Operations Research

Why the newsvendor orders too little

A retailer facing uncertain single-period demand and buying from a supplier at a wholesale price orders less than the two of them together would want. The reason is not irrationality but incentives: the retailer bears the whole cost of unsold stock while the supplier collects a margin on every unit ordered, so each party marks up its own cost and the combined markup, Spengler's (1950) double marginalization, depresses the order. Pasternack (1985) showed that a buyback credit for unsold units, priced correctly, realigns the retailer with the chain, and Cachon (2003) surveys the contracts that followed. Chapter 14 of Snyder and Shen's Fundamentals of Supply Chain Theory (2019) develops this as a Stackelberg game on the newsvendor model: the supplier sets contract terms, the retailer sets the order quantity. This mission formalizes the chapter's seven theorems, with the buyback allocation theorem, which shows that buyback both coordinates the chain and can divide its profit in any proportion, as the goal.

Setting

Demand $D$ is a random variable with law on $\mathbb{R}$ and mean $\mu$. The retail price is $r$; the supplier's and retailer's per-unit costs are $c_s$ and $c_r$ with $c = c_s + c_r < r$; lost sales cost the two parties goodwill penalties $p_s$ and $p_r$ with $p = p_s + p_r$; unsold units salvage for $v < c_r$ (ContractData). With $S(Q) = \mathbb{E}[\min\{Q, D\}]$ the expected sales (expSales) and $I(Q) = Q - S(Q)$ the expected leftover (expLeftover), a transfer payment $T(Q)$ from retailer to supplier determines the two profits (retailerProfit, supplierProfit),

$$\pi_r(Q) = (r - v + p_r)S(Q) - (c_r - v)Q - p_r\mu - T(Q), \qquad \pi_s(Q) = p_s S(Q) - c_s Q - p_s\mu + T(Q),$$

whose sum $\Pi(Q) = (r - v + p)S(Q) - (c - v)Q - p\mu$ (chainProfit) is independent of the contract. The chain-optimal quantity $Q_0$ maximizes $\Pi$; the retailer's and supplier's optimal quantities $Q^*_r$, $Q^*_s$ maximize their own profits. A contract coordinates the chain when $Q^*_r = Q^*_s = Q_0$, stated here as equality of the sets of maximizers, and is a coordinating contract type when some choice of its parameters does this with both profits positive.

The four contracts are their transfer payments. Wholesale price: $T = wQ$. Buyback: $T = wQ - b\,I(Q)$, the supplier crediting $b$ per unsold unit, with $0 \le b \le r - v + p_r$ and $w = w(b)$ of (14.22). Revenue sharing: the retailer keeps a fraction $\phi$ of sales and salvage revenue, $T = (w + (1-\phi)v)Q + (1-\phi)(r - v)S(Q)$, with $w = w(\phi)$ of (14.34). Quantity flexibility: the supplier reimburses the retailer's loss $w + c_r - v$ on unsold units up to $\delta Q$, $T = wQ - (w + c_r - v)\int_{(1-\delta)Q}^{Q} F(t)\,dt$, with $w = w(\delta)$ of (14.46). For buyback, $\lambda = (r - v + p_r - b)/(r - v + p)$ is the retailer's share of the chain profit (buybackShare), and $b_1 < b_2$ (buybackB1, buybackB2) are the credits at which one party earns everything.

Formalization targets

Goal: Theorem 14.5

Under buyback with $w(b)$ at the chain-optimal $Q_0$, the retailer's profit is decreasing and the supplier's increasing in $b \in [0, r - v + p_r]$, with $0 < b_1 < b_2 < r - v + p_r$ and

$$\pi_r(Q_0, w(b_1), b_1) = \Pi(Q_0), \qquad \pi_s(Q_0, w(b_2), b_2) = \Pi(Q_0),$$

the supplier losing money for $b < b_1$, both earning positive profit for $b_1 < b < b_2$, and the retailer losing money for $b > b_2$. This is buyback_allocation.

Supporting targets

Equation (14.8), $Q_0$ maximizes $\Pi$ iff $\bar F(Q_0) = (c - v)/(r - v + p)$; Theorem 14.1, the wholesale price contract coordinates iff $w = c_s - \frac{c-v}{r-v+p}p_s$, at which the supplier's profit is negative; Theorem 14.2, $Q^*_r < Q_0$ whenever $w > c_s$; Theorem 14.3, for IGFR demand the supplier's induced profit $\pi_s(Q, w(Q))$ is unimodal; the identities (14.27) and (14.28), $\pi_r = \lambda\Pi + \mu(\lambda p - p_r)$ and its complement under buyback; Theorem 14.4, buyback with $w(b)$ coordinates; Theorem 14.6, revenue sharing with $w(\phi)$ coordinates; Theorem 14.7, quantity flexibility with $w(\delta)$ makes $Q_0$ optimal for the retailer.

Significance

The chapter's theorems are the analytical basis of contract design in newsvendor supply chains. Theorem 14.1 and 14.2 make double marginalization precise: coordination by price alone is possible only at a price the supplier rejects, and any acceptable price makes the retailer under-order. Theorem 14.3 is what lets the supplier optimize the wholesale price at all, and is the reason the IGFR class of Lariviere and Porteus (2001) is standard in the field. Theorems 14.4 to 14.6 show that buyback and revenue sharing coordinate and, through the share $\lambda$, that the chain profit can be split arbitrarily, so a coordinating contract can be made acceptable to both parties. Theorem 14.7 shows the limits: quantity flexibility coordinates the retailer but not necessarily the supplier.

None of these results has a machine-checked proof. The book proves Theorems 14.1, 14.3, 14.4, 14.5, 14.6 and 14.7 and leaves 14.2 as an exercise. The formalization of the fractile characterization and of the affine profit identities is reusable for the many contract types (sales rebates, quantity discounts) the chapter cites but does not analyze.

Difficulty

The wholesale price results are first-order conditions on concave functions, and the difficulty is entirely in the analysis: $S(Q) = \mathbb{E}[\min\{Q, D\}]$ has derivative $\bar F(Q)$ for every $Q$ when $F$ is continuous, which is a differentiation under the integral that has to be carried out for a Lipschitz integrand and an arbitrary law, and the maximizers of the concave profit must then be identified with the solutions of the fractile equation, including existence by the intermediate value theorem. Theorem 14.1's "only if" direction requires that a coincidence of maximizer sets pins the fractile and hence the price, which is where strict monotonicity of $F$ enters.

Theorem 14.3 is the delicate one. The obvious approach, concavity of $\pi_s(Q, w(Q))$, fails: the book stresses the function is not concave in general. The proof is a sign-change argument on the derivative (14.16), which is $\bar F(Q)$ times a bracket that IGFR makes decreasing, minus a constant; one must show the derivative is positive near $0$, eventually negative, and crosses zero exactly once, and that the last needs both $\bar F$ strictly decreasing and the bracket positive at the crossing. Working with a density in Mathlib means relating the withDensity law to the distribution function throughout.

The buyback, revenue sharing and quantity flexibility theorems are algebra once the right identity is found: $\pi_r$ is an affine function of $\Pi$ with slope $\lambda$. The formalization must handle the boundary cases the book glosses over, where $\lambda = 0$ or $\lambda = 1$ and one party is indifferent among all quantities, so "the same $Q$ maximizes" holds only in one direction. Theorem 14.5 also needs $\Pi(Q_0) \le \mu(r - c)$, a Jensen-type bound $S(Q) \le \min\{Q, \mu\}$, for $b_1 > 0$, and the ordering $b_1 < b_2$ needs $\Pi(Q_0) > 0$, which the book's proof does not isolate.

Formalization scope

Demand is an arbitrary probability law on $\mathbb{R}$ with finite mean, not assumed nonnegative; the book's own examples use normal demand. Optimal quantities are maximizers over all of $\mathbb{R}$ (IsMaxOn … Set.univ), and coordination is the coincidence of maximizer sets, with the degenerate endpoints of the parameter ranges stated as one-directional. Where the book uses a density, continuity of the distribution function (NullSingletonClass) is assumed instead, except in Theorem 14.3, where the density $f$ is explicit, continuous on $[0, \infty)$ (so the exponential law is included), positive on $(0, \infty)$, and IGFR on $(0, \infty)$. Strict monotonicity of $F$ on $\mathbb{R}$ is never assumed, since it would rule out every nonnegative demand law. Theorem 14.1 assumes $p_s > 0$ and a positive chain optimum, $\bar F(0) > (c - v)/(r - v + p)$, both implicit in the book's proof.

The transfer payments are functions of $Q$ and the profits are defined for every $Q$, so the theorems compare values of one family of functions; the book's $\bar F$ is $1 - F$ with Mathlib's cdf, and the quantity flexibility integral is an interval integral. Theorem 14.5 takes $\Pi(Q_0) > 0$, $\mu > 0$ and $p_s, p_r > 0$ as hypotheses. Without positive goodwill costs its strict inequalities $0 < b_1$ and $b_2 < r - v + p_r$ fail (at $p_s = 0$, $b_1 = 0$; at $p_r = 0$, $b_2 = r - v + p_r$).

The definition module is shared by all eleven items. The equivalence (14.42)-(14.43) of revenue sharing and buyback, the supplier's stationarity under quantity flexibility (Problem 14.11) and the allocation results for revenue sharing (14.40)-(14.41) are natural extensions on the same definitions.

Selected references

• L. V. Snyder and Z.-J. M. Shen, Fundamentals of Supply Chain Theory, 2nd ed., Wiley, 2019, Chapter 14. https://doi.org/10.1002/9781119584445
• B. A. Pasternack, Optimal pricing and return policies for perishable commodities, Marketing Science 4(2), 1985. https://doi.org/10.1287/mksc.4.2.166
• G. P. Cachon, Supply chain coordination with contracts, in Handbooks in Operations Research and Management Science 11, 2003. https://doi.org/10.1016/S0927-0507(03)11006-7
• M. A. Lariviere and E. L. Porteus, Selling to the newsvendor: an analysis of price-only contracts, Manufacturing & Service Operations Management 3(4), 2001. https://doi.org/10.1287/msom.3.4.293.9971
• G. P. Cachon and M. A. Lariviere, Supply chain coordination with revenue-sharing contracts, Management Science 51(1), 2005. https://doi.org/10.1287/mnsc.1040.0215
• J. J. Spengler, Vertical integration and antitrust policy, Journal of Political Economy 58(4), 1950. https://doi.org/10.1086/256964

Batch quantities that cannot be chosen one site at a time

Chapter 9 of Axsäter's Inventory Control opens with the observation that in a multi-echelon system it is not optimal to choose batch quantities installation by installation: the batch at one site is the demand pattern of the next site upstream. Even with constant customer demand the exact optimum can be complicated, and the book's Example 9.4 shows a four-stage serial system whose optimal batch at one stage alternates between two values over time. Roundy (1985, 1986) showed that this complexity can be avoided at a guaranteed price: restrict every batch quantity to be a power of two times a common basic quantity, nested from stage to stage, and the best such policy costs at most 2 % more than the optimum. The book presents the result for a serial system and remarks that the same approach handles assembly and distribution systems and, in Sect. 7.3.1.2, joint replenishments. It is the capstone of Chapter 9 and the multi-echelon payoff of the powers-of-two analysis of Chapter 7.

Setting

A serial system has $N$ installations; installation $i$ produces item $i$ from one unit of item $i+1$, item $N$ is obtained from an outside supplier, and item 1 faces a constant, continuous final demand $d$. Lead-times are zero, shortages are not allowed, production is instantaneous, and each batch quantity $Q_i$ is constant over time. Installation $i$ has an ordering cost $A_i$ per batch and an echelon holding cost $e_i$ per unit and time unit, charged on the echelon stock (the stock at installation $i$ and everything downstream), so that the cost per time unit is the sum of $N$ single-item costs of the chapter 4 form,

$$C(Q) \;=\; \sum_{i=1}^{N}\Big(e_i\frac{Q_i}{2} + A_i\frac{d}{Q_i}\Big) \qquad\text{(Eq. 9.17).}$$

The book first works out the two-level case, where the optimum has $Q_2 = kQ_1$ for a positive integer $k$, the cost (9.9) is the EOQ cost with modified parameters $A_1 + A_2/k$ and $e_1 + ke_2$, and the best $k$ is found from $k^{*} = \sqrt{A_2e_1/(A_1e_2)}$ by a rounding rule.

For $N$ stages, Roundy's constraints (9.16) require $Q_i = 2^{k_i}Q_{i-1}$ with nonnegative integers $k_i$, so that every solution is nested. The relaxed constraints (9.18) require only $Q_{i-1} \le Q_i$; they are implied by (9.16), the relaxed problem is convex with linear constraints, and its solution $Q^{\mathrm{rel}}$ is computed by aggregating consecutive stages whose cost ratios $A_i/e_i$ decrease. Roundy's solution rounds $Q^{\mathrm{rel}}$ to $Q_i = 2^{m_i}q$ for a basic quantity $q$, chosen as in Proposition 7.2.

Formalization targets

Goal — Roundy's 98 % approximation

For $d > 0$, $A_i > 0$, $e_i > 0$ and any minimizer $Q^{\mathrm{rel}}$ of $C$ over positive batch quantities satisfying (9.18), there exist $q > 0$ and integers $m_1 \le \dots \le m_N$ such that $Q_i = 2^{m_i}q$ satisfies (9.16) and

$$C\big(2^{m}q\big) \;\le\; \frac{1}{\sqrt 2\,\ln 2}\,C\big(Q^{\mathrm{rel}}\big).$$

Supporting targets

The two-level results of Sect. 9.2.1: the equivalence of the installation and echelon cost forms (9.6) and (9.9); the optimal $Q_1$ and cost (9.10)-(9.11) for a given $k$; the closed form and convexity of $C(k)^2$ (9.12) and the real minimizer $k^{*}$ (9.13); and the integer rounding rule with its corollary that $A_1/e_1 \ge A_2/e_2$ forces $k = 1$. For $N$ stages: existence of the relaxed optimum; the aggregation lemma, that $A_i/e_i < A_{i-1}/e_{i-1}$ forces $Q^{\mathrm{rel}}_i = Q^{\mathrm{rel}}_{i-1}$; that (9.16) implies (9.18), so the relaxed optimum bounds every powers-of-two policy from below; and that rounding to the nearest power of two times $q$ is monotone and lands within a factor $\sqrt 2$.

Significance

The result itself. Roundy's theorem replaces an intractable lot-sizing problem by a closed-form computation with a provable 2 % guarantee, and the policies it produces are the nested, periodic schedules that production planning wants anyway. In Example 9.4 the rounding with $q = 1$ is already within 0.7 % of the relaxed bound. The two-level analysis has its own use: the modified parameters $A_1 + A_2/k$ and $e_1 + ke_2$ are what the Blackburn-Millen heuristic of Sect. 9.3.2 feeds to the Wagner-Whitin algorithm under time-varying demand, and the condition $A_1/e_1 \ge A_2/e_2$ tells when a two-stage system collapses to one stage.

Formalizing it. The book proves the bound in a paragraph that leans on three earlier results: Proposition 7.2 for the rounding, the Lagrangean relaxation for the lower bound, and the aggregation algorithm for the structure of $Q^{\mathrm{rel}}$. Formalizing it makes explicit what the paragraph glosses: that the multipliers vanish off tight constraints, that equal quantities round to equal quantities, and that the comparison class is the class of nested constant-batch policies. The published Proposition 7.2 (pot_two_percent) and the mean value $1/(\sqrt 2\ln 2)$ are cited as reference items. Nothing here is open; no statement has a machine-checked proof yet.

Difficulty

The obvious argument, rounding $Q^{\mathrm{rel}}$ item by item and invoking Proposition 7.2 for each, does not work: Proposition 7.2 bounds the rounded cost relative to each item's unconstrained optimum, and $Q^{\mathrm{rel}}_i$ is generally not that optimum. The proof must pass through the Lagrangean relaxation (9.19)-(9.20), under which $Q^{\mathrm{rel}}$ is the unconstrained optimum for the modified holding costs $e_i' = e_i - 2\lambda_i + 2\lambda_{i+1}$, apply Proposition 7.2 there, and transfer the bound back using complementary slackness: the multiplier $\lambda_i$ is positive only where $Q^{\mathrm{rel}}_i = Q^{\mathrm{rel}}_{i-1}$, and equal quantities round to equal quantities, so the correction terms $\lambda_i(Q_i - Q_{i-1})$ vanish for both $Q^{\mathrm{rel}}$ and its rounding. A solver therefore needs the KKT conditions for this convex program, or an equivalent direct argument through the aggregation structure (within an aggregate all quantities are equal and their sum is an EOQ problem). The two-level statements and the rounding lemma are elementary.

Formalization scope

Stages are indexed by Fin N; the cost is serialCost A e d Q = ∑ i, eoqCost (A i) d (e i) (Q i) with eoqCost imported from the chapter 4 definitions, and the two-level cost is eoqCost (A1 + A2/k) d (e1 + k e2) Q1 by definition, so the published eoq_optimal and eoq_cost_at_eoq apply to it directly. SerialNested and SerialPowerOfTwo are the constraints (9.18) and (9.16) on consecutive indices; for $N \le 1$ both hold vacuously and the goal is Proposition 7.2 for one item. potRound q Q is $\lfloor \log_2(Q/q) + \tfrac12\rfloor$.

All theorems assume $d, A_i, e_i > 0$ and positive batch quantities. The book says "nonnegative ordering and echelon holding costs"; strict positivity is needed for the relaxed problem to have a solution at all ($e_i = 0$ or $A_i = 0$ sends the optimal $Q_i$ to $\infty$ or $0$), and it is what the book's own examples satisfy. The relaxed optimum enters the goal as a hypothesis, with existence stated separately; uniqueness is not used. The constant is the exact $1/(\sqrt 2\ln 2)$.

Two readings are excluded. The bound is not against an arbitrary nested $Q$, for which it would be false, but against the relaxed optimum; and the comparison class is stated as nested constant-batch policies, the class the relaxed problem bounds directly, not the book's larger class of time-varying policies, which would need a dynamic model. The definitions are reusable for assembly systems and for the joint replenishment problem of Sect. 7.3.1.2, whose Roundy analysis is the same with a fictive item 0 of zero holding cost; contributions in that direction are welcome.

Selected references

• Sven Axsäter, Inventory Control, 3rd edition, International Series in Operations Research & Management Science 225, Springer, 2015, Sect. 9.2. DOI 10.1007/978-3-319-15729-0
• Robin Roundy, 98%-Effective Integer-Ratio Lot-Sizing for One-Warehouse Multi-Retailer Systems, Management Science 31(11), 1985, pp. 1416-1430. DOI 10.1287/mnsc.31.11.1416
• Robin Roundy, A 98%-Effective Lot-Sizing Rule for a Multi-Product, Multi-Stage Production/Inventory System, Mathematics of Operations Research 11(4), 1986, pp. 699-727. DOI 10.1287/moor.11.4.699
• John A. Muckstadt and Robin O. Roundy, Analysis of Multistage Production Systems, in: Handbooks in Operations Research and Management Science 4, Elsevier, 1993, pp. 59-131. DOI 10.1016/S0927-0507(05)80182-4
• Peter L. Jackson, William L. Maxwell and John A. Muckstadt, The Joint Replenishment Problem with a Powers-of-Two Restriction, IIE Transactions 17(1), 1985, pp. 25-32. DOI 10.1080/07408178508975268

Why the policy class is not up for debate

Every model in Chapters 5 and 6 of Axsäter's Inventory Control assumes at the outset that the ordering policy is of $(R,Q)$ or $(s,S)$ type. Section 6.2 asks whether better policies exist and answers, for the case in which there are no ordering costs but every order must be a multiple of a fixed batch quantity $Q$, that none do: Proposition 6.1, "an $(R,Q)$ policy is optimal", with a proof the book attributes to Chen (2000). For $Q = 1$ it is the optimality of an order-up-to-$S$ policy in the absence of ordering costs, and for continuous or Poisson demand it transfers to $(s,S)$ policies, which are then the same thing. The proposition is the one place in the book where a policy is compared against every feasible alternative rather than against other members of its own family, and its proof is short enough to be given in full, which makes it the natural capstone of Chapter 6.

Setting

Demand is compound Poisson: customers arrive according to a Poisson process with rate $\lambda$ and each demands an integral number of units, with sizes $D_0, D_1, \dots$ independent and identically distributed with law $f$ on the positive integers, independent of the arrival process. The book's standing assumption that not all demands are multiples of some integer larger than one is kept. Writing $T_n$ for the $n$-th arrival time, $N(t)$ for the number of arrivals by time $t$ and $S_n = D_0 + \dots + D_{n-1}$, the total demand by time $t$ is $S_{N(t)}$.

The replenishment lead-time $L$ is constant and $D(L)$, the demand over a lead-time, has law $D$. A holding cost $h > 0$ and a shortage cost $b_1 > 0$ per unit and time unit are charged. There are no ordering costs, but all orders must be multiples of a given batch quantity $Q \ge 1$ and can only be triggered by customer demands. A policy is therefore any rule $m$ that decides at each demand epoch how many batches to order; with initial position $y_0$ the inventory position evolves as $y_{n+1} = y_n - D_n + m_nQ$ (Eq. 6.22), and $y_t = y_{N(t)}$.

The standard argument of Sect. 5.3.2 gives the cost rate at time $t + L$ as

$$g(y_t), \qquad g(k) = -b_1(k - \mu') + (h+b_1)\sum_{j=1}^{k} j\,\Pr[D(L) = k - j],$$

the expected holding-plus-shortage cost rate of an inventory level $k - D(L)$ (Eq. 6.20), which is convex in $k$ with $g(k) \to \infty$ as $|k| \to \infty$. The band cost is $\bar g(y) = \sum_{j=1}^{Q} g(y+j)$ and $R$ denotes an integer minimizing $\bar g$. The $(R,Q)$ policy orders, as soon as the position is at or below $R$, the smallest number of batches that brings it above $R$; its position lives in the band $\{R+1, \dots, R+Q\}$ from the first order on. The performance measure is the long-run average cost rate $\frac{1}{T}\int_0^T g(y_{t-L})\,\mathrm{d}t$ as $T \to \infty$.

Formalization targets

Goal — Proposition 6.1

Almost surely, (1) for every policy $m$ and every $y_0$,

$$\liminf_{T\to\infty} \frac{1}{T}\int_0^T g\big(y^{m}_{t-L}\big)\,\mathrm{d}t \;\ge\; \frac{\bar g(R)}{Q},$$

and (2) the $(R,Q)$ policy attains it:

$$\frac{1}{T}\int_0^T g\big(y^{(R,Q)}_{t-L}\big)\,\mathrm{d}t \;\longrightarrow\; \frac{\bar g(R)}{Q}.$$

Supporting targets

Lemma 6.1, that $x \mapsto g(z + xQ)$ is convex and minimized at the representative of $z$ in the band; the closed form of the $(R,Q)$ position, $y_0 - S_n$ until the first order and the band representative of $y_0 - S_n$ afterwards; the uniform occupation of the band by the reduced process $y_t'$, the book's "the steady state distribution can be shown to be uniform", and its consequence that the long-run average of $g(y_t')$ is $\bar g(R)/Q$; Proposition 5.1 in the same ergodic form for the $(R,Q)$ policy; and the two halves of the goal as separate statements.

Significance

The result itself. Proposition 6.1 is what licenses the two-parameter policies on which the rest of the book's single-echelon theory is built, and it does so for the practically important case of batch ordering (pallets, containers, production lots). Its proof also explains why the policy works: the only quantity a policy controls is the residue class of the inventory position modulo $Q$, which no policy can influence, and the position within that class, which the $(R,Q)$ policy always sets to the cheapest possible value. The book extends the same reasoning to other cost structures and to periodic review.

Formalizing it. The proposition is a theorem about the class of all policies, and the book's proof is pathwise: Lemma 6.1 compares any policy with the reduced process instant by instant, and an ergodic statement about the reduced process does the rest. Formalizing it therefore forces the policy class, the demand process and the long-run average to be written down exactly, which the book never does. Nothing here is open; no statement has a machine-checked proof yet.

Difficulty

The pointwise comparison is elementary once Lemma 6.1 is available, and Lemma 6.1 is discrete convexity. The difficulty is entirely in the ergodic statement: that the reduced position $y_t' =$ (the band representative of $y_0 - S_{N(t)}$) spends a fraction $1/Q$ of the time at each point of the band, almost surely. In discrete time this is the convergence of occupation frequencies for an irreducible random walk on $\mathbb{Z}/Q\mathbb{Z}$ with step law $f$ modulo $Q$, where irreducibility is exactly the aperiodicity assumption on $f$, and the book's double-stochasticity argument (Eq. 5.33-5.34) identifies the uniform law as stationary. Passing to continuous time adds the exponential holding times: the time average is the arrival average weighted by i.i.d. holding times independent of the walk, which a strong law of large numbers turns back into the discrete statement. Mathlib has the strong law and the exponential law but no ergodic theorem for finite Markov chains, so that is the groundwork a solver must build. The naive route through the stationary distribution of the embedded chain at demand epochs is not enough on its own: for pure Poisson demand that chain is periodic, and the book itself notes it.

Formalization scope

A CompoundPoissonDemand on a probability space packages the rate, the size law with $f_0 = 0$ and the aperiodicity condition, and two sequences of random variables, the gaps and the sizes, with their laws (expMeasure lam, the given pmf), independence within each sequence, and independence between the sequences. Arrival times are partial sums of the gaps, count t is the supremum of $\{n : T_n \le t\}$, and cumDemand n is the partial sum of the sizes. A policy is a function m : ℕ → Ω → ℕ with no measurability requirement; ipPath and ipAt are the position after the $n$-th demand and at time $t$; rqIP and rqOrders are the $(R,Q)$ policy, with a theorem identifying rqOrders as a policy in the general sense. avgCost is $\frac{1}{T}\int_0^T g(y_{t-L})\,\mathrm{d}t$ with $y_s = y_0$ for $s < 0$. The cost function $g$ is sPolicyCost from the previous mission, applied to a DiscreteDemand that the hypotheses tie to the process as the law of the demand in $(0, L]$.

Conventions: $Q \ge 1$, $L \ge 0$, $h, b_1 > 0$; $R$ is any minimizer of $\bar g$ (it exists by the divergence of $g$, proved in the previous mission); $y_0$ is arbitrary. count and the interval integral take junk values on the null set where arrivals do not tend to infinity, which the almost-sure conclusions absorb. "$\liminf \ge c$" is stated as "for every $\varepsilon > 0$, eventually $\ge c - \varepsilon$", avoiding a liminf on $\mathbb{R}$ that could be junk.

Two readings that would trivialize the goal are excluded: the lower bound is over every rule, not over stationary or measurable ones, and the achievability half is a genuine limit, not a bound. The demand model and the occupation-frequency theorems are reusable for Proposition 10.1 and the batch-ordering models of Sect. 10.5; contributions establishing the ergodic theorem for irreducible chains on a finite cyclic group are welcome and would close most of this mission.

Selected references

• Sven Axsäter, Inventory Control, 3rd edition, International Series in Operations Research & Management Science 225, Springer, 2015, Sects. 5.3.1 and 6.2.1. DOI 10.1007/978-3-319-15729-0
• Fangruo Chen, Optimal Policies for Multi-Echelon Inventory Problems with Batch Ordering, Operations Research 48(3), 2000, pp. 376-389. DOI 10.1287/opre.48.3.376.12427
• Awi Federgruen and Yu-Sheng Zheng, An Efficient Algorithm for Computing an Optimal $(r,Q)$ Policy in Continuous Review Stochastic Inventory Systems, Operations Research 40(4), 1992, pp. 808-813. DOI 10.1287/opre.40.4.808
• Evan L. Porteus, Foundations of Stochastic Inventory Theory, Stanford University Press, 2002.

Two decisions that are usually taken separately

An $(R,Q)$ policy has two parameters. Chapter 4 of Axsäter's Inventory Control chooses the batch quantity $Q$ from a deterministic model, and Chapter 5 chooses the reorder point $R$ from a stochastic one with $Q$ held fixed; the book presents this two-step practice as an adequate approximation. Section 6.1 asks what is lost by it and shows how to optimize both parameters jointly in one stochastic model. For discrete demand the answer, due to Federgruen and Zheng (1992), is an algorithm of remarkable simplicity: increase $Q$ one unit at a time, keep the reorder point optimal along the way by a one-line rule, and stop at the first $Q$ for which the cost goes up. The claim that this stopping rule finds the global optimum over all pairs $(R, Q)$ is the capstone of Sect. 6.1.1.1 and the goal of this mission. The same idea is reused in the book for $(s,S)$ policies (Sect. 6.1.1.2) and, in continuous form, for normally distributed demand (Sect. 6.1.2).

Setting

An item is controlled by a continuous review $(R,Q)$ policy with integral reorder point $R$ and batch quantity $Q \ge 1$. Demand is discrete and stationary; the lead-time demand $D(L)$ takes values in the nonnegative integers with probabilities $p_j = \Pr[D(L) = j]$ and has a finite mean $\mu'$. The average demand per unit of time is $\mu$. Costs are a holding cost $h$ and a shortage cost $b_1$, both per unit and time unit, and an ordering cost $A$ per batch.

The building block is the cost of an $(S-1,S)$ policy that keeps the inventory position at a fixed integer $k$. By the standard argument of Sect. 5.3.2 the inventory level a lead-time later is $k - D(L)$, and the average holding and shortage cost rate is (Eq. 6.3)

$$g(k) \;=\; -b_1\,(k - \mu') + (h + b_1)\sum_{j=1}^{k} j\,\Pr[D(L) = k - j].$$

Under the $(R,Q)$ policy the inventory position is uniformly distributed on $\{R+1, \dots, R+Q\}$ (Proposition 5.1), so the total average cost rate is (Eq. 6.4)

$$C(R, Q) \;=\; \frac{A\mu}{Q} + \frac{1}{Q}\sum_{k=R+1}^{R+Q} g(k),$$

and $C(Q) = \min_R C(R,Q)$ (Eq. 6.5), attained at an optimal reorder point $R^{*}(Q)$. The ready rate $S_3(R) = \Pr[IL > 0] = \frac{1}{Q}\sum_{k=R+1}^{R+Q}\Pr[D(L) \le k-1]$ links the cost to service: raising the reorder point by one unit changes the cost by $-b_1 + (h + b_1)S_3(R+1)$ (Eq. 5.60).

Formalization targets

Goal — the Federgruen-Zheng stopping rule is optimal

Let $Q^{*} \ge 1$ be the smallest batch quantity with $C(Q^{*}+1) \ge C(Q^{*})$ and let $R^{*}$ be an optimal reorder point for $Q^{*}$. Then

$$C(R^{*}, Q^{*}) \;\le\; C(R, Q) \qquad\text{for all } R \in \mathbb{Z},\ Q \ge 1.$$

Supporting targets

The increment identity $g(k+1) - g(k) = -b_1 + (h+b_1)\Pr[D(L) \le k]$; convexity of $g$ on $\mathbb{Z}$ together with $g(k) \to \infty$ as $|k| \to \infty$; Eq. (5.60) and the convexity of $C(\cdot, Q)$ in $R$; Eq. (5.61), that the largest $R$ with $S_3(R) \le b_1/(h+b_1)$ is optimal for its $Q$; existence of an optimal $R$ for every $Q$; the recursion (6.6)-(6.7), $R^{*}(Q+1) \in \{R^{*}(Q) - 1, R^{*}(Q)\}$ chosen by comparing $g(R^{*}(Q))$ with $g(R^{*}(Q)+Q+1)$, and $C(Q+1) = C(Q)\frac{Q}{Q+1} + \min\{g(R^{*}(Q)), g(R^{*}(Q)+Q+1)\}\frac{1}{Q+1}$; the equivalence $C(Q+1) \ge C(Q) \iff \min\{\cdot\} \ge C(Q)$ and the monotonicity of that minimum in $Q$; and the existence of some $Q$ at which the costs stop decreasing.

Significance

The result itself. Joint optimization typically enlarges the batch and lowers the reorder point relative to the two-step procedure, and the book's Example 6.1 puts the resulting cost saving at a few percent. The Federgruen-Zheng procedure makes the exact joint optimum for discrete demand as cheap to compute as the two-step approximation, because each step of the recursion evaluates $g$ at two points. It is the exact benchmark against which the book's approximate techniques for normal demand (Sects. 6.1.2 and 6.1.3) are judged, and its structural core, that the optimal window of $Q$ consecutive inventory positions grows one neighbour at a time, is the discrete-convexity fact behind the whole of Sect. 6.1.

Formalizing it. The mathematics is settled. What the mission produces is a Lean development in which the steps the book marks "evident" and "obvious" are separate statements: that the recursion preserves optimality, that the marginal cost of enlarging the batch is monotone, and that a minimum over $R$ exists at all. None of the statements has a machine-checked proof yet.

Difficulty

The obvious first idea, to argue that $C(Q)$ is convex in $Q$ and stop at its first increase, does not work as stated: $C(Q)$ is a minimum over $R$ of a ratio and is not convex in general. What is true, and what the proof uses, is that the marginal cost $(Q+1)C(Q+1) - QC(Q)$ is nondecreasing, because it equals the smaller of the two $g$-values adjacent to the optimal window. Establishing the recursion (6.6) is where discrete convexity is needed: one must show that the best window of $Q+1$ consecutive positions is obtained from the best window of $Q$ by adding a neighbour, which fails for non-convex $g$. The window sums $R \mapsto \sum_{k=R+1}^{R+Q} g(k)$ are themselves convex in $R$ with increments $g(R+Q+1) - g(R+1)$, and the argument compares a competing window with the optimal one through the end terms. The remaining steps are finite algebra and an induction on $Q$ from $Q^{*}$.

Formalization scope

A DiscreteDemand is a function $p : \mathbb{N} \to \mathbb{R}$ with $p \ge 0$, $\sum p = 1$ and a summable first moment; $\Pr[D(L) \le k]$ is a finite sum over $0 \le j \le k$, empty for $k < 0$. The reorder point ranges over $\mathbb{Z}$ and the batch quantity over $\mathbb{N}$, with $Q \ge 1$ assumed in every statement because Lean's division by $0$ is $0$. Convexity of a function on $\mathbb{Z}$ is stated as nondecreasing increments, and divergence as Tendsto g (cocompact ℤ) atTop.

$C(Q)$ enters the goal as a function CQ together with the hypothesis that CQ Q is the least value of $R \mapsto C(R, Q)$; the stopping index $Q^{*}$ is characterized by the two conditions that define "the smallest $Q$ with $C(Q+1) \ge C(Q)$", and $R^{*}$ by optimality at $Q^{*}$. That these objects exist is the content of two separate items, so the goal is not vacuous: a minimum over $R$ exists for every $Q$ because $g \to \infty$, and the costs cannot decrease forever because the average of the $Q$ smallest values of $g$ tends to infinity. The positivity of $A$ and $\mu$ is the book's setting and is assumed where it appears.

The uniform inventory position of Proposition 5.1, which needs the book's assumption that not all demands are multiples of an integer larger than one, is taken as given in the cost formula (6.4), as the book does; the proposition itself is the subject of the next mission of this series. The definitions are reusable for the $(s,S)$ optimization of Sect. 6.1.1.2 and for the multi-echelon batch-ordering results of Sect. 10.5; contributions formalizing the Zheng-Federgruen $(s,S)$ algorithm on top of them are welcome.

Selected references

• Sven Axsäter, Inventory Control, 3rd edition, International Series in Operations Research & Management Science 225, Springer, 2015, Sects. 5.9.1 and 6.1.1. DOI 10.1007/978-3-319-15729-0
• Awi Federgruen and Yu-Sheng Zheng, An Efficient Algorithm for Computing an Optimal $(r,Q)$ Policy in Continuous Review Stochastic Inventory Systems, Operations Research 40(4), 1992, pp. 808-813. DOI 10.1287/opre.40.4.808
• Yu-Sheng Zheng and Awi Federgruen, Finding Optimal $(s,S)$ Policies Is About as Simple as Evaluating a Single Policy, Operations Research 39(4), 1991, pp. 654-665. DOI 10.1287/opre.39.4.654
• Paul Zipkin, Foundations of Inventory Management, McGraw-Hill, 2000.

Where the reorder point comes from

Chapter 4 of Axsäter's Inventory Control fixes the batch quantity $Q$ from a deterministic model, and Chapter 5 asks the question that deterministic models cannot answer: with demand random and a replenishment lead-time $L$, when should the next batch be ordered? Under a continuous review $(R,Q)$ policy the answer is a single number, the reorder point $R$, and Sections 5.3 through 5.9 are one sustained computation of what a given $R$ buys. The chapter's own capstone is Eq. (5.67): if backorders are charged at $b_1$ per unit and time unit and stock at $h$, the cost-minimizing reorder point is exactly the one whose fill rate is $b_1/(h+b_1)$. The book calls the relationship "even more striking" than its discrete counterpart, and it is used in practice in both directions: a backorder cost prescribes a service level, and a chosen service level reveals the backorder cost a planner is implicitly assuming (Eq. 5.68).

Setting

An order for a fixed batch quantity $Q > 0$ is triggered whenever the inventory position (stock on hand plus outstanding orders minus backorders) falls to the reorder point $R$, and it arrives $L$ time units later. Demand is continuous and normally distributed; the demand over a lead-time has mean $\mu'$ and standard deviation $\sigma' > 0$. Two modelling facts from the book are taken as the definition of the steady state. First (Sect. 5.3.1), the inventory position $IP$ is uniformly distributed on $[R, R+Q]$; the book proves this for compound Poisson demand (Proposition 5.1) and adopts it as an accurate approximation for continuous demand. Second (Sect. 5.3.2, Eq. 5.35), the inventory level a lead-time later is the inventory position now minus the demand in between,

$$IL(t+L) \;=\; IP(t) - D(t, t+L),$$

with the two terms independent. The law of $IL$ is therefore the image of the product of a uniform and a normal law under subtraction, and everything in the chapter is a functional of it:

• the distribution function $F(x) = \Pr[IL \le x]$ and its density $f$;
• the ready rate $S_3 = \Pr[IL > 0]$, which for continuous demand equals the fill rate $S_2$, the fraction of demand met from stock on hand;
• the expected cost rate $C(R) = \mathbb{E}\big[h\,(IL)^{+} + b_1\,(IL)^{-}\big]$, holding cost on positive stock and backorder cost on negative stock, Eq. (5.56).

The closed forms run through the standard normal loss function $G(x) = \int_x^\infty (v-x)\varphi(v)\,\mathrm{d}v$ of Eq. (5.40), published with the newsboy mission and reused here, and through its integral, the second loss function $H(x) = \int_x^\infty G(v)\,\mathrm{d}v$ of Eq. (5.64). Both are tabulated in the book's Appendix 2, and both recur in Chapters 6, 9 and 10.

Formalization targets

Goal — Eq. (5.67)

For $h, b_1, Q, \sigma' > 0$ and any $\mu'$, a reorder point $R$ minimizes $C$ over $\mathbb{R}$ if and only if

$$S_2(R) \;=\; S_3(R) \;=\; \frac{b_1}{h + b_1}.$$

The biconditional carries both halves of the book's sentence: the stationary point is the optimum ("the optimal $R$ is obtained for $\mathrm{d}C/\mathrm{d}R = 0$") and the optimum is stationary ("in the optimal solution we have $S_2 = S_3 = b_1/(h+b_1)$").

Supporting targets

In the order the chapter builds them: Eq. (5.41), $G' = \Phi - 1$, with $G$ decreasing and convex; Eq. (5.39), the distribution function by conditioning on the inventory position; Eq. (5.42), its closed form $F(x) = \frac{\sigma'}{Q}[G(\frac{R-x-\mu'}{\sigma'}) - G(\frac{R+Q-x-\mu'}{\sigma'})]$; Eq. (5.43), the density; Eq. (5.52), the fill rate $1 - \frac{\sigma'}{Q}[G(\frac{R-\mu'}{\sigma'}) - G(\frac{R+Q-\mu'}{\sigma'})]$; Eq. (5.55), the expected backorders $\mathbb{E}(B)$ covered by one batch, and Eq. (5.54), that $1 - \mathbb{E}(B)/Q$ is the same fill rate; integrability of the cost rate and the mean $\mathbb{E}(IL) = R + Q/2 - \mu'$; Eq. (5.63), $\mathbb{E}(IL)^{-} = \int_{-\infty}^0 F$; Eq. (5.64), the closed form of $H$ and $H' = -G$; Eq. (5.65), the cost $C = h(R + Q/2 - \mu') + (h+b_1)\frac{\sigma'^2}{Q}[H(\frac{R-\mu'}{\sigma'}) - H(\frac{R+Q-\mu'}{\sigma'})]$; Eq. (5.66), $\mathrm{d}C/\mathrm{d}R = -b_1 + (h+b_1)S_2$; and the convexity of $C$ in $R$.

Significance

The result itself. The reorder point is the one parameter of an $(R,Q)$ policy that stochastic demand actually decides, and Eq. (5.67) says that deciding it by cost and deciding it by service level are the same decision, with an explicit dictionary between the two. That is why the book can present service-level constraints (Sect. 5.7) and shortage costs (Sect. 5.9) as interchangeable ways of specifying the same thing, and why it warns, immediately after Eq. (5.67), that the equivalence is only valid when $Q$ is given: with an ordering cost and a joint optimization of $R$ and $Q$ it fails, which is Chapter 6's problem.

The intermediate formulas have independent standing. Eq. (5.42) is the single expression from which every service measure of the chapter is computed, and Eq. (5.65) is the cost function that Chapter 6 extends by an ordering cost, Eq. (6.10), and optimizes iteratively. The second loss function $H$ returns in the periodic-review fill rate of Eq. (5.86) and in the two-echelon batch-ordering model of Sect. 10.5.

Formalizing it. Nothing here is open; the value is that the steady-state model becomes an explicit measure, so that formulas the book obtains by manipulating integrals whose existence it never questions become theorems about that measure. Integrability of the cost rate is a target of its own for exactly that reason. None of the statements has a machine-checked proof yet.

Difficulty

The obvious route is the book's, and it is not the hard part: once $F$ is known in closed form, every later identity is calculus on $G$ and $H$. The work is upstream of that. The distribution function (5.39) is a conditioning argument over the product measure, a Fubini step in which the inner probability is a Gaussian tail; the density (5.43) is a derivative of a parameter-dependent integral; and Eq. (5.63) exchanges the order of two integrals over an unbounded region, which needs integrability of $IL$ itself. The derivative (5.66) is then obtained from the closed form (5.65), not by differentiating under an expectation, which is what makes the goal reachable: $C$ is a smooth function of $R$ with an explicitly increasing derivative, and Eq. (5.67) follows from strict convexity together with the fact that the fill rate is a continuous, strictly increasing function of $R$ ranging over $(0,1)$. A solver who starts from the expectation and tries to differentiate it directly will meet the kink of $x^{+}$ at $0$ and a dominated-convergence argument; the integrated route avoids both.

Formalization scope

The inventory position is rqPosition R Q, Lebesgue measure conditioned on $[R, R+Q]$; the lead-time demand is newsboyDemand m s, Mathlib's gaussianReal m (s^2).toNNReal from the newsboy mission, with $m = \mu'$ and $s = \sigma'$; and rqLevel R Q m s is the pushforward of their product under $(u, d) \mapsto u - d$. Two lemmas in the definition file record that both are probability measures when $Q > 0$. The distribution function, ready rate and cost are the measure of $(-\infty, x]$, the measure of $(0, \infty)$, and a Bochner integral against this law.

Every statement assumes $Q > 0$ and $\sigma' > 0$. At $Q = 0$ the conditioned measure is the zero measure and every integral is $0$, so a statement without the hypothesis would be true and empty; at $\sigma' = 0$ the demand is a point mass and $F$ has jumps. The mean $\mu'$ is unrestricted, as none of the formulas depends on its sign, and reorder points may be negative, which the book explicitly allows in Sect. 5.8. Costs $h$ and $b_1$ are positive in every statement that involves them, and $\mathbb{E}(B)$ is stated for $Q \ge 0$ because its closed form holds there.

Two trivializing readings are ruled out. The cost is not defined as a formula in $H$ but as an expectation, so the closed form (5.65) has content; and because Lean's Bochner integral of a non-integrable function is $0$, integrability of the cost rate is stated as a theorem rather than assumed, otherwise a zero cost would make every reorder point optimal. The goal quantifies optimality over all real competing reorder points, not a neighbourhood.

A complete development needs Gaussian tail integrals, differentiation of parameter-dependent integrals, Fubini on a product of a bounded interval with the line, and the strict monotonicity of the Gaussian distribution function. The loss functions $G$ and $H$ and the inventory-level law are reusable across the rest of the series; contributions that establish the same identities for an arbitrary continuous lead-time demand with a finite mean, where Eqs. (5.39), (5.63) and (5.66) hold verbatim, are welcome.

Selected references

• Sven Axsäter, Inventory Control, 3rd edition, International Series in Operations Research & Management Science 225, Springer, 2015, Sects. 5.3, 5.7, 5.8 and 5.9. DOI 10.1007/978-3-319-15729-0
• George Hadley and Thomson M. Whitin, Analysis of Inventory Systems, Prentice-Hall, 1963.
• Paul Zipkin, Foundations of Inventory Management, McGraw-Hill, 2000.
• Yu-Sheng Zheng, On Properties of Stochastic Inventory Systems, Management Science 38(1), 1992, pp. 87-103. DOI 10.1287/mnsc.38.1.87
• Kaj Rosling, Inventory Cost Rate Functions with Nonlinear Shortage Costs, Operations Research 50(6), 2002, pp. 1007-1017. DOI 10.1287/opre.50.6.1007.346

Motivation

Exact methods for the resource-constrained project scheduling problem — branch-and-bound over activity lists or over start-time assignments, and the lower-bound computations inside them — live or die by how much of the search space can be discarded before it is enumerated. The standard tool is constraint propagation: deducing, from the precedence, resource and time-window data, new precedence relations $i\to j$ that every feasible schedule must satisfy, and tighter time windows for the activities. Brucker and Knust's Section 3.6 (doi:10.1007/978-3-642-23929-8) presents the family of interval consistency tests — input, output, input-or-output and their negations — that constraint-programming schedulers apply at every node of the search, following Carlier and Pinson (An algorithm for solving the job-shop problem, Management Science 35, 1989, doi:10.1287/mnsc.35.2.164) and Baptiste, Le Pape and Nuijten (Constraint-Based Scheduling, Kluwer, 2001, doi:10.1007/978-1-4615-1479-4). Every test is an instance of one theorem, Theorem 3.7, and its cumulative-resource analogue, Theorem 3.8. Those two theorems, and the tests as their corollaries, are this mission.

Setting

The instance is the RCPSP of mission I: activities $0,\dots,n-1$ with integer processing times $p_i$, renewable resources $k$ with capacities $R_k$ and demands $r_{ik}$, and precedence arcs; a schedule is an integer start-time vector $S$, feasible when it meets the precedences and never exceeds a capacity. Section 3.6 adds three things.

Relations. A conjunction $i\to j$ holds in $S$ when $S_i+p_i\le S_j$. Two activities are parallel, $i\parallel j$, when they overlap for at least one time unit, and a disjunction $i-j$ is the negation of that: $i\to j$ or $j\to i$. The instance carries a set $C$ of conjunctions and a set $D$ of disjunctions that every feasible schedule must satisfy; initially $C_0$ is the precedence relation and $D_0$ the pairs whose combined demand exceeds some capacity, and propagation adds to them.

Disjunctive sets. A set $I$ of at least two activities is disjunctive when any two of its members are related by a disjunction or a conjunction, so no two are ever processed together: the activities of a unit-capacity resource, the jobs of a single machine, the operations of one job in a shop. Its total processing time is $P(I)=\sum_{i\in I}p_i$.

Time windows. Each activity has a head $r_i$ and a deadline $d_i$, and a feasible schedule has $r_i\le S_i$ and $S_i+p_i\le d_i$. An activity starts first in a set $J$ when no activity of $J$ starts earlier, and ends last when none completes later.

For a cumulative resource $k$ the work of activity $i$ is $w_i=r_{ik}p_i$ and $W(J)=\sum_{i\in J}w_i$.

Formalization targets

Goal — Theorem 3.7 (printed p. 169)

Let $I$ be a disjunctive set, $J\subseteq I$, and $J',J''$ proper subsets of $J$ with $J'\cup J''\ne\emptyset$. If

$$\max_{\substack{\nu\in J\setminus J',\ \mu\in J\setminus J''\\ \nu\ne\mu}}\bigl(d_\mu-r_\nu\bigr)<P(J),$$

then in every feasible schedule an activity from $J'$ starts first in $J$ or an activity from $J''$ ends last in $J$.

The first infeasibility test (printed p. 169)

If some nonempty $J\subseteq I$ has $\max_{\mu\in J}d_\mu-\min_{\nu\in J}r_\nu<P(J)$, no feasible schedule exists.

The input test (3.123) and the output test (3.124) (printed p. 171)

For $\Omega\subseteq I$ nonempty and $i\in I\setminus\Omega$: if $\max_{\mu\in\Omega\cup\{i\}}d_\mu-\min_{\nu\in\Omega}r_\nu<P(\Omega)+p_i$ then $i\to j$ for all $j\in\Omega$; symmetrically, if $\max_{\mu\in\Omega}d_\mu-\min_{\nu\in\Omega\cup\{i\}}r_\nu<P(\Omega)+p_i$ then $j\to i$ for all $j\in\Omega$.

The input-or-output test (printed p. 170)

For $i,j\in J\subseteq I$, $|J|\ge 2$: if $\max_{\mu\in J\setminus\{j\}}d_\mu-\min_{\nu\in J\setminus\{i\}}r_\nu<P(J)$ then $i$ starts first in $J$ or $j$ ends last in $J$, and $i\to j$ when $i\ne j$.

Theorem 3.8 (printed p. 186)

For a cumulative resource $k$, $J\subseteq I_k$ and proper subsets $J',J''$ of $J$: if $R_k\bigl(\max_{\mu\in J\setminus J''}d_\mu-\min_{\nu\in J\setminus J'}r_\nu\bigr)<W(J)$ then an activity from $J'$ starts first in $J$ or an activity from $J''$ ends last in $J$.

Significance

Theorem 3.7 is the single statement behind a whole toolbox. Every interval consistency test in the literature — the input and output tests that fix a new conjunction, the input-or-output test, the negation tests that only shrink a window — is the theorem with a particular choice of $J'$ and $J''$, and the book's Section 3.6.4 derives them one by one. A propagation engine that applies these tests to a fixpoint is what makes branch-and-bound for the job shop and the RCPSP practical; Carlier and Pinson's solution of the 10×10 job-shop instance is the historical demonstration. Theorem 3.8 extends the same reasoning from disjunctive to cumulative resources by replacing "no overlap" with "at most $R_k$ units per time unit", the energetic-reasoning viewpoint that the rest of Section 3.6.5 develops.

The results are elementary and proved; formalizing them fixes, once, what "feasible" means in the presence of the relation sets $C$ and $D$ and time windows, on top of the RCPSP model of mission I. That layer is reusable: the start-start distance matrix of Section 3.6.2 and the symmetric-triple rules of Section 3.6.3 are statements about the same schedules and the same relations. Nothing here is on the platform or in Mathlib.

Difficulty

The obvious argument for Theorem 3.7 is the correct one, and its difficulty is in the bookkeeping. If no activity of $J'$ starts first and none of $J''$ ends last, the first starter is some $\nu\in J\setminus J'$ and the last finisher some $\mu\in J\setminus J''$, and every activity of $J$ is processed inside $[S_\nu,\,S_\mu+p_\mu]\subseteq[r_\nu,d_\mu]$. Since the activities of a disjunctive set are pairwise non-overlapping, their total length $P(J)$ fits in that interval, contradicting (3.121). The formal work is the packing lemma: pairwise disjoint integer intervals inside an interval of length $L$ have total length at most $L$, which requires ordering the activities by start time and an induction that Mathlib does not supply.

The subtle point is the restriction $\nu\ne\mu$ in (3.121). The book allows it because "an activity which starts first cannot complete also last" when there are at least two activities with positive durations. The formal statement reads "starts first" and "ends last" with $\le$, which makes the theorem true without a positivity hypothesis: when the restriction empties the index set, $J\setminus J'=J\setminus J''=\{x\}$, the conclusion holds because $x$ cannot be both the unique first starter and the unique last finisher of a disjunctive set with two or more members. A solver should expect to handle that corner separately.

For the tests the extra step is turning "starts first" into a conjunction $i\to j$, which uses the disjunction between $i$ and $j$ together with positive processing times: with $p_j=0$ an activity could start at the same instant as $i$ without violating the disjunction, so the tests carry the positivity hypothesis that Theorem 3.7 itself does not need. Theorem 3.8 replaces the packing lemma by a work-counting lemma: over an interval of length $L$ a resource of capacity $R_k$ supplies at most $R_kL$ units, and every activity of $J$ consumes $r_{ik}p_i$ of them.

Formalization scope

Schedules are integer start-time vectors on Fin n, as in missions I and II, and a feasible schedule of this mission is one that is FeasibleSchedule for the RCPSP instance (mission I), respects the arcs of $C$ (RespectsArcs, mission II), satisfies the disjunctions of $D$ and lies within the time windows; these four hypotheses are the book's "feasible schedule" in Section 3.6 and are carried on every statement, although the arguments use only the last two, or, for Theorem 3.8, the resource constraint and the windows. The sets $C$ and $D$ are parameters, not derived from the instance, since propagation enlarges them.

Every inequality "$\max(\cdot)-\min(\cdot)<P$" is stated as the family of inequalities $d_\mu<r_\nu+P$ over the same index pairs. This is equivalent, avoids natural-number subtraction, and gives an empty index set the value the convention $\max\emptyset=-\infty$ would: the hypothesis is then vacuous. "Starts first" and "ends last" use $\le$. Proper-subset hypotheses $J'\subset J$, $J''\subset J$ are the book's; the first infeasibility test needs $J$ nonempty, and the input-or-output test needs $|J|\ge 2$.

A trivializing reading is ruled out on the disjunctive side by the nonemptiness hypotheses (an empty $J$ would make the infeasibility test's family vacuous and its conclusion false) and on the cumulative side by the observation that Theorem 3.8 with $J'=J''=\emptyset$ asserts infeasibility, which is the book's intended reading. Welcome contributions beyond the milestones: the input-negation and output-negation tests, the window-tightening rules of Section 3.6.4, and the SSD-matrix results of Section 3.6.2.

Selected references

• Peter Brucker and Sigrid Knust, Complex Scheduling, 2nd ed., Springer, 2012, Section 3.6. doi:10.1007/978-3-642-23929-8
• Jacques Carlier and Eric Pinson, An algorithm for solving the job-shop problem, Management Science 35 (1989). doi:10.1287/mnsc.35.2.164
• Philippe Baptiste, Claude Le Pape and Wim Nuijten, Constraint-Based Scheduling, Kluwer, 2001. doi:10.1007/978-1-4615-1479-4
• Ulrich Dorndorf, Erwin Pesch and Toàn Phan-Huy, Constraint propagation techniques for the disjunctive scheduling problem, Artificial Intelligence 122 (2000). doi:10.1016/S0004-3702(00)00040-0

Motivation

When several processors share a workload, the first question is how long the workload takes if it is spread out as well as possible. If a job may be interrupted and resumed later, possibly on another processor — preemption, the setting of operating systems, of communication links and of any resource that can be time-shared — the answer is a closed formula, and it is one of the oldest results in scheduling: McNaughton's wrap-around rule of 1959 (Scheduling with deadlines and loss functions, Management Science 6, doi:10.1287/mnsc.6.1.1) shows that on identical machines the optimal makespan is the larger of the longest job and the average load. Horvath, Lam and Sethi (A level algorithm for preemptive scheduling, Journal of the ACM 24, 1977, doi:10.1145/321992.321995) extended this to machines of different speeds, and Gonzalez and Sahni (Preemptive scheduling of uniform processor systems, Journal of the ACM 25, 1978, doi:10.1145/322047.322055) gave the fast algorithm with few preemptions. Brucker's Chapter 5 (doi:10.1007/978-3-540-69516-5) presents the level-algorithm version, and this mission formalizes the statements it proves about schedules.

Setting

There are $n$ jobs with processing requirements $p_1,\dots,p_n>0$ and $m$ uniform machines with speeds $s_1,\dots,s_m>0$: running job $i$ on machine $j$ for a period of length $\ell$ performs $s_j\ell$ units of its requirement, so the whole job would take $p_i/s_j$ time units there. Identical machines are the case $s_1=\dots=s_m=1$.

A preemptive schedule is a finite list of pieces, each a job, a machine, a start time and a stop time. It is feasible for the data $(s,p)$ when every piece lies in $[0,\infty)$, no two pieces on the same machine overlap, no two pieces of the same job overlap (a job is on at most one machine at any instant), and every job $i$ receives total work exactly $p_i$ over its pieces. Its makespan $C_{\max}$ is the largest stop time; the completion time $C_i$ of job $i$ is the largest stop time of one of its pieces. A schedule is nonpreemptive when every job consists of a single piece.

Following Section 5.1.2 the data are sorted, $p_1\ge\dots\ge p_n$ and $s_1\ge\dots\ge s_m$, with $n\ge m$, and one writes $P_j=\sum_{i\le j}p_i$, $S_j=\sum_{i\le j}s_i$. For a set $A$ of jobs, $h(A)=S_{|A|}$ if $|A|\le m$ and $h(A)=S_m$ otherwise: the largest combined speed that $|A|$ jobs can use at one instant.

Formalization targets

Goal — Theorem 5.8 (printed p. 127)

The optimal makespan of $Q\mid pmtn\mid C_{\max}$ is the bound (5.5):

$$w \;=\; \max\Bigl\{\max_{j=1}^{m-1}\frac{P_j}{S_j},\ \frac{P_n}{S_m}\Bigr\} ,$$

in the sense that some feasible preemptive schedule has makespan exactly $w$ and no feasible preemptive schedule has a smaller makespan.

The lower bound (5.5) (printed p. 125)

Every feasible preemptive schedule has makespan at least $w$.

$P\mid pmtn\mid C_{\max}$ (printed p. 108)

On identical machines, $LB=\max\{\max_i p_i,\ \tfrac1m\sum_i p_i\}$ is a lower bound on the makespan and is attained by some feasible preemptive schedule.

Condition (5.8) (printed p. 129)

The jobs can be scheduled preemptively within $[0,T]$ if and only if $\sum_{i\in A}p_i\le T\,h(A)$ for every set $A$ of jobs.

Theorem 5.7 (printed p. 121)

For $P\mid pmtn\mid\sum w_iC_i$ with nonnegative weights there is an optimal schedule without preemption.

Significance

Theorem 5.8 turns an optimization over an infinite family of schedules into a formula in the data, and the formula is tight in both directions: each of its terms is a resource bound that some schedule meets exactly. That is what makes preemptive makespan minimization one of the few parallel-machine problems that is solvable at all — its nonpreemptive counterpart $P2\parallel C_{\max}$ is NP-hard (p. 124) — and it is why the preemptive relaxation appears as a bound inside branch-and-bound methods for the nonpreemptive problems.

Condition (5.8) is the form in which the result is reused. It is a Hall-type condition, one inequality per set of jobs, and it is exactly what Section 5.1.2 needs to prove Theorem 5.9, which decides $Q\mid pmtn; r_i\mid L_{\max}$ by a maximum flow in an expanded network. Theorem 5.7 is the complementary statement for the other classical objective: for total weighted completion time preemption buys nothing, so the nonpreemptive solutions of Section 5.1.1 are optimal in the larger class too.

On status: every statement here is classical and proved, and the formalization adds a checked model of preemptive schedules. Mathlib has no scheduling material, and the platform's SchedulingAlgorithms series so far models only single-machine sequences (missions I, II, IV) and two-machine permutation flow shops (mission III), none of which allow a job to be split. The piece-list model of this mission is the first reusable object for preemptive and parallel-machine problems, and the later sections of Chapter 5 — $Q\mid pmtn; r_i\mid L_{\max}$, $P\mid pmtn\mid L_{\max}$ — are stated in it.

Difficulty

The obvious first idea for the goal is to run McNaughton's rule with the speeds ignored. It fails on uniform machines: filling machines one after another does not respect the constraint that a long job on a slow machine is not done when a short job on a fast one is. The correct idea is the level algorithm — always process the jobs of highest remaining requirement on the fastest free machines, sharing machines among tied jobs — and the difficulty is in the analysis rather than the idea. The proof of Theorem 5.8 has to show that the schedule it produces ends exactly at one of the terms of $w$: either no machine idles before the end, giving $P_n/S_m$, or the machines finish in speed order with the first $j$ jobs busy from time $0$, giving $P_j/S_j$. Making that case analysis rigorous requires tracking that the order of remaining requirements is preserved over time (the invariant (5.6)) and that ties are broken consistently.

A second, formal difficulty is that the level algorithm's output is defined by continuous-time events (the next completion, the next time two levels coincide), so producing an explicit finite list of pieces with the required properties is itself a construction. Any proof must build a concrete schedule; "the infimum of makespans equals $w$" is not the goal.

For the lower bound the trap is the opposite: it is tempting to argue only with total capacity $S_mT$, which gives $P_n/S_m$ but not $P_j/S_j$. The latter needs the rule that a job is on at most one machine at a time, so that $j$ jobs run at combined speed at most $S_j$; a model that let a job be split across machines simultaneously would make the theorem false, and the definition of feasibility rules it out explicitly.

Formalization scope

A schedule is a List of Pieces over jobs Fin n and machines Fin m, with real start and stop times. Feasibility is the three-part condition of the Setting, disjointness of two pieces meaning one stops no later than the other starts. Work is measured with the machine's speed, so the same definitions cover identical machines as the constant speed $1$. Pieces of length zero and unsorted lists are allowed; both are harmless.

The sorted orders are hypotheses Antitone p and Antitone s, the speeds and requirements are positive, $m\ge 1$ and, where the book assumes it, $n\ge m$. The book's normalization $s_1=1$ is not assumed: every statement here is invariant under scaling all speeds, and the book uses the normalization only for a running-time estimate. The bound $w$ is defined as the maximum of an explicit nonempty finite set, so no supremum of an empty or unbounded set occurs; $LB$ takes a proof that $n\ge 1$ so that $\max_i p_i$ is meaningful.

Two things are deliberately not stated. The level algorithm itself is not transcribed: Theorem 5.8 is stated as the existence of an optimal schedule of makespan $w$, which is what its proof establishes. And Theorem 5.9, the flow characterization for $Q\mid pmtn; r_i\mid L_{\max}$, is left for a later mission, since it needs release times and the expanded network on top of this model.

A trivializing reading is excluded by the existential form of the goal and of Theorem 5.7: each asserts that an optimal schedule exists, not merely that any optimal schedule has a property. A proof of the goal has to construct a schedule; a proof of Theorem 5.7 has to construct a nonpreemptive one that beats every preemptive competitor. Contributions welcome beyond the milestones: a general lemma that a feasible schedule can be normalized to sorted, positive-length pieces, and a proof that (5.8) for the sets $\{1,\dots,j\}$ is equivalent to $w\le T$.

Selected references

• Peter Brucker, Scheduling Algorithms, 5th ed., Springer, 2007, Chapter 5. doi:10.1007/978-3-540-69516-5
• Robert McNaughton, Scheduling with deadlines and loss functions, Management Science 6 (1959). doi:10.1287/mnsc.6.1.1
• E. C. Horvath, S. Lam and R. Sethi, A level algorithm for preemptive scheduling, Journal of the ACM 24 (1977). doi:10.1145/321992.321995
• Teofilo Gonzalez and Sartaj Sahni, Preemptive scheduling of uniform processor systems, Journal of the ACM 25 (1978). doi:10.1145/322047.322055