the museum visitor routing problem

Applied Mathematics and Computation 216 (2010) 719–729

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

The museum visitor routing problem

Vincent F. Yu a, Shih-Wei Lin b,*, Shuo-Yan Chou a

a Department of Industrial Management, National Taiwan University of Science and Technology, No. 43, Section 4, Keelung Road, Taipei 106, Taiwanb Department of Information Management, Chang Gung University, No. 259, Wen-Hwa 1st Road, Kwei-Shan, Taoyuan 333, Taiwan

a r t i c l e i n f o

Keywords:Museum visitor routing problemOpen shop scheduling problemSequence dependent setup timeSimulated annealing

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.01.066

* Corresponding author.E-mail address: [email protected] (S.-W. Li

a b s t r a c t

In the museum visitor routing problem (MVRP), each visitor group has some exhibit roomsof interest. The visiting route of a certain visitor group requires going through all the exhi-bit rooms that the group wants to visit. Routes need to be scheduled based on certain cri-teria to avoid congestion and/or prolonged touring time. In this study, the MVRP isformulated as a mixed integer program which is an extension of the open shop scheduling(OSS) problem in which visitor groups and exhibit rooms are treated as jobs and machines,respectively. The time each visitor group spends in an exhibit room is analogous to the pro-cessing time required for each job on a particular machine. The travel time required fromone exhibit room to another is modeled as the sequence-dependent setup time on amachine, which is not considered in the OSS problem. Due to the intrinsic complexity ofthe MVRP, a simulated annealing (SA) approach is proposed to solve the problem. Compu-tational results indicate that the proposed SA approach is capable of obtaining high qualityMVRP solutions within a reasonable amount of computational time and enables theapproach to be used in practice.

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1. Introduction

Museums are the centers for conservation, study and reflection of natural and cultural heritage. The public have theopportunities for the appreciation, enjoyment and understanding of such heritage by physically visiting the museum. Byaugmenting and improving collections and enhancing visitor services, museums are able to serve as a main facility to facil-itate the elevation of culture, art and tourism industries. With the help of advanced information and networking technolo-gies, museum services may be further strengthened by providing context-aware museum tour guide systems that utilizeremote sensors with presence detection mechanisms to provide in real-time customized contents to visitors with differentbackgrounds, interests and time constraints.

This paper describes a mathematical programming model for the routing and scheduling of museum visitors traveling ashomogeneous groups. The model is embedded in a prototype wireless context-aware museum tour guide system developedfor the National Palace Museum of Taiwan (NPMT), one of the top five museums in the world. A context-aware framework inwhich visitor groups in different contexts can be provided with information customized to their needs is established as fol-lows. Museum visitor groups provide their personal data, special needs and time constraints to the museum tour guide sys-tem. The system in return extracts customized information from the structured museum contents for individual visitorgroup to utilize during their visit. Such contextual data is classified by visitor groups’ demographic data, preferences andinterests, including ages, genders, education, professions, languages, media preference, time constraint, special subjects of

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720 V.F. Yu et al. / Applied Mathematics and Computation 216 (2010) 719–729

interest, mobile device, and current locations. The classification can be used to establish visitor ontology for further analysisand utilization [1–3].

As a part of the prototype of the museum tour guide system, a route guidance function needs to be implemented to auto-matically provide real-time routing information to visitor groups. Visitor groups are first given a visiting route based on theirspecific contexts as well as the overall context of all the visitor groups. As a visitor group tours around the museum, a loca-tion-detection device senses the visitor group’s present location and transmits this information to the museum tour guidesystem. The system then returns customized information to the visitor group’s handheld devices such as cellular phone,PDA (personal digital assistant), tablet PC, or other custom-made devices. The information customized to the specific visitorgroup, including information on the visitor group’s current location and subsequent direction, is presented to the visitorgroup on their handheld devices in real-time based on the visitor group’s context. The museum tour guide system is designedas a geographic information system (GIS), which is capable of performing data analysis and information selection and pro-vision based on geographic location [4,5].

While recommending visiting routes to visitor groups, it is necessary to route (or schedule) individual visitor groupsaccording to certain criteria to avoid congestion and/or prolonged tour time. In such a museum visitor routing problem(MVRP), each visitor group has a pre-determined collection of exhibit rooms to visit based on its context; the visiting se-quence is not restricted as long as all the exhibit rooms of interest are included in the scheduled route. In an analogous man-ner, the scheduling problems in the production environment, visitor groups are like jobs, exhibit rooms are like machines,and the visiting time a visitor group spends in an exhibit room is analogous to the processing time required for a job ona particular machine. This routing problem can therefore be modeled and solved as a variant of the open shop scheduling(OSS) problem. In an OSS problem, the processing of a job does not have to follow a definite sequence of operations. As longas all the required operations for the job are performed, the processing of the job is considered complete and accomplished.The open shop formulation allows more flexibility in scheduling, but it is difficult to develop rules that give optimum pro-cessing sequences for the jobs.

The objectives of the MVRP, in an analogy to the OSS, may be one of the following criteria: (1) minimizing the makespan,that is, minimizing the time the last visitor group leaves the museum; (2) minimizing the variation of flow time, that is, min-imizing the difference in total visiting time among all visitor groups; or (3) minimizing the maximum lateness, that is, eachvisitor group’s visit is scheduled so that its total visiting time is as closely as possible to the expected total visiting time pre-determined for each group. In this study, we adopted the first objective, i.e., minimizing the makespan.

The travel time between a pair of exhibit rooms depends on the physical distance between them, equivalent to a se-quence-dependent setup time in the OSS problem. In addition, each exhibit room has a limited space, thus it may not befeasible for two visitor groups to visit an exhibit room at the same time. The MVRP is therefore more complicated thanthe already difficult OSS problem. This research applied a simulated annealing (SA) approach to MVRP to find a (near) globaloptimum that minimizes the makespan of visitor groups.

To ensure continuous use of this museum tour guide system, a customer-centric approach needs to be utilized and imple-mented. The information provided by visitor groups, including personal data, information retrieved, web pages visited, timesspent on various exhibits, comments, routes taken and so on are all recorded and used as the basis for improving museumservices in the future. This data can be analyzed by data mining techniques to identify critical routing preference informationof the visitor groups. Museum layout, the exhibition change, special exhibition, seasonal effects, programs for special interestgroups can be designed based on such information. With these capabilities, the system only needs minimum input from avisitor group to automatically and intelligently recommend suitable routes for the group.

The rest of this paper is organized as follows. Section 2 gives the problem formulation and literature review. In Section 3,the proposed simulated annealing approach is presented. The results of computational experiments are provided in Section 4.Finally, conclusions and future research directions are given in Section 5.

2. Problem formulation and literature review

The MVRP is analogous to the OSS problem for production scheduling. The OSS problem can be formally defined as fol-lows. There are n jobs to be scheduled for processing on m machines. Each job has to be processed on each of the m ma-chines for a certain period of time. A relaxation allows some of these processing times to be zero. There are no restrictionson the processing sequence of each job through the machines. The planner/scheduler determines the processing sequenceof each job, that is, different jobs though require the same set of machining processes may have different processingsequences.

Similarly, the MVRP can be formally stated as follows. There are n visitor groups to be scheduled/routed to visit a collec-tion of m exhibit rooms in the museum. Each visitor group needs to visit every one of the m exhibit rooms exactly once. Thereare no restrictions on the routing sequence in which the visitor groups travel through the museum exhibits. The main dif-ference between the MVRP and the OSS problem is that there exist sequence-dependent setup times in the MVRP as visitorgroups need to walk some distance between exhibit rooms in a typical museum setting. Thus, different routes will be asso-ciated with different travel distances and thus tour times, which correspond to the OSS problem with different setup timesfor different scheduled sequences. The MVRP is therefore an extension of the OSS problem. Based on the OSS problem, wedeveloped a mathematical model in this research for the MVRP.

V.F. Yu et al. / Applied Mathematics and Computation 216 (2010) 719–729 721

The objective of the MVRP is minimizing the makespan, which is the difference between the time the first visitor groupleaves the museum entrance to start their visit and the time the last visitor group completes their visit and reaches the mu-seum exit. We assume that the entrance and the exit are at the same location, as seen in most museums. Let cik be the com-pletion time of visitor group i in exhibit room k; tij denote the time visitor group i spends on visiting exhibit room j; shk denotethe travel time required for a visitor group to walk from exhibit room h to exhibit room k, and s0k; sk0 denote the travel timerequired for a visitor group to walk from the museum entrance to exhibit room k, and from exhibit room k to the museumexit, for each k 2 f1;2; . . . ;mg.

The museum visitor routing problem with a makespan objective can be formulated as follows:

Minimize max16i6nfci0g

Subject to

cik � tik � shk þMð1� aihkÞP cih; i ¼ 1;2; . . . ;n; h ¼ 1;2; . . . ;m� 1; k ¼ hþ 1;2; . . . ;m; ð1Þcih � tih � skh þMaihk P cik; i ¼ 1;2; . . . ; n; h ¼ 1;2; . . . ;m� 1; k ¼ hþ 1;2; . . . ;m; ð2Þcjk � cik þMð1� xijkÞP tjk; i ¼ 1;2; . . . ;n� 1; j ¼ iþ 1; . . . ;n; k ¼ 1;2; . . . ;m; ð3Þcik � cjk þMxijk P tik; i ¼ 1;2; . . . ;n� 1; j ¼ iþ 1; . . . ; n; k ¼ 1;2; . . . ;m; ð4Þcik P s0k þ tik; i ¼ 1;2; . . . ; n; k ¼ 1;2; . . . ;m; ð5Þci0 P cik þ sk0; i ¼ 1;2; . . . ;n; k ¼ 1;2; . . . ;m; ð6Þaihk 2 f0;1g; i ¼ 1;2; . . . ; n; h; k ¼ 1;2; . . . ;m; ð7Þxijk 2 f0;1g; i ¼ 1;2; . . . ; n; k ¼ 1;2; . . . ;m: ð8Þ

The objective is to minimize the makespan. Let M be a large positive number. Constraints (1) and (2) ensure that eachvisitor group can visit only one exhibit room at a time. Constraints (3) and (4) ensure that each exhibit room is visited byat most one visitor group at a time. Constraints (5) and (6) make sure that the travel time from entrance to each exhibit roomand from each exhibit room to the exit is included in the model. Indicator variables are defined as follows. aihk ¼ 1, if visitorgroup i visits exhibit room h before visiting exhibit room k; aihk ¼ 0, otherwise. xijk ¼ 1, if visitor group i visits exhibit roombefore groupj does; xijk ¼ 0, otherwise.

For each pair of exhibit rooms, we assume that the travel time required from one room to the other is the same for allvisitor groups. Let cmax denote the makespan. A lower bound of MVRP, denoted by LB, when there are m exhibit rooms, nvisit groups, and preemption is not allowed can be derived as

cmax P LB ¼max maxj2f1;...;mg

Xn

i¼1

tij þ s0j þ sj0

!; max

i2f1;...;ng

Xm

j¼1

tij þ minh–k2f1;...;mg

ðs0h þ sk0Þ þ minS # A:jSj¼m�1

Xshk2S

Shk

!;

where A ¼ fshk : 1 6 h–k 6 mg.The OSS problem was shown to be NP-hard for m > 3 [6]. However, some OSS problems with special structures are proved

to be polynomially solvable. For instance, Adiri and Aizikowitz [7] developed a linear-time algorithm for the three-machineOSS problem, when one of the machines are dominated by another machine. Fiala [8] showed that when the sum of the pro-cessing times on one machine is large enough with respect to the maximal processing time, the OSS problem for arbitrary mcan be solved polynomially. Similarly, arbitrary m-machine OSS problems with one or two dominating machines are alsopolynomially solvable [9]. For the case of m = 2, Gonzalez and Sahni [6] proposed a polynomial time algorithm to solvethe two-machine OSS problem. Pinedo [10] proposed a simple rule called ‘‘the Longest Alternate Processing Time first”(LAPT), which was proved to give optimal schedules in polynomial time. See Brucker [11] for more polynomially solvablespecial cases of the OSS problem.

Exact solution approaches for the OSS problem, such as branch and bound algorithm [12,13], produced high quality solu-tions at the expense of computational time, and were only applicable to small scale instances. Global optimal solutions aredifficult to obtain when the problem size is large due to the complexity of the problem. Therefore, researchers have long beensearching for polynomial or pseudo-polynomial algorithms for the OSS problem. However, to date, there are only few effec-tive heuristic procedures for the general m-machine OSS problem in the literature. Röck and Schmidt [14] introduced a ma-chine aggregation algorithm based on the fact that two-machine cases are polynomially solvable. Guéret and Prins [15]proposed two fast heuristics with promising results. The first one is a list scheduling algorithm with two priorities. The sec-ond is based on the construction of matchings in a bipartite graph. Ramudhin and Marier [16] adapted the shifting bottleneckprocedure that was originally designed for job shop scheduling (JSS) problems to the OSS problem. Bräsel et al. [17] general-ized Wener and Winkler’s [18] insertion algorithm for JSS problems and proposed an efficient constructive insertion algo-rithm for the OSS problem. Liaw [19] provided an iterative improvement approach based on decomposition technique forthe OSS problem. In the last decade, many meta-heuristic approaches have been successfully applied to the OSS problem,including tabu search (TS) [20,21], genetic algorithm (GA) [22–24] and simulated annealing [25].

Some hybrid meta-heuristics have been developed for solving scheduling related problems. For example, Tavakkoli-Mog-haddam et al. [26] applied a hybrid method to the stochastic job shop scheduling problem. Jenabi et al. [27] proposed twohybrid meta-heuristics based on GA and SA respectively for the economic lot sizing and scheduling problem in flexible flow


lines with unrelated parallel machines over a finite planning horizon. Heinonen and Pettersson [28] applied a hybrid ant col-ony optimization (ACO) algorithm to the JSP. In the last few years, some powerful hybrid meta-heuristics were proposed forthe OSS problem. Liaw [29] developed an effective hybrid genetic algorithm (HGA) for the OSS problem that incorporates TSas a local improvement procedure into a basic GA. Blum [30] hybridized ant colony optimization with beam search algorithmto the OSS problem and obtained better solutions to existing benchmark instances. Yang et al. [31] applied a hybrid of par-ticle swarm optimization (PSO) and SA to solve the OSS problem. More recently, Shaa and Hsu [32] presented a new heuristicfor the OSS problem which hybridized PSO with beam search. These hybrid meta-heuristics are capable of obtaining (near)optimal solution, at the expense of computational time.

To the best of our knowledge, there are no previous studies that deal with the museum visitor routing problem, which isan extension of the OSS problem that also considers sequence-dependence setup times. In view of the success of meta-heu-ristics on tackling the OSS problem and the close relationship between MVRP problem and OSS problem, we proposed an S Abased meta-heuristic for the MVRP problem.

3. The proposed simulated annealing approach

Simulated annealing based heuristics have been successfully applied to a wide variety of hard combinatorial optimizationproblems [33–37]. The concept of simulated annealing was first introduced by Metropolis et al. [38], and later popularized byKirkpatrick et al. [39] and Cerny [40] in solving hard combinatorial optimization problems. The basic idea behind the sim-ulated annealing heuristic is taken from nature. Annealing is the cooling process of crystallization. Slow cooling of metal of-ten produces good and low energy state crystallization, whereas fast cooling produces poor crystallization. The optimizationprocedure of simulated annealing mimics the crystallization cooling procedure. SA starts its search with a random initialsolution. In each iteration, the algorithm finds a new solution from the pre-determined neighborhood of current solution.This new solution’s objective function value is then compared to that of the current solution to determine whether thenew solution is a better solution or not. If the new solution is better than the current solution, it is automatically acceptedand becomes the current solution from which the search continues. A new solution with worse objective function value thanthat of the current solution may also be accepted as the new current solution with a probability determined by the Metrop-olis criteria [38]. The idea is not to restrict the search algorithm moves in the directions that improve the objective functionvalue, but also allow moves that deteriorate the objective function value, with small probabilities. In principle, this may re-duce the chances of the search procedure being trapped at local minima.

For the application of the SA approach to the MVRP, the solution representation, the generation of initial solution, theneighborhood structure, the parameters used, and the procedure are discussed in the following sections.

3.1. Solution representation

For an MVRP with n visitor groups and m exhibit rooms, a solution can be represented as a string of nm entriesðp1; p2; . . . ; pnmÞ. An entry pi represents one visitor group and the value of pi ranges from 1 to nm . Thus, the solution repre-sentation is a permutation of numbers 1 through nm. A n�m lookup table is used to indicate which visit is under consid-eration when decoding a solution representation. For example, if there are four visitor groups and four exhibit rooms in theMVRP, each visit can be represented by a unique index value ranging from 1 to 16 as shown in Table 1. Given a solution rep-resentation, the visiting schedule/routes are derived as follows. The value of entry pi is used to schedule the ith visit. Supposethat the value of entry p3 is 6, then the third visit to be considered is the one that visitor group 2 visits exhibit room 2. If thevalue of entry p7 is 9, it means that the visit to be considered is the one that visitor group 3 visits exhibit room 1.

Consider the four visitor groups, four exhibit rooms instance given in Tables 2–4. Suppose a given solution string is[7,16,2,9,14,5,6,4,11,3,13,12,1,15,8,10]. Based on the solution string, an active visitor routing schedule generated by theone-pass heuristic is shown in Fig. 1.

3.2. Initial solution

The initial solution is obtained using a modified DS/LTRP dispatching rule for the OSS problem [19] in which the traveltime (setup time) is discarded for easy solution construction. Whenever an exhibit room is idle, the rule selects the visitorgroup that has the longest total remaining visit time on other exhibit rooms, among all idle groups that have not yet visited

Table 1An illustration of visit encoding for an MVRP instance with four visitor groups and four exhibit rooms.

Index value Exhibit room 1 Exhibit room 2 Exhibit room 3 Exhibit room 4

Visitor group 1 1 2 3 4Visitor group 2 5 6 7 8Visitor group 3 9 10 11 12Visitor group 4 13 14 15 16

Table 2Processing times (visiting times) for the MVRP instance with four visitor groups and four exhibit rooms.

Processing time Exhibit room 1 Exhibit room 2 Exhibit room 3 Exhibit room 4

Visitor group 1 4 5 5 6Visitor group 2 6 6 5 4Visitor group 3 6 4 7 3Visitor group 4 5 5 6 6

Table 3Travel time (setup time) required from exhibit room i to exhibit room j for the MVRP instance with four visitor groups and four exhibit rooms.

Travel time Exhibit room 1 Exhibit room 2 Exhibit room 3 Exhibit room 4

Exhibit room 1 – 2 1 2Exhibit room 2 2 – 2 2Exhibit room 3 1 2 – 2Exhibit room 4 2 2 2 –

Table 4Travel time (setup time) required from museum entrance to exhibit rooms and from exhibit rooms to the exit for the MVRP instance with four visitor groupsand four exhibit rooms.

Exhibit room 1 Exhibit room 2 Exhibit room 3 Exhibit room 4

Travel time from entrance to exhibit room 2 3 1 2Travel time from exhibit room to exit 2 3 1 2

Fig. 1. Active visitor routing schedule generated by the one-pass heuristic.


this exhibit room, is selected as the next visitor group for the exhibit room. If no such visitor group exists, the exhibit roomremains idle until a visitor group that has not visited this exhibit room completed their current visit. If more than one exhibitroom is idle at the same time, the rule selects the exhibit room with the longest total remaining visit time to schedule next.After the visit sequences for all exhibit rooms are determined, we have an initial solution and its objective function value canbe easily calculated.

3.3. Neighborhood

Let X be a solution to the MVRP. Its neighborhood, NðXÞ, is constructed either by insertion or swap at random. The inser-tion is carried out by randomly selecting an entry of X and inserting it into the position immediately preceding another ran-domly selected entry of X. The swap is performed by randomly selecting two entries of X, and then swapping their values.The swap is only applied to visits of the same visitor group or visits to the same exhibit room. The probability of carrying outinsertion and swap visits of the same visitor group, and swap visits to the same exhibit room is set to be 0.5, 0.25, and 0.25,respectively.


3.4. Parameters used

The proposed SA begins with six parameters, namely Iiter; T0; TF ;Nnon-improving;K , and a. Iiter denotes the number of iterationsthe search proceeds at a particular temperature. T0 and TF represent the initial temperature and final temperature, respec-tively. If the current temperature is lower than TF , the SA procedure is terminated. Nnon-improving is the maximum number oftemperature reductions during which the best objective function value obtained is not improved. K is the Boltzmann con-stant commonly used in SA heuristic. Lastly, 0 < a < 1 is a coefficient controlling the cooling schedule.

3.5. The SA procedure

In the beginning, the current temperature T is set to be T0, an initial solution X is generated according to the DS/LTRP rule,and the current best solution Xbest is set to be X. In each iteration, the next solution Y is sampled from NðXÞ and its objectivefunction value is evaluated. The current temperature T is decreased to aT after running Iiter iterations since the previous tem-perature reduction. Let obj(X) and obj(Y) denote the objective function value of X and Y, respectively, and D denote the dif-ference between them, i.e., D ¼ objðYÞ � objðXÞ . The probability of replacing X with Y, given that D > 0, is e�D=KT . Meanwhile,if D < 0 the probability of replacing X with Y is 1. Xbest keeps track of the best solution as the algorithm progresses. When T islower than TF , the algorithm is terminated. If Xbest is not improved in Nnon-improving successive reductions in temperature or thecurrent solution is equal to LB, the algorithm is also terminated. Following the termination of the SA procedure, the (near)global optimal schedule is Xbest with makespan Cbest ¼ objðXbestÞ.

4. Computation results

To our best knowledge, there are no benchmark instances for the MVRP. Therefore, we converted Taillard’s problem setfor OSS problem [22] into MVRP test problems to verify the efficiency of the proposed approach. This problem set consists ofsix different types of problems. There are 10 instances of each problem type, for a total of 60 different problems. These prob-lems are all square problems, i.e., n ¼ m , and range from small ones with 16 visits to large problems with 400 visits (Taillardobserved that open shop problems with n ¼ m are harder to solve than those with n� m). Since Taillard’s problem set wasintended for the OSS problem, the time values in those benchmark problems are not suitable for our computational exper-iment. Thus, for each Taillard’s problem, we generated new processing time values as visiting times, and sequence-depen-dent setup times as travel times. The visiting times and travel times are generated in a way similar to what was used byTaillard in generating the original OSS problem set. The generated visiting times are uniformly distributed over the interval[8,14], with an average �t ¼ 11. The generated travel times are uniformly distributed over the interval [2,10], with an averagetravel time �s ¼ 6. sk0 is set to be the same as s0k, implying that the entrance and exit are at the same location as seen in mostmuseums. The PASCAL codes used to generate visiting times and travel times are given in the Appendix.

The proposed SA approach is implemented in C, compiled with the option of maximum execution speed, and run on acomputer with an Intel Core 2 Dual T5850 processors running at 2.16 GHz and 1 GB of DRAM. There is no multi-core or mul-ti-threading programming so a single core on each computer is actually used.

Parameter selection may influence the quality of the results. Therefore, parameters Iiter; T0; TF ;K;Nnon-improving and a are cho-sen experimentally to ensure a compromise between the running time and the solution quality. After testing a few problemswith various combinations of parameters, the parameter values for SA are set to be Iiter ¼ 1000mn; T0 ¼50; TF ¼ 0:5;K ¼ 10;Nnon-improving ¼ 50 and a ¼ 0:95, where n is the number of visitor groups and m is the number of exhibitrooms to visit. The algorithm is terminated after at most 90 ðT0a90 ¼ 50ð0:95Þ90

< 0:5 ¼ TFÞ temperature reductions. The algo-

150

170

190

210

230

1 11 21 31 41 51 61 71 81Number of temperature reduction

Obj

ectiv

e fu

nctio

n va

lue

(mak

espa

n)

10_10_1

10_10_5

10_10_10

Fig. 2. Evolution of best solutions obtained in the 10 visitor groups and 10 exhibit rooms problems.


rithm may terminate earlier if the best solution obtained is not improved during 50 successive temperature reductions or thecurrent solution is found to be equal to LB. Three benchmark problems, 10_10_1, 10_10_5 and 10_10_10, are used to verify theconvergence trend of the proposed approach. The relationship between the objective function values of the best solutions andthe number of ‘‘temperature reductions” is displayed in Fig. 2. It can be seen that after 90 temperature reductions, the pro-

Table 5Comparison between various approaches for open shop scheduling problems.

Problem BKS TS-Liaw [21] SA-Liaw [25] HGA-Liaw [29] GA-Prins [24] Beam-ACO [30] PSO [32] Proposed approach Time used (s)

4 � 4_1 193 193* 193* 193* 193* 193* 193* 193* 0.014 � 4_2 236 236* 236* 236* 239 236* 236* 236* 0.064 � 4_3 271 271* 271* 271* 271* 271* 271* 271* 0.064 � 4_4 250 250* 250* 250* 250* 250* 250* 250* 0.414 � 4_5 295 295* 295* 295* 295* 295* 295* 295* 0.264 � 4_6 189 189* 189* 189* 189* 189* 189* 189* 0.004 � 4_7 201 201* 201* 201* 201* 201* 201* 201* 0.004 � 4_8 217 217* 217* 217* 217* 217* 217* 217* 0.014 � 4_9 261 261* 261* 261* 261* 261* 261* 261* 0.054 � 4_10 217 217* 217* 217* 221 217* 217* 217* 0.005 � 5_1 300 300* 300* 300* 301 300* 300* 300* 1.095 � 5_2 262 262* 262* 262* 263 262* 262* 262* 0.135 � 5_3 323 326 323* 323* 335 323* 323* 323* 0.845 � 5_4 310 310* 310* 310* 316 310* 310* 310* 0.955 � 5_5 326 326* 326* 326* 330 326* 326* 329 1.205 � 5_6 312 312* 312* 312* 312* 312* 312* 312* 0.585 � 5_7 303 303* 303* 303* 308 303* 303* 305 1.195 � 5_8 300 300* 300* 300* 304 300* 300* 300* 1.105 � 5_9 353 353* 353* 353* 358 353* 353* 353* 0.865 � 5_10 326 326* 326* 326* 328 326* 326* 326* 0.307 � 7_1 435 435* 435* 435* 436 435* 435* 435* 2.057 � 7_2 443 447 445 443* 447 443* 443* 443* 4.067 � 7_3 468 468* 468* 468* 472 468* 468* 468* 4.477 � 7_4 463 463* 466 463* 463* 463* 463* 463* 3.957 � 7_5 416 417 416* 416* 417 416* 416* 416* 3.337 � 7_6 451 459 456 451* 455 451* 451* 454 4.717 � 7_7 422 429 425 422* 426 422* 422* 422* 4.457 � 7_8 424 424* 424* 424* 424* 424* 424* 424* 3.197 � 7_9 458 458* 458* 458* 458* 458* 458* 458* 2.127 � 7_10 398 398* 398* 398* 398* 398* 398* 398* 2.1510 � 10_1 637 646 647 637* 637* 637* 637* 637* 22.7110 � 10_2 588 588* 589 588* 588* 588* 588* 588* 10.5110 � 10_3 598 601 604 598* 598* 598* 598* 598* 20.9610 � 10_4 577 577* 577* 577* 577* 577* 577* 577* 8.6610 � 10_5 640 64 640* 640* 640* 640* 640* 640* 20.3310 � 10_6 538 538* 538* 538* 538* 538* 538* 538* 3.4710 � 10_7 616 616* 616* 616* 616* 616* 616* 616* 11.9410 � 10_8 595 595* 595* 595* 595* 595* 595* 595* 13.3110 � 10_9 595 597 597 595* 595* 595* 595* 595* 15.1410 � 10_10 596 596* 596* 596* 596* 596* 596* 596* 11.4015 � 15_1 937 937* 937* 937* 937* 937* 937* 937* 42.3415 � 15_2 918 920 9189 918* 918* 918* 918* 918* 50.9515 � 15_3 871 871* 871* 871* 871* 871* 871* 871* 51.8915 � l5_4 934 934* 934* 934* 934* 934* 934* 934* 39.4315 � 15_5 946 9469 952 946* 946* 946* 946* 946* 93.8915 � 15_6 933 933* 933* 933* 933* 933* 933* 933* 52.9515 � 15_7 891 891* 891* 891* 891* 891* 891* 891* 59.9615 � 15_8 893 893* 893* 893* 893* 893* 893* 893* 49.7915 � 15_9 899 910 905 899* 899* 899* 899* 899* 105.6115 � 15_10 902 906 902* 902* 902* 902* 902* 902* 60.4120 � 20_1 1155 1155* 1159 1155* 1155* 1155* 1155* 1155* 248.4320 � 20_2 1241 1246 124612 124112 1241 1241* 1241* 1241* 315.2320 � 20_3 1257 1257* 1257* 1257* 1257* 1257* 1257* 1257* 136.7920 � 20_4 1248 1248* 1248* 1248* 1248* 1248* 1248* 1248* 214.7920 � 20_5 1256 1256* 1256* 1256* 1256* 1256* 1256* 1256* 195.4820 � 20_6 1204 1204* 1204* 1204* 1204* 1204* 1204* 1204* 201.6620 � 20_7 1294 1298 1296 1294* 1294* 1294* 1294* 1294* 242.6220 � 20_8 1169 1184 1189 1177 1169*1 1169* 1169* 1169* 382.2120 � 20_9 1289 1289* 1289* 1289* 1289* 1289* 1289* 1289* 164.1220 � 20_10 1241 1241* 1241* 1241* 1241* 1241* 1241* 1241* 178.07

* Indicates that BKS is attained.


cedure is still actively search for better solutions. If solution quality is extremely important, then a better solution can be ob-tained by additional computational effort. However, the solution presented herein is adequate for most applications.

Table 6Computational results for benchmark instances.

Problem n m Time seed LB RCPLEX CPLEX(UB)

InitialSolution

BestObj.

AverageObj.

Std.dev.

Percentagegapc

Running time(s)

4 � 4_1 1 4 4 1166510396 67 71a 72b 93 72b 72.0 0.00 1.4% 0.74 � 4_2 4 4 1624514147 68 68a 74b 75 74b 74.0 0.00 8.8% 0.64 � 4_3 4 4 1116611914 67 73a 76b 83 76b 76.0 0.00 4.1% 0.74 � 4_4 4 4 410579806 74 78a 79b 81 79b 79.0 0.00 1.3% 0.64 � 4_5 4 4 1036100146 70 69a 79b 84 79b 79.0 0.00 14.5% 0.64 � 4_6 4 4 597897640 71 74a 78b 81 78b 78.0 0.00 5.4% 0.64 � 4_7 4 4 1268670769 64 70a 71b 74 71b 71.0 0.00 1.4% 0.64 � 4_8 4 4 307928077 61 62a 68b 80 68b 68.0 0.00 9.7% 0.64 � 4_9 4 4 667545295 73 77a 83b 87 83b 83.0 0.00 7.8% 0.64 � 4_10 4 4 35780816 68 72a 76b 83 76b 76.0 0.00 5.6% 0.75 � 5_1 5 5 527556884 84 88a 89b 109 89b 91.1 0.74 1.1% 1.85 � 5_2 5 5 1046824493 76 77a 84b 103 84b 85.0 0.74 9.1% 1.95 � 5_3 5 5 1165033492 87 90a 97b 127 97b 97.9 0.79 7.8% 1.85 �;5_4 5 5 476292817 84 89a 94b 104 94b 94.8 0.40 5.6% 1.85 � 5_5 5 5 1181363416 77 84a 92b 114 92b 92.5 0.67 9.5% 1.95 � 5_6 5 5 897739730 82 86a 92b 109 92b 93.2 0.57 7.0% 1.85 � 5_7 5 5 577107303 83 89a 96b 111 96b 96.5 0.50 7.9% 1.85 � 5_8 5 5 1714191910 77 83a 90b 108 90b 91.1 0.67 8.4% 1.95 � 5_9 5 5 1813128617 87 87a 92b 109 92b 93.7 0.78 5.7% 1.95 � 5_10 5 5 808919936 82 86a 93b 112 93b 94.0 0.32 8.1% 1.87 � 7_1 7 7 1840686215 111 115a 121 139 119 122.5 1.29 3.5% 7.27 � 7_2 7 7 1026771938 102 107a 117 139 116 118.8 0.99 8.4% 7.37 � 7_3 7 7 609471574 111 121a 128 159 129 131.2 0.91 6.6% 7.27 � 7_4 7 7 1022295947 112 114a 126 154 126 127.3 0.78 10.5% 7.37 � 7_5 7 7 1513073047 111 117a 126 145 126 127.4 0.80 7.7% 7.27 � 7_6 7 7 1612211197 109 116a 128 148 128 129.4 0.85 10.3% 7.27� 7́_7 7 7 435024109 106 108a 122 139 121 123.2 0.98 12.0% 7.27 � 7_8 7 7 1760865440 108 112a 123 145 122 124.6 1.07 8.9% 7.37 � 7_9 7 7 122574075 110 112a 121 148 122 123.6 0.92 8.9% 7.37 � 7_10 7 7 248031774 109 110a 118 143 118 119.9 1.00 7.3% 7.110 � 10_1 10 10 1344106948 146 163 185 220 176 178.3 0.95 8.0% 27.010 � 10_2 10 10 425990073 143 154 177 191 165 167.8 1.22 7.1% 27.110 � 10_3 10 10 666128954 145 155 175 196 164 166.7 1.53 5.8% 27.710 � 10_4 10 10 442723456 141 147 173 186 162 164.9 1.20 10.2% 27.110 � 10_5 10 10 2033800800 148 160 180 199 170 173.1 1.20 6.3% 27.110 � 10_6 10 10 964467313 136 148 169 196 159 161.9 1.62 7.4% 27.210 � 10_7 10 10 1004528509 144 154 175 197 164 166.4 1.06 6.5% 27.110 � 10_8 10 10 1667495107 141 154 176 200 165 167.3 1.62 7.1% 27.110 � 10_9 10 10 1806968543 146 158 177 197 166 168.6 1.16 5.1% 27.110 � 10_10 10 10 938376228 141 146 171 190 161 163.6 1.40 10.3% 27.215 � 15_1 15 15 1561423441 218 241 283 300 255 259.5 1.99 5.8% 129.115 � 15_2 15 15 204120997 213 235 285 288 254 257.8 1.92 8.1% 129.215 � 15_3 15 15 801158374 209 232 274 297 246 248.2 1.56 6.0% 129.215 � l5_4 15 15 1502847623 218 239 291 307 253 257.8 2.06 5.9% 129.015 � 15_5 15 15 282791231 209 229 284 299 252 253.3 1.22 10.0% 130.015 � 15_6 15 15 1130361878 212 232 282 305 247 251.2 2.09 6.5% 130.315 �;15_7 15 15 379464508 208 234 279 290 252 254.5 1.43 7.7% 130.215 � 15_8 15 15 1760142791 211 231 289 294 246 252.0 2.26 6.5% 129.215 � 15_9 15 15 1993140927 210 233 288 298 247 251.7 2.24 6.0% 129.615 � 15_10 15 15 1678386613 211 234 287 302 253 256.0 1.92 8.1% 129.520 � 20_1 20 20 957638 273 317 33 379 335 339.9 2.35 5.7% 432.020 � 20_2 20 20 162587311 283 329 390 387 347 351.7 2.03 5.5% 430.720 � 20_3 20 20 965299017 280 319 424 410 342 346.6 1.96 7.2% 430.520 � 20_4 20 20 1158457671 275 325 419 401 346 350.0 2.41 6.5% 430.520 � 20_5 20 20 1191143707 277 317 408 386 342 346.5 2.40 7.9% 430.820 � 20_6 20 20 1826671743 278 323 413 376 339 342.8 1.67 5.0% 430.820 � 20_7 20 20 1591533998 284 324 407 393 348 351.9 1.74 7.4% 432.520 � 20_8 20 20 937297777 272 321 407 390 340 343.6 2.75 5.9% 430.720 � 20_9 20 20 687896268 280 316 392 405 336 341.2 2.44 6.3% 433.120 � 20_10 20 20 687034842 279 317 391 390 342 346.8 2.45 7.9% 432.1

a Indicates optimal solution to the relaxed problem.b Indicates optimal solution to the original problem.c Percentage gap is calculated as (Best Obj. � RCPLEX)/RCPLEX.


Before applying the proposed approach to MVRP, we tested it on OSS problem instances to verify its performance by ignoringthe setup times in MVRP. The well-known Taillard’s OSS problem set [22] is used to compare the performance of the proposedapproach with other existing heuristics, such as TS [21], SA [25], HGA [29], GA [24], beam-ACO [30] and PSO [32].

Each benchmark problem is solved 20 times. The best solutions obtained by other existing heuristics, the best objectivefunction values from 20 runs and the average computing time are listed in Table 5. Although the proposed approach is notspecifically designed for the OSS problems, it performs well against other existing approaches as it obtained all but three bestknown solutions. Since the computers used by various algorithms vary in speed, it is difficult to have a fair comparison of thecomputational time. Thus we only listed the computing time for the proposed approach.

After verifying the effectiveness of the proposed approach, it is applied to the MVRP benchmark problems. Each problemis solved 20 times. The best, average, and standard deviation of the objective function values from 20 runs, as well as theinitial solution values, LB and the solutions of relaxed problem obtained by CPLEX, denoted by RCPLEX, are listed in Table 6.The relaxed problems are obtained by relaxing the integer requirement of the indicative variables xihk. The relaxed problemsare solved by CPLEX with a time limit of 7200 s. CPLEX are able to obtain optimal solutions for relaxed problems of size up to7 � 7. The CPLEX column listed the MVRP solutions obtained by CPLEX within 7200 s, which can be regarded as upperbounds of the MVRP. Note that the 4 � 4 and 5 � 5 problems are solved optimally. CPLEX solutions to other problemsmay not be optimal due to time limit or computer memory restriction.

It can be seen that for smaller problems (4 � 4 and 5 � 5), the proposed SA approach is capable of quickly obtaining theoptimal solutions within 2 s. For other larger problems, the best solution and the average solution obtained from 20 runs arevery close to RCPLEX, indicating that the proposed SA approach is capable of producing quality MVRP solutions. The goodsolution quality can also be seen from the Percentage Gap column of Table 6. The percentage gap is calculated as (BestObj. – RCPLEX)/RCPLEX. For smaller problems of size 4 � 4 and 5 � 5, the Best Obj. is actually the optimal solution valueand the average percentage gap is 6.51%, indicating the good quality of the lower bound obtained by the relaxed problem.For problems of size up to 7 � 7, this gap is 7.14% so the solution obtained by the proposed algorithm is within 7.14% of theoptimal solution. Overall, the average gap is 7.07%. Thus, it is likely that the solutions obtained by the proposed algorithm onaverage are within about 7% of the optimal solutions. Furthermore, the small standard deviations (0.00–2.75) indicate thatthe proposed algorithm is relatively stable. The computational time required for the proposed SA ranges from less than 1 sfor small instances to about 7 min for the largest problems is acceptable for real-world applications.

5. Conclusions

In this study, the museum visitor routing problem is formulated as a mix integer program which is an extension of theopen shop scheduling problem. Due to the complexity of the problem, solving this problem by exact optimization algorithmsis not feasible given the current computer technology. Therefore, we proposed a simulated annealing approach to solve theMVRP. Computational experiments yielded encouraging results. The SA approach presented in this study is capable of gen-erating high quality solutions to the MVRP, with computer technology that is well within the reach of museums.

This study has made a step towards establishing an SA approach for the MVRP. There are several possible extensions tothis study in the future research. First, it may prove worthwhile to study the application of other meta-heuristics to theMVRP. Second, it may be possible to modify the proposed SA approach to solve more complex MVRPs. For example, someexhibit rooms may be able to accommodate more than one visitor group at the same time. Third, solving the MVRP withalternative performance criterion deserves further attention. Several other criteria such as earliness and tardiness relatedcriteria may prove both interesting and useful.

Acknowledgements

This research is supported in part by grants NSC 94-2422-H-011-002 and NSC 97-2410-H-182-020-MY2 from the Na-tional Science Council, Taiwan.

Appendix A.

PROGRAM MVRP;var

time_seed, nb_visitor_groups, nb_exhibits, i, j:INTEGER;s: ARRAY[0..20, 0..20] of INTEGER;t: ARRAY[0..20, 0..20] of INTEGER;

function unif(var seed : integer; low, high : integer): integer;(� generate a random number uniformly between low and high�)const

m = 2147483647;a =16807;


b =127773;c = 2836;

vark : integer;value_0_1 : double; (* floating point coded on 64 bits �)

begink := seed div b ;seed := a * (seed mod b) - k * c;if seed < 0 then seed := seed + m ;value_0_1 := seed / m ;unif := low + trunc(value_0_1� (high - low + 1))

end;

beginfor i := 1 to nb_nb_visit_groups dobegin

for j := 1 to nb_exhibits dobegin

t[i,j] := unif(time_seed, 8, 14);end;

end;

for i := 1 to nb_exhibits dobegin

s[0,i]:=unif(time_seed, 2, 10); s[i,0]:= s[0,i]; s[i,i]:=0;end;for i := 1 to nb_exhibits-1 dobegin

for j := i+1 to nb_exhibits dobegin

s[i,j] := unif(time_seed, 2, 10);s[j,i] := s[i,j];

end;end;

end.

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