5. Inventory Policies with Safety Stock

Chapter 4 calculated a buffer for demand over a protection period. A safety-stock number alone is not an operating rule: a supply chain also needs to know when it may order, how much it orders, and which inventory measure controls the decision.

This chapter applies the Normal-demand, fixed-lead-time model to two policies: continuous review with a fixed order quantity $(s,Q)$ and periodic review with an order-up-to target $(R,S)$. It then uses the policy equations to analyze stock components and to outline a simulation for cases that are harder to solve analytically.

Define the protection period

Let demand in one bucket have mean $\mu_d$ and standard deviation $\sigma_d$. Let $L$ be the fixed supplier lead time and $R$ the interval between scheduled reviews, both measured in the same buckets. For a chosen cycle service target $\alpha$, let $z_\alpha=\Phi^{-1}(\alpha)$.

If bucket demands are independent and have equal variance, demand over a protection window of $x_\tau$ periods has mean $\mu_dx_\tau$ and standard deviation $\sigma_d\sqrt{x_\tau}$. The buffer is:

$$SS(x_\tau)=z_\alpha\sigma_d\sqrt{x_\tau}$$

The protection window is the time during which demand can consume stock before the next replenishment opportunity is available. It differs by policy.

Continuous review with a fixed batch: $(s,Q)$

With continuous review, the inventory position can trigger an order at any time. Once an order is placed, the supplier lead time $L$ is the remaining exposure window. The safety stock and reorder point are:

$$SS_{s,Q}=z_\alpha\sigma_d\sqrt{L}, \qquad s=\mu_dL+SS_{s,Q}$$

When the inventory position reaches $s$, order the fixed batch $Q$. A baseline batch can come from the EOQ model in Chapter 2, but this does not mean EOQ has optimized shortage costs or the service target. The reorder point includes expected lead-time demand plus its safety buffer.

Reducing lead time lowers both expected pipeline inventory and the demand-uncertainty buffer. With all else fixed, halving $L$ multiplies this safety stock by $1/\sqrt{2}$, not by one half.

Periodic review with an order-up-to target: $(R,S)$

With periodic review, a planner who has just reviewed stock must wait until the next review opportunity and then still wait for delivery. The protection period is therefore $L+R$, giving:

$$SS_{R,S}=z_\alpha\sigma_d\sqrt{L+R}, \qquad S=\mu_d(L+R)+SS_{R,S}$$

At each review, calculate the inventory position and order the positive shortfall to $S$:

$$Q_{review}=\max(0,S-\text{inventory position})$$

The target is an inventory-position level, not an amount of physical stock expected to sit on the shelf. Compared with $(s,Q)$, increasing the review interval also lengthens the protection period and increases the required safety stock.

What about the review interval?

In a deterministic EOQ model, an interval can be derived from the EOQ by dividing batch size by demand. Under stochastic demand, changing $R$ also changes the safety stock needed for the $L+R$ protection window. The deterministic interval is a useful starting point, but it does not by itself minimize total cost with stochastic demand. Round practical schedules carefully, and optimize the service/cost trade-off using a model that includes the buffer.

A periodic review with a fixed batch: $(R,s,Q)$

Some operations review on a fixed schedule and can order only standard batches, for example a full case or truckload. If demand has crossed reorder point $s$ by the time of review, the inventory position may already be below $s$. This amount is called the undershoot. Its value depends on demand since the previous review and is random under stochastic demand.

That undershoot changes how much protection the threshold actually provides. Applying the $(R,S)$ or $(s,Q)$ formula unchanged does not fully optimize an $(R,s,Q)$ policy. For realistic demand, pack-size, and supplier constraints, simulation is often a more practical way to compare candidate settings.

Analyze the inventory, not just the target

Keep four quantities distinct:

Component	Meaning	Baseline measure
Safety stock	Buffer for demand variability over the policy's protection period	$SS_{s,Q}$ or $SS_{R,S}$
Cycle stock	Stock used to cover expected demand between replenishments	Average is $Q/2$ for $(s,Q)$ or $\mu_dR/2$ for $(R,S)$
In-transit stock	Units moving through the supplier lead time	Average is $\mu_dL$
On-hand stock	Physical units available at the location, before subtracting commitments	Typically safety stock plus cycle stock on average

For a steady $(R,S)$ policy, the maximum replenishment during one review interval is about $\mu_dR$, so the routine on-hand cycle range is approximately from $SS_{R,S}$ to $SS_{R,S}+\mu_dR$. Its average cycle stock is only half the range, $\mu_dR/2$. In-transit units are separate; do not add them to an on-hand threshold.

These are theoretical reference levels, not hard bounds. Actual on-hand stock can fall below safety stock during a demand spike, and startup or policy changes can place it outside the routine cycle range.

Worked stock analysis

A periodic-review item has mean demand of 40 units per day, lead time $L=4$ days, review period $R=1$ day, and a 95% cycle-service target. Suppose the demand variation produces about 50 units of safety stock for the five-day protection window.

• Expected demand over lead time and review window: $40(4+1)=200$ units.
• Order-up-to target: $S=200+50=250$ units of inventory position.
• Average cycle stock: $40(1)/2=20$ units; maximum routine cycle stock is about 40.
• Average in-transit stock: $40(4)=160$ units.
• Average on-hand stock: about $50+20=70$ units, not 250.

If current physical on-hand is 30, it is 20 units below the safety-stock reference. The pipeline may still contain stock, so inspect on-hand, in-transit, and the inventory position separately before deciding whether to expedite or place an additional order.

Simulate the policy you actually run

Analytical formulas answer questions under assumptions. A simulation can test how a candidate policy behaves when demand is sampled period by period, when shortages become backorders or lost sales, or when the policy has an undershoot. A useful simulation should make its assumptions visible and record more than one service metric.

For a periodic $(R,S)$ simulation, a planner can:

1. Specify the demand distribution, time bucket, lead time, review period, service target, and shortage treatment.
2. Generate demand for each period and initialize on-hand stock and the in-transit pipeline.
3. Receive shipments due at the start of a period, then apply that period's demand.
4. Record stockout events, fulfilled units, backorders or lost sales, and inventory levels.
5. At each review, calculate inventory position and order $\max(0,S-\text{position})$ for delivery after $L$ periods.
6. Discard initialization/warm-up periods and compare cycle service, period service, fill rate, average on-hand, and backlog across many runs.

The shortage rule changes the outcome. Under backorders, unfilled demand remains owed and later receipts can serve it; net inventory can become negative. Under lost sales, unmet demand disappears and physical on-hand cannot be negative. Do not compare policies simulated under different shortage rules as if their service results were directly comparable.

Normal demand can also generate negative values when variability is large relative to the mean. Clipping simulated demand at zero creates a truncated distribution and changes its mean and service results. Record this choice; for highly variable or discrete demand, consider a distribution that better fits the observations.

SQL practice: analyze simulation output

These exercises query a small deterministic snapshot of simulation results. They practice the analysis step after a simulation has run; they do not generate random demand or replace a period-by-period simulator. Open the Data tab to inspect the simulation rows and policy settings.

Practice: apply and analyze

Concept check

Quiz

Safety stock in inventory policies

1 / 6

Which protection period is used for safety stock in a continuous-review $(s,Q)$ policy with fixed lead time?

Pick an answer to see if you are right.

Rank the periodic-policy simulation

Put these steps in order for a periodic $(R,S)$ simulation with fixed lead time.

Quiz

Simulate an (R, S) policy

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Set up a period-by-period simulation and estimate service and inventory performance.

OrderDrag the handle to rearrange — top should be first.

1. ⋮⋮1At each review, calculate inventory position, order the shortfall to S, and schedule receipt after L periods.
2. ⋮⋮2Generate the demand stream and initialize on-hand inventory and shipments already in the pipeline.
3. ⋮⋮3Specify demand, lead time, review interval, target S, warm-up length, and backorder or lost-sales behavior.
4. ⋮⋮4Discard warm-up periods and summarize cycle service, period service, fill rate, average stock, and backlog.
5. ⋮⋮5For each period, receive due shipments, apply demand, and record stockout and fulfillment outcomes.

Drag items into the correct order (or use ↑↓), then check.

Takeaways

• For $(s,Q)$, protect demand over lead time $L$; for $(R,S)$, protect demand over $L+R$.
• Keep expected demand, safety stock, average cycle stock, maximum cycle range, and pipeline stock distinct.
• The deterministic EOQ is a batch-size benchmark; stochastic service requirements make review-period optimization more involved.
• Periodic fixed-batch policies can have random undershoot, making simulation useful.
• Match simulation shortage rules and service metrics to the actual business before comparing results.

Previous: Safety Stocks.

Next: Inventory Policy Simulation Lab — implement policy targets and simulate a seeded demand trace.