Interval SOC-Rule in Uncertain Inventory Management

• Interval SOC-rule is a cost allocation method for cooperative inventory management that uses interval arithmetic to handle demand uncertainty and enable equitable cost-sharing.
• The rule employs a proportional allocation formula with unique axiomatic properties, notably Inactive Agent Exemption and Transfer-Based Additivity, ensuring fairness and efficiency.
• A practical case study at Spanish airports demonstrates its computational simplicity and real-world efficacy in managing inconsistent, interval-based demand.

The interval SOC-rule is a cost allocation principle for cooperative inventory management under demand uncertainty, specifically developed for "interval inventory situations" in which the relevant parameters, such as demand rates, are represented by closed intervals rather than deterministic values. This rule extends classical deterministic allocation results to interval settings and enables robust and equitable sharing of joint ordering costs among coalition members. The interval SOC-rule is characterized both by its proportional allocation formula and by its unique axiomatic properties relative to other rules, most notably the interval Shapley rule. Its efficacy and real-world utility are demonstrated through rigorous theoretical analysis and an applied case paper in the coordination of perfume inventories at major Spanish airports (Gonçalves-Dosantos et al., 29 Jul 2025).

1. Mathematical Formulation and Foundation

The interval SOC-rule generalizes the deterministic allocation approach from classical EOQ-based inventory games to contexts where ordering parameters (typically each agent i's optimal order frequency, denoted $m_i$) are intervals reflecting uncertain demand. Let $N$ be the grand coalition of agents, $a$ the fixed ordering cost, and $m_i = [\underline{m}_i, \overline{m}_i]$ the interval-valued order frequency for agent $i$. The total minimum joint ordering cost under the interval setting is

$$w^{\mathcal{I}}(N) = 2a \sqrt{\sum_{i \in N} m_i^2}$$

For an agent $j \in N$, the interval SOC-rule allocates cost as

$$\Gamma_j(N, a, \{m_i\}) = \frac{2a \cdot m_j^2}{\sqrt{\sum_{i \in N} m_i^2}}$$

where both $m_j^2$ and the denominator are interpreted in the interval-arithmetic sense. The closed-form expression allows for straightforward computation, conditional on well-posedness under the chosen interval arithmetic.

2. Key Theoretical Properties

The interval SOC-rule is rigorously defined by, and is uniquely characterized with respect to, two key axiomatic properties:

• Inactive Agent Exemption (IAE): If an agent $j$ has order frequency $m_j = [0,0]$, then $\Gamma_j(N, a, \{m_i\}) = [0,0]$.
• Transfer-Based Additivity (TBA): When two interval inventory situations are combined, the allocation should satisfy:

$$\sqrt{\sum_{i \in N} (m_i^2 + \hat{m}_i^2)} \Psi_j(N, a, \{ \sqrt{m_i^2 + \hat{m}_i^2} \}) = \sqrt{\sum_{i \in N} m_i^2} \Psi_j(N, a, \{m_i\}) + \sqrt{\sum_{i \in N} \hat{m}_i^2} \Psi_j(N, a, \{\hat{m}_i\})$$

• Cross-Coalition Acceptability (CCA): For all $S \subseteq N$, $$\sum_{j \in S} \Gamma_j(N, a, \{m_i\}) \preceq 2a \sqrt{\sum_{j \in S} m_j^2}$$, preventing subgroups from paying more than their joint cost if forming a separate coalition.
• Efficiency: The sum of all agents’ allocations exactly matches the joint cost.

The rule stands out as the unique allocation method satisfying both IAE and TBA (Theorem 3.1 in (Gonçalves-Dosantos et al., 29 Jul 2025)).

3. Implementation and Computational Aspects

The interval SOC-rule's proportional structure leads to computational simplicity relative to rules that require averaging over all coalition orderings. Given a list of $n$ agents with interval-valued order frequencies, the allocation requires:

• Computing $m_j^2$ for each agent, typically as $[\underline{m}_j^2, \overline{m}_j^2]$;
• Summing these squares across all agents to form the denominator;
• Dividing (with proper interval handling), yielding the allocation interval for each agent.

Well-posedness of interval division is assumed; otherwise, tie-breaking or conservative arithmetic can be used. The rule is fully efficient and remains interpretable even as the number of agents increases.

4. Comparison with the Interval Shapley Rule

Both the interval SOC-rule and the interval Shapley rule are efficiency-preserving and coalition-acceptable, but diverge in several ways:

Rule	Allocation Expression	Axiomatic Basis	Computational Complexity
Interval SOC-rule	$\frac{2a m_j^2}{\sqrt{\sum_i m_i^2}}$	IAE, TBA	Closed-form, linear in $n$
Interval Shapley rule	$$[\phi_j(N, \underline{w}^{\mathcal{I}}),\; \phi_j(N, \overline{w}^{\mathcal{I}})]$$ (permutations)	Balanced Cost (BC)	Requires averaging over all coalitional orders

The interval SOC-rule yields, in applications, cost shares with shorter (i.e., more precise) intervals for high-demand agents compared to the Shapley rule. The interval Shapley rule distributes cost by averaging all marginal contributions, leading to possible higher computational costs and longer allocation intervals.

5. Practical Application: Case Study of Spanish Airports

The practical efficacy of the interval SOC-rule was demonstrated via a detailed case paper of collaborative perfume inventory management at seven Spanish airports. Passenger data (with ±30% seasonal adjustment) was used to define the interval demand for each airport, which translates to interval-valued optimal ordering frequencies and thus interval $m_i$. The total joint cost, $2a\sqrt{\sum_i m_i^2}$, was allocated according to the SOC-rule. Key observations include:

• High-demand airports (e.g., Madrid-Barajas) were allocated larger but more precise cost intervals.
• Low-demand airports received smaller shares, reflecting their lesser "order frequency."
• When compared to the interval Shapley rule, the SOC-rule produced allocation intervals of comparable fairness but generally shorter length, enhancing equity and robustness.

6. Relevance and Advantages Under Uncertainty

The interval SOC-rule is specifically tailored for environments where demand and supply are subject to significant uncertainty, as reflected in interval-based representations of inventory parameters. By distributing costs in direct proportion to interval-valued ordering activity, the rule balances fairness and precision, ensuring that agents “pay for what they might use” while accommodating real-world ambiguity. Its unique axiomatic foundation and computational tractability position it as a preferred method in multi-agent inventory collaborations under uncertain or variable demand (Gonçalves-Dosantos et al., 29 Jul 2025).

7. Summary and Impact

The interval SOC-rule provides a theoretically robust and practically tractable solution to cost allocation in cooperative inventory systems under uncertainty. Its unique axiomatic characterization, computational simplicity, and favorable empirical properties—demonstrated in large-scale applications such as airport retail inventory management—support its adoption for robust, fair, and efficient sharing of costs in interval inventory games. The rule’s proportional formulation, together with properties such as IAE, TBA, and CCA, collectively distinguish it from alternative allocation mechanisms, particularly the interval Shapley rule, making it a substantial contribution to the theory and practice of cooperative inventory management under uncertainty.