Interval Grey Number Research Articles

Dynamic Multi-Attribute Decision Analysis on Three-Parameter Interval Grey Number of Upper (Lower) Deviation Quasi-Normal Distribution

Aiming at the dynamic multi-attribute decision making problem where the weight of each decision stage and attribute weight are completely unknown and the attribute value is unknown distributed three-parameter interval grey number, a three-parameter interval grey number dynamic multiattribute grey target decision making method with attribute value following quasi-normal distribution is proposed. Firstly, the position relationship between the “center of gravity” point and the kernel of the three-parameter interval grey number is discussed. According to the characteristic that the attribute value obeys the quasi-normal distribution, a new weight is given to the “center of gravity” point, and a new distance measure formula of the three-parameter interval grey number is defined. Secondly, according to the principle of maximum entropy, the objective programming model is constructed to determine the stage weight and attribute weight. Then, the schemes are sorted according to the size of the comprehensive bull’s-eye-distance. Finally, an example is given to illustrate the effectiveness of the decision model.

Analysis on the related factors of China's technological innovation ability using greyness relational degree

PurposeIn order to make grey relational analysis applicable to the interval grey number, this paper discusses the model of grey relational degree of the interval grey number and uses it to analyze the related factors of China's technological innovation ability.Design/methodology/approachFirst, this paper gives the definitions of the lower bound domain, the value domain, the upper bound domain of interval grey number and the generalized measure and the generalized greyness of interval grey number. Then, based on the grey relational theory, this paper proposes the model of greyness relational degree of the interval grey number and analyzes its relationship with the classical grey relational degree. Finally, the model of greyness relational degree is applied to analyze the related factors of China's technological innovation ability.FindingsThe results show that the model of greyness relational degree has strict theoretical basis, convenient calculation and easy programming and can be applied to the grey number sequence, real number sequence and grey number and real number coexisting sequence. The relational order of the four related factors of China's technological innovation ability is research and development (R&D) expenditure, R&D personnel, university student number and public library number, and it is in line with the reality.Practical implicationsThe results show that the sequence values of greyness relational degree have large discreteness, and it is feasible and effective to analyze the related factors of China's technological innovation ability.Originality/valueThe paper succeeds in realizing both the model of greyness relational degree of interval grey number with unvalued information distribution and the order of related factors of China's technological innovation ability.

A grey Bayesian inference framework for structural damage assessment

Real-life civil structures often stay in a grey situation due to lack of knowledge and measurements. In a poor information condition, classic Bayesian inference is practically difficult to perform owing to intractable likelihood functions. Under such circumstance, a structure should be defined as a grey system. Thereby, a grey Bayesian inference framework has been proposed by incorporating the grey theory with approximate Bayesian computation for damage assessment purposes. Structural parameters are represented by interval grey variables, whose grey kernels and degrees of greyness are used to define probability distributions for sampling. A grey population Monte Carlo sampler has also been developed by introducing interval grey numbers and an evolutionary particle pool into the particle filtering process. A novel particle interchange mechanism is simultaneously proposed for effective particle interflow. Meanwhile, stochastic response surfaces are used to correlate structural parameters with responses for fast response computation. Also, likelihood calculation is replaced with a distance measure between simulated and measured samples. Lastly, the proposed inference strategy has been verified against both the numerical and experimental beams having the different damage scenarios. The damage locations and severities were successfully identified by the changes in the grey intervals and kernels.

Improved Model for Generating Incremental Product Innovation Schemes

To realize incremental product innovation, the key problems of existing products need to be identified and solved with feasible innovation schemes. This study examines the generation process of innovation schemes by integrating Quality Function Deployment (QFD) and the Theory of Inventive Problem Solving (TRIZ). This study has the following advantages: first, this study proposes new rules for distinguishing physical conflicts with technical conflicts and the method for calculating conflict importance after building House of Quality (HoQ) in QFD. Second, an improved integrated model of QFD and TRIZ is constructed, which generates innovation schemes by solving not only the conflicts between technical characteristics but also the problems of product parts. Third, the uncertainty and subjectivity of the data provided by decision-makers in the evaluation process of innovation schemes are processed by interval grey numbers. Moreover, a cone model is built to make the evaluation results more reliable. Last, a case study of water purifiers is conducted to illustrate the validity of the generation process of incremental product innovation schemes.

Research on node importance evaluation of complex products based on three-parameter interval grey number grey relational model

In the process of product development, the identification and evaluation of important nodes is of great significance for the effective control of complex product engineering change. In order to identify and evaluate important nodes accurately, this paper proposes a method to evaluate the importance of complex product nodes. Firstly, an engineering change expression method based on multi-stage complex network is proposed. Then, the evaluation index system of important nodes of complex products is constructed. A three parameter grey relational model based on subjective and objective weights is proposed to identify and evaluate the important nodes of complex products. Finally, an example of a large permanent magnet synchronous centrifugal compressor is analyzed. The example shows that the top nodes are node 4, 1, 7, 9 and 24. Compared with other experiments, the proposed method can effectively and reasonably evaluate the node importance of complex products.

A time-varying grey Riccati model based on interval grey numbers for China's clean energy generation predicting

PurposeIn order to accurately predict the uncertain and nonlinear characteristics of China's three clean energy generation, this paper presents a novel time-varying grey Riccati model (TGRM(1,1)) based on interval grey number sequences.Design/methodology/approachBy combining grey Verhulst model and a special kind of Riccati equation and introducing a time-varying parameter and random disturbance term the authors advance a TGRM(1,1) based on interval grey number sequences. Additionally, interval grey number sequences are converted into middle value sequences and trapezoid area sequences by using geometric characteristics. Then the predicted formula is obtained by using differential equation principle. Finally, the proposed model's predictive effect is evaluated by three numerical examples of China's clean energy generation.FindingsBased on the interval grey number sequences, the TGRM(1,1) is applied to predict the development trend of China's wind power generation, China's hydropower generation and China's nuclear power generation, respectively, to verify the effectiveness of the novel model. The results show that the proposed model has better simulated and predicted performance than compared models.Practical implicationsDue to the uncertain information and continuous changing of clean energy generation in the past decade, interval grey number sequences are introduced to characterize full information of the annual clean energy generation data. And the novel TGRM(1,1) is applied to predict upper and lower bound values of China's clean energy generation, which is significant to give directions for energy policy improvements and modifications.Originality/valueThe main contribution of this paper is to propose a novel TGRM(1,1) based on interval grey number sequences, which considers the changes of parameters over time by introducing a time-varying parameter and random disturbance term. In addition, the model introduces the Riccati equation into classic Verhulst, which has higher practicability and prediction accuracy.

Change impact analysis of complex product using an improved three-parameter interval grey relation model

Change impact evaluation of complex product plays an important role in controlling change cost and improving change efficiency of engineering change enterprises. In order to improve the accuracy of engineering change impact evaluation, this paper introduces three-parameter interval grey number to evaluate complex products according to the data characteristics. The linear combination of BWM and Gini coefficient method is used to improve the three-parameter interval grey number correlation model. It is applied to the impact evaluation of complex product engineering change. This paper firstly constructs a multi-stage complex network for complex product engineering change. Then the engineering change impact evaluation index system is determined. Finally, a case analysis was carried out with the permanent magnet synchronous centrifugal compressor in a large permanent magnet synchronous centrifugal unit to verify the effectiveness of the proposed method.

Exploiting fractional accumulation and background value optimization in multivariate interval grey prediction model and its application

In the context of small sample and poor information, the data often change rapidly and interact with multiple factors which make it a challenge to analyse and predict multivariate sequences efficiently. Due to the fact that real systems usually show fractional order characteristics, the aim of this paper is to consider the approach improving the forecast results of multivariate interval grey prediction model via exploiting fractional accumulation. Noting that fractional accumulation could disturb the exponential law, the connotation prediction method is introduced to balance the disturbance. Correspondingly, a fractional connotation prediction method is constructed. In addition, traditional background value coefficient is optimized by using the particle swarm optimization algorithm (PSO). Therefore, a multivariate interval grey fractional accumulative connotation prediction model with optimized background value coefficient is constructed, in which the interval grey number time series are transformed into kernel series and radius series. Finally, the developed model is applied to clean energy prediction in China to verify the feasibility and validity.

An Innovative Grey Approach for Group Multi-Criteria Decision Analysis Based on the Median of Ratings by Using Python

Some decision-making problems, i.e., multi-criteria decision analysis (MCDA) problems, require taking into account the attitudes of a large number of decision-makers and/or respondents. Therefore, an approach to the transformation of crisp ratings, collected from respondents, in grey interval numbers form based on the median of collected scores, i.e., ratings, is considered in this article. In this way, the simplicity of collecting respondents’ attitudes using crisp values, i.e., by applying some form of Likert scale, is combined with the advantages that can be achieved by using grey interval numbers. In this way, a grey extension of MCDA methods is obtained. The application of the proposed approach was considered in the example of evaluating the websites of tourism organizations by using several MCDA methods. Additionally, an analysis of the application of the proposed approach in the case of a large number of respondents, done in Python, is presented. The advantages of the proposed method, as well as its possible limitations, are summarized.

A comprehensive G-Lz-ADC effectiveness evaluation model for the single communication satellite system in the context of poor information

PurposeDuring the service period of communication satellite systems, their performance is often degraded due to the depletion mechanism. In this paper, the grey system theory is applied to the multi-state system effectiveness evaluation and the grey Lz-transformation ADC (availability, dependability and capability) effectiveness evaluation model is constructed to address the characteristics of the communication satellite system such as different constituent subsystems, numerous states and the inaccuracy and insufficiency of data.Design/methodology/approachThe model is based on the ADC effectiveness evaluation method, combined with the Lz transformation and uses the definite weighted function of the three-parameter interval grey number as a bridge to incorporate the possibility of system performance being greater than the task demand into the effectiveness solution algorithm. At the same time, using MATLAB (Matrix laboratory) to solve each state probability, the same performance level in the Lz transform is combined. Then, the system effectiveness is obtained by Python.FindingsThe results show that the G-Lz-ADC model constructed in this paper can accurately evaluate the effectiveness of static/dynamic systems and certain/uncertain system and also has better applicability in evaluating the effectiveness of the multi-state complex system.Practical implicationsThe G-Lz-ADC effectiveness evaluation model constructed in this paper can effectively reduce the complexity of traditional effectiveness evaluation models by combining the same performance levels in the Lz-transform and solving the effectiveness of the system with the help of computer programming, providing a new method for the effectiveness evaluation of the complex MSS. At the same time, the weaknesses of the system can be identified, providing a theoretical basis for improving the system’s effectiveness.Originality/valueThe possibility solution method based on the definite weighted function comparing the two three-parameter interval grey numbers is constructed, which compensates for the traditional calculation of the probability based on numerical values and subjective preferences of decision-makers. Meanwhile, the effectiveness evaluation model integrates the basic theories of three-parameter interval grey number and its definite weighted function, Grey−Markov, grey universal generating function (GUGF), grey multi-state system (GMSS), etc., which is an innovative method to solve the effectiveness of a multi-state instantaneous communication satellite system.

Importance Calculation of Customer Requirements for Incremental Product Innovation.

Incremental product innovation is achieved by finding and solving problems of existing products. The importance of customer requirements reflects the severity of existing product problems, which points out the direction for incremental product innovation. In this research, the calculation process of customer requirement importance mainly includes three steps. Firstly, from the perspective of customers, the improvement gap analytical method is used to obtain the improvement and original importance of customer requirements by measuring customer perceived satisfaction and dissatisfaction. Secondly, from the perspective of industry experts, an improved interval grey number ranking method is proposed to calculate the basic importance of customer requirements, which can deal with the inadequate problem of the data provided by experts due to the limited number of experts. Finally, a multi-dimensional vector cosine method, which avoids the interference of subjectivity of importance weight calculation to the final importance, is proposed to integrate the importance data provided by customers and experts. A case of a water purifier is considered to illustrate the validity of the proposed process. This research improves existing calculation methods and proposes an integrated calculation process from three dimensions to calculate the final importance of customer requirements effectively.

A novel grey multi-criteria three-way decisions model and its application

Three-way decision provides a new research perspective for solving multi-criteria decision-making problems. Under grey environment and grey decision idea, this paper proposes a novel grey multi-criteria three-way decisions model based on grey incidence analysis and TOPSIS. The model realizes the effective fusion of three-way decision and grey system theory, and further extends the totally ordered set evaluation-based three-way decisions. Firstly, we use grey incidence analysis and TOPSIS to construct a new conditional probability for three-way decisions, which not only considers the difference in distance, but also the difference in the shape of the evaluation sequence curve. Considering the multi-criteria information system and multi-criteria loss function matrix, we construct a calculation method of criteria weights and design three weighted aggregation operators for loss function with interval grey numbers. According to key steps and algorithm, the grey multi-criteria three-way decision rules can be obtained. Finally, two illustrative example of complex equipment supplier selection and movie recommendation are utilized to validate the applicability and interpretability of our proposed model.

A multi-objective multi-micro-swarm leadership hierarchy-based optimizer for uncertain flexible job shop scheduling problem with job precedence constraints

In realistic production scheduling, the processing time of operations and the due time of orders always fail to be precisely estimated as deterministic values due to fluctuating manufacturing environments and modest delay tolerance. When fabricating complex products that are assembled by multilevel parts, tree-structure dependencies between parts lead to hierarchical precedence constraints between corresponding jobs. Consequently, this paper studies an uncertain flexible job shop scheduling problem with job precedence constraints (U-FJSP-JPC). Uncertain processing time and due time are represented as interval grey number and trimmed triangular fuzzy number respectively. A tardiness index indicator is devised to assess the delay extent of grey completion time relative to fuzzy due time. To solve U-FJSP-JPC with minimizing three objectives simultaneously involving interval grey makespan, interval grey total machine workload and average tardiness index, this paper elaborately designs a novel multi-objective multiple-micro-swarm leadership hierarchy-based optimization algorithm (MOM2SLHO). This algorithm adopts a two-vector encoding scheme based on job and operation sequencing and a grey active decoding scheme based on heuristic machine assignment. In MOM2SLHO, the entire search agents are divided into multiple micro-swarms in which each one conducts an independent search based on leadership hierarchy and communicates with others by specific strategies. MOM2SLHO embodies an enhanced external grid archive to store and retrieve non-dominated Pareto optimal solutions. Extensive experiments and statistical analyses demonstrate that the proposed MOM2SLHO algorithm outperforms other well-known and state-of-art algorithms significantly for solving the studied U-FJSP-JPC.

Matrix form of interval multivariable gray model and its application

PurposeThe purpose of this paper is to make multivariable gray model to be available for the application on interval gray number sequences directly, the matrix form of interval multivariable gray model (IMGM(1,m,k) model) is constructed to simulate and forecast original interval gray number sequences in this paper.Design/methodology/approachFirstly, the interval gray number is regarded as a three-dimensional column vector, and the parameters of multivariable gray model are expressed in matrix form. Based on the dynamic gray action and optimized background value, the interval multivariable gray model is constructed. Finally, two examples and comparisons are carried out to verify the effectiveness of IMGM(1,m,k) model.FindingsThe model is applied to simulate and predict expert value, foreign direct investment, automobile sales and steel output, respectively. The results show that the proposed model has better simulation and prediction performance than another two models.Practical implicationsDue to the uncertainty information and continuous changing of reality, the interval gray numbers are used to characterize full information of original data. And the IMGM(1,m,k) model not only considers the characteristics of parameters changing with time but also takes into account information on lower, middle and upper bounds of interval gray numbers simultaneously to make better suitable for practical application.Originality/valueThe main contribution of this paper is to propose a new interval multivariable gray model, which considers the interaction between the lower, middle and upper bounds of interval numbers and need not to transform interval gray number sequences into real sequences. According to combining different characteristics of each bound of interval gray numbers, the matrix form of interval multivariable gray model is established to simulate and forecast interval gray numbers. In addition, the model introduces dynamic gray action to reflect the changes of parameters over time. Instead of white equation of classic MGM(1,m), the difference equation is directly used to solve the simulated and predicted values.

Development of the GM(1,1,⊗b) model with a trapezoidal possibility function and its application

PurposeThe purpose of this paper is to overcome the weakness of the traditional model, in which the grey action quantity is a real number and thus leads to a “unique solution” and to build the model with a trapezoidal possibility degree function.Design/methodology/approachUsing the system input and output block diagram of the model, the interval grey action quantity is restored under the condition of insufficient system influencing factors, and the trapezoidal possibility degree function is formed. Based on that, a new model able to output non-unique solutions is constructed.FindingsThe model satisfies the non-unique solution principle of the grey theory under the condition of insufficient information. The model is compatible with the traditional model in structure and modelling results. The validity and practicability of the new model are verified by applying it in simulating the ecological environment water consumption in the Yangtze River basin.Practical implicationsIn this study, the interval grey number form of grey action quantity is restored under the condition of insufficient system influencing factors, and the unique solution to the problem of the traditional model is solved. It is of great value in enriching the theoretical system of grey prediction models.Social implicationsTaking power consumption as an example, the accurate prediction of the future power consumption level is related to the utilization efficiency of the power infrastructure investment. If the prediction of the power consumption level is too low, it will lead to the insufficient construction of the power infrastructure and the frequent occurrence of “power shortage” in the power industry. If the prediction is too high, it will lead to excessive investment in the power infrastructure. As a result, the overall surplus of power supply will lead to relatively low operation efficiency. Therefore, building an appropriate model for the correct interval prediction is a better way to solve such problems. The model proposed in this study is an effective one to solve such problems.Originality/valueA new grey prediction model with its interval grey action quantity based on the trapezoidal possibility degree function is proposed for the first time.

Improved uncertain method for safety analysis of aircraft landing gear

There are various uncertain factors in most practical engineering applications, such as input loads, structural sizes, manufacturing tolerance, and initial and boundary conditions. The interval method and grey number theory are common methods to deal with uncertainty. In this article, the interval truncation method and grey number theory are improved. And a mixed method is proposed to represent the confidence interval of output result based on the improved interval truncation method and improved grey number theory. The proposed methods’ feasibility is verified by a stepped bar; the methods are applied to the analysis of aircraft landing gear safety uncertainty.

Drought grade assessment method based on grey cloud incidence clustering model

PurposeThe purpose of this paper is to establish a grey cloud incidence clustering model to assess the drought disaster degree under 15 indexes in 18 cities of Henan province.Design/methodology/approachThe grey incidence degree between each index and ideal index is used to determine the index weight and combined with the subjective weight, the comprehensive weight is given; the traditional possibility function is transformed into grey cloud possibility function by using the principle of maximum deviation and maximum entropy, which fully reflects the coexistence of multiple decision-making conclusions and constructs the grey cloud incidence clustering model.FindingsThe drought disaster degree of Henan province is divided into four grades under the selected 15 indexes. The drought grades show obvious regional differences. The risk levels of the east and southwest are higher, and the risk levels of the north and southeast are lower. This result is consistent with the study of drought disaster grades in Henan province, which shows the practicability and usefulness of the model.Practical implicationsIt provides an effective method for the assessment of drought disaster grade and the basis for formulating disaster prevention and mitigation plan.Originality/valueBy studying the method of multiattribute and multistage decision-making with interval grey number information. The objective weight model of index value is designed, and the subjective weight is given by experts. On the basis of the two, the comprehensive weight of subjective and objective combination is proposed, which effectively weakens the randomness of subjective weight and reasonably reflects the practicality of index decision-making. The time weight reflects the dynamic of the index. The traditional possibility function is transformed into the grey cloud possibility function, which effectively takes advantage of the grey cloud model in dealing with the coexistence of fuzzy information, grey information and random information. Thus, the conflict between the decision-making results and the objective reality is effectively solved. The interval grey number can make full use of the effective information and improve the accuracy of the actual information.

Development of a direct NGM(1,1) prediction model based on interval grey numbers

PurposeThe purpose of this paper is to construct an interval grey number NGM(1,1) direct prediction model (abbreviated as IGNGM(1,1)), which need not transform interval grey numbers sequences into real number sequences, and the Markov model is used to optimize residual sequences of IGNGM(1,1) model.Design/methodology/approachA definition equation of IGNGM(1,1) model is proposed in this paper, and its time response function is solved by recursive iteration method. Next, the optimal weight of development coefficients of two boundaries is obtained by genetic algorithm, which is designed by minimizing the average relative error based on time weighted. In addition to that, the Markov model is used to modify residual sequences.FindingsThe interval grey numbers’ sequences can be predicted directly by IGNGM(1,1) model and its residual sequences can be amended by Markov model. A case study shows that the proposed model has higher accuracy in prediction.Practical implicationsUncertainty and volatility information is widespread in practical applications, and the information can be characterized by interval grey numbers. In this paper, an interval grey numbers direct prediction model is proposed, which provides a method for predicting the uncertainty information in the real world.Originality/valueThe main contribution of this paper is to propose an IGNGM(1,1) model which can realize interval grey numbers prediction without transforming them into real number and solve the optimal weight of integral development coefficient by genetic algorithm so as to avoid the distortion of prediction results. Moreover, the Markov model is used to modify residual sequences to further improve the modeling accuracy.

The nonlinear time lag multivariable grey prediction model based on interval grey numbers and its application

The linear relationship of the original grey prediction model is too single, and the original grey prediction model does not consider the time delay of the effect of the current input parameters on the output parameters. In order to solve these problems, the interval grey number sequence is taken as the modelling sequence of the model, and the nonlinear parameter γ and the time-delay parameter τ are introduced into the multivariate grey prediction model, so as to construct the nonlinear time-delay multivariable grey prediction model for interval grey number. In view of the uncertain characteristics of the smog index data, this paper applies the improved model to the simulation and prediction of the smog index data. Compared with the original model, the results show that the prediction effect of the model proposed in this paper is superior to the original model in terms of its effectiveness and feasibility.

A grey approach to site selection for nursing homes: An extension of the fuzzy analytic hierarchy process

The construction of more nursing homes has become one of the most needed pension services in China, and the issue of site selection is one of the most important steps in their construction. The problem of site selection for nursing homes is a complex system engineering problem that involves not only economic interests but also social interests. Due to the limitations of human thinking in the evaluation process, the evaluation value of a nursing home site might be an interval grey number. Moreover, the evaluation indicator system for nursing home locations is a two-layer system that has been neglected in the literature. Therefore, the fuzzy analytical hierarchy process is extended to a new grey approach, i.e., the grey analytic hierarchy process, which can solve the evaluation problems for a two-layer indicator system under an interval grey environment. By constructing a three-point interval grey number, grey evaluation criteria are given to obtain a judgment matrix for interval grey numbers. Definitions of the initial weights, nongreyness weights and integrated weights are proposed to find the best evaluation object. Finally, the effectiveness of the method proposed by this paper is verified by comparative analyses of other grey methods.