Construction and Restrictions of Consistent Matrices by Macharis ’ Method

In 2004, Macharis et al. proposed a consistent comparison matrix. In this study, we provide a patch work for discussing the construction and restrictions of their consistent comparisonmatrix.Their study is important in academia andhas been cited by over 200 papers. However, in the article, the authors neither provide a detailed explanation on how to construct the proposed revised consistent comparison matrix nor mention any restriction when constructing it. In this study, we not only propose an algorithm to construct the consistentmatrix, but also prove that two restrictions exist when developing a sequence of their consistent comparison matrices, which are: (a) the changing of entries should follow a decreasing order with respect to the row index and (b) there is at most one entry in each row that can be revised by experts.The results of this study will help researchersproperly apply the consistent matrices proposed by Macharis et al. (2004) in the future.


Introduction
To provide reasonable related weights among alternatives (or criteria) is a critical factor during the process of multicriteria decision-making problems.Baek et al. [1] claimed that there are limitations to acquire sufficient information for the selected input parameters of statistical data in the area of life cycle assessment (LCA).Hence, there is a strong requirement for a consistent process to locate the input parameters for use in uncertainty analysis in the field of LCA when only partial information for a comparison matrix is known.Macharis et al. [2] discusses the advantages and shortcomings of the Preference Ranking Organization METHod for Enrichment Evaluations (PROMETHEE) to point out that it did not provide a systematical method to assign weights for the objectives.Hence, Macharis et al. [2] adopted Analytic Hierarchy Process (AHP) methods to incorporate AHP into PROMETHEE and then operational synergies can be obtained in multicriteria analysis.Because of its flexibility and easy implementation, AHP was applied to obtain relative weights for criteria and alternatives in many fields.
Second, based on the above construction, Macharis et al. [2] proposed a revised process, which involves developing a consistent comparison matrix and asking an expert to know which entries should be changed.Subsequently, the previous study provided an example and mentioned that a chosen expert wanted to change { 57 ,  26 ,  15 ,  24 } and explicitly added that  57 should be revised to 2. However, the preferred values for { 26 ,  15 ,  24 } was not provided based on the expert opinion.Finally, the previous authors did not change the values of { 26 ,  15 ,  24 }, after having revised the value of  57 , and instead changed the value of  45 from 5/2 to 2. Therefore, Macharis et al. [2] only demonstrated these two instances of entry changing and did not provide any procedure or algorithm in obtaining the new consistent matrices.There are new methods and progress for consistency, for example, Dong and Herrera-Viedma [3], Dong et al. [4], and Li et al. [5].Dong and Herrera-Viedma [3] constructed a new linear programming model to find interval numerical scale for linguistic term sets which are interval fuzzy preference relations.Dong et al. [4] developed a new consensus process to synthesize heterogeneous alternatives, attributes, and preference relations for complex and dynamic l multiple attribute group decision-making problems.Li et al. [5] provided a new framework to personalize individual semantics with hesitant fuzzy linguistic terms sets such that reflected individual personalized difference in understanding the meaning of comparative linguistic expressions in hesitant linguistic group decision-making.
The first goal of this paper is to provide an algorithm to help practitioners obtain a new consistent comparison matrix based on the changing of one predesigned entry.The second goal is to discuss the two restrictions during the revised process to further amend the questionable results in Macharis et al. [2].
Up until now, there have been 233 papers that cited Macharis et al. [2] in their references, which indicates that their paper is a popular research item.We classified those papers into the following four categories: (i) Authors developed their new multicriteria decisionmaking problems.For example, Alonso et al. [6] tried to solve the missing entries in pairwise preference values under incomplete fuzzy, multiplicative, interval-valued, and linguistic preference relations.Mutikanga et al. [7] applied multicriteria decision analysis as a strategic planning tool for water loss management.Salardini [8] combined AHP and grey relation analysis to arrange portfolio management by trade off risk and reward.Hashemian et al. [9] used a fuzzy hybrid group decision support system to measure the supplier evaluation process.Parajuli et al. [10] adopted multicriteria assessment of yellow, green, and woody biomasses to prescreen available biomass types to provide prepare for a sustainable system.Rahman et al. [11] developed a multicriteria decision analysis approach to decide the best inland waterway structure.
(ii) Other authors dealt with application-oriented problems and did not focus on how the consistent comparison matrix proposed by Macharis et al. [2] was constructed.For example, Ayoko et al. [12] applied multicriteria decision-making methods to enhance air quality in the environment of residential houses in Brisbane, Australia.Soltani and Marandi [13] used AHP and Analytic Network Process (ANP) to set a two-stage fuzzy multicriteria decision-making process of a hospital site selection.Wang et al. [14] adopted an AHP weighted aggregated data quality indicator approach to evaluate embodied energy of building materials.Alexakis and Sarris [15] studied landfill setting in Western, Crete by integrating geographical information systems and remote sensing analysis.Tsui and Wen [16]  Those papers only mentioned Macharis et al. [2] in their introduction and did not apply the consistent comparison matrix as proposed.This paper claims that the cited article only demonstrated the construction of two revised consistent comparison matrices but did not present the method on how to construct a revised consistent matrix proposed by their approach.After careful investigation, this paper focuses on the development of an algorithm to generate the matrix identical to the two examples proposed by Macharis et al. [2].This fills the gap that was previously overlooked in deriving a consistent revised comparison matrix.Furthermore, as the original and subsequent studies did not provide an examination for the matrix development, this paper puts forth two restrictions to execute a sequence of changes of entries proposed by experts.The first restriction is related to the order of changing.The previous paper claimed that it must follow a decreasing order via the column index.For example, the expert wanted to modify four entries as { 57 ,  The second restriction is related to the allowable number of entries that can be changed in a single row.This paper suggests that a maximum of one entry can be revised in each row, and following this restriction, the proper order should be { 57 ,  26 ,  15 } or { 57 ,  24 ,  15 }.
The main purpose of this paper is to offer an algorithm and show that this can obtain the three desired properties proposed by Macharis et al. [2] for the new consistent comparison matrix.
The proposed algorithm has four significant features: (a) A detailed item-by-item procedure to preserve the three desired properties proposed by Macharis et al. [2].
(b) A verified algorithm to prove this approach will derive a consistent and reciprocal matrix to help researchers incorporate AHP into PROMETHEE.
(c) Execution of the revision of { 57 ,  26 ,  15 ,  24 } to identify that the use of descending order via the column index is questionable, and to propose an improvement wherein the descending order should be applied on the row index instead.
(d) A further examination by the suggested two entries on the same row proposed by Macharis et al. [2] reveals that it will result in an endless revision process.Hence, a second restriction is recommended.
Our findings will help researchers realize how to revise an entry of a consistent comparison matrix to develop a new consistent revised comparison matrix proposed by Macharis et al. [2].
The following sections are organized as follows: Section 2 provides a brief review for Macharis et al. [2]; Section 3 discusses the proposed improvement that presents the approach to develop a new matrix, under a predesigned entry and the proof that it is reciprocal, is consistent, and satisfies the three desired properties; Section 4 sets out a numerical example to illustrate the algorithm and also to reveal two questionable assertions in Macharis et al. [2] with the discussion of the two restrictions; and Section 5 presents the main findings and recommendations for further research.

Brief Review for Macharis et al. [2]
The main shortcoming of AHP is that it needs a large number of pairwise comparisons between alternatives or criteria, which usually results in inconsistency in the comparison matrix.The main goal of Macharis et al. [2] is to reduce the number of pairwise comparison inputs in AHP without compromising consistency.Based on Saaty [31,32], to construct an n by n comparison matrix for n alternatives needs ( − 1)/2 times pairwise comparison for [  ] × , with   = 1 for  = 1, 2, . . ., .Macharis et al. [2] developed a method to construct a consistent matrix for comparison matrix for criteria that only needs  − 1 times pairwise comparison.
Such paper mentioned that the expert wanted to change the values of { 57 ,  26 ,  15 ,  24 } as arranged by the descending order via column index.
They further claimed that the expert wanted to change the value of  57 from 7/5 to 2. Table 2 shows the second comparison matrix, denoted as [  ] 8×8 .
Macharis et al. [2] pointed out that, in such table, the same entries of   =   , for 1 ≤ ,  ≤ 5, and for 6 ≤ ,  ≤ 8.Those identical values are marked by bold italic font in Table 2.
They further claimed that the expert wanted another revision for  45 from 5/2 to 2. The third comparison matrix is cited in Table 3.
However, Macharis et al. [2] did not explain how to develop the new revised comparison matrix after a specific entry is changed.One of the objectives of this paper is to provide a reasonable method to show that how to construct a revised comparison matrix that satisfies the requirements proposed by Macharis et al. [2].

Our Improvement
The original consistent comparison matrix is denoted as [  ] × and a specific entry, denoted as   , with  < , is assigned to a new value  and the new matrix is denoted as [  ] × .Based on the bold italic font entries in (1) If 1 ≤  ≤  and 1 ≤  ≤ , then   =   .(4) If  + 1 ≤  ≤  and  + 1 ≤  ≤ , then   =   .The proposed algorithm decomposes the matrix construction process into manageable steps.From this construction, we know that Properties (P1) and (P2) are satisfied.The following section illustrates the proof that the proposed new matrix is reciprocal, that is,     = 1, and consistent, that is,     =   , and then satisfies Property (P3)   = .
For Case 1, the following is presented: since the original comparison matrix, [  ] × is reciprocal.
For Case 2, it is derived that For Case 3, it is computed that For Case 4, it is evaluated as For all four cases,   = 1/  , for 1 ≤ ,  ≤  proving that the developed matrix is reciprocal.For Case A, For Case B, For Case C, For Case D, For Case E, For Case F, For Case G, For Case H, For all eight cases,     =   , for 1 ≤ , ,  ≤  proving that the developed matrix is consistent.
Proof.Since  + 1 ≤  ≤ , it is found that Theorems 1, 2, and 3 show that the developed matrix not only satisfies the previously proposed three desired properties but also is reciprocal and consistent.

Numerical Examples
The effectiveness of the proposed rules is tested using two numerical examples proposed by Macharis et al. [2].
To help readers and practitioners, the computer program used in this study can be accessed at http://bit.ly/2s4h3bC such that interested parties can apply this to adjust selected entries item by item to satisfy revisions provided by experts.
For completeness, this paper provides an algorithm to help interested readers in developing their own computer program.

Algorithm
Step .The original consistent comparison matrix is expressed as [  ] × .
Step .The expert decides to change an entry, denoted as   , with  <  and a specific value, say Δ.
Step .The new consistent comparison matrix will be expressed as [  ] × , after the following four steps.
For the two examples proposed by Macharis et al. [2], changing the entry (5,7) from 7/5 to 2 in the first example and (4,5) from 5/2 to 2 in the second example will yield the same results as Tables 2 and 3. Hence, there is no apparent need to list the findings again.Interested readers can apply the program using the link mentioned above.Instead, this section concentrates on the two questionable results in Macharis et al. [2] during the revised process.
Let us recall that Macharis et al. [2] mentioned that the expert wanted to change the values of { 57 ,  26 ,  15 ,  24 } that was arranged by the descending order of column index.Then, the expert wanted to alter  57 from 7/5 to 2. However, there was no setting of predesigned values for { 24 ,  15 ,  26 }.Thus, this paper assumes that the desired values for { 24 ,  15 ,  26 } are {1/2, 6, 2}.Applying the proposed algorithm produces the results as shown in Tables 4-6.
Table 7 shows that based on the changing order previously proposed, the satisfaction rate is up to 75% for the first three changes.However, the percentage drops to 50% on the fourth change.The preferred entry (2, 6) and entry (1, 5) are lost after the modification of entry (2,4).Therefore, the next criterion is provided to decide the order for changing.
The first criterion: the order of changes should be arranged by the descending order of row index.
After our criterion, the order of change should be arranged by the descending order of row index.Reconsidering the same set { 57 ,  26 ,  24 ,  15 } and the previous discussions, it is assumed that the expert wanted to change (a)  26 from 10/21 to 1/2 and (b)  24 from 1 (in Table 2, and 21/20 in Table 8) to 2. To clearly distinguish our results, they are denoted as [  ] 8×8 and [  ] 8×8 .The algorithm was applied twice and the results of such are presented in Tables 8 and 9.
Table 9 shows that the value of  26 does not satisfy the desired result 1/2.Hence, this paper executed the algorithm the third time to change  26 to 1/2, and list the findings in Table 10, denoted as [  ] 8×8 . and we know that [ (2+1)  ] 8×8 = [  ] 8×8 for  = 1, 2, . .., and [ (2)  ] 8×8 = [  ] 8×8 for  = 1, 2, . . . to imply an endless revision process such that the revision of two entries in the same row as in Macharis et al. [2] is questionable.A summary of results is given in the next criterion.
The second criterion: when the expert decides to select several entries to modify their values one by one, there must only be a maximum of one entry per row that can be modified.
The consistency approach proposed by Macharis et al. [2] will help researchers improve the consistency index for their comparison matrices to guarantee every decision maker having a consistent comparison matrix during the aggregation process, before the consensus reaching process starts to be iterated.
There are new methods and progress for consensus process, for example, Dong et al. [33], Li et al. [34], and Li et al. [35].Dong et al. [33] developed a self-management mechanism for noncooperative behaviors in large-scale group consensus reaching processes to decrease the evaluation values of experts with noncooperative behaviors.Li et al. [34] constructed a consensus procedure that preserves the consistency index for decision makers.They found an approximated adjustment interval for the optimal adjustment interval and then they verified that after several iterations to improve the group consensus level, then the group consensus level can pass the predesigned consensus level.Li et al. [35] studied a new consensus method to convert an expert from an unacceptable group to an acceptable group with the help of a selected moderator.Each time, one decision maker joins the acceptable group, and then after several iterations, the number of decision makers in the acceptable group attains the predesigned threshold.

Conclusion
The proposed derivation provides a patch work for incompleteness in Macharis et al. [2].A structured and transparent algorithm was developed to help researchers apply the consistent comparison matrix for AHP proposed by the previous study.Further investigations allowed this study to point out two new criteria for the selection of changing entries to avoid an endless revision process.These findings permit researchers to gain a deeper insight into the consistent comparison matrix problem in AHP.
26,  15 ,  24 } as arranged by the decreasing order of column index.The next section shows that the order of changing entries should still follow a decreasing order but through the row index.Consequently, this results to the first restriction where the order should be { 57 ,  26 ,  24 ,  15 } or { 57 ,  24 ,  26 ,  15 }.