PUBLICATIONS. Water Resources Research. A new selection metric for multiobjective hydrologic model. calibration RESEARCH ARTICLE 10.

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1 PUBLICATIONS Water Resources Research RESEARCH ARTICLE Key Points: New convex hull-based selection metric for multiobjective model calibration Biobjective model calibration Pareto front often approximated by convex curve CHC improves PADDS performance for MO hydrologic model calibration problems Correspondence to: M. Asadzadeh, Citation: Asadzadeh, M., B. A. Tolson, and D. H. Burn (2014), A new selection metric for multiobjective hydrologic model calibration, Water Resour. Res., 50, , doi: / 2013WR A new selection metric for multiobjective hydrologic model calibration Masoud Asadzadeh 1, Bryan A. Tolson 1, and Donald H. Burn 1 1 Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Canada Abstract A novel selection metric called Convex Hull Contribution (CHC) is introduced for solving multiobjective (MO) optimization problems with Pareto fronts that can be accurately approximated by a convex curve. The hydrologic model calibration literature shows that many biobjective calibration problems with a proper setup result in such Pareto fronts. The CHC selection approach identifies a subset of archived nondominated solutions whose map in the objective space forms convex approximation of the Pareto front. The optimization algorithm can sample solely from these solutions to more accurately approximate the convex shape of the Pareto front. It is empirically demonstrated that CHC improves the performance of Pareto Archived Dynamically Dimensioned Search (PA-DDS) when solving MO problems with convex Pareto fronts. This conclusion is based on the results of several benchmark mathematical problems and several hydrologic model calibration problems with two or three objective functions. The impact of CHC on PA-DDS performance is most evident when the computational budget is somewhat limited. It is also demonstrated that 1,000 solution evaluations (limited budget in this study) is sufficient for PA-DDS with CHC-based selection to achieve very high quality calibration results relative to the results achieved after 10,000 solution evaluations. Received 28 OCT 2013 Accepted 10 AUG 2014 Accepted article online 13 AUG 2014 Published online 3 SEP Introduction Multiobjective (MO) automatic calibration of hydrologic models has been an active research topic since Gupta et al. [1998] demonstrated a case study with two conflicting calibration objectives (simulation error metrics). Efstratiadis and Koutsoyiannis [2010] provide a thorough review of MO hydrologic model calibration literature and separate this literature between studies employing a pure Pareto-based approach, where a set of nondominated solutions is identified via MO optimization, and studies that aggregate multiple calibration objectives, where a unique compromise parameter set is identified on the basis of multiple criteria embedded in a scalar performance function. In model calibration exercises employing a pure Pareto-based approach, the modeller often must select a single good compromise solution from the set of nondominated solutions and methods for doing so are also discussed in Efstratiadis and Koutsoyiannis [2010]. In general though, this selection process is guided by the desire to achieve a good balance between multiple calibration objectives by marginally sacrificing all objective functions from their individual optima. This is easy to achieve for Pareto fronts that are bent or kinked in their central region close to the ideal point and are extended in an approximately linear fashion in the tails toward the individual optima. Figure 1a represents a set of nondominated points in the objective space of an example biobjective minimization problem with this desired behavior. As shown in Figure 1a, this type of Pareto front can be accurately approximated by a convex piecewise linear curve connecting a subset of nondominated points. However, in a calibration problem without this desirable characteristic, the convex approximation of the Pareto front may partially or totally misrepresent the Pareto front as in Figures 1b and 1c, respectively. For a general MO problem, the Pareto front shape is unknown before solving the problem. In contrast, for MO hydrologic model calibration problems, and in particular those that are well posed with a reasonable quality model and measured data set, we believe the Pareto front shape is usually convex or nearly convex in that the front can be closely approximated by convex piecewise linear curves or hyperplanes (see Figure 1a). This belief stems from the fact that there is only a single Pareto front solution for a MO calibration problem using a perfect hydrologic model (no structural errors, capable of exactly simulating all hydrologic processes), driven with error-free forcing data and calibrated to error-free observed data. While this ideal scenario is impossible to achieve in real calibration problems, as research improvements continue to push ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7082

2 Figure 1. Three sets of nondominated points in the objective space of example biobjective minimization problems and the Convex approximation of Pareto fronts. (a) convex curve accurately approximates the Pareto front, (b) convex curve misrepresents a portion of the Pareto front, (c) convex curve does not represent the whole Pareto front. modeling toward this ideal scenario, it becomes increasingly likely that the true Pareto front looks more like Figure 1a (largely convex) rather than Figure 1c. Studies by Xia et al. [2002], Fenicia et al. [2007b], and Lee et al. [2011] support our contention in that they show what happens to the Pareto front as models are improved. Xia et al. [2002] empirically demonstrated that more advanced or complex models result in a Pareto front with a smaller extent in objective space that are closer to the ideal point (no error) in the objective space. Also Fenicia et al. [2007b] noted that hydrologic model improvement can be identified when the Pareto front progressively moves toward the ideal point. They also showed that when the model simulates reality very precisely, the Pareto front between the simulation error metrics shrinks and therefore the final solution becomes less dependent on the error metric selection (objective function). Likewise, Lee et al. [2011] studied the impact of using more advanced models on modeling hydrologic events by calibrating a simple and an advanced model to minimize two different simulation error metrics for streamflow. They showed that using the more advanced model shrinks the Pareto front and moves it significantly toward the ideal point. Another interesting study by Kollat et al. [2012] provides insight into typical Pareto front shapes in MO calibration problems. Kollat et al. [2012] argued that meaningful conflict between error metrics happens in the presence of structural deficiencies in hydrologic models. They calibrated two different hydrologic models (two different levels of model complexity) for 392 catchments across the United States in a four-objective automatic calibration framework. They showed that, with proper numerical precision levels of the objective function values, 55% of MO model calibrations of the less complex model and 80% for the more complex model collapsed to 10 or fewer solutions on the final Pareto front meaning that, with appropriate numerical precision levels, the conflict between the four objective functions almost disappeared. They also showed that conflicting objectives in other calibration tasks are a sign of some structural error in the hydrologic model. So even if objective functions with full numerical precision are implemented, good quality hydrologic model calibrations are expected to have a Pareto front with a sharp bend in the middle and long tails that would disappear if proper numerical precision levels of objective functions are considered. The expectation of a convex or nearly convex Pareto front in MO hydrologic model calibration problems is also supported empirically based on dozens of studies in the literature. For example, of the 16 pure Paretobased MO calibration studies reviewed in Efstratiadis and Koutsoyiannis [2010, Table 1] depicting the Pareto front between two objectives, 15 of them show a Pareto front that is well approximated by a convex piecewise linear curve similar to that in Figure 1a [see Madsen, 2003, Vrugt et al., 2003a, Khu and Madsen, 2005, Schoups et al., 2005a, 2005b, Engeland et al., 2006, Bekele and Nicklow, 2007, Confesor and Whittaker, 2007, De Vos and Rientjes, 2007, Fenicia et al., 2007a, 2007b, Parajka et al., 2007, Tang et al., 2007, Khu et al., 2008, Moussa and Chahinian, 2009]. In addition to these studies, a quick, nonexhaustive search yielded nine additional studies [Yapo et al., 1998, Madsen, 2000, Xia et al., 2002, Tang et al., 2006, Pokhrel and Gupta, 2010, Zhang et al., 2010, Lee et al., 2011, Moussu et al., 2011, Pokhrel et al., 2012] showing a two-dimensional Pareto front that is well approximated by a convex piecewise linear curve. It is important to note that this empirical ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7083

3 evidence is limited to observing only biobjective Pareto fronts and we have not assessed the convexity of Pareto fronts in three or more dimensions. Coello [2001] noted that general MO algorithms such as NSGAII [Deb et al., 2002] and SPEA-2 [Zitzler et al., 2001] are insensitive to the shape of the Pareto front and therefore can be applied to general MO problems because the Pareto front shape is often unknown before solving a general MO problem. However, some studies including Feng et al. [1997], Cococcioni et al. [2007], and Chang et al. [2010], empirically demonstrated that designing MO algorithms with the knowledge that the given MO problem has a known, expected, or desired convex Pareto front can be beneficial for solving such problems. Feng et al. [1997] introduced a selection operator for MO algorithms specialized for solving biobjective optimization problems with convex Pareto front and solved the Time-Cost Trade-off Problem that aims to minimize the time and cost to complete the activities of a construction project and is expected to have a convex Pareto front. After each generation, the best piecewise linear convex polyline (connecting curve) of the population of solutions is formed in objective space and solutions closer to this convex polyline are given more chance to be selected as the parents for the new generation. Chang et al. [2010] applied this MO algorithm to solve a biobjective water allocation problem for minimizing water deficit in irrigation and public water sectors and showed that the final Pareto front has a convex shape. Also, Cococcioni et al. [2007] introduced Convex Hull Evolutionary Algorithm (CHEA) for solving a biobjective optimization problem where only solutions whose map to the objective space lies on the convex polyline (connecting curve) of nondominated solutions are desirable while the other nondominated solutions can be considered less valuable. MO hydrologic model calibration problems are almost exclusively solved by general MO algorithms that are designed to be insensitive to the shape of the Pareto front. Example popular MO algorithms are MO versions of the complex evolution algorithm in the studies by Yapo et al. [1998] and Vrugt et al. [2003a], NSGAII in the study by Khu and Madsen [2005], epsilon NSGAII and its parallel version in the studies by Tang et al. [2006, 2007], MOPSO in the study by Gill et al. [2006] and AMALGAM in the study by Zhang et al. [2010]. Other general MO algorithms recently applied to MO hydrologic model calibration include PA-DDS in the study by Asadzadeh and Tolson [2013] and BORG in the study by Hadka and Reed [2013]. The purpose of this study is to investigate if MO optimization algorithms can be enhanced by taking into account the convex or nearly convex shape of Pareto fronts that regularly occur in MO hydrologic model calibration. In this study, a novel selection metric called Convex Hull Contribution (CHC) is introduced in section 2.3 for solving MO problems with a known or expected convex Pareto front by MO algorithms that archive all nondominated solutions. As explained in section 2.4, CHC is implemented as the selection metric of the Pareto Archived Dynamically Dimensioned Search (PA-DDS) algorithm. Two numerical experiments are conducted in sections 3.1 and 3.2 to assess the impact of CHC on PA-DDS performance. The assessment approach explained in section 3.3 is used to statistically assess results of the two numerical experiments separately in sections 4.1 and 4.2. Also, as discussed in section 4.3, CHC can be utilized as a selection metric of MO algorithms with a bounded archive size with some minor adjustments. 2. Convex Hull in Multiobjective Optimization 2.1. Convex Hull Background Based on the definition, the convex hull of a finite set of given points ðy R m Þ in an m-dimensional space is the smallest convex set that contains all the given points [Barber et al., 1996]. A set is called convex if for each two points inside the set, all points on the line segment between them are inside the set. Figure 2 shows the convex hull (shaded area) of a set of given points (empty circles) in a two-dimensional space as the minimal (in area) convex polygon that contains all the given points. In a higher dimensional space, the convex hull forms a minimal (in size) convex shape that contains all the given points. A convex hull is bounded by its facets (dashed line segments in Figure 2) and all the given points inside the convex hull and on its facets satisfy the equation (1) where A is a l3m matrix, l is the number of convex hull facets, and m is the number of dimensions. Ax1b 0 (1) Each facet is a portion of a hyper-plane that divides the space into two half-spaces (two sides of the facet), one half-space contains the convex hull and the other one does not. Each row a i of matrix A is the unit ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7084

4 Figure 2. Convex Hull of a set of points in a two dimensional space. normal vector of facet i and points outward from the convex hull (i.e., arrows in Figure 2 that point to the half-space that does not contain the convex hull). The corresponding component of vector b is the facet s offset from the origin. Points on each facet i satisfy equation (2), points inside the convex hull satisfy equation (3) for all l facets and points outside the convex hull satisfy equation (4). X m j51 X m j51 a i; j y j 1b i 50 (2) a i; j y j 1b i < 0 8i51;...; l (3) X m j51 a i; j y j 1b i > 0 (4) Some of the given points form the vertices of the convex hull. If any of these points defining the convex hull is removed from the set of the given points, the convex hull size (area or hypervolume in two or higher-dimensional space, respectively) would decrease. However, removing any given point from inside the convex hull would not change the convex hull size Motivation of Using Convex Hull in Multiobjective Optimization Problems In any MO algorithm, selecting and perturbing a previously evaluated solution aims to find a new solution, typically in its neighborhood, that can improve the current approximation of the Pareto front. Feng et al. [1997] show that giving more selection priority to solutions that are closer to the convex approximation of the Pareto front can improve the performance of the optimization algorithm for solving MO problems with a convex Pareto front. A similar conclusion can be made from the study by Cococcioni et al. [2007] who replaced the nondominated sorting operator of NSGA-II [Deb et al., 2002] with the convex hull sorting operator and introduced the Convex Hull Evolutionary Algorithm (CHEA) for solving a biobjective optimization problem with an expected convex Pareto front. In CHEA, the first ranked solutions are those that form the convex approximation of the Pareto front of the whole population of solutions and the second ranked solutions are the ones that form the convex approximation front of the remaining solutions and so forth. After each generation, CHEA starts filling up the parent population with the first ranked convex hull solutions from the combined parents and mated solutions. The population size is fixed; therefore, if the number of first ranked solutions is more than the population size, some of them will be randomly removed from the new population. If the first ranked solutions are fewer than the population size, some second ranked convex hull solutions will be randomly added to the parent population. This process continues until the parent population is filled up. Therefore, CHEA gives the highest selection chance to vertices of the convex approximation of Pareto front. Cococcioni et al. [2007] demonstrated that CHEA performs better than NSGA-II for solving a biobjective optimization problem with convex Pareto front. ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7085

5 Goel et al. [2007] introduced a convex hull-based surface approximation of the Pareto front to better visualize the Pareto front generated by a MO algorithm for a MO problem and help the decision maker to select the compromise solution. The PAINT method in Hartikainen et al. [2012] was designed as a Pareto front interpolation tool to augment interactive multiobjective optimization methods. Interactive MO methods require the decision maker to iteratively adjust his/her preferences about the objectives to guide and adjust the Pareto front approximation in order to ultimately select a single solution [see Miettinen et al., 2008]. Our study is limited to no-preference MO methods, defined in Hwang and Masud [1979], where the decisionmaker preferences are not utilized until all optimization is complete. Although both methods in Goel et al. [2007] and Hartikainen et al. [2012] are applicable to nonconvex Pareto fronts, the methods were not developed or incorporated as a selection metric for MO optimization algorithms. Modifying these methods so they can be utilized as a selection method in MO optimization algorithms is beyond the scope of our work Convex Hull Contribution (CHC) Selection Metric A novel selection metric based on the Convex Hull concept is introduced and called Convex Hull Contribution (CHC). The calculation procedure for CHC as applied to MO algorithms is presented here for an unbounded archive size of nondominated solutions. Example MO algorithms with an unbounded archive size include those introduced in the studies by Fieldsend et al. [2003], Chen and Lee [2007], Yang [2007], Smith et al. [2008], Asadzadeh and Tolson [2009], and Kaylani et al. [2010]. In the first step of CHC calculation, the objective space is normalized by equation (5) to make CHC unbiased to various scales of objective functions. In equation (5), f i and fi N are the actual and normalized values of objective function i, respectively for all archived points in the m-dimensional objective space, and fi min and fi max are the minimum and maximum values of objective function i, respectively among all archived solutions. So the normalized objective function space changes as the optimization moves to the next iteration or generation; however, this change does not impact the selection since the metric is calculated for all archived solutions in each iteration or generation and therefore the metric represents the relative priority of each archived solution to be selected in each iteration or generation. fi N 5 f i2f min i f max i 2f min i 8i51;...; m (5) In Figure 3, a set of nondominated points in the normalized objective space of an example biobjective optimization problem is shown. The convex hull of all archived solutions, shaded area with vertical lines in Figure 3, in the normalized space is formed using the qhull code, available at: based on the work by Barberetal. [1996]. This code can measure the convex hull size (length, area, or hypervolume in one, two, or higherdimensional spaces, respectively). Also, it can identify the convex hull facets, vertices, and the unit normal outward vector for each facet. Using this information, all archived nondominated solutions are divided into the following four mutually exclusive and collectively exhaustive groups based on their position in the objective space: (i) Points inside the convex hull (ii) Vertices of top facet only (iii) Vertices of bottom facets only (iv) Vertices in the intersection of top and bottom facets Solutions in group (i) do not have any contribution to the convex hull (i.e., removing them from the set of nondominated solutions does not change the convex hull). CHC50 is assigned to these solutions and they are removed from the process of calculating CHC for other solutions. Also, CHC50 is assigned to solutions in group (ii) but they are not removed from the set of nondominated solutions in the process of calculating CHC for other solutions because removing them would change the size of convex hull. So solutions in groups (i) and (ii) become inactive archived solutions based on CHC. CHC is only calculated for solutions in group (iii). The calculation of CHC for an active solution such as x p with objective function values fx ð p Þthat represent point p in Figure 3 is shown in equation (6) where CH is the convex hull size (length, area, or hypervolume in one, two, or higher-dimensional spaces, respectively) of a set of nondominated points ND calculated by the qhull code mentioned above. So CHC is the difference in the convex hull size with and without an active solution such as x p in Figure 3. ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7086

6 Figure 3. Convex Hull Contribution (CHC) calculations for an active nondominated solution in the normalized objective space of an example biobjective minimization problem. CHCðx p Þ5CHðNDÞ2CHðND ffx ð p ÞgÞ (6) In biobjective optimization problems, solutions in group (iv) are the endpoints of the Pareto front (stars in Figure 3). In higher dimensions though, more than just endpoints of the Pareto front are at the intersection of top and bottom facets. These solutions are important for defining the convex approximation of the Pareto front; however, their contribution to the size of the convex hull (shaded gray area in Figure 3) is not consistent with that for solutions in group (iii). So instead of calculating CHC for solutions in group (iv), the CHC value of the closest active solution in the objective space from group (iii) is assigned to it. A relatively high CHC value for a solution indicates two things. First of all, the solution can be far from its neighboring active solutions in the objective space. Therefore, CHC measures the diversity of active solutions. Second, the solution can be far from the convex combination of its neighbouring solutions toward the Pareto optimal front. Therefore, CHC also measures the proximity of active solutions to the Pareto optimal front. A good characteristic of CHC is that it avoids extensively sampling from solutions on the extended tails of the Pareto front (i.e., near vertical or near horizontal lines in the biobjective space or hyperplanes in problems with more than two objective functions) because these solutions have negligible contribution to the convex hull size. The main difference between the proposed CHC-based selection and the selection in CHEA introduced by Cococcioni et al. [2007] is that the CHC-based selection considers the contribution of solutions to the convex hull and therefore it measures both the diversity and proximity of solutions while CHEA selection operator only considers the proximity of solutions. Also, to our knowledge, CHEA has not been applied to problems with more than two objective functions while results of this study suggest that the optimization algorithm that uses CHC-based selection can be applied to problems with at least three objective functions CHC Selection for PA-DDS PA-DDS introduced by Asadzadeh and Tolson [2009] is a single-solution based heuristic MO algorithm and is the MO version of DDS [Tolson and Shoemaker, 2007]. PA-DDS archives all nondominated solutions found during the search, so it does not suffer from the deterioration defined by Hanne [1999]. Based on a selection metric, PA-DDS uses the roulette wheel [De Jong, 1975] stochastic selection scheme to select one of the archived nondominated solutions per iteration and perturbs it as DDS does. If the perturbed solution is nondominated compared to all archived solutions or if it dominates an archived solution, the archive is updated to remove dominated solutions and include this recently perturbed solution, which will be perturbed in the next iteration. Otherwise, PA-DDS chooses another archived solution based on the selection metric. Interested readers are referred to Asadzadeh and Tolson [2013] for all algorithmic details of PA-DDS. In the original version of PA-DDS [Asadzadeh and Tolson, 2009], selection was based on the crowding distance metric as defined in Deb et al. [2002]; however, Asadzadeh and Tolson [2013] empirically demonstrated that selection based on hypervolume contribution (HVC), as defined in Knowles et al. [2003], significantly improves the performance of PA-DDS for solving general MO problems (i.e., Pareto fronts of any shape). In this study, CHC is implemented as the selection metric of PA-DDS for solving MO problems with a known or expected ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7087

7 convex Pareto front. This version of PA-DDS utilizes CHC to select for perturbation only the active nondominated solutions that form the convex approximation of the Pareto front. PA-DDS with HVC-based selection is used as a benchmark algorithm to assess the performance of PA-DDS with CHC-based selection. It should be noted here that the only difference between these two versions of PA-DDS is their selection metric. 3. Numerical Experiments Two numerical experiments are designed to assess the impact of CHC on the performance of PA-DDS. In the first numerical experiment, two variants of PA-DDS, with CHC or HVC-based selection, are compared for solving several mathematical MO problems with known convex Pareto optimal fronts. In the second numerical experiment, the performance of the two versions of PA-DDS is compared for solving several example multiobjective hydrologic model calibration problems Numerical Experiment 1: Mathematical Test Problems Mathematical MO test problems are designed with special characteristics to challenge MO algorithms. ZDT1 and ZDT4 introduced in the study by Zitzler et al. [2000] have two objective functions with convex Pareto optimal fronts. These two test problems with 10 decision variables are included in this experiment. A suite of MO test problems known as DTLZ with the user ability to control the number of decision variables and objective functions was introduced in the study by Deb et al. [2001]. The biobjective version of DTLZ2 (DTLZ2_M2) with 30 decision variables is given by equation (7) with the Pareto optimal front at x i 50:5 for all i52;...; 30, and therefore 11gðx m Þ51, and f1 21f [Deb et al., 2001]. As shown in Figure 4a, this Pareto optimal front is one quarter of the unit circle centered at the origin of the two-dimensional space (the objective functions in equation (7) are the formula of this circle in the polar coordinate system). So the Pareto optimal front of DTLZ2_M2 is a nonconvex curve; however, as shown in Figure 4b, if the range of x 1 is changed from [0, 1] to [2, 3] and the objective functions are moved so that the circle is centered around [1, 1], the Pareto optimal front of the modified DTLZ2_M2 problem becomes convex. f 1 ðxþ5ð11gþ cos ðx 1 p=2þ f 2 ðxþ5ð11gþ sin ðx 1 p=2þ g5 X30 i52 ðx i 20:5Þ 2 0 x i 1 for all i51;...; 30 (7) Similar changes are applied to the three-objective version of DTLZ2 (DTLZ2_M3) with the problem formulation in (8). The Pareto optimal front of DTLZ2_M3 corresponds to x i 50:5 for all i53;...; 30 and therefore 11g Figure 4. Pareto optimal front of DTLZ2 with two objective functions, (a) with original range of parameters, (b) with modified range of parameters to make the Pareto optimal front convex. ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7088

8 Table 1. SAC-SMA Parameters Name and Range, Lower and Upper Bounds as in Vrugt et al. [2003a] Name Lower Bound Upper Bound 1. UZTWM UZFWM LZTWM LZFPM LZFSM ADIMP UZK LZPK LZSK PCTIM ZPERC REXP PFREE ðx m Þ51andf1 21f 2 21f [Deb et al., 2001]. With the same logic above, DTLZ2_M3 is modified to have a convex Pareto optimal front by modifying the range of x 1 from [0, 1] to [21, 0] and x 2 from [0, 1] to [2, 3]. f 1 ðþ5 x ð11gþ cos ðx 1 p=2þcos ðx 2 p=2þ f 2 ðxþ5ð11gþ cos ðx 1 p=2þsin ðx 2 p=2þ f 3 ðxþ5ð11gþ sin ðx 1 p=2þ g5 X30 i53 ðx i 20:5Þ 2 Zhang et al. [2008] added some difficult features to DTLZ2 by rotating its decision variable space and extending its search space while adding a penalty to solutions outside the original search space. These difficulties are added to the convex version of DTLZ2_M2 and DTLZ2_M3 with 30 decision variables, and they are solved in numerical experiment 1. Li and Zhang [2009] designed MO test problems with controllable difficulty in the decision space (complexity of Pareto Solution shape). Three of these problems known as UF1, UF2, and UF3 that have convex optimal Pareto fronts are included in this numerical experiment. They have two objective functions and 30 decision variables. (8) 3.2. Numerical Experiment 2: Hydrologic Model Calibration Problems SAC-SMA: Leaf River Leaf River is a 1944 km 2 watershed located north of Collins, Mississippi and has been investigated in many single and multiobjective calibration studies including Sorooshian et al. [1993],Vrugt et al. [2003a], Tang et al. [2006],andKollat et al. [2012]. Sorooshian et al. [1993] modeled this watershed in the Sacramento Soil Moisture Accounting (SAC-SMA) to simulate the streamflow (cms) and suggested the calibration of 13 model parameters from the ranges reported in Table 1. The 13 parameters of SAC-SMA are calibrated within the parameter ranges in Table 1 to optimize two objective functions. The first objective is to maximize the Nash Sutcliffe coefficient, NSinequation(9),calculatedformeasuredm i and simulated s i flows in day i over the calibration period t. NSemphasizesonpeakflows[Gupta et al., 2009]. The second objective is to minimize the daily root mean squared error, DRMSE in equation (10), calculated for the Box-Cox transformed measured m T i and simulated s T i flows. The Box-Cox transformation is shown in equation (11), where y T is the transformed value of y and k is the power parameter of the transformation. Following Tang et al. [2006],k50:3 is used to increase the influence of low flow periods on the value of DRMSE. NS512 X t X t i51 ð m i51 i2s i Þ 2! 2 (9) m i 2 X t i51 mi t vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ux t DRMSE5t m T 2 i 2sT i (10) i51 y T 5 ðy11þ k 21 =k (11) Following Tang et al. [2006], we use a simulation period from 28 July 1952 to 30 September 1954 with model warm-up period from 28 July 1952 to 30 September 1952 and a short 2 year calibration period from 1 October 1952 to 30 September 1954 to decrease the computational demands of this experiment HYMOD: Leaf River Vrugt et al. [2003b] calibrated five parameters of a hydrologic model called HYMOD for simulating the measured flow data for the Leaf River watershed. In this study, the five parameters of the HYMOD reported in ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7089

9 Table 2. HYMOD Parameters Name and Range, Lower and Upper Bounds as in Vrugt et al. [2003b] Name Lower Bound Upper Bound 1. C max ðmmþ b exp Alpha R s ðdayþ R q ðdayþ Table 2 are calibrated in the exact same biobjective optimization problem formulation explained in the SAC-SMA calibration problem SWAT 2000: Town Brook Watershed Town Brook is a 37 km 2 subwatershed located upstream of the Cannonsville watershed, upstate New York. Tolson and Shoemaker [2007] calibrated 26 parameters, listed in Table 3, of the Soil and Water Assessment Tool version 2000 (SWAT2000) to adequately simulate flow (cms), total suspended sediment transport (kg), and total phosphorus delivery (kg) measurements in the Town Brook. They used the reduced Nash-Sutcliffe coefficient, ðrnsþ in equation (12) to measure the calibration quality where m i and s i denote the measured and simulated data at day i, respectively, and t is the total number of days in the calibration period (considered in the calculation of objective functions). Tolson and Shoemaker [2007] used a simulation period from 1 January 1996 to 30 September 2000 with a 639 day warm-up period resulting in a 1096 day calibration period (1 October 1997 to 30 September 2000). Due to the data quality and availability, the calibration period for the water quality constituents (total phosphorus delivery and sediment transport) was from 1 October 1998 to 30 September 2000 (731 days). Bias in equation (13) penalizes solutions that simulate the measured data with more bias than a specific threshold T. RNS range is ð21; 1Š and its higher value represents the better fit between X measured and simulated data (RNS51 is the perfect fit). t ð i51 RNS512 m i2s i Þ 2 X t m i2t 21X t 2 2maxð0; jbiasj2tþ (12) i51 Bias5 m i51 i X t s i51 i2 X t m i51 i X t m i51 i Tolson and Shoemaker [2007] aggregated RNS for the three measurements and solved the single objective optimization problem with a reasonably large (10,000) computational budget. T was set to 10% for the flow and 30% for the sediment and phosphorus delivery Table 3. SWAT2000 Parameters Name and Range, Lower and Upper Bounds as in Tolson and Shoemaker [2007] Name Lower Bound Upper Bound 1. SFTMP SMTMP SMFMX TIMP SURLAG APM CMN UBP PPERCO PHOSKD GW DELAY ALPHA BF GWQMN LAT TIME LAT SED ESCO ERGORGP SLSUBBSN a SLSOIL a CN2 a a 21. SOL Z a 22. SOL AWC Ksat a Clay a Rock a MUSLEadj a Decision variable value is multiplied by the default value of the corresponding SWAT parameter. (13) based on the difference in the measured data error. Asadzadeh and Tolson [2013] solved the three-objective version of this problem to maximize RNS for each of the flow, total suspended sediment, and total phosphorus data sets. This three-objective model calibration problem is solved in the second numerical experiment. Also, a twoobjective version of the problem is solved to be able to represent a more clear visual comparison of the results. Results of the three-objective version of the problem show a high correlation between maximizing RNS for total phosphorus and total sediment transport (correlation coefficient more than 77% in the aggregated results of the threeobjective version of this problem). Therefore, RNS is ignored for total sediment in the biobjective version of the problem SWAT2003: Mahantango Creek Experimental Watershed (MCEW) MCEW is a 420 km 2 benchmark research watershed located in Central Pennsylvania. It is a tributary of the Susquehanna River with 1100 mm long-term annual average rainfall. Zhang et al. [2010] modeled this watershed with 38% pasture, 34% forest, 26% mixed cropland, and 2% farmsteads in SWAT2003. Following ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7090

10 Table 4. SWAT2003 Parameters Name and Range, Lower and Upper Bounds as in Zhang et al. [2010] Name Lower Bound Upper Bound 1. SFTMP SMTMP SMFMX SMFMN TIMP ESCO SURLAG GW DELAY ALPHA BF GWQMN GW REVAP REVAPMN RCHRG DP CN2 a 80% 120% 15. CH K a 16. SOL AWC 80% 120% a Decision variable value is multiplied by the default value of the corresponding SWAT parameter. Zhang et al. [2010], the 16 parameters of the model that are listed in Table 4 are calibrated in a biobjective optimization problem to maximize the Nash Sutcliffe coefficient calculated for the daily flow at two monitoring stations named FD36 and WE38 for the simulation period from 1 January 1995 to 31 December The calibration period (for objective function calculation) is from 1 January 1997 to 31 December 1998, provided by X. Zhang (personal communication, 2012). This biobjective calibration problem is included as a case study in the second numerical experiment of this study Results Comparison Approach PA-DDS is a stochastic search algorithm and its performance varies by the initial random seed. Hence, its performance is assessed based on multiple optimization trials. Both versions of PA-DDS (with HVC and CHC-based selection) are applied to each MO problem in experiments 1 and 2 multiple times and the final result of each trial is assessed by three performance metrics introduced in section Then as explained in section 3.3.2, the first degree stochastic dominance concept is used to compare the empirical Cumulative Distribution Function (CDF) of each performance metric. Furthermore, the Wilcoxon rank sum test [Gibbons and Chakraborti, 1992] is applied to measure the statistical significance of the differences in CDFs (see section 3.3.3). Finally, the best and worst trials in each of the biobjective calibration problems are identified and compared as explained in section Each mathematical MO problem is solved 50 times with a large (25,000) and a relatively limited (2,500) number of solution evaluations. The 25,000 is considered large enough to achieve high quality results based on our experience in solving these problems. The MO model calibration problems are solved 10 times in experiment 2 and with a large (10,000) and a limited (1,000) number of solution evaluations similar to Asadzadeh and Tolson [2013]. The purpose of using the limited budget is to investigate the dependency of results to the computational budget. This is important for PA-DDS, which is designed to adjust its search strategy to the user-defined computational budget. Thus, if the relative effectiveness of selection metrics depends strongly on the computational budget, a heuristic might be required to switch between various selection metrics based on the budget and/or during the search MO Algorithm Comparison Performance Metrics The solution of a MO problem is a Pareto approximate front in the objective space and the corresponding set of decision variable values. MO performance metrics that assess the proximity and/or the diversity of a Pareto approximate front by a single number are referred to as unary performance metrics. Different unary performance metrics measure the proximity and diversity of a Pareto approximate front differently. A detailed list of MO performance metrics can be found in Coello et al. [2007]. Results of this study are evaluated using Normalized Hypervolume (NHV), Additive epsilon Indicator (e1 Indicator), and Inverse Generational Distance (IGD) unary performance metrics, which are detailed below. Hypervolume (HV) by Zitzler and Thiele [1998] is a popular performance metric that measures both proximity and diversity of a Pareto approximate front. It measures the volume of the partition of the objective space that is bounded between the Pareto approximate front and a reference point. HV is a complete unary performance metric in terms of weak dominance relation, that is, a preferred solution by HV is not weakly dominated by its opponent [Zitzler et al., 2003]. As suggested by Deb [2001], in this study, HV is calculated in the normalized objective space, equation (5), where all results are normalized to be inside the unit hypercube [0, 1] m and therefore all normalized HV values (or simply NHV) are less than or equal to 1. Higher values of NHV are desirable while lower values of the other two performance metrics are desirable. Therefore, 12NHV is reported so that lower values indicate improved performance for all three metrics. The unary performance metric additive epsilon indicator (e1 Indicator) [Zitzler et al., 2003] measures the smallest distance by which the Pareto approximate front must be shifted in the objective space to weakly dominate a reference set, which can be a subset of the Pareto optimal front. ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7091

11 Inverse Generational Distance ðigdþ was first proposed by Sato et al. [2004] as a performance metric that measures both diversity and proximity of a Pareto approximate front. IGD measures the average Euclidean distance in the objective space between each preidentified optimal point on a subset of the Pareto optimal front and its closest point on the Pareto approximate front. The publically available code developed for the CEC09 MOEA competition (see is used to calculate IGD. Also provided at the above link are the reference set of 1,000 points for UF1, UF2, and UF3 problems. This reference set of points is generated for ZDT1, ZDT4, R2_DTLZ2_2D, and R2_DTLZ2_3D in the same procedure used in CEC09 MOEA competition. IGD and e1 Indicator require the reference set that is unknown for the hydrologic model calibration problems. Okabe et al. [2004] noted that in this situation the most desired set of points should replace the unknown reference set. This reference set is created by merging results of all optimization trials for each calibration problem and identifying all nondominated points. It should be noted that IGD is not a complete performance metric (as defined in Zitzler et al. [2003]) with respect to weak dominance relation. Hence, a better IGD value for a Pareto approximate front does not guarantee that it is not weakly dominated by another Pareto approximate front. However, IGD is a popular performance metric and measures the overall distance between a Pareto approximate front and the reference set (example applications include Zhang et al. [2008]; Li and Zhang [2009]; and Hadka and Reed [2012]) and as such is utilized here along with the other performance metrics Stochastic Dominance Levy [1992] surveyed the history and application of stochastic dominance concept for comparing two random variables based on their CDFs. Although various degrees of stochastic dominance exist, the first degree stochastic dominance has been utilized in the scientific literature of optimization algorithm comparisons [e.g., Mugunthan et al., 2005; Carrano et al., 2011; Asadzadeh and Tolson, 2013]. Comparing two algorithms A and B based on CDFs, F A ðxþ and F B ðxþ of a performance metric (x) such that smaller values of x are preferred, A stochastically dominates B if and only if F A ðþf x B ðxþ for all possible values of x [Carrano et al., 2011]. When the two CDFs cross each other, first degree stochastic dominance does not hold and cannot identify the preferred algorithm result Statistical Significance Test The general form of the Wilcoxon rank sum test (see Gibbons and Chakraborti, 1992] is used to quantify the significance of the difference in pairwise CDF comparisons. This test has been used for comparing optimization algorithms in Tang et al. [2007], Hadka and Reed [2012], and Asadzadeh and Tolson [2013]. The null hypothesis of this test assumes that the two samples (i.e., performance metric values of two compared MO algorithms), A and B come from the same population such that F A ðþ5f x B ðxþ (i.e., no significant difference between algorithms A and B). The two-sided alternative hypothesis only assumes the two samples come from different populations such that F A ðþ6¼ x F B ðxþ. We utilize the first degree stochastic dominance concept and the Wilcoxon rank sum test to identify the preferable algorithm result. The algorithm A result is deemed clearly preferable to algorithm B result when A stochastically dominates B, and the p value of two-sided Wilcoxon rank sum test is smaller than 0.05 (significance level of the hypothesis test for 95% confidence level). However, when the Wilcoxon rank sum test suggests that F A ðþ6¼ x F B ðxþ and neither algorithm stochastically dominates the other (i.e., the CDFs exhibit some crossing behavior), the preferred algorithm result is not clear without additional subjective information. In such cases, the median of the two sets of data is used to determine the preferred algorithm result (i.e., median metric value critical versus the avoidance of extremely poor metric values being critical) The Best and Worst Pareto Approximate Front Comparison Each of the CDF plots ranks multiple trials of each MO algorithm from the best to the worst in each MO problem based on the corresponding performance metric. All the three performance metrics are considered equally important and therefore, the overall rank of each trial is calculated by taking the sum of the three ranks. The best and worst sum of three ranks (three performance metrics) is selected as the best and worst optimization trial, respectively. Pareto approximate fronts corresponding to the selected best trial of each MO algorithm for each biobjective calibration problem in experiment 2 are visually compared. Also, the worst trials are compared together. ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7092

12 Figure 5. UF3, empirical CDF plot based on e1 Indicator and final results (Pareto approximate fronts) of 50 independent trials of PA-DDS with CHC and HVC-based selections and with 2,500 and 25,000 solution evaluations. Conceptually, a vertical line at 0 represents perfect result. 4. Results and Discussion Results are presented in separate subsections for each of the two numerical experiments. In section 4.1, the performance of PA-DDS with CHC and HVCbased selections is compared for solving seven mathematical MO problems with known convex Pareto optimal front. In section 4.2, the performance of the two variants of PA-DDS is compared when solving hydrologic model calibration problems with expected convex Pareto front Results of Experiment 1 Both variants of PA-DDS are applied to ZDT1, ZDT4, UF1, UF2, UF3, R2_DTLZ2_2D biobjective problems and R2_DTLZ2_3D with three objective functions. Figure 5 shows a sample empirical CDF plot for e1 Indicator values, based on 50 independent trials for solving UF3. Based on this figure, at the limited budget of 2,500 solution evaluations, PA-DDS with CHC-based selection stochastically dominates PA-DDS with HVC-based selection since at any level of e1 Indicator, the former has higher probability of achieving better e1 Indicator values. But at the higher computational budget of 25,000 solution evaluations, the two CDFs cross each other and therefore neither of the two variants of PA-DDS stochastically dominates the other. In this case, the preferred selection metric is identified based on the smallest median of the performance metric values. So CHC is preferred to HVC at the higher budget in Figure 5. Similar visual CDF comparison is made for all MO problems solved in numerical Experiment 1 for all three MO performance metrics. Results of all these visual comparisons are summarized in Table 5 which includes the p value of the two sided Wilcoxon rank sum test that statistically measures the significance of the difference in the pairwise comparisons between CHC and HVC as selection metrics of PA-DDS. Table 5 shows that concentrating only on median metric values and ignoring statistical significance, CHC is preferred to HVC as the selection metric for PA-DDS in 19 out of 21 comparisons at the lower budget and 12 out of 21 comparisons at the higher computational budget. The first degree stochastic dominance analysis in Table 5. Statistical Performance Comparison of Results in Experiment 1 a Performance Metrics and Computational Budget 2,500 25,000 MOP NHV e1 Indicator IGD NHV e1 Indicator IGD ZDT1 <0.001 <0.001 < b c ZDT < <0.001 UF UF2 < UF3 <0.001 <0.001 < R2_DTLZ2_2D <0.001 <0.001 <0.001 < <0.001 b R2_DTLZ2_3D b < <0.001 a Numbers are p values of two-sided Wilcoxon rank sum test with null hypothesis that performance of PA-DDS with HVC and CHCbased selections are not different based on sample size 50. Bold p values highlight incidents that CHC is clearly preferred (CHC CDF stochastically dominates HVC CDF and p < 0.05). Underlined p values highlight incidents that HVC is clearly preferred (HVC CDF stochastically dominates CHC CDF and p < 0.05). Italicized p values highlight incidents that CHC is only preferred based on median metric value and p > b CDFs cross and CHC is preferred based on median metric value and p < c CDFs cross and HVC is preferred based on median metric value and p < ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7093

13 Table 6. Statistical Performance Comparison of Results in Experiment 2 a Performance Metrics and Computational Budget 1,000 10,000 Calibration Problem NHV e1 Indicator IGD NHV e1 Indicator IGD SAC-SMA HYMOD Town Brook_3D Town Brook_2D MCEW a Numbers are p values of two-sided Wilcoxon rank sum test with null hypothesis that performance of PA-DDS with HVC and CHCbased selections are not different based on sample size 10. Bold p values highlight incidents that CHC is clearly preferred (CHC CDF stochastically dominates HVC CDF and p < 0.05). Italicized p values highlight incidents that CHC is only preferred based on median metric value and p > 0.05 conjunction with the p values of the Wilcoxon rank sum test show that at the lower computational budget, CHC is clearly preferred to HVC in 10 out of 21 pairwise comparisons while there are no comparisons yielding HVC as the preferred selection metric. Similarly, at the higher budget, CHC is clearly preferred in 7 out of 21 comparisons and HVC is clearly preferred only in 5 out of 21 comparisons. Therefore, the preference of CHC as selection metric of PA-DDS is decreased by increasing the computational budget. The reason is that at the lower computational budget, it can be more beneficial to focus on a portion of the archived solutions (CHC) rather than to sample from all of them (HVC) Results of Experiment 2 PA-DDS with CHC-based selection is compared to PA-DDS with HVC-based selection for solving the real hydrologic model calibration problems introduced in section 3.2. Table 6 represents the result assessment summary based on the first degree stochastic dominance concept and the Wilcoxon rank sum test for all the three MO performance metrics used in this study. At the lower budget of 1,000 solution evaluations, CHC is preferred to HVC as the selection metric for PA-DDS in all 15 comparisons and 3 of these represent a statistically significant difference. At the higher budget of 10,000 solution evaluations, CHC is preferred to HVC in 11 out of 15 comparisons and one of these represents a statistically significant different results. As explained in section 3.3.4, the best and worst trials (out of 10) of both variants of PA-DDS are identified and visually compared to detect practical differences between the results of the two variants of PA-DDS. This comparison did not reveal any practically significant differences in the results of the SAC-SMA and HYMOD case studies. However as shown in Figure 6, some practically significant differences are detected in the results of SWAT2000 and SWAT2003 case studies. Panels A and B in Figure 6 show that the difference between the two variants of PA-DDS is marginal when solving the SWAT2000 case study with the relatively high computational budget. However, at the limited computational budget, PA-DDS with CHC-based selection is preferred since the corresponding Pareto approximate front is significantly closer to the aggregate Pareto front in both of the best and worst optimization trials. The aggregate Pareto front of multiple optimization trials is generated by merging all solutions obtained by those trials and filtering out the dominated solutions. Figure 6a shows that the best trial of the PA-DDS with CHC-based selection at the limited computational budget dominates the corresponding trial of PA-DDS with HVC-based selection and performs comparable to PA-DDS with HVC-based selection at the high computational budget. In the SWAT2003 case study (Figures 6c and 6d), comparing the best trials, no significant difference is observed in the results. However, the worst trial of PA-DDS with HVC-based selection at the limited computational budget performed very poorly compared to the other trials. This suggests that using CHC to focus on a portion of archived solutions can help PA-DDS escape from some poor local fronts. In summary, as shown in Table 7, CHC is preferred over HVC as the selection metric of PA-DDS in half of the comparisons made in the numerical experiments of this study. This preference is most evident when the computational budget is somewhat limited. For example, at the low computational budget, Table 7 shows CHC preferred or clearly preferred in seven of twelve comparisons while HVC is not preferred over CHC in any of the comparisons made. CHC makes PA-DDS sample only from an important subset of archived ASADZADEH ET AL. VC American Geophysical Union. All Rights Reserved. 7094