In: Statistics and Probability
A new edition of a
very popular textbook will be published a year from now. The
publisher currently has 1,000 copies on hand and is deciding
whether to do another printing before the new edition comes out.
The publisher estimates that demand for the book during the next
year is governed by the probability distribution in the file
P10_31.xlsx. (See Demand and Probability Table
Below)A production run incurs a fixed cost of $15,000 plus
a variable cost of $20 per book printed. Books are sold for $190
per book. Any demand that cannot be met incurs a penalty cost of
$30 per book, due to loss of goodwill. Up to 1,000 of any leftover
books can be sold to Barnes and Noble for $45 per book. The
publisher is interested in maximizing expected profit. The
following print-run sizes are under consideration: 0 (no production
run) to 16,000 in increments of 2,000. What decision would you
recommend? Use simulation with 1,000 replications.
Select02,0004,0006,0008,00010,00012,00014,00016,000
For your optimal
decision, the publisher can be 90% certain that the actual profit
associated with remaining sales of the current edition will be
between what two values? If needed, round your answers to whole
dollar amounts.
$ and $
Demand | Probability |
3000 | 0.20 |
4000 | 0.35 |
5000 | 0.25 |
6000 | 0.10 |
8000 | 0.05 |
10000 | 0.05 |
To simulate the demand, we get the following cumulative distribution and the random number intervals
Random number interval | ||||
Demand | Probability | Cumulative probability | From | To |
3000 | 0.2 | 0.2 | 0 | 0.2 |
4000 | 0.35 | 0.55 | 0.2 | 0.55 |
5000 | 0.25 | 0.8 | 0.55 | 0.8 |
6000 | 0.1 | 0.9 | 0.8 | 0.9 |
8000 | 0.05 | 0.95 | 0.9 | 0.95 |
10000 | 0.05 | 1 | 0.95 | 1 |
To simulate demand we use the following
Quantity sold = minimum(demand,print run)
Revenue = quantity sold * 190
Unsold quantity = print run - quantity sold
Revenue from unsold= min(unsold quantity,1000)*45
Cost of printing = 15000+print run * 20
Penalty = (demand - quantity sold) *30
Profit = (revenue+revenue from unsold) - (cost of printing +penalty)
= expected profit = average profit of 1000 simulations
s= sample standard deviation of profit
is the standard error of mean
90% confidence interval indicates a level of significance
Since the sample size is 1000 and it is greater than 30, using CLT, we can use normal distribution as the sampling distribution of mean.
The right tail critical value is
Using the standard normal table (or Excel function =NORM.INV(0.95,0,1)) we get
The 90% confidence interval for actual mean profit is
Prepare the following
copy the rows to make 1000 trials
Paste as values to avoid changes
Get this
Prepare a data table as below
get this
select the data table, then use data-->what if analysis--->data table to set
get this
The following print-run sizes are under consideration: 0 (no production run) to 16,000 in increments of 2,000. What decision would you recommend?
ans: 6000
Enter 6000 as the print run size and get the 90% confidence interval
For your optimal decision, the publisher can be 90% certain that the actual profit associated with remaining sales of the current edition will be between what two values?
ans: $761,921 and $784,149