A passenger aircraft seats 210 passengers in economy class, 7 abreast in rows 30 to 59. All seats are full for the current trip.
The in-flight menu for economy passengers has 3 main-course choices: curried mutton with rice, chicken supreme with brocolli, and pork belly with bean sprouts. Given that, on average, one-third of the passengers will prefer each main-course choice, the aircraft carries 70 meals of each type.
The meals are served from large trolleys that the servers push through the aircraft from row 30 abaft. Each passenger in turn is asked which meal they choose. Naturally, this depends on individual preference, but overall each choice is equally likely. Subject to availability, the passenger is then handed their selection.
Questions:
(1) In what row will at least one main-course choice likely become unavailable?
(2) How many passengers will likely receive a non-preferred meal?
There are 36 possible sets of preferences for each row. No given meal can run out before row 10, as there are 70 of each. It's an indication that the number of states of the system will get out of hand, taking up time and space to keep track of all of them for the thirty rows. So a simulation is in order.
totFirst=0; unsatisfied=0;rowlist=zeros(1,60);unslist=zeros(1,210);
for trial = 1:100000
unsthistime=0;
first=60;
pref=randi(3,30,7);
cart=[70,70,70];
for row=30:59
uns=0;
type="randperm";
for i= randperm(3)
asked(i)=length(find(pref(row-29,:)==i));
if cart(i)>=asked(i)
cart(i)=cart(i)-asked(i);
else
uns=uns+asked(i)-cart(i);
if first>row
first=row;
rowlist(first)=rowlist(first)+1;
end
end
unsatisfied=unsatisfied+uns;
unsthistime=unsthistime+uns;
while uns>0
r=randi(3);
if cart(r)>0
cart(r)=cart(r)-1;
uns=uns-1;
end
end
end
end
totFirst=totFirst+first;
unslist(unsthistime+1)=unslist(unsthistime+1)+1;
end
fprintf('%10d %10d %10.5f %10.5f %s\n',unsatisfied,trial,...
totFirst/trial,unsatisfied/trial,type);
% disp([totFirst/trial unsatisfied/trial])
for i=30:60
if rowlist(i)>60
disp([i rowlist(i)])
end
end
for i=1:210
if unslist(i)>0
disp([i unslist(i)])
end
end
In the above, a couple of choices had to be made:
1. If a passenger's choice was not available, the entree to be served was chosen at random.
2. Runs were done two ways: serving a given row always mutton first, pork last; or in random order.
The below results show, first for random order (randperm), then for entrees 1 through 3 each row.
The order of the numbers shown is: number of unsatisfied fliers, total number of trials, average first row in which something has run out, average number of unsatisfied customers, and type of run (whether each row was served in random order of entree or straight run order of entree).
Then a list broken down by number of times within the 100,000 flights that each given row was the first in which something ran out.
Then a list broken down by number of times within the 100,000 flights that the given number of passengers did not get their first choice meals.
For the first two runs each row was served mutton through pork. the next two runs, random order of entree.
>> subjectToAvailability
822633 100000 56.83830 8.22633 1:3
row times out of 100,000
51 184
52 697
53 2064
54 5169
55 10579
56 18161
57 25015
58 24825
59 12876
# not first choice flights out of 100,000 flights
1 394
2 3078
3 5650
4 7384
5 8306
6 8539
7 8470
8 8151
9 7734
10 7027
11 6402
12 5604
13 4880
14 4073
15 3340
16 2739
17 2173
18 1617
19 1252
20 957
21 694
22 500
23 327
24 249
25 141
26 118
27 70
28 49
29 27
30 23
31 12
32 10
33 5
34 4
35 0
36 1
>> subjectToAvailability
822027 100000 56.84589 8.22027 1:3
51 155
52 645
53 1970
54 5256
55 10726
56 17954
57 24997
58 24846
59 13023
1 391
2 3112
3 5624
4 7478
5 8322
6 8545
7 8480
8 8048
9 7768
10 7104
11 6255
12 5646
13 4743
14 4044
15 3423
16 2787
17 2132
18 1699
19 1215
20 949
21 716
22 477
23 362
24 236
25 151
26 113
27 68
28 45
29 24
30 17
31 5
32 6
33 5
34 6
35 3
36 1
>> subjectToAvailability
822195 100000 56.84147 8.22195 randperm
51 149
52 658
53 2088
54 5207
55 10514
56 18211
57 25078
58 24771
59 12935
1 363
2 3012
3 5680
4 7244
5 8327
6 8770
7 8506
8 8165
9 7660
10 7189
11 6405
12 5591
13 4793
14 4117
15 3313
16 2647
17 2091
18 1621
19 1304
20 957
21 688
22 516
23 341
24 249
25 144
26 113
27 85
28 47
29 22
30 15
31 12
32 5
33 2
34 4
35 1
36 0
37 0
38 1
>> subjectToAvailability
821794 100000 56.83554 8.21794 randperm
51 162
52 684
53 2114
54 5112
55 10748
56 18050
57 25139
58 24668
59 12896
1 382
2 3105
3 5731
4 7426
5 8239
6 8555
7 8508
8 8082
9 7536
10 7093
11 6487
12 5612
13 4963
14 4120
15 3343
16 2635
17 2093
18 1670
19 1241
20 939
21 712
22 475
23 368
24 219
25 167
26 105
27 53
28 54
29 41
30 13
31 12
32 11
33 6
34 2
35 2
It doesn't seem that randomizing the order of entrees served within a row makes a difference. The first row to run out of something averages row 56.8. On average, about 8.2 people do not get their desired preference.
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Posted by Charlie
on 2022-06-02 11:32:17 |
solution by simulation
