Inventory system with generalized triangular neutrosophic cost pattern incorporating maximum life-time-based deterioration and novel demand through PSO

At the manufacturing plant or while the products are being transferred from one supply layer to another, there is a considerable possibility of receiving damaged or faulty items mixed in with non-defective commodities. This research focuses on the non-defective and defective products that are shipped to retailers by their suppliers. The retailer reworks faulty items to make them non-defective, and the retailer receives a discount on the cost of purchasing defective items. The presented inventory system addresses the uncertainty in inventory costs and also considers the deterioration of items with prioritized maximum product life. In this study, our aim is to minimize the total inventory cost when demand rate as a function of quality and power pattern of time under crisp and generalized triangular neutrosophic environments. Based on the payment deal, interest charges are imposed only when the payment delay has passed a particular allowable time limit. The neutrosophic number, which provides three different types of membership functions representing truth, hesitation, and falseness, is used in the inventory model to handle the cost pattern’s uncertainty. A particle swarm optimization approach is used to analyze the proposed inventory model, and the results are validated using a numerical example and sensitivity analysis for various parameters.


Introduction
Regarding the wide range of different inventory models, researchers need to emphasize a few essential elements, including degradation, demand, defective goods, the quality of the products, and so on. It is abundantly clear that degradation is a time-dependent component; furthermore, it worsens with time, which results in a reduction in the amount of interest in the product. There are a variety of challenges that arise when products are kept in warehouses, including variations in quality, volume, and degradation of commodities. These challenges are specific to each type of commodity. As a consequence, the holding costs have a considerable influence on the value of the quantity that is being stored. The quality of items that have been held for an extended period decreases B G. Durga Bhavani bhavani.nitpy@gmail.com G. S. Mahapatra gs.mahapatra@nitpy.ac.in 1 Department of Mathematics, National Institute of Technology Puducherry, Karaikal 609609, India due to deterioration. The rate of degradation is relatively low in long-lasting products such as those made of iron or steel, as well as toys, consumer electronics, furniture, tools, jewellery, vehicles, and other types of durable goods. Concurrently, the pace of degradation faces quick changes in semi-durable things such as food items, pharmaceuticals, apparel, and cosmetics, among other examples. As a result, the study of the deterioration of goods in inventory systems plays a vital role since the economic order quantity (EOQ) model accounts for various degradation patterns. When a product is being manufactured or when it is being shipped to retailers, faulty items are discovered most frequently. In the case of some goods, such as clothing, footwear, and furniture, defective products significantly impact the total profit generated by the inventory system. For this reason, when modeling the inventory system, the researchers need to concentrate on defective products. In recent years, a significant number of models have been created, each of which takes into account the deterioration rate and defective rate. Several researchers (Sahoo et al. 2019;Rani et al. 2019;Nagare et al. 2020;Jaggi et al. 2014;Tiwari et al. 2018;Pathak et al. 2013;Roy et al. 2020) have considered deterioration and defective items (Wee et al. 2007;Das et al. 2017; for the study of different inventory models. Demand plays an important role in inventory management, the study of inventory systems cannot be anticipated without addressing demand rates. Due to various facts of fluctuating demand in different situations, the inventory systems have been developed by considering demand depending on: price and stock dependent (Pervin et al. 2019;Khan et al. 2020;Bhavani et al. 2022;Pervin et al. 2017, quadratic form (Dari and Sani 2020), simply constant demand (Chen et al. 2020), etc. The mathematical structures for dealing with various practical implications of uncertainties are interval numbers, fuzzy numbers, intuitionistic fuzzy numbers, and neutrosophic numbers. Since three independent membership functions represent the feature of neutrosophic numbers (Das and Tripathy 2020) to deal with uncertainties, such as truth, indeterminacy, and falsity membership function, it has practical advantages: a generalization of interval, fuzzy, and intuitionistic fuzzy numbers. Researchers can make use of neutrosophic numbers in order to account for the fact that the majority of the metrics in a real market are not accurate. In recent studies, a large number of researchers have been building their models in a variety of different uncertain contexts. De et al. (2020) developed an economic production quantity (EPQ) model for the nonrandom uncertain environment using the neutrosophic fuzzy approach. Dotoli et al. (2017) introduced a fuzzy technique for supply chain network under quantity discounts. Barman et al. (2021) developed an inventory model under a cloudyfuzzy uncertain environment. Bonilla-Enriquez et al. (2021) proposed a supply chain model by taking uncertain demand.
This paper considers an inventory system which imposes an interest after a specific maximum length of grace time to pay the payment. The model of the inventory that has been developed takes into consideration the rate of deterioration, which depends on the maximum life-time of the product. The demand rate of this inventory model depends on quality and power pattern time. These scenarios of the inventory system have been developed by considering the impreciseness of cost parameters, and the optimal total inventory cost is evaluated via particle swarm optimization (PSO) algorithm. The following is how this article is structured. Section 2 offers a review of the literature that highlights the reason for the study as well as the research gap. The inventory problem is mathematically formulated in Sect. 3. Section 4 provides a new ranking method for generalized triangular neutrosophic numbers (GTNNs). Section 5 develops the model formulation in a generalized triangular neutrosophic environment. Section 6 contains the optimization of the proposed inventory model using the PSO algorithm. In Sect. 7, a numerical example is explained, and then a sensitivity analysis of the optimal inventory policy for the system input parameters is presented, along with some significant managerial insights obtained from the results. Section 8 concludes with some findings and research directions for the future.

Literature review
Researchers consider various types of deterioration rates to develop their models based on the nature of the products. Most of the item's deterioration rate increases with time, such items are fruits, vegetables, flowers, food items, etc., and a few items have constant deterioration rates, such items are electronic goods, toys, plastic items, etc. Skouri et al. (2011) proposed an inventory model of time-dependent deterioration rate. Tadikamalla (1978) introduced an EOQ model using gamma distribution for representing the constant, increasing, and decreasing rates of deterioration with time. Pal et al. (2014) proposed an EPQ model for deteriorating items with two-parameter Weibull distribution. Wang and Lin (2012) developed the optimal replenishment strategy with deterioration, market demand, and price changes. An inventory model for deteriorating items with maximum life-time proposed by Wang et al. (2014). Pervin et al. (2018) developed an inventory model by considering stochastic deterioration. Sarkar et al. (2015) proposed an inventory model with a trade-credit policy with variable deterioration for fixed life-time products. In actual life, if a product has a maximum life duration, the degradation rate of the product decreases. This model takes into account the rate of degradation over the product's maximum life-time.
In the actual market, the demand for the majority of items is affected by their quality, such as footwear, apparel, household items, fruits and vegetables, etc. For each item, the demand rate is different. The demand rate for some items rises at the start of the cycle. The demand rate for some items remains constant throughout the cycle, whereas the demand rate for others increases at the end. The demand rate for cooked items such as bread, sweets, cakes, etc., increases at the beginning of the cycle because customers love just-made goods. Due to the expiration date, the demand rate decreases at the end of the cycle for goods such as fish, vegetables, fruits, etc. On the contrary, others have an increasing demand rate at the end of the cycle, such as household goods such as oil, sugar, milk, etc., and the demand rate is constant for furniture, electrical goods, etc. Khedlekar and Shukla (2013) developed a dynamic pricing model for logarithmic demand. Sanni and Chukwu (2016) introduced an EOQ model with quadratic demand trends and quasi-partial backlogging. Dutta Choudhury et al. (2015) presented an inventory model by taking two components demand. Prasad and Mukherjee (2016) developed an inventory model on stock and time-dependent demand. Wu et al. (2018) developed an inventory policy for trapezoidal-type demand patterns and maximum life-time under trade cred-its. Shaikh and Mishra (2019) developed an EOQ model for price-sensitive quadratic demand and inflationary conditions.  developed an inventory system considering demand rate as a quadratic decreasing function of time. Al-Amin Khan et al. (2020) presented an inventory model by considering advertisement and selling price-dependent demand. By considering above-mentioned all scenarios, the demand rate of this model depends on quality and power pattern of time-dependent demand. A typical phenomenon always arises in the inventory management system is a lag in the payment process. A delay in payment is offered by the supplier, where the retailer's purchase cost is paid at a later date without an interest charge. Liao et al. (2000) proposed an inventory model with deteriorating items under inflation and permissible delay in payment. Several researchers (Sarkar et al. 2014;Khanra et al. 2011;Bhaula et al. 2019;Huang 2007;Ahmad and Benkherouf 2019;Aggarwal and Jaggi 1995;Geetha and Uthayakumar 2010;Chen and Teng 2014) developed their inventory models under the condition of delay in payment. Teng et al. (2011) presented an inventory model under progressive payment strategy and also studied an EOQ model under a trade credit financing scheme (Teng et al. 2012). Mahapatra et al. (2012) presented an EPQ model by taking limited available intuitionistic fuzzy-type storage space. In this paper, we consider a delay in payment for two different situations based on time. Neutrosophic numbers explain the impreciseness of the systems. In real life, most of the parameters are uncertain, so in this situation, neutrosophic numbers play a crucial role in overcoming uncertainty. Many models are developed under a neutrosophic environment. Mullai and Surya (2018) developed a price break EOQ model with neutrosophic demand, and purchasing cost as triangular neutrosophic numbers. In the present paper, we consider the cost pattern of the inventory system as a generalized triangular neutrosophic number (GTNN).
In general, classic direct optimization methods are used to solve inventory problems; however, one of the disadvantages of these approaches is that these techniques frequently become stuck on the local optimal solution. PSO is a precious tool for finding inventory control solutions since it helps avoid some of the flaws of global optimization. Biuki et al. (2020) presented an inventory problem in optimizing through two hybrid metaheuristics as parallel and series combinations of genetic algorithm and PSO. Alejo-Reyes et al. (2020) introduced inventory model for supplier selection and order quantity allocation by using metaheuristic algorithms. Rau et al. (2018) proposed a multi-objective green cyclic inventory routing problem via discrete multi-swarm PSO method. AL-Khazraji et al. (2018) applied multi-objective PSO to optimize the production inventory control systems. Patne et al. (2018) presented a closed-loop supply chain network configuration model by using game-theoretic PSO. Dabiri et al. (2017) presented a bi-objective inventory routing problem with a step cost function by using multi-objective PSO. Kundu et al. (2017) presented EPQ model with fuzzy demand using compared hybrid particle swarm-genetic algorithm. Manatkar et al. (2016) presented an integrated inventory distribution optimization model for multiple products by using a novel hybrid multi-objective self-learning PSO. Srinivasan et al. (2016) applied PSO for optimizing a mathematical model with defective goods. A multi-item EPQ model with a production capacity restriction PSO algorithm developed by Pirayesh and Poormoaied (2015). Al Masud et al. (2014) presented an inventory model with the quality of the production process through PSO. Based on the literature review on the topics considered for this study as mentioned above scenarios, this inventory management work has been presented exclusively based on a comparison of contribution with the existing articles explored in Table 1.

Mathematical development of the proposed inventory system
In the context of a generalized triangular neutrosophic cost pattern, the contribution of this work is to determine the optimal cycle time in order to cut down on the overall cost of the inventory management system. To the best of our knowledge, this study explores all of the concepts listed, which have never been studied together in the literature earlier: 1. The demand rate of the item follows a power demand pattern on time and product quality. 2. The retailer's product quality depends on the supplier's product quality, i.e., q(r ) = (1 − e −ar ), where a > 0 and 0 < r < 1. The function q(r ) is continuously differentiable with the conditions q (r ) > 0, q (r ) < 0. 3. Deterioration of this inventory system reduces with product's maximum life-time. 4. After receiving products from the supplier, the retailer distinguishes faulty and non-defective items through screening, and the defective items are sent to rework. In addition, the retailer gets a discount on the purchase price of faulty products. 5. Because of allowing for a delay in payment, this deal is beneficial to both the retailer and the supplier during the payment process. 6. The impreciseness of cost parameters are taken as GTNNs, and we proposed a new ranking method to change neutrosophic numbers to crisp numbers. 7. The goal is to minimize the total inventory cost under the neutrosophic environment.
123 Table 1 Exclusive The simultaneous review of the assumptions considered in this study enables us to portray a more realistic inventory model that can be used in various circumstances that occur in real life.

Problem definition
In this study, we presented the mathematical modeling of a real-world phenomenon called the growing rate of deterioration with time. The study assumes that an item's maximum life-time is already known to its providers. In the proposed inventory management system, shortages are not permitted, and the maximum life period is taken into consideration for the item in each replenishment cycle. In the process of modeling the inventory system, the demand is the most crucial aspect. Although the inventory system is dependent on a number of factors, quality is one of the essential attributes. This study considers the demand rate depends on the quality of the product and the power pattern of time, i.e., The demand rate for the product increases with the quality of the product as shown in Fig. 1 The deterioration rate of the inventory system depends upon the maximum life-time of the product, i.e., θ(t) = 1 m−t f or 0 ≤ t ≤ T < m. When the credit period is less than or equal to T p , the inventory system considers payment delays that are permissible. During the credit period, the buyer is not required to pay any interest to the retailer; however, interest will be charged for any period beyond the period T p as shown in Fig. 2. In order to construct a realistic inventory system, GTNNs are considered for the inventory cost parameters rather than set crisp values.

Mathematical formulation
Based on these considerations, the inventory system can be represented by the differential equation in the time interval [0, T ] as given below: subject to boundary conditions: where Q is the initial inventory per each cycle.

Lemma 1
The product's quality q(r ) and demand rate D(q(r ), t) is increasing with supplier product's quality r for a > 0.
Proof Since, the product's quality q(r ) of the retailer is given by q Thus, the product's quality q(r ) increases with the supplier product's quality r . Also the demand rate of the given model is Therefore, the demand rate D(q(r ), t) is increasing with the supplier product's quality r for a > 0. Hence, the proof.

Lemma 2
The demand rate D(q(r ), t) is decreasing with time for n > 1, increasing with time for 0 < n < 1 and constant with time for n = 1.
Proof Since, demand rate of the given model is It shows that the demand rate D(q(r ), t) increasing with time for all t.
When n = 1, we have It shows that the demand rate D(q(r ), t) decreases with time for all t.
Hence, the proof.
By using the boundary conditions I (T ) = 0 and I (0) = f Q, we find the solution of the differential Eq. (1), during the time interval 0 ≤ t ≤ T given as follows: Again using boundary condition Now using Eqs. (2) and (3), the holding cost (HC), deterioration cost (DC), the purchasing cost (PC) and the ordering cost (OC) of the proposed inventory model can be obtained as given in the succeeding equations. The total holding cost of non-defective and defective items given by The deterioration cost (DC) of decaying goods in inventory system is given by The purchase cost of the inventory system consider per item and require to purchase the item for inventory The cost of ordering is fixed for each cycle of unit inventory model Damaged or defective commodities may qualify for a reduction from the supplier in the amount of γ off the total cost of the purchase. Therefore, the reduction in the cost of purchasing defective products is: The inspection cost of the inventory system is given by Rework cost of the defective items of the inventory system is given by .
This inventory system allows a delay in payment conditionally for the buyer under certain circumstances. Because of this, the two possible outcomes are that either the delay period is longer than the cycle time or the delay period is shorter than the cycle time. Now, we will analyze the interests due to delayed payments for both situations in order to determine the total inventory costs for the two cases that follow: Case I: (T ≤ T p ) Delay period is greater than or equal to the cycle time The retailer earns interest at a return rate I e per cycle if T ≤ T p , then the annual interest earned is given by The total inventory cost per unit time is obtained as follows: The total inventory cost under the delay period is greater than the cycle time I C 1 is to be optimized to the optimal cycle T * using PSO. Case II: (T p < T ) Delay period is less than the cycle time: The interest charged by the retailer per cycle is obtained by The interest earned during the time 0 to T p is given by In the situation T p < T , the total inventory cost per unit time (I C 2 ) is obtained as: The total inventory cost under the delay period is less than the cycle time is to be optimized (I C 2 ) to the optimal cycle T * using PSO.

Generalized triangular Neutrosophic number and its De-neutrosophication
In real-life scenarios, most of the parameters are imprecise, which means inexact, invalid, or inaccurate, to overcome this type of impreciseness. The GTNNs can explain the truth, hesitation, and falsity of the given parameters. Neutro-normal Let us consider three points p, q, r for which, πS( p) = 1, θS(q) = 1, ηS(r ) = 1, then theS is defined as neutro-normal.

Definition 3 Single-valued neutrosophic set
Definition 4 (α, β, γ ) cut: (α, β, γ ) cut of a neutrosophic number is defined asS Definition 5 Neutro-convex A neutrosophic setS is called neutro-convex if the following condition holds: (i). πS(λα , f orn 2 < x ≤ n 3 1, otherwise , and , f or p 2 < x ≤ p 3 1, otherwise Several methods are exist for defuzzification of fuzzy numbers in the literature, this paper proposed new defuzzification technique for GTNN by using Rouben's Roubens (1990) ranking function which is given as follows for any fuzzy numberÃ with membership function μÃ.

Definition 7
The defuzzification of neutrosophic numberS with truth membership function πS, indeterminacy membership function θS and falsity membership function ηS is defined by m 1 m 2 m 3 0 p 1 n 1 p 2 n 2 p 3 n 3 Truth Indeterminacy Falsity

Fig. 3 Graphical representation of GTNN
where R r is the Rouben's ranking function given in the equation (16). Based on Equation (17)

Inventory system under generalized Neutrosophic environment
In a real market, the cost parameters are unpredictable, and the decision-maker is faced with a dilemma. As a result, we attempt to express the inventory system by introducing a generalized neutrosophic set to represent the proposed inventory system's various inventory charges and rates. For this study, we have considered ordering cost (C 0 ), holding cost (C 1 ), deterioration cost (C 2 ), purchase cost (C 3 ), inspection cost (C 4 ) and rework cost (C 5 ) as a GTNNs. The representation of GTNNsC 0 ,C 1 ,C 2 ,C 3 ,C 4 andC 5 are as follows: C 0 =< (a 11 , a 12 , a 13 ; μ), (a 21 , a 22 , a 23 ; ν), (a 31 , a 32 , a 33 ; ζ ) >, converted into a deneutrosophic values such asC 0(D) ,C 1(D) , C 2(D) ,C 3(D) ,C 4(D) andC 5(D) . To obtain the total cost in the generalized neutrosophic domain ( I C 1 , I C 2 ), substituting the values of the deneutrosophic valuesC 0(D) ,C 1(D) , C 2(D) ,C 3(D) ,C 4(D) andC 5(D) in the total cost of both cases in Eqs. (12) and (15), we get the total cost of both cases in Eqs. (12) and (15) as follows: Now, we will find the optimal total cost in a generalized neutrosophic environment from Eqs. (19) and (20) through PSO.

Optimization of proposed inventory model using PSO
PSO is a nature-inspired optimization technique that can be used to solve even the most challenging optimization issues. The movement and intelligence of swarms are impacted in PSO, which was developed by Kennedy and Eberhart (1995). The swarms are considered as vector points in the domain space where the optimum value of a given objective function lies. The four critical vectors in PSO that: (i) x vector: records the current position, (ii) p-vector: records the personal best, (iii) vvector: control the velocity of the moving particle at each instance, and (iv) g-vector: records the direction toward the global best position. Four essential parameters control the movement of particles; a parameter representing the coefficient of inertia ('w ) along with a damping inertial coefficient 'wdic', and two constants, c 1 representing the acceleration of individual swarms, and c 2 representing the social acceleration. These parameters are varied according to the choice of the optimization problem. To find the optimal total cost of the proposed inventory model, we present Algorithm 1 of the variant of the PSO technique as follows:

Numerical solution
A numerical example of the presented inventory system is shown to assist the analytical derivation with the following configuration: assume that the ordering cost is $250 per order, the cost of raw material $16 per unit, the cost for holding the item is $3 per item per unit time, the cost of deteriorating per item is $2.5, and the selling price of each item is $25.
Because the system allows both defective and perfect things, let's assume the percentage of non-defective items is 0.9, and the item's maximum life-time is three years. Let the rework cost of the defective items is $1.8 per unit, the inspection cost is $2.2 per unit, and the faulty item's purchase cost be discounted by 30%. The supplier accepts payment within a delay period without taking any interest charges during which the customer earns 5% interest, but after the permissible delay period assuming the supplier's annual interest rate on stocks is 8%. Let us further consider the parameters settings for the PSO: m = 3 years, T p = 1 year(case I), T p = 0.5 (6 months) (case II), I e = 0.05, I c = 0.08, f = 0.9, n = 1, d = 55, r = 0.95, γ = 0.3, g = 1.2 and d = 2.5. We use a generalized triangular neutrosophic set to numerically demonstrate the influence of the cost parameters' imprecision on the proposed inventory system. (170, 250, 310; 0.75), (190, 260, 310 : 0.25), (180, 250, 300; 0.25 Table 2 shows the optimal solution of the suggested inventory model in a generalized neutrosophic environment that was obtained using the PSO for the inventory system.
The PSO is implemented to find the optimal total costs I C 1 and I C 2 incorporating the generalized neutrosophic cost parameters, the parameters of the PSO algorithm are w = 1, wdic = 0.99, c 1 = 2, c 2 = 0.2. From Table 2, we can observe 123 Algorithm 1 : PSO Algorithm for generalized neutrosophic inventory model Step I. Read: Swarm Population (SwPop), Maximum Iteration (MaxIt), particle(i).Velocity (V i ), particle(i).Position (P i ), particle(i).Best.Position (P B P,i ), Global Best.Position (P B P,G ), particle(i).Cost (C i ), particle(i).Best.Cost (C BC,i ), Global Best.Cost (C BC,G ), Step II. Set the objective function: (a) Obj Function = @(T , P) I C 1 f cn(T , P); (Optimum for I C 1 ), (b) Obj Function = @(T , P) I C 2 f cn(T , P); (Optimum for I C 2 ) Step III. Set the parameters of PSO for Inventory system: Step VII. Optimize and update for every iteration: (a) for i T = 1 to MaxIt, and for i = 1 to SwPop update the velocity and position: Step VIII. Set up the global best cost: (a) Best.Costs(i T ) = C BC,G (b) Set w = w * wdic and continue the iteration by going back to 6.
Step IX. Stop that the total cost is higher at n = 0.75(0 < n < 1) since the demand rate increases with time for 0 < n < 1, so the total cost of inventory systems increases. The total inventory cost is less at n = 2(n > 1) as the demand rate decreases over time for n > 1, so the total cost of inventory systems decreases. Decision makers face difficulty in estimating the values of the cost parameters in real-world market situations. The decision-maker can determine optimal values for cost parameters based on market conditions using neutrosophic figures. Neutrosophic numbers have membership functions for truth, hesitation, and falsity, which are helpful for overcoming imprecision in cost parameters. The decision-maker can determine values for cost parameters based on market conditions using neutrosophic numbers. Table 2 shows that I C 1 < I C 2 for all values on n, i.e., the total inventory cost of case I is less than the total inventory cost of case II since the delay period of case I is greater than the cycle time (T ), which helps to conclude that the total cost decreases with increasing delay period.

Sensitivity analysis
We undertake a sensitivity analysis based on the numerical example of the inventory system described in Table 2 for n = 1 case in the previous section by modifying the parameters by a specific percentage depending on the parameter threshold. We consider one parameter at a time for the sensitivity analysis and keep the other parameters fixed.

Table 3
Sensitivity analysis for the different inventory related cost  Observations from result of Table 3: 1. Total inventory cost functions I C 1 and I C 2 are highly sensitive to the purchasing cost (C 3(D) ). If the purchase cost (C 3(D) ) increases, the total cost of the inventory system increases dramatically and the cycle duration (T ) decreases moderately. If the purchase cost (C 3(D) ) increases, then the retailer reduces the percentage of purchase items so the cycle length (T ) decreases in the medium (Fig. 4a). 2. The ordering cost (C 0(D) ) is moderately sensitive to total inventory costs ( I C 1 , I C 2 ) as shown in (Fig.4b). Due to an increase in the ordering cost (C 0(D) ), the total inventory cost increases moderately and the cycle length increases (T ) much. The cycle length (T ) is highly sensitive to the ordering cost in both cases. Because if (C 0(D) ) increases, the retailer orders a bulk amount of goods at a time that causes the cycle length (T ) automatically increases. 3. The total inventory cost functions I C 1 and I C 2 are moderately sensitive to the holding cost (C 1(D) ). The holding cost increases by 20%, then the total inventory cost functions I C 1 and I C 2 increase in an analogous manner obtained from Fig. 4c. If the holding cost (C 1(D) ) increases in this model, then the cycle length (T ) of the inventory system moderately decreases. Because if holding costs increase, retailers try to reduce stock or send inventory as much as possible to customers, so that the cycle length (T ) decreases. 4. The total inventory costs are moderately sensitive to (C 4(D) ), and the cycle duration (T ) is less sensitive to inspection cost (C 4(D) ). Due to the increase in (C 4(D) ), the cycle length decreases and the total cost increases in this model (Fig. 4d). 5. Furthermore, I C 1 , I C 2 and T are less sensitive to the deterioration cost (C 2(D) ) and the rework cost (C 5(D) ) as shown in Table 3. Due to the increase in these costs, the total cost increases and the cycle length decreases, but it will not show much impact on the inventory system.
Observations from the result of Table 4: 1. The total inventory cost functions I C 1 and I C 2 are highly sensitive to the demand parameter d. If the average demand for the inventory system increases in this inventory model, the total cost increases and the cycle length (T ) decreases dramatically. Because if the demand increases, then item sales increase because of that the stock gets over soon so the cycle length (T ) decreases, as shown in Fig. 5a. 2. The maximum life-time of the product m is moderately sensitive to total inventory costs as shown in Fig. 5b and highly sensitive to cycle length (T ). Due to an increase in a maximum life-time, total inventory costs decrease moderately and the cycle length increases highly in this model. 3. The total inventory costs and cycle duration are moderately sensitive to the quality of the supplier's product r . Due to the increase in supplier product quality, the cycle length (T ) decreases and the total cost increases, as shown in Fig.5c. If the supplier's product quality increases, then the retailer's product quality also increases, and hence the cycle length (T ) decreases due to the increase in demand. Table 4 shows that the sensitivity analysis has taken changes of r from −5% to 5% because if a percentage of r increases by more than 5%, then r exceeds its permissible value. 4. The non-defective rate of item f is highly sensitive to total inventory cost functions ( I C 1 , I C 2 ) as shown in Fig. 5d and moderately sensitive to cycle length (T ). If the non-defective rate of items in the inventory system increases, then the total cost decreases and cycle length (T ) increases because of the relaxation of the rework cost. Table 4 shows that the sensitivity analysis has taken changes of f from −10% to 10% because if a percentage of f increases by more than 10%, then f exceeds its permissible value.
5. The total inventory costs and cycle length are less sensitive to discount on the purchase cost of defective items γ . If the discount on defective items increases from the supplier, then the total cost of the inventory system decreases in both cases but does not have much impact on the total cost.

Managerial perspective
Any product's value is usually time-dependent and the cost of such things will rise over time, whereas the price of a few select products will fall. Because most food items are perishable, their value will depreciate over time as the storage period is extended. A few select products' prices will rise as the storage period lengthens, similar to wine. This study persuaded decision-makers that perishable food should be stored in a healthier atmosphere to prolong shelf life and slow deterioration. From a managerial standpoint, this research increases the quality of the product. The demand for a well-maintained product will be stronger, resulting in more earnings during sales. Furthermore, management may have to invest higher holding costs to keep such a product. This indicates that we must strike a balance between these aspects in order to keep our total inventory cost low.
The inventory system's flow has preserved the efficiency of decaying items, cost trends and quality criteria. As a result, this research can be used to manage the inventory system regarding quality and deterioration rates.
As a result, the current inventory model was designed to represent these realistic qualities that can be used to forecast the components of the system. This model is vulnerable to the purchase cost, non-defective rate f and demand parameter d, as shown in the numerical analysis section, demonstrating the model's fundamental character. As a result, the decisionmaker should focus on choosing these parameters.
The study of generalized neutrosophic cost parameters helps the decision-maker decide the appropriate value for the uncertain cost parameters. The decision-maker can deal with items with varying demand rates from this innovative demand rate.
The total inventory cost is susceptible to the non-defective rate f , as shown in the sensitivity Table 4. Because the quality of the product acquired from the supplier affects the demand rate in this model, the decision-maker should have reasonable concern about the product quality.

Conclusion and future direction
An optimum order quantity inventory model without shortages in selecting quality goods, the time-dependent power pattern, and the permissible time delay is presented in this 123 Table 4 Sensitivity analysis for the different inventory-related parameters article. The inventory system is improved, the nature of objects for time-dependent deterioration factors is addressed, and the cost pattern is approximated for a real-world scenario. The proposed inventory management system includes various sophisticated and vital features to order and store products with varying uncertainties. The main benefit of this research work is that it can handle a wide range of product demand rates. This research on inventory management sending out defective goods for rework reduces the number of defective products. The environment benefits from the reduction of damaged items and waste. The merchant will undoubtedly focus on product quality to ensure that clients receive high-quality goods by considering the quality-dependent demand. The concept of impreciseness in the cost pattern is based on GTNNs. The inventory model with quality-dependent demand was addressed and formulated in this work, and the solutions were analyzed by changing various parameter values. A PSO algorithm is presented to determine the optimal total cost in two different time delay scenarios. In the generalized neutrosophic environment, PSO is utilized to find the solution and analyze the sensitivity of the inventory parameters.
This model does not permit shortages to occur through it frequently occurs in the inventory system. This is the disadvantage of the proposed inventory system which can be expanded in due course of time. Different inventory manage-ment models can be presented for multi-items with varying parameters based on this proposed study. More studies on inventory models can be performed in a probabilistic or probabilistic environment using various types of uncertainty, such as intuitionistic fuzzy, Hesitant fuzzy, Fermatean fuzzy, interval type-2 fuzzy, and many more.