined as a rough variable on the universal set of fuzzy variables, or a rough variable taking ‘fuzzy variable’ values. In order to further discuss the mathematical properties of fuzzy rough variables, this paper extends some inequalities to the context of fuzzy rough theory based on the chance measure and the expected value operator, involving the Markov inequality, the Chebyshev inequality, the Hölder inequality, the Minkowski inequality, and the Jensen inequality. After that, linearity, monotonicity, and continuity of critical values of fuzzy rough variables are also investigated.