Interval Ranking Research Articles

An extension of Pontryagin Maximum principle in interval environment and its application to inventory problem

The control theory is one of the most fundamental branches of engineering as it implicates solving abilities of numerous non-linear engineering design problems efficiently. Again, Pontryagin’s maximum principle is one of the most salient topics of control theory as it is involved in solving various important problems. However, in the current highly complex situation, most of such real-life problems appear to be highly uncertain/imprecise in nature. Consequently, in order to analyse such problems accurately, uncertainty/flexibility of such problems cannot be overestimated. Motivating from this fact and as a necessity, in this study, the Pontryagin’s maximum principles are extended in interval environment for interval valued control problems (IVCPs). In this context, an IVCP is defined along with different formal terminologies. Further, the necessary and sufficient optimality conditions (i.e., Pontryagin’s maximum principles) are extended for IVCPs using existing interval ranking proposed by Bhunia and Samanta (2014). Further, in the second part of this work, in order to test the effectiveness of the proposed theories, an economic order quantity (EOQ) model is developed by considering dynamic servicing strategies in interval environment. With the help of numerical example, the proposed extension of Pontryagin’s maximum principles for IVCPs is well validated.

Combinatorics of essential sets for positroids

Positroids are a family of matroids introduced by Postnikov in the study of non-negative Grassmannians. Postnikov identified several combinatorial objects in bijections with positroids, among which are bounded affine permutations. On the other hand, the notion of essential sets, introduced for permutations by Fulton, was used by Knutson in the study of the special family of interval rank positroids. We generalize Fulton's essential sets to bounded affine permutations. The bijection of the latter with positroids, allows the study of the relationship between them. From the point of view of positroids, essential sets are maximally dependent cyclic intervals. We define connected essential sets and prove that they give a facet description of the positroid polytope, as well as equations defining the positroid variety. We define a subset of essential sets, called core, which contains minimal rank conditions to uniquely recover a positroid. We provide an algorithm to retrieve the positroid satisfying the rank conditions in the core or any compatible rank condition on cyclic intervals.

Optimal resource allocation in hierarchical organizations under uncertainty: An interval-valued n-person cooperative game approach

In many real production scenarios, departmental organizations often exhibit a hierarchical structure, where departments cooperate with subordinate departments to optimize resource allocation and maximize their respective benefits. However, due to a lack of information or data, many model parameters in the allocation process cannot be precisely defined. In response to this challenge, an interval n-person hierarchical resource allocation model is proposed to achieve maximum economic benefit in uncertain environments. Based on the concepts of satisfactory degrees of comparing intervals and interval-valued cores of interval-valued n-person cooperative games, an auxiliary nonlinear programming model and method are developed to solve the interval-valued cores of such cooperative games. The approach explicitly considers the inclusion and/or overlap relations between intervals, whereas the traditional interval ranking method may not guarantee the existence of interval-valued cores. The proposed method offers cooperative opportunities under uncertain conditions. Finally, the feasibility and applicability of the models and methods are demonstrated through a numerical example and comparison with other methods.

Methodology for Reliable Assessment of Water Quality. V. Statistical Methods for Water Quality Research: Parametric Approach

The possibility of using a parametric tolerance interval to improve the accuracy of studying water quality indicators was analyzed. It was demonstrated that the proposed approach provides advantages over non-parametric interval rank, but not in all cases. The advantages of the new method due to the use of more complete experimental information were substantiated, and the reasons for the decrease in the accuracy of water quality assessment at its application were analysed.

Task allocation in multi-AUV dynamic game based on interval ranking under uncertain information

This paper presents a novel game allocation method for solving the task allocation problem in underwater multi-AUV dynamic confrontations. The proposed method addresses incomplete information using a multi-objective evaluation model and interval ranking. An improved particle swarm optimization (PSO) algorithm, incorporating a hybrid strategy, is introduced to determine the Nash equilibrium of the game model. Firstly, the paper introduces the problem of underwater multi-AUV confrontations and establishes a game model based on interval numbers. Secondly, it develops a multi-target interval evaluation model considering survival value, strike income, and ammunition consumption. Combined with the analytic hierarchy process (AHP), each objective in the evaluation model is weighted to adapt to the battlefield situation with changing confrontation tasks and decision preferences. TOPSIS, relative entropy, and probability degree sort the interval income for optimization. Finally, an improved PSO algorithm called GSPSO, combining the good point set theory (G) and speed control factor (S), solves the optimal allocation problem. Simulation results demonstrate that the probability degree-based ranking method achieves higher resolution incomes, aiding precise optimization. The algorithm improves convergence speed, accuracy, and global optimization ability by increasing population diversity and iterative effectiveness, ensuring real-time decision-making in complex game scenarios.

Pricing and dynamic service policy for an imperfect production system: Extended Pontryagin’s maximum principle for interval control problems

Pontryagin’s maximum principle is one of the most important topics of control theory. It plays a crucial role in solving variational problems in an alternative way. In this paper, Pontryagin’s maximum principle is extended in order to solve interval valued control problems (IVCPs). To serve this purpose, firstly all the formal definitions (viz. IVCP, interval ranking and optimizer of IVCP etc.) are presumed. Further, the necessary and sufficient condition, i.e., Pontryagin’s principles are extended in interval environment in order to solve an IVCP. The second part of this work presents an application of the proposed extension of Pontryagin’s maximum principle for IVCPs. Then, an imperfect production inventory problem is considered by assuming flexibility of all the involved parameters in interval environment. The proposed production problem mainly addresses a joint optimal policy for pricing, dynamic service investment and reworked/salvaged under interval flexibility. The IVCP related to the proposed model is solved using extended Pontryagin’s maximum principle and the interval optimization problem corresponding to the model is solved using improved c-r optimization technique. Further, a numerical example is taken and solved using the sparrow search algorithm (SSA) in order to illustrate the proposed theoretical results and validate the proposed model. Finally, sensitivity analyses are carried out w. r. to various system parameters in order to evaluate some implications to the manager.

Development of Bi-Objective Fuzzy Data Envelopment Analysis Model to Measure the Efficiencies of Decision-Making Units

The proposed bi-objective fuzzy data envelopment analysis (BOFDEA) model is a new approach to assess the performance efficiency of decision-making units (DMUs) in uncertain environments using α-cuts. The model is based on fuzzy data envelopment analysis (FDEA) and considers two objectives, and a solution method and ranking system are provided. Generally, the efficiency score obtained for a DMU using the α-cut approach is an interval. Intervals are partially ordered sets, due to which ranking intervals is a challenging task. The proposed BOFDEA model with α-cuts provides the efficiency of DMUs in the crisp form, not in the form of intervals. Due to this, ranking DMUs with the proposed method’s help becomes very easy and less computationally. The proposed model has been validated through numerical examples, and a real-world application in the education sector has been shown to demonstrate its practicality.

Ordinal-cardinal consensus analysis for large-scale group decision making with uncertain self-confidence

Consensus analysis is necessary for large-scale group decision making (LSGDM) for ensuring reasonable decision results. This paper offers a new method for LSGDM with uncertain self-confidence that follows ordinal-cardinal consensus analysis. For this purpose, a new interval ranking method is first proposed to compare alternatives. Then, an improved ordinal clustering method on criteria is introduced using the deviation between individual ranking positions. In view of opinion deviation, uncertain self-confidence deviation, and ranking deviation, the weights of decision makers (DMs) are defined. Similarly, the weights of clusters are determined by further combining cluster cardinality. Further, an ordinal-cardinal consensus procedure is offered, which contains two algorithms: the first algorithm is about the ordinal consensus improvement in view of three aspects: ranking adjustment, opinion adjustment, and uncertain self-confidence; the second algorithm studies the cardinal consensus improvement under the ordinal consensus requirement, which also mainly contains three aspects: opinion adjustment, the number of adjusted judgments, and uncertain self-confidence. Moreover, a new algorithm for LSGDM is presented. Finally, an example is provided to check the feasibility and efficiency of the new method, and a comparison analysis is also made.

Ranking of Z-Numbers Based on the Developed Golden Rule Representative Value

Real-world decision-making is based on human cognitive information, which is characterized by fuzziness and partial reliability. In order to better describe such information, Zadeh proposed the concept of Z-number. Ranking the Z-number is an indispensable step in solving the decision-making problem under the Z-number-based information. Golden rule representative value is a new concept introduced by Yager to rank interval values. This article expands it and proposes a new golden rule representative value for fuzzy numbers, and then, apply it to the ranking of the Z-number. Some new rules involving the centroid and fuzziness of fuzzy numbers are constructed to capture the preference of decision-makers. The Takagi–Sugeno–Kang fuzzy model is used to implement these rules. The obtained Rep function is used to construct a new golden rule representative value fuzzy subset of the Z-number and associate this new fuzzy subset with a scalar value. This fuzzy subset not only implies the fuzzy aspect of the Z-number but also contains the information in the hidden probability distribution. The scalar value is regarded as the golden rule representative value of the Z-number to participate in the ranking. The proposed method greatly retains the original information of the Z-number and can overcome the shortcomings of the existing methods. Some numerical examples are used to describe the specific process of the proposed method. The comparative analysis and discussion with existing methods clarify the advantages of the proposed method.

Multi-AUV Dynamic Maneuver Countermeasure Algorithm Based on Interval Information Game and Fractional-Order DE

The instability of the underwater environment and underwater communication brings great challenges to the coordination and cooperation of the multi-Autonomous Underwater Vehicle (AUV). In this paper, a multi-AUV dynamic maneuver countermeasure algorithm is proposed based on the interval information game theory and fractional-order Differential Evolution (DE), in order to highlight the features of the underwater countermeasure. Firstly, an advantage function comprising the situation and energy efficiency advantages is proposed on account of the multi-AUV maneuver strategies. Then, the payoff matrix with interval information is established and the payment interval ranking is achieved based on relative entropy. Subsequently, the maneuver countermeasure model is presented along with the Nash equilibrium condition satisfying the interval information game. The fractional-order DE algorithm is applied for solving the established problem to determine the optimal strategy. Finally, the superiority of the proposed multi-AUV maneuver countermeasure algorithm is verified through an example.

Threshold Effect of Industry Heterogeneity on Green Innovation Efficiency: Evidence From China

We scientifically evaluate the efficiency of green innovation in the industry, analyze its changes and development, and study the impact of industry heterogeneity on it. Which will help us grasp the efficiency of green innovation at the industry level, also provides a scientific basis for formulating relevant environmental regulations. Based on industry heterogeneity and green innovation efficiency, this first analyzes industry panel data from 2005 to 2014 in China for 35 industries. It then uses the entropy weight method to calculate the industry heterogeneity and green innovation efficiency of the industry in China. Then, environmental regulations are taken as threshold variables to empirically analyze the correlation between industry heterogeneity and green innovation efficiency. The study found a significant twofold threshold effect that links industry heterogeneity and green innovation efficiency, showing an “inverted N-type” result. Industry heterogeneity negatively impacts green innovation efficiency when the environmental regulation strength is at a higher interval rank and a lower level. However, industry heterogeneity negatively affects the intermediate level of green innovation efficiency.

Optimistic Ranking Pessimistic Ranking and Neutral Ranking of Generalized Fuzzy Numbers Using the Integral Mean Value

Fuzzy numbers ranking methods are one of the important tools for decision-makers in real-world problems. Often fuzzy numbers may not be normal, which are called generalized fuzzy numbers (G-fuzzy numbers). In this work, a simple and efficient method is given to rank fuzzy numbers. For every given G-fuzzy number we use interval arithmetic to assign two suitable intervals that are called pessimistic and optimistic expected ranking intervals, respectively. Since a decision-maker may be optimistic, pessimistic, or neutral, we use an index to choose a point of these intervals to rank G-fuzzy numbers according to the point of view of decision-makers. In this manner, our method can be used to rank all arbitrary G-fuzzy numbers without attention to certain cases. Some examples are given to compare the proposed method with other methods.

Anomaly Detection and Localization for Process Security Based on the Multivariate Statistical Method

Anomaly detection is very important for system monitoring and security since successful execution of these engineering tasks depends on access to validated data. The localization of the variable causing the fault is very essential. Indeed, the localization of the fault is defined as the ability to determine the source of the fault on a system. Generally, the identification of faults is linked to the detection procedure implemented. Therefore, it is very important to choose the adequate fault detection model to locate fault. For nonlinear uncertain systems, the most performed fault detection method is reduced rank interval kernel principal component analysis (RRIKPCA), which enhances the computational skill by downgrading the kernel matrix dimension. We have proposed in this article a new fault localization technique for uncertain systems, named partial RRIKPCA, which combines the benefits of the RRIKPCA technique and the principle of partial localization. The principal of this method involves selecting partially reduced rank data subsets and then building more accurate models with fewer PCs and isolating faults with higher precision. The proposed fault isolation method is applied for monitoring air quality monitoring network (AIRLOR) data.

A Novel Rank Learning Based No-Reference Image Quality Assessment Method

Recently, applying deep learning to no-reference image quality assessment (NR-IQA) has received significant attention. Especially in the last five years, an increasing interest has been drawn to the studies of rank learning since it can help mitigate the problem of small IQA datasets. However, on one hand, existing rank learning is not suitable for the authentically distorted images due to the lack of generated rank samples. On the other hand, the output of existing rank loss functions is uncontrollable, resulting in reduced performance. Motivated by these two limitations, we propose a novel rank learning based NR-IQA method, termed controllable list-wise ranking IQA (CLRIQA) in this paper. To be specific, we first present an imaging-heuristic approach, in which the over- and under-exposure is formulated as an inverse of the Weber-Fechner law, and fusion strategy and compression are adopted, to simulate the authentic distortion and generate the rank image samples. These samples are label-free yet associated with quality ranking information. Then we design a controllable list-wise ranking (CLR) loss function by setting an upper and lower bound of rank range and introducing an adaptive margin to tune rank interval. Finally, both the generated rank samples and proposed CLR are used to pre-train a convolutional neural network. Moreover, to obtain a more accurate prediction model, we take advantage of the IQA datasets to fine-tune the pre-trained network further. Various experiments are conducted on the IQA benchmark datasets, and experimental results demonstrate the effectiveness of the proposed CLRIQA method. The source code and network model can be downloaded at the following web address: <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><uri>https://github.com</uri> <inline-formula><tex-math notation="LaTeX">$/$</tex-math></inline-formula> GZHU-DVL <inline-formula><tex-math notation="LaTeX">$/$</tex-math></inline-formula> CLRIQA</i> .

Dominance and ranking interval in DEA parallel production systems

Dominance and ranking interval in DEA parallel production systems

Ranking interval type-2 fuzzy number based on a novel value-ambiguity ranking index and its application in risk analysis

Consequence of decision making problems under fuzzy domain mostly necessitate choosing the best alternative among the sets of alternatives. This requirement is generally fulfilled through the methodology called the ranking of fuzzy numbers. In this paper, an attempt under fuzzy domain to develop a coherent methodology of ordering fuzzy numbers is thrived. The ill-defined quantity ‘value’ and the amount of uncertainty in the ill-defined quantity ‘ambiguity’ of a fuzzy number constituted the important tools in ranking fuzzy numbers. As such, an effort has been made to employ these quantities in ranking interval type-2 fuzzy numbers constructing a novel value-ambiguity ranking index. A through comparative study with the existing method and the present method is executed to exhibit the essence of the proposed ranking method. The comparative study exhibit the superiority of the present method of ranking interval type-2 fuzzy numbers over the existing method. Further, the method is not just limited to interval type-2 fuzzy numbers, it can be broadly applied to arbitrary fuzzy numbers. Finally, a risk analysis problem has been discussed and the proposed novel value-ambiguity ranking index of ranking interval type-2 fuzzy numbers has been applied successfully.

A Novel Weapon System Effectiveness Assessment Method Based on the Interval-Valued Evidential Reasoning Algorithm and the Analytical Hierarchy Process

As one of the most essential parts of the military operation, the ability of weapon systems to achieve the operation mission is vital to the success of a military operation. Focusing on the problem of weapon system effectiveness assessment (WSEA) under interval uncertainty, this paper proposed WSEA method that utilized the interval-valued evidential reasoning (ER) algorithm, the analytical hierarchy process (AHP), and the two-grade interval ranking method to properly deal with interval uncertainty in the assessment as well as provide reliable assessment results. Firstly, the AHP is used to determine the weight of different attributes of the weapon system based on experts’ knowledge, which could enhance the subjectiveness of the assessment result. Then, the interval-valued ER algorithm is applied to combine the assessment of different attributes under interval uncertainty and provide the overall assessment result. Finally, the two-grade interval ranking method is utilized to compare and rank the assessment results of different weapons. Case study shows that the proposed method could provide reliable and accurate results for weapon effectiveness assessment. Comparison results and sensitivity analysis results further confirm that the proposed method is not only as effective as existing methods with precise data, but also has the ability to provide reliable results under uncertainty. In conclusion, by utilizing the interval-valued ER algorithm, the AHP and the two-grade interval ranking method, the proposed method provides a novel and effective way for weapon effectiveness assessment under uncertainty, especially interval uncertainty.

Application of Interval Information Comprehensive Ranking Model Based on Entropy Weight in River Water Quality Evaluation

Wang, Y.; Wu, Y., and Jiang, L., 2020. Application of interval information comprehensive ranking model based on entropy weight in river water quality evaluation. In: Hu, C. and Cai, M. (eds.), Geo-informatics and Oceanography. Journal of Coastal Research, Special Issue No. 105, pp. 137–140. Coconut Creek (Florida), ISSN 0749-0208.In this article, an interval information comprehensive ranking model based on entropy weight is established and applied to river water quality evaluation. In this model, the monitoring interval value of water quality index of river section is weighted with the interval data of standard grade. The weight of each index is calculated by using information entropy technology, and the water quality grade of each section is determined by comprehensive weighted ranking of each section and evaluation grade. The results show that the concept of the model is clear, the calculation is simple, and the evaluation results are objective and reasonable.

Experimental determination of solute redistribution behavior during solidification of additively manufactured 316L

Chemical measurement points obtained with energy dispersive X-ray spectroscopy (EDS) in scanning transmission electron microscopy (STEM) were processed by a modified weighted interval rank sort (WIRS) method in order to study the solute redistribution during solidification of an additively manufactured 316L stainless steel. Scheil-Gulliver calculations give a good first approximation, although the extent of segregation is lower experimentally than theoretically. These discrepancies are believed to be due to the fast solidification rate involved. STEM-EDS data processed by the modified WIRS method is a strong and reliable technique for characterizing the segregation profile for additively manufactured metals, and the proposed method is shown to reduce the experimental uncertainty associated with the quantification of minor elements.

Multi-modal and multi-route transportation problem for hazardous materials under uncertainty

ABSTRACT In this article, the multi-modal and multi-route transportation problem for hazardous materials under uncertainty is studied, and an interval 0–1 integer programming model is proposed to optimize the transportation risks and transportation costs during the transportation of hazardous materials. Factors such as freight rates and loading and unloading costs are different when the transportation modes, transportation time and transportation areas are different. Therefore, some parameters in the model are assumed as interval numbers. The proposed model is transformed to a deterministic model by the interval ranking method. In addition, a genetic algorithm is proposed to solve the model. Finally, a real-world case of the multi-modal and multi-route transportation problem for hazardous materials under uncertainty is studied to demonstrate the rationality and effectiveness of proposed model and algorithm.