5. Inventory Management

A shop sells a product that it buys from a supplier. If it orders too often it pays the fixed cost of placing and receiving an order again and again; if it orders too much at once it ties up cash and shelf space in stock that sits waiting to be sold. If demand is uncertain there is a second trade-off: too little stock means lost sales and unhappy customers, too much means leftovers that must be marked down or thrown away. How much should it order, and when?

These are the questions of inventory theory. Its models are among the oldest in OR and among the most widely deployed: every retailer, manufacturer and hospital pharmacy runs some version of the policies in this lecture, often inside an ERP system that computes them automatically. We look at two foundational models — the economic order quantity for steady, known demand, and the newsvendor for a single period of uncertain demand — and then at the policies that combine the two ideas.

The decision

An inventory policy answers two questions: how much to order, and when to order it. The relevant costs are:

an ordering cost $K$ per order, independent of its size (administration, transport, set-up of a production run);
a purchase cost $c$ per unit;
a holding cost $h$ per unit per unit of time, covering the cost of capital, storage, insurance and obsolescence — often expressed as a percentage of $c$ per year;
a shortage cost for demand that cannot be met from stock, either as a lost sale or as a backorder filled later.

Which costs matter depends on the setting. With steady demand, the trade-off is between ordering and holding costs. With uncertain demand, it is between holding (or overage) and shortage costs.

The economic order quantity

Assume demand occurs at a constant, known rate of $D$ units per year, replenishment is instantaneous, and no shortages are allowed. Then it is clearly optimal to order only when stock reaches zero, and to order the same quantity $Q$ each time. Inventory follows a sawtooth: it jumps to $Q$ , falls linearly to zero over a cycle of length $Q/D$ , and jumps back up.

The decision variable is the order quantity $Q \gt 0$ . Over a year,

the number of orders is $D/Q$ , so ordering costs $K D / Q$ ;
the average inventory is $Q/2$ , so holding costs $h Q / 2$ ;
purchasing costs $cD$ , which does not depend on $Q$ and can be ignored.

The annual cost to minimize is therefore

C(Q) = \frac{K D}{Q} + \frac{h Q}{2} .

The first term decreases in $Q$ and the second increases. $C$ is convex on $Q \gt 0$ , so setting the derivative to zero gives the minimum:

C'(Q) = -\frac{K D}{Q^2} + \frac{h}{2} = 0 .

(Economic order quantity)

The cost-minimizing order quantity and the resulting annual cost are

Q^* = \sqrt{\frac{2 K D}{h}}, \qquad C(Q^*) = \sqrt{2 K D h} .

At $Q^*$ the annual ordering cost and the annual holding cost are equal, each $\tfrac{1}{2}\sqrt{2KDh}$ .

This is the EOQ formula, usually attributed to Harris (1913). Its most useful property is robustness. Dividing $C(Q)$ by $C(Q^*)$ gives

\frac{C(Q)}{C(Q^*)} = \frac{1}{2}\left( \frac{Q}{Q^*} + \frac{Q^*}{Q} \right),

which is very flat near $Q/Q^* = 1$ . Ordering 25% more than optimal raises cost by only 2.5%. Equally, errors in the inputs are damped by the square root: if $K$ is overestimated by a factor of two, $Q^*$ is too large by a factor of $\sqrt{2} \approx 1.41$ , and cost rises by about 6%. This is why EOQ works well in practice even though its assumptions rarely hold exactly.

(EOQ)

A shop sells $D = 1200$ units a year. Each order costs $K = 100$ to place, and holding one unit for a year costs $h = 6$ . Then

Q^* = \sqrt{\frac{2 \cdot 1200 \cdot 100}{6}} = \sqrt{40\,000} = 200,

so the shop places $1200 / 200 = 6$ orders a year, one every two months. The annual cost is

C(200) = \frac{100 \cdot 1200}{200} + \frac{6 \cdot 200}{2} = 600 + 600 = 1200 = \sqrt{2 \cdot 1200 \cdot 100 \cdot 6} .

The cost curve is flat around the optimum:

Order quantity $Q$	150	200	250	300	400
Annual cost $C(Q)$	1250	1200	1230	1300	1500
Relative to optimum	+4.2%	—	+2.5%	+8.3%	+25%

If the supplier only delivers in cases of 250, the penalty is $30$ a year.

Lead times. If an order takes a known time $L$ to arrive, nothing changes except timing: place the order when the stock on hand equals the demand during the lead time, $r = DL$ . This reorder point becomes important once demand is uncertain.

The newsvendor model

Now consider a single selling period with uncertain demand. A newsvendor buys papers in the morning at cost $c$ each, sells them at price $p$ , and can return unsold copies at the end of the day for a salvage value $v$ , with $v \lt c \lt p$ . Demand $X$ is a random variable with distribution function $F$ . The decision is the order quantity $Q$ , made before demand is known. The same structure fits fashion goods, perishable food, seasonal products, and capacity decisions in general.

Two costs drive the decision:

the underage cost $c_u = p - c$ , the profit lost on each unit of demand that goes unmet;
the overage cost $c_o = c - v$ , the loss on each unit bought but not sold.

The expected cost of mismatch is

G(Q) = c_o\, \mathbb{E}\big[(Q - X)^+\big] + c_u\, \mathbb{E}\big[(X - Q)^+\big],

where $a^+ = \max\{a, 0\}$ , and maximizing expected profit is equivalent to minimizing $G$ .

Marginal analysis. Increase $Q$ by one small unit. With probability $F(Q)$ demand falls short of $Q$ , the extra unit is left over, and we lose $c_o$ . With probability $1 - F(Q)$ demand exceeds $Q$ , the extra unit is sold, and we save $c_u$ . Formally,

G'(Q) = c_o\, F(Q) - c_u \big(1 - F(Q)\big) .

This is increasing in $Q$ , so $G$ is convex, and the optimum is where the derivative vanishes.

(Critical ratio)

For continuous demand, the expected-profit-maximizing order quantity satisfies

F(Q^*) = \frac{c_u}{c_u + c_o} .

The right-hand side is the critical ratio, and $Q^*$ is the corresponding quantile of the demand distribution. For discrete demand, $Q^*$ is the smallest $Q$ with $F(Q) \ge c_u / (c_u + c_o)$ .

The critical ratio is the optimal probability of not running out. When underage is costly relative to overage (high margin, high salvage value), it is close to $1$ and we stock well above expected demand; when overage dominates, we stock below it. Ordering the mean demand is optimal only when $c_u = c_o$ and the distribution is symmetric.

For normal demand with mean $\mu$ and standard deviation $\sigma$ , $Q^* = \mu + z \sigma$ , where $z = \Phi^{-1}\big(c_u / (c_u + c_o)\big)$ and $\Phi$ is the standard normal distribution function. The expected shortage at any $Q = \mu + z\sigma$ is $\sigma\, \mathcal{L}(z)$ , where

\mathcal{L}(z) = \varphi(z) - z\,\big(1 - \Phi(z)\big)

is the standard normal loss function and $\varphi$ is the standard normal density. Expected leftovers are then $Q - \mu + \sigma\, \mathcal{L}(z)$ .

(Newsvendor with normal demand)

A bakery sells a speciality cake for $p = 10$ , which costs $c = 4$ to make; unsold cakes go to a food outlet for $v = 1$ . Daily demand is approximately normal with $\mu = 500$ and $\sigma = 100$ . Then

c_u = 10 - 4 = 6, \qquad c_o = 4 - 1 = 3, \qquad \frac{c_u}{c_u + c_o} = \frac{6}{9} = \frac{2}{3} .

The $2/3$ quantile of the standard normal distribution is $z = 0.4307$ , so

Q^* = 500 + 0.4307 \cdot 100 \approx 543 .

With $\varphi(0.4307) = 0.3636$ , the loss function is $\mathcal{L}(z) = 0.3636 - 0.4307 \cdot \tfrac{1}{3} = 0.2200$ . Hence

expected shortage: $100 \cdot 0.2200 = 22.0$ cakes;
expected sales: $500 - 22.0 = 478.0$ cakes;
expected leftovers: $543.07 - 478.0 = 65.1$ cakes.

Expected profit is $(p - c)\mu$ minus the expected cost of mismatch:

6 \cdot 500 - (6 \cdot 22.0 + 3 \cdot 65.1) = 3000 - 327.2 = 2672.8 .

Ordering the mean, $Q = 500$ , gives expected shortage and leftovers of $100\,\varphi(0) = 39.9$ each, and expected profit $3000 - 9 \cdot 39.9 = 2641.0$ . The value $3000$ is what perfect knowledge of demand would earn; the remaining $327$ is the cost of uncertainty, and the only way to reduce it further is to reduce $\sigma$ — better forecasting, or postponing production until demand is better known.

The probability of not running out, here $2/3$ , is not the same as the fraction of demand met. The bakery meets $478.0 / 500 \approx 95.6\%$ of demand on average, even though it sells out on one day in three.

Policies for repeated orders under uncertainty

Most items are reordered repeatedly with uncertain demand and a positive lead time. Three policy classes dominate practice. In each, decisions are based on the inventory position: stock on hand plus stock on order minus backorders.

Base-stock (order-up-to) policy. At each review, order enough to raise the inventory position to a fixed level $S$ . With periodic review, backordered demand and no fixed ordering cost, a base-stock policy is optimal, and $S$ is a newsvendor quantile of the demand over the lead time plus one review period.
(s, S) policy. At each review, if the inventory position is at or below $s$ , order up to $S$ ; otherwise do nothing. With a fixed ordering cost, policies of this form are optimal under fairly general conditions. Roughly, $S - s$ plays the role of the EOQ and $s$ protects against lead-time demand.
(Q, r) policy. With continuous review, whenever the inventory position falls to the reorder point $r$ , order a fixed quantity $Q$ . A good approximation takes $Q$ from the EOQ formula and sets $r$ from the lead-time demand distribution.

Safety stock

With uncertain demand, the reorder point is set above the expected lead-time demand. The excess is the safety stock. If demand over the lead time is normal with mean $\mu_L$ and standard deviation $\sigma_L$ , and we want to avoid a stockout in each cycle with probability $\alpha$ (the cycle service level), then

r = \mu_L + z_\alpha\, \sigma_L, \qquad \text{safety stock} = z_\alpha\, \sigma_L, \qquad z_\alpha = \Phi^{-1}(\alpha) .

If demand per period is independent with standard deviation $\sigma$ and the lead time is $L$ periods, then $\sigma_L = \sigma\sqrt{L}$ : safety stock grows with the square root of the lead time, not linearly.

(Safety stock for a (Q, r) policy)

Return to the shop of the EOQ example, with demand averaging $100$ units a month (1200 a year) but now with a monthly standard deviation of $30$ . The lead time is one month, so lead-time demand has $\mu_L = 100$ and $\sigma_L = 30$ . For a $95\%$ cycle service level, $z = 1.645$ , so

\text{safety stock} = 1.645 \cdot 30 = 49.3, \qquad r = 100 + 49.3 = 149.3 .

The policy is: when the inventory position falls to about $149$ units, order $Q = 200$ . The expected shortage per cycle is $30\, \mathcal{L}(1.645) = 0.63$ units, so the fraction of demand met from stock is $1 - 0.63/200 \approx 99.7\%$ . Holding the safety stock costs $6 \cdot 49.3 = 296$ a year. Raising the cycle service level to $99\%$ ( $z = 2.326$ ) needs $69.8$ units of safety stock, at $419$ a year.

The example shows the steep cost of high service levels: the last few percentage points require disproportionately more stock, because $z_\alpha$ grows rapidly as $\alpha$ approaches $1$ .

Extensions

Multi-echelon inventory. Stock is held at several levels — a central warehouse feeding regional depots feeding stores — and the question is where safety stock is most effective. Holding it centrally pools demand variability across locations.
Multiple items. Items share a budget, warehouse space or a joint ordering cost; Lagrangian methods adjust each item's EOQ or service level.
Perishability and obsolescence. Food, blood and medicines expire; fashion and electronics lose value quickly. These push towards newsvendor-type models.
Dynamic programming. Time-varying demand, capacity limits and non-stationary costs are handled by stochastic dynamic programming, which is also how the optimality of base-stock and (s, S) policies is proved.
Demand learning. In practice the demand distribution is itself estimated from data, often censored by stockouts — sales are observed, but lost demand is not.

Further reading: Silver, Pyke and Thomas, Inventory and Production Management in Supply Chains; Zipkin, Foundations of Inventory Management; Nahmias, Production and Operations Analysis.