Exercise 123Calculus › Single-variable calculus › Differentiation › Optimisation and curve sketchingL1Added 4 Nov 2026
The economic order quantity
#optimisation#economics#inventory#modelling
Statement
A shop sells items per year at a steady rate. Each order costs € whatever its size, and keeping one item in stock costs € per year. If the shop orders items each time, the average stock is . Find the order size that minimises the total annual cost.
Hint
The number of orders per year is .
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Solution
Total cost.
Minimum. . Since at and at and is the only critical point, it is the global minimum (equivalently, on and on ).
Answer. Order items at a time ( orders a year). The cost is € per year.
General formula. (the Wilson formula). Here .
Remark. Again, ordering cost and storage cost are equal at the optimum. Treating as a real number is harmless here, because the optimum happens to be an integer.