Determining units Ranking with the Best and the Worst Rank of Each unit as well as Interval Ranking . "

In this paper, we did unit ranking by determining the best and the worst rank that each unit could have. By applying the optimistic Model, we determine the best rank of each unit, then by using the Pessimistic model, we determine the worst rank of that unit. The highest rank of each unit is displayed by r°b and the lowest rank is displayed by r°w.


Introduction
Measuring the efficiency of a decision making unit (DMU) has long been considered as a difficult task because of dealing with complex economic and behavioral entities.This task becomes more difficult when it involves multiple inputs and multiple outputs.Data Envelopment Analysis (DEA) is apowerful managerial approach in measuring the relative efficiency of decision making unit with multiple inputs and outputs.Study

Data Envelopment Analysis and Decision Science
We determine the best and worst ranking each DMUo can possess and all possible choices from DEA weights, all DMUs which had been proposed and could be made.To accomplish this purpose, we need to put forward some symbolism and definitions.Determining by v=(v 1 …v n ) and u=(u 1 …u n ), s n and m n matrixes sequently from input weight vectors and output weight vectors that each DMU makes for giving choicefrom DEA, are related.

Preliminaries and notations
Definition 2.1.

Interval ranking
Now, after achieving the best or the highest rank of each unit as well as the worst or the lowest rank of that unit, we well have a ranking interval for that DMU.Now, if the best or the highest rank of a unit was equal to the worst or the lowest rank of that unit, we definitely assert that the rank of that DMU is what is gained from    and    calculation.In other words,    =rank DMU=    .
But, if the best or the highest rank of a unit was not equal to the worst or the lowest rank of that unit, to compare it with the rank of another unit, we use the following method.
For example, if the best or the highest rank of DMU j equals "a" and the worst and the lowest rank of DMU j equals "b", in addition, for DMU o , the best or the highest rank of DMU o equals "C" and the worst or the lowest rank of DMU o equals "b", then to compare the rank of these two units in order to see that which one has acquired a better rank, we use this method: For DMU j , we have: rank ∈ [a,b] , For DMUo, we have: rank ∈ [c,b] Now, we calculate the interval average for ranking interval of DMU o and DMU j .The one that has a better interval average, will acquire a better rank compared with the other one.However, if interval average for ranking intervals of the two unit was the same, we will compare the begining of the unit will be better and stays at a higher rank.If again, they were equal by any chance, we'll compare the and of the intervals; if they happened to be equal, we'll do the same procedure for other spots of the intervals till we reach to a different result and find a unit in a higher rank compared to its counterpart unit in the other interval.
in DEA was started by Charnes, Cooper and Rhodes in 1978 by introducing the first DEA model that known as CCR model [1].Then in 1984 Banker, Charnes and Cooper introduced the BCC model which was the variable return to scale version of the CCR model [2].Cross-efficiency score  ̅  is gained from DMUj evaluation and this value can be used to rank DMUs.Production and hosting by ISPACS GmbH.
) is a set of DMUs with exact crossefficiency scores, higher than DMUo when (v,u) are DEA weights that have chosen DMU.  (  ,  ) = {   ,  = 1 , … ,  |  ̅  <  ̅  },   (  ,  ) is defined the same way by simple analysis of DMU with cross-efficiency scores lower lower than DMUo, instead of them, with higher exact scores.In this manner, first by using    model that is stated bellow, we calculate the better rank for each unit: Now, using    model, we calculate the low rank or in other words, the worst rank that each unit may have: