Rank reversal problem related to wash criterion in analytic hierarchy process (AHP)

The rank reversal problem related to wash criterion in Analytic Hierarchy Process (AHP) was studied. The purpose of this paper is threefold. Firstly, the rank reversal problem did not come off when proportional adjustment of Liberatore and Nydick (2004) and Saaty and Vargas (2006) was applied. Secondly, the conditions that insure the occurrence of rank reversal by using the approach of Finan and Hurley (2002) and Lin, Chou, Chouhuang and Hsu (2008) were found. Thirdly, the study showed that the invariant phenomenon proposed by Jung, Wou, Li and Julian (2009) implied that the combined criterion is still a wash criterion. The findings will help researchers select the synthesizing method and keep the rank under some specific range when a wash criterion is deleted.


INTRODUCTION
A criterion, say J 0 is defined as a wash criterion if all alternatives have the same weight corresponding to J 0 (Finan and Hurley, 2002).They mentioned that sometimes during medical research some criteria (sub-criteria) in the second bottom level are wash criteria.If researchers are allowed to delete those wash criteria without influencing the final ranking, then the deletion of those wash criteria will simplify the process of constructing comparison matrices in the upper level.Deleting wash criteria will bring us a lot of economical benefits when assessing alternatives by Analytic Hierarchy Process (AHP).Hence, they started to pay attention to rank reversal problems with wash criteria.The theorem was proved for any threelevel hierarchy with a wash criterion which will not influence the final ranking of alternatives.Next, they mentioned that any hierarchy can be compressed into a three-level hierarchy.Finally, they constructed a fourlevel hierarchy problem with and without wash criteria to imply a rank reversal phenomenon.They then raised the question of "Whether or not a wash criterion could be deleted".It was announced that their findings were a *Corresponding author.E-mail: kehaojan@yahoo.com.
severe challenge to the legitimacy of AHP (Finan and Hurley, 2002).
Up to now, there have been several papers discussing the wash criteria for rank reversal problems in the AHP.The upper level relative weights should be reassessed after the deletion of wash criterion so that the researcher develops another AHP problem.This way whether or not rank reversal happened is irrelevant (Liberatore and Nydick, 2004;Saaty and Vargas, 2006).They also suggested that the upper level weight should be evaluated by experts again after the deletion of wash criterion.Lin et al. (2008) tried to revise Finan and Hurley (2002) to point out that their theorem is incomplete and the entries in the comparison matrix did not satisfy the 1-9 bound criterion (Saaty, 1980).After the modification of their entries, the rank reversal phenomenon disappeared.
Recently, Jung et al. (2009) tried to settle the dispute among them by discovering an invariant subspace for the final synthesizing weight.They asserted that the invariant phenomenon may be useful to highlight the character to decide the weight for multiple objective decision making problems by AHP.Finally, they found conditions to insure that the rank reversal problem will not occur for wash criterion.They also indicated that their findings will be useful to resolve the debate for rank reversal problems in

Criterion Goal
However, there still exist some questionable results.We revised their findings and pointed out the meaning of their approach and explained the original problem proposed in Jung et al. (2009) and then prepared our solution.The unnecessary assumption of the unchanged value being 0.5 in Jung et al. (2009) can be removed.

Analysis
Some of the theoretical concepts used in this paper are briefly introduced in this section.These include original version proposed by Saaty (1980) and modified version by various authors.

Wash criteria
There are four different approaches to deal with weights related to wash criteria: (a) with a wash criterion, (b) without a wash criterion (Finan and Hurley, 2002;Lin et al., 2008), (c) without a wash criterion (Liberatore and Nydick, 2004;Saaty and Vargas, 2006), and (d) invariant phenomenon (Jung et al. 2009).We adopted the decision problem in the paper proposed by Finan and Hurley (2002) with four-level hierarchy: (a) the top level: the goal, (b) the second top level: criteria, J and J ′ , (c) the second bottom level: sub- .Next, we considered the relative weight for comparison matrix after the deletion of the wash criterion.The total weight for the original case is computed as; and the total weight after the deletion of 0 J is computed as; Based on Equation (2), the relative weights without wash criterion 0 J must be adjusted to satisfy the constraint in which the total weight is one.

Two approaches without a wash criterion
There are two different approaches without a wash criterion: (a) applied by Finan and Hurley (2002) and Lin et al. (2008), the relative weight in the next upper level for J and J ′ did not need to be modified, and (b) applied by Liberatore and Nydick (2004) and Saaty and Vargas (2006), the relative weight in the next upper level for J and J ′ should be renormalized according to the remaining total weights.
For approach (a), Finan and Hurley (2002) and Lin et al. (2008) assumed that with or without a wash criteria, the weights of J and J ′ corresponding to the goal should be kept the same.The third row of the hierarchy did not change.They only modified the relative weights for 1 J and 2 J relative to J .If the wash criterion, 0 J is deleted, the total weight corresponding to J is left as 1 b − .So researchers revised the relative weight of 1 to the relative weights in Table 1 so that the total weight for 1 J and 2 J becomes one.For easy comparison, we have listed the detailed results in the following Table 2.
For approach (b) proposed by Liberatore and Nydick (2004) and Saaty and Vargas (2006), not only should the relative weights of 1 J and 2 J be revised, but also J and J ′ .Owing to the fact that after the deletion of wash criterion 0 J , the weight for J is changed from ( ) , the total weights corresponding to J and J ′ are left as ( ) , respectively.We have listed the detailed results in the following Table 3. Jung et al. (2009) claimed that they have discovered conditions to preserve the consistency of the final weight for 1 A and 2 A with or without wash criterion 0 J .Moreover, they predicted that their findings will cease the debate among Finan and Hurley (2002), Liberatore and Nydick (2004), Saaty and Vargas (2006) and Lin et al. (2008).We briefly quoted the results in Jung et al. (2009) for later discussion.

Goal
(2) without wash criterion 0 J , derived by Finan and Hurley (2002) ( ) These three different approaches have the same weight for A .Jung et al. (2009) believed that they have discovered some invariant phenomenon in AHP that deserves further investigation.They solved Equation ( 6) to find that; ( ) Consequently, Jung et al. (2009) then the final weight for 1 A will be the same for the three different approaches.It seems that they have discovered a six dimensional invariant subspace for rank reversal problem in AHP that deserves further study.Moreover, they discovered a six dimensional invariant subspace with or without wash criteria under the condition after their mathematical derivation when they announced that there is an eight dimensional problem with parameters, and h in (0,1) By applying three different approaches, the final weights are all the same.However, they could not provide any explanation from the operational research view point in AHP for their results in Equations ( 9) and ( 10).

Questionable results in Jung's findings and research revisions
If we assume that All the other derivations afterwards contain questionable results.Form Equation ( 6), the corrected expression should be; This implies Equation ( 7) and the revision of Equation ( 8) as; Therefore, the conditions that guarantee the validity of Equation ( 6 ( )h

RESULTS AND DISCUSSIONS
After the derivation of conditions to insure the validity of Equation ( 6), Jung et al. (2009) did not provide further explanation for their findings.In the following, we will provide a reasonable explanation for our findings.Let us recall Table 2, the left hand side of Equation ( 14) is the relative weight of 1 A corresponding to criterion J , so Equation ( 14) indicates that after deleting 0 J , the remaining criterion, , is a wash criterion.The same phenomenon happens for Table 3.Hence, for two different approaches: by Finan and Hurley (2002) and Lin et al. (2008), in Table 2, and by Liberatore and Nydick (2004) and Saaty and Vargas (2006) in Table 3, is a wash criterion, as well.

Further argument
Based on the aforementioned discussion, every criterion becomes a wash criterion when certain conditions are fulfilled.Therefore, the alternatives 1

A and 2
A have the same weight 0.5 which is a predictable result.Moreover, the three different approaches: (a) with a wash criterion, (b) without a wash criterion, proposed by Finan and Hurley (2002) and Lin et al. (2008) and (c) without a wash criterion, proposed by Liberatore and Nydick (2004) and Saaty and Vargas (2006), clearly implies that the final weight of 1 A and 2 A is the same weight of 0.5.
For completeness, we considered their numerical example for those strange combination of values,

Revision of Jung's approach
Jung's results Jung et al. (2009) tried to find conditions to guarantee that the weights are the same when derived by three different approaches as mentioned above.However, they used it to solve the following problem We must point out that the revised problem should be improved as;

Research revision
On the other hand, we referred to the findings in previous section to construct two new criteria:  17) is amended by Equations ( 3) and ( 5), so it yields that; ( ) ( ) From which we derive; ( ) Again by Equation ( 17), we obtain that;  We will summarize our findings in the next theorem.

Jan et al. 8305
Theorem 1 If we try to find conditions to guarantee that the final weights are the same by three different approaches then the unchanged value for 1 A must be 0.5 The original problem in Jung et al. (2009) is the rank reversal problem.However, they tried to find conditions to insure the weight will not change in two cases: (a) with a wash criterion, and (b) without a wash criterion.Unfortunately, our revision of their results and our improvement are all directed at different problem: fixing the weight.For example, with a wash criterion, say 0 J , we assumed that the final weights are ( ) . First, approach (1), the problem is to find conditions to insure that without wash criterion 0 can still be obtained.Similarly, if we study approach (2), the problem is finding conditions to insure that without wash criterion 0 J , then is still preserved.

Back to the rank reversal problem
In this section, we considered the rank reversal problem, so that the proposed problem should be improved as follows.If ( ) ( ) and are both valid.The rank reversal problem will not happen.
From Equation (3), we know that ( ) ( ) which can be simplified as: ( ) If we compare Equation ( 30) with (34), they are identical which also implies that J .Hence, the rank reversal problem will not occur as predicted.We have summarized our results in the next theorem.
. Therefore, the weights in the upper level should be revised proportionally to the total weight of ( ) = − .In the following, we use the existing data to get the new relative weights for J and J ′ .The revised weights for J and J ′ are renormalized as( ) expression.Based on our simplified expression, Equation ( alternatives if we follow the proportional adjustment to derive new relative weights of upper level suggested by approach (3) as (a) with wash criterion, 0 J , and (b) without wash criterion, 0
assumed that a and b are and h satisfy the following two conditions,