Abstract
One of the most widely used tools for quality engineers in quickly detecting assignable causes and monitoring processes is control charts. Duncan (in J Am Stat Assoc 51(274):228–242, 1956), in order to improve product quality and reduce the economic costs of the quality cycle, presented the first economic design of the \( \bar{X} \) control chart in the presence of multiple assignable causes. In his model and all the economic designs derived from it, it is assumed that after the occurrence of an assignable cause, no other assignable cause occurs until the correct alarm is issued. This assumption is unrealistic and impractical in production and service processes. Therefore, in this paper, we present a realistic and practical economic design in the presence of multiple assignable causes for the \(\bar{X}\) control chart under the Weibull shock model in industry. The numerical results of our model show well that in the previous models, the average cost per unit time of the quality cycle is severely underestimated compared to the actual value. Therefore, it is suggested that in order to eliminate the shortcomings of the previous economic design in the presence of multiple assignable causes, in future research, they should be redesigned based on our proposed model.
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Due to the increasing use of control charts to monitor and improve the quality of manufactured goods in production and industrial processes, the selection of parameters related to the design of control charts for managers and quality engineers is essential. Therefore, in this paper, we present a novel and practical model to determine the design parameters of control charts to minimize the costs associated with monitoring the production process and eliminating the deficiencies of previous models. It should be pointed out that more than 6 decades have passed since the first article presented by Duncan in this field. Based on his article, many types of research have been published in various journals. In Duncan (1971) and all the researches based on it, the economic design of control charts in the presence of multiple independent assignable causes has been performed with the assumption that after the occurrence of one assignable cause, another assignable cause will definitely not occur until the end of the quality cycle. Not only conclusive consideration of such an assumption is inconsistent with the independent assumption of the time of occurrence of assignable causes, but so far no study has been done on the conditions of occurrence of such an assumption in practice. Therefore, in this paper, after stating a fundamental error in the collection of papers published in these decades of Duncan's model, by presenting a new model, we calculate the probability of this assumption and prove its dependence on design parameters under the Weibull shock model. The optimal values of the design parameters for the \(\bar{X}\) control chart in our model are compared with the Chen and Yang models, where the results indicate a severe deviation of their model from reality. Our findings show that this deficiency in previous models severely underestimates the quality cycle costs in manufacturing processes (approximately 11.8% to 195.8% underestimation of the actual value). Therefore, according to the stated contents and the special origin of control chart applications in monitoring production processes, we believe that this manuscript is appropriate for publication in your journal’s scope.
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Appendix
Appendix
Proof of Theorem 1
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1.
From Eq. (2), \(w_{j} = j^{\frac{1}{k}} h_{1}\) is obtained. According to Fig. 1, the random variable X represents the time elapsed from the beginning of the quality cycle to the issuance of the true alarm. In order to summarize, we define the steps of proving the following equations:
$$ \lambda = \mathop \sum \limits_{i = 1}^{m} \lambda_{i} ,\quad \lambda_{ - i} = \lambda - \lambda_{i} $$(25)$$ \left\{ {{\mathbf{T}}_{{{\mathbf{ - i}}}} > X} \right\} = \{ T_{1} > X, \ldots ,T_{i - 1} > X, T_{i + 1} > X , \ldots ,T_{m} > X\} $$(26)Clearly with respect to Eq. (26) the occurrence of the cause of the ith documentable cause before the issuance of the true alarm and the absence of other attributable causes until the issuance of the true alarm ({Ii}) with the occurrence \(\{ T_{i} \le X ,T_{ - i} > X\}\) is equivalent, so using the law we have the total probability:
$$ \begin{aligned} P\left( {I_{i} } \right) & = P\left( {T_{i} \le X ,{\mathbf{T}}_{{{\mathbf{ - i}}}} > X} \right) \\ & = \mathop \sum \limits_{j = 1}^{\infty } P(T_{i} \le w_{j} ,{\mathbf{T}}_{{{\mathbf{ - i}}}} > w_{j} ,X = w_{j} ) \\ & = \mathop \sum \limits_{j = 1}^{\infty } \mathop \sum \limits_{l = 1}^{j} P\left( {w_{l - 1} < T_{i} \le w_{l} ,{\mathbf{T}}_{{{\mathbf{ - i}}}} > w_{j} ,X = w_{j} } \right) \\ \end{aligned} $$(27)According to the definition of variables as well as assumptions we have:
$$ P\left( {X = w_{j} |w_{l - 1} < T_{i} \le w_{l} ,{\mathbf{T}}_{{{\mathbf{ - i}}}} > w_{j} } \right) = \beta_{i}^{j - l} (1 - \beta_{i} ) $$(28)$$ P\left( {w_{l - 1} < T_{i} \le w_{l} } \right) = \left( {1 - {\text{e}}^{{ - \lambda_{i} h^{k} }} } \right) \cdot {\text{e}}^{{ - \lambda_{i} \left( {l - 1} \right)h_{1}^{k} }} $$(29)$$ P\left( {{\mathbf{T}}_{{{\mathbf{ - i}}}} > w_{j} |w_{l - 1} < T_{i} \le w_{l} } \right) = {\text{e}}^{{ - \lambda_{ - i} jh_{1}^{k} }} $$(30)From Eqs. (28) to (30) we have:
$$ \begin{aligned} P\left( {I_{i} } \right) & = \mathop \sum \limits_{j = 1}^{\infty } \mathop \sum \limits_{l = 1}^{j} \beta_{i}^{j - l} (1 - \beta_{i} )\left( {1 - {\text{e}}^{{ - \lambda_{i} h_{1}^{k} }} } \right).{\text{e}}^{{ - \lambda_{i} \left( {l - 1} \right)h_{1}^{k} }} {\text{e}}^{{ - \lambda_{ - i} jh_{1}^{k} }} \\ & = \mathop \sum \limits_{j = 1}^{\infty } (1 - \beta_{i} )(1 - {\text{e}}^{{ - \lambda_{i} h_{1}^{k} }} )\frac{{\left( {{\text{e}}^{{ - \lambda h_{1}^{k} }} } \right)^{j} - \left( {\beta_{i} {\text{e}}^{{ - \lambda_{ - i} h_{1}^{k} }} } \right)^{j} }}{{{\text{e}}^{{ - \lambda_{i} h_{1}^{k} }} - \beta_{i} }} \\ & = \frac{{\left( {1 - \beta_{i} } \right)\left( {1 - {\text{e}}^{{ - \lambda_{i} h_{1}^{k} }} } \right){\text{e}}^{{ - \lambda_{ - i} h_{1}^{k} }} }}{{(1 - e^{{ - \lambda h_{1}^{k} }} )(1 - \beta_{i} e^{{ - \lambda_{ - i} h_{1}^{k} }} )}} \\ \end{aligned} $$(31)Clearly \(\left\{ {I = 1} \right\} = \bigcup\nolimits_{i = 1}^{m} {I_{i} }\) and since we have \(I_{i} \cap I_{j} = \emptyset\) for every i ≠ j, therefore, \(P\left( {I = 1} \right) = \sum\nolimits_{i = 1}^{m} P \left( {I_{i} } \right)\).
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2.
Considering that for \(i = 1, 2, \ldots ,m\), \(\lim\nolimits_{{h \to 0^{ + } }} P\left( {I_{i} } \right) = \frac{0}{0}\), using the hopital rule, it can be easily shown that \(\lim\nolimits_{{h \to 0^{ + } }} P\left( {I_{i} } \right) = \lambda_{i} /\lambda\) and therefore \(\lim\nolimits_{{h \to 0^{ + } }} P\left( I \right) = 1\).
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3.
Clearly for \(i = 1, 2, \ldots ,m\), \(\lim\nolimits_{L \to + \infty } \beta_{i} = 1\) resulting in \(\lim\nolimits_{L \to + \infty } P\left( I \right) = 0\).
Proof of Lemma 1
Given that it is assumed that the time of taking and interpreting the sample is negligible, then the true alarm will occur in one of the sampling times, i.e. \(w_{1}\), \(w_{2}\), etc. Therefore, according to the Eqs. (28) to (30), we have
Proof of Lemma 2
Clearly if \(T = \min (T_{1} , \ldots ,T_{m} )\) then \(X_{1} = T\) and we have:
Therefore, it suffices to obtain \(f(t|I = 1)\) for \(w_{j - 1} < t < w_{j}\) as follows:
Now if we have \({\text{d}}t\, \to \,0\) then,
Therefore,
where the above relation \(\Gamma \left( {s,x} \right) = \mathop \int \nolimits_{x}^{\infty } t^{s - 1} {\text{e}}^{ - t} {\text{d}}t\) and \(f_{{T_{i} }} \left( t \right) = k\lambda_{i} t^{k - 1} {\text{e}}^{{ - \lambda_{i} t^{k} }}\).
Proof of Lemma 3
If \(Z_{2i}\) is the correction time of the ith attributable cause, then the average process correction time provided that only one attributable cause occurs until the true alarm is issued after identifying the attributable cause (\(E(X_{4} |I = 1)\)) Is equal to:
Proof of Lemma 4
If \(N_{{{\text{TST}}}}\) is equal to the number of samples taken from the process to the true alarm and \(N_{{{\text{INC}}}}\) is equal to the number of samples taken from the process in-control state, it is clearly \(N_{{{\text{OC}}}} = N_{{{\text{TST}}}} - N_{{{\text{INC}}}}\). Therefore, \(E\left( {N_{{{\text{OC}}}} |I = 1} \right) = E\left( {N_{{{\text{TST}}}} |I = 1} \right) - E(N_{{{\text{INC}}}} |I = 1)\), where,
and
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Shojaei, S.R., Bameni Moghadam, M. & Eskandari, F. Fundamental Changes in Theory of Duncan’s Model for Economic Design of Control Charts in the Presence of Multiple Assignable Causes. Iran J Sci Technol Trans Sci 46, 1613–1628 (2022). https://doi.org/10.1007/s40995-022-01365-8
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DOI: https://doi.org/10.1007/s40995-022-01365-8