Multiobjective planning of surface water resources by multiobjective genetic algorithm with constrained differential dynamic programming

Multiobjective Planning of Surface Water Resources

by Multiobjective Genetic Algorithm with Constrained

Differential Dynamic Programming

Chao-Chung Yang

; Liang-Cheng Chang

; Chao-Hsien Yeh

; and Chang-Shian Chen

Abstract: Owing to the conflict encountered between the two objectives of fixed cost in reservoir installation and operating cost in time-varying water deficit, multiobjective planning of surface water resources is a difficult job. Instead of combining these two objectives into just one objective using the weighting factor approach, this investigation proposes a novel method by integrating a multiobjective genetic algorithm共MOGA兲 with constrained differential dynamic programming 共CDDP兲. A MOGA is employed to generate the various combinations of reservoir capacity and estimate the noninferior solution set. However, applying this algorithm to solve the dynamics of the operating cost, the number of variables increasing with time will dramatically increase the use of computational resources. Conse-quently, the CDDP is herein adopted to distribute optimal releases among reservoirs to satisfy water demand as much as possible. Next, the effectiveness of the proposed methodology is verified by solving a multiobjective planning problem of surface water in southern Taiwan. This real application demonstrates that MOGA can be linked with CDDP to resolve a complex water resources problem. Additionally, the ability of MOGA on addressing multiple objectives simultaneously without converting to a weighted objective function provides the opportunity for significant advancement in multiobjective optimization. Finally, this investigation also proposes three suitable strategies of reservoir construction to decision makers with budget concerns through the analysis of all noninferior solutions.

DOI: 10.1061/共ASCE兲0733-9496共2007兲133:6共499兲

CE Database subject headings: Multiple objective analysis; Algorithms; Surface waters; Water resources.

Introduction

Many planning problems in water resources involve fixed charges attributed to the construction of new facilities or the expansion of existing facilities. Reservoirs, the most important hydraulic facili-ties in a water resource system, can significantly impact regional water conservation because of their ability to consistently distrib-ute water. They fulfill the demands of specified locations in a region where precipitation is unevenly distributed temporally and spatially. However, due to financial and environmental con-straints, only a limited number of reservoirs can be built in a river basin. Therefore, an appropriate policy is necessary to consider

the fixed costs and operating costs during the reservoir planning stage.

Past studies discuss several different methods used in water resources management and planning. To assess state-of-the-art optimization of reservoir system management and operation, Labadie共2004兲 reviewed various optimization methods, including multiobjective optimization models and application of heuristic programming methods using evolutionary and genetic algorithms. Also designed to solve the multireservoir problem, the “network flow programming,” was introduced by Khaliquzzaman and Chander共1997兲 for optimizing reservoir capacities through defin-ing the best zones for the reservoirs with unit cost in the objective function. However, the study did not separate the “cost” of reser-voir operation as an objective like this paper has done. Watkins and McKinney 共1998兲 used two decomposition algorithms to a conjunctive system of surface and groundwater with the cost function containing both discrete and nonlinear terms. With a dis-crete investment cost and a continuous operating cost in its objective function, their work failed to minimize a nonlinear pro-gramming formulation of these two terms because the reservoir capacity is preset rather than being a decision variable. Utilizing a penalty function, Hirad and Ramamurthy 共2000兲 proposed a method of converting two objectives, minimum cost and mini-mum water deficit, into a single objective function to determine the optimal multireservoir system design for water supply. None-theless, this methodology did not simultaneously consider the fixed costs for reservoirs installation and the time-varying operat-ing costs of water deficit.

Dynamic programming is capable of handling the many prob-1

Research Assistant Professor, Construction and Disaster Prevention Research Center, Feng Chia Univ., No. 100 Wenhwa Rd., Seatwen, Taichung, Taiwan 40724, R.O.C. E-mail: [email protected]

2

Professor, Dept. of Civil Engineering, National Chiao Tung Univ., 1001 Ta Hsueh Rd., Hsinchu, Taiwan 30010, R.O.C. E-mail: lcchang@ chang.cv.nctu.edu.tw

3

Associate Professor, Dept. of Water Resources Engineering, Feng Chia Univ., No. 100 Wenhwa Rd., Seatwen, Taichung, Taiwan 40724, R.O.C. E-mail: [email protected]

4

Associate Professor, Dept. of Water Resources Engineering, Feng Chia Univ., No. 100 Wenhwa Rd., Seatwen, Taichung, Taiwan 40724, R.O.C. E-mail: [email protected]

Note. Discussion open until April 1, 2008. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on December 23, 2004; approved on September 11, 2006. This paper is part of the Journal of Water Resources Planning and Management, Vol. 133, No. 6, November 1, 2007. ©ASCE, ISSN 0733-9496/2007/6-499–508/$25.00.

lems encountered during the decision-making process. Dynamic optimal control algorithms encounter difficulties in resolving a problem with an objective function that includes fixed costs due to the requirement of a separable objective function for each stage. Consequently, the multiobjective genetic algorithm, 共MOGA兲, is attractive because it does not stipulate the differen-tiability of the objective function. Further, the MOGA-based so-lution technique has two merits over conventional multiobjective programming approaches. First, it can generate both convex and concave points on the trade-off curve. Second, it can create large portions of the trade-off curve in a single iteration.

MOGA has been successfully employed in water resources for various purposes. For example, Cieniawski et al.共1995兲 presented an optimization method of MOGA to solve a multiobjective groundwater monitoring problem with the objectives of maximiz-ing reliability and minimizmaximiz-ing contaminated zones when first de-tected. In order to overcome the rising complexity when both location and sizing of detention dams are involved in a multiob-jective framework, Yeh and Labadie 共1997兲 utilized MOGA for the planning of a watershed-level detention dams system. Burn and Yulianti共2001兲 explored the capabilities of genetic algorithms for finding solutions to waste-load allocation problems with the objectives of focusing on the cost of treatment as well as the effect of water quality improvement. In order to design a water distribution network with the objectives of minimizing pipe net-work costs and maximizing reliability, Prasad and Park 共2004兲 presented a MOGA approach to produce a set of Pareto-optimal solutions. Kim and Heo 共2004兲 focused on the application of MOGA to the multireservoir system optimization. You et al. 共2004兲 adopted MOGA to solve the conflict between power gen-eration and water supply.

Even with various applications of MOGA in solving multiob-jective optimization problems, using this approach to overcome time-varying policies of operating cost will nonetheless signifi-cantly increase the use of computational resources. Therefore, the constrained differential dynamic programming共CDDP兲 method is used in order to calculate the operating cost, which is then sup-plied to MOGA for further processing.

Methodology

The basic planning problem of a water resources system that con-siders fixed costs and operating costs can be described as follows and is denoted as Problem A in this study共Problem A: Original form兲:

Objective

Min

a

៝,u˜ 兵Z1共a៝兲,Z2共u˜共a៝兲兲其 共1兲

Z1共a៝兲 = F共a៝兲 共2兲 Z₂共u˜共a៝兲兲 =

兺

a៝min_{艋 a៝ 艋 a៝}max _共6兲

0艋 s៝t艋 a៝, t = 1, ... ,n 共7兲

0艋 u៝t艋 U៝t

max_{, t = 1, . . . ,n 共8兲}

The original problem has two objectives, Z1 and Z2, called

fixed cost, and operating cost respectively, and are to be mini-mized. The fixed cost Z1 represents the design capacity cost,

denoted as vector a
to represent the reservoirs to be created and it is defined by a function F共a៝兲. The operating cost Z₂⫽cost of operating decisions identified by vector u˜ for each period, and

defined by function g共u˜兲. The vector u˜ represents the feasible decisions, such as reservoir outflow and spill, required to satisfy the physical and policy constraints imposed on operational proce-dures. This investigation assumes that the fixed cost rises linearly with reservoir size, and that the shortage index共SI兲 surrogates the operating costs. Proposed by the U.S. Army Corps of Engineers 共HEC 1966, 1975兲, the SI is often adopted to reflect the water deficit in Taiwan共Hsu 1995兲 and is utilized as a surrogate index for the objective function of operating cost in this study. Calcu-lated by the following equation, this index specifies the sum of the indicated values for all periods

SI =100 N

兺

冉

冊

target demand at time period i.

Constraint 共4兲 is the transition equation of a surface water system during time interval关t,t+1兴, and the reservoir level at end of the stage St+1 depends on the initial level state of reservoir St

and the decision vector u˜. As Constraint共5兲 represents the mass

balance or inequality of a surface system, the system limitations of state variables and decision variables are articulated by con-straints共6兲–共8兲.

An appropriate solution to a multiobjective problem is often difficult to obtain from the original form as expressed in Problem A. Therefore, merging multiple objective functions into a scalar function by weighting factors is usually employed for Problem A such that it can be solved by single objective optimization meth-ods, i.e., dynamic programming or nonlinear programming for our problem. Because dynamic programming needs a separable objective function for each stage t, it faces difficulties in over-coming this problem for the objective function containing the fixed costs formulated by vector a៝ which is independent of time, and it turns the weighted function into a nonseparable problem with time t 共Hsiao and Chang 2002兲. Although nonlinear pro-gramming can estimate the noninferior solutions for both sepa-rable and nonsepasepa-rable problems, the computation time becomes huge 共increases geometrically proportional to time step兲 as the operating time increases. Further, the weighting factor method can only be applied to the problem with a concave feasible solu-tion set where the objective funcsolu-tion has orthogonal characteris-tics共Hsiao and Chang 2002兲.

Rather than combining these two competing objectives with a weighting factor, this investigation presents a methodology em-ploying MOGA embedded with CDDP for Problem A through a two-stage formulation. To accomplish this, the original form is first modified into Problem B, comprised of a main form and a minor form. The main form is formulated to estimate noninferior solutions, while the minor form searches for optimal system op-eration Z₂* for all time stages under specific capacity decision a
provided by the main form. The mathematical expression is

cidated as follows共main form of Problem B兲 Objective Min a៝ 兵Z1共a៝兲,Z2 *_{共a៝兲其 共10兲} Z1共a៝兲 = F共a៝兲 Subject to

a៝min_{艋 a៝ 艋 a៝}max

共minor form of Problem B兲 Objective Z₂*共a៝兲 = Min u ˜

兺

Although still a multiobjective problem, the main form of Problem B becomes a type of problem without time-variant deci-sion variables and constraints, such that MOGA is performed to generate a set of noninferior solutions for the various combina-tions of reservoir capacities. Under the given vector a៝ from MOGA, the minor form of Problem B for the single objective

Z2共a៝兲 is expected to define the best system operation u˜* and its

optimal value Z₂*. As a៝ is a constant rather than a decision variable in the minor form of Problem B, the difficulty of a nonseparable problem for dynamic programming vanishes.

In the past, Murray and Yakowitz 共1981兲 presented a CDDP algorithm and applied it to a multireservoir control problem. They formulated the problem as a discrete optimal control problem with a linear transition function and linear constraints on the state and control variables. Their algorithm adapted the quadratic pro-gramming method into the DDP framework. As an extension of the same algorithm, Yakowitz 共1986兲 presented a stage-wise Kuhn–Tucker condition to ensure the convergence of the algo-rithm with the assumption of linear constraints. Chang et al. 共1992兲 and Hsiao and Chang 共2002兲 had successfully solved the problem of groundwater remediation by CDDP. From the above-mentioned studies, they mention that the CDDP outperforms conventional DP and mathematical programming algorithms in computational efficiency. They also point out the state and control vectors of the problem need not be discrete, implying that CDDP overcomes the “curse of dimensionality,” a serious limitation of conventional DP. CDDP can reduce a significant “working” dimensionality of the algorithm over that of mathematical pro-gramming algorithms 共Hsiao and Chang 2002兲. Based on those advantages, we adopt the CDDP instead of DP. The CDDP used herein is a modified procedure suggested by Murray and Yakow-itz 共1981兲. Within each iteration, quadratic programming is applied at each stage of the backward and forward sweep. The iterations are repeated until the solution converges. The more de-tailed discussion of the CDDP algorithm and application is pro-vided in Murray and Yakowitz共1981兲, Chang et al. 共1992兲, and Hsiao and Chang共2002兲.

As CDDP module is embedded in the structure of MOGA as a subroutine, CDDP is employed to distribute the release among reservoirs in every time step. Therefore, MOGA not only esti-mates noninferior solutions but also provides the input variables to CDDP 共the capacity design combination of reservoirs.兲 The comparison between the original multiobjective problem and its modified problem is illustrated in Fig. 1.

The integrated model proposed in this study has two critical features: First, the search for the noninferior solution set is achieved by MOGA at the main form, and second, CDDP calcu-lates the releases of the system associated with the combination of designed reservoirs scale at the minor form. Although the CDDP used herein is a procedure suggested by Murray and Yakowitz 共1981兲, the operation procedures of MOGA are modified from the study of the Pareto-optimal ranking method共Goldberg 1989兲 and elitist conservation共Yeh and Labadie 1997兲. Although this paper does not focus on the performance of various multiobjective ge-netic algorithms, it designs a test case to justify the utility of the proposed methodology as shown in the Appendix. With several trial runs of the MOGA for the test case, the parameters are iden-tified as those shown in Table 1.

The flow chart of our integrated model is shown in Fig. 2, and the detail operational procedures are described step by step as follows.

1. Select the potential scale of reservoirs. MOGA requires en-coding schemes that transform the decision varable vectors into a structure 共chromosome兲 that enables genetic opera-tions: Reproduction, crossover, and mutation. These genetic operations generate new sets of chromosomes共decision vari-ables兲 with, on average, enhanced performance. This step mainly focuses on encoding the decision variable as a chro-mosome, randomly generating an initial population of given size, 100 in this case. A chromosome represents a possible reservoir capacity design, and its length is determined by the number of bits required to represent a decision variable and the number of decision variables. In this investigation, each decision variable denotes the possible capacity of a reservoir such that each chromosome denotes a combination of the capacities of all reservoirs.

2. Define the system network and prepare hydrologic data. 3. Distribute the releases among reservoirs using CDDP in a

system network under the target of optimal distribution be-tween supply and demand. After the chromosomes共reservoir capacity兲 of the initial population have been determined as in Step 1, the release of the system in every period is calculated by the CDDP corresponding to each chromosome. This pro-cedure is repeated for all chromosomes in each generation. The CDDP is embedded in the MOGA to calculate the re-lease of the system under the target of optimal distribution between supply and demand. Finally, the release of the sys-tem for each chromosome is returned to the MOGA to mea-sure the operating cost.

4. Evaluate the fixed cost using the fixed cost-coefficient and reservoir capacity, and the operating cost using the shortage index. In this study, however, the SI surrogates the operating costs.

5. Search for the noninferior solution set. The values of the fixed cost共Z1兲 and operating cost 共Z2兲 can be identified for

every chromosome of one generation through Step 3 and 4, and MOGA finds the noninferior solution set based on these values at Step 5. A chromosome a1 is defined as inferior to chromosome a2 if the following condition holds.

Z1共a1兲 艌 Z1共a2兲, Z2共a1兲 艌 Z2共a2兲

If a1 is neither inferior nor superior to a2, then a1 and a2 are noninferior with respect to each other, or a1⬇a2. The non-inferior solution set is composed of chromosomes that are nondominated by any other chromosome in one generation. Additionally, all noninferior solutions from every generation are accumulatively recorded and updated in the indicated file to serve as stopping criteria for the MOGA.

6. Identify a final set of noninferior solutions through the pro-cedures of fitness calculation, elite, reproduction, crossover and mutation of MOGA until reaching converge condition. • Evaluate the fitness for each chromosome. The solutions of

the noninferior solutions set determined through Step 5 are assigned as Rank 1. The fitness of all feasible solutions is estimated by

fi= dmax− di_min 共12兲

where dmaxdenotes the maximum distance between all

fea-sible solutions and all noninferior solutions in Rank 1, i.e., Table 1. Key Parameters of MOGA

Parameter Value

Population size 100

Chromosome length 18 loci

Elitist set size 30

Crossover rate 0.7

Mutation rate 0.03

Stopping criterion The change ratio of the number of noninferior solutions sets over ten consecutive generations is

smaller than 5%

Fig. 1. Original multiobjective problem and its modified problem

dmax= max兵dij共i=1–pop,j=1–N兲其; di_min represents the minimal

distance between the feasible solution i and the noninferior solution j in Rank 1, dimin= min兵dij共j=1–N兲其; dij= distance

between the feasible solution i and any noninferior solution

j in Rank 1; fi= fitness of feasible solution i; pop represents

the total number of feasible solutions; and N = total number of noninferior solutions in Rank 1. Fig. 3 clearly demon-strates that the distance between any feasible solution of chromosomes and the members of Rank 1 is closer with respect to bigger fitness value compared to the other chromosomes.

• Store the elite solutions. This study proposed a new way to store the elite solutions to ensure that the set of noninferior solutions can proceed with the crossover step to avoid any component among this set from disappearing in the repro-duction process. This procedure also performs the function of diversity maintaining mechanism which is one of the key ingredients of a MOGA. If the number of solutions with Rank 1 is lower than the size of the elite set, then all of these solutions are included in this set. The rest of the elite set comprises feasible solutions with better fitness.

Other-wise, a portion of solutions with Rank 1 is chosen to com-pose the elite set. The number of the elite set in this study is 30, and thus the number of reproductions is 70. • Reproduce the best strings. This investigation undertakes

reproduction by tournament selection. The selection mechanism plays an important role in driving the search toward superior individuals and maintaining high genotypic diversity in the population. MOGA selects parents from a population of strings based on the fitness. In each tourna-ment selection, a group of five individuals are randomly selected from the population, and the fittest individual共s兲 is selected for reproduction. The procedure is repeated until the number of chromosomes required for crossover is met. • Crossover. Crossover involves randomly coupling the newly reproduced strings and exchanging information within a pair of strings. Crossover occurs with a constant probability of pcross for each pair of strings. In this work,

p_crosswas set to 0.7 with a uniform crossover operator.

• Mutation. Mutation restores lost or unexplored genetic ma-terial to the population to prevent the GA from converging prematurely to a local optimum. A mutation probability

p_mutat, p_mutat= 0.03 in this study, is specified, and mutation is applied randomly to individual genes. If a random number generated from a uniform distribution function is smaller than the mutation probability, then mutation is conducted by changing the binary value of the gene in the offspring strings produced by the crossover operation.

• Termination mechanism. A new population for the next generation is created after the mutation operation, and the noninferior solutions set is extracted as from in Steps关3–5兴. The stopping criterion in this study is based on the variation rate, which is defined as the change ratio in the noninferior solutions sets. The procedure finishes if the user-defined stopping criterion is met or the maximum allowed number of generations is reached; otherwise, Step 6 proceeds for another cycle共another generation兲.

Fig. 2. Flow chart of our proposed model, integration of the MOGA and CDDP, for the multiobjective planning of surface water

Fig. 3. Definition of fitness 共Z1and Z2 are objectives in minimal problem兲

Application

Description of Study Area

Located in southern Taiwan, the study area includes two major watersheds, the Tsengwen River and the Kaopin River, and two metropolitan areas, Tainan and Kaohsiung. Although Tainan is supplied by the Wushantou Reservoir with an effective storage capacity of 81.45⫻106_m3 _{and the Nanhwa Reservoir with an}

effective storage capacity of 149.46⫻106_m3_{, the water supply}

for the Kaohsiung area is completed by the Nanhwa Reservoir and the Kaopin River Weir. To raise the inflow for the two above-mentioned reservoirs, two diversions from weirs have been made: One from the Tongkou Weir to Wushantou reservoir, the other from the Chiahsien Weir to the Nanhwa Reservoir. Additionally, the Tongkou Weir receives water from the Tsengwen Reservoir which has an effective storage capacity of 581.23⫻106_m3_{. The}

water system diagram in southern Taiwan is shown in Fig. 4.

Problem Definition

To define how the fixed cost and operating cost affect one another, the proposed methodology is demonstrated to find appropriate facilities’ capacities and their operation procedures in order to meet future demands by 2011. The range of the storage capacities for the three above-mentioned reservoirs 共Nanhwa, Wushantou, and Tsengwen兲 is set between half and double of their original

capacities. The objective function and system dynamics in the main form of this problem are formulated in the following equa-tions共application problem; main form兲:

Objective Min Y៝ 兵Z1共Y៝兲,Z2 *_{共Y៝兲其 共13兲} Z1共Y៝兲 =

兺

where Y1, Y2, and Y3 denote the installation capacities of the Wushantou Reservoir, the Tsengwen Reservoir, and the Nanhwa Reservoir, respectively; the assumption of a linear function and the coefficient for the unit construction cost of reservoirs 共C=2.62. N.T. dollars/ton兲 are cited from Wu 共1997兲, and m 共m=3兲⫽total number of reservoirs.

With the shortage index 共Hsu 1995兲, the minor form of the problem discussed in this application is formulated as共application problem; minor form兲.

Fig. 4. Water system diagram for southern Taiwan

Objective Z2 *_{共Y៝兲 = Min} UO,WD 100 n

兺

再

兺

冋

册

冎

for known Y៝ 共16兲

Subject to:

Transition equation of reservoir

Si,t+1= Si,t+ RIi,t+ WDi,t− UOi,t− USi,t, i = 1, . . . ,m,

t = 1, . . . ,n 共17兲

Mass balance of weir

WIk,t+ RSk,t+ ROk,t= WQk,t+ WDk,t, k = 1, . . . ,g, t = 1, . . . ,n

共18兲 Water level

0艋 S_i,t艋 Y_i 共19兲

Capacity constraints

the upper limits of capacities

for reservoirs and pipelines 共20兲

supply capacity: UCj,t+ WDj,t艋 Dj,t 共21兲

nonnegativity: all variables are larger than or equal to zero, where, Dj,t= demand in the supply area j at time t; St+1 and St

denote the storage of the reservoir at time t + 1 and t, respectively; UO_t, US_t, and RI_t represent the amounts of outflow, spill, and inflow of reservoir at time t; WOt, WDt, and WIt= amounts of

outflow, diversion, and inflow of weir at time t; g = total number of weirs共g=3兲, and s=total number of demands 共s=2兲.

On the other hand, reservoirs are fully utilized due to the ex-treme temporal-varied distribution of precipitation as well as site/ capacity constraint in Taiwan. Even if the largest reservoir in Taiwan, the Tsengwen Reservoir, were expanded to twice its present capacity, it would be almost full once 共0.85 times兲 per year. For those reservoirs discussed in this paper, the average

ratios of annual inflow for the last ten years to reservoir capacity are 2.6, 1.7, and 4.3 for the Wushantou, Tsengwen, and Nanhwa Reservoirs, respectively. Therefore, final level constraint becomes a minor concern in this study.

Some key values or parameters in this paper’s optimization problem are cited from several papers and websites related to Taiwan’s reservoir system operations共Water Resources Planning Commission 1986; Chang and Yang 2002; Hsu 1995; Wu 1997兲. The sources of data or parameters for illustration are listed in Table 2.

Results

An integrated model is applied to the problem defined by Eqs. 共13兲–共21兲. The model estimates the noninferior solutions consist-ing of fixed cost and operatconsist-ing cost for the area of interest. The decision variables must be encoded as a chromosome before the MOGA is applied. For each decision variable, reservoir capacity is represented by six binary bits. As three decision variables 共res-ervoirs兲 are involved, a chromosome consists of a total of 18 loci. In the problem considered here, there are 100 chromosomes in each population, and the initial population is randomly generated. As indicated in Fig. 2, the CDDP is used repeatedly within each generation to simulate the operation of the system according to the reservoir’s scale via the chromosomes. The stopping criterion for MOGA is that the variation rate of noninferior solutions over ten consecutive generations should be under 5%. The computa-tions are implemented for 93.28 h on a PC Pentium III733 run-ning Microsoft Windows 98. Fig. 5 shows that the variation rate decreases from the initial value to convergence, and that the final noninferior solution appears in generation number 27.

Fig. 6 indicates the results with the initial and final noninferior solutions set, plotting the fixed cost against the shortage index. The trend of the results for the final noninferior solutions set are significant compared with the initial set. Solutions of the final noninferior solutions set with fixed costs range from 4,327,149,000 N.T. dollars to 3,098,960,000 N.T. dollars with the shortage index from 20.81 to 30.57. We choose one noninferior solution to conduct a more detailed analysis.

Figs. 7–9 display the operating results of the three reservoirs from one noninferior solution 共fixed cost=4,327,149,000 N.T. dollars, SI= 20.8兲, demonstrating that the water storage for every reservoir in the wet season共June–November兲 is larger than that in the dry season共December–May兲. This implies that the water stor-age of the reservoirs has seasonal variations.

When the noninferior solutions from the multiobjective prob-lem are identified, the decision maker’s preference has to be pro-vided for choosing the compromise solution from noninferior Table 2. Data Sources for Parameters

Data/parameter Source

Demand According to predictions made by Water Resources Planning Commission共1986兲.

Reservoir or weir Chang, and Yang共2002兲. Shortage index Hsu共1995兲.

Inflow Taiwan Water Resources Agency共http:// www.wra.gov.tw/兲

Cost coefficient Wu共1997兲.

Fig. 5. Variation rates by generation

Fig. 6. Initial and final noninferior solutions set

solutions. This investigation attempts to find an appropriate com-promise solution from all alternatives if the decision maker has no strong preference. Figs. 10–12 show the volumes of all noninfe-rior solutions for these three reservoirs, individually arranged in order according to shortage index performance. The y axis indicates reservoir volume, whereas the x axis indicates the non-inferior solutions which are numbered in order according to in-creasing shortage index. Obviously, the most congregated volume for the Nanhwa Reservoir is 202⫻106_m3_{in Fig. 12. On the other}

hand, the majority of Tainan’s water resources originate in the Tsengwen and Wushantou Reservoirs which together have a large volume and operate in series. Therefore, the appropriate volumes for the Tsengwen and Wushantou Reservoirs should be relative to each other. In Fig. 11, most noninferior solutions are located in and close proximity to the volumes 1,175⫻106_m3 _{and 940}

⫻106_m3_{. When the volume of the Tsengwen reservoir remains at}

940⫻106_m3_{, closely corresponding to the noninferior solutions}

between the 32nd and 65th, the related volume change of the Wushantou Reservoir in Fig. 10 is from 141⫻106_m3 _{falling to}

41⫻106_m3_{, and the shortage index is from 27 rising to 30.5.}

Although capacity extension facilitates the improvement in the shortage index, the shortage index is still larger than 27. Con-versely, when the volume of the Tsengwen Reservoir increases to 1,175⫻106_m3_{, closely corresponding to the 1st–21st noninferior}

solutions, the related volume change of the Wushantou Reservoir in Fig. 10 is from 141⫻106_m3 _{falling to 41⫻10}6_m3 _{and the}

shortage index rises from 20.8 to 25. The result is the shortage index is always less than 27. Also, the volume capacity of the

Fig. 7. Water storage in the Wushantou Reservoir 共fixed

cost= 4,327,149,000 N.T. dollars, SI= 20.8兲

Fig. 8. Water storage in the Nanhwa Reservoir 共fixed

cost= 4,327,149,000 N.T. dollars, SI= 20.8兲

Fig. 9. Water storage in the Tsengwen Reservoir 共fixed

cost= 4,327,149,000 N.T. dollars, SI= 20.8兲

Fig. 10. Noninferior solutions of the Wushantou Reservoir

Fig. 11. Noninferior solutions of the Tsengwen Reservoir

Fig. 12. Noninferior solutions of the Nanhwa Reservoir

Wushantou Reservoir becomes insignificant as long as the the Tsengwen Reservoir maintains a large volume capacity. This in-dicates that the reservoir extension upstream is better than that downstream for series operation in terms of unit cost investment. From the previous discussion, this study recommended three strategies for governmental authorities depending on the budget condition as follows:

1. The appropriate scales for Tsengwen, Wushantou, and Nan-hwa are 940⫻106_{, 41⫻10}6_{, and 202⫻10}6_m3_{, respectively,}

in the event of a constrained budget.

2. The appropriate scales for Tsengwen, Wushantou, and Nan-hwa are 1175⫻106_{, 141⫻10}6_{, and 202⫻10}6_m3_,

respec-tively, if there are no serious budget constraints.

3. Two construction options based on Strategy 1 can be selected if the initial budget is modest. One is the expansion of Wus-hantou to 141⫻106_m3_{, and the other is the expansion of}

Tsengwen to 1,175⫻106_m3_.

Conclusions

This investigation reveals MOGA’s ability to be linked with CDDP to resolve a complex water resources problem. Addition-ally, the power of MOGA to address multiple objectives simulta-neously without resorting to a weighted objective function provides the opportunity for significant advancement in multiob-jective optimization. Further, the use of MOGA not only shows its capability in generating the noninferior solutions set, but also pro-poses three suitable strategies of reservoir construction to decision-makers with budget concerns through the analysis of all noninferior solutions.

Acknowledgments

The writers would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC 90-2211-E-009-061. Appreciation also goes to Erik Avasalu, an English native speaker and current graduate student of Graduate Institute of Translation and Interpre-tation National Taiwan Normal University, for assistance in im-proving the English language in this paper.

Appendix. Multiobjective Problem for Test

With fundamental understanding of MOGA, we developed our algorithm consisting of various essential components in 1998. This algorithm, although 8 years old, is still suitable for solving multiobjective problems despite the appearance of newly devel-oped MOGAs, such as NSGAII. So the following test case has been designed to justify the utility of the proposed methodology. • Multiobjectives problem Objective min Z1= 2,500 − x1 2 _共22兲 min Z2= 50x1x2− x2 共23兲 Subject to 1艋 x1艋 50 共24兲 1艋 x2艋 10 共25兲 x1,x2苸 R

Fig. 13 displays all real feasible solutions and noninferior so-lutions sets to this test case. The population in each generation has 100 chromosomes, and the initial population is randomly generated. The distribution of problem solutions for these 100 chromosomes is shown in Fig. 14. In Fig. 14, although all solutions in the first generation located within the domain of feasible solutions are clearly observed, only three solutions are noninferior. The stopping criterion for MOGA is that the varia-tion rate of noninferior soluvaria-tions over ten consecutive genera-tions should be under 5%. Based on the stopping criterion, the final noninferior solutions set appears in generation number 30, and its distribution of problem solutions for 100 chromo-somes is presented in Fig. 15. Compared with Figs. 13–15, it demonstrates the most noninferior solutions exist in generation number 30 and the trend of the noninferior solutions set is significant.

Fig. 13. All real feasible solutions and noninferior solutions set in

test case

Fig. 14. Distribution of problem solutions to 100 chromosomes in

first generation

Fig. 15. Distribution of problem solutions to 100 chromosomes in

generation number 30

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