Title: Decision Making with Uncertainty
1Decision Making with Uncertainty
2 If a particular event has a probability x of
occurring on any particular occasion, then ...
3 if we examine a large number of occasions, we
would expect the event to occur on x of those
occasions.
4For example, if there is a10 probability of
buses on a particular service arriving late, we
would expect a late arrival once in every ten
journeys on those buses.
5To see how the principle can be applied in
practice, we will examine two situations.
61. The Wet Day and the Bus Stop
7We will consider someone who travels to the
University by bus, which is timetabled to run
every ten minutes. The last one which will
produce an arrival in time is the 8.30 from the
stop nearest the passengers home.
8It is found, however, that the traffic (and
possibly the vagaries of the drivers watch,
alarm-clock etc) causes the actual arrival time
of the buses to be somewhat variable as follows
9Time Probability2 min early 2 1 min
early 5 On time 25 1 min late 22 2
min late 17 3 min late 14 4 min late 10
5 min late 5
10It is a wet morning and the bus stop has no
adjacent bus shelter.
11At what time should we go to the stop to minimise
the mean time we have to stand there getting wet
but nevertheless arrive in time for our 9 am
lecture ?
12We will assume that the buses always appear and
that they are never so full that we cannot get on.
13If we go at 8.28 a.m., we will certainly catch
the timetabled 8.30 bus, but we will have to wait
as follows
140 minutes 2 of the time 0 minute contribution
to mean value1 minute 5 of the time 0.05
minute contribution to mean value2 minutes 25
of the time 0.5 minute ...3 minutes 22 of the
time 0.66 minute ..
154 minutes 17 of the time 0.68 minute ...5
minutes 14 of the time 0.7 minute ...6
minutes 10 of the time 0.6 minute ...7
minutes 5 of the time 0.35 minute ...
16What we wish to calculate is the mean waiting
time, i.e. the mean time for which we would have
waited on each occasion averaged over a large
number of visits to the bus stop.
17 This is the total of the numbers we have just
worked out.05 .5 .66 .68 .7 .6 .35
3.54 minutes
18and we will probably be feeling quite damp.
19If you are unhappy with this method of
calculating the mean, the following alternative
method may help.We assume a typical series of
100 journeys on the bus.
20We would expect2 2 journeys 2 minutes
earlyTotal wait 2 x 0 0 minutes5 5
journeys 1 minute early .. Total wait 5 x 1 5
minutes25 25 journeys on time..Total wait
25 x 2 50 minutesand so on
21Adding up the total waiting time gives 354
minutes waited in a total of 100
occasionsgiving a mean waiting time of 3.54
minutes.
22An alternative strategy might be to aim to catch
the 8.20 bus but to arrive at 8.19, so we will
miss it 2 of the time and have to catch the 8.30.
23We will suffer a good soaking on those wet
mornings, but...
24... our waiting time will otherwise be0 minutes
5 of the time 0 minutes contribution to mean
value1 minute 25 of the time 0.25 minute
...2 minutes 22 of the time 0.44 minute ..
253 minutes 17 of the time 0.51 minute ..4
minutes 14 of the time 0.56 minute ..5
minutes 10 of the time 0.5 minute ..6 minutes
5 of the time 0.3 minute ..
26When we miss the 8.20, we will have to wait for
the 8.30, for which on average we had to wait
3.54 minutes when we arrived at 8.28.
27If we arrive at 8.19 and nevertheless miss the
8.20, we will have to wait 9 minutes longer on
average for the 8.30 than we did when we arrived
at 8.28 ... a total of 9 3.54 12.54 minutes.
28This situation will occur 2 of the time, so it
will contribute 2 of 12.54 0.251 minutes to
the average.
29Our mean waiting time will therefore be 0.251
0.25 0.44 0.51 0.56 0.5 0.3 2.811
minutes, so, on average, we are better off.
30Over to you...At what time should we arrive at
the stop ?
31The second example sounds as if it might have to
do with buses also ...
32The second example sounds as if it might have to
do with buses also Just-in-time
33This term refers to a frequently-used method of
organising deliveries of raw materials to our
factory.
34It means that we require the supplier to deliver
the materials at precisely the instant that we
are ready to use them rather than at a somewhat
earlier time.
35It is a popular idea for the following reasons
361. We do not need a large stores area at our
factory to store large quantities of raw
materials until we are ready to use them.
372. We do not have to pay for the materials
before we need them.
38This is an advantage because either we would need
to borrow the money to pay for them (so we would
have to pay interest on it)
39 or we would have lost the opportunity to invest
the money and earn interest, or do something else
profitable with the money (an often-used term in
this respect is loss-of-opportunity cost).
40Suppliers, unfortunately, do occasionally
deliver late for various reasons (including their
lorry breaking down en route) and, if that
happens, production is disrupted if we really are
working just-in-time.
41There will be a cost associated with such
disruption, and we can calculate it by a similar
approach to the bus-stop example.
42We require a monthly supply of cocoa beans for
our chocolate factory. Our usual consignment
costs 30 000 and it is delivered by Sweettooth
Ltd whose reliability of delivery date is as
follows
43On Time 801 week late 152 weeks late 5
44If production is halted for want of beans, the
loss can be modelled as 1000 per weeks lost
production.The loss-of-opportunity rate is
0.5 per week on the money involved.
45How many weeks in advance should we order our
deliveries to be made ?
46We will first assume that we use just-in-time.
This will avoid the loss-of-opportunity cost
altogether but production will be lost for at
least 1 week 20 of the time and for a further
week 5 of the time.
47So we will incur mean costs of 1000 x 0.2
1000 x 0.05 250 per month.
48We will now assume we order one week early. We
incur loss-of-opportunity costs of 0.5 on the
30000 involved .005 x 30 000 150, and also
disruption costs of 1000 per month ..
49 ... but only 5 of the time, so this cost only
contributes 50 per month this time, giving a
mean total of 200 per month.
50Now we assume two weeks early. We now incur no
disruption costs but we now suffer
loss-of-opportunity costs for two weeks each
month, giving 1 of 30 000 300 per month.
51The cost argument suggests one week early will be
best.