The OR Formalization Drive

Motivation

When a system is controlled indefinitely and decisions are frequent — a router admitting packets, a queue accepting jobs, a machine being maintained — discounting future rewards is often unjustified, and what matters is the long-run average reward per period. Puterman's Chapter 8 (doi:10.1002/9780470316887) develops the theory of this criterion, and its central object is a single equation, the average reward optimality equation $0=\max_{a\in A_s}\{r(s,a)-g+\sum_jp(j\mid s,a)h(j)-h(s)\}$, whose unknowns are a scalar gain $g$ and a bias function $h$. For unichain models, in which every stationary policy generates a Markov chain with one recurrent class, this equation determines the optimal gain and an optimal stationary policy. The results go back to Howard (Dynamic Programming and Markov Processes, MIT Press, 1960) for the recurrent case and to Blackwell (Discrete dynamic programming, Annals of Mathematical Statistics 33, 1962, doi:10.1214/aoms/1177704593) and Derman for the general finite case; Puterman's Section 8.4 proves them through the discounted theory of mission II, by letting the discount factor tend to one.

Setting

The model is stationary (Assumption 8.0.1): a finite set $S$ of states, for each $s$ a finite nonempty set $A_s$ of actions, a reward $r(s,a)$ and transition probabilities $p(j\mid s,a)$, none depending on the decision epoch. A policy $\pi\in\Pi^{HR}$ may randomize and may depend on the whole history; the deterministic stationary policy $d^\infty$ applies the decision rule $d:S\to A$ at every epoch. Its transition matrix is $P_d(i,j)=p(j\mid i,d(i))$.

For a policy $\pi$, $v^\pi_{N+1}(s)=\mathbb E^\pi_s[\sum_{t=1}^Nr(X_t,Y_t)]$ is the expected reward over $N$ epochs. Since the limit of $N^{-1}v^\pi_{N+1}(s)$ need not exist (Example 8.1.1), the chapter works with the lim sup and lim inf average rewards $g^\pi_+(s)$ and $g^\pi_-(s)$, and with $g^*_\pm(s)=\sup_{\pi}g^\pi_\pm(s)$. A policy $\pi^*$ is average optimal when $g^{\pi^*}_-(s)\ge g^\pi_+(s)$ for all $s$ and $\pi$, the strongest of the three criteria of Section 8.1.2.

The optimality residual is $B(g,h)(s)=\max_{a\in A_s}\{r(s,a)-g+\sum_jp(j\mid s,a)h(j)-h(s)\}$, and the optimality equation is $B(g,h)=0$. A decision rule is $h$-improving when it attains $\max_{a\in A_s}\{r(s,a)+\sum_jp(j\mid s,a)h(j)\}$ at every state. A transition matrix is unichain when it consists of a single recurrent class plus a possibly empty set of transient states, and the MDP is unichain when $P_d$ is unichain for every deterministic decision rule.

Formalization targets

Goal — Theorem 8.4.5 (printed p. 361)

For a finite unichain model: (a) some deterministic stationary policy is average optimal; (b) the optimality equation $B(g^*,h^*)=0$ has a solution, and (d) its scalar satisfies $g^*_+(s)=g^*_-(s)=g^*$ for every $s$; (c) for every solution, every $h^*$-improving decision rule gives an average optimal stationary policy.

Theorem 8.4.1 (printed p. 356)

If $B(g,h)\le 0$ then $g\ge g^*_+$; if $B(g,h)\ge 0$ then $g\le\sup_{d}g^{d^\infty}_-\le g^*_-$; if $B(g,h)=0$ then $g^*_+=g^*_-=g$.

Theorem 8.4.3 (printed p. 358)

In a finite unichain model $B(g,h)=0$ has a solution, and every solution has the same $g$.

Theorem 8.4.4 (printed p. 361)

If $B(g^*,h^*)=0$ and $d^*$ is $h^*$-improving, then $(d^*)^\infty$ is average optimal.

Significance

Theorem 8.4.1(c) is what the source calls "one of the most important results for average reward models": a solution of the optimality equation with constant $g$ pins down the optimal gain under every criterion at once, so that in finite unichain models the three optimality criteria of Section 8.1.2 coincide. Theorem 8.4.3 guarantees such a solution exists, and Theorem 8.4.4 reads an optimal policy off it. Together, Theorem 8.4.5 reduces the infinite-horizon average reward problem over all history-dependent randomized policies to a finite system of equations in $(g,h)$, which is what policy iteration, value iteration and linear programming solve in Sections 8.5 to 8.8.

The results are classical and proved. Formalizing them fixes the chain-structure hypothesis in a checkable form and pins down which criterion "average optimal" means, two places where the literature is loose. The platform's MarkovDecisionProcesses series has the finite-horizon (mission I) and discounted (mission II) models; this mission adds the undiscounted stationary model, the gains, and the unichain classification, on which Chapter 9's multichain optimality equations and Chapter 10's sensitive discount optimality can be built.

Difficulty

The obvious argument for Theorem 8.4.3 is to take the discounted optimal value $v^*_\lambda$ of mission II and let $\lambda\uparrow 1$. It fails as stated because $v^*_\lambda$ blows up like $(1-\lambda)^{-1}$; what converges is the Laurent expansion $v^{d^\infty}_\lambda=(1-\lambda)^{-1}ge+h+o(1)$ of the value of a fixed stationary policy, Corollary 8.2.4, and that expansion needs the limiting matrix $P_d^*$ and the deviation matrix $H_{P_d}$ of a unichain chain. So the proof must first develop the Markov chain theory of Section 8.2 and Appendix A, choose a subsequence of discount factors along which one policy is discount optimal (possible because $D^{MD}$ is finite), and only then pass to the limit in the discounted optimality equation.

Theorem 8.4.1 looks elementary and hides the analytic step: iterating $ge\ge r_d+(P_d-I)h$ along an arbitrary history-dependent policy and dividing by $N$ requires the telescoping term $N^{-1}(P^\pi_N-I)h$ to vanish, which uses boundedness of $h$, and requires the reduction from history-dependent randomized to Markov randomized policies (Theorem 8.1.2). For Theorem 8.4.4 the step is Corollary 8.2.7, that $r_d-ge+(P_d-I)h=0$ forces the gain of $d^\infty$ to be $g$, which is the multiplication by $P_d^*$ that annihilates $(P_d-I)$.

The traps are in the definitions. Recurrence and the unichain property must be stated so that the source's Example 8.4.3 comes out as the book says — the policy using $a_{1,1}$ has the absorbing state $s_2$ as its single recurrent class — and the optimality residual must use the lim sup / lim inf gains, since a definition through a limit that need not exist would be a junk value on the policies of Example 8.1.1.

Formalization scope

State and action spaces are Fintypes and admissible actions are nonempty Finsets, as in missions I and II; the stationary model is a new structure because mission II's DiscountedMDP bundles a discount factor, and carries the same data otherwise. Policies are history-dependent and randomized, so "average optimal" has its full strength; a stationary policy is the deterministic one built from a decision rule. Expected total reward is defined by the policy evaluation recursion, as in the earlier missions, rather than through a measure on trajectories.

Gains are Filter.limsup and Filter.liminf of $N^{-1}v^\pi_{N+1}(s)$ on $\mathbb R$; these are the source's because the sequence is bounded by $\max|r|$, and the suprema $g^*_\pm$ over the nonempty family of policies are genuine real suprema for the same reason. The residual $B(g,h)$ is a Finset.sup' over the admissible actions. Recurrence is "every state reachable from $i$ reaches $i$" and unichain is "any two recurrent states communicate", the definitions of Appendix A for finite chains, applied to $P_d$ for every admissible deterministic decision rule.

Restrictions relative to the printed text, all noted in the items: Theorem 8.4.1 is stated for finite $S$ where the source says countable, since the chapter's standing assumption and the model are finite; the gain $g^{d^\infty}$ of a stationary policy in (8.4.5) is written as its lim inf gain, which equals it; and the chain $ge=g^*=g^*_+=g^*_-$ of (8.4.6) is stated through $g^*_+$ and $g^*_-$, since $g^*$ presupposes existing limits. Nothing is trivialized: the existential in Theorem 8.4.5(a) has to produce a decision rule, and $B(g,h)=0$ with a junk maximum is impossible since every $A_s$ is nonempty. Welcome contributions beyond the milestones: Theorem 8.1.2 (reduction to Markov policies), Corollary 8.2.7 (the gain of a stationary policy from the evaluation equations), and the equivalence of the three optimality criteria in finite models.

Selected references

• Martin L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, Wiley, 1994, Chapter 8. doi:10.1002/9780470316887
• Ronald A. Howard, Dynamic Programming and Markov Processes, MIT Press, 1960.
• David Blackwell, Discrete dynamic programming, Annals of Mathematical Statistics 33 (1962). doi:10.1214/aoms/1177704593
• Cyrus Derman, Finite State Markovian Decision Processes, Academic Press, 1970.
• Paul J. Schweitzer and Awi Federgruen, The functional equations of undiscounted Markov renewal programming, Mathematics of Operations Research 3 (1978). doi:10.1287/moor.3.4.308

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

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.

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.

Why orders swing more than sales

Procter & Gamble observed in the 1990s that the orders its distributors placed for diapers were far more variable than the retail sales of diapers, and that its own orders to suppliers were more variable still, although the end demand for diapers is about as stable as demand gets. The phenomenon, a growing amplification of variability as one moves upstream in a supply chain, is the bullwhip effect. Lee, Padmanabhan and Whang (1997) argued that it is not a symptom of irrational behaviour: four rational responses of an inventory manager to their own environment each produce it. Chapter 13 of Snyder and Shen's Fundamentals of Supply Chain Theory (2019) makes three of the four quantitative, following Chen, Drezner, Ryan and Simchi-Levi (2000) for demand signal processing, Lee et al. for the rationing game, and Cachon (1999) for order batching. This mission formalizes those three models and the theorems the chapter proves about them.

Setting

Demand signal processing. A retailer faces a demand process $D_t$, $t \in \mathbb{Z}$, that follows the stationary first-order autoregressive model

$$D_t = d + \rho D_{t-1} + \epsilon_t,$$

with a constant $d \ge 0$, a correlation constant $-1 < \rho < 1$, and errors $\epsilon_t$ that are independent $N(0, \sigma^2)$ variables, each independent of the demands before period $t$. In steady state every $D_t$ has the law $N\big(d/(1-\rho),\ \sigma^2/(1-\rho^2)\big)$. The retailer replenishes with a lead time of $L$ periods under a base-stock policy but does not know the demand parameters, so it estimates the lead-time demand from a moving average of the previous $m \ge 1$ demands:

$$\hat\mu^L_t = \frac{L}{m}\sum_{i=1}^m D_{t-i}, \qquad \hat\sigma^L_{et} = C\sqrt{\frac{1}{m}\sum_{i=1}^m e_{t-i}^2}, \qquad e_t = D_t - \hat\mu^1_t,$$

and sets the base-stock level $S_t = \hat\mu^L_t + z_\alpha \hat\sigma^L_{et}$, where $z_\alpha$ is a safety factor. The book writes the constant in $\hat\sigma^L_{et}$ as $C_{L\rho}$ and does not give its form; here it is a free parameter $C$. Each period the retailer orders $Q_t = S_t - S_{t-1} + D_{t-1}$, which may be negative. In Lean the process is the structure AR1Demand, whose fields are the parameters, the errors, the demands, the recursion, the independence properties and the stationary law; muHat, err, sigmaHat, baseStock and order are the five quantities above.

Order batching. $N$ retailers face independent $N(\mu, \sigma^2)$ demands in every period and each orders once every $R \ge 1$ periods, the order being its demand over the previous $R$ periods. The supplier's order in a given period is the total ordered by the retailers whose ordering day falls in that period. Three patterns are compared: random ordering, in which each retailer's day is uniform over the $R$ days, so the number $X$ of retailers ordering on a given day is binomial$(N, 1/R)$; positively correlated ordering, in which all retailers order on the same day, so $X = N$ with probability $1/R$ and $0$ otherwise; and balanced ordering, in which the retailers are spread as evenly as possible, so with $N = MR + k$, $0 \le k < R$, $X$ is $M+1$ with probability $k/R$ and $M$ otherwise. The structure BatchOrders P N R mu sigma carries the demands, the ordering count $X$ independent of them, and supplierOrder, the sum of the last $R$ demands of retailers $1, \dots, X$; each pattern enters a theorem as a hypothesis on the law of $X$.

Rationing game. Two identical retailers face single-period demand with distribution function $F$, holding cost $h$ and stockout penalty $p$, so the newsvendor quantity $Q^*$ satisfies $F(Q^*) = p/(h+p)$. With probability $r$ the supplier can deliver only $A_1 < 2Q^*$ units in total and allocates them pro rata to the orders, retailer 1 receiving $A_1 Q_1/(Q_1 + Q_2)$; with probability $1 - r$ supply is unlimited. Retailer 1's expected cost when the retailers order $Q_1$ and $Q_2$ is

$$g_1(Q_1) = (1-r)\,\mathrm{nv}(Q_1) + r\,\mathrm{nv}\!\Big(\frac{A_1 Q_1}{Q_1 + Q_2}\Big),$$

with $\mathrm{nv}$ the newsvendor cost; this is rationingCost.

Formalization targets

Goal: Theorem 13.2, demand signal processing

$$\frac{\mathrm{Var}[Q_t]}{\mathrm{Var}[D_t]} \;\ge\; 1 + \Big(\frac{2L}{m} + \frac{2L^2}{m^2}\Big)(1 - \rho^m),$$

with equality when $z_\alpha = 0$. This is bullwhip_signal_processing. The bound exceeds $1$ whenever $L > 0$, whatever the value of $\rho$: a lead time and a moving-average forecast are enough to produce the effect.

Supporting targets

The chapter's own route to the goal, each a milestone: the steady-state moments (13.2) to (13.4), $\mathbb{E}[D_t] = d/(1-\rho)$, $\mathrm{Var}[D_t] = \sigma^2/(1-\rho^2)$ and $\mathrm{Cov}[D_t, D_{t-k}] = \rho^k \mathrm{Var}[D_t]$; the identity $Q_t = (1 + L/m) D_{t-1} - (L/m) D_{t-m-1} + z_\alpha(\hat\sigma^L_{et} - \hat\sigma^L_{e,t-1})$; Lemma 13.1, $\mathrm{Cov}[D_{t-i}, \hat\sigma^L_{et}] = 0$ for $1 \le i \le m$; the vanishing of the cross term (13.12); and the variance of the demand part, $\big(1 + (2L/m + 2L^2/m^2)(1 - \rho^m)\big)\mathrm{Var}[D_t]$.

Order batching, Theorem 13.4: under the three patterns the supplier's order has mean $N\mu$ and

$$\mathrm{Var}[Q^c_t] \ge \mathrm{Var}[Q^r_t] \ge \mathrm{Var}[Q^b_t] \ge N\sigma^2,$$

through the three variance formulas $N\sigma^2 + \mu^2 N(R-1)$, $N\sigma^2 + \mu^2 N^2 (R-1)$ and $N\sigma^2 + \mu^2 k(R-k)$.

The rationing game, Theorem 13.3: if $Q > 0$ is a symmetric Nash equilibrium, that is, $Q$ minimizes $g_1$ over positive order quantities when the other retailer orders $Q$, then $Q > Q^*$.

Significance

The three theorems are the quantitative core of the chapter. Theorem 13.2 is the single-stage building block that Theorems 13.6 and 13.7 later iterate along a serial chain, giving the product-form and the exponential lower bounds on the amplification at stage $k$; its comparative statics, the bound decreasing in $m$ and increasing in $L$, are the basis of the remedies the chapter recommends (shorter lead times, smoother forecasts, sharing point-of-sale data). Theorem 13.4 ranks the ordering patterns and justifies the advice to balance ordering days when batching cannot be avoided. Theorem 13.3 shows that pro-rata rationing alone inflates orders; the book is careful to note that inflated orders are not by themselves inflated variances, and that the variance statement for this model is due to Rong, Shen and Snyder (2017).

None of these results has a machine-checked proof. The book's proofs of Theorems 13.2 and 13.4 are complete but informal, and the proof of Lemma 13.1 is omitted with a citation to Ryan's 1997 thesis; formalizing it requires a self-contained argument. The variance decomposition of $Q_t$ and the conditioning argument for Theorem 13.4 are reusable for the multistage results of Sect. 13.2.5, which are natural follow-up missions on the same definitions.

Difficulty

The obvious computation of $\mathrm{Var}[Q_t]$ expands the order into its demand part and its safety-stock part and hopes the cross term disappears. It does, but not for a reason visible in the formulas: $\hat\sigma^L_{et}$ is a square root of a sum of squares of forecast errors, a nonlinear function of $m + m$ demands, and its covariance with a single demand is zero only because the errors are jointly Gaussian with mean zero and $\hat\sigma$ is an even function of them, so the covariance is the expectation of an odd function of a centred Gaussian vector. That is Lemma 13.1, and the vanishing of the cross term needs two further covariances, $\mathrm{Cov}[D_{t-1}, \hat\sigma^L_{e,t-1}]$ and $\mathrm{Cov}[D_{t-m-1}, \hat\sigma^L_{et}]$, which the book reduces to the lemma through the recursion (the second reduction divides by $\rho$) but which hold for every $\rho$ by the same symmetry. A solver must set up the joint Gaussian structure of the demand vector and prove the odd-function argument; nothing in Mathlib does this directly.

The second obstacle is that the moments (13.2) to (13.4) are not assumed but derived: the structure carries the stationary law of each $D_t$ and the independence of $\epsilon_t$ from the past, and the autocovariance $\rho^k \mathrm{Var}[D_t]$ has to be obtained from the recursion by induction on the lag, with integrability supplied by the Gaussian laws.

For Theorem 13.4 the work is the conditioning on $X$: given $X = x$ the supplier's order is a sum of $xR$ independent normals, so its conditional mean is $xR\mu$ and conditional variance $xR\sigma^2$, and the total variance is $\mathbb{E}[\mathrm{Var}[Q \mid X]] + \mathrm{Var}[\mathbb{E}[Q \mid X]]$. The order is defined by a sum over retailers $i < X$, so the independence of $X$ from the demands has to be used through the indicator structure rather than through a conditional-expectation library result.

For Theorem 13.3 the argument is a first-order condition. It requires that the newsvendor cost be differentiable with derivative $(h+p)F(y) - p$, which holds when $F$ is continuous, and that the symmetric equilibrium be an interior minimizer, which is why $Q > 0$ and the minimization over $Q_1 > 0$ are hypotheses.

Formalization scope

Time is indexed by $\mathbb{Z}$ so that $D_{t-m-1}$ exists for every $t$. AR1Demand asserts the recursion for every outcome, the independence of the whole error family, the independence of $\epsilon_t$ from $(D_s)_{s < t}$, and the stationary law of every $D_t$; these are the "steady-state" assumptions the book makes in words. The structure is satisfiable: the stationary Gaussian AR(1) process on a full-measure set of error sequences has all these properties. The constant $C_{L\rho}$ is a free real parameter $C$; no theorem depends on its value.

The goal divides by $\mathrm{Var}[D_t]$, which is $\sigma^2/(1-\rho^2) > 0$ under the structure's hypotheses $\sigma > 0$ and $|\rho| < 1$, so the ratio is a genuine quotient. Mathlib's ProbabilityTheory.variance and covariance are used; both are the ordinary real quantities for square-integrable variables, which every variable here is, $\hat\sigma^L_{et}$ included.

In BatchOrders the demands are indexed by Fin N × Fin R, the count $X$ is a natural-valued random variable bounded by $N$ and independent of the demand family, and supplierOrder sums the $R$ demands of retailers $1, \dots, X$, the book's "without loss of generality" choice. The laws of $X$ are hypotheses on point probabilities P.real {ω | X ω = j}; with $R \ge 1$ each of the three families of hypotheses is satisfiable by a structure with the corresponding law. The subtractions $R - 1$ and $R - k$ are real.

In the rationing game the demand law is a probability measure on $\mathbb{R}$ whose distribution function is continuous and strictly increasing on $[0, \infty)$; the newsvendor loss is assumed integrable at every order quantity. The pro-rata allocation uses Lean's total division, which is never at $0$ in the theorem since $Q_1 + Q_2 > 0$.

Beyond the ten milestones, the multistage Theorems 13.6 and 13.7 and the centralized-information bound of Theorem 13.5 are welcome as extensions on the same AR1Demand.

Selected references

• L. V. Snyder and Z.-J. M. Shen, Fundamentals of Supply Chain Theory, 2nd ed., Wiley, 2019, Chapter 13. https://doi.org/10.1002/9781119584445
• H. L. Lee, V. Padmanabhan and S. Whang, Information distortion in a supply chain: the bullwhip effect, Management Science 43(4), 1997. https://doi.org/10.1287/mnsc.43.4.546
• F. Chen, Z. Drezner, J. K. Ryan and D. Simchi-Levi, Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting, lead times, and information, Management Science 46(3), 2000. https://doi.org/10.1287/mnsc.46.3.436.12069
• G. P. Cachon, Managing supply chain demand variability with scheduled ordering policies, Management Science 45(6), 1999. https://doi.org/10.1287/mnsc.45.6.843
• Y. Rong, Z.-J. M. Shen and L. V. Snyder, The impact of ordering behavior on order-quantity variability: a study of forward and reverse bullwhip effects, Naval Research Logistics 64(1), 2017. https://doi.org/10.1002/nav.21757

Pooling as a design principle

A firm that holds inventory in five warehouses needs more safety stock than one that holds the same inventory in one warehouse, because the demands of five regions do not all run high at once. Eppen (1979) made this precise for a multi-location newsvendor and gave it its name, the risk-pooling effect. Chapter 7 of Snyder and Shen's Fundamentals of Supply Chain Theory (2019) follows the same idea through three settings in which pooling happens without physical consolidation: two retailers who ship stock to each other after seeing demand (transshipments, after Tagaras 1989), and plants that can each make more than one product (process flexibility, after Jordan and Graves 1995). The chapter's capstone is the theorem of Simchi-Levi and Wei (2012) that, among designs in which every plant makes two products and every product is made at two plants, a single long chain through all of them is best. This mission formalizes the chapter's numbered results, with that theorem as its goal.

Setting

Risk pooling. $N$ distribution centers face normally distributed per-period demands $D_i \sim N(\mu_i, \sigma_i^2)$ with correlation coefficients $\rho_{ij}$, and each runs a base-stock policy with holding cost $h$ and backorder cost $p$ per unit per period, so its optimal expected cost is the optimal newsvendor cost optNvCost h p D, the infimum over base-stock levels $S$ of $\mathbb{E}[h(S - D)^+ + p(D - S)^+]$. Merging the centers gives one facing the total demand, normal with mean $\sum_i \mu_i$ and variance $\sigma_0^2 = \sum_i \sum_j \sigma_i \sigma_j \rho_{ij}$ (pooledVariance).

Transshipments. Two retailers $i, j$ with base-stock levels $S_i, S_j$ face independent demands. After demand is observed, under complete pooling the retailer with a surplus sends the retailer with a shortage $Y_{ji} = \min\{S_j - D_j,\ D_i - S_i\}$ units (transship), and nothing moves otherwise. The type-1 service level is the probability of no stockout, $\alpha^0_i = \Pr[D_i \le S_i]$ without and $\alpha_i = \Pr[D_i - S_i \le Y_{ji}]$ with transshipments; the type-2 service level is the fill rate, one minus expected unmet demand over expected demand, $\beta^0_i$ and $\beta_i$ likewise.

Process flexibility. A flexibility design on $n$ products and $n$ plants is a set $E$ of (product, plant) pairs, an edge $(i, j)$ meaning plant $j$ can make product $i$. Given a demand realization $d$ and a common plant capacity $C$, the performance $P(d, E)$ (perf) is the maximum sales obtainable by assigning production along the edges of $E$ without exceeding any capacity or demand, the linear program (7.22) to (7.26). A balanced system (BalancedSystem) has equal capacities and an exchangeable demand vector, one whose joint law is invariant under permutations of the products, and $[E] = \mathbb{E}[P(D, E)]$ is the expected performance (expPerf). The named designs are the dedicated design $D_n = \{(i, i)\}$, the long chain $C_n$ in which plant $j$ also makes product $j + 1$ (and plant $n$ makes product $1$), the open chain $L_k$ obtained from $C_k$ by deleting the edge $(1, k)$, and $L^n_k$, the open chain on the first $k$ pairs together with the dedicated edges of the rest. A 2-flexibility design (TwoFlex) is one in which every product has exactly two plants and every plant exactly two products; $C_n$ is one, and so is any union of disjoint shorter chains.

Formalization targets

Goal: Theorem 7.9

For a balanced system of size $n \ge 2$ with exchangeable demand,

$$C_n \in \arg\max_{A \in \mathcal{F}_2} [A],$$

that is, $C_n$ is a 2-flexibility design and $[A] \le [C_n]$ for every 2-flexibility design $A$. This is long_chain_optimal.

Supporting targets

The chapter's route to the goal: Lemma 7.5, supermodularity of sales in the flexible edges of the long chain for every realization, $P(d, E) + P(d, E \setminus \{\alpha, \beta\}) \ge P(d, E \setminus \{\alpha\}) + P(d, E \setminus \{\beta\})$ for $E \subseteq C_n$; Corollary 7.6, the same in expectation; Lemma 7.7, the increments $[L^n_{k+1}] - [L^n_k]$ are nondecreasing in $k$, ending with $[C_n] - [L^n_n]$; and Lemma 7.8, $[C_n] = n([L_n] - [L_{n-1}])$.

Risk pooling, Theorem 7.1: $g^*_C \le g^*_D$, the optimal cost of the merged center is at most the sum of the optimal costs of the separate ones, with the covariance inequality $\sqrt{\sum_i\sum_j \sigma_i\sigma_j\rho_{ij}} \le \sum_i \sigma_i$ as a separate lemma.

Transshipments, Theorems 7.2 to 7.4: $\alpha_i = \alpha^0_i + |\partial\mathbb{E}[Y_{ji}]/\partial S_i|$, $\beta_i = \beta^0_i + \mathbb{E}[Y_{ji}]/\mathbb{E}[D_i]$, and all four post-transshipment service levels are nondecreasing in $S_i$.

Significance

Theorem 7.9 is the analytical answer to a question that had been settled only by simulation: Jordan and Graves reported that one chain through all plants achieves nearly twice the sales benefit of three short chains with the same number of edges, and Simchi-Levi and Wei proved that no arrangement of the same edge budget does better. It is the justification for the chaining guideline used in automotive and semiconductor capacity planning, and Lemma 7.8, which expresses the long chain through open chains, is what makes the long chain's performance computable by a greedy pass. Theorem 7.1 is the quantitative basis for consolidation decisions and for postponement, since a generic product is pooled inventory. Theorems 7.2 to 7.4 quantify what transshipments buy in service, which is the argument for allowing them despite their cost.

None of these results has a machine-checked proof. The book proves Lemma 7.7, Lemma 7.8 and Theorem 7.9 in full given Lemma 7.5, which it cites to Simchi-Levi and Wei, and omits the proofs of Theorems 7.3 and 7.4 and the identity (7.30) behind Lemma 7.8. Formalizing Lemma 7.5 and (7.30) means formalizing the structure of maximum flows on a cycle, which is reusable for the later results of Simchi-Levi and Wei on the long chain's performance relative to full flexibility and for the multi-echelon flexibility models the chapter cites.

Difficulty

The obvious approach to Theorem 7.9 is to compare $C_n$ with an arbitrary 2-flexibility design directly. Nothing in the definitions supports that: the two designs share no structure beyond their degree sequences. The book's argument instead routes everything through the long chain's own edges. Lemma 7.5 gives supermodularity only for subsets of $C_n$, and the decomposition of an arbitrary 2-flexibility design into disjoint cycles, each a relabeled long chain on a subsystem, is what allows the comparison. A solver must therefore prove that a 2-regular bipartite graph is a disjoint union of even cycles, that exchangeability makes every relabeling of a cycle worth the same as $C_{n_j}$ on its subsystem, and that the performance of a disjoint union is the sum of the performances of its parts.

Lemma 7.5 itself is where the combinatorics lives. It says that on the cycle $C_n$ the maximum flow is supermodular in the flexible edges, and the proof in Simchi-Levi and Wei goes through the structure of augmenting paths on a cycle. The natural first idea, that supermodularity follows from some general property of maximum flows, is false: maximum flow is not supermodular in arbitrary edge sets, and the lemma is specific to subsets of a single cycle.

Lemma 7.7 is where exchangeability is used, and it is used in a way that is easy to state and tedious to formalize: removing the edge $(2, 1)$ from $L^n_{k+1}$ leaves a design that is $L^n_k$ only after the pair $1$ is moved to the end, so the argument needs the invariance of $[E]$ under relabeling the products and plants by a common permutation. The book notes that Lemma 7.7, unlike Lemma 7.5, is false realization by realization.

For the transshipment theorems, the book differentiates a density formula by Leibniz's rule. Under the weaker hypothesis stated here, laws without atoms and with finite means, the derivative of $\mathbb{E}[Y_{ji}]$ in $S_i$ has to be obtained by dominated convergence from the pointwise derivative of a piecewise-linear function whose kinks lie on null sets.

Formalization scope

perf is a supremum over a set of reals, nonempty because $y = 0$ is feasible when $d \ge 0$ and $C \ge 0$, and bounded by $\sum_i d_i$; the demand is nonnegative for every outcome and the capacity nonnegative in BalancedSystem, and Lemma 7.5 carries these as hypotheses. The supremum is attained, but the definition does not assert it. Expected performance is a Lebesgue integral; the demand is integrable by assumption and $P(d, E)$ is $1$-Lipschitz in $d$, so the integrand is integrable, and a solver must prove this measurability rather than assume it.

Exchangeability is the equality of the laws of $(D_{\sigma(i)})_i$ and $(D_i)_i$ for every permutation $\sigma$. Designs are finite sets of pairs of Fin n; the chains are defined with finRotate, so indices wrap modulo $n$ and the closing edge of $C_n$ is $(1, n)$ in the book's numbering, which is the edge its proofs and Figure 7.3(c) use. Lemma 7.8 involves open chains on subsystems of sizes $n$ and $n - 1$; these are designs on Fin k evaluated on the first $k$ coordinates of the demand (subDemand, subPerf).

Theorem 7.1 states the optimal costs as infima of the newsvendor cost over all base-stock levels, on Mathlib's gaussianReal; a nonpositive pooled variance gives a degenerate law, for which the inequality still holds, so the statement is not trivialized by that convention. The transshipment theorems take the two demand laws as probability measures on $\mathbb{R}$ with no atoms (Theorem 7.2) and finite, positive means; the quantity $Y_{ji}$ is defined for all outcomes and the service levels are probabilities and expectations under the product law.

The definition module is shared by all eleven items. Beyond the milestones, formalizing the identity (7.30) as its own lemma and the disjoint-union additivity of perf would be natural contributions.

Selected references

• L. V. Snyder and Z.-J. M. Shen, Fundamentals of Supply Chain Theory, 2nd ed., Wiley, 2019, Chapter 7. https://doi.org/10.1002/9781119584445
• G. D. Eppen, Effects of centralization on expected costs in a multi-location newsboy problem, Management Science 25(5), 1979. https://doi.org/10.1287/mnsc.25.5.498
• G. Tagaras, Effects of pooling on the optimization and service levels of two-location inventory systems, IIE Transactions 21(3), 1989. https://doi.org/10.1080/07408178908966208
• W. C. Jordan and S. C. Graves, Principles on the benefits of manufacturing process flexibility, Management Science 41(4), 1995. https://doi.org/10.1287/mnsc.41.4.577
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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