Interval Inventory Modeling

• Interval inventory situations are a modeling framework that uses closed demand intervals to capture uncertainty in supply chain environments.
• The approach extends classical inventory models like the EOQ by incorporating interval-valued variables to optimize order quantities and procurement cycles.
• Cost allocation rules such as the interval SOC-rule and Shapley rule ensure fair and robust distribution of joint costs among cooperative agents.

Interval inventory situations constitute a modeling and decision-theoretic framework in which core system parameters—especially demand—are represented as closed intervals to directly account for operational uncertainty. This approach generalizes classical inventory theory—including the Economic Order Quantity (EOQ) model and cooperative cost-sharing structures—to settings characterized by demand ambiguity, thereby providing robust methodologies for both decision-making and equitable cost allocation. Interval inventory models have found particularly strong relevance in collaborative environments and sectors such as supply chain coordination, where agents (e.g., retailers, airports) face shared procurement or replenishment questions under incomplete demand information (Gonçalves-Dosantos et al., 29 Jul 2025).

1. Interval Inventory Modeling: Motivation and Formulation

In interval inventory situations, each agent or product is assigned a demand interval: $$d_{(i)} = [\underline{d}_{(i)}, \overline{d}_{(i)}]$$ where $\underline{d}_{(i)}$ and $\overline{d}_{(i)}$ give lower and upper bounds for agent $i$'s demand over the planning horizon (Gonçalves-Dosantos et al., 29 Jul 2025). As a consequence, principal decision variables—order quantities, number of procurement cycles, and replenishment timings—are mapped to interval values. This “intervalization” is a direct response to volatility, nonstationarity, or lack of precise forecasting, all of which are pervasive in modern supply chains.

The interval framework leads to an interval-valued objective function. For instance, in cooperative multi-agent EOQ systems with joint ordering: $$m_{(i)} = [\underline{m}_{(i)}, \overline{m}_{(i)}]$$ denotes the optimal number of orders for agent $i$, itself an interval dependent on $d_{(i)}$. The minimal joint ordering cost for a coalition $N$ becomes: $w^{I}(N) = 2a \sqrt{\sum_{i \in N} m_{(i)}^2}$ where $a$ is the fixed ordering cost (Gonçalves-Dosantos et al., 29 Jul 2025).

This modeling ensures that analysis, planning, and subsequent allocation of costs are robust to the full range of plausible demand realizations.

2. Allocation Rules for Cooperative Interval Inventory Decisions

Interval inventory situations raise nontrivial questions about how to equitably allocate joint costs when both total cost and individual contributions are themselves interval-valued. Two principal rules are central:

Interval SOC-Rule (Share of Cost Rule)

This method generalizes the deterministic proportional core allocation to the interval setup. Each agent receives an allocation—typically an interval—proportional to the square of their number of orders: $$\Gamma_{i}(N, a, \{m_{j}\}) = \frac{2a\, m_i^2}{\sqrt{\sum_{j\in N} m_j^2}}$$ where $m_{i}$ is agent $i$'s (possibly interval) optimal order number. The entire cost is allocated, and (Crucially) an agent with $m_{j} = [0, 0]$ receives zero allocation (Inactive Agent Exemption, IAE), and allocation is additive over independent inventory situations (Transfer-Based Additivity, TBA) (Gonçalves-Dosantos et al., 29 Jul 2025).

Interval Shapley Rule

The interval Shapley value is constructed as the set of all possible classical Shapley allocations for the extreme points (lower and upper bounds) of the interval game: $$Sh_i(N, a, \{m_j\}) = [ \varphi_i(N, \underline{w}^I),\; \varphi_i(N, \overline{w}^I) ]$$ where $\varphi_i$ is the classical Shapley value: $$\varphi_i(N, w^I) = \frac{1}{n!} \sum_{\sigma \in \mathcal{P}(N)} \big(w^I (P^\sigma(i) \cup \{i\}) - w^I (P^\sigma(i))\big)$$ with $P^\sigma(i)$ the set of agents preceding $i$ in permutation $\sigma$.

This rule provides each agent with an interval reflecting their average marginal cost contribution under all orderings, capturing an axiomatic fairness notion, especially under monotonic interval games (where marginal contribution is nondecreasing with coalition size) (Gonçalves-Dosantos et al., 29 Jul 2025).

3. Theoretical Characterization and Properties

Efficiency and Stability

Both methods are efficient: $$\sum_{i \in N} \Gamma_{i}(N, a, \{m_{j}\}) = 2a \sqrt{\sum_{j\in N} m_{j}^2}$$ and every agent's allocation lies within the individual cost interval.

• SOC-rule is uniquely characterized by IAE and TBA, ensuring allocations are robust and swiftly computable, with allocations respecting coalition stability (Cross-Coalitional Acceptable, CCA property).
• Interval Shapley rule is uniquely defined by the Balanced Cost (BC) property: the marginal impact on one agent's allocation due to changes in another's interval is symmetric and consistent across permutations in the agent set.

Both are in the core for convex (or concave) settings, implying no subgroup has an incentive to deviate (Gonçalves-Dosantos et al., 29 Jul 2025).

4. Empirical Application: Spanish Airports Case Study

A practical case evaluates the allocations for pooled duty-free perfume inventory management across seven Spanish airports, with passenger-based monthly demand intervals: $$\text{For } i \text{ (Madrid)}: d_{(i)} = [175{,}000,\, 325{,}000]$$ and similar intervals for the others.

• The minimal cost, $w^I(N)$, was calculated.
• Individual allocations via SOC and Shapley methods were compared to the costs if airports operated independently.

Table: Selected outcomes for Madrid

Metric	Interval Lower Bound	Interval Upper Bound
Individual Cost	76,470	110,232
SOC Allocation	54,608	90,577
Shapley Allocation	56,088	90,591

Both allocation rules reduced the interval width compared to the individual approach, and the interval SOC and Shapley rules produced closely similar results (Gonçalves-Dosantos et al., 29 Jul 2025).

5. Implications and Advancements

Decision Robustness and Fairness

Interval-based modeling promotes robust decision-making by ensuring that cost allocations remain fair and stable over all plausible realizations of uncertain demand. This is particularly important in environments susceptible to shocks or high volatility.

Computational Considerations

• The SOC-rule is computationally tractable, relying only on order intervals.
• The interval Shapley rule, while more intensive due to required averaging over all permutations and interval bounds, provides an axiomatized fairness allocation, which can be approximated via random sampling for large systems.

Policy Applicability

• Facilitates collaborative procurement among agents with heterogeneous and uncertain demand profiles.
• The reduction in the interval width of allocated costs provides actionable predictability for budgeting and planning in uncertain environments.
• Applicability spans from classic retail and supply chain pooling to more general resource-sharing systems under parameter uncertainty.

6. Research Directions

• Further Generalization: Incorporating fuzzy intervals or probabilistic intervals to capture higher-order uncertainties.
• Algorithmic Refinement: Improving computational efficiency for large cooperative agent sets, especially for the interval Shapley rule.
• Dynamics and Extensions: Adapting allocation rules to multi-period and time-varying interval demand settings, as well as integrating transportation, lead time variability, or multi-product systems.
• Core-Like Solution Concepts: Developing and analyzing new solution concepts beyond the core, tailored to interval and robust cooperative games.

7. Summary and Future Outlook

Interval inventory situations provide a rigorous, flexible, and interpretable foundation for cooperative decision-making in the presence of demand uncertainty. The interval SOC and Shapley rules serve as principled allocation mechanisms, each supported by uniqueness theorems and stability properties. The practical efficacy of these models, validated by real-world case studies, suggests wide relevance for robust and fair cost-sharing in modern supply chains and collaborative inventory systems (Gonçalves-Dosantos et al., 29 Jul 2025). Future research is poised to deepen both the mathematical underpinnings and practical reach of interval-based inventory analysis.