Pub Date : 2022-12-04DOI: 10.18187/pjsor.v18i4.4095
E. Martinez, Diego Gafuri Silva, Larissa Intrebartoli Resende, Elisângela Aparecida da Silva Lizzi, J. Achcar
Spatial analysis techniques are used in the data analysis of ecological studies, which consider geographical areas as observation units. In this article, we propose a Bayesian bivariate spatial shared component model to mapping the breast and cervical cancer mortality in Southern Brazil, based on the models introduced by Knorr-Held and Best (2001) and Held et al. (2005). Markov Chain Monte Carlo (MCMC) methods were used to spatially smooth the standardized mortality ratios (SMR) for both diseases. Local Indicator of Spatial Association (LISA) was used to verify the existence of spatial clusters in specific geographical areas. This study was carried out using secondary data obtained from publicly available health information systems.
空间分析技术用于生态研究的数据分析，将地理区域视为观测单位。在这篇文章中，我们基于Knorr-Held和Best（2001）以及Held等人（2005）引入的模型，提出了一个贝叶斯双变量空间共享分量模型来绘制巴西南部乳腺癌和宫颈癌癌症死亡率。Markov Chain Monte Carlo（MCMC）方法用于对两种疾病的标准化死亡率（SMR）进行空间平滑。空间关联局部指标（LISA）用于验证特定地理区域中空间集群的存在。这项研究是利用从公共卫生信息系统获得的二次数据进行的。
{"title":"Bayesian bivariate spatial shared component model: mapping breast and cervical cancer mortality in Southern Brazil","authors":"E. Martinez, Diego Gafuri Silva, Larissa Intrebartoli Resende, Elisângela Aparecida da Silva Lizzi, J. Achcar","doi":"10.18187/pjsor.v18i4.4095","DOIUrl":"https://doi.org/10.18187/pjsor.v18i4.4095","url":null,"abstract":"Spatial analysis techniques are used in the data analysis of ecological studies, which consider geographical areas as observation units. In this article, we propose a Bayesian bivariate spatial shared component model to mapping the breast and cervical cancer mortality in Southern Brazil, based on the models introduced by Knorr-Held and Best (2001) and Held et al. (2005). Markov Chain Monte Carlo (MCMC) methods were used to spatially smooth the standardized mortality ratios (SMR) for both diseases. Local Indicator of Spatial Association (LISA) was used to verify the existence of spatial clusters in specific geographical areas. This study was carried out using secondary data obtained from publicly available health information systems.","PeriodicalId":19973,"journal":{"name":"Pakistan Journal of Statistics and Operation Research","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42171702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-04DOI: 10.18187/pjsor.v18i4.4149
L. Sapkota, Vijay Kumar
This paper proposes the 4-parameter odd Lomax generalized exponential distribution for the study of engineering and COVID-19 data. The statistical and mathematical properties of this distribution such as a linear representation of the probability density function, survival function, hazard rate function, moments, quantile function, order statistics, entropy, mean deviation, characteristic function, and average residual life function are established. The estimates of parameters of the proposed distribution are obtained using maximum likelihood estimation (MLE), Maximum product spacings (MPS), least-square estimation (LSE), and Cramer-Von-Mises estimation (CVME) methods. A Monte-Carlo simulation experiment is carried out to study the MLEs. The applicability of the proposed distribution is evaluated using two real datasets related to engineering and COVID-19. All the computational work was performed in R programming software.
{"title":"Odd Lomax Generalized Exponential Distribution: Application to Engineering and COVID-19 data","authors":"L. Sapkota, Vijay Kumar","doi":"10.18187/pjsor.v18i4.4149","DOIUrl":"https://doi.org/10.18187/pjsor.v18i4.4149","url":null,"abstract":"This paper proposes the 4-parameter odd Lomax generalized exponential distribution for the study of engineering and COVID-19 data. The statistical and mathematical properties of this distribution such as a linear representation of the probability density function, survival function, hazard rate function, moments, quantile function, order statistics, entropy, mean deviation, characteristic function, and average residual life function are established. The estimates of parameters of the proposed distribution are obtained using maximum likelihood estimation (MLE), Maximum product spacings (MPS), least-square estimation (LSE), and Cramer-Von-Mises estimation (CVME) methods. A Monte-Carlo simulation experiment is carried out to study the MLEs. The applicability of the proposed distribution is evaluated using two real datasets related to engineering and COVID-19. All the computational work was performed in R programming software.","PeriodicalId":19973,"journal":{"name":"Pakistan Journal of Statistics and Operation Research","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46058411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-04DOI: 10.18187/pjsor.v18i4.4048
G. Hamedani, A. Roshani
As we mentioned in our previous works, sometimes in real life cases, it is very difficult to obtain samples from a continuous distribution. The observed values are generally discrete due to the fact that they are not measured in continuum. In some cases, it may be possible to measure the observations via a continuous scale, however, they may be recorded in a manner in which a discrete model seems more suitable. Consequently, the discrete models are appearing quite frequently in applied fields and have attracted the attention of many researchers. Characterizations of distributions are important to many researchers in the applied fields. An investigator will be vitally interested to know if their model fits the requirements of a particular distribution. To this end, one will depend on the characterizations of this distribution which provide conditions under which the underlying distribution is indeed that particular distribution. Here, we present certain characterizations of 14 recently introduced discrete distributions.
{"title":"Characterizations of Fourteen (2021-2022) Proposed Discrete Distributions","authors":"G. Hamedani, A. Roshani","doi":"10.18187/pjsor.v18i4.4048","DOIUrl":"https://doi.org/10.18187/pjsor.v18i4.4048","url":null,"abstract":"As we mentioned in our previous works, sometimes in real life cases, it is very difficult to obtain samples from a continuous distribution. The observed values are generally discrete due to the fact that they are not measured in continuum. In some cases, it may be possible to measure the observations via a continuous scale, however, they may be recorded in a manner in which a discrete model seems more suitable. Consequently, the discrete models are appearing quite frequently in applied fields and have attracted the attention of many researchers. \u0000Characterizations of distributions are important to many researchers in the applied fields. An investigator will be vitally interested to know if their model fits the requirements of a particular distribution. To this end, one will depend on the characterizations of this distribution which provide conditions under which the underlying distribution is indeed that particular distribution. Here, we present certain characterizations of 14 recently introduced discrete distributions.","PeriodicalId":19973,"journal":{"name":"Pakistan Journal of Statistics and Operation Research","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48665676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-04DOI: 10.18187/pjsor.v18i4.3869
A. Amin
In this paper we extend autoregressive models to fit time series that have three layers of seasonality, i.e. triple seasonal autoregressive (TSAR) models, and we introduce the Bayesian inference for these TSAR models. Assuming the TSAR model errors are normally distributed and employing three priors, i.e. Jeffreys', g, and normal-gamma priors, on the model parameters, we derive the marginal posterior distributions of the TSAR model parameters. In particular, we show that the marginal posterior distributions to be multivariate t and gamma distributions for the model coefficients and precision, respectively. We evaluate the efficiency of the proposed Bayesian inference using simulation study, and we then apply it to real-world hourly electricity load time series datasets in six European countries.
{"title":"Bayesian Inference of Triple Seasonal Autoregressive Models","authors":"A. Amin","doi":"10.18187/pjsor.v18i4.3869","DOIUrl":"https://doi.org/10.18187/pjsor.v18i4.3869","url":null,"abstract":"In this paper we extend autoregressive models to fit time series that have three layers of seasonality, i.e. triple seasonal autoregressive (TSAR) models, and we introduce the Bayesian inference for these TSAR models. Assuming the TSAR model errors are normally distributed and employing three priors, i.e. Jeffreys', g, and normal-gamma priors, on the model parameters, we derive the marginal posterior distributions of the TSAR model parameters. In particular, we show that the marginal posterior distributions to be multivariate t and gamma distributions for the model coefficients and precision, respectively. We evaluate the efficiency of the proposed Bayesian inference using simulation study, and we then apply it to real-world hourly electricity load time series datasets in six European countries.","PeriodicalId":19973,"journal":{"name":"Pakistan Journal of Statistics and Operation Research","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46391140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-04DOI: 10.18187/pjsor.v18i4.3655
Hossein Pasha-Zanoosi
In this article, the reliability inference for a multicomponent stress-strength (MSS) model, when both stress and strength random variables follow inverse Topp-Leone distributions, was studied. The maximum likelihood and uniformly minimum variance unbiased estimates for the reliability of MSS model were obtained explicitly. The exact Bayes estimate of MSS reliability was derived the under squared error loss function. Also, the Bayes estimate was obtained using the Monte Carlo Markov Chain method for comparison with the aforementioned exact estimate. The asymptotic confidence interval was determined under the expected Fisher information matrix. Furthermore, the highest probability density credible interval was established through using Gibbs sampling method. Monte Carlo simulations were implemented to compare the different proposed methods. Finally, a real life example was presented in support of the suggested procedures.
{"title":"Estimation of Multicomponent Stress-strength Reliability under Inverse Topp-Leone Distribution","authors":"Hossein Pasha-Zanoosi","doi":"10.18187/pjsor.v18i4.3655","DOIUrl":"https://doi.org/10.18187/pjsor.v18i4.3655","url":null,"abstract":"In this article, the reliability inference for a multicomponent stress-strength (MSS) model, when both stress and strength random variables follow inverse Topp-Leone distributions, was studied. The maximum likelihood and uniformly minimum variance unbiased estimates for the reliability of MSS model were obtained explicitly. The exact Bayes estimate of MSS reliability was derived the under squared error loss function. Also, the Bayes estimate was obtained using the Monte Carlo Markov Chain method for comparison with the aforementioned exact estimate. The asymptotic confidence interval was determined under the expected Fisher information matrix. Furthermore, the highest probability density credible interval was established through using Gibbs sampling method. Monte Carlo simulations were implemented to compare the different proposed methods. Finally, a real life example was presented in support of the suggested procedures. ","PeriodicalId":19973,"journal":{"name":"Pakistan Journal of Statistics and Operation Research","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47403754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-10DOI: 10.18187/pjsor.v18i3.3937
Wedad H. Aljuhani, Hadeel S. Klakattawi, L. Baharith
In this paper, a new five-parameter model called alpha power exponentiated new Weibull-Pareto distribution is introduced based on a new developing technique. We derived some properties relating to the proposed distribution, including moments, moment generating function, quantile function, mean residual life and mean waiting time, and order statistics of the new model. The model parameters are estimated using the maximum likelihood method. Some simulation studies are performed to investigate the effectiveness of the estimates. Finally, we used three real-life data sets to show the flexibility of the introduced distribution.
{"title":"Alpha Power Exponentiated New Weibull-Pareto Distribution: Its Properties and Applications","authors":"Wedad H. Aljuhani, Hadeel S. Klakattawi, L. Baharith","doi":"10.18187/pjsor.v18i3.3937","DOIUrl":"https://doi.org/10.18187/pjsor.v18i3.3937","url":null,"abstract":"In this paper, a new five-parameter model called alpha power exponentiated new Weibull-Pareto distribution is introduced based on a new developing technique. We derived some properties relating to the proposed distribution, including moments, moment generating function, quantile function, mean residual life and mean waiting time, and order statistics of the new model. The model parameters are estimated using the maximum likelihood method. Some simulation studies are performed to investigate the effectiveness of the estimates. Finally, we used three real-life data sets to show the flexibility of the introduced distribution.","PeriodicalId":19973,"journal":{"name":"Pakistan Journal of Statistics and Operation Research","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43450768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-10DOI: 10.18187/pjsor.v18i3.2902
Rania Hassan Abd El Khaleq
We introduce a new extension of the Fréchet distribution for modeling the extreme values. The new model generalizes eleven distributions at least, five of them are quite new. Some important mathematical properties of the new model are derived. We assess the performance of the maximum likelihood estimators (MLEs) via a simulation study. The new model is better than some other important competitive models in modeling the breaking stress data, the glass fibers data and the relief time data.
{"title":"The Generalized Odd Log-Logistic Fréchet Distribution for Modeling Extreme Values","authors":"Rania Hassan Abd El Khaleq","doi":"10.18187/pjsor.v18i3.2902","DOIUrl":"https://doi.org/10.18187/pjsor.v18i3.2902","url":null,"abstract":"We introduce a new extension of the Fréchet distribution for modeling the extreme values. The new model generalizes eleven distributions at least, five of them are quite new. Some important mathematical properties of the new model are derived. We assess the performance of the maximum likelihood estimators (MLEs) via a simulation study. The new model is better than some other important competitive models in modeling the breaking stress data, the glass fibers data and the relief time data.","PeriodicalId":19973,"journal":{"name":"Pakistan Journal of Statistics and Operation Research","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47322076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-10DOI: 10.18187/pjsor.v18i3.3766
A. Elshahhat, M. K. Rastogi
Chen's model with bathtub shape provides an appropriate conceptual for the hazard rate of various industrial products and clinical cases. This article deals with the problem of estimating the model parameters, reliability and hazard functions of a three-parameter Chen distribution based on progressively Type-II censored sample have been obtained. Based on the s-normal approximation to the asymptotic distribution of the maximum likelihood estimates and log-transformed maximum likelihood estimates, the approximate confidence intervals for the unknown parameters, and any function of them, are constructed. Using independent gamma conjugate priors, the Bayes estimators of the unknown parameters and reliability characteristics are derived under different versions of a symmetric squared error loss functions. However, the Bayes estimators are obtained in a complex form, so we have been used Metropolis-Hastings sampler procedure to carry out the Bayes estimates and also to construct the corresponding credible intervals. To assess the performance of the proposed estimators, numerical results using Monte Carlo simulation study were reported. To determine the optimum censoring scheme among different competing censoring plans, some optimality criteria have been considered. A practical example using real-life data set, representing the survival times of head and neck cancer patients, is discussed to demonstrate how the applicability of the proposed methods in real phenomenon.
{"title":"Bayesian Life Analysis of Generalized Chen's Population Under Progressive Censoring","authors":"A. Elshahhat, M. K. Rastogi","doi":"10.18187/pjsor.v18i3.3766","DOIUrl":"https://doi.org/10.18187/pjsor.v18i3.3766","url":null,"abstract":"Chen's model with bathtub shape provides an appropriate conceptual for the hazard rate of various industrial products and clinical cases. This article deals with the problem of estimating the model parameters, reliability and hazard functions of a three-parameter Chen distribution based on progressively Type-II censored sample have been obtained. Based on the s-normal approximation to the asymptotic distribution of the maximum likelihood estimates and log-transformed maximum likelihood estimates, the approximate confidence intervals for the unknown parameters, and any function of them, are constructed. Using independent gamma conjugate priors, the Bayes estimators of the unknown parameters and reliability characteristics are derived under different versions of a symmetric squared error loss functions. However, the Bayes estimators are obtained in a complex form, so we have been used Metropolis-Hastings sampler procedure to carry out the Bayes estimates and also to construct the corresponding credible intervals. To assess the performance of the proposed estimators, numerical results using Monte Carlo simulation study were reported. To determine the optimum censoring scheme among different competing censoring plans, some optimality criteria have been considered. A practical example using real-life data set, representing the survival times of head and neck cancer patients, is discussed to demonstrate how the applicability of the proposed methods in real phenomenon.","PeriodicalId":19973,"journal":{"name":"Pakistan Journal of Statistics and Operation Research","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44025244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-10DOI: 10.18187/pjsor.v18i3.3657
M. Y. Hmood, Amjed Hibatallah
In this paper, we propose a method using continuous wavelets to study the multivariate fractional Brownian motion through the deviations of the transformed random process to find an efficient estimate of Hurst exponent using eigenvalue regression of the covariance matrix. The results of simulations experiments shown that the performance of the proposed estimator was efficient in bias but the variance get increase as signal change from short to long memory the MASE increase relatively. The estimation process was made by calculating the eigenvalues for the variance-covariance matrix of Meyer’s continuous wavelet details coefficients.
{"title":"Continuous wavelet estimation for multivariate fractional Brownian motion","authors":"M. Y. Hmood, Amjed Hibatallah","doi":"10.18187/pjsor.v18i3.3657","DOIUrl":"https://doi.org/10.18187/pjsor.v18i3.3657","url":null,"abstract":" In this paper, we propose a method using continuous wavelets to study the multivariate fractional Brownian motion through the deviations of the transformed random process to find an efficient estimate of Hurst exponent using eigenvalue regression of the covariance matrix. The results of simulations experiments shown that the performance of the proposed estimator was efficient in bias but the variance get increase as signal change from short to long memory the MASE increase relatively. The estimation process was made by calculating the eigenvalues for the variance-covariance matrix of Meyer’s continuous wavelet details coefficients.","PeriodicalId":19973,"journal":{"name":"Pakistan Journal of Statistics and Operation Research","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41439154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-10DOI: 10.18187/pjsor.v18i3.3911
Joseph Ackora Prah, Valentine Acheson, B. Barnes, I. Takyi, E. Owusu-Ansah
The Transportation Problem (TP) is a mathematical optimization technique which regulates the flow of items along routes by adopting an optimum guiding principle to the total shipping cost. However, instances including road hazards, traffic regulations, road construction and unexpected floods sometimes arise in transportation to ban shipments via certain routes. In formulating the TPs, potential prohibited routes are assigned a large penalty cost, M, to prevent their presence in the model solution. The arbitrary usage of the big M as a remedy for this interdiction does not go well with a good solution. In this paper, a two-phase method is proposed to solve a TP with prohibited routes. The first phase is formulated as an All-Pairs Least Cost Problem (APLCP) which assigns respectively a non-discretionary penalty cost M*ij <= M to each of n prohibited routes present using the Floyd¢s method. At phase two, the new penalty values are substituted into the original problem respectively and the resulting model is solved using the transportation algorithm. The results show that, setting this modified penalty cost ( M*) logically presents a good solution. Therefore, the discretionary usage of the M <= ∞ is not a guarantee for good model solutions. The modified cost M*<= M so attained in the sample model, is relatively less than the Big M ( <= ∞) and gives a good solution which makes the method reliable.
{"title":"A 2-Phase Method for Solving Transportation Problems with Prohibited Routes","authors":"Joseph Ackora Prah, Valentine Acheson, B. Barnes, I. Takyi, E. Owusu-Ansah","doi":"10.18187/pjsor.v18i3.3911","DOIUrl":"https://doi.org/10.18187/pjsor.v18i3.3911","url":null,"abstract":"The Transportation Problem (TP) is a mathematical optimization technique which regulates the flow of items along routes by adopting an optimum guiding principle to the total shipping cost. However, instances including road hazards, traffic regulations, road construction and unexpected floods sometimes arise in transportation to ban shipments via certain routes. In formulating the TPs, potential prohibited routes are assigned a large penalty cost, M, to prevent their presence in the model solution. The arbitrary usage of the big M as a remedy for this interdiction does not go well with a good solution. In this paper, a two-phase method is proposed to solve a TP with prohibited routes. The first phase is formulated as an All-Pairs Least Cost Problem (APLCP) which assigns respectively a non-discretionary penalty cost M*ij <= M to each of n prohibited routes present using the Floyd¢s method. At phase two, the new penalty values are substituted into the original problem respectively and the resulting model is solved using the transportation algorithm. The results show that, setting this modified penalty cost ( M*) logically presents a good solution. Therefore, the discretionary usage of the M <= ∞ is not a guarantee for good model solutions. The modified cost M*<= M so attained in the sample model, is relatively less than the Big M ( <= ∞) and gives a good solution which makes the method reliable.","PeriodicalId":19973,"journal":{"name":"Pakistan Journal of Statistics and Operation Research","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41730768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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