Dependence-structure sensitivity of optimal inventory-pooling policy choice beyond two demand sources (vine-copula extension of Derbel–Hachicha–Aljuaid)

Derbel, Hachicha and Aljuaid (2021) showed by simulation that, in a two-location inventory-pooling system with beta marginal demands, the choice among three pooling policies (inventory centralization, regular lateral transshipments, and an independent decentralized system) is sensitive to the copula used to model the dependence between the two demands: the optimal policy diverged across four copula families (Gaussian, Gumbel, Clayton, Frank) in about 9% of scenarios, and misspecifying the marginals as normal added a 2.2%–4.3% decision error. Their study is confined to two decentralized demand locations and two centralized stocking points (n = m = 2), where a bivariate copula suffices, and assumes demands are independent of lead times and cost variables. The open problem is to extend this copula-based sensitivity analysis of pooling-policy choice to systems with more than two demand sources, which requires vine copulas (pair-copula constructions) for the n-variate demand vector, to characterize how the degree and pattern of dependence-induced divergence in the optimal policy scale with the number of locations; characterizing how dependence between demands and lead times or cost variables affects that divergence is a further direction named in the source.

Background

Inventory pooling is the coordination of stock across several demand locations so that shared or reallocated inventory can cover aggregate demand variability: the two classical mechanisms are centralization of stocking at fewer facilities and lateral transshipments, in which an overstocked location ships to an understocked one. In the multi-location newsvendor model (a single-period stocking problem with random demand and costs for over- and under-stocking), Eppen (1979) showed that centralizing orders reduces expected cost when demands are independent normal random variables — the classical risk-pooling effect. Whether pooling still pays off under general statistical dependence between demands was settled by Corbett and Rajaram (2006), who generalized the pooling effect to non-normal dependent demand. A copula is a joint distribution function with standard uniform marginals that couples arbitrary marginal distributions into a multivariate distribution (Sklar's theorem); unlike linear correlation, a copula can encode non-linear and tail dependence, meaning co-movement of extreme values. Common bivariate families are the Gaussian (elliptical, linear-type dependence without tail dependence) and the Archimedean Gumbel, Clayton, and Frank families, which capture, respectively, upper-tail, lower-tail, and symmetric dependence. Aydın, Güler and Kayış (2012), Wanke (2014), and Silbermayr, Jammernegg and Kischka (2017) used copulas to analyze pooling benefits and their sensitivity to the dependence structure, under service or environmental constraints. Derbel, Hachicha and Aljuaid (2021) pushed this further by studying the joint sensitivity of the pooling-policy choice itself to both the copula and the marginal distributions. Their setting is a continuous-review cost model (inventory positions are monitored continuously rather than at fixed review periods) in the consolidation-effect tradition (the consolidation effect is the cost reduction obtained when stocks are held jointly rather than separately): $n$ decentralized demand locations (markets) are served by $m$ centralized stocking points, and the total cost is the sum of cycle stock cost, safety stock cost, and distribution cost under a demand-satisfaction (service-level) constraint. Two allocation rules are compared — the Tyagi–Das rule, in which demands are pooled by centralization of inventory, and the Ballou–Burnetas rule, under which regular transshipments or an independent system arise — yielding three competing policies: inventory centralization, regular transshipment, and an independent system. In a numerical study with $n = m = 2$, beta marginal demands with varying skewness, four copulas (Gaussian, Gumbel, Clayton, Frank), and roughly 10,000 scenarios optimized by a differential-evolution algorithm (a population-based stochastic search heuristic), they found: (i) the optimal pooling policy differed across the four copulas in about 9% of scenarios; (ii) wrongly assuming normal marginals while the dependence model is adequate added a 2.2%–4.3% decision error, growing with demand asymmetry; and (iii) high positive demand correlation favored the independent system, while weak correlation favored transshipment. A structural limitation remains: bivariate copulas cannot represent the dependence among three or more demands. Vine copulas (also called pair-copula constructions; the canonical and drawable variants are jointly referred to as C-vine/D-vine or CD-vine) build a high-dimensional joint distribution as a cascade of bivariate copulas, and are the standard tool for dependence modeling in dimension three and higher. The demands in the 2021 study were also generated independently of lead times and of all cost variables, an assumption the authors themselves flag as unrealistic.

Problem Statement

In the continuous-review inventory-pooling framework of Derbel, Hachicha and Aljuaid (2021) — $n$ decentralized demand locations served by $m$ centralized stocking points, total cost equal to cycle stock cost plus safety stock cost plus distribution cost under a demand-satisfaction constraint, with the three competing pooling policies of inventory centralization (Tyagi–Das allocation rule), regular lateral transshipments versus an independent system (Ballou–Burnetas allocation rule) — extend the copula-based sensitivity analysis of the optimal pooling-policy choice beyond the case $n = m = 2$ that the source studied. Specifically: (1) model the demand vector of the $n > 2$ decentralized locations (and, where applicable, $m > 2$ stocking points) with vine copulas — pair-copula constructions such as C-vine and D-vine, the CD-vine tool the source proposes for the case where the number of centralized and decentralized locations exceeds two — over flexible marginals such as the beta family used in the source, specifying how the vine structure and bivariate pair families are selected; (2) quantify, over the scenario space of marginal shapes, dependence strengths, and cost parameters, how frequently and how strongly the chosen optimal policy (centralization versus regular transshipment versus independent system) differs across plausible dependence structures and vine specifications, and determine whether the cross-copula divergence rate observed at $n = m = 2$ (about 9% of scenarios, with an additional 2.2%–4.3% error from normal-marginal misspecification) increases, as the source's authors expect, or exhibits different patterns as $n$ and $m$ grow; (3) characterize how this divergence depends on marginal misspecification and demand skewness as dimension grows; and (4) as a further direction suggested separately by the source, relax the source's assumption that demands are independent of lead times and of the other variables entering the cost program, by incorporating dependence between demands and lead times (and other cost variables) into the optimization, and determine how such dependence changes the optimal pooling-policy selection. Components (1)–(3) share one scientific objective: characterizing the robustness of the pooling-policy decision to dependence-modeling choices in inventory-pooling networks as the number of demand sources grows beyond two; component (4) is a related additional direction named in the source, addressing the robustness of the policy choice to dependence between demands and lead times or cost variables.

References

1. Derbel, M.; Hachicha, W.; Aljuaid, A.M. (2021). Sensitivity Analysis of the Optimal Inventory-Pooling Strategies According to Multivariate Demand Dependence. Symmetry 13(2), 328. DOI 10.3390/sym13020328. https://doi.org/10.3390/sym13020328
2. Eppen, G.D. (1979). Note—Effects of Centralization on Expected Costs in a Multi-Location Newsboy Problem. Management Science 25(5), 498–501. DOI 10.1287/mnsc.25.5.498. https://doi.org/10.1287/mnsc.25.5.498
3. Corbett, C.J.; Rajaram, K. (2006). A Generalization of the Inventory Pooling Effect to Nonnormal Dependent Demand. Manufacturing & Service Operations Management 8(4), 351–358. DOI 10.1287/msom.1060.0117. https://doi.org/10.1287/msom.1060.0117
4. Ballou, R.H.; Burnetas, A. (2003). Planning Multiple Location Inventories. Journal of Business Logistics 24(2), 65–89. DOI 10.1002/j.2158-1592.2003.tb00046.x. https://doi.org/10.1002/j.2158-1592.2003.tb00046.x
5. Aydın, B.; Güler, K.; Kayış, E. (2012). A Copula Approach to Inventory Pooling Problems with Newsvendor Products. In Handbook of Newsvendor Problems: Models, Extensions and Applications (International Series in Operations Research & Management Science), Springer, pp. 81–101. DOI 10.1007/978-1-4614-3600-3_3. https://doi.org/10.1007/978-1-4614-3600-3_3
6. Wanke, P.F. (2014). Consolidation effects: assessing the impact of tail dependence on inventory pooling using copulas. International Journal of Inventory Research 2(3), 174. DOI 10.1504/ijir.2014.069188. https://doi.org/10.1504/ijir.2014.069188
7. Silbermayr, L.; Jammernegg, W.; Kischka, P. (2017). Inventory pooling with environmental constraints using copulas. European Journal of Operational Research 263(2), 479–492. DOI 10.1016/j.ejor.2017.04.060. https://doi.org/10.1016/j.ejor.2017.04.060
8. Huerta, M.; Leiva, V.; Rojas, F.; Wanke, P.; Cabezas, X. (2023). A Methodology for Consolidation Effects of Inventory Management with Serially Dependent Random Demand. Processes 11(7), 2008. DOI 10.3390/pr11072008. https://doi.org/10.3390/pr11072008
9. Yang, C.; Hu, Z.; Zhou, S.X. (2021). Multilocation Newsvendor Problem: Centralization and Inventory Pooling. Management Science 67(1), 185–200. DOI 10.1287/mnsc.2019.3532. https://doi.org/10.1287/mnsc.2019.3532
10. Govindarajan, A.; Sinha, A.; Uichanco, J. (2021). Distribution-Free Inventory Risk Pooling in a Multilocation Newsvendor. Management Science 67(4), 2272–2291. DOI 10.1287/mnsc.2020.3719. https://doi.org/10.1287/mnsc.2020.3719
11. Yilmaz, O. (2025). A Literature Review on Inventory Pooling with Applications. Sustainability 17(2), 797. DOI 10.3390/su17020797. https://doi.org/10.3390/su17020797