An Inventory System with Two Supply Modes and Lévy Demand

TL;DR

The paper develops a two-mode replenishment inventory model driven by a general spectrally positive Lévy demand process and randomized discounted replenishment opportunities, proving that a hybrid barrier policy is optimal under suitable convex holding-cost assumptions.The authors leverage scale-function methods from fluctuation theory to obtain semi-explicit cost and value-function formulas for the policy, and establish existence of optimal barriers via analysis of Gamma and gamma functions tied to the holding-cost slope.A verification lemma ensures optimality of the identified policy among all admissible policies, with a discussion of a pure-discount policy in cases where the holding-cost slope condition fails; a numerical study corroborates the theory and illustrates barrier selection and cost savings.The work contributes a flexible, analytically tractable framework for inventory control with random discount opportunities and Lévy-demand, enabling efficient computation of optimal policies and quantitative evaluation of hybrid strategies in practical settings.

Abstract

This study considers a continuous-review inventory model for a single item with two replenishment modes. Replenishments may occur continuously at any time with a higher unit cost, or at discrete times governed by Poisson arrivals with a lower cost. From a practical standpoint, the model represents an inventory system with random deal offerings. Demand is modeled by a spectrally positive Lévy process (i.e., a Lévy process with only positive jumps), which greatly generalizes existing studies. Replenishment quantities are continuous and backorders are allowed, while lead times, perishability, and lost sales are excluded. Using fluctuation theory for spectrally one-sided Lévy processes, the optimality of a hybrid barrier policy incorporating both kinds of replenishments is established, and a semi-explicit expression for the associated value function is computed. Numerical analysis is provided to support the optimality result.

Figures (6)

• Figure 1: Sample path of a standard Brownian motion (blue) and the replenished process (black). Discounted replenishment opportunities (denoted by $T(1)$, $T(2)$, $T(3)$) are shown as vertical dashed lines. At $T(1)$ and $T(3)$, the inventory level is below $b$, triggering replenishment. At $T(2)$ the inventory level exceeds $b$, so no replenishment occurs.
• Figure 2: Plot of $b \mapsto \Gamma(a, b)$ for $a = a^* - 0.010, a^* - 0.005, a^* + 0.005, a^* + 0.010$ (blue dashed) and $b \mapsto \Gamma(a^*, b)$ (red solid).
• Figure 3: Left: Plot of $v_{a^*, b}$ (blue dashed) for $b = b^* - 0.2, b^*, b^* + 0.2, b^* + 0.4$, with $(a^*, v_{a^*, b}(a^*))$ (lime triangle), $(a^*, v_{a^*, b^*}(a^*))$ (red triangle), $(b, v_{a^*, b}(b))$ (lime square), and $(b^*, v_{a^*, b^*}(b^*))$ (red square). Right: Plot of $v_{a, b^*}$ (blue dashed) for $a = a^* - 0.2, a^* - 0.1, a^*, a^* + 0.1, a^* + 0.2$, with $(a, v_{a, b^*}(a))$ (lime triangle), $(a^*, v_{a^*, b^*}(a^*))$ (red triangle), $(b^*, v_{a, b^*}(b^*))$ (lime square), and $(b^*, v_{a^*, b^*}(b^*))$ (red square). The optimal total cost function $v_{a^*, b^*}$ is indicated by red curves.
• Figure 4: Left: Plot of $v_{a^*, b}$ for $b = b^* - 0.2, b^* - 0.1, b^*, b^* + 0.1, b^* + 0.2$. Right: Plot of $v_{a, b^*}$ for $a = a^* - 1, a^* - 0.5, a^*, a^* + 0.5, a^* + 1$.
• Figure 5: Value functions $v_{a^*, b^*}$ for $K_c \to K_p$ (blue dashed), with $(a^*, v_{a^*, b^*}(a^*))$ and $(b^*, v_{a^*, b^*}(b^*))$ marked by lime triangles and squares, respectively. The classical value function $v_{a^\ddagger}$ from yamazaki_inventory_2017 is shown in red.

Theorems & Definitions (37)

• Lemma 3.1: inventory-holding cost
• Lemma 3.2: replenishment cost
• Lemma 3.3
• Definition 4.1
• Lemma 4.1
• Lemma 4.2
• Proposition 4.3
• proof
• Lemma 5.1
• Lemma 5.2