This is the course where the answer is a formula, which makes it the course where the formula most needs taking apart. The economic order quantity is √(2KD/h) and it arrives here from a quadratic inequality whose two roots collide — the same discriminant Algebra uses, applied to a budget — so that existence, uniqueness and optimality come out of one argument with no calculus in it. The claim that the cost curve is flat either side of the optimum is then computed rather than repeated: ordering t times the best quantity costs (t + 1/t)/2 times the minimum, an expression containing none of the data, so “within one per cent” is 13.2% below to 15.2% above in every warehouse there has ever been. After that the assumptions come off one at a time — a price that depends on the order, a run that takes time, a shortage that was planned, and finally a demand that is a random variable rather than a rate.
Overview
By the end you can derive the economic order quantity without calculus, say how much it costs to miss it and over what range, adjust it for price breaks, a finite production rate and planned shortages, and choose a quantity, a reorder point or a base-stock level when demand is a distribution — stating in each case which service level you have measured.
Constant demand, instantaneous replenishment, no shortages and a price independent of the order, each written down as an assumption rather than absorbed into a formula — with the sawtooth drawn so that Q/2 is something seen rather than remembered, and each of the four removed in turn later in the course.
The budget question K·D/Q + h·Q/2 ≤ T turned into h·Q²/2 − T·Q + K·D ≤ 0 with the positivity condition stated, its three regimes read off T² − 2hKD, and the collision of the roots at T = √(2hKD) producing Q* = √(2KD/h) together with the proof that nothing beats it.
C(t·Q*)/C* = (t + 1/t)/2, the second quadratic t² − 2(1 + e)t + 1 ≤ 0 it produces, and the band (1 + e) ± √(e(2 + e)) — which on any instance at all is 13.2% below to 15.2% above for one per cent, and exactly half to double for twenty-five.
All-units breaks, the genuine discontinuity they put in the total-cost curve, one candidate per band with the band floor where its own economic order quantity is unreachable, and a comparison of surd-valued totals settled by exact arithmetic rather than by two printed decimals.
The factor 1 − D/P derived from the geometry of a tilted sawtooth, the further division by (h + π)/π that a priced backorder buys, all four named models produced from one expression, and the two refusals — P ≤ D and π = 0 — reported as statements about the model rather than as errors.
The critical ratio cᵤ/(cᵤ + cₒ) read as a probability, the reorder point as a quantile of lead-time demand built by convolution, the base-stock level over R + L periods, and cycle service printed beside fill rate every time because on one worked policy they are 68.75% and 99.625%.
Lessons
The plain model first, because everything else on the course is it with one assumption removed, and the derivation immediately after it rather than inside it — the crossing of the two cost curves is where the answer is, and the discriminant is why it is a minimum, and separating those two makes both clearer. Then the assumptions come off in order of how much they change the shape of the answer: a price break breaks the curve into pieces, a production rate and a planned shortage rescale it, and a random demand replaces it. The random half is ordered by how much machinery it needs — one season, then a lead time, then a review interval — and it ends on the misconception this course would keep if it could keep only one.
From algebra, quadratics in full — the quadratic formula, the discriminant and what its sign says, quadratic inequalities and the fact that an upward parabola is below zero exactly between its roots, and radical expressions kept in surd form, because the economic order quantity is one and this course carries it exactly to the point of printing. From discrete mathematics, Discrete Probability through variance: random variables, expected value, linearity of expectation, and the distribution of a sum of independent variables, which is what a lead time of several periods is. Nothing from earlier on this path is required to follow the arguments, although Linear Programming Models is worth having met for what it says about naming a model's assumptions before solving it. There is no calculus here and none is wanted: where a textbook differentiates the cost, this course lowers a budget until two roots collide.
Say which of the four assumptions your situation breaks before choosing a model. Constant demand, instant arrival, no shortages, a price that does not depend on the quantity: there is a lesson for each, and picking the model by which formula you remember is how a plain economic order quantity ends up answering a question about a machine that takes three days to produce a batch.
Keep the root until the last moment. √(2KD/h) is either a whole number or an irrational one, and the difference is a property of the data rather than of the method. Rounding early makes a merge of two roots look like a near-miss, makes two candidate costs look separated when they are not, and makes an exact statement into a decimal the reader cannot check.
Price the mistake rather than warning about it. Every panel on this course computes the wrong answer beside the right one — the band floor against the band's own quantity, b = 0 against the optimal backorder level, the lead-time-only base-stock level against R + L — because a reader who has seen 34.38% printed where 90% was asked for does not make that mistake again, and a reader who has read a caution about it does.
Never write “service level” without saying which one. Cycle service and fill rate are computed from the same policy and on the worked instance of “Reorder Points, and Two Numbers Called Service” they are 68.75% and 99.625%. Both are correct, both are in general use, and a target quoted without its definition has specified nothing at all.
Calculus, anywhere. The economic order quantity is found here by the discriminant of a quadratic, and the flatness of the cost curve by the discriminant of a second one. That is a complete argument rather than a workaround: it proves existence, uniqueness and optimality at once, where a derivative gives a stationary point and then needs a second test to say what kind it is.
Incremental quantity discounts. Only all-units pricing is built, because it is the one whose cost curve is discontinuous and therefore the one whose answer cannot be found by a single formula. Incremental pricing has a continuous curve and is a straightforward variation; the reason for choosing the harder one is that the candidate-per-band argument is what this course is teaching.
Joint optimisation of the order quantity and the reorder point. The two are treated as separate decisions here, which is the standard working assumption and is close to optimal, and it lets the risk question be asked cleanly: the exposure window is the lead time, and the order quantity does not appear in it. Solving the pair together needs an iteration that would obscure exactly that.
Continuous demand distributions, and the normal approximation to lead-time demand. Every distribution on this course is discrete with rational probabilities, so every convolution, quantile and service level is an exact fraction. The standard textbook route — a normal lead-time demand and a safety factor read from a table — would replace those with decimals from an approximation this path has not built, and would hide the overshoot that makes a discrete target interesting.
Multi-echelon inventory, perishability, and lost sales as distinct from backorders. All three are real and all three change the model rather than its parameters: a second stocking location couples two policies, a shelf life makes the cost depend on age, and a customer who leaves rather than waits removes the backorder term this course prices. Naming them is as far as it goes.