Neutrosophic Inventory Management: A Cost-Effective Approach

1 Introduction

Operations research, often abbreviated as OR, is a multidisciplinary field at the intersection of mathematics, statistics, and decision science. The field of OR is incredibly vast and fascinating. It encompasses numerous applications, making it challenging to fully comprehend. As researchers, we have observed that the most significant issue across all these applications is uncertainty, as highlighted in Table 1.

Now, as we know, our article primarily focuses on IM. Therefore, we have proceeded to discuss IM further. Operations research is intricately connected to inventory management by providing a systematic approach to optimize key inventory decisions. Through mathematical models, simulations, and analysis, operations research enables businesses to determine optimal reorder points, order quantities, and inventory policies, accounting for factors like demand variability, lead times, and cost structures. This connection empowers organizations to minimize holding and ordering costs while maintaining desired service levels, enhancing overall supply chain efficiency and profitability through data-driven decision-making in the realm of inventory management.

The last few years have witnessed a growing body of research focusing on various dimensions of inventory control and management. Numerous studies have explored complex facets with an emphasis on dealing with demand that changes depending on pricing and partial backlog, Das et al. (2020) integrated preservation technology into an inventory control system. In 2020, Mashud contributed by introducing an EOQ framework for a failing IM that considers various forms of demands and full backlog. In the year 2021, Khan and Sarkar investigated the landscape of risk transfer in the supply chain, integrating price and inventory decisions and handling shortages with great care. To tackle difficult inventory control situations, Setiawan et al. (2021) provided helpful information on how to handle exponential and quadratic demand in the context of Weibull deterioration. In the following year, Sharma et al. (2022) published their work, which developed a model that considers demand-driven production and accounts for time and stock-related demand for items with declining inventory. An adaptive IM that is specifically designed for pharmaceutical distribution, with dynamic discreteness and the ability to account for both deterministic and stochastic demand, was presented by Antic et al. (2022). Duary et al. (2022) extended the scope by considering payment timing and inventory discounts in a model for deteriorating items, thoughtfully incorporating capacity constraints and partially backlogged shortages. The most recent addition, by Jani et al. (2023), offers a decision support system tailored for retailer's deterioration control, factoring in trade credit dynamics and the presence of shortages. Farahbakhsh and Kheirkhah (2023) presented a useful genetic algorithm-Taguchi-based method for solving the multi-period inventory route problem. In the same year, Dash et al. (2023) explored the coordination of a single-manufacturer multi-retailer supply chain, taking into account price and green-sensitive demand under stochastic lead time. Miriam et al. (2023) focused on decision-making processes in customer-centric IM. Moving into (2024), Das and Samanta presented an EOQ model for a two-warehouse system during lockdown, considering linear time-dependent demand. Lastly, Nazabadi et al. (2024) using agent-based and reinforcement learning models came up with a joint policy for production, maintenance, and product quality that works for a multi-machine production system. Together, these studies form a comprehensive mosaic of inventory management research, contributing valuable perspectives to this intricate and dynamic field. To avoid stockouts or excess inventory, classical IM must accurately predict future demand to strike a balance between service levels, costs, and reorder points, it must optimize safety stock levels and reorder points etc. In the sun case for a better alternative to classical inventory models (IM), fuzzy logic is useful in situations, where there are numerous complicated factors. In 1965, fuzzy theory was introduced by Lotfi A. Zadeh, which significantly improved its ability to aid in better decision-making.

The fuzzy inventory model (FIM), first proposed by Dubois and Parade (1988), achieves a flexible method of inventory management compared to classical IM. In 2011, Leopoldo Eduardo Cárdenas-Barron used analytical geometry and algebra to create EPQ/EOQ IMs including dual backorder costs. Based on this, Sulak (2015) proposed an EOQ model under a fuzzy set that handled defective shortages and products, offering a more realistic approach to inventory management. In 2019, Gani and colleagues used the AGM inequality approach to determine EOQ/EPQ in a fuzzy situation. This helped with inventory estimations. With a focus on variable holding costs, Alfares and Ghaithan (2019) investigated EOQ and EPQ production-IM, an essential part of contemporary supply chain dynamics. Additional research by Thinakaran et al. (2019) examined partial backorders in EOQ and EPQ IMs. A typical IM issue that may be rather difficult for corporations was shed light on here. To enhance EOQ/EPQ IM formulations with two backorder prices, Lin (2019) utilized algebra and analytic geometry. Gani and Rafi (2020), enhanced EOQ/EPQ calculation using algebraic and AGM inequality approaches to streamline decision-making in a fuzzy context.

Later, Das (2020) introduced the multi-item IM with lead time in his research. In a fuzzy setting, demand influences both production cost and setup cost. In 2022, 2 years after this idea was first proposed, Das further expanded the proposed multi-objective IM in a fuzzy context. In 2022, a new approach was launched to handle production faults in EOQ/EPQ IM with shortages utilizing fuzzy methodologies. This method significantly improved the accuracy and resilience of inventory management. As a group, these academics have tackled many of the challenges and unknowns that contemporary companies confront by making great theoretical and practical advances in the field of inventory management. So far, we have discussed the FIM, but some challenges in FIM. Challenges in FIM involve handling fluctuating demand patterns, addressing imprecise or incomplete data, optimizing inventory allocation and distribution across complex supply chains, and adapting to unforeseen disruptions, whereas extended fuzzy inventory methodologies refine traditional approaches by integrating higher-order fuzzy reasoning, multiple granularities, and dynamic adjustments, enabling more adaptive, resilient, and agile inventory control strategies in volatile and uncertain environments.

However, in 1999, F. Smarandache outlined the neutrosophic set and its unique features that distinguish it apart from the classical and fuzzy models. In the context of this progression, we will now proceed to discuss the neutrosophic inventory model (NIM).

2 Proposed Neutrosophic Inventory Management Model

3 Numerical Analysis and Result Discussion

The purpose of introducing this section is to explain the validation of our proposed model. Additionally, we have attempted to compare our proposed model with some of the existing models. This comparison helps demonstrate the superiority of our proposed model, and we have used two examples to illustrate this. Specifically, we conducted two case studies using existing literature datasets (such as those of Rajput et al. (2019), Saranya and Varadarajan (2018) and Sen and Malakar (2015). In Example 3.1, our goal is to establish the approach of our proposed model by considering the existing dataset and comparing it with the current existing method. Furthermore, it is evident that from Example 3.1, our proposed method not only addresses existing problems but also solves a new type of environment, as discussed below in Example 3.2.

Example 3.1: Comparison with the existing method: As per Rajput et al.’s (2019) consideration, A manufacturing facility must establish an EOQ model to maximize the product's total cost. The cycle length is 6 months, with ordering costs of Rs. 20, holding costs of Rs. 04, and shortage costs of Rs. 10 per unit. We use an SVTpN membership function to capture the data’s inherent uncertainty when addressing this challenge using Neutrosophic parameters. For every feasible cost, the corresponding membership functions are specified as follows: i ˜ neu = 〈 ( 1 , 3 , 5 , 6 ) ; ( 1 , 0 , 0 ) 〉 , k ˜ neu = 〈 ( 8 , 9 , 11 , 12 ) ; ( 1 , 0 , 0 ) 〉 , and n ˜ neu = 〈 ( 15 , 18 , 22 , 25 ) ; ( 1 , 0 , 0 ) 〉 .

Solution: In Example 3.1, we are demonstrating that our proposed method not only solves the new type of problem but also addresses the problem solved by the existing method, as shown in Table 2. In this table, we can see that the Total Cost (TC) of the existing method is equal to the TC of our proposed method.

In addition to tabular comparison, we also conducted a pictorial comparison study with some of the existing methods such as those of Rajput et al. (2019), Saranya and Varadarajan (2018), and Sen and Malakar (2015).

In Figures 1 and 2, we have attempted to compare two approaches: the first using clusters and the second using a line graph. In Figure 1, we can clearly see that as we changed the demand in increasing order, the same trend was observed in all other clusters. This indicates that demand plays a significant role in terms of total cost, a finding consistent with what other authors have observed in their studies. Additionally, in Figure 2, we aim to demonstrate that our proposed method is either equivalent to or provides a better solution than other methods. The dominance of the yellow-colored graph suggests that our proposed method provides the least value in comparison to the other method. Finally, after doing the logical comparison, i.e. the classical total cost is greater than Rajput et al. (2019) proposed method but it is equal to our proposed method.

Figure 1

Clustered comparison with existing methods.

Figure 2

Scatter representation of existing methods.

In Example 3.1, many authors have proposed different methods to solve Rajput’s numerical problem. In the comparison study of tabular, pictorial, and logical methods, it is observed that our proposed method provides an optimal solution similar to that of Rajput et al. (2019). Our proposed method not only solves existing problems but also solves a new type of environment, which is discussed below in Example 3.2.

Example 3.2: A manufacturing facility must create an EOQ model to maximize the product's total cost. The cycle length is 6 months, with ordering costs of Rs. 20, holding costs of Rs. 04, and shortage costs of Rs. 10 per unit. We use an SVTpN membership function to capture the data’s inherent uncertainty when addressing this challenge using Neutrosophic parameters by considering the six cases discussed in Table 3.

Solution: After implementing equations (A5), (A6), and (A7) of our proposed method, the final total cost (TC) that we obtained is displayed in Table 4.

We compared our proposed method with some of the existing methods such as those of Rajput et al. (2019), Saranya and Varadarajan (2018), and Sen and Malakar (2015) using both tabular and pictorial comparison studies.

In Figures 3 and 4, we can see that both figures are represented in green colour. This indicates that only our proposed method can handle this type of problem. Additionally, we observe that representations in red, blue, and light black colours do not exist in either figure. This suggests that all other existing methods are unable to handle similar uncertain situations. Hence, it is clear that our proposed method not only addresses existing numerical challenges but also tackles new types of uncertain problem types. Moreover, from the above comparison, it is clear that our proposed method is superior to some of the existing methods. Now, we are going to perform a logical comparison of Examples 3.1 and 3.2, as discussed in Table 5.

Figure 3

Column representation (ref Table 4).

Figure 4

Graphical interpretation of Table 4.

4 Sensitive Analysis

A manufacturing facility develops an EOQ model to maximize the product's total cost. The cycle length is 6 months, and the ordering price is Rs. 〈 ( 15 , 18 , 22 , 25 ) ; ( 0.99 , 0.97 , 0.70 ) 〉 per unit, the holding cost is Rs. 〈 ( 1 , 3 , 5 , 6 ) ; ( 0.99 , 0.98 , 0.73 ) 〉 per unit, and the shortage cost is Rs. 〈 ( 8 , 9 , 11 , 12 ) ; ( 0.98 , 0.91 , 0.71 ) 〉 per unit.

Table 6 provides insights into optimizing total costs in a manufacturing setting, specifically focusing on cost-effective IM using neutrosophic theory. This sensitive analysis is crucial for decision-making in IM. Therefore, optimizing these parameters ( i ˜ neu , k ˜ neu , and n ˜ neu ) can lead to substantial savings or expenditures, depending on their values.

Based on the information from Table 6, we can conclude that the model is extremely sensitive to the holding cost, shortage cost, and total cost parameters.

• The model is highly sensitive to holding cost i ˜ neu = 〈 ( 1 , 3 , 5 , 6 ) ; ( 0.99 , 0.98 , 0.73 ) 〉 ; i.e. If we increase the percentage in the holding cost parameter (i.e. 13, 29, and 37%), we observe that the optimal order quantity decreases, while the optimal shortage quantity increases and the total cost also increases. Conversely, if we decrease the percentage in the holding cost (i.e. −13, −29, and −37%), we find that the optimal order quantity increases, the optimal shortage quantity decreases and the total cost decreases as well.

• Similarly, if we increase the percentage in the shortage cost k ˜ neu parameter (i.e. 13, 29, and 37%), we observe that the optimal order quantity decreases, and the optimal shortage quantity also decreases, while the total cost increases. Conversely, if we decrease the percentage (i.e. −13, −29, and −37%) in the shortage cost, we find that the optimal order quantity increases, the optimal shortage quantity increases, and the total cost decreases as well.

• Changing the n ˜ neu ordering cost parameter will have an effect, with an increase of 13, 29, and 37% the observations raise in overall cost, optimal shortage quantity, and ideal order quantity, respectively. On the other hand, decreasing the proportion of the ordering cost by −13, −29, or −37% results in a reduction of the overall cost, optimal shortage quantity, and optimal order quantity.

5 Conclusion

The investigation of neutrosophic set theories defines a potential use in handling inventories in our research study. Current FIMs have problems with handling uncertainty, inaccurate data, and imprecise timing, in contrast to classical models that have optimized order amounts, timing, and safety stock levels. To address these challenges, our novel approach focuses on the uncertainty associated with holding costs, shortage costs, and ordering costs, utilizing trapezoidal neutrosophic numbers. We improve decision-making procedures and offer more reliable inventory optimization solutions by utilizing neutrosophic reasoning. Notably, our method demonstrates promising results in managing inventory under uncertainty a milestone in the literature of neutrosophic sets. Additionally, Our research contributes valuable insights to enhance inventory management strategies and moderate uncertainties in EPQ operations through graphical, logical, and tabular comparisons. Despite our significant progress, several avenues for future exploration and improvement remain. These include investigating Dynamic Demand and Supply, Multi-Objective Optimization, and conducting Empirical Validation. However, it's crucial to acknowledge the limitations of our model. Its success hinges on accurate data related to costs, demand, and lead times. Obtaining precise data, especially under conditions of uncertainty, remains a challenge. Like any model, our approach is based on certain assumptions, and ongoing validation against various scenarios is essential.