Incorporating Decision-Maker Preferences into the PADDS Multi- Objective Optimization Algorithm for the Design of Water Distribution Systems

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1 Incorporating Decision-Maker Preferences into the PADDS Multi- Objective Optimization Algorithm for the Design of Water Distribution Systems Bryan A. Tolson 1, Mohammadamin Jahanpour 2 1,2 Department of Civil and Environmental Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada, N2L3G1 2 mjahanpo@uwaterloo.ca ABSTRACT For many multiple objective (MO) water distribution system (WDS) design problems, decisionmakers know a priori some preferences that define a region of interest in objective space. For computationally intensive design problems, incorporating such preferences has the potential to save computation time if the preferences can appropriately guide the algorithm to the region of interest. MO algorithms have been specifically adapted and modified in order to incorporate such preferences. Unfortunately this complicates the original algorithms and can add additional algorithm parameters to the search. In this paper, the objective functions, rather than the algorithm, are specially adapted to incorporate preferences defining a region of interest in objective space and the Pareto Archived Dynamically Dimensioned Search (PADDS) algorithm is utilized for optimization. For two benchmark MO WDS design problems, results show the objective function adaptations work well. Results show the new formulation yields new best known Pareto front solutions to the design problems. Keywords: Water distribution system, Multiobjective optimization, preference information from decision-makers 1. Introduction The multiobjective design of water distribution systems (WDS) are challenging problems for multiobjective optimization algorithms such as multiobjective evolutionary algorithms (MOEAs). The design cost of the network is often among the multiple objectives in this context and millions of dollars could be saved taking design decisions on the basis of a superior approximate Pareto front (PF). In many cases, designers are aware of limits on objective functions like total design cost as projects can have a maximum tolerable budget for example. Conceptually, MO algorithms should be able to utilize such information to hone in and focus on returning approximate Pareto front solutions from the region of interest. This approach would ideally save computation time as well since the algorithm is not tasked with approximating all parts of the Pareto front. Wang et al. [1] set up an archive of 12 benchmark bi-objective WDS design problems from previously published WDS optimization literature. The design objectives in each benchmark problem are to minimize total pipe material costs and maximize the system resilience calculated according to Wang et al. [1]. The benchmark problems range in network size from 8 to 3032 pipes and 8 to 567 integer decision variables associated with defining the pipe diameters of some or all pipes in the network. Wang et al. [1] aimed to obtain the best-known PFs for those benchmark problems given extensive computational budgets and applied five different MOEAs, where are AMALGAM [2], Borg [3], NSGA-II [4], epsilon-moea, and epsilon-nsga-ii. Jahanpour et al [5] used Pareto archived dynamically dimensioned search (PADDS) algorithm with two different selection metrics to solve

2 the 12 water distribution system design benchmark problems. Convex hull contribution (CHC) and hypervolume contribution (HVC) were the PADDS selection metrics compared in their study. Their results augmented the previously best-known PFs in nine benchmark problems with new PF solutions to define updated best known PFs. They reported the superiority of CHC selection metric compared to HVC. Multi-objective (MO) optimization algorithm comparison studies like those noted previously almost invariably set out to approximate the entire Pareto front. Unfortunately, that is inconsistent with practical engineering design where engineers/decision-makers can typically formulate acceptable limits (i.e., preferences) on objectives such as total system cost, reliability or resilience in advance of any optimization. For especially large problems in terms of the size of decision-space, ignoring preference information from decision-makers forces MO algorithms to needlessly waste computation time discovering Pareto solutions in effectively useless regions of decision space (e.g., costs exceed realistic maximum budget for the project). Our study investigates how to best incorporate these practical decision-maker preferences into an MO optimization algorithm and then demonstrates the impact on algorithm performance in terms of Pareto front quality in the region of interest. Some researchers have investigated incorporating decision-maker preferences into the MO search process. Reynoso-Meza et al. [6] provide an overview and then comparison of techniques to incorporate preferences into multi-objective optimization. A priori methods focus on specifying preferences before any optimization in order to guide the algorithm. The most common a priori approach seems to be the construction of specialized or adapted MOEAs [7]. The main disadvantage of existing approaches is that they are add substantial complexity to existing MOEAs to define new algorithms and also can introduce additional algorithm parameters which may require tuning. This study investigates how to respect decision-maker preferences without defining a new specialized algorithm and based on the excellent results in [5] for PADDS, we demonstrate our approach utilizing the PADDS algorithm. PADDS is a single-solution based heuristic multiobjective algorithm and is the multiobjective version of DDS [7]. Interested readers are referred to Asadzadeh and Tolson [8] for all algorithmic details of PADDS. Asadzadeh et al. [9] provide the following summary description of the algorithm. PADDS archives all nondominated solutions found during the search, so it does not suffer from deterioration defined by Hanne [10] to occur for a bounded size archive if in a generation some nondominated solutions are discarded from the archive and later on in the search, some worse solutions are generated and archived. Unlike most MO algorithms, and MOEAs in particular, PADDS searches from only the current set of non-dominated solutions. A key aspect of our proposed approach is the distinct way we separate the handling of hard constraints (e.g., nodal pressure requirements) from preference-based constraints that define the region of interest in objective space. Past work on incorporating preferences handles hard constraints in the same manner as preferences [7]. Our approach builds on the constraint-handling concepts [8] built into genetic algorithms using tournament selection. The key to the approach of [8] is that the objective function is defined such that any infeasible solution always has an objective function value that is worse than the objective function value of any feasible solution. The only other requirement for the constraint handling in [8] is to quantify the relative magnitude of constraint violations for infeasible solutions so that the relative quality of two infeasible solutions can be compared.

3 Section 2 will describe our new objective function formulation and in particular, how we utilize the constraint-handling approach in [8] in an MO problem and then how it is extended for preferences. Section 3 will present results for two bi-objective design problems and conclusions will be provided in Section Objective Functions for Constrained MO Optimization with Preferences Preferences in this study are defined to be a maximum of one threshold per design objective that separates desirable solutions from undesirable solutions. For example, a limit of the allowable budget for a WDS upgrade design problem with total design cost as an objective. Constraints, hard constraints like minimum required pressures across the WDS, can be handled in MO problems following the approach in [8]. Specifically, infeasible solutions must be assigned ***. A novel, parameter-free objective function formulation is presented that accounts for preference information and then implement this for the PADDS algorithm. Based on the original of Wang [1], for a design solution X, objective functions in Eq. 1 are evaluated. 1 2 (1) In this paper, values of Cost and Resiliency are called objective function values (OFV). Logical boundaries apply to both OFVs as described in Eq (2) The new objective space (shown in Figure 1) is divided into three sub-regions: desired, feasible nondesired, and non-feasible. Desired region is defined by vector, where and are the upper boundaries for the cost and negative resiliency, respectively. is defined by user-defined limits on the two objective functions. Eq. 4 shows boundaries for OFVs inside the desired region and also boundaries for elements of Z. Where (3) The concept of desirability is not to be mistaken with the concept of feasibility. Desirability constraints are essentially soft-constraints that can be violated in the event that no feasible solutions are found that satisfy the desirability constraints. A solution is considered feasible if the model does not show any violation of constraints after simulation of the solution. Constrains include but are not limited to maintaining nodal pressures and flow velocities within certain ranges. A feasible solution can be desired or non-desired based on its objective function values. Not all feasible solutions are classified as desired. A feasible solution with objective function values outside of the desired region

4 of objectives space, classifies as a feasible non-desired solution. On the other hand, a desirable solution has to be feasible. Figure 1. Decision Space and Desired Region In the proposed decision space, R1 is the desired region; a solution in R1 has OFVs less (better) than or equal to the predefined limits (Z). R2 is the undesired yet feasible region. A feasible solution if not in R1 will be shifted to R2. As shown in Figure 1, 1 : 1 and 2 : 2. R3 is a half-line from point (1, 2Cmax) extending indefinitely, crossing point (2, 3Cmax). In the problems under consideration in this study, feasible solutions may only occur inside A:1 region. That is because of the logical fact that network resiliency ranges from 0 to 1, and that Cmax is actually the most expensive network design cost possible (obtained by simulating the network having biggest possible pipes assigned). R4 is a point located far away from (-1, 0). Solutions shift to R4 only if they are dominated or they cause EPANET solver crash. To deal with infeasibility of solutions, OFVs are shifted in the decision space. The shifts size is a function of the phase in which PADDS is. Phases are explained in the next sections. Shifts are included in the quantities PADDS is set to minimize. These quantities, called F1 and F2 in Eq. 3, could be thought of as the fitness functions. 1 2 (4) In Eq. 3, and are penalty values added to OFVs. Next, we explain how values of the shifts, which essentially serve as penalty values, are calculated based on the phases.

5 3. Phase 1 Phase one is the first phase in which the optimization initiates. Being in this phase means all solutions so far have been infeasible and thus were moved to R3 or R4. In this phase, effort is focused on finding the first feasible solution. Newly generated solutions in this phase are checked for feasibility. While in Phase 1, OFVs are always set to zero and shifts are calculated by Eq. 6: 1 2 (5) If a solution is infeasible, there is no need to calculate any of OFVs because for solution in R3, only amount of constraint violation accounts. OFVs of an infeasible solution are set to zero and the solution is shifted to R3. There will be always only one solution in R3. In the next cycles of the search, this solution will be dominated and replaced by another solution with less amount of infeasibility. This strategy makes PADDS to gradually minimize the violation and try to transfer to R2 eventually (if it ever happens). While on R3, PADDS works as a single objective optimization algorithm trying to minimize the violation value. If a solution does not violate any constraint it becomes the first feasible solution found. This would be the end of phase 1. OFVs are then calculated for the feasible solution and Phase 2 begins. It was noted that only a small portion of the budget is spent to transfer from Phase 1 to Phase 2 (i.e., to find the first feasible solution). This portion found to be always less than %0.5 of the budget in networks 1 to 10. For cases 11 and 12, which are the largest problems, this portion was 0.5% and %3.2 respectively. 4. Phase 2 Being in this phase means that the first feasible solution has already been found. A feasible solution will be inside either R2 or R1 and thus dominates its origin solution on R3 immediately. This solution becomes the origin of the rest of the search. Since it is a feasible solution, it will dominate any next solution which is not feasible. So while in Phase 2, if a new solution is infeasible there is no need to calculate any of its OFVs; it will be discarded. If a new solution is feasible then it should be investigated whether it stays in R1 or should shift to R2. OFVs for the new solutions need to be evaluated one by one. If any OFV is undesired, below check is performed: If there is solution in R1 already, the new solution should be discarded as it will be however dominated by the solution in R1. If there is no solution in R1 yet, the other OFVs need to be evaluated and then the solution will be placed on R2. 1 (6) In case all OFVs are desired then the solution is an R1 solution and this is end of Phase Phase 3 Being in Phase 3 means that there exists at least one feasible solution with OFVs inside the desired boundaries. Phase 3 is the final phase. While in this phase, any new solution that is infeasible or undesired is discarded. Therefore, as soon as any piece of evidence for infeasibility or being undesired is found, there is no need to evaluate further OFVs. Solutions in this phase do not shift.

6 Figure 2. Flowchart for solution evaluation process 6. Results The proposed objective space formulation is used to resolve the 2 benchmark design problems for example decision-maker preferences using the PADDS algorithm. The computational budget (number of maximal solution evaluations) used was equal to the one in [1] and [5]. Table 1 summarizes the problem specifications and PADDS computational budgets used in this study for each benchmark problem. Network # of PADDSDV range SS size NSE Cost max DVs TRN 8 [0, 9] EXN 567 [0, 10] DV = decision variables; SS = search space; NSE = number of solution evaluations (computational budget); CE = cost epsilon; RY = resiliency epsilon; Cost max = maximum cost in millions of dollars corresponding to the extreme design (highest cost decision variable option selected). Table 1. Specifications for benchmark design problems.

7 The decision-maker preferences were defined to enclose the knee part of the best PFs from [5]. This is based on the general strategy that the desirability of solution is highest when it is a minimum distance from the Utopia point (point with all objective functions at their minimum). Since all the problems have convex PFs, the solutions on the knee part of PF are the closest ones to the Utopia. In this work, there were 2 network problems and 2 PADDS algorithms and 2 computational budget groups resulting in 12 2=24 sets of algorithm results, each containing 30 optimization trials. Each of the 24 sets of optimization trials was post-processed as follows. First, all obtained solutions of the trial set were rounded according to the epsilon precisions (mentioned in Table 1). Second, solutions from all 30 trials were merged into one list of unique and non-dominated solutions. The remaining solutions form an approximate aggregate PF for the respective set of optimization trials. Results in Fig. 1 and Fig 3 demonstrate that the formulation is robust for most example decisionmaker preferences. For example, for the smaller networks, the best known Pareto Front (PF) solutions are also identified with the new formulation in the desired region. Furthermore, this new formulation enhances the algorithm results (for an equivalent computational budget) for the larger network problems. For example, this new formulation generates new non-dominated solutions relative to the solutions available in the aforementioned comparison studies for these benchmarks. Figure 3 shows new dominating solutions from solving network EXN, which is the largest benchmark problem, with the proposed objective-restricted PADDS with CHC and HVC selection metrics compared to the best known PF taken from [5]. All solutions obtained by objectiverestricted PADDS-CHC (Figure 1), were nondominated and augmented the best-known PF for EXN network. Figure 3. Objective-Restricted PADDS-CHC Versus the Best Known PF from [5]

8 7. Conclusion The new formulation for PADDS works very well in initial tests. The approach yields new best known tradeoff solutions in an example network that has been optimized hundreds of times with seven other MO algorithms. Nonetheless, future work is required to confirm the performance advantage extends to other problems and then compare to other MOEAs with preference-handling built in. 8. References [1] Q. Wang, M. Guidolin, D. Savic and Z. Kapelan, "Two-Objective Design of Benchmark Problems of a Water Distribution System via MOEAs: Towards the Best-Known Approximation of the True Pareto Front," J. Water Resour. Plann. Manage., vol. 141, (3), pp , 03/01; 2015/12, [2] J. A. Vrugt and B. A. Robinson, "Improved evolutionary optimization from genetically adaptive multimethod search," Proceedings of the National Academy of Sciences, vol. 104, (3), pp , January 16, [3] D. Hadka and P. Reed, "Borg: An auto-adaptive many-objective evolutionary computing framework," Evol. Comput., vol. 21, (2), pp , [4] K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," IEEE Transactions on Evolutionary Computation, vol. 6, (2), pp , [5] M. Jahanpour, B. Tolson and J. Mai, "Improving Best-Known Pareto Fronts for Water Distribution Systems Benchmark Design Problems Using PADDS," J. Water Resour. Plann. Manage., vol. Under Review, [6] G. Reynoso-Meza, V. H. Alves Ribeiro and E. P. Carreño-Alvarado, "A Comparison of Preference Handling Techniques in Multi-Objective Optimisation for Water Distribution Systems," Water, vol. 9, (12), pp. 996, [7] G. Reynoso-Meza, J. Sanchis, X. Blasco and S. García-Nieto, "Physical programming for preference driven evolutionary multi-objective optimization," Applied Soft Computing, vol. 24, pp , [8] K. Deb, "An efficient constraint handling method for genetic algorithms," Comput. Methods Appl. Mech. Eng., vol. 186, (2), pp , 2000.