Investigating a Sustainable Inventory System with Controlled Non-Instantaneous Deterioration for Green Products via the Dragonfly Algorithm

Abstract

Sustainability is essential in addressing the environmental impacts of supply chains, a significant source of global emissions. This study develops an inventory model to optimize retailer profit by integrating joint pricing, environmental investment, ordering costs, preservation technology, and replenishment timing for non-instantaneously decaying items. Demand depends on stock and selling price, while an algorithm optimizes variables such as selling price, preservation investment, emission costs, ordering costs, and replenishment cycles. The dragonfly algorithm (DA) is employed to find optimal solutions, with numerical analysis demonstrating the model’s application. To justify the results, we have used an updated version of the dragonfly algorithm. Managerial insights highlight the practical relevance of the proposed framework.

1. Introduction

The development of an industry must be focused on sustainable growth. The cause of global warming is increasing carbon emissions. The majority of wealthy nations are making an effort to lower carbon emissions by implementing innovative technology (Lin [1]; Annadurai and Uthayakumar [2]; Lee et al. [3]; Chang et al. [4]). Regardless of the order level, the price of the products is always the same and the demand rate has a fixed value. Demand for flowers is ascertained by pricing policies. The sales rate of the greenhouse store suffers when selling prices are raised. One aspect can affect the level of demand in the market for easily accessible sustainable inventories. Additional consumers are drawn to an array with more products. Items of this kind are particularly prevalent in greenhouse farms. Both the maximum number of flowers produced and the rates of carbon emissions rise with demand. Environmental concerns should be taken into account when determining the business period and price for a degrading item because they have a bigger impact on business. Products that are not meant for continuing use might spoil, alter, or decay; this is known as degradation. Flowers, wine, fuel, seasonal goods, and seafood are a few examples.

According to conventional deteriorating inventory models, the producer should pay the purchase price at the time of placing an order. In order to prevent order cancelations, the buyer may, if desired by the wholesaler, make payments in advance of the time of shipment (Teng et al. [5]). The best approach to boost a specific product’s sales is through payment. To maximize their profit from the producer, customers could choose to pay the purchase price in multiple installments. Setting up a new process to create a particular product should incur additional costs for the producer. Initially, the producer merely tries to recoup a small portion of the purchase price. These kinds of programs are typically thought to make sure the customer does not cancel the order. Previous research has looked into environmentally carbon-emitting long-term supply networks with a decreasing price strategy (Dye and Yang, [6]). The expense of keeping deteriorating goods, including seasonal goods, goes up since it is crucial to maintain the goods’ quality and guard against deterioration. In actuality, because there is no deterioration in the winter, the majority of objects keep their original or good state. The non-instantaneous rate of deterioration is the term used to describe these processes. In a greenhouse flower farm, non-instantaneous deterioration takes place; during the winter, flowers remain fresh, but during the summer, they begin to wilt.

Because greenhouse emissions and environmental degradation occur at a faster rate than when they were first produced, greenhouse flower products lose quality. Purchasing high-quality monitoring equipment for the greenhouses lowers the rate of environmental pollution and degradation. Within the framework of a greenhouse flower farm, profitability also depends on controlled order costs (Chang et al. [4]; Lee et al. [1]; Annadurai and Uthayakumar [2]). It has been demonstrated that controlling the order cost may have a direct or indirect impact on lot size. Order costs can be managed in real-world scenarios in a number of ways, such as by purchasing equipment or providing training to greenhouse workers. A sustainable model for an inventory system that is nonlinear and does not degrade instantly has not been developed by many studies in the literature. In order to lessen the rate at which emissions are released into the environment and degradation, none of the current models show the effects of using cost-nonlinear, inventory-level demand in flower farms with greenhouses. The question of how to create an environmentally friendly inventory model that takes into account deterioration and reduction in environmental emissions in a greenhouse flower farm emerges in light of these variables. A retailer who sells greenhouse-related products aims to strike the best possible balance between retail selling price, investment costs, and sustainable inventory cycle time. This study integrates the idea of ordering strategies, where demand is inventory-level and nonlinear in price and of a green level for adjustable carbon emissions in the environment, degradation, and ordering cost for monitoring a non-instantaneous declining sustainable inventory supply chain. This strategy creates a new method of problem-solving for the greenhouse flower farm sector.

The following is how the remainder of this paper is structured. A review of the pertinent literature is given in Section 2. Assumptions and helpful notation are included in Section 3. The model formulation and solution process are covered in Section 4. A sensitivity analysis and case study are presented in Section 5 and Section 6. In Section 7, some last thoughts, constraints, and recommendations for further research are presented.

2. Literature Survey

These two sections comprise inventory management models and environmentally friendly inventory models. While demand and degradation are assumed to serve separate purposes in the first section, the second section shows an environmentally friendly inventory system that accounts for emissions of carbon.

2.1. Models Related to Inventory

Harris examined a fixed demand rate in his examination of a conventional inventory system. Baker developed the Harris model by Baker and Urban’s [7] model based on the notion that inventory level influences demand rate. Hou and Lin [8] enhanced an inventory model under inflation, depending on selling price and inventory level. Within a limited planning timeframe, they computed the replenishment time using the Discount Cash Flow (DCF) technique. They also found a useful technique for calculating the length of the cycle, the selling price, and the overall amount of replenishment. Chang et al. [4] modified a conventional system of inventory with modifiable lead times and ordering fees. They found that the pricing was impacted by the lead time for customers. Next, Lee et al. [1] used a computational algorithmic technique to optimize decisions by looking at a framework with a lower-order cost and reserve discounts over lead time. In an effort to boost sales, Alfares [8] looked at an overall inventory system where the maximum availability of an item and level of inventory is determined by the rate of demand. To determine the order amount and cycle duration, they also looked at the current inventory system, which covers the total inventory. Min and Zhou [9] studied decaying, price- and inventory level-dependent rate, and capacity-limited backlogging. They looked into how a ceiling can affect inventory levels.

By adding a trade credit policy, Geetha and Uthayakumar [10] enhanced a system of inventories with gradual degradation. They disclosed that credit policies and non-instantaneous deterioration depend on shortages. They used a variety of theorems to describe the optimal solutions and came up with a minimal quantity of goods and time needed for restocking. Deane and Agarwal [11], to determine the maximum profit at specific frequencies, created a model of display frequencies using scheduled internet ads. They detailed the proportionate area occupied by the fixed-display-frequency paradigm. The Deane and Agarwal [11] model was enhanced by Shah et al. [12]. Pando et al. [13] introduced a policy related to inventory by accounting for an accumulation and rate of demand that depends on the level inventory. They used theoretical proof to enhance the per unit profit. Maihami and Abadi [14] developed a pricing model for a not instantly degrading item, wherein the buyer’s demand function is dependent upon price-time under shortages. They made decisions about the price of sale, cycle duration, and order quantity at the same time in order to maximize overall profit. Pal and Mahapatra [15] created a model of continuously declining inventory in which price and stock levels influence demand. Yang [16] added a zero-end relaxed condition to an existing model with holding costs and stock-dependent demand. In order to evaluate the effect of the backorder inventory policy, Maihami and Karimi [17] made certain assumptions. They then used these data to show a non-instantaneous degrading item system of inventory with a price-dependent demand. The best selling price, order quantity, and replenishment schedule were determined. Using a damaged product and a backorder price discount, an optimization of the production–inventory policy was extended by Annadurai and Uthayakumar [2]. Chang et al. [18] improved a delay credit model. To overcome the order quantity problem, they also looked into the effects of non-instantaneously degrading products with related credits. Teng et al. [5] modified the ordering amount management system with time limitations and payment in advance for depreciating goods. They employed both theoretical and numerical techniques to establish the lowest and ideal cycle times.

Finding the minimal cost/maximum profit for producers relies heavily on controlled deterioration. Dye and Hsieh [19] developed a model where they considered a constant carrying cost and a backordering scenario decrease in deterioration rate with extra investment. In a backlog scenario, they calculated that the increase in instantaneous deterioration rate varies with time, along with the expenditure cost and replacement time. Dye [20] modified a model that included a non-instant degradation rate where backlogging situations appear. The goal of the Dye [20] problem is used to ascertain how inventory choices affect the cost of preservation investments. Fractional programming produced some of the data that were used to demonstrate the biggest profit globally. He and Huang [21] looked at a scenario where the seasonal deterioration of products was managed by paying for high-tech equipment preservation. They also looked at a few comparable papers before taking into account both the seasonal and deteriorating properties at the same time. Assuming an instantaneous production model, Hsieh and Dye [22] expanded on the He and Huang [21] model. They also calculated the total cost, additional investment cost, and business period. Building on the study of Hsieh and Dye [22], Mishra [23] examined a product that degrades instantly for a demand that is trapezoidal. Hsieh and Dye [22] discussed how the cost was impacted overall by the preservation investment. Subsequently, Liu et al. [24] expanded on Hsieh and Dye’s [22] work by presuming a preservation strategy and dynamic pricing approach for goods that are deteriorating and where the price–quality relationship exists. Alfares and Ghaithan [25] studied an inventory model with quantity discount with price-dependent demand and a time-varying holding cost. Once more, to assess the impact of preservation investment, Mishra [26] employed a systematic procedure to study an instantly decaying manufacturing system. Bardhan et al. [27] extended the replenishment strategy and preservation investment model for non-instantaneous degradation in the presence of inventory level-dependent nonlinear demand. Both with and without preservation expenditure, their solution is contrasted. Sustainability in inventory and production management has gained significant attention in recent years due to the growing emphasis on environmental conservation and the need for efficient resource utilization. With increasing awareness of the environmental impact of industrial operations, researchers have explored various approaches to incorporate sustainability into inventory and supply chain management. This has led to the development of innovative models and optimization techniques aimed at balancing economic, environmental, and social objectives. Several studies have addressed sustainable inventory management by considering the integration of trade credit policies, carbon emission constraints, and backordering systems. Sarkar et al. [28] analyzed inventory systems under partial backordering and multi-trade-credit periods to evaluate their environmental impact. Similarly, Taleizadeh et al. [29] focused on sustainable economic production quantity models that incorporate shortages to enhance sustainability in inventory systems. Extending this research, Tiwari et al. [30] proposed sustainable ordering policies for deteriorating items under carbon emission and trade credit considerations. The incorporation of carbon emissions into supply chain models has been a key focus, particularly in systems involving deteriorating and imperfect-quality items. Tiwari et al. [31] and Ahmadini et al. [32] explored inventory systems that account for carbon emission compliance and green production strategies, highlighting the trade-offs between environmental and operational objectives. Furthermore, Manna et al. [33] investigated green production inventory problems with selling price- and green-level-sensitive demand, utilizing metaheuristic algorithms to solve complex optimization problems. In perishable product supply chains, sustainability challenges are further exacerbated by the need to minimize waste while maintaining customer satisfaction. Goli et al. [34] addressed these issues using hybrid optimization algorithms, while Mashud et al. [35] developed sustainable inventory models for green warehouse farms with controllable carbon emissions. Pan et al. [36] expanded this scope by optimizing perishable product supply chain networks through hybrid metaheuristic approaches. Metaheuristic algorithms have emerged as powerful tools for solving complex sustainable inventory problems, as evidenced by recent studies. Sadeghi et al. [37] utilized the Grey Wolf and Whale Optimization algorithms for stochastic inventory management in a two-level supply chain of reusable products. Das et al. [38] employed the Water Cycle Algorithm to model inventory systems with nonlinear demand patterns influenced by green levels.

This paper aims to build on these advancements by exploring innovative strategies for sustainable inventory and production systems, incorporating novel optimization techniques and addressing practical challenges in green supply chain management. The objective is to provide actionable insights for decision-makers to achieve sustainability goals while maintaining operational efficiency.

2.2. Inventory Model with Sustainability

A study by Benjaafar et al. [39] examined carbon-footprint-conscious sustainable supply chain policies. They expanded their approach by incorporating operational modifications for lowering carbon emissions. They also calculated the effect of greenhouse emissions on the supply chain using their model. Without raising prices, Chen et al. [40] expanded a low-carbon inventory model that is sustainable. They applied the sustainable inventory problem, with the requirement that order amounts be reduced in order to minimize carbon emissions. The impact of deterioration was not addressed. According to their calculations, the savings from reducing carbon emissions outweigh the expenses. Toptal et al. [41] created a policy framework for the emissions of a carbon regulatory inventory. Along with the investment strategy for reducing emissions under various emission strategies, they examined the retailer’s collaborative guidelines for sustainable inventory management. Additionally, they expanded their methodology to include lot size into an emission reduction requirement in a trade–tax–cap combination method. For a long-term two-state strategy, Lou et al. [42] investigated a co2-related pollution trade with investments in eco-friendly technologies to limit emissions. Retail prices and environmental investment cost were also examined. In order to keep the fewest possible price combinations, tax exemptions are preserved and it was found that the lowest carbon items demand the highest costs. Research on an immediate sustainable inventory that incorporates cycle length and emission limits was carried out by Manna et al. [43]. They conducted thorough treatments to identify the consequences of emissions and analyzed their models under various environmental regulations.

An environmentally friendly supply chain with a credit policy was extended by Qin et al. [44] using a carbon tax and cap–trade scheme. In addition to discussing endogenous and exogenous credit strategies, they arrived at three distinct answers. In order to determine the highest profit from lowering emissions under a carbon tax policy, Datta [45] and Hovelaque and Bironneau [46] introduced a model of supply chain inventory. Lin [1] conducted a study on environmentally friendly inventory models with backordering, focusing on investment reduction for transportation emission costs. Tiwari et al. [47] investigated a sustainable deteriorating inventory management system with an ideal rate of emission. Tiwari et al. [48] solved a multi-item sustainability manufacturing for green production model that took trade financing and shortages into account. In order to create a second-generation biofuel, a carbon-emitting efficient supply chain system was developed by Ahmed and Sarkar [49]. Alharbi [50] developed a manufacturing inventory model that takes into account a product’s warranty, degree of greenness, and carbon emissions. This study evaluates how an item’s greenness, warranty, and technology, which reduce carbon emissions into the atmosphere, affect total profit in order to assist decision-makers in making better-informed choices regarding pricing and restocking. Research on the relationship between order cost, price nonlinearity, stock-dependent demand, regulated non-instantaneous deterioration systems, and environmentally conscious supply chain inventory policy is lacking. This research provides a practical, ecological supply chain inventory policy that accounts for greenhouse product environmental laws and non-instantaneous decay. Bardhan et al. [27] presents a method that is utilized in the adjustable non-instantaneous degrading inventory system. It is therefore challenging to remain profitable in today’s climate without taking sustainability into account. In a variety of corporate and economic contexts, supply chain operations are significant producers of emissions into the environment. Greenhouse companies have the option to invest more money in system improvements in order to reduce emissions and deteriorating components. Thus, many practitioners and researchers are interested in this environmentally conscious inventory management with limited carbon emissions. The increasing focus on sustainability has profoundly influenced inventory and production management practices, driving the need for innovative models that balance environmental and economic objectives. Sustainable inventory management incorporates strategies to minimize environmental impacts, optimize resource utilization, and align with global sustainability goals. Researchers have explored diverse approaches, including collaborative investments, demand-responsive models, and carbon emission reduction strategies, to develop efficient and environmentally responsible inventory systems. One prominent area of research is the integration of economic production quantity models with sustainability considerations. Taleizadeh et al. [51] analyzed sustainable economic production quantity models, emphasizing the importance of addressing shortages in inventory systems. Similarly, Dey et al. [52] proposed an integrated inventory model that combines selling price-dependent demand, setup cost reduction, and investment in variable safety factors, showcasing the multifaceted nature of sustainable inventory management. The impact of carbon emissions on production and inventory decisions has been extensively studied. Lu et al. [53] introduced a Stackelberg game approach for a sustainable production–inventory model with collaborative technology investments to reduce carbon emissions. In parallel, Sepehri et al. [54] developed a model incorporating imperfect-quality items, preservation technologies, and quality improvement investments to achieve sustainability goals. Comprehensive reviews and bibliometric analyses have further highlighted the evolution and trends in sustainable inventory management. Becerra et al. [55,56] reviewed green supply chain models, emphasizing the integration of sustainability into inventory practices and identifying opportunities for further research. Salas-Navarro et al. [57] conducted a bibliometric analysis of inventory models in sustainable supply chains, shedding light on emerging themes and research gaps. Specific challenges, such as managing deteriorating items and composite demand patterns, have also been addressed in recent studies. Pervin et al. [58] explored sustainable inventory models for non-instantaneously deteriorating items with composite demand, focusing on environmental impacts. These studies demonstrate the growing recognition of sustainability as a critical component of modern inventory management. San-José et al. [59] extended these efforts by examining inventory models for deteriorating items under power demand and carbon emission taxes. This paper builds on these foundational works to advance sustainable inventory management practices. By integrating novel approaches, such as collaborative investments, demand-driven strategies, and environmental impact considerations, this research aims to provide actionable insights for practitioners and policymakers seeking to balance economic efficiency with environmental responsibility.

It is challenging to succeed in the modern world without taking sustainability into account. The repercussions of climate change are increasingly apparent in day-to-day living. One of the main sources of pollutants in the environment is the supply chain. This consequence increases the significance of creating a supply chain inventory model in order to reduce emissions. In an attempt to maximize retailer profit, a replenishment problem for a non-instantaneously decaying item is addressed, based on cooperative prices, dynamical expenditures for sustainable cost, ordering cost, technology for preservation cost, and optimum replacement timing. In this case, the stock and selling price are the variables that affect the rate of demand. The price of sales, the amount invested in preservation technology, the cost of environmental emissions, the cost of orders, and the replenishment cycle time are all determined by a supply chain inventory problem-solving algorithm. After a theoretical analysis of the ideal characteristics of the suggested model, potential solutions are put forth. The model’s application is demonstrated by a numerical study. Practices are given managerial implications. The results of the inquiry demonstrate the practical applications of the suggested paradigm.

3. Notation

The following glossary of terms and underlying presumptions, which presented in

Table 1. Notations of the model.

Symbols	Units	Common Interpretations
A	USD/order	Ordering cost
$A_0$	USD/order	Fixed ordering cost
$d(p,g,I(t))$		Demand of the product
$\xi$	%	Investment of preservation
${\varphi }_t$	%	Emission cost related to the environment
${\varphi }_t^0$		Average emission cost
I(t)	units	$\mathrm{Inventory} \mathrm{level} \mathrm{throughout} \mathrm{the} \mathrm{duration} 0\le t\le T$
$S$	Kg/unit	Carbon cap
$C_h$	$/unit/time unit	Cost of unit holding per time unit
$C_d$	$/time unit	Deterioration cost
Q	units	Arranging the sizes in every cycle
$\varpi$	$/unit/time	Cost of opportunity against capital investment
$C_p$	$/unit	Unit cost of purchase
${\theta }_0$		Deterioration rate
$\pi (g,p,T)$	$/time unit	Profit per unit time
		Decision variables
T	Time unit	Cycle length.
$p$	USD/unit	Price of sale per unit
$g$	--	Green level

. aids in precise model analysis.

Assumption

There must be just one non-instantaneously decaying object for the study.

Lead time is negligible.

There is often a shift in the economic order quantity (EOQ) when ordering costs are reduced. Order-to-quantity (EOQ) ratios show how best to balance holding and ordering costs while minimizing overall inventory expenses. A greater EOQ—that is, fewer, larger orders—is typically the result of a reduced ordering cost. The reorder point is influenced by the ordering cost decision K, which is a financial investment. Hence, $L(K)$ is the investment of capital in order to reduce the ordering cost $L(K)={\displaystyle \frac{1}{\psi }}\log \left({\displaystyle \frac{K_0}{K}}\right)T$ for $0<K\le K_0$, where $T$ is the total cycle time (Chang et al. [4]; Lee et al. [3]; Annadurai and Uthayakumar [2]). The capital investment cost is ${\displaystyle \frac{\mu }{\psi }}\log \left({\displaystyle \frac{A_0}{A}}\right)T$.

In this case, a product’s price is determined by demand rather than being fixed. This pricing technique, which is frequently observed in dynamic pricing models, modifies the product’s price in reaction to variations in demand. For instance, the price can go up during times of high demand and down during times of low demand. A product’s sustainability or friendliness to the environment is measured by its “green level.” The utilization of recycled materials, energy efficiency, carbon footprint, and total environmental impact are a few examples of this. For customers that care about the environment, the green factor is frequently a major selling feature and can have an impact on decisions about what to buy. A company’s inventory level is the amount of a product that it keeps on hand. Maintaining a balance between inventory holding costs and consumer demand is essential. Lead times, demand fluctuations, and ordering costs are a few examples of the variables that can impact inventory levels. There is a complicated link between these variables. In the event that demand is insufficient to warrant a price increase, inventory levels may increase even though a higher green level may be justified by a higher price. To maintain a balance between supply and demand, pricing tactics may need to be modified if demand is high but the green level is low. The rate of demand is $D(p,\hspace{0.17em}g,I(t))=d(p,g){\left[I(t)\right]}^{\eta },0<t\le T$, $I(t)$ is the level of stock, and $0<\eta \le 1$ is the nonlinear stock-sensitive parameter (Bardhan et al. [27]). In addition, $d(p,g)\hspace{0.17em}(>0)$ is a nonlinear function of the price and green level of the product, i.e., $d(p,g)\hspace{0.17em}=\hspace{0.17em}a{p}^{-b}+c{g}^{\epsilon }$, where $a>0$, $b,\epsilon >0$.

In inventory management, the term “nonlinear holding costs” describes circumstances in which the cost of carrying or holding inventory fluctuates according to the amount of inventory retained. Regardless of the quantity maintained, the holding cost per unit of inventory in a linear holding cost model stays constant. On the other hand, if the inventory level fluctuates, the holding cost per unit in a nonlinear model could go up or down. Mathematically, the holding cost of the inventory system is taken as h ${\left[I(t)\right]}^{\sigma }$, where $h,\hspace{0.17em}\sigma >0$ according to Yang [16].

The rate of deterioration is taken as constant $({\theta }_0)$.

The allocation of funds for the upkeep and protection of assets—typically tangible assets like machinery, infrastructure, or natural resources—is referred to as preservation investment. Preserving these assets is an investment that aims to keep them functional, prolong their useful life, and stop degradation. Maintaining assets in excellent functioning order requires routine maintenance and repairs to stop degradation. Preventive, predictive, and corrective maintenance are a few examples of this. The preservation investment $P(\xi )\hspace{0.17em}$ is taken into account. Here, preservation investment is taken as $P(\xi )\hspace{0.17em}=\hspace{0.17em}1\hspace{0.17em}-\hspace{0.17em}{e}^{-\lambda \xi }$, where $\xi$ is the investment cost of preservation and $\lambda >0.$ $P(\xi )\hspace{0.17em}=\hspace{0.17em}1\hspace{0.17em}-\hspace{0.17em}{e}^{-\lambda \xi }$ is a continuous function that is twice differentiable and concave with respect to the preservation investment, where $P(0)\hspace{0.17em}=\hspace{0.17em}0$, $\underset{\xi \to \infty }{\lim }P(\xi )\hspace{0.17em}=\hspace{0.17em}1$, and $P”(\xi )\hspace{0.17em}=\hspace{0.17em}-{\lambda }^2\hspace{0.17em}{e}^{-\lambda \xi }<0$ (Dye [20]; Bardhan et al. [27], Mishra et al. [60]).

The wide range of techniques, tools, and policies aimed at lowering the amount of toxic emissions released into the atmosphere are collectively referred to as emission reduction technologies. Pollutants including greenhouse gases, particulate matter, sulfur dioxide, nitrogen oxides, and volatile organic compounds are a few examples of what can be released during these emissions. Technologies for reducing emissions are essential for reducing air pollution, enhancing public health, and halting global warming. In order to develop novel solutions that can further cut emissions and build a more sustainable future, research and development in this area must continue. The emission of carbon to the environment is taken as ${\varphi }_t$. $\nu ({\varphi }_t)$ is the reduction investment against emission, which is expressed as $\nu ({\varphi }_t)={\displaystyle \frac{1}{\delta }}\log \left({\displaystyle \frac{{{\varphi }_t}^0}{{\varphi }_t}}\right)T$ for $0<{\varphi }_t\le {{\varphi }_t}^0$, where $\delta >0$ throughout the cycle $T$. Total emission investment throughout the business period is $\rho \nu ({\varphi }_t)={\displaystyle \frac{\rho }{\delta }}\log \left({\displaystyle \frac{{{\varphi }_t}^0}{{\varphi }_t}}\right)T$, where $\rho$ is the cost of opportunity (Chang et al. [4]; Lee et al. [3]; Annadurai and Uthayakumar [2])).

Deteriorated units are not repaired during an inventory cycle. Here, it is assumed that $T>t_1$.

A shortfall arises in an inventory system when the amount of a product that customers demand is greater than the amount of that product that is on hand. Inaccurate demand forecasts, unforeseen spikes in demand, production or delivery delays, or poor inventory management techniques are just a few of the variables that might cause shortages. It is not acceptable to be short on supplies.

4. Mathematical Formulation

Creating equations or relationships that represent the connections between selling price, green level, inventory level, and demand is the first step in creating an inventory model that includes these components. Equations that illustrate how variations in the selling price and green level impact demand as well as how demand responds to shifts in the level of inventory might be included in this. Such a model would aim to maximize profitability, minimize environmental effects, and satisfy customer demand by optimizing inventory management decisions, including pricing, green product design, and inventory levels. The model would probably be developed and solved using sophisticated mathematical and analytical approaches, taking into consideration the intricate connections between various components. Let us consider that a green product seller ordered Q units for starting his business. The product loses its freshness after a certain time period $t=t_1$. The inventory level reduces due to the consumption of the customers and deterioration throughout the business period and it reaches to zero at $t=T$. The solution process of the inventory system is presented in Figure 1.

Hence, the inventory system follows the following governing differential equations.

$${\displaystyle \frac{dI\left(t\right)}{dt}}=-d\left(p,g\right){\left[I\left(t\right)\right]}^{\eta }, 0<t\le t_1$$

(1)

$${\displaystyle \frac{dI\left(t\right)}{dt}}=-\left[1-P\left(\xi \right)\right]{\theta }_{0}I\left(t\right)-d\left(p,g\right){\left[I\left(t\right)\right]}^{\eta }, t_1<t\le T$$

(2)

With the boundary conditions $I\left(0\right)=Q, I\left(T\right)=0.$ Also, $I\left(t\right)$ is continuous at $t=t_1$.

Solving Equations (1) and (2) and using the boundary condition $I(0)=Q$ and $I(T)=0$, we obtain

$$I(t)={\left[Q^{1-\eta }-k_{1}t\right]}^{{\displaystyle \frac{1}{1-\eta }}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0<t\le t_1$$

(3)

$$I(t)={\left[{\displaystyle \frac{k_1}{k_3}}\left\{e^{k_3\left(T-t\right)}-1\right\}\right]}^{{\displaystyle \frac{1}{1-\eta }}},t_1<t\le T$$

(4)

where $k_1=d(p,g)\left(1-\eta \right),k_2=\left[1-P\left(\xi \right)\right]{\theta }_0\hspace{0.17em}\mathrm{and} k_3=\left(1-\eta \right)k_2$.

Then, by the continuity condition of $I(t)$ at $t=\gamma$ we obtain

$$Q={\left[k_1{t}_1+{\displaystyle \frac{k_1}{k_3}}\left\{e^{k_3\left(T-t_1\right)}-1\right\}\right]}^{{\displaystyle \frac{1}{1-\eta }}}$$

(5)

If we expand the exponential part of Equation (5), then we have

$$Q={\left(k_{1}T\right)}^{{\displaystyle \frac{1}{1-\eta }}}$$

(6)

The following list includes the inventory system’s related costs:

Ordering cost (OC) = ${\displaystyle \frac{\mu }{\psi }}\log \left({\displaystyle \frac{A_o}{A}}\right)T$;

Purchase cost (PC) = $C_{p}Q$;

$$\begin{array}{cc}\hfill \mathrm{Holding} \mathrm{cost} (\mathrm{HC})& = C_h\left[{\displaystyle \underset{0}{\overset{t_1}{\int }}{\left[I\left(t\right)\right]}^{\sigma }dt+{\displaystyle \underset{t_1}{\overset{T}{\int }}{\left[I\left(t\right)\right]}^{\sigma }dt}}\right];\hfill \\ & ={\displaystyle \frac{c_h}{k_1{k}_4}}\left[\left\{Q^{k_4\left(1-\eta \right)}-{\left(Q^{1-\eta }-k_1{t}_1\right)}^{k_4}\right\}+{\left\{k_1(T-t_1)\right\}}^{k_4}\right]\hfill \end{array}$$

where $k_4=1+{\displaystyle \frac{\sigma }{1-\eta }}$.

Deterioration cost (DC) = $c_d\left[k_2{\displaystyle \underset{t_1}{\overset{T}{\int }}I(t)dt}\right]={\displaystyle \frac{c_d{k}_2}{k_1{k}_5}}{\left[k_1\left(T-t_1\right)\right]}^{k_5}$;

Preservation cost (PRC) (PRC) = $\xi T$;

Carbon allowance income (CAI) = ${\varphi }_{t}ST$;

Environmental emission cost (EC) = ${\varphi }_{t}mQ$;

Environmental emission reduction cost (EERC) = ${\displaystyle \frac{\rho }{\delta }}\log \left({\displaystyle \frac{{\varphi }_o}{{\varphi }_t}}\right)T$;

Sales revenue (SR) = $p(Q-Q_d)$.

Therefore, the system’s profit is as follows:

$$TP=SR+CAI-OC-PC-HC-DC-PRC-EC-EERC$$

(7)

Consequently, average profit is determined by

$$ATP(p,g,T)={\displaystyle \frac{TP}{T}}.$$

(8)

i.e.,

$$ATP\left(p,g,T\right)={\displaystyle \frac{1}{T}}\left[\begin{array}{l}p\left(Q-Q_d\right)+{\varphi }_{t}ST-{\displaystyle \frac{\mu }{\psi }}\log \left({\displaystyle \frac{A_o}{A}}\right)T-C_{p}Q-{\displaystyle \frac{c_h}{k_1{k}_4}}\left[\left\{Q^{k_4\left(1-\eta \right)}-{\left(Q^{1-\eta }-k_1{t}_1\right)}^{k_4}\right\}+{\left\{k_1(T-t_1)\right\}}^{k_4}\right]\\ -{\displaystyle \frac{c_d{k}_2}{k_1{k}_5}}{\left[k_1\left(T-t_1\right)\right]}^{k_5}-\xi T-{\varphi }_{t}mQ-{\displaystyle \frac{\rho }{\delta }}\log \left({\displaystyle \frac{{\varphi }_o}{{\varphi }_t}}\right)T\end{array}\right]$$

(9)

It is now necessary to optimize the objective function (9) with respect to the choice variables. The process for solving the objective function will be covered in the next section.

5. Solution Methodology

The objective function presented in Equation (9) is highly nonlinear in nature. It is quite difficult to solve the problem analytically. Due to this reason, we have applied a popular metaheuristic algorithm, viz. the dragonfly algorithm. A detailed discussion about the dragonfly algorithm is given below.

The dragonfly algorithm (DA) is a novel and intriguing nature-inspired metaheuristic optimization algorithm that was created by Mirjalili [61] in the year 2016. It may be applied to a wide range of optimization tasks. Little flying carnivorous insects, dragonflies seek and consume a broad range of tiny insects, including mosquitoes, ants, butterflies, and bees (Mirjalili [61], Amroune et al. [62], Babayigit [63]). Dragonflies come in 3000 different species, and they go through two life stages: nymph and adult (Mirjalili [61], Jafari and Chaleshtari [64]). The foundation of the DA is the swarming behaviors of dragonflies, which are inherently both static (feeding) and dynamic (migratory) (Mirjalili [61], Baiche et al. [65]). The DA’s exploration and exploitation stages are represented by the static and dynamic swarms, respectively, During the exploitation phase, swarms of dragonflies migrate across great distances in one direction to divert attention from potential adversaries due to their vast numbers. However, during the exploration stage, dragonflies form small groups and circle a limited region in search of food and to draw in flying prey.

The swarm behaviors of dragonflies are modeled using five fundamental primitive principles (Mirjalili [61], Hariharan et al. [66], Rahman and Rashid [67]). The position of the current individual, Pj, is the position of the jth neighboring individual, and M is the number of neighboring individuals in the following equations.

The mathematical model of separation is as follows: it depicts the static collision avoidance that people use to avoid colliding with other people in the neighborhood.

$$S_i=-{\displaystyle \sum _{j-1}^{M}P-P_j}$$

The alignment shows how quickly the person moves in relation to other members of the same group who live nearby. As an illustration of the alignment, consider the following:

$$A_i={\displaystyle \frac{{\displaystyle \sum _{j-1}^M{V}_j}}{M}}$$

where $V_j$ denotes the velocity of the j-th individual.

Individuals tend to gravitate towards the center of the swarm group, which is represented by cohesiveness. Its definition is as follows:

$$C_i={\displaystyle \frac{{\displaystyle \sum _{j-1}^M{P}_j}}{M}}-P$$

Mathematically, attraction to the food supply (F) is represented by $F_i=F_p-F$, where F_i represents the food source of the i-th individual and F_p is the position of the food source.

Mathematically, the enemies’ distraction is represented by $E_i=E_p+P$, where Ei denotes the enemy’s position as that of the i-th individual, with E_p representing the enemy’s location.

The step vector DP and the position vector P are taken into consideration when updating the positions of artificial dragonflies within their search space. The velocity vector in the PSO method and the step vector DP are comparable. It is provided and updated as follows:

$$\Delta P_i^{t+1}=\left(s{S}_i+a{A}_i+c{C}_i+f{F}_i+e{E}_i\right)+\omega \Delta P_i^t$$

where S1 is the i-th individual’s separation; s is the separation weight; an is the alignment weight; Ai is the i-th individual’s alignment; c is the cohesion weight; C_i is the i-th individual’s cohesion; f is the food factor; Fi is the i-th individual’s food source; e is the enemy factor; Ei is the i-th individual’s enemy position; w is the inertia weight; and t is the iteration number. Next, the i-th dragonfly’s position at t þ 1 is updated in the manner described below: $P_i^{t+1}=P_i^t+\Delta P_i^{t+1}$.

High alignment and low cohesion weights guarantee exploration, but low alignment and high cohesion weights guarantee exploitation (Rahman and Rashid, 2019 [67]). By adjusting the weights s, a, c, f, e, and w adaptively, the convergence rate of the DA can be managed. When there are no neighboring solutions, random walk (Levy flight) is applied to improve the artificial dragonflies’ exploration, randomness, and exploitation. As a result, at iteration t þ 1, the location of the i-th dragonfly is modified as follows: $P_i^{t+1}=P_i^t+Levy\hspace{0.17em}(d)\times P_i^t$, where d represents the dimension of the position vectors. Levy flight is calculated by

$$Levy\hspace{0.17em}(d)=0.01\times {\displaystyle \frac{r_1\times \sigma }{{\left|\sigma \right|}^{\frac{1}{\beta }}}}$$

where r1 and r2 are random vectors uniformly distributed in the range [0, 1], b is a constant, and c r represents the gamma function. c is calculated as follows:

$$\sigma ={\left({\displaystyle \frac{\mathrm{\Gamma }\left(1+\beta \right)\times \sin \left(\frac{\pi \beta }{2}\right)}{\mathrm{\Gamma }\left(\frac{1+\beta }{2}\right)\times \beta \times 2^{\left(\frac{\beta -1}{2}\right)}}}\right)}^{{\displaystyle \frac{1}{\beta }}}$$

$$\mathrm{\Gamma }\left(a\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}\left(e^{-t}{t}^{a-1}\right)}dt$$

When a is an integer, we have

$$\mathrm{\Gamma }\left(a\right)=\left(a-1\right)!$$

6. Numerical Explanation

We chose the dragonfly algorithm (DA) for solving the inventory problem due to its robust ability to handle complex, nonlinear, and multi-objective optimization challenges often encountered in sustainable inventory systems. The DA is inspired by the swarming behavior of dragonflies, which balances exploration and exploitation through key mechanisms such as attraction, alignment, and cohesion. This dynamic adaptability makes it particularly suitable for addressing the intricate trade-offs in inventory management, such as minimizing costs, mitigating non-instantaneous product deterioration, and maximizing sustainability in green product systems. Moreover, the DA’s flexibility allows it to be tailored to specific problem characteristics, such as incorporating constraints related to environmental goals, deterioration control, and demand variability. Its efficiency in navigating vast solution spaces and avoiding premature convergence ensures that it can identify optimal or near-optimal solutions, making it an excellent choice for tackling real-world inventory optimization problems.

We conducted a comprehensive comparison of different versions of the dragonfly algorithm (DA) to solve the problem that we have studied in this research. The optimization challenge required balancing multiple objectives, including minimizing costs and environmental impact while managing product deterioration and sustainability constraints. We evaluated the standard DA alongside advanced variants such as the Chaotic Dragonfly Algorithm (CDA) [68], Adaptive Dragonfly Algorithm (ADA) [69], and Opposition-Based Dragonfly Algorithm (OBDA) [70]. Each variant was tested for its efficiency in exploring complex solution spaces, maintaining diversity, and achieving convergence. It is observed that each variant of dragonfly algorithm is equally efficient for solving this particular inventory problem.

A single numerical case is examined in order to validate the suggested model. The following are the various system parameters’ numerical values.

Example: a list of values for the several inventory parameters that are connected to the proposed model and results are presented in

Table 2. Finding the best answers to solve the objective function.

Variables	Optimal Values Obtain from DA	Optimal Values Obtain from CDA	Optimal Values Obtain from ADA	Optimal Values Obtain from OBDA
g	86.58237%	86.58237%	86.58237%	86.58237%
$T$	5.476035 months	5.476035 months	5.476035 months	5.476035 months
$Q$	332.2873	332.2873	332.2873	332.2873
$p$	USD 32.17468	USD 32.17468	USD 32.17468	USD 32.17468
$ATP$	USD 337.3759	USD 337.3759	USD 337.3759	USD 337.3759

.

$$\begin{gathered} a=120, b=2.7, c=2.1, \alpha =0.5, \beta =0.5, \gamma =0.07, d=0.1, \xi =0.05, P=200, \delta =0.07, \\ \kappa =15, \eta =0.2, p_c=\mathrm{USD} 20/\mathrm{unit}, \mu =2,h=\mathrm{USD} 2/\mathrm{unit}/\mathrm{month}, K_0=\mathrm{USD} 500/\mathrm{order}, t_1=\mathrm{USD} 0.2 \mathrm{months}. \end{gathered}$$

The concavity of the objective function is also visually illustrated in Figure 2, Figure 3 and Figure 4 by considering two decision factors simultaneously and maintaining one decision variable at its optimal levels.

7. Post-Optimality Analyses

Sensitivity studies are conducted by changing the range of parameter values, expressed in terms of numerical Example 1, from −20 to +20 percent. This is carried out to illustrate how various inventory management system parameters affect the overall cycle duration (T), profit per unit (ATP), starting stock level, and maximum green level. The results of these experiments are idealized and presented in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11.

From Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, it is observed that the unit profit ($ATP$) is highly sensible for the changing demand parameters ‘$a$’ and ‘$b$’, whereas ($ATP$) has the reverse effect on the changes in the parameter value (b). According to the changes in other parameters like $\mu$, $c_h$, and $K_o$, the profit per unit ($ATP$) is insensitive, whereas it is a little bit sensitive with respect to $c$ and $\delta$.

According to Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, one may conclude that the cycle length ($T$) rapidly changes due to the changes in the value of parameter $b$ directly. On the other hand, cycle length ($T$) is equally sensitive due to the changes in the demand parameter $a$. It is less sensitive with respect to the other parameters.

The initial stock ($Q$) is equally sensitive due to the changes in the parameter ordering cost. It is less sensitive with respect to the changes in the parameters $c_h$,$\delta$, and $a$, respectively. It is insensitive with respect to the parameters $\alpha$,$\mu$, and $b$.

Managerial Implication

Businesses can make well-informed decisions to maximize profitability, sustainability, and customer pleasure by utilizing managerial insights regarding the interplay between demand-dependent price, green level, and inventory level. The following are some salient observations.

Businesses can choose the best pricing strategies by knowing how demand reacts to changes in price, green level, and inventory level. For instance, raising the price of environmentally friendly products might be possible in times of great demand, but if the level of greenness is low, buyers would not be as eager to pay a premium. With this knowledge, managers can dynamically modify prices according to inventory levels and environmentally friendly product attributes.

Demand-dependent pricing and environmentally friendly product characteristics make inventory management more challenging. Companies must strike a compromise between lowering holding costs and the possibility of stockouts and keeping enough inventory on hand to meet demand. Businesses can successfully achieve this equilibrium with the use of insights from inventory optimization and demand forecasting algorithms.

Gaining insight into consumer preferences for eco-friendly items and their price sensitivity might be beneficial. For instance, while some customers may value reduced costs, others may be prepared to pay more for goods with higher green levels. Businesses can adjust their product offers and pricing strategies to different market segments by analyzing consumer behavior.

Although it may result in higher production costs, making items more environmentally friendly can benefit sustainability. Managers have to balance the potential impact on profitability with the environmental benefits of increasing green levels. Businesses can identify the ideal degree of greenness that optimizes sustainability and profitability with the aid of insights from demand analysis.

8. Conclusions

Certain degrading products in practice have a seasonal need. A vendor of greenhouses may ask for a deposit that is equal to a portion of the total cost. The pricing, which is meant to entice buyers to buy more goods, is connected with the seasonal demand rate. For a non-instantaneously deteriorating item, the greenhouse retailer wants to control the rate of degradation, environmental pollution, and order cost in order to maximize retailer profit. The replenishment problem was studied in this study in order to maximize merchant profit. It involved cooperative pricing, dynamic order cost investment, environmental cost, technology for preservation cost, and the best times to replace commodities that do not degrade instantly. In this case, the demand rate is influenced by the selling price and the stock. An algorithm determines the best answer to the supply chain inventory problem, which determines pricing, preservation technology costs, ecological emissions costs, order costs, and the replenishing cycle time. Due to the highly nonlinear nature of the objective function, it is unable to solve analytically. However, we have used a popular metaheuristic algorithm (DA) and its different variants. It is observed that each variant is equally efficient for solving this particular problem. In addition, we have shown the concavity of the objective function graphically with the help of MATLAB software (2018b).

Specific Future Research Directions

We will propose targeted areas for further investigation, such as the following:

Developing dynamic models that account for seasonal variations in demand and their impact on green investment and inventory policies.

Exploring multi-echelon supply chain models to assess how collaboration among growers, distributors, and retailers can optimize green practices across the supply chain.

Incorporating customer behavior studies to refine the relationship between green levels, pricing, and demand elasticity in specific markets.

This model can be expanded to include other sustainability issues under the heading of shortages, such as stochastic demand. Future iterations of this model could incorporate the uncertainty in the inventory parameters and be applied to more imprecise contexts, like Type-2 interval, fuzzy, and interval.