An Integrated Decision-Making Framework Based on Many-Objective Brain Storming Optimization for Urban Drainage System Design

Many cities around the world face the flooding management problems as extreme rainfall events have become more frequent. It’s of great practical significance to design an effective urban drainage system (UDS) to improve the stormwater runoff quality. An integrated decision-making analysis framework based on many-objective brain storm optimization is proposed to make the optimal design of UDS. Firstly, a generic mathematical model is presented considering five objectives simultaneously. Secondly, an effective many-objective brain storm optimization based on local region dominance is designed as the solver of the model and a set of Pareto-optimal solutions are obtained. Thirdly, two kinds of multi-attribute decision analysis methods are introduced to rank the Pareto-optimal solutions. Fourthly, the comparison experiments of a case study with three multi-objective swarm intelligence algorithms show the promising performance of the integrated decision-making framework and the corresponding optimization algorithm.

nomical and effective UDS to cope with the extreme rainfall 23 events. 24 The design of UDS is a daunting task due to its inher- 25 ent characteristics such as hydraulic complexity together 26 with the conflicting objectives. This leads to an increasing 27 The associate editor coordinating the review of this manuscript and approving it for publication was Shaoyong Zheng . number of studies regarding the design of UDS as a multi-28 objective optimization problem (MOP). Most traditional opti-29 mization methods (e.g., linear or nonlinear programming and 30 dynamic programming) can't handle it well for its nonlinear- 31 ity, multimodal and multidimensional [4]. Swarm intelligence 32 (SI) algorithms, such as particle swarm optimization (PSO), 33 brain storm optimization (BSO) [5] and Evolutionary algo-34 rithm (EA), have been proved effective and efficient in many 35 complex multi-objective engineering optimization [6], [7]. 36 Accordingly, multi-objective SI algorithms are applied as the 37 solver of the optimal design of UDS. Muleta et al. presented 38 the Non-dominated Sorting Genetic Algorithm II (NSGA-II) 39 to solve the optimal pipe size of drainage system with two 40 objectives including both the overflow volume and the cost 41 of pipes [8]. Duan  is rarely mentioned in other related works. Specially, the 101 factors that are difficult to establish the mathematical 102 expression are also considered in this process. 103 The article is organized as follows: The design of urban 104 drainage system and the proposed framework are introduced 105 in Section II, which contains system modelling, optimiza-106 tion and decision-making. The model formulation of the 107 UDS problem is described in detail in Section III. And the 108 many-objective BSO algorithm named LRD-BSO is pre-109 sented in Section IV. Two kinds of multi-attribute decision 110 analysis methods are discussed in Section V. A study case 111 of the proposed framework and the experimental study of 112 LRD-BSO are shown in Section VI. The conclusion is fol-113 lowed in Section VII.

116
Many factors should be considered by engineers in the pro-117 cess of UDS design. It has been widely taken as a MOP, 118 in which the points that need to be considered are regarded 119 as the optimization objectives. Most of the existing multi-120 objective UDS optimization models only consider two or 121 three objectives. But in fact, there are a range of criteria can be 122 used to assess the performance of the designed UDS from dif-123 ferent angles, including the flood volume, flood damage, the 124 peak runoff, the flood duration, the cost of storage tanks, the 125 pipe material cost, and the costs of land use, construction as 126 well as the maintenance of the UDS. These models with lim-127 ited objective number may not be enough to design the UDS 128 with commendable overall performance. Trade-off among 129 more optimization objectives can better reflect the planning 130 objectives of the engineers in real-word UDS design. There-131 fore, a many-objective UDS optimization model considered 132 five optimization objectives is presented in this article.

133
With the increase of the number of objectives, the tradi-134 tional multi-objective SI algorithms can't solve the corre-135 sponding UDS optimization problem well. Therefore, it is 136 necessary to design an effective many-objective optimization 137 algorithm to find the Pareto-optimal solution set with good 138 distribution and great convergent performance. The solutions 139 in the set are also called noninferior solution, which means 140 the solutions are equally well. It's complicated for the deci-141 sion makers to select the most suitable solution from the 142 Pareto-optimal solution set. Furthermore, apart from the opti-143 mization objectives considered in the optimization model, 144 there are also some aspects that are difficult to be mathe-145 matically represented, such as the construction safety and 146 citizen satisfaction, which also should be involved in the 147 decision-making process. As a result of this problem, the 148 multi-attribute decision-making methods are introduced to 149 consider not only the optimization objectives but also these 150 necessary aspects in the final decision-making.

151
The proposed framework of the UDS design, includes three 152 components, as shown in Fig. 1. The first component is model 153 formulation. The main aim is to assess the overall benefits 154 of the designed UDS more comprehensively. The second 155 component is many-objective optimization algorithm design.

156
The aim is to get the Pareto-optimal solution set with excel-157 lent performance in term of convergence and diversity. The where D i is the diameter of ith pipe, C P (D i ) represents the ith 179 pipe cost per unit length, and L i is the length of ith pipe, N P 180 indicates the total number of pipes in the designed UDS, C o is 181 other implement cost. The total storage tanks cost is defined 182 as: where DF i represents the cost of ith treatment device, N m is ing equation is used to calculate the flood damage loss [18]: where w 0 , w 1 and w 2 are the coefficients about the local land, 196 DEP indicates the water depth and DUR is the flood duration. 197 Finally, the intended economic loss caused by flood is defined 198 as: where η 0 and η 1 are the parameters of local economic 201 and VOL indicates the flood volume. Overall, the neces-202 sary inputs of the UDS optimization model are presented in 203 Table 1.

205
The typical multi-objective SI optimization algorithms, such 206 as NSGA-II [19], have widely used as the optimization solver 207 of UDS design problem, and make great performance in the 208 case of two or three optimization objectives [20]. But, due to 209 the existence of dominance resistance, they can't work well 210 when handling the many-objective UDS optimization model 211 discussed in this article.

212
BSO is a new SI algorithm proposed in recent years, and it 213 uses all possible individuals to update the population, so as to 214 good diversity, which is especially beneficial to solve multi-215 objective optimization problems. Some multi-objective BSO 216 algorithms have been proposed [21], [22], [23], [24], and 217 perform well on a wide range of MOP. And a novel many-218 objective algorithm based on BSO was proposed to deal with 219 the many-objective UDS optimization model.

221
Inspired by the human brainstorming conference, BSO is 222 guided by the cluster centers and other individuals accord-223 ing to a certain probability, which can balance convergence 224 and diversity greatly. The main process of the BSO includes 225 three important operations [25]: clustering, disruption, and 226 creation, which is shown in Fig. 2. Clustering is a kind of 227 techniques that divides individuals into several groups (clus-228 ters), and the individuals being similar (or related) in the 229 same cluster, which could refine a search area. Without los-230 ing generality, these solutions are clustered by the k-means 231 algorithm, and the solution with best fitness in each cluster is 232 regarded as its cluster center. A probability value is used to 233 replacing a cluster center by a randomly generated solution 234 in the disruption operation. This could avoid the premature 235 convergence and the local optima. A new individual can be 236 generated by one or two cluster(s). One cluster could refine a 237 search region and improve the exploitation ability. Two clus-238 ters could improve the diversity of population. The creation 239 process is described as the following formula: where X new is the new generated solution, N (0, 1) denotes 242 a random value following the standard normal distribution, 243 X selected denotes the selected solution in current iteration as 244 shown in Formula (7), In Formula (7), X 1i and X 2i represent two selected solutions, 248 randis a random value obeying uniform distribution within 249 0 to 1.ξ represents the current step length as shown in For- In Formula (8) 266 Algorithm 1 Main Framework of the Proposed LRD-BSO 1: Input: the maximal number of iterations t max , the probability P one of selecting one individual to generate new individual, the probability P Center of selecting the cluster center to generate new individual; 2: Output: final population P t ; 3: / * Initialization * / 4: Initialize the population P 0 ; Generate reference point set V ; Cluster the reference point set V ; t ← 0; 5: / * Main Loop * / 6: while t < t max do 7: Q t ←Offspring-Generation(V, P t , P one , P Center ); 8: R t = P t ∪ Q t ; 9: P t+1 ← Environment-Selection(R t ); 10: t = t + 1; 11: end while As shown in Algorithm 1, the framework of the LRD-BSO 267 includes three main parts: initialization, offspring genera-268 tion and environment selection. Note that the proposed new 269 dominance relation and two offspring generation manners 270 are applied to the environment selection as well as offspring 271 generation operations, respectively. The details of above three 272 operations are as follows. The initialization procedure of LRD-BSO contains two main 275 aspects. One is the initialization of parent population P 0 , the 276 other is the generation of reference points. To be specific, 277 the initial parent population P 0 is randomly sampled via a 278 uniform distribution. The reference points are generated by 279 the Normal-Boundary Intersection (NBI) method proposed 280 by Das and Dennis [27].
The generation of offspring population Q t includes two steps. 283 First is the reference-point-guided clustering: The reference 284 point set V are grouped by k-means algorithm. And the N ×N 285 angle matrix between solution vectors (from origin to solu-286 tions in P t ) and reference vectors (from origin to reference 287 points in V ) are calculated. Each solution is allocated to the 288 reference points based on the angle value. Specifically, v 1 , v 2 , 289 v 3 and v 4 in Fig.3 are vectors from origin point to reference 290 point and F is the vector from origin point to a certain solu-291 tion. Obviously, the θ 3 is minimum among θ 1 , θ 2 , θ 3 , and θ 4 292 for the solution vector F, so this solution is allocated to the 293 reference point corresponding to v 3 . 294 Afterward, the clustering result of each solution in P t is the 295 same as the reference point it belongs. This clustering method 296 makes the clusters have better spatial distribution.

297
Second is offspring generation: Two ways, that is, inter-298 cluster and intra-cluster generation, are designed for this 299 purpose according to the probability p m as displayed in 300 Fig. 4. The inter-cluster generation is adopted to improve the 301 exploitation of the LRD-BSO algorithm. The new solutions 302 VOLUME 10, 2022   i ← 1; 10: while |P t+1 | + F LRD,i < N 11: In this section, next generation population P t+1 with N elite 314 solutions is obtained from the combined population R t with 315 2N solutions (P t and Q t ). The specific environment selection 316 process includes three parts, and the details are displayed in 317 Algorithm 3. The Pareto dominance level of solutions in combined pop-320 ulation R t are calculated by the Pareto non-dominated sort-321 ing [19]. For the population For the problems with high number of objec-324 tives, |F 1 | is almost always greater than N , which means that 325 S t = F 1 [28]. So, the solutions in F 1 should make a further 326 distinction through the LRD relation proposed by us. it is a corner point [29]. Take the corner point as the center, 331 each solution of current population is assigned to the corner 332 point with the smallest Euclidean distance with it, as follows: 333 where X is a solution, X corner,i denotes ith corner point and the 335 number of corner points is equal to the number of objectives. 336 So that the objective space is divided into several subspaces. 337 Then, the convergence degree of solution X is defined as 338 follows [30] 339 tions X 1 and X 2 ; I(X 1 ) is the subspace index which X 1 345 belongs, obtained from above (9); and X 1 is LPD-dominance 346 X 2 (denoted by X 1 ≺ LRD X 2 ), if the following two conditions 347 hold true:

349
It's noted that, only the solutions are assigned to the same 350 subspace can make the LRD relation comparison. In the final selection from the F LRD,τ , we are inclined to 361 delete the solutions located in the denser distribution area.

362
For the MaOP, the Euclidean distance has been proved to be 363 unable to measure the similarity of solutions well [31]. So, 364 angle crowded distance is introduced in LRD-BSO to make it.

365
The difference between the Euclidean crowded distance and 366 the angle crowded distance is shown in Fig. 5. The solid dot 367 8 should be deleted based on the Euclidean distance, which 368 is shown in Fig. 5(a). And the solid dot 11 should be deleted 369 according to the angle crowded distance shown in Fig. 5(b).

370
Obviously, the solid dot 11 locates at a denser area than the 371 solid dot 8.

372
Accordingly, the angle crowded distance may provide a 373 more promising way to describe the solution distribution.

V. DECISION-MAKING PROCESS
where X is the decision matrix, x ij is the jth attribute value of 401 ith alternative,x + j represents the maximum value of jth benefit 402 attribute as well as x − j denotes the minimum value of jth cost 403 attribute, max and min indicate both the benefit and cost 404 attribute set andr ij denotes the normalized value. 3) Calculate the attribute weight sum of each alternative 409 according to the Formula (14).
where A i is the attributes weight sum of the ith alternative. 412 In the SAWM, the weight sum A i denotes the quality of ith 413 alternative. The higher the better.  (17) and (18):  (19) and (20): 442 best. The conceptualization of UDS case is shown in Fig.6. The

479
The hydrologic performance of the subbasin is simu-480 lated by an urban hydrologic simulation program named 481 LANDSTORM [37], and twenty-three years of recorded rain-482 fall data from the studied area as well as the land characteris-483 tics are used as the simulation input. Due to the model outputs 484 is obtained from the simulation software, it is necessary to 485 approximate the simulation outputs by regression models. 486 To this end, numerous models are attempted for each of the 487 output, and the model with respectable fit performance is cho-488 sen. The objective function and constraint function expres-489 sions are displayed in Table 2 and Table 3, respectively.  Table 4.

500
The indicator HV [40] is utilized to evaluate the conver-501 gence and diversity performance of these four algorithms. 502 The HV values (the higher the better) of the four algorithms 503 are shown in Table 5 and Fig. 7. It can be seen that the 504     As for the NSGA-II and MOPSO, crowd distance and 516 grid method can't work well in the high-dimensional objec-517 tive space to maintain the diversity of solutions. While the 518 novel offspring generation process in LRD-BSO ensures both 519 exploration and exploitation, which will promote the pro-520 duction of the high-quality new solutions. Moreover, the 521 angle-based selection strategy is beneficial to improve the 522 selection pressure and maintain the diversity of solutions. 523 Therefore, it's reasonable that the proposed LRD-BSO pos-524 sesses remarkable performance compared with other three 525 multi-objective EAs. To make the decision makers easily understand and study, 528 the parallel axis plot is used to display the distribution of 529 Pareto-optimal solution set. Two design examples of the UDS 530 FIGURE 8. The parallel axis plot presents the trade-off among five optimization objectives in UDS optimal design, in which each axis represents an optimization objective and each line passing through the five coordinate axes denotes a Pareto-optimal solution. a) the Pareto-optimal solutions that the drainage network cost lies between 65000$ and 70000$ are marked in red, b) the Pareto-optimal solutions that the storage cost lies between 300$ and 800$ are marked in red.
problem are provide to explain the meaning of the parallel 531 axis plot here.

532
The first example shows the feasible solutions (marked in 533 red) when the budget of drainage network cost is from 65000$ 534 to 70000$, which is shown in Fig.8 (a). From Fig. 8, the deci-535 sion makers can select the solution that satisfies their need 536 from the feasible solutions further. The second example gives 537 the solutions, marked as red in Fig. 8(b), when the storage cost 538 is given priority, from 300$ to 800$. The decision makers can 539 also choose the proper solution from the marked solutions.  In this study case, the citizen satisfaction is also considered 549 in the decision-making process besides the five objectives 550 described in the optimization model. Assumed that the citizen 551 satisfaction is measured by scoring between 0 and 1. The goal 552 of the decision maker is to minimize the five optimization 553 objectives and improve citizen satisfaction. So, the former are 554 five cost attributes and the latter is benefit attribute.

555
When the decision makers have certain prior knowledge 556 for the USDS design problem, SAWM can make the suitable 557 decision. They can provide the weight of each attribute. For 558 example, the weight vector is randomly set asM . According 559 to the procedures of SAWM, the weighted sum value A i of 560 each solution is obtained and the solution with highest value 561 shown in Table 6. The final output depends on the weight 562 vector, and different weight vector means the different design 563 requirements.

564
For the decision makers who can't give the relatively 565 weight of different objectives, TOPSIS can help them make 566 decisions. According to the step of TOPSIS mentioned in 567 section 5, set all attributes to equal weight, the relative dis-568 tance C i to the ideal solution of each alternative is obtained 569 and the solution with maximum C i is the optimal scheme in 570 this decision-making process, which is shown in the Table 7. 571 The above multi-attribute decision-making process not 572 only consider the five objectives of UDS optimization model, 573 but also the citizen satisfaction that can't be expressed math-574 ematically. For the SAWM method, the decision makers can 575 participate in the design of the UDS by determine the weight 576 vector, which might reduce the bad procedural results, resis-577 tance and conflicts when decision-makers feel undervalued 578 and neglected. On the other hand, TOPSIS can help decision-579 making when the decision-makers do not give weight vector, 580 and the result can provide reference for the final UDS design. 581 It must be noted that the decision-making results are based on 582 a specific case study and no general conclusion can therefore 583 be made. However, the proposed framework can be applied 584 to different case studies with different situation.