Trapezoidal Distribution Research Articles

An EOQ inventory model for deteriorating items with varying trapezoidal type demand rate and Weibull distribution deterioration

In a recent paper, Cheng and Wang's (2009) presented an inventory model for deteriotrating items with trapezoidal type demand rate. The present study focuses on determining the EOQ (Economic Order Quantity) model of an inventory system for a deteriorating item. Deterioration considered in most of the EOQ models is constant, while in most of the practical cases the deterioration rate increases with time. This work is motivated by Cheng and Wang's (2009) paper extending their model to allow for a Weibull distribution deterioration rate when the demand is taken as the trapezoidal function of time. Mathematical models are derived for three different cases. Our approach is to find minimum variable cost per unit time. Numerical examples are also given to illustrate the theoretical results.

Scheduling in polling systems in heavy traffic

We consider the classical cyclic polling model with Poisson arrivals and with gated service at all queues, but where the local scheduling policies are not necessarily First-Come- First-Served (FCFS). More precisely, we study the waitingtime performance of polling models where the local service order is Last-Come-First-Served (LCFS), Random-Orderof- Service (ROS) or Processor Sharing (PS). Under heavytraffic conditions the waiting times turn out to converge to products of generalized trapezoidal distributions and a gamma distribution.

An Analytic Solution to the Settlement under the Trapezoidal Distribution of the Strip Load of the Soft Soil Foundation

In order to manage and control the settlement along the highway, and meet the needs of the project, it is necessary to study the settlement calculation and prediction methods. Based on the layer-wise summation method, the derivation of the analytic solution of the final settlement can be done through the use of the calculation mode of additional stress and the Integral transform. With an illustrative example, this paper compares the differences of the value of the settlement derived respectively by the specification method and the analytic method. The graphs give an intuitive indication of the performance of the methods, and enable determination of the cubic capacity of settlement of the whole project, laying the foundation for the calculation of the ripped-rock quantity.

Thermal performance analysis of porous-microchannel heat sinks with different configuration designs

Three-dimensional models of porous-microchannel heat sinks (porous-MCHSs) with different configuration designs, such as rectangular, outlet enlargement, trapezoidal, thin rectangular, block, and sandwich distributions, are verified in this work. Hydraulic and thermal performances of the porous-MCHSs with various configuration designs are investigated from the pumping power, heat transfer coefficient, and temperature control effectiveness, results with Reynolds number ranging from 45 to 1350. The results reveal that the thermal performances can be improved using the porous configuration designs and can increase with a large Reynolds number. Both the sandwich and the trapezoidal distribution designs have the best heat transfer efficiency, cooling performance, and convective performance. In particular, the thermal performance of the rectangular, outlet enlargement, thin rectangular, or block distribution designs are not necessarily better than the nonporous medium for lower pumping power. In addition, adding a porous medium to the channel leads to a significant increase in the pressure drop. Among the six porous configuration designs, the lowest pressure drop was observed for a sandwich distribution design. Hence, the sandwich distribution design is the best porous-MCHS design when considering the thermal performance along with the pressure drop.

Estimation of Random Variable Distribution Parameters by the Monte Carlo Method

Abstract The paper is concerned with issues of the estimation of random variable distribution parameters by the Monte Carlo method. Such quantities can correspond to statistical parameters computed based on the data obtained in typical measurement situations. The subject of the research is the mean, the mean square and the variance of random variables with uniform, Gaussian, Student, Simpson, trapezoidal, exponential, gamma and arcsine distributions.

Proposed Guidelines for the Selection of Trapezoidal and Triangular Distributions for an Uncertainty Evaluation

Estimating measurement uncertainty involves categorizing sources of uncertainty into Type A and Type B components. Components evaluated as Type A are assumed to have a normal probability distributi...

An informational distance for estimating the faithfulness of a possibility distribution, viewed as a family of probability distributions, with respect to data

An acknowledged interpretation of possibility distributions in quantitative possibility theory is in terms of families of probabilities that are upper and lower bounded by the associated possibility and necessity measures. This paper proposes an informational distance function for possibility distributions that agrees with the above-mentioned view of possibility theory in the continuous and in the discrete cases. Especially, we show that, given a set of data following a probability distribution, the optimal possibility distribution with respect to our informational distance is the distribution obtained as the result of the probability-possibility transformation that agrees with the maximal specificity principle. It is also shown that when the optimal distribution is not available due to representation bias, maximizing this possibilistic informational distance provides more faithful results than approximating the probability distribution and then applying the probability-possibility transformation. We show that maximizing the possibilistic informational distance is equivalent to minimizing the squared distance to the unknown optimal possibility distribution. Two advantages of the proposed informational distance function is that (i) it does not require the knowledge of the shape of the probability distribution that underlies the data, and (ii) it amounts to sum up the elementary terms corresponding to the informational distance between the considered possibility distribution and each piece of data. We detail the particular case of triangular and trapezoidal possibility distributions and we show that any unimodal unknown probability distribution can be faithfully upper approximated by a triangular distribution obtained by optimizing the possibilistic informational distance.

Effects of Ge profile on thermal characteristics of SiGe heterojunction bipolar transistor with non-uniform doping profile in base region

As is well known, base region is the design focus of bipolar junction transistor (BJT). Generally, the doping distribution in base is non-uniform. In this paper, the effects of different Gaussian dopings in the base on bulk temperature distribution, temperature dependences of current gain and cut-off frequency are first studied. It is found that current gain and cut-off frequency of BJT have positive temperature coefficients, and the temperature in bulk is high. Then, the effect of Ge composition distribution on these device parameters is investigated. It is found that the SiGe heterojunction bipolar transistors (HBTs) with box Ge composition distribution and trapezoidal Ge composition distribution have negative temperature coefficients of current gain and cut-off frequency, and have good bulk temperature distributions. Furthermore, the SiGe HBT with trapezoidal Ge profile has higher current gain and cut-off frequency, and its temperature insensitivity is kept. The good trade-off is made among the magnitudes of current gain and cut-off frequency, temperature sensitivity and bulk temperature distribution.

Reply to the Discussion by H. Jiang and B. Fu of “Confined masonry walls subjected to combined axial loads and out-of-plane uniform pressures” 1Appears in the Canadian Journal of Civil Engineering, 39(11): 1238–1239 [doi: 10.1139.l2012-105

Both horizontal and vertical in-plane displacements of confining elements of masonry walls are reported to show that arching action mechanism is a two-direction mechanism. The spring-strut model is a simplified model that only considers equilibrium of forces in the main arching action direction. For the case of the walls studied, with a wall aspect ratio of about 0.75, the main direction is the vertical one. The two-direction arching mechanism is included in the model only for the calculation of the in-plane stiffness (K) of the confining elements. The stiffness (K) is an “equivalent” stiffness represented by a single spring located at the top of the wall. This in-plane stiffness is calculated modeling the four confining elements as a frame. After the formation of the wall final cracking pattern, arching action forces are transfered from the wall segments to the confining elements in two directions. For this reason, the frame is loaded with two triangular and two trapezoidal load distributions. Maximum value of these loads is represented by FT. The frame is also loaded with the corresponding axial load (P), if any. Boundary conditions considered are consisted with those used during testing. A structural analysis of the frame is carried out to determine the total in-plane vertical displacements at different points in the top confining element. Nine equally spaced points are used along this element. For wall specimen S-1, these vertical displacements are upwards; for this reason, an individual stiffness is calculated at each point dividing the force FT by the corresponding in-plane vertical displacement. The average value is defined as the “equivalent” in-plane stiffness (K). For specimens S-2 and S-3 the vertical in-plane displacements are downward; for this reason, the stiffness K is calculated assuming that the confining elements are very rigid and therefore vertical in-plane displacements are neglected. In this case, the stiffness K is calculated as the compressive force that causes crushing of the masonry divided by the corresponding axial deformation of the wall segments. The in-plane stiffness (K) of the confining elements, as presented in this work, depends on cross section properties, material properties, boundary conditions, and axial load. For the walls studied, gross cross sections were equal, modulus of elasticity varied among walls, boundary conditions were equal, and axial load changed from wall to wall. For these reasons, changes in the in-plane stiffness are related not only to the modulus of elasticity of concrete but also to the axial load. The experimental results and analytical models sections are consistent with this matter. The authors believe that the discussers misunderstood those sections. For the walls studied, cracking maximum pressure depends on boundary conditions, flexural tensile strength of masonry perpendicular to bed joints, out-of-plane load pattern (uniform pressure), and axial load. All these variables are included in the analytical models developed in the work. The authors did not consider it necessary to explain, for example, that cracking pressure increases as axial load or tensile strength increases.

Analytical approach for the prediction of steady state tyre forces and moments under different normal pressure distributions

In general, all primary as well as disturbing forces acting on a vehicle (with the exception of aerodynamic forces) are generated at the tyre–road interface. The small size pneumatic tyres presented in this work are mostly used in lightweight vehicles like scooters, motor cycles and 3-wheelers that are extensively used for intercity transportation applications in India and other developing countries. Analytical approach is more advantage than experimental work due to many factors like it involves less cost, minimum time and little effort. Hence, Analytical tyre models are developed based on various normal pressure distributions with tyre contact patch as rectangle at the middle with part of circles at the ends. The tyre longitudinal and lateral force characteristic equations for proposed tyre models are developed. In addition, the equations for self – aligning moment and overturning moment characteristics for all tyre models are also derived. Results of proposed tyre models in the present study are compared with well-established earlier tyre models [4–6]. From the force and moment characteristics of all tyre models, unsymmetrical trapezoidal pressure distribution tyre model gives better performance than other models due to which better lateral stability of the vehicle can be obtained.

Glass Unit Corner Loading—Key Parameter in Durability

Abstract Bent glass has garnered increased interest on the part of architects for the realization of curved glass façades. As one method of bending of glass façade units, cold bending is an economically efficient procedure for manufacturing; however, it introduces permanent stresses in the glazing structure, especially in the corner zones of the glass units for warped designs. In a similar manner, high stresses in the corner zones are also generated in general by constant surface loads acting on the panes of the glass unit, which can be explained by thin plate theory. Thus it can be expected that these unsteady loads, e.g., evoked by wind and/or snow loads, unfavorably interfere with the permanent stresses in the adhesives of both the structural glazing sealant and the insulating glass sealant from a durability point of view. The existence of these corner loads is not adequately accounted for by the ETAG 002 guideline for structural glazing applications, which postulates a trapezoidal load distribution in the bonding with diminishing stresses in the corner zones. This paper presents numerical results of a parametric study of pressure-loaded glass units, with a focus on corner loads and stresses. The results show that the stress levels in the corner zones might be significantly higher than the design stress values used for sizing the bonding.

Indirect parameter elicitation procedures for some distributions with bounded support – with applications in Program Evaluation and Review Technique (PERT)

The introduction of the Program Evaluation and Review Technique (PERT) dates back to the 1960s and has found wide application since then in the planning of construction projects. Difficulties with the interpretation of the parameters of the beta distribution let Malcolm et al. (Malcolm, D.G., et al., 1959. Application of a technique for research and development program evaluation. Operations Research, 7, 646–649) to elicit them indirectly via the classical expressions for the PERT mean and variance for an activity's completion time. These expressions are specified, given a lower and upper bound estimates a and b and a most likely estimate θ for the activity's duration. Despite a by now 50 -year controversy exemplified by current articles still questioning the PERT mean and variance elicitation approach, their use is still prevalent in current operations research and industrial engineering college text books. In this article an overview is presented of some alternative approaches that have been suggested, including a recent approach that allows for a direct modal range estimation combined with an indirect elicitation of bound and tail parameters of generalised trapezoidal uniform distributions describing activity uncertainty. The potential effect of eliciting a modal range, rather than the narrower point estimate of a single most likely value, shall be demonstrated in an illustrative Monte Carlo analysis for the completion time of an 18-node activity network.

Ac loss analyses of superconducting power transmission cables considering their three-dimensional geometries

Numerical electromagnetic field analyses of superconducting power transmission cables consisting of coated conductors were made considering their three-dimensional spiraled structures. Thin strips of superconductor were wound spirally around a round former to represent a layer in a cable. The analyses were made for two-layer cables where the spiral pitch of the inner layer and that of the outer layer are different. A non-uniform lateral critical current density distribution, a trapezoidal distribution with shoulders, was assumed across each coated conductor. The calculated ac losses were compared with measured ones. The influence of the relative positions between the inner-layer coated conductors and the outer-layer coated conductors on the current distribution across each coated conductor as well as ac losses was discussed. The influence of the current distribution between layers, determined by the spiral pitches, on the ac losses was also studied.

Numerical analysis of bearing reinforcement earth (BRE) wall

This paper presents a numerical simulation of the bearing reinforcement earth wall by PLAXIS 2D. The bearing reinforcement was regarded as a cost-effective earth reinforcement. The model parameters for the simulation were obtained from the conventional laboratory tests and back analyses from the laboratory pullout tests of the bearing reinforcement. The simplified method for modeling the bearing reinforcement, which converts the contribution of friction and bearing resistance to the equivalent friction resistance, is introduced. This method is considered to be acceptable and practical in working state with sufficient factor of safety and small pullout displacement. The bearing reinforcement is modeled as the geotextile and the equivalent friction resistance is represented by the soil/reinforcement interface parameter, R, which was obtained from a back analysis of the laboratory pullout test results. The R values are 0.65 and 0.75 for the bearing reinforcement with 2 and 3 transverse members, respectively. The change in bearing stresses, settlements, lateral earth pressures and tensions in the reinforcements during and after construction is simulated. Overall, the simulated test results are in good agreement with the measured ones. The simulated results show that the BRE wall behaves as a rigid body, retaining the unreinforced backfill. The simulated bearing stress presents a trapezoid distribution shape as generally assumed by the conventional method of examination of the external stability of MSE walls. The simulated settlement is almost uniform due to a high stiffness of the rigid foundation and the bearing reinforcements. The maximum lateral movement occurs at about the mid-height of the wall, resulting in the bi-linear maximum tension plane. The knowledge gained from this study can be applied to other BRE walls with different wall heights, foundations and features of bearing reinforcements.

Impact of GOS Misclassification on Ordinal Outcome Analysis of Traumatic Brain Injury Clinical Trials

This study extends our previous investigation regarding the effect of nondifferential dichotomous Glasgow Outcome Scale (GOS) misclassification in traumatic brain injury (TBI) clinical trials to the effect of GOS misclassification on ordinal analysis in TBI clinical trials. The impact of GOS misclassification and ordinal outcome analysis was explored via probabilistic sensitivity analyses using TBI patient datasets from the IMPACT database (n = 9205). Three patterns of misclassification were explored given the pre-specified misclassification distributions. For the random pattern, we specified a trapezoidal distribution (minimum: 80%, mode: 85%, and 95%, maximum: 100%) for both sensitivity and specificity; for the upward pattern, the same trapezoidal distribution for sensitivity but with a perfect specificity; and for the downward pattern, the same trapezoidal distribution for specificity but with a perfect sensitivity. The conventional 95% confidence intervals and simulation intervals, which accounts for the misclassification and random errors together, were reported. The results showed that given the specified misclassification distributions, the misclassification with a random or upward pattern would have caused a slightly underestimated outcome in the observed data. However, the misclassification with a downward pattern would have resulted in an inflated estimation. Thus the sensitivity analysis suggests that the nondifferential misclassification can cause uncertainties on the primary outcome estimation in TBI trials. However, such an effect is likely to be small when ordinal analysis is applied, compared with the impact of dichotomous GOS misclassifications. The result underlines that the ordinal GOS analysis may gain from both statistical efficiency, as suggested by several recent studies, and a relatively smaller impact from misclassification as compared with conventional binary GOS analysis.

Model Experimental Research on Anti-Sliding Characteristics of the Frame Anti-Sliding Piles

Through analysis of model test results, some of the basic sliding piles features of the frame anti-slide pile can be obtained. Under the thrust of landslide, a complex parabola and line function could be used to describe the relationship of load and lateral displacement of pile head. At the same time, there was a big crack in front of the pile. It was shown by monitoring the soil pressure that pushing force acting on the back frame pile was trapezoidal distribution and on the leading frame pile was parabola distribution. The soil pressure value was larger at back piles central and around sliding surface and smaller at the head of piles. the landslide thrust acting on the back piles more larger than the front row piles, the central of piles’ maximum soil pressure minimum 1:3 ratio of the middle pile at the before and after row, the ratio between the changes with the load. Frame pile of the inflection point after another figure appeared, at the top of the link beam and ring beam structure of the entire framework of the pile Showed a synergistic effect of spatial, compare to other forms of single row of piles and others forms, it can significantly reduce the size, to achieve the economic security landslide treatment goal, there is wide application prospect.

Ac loss reduction of multilayer superconducting power transmission cables by using narrowcoated conductors

The ac loss characteristics of coated conductors are dominated by the magnetic fieldcomponent normal to their superconductor layer. Multilayer cables as well asmonolayer cables consisting of 4 mm-wide coated conductors (named 4 mm cables)and those consisting of 2 mm-wide coated conductors (named 2 mm cables) weredesigned, and numerical electromagnetic field analyses were performed in their crosssections to calculate their ac losses. Trapezoidal lateral critical current densityJc distributions with shoulders as well as uniform ones were assumed incoated conductors for the analyses. The former models the degradedJc near the edges of coated conductors. In the case of the monolayer, thecalculated ac losses of the 2 mm cables were comparable to those of the 4 mmcables. In the cases of the multilayers, the calculated ac losses of the 2 mmcables were obviously less than those of the 4 mm cables. The degradedJc near the edges of coated conductors more seriously affects the ac loss characteristics of the 2 mmcables than those of the 4 mm cables. However, even if we consider the influence of the degradedJc near the edges of coated conductors, 2 mm-wide coated conductors are more profitable than4 mm-wide coated conductors in multilayer cables from the viewpoint of ac loss reduction.

Continuous review inventory model for fuzzy price dependent demand

In this paper, a continuous review inventory system has been analysed with an objective to determine an (r, Q) policy which minimises the cost function. The present study considers the demand rate, holding cost and shortage cost are sensitive to imprecise selling price. The proposed model assumes that shortages are allowed and completely backlogged and the lead time is fuzzy in nature. The arithmetical operations of fuzzy total cost are performed under function principle and the manager's k-preference integration representation method is applied to defuzzify the fuzzy total cost. Trapezoidal distribution of fuzzy number is used to work out the shortages. Then the model is solved jointly for (r, Q) policy through generalised reduced gradient (GRG) technique. A rigorous numerical study has been carried out and explained as managerial view point.

Derivation of isosceles trapezoidal distributions

The Guide to the Expression of Uncertainty in Measurement (GUM) (ISO 1995, Geneva: International Organization for Standardization) notes that an isosceles trapezoidal distribution often expresses the state of knowledge probabilities better than a rectangular distribution. The GUM lends further support to an isosceles trapezoidal distribution by interpreting it as a rectangular distribution whose inexactly known half-width may be represented by a (narrower) rectangular distribution. This interpretation is not exactly correct. It turns out that if the inexact knowledge about the half-width is represented by a rectangular distribution, then the resulting distribution is a variation of the isosceles trapezoid whose sloping sides are curved, an isocurvilinear trapezoidal distribution. The isosceles trapezoid can be regarded as an approximation to the isocurvilinear trapezoid. Therefore the GUM's interpretation is approximately correct. Question: does a probability distribution exist for the inexactly known half-width for which the resulting distribution is the isosceles trapezoid recommended in the GUM? We show that the isosceles trapezoidal distribution results when the inexact knowledge about the half-width is represented by a truncated right triangular distribution. The truncated right triangular distribution looks like a modification of the rectangular distribution whose top is sloping. The required truncated right triangular distribution has the same midpoint and the same second moment about the midpoint as the rectangular distribution on the same interval. It is often difficult for a metrologist to express the state of inexact knowledge about the half-width in terms of a well-defined probability distribution function. However, a metrologist can be expected to specify approximate limits for the inexactly known half-width. At least two probability distributions exist for the half-width for which the resulting distribution is isosceles trapezoid exactly or approximately. Therefore the isosceles trapezoidal distribution recommended in the GUM may be used as an approximation when only approximate limits for the half-width can be specified.

AC loss characteristics of superconducting power transmission cables: gap effect andJc distribution effect

Four groups of superconducting power transmission cables composed of coated conductors with 4.0 mm widthand 2 µm superconductor thickness have been designed, including mono-layer, two-layer, four-layerand six-layer cables. In each group, three types of cables have been constructed with gapsof different size between adjacent coated conductors, which are classified into small gap,medium gap and large gap. Moreover, different lateral critical current density(Jc) distributions, specifically a uniform distribution and a trapezoidal distribution with a slopingshoulder, have been assumed while calculating the AC losses of these cables numerically byusing a one-dimensional FEM model. Numerical results show that AC losses in mono-layercables are significantly influenced by gaps between coated conductors as well as lateralJc distribution, while cables with many layers are hardly affected by them. Thisproves that small gaps between coated conductors are not absolutely essentialto reduce AC losses in multi-layer cables, and a sloping shoulder of theJc distribution is more allowable in multi-layer cables than in mono-layer cables.The AC loss distributions among layers in six-layer cables are presentedand the reasons for different influences of the gap as well as the lateralJc distribution in mono-layer and multi-layer cables are discussed.