Tolson, B. A., M. Asadzadeh, H. R. Maier, and A. Zecchin (2009), Hybrid discrete dynamically dimensioned search (HD-DDS) algorithm for water distribution system design optimization, Water Resources Research, 45, W12416, doi:10.1029/2008WR007673. 1
Hybrid Discrete Dynamically Dimensioned Search (HD-DDS) Algorithm for
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Water Distribution System Design Optimization
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Abstract
Bryan A. Tolson Department of Civil Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada (
[email protected])
Masoud Asadzadeh Department of Civil Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada (
[email protected])
Aaron Zecchin School of Civil, Environmental and Mining Engineering, University of Adelaide, Adelaide, SA 5005 Australia (
[email protected])
Holger R. Maier School of Civil, Environmental and Mining Engineering, University of Adelaide, Adelaide, SA 5005 Australia (
[email protected])
21
The Dynamically Dimensioned Search (DDS) continuous global optimization algorithm by
22
Tolson and Shoemaker [2007] is modified to solve discrete, single-objective, constrained Water
23
Distribution System (WDS) design problems. The new global optimization algorithm for WDS
24
optimization is called Hybrid Discrete Dynamically Dimensioned Search (HD-DDS) and
25
combines two local search heuristics with a discrete DDS search strategy adapted from the
26
continuous DDS algorithm. The main advantage of the HD-DDS algorithm compared with other
27
heuristic global optimization algorithms, such as genetic and ant colony algorithms, is that its
28
searching capability (i.e. the ability to find near globally optimal solutions) is as good, if not
29
better, while being significantly more computationally efficient. The algorithm’s computational
30
efficiency is due to a number of factors, including the fact that it is not a population-based
31
algorithm and only requires computationally expensive hydraulic simulations to be conducted for
32
a fraction of the solutions evaluated. This paper introduces and evaluates the algorithm by
33
comparing its performance with that of three other algorithms (specific versions of the Genetic
34
Algorithm, Ant Colony Optimization, and Particle Swarm Optimization) on four WDS case
35
studies (21- to 454-dimensional optimization problems) on which these algorithms have been 1
1
found to perform well. The results obtained indicate that the HD-DDS algorithm outperforms the
2
state-of-the-art existing algorithms in terms of searching ability and computational efficiency. In
3
addition, the algorithm is easier to use, as it does not require any parameter tuning and
4
automatically adjusts its search to find good solutions given the available computational budget.
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Keywords: Water Distribution Systems; Discrete Optimization; Global optimization; Heuristic
7
optimization; Constraint-handling
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1.
Introduction
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The optimal design and rehabilitation of Water Distribution Systems (WDSs) is an important
12
research area as improved optimization methods save substantial infrastructure capital and
13
operational costs. Historically, traditional optimization methods such as linear programming
14
[Schaake and Lai, 1969; Alperovits and Shamir, 1977; Bhave and Sonak, 1992], nonlinear two
15
phase decomposition methods [Fujiwara and Khang, 1990; Eiger et al., 1994] and nonlinear
16
programming [Varma et al., 1997] have been applied to a continuous version of the WDS
17
optimization problem. These methods are sophisticated in terms of their use of the fundamental
18
hydraulic equations to recast the form of the optimization problem and to yield gradient and
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hessian information, however, their inability to restrict the search space to discrete pipe sizes is a
20
significant practical limitation [Cunha and Sousa, 1999]. Reca and Martinez [2006] provide a
21
more detailed review of classical optimization techniques as applied to WDS optimization
22
problems.
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The majority of current single-objective WDS optimization literature report using heuristic
24
global optimization algorithms, including evolutionary algorithms, with great success. Genetic
25
Algorithms (GAs) are probably the most well known combinatorial evolutionary algorithm.
26
Example applications of GAs to WDS optimization include Simpson et al. [1994], Savic and
27
Walters [1997], Wu et al. [2001] and Tolson et al. [2004]. Ant Colony Optimization (ACO) 2
1
algorithms have also received attention in the recent WDS literature [Maier et al., 2003; Zecchin
2
et al. 2006; Zecchin et al. 2007]. Other new and promising approaches applied to WDS
3
optimization include the Shuffled Frog Leaping Algorithm (SFLA) in Eusuff and Lansey [2003],
4
the Harmony Search (HS) Algorithm in Geem [2006], the Cross Entropy (CE) method in
5
Perelman and Ostfeld [2007], Particle Swarm Optimization (PSO) in Suribabu and Neelakantan
6
[2006] and another PSO variant in Montalvo et al. [2008]. Additionally, more traditional heuristic
7
search strategies that are not population-based, such as Simulated Annealing [Cunha and Sousa,
8
1999] and Tabu Search [Lippai et al., 1999; Cunha and Ribeiro, 2004] continue to be applied to
9
some WDS optimization problems.
10
The specific WDS optimization problem we address is to determine the pipe diameters from
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a discrete set of available options such that the total pipe material cost is minimized and pressure
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constraints are met across the network. All other network characteristics are known. This is a
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classical WDS design problem that the majority of the above WDS optimization references also
14
solve.
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One of the major problems associated with the use of heuristic global optimization
16
algorithms is that their performance, both in terms of computational efficiency and their ability to
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find near globally optimal solutions, can be affected significantly by the settings of a number of
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parameters that control their searching behavior (e.g. population size, probability of mutation,
19
probability of crossover in the case of GAs), as well as penalty functions that are commonly used
20
to account for system constraints. In accordance with the No Free Lunch Theorem [Wolpert and
21
MacReady, 1997], the set of parameters that results in optimal performance will vary with the
22
characteristics of each optimization problem. Consequently, values of these parameters are
23
generally obtained by trial and error for different case studies.
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The algorithm parameter characteristics for eight recent WDS optimization studies are
25
summarized in Table 1. As can be seen, the reported number of total parameters (algorithm +
26
penalty) in these eight algorithms ranges from three to eight, and seven of these eight studies
27
report that a subset of algorithm parameters were either experimented with or modified for each 3
1
of the case studies they were applied to. However, such algorithm parameter setting experiments
2
can be extremely computationally expensive, as many of the parameters are dependent. In
3
addition, this problem is exacerbated for problems with long WDS simulation times, particularly
4
those requiring extended period simulations (e.g. when water quality considerations are
5
important). In addition to requiring a large amount of computational time, there is no well
6
accepted methodology for conducting these parameter setting experiments, which are therefore
7
typically implemented on an ad-hoc basis. For example, the methodology used to determine the
8
the optimal parameter settings for many of the studies in Table 1 is not described.
9
The issue of heuristic optimization parameter tuning has been investigated in a number of
10
recent studies. In relation to GAs, one approach is to self adapt the parameters as part of the
11
optimization procedure itself [e.g. Srinivasa et al., 2007]. Alternative approaches are based on
12
parameterless GA calibration methodologies [e.g. Lobo and Goldberg, 2004, Minsker, 2005] and
13
GA convergence behavior associated with genetic drift [e.g. Rogers and Prugel-Bennett, 1999,
14
Gibbs et al., 2008]. Gibbs et al. [2009] compared the performance of the above approaches on
15
the Cherry Hills-Brushy Plains WDS optimization problem [Bocelli et al., 1998] and found that
16
the approach based on genetic drift performed best overall. In relation to ACO, Zecchin et al.
17
[2005] used a mixture of theoretical and sensitivity analyses to derive expressions for seven ACO
18
parameters, which have been shown to perform well for a number of benchmark WDS
19
optimization problems [Zecchin et al., 2007]. One approach for eliminating penalty parameters
20
or multipliers in single objective WDS optimization problems is to convert hydraulic constraints
21
into objectives and therefore solve a multi-objective optimization problem without hydraulic
22
constraints [e.g. Wu and Simpson, 2002; Farmani et al., 2005].
23
Despite these efforts, common practice in many current heuristic optimization methods for
24
WDS design (see Table 1) still involves case study specific experimentation for tuning algorithm
25
parameters. As discussed previously, such experimentation is undesirable, as it has the potential
26
to increase computational effort significantly.
27
experimentation is sufficient and what impact limited experimentation has on algorithm
In addition, it is unclear how much
4
1
performance for a particular problem. In order to address these issues, the Hybrid Discrete
2
Dynamically Dimensioned Search (HD-DDS) algorithm is introduced in this paper. Not only is
3
the performance of HD-DDS reliant on only one parameter, but the value of this parameter does
4
not have to be adjusted for different case studies. This is in contrast to existing heuristic
5
optimization methods (Table 1).
6
Dimensioned Search (DDS) algorithm introduced by Tolson and Shoemaker [2007] for
7
continuous optimization problems and can be used for solving constrained, single-objective
8
combinatorial WDS optimization.
The HD-DDS algorithm builds on the Dynamically
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The remainder of the paper is organized as follows. Section 2.1 describes the HD-DDS
10
algorithm in detail. The optimization algorithms with which HD-DDS is compared in order to
11
evaluate its utility are outlined in section 2.2, and the WDS benchmarks to which it is applied
12
(ranging from 21- to 454-dimensional problems) are introduced in section 2.3. Results are
13
presented in section 3 and are followed by a discussion in section 4 and conclusions in section 5.
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2.
Methodology
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2.1. Components of the Hybrid Discrete Dynamically Dimensioned Search Algorithm
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The Hybrid Discrete Dynamically Dimensioned Search (HD-DDS) Algorithm for WDS
18
design optimization is described in the following sections. The HD-DDS algorithm utilizes global
19
and local search strategies, as such a hybrid approach has been shown to be successful previously
20
[e.g. Broad et al., 2006]. Sections 2.1.1 through 2.1.4 describe how each strategy functions
21
independently and then section 2.1.5 describes how they are combined to form the overall HD-
22
DDS algorithm.
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2.1.1.
Discrete Dynamically Dimensioned Search Algorithm
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The discrete Dynamically Dimensioned Search (discrete DDS) algorithm is the most
25
important component of the HD-DDS algorithm and is a discrete adaptation of the DDS 5
1
algorithm recently introduced by Tolson and Shoemaker [2007] for continuous optimization
2
problems. DDS was designed as a simple and parsimonious algorithm (it has only one algorithm
3
parameter) to solve computationally intensive environmental simulation model automatic
4
calibration problems. One DDS design goal was to have the algorithm automatically adjust and
5
exhibit good performance within the user’s timeframe for optimization (maximum number of
6
objective function evaluations) rather than require the user to modify and/or experiment with
7
algorithm parameters to match their timeframe. A related DDS design goal was to eliminate the
8
need for algorithm parameter adjustment when the case study or number of decision variables
9
changes. While it is acknowledged that this is unlikely to result in the identification of globally
10
optimal solutions, the DDS algorithm is simple to use in practice, while being able to consistently
11
find near globally optimal solutions. In fact, Tolson and Shoemaker [2007] demonstrate better
12
overall performance of DDS relative to other benchmark automatic calibration algorithms on
13
optimization problems ranging from 6- to 30-dimensions with 1000 to 10,000 objective function
14
evaluations per optimization trial while using the same DDS algorithm parameter value.
15
The parsimonious nature of DDS provides an attractive alternative for discrete WDS
16
optimization problems given the very recent set of single objective WDS optimization algorithms
17
reviewed in section 1 (see Table 1), all of which have from three to eight algorithm parameters of
18
which a subset is usually modified and even optimized for different case studies. The discrete
19
DDS algorithm is identical to the original DDS algorithm except for two modifications. The first
20
of these enables the proposed algorithm to sample discrete valued candidate solutions, whereas
21
the second is the addition of a new algorithm stopping criterion. The paragraphs below describe
22
the DDS algorithm (largely from Tolson and Shoemaker [2007]) and are followed by a
23
description of the two modifications that distinguish discrete DDS from DDS.
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The DDS algorithm is a simple, stochastic, single-solution based, heuristic, global search
25
algorithm that was developed for the purpose of finding a good approximation of the globally
26
optimal solution within a specified maximum number of objective function evaluations. The
27
algorithm is designed to scale the search to a user-specified number of maximum objective 6
1
function evaluations. In short, DDS searches globally at the start of the search, transitioning to a
2
more local search as the number of function evaluations approaches the maximum allowable
3
limit. The adjustment from global to local search is achieved by dynamically and probabilistically
4
reducing the number of dimensions in the neighborhood (i.e. the set of decision variables
5
modified from their best value). Candidate solutions are sampled from the neighborhood by
6
perturbing only the randomly selected decision variables from the current solution. These
7
perturbation magnitudes are randomly sampled from a normal distribution with a mean of zero
8
for the continuous version of DDS. These features of the DDS algorithm ensure that it is as
9
simple and parsimonious as possible. DDS is a greedy type of algorithm since the current
10
solution, also the best solution identified so far, is never updated with a solution that has an
11
inferior value of the objective function. The algorithm is unique compared with current
12
optimization algorithms because of the way the neighborhood is dynamically adjusted by
13
changing the dimension of the search. The DDS perturbation variances remain constant and the
14
number of decision variables perturbed from their current best value decreases as the number of
15
function evaluations approaches the maximum function evaluation limit. This key feature of
16
DDS was motivated by experience with manual calibration of watershed models where early in
17
the calibration exercise relatively poor solutions suggested the simultaneous modification of a
18
number of decision variables but as the calibration results improved, it became necessary to only
19
modify one or perhaps a few decision variables simultaneously so that the current gain in
20
calibration results was not lost.
21
The discrete DDS algorithm pseudocode is given in Figure 1, and the changes relative to the
22
original DDS algorithm are in Steps 3 and 5. The only user-defined algorithm parameter is the
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scalar neighborhood size perturbation parameter (r), which defines the standard deviation of the
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random perturbation size as a fraction of the decision variable range. In the discrete DDS
25
algorithm, the decision variables are integers from 1 to the number of discrete options for each
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decision variable (ximax). The objective function must translate or map these option numbers to
27
discrete pipe diameters in the WDS optimization case. Consistent with the continuous version of 7
1
DDS, a default value of the r parameter for discrete DDS is recommended as 0.2 (and used in this
2
study). In the original DDS algorithm, the perturbation magnitude for each decision variable is
3
sampled from a normal probability distribution. In Step 3 of discrete DDS (Figure 1), the
4
perturbation magnitude is randomly sampled from a discrete probability distribution that
5
approximates a normal distribution (see example probability mass functions in Figure 2). In the
6
continuous and discrete DDS algorithms, r = 0.2 yields a sampling range that practically spans
7
the normalized decision variable range for a current decision variable value halfway between the
8
decision variable bounds. This sampling region size is designed to allow the algorithm to escape
9
regions around poor local minima. The r parameter is a scaling factor assigning the same relative
10
variation to each decision variable (relative to the decision variable range).
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The one-dimensional decision variable perturbations in Step 3 of the DDS algorithm can
12
generate new decision variable values outside of the decision variable bounds (e.g. 1 and ximax).
13
In order to ensure that each one-dimensional perturbation results in a new decision variable that
14
respects the bounds, the DDS and discrete DDS algorithms define reflecting boundaries (See Step
15
3 of Figure 1). In the discrete DDS algorithm, the candidate decision variable values are initially
16
treated as continuous random variables and the reflecting boundaries for the decision variables
17
within the algorithm are defined to be 0.5 and 0.5 + ximax. Once a candidate decision variable
18
value within these reflecting boundaries is sampled, it is rounded to the nearest integer value to
19
represent the discrete option number (e.g. 1, 2, 3, … , ximax). This reflecting boundary approach
20
allows decision variable values to more easily approach their minimum or maximum values in
21
comparison with a simple perturbation resampling approach (truncated normal distribution) for
22
ensuring decision variable boundaries are respected. In the event that a perturbed decision
23
variable has a candidate discrete option number that is the same as the option number of the
24
current best solution (i.e. the decision variable is not perturbed), a new discrete option number is
25
sampled from a discrete uniform probability distribution. The resultant probability mass function
26
for the decision variable option number is given for four examples in Figure 2 based on Step 3 of
27
Figure 1. 8
1
The maximum number of objective function evaluations (M) is an algorithm input (like the
2
initial solution) rather than an algorithm parameter because it should be set according to the
3
problem specific available (or desired) computational time [see Gibbs et al., 2008]. The value of
4
M therefore depends on the time to compute the objective function and the available
5
computational resources or time. Except for the most trivial objective functions, essentially 100%
6
of discrete DDS execution time is associated with the objective function evaluation. Recall that
7
discrete DDS scales the search strategy from global in the initial stages of the search to more
8
local in the final stages of the search regardless of whether M is 1000 or 1,000,000 function
9
evaluations. In the absence of a specific initial solution, discrete DDS is initialized to the best of a
10
small number (maximum of 0.005M and 5) of uniformly sampled random solutions.
11
In the continuous DDS algorithm, depending on the problem dimension and maximum
12
objective function evaluation limit (M), a significant proportion of function evaluations in the
13
latter half of the search would evaluate a change in only one decision variable relative to the
14
current best solution. Given the general constrained WDS problem, we know a priori that the
15
feasible high quality solutions discrete DDS (or any other algorithm) identifies by the later stages
16
of the search will be very close to infeasibility. We also know that the only way to minimize the
17
objective function we define for discrete DDS is to select a smaller pipe diameter and thus reduce
18
nodal pressures somewhere in the network. Being close to infeasibility means that the original
19
DDS perturbation strategy that happens late in the search (changing only one decision variable)
20
will quickly become futile and all possible single pipe diameter reductions will result in
21
infeasible solutions. Therefore, a new primary stopping condition was added to discrete DDS in
22
Step 5 to stop the search before all M function evaluations are utilized (and thus save the
23
remaining computation time for more productive search strategies) based on the probability of
24
decision variable perturbation, P(i).
25
The number of dimensions selected for perturbation at each discrete DDS iteration, as
26
determined by the first two bullets of Step 2 in Figure 1, follows a binomial probability
27
distribution parameterized by P(i). It is desired that more than one dimension is selected for 9
1
perturbation at each iteration, which implies that an appropriate termination point is when the
2
expected value of the binomial random variable is one. The corresponding probability value at
3
this expected value is 1/D (D is the number of decision variables) leading to a termination
4
criterion of P(i) < 1/D.
5
For the moderately sized WDS case studies solved (21-42 dimensions) with computational
6
budgets ranging from 10,000 to 1,000,000 objective function evaluations, this new stopping
7
condition terminates discrete DDS after 52% to 80% of M allowable function evaluations are
8
conducted. In networks with an order of magnitude of more decision variables (more than 200),
9
for any computational budget fewer than 10,000,000 function evaluations, discrete DDS will only
10
stop after 90% or more of M allowable function evaluations are utilized. The complete HD-DDS
11
algorithm and its other component search strategies (sections 2.1.3, 2.1.4, and 2.1.5) define how
12
the remaining computational budget is utilized after an initial optimization with discrete DDS
13
terminates.
14
A preliminary version of discrete DDS was first defined in Tolson et al. [2008]. However,
15
discrete DDS in Figure 1 of this study differs from Tolson et al. [2008] in two ways. First, we use
16
a slightly modified neighborhood definition more consistent with the original DDS algorithm in
17
Tolson and Shoemaker [2007]. Another distinction is that Tolson et al. [2008] allowed the
18
preliminary discrete DDS to utilize all M function evaluations rather than stopping the algorithm
19
early.
20
2.1.2. Constraint-Handling with Discrete DDS
21
Discrete DDS is not a population based algorithm and only one solution (the best found so
22
far) influences the region of decision variable space that is sampled. As shown in Step 5 of Figure
23
1, discrete DDS (like DDS) is not impacted by objective function scaling. Only the relative ranks
24
of the best solution found so far and the candidate solution determine whether the best solution
25
should be updated. Any update to the best solution moves it to a different point in decision space,
26
around which future candidate solutions are centered. This aspect of DDS and discrete DDS 10
1
makes handling constraints very straightforward without the need for penalty function
2
parameters. The constraint handling approach described below is equivalent to the method
3
described in Deb [2000] for constraint handling in GAs using tournament selection (where
4
objective function scaling also has no impact on the optimization). The key to the approach in
5
Deb [2000] is that the objective function is defined such that any infeasible solution always has
6
an objective function value that is worse than the objective function value of any feasible
7
solution. In a WDS design problem where costs are to be minimized subject to minimum pressure
8
requirements across the network, the worst cost feasible solution is known before any
9
optimization as the cost of the solution specifying all pipes at their maximum diameter. The only
10
other requirement for the constraint handling technique in Deb [2000] is to quantify the relative
11
magnitude of constraint violations for infeasible solutions so that the relative quality of two
12
infeasible solutions can be compared. The steps to evaluate the objective function in discrete
13
DDS are outlined in Figure 3.
14
The evaluation of the objective function in Figure 3 has a penalty term, but there is no need to
15
scale it to a different magnitude with a case study specific penalty multiplier and/or exponent.
16
Instead, the objective function definition implements the following logic:
17
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assigned a better objective function value.
20 21 22
Between two infeasible solutions, the one with the least total pressure violation is always
Between an infeasible and a feasible solution, the feasible one is always assigned a better objective function value.
Between two feasible solutions, the one which is less costly is always assigned a better objective function value.
23
In addition to the benefit of requiring no case study specific penalty parameter values
24
(which generally require experimentation to determine), our definition of the objective function
25
yields the significant benefit of eliminating the need to evaluate the hydraulics of a large number
26
of solutions generated by discrete DDS. The multi-objective optimization study by Prasad and
11
1
Park [2004] is the only other WDS study that we are aware of to utilize the constraint handling
2
strategy in Deb [2000].
3
2.1.3. One-pipe Change Local Search (L1)
4
This local search technique starts at a feasible solution and cycles through all the possible
5
ways to perturb the current best solution by reducing the diameter of one pipe at a time (thus
6
saving costs). Each perturbed solution is evaluated for cost and feasibility. The L1 search
7
continues until it has evaluated a maximum number of objective function evaluations or until it
8
has enumerated all possible one-pipe changes without finding any improvement to the current
9
best solution and therefore confirms that a local minimum has been located. The solution is a
10
local minimum in that no better solution exists that differs from the current best solution in only
11
one decision variable (pipe diameter). Pseudocode for implementing L1 will be archived with this
12
paper online1.
13
Note that L1 is implemented such that it is always evaluating one-pipe changes relative to
14
the current best known solution. In other words, whenever L1 finds a new best solution, the
15
current best solution is updated and L1 searches in the vicinity of the updated current best
16
solution. Therefore, the order of enumeration (starting at pipe D instead of pipe 1) can change
17
the final best solution returned by L1.
18
2.1.4. Two-pipe Change Local Search (L2)
19
This local search technique starts at a feasible solution and cycles through all the possible
20
ways to perturb the current best solution by only two pipes where one pipe has the diameter
21
increased and the other pipe has the diameter decreased. In this study, the L2 search is only
22
initiated at feasible solutions that are already confirmed to be a local minimum with respect to L1.
23
As Gessler [1985] suggested in his pipe design enumeration strategy, solutions that have a higher
1
Auxiliary material is available at ftp://ftp.agu.org/***/L1.eps.
12
1
cost than the current best solution are not evaluated for their hydraulics in the L2 search. The L2
2
search continues until it has evaluated a maximum number of objective function evaluations, or it
3
has enumerated all possible two-pipe changes without finding any improvement to the current
4
best solution and therefore confirms that a local minimum has been located. The solution is a
5
local minimum in that no better solution exists that differs from the current best solution in only
6
two decision variables (pipe diameters). Pseudocode for implementing L2 will be archived with
7
this paper online1.
8
Note that L2 is implemented such that it enumerates all possible two-pipe changes from the
9
current best solution to identify the best two-pipe change. This is a different enumeration
10
approach in comparison to L1. Only after all two-pipe change possibilities are enumerated in L2 is
11
the current best solution updated and the enumeration process repeated. Therefore, this local
12
search method is defined so that different orders of enumeration do not affect the results if L2
13
converges to a local minimum.
14
In L2, the solution being enumerated only has the objective function evaluated if it is less
15
expensive than the current best L2 solution. This is quickly determined based only on comparing
16
the costs associated with the two decision variables (pipe diameters) being changed relative to the
17
corresponding costs in the current best L2 solution.
18
2.1.5. HD-DDS Algorithm Definition
19
The explicit design goals that guided our development of the Hybrid Discrete Dynamically
20
Dimensioned Search (HD-DDS) Algorithm were, in highest to lowest priority, to develop an
21
algorithm that 1) reliably returned good quality approximations of the global optimum 2) was
22
capable of solving WDN design problems to optimality and 3) could achieve these goals in a way
23
that was computationally efficient. In addition, the algorithm was designed to be parsimonious in
24
the number of algorithm parameters such that achieving the above goals did not necessarily
1
Auxiliary material is available at ftp://ftp.agu.org/***/L2.eps.
13
1
require parameter tuning experiments for each case study. The definition of a good quality
2
approximation is clearly specific to each case study and even the computational budget used to
3
solve the problem, but in general, is taken to be a solution whose cost is small enough to satisfy
4
the designer such that further optimization would be deemed unnecessary.
5
characteristic that the algorithm inherits from the continuous DDS algorithm is the ability to
6
effectively scale to various computational budgets such that it is capable of generating good
7 8 9 10 11 12 13 14 15
results without algorithm parameter tuning. The HD-DDS algorithm component is designed to overcome the shortcomings of other individual component search strategies. First and foremost, as described in section 2.1.1, the discrete DDS search is predictably unproductive with the original DDS algorithm stopping criterion, and thus HD-DDS invokes alternative search strategies after discrete DDS reaches this unproductive stage (when P(i) < 1/D). Unlike other hybridized algorithms (such as a ACO + a local search), the point at which to terminate a global search in favour of a new search strategy in HD-DDS is known a priori and thus does not require the definition of various algorithm specific convergence measures. The component search combination strategy defining HD-DDS is given in
16 17
Figure 4. Initially (Step 0 in
A design
18 19
Figure 4), HD-DDS requires all problem inputs to be specified. This includes the user-
20
defined computational budget, Mtotal, which can be expended in solving the problem. Mtotal is a
21
quantity that users should determine based on their problem specific available computation time
22
and the average computation time required for an objective function evaluation that evaluates
23
both cost and hydraulics. As discussed later in section 4, because HD-DDS can skip the
24
hydraulics evaluation for a large number of candidate solutions, HD-DDS will terminate much
25
quicker than what would be estimated in this way. The only algorithm parameter in HD-DDS is
26
the discrete DDS parameter r, which has a default of 0.2 that is utilized for all HD-DDS analyses
27 28
presented in this paper. The first search strategy implemented in Step 1 of
29 30
Figure 4 is a global search with discrete DDS for a maximum of M objective function
31
evaluations, where M in HD-DDS is a variable tracking the remaining available number of 14
1
objective function evaluations. The second search strategy in Step 2 of HD-DDS is defined to be
2
the L1 local search (one-pipe change), which polishes the discrete DDS solution to a local
3
minimum with respect to L1. Results show this enumerative search is typically very quick (often
4 5
less than D evaluations of EPANET2) when polishing a discrete DDS solution. Step 3 in HD-DDS (
6 7 8 9 10 11
Figure 4) is defined as a second independent discrete DDS search from a new initial solution followed by another L1 search to polish the second discrete DDS solution. This second discrete DDS search has a smaller M than the first and therefore will utilize fewer objective function evaluations. At the end of Step 3, a second potentially new solution that is a local minimum with respect to L1 has been located, which is referred to as xBbest. Step 4 in HD-DDS (
12 13
Figure 4) implements the first L2 local search to polish the best of the two HD-DDS
14
solutions found so far. With very large networks, L2 can be extremely slow to converge and
15
confirm that a local minimum has been reached.
16
reasonably good quality solution, L2 typically converges in a few thousand objective function
17
evaluations in HD-DDS for case studies with up to 42 decision variables. Step 5 in HD-DDS
18
implements the second L2 local search to polish the other HD-DDS solution that is confirmed to
19
be a local minimum with respect to L1.
However, as results will show, with a
20
HD-DDS can terminate when either the total computational budget (Mtotal) is exceeded or
21
when all five steps are completed in fewer than Mtotal objective function evaluations. For a fixed
22
and reasonably large computational budget (e.g. 100,000), as the network size decreases (number
23
of decision variables decrease), the likelihood that HD-DDS will be able to complete all five
24
steps increases. However, as network size increases, it becomes more and more likely that HD-
25
DDS will terminate without a second discrete DDS search. In other words, HD-DDS reduces to
26
only Steps 1 and 2 when the number of decision variables becomes extremely large because the
27
user defined computation limit will be exceeded before Step 2 finishes. This behavior is based on
28
the assumption that for a fixed and practical computational budget, the globally optimal solution
15
1
to an extremely large problem is virtually impossible to find, and the best approach would be to
2
conduct the longest possible global search.
3
HD-DDS was designed to always perform a second global search (Step 3) before the first
4
L2 local search polishing step. The main purpose of a second global search is to act as a safety
5
mechanism to guard against a very poor solution from the first discrete DDS trial. To maximize
6
the chances the safety mechanism works, this second global search requires the largest possible
7
computation budget and therefore must be executed prior to L2 (which can take an indeterminate
8
amount of time). Furthermore, our belief in general is that the best chance to significantly
9
improve upon a particularly poor solution is to conduct a new global search rather than polish the
10
particularly poor solution. This design choice has no impact on final solution quality in cases
11
where the computation budget is large enough and/or the problem dimension is small enough to
12
allow all five steps in HD-DDS terminate completely in fewer than Mtotal objective function
13
evaluations. There are some situations (particularly with problems having hundreds of decision
14
variables) where the choice to conduct a second global search (followed by L1) uses all or most of
15
the remaining computational budget and thus precludes or reduces computation time available for
16
the L2 local search. In these situations with such large networks, the L2 local search can require
17
an incredible number of objective function evaluations to enumerate all possible two-pipe
18
changes such that L2 terminates before it confirms the solution is a local minimum. Despite this,
19
the optimal design choice between L2 or second global search is certainly case study dependent.
20
Results will show that our design choice has very little impact on HD-DDS results.
21
2.2. Benchmark Optimization Algorithms
22
The main focus of this study is the introduction of the HD-DDS algorithm and its
23
performance comparison with alternative algorithms. The design case studies and benchmark
24
algorithms were selected from the literature based on whether the WDS case study could be
25
replicated exactly, and whether the algorithm results were presented for multiple optimization
26
trials. Exact replication of a WDS case study meant that comparative algorithm results must have 16
1
been generated with EPANET2 using metric units. Most WDS optimization algorithms are
2
stochastic in that they can and often do generate a different final solution for a different random
3
seed and/or different initial solution. Therefore, objectively assessing relative algorithm
4
performance requires multiple independent optimization trials.
5
Three recent studies that meet the above criteria and thus provide the majority of the
6
comparative algorithm results used in this study are Zecchin et al. [2007], Reca and Martinez
7
[2006], and Montalvo et al. [2008]. Zecchin et al. [2007] test the performance of five different
8
Ant Colony Optimization (ACO) algorithms on some standard WDS case studies. They report
9
that the MMAS ACO algorithm outperforms all other algorithms applied to their case studies in
10
the literature. Reca and Martinez [2006] apply a GA called GENOME to a much larger scale
11
WDS optimization problem. Montalvo et al. [2008] introduce a Particle Swarm Optimization
12
(PSO) algorithm variant to some standard WDS case studies and show that their PSO variant
13
outperforms a standard discrete PSO algorithm. Details of the WDS case studies selected for HD-
14
DDS testing from the above studies are given in the following section.
15
2.3. Benchmark WDS Design Studies
16
The first three example WDS design problems in this study, the New York Tunnels, Doubled
17
New York Tunnels, and Hanoi problems, are equivalent to those used in Zecchin et al. [2007].
18
For complete details of these three case studies, readers are referred to Zecchin et al. [2005]. The
19
GoYang WDS problem was introduced by Kim et al. [1994] and is also utilized as our fourth case
20
study. The fifth and final case study is the Balerma WDS problem, which was first proposed by
21
Reca and Martinez [2006]. The next few paragraphs provide an overview of each WDS case
22
study, and readers should consult the references given for case study details (e.g. discrete pipe
23
diameter options, pipe length, nodal head requirements etc.). Note that all EPANET2 units are
24
metric in each of the case studies.
25
The New York Tunnels Problem (NYTP), originally considered in Schaake and Lai [1969],
26
involves the rehabilitation of an existing WDS by specifying design options for the 21 tunnels in 17
1
the system. There are 16 design options for each tunnel (parallelization with one of 15 tunnel
2
sizes or a do-nothing option), thus defining a search space size of 1621 (approximately
3
1.93×1025). The Doubled New York Tunnels problem (NYTP2) was originally proposed by
4
Zecchin et al. [2005]. This network has 42 pipes to be sized, and each has 16 options. This
5
defines a search space of 1642 (approximately 3.74×1050) for NYTP2.
6
The Hanoi Problem (HP) was first reported on in Fujiwara and Khang [1990] and has a
7
larger search space size (2.87x1026 solutions) than the NYTP based on 32 pipes to be sized and 6
8
discrete pipe diameters. HP problem details are available in Wu et al. [2001]. The GoYang
9
problem (GYP) represents a South Korean WDS and was first proposed by Kim et al. [1994] and
10
has 30 pipes to be sized with 8 diameter options per pipe to define a search space size of 830
11
(approximately 1.24×1027).
12
The Balerma irrigation network problem (BP) from Reca and Martinez [2006] is a large and
13
complex network that has 443 demand nodes, 454 pipes, 8 loops, and 4 reservoirs. Each of the
14
454 pipes to be sized has ten possible diameters and thus defines a search space size of 10454.
15
Based on a review of the current literature, the best current known feasible solutions (using
16
EPANET2) are $38.638 million for the NYTP [Maier et al., 2003], $77.276 million for the
17
NYTP2 [Zecchin et al., 2005], $6.081 million for the HP [Perelman and Ostfeld 2007], and
18
€2.302 million for the BP [Reca and Martinez 2006].
19
algorithm results for GYP with EPANET2 as the hydraulic solver are unavailable and thus HD-
20
DDS is not compared to any other algorithm for the GYP.
21
2.4. Optimization Model Formulation for HD-DDS
22
Currently, published optimization
The HD-DDS algorithm solves the following optimization model for each WDS case study:
23
Min F( x ) ,
24 25
x
s.t.
(1)
xd 1, 2, , xdmax , d = 1, …, D
18
1
where F(x) is the objective function value for decision vector x = [x1 … xD] as determined by the
2
procedure in Figure 3, and xd is the integer-valued pipe diameter option number for pipe d and is
3
between option 1 (the smallest diameter) and the maximum diameter option, x dmax , for all D pipes
4
in the network to be sized. The minimum required heads for the nodes in the network (e.g. himin
5
for node i in Figure 3) and nonlinear cost equations vary with each case study and must be
6
specified (readers can find these case study specific values in the case study references noted in
7
section 2.3).
8
2.5. Outline of Algorithm Comparisons
9
The performance of different algorithms is generally compared for a similar computational
10
budget as measured by the total number of objective function evaluations utilized in the
11
optimization run. The computational budget we report for all algorithms is measured as the
12
budget required for the algorithms to terminate. Although this is not the measure that is reported
13
in Zecchin et al. [2007] for the MMAS ACO algorithm, the measure used in our study reflects the
14
fact that in any new problem, with an unknown best solution, the user will not generally have any
15
knowledge or reason to stop the algorithm before it terminates normally.
16
Due to the stochastic nature of most heuristic global optimization algorithms, their relative
17
performance must be assessed over multiple independent optimization trials, each initialized to
18
independent populations or solutions. Previously published algorithm results are compared to
19
HD-DDS results using 10 to 50 optimization trials. Optimization algorithm comparisons cover
20
21-, 30-, 32-, 42- and 454-dimensional (i.e. the number of decision variables or pipes to be sized)
21
problems, and the size of the search spaces ranges from 1.93x1025 to 1.0x10454 possible solutions.
22
The maximum number of objective function evaluations, Mtotal, per HD-DDS optimization trial
23
varies from 10,000 to 10,000,000. Given that average algorithm performance across multiple
24
independent optimization trials does not provide a complete picture of results, the distribution or
25
range of the HD-DDS and benchmark algorithm solutions are also assessed.
19
1
The initial solution for HD-DDS is generated as described in section 2.1.1, and the
2
neighborhood size parameter, r, is set to the default value of 0.2 for all HD-DDS trials (no
3
parameter setting experimentation was performed on r). In some of the case studies, L1 or L2
4
local searches initiated in HD-DDS before the remaining available computational budget is
5
exceeded were typically allowed to terminate at a local minimum even if that meant exceeding
6
Mtotal by a few thousand objective function evaluations on average. In contrast, comparative
7
MMAS ACO algorithm results from Zecchin et al. [2007] are based on MMAS ACO algorithm
8
parameters that were optimized independently for each case study using millions of EPANET2
9
simulations. Similarly, the comparative results for the GENOME GA and PSO variant algorithms
10
were also derived from some experimental optimization runs to identify good algorithm
11
parameters.
12 13
3.
Results
14 15
The results are presented here in two sections. Section 3.1 assesses the importance and effect
16
of each step of the HD-DDS algorithm for four case studies. Comparative algorithm performance
17
results are presented in section 3.2
18
3.1. HD-DDS Component Performance Assessment
19
For all HD-DDS optimization trials, at the end of each HD-DDS step in
20 21
Figure 4, the best HD-DDS solution found so far and the corresponding number of objective
22
function evaluations are recorded. This information is shown in Figure 5 for the NYTP, NYTP2,
23
HP, and GYP benchmarks. In addition to presenting results from all individual optimization
24
trials, the average cost of the best solution found so far is shown in Figure 5 where objective
25
function values are plotted against the average number of function evaluations for clarity because
26
for Steps 2, 3, 4 and 5, the number of utilized function evaluations varies between optimization 20
1
trials. The difference between each step on the x-axis in Figure 5 shows the average
2
computational requirements of each Step in HD-DDS.
3
For each case study in Figure 5, Step 2 in HD-DDS (the L1, one-pipe change local search)
4
requires a negligible number (~10) of objective function evaluations to determine that discrete
5
DDS terminated at a local minimum with respect to L1. In fact, in all HD-DDS optimization
6
trials, L1 never improved upon a discrete DDS solution, indicating that discrete DDS always
7
terminated at a local minimum with respect to L1. With a much smaller computational budget
8
and/or much larger network (e.g. see BP results for Mtotal ≤ 10,000 in section 3.2.1), discrete DDS
9
will not always terminate at a local minimum with respect to L1, and thus L1 can improve upon
10
discrete DDS solutions.
11
The computational budget for the second discrete DDS search (Step 3) is significantly
12
smaller than that for the first discrete DDS search for all four case study results in Figure 5 (21%
13
- 41% of the budget of the first discrete DDS search). Despite the substantial decrease in
14
computational budget, results of Steps 2 and 3 in Figure 5 show that this second, quicker discrete
15
DDS search does sometimes improve upon the first DDS search result (for all four case studies,
16
Step 3 improves best solution in 20% to 32% of optimization trials). Most notable in Figure 5 is
17
that the second DDS search operates to eliminate the worst cost solutions from Step 2 in all four
18
case studies.
19
Results in Figure 5, at each Step of HD-DDS, also present a count of the number of
20
optimization trials that have located the best known solution for each of the case studies. Discrete
21
DDS (Steps 1 and 3 of HD-DDS) generally does not find the best known solution (except for a
22
small number of trials in NYTP and GYP). Results for Steps 4 and 5 of HD-DDS in Figure 5
23
clearly show that the L2 (two-pipe change) local search is capable of polishing discrete DDS
24
solutions (from Steps 1 and 2) and returning the best known solution (with 85% frequency for
25
NYTP and NYTP2). The average computational burden and percentage of optimization trials
26
with improved solutions due to L2 in Step 4 (which terminates with a local minimum with respect
27
to L2) was 1421, 9031, 3250, and 586 function evaluations and 90%, 56%, 100%, and 66% for 21
1
NYTP, NYTP2, HP, and GYP case studies, respectively. The average computational burden for
2
Step 5 (second L2 search which terminates with a local minimum with respect to L2) increases to
3
2302, 15403, 4487, and 649 function evaluations on average for the NYTP, NYTP2, HP, and
4
GYP case studies, respectively, because the second L2 search is typically starting from a lower
5
quality discrete DDS solution. Results for NYTP and NYTP2 in Figure 5 demonstrate that Step 5
6
can improve results indicating that a second L2 search starting from a lower quality initial
7
solution can be fruitful. The percentage of optimization trials with improved solutions due to L2
8
in Step 5 is 32%, 25%, 14%, and 6% for the NYTP, NYTP2, HP, and GYP case studies,
9
respectively.
10
3.2. HD-DDS Performance Relative to Benchmark Algorithms
11
The first comparative results are for HD-DDS and the MMAS ACO algorithm results in
12
Zecchin et al. [2007] for the NYTP, HP and NYTP2. Figure 6 shows the empirical cumulative
13
distribution function (CDF) for HD-DDS and MMAS ACO of the final best objective function
14
values (costs) from all optimization trials. HD-DDS results stochastically dominate MMAS ACO
15
results in all three case studies because for any desirable cost target identified by a decision-
16
maker, HD-DDS always has an equal or higher probability of satisfying the cost target than
17
MMAS ACO. The near vertical lines for NYTP and NYTP2 indicate that HD-DDS yields the
18
best known solution with a high reliability (more than 80%).
19
optimization trials were better than the best MMAS ACO solution. HD-DDS avoided the worst
20
solutions identified by MMAS ACO. The superior performance of HD-DDS over MMAS ACO
21
is even more noteworthy considering there was no algorithm parameter experimentation with
22
HD-DDS, and there were extensive case study specific parameter setting experiments conducted
23
for MMAS ACO, as discussed previously.
24
3.2.1. Large Scale WDS: Balerma Case Study
25
The Balerma network has 454 pipes to be sized (D=454), and as a result, Step 1 in HD-DDS (see
For HP, eight HD-DDS
22
1 2
Figure 4) uses more than 96% of any of the computational budgets specified in our study
3
(1,000 ≤ Mtotal ≤ 10,000,000). Note that Step 1 of HD-DDS (the first discrete DDS search)
4
terminates only when P(i), as calculated in Step 2 of discrete DDS (see Figure 1), is less than 1/D.
5
The remaining 4% or less of the computational budget is utilized by Step 2 (L1 search) and then
6
any remaining budget is dedicated to Step 3 of HD-DDS (second global search followed by
7
another typically incomplete L1 search).
8
Figure 7 shows all of the HD-DDS results generated for Balerma as the number of
9
objective function evaluations are increased from 1000 to 10,000,000. All HD-DDS results are
10
based on 10 optimization trials. As expected, Figure 7 shows that HD-DDS performance
11
improves with a larger computational budget. It is important to recall that HD-DDS scales the
12
search to the user input Mtotal, and thus separate independent optimization trials are used to
13
generate results for different computational budgets. For example, the best solution after 10,000
14
objective function evaluations in HD-DDS with Mtotal=100,000 will not be equal to the final best
15
solution in HD-DDS with Mtotal=10,000.
16
Comparative algorithm performance is also shown in Figure 7 using results for MSATS
17
and GENOME GA reported in Reca et al. [2008] and GENOME GA results reported in Reca and
18
Martinez [2006]. For the same computational budget (10,000,000 objective function evaluations),
19
HD-DDS clearly outperforms the GENOME GA, as the worst HD-DDS solution is better than
20
the best GENOME GA solution by nearly 200,000 Euros. Even with 1/100th of the computational
21
budget (Mtotal=100,000), HD-DDS still outperforms the GENOME GA as the worst HD-DDS
22
result costs nearly 100,000 Euros less than the best GENOME GA result after 10,000,000
23
function evaluations.
24
In the interest of determining the best possible solution, the best HD-DDS solution with
25
Mtotal=10,000,000 (cost of €1,956,226) was passed onto the L2 local search to polish the solution.
26
After an additional 20 million EPANET2 objective function evaluations in L2, the minimum cost
23
1
solution improved to €1,940,923 (also shown in Figure 7). Note that L2 was manually terminated
2
prior to confirmation a local solution was identified.
3
The HD-DDS results for Mtotal=10,000 and Mtotal=1000 in Figure 7 demonstrate that even
4
with a severely restricted computational budget, HD-DDS can generate reasonable solutions, all
5
of which are feasible. These HD-DDS results are much better than the GENOME GA and
6
MSATS results from Reca et al. [2008] after 45,000 objective function evaluations. Note that the
7
HD-DDS results for Mtotal=1,000 utilized an average of 2,900 objective function evaluations
8
rather than 1000 because the L1 one-pipe change search was only terminated when it returned a
9
local solution with respect to L1 (which required approximately 1900 additional function
10
evaluations). In fact, the L1 search drastically improved the average discrete DDS solution quality
11
from 5.021 to 3.080 million Euros. Application of only L1 without the preliminary discrete DDS
12
search was ineffective in large part because without the discrete DDS solution, L1 could not
13
quickly identify a feasible solution.
14
Some final tests were performed to compare HD-DDS performance if L2 was performed
15
instead of the second global search. Although performing L2 yields reliable but very small
16
improvements (0.1%-0.2% on average) in the best cost solution after Step 2 of HD-DDS in
17
Figure 4, performing the second global search is capable of yielding significantly larger best cost
18
improvements (more than 2%) much less frequently. Therefore, as designed the algorithm
19
forgoes very small improvements under L2 for the significantly higher but less frequent
20
improvements capable with second global search.
21
3.2.2. HD-DDS Performance Comparison Summary
22
Table 2 summarizes and compares the results of HD-DDS and other algorithms previously
23
noted in sections 3.2 and 3.2.1. The main difference in addition to compressing all previous
24
graphical results into one table is that the solution quality is also measured with respect to the
25
percent deviation from the best known solution. Table 2 also includes new results for a few other
26
algorithms and HD-DDS computational budgets. Results for other algorithms in Table 2 are only 24
1
included where it was possible to confirm that the algorithms were applied to the exact same
2
optimization formulation (e.g. EPANET2 was used with metric units).
3
Results in Table 2 pair each HD-DDS result with another comparative algorithm result,
4
where HD-DDS is typically applied with a similar number of objection function evaluations, and
5
demonstrate the excellent overall performance of HD-DDS. In all algorithm comparisons, the
6
median best costs found by HD-DDS are always equal to or lower than the costs obtained using
7
the other algorithms. Importantly, the maximum cost solutions found by HD-DDS are always
8
better than those found by the comparative algorithms for all case studies (HD-DDS is 1.2% to
9
21.3% closer to the best known solution). For example, the worst HD-DDS solution for NYTP2
10
is nearly 2 million dollars less than the worst MMAS ACO solution and the worst HD-DDS
11
solution for NYTP is just over 8 million dollars less than the worst PSO variant solution. The
12
minimum best costs found by HD-DDS are equal to or lower than the costs obtained using all
13
other algorithms in all comparisons. With the exception of the 150,000 x 10 results for HP, Table
14
2 shows HD-DDS always returns the best known solution with a higher frequency than other
15
algorithms. For example, HD-DDS finds the best known NYTP solution in 86% of optimization
16
trials compared to 30% for the PSO variant despite HD-DDS using 30,000 fewer objective
17
function evaluations.
18
The best HD-DDS solutions for NYTP and NYTP2 are the same as the best known
19
solutions found in Zecchin et al. [2007]. The best HD-DDS solution for HP is the same as the
20
best known solution as reported in Perelman and Ostfeld [2007]. The best HD-DDS solution for
21
BP is a new best known solution. All of these HD-DDS identified best known solutions will be
22
archived with this paper online1.
23 24
4.
Discussion
25
1
Auxiliary material is available at ftp://ftp.agu.org/***/Archive_best_solutions.txt.
25
1
The default value of the neighborhood perturbation size parameter, r, of 0.2 produced
2
excellent results compared to all other algorithms for the WDS case studies reported on in section
3
3. These good results cover 21- to 454-dimensional problems and are based on computational
4
budgets ranging from 1,000 to 10,000,000 objective function evaluations. Therefore, the default
5
value for r appears robust and is suggested for future HD-DDS applications. Results also showed
6
that each search strategy (or step) in HD-DDS played an important role in at least one case study.
7
HD-DDS has a very large computational efficiency advantage over most other WDS
8
optimization algorithms that is not obvious from results reported in section 3. For consistency
9
with previous studies, the computational budget of HD-DDS was defined with respect to the total
10
number of objective function evaluations. However, the objective function evaluation strategy
11
with constraint handling (see Figure 3) and the computationally efficient implementation of L2
12
enable HD-DDS to evaluate the solution quality to a sufficient level without simulating the
13
hydraulics of the solution (e.g. an EPANET2 simulation). Therefore, even though, for example,
14
HD-DDS utilized approximately 46,000 objective function evaluations to optimize the NYTP
15
with Mtotal=50,000, EPANET2 simulations were not required for approximately 33,000 or 72% of
16
all HD-DDS solutions evaluated. Based on actual run times for HD-DDS, results show that our
17
HD-DDS algorithm as implemented runs 50% faster than it would have if EPANET2 was used to
18
evaluate all 46,000 solutions identified in HD-DDS. Balerma results for Mtotal=100,000 were
19
similar in that EPANET2 simulations were not required for approximately 71% of all HD-DDS
20
solutions evaluated. Based on HD-DDS run times for Balerma, results show that our HD-DDS
21
algorithm as implemented runs 67% faster than it would have if EPANET2 was used to evaluate
22
all solutions identified in HD-DDS. The relative computational efficiency gain of HD-DDS over
23
other optimization algorithms that must evaluate hydraulics of every candidate solution becomes
24
larger as the computational demand of the hydraulics simulation increases. If the EPANET2
25
evaluations accounted for nearly 100% of HD-DDS computation time then HD-DDS could be
26
more than 70% faster than other WDS optimization algorithms like MMAS ACO [Zecchin et al.,
26
1
2007] and the roulette wheel selection based GENOME GA [Reca and Martinez, 2006] in
2
evaluating the same number of objective functions.
3
To our knowledge, this paper was the first time the constraint handling strategy in Deb
4
[2000] was used in a single objective WDS optimization application, and overall, the excellent
5
results of HD-DDS suggest the strategy works very well. Every discrete DDS optimization trial
6
we performed always returned a final solution that was feasible, even in the HP for which
7
multiple studies report difficulty in locating any feasible solution due to its small feasible search
8
space [Euseff and Lansey, 2003; Zecchin et al., 2005 and Zecchin et al., 2007]. In order to further
9
demonstrate the excellent performance of HD-DDS with our constraint handling approach for
10
HP, 10 independent trials initialized at the most infeasible solution (all pipes at their minimum
11
diameter) with Mtotal=10,000 were conducted. Each of these runs were terminated at exactly
12
10,000 function evaluations, and all returned feasible solutions with an average cost of $6.299
13
million and a worst cost of $6.375 million.
14
For a fixed computational budget and specific case study, it is possible that there are better
15
ways to combine discrete DDS, one-pipe and two-pipe change algorithms than the HD-DDS
16
algorithm presented. In other words, we do not claim that HD-DDS is the optimal way to
17
configure these three search strategies across all case studies and all computational budgets. The
18
optimal configuration of these three search strategies is almost certainly case study and
19
computational budget specific. Instead, we have demonstrated a very robust and parsimonious
20
way to combine these strategies. HD-DDS performs two independent global searches before
21
spending an unknown amount of effort to polish the best solution with the two-pipe change local
22
search. Our results show that relative to available benchmark algorithm performance, for similar
23
computational budgets, HD-DDS performs equivalent to or better than all algorithms in our
24
comparison in the five case studies considered here. Therefore, experimenting with alternative
25
ways to combine the search strategies in HD-DDS is unnecessary. Instead of experimenting with
26
HD-DDS component configurations to improve upon HD-DDS performance for a new WDS
27
problem, it is recommended that HD-DDS users utilize the available time they have for 27
1
performing multiple independent HD-DDS optimization trials or implement alternative local
2
search strategies.
3
In practice, when users apply the HD-DDS algorithm as we suggest, results have shown they
4
will have a HD-DDS solution much more quickly (by 50-70%) than the worst case we
5
recommend they plan for (e.g. under the worst case assumption that all solutions will have their
6
hydraulics evaluated). In deciding how to utilize any available remaining computational budget
7
after their first HD-DDS trial terminates, we suggest that if the user is satisfied with the current
8
HD-DDS solution quality (cost), and HD-DDS terminated before the L2 local searches both
9
converge, we suggest they give HD-DDS the extra time to polish the available solutions with L2.
10
In any other cases (the user is dissatisfied with best cost returned or HD-DDS terminates because
11
all five algorithm steps were completed), we would suggest the user perform a second HD-DDS
12
optimization trial with the remaining budget.
13 14
5.
Conclusions and Future Work
15 16
For the range of WDS benchmark case studies considered in this study, numerical results
17
demonstrate that the HD-DDS algorithm exhibits superior overall performance in comparisons to
18
the MMAS ACO [Zecchin et al., 2007], GENOME GA [Reca and Martinez, 2006], and PSO
19
variant [Montalvo et al., 2008] algorithms.
20
evaluations HD-DDS stochastically dominates MMAS ACO results in all three case studies for
21
which their performance is compared. This is achieved despite the fact that no parameter tuning
22
was conducted in HD-DDS while MMAS ACO parameters were specifically tuned to each case
23
study (involving millions of EPANET2 simulations). The worst HD-DDS result was better than
24
the best GENOME GA result for the 454 decision variable Balerma network even though the
25
GENOME GA utilized 100 times more objective function evaluations. In addition, HD-DDS
26
found a new best solution to the Balerma problem. HD-DDS found the best known solutions
27
more frequently and easily avoided the worst solutions returned by the PSO variant [Montalvo et
For the same number of objective function
28
1
al., 2008] despite the fact that PSO algorithm parameters were determined with preliminary
2
tuning experiments. Furthermore, because the evaluation of many of the candidate solutions
3
identified by HD-DDS does not require simulating network hydraulics (e.g. 50%-70% of
4
objective function evaluations in this study), the actual HD-DDS computation time would be
5
much less (by nearly 50%-70%) than that of the comparative algorithms in this study. This
6
computational advantage of HD-DDS extends over any optimization algorithm requiring that
7
network hydraulics be simulated for all candidate solutions. The parameter-free constraint
8
handling approach based on Deb [2000] was successful.
9
HD-DDS can also be applied to other types of constrained discrete optimization problems in
10
water resources and environmental management such as watershed best management practice
11
optimization [e.g. Arabi et al., 2006], groundwater management and monitoring problems [e.g.
12
Reed et al., 2000], and sorptive barrier design [Matott et al., 2006]. HD-DDS application to new
13
problem types like these requires users consider whether changes are necessary in the two local
14
search types defined here. In the most general interpretation of L1 and L2, HD-DDS is a general
15
methodology that is not specific to WDS problems. L1 enumerates all solutions that differ from
16
the current best solution by a single decision variable (for WDS design this can be very efficient).
17
L2 enumerates all solutions that differ from the current best solution by only two decision
18
variables (again, for WDS design this can be very efficient). Furthermore, for design problems
19
with more complex constraint sets than those considered in this paper, Deb [2000] shows how
20
normalizing all constraint violations enables the use of his constraint handling technique.
21
It is important to note that the benchmark WDS case studies solved here (which minimize
22
cost subject to some design constraints) are gross simplifications of real-world WDS design
23
problems. Walski [2001] discusses why real world problems need to be solved by minimizing net
24
benefits and thus considering multiple different objectives. Our current work is extending the
25
HD-DDS methodology to multi-objective optimization benchmarks that are better representations
26
of real-world WDS design problems. Now that HD-DDS has been shown to be effective relative
27
to benchmark single-objective optimization algorithms, future studies should focus on the 29
1
application of HD-DDS to real-world WDS design problems as formulated by practicing
2
engineers designing the system. In such a study, we would expect that the one-pipe and/or
3
especially the two-pipe change local searches could be modified or replaced with alternate and
4
perhaps case study specific local search strategies that replicate the logic practicing engineers
5
employ when evaluating alternative system designs by trial and error. For example, the two-pipe
6
change local search could be replaced with the grouping method search strategy that Gessler
7
[1985] suggested for large real-world WDS design problems. A more promising and incredibly
8
efficient deterministic local search strategy that could replace or even precede L2 in HD-DDS for
9
polishing discrete DDS solutions, especially for case studies with hundreds of decision variables,
10
is the cellular automaton network design algorithm (CANDA) for WDS optimization introduced
11
by Keedwell and Khu [2006]. CANDA enables expert designers to encapsulate their case study
12
knowledge in the optimization procedure.
13
The HD-DDS algorithm does not require practicing WDS engineers to experiment and
14
identify good optimization algorithm parameters and instead gives them a robust optimization
15
tool with which they can experiment and solve multiple optimization problems that have different
16
design problem characteristics that practicing WDS engineers are ultimately interested in. These
17
would include different design constraint sets, different objectives and different future scenarios
18
leading to different nodal demand scenarios. An additional benefit of HD-DDS is that it scales
19
the search strategy to the user input computational budget.
20
Matlab source codes for HD-DDS are available by emailing the first author and will
21
eventually be available at http://www.civil.uwaterloo.ca/btolson/softare. The EPANET2 input
22
files of the lowest cost solutions found by HD-DDS for each of the five case studies in this paper
23
are available by emailing the first author. Future researchers can replicate our case studies
24
exactly by using these input files in conjunction with the EPANET2 Programmer's Toolkit.
25
30
1
Acknowledgements
2
This research was supported with funding from Dr. Tolson’s NSERC Discovery Grant. We
3
thank the reviewers of this manuscript for their insightful comments which definitely improved
4
the presentation of our findings.
5 6
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34
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4
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12 13 14
35
1 2
Table 1. Characteristics of various optimization algorithms recently applied to WDS optimization.
3 Optimization Algorithm
Application Reference
GA (GENOME)
Reca and Martinez [2006] Montalvo et al. [2008] Eusuff and Lansey [2003] Geem [2006] Reca et al. [2008] Suribabu and Neelakantan [2006]
PSO variant SFLANET HS MSATS7 PSO MMAS ACO CE HD-DDS
4 5 6 7 8 9 10 11 12 13
Zecchin et al. [2007] Perelman and Ostfeld [2007] This study
# of Reported Algorithma + penalty functionb parameters
Do algorithm parameter sets vary by case study?d
Do authors report parameter values determined by case study specific experimentation/optimization?d
8
NO
YES (for 1 of 3 case studies)
8
YES
YES
6
YES
YES
5 5
YES NOc
YES YES
5
YES
YES
4
YES
YES
3
YES
NO
1
NO
NO
a) Algorithm parameter counts do not include stopping criteria for algorithms (such as max. number of generations, max. number of objective function evaluations) since these can be specified based on project specific timelines for when a result is needed. b) All studies (this one excluded) report using a standard penalty function. All studies with a penalty function report either 1 or 2 penalty parameters except Perelman and Ostfeld [2007] who do not report on the form of the penalty function used. c) The authors report that parameters were only transferred from the first to the second case study because the extreme computational burden of the second case study precluded parameter experiments. d) Usually only a subset of reported algorithm parameters are varied/optimized by case study.
14 15 16 17 18
36
1 2 3 4
Table 2. Summary of HD-DDS and other algorithm performance for the five WDS case studies investigated in this study. WDS Case Study
NYTP
HP
NYTP2
BPc
5 6 7 8 9 10 11 12 13
Algorithm (see Table 1 for references)
Objective Function Evaluationsa x number of optimization trials
% of trials with best known solution found
–MMAS ACO
50,000 x 20
HD-DDS
Best Cost (monetary units x 106) and % deviation from best known solution (in brackets) Minimum
Median
Maximum
60
38.638 (0.0)
38.638 (0.0)
39.415 (2.0)
50,000 x 50
86
38.638 (0.0)
38.638 (0.0)
38.769 (0.3)
PSO variant
80,000 x 2000
30
38.638 (0.0)
38.83 (0.5)
47.0b (21.6)
–MMAS ACO
100,000 x 20
0
6.134 (0.9)
6.386 (5.0)
6.635 (9.1)
HD-DDS
100,000 x 50
8
6.081 (0.0)
6.252 (2.8)
6.408 (5.4)
PSO variant
80,000 x 2000
5
6.081 (0.0)
6.31 (3.8)
6.55b (7.7)
CE
97,000 x 1
-
6.081 (0.0)
-
-
GENOME GA
150,000 x 10
10
6.081 (0.0)
6.248 (2.7)
6.450 (6.1)
HD-DDS
150,000 x 50
2
6.081 (0.0)
6.260 (2.9)
6.393 (5.1)
–MMAS ACO
300,000 x 20
5
77.275 (0.0)
78.199 (1.2)
79.353 (2.7)
HD-DDS
300,000 x 20
85
77.275 (0.0)
77.275 (0.0)
77.434 (0.2)
GENOME GA
10,000,000 x 10
0
2.302 (18.7)
2.334 (20.3)
2.35 (21.1)
HD-DDS
100,000 x 10
0
2.099 (8.2)
2.165 (11.6)
2.212 (14.0)
MSATSd
45,000 x 1
-
3.298 (69.9)
-
-
HD-DDS
10,000 x 10
0
2.660 (37.0)
2.759 (42.2)
2.897 (49.3)
a) Unlike all other algorithms, the majority of HD-DDS objective function evaluations do not require evaluating the hydraulics with EPANET2. See discussion in section 4. b) Based on 100 (not 2000) optimization trials (Figure 3 for HP, Figure 5 for NYTP in Montalvo et al. [2008]). c) The best known solution to the Balerma network based on 30 million objective function evaluations is 1.9409 million Euro (see section 3.2.1). d) MSATS (mixed simulated annealing tabu search) is the best of 4 metaheuristics on this problem from Reca et al. [2008].
37
1 2
Figure 1. Discrete Dynamically Dimensioned Search (Discrete DDS) algorithm.
3 4 5 6 7
Figure 2. Example discrete DDS probability mass functions for candidate option numbers (xinew) for a single decision variable with 16 possible options (A and B) and 6 possible options (C and D) under various values for xibest. Default Discrete DDS r parameter of 0.2.
8 9 10 11 12 13 14
Figure 3. Evaluating the objective function in HD-DDS. Note that x is the new solution to be evaluated, xbest is the current best solution, cost(x) calculates the cost of the network based on the diameter and length of pipes, F(x) is the objective function, H(x) is the summation of pressure violations at all nodes in the network, hi(x) is the head at node i, himin is the minimum required head at node i and xmax is the solution with all pipes at their maximum diameter.
15 16 17
Figure 4. The Hybrid Discrete Dynamically Dimensioned Search (HD-DDS) algorithm.
18 19
Figure 5. Progress as of the end of each step of the HD-DDS algorithm (see
20 21 22 23 24 25 26
Figure 4) versus average number of function evaluations for various WDS case studies and corresponding total computational budget input to HD-DDS. 50 optimization trials are shown for each case study except for NYTP2 where only 20 optimization trials are shown. The numbers in brackets count the number of trials where the best solution so far is equal to the best known solution.
27 28 29 30 31 32 33
Figure 6. Empirical CDF of best solutions from HD-DDS and MMAS ACO algorithm for the A) NYTP, B) HP and C) NYTP2 for approximately the same number of objective function evaluations (see the figure for Mtotal in brackets). 20 optimization trials are shown for MMAS ACO. 50 optimization trials for NYTP and HP HD-DDS results are shown. 20 optimization trials for NYTP2 are shown.
34 35 36 37 38
Figure 7. HD-DDS performance with different computational budgets (Mtotal) compared to other algorithm performance on Balerma network. HD-DDS results show all 10 optimization trials. Other algorithm results are for a single trial or show the range of results from multiple trials.
39 40 38
1
39
Tolson, B. A., M. Asadzadeh, H. R. Maier, and A. Zecchin (2009), Hybrid discrete dynamically dimensioned search (HD-DDS) algorithm for water distribution system design optimization, Water Resources Research, 45, W12416, doi:10.1029/2008WR007673. STEP 0. Define discrete DDS inputs which are as follows: • maximum number of objective function evaluations, M • neighborhood perturbation size parameter, r (0.2 is default) • vector with number of discrete options for all D decision variables, xmax. Note that xmin = [1, 1, …, 1] • initial solution, x0 = [x1, …, xD], respecting decision variable bounds STEP 1. Set objective function evaluation counter to 1, i = 1, and evaluate objective function F at initial solution, F(x0): • Fbest = F(x0), and xbest = x0 STEP 2. Randomly select J of the D decision variables for inclusion in neighborhood, {N}: • calculate probability each decision variable is included in {N} as a function of i: P(i) = 1–ln(i)/ln(M) • FOR d = 1, … D decision variables, add d to {N} with probability P(i) • IF {N} empty, select one random d for {N} STEP 3. FOR j = 1, …, J decision variables in {N}, perturb xjbest by sampling from a discrete probability distribution. This discrete distribution approximates a normal probability distribution as follows: • Sample a standard normal random variable, N(0,1) • xjnew = xjbest + σjN(0,1), where σj = r(xjmax - xjmin) • IF xjnew < (xjmin – 0.5), reflect perturbation at xjmin – 0.5: xjnew = (xjmin - 0.5) + ([xjmin - 0.5] - xjnew) = 2xjmin - xjnew - 1 IF xjnew > (xjmax + 0.5), set xjnew = xjmin • IF xjnew > (xjmax + 0.5), reflect perturbation at xjmax + 0.5: xjnew = (xjmax + 0.5) - (xjnew - [xjmax + 0.5]) = 2xjmax - xjnew + 1 IF xjnew < (xjmin - 0.5), set xjnew = xjmax • Round xjnew to the nearest integer representing the discrete option number • IF xjnew = xjbest, sample xjnew from a discrete uniform probability distribution, U(xjmax, xjmin), until xjnew ≠ xjbest STEP 4. Evaluate F(xnew) and update current best solution if necessary: • IF F(xnew) ≤ Fbest, update new best solution: Fbest = F(xnew) and xbest = xnew STEP 5. Update objective function evaluation counter, i = i+1, and check stopping criterion: • IF (P(i) < 1/D) OR IF (i = M), STOP, save Fbest & xbest • ELSE, set xnew = xbest, and go to STEP 3
1
Probability
0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
A) xibest = 8
1
0.30
3
4
best
B) xi
0.25 Probability
2
5 6 7 8 9 10 11 12 13 14 15 Option # for Decision Variable xi
= 14
0.20 0.15 0.10 0.05 0.00
0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00
2
3
4
5 6 7 8 9 10 11 12 13 14 15 Option # for Decision Variable xi
C) xibest = 4
1 2 3 4 5 6 Option # for Decision Variable xi
Probability
Probability
1
0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00
D) xibest = 5
1 2 3 4 5 6 Option # for Decision Variable xi
2
Evaluate cost(x) IF [cost(x) > cost(xbest) AND H(xbest) = 0] F(x) = cost(x) (no EPANET run) ELSE evaluate H(x) by running EPANET, then
H( x ) =
∑ max[0,
# nodes i =1
himin − hi ( x )
]
IF H(x) = 0, F(x) = cost(x) IF H(x) > 0, F(x) = cost(xmax) + H(x)
3
STEP 0. Define case study and algorithm inputs: • maximum number of objective function evaluations, Mtotal • current available computational budget, M, as M=Mtotal • network inputs (layout, available pipe diameters, pipe costs, etc.) • vectors of decision variables (pipe diameter option numbers) for the best solutions found so far, xAbest and xBbest defined initially as empty sets, xAbest = [ ], xBbest = [ ] STEP 1. Perform global search with discrete DDS: • Run the Discrete DDS with M as function evaluation limit • Return the best solution of this step, xbest and the corresponding objective function value, F(xbest) • Return the computational effort in this step, m, and update the available computational budget M = M-m STEP 2. Perform a fast local search, L1, which changes current best solution by only one pipe at a time: • Run L1 initialized at xbest • Check if update needed for best solution, xbest, and F(xbest) • Return the computational effort in this step, m, and update the available computational budget M = M-m • IF xAbest is empty, set xAbest = xbest • ELSE IF xBbest is empty set xBbest = xbest • IF M ≤ 0, STOP HD-DDS • ELSE IF xBbest is not empty Go to STEP 4 • ELSE, Go to STEP 3 STEP 3. Perform the second independent global search with discrete DDS followed by L1: • Go to STEP 1. STEP 4. Perform a slower local search, L2, which changes current best solution by only two-pipes at a time: • Run L2 initialized at the best of xAbest and xBbest (this is referred to as L2') • Check if update needed for xAbest, xBbest, xbest, and F(xbest) • Return the computational effort in this step, m, and update the available computational budget M = M- m • IF M ≤ 0 or xBbest = xAbest STOP HD-DDS • ELSE, Go to STEP 5 STEP 5. Perform another L2 local search: • Run L2 initialized at the worst of xAbest and xBbest (this is referred to as L2'') • Check if update needed for xAbest, xBbest, xbest, and F(xbest) • Return the computational effort in this step, m and update the available computational budget M = M- m STOP HD-DDS • Return xbest as the best of xAbest and xBbest, corresponding F(xbest) and total objective functions evaluated (Mtotal – M) • Report if xbest is local optimum with respect to L1 or L2.
4
6.8 6.7
STEP1 STEP2 STEP3 STEP4 STEP5 Average
C) HP, Mtotal = 100,000
6.6 6.5 6.4 6.3 6.2 6.1 (0) 6.0 70000
(0)
(4)
(4)
80000 90000 100000 Average number of function evaluations
Objective function (Cost $×10 6)
99 97 95 93 91 89 87 85 83 81 79 77 (0) 75 220000
Objective function (Cost Won×106)
Objective function (Cost $×10 6)
Objective function (Cost $×10 6)
50 A) NYTP, Mtotal = 50,000 STEP1 49 STEP2 48 STEP3 47 STEP4 46 STEP5 Average 45 44 43 42 41 40 39 (4) (5) (29 (43) 38 28000 33000 38000 43000 48000 Average number of function evaluations
B) NYTP2, Mtotal = 300,000
(0) (13)
STEP1 STEP2 STEP3 STEP4 STEP5 Average
(17)
240000 260000 280000 300000 Average number of function evaluations
320000
178.0 D) GYP, Mtotal = 10,000 STEP1 177.9 STEP2 177.8 STEP3 STEP4 177.7 STEP5 177.6 Average 177.5 177.4 177.3 177.2 177.1 177.0 (2) (2) (15) (16) 176.9 7000 8000 9000 10000 11000 Average number of function evaluations
5
Probability of equal or better solution
1.0
A) NYTP, Mtotal = 50,000
0.9 0.8 0.7 0.6 0.5 0.4 0.3
HD-DDS
0.2
MMAS
0.1 0.0 38.5
38.6
38.7
38.8
38.9
39.0
39.1
39.2
39.3
39.4
39.5
6
Probability of equal or better solution
Objective function (Cost $×10 )
1.0
B) HP, Mtotal = 100,000
0.9 0.8 0.7 0.6 0.5 0.4
HD-DDS
0.3 0.2
MMAS
0.1 0.0 6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6
Probability of equal or better solution
Objective function (Cost $×10 ) 1.0 0.9
C) NYTP2, Mtotal = 300,000
0.8 0.7 0.6 0.5 0.4 0.3
HD-DDS
0.2
MMAS
0.1 0.0 77.0
77.5
78.0
78.5
79.0
79.5
80.0
6
Objective function (Cost $×10 )
6
3.8
HD-DDS (1,000) 6
Objective Function (Cost €×10 )
3.6
Reca et al. [2007]
HD-DDS (10,000)
3.4
HD-DDS (100,000)
3.2
HD-DDS (1,000,000) HD-DDS (10,000,000)
3.0
Best HD-DDS + two-pipe
2.8
MSATS
2.6
GENOME GA
2.4
Reca and Martinez [2006]
2.2 2.0 1.8 1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
Number of objective function evaluations
7
Start L1 at feasible solution xini with maximum of M objective function evaluations. Initialize the following: o current internal best solution, xbest = xini o previous internal best solution to empty set, xpbest = [ ] o iteration (objective function evaluation) counter, i = 0 WHILE L1 is not converged (xbest xpbest) : o xpbest = xbest o FOR j = 1 to D decision variables o xtest = xbest o WHILE pipe j can have diameter decreased from xjbest (and thus a reduced cost): Decrease the diameter of pipe j by 1 discrete option, xjtest = xjbest-1 Evaluate objective function, F(xtest) IF xtest is feasible Update xbest by xtest Update the iteration counter, i = i+1 IF i = M, STOP L1 BREAK inner loop when xtest is infeasible because smaller pipe j would also be infeasible STOP L1: Return xbest, F(xbest), i, and whether L1 is converged to a local solution such that no further improvement is possible by changing one pipe at a time.
Outline of fast local search L1 which can identify a local minimum such that that no further improvement is possible by changing one pipe at a time.
Start L2 at feasible solution xini with maximum of M objective function evaluations. Initialize the following: o current internal best solution, xbest = xini o previous internal best solution to empty set, xpbest = [ ] o iteration (objective function evaluation) counter, i = 0 WHILE L2 is not converged (xbest xpbest) and computational budget is not exceeded (i < M): o xpbest = xbest o R = max(x1best, x2best, … , xDbest), which is the maximum number of pipe diameter reductions to consider o FOR r = 1 to R x FOR j = 1 to D decision variables o xtest = xpbest o IF (xjpbest - r) > minimum diameter option number of pipe j (and thus a reduced cost) FOR k = 1 to D o xtest = xpbest o xjtest = xjpbest – r o IF k j evaluate increased diameters for pipe k starting at largest diameter Set xktest = xkmax + 1 WHILE xktest can have diameter decreased by one option and xktest > xkpbest + 1: o Decrease the diameter of pipe k by 1, xktest = xktest – 1 o Calculate cost(xtest). No EPANET run. Note: xtest differs from xpbest in the jth and kth pipes only o IF cost(xtest) < cost(xbest) x Evaluate objective function, F(xtest) x IF xtest is feasible o Update xbest by xtest x Update the iteration counter, i = i+1 x IF i = M, STOP L2 x BREAK inner loop when xtest becomes infeasible because smaller diameters for pipe k would also be infeasible STOP L2: Return xbest , F(xbest), i, and whether L2 is converged such that no further improvement is possible by changing two-pipes at a time.
Outline of local search L2 for constrained WDS optimization problem that can identify a local minimum such that no further improvement is possible by changing two decision variables (pipes) at a time.
