1560 HYBRID GENETIC ALGORITHM PARAMETER EFFECTS FOR OPTIMIZATION OF CONSTRUCTION RESOURCE ALLOCATION PROBLEM Jin-Lee KIM 1, M. ASCE 1 Assistant Professor, Department of Civil Engineering and Construction Engineering Management, California State University, 1250 Bellflower Blvd., Long Beach, CA 90840, Phone: (562) 985-1679, Fax: (562) 985-2380, Email: jinlee.kim@csulb.edu ABSTRACT The optimal solutions for the resource allocation problem are of great significant to project planners for distributing their available resources into the activities most effectively. Many studies have been undertaken to solve the resource-constrained project scheduling problems using genetic algorithms, which have been proven as an effective and efficient optimization tool to solve difficult and complex problems. One of the trends in the genetic algorithm research study is to develop a hybrid meta-heuristic method using artificial intelligence and biologically-inspired techniques. In an effort to address this issue, the author developed a new hybrid genetic algorithm to solve the construction resource-constrained project scheduling problems. This paper evaluates the parameter effects of the hybrid genetic algorithm for optimization because optimal settings of the genetic algorithm parameters such as population size, crossover probability, and mutation probability, are critical conditions in producing the best value for the outcomes. KEY WORDS Optimization, Heuristics, Resource allocation, Scheduling, Algorithm INTRODUCTION Construction scheduling must include resource allocation to avoid waste and shortage of resources on an actual construction jobsite, since resources for construction activities are limited in the real world. Thus, the resource-constrained project scheduling problem attempts to allocate the available resources to construction activities so as to find the shortest duration of a project within the constraints of precedence relationships. The optimal solutions for the resource-constrained project scheduling problem are of great significant to project planners for distributing their available resources into the activities most effectively. Many studies have been undertaken to solve the resource-constrained project scheduling problem using Genetic algorithms (GAs), which have been proven as an effective and efficient optimization tool to solve difficult and complex problems. Numerous studies have been done to solve the standard resource-constrained project scheduling problem using a GA. Specifically, several works of research conducted by Brucker et al. (1998), Demeulemeester and Herroelen (1997), Mingozzi et al. (1998), Sprecher and Drexl (1998), Brucker et al. (1999), and Klein and Scholl (1999) have been considered to be the currently most powerful exact procedures for solving the resourceconstrained project scheduling problem. One of the trends in the GA research study is to develop a new hybrid meta-heuristic method using artificial intelligence and biologically-
1561 inspired techniques. The concept of a hybrid genetic algorithm has been successfully applied to many engineering optimization problems, such as aerodynamic design (Foster and Dulikravitch 1997), signal analysis (Sabatini 2000), and water resources planning and management (Espinoza et al. 2005), and others (Lobo and Goldberg 1997; Goldberg and Voessner 1999; Kapelan et al. 2000; Sinha and Goldberg 2001; Kapelan et al. 2002; and Hsiao and Chang 2002). Like general genetic algorithm, optimal settings of the GA parameters such as population size, crossover probability, and mutation probability, are critical conditions in producing the best value for the outcomes. In an effort to address this issue, the author developed a new hybrid genetic algorithm to solve the construction resource-constrained project scheduling problems. This paper evaluates the parameter effects of the hybrid genetic algorithm for optimization of construction resource-constrained project scheduling problems. The following section briefly introduces the problem definition of the resource-constrained project scheduling problem, followed by the hybrid genetic algorithm, developed in the previous stage of this research. Next, the methodology for selecting various input parameters that affect the performance of the hybrid algorithm is examined in the experimental design section, followed by the concluding remarks and future studies. RESOURCE ALLOCATION PROBLEM DEFINITION A project that includes a finite set of activities is considered here, where N activities labeled i = 1,, n are given. The objective function for a hybrid genetic algorithm to the resourceconstrained project scheduling problem is to minimize the project duration when constrained by precedence relationships among project activities and the availability of renewable resources. The mathematical expressions of the objective function associated with resource and precedence constraints are given as follows (Kim and Ellis 2009): Minimize f ( i) = max { ti + d i i = 1,2,... n} (1) subject to t t d j S (2) P M N p = 1 m = 1 i = 1 j i i, i Η RR imt P M p = 1 m = 1 Ξ RA mt f ( i) 0 (4) HYBRID GENETIC ALGORITHM This section introduces the hybrid genetic algorithm (HGA) for an optimal solution to the resource-constrained project scheduling problem of construction project networks. Figure 1 shows the operation of HGA. The elitist genetic algorithm (EGA), which is used for base platform, employs four basic operators: elite selection, roulette wheel selection, one-point crossover, and uniform mutation. The initial population of possible solutions to the resourceconstrained project scheduling problems is created to apply the algorithm in the very first step of global search. A fitness value of an individual in an initial population is calculated using three different schedule generation schemes. Evaluation of local search is achieved before the move to the selection operator embedded in the EGA. If local search is needed, it (3)
1562 occurs following the selection operator. Otherwise, local search is not implemented. The selection of the parent individuals is made through the elitist roulette wheel selection operator for the next generation. Using the parent individuals obtained from the selection operator, one-point crossover operator is performed by exchanging parent individual segments and then recombining them to produce two resulting offspring individuals. The uniform mutation operator is performed to play the role of random local search, which searches much smaller portion than the random walk algorithm. Figure 1: Operation of hybrid genetic algorithm (After Kim and Ellis 2009) EXPERIMENTAL DESIGN FOR PARAMETER VALUES The fifteen numbers of combinations are designed based on various input parameter values for the hybrid genetic algorithm in order to show the overall performances, which measure the average percentage deviation and the average CPU runtime. All of input parameter values of local search are set as follows: Variation threshold, local search probability, adaptive parameter, maximum number of local search iterations, and local search proportion are set to 0.75, 0.2, 0.5, 20, and 0.2, respectively, for the hybrid genetic algorithm. The population size varies 30, 50, and 100 for each schedule of 1000 and 5000. The crossover probability varies ranging from 0.5 to 0.9 by the increase of 0.1, which produces
1563 five different combinations per each population size. Transformation power and mutation probability are set to 1.6 and 0.03, respectively, for the comparison purpose. Then, the hybrid genetic algorithm is tested with 30-activity instance sets of 480 problems using the serial schedule generation scheme as a function of the uniquely generate schedules 1000 and 5000, respectively. We compare the results obtained from the hybrid genetic algorithm to those obtained from state-of-the-art algorithms. The number of uniquely generated schedules is adopted as the stopping criteria in order to form the basis for the comparison. A unique schedule means that it is possible for several individuals to have the same fitness value (makespan, the term used in many resource-constrained project scheduling problem studies), but their starting time should be totally different. This measurement is reasonable because of the assumption that the computation effort for constructing one schedule is similar in most heuristics. It is independent of the computer platform so hybrid genetic algorithm can be tested with the original implementation and the best configuration of various input parameters. It is also independent of compilers and implementation skills so that heuristic concepts can be evaluated rather than program codes. It is notable that this stopping criterion has a few drawbacks: it cannot be applied to all heuristics and different heuristics may require different computation times to generate one schedule (Kolisch and Hartmann 2005). Regardless of these drawbacks, the limitation of the number of schedules is the best criterion available for such a broad comparison. Therefore, 1,000 and 5,000 unique schedules are selected as stopping criteria in order to make use of the benchmark results presented in Kolisch and Hartmann (2005). It is very important to note that if one makes use of the results in their works, a schedule generated from hybrid genetic algorithm should be a unique schedule. It is necessary to set the number of a unique schedule as a termination condition, but it is not necessary to set the number of generations as a termination condition in the input parameter value of the hybrid genetic algorithm. In this experiment, the average deviation method is used to compare the performance of the hybrid genetic algorithm to that of the existing algorithms. The average deviation is the precision of measurement. The average deviations for the existing algorithms are known and they are relatively easy to calculate even though standard deviation is a more accurate method of finding the error margin. The average deviation is an estimate of how far off the observed values are from the average value with the assumption that the results computed from the hybrid genetic algorithm are accurate. As shown in Eq. 5, the average deviation can be calculated by finding the average of the percentage of the deviations, subtracting the average values from the observed values. n = i= 1 ( β i α i ) α i n 100 (5)
1564 where, = the average percentage deviation of a problem instance i, i = 1, 2,.., n, α i = the optimal value for 30-activity instance sets, α i = the optimal value for 30-activity instance sets, β i = the observed value obtained from the hybrid genetic algorithm, and N = the number of the problem instance. All optimal solutions for the set with 30 non-dummy activities are known. Tables 1 and 2 show the computation results obtained from the hybrid genetic algorithm for 1000 and 5000 unique schedules, respectively. Test No. Test No. Table 1. Results for 30-activity Instance Sets of 480 Problems with 1000 Schedules Parameters for Combinations Unique Schedules Population size Cp Average fitness values Experimental results Average CPU runtime (sec.) Average deviation (%) from PSPLIB (ave_opt: 58.99) 1 1000 30 0.5 60.08 0.52 1.84 2 1000 30 0.6 60.01 0.54 1.74 3 1000 30 0.7 60.01 0.44 1.73 4 1000 30 0.8 60.02 0.49 1.74 5 1000 30 0.9 60.09 0.46 1.86 6 1000 50 0.5 60.05 0.55 1.79 7 1000 50 0.6 60.06 0.49 1.81 8 1000 50 0.7 59.97 0.46 1.66 9 1000 50 0.8 60.05 0.46 1.79 10 1000 50 0.9 60.08 0.48 1.85 11 1000 100 0.5 60.10 0.52 1.88 12 1000 100 0.6 60.06 0.52 1.81 13 1000 100 0.7 60.04 0.51 1.77 14 1000 100 0.8 60.06 0.50 1.82 15 1000 100 0.9 60.02 0.53 1.75 Table 2. Results for 30-activity Instance Sets of 480 Problems with 5000 Schedules Parameters for Combinations Unique Schedules Population size Cp Average fitness values Experimental results Average CPU runtime (sec.) Average deviation (%) from PSPLIB (ave_opt: 58.99) 1 5000 30 0.5 59.77 2.41 1.32 2 5000 30 0.6 59.76 1.91 1.31 3 5000 30 0.7 59.80 1.82 1.37 4 5000 30 0.8 59.74 1.87 1.27 5 5000 30 0.9 59.72 1.84 1.24 6 5000 50 0.5 59.55 2.67 0.95 7 5000 50 0.6 59.63 1.92 1.08 8 5000 50 0.7 59.59 1.85 1.02 9 5000 50 0.8 59.63 1.93 1.08 10 5000 50 0.9 59.59 1.86 1.02 11 5000 100 0.5 59.55 2.08 0.95 12 5000 100 0.6 59.50 2.13 0.87 13 5000 100 0.7 59.56 2.07 0.96 14 5000 100 0.8 59.55 1.88 0.95 15 5000 100 0.9 59.53 2.05 0.91
1565 The hybrid genetic algorithm produces consistent and correct solutions for the resource-constrained project scheduling problem with a little difference among the average fitness values regardless of the combination considered for 30-activity instance sets of 480 problems. The hybrid genetic algorithm seems likely that it produces comparable solutions for all the problem sets because the location of average deviation obtained from hybrid genetic algorithm takes a middle position among other heuristics. Parameter effect analysis is conducted to examine the behaviors of the hybrid genetic algorithm according to either the population size or the crossover probability. Figure 2 shows the behaviors of average fitness values according to the different population size using results for 30-activity instance sets of 480 problems. It is shown that the hybrid genetic algorithm produces consistent solutions for the resource-constrained project scheduling problems with a little difference among the average fitness values regardless of the combination considered. However, it is observed that as the population size increases, the hybrid genetic algorithm is likely to obtain better solutions. Figure 3 also shows the behaviors of average fitness values according to the different crossover probability. It is observed that as the crossover probability increases, the hybrid genetic algorithm is not likely to show similar patterns as population size. Therefore, it can be concluded that the selection of the population size is a more critical element than that of the crossover probability. Figure 2. Comparison of average fitness values by population sizes
1566 Figure 3. Comparison of average fitness values by crossover probabilities Figure 4 shows the behaviors of the average CPU runtime in seconds per instance according to the different population size using results for 30-activity instance sets of 480 problems. It was shown that the hybrid genetic algorithm requires a similar computation time to obtain a fitness value for a resource-constrained project scheduling problems with little difference among the average CPU time in seconds regardless of the population size. Figure 5 also shows the behaviors of the average CPU runtime in seconds per instance according to the different crossover probability. It was observed that as the crossover probability increases, the hybrid genetic algorithm is likely to require less computation time to obtain a fitness value for a resource-constrained project scheduling problems. Therefore, it can be concluded that the selection of the crossover probability with a high value is likely to decrease the computation time of the hybrid genetic algorithm. Figure 4. Comparison of average CPU runtime by population sizes
1567 Figure 5. Comparison of average CPU runtime by crossover probabilities CONCLUDING REMARKS This paper presented the evaluation of parameter effects on performance of the hybrid genetic algorithm that searches the optimal and/or near optimal solutions to the construction resource allocation problems. The parameters considered in this paper include population size, crossover and mutation probability. As the population size increases, the hybrid genetic algorithm is likely to obtain better solutions. However, as the crossover probability increases, the hybrid genetic algorithm is not likely to show similar patterns as population size, which means that the selection of the population size is a more critical element than that of the crossover probability. The hybrid genetic algorithm requires a similar computation time to obtain a fitness value with little difference among the average CPU time in seconds regardless of the population size. The selection of the crossover probability with a high value is likely to decrease the computation time of the hybrid genetic algorithm. A trial-and-error calibration approach is generally used to determine the best configuration of the parameter values for any arbitrary genetic algorithm. The algorithm outcomes using this approach sometimes result in an inconsistent and time-consuming process. Therefore, the outcomes of this study will be further used to propose a design and analysis methodology that aims to develop response surface design and to employ the response optimization method to jointly optimize two responses, which are minimum project duration and minimal algorithm runtime. The response optimization method, if developed, will be a valuable tool for GA users to control the design variables simultaneously, as it compromises the two responses.
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