Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6380521, 18 pages

https://doi.org/10.1155/2017/6380521

## A Two-Stage Queue Model to Optimize Layout of Urban Drainage System considering Extreme Rainstorms

^{1}Shanghai Maritime University School of Economics and Management, Shanghai 201306, China^{2}Tongji University School of Economics and Management, Shanghai 200092, China

Correspondence should be addressed to Wenfa Hu

Received 27 August 2016; Revised 31 December 2016; Accepted 5 February 2017; Published 19 March 2017

Academic Editor: Mauro Gaggero

Copyright © 2017 Xinhua He and Wenfa Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Extreme rainstorm is a main factor to cause urban floods when urban drainage system cannot discharge stormwater successfully. This paper investigates distribution feature of rainstorms and draining process of urban drainage systems and uses a two-stage single-counter queue method to model urban drainage system. The model emphasizes randomness of extreme rainstorms, fuzziness of draining process, and construction and operation cost of drainage system. Its two objectives are total cost of construction and operation and overall sojourn time of stormwater. An improved genetic algorithm is redesigned to solve this complex nondeterministic problem, which incorporates with stochastic and fuzzy characteristics in whole drainage process. A numerical example in Shanghai illustrates how to implement the model, and comparisons with alternative algorithms show its performance in computational flexibility and efficiency. Discussions on sensitivity of four main parameters, that is, quantity of pump stations, drainage pipe diameter, rainstorm precipitation intensity, and confidence levels, are also presented to provide guidance for designing urban drainage system.

#### 1. Introduction

A storm drainage system is a critical infrastructure for a city to support its development, but many cities in China are suffering severe flood problems because of inadequate investment in their drainage system [1, 2]. An urban flood may seriously ruin buildings, block traffic, and even threaten resident’s life. Although there are many factors to cause urban floods, inadequate drainage capacity and improper drainage layout are main factors. Human behavior as well as economy varies in different districts, whose hydrological characteristics change, and a rainstorm would cause different consequences [3].

An urban drainage system usually consists of groundwater convergence points, drainage pipes, and stormwater pump stations and outlets. Its drainage capacity is determined by complicated combination of the above components. The draining operation process can be regarded as a serial two-stage queuing system with three main activities: in the first queue stage, the drainage system is designed to collect stormwater runoff from the road surface and right-of-way. Stormwater collection is here accommodated through the use of drainage inlets, which serve as the mechanism whereby surface water enters underground storm drainage pipes. Drainage inlet locations are often established by the roadway and on side streets to reduce the spread of water onto the roadway surface. Upon reaching the drainage pipes, stormwater is conveyed along to its discharge point. In the second queue state, the drainage system is to discharge stormwater from the drainage pipes by stormwater pump station to an adequate receiving body such as a stream, creek, river, lake, or other piped systems. A stormwater pump station is often required to drain depressed sections of drainage pipes where gravity drainage is impossible or not economically justifiable.

Many researchers have studied the queuing theory of serial multiple service stations. Paper [4] presented a two-stage queue model in solving the state probabilities of the multiqueue problem. Paper [5] developed an retrial queue model to solve a repeated drainage problem. Paper [6] focused on a two-stage queuing network for outfitting process in shipbuilding. Paper [7] extended the cascaded queuing system to infinite servers queue system to solve logistics industry problems. It is a reasonable approach to adopt serial multiple queue model for this urban drainage system layout problem. Many researchers also developed queue models to optimize and design facility layout. Paper [8] presented an innovative approach to solve a layout problem with a multiobjective nonlinear mixed integer programming model considering queening congestion. Paper [9] developed a queuing model of ambulance service to calculate key performance indicators which both managers and patients focused on. Paper [10] proposed a multipriority coverage layout model based on queuing theory.

But a facility layout system often faces uncertain factors which might result in stochastic demands; how to use queue models to design a facility layout in an uncertain environment attracts many researchers. Papers [11], [12], and [13] considered influences of stochastic factors on the multiobjective queuing location-allocation problem, respectively. Papers [14–16] discussed the uncertain problems of fuzzy factors on location by fuzzy queuing models.

When designing a stochastic facility system, its objectives and constraints are stochastic; how to solve this kind of nonlinear programing problem is a challenge which few literatures observed. Paper [17] developed a queuing model considering stochastic custom demands and unmovable service counter in a facility layout problem and then solved it by genetic algorithm with expectation value. Paper [18] established a biobjective model for bidirectional facility network under uncertain supplies and then presented a solution method combined with queuing theory and fuzzy programming.

However, previous studies mainly focus on such problems as optimization of queuing layout routines or assignment of customs; they have never tried to solve complex problems such as the following: () effects of extreme demands from customers, for example, extreme climate; () facility layout as a strategic decision problem because a drainage facility often operates for a long time; () priority of custom demands in various areas where a rainstorm might have different damage and losses in different urban neighborhoods.

This paper considers characteristics of extreme rainstorm and urban flood, stochastic demand for drainage, drainage system capability, fuzzy sojourn time of the system, and draining priority in different areas and then discusses the problem of drainage facility layout under extreme rainstorms. A biobjective, drainage system cost and drainage sojourn time model will be developed, and a stochastic and fuzzy generic algorithm is presented to optimize stormwater impacts on drainage facilities.

The remainder is structured as follows. The mathematical modeling of urban drainage system layout under extreme rainstorm is introduced in Section 2, including the identification of parameters and decision variables, assumptions, methodology for stochastic and fuzzy problems, and constraints. A solution method of improved genetic algorithm for nondeterministic problems with hybrid constraints of randomness and fuzziness is described in Section 3. An example illustrating the parameter setting and flexibility of the proposed algorithm is provided in Section 4. Sensitivities of several parameters and their patterns are discussed in Section 5. Finally, some conclusions are provided in Section 6.

#### 2. Mathematical Modeling

An urban drainage system is designed to collect stormwater runoff from the roadway surface through inlets, to convey it through drainage pipes, and to discharge it to a receiving body by pump stations. The draining process is regarded as a stochastic two-stage queuing system in this paper, where stormwater, as customers, arrives at the drainage system in uncertain time and stochastic quantity. The stochastic and fuzzy features increase complexity in designing urban drainage system. A mathematical model of urban drainage system with stochastic and fuzzy parameters is developed as follows.

##### 2.1. Parameters and Decision Variables

The model parameters and their notations and decision variables are listed in Notations.

Both and are set of discrete points in a draining district. is determined by , , , and . A smaller means the construction and operation cost of urban drainage system is less. is drainage pipe length which is an equivalent distance of drainage pipe from point to pump station . is water capacity of drainage pipe whose diameter is . is total sojourn time of stormwater collected, conveyed, and discharged by urban drainage system, determined by the longest sojourn time . The smaller is, the more efficient the drainage system is. Sojourn time is a fuzzy value, and its mean value is affected by , , and . is demand rate at which stormwater at point receives service from drainage pipe, and it is relevant to precipitation intensity at point .

There are three decision variables, , , and , in this model. and are 0-1 variables, and is a continuous variable. indicates whether pump station is constructed at point or not, indicates whether pump station provides service to point or not, and is service ability, namely, draining capability, of pump station .

##### 2.2. Assumptions

An extreme rainstorm often shows such characteristics as outburst, intensive precipitation, uneven geographical distribution, and short duration. If such intensive stormwater is not drained out in time, flood or waterlog will appear. When designing a drainage system, all rainfall, equivalent to precipitation in its covered area, is a critical factor. Although precipitation in urban district can be obtained from historical weather data, it is hardly predicted accurately because of its uncertain and random weather conditions. The precipitation duration and intensity are also regarded as stochastic variables. Here are main assumptions in modeling urban drainage system in this paper.

*Assumption 1. *Precipitation intensity of an extreme rainstorm in an urban district is subject to a stationary Poisson distribution and the duration is stochastic [19, 20].

When precipitation intensity at point is subject to a Poisson distribution, its distribution function of precipitation intensity is

The parameter in the distribution function could be determined by the maximum likelihood estimation. If there is a population and sample , then its maximum likelihood function [21] is

The parameter would be replaced by the maximum likelihood estimation :

This distribution function adaptability then can be tested by chi-square distribution method which has been broadly applied in fit test [22, 23]. The test hypotheses are as follows:The population is subject to Poisson distribution.The population is not subject to Poisson distribution.

All values from are divided into subsets: , which are pairwise disjoint. Observation of samples reveals that belongs to and its occurrence frequency is , where and are, respectively, the lower and upper bounds of group . Then, the test probability and the fitting value are, respectively, obtained:

There are parameters to be estimated, and their confidence level is . If , then the hypothesis will be accepted and its expectation and variance are and , respectively; otherwise, the hypothesis will be accepted.

*Assumption 2. *All stormwater which is drained by drainage pipes comes from inlets, and its volume is equal to quantity of all precipitation.

Stormwater quantity at point is multiple of its precipitation intensity, precipitation area, and precipitation duration; that is,where is the rainfall area covered by drainage pipes (pipe diameter is ) and is precipitation duration.

*Assumption 3. *Stormwater flows in separate queues inside drainage pipes and its flowing sequence will not change until arriving at its pump station. The arrival time of stormwater which flows from drainage pipes to its pump station is stochastic, and only the nearest pump station will provide service to it.

*Assumption 4. *Urban drainage system consists of several drainage pipes and pump stations, and the whole draining procedure in which stormwater is processed by drainage system is regarded as a serial two-stage queue system, namely, queue system, subject to first-come first-served (FCFS) order.

The service counter in the first stage of this queue system is the drainage pipe, and its serviced object is stormwater runoff collected by inlets and its service capacity is drainage pipe’s water capacity. In the second stage, the service counter is the pump station and its serviced object is stormwater conveyed inside drainage pipes which finally is discharged into rivers or other bigger pipes, and its service capacity is pump station’s operation capacity.

The service time of drainage pipe is regarded as an independent random variable which is subject to a negative exponential distribution [24]. The service capacity of drainage pipe is , which is relevant to its diameter: where is the pipe diameter and is the average flowing velocity inside drainage pipe .

A drainage pipe only receives service from one pump station . If service capacity of the pump station is , the service time and the arrival time in two stages are independent, the demand rate of stormwater at point is , and the average arrival rate of stormwater arriving at pump station is , then, service intensities in two stages are and , respectively:

*Assumption 5. *Each pump station in the urban drainage system is independent but each one has an independent drainage capacity. The sojourn time of a pump station discharging stormwater is fuzzy.

A deterministic sojourn time of stormwater collected from point , conveyed by drainage pipe and discharged by pump station in urban drainage system, can be derived:

However, there are many factors affecting the sojourn time; it is regarded as a fuzzy variable in this paper. Since triangular fuzzy number has been proven to be excellent by previous literatures in solving fuzzy simulation problems [25, 26], the sojourn time is regarded as a triangular fuzzy number and it can be transferred to a possibilistic value by this method. The total sojourn time of stormwater from point processed by pump station can be represented by a triangular fuzzy number :

Its fuzzy membership function is

Considering confidence level of fuzzy number, its -cut set is :

The upper possibilistic mean value and the lower possibilistic mean value of drainage sojourn time in -cut set are obtained:

The clear possibilistic mean value of the fuzzy numbers is obtained from and [27]. Therefore the possibilistic mean value of drainage sojourn time is as follows:

The possibilistic variance of the total drainage sojourn time is as follows:

*Assumption 6. *If a region at point is more important than other regions, a smaller restriction value will be applied to reduce its sojourn time of stormwater. is a stochastic variable and subject to normal distribution. When there are experts, each one ranks a restricted sojourn time of stormwater at point ; then, its expectation and variance can be obtained, and its restriction value of sojourn at point can be described by this normal distribution function .

One of the service time constraints is the regional drainage priority based on their importance. The same rainstorm has varied impacts on different districts because their characteristics such as population, traffic, and economics are totally different. In order to reduce potential flood or waterlog in key regions such as commercial centers, major communities, busy roads, and political and culture centers, a regional priority is introduced to adjust their drainage sojourn time as well as drainage service capability in different regions. An expert ranking method or weighting method is often used in choosing key regions. So several experts can be asked to choose a constraint value to adjust regional sojourn time.

*Assumption 7. *Total cost of urban drainage system includes () construction cost of drainage pipes, () construction cost of pump stations, () equipment cost of pump stations, and () operation cost of drainage system. Among them, construction and equipment cost of pump stations is supposed to be proportional to its drainage capacity, and pipe’s construction cost is proportional to its length.

##### 2.3. Methodology for Modeling a Stochastic and Fuzzy System

An urban drainage system incorporates randomness and fuzziness because of its uncertain environment, so its model consists of mixed conditions with a lot of random and fuzzy parameters. After considering this kind of mixed conditions, a methodology to model a stochastic and fuzzy system is presented as follows:where represents possibility of an event in and represents probability of an event in . is a decision vector, is a stochastic parameter vector, and is a fuzzy parameter vector. is an objective function, and is a constraint function.

Under mixed conditions, the objective function is and its value is . The constraint function is , . The position of randomness and fuzziness in the same function can be exchanged. Therefore, is equivalent to .

##### 2.4. A Biobjective Model of Urban Drainage System

The two objectives in this biobjective model of urban drainage system are total sojourn time of stormwater and total cost . The total sojourn time is the maximum of sojourn time of stormwater from all points . The total cost consists of construction cost, equipment cost, and operation cost. Considering stochastic rainstorm demand, fuzzy sojourn time, and its restriction value, the biobjective model with random and fuzzy constraints for urban drainage system is developed:

The objective function (16) is to minimize total sojourn time of stormwater processed by drainage system, and constraint function (18) represents the possibility of total sojourn time including waiting time and service time which is not less than . The objective function (17) is to minimize the expected value of total cost that the stormwater is processed by drainage system, and constraint (19) is to ensure cost possibility is not less than and denotes the total cost components.

Constraint (20) is to ensure probability of total sojourn time of drainage system subject to its restriction value is not less than and its possibility is guaranteed to be not less than . Constraint (21) denotes a method to estimate length of drainage pipes. Constraint (22) is to ensure stormwater will not exceed the maximum water capacity of drainage pipes. Constraint (23) ensures that probability of balance between service ability of pump stations and drainage demands is not less than . Constraint (24) denotes that stormwater demanding service of drainage system appears only after establishment of a pump station at location is finished. Constraints (25) and (26) indicate that stormwater at point is only served by its nearest drainage pump station. Constraint (27) is a given quantity of pump stations to be constructed in this system. Constraint (28) denotes the average arrival rate of stormwater to pump station in unit time. Constraint (29) is property of two decision variables.

#### 3. An Improved Genetic Algorithm for Stochastic and Fuzzy System

Since a stochastic and fuzzy system demonstrates complex nonlinear characteristics in its objectives and constraints which are not easy to be converted into deterministic variables, it is difficult to solve a large-scale nonlinear problem by traditional algorithms. Fortunately, many heuristic algorithms such as genetic algorithm have been developed by researchers to solve deterministic problems [28] as well as nonlinear problems [29].

In order to apply genetic algorithm in stochastic and fuzzy urban drainage system, its nondeterministic variables would be sampled based on probability and possibility distribution method and then a simulation model could be developed [30]. After improvement of genetic algorithm, nonlinear problems in urban drainage system would be solved.

##### 3.1. Method of Simulating Nondeterministic Variables

For a given decision vector , is equal to

In order to verify the above equation, a fuzzy vector generates evenly an which satisfies ; namely, is extracted from level -cut set of , or is extracted from a super geometry body which includes level -cut set. The condition determines whether is accepted or not.

After is selected, is verified in the following procedure. There are individual stochastic variables generated from probability distribution . After times tests of , if times are observed and , then is accepted. Otherwise, should be selected again and the procedure would be repeated. After times of selecting , , and then the given vector is regarded as unacceptable.

When calculating result of , if and , then and .

The objective function in urban drainage system is to maximize value in its inequality when the decision vector is given. Its solution procedure is described as follows. is evenly generated from and . There are independent stochastic variables, , generated from and a sequence is obtained, where , .

It is supposed that the integer part of is , ), and the th biggest element of is . If , then . After repeating times of choices, is regarded as the objective value at point .

##### 3.2. Improved Genetic Algorithm for Urban Drainage System

Genetic algorithm is a high parallel, random, and adaptive intelligent optimization algorithm [31]. An improved genetic algorithm can simulate stochastic and fuzzy characteristics in its process of producing population, crossover, and mutation operations and calculating value of chromosome [32]. An improved genetic algorithm is redesigned to solve urban drainage system as follows.

*(1) Coding*. The decision variables and are coded by a mixed coding method which is combined with floating point and binary code. The code length is , where represent amount of stormwater inlets points and the amount of pump stations, respectively.

The first digits in the code are a floating point number, representing decision variable . If , then , , , which denotes that in the point a pump station will not be constructed. If , then , which denotes that a pump station will be constructed in point .

The th to the th digits in the code are binary numbers, which represent decision variable . If , it denotes that stormwater will be discharged by pump station . If , it denotes that stormwater will not be discharged by pump station . Each chromosome can be decoded, and its genotype and phenotype can be exchanged.

*(2) Initialization*. Several random points are generated from the feasible domain of decision vectors when chromosome is initialized. Then, its feasibility would be tested by a random and fuzzy simulation method. If it is feasible, the chromosome will be selected. Otherwise, a new random point should be generated until a feasible solution is obtained. After repeating times, there are initial feasible chromosomes obtained. Target values of all chromosomes would be obtained by the random and fuzzy simulation method.

*(3) Setting Fitness Function*. A probability value is used in each chromosome which is determined based on order of an evaluation function . Roulette wheel selection method is applied to make the possibility of each selected chromosome be proportional to the adaptability of other chromosomes in the same population, meaning that a more adaptable chromosome would be selected firstly and to produce possibly more offspring.

The evaluation function iswhere and usually . is ordered from good to bad according to values of objective functions. A better chromosome is obviously assigned a smaller number.

*(4) Selection*. Cumulative probability is calculated based on evaluation function of each chromosome .

A random number is generated from the interval . If , then the th chromosome is selected, and times of duplicated chromosomes are available after repeating times.

*(5) Crossover*. The probability of crossover operation is and the following process is repeated from to* popsize*: A random number is generated from the interval . If , chromosome is selected as a parent to participate in crossover operation. After selection, the parents are named , and they are divided into pairwise groups . If the number of parents is odd, one of them is deleted randomly. A random number is generated from and crossover is operated in the number of the first digits of each parent. Therefore, two children are generated from the first digits. The crossover operation is expressed as follows:

The numbers from the th digit to th digits of each parent are crossed as follows: A random genetic position is regarded as the cross point, where the same genetic positions at two parents are exchanged. Then, two children are generated from the th digit to th digit, respectively, in probability .

Combination of and and and will generate two new offspring and . The feasibility of offspring is tested by a random and fuzzy simulation method. If its test is feasible, it will be replaced by its parents. Otherwise, a new crossover operation will repeat until two feasible offspring are obtained or the iteration reaches a limit cycle.

*(6) Mutation*. The probability of mutation operation and following process is repeated from to* popsize*: A random number is generated from the interval . If , then the chromosome is selected as a parent to participate in mutation operation.

Mutation is operated in the number of first digits of each parent as follows: A genetic position is randomly selected as the mutation position, its genetic value is replaced by a random value generated from , and the number of its first digits is regarded as its offspring .

Mutation is operated at the number of the th digit to th digit of each parent as follows: A genetic position is also randomly selected as the mutation position, where 1 is replaced by 0 and 0 is replaced by 1. Then, its offspring is generated at the th digit to th digit.

Combination of and is to generate a new offspring . The feasibility of offspring is tested by a random and fuzzy simulation method. If its test is feasible, it will be replaced by its parents. Otherwise, a new mutation operation will repeat until a feasible offspring is obtained or the iteration reaches a limit cycle.

() The above process of selection, crossover, and mutation will repeat until they meet a termination criterion. Finally, all optimal solutions of decision variables, sojourn time, and drainage cost are obtained, respectively.

#### 4. Illustrative Example

Since a stochastic and fuzzy problem is so complicated that it hardly validates the proposed algorithm by any direct approaches, after reviewing the methods in [33, 34], this stochastic and fuzzy problem would be simplified as a simulated deterministic problem. In order to verify this algorithm’s validity to solve the large-scale nonlinear complex model, a numerical example is chosen and several tests are validated and discussed.

##### 4.1. Description of Example and Setting of Parameters

A square district in Shanghai is chosen as a numerical example to illustrate how to optimize drainage system layout by this algorithm. The district is 3200-meter-long and 3200-meter-wide, but it is suffering frequent extreme rainstorms which result in more losses. In order to reduce urban floods, local authority wish to rezone drainage area and redesign its drainage system soon. It is divided into 64 drainage blocks whose length is 400 meters, shown in Figure 1. Drainage block’s centers are supposed to be drainage points and all groundwater inside a block only flows to its drainage point where an inlet is installed. A drainage pipe is laid underground to connect with an inlet and a pump station. All groundwater is collected by inlets, conveyed by underground drainage pipes to pump stations. Any drainage point could be a candidate position of pump station. Stormwater from one drainage point only flows to one pump station through a drainage pipe; its flowing direction would never change.