
Citation: Agustín Halty, Rodrigo Sánchez, Valentín Vázquez, Víctor Viana, Pedro Piñeyro, Daniel Alejandro Rossit. Scheduling in cloud manufacturing systems: Recent systematic literature review[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7378-7397. doi: 10.3934/mbe.2020377
[1] | Jawad Saleemi . COVID-19 and liquidity risk, exploring the relationship dynamics between liquidity cost and stock market returns. National Accounting Review, 2021, 3(2): 218-236. doi: 10.3934/NAR.2021011 |
[2] | Yanting Xu, Tinghui Li . Measuring digital economy in China. National Accounting Review, 2022, 4(3): 251-272. doi: 10.3934/NAR.2022015 |
[3] | Jiehua Ma, Zhenghui Li . Measuring China's urban digital economy. National Accounting Review, 2022, 4(4): 329-361. doi: 10.3934/NAR.2022019 |
[4] | Rashesh Vaidya . NEPSE in Bollinger Bands. National Accounting Review, 2021, 3(4): 439-451. doi: 10.3934/NAR.2021023 |
[5] | Mustafa Tevfik Kartal . Do activities of foreign investors affect main stock exchange indices? Evidence from Turkey before and in time of Covid-19 pandemic. National Accounting Review, 2020, 2(4): 384-401. doi: 10.3934/NAR.2020023 |
[6] | Cheng Li . Unequal wealth of nations: Evidence from the World Bank wealth accounts. National Accounting Review, 2024, 6(3): 384-406. doi: 10.3934/NAR.2024018 |
[7] | Jinhui Zhu, Mengxin Wang, Changhong Zhang . Impact of high-standard basic farmland construction policies on agricultural eco-efficiency: Case of China. National Accounting Review, 2022, 4(2): 147-166. doi: 10.3934/NAR.2022009 |
[8] | Tinghui Li, Zimei Huang, Benjamin M Drakeford . Statistical measurement of total factor productivity under resource and environmental constraints. National Accounting Review, 2019, 1(1): 16-27. doi: 10.3934/NAR.2019.1.16 |
[9] | Francesco Scalamonti . A quantitative and qualitative macroeconomic and sociopolitical outlook of the MEDA transitional economies: development-paths, governance climate, and sociocultural factors. National Accounting Review, 2024, 6(3): 407-448. doi: 10.3934/NAR.2024019 |
[10] | Daniel Francois Meyer . Economic sectoral diversification: A case study of the Gauteng provincial region, South Africa. National Accounting Review, 2023, 5(4): 356-372. doi: 10.3934/NAR.2023021 |
Probability distributions are very important in modeling and fitting random phenomena in all areas of life. In the literature on distribution theory, there are various probability distributions for analyzing and predicting multiple kinds of data in many sectors, including life, biology, medical science, insurance, finance, engineering, and industry [1,2,3,4,5,6,7,8,9]. Based on existing findings, industrial data often exhibits a thick right tail, and many authors have developed several well-known right-skewed families. Afify et al. [10] defined the power-modified Kies-exponential distribution. Coşkun et al. [11] introduced the modified-Lindley distribution, and Gómez et al. [12] proposed the power piecewise exponential model. In addition, Dhungana and Kumar [13] proposed an exponentiated odd Lomax exponential distribution, while Hassan et al. [14] introduced the alpha power transformed extended exponential distribution. In the same line, Karakaya et al. [15], presented a unit-Lindley distribution, and Tung et al. [16] developed the Arcsine-X family of distributions.
To bring further flexibility to these generated distributions, various approaches of well-known models have been defined and used in several applied sciences to allow the smoothing parameter to vary across different locations in the data space. One of the new model-generating techniques is the error function (EF) transformation, which was first proposed by Fernández and De Andrade [17]. The cumulative distribution function (CDF) and the corresponding probability distribution function (PDF) of the EF transformation are as follows:
Δ(y)=erf(H(y)1−H(y)),y∈R, | (1.1) |
and
δ(y)=2h(y)√π(1−H(y))2 exp{−(H(y)1−H(y))2}. | (1.2) |
The EF transformation is a novel method for generalizing a given model, which transforms a distribution without adding any parameters. It is a modified version of traditional probability distributions for the relative importance or worth of data points. This strategy improves flexibility, allowing analysts to better explain real-world scenarios in which traditional random sampling fails to capture the underlying data structure. The derivation of the new attractive EF transformation to modify the existing distribution helps the fitting power of the existing distributions. The proposed method has many applications that extend to fitting, especially in industrial domains. However, recent works considering the EF technique, such as [18,19].
The inverse Weibull (IW) distribution is widely used in reliability and lifetime modeling for mortality rates, especially when studying extreme events. Since it captures tail behavior effectively, it is effective in understanding the upper quantiles of life expectancy or survival time. The CDF of the IW distribution, denoted as G(x), is defined as follows:
G(x)=e−θx−β,; x, θ,β>0. | (1.3) |
In reference to G(x) as stated in Eq (1.3), the PDF g(x) is formulated as:
g(x)=θβ x−(β+1) e−θx−β. | (1.4) |
The IW model has undoubtedly established itself as a crucial tool for data modeling across nearly all sectors. However, despite its widespread use and advantages, the IW distribution is constrained by its inherent limitations. One of the primary constraints of the IW distribution is its capacity to represent solely monotonic forms of hazard functions, as it can only model situations where the hazard rate increases or decreases consistently over time. More papers have used the IW model for many different statistical models, such as the following : Alzeley et al. [20] discussed statistical inference under censored data for IW model, Hussam et al. [21] discussed fuzzy vs. traditional reliability models, Ahmad et al. [22] derived the new cotangent IW model, Mohamed et al. [23] discussed Bayesian and E-Bayesian estimation for an odd generalized exponential IW model. Abdelall et al. [24] introduced a new extension of the odd IW model. Al Mutairi et al. [25] obtained Bayesian and non-Bayesian inference based on a jointly type-Ⅱ hybrid censoring model. Hassan et al. [26] discussed the statistical analysis of IW based on step-stress partially accelerated life testing. Alsadat et al. [27] presented novel Kumaraswamy power IW distribution with data analysis related to diverse scientific areas.
In this paper, we focus on providing a new form of the IW distribution for analyzing the datasets of different areas and highlighting specific characteristics. We extend this distribution by using the approach discussed in equation (1.1), and the resultant distribution is named the error function inverse Weibull (EF-IW) model. This heightened flexibility allows for a better fit to datasets with diverse kurtotic characteristics, enhancing the model's applicability across various scenarios. Further, the key objectives of the current study are as follows.
(1) The primary objective was extending the EF-IW distribution using the error function method, allowing for the derivation and investigation of its essential mathematical characteristics.
(2) The second main goal was to estimate the models' parameters using two different estimation methods, such as the maximum likelihood estimator (MLE) and Bayesian estimator, under different loss functions via Metropolis-Hastings (MH) algorithms. We conduct a detailed simulation study to demonstrate the behavior of derived estimators and pinpoint the most efficient estimation method.
(3) Two data sets from the industry field are utilized to illustrate the applicability and utilization of the proposed distribution.
The following is the organization of the study. Section 2 introduces the model description and the extension distribution, while Section 3 discusses various statistical properties such as moments, quantiles, and moment-generating functions. In Section 4, parameters are estimated using two different estimation methods. The performance of the EF-IW distribution using simulation is carried out and illustrated using three real industrial data sets in Sections 5 and 6, respectively. Finally, Section 7 presents the concluding remark of the paper.
Here, we provide the inverse Weibull distribution as a classical distribution. Plugging Eqs (1.3) and (1.4) into Eqs (1.1) and (1.2) gives the CDF and PDF of the new EF-IW model:
Ξ(z)=erf(e−θz−β1−e−θz−β),z,θ,β>0, | (2.1) |
and
ξ(z)=2θβ z−(β+1) e−θz−β√π(1−e−θz−β)2 exp{−(e−θz−β1−e−θz−β)2}, | (2.2) |
where erf(x)=2√π∫x0e−z2dz. The plots of the EF-IW PDF for some parameter values given in Figure 1 reveal that this function can be decreasing, unimodal, and skewed depending on the parameter values.
Suppose the random variable Z has a CDF denoted by Ξ(z). Then, its survival function (SF) and hazard rate function (HRF) can then be expressed as
S(z)=1−erf(e−θz−β1−e−θz−β), | (2.3) |
and
h(z)=2θβ z−(β+1) e−θz−β√π(1−e−θz−β)2[1−erf(t)] exp{−t}, | (2.4) |
with t=(e−θz−β1−e−θz−β)2.
Next, the cumulative hazard rate function (CHRF) and reversed hazard rate function (RHRF) of the random variable Z can be expressed as
H(z)=−log[1−erf(e−θz−β1−e−θz−β)], | (2.5) |
and
R(z)=2θβ z−(β+1) e−θz−β√π(1−e−θz−β)2 erf(e−θz−β1−e−θz−β). | (2.6) |
Figure 2 shows HRF plots of EF-IW for different sets of parameter values. It has increasing, unimodal, and decreasing shapes.
The quantile function Ξ−1(u) holds significant importance in simulation studies across various disciplines due to its ability to generate random variables with desired distribution characteristics.The quantile function of the new EF-IW model can be expressed as
Ξ−1(u)=[−1θlog(erf−1(u)1+erf−1(u))]−1/β,0≤u≤1, | (3.1) |
where erf−1(x)=Φ−1(x) is the standard normal quantile function.
Proof. By setting the Eq (2.1) equal u, we get
erf(e−θz−β1−e−θz−β)=u,e−θz−β1−e−θz−β=erf−1(u),e−θz−β(1+erf−1(u))=erf−1(u),e−θz−β=erf−1(u)1+erf−1(u),θz−β=−log(erf−1(u)1+erf−1(u)),z=[−1θlog(erf−1(u)1+erf−1(u))]−1/β. |
The quantile function can be used to compute the first, second, and third quantiles by replacing u with 14, 12, and 34.
Additionally, the Bowleys skewness (N) and Moors kurtosis (M) of the EF-IW model are described as
N=Ξ−1(1/4)+Ξ−1(3/4)−2Ξ−1(1/2)Ξ−1(3/4)−Ξ−1(1/4), |
and
M=Ξ−1(7/8)−Ξ−1(5/8)+Ξ−1(3/8)−Ξ−1(1/8)Ξ−1(6/8)−Ξ−1(2/8). |
In this part, we provide a series representation of the EF-IW CDF and PDF by employing the erf series, see Fernández and De Andrade [17] and Ajongba et al. [18],
erf(t)=2√(π)∞∑l=0(−1)lt2l+1l!(2l+1), |
and by applying the expansion
t1−t=∞∑j=0tj, |t|<1, |
the corresponding CDF of the EF-IW distribution can be rewritten as:
Ξ(z)=2√(π)∞∑l=0(−1)ll!(2l+1) [∞∑j=0 e−θz−β]2l+1. |
Now, consider the series expansion
[∞∑j=0ajtj]k=∞∑n=0Dk,n tn, |
where Dk,0=ak0 and Dk,n=1n a0n∑s=1 (sk−n+s) as Dk,n−s, n≥1.
Consequently, the EF-IW CDF takes the expression
Ξ(z)=2√π∞∑l=0∞∑n=0(−1)lD2l+1,nl!(2l+1) e−θ(n+2l+1)z−β=∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β, |
with Cl,n=2(−1)lD2l+1,n√πl!(2l+1), D2l+1,n=1nn∑s=1[2s(l+1)−n] D2l+1,n−s and D2l+1,0=1.
Similarly, the density of the recommended EF-IW model becomes
ξ(z)=θβ∞∑l=0∞∑n=0 Hl,n z−β−1 e−θ(n+2l)z−β, |
with Hl,n=Cl,n(n+2l+1).
One of the efficient statistical criteria that can calculate symmetry, spread-ness, and asymmetry is the ordinary moment. The r-th moment of the EF-IW distribution, whose PDF is given in Eq (2.2), can be determined as follows:
μ′r=θ∞∑l=0∞∑n=0Hl,n Γ(1−rβ)[θ(2l+n)]1−rβ, | (3.2) |
where Γ(.) represents the gamma function.
Thus, for r=1 and r=2, the mean (μ′1) and second moment (μ′2) of the EF-IW distribution are defined, respectively, as
μ′1=θ∞∑l=0∞∑n=0Hl,n Γ(1−1β)[θ(2l+n)]1−1β, |
and
μ′2=θ∞∑l=0∞∑n=0Hl,n Γ(1−2β)[θ(2l+n)]1−2β. |
The variance (Varz) with a corresponding coefficient of variation (CV) for the EF-IW model are obtained to be
VarZ=μ′2−μ′21, |
and
CV=VarZμ′1. |
Table 1 defined various proposed mathematical characteristics of the suggested EF-IW. In addition, Figure 3 shows the 3D plots of these statistical properties.
β | μ1 | VarZ | CV | N | M | |
θ=0.4 | 0.3 | 0.0874 | 0.0234 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.2287 | 0.0351 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.3488 | 0.0369 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.4412 | 0.034 | 0.4179 | 0.5665 | 0.0143 | |
θ=0.6 | 0.3 | 0.3375 | 0.3492 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.4495 | 0.1355 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.5473 | 0.0908 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.6186 | 0.0668 | 0.4179 | 0.5665 | 0.0143 | |
θ=0.8 | 0.3 | 0.8806 | 2.3770 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.7260 | 0.3535 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.7534 | 0.1721 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.7862 | 0.1080 | 0.4179 | 0.5665 | 0.0143 | |
θ=1.2 | 0.3 | 3.4022 | 35.479 | 1.7508 | 4.3227 | 30.106 |
0.6 | 1.4270 | 1.3659 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 1.1822 | 0.4238 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 1.1022 | 0.2122 | 0.4179 | 0.5665 | 0.0143 | |
θ=1.5 | 0.3 | 7.1581 | 157.05 | 1.7508 | 4.3227 | 30.106 |
0.6 | 2.0699 | 2.8737 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 1.5149 | 0.6958 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 1.3274 | 0.3078 | 0.4179 | 0.5665 | 0.0143 |
The moment generating function (MGF), M(t) of the KMIW model is derived as
M(t)=θ∞∑l=0∞∑n=0∞∑r=0Hl,n trr! Γ(1−rβ)[θ(2l+n)]1−rβ. |
The PDF of the rth-order statistics for a sample of size m taken from the EF-IW model is expressed as follows:
k(r)(z)=m!ξ(z)(r−1)!(m−r)![Ξ(z)]r−1[1−Ξ(z)]m−r=m!(r−1)!(m−r)! θβ∞∑l=0∞∑n=0 Hl,n z−β−1 e−θ(n+2l)z−β [∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β]r−1×[1−∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β]m−r. |
In a special case, the PDF of the minimum 1th and maximum mth order statistics of the EF-IW distribution can be given below as
k(1)(z)=mθβ∞∑l=0∞∑n=0 Hl,n z−β−1 e−θ(n+2l)z−β [1−∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β]1−r, |
and
k(m)(z)=mθβ∞∑l=0∞∑n=0 Hl,n z−β−1 e−θ(n+2l)z−β [∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β]m−1. |
The corresponding CDF of the EF-IW model can be written as
K(r)(z)=m∑k=0Ξk(z)[1−Ξ(z)]m−k=m∑k=0[∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β]k [1−∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β]m−k. |
In this part of the study, we estimate the models' parameters η=(β,θ) using two different estimation methods. For this purpose, the maximum likelihood and Bayesian estimators are the estimation methods used.
Assuming {z1,z2,…,zm} are the observed values of a random sample {Z1,Z2,…,Zm} from the EF-IW distribution with vector of parameters η=(β,θ), the log-likelihood function can be obtained to be
LL(z)=m∑i=1logξ(z)=m∑i=1log(2θβ z−(β+1) e−θz−β√π(1−e−θz−β)2 exp{−(e−θz−β1−e−θz−β)2})∝mlogθ+mlogβ−2m∑i=1log(1−e−θz−βi)−θm∑i=1z−βi−(β+1)m∑i=1logzi−m∑i=1(e−θz−βi1−e−θz−βi)2. | (4.1) |
With the vector of the parameters η=(β,θ), the corresponding partial derivatives of Eq (4.1) are obtained as:
∂LL(z;ϑ)∂θ=mθ−2m∑i=1z−βie−θz−βi1−e−θz−βi−m∑i=1z−βi+2[m∑i=1z−βi e−3θz−βi(1−e−θz−βi)3+m∑i=1z−βi e−2θz−βi(1−e−θz−βi)2], | (4.2) |
and
∂LL(z;ϑ)∂β=mβ−m∑i=1logz−βi−θm∑i=1z−βilogzi+2θ[m∑i=1z−βilogzi e−3θz−βi(1−e−θz−βi)3+m∑i=1z−βilogzi e−2θz−βi(1−e−θz−βi)2]. | (4.3) |
The parameter estimates for the parameters η=(β,θ) can be obtained by solving the above non-linear equations with respect to the parameters. It might be difficult to obtain a precise solution to the derived equations, and thus one option to optimize them is to use techniques like the Newton-Raphson algorithm. We used the R software's optimize function in this case.
We proceed based on the information available on the unknown parameters obtained from the opinions of the researchers. The interpretation of the informative prior is rarely precise enough to determine a single prior distribution. However, there are laws calibrated according to the distribution of observations, called the conjugate prior or the gamma prior. For more details, see Xu [28] and Zhuang [29]. Assuming that the unknown parameters β and θ are random variables that have a Gamma distribution with PDF expressed as
π1(θ)=ba11Γ(a1) θa1−1 e−b1θ, θ,a1,b1>0, |
and
π1(β)=ba22Γ(a2s) βa2−1 e−b2β, β,a2,b2>0. |
Henceforth, the joint prior PDF of η=(β,θ) can be derived as
π(ϑ)∝θa1−1 βa2−1 e−b1θ−b2β. |
Next, the joint posterior PDF of η=(β,θ) is
π∗(ϑ∣z)=L(ϑ))π(ϑ)∣z)∝θm+a1−1 βm+a2−1 eb1θ−b2βm∏i=1=z−(β+1)i e−θz−βi(1−e−θz−βi)2 exp{−(e−θz−βi1−e−θz−βi)2}. |
The Bayes estimates of the parametric function η=(β,θ) under the assumption of the square error loss function (BSE) is the posterior mean of η. The BSE is
ˆfSE=∫ηf π∗(ϑ∣z)dη. | (4.4) |
Now, the Bayes estimator under linear exponential loss function (BLI), can be written f=eδ(η−ˆη)−δ(η−ˆη). The BLI is
ˆfLI=−1δlog(∫ηe−δf π∗(η∣z)dη). | (4.5) |
In the end, the Bayes estimator under general entropy loss function (BGE), defined as f=(ˆηη)δ−δlog(ˆηη)−1, is
ˆfGE=(∫ηf−δ π∗(η∣z)dη)−1/δ, | (4.6) |
with δ≠0. It is difficult to obtain analytical expressions of Eqs (4.4)–(4.6). To solve this issue, we have considered the Metropolis Hasting (MH) algorithm for this purpose.
In this section, a detailed simulation study is carried out to examine the behavior of two derived estimators using the R software to evaluate the efficiency of the recommended estimators. The results are presented for various sample sizes m={30,60,80,100} from the proposed EF-IW distribution and several parameter values of η=(β,θ) (Set 1: (0.5, 0.75), Set 2: (0.8, 1.25), and Set 3: (1.2, 1.5)) to provide more accurate and comprehensive results. The Monte Carlo simulations are repeated 1000 times, and the estimates are assessed based on the mean estimate (AEs) and mean squared errors (MSEs). The empirical results are illustrated in Tables (2)–(4), and in this simulation, we choose δ=1.5 to compute the BLI and BGE. To check that the iterative non-linear method converges to the MLEs, we have applied the Newton Raphson technique with some other initial estimates, and it converges to the same set of estimates, which ensures that the estimates obtained via the suggested Newton Raphson method converges to the MLEs. The following conclusions are drawn from these tables.
m | MLE | BSE | BLI | BGE | |||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | ||
30 | θ | 0.4919 | 0.0039 | 0.4502 | 0.0035 | 0.4505 | 0.0037 | 0.4496 | 0.0039 |
β | 0.7802 | 0.0115 | 0.7496 | 0.0008 | 0.7497 | 0.0101 | 0.7494 | 0.0103 | |
60 | θ | 0.4928 | 0.0018 | 0.5093 | 0.0010 | 0.5093 | 0.0013 | 0.5085 | 0.0015 |
β | 0.7681 | 0.0051 | 0.7395 | 0.0007 | 0.7396 | 0.0009 | 0.7393 | 0.0101 | |
80 | θ | 0.5007 | 0.0011 | 0.4888 | 0.0006 | 0.4888 | 0.0008 | 0.4886 | 0.0010 |
β | 0.7586 | 0.0034 | 0.7882 | 0.0005 | 0.7883 | 0.0008 | 0.7880 | 0.0009 | |
100 | θ | 0.4943 | 0.0010 | 0.5091 | 0.0004 | 0.5093 | 0.0005 | 0.5087 | 0.0008 |
β | 0.7591 | 0.0024 | 0.7577 | 0.0004 | 0.7581 | 0.0006 | 0.7573 | 0.0008 |
m | MLE | BSE | BLI | BGE | |||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | ||
30 | θ | 0.7998 | 0.0055 | 0.8643 | 0.0039 | 0.8645 | 0.0041 | 0.4816 | 0.0043 |
β | 1.2981 | 0.0309 | 1.3373 | 0.0121 | 1.3384 | 0.0123 | 1.3364 | 0.0125 | |
60 | θ | 0.8001 | 0.0024 | 0.8358 | 0.0015 | 0.8359 | 0.0017 | 0.8357 | 0.0019 |
β | 1.2646 | 0.0148 | 1.1742 | 0.0075 | 1.1746 | 0.0078 | 1.1738 | 0.0079 | |
80 | θ | 0.7993 | 0.0023 | 0.7856 | 0.0014 | 0.7859 | 0.0016 | 0.7852 | 0.0018 |
β | 1.2641 | 0.0075 | 1.2241 | 0.0017 | 1.2244 | 0.0020 | 1.2239 | 0.0021 | |
100 | θ | 0.7948 | 0.0014 | 0.8046 | 0.0007 | 0.8048 | 0.0009 | 0.8044 | 0.0011 |
β | 1.2673 | 0.0072 | 1.2320 | 0.0011 | 1.2322 | 0.0013 | 1.2319 | 0.0015 |
m | MLE | BSE | BLI | BGE | ||||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | |||
30 | θ | 1.2240 | 0.0188 | 1.2206 | 0.0051 | 1.2217 | 0.0053 | 1.2196 | 0.0055 | |
β | 1.5799 | 0.0559 | 1.6200 | 0.0245 | 1.6225 | 0.0248 | 1.6184 | 0.0249 | ||
60 | θ | 1.2049 | 0.0065 | 1.2310 | 0.0032 | 1.2316 | 0.0034 | 1.2306 | 0.0036 | |
β | 1.5556 | 0.0215 | 1.5530 | 0.0058 | 1.5538 | 0.0061 | 1.5526 | 0.0062 | ||
80 | θ | 1.2035 | 0.0040 | 1.2276 | 0.0023 | 1.2280 | 0.0025 | 1.2272 | 0.0028 | |
β | 1.5524 | 0.0159 | 1.4679 | 0.0032 | 1.4685 | 0.0034 | 1.4676 | 0.0035 | ||
100 | θ | 1.2139 | 0.0041 | 1.2160 | 0.0019 | 1.2165 | 0.0022 | 1.2157 | 0.0024 | |
β | 1.5158 | 0.0113 | 1.4818 | 0.0022 | 1.4822 | 0.0023 | 1.4815 | 0.0025 |
(1) All estimation approaches produce estimates that converge toward the true parameter values as the sample size increases, which confirm that they are consistent and asymptotically unbiased.
(2) In most cases, the value of MSEs decreases as the value of m increases.
(3) As m increases, the Bayes estimates tends to perform efficiently based on MSE as an optimal criterion. On the contrary, BSE is more appropriate than BLI and BGE.
(4) Figure (4) ensures the same conclusion.
In this section, we utilized two data sets from the industrial field to show the EF-IW model introduced in Section 2. We demonstrate the flexibility of this new distribution by analyzing two real-world datasets drawn from industrial areas in the Kingdom of Saudi Arabia (KSA).
The data set represents the quarterly evolution of the number of foreign licenses in the construction sector in KSA. It was obtained from https://datasaudi.sa/en/sector/construction#real-sector-indicators. The values of the data set are summarized in Table (5).
8 | 6 | 8 | 16 | 23 | 20 | 28 | 40 |
43 | 50 | 54 | 32 | 52 | 29 | 33 | 42 |
41 | 52 | 56 | 79 | 155 | 84 | 95 | 111 |
136 | 161 | 204 | 241 |
The second application introduced the scale efficiency of the construction industry in KSA between 2013 and 2022. The suggested data set was considered by Yu et al. [30], and the values are presented in Table (6).
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
Mecca | 9.39 | 9.71 | 9.83 | 9.96 | 9.97 | 9.95 | 9.98 | 9.97 | 10.005 | 9.96 |
Eastern | 8.92 | 9.23 | 9.43 | 9.56 | 9.58 | 9.71 | 9.78 | 9.72 | 9.82 | 9.87 |
Al-Madinah | 7.46 | 7.47 | 7.81 | 8.52 | 8.62 | 8.61 | 8.73 | 8.43 | 8.74 | 8.77 |
Jizan | 6.66 | 6.69 | 6.84 | 7.64 | 7.71 | 7.75 | 7.75 | 7.68 | 7.82 | 7.82 |
Al-Qassim | 6.6 | 6.62 | 6.67 | 7.47 | 7.51 | 7.53 | 7.67 | 7.6 | 7.73 | 7.73 |
Tabuk | 5.31 | 5.46 | 5.66 | 6.41 | 6.54 | 6.52 | 6.54 | 6.43 | 6.67 | 6.6 |
Ha'il | 4.23 | 4.27 | 4.29 | 5.31 | 5.47 | 5.47 | 5.59 | 5.14 | 5.62 | 5.72 |
This recommended data set is about the efficiency of the pure technical construction industry between 2013 and 2022 in the KSA. The proposed data was considered by Yu et al. [30], and its records can be reported in Table 7.
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
Eastern | 2.55 | 3.90 | 4.59 | 6.37 | 7.11 | 7.38 | 7.55 | 7.17 | 7.89 | 8.54 |
Al Madinah | 3.26 | 3.46 | 3.47 | 4.99 | 6.38 | 6.42 | 6.81 | 6.16 | 6.77 | 7.21 |
Asir | 3.41 | 3.81 | 3.98 | 4.65 | 5.47 | 5.74 | 5.92 | 6.17 | 6.13 | 6.53 |
Jizan | 3.42 | 3.39 | 3.62 | 4.46 | 5.37 | 5.71 | 5.56 | 5.49 | 5.64 | 5.80 |
Al-Qassim | 3.43 | 3.45 | 3.37 | 4.11 | 4.46 | 4.81 | 5.10 | 5.07 | 5.24 | 5.45 |
Tabuk | 2.99 | 2.78 | 2.96 | 3.96 | 4.48 | 4.96 | 4.82 | 4.75 | 4.89 | 5.13 |
Ha'il | 2.89 | 2.59 | 2.73 | 3.59 | 4.19 | 4.59 | 4.52 | 4.50 | 4.70 | 4.75 |
Al Jawf | 2.29 | 2.75 | 2.48 | 3.35 | 4.22 | 4.42 | 4.55 | 4.44 | 4.63 | 4.71 |
Najran | 2.83 | 2.92 | 2.62 | 3.33 | 4.02 | 4.38 | 4.47 | 4.44 | 4.61 | 4.8 |
Northern Borders | 1.51 | 1.51 | 1.6 | 2.79 | 3.95 | 4.04 | 3.99 | 4.08 | 4.4 | 4.48 |
Table (8) presents a statistical summary of the three data sets. Furthermore, Figure 5 shows several significant plots (scaled total time on test (TTT), quantile-quantile (Q-Q), and box plots) derived from the three industrial datasets. These plots help analyze the historical performance of the industrial sectors.
Data | Q1 | Q2 | μ′1 | Q3 | CV | N | M |
1 | 28.75 | 46.50 | 67.82 | 27.60 | 54.57 | 1.31 | 0.83 |
2 | 6.55 | 7.72 | 7.67 | 9.35 | 0.35 | -0.18 | -0.96 |
3 | 3.445 | 4.475 | 4.521 | 5.39 | 0.35 | 0.3589 | -0.0846 |
Additionally, we would like to select the more appropriate fitting model for the two proposed data sets. We consider several renowned competitive probability distributions to compare with the results of EF-IW, including the inverse Weibull (IW), error function Weibull (EF-W), error function exponential (EF-E), power Burr X (PBX), and generalized exponential (GE) models.
Akaike information criterion (A), Bayesian information criterion (B), Hannan-Quin information criterion (C), correction Akaike information criterion (D), Kolmogorov-Smirnov (KS) statistics with its associated P-values are considered when comparing the model and recommending the best model. By calculating and comparing the proposed measures, we gain a clear understanding of the relative performance of each model. Models with the lowest values for these statistics will be considered for the best fit of the given data set. This approach reflects the strengths of the new distribution in terms of its suitability for different data structures and ensures that the model selection process takes into account both the complexity of the model and the goodness of fit across multiple aspects of the data distribution. Table (9) summarizes the final estimates of the unknown parameters with their corresponding log-likelihood (LL). Consequently, the recommended EF-IW model emerges as the most favorable distribution for modeling the three data sets. Henceforth, the empirical v.s. the fitted (PDF and CDF) plots for the proposed model with its competitors are generated and reported in Figures (6)–(8) using the two data sets. These visual plots demonstrate that the EF-IW distribution works well with the three data sets.
Data | Model | ˆθ | ˆβ | KS | P-value | LL | A | B | C | D |
EF-IW | 10.609 | 0.5686 | 0.1425 | 0.6197 | -144.601 | 293.202 | 295.866 | 294.016 | 293.682 | |
IW | 8.5601 | 0.6934 | 0.2135 | 0.1557 | -152.624 | 309.269 | 311.934 | 300.958 | 300.624 | |
1 | EF-W | 224.821 | 0.7022 | 0.1961 | 0.2316 | -147.318 | 298.637 | 301.301 | 299.451 | 299.117 |
EF-E | 0.0047 | 0.3475 | 0.0023 | -149.955 | 301.911 | 303.243 | 302.318 | 302.064 | ||
PBX | 0.0771 | 0.5983 | 0.1575 | 0.4901 | -145.356 | 294.713 | 297.378 | 295.603 | 295.269 | |
GE | 0.0188 | 1.4887 | 0.1495 | 0.5582 | -145.041 | 294.082 | 296.746 | 294.897 | 294.562 | |
EF-IW | 169.011 | 2.4633 | 0.1114 | 0.3500 | -132.473 | 268.947 | 273.444 | 270.734 | 269.126 | |
IW | 2168.36 | 4.0579 | 0.1697 | 0.0354 | -146.885 | 297.771 | 302.268 | 299.557 | 297.950 | |
2 | EF-W | 10.415 | 3.6140 | 0.1468 | 0.0978 | -132.736 | 269.472 | 273.969 | 271.259 | 269.651 |
EF-E | 0.0709 | 0.4251 | <0.0001 | -179.247 | 360.494 | 362.743 | 361.387 | 360.553 | ||
PBX | 0.0407 | 1.5378 | 0.2102 | 0.0041 | -146.950 | 297.936 | 302.433 | 299.723 | 298.115 | |
GE | 0.6226 | 69.883 | 0.1399 | 0.1290 | -139.298 | 282.596 | 287.093 | 284.382 | 282.775 | |
EF-IW | 11.193 | 1.5481 | 0.0756 | 0.6168 | -177.432 | 358.864 | 364.074 | 360.973 | 358.988 | |
IW | 25.075 | 2.5284 | 0.1395 | 0.0408 | -198.317 | 400.634 | 405.845 | 402.743 | 400.758 | |
3 | EF-W | 1.9167 | 7.5766 | 0.1301 | 0.0676 | -185.737 | 375.475 | 380.686 | 377.584 | 375.599 |
EF-E | 0.1131 | 0.3139 | <0.0001 | -210.347 | 422.694 | 425.300 | 423.749 | 422.735 | ||
PBX | 0.0639 | 1.7021 | 0.0909 | 0.3791 | -177.673 | 359.347 | 364.557 | 361.455 | 359.470 | |
GE | 0.7435 | 16.358 | 0.1060 | 0.2107 | -180.239 | 364.478 | 69.688 | 366.587 | 364.602 |
Finally, the estimates of the model parameters using the Bayesian technique under several loss functions of the EF-IW distribution by applying the three data sets are computed and reported in Table (10). Also, Figures (9)–(11) show the histogram and trace plots of MH results.
Data | Par | Bayes | ||
BSE | BLI | BGE | ||
1 | θ | 10.468 | 10.470 | 10.468 |
β | 0.5765 | 0.5766 | 0.5764 | |
2 | θ | 168.989 | 168.990 | 168.989 |
β | 2.457 | 2.457 | 2.457 | |
3 | θ | 11.692 | 11.689 | 11.694 |
β | 1.5635 | 1.5637 | 1.5634 |
This study introduces a new probability distribution, and its mathematical properties are thoroughly explored. The new model is named the error function inverse Weibull distribution. The model parameters are estimated using two different estimation methods, and extensive simulation studies are conducted to identify the most efficient estimation technique. To demonstrate the versatility and practical usefulness of the EF-IW distribution, the new distribution is applied to three datasets, demonstrating its ability to adapt to varied data properties. The findings of these applications show that the EF-IW distribution surpasses considered competitive probability distributions previously studied in the literature, giving more accurate and efficient outcomes in terms of fit and prediction. These findings show the novel distribution's potential as a robust tool for modeling data across several domains, providing a promising alternative to established models.
Future work on the EF-IW distribution may include expanding modifications, estimation, and applications. Some potential directions include the following
(1) New extended forms of the EF-IW distribution can be proposed, such as truncation, zero-inflation, and Neutrosophic extension for imprecise datasets.
(2) The progressive censoring type may also be used to obtain the model parameter estimations.
(3) Future studies should focus on the utilization of the EF-IW distribution to handle ranked set sampling data, which is frequently seen in survival and reliability analysis studies. Enhancing the distribution applicability and usefulness will require developing parameter estimation approaches for censored and uncensored data with a cure fraction.
All authors contributed equally to this paper. Badr Aloraini and Abdulaziz S. Alghamdi did the writing and mathematics, Mohammad Zaid Alaskar and Maryam Ibrahim Habadi did the revising, editing, and validating.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446).
All authors declare no conflicts of interest in this paper.
[1] |
L. Atzori, A. Iera, G. Morabito, The internet of things: A survey, Comp. Netw., 54 (2010), 2787-2805. doi: 10.1016/j.comnet.2010.05.010
![]() |
[2] | P. Mell, T. Grance, The nist definition of cloud computing, 2011. |
[3] | P. Wang, R. X. Gao, Z. Fan, Cloud computing for cloud manufacturing: Benefits and limitations, J. Manufac. Sci. Eng., 137 (2015), 1-9. |
[4] | B. Li, L. Zhang, S. Wang, F. Tao, J. W. Cao, X. D. Jiang, et al., Cloud manufacturing: A new service-oriented networked manufacturing model, Compu. Inte. Manufac. Sys., 16 (2010), 1-7. |
[5] |
X. Xu, From cloud computing to cloud manufacturing, Compu. Inte. Manufac. Sys., 28 (2012), 75-86. doi: 10.1016/j.rcim.2011.07.002
![]() |
[6] | Y. Yang, Y. D. Cai, Q. Lu, Y. Zhang, S. Koric, C. Shao, High-performance computing based big data analytics for smart manufacturing, In: ASME 2018 13th International Manufacturing Science and Engineering Conference. American Society of Mechanical Engineers Digital Collection, (2018). |
[7] | L. Wang, X. V. Wang, Cloud-based cyber-physical systems in manufacturing, 1st edition, SpringerVerlag, 2018. |
[8] |
Y. Liu, L. Wang, X. Wang, X. Xu, P. Jiang, Cloud manufacturing: key issues and future perspectives, Int. J. Compu. Inte. Manufac., 32 (2019), 858-874. doi: 10.1080/0951192X.2019.1639217
![]() |
[9] |
Y. Liu, L. Wang, X. V. Wang, Cloud manufacturing: Latest advancements and future trends, Proc. Manufac., 25 (2018), 62-73. doi: 10.1016/j.promfg.2018.06.058
![]() |
[10] |
D. Wu, D. W. Rosen, L. Wang, D. Schaefer, Cloud-based design and manufacturing: A new paradigm in digital manufacturing and design innovation, Compu. Aided Des., 59 (2015), 1-14. doi: 10.1016/j.cad.2014.07.006
![]() |
[11] |
Y. Liu, L. Wang, X. V. Wang, X. Xu, L. Zhang, Scheduling in cloud manufacturing: State-of-theart and research challenges, Cinter. J. Prod. Res., 57 (2019), 4854-4879. doi: 10.1080/00207543.2018.1449978
![]() |
[12] | L. Monostori, Cyber-physical production systems: Roots, expectations and r & d challenges, Proc. CIRP, 17 (2019), 9-13. |
[13] |
D. A. Rossit, F. Tohmé, M. Frutos, Industry 4.0: Smart scheduling, Int. J. Prod. Res., 57 (2019), 3802-3813. doi: 10.1080/00207543.2018.1504248
![]() |
[14] | J. Lee, B. Bagheri, H. Kao, A cyber-physical systems architecture for industry 4.0-based manufacturing systems, Manufac. Let., 3 (2018), 18-23. |
[15] | J.Wang, L. Zhang, L. Duan, R. X. Gao, A new paradigm of cloud-based predictive maintenance for intelligent manufacturing, J. Intel. Manufac., 28 (2019), 1125-1137. |
[16] |
Y. Zhang, Y. Cheng, X. V. Wang, R. Y. Zhong, Y. Zhang, F. Tao, Data-driven smart production line and its common factors, Int. J. Adv. Manufac. Tech., 103 (2019), 1211-1223. doi: 10.1007/s00170-019-03469-9
![]() |
[17] |
D. A. Rossit, F. Tohmé, M. Frutos, Production planning and scheduling in cyber-physical produc-tion systems: A review, Int. J. Compu. Inte. Manufac., 32 (2019), 385-395. doi: 10.1080/0951192X.2019.1605199
![]() |
[18] |
J. Wang, K. Wang, Y. Wang, Z. Huang, R. Xue, Deep boltzmann machine based condition prediction for smart manufacturing, J. Amb. Intel. Hum. Compu., 10 (2019), 851-861. doi: 10.1007/s12652-018-0794-3
![]() |
[19] |
J. K. Lenstra, A. R. Kan, P. Brucker, Complexity of machine scheduling problems, An. Dis. Math., 1 (1977), 343-362. doi: 10.1016/S0167-5060(08)70743-X
![]() |
[20] | M. Pinedo, Scheduling, 5th edition, Springer-Verlag, 2016. |
[21] |
A. Dolgui, D. Ivanov, S. P Sethi, B. Sokolov, Scheduling in production, supply chain and industry 4.0 systems by optimal control: Fundamentals, state-of-the-art and applications, Int. J. Prod. Res., 57 (2019), 411-432. doi: 10.1080/00207543.2018.1442948
![]() |
[22] | D. A. Rossit, F. Tohmé, M. Frutos, A data-driven scheduling approach to smart manufacturing, J. Indus. Infor. Int., 15 (2019), 69-79. |
[23] |
D. A. Rossit, F. Tohmé., Scheduling research contributions to smart manufacturing, Manufac. Let., 15 (2018), 111-114. doi: 10.1016/j.mfglet.2017.12.005
![]() |
[24] |
H. Akbaripour, M. Houshmand, T. VanWoensel, N. Mutlu, Cloud manufacturing service selection optimization and scheduling with transportation considerations: Mixed-integer programming models, Int. J. Adv. Manufac. Tech., 95 (2018), 43-70. doi: 10.1007/s00170-017-1167-3
![]() |
[25] | Y. Liu, L. Zhang, L. Wang, Y. Xiao, X. Xu, M. Wang, A framework for scheduling in cloud manufacturing with deep reinforcement learning, in 2019 IEEE 17th International Conference on Industrial Informatics (INDIN), 1 (2019), 1775-1780. |
[26] | S. Lin, Y. Laili, Y. Luo, Integrated optimization of supplier selection and service scheduling in cloud manufacturing environment, in 2018 4th International Conference on Universal Village, (2018), 1-6. |
[27] |
H. Zhu, M. Li, Y. Tang, Y. Sun, A deep-reinforcement-learning-based optimization approach for real-time scheduling in cloud manufacturing, IEEE Access, 8 (2020), 9987-9997. doi: 10.1109/ACCESS.2020.2964955
![]() |
[28] | M. Petticrew, H. Roberts, Systematic reviews in the social sciences: A practical guide, John Wiley & Sons, (2008). |
[29] | R. B. Briner, D. Denyer, Systematic review and evidence synthesis as a practice and scholarship too, Handb. Evid. Manag. Comp. Class. Res., (2012), 112-129. |
[30] | D. Denyer, D. Tranfield, Producing a systematic review, (2009). |
[31] |
J. Delaram, O. F. Valila, A mathematical model for task scheduling in cloud manufacturing systems focusing on global logistics, Proc. Manufact., 17 (2018), 387-394. doi: 10.1016/j.promfg.2018.10.061
![]() |
[32] | T. Suma, R. Murugesan, Study on multi-task oriented service composition and optimization problem of customer order scheduling problem using fuzzy min-max algorithm, Int. J. Mecha. Eng. Tech., 10 (2019), 219-231. |
[33] |
B. Vahedi-Nouri, R. Tavakkoli-Moghaddam, M. Rohaninejad, A multi-objective scheduling model for a cloud manufacturing system with pricing, equity, and order rejection, IFAC-Paper, 52 (2019), 2177-2182. doi: 10.1016/j.ifacol.2019.11.528
![]() |
[34] |
L. Zhang, C. Yu, T. N. Wong, Cloud-based frameworks for the integrated process planning and scheduling, Int. J. Compu. Inte. Manufac., 32 (2019), 1192-1206. doi: 10.1080/0951192X.2019.1690682
![]() |
[35] | D. Wang, Y. Yu, Y. Yin, T. C. E. Cheng, Multi-agent scheduling problems under multitasking, Int. J. Produc. Res., (2020), 1-31. |
[36] |
Y. Liu, L. Wang, Y. Wang, X. V. Wang, L. Zhang, Multi-agent-based scheduling in cloud manufacturing with dynamic task arrivals, Proc. CIRP, 72 (2018), 953-960. doi: 10.1016/j.procir.2018.03.138
![]() |
[37] |
J. Xiao, W. Zhang, S. Zhang, X. Zhuang, Game theory-based multi-task scheduling in cloud manufacturing using an extended biogeography-based optimization algorithm, Concur. Eng., 27 (2019), 314-330. doi: 10.1177/1063293X19882744
![]() |
[38] | J. Chen, G. Q Huang, J. Wang, C. Yang, A cooperative approach to service booking and scheduling in cloud manufacturing, Eur. J. Oper. Res., 273(3) (2019), 861-873. |
[39] | Z. Liu, Z. Wang, C. Yang, Multi-objective resource optimization scheduling based on iterative double auction in cloud manufacturing, Adv. Manufac., 7(4) (2019), 374-388. |
[40] | T. Bai, S. Liu, L. Zhang, A manufacturing task scheduling method based on public goods game on cloud manufacturing model, in 2018 4th International Conference on Universal Village (UV), (2018), 1-6. |
[41] | Z. Liu, Z.Wang, A novel truthful and fair resource bidding mechanism for cloud manufacturing, IEEE Access, 8 (2019), 28888-28901. |
[42] |
L. Zhou, L. Zhang, Y. Laili, C. Zhao, Y. Xiao, Multi-task scheduling of distributed 3d printing services in cloud manufacturing, Int. J. Adv. Manufac. Tech., 96 (2018), 3003-3017. doi: 10.1007/s00170-017-1543-z
![]() |
[43] |
A. Simeone, A. Caggiano, B. N. Deng, Y. Zeng, L. Boun, Resource efficiency optimization engine in smart production networks via intelligent cloud manufacturing platforms, Proc. CIRP, 78 (2018), 19-24. doi: 10.1016/j.procir.2018.10.003
![]() |
[44] |
P. Helo, D. Phuong, Y. Hao, Cloud manufacturing-scheduling as a service for sheet metal manufacturing, Comp. Oper. Res., 110 (2019), 208-219. doi: 10.1016/j.cor.2018.06.002
![]() |
[45] | T. Suma, R. Murugesan, Artificial immune algorithm for subtask industrial robot scheduling in cloud manufacturing, In J. Phys. Conf. Ser, 1000 (2018), 1-8. |
[46] |
L. Zhou, L. Zhang, C. Zhao, Y. Laili, L. Xu, Diverse task scheduling for individualized requirements in cloud manufacturing, Enter. Infor. Sys., 12 (2018), 300-318. doi: 10.1080/17517575.2017.1364428
![]() |
[47] | M. Yuan, X. Cai, Z. Zhou, C. Sun, W. Gu, J. Huang, Dynamic service resources scheduling method in cloud manufacturing environment, Int. J. Produc. Res., 11 (2019), 1-18. |
[48] |
W. He, G. Jia, H. Zong, J. Kong, Multi-objective service selection and scheduling with linguistic preference in cloud manufacturing, Sustainability, 11 (2019), 2619. doi: 10.3390/su11092619
![]() |
[49] | E. Jafarnejad-Ghomi, A. M. Rahmani, N. N. Qader, Service load balancing, scheduling, and logistics optimization in cloud manufacturing by using genetic algorithm, Concu. Compu. Prac. Exper., 31 (2019), 5329. |
[50] |
Y. Hu, F. Zhu, L. Zhang, Y. Lui, Z. Wang, Scheduling of manufacturers based on chaos optimization algorithm in cloud manufacturing, Robo. Comp. Inte. Manufac., 58 (2019), 13-20. doi: 10.1016/j.rcim.2019.01.010
![]() |
[51] | F. Zhang, J. Hui, B. Zhu, Y. Guo, An improved firefly algorithm for collaborative manufacturing chain optimization problem, in Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 233 (2019), 1711-1722. |
[52] |
W. Zhang, J. Ding, Y. Wang, S. Zhang, Z. Xiong, Multi-perspective collaborative scheduling using extended genetic algorithm with interval-valued intuitionistic fuzzy entropy weight method, J. Manufac. Sys., 53 (2019), 249-260. doi: 10.1016/j.jmsy.2019.10.002
![]() |
[53] |
A. Elgendy, J. Yan, M. Zhang, Integrated strategies to an improved genetic algorithm for allocating and scheduling multi-task in cloud manufacturing environment, Proc. Manufac., 39 (2019), 1872- 1879. doi: 10.1016/j.promfg.2020.01.251
![]() |
[54] | Y. Du, J. L.Wang, L. Lei, Multi-objective scheduling of cloud manufacturing resources through the integration of cat swarm optimization and firefly algorithm, Adv. Prod. Eng. Manag., 14 (2019). |
[55] | H. Zhang, C. Ma, S. Zhang, S. Liu, Research on the fjss problem with discrete equipment capability in cloud manufacturing environment, Int. J. Inter. Manufac. Ser., 6 (2019), 123-138. |
[56] |
F. Li, L. Zhang, T. W. Liao, Y. Liu, Multi-objective optimisation of multi-task scheduling in cloud manufacturing, Int. J. Prod. Res., 57 (2019), 3847-3863. doi: 10.1080/00207543.2018.1538579
![]() |
[57] |
E. Jafarnejad-Ghomi, A. M. Rahmani, N. N. Qader, Service load balancing, task scheduling and transportation optimisation in cloud manufacturing by applying queuing system, Enter. Infor. Sys., 13 (2019), 865-894. doi: 10.1080/17517575.2019.1599448
![]() |
[58] | Y. Li, G. Luo, Solving flexible job shop scheduling problem in cloud manufacturing environment based on improved genetic algorithm, in IOP Conference Series: Materials Science and Engineering, 612 (2019). |
[59] | Y. Shi, L. Luo, H. Guang, Research on scheduling of cloud manufacturing resources based on bat algorithm and cellular automata, in 2019 IEEE International Conference on Smart Manufacturing, Industrial & Logistics Engineering (SMILE), (2019), 174-177. |
[60] | Y. Laili, S. Lin, D. Tang, Multi-phase integrated scheduling of hybrid tasks in cloud manufacturing environment, Rob. Compu.Inte. Manufac., 61 (2020). |
[61] |
M. M. Fazeli, Y. Farjami, M. Nickray, An ensemble optimisation approach to service composition in cloud manufacturing, Int. J. Comp. Int. Manufac., 32 (2019), 83-91. doi: 10.1080/0951192X.2018.1550679
![]() |
[62] |
J. Ding, Y. Wang, S. Zhang, W. Zhang, Z. Xiong, Robust and stable multi-task manufacturing scheduling with uncertainties using a two-stage extended genetic algorithm, Enter. Infor. Systems, 13 (2019), 1442-1470. doi: 10.1080/17517575.2019.1656290
![]() |
[63] | F. Li, W. Liao, W. Cai, L. Zhang, Multi-task scheduling in consideration of fuzzy uncertainty of multiple criteria in service-oriented manufacturing, IEEE Trans. Fuz. Sys., (2020). |
[64] |
S. Chen, S. Fang, R. Tang, A reinforcement learning based approach for multi-projects scheduling in cloud manufacturing, Int. J. Prod. Res., 57 (2019), 3080-3098. doi: 10.1080/00207543.2018.1535205
![]() |
[65] | T. Dong, F. Xue, C. Xiao, J. Li, Task scheduling based on deep reinforcement learning in a cloud manufacturing environment, Concur. Comp. Prac. Exper., 32 (2020), e5654. |
[66] | C. Morariu, O. Morariu, S. Raileanu, T. Borangiu, Machine learning for predictive scheduling and resource allocation in large scale manufacturing systems, Compu. Indus., 120 (2020), e5654. |
[67] | L. Zhou, L. Zhang, L. Ren, Simulation model of dynamic service scheduling in cloud manufacturing, in IECON 2018-44th Annual Conference of the IEEE Industrial Electronics Society, (2018), 4199-4204. |
[68] |
L. Zhou, L. Zhang, B. R. Sarker, Y. Laili, L. Ren, An event-triggered dynamic scheduling method for randomly arriving tasks in cloud manufacturing, Int. J. Compu. Inte. Manufac., 31 (2018), 318- 333. doi: 10.1080/0951192X.2017.1413252
![]() |
[69] | W. He, G. Jia, H. Zong, T. Huang, Multi-objective cloud manufacturing service selection and scheduling with different objective priorities, Sustainability, 11 (2019). |
[70] |
Y. Wang, P. Zheng, X. Xu, H. Yang, J. Zou, Production planning for cloud-based additive manufacturing-a computer vision-based approach, Robo. Compu. Inte. Manufac., 58 (2019), 145-157. doi: 10.1016/j.rcim.2019.03.003
![]() |
[71] | L. Zhou, L. Zhang, Y. Fang, Logistics service scheduling with manufacturing provider selection in cloud manufacturing, Robo. Compu. Inte. Manufac., 65 (2020). |
[72] |
J.Wang, Y. Ma, L. Zhang, R. X. Gao, D.Wu, Deep learning for smart manufacturing: Methods and applications, J. Manufac. Sys., 48 (2018), 144-156. doi: 10.1016/j.jmsy.2018.01.003
![]() |
[73] | K. Deb, Multi-objective optimization using evolutionary algorithms, John Wiley & Sons (2001). |
[74] |
G. E. Vieira, J. W. Herrmann, E. Lin, Rescheduling manufacturing systems: A framework of strategies, policies, and methods, J. Schedu., 6 (2003), 39-62. doi: 10.1023/A:1022235519958
![]() |
[75] | F. Bonomi, R. Milito, J. Zhu, S. Addepalli, Fog computing and its role in the internet of things, in Proceedings of the first edition of the MCC workshop on Mobile cloud computing, (2012), 13-16. |
[76] | S. Yi, C. Li, Q. Li, A survey of fog computing: concepts, applications and issues, in Proceedings of the 2015 workshop on mobile big data, (2015), 37-42. |
[77] | F. Al-Haidari, M. Sqalli, K. Salah, Impact of cpu utilization thresholds and scaling size on autoscaling cloud resources, in 2013 IEEE 5th International Conference on Cloud Computing Technology and Science, 2 (2013), 256-261. |
[78] | K. Salah, J. M. A. Calero, S. Zeadally, S. Al-Mulla, M. Alzaabi, Using cloud computing to implement a security overlay network, IEEE Secu. Pri., 11 (2012), 44-53. |
[79] | C. Xu, G. Zhu, Intelligent manufacturing lie group machine learning: Real-time and efficient inspection system based on fog computing, J. Intel. Manufac., 11 (2020), 1-13. |
[80] | K. Salah, A queueing model to achieve proper elasticity for cloud cluster jobs, in 2013 IEEE Sixth International Conference on Cloud Computing, (2013), 755-761. |
[81] |
S. El-Kafhali, K. Salah, Efficient and dynamic scaling of fog nodes for iot devices, J. Supercomp., 73 (2017), 5261-5284. doi: 10.1007/s11227-017-2083-x
![]() |
β | μ1 | VarZ | CV | N | M | |
θ=0.4 | 0.3 | 0.0874 | 0.0234 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.2287 | 0.0351 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.3488 | 0.0369 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.4412 | 0.034 | 0.4179 | 0.5665 | 0.0143 | |
θ=0.6 | 0.3 | 0.3375 | 0.3492 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.4495 | 0.1355 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.5473 | 0.0908 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.6186 | 0.0668 | 0.4179 | 0.5665 | 0.0143 | |
θ=0.8 | 0.3 | 0.8806 | 2.3770 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.7260 | 0.3535 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.7534 | 0.1721 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.7862 | 0.1080 | 0.4179 | 0.5665 | 0.0143 | |
θ=1.2 | 0.3 | 3.4022 | 35.479 | 1.7508 | 4.3227 | 30.106 |
0.6 | 1.4270 | 1.3659 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 1.1822 | 0.4238 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 1.1022 | 0.2122 | 0.4179 | 0.5665 | 0.0143 | |
θ=1.5 | 0.3 | 7.1581 | 157.05 | 1.7508 | 4.3227 | 30.106 |
0.6 | 2.0699 | 2.8737 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 1.5149 | 0.6958 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 1.3274 | 0.3078 | 0.4179 | 0.5665 | 0.0143 |
m | MLE | BSE | BLI | BGE | |||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | ||
30 | θ | 0.4919 | 0.0039 | 0.4502 | 0.0035 | 0.4505 | 0.0037 | 0.4496 | 0.0039 |
β | 0.7802 | 0.0115 | 0.7496 | 0.0008 | 0.7497 | 0.0101 | 0.7494 | 0.0103 | |
60 | θ | 0.4928 | 0.0018 | 0.5093 | 0.0010 | 0.5093 | 0.0013 | 0.5085 | 0.0015 |
β | 0.7681 | 0.0051 | 0.7395 | 0.0007 | 0.7396 | 0.0009 | 0.7393 | 0.0101 | |
80 | θ | 0.5007 | 0.0011 | 0.4888 | 0.0006 | 0.4888 | 0.0008 | 0.4886 | 0.0010 |
β | 0.7586 | 0.0034 | 0.7882 | 0.0005 | 0.7883 | 0.0008 | 0.7880 | 0.0009 | |
100 | θ | 0.4943 | 0.0010 | 0.5091 | 0.0004 | 0.5093 | 0.0005 | 0.5087 | 0.0008 |
β | 0.7591 | 0.0024 | 0.7577 | 0.0004 | 0.7581 | 0.0006 | 0.7573 | 0.0008 |
m | MLE | BSE | BLI | BGE | |||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | ||
30 | θ | 0.7998 | 0.0055 | 0.8643 | 0.0039 | 0.8645 | 0.0041 | 0.4816 | 0.0043 |
β | 1.2981 | 0.0309 | 1.3373 | 0.0121 | 1.3384 | 0.0123 | 1.3364 | 0.0125 | |
60 | θ | 0.8001 | 0.0024 | 0.8358 | 0.0015 | 0.8359 | 0.0017 | 0.8357 | 0.0019 |
β | 1.2646 | 0.0148 | 1.1742 | 0.0075 | 1.1746 | 0.0078 | 1.1738 | 0.0079 | |
80 | θ | 0.7993 | 0.0023 | 0.7856 | 0.0014 | 0.7859 | 0.0016 | 0.7852 | 0.0018 |
β | 1.2641 | 0.0075 | 1.2241 | 0.0017 | 1.2244 | 0.0020 | 1.2239 | 0.0021 | |
100 | θ | 0.7948 | 0.0014 | 0.8046 | 0.0007 | 0.8048 | 0.0009 | 0.8044 | 0.0011 |
β | 1.2673 | 0.0072 | 1.2320 | 0.0011 | 1.2322 | 0.0013 | 1.2319 | 0.0015 |
m | MLE | BSE | BLI | BGE | ||||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | |||
30 | θ | 1.2240 | 0.0188 | 1.2206 | 0.0051 | 1.2217 | 0.0053 | 1.2196 | 0.0055 | |
β | 1.5799 | 0.0559 | 1.6200 | 0.0245 | 1.6225 | 0.0248 | 1.6184 | 0.0249 | ||
60 | θ | 1.2049 | 0.0065 | 1.2310 | 0.0032 | 1.2316 | 0.0034 | 1.2306 | 0.0036 | |
β | 1.5556 | 0.0215 | 1.5530 | 0.0058 | 1.5538 | 0.0061 | 1.5526 | 0.0062 | ||
80 | θ | 1.2035 | 0.0040 | 1.2276 | 0.0023 | 1.2280 | 0.0025 | 1.2272 | 0.0028 | |
β | 1.5524 | 0.0159 | 1.4679 | 0.0032 | 1.4685 | 0.0034 | 1.4676 | 0.0035 | ||
100 | θ | 1.2139 | 0.0041 | 1.2160 | 0.0019 | 1.2165 | 0.0022 | 1.2157 | 0.0024 | |
β | 1.5158 | 0.0113 | 1.4818 | 0.0022 | 1.4822 | 0.0023 | 1.4815 | 0.0025 |
8 | 6 | 8 | 16 | 23 | 20 | 28 | 40 |
43 | 50 | 54 | 32 | 52 | 29 | 33 | 42 |
41 | 52 | 56 | 79 | 155 | 84 | 95 | 111 |
136 | 161 | 204 | 241 |
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
Mecca | 9.39 | 9.71 | 9.83 | 9.96 | 9.97 | 9.95 | 9.98 | 9.97 | 10.005 | 9.96 |
Eastern | 8.92 | 9.23 | 9.43 | 9.56 | 9.58 | 9.71 | 9.78 | 9.72 | 9.82 | 9.87 |
Al-Madinah | 7.46 | 7.47 | 7.81 | 8.52 | 8.62 | 8.61 | 8.73 | 8.43 | 8.74 | 8.77 |
Jizan | 6.66 | 6.69 | 6.84 | 7.64 | 7.71 | 7.75 | 7.75 | 7.68 | 7.82 | 7.82 |
Al-Qassim | 6.6 | 6.62 | 6.67 | 7.47 | 7.51 | 7.53 | 7.67 | 7.6 | 7.73 | 7.73 |
Tabuk | 5.31 | 5.46 | 5.66 | 6.41 | 6.54 | 6.52 | 6.54 | 6.43 | 6.67 | 6.6 |
Ha'il | 4.23 | 4.27 | 4.29 | 5.31 | 5.47 | 5.47 | 5.59 | 5.14 | 5.62 | 5.72 |
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
Eastern | 2.55 | 3.90 | 4.59 | 6.37 | 7.11 | 7.38 | 7.55 | 7.17 | 7.89 | 8.54 |
Al Madinah | 3.26 | 3.46 | 3.47 | 4.99 | 6.38 | 6.42 | 6.81 | 6.16 | 6.77 | 7.21 |
Asir | 3.41 | 3.81 | 3.98 | 4.65 | 5.47 | 5.74 | 5.92 | 6.17 | 6.13 | 6.53 |
Jizan | 3.42 | 3.39 | 3.62 | 4.46 | 5.37 | 5.71 | 5.56 | 5.49 | 5.64 | 5.80 |
Al-Qassim | 3.43 | 3.45 | 3.37 | 4.11 | 4.46 | 4.81 | 5.10 | 5.07 | 5.24 | 5.45 |
Tabuk | 2.99 | 2.78 | 2.96 | 3.96 | 4.48 | 4.96 | 4.82 | 4.75 | 4.89 | 5.13 |
Ha'il | 2.89 | 2.59 | 2.73 | 3.59 | 4.19 | 4.59 | 4.52 | 4.50 | 4.70 | 4.75 |
Al Jawf | 2.29 | 2.75 | 2.48 | 3.35 | 4.22 | 4.42 | 4.55 | 4.44 | 4.63 | 4.71 |
Najran | 2.83 | 2.92 | 2.62 | 3.33 | 4.02 | 4.38 | 4.47 | 4.44 | 4.61 | 4.8 |
Northern Borders | 1.51 | 1.51 | 1.6 | 2.79 | 3.95 | 4.04 | 3.99 | 4.08 | 4.4 | 4.48 |
Data | Q1 | Q2 | μ′1 | Q3 | CV | N | M |
1 | 28.75 | 46.50 | 67.82 | 27.60 | 54.57 | 1.31 | 0.83 |
2 | 6.55 | 7.72 | 7.67 | 9.35 | 0.35 | -0.18 | -0.96 |
3 | 3.445 | 4.475 | 4.521 | 5.39 | 0.35 | 0.3589 | -0.0846 |
Data | Model | ˆθ | ˆβ | KS | P-value | LL | A | B | C | D |
EF-IW | 10.609 | 0.5686 | 0.1425 | 0.6197 | -144.601 | 293.202 | 295.866 | 294.016 | 293.682 | |
IW | 8.5601 | 0.6934 | 0.2135 | 0.1557 | -152.624 | 309.269 | 311.934 | 300.958 | 300.624 | |
1 | EF-W | 224.821 | 0.7022 | 0.1961 | 0.2316 | -147.318 | 298.637 | 301.301 | 299.451 | 299.117 |
EF-E | 0.0047 | 0.3475 | 0.0023 | -149.955 | 301.911 | 303.243 | 302.318 | 302.064 | ||
PBX | 0.0771 | 0.5983 | 0.1575 | 0.4901 | -145.356 | 294.713 | 297.378 | 295.603 | 295.269 | |
GE | 0.0188 | 1.4887 | 0.1495 | 0.5582 | -145.041 | 294.082 | 296.746 | 294.897 | 294.562 | |
EF-IW | 169.011 | 2.4633 | 0.1114 | 0.3500 | -132.473 | 268.947 | 273.444 | 270.734 | 269.126 | |
IW | 2168.36 | 4.0579 | 0.1697 | 0.0354 | -146.885 | 297.771 | 302.268 | 299.557 | 297.950 | |
2 | EF-W | 10.415 | 3.6140 | 0.1468 | 0.0978 | -132.736 | 269.472 | 273.969 | 271.259 | 269.651 |
EF-E | 0.0709 | 0.4251 | <0.0001 | -179.247 | 360.494 | 362.743 | 361.387 | 360.553 | ||
PBX | 0.0407 | 1.5378 | 0.2102 | 0.0041 | -146.950 | 297.936 | 302.433 | 299.723 | 298.115 | |
GE | 0.6226 | 69.883 | 0.1399 | 0.1290 | -139.298 | 282.596 | 287.093 | 284.382 | 282.775 | |
EF-IW | 11.193 | 1.5481 | 0.0756 | 0.6168 | -177.432 | 358.864 | 364.074 | 360.973 | 358.988 | |
IW | 25.075 | 2.5284 | 0.1395 | 0.0408 | -198.317 | 400.634 | 405.845 | 402.743 | 400.758 | |
3 | EF-W | 1.9167 | 7.5766 | 0.1301 | 0.0676 | -185.737 | 375.475 | 380.686 | 377.584 | 375.599 |
EF-E | 0.1131 | 0.3139 | <0.0001 | -210.347 | 422.694 | 425.300 | 423.749 | 422.735 | ||
PBX | 0.0639 | 1.7021 | 0.0909 | 0.3791 | -177.673 | 359.347 | 364.557 | 361.455 | 359.470 | |
GE | 0.7435 | 16.358 | 0.1060 | 0.2107 | -180.239 | 364.478 | 69.688 | 366.587 | 364.602 |
Data | Par | Bayes | ||
BSE | BLI | BGE | ||
1 | θ | 10.468 | 10.470 | 10.468 |
β | 0.5765 | 0.5766 | 0.5764 | |
2 | θ | 168.989 | 168.990 | 168.989 |
β | 2.457 | 2.457 | 2.457 | |
3 | θ | 11.692 | 11.689 | 11.694 |
β | 1.5635 | 1.5637 | 1.5634 |
β | μ1 | VarZ | CV | N | M | |
θ=0.4 | 0.3 | 0.0874 | 0.0234 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.2287 | 0.0351 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.3488 | 0.0369 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.4412 | 0.034 | 0.4179 | 0.5665 | 0.0143 | |
θ=0.6 | 0.3 | 0.3375 | 0.3492 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.4495 | 0.1355 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.5473 | 0.0908 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.6186 | 0.0668 | 0.4179 | 0.5665 | 0.0143 | |
θ=0.8 | 0.3 | 0.8806 | 2.3770 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.7260 | 0.3535 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.7534 | 0.1721 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.7862 | 0.1080 | 0.4179 | 0.5665 | 0.0143 | |
θ=1.2 | 0.3 | 3.4022 | 35.479 | 1.7508 | 4.3227 | 30.106 |
0.6 | 1.4270 | 1.3659 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 1.1822 | 0.4238 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 1.1022 | 0.2122 | 0.4179 | 0.5665 | 0.0143 | |
θ=1.5 | 0.3 | 7.1581 | 157.05 | 1.7508 | 4.3227 | 30.106 |
0.6 | 2.0699 | 2.8737 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 1.5149 | 0.6958 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 1.3274 | 0.3078 | 0.4179 | 0.5665 | 0.0143 |
m | MLE | BSE | BLI | BGE | |||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | ||
30 | θ | 0.4919 | 0.0039 | 0.4502 | 0.0035 | 0.4505 | 0.0037 | 0.4496 | 0.0039 |
β | 0.7802 | 0.0115 | 0.7496 | 0.0008 | 0.7497 | 0.0101 | 0.7494 | 0.0103 | |
60 | θ | 0.4928 | 0.0018 | 0.5093 | 0.0010 | 0.5093 | 0.0013 | 0.5085 | 0.0015 |
β | 0.7681 | 0.0051 | 0.7395 | 0.0007 | 0.7396 | 0.0009 | 0.7393 | 0.0101 | |
80 | θ | 0.5007 | 0.0011 | 0.4888 | 0.0006 | 0.4888 | 0.0008 | 0.4886 | 0.0010 |
β | 0.7586 | 0.0034 | 0.7882 | 0.0005 | 0.7883 | 0.0008 | 0.7880 | 0.0009 | |
100 | θ | 0.4943 | 0.0010 | 0.5091 | 0.0004 | 0.5093 | 0.0005 | 0.5087 | 0.0008 |
β | 0.7591 | 0.0024 | 0.7577 | 0.0004 | 0.7581 | 0.0006 | 0.7573 | 0.0008 |
m | MLE | BSE | BLI | BGE | |||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | ||
30 | θ | 0.7998 | 0.0055 | 0.8643 | 0.0039 | 0.8645 | 0.0041 | 0.4816 | 0.0043 |
β | 1.2981 | 0.0309 | 1.3373 | 0.0121 | 1.3384 | 0.0123 | 1.3364 | 0.0125 | |
60 | θ | 0.8001 | 0.0024 | 0.8358 | 0.0015 | 0.8359 | 0.0017 | 0.8357 | 0.0019 |
β | 1.2646 | 0.0148 | 1.1742 | 0.0075 | 1.1746 | 0.0078 | 1.1738 | 0.0079 | |
80 | θ | 0.7993 | 0.0023 | 0.7856 | 0.0014 | 0.7859 | 0.0016 | 0.7852 | 0.0018 |
β | 1.2641 | 0.0075 | 1.2241 | 0.0017 | 1.2244 | 0.0020 | 1.2239 | 0.0021 | |
100 | θ | 0.7948 | 0.0014 | 0.8046 | 0.0007 | 0.8048 | 0.0009 | 0.8044 | 0.0011 |
β | 1.2673 | 0.0072 | 1.2320 | 0.0011 | 1.2322 | 0.0013 | 1.2319 | 0.0015 |
m | MLE | BSE | BLI | BGE | ||||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | |||
30 | θ | 1.2240 | 0.0188 | 1.2206 | 0.0051 | 1.2217 | 0.0053 | 1.2196 | 0.0055 | |
β | 1.5799 | 0.0559 | 1.6200 | 0.0245 | 1.6225 | 0.0248 | 1.6184 | 0.0249 | ||
60 | θ | 1.2049 | 0.0065 | 1.2310 | 0.0032 | 1.2316 | 0.0034 | 1.2306 | 0.0036 | |
β | 1.5556 | 0.0215 | 1.5530 | 0.0058 | 1.5538 | 0.0061 | 1.5526 | 0.0062 | ||
80 | θ | 1.2035 | 0.0040 | 1.2276 | 0.0023 | 1.2280 | 0.0025 | 1.2272 | 0.0028 | |
β | 1.5524 | 0.0159 | 1.4679 | 0.0032 | 1.4685 | 0.0034 | 1.4676 | 0.0035 | ||
100 | θ | 1.2139 | 0.0041 | 1.2160 | 0.0019 | 1.2165 | 0.0022 | 1.2157 | 0.0024 | |
β | 1.5158 | 0.0113 | 1.4818 | 0.0022 | 1.4822 | 0.0023 | 1.4815 | 0.0025 |
8 | 6 | 8 | 16 | 23 | 20 | 28 | 40 |
43 | 50 | 54 | 32 | 52 | 29 | 33 | 42 |
41 | 52 | 56 | 79 | 155 | 84 | 95 | 111 |
136 | 161 | 204 | 241 |
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
Mecca | 9.39 | 9.71 | 9.83 | 9.96 | 9.97 | 9.95 | 9.98 | 9.97 | 10.005 | 9.96 |
Eastern | 8.92 | 9.23 | 9.43 | 9.56 | 9.58 | 9.71 | 9.78 | 9.72 | 9.82 | 9.87 |
Al-Madinah | 7.46 | 7.47 | 7.81 | 8.52 | 8.62 | 8.61 | 8.73 | 8.43 | 8.74 | 8.77 |
Jizan | 6.66 | 6.69 | 6.84 | 7.64 | 7.71 | 7.75 | 7.75 | 7.68 | 7.82 | 7.82 |
Al-Qassim | 6.6 | 6.62 | 6.67 | 7.47 | 7.51 | 7.53 | 7.67 | 7.6 | 7.73 | 7.73 |
Tabuk | 5.31 | 5.46 | 5.66 | 6.41 | 6.54 | 6.52 | 6.54 | 6.43 | 6.67 | 6.6 |
Ha'il | 4.23 | 4.27 | 4.29 | 5.31 | 5.47 | 5.47 | 5.59 | 5.14 | 5.62 | 5.72 |
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
Eastern | 2.55 | 3.90 | 4.59 | 6.37 | 7.11 | 7.38 | 7.55 | 7.17 | 7.89 | 8.54 |
Al Madinah | 3.26 | 3.46 | 3.47 | 4.99 | 6.38 | 6.42 | 6.81 | 6.16 | 6.77 | 7.21 |
Asir | 3.41 | 3.81 | 3.98 | 4.65 | 5.47 | 5.74 | 5.92 | 6.17 | 6.13 | 6.53 |
Jizan | 3.42 | 3.39 | 3.62 | 4.46 | 5.37 | 5.71 | 5.56 | 5.49 | 5.64 | 5.80 |
Al-Qassim | 3.43 | 3.45 | 3.37 | 4.11 | 4.46 | 4.81 | 5.10 | 5.07 | 5.24 | 5.45 |
Tabuk | 2.99 | 2.78 | 2.96 | 3.96 | 4.48 | 4.96 | 4.82 | 4.75 | 4.89 | 5.13 |
Ha'il | 2.89 | 2.59 | 2.73 | 3.59 | 4.19 | 4.59 | 4.52 | 4.50 | 4.70 | 4.75 |
Al Jawf | 2.29 | 2.75 | 2.48 | 3.35 | 4.22 | 4.42 | 4.55 | 4.44 | 4.63 | 4.71 |
Najran | 2.83 | 2.92 | 2.62 | 3.33 | 4.02 | 4.38 | 4.47 | 4.44 | 4.61 | 4.8 |
Northern Borders | 1.51 | 1.51 | 1.6 | 2.79 | 3.95 | 4.04 | 3.99 | 4.08 | 4.4 | 4.48 |
Data | Q1 | Q2 | μ′1 | Q3 | CV | N | M |
1 | 28.75 | 46.50 | 67.82 | 27.60 | 54.57 | 1.31 | 0.83 |
2 | 6.55 | 7.72 | 7.67 | 9.35 | 0.35 | -0.18 | -0.96 |
3 | 3.445 | 4.475 | 4.521 | 5.39 | 0.35 | 0.3589 | -0.0846 |
Data | Model | ˆθ | ˆβ | KS | P-value | LL | A | B | C | D |
EF-IW | 10.609 | 0.5686 | 0.1425 | 0.6197 | -144.601 | 293.202 | 295.866 | 294.016 | 293.682 | |
IW | 8.5601 | 0.6934 | 0.2135 | 0.1557 | -152.624 | 309.269 | 311.934 | 300.958 | 300.624 | |
1 | EF-W | 224.821 | 0.7022 | 0.1961 | 0.2316 | -147.318 | 298.637 | 301.301 | 299.451 | 299.117 |
EF-E | 0.0047 | 0.3475 | 0.0023 | -149.955 | 301.911 | 303.243 | 302.318 | 302.064 | ||
PBX | 0.0771 | 0.5983 | 0.1575 | 0.4901 | -145.356 | 294.713 | 297.378 | 295.603 | 295.269 | |
GE | 0.0188 | 1.4887 | 0.1495 | 0.5582 | -145.041 | 294.082 | 296.746 | 294.897 | 294.562 | |
EF-IW | 169.011 | 2.4633 | 0.1114 | 0.3500 | -132.473 | 268.947 | 273.444 | 270.734 | 269.126 | |
IW | 2168.36 | 4.0579 | 0.1697 | 0.0354 | -146.885 | 297.771 | 302.268 | 299.557 | 297.950 | |
2 | EF-W | 10.415 | 3.6140 | 0.1468 | 0.0978 | -132.736 | 269.472 | 273.969 | 271.259 | 269.651 |
EF-E | 0.0709 | 0.4251 | <0.0001 | -179.247 | 360.494 | 362.743 | 361.387 | 360.553 | ||
PBX | 0.0407 | 1.5378 | 0.2102 | 0.0041 | -146.950 | 297.936 | 302.433 | 299.723 | 298.115 | |
GE | 0.6226 | 69.883 | 0.1399 | 0.1290 | -139.298 | 282.596 | 287.093 | 284.382 | 282.775 | |
EF-IW | 11.193 | 1.5481 | 0.0756 | 0.6168 | -177.432 | 358.864 | 364.074 | 360.973 | 358.988 | |
IW | 25.075 | 2.5284 | 0.1395 | 0.0408 | -198.317 | 400.634 | 405.845 | 402.743 | 400.758 | |
3 | EF-W | 1.9167 | 7.5766 | 0.1301 | 0.0676 | -185.737 | 375.475 | 380.686 | 377.584 | 375.599 |
EF-E | 0.1131 | 0.3139 | <0.0001 | -210.347 | 422.694 | 425.300 | 423.749 | 422.735 | ||
PBX | 0.0639 | 1.7021 | 0.0909 | 0.3791 | -177.673 | 359.347 | 364.557 | 361.455 | 359.470 | |
GE | 0.7435 | 16.358 | 0.1060 | 0.2107 | -180.239 | 364.478 | 69.688 | 366.587 | 364.602 |
Data | Par | Bayes | ||
BSE | BLI | BGE | ||
1 | θ | 10.468 | 10.470 | 10.468 |
β | 0.5765 | 0.5766 | 0.5764 | |
2 | θ | 168.989 | 168.990 | 168.989 |
β | 2.457 | 2.457 | 2.457 | |
3 | θ | 11.692 | 11.689 | 11.694 |
β | 1.5635 | 1.5637 | 1.5634 |