Multi-tier interactive genetic algorithms for the optimization of long ...

Advances in Water Resources 34 (2011) 1343–1351

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Multi-tier interactive genetic algorithms for the optimization of long-term reservoir operation Kuo-Wei Wang a, Li-Chiu Chang b, Fi-John Chang a,⇑ a b

Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei 10617, Taiwan, ROC Department of Water Resources and Environmental Engineering, Tamkang University, New Taipei City 25137, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 22 February 2011 Received in revised form 18 July 2011 Accepted 18 July 2011 Available online 24 July 2011 Keywords: Multi-tier interactive genetic algorithm (MIGA) Optimization Reservoir operation Decomposition

a b s t r a c t Genetic algorithms (GAs) are well known optimization methods. However, complicated systems with high dimensional variables, such as long-term reservoir operation, usually prevent the methods from reaching optimal solutions. This study proposes a multi-tier interactive genetic algorithm (MIGA) which decomposes a complicated system (long series) into several small-scale sub-systems (sub-series) with GA applied to each sub-system and the multi-tier (key) information mutually interacts among individual sub-systems to find the optimal solution of long-term reservoir operation. To retain the integrity of the original system, over the multi-tier architecture, an operation strategy is designed to concatenate the primary tier and the allocation tiers by providing key information from the primary tier to the allocation tiers when initializing populations in each sub-system. The Shihmen Reservoir in Taiwan is used as a case study. For comparison, three long-term operation results of a sole GA search and a simulation based on the reservoir rule curves are compared with that of MIGA. The results demonstrate that MIGA is far more efficient than the sole GA and can successfully and efficiently increase the possibility of achieving an optimal solution. The improvement rate of fitness values increases more than 25%, and the computation time dramatically decreases 80% in a 20-year long-term operation case. The MIGA with the flexibility of decomposition strategies proposed in this study can be effectively and suitably used in long-term reservoir operation or systems with similar conditions. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The water issues in the current and foreseeable future will stay increasingly challenging and become more and more intertwined with development sectors, such as agriculture, energy, and industry. The effects of global climate change may add further to this challenge; thus water shortage problems require greater attention and more actions. Most tropical and sub-tropical regions where Taiwan is located are characterized by vast seasonal and annual variations in rainfall, which are usually a composite of eccentric short-term variations. Such variability increases the demand for infrastructure development and the need for water resources management in a cost-effective and sustainable way. The system analysis approach has been established as one of the most important advances in the field of water resources management, and has had great potential in providing appropriate support for effective management in this emerging context. Applications of system analysis models in various water resources problems are extensive. A large number of computer-based analytical tools, from simulation to optimization, are available for formulating, analyzing and ⇑ Corresponding author. Tel.: +886 2 23639461; fax: +886 2 23635854. E-mail address: [email protected] (F.-J. Chang). 0309-1708/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2011.07.004

solving water resources planning and operational problems [16,18,21,25]. In the early stages, most optimization methods for water resources systems analysis involved linear programming (LP), dynamic programming (DP), and various nonlinear programming methods. Numerous water resources studies have utilized these methods along with additional techniques for a wide variety of problems [10,12,20,22]. Rapid developments have taken place in this field, and high-speed computers have contributed to its development. Genetic algorithms (GAs), introduced by Holland [11] and developed by Goldberg [9], are one of those developments that offer a powerful optimization approach. GAs, analogous to Darwinian natural selection, combine an artificial survival of the fittest and the natural genetic operators in an attempt to find the optimal solution in a given solution space. They have become increasingly popular for solving complicated optimization problems in a large number of different disciplines. In recent decades, GAs have become popular in global optimization application of reservoir planning and management [2– 7,13,24]. The increasing complication of modern engineering fields has brought new problems involving huge numbers of variables and constraints. Due to the complication with high-dimensionality of the problems, it is extremely difficult to obtain suitable solutions

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to such large-scale problems. How to decompose a large-scale engineering problem into smaller-scale sub-problems to accelerate the search of numerical solutions and enhance the robustness of the search is a crucial and very challenging issue. The idea of decomposing a complicated and/or high-dimensional problem into sub-problems was elaborated on by Mausser and Lawrence [17], Sprecher [23], and Palpant et al. [19]. These studies presented different methodologies to improve their solutions to the problems through producing specific sub-problems and then solving the sub-problems by an exact or heuristic solution procedure. From such a point of view, an effective way to find a suitable decomposition strategy for a high-dimensional problem and then interactively solved by GA is extremely important and highly demanded. GA codes decision variables of an optimization problem to one artificial chromosome in the population, and then a search starts with an initial population generated randomly with a large number of chromosomes and progresses to improve the fitness of solutions through iterations by implementing GA operators. As the number of decision variables increases, it requires huge number of computation in each generation and easily causes the curse of dimensionality. The search process would be ineffective iterations of GA operators within the vast chromosomes and result in very slow evolution between consecutive generations, consequently sharply decreasing the possibility of converging to an optimal solution. A direct solution to the problem with a GA formulation on this whole string could be very time consuming, while the possibility of converging to a local optimal solution is high. GA incorporating sequential decomposition procedure to provide a faster and more effective search mechanism should be very useful. Anelli et al. [1] developed a method to decompose arbitrarily shaped morphological binary structuring elements into chains of elementary factors using GAs. Debels and Vanhoucke [8] proposed a decomposition-based GA for solving the resource-constrained project scheduling problem. Kato and Sakawa [14] presented a detailed treatment of GAs with decomposition procedures developed for large-scale multidimensional knapsack problem (MKP) with block angular structures through a triple string representation and the corresponding decoding algorithm. Kim and Weissman [15] decomposed one job into n independent sub-tasks and solved by a GA-base algorithm. Most of the above studies have presented how their problems could be decomposed into independent sub-problems and the effectiveness through their decomposition strategies. For long-term reservoir operation, the optimal reservoir operation is commonly searched year by year, and the initial reservoir storage is usually set as the historical reservoir storage. The initial storage of the next consecutive year is then set the same as the reservoir storage at the end of the preceding year, and a year-overyear search is continuously operated and recorded [4,7]. In such way, the long-term (n years) operation problem can be easily broken into n independent single year operation problem, while the search domain for initial storage is limited to certain ranges and the over year conditions could not be adequately delivered over time and effectively integrated as a whole situation. In this study, a multi-tier interactive genetic algorithm (MIGA) is developed and its efficiency and effectiveness for searching optimal solutions of long-term (n years) reservoir operation is investigated.

search the optimal solutions of the entire system and sub-systems through genetic algorithms. There are two kinds of tiers, primary tier and allocation tier, dealing with an entire system and its sub-systems, shown in Fig. 1. The allocation tier is pulled out from the primary tier and manages the sub-systems in view of small scale by performing the detailed search. By breaking up a complicated system into tiers, the system or the sub-systems can be decomposed into several small-scale sub-systems by only adding another allocation tier, rather than rewriting the entire application over. We would like to note that a sub-system can be further decomposed into several sub-systems to form another allocation tier; consequently it is a multi-tier architecture. The MIGA manages the entire system in view of large scale and supplies key information over the sub-systems. The key information relates the sub-systems to the entire system and helps generating initial populations for GA search in sub-systems. In the MIGA, the GAs play as the search engines in the primary tier and the allocation tier(s). In view of large scale, the system has key variables searched by one GA (hereinafter denoted pGA) in the primary tier; while in view of small scale, each sub-system has its own variables searched by another GA (hereinafter denoted aGA) in the allocation tier(s). Therefore, there are one pGA in the primary tier and several aGAs in the allocation tier(s) (the number of the allocation tiers depends on the way of decomposition or the property of the system). However, pGA and aGAs can be implemented by different GA algorithms. The operation procedure of MIGA is addressed as follows. 1. Decomposition: The entire system is decomposed into a number of sub-systems based on the property of the system, and the entire system and sub-systems are assigned to the primary tier and allocation tiers, respectively. 2. pGA search: (a) Randomly initialize a population P0 of size M for pGA, and then go to Step 3. (b) Combine the fitness values of all aGAs as the fitness value of pGA and then implement pGA operator. 3. Transmission of key information: The key information is transmitted from the primary tier to the allocation tier or from the preceding allocation tier to the following (allocation) tier. 4. aGA search: (a) Initialize a population Pi of size Mi for the ith aGA subject to their corresponding key information. (b) Evaluate their own fitness values and then implement aGA operators.

2. Methodology In the real world, complicated systems usually have a great number of variables that cause great difficulty for optimization methods to search the optimal solutions due to dimension complication. The main concept of MIGA is to decompose a large-scale system into several small-scale sub-systems and interactively

Fig. 1. Decomposition concept of MIGA.

K.-W. Wang et al. / Advances in Water Resources 34 (2011) 1343–1351

(c) Repeat Step 4(b) until the stop criterion of each aGA is reached. 5. Termination condition: Go to Step 2(b) until the stop criterion of pGA is reached. In searching the optimal solution for a complicated system with a large number of variables, it is not uncommon to find some of the search processes repeatedly proceed without any influence to the objective. Those redundant search processes could be eliminated to increase the effectiveness. We implement a search strategy (denoted MIGAI): it is not necessary to re-execute the aGA search for a specific sub-system if the values of the key information (i.e., the initial and final storage variables of the sub-system) in the current aGA remain the same as those in the preceding generation. That means the connectors remain the same for the sub-system and no further improvement would be obtained, consequently its search processes become redundant. The aGA search result would be set the same as that of the preceding generation. Moreover, if the pGA searches for the optimal solution, the key information still have chances to change its variable values. Consequently, the aGAs search new optimal solutions, and MIGA can obtain a new optimal solution. In sum, we propose MIGA that decomposes a large-scale system into several small-scale sub-systems and interactively searches the optimal solutions to the entire system and sub-systems through genetic algorithms over the multi-tier architecture, which dramatically reduces the search time and increases the possibility of obtaining the optimal solution to the entire system. In addition MIGAI, a search strategy that eliminates redundant search in sub-systems, would further increase the efficiency and effectiveness. 3. Application The Shihmen Reservoir, located on the mid-stream reach of the Tahan River in northern Taiwan, is used as a study case (see Fig. 2). One of the most important water resources facilities in northern Taiwan, this reservoir has operated for over 40 years and served five purposes: irrigation, hydropower generation, municipal and industrial water supply (public water supply), flood prevention, and recreation. In this study, the long-term optimal water releases of the Shihmen Reservoir are investigated. Because serious drought events occurred from 2002 to 2005 in the study area, these years are investigated in all study cases. A 10-day period, the traditional time frame in Chinese agricultural society, is used as a time step. According to this scale, each month has three time steps, and each year has 36 time steps (variables). Case I: five consecutive years (2001–2006) with 180 variables (i.e. 36  5 = 180); Case II: 10 consecutive years (1996–2006) with 360 variables (i.e. 36  10 = 360); Case III: 20 consecutive years (1986–2006) with 720 variables (i.e. 36  20 = 720). To reduce the over-year effort, the annual evaluation is set from the 13th 10-day of the current year up to the 12th 10-day of the next year (e.g., the 13th 10-day of the year 2001 up to the 12th 10-day of the year 2002), because this time frame has relatively slighter variation of reservoir storages than other time frames in Taiwan. The initial value of reservoir storage in each year is set the same as that of reservoir storage at the end of the preceding year, and the MIGA search is performed and recorded for Cases I, II and III. 3.1. The reservoir operation formulation The purpose of long-term reservoir operations is to best satisfy water demands in the downstream area of the Shihmen Reservoir, and the decision variables are the reservoir storages for the longterm operations. After finding the reservoir storages, the amount of water release from the reservoir can be calculated, and then

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the average 10-day water shortage rate can be evaluated based on water release and water demand. The average 10-day water shortage rate, defined as Eq. (1), was adopted as the criterion for evaluating the reservoir performance; therefore, the objective function is defined as Eq. (2)

  Dðk; jÞ  Rðk; jÞ ;0 Dðk; jÞ ! n 36 X X 1 SRðk; jÞ Minimize Z ¼ SR ¼ n  36 k¼1 j¼1

SRðk; jÞ ¼ max

ð1Þ ð2Þ

where SR(k, j), R(k, j) and D(k, j) are the average 10-day water shortage rate in the jth 10-day of the kth year, the total water release and total water demand (including public and irrigation water demand) between the beginning of jth 10-day and the end of jth 10-day of the kth year, respectively; j denotes the jth 10-day; k denotes the kth year; n denotes the total number of evaluation years. SR reflects whether the water release can satisfy the water demand in every time period and spans the range from 0 to 1. For long-term reservoir operation, SR represents the entire water allocation status. Cases I, II, III have 180, 360, and 720 variables, respectively; and three GA search strategies (the sole GA, MIGA, MIGAI) are applied to each case. The initial reservoir storage of the first year is given by the historical operation record for three cases. The sole GA search strategy is used for comparison purpose. The merits of the sole GA and the MIGA search strategies are addressed as follows. 3.1.1. Multi-tier interactive genetic algorithm (MIGA) In the MIGA, we take a half year as a large scale and 10-day as a small scale for n years; therefore, the pGA has 2n key variables and each aGA has 17 variables. The key variables of pGA supply the initial and final storages for related aGA. For instance, Case I, which includes five-year reservoir operation with 180 variables, can be divided into 10 half-yearly variables, searched by one pGA, and 10 aGAs, each has 17 variables, shown in Fig. 3. The MIGA has decomposed the entire time interval of reservoir operations (all 10-day operations of n years) into several sub-time intervals (10day operations of a half year). Therefore, Eq. (2) needed to be decomposed into Eq. (3), the average 10-day water shortage rate of 18 10-day periods, for each aGA. Eq. (4) shows the objective function of pGA for summarizing the average 10-day water shortage rate of 2n aGAs

Minimize zðiÞ ¼ Minimize Z ¼

18 1 X SRði; jÞ 18 j¼1

2n 1 X zðiÞ 2n i¼1

ð3Þ ð4Þ

where SR(i, j) is the average 10-day water shortage rate in the jth 10-day of the ith aGA. 3.1.2. Constraints The constraints are the reservoir water balance, the bounds of storage and the positive condition of release. These constraints are expressed as follows:

Rði; jÞ ¼ Sði; j  1Þ  Sði; jÞ  Eði; jÞ þ Iði; jÞ

ð5Þ

Smin 6 Sði; jÞ 6 Smax

ð6Þ

Rði; jÞ P 0

ð7Þ

where Smin and Smax are the minimum and maximum storages; S(i, j) is the reservoir storage in the jth 10-day of the ith year (sub-system); R(i, j), E(i, j) and I(i, j) are the total water release, total evaporation and total inflow of the reservoir between the beginning of jth 10-day and the end of jth 10-day of the ith year (sub-system), respectively.

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Fig. 2. Locations of the Shihmen Reservoir and the Tahan River.

x1

x2

x3

…

x180

Multi-tier Interactive Genetic Algorithm Primary Tier

x18 (1,18)

x36 (1,36)

x54 (2,18)

x72 (2,36)

x180 (5,36)

…

Allocation Tier x1

x19

x37

x55

x163

x2

x20

x38

x56

x164

x3

x21

x39

x57

…

x165

…

…

…

…

…

x17

x35

x53

x71

x179

Fig. 3. Decomposition strategy of MIGA applied to reservoir operation as a case study.

3.2. Setting parameters of the genetic algorithms The huge search domain of a complicated system could make the GA search procedure easily fall into local optima and extremely difficult to converge to an optimal solution. To avoid these situations, it is crucial to find the suitable initial sets of feasible values for the decision variables. In this case, the long-term reservoir operations are very dependent on the distribution and amount of

inflow during the evaluation years. Therefore, the reasonable water releases, sketchily estimated from the inflows and the demands, can provide potential values to calculate the suitable reservoir storages. The initial populations of the sole GA and pGA are divided into two different search spaces: (a) 20% of the initial populations are distributed randomly from the abovementioned suitable initial sets, and (b) the remaining 80% are spread randomly between their corresponding Smin and Smax. For each aGA, the similar initialization

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K.-W. Wang et al. / Advances in Water Resources 34 (2011) 1343–1351 Table 1 Parameters of the sole GA and MIGA. Case

Case I (5 years )

Method

Sole GA

GA type Variable number

CGAa 180

Population size Generation number Constraint method

20,000 20,000 Penalty function 4  108

Number of computationb,c Selection Crossover Crossover rate Mutation rate a b c

Case II (10 years ) MIGA

Sole GA

pGA

aGA

GA 10 (5  2) 100 100 None

CGA 17

CGA 360

200 200 Penalty function

60,000 50,000 Penalty function 3  109

4  108

Case III (20 years )

MIGA

Sole GA

pGA

aGA

GA 20 (10  2) 200 100 None

CGA 17

CGA 720

200 200 Penalty function

60,000 50,000 Penalty function 3  109

8  108

MIGA pGA

aGA

GA 40 (20  2) 200 100 None

CGA 17 200 200 Penalty function

8  108

Roulette wheel selection with elite strategy Linear interpolation method 0.80 0.05

CGA: constrained GA. Computation number of sole GA = population size  generation number. Computation number of MIGA = computation number of pGA  computation number of aGA.

is used, but the initial and final storages provided from pGA should be considered. Table 1 presents the setting parameters of the sole GA and MIGA for three cases. The computation means the number of population plus the number of generation in each GA search, and furthermore, the number of computation of MIGA means the number of computation of pGA plus the number of computation of aGA. Additionally, the constrained genetic algorithm [4] is applied to aGAs and the sole GA due to Eqs. (5)–(7). As shown, the sole GA requires a larger population size and generation number because of large numbers of variables for all cases. In comparison, the relatively small number of pGA variables require a much smaller population size and generation number. Furthermore, the number of

aGA variables is also comparatively small as compared with those of the sole GA. Consequently MIGA can get the optimal solution much easier than the sole GA. For comparison purpose, a simulation based on the M-5 rule curves is used to evaluate the model performances in Cases I, II and III. Fig. 4 shows the current operating M-5 rule curves of the Shihmen Reservoir. The upper, lower and critical limit rule curves are used for the multi-purpose reservoir operations. Based on the operating rule curves, when the water level is below the critical limit, agricultural water release must be cut off by 50%, and public water release must be cut off by 20%. More detailed description regarding the M-5 rule curves can be found in Chang et al. [4].

250 Upper limit curve Lower limit curve Critical lower limit curve

Storages(10 6 m3)

200

150

100

50

0

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

10-day Fig. 4. M-5 rule curves of the Shihmen Reservoir.

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103 the Water Demand in the Downstream Area

inflow (106 m3)

102

101

wet season 100 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

10-day Fig. 5. 10-Day inflow boxplot of the reservoir based on 20-year data (1986–2006) and the water demand in the downstream area (y-axis is of log scale).

4. Results The data used in this study, including water demand, inflow and reservoir storage in the investigation periods, were extracted from the annual reports of the Water Resources Agency, Taiwan. Fig. 5 shows the 10-day inflow boxplot of the Shihmen Reservoir based on 20-year data (1986–2006) and the water demand in downstream area. As shown, a relatively large inflow variation occurs in the wet season (16th to 30th 10-day periods), where the inflow in most of the time is great than the water demand, while during the dry season the inflow is usually less than the water demand. Through formulating the reservoir operation for water supply as an optimization problem, the sole GA, MIGA and MIGAI are used to search the storage hydrographs (coupled with water release histograms) of reservoir operations to fulfill the minimization of the 10-day shortage subject to constraints. In Case I, despite 180 variables, the sole GA can still well search the optimal solution to suitably control the storages of the Shihmen Reservoir and thus satisfy water demands in the downstream area. In Case II, it becomes difficult for the sole GA to search the optimal solution. In Case III, due to the huge number of decision variables, the sole GA cannot reach the optimal solution no matter how we set up the GA parameters. This appears the problem of the curse of dimensionality. The re-

sults of the M-5 rule curves and Cases I, II and III obtained by the sole GA, MIGA and MIGAI are shown in Table 2 and briefly summarized as follows. The SR obtained by the sole GA, MIGA and MIGAI are smaller than those calculated by the M-5 rule curves operation in all cases except the sole GA in Case III. That means the optimization methods can obtain better results than the current used operating curves if properly formulate the system and implement appropriate search methods. Moreover, the results obtained by MIGA and MIGAI are better than those obtained by the sole GA in all cases. For instance, the SR obtained by the sole GA, MIGA and MIGAI are 4.72, 4.60, 4.45 in Case II; accordingly, and the execution time by the sole GA, MIGA and MIGAI are 295, 64, 58 hours in Case III. Results indicate that MIGA and MIGAI are far more efficient than the sole GA in terms of execution time and improvement rate. Moreover, the efficiency of MIGAI is better than that of MIGA for long-term operation in Cases II and III. That means we could increase the effectiveness as the redundant search processes in some sub-systems were skipped. Fig. 6 displays the SR of the sole GA and MIGAI in Case I, II and III, respectively. The results clearly indicate that MIGAI has better performance than the sole GA in all three cases. Fig. 6(c) shows that the sole GA does not converge to the optimal solution in this

Table 2 Search results of the 10-day water shortage rate evaluated by M-5 rule curves, the sole GA, MIGA and MIGAI. Case

Case I (5 years)

Method

M-5

Sole GA

MIGA

MIGAI

M-5

Case II (10 years) Sole GA

MIGA

MIGAI

M-5

Sole GA

MIGA

MIGAI

SR (%) IR (%) Computation time (h)

9.89 – –

8.91 9.9 9.5

8.86 10.4 7

8.86 10.4 7

4.94 – –

4.72 4.4 157

4.60 7.0 29.5

4.45 9.9 16

4.57 – –

5.40 18.1 294.5

4.19 8.3 64

3.88 15.1 58

SR: average 10-day water shortage rate. IR: improvement rate compared with M-5 rule curves.

Case III (20 years)

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Annual average of 10-day water shortage rate (%)

100

sole GA

MIGA I

10

1

0.1 2001-2002

2002-2003

2003-2004 Year

2004-2005

2005-2006

(a) Case I

Annual average of 10-day water shortage rate (%)

100

sole GA

MIGA I

10

1

0.1

Year

(b) Case II

Annual average of 10-day water shortage rate (%)

100

sole GA

MIGA I

10

1

0.1

Year

(C) Case III Fig. 6. Annual average of 10-day water shortage rate (y-axis is of log scale).

long-term operation where high SR values present in most of the 20 years, while MIGAI can effectively solve the long-term operation problem and provide suitable results. To learn the efficiency and effectiveness of model performance, all models were executed on Intel Core 2 Duo CPU E8400 3.0 GHz type PC. Figs. 7 and 8 show the fitness values and their corresponding search time in the search processes. It appears that the fitness value evaluated by MIGA is close to the optimal fitness value within limited computation time, whereas the sole GA does not con-

verge to the optimal fitness value even after a very long search time (ex. 294.5 hours in Case III). The improvement rate of fitness values increases more than 25% [(1  (3.88/5.4))  100% = 28%], and the computation time dramatically decreases 80% [(1  (58/ 294.5))  100% = 80%] in a 20-year long-term operation case. We also find that MIGA and MIGAI can quickly converge before completing the preset computation numbers. For instance, the MIGA and MIGAI reached the curved points (near the optimal solutions) less than 1 hours for Case I, and 6.5 hours for Case III. Results indi-

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Fig. 7. Performance comparison of the search processes for Case I (5-year).

Fig. 8. Performance comparison of the search processes for Case III (20-year).

cate MIGA and MIGAI perform better, in terms of small fitness values and less execution time, than the sole GA under similar computation numbers. The sole GA could not reach an optimal solution even under very large population size and generation number. In contrast, MIGA and MIGAI require only very small generation numbers and population sizes to quickly reach optimal solutions. 5. Conclusion Genetic algorithms offer a powerful optimization approach and have become popular in reservoir planning and management; however, increasing numbers of variables require huge number of computation, which easily causes the curse of dimensionality and results in very slow evolution between consecutive generations, consequently sharply decreasing the possibility of converging to an optimal solution. In this study, a multi-tier interactive genetic algorithm (MIGA) is proposed to decompose a complicated system into several small-scale sub-systems by variable decomposition strategy accompanied with multi-tier GA search, and then to produce the optimal solution by integrating the individual optimal solutions of subsets into the desired optimal solution for the whole variable set. The number of variables is reduced to a comparatively small scale in each sub-system so that the optimization search re-

quires a comparatively small population size and generation number for individual search; consequently, the search process becomes much more effective and the likelihood of reaching the optimization solution can be increased. Long-term reservoir operation usually involves a large number of decision variables. The proposed methods have been applied to searching the optimal solution of long-term reservoir operation for three variable sets, 180, 360, and 720 variables of Cases I, II, and III, respectively. Results indicate that MIGA is far more efficient than the sole GA in terms of execution time and improvement rate. For instance in Case III, the sole GA could not reach an optimal solution even under very large population size and generation number. In contrast, MIGA requires only a very small generation number and population size to quickly reach an optimal solution. The results demonstrate that MIGA can successfully and efficiently increase the possibility of achieving an optimal solution. Therefore, the MIGA search method can be suitably used in long-term reservoir operation or systems with similar conditions. In addition, the efficiency of MIGA search will be greatly increased if parallel computing techniques can be applied. To extend our research, the flexibility and usefulness of MIGA could be explored in the following ways: (1) a complicated system can be non-uniformly decomposed into multi-tier small-scale sub-

K.-W. Wang et al. / Advances in Water Resources 34 (2011) 1343–1351

systems; (2) search engines can be replaced by other search methods; and (3) more strategies or considerations can be applied to the optimization search process in aGAs because of the decomposition strategy. These advantages will stand on the knowledge of professionalism or the characteristics of the problem itself. Acknowledgments It is very much appreciated that data used in this study were extracted from the annual reports of the Water Resources Agency, Taiwan. The authors are indebted to the editors and reviewers for their valuable comments and suggestions. References [1] Anelli G, Broggi A, Destri G. Decomposition of arbitrarily shaped binary morphological structuring elements using genetic algorithms. IEEE Trans Pattern Anal Mach Intell 1998;20(2):217–24. [2] Cai X, McKinney DC, Lasdon LS. Solving nonlinear water management models using a combined genetic algorithm and linear programming approach. Adv Water Resour 2001;24(6):667–76. [3] Chang LC. Guiding rational reservoir flood operation using penalty-type genetic algorithm. J Hydrol 2008;354:65–74. [4] Chang LC, Chang FJ, Wang KW, Dai SY. Constrained genetic algorithms for optimizing multi-use reservoir operation. J Hydrol 2010;390:66–74. [5] Chaves P, Chang FJ. Intelligent reservoir operation system based on evolving artificial neural networks. Adv Water Resour 2008;31:926–36. [6] Chen YH, Chang FJ. Evolutionary artificial neural networks for hydrological systems forecasting. J Hydrol 2009;367:125–37. [7] Chiu YC, Chang LC, Chang FJ. Using a hybrid genetic algorithm–simulated annealing algorithm for fuzzy programming of reservoir operation. Hydrol Process 2007;21:3162–72. [8] Debels D, Vanhoucke M. A decomposition-based genetic algorithm for the resource-constrained project-scheduling problem. Oper Res 2007;55(3):457–69.

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