CoCalc -- ModSim Project 1.tex

Project: Tiger Jewell-Alibhai - f19_modsim
Path: projects/project-1 / ModSim Project 1 / ModSim Project 1.tex
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    % Document parameters
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    \title{ModSim Project 1}
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    \begin{document}
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    \maketitle
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    Modeling Flight Delays
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    The Question
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What is the best way to increase the number of flights without delays?
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We will model airplane traffic between several airports and test two
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different modeling strategies to avoid flight delays and maintain flight
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turnaround efficiency.
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    \begin{Verbatim}[commandchars=\\\{\}]
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{\color{incolor}In [{\color{incolor}1}]:} \PY{c+c1}{\PYZsh{} Configure Jupyter so figures appear in the notebook}
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        \PY{o}{\PYZpc{}}\PY{k}{matplotlib} inline
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        \PY{c+c1}{\PYZsh{} Configure Jupyter to display the assigned value after an assignment}
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        \PY{o}{\PYZpc{}}\PY{k}{config} InteractiveShell.ast\PYZus{}node\PYZus{}interactivity=\PYZsq{}last\PYZus{}expr\PYZus{}or\PYZus{}assign\PYZsq{}
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        \PY{c+c1}{\PYZsh{} import functions from the modsim library}
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        \PY{k+kn}{from} \PY{n+nn}{modsim} \PY{k}{import} \PY{o}{*}
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        \PY{c+c1}{\PYZsh{} set the random number generator}
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        \PY{n}{np}\PY{o}{.}\PY{n}{random}\PY{o}{.}\PY{n}{seed}\PY{p}{(}\PY{l+m+mi}{7}\PY{p}{)}
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        \PY{k+kn}{import} \PY{n+nn}{random}
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        \PY{k+kn}{import} \PY{n+nn}{pandas} \PY{k}{as} \PY{n+nn}{pd}
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        \PY{k+kn}{import} \PY{n+nn}{datetime}
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        \PY{k+kn}{from} \PY{n+nn}{dateutil}\PY{n+nn}{.}\PY{n+nn}{parser} \PY{k}{import} \PY{n}{parse}
319
        \PY{k+kn}{import} \PY{n+nn}{math}
320
        \PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np}
321
\end{Verbatim}
322

323
    Below is data collected in 2008 which details flights and delays. This
324
data was narrowed to include only Delta (DL) and United (UA) flights
325
between airports LAX, JFK, ATL, IAD, SEA. By using only flights between
326
specific airports, we reduce the likelihood that the data is influenced
327
primarily by the airport or the airline.
328

329
    \begin{Verbatim}[commandchars=\\\{\}]
330
{\color{incolor}In [{\color{incolor}2}]:} \PY{n}{trips} \PY{o}{=} \PY{n}{pd}\PY{o}{.}\PY{n}{read\PYZus{}csv}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{2008.csv}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
331
\end{Verbatim}
332

333
\begin{Verbatim}[commandchars=\\\{\}]
334
{\color{outcolor}Out[{\color{outcolor}2}]:}        Year  Month  DayofMonth  DayOfWeek  DepTime  ArrTime UniqueCarrier  \textbackslash{}
335
        0      2008      1           1          2    613.0   1407.0            UA   
336
        1      2008      1           2          3    615.0   1435.0            UA   
337
        2      2008      1           3          4    607.0   1454.0            UA   
338
        3      2008      1           4          5    618.0   1523.0            UA   
339
        4      2008      1           5          6    615.0   1416.0            UA   
340
        {\ldots}     {\ldots}    {\ldots}         {\ldots}        {\ldots}      {\ldots}      {\ldots}           {\ldots}   
341
        10152  2008      2          29          5   2128.0   2311.0            DL   
342
        10153  2008      2          29          5   1858.0   2041.0            DL   
343
        10154  2008      2          29          5   1455.0   1646.0            DL   
344
        10155  2008      2          29          5    824.0   1002.0            DL   
345
        10156  2008      2          29          5    957.0   1147.0            DL   
346
        
347
               ActualElapsedTime  AirTime  ArrDelay  {\ldots}  Origin Dest Distance  \textbackslash{}
348
        0                  294.0    278.0     -24.0  {\ldots}     LAX  JFK     2475   
349
        1                  320.0    298.0       4.0  {\ldots}     LAX  JFK     2475   
350
        2                  347.0    299.0      23.0  {\ldots}     LAX  JFK     2475   
351
        3                  365.0    284.0      52.0  {\ldots}     LAX  JFK     2475   
352
        4                  301.0    282.0     -15.0  {\ldots}     LAX  JFK     2475   
353
        {\ldots}                  {\ldots}      {\ldots}       {\ldots}  {\ldots}     {\ldots}  {\ldots}      {\ldots}   
354
        10152              103.0     77.0      -2.0  {\ldots}     ATL  IAD      533   
355
        10153              103.0     79.0       0.0  {\ldots}     ATL  IAD      533   
356
        10154              111.0     78.0       5.0  {\ldots}     ATL  IAD      533   
357
        10155               98.0     78.0      -5.0  {\ldots}     ATL  IAD      533   
358
        10156              110.0     82.0      -2.0  {\ldots}     ATL  IAD      533   
359
        
360
               TaxiIn  TaxiOut  CarrierDelay  WeatherDelay  NASDelay  SecurityDelay  \textbackslash{}
361
        0         3.0     13.0           NaN           NaN       NaN            NaN   
362
        1         3.0     19.0           NaN           NaN       NaN            NaN   
363
        2         8.0     40.0           0.0           0.0      23.0            0.0   
364
        3         3.0     78.0           0.0           0.0      52.0            0.0   
365
        4         4.0     15.0           NaN           NaN       NaN            NaN   
366
        {\ldots}       {\ldots}      {\ldots}           {\ldots}           {\ldots}       {\ldots}            {\ldots}   
367
        10152     8.0     18.0           NaN           NaN       NaN            NaN   
368
        10153     7.0     17.0           NaN           NaN       NaN            NaN   
369
        10154     5.0     28.0           NaN           NaN       NaN            NaN   
370
        10155     4.0     16.0           NaN           NaN       NaN            NaN   
371
        10156     7.0     21.0           NaN           NaN       NaN            NaN   
372
        
373
               LateAircraftDelay  
374
        0                    NaN  
375
        1                    NaN  
376
        2                    0.0  
377
        3                    0.0  
378
        4                    NaN  
379
        {\ldots}                  {\ldots}  
380
        10152                NaN  
381
        10153                NaN  
382
        10154                NaN  
383
        10155                NaN  
384
        10156                NaN  
385
        
386
        [10157 rows x 21 columns]
387
\end{Verbatim}
388
            
389
    The Model
390

391
To model flights and delays, we will use a state object which keeps a
392
list of planes and also keeps track of ticks with the time variable.
393
These variables are global but change throughout, so putting them in the
394
state object makes sense. To simulate the planes themselves, a Plane
395
class is created, which contains any variables for the planes and
396
several functions to update them.
397

398
Our model, obviously, is more simple than a real-life airport system. We
399
have limited our traffic to only a few airports, and a small number of
400
planes. We have also decided to focus on airport delays--effectively
401
ignoring in-flight delays due to weather, diversions, or other
402
spontaneous circumstances.
403

404
    \begin{Verbatim}[commandchars=\\\{\}]
405
{\color{incolor}In [{\color{incolor}3}]:} \PY{n}{planes} \PY{o}{=} \PY{p}{[}\PY{p}{]}
406
        \PY{n}{time} \PY{o}{=} \PY{l+m+mi}{0}
407
        \PY{n}{state} \PY{o}{=} \PY{n}{State}\PY{p}{(}\PY{n}{planes} \PY{o}{=} \PY{n}{planes}\PY{p}{,}\PY{n}{time} \PY{o}{=} \PY{n}{time}\PY{p}{)}
408
\end{Verbatim}
409

410
\begin{Verbatim}[commandchars=\\\{\}]
411
{\color{outcolor}Out[{\color{outcolor}3}]:} planes    []
412
        time       0
413
        dtype: object
414
\end{Verbatim}
415
            
416
    \begin{Verbatim}[commandchars=\\\{\}]
417
{\color{incolor}In [{\color{incolor}4}]:} \PY{k}{class} \PY{n+nc}{Plane}\PY{p}{:}
418
            
419
            \PY{k}{def} \PY{n+nf}{\PYZus{}\PYZus{}init\PYZus{}\PYZus{}}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,} \PY{n}{airline}\PY{p}{,} \PY{n}{inFlight}\PY{p}{,} \PY{n}{distance}\PY{p}{,} \PY{n}{target}\PY{p}{)}\PY{p}{:}         \PY{c+c1}{\PYZsh{}\PYZsh{} Initializes an instance of the Plane class}
420
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{airline} \PY{o}{=} \PY{n}{airline}
421
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{inFlight} \PY{o}{=} \PY{n}{inFlight}
422
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{distance} \PY{o}{=} \PY{n}{distance}
423
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{target} \PY{o}{=} \PY{n}{target}
424
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{wait} \PY{o}{=} \PY{l+m+mi}{0}
425
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{data} \PY{o}{=} \PY{p}{[}\PY{p}{]}
426
                
427
            \PY{k}{def} \PY{n+nf}{move}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{)}\PY{p}{:}           \PY{c+c1}{\PYZsh{}\PYZsh{}the plane\PYZsq{}s movement tracker, which moves the plane towards its target by one unit every tick}
428
                \PY{k}{if} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{distance} \PY{o}{\PYZgt{}} \PY{l+m+mi}{0}\PY{p}{:}
429
                    \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{data}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n+nb}{str}\PY{p}{(}\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{distance}\PY{p}{)}\PY{p}{)}
430
                    \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{distance} \PY{o}{\PYZhy{}}\PY{o}{=} \PY{l+m+mi}{1}
431
                    \PY{k}{return} \PY{k+kc}{True}
432
                \PY{k}{else}\PY{p}{:}
433
                    \PY{k}{return} \PY{k+kc}{False}
434
            
435
            \PY{k}{def} \PY{n+nf}{delay}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{)}\PY{p}{:}            \PY{c+c1}{\PYZsh{}\PYZsh{}the plane\PYZsq{}s delay timer at airports, which counts down tick by one second if it is at an airport}
436
                \PY{k}{if} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{wait} \PY{o}{\PYZgt{}} \PY{l+m+mi}{0}\PY{p}{:}
437
                    \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{data}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{)}
438
                    \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{wait} \PY{o}{\PYZhy{}}\PY{o}{=} \PY{l+m+mi}{1}
439
                    \PY{k}{return} \PY{k+kc}{True}
440
                \PY{k}{else}\PY{p}{:}
441
                    \PY{k}{return} \PY{k+kc}{False}
442
            
443
            \PY{k}{def} \PY{n+nf}{go\PYZus{}to}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,} \PY{n}{target}\PY{p}{)}\PY{p}{:}     \PY{c+c1}{\PYZsh{}\PYZsh{}sets a new target airport for the plane, while also calculating the distance from its current location to the target}
444
                \PY{n}{temp} \PY{o}{=} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{target}
445
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{target} \PY{o}{=} \PY{n}{target}
446
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{distance} \PY{o}{=} \PY{n}{flight\PYZus{}time}\PY{p}{(}\PY{n}{temp}\PY{p}{,}\PY{n}{target}\PY{p}{)}
447
            
448
            \PY{c+c1}{\PYZsh{}\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}Getters\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZsh{}\PYZsh{}}
449
            \PY{k}{def} \PY{n+nf}{getAirline}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{)}\PY{p}{:}
450
                \PY{k}{return} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{airline}
451
            \PY{k}{def} \PY{n+nf}{getInFlight}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{)}\PY{p}{:}
452
                \PY{k}{return} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{inFlight}
453
            \PY{k}{def} \PY{n+nf}{getDistance}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{)}\PY{p}{:}
454
                \PY{k}{return} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{distance}
455
            \PY{k}{def} \PY{n+nf}{getTarget}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{)}\PY{p}{:}
456
                \PY{k}{return} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{target}
457
            \PY{k}{def} \PY{n+nf}{getData}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{)}\PY{p}{:}
458
                \PY{k}{return} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{data}
459
            \PY{k}{def} \PY{n+nf}{getWait}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{)}\PY{p}{:}
460
                \PY{k}{return} \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{wait}
461
            
462
            \PY{c+c1}{\PYZsh{}\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}Setters\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZsh{}\PYZsh{}}
463
            \PY{k}{def} \PY{n+nf}{setAirline}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,}\PY{n}{airline}\PY{p}{)}\PY{p}{:}
464
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{airline} \PY{o}{=} \PY{n}{airline}
465
            \PY{k}{def} \PY{n+nf}{setInFlight}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,}\PY{n}{inFlight}\PY{p}{)}\PY{p}{:}
466
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{inFlight} \PY{o}{=} \PY{n}{inFlight}
467
            \PY{k}{def} \PY{n+nf}{setDistance}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,}\PY{n}{distance}\PY{p}{)}\PY{p}{:}
468
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{distance} \PY{o}{=} \PY{n}{distance}
469
            \PY{k}{def} \PY{n+nf}{setTarget}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,}\PY{n}{target}\PY{p}{)}\PY{p}{:}
470
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{target} \PY{o}{=} \PY{n}{target}
471
            \PY{k}{def} \PY{n+nf}{setWait}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,} \PY{n}{wait}\PY{p}{)}\PY{p}{:}
472
                \PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{wait} \PY{o}{=} \PY{n}{wait}
473
                
474
        \PY{k}{def} \PY{n+nf}{flight\PYZus{}time}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{y}\PY{p}{)}\PY{p}{:}     \PY{c+c1}{\PYZsh{}Outside the plane class, flight time calculates the time/distance in ticks between any of the 5 airports in the simulation}
475
            \PY{k}{if} \PY{p}{(}\PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)} \PY{o+ow}{or} \PY{p}{(}\PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{:}
476
                \PY{k}{return} \PY{l+m+mi}{51}
477
            \PY{k}{elif} \PY{p}{(}\PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)} \PY{o+ow}{or} \PY{p}{(}\PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{:}
478
                \PY{k}{return} \PY{l+m+mi}{21}
479
            \PY{k}{elif} \PY{p}{(}\PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)} \PY{o+ow}{or} \PY{p}{(}\PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{:}
480
                \PY{k}{return} \PY{l+m+mi}{28}
481
            \PY{k}{elif} \PY{p}{(}\PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)} \PY{o+ow}{or} \PY{p}{(}\PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{:}
482
                \PY{k}{return} \PY{l+m+mi}{57}
483
            \PY{k}{elif} \PY{p}{(}\PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)} \PY{o+ow}{or} \PY{p}{(}\PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{:}
484
                \PY{k}{return} \PY{l+m+mi}{59}
485
            \PY{k}{elif} \PY{p}{(}\PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)} \PY{o+ow}{or} \PY{p}{(}\PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{:}
486
                \PY{k}{return} \PY{l+m+mi}{35}
487
            \PY{k}{elif} \PY{p}{(}\PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)} \PY{o+ow}{or} \PY{p}{(}\PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{:}
488
                \PY{k}{return} \PY{l+m+mi}{66}
489
            \PY{k}{elif} \PY{p}{(}\PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)} \PY{o+ow}{or} \PY{p}{(}\PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{:}
490
                \PY{k}{return} \PY{l+m+mi}{17}
491
            \PY{k}{elif} \PY{p}{(}\PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)} \PY{o+ow}{or} \PY{p}{(}\PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{:}
492
                \PY{k}{return} \PY{l+m+mi}{70}
493
            \PY{k}{elif} \PY{p}{(}\PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)} \PY{o+ow}{or} \PY{p}{(}\PY{n}{y} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{and} \PY{n}{x} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{:}
494
                \PY{k}{return} \PY{l+m+mi}{76}
495
            \PY{k}{else}\PY{p}{:}
496
                \PY{k}{return} \PY{k+kc}{False}
497
            
498
        \PY{k}{def} \PY{n+nf}{delay\PYZus{}factor}\PY{p}{(}\PY{n}{baseNum}\PY{p}{,} \PY{n}{margin}\PY{p}{)}\PY{p}{:}     \PY{c+c1}{\PYZsh{}\PYZsh{}Adds an element of randomness to the delay, which can be adjusted with the base number and the amout it can deviate}
499
            \PY{n}{rnd} \PY{o}{=} \PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{margin}\PY{o}{*}\PY{l+m+mi}{2}\PY{p}{)}
500
            \PY{k}{return} \PY{n+nb}{int}\PY{p}{(}\PY{p}{(}\PY{n}{baseNum} \PY{o}{\PYZhy{}} \PY{p}{(}\PY{n}{margin}\PY{p}{)}\PY{p}{)} \PY{o}{+} \PY{n}{rnd}\PY{p}{)}
501
\end{Verbatim}
502

503
    \begin{Verbatim}[commandchars=\\\{\}]
504
{\color{incolor}In [{\color{incolor}5}]:} \PY{n}{plane1} \PY{o}{=} \PY{n}{Plane}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{UA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{k+kc}{False}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
505
        \PY{n}{plane2} \PY{o}{=} \PY{n}{Plane}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{DL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{k+kc}{False}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
506
        \PY{n}{plane3} \PY{o}{=} \PY{n}{Plane}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{UA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{k+kc}{False}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
507
        \PY{n}{plane4} \PY{o}{=} \PY{n}{Plane}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{DL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{k+kc}{False}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
508
        \PY{n}{plane5} \PY{o}{=} \PY{n}{Plane}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{UA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{k+kc}{False}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
509
        \PY{n}{plane6} \PY{o}{=} \PY{n}{Plane}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{DL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{k+kc}{False}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
510
        \PY{n}{plane7} \PY{o}{=} \PY{n}{Plane}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{UA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{k+kc}{False}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
511
        \PY{n}{plane8} \PY{o}{=} \PY{n}{Plane}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{DL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{k+kc}{False}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
512
        \PY{n}{plane9} \PY{o}{=} \PY{n}{Plane}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{UA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{k+kc}{False}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
513
        \PY{n}{plane10} \PY{o}{=} \PY{n}{Plane}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{DL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{k+kc}{False}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
514
        \PY{n}{plane11} \PY{o}{=} \PY{n}{Plane}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{UA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{k+kc}{False}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
515
        \PY{n}{state}\PY{o}{.}\PY{n}{planes}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{plane1}\PY{p}{)}
516
        \PY{n}{state}\PY{o}{.}\PY{n}{planes}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{plane2}\PY{p}{)}
517
        \PY{n}{state}\PY{o}{.}\PY{n}{planes}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{plane3}\PY{p}{)}
518
        \PY{n}{state}\PY{o}{.}\PY{n}{planes}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{plane4}\PY{p}{)}
519
        \PY{n}{state}\PY{o}{.}\PY{n}{planes}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{plane5}\PY{p}{)}
520
        \PY{n}{state}\PY{o}{.}\PY{n}{planes}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{plane6}\PY{p}{)}
521
        \PY{n}{state}\PY{o}{.}\PY{n}{planes}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{plane7}\PY{p}{)}
522
        \PY{n}{state}\PY{o}{.}\PY{n}{planes}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{plane8}\PY{p}{)}
523
        \PY{n}{state}\PY{o}{.}\PY{n}{planes}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{plane9}\PY{p}{)}
524
        \PY{n}{state}\PY{o}{.}\PY{n}{planes}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{plane10}\PY{p}{)}
525
        \PY{n}{state}\PY{o}{.}\PY{n}{planes}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{plane11}\PY{p}{)}
526
\end{Verbatim}
527

528
    For comparison we are using two different models for airlines, assuming
529
each has only 2 planes, going between 2 airports.
530

531
Delta Airlines (DL) will be using a model where 1 plane is kept in
532
reserve. Any time delta experiences a significant delay (variable
533
maxDelay), the reserve plane will be called in to replace the original,
534
instantly resetting the delay to 0.
535

536
United Airlines (UA) will be using a model where all planes are always
537
in service, flying opposite directions between the 2 airports. Since
538
there is no reserve plane, United makes turnarounds longer to maintain
539
planes and reduce the impact of delays. However, if one of their planes
540
exceeds a significant delay (variable maxDelay), the flight is
541
cancelled, and the plane must wait until the next scheduled flight.
542
Since the planes fly between two airports, this means two previously
543
scheduled flights are cancelled.
544

545
The data will be obtained in the form of a ratio, comparing the number
546
of successful flights for each airline. The variables for maximum delays
547
and turnarounds are designed to be as close to real life as possile
548
based on research. Running the simulation usually takes upwards of 2
549
minutes because of the vast quantity of data being processed. We
550
experimented with smaller time scales and numbers but this resulted in
551
very varied outputs.
552

553
    \begin{Verbatim}[commandchars=\\\{\}]
554
{\color{incolor}In [{\color{incolor}6}]:} \PY{k}{def} \PY{n+nf}{run\PYZus{}simulation}\PY{p}{(}\PY{n}{numPlanes}\PY{p}{,} \PY{n}{air1}\PY{p}{,}
555
                           \PY{n}{air2}\PY{p}{)}\PY{p}{:}  \PY{c+c1}{\PYZsh{} The run\PYZus{}simulation function runs the simulation}
556
            \PY{n}{state}\PY{o}{.}\PY{n}{time} \PY{o}{=} \PY{l+m+mi}{0}
557
            \PY{n}{DL} \PY{o}{=} \PY{l+m+mi}{0}
558
            \PY{n}{UA} \PY{o}{=} \PY{l+m+mi}{0}
559
            \PY{n}{planes} \PY{o}{=} \PY{n}{state}\PY{o}{.}\PY{n}{planes}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{:}\PY{p}{(}\PY{p}{(}\PY{n}{numPlanes} \PY{o}{*} \PY{l+m+mi}{2}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
560
            \PY{k}{for} \PY{n}{x} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{100000}\PY{p}{)}\PY{p}{:}
561
                \PY{n}{state}\PY{o}{.}\PY{n}{time} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
562
                \PY{n}{DL} \PY{o}{+}\PY{o}{=} \PY{n}{sim1}\PY{p}{(}\PY{n}{planes}\PY{p}{,} \PY{n}{air1}\PY{p}{,} \PY{n}{air2}\PY{p}{,} \PY{l+m+mi}{22}\PY{p}{)}
563
                \PY{n}{UA} \PY{o}{+}\PY{o}{=} \PY{n}{sim2}\PY{p}{(}\PY{n}{planes}\PY{p}{,} \PY{n}{air1}\PY{p}{,} \PY{n}{air2}\PY{p}{,} \PY{l+m+mi}{22}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{)}
564
            \PY{k}{return} \PY{p}{[}\PY{n}{DL}\PY{p}{,} \PY{n}{UA}\PY{p}{,} \PY{n}{DL} \PY{o}{/} \PY{n}{UA}\PY{p}{]}
565
        
566
        
567
        \PY{k}{def} \PY{n+nf}{sim1}\PY{p}{(}\PY{n}{planes}\PY{p}{,} \PY{n}{air1}\PY{p}{,} \PY{n}{air2}\PY{p}{,}
568
                 \PY{n}{maxDelay}\PY{p}{)}\PY{p}{:}  \PY{c+c1}{\PYZsh{} Sim1 implements Delta\PYZsq{}s reserve plane model}
569
            \PY{n}{success} \PY{o}{=} \PY{l+m+mi}{0}
570
            \PY{k}{for} \PY{n}{plane} \PY{o+ow}{in} \PY{n}{planes}\PY{p}{:}
571
                \PY{k}{if} \PY{n}{plane}\PY{o}{.}\PY{n}{getAirline}\PY{p}{(}\PY{p}{)} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{DL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:}
572
                    \PY{k}{if} \PY{o+ow}{not} \PY{p}{(}\PY{n}{plane}\PY{o}{.}\PY{n}{delay}\PY{p}{(}\PY{p}{)}\PY{p}{)}\PY{p}{:}
573
                        \PY{k}{if} \PY{n}{plane}\PY{o}{.}\PY{n}{getWait}\PY{p}{(}\PY{p}{)} \PY{o}{\PYZgt{}} \PY{n}{maxDelay}\PY{p}{:}
574
                            \PY{n}{plane}\PY{o}{.}\PY{n}{setWait}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{)}
575
                        \PY{k}{if} \PY{o+ow}{not} \PY{p}{(}\PY{n}{plane}\PY{o}{.}\PY{n}{move}\PY{p}{(}\PY{p}{)}\PY{p}{)}\PY{p}{:}
576
                            \PY{n}{success} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
577
                            \PY{k}{if} \PY{p}{(}\PY{n}{plane}\PY{o}{.}\PY{n}{getTarget}\PY{p}{(}\PY{p}{)} \PY{o}{==} \PY{n}{air1}\PY{p}{)}\PY{p}{:}
578
                                \PY{n}{plane}\PY{o}{.}\PY{n}{go\PYZus{}to}\PY{p}{(}\PY{n}{air2}\PY{p}{)}
579
                            \PY{k}{else}\PY{p}{:}
580
                                \PY{n}{plane}\PY{o}{.}\PY{n}{go\PYZus{}to}\PY{p}{(}\PY{n}{air1}\PY{p}{)}
581
                            \PY{n}{plane}\PY{o}{.}\PY{n}{setWait}\PY{p}{(}\PY{n}{delay\PYZus{}factor}\PY{p}{(}\PY{l+m+mi}{15}\PY{p}{,} \PY{l+m+mi}{9}\PY{p}{)}\PY{p}{)}
582
            \PY{k}{return} \PY{n}{success}
583
        
584
        
585
        \PY{k}{def} \PY{n+nf}{sim2}\PY{p}{(}\PY{n}{planes}\PY{p}{,} \PY{n}{air1}\PY{p}{,} \PY{n}{air2}\PY{p}{,} \PY{n}{maxDelay}\PY{p}{,}
586
                 \PY{n}{addTurn}\PY{p}{)}\PY{p}{:}  \PY{c+c1}{\PYZsh{} Sim2 implements United\PYZsq{}s model}
587
            \PY{n}{success} \PY{o}{=} \PY{l+m+mi}{0}  \PY{c+c1}{\PYZsh{} that operates without reserve planes}
588
            \PY{k}{for} \PY{n}{plane} \PY{o+ow}{in} \PY{n}{planes}\PY{p}{:}
589
                \PY{k}{if} \PY{n}{plane}\PY{o}{.}\PY{n}{getAirline}\PY{p}{(}\PY{p}{)} \PY{o}{==} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{UA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:}
590
                    \PY{k}{if} \PY{o+ow}{not} \PY{p}{(}\PY{n}{plane}\PY{o}{.}\PY{n}{delay}\PY{p}{(}\PY{p}{)}\PY{p}{)}\PY{p}{:}
591
                        \PY{k}{if} \PY{o+ow}{not} \PY{p}{(}\PY{n}{plane}\PY{o}{.}\PY{n}{move}\PY{p}{(}\PY{p}{)}\PY{p}{)}\PY{p}{:}
592
                            \PY{n}{success} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
593
                            \PY{k}{if} \PY{p}{(}\PY{n}{plane}\PY{o}{.}\PY{n}{getTarget}\PY{p}{(}\PY{p}{)} \PY{o}{==} \PY{n}{air1}\PY{p}{)}\PY{p}{:}
594
                                \PY{n}{plane}\PY{o}{.}\PY{n}{go\PYZus{}to}\PY{p}{(}\PY{n}{air2}\PY{p}{)}
595
                            \PY{k}{else}\PY{p}{:}
596
                                \PY{n}{plane}\PY{o}{.}\PY{n}{go\PYZus{}to}\PY{p}{(}\PY{n}{air1}\PY{p}{)}
597
                            \PY{n}{delay} \PY{o}{=} \PY{n}{delay\PYZus{}factor}\PY{p}{(}\PY{l+m+mi}{15}\PY{p}{,} \PY{l+m+mi}{9}\PY{p}{)}
598
                            \PY{n}{plane}\PY{o}{.}\PY{n}{setWait}\PY{p}{(}\PY{n}{delay} \PY{o}{+} \PY{n}{addTurn}\PY{p}{)}
599
                            \PY{k}{if} \PY{n}{delay} \PY{o}{\PYZgt{}} \PY{n}{maxDelay}\PY{p}{:}
600
                                \PY{n}{success} \PY{o}{\PYZhy{}}\PY{o}{=} \PY{l+m+mi}{2}
601
            \PY{k}{return} \PY{n}{success}
602
        
603
        
604
        \PY{n}{test1} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}
605
                               \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}  \PY{c+c1}{\PYZsh{} This section collects data from run\PYZus{}simulation}
606
        \PY{n}{test2} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}
607
            \PY{l+m+mi}{2}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}  \PY{c+c1}{\PYZsh{} and creates lists to store all the different datasets}
608
        \PY{n}{test3} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
609
        \PY{n}{test4} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
610
        \PY{n}{test5} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
611
        \PY{n}{test6} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
612
        \PY{n}{test7} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
613
        \PY{n}{test8} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
614
        \PY{n}{test9} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
615
        \PY{n}{test10} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{5}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
616
        \PY{n}{test11} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{5}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
617
        \PY{n}{test12} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{5}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
618
        \PY{n}{test13} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{6}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{IAD}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
619
        \PY{n}{test14} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{6}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ATL}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{LAX}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
620
        \PY{n}{test15} \PY{o}{=} \PY{n}{run\PYZus{}simulation}\PY{p}{(}\PY{l+m+mi}{6}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{JFK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{SEA}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
621
        
622
        \PY{n}{tests} \PY{o}{=} \PY{p}{[}
623
            \PY{n}{test1}\PY{p}{,} \PY{n}{test2}\PY{p}{,} \PY{n}{test3}\PY{p}{,} \PY{n}{test4}\PY{p}{,} \PY{n}{test5}\PY{p}{,} \PY{n}{test6}\PY{p}{,} \PY{n}{test7}\PY{p}{,} \PY{n}{test8}\PY{p}{,} \PY{n}{test9}\PY{p}{,} \PY{n}{test10}\PY{p}{,}
624
            \PY{n}{test11}\PY{p}{,} \PY{n}{test12}\PY{p}{,} \PY{n}{test13}\PY{p}{,} \PY{n}{test14}\PY{p}{,} \PY{n}{test15}
625
        \PY{p}{]}
626
        
627
        \PY{n}{DL\PYZus{}Flights} \PY{o}{=} \PY{p}{[}\PY{p}{]}
628
        \PY{k}{for} \PY{n}{test} \PY{o+ow}{in} \PY{n}{tests}\PY{p}{:}
629
            \PY{n}{DL\PYZus{}Flights}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{test}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{)}
630
        
631
        \PY{n}{UA\PYZus{}Flights} \PY{o}{=} \PY{p}{[}\PY{p}{]}
632
        \PY{k}{for} \PY{n}{test} \PY{o+ow}{in} \PY{n}{tests}\PY{p}{:}
633
            \PY{n}{UA\PYZus{}Flights}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{test}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
634
        
635
        \PY{n}{ratio} \PY{o}{=} \PY{p}{[}\PY{p}{]}
636
        \PY{k}{for} \PY{n}{test} \PY{o+ow}{in} \PY{n}{tests}\PY{p}{:}
637
            \PY{n}{ratio}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{test}\PY{p}{[}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
638
        
639
        \PY{n}{num\PYZus{}planes} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{4}\PY{p}{,} \PY{l+m+mi}{4}\PY{p}{,} \PY{l+m+mi}{4}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{,} \PY{l+m+mi}{6}\PY{p}{,} \PY{l+m+mi}{6}\PY{p}{,} \PY{l+m+mi}{6}\PY{p}{]}
640
        \PY{n}{flight\PYZus{}length} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{5}\PY{p}{]}
641
\end{Verbatim}
642

643
\begin{Verbatim}[commandchars=\\\{\}]
644
{\color{outcolor}Out[{\color{outcolor}6}]:} [1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5]
645
\end{Verbatim}
646
            
647
    The Results
648

649
The ratios of Delta's successful flights versus United's succesful
650
flights are shown below. For each flight path, there are four
651
ratios--each representing a test with a different number of planes. For
652
reference, the flight paths are in order of shortest time to longest.
653

654
    \begin{Verbatim}[commandchars=\\\{\}]
655
{\color{incolor}In [{\color{incolor}7}]:} \PY{n+nb}{print}\PY{p}{(}\PY{n}{ratio}\PY{p}{)}
656
\end{Verbatim}
657

658
    \begin{Verbatim}[commandchars=\\\{\}]
659
[0.749, 0.6914498141263941, 0.6605504587155964, 0.986452998513134, 0.9151234567901234, 0.8929752066115703, 1.1055034550839091, 1.0265650265650266, 1.0106051154086089, 1.1862637908756586, 1.0993880956795847, 1.1006625891946993, 1.2303370786516854, 1.1479473276529821, 1.1163366336633664]
660

661
    \end{Verbatim}
662

663
    \begin{Verbatim}[commandchars=\\\{\}]
664
{\color{incolor}In [{\color{incolor}8}]:} \PY{c+c1}{\PYZsh{} Creates a bubble plot to plot the ratio of successful flights}
665
        \PY{n}{data} \PY{o}{=} \PY{p}{\PYZob{}}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{a}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{n}{num\PYZus{}planes}\PY{p}{,}
666
                \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{c}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{blue}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}
667
                \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{d}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{p}{(}\PY{n}{flight\PYZus{}length}\PY{p}{)}\PY{p}{\PYZcb{}}
668
        \PY{n}{data}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{b}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]} \PY{o}{=} \PY{n}{ratio}
669
        \PY{n}{data}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{d}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{abs}\PY{p}{(}\PY{n}{data}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{d}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)} \PY{o}{*} \PY{l+m+mi}{15}
670
        
671
        \PY{n}{plt}\PY{o}{.}\PY{n}{scatter}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{a}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{b}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{c}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{c}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{s}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{d}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{data}\PY{o}{=}\PY{n}{data}\PY{p}{)}
672
        \PY{n}{plot}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{6}\PY{p}{]}\PY{p}{,} \PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n}{color}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{purple}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{linestyle}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{\PYZhy{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{linewidth}\PY{o}{=}\PY{l+m+mi}{2}\PY{p}{)}
673
        \PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Number of Planes}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
674
        \PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Ratio of Successful Flights}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
675
        \PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Plot 1: Ratios of Successful Flights v Number of Planes}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
676
        \PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
677
\end{Verbatim}
678
\texttt{\color{outcolor}Out[{\color{outcolor}8}]:}
679
    
680
    \begin{center}
681
    \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{ModSim Project 1_files/ModSim Project 1_13_0.png}
682
    \end{center}
683
    { \hspace*{\fill} \\}
684
    
685

686
    This graph shows the ratio of successful flights from United to those of
687
Delta. The higher the ratio, the more successful Delta is. The line at y
688
= 1 represents the point at which Delta begins having more successful
689
flights than United. The size of each bubble represents the length of
690
the flight.
691

692
    \begin{Verbatim}[commandchars=\\\{\}]
693
{\color{incolor}In [{\color{incolor}9}]:} \PY{c+c1}{\PYZsh{} Creates a bubble plot to plot the number of successful flights for each airline}
694
        \PY{n}{data} \PY{o}{=} \PY{p}{\PYZob{}}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{a}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{n}{num\PYZus{}planes}\PY{p}{,}
695
                \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{c}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{maroon}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}
696
                \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{d}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{n}{flight\PYZus{}length}\PY{p}{\PYZcb{}}
697
        \PY{n}{data}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{b}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]} \PY{o}{=} \PY{n}{DL\PYZus{}Flights}
698
        \PY{n}{data}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{d}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{abs}\PY{p}{(}\PY{n}{data}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{d}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)} \PY{o}{*} \PY{l+m+mi}{20}
699
        
700
        \PY{n}{plt}\PY{o}{.}\PY{n}{scatter}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{a}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{b}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{c}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{c}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{s}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{d}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{data}\PY{o}{=}\PY{n}{data}\PY{p}{)}
701
        
702
        \PY{n}{data} \PY{o}{=} \PY{p}{\PYZob{}}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{a}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{n}{num\PYZus{}planes}\PY{p}{,}
703
                \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{c}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{navy}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}
704
                \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{d}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{n}{flight\PYZus{}length}\PY{p}{\PYZcb{}}
705
        \PY{n}{data}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{b}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]} \PY{o}{=} \PY{n}{UA\PYZus{}Flights}
706
        \PY{n}{data}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{d}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{abs}\PY{p}{(}\PY{n}{data}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{d}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)} \PY{o}{*} \PY{l+m+mi}{20}
707
        
708
        \PY{n}{plt}\PY{o}{.}\PY{n}{scatter}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{a}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{b}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{c}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{c}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{s}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{d}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{data}\PY{o}{=}\PY{n}{data}\PY{p}{)}
709
        
710
        \PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Number of Planes}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
711
        \PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Number of Successful Flights}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
712
        \PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Plot 2: Number of Successful Flights v Number of Planes}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
713
        \PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
714
\end{Verbatim}
715
\texttt{\color{outcolor}Out[{\color{outcolor}9}]:}
716
    
717
    \begin{center}
718
    \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{ModSim Project 1_files/ModSim Project 1_15_0.png}
719
    \end{center}
720
    { \hspace*{\fill} \\}
721
    
722

723
    This graph shows the individual number of flights from United (blue) and
724
Delta (red) that are successful. The size of each bubble represents the
725
length of the flights.
726

727
    The Interpretation
728

729
The data in these two plots shows most clearly that given a certain time
730
period, more shorter flights can be completed than longer ones. This is
731
no surprise, and more interesting is the ratios of United flights to
732
Delta flights. With less planes, it is clear that United's model of
733
using all of its planes works well, even if they have a slightly longer
734
turnaround. However, as the airlines' fleets grow, Delta gains the
735
advantage. This can best be explained by the fact that Delta constantly
736
has 1 reserve plane, and so the ratio of reserve planes:in service
737
planes decreases, while the advantage of having a reserve plane doesn't.
738

739
Plot 1 shows the ratio increasing throughout (shifting towards Delta's
740
side) but levelling off near the end. This could imply that Delta's
741
strategy works better with more planes, but only up to a point, after
742
which it is still beneficial, but stops improving relative to United.
743

744
Plot 2 shows a very significant change in the number of successful short
745
flights, and a smaller change in the longer flights. This can be
746
attributed to the fact that there are more shorter flights so any
747
imbalances will compound much faster. However, it should also be noted
748
that Delta's method definitely seems more effective with shorter
749
flights. This can be seen in both plots, as in plot 1, the shorter
750
flights (smaller bubbles) have a much larger ratio, and in the second
751
plot, delta's short flights exceed those of United earlier than the long
752
flights.
753

754
In conclusion, United's model tends to be better suited to longer
755
flights and fewer planes, while the opposite is true for Delta.
756

757

758
    % Add a bibliography block to the postdoc
759
    
760
    
761
    
762
    \end{document}
763

764