Declining sales prices and holding costs
A hamster farm has 500 hamsters to sell. The farmer sells at the end of each week, where is a constant factor of 500 (e.g. 5,100,etc.). Price is 10 dollars in week 1 and drops by 5% each subsequent week. There is neither birth nor death.
(i) State the total number of weeks to sell all in terms of . mark
(ii) Show the total proceeds are dollars. marks
(iii) Rearing costs 0.50 dollars per hamster per week. Find the total rearing cost when all are sold. marks
(iv) If other costs are negligible, find the least for profit. marks
[2012/PJC/Prelim/I/9]
Solution
Weeks to sell all hamsters
Find the number of weeks needed to sell all 500 hamsters.
Hint
k is restricted to positive divisors of 500.
Answer · (i)
weeks.
Divide stock by batch size
Keep the admissibility condition for later optimization.
Total proceeds
Find the total sales proceeds.
Hint
There are 500/k sales weeks; the ratio applies to price, not stock.
Answer · (ii)
dollars.
Identify weekly earnings
The first week earns 10k, and subsequent earnings multiply by 0.95.
Sum the correct number of weeks
Use N=500/k, not 500 terms.
Total rearing cost
Find the total rearing cost, including each selling week.
Hint
Does the last batch cost nothing to rear, or does it remain until the end of its selling week?
Answer · (iii)
dollars.
Count the animals during week j
Before week j’s sale,500−(j−1)k animals remain.
Identify the first and last costs
The last week still contains k hamsters.
Sum the AP
Using zero as the last weekly cost would incorrectly omit the final week.
Smallest profitable batch size
Find the least permitted batch size that makes a profit.
Hint
22 is an integer, but does it divide 500?
Answer · (iv)
.
must divide .
Subtract cost from revenue
Profit must be strictly positive.
Test the allowed smaller sizes
These are every positive divisor of 500 below 25. For the source calculator's numerical guide, enter in Y=, set TblStart to 1 and the table step to 1, then inspect the sign change: the formula is about at 21 and at 22. These are not allowed batch sizes, so use the table only to guide the search and check the permitted divisors.
Check the first profitable allowed batch
Every smaller allowed batch gives a loss, so 25 is the least permitted batch size that makes a profit.
Keep the allowed batch sizes in the optimisation
The continuous/integer calculator threshold is not automatically an allowed divisor of 500.