Project Cost Prediction Model Based on Fuzzy ... - Semantic Scholar
JOURNAL OF NETWORKS, VOL. 8, NO. 7, JULY 2013
1595
Project Cost Prediction Model Based on Fuzzy Theory Hailing Sun and Tianbao Su Henan University of Urban Construction, Pingdingshan, Henan Province, 467036, P. R. China Email: [email protected]
Abstract—Project cost is the main factor contributing to the cost of construction, which accounts for a large proportion of the total investment. So strengthening the management and control of the project cost will be beneficial for the improvement of international competitiveness and conservation of social resource for construction enterprises. The article applies the basic principles of fuzzy mathematics to project cost prediction, and makes use of the evaluation information of typical project characteristics to carry out a systematic analysis of a prediction model of project cost. This calculation method reflects the similarities between projects and provides an efficient way for effective project cost prediction by a quantitative research and contrast on to-be-built and built projects. Index Terms—Fuzzy Theory, Project Cost, Prediction Model, System Analysis
I.
INTRODUCTION
The project cost, also called a total investment of the construction project, generally refers to the full cost of the project from the feasibility study stage to the completion stage. It mainly consists of fixed assets investment and current assets investment and the former is composed of construction and installation cost, equipment and instrument cost, construction and other expenses, loan interest during construction and reserve funds, etc. Project cost is the main factor to decide the cost of construction which accounts for a large proportion of the total investment. So enhancing the project cost management and controlling the project cost reasonably not only help construction enterprises to enhance their competitiveness in the international market, but also benefit social resource saving. In the face of the international market competition, the project cost management departments are formulating a series of laws and regulations matched to the model of "bill of quantities" and "static control, dynamic management", and commonly adopt the international general mode such as "bill of quantities" to gradually improve the ability of project cost consulting companies and engineering guarantee system. Project cost management information system will be designed to collect and sending messages through the internet and casted as the software platform facing the whole process
© 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.7.1595-1600
of project cost management to provide efficient, quick and accurate information for all the engineering participants. According to the whole life cycle theory, a project needs to go through the stages of feasibility study, design, tender and bid, construction and completion acceptance, etc, and the corresponding cost files are investment estimation, design budgetary estimation, tender price, settlement and project final accounts traditional project cost valuation mode. Although this traditional valuation method is accurate and reliable, it needs great workload and long period. It is urgent to find out a reliable method to quickly and accurately predict the project cost to meet the needs of the dynamic competitive environment especially in the pre-construction stage. For this situation, the BP neural network, GST method and grey system theory have been widely used in recent years [1]. Nowadays fuzzy theory is widely used in many fields. We can find fuzzy sets to describe the ambiguous, inaccurate phenomenon, Using the concept to make judgment, evaluation or reasoning. Such as fuzzy clustering analysis and fuzzy pattern recognition, fuzzy comprehensive evaluation, Fuzzy Logic and Fuzzy Predictive fuzzy control, fuzzy information processing. These methods constitute a kind of fuzzy systems theory, constitutes a prototype of a speculative mathematics, it has been in the medical, meteorological, psychological, economic management, petroleum, geological, environmental, biological, agricultural, forestry, chemical, language, controlling, remote sensing, education, sports and other specific research. The most important fields of application of fuzzy mathematics should be computer smart. It has been used in expert systems and knowledge engineering, to see a very important role to play in the various fields, and has huge economic benefits. In 1923, Bertrand Russell, the great philosopher, pointed out: "law of excluded middle used for accurate symbol is correct; but when symbol is fuzzy, the law of excluded middle is inappropriate. In fact, all the symbols are fuzzy......”. This is the philosophical foundation of the fuzzy theory. Fuzzy mathematics is founded by cybernetics expert, LA Zadeh, a professor in the University of California. In 1965, he published a paper entitled "Fuzzy Set Theory" to declare the birth of fuzzy mathematics, a new mathematical branch [2].
1596
Predictive activities generally exist in every field of human society, especially closely related to social science and natural science. Project cost prediction refers to speculate on the project cost based on experience, statistical data and mathematical models. The effective data of prediction can be provided for construction enterprises to make decisions and project management department to prepare the project cost plan. In the classical prediction method, input and output data as well as its corresponding relationship is clear. However, due to the incompleteness of the project cost data, the traditional prediction method failed to function well and fuzzy theory becomes the best choice in project cost prediction. The principle of the fuzzy prediction is: no two projects are the in the world for their oneness and complexity. To estimate the to-be-built project cost, we should first find 5-7 typical projects from similar projects ever built, then take the closest project as raw materials and estimate the to-be-built project cost by using the prediction model [3]. The process of cost prediction is as follows: A. Making prediction plans The main content includes: organization and arrangement; supporting departments, time schedule and scope of materials collection, etc. B. Collecting and sorting prediction materials Prediction materials consist of the data from longitudinal and transverse. Longitudinal materials, the base of analyzing the development trend, are all kinds of historical data of material consumption and price. Transverse materials, the cost materials of the similar projects, can provide differences between the prediction projects and similar projects. C. Choosing forecast methods Prediction generally includes qualitative and quantitative methods. Qualitative methods mainly include experts meeting method, subjective probability method and Delphi method, etc; Quantitative methods mainly cover the following: moving average method, exponential smoothing method and regression analysis method, etc. D. Preliminary prediction The project cost is preliminarily estimated according to the qualitative and quantitative prediction of some transverse cost materials. E. Predicting factors influencing the project cost According to the specific conditions of the project, we should initially predict the critical factors affecting the project cost and then determine the cost construction of the project [4]. F. Analyzing prediction error Cost prediction usually happens before the implementation of the construction, which is seldom consistent with the actual cost and causes prediction errors. The prediction error reflects the extent of prediction accuracy. If the error is larger, we should analyze the causes and accumulate experience in prediction. Fussy mathematics was adopted to analyze the project cost prediction based on completed projects. According
© 2013 ACADEMY PUBLISHER
JOURNAL OF NETWORKS, VOL. 8, NO. 7, JULY 2013
to the theory of fuzzy membership function and nearness principles, the basic elements of project cost were combined organically to make reasonable predictions for similar project cost. Facts prove that this method has strong rationality and applicability in engineering practices [5]. II.
PROPOSED MODEL
A. Membership The fuzziness of the projects roots in the middle transition of the differences between the various projects, namely there exists difference in “both this and that” and can still be compared. The objectively existing membership function that reveals engineering characteristics can be accurately shown. Professor L.A.Zadeh’s hypothesis theory considers that supposing the domain of the prediction is U, the fuzzy set of the prediction results is Y , Y F (U ) , Y can be described by membership function , y : U [0,1] . Obviously, predicted results can be directly determined by the membership degree. While membership μ equals 1, which shows the degree of y belongs to U is much higher; instead while U is more close to 0, which says the degree of that is much lower [6]. B. Operation between Fuzzy Sets Fuzzy mathematics was founded by cybernetics expert, Professor LA Zadeh (LAZadeh 1921-1986] ). In 1965, he published a paper entitled by “Fuzzy Set Theory” (“Fuzzy Sets”) which declared the birth of fuzzy mathematics. professor LA Zadeh had dedicated to the research of “computer” and “system”, and committed to thinking why computer can not think as flexible as human being’s brain. Although the computer memory is very huge, however, it can when it can’t deal with the fuzzy message. Why computer can not deal with fuzzy information? The reason is that traditional mathematics, such as Cantor set theory (Cantor's Set), can not describe the phenomenon of “black or white”. A set is a description of the human being’s brain thinking about mathematical methods of identification and classification of the overall objective things [7]. Suppose A and B as subsets, inner product of A and B is that the smaller membership value of the two elements is chosen and then takes greater value as the final operation results; correspondingly, for outer product of A and B, the opposite is done. For example:
1 1 0.75 0.8 0.5 t1 t2 t3 t4 t5
(1)
1 0.85 0.75 0.7 0.75 t1 t2 t3 t4 t5
(2)
A B
Inner product between A and B is
A B (1 1) (1 0.85) (0.75 0.75) (0.8 0.7) (0.5 0.75) 1 0.85 0.75 0.7 0.5 1
(3)
JOURNAL OF NETWORKS, VOL. 8, NO. 7, JULY 2013
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Outer product between A and B is
A B (1 1) (1 0.85) (0.75 0.75) (0.8 0.7) (0.5 0.75)
(4)
1 1 0.75 0.8 0.75 0.7 C. Nearness Nearness can reflect the closeness of the two fussy sets. The purpose of calculating the nearness of the two projects is to decompose the elements according to their main features and to furthest reflect the similarity of the two projects while we predict the cost. Suppose mapping relationship F (U ) F (U ) 0,1 ( A, B) is nearness. If it meets: (1) ( A, B) 1
(5)
(2) (,U ) 0
(6)
(3) ( A, B) ( B, A)
(7)
(4) A B C ( A, C) ( A, B) ( B, C)
(8)
Suppose n known typical projects is A1 , A2 ,..., An , its characteristics T {t1 , t2 ,, tm ,| t j is chauacteristic value} , character vector set is ) i 1, 2,3, Ai (t j │
, nj│ ; j 1, 2,3, the nearness is AP and AQ is: m
( Ap , Aq )
, m} ,
Ap
( k ) Aq ( k ))
(
Ap
( k ) Aq ( k ))
k 1
(9)
( p, q 1, 2,3,..., n) The cost prediction is determined through the decomposition of the characteristic quantity (or characteristic element) and comprehensive analysis. Suppose the influence weight of the project characteristic value on the project cost is 1 , 2 ,..., m , and
m
k 1
k
1,
the formula (9) can be deformed as: m
( Ap , Aq )
( k 1 M
k
( k 1
k
Ap
Ap
1 (continuous state)
|
( A, B) 1
A
( X ), B ( X ) | dx
(11)
(12)
Formulas (11) and (12) show the average value of the same parts in curves A ( X ) and B ( X ) [9]. When formula (9) is used to determine the nearness of sets A and B, and membership function is continuous value, it becomes as:
min{ ( A, B) max{
A
( X ), B ( X )}dx
A
( X ), B ( X )}dx
k1 k2
(13)
( k ) Aq ( k )) ( k ) Aq ( k ))
surrounded by the curves A ( x) and B ( x) . The project characteristics has different influences on the project cost, and according to the statistical analysis of the typical project, it is scientific to use cost ratio to determine the weight indexes of their influences. At the same time, we can see that formula (9) or (10) is the amendment of Hamming nearness, and formula (10) is chosen as the calculation basis of fussy nearness in cost prediction [10]. In actual cost calculation, for being more practical and simplified, we often use the deformation of the formula (10), the formula put forward by Peizhuang Wang, Professor of Guangzhou University, to calculate the nearness. Namely:
1 ( A, B) [ A B (1 A B)] 2 (10)
( p, q 1, 2,3,..., n) In the formula (10), the weight is usually determined by the relationship between its actual cost and the prediction cost based on the analysis of the typical engineering characteristics. As the nearness reflects the similarity of fussy sets and can better describe the relationship between the same subsets and different subsets. Usually, while considering the choice of the nearness calculation formula, Hamming nearness or the formula (9) is often adopted. Generally
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1 m | A ( k ) B | n k 1 (discrete state)
( A, B) 1
In formula (13), k1 is the areas of the same parts in curves A ( X ) and B ( X ) , k 2 is the maximum area
( k 1 M
speaking, Hamming distance is a better model to reflect the difference of the various sets [8]. Suppose A ( x) and B ( x) respectively represents the membership function of fussy sets A and B, Hamming nearness can be shown in the following formulas.
(14)
D. The Procedures of Project Cost Prediction based on Fussy Theory 1. Preparation stage Find out 3-5 typical built engineering of the same type and list all the names of elements in the typical project sets. 2. Implementation stage We often choose more complex and costly elements for reference from the similar elements, and set its membership as 1, and other elements are assigned membership values from 0 to 1 compared with the benchmark element according to the engineering actual conditions and experiences, so the table of the fuzzy
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relation coefficient of the contrast engineering is preliminarily established. 3. The verification stage Firstly, Calculating the nearness among the known typical project respectively and setting their corresponding fuzzy relation coefficients of unit construction set as Ta1 , Ta 2 , Ta 3 according to the sequence of size. Secondly, Calculating the adjustment coefficient of the typical engineering respectively. Thirdly, Checking the accuracy of the typical project and determining the membership function value of each element finally. 4. Delivery decision stage After the precision is confirmed, we will regard the membership function in the table of contrast project fuzzy relation coefficient as the final membership function value, the basis of direct fee of project prediction [11]. E. Mathematical Model for Project Cost Prediction
A1 , A2 ,..., Ai (i 1, 2,..., n) represents n typical projects respectively. T represents the characteristic vector set of project cost, which can describe the project structure feature, namely: T {t1 , t2 ,..., t j } ( j 1, 2,..., m)
(15)
and is the adjustment coefficient in formula (8) [12]. In formula (18), formula (9) or (10) is used to conduct self-estimation on A project cost. The result is that e*A equals the actual cost, namely e*A eA . Suppose eA , eB , eC respectively represents the cost of the similar typical project A, B and C, and their fussy nearness to project A can be shown:
1 ( A, A), 2 ( A, B), 3 ( A, C)
Suppose 1 2 3 , the project cost prediction value of project A is:
e*A [1eA (1 1 ) 2 eB (1 1 )(1 2 ) 1 3
3eC (1 1 )(1 2 )(1 3 )(eA eB eC )]
tij tj
e*A 1eA (1 1 ) 2 eB (1 1 )(1 2 ) 1 3
3 eC (1 1 )(1 2 )(1 3 )(eA eB eC )
t j : represents the name of the characteristic element Ti : represents the fussy sub-set corresponding to ith typical project tij : represents the membership function value of jth characteristic element in ith typical project The fussy sub-set of to-be-built project is : T*
t *j t1* t2* t1 t2 tj
e [1 E1 (1 1 ) 2 E2 (1 1 )(1 2 ) 3 E3 * B
1 (1 1 )(1 2 )(1 3 )( E1 E2 E3 )] 3
© 2013 ACADEMY PUBLISHER
(18)
(22)
In a word, the prediction model based on nearness (9) or (10) can guarantee the project cost prediction value equals to their actual value, which indicates that the model is successful [13]. The adjustment coefficient will be calculated according to the following empirical formula in project cost prediction.
1
1 T' T' [1.8( 1 0.8( 1) m Ta1 Ta 2
T' 0.4( 1)] Ta 3
(17)
t *j ( j 1, 2,3, , m) represents the membership function value corresponding to jth characteristic element of to-be-built project. Again, assume represents the fussy nearness between the built project Ai (i 1, 2,3, , m) and to-be-built project B, which is arranged from big to small, namely, 1 , 2 ,..., n , E1 , E2 ,..., En and the corresponding unit cost of the project is A1 , A2 ,..., An . Choose three known projects with big nearness as estimated benchmark, and meet 1 2 3 ,and the unit cost evaluation value eB* of to-be-built project B is:
(21)
Since 1 ( A, A) 1 , the value of the above three formula is zero. After calculation, we can draw a conclusion that:
e*A eA (16)
(20)
Set equals to 1,
The fussy sub-set of T , by using Zadeh mark, is
t t Ti i1 i 2 t1 t2
(19)
(23)
m : is the number of the elements of fussy set T , Ta1 , Ta 2 , Ta3 : indicates the typical project fuzzy relation coefficient corresponding to to-be-built project respectively and a1 , a2 , a3 , and the range of values
t
ij
Ti
j
is
max tij
concluded
to
the
interval
j 1
[0,1](i 1, 2,..., m) [14]. III.
NUMERICAL RESULTS
A to-be-built project is a small hydropower station, which is named by E, please predict the C20 concrete cost. A,B,C,D is typical engineering which has been established.
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A. List out the Character Vector Set of the Project According to the specific situation of the projects, list out the name of each element of project sets, and these elements should describe the representative characteristics of this project summarily (in tableI). B. Determine the Membership Function Value According to the determined methods of membership function value that are mentioned previously, please determine to the membership function value of each character vector set of the project ( t j ) (in tableI).
t
C. Calculate the
j
of each Typical Project and
To-Be-Built Project For The Engineering of A: tAj 1 1 0.85 0.9 0.6 4.35,( j 1, 2,
,5)
For The Engineering of B: tBj 1 0.95 0.85 0.8 0.85 4.45,( j 1, 2, For The Engineering of C: tCj 1 0.95 1 1 0.9 4.85,( j 1, 2,
,5)
The Engineering of D: tDj 1 0.9 0.8 0.6 0.6 3.9,( j 1, 2, For The Engineering of E: tEj 1 0.95 1 1 1 4.95,( j 1, 2, Because
t
Ej
,5) For
,5)
,5)
4.95 is maximal, so
TE 1, TA 0.88, TB 0.9, TC 0.98, TD 0.79 TABLE I.
THE RELATED CALCULATION OF THE SMALL HYDROPOWER STATION CALCULATION
Hydrological condition
Geologica l condition
t1
t2
The layout of fields and dams t3
t4
The flood routin g metho -ds t5
A
1
1
0.85
0.9
0.6
B
1
0.95
0.85
0.8
0.85
C
1
0.95
1
1
0.9
D
1
0.9
0.8
0.6
0.6
E
1
0.95
1
1
1
C o d e
The size of unit s
The amo unt of C20
Remarks
/m3 4735 typical 5 project 4837 typical 0 project 5131 typical 0 project 4578 typical 0 project To-be- bulit project
D. Test the Reliability of Engineering Firstly, According to the formula 16,List out the fuzzy subset of the Typical Engineering Secondly, According to the formula 14,Calculate the fuzzy nearness of the typical Engineering and other Typical Engineering by turns; ( A, B) 0.58 ,
( A, C) 0.5 , ( A, D) 0.7 ; Thirdly, Sort the fuzzy nearness (from the biggest down to the smallest): 1 0.7 , 2 0.58 , 3 0.55 ;
© 2013 ACADEMY PUBLISHER
The relevant Engineering Quantity: E1 45780 , E2 48370 , E3 51310 ; Fourthly, According to the formula 23,Please calculate to the Adjustment Coefficient 1.03 ; Fifthly, According to the formula 18,Please calculate to the typical project of engineering quantity of A e*A 48174m3 ; Sixthly, Checking the error. 48174 47355 Error: 100% =1.7%
1595
Project Cost Prediction Model Based on Fuzzy Theory Hailing Sun and Tianbao Su Henan University of Urban Construction, Pingdingshan, Henan Province, 467036, P. R. China Email: [email protected]
Abstract—Project cost is the main factor contributing to the cost of construction, which accounts for a large proportion of the total investment. So strengthening the management and control of the project cost will be beneficial for the improvement of international competitiveness and conservation of social resource for construction enterprises. The article applies the basic principles of fuzzy mathematics to project cost prediction, and makes use of the evaluation information of typical project characteristics to carry out a systematic analysis of a prediction model of project cost. This calculation method reflects the similarities between projects and provides an efficient way for effective project cost prediction by a quantitative research and contrast on to-be-built and built projects. Index Terms—Fuzzy Theory, Project Cost, Prediction Model, System Analysis
I.
INTRODUCTION
The project cost, also called a total investment of the construction project, generally refers to the full cost of the project from the feasibility study stage to the completion stage. It mainly consists of fixed assets investment and current assets investment and the former is composed of construction and installation cost, equipment and instrument cost, construction and other expenses, loan interest during construction and reserve funds, etc. Project cost is the main factor to decide the cost of construction which accounts for a large proportion of the total investment. So enhancing the project cost management and controlling the project cost reasonably not only help construction enterprises to enhance their competitiveness in the international market, but also benefit social resource saving. In the face of the international market competition, the project cost management departments are formulating a series of laws and regulations matched to the model of "bill of quantities" and "static control, dynamic management", and commonly adopt the international general mode such as "bill of quantities" to gradually improve the ability of project cost consulting companies and engineering guarantee system. Project cost management information system will be designed to collect and sending messages through the internet and casted as the software platform facing the whole process
© 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.7.1595-1600
of project cost management to provide efficient, quick and accurate information for all the engineering participants. According to the whole life cycle theory, a project needs to go through the stages of feasibility study, design, tender and bid, construction and completion acceptance, etc, and the corresponding cost files are investment estimation, design budgetary estimation, tender price, settlement and project final accounts traditional project cost valuation mode. Although this traditional valuation method is accurate and reliable, it needs great workload and long period. It is urgent to find out a reliable method to quickly and accurately predict the project cost to meet the needs of the dynamic competitive environment especially in the pre-construction stage. For this situation, the BP neural network, GST method and grey system theory have been widely used in recent years [1]. Nowadays fuzzy theory is widely used in many fields. We can find fuzzy sets to describe the ambiguous, inaccurate phenomenon, Using the concept to make judgment, evaluation or reasoning. Such as fuzzy clustering analysis and fuzzy pattern recognition, fuzzy comprehensive evaluation, Fuzzy Logic and Fuzzy Predictive fuzzy control, fuzzy information processing. These methods constitute a kind of fuzzy systems theory, constitutes a prototype of a speculative mathematics, it has been in the medical, meteorological, psychological, economic management, petroleum, geological, environmental, biological, agricultural, forestry, chemical, language, controlling, remote sensing, education, sports and other specific research. The most important fields of application of fuzzy mathematics should be computer smart. It has been used in expert systems and knowledge engineering, to see a very important role to play in the various fields, and has huge economic benefits. In 1923, Bertrand Russell, the great philosopher, pointed out: "law of excluded middle used for accurate symbol is correct; but when symbol is fuzzy, the law of excluded middle is inappropriate. In fact, all the symbols are fuzzy......”. This is the philosophical foundation of the fuzzy theory. Fuzzy mathematics is founded by cybernetics expert, LA Zadeh, a professor in the University of California. In 1965, he published a paper entitled "Fuzzy Set Theory" to declare the birth of fuzzy mathematics, a new mathematical branch [2].
1596
Predictive activities generally exist in every field of human society, especially closely related to social science and natural science. Project cost prediction refers to speculate on the project cost based on experience, statistical data and mathematical models. The effective data of prediction can be provided for construction enterprises to make decisions and project management department to prepare the project cost plan. In the classical prediction method, input and output data as well as its corresponding relationship is clear. However, due to the incompleteness of the project cost data, the traditional prediction method failed to function well and fuzzy theory becomes the best choice in project cost prediction. The principle of the fuzzy prediction is: no two projects are the in the world for their oneness and complexity. To estimate the to-be-built project cost, we should first find 5-7 typical projects from similar projects ever built, then take the closest project as raw materials and estimate the to-be-built project cost by using the prediction model [3]. The process of cost prediction is as follows: A. Making prediction plans The main content includes: organization and arrangement; supporting departments, time schedule and scope of materials collection, etc. B. Collecting and sorting prediction materials Prediction materials consist of the data from longitudinal and transverse. Longitudinal materials, the base of analyzing the development trend, are all kinds of historical data of material consumption and price. Transverse materials, the cost materials of the similar projects, can provide differences between the prediction projects and similar projects. C. Choosing forecast methods Prediction generally includes qualitative and quantitative methods. Qualitative methods mainly include experts meeting method, subjective probability method and Delphi method, etc; Quantitative methods mainly cover the following: moving average method, exponential smoothing method and regression analysis method, etc. D. Preliminary prediction The project cost is preliminarily estimated according to the qualitative and quantitative prediction of some transverse cost materials. E. Predicting factors influencing the project cost According to the specific conditions of the project, we should initially predict the critical factors affecting the project cost and then determine the cost construction of the project [4]. F. Analyzing prediction error Cost prediction usually happens before the implementation of the construction, which is seldom consistent with the actual cost and causes prediction errors. The prediction error reflects the extent of prediction accuracy. If the error is larger, we should analyze the causes and accumulate experience in prediction. Fussy mathematics was adopted to analyze the project cost prediction based on completed projects. According
© 2013 ACADEMY PUBLISHER
JOURNAL OF NETWORKS, VOL. 8, NO. 7, JULY 2013
to the theory of fuzzy membership function and nearness principles, the basic elements of project cost were combined organically to make reasonable predictions for similar project cost. Facts prove that this method has strong rationality and applicability in engineering practices [5]. II.
PROPOSED MODEL
A. Membership The fuzziness of the projects roots in the middle transition of the differences between the various projects, namely there exists difference in “both this and that” and can still be compared. The objectively existing membership function that reveals engineering characteristics can be accurately shown. Professor L.A.Zadeh’s hypothesis theory considers that supposing the domain of the prediction is U, the fuzzy set of the prediction results is Y , Y F (U ) , Y can be described by membership function , y : U [0,1] . Obviously, predicted results can be directly determined by the membership degree. While membership μ equals 1, which shows the degree of y belongs to U is much higher; instead while U is more close to 0, which says the degree of that is much lower [6]. B. Operation between Fuzzy Sets Fuzzy mathematics was founded by cybernetics expert, Professor LA Zadeh (LAZadeh 1921-1986] ). In 1965, he published a paper entitled by “Fuzzy Set Theory” (“Fuzzy Sets”) which declared the birth of fuzzy mathematics. professor LA Zadeh had dedicated to the research of “computer” and “system”, and committed to thinking why computer can not think as flexible as human being’s brain. Although the computer memory is very huge, however, it can when it can’t deal with the fuzzy message. Why computer can not deal with fuzzy information? The reason is that traditional mathematics, such as Cantor set theory (Cantor's Set), can not describe the phenomenon of “black or white”. A set is a description of the human being’s brain thinking about mathematical methods of identification and classification of the overall objective things [7]. Suppose A and B as subsets, inner product of A and B is that the smaller membership value of the two elements is chosen and then takes greater value as the final operation results; correspondingly, for outer product of A and B, the opposite is done. For example:
1 1 0.75 0.8 0.5 t1 t2 t3 t4 t5
(1)
1 0.85 0.75 0.7 0.75 t1 t2 t3 t4 t5
(2)
A B
Inner product between A and B is
A B (1 1) (1 0.85) (0.75 0.75) (0.8 0.7) (0.5 0.75) 1 0.85 0.75 0.7 0.5 1
(3)
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Outer product between A and B is
A B (1 1) (1 0.85) (0.75 0.75) (0.8 0.7) (0.5 0.75)
(4)
1 1 0.75 0.8 0.75 0.7 C. Nearness Nearness can reflect the closeness of the two fussy sets. The purpose of calculating the nearness of the two projects is to decompose the elements according to their main features and to furthest reflect the similarity of the two projects while we predict the cost. Suppose mapping relationship F (U ) F (U ) 0,1 ( A, B) is nearness. If it meets: (1) ( A, B) 1
(5)
(2) (,U ) 0
(6)
(3) ( A, B) ( B, A)
(7)
(4) A B C ( A, C) ( A, B) ( B, C)
(8)
Suppose n known typical projects is A1 , A2 ,..., An , its characteristics T {t1 , t2 ,, tm ,| t j is chauacteristic value} , character vector set is ) i 1, 2,3, Ai (t j │
, nj│ ; j 1, 2,3, the nearness is AP and AQ is: m
( Ap , Aq )
, m} ,
Ap
( k ) Aq ( k ))
(
Ap
( k ) Aq ( k ))
k 1
(9)
( p, q 1, 2,3,..., n) The cost prediction is determined through the decomposition of the characteristic quantity (or characteristic element) and comprehensive analysis. Suppose the influence weight of the project characteristic value on the project cost is 1 , 2 ,..., m , and
m
k 1
k
1,
the formula (9) can be deformed as: m
( Ap , Aq )
( k 1 M
k
( k 1
k
Ap
Ap
1 (continuous state)
|
( A, B) 1
A
( X ), B ( X ) | dx
(11)
(12)
Formulas (11) and (12) show the average value of the same parts in curves A ( X ) and B ( X ) [9]. When formula (9) is used to determine the nearness of sets A and B, and membership function is continuous value, it becomes as:
min{ ( A, B) max{
A
( X ), B ( X )}dx
A
( X ), B ( X )}dx
k1 k2
(13)
( k ) Aq ( k )) ( k ) Aq ( k ))
surrounded by the curves A ( x) and B ( x) . The project characteristics has different influences on the project cost, and according to the statistical analysis of the typical project, it is scientific to use cost ratio to determine the weight indexes of their influences. At the same time, we can see that formula (9) or (10) is the amendment of Hamming nearness, and formula (10) is chosen as the calculation basis of fussy nearness in cost prediction [10]. In actual cost calculation, for being more practical and simplified, we often use the deformation of the formula (10), the formula put forward by Peizhuang Wang, Professor of Guangzhou University, to calculate the nearness. Namely:
1 ( A, B) [ A B (1 A B)] 2 (10)
( p, q 1, 2,3,..., n) In the formula (10), the weight is usually determined by the relationship between its actual cost and the prediction cost based on the analysis of the typical engineering characteristics. As the nearness reflects the similarity of fussy sets and can better describe the relationship between the same subsets and different subsets. Usually, while considering the choice of the nearness calculation formula, Hamming nearness or the formula (9) is often adopted. Generally
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1 m | A ( k ) B | n k 1 (discrete state)
( A, B) 1
In formula (13), k1 is the areas of the same parts in curves A ( X ) and B ( X ) , k 2 is the maximum area
( k 1 M
speaking, Hamming distance is a better model to reflect the difference of the various sets [8]. Suppose A ( x) and B ( x) respectively represents the membership function of fussy sets A and B, Hamming nearness can be shown in the following formulas.
(14)
D. The Procedures of Project Cost Prediction based on Fussy Theory 1. Preparation stage Find out 3-5 typical built engineering of the same type and list all the names of elements in the typical project sets. 2. Implementation stage We often choose more complex and costly elements for reference from the similar elements, and set its membership as 1, and other elements are assigned membership values from 0 to 1 compared with the benchmark element according to the engineering actual conditions and experiences, so the table of the fuzzy
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relation coefficient of the contrast engineering is preliminarily established. 3. The verification stage Firstly, Calculating the nearness among the known typical project respectively and setting their corresponding fuzzy relation coefficients of unit construction set as Ta1 , Ta 2 , Ta 3 according to the sequence of size. Secondly, Calculating the adjustment coefficient of the typical engineering respectively. Thirdly, Checking the accuracy of the typical project and determining the membership function value of each element finally. 4. Delivery decision stage After the precision is confirmed, we will regard the membership function in the table of contrast project fuzzy relation coefficient as the final membership function value, the basis of direct fee of project prediction [11]. E. Mathematical Model for Project Cost Prediction
A1 , A2 ,..., Ai (i 1, 2,..., n) represents n typical projects respectively. T represents the characteristic vector set of project cost, which can describe the project structure feature, namely: T {t1 , t2 ,..., t j } ( j 1, 2,..., m)
(15)
and is the adjustment coefficient in formula (8) [12]. In formula (18), formula (9) or (10) is used to conduct self-estimation on A project cost. The result is that e*A equals the actual cost, namely e*A eA . Suppose eA , eB , eC respectively represents the cost of the similar typical project A, B and C, and their fussy nearness to project A can be shown:
1 ( A, A), 2 ( A, B), 3 ( A, C)
Suppose 1 2 3 , the project cost prediction value of project A is:
e*A [1eA (1 1 ) 2 eB (1 1 )(1 2 ) 1 3
3eC (1 1 )(1 2 )(1 3 )(eA eB eC )]
tij tj
e*A 1eA (1 1 ) 2 eB (1 1 )(1 2 ) 1 3
3 eC (1 1 )(1 2 )(1 3 )(eA eB eC )
t j : represents the name of the characteristic element Ti : represents the fussy sub-set corresponding to ith typical project tij : represents the membership function value of jth characteristic element in ith typical project The fussy sub-set of to-be-built project is : T*
t *j t1* t2* t1 t2 tj
e [1 E1 (1 1 ) 2 E2 (1 1 )(1 2 ) 3 E3 * B
1 (1 1 )(1 2 )(1 3 )( E1 E2 E3 )] 3
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(18)
(22)
In a word, the prediction model based on nearness (9) or (10) can guarantee the project cost prediction value equals to their actual value, which indicates that the model is successful [13]. The adjustment coefficient will be calculated according to the following empirical formula in project cost prediction.
1
1 T' T' [1.8( 1 0.8( 1) m Ta1 Ta 2
T' 0.4( 1)] Ta 3
(17)
t *j ( j 1, 2,3, , m) represents the membership function value corresponding to jth characteristic element of to-be-built project. Again, assume represents the fussy nearness between the built project Ai (i 1, 2,3, , m) and to-be-built project B, which is arranged from big to small, namely, 1 , 2 ,..., n , E1 , E2 ,..., En and the corresponding unit cost of the project is A1 , A2 ,..., An . Choose three known projects with big nearness as estimated benchmark, and meet 1 2 3 ,and the unit cost evaluation value eB* of to-be-built project B is:
(21)
Since 1 ( A, A) 1 , the value of the above three formula is zero. After calculation, we can draw a conclusion that:
e*A eA (16)
(20)
Set equals to 1,
The fussy sub-set of T , by using Zadeh mark, is
t t Ti i1 i 2 t1 t2
(19)
(23)
m : is the number of the elements of fussy set T , Ta1 , Ta 2 , Ta3 : indicates the typical project fuzzy relation coefficient corresponding to to-be-built project respectively and a1 , a2 , a3 , and the range of values
t
ij
Ti
j
is
max tij
concluded
to
the
interval
j 1
[0,1](i 1, 2,..., m) [14]. III.
NUMERICAL RESULTS
A to-be-built project is a small hydropower station, which is named by E, please predict the C20 concrete cost. A,B,C,D is typical engineering which has been established.
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A. List out the Character Vector Set of the Project According to the specific situation of the projects, list out the name of each element of project sets, and these elements should describe the representative characteristics of this project summarily (in tableI). B. Determine the Membership Function Value According to the determined methods of membership function value that are mentioned previously, please determine to the membership function value of each character vector set of the project ( t j ) (in tableI).
t
C. Calculate the
j
of each Typical Project and
To-Be-Built Project For The Engineering of A: tAj 1 1 0.85 0.9 0.6 4.35,( j 1, 2,
,5)
For The Engineering of B: tBj 1 0.95 0.85 0.8 0.85 4.45,( j 1, 2, For The Engineering of C: tCj 1 0.95 1 1 0.9 4.85,( j 1, 2,
,5)
The Engineering of D: tDj 1 0.9 0.8 0.6 0.6 3.9,( j 1, 2, For The Engineering of E: tEj 1 0.95 1 1 1 4.95,( j 1, 2, Because
t
Ej
,5) For
,5)
,5)
4.95 is maximal, so
TE 1, TA 0.88, TB 0.9, TC 0.98, TD 0.79 TABLE I.
THE RELATED CALCULATION OF THE SMALL HYDROPOWER STATION CALCULATION
Hydrological condition
Geologica l condition
t1
t2
The layout of fields and dams t3
t4
The flood routin g metho -ds t5
A
1
1
0.85
0.9
0.6
B
1
0.95
0.85
0.8
0.85
C
1
0.95
1
1
0.9
D
1
0.9
0.8
0.6
0.6
E
1
0.95
1
1
1
C o d e
The size of unit s
The amo unt of C20
Remarks
/m3 4735 typical 5 project 4837 typical 0 project 5131 typical 0 project 4578 typical 0 project To-be- bulit project
D. Test the Reliability of Engineering Firstly, According to the formula 16,List out the fuzzy subset of the Typical Engineering Secondly, According to the formula 14,Calculate the fuzzy nearness of the typical Engineering and other Typical Engineering by turns; ( A, B) 0.58 ,
( A, C) 0.5 , ( A, D) 0.7 ; Thirdly, Sort the fuzzy nearness (from the biggest down to the smallest): 1 0.7 , 2 0.58 , 3 0.55 ;
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The relevant Engineering Quantity: E1 45780 , E2 48370 , E3 51310 ; Fourthly, According to the formula 23,Please calculate to the Adjustment Coefficient 1.03 ; Fifthly, According to the formula 18,Please calculate to the typical project of engineering quantity of A e*A 48174m3 ; Sixthly, Checking the error. 48174 47355 Error: 100% =1.7%
