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An improved artificial bee colony algorithm for pavement resurfacing problem
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ScienceDirect International Journal of Pavement Research and Technology xxx (2018) xxx–xxx www.elsevier.com/locate/IJPRT
An improved artificial bee colony algorithm for pavement resurfacing problem Tapas Ranjan Panda a,1, Aravind Krishna Swamy b,⇑ a
Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India b Department of Civil Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India Received 31 July 2016; received in revised form 30 March 2018; accepted 12 April 2018
Abstract Pavement resurfacing is a maintenance activity that is undertaken to enhance the service life of pavement. This pavement resurfacing activity involves laying a new layer of asphalt concrete over existing pavement after certain time. Due to engineering factors, economic variables and uncertainty in forecasting, the pavement resurfacing decision process is a complicated activity. Various optimization approaches currently used are simplified models for finding optimal frequency and resurfacing intensity within pavement maintenance framework. In this research, artificial bee colony algorithm is proposed to solve this pavement resurfacing optimization problem. This algorithm mimics the collective behaviour of bees while searching for nectar. In this approach, various scenarios are generated, optimality of each case is evaluated, and the information thus generated is used in subsequent evaluation until global optimality is reached. The effectiveness of proposed method is demonstrated through a numerical example. The solution obtained is similar to the exact solution reported in literature. The results indicate optimal resurfacing values can be obtained with little computational effort with proposed approach. The main advantage of proposed algorithm is removal of specification of trigger values for maintenance decision. Ó 2018 Chinese Society of Pavement Engineering. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Pavement; Resurfacing; Artificial bee colony; Life cycle cost; Optimization
1. Introduction Under the combined action of traffic loading and climatic factors, the pavement deteriorates over a period of time. Beyond a certain level of deterioration, the pavement becomes unserviceable and the whole pavement has to be reconstructed. To avoid this expensive reconstruction process, maintenance activity like resurfacing is taken up. Camahan et al. [1] reported that resurfacing is less expensive treatment that decreases overall life cycle cost while increasing service life of pavement. Once the pavement is
⇑ Corresponding author. Fax: +91 11 2658 1117.
E-mail address: [email protected] (A.K. Swamy). Formerly Master’s Student. Peer review under responsibility of Chinese Society of Pavement Engineering. 1
resurfaced, serviceability level increases to a certain extent. After this resurfacing, the deterioration process restarts and continues until next cycle of resurfacing. It is generally accepted that roughness of pavement is an indicator of pavement serviceability/pavement deterioration. Any increase in roughness leads to reduction in ride quality of pavement. Fig. 1 shows changes in pavement roughness with time. Due to simultaneous action of traffic and environment, the deterioration process (or roughness increase) is continuous and nonlinear in nature. Various trends like power form [2], polynomial form [3], and exponential form [4,5] have been used to describe this roughness progression. On the other hand, resurfacing activity leads to sudden drop in roughness level. This decision on pavement resurfacing is made by highway agencies based on network condition, budget constraints, and resources availability. Often engineer’s experience, subjective judgement, and historical
https://doi.org/10.1016/j.ijprt.2018.04.001 1996-6814/Ó 2018 Chinese Society of Pavement Engineering. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: T.R. Panda, A.K. Swamy, An improved artificial bee colony algorithm for pavement resurfacing problem, Int. J. Pavement Res. Technol. (2018), https://doi.org/10.1016/j.ijprt.2018.04.001
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Fig. 1. Variation of pavement roughness during service life of pavement.
practice plays important role in this decision process [6]. This continuous increase in roughness and sudden drop in roughness results in a saw-tooth trend in pavement roughness during its service life. Exact shape/trajectory of this saw-tooth curve depends on traffic, climate, construction quality, and maintenance interventions [7]. Several optimization approaches have been proposed that aid in decision process of pavement resurfacing activity. These optimization approaches ranges from single pavement rehabilitation event in finite horizon to multiple pavement maintenance activities over infinite horizon [8–11]. Several researchers [10,12] have used deterministic approach while Pasupathy et al. [7] have used stochastic approach. Even though deterministic approach aids in arriving at exact solution for pavement resurfacing optimization problem, it deviates from real situation significantly [13]. In this research, Artificial Bee Colony (ABC) algorithm is used to solve this pavement resurfacing problem. The proposed methodology accommodates various factors like user costs, agency costs, inflation rate, interest rate, and achievable roughness levels during pavement resurfacing cycles. The frequency and thickness of resurfacing are obtained as solution while ensuring maximum cost effectiveness. Within ABC algorithm framework, various improvements like generation of multiple colonies (i.e. possible solutions), achieving global solution are proposed in this article. The main advantage of proposed ABC algorithm for pavement resurfacing problem is elimination of trigger roughness level specification beforehand. This paper contains 6 sections of which this is first one. Section 2 presents a review of the relevant literature in pavement resurfacing optimization and ABC. Section 3 describes the model formulation used in optimization of pavement resurfacing. Section 4 describes the proposed solution procedure using ABC algorithm while a case study is presented in Section 5. Finally conclusions of the paper are presented in Section 6. 2. Literature review Control theory has been used by various researchers to optimize pavement maintenance activities [8,14,15]. Control theory essentially assumes that pavement condition
changes continuously over time. However, any maintenance activity leads to discontinuous pavement serviceability function. Pasupathy et al. [7] reported that control theory problems with discontinuous response is computationally interactable and is quite cumbersome. To handle this discontinuity issue, Tsunokawa and Schofer [16] suggested saw-tooth pavement roughness curve. In this approach, the saw-tooth curve is replaced by a continuous function that passes through the midpoints of the spikes and discrete resurfacing activity is replaced by continuous resurfacing rate. Thus, solution obtained using the approach proposed by Tsunokawa and Schofer [16] will yield approximate results. Tsunokawa et al. [17] proposed gradient search methods to obtain optimal pavement maintenance strategy. However, these gradient search methods have inherent disadvantage of getting trapped in local minima with wrong initial guess. Li and Madanat [10] used the concept of saw-tooth roughness profile to optimize the frequency and intensity of pavement resurfacing actions under steady-state conditions. They provided a simple, practical, and robust method that uses the concept of trigger roughness level which initiates resurfacing activity. The specification of predefined trigger roughness level essentially requires one to have prior information regarding optimality. Since this trigger roughness level is a part of optimization process, such an exercise might not be meaningful. If such an assumption has to be made, one has to resort to extensive parametric study before resorting to optimization. Ouyang and Madanat [12] proposed an analytical solution to the pavement resurfacing problem for a finite horizon using saw-tooth roughness curve. Sathaye and Madanat [18] extended this concept to multiple facility optimization problem. To obtain feasible solution, most of the work discussed above assume (i) optimization problem as deterministic, (ii) approximate discrete changes as continuous phenomenon, and (iii) certain functional form for deterioration function. Thus, solution obtained by these optimization approaches deviate from real situation significantly [13]. Under such circumstances, metaheuristic approaches have proved to be effective. The goal of present research is to apply one such metaheuristic optimization algorithm to solve this pavement resurfacing problem. Bio-inspired metaheuristic algorithms are very popular for solving complex engineering problems. Ant colony optimization [19], genetic algorithm [20], particle swarm optimization [21], ABC algorithm [22] are some examples for such bio-inspired optimization algorithms. All these algorithms mimic the collective behaviour of insect/animal groups through complex interaction of individuals without supervision. These optimization approaches offer advantages like scalability, adaptation, speed, modularity and efficiency [23]. Karaboga [22] introduced ABC algorithm for solving multimodal, multi-dimensional numerical unconstrained optimization problem. This algorithm was inspired by the intelligent nectar foraging behaviour of honeybees in a colony. In a real bee colony, tasks like identification of poten-
Please cite this article in press as: T.R. Panda, A.K. Swamy, An improved artificial bee colony algorithm for pavement resurfacing problem, Int. J. Pavement Res. Technol. (2018), https://doi.org/10.1016/j.ijprt.2018.04.001
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Fig. 2. Flowchart of proposed ABC algorithm with multiple colonies.
tial sources of nectar, employing number of bees to bring nectar from a source to hive, and storage of nectar are performed by efficient division of labour, exchange of information and self-organization. To perform all these tasks, bees in a particular colony perform different roles at different stages. In the initial phase of the foraging process, the bees start to explore the environment randomly in order to find a nectar source. These bees looking for new sources of nectar are referred to as scouts. After finding a nectar source, scouts become employed bees and start exploiting the identified nectar source. The employed bee returns to the hive with the nectar and unloads the nectar. After unloading the nectar, employed bee shares information about current nectar source with other bees. Based on the information shared by the employed bees, onlooker bees (those waiting in hive) decide on current source or new nectar source to exploit. The employed bee whose nectar source is discarded becomes the scout bee to search new nectar source. This process is continued until all the
requirements regarding nectar source, and storage are met. In an engineering optimization problem, the position of a nectar source represents a possible solution to the optimization problem, and the nectar amount of a nectar source corresponds to the fitness of the associated solution. ABC algorithm has been successfully applied in various fields including signal processing [24], parameter optimization of a multi-pass milling process [25], inverse problem in heat conduction [26], and behaviour of confined plasma in a nuclear fusion device [27]. In civil engineering, ABC algorithm has been applied in the area of subway route location [28], travelling salesman problem [29], structural optimization of planar and space trusses [30]. 3. Proposed methodology This section presents ABC based optimization approach to solve the problem of pavement resurfacing. In this proposed approach, the exploration and the exploitation capa-
Please cite this article in press as: T.R. Panda, A.K. Swamy, An improved artificial bee colony algorithm for pavement resurfacing problem, Int. J. Pavement Res. Technol. (2018), https://doi.org/10.1016/j.ijprt.2018.04.001
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bility of the algorithm is enhanced by employing multiple colonies that are spread throughout the search space. Overall optimization framework consists of several steps i.e. (i) input regarding solution space, (ii) subdivision of solution space into smaller sub-search spaces, (iii) mapping these sub-search spaces to individual bee colonies, (iv) computation of optimal solution within sub-search spaces, (v) comparison of potential global best solution with optimal solution in sub-search space, and (vi) selection of global optimal solution. The proposed algorithm is presented in the Fig. 2 and explained in detail below. In the first step, approximate range of decision variables (i.e. resurfacing thickness and resurfacing frequency) within which output is expected is specified as solution space limits. This specification regarding solution space limits is essential to avoid excessive computational time and effort. These limits on solution space may be specified based on intuition, historical data, available expertise, and theoretical justifications. In the second step, this solution space is divided into smaller sub-search spaces. These small sub-search spaces are assigned to individual bee colonies. The number of smaller sub-search spaces (i.e. number of colonies) depends on nature of search space. A relatively smooth objective response surface requires less number of colonies. The number of bees in a colony is kept to a minimum without any compromise on quality of solution. Further, 50% of bees in a particular colony are designated as employer bees and are spread throughout the assigned sub-search space. Each bee is assigned to one nectar source (i.e. potential optimal values) within sub-search space. The number of onlooker bees on a particular solution depends upon the fitness of the solution. In the third step, nectar sources (XijN) are initialised in the sub-search space as in Eq. (1). This equation controls the production of neighbour nectar sources within boundaries of sub-search space. The fitness of nectar sources (F (xi)) is evaluated using Eq. (2) [25]. The fitness of each source is determined using objective function value (Fi). This fitness defines the quality of the solution obtained in each iteration. These fitness values aid in identifying the quality of potential nectar sources within sub-search space. Depending on the quality of nectar sources, some nectar sources are selected for subsequent steps. New random nectar sources (i.e. possible solutions) are generated near the solution obtained in previous step using Eq. (3). Further, greedy selection process is applied between the solutions for the employed bees to find better solution. Probability for each solution is calculated by comparing fitness of a solution (Fi) to the total fitness of all solutions using Eq. (4). This equation essentially helps in finding quality of potential nectar sources and ranking the same. The numerical values of potential nectar sources having higher probability are used for subsequent interaction. New nectar sources (i.e. possible solutions) are generated depending on the probability of the previous solutions within each small sub-search space. After comparison of all solutions within each small sub-search space, the local best solution
is memorized. This process is repeated until the termination criterion is reached. max min X ijN ¼ X min jN þ ;iN ðX jN X jN Þ
ð1Þ
where ;ijN = random number between [0,1]; i = 1 to number of nectar sources within each sub-search space; j = 1 to number of decision variables (in this research maximum value of 2); N = 1 to number of sub-search spaces. ( 1 þ jF i j if F i < 0 F ðxi Þ ¼ ð2Þ 1 if F i P 0 1þF i vijN ¼ X ijN þ ;ijN ðX ijN X kjN Þ where ; k 2 f1; 2; . . . Number of employed beesg j 2 f1; 2; . . . Number of optimization parameterg are domly chosen indexes. Fi p i ¼ Pn
i¼1 F i
ð3Þ and ranð4Þ
where n = number of nectar sources. In the first sub-search space, local best solution is designated as global best solution. For subsequent sub-search spaces, local best solution in current sub-search space is compared with potential global best solution and better one is designated as potential global best solution. Updated potential global best solution is fed back for the next colony. The above procedure is repeated for all sub-search spaces and final potential global best solution is designated as optimal solution. 4. Numerical example To demonstrate the applicability of the proposed optimization methodology to the pavement resurfacing problem, a numerical example is presented in this section. 4.1. Problem statement It is assumed that resurfacing activity will be taken up on 1 km long lane of a highway pavement. Expressions for deterioration function F(s), improvement function G (w, s), user cost function C(s) and agency cost function M (w) are presented in Eqs. (5)–(8), respectively. Here serviceability is expressed in terms of Quality Index (QI), the time is taken in years, resurfacing thickness is in millimetres and cost is taken in dollars per km of pavement. Based on a field study mentioned by Tsunokawa and Schofer [16] road deterioration rate is a function of serviceability and given in Eq. (5). After an overlay with a thickness w, the serviceability of resurfaced pavement improves as given in Eq. (6). F ðsÞ ¼ f 1 s
pffiffiffiffi Ds ¼ Gðw; sÞ ¼ g1 w þ g2 s þ g3
ð5Þ ð6Þ
CðsÞ ¼ c1 s þ c2
ð7Þ
MðwÞ ¼ m1 w þ m2
ð8Þ
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where s = serviceability of the pavement and, f 1 ; gi ; ci , mi are constants. Deterioration parameter f1 was assumed to be 0.05 per year. The improvement function parameters i.e. g1 ; g2 ; g3 are taken as 5.0, 0.78, 66, respectively. The numerical values of c1 and c2 are taken as 1000 $/QI/year and 0 $/year, respectively. The variable and fixed cost parameters for a resurfacing on that 1 km lane are assumed to be m1 = 11,000 $/mm, m2 = 150,000 $. Rate of variable user costs per one QI unit on this facility was taken as 0.0002 $/QI/ veh, and the annual traffic volume in that lane was assumed as 5 106 veh/year. This would indicate that pavement at good condition (QI = 25) will deteriorate to poor condition (QI = 100) in about 28 years. All these values have been taken from published literature [10,16]. The life cycle cost of the pavement can be calculated by Net Present Value (NPV) approach [16]. The NPV of the pavement includes both agency and user cost. The rehabilitation activity cost is the major contributor towards the agency cost. The user cost is contributed by two major components i.e. vehicle related and non-vehicle related. The vehicle related cost includes vehicle maintenance cost, fuel cost etc. that are dependent on the pavement roughness. The cost of delay during rehabilitation activity is accounted towards non-vehicle related cost. Life cycle cost of the pavement is calculated by adding the continuous discounted user and agency costs for one cycle of resurfacing. User cost occurs throughout the cycle while agency cost occurs at the end of the cycle for the resurfacing. The final NPV is calculated by adding discounted costs of all the cycles. Objective is to minimize the total lifecycle cost of pavement. Additional details regarding this problem can be found elsewhere [10,16]. The sum of cost rate (as a function of roughness) incurred by user and agency during each maintenance cycle is given in Eq. (9). The total cost incurred by user and agency during pavements life cycle can be obtained by summation of incurred cost during each cycle. The same is given in Eq. (10). Z t n Jn ¼ CðsðtÞÞert dt þ Mðwn Þertn ð9Þ tn1
NPV ¼
N X Jn
ð10Þ
1
Z
s1 s0
ds ¼ f 1s
Z
5
t
dt ¼> s1 ¼ s0 ef 1 t
ð12Þ
0
Eq. (12) explains that if s0 is the initial pavement condition, after a time the condition of pavement will be s0 ef 1 t . Substituting Eqs. (7) and (12) in Eq. (9) gives the final equation for the lifecycle cost of pavement and the same is presented in Eq. (13). N N Z tn X X Jn ¼ c1 sn1 ef 1 t ert dt þ ðm1 wi þ m2 Þertn ð13Þ 1
1
tn1
4.2. Constraints on design variables The formulated problem was solved using proposed ABC algorithm. Based on theoretical and practical considerations, limits on resurfacing frequency and resurfacing thickness were specified. A typical flexible pavement is designed for a useful life between 15 and 20 years. Any maintenance activity can be taken up only after new pavement is actually constructed and opened for traffic. Further, maintenance cost would be incurred only after pavement is opened for traffic. Recent long life pavement (life greater than 50 years) design methodologies adopt periodic resurfacing of 25 mm to 200 mm every 20 years [31]. Thus, 0 year and 40 year was specified as lower limit and upper limit on resurfacing frequency, respectively. The lower and upper limits on resurfacing thickness were taken as 0 mm and 200 mm, respectively. 5. Optimization of control parameters The proposed ABC algorithm required input regarding control parameters i.e. colony size, number of employed bees, and termination criteria. As these control parameters affect the quality of solution significantly, a parametric study was conducted initially. The colony size was varied between 10 and 200, and the algorithm was run for 5000 iterations. The number of employed bees was taken as half of the colony size. The variation of function value with number of iterations, for different colony sizes is presented in Fig. 3. It is clear from Fig. 3 that function value converged at a faster rate with a larger colony size. After 100
where CðsðtÞÞ = user cost as a function of pavement serviceability (s) accounting delay and vehicle maintenance costs at the time t; M(wn) = agency cost incurred to resurface for the nth cycle; r = discount rate (7% for life cycle cost calculation); N = number of resurfacing cycles. The rate of pavement deterioration is a function of serviceability (s) at that time as mentioned by Tsunokawa and Schofer [16]. Hence: ds ¼ F ðsðtÞÞ dt So substituting Eq. (5) in Eq. (9)
ð11Þ Fig. 3. Variation of function value with number of iterations and colony size.
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iterations, the function value exhibited stable region with little variation. This stable region was consistent for all colony sizes. Thus, colony size of 100 employed bees with 50 onlooker bees was adopted in this study.
5.1. Solution In general, pavement resurfacing cycle refers to small segment ending with resurfacing during its service life. The resurfacing problem was solved for different number of resurfacing cycles and the resulting thickness and resurfacing frequency are presented in Table 1. As seen in Table 1, lower resurfacing thickness with more frequent resurfacing is recommended. Tsunokawa and Schofer [16] reported that steady state optimal strategy would be to have 96 mm of resurfacing every 10 years. Li and Madanat [10] solved the same problem through exact solution approach and reported 154 mm resurfacing thickness and resurfacing frequency 26 years for trigger roughness value of 73. If 108 years of design life is assumed, total overlay thickness of 1036 mm is obtained using approach proposed by Tsunokawa and Schofer [16]. For the same problem, total overlay thickness of 639 mm is obtained using approach proposed by Li and Madanat [10]. These values are higher than the solution obtained in present research
using ABC algorithm. The differences between solution obtained using ABC approach and solution reported in literature can be attributed to (i) use of suboptimal trigger values for resurfacing, (ii) flexibility with proposed ABC algorithm to have different strategies for different maintenance cycles, and (iii) numerical accuracy of proposed ABC algorithm. In case of steady state condition (as in case of published literature), it is assumed that overlay of certain thickness at regular frequencies is sufficient to minimize overall life cycle cost of pavement. Also, the roughness at which overlay has to be undertaken is assumed beforehand. However, both are not true in a realistic pavement resurfacing problem.
5.2. Effect of initial roughness In this research, the effect of initial roughness level on the lifecycle cost of the pavement was evaluated. The pavement life in this case was taken as five resurfacing cycles. The resurfacing frequency, and resurfacing thickness obtained for various initial roughness conditions using ABC algorithm are presented in Table 2. In general, more frequent but lower resurfacing thickness is obtained as solution using proposed ABC algorithm. This is in line with generally accepted criteria that more frequent mainte-
Table 1 Solution of resurfacing problem for different cycles by ABC algorithm. No. of cycles
Resurfacing thickness (mm) w2
w3
w4
3 4 5 6
115 115 115 115
115 115 115 115
116 116 116 116
Resurfacing frequency (years) w5 104 104 104
w6
110 110
t1
t2
t3
t4
t5
t6
117
28 28 28 28
20 20 20 20
20 20 20 20
17 17 17
23 23
20
Table 2 Variation of resurfacing frequency and thickness with initial roughness level. Parameter
Initial roughness level
ABC algorithm
Resurfacing frequency (years)
20 30 40 50 20 30 40 50
28 20 15 13 115 144 129 120
Resurfacing thickness (mm)
QI QI QI QI QI QI QI QI
Li and Madanat [10]
20 16 17 19 115 80 79 78
20 11 12 11 116 71 108 94
17 11 12 19 104 105 108 107
23 11 10 15 110 104 72 100
26 20 16 13 154 116 83 55
Table 3 Variation of NPV with initial roughness level. Initial roughness level
20 30 40 50
QI QI QI QI
Computed NPV ($)
Percentage difference
ABC algorithm
Li and Madanat (10)
590390 819499 979327 986682
600000 820000 1010000 1170000
1.60 0.06 3.03 15.66
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nance is better approach. As seen in Table 2, numerical values of resurfacing thickness and frequency are different for various resurfacing cycles (at a given initial roughness level). Also, the numerical values obtained in this study (for both overlay thickness and resurfacing frequency) are different from those recommended by Li and Madanat [10]. The reasons for these differences between two studies can be attributed to relaxation of steady state condition requirement. In other words, assumption of steady state condition [as in Li and Madanat [10]] implies that trigger roughness and overlay thickness will be same for all resurfacing cycles. The NPV’s obtained for different initial roughness values are presented in Table 3. From Table 3, it is clear that increase in initial roughness level results in corresponding increase in NPV. This is due to fact that pavement at poor condition initially will require more maintenance over a period of time. Such maintenance activities over a period of time require more investment from agency. The values obtained using ABC algorithm (except initial roughness = 50 QI) are in close agreement with those obtained by Li and Madanat [10]. The difference in numerical values can be attributed to approximations used in both approaches. 6. Conclusion This article offers ABC methodology based solution to the question ‘‘when transportation agencies should schedule resurfacing maintenance activity under given constraints?” The proposed ABC algorithm mimics the collective behaviour of bees searching for best nectar source while solving an optimization problem. The proposed method accounts various factors like resource availability, economic considerations, engineering considerations while optimizing for resurfacing intensity and frequency. The proposed algorithm offers flexibility to designer in terms of having different maintenance strategies for various maintenance cycles. This flexibility helps in realistic modelling of the pavement resurfacing with little computational effort. The solution obtained using proposed approach slightly deviated from exact solution approach reported in literature due to removal of several assumptions. References [1] J.V. Camahan, W.J. Davis, M.Y. Shahin, P.L. Keane, M.I. Wu, Optimal maintenance decisions for pavement management, J. Transp. Eng. 113 (5) (1987) 554–572. [2] A. Shekharan, Solution of pavement deterioration equations by genetic algorithms, Transp. Res. Rec. 1699 (2000) 101–106. [3] K.D. Johnson, K.A. Cation, Performance prediction development using three indexes for North Dakota pavement management system, Transp. Res. Rec. 1344 (1992) 22–30. [4] G. Lamptey, Optimal Scheduling of Pavement Preventive Maintenance Using Life Cycle Cost Analysis (MS thesis), Purdue University, West Lafayette, Indiana, USA, 2004.
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Speed Railway, Heavy Haul Railway and Urban Rail Transit, Beijing Jiaotong University, Beijing, pp. 747–751. [29] W.H. Li, W.J. Li, Y. Yang, H.Q. Liao, J.L. Li, X.P. Zheng, Artificial bee colony algorithm for traveling salesman problem, Adv. Mater. Res. 314 (2011) 2191–2196.
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ScienceDirect International Journal of Pavement Research and Technology xxx (2018) xxx–xxx www.elsevier.com/locate/IJPRT
An improved artificial bee colony algorithm for pavement resurfacing problem Tapas Ranjan Panda a,1, Aravind Krishna Swamy b,⇑ a
Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India b Department of Civil Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India Received 31 July 2016; received in revised form 30 March 2018; accepted 12 April 2018
Abstract Pavement resurfacing is a maintenance activity that is undertaken to enhance the service life of pavement. This pavement resurfacing activity involves laying a new layer of asphalt concrete over existing pavement after certain time. Due to engineering factors, economic variables and uncertainty in forecasting, the pavement resurfacing decision process is a complicated activity. Various optimization approaches currently used are simplified models for finding optimal frequency and resurfacing intensity within pavement maintenance framework. In this research, artificial bee colony algorithm is proposed to solve this pavement resurfacing optimization problem. This algorithm mimics the collective behaviour of bees while searching for nectar. In this approach, various scenarios are generated, optimality of each case is evaluated, and the information thus generated is used in subsequent evaluation until global optimality is reached. The effectiveness of proposed method is demonstrated through a numerical example. The solution obtained is similar to the exact solution reported in literature. The results indicate optimal resurfacing values can be obtained with little computational effort with proposed approach. The main advantage of proposed algorithm is removal of specification of trigger values for maintenance decision. Ó 2018 Chinese Society of Pavement Engineering. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Pavement; Resurfacing; Artificial bee colony; Life cycle cost; Optimization
1. Introduction Under the combined action of traffic loading and climatic factors, the pavement deteriorates over a period of time. Beyond a certain level of deterioration, the pavement becomes unserviceable and the whole pavement has to be reconstructed. To avoid this expensive reconstruction process, maintenance activity like resurfacing is taken up. Camahan et al. [1] reported that resurfacing is less expensive treatment that decreases overall life cycle cost while increasing service life of pavement. Once the pavement is
⇑ Corresponding author. Fax: +91 11 2658 1117.
E-mail address: [email protected] (A.K. Swamy). Formerly Master’s Student. Peer review under responsibility of Chinese Society of Pavement Engineering. 1
resurfaced, serviceability level increases to a certain extent. After this resurfacing, the deterioration process restarts and continues until next cycle of resurfacing. It is generally accepted that roughness of pavement is an indicator of pavement serviceability/pavement deterioration. Any increase in roughness leads to reduction in ride quality of pavement. Fig. 1 shows changes in pavement roughness with time. Due to simultaneous action of traffic and environment, the deterioration process (or roughness increase) is continuous and nonlinear in nature. Various trends like power form [2], polynomial form [3], and exponential form [4,5] have been used to describe this roughness progression. On the other hand, resurfacing activity leads to sudden drop in roughness level. This decision on pavement resurfacing is made by highway agencies based on network condition, budget constraints, and resources availability. Often engineer’s experience, subjective judgement, and historical
https://doi.org/10.1016/j.ijprt.2018.04.001 1996-6814/Ó 2018 Chinese Society of Pavement Engineering. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: T.R. Panda, A.K. Swamy, An improved artificial bee colony algorithm for pavement resurfacing problem, Int. J. Pavement Res. Technol. (2018), https://doi.org/10.1016/j.ijprt.2018.04.001
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Fig. 1. Variation of pavement roughness during service life of pavement.
practice plays important role in this decision process [6]. This continuous increase in roughness and sudden drop in roughness results in a saw-tooth trend in pavement roughness during its service life. Exact shape/trajectory of this saw-tooth curve depends on traffic, climate, construction quality, and maintenance interventions [7]. Several optimization approaches have been proposed that aid in decision process of pavement resurfacing activity. These optimization approaches ranges from single pavement rehabilitation event in finite horizon to multiple pavement maintenance activities over infinite horizon [8–11]. Several researchers [10,12] have used deterministic approach while Pasupathy et al. [7] have used stochastic approach. Even though deterministic approach aids in arriving at exact solution for pavement resurfacing optimization problem, it deviates from real situation significantly [13]. In this research, Artificial Bee Colony (ABC) algorithm is used to solve this pavement resurfacing problem. The proposed methodology accommodates various factors like user costs, agency costs, inflation rate, interest rate, and achievable roughness levels during pavement resurfacing cycles. The frequency and thickness of resurfacing are obtained as solution while ensuring maximum cost effectiveness. Within ABC algorithm framework, various improvements like generation of multiple colonies (i.e. possible solutions), achieving global solution are proposed in this article. The main advantage of proposed ABC algorithm for pavement resurfacing problem is elimination of trigger roughness level specification beforehand. This paper contains 6 sections of which this is first one. Section 2 presents a review of the relevant literature in pavement resurfacing optimization and ABC. Section 3 describes the model formulation used in optimization of pavement resurfacing. Section 4 describes the proposed solution procedure using ABC algorithm while a case study is presented in Section 5. Finally conclusions of the paper are presented in Section 6. 2. Literature review Control theory has been used by various researchers to optimize pavement maintenance activities [8,14,15]. Control theory essentially assumes that pavement condition
changes continuously over time. However, any maintenance activity leads to discontinuous pavement serviceability function. Pasupathy et al. [7] reported that control theory problems with discontinuous response is computationally interactable and is quite cumbersome. To handle this discontinuity issue, Tsunokawa and Schofer [16] suggested saw-tooth pavement roughness curve. In this approach, the saw-tooth curve is replaced by a continuous function that passes through the midpoints of the spikes and discrete resurfacing activity is replaced by continuous resurfacing rate. Thus, solution obtained using the approach proposed by Tsunokawa and Schofer [16] will yield approximate results. Tsunokawa et al. [17] proposed gradient search methods to obtain optimal pavement maintenance strategy. However, these gradient search methods have inherent disadvantage of getting trapped in local minima with wrong initial guess. Li and Madanat [10] used the concept of saw-tooth roughness profile to optimize the frequency and intensity of pavement resurfacing actions under steady-state conditions. They provided a simple, practical, and robust method that uses the concept of trigger roughness level which initiates resurfacing activity. The specification of predefined trigger roughness level essentially requires one to have prior information regarding optimality. Since this trigger roughness level is a part of optimization process, such an exercise might not be meaningful. If such an assumption has to be made, one has to resort to extensive parametric study before resorting to optimization. Ouyang and Madanat [12] proposed an analytical solution to the pavement resurfacing problem for a finite horizon using saw-tooth roughness curve. Sathaye and Madanat [18] extended this concept to multiple facility optimization problem. To obtain feasible solution, most of the work discussed above assume (i) optimization problem as deterministic, (ii) approximate discrete changes as continuous phenomenon, and (iii) certain functional form for deterioration function. Thus, solution obtained by these optimization approaches deviate from real situation significantly [13]. Under such circumstances, metaheuristic approaches have proved to be effective. The goal of present research is to apply one such metaheuristic optimization algorithm to solve this pavement resurfacing problem. Bio-inspired metaheuristic algorithms are very popular for solving complex engineering problems. Ant colony optimization [19], genetic algorithm [20], particle swarm optimization [21], ABC algorithm [22] are some examples for such bio-inspired optimization algorithms. All these algorithms mimic the collective behaviour of insect/animal groups through complex interaction of individuals without supervision. These optimization approaches offer advantages like scalability, adaptation, speed, modularity and efficiency [23]. Karaboga [22] introduced ABC algorithm for solving multimodal, multi-dimensional numerical unconstrained optimization problem. This algorithm was inspired by the intelligent nectar foraging behaviour of honeybees in a colony. In a real bee colony, tasks like identification of poten-
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Fig. 2. Flowchart of proposed ABC algorithm with multiple colonies.
tial sources of nectar, employing number of bees to bring nectar from a source to hive, and storage of nectar are performed by efficient division of labour, exchange of information and self-organization. To perform all these tasks, bees in a particular colony perform different roles at different stages. In the initial phase of the foraging process, the bees start to explore the environment randomly in order to find a nectar source. These bees looking for new sources of nectar are referred to as scouts. After finding a nectar source, scouts become employed bees and start exploiting the identified nectar source. The employed bee returns to the hive with the nectar and unloads the nectar. After unloading the nectar, employed bee shares information about current nectar source with other bees. Based on the information shared by the employed bees, onlooker bees (those waiting in hive) decide on current source or new nectar source to exploit. The employed bee whose nectar source is discarded becomes the scout bee to search new nectar source. This process is continued until all the
requirements regarding nectar source, and storage are met. In an engineering optimization problem, the position of a nectar source represents a possible solution to the optimization problem, and the nectar amount of a nectar source corresponds to the fitness of the associated solution. ABC algorithm has been successfully applied in various fields including signal processing [24], parameter optimization of a multi-pass milling process [25], inverse problem in heat conduction [26], and behaviour of confined plasma in a nuclear fusion device [27]. In civil engineering, ABC algorithm has been applied in the area of subway route location [28], travelling salesman problem [29], structural optimization of planar and space trusses [30]. 3. Proposed methodology This section presents ABC based optimization approach to solve the problem of pavement resurfacing. In this proposed approach, the exploration and the exploitation capa-
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bility of the algorithm is enhanced by employing multiple colonies that are spread throughout the search space. Overall optimization framework consists of several steps i.e. (i) input regarding solution space, (ii) subdivision of solution space into smaller sub-search spaces, (iii) mapping these sub-search spaces to individual bee colonies, (iv) computation of optimal solution within sub-search spaces, (v) comparison of potential global best solution with optimal solution in sub-search space, and (vi) selection of global optimal solution. The proposed algorithm is presented in the Fig. 2 and explained in detail below. In the first step, approximate range of decision variables (i.e. resurfacing thickness and resurfacing frequency) within which output is expected is specified as solution space limits. This specification regarding solution space limits is essential to avoid excessive computational time and effort. These limits on solution space may be specified based on intuition, historical data, available expertise, and theoretical justifications. In the second step, this solution space is divided into smaller sub-search spaces. These small sub-search spaces are assigned to individual bee colonies. The number of smaller sub-search spaces (i.e. number of colonies) depends on nature of search space. A relatively smooth objective response surface requires less number of colonies. The number of bees in a colony is kept to a minimum without any compromise on quality of solution. Further, 50% of bees in a particular colony are designated as employer bees and are spread throughout the assigned sub-search space. Each bee is assigned to one nectar source (i.e. potential optimal values) within sub-search space. The number of onlooker bees on a particular solution depends upon the fitness of the solution. In the third step, nectar sources (XijN) are initialised in the sub-search space as in Eq. (1). This equation controls the production of neighbour nectar sources within boundaries of sub-search space. The fitness of nectar sources (F (xi)) is evaluated using Eq. (2) [25]. The fitness of each source is determined using objective function value (Fi). This fitness defines the quality of the solution obtained in each iteration. These fitness values aid in identifying the quality of potential nectar sources within sub-search space. Depending on the quality of nectar sources, some nectar sources are selected for subsequent steps. New random nectar sources (i.e. possible solutions) are generated near the solution obtained in previous step using Eq. (3). Further, greedy selection process is applied between the solutions for the employed bees to find better solution. Probability for each solution is calculated by comparing fitness of a solution (Fi) to the total fitness of all solutions using Eq. (4). This equation essentially helps in finding quality of potential nectar sources and ranking the same. The numerical values of potential nectar sources having higher probability are used for subsequent interaction. New nectar sources (i.e. possible solutions) are generated depending on the probability of the previous solutions within each small sub-search space. After comparison of all solutions within each small sub-search space, the local best solution
is memorized. This process is repeated until the termination criterion is reached. max min X ijN ¼ X min jN þ ;iN ðX jN X jN Þ
ð1Þ
where ;ijN = random number between [0,1]; i = 1 to number of nectar sources within each sub-search space; j = 1 to number of decision variables (in this research maximum value of 2); N = 1 to number of sub-search spaces. ( 1 þ jF i j if F i < 0 F ðxi Þ ¼ ð2Þ 1 if F i P 0 1þF i vijN ¼ X ijN þ ;ijN ðX ijN X kjN Þ where ; k 2 f1; 2; . . . Number of employed beesg j 2 f1; 2; . . . Number of optimization parameterg are domly chosen indexes. Fi p i ¼ Pn
i¼1 F i
ð3Þ and ranð4Þ
where n = number of nectar sources. In the first sub-search space, local best solution is designated as global best solution. For subsequent sub-search spaces, local best solution in current sub-search space is compared with potential global best solution and better one is designated as potential global best solution. Updated potential global best solution is fed back for the next colony. The above procedure is repeated for all sub-search spaces and final potential global best solution is designated as optimal solution. 4. Numerical example To demonstrate the applicability of the proposed optimization methodology to the pavement resurfacing problem, a numerical example is presented in this section. 4.1. Problem statement It is assumed that resurfacing activity will be taken up on 1 km long lane of a highway pavement. Expressions for deterioration function F(s), improvement function G (w, s), user cost function C(s) and agency cost function M (w) are presented in Eqs. (5)–(8), respectively. Here serviceability is expressed in terms of Quality Index (QI), the time is taken in years, resurfacing thickness is in millimetres and cost is taken in dollars per km of pavement. Based on a field study mentioned by Tsunokawa and Schofer [16] road deterioration rate is a function of serviceability and given in Eq. (5). After an overlay with a thickness w, the serviceability of resurfaced pavement improves as given in Eq. (6). F ðsÞ ¼ f 1 s
pffiffiffiffi Ds ¼ Gðw; sÞ ¼ g1 w þ g2 s þ g3
ð5Þ ð6Þ
CðsÞ ¼ c1 s þ c2
ð7Þ
MðwÞ ¼ m1 w þ m2
ð8Þ
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where s = serviceability of the pavement and, f 1 ; gi ; ci , mi are constants. Deterioration parameter f1 was assumed to be 0.05 per year. The improvement function parameters i.e. g1 ; g2 ; g3 are taken as 5.0, 0.78, 66, respectively. The numerical values of c1 and c2 are taken as 1000 $/QI/year and 0 $/year, respectively. The variable and fixed cost parameters for a resurfacing on that 1 km lane are assumed to be m1 = 11,000 $/mm, m2 = 150,000 $. Rate of variable user costs per one QI unit on this facility was taken as 0.0002 $/QI/ veh, and the annual traffic volume in that lane was assumed as 5 106 veh/year. This would indicate that pavement at good condition (QI = 25) will deteriorate to poor condition (QI = 100) in about 28 years. All these values have been taken from published literature [10,16]. The life cycle cost of the pavement can be calculated by Net Present Value (NPV) approach [16]. The NPV of the pavement includes both agency and user cost. The rehabilitation activity cost is the major contributor towards the agency cost. The user cost is contributed by two major components i.e. vehicle related and non-vehicle related. The vehicle related cost includes vehicle maintenance cost, fuel cost etc. that are dependent on the pavement roughness. The cost of delay during rehabilitation activity is accounted towards non-vehicle related cost. Life cycle cost of the pavement is calculated by adding the continuous discounted user and agency costs for one cycle of resurfacing. User cost occurs throughout the cycle while agency cost occurs at the end of the cycle for the resurfacing. The final NPV is calculated by adding discounted costs of all the cycles. Objective is to minimize the total lifecycle cost of pavement. Additional details regarding this problem can be found elsewhere [10,16]. The sum of cost rate (as a function of roughness) incurred by user and agency during each maintenance cycle is given in Eq. (9). The total cost incurred by user and agency during pavements life cycle can be obtained by summation of incurred cost during each cycle. The same is given in Eq. (10). Z t n Jn ¼ CðsðtÞÞert dt þ Mðwn Þertn ð9Þ tn1
NPV ¼
N X Jn
ð10Þ
1
Z
s1 s0
ds ¼ f 1s
Z
5
t
dt ¼> s1 ¼ s0 ef 1 t
ð12Þ
0
Eq. (12) explains that if s0 is the initial pavement condition, after a time the condition of pavement will be s0 ef 1 t . Substituting Eqs. (7) and (12) in Eq. (9) gives the final equation for the lifecycle cost of pavement and the same is presented in Eq. (13). N N Z tn X X Jn ¼ c1 sn1 ef 1 t ert dt þ ðm1 wi þ m2 Þertn ð13Þ 1
1
tn1
4.2. Constraints on design variables The formulated problem was solved using proposed ABC algorithm. Based on theoretical and practical considerations, limits on resurfacing frequency and resurfacing thickness were specified. A typical flexible pavement is designed for a useful life between 15 and 20 years. Any maintenance activity can be taken up only after new pavement is actually constructed and opened for traffic. Further, maintenance cost would be incurred only after pavement is opened for traffic. Recent long life pavement (life greater than 50 years) design methodologies adopt periodic resurfacing of 25 mm to 200 mm every 20 years [31]. Thus, 0 year and 40 year was specified as lower limit and upper limit on resurfacing frequency, respectively. The lower and upper limits on resurfacing thickness were taken as 0 mm and 200 mm, respectively. 5. Optimization of control parameters The proposed ABC algorithm required input regarding control parameters i.e. colony size, number of employed bees, and termination criteria. As these control parameters affect the quality of solution significantly, a parametric study was conducted initially. The colony size was varied between 10 and 200, and the algorithm was run for 5000 iterations. The number of employed bees was taken as half of the colony size. The variation of function value with number of iterations, for different colony sizes is presented in Fig. 3. It is clear from Fig. 3 that function value converged at a faster rate with a larger colony size. After 100
where CðsðtÞÞ = user cost as a function of pavement serviceability (s) accounting delay and vehicle maintenance costs at the time t; M(wn) = agency cost incurred to resurface for the nth cycle; r = discount rate (7% for life cycle cost calculation); N = number of resurfacing cycles. The rate of pavement deterioration is a function of serviceability (s) at that time as mentioned by Tsunokawa and Schofer [16]. Hence: ds ¼ F ðsðtÞÞ dt So substituting Eq. (5) in Eq. (9)
ð11Þ Fig. 3. Variation of function value with number of iterations and colony size.
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iterations, the function value exhibited stable region with little variation. This stable region was consistent for all colony sizes. Thus, colony size of 100 employed bees with 50 onlooker bees was adopted in this study.
5.1. Solution In general, pavement resurfacing cycle refers to small segment ending with resurfacing during its service life. The resurfacing problem was solved for different number of resurfacing cycles and the resulting thickness and resurfacing frequency are presented in Table 1. As seen in Table 1, lower resurfacing thickness with more frequent resurfacing is recommended. Tsunokawa and Schofer [16] reported that steady state optimal strategy would be to have 96 mm of resurfacing every 10 years. Li and Madanat [10] solved the same problem through exact solution approach and reported 154 mm resurfacing thickness and resurfacing frequency 26 years for trigger roughness value of 73. If 108 years of design life is assumed, total overlay thickness of 1036 mm is obtained using approach proposed by Tsunokawa and Schofer [16]. For the same problem, total overlay thickness of 639 mm is obtained using approach proposed by Li and Madanat [10]. These values are higher than the solution obtained in present research
using ABC algorithm. The differences between solution obtained using ABC approach and solution reported in literature can be attributed to (i) use of suboptimal trigger values for resurfacing, (ii) flexibility with proposed ABC algorithm to have different strategies for different maintenance cycles, and (iii) numerical accuracy of proposed ABC algorithm. In case of steady state condition (as in case of published literature), it is assumed that overlay of certain thickness at regular frequencies is sufficient to minimize overall life cycle cost of pavement. Also, the roughness at which overlay has to be undertaken is assumed beforehand. However, both are not true in a realistic pavement resurfacing problem.
5.2. Effect of initial roughness In this research, the effect of initial roughness level on the lifecycle cost of the pavement was evaluated. The pavement life in this case was taken as five resurfacing cycles. The resurfacing frequency, and resurfacing thickness obtained for various initial roughness conditions using ABC algorithm are presented in Table 2. In general, more frequent but lower resurfacing thickness is obtained as solution using proposed ABC algorithm. This is in line with generally accepted criteria that more frequent mainte-
Table 1 Solution of resurfacing problem for different cycles by ABC algorithm. No. of cycles
Resurfacing thickness (mm) w2
w3
w4
3 4 5 6
115 115 115 115
115 115 115 115
116 116 116 116
Resurfacing frequency (years) w5 104 104 104
w6
110 110
t1
t2
t3
t4
t5
t6
117
28 28 28 28
20 20 20 20
20 20 20 20
17 17 17
23 23
20
Table 2 Variation of resurfacing frequency and thickness with initial roughness level. Parameter
Initial roughness level
ABC algorithm
Resurfacing frequency (years)
20 30 40 50 20 30 40 50
28 20 15 13 115 144 129 120
Resurfacing thickness (mm)
QI QI QI QI QI QI QI QI
Li and Madanat [10]
20 16 17 19 115 80 79 78
20 11 12 11 116 71 108 94
17 11 12 19 104 105 108 107
23 11 10 15 110 104 72 100
26 20 16 13 154 116 83 55
Table 3 Variation of NPV with initial roughness level. Initial roughness level
20 30 40 50
QI QI QI QI
Computed NPV ($)
Percentage difference
ABC algorithm
Li and Madanat (10)
590390 819499 979327 986682
600000 820000 1010000 1170000
1.60 0.06 3.03 15.66
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nance is better approach. As seen in Table 2, numerical values of resurfacing thickness and frequency are different for various resurfacing cycles (at a given initial roughness level). Also, the numerical values obtained in this study (for both overlay thickness and resurfacing frequency) are different from those recommended by Li and Madanat [10]. The reasons for these differences between two studies can be attributed to relaxation of steady state condition requirement. In other words, assumption of steady state condition [as in Li and Madanat [10]] implies that trigger roughness and overlay thickness will be same for all resurfacing cycles. The NPV’s obtained for different initial roughness values are presented in Table 3. From Table 3, it is clear that increase in initial roughness level results in corresponding increase in NPV. This is due to fact that pavement at poor condition initially will require more maintenance over a period of time. Such maintenance activities over a period of time require more investment from agency. The values obtained using ABC algorithm (except initial roughness = 50 QI) are in close agreement with those obtained by Li and Madanat [10]. The difference in numerical values can be attributed to approximations used in both approaches. 6. Conclusion This article offers ABC methodology based solution to the question ‘‘when transportation agencies should schedule resurfacing maintenance activity under given constraints?” The proposed ABC algorithm mimics the collective behaviour of bees searching for best nectar source while solving an optimization problem. The proposed method accounts various factors like resource availability, economic considerations, engineering considerations while optimizing for resurfacing intensity and frequency. The proposed algorithm offers flexibility to designer in terms of having different maintenance strategies for various maintenance cycles. This flexibility helps in realistic modelling of the pavement resurfacing with little computational effort. The solution obtained using proposed approach slightly deviated from exact solution approach reported in literature due to removal of several assumptions. References [1] J.V. Camahan, W.J. Davis, M.Y. Shahin, P.L. Keane, M.I. Wu, Optimal maintenance decisions for pavement management, J. Transp. Eng. 113 (5) (1987) 554–572. [2] A. Shekharan, Solution of pavement deterioration equations by genetic algorithms, Transp. Res. Rec. 1699 (2000) 101–106. [3] K.D. Johnson, K.A. Cation, Performance prediction development using three indexes for North Dakota pavement management system, Transp. Res. Rec. 1344 (1992) 22–30. [4] G. Lamptey, Optimal Scheduling of Pavement Preventive Maintenance Using Life Cycle Cost Analysis (MS thesis), Purdue University, West Lafayette, Indiana, USA, 2004.
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Please cite this article in press as: T.R. Panda, A.K. Swamy, An improved artificial bee colony algorithm for pavement resurfacing problem, Int. J. Pavement Res. Technol. (2018), https://doi.org/10.1016/j.ijprt.2018.04.001