مرزهاي دوگانه در تحليل پوششي داده هاي نامحدب با داده هاي بازه اي:ارزيابي كارايي و ناكارايي و تحليل پايداري

Dual Frontiers in Non-convex Data Envelopment Analysis: Efficiency and In-efficiency Assessment and Stability Analysis

ﻣﺪلﻫﺎي اﺳﺎﺳﯽ ﺗﺤﻠﯿﻞ ﭘﻮﺷﺸﯽ دادهﻫﺎ ﺑﻪ ﻃﻮر ذاﺗﯽ ارزﯾﺎﺑﯽ واﺣﺪﻫﺎي ﺗﺼﻤﯿﻢﮔﯿﺮﻧﺪه را ﺑﺎ دﯾﺪﮔﺎه ﺧﻮشﺑﯿﻨﺎﻧﻪ اﻧﺠﺎم ﻣﯽدﻫﻨﺪ، ﺑﻪ اﯾﻦ ﻣﻔﻬﻮم ﮐﻪ ارزﯾﺎﺑﯽ ﻋﻤﻠﮑﺮد ﻫﺮ واﺣﺪ ﺗﺼﻤﯿﻢﮔﯿﺮﻧﺪه از ﻃﺮﯾﻖ ﻣﻘﺎﯾﺴﻪ آن واﺣﺪ ﺑﺎ ﻣﺮز ﮐﺎراﯾﯽ اﻧﺠﺎم ﻣﯽﺷﻮد. ﻣﺮز ﮐﺎراﯾﯽ در واﻗﻊ ﻣﺮز ﻣﺘﺸﮑﻞ از ﻫﻤﻪ واﺣﺪﻫﺎﯾﯽ اﺳﺖ ﮐﻪ ﻧﺸﺎندﻫﻨﺪه ﺑﻬﺘﺮﯾﻦ ﻋﻤﻠﮑﺮد ﻫﺴﺘﻨﺪ. اﮔﺮ ﯾﮏ ﻓﻌﺎﻟﯿﺖ روي ﻣﺮز ﮐﺎراﯾﯽ ﻗﺮار داﺷﺘﻪ ﺑﺎﺷﺪ ﮐﺎﻣﻼ ﮐﺎرا و در ﻏﯿﺮ اﯾﻦ ﺻﻮرت ﻏﯿﺮﮐﺎرا ﻧﺎﻣﯿﺪه ﻣﯽﺷﻮد. ﺑﻪ ﻣﻨﻈﻮر اراﺋﻪ ﯾﮏ ارزﯾﺎﺑﯽ دﻗﯿﻘﺘﺮ ﻣﯽﺗﻮان وﺿﻌﯿﺖ واﺣﺪﻫﺎي ﺗﺼﻤﯿﻢﮔﯿﺮﻧﺪه را ﺑﺎ دﯾﺪﮔﺎه ﺑﺪﺑﯿﻨﺎﻧﻪ ﻧﯿﺰ ﻣﻮرد ﺑﺮرﺳﯽ ﻗﺮار داد، ﺑﻪ اﯾﻦ ﻣﻔﻬﻮم ﮐﻪ ﻣﺮزي ﺗﺤﺖ ﻋﻨﻮان ﻣﺮز ﻧﺎﮐﺎراﯾﯽ ﻣﺘﺸﮑﻞ از ﻫﻤﻪ واﺣﺪﻫﺎﯾﯽ ﮐﻪ ﺑﺪﺗﺮﯾﻦ ﻋﻤﻠﮑﺮد را دارﻧﺪ، ﺗﺸﮑﯿﻞ داده و ﺳﭙﺲ ﻋﻤﻠﮑﺮد ﻫﺮ واﺣﺪ ﺗﺼﻤﯿﻢﮔﯿﺮﻧﺪه را ﻧﺴﺒﺖ ﺑﻪ آن ارزﯾﺎﺑﯽ ﻧﻤﻮد، ﺑﻪ اﯾﻦ ﺻﻮرت ﮐﻪ ﻫﺮ ﭼﻪ ﯾﮏ واﺣﺪ ﺗﺼﻤﯿﻢﮔﯿﺮﻧﺪه ﺑﻪ ﻣﺮز ﻧﺎﮐﺎراﯾﯽ ﻧﺰدﯾﮑﺘﺮ ﺑﺎﺷﺪ، ﻧﺎﮐﺎراﺗﺮ ﺗﻠﻘﯽ ﻣﯽﺷﻮد. ﺑﻪ ﺻﻮرت ﻣﺸﺎﺑﻪ ﯾﮏ ﻓﻌﺎﻟﯿﺖ را ﮐﺎﻣﻼ ﻧﺎﮐﺎرا ﮔﻮﯾﯿﻢ اﮔﺮ روي ﻣﺮز ﻧﺎﮐﺎراﯾﯽ ﻗﺮار داﺷﺘﻪ ﺑﺎﺷﺪ. در ﻏﯿﺮ اﯾﻦ ﺻﻮرت آن را ﻏﯿﺮ ﻧﺎﮐﺎرا ﻣﯽﻧﺎﻣﯿﻢ. در اﯾﻦ ﻣﻘﺎﻟﻪ ﺑﺎ در ﻧﻈﺮ ﮔﺮﻓﺘﻦ اﯾﻦ ﻓﺮض ﮐﻪ ﻣﺠﻤﻮﻋﻪ اﻣﮑﺎن ﺗﻮﻟﯿﺪ ﻧﺎﻣﺤﺪب اﺳﺖ، ﺑﻪ ﺗﺤﻠﯿﻞ ﮐﺎراﯾﯽ و ﻧﺎﮐﺎراﯾﯽ واﺣﺪﻫﺎي ﺗﺤﺖ ﺑﺮرﺳﯽ ﭘﺮداﺧﺘﻪ و آﻧﻬﺎ را در دو رده ﮐﺎﻣﻼ )ﻧﺎ(ﮐﺎرا و ﻏﯿﺮ )ﻧﺎ(ﮐﺎرا اﻓﺮاز ﻣﯽﮐﻨﯿﻢ. ﺳﭙﺲ ﻣﻔﻬﻮم ﭘﺎﯾﺪاري اﻓﺮاز را در ﺗﺤﻠﯿﻞ ﮐﺎراﯾﯽ و ﻧﺎﮐﺎراﯾﯽ ﻣﻮرد ﺑﺮرﺳﯽ ﻗﺮار ﻣﯽدﻫﯿﻢ. در ﻧﻬﺎﯾﺖ ﺑﺎ ﻓﺮض اﯾﻦ ﮐﻪ واﺣﺪﻫﺎي ﺗﺤﺖ ارزﯾﺎﺑﯽ داراي ورودي و ﺧﺮوﺟﯽ ﺑﺎزهاي ﻫﺴﺘﻨﺪ، ﺑﻪ ارزﯾﺎﺑﯽ ﮐﺎراﯾﯽ و ﻧﺎﮐﺎراﯾﯽ آنﻫﺎ ﭘﺮداﺧﺘﻪ و ﺳﭙﺲ ﺑﺮ اﺳﺎس ﻣﻘﺪار ﺑﻬﯿﻨﻪ ﻣﺪلﻫﺎي ارزﯾﺎﺑﯽ ﻋﻤﻠﮑﺮد واﺣﺪﻫﺎي ﺗﺼﻤﯿﻢ ﮔﯿﺮﻧﺪه را ﺑﻪ ﺳﻪ زﯾﺮﻣﺠﻤﻮﻋﻪ اﻓﺮاز ﮐﺮده و ﻣﻮﺿﻮع ﭘﺎﯾﺪاري را ﺑﺮاي اﻓﺮازﻫﺎي ﺣﺎﺻﻞ ﺑﺮرﺳﯽ ﻣﯽﮐﻨﯿﻢ. ﺑﻪ اﯾﻦ ﺻﻮرت ﮐﻪ ﺑﺮاي ﻫﺮ واﺣﺪ ﺗﺼﻤﯿﻢﮔﯿﺮﻧﺪه ﺑﺮ ﺣﺴﺐ اﯾﻨﮑﻪ اﯾﻦ واﺣﺪ در ﮐﺪاﻣﯿﮏ از زﯾﺮﻣﺠﻤﻮﻋﻪﻫﺎي اﻓﺮاز ﻗﺮار ﻣﯽﮔﯿﺮد، ﺷﻌﺎع ﭘﺎﯾﺪاري را ﺑﺮاي ﺣﻔﻆ ﻣﻮﻗﻌﯿﺖ ﻓﻌﻠﯽ آن ﺑﻪدﺳﺖ ﻣﯽآورﯾﻢ.

Basic models of Data Envelopment Analysis intrinsically evaluate the decision making units with an optimistic point of view, in the sense that the efficiency status of each unit is evaluated by means of calculating its distance from the efficiency frontier. The efficiency frontier is in fact composed of all units indicating the best practice, in the sense that for each one there exist no other (virtual) unit with a better performance. A unit located on this frontier is called fully efficient and non-efficient, otherwise. In order to provide a more precise assessment, one can evaluate units with a pessimistic point of view, in the sense that a frontier consisting of the worst performance, called the in-efficient frontier is formed and then each unit is evaluated with respect to its distance from this frontier, in a way that the closer the unit to the in-efficient frontier, the more in-efficient it is. In this paper, assuming that the production technology is non-convex, we perform efficiency and in-efficiency evaluation and then, based on the optimal value of the corresponding (in-) efficiency model, we partition all units in two subsets called (in-)efficient and non- (in-)efficient units. Then we investigate the concept of stability of the obtained partitions, by means of presenting related multi objective programs. In the next step, assuming that the input and output data of all units are real intervals, we deal with the efficiency and in-efficiency analysis of units and partition them into three subsets, in each case.