Dec 1998 QT211: Quantitative Analysis For Management (Question 2)

A baker's shop wishes to decide how many cream cakes it should make each day. Each cream cake costs $1 to make and sells for $2.50. Cakes are made in batches of 10, so the shopkeeper wants to use simulation to decide whether to make 50 or 60 cakes per day. If any cream cakes are left over at the end of the day, they are thrown away. Based on experience with other cakes, the shopkeeper believes that the estimated numbers of new customers on each day are expected to follow the probabillity distribution given below:

# new customers	Probability
20	0.10
40	0.22
60	0.28
80	0.24
100	0.16

Each customer buys one cake each. If the shop sells all of its cakes on a particular day, then only half of ths customers who arrive subsequently and are unable to buy a cake will return the following day, the other half will not return at all.

(a)

Explain carefully how the above probability distribution can be used with the following sequence of random digits to simulate the number of new customers who wish to buy cakes each day over a ten-day period.

48 23 12 62 06 76 08 54 17 86

[6]

Random numbers in the range 00, 01, 02, up to 99, are assigned to the table according to the cumulative probablility values (1 mark) associated with each level of demand, so that demand of 20 has numbers 00-09, demand of 40 has 10-31 and so on (1 mark), as listed below.

Demand	Probability	Cumulative Probability	Random Values
20	0.10	0.10	00-09
40	0.22	0.32	10-31
60	0.28	0.60	32-59
80	0.24	0.84	60-83
100	0.16	1.00	84-99

Deduct one mark for each error (minimum zero marks). Candidates who correctly use the alternative allocation of 01, 02, .., 99, 00 should also receive full credit.

(b)

Determine on each day

how many new customers buy cakes
how many of the previous days customers return to buy cakes
the total number of customers who buy cakes
the total number of cakes that are wasted on each day

for each of the following

(i) if the shopkeeper bakes 50 cakes per day
(ii) if the shopkeeper bakes 60 cakes per day

[10]

2 marks should be awarded for correctly allocating the number of new customers per day, and 1 mark should be awarded for each other column. This marking will have to be done with some care, as errors, in any one column will necessary lead to errors in other columns, and any such error should only be penalised once.

(c)

How many cakes should the shopkeeper bake?

Note: 'new customers' are those who are attempting to buy a cake for the first time on a particular day. 'Previous days customers' are those customers who return to the shop because they are unable to buy a cake on the previous day because the shop had sold out of cakes.

[4]

Profit calculations. If 50 cakes are baked per day then the total profit is 450 x (2.5 - 1.0) - 50 x 1.0 = $625 (2 marks), and if 60 cakes are baked per day then the total profit is 480 x (2.5 - 1.0) - 120 x 1.0 = $600 (2 marks). So the baker should bake 50 cakes per day.

Again, this final part should be marked with care so that earlier errors are not penalised twice.