DNBMA: A Double Normalization-Based Multi-Aggregation Method

DNBMA: A Double Normalization-Based Multi-Aggregation Method Huchang Liao1,2, Xingli Wu1,*, Francisco Herrera2,3 1 2

Business School, Sichuan University, Chengdu 610064, China

Department of Computer Science and Artificial Intelligence, University of Granada, E-18071 Granada, Spain 3 Faculty

of Computing and Information Technology, King Abdulaziz University, Jeddah, Saudi Arabia [email protected]; [email protected]; [email protected]

Abstract. We propose a double normalization-based multi-aggregation method to deal with the multi-criteria decision making problems considering the benefit, cost and target criterion values. To do so, we introduce an enhanced target-based linear normalization formula and a target-based vector normalization formula. Given that different normalization techniques maintain special advantages and disadvantages, we combine them with three aggregation models to describe the alternatives’ performance from different aspects. Then, a new integration approach is developed to integrate three types of subordinate utility values and ranks to derive the final ranking. The selected alternative not only has a comprehensive performance but does not perform badly under each criterion. Finally, the proposed method is highlighted by a case study of selecting an optimal innovation enterprise. Keywords: Multi-criteria decision making, double normalization-based multi-aggregation method, targetbased linear normalization, target-based vector normalization

1

Introduction

Multi-Criteria Decision Making (MCDM) is a process of ranking a finite set of alternatives based on multiple criteria. There are basically two types of techniques to handle the MCDM problems: the outranking methods and the multi-criterion value methods [1]. The former is based on pairwise comparisons of alternatives under each criterion, which is limited in dealing with massive alternatives due to the complicated calculation. The latter composes a simple process of aggregating the criterion values to rank alternatives, which includes the normalization process and the aggregation process. There are mainly three kinds of normalization techniques: the linear normalization model, the vector normalization model and the non-normalization model. Jahan and Edwards [2] illustrated that different results can be derived by different normalization models. The multi-criterion value methods are various from different normalization and aggregation tools. Simple Multi-Attribute Rating Technique (SMART) is a rather simple MCDM method which uses the linear normalization model to eliminate the different dimensions among criteria and employs the weighted arithmetic aggregation operator to integrate the normalized criterion values [3]. Considering that the weighted arithmetic aggregation operator, the geometric weighted aggregation operator and the weighted maximum operator have different effects in representing the performance of objects, the MULTIplicative Multi-Objective Optimization by Ratio Analysis (MULTIMOORA) method [4], based on the vector normalization, applies these operators respectively to derive three kinds of utility values and then to yield the subordinate rankings of alternatives. Based on the vector normalization and the weighted arithmetic aggregation, the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method [5] determines the compromise solution which is nearest to the positive ideal solution by calculating the distance of

*

Corresponding author. ORCID: 0000-0001-8278-3384 (Huchang Liao); 0000-0001-7435-5240 (Xingli Wu) 1

each alternative from the reference point. Opricovic and Tzeng [6] claimed that the solution selected by TOPSIS may not be closest to the ideal one since it ignores the relative importance between the distance from the ideal point and that from the negative-ideal point, and then proposed the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method [7] to further compute the “individual regret value” by the weighted maximum formula after deriving the “group utility” based on linear normalization and weighted arithmetic aggregation. However, the subordinate ranks are not taken into consideration in the VIKOR method when integrating two types of utility values, which makes the result with low robustness. Given that the criteria include benefit, cost, and target values in practice, Jahan et al. [8, 9] extended the linear normalization to the target-based linear normalization. On this basis, the target-based TOPSIS method [8], the target-based VIKOR method [9] and the target-based MULTIMOORA method [10] were proposed. In conclusion, the common defect of the existing methods is that they eliminate the criterion dimensions only based on one normalization method, which may bias the results since all the normalization methods loss the original information more or less from different aspects. Furthermore, calculating the utility values by different aggregation operators is useful, but there still is a challenge to integrate the subordinate utility values and the ranks of alternatives at the same time to derive the final ranking. This paper aims to propose a new MCDM method, named Double Normalization-Based Multi-Aggregation (DNBMA), to solve these problems. The paper is highlighted by the following innovative work: (1) We introduce an improved target-based linear normalization formula and a target-based vector normalization formula. (2) After analyzing the advantages and disadvantages of the target-based linear and vector normalization techniques, respectively, we make a suitable combination on two kinds of normalized values and three types of aggregation models to derive the subordinate utility values and ranks. It can reduce the information loss caused by one normalization technique. (3) We propose a new aggregation formula to derive the final ranking of alternatives. It considers the subordinate utility values and the ranks of alternative simultaneously, and their relative importance is also taken into consideration. In this way, the result is more robustness than the ranking which is only integrated by subordinate utility values or subordinate ranks. The paper is organized as follows: Section 2 presents the target-based linear and vector normalization formulas. The DNBMA method is proposed in Section 3. Section 4 illustrates the method by an example. Final concluding remarks are pointed out in Section 5.

2

The Target-Based Normalization Techniques

A general MCDM problem contains a set of alternatives

A  a1 , a2 ,..., am 

and a set of criteria

T C  c1 , c2 ,..., cn  with the weight vector W  (1 , 2 ,..., n ) . The decision matrix is composed as X  ( xij )mn

where xij is the value of alternative ai with respect to criterion c j . Given that the non-normalization tool can only be applied to special problems, in the following, we start our investigation by introducing two commonly used linear and vector normalization formulas. Afterwards, we illustrate their advantages and disadvantages based on some examples. 2.1

The Target-Based Linear Normalization

Considering cost, benefit, and target-based criteria at the same time for a MCDM problem, based on the distance of each judgment to target value, Jahan et al. [9] proposed a linear normalization formula as:

2

yij1  1 

where



xij  rj





max max xij , rj  min min xij , rj i

i

(1)



xij if c j is in benefit type, rj  min xij if c j rj is the target value on criterion c j , especially, rj  max i i

is in cost type. Motivated by the simplified form of linear normalization [10], we improve the target-based linear normalization formula as: yij1  1 

xij  rj

(2)

max xij  rj i

The linear normalization given as Eq. (2) can reflect the closeness between each alternative and the target solution under each criterion. The normalization values are the same for different convertible units ( ij   xij   ,

  0 ) with the same criterion function given that ij1  1 

ij   j



max ij   j i

( xij   )  ( rj   ) max ( xij   )  ( rj   )

=yij1

i

where  j is the target value on criterion c j and ij1 is the linear normalized value of ij . Thus, it is reasonable to aggregate the values of all criteria of an alternative directly because the normalized values only represent the normalized distances between the judgments of alternatives and the ideal solution. However, it loses the size of value itself, which would bias the result. This defect can be illustrated by Example 1. Example 1. Suppose that there are three projects a1 , a2 and a3 against the internal rate of return c1 (in %) and the payback period c2 (in years), and the decision matrix is given as:

1 5  D1  6 5.5 11 6  1 1 1 1 1  1 , y122  0.5 and y32  0 . If the weight vector of By Eq. (2), we get y11  0 , y21  0.5 , y31  1 , y12 criteria is w  (0.5, 0.5)T , based on the weighted arithmetic aggregation operator, we obtain y1  0.5 , y2  0.5 and y3  0.5 , then a1 a2 a3 . However, we could not accept this result. There are great differences on the values of c1 and x11 is so inferior that we cannot select a1 , while there are small differences on the values of c2 and x32 is not so bad. Thus, the linear normalization is unable to describe the real differences between different data. 2.2

The Target-Based Vector Normalization

The vector normalization, employed in the MULTIMOORA method [4] and the TOPSIS method [5], is shown as   xij  2  yij    1  xij 

 x  m

i 1

2

(3)

 x  m

i 1

if c j is a benefit criterion

ij

2

ij

if c j is a cost criterion

The vector normalization aims to normalize the values of all alternatives with respect to criteria to the interval [0,1] . The dimensionless number yij2 can maintain the size of the original value xij compared with the linear normalization formulas. Brauers et al. [11] proved that the vector normalization formula is a robust option. But it fails to eliminate the evaluation units of criteria essentially in two aspects: (1) On the one hand, it cannot eliminate the influences of different convertible units of the same criterion function on the result of a MCDM method, such 0 0 as the length xij [m] or ij [km] , and the temperature xij [C ] or ij [ F ] . These ‘‘convertible’’ units are related as ij   xij   ,   0 . The normalized value yij2 N is different with respect to different evaluation units of a criterion function [6]. That is to say,

3

yij2  xij

 x  m

i 1

ij

2

but

ij2  ij

   m

i 1

ij

2



 x

ij



  x m

i 1

ij

2 2    , and if   0 , then yij  ij . (2) On the other hand, it is unable to eliminate the 2

influence of different units of different criteria on the result of a MCDM method which integrates the information based on the fully compensated aggregation operator. This defect can be verified by Example 2. Example 2. Suppose that there are three production lines a1 , a2 and a3 against the cost c1 (million) and the production c2 (number of packages), and the decision matrix is given as:

 43 1100  D2   42 1050   41 900  2  0.42 , y31 2  0.44 , y122  0.62 , y22 2  0.59 and y32 2  0.51 . If the By Eq. (3), we get y112  0.41 , y21 weight vector of criteria is w  (0.5, 0.5)T , based on the weighted arithmetic aggregation function, we obtain

y1  0.515 , y2  0.505 and y3  0.475 . Then a1

a2

a3 . However, the fact is that the performance of a2

is the best and a1 is as bad as a3 . The result is misleading since the differences of alternatives on cost are decreased by the vector normalization which is only able to measure the differences between numbers but ignores the unit differences. In fact, there is a big separation between 42 million and 41 million of cost but a small division between 1100 and 1050 of production packages. Thus, a2 is superior to a1 . In conclusion, the vector normalization is not suitable to aggregate the values of an alternative under all criteria based on the completely compensated arithmetic aggregation operator. To fill the gap of normalizing all the benefit, cost and target-based criterion values by the vector normalization, we introduce a target-based vector normalization formula based on the distance of each judgment to the target value, shown as:

yij2  1 

xij  rj

 x   r  m

i 1

3

2

ij

(4) 2

j

The Double Normalization-Based Multi-Aggregation Method

The MULTIMOORA method employs three different aggregation methods to calculate the utility values respectively based on the vector normalization. However, it does not take into account the matching of normalization and aggregation methods. Consequently, the selected alternative is not always nearest to the ideal one. Given that both the target-based linear and vector normalization methods have their advantages and disadvantages, we combine them with different aggregation operators to obtain the different utility values of alternatives. This section aims to propose a DNBMA method to deal with the defects of the existing multi-criterion value methods and obtain a decision result with high reliability and robustness. 3.1

The Subordinate Aggregation Models

In the following, we develop three kinds of aggregation operators based on the two target-based normalization techniques. 3.1.1

The Complete Compensatory Model (CCM)

To measure the closeness to the ideal solution, Zeleny [12] proposed a measurement r ( x; p) , used as an aggregation function, to measure the regret from alternative ai to the ideal solution a* .

n Ri , p    j xij  rj  j 1



4



1 p

p

  , 1 p   

(5)

where  j is the weight of c j . As we know, with the increase of p , the weight of the larger value  j xij  rj

becomes greater and greater.

The measurements r ( x; p) of p  1 and p   are used in the VIKOR method, and the measurement r ( x; p) of p  2 is used in the classical TOPSIS method as well. Since each criterion has a weight, there is no reason to add a weight to a bigger one. Thus, we employ the measurement r ( x; p) of p  1 as the first aggregation function of the proposed method. From Section 2, we can find that the target-based linear normalization is superior to the target-based vector normalization to combine with the linear aggregation operator to aggregate the values of an alternative under all criteria. Thus, we define the CCM based on the arithmetic weighted aggregation operator as: n

u1 (ai )    j yij1

(6)

j 1

The alternatives are ranked by u1 (ai ) (i  1, 2,..., m) in descending order and we get the first type of ranks r1 (ai ) (i  1, 2,..., m) . Here we let the ranks obtained in this paper be the Besson’s mean ranks [13]: If an object ai ranks the u th position, then r (ai )  u ; if both ai and at rank the u th position, then r (ai )  r (at )  (u  u  1) 2  u  0.5 . For example, if a1 prefers to a2 , and a2 is indifferent to a3 , then r (a1 )  1 and r (a2 )  r (a3 )  2.5 . 3.1.2

The Un-compensatory Model (UCM)

To avoid the selected solution having an extremely poor performance under a criterion, we employ the measurement r ( x; p) with p   and the linear normalized values to compose the second aggregation function, shown as Eq. (7). u2 (ai )  max  j (1  yij1 ) (7) j

The alternatives are ranked by u2 (ai ) (i  1, 2,..., m) in ascending order and we get the second type of ranks r2 (ai ) (i  1, 2,..., m) .

3.1.3

The Incomplete Compensatory Model (ICM)

Since the linear normalization is unable to reflect the quality of original values, the results would be misleading by the above two aggregation functions in some cases as illustrated by Example 3. Example 3. Suppose that there are two decision matrices of two MCDM problems with three types of products a1 , a2 and a3 against the reliability c1 and the price c2 given as:

90% 100  75% 100    D3  94% 105  , D4  85% 105  98% 110  95% 110  By Eqs. (6)-(7), we obtain the same results regarding to D3 and D4 that a2 is the optimal solution and a1 a3 . It is easy to accept the result of D3 but hard to accept the result of D4 since the alternatives associated to D4 have a big gap in quality, but a small gap in price. There is a fact that the alternative a1 is of extremely poor quality but not good price, the alternative a2 is of medium quality and slightly bad price, and the alternative a3 is of extremely good quality and not so bad price. According to our intuition, we may select a3 and deem a1 as the worst. The above two aggregation functions fail to consider the size of the value itself. To solve this defect and make the result more reliable, we employ the vector normalized values combined with the multiplicative form to propose the third aggregation function as Eq. (8). j

u3 (ai )  ( yij2 ) j

(8) The multiplicative formula can reflect people's preferences that the former case is superior to the latter case. That is, the good performance of an alternative cannot fully compensate for poor performance. The alternatives are ranked by u2 (ai ) (i  1, 2,..., m) in ascending order and we obtain the third type of ranks r3 (ai ) (i  1, 2,..., m) .

5

3.2

The Subordinate Utilities and Ranks Integration

In the final phase, we need to obtain a comprehensive ranking of the alternatives by integrating the results of the above three models. The three models can be deemed as three criteria: CCM (denoted by C1 ), UCM (denoted by

C2 ) and ICM (denoted by C3 ). Each alternative ai have two kinds of evaluation values: the utility value u y (ai ) and the rank ry (ai ) with respect to each criterion Cy ( y  1, 2,3) . Obviously, this is a MCDM problem composed by two decision matrixes: the utility value decision matrix D(u )  u y (ai )  decision matrix D(r )  ry (ai ) 

m3

m3

and the ranking

. u y (ai ) can be normalized by the vector normalization formula:

u yN (ai ) 

u y (ai )

 u m

i 1

y

(ai ) 

, y  1, 2,3

(9)

2

We then define the integrated score by a weighted Euclidean distance formula as Eq. (10). 2

2

2 2 2  m  r1 (ai )  1   r2 (ai )   m  r3 (ai )  1  N N Si    u1N (ai )   (1   )      u2 (ai )   (1   )      u3 (ai )   (1   )   m ( m  1) 2 m ( m  1) 2      m(m  1) 2 

2

(10)

where  is the coefficient to highlight the importance between the subordinate utility value and the subordinate rank. The final rank set R  r (a1 ), r (a2 ),..., r (am ) is determined in descending order of Si (i  1, 2,..., m) . The DNBMA method is summarized as follows: Step 1. Calculate the target-based linear normalization values by Eq. (2) and the target-based vector normalization values by Eq. (4). Go to next step. Step 2. Compute the utility values u1 (ai ) , u2 (ai ) and u3 (ai ) (i  1, 2,..., m) based on the CCM (as Eq. (6)), the UCM (as Eq. (7)) and the ICM (as Eq. (8)), respectively, and then determine the three types of subordinate ranks ry (ai ) ( y  1, 2,3; i  1, 2,..., m) . Go to next step. Step 3. Normalize the utility values u y (ai ) ( y  1, 2,3; i  1, 2,..., m) by Eq. (9). Go to next step. Step 4. Integrate the subordinate normalized utility values and the subordinate ranks by Eq. (10). Determine the final ranking and ends the algorithm.

4

A Case Study and Some Comparative Analyses

To promote the innovation of small and medium iron and steel enterprises, one city decides to choose an optimal green enterprise to reward. There are four candidates a1 , a2 , a3 , a4 . The evaluation criteria are R & D investment accounting for the proportion of total investment ( c1 , target criterion in %), the number of developers ( c2 , benefit criterion in number), sales revenue of new products ( c3 , benefit criterion in 106 million), and comprehensive energy consumption ( c4 , cost criterion in number). Suppose that the target value of criterion c1 is 6%. The decision matrix D is determined as:

1  15.4 10 0.8 10.9 25 16.5 1.1   D  1.8 14 5.7 0.9     4.5 8 10.3 0.95 We solve the case by the DNBMA method. The target-based linear normalized values are computed by Eq. (2), shown in Table 2 and the target-based vector normalized values are calculated by Eq. (4), shown in Table 3. Table 2. The target-based linear normalized values

a1

c1

c2

c3

c4

0

0.12

0

0.5

6

a2

0.48

1

1

0

a3

0.55

0.35

0.31

1

a4

0.84

0

0.61

0.75

Table 3. The target-based vector normalized values

c1

c2

c3

c4

a1

0.54

0.63

0.4

0.95

a2

0.76

1

1

0.91

a3

0.79

0.73

0.59

1

a4

0.93

0.58

0.76

0.98

Suppose that the criterion weights are the same as  j  0.25 ,

j  1, 2,3, 4 . By Eq. (6), we obtain

u1 (a1 )  0.16 , u1 (a2 )  0.62 , u1 (a3 )  0.55 and u1 (a4 )  0.55 . Then R1  {4,1, 2.5, 2.5} . By Eq. (7), we obtain u2 (a1 )  0.25 , u2 (a2 )  0.25 , u2 (a3 )  0.17 and u2 (a4 )  0.25 . Then R2  {3,3,1,3} . By Eq. (8), we obtain u3 (a1 )  0.6 , u3 (a2 )  0.92 , u3 (a3 )  0.76 and u3 (a4 )  0.8 . Then R3  {4,1,3, 2} . Let   0.5 . According to Eqs. (9) and (10), we obtain the values S1  0.02 , S2  0.59 , S3  0.53 and S4  0.41 . Thus we get the ranking relation a2 a3 a4 a1 , which shows that a2 is an optimal innovation enterprise. a2 have good performances on all criteria except criterion c4 . The government can encourage a2 to reduce the comprehensive

energy consumption. The enterprise a1 invests much money for innovation, but the sales revenue of new products is low. It can introduce more developers to obtain the advanced technology. To make comparison, we also solve the case by other MCDM methods. The results are shown in Table 4. Table 4. The results derived by different Target-based MCDM methods Methods

Utility values

Rankings

c1

c2

c3

c4

Target-based TOPSIS [8]

0.25

0.58

0.55

0.51

a2

a3

a4

a1

Target-based VIKOR [9]

0

0.5

0.87

0.36

a3

a2

a4

a1

Target-based MULTIMOORA

0.47,0.16,0.46

0.77,0.16,0.71

0.69,0.13,0.66

0.68,0.16,0.65

a2

a3

a4

a1

-0.02

0.59

0.53

0.41

a2

a3

a4

a1

[10] The proposed DNBMA method

Comparative analysis: We obtain different results derived by the target-based VIKOR method. From calculation process, we find that the “individual regret value” has a huge impact on the utility values. When using the linear normalization formula, the “individual regret value” of a3 is significantly superior to a1 , which leads to the final result a3

a1 . Besides, the result of the target-based VIKOR is sensitive to the threshold depicting the

relative importance between the “group utility” and “individual regret value”, and it is hard for decision maker to select a suitable threshold. Despite that the same ranks are derived by the target-based TOPSIS method, the targetbased MULTIMOORA method and the proposed DNBMA method, but there are different utility values since different normalization and aggregation techniques are employed. The target-based TOPSIS method only calculates the “group utility” by the arithmetic weighted aggregation operator based on the linear normalization but ignores the “individual regret value”. There is a fact that the same results are obtained by the ratio system model and the full multiple form model of the target-based MULTIMOORA method due to the same normalization values are utilized. In DNBMA method, we obtain different normalized valued from the linear and the vector normalization formula. Besides, the different subordinate utility values are calculated by three aggregation models. After integrating these utility values and subordinate ranks, we obtain a robust ranking result.

7

5

Conclusions

We proposed a new MCDM method named DNBMA, which can handle the benefit, cost and target-based criteria at the same time. The proposed method is based on two normalization tools: the target-based linear and the targetbased vector normalization formulas, and consists three aggregation models: the CCM with arithmetic weighted aggregation, the UCM with weighted maximization formula and ICM with the geometric weighted aggregation. These subordinate methods depict the performance of alternatives from different aspects, which make the DNBMA method robust. After making comparative analysis with other MCDM methods based on a case study, the advantages of the proposed method were highlighted. As future studies, the DNBMA method can be combined with different fuzzy information, such as the hesitant fuzzy linguistic term set to deal with the subjective MCDM problems.

Acknowledgements The work was supported by the National Natural Science Foundation of China (71501135, 71771156), the Fundamental Research Funds for the central Universities (No. YJ201535), and the Scientific Research Foundation for Excellent Young Scholars at Sichuan University (No. 2016SCU04A23).

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