Intelligent Process Control Using Control Charts—I: Control Charts for Variables

Abstract

Shewhart’s control charts are used when you have enough and exact observed data. In case of incomplete and vague data, they can be still used by the help of the fuzzy set theory. In this chapter, we develop the fuzzy control charts for variables, which are namely $\overline{X}$ and R and $\overline{X}$ and S charts. Triangular fuzzy numbers have been used in the development of these charts. Unnatural patterns have been examined under fuzziness. Besides, fuzzy EWMA charts have been also developed in this chapter. For each fuzzy case, we present a numerical example.

Appendices

Appendix A

Table of coefficients for control charts for variables.

Observations in subgroup, n	A 2	A 3	c4	B 3	B 4	B 5	B 6	d 2	d 3	D 1	D 2	D 3	D 4	E 2
2	1.880	2.659	0.798	0.000	3.267	0.000	2.606	1.128	0.853	0.000	3.686	0.000	3.267	2.660
3	1.023	1.954	0.886	0.000	2.568	0.000	2.276	1.693	0.888	0.000	4.358	0.000	2.574	1.772
4	0.729	1.628	0.921	0.000	2.266	0.000	2.088	2.059	0.880	0.000	4.698	0.000	2.282	1.457
5	0.577	1.427	0.940	0.000	2.089	0.000	1.964	2.326	0.864	0.000	4.918	0.000	2.114	1.290
6	0.483	1.287	0.952	0.030	1.970	0.029	1.874	2.534	0.848	0.000	5.078	0.000	2.004	1.184
7	0.419	1.182	0.959	0.118	1.882	0.113	1.806	2.704	0.833	0.204	5.204	0.076	1.924	1.109
8	0.373	1.099	0.965	0.185	1.815	0.179	1.751	2.847	0.820	0.388	5.306	0.136	1.864	1.054
9	0.337	1.032	0.969	0.239	1.761	0.232	1.707	2.970	0.808	0.547	5.393	0.184	1.816	1.010
10	0.308	0.975	0.973	0.284	1.716	0.276	1.669	3.078	0.797	0.687	5.469	0.223	1.777	0.975
11	0.285	0.927	0.975	0.321	1.679	0.313	1.637	3.173	0.787	0.811	5.535	0.256	1.744	0.945
12	0.266	0.886	0.978	0.354	1.646	0.346	1.610	3.258	0.778	0.922	5.594	0.283	1.717	0.921
13	0.249	0.850	0.979	0.382	1.618	0.374	1.585	3.336	0.770	1.025	5.647	0.307	1.693	0.899
14	0.235	0.817	0.981	0.406	1.594	0.399	1.563	3.407	0.763	1.118	5.696	0.328	1.672	0.881
15	0.223	0.789	0.982	0.428	1.572	0.421	1.544	3.472	0.756	1.203	5.741	0.347	1.653	0.864
16	0.212	0.763	0.984	0.448	1.552	0.440	1.526	3.532	0.750	1.282	5.782	0.363	1.637	0.849
17	0.203	0.739	0.985	0.466	1.534	0.458	1.511	3.588	0.744	1.356	5.820	0.378	1.622	0.836
18	0.194	0.718	0.985	0.482	1.518	0.475	1.496	3.640	0.739	1.424	5.856	0.391	1.608	0.824
19	0.187	0.698	0.986	0.497	1.503	0.490	1.483	3.689	0.734	1.487	5.891	0.403	1.597	0.813
20	0.180	0.680	0.987	0.510	1.490	0.504	1.470	3.735	0.729	1.549	5.921	0.415	1.585	0.803
21	0.173	0.663	0.988	0.523	1.477	0.516	1.459	3.778	0.724	1.605	5.951	0.425	1.575	0.794
22	0.167	0.647	0.988	0.534	1.466	0.528	1.448	3.819	0.720	1.659	5.979	0.434	1.566	0.786
23	0.162	0.633	0.989	0.545	1.455	0.539	1.438	3.858	0.716	1.710	6.006	0.443	1.557	0.778
24	0.157	0.619	0.989	0.555	1.445	0.549	1.429	3.895	0.712	1.759	6.031	0.451	1.548	0.770
25	0.153	0.606	0.990	0.565	1.435	0.559	1.420	3.931	0.708	1.806	6.056	0.459	1.541	0.763

Appendix B

The equations to compute sample area outside the control the limits.

$$ \begin{aligned} A_{out}^{U} = & \, \frac{1}{2}\left[ {\left( {d^{\alpha } - {UCL}_{4}^{\alpha } } \right) + \left( {d^{t} - {UCL}_{4}^{t} } \right)} \right]\left( {\hbox{max} \left( {t - \alpha ,0} \right)} \right) \\ & +\, \frac{1}{2}\left[ {\left( {d^{z} - a^{z} } \right) + \left( {c - b} \right)} \right]\left( {\hbox{min} \left( {1 - t,1 - \alpha } \right)} \right) \\ \end{aligned} $$

(2.104)

where,

$$ t = \frac{{{UCL}_{4} - a}}{{\left( {b - a} \right) + \left( {c - b} \right)}}\,{\text{and}}\,z = \hbox{max} \left( {t,\alpha } \right) $$

$$ A_{out}^{U} = \frac{1}{2}\left[ {\left( {d^{\alpha } - {UCL}_{4}^{\alpha } } \right) + \left( {c - {UCL}_{3} } \right)} \right]\left( {1 - \alpha } \right) $$

(2.105)

$$ A_{out}^{U} = \frac{1}{2}\left( {d^{\alpha } - {UCL}_{4}^{\alpha } } \right)\left( {\hbox{max} \left( {t - \alpha ,0} \right)} \right) $$

(2.106)

where

$$ t = \frac{{{UCL}_{4} - d}}{{\left( {{UCL}_{4} - {UCL}_{3} } \right) - \left( {d - c} \right)}} $$

$$ A_{out}^{U} = \frac{1}{2}\left[ {\left( {c - {UCL}_{3} } \right) + \left( {d^{z} - {UCL}_{4}^{z} } \right)} \right]\left( {\hbox{min} \left( {1 - t,1 - \alpha } \right)} \right) $$

(2.107)

where

$$ t = \frac{{{UCL}_{4} - d}}{{\left( {{UCL}_{4} - {UCL}_{3} } \right) - \left( {d - c} \right)}}\,{\text{and}}\,z = \hbox{max} \left( {t,\alpha } \right) $$

$$ \begin{aligned} A_{out}^{U} & = \frac{1}{2}\left[ {\left( {d^{{z_{2} }} - {UCL}_{4}^{{z_{2} }} } \right) + \left( {d^{{t_{1} }} - {UCL}_{4}^{{t_{1} }} } \right)} \right]\left( {\hbox{min} \left( {\hbox{max} \left( {t_{1} - \alpha ,0} \right),t_{1} - t_{2} } \right)} \right) \\ & \quad + \frac{1}{2}\left[ {\left( {d^{{z_{1} }} - a^{{z_{1} }} } \right) + \left( {c - b} \right)} \right]\left( {\hbox{min} \left( {1 - t_{1} ,1 - \alpha } \right)} \right) \\ \end{aligned} $$

where

$$ \begin{aligned} t_{1} & = \frac{{{UCL}_{4} - a}}{{\left( {b - a} \right) + \left( {{UCL}_{4} - {UCL}_{3} } \right)}}, \\ t_{2} & = \frac{{{UCL}_{4} - d}}{{\left( {{UCL}_{4} - {UCL}_{3} } \right) - \left( {d - c} \right)}}, \\ \end{aligned} $$

(2.108)

$$ z_{1} = \hbox{max} \left( {\alpha ,t_{1} } \right), \,{\text{and}}\,z_{2} = \hbox{max} \left( {\alpha ,t_{2} } \right) $$

$$ A_{out}^{U} = 0 $$

(2.109)

$$ A_{out}^{U} = \frac{1}{2}\left[ {\left( {d^{\alpha } - a^{\alpha } } \right) + \left( {c - b} \right)} \right]\left( {1 - \alpha } \right) $$

(2.110)

$$ \begin{aligned} A_{out}^{L} & = \frac{1}{2}\left[ {\left( {{LCL}_{1}^{\alpha } - a^{\alpha } } \right) + \left( {{LCL}_{1}^{t} - a^{t} } \right)} \right]\left( {\hbox{max} \left( {t - \alpha ,0} \right)} \right) \\ & \quad + \frac{1}{2}\left[ {\left( {d^{z} - a^{z} } \right) + \left( {c - b} \right)} \right]\left( {\hbox{min} \left( {1 - t,1 - \alpha } \right)} \right) \\ \end{aligned} $$

(2.111)

where

$$ t = \frac{{d - {LCL}_{1} }}{{\left( {{LCL}_{2} - {LCL}_{1} } \right) + \left( {d - c} \right)}}\,{\text{and}}\,z = \hbox{max} \left( {\alpha ,t} \right) $$

$$ A_{out}^{L} = \frac{1}{2}\left[ {\left( {d^{\alpha } - a^{\alpha } } \right) + \left( {c - b} \right)} \right]\left( {1 - \alpha } \right) $$

(2.112)

$$ A_{out}^{L} = \frac{1}{2}\left[ {\left( {{LCL}_{1}^{\alpha } - a^{\alpha } } \right) + \left( {{LCL}_{2} - b} \right)} \right]\left( {1 - \alpha } \right) $$

(2.113)

$$ \begin{aligned} A_{out}^{L} & = \frac{1}{2}\left[ {\left( {{LCL}_{1}^{{z_{2} }} - a^{{z_{2} }} } \right) + \left( {{LCL}_{1}^{{t_{1} }} - a^{{t_{1} }} } \right)} \right]\left( {\hbox{min} \left( {\hbox{max} \left( {t_{1} - \alpha ,0} \right),t_{1} - t_{2} } \right)} \right) \\ & \quad + \frac{1}{2}\left[ {\left( {d^{{z_{1} }} - a^{{z_{1} }} } \right) + \left( {c - b} \right)} \right]\left( {\hbox{min} \left( {1 - t,1 - \alpha } \right)} \right) \\ \end{aligned} $$

(2.114)

where

$$ \begin{aligned} t_{1} & = \frac{{d - {LCL}_{1} }}{{\left( {{LCL}_{2} - {LCL}_{1} } \right) + \left( {d - c} \right)}}, \\ t_{2} & = \frac{{a - {LCL}_{1} }}{{\left( {{LCL}_{2} - {LCL}_{1} } \right) - \left( {b - a} \right)}} \\ \end{aligned} $$

$$ z_{1} = \hbox{max} \left( {\alpha ,t_{1} } \right),\,{\text{and}}\,z_{2} = \hbox{max} \left( {\alpha ,t_{2} } \right) $$

$$ A_{out}^{L} = \frac{1}{2}\left[ {\left( {{LCL}_{1}^{z} - a^{z} } \right) + \left( {{LCL}_{2} - b} \right)} \right]\left( {\hbox{min} \left( {1 - t,1 - \alpha } \right)} \right) $$

(2.115)

where

$$ t = \frac{{a - {LCL}_{1} }}{{\left( {{LCL}_{2} - {LCL}_{1} } \right) - \left( {b - a} \right)}},\,{\text{and}}\,z = \hbox{max} \left( {\alpha ,t} \right) $$

$$ A_{out}^{L} = 0 $$

(2.116)

$$ A_{out}^{L} = \frac{1}{2}\left[ {\left( {d^{\alpha } - a^{\alpha } } \right) + \left( {c - b} \right)} \right]\left( {1 - \alpha } \right) $$

(2.117)