Optimizing Inventory Management with Seasonal Demand Forecasting in a Fuzzy Environment

Demand forecasting stands as a pivotal element in shaping business strategies, offering organizations the means to optimize operations, cut costs, and meet consumer expectations [2]. Mishra and Jain [9] put forward a decentralized supply chain optimization model that incorporates blockchain and uses an iterative strategy to calculate overall costs for retailers and suppliers. By establishing its uniqueness and optimal results through theoretical analysis, the model efficiently determines optimal replenishment cycles using Wolfram Mathematica 13.0., guiding decision-makers with managerial insights for enhanced supply chain management efficiency and resilience. The continuous expansion of ML techniques provides a fertile ground for researchers aiming to enhance the accuracy of demand forecasting [3]. Early research by Persinger and Levesque [10] delved into the relationship between weather conditions and individuals' moods, establishing that weather significantly influences consumer behaviour. Wright and Schultz [11] emphasized the critical role of demand forecasting models in predicting overstocking and understocking situations, particularly during fluctuations in consumer demand. Notable festive seasons, such as Christmas and Diwali, witness a surge in demand for e-commerce giants like Amazon and Flipkart as consumers actively seek gifts and festive supplies. Accurate demand forecasting becomes imperative during such peak periods, emphasizing the need for dynamic models that surpass fixed demand predictions [12]. Decision trees, recognized as powerful data mining techniques, have been extensively employed in various sectors for demand forecasting [13]. In the field of defective damaging goods, Ghare [14] first developed research on damaging items in inventory systems, with a focus on constant deterioration ratios. Wee et al. [15] conducted subsequent research and built a production inventory model particularly for damaging seasonal goods. The consideration of partial payment and trade credit policies in a non-instantaneous disintegrating concept of inventory was introduced by Lashgari et al. [16]. The reality of imperfect products, either due to manufacturing errors or deterioration, led to economic production quantity (EPQ) models for imperfect quality items [17]. Further investigations delved into imperfect quality inventory systems, integrating stochastic processes to model defective products [18].

Carbon emissions have become a significant concern in recent years, prompting researchers to incorporate environmental considerations into inventory models. Mishra [19] provides a supply chain inventory model for degrading items that takes into account carbon emission-dependent demand, advanced payment methods, and the impact of carbon taxes and limits on a constrained planning horizon. It seeks to balance economic and environmental considerations, providing insights for businesses and potential relevance for government policies. Hua et al. [20] built an economic order quantity (EOQ) model which incorporates carbon costs associated with storage and shipment. Carbon taxes and emissions reduction policies have been explored to understand their impact on inventory costs [21]. Green inventory models were studied under settings of carbon emission penalty fees, the cap-and-trade scheme systems, and severe emission limit laws [22].

Fuzzy methods have gained attention in inventory management problems due to their ability to generate more relevant solutions. Early applications of fuzzy set theory in inventory models include analyses of economic order quantity [23, 24]. Extensions into fuzzy overall cost functions, considering fuzzy demand rates and proportions of defective products, have been explored [25, 26]. Fuzzy inventory estimation methods for degrading items with time-dependent demand and backlog rates considered as fuzzy numbers have also been developed [27]. The integration of fuzzy concepts into inventory models, considering imperfect products, payment delays, variable demand and partial backlog, has been investigated [28]. Mishra et al. [29] present a fuzzified supply chain finite planning horizon model, addressing deteriorating materials. The model uses fuzzy parameters, such as deterioration cost, and applies defuzzification methods with finite planning horizon. Mishra et al. [30] provided research on the fuzziness of supplier-retailer supply coordination. The research explores credit terms in the context of managing items are degrading due to time-quadratic demand and partial backlog throughout all cycles across the finite planning horizon. Singh and Mishra [31] provides an inventory model that uses artificial intelligence to estimate demand, with a focus on imperfect deteriorating items and partial backlog concerns. The model also incorporates the impact of carbon emissions.

The main innovation of this study is in crafting a distinctive inventory system supported by machine learning and fuzzy logic. This system is custom designed to confront the difficulties arising from defective degrading goods amid carbon emissions within a limited planning timeframe. Unlike traditional inventory models, this study addresses uncertainties surrounding defective percentages and deterioration rates by treating them as fuzzy variables. Through the application of a machine learning technique, specifically the time series prediction or (production): the model aims to enhance the precision of variable demand forecasts for degrading items over a limited planning horizon.

The importance of this research work is evident in its potential to revolutionize demand forecasting in businesses. By moving beyond fixed demand assumptions and incorporating machine learning-based monthly predicted demand, organizations can achieve more precise and reliable inventory management. The numerical experiment conducted in the study demonstrates a substantial reduction in overall costs when utilizing seasonal forecasted demand. Moreover, the model's incorporation of carbon emissions costs underscores a commitment to sustainability and environmental responsibility.

The research's overarching contribution is its holistic approach, fuzzy variables, integrating machine learning techniques and considerations for carbon emissions. This comprehensive framework not only improves demand forecasting accuracy but also provides businesses with a strategic means to minimize their ecological footprint. Ultimately, the research aims to identify optimal policies that minimize overall costs while simultaneously addressing the complexities associated with deteriorating products, thereby offering a valuable contribution to the field of inventory management over the finite planning horizon.

4.1 Environmental benefits of carbon emissions integration in inventory control

The decision of including the cost of carbon emissions in inventory management has many environmental gains. This integration encourages organisations to implement the practices that are environmentally friendly across its operations since the environmental cost attached to the inventory activities are supported. Particularly, it promotes the improvements in motion-related measures including the optimisation of routes to avoid gross fuel consumption and the purchase of effective energy-efficient resources used in the storage of goods. Also, the incorporation of carbon emissions costs can actually result to the use of suppliers and production processes with a relatively low levels of carbon emissions hence enhancing the use of a cleaner supply chain.

Further, it can lead to the optimization of inventory and waste as firms look to integrate sustainability strategy in inventory control so that the production and disposal of extra and obsolete products are eliminated leading to fewer emission values. It also aids organisations to conform to environment legislation and policies which contributes to its sustainable status and may provide certain firm’s with an edge in regions where sustainability is becoming increasingly more of a concern and criterion to consumers. When such costs are incorporated, firms help in creating organizational awareness towards the negative effects of their operations on the environment while at the same time driving down the general climate change.

4.2 Rationale for selecting the decision tree classifier in demand forecasting

The decision tree classifier is chosen to be used for demand forecasting by means of selecting the Machine Learning technique because of several benefits. First, decision trees practically allow for evaluating interpretability on a high level because the decision rules and their results are easily presented in a tree-like structure. This aspect enables one to grasp how different aspects affect the demand forecast; it assists the stakeholders to develop significant data insights and make the required modifications to their strategies.

In addition to this, decision trees do not assume the normality of the data and have the ability to deal with numerical as well as categorical data for unique datasets, which are always involved in the demand for estimation, for example, past sales data, seasonal variation, and promotional influences. This is especially helpful when it comes to capturing the interaction effects of features since, unlike linear methods, tree-based models can capture a quadratic and higher order relationships with the variables.

Moreover, decision trees have high tolerance to outliers and noisy data that is typical for real data sets. It isolates data into homogenous subsets, thereby; the existence of anomalies or irregularities does not undermine the model’s predictive capability. Also, decision trees are less sensitive to data pre-processing than most of the algorithms hence data pre-processing is done in a minimal rate thus minimizing the chances of high bias of the data.

It is also noteworthy that decision trees are extremely efficient in terms of learning and prediction times. The fact that building block models can be done relatively quickly is an advantage since some of the institutional demands for dynamic demand forecasting may necessitate frequent updates. Furthermore, decision trees can be replaced, added or integrated with some ensembling algorithms like random forest or gradient boosting in order to improve the efficiency of prediction or to avoid some problems like overfitting or interpretability.

Concisely, the merits such the decision tree classifier’s interpretability, flexibility, noise tolerance and efficiency put it in a good standing to serve as a viable and effective predictor for demand, and by extension a catalyst in constructing an effective inventory management system.

This article explores a sustainable supply model designed for imperfect diminishing products within a retail setting. At the onset, the retailer acquires a quantity ($Q_{i+1}$) of products. The stock level experiences a reduction due to both demand and degradation throughout the period. A partial backlogging shortage occurs at a rate of β and persists until time T. By the conclusion of the cycle, the supply has exceeded its maximum deficit level. To rectify this backlog, the retailer initiates the replenishment of products. The dynamics of the stock level during the time intervals [0,t₀], [t₀,t₁] are governed by the following set of differential equations:

The suggested approach in this work focuses on accurate demand forecasting using artificial intelligence (machine learning). While many academics believe that demand is predictable, the presence of unpredictable swings implies that an approach based on machine learning is more suited for demand prediction. For this study, time series method is chosen due to its simplicity and effectiveness in ML techniques. The primary objective is to ascertain the precise seasonal demand for deteriorating products. The flowchart illustrating the methodology for demand forecasting is presented above.

PYTHON code (version 3.10.0) is used to validate the forecasted inventory system by predicting seasonal demand for deteriorating products. Before executing the Python code, ensure that essential packages like Pandas for data manipulation and sklearn for handling time series data are installed.

Ultimately, by inputting the parameter (month): the seasonally demand for deteriorating products is obtained. As we can see the Figure 1.

Inventory level with boundary condition $I L_{i+1}\left(s_{i+1}\right)=0$,

$\begin{gathered}\frac{d I L_{i+1}(t)}{d t}+(\theta) I L_{i+1}(t)=-(1-k) D \\ t_i<t<s_{i+1}\end{gathered}$       (1)

where $\left\{1,2,3 \ldots \ldots \ldots \ldots \ldots \ldots \ldots n_1\right\}$ ,

$\begin{aligned} \frac{d I L_{i+1}(t)}{d t}= & -(1-k) D-(\theta) I L_{i+1}(t) \\ & t_i<t<s_{i+1}\end{aligned}$          (2)

$I L_{i+1}(t)=\int_t^{s_{i+1}}-(1-k) D e^{\theta(u-t)} d u$              (3)

$I L_{i+1}(t)=\theta(1-k) D\left[1-e^{\theta\left(s_{i+1}-t\right)}\right]$        (4)

1a.png

1b.png

Figure 1. Demand forecasting

The current shortage level, denoted as $S_{i+1}(t)$ under the boundary condition $S_{i+1}\left(s_i\right)=0$, is defined by the subsequent differential equation:

$\frac{d I L S_{i+1}(t)}{d t}=D \beta$     (5)

where, $s_i<t<t_i$ ,

$I L S_{i+1}(t)=\int_{s_i}^t D \beta d t=D \beta\left(s_i-t\right)$      (6)

Therefore, the overall inventory quantity maintained throughout the interval $\left[t_i, s_{i+1}\right]$,

$R_{i+1}=\int_{t_i}^{s_{i+1}}\left\{\int_t^{s_{i+1}}-(1-\mathrm{k}) \mathrm{De}^{\theta(u-t)} \mathrm{du}\right\} d t$     (7)

Eq. (7) can be reformulated as follows by adjusting the integration position and omitting the higher-order terms of $\alpha^2$.

$R_{i+1}=\frac{-(1-\mathrm{k})}{\theta} \int_{t_i}^{s_{i+1}}\left\{e^{\theta\left(s_{i+1}-t\right)}-1\right\} D d t$     (8)

Customers are waiting for the complete amount of that quantity i.e., the quantity of deficit throughout the timeframe  $\left[s_i, t_i\right]$.

After rearranging the ordering, $S_{i+1}$ can be given as:

$\begin{aligned} & S_{i+1}=\int_{s_i}^{t_i} I L S_{i+1}(t) d t = \int_{s_i}^{t_i}\left\{\int_{s_i}^t D \beta\left(s_i-u\right) d u\right\} d t = \frac{-1}{2} \int_{s_i}^{t_i}\left(D \beta\left(s_i-t\right)\right)^2 d t\end{aligned}$     (9)

The total order quantity for a finite planning horizon:

$$\begin{aligned}& \mathrm{Q}=\sum_{i=1}^n Q_{i+1}=\sum_{i=1}^n\left\{R_{i+1}+S_{i+1}\right\} \\& Q_{i+1}=\frac{-(1-\mathrm{k})}{\theta} \int_{t_i}^{s_i+1}\left\{e^{\theta\left(s_{i+1}-t\right)}-1\right\} D d t+\frac{1}{3}\left(D \beta\left(s_i-t_i\right)\right)^3\end{aligned}$$

The overall number of degraded components at each refill is as follows:

$\begin{gathered}D_{i+1}=\int_{s_i}^{t_i} \theta I L_{i+1}(t) d t=\left\{\theta^2 \int_{t_i}^{s_i}(1-k) D\left[1-e^{\theta\left(s_{i+1}-t\right)}\right] d t\right\}\end{gathered}$            (10)

In the case of Fast-Moving Consumer Goods (FMCG) or basic necessities: Consumers cannot delay their purchases, resulting in only a portion, β, of the demand being held back during stockouts. Consequently, the remaining fraction (1-β) is lost.

The quantity that was lost throughout the interval $\left[s_i, t_i\right]$ is given as:

$\begin{gathered}L_{i+1}=\int_{s_i}^{t_i}\{D-D \beta\} d t=\int_{s_i}^{t_i}\{(1-\beta) D\} d t =(1-\beta) D\left(t_i-s_i\right)\end{gathered}$   (11)

Cost of carbon emissions during the period $\left[t_i, s_{i+1}\right]$ can be expressed as:

$C e=\sum_{i=0}^{n_1-1} c^{\wedge}+\widehat{P}_r * R_{i+1}+\widehat{h_c} \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t$

$\begin{gathered}C e=\sum_{i=0}^{n_1-1} c^{\wedge}+\hat{P}_r * Q_{i+1} +\widehat{h_c} \int_{t_i}^{s_{i+1}}\left\{\int_t^{s_i+1}-(1-k) D e^{\theta(u-t)} d u\right\} d t\end{gathered}$

According to Mishra [9]: The overall cost of carbon emissions throughout the interval $\left[t_i, s_{i+1}\right]$ can be expressed as:

$\left.C e=\tau\left\{\sum_{i=0}^{n_1-1} c^{\wedge}+\hat{P}_r\left\{\theta \int_{t_i}^{s_i}(1-k) D\left[1-e^{\theta\left(s_{i+1}-t\right)}\right] d t\right\}-(1-k) D * \widehat{h_c} * \theta \int_{t_i}^{s_i+1}\left\{e^{\theta\left(s_{i+1}-t\right)}-1\right) d t\right\}\right\}$    (12)

The transportation expenses for the retailer take into account both variable transportation costs and fixed costs, as well as carbon emissions resulting from FEC during refrigeration. Eq. (13) consequently outlines the transportation cost as follows.

$c=F_c+2 d v_c C_1+d * v_c C_2 \int_{t_i}^{t_{i+1}} D e^{\theta\left(t-t_i\right)} d t+2 d e_1+d e_2 \int_{t_i}^{t_{i+1}} D e^{\theta\left(t-t_i\right)} d t$    (13)

Total cost = Replenishment Cost + Inventory Holding Cost + Acquisition Cost + Depreciation Cost + Storage Expense + Lost Sales Cost + Carbon Emission Cost + Transportation Expense

$$\begin{aligned}& T C\left(t_i, s_i, n_1\right)=n_1 * O_r+\sum_{i=0}^{n_1-1} H \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t+\sum_{i=0}^{n_1-1} W_h * Q_{i+1}+\sum_{i=0}^{n_1-1} D_t * \theta \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t+\sum_{i=0}^{n_1-1} s \int_{s_i}^{t_i} I L S_{i+1}(t) d t+\sum_{i=0}^{n_1-1} l \int_{s_i}^{t_i}(1-\beta) D\left(t_i-s_i\right) d t \\& +\sum_{i=0}^{n_1-1} c^{\wedge}+\widehat{P}_r Q_{i+1}+\widehat{h_c} \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t+F_c+2 d v_c C_1+d * v_c C_2 \int_{t_i}^{t_{i+1}} D e^{\theta\left(t-t_i\right)} d t+2 d e_1+d e_2 \int_{t_i}^{t_{i+1}} D e^{\theta\left(t-t_i\right)} d t \\& T C\left(t_i, s_i, n_1\right)=n_1 * O_r \\& +\sum_{i=0}^{n_1-1} H \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t+\sum_{i=0}^{n_1-1} W_h * Q_{i+1}+\sum_{i=0}^{n_1-1} D_t * \theta \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t+\sum_{i=0}^{n_1-1} s \int_{s_i}^{t_i} I L S_{i+1}(t) d t+\sum_{i=0}^{n_1-1} l \int_{s_i}^{t_i}(1-\beta) D\left(t_i-s_i\right) d t \\& +\sum_{i=0}^{n_1-1} \tau c^{\wedge}+\tau \widehat{P}_r Q_{i+1}+\tau \widehat{h_c} \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t+F_c+2 d v_c C_1+d * v_c C_2 \int_{t_i}^{t_{i+1}} D e^{\theta\left(t-t_i\right)} d t+2 d e_1+d e_2 \int_{t_i}^{t_{i+1}} D e^{\theta\left(t-t_i\right)} d t \\& T C\left(t_i, s_i, n_1\right)=n_1 * O_r+\sum_{i=0}^{n_1-1}\left\{H+\tau \widehat{h_c}+D_t * \theta\right\} \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t+\left\{W_h+\tau \widehat{P}_r\right\} \sum_{i=0}^{n_1-1} Q_{i+1}+\sum_{i=0}^{n_1-1} s \int_{s_i}^{t_i} D \beta\left(s_i-t\right) d t \\& +\sum_{i=0}^{n_1-1} l \int_{s_i}^{t_i}(1-\beta) D\left(t_i-s_i\right) d t+\tau c^{\wedge}+F_c+2 d v_c C_1+d * v_c C_2 \int_{t_i}^{t_{i+1}} D e^{\theta\left(t-t_i\right)} d t+2 d e_1+d e_2 \int_{t_i}^{t_{i+1}} D e^{\theta\left(t-t_i\right)} d t\end{aligned}$$

$\begin{aligned} & T C\left(t_i, s_i, n_1\right)=n_1 * O_r+\sum_{i=0}^{n_1-1}\left\{H+\tau \widehat{h_c}+D_t * \theta\right\} \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t+\left\{W_h+\tau \hat{P}_r\right\} \\ & \sum_{i=0}^{n_1-1}\left(\frac{-(1-k)}{\theta} \int_{t_i}^{s_{i+1}}\left\{e^{\theta\left(s_{i+1}-t\right)}-1\right\} D d t+\frac{1}{3}\left(D \beta\left(s_i-t_i\right)\right)^3\right)+\sum_{i=0}^{n_1-1} s \int_{s_i}^{t_i} D \beta\left(s_i-t\right) d t \\ & +\sum_{i=0}^{n_1-1} l \int_{s_i}^{t_i}(1-\beta) D\left(t_i-s_i\right) d t+\tau c^{\wedge}+F_c+2 d * v_c C_1+d * v_c C_2 \int_{t_i}^{t_{i+1}} D(t) e^{\theta\left(t-t_i\right)} d t +2 d e_1+d e_2 \int_{t_i}^{t_{i+1}} D(t) e^{\theta\left(t-t_i\right)} d t\end{aligned}$   (14)

$\begin{aligned} & T C\left(t_i, s_i, n_1\right)=n_1 * O_r+\sum_{i=0}^{n_1-1}(1-k)\left\{H+\tau \widehat{h_c}+D_t * \theta\right\} \int_{t_i}^{s_{i+1}}\left[D \beta\left(s_i-t\right)\right] d t+\left\{W_h+\tau \widehat{P}_r\right\} \\ & \sum_{i=0}^{n_1-1}\left(\frac{-(1-k)}{\theta} \int_{t_i}^{s_{i+1}}\left\{e^{\theta\left(s_{i+1}-t\right)}-1\right\} D d t+\frac{1}{3}\left(D \beta\left(s_i-t_i\right)\right)^3\right)+\sum_{i=0}^{n_1-1} s \int_{s_i}^{t_i} D \beta\left(s_i-t\right) d t \\ & +\sum_{i=0}^{n_1-1} l \int_{s_i}^{t_i}(1-\beta) D\left(t_i-s_i\right) d t+\tau c^{\wedge}+F_c+2 d v_c C_1+\left(d e_2+d * v_c C_2\right) \int_{t_i}^{t_{i+1}} D e^{\theta\left(t-t_i\right)} d t\end{aligned}$        (15)

6.1 Fuzzification of model

In inventory systems, pinpointing precise values for known parameters poses a challenge for decision-makers, introducing uncertainty into key parameters. Consequently, the defective percentage in quantity (k) and deterioration rate (θ) are treated as trapezoidal fuzzy interval types. Fuzzy arithmetic operations for trapezoidal fuzzy numbers are concisely explained. Building upon these basic definitions and results, the proposed model is then fuzzified. Let $\left(\tilde{\theta}_1, \tilde{\theta}_2, \tilde{\theta}_3, \tilde{\theta}_4\right)$ and  $\left(\tilde{k}_1, \tilde{k}_2, \tilde{k}_3, \tilde{k}_4\right)$ represent trapezoidal fuzzy numbers, as depicted in Figure 2. Consequently, the succinct total average expense function is converted into a fuzzy cost function. Since the deterioration rate (θ) and the defective quantity percentage (k) are both represented as trapezoidal fuzzy figures, the total cost (TC) is also treated as a trapezoidal fuzzy figure.

2.png

Figure 2. Displays the trapezoidal fuzzy

$T C=\left(\widetilde{T C}_1, \widetilde{T C}_2, \widetilde{T C}_3, \widetilde{T C}_4\right)$

$T C_d=\frac{1}{4}\left(\widetilde{T C}_1+\widetilde{T C}_2+\widetilde{T C}_3+\widetilde{T C}_4\right)$

$$\begin{aligned}& \widetilde{T C R}_i\left(t_i, s_i, n_1\right)=n_1 * O_r+\sum_{i=0}^{n_1-1}\left(1-\tilde{\mathrm{k}}_l\right)\left\{H+\tau \widetilde{h_c}+D_t * \widetilde{\theta_i}\right\} \int_{t_i}^{s_{i+1}}\left[D \beta\left(s_i-t\right)\right] d t \\& +\left\{W_h+\tau \widehat{P}_r\right\} \sum_{i=0}^{n_1-1}\left(\frac{-\left(1-\widetilde{\mathrm{k}}_i\right)}{\widetilde{\theta}_i} \int_{t_i}^{s_{i+1}}\left\{e^{\tilde{\theta}_i\left(s_{i+1}-t\right)}-1\right\} D d t+\frac{1}{3}\left(D \beta\left(s_i-t_i\right)\right)^3\right) \\& +\sum_{i=0}^{n_1-1} s \int_{s_i}^{t_i} D \beta\left(s_i-t\right) d t+\sum_{i=0}^{n_1-1} l \int_{s_i}^{t_i}(1-\beta) D\left(t_i-s_i\right) d t+\tau c^{\wedge}+F_c+2 d v_c C_1\left(d e_2+d * v_c C_2\right) \int_{t_i}^{t_{i+1}} D(t) e^{\widetilde{\theta_l\left(t-t_i\right)}} d t\end{aligned}$$

The goal is to discover the basic values of t_i and s_i in order to lower the total variable cost (TC) of stock control and management. Figure 2 exhibits the trapezoidal fuzzy numbers denoting the deterioration rate (θ) and the defective quantity percentage (k). The requirements to shows the $\widetilde{T C}{ }_d$ to be minimum are given below:

$\frac{\partial \widetilde{T C R}_d\left(t_i, s_i, n_1\right)}{\partial t_i}=0$

$\frac{\partial \widetilde{T C R}_d\left(t_i, s_i, n_1\right)}{\partial s_i}=0$

$$\begin{aligned}& \frac{\partial \widetilde{T C R}_d\left(t_i, s_i, n_1\right)}{\partial s_i}=\left\{H+\tau \widehat{h_c}+D_t * \widetilde{\theta}_l\right\} \int_{t_i}^{s_{i+1}}[D \beta] d t+\left\{W_h+\tau \widehat{P}_r\right\} \sum_{i=0}^{n_1-1}\left(\left(D \beta\left(s_i-t_i\right)\right)^2\right)+\sum_{i=0}^{n_1-1} s \int_{s_i}^{t_i} D \beta d t+\sum_{i=0}^{n_1-1} l \int_{s_i}^{t_i}(\beta-1) D d t\end{aligned}$$        (16)

$$\begin{aligned}& \frac{\partial \widetilde{T C R}_d\left(t_i, s_i, n_1\right)}{\partial t_i}=\left\{H+\tau \widehat{h_c}+D_t * \widetilde{\theta}_l\right\}\left[D \beta\left(t_i-s_i\right)\right]+\left\{W_h+\tau \widehat{P}_r\right\} \\& \sum_{i=0}^{n_1-1}\left(\frac{-\left(1-\tilde{\mathrm{k}}_i\right)}{\widetilde{\theta}_t}\left(1-e^{{\tilde{\mathrm{\theta_l}}\left(s_{i+1}-t_i\right)}}\right)-\left(D \beta\left(s_i-t_i\right)\right)^2\right)+\sum_{i=0}^{n_1-1} l \int_{s_i}^{t_i}(1-\beta) D d t+l(1-\beta) D\left(t_i-s_i\right)-\left(\mathrm{d} e_2+\mathrm{d}\right.\left.* v_c C_2\right) * \widetilde{\theta}_i \int_{t_i}^{t_{i+1}} D e^{\tilde{\mathrm{{\theta}_l}}\left(\mathrm{t}-\mathrm{t}_i\right)} \mathrm{d} t-\left(\mathrm{d} e_2+\mathrm{d} * v_c C_2\right) D\end{aligned}$$     (17)

$\frac{\partial^2 \widetilde{T C R}_d\left(t_i, s_i, n_1\right)}{\partial s_i{ }^2}=\left\{W_h+\tau \hat{P}_r\right\} \sum_{i=0}^{n_1-1}\left(\left(D \beta\left(s_i-t_i\right)\right)\right)-s D \beta+l(\beta-1) D$         (18)

$\begin{aligned} & \frac{\partial^2 \widetilde{T C R}_d\left(t_i, s_i, n_1\right)}{\partial t_i^2}=\left\{H+\tau \widetilde{h_c}+D_t * \widetilde{\theta_t}\right\}[D \beta]+\left\{W_h+\tau \widehat{P}_r\right\} \sum_{i=0}^{n_1-1}\left(-\left(1-\mathrm{k} \backslash \widetilde{\mathrm{k}_t}\right)\left(e^{\widetilde{\theta}_l\left(s_{i+1}-t_i\right)}\right)+2 D \beta\left(s_i-t_i\right)\right) \\ & \quad+\sum_{i=0}^{n_1-1} l(1-\beta) D+l(1-\beta) D+\left(\mathrm{d} e_2+\mathrm{d} * v_c C_2\right) \widetilde{\theta}_l^2 \int_{t_i}^{t_{i+1}} D e^{\widetilde{\theta}_l\left(\mathrm{t}-\mathrm{t}_i\right)} \mathrm{d} t+D\left(\mathrm{~d} e_2+\mathrm{d} * v_c C_2\right) \widetilde{\theta}_l\end{aligned}$              (19)

The essential condition is that the Hessian matrix with $\widetilde{T C}_d$ must be positive definite for $\widetilde{T C R}_d$ to achieve its minimum, with n₁ held constant. Additionally, the theorem establishes the positivity of $\widetilde{T C}_d$. Consequently, through an iterative process and leveraging Mathematica software, one can compute the optimal values of t_i and s_i for a specified positive intege  using the Eqs. (18) and (19).

Hessian matrix

$\nabla T_R\left(t_j, s_j, n_1\right)=$$\left[\begin{array}{ccccccccc}\frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_1{ }^2} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_1 \partial s_1} & 0 & 0 & \ldots & 0 & 0 & 0 \\ \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_2 \partial t_1} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_1{ }^2} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_1 \partial t_2} & 0 & \ldots & \ldots & 0 & 0 & 0 \\ 0 & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_2 \partial s_1} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_2{ }^2} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_2 \partial s_2} & \ldots & \ldots & 0 & 0 & 0 \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & 0 \\ 0 & 0 & 0 & 0 & \ldots & \ldots & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_{n_1-1} \partial s_{n_{1-1}}} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_{n_{1-1}}{ }^2} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial s_{n_{1-1}} \partial t_{n_1}} \\ 0 & 0 & 0 & 0 & \ldots & \ldots & 0 & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_{n_1} \partial s_{n_{1-1}}} & \frac{\partial^2 T_R\left(t_j, s_j, n_1\right)}{\partial t_{n_1}{ }^2}\end{array}\right]$

Theorem: if t_i and s_i satisfy the inequality,

(i) $\frac{\partial^2 \widetilde{T C} R_d\left(t_i, s_i, n_1\right)}{\partial t_i{ }^2} \geq 0$,

(ii) $\frac{\partial^2 \widetilde{T C R}_d\left(t_i, s_i, n_1\right)}{\partial s_i^2} \geq 0$,

(iii) $\frac{\partial^2 \widetilde{T C R}_d\left(t_i, s_i, n_1\right)}{\partial t_i{ }^2}-\left|\frac{\partial \widetilde{T C } R_d\left(t_i, s_i, n_1\right)}{\partial t_i \partial s_i}\right| \geq 0$,

(iv) $\frac{\partial^2 \widetilde{T C} R_d\left(t_i, s_i, n_1\right)}{\partial s_i^2}-\left|\frac{\partial \widetilde{T C R}_d\left(t_i, s_i, n_1\right)}{\partial s_i \partial t_i}\right| \geq 0$,

for all i=1,2,...,n then TC will be positive definite.

$$\begin{aligned}&\widetilde{Q_{l+1}}=\frac{-\left(1-\widetilde{\mathrm{k}_1}\right)}{\widetilde{\theta_1}} \int_{t_i}^{s_{i+1}}\left\{e^{\widetilde{\theta_1}\left(s_{i+1}-t\right)}-1\right\} D d t+\frac{-\left(1-\widetilde{\mathrm{k}_2}\right)}{\widetilde{\theta_2}} \int_{t_i}^{s_{i+1}}\left\{e^{\widetilde{\theta_2}\left(s_{i+1}-t\right)}-1\right\} D d t+\frac{-\left(1-\widetilde{\mathrm{k}_3}\right)}{\widetilde{\theta_3}} \int_{t_i}^{s_{i+1}}\left\{e^{\widetilde{\theta_3}\left(s_{i+1}-t\right)}-1\right\} D d t \\& +\frac{-\left(1-\widetilde{\mathrm{k}_4}\right)}{\widetilde{\theta_4}} \int_{t_i}^{s_{i+1}}\left\{e^{\widetilde{\theta_4}\left(s_{i+1}-t\right)}-1\right\} D d t+\left(D \beta\left(s_i-t_i\right)\right)^3\end{aligned}$$

$\begin{gathered}\widetilde{T C} s_i\left(t_i, s_i, n_1\right)=n_1{ }^* * S_s+P_s+\left(\frac{-\left(1-\widetilde{\mathrm{k}_1}\right)}{\widetilde{\theta_1}} \int_{t_i}^{s_{i+1}}\left\{e^{\widetilde{\theta_1}\left(s_{i+1}-t\right)}-1\right\} D d t+\frac{-\left(1-\widetilde{\mathrm{k}_2}\right)}{\widetilde{\theta_2}}\right. \\ \left.\int_{t_i}^{s_{i+1}}\left\{e^{\widetilde{\theta_2}\left(s_{i+1}-t\right)}-1\right\} D d t+\frac{-\left(1-\widetilde{\mathrm{k}_3}\right)}{\widetilde{\theta_3}} \int_{t_i}^{s_{i+1}}\left\{e^{\widetilde{\theta_3}\left(s_{i+1}-t\right)}-1\right\} D d t+\frac{-\left(1-\widetilde{\mathrm{k}_4}\right)}{\widetilde{\theta_4}} \int_{t_i}^{s_{i+1}}\left\{e^{\widetilde{\theta_4}\left(s_{i+1}-t\right)}-1\right\} D d t+\left(D \beta\left(s_i-t_i\right)\right)^3\right)\end{gathered}$

Illustrative scenario: $D=40, O_r=0.5, \widetilde{k_1}=0.01$, $\widetilde{k_2}=0.02, \widetilde{k_3}=0.03, \widetilde{k_4}=0.04, H=420, \tau=0.2, \widehat{h_c}=$ $0.1, D_t=1, \widetilde{\theta_1}=0.001, \widetilde{\theta_2}=0.002, \widetilde{\theta_3}=0.003, \widetilde{\theta_4}=$ $0.004, \beta=0.1, W_h=0.3, \hat{P}_r=0.2, c^{\wedge}=0.1, F_c=1.2$, $d=2, e_1=2, C_1=0.04, e_2=2, v_c=3, C_2=0.2, S_s=$ $430, P_s=0.001, l=0.02, s=2$. Eqs. (18) and (19), depicting nonlinear systems, were resolved utilizing Mathematica version 12 mathematical software, employing a numerical iterative technique to solve the nonlinear differential equation.The optimal condition of the overall system cost and replenishment cycles may be noticed in Tables 2-4, Figure 3, and Figure 4, respectively, for all the values specified in example 1.

Table 2. Optimal replenishment time interval, total cost table of retailer, supplier, and total quantity for example 1

Table 3. Optimal time interval for shortage, total cost table of retailer, supplier, and total quantity for example 1

Table 4. Total cost table of retailer for example 1

3.png

Figure 3. Graphical representation of the replenishment time and shortage time

4.png

Figure 4.Graphical representation of the optimal values of total cost of retailer

The most optimal solution occurs at node 6, where $t_1=$ $1.54979, t_2=2,03524, t_3=2.52702, t_4=3.01879, t_5=$ $3.51056, t_6=4.00234$ and $s_1=0, s_2=1.54113, s_3=$ $2.03291, s_4=2.52468, s_5=3.01645, s_6=3.50822$, and the total cost is 35.0379.

In summary, the present paper introduces a more practical inventory model designed for imperfectly decaying items, which incorporates a Machine Learning technique for seasonal demand prediction. The incorporation of deterioration rate and defective percentage quantity as fuzzy variables addresses inherent uncertainties, allowing for permitted shortages that are partially backlogged over the finite planning horizon.

Recognizing the pivotal role of product demand in business operations, particularly its seasonal variations, the study utilizes a time series predicting method to analyse seasonal forecasted demand. The results highlight the generation of direct month-wise predicted demand, offering managers valuable insights to enhance the management of inventory is based on expected demand. The Sign Distance method is used for defuzzification, aiding in the determination of optimal replenishment periods and ordering quantities that minimize total average cost, including emission costs over the finite planning horizon.

The numerical example confirms the robustness of the mathematical model, unveiling a significant decrease in overall costs when employing projected seasonal demand as opposed to fixed demand. Sensitivity analysis pinpoints critical parameters demanding heightened managerial attention. Graphical representation illustrates the curvature of the total cost function, emphasizing its highly non-linear nature.

To enhance the model's versatility, future extensions could explore alternative demand forecasting approaches, such as decision tree methods forecasting. Additionally, a comparative study across different forecasting methods could provide valuable insights. Further extensions may involve incorporating different fuzzy variables to account for increased uncertainty in parameter estimation.