13.4 Safety stock and \((s, S)\) policies

When demand in the lead time \(L\) is random with mean \(\mu _L\) and standard deviation \(\sigma _L\), a reorder point \(r = \mu _L + z\sigma _L\) gives a probability \(\Phi (z)\) of not running out in a cycle (normal approximation); the safety stock \(z\sigma _L\) is the price of that service level. Combined with an order quantity near the EOQ, this is the continuous-review \((r, Q)\) policy.

Under periodic review with a fixed ordering cost, demand random and independent from period to period, and linear holding and backorder costs, the optimal policy has a remarkably simple form: if the stock (on hand plus on order) has fallen to \(s\) or below, order up to \(S\); otherwise order nothing. Scarf (1960) proved this by showing that the dynamic programming value function (Theorem 11.1.1) is \(K\)-convex, a weakening of convexity that survives the recursion. Finding the best \(s\) and \(S\) is a numerical task, by dynamic programming over the stock level or by simulation (Software 13.5.3); the EOQ formula with \(s\) set for the desired service usually comes close.

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