Appendix A — Notation
A single reference for the symbols used throughout the book. Each entry gives the symbol, what it denotes, its unit of measure, and where it is defined.
A.1 Conventions
Time. The models are written in a generic time unit. Whether that unit is a day, a week, or a year is a matter of convention; only two rules matter. Every rate in a model must use the same unit, and every price quoted per period must be converted to that unit before it enters the model. Carrying charges and positioning costs are quoted per year by universal accounting convention, so a model running in days divides them by 365 on entry.
Time averages. An overbar denotes a time average of a time-persistent variable:
\[ \overline{X} \;=\; \frac{1}{T}\int_{0}^{T} X(t)\,\mathrm{d}t \]
Under the conditions discussed in Section 1.3, these averages converge as \(T \to \infty\) to the corresponding steady-state quantities. An overbar on a quantity indexed by demand transaction rather than by time denotes an average over transactions, and the definition is given where that quantity is introduced.
Functions of time. A symbol written \(X(t)\) is a quantity that varies over time and can, in principle, be observed at any instant. A symbol without the argument denotes either a parameter or the corresponding average.
Positive and negative parts. For a real quantity \(x\), \([x]^{+} = \max(x, 0)\) and \([x]^{-} = \max(-x, 0)\).
A.2 State Variables
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(I(t)\) | Inventory on hand | units | Section 1.3 |
| \(\mathit{IO}(t)\) | Inventory on order | units | Section 1.3 |
| \(B(t)\) | Units backordered | units | Section 1.3 |
| \(I_{a}(t)\) | Inventory actually on hand | units | Section 2.3.2 |
| \(I_{r}(t)\) | Inventory on hand according to the record | units | Section 2.3.2 |
| \(D(t)\) | Record discrepancy, \(I_{a}(t) - I_{r}(t)\) | units | Section 2.3.2 |
| \(\mathit{IN}(t)\) | Net inventory, \(I(t) - B(t)\) | units | Section 1.3 |
| \(\mathit{IP}(t)\) | Inventory position, \(I(t) + \mathit{IO}(t) - B(t)\) | units | Section 1.3 |
| \(\mathit{SO}(t)\) | Stockout indicator, \(1\) when \(I(t) \le 0\) | dimensionless | Section 1.3 |
| \(\mathit{RR}(t)\) | Ready indicator, \(1 - \mathit{SO}(t)\) | dimensionless | Section 1.3 |
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(\bar{I}\) | Average inventory on hand | units | Section 1.3 |
| \(\bar{B}\) | Average units backordered | units | Section 1.3 |
| \(\overline{N}\) | Average order rate | orders per unit time | Section 1.6.1 |
| \(\overline{\mathit{IO}}\) | Average inventory on order | units | Section 1.3 |
| \(\overline{\mathit{IN}}\) | Average net inventory | units | Section 1.3 |
| \(\overline{\mathit{IP}}\) | Average inventory position | units | Section 1.3 |
| \(\overline{\mathit{SO}}\) | Proportion of time out of stock | dimensionless | Section 1.3 |
A.3 Demand and Time
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(\lambda\) | Demand rate | units per unit time | Section 1.3 |
| \(D_i\) | Size of the \(i\)th demand transaction | units | Section 1.3 |
| \(n\) | Number of demand transactions observed | transactions | Section 1.3 |
| \(L\) | Replenishment lead time | time | Section 1.3 |
| \(p\) | Production or replenishment rate | units per unit time | Chapter 3 |
| \(T_{p}\) | Time to replenish, \(Q/p\) | time | Chapter 3 |
| \(\overline{L}\) | Average lead time | time | Section 1.3 |
| \(\overline{W}\) | Average time a customer waits | time | Section 1.3 |
| \(\overline{T}_{s}\) | Average time a unit spends on hand | time | Section 1.3 |
| \(\mathit{TO}\) | Inventory turnover ratio, \(\lambda/\bar{I}\) | per unit time | Section 1.3 |
| \(\lambda_{\ell}\) | Rate of lost demand | units per unit time | Section 1.3 |
| \(T\) | Length of the observation interval | time | Section 1.3 |
| \(t_{F}\) | Basic forecast period, the interval a forecasting system works in | time | Section 8.15 |
| \(\hat{D}_{t}\) | Forecast of \(D_{t}\), the demand in one forecast period, revised at the end of period \(t\) | units | Equation 8.79 |
| \(\mathit{MAD}_{t}\) | Smoothed mean absolute deviation of the one period ahead forecast error | units | Equation 8.80 |
| \(\eta\) | Scaling exponent carrying a period standard deviation over a lead time, \(\tfrac{1}{2} \le \eta \le 1\) | dimensionless | Equation 8.81 |
A.4 Cost Parameters
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(c\) | Unit cost | dollars per unit | Section 1.4.3 |
| \(i\) | Carrying charge | dollars per dollar of value per unit time | Section 1.4.4 |
| \(h\) | Holding cost, \(h = ic\) | dollars per unit per unit time | Section 1.4.4 |
| \(k\) | Fixed order cost | dollars per order | Section 1.4.5 |
| \(b\) | Backorder cost | dollars per unit per unit time | Section 1.4.6 |
| \(\pi\) | Stockout cost | dollars per unit | Section 1.4.6 |
| \(f\) | Stocking position cost | dollars per item per location per unit time | Section 1.4.7 |
| \(\gamma\) | Carrying charge taken common to every item in an aggregate development | dollars per dollar of value per unit time | Section 4.4 |
| \(K\) | Major setup cost, incurred once per order regardless of its contents | dollars per order | Section 4.5 |
A.5 Performance Measures
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(\overline{\mathit{RR}}\) | Ready rate, the proportion of time stock is on hand | dimensionless | Section 1.6 |
| \(\overline{\mathit{FR}}\) | Transaction fill rate | dimensionless | Section 1.6 |
| \(\overline{\mathit{FR}}_{u}\) | Unit fill rate | dimensionless | Section 1.6 |
| \(Y_i\) | Indicator that the \(i\)th demand is filled immediately | dimensionless | Section 1.6 |
| \(\mathit{CSL}\) | Cycle service level | dimensionless | Section 1.6 |
| \(\mathit{TC}\) | Total cost rate, Equation 1.15 | dollars per unit time | Section 1.6.1 |
| \(C\) | Relevant cost rate, the part of \(\mathit{TC}\) that varies with the policy, Equation 1.16 | dollars per unit time | Section 1.6.1 |
| \(\alpha\) | Type 1 service target, the cycle service level a policy is set to reach | dimensionless | Section 8.7 |
| \(\beta\) | Type 2 service target, the fill rate a policy is set to reach | dimensionless | Section 8.7 |
| \(\pi^{*}\) | Implied stockout cost, the \(\pi\) at which the cost-minimizing policy meets a stated target | dollars per unit | Section 8.7.6 |
A.6 Policy Parameters
These are the decision variables. Each is introduced with the policy that uses it; they are collected here for reference.
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(Q\) | Order quantity | units | Chapter 3 |
| \(r\) | Reorder point | units | Chapter 8 |
| \(s\) | Reorder point in an \((s, S)\) policy | units | Chapter 8 |
| \(S\) | Order-up-to level, and the base-stock level when \(Q = 1\) | units | Section 8.4 |
| \(R\) | Review interval | time | Section 8.13 |
A.7 Demand Classification and Record Quantities
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(\mathit{ADI}\) | Average demand interval, periods per period with a demand | periods | Section 2.2.6 |
| \(\mathit{CV}^{2}\) | Squared coefficient of variation of the non-zero demands, with the sample standard deviation | dimensionless | Section 2.2.6 |
| \(p\) | Probability that a verified record becomes wrong in one period. Not the replenishment rate of Table A.3 | per period | Section 2.3.7 |
| \(n\), \(A(n)\) | Periods between counts, and the average record accuracy over that count cycle | periods, dimensionless | Section 2.3.7 |
| \(m\), \(e\), \(\hat{a}\) | Records to count in a sample, the margin of error, and a prior estimate of accuracy | records, dimensionless, dimensionless | Section 2.3.4 |
A.8 Deterministic Lot Sizing Quantities
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(\hat{I}\), \(\hat{B}\) | Peak on-hand level and maximum backorder level in a cycle | units | Section 3.1 |
| \(T_1, \ldots, T_4\) | Lengths of the four segments of a cycle, summing to \(T = Q/\lambda\) | time | Section 3.1.2 |
| \(m\) | Number of price levels in a discount schedule | levels | Section 3.8.1 |
| \(q_j\), \(c_j\) | Break point beginning level \(j\), and the unit price on that level | units, dollars per unit | Section 3.8.1 |
| \(R_j\) | Under an incremental schedule, the cost of filling every interval below level \(j\) | dollars | Section 3.8.2 |
| \(k_j\) | Effective ordering cost on level \(j\) of an incremental schedule, \(k + R_j - c_j q_j\) | dollars per order | Section 3.8.2 |
| \(H_j\) | The fixed component \(R_j - c_j q_j\), as a worksheet carries it | dollars | Section 3.10.4 |
A.9 Single Period Quantities
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(X\), \(F(x)\) | Demand over the single period, and its distribution function \(P\{X \le x\}\) | units | Section 7.5 |
| \(c_{o}\) | Overage cost, the loss per unit left over, \(c - u\) | dollars per unit | Section 7.5 |
| \(c_{u}\) | Underage cost, the loss per unit of unmet demand, \(s - c\) | dollars per unit | Section 7.5 |
| \(s\), \(u\) | Selling price and salvage value per unit. Not the \(s\) of an \((s, S)\) policy | dollars per unit | Section 7.5.1 |
| \(Q^{*}\), \(S^{*}\) | Optimal order quantity, and the order-up-to level when stock is already on hand | units | Section 7.5 |
| \(c_{u}/(c_{u}+c_{o})\) | The critical ratio, the probability of no shortage at the optimum | dimensionless | Section 7.5 |
A.10 Continuous and Periodic Review Quantities
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(\theta\) | \(E[D(L)]\), the mean lead time demand. Not the multiplier of Table A.11 | units | Section 8.3 |
| \(\sigma_{D(L)}\) | Standard deviation of lead time demand | units | Section 8.3 |
| \(w\) | \(b/(b+h)\), the critical ratio the base-stock level is read against | dimensionless | Section 8.4 |
| \(C(s)\) | Cost rate of a base-stock policy at level \(s\), holding plus backorder | dollars per unit time | Section 8.4 |
| \(\mathit{ss}\) | Safety stock, \(r - \theta\) | units | Section 8.7 |
| \(z\) | Safety factor, safety stock in standard deviations of lead time demand | dimensionless | Section 8.7 |
| \(Q_{k}\), \(C_{k}\) | Order quantity and cost of the deterministic model with backorders, the lower bounds of the search | units, dollars per unit time | Section 8.9 |
| \(C_{\infty}\) | \(\sqrt{bh}\,\sigma_{D(L)}\), a reference cost built from lead time demand alone | dollars per unit time | Section 8.9 |
| \(Y\), \(\lambda_{e}\) | Size of a demand lot, and the rate at which lots arrive | units, lots per unit time | Section 8.11 |
| \(U\) | Undershoot, the amount by which a demand carries the position below \(s\) | units | Section 8.12 |
| \(\tau\) | \(R + L\), the protection interval a periodic policy must cover | time | Section 8.13 |
| \(\bar{G}\), \(\bar{G}^{1}\) | Distribution and first order loss function of \(D(L+U)\), averaged over a review interval | Section 8.13.2 |
A.11 Multi-Item Quantities
Items in a multi-item problem are indexed by \(j\), and any symbol of the previous tables carries that subscript when it refers to one item. In Chapter 4 the letter \(n\) counts items rather than demand transactions.
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(n\) | Number of items sharing the resource or the setup; in Section 4.6 it counts stages or regional warehouses instead | items, or locations | Chapter 4 |
| \(a_j\) | Amount of the shared resource one unit of item \(j\) consumes | resource units per unit | Section 4.2 |
| \(A\) | Amount of the shared resource available | resource units | Section 4.2 |
| \(\theta\) | Lagrange multiplier on the shared-resource constraint. See Section A.18 | varies with the resource | Section 4.2.2 |
| \(R(\theta)\) | Resource function, the amount consumed at \(\theta\) less the amount available | resource units | Section 4.2.3 |
| \(v_j\) | Volume one unit of item \(j\) occupies, the space-constraint case of \(a_j\) | cubic units per unit | Section 4.2 |
| \(\overline{\mathit{OF}}\) | Order frequency for one item, \(\lambda/Q\) | orders per unit time | Section 4.4.1 |
| \(\bar{I}^{a}\) | Aggregate investment in cycle stock | dollars | Section 4.4.2 |
| \(N^{a}\) | Aggregate replenishments across all items | orders per unit time | Section 4.4.2 |
| \(\overline{\mathit{OC}}^{a}\) | Aggregate ordering cost rate | dollars per unit time | Section 4.4.4 |
| \(J^{*}\) | Variety index, the effective number of items | items | Section 4.4.4 |
| \(C^{a}\) | Aggregate purchase cost rate, \(\sum_j c_j\lambda_j\) | dollars per unit time | Section 4.4.4 |
| \(\lambda^{a}\) | Aggregate demand rate, \(\sum_j \lambda_j\) | units per unit time | Section 4.4.4 |
| \(c^{w}\) | Demand-weighted average unit cost | dollars per unit | Section 4.4.4 |
| \(K^{w}\) | Purchase-weighted average ordering cost | dollars per order | Section 4.4.4 |
| \(K\) | Major setup cost, incurred once whenever an order is placed at all | dollars per order | Section 4.5 |
| \(k_j\) | Minor setup cost, incurred for each item included on that order | dollars per order | Section 4.5 |
| \(T\) | Base period between order opportunities | time | Section 4.5.1 |
| \(m_j\) | Number of base periods between replenishments of item \(j\) | dimensionless | Section 4.5.1 |
| \(\ell_j\) | Exponent in a power-of-two multiplier, \(m_j = 2^{\ell_j}\); the subscript is dropped while following one item | dimensionless | Section 4.5.2 |
| \(C(T_j)\) | Relevant cost rate written as a function of the reorder interval | dollars per unit time | Section 4.5.3 |
| \(g\) | Holding cost coefficient in the interval form, \(h\lambda/2\) | dollars per unit time squared | Section 4.5.3 |
A.12 Multi-Echelon Quantities
Stages in a serial system are numbered so that stage \(i\) supplies stage \(i-1\), with stage 1 facing the customer. In a distribution system the central warehouse is indexed 0.
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(h_i\) | Installation holding cost rate at stage \(i\), the total value accumulated through that stage | dollars per unit per unit time | Section 4.6.1 |
| \(h'_i\) | Echelon holding cost rate, the value added at stage \(i\) alone. In a serial chain \(h'_i = h_i - h_{i+1}\); in a distribution system \(h'_i = h_i - h_0\) | dollars per unit per unit time | Section 4.6.1 |
| \(k_i\) | Fixed ordering cost at stage \(i\). There is no major setup in these sections, so this is the same \(k\) as Section 1.4.5 | dollars per order | Section 4.6.2 |
| \(g_i\) | The echelon holding coefficient at stage \(i\): \(\tfrac{1}{2}\lambda h'_i\) in a serial chain, where every stage sees the same rate, and \(\tfrac{1}{2}\lambda_i h'_i\) in a distribution system | dollars per unit time squared | Section 4.6.2 |
| \(T_i\) | Reorder interval at stage \(i\) | time | Section 4.6.2 |
| \(\bar{I}_i\) | Average on-hand stock at stage \(i\) | units | Section 4.6.1 |
| \(\bar{I}^{\,e}_i\) | Average echelon stock at stage \(i\) | units | Section 4.6.1 |
| \(G_r\) | The \(r\)th block of consecutive stages sharing one reorder interval | Section 4.6.2 | |
| \(M\) | Number of blocks in the partition | blocks | Section 4.6.2 |
| \(k(G_r)\), \(g(G_r)\) | Block totals, \(\sum_{i \in G_r}k_i\) and \(\sum_{i \in G_r}g_i\) | Section 4.6.2 | |
| \(T(r)\) | The interval shared by every stage in block \(G_r\) | time | Section 4.6.2 |
| \(\mathcal{C}^{0}\) | The pinned set: the central warehouse together with the regions sharing its interval. Set in a calligraphic capital, to keep it apart from the cost rates written \(C\) | Section 4.6.3 |
A.13 Repairable Item Quantities
Chapter 9 holds a fixed population of repairable units circulating between a depot and the storerooms it supports. The depot is indexed 0 and the storerooms \(j = 1, \ldots, J\), which continues the convention of Section A.12. This book says storeroom where the repairable literature says base, because base-stock is already the name of a policy; see Section 9.2.2.
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(J\) | Number of storerooms one depot supports | locations | Section 9.2.3 |
| \(\lambda_j\) | Failure rate at storeroom \(j\), assumed Poisson. A failure is simultaneously a demand and a carcass | units per unit time | Section 9.2.3 |
| \(\phi_j\) | Fraction of failures at storeroom \(j\) that storeroom \(j\) repairs itself. Written \(r_I\) in Sherbrooke (1986), which collides with the reorder point | Section 9.2.3 | |
| \(T_j\) | Average repair time at storeroom \(j\), for the carcasses it keeps | time | Section 9.2.3 |
| \(T_0\) | Average repair time at the depot | time | Section 9.2.3 |
| \(O_j\) | Average order and ship time, depot to storeroom \(j\), when the depot has stock | time | Section 9.2.3 |
| \(\lambda_{j0}\) | \((1-\phi_j)\lambda_j\), the rate at which storeroom \(j\) asks the depot for units | units per unit time | Section 9.2.3 |
| \(\lambda_0\) | \(\sum_j \lambda_{j0}\), the depot’s demand rate. Poisson, by thinning and superposition | units per unit time | Section 9.2.3 |
| \(S_j\), \(S_0\) | Stock level at storeroom \(j\) and at the depot, the units held when nothing is in resupply | units | Section 9.2.3 |
| \(X_j\), \(X_0\) | Units in resupply for storeroom \(j\), and units on the depot’s benches | units | Section 9.3 |
| \(\mu_j\), \(\mu_0\) | \(E[X_j]\) and \(E[X_0]\). Both are exact; only the distribution of \(X_j\) is approximated | units | Section 9.3.3 |
| \(\sigma_j^{2}\) | \(\mathit{Var}[X_j]\). METRIC sets this equal to \(\mu_j\); VARI-METRIC does not | units squared | Section 9.8.2 |
| \(W_j\), \(\bar{W}\) | Wait a request suffers at the depot, and its mean. The mean is the same for every storeroom | time | Section 9.6.2 |
| \(B_{0j}\) | Units the depot owes storeroom \(j\). Binomial, conditional on the depot’s total | units | Section 9.8.1 |
| \(f_j\) | \(\lambda_{j0}/\lambda_0\), storeroom \(j\)’s share of the depot’s demand, and the binomial’s success probability | Section 9.8.1 | |
| \(E[B^{2}(s)]\) | Second moment of the backorder function at stock level \(s\). Not a loss function, though Equation 9.25 builds it from two | units squared | Section 9.8.3 |
| \(\mathit{EBO}\) | Total expected backorders at the storerooms, the quantity Section 9.9 minimizes | units | Section 9.9.1 |
| \(\delta(S)\) | Delta value: backorders removed per dollar by the next unit | units per dollar | Section 9.9.2 |
A.14 Dynamic Lot Sizing Quantities
Chapter 5 plans one item over a finite horizon of \(N\) periods indexed \(t = 1, \ldots, N\). Every quantity below carries a period subscript, and the subscript is dropped in the many cases where the costs hold still. Notice that \(t\) is a period number here and a point in continuous time elsewhere in the book.
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(N\) | Number of periods in the planning horizon | periods | Section 5.1.1 |
| \(d_t\) | Requirement in period \(t\). This is the discrete counterpart of the rate \(\lambda\), and it replaces it for the whole of Chapter 5 | units | Section 5.1.1 |
| \(d[t,u]\) | Requirement of periods \(t\) through \(u\) inclusive, and zero when \(u < t\) | units | Section 5.1.1 |
| \(\bar{d}\) | Average requirement per period over the horizon | units per period | Section 5.6.1 |
| \(k_t\) | Order cost in period \(t\). The same \(k\) as Section 1.4.5, given a period subscript | dollars per order | Section 5.1.1 |
| \(c_t\) | Unit purchase cost in period \(t\) | dollars per unit | Section 5.1.1 |
| \(h_t\) | Cost of carrying one unit from the end of period \(t\) into period \(t+1\) | dollars per unit per period | Section 5.1.1 |
| \(Q_t\) | Order quantity received in period \(t\) | units | Section 5.2 |
| \(I_t\) | Inventory at the end of period \(t\), with \(I_0 = 0\). Unlike the \(I(t)\) of Section A.2 this is a period-end balance and not a continuous level | units | Section 5.2 |
| \(y_t\) | One when an order is placed in period \(t\), zero otherwise | Section 5.2 | |
| \(\hat{c}(t,u)\) | Cost of one unit bought in period \(t\) and used in period \(u\), which is \(c_t\) plus the holding rates of periods \(t\) through \(u-1\) | dollars per unit | Section 5.3.2 |
| \(w(t,u)\) | Window cost: the cost of a replenishment placed in period \(t\) that covers the requirements of periods \(t\) through \(u\) | dollars | Section 5.3.2 |
| \(V(u)\) | Least cost of covering periods 1 through \(u\) and arriving at period \(u+1\) empty, with \(V(0) = 0\) | dollars | Section 5.4.2 |
| \(S(u)\) | The period in which the last order of that least-cost plan starts | period | Section 5.4.2 |
| \(M\) | The large constant in the linking constraint of the mixed integer program. Not the block count of Table A.12 | units | Section 5.4.3 |
| \(VC\) | Variability coefficient, the variance of the requirements over the square of their mean | Section 5.6.1 | |
| \(W\) | Length of the planning window under a rolling horizon | periods | Section 5.7.1 |
Relevant cost is written \(TRC\) throughout Chapter 5 and means setup plus carrying, with the purchase term left out. Section 5.2.1 gives the one case in which it cannot be left out.
A.15 Requirements Planning Quantities
Chapter 6 plans every item of a product structure over a common horizon of periods. Three of its quantities are the same objects met earlier under other names, and the table says so where that is the case.
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(G_t\) | Gross requirement in period \(t\): what the item is needed for, from a master schedule for an end item and from Equation 6.1 for everything else | units | Section 6.3 |
| \(\mathit{SR}_t\) | Scheduled receipt in period \(t\), an order already placed in a previous planning cycle. Not the \(S(u)\) of Table A.14, which is a Wagner-Whitin pointer | units | Section 6.3 |
| \(I_t\) | Projected on hand at the end of period \(t\). Deliberately the same symbol as Table A.14, because it is the same quantity: Equation 6.2 is Equation 5.1 with the scheduled receipts added | units | Section 6.3 |
| \(\mathit{NR}_t\) | Net requirement in period \(t\), floored at zero. Not the \(N\) of Table A.14, which is a horizon length | units | Section 6.3 |
| \(\mathit{POR}_t\) | Planned order receipt in period \(t\): what the lot sizing rule decides should arrive | units | Section 6.3 |
| \(\mathit{Rel}_t\) | Planned order release in period \(t\), the receipt moved back by the lead time. This row is the record’s output and the next level’s input | units | Section 6.3 |
| \(L\) | Lead time, in periods. The same \(L\) as Table A.1 | periods | Section 6.3 |
| \(\mathit{SS}\) | Safety stock, added to the gross requirement before netting | units | Section 6.3.2 |
| \(q_{ij}\) | Quantity per: units of item \(j\) in one unit of item \(i\). Equal to one throughout a distribution network | Section 6.2.2 |
The cumulative lead time is the sum of \(L\) down the longest path of a product structure. Section 6.3.3 shows why a horizon shorter than it cannot produce a plan.
A.16 Distribution and Loss Function Quantities
From Chapter 7 onward demand is a random variable, and the cost expressions need the expected amount by which it exceeds a stock level rather than its plain expectation. Section C.3 derives these functions and tabulates them for the distributions this book uses.
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(X\) | A random variable, usually a demand over a stated interval | units | Section C.3 |
| \((X-b)^{+}\) | \(\max(X-b, 0)\), the amount by which \(X\) exceeds \(b\) | units | Section C.3 |
| \(g(x)\) | Probability mass function when \(X\) is discrete, probability density function when it is continuous | Section C.3 | |
| \(G(b)\) | \(P\{X \le b\}\), the cumulative distribution function | Section C.3 | |
| \(G^{0}(b)\) | \(1 - G(b)\). For a discrete \(X\) this is \(P\{X > b\}\) and not \(P\{X \ge b\}\) | Equation C.9 | |
| \(G^{1}(b)\) | First order loss function, \(E[(X-b)^{+}]\): the expected shortage against a stock level of \(b\) | units | Equation C.9 |
| \(G^{2}(b)\) | Second order loss function. Defined with a factor of \(\tfrac{1}{2}\) and differently for discrete and continuous \(X\) | units squared | Equation C.10 |
| \(G^{3}(b)\) | Third order loss function. Needed for the variance of a shortage when the stock level is itself random | units cubed | Equation C.22 |
| \(H^{1}(b)\), \(H^{2}(b)\), \(H^{3}(b)\) | Partial expectations, the integrals of \(xg(x)\), \(x^{2}g(x)\) and \(x^{3}g(x)\) above \(b\) | units, units squared, units cubed | Equation C.17 |
| \(c_{X}\) | Coefficient of variation, \(\sigma_{X}/E[X]\) | dimensionless | Section B.1.1 |
| \(\mathit{VMR}\) | Variance to mean ratio, \(\mathit{Var}[X]/E[X]\). The diagnostic that selects a discrete family | units | Equation C.1 |
| \(S\) | A random sum, \(\sum_{i=1}^{N}X_{i}\), of a random number of random quantities | units | Equation B.26 |
| \(N\), \(N(t)\) | The number of terms in a random sum. \(N(t)\) counts demand occurrences in an interval of length \(t\) | Section B.5 | |
| \(D(L)\) | Demand over a lead time, the random sum \(\sum_{i=1}^{L}D_{i}\). Written \(\mathit{LTD}\) in much of the literature | units | Equation B.30 |
| \(\phi(z)\), \(\Phi(z)\) | Standard normal density and distribution function | Section C.3.4 | |
| \(\Phi^{0}(z)\), \(\Phi^{1}(z)\), \(\Phi^{2}(z)\), \(\Phi^{3}(z)\) | The standard normal complement and loss functions, all dimensionless. Scaled by \(\sigma\), \(\sigma^{2}\) and \(\sigma^{3}\) to give \(G^{1}\), \(G^{2}\) and \(G^{3}\) | Equation C.19 |
Notice the superscripts. They are orders of a single family and not powers: \(G^{0}\) is a probability, \(G^{1}\) is in units, \(G^{2}\) is in units squared and \(G^{3}\) is in units cubed, each obtained by integrating or summing the one before it. Thus, the unit of \(G^{n}\) is the unit of the demand raised to the \(n\)th power, which is the quickest check that a loss function has been scaled correctly.
A.17 Stochastic Process Quantities
Section B.3 and Section B.4 model demand as a stream of events in time rather than as a quantity, and Chapter 8 works with both views. The symbols below belong to the stream.
| Symbol | Meaning | Unit | Defined in |
|---|---|---|---|
| \(N(t)\) | Number of demand events in an interval of length \(t\) | events | Equation B.15 |
| \(T_{i}\) | Time between the \((i-1)\)st and \(i\)th events | time | Section B.3.1 |
| \(\mu_{T}\), \(\sigma_{T}^{2}\) | Mean and variance of the time between events | time, time squared | Section B.4 |
| \(m(t)\) | Renewal function, \(E[N(t)]\) | events | Equation B.21 |
| \(S_{n}\) | Time of the \(n\)th event, \(T_{1} + \cdots + T_{n}\) | time | Equation B.20 |
| \(D(L)\) | Demand over a lead time. Written \(\mathit{LTD}\) in much of the literature | units | Equation B.30 |
Two of these are worth keeping apart from symbols that look like them. \(\overline{L}\) in Table A.3 is the average lead time and \(\overline{\mathit{IO}}\) in Table A.2 is the average number of units on order; neither is the \(\overline{L}\) that the queueing literature writes for the average number in a system, which this book writes as a named average instead. And \(N(t)\) counts demand events, while \(\overline{N}\) in Table A.2 is the average rate at which orders are placed. The two are different streams.
A.18 Symbols That Carry More Than One Meaning
A book this long cannot give every quantity its own letter, and five symbols carry more than one meaning. The context always decides, and the collisions are listed here so that nobody has to discover them.
- \(\theta\) is the Lagrange multiplier of Section 4.2, the mean lead time demand \(E[D(L)]\) throughout Chapter 8, the price of money in the relaxation of Section 9.11.9, and the error factor on a parameter in Section 1.5. The second is by far the most frequent.
- \(\beta\) is the Type 2 service target in Section 8.7, the scale parameter of the gamma distribution in Section C.2, and \((1-p)/p\) in the negative binomial fit of Section 9.8.4.
- \(\alpha\) is the Type 1 service target in Section 8.7, the shape parameter of the gamma distribution, and the smoothing constant of Section 8.15.
- \(T\) is the observation interval in Section 1.3, the cycle length \(Q/\lambda\) in Chapter 3, the base period in Section 4.5, and a repair time when it carries a location subscript in Chapter 9.
- \(S\) is an order-up-to or base-stock level in every policy chapter, a random sum in Section B.5, and, written \(S(u)\), the traceback pointer of Section 5.4.2.
A.19 A Note on Other Sources
Inventory theory has an unfortunate amount of notational variation, and a reader moving between this book and the literature will meet the same quantity under several names. Two differences recur.
The fixed order cost is written \(k\) here and \(A\) in much of the classical literature; the unit cost is written \(c\) here and \(v\) in Silver et al. (2016) and the practitioner literature that follows it. And Simulation Modeling using the KSL writes \(\mathit{BO}(t)\) for the quantity this book calls \(B(t)\); the definition is identical.
Three more recur throughout Chapter 8. The replenishment lead time is \(L\) here and \(\tau\) in a large part of the stochastic inventory literature; this book uses \(\tau\) for the protection interval \(R + L\) of Section 8.13. Demand over that lead time is \(D(L)\) here and \(\mathit{LTD}\) or \(X\) elsewhere. And the backorder cost is \(b\) here, which collides with the scale parameter some sources write \(b\) for in the gamma distribution; this book writes that scale as \(\beta\) throughout Section C.2.
The forecasting literature differs again. Section 8.15 writes the one period forecast as \(\hat{D}_{t}\) and the scaling exponent as \(\eta\), where Silver et al. (2016) writes \(\hat{a}_{t}\) and \(c\). The exponent is renamed here because \(c\) is the unit cost throughout this book. You should check the symbol list of any source before carrying a formula across.