Order once for a season whose demand is a random variable, and the decision is a balance between two unit costs: cᵤ for every unit short and cₒ for every unit left. The rule is the critical ratio cᵤ/(cᵤ + cₒ), and the answer is the smallest quantity whose cumulative probability reaches it. The ratio is a probability, not a quantity, which is the single most useful thing to know about it — and the panel tabulates the whole expected-cost curve beside the rule so that the rule is checked rather than applied.
Core ideas
Three ideas. The first is what makes this model different from every earlier one on the course, the second is the rule, and the third is the reading of the rule that stops it being misused.
Every earlier model on this course repeats: order, deplete, order again, and the fixed cost K is what stops the batches being tiny. Here there is one order and one season. Nothing repeats, K is paid once whatever is decided and drops out of the comparison, and what is traded off instead is the cost of a unit that was not there against the cost of a unit that was not wanted.
Order Q and, for each possible demand d, you are left with Q − d units if d is below Q and short by d − Q if it is above. Weighting those by their probabilities gives cₒ·E[over] + cᵤ·E[under], a sum of finitely many rational terms. Raising Q by one buys one more unit of protection at expected benefit cᵤ·P(D > Q) and expected cost cₒ·P(D ≤ Q), so it is worth doing exactly while P(D ≤ Q) is below the critical ratio.
cᵤ/(cᵤ + cₒ) = 7/10 does not mean order seven tenths of something. It means “be willing to have stock left over 70% of the time”, and the order quantity is whatever level of demand that corresponds to on this distribution. Change the distribution and keep the costs and the ratio is unchanged while the quantity moves; that is the clearest evidence of what kind of object it is.
The material
The costs, the expected-cost sum, the rule, and the three instances that show what the ratio is doing.
Demand D for a single season is a random variable with a known distribution. A quantity Q is ordered before the season and cannot be changed. Each unit short costs cᵤ, the underage cost; each unit left over costs cₒ, the overage cost.
The expected cost is C(Q) = cₒ·E[(Q − D)⁺] + cᵤ·E[(D − Q)⁺], and the problem is to choose Q to minimise it.
Both costs are usually margins rather than prices. If a unit is bought for 3 and sold for 10, the overage cost is the 3 that was spent and not recovered, and the underage cost is the 7 of margin that was not earned. Working out which is which is most of the modelling; once they are numbers, the rest is one comparison.
The optimal order quantity is the smallest Q with P(D ≤ Q) ≥ cᵤ/(cᵤ + cₒ).
Raising the order from Q to Q + 1 saves cᵤ whenever demand exceeds Q, which happens with probability 1 − P(D ≤ Q), and costs cₒ otherwise. The change in expected cost is therefore cₒ·P(D ≤ Q) − cᵤ·(1 − P(D ≤ Q)), which is negative exactly while P(D ≤ Q) is below the ratio.
The steps are of decreasing benefit and then of increasing cost, so the expected-cost curve falls and then rises with one minimum, and the first quantity at which the cumulative distribution reaches the ratio is it. Nothing in that argument needs the distribution to have any particular shape, and nothing needs it to be continuous.
The window runs past the top of the demand distribution on purpose. A reader's first instinct is to try ordering nothing and to try ordering more than demand could ever be, and both are priced here: Q = 6 costs 117/10 and never runs short, Q = 0 costs 147/10 and never has anything left. “The crossing is the minimum” is then a statement about the whole line rather than about the five points where demand has mass.
Every figure on that table is exact. Expectations over a finite distribution with rational probabilities are ratios of whole numbers, there is no square root anywhere in this model, and nothing on the page rounds — which is worth saying explicitly, because the two quantities this course does round, the economic order quantity and its cost, are the headline of the lessons before it.
Demand 8 to 12 with probabilities 1/8, 1/4, 1/4, 1/4, 1/8, underage 2, overage 5. The ratio is 2/7 ≈ 28.6%, and the cumulative distribution reaches it at Q = 9 for an expected cost of 23/8.
The mean demand is exactly 10 and the order is 9. When leftovers cost more than shortages, the right answer is to order below the mean and be short some of the time on purpose — which is the clearest possible demonstration that the mean is not the answer.
Demand 0 to 4 with probabilities 1/2, 1/4, 1/8, 1/16, 1/16, underage 20, overage 1. The ratio is 20/21 ≈ 95.2%, and the cumulative distribution does not reach it until the very top of the support: Q = 4, costing 49/16.
Mean demand is 15/16, less than one unit, and the order is four. A stockout costing twenty times a leftover pushes the order to cover the whole of the distribution, and the wide window confirms that 5 and 6 are dearer — 65/16 and 81/16 — so the answer really is the top of the support rather than a runaway.
Three instances, three ratios, three answers relative to the mean: above it, below it, and far above it. Nothing about the distributions explains that pattern and everything about the two unit costs does, which is the whole content of the rule.
The panel computes the answer twice. Once by the critical ratio, walking the cumulative distribution until it reaches the threshold; and once by tabulating the expected cost at every whole quantity in the window and taking the cheapest. It then prints whether the two agree, and says so in red if they do not. They agree on all three worked instances.
That is a habit worth carrying rather than a feature of this page. A rule derived by an argument about one step at a time is exactly the kind of rule that is right in general and mis-stated in a particular textbook — with ≥ written as >, or the ratio inverted — and a tabulation of the objective is the cheapest possible refutation. The panel also refuses a distribution whose probabilities do not sum to one, rather than rescaling it: a distribution that does not sum to one is a typo, and normalising it silently answers a question nobody asked.
Interactive
the critical ratio, and the cost curve that has to agree with it
Method
Four steps. The first is the modelling and it is where the mistakes are; the rest is arithmetic.
cᵤ is what one unit of unmet demand costs — usually lost margin, sometimes a penalty. cₒ is what one unsold unit costs — usually the purchase price less any salvage. Swapping them inverts the ratio and produces a confidently wrong answer in the opposite direction, which no later arithmetic will reveal.
cᵤ/(cᵤ + cₒ) is between zero and one and it is the fraction of the time you are willing to be left holding stock. Write it as a fraction, not a percentage: 7/10 is exact and 70% invites rounding at the comparison step.
Smallest Q with P(D ≤ Q) ≥ the ratio. The inequality is not strict, so a cumulative probability landing exactly on the ratio is reached. Ties here mean two quantities cost the same, not that the rule is ambiguous.
The expected cost at Q − 1, Q and Q + 1 should fall then rise. This is three sums over a finite distribution and it catches an inverted ratio immediately, which is the one error the rule cannot catch by itself.
Three distributions and three pairs of unit costs. The only thing that changes between them is the ratio, and it moves the answer from above the mean to below it and back again.
Three answers, three positions relative to the mean, and one rule. The distributions in A and C have the same support and the same shape of story; what separates an order of 3 from an order of 4 is that a stockout costs twenty times a leftover in one and a little over twice as much in the other. Anyone who reaches for the mean is answering a question about demand when the question was about costs.
Every number above is a ratio of whole numbers, and there is nothing on this page to round. That is a property of the model rather than of the worked example: the expected cost is a finite sum of products of rational probabilities and rational costs, so no root can arise. The two quantities this course does round belong to the earlier lessons, and it is worth being able to say which is which without checking.
For a rehearsal, keep distribution A and raise the overage cost from 3 to 7. The supplied first move is that the ratio becomes 1/2. Work out which quantity the cumulative distribution reaches it at, and predict whether the expected cost there is above or below 37/10 before the panel prices it.
You should be able to identify the underage and overage costs in a described situation, form the ratio, find the smallest quantity whose cumulative probability reaches it, confirm by pricing either side, and say why the answer can be above or below the mean depending on nothing but the two costs.
Something to change about this lesson? Write it down, tick it off when it is handled, and export the list when you want to act on it. Kept in this browser only: nothing is sent anywhere, and nobody is notified.
No recommendations yet for this lesson.