The EOQ knew its demand and ordered forever.
The newsvendor did not know its demand, and ordered once.
Now: the item is stocked permanently, demand arrives at random forever, and a replenishment decision is available at every instant.
The newsvendor removed the assumption that demand is known and kept the assumption that there is one decision. This chapter removes the second one.
Let \(r\) represent the reorder point and let \(Q\) represent the order quantity.
Whenever the inventory position reaches or falls below \(r\), order \(Q\) units.
It is the policy most stock rooms actually run, under the name of a two-bin system or a min-max when the parameters are set by hand.
Three details of its operation matter later.
Recall that an inventory system carries more than one level.
Notice that \(\mathit{IN}\) can be negative and \(I\) and \(B\) cannot.
The comparison is against the position
A rule watching \(I(t)\) would order again when the shelf is empty and an order is in transit, and again at the next demand, until the first delivery arrived.
Thus, it would order the same shortage several times over. The position counts what is already coming, so an order placed is an order not placed again.
The utility storeroom carries a pad-mount transformer. Demand runs at \(\lambda = 45\) units a year, one unit at a time. The supplier quotes two months, which the ledger carries as eight weeks. The storekeeper runs \(r = 4\), \(Q = 5\), and starts with 6 on the shelf, none on order, nothing owed.
| Question | Answer |
|---|---|
| What is stocked? | One item, the transformer |
| What is the demand process? | Random arrivals of one unit each, about 45 a year |
| When is inventory reviewed? | Continuously; the position is known after every transaction |
| What triggers replenishment, and how much? | The position reaching \(r = 4\) or below; order \(Q = 5\) |
| What happens to unmet demand? | Backordered; a utility crew waits for the transformer |
| What costs are incurred, and when? | Not needed to run the ledger; they come at the end of today |
| Week | Event | \(I\) | \(\mathit{IO}\) | \(B\) | \(\mathit{IN}\) | \(\mathit{IP}\) | What happened |
|---|---|---|---|---|---|---|---|
| 0 | start | 6 | 0 | 0 | 6 | 6 | Six on the shelf, nothing on order or owed |
| 1 | demand | 5 | 0 | 0 | 5 | 5 | Filled; \(5 > 4\), no order |
| 2 | demand, order 5 | 4 | 5 | 0 | 4 | 9 | \(\mathit{IP}\) reaches \(r\); order arrives week 10; \(\mathit{IP} = 4 + 5\) |
| 3 | demand | 3 | 5 | 0 | 3 | 8 | Filled from the shelf |
| 5 | demand | 2 | 5 | 0 | 2 | 7 | Filled from the shelf |
| 6 | demand | 1 | 5 | 0 | 1 | 6 | Filled from the shelf |
| 7 | demand | 0 | 5 | 0 | 0 | 5 | Shelf empty. Order? No: \(\mathit{IP} = 5 > 4\) |
| 8 | backordered, order 5 | 0 | 10 | 1 | \(-1\) | 9 | Unfilled, yet \(\mathit{IP}\) falls to 4; arrives week 16 |
| 9 | backordered | 0 | 10 | 2 | \(-2\) | 8 | Two owed |
| 10 | receipt of 5 | 3 | 5 | 0 | 3 | 8 | Backorders filled first; \(\mathit{IP}\) unchanged |
| 11 | demand | 2 | 5 | 0 | 2 | 7 | Filled from the shelf |
| 13 | demand | 1 | 5 | 0 | 1 | 6 | Filled from the shelf |
Week 7. The shelf is empty and the policy does not order. The position knows five transformers are coming, and the shelf does not.
Week 8. A demand that cannot be filled still lowers the position. If a system decrements the position only when it ships, it under-orders when it is already short.
Week 10. A delivery does not change the inventory position. It converts units on order into units on hand, and the position counts both.
The check. Every row must satisfy \(\mathit{IP}(t) = I(t) + \mathit{IO}(t) - B(t)\).
Week 8 is the first row on which a backorder enters the sum:
\[ 0 + 10 - 1 = 9 \]
That check is worth performing on every ledger, because it catches an order recorded in the wrong column immediately.
A band of width \(Q\). The position falls one unit at a time from just above \(r + Q\) down to \(r\), and an order restores it. In the ledger it ran over 5, 6, 7, 8, 9, which is \(r + 1\) through \(r + Q\).
One exposure. Everything that can go wrong happens during a lead time. Between placing an order and receiving it, nothing the storekeeper does changes the outcome. Thus, the only random quantity the policy is exposed to is the demand over that interval.
Two questions. The shelf and the position answer different questions. The position decides when to order. The shelf decides whether a customer is served. A model that confuses them will get the ordering right and the service wrong, or the reverse.
A policy is judged by long-run time averages of the ledger’s columns. Over the 13 weeks of the ledger they are:
\[ \begin{aligned} \bar{I} &\approx \frac{6+5+4+3(2)+2+1+0(3)+3+2(2)}{13} = \frac{31}{13} = 2.38 &&\text{area under } I(t) \text{, over 13 weeks}\\[2pt] \bar{B} &\approx \frac{1(1) + 2(1)}{13} = \frac{3}{13} = 0.23 &&\text{one owed in week 8, two in week 9}\\[2pt] \overline{\mathit{SO}} &\approx \frac{3}{13} = 0.23 &&\text{shelf empty from week 7 to week 10}\\[2pt] \overline{\mathit{OF}} &= \frac{\lambda}{Q} = \frac{45}{5} = 9 \text{ orders a year} &&\text{no analysis needed} \end{aligned} \]
Each term of \(\bar{I}\) is a level times the weeks it was held. The level 3 was held two weeks (3 to 5), the level 0 three weeks (7 to 10), and the level 2 at the end two weeks (11 to 13).
Every order is for \(Q\) units and all demand is eventually filled, so orders are placed at the rate demand consumes them. Over 13 weeks that is \(9 \times 13/52 = 2.25\) orders; the ledger placed 2.
Notice that (a) counts cycles, (b) counts demands, (c) counts time, and (d) and (e) are not proportions at all.
An organization that has agreed on a number without agreeing on which of the five it measures has agreed on nothing.
The rest of today takes the two pairs most often confused: (b) against (c), and (d) against (e). Measures (a) and (b) come back as targets a planner sets.
The ready rate \(\overline{\mathit{RR}} = 1 - \overline{\mathit{SO}}\) is the proportion of time stock is on the shelf, measure (c).
The fill rate \(\overline{\mathit{FR}}\) is the proportion of demands filled from the shelf on arrival, measure (b).
On the ledger they differ:
\[ \overline{\mathit{RR}} \approx 1 - \tfrac{3}{13} = 0.77, \qquad \overline{\mathit{FR}} \approx \tfrac{8}{10} = 0.80 \]
Recall that PASTA says a Poisson arrival finds the system in its time-stationary state. Thus, for Poisson demand
\[ \overline{\mathit{FR}} = \overline{\mathit{RR}} = 1 - \overline{\mathit{SO}} \]
Under any other arrival process this fails in the direction that matters. Demand in bursts is more likely to find the shelf empty than a random instant is, because the burst that emptied the shelf is still arriving.
Little’s Law, with the backordered demands as the system:
\[ \overline{W} = \frac{\bar{B}}{\lambda} \]
On the ledger, the area under \(B(t)\) is 3 unit-weeks, and the two backordered demands waited 2 weeks and 1 week. The area is the total wait, so
\[ \overline{W} \approx \frac{3/13}{10/13} = 0.3 \text{ weeks per demand} \]
For \(\bar{B} = 0.12\) units at \(\lambda = 45\) a year:
\[ \begin{aligned} \overline{W} &= \frac{0.12}{45} = 0.00267 \text{ years}\\ &= 0.00267 \text{ years} \times \frac{365 \text{ days}}{1 \text{ year}} = 0.97 \text{ days} \end{aligned} \]
Does a crew that waits, wait about a day?
No. The average is over all demands. Most wait no time at all, and the few that wait, wait much longer than a day.
Let \(C(r, Q)\) represent the relevant cost rate, with \(k\) the cost per order, \(h\) the holding cost per unit per year, and \(b\) the backorder cost per unit per year:
\[ C(r, Q) = k\,\frac{\lambda}{Q} + h\,\bar{I} + b\,\bar{B} \]
| Term | Units |
|---|---|
| \(k\lambda/Q\) | dollars per order \(\times\) orders per year |
| \(h\bar{I}\) | dollars per unit per year \(\times\) units |
| \(b\bar{B}\) | dollars per unit per year \(\times\) units |
Notice the units of the last term. \(b\) charges for how long a demand waits, not merely that it waited. A model that charges a flat \(\pi\) per unit short is a different model, and the classical approximation uses it.
For the transformer, \(h = 0.25(3{,}800) = \$950\) per unit per year, and the utility judges \(b = \$8{,}550\) per unit per year in rescheduled work and reliability penalties.
\(b\) is the parameter no ledger contains. The usual response is to drop it and impose a service target instead.
Imposing a service target does not remove \(b\) from the model. A service target is not an escape from \(b\); it is a statement about \(b\) made without saying so.
Recall from the multi-item chapter that a firm’s observed order quantities imply a ratio of its own cost parameters whether or not the firm ever estimated them. The same argument runs here, and later in the chapter we recover the \(b\) that a service target implies and ask whether anyone believes it.
The \((r, Q)\) policy compares \(r\) against the position, not the shelf. The position counts what is already coming, so an order placed is not placed again.
The ledger shows it. At week 7 the shelf was empty and nothing was ordered; at week 8 an unfilled demand triggered an order; at week 10 a delivery left the position unchanged.
A policy is judged by time averages of the ledger’s columns. Only the order frequency, \(\lambda/Q\), needs no analysis.
The ready rate counts time and the fill rate counts demands. Poisson demand makes them equal; under any other arrival process they can come apart.
Backorders and waiting time are one target in two units. Little’s Law, with no distributional assumption.
Read before next session: Lead Time Demand and The Base-Stock Model.
Today’s material was One Item, Watched Continuously and What to Measure.
Next session:
Bring the newsvendor’s critical ratio with you. You will use it on the transformer.