Extensions and characterization of optimal maintenance policies for multistate partially observed Markovian systems

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Fadel AlDurgam, Mohammad Mansour
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Saudi Digital Library
Abstract
This research aims at modeling and characterizing optimal maintenance policies for complex deteriorating systems. Such systems widely exist in manufacturing enterprises where it has been gaining the interest of researchers for many decades. Representing complex systems as multi-state systems has been the trend in the literature (Derman, 1962), (Hopp and Wu, 1990) and (Maillart, 2006). Markov Decision Process (MDP) and Partially Observed Markov Decision Process (POMDP) have been used as the general frame to model and represent such systems. The main objective of this work is to characterize optimal maintenance policies for multistate systems over the systems state occupancy vectors ordered by the first order stochastic dominance. New set of conditions to guarantee the existence of optimal threshold-type maintenance policies are provided. The main advantage of the developed conditions is ensuring the first order stochastic dominance to survive conditioning. As a new approach, this is achieved by developing new relations with other useful partial orders which were not considered for this problem before, namely, the reverse hazard rate and the component wise dominance partial orders. For the sake of illustration, a two-state model is provided first, and then it is extended to the case of n-states. In order to link maintenance, operation and quality, a new model within the POMDP framework is formulated. The model uses Overall Systems Effectiveness (OSE) as a criterion. OSE combines systems availability, process rate and quality rate in a composite criterion. This provides a mechanism that ties maintenance and operation through the process rate. In such situation the optimal action will be a maintenance action coupled with a specific system speed that reflects the process rate. ﻷﺳﻢ ﻣﺤﻤﺪ ﺑﻦ ﻣﻨﺼﻮر ﻓﺎﺿﻞ اﻟﻀﺮﻏﺎم : ﺗﻤﺜﻴﻞ ﺳﻴﺎﺳﺎت اﻟﺼﻴﺎﻧﺔ اﻟﻤﺜﻠﻰ ﻟﻸﻧﻈﻤﺔ اﻟﻤﺎرآﻮﻓﻴﺔ ﻣﺘﻌﺪدة اﻟﺤﺎﻻت ﺟﺰﺋﻴﺔ اﻟﻤﺸﺎهﺪة ﻋﻨﻮان اﻷﻃﺮوﺣﺔ : اﻟﺘﺨﺼﺺ اﻟﺪﻗﻴﻖ هﻨﺪﺳﺔ ﻧﻈﻢ : ﻳﻮﻧﻴﻮ 2009 اﻟﺘﺎرﻳﺦ : ﻳﻬﺪف هﺬا اﻟﺘﺮآﻴﺐ اﻟﺒﺤﺚ إﻟﻰ ﻧﻤﺬﺟﺔ و ﺗﻤﺜﻴﻞ ﺳﻴﺎﺳﺎت اﻟﺼﻴﺎﻧﺔ اﻟﻤﺜﻠﻰ ﻟﻸﻧﻈﻤﺔ ﻣﻌﻘﺪة اﻟﺰﻣﻦ و اﻟﺘﻲ ﺗﺘﻬﺎﻟﻚ ﻣﻊ ﻣﺮور . ﺗﻮﺟﺪ ﻣﺜﻞ هﺬﻩ اﻷﻧﻈﻤﺔ ﻓﻲ اﻟﻤﺆﺳﺴﺎت اﻟﺼﻨﺎﻋﻴﺔ و ﻏﻴﺮهﺎ و ﻗﺪ اﺳﺘﺤﻮذت ﻋﻠﻰ اهﺘﻤﺎم اﻟﺒﺎﺣﺜﻴﻦ ﻓﻲ هﺬا اﻟﻤﺠﺎل ﻟﻌﺪة ﻋﻘﻮد . إن اﻷﺳﻠﻮب اﻟﻤﺘﺒﻊ ﻟﻠﻨﻤﺬﺟﺔ اﻟﺮﻳﺎﺿﻴﺔ ﻟﻤﺜﻞ هﺬﻩ اﻷﻧﻈﻤﺔ اﻟﺤﺎﻻت هﻮ ﺗﻤﺜﻴﻠﻬﺎ آﺄﻧﻈﻤﺔ ﻣﺘﻌﺪدة . ﺑﺎﺳﺘﺨﺪام اﻟﻨﻤﺎذج اﻟﻤﺎرآﻮﻓﻴﺔ آﺎﻣﻠﺔ وﺟﺰﺋﻴﺔ اﻟﻤﺸﺎهﺪة . ان اﻟﻬﺪف اﻟﺮﺋﻴﺲ ﻟﻬﺬا اﻟﺒﺤﺚ هﻮ ﻧﻤﺬﺟﺔ و ﺗﻤﺜﻴﻞ ﻗﺮارات اﻟﺼﻴﺎﻧﺔ اﻟﻤﺜﻠﻰ ﻟﻸﻧﻈﻤﺔ ﻣﺘﻌﺪدة اﻟﺤﺎﻻت و اﻟﻤﺸﺎهﺪة ﺟﺰﺋﻴﺎ ﻋﻠﻰ ﻋﻨﺎﺻﺮ اﻟﻤﺠﺎل )ﻟﻠﻨ ﻣﺼﻔﻮﻓﺎت أﺣﺎدﻳﺔ اﻟﺒﻌﺪ ﺗﻤﺜﻞ اﻟﺤﺎﻟﺔ اﻟﻤﺘﻮﻗﻌﺔ ﻈﺎم (اﻟﺮﺗﺒﺔ و اﻟﻤﺮﺗﺒﺔ وﻓﻖ اﻟﻬﻴﻤﻨﺔ اﻟﻌﺸﻮاﺋﻴﺔ أﺣﺎدﻳﺔ . ﺗﻢ اﺳﺘﺤﺪاث ﺑﻌﺾ اﻟﺸﺮوط اﻟﺠﺪﻳﺪة و اﻟﺘﻲ ﺗﻀﻤﻦ وﺟﻮد ﺳﻴﺎﺳﺎت ﺻﻴﺎﻧﺔ ﻣﺜﻠﻰ و ﻣﻨﺘﻈﻤﺔ . إن هﺬﻩ اﻟﺸﺮوط ﺗﺘﻤﻴﺰ ﺑﻀﻤﺎن اﺳﺘﻤﺮارﻳﺔ و ﺑﻘﺎء ﻋﻨﺎﺻﺮ اﻟﻤﺠﺎل ﻣﺮﺗﺒﺔ وﻓﻖ اﻟﻬﻴﻤﻨﺔ اﻟﻌﺸﻮاﺋﻴﺔ أﺣﺎدﻳﺔ اﻟﺮﺗﺒﺔ ﻋﻠﻰ ﻣﺪى اﻹﻃﺎر اﻟﺰﻣﻨﻲ ﻟﻠﺤﻞ .آﺄﺳ ﻠﻮب ﺟﺪﻳﺪ ﻟﺪراﺳﺔ هﺬﻩ اﻟﻤﺴﺄﻟﺔ، ﺗﻤﺖ دراﺳﺔ اﻟﻌﻼﻗﺔ ﺑﻴﻦ اﻟﺘﺮﺗﻴﺐ اﻟﺠﺰﺋﻲ أﺣﺎدي اﻟﺮﺗﺒﺔ و ﺑﻌﺾ اﻟﺘﺮﺗﻴﺒﺎت اﻟﺠﺰﺋﻴﺔ اﻷﺧﺮى و اﻟﺘﻲ ﻟﻢ ﺗﺆﺧﺬ ﺑﻌﻴﻦ اﻻﻋﺘﺒﺎر ﻣﻦ ﻗﺒﻞ ﻟﺤﻞ هﺬﻩ اﻟﻤﺴﺄﻟﺔ، ﻣﺜﻞ ﺗﺮﺗﻴﺐ اﻟﺨﻄﻮرة اﻟﻌﻜﺴﻴﺔ و هﻴﻤﻨﺔ اﻟﻌﻨﺎﺻﺮ اﻟﻜﻠﻴﺔ . و ﺑﻬﺪف اﻹﻳﻀﺎح ﺗﻢ ﻋﺮض ﻧﻤﻮذج رﻳﺎﺿﻲ ﻟﻨﻈﺎم ذو اﻟﻌﺎﻣﺔ ﺣﺎﻟﺘﻴﻦ وﻣﻦ ﺛﻢ ﻃﻮر ﻟﻠﺤﺎﻟﺔ . ﺗﻢ رﺑﻂ ﺳﻴﺎﺳﺎت اﻟﺼﻴﺎﻧﺔ واﻹﻧﺘﺎج و اﻟﺠﻮدة ﻋﻦ ﻃﺮﻳﻖ ﻧﻤﻮذج رﻳﺎﺿﻲ ﻣﺎرآﻮﻓﻲ ﻳﻬﺪف إﻟﻰ ﺗﻌﻈﻴﻢ اﻟﻜﻔﺎء ﻟﻠﻨﻈﺎم ة اﻟﻜﻠﻴﺔ . إن ﻣﺼﻄﻠﺢ " اﻟﻜﻔﺎء اﻟﻜﻠﻴﺔ ة "اﻟﺠﻮدة ﻟﻸﻧﻈﻤﺔ ﻳﻤﺜﻞ ﻣﻘﻴﺎس ﻣﺘﻜﺎﻣﻞ ﻟﻤﺪى ﺟﺎهﺰﻳﺔ اﻟﻨﻈﻢ ﻟﻺﻧﺘﺎج ﻋﻠﻰ ﻣﺪار اﻟﺴﺎﻋﺔ و ﻣﻌﺪل . اﺳﺘﺨﺪم اﻟﻨﻤﻮذج ﻹﻳﺠﺎد ﺳﻴﺎﺳﺎت اﻟﺼﻴﺎﻧﺔ واﻹﻧﺘﺎج اﻟﻤﺜﻠﻰ ﻟﻸﻧﻈﻤﺔ وﺗﻢ إﺛﺒﺎت إن اﻟﺪاﻟﺔ اﻟﻬﺪﻓﻴﺔ ﻟﻬﺬا اﻟﻨﻤﻮذج داﻟﺔ ﻣﻘﻌﺮة ﻣﻜﻮﻧﺔ ﻣﻦ ﻗﻄﻊ ﺧﻄﻴﺔ ﻣﺴﺘﻘﻴﻤﺔ . ﺗﻤﺖ أﻳﻀﺎ ﻧﻤﺬﺟﺔ وﺗﻤﺜﻴﻞ اﺛﺮ اﻟﺨﻄﺄ ﻓﻲ اﻟﻘﻴﺎس ﻋﻠﻰ ﺳﻴﺎﺳﺎت اﻟﺼﻴﺎﻧﺔ اﻟﻤﺜﻠﻰ ﻟﻠﻨﻈﻢ اﻟﻤﺎرآﻮﻓﻴﺔ ﻣﺘﻌﺪدة اﻟﺤﺎﻻت و اﻟﻤﺸﺎهﺪة ﺟﺰﺋﻴﺎ . وﻗﺪ اﺷُﺘﻘﺖ ﻣﻌﺎدﻟﺔ- ﺑﻴﺰ ﺑﺎﺳﺘﺨﺪام ﻗﺎﻧﻮن - ﻟﺘﻮﻗﻊ ﺣﻠﺔ اﻟﻨﻈﺎم اﻟﻤﻘﺘﺮح و اﻟﻤﻤﺜﻞ ﺑﺴﻠﺴﻠﺔ ﻣﺎرآﻮﻓﻴﺔ ﺛﻼﺛﻴﺔ اﻟﻄﺒﻘﺎت و ﺟﺰﺋﻴﺔ اﻟﻤﺸﺎهﺪة .اﻟ وﺗﻢ إﺛﺒﺎت أن ﻣﺘﺼﻠﺔ ﺪاﻟﺔ اﻟﻬﺪﻓﻴﺔ ﻟﻬﺬا اﻟﻨﻤﻮذج ﺑﻮﺟﻮد أﺧﻄﺎء اﻟﻘﻴﺎس داﻟﺔ ﻣﻘﻌﺮة ﻣﻜﻮﻧﺔ ﻣﻦ ﻗﻄﻊ ﺧﻄﻴﺔ ﻣﺴﺘﻘﻴﻤﺔ و . و ﻓﻲ اﻟﻨﻬﺎﻳﺔ ُدرﺳﺖ اﻟﻌﻼﻗﺔ ﺑﻴﻦ ﻧﻮﻋﻴﺔ اﻟﻤﻌﻠﻮﻣﺎت أو اﻟﻘﺮاءات اﻟﻤﻘﺎﺳﺔ ﻟﻼﺳﺘﺪﻻل ﻋﻠﻰ ﺣﺎﻟﺔ اﻟﻨﻈﺎم و أﺛﺮ اﻟﺨﻄﺄ ﻓﻲ اﻟﻘﻴﺎس ﻋﻠﻰ ﺳﻴﺎﺳﺎت اﻟﺼﻴﺎﻧﺔ اﻟﻤﺜﻠﻰ . درﺳﺖ اﻷﻃﺮوﺣﺔ ﺣﺎﻟﺔ وﺟﻮ د رأي اﻟﺨﺒﺮاء آﺄﺣﺪ ﻣﻌﻄﻴﺎت اﻟﻤﺴﺄﻟﺔ ﺑﺤﻴﺚ ﻳﻜﻮن رأي ﺧﺒﺮاء اﻟﺼﻴﺎﻧﺔ هﻮ اﻷﻓﻀﻞ ﻟﻠﺮﺑﻂ ﺑﻴﻦ ﻧﻮﻋﻴﺔ اﻟﻤﺸﺎهﺪات أو اﻟﻘﺮاءات اﻟﻤﺄﺧﻮذة ﻣﻦ اﻟﻨﻈﺎم و اﻟﺤﺎﻟﺔ اﻟﺤﻘﻴﻘﻴﺔ ﻟﻠﻨﻈﺎم . ﻟﺬا ﺗﻢ اﻓﺘﺮاض وﺟﻮد ﻋﻤﻠﻴﺔ اﺗﺨﺎذ ﻗﺮار ﻟﻨﻈﺎم ﻣﺎرآﻮﻓﻲ ﻣﺸﺎهﺪ ﺟﺰﺋﻴﺎ و ذﻟﻚ ﺑﻮﺟﻮد ﻣﺸﺎهﺪات ﻏﻴﺮ واﺿﺤﺔ ﺛﻢ ﺗﻢ اﻗﺘ ﺮاح اﻗﺘﺮان ﻋﻀﻮﻳﺔ ﻟﻠﻤﺸﺎهﺪات ﻏﻴﺮ اﻟﻮاﺿﺤﺔ ﻣﻦ اﺟﻞ إﻋﺎدة ﺗﻜﻮﻳﻦ اﻟﻤﺼﻔﻮﻓﺔ اﻟﺘﻲ ﺗﻤﺜﻞ اﻟﻌﻼﻗﺔ ﺑﻴﻦ اﻟﻤﺸﺎهﺪات و اﻟﺤﺎﻟﺔ اﻟﺤﻘﻴﻘﻴﺔ ﻟﻠﻨﻈﺎم و ﻗﺪ ﻋﺮﺿﺖ ﺑﻌﺾ اﻷﻣﺜﻠﺔ ﻟﺘﻮﺿﻴﺢ هﺬﻩ اﻟﺤﺎﻟﺔ. اﻟﻤﺴﺘﻘﺒﻠﻴﺔ ﺧﺘﻤﺖ اﻷﻃﺮوﺣﺔ ﺑﻤﻠﺨﺺ ﻟﻨﺘﺎﺋﺞ اﻟﺒﺤﺚ و ﻣﻘﺘﺮﺣﺎت ﻟﻠﺒﺤﻮث .
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