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Bee Inspired Novel Optimization Algorithm and Mathematical Model for Effective and Efficient Route Planning in Railway System

Bee Inspired Novel Optimization Algorithm and Mathematical Model for Effective and Efficient Route Planning in Railway System

  • Kah Huo Leong, 
  • Hamzah Abdul-Rahman, 
  • Chen Wang, 
  • Chiu Chuen Onn, 
  • Siaw-Chuing Loo
PLOS
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Correction

23 May 2017: The PLOS ONE Staff (2017) Correction: Bee Inspired Novel Optimization Algorithm and Mathematical Model for Effective and Efficient Route Planning in Railway System. PLOS ONE 12(5): e0178583. https://doi.org/10.1371/journal.pone.0178583 View correction

Abstract

Railway and metro transport systems (RS) are becoming one of the popular choices of transportation among people, especially those who live in urban cities. Urbanization and increasing population due to rapid development of economy in many cities are leading to a bigger demand for urban rail transit. Despite being a popular variant of Traveling Salesman Problem (TSP), it appears that the universal formula or techniques to solve the problem are yet to be found. This paper aims to develop an optimization algorithm for optimum route selection to multiple destinations in RS before returning to the starting point. Bee foraging behaviour is examined to generate a reliable algorithm in railway TSP. The algorithm is then verified by comparing the results with the exact solutions in 10 test cases, and a numerical case study is designed to demonstrate the application with large size sample. It is tested to be efficient and effective in railway route planning as the tour can be completed within a certain period of time by using minimal resources. The findings further support the reliability of the algorithm and capability to solve the problems with different complexity. This algorithm can be used as a method to assist business practitioners making better decision in route planning.

Introduction

Railway system (RS) is growing in popularity for major cities around the world as it provides significant transit capacity and become an essential infrastructure that is needed to serve growing transportation demands. One of the main reasons RS is developed in the city is to reduce the traveling cost but due to expansion of the transit networks and structures, often, unplanned travel will cause unnecessary waste of time and passengers congestion in the stations [1]. Transit planning can be an extremely complex due to multiple variables that will affect the quality of the solution [2, 3]. For instance, departure time, train intervals, operating characteristics, minimal and maximum headway of lines. There are many optimization algorithms to solve routing problems evolved such as ant colony optimization, greedy algorithm, genetic algorithm, termite colony algorithm and bat algorithm but very limited literature is found related to Railway Traveling Salesman Problem (RTSP) [4].

In Malaysia, railway transport is one of the most commonly used public transportation by many Malaysians, business practitioners and travellers to travel from one destination to another daily. LRT, KL Monorail, Airport Express Rail Link and KTM Commuter are the lines that connect in the Malaysian RS (Fig 1). According to Brenda Ch’ng [5], RS in Malaysia has a daily ridership of more than half a million in year 2014 and the number is expected to doublerf when new lines are ready in the future. People tend to use this mode of transport to travel in the city in order to save time and costs by avoiding traffic congestion, spending time looking for parking bay and traveling on toll roads. When the RS network is expanded, choosing the shortest route to multiple destinations will be difficult due to the complexity of the network design and structure.

Background

Traveling Salesman Problem (TSP)

TSP is a mathematical problem introduced by Sir William Rowan Hamilton and Thomas Penyngton Kirkman in 18th century and later promoted by Hassler, Whitney & Merrill at Princeton [6]. TSP is to find the shortest possible route that enable the traveller visits each city exactly once and returns to the origin city [7]. There is a growing body of literature that recognises the importance of TSP and it is not only studied by mathematicians and computer scientists but researchers from other industries to find out how TSP can help in operations planning and decision making [8]. TSP is one of the most well-known routing problems that researchers are still looking for the best solutions yet [9]. In TSP, a salesman was given a set of cities along with the travel cost between these cities. The salesman’s task is to complete a tour by visiting all of those cities exactly once and then return to the point where the salesman started his journey. The primary objective is to find the best possible solution with the minimum cost to travel [10].

Solving TSP by using exhaustive search methods is possible but not practical because it will be very costly to generate solutions when possible routes exponentially increased. Due to this reason, no efficient solution to the general case of TSP (for all variants) has been found yet [9]. Route planning in railway systems is one of the variants of TSP. This suggests that the route planning in railway system requires a different set of variables in order to develop a technique or algorithm to solve the TSP for railway system. Unfortunately, the technique or algorithm to solve TSP for railway system is yet available in the existing body of knowledge. Therefore, no current TSP technique or algorithm can be applied in solving the route planning in railway systems. Therefore, this study attempts to bridge the gap by developing a TSP solution for route planning in railway system. In spite of the computational difficulty of the problem, various known techniques have been introduced by researchers to generate the best solution to the problem. These techniques can be classified into 2 categories that are exact and approximation algorithms [11].

Swarm Intelligence (SI)

SI is known as an efficient meta-heuristic approximation algorithm used by researchers to solve TSP and optimization related problems [12]. SI was introduced in 1989 by Beni and Wang [13] and later defined by Bonabeau as the emergent collective intelligence of groups [12]. It is broadly defined as a group of individuals acting collectively in ways that seem intelligent and often inspired from natural or artificial process [7]. It constituted a swarm of simple agents interacts locally with each other and environment to discover the unknown knowledge [14]. Examples of swarm intelligence based models are ant colony optimization, particle swarm optimization, bee colony optimization and artificial immune system. A modified bee colony optimization algorithm is proposed in this paper to solve Railway Traveling Salesman Problem (RTSP).

Bee algorithm as approximation method.

Meta-heuristic and approximation method are defined as the upper-level general methodologies and it can be used as a guidance of strategies to design underlying heuristics to solve specific optimization problems [15]. Intensification and diversification are the two important characteristics of meta-heuristic methods. Intensification is selecting the best candidates from the best solutions gathered and diversification is making sure that the algorithm works efficiently to explore the search space randomly [16].

Bee inspired algorithm such as Bee Colony Optimization (BCO) has been successfully used to solve many problems in engineering, operations and management related fields [17]. The general idea of BCO is constructing multi agent system that consists of artificial bees in a colony, where they will find the best solution during the process of collecting nectars. Bee behaviour in nature has inspired researchers to design various algorithms and solutions to solve difficult combinatorial optimization problems such as TSP [18]. Although various social insect species based algorithms have successfully solved various complex problems, Teodorovic [18] claimed that bee behaviour in nature has inspired more significant solutions to the problems. According to Aghazadeh and Meybodi [19], bees will adapt their behaviour according to the environment to accomplish a task by using collective intelligence. Basically, all the insect colonies have their own division of work based on their system and this applies to bee colonies as well.

For instance, honeybee colony is distributed in multiple directions for long distances at a same time in order to find more food sources [20]. The deployment of its foragers to better fields is the success criteria of the bee colony. The bee colony follows the rules that if the flower was patched with plenty amount of nectar then the flower will be visited by more bees and vice versa. Baykasoglu, Ozbakir and Tapkan [21] identified food and foragers are the two important criteria in a bee system (Table 1).

According to Teodorovic, Davidovic and Selmic [22], new node will be analysed in the BCO algorithm and added to the partial TSP tour identified in every single step (Table 2). This process is done by a random manner with a certain probabilities. In backward pass process, each bee will decide whether to abandon the partial solution that is generated or keep it. The bees will expand the previous generated partial solution after the selection has been made by a predefined number of nodes during next forward pass, followed by the second backward pass and return to the hive. The decision process will be repeated until complete solution is obtained.

In recent studies conducted by Nikolic and Teodorovic [23], they highlighted that in order to design an effective transit network, several issues need to be solved in order to increase number of satisfied riders and at the same time reduce the total time to complete a tour. The optimal solution of transit network design issue is difficult to find which makes it falls under the class of hard combinatorial optimization problem and difficult to be solved without a proper method applied. Therefore, Eq 1 was introduced in the study to calculate the total travel time of all passengers in the network, (1) where:

TT–total in-vehicle time of all served passengers.

TW–total waiting time of all served passengers.

TTR–total time penalties for all passengers’ transfers (usually time penalty is equal to 5 min per transfer).

Eq 2 is used to calculate the bee’s partial solutions as described in the BCO model: (2) Where Tb denotes the total travel time generated by the bth bee while Ob denotes the normalized value of the total travel time. Tmin and Tmax are the smallest and largest total travel time in the transit networks generated by all bees. To calculate the loyalty of the bee to the previous solution that is generated, Eq (3) is used.

(3)

Omax indicates the maximal normalized value of all bees (Ob). By using this formula (3), a bee can decide whether to become a follower or not. The higher Ob value, the higher the probability of the bee to become a follower and become loyal to the generated solution [1].

Exact method.

Exact methods are the techniques used by researchers before the heuristic method was introduced and can be considered as the traditional method in solving TSP. Some of the exact methods that are widely used to solve TSP are brute force, dynamic and linear programming [10]. A recent study by Sahalot and Shrimali [24] found that brute force method is a common method used when developing a solution to solve TSP related cases. The brute force method is basically made up of processes that generate all possible tours and calculate every tours distance. The best tour will be the one that with shortest tour identified by using mathematical method (Table 3).

Awuni [25] identified that brute force approach returned best and most accurate solution all the time, but it is only worked to problems that involve less than 10 cities. Typically, a computer can compute all possible path and distances in a couple of second if the cities are less than 10 where up to 3,628,800 possible routes will be analysed. If only the problem add one more city, the number of possible route will rise by 1000% and this will increase the server load significantly, which is not feasible to be implemented in super computer. Hence, heuristic methods such as bee, ant and genetic algorithms are used to generate the best possible solutions.

Method

The algorithm is designed to address the prevalent issues of choosing the best route to multiple destinations via RS before going back to the starting point, which can be considered as a variant of TSP. Since exact method capable of generating very reliable solutions in RTSP, solutions generated by using brute force and constraints based on Eq (1) are used as a benchmark for verification purposes. 10 test cases are used to evaluate and compare the solutions generated by the algorithm to the exact solutions. Brute force method is used to search all possible routes that can reach the desired stations before proposing an optimum route at the end of the analysis. This method is slow but accurate in getting the best optimum route to the stations desired provided enough time is given to analyse all paths in the network. In terms of algorithmic complexity, this method is easy to implement but it will be very time consuming depending on the complexity of the RS design.

Thorough observation survey has been conducted to obtain the real travel time to every station in Malaysia RS for algorithm verification usage [26]. The observation approach includes the process of timing and recording the time taken from one station to the next station in minute. To increase the reliability of the data obtained from the observation approach, two observers have been assigned to perform the observation. The first observer controlled the stopwatch and the other observer recorded the time observed. To ensure the data has the consistency, the observers have gone to each station 3 times during different hour of the day to verify the data collected. Variables defined in this survey are travel time from one station to the next closest stations, transit lines, operators, stations name and type. Besides, part of the data required in the verification process is obtained and verified with information obtained from Myrapid official web site (http://www.myrapid.com.my) and google map tool (https://www.google.com/maps).

Data obtained is used to generate 10 RTSP test cases with different settings to examine the reliability of the solutions generated by the algorithm. In the test cases, we have assumed that a salesman has to plan his tour so he manages to attend multiple meetings in Klang Valley by using RS starting from the station nearest to his office and then returning back to the same station at the end of the tour. To avoid analysis error and minimize bias, a simple PHP random function has been used to randomly pick the desired stations included in the tour (Table 4). There are 3 levels of complexity defined in the cases created. 5 cases involve 4 stations, 2 cases with 5 stations and another 3 with 6 stations in the tour. The same PHP function has been used to select initial station in all cases to avoid bias in the research.

The solutions obtained from both exact and heuristic methods were compared to ascertain the accuracy and reliability of the output. This verification approach is limited by the fact that obtaining exact solutions for complex RS is not likely but still, this approach will help as sanity check for algorithm proposed [27].

Results- Proposed Novel Bee Inspired Algorithm

Fig 2 shows the conceptual view of the solution and the algorithm designed is presented in Table 5. Mathematical model to present the five constraints considered in finding the optimum route (Table 6).

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Fig 2. Bee foraging concept used in the algorithm (Table 4) (Source: Adapted from Teodorovic, Davidovic and Selmic [22]).

https://doi.org/10.1371/journal.pone.0166064.g002

There are three major groups of bees in the bee foraging model referred. The groups are the employed bees, onlookers and scouts group [28]. These bees have their own tasks to find nectar around the hive. The information of the food source around the hive gathered by the employed bee will be shared with the onlooker bees. The onlooker bees will evaluate this information to start a neighbourhood search by using a probabilistic approach while a scout bee will perform a random search in order to find a food source [29]. The bees shared the information about their food source by performing a dance, known as the “waggle dance”. One study by Chauhan and Butani [30], shows that the onlooker bee will evaluate the information gathered from the employed bees and select the food source with a greater nectar amount. After that, the bees will memorize the new position of a higher nectar food source and forget or abandon the old one. Solution to RTSP can be easily obtained by using the novel optimization algorithm (Table 5) and mathematical model (Table 6) designed based on the bee foraging concept (Fig 2).

The objective function (E1) is used to find optimum route that have minimum traveling time to multiple destination before returning to the first station. Total tour traveling time is summation of TSG and XSG, where SG represents as node S to node G. This summation repeats until m, where m represents number of station to visit.

There are the 5 constraints to be considered in the mathematical model presented (Table 6):

First Constraint (E1)

The first constraint is to check if the number of routes is less than 6, probability (a) will be used. All the routes will be considered and the travel time for each route will be calculated. However, if there are more or equal than 6 routes, probability (b) will be used where only 50 percent of the potential routes will be analyzed. After determining the number of routes to be analyzed, constraints (E2) will be investigated. Number of possible routes is denoted as and number of routes that will be assigned to put into the set L.

Second Constraint (E2)

Tw is transiting time from one station to another in different line via the interchange

The second constraint (E2) is used to check whether transiting from node S to node G consumes any time. If the transit consume more than or equal to 1 minute is satisfied, then extra time will be added to the travel time from previous station to this interchange station

Third Constraint (E3)

The third constraint (E3) is to identify whether more than 1 route can lead to any G is identified. If condition more than 1 route that can lead to any G is satisfied, then extra 1 minute will be added for each station passed between nodes S to G. If the condition is not satisfied, station count in between 2 stations will be ignored. If the travel time for identified routes connected to G is same, optimum route will be selected randomly.

Fourth Constraint (E4)

Where i = 1, 2,…, n

i represent as number of each possible route that had been choose by using first constraint (E1)

The fourth constraint (E4) is used to select the route with the shortest travelling time among routes connected to the station. In the container, set L’ is known as numbers of possible routes and the route with minimum traveling time to be chosen.

Fifth Constraint (E5)

The last constraint is to decide the route and the travel time between node S to node G to be included in the optimum solution after potential routes comparison has been done by using third constraint (E3). If condition is satisfied and the route is chosen, then XSG is equal to 1.

Computational Results

The computational study evaluates the robustness of the algorithm through test case and numerical case study. In test case section, the algorithm is compared to the optimum route identified using exact method. In the following section, a representative application of the algorithm developed compared with exact and greedy methods generated from a TSP solver is presented via a numerical case study.

Test Case

In this test case, a salesman is required to attend multiple meetings at different locations, Bank Negara, Bandar Raya and KLCC via RS before returning to his office at Taman Jaya (Fig 3). The proposed algorithm has been used to identify the optimum route to the stations required before returning to first station. Table 7 demonstrated how the algorithm is applied to obtain the optimum route of the tour without using any computers.

Table 8 compares the results obtained from the algorithm and the brute force methods. It can be seen from the Table 8 that the result generated by the algorithm matched the optimum route identified using exact method.

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Table 8. Comparison of solutions between the proposed algorithm and exact method.

https://doi.org/10.1371/journal.pone.0166064.t008

Comparing the results from Table 9 below, it shows 9 out of the 10 solutions obtained from the test cases matched the exact solutions. With only one case (case 7) not matching the exact solution’s result, these results suggested that algorithm developed has the potential to perform better and generate reliable results as exact method when it is applied practically to solve TSP. This also further support the idea of using bee intelligence to solve complex problems can be productive.

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Table 9. Comparison of 10 test cases solutions generated by using exact and proposed algorithm.

https://doi.org/10.1371/journal.pone.0166064.t009

These findings could not be extrapolated to all RS in the world due to various constraints that will affect the reliability of the solutions. For instance, time required to transit to another line, stopping and waiting time in each station might is different if it is controlled by human, congestion of the station and train schedules. However, the tests were successful as it demonstrated that the effectiveness of the algorithm in solving the problem discussed. Further studies, which take these variables into account, will need to be undertaken to enhance the algorithm so it can be applied in different RS without much modification.

The research introduced a new algorithm and method that can be beneficial to business practitioners in enhancing the supply chain and RS transportation users who travel in complex network with hundreds of stations and interchanges such as RS in London, New York, China, Japan, India and Germany. Besides of serving as a future reference on the subject of swarm and collective intelligence in transportation planning and scheduling, the potential of using swarm intelligence in solving complex approximation and routing problems is uncovered. The research demonstrated that bee concept works effectively in RS route planning and could be groundbreaking approaches that will change the way people solve TSP and other operations science problems related to rail freight.

Numerical Case Study

In order to test the effectiveness of the algorithm in solving TSP, we have used a TSP solver [31] to create 100 cases with different number of vertices and then compared the solutions with the one generated by using the proposed algorithm. In this paper, we present one example taken from the 100 cases known as case 1 to show how the comparison is done among exact method, greedy method and our proposed algorithm.

Due to practicality and time complexity issues in generating TSP exact solutions with high number of vertices, the solver only allows up to 9 vertices in a graph (Fig 4). Awuni [25] claims the same in his TSP research paper where the brute force algorithm has to perm 10! to compare all routes before returning the solution and the number increases 1000% if an additional vertex is added into a graph. The main objective is not to beat the current best optimization algorithm in solving TSP but to examine the capability of the algorithm in solving TSP when all the constraints proposed are eliminated.

The solutions generated by the TSP solver for the graph with 9 vertices are shown in Fig 5 (for greedy algorithm or heuristics methods) and Fig 6 (for Brute-Force algorithm or exact methods).

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Fig 5. TSP solver solution for case 1 using greedy algorithm (Heuristics methods).

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Fig 6. TSP solver solution for case 1 using Brute-Force algorithm (Exact methods).

https://doi.org/10.1371/journal.pone.0166064.g006

Solving case 1 using proposed algorithm first requires consideration of constraints proposed in the algorithm. Since there is no interchange involved, thus the constraints proposed in the algorithm are not applicable in solving TSP. We have eliminated all constraints used in the algorithm so that it is in comparable term with TSP solver solutions (Fig 5 and Fig 6) to test how efficient is our proposed algorithm in solving TSP. Fig 7 is a redrawn diagram of case 1.

The proposed bee algorithm to solve case 1 is shown in Table 10. The process 1 to 18 of the proposed bee algorithm can be found in S1 Appendix together with the description of each process. The solution by using proposed algorithm is portrayed in Fig 8 and the comparison of case 1 results generated by exact methods, greedy methods and proposed algorithms is presented in Table 11.

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Table 11. Comparison of case 1 results generated by exact, greedy and proposed algorithm.

https://doi.org/10.1371/journal.pone.0166064.t011

All 100 solutions generated by the TSP solver and proposed algorithm in the case study are presented in S2 Appendix. The method proposed managed to generate 80 accurate solutions and the remaining 20 close to the exact solutions. The results obtained from the comparisons are summarized in Fig 9.

Fig 9 shows the results generated by proposed bee algorithm are comparable to the exact solutions with average 80% accuracy. Comparing the results, it can be seen that the increased number of vertices in the cases did not affect the efficiency of the algorithm. These results further support the idea of using the bee inspired algorithm to generate optimal solutions under different environment and constraints.

Conclusion

Travel planning in the business world is no longer uncommon and often related to one of the most important optimization problem, the Traveling Salesman Problem (TSP). Companies are starting to rely on early planning to achieve the objectives and even tourists can gain wide range of benefits from planning the optimum tour. A novel and verified optimization algorithm and mathematical model based on bee foraging cycle presented in this paper showed how it can be used to solve RTSP, a variant of TSP that received little attention from the researchers efficiently and effectively. This study and analysis also strengthened the idea that the heuristic method used can generate highly reliable solutions. The algorithm can be easily customised and implemented comparing to the exact methods that require higher computational time and resources. The algorithm can be replicated and applied on any RS to solve TSP related cases with different constraints and complexity. Findings of the research can be served as a base for future studies and extend the implication of swarm intelligence in solving TSP, enhancing the supply chain and tour planning. Given the complexity of route planning in cities with hundreds of stations connected in the network such as Tokyo, Seoul, London and New York, there is an opportunity for the use of collective intelligence to enhance the algorithm by using the Knowledge Discovery in Database (KDD) techniques and machine learning theory. Most importantly, the findings help to uncover critical areas in the RTSP that many researchers have not explored and provide opportunity to advance the understanding how swarm intelligence such as bees can be used in route planning.

Supporting Information

S1 Appendix. Flow chart of proposed bee algorithm.

https://doi.org/10.1371/journal.pone.0166064.s001

(DOCX)

S2 Appendix. TSP solutions generated by the TSP solver and proposed algorithm in the experiments conducted.

https://doi.org/10.1371/journal.pone.0166064.s002

(DOCX)

Acknowledgments

The authors wish to thank the Center of Building, Construction & Tropical Architecture of the Faculty of Built Environment, Center of Transport Research, Department of Civil Engineering, Faculty of Engineering, and University of Malaya (UM), Malaysia for their financial support to carry out this research project. The project was funded under the project LL026-16SUS.

Author Contributions

  1. Conceptualization: KHL HAR CW SCL CCO.
  2. Data curation: KHL SCL.
  3. Formal analysis: KHL HAR CW SCL.
  4. Funding acquisition: SCL CCO.
  5. Investigation: KHL HAR CW SCL.
  6. Methodology: KHL CW SCL.
  7. Project administration: KHL SCL CCO.
  8. Resources: KHL HAR CW SCL.
  9. Software: KHL SCL.
  10. Supervision: HAR CW CCO.
  11. Validation: KHL SCL CW.
  12. Visualization: KHL SCL.
  13. Writing – original draft: KHL SCL.
  14. Writing – review & editing: KHL SCL.

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