Dimension Of Problem Research Articles

Nonconvex Quadratically-Constrained Feasibility Problems: An Inside-Ellipsoids Outside-Sphere Model

This paper proposes a new approach for solving Quadratically Constrained Feasibility Problems (QCFPs). We introduce an isomorphic mapping (one-to-one and onto correspondence), which equivalently converts the QCFP to an optimization problem called the Inside-Ellipsoids Outside-Sphere Problem (IEOSP). This mapping preserves the convexity of convex constraints, but it converts all non-convex constraints to convex ones. The QCFP is a feasibility problem with non-convex constraints, while the IEOSP is an optimization problem with a convex feasible region and a non-convex objective function. It is shown that the global optimal solution of IEOSP is a feasible solution of the QCFP. Comparing the structures of QCFP and the proposed IEOSP, the second model only has one extra variable compared to the original QCFP because it employs one slack variable for the mapping. Thus, the problem dimension approximately remains unchanged. Due to the convexity of all constraints in IEOSP, it has a well-defined feasible region. Therefore, it can be solved much easier than the original QCFP. This paper proposes a solution algorithm for IEOSP that iteratively solves a convex optimization problem. The algorithm is mathematically shown to reach either a feasible solution of the QCFP or a local solution of the IEOSP. To illustrate our theoretical developments, a comprehensive numerical experiment is performed, and 500 different QCFPs are studied. All these numerical experiments confirm the promising performance and applicability of our theoretical developments in the current paper.

A systematic in vitro study of the effect of normoglycaemic and hyperglycaemic conditions on the biochemical and cellular interactions of clinically-available wound dressings with different physicochemical properties.

Diabetic foot, leg ulcers and decubitus ulcers affect millions of individuals worldwide leading to poor quality of life, pain and in several cases to limb amputations. Despite the global dimension of this clinical problem, limited progress has been made in developing more efficacious wound dressings, the design of which currently focusses on wound protection and control of its exudate volume. The present in vitro study systematically analysed seven types of clinically-available wound dressings made of different biomaterial composition and engineering. Their physicochemical properties were analysed by infrared spectroscopy, swelling and evaporation tests and variable pressure scanning electron microscopy. These properties were linked to the interactions with inflammatory cells in simulated normoglycaemic and hyperglycaemic conditions. It was observed that the swelling behaviour and evaporation prevention at different glucose levels depended more on the engineering of the fibres than on the hydrophilicity and hydrophobicity of their biomaterials. Likewise, the data show that the engineering of the dressings as either non-woven or woven or knitted fibres seems to determine the swelling behaviour and interactions with inflammatory cells more than their polymer composition. Dressings presenting absorbent layers made of synthetic, non-woven fibres supported the adhesion of monocytes macrophages and stimulate the release of factors known to play a role in the chronic inflammation. Non-woven absorbent layers based on carboxymethyl cellulose mainly stimulating the iNOS, an enzyme producing free radicals; in the case of Kerracel this was combined with a swelling of fibres preventing the penetration of cells. Kaltostat, an alginate-based wound dressing, showed the higher level of swelling and supporte the adhesion of inflammatory cells with limited activation. Knitted dressings showed a limited adhesion of inflammatory cells. In conclusion, this work offers insights about the interactions of these wound dressings with inflammatory cells upon exudate changes thus providing further criteria of choice to clinicians.

Partisipasi Masyarakat dalam Kualitas Penyusunan Rencana Pembangunan Jangka Menengah (RPJM) Desa Panduman Tahun 2020-2025

This study aims to determine the level of participation/involvement of the village community in the process of preparing the Panduman Village RPJM 2020-2025. Using a qualitative method, the data was collected by conducting interviews with informants and reviewing the Panduman Village RPJM document. The data was analyzed to see the level of community participation, including the quality of program proposals against the 4 dimensions of Panduman Village's problems and potentials. The results showed that; The Panduman Village community actively involved themselves in the process of preparing the Panduman RPJMDes for the 2020-2025 period. Village community aspirations always focus on physical infrastructure development. The village community has been able to evaluate the performance of the previous village government, including identifying 4 dimensions of village problems and potentials. However, the alternative solutions proposed by the community are still unable to answer the problems and manage the village's potential in a sustainable manner.

Social conditions of smallholder dairy farmers influence their environmental decisions

The objective of this study is to characterize the social conditions for making agricultural decisions. Particular attention is paid to sustainable agricultural practices in dairy farming. The theoretical framework has been developed around two major explicatory perspectives: sustainable development and quality of life. In the study, we assume that the regional dimension of problems related to the sustainable development of the dairy sector requires solutions adequate to the needs of the regions to create sustainable milk production systems. We put forward a thesis on the impact of objective living conditions of farmers on the readiness to introduce changes in the farm and to apply sustainable agricultural practices. The study has been carried out using the survey method, conducted in a face-to-face interview with fifty farmers engaged in milk production. This study contributes to revealing the role of social sustainability criteria for adopting environmentally friendly farming practices. The novelty of the approach is the use of in-depth sociological analysis in research on the sustainable development of agriculture in small farms (1–5 cows). The results have revealed a significant positive relationship between the assessment of farmers’ well-being and the willingness to adopt sustainable agricultural practices. The results also suggest that the analysis of social determinants of agricultural decisions is essential for the implementation of the assumptions of sustainable agriculture. Here we demonstrate that the criteria of subjective well-being can be considered important factors in a positive attitude towards sustainable development and openness to changes in agricultural production.

Reduced order modeling of blood perfusion in parametric multipatch liver lobules

In this paper, we present a computationally efficient reduced order model for obtaining blood perfusion profiles within parametric functional units of the liver called ‘lobules’. We consider Darcy’s equation in two-dimensional hexagonal lobule domains with six flow inlets and one outlet, whose positions are parameterized to represent varying lobule geometries. To avoid the meshing effort for every new lobule domain, we map the parametric domain onto a single reference domain. By making use of the contra-variant Piola mapping, we represent solutions of the parametric domains in the reference domain. We then construct a reduced order model via proper orthogonal decomposition (POD). Additionally, we employ the discrete empirical interpolation method (DEIM) to treat the non-affine parameter dependence that appears due to the geometric mapping. For sampling random shapes and sizes of lobules, we generate Voronoi diagrams (VD) from Delaunay triangulations and use an energy minimization problem to control the packing of the lobule structures. To reduce the dimension of the parameterized problem, we exploit the mesh symmetry of the full lobule domain to split the full domain into six rotationally symmetric subdomains. We then use the same set of reduced order basis (ROB) functions within each subdomain for the construction of the reduced order model. We close our study by a thorough investigation of the accuracy and computational efficiency of the resulting reduced order model.

GENERALIZED DATASET OF GEOLOGICAL AND GEOPHYSICAL INFORMATION ON THE EASTERN SECTOR OF THE RUSSIAN ARCTIC FOR MACHINE LEARNING – BASED ANALYSIS

The article presents a practical approach to the geological and geophysical spatial data collection and preliminary processing to use in machine learning models for geophysical applications. According to the established principles for estimating efforts in data analysis, which are confirmed by the results of surveys among specialists, this stage is viewed as major time and resource-consuming, amounting up to 80% in total volume of data analysis for a hypothesis testing project. The paper focuses on creating a consistent data set that integrates geological and geophysical information on a given region. We consider problems of different sources in the geodata representation to be related to their format (vector/raster), scale, type of attribute information (quantitative/qualitative) and their availability. The algorithm formalization and synthesis for combining geospatial data and converting them into quantitative vectors is a critical aspect. Combining various data draws on the concept of neighborhood fitting in with the data selection techniques and data consolidation strategy. The paper presents the general architecture of the software and hardware complex which includes a module for data collection and transformation in Python using the Pandas library, a data storage system based on the PostgreSQL DBMS (Database Management System) with the PostGIS extension. It is shown that for the considered class of problems in geophysics, it is sufficient to use a relational DBMS for data storing and processing. If the problem dimension increases, it is proposed to use the Big Data technology based on Apache Hadoop for scaling the system. A practical application of the proposed approach is demonstrated as results of data collection for the Caucasus region and eastern sector of the Russian Arctic. Based on the prepared data, experiments were carried out using machine learning models for recognition of locations of potential strong earthquakes and for sensitivity estimation of several geophysical features of these regions. The article presents the experimental results and evaluation of their efficiency.

Mini-Workshop: Mathematics of Entropic AI in the Natural Sciences

The mathematical framework of approximate entropic learning introduced very recently promises to provide robust, cheap and efficient ways of machine learning in the so called “small data” regime, when the underlying learning task is highly-underdetermined, due to a large problem dimension and relatively small data statistics size. Such “small data” learning challenges are particularly common in the natural sciences (e.g., in geosciences, in climate research, in economics, and in biomedicine), imposing considerable difficulties for the numerics of common “data-hungry” Artificial Intelligence (AI) tools like Deep Learning (DL). The aim of this workshop will be to bring together experts in the emergent fields of entropic and DL mathematics/numerics, with some lead experts applying AI in the domain disciplines. The goal will be to detect and to discuss the commonalities in the challenges and in their mathematical solutions, as well as to discuss and fine-tune common mathematical problem formulations that are motivated by the AI applications in natural sciences. The establishment of a common mathematical framework for such small-data machine learning tasks would not only bolster future methodological developments but would also lay solid foundations to further in-depth rigorous analysis and theoretically founded interpretation of these methods and their results.

ТЕКСТОВІ ЗАДАЧІ ЯК ЗАСІБ ФОРМУВАННЯ У ЗДОБУВАЧІВ ПОЧАТКОВОЇ ОСВІТИ ЗНАНЬ ПРО БЕЗПЕЧНУ ПОВЕДІНКУ НА УРОКАХ МАТЕМАТИКИ

The article substantiates the relevance of the problem of preserving educational institutions as safe spaces for learning and education. A number of regulatory and legal acts of Ukraine were announced, detailing the content, forms and methods of organizational and educational work in the relevant direction. It was established that the formation of knowledge about safe behavior among students must be carried out directly and indirectly ‒ by means of special educational disciplines and interdisciplinary integration. A number of studies of Ukrainian scientists were noted, in which the problem of forming the basics of safe behavior among students of education is raised. The scientific work of Ukrainian and Polish scientists regarding the integration of «security topics» into the content of mathematics lessons is briefly characterized. The conceptual dimension of the security problem has been clarified: the definitions of the terms «security», «human security», «safety of the educational environment», «safe educational environment» available in the scientific space have been clarified. It is noted that among the directions of the organization of a safe educational environment, the pedagogical one occupies an important place ‒ it concerns the formation of students’ knowledge, abilities and skills regarding safe behavior. The concept of «safe behavior» has been specified. The content of one of the Standard educational programs for primary school was analyzed for the purpose of the reflection of issues of safe behavior in its content: «expected learning outcomes» from certain educational fields have been updated, which reflect the focus of the content of primary education on the formation of students’ safe behavior skills; relevant thematic profiles are defined. On the basis of the analysis of the studied sources, the potential possibilities of textual problems as a means of forming knowledge about safe behavior among students of education have been revealed. Variants of textual problems that can be used in mathematics lessons in order to implement the specified pedagogical process are represented (among them are simple tasks for finding the sum and remainder and for increasing and decreasing a number by several units; simple difference comparison problems; simple tasks for finding the product and quotient; composed tasks for finding the sum and the difference;composed tasks for finding the sum and the difference, which contain the relation «more than», «more than», «less than», «less than»;composed tasks for finding the third term; complex problems, which include finding a part of a number and a number by its part; etc.).

Competition of tribes and cooperation of members algorithm: An evolutionary computation approach for model free optimization

Metaheuristic algorithms are a series of intelligent algorithms that are inspired by natural phenomenon and achieve the optimal searching via utilizing behaviors from nature. The rising complexity and dimensions of practical engineering problem in reality, a significant number of metaheuristic algorithms are promoted and applied in various of fields. Inspired by ancient tribes competition and members cooperative behavior, this paper proposes the Competition of Tribes and Cooperation of Members Algorithm (CTCM). Subsequent experiments are conducted on 23 benchmark test functions and exhaustively compared with other state-of-the-art algorithms, including particle swarm optimization (PSO), grey wolf optimizer (GWO), sparrow search algorithm (SSA), egret swarm optimization (ESOA), beetle antennae search (BAS) and whale optimization (WOA). The standard deviation and average, as well as statistic test are utilized to compare the performance of each algorithms, which reveal that CTCM outperforms in most kinds of problem. And from the result of Wilcoxon and Friedman rank test, the CTCM achieve the first place in all categories of problems, which indicate that CTCM possesses strong global optimization search capability and stability, and has faster convergence speed. The superiority of CTCM is then proofed on practical engineering optimization problems, in which CTCM achieve all the optimal solution for each engineering problem.

POD and mesh-adaptivity based reduce order model: Application to solve PDEs and inviscid flows

Time efficiency is a critical concern in Computational Fluid Dynamics (CFD) simulations of industrial applications. Despite the extensive research to improve the underlying numerical schemes to reduce the processing time, many CFD applications still need to improve on this issue. Reduced-Order Models (ROM) appeared as a promising alternative for significantly enhancing cost-effectiveness. The dimension of the initial problem is reduced significantly while keeping an acceptable level of accuracy. Proper Orthogonal Decomposition (POD), a data analysis technique, is widely used to construct a ROM to solve Euler and Navier-Stokes equations in CFD. Many aspects, however, need to be improved. In this paper, first, a general upper-bound error formula is derived, and second, a mesh-adaptivity based algorithm for snapshot locations in parameter space is provided (a sensitive step for POD-based ROM) in the context of Partial Differential Equations (PDE). The method’s efficiency is demonstrated through numerical tests by comparison to exact solutions and numerical results of a finite volume scheme on inviscid flow over a NACA0012.

Multi-input Fourier neural network and its sparrow search optimization

In engineering applications, the back-propagation (BP) neural network often encounters many limitations due to its slow convergence and high noise sensitivity, and meanwhile the reported Fourier neural networks have no ability to extract the features of multi-attribute input data. Hereby, This work proposes a gradient descent-based multi-input Fourier neural network after integrating the multi-layer perceptron with an overlapping Fourier neural network. Thereafter, related to the difficulty of deciding the global optimal parameter settings, an improved sparrow search algorithm is developed to optimize the parameter settings and solve high dimensional function optimization problems, after the Cat chaotic map and the mechanisms of population-size adjustment and parameter adaptiveness are designed to promote the sparrow search algorithm's ability to balance global exploration and local exploitation. The theoretical analysis shows that the improved algorithm's computational complexity is decided by its population size and the optimization problem's dimension. Numerically comparative experiments have validated that not only the acquired Fourier neural network can effectively extract the features of multi-attribute data with strong generalization ability, but also the improved algorithm has significant advantages in coping with high dimensional function optimization problems.

Super-resolution localization and orientation estimation of multiple dipole sound sources: From a maximum likelihood framework to wind tunnel validation

This paper investigates the identification of multiple dipole sound sources using sound pressures measured from a microphone array. The problem is addressed in the maximum likelihood (ML) framework, where the locations, orientations, and powers of multiple dipole sound sources are unknown parameters to be estimated. By the consistency property of ML, the estimated parameters converge to their actual values, which implies an asymptotically perfect spatial resolution, if a sufficiently high signal-to-noise ratio can be achieved. In order to reduce the dimension of the optimization problem of ML, the contribution of each dipole source to the measured pressures is assumed to be a latent variable and the ML problem is equivalently solved via the expectation–maximization (EM) algorithm, which iteratively and sequentially updates each source contribution and the associated sound source parameters. The number of sound sources can also be determined by the model selection approaches which add a penalty of model dimension to the ML objective function. The proposed method is assessed via a laboratory experiment where the sound field is produced by dipole speakers and a wind tunnel experiment where airframe aerodynamic noise is generated at a high Reynolds number. Experimental results show that the proposed method outperforms existing approaches in the sense of higher spatial resolution, more accurate localization, and the capacity to identify the orientations of multiple dipole sound sources.

Decomposition of a parallel automaton into a net of sequential automata and low power state assignment of them at asynchronous implementation

Objectives. The problems of decomposition of a parallel automaton into a net of sequential automata at synchronous realization and low power race free state assignment of them are considered. The objective of the paper is to investigate the possibilities of applying decomposition in state assignment of partial states in order to decrease the problem dimension taking into account the peculiar properties of the asynchronous realization.Methods. The given parallel automaton is decomposed into a net of sequential asynchronous automata whose states are assigned then with ternary vectors. The power consumption lowering of the designed device is achieved by lowering the intensity of its memory elements switching that is appreciated by probabilities of transitions between the states of the automaton. The state assignment is reduced to the problem of minimal weighted cover. The probabilities of transitions between sets are calculated by means of solving a system of linear equations according to the Chapmann – Kolmogorov method.Results. A method to construct a net of sequential asynchronous automata that realizes the given parallel automaton is described. The paper touches upon the problem of minimization of interconnections in the net.Conclusion. Applying parallel automaton decomposition allows decreasing the dimension of the laborious problem of state assignment. The proposed method is intended for application in computer aided systems for design of discrete devices.

Bridging human rights and social work education: the triple mandate in Austrian and Australian teaching contexts

ABSTRACT This article explores social work’s mandates, delineating the profession’s obligations across three key domains: the client-centric mandate, the state and its agency-driven regulatory and policy mandate and the mandate grounded in the profession’s scientific and ethical standards, including the commitment to human rights—social works triple mandate. Drawing on the author’s experience as a senior lecturer in Austria and Australia, the analysis highlights how the socio-political environments and educational frameworks can shape social work education. The application of the triple mandate is illustrated through a systematic set of W-questions to identify different dimensions of social problems within a broader systemic framework. This approach, grounded in scientific realism—which holds that reality exists independently of our perceptions and aligns with emergent systemism by scientifically examining social interactions to explain complex social systems—equips emerging social workers with the analytical and practical skills to transform private troubles into public issues. A scenario where a school student experiences alienation, leading to absenteeism, exemplifies the connection between theory and action. The discussion concludes by asserting that integrating science-based social work practices with ethical principles empowers social workers to navigate complex professional challenges and strengthens their commitment to human rights and social justice.

“Periphery-Centric” Approach as a Tactic for Everyday Digital In- and Exclusion of Indonesian Villages

Much has been said about the importance of digital inclusion in reducing the digital divide and ensuring equal access and use of ICTs for all. Generally, inclusivity has a positive connotation, meaning that no one should be left behind by digitalization processes. However, the inclusion of marginalized communities into the digital system could lead to new exclusions within the new system because it amplifies the pre-existing social inequalities that these communities face, creating digital inequality. By observing the implementation of School of Community Networks in 10 Indonesian villages, this article attempts to describe those inequalities and explain how rural communities actively design a strategy to make their involvement in Indonesia’s digitalization more meaningful. The conceptual framework developed in this study adopts a “periphery-centric” approach, aligning with the user-centric approach, as it examines from the perspective of marginalized communities how they perceive, understand, and utilize digital technology by generating new forms of innovation that have real impacts on their community, such as tailored applications for public services and local internet infrastructure to reach remote areas. The approach poses the problem and strategic dimension of the issue and puts forward the ownership and locality of these innovations to overcome digital inequality. With the meaningful use of digital technology, the inclusiveness of socially less-advantaged groups into the digital system does not immediately create new inequality; rather, they make themselves the center of their respective bottom-up innovation projects.

A gradient-enhanced univariate dimension reduction method for uncertainty propagation

The univariate dimension reduction (UDR) method stands as a way to estimate the statistical moments of the output that is effective in a large class of uncertainty quantification (UQ) problems. UDR's fundamental strategy is to approximate the original function using univariate functions so that the UQ cost only scales linearly with the dimension of the problem. Nonetheless, UDR's effectiveness can diminish when uncertain inputs have high variance, particularly when assessing the output's second and higher-order statistical moments. This paper proposes a new method, gradient-enhanced univariate dimension reduction (GUDR), that enhances the accuracy of UDR by incorporating univariate gradient function terms into the UDR approximation function. Theoretical results indicate that the GUDR approximation is expected to be one order more accurate than UDR in approximating the original function, and it is expected to generate more accurate results in computing the output's second and higher-order statistical moments. Our proposed method uses a computational graph transformation strategy to efficiently evaluate the GUDR approximation function on tensor-grid quadrature inputs, and uses the tensor-grid input-output data to compute the statistical moments of the output. With an efficient automatic differentiation method to compute the gradients, our method preserves UDR's linear scaling of computation time with problem dimension. Numerical results show that the GUDR is more accurate than UDR in estimating the standard deviation of the output and has a performance comparable to the method of moments using a third-order Taylor series expansion.

Graph-accelerated non-intrusive polynomial chaos expansion using partially tensor-structured quadrature rules for uncertainty quantification

Recently, the graph-accelerated non-intrusive polynomial chaos (NIPC) method has been proposed for solving uncertainty quantification (UQ) problems. This method leverages the full-grid integration-based NIPC method to address UQ problems while employing the computational graph transformation approach, AMTC, to accelerate the tensor-grid evaluations. This method exhibits remarkable efficacy on a broad range of low-dimensional (three dimensions or less) UQ problems featuring multidisciplinary models. However, it often does not scale well with problem dimensions due to the exponential increase in the number of quadrature points when using the full-grid quadrature rule. To expand the applicability of this method to a broader range of UQ problems, this paper introduces a new framework for generating a tailored, partially tensor-structured quadrature rule to use with the graph-accelerated NIPC method. This quadrature rule, generated through the designed quadrature approach, possesses a tensor structure that is tailored for the computational model. The selection of the tensor structure is guided by an analysis of the computational graph, ensuring that the quadrature rule effectively capitalizes on the sparsity within the computational graph when paired with the AMTC method. This method has been tested on one 4D and one 6D UQ problem, both originating from aircraft design scenarios and featuring multidisciplinary models. Numerical results show that, when using with graph-accelerated NIPC method, our approach generates a partially tensor-structured quadrature rule that outperforms the full-grid Gauss quadrature and the designed quadrature methods (more than 40% reduction in computational costs) in both of the test problems.

Palliative care knowledge among Peruvian physicians

Introduction: Palliative Medicine (PM) is a vital branch of medicine that focuses on the relief of symptoms while facing a life-threatening illness. In Peru, PM education is not included in the medical school curriculum and there is currently no fellowship for adequate PM training.Objective: To evaluate Palliative Medicine knowledge among Peruvian graduated physicians who are currently training or practicing in Peru.Methods: An observational, cross sectional and analytical study was conducted in graduated physicians from Peruvian medical schools who have or have not done a residency training and are currently residing and practicing medicine in Peru. Peruvian physicians were provided with an online link to assess their palliative medicine knowledge, primarily utilizing the Palliative Care Knowledge Test Spanish Version (PCKT-SV) which was our primary outcome variable.Results: A total of 120 physicians answered the questionnaire from different parts of Peru. Most of them were male (55.83%) and the range of age was between 25-40 years (81.67%). Regarding knowledge levels about palliative care, we found a higher correct answers for philosophy (83.75%), followed by dyspnea (68.12%). The least proportion of correct answers were present in the dimensions of psychiatry problems (54.58%), pain (56.39%) and gastrointestinal problems (58.33%).Conclusion: Peruvian physicians exhibit a commendable level of knowledge regarding palliative care, but issues arise in the comprehensive management of palliative patients since most of the medical practitioners lack specific expertise in this domain. Furthermore, the deficiency of undergraduate and postgraduate education on the subject is noteworthy.

Mental Health and Quality of Life of Patients with Differentiated Thyroid Cancer Pre and Post Radioactive Iodine Treatment: A Prospective Study.

Background: Although patients with differentiated thyroid cancer (DTC) have a good prognosis, their long-term clinical course can influence their mental health and their health-related quality of life (HRQoL). However, few studies have evaluated the psychological factors that influence subsequent HRQoL in this population, particularly during the initial treatment stage. Methods: In this 1-month cohort study, we evaluated depressive and anxiety symptoms and HRQoL of patients with DTC and examined possible predictors of further HRQoL impairment. Results: In total, 181 patients completed questionnaires where they self-rated their psychological status (the Chinese Health Questionnaire [CHQ], Taiwanese Depression Questionnaire [TDQ]) and HRQoL (the 36-item Short Form Health Survey [SF-36]) at baseline and 1 month after radioactive iodine (RAI) therapy. Compared with the general Taiwanese population, patients with DTC reported a worse HRQoL in all dimensions of the SF-36. Multivariate regression models indicated that anxiety and depressive symptoms were inversely correlated with some dimensions (physical functioning, bodily pain, and general health perceptions for the CHQ; role limitations due to physical problems and social functioning for the TDQ). However, psychiatric follow-up and treatment history were significantly associated with physical functioning and role limitations owing to the physical problem dimensions of HRQoL. Conclusions: In conclusion, although anxiety and depressive symptoms may negatively affect certain HRQoL domains, psychiatric follow-up can improve the physical dimensions.

CMAA–AHP: combinatorial multicriteria acceptability analysis with the analytic hierarchy process

AbstractCombinatorial multi-criteria acceptability analysis (CMAA) is a framework for supporting multicriteria group decisions that provides both a detailed analysis of the effects of individual decision-maker inputs as well as interactive guidance for a consensus-building process. The analytical hierarchy process (AHP) is a widely-used model of decision-maker evaluations that is based on pairwise comparisons. The goal of this work is to show how CMAA can be integrated with AHP in order to make its benefits available to AHP users. We use a minimal input format for AHP which avoids a problem with inconsistency and also reduces the cognitive load on the decision-makers. We extend the CMAA method by introducing new judgement and preference sensitivity variables, which provide helpful insights for the facilitator of the group decision. An example illustrates the combined CMAA–AHP method and its ability to deliver consensus in a very small number of iterations. Monte Carlo simulation is used to study the convergence behavior of the method for a range of problem dimensions. It was found that the mean number of steps to reach consensus grows linearly with the number of alternatives and criteria. We consider two previously published group decisions that use the standard AHP approach of averaging decision-maker judgements and preferences. In both cases, CMAA–AHP delivers the same rankings based on the original input. However, the new method also provides insight into each decision and would have been able to guide each group to consensus within a small number of resolution steps.