Asymmetric Deviation Research Articles

A Markovian Gauss inequality for asymmetric deviations from the mode of symmetric unimodal distributions

For a random variable with a unimodal distribution and finite second moment Gauß (1823) proved a sharp bound on the probability of the random variable to be outside a symmetric interval around its mode. An alternative proof for it under the assumption of a finite r r -th absolute moment is given ( r ≥ 1 r\geq 1 ), based on the Khintchine representation of unimodal random variables. A special instance of the resulting Narumi–Gauß inequality is the one with finite first absolute moment, which might be called a Markovian Gauß inequality. For symmetric unimodal distributions with finite second moment Semenikhin (2019) generalized the Gauß inequality to arbitrary intervals. For the class of symmetric unimodal distributions with finite first absolute moment we construct a Markovian version of it. Related inequalities of Volkov (1969) and Sellke and Sellke (1997) will be discussed as well.

A novel method to study the long period three-dimensional vibration characteristics of herringbone gear with asymmetry pitch deviation

Most of the existing dynamic models of herringbone gear assume that the three major internal excitations, namely, the comprehensive mesh stiffness (CMS), the comprehensive mesh error (CME), and the meshing impact force (MIF), are the same at each meshing period. However, due to the inevitable asymmetric pitch deviation in the machining and installation process, the three internal excitations are coupled with each other and vary in a long period. In addition, since the driving gear usually adopts an axially floating support, this in turn gives rise to a unique axial displacement (AD) excitation of the herringbone gear. Firstly, an improved loaded tooth contact analysis (LTCA) model of herringbone gear with asymmetric pitch deviation is established, and the coupling relationship between multi-source excitation is explored. Secondly, a nonlinear dynamic model of herringbone gear with asymmetric pitch deviation is proposed, and the above multi-source excitation is introduced into the model to study the effects of load, speed, and asymmetric MIF on the long period three-dimensional (3-D) vibration characteristics of herringbone gear. Finally, the correctness of the theoretical simulation results is verified by experiments. The proposed novel method can predict the 3-D vibration characteristics of herringbone gear more realistically and effectively.

Pinball boosting of regression quantiles

An algorithm for boosting regression quantiles using asymmetric least absolute deviations, better known as pinball loss, is proposed. Existing approaches for boosting regression quantiles are essentially equal to least squares boosting of regression means with the single difference that their working residuals are based on pinball loss. All steps of our boosting algorithm are embedded in the well-established framework of quantile regression, and its main components – sequential base learning, fitting, and updating – are based on consistent scoring rules for regression quantiles. The Monte Carlo simulations performed indicate that the pinball boosting algorithm is competitive with existing approaches for boosting regression quantiles in terms of estimation accuracy and variable selection, and that its application to the study of regression quantiles of hedonic price functions allows the estimation of previously infeasible high-dimensional specifications.

Robust opportunistic optimal energy management of a mixed microgrid under asymmetrical uncertainties

Energy management within microgrids under the presence of large number of renewables such as photovoltaics is complicated due to uncertainties involved. Randomness in energy production and consumption make both the prediction and optimality of exchanges challenging. In this paper, we evaluate the impact of uncertainties on optimality of different robust energy exchange strategies. To address the problem, we present AIROBE, a data-driven system that uses machine-learning-based predictions of energy supply and demand as input to calculate robust energy exchange schedules using a multiband robust optimization approach to protect from deviations. AIROBE allows the decision maker to tradeoff robustness with stability of the system and energy costs. Our evaluation shows, how AIROBE can deal effectively with asymmetric deviations and how better prediction methods can reduce both the operational cost while at the same time may lead to increased operational stability of the system.

Penalized enhanced portfolio replication with asymmetric deviation measures

Passive investment strategies, such as those implemented by Exchange Traded Funds (ETFs), have gained increasing popularity among investors. In this context, smart beta products promise to deliver improved performance or lower risk through the implementation of systematic investing strategies, and they are also typically more cost-effective than traditional active management. The majority of research on index replication focuses on minimizing tracking error relative to a benchmark index, implementing constraints to improve performance, or restricting the number of assets included in portfolios. Our focus is on enhancing the benchmark through a limited number of deviations from the benchmark. We propose a range of innovative investment strategies aimed at minimizing asymmetric deviation measures related to expectiles and quantiles, while also controlling for the deviation of portfolio weights from the benchmark composition through penalization. This approach, as compared to traditional minimum tracking error volatility strategies, places a greater emphasis on the overall risk of the portfolio, rather than just the risk relative to the benchmark. The use of penalization also helps to mitigate estimation risk and minimize turnover, as compared to strategies without penalization. Through empirical analysis using simulated and real-world data, we critically examine the benefits and drawbacks of the proposed strategies in comparison to state-of-the-art tracking models.

Cerebral Infarction Due to Polycythemia Vera: A Case Report

Introduction: Changes in blood viscosity can cause thrombosis in the brain, causing a stroke. Polycythemia vera is one of the causes of blood hyperviscosity. This study aimed to present cases of cerebral infarction related to polycythemia vera.
 Case presentation: A 55-year-old woman came to the emergency room with complaints of sudden weakness in the left limbs. One week before admission, the patient complained of headaches accompanied by spinning dizziness. Her left limbs often tingled and became weaker. There was an asymmetric facial deviation to the right, and the patient had difficulty communicating with unclear articulation. Specific examination of neurology shows dysarthria, left-sided motor type facial paralysis, examination of motor strength of the upper and lower extremities on the left 2 and 5 on the right, and the left upper limb often feels tingling. On the left limb, normal physiological biceps reflex was found, positive Chaddock pathological reflex. Morphological examination of peripheral blood showed normocytic normochromic with erythrosis and thrombocytosis and found giant platelets. This patient was diagnosed with an ischemic stroke caused by cerebral infarction, with polycythemia vera etiology, and accompanied by comorbid diseases of diabetes mellitus and uncontrolled hypertension.
 Conclusion: In patients with high-risk criteria for thrombosis, the management of phlebotomy, administration of low-dose aspirin, and cytoreduction in the form of hydroxyurea aims to achieve a hematocrit <45% so as to prevent worsening and recurrence of stroke.

The influence of induced gait asymmetry on joint reaction forces

Chronic injury- or disease-induced joint impairments result in asymmetric gait deviations that may precipitate changes in joint loading associated with pain and osteoarthritis. Understanding the impact of gait deviations on joint reaction forces (JRFs) is challenging because of concurrent neurological and/or anatomical changes and because measuring JRFs requires medically invasive instrumented implants. Instead, we investigated the impact of joint motion limitations and induced asymmetry on JRFs by simulating data recorded as 8 unimpaired participants walked with bracing to unilaterally and bilaterally restrict ankle, knee, and simultaneous ankle + knee motion. Personalized models, calculated kinematics, and ground reaction forces (GRFs) were input into a computed muscle control tool to determine lower limb JRFs and simulated muscle activations guided by electromyography-driven timing constraints. Unilateral knee restriction increased GRF peak and loading rate ipsilaterally but peak values decreased contralaterally when compared to walking without joint restriction. GRF peak and loading rate increased with bilateral restriction compared to the contralateral limb of unilaterally restricted conditions. Despite changes in GRFs, JRFs were relatively unchanged due to reduced muscle forces during loading response. Thus, while joint restriction results in increased limb loading, reductions in muscle forces counteract changes in limb loading such that JRFs were relatively unchanged.

Effects of asymmetric stroke deviation on the aerodynamic performance of flapping wing

The kinematics of insect flapping flight are complex and asymmetric, which are contributed to their superior flying capabilities, and the design of novel flapping micro air vehicles can draw inspiration from relevant researches. Previous studies usually focus on the wing with asymmetric stroke or pitch motions. A trajectory with asymmetric deviation motion, named as “pear-shaped” pattern, is proposed in current work. The hovering aerodynamics and vortex dynamics of a rigid flapping wing have been numerically investigated by comparing with that of “line-shaped” pattern with no deviation. In order to have a better insight into the influences of the asymmetric deviation, we change the kinematic parameters, that is, stroke amplitude, pitching amplitude, deviation amplitude, and phase lag between stroke and pitch angles. The results show that the wing with asymmetric deviation exhibits superior capability in lift enhancement for most of the cases analyzed, which is accompanied by the extra power cost and slight reduction in efficiency. The asymmetric deviation in cases with high stroke amplitude or low pitching amplitude may be considered as a cost-saving strategy, subject to slight damage on lift generation (if acceptable). Additionally, the asymmetric deviation brings a strong asymmetry into the instantaneous forces during one flapping cycle. The underlying lift-enhancing mechanism is explored by examining the dominant vortex structures in the adjacent flow field of the wing, which is mainly attributed to the changes in the effective angle of attack, increasing with downward deviation and decreasing with upward deviation.

A Computed Tomographic Study to Document Anatomical Variations of First Cervical Vertebra: Prevalence, Classification and Possible Clinical Implications

Background - First cervical vertebra has different anatomical features as compared to other cervical vertebrae. It holds globe of skull and lacks body and spine. It is composed of 2 lateral masses linked by anterior and posterior arches. Posterior atlanto-occipital membrane is attached to posterior arch. Lateral edge of this membrane sometimes ossifies thus converting groove into canal. Consequently, neurovascular groove gets converted into a bony ring “ponticulus posticus”. It is considered as potential cause of neck pain and headache. Knowledge of this variation is important during procedures like instrumentation of atlas including C1 lateral mass screw placement which requires mobilization of C2 nerve root and screw insertion inferior to bony arch of atlas directly into lateral mass. In patients with ponticulus posticus, if not identified pre-operatively may give impression of wide bony arch for screw placement, which may result in vertebral artery injury due to a breach of this variation. Congenital defects of atlantal arch a developmental failure of chondrogenesis is a rare anomaly. These defects are a benign variation discovered incidentally. Detection of these anomalies is clinically important as they can cause acute neurologic deficits, which is associated with neck extension. Currarino et al proposed classification of posterior arch anomalies Type A Failure of posterior midline fusion of the 2 hemi arches. Type B Unilateral cleft. Type C Bilateral cleft with persistent dorsal part of arch. Type D Absence of posterior arch with persistent posterior tubercle. Type E Absence of entire posterior arch Objective Objectives this study is to find the prevalence of Ponticulus posticus Congenital anomalies of atlantal arch and classify them Material and Method- 250 computerized tomography (CT) head and neck in axial and sagittal sections from Picture Archiving and Communication System (PACS) of Dr RMLIMS, Lucknow was studied as CT is the best method to study the bony landmarks and any anomaly. Prevalence of variations was calculated. Data was analyzed statistically using SPSS version 21 by Chi-square test. Result- Prevalence of ponticulus posticus was 32.4%. Males had higher prevalence (20.4%) than females (12%). Incomplete ponticulus posticus (24.4%) was more than complete variant (8%). There was predominance of left sided ponticulus posticus(12% vs 8.4%) Deficient posterior arch was found in 9(3.6%) cases. 6 were of type A 3 were of type B. Conclusion- This study will help in determining cause of neurological deficit in patients due to presence of above variations. Significance Accurate evaluation of anatomic landmarks in patient with migraine or neck ache is important because changes in anatomy of atlas may be index pointing towards underlying disease process like vertebrobasilar insufficiency. Asymmetrical deviations of atlas in form, size, shape, and contour are often multiple and confusing. Therefore, they must not only be studied with care and precision and meticulously recorded. In our future project we plan to carry out computed tomographic study to find association of these variations in patients with migraine and cervical pain.

Determination of the regularities of the soil punching process by the working body with the asymetric tip

The presence of analytical dependencies describing the process of static soil puncture by a working body with a conical asymmetric tip is necessary to create installations with the ability to control the trajectory of the soil puncture. The paper considers the features of the process of interaction of an asymmetric conical tip with the ground. Analytical relationships were obtained to determine its reactions during a static puncture, the deviation of the head trajectory from a straight line, to determine the size of the soil compaction zone and the magnitude of the destructive force that acts on adjacent communications and other underground objects. It was found that with an increase in the value of the displacement of the top of the cone, for example, from its axis from 0.02 m to 0.08 m with a borehole diameter of 0.2 m, the value of soil resistance increases almost four times. The greatest resistance is achieved when piercing a hard sandy sand. It was found that with an increase in the displacement of the tip of the tip cone, the deviation of the trajectory increases. The piercing head achieves the greatest deviation from the straight trajectory of movement with a sharper cone and a greater asymmetric deviation of its top, and, for example, in hard sandy loam can be up to 0.17 m with a span of 10 m. It was found that the size of the soil destruction zone will be almost 1.8 times larger than the tip in the form of a symmetrical cone and reaches from 8 to 12 borehole diameters, depending on the type of soil. The maximum pressure on adjacent objects can reach from 0.06 MPa in hard-plastic clay to 0.09 MPa in hard sandy loam. The calculated dependences obtained for determining the power and technological parameters depending on the geometric dimensions of the asymmetric tip of the working body can be used to create installations with a controlled static puncture for use in the most common soil conditions.

Leaf sodium, potassium and electrolyte accumulation capacity of plant species from salt-affected coastal habitats of the Baltic Sea: Towards a definition of Na hyperaccumulation

The aim of the present study was to compare a relatively large number of native plant species from salt-affected coastal habitats with respect to their Na+ and K+ accumulation capacity and electrolyte level in leaves. Sampling sites (77 in total) were chosen during field survey in coastal habitats located at geographically distant places along the Baltic Sea in the territory of Latvia, Estonia, Denmark and Sweden. The main criterion for selection of sampling sites was minimum average soil electrical conductivity (EC) above 200 mS m–1. Leaf samples of 102 taxa (species) from 34 families were collected, with a total number of samples reaching 502, and their water content, Na+ and K+ concentration, and electrical conductivity in water extracts were analyzed. High intraspecies and interspecies variability in analyzed parameters was found. Frequency distribution of Na+ concentration showed highly significant positive asymmetry and deviation from normality. Data on K+ concentration and tissue EC level had asymmetric distribution and deviation from normality but to a lesser extent than that for Na+. Correlation between tissue Na+ concentration and substrate EC at sampling sites was low on individual sample basis but increased to moderate level when mean values of species were analyzed. According to accumulation characteristics of Na+ and K+ in leaves, all species corresponded to one of three general groups: Na+ excluders, regulating EC by changes in K+ concentration; K+ excluders, regulating EC by changes in Na+ concentration; and tight EC regulators; but these features were evident only when intraspecies variability was considered. Na+ concentrations in a range 18 to 30 g kg–1 (on dry biomass basis) and 0.24 to 0.30 mol L–1 (on tissue water basis) can be used as approximate values of a "normal" vs "hyperaccumulation" Na+ range in leaves of coastal plant species of the Baltic Sea.

Noisy Density Evolution With Asymmetric Deviation Models

This paper considers low-density parity-check (LDPC) decoders affected by deviations introduced by the electronic device on which the decoder is implemented. Noisy density evolution (DE) that allows to theoretically study the performance of these LDPC decoders can only consider symmetric deviation models due to the all-zero codeword assumption. A novel DE method is proposed that admits the use of asymmetric deviation models, thus widening the range of faulty implementations that can be analyzed. DE equations are provided for three noisy decoders: belief propagation, Gallager B, and quantized min-sum (MS). Simulation results confirm that the proposed DE accurately predicts the performance of LDPC decoders with asymmetric deviations. Furthermore, asymmetric versions of the Gallager B and MS decoders are proposed to compensate the effect of asymmetric deviations. The parameters of these decoders are then optimized using the proposed DE, leading to better ensemble thresholds and improved finite-length performance in the presence of asymmetric deviations.

Tail risks in large portfolio selection: penalized quantile and expectile minimum deviation models

Accurate estimation and optimal control of tail risk is important for building portfolios with desirable properties, especially when dealing with a large set of assets. In this work, we consider optimal asset allocation strategies based on the minimization of two asymmetric deviation measures, related to quantile and expectile regression, respectively. Their properties are discussed in relation with the ‘risk quadrangle’ framework introduced by Rockafellar and Uryasev [The fundamental risk quadrangle in risk management, optimization and statistical estimation. Surv. Oper. Res. Manag. Sci., 2013, 18(1–2), 33–53], and compared to traditional strategies, such as the mean–variance portfolio. In order to control estimation error and improve the out-of-sample performance of the proposed models, we include ridge and elastic-net regularization penalties. Finally, we propose quadratic programming formulations for the optimization problems. Simulations and real-world analyses on multiple datasets allow to discuss pros and cons of the different methods. The results show that the ridge and elastic-net allocations are effective in improving the out-of-sample performance, especially in large portfolios, compared to the un-penalized ones.

A graphical approach to assess the goodness-of-fit of random-effects linear models when the goal is to measure individual benefits of medical treatments in severely ill patients

BackgroundTwo-dimensional personalized medicine (2-PM) models are tools for measuring individual benefits of medical treatments for chronic diseases which have potential applications in personalized medicine. These models assume normality for the distribution of random effects. It is necessary to examine the appropriateness of this assumption. Here, we propose a graphical approach to assessing the goodness-of-fit of 2-PM models with continuous responses.MethodsWe propose benefit quantile-quantile (BQQ) plots which compare the empirical quantiles of individual benefits from a patient sample predicted through an empirical Bayes (EB) approach versus the quantiles of the theoretical distribution of individual benefits derived from the assumption of normality for the random effects. We examine the performance of the approach by conducting a simulation study that compared 2-PM models with non-normal distributions for the random effects versus models with comparable normal distributions. Cramer-von Mises discrepancies were used to quantify the performance of the approach. The approach was illustrated with data from a clinical trial of imipramine for patients with depression.ResultsSimulations showed that BQQ plots were able to capture deviations from the normality assumption for the random effects and did not show any asymmetric deviations from the y = x line when the random effects were normally distributed. For the depression data, the points of the BQQ plot were scattered around closely to the y = x line, without presenting any asymmetric deviations. This implied the adequacy of the normality assumption for the random effects and the goodness-of-fit of the 2-PM model for the imipramine data.ConclusionBQQ plots are sensitive to violations of the normality assumption for the random effects, suggesting that the approach is a useful tool for examining the goodness-of-fit of random-effects linear models when the goal is to measure individual treatment benefits.

Tail Risks in Large Portfolio Selection: Penalized Quantile and Expectile Minimum Deviation Models

Accurate estimation and optimal control of tail risk is important for building portfolios with desirable properties, especially when dealing with a large set of assets. In this work, we consider optimal asset allocations strategies based on the minimization of two asymmetric deviation measures, related to quantile and expectile regression, respectively. Their properties are discussed in relation with the `risk quadrangle' framework introduced by Rockafellar et al (2013) and compared to traditional strategies, such as the mean-variance portfolio. In order to control estimation error and improve the out-of-sample performances of the proposed models, we include ridge and elastic-net regularization penalties. Finally, we propose quadratic programming formulations for the optimization problems. Simulations and real-world analyses on multiple datasets, allow to discuss pros and cons of the different methods. The results show that the ridge and elastic-net allocations are effective in improving the out-of-sample performances, especially in large portfolios, compared to the un-penalized ones.

Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value

In various disordered systems or non-equilibrium dynamical models, the large deviations of some observables have been found to display different scalings for rare values bigger or smaller than the typical value. In the present paper, we revisit the simpler observables based on independent random variables, namely the empirical maximum, the empirical average, the empirical non-integer moments or other additive empirical observables, in order to describe the cases where asymmetric large deviations already occur. The unifying starting point to analyze the large deviations of these various empirical observables is given by the Sanov theorem for the large deviations of the empirical histogram: the rate function corresponds to the relative entropy with respect to the true probability distribution and it can be optimized in the presence of the appropriate constraints. Finally, the physical meaning of large deviations rate functions is discussed from the renormalization perspective.

Fano-resonance collapse induced terahertz magnetic dipole oscillation in complementary meta-atoms via local symmetry breaking

A novel phenomenon is observed in the meta-atoms composed of a complementary rectangular double split-ring resonator (SRR). An intrinsic Fano-resonance collapses with the outer SRR deformed asymmetrically. Alternatively, a trapped mode emerges at an adjacent frequency region, of which its strength grows up with increasing the asymmetric deviation. However, the asymmetric deformation in the inner SRR has influence neither on the evolution of this intrinsic Fano-resonance nor on the excitation of the aforementioned trapped mode. The results of electromagnetic field simulation indicate that an interference of two magnetic dipoles leads to the intrinsic Fano-resonance on the outer SRR. The asymmetric deviation destructs coherent interference so that the Fano-resonance collapses. To the trapped mode, the surface current passes through the metal gap of the outer SRR, leading to a couple of antiparallel currents, which results in a couple of magnetic dipole oscillations. The intrinsic modes are kept constant, even though the inner SRR is asymmetrically deformed. The outer SRR plays the role of a Faraday cage, which electromagnetically shields the trapped mode on the inner SRR.

Asymmetry in Luticola species

Morphological asymmetry in raphid pennate diatoms is detectable not only by valve outline and specific frustule features, but also in their ontogeny. All raphid pennate diatoms exhibit cryptic lateral polarity, which is associated with the asymmetric formation of the valve sides along the apical axis. Differences in the relative widths of primary and secondary valve sides have proven taxonomically important for separating dorsiventral genera within the Cymbellales. However, rigorous studies examining subtle asymmetric differences in valve outline are scarce. In the present geometric morphometric study of 23 Luticola species, the total asymmetry in valve outline of natural populations was decomposed into components of directional asymmetry (DA) and fluctuating asymmetry (FA) to test whether cryptic directional asymmetry has a significant effect on valve outlines across the genus as a whole. The patterns of asymmetric variation provide a view on average asymmetry of populations (evaluated by DA) and individual asymmetric deviations (associated with FA). Subtle, but statistically significant, average irregularities between the primary and secondary sides of Luticola species (i.e., the effect of DA) were found in all but two species. Patterns of DA between valve sides were quite similar in 17 of the analysed species. This likely indicates common developmental patterns in their morphogenesis. The effect of FA, which has often been used as a proxy of developmental instability of populations, was significant in all Luticola species. Since FA scores of individual cells were not correlated with size diminution of most species, FA might reflect the effect of environmental factors but not life cycle changes. The methodological framework presented in this study could be valuable for future work focused on evolutionarily and/or environmentally related shape aberrations and valve variation in pennate diatoms.

Multiband Simulations of Multistream Polarimetric Microwave Radiances Over Aspherical Hydrometeors

AbstractA numerical precision assessment and simulation results of the Unified Microwave Radiative Transfer (UMRT) model incorporating aspherical frozen hydrometeors based on the NASA Goddard Space Flight Center (NASA/GSFC) OpenSSP database are presented. It is shown that UMRT maintains unconditional numerical stability and computational efficiency for absorbing and scattering clouds. UMRT requires symmetry of the transition matrix for the discretized planar‐stratified radiative transfer equation to realize numerically stable and accurate matrix operations as required by the discrete ordinate eigenanalysis method. UMRT has heretofore been restricted to spherical polydispersive hydrometeors. In this study, the necessary block‐diagonal structure of the full Stokes matrix for randomly oriented OpenSSP aspherical hydrometeors is shown to be maintained, albeit with small asymmetric deviations, which introduce small asymmetric components into the transition matrix that are negligible for most remote sensing applications. An upper bound of the brightness temperature error calculated by neglecting the asymmetric components of the transition matrix under even extreme atmospheric conditions is shown to be small. Hence, the OpenSSP hydrometeor database can be reliably used within the UMRT model. Block‐diagonal Stokes matrix elements along with other single‐scattering hydrometeor parameters were subsequently used in radiative simulations of multistream dual‐polarization radiances for a simulated hurricane event to demonstrate the inherent numerical stability and utility of the enhanced UMRT model. An intercomparison of computed upwelling radiances for a multiphase distribution of aspherical OpenSSP hydrometeors versus a mass‐equivalent Mie hydrometeor polydispersion for key sensing frequencies from 10 to 874 GHz shows the considerable impact of complex (vs. simple spherical) hydrometeors on predicted microwave radiances.

A multiple objective framework for optimal asymmetric tolerance synthesis of mechanical assemblies with degrading components

In order to produce the mechanical assemblies with the high quality, the finest functionality, and the low cost, the optimal tolerance synthesis can be a useful tool in the design stage. The degradation of components due to some operational or environmental factors (such as the thermal cycling, the mechanical deformation, and the wear) can lead to dimensional variations in components and fall-off of the functionality of the product. In addition, different manners of the degradation on the internal and external dimensions can cause asymmetric deviations in the dimensions. On the other hand, the effect of degradation on the product quality has not been considered in most researches. In this paper, a new multi-objective framework is proposed for optimal asymmetric tolerance synthesis of mechanical assemblies with degrading components. In this method, the optimal tolerances are allocated based on ensuring the fulfillment of the product’s functional requirements, maximizing the product quality, and minimizing the total cost over the lifetime of the product. To incorporate the degradation effect into the loss function concept, the present worth of the expected quality loss (PWL) is formulated in terms of asymmetric tolerances. Accordingly, the functional process capability and manufacturing cost are developed based on asymmetric tolerances and the degradation effects. In order to extract Pareto fronts of optimal solutions, the elitist Non-dominated Sorting Genetic Algorithm II as an evolutionary generating methodology is utilized. In solving multi-criteria tolerance synthesis problem by a generating method, selecting the best tolerances from the obtained optimal Pareto solutions is a significant challenge. In this paper, to find the best asymmetric tolerances from Pareto solutions, a combined Shannon’s entropy-based TOPSIS algorithm is used. Finally, a bi-directional non-back drivable roller clutch assembly as an industrial case study is considered to illustrate the efficiency of the proposed method, and the obtained results are compared and discussed for verification.