Improved TOPSIS Model and its Application in the Evaluation of NCAA Basketball Coaches

Traditional TOPSIS model has some disadvantages, such as correlations between criteria, uncertainty in obtaining the weights only by objective methods or subjective methods and possibility of alternative closed to ideal point and nadir point concurrently, and many solutions have been proposed regarding these disadvantages. This paper presents a more systematic TOPSIS model, in which the correlations between criteria were overcome by a new method on evaluation index system based on R cluster analysis. It also proposes a combination weighting method which has considered subjective potency of human and the variance in the data. Besides, the possibility of alternative closed to ideal point and nadir point concurrently was avoided by vertical projection method and the measurement of similarity to solution was simplified by vertical projection distance. The feasibility and validity of this improved TOPSIS model were testified by the evaluation of NCAA basketball coaches after 1939.


Introduction
TOPSIS (technique for order preference by similarity to solution), firstly presented by Hwang and Yoon, is a simple and efficient multiple criteria method to identify solutions from a finite set of alternatives.However, there are some main disadvantages in traditional TOPSIS model: correlations between criteria, uncertainty in obtaining the weights only by objective methods or subjective methods and possibility of alternative closed to ideal point and nadir point concurrently (Li et al., 2011).This paper uses R cluster analysis to overcome the disadvantage-correlations between criteria.In multi-attribute decision making problem, in order not to miss some important criteria, primarily the decision maker would determine as many criteria as possible, which would cause correlations between criteria.Nevertheless, the evaluation system may result in creating too many criteria, which are highly correlated with each other, and can cause inaccuracy in results of the evaluation.Hence, it is necessary to explore the correlations between these variables and classify them into different categories.R cluster analysis, however, can achieve the goals.
Different weighting methods can be used to obtain the weights in TOPSIS model.These weighting methods can be summarized into three kinds, objective weighting method (Chen et al., 2010;Zhang et al., 2007;Lu, 2003;Yang et al., 2008), subjective weighting method (Luo et al., 2011) and combination weighting method (Li et al., 2011;Zhang et al., 2009).Most papers concerning TOPSIS model have a preference for objective weighting method.W.-H. SU demonstrates that the excessive pursuit of objectivity about weighting methods ignores the subjective potency of human.The final weights may be unconscionable (Su, 2011).L.-P.YU et al. argue that objective weighting methods cannot be entirely ensured.The future trend seems to conduct more combination weighting methods which combine subjective weighting methods and objective weighting methods (Yu et al., 2009).This paper proposes a combination weighting method which combines subjective weighting method (Analytic Hierarchy Process) and objective weighting method (Principal Component Analysis).
Euclid distance was replaced by Chi-square or Mahalanobis distance (Li et al., 2011;Wang et al., 2012) to overcome the possibility of alternatives closed to ideal point and nadir point concurrently.Grey system theory was also used to solve the problem (Chen et al., 2010;Sun et al., 2005).Some papers developed new evaluation methods (Liu et al., 1996).X.-Y.HUA et al. proposed a revised vertical projection method.In their paper, they defined a new distance: vertical projection distance and used vertical projection distance to replace Euclid distance (Hua et al., 2004).This method is simple and efficient, so in this paper, we use the vertical projection method for reference.
Finally, an improved TOPSIS model was used to evaluate NCAA basketball coaches after 1939 in the paper, whose feasibility and validity have been testified.

TOPSIS Model
TOPSIS (technique for order preference by similarity to solution) is a multiple criteria method to identify solutions from a finite set of alternatives.It first determines the ideal point and nadir point, then ranks all the alternatives according to their similarity to the ideal point.
The idea of TOPSIS can be expressed in the following steps (Olson, 2004).
1) Obtain performance data for n alternatives over k criteria.Raw data are usually standardized.
2) Develop a set of importance weights k ω for each of the criteria.
3) Identify the ideal alternative (extreme performance on each criterion) and the nadir alternative (reverse extreme performance on each criterion).
4) Develop a distance measure over each criterion to both ideal point( * C ) and nadir point( 0 C ).
5) For each alternative, determine a ratio * i f which represents the similarity to solution.
6) Rank order alternatives by maximzing the ratio * i f in step 5.

Improved TOPSIS Model
In the improved TOPSIS model, R cluster analysis was used to overcome correlations between criteria.A combination weighting method which combines subjective weighting method (Analytic Hierarchy Process) and objective weighting method (Principal Component Analysis) was presented.Vertical projection distance was used to replace Euclid distance which simplified the measurement of similarity to solution and the possibility of alternative closed to ideal point and nadir point concurrently was avoided.

Normalization
Y. P. LIAO et al. demonstrate that the vector normalization method was the best normalization method for TOPSIS model.It could deal with the general multi-attribute decision making (MADM) problems with various problem sizes, data ranges and attribute types effectively (Liao et al., 2012).
Use vector normalization method to standardized raw data.
where ij b is a standardized measure converted from raw measure ij a .In practice, comparing one with another and constructing a judgment matrix according to the 1-9 scale may cause mistakes in the order of priorities.Y. -H.HOU et al. present that 1-9 scale is not fit for generating accurate weights, and exponential scale is the most accurate scale (Hou et al., 1995).Therefore, in this paper the exponential scale is used.

Combination Weighting Method
The random consistency index (R.I.) of exponential scale is as follows: Table 2. Random consistency index (R.I.) of exponential scale (Wei, 2002) n 3 4 5 6 7 8 9 RI 0.32 0.53 0.66 0.75 0.81 0.86 0.89 2) Objective weighting method: Principal Component Analysis (Wu et al., 2009) PCA (Principal Component Analysis) is the simplest of the true eigenvector-based multivariate analyses.Often, its operation can be considered as revealing the internal structure of the data in a way that best explains the variance in the data.So, this paper chooses PCA as an objective weighting method.The steps of PCA are as follows:

(a). Standardize raw data
There are n criteria ( 1 2 , , , n x x x  ) and m alternatives.The value of number i alternative in number j criterion is ij a , ij a is standardized to be ij a  .
The weight of criterion j x is j ω . 1 ( 1,2, , ) Where j c is the coefficient of j x  in formula (7).
For a criterion, the average value of its weights obtained from subjective weighting method and objective weighting method is its combination weight.

Weighted Matrix
( ) Where [ ]  is the final weight vector and ij b is the standardized measure converted from raw measure ij a .nadir point:

Determining Ideal Point and Nadir Point
Where * j c is the value of number j attribute of ideal point * C , 0 j c is the value of number j attribute of nadir point 0 C .

Measuring Similarity by Vertical Projection Distance
3) Traditional method: measuring similarity to solution by Euclid distance Similarity to solution is alternative's similarity to ideal point.

Calculate the distances of alternatives between ideal point and nadir point. The distance between alternative and ideal point
The distance between alternative and nadir point 0 C is 0 i s : ( ) Calculate the similarity to solution * i f : ( 1,2, , ) 4) New method: measure similarity to solution by vertical projection distance (Hua et al., 2004) Figure 1.Presentation of vertical projection distance In Figure 1, P and N represent ideal point and nadir point.The vertical projection distance from X to Y represents the distance between two planes whose normal vector is straight line PN.One of them crosses X and the other crosses Y.
Vertical projection distance reflects the similarity to solution, X. -Y.HUA et al. demonstrate that when the vertical projection distance between an alternative and the ideal point is shorter, the vertical projection distance between the alternative and the nadir point will be longer (Hua et al., 2004).
Some key and basic steps involved vertical projection method are as follows: In order to simplify calculation, translate the origin of coordinate to the ideal point firstly.The matrix after translation is After translation, the ideal point is{ } 0, 0, , 0  , the nadir point is X. -Y.HUA et al. demonstrate that the vertical projection distance between a alternative and the ideal point can be expressed as follows: The smaller the value of i P is, the better the alternative will be (Hua et al., 2004).The proofs of the conclusions discussed before can be found in paper by Hua et al.

Screening Criteria
The National Collegiate Athletic Association (NCAA) is a non-profit association that organizes the athletic programs of many colleges and universities in the United States and Canada.NCAA basketball is one of the most popular programs.This paper evaluated all the NCAA basketball coaches after 1939.Primarily, according to the related website's statistical data, 10 criteria were confirmed.There are obvious correlations among the 10 criteria.R cluster analysis was used to classify the 10 criteria.Then we selected the most representative criterion in each group.The cluster tree is shown in Figure 2. According to intuitive judgment, we can see that there is high correlation among 1, 2, 3, 4. 2 games, 3 wins, 4 losses will increase as 1 years increase.According to NCAA basketball competition system, 6, 7, 8 are of high correlation and there is also high correlation between 9, 10.So the result of R cluster analysis is reasonable.
In the first, second and third group, we selected criteria 1, 7, 5.In the fourth group, a new criterion named FFNC is created.9 and 10 are added according to the proportion of 1:4.
Finally 1 2 3 4 , , , x x x x are used to represent the four criteria: Yrs, W-L%, CTRN and FFNC.

1) Analytic Hierarchy Process
The hierarchy for obtaining the weights of criteria is shown below in Figure 3.All the consistency ratios do not exceed 0.10.To synthesize the overall weight scale we multiply as follows: The final weights of the four criteria 1 2 3 4 , , , x x x x by AHP are (0.0768 0.1803 0.2102 0.5327).

3) Combination weights
The final weights of 1 x , 2 x , 3 x , 4 x are as follows: According to table 8, the 8 greatest coaches are all in this paper's ranking.
This paper only considers some quantifiable factors, but Bleacher Report and ESPN considered some factors that cannot be quantified.So there are some differences about the ranking.
The improved TOPSIS model proposed in this paper has three advantages: 1) This paper proposes a new method on evaluation index system based on R cluster analysis, which overcome the correlations between criteria.
2) This paper presents a combination weighting method which combines subjective weighting method (Analytic Hierarchy Process) and objective weighting method (Principal Component Analysis).This weighting method considered subjective potency of human and the variance in the data at the same time.This method makes final weight more reasonable.
3) This paper introduces vertical projection method for reference.The possibility of alternative closed to ideal point and nadir point concurrently was avoided.The measurement of similarity to solution was simplified.

Conclusion
The main purpose of this paper is to propose a more systematic TOPSIS model, which is easy to apply in different fields.This paper uses its application on the evaluation of NCAA basketball coaches as an example.This improved TOPSIS model can also be easily applied to other fields, including the evaluation of other kinds of coaches, employee performance review and supplier selection, etc.
In the improved TOPSIS model, the correlations between criteria were overcome by a new method on the evaluation index system based on R cluster analysis.The paper also proposes a combination weighting method which considered both subjective potency of human and the variance in the data.The possibility of alternative closed to ideal point and nadir point concurrently was avoided by vertical projection method and the measurement of the similarity to solution was simplified.The feasibility and validity of this improved model are testified by the evaluation of NCAA basketball coaches, however, the evaluation was based on 10 criteria, which are quantifiable factors which can be collected easily, and may not be comprehensive enough.The final results will be more accurate if more criteria are obtained and the subjective factors are quantified.

1)
Subjective weighting method: Analytic Hierarchy Process Some basic key steps involve Analytic Hierarchy Process are (Ovidya et al., 2006): (a).State the problem and identify the criteria that influence the problem.(b).Structure the problem in a hierarchy of different levels constituting goal, criteria, sub-criteria and alternatives.(c).Compare each element in the corresponding level and calibrate them on the numerical scale.(d).Perform calculations to find the maximum Eigen value, consistency index CI, consistency ratio CR, and normalized values for each criteria/alternative. (e).If the maximum Eigen value, CI and CR are satisfactory, then decision is taken based on the normalized values; else the procedure is repeated till these values lie in a desired range.
r is the correlation coefficient of criteria i and j .(c).Calculate the eigenvalue and eigenvector.Calculate the eigenvalue of correlation coefficient matrix R ,

Figure
Figure 3.The hierarchy Table 1 outlines the exponential scale.

Table 4 .
Structures of the 4 principal components