Chap4

‫اﻟﺒﺎب اﻟﺮاﺑﻊ‬ ‫ﺧﺮاﺋﻂ اﻟﺘﺤﻜﻢ ﻟﻠﺨﻮاص‬ ‫‪Control Charts for Attributes‬‬

‫د‪ .‬ﻣﺤﻤﺪ ﻋﻴﺸـــﻮﻧﻲ‬ ‫أﺳﺘﺎذ ﻣﺴﺎﻋﺪ – ﻗﺴﻢ اﻟﺘﻘﻨﻴﺔ اﻟﻤﻴﻜﺎﻧﻴﻜﻴﺔ ‪٢٠٠٤ -‬‬ ‫‪Email : [email protected]‬‬ ‫‪http://hctmetrology.tripod.com/quality‬‬

‫ﻣﻘﺪﻣﺔ ﻋﻦ ﺧﺮاﺋﻂ اﻟﺘﺤﻜﻢ ﻟﻠﺨﻮاص‬ ‫“‬

‫ﺧﺮاﺋﻂ اﻟﺘﺤﻜﻢ ﻟﻠﺨﻮاص هﻲ أداة ﺕﻘﻨﻴﺔ ﻟﻠﻀﺒﻂ اﻻﺡﺼﺎﺋﻲ ﻟﺠﻮدة‬ ‫اﻟﻤﻨﺘﺠﺎت‪ ،‬ﺕﻘﻮم ﻋﻠﻰ ﻗﻴﺎﺳﺎت ﻋﺎﻣﺔ ﻟﻤﺪى ﻣﻄﺎﺑﻘﺔ اﻟﻮﺡﺪات‬ ‫اﻟﻤﻨﺘﺠﺔ ﻣﻊ اﻟﻤﻮاﺹﻔﺎت اﻟﻘﻴﺎﺳﻴﺔ ﻣﻦ ﻋﺪﻣﻪ‪.‬‬

‫“‬

‫ﻧﻘﻮم ﺑﺘﺴﺠﻴﻞ ﺑﻴﺎﻧﺎت اﻟﺠﻮدة ﻋﻠﻰ ﺵﻜﻞ اﻋﺪاد ﻟﻠﻘﻄﻊ اﻟﻤﻄﺎﺑﻘﺔ‬ ‫‪ conforming‬أو ﻏﻴﺮ اﻟﻤﻄﺎﺑﻘﺔ )ﻣﻨﺘﺞ ﻣﻌﻴﺐ( ‪.non conforming‬‬

‫“‬

‫هﺬﻩ اﻟﺨﺮاﺋﻂ ﻋﻠﻰ ﻧﻮﻋﻴﻦ‪:‬‬ ‫ﺧﺮیﻄﺔ ﻧﺴﺒﺔ اﻟﻤﻌﻴﺐ ‪p chart‬‬ ‫‪c chart‬‬ ‫ﺧﺮیﻄﺔ ﻋﺪد اﻟﻌﻴﻮب‬

‫‪.١‬‬ ‫‪.٢‬‬

‫‪١‬‬

‫ﺧﺮیﻄﺔ ﻧﺴﺒﺔ اﻟﻤﻌﻴﺐ‬

‫‪p chart‬‬

‫“ ﺕﻘﻮم هﺬﻩ اﻟﺨﺮاﺋﻂ ﻋﻠﻰ دراﺳﺔ ﻗﻴﺎس اﻟﺼﻔﺎت و ﺧﺼﺎﺋﺺ‬ ‫اﻟﻤﻨﺘﺞ و ذﻟﻚ ﺑﺘﺤﺪیﺪ اﻟﻨﺴﺒﺔ اﻟﻤﺌﻮیﺔ اﻟﻐﻴﺮ ﻣﻄﺎﺑﻘﺔ‬ ‫ﻟﻠﻤﻮاﺹﻔﺎت )أو اﻟﻤﻌﻴﺒﺔ(‪.‬‬ ‫“ ﻣﺜﺎل ‪:‬‬ ‫ﻋﺪد اﻟﻜﺮاﺳﻲ اﻟﺘﺎﻟﻔﺔ ﻓﻲ اﻟﻘﺎﻋﺔ = ‪٥‬‬ ‫اﻟﻌﺪد اﻹﺝﻤﺎﻟﻲ ﻟﻠﻜﺮاﺳﻲ اﻟﻤﻔﺤﻮﺹﺔ )اﻟﻤﻮﺝﻮدة ﻓﻲ اﻟﻘﺎﻋﺔ( = ‪٥٠‬‬ ‫ﻧﺴﺒﺔ اﻟﻜﺮاﺳﻲ اﻟﻤﻌﻴﺒﺔ = ‪%١٠ = ١٠٠ * ٥٠ /٥‬‬ ‫“ اﻟﻘﻄﻌﺔ اﻟﻤﻔﺤﻮﺹﺔ ‪ :‬ﻣﻄﺎﺑﻘﺔ أو ﻏﻴﺮ ﻣﻄﺎﺑﻘﺔ‬ ‫‪٣‬‬

‫ﺧﺮیﻄﺔ ﻧﺴﺒﺔ اﻟﻤﻌﻴﺐ‬ ‫“‬

‫‪.١‬‬ ‫‪.٢‬‬ ‫‪.٣‬‬ ‫‪.٤‬‬ ‫‪٤‬‬

‫‪٢‬‬

‫‪p chart‬‬

‫ﺕﺆﺧﺬ ﻋﻴﻨﺎت ﻣﻦ ﺧﻂ اﻻﻧﺘﺎج ﻋﻠﻰ ﻓﺘﺮات ﻣﺨﺘﻠﻔﺔ و ﺕﻔﺘﺶ ﻋﻠﻰ‬ ‫ﺝﻮدة اﻟﻤﻨﺘﺞ ﺑﺤﺴﺎب ﻋﺪد اﻟﻮﺡﺪات اﻟﻤﻌﻴﺒﺔ )‪(#nonconforming items‬‬ ‫و ﻣﻦ ﺙﻢ و ﻗﺼﺪ اﻧﺸﺎء ﺧﺮیﻄﺔ ﻧﺴﺒﺔ اﻟﻤﻌﻴﺐ ﻧﻘﻮم ﺑﻤﺎ یﻠﻲ‪:‬‬ ‫ﻋﺪد اﻟﻮﺡﺪات اﻟﻤﻌﻴﺒﺔ ﻓﻲ آﻞ ﻋﻴﻨﺔ‬

‫ﺡﺴﺎب ﻧﺴﺒﺔ اﻟﻤﻌﻴﺐ ﻓﻲ آﻞ ﻋﻴﻨﺔ‬ ‫ﺡﺴﺎب ﺡﺪود اﻟﻀﺒﻂ ﻟﻠﻨﺴﺒﺔ‬ ‫رﺳﻢ ﺧﺮیﻄﺔ ﻧﺴﺒﺔ اﻟﻤﻌﻴﺐ ﻣﻊ ﺡﺪود اﻟﻀﺒﻂ‬ ‫دراﺳﺔ اﺳﺒﺎب أي اﻧﺤﺮاﻓﺎت ﻗﺪ ﻧﻼﺡﻈﻬﺎ‪.‬‬

‫اﻟﻌﺪد اﻻﺝﻤﺎﻟﻲ ﻟﻠﻮﺡﺪات ﻓﻲ آﻞ ﻋﻴﻨﺔ‬

‫=‪p‬‬

‫‪p Chart‬‬

‫ﺣﺴﺎب ﺣﺪود اﻟﻀﺒﻂ‬ ‫اﻟﺤﺪ اﻷﻋﻠﻰ ﻟﻠﻀﺒﻂ‬

‫‪Upper Control Limit‬‬

‫اﻟﺤﺪ اﻷدﻧﻰ ﻟﻠﻀﺒﻂ‬

‫‪Lower Control Limit‬‬

‫اﻻﻧﺤﺮاف اﻟﻤﻌﻴﺎري ﻟﻨﺴﺒﺔ اﻟﻤﻌﻴﺐ ‪σp‬‬

‫‪Control Limits‬‬

‫) ‪p (1 − p‬‬ ‫‪n‬‬

‫‪= p + z‬‬

‫) ‪p (1 − p‬‬ ‫‪n‬‬

‫‪= p − z‬‬

‫‪p‬‬

‫‪p‬‬

‫‪UCL‬‬

‫‪LCL‬‬

‫ﻣﺘﻮﺳﻂ ﻧﺴﺒﺔ اﻟﻤﻌﻴﺐ ﻓﻲ اﻟﻌﻴﻨﺎت ‪p‬‬ ‫‪s‬‬

‫ﻳﻤﺜﻞ ‪ z‬ﻡﻌﺎﻡﻞ ﺽﺮب ﻧﺴﺘﻌﻤﻠﻪ آﺎﻟﺘﺎﻟﻲ‪:‬‬

‫‪∑ xi‬‬

‫;‪• z = 2 for 95.5% limits‬‬

‫‪∑ ni‬‬

‫‪i =1‬‬ ‫‪s‬‬

‫= ‪p‬‬

‫‪i =1‬‬

‫‪• z = 3 for 99.7% limits‬‬ ‫‪٥‬‬

‫ﻡﺜﺎل ﻋﻤﻠﻲ ﻟﺨﺮﻳﻄﺔ ﻧﺴﺒﺔ اﻟﻤﻌﻴﺐ‬ ‫“‬

‫ﺵﺮآﺔ ﺹﻨﺎﻋﻴﺔ ﺕﺼﻨﻊ ﻗﻄﻊ ﻣﻴﻜﺎﻧﻴﻜﻴﺔ ﻟﻤﺤﺮآﺎت اﻟﺪیﺰل‪ .‬أﺧﺬت‬ ‫‪ ١٠‬ﻋﻴﻨﺎت ﻣﻦ ﺧﻂ اﻻﻧﺘﺎج‪ ،‬ﺕﺤﺘﻮي آﻞ واﺡﺪة ﻋﻠﻰ ‪١٠٠‬‬ ‫ﻗﻄﻌﺔ و ﺕﻢ اﻟﺘﻔﺘﻴﺶ ﻋﻨﻬﺎ ﺡﺴﺐ ﻣﻮاﺹﻔﺎت ﻣﻌﻴﻨﺔ و رﺹﺪت‬ ‫أﻋﺪاد اﻟﻘﻄﻊ اﻟﻤﻌﻴﺒﺔ ﻋﻠﻰ اﻟﺠﺪول اﻟﺘﺎﻟﻲ‪:‬‬

‫“‬

‫هﻞ ﻧﻈﺎم اﻟﺘﺼﻨﻴﻊ ﻣﻨﻀﺒﻂ اﺡﺼﺎﺋﻴﺎ أم ﻻ ؟‬

‫‪٦‬‬

‫‪٣‬‬

‫‪p chart‬‬

‫‪10‬‬

‫‪9‬‬

‫‪8‬‬

‫‪7‬‬

‫‪6‬‬

‫‪5‬‬

‫‪4‬‬

‫‪3‬‬

‫‪2‬‬

‫‪1‬‬

‫اﻟﻌﻴﻨﺔ‬

‫‪4‬‬

‫‪3‬‬

‫‪6‬‬

‫‪2‬‬

‫‪1‬‬

‫‪4‬‬

‫‪8‬‬

‫‪3‬‬

‫‪2‬‬

‫‪5‬‬

‫ﻋﺪد اﻟﻘﻄﻊ‬ ‫اﻟﻤﻌﻴﺒﺔ‬

‫ﻡﺜﺎل ﻋﻤﻠﻲ‬

‫‪p chart‬‬

‫‪٢‬‬

‫‪m = 10‬‬ ‫ﻋﺪد اﻟﻌﻴﻨﺎت‬ ‫ﻋﺪد اﻟﻘﻄﻊ ﻓﻲ آﻞ ﻋﻴﻨﺔ ‪n = 100‬‬ ‫‪10‬‬

‫‪9‬‬

‫‪8‬‬

‫‪7‬‬

‫‪6‬‬

‫‪5‬‬

‫‪4‬‬

‫‪3‬‬

‫‪2‬‬

‫‪1‬‬

‫اﻟﻌﻴﻨﺔ‬

‫‪4‬‬

‫‪3‬‬

‫‪6‬‬

‫‪2‬‬

‫‪1‬‬

‫‪4‬‬

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‫‪3‬‬

‫‪2‬‬

‫‪5‬‬

‫ﻋﺪد اﻟﻤﻌﻴﺐ‬

‫‪ 0.05 0.02 0.03 0.08 0.04 0.01 0.02 0.06 0.03 0.04‬ﻧﺴﺒﺔ اﻟﻤﻌﻴﺐ‬ ‫‪m‬‬

‫ﻣﺘﻮﺳﻂ ﻧﺴﺒﺔ اﻟﻤﻌﻴﺐ ﻓﻲ آﻞ اﻟﻌﻴﻨﺎت‬

‫‪= 0 . 038‬‬

‫‪∑ pˆ i‬‬

‫‪i=1‬‬

‫‪m‬‬

‫= ‪p‬‬

‫‪٧‬‬

‫ﻡﺜﺎل ﻋﻤﻠﻲ‬

‫‪٣‬‬

‫) ‪p (1 − p‬‬ ‫‪n‬‬

‫‪= p + z‬‬

‫‪p‬‬

‫‪UCL‬‬

‫) ‪p (1 − p‬‬ ‫‪n‬‬

‫‪= p − z‬‬

‫‪p‬‬

‫‪LCL‬‬

‫‪i=1‬‬ ‫‪s‬‬

‫= ‪p‬‬

‫‪z=3‬‬

‫‪p chart‬‬

‫ﺣﺴﺎب ﺣﺪود اﻟﻀﺒﻂ‬

‫‪s‬‬

‫‪i‬‬

‫‪∑ x‬‬

‫‪i‬‬

‫‪∑ n‬‬

‫‪i=1‬‬

‫)‪0.038(1 − 0.038‬‬ ‫‪= 0.095‬‬ ‫‪100‬‬

‫‪UCL = 0.038 + 3‬‬ ‫‪CL = 0.038‬‬

‫)‪0.038(1 − 0.038‬‬ ‫‪= −0.02 → 0‬‬ ‫‪100‬‬ ‫‪٨‬‬

‫‪٤‬‬

‫‪LCL = 0.038 + 3‬‬

‫رﺱﻢ ﺥﺮﻳﻄﺔ ﻧﺴﺒﺔ اﻟﻤﻌﻴﺐ‬

‫‪p chart‬‬

‫‪٤‬‬

‫‪P Chart‬‬ ‫‪0.10‬‬

‫‪3.0SL=0.09536‬‬

‫‪P=0.03800‬‬

‫‪- 3.0SL=0.000‬‬

‫اﻟﻨﺴﺒﺔ‬ ‫‪Proporti on‬‬

‫‪0.05‬‬

‫‪0.00‬‬ ‫‪10‬‬

‫‪9‬‬

‫‪8‬‬

‫‪7‬‬

‫‪6‬‬

‫‪5‬‬

‫‪4‬‬

‫‪3‬‬

‫‪2‬‬

‫‪1‬‬

‫‪0‬‬

‫رﻗﻢ اﻟﻌﻴﻨﺔ‬

‫‪Sampl e Number‬‬

‫‪٩‬‬

‫ﺧﺮیﻄﺔ ﻋﺪد اﻟﻌﻴﻮب ﻓﻲ اﻟﻮﺡﺪة ‪c chart‬‬ ‫“‬

‫ﺧﺮیﻄﺔ ﻋﺪد اﻟﻌﻴﻮب هﻲ إﺡﺪى أهﻢ ﺧﺮاﺋﻂ اﻟﺘﺤﻜﻢ )ﺽﺒﻂ اﻟﺠﻮدة(‬ ‫ﻟﻠﺨﻮاص‪:‬‬ ‫•‬

‫“‬

‫‪١٠‬‬

‫‪٥‬‬

‫ﻋﺒﺎرة ﻋﻦ أﻋﺪاد ﺡﻘﻴﻘﻴﺔ )ﻻ یﻤﻜﻦ ﺡﺴﺎﺑﻬﺎ ﺑﺎﻟﻨﺴﺒﺔ اﻟﻤﺌﻮیﺔ( ‪.‬‬

‫ﺕﺒﻴﻦ هﺬﻩ اﻟﺨﺮیﻄﺔ ﻋﺪد اﻟﻌﻴﻮب‬ ‫وﺡﺪة ﻣﻦ اﻟﻤﻨﺘﺞ ‪:‬‬

‫))‪(nonconformities (defects‬‬

‫ﻓﻲ آﻞ‬

‫•‬

‫اﻟﻮﺡﺪة ﻗﺪ ﺕﻜﻮن ﻣﺜﻼ آﺮﺳﻲ‪ ،‬ﺹﻔﻴﺤﺔ ﺹﻠﺐ أو ﺳﻴﺎرة ‪...‬‬

‫•‬

‫ﺡﺠﻢ اﻟﻮﺡﺪة یﻜﻮن ﺙﺎﺑﺖ‪.‬‬

‫•‬

‫ﻡﺜﺎل‪ :‬ﺡﺴﺎب ﻋﺪد اﻟﻌﻴﻮب )ﺧﺪوش‪ ،‬ﻣﺴﺎﻣﻴﺮ ﻏﻴﺮ ﻣﺜﺒﺘﺔ اﻟﺦ‪ (..‬ﻓﻲ آﻞ‬ ‫آﺮﺳﻲ ﻣﻦ ﻋﻴﻨﺔ ﺕﺤﺘﻮي ﻋﻠﻰ ‪ ١٠٠‬آﺮﺳﻲ‬

‫‪.‬‬

‫‪c Chart‬‬

‫ﺣﺴﺎب ﺣﺪود اﻟﻀﺒﻂ‬

‫‪Control Limits‬‬

‫اﻟﺤﺪ اﻷﻋﻠﻰ ﻟﻠﻀﺒﻂ‬

‫‪c‬‬

‫‪UCL c = c + 3z‬‬

‫اﻟﺤﺪ اﻷدﻧﻰ ﻟﻠﻀﺒﻂ‬

‫ﻋﺪد اﻟﻌﻴﻮب ﻓﻲ آﻞ وﺡﺪة ‪ci‬‬

‫‪c‬‬ ‫ﻳﻤﺜﻞ ‪ z‬ﻡﻌﺎﻡﻞ ﺽﺮب ﻧﺴﺘﻌﻤﻠﻪ‬ ‫ﻟﺘﺤﺪﻳﺪ اﻟﺤﺪود اﻟﻤﺮاد ﺕﺤﻘﻴﻘﻬﺎ‪:‬‬ ‫;‪• z = 2 : for 95.5% limits‬‬ ‫‪• z = 3 : for 99.7% limit‬‬

‫‪Upper Control Limit‬‬

‫‪Lower Control Limit‬‬

‫‪LCL c = c − 3z‬‬

‫ﻡﺘﻮﺱﻂ ﻋﺪد اﻟﻌﻴﻮب ﻓﻲ آﻞ اﻟﻮﺣﺪات‬ ‫‪k‬‬ ‫‪∑ ci‬‬ ‫‪c = i =1‬‬

‫‪k‬‬

‫‪١١‬‬

‫ﻡﺜﺎل ﻋﻤﻠﻲ ﻟﺨﺮﻳﻄﺔ ﻋﺪد اﻟﻌﻴﻮب‬ ‫“‬

‫“‬

‫‪١٢‬‬

‫‪٦‬‬

‫‪c chart‬‬

‫ﺵﺮآﺔ وودﻻﻧﺪ ﺕﺼﻨﻊ ورق ﻟﻄﺒﺎﻋﺔ اﻟﺠﺮاﺋﺪ‪ .‬ﻓﻲ ﺁﺧﺮ ﻣﺮﺡﻠﺔ‬ ‫اﻻﻧﺘﺎج ﻗﺎم ﻣﻔﺘﺶ اﻟﺠﻮدة ﻟﺪى اﻟﺸﺮآﺔ ﺑﺎﻟﺘﻔﺘﻴﺶ ﻋﻦ ﺝﻮدة‬ ‫اﻟﻮرق ﺑﺈﺝﺮاء ﻗﻴﺎﺳﺎت ﻟﺨﺼﺎﺋﺺ اﻟﺠﻮدة ﻋﻠﻰ ‪ 5‬ﻟﻔﺎت ﻣﻦ‬ ‫اﻟﻮرق اﻟﻤﺼﻨﻊ و رﺹﺪ اﻟﻨﺘﺎﺋﺞ اﻟﻤﺠﺪوﻟﺔ أدﻧﺎﻩ‪.‬‬ ‫اﻟﻤﻄﻠﻮب ‪ :‬ﻋﻦ ﻃﺮیﻖ ﺧﺮیﻄﺔ اﻟﺘﺤﻜﻢ ﻟﻌﺪد اﻟﻌﻴﻮب ادرس‬ ‫اﺳﺘﻘﺮار اﻟﻌﻤﻠﻴﺔ اﻟﺘﺼﻨﻴﻌﻴﺔ ﻟﻠﺸﺮآﺔ )هﻞ ﻧﻈﺎم اﻟﺘﺼﻨﻴﻊ ﻣﻨﻀﺒﻂ‬ ‫)اﺡﺴﺐ ب‪(z = 2 :‬‬ ‫اﺡﺼﺎﺋﻴﺎ أم ﻻ ؟(‬ ‫‪5‬‬

‫‪4‬‬

‫‪3‬‬

‫‪2‬‬

‫‪1‬‬

‫اﻟﻠﻔﺔ‬

‫‪24‬‬

‫‪22‬‬

‫‪17‬‬

‫‪21‬‬

‫‪16‬‬

‫ﻋﺪد اﻟﻌﻴﻮب‬

‫‪c Chart‬‬

‫ﺣﺴﺎب ﺣﺪود اﻟﻀﺒﻂ‬

‫‪Control Limits‬‬

‫ﻣﺘﻮﺳﻂ ﻋﺪد اﻟﻌﻴﻮب ﻓﻲ آﻞ اﻟﻮﺡﺪات‬ ‫‪c = (16+21+17+22+24) / 5 = 20‬‬

‫‪c = 20‬‬

‫‪z=2‬‬

‫اﻟﺤﺪ اﻷﻋﻠﻰ ﻟﻠﻀﺒﻂ‬

‫‪Upper Control Limit‬‬

‫‪UCLc = c + z c = 28.94‬‬ ‫اﻟﺤﺪ اﻷدﻧﻰ ﻟﻠﻀﺒﻂ‬

‫‪c = 11.06‬‬

‫‪Lower Control Limit‬‬

‫‪LCLc = c – z‬‬

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‫رﺱﻢ ﺥﺮﻳﻄﺔ ﻋﺪد اﻟﻌﻴﻮب‬

‫‪c chart‬‬

‫ﻋﺪد اﻟﻌﻴﻮب‬

‫رﻗﻢ اﻟﻌﻴﻨﺔ )اﻟﻠﻔﺔ(‬

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‫ – ﺣﺎﻟﺔ دراﺱﻴﺔ ﻋﻦ ﺏﻨﻚ‬١ ‫ﺕﺪرﻳﺐ‬ Construction of p chart Example 1

Sample Number

Wrong Account Number

1 2 3 4 5 6 7 8 9 10 11 12

15 12 19 2 19 4 24 7 10 17 15 3

The operations manager of the booking services department of Hometown Bank is concerned about the number of wrong customer account numbers recorded by Hometown personnel. Each week a random sample of 2,500 deposits is taken, and the number of incorrect account numbers is recorded. The records for the past 12 weeks are shown in the following table.

Is the process out of control? (Use 3-sigma control limits.)

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٢ ‫ﺕﺪرﻳﺐ‬ Construction of c chart Example 2 Surface defects have been counted on 25 rectangular steel plates, and the data are shown in the table. Construct a c control chart for nonconformities using this data to study if the process is under control

Plate No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

No. of Nonconformities 1 0 4 3 1 2 5 0 2 1 1 0 8 0 2 1 3 5 4 6 3 1 0 2 4

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‫ﺟﺰاآﻢ اﷲ ﺥﻴﺮا ﻋﻠﻰ ﺣﺴﻦ اﻻﺱﺘﻤﺎع‬

‫هﻞ ﻡﻦ أﺱﺌﻠﺔ ؟‬ ‫‪٢٠٠٠‬‬

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