445. (Q, r)

Continuous review, fixed order quantity. The most common stochastic-demand policy in textbooks and practice.

Decision rule: monitor inventory position continuously. When it drops to (or below) the reorder point , order a fixed quantity .

Two parameters:

• = order quantity (a fixed batch every time)
• = reorder point (the trigger level)

445.0.1. Setup

Random demand with rate per unit time and standard deviation . Lead time (constant for now). Holding cost , fixed order cost , plus a service-level requirement (e.g., for 95% cycle service level).

Inventory position . Continuous review means we always know it.

445.0.2. Inventory profile

Sawtooth shape, but with random consumption rate:

• Inventory drains stochastically.
• When position hits , a Q-unit order is placed; arrives time later.
• During lead time, demand consumes random draw from a distribution with mean and std .
• Service level (probability of no stockout per cycle) = .

445.0.3. Set — the reorder point

To meet cycle service level :

where:

• (expected demand during lead time)
• (std of lead-time demand, assuming i.i.d. periods)
• (standard-normal quantile for the desired service level)

The term is the safety stock — buffer for demand variability.

445.0.4. Set — the order quantity

When fixed cost is significant, use the EOQ formula:

(Approximate — exact joint optimization of and exists but EOQ is within a few % of optimal.)

445.0.5. Final formulas

Average inventory (approximation): , so total holding cost .