How much to order, and when: the oldest formula in the field?
The course closes on the oldest applied problem in the field, the one Ford Whitman Harris wrote a formula for in 1913, and shows how many of the threads of this course meet inside a warehouse.
Inventory exists to absorb the mismatch between how supply arrives and how demand draws it down. Hold too much and you pay to store it, insure it, and watch it go stale; hold too little and you stock out, losing the sale and sometimes the customer. Every inventory model is a way of pricing that trade-off precisely instead of by feel, and it is the problem operations research was arguably invented on, decades before it had the name: Ford Whitman Harris published the economic order quantity formula in 1913; he had worked it out as an engineer at Westinghouse.
Three questions organise the whole area. How much to order at once: the batch-size trade-off in the figure. When to reorder. You cannot wait until the shelf is empty, because resupply takes time, so you trigger a new order once stock falls to a level that covers expected demand over that lead time. How much extra to carry as a cushion, safety stock, sized against demand variability and the service level you are willing to promise, which turns out to be topic 22's critical fractile wearing a different hat.
Scale this from one shelf to a chain of suppliers, factories, warehouses and retailers and a new problem appears that no single stage can see: the bullwhip effect. Small wobbles in retail demand get amplified at every step upstream, so the factory sees wild swings even when the shop floor is nearly steady, and one of its documented causes is a rationing scramble that is, precisely, a game in topic 11's sense.
Economic order quantity. Demand D per year is steady, each order costs a fixed K to place regardless of size, holding one unit for a year costs h. Ordering Q at a time means D/Q orders a year and an average of Q/2 units on hand, so annual cost is
C(Q) = (D/Q)·K + (Q/2)·h minimised at Q* = √(2KD / h)
Setting C′(Q) = 0 gives it in one line, and at Q* the two terms are equal: ordering cost exactly matches holding cost, which is the fastest way to sanity-check any EOQ answer. With K = 50, D = 3600, h = 4 the formula gives √90000 = 300, an integer scan of C(Q) confirms 300 is the minimum, and C(300) = 600 + 600 = 1200. The cost curve's flatness is not incidental: because C is the sum of a term in 1/Q and a term in Q, it rises only gently on either side of the optimum: ordering 20% above EOQ costs about 1.7% more (1,220 against 1,200), 20% below about 2.5% more (1,230), which is why rounded, practical order sizes lose almost nothing.
Reorder point with uncertain lead-time demand. If demand over the resupply lead time is roughly normal with mean μ and standard deviation σ, then to hold the probability of stocking out before resupply down to 1 − α,
reorder point = μ + z_α · σ safety stock = z_α · σ
where zα is the standard-normal quantile for service level α: the same critical-fractile idea as the newsvendor, since α = Cu/(Cu+Co) with Cu the stockout cost and Co the holding cost. For μ = 300, σ = 50 and a 97.5% target, z = 1.96, so the reorder point is 300 + 1.96·50 = 398 and the safety stock is 98 units: the price of moving from a coin-flip service level to 97.5%. Verification: a script recomputes this.
Optimal policy structure. When each order also carries a fixed cost, Scarf (1960) proved that the optimal dynamic policy is (s, S): let stock fall to s, then order back up to S. That is a threshold rule produced by the same dynamic programming as topics 08 and 20 (the proof turns on a property called K-convexity) so inventory control is, underneath, a Bellman equation with a fixed charge in it.
The bullwhip effect was quantified by Lee, Padmanabhan & Whang (“Information Distortion in a Supply Chain: The Bullwhip Effect,” Management Science 43(4), 546–558 (1997)), who identified four structural causes: demand-forecast updating, order batching, price promotions, and a rationing game in which retailers inflate orders to win a larger share of a shortage, then cancel. That last one is a genuine game in the sense of topics 11–12, and the fix is a mechanism-design fix in the sense of topic 14: allocate shortages by past sales rather than by current orders, and the incentive to inflate disappears.
Every retailer you have ever bought from Continuous-review (s, S) and periodic base-stock policies, sized by EOQ-style batching and critical-fractile safety stock, are what replenishment systems at large retailers and distributors actually run, per stock-keeping unit, per location, nightly. Zara is the standard counter-example taught alongside them: it deliberately shortens and speeds its supply chain so that lead times are small, which shrinks the safety stock it must carry and damps the bullwhip it would otherwise amplify: a supply-chain design choice made to change the inputs to these formulas rather than to solve them harder.
Where this course's threads meet This last topic deliberately has lower quantum cross-link density than the rest of the page, but it is where the operations-research threads themselves converge. The batch size is a calculus optimum (topic 01's spirit); the reorder point is topic 22's newsvendor; the (s, S) policy is topics 08 and 20's dynamic programming; the bullwhip's rationing game is topics 11–12; and its fix is topic 14's mechanism design. A warehouse is where the whole feasible region shows up at once.
Check yourself
In the economic order quantity model, what is true at the optimal order size?
At Q* the two terms are equal (600 + 600 = 1,200 in the example), which is the fastest sanity check of any EOQ answer. Doubling demand multiplies Q* by only the square root of 2.