The vendor managed inventory (VMI) is the most commonly known marketing and distribution methodology among the service provider and the retailer in a supply chain environment. The VMI framework's critical feature is to satisfy the demand of the products or items by the supplier rapidly generated from the retailers with the number of orders (order quantities). Due to the business's changing conditions, necessities of the particular product, production cycle, and manufacturing expenses, the VMI system's demand and order quantity are highly uncertain. Therefore, the total cost of a VMI system possesses higher-order uncertainties for a real-world scenario. This paper proposes a novel interval type-2 fuzzy vendor managed inventory (IT2FVMI) system, in which demand and order quantity are represented by the interval type-2 fuzzy numbers. The proposed IT2FVMI model aims to minimize the overall cost for the single vendor-retailer business, merchandise of multi-product, and a centralized warehouse of retailers' inventory managed by the vendor. Since the proposed model is an NP-hard, a particle swarm optimization (PSO) based solution approach is developed for solving it appropriately. Moreover, we have also formulated the classical/crisp VMI model and type-1 fuzzy vendor managed inventory (T1FVMI) model via considering demand and order quantity as the deterministic and the type-1 fuzzy numbers, respectively. The proposed solution technique is capable of solving both crisp VMI and T1FVMI problems. An appropriate real-world application is considered to conduct experimental simulations for the five test problems that displayed the various situations, and the minimum costs of the models are obtained. All the three models are thoroughly analyzed, and from comparison, it is demonstrated that the proposed IT2FVMI model outperforms both T1FVMI and crisp VMI models by providing a more trustworthy solution in terms of minimum cost, statistical analysis, and significantly faster convergence.

供应商管理库存 (VMI) 是供应链环境中服务提供商和零售商之间最广为人知的营销和分销方法。VMI 框架的关键特性是满足供应商从零售商处快速生成的具有订单数量（订单数量）的产品或物品的需求。由于业务条件的变化、特定产品的必要性、生产周期和制造费用，VMI 系统的需求和订单数量具有高度不确定性。因此，VMI 系统的总成本对于实际场景具有更高阶的不确定性。本文提出了一种新的区间 2 型模糊供应商管理库存 (IT2FVMI) 系统，其中需求和订单数量由区间 2 型模糊数表示。提议的 IT2FVMI 模型旨在最小化单一供应商-零售商业务、多产品商品以及由供应商管理的零售商库存集中仓库的总体成本。由于所提出的模型是 NP-hard，因此开发了一种基于粒子群优化 (PSO) 的解决方法来适当地解决它。此外，我们还分别通过将需求和订单数量作为确定性和类型 1 模糊数来制定经典/清晰 VMI 模型和类型 1 模糊供应商管理库存 (T1FVMI) 模型。所提出的求解技术能够解决清晰的 VMI 和 T1FVMI 问题。一个适当的实际应用被认为是对显示各种情况的五个测试问题进行实验模拟，并获得模型的最小成本。对所有三个模型进行了彻底分析，通过比较，表明所提出的 IT2FVMI 模型在最低成本、统计分析和显着更快的收敛方面提供了更值得信赖的解决方案，从而优于 T1FVMI 和清晰的 VMI 模型。