In this paper, an Economic Production Quantity model for deteriorating items with time-dependent demand and shortages including partially back-ordered is developed under a cloudy-fuzzy environment. At first, we develop a crisp model by considering linearly time-dependent demand with constant deterioration rate, constant inflation rate and shortages under partially back-ordered, then we fuzzify the model to archive a decision under the cloudy-fuzzy (extension of fuzziness) demand rate, inflation rate, deterioration rate and the partially back-ordered rate which are followed by their practical applications. In this model, we assume ambiances where cloudy normalized triangular fuzzy number is used to handle the uncertainty in information which is coming from the data. The main purpose of our study is to defuzzify the total inventory cost by applying Ranking Index method of fuzzy numbers as well as cloudy-fuzzy numbers and minimize the total inventory cost of crisp, fuzzy, and cloudy-fuzzy model. Finally, a comparative analysis among crisp, fuzzy and cloudy-fuzzy total cost is carried out in this paper. Numerical example, sensitivity analysis, and managerial insights are elaborated to justify the usefulness of the new approach. A comparative inquiry of the numerical result with a new existing paper is also carried out. This paper ends with a conclusion along with advantages and limitations of our solution approach, and an outlook towards possible future studies.

中文翻译：

---

多云-模糊环境下的带通货膨胀的滞销库存模型

本文建立了一种在多云，模糊的环境下，针对具有时变需求和短缺（包括部分缺货）的变质物品的经济生产数量模型。首先，我们通过考虑线性的，与时间相关的需求（具有恒定的恶化率，恒定的通货膨胀率和部分缺货的情况下的短缺）来开发清晰的模型，然后对模型进行模糊处理，以在多云-模糊（模糊性的扩展）下归档决策。需求率，通货膨胀率，恶化率和部分滞后订货率，随后是它们的实际应用。在此模型中，我们假设使用多云归一化三角模糊数来处理来自数据的信息不确定性的环境。我们研究的主要目的是通过应用模糊数和多云-模糊数的排名指数方法对总库存成本进行模糊化处理，并最大限度地减少清晰，模糊和多云-模糊模型的总库存成本。最后，对脆性，模糊性和模糊性总成本进行了比较分析。详细阐述了数值示例，敏感性分析和管理洞察力，以证明新方法的有用性。还与新的现有论文进行了数值结果的比较查询。本文最后给出了结论，以及我们解决方案方法的优点和局限性，并对未来可能的研究进行了展望。最后，对脆性，模糊性和模糊性总成本进行了比较分析。详细阐述了数值示例，敏感性分析和管理洞察力，以证明新方法的有用性。还与新的现有论文进行了数值结果的比较查询。本文最后总结了我们的解决方案方法的优点和局限性，并对未来可能的研究进行了展望。最后，对脆性，模糊性和模糊性总成本进行了比较分析。详细阐述了数值示例，敏感性分析和管理洞察力，以证明新方法的有用性。还与新的现有论文进行了数值结果的比较查询。本文最后给出了结论，以及我们解决方案方法的优点和局限性，并对未来可能的研究进行了展望。