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Evaluating the impact of items and cooperation in inventory models with exemptable ordering costs

M. Gloria Fiestras-Janeiro, Ignacio García-Jurado, Ana Meca, Manuel A. Mosquera

TL;DR

This paper addresses EOQ inventory problems in which suppliers waive ordering costs when orders exceed a discount threshold, and analyzes the resulting cooperative behavior among items and firms. It develops four model families (one-item, multi-item, multi-firm, and multi-firm-item EOQ with exemptable costs) and derives optimal replenishment policies, then introduces game-theoretic cost-sharing mechanisms: the Shapley value to assess item-level impact in multi-item settings, and the hd-proportional rule to achieve coalitionally stable cost allocations for multi-firm cases. A two-phase Shapley-proportional rule extends these ideas to multi-firm-item coalitions, with rigorous axiomatic characterization showing stability and fairness within each stage and across unions. The framework is illustrated with a data-driven case study, revealing how different sharing rules yield varying item rankings and allocations, and highlighting the practical value for inventory coordination under supplier discounts. Overall, the work provides a principled, solvable approach to joint ordering and cost allocation in competitive supply chains facing exemptable ordering costs.

Abstract

In this paper we introduce and analyse, from a game theoretical perspective, several multi-agent or multi-item continuous review inventory models in which the buyers are exempted from ordering costs if the price of their orders is greater than or equal to a certain amount. For all models we obtain the optimal ordering policy. We first analyse a simple model with one firm and one item. Then, we study a model with one firm and several items, for which we design a procedure based on cooperative game theory to evaluate the impact of each item on the total cost. Then, we deal with a model with several firms and one item for each firm, for which we characterise a rule to allocate the total cost among the firms in a coalitionally stable way. Finally, we discuss a model with several firms and several items, for which we characterise a rule to allocate the total cost among the firms in a coalitionally stable way and to evaluate the impact of each item on the cost that would be payable to each firm when using the allocation rule. All the concepts and results of this article are illustrated using data from a case study.

Evaluating the impact of items and cooperation in inventory models with exemptable ordering costs

TL;DR

This paper addresses EOQ inventory problems in which suppliers waive ordering costs when orders exceed a discount threshold, and analyzes the resulting cooperative behavior among items and firms. It develops four model families (one-item, multi-item, multi-firm, and multi-firm-item EOQ with exemptable costs) and derives optimal replenishment policies, then introduces game-theoretic cost-sharing mechanisms: the Shapley value to assess item-level impact in multi-item settings, and the hd-proportional rule to achieve coalitionally stable cost allocations for multi-firm cases. A two-phase Shapley-proportional rule extends these ideas to multi-firm-item coalitions, with rigorous axiomatic characterization showing stability and fairness within each stage and across unions. The framework is illustrated with a data-driven case study, revealing how different sharing rules yield varying item rankings and allocations, and highlighting the practical value for inventory coordination under supplier discounts. Overall, the work provides a principled, solvable approach to joint ordering and cost allocation in competitive supply chains facing exemptable ordering costs.

Abstract

In this paper we introduce and analyse, from a game theoretical perspective, several multi-agent or multi-item continuous review inventory models in which the buyers are exempted from ordering costs if the price of their orders is greater than or equal to a certain amount. For all models we obtain the optimal ordering policy. We first analyse a simple model with one firm and one item. Then, we study a model with one firm and several items, for which we design a procedure based on cooperative game theory to evaluate the impact of each item on the total cost. Then, we deal with a model with several firms and one item for each firm, for which we characterise a rule to allocate the total cost among the firms in a coalitionally stable way. Finally, we discuss a model with several firms and several items, for which we characterise a rule to allocate the total cost among the firms in a coalitionally stable way and to evaluate the impact of each item on the cost that would be payable to each firm when using the allocation rule. All the concepts and results of this article are illustrated using data from a case study.
Paper Structure (6 sections, 6 theorems, 47 equations, 2 figures, 6 tables)

This paper contains 6 sections, 6 theorems, 47 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. A basic EOQ problem with exemptable ordering costs
  3. A problem with multiple items
  4. A problem with multiple firms
  5. A problem with multiple firms and items
  6. Concluding remarks and further research

Key Result

Proposition 2

Let $P = (N,\{d_i\}_{i\in N},\{h_i\}_{i\in N},\{c_i\}_{i\in N},a,B)$ be a multi-firm EOQ problem with exemptable ordering costs and let $(N, c^P)$ be its associated cost game. Then, $(N, c^P)$ is strictly subadditive.

Figures (2)

  • Figure 1: The function $CPT$ for $d=15$, $h=8$, $a=10$, $A=10$.
  • Figure 2: Shapley value vs hd-proportional rule: items/firms renumbered in increasing order of their Shapley values.

Theorems & Definitions (15)

  • Example 1
  • Proposition 2
  • Example 3
  • Example 4
  • Example 5
  • Definition 6
  • Definition 7
  • Proposition 8
  • Example 9
  • Theorem 10
  • ...and 5 more