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Supply Meets Demand: Inventory Optimization
Quantitative Analysis · Chapter 4: Inventory Management and QueuingSection 4.3 · lesson 14 of 5512 min · free with a free account
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In this lesson
The Economic Order Quantity model, its extensions, and the reorder point that goes with it.
- Derive EOQ from the ordering-cost and holding-cost trade-off
- State the five assumptions the basic model rests on
- Extend EOQ to gradual production, quantity discounts and backorders
- Set a reorder point and size safety stock for a chosen service level
Transcript
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Inventory Management and Queuing · Section 4.3
Supply Meets Demand
Inventory is money standing still. Every unit on a shelf is cash you have already spent, waiting to be sold. So the obvious move is to hold as little as possible. But that is wrong too. Order in small batches, and you pay to place orders again and again. You also risk running out, which costs you the sale. This section finds the order quantity that sits between those two mistakes. It is called the Economic Order Quantity, or E O Q. We derive it from the cost trade-off, then relax its assumptions one at a time. Gradual production, quantity discounts, planned backorders, and uncertain demand. Then we work a full numerical example, from parameters to a final decision.
The Inventory Problem
Why Too Much and Too Little Both Cost Money
Inventory exists because supply and demand are rarely synchronised. Goods arrive in batches; customers arrive one at a time. Order too much, and you pay to store goods you do not need yet. Order too little, and you pay for constant reordering, and you risk a stockout. The goal is the quantity that minimises total cost at an acceptable service level. The classical model rests on five simplifying assumptions. Demand is constant and known, at D units per year, and replenishment is instantaneous. No stockouts are permitted, so orders land exactly when inventory hits zero. Each order costs a fixed amount S, whatever its size. And each unit in stock costs a constant amount h, per unit per year. Under those rules inventory traces a perfect sawtooth. It jumps to Q when the order lands, then falls in a straight line to zero. None of that holds exactly. Every one of the five is relaxed later on.
The Cost Trade-off
Two Costs Pulling in Opposite Directions
Annual inventory cost splits into two components that fight each other. Ordering cost is D over Q, times S. Annual demand divided by order size gives the number of orders; multiply by the cost per order. That rises as Q gets smaller, because you order more often. Holding cost is Q over two, times h. Under the sawtooth, average inventory is half the order quantity, and each unit costs h to hold. That one rises as Q gets larger. One curve falls, the other rises, so their sum has a minimum. Calculus puts that minimum exactly where the two components are equal. Set them equal, solve for Q, and out drops the E O Q: the square root of two D S over h. That equality is also a free check. If ordering and holding cost do not match at your answer, your answer is wrong.
Companion Formulas
Three Numbers That Follow Immediately
Once you know Q star, three more numbers follow almost for free. Cycle time, T star, is Q star divided by D. That is the gap between consecutive orders, in years; multiply by three hundred and sixty-five for days. Number of orders per year, N star, is D divided by Q star. It is simply the reciprocal of the cycle time. Minimum total cost, T C star, is the square root of two D S h. Look at what is inside that root: demand, ordering cost, and holding cost. There is no purchase price in it at all. That is deliberate. If the price is fixed, buying D units a year costs the same whatever the batch size. Purchase price only matters when it depends on order size, which is the discount case, coming next.
Two Extensions
EOQ Times an Adjustment Factor
The basic model assumes an idealised world, and real operations rarely match it. Two of the standard extensions keep the square root intact, and simply multiply it by a correction factor. First, the Economic Production Quantity, or E P Q. Here stock is produced gradually rather than delivered in one shipment, as on an in-house line. Inventory builds while production runs, and depletes only after the run ends. Let p be the production rate and d the demand rate, with p greater than d. The batch size is the basic E O Q, times the square root of p over p minus d. Check the limits. As p grows very large, production is effectively instant, and the factor goes to one. As d creeps towards p, the factor grows without bound; production that barely keeps pace needs enormous batches. Second, backorders, for when customers are willing to wait. The firm pays a backorder cost b per unit per year of waiting, and now decides two things: the order quantity, and the maximum backorder level. Order quantity becomes the basic E O Q, times the square root of h plus b, over b. And as b grows very large, waiting is effectively forbidden, and you are back to plain E O Q.
Quantity Discounts
When the Price Depends on the Order Size
Suppliers often cut the unit price for larger orders. That turns a two-way trade-off into a three-way one: ordering, holding, and purchase cost. The procedure is mechanical, and worth memorising. Step one. For each price tier, compute the E O Q using that tier's holding cost. Holding cost is often quoted as a percentage of unit price, so each tier has its own h. Step two. Check feasibility. Is that tier's E O Q actually inside that tier's quantity range? If it is not, use the nearest boundary quantity for that tier instead. Step three. Compute total annual cost at every feasible candidate. Ordering, plus holding, plus purchase, where purchase is D times the price at that tier. Step four. Take the candidate with the lowest total annual cost. Step two is where most mistakes happen. An E O Q outside its own tier is not a real option.
When To Order
Reorder Point and Safety Stock
E O Q answers how much to order. It says nothing at all about when. Suppliers do not deliver instantly, so orders must be triggered while stock is still on hand. You need enough left to cover demand through the supplier's lead time. That trigger level is the reorder point. Under certainty it is simple: daily demand d, times lead time L. That is just demand during the lead time, expressed in units. In practice, demand during the lead time wobbles. A random spike while you are waiting on a delivery is exactly how stockouts happen. So firms carry a buffer, called safety stock, equal to z times sigma d L. Sigma d L is the standard deviation of demand during the lead time. And z is the service-level score, read from the standard normal table. The reorder point then becomes d times L, plus safety stock, and the buffer absorbs the spike.
Service Level
A Managerial Choice, Not a Formula
Service level is a managerial choice, not a calculation. It is the probability that demand during the lead time does not exceed the stock on hand. Ninety per cent uses a z of one point two eight; ninety-five per cent needs one point six five. Ninety-seven point five per cent needs one point nine six, and ninety-nine per cent needs two point three three. Ninety-nine point nine per cent needs three point zero nine. Safety stock is proportional to z, so those scores translate straight into cost. Against a ninety per cent baseline, ninety-five per cent costs twenty-nine per cent more safety stock. Ninety-seven point five costs fifty-three per cent more, and ninety-nine per cent costs eighty-two per cent more. Ninety-nine point nine costs a hundred and forty-one per cent more. Look at that last step: going from ninety-nine to ninety-nine point nine adds fifty-nine percentage points. That is the diminishing return. Most firms sit between ninety-five and ninety-nine, saving the extremes for critical or high-margin items.
Worked Example
A Fast-Moving SKU, Step by Step
Now to the numbers. A regional distributor sells a fast-moving item. Annual demand is twelve thousand units. Each order costs one hundred and fifty dollars to place. Holding costs four dollars per unit per year. The firm works three hundred days a year, and the supplier's lead time is ten days. Step one is the E O Q itself. Two, times twelve thousand, times one hundred and fifty, is three million six hundred thousand. Divide by the holding cost of four, and you get nine hundred thousand. The square root of nine hundred thousand is about nine hundred and forty-nine units. Step two. Orders per year is annual demand divided by that quantity. Twelve thousand over nine hundred and forty-nine is about twelve point six five orders a year.
Worked Example
Cycle Time, Reorder Point, Minimum Cost
Step three. Cycle time is the working year divided by the number of orders. Three hundred days over twelve point six five is about twenty-three point seven days. So roughly one order every twenty-four working days. Step four. The reorder point, with no uncertainty yet. Daily demand is twelve thousand over three hundred days, which is forty units a day. Lead time is ten days, so demand during the lead time is four hundred units. The distributor reorders the moment stock on hand falls to four hundred. Step five. Minimum total cost. Two, times twelve thousand, times one hundred and fifty, times four, is fourteen million four hundred thousand. Its square root is about three thousand seven hundred and ninety-five dollars a year. And here is the promised check. Ordering and holding cost are each about eighteen hundred and ninety-seven dollars, and they add to that same total.
Taking The Discount
Does a Lower Price Justify a Bigger Order?
Now the supplier offers a discount for bulk. Below five hundred units the price is ten dollars each; at five hundred or more, nine dollars fifty. Demand, ordering cost and holding cost are all unchanged. Step one. The unconstrained E O Q is still nine hundred and forty-nine units. That is above five hundred, so it sits in Tier two and is feasible at the lower price. Step two. What is the best we can do inside Tier one? Total cost is convex, and it rises as you move away from the E O Q. So the best feasible Tier one quantity is the boundary itself, four hundred and ninety-nine units. Step three. Cost both candidates in full, purchase price included. Tier one totals about one hundred and twenty-four thousand, six hundred and five dollars. Tier two totals about one hundred and seventeen thousand, seven hundred and ninety-five dollars. So Tier two wins by roughly six thousand eight hundred and ten dollars a year. Sanity check the direction. Saving fifty cents a unit on twelve thousand units is six thousand dollars of purchase savings. That more than offsets the higher cost of carrying a larger average inventory. The numbers confirm the intuition.
Pitfalls And Summary
What EOQ Gets Right, and Where It Bites
Five things to carry out of this section. First, E O Q is surprisingly robust, because the total cost curve is flat near its minimum. Ordering twenty or thirty per cent away from it costs very little, so do not chase the third decimal place. Second, holding cost is almost always understated. It covers capital cost, warehousing, insurance, obsolescence and shrinkage, not just the rent on the shelf. Third, ordering cost is not just paperwork. Receiving, inspection, setup and accounting are each genuinely fixed per order. Fourth, E O Q assumes a stable world. For seasonal goods, new products, or an item in a price war, use a dynamic model, or recalculate far more often. Fifth, quantity discounts can mislead. A discount only pays if it survives the extra carrying cost of a bigger average inventory. How much comes from the square root formula; when comes from the reorder point; and how much buffer is a service-level choice you pay for.
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