d2017年美国大学生数学建模竞赛o奖论文全部d67316-校苑数模_W

1、Team Control Number67316For office use onlyFor office use onlyT1 T2 T3T4 F1 F2 F3F4 Problem ChosenD2017MCM/ICMSummary Sheet(Your teams summary should be included as the first page of your electronic submission.)Type a summary of your results on this page. Do not include the name of your school, advi

2、sor, or team members on this page.SummaryBottlenecks that passengers take time to take care of their carry-on properties before X-ray scanning and that the structure of the checkpoints is not satisfying are spotted, and detailed recommendations for the airport security management to raise throughput

3、 of the security checkpoints, to improve the passengers satisfaction and to keep the cost relatively low are given with cultural factors quantied and their impacts on the models discussed.We divided the security check process into two phases and regard the entire as two queueing models in series. By

4、 analyzing the given data, the document check i s found to be a Poisson queueing (/ ) in Kendall notation, and Erlangian process (/) is concerned in the scan check process. Numeric solution of ( / ) is introduced using simulation tech- nique.Assumed that arrivals of passengers for a ight obey normal

5、 distribution, the varying passenger ow in a period of time can be generated from the real data and used for our sim- ulation. We also believe that the mean and the variance of the distribution is culture-re- lated. We also nd that the passenger ow can be changed by recommending the passengers their

6、 arrivals, and thus better result of the passengers waiting time can be achieved.We also suggested several methods to improve the design of the checkpoint, including shortening the distances between identical checkpoints and more rational human resource allocation.Virtual queueing is recommended as

7、an approach to improve passengers experience, and modify the conventional First In First Service queueing discipline to partial priority queueing discipline as well. A partial priority queueing discipline is put forward to reduce the remaining time variance of the passengers and to decrease the numb

8、er of passengers that missed their ights, thus better passenger satisfaction is reached.We also introduced culture-related factors for passenger arrival recommendation and pri- ority queueing discipline. For the latter, an “acceptability factor” named is used to denote the acceptability of strict pr

9、iority discipline. And the examples of dierent cultures are given to illustrate this idea.Validation of each model are made in our essay to make them convincing. We later assess the models and give a complete guide for the security managers to optimize the airport secu- rity check workow. Weaknesses

10、 and further work that is not implemented in our essay are also pointed out.Team #67316Page 1 of 20Time Counts! Less Waiting & Better AirportsIntroductionBackground1.1Airport security check has been improved ever since the 911 attack. Although enhanced secu-rity means safer ights, however the compli

11、cated procedure may also increase the passengers waiting time and add cost to the U.S. Transportation Security Agency (TSA). Under some extreme circumstances, passengers have to wait for hours (and they are often recommended to be earlier for 23 hours, which often lead to confusion) (Hetter, 2016).

12、Thus, to shorten the passengers waiting and designing amore ecient security check procedure is vitally important. TSA is now in controversy for causing long queues waiting for security check. We, the Internal Control Management (ICM) team, trying to nd a solution, faces the problems below:1.2.3.Iden

13、tify the bottlenecks of the current security check workow.Improve the process with modications, and illustrate how the modications work. Find how to allow the modied process to be compatible with dierent culture back- grounds and lower the variance of the passengers waiting time.Make suggestions on

14、the policy for the security manager, with concern of the former requirements and corresponding models.4.1.2Analysis and Approach OverviewFor problem 1, we divide the security check process in two parts: Phase 1, document check;and Phase 2, luggage and body scanning. The former is a Poisson queue, wh

15、ile the latter concerns an Erlangian model. Simulation is practiced as a means of solving multi-server Er- langian model. By testing the total waiting times sensitivity to changes on numbers of parallel servers in the two phases respectively, the bottlenecks of the workow can be spotted.To solve pro

16、blem 2, considering the inuential factors of a queueing process, modicationswill be put forward to optimize and avoid congestions.The current TSA recommended passengers arrival time is used to build a model of the passenger arrival behavior at an airport, and we assume the arrivals in time for one c

17、ertain ight obey normal distribution, and in a small time interval, the arrivals of all passengers for all ights obey an exponential distribution. We will modify the arrival recommendation strat- egy to inuence passengers arrival behavior.Another direction to improve the current process is to provid

18、e more robust security check service with greater capacity. A few suggestions and their verication or explanation will be given.Team #67316Page 2 of 20Besides the performance, justiability also counts. Virtual queueing under other disciplines(Zhao, et al., 2016) will be a good practice. Queueing dis

19、cipline modication is culture-sensitive, and thus lead to the discussion of the third problem.The following diagram illustrate the above ideas in a more visual way:Figure 1 Optimizing directions and our modications,where “service pattern” concerns the service rate and number of servers.Assumptions1.

20、2.Individual variability is not considered for the servers, the checkpoint structures, etc.Although the number of lanes opened in the scan check process is dynamic, we assume that the service capability is always at its maximum. That is to say, spare lanes will be open, so long as the arrival exceed

21、s the current capability.Assume that every passenger will choose to wait in the queue that minimizes their waiting time at every checkpoint.Points in time that passengers arrive at the airport for a certain ight obey normaldistribution.TSA Pre-check wont contribute much to the congestion compared wi

22、th the normal one. Almost everyone would arrive at the airport for at least half an hour..See 4.1 Security Check ProcessOverview, and 4.3 Queueing Model Specication for de-tailed interpretation and assumptions of the given datasheet.Symbols and NotationsSymbol or NotationSpecicationKendall no

23、tation, whereanddenote the inter-arrival-time distribution, the service time distribution, the number of parallel servers,Team #67316Page 3 of 20Denote an exponential distribution for inter-arrival time and service time,i.e. a Poisson distribution for arrival and remove rate.Estimated value of .Queu

24、eing Model for Security Check ProcessSecurity Check Process -Overview4.1Figure 2 TSA Security checkpointThe detailed security check process of one single checkpoint given in the diagram above. Ac-cording to TSAs policy, we classify the security processes in two types (pre-check included and no pre-c

25、heck concerned process), and divide the each process in two queueing phases. In Phase 1, the passengers identity documents will be checked, after which they enter Phase 2, where luggage and body screening will be accepted. Two dierent types are in essence the same queueing with dierent parameters (n

26、umber of servers, service time, etc.). And the entire security check process can be seen as two queueing models in series.To better clarify the whole process, a procedure sequence diagram is given below:0Normal distribution with mean , and variance.Arriivaall and seerrvviiccee rraatteess ooff tthee

27、sdcoacnucmheecnkt pcrhoeccekssp. rocess.Erlang distribution type .queueing discipline, restriction on system capacity, and the source of thearrival (usually innity) respectively. (Taha, 2014)Team #67316Page 4 of 20ABCDEFGHFigure 3 Procedure sequence diagramA: time waiting for the document check, B:

28、document checking timeC: time waiting for sending luggage for X-ray scanning, D: X-ray scanning time, E: other possible luggage checksF: time waiting for millimeter wave scan, G: body scan time, H: other possible body checks The yellow shaded part is Phase 1 as we introduced in previous paragraphs,

29、and the rest Phase 2.The given Excel datasheet contains time records of the airport checkpoints (whose dier-ences denote the inter-arrival time), time taken of the ID check process of previous checked passenger (i.e. B in the diagram), millimeter wave scan timestamps (dierences of which are the mill

30、imeter wave scanning time, shown as G in the diagram), timestamps that luggage getting out of the X-ray scan (dierences denote E), and time to get scanned property (D, E, F, G, H).By analyzing the given datasheet, patterns of the airport security check behaviors, such as the distribution of passenge

31、rs inter-arrival time and service time at each section, can be dis- covered. Therefore a complete queueing model can be specied.4.2 Significance Test on Two TSA OfficersThe given datasheet involves timing of two dierent TSA ocers, the signicance test here isto verify that individual variability does

32、 not contribute much to the service time.Suppose that variances of the service time of the two TSA ocers are equal, 2 = 2= 2.12We make the null hypothesis 0 that service time expectations of both ocers are the same,i.e. 1 = 2. The combined sample variance of the two ocers, 2 = 12.927.|12At 5% level,

33、= 1.375 0.025(14) = 2.145, the null hypothesis is not rejected. 1/1 |+1/2Assume that the document check service time obeys the exponential distribution. The pa-rameter of the distribution, i.e. service rate, is estimated, = 0.09465 .4.3 Queueing Model SpecificationThe two phases of the process can b

34、e specied as a series of queues by giving the arrival andservice time distribution pattern.Team #67316Page 5 of 204.3.1Arrival3020251520151010550001020304050020406080Figure 4 Frequency histogram of non-pre-check arrivalsFigure 5 Frequency histogram of pre-check arrivalsAccording to the histograms ab

35、ove, we suppose that the time intervals of both the regular andpre-check arrivals obey exponential distribution ()= eMaximum likelihood estimation of the parameter : . () = ( ) = e= e=1=1=1 =1d ln()ln ( ) = ln ,=1= d0Therefore =and the expectation, 1 , = = , is an unbi-11=() =1 ased estimation. Thus

36、 the arrival rates of the regular and pre-check procedures are estimatedto be 1 = 0.077244 and 2= 0.10882016 , respectively.Here, a goodness of t test is done, and each of the procedures are classied into 10 classes.Lets take the regular (without pre-check) procedure as an example:True Frequency2113

37、63001101 Predicted Probability0.4610.2480.1340.0720.0390.0210.0110.0060.0030.002Predicted Frequency21.20411.4306.1613.3211.7900.9650.5200.2800.1510.081Table 1 Regular procedure, non-pre-check, true frequencies and predicted frequencies10 ( 2= =1= 15.898 2(9) = 16.919 )0.05Thus we consider the above

38、as an exponential distribution.And similarly, t for the pre-check procedure, 2 = 15.599 0; 0 )()with the expectation () = and variance () = 2 . Parameter and is related with and the service rate (Gross, et al., 1985):= =Fit the distribution for the total service time in Phase 2 (time to get scanned

39、property),our results are, = 4, = 0.0357.Therefore, the Kendall notation of Phase 2 queueing is ( / /) , where the input is identical to the output of Phase 1, is an Erlang type = 4, = 0 .0357 distribution, and the number of parallel servers (i.e. the number of X-ray scanners and millimeter wave sca

40、nners) is depend on specic airport.( / /) queueing does not have a good analytical solution; our numerical solution using simulation technique will be introduced later.4.4 Section SummaryThe airport security check process consists of two phases, rst of which is an ( / /) queue- ing process, and the

41、second is an ( / / ) queueing. Arrival rate of the second is the service rate of the rst. Parameters of the distributions can be estimated using the given data.Team #67316Page 7 of 20Queueing Simulation and Bottlenecks Spotting5.1Estimating ExpensesAccording to (L; PayS), income of a

42、 security ocer is $28,624$58,987 and of an airline security screener is $23,262$54,015. That is the human resource expenses of the security check. Costs of security devices are mostly one-o consumptions. The price of an airport X-ray baggage scanner is at $20,00050,000 and of a millimeter wave full

43、body scanner is at $100,000 (A). And the maintenance cost is mainly from the electricity. Another fact is that one single lane of the scan check process requires 4 security screeners on average. Therefore, we drew a conclusion that the cost of one lane (server) in Phase 2, is about the cos

44、t of 4 desks (servers) at Phase 1.5.2 Basic IdeasA MATLAB program is written to simulate the entire security check process. To build up thenumeric solution, Markov chains are used for Erlangian queueing. (Zeng, et al., 2011)Figure 7 MATLAB simulation specicationTeam #67316Page 8 of 20Since there is

45、no analytical solution of an ( / /) (Phase 2) queueing model, a simulation of the entire security check process is put forward, and both Phase 1 and Phase 2 are concerned in the model in series.Given the parameters of each probability distribution at the two phases, we generate series of random vari

46、ates to simulate the process. First, random variates obeying exponential distri- bution is generated as peoples inter-arrival time. We then calculate and store the arrival timestamps, service time (generated random variablesobeying another exponential distribution) and waiting time (worked out with

47、the previous passengers waiting time, service time and the arrival interval) of each passenger in a table. Besides getting results of waiting time spans, the removal rate of Phase 1 is also observed, which is used as the arrival of Phase 2. After working on Phase 1, we do the same on Phase 2, but th

48、e new service is Erlangian. Finally, expectations of total waiting time are evaluated.To spot the bottlenecks in the process, we tested the improvements of total waiting time expectations when 4 document check servers are added (row “Document Check Added” in the table below), and 1 scan check server is added (row “Scan Check Added”).5.3 Spot the Bottlenecks!The simulated security process has a changeable arrival rate , and both of the initial