An Improved Tabu Search Heuristic for Static Dial-A-Ride Problem
Songguang Ho, Sarat Chandra Nagavarapu, Ramesh Ramasamy Pandi, Justin Dauwels
aa r X i v : . [ c s . A I] F e b An Improved Tabu Search Heuristic for Static Dial-A-Ride Problem
Songguang Ho, Sarat Chandra Nagavarapu, Ramesh Ramasamy Pandi and Justin Dauwels
Abstract — Multi-vehicle routing has become increasingly im-portant with the rapid development in autonomous vehicletechnology. Dial-a-ride problem, a variant of vehicle routingproblem (VRP), deals with the allocation of customer requeststo vehicles, scheduling the pick-up and drop-off times and thesequence of serving those requests by ensuring high customersatisfaction with minimized travel cost. In this paper, wepropose an improved tabu search (ITS) heuristic for staticdial-a-ride problem (DARP) with the objective of obtaininghigh quality solutions in short time. Two new techniques,construction heuristic and time window adjustment are pro-posed to achieve faster convergence to global optimum. Variousnumerical experiments are conducted for the proposed solutionmethodology using DARP test instances from the literature andthe convergence speed up is validated.
Index Terms — Tabu search, dial-a-ride problem, heuristic,convergence.
I. I
NTRODUCTION
Dial-A-Ride Problem (DARP) addresses the issue of door-to-door transportation service for the customers with highcustomer satisfaction. Now-a-days, transportation serviceshave increasing need in our daily life, and it started todirectly impact our environment as well as quality of living.According to a study conducted by University of BritishColumbia, the road pricing or pay-per-use is the most ef-fective way to reduce emissions and traffic [1]. DARP hasmany applications ranging from taxi services to autonomouscargo and ground operations at the airports.DARP is an extension of pick-up and delivery problemunder the class of vehicle routing problem (VRP) [2]. Itis a combinatorial optimization problem with an objectivefunction to minimise the overall cost while satisfying aspecific set of constraints such as time-window, maximumwaiting time and maximum ride time to ensure high-qualitycustomer service. In this problem, a set of customers makes arequest for pick-up and drop-off at certain locations within apredefined time-window. An approach to solve DARP basedon dynamic programming has been proposed in [3], in whichdivide and conquer method is used to solve the problem.However, the exponential relation between computational
Songguang Ho is a research associate, Sarat ChandraNagavarapu is a research fellow and Ramesh Ramasamy Pandiis a phd candidate with the School of Electrical and ElectronicEngineering, Nanyang Technological University, 50 NanyangAvenue, Singapore, 639798. [email protected],[email protected],[email protected]
Justin Dauwels is an associate professor with the School of Electricaland Electronic Engineering, Nanyang Technological University, 50 NanyangAvenue, Singapore, 639798. [email protected]
The research was partially supported by the ST Engineering − NTUCorporate Lab through the National Research Foundation (NRF) corporatelab@university scheme. complexity and instance size severely limits the applicabilityto small scale instances.Branch and cut is a classical way of solving mathematicaloptimization problems. A new branch-and-cut method [4]was proposed which employs cutting planes to producemathematical formulation for DARP. The problem was ini-tially formulated using Mixed Integer Linear Programming(MILP). Later Cordeau et. al. [5] have revisited the mathe-matical formulation for DARP and the research communityhas addressed this version as standard DARP. A modifiedapproach based on branch-and-cut algorithm [6] for pick-up and delivery problems with time windows was solvedfor larger instances to achieve optimality. Recently, anothervariant of branch-and-cut [7] with less compact modelinghelped to solve the previously unsolved benchmark instancesfor the heterogeneous-DARP to optimality within a matter ofseconds.Though the exact methods are useful to optimally solve theproblem, the computational complexity of such methods isvery high. Therefore, heuristic and meta-heuristic approacheshave become widely used techniques to solve DARP. Onesuch technique proposed in [8], which has pioneered theheuristic approaches. A sequential insertion algorithm [9] hasbeen designed for static DARP, which analyses the problemcomplexity, while offering flexibility to users. The literaturealso has archives of parallel insertion heuristics using thedistributed computing technologies [10].Dynamic fuzzy logic, a computationally efficient heuristicmethod was adopted to solve DARP [11]. The tabu searchheuristic [5] aims to progressively explore the neighborhoodstructure from an initial solution while forbidding somesolutions with similar attributes of recently visited solutions.It can be described as cleverly guided local search that isefficient in exploring the search space at the hope of findingglobal optimal solutions. However, the main drawback isthe time spent while getting stuck in sub-optimal solutions.In order to avoid that, a parallel tabu search heuristic wasproposed in [12]. When infinite penalty is considered, thebest solution is obtained irrespective of irrelevance in choiceof initial static solution for the dynamic dial a ride problem.A two phase heuristics approach [13] based on insertionand improvement phase was proposed for DARP by Beaudryet al. From the solution obtained by insertion heuristics, thealgorithms selects best non-tabu solution progressively whileperforming both inter and intra-route neighborhood evalu-ations. Recently, another variation of tabu search, namedas granular tabu search algorithm [14] that produces goodsolutions in short amount of time within 2-3 mins hasbeen introduced. Also, this new method produces betteresults when compared to the classical tabu search, geneticalgorithm and variable neighborhood search techniques.In this paper, we propose an improved tabu search (ITS)heuristic for dial-a-ride problem. We assume that the routingtime and cost from each vertex to every other vertex areknown apriori. The major contributions of the paper aresummarized as follows: • Determining a computationally faster and reliable vari-ant of tabu search for DARP by thorough investigationof various neighborhood evaluation and insertion tech-niques. • Proposed a new construction heuristic for tabu searchto obtain good initial solution rapidly. • Designed a time window adjustment technique for fastersolution convergence. • Implemented and tested the proposed techniques usingvarious DARP test instances to verify the accelerationin convergence.The remainder of the paper is organized as follows: Sec-tion II briefly discusses the DARP mathematical formulation.Section III details tabu search heuristic and several variantsof tabu search using neighborhood evaluation and insertiontechniques. Section IV presents the proposed ITS heuristicmethod. Section V illustrates the convergence analysis resultsfor the proposed algorithm. Conclusions are provided inSection VI.II. P
ROBLEM F ORMULATION FOR
DARPDial-a-ride problem (DARP) is a variant of VRP thatinvolves dispatching of a fleet of vehicles to transport cus-tomers between their desired pick up and drop off locationswithin specified time windows. The aim is to minimise theover-all transportation cost of the vehicle by evading thelongest route in tandem with providing superior passengercomfort and safety. DARP is mathematically formulated asan optimization problem with an objective function subjectedto several constraints.In dial-a-ride problem, n customer requests are servedusing m vehicles. Each request i consists of time windoweither for departure or arrival vertex. The objective is tominimise the travel cost subject to several constraints. Let S = { s , s , · · · } denotes the solution space. All solutions s i ∈ S need to satisfy three basic constraints. Every route for avehicle k starts and ends at the depot and the departure vertex v i and arrival vertex v i + n must belong to the same route,and the arrival vertex v i + n is visited after departure vertex v i . Any solution that violate these basic set of constraintsbecomes infeasible. In addition, several other constraintsneed to be satisfied; the load of vehicle k cannot exceedpreset load bound Q k at any time; the total route duration ofa vehicle k cannot exceed preset duration bound T k ; the ridetime of any passenger cannot exceed the ride time bound L ;the time window set by the customer must not be violated.Four major constraints exist in dial-a-ride problem: load,duration, time window and ride time constraints. Load con-straint violation q ( s ) occurs when the number of passengerin a vehicle k exceeds its load limit Q k ; duration constraint violation d ( s ) happens when a vehicle k exceeds its durationlimit T k ; time window constraint violation w ( s ) appearswhen the time constraint is violated; ride time constraintviolation t ( s ) occurs when a passenger is transported for alonger time than ride time limit L . The constraints are givenby Eq. (1-4). q ( s ) = X ∀ k max( q k,max − Q k , , (1) d ( s ) = X ∀ k max( d k − T k , , (2) w ( s ) = X ∀ i (cid:2) max( B i − l i ,
0) + max( B i + n − l i + n , (cid:3) , (3) t ( s ) = X ∀ i max( L i − L, . (4)The next section presents the variants of tabu search basedon the neighborhood evaluation and insertion techniques.III. T ABU S EARCH H EURISTIC
Tabu search is a higher level heuristic procedure to solveoptimization problems, as originally defined by [15]. Themethodology is efficient at escaping local optimal solutionswith structured memories and tabu list. During the neighbor-hood transitions, cycling should be avoided to intelligentlyexplore the search space for global optimal solutions. Inten-sification and diversification strategies have to be properlyemployed in order to attain optimal solutions. In tabu search,the recent neighborhood transition is recorded in a tabu list.On subsequent transitions, the recorded moves in tabu list isnot considered. In this way, cycling of moves gets avoided.However, for some cases, if the objective function is belowthe best obtained cost, then aspiration level performs themove even if it is in tabu list.[5] has successfully superimposed the methodology onlocal search with neighborhood reduction to solve DARP. Inthis work, the objective function is considered as travel costwith weighted penalties for additional constraints. At thiscase, the method could efficiently explore the search spacethrough some infeasible solutions in the hope of findingglobal optimal solutions. f ( s ) = c ( s ) + αq ( s ) + βd ( s ) + γw ( s ) + τ t ( s ) . (5)The objective function f ( s ) is given by Eq. 1, where c ( s ) is the travel cost and α , β , γ , τ are the penalty coefficientswhich are initialized to 1. These penalty coefficients changeperiodically once the optimization process begins. In tabusearch, the moves that result in both feasible and infeasiblesolutions are accepted during the optimization process. Whena solution violates the constraints and become infeasible,the penalty coefficient for those constraints are increased bya factor (1 + δ ) and decreased by the same factor, whenthe constraint is not violated. In this way, the algorithmintelligently explores the search space in the direction wherethe constraints that are violated during the previous movesare relaxed.n this paper, construction of initial solution, route op-timization, intensification, diversification and the evaluationfunctions are adopted from Cordeau et al. [5]. A. Neighborhood Evaluation
An important aspect of Tabu search heuristic is the proce-dure for neighborhood evaluation technique. The objective ofneighborhood evaluation is to reduce the constraint violationsand assess the feasibility of the solution. There are threeobjectives in neighborhood evaluation: Reduction of time-window constraint associated with requests (R1); Reductionof route duration constraint associated with vehicle (R2);Reduction of Ride Time Constraint associated with requests(R3). Fig. 1 depicts the three level neighborhood evaluationmethod for tabu search.
Fig. 1. Neighborhood evaluation for tabu search.
There are three neighborhood evaluation techniques: i)1-level ii) 2-level and iii) 3-level, which are ‘steps(1-2)’,‘steps(1-6)’ and ‘full procedure’ adopted from [5] in theirrespective order. The objectives for each level is substrate-based i.e., each level is built upon subsequent levels. 1-levelhas the objective R1, 2-Level has the objective R1 & R2 and3-Level has objective R1, R2 & R3.
B. Insertion Techniques
The size of neighborhood directly influence the computa-tional complexity required during the optimization process.There are two insertion techniques carried out in this paper: i)one-step insertion and ii) two-step insertion. The conventionwas adopted due to nature of the neighborhood transition,which are: a) Single paired insertion (SPI) [16] and b)Neighborhood Reduction [5]. Fig. 2 illustrates the insertionof departure and arrival vertices into the routes.
Fig. 2. Insertion technique.
For one-step insertion, the algorithm attempts to moveeach vertex pair from one vehicle to another in a singletransition. After insertion of the vertex pair, the neighborhoodsolutions are evaluated using the objective function in Eq.5. The neighbor with minimum cost is selected as the nextmove. Though the search process is very expensive, this typeof move has the greatest potential for improvement in theobjective function [16].For two-step insertion, the algorithm attempts to moveeach vertex pair from one vehicle to another in two sequentialsteps. For each request, critical vertex indicates pick-up ordrop-off point that consists of narrower time window whencompared to the non-critical vertex. The first step is theinsertion of critical vertex into its best position, which giveswith least value of objective function in Eq. 5. While holdingthe current position, the second step is the insertion of non-critical vertex into its best position. The second step consistsof two possibilities: i) if critical vertex is departure node, theninsert non-critical vertex only after the critical vertex, ii) ifcritical vertex is arrival node, then insert non-critical vertexonly before the critical node. This technique significantlyreduce the neighborhood from moves O ( r ) to O ( r ) , where r is number of vertices in route k . One Step Insertion Two Step Insertion1-Level Evaluation TS TS TS TS TABLE IC
OMPARISON OF THE MEDIAN OF TRAVEL COST : T
ABU S EARCH (TS)[5] VS I MPROVED T ABU S EARCH (ITS).
Based on the mentioned neighborhood evaluation andinsertion techniques, the possible combinations are listed inTable I. Each method is represented using the conventionTS NI , where N represents the neighborhood evaluation leveland I represents the insertion step. x10 Time (ms) G ap ( % ) TS TS TS TS TS TS Fig. 3. Convergence Analysis for TS using R1a. Time (ms)
70 80 90 G ap ( % ) TS TS TS TS TS TS Fig. 4. Convergence Analysis for TS using R3a. (cid:3) 3(cid:4)(cid:5)(cid:6)(cid:7) Time (ms) (cid:8)
60 70 80 90100110 G ap ( % ) TS TS TS TS TS TS Fig. 5. Convergence Analysis for TS using R6a.
Various benchmark instances for standard DARP are pro-vided by Cordeau and Laporte in [5]. In these instances, thenumber of requests vary between 24 to 144 and the fleetsize varies between 3 to 13. The capacity of each vehicle isset to 6, the maximum passenger ride time is 90 min, themaximum route duration is 480 min and the route planninghorizon is 24 hrs.The costs of the benchmark solutions provided by Parraghand Schmid [17] for the instances R1a, R3a and R6a are190.02 , 532.00 and 785.26 respectively. Figs. 3, 4, 5 illus-trate the convergence of each of these tabu search variantsvalidated using these instances. Each colored line in the plotcorresponds to a distinct variant of tabu search as listed inTable I. The x-axis of the plots corresponds to the simulationrun time and the y-axis represents the gap (%) as givenby (6). We plot the median of the gap recorded from five independent simulations ran for a duration of five minutes.Where, ‘BKS’ corresponds to the best known solution forthe problems. We decided to consider the median in orderto restrict the effect of outliers on the analysis.Gap (%) = cost − BKSBKS × . (6)From the convergence plots, the following inferencesare observed: 1) Two-step insertion technique has fasterconvergence than One-step insertion. 2) One-level achievesinitial solution in the least time, followed by Three-leveland Two-level neighborhood evaluation techniques. 3) Three-level neighborhood evaluation technique converges deeperover time when compared to other techniques, which isfollowed by Two-level, while One-level is the slowest.We consider that TS as better variant as it convergesto optimality in a lesser time, while employing three-levelneighborhood evaluation and two-step insertion techniques.However, TS takes more time to obtain a feasible initialsolution. In order to address this issue, new techniques areproposed in the next section.IV. I MPROVED T ABU S EARCH (ITS)Tabu search heuristic has been extensively used in dial-a-ride problem due to its comparatively faster execution.However, the time required is still significant, especiallyfor larger instances. So, there is a need to optimize thealgorithm in order to obtain good results in a reasonabletime. From the analysis presented in Section III, it is clearthat the algorithm takes longer time to converge and toobtain a feasible initial solution. In order to overcome thisissue, two new methodologies are proposed: a) constructionheuristic (CH) and b) time window adjustment (TW). Themain objective of the improved tabu search is to obtain highquality solutions within a short time. The Sections IV-A andIV-B detail the proposed methodologies.
A. Construction Heuristic
The convergence analysis presented for tabu search in Sec-tion III indicates that the time taken to find the first feasiblesolution for a problem highly depends on the quality of initialsolution. In [5], a random initial solution is generated tostart the search for global optimal solution. However, suchmethodology has higher tendency to random seed, and ittakes longer time to find first feasible solution.In this paper, a new construction heuristic is proposed tofind high quality solutions more rapidly. In this method, anempty set of routes is created, and requests are sorted ran-domly. After the initial preparation, each request is insertedsequentially into the position that attains minimum objectivefunction. Therefore, the objective function is formulated as f ( s ) in 5, with α = β = γ = τ = 1. The details of theproposed construction heuristic technique are described usingAlgorithm 1.The next section discusses the proposed time windowadjustment. lgorithm 1 Construction Heuristic (CH)
Require:
Number of request ( n ), number of vehicle ( m ), allconstraints Ensure:
Initial solution for ITS Parameter initialization: set α = β = γ = τ = 1. random list = sort requests in random order. Initialize the vehicles with empty set of routes. for all requests in random list do for all vehicles do Try inserting request in all possible positions. end for Select an insertion with least f ( s ) . Update the solution. end for
B. Time Window Adjustment
According to the benchmark instances from [5], the DARPproblem is modeled as set of requests with time-windowconstraints for optimal allocation to vehicles. The objectiveis to minimise the travel cost, while satisfying the constraintssuch as time window and ride time constraints associatedwith requests, and route duration constraint associated withvehicles.
Fig. 6. Ride time L i for request i . The DARP formulation given by [5] considers either in-bound or out-bound request. Here, one vertex is alwayscritical, which has narrower time window and the other isnon-critical with time window usually set to be between [0 , T ] , where T denotes the end of the day. In DARP, the ridetime constraint is important for high user satisfaction. Theservice for a request starts at time B i . The service time at thepick-up and drop-off point is indicated by d i . The ride timeof the user L i as shown in Fig. 6 is bounded by the ride timeconstraint L . In this work, a new time window adjustment(TW) method is proposed to improve the convergence of thetabu search heuristic towards global optimal solution. Themethodology is illustrated using Figs. 7 and 8. Initially, theservice of this request (out-bound) consist of relaxed timewindow (at pick up point). When the time window [0 , T ] of non critical vertex is constrained as per the proposedmethodology, the direction of search is intensified towardsmore feasible region in the search space. Fig. 7. Time window adjustment for departure vertex.Fig. 8. Time window adjustment for arrival vertex.
To adjust the time window of departure vertex (non crit-ical), the earliest service time of departure vertex e i shouldnot be earlier than d i + L from earliest service time of arrivalvertex e i + n ; the latest service time of departure vertex l i should not be later than l i + n − d i . The adjustment made todeparture vertex are given as follows: e i,new = max { e i,old , e i + n − d i − L } , (7) l i,new = min { l i,old , l i + n − d i } . (8)In Eq. 7 and 8, e i,old (= ) and l i,old (= T ) are the earliestand latest time windows associated with the departure vertexthat are adjusted to obtain e i,new and l i,new respectively.Similarly, to adjust the time window of arrival vertex,the earliest service time of departure vertex e i + n shouldnot be earlier than e i + d i , while the latest service time ofdeparture vertex l i + n should not be later than l i + L + d i .The adjustments made to arrival vertex are given by Eq. 9and 10. e i + n,new = max { e i + n,old , e i + d i } , (9) l i + n,new = min { l i + n,old , l i + d i + L } . (10)In Eq. 9 and 10, e i + n,old (= ) and l i + n,old (= T ) are theearliest and latest time windows associated with the arrivalvertex that are adjusted to obtain e i + n,new and l i + n,new respectively.The improved tabu search (ITS) is obtained by incorporat-ing the two proposed techniques, i.e., construction heuristic(CH) and time window adjustment (TW) into the tabu searchvariant T S . ITS ( T S CH + T W ) ) is tested against thebenchmark of [7] and the simulation results are presentedin the next section.V. S IMULATION R ESULTS
The proposed ITS heuristic (TS ) is implementedin C++. Simulations have been carried out on a computerrunning 2.1 GHz Intel Xeon E5-2620 v4 processor with128 GB RAM. The parameters suggested by [5] have beenfollowed to conduct a fair comparison of the results obtainedusing the proposed heuristic with the existing tabu search [5] Time (ms) G ap ( % ) TSITS (a) R1a test instance Time (ms) G ap ( % ) TSITS (b) R1b test instance Time (ms) G ap ( % ) TSITS (c) R2a test instance Time (ms) G ap ( % ) TSITS (d) R2b test instance Time (ms) G ap ( % ) TSITS (e) R3a test instance Time (ms) G ap ( % ) TSITS (f) R3b test instance Time (ms) G ap ( % ) TSITS (g) R4a test instance Time (ms) G ap ( % ) TSITS (h) R4b test instance Time (ms) G ap ( % ) TSITS (i) R5a test instance Time (ms) G ap ( % ) TSITS (j) R5b test instance Time (ms) G ap ( % ) TSITS (k) R6a test instance Time (ms) G ap ( % ) TSITS (l) R6b test instance Time (ms) G ap ( % ) TSITS (m) R7a test instance Time (ms) G ap ( % ) TSITS (n) R7b test instance Time (ms) G ap ( % ) TSITS (o) R8a test instance Time (ms) G ap ( % ) TSITS (p) R8b test instance Time (ms) G ap ( % ) TSITS (q) R9a test instance Time (ms) G ap ( % ) TSITS (r) R9b test instance Time (ms) G ap ( % ) TSITS (s) R10a test instance Time (ms) G ap ( % ) TSITS (t) R10b test instanceFig. 9. Convergence analysis for improved tabu search (ITS) w.r.t. tabu search (TS) [5] using various test instances. est Instance BKS Tabu Search (TS) ([5]) Improved Tabu Search (ITS)1 sec 2 sec 5 sec 15 sec 30 sec 60 sec 1 sec 2 sec 5 sec 15 sec 30 sec 60 secR1a 190.02 [17] 193.26 191.66 191.11 191.05 190.79 190.02 193.77 191.88 191.66 191.11 190.02 190.02R2a 301.34 [17] 373.23 332.21 318.96 315.31 314.20 314.18 334.22 328.48 321.53 320.11 313.40 313.13R3a 532.00 [17] - 863.26 644.51 592.15 578.66 578.66 750.64 670.69 632.84 593.06 582.25 582.25R4a 570.25 [17] - - 935.45 698.30 661.68 635.35 - 906.90 770.07 679.35 658.53 640.39R5a 626.93 [7] - - - 918.30 780.16 729.24 - - 906.06 795.77 753.48 721.48R6a 785.26 [17] - - - 1416.25 1171.76 977.61 - - - 1143.63 1034.10 945.55R7a 291.71 [17] 315.87 310.79 305.58 305.55 302.81 300.68 310.89 310.74 307.63 305.52 301.83 300.30R8a 487.84 [17] - - 614.78 548.96 544.45 544.45 - 642.78 574.11 549.66 536.74 536.74R9a 658.31 [17] - - - - - 989.98 - - - 1013.78 882.53 815.50R10a 851.82 [7] - - - - 1357.03 1120.59 - - - 1262.90 1128.65 1043.75R1b 164.46 [17] 174.55 173.21 171.81 169.14 168.80 168.35 173.39 173.19 171.67 169.71 168.70 167.98R2b 295.66 [17] 366.00 334.91 322.87 320.08 314.75 314.75 345.04 334.01 322.25 317.70 314.76 314.28R3b 484.83 [17] - 816.63 604.61 548.84 531.89 530.54 738.39 650.30 571.50 546.19 536.46 536.46R4b 529.33 [17] - - 904.96 651.59 607.58 602.90 - - 731.19 651.49 625.01 605.95R5b 577.29 [7] - - - 859.20 724.33 665.43 - - 888.87 759.56 698.75 670.99R6b 730.69 [7] - - - - 1096.04 926.58 - - - 1068.33 980.66 897.41R7b 248.21 [17] 285.32 271.17 270.71 267.24 264.60 262.29 276.14 275.04 268.66 267.47 262.76 261.49R8b 458.73 [7] - 711.44 559.40 518.31 516.46 513.56 655.42 600.62 541.42 514.95 514.55 512.68R9b 593.49 [17] - - - 842.19 720.25 694.87 - - 903.30 748.44 712.13 674.22R10b 785.68 [7] - - - - 1342.78 1086.40 - - - 1287.53 1132.89 1029.22
TABLE IIC
OMPARISON OF THE MEDIAN OF TRAVEL COST : T
ABU S EARCH (TS) ([5]) VS I MPROVED T ABU S EARCH (ITS).Test Instance BKS First Feasible SolutionTabu Search (TS) ([5]) Improved Tabu Search (ITS)Cost Gap (%) Time (ms) Cost Gap (%) Time (ms)R1a 190.02 [17] 248.05 30.54 62 228.85 20.43 16R2a 301.34 [17] 435.34 44.47 641 412.74 36.97 250R3a 532.00 [17] 935.17 75.78 1672 765.95 43.98 938R4a 570.25 [17] 1022.51 79.31 3704 906.90 59.04 1985R5a 626.93 [7] 1210.45 93.08 6766 996.00 58.87 3610R6a 785.26 [17] 1530.40 94.89 11642 1269.81 61.71 7469R7a 291.71 [17] 433.79 48.71 266 407.89 39.83 93R8a 487.84 [17] 799.18 63.82 2562 731.00 49.84 1031R9a 658.31 [17] 1011.51 53.65 53051 1030.78 56.58 1751R10a 851.82 [7] 1567.82 84.06 21533 1431.05 68.00 6954R1b 164.46 [17] 237.80 44.59 63 233.73 42.12 15R2b 295.66 [17] 447.23 51.26 672 414.94 40.35 235R3b 484.83 [17] 933.77 92.60 1688 784.44 61.80 859R4b 529.33 [17] 1000.97 89.10 4094 860.97 62.65 2062R5b 577.29 [7] 984.23 70.49 10141 933.52 61.71 3844R6b 730.69 [7] 1342.25 83.70 15329 1243.52 70.18 6376R7b 248.21 [17] 384.93 55.08 328 373.82 50.60 93R8b 458.73 [7] 828.88 80.69 1797 745.12 62.43 781R9b 593.49 [17] 1211.39 104.11 5938 1068.24 79.99 2797R10b 785.68 [7] 1411.05 79.60 25424 1374.35 74.92 11329TABLE IIIC
OMPARISON OF THE INITIAL TRAVEL COST : TS ([5]) VS ITS. method. Alongside the variant of TS ; TS , a hybridof TS with both construction heuristic and time windowadjustment have been tested.In Figs. 9(a) - 9(t), we depict the progression of thesolution towards the benchmark during the first sixty secondsof execution. In these plots, the blue and red lines correspondto the Cordeau’s tabu search [5] and the proposed ITSheuristic respectively. The plots show the median of gapobtained from fifteen independent simulations run for sixtyseconds. From these plots, it is clearly evident that theproposed improved tabu search (ITS) heuristic outperformstabu search (TS) [5] for all the DARP test instances. The following inference is made from these numericalexperiments: Construction heuristic (CH) along with the timewindow adjustment (TW) not only contributes to significantspeed up in the convergence, but also finds good feasiblesolution in a shorter time.In Table II, we compare the travel cost of TS and ITSat various time instances {
1, 2, 5, 15, 30, 60 } sec for thebenchmark instances. The presentation style is adopted from[14]. It is observed that the solution quality of ITS is alwaysbetter when compared to TS, especially during the first threeminutes of the execution. Based on the best known solutionslisted in the second column labeled with ‘BKS’ in Table II,t can be concluded that the proposed method attains nearoptimal solutions in shorter time. As mentioned in SectionI, the proposed construction heuristic of the improved tabusearch (ITS) helps to produce a good feasible initial solutionrapidly. Table III provides an empirical evidence for thisclaim and presents a comparison of first feasible solutionsfor various test instances and the time at which they areobtained. The next section concludes the paper.VI. C ONCLUSION
In this paper, an improved tabu search heuristic has beenproposed to solve static dial-a-ride problem. Several variantsof tabu search method based on the neighborhood evaluationand insertion techniques have been tested to analyse theconvergence behavior. In the existing tabu search heuristic,two performance bottle necks have been identified: i) longerrun time requirement to obtain first feasible solution andii) slower convergence to the global optimum. To addressthese, two new techniques, i.e., construction heuristic andtime window adjustment have been proposed to improve theperformance. Simulation results for various test instancesshow that the proposed ITS heuristic not only improvesthe convergence, but also finds high quality solution faster.Moreover, the approach can be extended to dynamic DARP.Some other possible directions for future work could be theparallelization of the developed tabu search algorithm usingGPUs to reduce the computation time.R
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