Introduction
The rapid growth of electric vehicles (EVs) has increased the need for reliable charging and battery-swapping services. The International Energy Agency projects that the global EV fleet could reach about 250 million vehicles by 2030 under its Stated Policies Scenario, excluding two- and three-wheelers ([19 ] ). Battery swapping offers faster energy replenishment than plug-in charging, but its infrastructure deployment has lagged behind EV adoption ([30 , 8 ] ). A central operational challenge is to maintain sufficient ready-battery inventory despite limited station capacity, volatile demand, and recharging delays. One way to improve battery availability is to combine centralized replenishment with local charging. We refer to this hybrid configuration as “Swap-Locally, Charge-Adaptively (SLCA)” as illustrated in Figure 1 . Ready batteries can be supplied from a centralized charging facility, while depleted batteries retained at the swapping station can be recharged locally and returned to the ready-battery inventory. Related centralized charging-and-swapping configurations have been studied in the literature [26 , 24 , 32 ] . Charging and battery-swapping infrastructure has also received policy support in cities such as Shanghai and Shenzhen [41 , 46 ] .
Figure 1. Schematic of the SLCA operating system
Our operational model captures a distinctive feature of this hybrid system: central replenishment is actively controlled, whereas local charging provides automatic recovery. After the station chooses its ready-battery level, a fixed fraction of the depleted batteries remaining on site is recharged before the next decision epoch. Replenishing more ready batteries therefore reduces immediate shortage exposure but leaves fewer depleted batteries available for local recovery, creating an intertemporal trade-off between current service and future endogenous supply. This trade-off becomes especially difficult to manage when demand is serially dependent but its probability law is not reliably known. A central question is therefore how the recovery intensity changes the value of centralized replenishment as the remaining horizon evolves. Although correlated demand and evolving forecasts have long been studied in dynamic inventory models [36 , 37 , 7 , 15 , 12 , 10 ] , these approaches typically require a specified stochastic demand model. Such a model may be difficult to estimate and sensitive to misspecification [5 , 13 , 14 ] , particularly for newly deployed swapping stations with limited historical data.
Distributionally robust optimization (DRO) provides a natural way to address this form of model uncertainty by optimizing against the worst-case distribution within an ambiguity set constructed from available information [38 , 6 , 11 ] . Such sets may be defined through moments [38 , 17 , 34 ] or through distances from a reference distribution [16 , 22 ] . In multistage problems, however, a joint ambiguity set alone does not generally provide a dynamically consistent model. Distributional restrictions may couple uncertainty across histories, preventing the problem from decomposing after new demand is observed and causing continuation decisions obtained by reoptimization to differ from those implied by the original plan, i.e., time inconsistency [35 , 52 ] . This motivates ambiguity models that specify how admissible conditional distributions evolve with the observed demand history.
Several studies develop dynamic programming approaches for distributionally robust multistage problems [1 , 31 , 27 ] . A key requirement for such formulations is rectangularity : after new information is observed, the admissible joint distributions can be decomposed into stagewise conditional ambiguity sets [20 , 43 , 53 ] . This structure ensures time consistency and allows the worst-case expectation to be evaluated recursively through dynamic programming. Even under rectangularity, however, multistage inventory problems remain difficult to analyze, and most existing approaches rely on numerical methods. Although structural policies such as nonstationary or history-dependent ( s , S ) (s, S) ( s , S ) [2 , 23 ] , explicit finite-horizon policies under dynamically dependent demand remain scarce.
We therefore model demand through stagewise conditional-mean information. Specifically, the ambiguity set is constructed so that, after observing the latest demand, the next conditional distribution may be any distribution on the bounded support whose mean equals that observation. Equivalently, the demand process satisfies the martingale condition
E [ D ~ t ∣ F t + 1 ] = D ~ t + 1 , t ∈ [ T ] , \mathbb{E}[\widetilde{D}_t \mid\mathcal{F}_{t+1}] = \widetilde{D}_{t+1}, \qquad t \in[T], E [ D t ∣ F t + 1 ] = D t + 1 , t ∈ [ T ] , with D ~ T + 1 : = μ \widetilde{D}_{T+1} := \mu D T + 1 := μ
We make three main contributions.
We develop a time-consistent distributionally robust inventory model that integrates central replenishment with automatic local recovery under martingale demand ambiguity. Using only sequentially updated conditional-mean information, the model accommodates temporally dependent demand without requiring a fully specified distribution. We establish the existence of an optimal order-up-to policy for every finite horizon.
We characterize how automatic recovery shapes optimal replenishment decisions. Two critical recovery rates identify four operating regimes and determine how replenishment incentives evolve with the planning horizon. We derive exact two-period solutions and, under a regular penalty condition, explicit finite-horizon policies and worst-case demand distributions. In particular, central replenishment can remain beneficial at low recovery rates but becomes unnecessary in all nonterminal periods beyond a critical threshold. We also provide approximation policies with bounded optimality gaps for the complementary case.
We investigate how directional demand dependence changes replenishment behavior. Under submartingale ambiguity, sufficiently long horizons can make full-capacity replenishment optimal, whereas supermartingale ambiguity reveals that stronger local recovery can sometimes encourage central replenishment. Numerical experiments further demonstrate the value and limitations of M-DRO: it is particularly competitive under highly dispersed demand and limited historical information, while its advantage depends on the reliability of conditional-mean information and the operating environment.
§ 2 reviews related literature, and § 3 presents the model. § 4 –§ 6 develop the analytical results, followed by numerical experiments in § 7 and conclusions in § 8 . Proofs and supplementary results appear in the Online Appendix.
Notations: For any positive integer n n n [ n ] : = { 1 , … , n } [n] := \{1,\ldots,n\} [ n ] := { 1 , … , n } P 0 ( Z ) \mathcal{P}_0(\mathcal{Z}) P 0 ( Z ) Z \mathcal{Z} Z P \mathbb{P} P Q \mathbb{Q} Q E P [ ⋅ ] \mathbb{E}_{\mathbb{P}}[\cdot] E P [ ⋅ ] E Q [ ⋅ ] \mathbb{E}_{\mathbb{Q}}[\cdot] E Q [ ⋅ ] δ z \delta_z δ z z z z ( z ) + : = max { z , 0 } (z)_+ := \max\{z,0\} ( z ) + := max { z , 0 }
Related Literature
Highly relevant literature can be divided into two streams: the operations management research of EV battery swapping stations, and the discussion of distributionally robust optimization in inventory management, both in static and dynamic.
Battery-Swapping Inventory and Recovery Operations
EVs with battery-swapping services have been experiencing a resurgence in recent years [28 , 50 , 57 ] . Battery-swapping research studies network design, inventory, and charging operations. For example, Qi et al. [32 ] develop a joint location and repairable-inventory model and show that centralized charging can achieve significant cost efficiencies while maintaining service performance. Liang et al. [25 ] study the real-time operation of battery swapping–charging systems and propose a multi-agent deep reinforcement learning framework. Their approach coordinates swapping and charging decisions under stochastic demand and system dynamics. Shaker et al. [40 ] investigate the joint planning of electric-vehicle battery swapping stations and the distribution grid under centralized charging. They develop an integrated planning model that captures the interactions between station deployment, battery charging, and grid constraints. These studies address different planning and operating decisions, including dynamic charging control. The single-station benchmark here abstracts from network allocation and charging schedules to isolate how automatic recovery changes replenishment geometry. To account for demand uncertainty, these works often formulate two-stage stochastic optimization models for scheduling and routing decisions (see Schneider et al. [39 ] , Sun et al. [48 ] , Ni et al. [29 ] , He et al. [18 ] ). Some studies model inventory control problems as Markov decision processes (MDP). However, solving the resulting DP via backward induction often becomes computationally intractable due to the curse of dimensionality. Consequently, most of this literature resorts to approximate solution methods (see Asadi and Pinkley [4 ] , Asadi and Nurre Pinkley [3 ] ). Our work provides an explicit finite-horizon characterization of the optimal policy under the regular penalty condition. Its unit of analysis is a single station with immediately available central supply and a fixed recovery rule. Central-facility capacity, transportation lead times, multi-station allocation, and charging queues are outside this benchmark.
Distributionally Robust Optimization in Inventory Management
A conventional approach to decision making under uncertainty is stochastic optimization. Common stochastic optimization methods include sample average approximation (Shapiro and Kleywegt [45 ] ) and stochastic approximation methods (Shapiro and Homem-de Mello [44 ] ). Moreover, the distribution of random variables can be difficult to predict or fit accurately, and imposing a probability distribution can result in biased decisions or overfitting. DRO addresses this uncertainty by optimizing over a specified set of probability laws; its guarantees depend on the information used to construct that set. In practice, Qiu et al. [33 ] develop a distributionally robust optimization framework for multi-product inventory decisions under budget constraints, accounting for both demand and yield uncertainties. Their approach provides inventory policies that are robust to distributional ambiguity while controlling total investment cost. Kang et al. [21 ] use distributional information to reduce conservatism in robust linear optimization while retaining probabilistic feasibility guarantees, and apply the method to inventory control with quality-of-service constraints. Yin et al. [54 ] study a distributionally robust multi-period humanitarian relief network design problem that integrates facility location, inventory, allocation, and evacuation decisions. Their model enhances network resilience by jointly accounting for uncertainty across multiple planning stages. For more details, Zhang et al. [55 ] and Zhang et al. [56 ] provide a comprehensive survey of robust optimization approaches in inventory management, reviewing modeling frameworks, uncertainty sets, and solution methods. These papers synthesize theoretical developments and applications, highlighting how robust optimization addresses demand, cost, and supply uncertainties in inventory systems.
For multistage decisions, the distinction between a joint ambiguity set and a rectangular family of conditional laws is central. Robust dynamic programming provides the decomposition for rectangular transition uncertainty ([20 ] ), while distributional constraints linking histories require separate attention to dynamic consistency ([43 ] ). We use this conditional framework to analyze the inventory coupling created by automatic local recovery. Our analysis proceeds from one-dimensional conditional moment problems to demand envelopes, recursive demand points and inventory thresholds, and finite-horizon closed-form policies.
Model Formulation
We formulate a finite-horizon inventory problem with central replenishment, automatic local recovery, and rectangular conditional-mean ambiguity. The Bellman formulation is valid at every finite horizon; the explicit closed-form representations developed later require additional regime-specific conditions.
System dynamics and demand information
We consider T T T t t t T , T − 1 , … , 1 T,T-1,\ldots,1 T , T − 1 , … , 1 t t t y t y_t y t x t ∈ [ y t , U ] x_t \in[y_t,U] x t ∈ [ y t , U ] U > 0 U>0 U > 0 x t x_t x t U − x t U-x_t U − x t x t − y t x_t-y_t x t − y t D ~ : = ( D ~ T , … , D ~ 1 ) \widetilde{D}:=(\widetilde{D}_T,\ldots,\widetilde{D}_1) D := ( D T , … , D 1 ) D : = ( D T , … , D 1 ) D:=(D_T,\ldots,D_1) D := ( D T , … , D 1 )
Unmet demand is lost and incurs a unit penalty b b b h h h c c c [9 , 42 , 49 ] ). Local charging provides automatic recovery. After x t x_t x t U − x t U-x_t U − x t α ∈ [ 0 , 1 ) \alpha\in[0,1) α ∈ [ 0 , 1 ) w w w α = 1 \alpha=1 α = 1 β : = 1 − α \beta:= 1 - \alpha β := 1 − α
C ( y , x , D ) : = c ( x − y ) + w α ( U − x ) + b ( D − x ) + + h ( x − D ) + , C(y,x,D) := c(x-y) + w\alpha(U-x) + b(D-x)_{+} + h(x-D)_{+}, C ( y , x , D ) := c ( x − y ) + w α ( U − x ) + b ( D − x ) + + h ( x − D ) + , with the positive-part notation defined above.
Figure 2 summarizes the within-period sequence. Once x t x_t x t α ( U − x t ) \alpha(U-x_t) α ( U − x t ) D t D_t D t x t x_t x t t t t y t − 1 = R ( x t , D t ) : = ( x t − D t ) + + α ( U − x t ) y_{t-1}=R(x_t,D_t):=(x_t-D_t)_{+}+\alpha(U-x_t) y t − 1 = R ( x t , D t ) := ( x t − D t ) + + α ( U − x t ) ( x , D ) ∈ [ 0 , U ] 2 (x,D)\in[0,U]^2 ( x , D ) ∈ [ 0 , U ] 2 R ( x , D ) ∈ [ 0 , U ] R(x,D)\in[0,U] R ( x , D ) ∈ [ 0 , U ] μ T = μ \mu_T=\mu μ T = μ
Figure 2. Sequence of Events
is given. For t < T t<T t < T D t + 1 D_{t+1} D t + 1 D T + 1 : = μ D_{T+1}:=\mu D T + 1 := μ μ t : = D t + 1 \mu_t:=D_{t+1} μ t := D t + 1 D t D_t D t ( y t − 1 , μ t − 1 ) = ( R ( x t , D t ) , D t ) (y_{t-1},\mu_{t-1})=(R(x_t,D_t),D_t) ( y t − 1 , μ t − 1 ) = ( R ( x t , D t ) , D t ) ( y t , μ t ) (y_t,\mu_t) ( y t , μ t )
Set D ~ T + 1 ≡ μ \widetilde D_{T+1}\equiv\mu D T + 1 ≡ μ F T + 1 : = { ∅ , Ω } \mathcal{F}_{T+1}:=\{\varnothing,\Omega\} F T + 1 := { ∅ , Ω } t ∈ [ T ] t\in[T] t ∈ [ T ] F t : = σ ( D ~ T , … , D ~ t ) \mathcal{F}_t:=\sigma(\widetilde D_T,\ldots,\widetilde D_t) F t := σ ( D T , … , D t ) π = ( π T , … , π 1 ) \pi=(\pi_T,\ldots,\pi_1) π = ( π T , … , π 1 ) π t : [ 0 , U ] T − t → [ 0 , U ] \pi_t:[0,U]^{T-t}\to[0,U] π t : [ 0 , U ] T − t → [ 0 , U ] x t π = π t ( D ~ T , … , D ~ t + 1 ) x_t^\pi=\pi_t(\widetilde D_T,\ldots,\widetilde D_{t+1}) x t π = π t ( D T , … , D t + 1 ) y t π ≤ x t π ≤ U y_t^\pi\le x_t^\pi\le U y t π ≤ x t π ≤ U y t − 1 π = R ( x t π , D ~ t ) y_{t-1}^\pi=R(x_t^\pi,\widetilde D_t) y t − 1 π = R ( x t π , D t ) y ∈ [ 0 , U ] y\in[0,U] y ∈ [ 0 , U ] A ( y ) \mathcal{A}(y) A ( y ) y T π = y y_T^\pi=y y T π = y
We define the joint martingale ambiguity set as
M A R ( μ ) : = { P ∈ P 0 ( [ 0 , U ] T ) : E P [ D ~ t ∣ F t + 1 ] = D ~ t + 1 , t ∈ [ T ] } . \mathcal{MAR}(\mu):=\left\{\mathbb{P}\in\mathcal{P}_0([0,U]^T):\mathbb{E}_{\mathbb{P}}[\widetilde D_t\mid\mathcal{F}_{t+1}]=\widetilde D_{t+1},\quad t\in[T]\right\}. M A R ( μ ) := { P ∈ P 0 ([ 0 , U ] T ) : E P [ D t ∣ F t + 1 ] = D t + 1 , t ∈ [ T ] } . For a one-period reference mean m ∈ [ 0 , U ] m\in[0,U] m ∈ [ 0 , U ]
M ( m ) : = { Q ∈ P 0 ( [ 0 , U ] ) : E Q [ D ~ ] = m } . \mathcal{M}(m):=\left\{\mathbb{Q}\in\mathcal{P}_0([0,U]):\mathbb{E}_{\mathbb{Q}}[\widetilde D]=m\right\}. M ( m ) := { Q ∈ P 0 ([ 0 , U ]) : E Q [ D ] = m } . Under any P ∈ M A R ( μ ) \mathbb{P} \in\mathcal{MAR}(\mu) P ∈ M A R ( μ ) D ~ t \widetilde{D}_t D t ( D T , … , D t + 1 ) (D_T,\ldots,D_{t+1}) ( D T , … , D t + 1 ) M ( D t + 1 ) \mathcal{M}(D_{t+1}) M ( D t + 1 ) D ~ T \widetilde{D}_T D T M ( μ ) \mathcal{M}(\mu) M ( μ )
For initial inventory y y y μ \mu μ T T T
J T ( y , μ ) : = inf π ∈ A ( y ) sup P ∈ M A R ( μ ) E P [ ∑ t = 1 T C ( y t π , x t π , D ~ t ) ] . J_T(y,\mu) := \inf_{\pi\in\mathcal{A}(y)} \sup_{\mathbb{P} \in\mathcal{MAR}(\mu)} \mathbb{E}_{\mathbb{P}}\left[\sum_{t=1}^{T} C(y_t^\pi,x_t^\pi,\widetilde{D}_t)\right]. J T ( y , μ ) := π ∈ A ( y ) inf P ∈ M A R ( μ ) sup E P [ t = 1 ∑ T C ( y t π , x t π , D t ) ] . where y T π = y y_T^\pi= y y T π = y y t − 1 π = R ( x t π , D ~ t ) y_{t-1}^\pi= R(x_t^\pi,\widetilde{D}_t) y t − 1 π = R ( x t π , D t )
Bellman recursion and analytical foundations
For each t ∈ { 0 , 1 , … , T } t \in\{0,1,\ldots,T\} t ∈ { 0 , 1 , … , T } V ^ t ( y , μ ) \widehat{V}^t(y,\mu) V t ( y , μ ) t t t y y y μ \mu μ V ^ 0 ( y , μ ) = 0 \widehat{V}^0(y,\mu)=0 V 0 ( y , μ ) = 0 [ 0 , U ] 2 [0,U]^2 [ 0 , U ] 2 t ∈ [ T ] t \in[T] t ∈ [ T ]
V ^ t ( y , μ ) = min x ∈ [ y , U ] { c ( x − y ) + w α ( U − x ) + sup Q ∈ M ( μ ) E Q [ b ( D ~ − x ) + + h ( x − D ~ ) + + V ^ t − 1 ( R ( x , D ~ ) , D ~ ) ] } , ∀ t ∈ [ T ] . \widehat{V}^t(y,\mu)=\min_{x\in[y,U]}\left\{c(x-y)+w\alpha(U-x)+\sup_{Q\in\mathcal{M}(\mu)}\mathbb{E}_{Q}\left[b(\widetilde{D}-x)_+ + h(x-\widetilde{D})_+ + \widehat{V}^{t-1}(R(x,\widetilde{D}),\widetilde{D})\right]\right\},\quad\forall t\in[T]. V t ( y , μ ) = x ∈ [ y , U ] min { c ( x − y ) + w α ( U − x ) + Q ∈ M ( μ ) sup E Q [ b ( D − x ) + + h ( x − D ) + + V t − 1 ( R ( x , D ) , D ) ] } , ∀ t ∈ [ T ] . After demand D D D ( R ( x , D ) , D ) (R(x,D),D) ( R ( x , D ) , D )
Proposition 1. For every ( y , μ ) ∈ [ 0 , U ] 2 (y,\mu)\in[0,U]^2 ( y , μ ) ∈ [ 0 , U ] 2
J T ( y , μ ) = V ^ T ( y , μ ) . J_T(y,\mu)=\widehat{V}^T(y,\mu). J T ( y , μ ) = V T ( y , μ ) . Proposition 1 also establishes attainment of the Bellman minimum and the conditional suprema. By rectangularity, the attaining conditional kernels can be concatenated into an attaining joint distribution in M A R ( μ ) \mathcal{MAR}(\mu) M A R ( μ )
Corollary 1 (Time consistency). There exists an optimal deterministic Markov policy whose decision at each state ( y , μ ) (y,\mu) ( y , μ ) t ∈ [ T ] t\in[T] t ∈ [ T ] ( y , μ ) ∈ [ 0 , U ] 2 (y,\mu)\in[0,U]^2 ( y , μ ) ∈ [ 0 , U ] 2 t t t V ^ t ( y , μ ) \widehat{V}^t(y,\mu) V t ( y , μ )
Under the conditional mean and support restrictions, the worst-case expectation in the Bellman recursion can be expressed through a concave envelope [42 ] . Moreover, a worst-case distribution can be chosen with support on at most two demand points.
Lemma 1. For any continuous φ : [ 0 , U ] → R \varphi: [0,U] \to\mathbb{R} φ : [ 0 , U ] → R μ ∈ [ 0 , U ] \mu\in[0,U] μ ∈ [ 0 , U ] Q ∗ ∈ arg max Q ∈ M ( μ ) E Q [ φ ( D ~ ) ] \mathbb{Q}^{*} \in\operatorname*{arg\,max}_{Q\in\mathcal{M}(\mu)} \mathbb{E}_{Q}[\varphi(\widetilde{D})] Q ∗ ∈ arg max Q ∈ M ( μ ) E Q [ φ ( D )] D − ≤ μ ≤ D + D^{-} \le\mu\le D^{+} D − ≤ μ ≤ D +
For D − < D + D^{-} < D^{+} D − < D +
Q ∗ ( D − ) = D + − μ D + − D − , Q ∗ ( D + ) = μ − D − D + − D − . \mathbb{Q}^{*}(D^{-}) = \frac{D^{+}-\mu}{D^{+}-D^{-}}, \qquad\mathbb{Q}^{*}(D^{+}) = \frac{\mu-D^{-}}{D^{+}-D^{-}}. Q ∗ ( D − ) = D + − D − D + − μ , Q ∗ ( D + ) = D + − D − μ − D − . If D − = D + = μ D^{-} = D^{+} = \mu D − = D + = μ Q ∗ = δ μ \mathbb{Q}^{*} = \delta_{\mu} Q ∗ = δ μ
For a fixed post-replenishment level x x x
F t x ( D ) : = b ( D − x ) + + h ( x − D ) + + V ^ t − 1 ( R ( x , D ) , D ) , t ∈ [ T ] . F_{t}^{x}(D) := b(D-x)^{+} + h(x-D)^{+} + \widehat{V}^{t-1}(R(x,D),D), \qquad t \in[T]. F t x ( D ) := b ( D − x ) + + h ( x − D ) + + V t − 1 ( R ( x , D ) , D ) , t ∈ [ T ] . By Lemma 1, the worst-case expectation of F t x F_{t}^{x} F t x − c y -cy − cy
B t ( x , μ ) : = c x + w α ( U − x ) + sup Q ∈ M ( μ ) E Q [ F t x ( D ~ ) ] , t ∈ [ T ] . B_{t}(x,\mu) := cx + w\alpha(U-x)
+ \sup_{Q\in\mathcal{M}(\mu)} \mathbb{E}_{Q}[F_{t}^{x}(\widetilde{D})], \qquad t \in[T]. B t ( x , μ ) := c x + w α ( U − x ) + Q ∈ M ( μ ) sup E Q [ F t x ( D )] , t ∈ [ T ] . The Bellman equation can therefore be written as
V ^ t ( y , μ ) = − c y + min x ∈ [ y , U ] B t ( x , μ ) , t ∈ [ T ] . \widehat{V}^{t}(y,\mu) = -cy + \min_{x\in[y,U]} B_{t}(x,\mu), \qquad t \in[T]. V t ( y , μ ) = − cy + x ∈ [ y , U ] min B t ( x , μ ) , t ∈ [ T ] . To characterize the replenishment policy analytically, we impose the following strict ordering on the four unit costs. No ordering is imposed on the recovery thresholds introduced below.
Assumption 1. $0 < h < w < c < b$.
The ordering reflects inexpensive inventory carryover, local charging that is cheaper than central replenishment, and a shortage penalty exceeding the central replenishment cost. Define the two critical recovery rates
α h : = h w , α b : = b − c c − w . \alpha_{h} := \frac{h}{w}, \qquad\alpha_{b} := \frac{b-c}{c-w}. α h := w h , α b := c − w b − c . As we show in the subsequent analysis, these two thresholds play distinct roles: α h \alpha_{h} α h α b \alpha_{b} α b 0 < α h < 1 0 < \alpha_{h} < 1 0 < α h < 1 α b > 0 \alpha_{b} > 0 α b > 0 α b \alpha_{b} α b α \alpha α
Unless otherwise specified, the subsequent analysis is based on Assumption 1. The standing cost ordering also guarantees a finite-horizon order-up-to structure before any explicit envelope construction is imposed.
Proposition 2. For every finite t t t B t ( ⋅ , μ ) B_t(\cdot,\mu) B t ( ⋅ , μ ) [ 0 , U ] [0,U] [ 0 , U ]
χ t ( μ ) : = max a r g m i n x ∈ [ 0 , U ] B t ( x , μ ) . \chi_t(\mu) := \max\mathop{\rm arg\,min}_{x\in[0,U]} B_t(x,\mu). χ t ( μ ) := max arg min x ∈ [ 0 , U ] B t ( x , μ ) . Then x t ∗ ( y , μ ) = max { y , χ t ( μ ) } x_t^*(y,\mu)=\max\{y,\chi_t(\mu)\} x t ∗ ( y , μ ) = max { y , χ t ( μ )} ( y , μ ) ∈ [ 0 , U ] 2 (y,\mu)\in[0,U]^2 ( y , μ ) ∈ [ 0 , U ] 2
Here χ t \chi_t χ t α = α b \alpha=\alpha_b α = α b x t ∗ = y x_t^*=y x t ∗ = y χ t \chi_t χ t
Two-Period Envelope Geometry and Policy Structure
This section examines how automatic local recovery affects optimal replenishment decisions. We characterize the optimal policies and corresponding worst-case demand distributions for the terminal- and two-period problems. These results provide the foundation for the finite-horizon analysis developed in § 5 .
Terminal-period benchmark
The terminal-period problem is common to all four recovery regimes. Define
Φ 1 : = c + h − α w , K 1 : = b + h , ρ 1 : = Φ 1 K 1 , q ˉ 2 : = ρ 1 U ∈ ( 0 , U ) . \Phi_1 := c+h-\alpha w,\qquad K_1 := b+h,\qquad\rho_1 := \frac{\Phi_1}{K_1},\qquad\bar q_2 := \rho_1 U\in(0,U). Φ 1 := c + h − α w , K 1 := b + h , ρ 1 := K 1 Φ 1 , q ˉ 2 := ρ 1 U ∈ ( 0 , U ) . where the strict bounds follow from Assumption 1. This threshold determines the terminal-period replenishment policy and plays a central role in the subsequent two-period analysis.
Proposition 3. For ( y 1 , μ 1 ) ∈ [ 0 , U ] 2 (y_1,\mu_1)\in[0,U]^2 ( y 1 , μ 1 ) ∈ [ 0 , U ] 2 χ 1 ( μ 1 ) = U 1 { μ 1 ≥ q ˉ 2 } \chi_1(\mu_1)=U\mathbf{1}_{\{\mu_1\ge\bar q_2\}} χ 1 ( μ 1 ) = U 1 { μ 1 ≥ q ˉ 2 }
x 1 ∗ ( y 1 , μ 1 ) = max { y 1 , χ 1 ( μ 1 ) } , Q 1 ∗ = ( 1 − μ 1 U ) δ 0 + μ 1 U δ U . x_1^*(y_1,\mu_1)=\max\{y_1,\chi_1(\mu_1)\},\qquad Q_1^*=\left(1-\frac{\mu_1}{U}\right)\delta_0+\frac{\mu_1}{U}\delta_U. x 1 ∗ ( y 1 , μ 1 ) = max { y 1 , χ 1 ( μ 1 )} , Q 1 ∗ = ( 1 − U μ 1 ) δ 0 + U μ 1 δ U . The corresponding optimal value is
V ^ 1 ( y 1 , μ 1 ) = − c y 1 + ( c + h ) U − h μ 1 − ( U − x 1 ∗ ( y 1 , μ 1 ) ) ( Φ 1 − K 1 μ 1 U ) . \widehat V^1(y_1,\mu_1)=-cy_1+(c+h)U-h\mu_1-\left(U-x_1^*(y_1,\mu_1)\right)\left(\Phi_1-K_1\frac{\mu_1}{U}\right). V 1 ( y 1 , μ 1 ) = − c y 1 + ( c + h ) U − h μ 1 − ( U − x 1 ∗ ( y 1 , μ 1 ) ) ( Φ 1 − K 1 U μ 1 ) . With two periods remaining, realized demand D D D D = q ˉ 2 D=\bar q_2 D = q ˉ 2 F 2 x ( D ) F_2^x(D) F 2 x ( D )
By Lemma 1, an attaining worst-case distribution can be supported on at most two demand points. For 0 ≤ D − ≤ μ ≤ D + ≤ U 0\le D^- \le\mu\le D^+ \le U 0 ≤ D − ≤ μ ≤ D + ≤ U D − < D + D^-<D^+ D − < D +
ℓ D − , D + x ( μ ) : = D + − μ D + − D − F 2 x ( D − ) + μ − D − D + − D − F 2 x ( D + ) . \ell^x_{D^-,D^+}(\mu):=\frac{D^+-\mu}{D^+-D^-}F_2^x(D^-)+\frac{\mu-D^-}{D^+-D^-}F_2^x(D^+). ℓ D − , D + x ( μ ) := D + − D − D + − μ F 2 x ( D − ) + D + − D − μ − D − F 2 x ( D + ) . Consequently,
sup Q ∈ M ( μ ) E Q [ F 2 x ( D ~ ) ] = sup 0 ≤ D − ≤ μ ≤ D + ≤ U D − < D + ℓ D − , D + x ( μ ) . \sup_{Q\in\mathcal{M}(\mu)} \mathbb{E}_{Q}\left[F_2^x(\widetilde{D})\right] = \sup_{\substack{0\leq D^{-}\leq\mu\leq D^{+}\leq U\\D^{-}<D^{+}}} \ell_{D^{-},D^{+}}^x(\mu). Q ∈ M ( μ ) sup E Q [ F 2 x ( D ) ] = 0 ≤ D − ≤ μ ≤ D + ≤ U D − < D + sup ℓ D − , D + x ( μ ) . The maximizing endpoints determine an attaining worst-case distribution.
Two-period worst-case demand distributions
Recall that the terminal-period order-up-to target switches at q ˉ 2 \bar{q}_2 q ˉ 2 D D D F 2 x ( D ) F_2^x(D) F 2 x ( D ) x ∈ [ 0 , U ] x\in[0,U] x ∈ [ 0 , U ]
F 2 x ( D ) ≤ { ℓ 0 , q ˉ 2 x ( D ) , 0 ≤ D ≤ q ˉ 2 , ℓ q ˉ 2 , U x ( D ) , q ˉ 2 ≤ D ≤ U . F_2^x(D)\leq
\begin{cases}
\ell_{0,\bar{q}_2}^x(D), & 0\leq D\leq\bar{q}_2,\\
\ell_{\bar{q}_2,U}^x(D), & \bar{q}_2\leq D\leq U.
\end{cases} F 2 x ( D ) ≤ { ℓ 0 , q ˉ 2 x ( D ) , ℓ q ˉ 2 , U x ( D ) , 0 ≤ D ≤ q ˉ 2 , q ˉ 2 ≤ D ≤ U . Thus, any demand realization can be replaced by a mean-preserving mixture of the endpoints of its respective interval without decreasing the expected cost. It follows that an attaining worst-case distribution can be supported on { 0 , q ˉ 2 , U } \{0,\bar{q}_2,U\} { 0 , q ˉ 2 , U }
It remains to determine whether q ˉ 2 \bar{q}_2 q ˉ 2 F 2 x ( q ˉ 2 ) ≤ ℓ 0 , U x ( q ˉ 2 ) F_2^x(\bar{q}_2)\leq\ell_{0,U}^x(\bar{q}_2) F 2 x ( q ˉ 2 ) ≤ ℓ 0 , U x ( q ˉ 2 ) F 2 x F_2^x F 2 x [ 0 , U ] [0,U] [ 0 , U ] q ˉ 2 \bar{q}_2 q ˉ 2 Φ 2 : = c + ( 1 + β ) ( h − α w ) \Phi_2:=c+(1+\beta)(h-\alpha w) Φ 2 := c + ( 1 + β ) ( h − α w ) K 2 : = b + h + β Φ 1 − c K_2:=b+h+\beta\Phi_1-c K 2 := b + h + β Φ 1 − c
Δ 2 ( x ) : = F 2 x ( q ˉ 2 ) − ℓ 0 , U x ( q ˉ 2 ) = { ( 1 − q ˉ 2 U ) ( β Φ 1 U − K 2 x ) , x ≤ q ˉ 2 , ( β K 1 − K 2 ) q ˉ 2 U ( U − x ) , x ≥ q ˉ 2 . \Delta_2(x):=F_2^x(\bar{q}_2)-\ell_{0,U}^x(\bar{q}_2)=
\begin{cases}
\left(1-\dfrac{\bar{q}_2}{U}\right)(\beta\Phi_1U-K_2x), & x\leq\bar{q}_2,\\
(\beta K_1-K_2)\dfrac{\bar{q}_2}{U}(U-x), & x\geq\bar{q}_2.
\end{cases} Δ 2 ( x ) := F 2 x ( q ˉ 2 ) − ℓ 0 , U x ( q ˉ 2 ) = ⎩ ⎨ ⎧ ( 1 − U q ˉ 2 ) ( β Φ 1 U − K 2 x ) , ( β K 1 − K 2 ) U q ˉ 2 ( U − x ) , x ≤ q ˉ 2 , x ≥ q ˉ 2 . When Δ 2 ( x ) > 0 \Delta_2(x)>0 Δ 2 ( x ) > 0 q ˉ 2 \bar{q}_2 q ˉ 2 U U U { 0 , U } \{0,U\} { 0 , U }
The sign of Δ 2 ( x ) \Delta_2(x) Δ 2 ( x ) K 2 > β K 1 K_2>\beta K_1 K 2 > β K 1 Δ 2 ( x ) \Delta_2(x) Δ 2 ( x ) x ˉ 2 : = β Φ 1 U / K 2 < q ˉ 2 \bar{x}_2:=\beta\Phi_1U/K_2<\bar{q}_2 x ˉ 2 := β Φ 1 U / K 2 < q ˉ 2 q ˉ 2 \bar{q}_2 q ˉ 2 x < x ˉ 2 x<\bar{x}_2 x < x ˉ 2 { 0 , U } \{0,U\} { 0 , U } x ≥ x ˉ 2 x\geq\bar{x}_2 x ≥ x ˉ 2 K 2 ≤ β K 1 K_2\leq\beta K_1 K 2 ≤ β K 1 Δ 2 ( x ) ≥ 0 \Delta_2(x)\geq0 Δ 2 ( x ) ≥ 0 x ∈ [ 0 , U ] x\in[0,U] x ∈ [ 0 , U ] q ˉ 2 \bar{q}_2 q ˉ 2 x x x
Lemma 2 (Two-period worst-case expectation). For every x , μ ∈ [ 0 , U ] x,\mu\in[0,U] x , μ ∈ [ 0 , U ] F 2 x F_2^x F 2 x
If K 2 > β K 1 K_2>\beta K_1 K 2 > β K 1
sup Q ∈ M ( μ ) E Q [ F 2 x ( D ~ ) ] = { ℓ 0 , q ˉ 2 x ( μ ) , 0 ≤ x < x ˉ 2 , 0 ≤ μ ≤ q ˉ 2 , ℓ q ˉ 2 , U x ( μ ) , 0 ≤ x < x ˉ 2 , q ˉ 2 ≤ μ ≤ U , ℓ 0 , U x ( μ ) , x ˉ 2 ≤ x ≤ U , 0 ≤ μ ≤ U . (1) \sup_{Q\in\mathcal{M}(\mu)}\mathbb{E}_{Q}\left[F_2^x(\widetilde{D})\right]=
\begin{cases}
\ell_{0,\bar{q}_2}^x(\mu), & 0\leq x<\bar{x}_2,\quad0\leq\mu\leq\bar{q}_2,\\
\ell_{\bar{q}_2,U}^x(\mu), & 0\leq x<\bar{x}_2,\quad\bar{q}_2\leq\mu\leq U,\\
\ell_{0,U}^x(\mu), & \bar{x}_2\leq x\leq U,\quad0\leq\mu\leq U.
\end{cases} \tag*{(1)} Q ∈ M ( μ ) sup E Q [ F 2 x ( D ) ] = ⎩ ⎨ ⎧ ℓ 0 , q ˉ 2 x ( μ ) , ℓ q ˉ 2 , U x ( μ ) , ℓ 0 , U x ( μ ) , 0 ≤ x < x ˉ 2 , 0 ≤ μ ≤ q ˉ 2 , 0 ≤ x < x ˉ 2 , q ˉ 2 ≤ μ ≤ U , x ˉ 2 ≤ x ≤ U , 0 ≤ μ ≤ U . (1) If K 2 ≤ β K 1 K_2\leq\beta K_1 K 2 ≤ β K 1
sup Q ∈ M ( μ ) E Q [ F 2 x ( D ~ ) ] = { ℓ 0 , q ˉ 2 x ( μ ) , 0 ≤ μ ≤ q ˉ 2 , ℓ q ˉ 2 , U x ( μ ) , q ˉ 2 ≤ μ ≤ U , x ∈ [ 0 , U ] . (2) \sup_{Q\in\mathcal{M}(\mu)}\mathbb{E}_{Q}\left[F_2^x(\widetilde{D})\right]=
\begin{cases}
\ell_{0,\bar{q}_2}^x(\mu), & 0\leq\mu\leq\bar{q}_2,\\
\ell_{\bar{q}_2,U}^x(\mu), & \bar{q}_2\leq\mu\leq U,
\end{cases}
\qquad x\in[0,U]. \tag*{(2)} Q ∈ M ( μ ) sup E Q [ F 2 x ( D ) ] = { ℓ 0 , q ˉ 2 x ( μ ) , ℓ q ˉ 2 , U x ( μ ) , 0 ≤ μ ≤ q ˉ 2 , q ˉ 2 ≤ μ ≤ U , x ∈ [ 0 , U ] . (2) Lemma 2 identifies two possible worst-case demand structures. When K 2 > β K 1 K_2 > \beta K_1 K 2 > β K 1 q ˉ 2 \bar{q}_2 q ˉ 2 x < x ˉ 2 x < \bar{x}_2 x < x ˉ 2 { 0 , U } \{0,U\} { 0 , U } x ≥ x ˉ 2 x \ge\bar{x}_2 x ≥ x ˉ 2 K 2 ≤ β K 1 K_2 \le\beta K_1 K 2 ≤ β K 1 q ˉ 2 \bar{q}_2 q ˉ 2 x x x 0 0 0 U U U μ \mu μ
Despite these differences in the worst-case distributions, the optimal replenishment decisions satisfy a common boundary.
Corollary 2 (Two-period replenishment boundary). The following statements hold:
If α < α b \alpha< \alpha_b α < α b μ 2 = U \mu_2 = U μ 2 = U y 2 < U y_2 < U y 2 < U x 2 ∗ ( y 2 , U ) = U x_2^*(y_2,U) = U x 2 ∗ ( y 2 , U ) = U
If α ≥ α b \alpha\ge\alpha_b α ≥ α b ( y 2 , μ 2 ) ∈ [ 0 , U ] 2 (y_2,\mu_2) \in[0,U]^2 ( y 2 , μ 2 ) ∈ [ 0 , U ] 2 x 2 ∗ ( y 2 , μ 2 ) = y 2 x_2^*(y_2,\mu_2) = y_2 x 2 ∗ ( y 2 , μ 2 ) = y 2 α > α b \alpha> \alpha_b α > α b
Corollary 2 establishes α b \alpha_b α b α < α b \alpha< \alpha_b α < α b
Two-Period Policy Characterization
By Corollary 2, no replenishment is optimal when α ≥ α b \alpha\ge\alpha_b α ≥ α b α < α b \alpha< \alpha_b α < α b ρ 2 : = Φ 2 / K 2 ∈ ( 0 , 1 ) \rho_2 := \Phi_2/K_2 \in(0,1) ρ 2 := Φ 2 / K 2 ∈ ( 0 , 1 )
Theorem 1 (Two-period policy for α < α b \alpha< \alpha_b α < α b For every ( y 2 , μ 2 ) ∈ [ 0 , U ] 2 (y_2,\mu_2) \in[0,U]^2 ( y 2 , μ 2 ) ∈ [ 0 , U ] 2 x 2 ∗ ( y 2 , μ 2 ) = max { y 2 , χ 2 ( μ 2 ) } x_2^*(y_2,\mu_2) = \max\{y_2,\chi_2(\mu_2)\} x 2 ∗ ( y 2 , μ 2 ) = max { y 2 , χ 2 ( μ 2 )} χ 2 \chi_2 χ 2
If K 2 > β K 1 K_2 > \beta K_1 K 2 > β K 1
χ 2 ( μ ) = { 0 , 0 ≤ μ < ρ 2 q ˉ 2 , x ˉ 2 , ρ 2 q ˉ 2 ≤ μ < ρ 2 U , U , ρ 2 U ≤ μ ≤ U . \chi_2(\mu) =
\begin{cases}
0, & 0 \le\mu< \rho_2\bar{q}_2,\\
\bar{x}_2, & \rho_2\bar{q}_2 \le\mu< \rho_2 U,\\
U, & \rho_2 U \le\mu\le U.
\end{cases} χ 2 ( μ ) = ⎩ ⎨ ⎧ 0 , x ˉ 2 , U , 0 ≤ μ < ρ 2 q ˉ 2 , ρ 2 q ˉ 2 ≤ μ < ρ 2 U , ρ 2 U ≤ μ ≤ U . Q 2 ∗ ( ⋅ ∣ y 2 , μ 2 ) = { ( 1 − μ 2 q ˉ 2 ) δ 0 + μ 2 q ˉ 2 δ q ˉ 2 , x 2 ∗ < x ˉ 2 , ( 1 − μ 2 U ) δ 0 + μ 2 U δ U , x 2 ∗ ≥ x ˉ 2 . Q_2^*(\cdot\mid y_2,\mu_2) =
\begin{cases}
\left(1-\frac{\mu_2}{\bar{q}_2}\right)\delta_0+\frac{\mu_2}{\bar{q}_2}\delta_{\bar{q}_2}, & x_2^* < \bar{x}_2,\\
\left(1-\frac{\mu_2}{U}\right)\delta_0+\frac{\mu_2}{U}\delta_U, & x_2^* \ge\bar{x}_2.
\end{cases} Q 2 ∗ ( ⋅ ∣ y 2 , μ 2 ) = { ( 1 − q ˉ 2 μ 2 ) δ 0 + q ˉ 2 μ 2 δ q ˉ 2 , ( 1 − U μ 2 ) δ 0 + U μ 2 δ U , x 2 ∗ < x ˉ 2 , x 2 ∗ ≥ x ˉ 2 . If K 2 ≤ β K 1 K_2 \le\beta K_1 K 2 ≤ β K 1
χ 2 ( μ ) = { 0 , 0 ≤ μ < ρ 2 q ˉ 2 , q ˉ 2 , ρ 2 q ˉ 2 ≤ μ < U − K 2 − Φ 2 b + h − c ( U − q ˉ 2 ) , U , otherwise . \chi_2(\mu) =
\begin{cases}
0, & 0 \le\mu< \rho_2\bar{q}_2,\\
\bar{q}_2, & \rho_2\bar{q}_2 \le\mu< U-\frac{K_2-\Phi_2}{b+h-c}(U-\bar{q}_2),\\
U, & \text{otherwise}.
\end{cases} χ 2 ( μ ) = ⎩ ⎨ ⎧ 0 , q ˉ 2 , U , 0 ≤ μ < ρ 2 q ˉ 2 , ρ 2 q ˉ 2 ≤ μ < U − b + h − c K 2 − Φ 2 ( U − q ˉ 2 ) , otherwise . Q 2 ∗ ( ⋅ ∣ y 2 , μ 2 ) = { ( 1 − μ 2 q ˉ 2 ) δ 0 + μ 2 q ˉ 2 δ q ˉ 2 , 0 ≤ μ 2 ≤ q ˉ 2 , U − μ 2 U − q ˉ 2 δ q ˉ 2 + μ 2 − q ˉ 2 U − q ˉ 2 δ U , q ˉ 2 < μ 2 ≤ U . Q_2^*(\cdot\mid y_2,\mu_2)=
\begin{cases}
\left(1-\frac{\mu_2}{\bar q_2}\right)\delta_0+\frac{\mu_2}{\bar q_2}\delta_{\bar q_2}, & 0\le\mu_2\le\bar q_2,\\
\frac{U-\mu_2}{U-\bar q_2}\delta_{\bar q_2}+\frac{\mu_2-\bar q_2}{U-\bar q_2}\delta_U, & \bar q_2<\mu_2\le U.
\end{cases} Q 2 ∗ ( ⋅ ∣ y 2 , μ 2 ) = { ( 1 − q ˉ 2 μ 2 ) δ 0 + q ˉ 2 μ 2 δ q ˉ 2 , U − q ˉ 2 U − μ 2 δ q ˉ 2 + U − q ˉ 2 μ 2 − q ˉ 2 δ U , 0 ≤ μ 2 ≤ q ˉ 2 , q ˉ 2 < μ 2 ≤ U . In both cases, the optimal value is
V ^ 2 ( y 2 , μ 2 ) = − c y 2 + B 2 ( x 2 ∗ ( y 2 , μ 2 ) , μ 2 ) . \hat V^2(y_2,\mu_2)=-cy_2+B_2(x_2^*(y_2,\mu_2),\mu_2). V ^ 2 ( y 2 , μ 2 ) = − c y 2 + B 2 ( x 2 ∗ ( y 2 , μ 2 ) , μ 2 ) . The two cases differ in their intermediate order-up-to targets and worst-case demand distributions. When K 2 > β K 1 K_2>\beta K_1 K 2 > β K 1 x ˉ 2 \bar x_2 x ˉ 2 { 0 , U } \{0,U\} { 0 , U } x ˉ 2 \bar x_2 x ˉ 2 K 2 ≤ β K 1 K_2\le\beta K_1 K 2 ≤ β K 1 q ˉ 2 \bar q_2 q ˉ 2 K 2 = β K 1 K_2=\beta K_1 K 2 = β K 1 x ˉ 2 = q ˉ 2 \bar x_2=\bar q_2 x ˉ 2 = q ˉ 2 ρ 2 U \rho_2U ρ 2 U
Together with Corollary 2, the preceding theorem completes the two-period characterization under Assumption 1. The condition K 2 > β K 1 K_2>\beta K_1 K 2 > β K 1 α b \alpha_b α b
Finite-Horizon Policies under the Regular Penalty Condition
We extend the two-period analysis to arbitrary finite horizons. The coefficients introduced in § 4 generalize to
Φ t : = c + ∑ k = 0 t − 1 β k ( h − α w ) , t ≥ 1 , K 1 : = b + h and K t : = b + h + β Φ t − 1 − c , t ≥ 2. (3) \begin{aligned}
\Phi_t&:=c+\sum_{k=0}^{t-1}\beta^k(h-\alpha w), && t\ge1,\\
K_1&:=b+h && \text{and} \quad K_t:=b+h+\beta\Phi_{t-1}-c, && t\ge2. \tag*{(3)}
\end{aligned} Φ t K 1 := c + k = 0 ∑ t − 1 β k ( h − α w ) , := b + h t ≥ 1 , and K t := b + h + β Φ t − 1 − c , t ≥ 2. (3) These coefficients satisfy
Φ t + 1 − Φ t = β t ( h − α w ) , t ≥ 1 , \Phi_{t+1}-\Phi_t=\beta^t(h-\alpha w), \qquad t\ge1, Φ t + 1 − Φ t = β t ( h − α w ) , t ≥ 1 , K t − Φ t = b − c − α ( c − w ) , t ≥ 2. K_t-\Phi_t=b-c-\alpha(c-w), \qquad t\ge2. K t − Φ t = b − c − α ( c − w ) , t ≥ 2. The first difference determines whether Φ t \Phi_t Φ t α h = h / w \alpha_h=h/w α h = h / w α b = ( b − c ) / ( c − w ) \alpha_b=(b-c)/(c-w) α b = ( b − c ) / ( c − w ) Table 1 .
Table 1. Recovery-rate regimes
The relative ordering of α h \alpha_h α h α b \alpha_b α b α ∈ [ 0 , 1 ) \alpha\in[0,1) α ∈ [ 0 , 1 )
The two-period analysis in § 4 identifies two possible worst-case envelope structures, distinguished by the sign of K 2 − β K 1 K_2-\beta K_1 K 2 − β K 1 K 2 > β K 1 K_2>\beta K_1 K 2 > β K 1 regular penalty condition :
b − c > w − 2 h w . (4) b-c>w-2\sqrt{hw}. \tag*{(4)} b − c > w − 2 h w . (4) Under Assumption 1, this condition is necessary and sufficient for K 2 > β K 1 K_2>\beta K_1 K 2 > β K 1 α ∈ [ 0 , 1 ) \alpha\in[0,1) α ∈ [ 0 , 1 )
K 2 − β K 1 = ( h − α w ) 2 + α ( b − c − w + 2 h w ) . K_2-\beta K_1=(\sqrt{h}-\alpha\sqrt{w})^2+\alpha(b-c-w+2\sqrt{hw}). K 2 − β K 1 = ( h − α w ) 2 + α ( b − c − w + 2 h w ) . For t ≥ 3 t\ge3 t ≥ 3
K t − β K t − 1 = α ( b + h − c ) + β { h + α ( c − w ) } > 0. K_t-\beta K_{t-1}=\alpha(b+h-c)+\beta\{h+\alpha(c-w)\}>0. K t − β K t − 1 = α ( b + h − c ) + β { h + α ( c − w )} > 0. The regular penalty condition is economically reasonable and covers a broad range of cost configurations. It holds automatically whenever h ≥ w / 4 h\ge w/4 h ≥ w /4 h < w / 4 h<w/4 h < w /4 b − c b-c b − c w − 2 h w < w w-2\sqrt{hw}<w w − 2 h w < w [32 ] , customer waiting and abandonment Xie et al. [51 ] , and explicit waiting costs Sun et al. [47 ] . As illustrated in Figure 3 , the condition excludes only a restricted portion of the normalized cost-parameter space, where relatively low holding costs coincide with a small shortage-penalty premium.
Figure 3. Regular and complementary penalty regions
Throughout this section, Assumption 1 and the regular penalty condition hold. Our analysis is divided according to the sign of K t − Φ t = b − c − α ( c − w ) K_t-\Phi_t=b-c-\alpha(c-w) K t − Φ t = b − c − α ( c − w ) § 5.1 derives explicit finite-horizon policies for α < α b \alpha<\alpha_b α < α b § 5.2 establishes the optimality of no replenishment for α ≥ α b \alpha\ge\alpha_b α ≥ α b
Finite-horizon characterization in Regimes I and II
We first consider α < α b \alpha<\alpha_b α < α b K t − Φ t = b − c − α ( c − w ) > 0 K_t-\Phi_t=b-c-\alpha(c-w)>0 K t − Φ t = b − c − α ( c − w ) > 0
For t ≥ 2 t \ge2 t ≥ 2 ρ t : = Φ t / K t ∈ ( 0 , 1 ) \rho_t := \Phi_t/K_t \in(0,1) ρ t := Φ t / K t ∈ ( 0 , 1 ) ρ 1 \rho_1 ρ 1 1 ≤ j ≤ t 1 \le j \le t 1 ≤ j ≤ t
q t , j : = U ∏ k = j t − 1 ρ k , (5) q_{t,j} := U \prod_{k=j}^{t-1} \rho_k, \tag*{(5)} q t , j := U k = j ∏ t − 1 ρ k , (5) where an empty product equals one. Initialize the inventory thresholds by s 1 , 0 = 0 s_{1,0}=0 s 1 , 0 = 0 s 1 , 1 = U s_{1,1}=U s 1 , 1 = U t ≥ 2 t \ge2 t ≥ 2 s t , 0 = 0 s_{t,0}=0 s t , 0 = 0 s t , t = U s_{t,t}=U s t , t = U
s t , j : = Φ t − 1 K t ( s t − 1 , j − α U ) , j = 1 , … , t − 1. (6) s_{t,j} := \frac{\Phi_{t-1}}{K_t}(s_{t-1,j}-\alpha U), \qquad j=1,\ldots,t-1. \tag*{(6)} s t , j := K t Φ t − 1 ( s t − 1 , j − α U ) , j = 1 , … , t − 1. (6) In particular, q 2 , 1 = q ˉ 2 q_{2,1}=\bar q_2 q 2 , 1 = q ˉ 2 s 2 , 1 = x ˉ 2 s_{2,1}=\bar x_2 s 2 , 1 = x ˉ 2
a t : = min { j ∈ { 1 , … , t } : s t , j > 0 } . a_t := \min\{j \in\{1,\ldots,t\}:s_{t,j}>0\}. a t := min { j ∈ { 1 , … , t } : s t , j > 0 } . The regular penalty condition guarantees, for all finite t t t
0 < q t , 1 < ⋯ < q t , t = U , 0 < s t , a t < ⋯ < s t , t = U , s t , j < q t , j , a t ≤ j < t . (7) \begin{aligned}
0 < q_{t,1} < \cdots< q_{t,t} &= U, \qquad0 < s_{t,a_t} < \cdots< s_{t,t}=U,\\
s_{t,j} &< q_{t,j}, \qquad a_t \le j < t. \tag*{(7)}
\end{aligned} 0 < q t , 1 < ⋯ < q t , t s t , j = U , 0 < s t , a t < ⋯ < s t , t = U , < q t , j , a t ≤ j < t . (7) Thus only the positive inventory thresholds can enter the optimal policy. For x < U x<U x < U
J t ( x ) : = min { j ∈ { a t , … , t } : x < s t , j } , J t ( U ) : = t . J_t(x):=\min\{j\in\{a_t,\ldots,t\}:x<s_{t,j}\}, \qquad J_t(U):=t. J t ( x ) := min { j ∈ { a t , … , t } : x < s t , j } , J t ( U ) := t . For fixed x x x F t x ( D ~ ) F_t^x(\widetilde{D}) F t x ( D ) F t x F_t^x F t x 0 , q t , J t ( x ) , … , q t , t = U 0,q_{t,J_t(x)},\ldots,q_{t,t}=U 0 , q t , J t ( x ) , … , q t , t = U x ∈ [ χ t ( μ ) , U ] x\in[\chi_t(\mu),U] x ∈ [ χ t ( μ ) , U ] j = J t ( x ) j=J_t(x) j = J t ( x )
B t ( x , μ ) = ( c + t h ) U + ( ( t − 1 ) c − t h ) μ − ∑ k = j + 1 t ( s t , k − s t , k − 1 ) ( Φ t − K t μ q t , k ) − ( s t , j − x ) ( Φ t − K t μ q t , j ) . (8) B_t(x,\mu)=(c+th)U+((t-1)c-th)\mu-\sum_{k=j+1}^{t}(s_{t,k}-s_{t,k-1})\left(\Phi_t-K_t\frac{\mu}{q_{t,k}}\right)-(s_{t,j}-x)\left(\Phi_t-K_t\frac{\mu}{q_{t,j}}\right). \tag*{(8)} B t ( x , μ ) = ( c + t h ) U + (( t − 1 ) c − t h ) μ − k = j + 1 ∑ t ( s t , k − s t , k − 1 ) ( Φ t − K t q t , k μ ) − ( s t , j − x ) ( Φ t − K t q t , j μ ) . (8) Theorem 2 (Finite-horizon characterization in Regimes I and II). For α < α b \alpha<\alpha_b α < α b T T T
χ t ( μ ) = { 0 , 0 ≤ μ < ρ t q t , a t , s t , j , ρ t q t , j ≤ μ < ρ t q t , j + 1 , j = a t , … , t − 1 , U , ρ t U ≤ μ ≤ U . (9) \chi_t(\mu)=
\begin{cases}
0, & 0\le\mu<\rho_tq_{t,a_t},\\
s_{t,j}, & \rho_tq_{t,j}\le\mu<\rho_tq_{t,j+1},\quad j=a_t,\ldots,t-1,\\
U, & \rho_tU\le\mu\le U.
\end{cases} \tag*{(9)} χ t ( μ ) = ⎩ ⎨ ⎧ 0 , s t , j , U , 0 ≤ μ < ρ t q t , a t , ρ t q t , j ≤ μ < ρ t q t , j + 1 , j = a t , … , t − 1 , ρ t U ≤ μ ≤ U . (9) For every t ∈ [ T ] t\in[T] t ∈ [ T ] ( y t , μ t ) ∈ [ 0 , U ] 2 (y_t,\mu_t)\in[0,U]^2 ( y t , μ t ) ∈ [ 0 , U ] 2
x t ∗ ( y t , μ t ) = max { y t , χ t ( μ t ) } , x_t^*(y_t,\mu_t)=\max\{y_t,\chi_t(\mu_t)\}, x t ∗ ( y t , μ t ) = max { y t , χ t ( μ t )} , V ^ t ( y t , μ t ) = − c y t + B t ( x t ∗ ( y t , μ t ) , μ t ) . \widehat{V}_t(y_t,\mu_t)=-cy_t+B_t(x_t^*(y_t,\mu_t),\mu_t). V t ( y t , μ t ) = − c y t + B t ( x t ∗ ( y t , μ t ) , μ t ) . Moreover, with q t ∗ : = q t , J t ( x t ∗ ) q_t^*:=q_{t,J_t(x_t^*)} q t ∗ := q t , J t ( x t ∗ )
Q t ∗ ( ⋅ ∣ y t , μ t ) = ( 1 − μ t q t ∗ ) δ 0 + μ t q t ∗ δ q t ∗ , \mathbb{Q}_t^*(\cdot\mid y_t,\mu_t)=\left(1-\frac{\mu_t}{q_t^*}\right)\delta_0+\frac{\mu_t}{q_t^*}\delta_{q_t^*}, Q t ∗ ( ⋅ ∣ y t , μ t ) = ( 1 − q t ∗ μ t ) δ 0 + q t ∗ μ t δ q t ∗ , where q t ∗ ≥ max { x t ∗ , μ t } q_t^*\ge\max\{x_t^*,\mu_t\} q t ∗ ≥ max { x t ∗ , μ t }
The optimal post-replenishment level is nondecreasing in both current inventory and the conditional mean. Its possible positive targets are given by the inventory thresholds, while the corresponding demand contacts determine the positive support of the worst-case conditional laws. Every positive worst-case realization exhausts the ready inventory. Since zero is absorbing under martingale demand, the conditional laws can be concatenated into an attaining joint martingale distribution supported on at most T + 1 T+1 T + 1
For strictly positive recovery, the number of positive interior inventory thresholds also stabilizes as the planning horizon increases. Let N t : = t − a t N_t:=t-a_t N t := t − a t 0 < α < α b 0<\alpha<\alpha_b 0 < α < α b
Φ ∞ = c + h − α w α , λ ∞ = Φ ∞ Φ ∞ + b − c − α ( c − w ) , \Phi_\infty=c+\frac{h-\alpha w}{\alpha},\qquad
\lambda_\infty=\frac{\Phi_\infty}{\Phi_\infty+b-c-\alpha(c-w)}, Φ ∞ = c + α h − α w , λ ∞ = Φ ∞ + b − c − α ( c − w ) Φ ∞ , and
ζ : = log ( α / ( 1 − β λ ∞ ) ) log λ ∞ . \zeta:=\frac{\log(\alpha/(1-\beta\lambda_\infty))}{\log\lambda_\infty}. ζ := log λ ∞ log ( α / ( 1 − β λ ∞ )) . Proposition 4 (Stabilization of inventory thresholds). For 0 < α < α b 0 < \alpha< \alpha_b 0 < α < α b N t N_t N t
N ∞ = { ⌈ ζ ⌉ , α ≤ α h , 1 + ⌊ ζ ⌋ , α h < α < α b . N_\infty=
\begin{cases}
\lceil\zeta\rceil, & \alpha\le\alpha_h,\\
1+\lfloor\zeta\rfloor, & \alpha_h < \alpha< \alpha_b.
\end{cases} N ∞ = { ⌈ ζ ⌉ , 1 + ⌊ ζ ⌋ , α ≤ α h , α h < α < α b . For α ≤ α h \alpha\le\alpha_h α ≤ α h N t + 1 ∈ { N t , N t + 1 } N_{t+1} \in\{N_t,N_t+1\} N t + 1 ∈ { N t , N t + 1 } { N t } \{N_t\} { N t }
Thus, although the planning horizon increases, the number of positive interior replenishment targets eventually remains constant. The two regimes differ in how this stabilization occurs: the number of thresholds is nondecreasing when 0 < α ≤ α h 0 < \alpha\le\alpha_h 0 < α ≤ α h α > α h \alpha> \alpha_h α > α h α = 0 \alpha= 0 α = 0 N t = t − 1 N_t = t - 1 N t = t − 1
Finite-horizon policy in Regimes III and IV
We next consider α ≥ α b \alpha\ge\alpha_b α ≥ α b
Theorem 3 (Finite-horizon policy in Regimes III and IV). For α ≥ α b \alpha\ge\alpha_b α ≥ α b T T T ( y , μ ) ∈ [ 0 , U ] 2 (y,\mu) \in[0,U]^2 ( y , μ ) ∈ [ 0 , U ] 2
x t ∗ ( y , μ ) = y , t = 2 , … , T . x_t^*(y,\mu)=y,\qquad t=2,\ldots,T. x t ∗ ( y , μ ) = y , t = 2 , … , T . The nonterminal action is unique when α > α b \alpha> \alpha_b α > α b
x 1 ∗ ( y , μ ) ∈ { { y } , μ < ρ 1 U , [ y , U ] , μ = ρ 1 U , { U } , μ > ρ 1 U . x_1^*(y,\mu) \in
\begin{cases}
\{y\}, & \mu< \rho_1 U,\\
[y,U], & \mu= \rho_1 U,\\
\{U\}, & \mu> \rho_1 U.
\end{cases} x 1 ∗ ( y , μ ) ∈ ⎩ ⎨ ⎧ { y } , [ y , U ] , { U } , μ < ρ 1 U , μ = ρ 1 U , μ > ρ 1 U . Unlike Regimes I and II, the optimal nonterminal policy is independent of both the conditional mean and the remaining horizon. This distinction follows from the replenishment margin b − c − α ( c − w ) ≤ 0 b-c-\alpha(c-w) \le0 b − c − α ( c − w ) ≤ 0 h − α w h-\alpha w h − α w
Together, Theorems 2 and 3 characterize the optimal replenishment policies across the four recovery regimes under the regular penalty condition. When α < α b \alpha< \alpha_b α < α b α > 0 \alpha> 0 α > 0 α ≥ α b \alpha\ge\alpha_b α ≥ α b α h \alpha_h α h
The complementary case K 2 ≤ β K 1 K_2 \le\beta K_1 K 2 ≤ β K 1 § 5 . We develop two regime-specific approximation policies: a conditional-mean threshold policy in Regime II and a no-replenishment policy in Regime IV. Both recover the exact terminal- and two-period solutions and admit uniform bounds on their worst-case optimality gaps. In particular, their absolute gaps remain bounded independently of the planning horizon, while their relative gaps converge to zero at rate O ( 1 / t ) O(1/t) O ( 1/ t )
Submartingale and Supermartingale Demand
The martingale model specifies the next conditional mean as the latest observed demand. We extend this restriction to allow the conditional mean to lie above or below that reference level. The inventory dynamics and operating costs remain unchanged, but the admissible demand distributions now differ in their conditional means as well as their shapes. This section retains Assumption 1; the regular penalty condition is not imposed. We continue to use the algebraic coefficients Φ t , K t \Phi_t, K_t Φ t , K t (3) and the ratio ρ t = Φ t / K t \rho_t = \Phi_t/K_t ρ t = Φ t / K t
Directional ambiguity and order-up-to structure
For a reference level μ ∈ [ 0 , U ] \mu\in[0,U] μ ∈ [ 0 , U ]
M ↑ ( μ ) : = { Q ∈ P 0 ( [ 0 , U ] ) : E Q [ D ~ ] ≥ μ } , \mathcal{M}^{\uparrow}(\mu) := \left\{Q \in\mathcal{P}_0([0,U]) : \mathbb{E}_Q[\widetilde{D}] \ge\mu\right\}, M ↑ ( μ ) := { Q ∈ P 0 ([ 0 , U ]) : E Q [ D ] ≥ μ } , M ↓ ( μ ) : = { Q ∈ P 0 ( [ 0 , U ] ) : E Q [ D ~ ] ≤ μ } . \mathcal{M}^{\downarrow}(\mu) := \left\{Q \in\mathcal{P}_0([0,U]) : \mathbb{E}_Q[\widetilde{D}] \le\mu\right\}. M ↓ ( μ ) := { Q ∈ P 0 ([ 0 , U ]) : E Q [ D ] ≤ μ } . They correspond to submartingale and supermartingale demand, respectively. As in § 3 , μ T = μ \mu_T = \mu μ T = μ μ t = D t + 1 \mu_t = D_{t+1} μ t = D t + 1 t < T t < T t < T
M A R ↑ ( μ ) : = { P ∈ P 0 ( [ 0 , U ] T ) : E P [ D ~ t ∣ F t + 1 ] ≥ D ~ t + 1 , t ∈ [ T ] } , \mathcal{MAR}^{\uparrow}(\mu) := \left\{P \in\mathcal{P}_0([0,U]^T) : \mathbb{E}_P[\widetilde{D}_t \mid\mathcal{F}_{t+1}] \ge\widetilde{D}_{t+1},\ t \in[T]\right\}, M A R ↑ ( μ ) := { P ∈ P 0 ([ 0 , U ] T ) : E P [ D t ∣ F t + 1 ] ≥ D t + 1 , t ∈ [ T ] } , M A R ↓ ( μ ) : = { P ∈ P 0 ( [ 0 , U ] T ) : E P [ D ~ t ∣ F t + 1 ] ≤ D ~ t + 1 , t ∈ [ T ] } , \mathcal{MAR}^{\downarrow}(\mu) := \left\{P \in\mathcal{P}_0([0,U]^T) : \mathbb{E}_P[\widetilde{D}_t \mid\mathcal{F}_{t+1}] \le\widetilde{D}_{t+1},\ t \in[T]\right\}, M A R ↓ ( μ ) := { P ∈ P 0 ([ 0 , U ] T ) : E P [ D t ∣ F t + 1 ] ≤ D t + 1 , t ∈ [ T ] } , where D ~ T + 1 ≡ μ \widetilde{D}_{T+1} \equiv\mu D T + 1 ≡ μ V ^ t ↑ \widehat{V}^{\uparrow}_t V t ↑ V ^ t ↓ \widehat{V}^{\downarrow}_t V t ↓ V ^ 0 ↑ = V ^ 0 ↓ = 0 \widehat{V}^{\uparrow}_0 = \widehat{V}^{\downarrow}_0 = 0 V 0 ↑ = V 0 ↓ = 0
Proposition 5 (Directional dynamic programming). For either ⋄ ∈ { ↑ , ↓ } \diamond\in\{\uparrow,\downarrow\} ⋄ ∈ { ↑ , ↓ }
V ^ t ⋄ ( y , μ ) = min x ∈ [ y , U ] { c ( x − y ) + w α ( U − x ) + sup Q ∈ M ⋄ ( μ ) E Q [ b ( D ~ − x ) + + h ( x − D ~ ) + + V ^ t − 1 ⋄ ( R ( x , D ~ ) , D ~ ) ] } . (10) \widehat{V}^{\diamond}_t(y,\mu) = \min_{x \in[y,U]} \left\{ c(x-y) + w\alpha(U-x) + \sup_{Q \in\mathcal{M}^{\diamond}(\mu)} \mathbb{E}_Q\left[b(\widetilde{D}-x)_+ + h(x-\widetilde{D})_+ + \widehat{V}^{\diamond}_{t-1}(R(x,\widetilde{D}),\widetilde{D})\right]\right\}. \tag*{(10)} V t ⋄ ( y , μ ) = x ∈ [ y , U ] min { c ( x − y ) + w α ( U − x ) + Q ∈ M ⋄ ( μ ) sup E Q [ b ( D − x ) + + h ( x − D ) + + V t − 1 ⋄ ( R ( x , D ) , D ) ] } . (10) For each fixed action, the supremum is attained by a distribution supported on at most two demand points.
Corollary 3. For every finite t t t ( y , μ ) ∈ [ 0 , U ] 2 (y,\mu) \in[0,U]^2 ( y , μ ) ∈ [ 0 , U ] 2
V ^ t ( y , μ ) ≤ V ^ ↑ t ( y , μ ) , V ^ t ( y , μ ) ≤ V ^ ↓ t ( y , μ ) . \widehat{V}^{t}(y,\mu) \le\widehat{V}^{t}_{\uparrow}(y,\mu), \qquad\widehat{V}^{t}(y,\mu) \le\widehat{V}^{t}_{\downarrow}(y,\mu). V t ( y , μ ) ≤ V ↑ t ( y , μ ) , V t ( y , μ ) ≤ V ↓ t ( y , μ ) . Furthermore, V ^ ↑ t ( y , μ ) \widehat{V}^{t}_{\uparrow}(y,\mu) V ↑ t ( y , μ ) μ \mu μ V ^ ↓ t ( y , μ ) \widehat{V}^{t}_{\downarrow}(y,\mu) V ↓ t ( y , μ ) μ \mu μ
The opposite monotonicities reflect the admissible mean intervals: increasing μ \mu μ M ↑ ( μ ) \mathcal{M}^{\uparrow}(\mu) M ↑ ( μ ) M ↓ ( μ ) \mathcal{M}^{\downarrow}(\mu) M ↓ ( μ ) M ↑ ( U ) = { δ U } \mathcal{M}^{\uparrow}(U) = \{\delta_U\} M ↑ ( U ) = { δ U } M ↓ ( 0 ) = { δ 0 } \mathcal{M}^{\downarrow}(0) = \{\delta_0\} M ↓ ( 0 ) = { δ 0 }
Write B ⋄ , t ( x , μ ) B_{\diamond,t}(x,\mu) B ⋄ , t ( x , μ ) (10) after separating − c y -cy − cy B t B_t B t § 3 . The convexity argument underlying Proposition 2 applies to both conditional mean restrictions.
Proposition 6. For either ⋄ ∈ { ↑ , ↓ } \diamond\in\{\uparrow,\downarrow\} ⋄ ∈ { ↑ , ↓ } t t t B ⋄ , t ( ⋅ , μ ) B_{\diamond,t}(\cdot,\mu) B ⋄ , t ( ⋅ , μ ) [ 0 , U ] [0,U] [ 0 , U ]
χ t ⋄ ( μ ) : = max a r g m i n x ∈ [ 0 , U ] B ⋄ , t ( x , μ ) . \chi^{\diamond}_t(\mu) := \max\mathop{\rm arg\,min}_{x\in[0,U]} B_{\diamond,t}(x,\mu). χ t ⋄ ( μ ) := max arg min x ∈ [ 0 , U ] B ⋄ , t ( x , μ ) . Then an optimal replenishment decision is
x ⋄ , t ∗ ( y , μ ) = max { y , χ t ⋄ ( μ ) } . x^{*}_{\diamond,t}(y,\mu) = \max\{y,\chi^{\diamond}_t(\mu)\}. x ⋄ , t ∗ ( y , μ ) = max { y , χ t ⋄ ( μ )} . The directional models therefore preserve the order-up-to structure, but their explicit policies differ markedly.
Submartingale demand: upward drift
Under upward ambiguity, U U U δ U \delta_U δ U M ↑ ( μ ) \mathcal{M}^{\uparrow}(\mu) M ↑ ( μ ) α < α b \alpha< \alpha_b α < α b
Recall ρ t = Φ t / K t \rho_t = \Phi_t/K_t ρ t = Φ t / K t
ω t : = 1 − ( t h − ( t − 1 ) c ) + K t , C t ↑ : = t c U + ( t h − ( t − 1 ) c ) + ( 1 − ρ t ) U . \omega_t := 1 - \frac{\left(th-(t-1)c\right)_+}{K_t}, \qquad C^{\uparrow}_t := tcU + \left(th-(t-1)c\right)_+(1-\rho_t)U. ω t := 1 − K t ( t h − ( t − 1 ) c ) + , C t ↑ := t c U + ( t h − ( t − 1 ) c ) + ( 1 − ρ t ) U . Theorem 4 (Finite-horizon policy under submartingale demand). For α < α b \alpha< \alpha_b α < α b t t t
χ t ↑ ( μ ) = { ω t U , 0 ≤ μ < ρ t U , U , ρ t U ≤ μ ≤ U . (11) \chi^{\uparrow}_t(\mu) =
\begin{cases}
\omega_t U, & 0 \le\mu< \rho_t U,\\
U, & \rho_t U \le\mu\le U.
\end{cases} \tag*{(11)} χ t ↑ ( μ ) = { ω t U , U , 0 ≤ μ < ρ t U , ρ t U ≤ μ ≤ U . (11) An optimal action is
x ↑ , t ∗ ( y , μ ) = max { y , χ t ↑ ( μ ) } , x^{*}_{\uparrow,t}(y,\mu) = \max\{y,\chi^{\uparrow}_t(\mu)\}, x ↑ , t ∗ ( y , μ ) = max { y , χ t ↑ ( μ )} , and the corresponding value is
V ^ ↑ t ( y , μ ) = − c y + C t ↑ + ( Φ t − K t ) ( x ↑ , t ∗ ( y , μ ) − ω t U ) + K t ( 1 − μ U ) ( x ↑ , t ∗ ( y , μ ) − ω t U ) + . \widehat{V}^{t}_{\uparrow}(y,\mu) = -cy + C^{\uparrow}_t + (\Phi_t-K_t)\left(x^{*}_{\uparrow,t}(y,\mu)-\omega_t U\right)
+ K_t\left(1-\frac{\mu}{U}\right)\left(x^{*}_{\uparrow,t}(y,\mu)-\omega_t U\right)_+. V ↑ t ( y , μ ) = − cy + C t ↑ + ( Φ t − K t ) ( x ↑ , t ∗ ( y , μ ) − ω t U ) + K t ( 1 − U μ ) ( x ↑ , t ∗ ( y , μ ) − ω t U ) + . For α ≥ α b \alpha\ge\alpha_b α ≥ α b
x ↑ , 2 ∗ ( y , μ ) = y , ( y , μ ) ∈ [ 0 , U ] 2 . x^*_{\uparrow,2}(y,\mu)=y,\qquad(y,\mu)\in[0,U]^2. x ↑ , 2 ∗ ( y , μ ) = y , ( y , μ ) ∈ [ 0 , U ] 2 . This action is unique when α > α b \alpha>\alpha_b α > α b x ↑ , 2 ∗ = y x^*_{\uparrow,2}=y x ↑ , 2 ∗ = y
The lower target ω t U \omega_tU ω t U ( t − 1 ) c ≥ t h (t-1)c\ge th ( t − 1 ) c ≥ t h c > h c>h c > h
t ↑ : = ⌈ c c − h ⌉ . t_{\uparrow}:=\left\lceil\frac{c}{c-h}\right\rceil. t ↑ := ⌈ c − h c ⌉ . The two branches of (11) then coincide for every t ≥ t ↑ t\ge t_{\uparrow} t ≥ t ↑
Corollary 4. Under α < α b \alpha<\alpha_b α < α b t ≥ t ↑ t\ge t_{\uparrow} t ≥ t ↑
x ↑ , t ∗ ( y , μ ) = U , V ^ t ↑ ( y , μ ) = t c U − c y , x^*_{\uparrow,t}(y,\mu)=U,\qquad\hat{V}^{\uparrow}_t(y,\mu)=tcU-cy, x ↑ , t ∗ ( y , μ ) = U , V ^ t ↑ ( y , μ ) = t c U − cy , for all ( y , μ ) ∈ [ 0 , U ] 2 (y,\mu)\in[0,U]^2 ( y , μ ) ∈ [ 0 , U ] 2
In particular, t ↑ = 2 t_{\uparrow}=2 t ↑ = 2 c ≥ 2 h c\ge2h c ≥ 2 h α ≥ α b \alpha\ge\alpha_b α ≥ α b
Thus the two-period submartingale policy shares the replenishment boundary of Corollary 2. At longer horizons, Proposition 6 guarantees an order-up-to policy, but does not establish no replenishment throughout the region α ≥ α b \alpha\ge\alpha_b α ≥ α b
Supermartingale demand: downward drift
Under downward ambiguity, zero is absorbing because M ↓ ( 0 ) = { δ 0 } \mathcal{M}^{\downarrow}(0)=\{\delta_0\} M ↓ ( 0 ) = { δ 0 }
The terminal target illustrates the difference from upward ambiguity. Let
α d : = b b + h ∈ ( 0 , 1 ) . \alpha_d:=\frac{b}{b+h}\in(0,1). α d := b + h b ∈ ( 0 , 1 ) . Its largest optimal order-up-to target is
χ ↓ 1 ( μ ) = { 0 , 0 ≤ μ < ρ 1 U , α d U , ρ 1 U ≤ μ ≤ U . \chi_{\downarrow1}(\mu)=
\begin{cases}
0, & 0\le\mu<\rho_1U,\\
\alpha_dU, & \rho_1U\le\mu\le U.
\end{cases} χ ↓ 1 ( μ ) = { 0 , α d U , 0 ≤ μ < ρ 1 U , ρ 1 U ≤ μ ≤ U . Even at high reference demand, the terminal target remains below capacity because the worst-case distribution may select a lower mean.
A different effect arises when recovery is sufficiently strong. If α > α d \alpha>\alpha_d α > α d α U > α d U \alpha U>\alpha_dU α U > α d U α > α b \alpha> \alpha_b α > α b
For α > max { α b , α d } \alpha> \max\{\alpha_b,\alpha_d\} α > max { α b , α d }
z 2 ↓ : = U ( 1 − α d α ) , z_2^\downarrow:= U\left(1-\frac{\alpha_d}{\alpha}\right), z 2 ↓ := U ( 1 − α α d ) , γ 0 : = c + ( 2 − α ) ( h − α w ) , \gamma_0 := c+(2-\alpha)(h-\alpha w), γ 0 := c + ( 2 − α ) ( h − α w ) , γ 1 : = c − b − α h − α β w , \gamma_1 := c-b-\alpha h-\alpha\beta w, γ 1 := c − b − α h − α β w , μ 2 † : = U γ 0 γ 0 − γ 1 . \mu_2^\dagger:= U\frac{\gamma_0}{\gamma_0-\gamma_1}. μ 2 † := U γ 0 − γ 1 γ 0 . The standing cost ordering gives γ 0 > 0 > γ 1 \gamma_0>0>\gamma_1 γ 0 > 0 > γ 1 μ 2 † ∈ ( 0 , U ) \mu_2^\dagger\in(0,U) μ 2 † ∈ ( 0 , U )
Proposition 7 (Two-period recovery suppression under downward drift ). Under α > max { α b , α d } \alpha>\max\{\alpha_b,\alpha_d\} α > max { α b , α d }
χ 2 ↓ ( μ ) = { 0 , 0 ≤ μ < μ 2 † , z 2 ↓ , μ 2 † ≤ μ ≤ U . \chi_2^\downarrow(\mu)=
\begin{cases}
0, & 0\le\mu<\mu_2^\dagger,\\
z_2^\downarrow, & \mu_2^\dagger\le\mu\le U.
\end{cases} χ 2 ↓ ( μ ) = { 0 , z 2 ↓ , 0 ≤ μ < μ 2 † , μ 2 † ≤ μ ≤ U . Consequently,
x ↓ , 2 ∗ ( y , μ ) = max { y , χ 2 ↓ ( μ ) } . x_{\downarrow,2}^*(y,\mu)=\max\{y,\chi_2^\downarrow(\mu)\}. x ↓ , 2 ∗ ( y , μ ) = max { y , χ 2 ↓ ( μ )} . Thus α b \alpha_b α b
We next obtain an explicit finite-horizon policy for α < α b \alpha<\alpha_b α < α b α ≤ min { α h , α d } \alpha\le\min\{\alpha_h,\alpha_d\} α ≤ min { α h , α d } q t , j q_{t,j} q t , j (5) but uses a different recursion for inventory targets. Initialize s 1 , 0 ↓ = 0 s_{1,0}^\downarrow=0 s 1 , 0 ↓ = 0 s 1 , 1 ↓ = α d U s_{1,1}^\downarrow=\alpha_dU s 1 , 1 ↓ = α d U t ≥ 2 t\ge2 t ≥ 2 s t , 0 ↓ = 0 s_{t,0}^\downarrow=0 s t , 0 ↓ = 0
s t , j ↓ : = min { U , [ Φ t − 1 K t ( s t − 1 , j ↓ − α U ) ] + } , j = 1 , … , t − 1 , s_{t,j}^\downarrow:=\min\left\{U,\left[\frac{\Phi_{t-1}}{K_t}\left(s_{t-1,j}^\downarrow-\alpha U\right)\right]_+\right\},\qquad j=1,\ldots,t-1, s t , j ↓ := min { U , [ K t Φ t − 1 ( s t − 1 , j ↓ − α U ) ] + } , j = 1 , … , t − 1 , s t , t ↓ : = min { U , [ Φ t − 1 K t ( s t − 1 , t − 1 ↓ − α U ) + b K t U ] + } . (12) s_{t,t}^\downarrow:=\min\left\{U,\left[\frac{\Phi_{t-1}}{K_t}\left(s_{t-1,t-1}^\downarrow-\alpha U\right)+\frac{b}{K_t}U\right]_+\right\}. \tag*{(12)} s t , t ↓ := min { U , [ K t Φ t − 1 ( s t − 1 , t − 1 ↓ − α U ) + K t b U ] + } . (12) With V ^ 0 ↓ ≡ 0 \widehat V_0^\downarrow\equiv0 V 0 ↓ ≡ 0 x ∈ [ χ t ↓ ( μ ) , U ] x\in[\chi_t^\downarrow(\mu),U] x ∈ [ χ t ↓ ( μ ) , U ]
B ↓ , t ( x , μ ) = ( c + t h ) U + ( V ^ t − 1 ↓ ( 0 , U ) U − t h ) μ − Φ t ( U − max { x , s t , t ↓ } ) − ∑ j = 1 t ( s t , j ↓ − max { x , s t , j − 1 ↓ } ) + ( Φ t − K t μ q t , j ) . (13) \begin{aligned}
B_{\downarrow,t}(x,\mu)
={}&(c+th)U+\left(\frac{\widehat V_{t-1}^\downarrow(0,U)}{U}-th\right)\mu\\
&-\Phi_t\left(U-\max\{x,s_{t,t}^\downarrow\}\right)\\
&-\sum_{j=1}^{t}\left(s_{t,j}^\downarrow-\max\{x,s_{t,j-1}^\downarrow\}\right)
+\left(\Phi_t-K_t\frac{\mu}{q_{t,j}}\right). \tag*{(13)}
\end{aligned} B ↓ , t ( x , μ ) = ( c + t h ) U + ( U V t − 1 ↓ ( 0 , U ) − t h ) μ − Φ t ( U − max { x , s t , t ↓ } ) − j = 1 ∑ t ( s t , j ↓ − max { x , s t , j − 1 ↓ } ) + ( Φ t − K t q t , j μ ) . (13) Theorem 5 (Finite-horizon policy under downward drift ). Under α < α b \alpha<\alpha_b α < α b α ≤ min { α h , α d } \alpha\le\min\{\alpha_h,\alpha_d\} α ≤ min { α h , α d } t t t
χ t ↓ ( μ ) = { 0 , 0 ≤ μ < ρ t q t , 1 , s t , j ↓ , ρ t q t , j ≤ μ < ρ t q t , j + 1 , j = 1 , … , t − 1 , s t , t ↓ , ρ t U ≤ μ ≤ U . (14) \chi_t^\downarrow(\mu)=
\begin{cases}
0, & 0\le\mu<\rho_tq_{t,1},\\
s_{t,j}^\downarrow, & \rho_tq_{t,j}\le\mu<\rho_tq_{t,j+1},\quad j=1,\ldots,t-1,\\
s_{t,t}^\downarrow, & \rho_tU\le\mu\le U.
\end{cases} \tag*{(14)} χ t ↓ ( μ ) = ⎩ ⎨ ⎧ 0 , s t , j ↓ , s t , t ↓ , 0 ≤ μ < ρ t q t , 1 , ρ t q t , j ≤ μ < ρ t q t , j + 1 , j = 1 , … , t − 1 , ρ t U ≤ μ ≤ U . (14) An optimal action and its value are
x ↓ , t ∗ ( y , μ ) = max { y , x t ↓ ( μ ) } , x^{*}_{\downarrow,t}(y,\mu)=\max\{y,x^{\downarrow}_{t}(\mu)\}, x ↓ , t ∗ ( y , μ ) = max { y , x t ↓ ( μ )} , V ^ t ↓ ( y , μ ) = − c y + B ↓ , t ( x ↓ , t ∗ ( y , μ ) , μ ) . \widehat{V}^{\downarrow}_{t}(y,\mu)=-cy+B_{\downarrow,t}(x^{*}_{\downarrow,t}(y,\mu),\mu). V t ↓ ( y , μ ) = − cy + B ↓ , t ( x ↓ , t ∗ ( y , μ ) , μ ) . Some inherited targets may be zero, so the number of positive interior targets can be smaller than the number of recursive candidates. Define
a t ↓ : = min { j ∈ { 1 , … , t } : s t , j ↓ > 0 } , N t ↓ : = t − a t ↓ . a^{\downarrow}_{t}:=\min\{j\in\{1,\ldots,t\}:s^{\downarrow}_{t,j}>0\},\qquad N^{\downarrow}_{t}:=t-a^{\downarrow}_{t}. a t ↓ := min { j ∈ { 1 , … , t } : s t , j ↓ > 0 } , N t ↓ := t − a t ↓ . The number of distinct positive target levels is N t ↓ + 1 N^{\downarrow}_{t}+1 N t ↓ + 1
Corollary 5 (Stabilization of supermartingale targets). Under the conditions of Theorem 5 and α > 0 \alpha>0 α > 0 { N t ↓ } \{N^{\downarrow}_{t}\} { N t ↓ }
N t + 1 ↓ ∈ { N t ↓ , N t ↓ + 1 } . N^{\downarrow}_{t+1}\in\{N^{\downarrow}_{t},N^{\downarrow}_{t}+1\}. N t + 1 ↓ ∈ { N t ↓ , N t ↓ + 1 } . It stabilizes after a finite horizon at the value N ∞ N_{\infty} N ∞
T ↓ s t a b = min { t ≥ N ∞ + 1 : s t , t − N ∞ ↓ > 0 } , N t ↓ = N ∞ ( t ≥ T ↓ s t a b ) . T^{\mathrm{stab}}_{\downarrow}=\min\{t\geq N_{\infty}+1:s^{\downarrow}_{t,t-N_{\infty}}>0\},\qquad N^{\downarrow}_{t}=N_{\infty}\quad(t\geq T^{\mathrm{stab}}_{\downarrow}). T ↓ stab = min { t ≥ N ∞ + 1 : s t , t − N ∞ ↓ > 0 } , N t ↓ = N ∞ ( t ≥ T ↓ stab ) . Thus the positive supermartingale target levels eventually stabilize in number, whereas upward ambiguity leads to uniform capacity replenishment once t ≥ t ↑ t\geq t_{\uparrow} t ≥ t ↑
Numerical Study
We evaluate M-DRO against five benchmark policies using conditional Beta demand processes with varying dispersion and departures from the martingale mean assumption. We first compare out-of-sample costs and service levels across demand conditions and historical sample sizes, then assess robustness to changes in demand distributions between training and deployment. Finally, we examine how central replenishment costs, planning horizons, and automatic recovery rates affect operating performance and replenishment decisions.
Experimental Setting and Benchmark Policies
We consider a baseline station with capacity U = 25 U=25 U = 25 T = 24 T=24 T = 24 U / 2 U/2 U /2 [32 ] , we generate heterogeneous operating instances by independently sampling the cost and recovery parameters from uniform distributions over
h ∈ [ 0.5 , 20 ] , w ∈ [ 30 , 50 ] , c ∈ [ 40 , 110 ] , b ∈ [ 100 , 120 ] , α ∈ [ 0.02 , 0.8 ] . h\in[0.5,20],\quad w\in[30,50],\quad c\in[40,110],\quad b\in[100,120],\quad\alpha\in[0.02,0.8]. h ∈ [ 0.5 , 20 ] , w ∈ [ 30 , 50 ] , c ∈ [ 40 , 110 ] , b ∈ [ 100 , 120 ] , α ∈ [ 0.02 , 0.8 ] . We retain samples satisfying Assumption 1 and the regular penalty condition (4) , ensuring that the analytical policies apply to all generated instances. These instances span all four recovery regimes.
We generate temporally dependent demand using a conditional Beta distribution. For each replication, we independently draw p B ∼ Unif ( 0 , 1 ) p_B \sim\operatorname{Unif}(0,1) p B ∼ Unif ( 0 , 1 ) p B p_B p B m m m
q δ ( m ) = ( 1 − δ ) m U + δ p B , q_{\delta}(m) = (1-\delta)\frac{m}{U} + \delta p_B, q δ ( m ) = ( 1 − δ ) U m + δ p B , D ~ j + 1 c o n t ∣ ( D ~ j = m , p B ) ∼ U Beta ( κ q δ ( m ) , κ [ 1 − q δ ( m ) ] ) . \widetilde{D}^{\mathrm{cont}}_{j+1} \mid(\widetilde{D}_j=m,p_B) \sim U\operatorname{Beta}\left(\kappa q_{\delta}(m),\kappa[1-q_{\delta}(m)]\right). D j + 1 cont ∣ ( D j = m , p B ) ∼ U Beta ( κ q δ ( m ) , κ [ 1 − q δ ( m )] ) . where j j j D ~ j \widetilde{D}_j D j j j j t t t δ \delta δ p B U p_B U p B U E [ D ~ j + 1 ∣ D ~ j = m , p B ] − m = δ ( p B U − m ) \mathbb{E}[\widetilde{D}_{j+1}\mid\widetilde{D}_j=m,p_B]-m=\delta(p_B U-m) E [ D j + 1 ∣ D j = m , p B ] − m = δ ( p B U − m ) κ > 0 \kappa>0 κ > 0 U 2 q δ ( m ) [ 1 − q δ ( m ) ] / ( κ + 1 ) U^2q_{\delta}(m)[1-q_{\delta}(m)]/(\kappa+1) U 2 q δ ( m ) [ 1 − q δ ( m )] / ( κ + 1 ) 1 / 4 1/4 1/4 0 ≤ δ ≤ 1 0\le\delta\le1 0 ≤ δ ≤ 1 δ ∈ { 0.075 , 0.15 , 0.30 } \delta\in\{0.075,0.15,0.30\} δ ∈ { 0.075 , 0.15 , 0.30 } κ ∈ { 0.125 , 0.25 , 0.5 , 0.75 , 1 , 1.5 , 2 , 3 , 4 , 6 , 8 } \kappa\in\{0.125,0.25,0.5,0.75,1,1.5,2,3,4,6,8\} κ ∈ { 0.125 , 0.25 , 0.5 , 0.75 , 1 , 1.5 , 2 , 3 , 4 , 6 , 8 } κ \kappa κ
We compare M-DRO with five benchmark policies that differ in their use of demand information and replenishment decision rules. All policies observe the same current inventory and latest demand, but differ in whether and how they use historical observations, as summarized in Table 2 .
Table 2. Policies compared in the numerical study
Specifically, rolling CE uses the latest observed demand as a point forecast, allowing us to isolate the value of accounting for demand uncertainty without historical fitting. IID DP and Conditional DP assess the benefits of estimating unconditional and state-dependent demand distributions, respectively. Fixed Base-stock and Myopic Newsvendor provide simpler alternatives that do not optimize over the full planning horizon. Together, these benchmarks distinguish the effects of distribution estimation, temporal dependence, and dynamic optimization.
All historical fitting procedures are fixed before out-of-sample evaluation, and no policy has access to the true demand-generating parameters.
We generate 80 independent operating instances, reuse them across demand settings, and evaluate all policies on 250 test paths per instance. The data-driven benchmarks are trained using N ∈ { 24 , 48 , 72 , 96 } N \in\{24, 48, 72, 96\} N ∈ { 24 , 48 , 72 , 96 }
Out-of-Sample Performance
We first examine out-of-sample performance when training and test data follow the same demand distribution. Figure 4 compares the six policies in terms of operating costs and fill rates. Across all tested values of δ \delta δ κ ≤ 2 \kappa\le2 κ ≤ 2 κ = 3 \kappa= 3 κ = 3 δ = 0.075 \delta= 0.075 δ = 0.075
Figure 4. Out-of-sample cost savings and fill rates under persistent Beta demand. (N = 48 N = 48 N = 48 T = 24 T = 24 T = 24
For instance, at δ = 0.075 \delta= 0.075 δ = 0.075 κ = 0.25 \kappa= 0.25 κ = 0.25 N = 48 N = 48 N = 48 κ \kappa κ δ = 0.075 \delta= 0.075 δ = 0.075 κ = 3 \kappa= 3 κ = 3
Table 3 compares M-DRO and Rolling CE, which require no historical fitting, with Conditional DP, the best-performing fitted benchmark in these settings, trained on different numbers of historical demand observations. Increasing the sample size consistently reduces Conditional DP’s operating cost, but its relative performance depends strongly on demand dispersion. For all κ ≤ 2 \kappa\le2 κ ≤ 2 κ = 0.125 \kappa= 0.125 κ = 0.125 N N N κ = 8 \kappa= 8 κ = 8 κ = 3 \kappa= 3 κ = 3
Table 3. Effect of historical sample size on operating cost. δ = 0.15 \delta= 0.15 δ = 0.15
These findings suggest that the value of historical demand estimation depends on demand variability as well as sample size. Under highly dispersed demand, even substantial increases in historical observations do not necessarily make fitted policies preferable to M-DRO. When demand is more concentrated, distribution estimation becomes more effective, although Rolling CE may still provide a better alternative without historical fitting. For inventory managers, this highlights the importance of selecting replenishment policies according to the demand environment rather than simply increasing the amount of historical data.
Robustness Testing
We further examine policy robustness to demand shifts across heterogeneous historical environments. In each of 80 independent replications, we draw δ ∼ Unif ( 0.075 , 0.30 ) \delta\sim\operatorname{Unif}(0.075, 0.30) δ ∼ Unif ( 0.075 , 0.30 ) κ ∼ Unif ( 0.5 , 2 ) \kappa\sim\operatorname{Unif}(0.5, 2) κ ∼ Unif ( 0.5 , 2 ) N = 48 N = 48 N = 48 κ / 2 \kappa/2 κ /2 2 κ 2\kappa 2 κ 4 κ 4\kappa 4 κ δ + 0.10 \delta+ 0.10 δ + 0.10 δ − 0.05 \delta- 0.05 δ − 0.05 δ + 0.10 \delta+ 0.10 δ + 0.10 κ / 2 \kappa/2 κ /2 2 κ 2\kappa 2 κ δ = 1 \delta= 1 δ = 1
N = 48 N = 48 N = 48 T = 24 T = 24 T = 24
Table 4 shows that M-DRO maintains the lowest sample mean cost under increased dispersion and the tested moderate changes in mean reversion, including both joint shifts. Its advantage over Rolling CE nevertheless narrows as demand becomes more concentrated: when κ \kappa κ κ \kappa κ δ = 1 \delta= 1 δ = 1
Table 4. Operating costs under demand distribution shifts
These patterns distinguish robustness to demand variability from robustness to changes in temporal dependence. When recent demand remains informative, M-DRO avoids fitting a potentially unstable conditional distribution and is competitive under dispersion shifts. As demand concentrates, Rolling CE benefits from more predictable outcomes. When lag dependence disappears, even a previously fitted IID model can be preferable to policies that condition heavily on the latest observation. For managers, the central question is therefore not merely whether demand has changed, but whether recent demand remains a reliable signal for replenishment decisions.
Sensitivity Analysis and Managerial Insights
We further examine how central replenishment costs, planning horizons, and automatic recovery rates affect operating performance. All experiments use N = 48 N = 48 N = 48 δ = 0.15 \delta= 0.15 δ = 0.15 Figure 5 varies c ∈ { 50 , 60 , 70 , 80 , 90 } c \in\{50, 60, 70, 80, 90\} c ∈ { 50 , 60 , 70 , 80 , 90 } T ∈ { 12 , 24 , 48 } T \in\{12, 24, 48\} T ∈ { 12 , 24 , 48 } κ = 2 \kappa= 2 κ = 2 c c c
Figure 5. Sensitivity to central replenishment cost and planning horizon.
Figure 5 shows that higher central replenishment costs increase total operating costs and encourage greater reliance on local recovery. This behavior is consistent with the replenishment threshold α b = ( b − c ) / ( c − w ) \alpha_b = (b-c)/(c-w) α b = ( b − c ) / ( c − w ) c c c c c c c = 60 c = 60 c = 60
Figure 6 examines the interaction between recovery intensity and demand dispersion across six values of κ \kappa κ α \alpha α κ ≤ 1 \kappa\le1 κ ≤ 1 κ = 2 \kappa= 2 κ = 2 α ≥ 0.65 \alpha\ge0.65 α ≥ 0.65 κ = 4 \kappa= 4 κ = 4 α = 0.30 \alpha= 0.30 α = 0.30 κ = 8 \kappa= 8 κ = 8
Figure 6. Recovery intensity and mean operating cost.
Together, Figure 5 and Figure 6 reveal two important operational trade-offs. First, substituting local recovery for costly central replenishment can reduce central supply requirements but may compromise service availability. Second, stronger recovery does not uniformly reduce operating costs or preserve the advantage of M-DRO across demand environments. For inventory managers, effective replenishment planning therefore requires balancing recovery capability, central supply costs, and demand uncertainty rather than optimizing each consideration in isolation.
Concluding Remarks
This paper studies dynamic inventory control for battery-swapping operations with central replenishment and automatic local recovery. Motivated by the limited availability of historical demand data and the difficulty of predicting temporally dependent demand, we develop a time-consistent distributionally robust inventory model under martingale ambiguity. We establish the finite-horizon order-up-to structure and identify two critical recovery rates that characterize the interplay between central replenishment and local recovery. Under a regular penalty condition, we derive explicit finite-horizon policies and worst-case demand distributions, revealing when central replenishment is beneficial and when it becomes unnecessary. Extensions to submartingale and supermartingale demand further reveal distinct replenishment behaviors under directional demand uncertainty.
Numerical experiments demonstrate that M-DRO is particularly competitive when demand is highly dispersed and historical information is limited. However, fitted or rolling-horizon policies may perform better when demand becomes more predictable. The results also highlight the trade-off between central replenishment, local recovery, and service availability. Future research could extend the model to more general demand dependence, data-driven conditional-mean updates, and operational features such as replenishment lead times and stochastic recovery capacity, while preserving tractable policy structures.
Online Appendix: Proofs and Numerical Implementation
Unprefixed references refer to the main manuscript. Notation follows § 3 ; numerical implementation appears in the final section.
Dynamic-Programming Foundations
Proof of Proposition 1 and Corollary 1. On the Borel demand space [ 0 , U ] T [0,U]^{T} [ 0 , U ] T M A R ( μ ) \mathcal{MAR}(\mu) M A R ( μ ) M ( D t + 1 ) \mathcal{M}(D_{t+1}) M ( D t + 1 ) δ D t + 1 \delta_{D_{t+1}} δ D t + 1 δ μ \delta_{\mu} δ μ M A R ( μ ) \mathcal{MAR}(\mu) M A R ( μ ) y ↦ [ y , U ] y \mapsto[y,U] y ↦ [ y , U ] m ↦ M ( m ) m \mapsto\mathcal{M}(m) m ↦ M ( m ) δ 0 \delta_0 δ 0 δ U \delta_U δ U [20 ] ; Shapiro [43 ] ), induction from V ^ 0 = 0 \widehat{V}^{0}=0 V 0 = 0 x t ∗ ( y , m ) x_t^{*}(y,m) x t ∗ ( y , m ) Q t ∗ ( x , m ) Q_t^{*}(x,m) Q t ∗ ( x , m ) π ∗ \pi^{*} π ∗
E [ C ( y t , x t ∗ , D ~ t ) + V ^ t − 1 ( y t − 1 , D ~ t ) ∣ F t + 1 ] ≤ V ^ t ( y t , μ t ) . \mathbb{E}\left[C(y_t,x_t^{*},\widetilde{D}_t)+\widehat{V}^{t-1}(y_{t-1},\widetilde{D}_t)\mid\mathcal{F}_{t+1}\right]\leq\widehat{V}^{t}(y_t,\mu_t). E [ C ( y t , x t ∗ , D t ) + V t − 1 ( y t − 1 , D t ) ∣ F t + 1 ] ≤ V t ( y t , μ t ) . Iterated conditioning yields sup P E P ∑ t C t π ∗ ≤ V ^ T ( y , μ ) \sup_{P}\mathbb{E}_{P}\sum_t C_t^{\pi^{*}}\leq\widehat{V}^{T}(y,\mu) sup P E P ∑ t C t π ∗ ≤ V T ( y , μ ) π \pi π y y y Q t ∗ ( x t π , μ t ) Q_t^{*}(x_t^{\pi},\mu_t) Q t ∗ ( x t π , μ t ) M A R ( μ ) \mathcal{MAR}(\mu) M A R ( μ ) sup P E P ∑ t C t π ≥ V ^ T ( y , μ ) \sup_{P}\mathbb{E}_{P}\sum_t C_t^{\pi}\geq\widehat{V}^{T}(y,\mu) sup P E P ∑ t C t π ≥ V T ( y , μ ) π ∗ \pi^{*} π ∗ □ \square □
Proof of Lemma 1. Jensen’s inequality bounds the expectation by the least concave majorant at μ \mu μ [42 ] ). The mean-matching Dirac or two-point law attains the bound. □ \square □
Lemma EC.1 (Inventory convexity). If b + h ≥ c b+h\geq c b + h ≥ c
V t ( y ′ , μ ) − V t ( y , μ ) ≥ − c ( y ′ − y ) , y ′ ≥ y . (15) V^{t}(y',\mu)-V^{t}(y,\mu)\geq-c(y'-y),\qquad y'\geq y. \tag*{(15)} V t ( y ′ , μ ) − V t ( y , μ ) ≥ − c ( y ′ − y ) , y ′ ≥ y . (15) Proof. Write V V V V t ( y , μ ) + c y = min x ∈ [ y , U ] B t ( x , μ ) V^{t}(y,\mu)+cy=\min_{x\in[y,U]}B_t(x,\mu) V t ( y , μ ) + cy = min x ∈ [ y , U ] B t ( x , μ ) V t − 1 ( ⋅ , D ) V^{t-1}(\mathord{\cdot},D) V t − 1 ( ⋅ , D ) x x x x = D x=D x = D r = α ( U − D ) r=\alpha(U-D) r = α ( U − D ) v = ∂ y + V t − 1 ( r , D ) ≥ − c v=\partial_y^{+}V^{t-1}(r,D)\geq-c v = ∂ y + V t − 1 ( r , D ) ≥ − c b + h + v ≥ b + h − c ≥ 0 b+h+v\geq b+h-c\geq0 b + h + v ≥ b + h − c ≥ 0 B t ( ⋅ , μ ) B_t(\mathord{\cdot},\mu) B t ( ⋅ , μ ) y = λ y 1 + ( 1 − λ ) y 2 y=\lambda y_1+(1-\lambda)y_2 y = λ y 1 + ( 1 − λ ) y 2 x i ≥ y i x_i\ge y_i x i ≥ y i min x ≥ y B t ( x , μ ) \min_{x\ge y}B_t(x,\mu) min x ≥ y B t ( x , μ ) y y y c y cy cy □ \square □
Proof of Proposition 2. Convexity of B t ( ⋅ , μ ) B_t(\cdot,\mu) B t ( ⋅ , μ ) χ t ( μ ) \chi_t(\mu) χ t ( μ ) [ y , U ] [y,U] [ y , U ] max { y , χ t ( μ ) } \max\{y,\chi_t(\mu)\} max { y , χ t ( μ )} □ \square □
The Terminal and Two-Period Problems
Proof of Proposition 3. For fixed x x x ( 1 − μ / U ) δ 0 + ( μ / U ) δ U (1-\mu/U)\delta_0+(\mu/U)\delta_U ( 1 − μ / U ) δ 0 + ( μ / U ) δ U − c y + w α U + b μ + x ( Φ 1 − K 1 μ / U ) -cy+w\alpha U+b\mu+x(\Phi_1-K_1\mu/U) − cy + w α U + b μ + x ( Φ 1 − K 1 μ / U ) μ = ρ 1 U \mu=\rho_1U μ = ρ 1 U [ y , U ] [y,U] [ y , U ] □ \square □
Proof of Lemma 2. Write q = q ˉ 2 q=\bar q_2 q = q ˉ 2 s = x ˉ 2 s=\bar x_2 s = x ˉ 2 F 2 x F_2^x F 2 x x , q x,q x , q x < q x<q x < q
ℓ 0 , U x ( q ) − F 2 x ( q ) = K 1 ( b + 2 h − α w ) U Φ 1 x ( q − x ) ≥ 0 ; \ell_{0,U}^{x}(q)-F_2^x(q)=\frac{K_1(b+2h-\alpha w)}{U\Phi_1}x(q-x)\ge0; ℓ 0 , U x ( q ) − F 2 x ( q ) = U Φ 1 K 1 ( b + 2 h − α w ) x ( q − x ) ≥ 0 ; F 2 x F_2^x F 2 x [ q , U ] [q,U] [ q , U ] K 2 ( s − x ) / q K_2(s-x)/q K 2 ( s − x ) / q x < s x<s x < s s ≤ x < q s\le x<q s ≤ x < q K 2 > β K 1 K_2>\beta K_1 K 2 > β K 1
ℓ 0 , U x ( q ) − F 2 x ( q ) = K 2 ( 1 − q / U ) ( x − s ) ≥ 0. \ell_{0,U}^{x}(q)-F_2^x(q)=K_2(1-q/U)(x-s)\ge0. ℓ 0 , U x ( q ) − F 2 x ( q ) = K 2 ( 1 − q / U ) ( x − s ) ≥ 0. When K 2 ≤ β K 1 K_2\le\beta K_1 K 2 ≤ β K 1 s ≥ q s\ge q s ≥ q x ≥ q x\ge q x ≥ q q q q D = x D=x D = x b + h − c > 0 b+h-c>0 b + h − c > 0
ℓ 0 , U x ( q ) − F 2 x ( q ) = ( K 2 − β K 1 ) q U ( U − x ) . \ell_{0,U}^{x}(q)-F_2^x(q)=(K_2-\beta K_1)\frac{q}{U}(U-x). ℓ 0 , U x ( q ) − F 2 x ( q ) = ( K 2 − β K 1 ) U q ( U − x ) . The sign determines the corresponding least concave majorant: all chosen chords dominate and touch F 2 x F_2^x F 2 x □ \square □
Proof of Corollary 2. At μ = U \mu=U μ = U U U U 2 c U − c y + [ b − c − α ( c − w ) ] ( U − x ) 2cU-cy+[b-c-\alpha(c-w)](U-x) 2 c U − cy + [ b − c − α ( c − w )] ( U − x ) x = U x=U x = U α < α b \alpha<\alpha_b α < α b α ≥ α b \alpha\ge\alpha_b α ≥ α b η = Φ 2 − K 2 = ( c − w ) ( α − α b ) ≥ 0 \eta=\Phi_2-K_2=(c-w)(\alpha-\alpha_b)\ge0 η = Φ 2 − K 2 = ( c − w ) ( α − α b ) ≥ 0 η \eta η x = y x=y x = y α > α b \alpha>\alpha_b α > α b □ \square □
Proof of Theorem 1. Set q = q ˉ 2 q=\bar q_2 q = q ˉ 2 s = x ˉ 2 s=\bar x_2 s = x ˉ 2 η = Φ 2 − K 2 < 0 \eta=\Phi_2-K_2<0 η = Φ 2 − K 2 < 0 K 2 > β K 1 K_2>\beta K_1 K 2 > β K 1 (1) gives slopes Φ 2 − K 2 μ / q \Phi_2-K_2\mu/q Φ 2 − K 2 μ / q x < s , μ ≤ q x<s,\mu\le q x < s , μ ≤ q η \eta η x < s , μ > q x<s,\mu>q x < s , μ > q Φ 2 − K 2 μ / U \Phi_2-K_2\mu/U Φ 2 − K 2 μ / U x > s x>s x > s μ = ρ 2 q , ρ 2 U \mu=\rho_2q,\rho_2U μ = ρ 2 q , ρ 2 U { 0 , q } \{0,q\} { 0 , q } x ∗ < s x^*<s x ∗ < s K 2 ≤ β K 1 K_2\le\beta K_1 K 2 ≤ β K 1 ( x < q , μ ≤ q ) (x<q,\mu\le q) ( x < q , μ ≤ q ) ( x < q , μ > q ) (x<q,\mu>q) ( x < q , μ > q ) ( x > q , μ ≤ q ) (x>q,\mu\le q) ( x > q , μ ≤ q ) ( x > q , μ > q ) (x>q,\mu>q) ( x > q , μ > q )
Φ 2 − K 2 μ / q , η , Φ 2 − β Φ 1 μ / q , η + ( b + h − c ) ( U − μ ) / ( U − q ) . \Phi_2-K_2\mu/q,\qquad\eta,\qquad\Phi_2-\beta\Phi_1\mu/q,\qquad\eta+(b+h-c)(U-\mu)/(U-q). Φ 2 − K 2 μ / q , η , Φ 2 − β Φ 1 μ / q , η + ( b + h − c ) ( U − μ ) / ( U − q ) . The third is positive at μ = q \mu=q μ = q μ c = q + [ h + α ( c − w ) ] ( U − q ) / ( b + h − c ) \mu_c=q+[h+\alpha(c-w)](U-q)/(b+h-c) μ c = q + [ h + α ( c − w )] ( U − q ) / ( b + h − c ) 0 0 0 q q q ρ 2 q \rho_2q ρ 2 q q q q U U U μ c \mu_c μ c x ≥ y x\ge y x ≥ y V ~ 2 = − c y + B 2 ( x ∗ , μ ) \widetilde V^2=-cy+B_2(x^*,\mu) V 2 = − cy + B 2 ( x ∗ , μ ) □ \square □
Finite-Horizon Characterization
Throughout this section, Assumption 1, the regular penalty condition (4) , and α < α b \alpha< \alpha_b α < α b
Lemma EC.2 (Automatic grid ordering). The grids in (5) –(6) satisfy (7) at every finite horizon.
Proof. Since K t − Φ t = b − c − α ( c − w ) > 0 K_t-\Phi_t=b-c-\alpha(c-w)>0 K t − Φ t = b − c − α ( c − w ) > 0 t ≥ 2 t\ge2 t ≥ 2 0 < ρ t < 1 0<\rho_t<1 0 < ρ t < 1 K 2 > β K 1 K_2>\beta K_1 K 2 > β K 1 0 < s 2 , 1 < q 2 , 1 < U 0<s_{2,1}<q_{2,1}<U 0 < s 2 , 1 < q 2 , 1 < U s t , t − 1 = β Φ t − 1 U / K t < U s_{t,t-1}=\beta\Phi_{t-1}U/K_t<U s t , t − 1 = β Φ t − 1 U / K t < U K t − β Φ t − 1 = b + h − c > 0 K_t-\beta\Phi_{t-1}=b+h-c>0 K t − β Φ t − 1 = b + h − c > 0 t ≥ 3 t\ge3 t ≥ 3 K t > β K t − 1 K_t>\beta K_{t-1} K t > β K t − 1
s t , j = Φ t − 1 K t ( s t − 1 , j − α U ) < β Φ t − 1 K t q t − 1 , j < Φ t − 1 K t − 1 q t − 1 , j = q t , j , s_{t,j}=\frac{\Phi_{t-1}}{K_t}(s_{t-1,j}-\alpha U)<\frac{\beta\Phi_{t-1}}{K_t}q_{t-1,j}<\frac{\Phi_{t-1}}{K_{t-1}}q_{t-1,j}=q_{t,j}, s t , j = K t Φ t − 1 ( s t − 1 , j − α U ) < K t β Φ t − 1 q t − 1 , j < K t − 1 Φ t − 1 q t − 1 , j = q t , j , by induction and q t − 1 , j ≤ U q_{t-1,j}\le U q t − 1 , j ≤ U s t , t − 1 / q t , t − 1 = β K t − 1 / K t < 1 s_{t,t-1}/q_{t,t-1}=\beta K_{t-1}/K_t<1 s t , t − 1 / q t , t − 1 = β K t − 1 / K t < 1 □ \square □
Lemma EC.3 (Secant recursion). Assume the ( t − 1 ) (t-1) ( t − 1 ) Γ t , j ( x ) = [ F t x ( q t , j ) − F t x ( 0 ) ] / q t , j \Gamma_{t,j}(x)=[F_t^x(q_{t,j})-F_t^x(0)]/q_{t,j} Γ t , j ( x ) = [ F t x ( q t , j ) − F t x ( 0 )] / q t , j a t ≤ j < t a_t\le j<t a t ≤ j < t 0 ≤ x ≤ q t , j 0\le x\le q_{t,j} 0 ≤ x ≤ q t , j
Γ t , j + 1 ( x ) − Γ t , j ( x ) = K t ( 1 q t , j − 1 q t , j + 1 ) ( x − s t , j ) . (16) \Gamma_{t,j+1}(x)-\Gamma_{t,j}(x)=K_t\left(\frac{1}{q_{t,j}}-\frac{1}{q_{t,j+1}}\right)(x-s_{t,j}). \tag*{(16)} Γ t , j + 1 ( x ) − Γ t , j ( x ) = K t ( q t , j 1 − q t , j + 1 1 ) ( x − s t , j ) . (16) Moreover, F t x ( 0 ) F_t^x(0) F t x ( 0 ) [ 0 , U ] [0,U] [ 0 , U ] F t x ( q t , j ) F_t^x(q_{t,j}) F t x ( q t , j ) [ 0 , q t , j ] [0,q_{t,j}] [ 0 , q t , j ] j j j
Proof. Write P = Φ t − 1 P=\Phi_{t-1} P = Φ t − 1 M = K t − 1 M=K_{t-1} M = K t − 1 A = ( c + ( t − 1 ) h − P ) U A=(c+(t-1)h-P)U A = ( c + ( t − 1 ) h − P ) U
F t x ( 0 ) = A + ( P − c ) α U + { h + β ( P − c ) } x . (17) F_t^x(0)=A+(P-c)\alpha U+\{h+\beta(P-c)\}x. \tag*{(17)} F t x ( 0 ) = A + ( P − c ) α U + { h + β ( P − c )} x . (17) For active j < t j<t j < t s t − 1 , j > α U s_{t-1,j}>\alpha U s t − 1 , j > α U q t , j = ρ t − 1 q t − 1 , j q_{t,j}=\rho_{t-1}q_{t-1,j} q t , j = ρ t − 1 q t − 1 , j s t − 1 , j s_{t-1,j} s t − 1 , j x ≤ q t , j x\le q_{t,j} x ≤ q t , j α ( U − x ) \alpha(U-x) α ( U − x )
ζ j = ( t − 2 ) c − ( t − 1 ) h + M { s t − 1 , j q t − 1 , j + ∑ k = j + 1 t − 1 s t − 1 , k − s t − 1 , k − 1 q t − 1 , k } , j < t , \zeta_j=(t-2)c-(t-1)h+M\left\{\frac{s_{t-1,j}}{q_{t-1,j}}+\sum_{k=j+1}^{t-1}\frac{s_{t-1,k}-s_{t-1,k-1}}{q_{t-1,k}}\right\},\qquad j<t, ζ j = ( t − 2 ) c − ( t − 1 ) h + M ⎩ ⎨ ⎧ q t − 1 , j s t − 1 , j + k = j + 1 ∑ t − 1 q t − 1 , k s t − 1 , k − s t − 1 , k − 1 ⎭ ⎬ ⎫ , j < t , and ζ t = ( t − 2 ) c − ( t − 1 ) h + P \zeta_t=(t-2)c-(t-1)h+P ζ t = ( t − 2 ) c − ( t − 1 ) h + P j j j
F t x ( q t , j ) = b q t , j + A + q t , j ζ j − c α U + ( − b + α c ) x , x ≤ q t , j . (18) F_t^x(q_{t,j})=bq_{t,j}+A+q_{t,j}\zeta_j-c\alpha U+(-b+\alpha c)x,\qquad x\le q_{t,j}. \tag*{(18)} F t x ( q t , j ) = b q t , j + A + q t , j ζ j − c α U + ( − b + α c ) x , x ≤ q t , j . (18) Since K t = b + h − c + β P K_t=b+h-c+\beta P K t = b + h − c + β P Γ t , j ( x ) = b + ζ j − P α U + K t x q t , j \Gamma_{t,j}(x)=b+\zeta_j-\frac{P\alpha U+K_tx}{q_{t,j}} Γ t , j ( x ) = b + ζ j − q t , j P α U + K t x
ζ j + 1 − ζ j = − P s t − 1 , j ( 1 q t , j − 1 q t , j + 1 ) . \zeta_{j+1}-\zeta_j=-Ps_{t-1,j}\left(\frac{1}{q_{t,j}}-\frac{1}{q_{t,j+1}}\right). ζ j + 1 − ζ j = − P s t − 1 , j ( q t , j 1 − q t , j + 1 1 ) . The inventory recursion now yields (EC.2). The last pair follows by substituting ζ t \zeta_t ζ t s t , t − 1 = β P U / K t s_{t,t-1}=\beta PU/K_t s t , t − 1 = β P U / K t □ \square □
Lemma EC.4 (Global adjacent-contact dominance). Suppose Theorem 2 holds at horizon t − 1 t-1 t − 1 j = 1 , … , t j=1,\ldots,t j = 1 , … , t
Γ t , j ( x ) : = F t x ( q t , j ) − F t x ( 0 ) q t , j . \Gamma_{t,j}(x):=\frac{F_t^x(q_{t,j})-F_t^x(0)}{q_{t,j}}. Γ t , j ( x ) := q t , j F t x ( q t , j ) − F t x ( 0 ) . Then, for every x ∈ [ 0 , U ] x \in[0,U] x ∈ [ 0 , U ]
Γ t , j + 1 ( x ) ≥ Γ t , j ( x ) whenever x ≥ max { 0 , s t , j } . (19) \Gamma_{t,j+1}(x) \ge\Gamma_{t,j}(x) \qquad\text{whenever } x \ge\max\{0,s_{t,j}\}. \tag*{(19)} Γ t , j + 1 ( x ) ≥ Γ t , j ( x ) whenever x ≥ max { 0 , s t , j } . (19) In particular, all demand contacts discarded by J t ( x ) J_t(x) J t ( x ) 0 0 0 q t , J t ( x ) q_{t,J_t(x)} q t , J t ( x )
Proof. Put n = t − 1 n=t-1 n = t − 1 P = Φ n P=\Phi_n P = Φ n p j = q n , j p_j=q_{n,j} p j = q n , j z j = ( s n , j ) + z_j=(s_{n,j})_+ z j = ( s n , j ) + z 0 = 0 , z n = U z_0=0,z_n=U z 0 = 0 , z n = U m j = ρ n p j = q t , j m_j=\rho_n p_j=q_{t,j} m j = ρ n p j = q t , j j ≤ n j\le n j ≤ n m n + 1 = U m_{n+1}=U m n + 1 = U ϑ j = m j / m j + 1 \vartheta_j=m_j/m_{j+1} ϑ j = m j / m j + 1
V ^ n ( y , m j ) = C j − c y + P H j ( y ) , V ^ n ( y , 0 ) = C 0 + ( P − c ) y , (20) \widehat{V}^{n}(y,m_j)=C_j-cy+PH_j(y), \qquad\widehat{V}^{n}(y,0)=C_0+(P-c)y, \tag*{(20)} V n ( y , m j ) = C j − cy + P H j ( y ) , V n ( y , 0 ) = C 0 + ( P − c ) y , (20) where C j = V ^ n ( 0 , m j ) C_j=\widehat{V}^{n}(0,m_j) C j = V n ( 0 , m j ) C 0 = V ^ n ( 0 , 0 ) C_0=\widehat{V}^{n}(0,0) C 0 = V n ( 0 , 0 )
H j ( y ) = ∑ k = j + 1 n ( 1 − p j p k ) { ( y − z k − 1 ) + − ( y − z k ) + } , H n + 1 = 0. H_j(y)=\sum_{k=j+1}^{n}\left(1-\frac{p_j}{p_k}\right)\{(y-z_{k-1})_+-(y-z_k)_+\}, \qquad H_{n+1}=0. H j ( y ) = k = j + 1 ∑ n ( 1 − p k p j ) {( y − z k − 1 ) + − ( y − z k ) + } , H n + 1 = 0. Each H j H_j H j y ≤ z j y\le z_j y ≤ z j
H j ( y ) = ( 1 − ϑ j ) ( y − z j ) + + ϑ j H j + 1 ( y ) , (21) H_j(y)=(1-\vartheta_j)(y-z_j)_+ + \vartheta_j H_{j+1}(y), \tag*{(21)} H j ( y ) = ( 1 − ϑ j ) ( y − z j ) + + ϑ j H j + 1 ( y ) , (21) C j = ( 1 − ϑ j ) C 0 + ϑ j C j + 1 + P ( 1 − ϑ j ) z j . (22) C_j=(1-\vartheta_j)C_0+\vartheta_j C_{j+1}+P(1-\vartheta_j)z_j. \tag*{(22)} C j = ( 1 − ϑ j ) C 0 + ϑ j C j + 1 + P ( 1 − ϑ j ) z j . (22) We also have
z j ≤ α U + β m j + 1 ( j ≤ n ) . (23) z_j\le\alpha U+\beta m_{j+1}\qquad(j\le n). \tag*{(23)} z j ≤ α U + β m j + 1 ( j ≤ n ) . (23) Equality holds for j = n j=n j = n j < n j<n j < n z j ≤ β λ n q n , j + 1 z_j\le\beta\lambda_n q_{n,j+1} z j ≤ β λ n q n , j + 1
β ( λ n − ρ n ) + = β n ( α w − h ) + K n ≤ α . \beta(\lambda_n-\rho_n)_+=\frac{\beta^n(\alpha w-h)_+}{K_n}\le\alpha. β ( λ n − ρ n ) + = K n β n ( α w − h ) + ≤ α . If α w > h \alpha w>h α w > h K n = ( K n − Φ n ) + c − w + h / α + β n ( w − h / α ) > β n ( w − h / α ) K_n=(K_n-\Phi_n)+c-w+h/\alpha+\beta^n(w-h/\alpha)>\beta^n(w-h/\alpha) K n = ( K n − Φ n ) + c − w + h / α + β n ( w − h / α ) > β n ( w − h / α ) q n , j + 1 ≤ U q_{n,j+1}\le U q n , j + 1 ≤ U
Fix j ≤ n j\le n j ≤ n a = m j a=m_j a = m j d = m j + 1 d=m_{j+1} d = m j + 1 ϑ = a / d \vartheta=a/d ϑ = a / d z = z j z=z_j z = z j s = α ( U − x ) s=\alpha(U-x) s = α ( U − x )
E j ( x ) = ( 1 − ϑ ) F t x ( 0 ) + ϑ F t x ( d ) − F t x ( a ) = a { Γ t , j + 1 ( x ) − Γ t , j ( x ) } . E_j(x)=(1-\vartheta)F_t^x(0)+\vartheta F_t^x(d)-F_t^x(a)=a\{\Gamma_{t,j+1}(x)-\Gamma_{t,j}(x)\}. E j ( x ) = ( 1 − ϑ ) F t x ( 0 ) + ϑ F t x ( d ) − F t x ( a ) = a { Γ t , j + 1 ( x ) − Γ t , j ( x )} . Using (EC.6)–(EC.8), for x ≤ a x\le a x ≤ a
E j ( x ) = ( 1 − ϑ ) [ ( b + h − c ) x + P { s + x − z − ( s − z ) + } ] . E_j(x)=(1-\vartheta)\left[(b+h-c)x+P\{s+x-z-(s-z)_+\}\right]. E j ( x ) = ( 1 − ϑ ) [ ( b + h − c ) x + P { s + x − z − ( s − z ) + } ] . If s ≥ z s\ge z s ≥ z ( 1 − ϑ ) K t ( x − r j ) (1-\vartheta)K_t(x-r_j) ( 1 − ϑ ) K t ( x − r j ) r j = P ( z − α U ) / K t = s t , j < a r_j=P(z-\alpha U)/K_t=s_{t,j}<a r j = P ( z − α U ) / K t = s t , j < a r j > 0 r_j>0 r j > 0 a ≤ x ≤ d a\le x\le d a ≤ x ≤ d y = s + x − a y=s+x-a y = s + x − a δ = ϑ ( d − x ) \delta=\vartheta(d-x) δ = ϑ ( d − x )
E j ( x ) = ( b + h − c ) δ + P Q , Q = ( 1 − ϑ ) { s + x − z − ( y − z ) + } + ϑ { H j + 1 ( s ) − H j + 1 ( y ) } . E_j(x)=(b+h-c)\delta+PQ,\qquad Q=(1-\vartheta)\{s+x-z-(y-z)_+\}+\vartheta\{H_{j+1}(s)-H_{j+1}(y)\}. E j ( x ) = ( b + h − c ) δ + P Q , Q = ( 1 − ϑ ) { s + x − z − ( y − z ) + } + ϑ { H j + 1 ( s ) − H j + 1 ( y )} . For y ≥ z y\ge z y ≥ z Q ≥ δ Q\ge\delta Q ≥ δ y < z y<z y < z E j E_j E j x = a x=a x = a y = z y=z y = z x = d x=d x = d E j ( d ) = P ( 1 − ϑ ) ( β d + α U − z ) ≥ 0 E_j(d)=P(1-\vartheta)(\beta d+\alpha U-z)\ge0 E j ( d ) = P ( 1 − ϑ ) ( β d + α U − z ) ≥ 0 x ≥ d x\ge d x ≥ d y a = s + x − a y_a=s+x-a y a = s + x − a y d = s + x − d y_d=s+x-d y d = s + x − d
E j ( x ) = P [ ( 1 − ϑ ) { s + x − z − ( y a − z ) + } + ϑ { H j + 1 ( y d ) − H j + 1 ( y a ) } ] . E_j(x)=P\left[(1-\vartheta)\{s+x-z-(y_a-z)_+\}+\vartheta\{H_{j+1}(y_d)-H_{j+1}(y_a)\}\right]. E j ( x ) = P [ ( 1 − ϑ ) { s + x − z − ( y a − z ) + } + ϑ { H j + 1 ( y d ) − H j + 1 ( y a )} ] . This is nonnegative by the 1-Lipschitz bound when y a ≥ z y_a\ge z y a ≥ z P ( 1 − ϑ ) ( β x + α U − z ) ≥ 0 P(1-\vartheta)(\beta x+\alpha U-z)\ge0 P ( 1 − ϑ ) ( β x + α U − z ) ≥ 0
Thus E j ( x ) ≥ 0 E_j(x)\ge0 E j ( x ) ≥ 0 x ≥ max { 0 , s t , j } x\ge\max\{0,s_{t,j}\} x ≥ max { 0 , s t , j }
Lemma EC.5 (Automatic finite-horizon worst-case expectation). Under the ( t − 1 ) (t-1) ( t − 1 ) F t x F_t^x F t x 0 , q t , J t ( x ) , … , q t , t = U 0,q_{t,J_t(x)},\ldots,q_{t,t}=U 0 , q t , J t ( x ) , … , q t , t = U x ∈ [ 0 , U ] x\in[0,U] x ∈ [ 0 , U ] μ \mu μ
Proof. Fix x x x J = J t ( x ) J=J_t(x) J = J t ( x ) q t , J q_{t,J} q t , J J < t J<t J < t x < s t , J < q t , J x<s_{t,J}<q_{t,J} x < s t , J < q t , J m t , j m_{t,j} m t , j
m t , J − Γ t , J = K t q t , J ( x − s t , J ) ≤ 0 , m t , j + 1 − m t , j = − K t q t , j + 1 ( s t , j + 1 − s t , j ) < 0. m_{t,J}-\Gamma_{t,J}=\frac{K_t}{q_{t,J}}(x-s_{t,J})\le0,\qquad m_{t,j+1}-m_{t,j}=-\frac{K_t}{q_{t,j+1}}(s_{t,j+1}-s_{t,j})<0. m t , J − Γ t , J = q t , J K t ( x − s t , J ) ≤ 0 , m t , j + 1 − m t , j = − q t , j + 1 K t ( s t , j + 1 − s t , j ) < 0. The polygon is concave; for J = t J=t J = t D = x D=x D = x R ( x , D ) R(x,D) R ( x , D ) F t x F_t^x F t x 0 0 0 2 K t − 1 / q t − 1 , j ≥ 0 2K_{t-1}/q_{t-1,j}\ge0 2 K t − 1 / q t − 1 , j ≥ 0 D = x D=x D = x b + h − c > 0 b+h-c>0 b + h − c > 0 F t x F_t^x F t x
Proof of Theorem 2. We use induction on t t t t − 1 t-1 t − 1 F t x F_t^x F t x x ∈ [ 0 , U ] x\in[0,U] x ∈ [ 0 , U ]
For x < U x<U x < U J = J t ( x ) J=J_t(x) J = J t ( x )
∂ B t ( x , μ ) ∂ x = Φ t − K t min { μ / q t , J , 1 } . \frac{\partial B_t(x,\mu)}{\partial x}=\Phi_t-K_t\min\{\mu/q_{t,J},1\}. ∂ x ∂ B t ( x , μ ) = Φ t − K t min { μ / q t , J , 1 } . For the origin chord this follows from (EC.3)–(EC.4); for a chord between positive contacts the slope is Φ t − K t = c + α ( c − w ) − b < 0 \Phi_t-K_t=c+\alpha(c-w)-b<0 Φ t − K t = c + α ( c − w ) − b < 0 q t , J q_{t,J} q t , J x x x μ = ρ t q t , J \mu=\rho_tq_{t,J} μ = ρ t q t , J χ t ( μ ) \chi_t(\mu) χ t ( μ ) (9) , and imposing x ≥ y x\ge y x ≥ y x t ∗ ( y , μ ) = max { y , χ t ( μ ) } x_t^*(y,\mu)=\max\{y,\chi_t(\mu)\} x t ∗ ( y , μ ) = max { y , χ t ( μ )}
At capacity, the envelope is the endpoint chord and B t ( U , μ ) = ( c + t h ) U + ( ( t − 1 ) c − t h ) μ B_t(U,\mu)=(c+th)U+((t-1)c-th)\mu B t ( U , μ ) = ( c + t h ) U + (( t − 1 ) c − t h ) μ x t ∗ < U x_t^*<U x t ∗ < U μ < ρ t q t , J t ( x t ∗ ) \mu<\rho_tq_{t,J_t(x_t^*)} μ < ρ t q t , J t ( x t ∗ ) x t ∗ x_t^* x t ∗ U U U (8) on [ χ t ( μ ) , U ] [\chi_t(\mu),U] [ χ t ( μ ) , U ]
For x t ∗ < U x_t^*<U x t ∗ < U μ < ρ t q t , J t ( x t ∗ ) < q t , J t ( x t ∗ ) \mu<\rho_tq_{t,J_t(x_t^*)}<q_{t,J_t(x_t^*)} μ < ρ t q t , J t ( x t ∗ ) < q t , J t ( x t ∗ ) x t ∗ < q t , J t ( x t ∗ ) x_t^*<q_{t,J_t(x_t^*)} x t ∗ < q t , J t ( x t ∗ ) x t ∗ = U x_t^*=U x t ∗ = U q t ∗ = U ≥ μ q_t^*=U\ge\mu q t ∗ = U ≥ μ q t ∗ ≥ max { x t ∗ , μ } q_t^*\ge\max\{x_t^*,\mu\} q t ∗ ≥ max { x t ∗ , μ }
Every positive realization under that law exhausts ready inventory, leaving R ( x t ∗ , q t ∗ ) = α ( U − x t ∗ ) R(x_t^*,q_t^*)=\alpha(U-x_t^*) R ( x t ∗ , q t ∗ ) = α ( U − x t ∗ ) M ( 0 ) = { δ 0 } \mathcal{M}(0)=\{\delta_0\} M ( 0 ) = { δ 0 } T + 1 T+1 T + 1
Proof of Proposition 4. Set λ t = Φ t − 1 / K t = Φ t − 1 / ( b + h − c + β Φ t − 1 ) \lambda_t=\Phi_{t-1}/K_t=\Phi_{t-1}/(b+h-c+\beta\Phi_{t-1}) λ t = Φ t − 1 / K t = Φ t − 1 / ( b + h − c + β Φ t − 1 ) Φ t − 1 \Phi_{t-1} Φ t − 1 λ ∞ ∈ ( 0 , 1 ) \lambda_\infty\in(0,1) λ ∞ ∈ ( 0 , 1 ) λ t \lambda_t λ t α < α h \alpha<\alpha_h α < α h α > α h \alpha>\alpha_h α > α h ℓ \ell ℓ r t , ℓ = s t , t − ℓ / U r_{t,\ell}=s_{t,t-\ell}/U r t , ℓ = s t , t − ℓ / U r t , 1 = β λ t r_{t,1}=\beta\lambda_t r t , 1 = β λ t r t , ℓ = λ t ( r t − 1 , ℓ − 1 − α ) r_{t,\ell}=\lambda_t(r_{t-1,\ell-1}-\alpha) r t , ℓ = λ t ( r t − 1 , ℓ − 1 − α )
r t , ℓ > 0 ⟺ ∑ k = 1 ℓ − 1 ( ∏ m = 1 k λ t − ℓ + m ) < β α . (24) r_{t,\ell}>0\quad\Longleftrightarrow\quad\sum_{k=1}^{\ell-1}\left(\prod_{m=1}^{k}\lambda_{t-\ell+m}\right)<\frac{\beta}{\alpha}. \tag*{(24)} r t , ℓ > 0 ⟺ k = 1 ∑ ℓ − 1 ( m = 1 ∏ k λ t − ℓ + m ) < α β . (24) Positive ages form an initial block. Their limit test is λ ∞ ℓ − 1 > α / ( 1 − β λ ∞ ) \lambda_\infty^{\ell-1}>\alpha/(1-\beta\lambda_\infty) λ ∞ ℓ − 1 > α / ( 1 − β λ ∞ ) ℓ − 1 < ζ \ell-1<\zeta ℓ − 1 < ζ α ≤ α h \alpha\le\alpha_h α ≤ α h N ∞ = ⌈ ζ ⌉ N_\infty=\lceil\zeta\rceil N ∞ = ⌈ ζ ⌉ α > α h \alpha>\alpha_h α > α h ζ \zeta ζ N ∞ = 1 + ⌊ ζ ⌋ N_\infty=1+\lfloor\zeta\rfloor N ∞ = 1 + ⌊ ζ ⌋ N ∞ + 1 N_\infty+1 N ∞ + 1
Recovery-Dominant Regimes III and IV
Proof of Theorem 3. Put
η = c + α ( c − w ) − b = ( c − w ) ( α − α b ) ≥ 0 , ν t = c − η 1 − ( − α ) t − 1 1 + α , M t = Φ t − c + ν t . \eta=c+\alpha(c-w)-b=(c-w)(\alpha-\alpha_b)\ge0,\qquad\nu_t=c-\eta\frac{1-(-\alpha)^{t-1}}{1+\alpha},\qquad M_t=\Phi_t-c+\nu_t. η = c + α ( c − w ) − b = ( c − w ) ( α − α b ) ≥ 0 , ν t = c − η 1 + α 1 − ( − α ) t − 1 , M t = Φ t − c + ν t . Then ν 1 = c \nu_1=c ν 1 = c ν t = b + α w − α ν t − 1 \nu_t=b+\alpha w-\alpha\nu_{t-1} ν t = b + α w − α ν t − 1 0 < ν t ≤ c 0<\nu_t\le c 0 < ν t ≤ c M t > 0 M_t>0 M t > 0 h ≥ α w h\ge\alpha w h ≥ α w M t ≥ ν t M_t\ge\nu_t M t ≥ ν t Φ t − c > ( h − α w ) / α \Phi_t-c>(h-\alpha w)/\alpha Φ t − c > ( h − α w ) / α ν t ≥ c − η > w \nu_t\ge c-\eta>w ν t ≥ c − η > w q = ρ 1 U q=\rho_1U q = ρ 1 U
v t , 0 ( y ) = t h U − ( Φ t − c ) ( U − y ) , v t , U ( y ) = U ∑ r = 1 t ν r − ν t y , v_{t,0}(y)=thU-(\Phi_t-c)(U-y),\qquad v_{t,U}(y)=U\sum_{r=1}^{t}\nu_r-\nu_t y, v t , 0 ( y ) = t h U − ( Φ t − c ) ( U − y ) , v t , U ( y ) = U r = 1 ∑ t ν r − ν t y , s t ∘ = U M t ( Φ 1 − α ∑ r = 1 t − 1 M r ) + , τ ( μ ) = min { ( 1 − ρ 1 ) μ q , 1 − μ U } . s_t^\circ=\frac{U}{M_t}\left(\Phi_1-\alpha\sum_{r=1}^{t-1}M_r\right)_+,\qquad\tau(\mu)=\min\left\{\frac{(1-\rho_1)\mu}{q},1-\frac{\mu}{U}\right\}. s t ∘ = M t U ( Φ 1 − α r = 1 ∑ t − 1 M r ) + , τ ( μ ) = min { q ( 1 − ρ 1 ) μ , 1 − U μ } . Regularity gives s 2 ∘ = x ˉ 2 < q s_2^\circ=\bar{x}_2<q s 2 ∘ = x ˉ 2 < q s t ∘ = M t − 1 ( s t − 1 ∘ − α U ) + / M t s_t^\circ=M_{t-1}(s_{t-1}^\circ-\alpha U)_+/M_t s t ∘ = M t − 1 ( s t − 1 ∘ − α U ) + / M t 0 ≤ s t ∘ < q 0\le s_t^\circ<q 0 ≤ s t ∘ < q
V ^ t ( y , μ ) = ( 1 − μ U ) v t , 0 ( y ) + μ U v t , U ( y ) + M t ( s t ∘ − y ) + τ ( μ ) . (25) \widehat{V}^t(y,\mu)=\left(1-\frac{\mu}{U}\right)v_{t,0}(y)+\frac{\mu}{U}v_{t,U}(y)+M_t(s_t^\circ-y)_+\tau(\mu). \tag*{(25)} V t ( y , μ ) = ( 1 − U μ ) v t , 0 ( y ) + U μ v t , U ( y ) + M t ( s t ∘ − y ) + τ ( μ ) . (25) The base is Proposition 3. Inductively F t x F_t^x F t x [ 0 , q ] [0,q] [ 0 , q ] [ q , U ] [q,U] [ q , U ] x ≤ q x\le q x ≤ q q q q
M t − 1 ( s t − 1 ∘ − α U + α x ) + − ( M t + α M t − 1 ) x , M_{t-1}(s_{t-1}^\circ-\alpha U+\alpha x)_+-(M_t+\alpha M_{t-1})x, M t − 1 ( s t − 1 ∘ − α U + α x ) + − ( M t + α M t − 1 ) x , whose positive part is M t ( s t ∘ − x ) + M_t(s_t^\circ-x)_+ M t ( s t ∘ − x ) + x ≥ q x\ge q x ≥ q 0 , q , U 0,q,U 0 , q , U x < s t ∘ x<s_t^\circ x < s t ∘ Φ t − M t μ / q \Phi_t-M_t\mu/q Φ t − M t μ / q Φ t − M t \Phi_t-M_t Φ t − M t Φ t − M t μ / U \Phi_t-M_t\mu/U Φ t − M t μ / U c − ν t ≥ 0 c-\nu_t\ge0 c − ν t ≥ 0 x = y x=y x = y α > α b \alpha>\alpha_b α > α b
Substitution proves (EC.11), with adjacent-contact laws attaining it. Since α > 0 \alpha> 0 α > 0 M t M_t M t s t ∗ s_t^* s t ∗
Approximation Policies for the Complementary Case
Under Assumption 1, the complementary condition K 2 ≤ β K 1 K_2 \le\beta K_1 K 2 ≤ β K 1 h < w / 4 h < w/4 h < w /4 0 < b − c ≤ w − 2 h w 0 < b-c \le w - 2\sqrt{hw} 0 < b − c ≤ w − 2 h w q ˉ 2 \bar{q}_2 q ˉ 2 (3) .
In Regime II, define the cost-based threshold
θ : = h + α ( c − w ) b + h − c ∈ ( 0 , 1 ) , ρ : = min { θ , ρ 1 } . \theta:= \frac{h+\alpha(c-w)}{b+h-c} \in(0,1), \qquad\rho:= \min\{\theta,\rho_1\}. θ := b + h − c h + α ( c − w ) ∈ ( 0 , 1 ) , ρ := min { θ , ρ 1 } . Use the policy
x ~ t ( y , μ ) = { x t ∗ ( y , μ ) , t = 1 , 2 , y , t ≥ 3 , μ < θ U , U , t ≥ 3 , μ ≥ θ U . (26) \widetilde{x}_t(y,\mu) =
\begin{cases}
x_t^*(y,\mu), & t=1,2,\\
y, & t\ge3,\ \mu<\theta U,\\
U, & t\ge3,\ \mu\ge\theta U.
\end{cases} \tag*{(26)} x t ( y , μ ) = ⎩ ⎨ ⎧ x t ∗ ( y , μ ) , y , U , t = 1 , 2 , t ≥ 3 , μ < θ U , t ≥ 3 , μ ≥ θ U . (26) Relative to capacity continuation, action x x x ( U − x ) [ ( b + h − c ) μ / U − h − α ( c − w ) ] (U-x)[(b+h-c)\mu/U-h-\alpha(c-w)] ( U − x ) [( b + h − c ) μ / U − h − α ( c − w )] θ U \theta U θ U
In Regime IV, use no replenishment in every nonterminal period:
x ~ t ( y , μ ) = { x 1 ∗ ( y , μ ) , t = 1 , y , t ≥ 2. (27) \widetilde{x}_t(y,\mu) =
\begin{cases}
x_1^*(y,\mu), & t=1,\\
y, & t\ge2.
\end{cases} \tag*{(27)} x t ( y , μ ) = { x 1 ∗ ( y , μ ) , y , t = 1 , t ≥ 2. (27) Let V ~ t \widetilde{V}^t V t g : = β K 1 − K 2 ≥ 0 g:=\beta K_1-K_2\ge0 g := β K 1 − K 2 ≥ 0
Proposition EC.1 (Finite-horizon approximation guarantees). Under Assumption 1, let K 2 ≤ β K 1 K_2\le\beta K_1 K 2 ≤ β K 1 t = 1 , 2 t=1,2 t = 1 , 2 μ ∈ { 0 , U } \mu\in\{0,U\} μ ∈ { 0 , U } ( y , μ ) ∈ [ 0 , U ] 2 (y,\mu)\in[0,U]^2 ( y , μ ) ∈ [ 0 , U ] 2
In Regime II, define the dimensionless constant
K I I : = β Φ 2 ( 1 − ρ ) h ( 1 − ρ ) + c ρ . (28) \mathcal{K}_{\mathrm{II}}:=\frac{\beta\Phi_2(1-\rho)}{h(1-\rho)+c\rho}. \tag*{(28)} K II := h ( 1 − ρ ) + c ρ β Φ 2 ( 1 − ρ ) . (28) For every t ≥ 3 t\ge3 t ≥ 3
1 ≤ V ~ t ( y , μ ) V ^ t ( y , μ ) ≤ 1 + K I I t , 0 ≤ V ~ t ( y , μ ) − V ^ t ( y , μ ) ≤ β Φ 2 ( 1 − ρ ) U . (29) 1\le\frac{\widetilde{V}^t(y,\mu)}{\widehat{V}^t(y,\mu)}\le1+\frac{\mathcal{K}_{\mathrm{II}}}{t}, \qquad0\le\widetilde{V}^t(y,\mu)-\widehat{V}^t(y,\mu)\le\beta\Phi_2(1-\rho)U. \tag*{(29)} 1 ≤ V t ( y , μ ) V t ( y , μ ) ≤ 1 + t K II , 0 ≤ V t ( y , μ ) − V t ( y , μ ) ≤ β Φ 2 ( 1 − ρ ) U . (29) In Regime IV, for every t ≥ 1 t\ge1 t ≥ 1
0 ≤ V ~ t ( y , μ ) − V ^ t ( y , μ ) ≤ g min { ( 1 − ρ 1 ) μ , ρ 1 ( U − μ ) } ≤ g U 4 . (30) 0\le\widetilde{V}^t(y,\mu)-\widehat{V}^t(y,\mu)\le g\min\{(1-\rho_1)\mu,\rho_1(U-\mu)\}\le\frac{gU}{4}. \tag*{(30)} 0 ≤ V t ( y , μ ) − V t ( y , μ ) ≤ g min {( 1 − ρ 1 ) μ , ρ 1 ( U − μ )} ≤ 4 g U . (30) In particular, the policy is optimal at every horizon if g = 0 g=0 g = 0
Both policies avoid the demand grid. The Regime II relative error is O ( 1 / t ) O(1/t) O ( 1/ t ) t t t
For Regime IV’s relative bound, define
η : = c + α ( c − w ) − b ≥ 0 , ν r : = c − η 1 − ( − α ) r − 1 1 + α > 0. (31) \eta:=c+\alpha(c-w)-b\ge0,\qquad\nu_r:=c-\eta\frac{1-(-\alpha)^{r-1}}{1+\alpha}>0. \tag*{(31)} η := c + α ( c − w ) − b ≥ 0 , ν r := c − η 1 + α 1 − ( − α ) r − 1 > 0. (31) For t ≥ 2 t \ge2 t ≥ 2
1 ≤ V ~ t ( y , μ ) V ^ t ( y , μ ) ≤ 1 + g ρ 1 ( 1 − ρ 1 ) t h ( 1 − ρ 1 ) + ρ 1 ∑ r = 1 t − 1 ν r . (32) 1 \le\frac{\widetilde{V}^{t}(y,\mu)}{\widehat{V}^{t}(y,\mu)} \le1+\frac{g\rho_{1}(1-\rho_{1})}{th(1-\rho_{1})+\rho_{1}\sum_{r=1}^{t-1}\nu_{r}}. \tag*{(32)} 1 ≤ V t ( y , μ ) V t ( y , μ ) ≤ 1 + t h ( 1 − ρ 1 ) + ρ 1 ∑ r = 1 t − 1 ν r g ρ 1 ( 1 − ρ 1 ) . (32) The excess is O ( 1 / t ) O(1/t) O ( 1/ t )
Proof of Proposition EC.1(1). Put p = μ / U p=\mu/U p = μ / U a = h + α ( c − w ) a=h+\alpha(c-w) a = h + α ( c − w ) d = b − c − α ( c − w ) > 0 d=b-c-\alpha(c-w)>0 d = b − c − α ( c − w ) > 0 L = b + h − c = a + d L=b+h-c=a+d L = b + h − c = a + d
Λ ( μ ) = h ( U − μ ) + c μ , C t c a p ( y , μ ) = t Λ ( μ ) + c ( U − y ) − c μ . (33) \Lambda(\mu)=h(U-\mu)+c\mu,\qquad C_{t}^{\mathrm{cap}}(y,\mu)=t\Lambda(\mu)+c(U-y)-c\mu. \tag*{(33)} Λ ( μ ) = h ( U − μ ) + c μ , C t cap ( y , μ ) = t Λ ( μ ) + c ( U − y ) − c μ . (33) Capacity replenishment incurs this cost. The absorbing endpoint law { 0 , U } \{0,U\} { 0 , U } V ^ t ( y , 0 ) = t h U − ( Φ t − c ) ( U − y ) \widehat{V}^{t}(y,0)=thU-(\Phi_{t}-c)(U-y) V t ( y , 0 ) = t h U − ( Φ t − c ) ( U − y ) V ^ t ( y , U ) = t c U − c y \widehat{V}^{t}(y,U)=tcU-cy V t ( y , U ) = t c U − cy Φ t − K t p \Phi_{t}-K_{t}p Φ t − K t p
V ^ t ( y , μ ) ≥ C t c a p ( y , μ ) − ( U − y ) ( Φ t − K t p ) + . (34) \widehat{V}^{t}(y,\mu)\ge C_{t}^{\mathrm{cap}}(y,\mu)-(U-y)(\Phi_{t}-K_{t}p)_{+}. \tag*{(34)} V t ( y , μ ) ≥ C t cap ( y , μ ) − ( U − y ) ( Φ t − K t p ) + . (34) For W 0 ( y , μ ) = c ( U − y ) − c μ W_{0}(y,\mu)=c(U-y)-c\mu W 0 ( y , μ ) = c ( U − y ) − c μ
E [ C ( y , x , D ) + W 0 ( R ( x , D ) , D ) ] − Λ ( μ ) − W 0 ( y , μ ) = L E ( D − x ) + − a ( U − x ) . (35) \mathbb{E}[C(y,x,D)+W_{0}(R(x,D),D)]-\Lambda(\mu)-W_{0}(y,\mu)=L\mathbb{E}(D-x)_{+}-a(U-x). \tag*{(35)} E [ C ( y , x , D ) + W 0 ( R ( x , D ) , D )] − Λ ( μ ) − W 0 ( y , μ ) = L E ( D − x ) + − a ( U − x ) . (35) The endpoint law maximizes the right side at ( U − x ) ( L p − a ) (U-x)(Lp-a) ( U − x ) ( L p − a ) V ~ t ≤ C t c a p − ( U − y ) ( a − L p ) + \widetilde{V}^{t}\le C_{t}^{\mathrm{cap}}-(U-y)(a-Lp)_{+} V t ≤ C t cap − ( U − y ) ( a − L p ) +
λ ~ 1 = max { K 1 , Φ 1 / θ } , λ ~ t = L α + ( λ ~ 1 − L / α ) β t − 1 . (36) \widetilde{\lambda}_{1}=\max\{K_{1},\Phi_{1}/\theta\},\qquad\widetilde{\lambda}_{t}=\frac{L}{\alpha}+(\widetilde{\lambda}_{1}-L/\alpha)\beta^{t-1}. \tag*{(36)} λ 1 = max { K 1 , Φ 1 / θ } , λ t = α L + ( λ 1 − L / α ) β t − 1 . (36) Then λ ~ t = L + β λ ~ t − 1 \widetilde{\lambda}_{t}=L+\beta\widetilde{\lambda}_{t-1} λ t = L + β λ t − 1 Φ t = a + β Φ t − 1 \Phi_{t}=a+\beta\Phi_{t-1} Φ t = a + β Φ t − 1 r t = Φ t / λ ~ t ≤ θ r_{t}=\Phi_{t}/\widetilde{\lambda}_{t}\le\theta r t = Φ t / λ t ≤ θ
V ~ t ≤ C t c a p − ( U − y ) max { ( a − L p ) + , ( Φ t − λ ~ t p ) + } , t ≥ 2. (37) \widetilde{V}^{t}\le C_{t}^{\mathrm{cap}}-(U-y)\max\{(a-Lp)_{+},(\Phi_{t}-\widetilde{\lambda}_{t}p)_{+}\},\qquad t\ge2. \tag*{(37)} V t ≤ C t cap − ( U − y ) max {( a − L p ) + , ( Φ t − λ t p ) + } , t ≥ 2. (37) At t = 1 t=1 t = 1 λ ~ 1 ≥ K 1 \widetilde{\lambda}_{1}\ge K_{1} λ 1 ≥ K 1 t = 2 t=2 t = 2 t ≥ 3 t\ge3 t ≥ 3 p < θ p<\theta p < θ x = y x=y x = y sup Q ∈ M ( μ ) E Q q ( D ) \sup_{Q\in\mathcal{M}(\mu)}\mathbb{E}_{Q}q(D) sup Q ∈ M ( μ ) E Q q ( D )
q ( D ) = L ( D − y ) + − a ( U − y ) − ( U − R ( y , D ) ) ( Φ t − 1 − λ ~ t − 1 D / U ) + . q(D)=L(D-y)_{+}-a(U-y)-(U-R(y,D))(\Phi_{t-1}-\widetilde{\lambda}_{t-1}D/U)_{+}. q ( D ) = L ( D − y ) + − a ( U − y ) − ( U − R ( y , D )) ( Φ t − 1 − λ t − 1 D / U ) + . For r = r t − 1 r=r_{t-1} r = r t − 1 q q q [ 0 , r U ] [0,rU] [ 0 , r U ] [ r U , U ] [rU,U] [ r U , U ] q ( 0 ) = − ( U − y ) Φ t q(0)=-(U-y)\Phi_{t} q ( 0 ) = − ( U − y ) Φ t q ( r U ) ≤ ( U − y ) ( L r − a ) q(rU)\le(U-y)(Lr-a) q ( r U ) ≤ ( U − y ) ( L r − a ) q ( U ) = ( U − y ) d q(U)=(U-y)d q ( U ) = ( U − y ) d 0 , r U 0,rU 0 , r U ( U − y ) ( − Φ t + λ ~ t D / U ) (U-y)(-\Phi_{t}+\widetilde{\lambda}_{t}D/U) ( U − y ) ( − Φ t + λ t D / U ) r U , U rU,U r U , U ( U − y ) ( − a + L D / U ) (U-y)(-a+LD/U) ( U − y ) ( − a + L D / U ) D = r U D=rU D = r U Φ t = a + β Φ t − 1 \Phi_{t}=a+\beta\Phi_{t-1} Φ t = a + β Φ t − 1 λ ~ t = L + β λ ~ t − 1 \widetilde{\lambda}_{t}=L+\beta\widetilde{\lambda}_{t-1} λ t = L + β λ t − 1 ( λ ~ t > L ) (\widetilde{\lambda}_{t}>L) ( λ t > L ) global majorant
q ( D ) ≤ ( U − y ) min { − Φ t + λ ~ t D / U , − a + L D / U } . (38) q(D)\le(U-y)\min\{-\Phi_{t}+\widetilde{\lambda}_{t}D/U,-a+LD/U\}. \tag*{(38)} q ( D ) ≤ ( U − y ) min { − Φ t + λ t D / U , − a + L D / U } . (38) Because the minimum of these two affine functions is concave, Jensen’s inequality bounds the worst-case expectation by its value at μ \mu μ p ≥ θ p\ge\theta p ≥ θ
Put A t = β Φ t − 1 A_{t}=\beta\Phi_{t-1} A t = β Φ t − 1 δ ~ t = λ ~ t − K t = β ( λ ~ t − 1 − Φ t − 1 ) > 0 \widetilde{\delta}_{t}=\widetilde{\lambda}_{t}-K_{t}=\beta(\widetilde{\lambda}_{t-1}-\Phi_{t-1})>0 δ t = λ t − K t = β ( λ t − 1 − Φ t − 1 ) > 0 ρ = r 1 ≤ r t − 1 ≤ r t ≤ θ \rho=r_{1}\le r_{t-1}\le r_{t}\le\theta ρ = r 1 ≤ r t − 1 ≤ r t ≤ θ A t / ( A t + δ ~ t ) = r t − 1 A_{t}/(A_{t}+\widetilde{\delta}_{t})=r_{t-1} A t / ( A t + δ t ) = r t − 1
0 ≤ V ~ t − V ^ t ≤ ( U − y ) min { δ ~ t p , A t ( 1 − p ) , ( Φ t − K t p ) + } . (39) 0\le\widetilde{V}^{t}-\widehat{V}^{t}\le(U-y)\min\{\widetilde{\delta}_{t}p,A_{t}(1-p),(\Phi_{t}-K_{t}p)_{+}\}. \tag*{(39)} 0 ≤ V t − V t ≤ ( U − y ) min { δ t p , A t ( 1 − p ) , ( Φ t − K t p ) + } . (39) The first two terms meet at p = r t − 1 p=r_{t-1} p = r t − 1 A t ( 1 − r t − 1 ) ≤ β Φ 2 ( 1 − ρ ) A_{t}(1-r_{t-1})\le\beta\Phi_{2}(1-\rho) A t ( 1 − r t − 1 ) ≤ β Φ 2 ( 1 − ρ ) p < ρ t p<\rho_{t} p < ρ t v = U − y v=U-y v = U − y U { t h ( 1 − p ) + ( t − 1 ) c p } + v ( c − Φ t + K t p ) > 0 U\{th(1-p)+(t-1)cp\}+v(c-\Phi_t+K_t p)>0 U { t h ( 1 − p ) + ( t − 1 ) c p } + v ( c − Φ t + K t p ) > 0 v v v v = U v=U v = U D t ( p ) = t h + c − Φ t + { ( t − 1 ) c − t h + K t } p D_t(p)=th+c-\Phi_t+\{(t-1)c-th+K_t\}p D t ( p ) = t h + c − Φ t + {( t − 1 ) c − t h + K t } p D t ( 0 ) > 0 D_t(0)>0 D t ( 0 ) > 0 D t ( 1 ) = t c + d > 0 D_t(1)=tc+d>0 D t ( 1 ) = t c + d > 0 δ ~ t p / D t ( p ) \tilde{\delta}_{tp}/D_t(p) δ ~ tp / D t ( p ) A t ( 1 − p ) / D t ( p ) A_t(1-p)/D_t(p) A t ( 1 − p ) / D t ( p ) r = r t − 1 r=r_{t-1} r = r t − 1
V ~ t − V ^ t V ^ t ≤ A t ( 1 − r ) ( 1 − r ) ( t h + c − Φ t ) + r ( t c + d ) ≤ β Φ 2 ( 1 − ρ ) t { h ( 1 − ρ ) + c ρ } = K I I t . \frac{\widetilde{V}^t-\widehat{V}^t}{\widehat{V}^t}
\le\frac{A_t(1-r)}{(1-r)(th+c-\Phi_t)+r(tc+d)}
\le\frac{\beta\Phi_2(1-\rho)}{t\{h(1-\rho)+c\rho\}}
=\frac{K_{\mathrm{II}}}{t}. V t V t − V t ≤ ( 1 − r ) ( t h + c − Φ t ) + r ( t c + d ) A t ( 1 − r ) ≤ t { h ( 1 − ρ ) + c ρ } β Φ 2 ( 1 − ρ ) = t K II . Here Φ t < c \Phi_t<c Φ t < c r ≥ ρ r\ge\rho r ≥ ρ A t ≤ β Φ 2 A_t\le\beta\Phi_2 A t ≤ β Φ 2 t = 1 , 2 t=1,2 t = 1 , 2
Proof of Proposition EC.1(2). Use η , ν t \eta,\nu_t η , ν t M t = Φ t − c + ν t M_t=\Phi_t-c+\nu_t M t = Φ t − c + ν t v t , 0 , v t , U v_{t,0},v_{t,U} v t , 0 , v t , U 0 < ν t ≤ c 0<\nu_t\le c 0 < ν t ≤ c M t > 0 M_t>0 M t > 0 M t − β M t − 1 = K 1 − ν t − 1 > 0 M_t-\beta M_{t-1}=K_1-\nu_{t-1}>0 M t − β M t − 1 = K 1 − ν t − 1 > 0
V b d t ( y , μ ) = ( 1 − μ / U ) v t , 0 ( y ) + ( μ / U ) v t , U ( y ) , τ ( μ ) = min { ( 1 − ρ 1 ) μ / q ˉ 2 , 1 − μ / U } . V_{\mathrm{bd}}^t(y,\mu)=(1-\mu/U)v_{t,0}(y)+(\mu/U)v_{t,U}(y),\qquad\tau(\mu)=\min\{(1-\rho_1)\mu/\bar{q}_2,1-\mu/U\}. V bd t ( y , μ ) = ( 1 − μ / U ) v t , 0 ( y ) + ( μ / U ) v t , U ( y ) , τ ( μ ) = min {( 1 − ρ 1 ) μ / q ˉ 2 , 1 − μ / U } . The complementary two-period value is V ^ 2 = V b d 2 + Ω 2 ( y ) τ ( μ ) \widehat{V}^2=V_{\mathrm{bd}}^2+\Omega_2(y)\tau(\mu) V 2 = V bd 2 + Ω 2 ( y ) τ ( μ )
Ω 2 ( y ) = { β Φ 1 U − K 2 y , y ≤ q ˉ 2 , g ρ 1 ( U − y ) / ( 1 − ρ 1 ) , y ≥ q ˉ 2 . \Omega_2(y)=
\begin{cases}
\beta\Phi_1U-K_2y, & y\le\bar{q}_2,\\
g\rho_1(U-y)/(1-\rho_1), & y\ge\bar{q}_2.
\end{cases} Ω 2 ( y ) = { β Φ 1 U − K 2 y , g ρ 1 ( U − y ) / ( 1 − ρ 1 ) , y ≤ q ˉ 2 , y ≥ q ˉ 2 . For V 2 = V b d 2 + K 2 ( q ˉ 2 − y ) + + τ ( μ ) V^2=V_{\mathrm{bd}}^2+K_2(\bar{q}_2-y)_++\tau(\mu) V 2 = V bd 2 + K 2 ( q ˉ 2 − y ) + + τ ( μ )
0 ≤ V ^ 2 − V 2 ≤ g ρ 1 U τ ( μ ) . (40) 0\le\widehat{V}^2-V^2\le g\rho_1U\tau(\mu). \tag*{(40)} 0 ≤ V 2 − V 2 ≤ g ρ 1 U τ ( μ ) . (40) Extend V ‾ 2 \underline{V}^2 V 2 s 2 = q ˉ 2 s_2=\bar{q}_2 s 2 = q ˉ 2 s t = ( M t − 1 / M t ) ( s t − 1 − α U ) + s_t=(M_{t-1}/M_t)(s_{t-1}-\alpha U)_+ s t = ( M t − 1 / M t ) ( s t − 1 − α U ) + 0 ≤ s t ≤ q ˉ 2 0\le s_t\le\bar{q}_2 0 ≤ s t ≤ q ˉ 2
V ‾ t = V b d t + M t ( s t − y ) + + τ ( μ ) , t ≥ 2. (41) \underline{V}^t=V_{\mathrm{bd}}^t+M_t(s_t-y)_++\tau(\mu),\qquad t\ge2. \tag*{(41)} V t = V bd t + M t ( s t − y ) + + τ ( μ ) , t ≥ 2. (41) Indeed the induction integrand is convex in demand on each of [ 0 , q ˉ 2 ] [0,\bar{q}_2] [ 0 , q ˉ 2 ] [ q ˉ 2 , U ] [\bar{q}_2,U] [ q ˉ 2 , U ] x ≤ q ˉ 2 x\le\bar{q}_2 x ≤ q ˉ 2 q ˉ 2 \bar{q}_2 q ˉ 2
M t − 1 ( s t − 1 − α U + α x ) + − ( M t + α M t − 1 ) x , M_{t-1}(s_{t-1}-\alpha U+\alpha x)_+-(M_t+\alpha M_{t-1})x, M t − 1 ( s t − 1 − α U + α x ) + − ( M t + α M t − 1 ) x , whose positive part is M t ( s t − x ) + M_t(s_t-x)_+ M t ( s t − x ) + x ≥ q ˉ 2 x\ge\bar{q}_2 x ≥ q ˉ 2 0 , q ˉ 2 , U 0,\bar{q}_2,U 0 , q ˉ 2 , U x < s t x<s_t x < s t 0 , U 0,U 0 , U Φ t − M t μ / q ˉ 2 \Phi_t-M_t\mu/\bar{q}_2 Φ t − M t μ / q ˉ 2 Φ t − M t \Phi_t-M_t Φ t − M t Φ t − M t μ / U \Phi_t-M_t\mu/U Φ t − M t μ / U c − ν t ≥ 0 c-\nu_t\ge0 c − ν t ≥ 0 x = y x=y x = y V ‾ 2 ≤ V ^ 2 = V ~ 2 ≤ V 2 + g ρ 1 U τ ( μ ) \underline{V}^2\le\widehat{V}^2=\widetilde{V}^2\le V^2+g\rho_1U\tau(\mu) V 2 ≤ V 2 = V 2 ≤ V 2 + g ρ 1 U τ ( μ ) τ \tau τ E Q [ τ ( D ~ ) ] ≤ τ ( μ ) \mathbb{E}_Q[\tau(\widetilde{D})]\le\tau(\mu) E Q [ τ ( D )] ≤ τ ( μ ) Q ∈ M ( μ ) Q\in\mathcal{M}(\mu) Q ∈ M ( μ )
V ‾ t ≤ V ^ t ≤ V ~ t ≤ V t + g ρ 1 U τ ( μ ) , \underline{V}^t\le\widehat{V}^t\le\widetilde{V}^t\le V^t+g\rho_1U\tau(\mu), V t ≤ V t ≤ V t ≤ V t + g ρ 1 U τ ( μ ) , proving (EC.16). Moreover V ^ t ≥ V b d t ( y , μ ) ≥ t h ( U − μ ) + μ ∑ r = 1 t − 1 ν r > 0 \widehat{V}^t\ge V_{\mathrm{bd}}^t(y,\mu)\ge th(U-\mu)+\mu\sum_{r=1}^{t-1}\nu_r>0 V t ≥ V bd t ( y , μ ) ≥ t h ( U − μ ) + μ ∑ r = 1 t − 1 ν r > 0 t ≥ 2 t\ge2 t ≥ 2 μ = q ˉ 2 \mu=\bar{q}_2 μ = q ˉ 2 ν r → ( b + α w ) / ( 1 + α ) > 0 \nu_r\to(b+\alpha w)/(1+\alpha)>0 ν r → ( b + α w ) / ( 1 + α ) > 0 g = 0 g=0 g = 0
For ⋄ ∈ { ↑ , ↓ } \diamond\in\{\uparrow,\downarrow\} ⋄ ∈ { ↑ , ↓ } F ⋄ , t x ( D ) = b ( D − x ) + + h ( x − D ) + + V ^ ⋄ t − 1 ( R ( x , D ) , D ) F_{\diamond,t}^{x}(D)=b(D-x)_{+}+h(x-D)_{+}+\hat{V}_{\diamond}^{t-1}(R(x,D),D) F ⋄ , t x ( D ) = b ( D − x ) + + h ( x − D ) + + V ^ ⋄ t − 1 ( R ( x , D ) , D ) B ⋄ , t ( x , μ ) B_{\diamond,t}(x,\mu) B ⋄ , t ( x , μ ) c x + α w ( U − x ) cx+\alpha w(U-x) c x + α w ( U − x ) V ^ ⋄ t = − c y + min x ∈ [ y , U ] B ⋄ , t ( x , μ ) \hat{V}_{\diamond}^{t}=-cy+\min_{x\in[y,U]}B_{\diamond,t}(x,\mu) V ^ ⋄ t = − cy + min x ∈ [ y , U ] B ⋄ , t ( x , μ )
Proof of Proposition 5 and Corollary 3. Write H = cav φ H=\operatorname{cav}\varphi H = cav φ
sup Q ∈ M ↑ ( μ ) E Q φ = max m ∈ [ μ , U ] H ( m ) , sup Q ∈ M ↓ ( μ ) E Q φ = max m ∈ [ 0 , μ ] H ( m ) . \sup_{Q\in\mathcal{M}^{\uparrow}(\mu)}\mathbb{E}_{Q}\varphi=\max_{m\in[\mu,U]}H(m),\qquad\sup_{Q\in\mathcal{M}^{\downarrow}(\mu)}\mathbb{E}_{Q}\varphi=\max_{m\in[0,\mu]}H(m). Q ∈ M ↑ ( μ ) sup E Q φ = m ∈ [ μ , U ] max H ( m ) , Q ∈ M ↓ ( μ ) sup E Q φ = m ∈ [ 0 , μ ] max H ( m ) . Continuity of H H H δ U \delta_{U} δ U δ 0 \delta_{0} δ 0 μ \mu μ □ \square □
Proof of Proposition 6. Inventory convexity gives the constrained largest minimizer max { y , χ ⋄ ( μ ) } \max\{y,\chi_{\diamond}(\mu)\} max { y , χ ⋄ ( μ )} □ \square □
Proof of Theorem 4 and Corollary 4. For the α < α b \alpha<\alpha_{\mathrm b} α < α b θ t = t h − ( t − 1 ) c \theta_t=th-(t-1)c θ t = t h − ( t − 1 ) c U U U V ^ ↑ t ( y , U ) = t c U − c y \hat{V}_{\uparrow}^{t}(y,U)=tcU-cy V ^ ↑ t ( y , U ) = t c U − cy x ↑ , t ∗ ( y , U ) = U x_{\uparrow,t}^{*}(y,U)=U x ↑ , t ∗ ( y , U ) = U t ≥ 2 t\geq2 t ≥ 2 U − x U-x U − x b − c − α ( c − w ) > 0 b-c-\alpha(c-w)>0 b − c − α ( c − w ) > 0
We first handle θ t ≤ 0 \theta_t\leq0 θ t ≤ 0 c > h c>h c > h t ≥ ⌈ c / ( c − h ) ⌉ t\geq\lceil c/(c-h)\rceil t ≥ ⌈ c / ( c − h )⌉ U U U t c U − c y + { b − c − α ( c − w ) } ( U − x ) tcU-cy+\{b-c-\alpha(c-w)\}(U-x) t c U − cy + { b − c − α ( c − w )} ( U − x ) U U U m r = E [ D ~ r ] m_r=\mathbb{E}[\tilde{D}_r] m r = E [ D ~ r ] c ( U − y ) + t h U + ( c − h ) ∑ r = 2 t m r − h m 1 c(U-y)+thU+(c-h)\sum_{r=2}^{t}m_r-hm_1 c ( U − y ) + t h U + ( c − h ) ∑ r = 2 t m r − h m 1 μ ≤ m t ≤ ⋯ ≤ m 1 ≤ U \mu\leq m_t\leq\cdots\leq m_1\leq U μ ≤ m t ≤ ⋯ ≤ m 1 ≤ U c ( U − y ) + t h U + { ( t − 1 ) c − t h } m 1 ≤ t c U − c y c(U-y)+thU+\{(t-1)c-th\}m_1\leq tcU-cy c ( U − y ) + t h U + {( t − 1 ) c − t h } m 1 ≤ t c U − cy θ t ≤ 0 \theta_t\leq0 θ t ≤ 0
For θ t > 0 \theta_t>0 θ t > 0
V ^ ↑ t ( y , μ ) = − c y + C t ↑ + ( Φ t − K t μ U ) + ( y − ω t U ) + − θ t ( μ − ρ t U ) + . (42) \hat{V}_{\uparrow}^{t}(y,\mu)=-cy+C_t^{\uparrow}+\left(\Phi_t-K_t\frac{\mu}{U}\right)_{+}(y-\omega_tU)_{+}-\theta_t(\mu-\rho_tU)_{+}. \tag*{(42)} V ^ ↑ t ( y , μ ) = − cy + C t ↑ + ( Φ t − K t U μ ) + ( y − ω t U ) + − θ t ( μ − ρ t U ) + . (42) For t = 1 t=1 t = 1 [ μ , U ] [\mu,U] [ μ , U ]
For the induction step, write P = Φ t − 1 P=\Phi_{t-1} P = Φ t − 1 M = K t − 1 M=K_{t-1} M = K t − 1 q = P / M q=P/M q = P / M x ≥ ω t U x\geq\omega_tU x ≥ ω t U x ˉ = x / U \bar{x}=x/U x ˉ = x / U a = θ t − 1 / M − β ( 1 − x ˉ ) a=\theta_{t-1}/M-\beta(1-\bar{x}) a = θ t − 1 / M − β ( 1 − x ˉ ) θ t > 0 \theta_t>0 θ t > 0 t ≥ 2 t\geq2 t ≥ 2 h > c / 2 > w / 2 h>c/2>w/2 h > c /2 > w /2 K 2 > β K 1 K_2>\beta K_1 K 2 > β K 1 t ≥ 3 t\geq3 t ≥ 3 K t > β K t − 1 K_t>\beta K_{t-1} K t > β K t − 1 K t > β M K_t>\beta M K t > β M θ t − 1 = θ t + c − h \theta_{t-1}=\theta_t+c-h θ t − 1 = θ t + c − h P ≥ θ t − 1 P\geq\theta_{t-1} P ≥ θ t − 1 0 ≤ a ≤ q 0\leq a\leq q 0 ≤ a ≤ q
c x + α w ( U − x ) + F ↑ , t x ( U ) = t c U + ( K t − Φ t ) ( U − x ) , F ↑ , t x ( 0 ) − F ↑ , t x ( U ) = K t ( x − ω t U ) . (43) \begin{aligned}
cx+\alpha w(U-x)+F_{\uparrow,t}^{x}(U)&=tcU+(K_t-\Phi_t)(U-x),\\
F_{\uparrow,t}^{x}(0)-F_{\uparrow,t}^{x}(U)&=K_t(x-\omega_tU). \tag*{(43)}
\end{aligned} c x + α w ( U − x ) + F ↑ , t x ( U ) F ↑ , t x ( 0 ) − F ↑ , t x ( U ) = t c U + ( K t − Φ t ) ( U − x ) , = K t ( x − ω t U ) . (43) For z ∈ [ 0 , 1 ] z\in[0,1] z ∈ [ 0 , 1 ]
E ( z ) : = ( 1 − z ) F ↑ , t x ( 0 ) + z F ↑ , t x ( U ) − F ↑ , t x ( U z ) U = ( b + h − c ) min { z ( 1 − x ˉ ) , x ˉ ( 1 − z ) } − θ t − 1 min { z ( 1 − q ) , q ( 1 − z ) } + P a ( 1 − z ) − M ( q − z ) + ( a − min { z , x ˉ } ) + . \begin{aligned}
E(z)&:=\frac{(1-z)F_{\uparrow,t}^{x}(0)+zF_{\uparrow,t}^{x}(U)-F_{\uparrow,t}^{x}(Uz)}{U}\\
&=(b+h-c)\min\{z(1-\bar{x}),\bar{x}(1-z)\}-\theta_{t-1}\min\{z(1-q),q(1-z)\}\\
&\quad+Pa(1-z)-M(q-z)_{+}(a-\min\{z,\bar{x}\})_{+}.
\end{aligned} E ( z ) := U ( 1 − z ) F ↑ , t x ( 0 ) + z F ↑ , t x ( U ) − F ↑ , t x ( U z ) = ( b + h − c ) min { z ( 1 − x ˉ ) , x ˉ ( 1 − z )} − θ t − 1 min { z ( 1 − q ) , q ( 1 − z )} + P a ( 1 − z ) − M ( q − z ) + ( a − min { z , x ˉ } ) + . With u = 1 − x ˉ u=1-\bar{x} u = 1 − x ˉ θ t − 1 = M ( a + β u ) \theta_{t-1}=M(a+\beta u) θ t − 1 = M ( a + β u )
{ z { M ( q − z ) + u ( K t − β M ) } , 0 ≤ z ≤ min { a , x ˉ } , M a ( q − z ) + u z ( K t − β M ) , a ≤ z ≤ min { x ˉ , q } , a ≤ x ˉ , u ( K t z − β P ) , q ≤ z ≤ x ˉ , ( 1 − z ) ( K t x ˉ − β P ) , max { x ˉ , q } ≤ z ≤ 1. \begin{cases}
z\{M(q-z)+u(K_t-\beta M)\}, & 0\le z\le\min\{a,\bar{x}\},\\
Ma(q-z)+uz(K_t-\beta M), & a\le z\le\min\{\bar{x},q\},\quad a\le\bar{x},\\
u(K_tz-\beta P), & q\le z\le\bar{x},\\
(1-z)(K_t\bar{x}-\beta P), & \max\{\bar{x},q\}\le z\le1.
\end{cases} ⎩ ⎨ ⎧ z { M ( q − z ) + u ( K t − β M )} , M a ( q − z ) + u z ( K t − β M ) , u ( K t z − β P ) , ( 1 − z ) ( K t x ˉ − β P ) , 0 ≤ z ≤ min { a , x ˉ } , a ≤ z ≤ min { x ˉ , q } , a ≤ x ˉ , q ≤ z ≤ x ˉ , max { x ˉ , q } ≤ z ≤ 1. Each expression is nonnegative because K t > β M K_t>\beta M K t > β M K t x ˉ − β P ≥ b + h − c − θ t > 0 K_t\bar{x}-\beta P\ge b+h-c-\theta_t>0 K t x ˉ − β P ≥ b + h − c − θ t > 0 x ˉ < q \bar{x}<q x ˉ < q [ x ˉ , q ] [\bar{x},q] [ x ˉ , q ] x ≥ ω t U x\ge\omega_tU x ≥ ω t U [ μ , U ] [\mu,U] [ μ , U ] μ \mu μ
B ↑ , t ( x , μ ) = t c U + ( K t − Φ t ) ( U − x ) + K t ( 1 − μ U ) ( x − ω t U ) , x ≥ ω t U . (44) B_{\uparrow,t}(x,\mu)=tcU+(K_t-\Phi_t)(U-x)+K_t\left(1-\frac{\mu}{U}\right)(x-\omega_tU),\qquad x\ge\omega_tU. \tag*{(44)} B ↑ , t ( x , μ ) = t c U + ( K t − Φ t ) ( U − x ) + K t ( 1 − U μ ) ( x − ω t U ) , x ≥ ω t U . (44) For x < ω t U x<\omega_tU x < ω t U δ U \delta_U δ U
B ↑ , t ( x , μ ) ≥ t c U + ( K t − Φ t ) ( U − x ) > B ↑ , t ( ω t U , μ ) . B_{\uparrow,t}(x,\mu)\ge tcU+(K_t-\Phi_t)(U-x)>B_{\uparrow,t}(\omega_tU,\mu). B ↑ , t ( x , μ ) ≥ t c U + ( K t − Φ t ) ( U − x ) > B ↑ , t ( ω t U , μ ) . Therefore minimization only requires (EC.30), whose slope is Φ t − K t μ / U \Phi_t-K_t\mu/U Φ t − K t μ / U θ t > 0 \theta_t>0 θ t > 0 ( y , μ ) = ( 0 , 0 ) (y,\mu)=(0,0) ( y , μ ) = ( 0 , 0 ) c > h c>h c > h
□
Proof of the recovery-dominant assertion in Theorem 4. Write k = b / ( b + h ) k=b/(b+h) k = b / ( b + h ) η = c + α ( c − w ) − b = ( c − w ) ( α − α b ) ≥ 0 \eta=c+\alpha(c-w)-b=(c-w)(\alpha-\alpha_b)\ge0 η = c + α ( c − w ) − b = ( c − w ) ( α − α b ) ≥ 0 R ( 0 , D ) = α U R(0,D)=\alpha U R ( 0 , D ) = α U F ↑ , 2 0 ( D ) = b D + V ^ ↑ 1 ( α U , D ) F_{\uparrow,2}^{0}(D)=bD+\widehat{V}_{\uparrow}^{1}(\alpha U,D) F ↑ , 2 0 ( D ) = b D + V ↑ 1 ( α U , D ) D = ρ 1 U D=\rho_1U D = ρ 1 U b b b α ≤ k \alpha\le k α ≤ k 2 b − ( b + h ) α > b − h 2b-(b+h)\alpha>b-h 2 b − ( b + h ) α > b − h α > k \alpha>k α > k ρ 1 U \rho_1U ρ 1 U b − h > 0 b-h>0 b − h > 0 δ U \delta_U δ U
At terminal reference U U U V ^ ↑ 1 ( R ( x , U ) , U ) = c { U − α ( U − x ) } \widehat{V}_{\uparrow}^{1}(R(x,U),U)=c\{U-\alpha(U-x)\} V ↑ 1 ( R ( x , U ) , U ) = c { U − α ( U − x )} ∂ x + B ↑ , 2 ( 0 , μ ) = c − α w − b + α c = η ≥ 0 \partial_x^+B_{\uparrow,2}(0,\mu)=c-\alpha w-b+\alpha c=\eta\ge0 ∂ x + B ↑ , 2 ( 0 , μ ) = c − α w − b + α c = η ≥ 0 B ↑ , 2 ( ⋅ , μ ) B_{\uparrow,2}(\cdot,\mu) B ↑ , 2 ( ⋅ , μ ) x ↑ , 2 ∗ ( y , μ ) = y x_{\uparrow,2}^*(y,\mu)=y x ↑ , 2 ∗ ( y , μ ) = y α > α b \alpha>\alpha_b α > α b
□
Proof of Proposition 7. Put k = α d = b / ( b + h ) k=\alpha_d=b/(b+h) k = α d = b / ( b + h ) z = z 2 = U ( 1 − k / α ) > 0 z=z_2=U(1-k/\alpha)>0 z = z 2 = U ( 1 − k / α ) > 0 η = c + α ( c − w ) − b > 0 \eta=c+\alpha(c-w)-b>0 η = c + α ( c − w ) − b > 0 0 ≤ x ≤ z 0\le x\le z 0 ≤ x ≤ z R ( x , D ) ≥ α ( U − x ) ≥ α ( U − z ) = k U R(x,D)\ge\alpha(U-x)\ge\alpha(U-z)=kU R ( x , D ) ≥ α ( U − x ) ≥ α ( U − z ) = k U b − ( b + h ) R ( x , D ) / U ≤ 0 b-(b+h)R(x,D)/U\le0 b − ( b + h ) R ( x , D ) / U ≤ 0 V ^ ↓ 1 ( R ( x , D ) , D ) = α w { U − R ( x , D ) } + h R ( x , D ) \widehat{V}_{\downarrow}^{1}(R(x,D),D)=\alpha w\{U-R(x,D)\}+hR(x,D) V ↓ 1 ( R ( x , D ) , D ) = α w { U − R ( x , D )} + h R ( x , D ) F ↑ , 2 x F_{\uparrow,2}^{x} F ↑ , 2 x α w − 2 h \alpha w-2h α w − 2 h D < x D<x D < x b b b D > x D>x D > x b + 2 h − α w > 0 b+2h-\alpha w>0 b + 2 h − α w > 0 Δ ( x ) = b U − ( b + 2 h − α w ) x \Delta(x)=bU-(b+2h-\alpha w)x Δ ( x ) = b U − ( b + 2 h − α w ) x [ 0 , z ] [0,z] [ 0 , z ] q = α − k ∈ ( 0 , 1 − k ) q=\alpha-k\in(0,1-k) q = α − k ∈ ( 0 , 1 − k )
α Δ ( z ) U = k b − q ( 2 h − α w ) , q ( 2 h − α w ) ≤ h ( α − k ) ( 2 − α ) ≤ h 2 b + h < b 2 b + h = k b . \frac{\alpha\Delta(z)}{U}=kb-q(2h-\alpha w),\qquad q(2h-\alpha w)\le h(\alpha-k)(2-\alpha)\le\frac{h^2}{b+h}<\frac{b^2}{b+h}=kb. U α Δ ( z ) = k b − q ( 2 h − α w ) , q ( 2 h − α w ) ≤ h ( α − k ) ( 2 − α ) ≤ b + h h 2 < b + h b 2 = k b . The second inequality uses monotonicity of ( α − k ) ( 2 − α ) (\alpha-k)(2-\alpha) ( α − k ) ( 2 − α ) [ k , 1 ] [k,1] [ k , 1 ] [ 0 , z ] [0,z] [ 0 , z ] ( 1 − μ / U ) δ 0 + ( μ / U ) δ U (1-\mu/U)\delta_0+(\mu/U)\delta_U ( 1 − μ / U ) δ 0 + ( μ / U ) δ U
∂ x B ↓ , 2 ( x , μ ) = γ 0 − ( μ / U ) ( γ 0 − γ 1 ) , { γ 0 = c + ( 2 − α ) ( h − α w ) , γ 1 = c − b − α h − α β w . \partial_x B_{\downarrow,2}(x,\mu)=\gamma_0-(\mu/U)(\gamma_0-\gamma_1),\qquad
\begin{cases}
\gamma_0=c+(2-\alpha)(h-\alpha w),\\
\gamma_1=c-b-\alpha h-\alpha\beta w.
\end{cases} ∂ x B ↓ , 2 ( x , μ ) = γ 0 − ( μ / U ) ( γ 0 − γ 1 ) , { γ 0 = c + ( 2 − α ) ( h − α w ) , γ 1 = c − b − α h − α β w . Since α ( 2 − α ) ≤ 1 \alpha(2-\alpha)\le1 α ( 2 − α ) ≤ 1 γ 0 > c − w > 0 \gamma_0>c-w>0 γ 0 > c − w > 0 γ 1 < c − b < 0 \gamma_1<c-b<0 γ 1 < c − b < 0 μ 2 † \mu_2^\dagger μ 2 † x = z x=z x = z ( 0 , U ) (0,U) ( 0 , U ) 0 < μ < U 0<\mu<U 0 < μ < U D = 0 D=0 D = 0 γ 0 \gamma_0 γ 0 R ( x , U ) < k U R(x,U)<kU R ( x , U ) < k U D = U D=U D = U c − α w − b + α c = η > 0 c-\alpha w-b+\alpha c=\eta>0 c − α w − b + α c = η > 0
∂ x + B ↓ , 2 ( z , μ ) = ( 1 − μ / U ) γ 0 + ( μ / U ) η > 0. \partial_x^+ B_{\downarrow,2}(z,\mu)=(1-\mu/U)\gamma_0+(\mu/U)\eta>0. ∂ x + B ↓ , 2 ( z , μ ) = ( 1 − μ / U ) γ 0 + ( μ / U ) η > 0. Convexity now yields largest minimizer 0 0 0 μ < μ 2 † \mu<\mu_2^\dagger μ < μ 2 † z z z μ ≥ μ 2 † \mu\ge\mu_2^\dagger μ ≥ μ 2 † [ 0 , z ] [0,z] [ 0 , z ] x ≥ y x\ge y x ≥ y
Proof of Theorem 5. At t = 1 t=1 t = 1 x x x h x + { b − ( b + h ) x / U } m hx+\{b-(b+h)x/U\}m h x + { b − ( b + h ) x / U } m 0 ≤ m ≤ μ 0\le m\le\mu 0 ≤ m ≤ μ m = μ m=\mu m = μ x < b U / ( b + h ) x<bU/(b+h) x < b U / ( b + h ) m = 0 m=0 m = 0 x > b U / ( b + h ) x>bU/(b+h) x > b U / ( b + h ) Φ 1 − K 1 μ / U \Phi_1-K_1\mu/U Φ 1 − K 1 μ / U Φ 1 \Phi_1 Φ 1 s 1 , 1 ↓ = b U / ( b + h ) s^\downarrow_{1,1}=bU/(b+h) s 1 , 1 ↓ = b U / ( b + h )
For the induction step, write n = t − 1 n=t-1 n = t − 1 P = Φ n P=\Phi_n P = Φ n z j = s n , j ↓ z_j=s^\downarrow_{n,j} z j = s n , j ↓ r j = P K t ( z j − α U ) r_j=\frac{P}{K_t}(z_j-\alpha U) r j = K t P ( z j − α U ) j = 1 , … , n j=1,\ldots,n j = 1 , … , n A r : = ( V ^ ↓ r − 1 ( 0 , U ) / U − r h ) + A_r:=(\widehat V^{r-1}_{\downarrow}(0,U)/U-rh)_+ A r := ( V ↓ r − 1 ( 0 , U ) / U − r h ) + s r , r ↓ < U s^\downarrow_{r,r}<U s r , r ↓ < U r = 1 r=1 r = 1
Ordering. The ratios ρ s \rho_s ρ s Φ s \Phi_s Φ s K s − Φ s = b − c − α ( c − w ) > 0 K_s-\Phi_s=b-c-\alpha(c-w)>0 K s − Φ s = b − c − α ( c − w ) > 0 s ≥ 2 s\ge2 s ≥ 2 K 1 − Φ 1 K_1-\Phi_1 K 1 − Φ 1 α c \alpha c α c
0 < q s , 1 < ⋯ < q s , s = U , q s , j q s , j + 1 = ρ j , 0 ≤ s s , 1 ↓ ≤ ⋯ ≤ s s , s ↓ ≤ U , s s , j ↓ ≤ q s , j . 0<q_{s,1}<\cdots<q_{s,s}=U,\qquad
\frac{q_{s,j}}{q_{s,j+1}}=\rho_j,\qquad
0\le s^\downarrow_{s,1}\le\cdots\le s^\downarrow_{s,s}\le U,\qquad
s^\downarrow_{s,j}\le q_{s,j}. 0 < q s , 1 < ⋯ < q s , s = U , q s , j + 1 q s , j = ρ j , 0 ≤ s s , 1 ↓ ≤ ⋯ ≤ s s , s ↓ ≤ U , s s , j ↓ ≤ q s , j . For inherited switches, r j ≤ q t , j r_j\le q_{t,j} r j ≤ q t , j K t ≥ K t − 1 K_t\ge K_{t-1} K t ≥ K t − 1 t ≥ 3 t\ge3 t ≥ 3 t = 2 t=2 t = 2
K 2 − K 1 ( b b + h − α ) = 2 h − α w + α ( b − c ) + α 2 w > 0. K_2-K_1\left(\frac{b}{b+h}-\alpha\right)=2h-\alpha w+\alpha(b-c)+\alpha^2w>0. K 2 − K 1 ( b + h b − α ) = 2 h − α w + α ( b − c ) + α 2 w > 0. Positive targets are strictly ordered: the recursion preserves positive inherited differences, and the new highest target strictly exceeds the last inherited target, which is below U U U ρ s \rho_s ρ s
β q t , j + 1 + α U ≥ z j , 1 ≤ j ≤ n . \beta q_{t,j+1}+\alpha U\ge z_j,\qquad1\le j\le n. β q t , j + 1 + α U ≥ z j , 1 ≤ j ≤ n . The highest target satisfies s s , s ↓ ≥ b U / ( b + h ) s^\downarrow_{s,s}\ge bU/(b+h) s s , s ↓ ≥ b U / ( b + h ) k = b / ( b + h ) k=b/(b+h) k = b / ( b + h ) t t t k U kU k U U { k c − α ( 1 − k ) P } / K t ≥ 0 U\{kc-\alpha(1-k)P\}/K_t\ge0 U { k c − α ( 1 − k ) P } / K t ≥ 0 α P ≤ h + α ( c − w ) < b c / h \alpha P\le h+\alpha(c-w)<bc/h α P ≤ h + α ( c − w ) < b c / h s s , s ↓ ≥ α U s^\downarrow_{s,s}\ge\alpha U s s , s ↓ ≥ α U
Adjacent-contact comparisons. As in the proof of Lemma EC.4, put p j = q n , j p_j=q_{n,j} p j = q n , j m j = ρ n p j = q t , j m_j=\rho_n p_j=q_{t,j} m j = ρ n p j = q t , j m n + 1 = U m_{n+1}=U m n + 1 = U ϑ j = m j / m j + 1 \vartheta_j=m_j/m_{j+1} ϑ j = m j / m j + 1
V ^ ↓ n ( y , m j ) = C j − c y + P H j ( y ) , V ^ ↓ n ( y , 0 ) = C 0 + ( P − c ) y , \widehat V^n_{\downarrow}(y,m_j)=C_j-cy+PH_j(y),\qquad
\widehat V^n_{\downarrow}(y,0)=C_0+(P-c)y, V ↓ n ( y , m j ) = C j − cy + P H j ( y ) , V ↓ n ( y , 0 ) = C 0 + ( P − c ) y , where C j = V ^ ↓ n ( 0 , m j ) C_j=\widehat{V}_{\downarrow}^{n}(0,m_j) C j = V ↓ n ( 0 , m j ) C 0 = V ^ ↓ n ( 0 , 0 ) C_0=\widehat{V}_{\downarrow}^{n}(0,0) C 0 = V ↓ n ( 0 , 0 )
H j ( y ) = ∑ ℓ = j + 1 n ( 1 − p j p ℓ ) { ( y − z ℓ − 1 ) + − ( y − z ℓ ) + } + ( y − z n ) + , H n + 1 ( y ) = ( y − z n ) + . H_j(y)=\sum_{\ell=j+1}^{n}\left(1-\frac{p_j}{p_\ell}\right)\{(y-z_{\ell-1})_+-(y-z_\ell)_+\}+(y-z_n)_+,\qquad H_{n+1}(y)=(y-z_n)_+. H j ( y ) = ℓ = j + 1 ∑ n ( 1 − p ℓ p j ) {( y − z ℓ − 1 ) + − ( y − z ℓ ) + } + ( y − z n ) + , H n + 1 ( y ) = ( y − z n ) + . The functions H j H_j H j y ≤ z j y\le z_j y ≤ z j a = m j a=m_j a = m j d = m j + 1 d=m_{j+1} d = m j + 1 ϑ = a / d \vartheta=a/d ϑ = a / d z = z j z=z_j z = z j s = α ( U − x ) s=\alpha(U-x) s = α ( U − x )
E j ( x ) = ( 1 − ϑ ) F ↓ , t x ( 0 ) + ϑ F ↓ , t x ( d ) − F ↓ , t x ( a ) . E_j(x)=(1-\vartheta)F_{\downarrow,t}^{x}(0)+\vartheta F_{\downarrow,t}^{x}(d)-F_{\downarrow,t}^{x}(a). E j ( x ) = ( 1 − ϑ ) F ↓ , t x ( 0 ) + ϑ F ↓ , t x ( d ) − F ↓ , t x ( a ) . The three-range calculation in the proof of Lemma EC.4 applies unchanged: for x ≤ a x\le a x ≤ a
E j ( x ) = ( 1 − ϑ ) [ ( b + h − c ) x + P { s + x − z − ( s − z ) + } ] , E_j(x)=(1-\vartheta)\left[(b+h-c)x+P\{s+x-z-(s-z)_+\}\right], E j ( x ) = ( 1 − ϑ ) [ ( b + h − c ) x + P { s + x − z − ( s − z ) + } ] , which is negative exactly when r j > 0 r_j>0 r j > 0 x < r j x<r_j x < r j a ≤ x ≤ d a\le x\le d a ≤ x ≤ d β d + α U ≥ z \beta d+\alpha U\ge z β d + α U ≥ z E j ( x ) ≥ 0 E_j(x)\ge0 E j ( x ) ≥ 0 x ≥ d x\ge d x ≥ d s + x − a ≥ z s+x-a\ge z s + x − a ≥ z E j ( x ) = P ( 1 − ϑ ) ( β x + α U − z ) ≥ 0 E_j(x)=P(1-\vartheta)(\beta x+\alpha U-z)\ge0 E j ( x ) = P ( 1 − ϑ ) ( β x + α U − z ) ≥ 0
E j ( x ) < 0 ⟺ r j > 0 and x < r j ; E j ( x ) ≥ 0 otherwise . (45) E_j(x)<0\quad\Longleftrightarrow\quad r_j>0\text{ and }x<r_j;\qquad E_j(x)\ge0\text{ otherwise}. \tag*{(45)} E j ( x ) < 0 ⟺ r j > 0 and x < r j ; E j ( x ) ≥ 0 otherwise . (45) Concave envelope. Let J J J x < r J x<r_J x < r J J = t J=t J = t Γ j ( x ) = { F ↓ , t x ( q t , j ) − F ↓ , t x ( 0 ) } / q t , j \Gamma_j(x)=\{F_{\downarrow,t}^{x}(q_{t,j})-F_{\downarrow,t}^{x}(0)\}/q_{t,j} Γ j ( x ) = { F ↓ , t x ( q t , j ) − F ↓ , t x ( 0 )} / q t , j E j ( x ) = q t , j { Γ j + 1 ( x ) − Γ j ( x ) } E_j(x)=q_{t,j}\{\Gamma_{j+1}(x)-\Gamma_j(x)\} E j ( x ) = q t , j { Γ j + 1 ( x ) − Γ j ( x )} q t , J q_{t,J} q t , J J < t J<t J < t j ≥ J j\ge J j ≥ J ℓ j ( x ) \ell_j(x) ℓ j ( x ) q t , j q_{t,j} q t , j q t , j + 1 q_{t,j+1} q t , j + 1
ℓ J ( x ) − Γ J ( x ) = K t q t , J ( x − r J ) < 0 , ℓ j + 1 ( x ) − ℓ j ( x ) = − K t q t , j + 1 ( r j + 1 − r j ) ≤ 0. \ell_J(x)-\Gamma_J(x)=\frac{K_t}{q_{t,J}}(x-r_J)<0,\qquad\ell_{j+1}(x)-\ell_j(x)=-\frac{K_t}{q_{t,j+1}}(r_{j+1}-r_j)\le0. ℓ J ( x ) − Γ J ( x ) = q t , J K t ( x − r J ) < 0 , ℓ j + 1 ( x ) − ℓ j ( x ) = − q t , j + 1 K t ( r j + 1 − r j ) ≤ 0. Thus the polygon through 0 , q t , J , … , q t , t 0,q_{t,J},\ldots,q_{t,t} 0 , q t , J , … , q t , t J = t J=t J = t D = x D=x D = x R ( x , D ) = z j R(x,D)=z_j R ( x , D ) = z j 2 K n / q n , j > 0 2K_n/q_{n,j}>0 2 K n / q n , j > 0 D = x D=x D = x b + h − c > 0 b+h-c>0 b + h − c > 0
Minimization and value. On a retained positive-contact face, each contact clears inventory and has continuation inventory below its target. The fixed-action slope is therefore c + α ( c − w ) − b < 0 c+\alpha(c-w)-b<0 c + α ( c − w ) − b < 0 q t , j q_{t,j} q t , j Φ t − K t μ / q t , j \Phi_t-K_t\mu/q_{t,j} Φ t − K t μ / q t , j Φ t > 0 \Phi_t>0 Φ t > 0
To locate the last change, first suppose z n < U z_n<U z n < U
F ↓ , t x ( U ) − F ↓ , t x ( 0 ) = K t { P K t ( z n − α U ) + b K t U − x } . F_{\downarrow,t}^{x}(U)-F_{\downarrow,t}^{x}(0)=K_t\left\{\frac{P}{K_t}(z_n-\alpha U)+\frac{b}{K_t}U-x\right\}. F ↓ , t x ( U ) − F ↓ , t x ( 0 ) = K t { K t P ( z n − α U ) + K t b U − x } . This yields the stated highest target, clipped at capacity. If z n = U z_n=U z n = U U U U ( β P + b − K t ) U / K t = ( c − h ) U / K t ≥ 0 (\beta P+b-K_t)U/K_t=(c-h)U/K_t\ge0 ( β P + b − K t ) U / K t = ( c − h ) U / K t ≥ 0 r j r_j r j U U U ρ t q t , j \rho_t q_{t,j} ρ t q t , j χ t ↓ ( μ ) \chi_t^{\downarrow}(\mu) χ t ↓ ( μ ) x ≥ y x \ge y x ≥ y
At x = U x=U x = U t h U thU t h U V ^ ↓ t − 1 ( 0 , U ) \widehat{V}_{\downarrow}^{t-1}(0,U) V ↓ t − 1 ( 0 , U )
B ↓ , t ( U , μ ) = ( c + t h ) U + ( V ^ ↓ t − 1 ( 0 , U ) U − t h ) μ . B_{\downarrow,t}(U,\mu)=(c+th)U+\left(\frac{\widehat{V}_{\downarrow}^{t-1}(0,U)}{U}-th\right)\mu. B ↓ , t ( U , μ ) = ( c + t h ) U + ( U V ↓ t − 1 ( 0 , U ) − t h ) μ . For x ≥ χ t ↓ ( μ ) x \ge\chi_t^{\downarrow}(\mu) x ≥ χ t ↓ ( μ ) μ ≤ q t , j \mu\le q_{t,j} μ ≤ q t , j Φ t − K t μ / q t , j \Phi_t-K_t\mu/q_{t,j} Φ t − K t μ / q t , j Φ t \Phi_t Φ t (13) on this action interval. Evaluating it at x = max { y , χ t ↓ ( μ ) } x=\max\{y,\chi_t^{\downarrow}(\mu)\} x = max { y , χ t ↓ ( μ )} z n < U z_n<U z n < U V ^ ↓ n ( 0 , U ) = ( c + n h ) U − P ( U − z n ) \widehat{V}_{\downarrow}^{n}(0,U)=(c+nh)U-P(U-z_n) V ↓ n ( 0 , U ) = ( c + nh ) U − P ( U − z n )
A t = ( c − h − P ( 1 − z n / U ) ) + = K t ( P ( z n / U − α ) + b K t − 1 ) + . A_t=(c-h-P(1-z_n/U))_+=K_t\left(\frac{P(z_n/U-\alpha)+b}{K_t}-1\right)_+. A t = ( c − h − P ( 1 − z n / U ) ) + = K t ( K t P ( z n / U − α ) + b − 1 ) + . It vanishes when the highest target is below U U U z n = U z_n=U z n = U U U U □ \square □
Proof of Corollary 5. Use λ t \lambda_t λ t u t = s t , t ⋆ / U u_t=s_{t,t}^{\star}/U u t = s t , t ⋆ / U
f t ( u ) = Φ t − 1 ( u − α ) + b K t , f t ( u ) − u = ( c − h ) u + ( b − α Φ t − 1 ) ( 1 − u ) K t . f_t(u)=\frac{\Phi_{t-1}(u-\alpha)+b}{K_t},\qquad f_t(u)-u=\frac{(c-h)u+(b-\alpha\Phi_{t-1})(1-u)}{K_t}. f t ( u ) = K t Φ t − 1 ( u − α ) + b , f t ( u ) − u = K t ( c − h ) u + ( b − α Φ t − 1 ) ( 1 − u ) . Since b − α Φ t − 1 ≥ b − h − α ( c − w ) > 0 b-\alpha\Phi_{t-1}\ge b-h-\alpha(c-w)>0 b − α Φ t − 1 ≥ b − h − α ( c − w ) > 0 u t u_t u t u 1 = b / ( b + h ) ≥ α u_1=b/(b+h)\ge\alpha u 1 = b / ( b + h ) ≥ α α > 0 \alpha>0 α > 0 K t K_t K t min { c − h , b − h − α ( c − w ) } / sup r ≥ 2 K r > 0 \min\{c-h,b-h-\alpha(c-w)\}/\sup_{r\ge2}K_r>0 min { c − h , b − h − α ( c − w )} / sup r ≥ 2 K r > 0 u t u_t u t ℓ ≥ 1 \ell\ge1 ℓ ≥ 1 r t , ℓ = s t , t − ℓ ⋆ / U r_{t,\ell}=s_{t,t-\ell}^{\star}/U r t , ℓ = s t , t − ℓ ⋆ / U r t , 0 = u t r_{t,0}=u_t r t , 0 = u t
r t , ℓ ↓ = λ t ( r t − 1 , ℓ − 1 ↓ − α ) + . r_{t,\ell}^{\downarrow}=\lambda_t(r_{t-1,\ell-1}^{\downarrow}-\alpha)_+. r t , ℓ ↓ = λ t ( r t − 1 , ℓ − 1 ↓ − α ) + . Since u t u_t u t λ t \lambda_t λ t N t + 1 ↓ ∈ { N t ↓ , N t ↓ + 1 } N_{t+1}^{\downarrow}\in\{N_t^{\downarrow},N_t^{\downarrow}+1\} N t + 1 ↓ ∈ { N t ↓ , N t ↓ + 1 }
The limiting age recursion starts at one and uses λ ∞ \lambda_\infty λ ∞ ℓ \ell ℓ
λ ∞ ℓ − 1 > α 1 − β λ ∞ . \lambda_\infty^{\ell-1}>\frac{\alpha}{1-\beta\lambda_\infty}. λ ∞ ℓ − 1 > 1 − β λ ∞ α . Exactly ages 1 , … , N ∞ 1,\ldots,N_\infty 1 , … , N ∞ s t , t − N ∞ ↓ > 0 s_{t,t-N_\infty}^{\downarrow}>0 s t , t − N ∞ ↓ > 0 T ↓ s t a b T_{\downarrow}^{\mathrm{stab}} T ↓ stab □ \square □
Numerical Implementation
The master seed is 2026102004.
Demand and histories. The conditional Beta law is discretized on { 0 , … , U } \{0,\ldots,U\} { 0 , … , U } z ∈ [ k , k + 1 ] z\in[k,k+1] z ∈ [ k , k + 1 ] k + 1 k+1 k + 1 z − k z-k z − k k k k ( 1 − δ ) m + δ p B U (1-\delta)m+\delta p_B U ( 1 − δ ) m + δ p B U p B p_B p B c c c N ∈ { 24 , 48 , 72 , 96 } N \in\{24,48,72,96\} N ∈ { 24 , 48 , 72 , 96 }
Policies. M-DRO uses the analytical policy. Rolling CE optimizes assuming constant future demand equal to the latest observation. For training pairs ( m i , d i ) (m_i,d_i) ( m i , d i ) n m d = ∑ i 1 { m i = m , d i = d } n_{md} = \sum_i 1\{m_i=m,d_i=d\} n m d = ∑ i 1 { m i = m , d i = d } n m = ∑ d n m d n_m = \sum_d n_{md} n m = ∑ d n m d p d I I D p_d^{\mathrm{IID}} p d IID P m d c o n d P_{md}^{\mathrm{cond}} P m d cond
p d I I D = ∑ i 1 { d i = d } + 0.5 / ( U + 1 ) N + 0.5 , P m d c o n d = n m d n m ( n m > 0 ) . p_d^{\mathrm{IID}} = \frac{\sum_i 1\{d_i=d\} + 0.5/(U+1)}{N+0.5}, \qquad P_{md}^{\mathrm{cond}} = \frac{n_{md}}{n_m} \quad(n_m>0). p d IID = N + 0.5 ∑ i 1 { d i = d } + 0.5/ ( U + 1 ) , P m d cond = n m n m d ( n m > 0 ) . If n m = 0 n_m=0 n m = 0 S ( m ) = arg min r : n r > 0 ∣ r − m ∣ S(m)=\arg\min_{r:n_r>0}|r-m| S ( m ) = arg min r : n r > 0 ∣ r − m ∣ P m d c o n d = ∑ r ∈ S ( m ) n r d / ∑ r ∈ S ( m ) n r P_{md}^{\mathrm{cond}}=\sum_{r\in S(m)}n_{rd}/\sum_{r\in S(m)}n_r P m d cond = ∑ r ∈ S ( m ) n r d / ∑ r ∈ S ( m ) n r ℓ = 3 \ell=3 ℓ = 3 s = 2 s=2 s = 2
p d p o o l = ∑ i 1 { d i = d } + 0.05 N + 0.05 ( U + 1 ) , p_d^{\mathrm{pool}} = \frac{\sum_i 1\{d_i=d\}+0.05}{N+0.05(U+1)}, p d pool = N + 0.05 ( U + 1 ) ∑ i 1 { d i = d } + 0.05 , P m d N V = ∑ i e − ( m − m i ) 2 / ( 2 ℓ 2 ) 1 { d i = d } + s p d p o o l ∑ i e − ( m − m i ) 2 / ( 2 ℓ 2 ) + s . P_{md}^{\mathrm{NV}} = \frac{\sum_i e^{-(m-m_i)^2/(2\ell^2)}1\{d_i=d\}+s p_d^{\mathrm{pool}}}{\sum_i e^{-(m-m_i)^2/(2\ell^2)}+s}. P m d NV = ∑ i e − ( m − m i ) 2 / ( 2 ℓ 2 ) + s ∑ i e − ( m − m i ) 2 / ( 2 ℓ 2 ) 1 { d i = d } + s p d pool . It orders up to the smallest fitted ( b − c + α w ) / ( b + h ) (b-c+\alpha w)/(b+h) ( b − c + α w ) / ( b + h ) { 0 , 0.5 , … , U } \{0,0.5,\ldots,U\} { 0 , 0.5 , … , U } U / 2 U/2 U /2
Evaluation. Each replication generates 250 test paths shared by all policies. Common random numbers couple settings; horizons 12, 24, and 48 share path prefixes. For Table 4 , each replication draws δ ∼ Unif ( 0.075 , 0.30 ) \delta\sim\operatorname{Unif}(0.075,0.30) δ ∼ Unif ( 0.075 , 0.30 ) κ ∼ Unif ( 0.5 , 2 ) \kappa\sim\operatorname{Unif}(0.5,2) κ ∼ Unif ( 0.5 , 2 ) p B p_B p B N = 48 N=48 N = 48 κ / 2 \kappa/2 κ /2 2 κ 2\kappa 2 κ 4 κ 4\kappa 4 κ δ + 0.10 \delta+0.10 δ + 0.10 δ − 0.05 \delta-0.05 δ − 0.05 δ + 0.10 \delta+0.10 δ + 0.10 κ / 2 \kappa/2 κ /2 2 κ 2\kappa 2 κ δ = 1 \delta=1 δ = 1 p B p_B p B T T T T − 1 T-1 T − 1 1 − lost demand / total demand 1-\text{lost demand}/\text{total demand} 1 − lost demand / total demand t t t 100 ( 1 − C ‾ M − D R O / C ‾ π ) 100(1-\overline{C}^{\mathrm{M-DRO}}/\overline{C}^{\pi}) 100 ( 1 − C M − DRO / C π )
