The Economic Order Quantity

Balance Is the Certificate

EOQ does not need calculus for this pinned instance. The balance of ordering and holding cost gives an AM-GM certificate for the model optimum.

highlighted = computed this step

Balance Point

At the EOQ, ordering cost and holding cost both equal 4. Motivation: the optimum is where the two opposing cost pressures meet.

KD/Q=hQ/2=4KD/Q=hQ/2=4
EOQ balanceEOQ balanceordering cost and holding cost; order quantity Q and annual costSegments interpolate between exact evaluated points; marked points are recomputed.2481632024681012141618EOQ (8,4)

AM-GM Certificate

The AM-GM lower bound is 8. Why: total cost is at least twice the geometric mean of the two positive components, so the lower bound is a certificate.

C(Q)≥8C(Q)\ge 8
AM-GM certificateThe AM-GM bound and equality quantity are recomputed from EOQ.balance certificateorderingholdingboundQEOQ4488

Equality Condition

Equality holds at order quantity 8. Interpretation: the AM-GM bound is tight exactly when ordering cost equals holding cost, so the model optimum is certified.

Q∗=8Q^*=8
AM-GM certificateThe AM-GM bound and equality quantity are recomputed from EOQ.balance certificateorderingholdingboundQEOQ4488

Diagram note

Honesty note: EOQ is exact only for the idealized continuous constant-demand model: known steady demand, infinitely divisible order quantity, zero lead time, and no capacity. EOQ plot segments interpolate between exact evaluated points; only the marked points are recomputed. Pixel positions are rounded for layout; every number shown is exact.

balance certifies the EOQ model optimum\text{balance certifies the EOQ model optimum}
EOQ balanceEOQ balanceordering cost and holding cost; order quantity Q and annual costSegments interpolate between exact evaluated points; marked points are recomputed.2481632024681012141618EOQ (8,4)