Overview

Policy dimensions

Stochastic-demand inventory policies sit at the intersection of five design choices. The choices made along these dimensions name the policy.

1. Inventory review

Periodic review (R-policies): inventory checked at regular intervals $𝑅$ . Orders are placed only at review points.
Continuous review (s-policies): inventory monitored continuously. Orders are triggered immediately when the level falls below a threshold.

2. Order quantity

Fixed order quantity (Q): a predetermined, constant quantity is ordered each time (lot-for-lot replenishment).
Order-up-to level (S): inventory replenished up to a pre-defined maximum $𝑆$ , adapting to the current stock gap.
Reorder-point shorthand (s): the trigger level — used in both Q-style and S-style policies.

3. Flexibility

Q-policies: rigid — order size is always the same.
S-policies: responsive — order size adapts to the actual inventory level.

4. Lead time

Constant: time between placing an order and receiving it is fixed.
Variable: lead time fluctuates, requiring extra safety stock (see [stocks/safety_stock.typ](../stocks/safety_stock.typ)).

5. Demand

Deterministic: known and constant — yields the EOQ family.
Stochastic: random — yields the policies in this section.

Policy comparison table

Policy	Review type	Trigger	Order size
$(𝒔, 𝑸)$ — see [q_r.typ](q_r.typ)	Continuous	Inventory $\leq 𝒔$	Fixed batch $𝑸$
$(𝒔, 𝑺)$ — see [s_S.typ](s_S.typ)	Continuous	Inventory $\leq 𝒔$	Order-up-to $𝑺$
$(𝒏 𝑸, 𝒓)$ — see [nQ_r.typ](nQ_r.typ)	Continuous	Inventory $\leq 𝒓$	Smallest $𝑛 𝑸$ above $𝒓$
$(𝑺 - 1, 𝑺)$ — see [base_stock.typ](base_stock.typ)	Continuous	Every consumption	One unit (one-for-one)
$(𝑹, 𝑺)$ — see [R_S.typ](R_S.typ)	Periodic (every $𝑹$ )	Always at review	Order-up-to $𝑺$
$(𝑹, 𝒔, 𝑺)$ — see [R_s_S.typ](R_s_S.typ)	Periodic (every $𝑹$ )	If inventory $\leq 𝒔$ at review	Order-up-to $𝑺$
$(𝑹, 𝒏 𝑸, 𝒔)$ — see [R_nQ_s.typ](R_nQ_s.typ)	Periodic (every $𝑹$ )	If inventory $\leq 𝒔$ at review	Smallest $𝑛 𝑸$ above $𝒔$

A policy is named by listing its parameters in sorted order: review interval $𝑅$ (if periodic), reorder point $𝑠$ or $𝑟$ , target level $𝑆$ , fixed quantity $𝑄$ , with $𝑛$ for integer multipliers.

The (s, Q) policy is also widely written (Q, r) — same thing, different naming convention. Likewise (R, s) is a family (specifies trigger but not order rule); concrete instances are (R, s, Q) and (R, s, S).