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Method and Application of Dynamic Comprehensive Evaluation
Systems Engineering - Theory & Practice Volume 27, Issue 10, October 2007 Online English edition of the Chinese language journal Cite this article as: SETP, 2007, 27(10): 154–158
Method and Application of Dynamic Comprehensive Evaluation GUO Ya-jun1 ,∗ , YAO Yuan2 , YI Ping-tao1 1. Department of Management Science and Engineering, School of Business Administration, Northeastern University, Shenyang 110004, China 2. Department of Management Engineering, Shenyang Institute of Engineering, Shenyang 110136, China
Abstract: Decreasing dimensions is the key of dynamic comprehensive evaluation problems, which turns multidimensional time series to two-dimensional data. This article extends the ordered weighted averaging (OWA) operator to aggregating data in dynamic comprehensive evaluation problems and presents two new operator (time ordered weighted averaging (TOWA) operator and time ordered weighted geometric averaging (TOWGA) operator). Then, two new methods are presented, which are twice-weighted evaluation and prefix comprehensive evaluation. And the article introduces the mathematical programming model to solve TOWA and TOWGA weighting vector (time weighting vector). Finally, an applied example was given to illustrate the effectiveness of the proposed method. Key Words: dynamic comprehensive evaluation; TOWA operator; TOWGA operator; programming model
1 Introduction In practical economic management and evaluation, such problems often happen: to the same evaluation object, as time passes and data accumulates, people own a large number of two-dimensional data series by time order, called multidimensional time series. These problems are defined as dynamic comprehensive evaluation problems[1] , which are supported by the multidimensional time series and have dynamic parameters. At present, the research on the dynamic evaluation method has two classifications: One is confirming the evaluation index’s weight at different times; and the other is dynamically dealing with the comprehensive evaluation processing[2−4] , which at different times evaluation object’s attribute keeps changing, so evaluation index needs to be adjusted. Dynamic comprehensive evaluation is a sort of decision-makings, which is of applied value and practice. Some scholars have been focusing on this problem, but the research results are still rare. In this article, the method of decreasing dimensions is used to aggregate data, multidimensional time series are turned to two-dimensional data, the existing static comprehensive evaluation method is applied to aggregate data and evaluation. And the article will confirm the time weighing using the TOWA operator and the TOWGA operator.
OWA operator. Later, other scholars developed the ordered weighted geometric averaging (OWGA) operator and the induced ordered weighted geometric averaging (IOWGA) operator[7−8] . This article presents the TOWA operator and the TOWGA operator, which are defined as Definition 2.1 A TOWA operator of dimension n is a mapping F that has an associated weighting vector W = n T (w1 , w2 , · · · , wn ) having the properties of wj = 1, j=1
wj ∈ [0, 1], j ∈ N , N = {1, 2, · · · , n}, such that F (u1 , a1 , · · · , un , an ) =
n
wj bj
(1)
j=1
where ui , ai (i ∈ N ) is called TOWA tuples, within these pairs ui is the time order inducing value and ai is the argument value; bj is the jth time of collection of the aggregated objects ai . Definition 2.2 A TOWGA operator of dimension n is a mapping G that has an associated weighting vector W = n T (w1 , w2 , · · · , wn ) having the properties of wj = 1, j=1
2 Aggregate operator In 1988, Yager introduced the ordered weighted averaging (OWA) operator[5] . The OWA operator, between the maximum operator and the minimum operator, is a useful tool in aggregating multiattribute making information. Since its appearance, the OWA operator has been used in a wide range of applications, such as, decision making, image process, and so on. In 1999, Yager developed the induced ordered weighted averaging (IOWA) operator[6] based on the
wj ∈ [0, 1], j ∈ N , N = {1, 2, · · · , n}, such that G (u1 , a1 , · · · , un , an ) =
n j=1
w
bj j
(2)
where ui , ai (i ∈ N ) is called TOWGA tuples within these pairs ui is the time order inducing value and ai is the argument value; bj is the jth time of the collection of the aggregated objects ai .
Received date: May 22, 2006 ∗ Corresponding author: Tel: +86-024-83672613; E-mail: [email protected] Foundation item: Supported by the National Natural Science Foundation of China (No.70472032) c 2007, Systems Engineering Society of China. Published by Elsevier BV. All rights reserved. Copyright
GUO Ya-jun, et al./Systems Engineering – Theory & Practice, 2007, 27(10): 154–158
The essential of these two operators is that the time order inducing value ui is an index function, ui is re-ordered by some time sequence, relevant argument value ai is aggregated by a new ui -sequence, and in particular wj is not associated with ai ’s value and position, but rather wj is associated with the jth position of the ui -sequence.
3 Dynamic comprehensive evaluation method In the multiattribute dynamic comprehensive evaluation, multidimensional time series is of a three-dimensional structure, which is composed of index dimension, evaluation object dimension and time dimension. Basing on the Reference [9], this article presents two new methods (twiceweighted evaluation and prefix comprehensive evaluation) to aggregate index dimension and time dimension. Both methods use the TOWA operator and the TOWGA operator. Their time order inducing value ui (i ∈ N ) is ordered by time sequence; their argument value ai (i ∈ N ) is the index value. 3.1 Twice-weighted evaluation Twice-weighted evaluation is the two-step comprehensive evaluation method. The first step emphasizes the function of the evaluation index at different times; the second step emphasizes the function of the time factor, based on the first step. To be exact the subjective weighted method or objective weighted method is used to calculate index xj ’s weighting values vj (tk ) (j = 1, 2, · · · , m; k = 1, 2, · · · , p) at tk (k = 1, 2, · · · , p) time, and the selected comprehensive evaluation model is used to obtain the system (or evaluation object) si ’s comprehensive evaluation values yi (tk ) at tk time. To emphasize the time series’ influence on systems, this article introduces the TOWA operator (or the TOWGA operator) to define the final evaluation results as hi
= F (t1 , yi (t1 ) , t2 , yi (t2 ) , · · · , tn , yi (tp )) p = wk bik (3) k=1
(or hi = G (t1 , yi (t1 ) , t2 , yi (t2 ) , · · · , tn , yi (tp )) p k = bw ik ) k=1
where i = 1, 2, · · · , n, hi is the systemic final evaluaT tion value, W = (w1 , w2 , · · · , wp ) is the time weighting vector, bik is the TOWA (or TOWGA) tuples’ yi (tk ) (k = 1, 2, · · · , p) at k time. 3.2 Prefix comprehensive evaluation Prefix comprehensive evaluation first aggregates the data in time dimension. This article introduces the TOWA operator (or the TOWGA operator) for implementation. xij
= =
F (t1 , xij (t1 ) , t2 , xij (t2 ) , · · · , tp , xij (tp )) p wk bijk (4) k=1
(or xij = G (t1 , xij (t1 ) , t2 , xij (t2 ) , · · · , tp , xij (tp )) p k = bw ijk ) k=1
Table 1. 0.1–0.9 ratio scale for time-degree
Values 0.1 0.3 0.5 0.7 0.9 0.2, 0.4, 0.6, 0.8
Explanation Strong attention to recent data Moderate attention to recent data The same attention to every phase Moderate attention to distant data Strong attention to distant data Intermediate values between the two adjacent judgments
where xij is the aggregated index value, W = (w1 , w2 , · · · , wp )T is the time weighting vector, bik is the TOWA (or TOWGA) tuples’ xij (tk ) (k = 1, 2, · · · , p) at k time. Thus, the multidimensional time series {xij (tk )} has turned into a two-dimensional data {xij }, that is, the problem has transferred from the dynamic comprehensive evaluation to the static evaluation. 3.3
Obtaining time weighting vector
Throughout these two methods mentioned earlier, the scientifically confirming time weighting vector W = T (w1 , w2 , · · · , wp ) is the key to obtain logical evaluation results. The time weighting vector W characterizes the degree to weightiness at different times. W can be obtained through different weighted methods based on different rules. Before presenting mathematical programming model to confirm W = (w1 , w2 , · · · , wp )T , for one thing it introduces two characterizing measures associated with the time weighting vector of a TOWA operator (or a TOWGA operator). The first one, the time weighting vector’s entropy I is the measure of dispersion of the aggregation and defined as I=−
p
wk ln wk
k=1
Entropy is a term from thermodynamics, and is called average information content in information theory. It is a measure of information content. If the value of entropy is greater, the information content is smaller. Here I characterizes the degree to the weighting information content in the process of a sample’s aggregation. Furthermore, time-degree λ extends Yager’s orness concept[5] and defines it as λ=
p p−k k=1
p−1
wk T
Particularly, when w = (1, 0, · · · , 0) , then λ = 1; T When w = (0, 0, · · · , 1) , then λ = 0; When w = T 1 1 1 , then λ = 0.5. p, p, · · · , p Time-degree λ characterizes the degree to time series in the process of operator aggregation (Table 1). When λ is close to 0, it indicates that the valuator pays more attention to recent data from the evaluation time tp , and it mainly deals with the perfect tenses’ dynamic comprehensive evaluation
GUO Ya-jun, et al./Systems Engineering – Theory & Practice, 2007, 27(10): 154–158
Table 2. yi (tk ) of 12 provinces/cities in western region during 1999–2004
Province/City Neimenggu Guangxi Chongqing Sichuan Guizhou Yunnan Xizang Shanxi Gansu Qinghai Ningxia Xinjiang
1999 2.376 2.487 2.680 2.549 1.813 2.300 1.425 2.915 2.151 2.128 2.384 2.962
2000 2.251 2.459 2.516 2.441 1.840 2.149 1.396 2.674 1.905 2.001 2.253 2.821
2001 2.716 3.109 3.239 3.039 2.291 2.648 1.516 3.245 2.355 2.294 2.691 3.364
2002 2.717 2.959 3.241 3.005 2.301 2.732 1.508 3.365 2.358 2.267 2.487 3.264
2003 2.915 2.802 3.352 3.019 2.276 2.553 1.566 3.430 2.293 2.212 2.705 3.176
2004 2.919 2.319 2.930 2.415 1.749 2.137 1.647 2.888 1.923 1.957 2.450 2.793
problem; When λ is close to 1, it indicates that the valuator pays more attention to distant data from evaluation time tp , and it mainly deals with the future tenses’ dynamic comprehensive evaluation problem; when λ = 0.5, it indicates that the valuator pays the same attention to every phase, with no preference. The approach to obtain wk determines the specialty of the TOWA (or TOWGA) operators, having maximal entropy of the TOWA (or TOWGA) time weighting vector for a given level of time-degree λ. This approach is based on the solution of the following mathematical programming problem ⎫ p ⎪ ⎪ ⎪ max − wk ln wk ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ p ⎪ p−k ⎪ ⎬ wk s.t. λ = (5) p−1 k=1 ⎪ ⎪ p ⎪ ⎪ ⎪ ⎪ wk = 1, wk ∈ [0, 1] ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎭ k = 1, 2, · · · , p 3.4 Distinction between TOWA and TOWGA It is clear from Eqs.(1) and (2) that the TOWA operator is additive and the TOWGA operator is multiplicative, so the two operators emphasize on different aspects in the pro-
cess of data aggregation. The TOWA operator gives prominence to the system development’s functionality, and thus allows the same object’s index values from different phases to strongly complement to each other. The TOWGA operator gives prominence to the system development’s proportionality, namely, emphasizes that the objects coordinate and develop, also it is the cannikin law’s typical exhibition and can the prevent system development’s “short board” phenomenon. In data aggregation the TOWA operator and TOWGA operator have their own advantages and disadvantages, so they should be used in specific conditions. If one model is composed of two operators, it has both complementarity and proportionality and it will be more rational[10] .
4
Example
The methods mentioned earlier were used to evaluate and analyze, during 1999–2004, the macroeconomic developmental status of 12 provinces/cities in the western region of China. On the basis of availability from [11], we select eight benefit indexes, which are per capita GDP, per capita total investment in fixed assets, per capita total retail sale of consumer goods, residents consumption level, number of hospital beds per 100 000 population, number of students in regular institutions of higher education per 10000 population, the ratio of government revenue to government expenditure by region, per capita business volume of post and telecommunications. To get the best results, the following steps are involved: Step 1 The first aggregation. Use “extremum” method to pretreat the original data, select “entropy” method to confirm the weighting vector at different times, and aggregate by using the linear model[9] . Then the evaluation results yi (tk ), at different times, are obtained (Table 2). Step 2 Obtain the time weighting vector. In the experts’ opinion time-degree λ = 0.1 is logical, then use Eq.(5) to calculate the time weighting vector W = (0.0029, 0.0086, 0.0255, 0.0755, 0.2238, 0.6637)T . Step 3 The second aggregation. Use the TOWA (or TOWGA) operator to aggregate the first evaluation results, yi (tk ). Calculate the final values hi using Eq.(3). Then sort using hi (Table 3).
Table 3. Results of macroeconomic developmental status of 12 provinces/cities in western region
Province/City Neimenggu Guangxi Chongqing Sichuan Guizhou Yunnan Xizang Shanxi Gansu Qinghai Ningxia Xinjiang Order
Value Basing on TOWA operator Basing on TOWGA operator 2.890 7.420 2.497 7.132 3.051 7.486 2.611 7.202 1.922 6.745 2.289 7.002 1.612 6.544 3.052 7.477 2.050 6.844 2.047 6.849 2.513 7.168 2.929 7.409 ShanxiChongqingXinjiangNeimengguSichuan ChongqingShanxiNeimengguXinjiangSichuan NingxiaGuangxiYunnanGansuQinghai NingxiaGuangxiYunnanQinghaiGansuGuizhou GuizhouXizang Xizang
GUO Ya-jun, et al./Systems Engineering – Theory & Practice, 2007, 27(10): 154–158
From Table 3, the macroeconomic developmental statuses of 12 provinces/cities in the western region have been classified into three levels by the results. The first level is Shanxi, Chongqing, Xinjiang and Neimenggu, which have the advantage of developing functionality and proportionality compared to the other provinces/cities. The second level is Sichuang, Ningxia, Guangxi and Yunnan, which have moderate developing functionality and proportionality. The third level is Gansu, Qinghai, Guizhou and Xizang, which have the disadvantage of developing functionality and proportionality when compared with the other provinces/cities. As a whole, two orders are accordant, which illustrates that 12 provinces/cities have no big waves during 1999–2004.
5 Conclusions Dynamic comprehensive evaluation is more complex than static comprehensive evaluation, and its key is decreasing dimensions. To this characteristic, this article presents two new dynamic comprehensive evaluation methods with the TOWA (or TOWGA) operator. The TOWA (or TOWGA) operator has good flexibility, so experts/valuators can obtain special time-degree, based on special problems, to calculate the time weighting vector. Thus, the methods, which are presented, are fit for most dynamic evaluation problems, such as, economy system, performance appraisal, and so on.
References [1] Ren R E, Wang H W. Multi-variable data analysis. National Defense Industry Press, Beijing, 1997.
[2] Chen Y T, Chen G H, Li M J. Classification & research advancement of comprehensive evaluation methods. Journal of Management Sciences, 2004, 7(2): 69–79. [3] Guo Y J. New theory and method of dynamic comprehensive evaluation. J of Management Sciences, 2002, 5(2): 49–52. [4] Guo Y J, Pan J M, Gao Z Q. Dynamic comprehensive evaluation method supported by multidimensional time series. Journal of Northeastern University (Natural Science), 2001, 22(4): 464–467. [5] Yager R R. On ordered weight averaging aggregation operators in multi-criteria decision making. IEEE Trans on Systems, Man, and Cybernetics, 1988, 18(1): 183–190. [6] Chiclana F, Herrera F, Herrera-Viedma E. Integrating multiplicative preference relations in a multipurpose decisionmaking model based on fuzzy preference relations. Fuzzy Sets and Systems, 2001, 122(2): 277–291. [7] Yager R R. Induced ordered weight averaging operators. IEEE Trans on Systems, Man, and Cybernetics, 1999, 29(2): 141–150. [8] Chen H Y, Sheng Z H. A kind of new combination forecasting method based on induced ordered weighted geometric averaging (IOWGA) operator. Journal of Industrial Engineering and Engineering Management, 2005, 15(4): 36–39. [9] Guo Y J. Theory and methodology of synthesis evaluation. Science Press, Beijing, 2002. [10] Guo Y J, Zhong T L. Multiple attribute evaluation method taking account of functionality and proportionality and its application. China Soft Science, 2001: 104–106. [11] National bureau of statistics of China, 1999–2005 China Statistics Yearbook, China Statistics Press, Beijing.
Method and Application of Dynamic Comprehensive Evaluation GUO Ya-jun1 ,∗ , YAO Yuan2 , YI Ping-tao1 1. Department of Management Science and Engineering, School of Business Administration, Northeastern University, Shenyang 110004, China 2. Department of Management Engineering, Shenyang Institute of Engineering, Shenyang 110136, China
Abstract: Decreasing dimensions is the key of dynamic comprehensive evaluation problems, which turns multidimensional time series to two-dimensional data. This article extends the ordered weighted averaging (OWA) operator to aggregating data in dynamic comprehensive evaluation problems and presents two new operator (time ordered weighted averaging (TOWA) operator and time ordered weighted geometric averaging (TOWGA) operator). Then, two new methods are presented, which are twice-weighted evaluation and prefix comprehensive evaluation. And the article introduces the mathematical programming model to solve TOWA and TOWGA weighting vector (time weighting vector). Finally, an applied example was given to illustrate the effectiveness of the proposed method. Key Words: dynamic comprehensive evaluation; TOWA operator; TOWGA operator; programming model
1 Introduction In practical economic management and evaluation, such problems often happen: to the same evaluation object, as time passes and data accumulates, people own a large number of two-dimensional data series by time order, called multidimensional time series. These problems are defined as dynamic comprehensive evaluation problems[1] , which are supported by the multidimensional time series and have dynamic parameters. At present, the research on the dynamic evaluation method has two classifications: One is confirming the evaluation index’s weight at different times; and the other is dynamically dealing with the comprehensive evaluation processing[2−4] , which at different times evaluation object’s attribute keeps changing, so evaluation index needs to be adjusted. Dynamic comprehensive evaluation is a sort of decision-makings, which is of applied value and practice. Some scholars have been focusing on this problem, but the research results are still rare. In this article, the method of decreasing dimensions is used to aggregate data, multidimensional time series are turned to two-dimensional data, the existing static comprehensive evaluation method is applied to aggregate data and evaluation. And the article will confirm the time weighing using the TOWA operator and the TOWGA operator.
OWA operator. Later, other scholars developed the ordered weighted geometric averaging (OWGA) operator and the induced ordered weighted geometric averaging (IOWGA) operator[7−8] . This article presents the TOWA operator and the TOWGA operator, which are defined as Definition 2.1 A TOWA operator of dimension n is a mapping F that has an associated weighting vector W = n T (w1 , w2 , · · · , wn ) having the properties of wj = 1, j=1
wj ∈ [0, 1], j ∈ N , N = {1, 2, · · · , n}, such that F (u1 , a1 , · · · , un , an ) =
n
wj bj
(1)
j=1
where ui , ai (i ∈ N ) is called TOWA tuples, within these pairs ui is the time order inducing value and ai is the argument value; bj is the jth time of collection of the aggregated objects ai . Definition 2.2 A TOWGA operator of dimension n is a mapping G that has an associated weighting vector W = n T (w1 , w2 , · · · , wn ) having the properties of wj = 1, j=1
2 Aggregate operator In 1988, Yager introduced the ordered weighted averaging (OWA) operator[5] . The OWA operator, between the maximum operator and the minimum operator, is a useful tool in aggregating multiattribute making information. Since its appearance, the OWA operator has been used in a wide range of applications, such as, decision making, image process, and so on. In 1999, Yager developed the induced ordered weighted averaging (IOWA) operator[6] based on the
wj ∈ [0, 1], j ∈ N , N = {1, 2, · · · , n}, such that G (u1 , a1 , · · · , un , an ) =
n j=1
w
bj j
(2)
where ui , ai (i ∈ N ) is called TOWGA tuples within these pairs ui is the time order inducing value and ai is the argument value; bj is the jth time of the collection of the aggregated objects ai .
Received date: May 22, 2006 ∗ Corresponding author: Tel: +86-024-83672613; E-mail: [email protected] Foundation item: Supported by the National Natural Science Foundation of China (No.70472032) c 2007, Systems Engineering Society of China. Published by Elsevier BV. All rights reserved. Copyright
GUO Ya-jun, et al./Systems Engineering – Theory & Practice, 2007, 27(10): 154–158
The essential of these two operators is that the time order inducing value ui is an index function, ui is re-ordered by some time sequence, relevant argument value ai is aggregated by a new ui -sequence, and in particular wj is not associated with ai ’s value and position, but rather wj is associated with the jth position of the ui -sequence.
3 Dynamic comprehensive evaluation method In the multiattribute dynamic comprehensive evaluation, multidimensional time series is of a three-dimensional structure, which is composed of index dimension, evaluation object dimension and time dimension. Basing on the Reference [9], this article presents two new methods (twiceweighted evaluation and prefix comprehensive evaluation) to aggregate index dimension and time dimension. Both methods use the TOWA operator and the TOWGA operator. Their time order inducing value ui (i ∈ N ) is ordered by time sequence; their argument value ai (i ∈ N ) is the index value. 3.1 Twice-weighted evaluation Twice-weighted evaluation is the two-step comprehensive evaluation method. The first step emphasizes the function of the evaluation index at different times; the second step emphasizes the function of the time factor, based on the first step. To be exact the subjective weighted method or objective weighted method is used to calculate index xj ’s weighting values vj (tk ) (j = 1, 2, · · · , m; k = 1, 2, · · · , p) at tk (k = 1, 2, · · · , p) time, and the selected comprehensive evaluation model is used to obtain the system (or evaluation object) si ’s comprehensive evaluation values yi (tk ) at tk time. To emphasize the time series’ influence on systems, this article introduces the TOWA operator (or the TOWGA operator) to define the final evaluation results as hi
= F (t1 , yi (t1 ) , t2 , yi (t2 ) , · · · , tn , yi (tp )) p = wk bik (3) k=1
(or hi = G (t1 , yi (t1 ) , t2 , yi (t2 ) , · · · , tn , yi (tp )) p k = bw ik ) k=1
where i = 1, 2, · · · , n, hi is the systemic final evaluaT tion value, W = (w1 , w2 , · · · , wp ) is the time weighting vector, bik is the TOWA (or TOWGA) tuples’ yi (tk ) (k = 1, 2, · · · , p) at k time. 3.2 Prefix comprehensive evaluation Prefix comprehensive evaluation first aggregates the data in time dimension. This article introduces the TOWA operator (or the TOWGA operator) for implementation. xij
= =
F (t1 , xij (t1 ) , t2 , xij (t2 ) , · · · , tp , xij (tp )) p wk bijk (4) k=1
(or xij = G (t1 , xij (t1 ) , t2 , xij (t2 ) , · · · , tp , xij (tp )) p k = bw ijk ) k=1
Table 1. 0.1–0.9 ratio scale for time-degree
Values 0.1 0.3 0.5 0.7 0.9 0.2, 0.4, 0.6, 0.8
Explanation Strong attention to recent data Moderate attention to recent data The same attention to every phase Moderate attention to distant data Strong attention to distant data Intermediate values between the two adjacent judgments
where xij is the aggregated index value, W = (w1 , w2 , · · · , wp )T is the time weighting vector, bik is the TOWA (or TOWGA) tuples’ xij (tk ) (k = 1, 2, · · · , p) at k time. Thus, the multidimensional time series {xij (tk )} has turned into a two-dimensional data {xij }, that is, the problem has transferred from the dynamic comprehensive evaluation to the static evaluation. 3.3
Obtaining time weighting vector
Throughout these two methods mentioned earlier, the scientifically confirming time weighting vector W = T (w1 , w2 , · · · , wp ) is the key to obtain logical evaluation results. The time weighting vector W characterizes the degree to weightiness at different times. W can be obtained through different weighted methods based on different rules. Before presenting mathematical programming model to confirm W = (w1 , w2 , · · · , wp )T , for one thing it introduces two characterizing measures associated with the time weighting vector of a TOWA operator (or a TOWGA operator). The first one, the time weighting vector’s entropy I is the measure of dispersion of the aggregation and defined as I=−
p
wk ln wk
k=1
Entropy is a term from thermodynamics, and is called average information content in information theory. It is a measure of information content. If the value of entropy is greater, the information content is smaller. Here I characterizes the degree to the weighting information content in the process of a sample’s aggregation. Furthermore, time-degree λ extends Yager’s orness concept[5] and defines it as λ=
p p−k k=1
p−1
wk T
Particularly, when w = (1, 0, · · · , 0) , then λ = 1; T When w = (0, 0, · · · , 1) , then λ = 0; When w = T 1 1 1 , then λ = 0.5. p, p, · · · , p Time-degree λ characterizes the degree to time series in the process of operator aggregation (Table 1). When λ is close to 0, it indicates that the valuator pays more attention to recent data from the evaluation time tp , and it mainly deals with the perfect tenses’ dynamic comprehensive evaluation
GUO Ya-jun, et al./Systems Engineering – Theory & Practice, 2007, 27(10): 154–158
Table 2. yi (tk ) of 12 provinces/cities in western region during 1999–2004
Province/City Neimenggu Guangxi Chongqing Sichuan Guizhou Yunnan Xizang Shanxi Gansu Qinghai Ningxia Xinjiang
1999 2.376 2.487 2.680 2.549 1.813 2.300 1.425 2.915 2.151 2.128 2.384 2.962
2000 2.251 2.459 2.516 2.441 1.840 2.149 1.396 2.674 1.905 2.001 2.253 2.821
2001 2.716 3.109 3.239 3.039 2.291 2.648 1.516 3.245 2.355 2.294 2.691 3.364
2002 2.717 2.959 3.241 3.005 2.301 2.732 1.508 3.365 2.358 2.267 2.487 3.264
2003 2.915 2.802 3.352 3.019 2.276 2.553 1.566 3.430 2.293 2.212 2.705 3.176
2004 2.919 2.319 2.930 2.415 1.749 2.137 1.647 2.888 1.923 1.957 2.450 2.793
problem; When λ is close to 1, it indicates that the valuator pays more attention to distant data from evaluation time tp , and it mainly deals with the future tenses’ dynamic comprehensive evaluation problem; when λ = 0.5, it indicates that the valuator pays the same attention to every phase, with no preference. The approach to obtain wk determines the specialty of the TOWA (or TOWGA) operators, having maximal entropy of the TOWA (or TOWGA) time weighting vector for a given level of time-degree λ. This approach is based on the solution of the following mathematical programming problem ⎫ p ⎪ ⎪ ⎪ max − wk ln wk ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ p ⎪ p−k ⎪ ⎬ wk s.t. λ = (5) p−1 k=1 ⎪ ⎪ p ⎪ ⎪ ⎪ ⎪ wk = 1, wk ∈ [0, 1] ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎭ k = 1, 2, · · · , p 3.4 Distinction between TOWA and TOWGA It is clear from Eqs.(1) and (2) that the TOWA operator is additive and the TOWGA operator is multiplicative, so the two operators emphasize on different aspects in the pro-
cess of data aggregation. The TOWA operator gives prominence to the system development’s functionality, and thus allows the same object’s index values from different phases to strongly complement to each other. The TOWGA operator gives prominence to the system development’s proportionality, namely, emphasizes that the objects coordinate and develop, also it is the cannikin law’s typical exhibition and can the prevent system development’s “short board” phenomenon. In data aggregation the TOWA operator and TOWGA operator have their own advantages and disadvantages, so they should be used in specific conditions. If one model is composed of two operators, it has both complementarity and proportionality and it will be more rational[10] .
4
Example
The methods mentioned earlier were used to evaluate and analyze, during 1999–2004, the macroeconomic developmental status of 12 provinces/cities in the western region of China. On the basis of availability from [11], we select eight benefit indexes, which are per capita GDP, per capita total investment in fixed assets, per capita total retail sale of consumer goods, residents consumption level, number of hospital beds per 100 000 population, number of students in regular institutions of higher education per 10000 population, the ratio of government revenue to government expenditure by region, per capita business volume of post and telecommunications. To get the best results, the following steps are involved: Step 1 The first aggregation. Use “extremum” method to pretreat the original data, select “entropy” method to confirm the weighting vector at different times, and aggregate by using the linear model[9] . Then the evaluation results yi (tk ), at different times, are obtained (Table 2). Step 2 Obtain the time weighting vector. In the experts’ opinion time-degree λ = 0.1 is logical, then use Eq.(5) to calculate the time weighting vector W = (0.0029, 0.0086, 0.0255, 0.0755, 0.2238, 0.6637)T . Step 3 The second aggregation. Use the TOWA (or TOWGA) operator to aggregate the first evaluation results, yi (tk ). Calculate the final values hi using Eq.(3). Then sort using hi (Table 3).
Table 3. Results of macroeconomic developmental status of 12 provinces/cities in western region
Province/City Neimenggu Guangxi Chongqing Sichuan Guizhou Yunnan Xizang Shanxi Gansu Qinghai Ningxia Xinjiang Order
Value Basing on TOWA operator Basing on TOWGA operator 2.890 7.420 2.497 7.132 3.051 7.486 2.611 7.202 1.922 6.745 2.289 7.002 1.612 6.544 3.052 7.477 2.050 6.844 2.047 6.849 2.513 7.168 2.929 7.409 ShanxiChongqingXinjiangNeimengguSichuan ChongqingShanxiNeimengguXinjiangSichuan NingxiaGuangxiYunnanGansuQinghai NingxiaGuangxiYunnanQinghaiGansuGuizhou GuizhouXizang Xizang
GUO Ya-jun, et al./Systems Engineering – Theory & Practice, 2007, 27(10): 154–158
From Table 3, the macroeconomic developmental statuses of 12 provinces/cities in the western region have been classified into three levels by the results. The first level is Shanxi, Chongqing, Xinjiang and Neimenggu, which have the advantage of developing functionality and proportionality compared to the other provinces/cities. The second level is Sichuang, Ningxia, Guangxi and Yunnan, which have moderate developing functionality and proportionality. The third level is Gansu, Qinghai, Guizhou and Xizang, which have the disadvantage of developing functionality and proportionality when compared with the other provinces/cities. As a whole, two orders are accordant, which illustrates that 12 provinces/cities have no big waves during 1999–2004.
5 Conclusions Dynamic comprehensive evaluation is more complex than static comprehensive evaluation, and its key is decreasing dimensions. To this characteristic, this article presents two new dynamic comprehensive evaluation methods with the TOWA (or TOWGA) operator. The TOWA (or TOWGA) operator has good flexibility, so experts/valuators can obtain special time-degree, based on special problems, to calculate the time weighting vector. Thus, the methods, which are presented, are fit for most dynamic evaluation problems, such as, economy system, performance appraisal, and so on.
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