433. EPQ
433.1. EPQ (Economic Production Quantity)
Relax one dimension from basic EOQ: replenishment is no longer instantaneous. Instead of receiving all at once, you produce at finite rate units/year while demand drains stock simultaneously.
433.1.1. Setup
New variables (beyond basic EOQ):
- = production rate (units/year), with (otherwise demand outpaces supply)
- = demand rate (units/year) — same as in basic EOQ; renamed here to emphasize it’s a rate, not annual quantity
The inventory profile in each cycle:
- Buildup phase (length ): produce at rate , demand at rate — net accumulation rate . Inventory rises to a peak .
- Drawdown phase (length ): production stops, demand continues at rate — inventory drains from to 0.
Peak inventory reached at end of buildup:
The peak is less than (the case for basic EOQ where ) because some units are consumed during production.
Average on-hand inventory (triangle, height ):
The factor shrinks the average vs basic EOQ.
433.1.2. Cost model
Total relevant cost (drop the constant term):
Only the holding term changes from basic EOQ — multiplied by . As (instant production), the factor and we recover basic EOQ.
433.1.3. Derive
Same calculus as basic EOQ: differentiate, set to zero.
Take the square root and rewrite :
The first factor is the basic EOQ; the second is a multiplier that grows the production batch when the production rate is close to demand (long buildup, little overlap with drawdown).
433.1.4. Final formulas
Sanity check: as (instantaneous), the multiplier and we recover basic EOQ exactly. As (production barely keeps up), — you should run the line continuously.
Example
Given (shared EOQ params + a finite production rate):
- Annual demand: units/year
- Setup cost: = $50 / setup
- Holding cost: = $2 / unit / year
- Production rate: units/year (twice the demand rate)
Step 1 — multiplier from finite production
Step 2 — production batch size
Step 3 — peak inventory and cycle
Of , the buildup takes days; drawdown takes the remaining days.
Total cost:
Compare to basic EOQ on the same demand and setup costs:
- Basic EOQ (): , $1549
- EPQ (): (larger), $1095 (29% lower)
EPQ is cheaper because some units are consumed during production — average on-hand is lower than the basic EOQ assumes.