The Economic Order Quantity, and What a Wrong One Costs
Manuel D. Rossetti, PhD P.E.
Agenda
- The classical economic order quantity, worked end to end on one item
- When to place the order, and the case where the obvious answer is wrong
- What a wrong order quantity costs, derived and then priced
- Why that cost licenses rounding, case quantities and delivery schedules
Same assumptions as last session: demand known and constant, replenishment instantaneous, no shortages permitted.
Where We Left Off
Last session the general cycle reduced to four classical models. Today we work inside the simplest. Its cost rate has three terms:
\[ \mathit{TC}(Q) = \underbrace{\frac{k\lambda}{Q}}_{\text{ordering}} + \underbrace{c\lambda}_{\text{purchase}} + \underbrace{\frac{hQ}{2}}_{\text{holding}} \]
\[ \begin{aligned} \frac{d\,\mathit{TC}}{dQ} &= -\frac{k\lambda}{Q^{2}} + \frac{h}{2} = 0 &&\text{the purchase term has no } Q \text{ in it}\\ Q^{*} &= \sqrt{\frac{2k\lambda}{h}} &&\text{solve for } Q\\ \mathit{TC}^{*} &= \sqrt{2k\lambda h} + c\lambda &&\text{substitute back} \end{aligned} \]
At \(Q^{*}\) the ordering and holding terms are equal. If they are not, \(Q\) is not optimal.
The Trade-Off, Drawn
\(\lambda = 2400\) per year, \(k = \$75\), \(h = \$4\).
Ordering falls steeply: a small order means many orders a year.
Holding rises in a straight line, since average stock is \(Q/2\).
They cross at \(Q^{*} = 300\), the equal-terms property made visible.
Notice that the total is flat near its minimum. That flatness is the second half of today.
The Item
A product is demanded at \(\lambda = 220\) units per year and costs \(c = \$1{,}200\) per unit. An order costs \(k = \$800\), and the carrying charge is \(i = 0.18\) per year. No shortages are allowed.
| Question | Answer |
|---|---|
| What is stocked? | One item, bought from an outside supplier |
| What is the demand process? | Constant at 220 units per year, known in advance |
| When is inventory reviewed? | Continuously |
| What triggers replenishment, and how much? | The shelf reaching zero; \(Q\) units, which is the decision |
| What happens to unmet demand? | None occurs; shortages are not allowed |
| What costs are incurred, and when? | \(k = \$800\) per order; \(i = 0.18\) per year on the value held |
The EOQ, Worked End to End
\[ \begin{aligned} h &= ic = (0.18)(1200) = \$216 \text{ per unit per year}\\[4pt] Q^{*} &= \sqrt{\frac{2k\lambda}{h}} = \sqrt{\frac{2(800)(220)}{216}} = \sqrt{1629.63} = 40.37 \text{ units}\\[4pt] T^{*} &= \frac{Q^{*}}{\lambda} = \frac{40.37}{220} = 0.1835 \text{ year} = 66.98 \text{ days}\\[4pt] \sqrt{2k\lambda h} &= \sqrt{2(800)(220)(216)} = \$8{,}719.63\\[4pt] \mathit{TC}^{*} &= \$8{,}719.63 + (1200)(220) = \$272{,}719.63 \end{aligned} \]
The check. The two relevant terms must be equal at \(Q^{*}\).
\[ \begin{aligned} \frac{k\lambda}{Q^{*}} &= \frac{(800)(220)}{40.37} = \$4{,}359.82\\[4pt] \frac{hQ^{*}}{2} &= \frac{(216)(40.37)}{2} = \$4{,}359.82 \end{aligned} \]
Equal, so \(Q^{*}\) is optimal. Perform this check every time; unequal halves mean \(Q\) is not optimal.
What the Decision Can Move
| Component | Amount | Share |
|---|---|---|
| Purchase, \(c\lambda\) | $264,000.00 | 96.8% |
| Ordering, \(k\lambda/Q^{*}\) | $4,359.82 | 1.6% |
| Holding, \(hQ^{*}/2\) | $4,359.82 | 1.6% |
| Total | $272,719.63 |
Notice that the purchase term is 97% of the total and is untouched by the decision: we buy 220 units a year whatever \(Q\) is.
The $8,719.63 is the entire quantity the lot sizing decision can influence. Comparing policies on total cost, with the purchase term in, makes every alternative look nearly identical.
When Do We Order?
With demand and lead time both known, the timing is exact: the order arrives at the instant stock would otherwise reach \(-\hat{B}\).
- Let \(L\) represent the replenishment lead time, known and constant.
- Let \(r\) represent the reorder point, the inventory position at which the order is placed.
When \(L \le T\), demand over the lead time is \(\lambda L\), so
\[ r = \lambda L - \hat{B}, \qquad\text{and with no backorders}\qquad r = \lambda L \]
For the item, with a seven day lead time: \(r = 220\left(\dfrac{7}{365}\right) = 4.22\) units. Seven days is well under the 67 day cycle.
\(r\) is a level of inventory position, not of stock on hand. A rule that triggers on the shelf alone orders again, and again, while the first order is still in transit.
A Lead Time Longer Than a Cycle
When \(L > T\), orders are already outstanding when the next one is placed. They cover whole cycles of the lead time, and only the remainder is uncovered:
\[ \begin{aligned} &\left\lfloor L/T \right\rfloor &&\text{whole cycles inside the lead time, each already ordered}\\ &L - \left\lfloor L/T \right\rfloor T &&\text{the part of the lead time not yet covered}\\ r &= \left(L - \left\lfloor \frac{L}{T} \right\rfloor T\right)\lambda - \hat{B} &&\text{demand over that part} \end{aligned} \]
The worksheet’s item. \(\lambda = 100\) per month, \(T^{*} = 1.5492\) months, and the supplier quotes \(L = 4\) months. Why can \(r = \lambda L = 400\) not be right?
It is more than twice the order quantity. Instead:
\[ \left\lfloor \frac{4}{1.5492} \right\rfloor = 2, \qquad 4 - 2(1.5492) = 0.9016 \text{ month}, \qquad r = 100(0.9016) = 90.16 \text{ units} \]
The Reorder Point Against the Lead Time
The item of today’s example, \(T^{*} = 66.98\) days.
| \(L\) (days) | \(\lfloor L/T \rfloor\) | \(r\) |
|---|---|---|
| 7 | 0 | 4.22 |
| 60 | 0 | 36.16 |
| 70 | 1 | 1.82 |
Notice that ten more days of lead time drops \(r\) from 36.16 to 1.82. The reorder point is not monotone in the lead time.
Where the Penalty Formula Comes From
No warehouse orders 40.37 of anything. How much is lost by not ordering \(Q^{*}\)? Let \(C(Q) = k\lambda/Q + hQ/2\) represent the relevant cost rate and \(C^{*} = C(Q^{*})\) its minimum.
\[ \begin{aligned} Q^{*2} &= \frac{2k\lambda}{h} &&\text{square the EOQ}\\ k\lambda &= \frac{hQ^{*2}}{2} &&\text{solve for } k\lambda\\ C(Q) &= \frac{hQ^{*2}}{2Q} + \frac{hQ}{2} = \frac{h}{2}\left(\frac{Q^{*2}}{Q} + Q\right) &&\text{substitute for } k\lambda\\ C^{*} &= \frac{h}{2}\left(Q^{*} + Q^{*}\right) = hQ^{*} &&\text{set } Q = Q^{*}\\ \frac{C(Q)}{C^{*}} &= \frac{1}{2Q^{*}}\left(\frac{Q^{*2}}{Q} + Q\right) = \frac{1}{2}\left(\frac{Q}{Q^{*}} + \frac{Q^{*}}{Q}\right) &&\text{divide} \end{aligned} \]
Notice that \(k\), \(\lambda\), \(h\) and \(c\) have all cancelled. The penalty depends only on the ratio \(Q/Q^{*}\).
Wrong by a Fifth, Wrong by Half
Order 50% too much, \(Q/Q^{*} = 3/2\):
\[ \frac{C}{C^{*}} = \frac{1}{2}\left(\frac{3}{2} + \frac{2}{3}\right) = \frac{1}{2}\left(\frac{9+4}{6}\right) = \frac{13}{12} = 1.083 \]
That is 8.3% of the relevant cost, about $727 a year for today’s item (\(0.0833 \times \$8{,}719.63\)).
The formula is symmetric in \(Q/Q^{*}\) and its reciprocal, so half and double cost the same.
| \(Q/Q^{*}\) | \(C/C^{*}\) | Penalty | Today’s item |
|---|---|---|---|
| 1/2 | 1.2500 | 25.0% | $2,179.91 |
| 4/5 | 1.0250 | 2.5% | $217.99 |
| 6/5 | 1.0167 | 1.7% | $145.33 |
| 3/2 | 1.0833 | 8.3% | $726.64 |
| 2 | 1.2500 | 25.0% | $2,179.91 |
Being wrong by half is expensive; being wrong by a fifth is not.
Rounding Today’s Item
\(Q^{*} = 40.37\) units, relevant cost $8,719.63 a year. Three ways a warehouse would actually order it:
| Order | \(Q\) | \(Q/Q^{*}\) | \(C/C^{*}\) | Extra per year |
|---|---|---|---|---|
| Three cases of 12 | 36 | 0.8918 | 1.0066 | $57.26 |
| Four cases of 12 | 48 | 1.1890 | 1.0150 | $131.03 |
| Every ten weeks on the truck, \(220(10/52)\) | 42.31 | 1.0480 | 1.0011 | $9.60 |
Notice the last row. Ordering every ten weeks costs 0.11%, nine dollars and sixty cents a year, to put the item on a truck that was leaving anyway.
What Flatness Licenses
All of these move \(Q\) off its optimum, and all of them are nearly free:
- round to a whole number, a case, a pallet;
- fit a delivery schedule you already run;
- put several items on one shared cycle;
- round a cycle length to a power of two.
And what it does not license
Flatness is a statement about the relevant cost, which excludes the purchase term.
Introduce a quantity discount and the purchase term stops being constant. It becomes the thing the decision moves, it is far larger than the relevant cost, and it is discontinuous at each price break.
What We Did Today
The EOQ worked end to end. \(h = ic\), then \(Q^{*}\), then \(T^{*}\), then \(\mathit{TC}^{*}\), then the equal-halves check.
The purchase term dominates the total and is irrelevant to the decision. 97% of $272,719.63 could not be moved by any choice of \(Q\).
The reorder point reads the position, not the shelf, and it is not monotone in the lead time. \(r = \lambda L\) is wrong past the first cycle.
The penalty depends only on \(Q/Q^{*}\). Wrong by a fifth costs about 2%; wrong by half costs 25%.
For Next Time
Read before next session: Quantity Discounts, and Performance Measures for the Cycle.
Today’s material was The Classical Economic Order Quantity, Lead Time and the Reorder Point and Sensitivity of the Order Quantity.
Next session:
- All-units discounts, where the whole order reprices at a break, and why the break quantities themselves become candidates
- Incremental discounts, where only the units above a break reprice, and why they encourage larger orders
- The service measures the general cycle delivers, which we get for nothing from the four segments
The flat cost curve does not survive a price schedule. Come prepared to stop trusting it.