INTEGRATED TIMETABLE AND SCHEDULING OPTIMIZATION WITH MULTIOBJECTIVE EVOLUTIONARY ALGORITHM. Michal Weiszer 1

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1 The International Journal o TRANSPORT & LOGISTICS Medzinárodný časopis DOPRAVA A LOGISTIKA ISSN 45-07X INTEGRATED TIMETABLE AND SCHEDULING OPTIMIZATION WITH MULTIOBJECTIVE EVOLUTIONARY ALGORITHM Michal Weiszer Technical University in Košice, Faculty BERG, Institute o Industrial and Transport logistic, Slovaia, tel.: , michal.weiszer@tue.s Abstract: This paper is discussing the integration o timetabling and vehicle scheduling stage o the transportation planning process with implementation o evolutionary multiobjective genetic algorithm (NSGA-II). Test case in public transport with minimization o the transer time o the passengers in transer node along with minimization o the number o vehicles needed to operate such timetable is presented. Other applications in the reight transport ield are also discussed. Developed solution proposes that it is able to optimize conlicting objectives o passengers or customers and transportation company simultaneously. Key words: multiobjective optimization, genetic algorithm, timetable, vehicle scheduling INTRODUCTION Planning process o transportation service typically consists o several distinctive stages. For example, planning process o public transport service has 4 major parts: networ design (routing), timetabling, vehicle scheduling, crew scheduling and rostering. Similar pattern with separate stages can be observed also in reight transportation planning. In real world sized transportation systems, each planning stage is a complex tas on its own. Thereore, the stages are executed sequentially, usually in direction rom the strategical level to the operational level. The stages are interconnected and the output o one stage is an input o the next stage. Moreover, the stages have dierent, oten conlicting objectives. While some stages o the planning process ocus more on the customer (passenger in case o public transport), other tend to concentrate more on the transport company's view. Typical example can be maximizing transport service quality mainly determined by networ design and timetabling in public transport, and simultaneously minimization o cost related to vehicle and crew scheduling. One can see, that ideally the whole planning process should be optimized to a global optimality instead o local objectives o separate planning stages. In the transportation research ield, there are ew wors integrating timetabling and scheduling. Examples include Periodic Event Scheduling problem or railway with partial integration o other planning aspects in [], an approach using iterated local search in [2] and [3]. The objective o this paper is to discuss integrated planning approach employing multiobjective evolutionary algorithm. Rest o this paper is organized as ollows: in next 47

2 section, basic theory on multiobjective optimization and multiobjective genetic algorithm NSGA-II is presented. Thereater, application o this approach both in public and reight transportation is discussed. 2 MULTIOBJECTIVE OPTIMIZATION AND EVOLUTIONARY ALGORITHMS In contrast to single-objective optimization, multiobjective optimization deals with several objective unctions. General multiobjective optimization problem can be deined as ollows [4]: Minimize = [ ( x), 2, K, ] Subject to: 0, j =,2, K, m, g j h l = 0, l =,2, K, e. where is the number o objective unctions, m is the number o inequality n constraints, and e is the number o equality constraints. A solution x R is a vector o n decision variables in the solution space X: x = {x, x 2,..., x n }. The objective is to ind a vector * * * x* that minimizes a given set o K objective unctions ( x ) = { ( x ), K, K ( x )}. Usually, there is not just a single solution to the multiobjective optimization problem, but more solutions can be optimal. Then the goal o multiobjective optimization is to ind possibly all solutions each o which minimizes the objective unctions at an acceptable level. The most used deinition o optimality, that is when the solution is deined as optimal, is Pareto optimality. 2. Concept o Pareto optimality A solution x X is Pareto optimal i there is no other solution x 2 X such that (x 2 ) (x ), and i (x 2 ) < i (x ) or at least one unction [4]. This Pareto optimal solution in the objective space Z is called non-dominated. A Pareto-optimal solution can not be improved in any objective without worsening in at least one other objective. The all Pareto optimal solutions in solution space X constitute the Pareto optimal set. The corresponding values o the objective unctions o the Pareto optimal solutions in the objective space constitute Pareto ront. Assume two objective unctions,, 2. Then Fig. shows the Pareto ront which is highlighted in objective space. For example, solution i is dominated by solutions c, d and e. Solutions, g and h are dominated by only a single solution a, b, b, respectively. The Pareto optimality concept means that all Pareto optimal solutions are equally optimal, i.e. we can not say solution b is better than solution a, or example. 48

3 Fig. Pareto ront in objective space There are several approaches how to tacle multiobjective optimization problem. Some o the most well nown include Weighted sum method, Lexicographic method, in which the objective unctions are arranged in order o importance, Weighted product method, or Multiobjective evolutionary algorithms. Introduction an review o multiobjective evolutionary optimization can be ound in [5] and [6]. 2.2 Fast Non-dominated Sorting Genetic Algorithm (NSGA-II) One o the evolutionary algorithms adapted to multiobjective optimization is well tested and computationally eicient Fast Non-dominated Sorting Genetic Algorithm NSGA-II by Deb [7]. Evolutionary algorithms are broad category o optimization algorithms (including Genetic algorithm and Dierential evolution) which use a population based search. The population iteratively evolves with each new generation. Traditional evolutionary algorithms are singleobjective in which the ittest individual (with highest objective unction value) represents the single suboptimal solution. The act that evolutionary algorithms wor with multiple solutions at time mae them very suitable or multiobjective optimization. NSGA-II evaluates individual solutions by dominance ran and crowding distance instead o objective unction value. Crowding distance promotes search o solutions uniormly spread along the Pareto ront. The crowding distance is used in ollowing manner (Fig. 2): Step : The population is raned by dominance rule and non-dominated ronts F, F 2,...,F R are identiied. For each ront F j, j =,, R repeat Steps 2 and 3. Step 2: The solutions in ront F j are sorted in ascending order. The sorting is repeated or l = F each objective unction. Let j and x i, represent the i-th solution in the sorted list with respect to the objective unction.. Assign cd (x, ) = and cd (x l, ) =, and then assign or i = 2,, l- : cd ( x i, ) = ( x i+, max ) ( x min i, ) Step 3: To compute the total crowding distance cd(x) o a solution x, the solution s crowding cd = cd distances with respect to each objective are summed,. 49

4 This crowding distance measure is used in crowded tournament selection operator. That is, between two solutions with diering nondomination rans, we preer the solution with the lower (better) ran. Otherwise, i both solutions belong to the same ront, then we preer the solution that has higher crowding distance. A solution with a higher value o this distance measure is less crowded by other solutions [7]. Fig. 2 Crowding distance calculation [5] 3 INTEGRATED TIMETABLING AND SCHEDULING IN PUBLIC TRANSPORTATION In this section, integrated model or timetabling and vehicle scheduling in public transportation is presented. Multiobjective evolutionary algorithm enables that individual objective unctions are optimized simultaneously. The basic structure o the model is depicted in Fig. 3. Decision variables Multiobjective evolutionary algorithm Timetabling (timetable) Timetable (schedule) Vehicle scheduling Fig. 3 Structure o the integrated model Decision variables determine the timetables o the lines. The timetable is constructed rom decision variables according to the type o timetable. I the timetable is periodical, only the oset o the irst departure is needed and the other departures are calculated by adding 50

5 multiples o headway to the irst departure. In other cases, the decision variables can be exact departure times or osets rom initial timetable. Constructed timetable serves as an input to the vehicle scheduling. The vehicle scheduling can be done by some scheduling model such as assignment model, set partitioning model, time-space networs or by heuristics. In the same time, timetables and vehicle schedules are evaluated and the values o the objective unctions are passed over to the evolutionary algorithm. The evolutionary algorithm calculates the dominance rans and crowding distances and then iteratively repeats the cycle in order to ind Pareto optimal values o the decision variables. The objective unction o the timetabling can consist o: transer time o the passengers, waiting time on the stops, other measures such as evenness o the headway, their combination. The vehicle schedules can be evaluated in terms o: number o vehicles needed, length o deadhead trips, waiting time, their combination. 3. Test case with one transer node Simple example with 4 lines and periodic timetable presented in [8] has two objective unctions. The irst objective unction returns the traner time o passengers: = α ( ti t jl ) c Minimize i j l Subject to: ( t i t jl ) i, j,, l, α = () α 0, i, j (2) + ti, t jl Z i, j,, l (3) t i, t jl 0, 60) i, j,, l (4) l Where the transer time is a dierence between the arrival o -th bus/tram o i-th route and the arrival o l-th bus/tram o j-th route, multiplied by number o transering passengers and summed over all buses/trams o all routes. The term α can be either 0 or according to the possibility o transer between routes i and j. Constraint () ensures that transer time is greater or equal than minute, constraint (2) states that α can tae binary values and constraint (3) ensures that the arrivals have integer values. The constraint (4) ensures that the arrivals are within the time period o 60 minutes. The second objective unction 2 evaluates the vehicle schedule in terms o number o vehicles needed to operate the given timetable. Since no interlining is allowed or the vehicles, number o vehicles N to serve one line with headway h i can be calculated by ollowing ormula: l N = AB + l BA + r h i A + B i, i+ ri, i+ i, i Where l AB, l BA are travel times between terminal stops A and B, + i, i+, is the dierence between the arrival t A i, resp. t B i+ and departure t A i+, resp. t B i rom the terminal. 5 A r B r c l

6 The algorithm ound two solutions in terms o objective unctions. The Pareto ront o only two points is shown in Fig. 5. The irst point has transer time = 60 minutes and number o vehicles is 2 = 2. The second point has worse transer time = 64 minutes but lower number o vehicles 2 = 20. Fig. 4 Pareto ront or test case 4 FURTHER EXTENSIONS IN FREIGHT TRANSPORTATION Integrated approach presented above can be applied also in reight transportation. Timetabling and vehicle scheduling in public transportation have parallels in reight transport planning. Especially, this is true when vehicles operate in lines. One example that can illustrate this methodology, is transportation o ready mixed concrete rom concrete batch plant to the construction site. Usually, total volume o concrete which has to be transported in certain time period is nown in advance. Since the capacity o the vehicles is much smaller than the total volume o the concrete, several deliveries are needed. Depending on the route length and total concrete volume, vehicles can mae several deliveries. This transportation resembles one line with certain headway. Other actors such as variable travel times due to the traic conditions or variable unloading time at the construction site can complicate the process. The ey to the concrete transportation problem is the timetable. There are several objectives that should be satisied: minimization o used vehicles, minimization o total duration o process, minimization o waiting times o vehicles in the construction site, since no or limited simultaneous unloading is allowed, minimization o idle time o construction sta due to the waiting or the concrete arrival. Similarly as in public transport, there are conlicting objectives o the transportation provider and the customer. Thereore, simultaneous optimization o objectives is desirable. The use o Pareto based optimization can help the transportation planner to better explore the alternatives available and to choose the suitable trade-o choice. 52

7 5 CONSLUSION Optimization techniques or solving large and computationally hard problems lie evolutionary algorithms enable integrate previously separate stages o the transportation planning process. Furthermore, employing multiobjective approach allows to tae multiple objectives into account. This way the conlicting objectives o dierent staeholders such as passengers or customers and transportation provider can be optimized simultaneously. Exploring Pareto optimal set helps the decision maer to investigate various scenarios in comparison with traditional methods such as weighted sum method. Further wor will be concentrated on applying this method to real world problems and employing advanced vehicle scheduling techniques such as time-space networs or heuristics. Reerences [] LIEBCHEN, C., MÖHRING, R., H. The Modeling Power o the Periodic Event Scheduling Problem : Railway Timetables and Beyond. In: Algorithmic Methods or Railway Optimization : Lecture Notes in Computer Science, 2007, vol. 4359, p ISSN [2] GUIHAIRE, V., HAO J., K. Transit networ re-timetabling and vehicle scheduling. In: Communications in Computer and Inormation Science (CCIS). Vol. 4, Berlin: Springer, 2008, p , ISBN [3] PETERSEN, H.L., et al., The Simultaneous Vehicle Scheduling and Passenger Service Problem, Technical University o Denmar, Technical report. [4] MARLER, R.T., ARORA, J.S., Survey o multi-objective optimization methods or engineering, In: Structural and Multidisciplinary Optimization. Vol. 26, Berlin: Springer, 2004, p [5] DEB, K. Introduction to Evolutionary Multiobjective Optimization. In: Multiobjective Optimization : Lecture Notes in Computer Science, 2008, vol. 5252, p ISSN [6] KONAK, A., COIT, D., W., SMITH, A., E., Multiobjective optimization using genetic algorithms: A tutorial, In: Reliability Engineering & System Saety, Special Issue - Genetic Algorithms and Reliability, 2006, Vol. 9, Issue 9, p , ISSN [7] DEB, K., et. al., A ast and elitist multiobjective genetic algorithm: NSGA-II, In: IEEE Transactions on Evolutionary Computation, 2002, vol.6, no.2, p.82-97, ISSN X [8] WEISZER, M., FEDORKO, G., ČUJAN, Z., Multiobjective evolutionary algorithm or integrated timetable optimization with vehicle scheduling aspects, In: Perner's Contacts, 200, Vol. 5, no. 4 (200), p , ISSN X Recenzia/Review: doc. Ing. Gabriel Fedoro, PhD. 53