Fuzzy Clustering Chaotic-based Differential Evolution ...

2012 (第 16 屆)營建工程與管理學術研討會

Fuzzy Clustering Chaotic-based Differential Evolution for Construction Project Resource Leveling Min-Yuan Cheng1, Duc-Hoc Tran2* and Nhat-Duc Hoang3

Abstract In construction industry, project schedules are commonly generated by network scheduling techniques, e.g. Critical Path Method. However, these schedules oftentimes result in significant resource fluctuations that are impractical and costly for contractors to carry out. Thus, construction managers are required to perform resourceleveling processes to smooth out the resource profile. In this research, a novel optimization model, named as Fuzzy Clustering Chaotic-based Differential Evolution for solving Resource Leveling (FCDE-RL), is introduced. Fuzzy Clustering Chaotic-based Differential Evolution (FCDE) is developed by integrating original Differential Evolution with fuzzy c-means clustering and chaotic techniques to tackle complex optimization problems. Chaotic was exploited to prevent the new approach from premature convergence. Meanwhile, fuzzy c-means clustering acts as several multi-parent crossover operators to utilize the information of the population efficiently to enhance the convergence. Experimental studies revealed that the new optimization model is a promising alternative to assist project managers in dealing with construction project resource leveling. Key words: Resource Leveling; Fuzzy Clustering, Chaotic, Differential Evolution; Construction Management.

1. Introduction In today’s market condition, the survivability of a construction company essentially depends upon its capability of managing resources [1]. Improper resource management may increase the operational expense or even gives rise to financial and scheduling problems. The excess requirement of resource in the construction site may lead to the extension of project duration. As the contractor cannot accomplish the project by the prespecified date, the owner may suffer from financial loss due to the non-availability of the facility [2]. Moreover, construction delays often bring about disputes among parties, higher overhead costs, degradation of reputation, and occasionally result in project failure [3, 4]. Hence, resource management is a crucial task that needs to be implemented thoroughly in the planning phase. Construction resources basically consist of manpower, equipment, materials, money, and expertise; efficient management of these resources holds the key to the successful execution of any project [2]. However, construction schedules, generated by network scheduling techniques, often bring about undesirable resource fluctuations that are impractical, inefficient, and costly for the contractors to implement [5]. Thus, construction managers mandatorily need to perform schedule-adjusting process to reduce unnecessary fluctuations in resource utilization during the project execution. Needless to say, the fluctuations of resource are troublesome for the contractor [6]. The reason is that it is expensive to hire and to lay off workers on a short-term basis according to the fluctuations in the resource profile. Additionally, if the resources cannot be managed efficiently, they may exceed the supply capability of the contractor and lead to schedule delay. Finally, the contractor must maintain a number of idle resources during the time of low demand. These facts undoubtedly cause profit damage for construction companies. The process of smoothing out resources is well known as resource leveling and has been studied extensively by many researchers [5, 7, 8]. In resource leveling, the objective is to minimize the peak demand and fluctuations in the pattern of resource usage [9]. This process aims to minimize variation in resource profile by shifting noncritical activities within their available floats and keep the project duration unchanged. Resource leveling of construction project can be solved by a variety of methods, ranging from mathematical methods, heuristics to evolutionary approaches (e.g. Genetic Algorithm, Particle Swarm Optimization, Differential Evolution, etc.). Recently, Differential Evolution (DE) [10, 11] has increasingly drawn interest of researchers who have explored the capability of this algorithm in a wide range of problems. DE is a population-based stochastic search engine, which is efficient and effective for global optimization in the continuous domain. It uses mutation, crossover, and selection operators at each generation to move its population toward the global optimum. Superior performance of DE over other algorithms has been verified in many reported research works [10, 12, 13]. Despite of aforementioned advantages, original DE or many of its variants still have to face some drawbacks. DE 1

National Taiwan University of Science and Technology, Professor, [email protected] National Taiwan University of Science and Technology, M.Sc. Student, [email protected], *Corresponding author 3 National Taiwan University of Science and Technology, PhD Student, [email protected] 2

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2012 (第 16 屆)營建工程與管理學術研討會 does not guarantee the convergence to the global optimum. It is easily trapped into local optima resulting in a low optimizing precision or even failure [14]. In DE, population may not be distributed over search space, and individuals may be trapped in local solution. It may require more generations to converge toward optimal or near-optimal solution [15]. DE has been shown to have certain weaknesses, especially if the global optimum should be located using a limited number of fitness function evaluations. It is good at exploring the search space and locating the region of global minimum slow at exploitation of the solution [16]. Chaos can be considered as an irregular motion, seemingly unpredictable random behavior under deterministic conditions. Chaotic systems are sensitive to small differences in initial condition may produce huge changes in outcomes. It is extremely sensitive to the initial conditions, and its property sometimes referred to as the instability in the so-called butterfly effect or Liapunove’s sense [17]. The inherent characteristics of chaotic systems provide an efficient approach for maintaining the population diversity in search algorithms. Clustering is one of the most important and the most challenging of classifying algorithms. A successful clustering is able to reliably find true natural groupings in the data set. A soft clustering approach, fuzzy c-means clustering, is introduced to DE to help track the evolution of search algorithm by introducing cluster centers to the populations. Fuzzy c-means clustering is the process of dividing a set of objects into groups or clusters of similarities thereby speeding up the optimization search in DE. Therefore, the objective of this research is to utilize fuzzy c-means clustering and chaotic techniques to cope with the difficulties in original DE. Chaotic sequences have been adopted instead of random sequences and very interesting and somewhat good results have been exploited to prevent the new approach from premature convergence. Meanwhile, fuzzy c-means clustering acts as several multi-parent crossover operators to utilize the information of the population efficiently to make the algorithms converge faster. The paper is organized as follows: In section 2, literature related to the establishment of the new optimization model is briefly reviewed. In section 3 and 4, the overall pictures of the newly proposed optimization model are presented in details. The performance of the newly developed model is demonstrated using numerical experiment and result comparisons in section 5. Conclusions are mentioned in the last section of this article.

2. Literature review 2.1 Resource Leveling In the resource-leveling problem, the objective is to reduce the peak resource demand and smooth out day-today consumption within the required project duration, and with the assumption of unlimited resource availability. Thus, the resource leveling can be formulized as an optimization problem within which the following cost function is minimized [9, 18]: T

f   ( yi  yu )2

(1)

i 1

where T denotes the project duration; yi represents the total resource requirements of the activities performed at time unit i; and yu represents a uniform resource level given by T

yu 

y i 1

(2)

i

T According to Son and Skibniewski [9], Eq. (1) can be rewritten as follows T

T

f   yi2  2 yu  yi  yu2 i 1

(3)

i 1

T

Since activity duration and rate of resource for each activity are fixed, yu and

y i 1

i

are constant. Thus, the cost

function can be expressed as T

f   yi2

(4)

i 1

The Eq. (4), in essence, is equivalent to the minimum moment of resource histogram around the time axis as mentioned in previous works [18, 19]. Moreover, the objective function of the resource-leveling problem needs some modification. It is because the optimization process may yield several scheduling solutions or, in other words, resource profiles that have the same minimum moment of resource demand [9]. Although the cost function values are identical, their resource fluctuations can be different. Hence, to identify the most preferable resource profile, the deviations between resource consumption in consecutive time periods [20] and the peak of resource demand should be taken into account [9]. In the research, a modified objective function for resourceleveling optimization model is presented later on.

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2012 (第 16 屆)營建工程與管理學術研討會

2.2 Chaos Approach Chaos theory is a scientific theory describing erratic behavior in certain nonlinear dynamical systems. Chaotic mappings can be considered traveling particles within a limited range occurred in a deterministic nonlinear dynamic system. There is no definite regularity for such a traveling path. Such a movement is very similar to a random process, but extremely sensitive to the initial condition [21]. Chaotic sequences have been proven easy and fast to generate and store, there is no need for storage for long sequences [15]. Moreover, these sequences are deterministic and reproducible. Many researchers have adopted chaotic sequences instead of random sequences [22, 23]. The one dimensional logistic map is one of the simplest systems with density of periodic orbits

X n1   (1  X n )

In this equation, X n is the

n

th

(5)

chaotic number where n denotes the iteration number. Obviously, X n  (0,1)

under conditions that initial X 0  (0,1) and that X 0 {0.0,0.25,0.5,0.75,1.0} . The variation of control parameter  in Eq. (5) will directly impact the behavior of X greatly. The domain area of control parameter  has often been defined as [0, 4] . In the experiments   4 has been used. The logistic map that generate chaotic sequences in DE, named CDE which ensures the individual in population to be spread in the search space as much as possible for population diversity used in experiments. Incorporating chaotic map into DE is proven to enhance the global convergence by escaping the suboptimal solution. Figure 1 shows the main steps of generating chaotically population.

2.3 Fuzzy c-means Clustering Clustering is a process that aims at decomposing a given set of objects into subgroups or clusters based on similarity. The aim is to divide the data-set in such way that objects belonging to the same cluster are similar as possible, whereas objects belonging to different clusters are as dissimilar as possible. Clustering algorithms can be divided into main categories: crisp (or hard) clustering procedures where each data is assigned to exactly one cluster, fuzzy clustering techniques where every data point belongs to every cluster with a specific algorithm degree of membership [24]. Many clustering algorithms are introduced in the literature. The fuzzy c-means (FCM) clustering [25] is employed in this study. The FCM clustering technique adopted in DE, named FDE, could easily conduct an efficient convergence of DE. FCM introduced in this study was intended to track the main stream of population movement during DE evolution. Each cluster centers could be treated approximately as one of the items in the main stream of evolution, and replaced for population as candidate individuals. The FDE algorithm is illustrated in Figure 2. Where m is clustering period, NP is the population size, and k, the number of centroid [26], is an integer number from [2, NP ] . Population

Population Percent to chaos Randomly select From (40-60%)

Clustering period m

CF = (0.4 ~ 0.6) NP Mod (g,m) = 0 2

 g   g  threshold      2   G max   G max 

Generate randomly k from [2,sqrt(NP)]

Rand() ≤ threshold

Choose randomly k parents (SetA) from population P

Yes

No

Mapping population to chaos feasible region (0,1) No

Adopt the fuzzy c-means clustering to create k centroid points (SetB)

Chaos fuctions

From the combined (SetA+SetB) choose k best solutions and put them in SetC

Mapping to feasible region X ig, j 1  X imin  cmig 1 ( X imax  X imin )

Update the population P as P=P – SetB + SetC

Population

Population

Figure 1 Chaotic approach

Figure 2 Fuzzy c-means clustering algorithm

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2012 (第 16 屆)營建工程與管理學術研討會

3. Fuzzy c-means Clustering Chaotic-based Differential Evolution (FCDE) In this section, the proposed FCDE optimization algorithm is described in details. The FCDE is the core search engine in the FCDE-RL model. It is noticed that our algorithm is developed based on standard Differential Evolution [10, 11] by integrating original DE with fuzzy c-means clustering and chaotic techniques. Chaos approach effectively exploits the whole search space and provides the necessary diversity in the DE population. Consequently, this operation incurs additional time and iteration to search the global optimum. On the contrary, fuzzy c-means clustering technique enhances the convergence speed of the algorithm by introducing the cluster centers. These moving centers provide direction for search of the global optimum improving the overall efficiency of the search algorithm. FCDE model, exploits the inherent characteristics of both chaos algorithm and fuzzy clustering and integrates it with differential evolution to improve the overall search capabilities of DE in finding the optimal solutions for a given search space. The overall picture of the proposed algorithm is illustrated in Figure 3. 1. Initialization g =1

3. Crossover 4. Selection 5. Chaos operator

g=g+1

2. Mutation

No

6. Fuzzy c-means clustering approach

7. Stopping Condition Yes Optimal Solution

Figure 3 Fuzzy Clustering Chaotic based Differential Evolution (FCDE)

3.1 Initialization FCDE commences the search process by randomly generating population size NP, Maximum of generation

Gmax number of D-dimensional parameter vectors X i , g where i  1,2,..., NP and g indicates the current generation. In the original DE algorithm, NP does not change during the optimization process [11]. Moreover, the initial population (at g = 0) is expected to cover the entire search space uniformly. Hence, we can simply generate these individuals as follow:

X i ,0  LB  rand [0,1]*(UB  LB)

(6) at the first generation. rand [0,1] denotes a uniformly distributed

where X i ,0 is the decision variable i random number between 0 and 1. LB and UB are two vectors of lower bound and upper bound for any decision variable.

3.2 Mutation For each target vector, a mutant vector is created by three randomly chosen vectors in the current population. It is worth noticing that the magnitude of the mutation scale factor (F) influences the search step of mutation operator. This process is mathematically shown as follow:

Vi , g 1  X r1,g  F ( X r 2,g  X r 3,g )

(7) where r1, r2, and r3 are three random indexes lying between 1 and NP. These three randomly chosen integers are also selected to be different from the index i of the target vector. F denotes the mutation scale factor, which controls the amplification of the differential variation between X r 2, g and X r 3, g . Vi , g 1 represents the newly created mutant vector.

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2012 (第 16 屆)營建工程與管理學術研討會

3.3 Crossover The crossover stage aims to diversify the current population by exchanging components of target vector and mutant vector. In this stage, a new vector, named as trial vector, is created. The trial vector is also called the offspring. The trial vector can be formed as follow:

 V j ,i , g 1 , if rand j  Cr or j  rnb(i ) U j ,i , g 1     X j ,i , g , if rand j  Cr and j  rnb(i )

(8)

where Uj,i,g+1 is the trial vector. j denotes the index of element for any vector. randj is a uniform random number lying between 0 and 1. Cr is the crossover probability, which is needed to be determined by the user. rnb(i) is a randomly chosen index of {1,2,..., NP} which guarantees that at least one parameter from the mutant vector (Vj,i,g+1)

3.4 Selection In this stage, the trial vector is compared to the target vector [11]. If the trial vector can yield a lower objective function value than its parent, then the trial vector replaces the position of the target vector. The selection operator is expressed as follow:  U i , g if f (U i , g )  f ( X i , g ) X i , g 1     X i , g if f (U i , g )  f ( X i , g )

(9)

3.5 Chaos Operator If the probability condition is satisfied, a percentage of the population is selected to do chaos. Mapping population to chaotic feasible region (0,1) and performs the logistic map following equation (5). Afterwards, map the chaotic variables to feasible region according to equation (10):

X gj ,i  X min  cmkj ( X max  X min j j j )

j  1,2,..., D

(10)

j k

where cm are chaotic variables according to chaotic formula.

3.6 Fuzzy c-means clustering approach Initially, the period of the clustering operator specified in the algorithm is 10. Consequently, the number of clusters defined randomly in range of [2, NP ] . The corresponding centroids of each cluster are added to the population as feasible solutions.

3.7 Stopping Condition The optimization process terminates when the stopping criterion is met. The user can set the type of this condition. Commonly, maximum generation (Gmax) or maximum number of function evaluations (NFE) can be used as the stopping condition. When the optimization process terminates, the final optimal solution is readily presented to the user.

4. Fuzzy c-means Clustering Chaotic-based Differential Evolution for Resource Leveling This section of the article is dedicated in describing the FCDE-RL optimization model (see Figure 4). It is noticed that the FCDE-RL is developed based on the FCDE as the searching engine. The objective of this optimization model is to minimize daily fluctuations in resource utilization without altering the total project duration. In this study, we consider the case that the resource leveling is accomplished by minimizing the fluctuations between resource requirements and a desirable uniform resource level. The model requires inputs of project information including activity relationship, activity duration and resource demand. In addition, the user also needs to provide parameter setting for the search engine, such as maximum number of searching generation (Gmax), the population size (NP), the chaotic percentage (CF), and the period clustering (m). With these inputs, the scheduling module can carry out calculation process to obtain critical path method (CPM) based schedule, early start and late start of each activity. With all the necessary information provided, the model is capable of operating automatically without any human intervention.

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2012 (第 16 屆)營建工程與管理學術研討會 Project information Optimizer’s parameter setting

Scheduling module

FCDE Optimizer Parameter Generation

Population of NP solutions [X1, 1, X1, 2,…, X1, D ] ... [XNP, 1, XNP, 2,…, XNP, D ]

Mutation Crossover Selection Chaos Fuzzy c-means clustering

No

Searching Termination Yes

Optimal Project Schedule Optimal Activity Start-Time [X1, X2, …, XD]

Figure 4 Fuzzy Clustering Chaotic based Differential Evolution for Resource Leveling (FCDE-RL) Before the searching process can commence, an initial population of feasible solutions is created using a uniform random generator. A solution for the resource-leveling problem is represented as a vector with D elements as follow:

X  [ X i ,1, X i , 2 ,..., X i , D ]

(11) where D is the number of decision variable of the problem at hand. It is obvious that D is also the number of activities in the project network. The index i denotes the ith individual in the population. The vector X represents the start time of D activities in the network. Since original DE operates with real-value variables, a function is employed to convert those activities’ start times from real values to integer values within the feasible domain.

X i , j  Round ( LB( j )  rand[0,1]  (UB( j )  LB( j )))

(12) where X i , j is the start time of activity j at the individual ith. rand[0,1] denotes a uniformly distributed random number between 0 and 1. LB(j) and UB(j) are early start and late start of the activity j. The search engine (FCDE) takes into account the result obtained from scheduling module and shifts noncritical activities within their float times to seek for an optimal project schedule. In the research, the following objective function and constraints are employed: T

T 1

k 1

k 1

f    ( yk )2    yk 1  yk    ymax

(13)

Subject to

STi  ESi  TFi ; STi  0 ; i  1,2,..., D (14) where T denotes the project duration; yk represents the total resource requirements of the activities performed at time unit k;. (yk+1 - yk) measures the different of resource usage between two consecutive time periods. ymax denotes the peak of resource demand throughout the project execution. α, β, and γ are weighting coefficients. STi is the start time of activity i. ESi and TFi represent early start and total float of activity i, respectively. D is the number of activities in the network. After the searching process terminates, an optimal solution is identified. The project schedule and its corresponding resource histogram are then constructed based on the optimal activities start time. 5. Experimental Results In this section, a construction project adapted from Sear et al. [27] is used to demonstrate the capability of the newly developed FCDE-RL model. The project consists of 44 activities and the total project duration is determined to be 70 days (see Table 1). In this study, the resource of interest is manpower. The resource profile of project before resource-leveling process is shown in the Figure 5.

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2012 (第 16 屆)營建工程與管理學術研討會 Table 1 Project information Activity ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

Duration

Predecessors

0 10 5 15 3 10 15 7 3 3 2 3 2 2 3 1 1 1 4 2 2 1 3 1 4 2 25 3 3 3 3 3 2 0 4 3 3 3 5 1 3 1 6 0

1 1 1 1 1 2 3 5 5 5 9, 10, 11 10 8, 12 12, 13 14 15 16 7, 9, 17, 18 15, 18 19 20 21 22 23, 24 25 6 23 23 26 30 30 27, 29, 30 32 33 34, 35 36 28, 31, 37 28, 31, 37 36 38, 39, 40 41 42 43

Daily Resource Demand 0 5 2 3 2 2 6 10 6 2 2 6 1 5 2 6 7 7 13 9 4 6 8 3 8 7 10 6 2 9 10 3 4 0 1 12 12 3 8 2 10 3 3 0

Early Start (ES) 0 0 0 0 0 0 10 5 3 3 3 15 6 18 18 20 21 21 25 22 29 24 31 25 34 38 10 34 34 40 43 43 43 46 45 49 52 55 55 52 60 63 64 70

Late Start (LS) 0 0 9 3 12 8 10 14 22 15 16 18 19 21 21 23 24 24 25 30 29 32 31 33 34 38 18 52 40 40 52 46 43 49 45 49 52 57 55 59 60 63 64 70

5.1 Optimization result of FCDE-RL In this section, FCDE-RL model is applied to reduce significant resource fluctuations. Parameter setting for FCDE optimizer is shown in the Table 2. The project’s resource profile after being optimized by FCDE-RL is depicted in the Figure 6. The optimal solution or optimal activities’ start times are listed in the Table 3. The optimal results obtained from the new model are listed as followed: Fitness = 9486, Mx = 9201, RDmax = 23, RVmax = 7, and CRV = 55. Where: Fitness  

T 1 1 T 1 T 2 M  ( yk ) 2 ; RDmax  ymax ( y )   y  y    y ;   k  x k 1 k max 2 2 k 1 k 1 k 1 T 1

RVmax  max[ y2  y1 , y3  y2 ,..., yT  yT 1 ] ; CRV   ( yk 1  yk ) k 1

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2012 (第 16 屆)營建工程與管理學術研討會

25

35

Manpower

Manpower 30

20

Resource demand

Resource demand

25

20

15

15

10

10

5 5

0

0 0

10

20

30

40

50

60

70

0

10

20

30

Period

40

50

60

70

Period

Figure 5 Project resource profile before resourceleveling process

Figure 6 Project resource profile after being optimized by FCDE-RL

Table 2 FCDE-RL’s parameter setting Input parameters Number of decision variables Population size The crossover probability Percentage of population to chaos Period clustering

Notation

Number of centroid in clustering Maximum generation Weighting coefficient 1 Weighting coefficient 2 Weighting coefficient 2

Setting

D

44

NP

8xD

CR

0.9

CF

40-60%

m

0.1

k

[2, NP ]

Gmax

3000 1 1 10

Table 3 Optimal Start Time (ST) for all activities found by FCDE-RL Activity ID 1 2 3 4 5 6 7 8 9 10 11

5.1

Optimal ST 0 0 0 0 0 0 10 8 5 15 3

Activity ID 12 13 14 15 16 17 18 19 20 21 22

Optimal ST 15 15 19 18 23 22 21 25 29 29 24

Activity ID 23 24 25 26 27 28 29 30 31 32 33

Optimal ST 31 31 34 38 18 43 37 40 46 43 43

Activity ID 34 35 36 37 38 39 40 41 42 43 44

Optimal ST 48 45 49 52 55 55 58 60 63 64 70

Result comparisons

The results comparison between FCDE-RL and project management software, Microsoft Project 2007, is shown in the table 5. Obviously, performance of the new model is significantly better than that of the commercial software in terms of Mx, RDmax, RVmax and CRV. This means that the new model has reduced the resource fluctuation considerably.

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2012 (第 16 屆)營建工程與管理學術研討會 Table 4 Result comparison between FCDE-RL and Microsoft Project 2007 Methods FCDE-RL Microsoft Project 2007

Mx 9201 9717

RDmax 23 24

RVmax 7 14

CRV 55 125

To better verify the performance of the proposed model (FCDE-RL), three different algorithms are used for performance comparison: standard DE (DE) [28], Genetic Algorithm (GA), and Particle Swarm Optimization (PSO). To evaluate the stability and accuracy of each algorithm, the optimization performance is expressed in terms of best result found (best), average result (avg), standard deviation (std), and worst result (worst) after 20 times of running (see Table 5). Observing from Table 5, the performance of the newly developed model is competitive in terms of accuracy and stability. The proposed algorithm achieves the best results in all of the metrics for performance measurement. FCDE-RL successfully diminishes the moment of resource histogram, maximum resource demand, and deviation of resource between consecutive periods. Since the average and standard deviation of results obtained from FCDE-RL are also smaller than that of other algorithms, the stability and accuracy of the new model are demonstrated. Table 5 Result comparison between FCDE-RL and benchmarked algorithms Performance Measurement

Fitness

Mx

RDmax

RVmax

CRV

Best Avg Std Worst Best Avg Std Worst Best Avg Std Worst Best Avg Std Worst Best Avg Std Worst

PSO

GA

DE1

DE2

DE3

FCDE-RL

9591.0 9682.9 79.8 9940.0 9266.0 9320.3 56.5 9513.0 24.0 27.4 2.6 32.0 7.0 8.5 0.9 10.0 61.0 89.1 12.1 110.0

9579.0 9609.3 14.1 9630.0 9251.0 9278.5 12.7 9303.0 23.0 23.8 0.7 26.0 7.0 9.3 1.2 12.0 77.0 92.8 9.0 109.0

9548.0 9610.6 44.1 9713.0 9235.0 9280.8 26.8 9323.0 23.0 24.9 1.7 29.0 7.0 9.1 1.0 10.0 68.0 81.3 10.8 109.0

9488.0 9508.5 17.1 9566.0 9201.0 9219.1 11.3 9251.0 23.0 23.0 0.0 23.0 7.0 7.2 0.6 9.0 51.0 59.3 9.9 85.0

9488.0 9508.1 10.2 9522.0 9201.0 9216.9 10.0 9231.0 23.0 23.4 0.5 24.0 7.0 7.2 0.6 9.0 51.0 57.2 3.5 65.0

9486.0 9494.5 7.5 9504.0 9201.0 9206.5 7.6 9221.0 23.0 23.0 0.0 23.0 7.0 7.0 0.0 7.0 51.0 56.3 2.8 65.0

6. Conclusions This paper presents the use of FCDE in solving resource leveling problem. The integration of two contrasting algorithms, fuzzy clustering and chaos, has proven to be effective in eliminating the drawbacks of original DE. Chaos algorithm’s randomness enhanced the population diversity and avoided being trapped at local optima, while fuzzy c-means clustering improved the convergence speed of the search algorithm provided by the moving cluster centers. Experimental results and results comparisons have demonstrated that FCDE-RL can deliver accurate and stable results. Improved performance of FCDE presents an alternative tool in solving large complex optimization problems in construction management.

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2012 (第 16 屆)營建工程與管理學術研討會

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