revenue and cost

Question

A retail outlet sells 2000 pairs of pants a year. It costs $2 to store a pair of pants
per year. Each reorder costs $15, plus an additional $4 per pair of pants. How many
times per year should the store order pants and in what lot size to minimize
inventory costs?

Answer 1

if inventory costs= storage costs then minimizing those costs means reorder 2000 times a year

but that can't be what you mean

as that costs 19x2,000 = $38,000 per year

so you must mean minimize sum of storage & reordering costs

then try 1 recorder per year costing 15 + 4x2000 = $8,015 plus storage costs

= 8015 +maybe 2000 (1000x2) = $10,015 total

compare with 2 reorders per year = 30 +2x4x1000 plus maybe 500x2 = $9,030

try 4 reorders per year = 60 +4x4x500 +250x2= $8,560

8 per year = 8x15 + 8x4x250 +125x2 = $8,620

about 4 reorders seems to minimize total costs as it appears at or near bottom of a U shaped cost curve

y reorders per year cost = 15y + 15y(2000/y) +(1000/y)2 = 3000 + 2000/y +15y = TC

take the derivative = TC' = 15 - 2000/y^2 = 0

15y^2 = 2000

y^2 = 2000/15 = 400/3 = 133 1/3

y = sqr(400/3) = 20/sqr3 = 20(sqr3)/3 = about

11.5 reorders per year

go with somewhere between 4 and 12 reorders per year