Modelling of an imprecise sustainable production control problem with interval valued demand via improved centre-radius technique and sparrow search algorithm

Abstract

The modelling and optimization of a manufacturing systems in the context of sustainable production under uncertainty remain a pivotal focus in control theory. The goal of this research is to develop a robust decision-making framework for a production-inventory system characterised by imperfect production with reworking processes, and interval valued non-linear demand rate which is dependent on green level, selling price, warranty period, and time. This study also considers the impact of carbon emission regulation taxes to demonstrate how CO2 emission control influences the best-found policy of the proposed system. To fulfil the goal, an interval-valued optimal control problem (IVOCP) is constructed using generalised variational principle and corresponding highly nonlinear interval maximization problem is obtained. To tackle this interval optimisation problem, an improved c-r optimisation technique and the meta-heuristic algorithm Sparrow Search algorithm (SSA) are employed. The best-found solution for the corresponding problem is numerically illustrated through four distinct scenarios based on the presence of green investment levels and warranty periods in the demand rate. The obtained best found results are compared by some other metaheuristic algorithms. Additionally, statistical tests and non-parametric tests are conducted to assess the effectiveness, consistency, and stability of the algorithms. Furthermore, sensitivity analyses have been made to observe how inventory system parameters impact the optimal policy. Based on these analyses, managerial insights are derived to aid in decision-making processes.

Introduction

In the current competitive and unpredictable landscape, modelling of real-life optimisation problems, especially inventory control problems with uncertainty, opens a new horizon for researchers for innovations and developments. Though there is no well-established deterministic technique for modelling the optimality of such inventory control problems in an imprecise environment, a number of scholars have merely used either the stochastic/fuzzy-stochastic or interval approach. Among these, the interval technique is easier and more efficient. In this approach, all the fluctuating or imprecise parameters, viz., production rate, demand rate, defective rate, inventory cost components, etc., of an inventory control problem are presented in terms of closed intervals during the mathematical formulation of it. In the area of inventory control in an interval environment, several notable scholarly articles taking different imprecise parameters in the form of interval were accomplished by the numerus researchers—Bhunia and Shaikh¹, Bhunia et al.², Ruidas et al.³, Rahman et al.⁴, and others. On the other side, if either the demand rate, production rate, or holding cost of an inventory control problem is an unknown function of time, then the corresponding optimisation problem leads to an optimal control problem. In this area, Das et al.^5,6,7 studied a number of inventory control problems by making several realistic assumptions on inventory parameters using control theory.

Imperfect production and rework policies are significant aspects in manufacturing and production settings, particularly when output quality may not initially meet desired standards. Imperfect production occurs when items are manufactured but fail to meet quality specifications due to reasons like equipment malfunctions, human error, raw material defects, or process variability. Understanding the causes of imperfection is crucial for implementing effective quality improvement measures. Handling imperfect items involves developing rework policies to address defects and bring items up to acceptable quality levels.

The concept of the greenness of a product holds significant importance for human beings throughout the world. The greenness of a product refers to its sustainability and environmental impact throughout its lifecycle. Products categorised at the green level typically prioritise eco-friendliness, minimal waste generation, energy efficiency, and responsible manufacturing processes. These products aim to reduce carbon footprints, conserve natural resources, and mitigate pollution. Mathematically, it can be defined as a real number that lies between zero and one and quantifies the level of harmlessness or eco-friendliness associated with the product. A product achieves total greenness when the value of equals or closely approaches 1, though truly achieving this is rare. Due to the increase in environmental awareness among most people, different industrial sectors are keen to launch green products on the market. So, in such scenarios, the green level becomes a key parameter to affect the customers’ demand rate. Also, the production cost has been slightly increased due to the impact of the green level on the production.

Over the last few decades, most companies and manufacturing firms have adopted various business policies to sustain their identity in the current competitive and unpredictable marketing scenario. Among those, providing a warranty period to the customers on the manufactured product is one of the fascinating business policies. This policy not only stimulates market demand but also mitigates risk for consumers. The warranty period stands as a pivotal safeguard for consumers and a testament to product reliability. It delineates the timeframe within which manufacturers or sellers assure the quality, functionality, and performance of a product or service. Familiarity with warranty terms is indispensable for consumers, outlining their entitlements and avenues for resolution in cases of defects or failures. In this regard, one can refer to televisions, refrigerators, washing- machines, water- purifiers, laptops, desktops, mobile phones, etc. as some popular examples of goods for which warranty facilities are available.

The emission of harmful greenhouse gases from production industries becomes a severe problem all over the globe. The harmful greenhouse gases are emitted during various stages of the inventory management process, including production, transportation, storage, and disposal of inventory items. These emissions contribute to climate change and environmental fatalities. To reduce carbon emissions associated with inventory, businesses and governments implemented various regulations in different sectors of industry. One such notable regulation is the carbon tax regulation. According to this regulation, manufacturers have to pay emission taxes to the government for reckless emissions. Sometimes, manufacturers contemplate either absorbing the emission tax costs or transitioning to emission-free production processes. Emission reduction strategies include improving production efficiency through updated processes, increasing renewable energy usage while reducing reliance on adopting eco-friendly design practices and fossil fuels, curbing emissions from transportation, and setting ambitious emission reduction targets. Existing literature places significant emphasis on the effects of the emission tax on the best-found policies for the production and supply chain systems.

Optimality of production control problems with dynamic rate of demand is still a challenging task for industrialists, system analysts, and decision-makers. The optimality of interval valued production control problem (IVPCP) is even more complicated under uncertain conditions, for which strong optimization methods should be adopted. To study optimal policies of such problems, researchers used center-radius optimization technique hybridized with some metaheuristic algorithms. Among those, the Sparrow Search Algorithm (SSA) (Xue & Shen⁸), which is a swarm-based metaheuristic with its strong exploration and exploitation balance. Although SSA has been reported to be effective in several optimization problems, some other intelligent algorithms have emerged as better alternatives.

To guide the scope of this work, the following research questions are formulated:

• 1. How can an interval-valued optimal control framework be effectively constructed to address uncertainty in demand influenced by green investments and warranty policies?

• 2. How do improved optimization techniques, such as the improved c-r algorithm and the Sparrow Search Algorithm (SSA), perform in solving the resulting nonlinear interval optimization problem compared to other metaheuristic methods?

• 3. What insights can be drawn from sensitivity analyses and statistical evaluations regarding the robustness and managerial applicability of the proposed inventory model?

To answer the above-mentioned questions, in this work, a green production inventory problem with imperfect production under reworking in interval uncertainty has been studied. Here, production rate is taken as an interval and unknown function of time, customers’ demand rate follows the formula, which is an interval-valued non-linear function of green level, selling price, warranty period, and time, and another important concept—carbon emission regulation tax—is considered to show the CO2 emission control effect on the best-found policy of the proposed system. The corresponding interval optimisation problem has been obtained by forming an IVOCP using interval mathematics and the generalised variational principle for solving an IVOCP. Then the said interval optimisation problem has been solved with improved c-r optimisation techniques and different meta-heuristic algorithms, viz., SSA, EOA, AEFA, GWOA, TLBOA, and WOA. The best-found strategy of the proposed problem is illustrated numerically by forming four various cases depending up on the presence of green investment level and warranty period facility in the demand rate. Further, statistical tests and non-parametric test are performed to test the efficiency, consistency and stability of the algorithms. Finally, to observe the effect of inventory system parameters on the best-found strategy, sensitivity analyses are executed.

Literature review

The study of production control under interval uncertainty has gained significant attention in control theory and supply chain management, particularly in inventory optimization where uncertain production rates, fluctuating demand, reworking strategies, and sustainability constraints play crucial roles. In this section, a brief study of literature has been made based on some key features related to the proposed work to find out the comprehensive research gaps.

Production control under interval uncertainty

Manufacturing systems often encounter uncertainties in several inventory components viz. production rates, demand fluctuations, and defect rates, which are influenced by machine breakdowns, supplier reliability, and market conditions. Traditionally, probabilistic and fuzzy-based techniques have been used to manage these uncertainties, but interval mathematics provides a more adaptable approach by representing uncertain parameters as closed intervals. In this area, Rahman et al.⁹ developed an interval-valued demand-led inventory model for all-units discounts and deterioration. They disclosed how pricing and discount policies induce optimal inventory decisions under uncertainty. Das et al.⁵ emerged control theory in the inventory problem with carbon emission investment policies. They described how firms can make the optimal investment choices for sustainable production even in cases when the conditions are not certain. Nonetheless, these studies have no inclusion of dynamic service policies and rework strategies, which are vital for maximizing production efficiency. To improve the performance of interval-based production control models, Manna & Bhunia¹⁰ studied a green production inventory system. They optimized selling price and green level-dependent demand using advanced algorithms. Additionally, Manna et al.¹¹ proposed a carbon-emitted production inventory system under uncertainty and used these algorithms for environmentally friendly decisions. These studies show how environmental issues, like carbon taxes and green investments, affect production control rules under uncertainty. Recently, Rahman et al.¹² developed an interval-based decision-making method for warranty-linked demand models. They incorporated carbon tax policies to analyse how environmental policy affects production control. These papers show how critical it is to take interval uncertainty into account while making pricing, warranty, and tax decisions within inventory systems. Furthermore, Das et al.⁷ applied extended Pontryagin’s maximum principle for solving a production control problem under interval uncertainty.

Imperfect production and reworking

Imperfect production is an inevitable aspect of manufacturing, leading to defective products due to machine inefficiencies, human errors, or raw material inconsistencies. Managing these defects through reworking policies is critical for maintaining production efficiency and minimizing waste. In this area, Sana¹³ presented the optimality of a production system addressing imperfect production. Krishnamoorthi and Panayappan¹⁴ introduced a production model tailored for imperfect production systems, considering reworking and backlogging. Sivashankari and Panayappan¹⁵ developed an EPQ model that accounts for defective production with reworking, shortages, and scrap. Li et al.¹⁶ proposed a stochastic EPQ model for a two-state manufacturing process, incorporating reworking, deterioration, and backordering. Taleizadeh et al.¹⁷ explored outsourcing rework of defective items within the EPQ inventory model, focusing on backordered items. Sanjai and Periyasamy¹⁸ solved an inventory problem considering imperfect production with reworking and shortages. Mokhtari et al.¹⁹ developed an extension of a multi-product EPQ model, demonstrating the impact of reworking strategies on production and inventory management. Gautam et al.²⁰ examined optimal strategies for a defective production system with a reworking option, considering price and advertisement-dependent demand. Narang and De²¹ developed an AI-based approach using genetic algorithms to optimize rework strategies with advertisement and time-sensitive demand, demonstrating the application of intelligent decision-making. On the contrary, Nobil et al.²² developed an EPQ model with strict inspection policies, guaranteeing quality compliance before rework operations. These papers demonstrate advancements in rework policies through market-driven strategies. However, these models still lack a comprehensive interval-based framework that integrates economic production rates, green investment costs, and sustainability regulations into a unified optimization strategy.

Greenness and carbon emission tax in production systems

As industries shift towards sustainable manufacturing, the greenness of a product has become a key factor influencing consumer preferences, regulatory compliance, and production cost optimization. Green investment strategies, such as the implementation of cleaner production technologies and eco-friendly supply chains, have been explored in several studies. In this context, Chung and Wee²³ studied the optimality of a supply chain problem with the remanufacturing of a green product for a short life-cycle. Then, Swami and Shah²⁴ investigated a modelling problem for green products with channel coordination. After that, Govindan et al.²⁵ conducted a scholarly review on the modelling and optimality of a green production supply chain using multi-criteria decision-making approaches. Basiri and Heydari²⁶ formulated a green supply chain model for sustainability with coordination. Soysal et al.²⁷ studied the modelling and optimality of a routing problem considering greenness and horizontal collaboration for perishable products. Rezaee²⁸ studied an inventory control problem for green production under uncertainty. Dev et al.²⁹ investigated the effect of the diffusion of green products in production planning systems in industry. Some other notable works on the modelling of green production with several realistic assumptions were accomplished by Mishra et al.³⁰, Sarkar & Bhuniya³¹, Manna et al.¹¹, Sepehri and Gholamian³², Ali et al.³³, Sheykhzadeh et al.³⁴, and others. However, these models fail to integrate green investment within an interval-valued production optimization framework, which is necessary for real-world industrial applications.

In addition to carbon emission reduction, sustainability in inventory/supply chains has been another prominent area of study. Notably, Zhang et al.³⁵ conducted an analysis on how emission tax influences the best-found policies of a multi-item-based manufacturing system. Tiwari and colleagues³⁶ investigated the sustainability of an inventory management system involving defective quality items while reducing carbon emissions. Xu et al.³⁷ studied spatial and temporal variations in carbon emissions using multi-resolution emission inventory modelling. Yu et al.³⁸ developed an inventory model for perishable products with carbon emissions explicitly incorporated in production planning. Mashud et al.³⁹ designed an inventory model with emphasis on sustainability, considering controllable carbon emissions in green warehouse operations. Taleizadeh et al.⁴⁰ designed a sustainable inventory model that addresses price-linked demand, carbon emissions, partial trade credit, and partial backordering. Sarkar et al.⁴¹ proposed a circular economy-driven two-stage supply chain model for waste minimization, showcasing how green supply chain strategies can contribute to environmentally sustainable production models. Furthermore, Garai & Sarkar⁴² examined the economically independent reverse logistics of a closed-loop supply chain, demonstrating the feasibility of integrating sustainability measures with interval uncertainty models. Parida et al.⁴³ explored two-warehouse-based inventory models for sustainability in various fuzzy environments while optimising carbon emissions. San-José et al.⁴⁴ examined optimal inventory policies for spoiled products under carbon tax regulations, demonstrating how economic incentives aid in emission control. These studies reveal the significance of sustainability, reducing emissions, and environmental supply chain practices in modern production and stock management.

Warranty policy related production system

In the area of inventory, numerous researchers have accomplished their scholarly contributions on warranty facility-based inventory and supply chain problems. Khawam et al.⁴⁵ went through the optimality of a warranty-linked inventory problem for Hitachi global storage technologies. Huang et al.⁴⁶ examined the effect of a replacement warranty on an inventory control problem. Khawam⁴⁷ made a scholarly contribution on warranty-based inventory management. Sana⁴⁸ developed a preventive maintenance and buffer stock-based inventory model with warranty in an imperfect production system. Giri and Maiti⁴⁹ established a multi-manufacturer and single-retailer-based model for a supply chain with a warranty period and price-dependent demand. Hajej et al.⁵⁰ studied the optimality of a production inventory problem with maintenance strategy and warranty periods under a lease contract. Banu and Mondal⁵¹ investigated an integrated model for a manufacturer considering warranty-linked credit periods under two different manufacturing policies. Afsahi and Shafiee⁵² studied the optimality of a stochastic inventory model with different decision-making based on the base-warranty and extended-warranty of the products. Afsahi et al.⁵³ studied the optimality of joint optimisation problems with base and extended warranty facilities. Recently, Wang et al.⁵⁴, Maji et al.⁵⁵, Manna et al.⁵⁶, Rahman et al.¹², Wang et al.⁵⁷, Das et al.⁵⁸, and others accomplished their works by taking the warranty policy into account.

Metaheuristic approaches for interval-valued optimization

Traditional optimization methods, such as analytical models and heuristic-based algorithms, often struggle with complex, large-scale optimization problems, particularly in interval-valued production inventory systems. Recent advancements in metaheuristic algorithms have demonstrated superior performance in solving these challenges (Yadav⁵⁹; Mirjalili & Lewis⁶⁰). Several well-known metaheuristics algorithms have been applied to production inventory control, including:

• Sparrow Search Algorithm (SSA) (Xue & Shen⁸).

• Teaching–Learning-Based Optimization (TLBO) (Rao et al.⁶¹).

• Whale Optimization Algorithm (WOA) (Mirjalili & Lewis⁶⁰).

• Grey Wolf Optimizer (GWO) (Mirjalili et al.⁶²).

• Artificial Electric Field Algorithm (AEFA) (Yadav⁵⁹).

• Equilibrium Optimizer (EO) (Faramarzi et al.⁶³).

• Beetle Antennae Search (BAS) (Zhang et al.⁶⁴).

• Competition of Tribes and Cooperation of Members Algorithm (CoTCMA) (Chen et al.⁶⁵).

• Egret Swarm Optimization Algorithm (ESOA) (Chen et al.⁶⁶).

Swarm-based metaheuristics such as the Sparrow Search Algorithm (SSA) (Xue & Shen⁸) and Beetle Antennae Search (BAS) (Zhang et al.⁶⁴) have shown remarkable performance in real-time optimization. Further, SSA is a fast, adaptive, and robust swarm-based optimizer that efficiently balances exploration and exploitation while avoiding local optima with a simple and flexible structure. Despite these advancements, no study has explored SSA hybridized with the centre-radius optimization technique in the context of interval-based uncertain optimization. One of the leading methods in this regard is the Beetle Antennae Search (BAS) algorithm (Zhang et al.⁶⁴), described as a model-free, single-particle optimization algorithm with strong global convergence properties. BAS has successfully been applied in numerous applications, ranging from robust control of biped robots (Khan et al.⁶⁷), real-time robotic arm trajectory planning (Khan et al.⁶⁸), to non-convex tax-aware portfolio optimization (Khan et al.⁶⁹). In addition, others evolutionary computation method, the Egret Swarm Optimization Algorithm (ESOA) (Chen et al.⁶⁶) and the Competition of Tribes and Cooperation of Members Algorithm (CoTCMA) (Chen et al.⁶⁵) have been introduced for model-free optimization with varying computational efficiency and convergence characteristics. Furthermore, multi-objective scheduling problems, including, hybrid flows hop scheduling with limited waiting time (Yılmaz et al.⁷⁰), multi-product disassembly line balancing (Yeni et al.⁷¹) and lot streaming and worker transfers (Yılmaz et al.⁷²) highlighted the increasing need for efficient decision-making models. Additionally, lean management-based in-plant logistics optimization (Büyüközkan et al.⁷³) further reinforces the importance of adaptive and computationally efficient techniques in production environments.

To show the novelty and significant contributions of the model at a glance, a key term wise comparison of the existing works with the present work has been depicted in Table 1.

Table 1 A comparative literature review of the scholarly works versus present work.

Research gap and contribution

Although extensive work has been done on inventory systems under uncertainty using probabilistic, fuzzy, and interval-based approaches, several limitations have been remained in those approches. Most existing studies (cf. Table 1) either consider imperfect production, unknown production rates, and variable demand rates under the effects of warranty period, sustainability aspects such as green investment and carbon emission taxes. However, those works fails to integrate those characteristics together within a comprehensive interval uncertainty framework. Furthermore, very few researchers employed the hybridization of the metaheuristic algorithms Sparrow Search Algorithm (SSA) and improved Centre-Radius (c-r) optimisation technique in solving an interval optimization problem for a green production-inventory model involving rework and environmental taxation. This hybrid algorithm still can be applied in the production modelling problems under production controlling scenario in interval environment. These limitations make a significant research gap in the existing literature.

To make up these gaps, the present work has incorporated green investment and warranty period to the joint effects on the best-found policy of the of the proposed model. Here, the best-found strategy of the model has been investigated in four different cases based on the greenness and warranty period. The main novelty of this work is highlighted as follows:

• Comprehensive demand structure: The demand rate of the present model is taken as a non-linear interval-valued function of selling price, green investment level, warranty period, and time, capturing realistic market dynamics under uncertainty.

• Hybrid metaheuristic optimization approach: A novel solution strategy is developed by combining an enhanced centre-radius (c-r) optimisation technique with the Sparrow Search Algorithm (SSA). Performance is benchmarked against six other state-of-the-art metaheuristics: AEFA, EOA, GWOA, TLBOA, WOA, and SSA.

• Scenario-based numerical illustration: The model is tested under four practical scenarios, reflecting different levels of green investment and warranty periods, to demonstrate its adaptability.

Problem description, notation and assumptions

Problem description

With the competitive market scenario, innovation and on-time product release are important tactics for firms to maintain market share and expand customer base. A scooter-making firm aims to tap into the kids’ market through the introduction of a new version of mini electric scooters for children. But introducing a new product, particularly in the electronics industry, has a number of uncertainties. Production flaws, demand volatility, rising costs, as well as regulation by environmental agencies are some of the variables that bring uncertainty to the venture, which can render the business as lucrative as well as unsustainable. Thus, the production manager, being conscious of these uncertainties, seeks to make a risk-conscious and a best decision by simulating the production process with interval uncertainty. Rather than using single-point estimates, the manager wishes to consider best-case and worst-case scenarios for joint effect of greenness and warranty period of the items on demand to allow a more flexible and resilient manufacturing policy.

Keeping the above realistic situation in mind, in this work, the modelling and optimality of a realistic production inventory problem for a mini electric scooter under interval uncertainty are examined. In the modelling part of the proposed problem, it is assumed that the production system produces an imperfect product with reworking, and the production rate is assumed to be an unknown function of time that fluctuates within an interval. Also, the product has been produced by keeping the greenness and warranty period of the item in mind so that the customers’ demand is positively affected by these two factors. Additionally, pricing negatively affects product demand. Taking into account these factors’ impact on customer demand, this study treats customer demand as a function that varies within a certain range based on warranty period, green level, selling price, and time. In this regard, some extra costs, viz., warranty costs and production costs due to green levels, have to be incurred by the manufacturer. Furthermore, the manufacturer paid the carbon emission taxes to the government, and as a consequence, there arises another cost, viz., the carbon emission cost, which is dependent on the production rate. With consideration of all of the above-mentioned characteristics, the problem was modelled mathematically by interval mathematics, and an IVOCP was obtained. Then the IVOCP has been solved by the interval variational principle, and finally, an interval optimisation problem has been obtained. The modified C-R optimisation technique and the Sparrow Search Algorithm (SSA) have studied the optimality of this problem. The numerical illustration of this study is of another interest because numerical simulations divided the modelling into the following parts:

Case-I	Modelling without both green level (g) and warranty period (${t_w}$)
Case-II	Modelling with green level (g) but without warranty period (${t_w}$)
Case-III	Modelling without green level (g) but with warranty period (${t_w}$)
Case-IV	Modelling with both green level (g) and warranty period (${t_w}$)

The required assumptions and notation used in the work are presented in the later successive sub-sections respectively.

Notation

Notation	Description
$\left[ {{D_L}\left( {{s_p},g,t,{t_w}} \right),{D_U}\left( {{s_p},g,t,{t_w}} \right)} \right]$	Demand rate of the customers in interval form
$\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right]$	Production rate in interval form
$\left[ {{x_L}\left( t \right),{x_U}\left( t \right)} \right]$	Inventory level at time t in interval form
r	Reworking cost per unit($/unit)
$\left[ {{\eta _L},{\eta _U}} \right]$	Imperfect rate of production in interval form
$\left[ {CE{T_L},CE{T_U}} \right]$	Rate of carbon emission in interval form at time t (ton/time unit)
$\left[ {{c_{mL}},{c_{mU}}} \right]$	Carbon emission investment rate in interval form
$\left[ {C{p_L},C{p_U}} \right]$	Production cost in interval form ($/unit)
$\left[ {{h_{cL}},\,{h_{cU}}} \right]$	Holding cost in interval form ($/unit/time unit)
$\left[ {{A_L},{A_U}} \right]$	Setup cost in interval form ($/cycle)
$\left[ {A{P_L}\left( {{s_p},g,{t_w},T} \right),A{P_U}\left( {{s_p},g,{t_w},T} \right)} \right]$	Interval valued average profit function ($/time unit)
Decision variables
${s_p}$	Selling price of the products ($/unit)
g	Green level of the products
${t_w}$	Warranty period of the product (month)
T	Business period (month)

Assumptions

• (i) An interval-valued production rate is a reliable technique for managing unpredictability and uncertainty in production processes in a manufacturing system. It specifies the production rate as an interval or range instead of a single, fixed value. When maintaining accurate control over the production rate is difficult due to factors like workforce productivity, equipment efficiency, raw material availability, or external market demands, it is very helpful. In the present system, the rate of production is considered as unknown function of time which is fluctuated with in an interval i.e., $\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right].$ Also, the production process is imperfect and the rate of imperfectness is lying with in the interval $\left[ {{\eta _L},{\eta _U}} \right],\,\,0<{\eta _L}<{\eta _U}<<1.$

• (ii) The imperfectly produced items are reworked to make them perfectly product and per unit reworking cost is considered as constant $r.$

• (iii) In sustainable production control problem, the variable factors- selling price, warranty, and green level have a significant impact in product’s demand. When these factors are taken into consideration within an interval framework, each of these factors may impact demand differently. The market for green products is expanding along with people’s awareness of environmental challenges. The green level of a product can significantly influence consumers’ purchasing decisions. The effect of green level on customers demand is shown graphically in Fig. 1. Additionally, a longer or more thorough guarantee boosts customer confidence in the product’s quality and longevity, which could lead to an increase in demand. The effect of warranty period on customers demand is shown graphically in Fig. 2. Keeping the realistic effects of these factors, in this work, the demand rate is taken as interval valued and nonlinear function of selling price, product’s green level, warranty period, and time. In this regard, the demand rate is expressed mathematically as:

$$\begin{gathered} \left[ {{D_L}\left( {{s_p},g,t,{t_w}} \right),{D_U}\left( {{s_p},g,t,{t_w}} \right)} \right]=\left[ {{\alpha _{0L}},{\alpha _{0U}}} \right] - \left[ {{\alpha _{1L}},{\alpha _{1U}}} \right]s_{p}^{a}+\left[ {{\alpha _{2L}},{\alpha _{2U}}} \right]g{}^{b}+\left[ {{\alpha _{3L}},{\alpha _{3U}}} \right]t_{w}^{c} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left[ {{\alpha _{4L}},{\alpha _{4U}}} \right]t\,\,\,{\text{where}}\,a,b,c,{\alpha _{iL}}>0\,{\text{for}}\,i=0,1,2,3,4 \hfill \\ \end{gathered}$$

Fig. 1

Effect of green level in demand rate.

Fig. 2

Effect of warranty period in demand rate.

• (iv) Production cost is dependent on quality of the raw material. If manager want to produce green products, then per unit production cost will be dependent on green level of the products. How the green level affects the production cost is shown graphically in Fig. 3. Thus, the per unit production cost is taken as an increasing function of production rate and green level of the products which is given by.

$$\left[ {{c_{pL}},{c_{pU}}} \right] = \left[ {{\lambda _{0L}},{\lambda _{0U}}} \right] + \left[ {{\lambda _{1L}},{\lambda _{1U}}} \right]\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right] + \left[ {{\lambda _{2L}},{\lambda _{2U}}} \right]g{}^d,\,{\rm{where}}\,\,d,\,{\lambda _{iL}}\,{\rm{positive}}\,{\rm{for}}\,i = 0,1,2$$

Fig. 3

Effect of green level on per unit production cost.

• (v) Due to the carbon emission during production process, Manufacturers pay carbon emission tax to the Government and the carbon emission rate of the production process is taken as linear function of production rate. Mathematically it is expressed as.

$\left[ {CE{T_L}\left( {\left[ {{P_L},{P_U}} \right]} \right),\,CE{T_U}\left( {\left[ {{P_L},{P_U}} \right]} \right)} \right]=\left[ {{a_{mL}},\,{a_{mU}}} \right]\left[ {{P_L},{P_U}} \right],\,\,\,0 \leqslant t \leqslant T,$ where ${a_{mL}}>0.$

• (vi) Per unit warranty cost is dependent on the time period of warranty. If length of warranty is greater then warranty cost will be higher and if length is lesser then cost will be comparatively lower. The effect of warranty period on warranty cost is presented graphically in Fig. 4. Here, the per unit warranty cost is considered as an interval-valued function of the warranty period and the mathematical expression of the per unit warranty cost is given by

$$\left[ {{W_L}\left( {{t_w}} \right),{W_U}\left( {{t_w}} \right)} \right]=\left[ {{\gamma _{0L}},{\gamma _{0U}}} \right]+\left[ {{\gamma _{1L}},{\gamma _{1U}}} \right]t_{w}^{\delta }\,\,{\text{where}}\,\,\delta ,{\gamma _{iL}}>0\,{\text{for}}\,\,i=0,1.$$

Fig. 4

Effect of warranty period on per unit warranty cost.

Mathematical formulation of the proposed model

The proposed model is formulated according to the notation and assumptions mentioned in subsection 3.2 and 3.3 respectively.

In this system, the manufacturing firm produce imperfect items at a rate $\left[ {{\eta _L},{\eta _U}} \right]\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right]$ and hence the rate of perfect-quality items is $\left( {1 - \left[ {{\eta _L},{\eta _U}} \right]} \right)\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right]$ up to a certain time period. Initially, the stock level increases and reaches its maximum position in a certain time during the time period $\left[ {0,T} \right]$due to the higher manufacturing rate than the demand. After that time onwards, the stock level decreases due to the dominance of customers’ demand over the manufacturing rate and finally it becomes zero at the end of the cycle length.

Therefore, interval boundary value problem (IBVP) of the stock level of the manufacturer is as follows:

$$\frac{{d\left[ {{x_L}\left( t \right),{x_U}\left( t \right)} \right]}}{{dt}}=\,\left( {1 - \left[ {{\eta _L},{\eta _U}} \right]} \right)\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right] - \left[ {{D_L}\left( {{s_p},g,t,{t_w}} \right),{D_U}\left( {{s_p},g,t,{t_w}} \right)} \right],{\text{ }}0 \leqslant t \leqslant T$$

(1)

subject to $\left[ {{x_L}\left( 0 \right),{x_U}\left( 0 \right)} \right]=\left[ {0,0} \right]$ and $\left[ {{x_L}\left( T \right),{x_U}\left( T \right)} \right]=\left[ {0,0} \right].$

Now, the interval-valued sales revenue and different interval-valued cost components are calculated as follows:

(i) Total sales revenue is given by

$$\begin{gathered} \left[ {S{R_L}\left( {{s_p},g,{t_w},\,T} \right),S{R_U}\left( {{s_p},g,{t_w},\,T} \right)} \right]=\int_{0}^{T} {\,{s_p}\left[ {{D_L}\left( {{s_p},g,{t_w},\,T} \right),{D_U}\left( {{s_p},g,{t_w},\,T} \right)} \right]\,} \,dt \hfill \\ ={s_p}\left[ {\left( {{\alpha _{0L}} - {\alpha _{1U}}s_{p}^{a}+{\alpha _{2L}}{g^b}+{\alpha _{3L}}t_{w}^{c}+\frac{{{\alpha _{4L}}T}}{2}} \right)T,\,\,\left( {{\alpha _{0U}} - {\alpha _{1L}}s_{p}^{a}+{\alpha _{2U}}{g^b}+{\alpha _{3U}}t_{w}^{c}+\frac{{{\alpha _{4U}}T}}{2}} \right)T} \right] \hfill \\ \end{gathered}$$

(2)

(ii) Total manufacturer’s production cost is given by

$$\begin{gathered} \left[ {P{C_L},\,P{C_U}} \right]=\,\int_{0}^{T} {\left[ {{C_{pL}}\left( {\left[ {{P_L},{P_U}} \right],g} \right),{C_{pU}}\left( {\left[ {{P_L},{P_U}} \right],g} \right)} \right]\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right]} \,dt \hfill \\ =\int_{0}^{T} {\left\{ {\left( {\left[ {{\lambda _{0L}},{\lambda _{0U}}} \right]+\left[ {{\lambda _{2L}},{\lambda _{2U}}} \right]{g^d}} \right)\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right]+\left[ {{\lambda _{1L}},{\lambda _{1U}}} \right]{{\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right]}^2}} \right\}\,dt} \hfill \\ \end{gathered}$$

(3)

(iii) Total holding cost of the system is

$$\left[ {H{C_L},H{C_U}} \right]=\left[ {{h_{cL}},{h_{cU}}} \right]\int\limits_{0}^{T} {\left[ {{x_L}(t),{x_U}(t)} \right]dt}$$

(4)

(iv) Total tax due to emission is obtained as

$$\left[ {{C_{taxL}},{C_{taxU}}} \right]=\left[ {{c_{mL}},{c_{mU}}} \right]\int_{0}^{T} {\left[ {{a_{mL}},\,{a_{mU}}} \right]\left[ {{P_L},{P_U}} \right]dt}$$

(5)

(v)Total cost due to reworking is as follows:

$$\left[ {R{C_L},\,R{C_U}} \right]=\int_{0}^{T} {r\left[ {{\eta _L},{\eta _U}} \right]\left[ {{P_L}\left( t \right),\,{P_U}\left( t \right)} \right]\,dt}$$

(6)

(vi) The extra revenue due to reworking process can be calculated as

$$\left[ {ES{R_L},ES{R_U}} \right]={s_p}\left[ {{\eta _L},{\eta _U}} \right]\int\limits_{0}^{T} {\left[ {{P_L}\left( t \right),\,{P_U}\left( t \right)} \right]dt}$$

(7)

(iv) Total warranty cost is given by

$$\begin{gathered} \left[ {T{W_L},\,T{W_U}} \right]=\,\int_{0}^{T} {\left[ {{W_L}(\,{t_w}),{W_U}(\,{t_w})} \right]\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right]} \,dt \hfill \\ =\int_{0}^{T} {\left( {\left[ {{\gamma _{0L}},{\gamma _{0U}}} \right]+\left[ {{\gamma _{1L}},{\gamma _{1U}}} \right]t_{w}^{\delta }} \right)\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right]\,\,dt} \hfill \\ \end{gathered}$$

(8)

Thus, the total interval-valued profit of the manufacturer (using (2)-(8)) is obtained as:

$\begin{gathered} \left[ {T{P_L}\left( {{s_p},g,{t_w},\,T} \right),\,T{P_U}\left( {{s_p},g,{t_w},\,T} \right)} \right]=\,<Sales\,revenue> - <Carbon\,\,emission\,\,tax> - <Ordering\,cost> \hfill \\ - <Production\,\,cost> - <Holding\,\,cost> - <Rework\,\,cost>+<Extra\,\,sales\,\,revenue> - <Service\,cost>\, \hfill \\ =\left[ {S{R_L}\left( {{s_p},g,{t_w},\,T} \right),S{R_U}\left( {{s_p},g,{t_w},\,T} \right)} \right] - \left[ {{C_{taxL}}\left( T \right),{C_{taxU}}\left( T \right)} \right] - \left[ {{A_L},{A_U}} \right] - \left[ {P{C_L},P{C_U}} \right] - \left[ {H{C_L},H{C_U}} \right] - \left[ {R{C_L},\,R{C_U}} \right] \hfill \\ +\left[ {ES{R_L},ES{R_U}} \right] - \left[ {T{W_L},\,T{W_U}} \right] \hfill \\ =\left[ \begin{gathered} {s_p}\left( {{\alpha _{0L}} - {\alpha _{1U}}s_{p}^{a}+{\alpha _{2L}}{g^b}+{\alpha _{3L}}t_{w}^{c}+\frac{{{\alpha _{4L}}T}}{2}} \right)T - {A_U},\, \hfill \\ \,{s_p}\left( {{\alpha _{0U}} - {\alpha _{1L}}s_{p}^{a}+{\alpha _{2U}}{g^b}+{\alpha _{3U}}t_{w}^{c}+\frac{{{\alpha _{4U}}T}}{2}} \right)T - {A_L} \hfill \\ \end{gathered} \right] \hfill \\ - \int_{0}^{T} {\left[ \begin{gathered} \left( {{\lambda _{0L}}+{\lambda _{2L}}{g^d}+\left( {{\gamma _{0L}}+{\gamma _{1L}}t_{w}^{\delta }} \right)+r{\eta _L}+{c_{mL}}{a_{mL}}} \right){P_L}\left( t \right)+{\lambda _{1L}}{P_L}^{2}\left( t \right) - {s_p}\,{\eta _U}{P_U}\left( t \right)+{h_{cL}}{x_L}\left( t \right), \hfill \\ \,\left( {{\lambda _{0U}}+{\lambda _{2U}}{g^d}+\left( {{\gamma _{0U}}+{\gamma _{1U}}t_{w}^{\delta }} \right)+r{\eta _U}+{c_{mU}}{a_{mU}}} \right){P_U}\left( t \right)+{\lambda _{1U}}{P_U}^{2}\left( t \right) - {s_p}\,{\eta _L}{P_L}\left( t \right)+{h_{cU}}{x_U}\left( t \right) \hfill \\ \end{gathered} \right]\,dt} \hfill \\ \end{gathered}$

Hence, manufacturer’s average profit is written as:

$$\left[ {A{P_L}\left( {{s_p},g,{t_w},\,T} \right),A{P_U}\left( {{s_p},g,{t_w},\,T} \right)} \right]=\frac{{\left[ {T{P_L}\left( {{s_p},g,{t_w},\,T} \right),T{P_U}\left( {{s_p},g,{t_w},\,T} \right)} \right]}}{T}$$

(9)

Corresponding IVOCP and its solution

Now, the from the mathematical formulation of proposed model, following IVOCP is considered to maximize the interval-valued average profit function (9).

$$\begin{gathered} {\text{Minimize}}\,\,\left[ {{f_L},{f_U}} \right]=\frac{1}{T}\int_{0}^{T} {\left[ \begin{gathered} \left( {{\lambda _{0L}}+{\lambda _{2L}}{g^d}+{\gamma _{0L}}+{\gamma _{1L}}t_{w}^{\delta }+r{\eta _L}+{c_{mL}}{a_{mL}}} \right){P_L}\left( t \right)+{\lambda _{1L}}{P_L}^{2}\left( t \right) - {s_p}\,{\eta _U}{P_U}\left( t \right)+{h_{cL}}{x_L}\left( t \right), \hfill \\ \,\left( {{\lambda _{0U}}+{\lambda _{2U}}{g^d}+{\gamma _{0U}}+{\gamma _{1U}}t_{w}^{\delta }+r{\eta _U}+{c_{mU}}{a_{mU}}} \right){P_U}\left( t \right)+{\lambda _{1U}}{P_U}^{2}\left( t \right) - {s_p}\,{\eta _L}{P_L}\left( t \right)+{h_{cU}}{x_U}\left( t \right) \hfill \\ \end{gathered} \right]\,dt} \hfill \\ {\text{subject to the conditions}} \hfill \\ \left\{ {\begin{array}{*{20}{c}} {\frac{{d\left[ {{x_L}\left( t \right),{x_U}\left( t \right)} \right]}}{{dt}}=\,\left( {1 - \left[ {{\eta _L},{\eta _U}} \right]} \right)\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right] - \left[ {{D_L}\left( {{s_p},g,t,{t_w}} \right),{D_U}\left( {{s_p},g,t,{t_w}} \right)} \right],} \\ {\left[ {{x_L}\left( 0 \right),{x_U}\left( 0 \right)} \right]=\left[ {0,0} \right]\,and\,\left[ {0,0} \right]=\,\left[ {{x_L}\left( T \right),{x_U}\left( T \right)} \right].\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \end{array}} \right. \hfill \\ \end{gathered}$$

(10)

Substituting the value of production rate $\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right]$in (10) and incorporating interval valued Hamiltonian $\left[ {{H_L},\,{H_U}} \right]$, IVOCP (10) takes the form:

$$\begin{gathered} {\text{Minimize}}\,\,\left[ {{f_L}\left( {t,{x_L}\left( t \right),{{\dot {x}}_L}\left( t \right)} \right),{f_U}\left( {t,{x_L}\left( t \right),{{\dot {x}}_L}\left( t \right)} \right)} \right] \hfill \\ =\frac{1}{T}\int_{0}^{T} {\left[ \begin{gathered} \left( {{\lambda _{0L}}+{\lambda _{2L}}{g^d}+{\gamma _{0L}}+{\gamma _{1L}}t_{w}^{\delta }+r{\eta _L}+{c_{mL}}{a_{mL}}} \right)\left( {\frac{{{{\dot {x}}_L}\left( t \right)+{D_L}\left( {{s_p},g,t,{t_w}} \right)}}{{1 - {\eta _U}}}} \right)+{\lambda _{1L}}{\left( {\frac{{{{\dot {x}}_L}\left( t \right)+{d_L}\left( {{s_p},g,t,{t_w}} \right)}}{{1 - {\eta _U}}}} \right)^2} \hfill \\ +{h_{cL}}{x_L}\left( t \right) - {s_p}\,{\eta _U}\left( {\frac{{{{\dot {x}}_U}\left( t \right)+{D_U}\left( {{s_p},g,t,{t_w}} \right)}}{{1 - {\eta _L}}}} \right),\, \hfill \\ \left( {{\lambda _{0U}}+{\lambda _{2U}}{g^d}+{\gamma _{0U}}+{\gamma _{1U}}t_{w}^{\delta }+r{\eta _U}+{c_{mU}}{a_{mU}}} \right)\,\left( {\frac{{{{\dot {x}}_U}\left( t \right)+{D_U}\left( {{s_p},g,t,{t_w}} \right)}}{{1 - {\eta _L}}}} \right)+{\lambda _{1U}}\,{\left( {\frac{{{{\dot {x}}_U}\left( t \right)+{D_U}\left( {{s_p},g,t,{t_w}} \right)}}{{1 - {\eta _L}}}} \right)^2} \hfill \\ +{h_{cU}}{x_U}\left( t \right) - {s_p}\,{\eta _L}\left( {\frac{{{{\dot {x}}_L}\left( t \right)+{D_L}\left( {{s_p},g,t,{t_w}} \right)}}{{1 - {\eta _U}}}} \right) \hfill \\ \end{gathered} \right]\,dt} \hfill \\ =\frac{1}{T}\int_{0}^{T} {\left[ {{H_L}\left( {t,{x_L}\left( t \right),{{\dot {x}}_L}\left( t \right)} \right),{H_U}\left( {t,{x_L}\left( t \right),{{\dot {x}}_L}\left( t \right)} \right)} \right]\,dt} \hfill \\ {\text{subject}}\,{\text{to}}\left[ {{x_L}\left( 0 \right),{x_U}\left( 0 \right)} \right]=\left[ {{x_L}\left( T \right),{x_U}\left( T \right)} \right]=\left[ {0,0} \right]. \hfill \\ \end{gathered}$$

(11)

$\begin{gathered} {\text{where}}\,\,{H_L}\left( {t,{x_L}\left( t \right),{{\dot {x}}_L}\left( t \right)} \right)=\left( {{\lambda _{0L}}+{\lambda _{2L}}{g^d}+{\gamma _{0L}}+{\gamma _{1L}}t_{w}^{\delta }+r{\eta _L}+{c_{mL}}{a_{mL}}} \right)\left( {\frac{{{{\dot {x}}_L}\left( t \right)+{D_L}\left( {{s_p},g,t,{t_w}} \right)}}{{1 - {\eta _U}}}} \right) \hfill \\ +{\lambda _{1L}}{\left( {\frac{{{{\dot {x}}_L}\left( t \right)+{D_L}\left( {{s_p},g,t,{t_w}} \right)}}{{1 - {\eta _U}}}} \right)^2}+{h_{cL}}{x_L}\left( t \right) - {s_p}\,{\eta _U}\left( {\frac{{{{\dot {x}}_U}\left( t \right)+{D_U}\left( {{s_p},g,t,{t_w}} \right)}}{{1 - {\eta _L}}}} \right) \hfill \\ {\text{and}}\,\,{H_U}\left( {t,{x_L}\left( t \right),{{\dot {x}}_L}\left( t \right)} \right)=\left( {{\lambda _{0U}}+{\lambda _{2U}}{g^d}+{\gamma _{0U}}+{\gamma _{1U}}t_{w}^{\delta }+r{\eta _U}+{c_{mU}}{a_{mU}}} \right)\,\left( {\frac{{{{\dot {x}}_U}\left( t \right)+{D_U}\left( {{s_p},g,t,{t_w}} \right)}}{{1 - {\eta _L}}}} \right) \hfill \\ +{\lambda _{1U}}\,{\left( {\frac{{{{\dot {x}}_U}\left( t \right)+{D_U}\left( {{s_p},g,t,{t_w}} \right)}}{{1 - {\eta _L}}}} \right)^2}+{h_{cU}}{x_U}\left( t \right) - {s_p}\,{\eta _L}\left( {\frac{{{{\dot {x}}_L}\left( t \right)+{D_L}\left( {{s_p},g,t,{t_w}} \right)}}{{1 - {\eta _U}}}} \right) \hfill \\ \end{gathered}$

To convert the IVOCP (11) into the corresponding deterministic optimal control problem, Theorem 1 is utilized.

Theorem 1

The IVOCP (11) is equivalent to the deterministic optimal control problem (12).

$$\begin{gathered} {\text{Minimize}}\,\,{f_c}\left( {t,{x_L}\left( t \right),{x_U}\left( t \right),{{\dot {x}}_L}\left( t \right),{{\dot {x}}_U}\left( t \right)} \right)=\frac{1}{T}\int_{0}^{T} {{H_c}\left( {t,{x_L}\left( t \right),{x_U}\left( t \right),{{\dot {x}}_L}\left( t \right),{{\dot {x}}_U}\left( t \right)} \right)\,dt} \hfill \\ {\text{subject}}\,\,{\text{to}}\,\,\left[ {{x_L}(0),{x_U}(0)} \right]=\left[ {0,0} \right]\,\,and\,\,\left[ {{x_L}(T),{x_U}(T)} \right]=\left[ {0,0} \right]. \hfill \\ \end{gathered}$$

(12)

where ${f_c}=\frac{{{f_L}+{f_U}}}{2}$ and ${H_c}=\frac{{{H_L}+{H_U}}}{2}$

Proof

Proof is similar as Theorem 1 of the work as Das et al.⁵⁸

Inventory level of the system at any time t can be calculated using Theorem 2.

Theorem 2

The inventory level of the system is given by the following expression.

$$\left[ {{x_L}\left( t \right),{x_U}\left( t \right)} \right]=\left[ {\left( {\frac{{{\alpha _{4L}}}}{2} - \frac{{{{\left( {1 - {\eta _U}} \right)}^2}{h_{cL}}}}{{4{\lambda _{1L}}}}} \right)t\left( {T - t} \right),\,\left( {\frac{{{\alpha _{4U}}}}{2} - \frac{{{{\left( {1 - {\eta _L}} \right)}^2}{h_{cU}}}}{{4{\lambda _{1U}}}}} \right)t\left( {T - t} \right)} \right],\,\,\,0 \leqslant t \leqslant T.$$

(13)

Proof

Since, the IVOCP (10) is equivalent to (12) (cf. Theorem 1), using Euler-Lagrange’s equation on (12), we have.

$\left\{ \begin{gathered} \frac{{\partial {H_c}}}{{\partial {x_L}}} - \frac{d}{{dt}}\left( {\frac{{\partial {H_c}}}{{\partial {{\dot {x}}_L}}}} \right)=0\, \hfill \\ \frac{{\partial {H_c}}}{{\partial {x_U}}} - \frac{d}{{dt}}\left( {\frac{{\partial {H_c}}}{{\partial {{\dot {x}}_U}}}} \right)=0 \hfill \\ \end{gathered} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}} {{h_{cL}} - \frac{{2{\lambda _{1L}}}}{{{{\left( {1 - {\eta _U}} \right)}^2}}}\left( {\frac{{{d^2}{x_L}}}{{d{t^2}}}+{\alpha _{4L}}} \right)=0\,} \\ {{h_{cU}} - \frac{{2{\lambda _{1U}}}}{{{{\left( {1 - {\eta _L}} \right)}^2}}}\left( {\frac{{{d^2}{x_U}}}{{d{t^2}}}+{\alpha _{4U}}} \right)=0} \end{array}} \right.$

Applying $\left[ {{x_L}(0),{x_U}(0)} \right]=\left[ {0,0} \right]$and$\left[ {{x_L}\left( T \right),{x_U}\left( T \right)} \right]=\left[ {0,0} \right]$, inventory at time t is given by.

$\left\{ \begin{gathered} {x_L}\left( t \right)=\left( {\frac{{{\alpha _{4L}}}}{2} - \frac{{{{\left( {1 - {\eta _U}} \right)}^2}{h_{cL}}}}{{4{\lambda _{1L}}}}} \right)\left( {T - t} \right)t \hfill \\ {x_U}\left( t \right)=\left( {\frac{{{\alpha_{4U}}}}{2} - \frac{{{{\left( {1 - {\eta _L}} \right)}^2}{h_{cU}}}}{{4{\lambda _{1U}}}}} \right)\left( {T - t} \right)t,\,\,\,\,\,\,t \in \left[ {0,T} \right] \hfill \\ \end{gathered} \right.$

Here $\left[ {\begin{array}{*{20}{c}} {\frac{{{\partial ^2}{H_c}}}{{\partial \dot {x}_{L}^{2}}}}&{\frac{{{\partial ^2}{H_c}}}{{\partial {{\dot {x}}_L}\partial {{\dot {x}}_U}}}} \\ {\frac{{{\partial ^2}{H_c}}}{{\partial {{\dot {x}}_U}\partial {{\dot {x}}_L}}}}&{\frac{{{\partial ^2}{H_c}}}{{\partial \dot {x}_{U}^{2}}}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {\frac{{2{\lambda _{1L}}}}{{T{{\left( {1 - {\eta _U}} \right)}^2}}}}&0 \\ 0&{\frac{{2{\lambda _{1U}}}}{{T{{\left( {1 - {\eta _L}} \right)}^2}}}} \end{array}} \right]$ is positive definite matrix.

Therefore, it is clear from the Legendre’s condition, (12) attains its minimum value along (13).

The economic rate of production of the system at time t has the following form

$$\left[ {{P_L}\left( t \right),{P_U}\left( t \right)} \right]=\left[ \begin{gathered} \frac{1}{{1 - {\eta _U}}}\left\{ {\left( {\frac{{{\alpha _{4L}}}}{2} - \frac{{{{\left( {1 - {\eta _U}} \right)}^2}{h_{cL}}}}{{4{\lambda _{1L}}}}} \right)\left( {T - 2t} \right)+{D_L}\left( {{s_p},g,t,{t_w}} \right)} \right\}, \hfill \\ \frac{1}{{1 - {\eta _L}}}\left\{ {\left( {\frac{{{\alpha_{4U}}}}{2} - \frac{{{{\left( {1 - {\eta _L}} \right)}^2}{h_{cU}}}}{{4{\lambda _{1U}}}}} \right)\left( {T - 2t} \right)+{D_U}\left( {{s_p},g,t,{t_w}} \right)} \right\} \hfill \\ \end{gathered} \right],\,\,\,\,\,\,0<t \leqslant T.$$

(14)

Various cost components of the proposed system are calculated as follows:

The holding cost is calculated as

$$\left[ {H{C_L}\left( T \right),\,H{C_U}\left( T \right)} \right]=\left[ {\frac{{{h_{cL}}{T^3}}}{6}\left( {\frac{{{\alpha _{4L}}}}{2} - \frac{{{{\left( {1 - {\eta _U}} \right)}^2}{h_{cL}}}}{{4{\lambda _{1L}}}}} \right),\,\frac{{{h_{cU}}{T^3}}}{6}\left( {\frac{{{\alpha _{4U}}}}{2} - \frac{{{{\left( {1 - {\eta _L}} \right)}^2}{h_{cU}}}}{{4{\lambda _{1U}}}}} \right)} \right]$$

(15)

The production cost is given by

$$\begin{gathered} \left[ {P{C_L}\left( {{s_p},g,{t_w},\,T} \right),\,P{C_U}\left( {{s_p},g,{t_w},\,T} \right)} \right] \hfill \\ =\left[ \begin{gathered} \frac{{\left( {{\lambda _{0L}}+{\lambda _{2L}}{g^d}} \right)T}}{{1 - {\eta _U}}}\left( {{\alpha _{0L}} - {\alpha _{1U}}s_{p}^{a}+{\alpha _{2L}}{g^b}+{\alpha _{3L}}t_{w}^{c}+\frac{{{\alpha _{4L}}T}}{2}} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{{T{\lambda _{1L}}}}{{3{{\left( {1 - {\eta _U}} \right)}^2}}}\left\{ {3{{\left( {{\alpha _{0L}} - {\alpha _{1U}}s_{p}^{a}+{\alpha _{2L}}{g^b}+{\alpha _{3L}}t_{w}^{c}+\frac{{{\alpha _{4L}}T}}{2}} \right)}^2}+\frac{{{{\left( {1 - {\eta _U}} \right)}^4}{T^2}{h_{cL}}^{2}}}{{16{\lambda _{1L}}^{2}}}} \right\}, \hfill \\ \frac{{\left( {{\lambda _{0U}}+{\lambda _{2U}}{g^d}} \right)T}}{{1 - {\eta _L}}}\left( {{\alpha _{0U}} - {\alpha _{1L}}s_{p}^{a}+{\alpha _{2U}}{g^b}+{\alpha _{3U}}t_{w}^{c}+\frac{{{\alpha _{4U}}T}}{2}} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{{T{\lambda _{1U}}}}{{3{{\left( {1 - {\eta _L}} \right)}^2}}}\left\{ {3{{\left( {{\alpha _{0U}} - {\alpha _{1L}}s_{p}^{a}+{\alpha _{2U}}{g^b}+{\alpha _{3U}}t_{w}^{c}+\frac{{{\alpha _{4U}}T}}{2}} \right)}^2}+\frac{{{{\left( {1 - {\eta _L}} \right)}^4}{T^2}{h_{cU}}^{2}}}{{16{\lambda _{1U}}^{2}}}} \right\} \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered}$$

(16)

Total reworking cost of detective products of the manufacturing process is given by

$$\left[ {R{C_L}\left( {{s_p},g,{t_w},\,T} \right),R{C_U}\left( {{s_p},g,{t_w},\,T} \right)} \right]=\left[ \begin{gathered} \frac{{rT{\eta _L}}}{{1 - {\eta _U}}}\left( {{\alpha _{0L}} - {\alpha _{1U}}s_{p}^{a}+{\alpha _{2L}}{g^b}+{\alpha _{3L}}t_{w}^{c}+\frac{{{\alpha _{4L}}T}}{2}} \right), \hfill \\ \,\,\frac{{rT{\eta _U}}}{{1 - {\eta _L}}}\left( {{\alpha _{0U}} - {\alpha _{1L}}s_{p}^{a}+{\alpha _{2U}}{g^b}+{\alpha _{3U}}t_{w}^{c}+\frac{{{\alpha _{4U}}T}}{2}} \right) \hfill \\ \end{gathered} \right]$$

(17)

The extra manufacturer’s revenue from the selling of reworked products is given by

$$\left[ {ES{R_L}\left( {{s_p},g,{t_w},\,T} \right),ES{R_U}\left( {{s_p},g,{t_w},\,T} \right)} \right]=\left[ \begin{gathered} \frac{{{s_p}T{\eta _L}}}{{1 - {\eta _U}}}\left( {{\alpha _{0L}} - {\alpha _{1U}}s_{p}^{a}+{\alpha _{2L}}{g^b}+{\alpha _{3L}}t_{w}^{c}+\frac{{{\alpha _{4L}}T}}{2}} \right), \hfill \\ \,\,\frac{{{s_p}T{\eta _U}}}{{1 - {\eta _L}}}\left( {{\alpha _{0U}} - {\alpha _{1L}}s_{p}^{a}+{\alpha _{2U}}{g^b}+{\alpha _{3U}}t_{w}^{c}+\frac{{{\alpha _{4U}}T}}{2}} \right) \hfill \\ \end{gathered} \right]$$

(18)

Total warranty cost is given by

$$\begin{gathered} \left[ {T{W_L}\left( {{s_p},g,{t_w},\,T} \right),\,T{W_U}\left( {{s_p},g,{t_w},\,T} \right)} \right] \hfill \\ =\left[ \begin{gathered} \frac{{\left( {{\gamma _{0L}}+{\gamma _{0L}}t_{w}^{\delta }} \right)T}}{{1 - {\eta _U}}}\left( {{\alpha _{0L}} - {\alpha _{1U}}s_{p}^{a}+{\alpha _{2L}}{g^b}+{\alpha _{3L}}t_{w}^{c}+\frac{{{\alpha _{4L}}T}}{2}} \right), \hfill \\ \frac{{\left( {{\gamma _{0U}}+{\gamma _{0U}}t_{w}^{\delta }} \right)T}}{{1 - {\eta _L}}}\left( {{\alpha _{0U}} - {\alpha _{1L}}s_{p}^{a}+{\alpha _{2U}}{g^b}+{\alpha _{3U}}t_{w}^{c}+\frac{{{\alpha _{4U}}T}}{2}} \right) \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered}$$

(19)

Total carbon emission tax is calculated as:

$$\left[ {{C_{taxL}}\left( {{s_p},g,{t_w},\,T} \right),{C_{taxU}}\left( {{s_p},g,{t_w},\,T} \right)} \right]=\left[ \begin{gathered} \frac{{{c_{mL}}{a_{mL}}T}}{{1 - {\eta _U}}}\left( {{\alpha _{0L}} - {\alpha _{1U}}s_{p}^{a}+{\alpha _{2L}}{g^b}+{\alpha _{3L}}t_{w}^{c}+\frac{{{\alpha _{4L}}T}}{2}} \right), \hfill \\ \,\,\frac{{{c_{mU}}{a_{mU}}T}}{{1 - {\eta _L}}}\left( {{\alpha _{0U}} - {\alpha _{1L}}s_{p}^{a}+{\alpha _{2U}}{g^b}+{\alpha _{3U}}t_{w}^{c}+\frac{{{\alpha _{4U}}T}}{2}} \right) \hfill \\ \end{gathered} \right]$$

(20)

Using (15)-(20), the minimum value of (10) is as follows:

$$\begin{gathered} {\left[ {{f_L}\left( {{s_p},g,{t_w},\,T} \right),{f_U}\left( {{s_p},g,{t_w},\,T} \right)} \right]_{\hbox{min} }} \hfill \\ =\frac{1}{T}\left[ \begin{gathered} P{C_L}\left( {{s_p},g,{t_w},\,T} \right)+H{C_L}\left( T \right)+R{C_L}\left( {{s_p},g,{t_w},\,T} \right) - ES{R_U}\left( {{s_p},g,{t_w},\,T} \right) \hfill \\ +T{W_L}\left( {{s_p},g,{t_w},\,T} \right)+{C_{taxL}}\left( {{s_p},g,{t_w},\,T} \right) \hfill \\ \,P{C_U}\left( {{s_p},g,{t_w},\,T} \right)+H{C_U}\left( T \right)+R{C_U}\left( {{s_p},g,{t_w},\,T} \right) - ES{R_L}\left( {{s_p},g,{t_w},\,T} \right) \hfill \\ +T{W_U}\left( {{s_p},g,{t_w},\,T} \right)+{C_{taxU}}\left( {{s_p},g,{t_w},\,T} \right) \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered}$$

(21)

Hence, manufacturer’s average profit is given by

$$\begin{gathered} \left[ {A{P_L}({s_p},g,{t_w},\,T),A{P_U}({s_p},g,{t_w},\,T)} \right]=\left[ {S{R_L}\left( {{s_p},g,{t_w},\,T} \right),S{R_U}\left( {{s_p},g,{t_w},\,T} \right)} \right] - \left[ {{A_L},{A_U}} \right] \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - {\left[ {{f_L}\left( {{s_p},g,{t_w},\,T} \right),{f_U}\left( {{s_p},g,{t_w},\,T} \right)} \right]_{\hbox{min} }} \hfill \\ \end{gathered}$$

(22)

The corresponding problem related to the proposed system is given by

$$\left\{ \begin{gathered} {\text{Maximize}}\,\,\left[ {A{P_L}\left( {{s_p},g,{t_w},\,T} \right),A{P_U}\left( {{s_p},g,{t_w},\,T} \right)} \right] \hfill \\ {\text{subject}}\,\,{\text{to}}\,\,{s_p}>0,\,\,0<g<1,\,\,{t_w}>0,\,\,T>0. \hfill \\ \end{gathered} \right.$$

(23)

Now, the goal is to find the best-found values of the decision variables viz., green level of the product$\left( g \right),$ selling price $\left( {{s_p}} \right),$warranty period $\left( {{t_w}} \right)$ and cycle length $\left( T \right)$ which maximize $\left[ {A{P_L}\left( {{s_p},g,{t_w},\,T} \right),A{P_U}\left( {{s_p},g,{t_w},\,T} \right)} \right]$.

Solution methodology

In this section, we have discussed the systematic solution procedure to solve the corresponding interval maximization problem (23) of the proposed system. The solution steps are two folded: (i) improved c-r optimization for converting (23) into crisp one (ii) metaheuristic algorithm for finding best-found optimal policy.

Improved c-r optimization

To address and solve the problem (23), the improved c-r optimization technique is employed. This method, introduced by Rahman et al.⁴ and modified by Manna and Bhunia, ¹⁰builds upon the interval ranking concept put forth by Bhunia and Samanta⁸¹. In this regard, the essential definitions are presented to define the improved c-r optimization approach used for converting Problems 23.

Definition 1

⁸¹. Let ${M_1}=\left[ {{m_{1L}},{m_{1U}}} \right] \cong \left\langle {{m_{1c}},{m_{1r}}} \right\rangle$and ${M_2}=\left[ {{m_{2L}},{m_{2U}}} \right] \cong \left\langle {{m_{2c}},{m_{2r}}} \right\rangle$ be two intervals. Then ranking between ${M_1}$and ${M_2}$ can be defined as

$({\text{i}})\,\,\,\,\,\,\,\,\,\,{M_1}{ \geqslant _{\hbox{max} }}{M_2} \Leftrightarrow \,\left\{ {\begin{array}{*{20}{c}} {{m_{1c}}>{m_{2c}}\,{\text{if}}\,{m_{1c}} \ne {m_{2c}}\,} \\ {{m_{1r}} \leqslant {m_{2r}}\,{\text{if }}{m_{1c}}={m_{2c}}.} \end{array}} \right.$

Where ${m_{1c}}=\frac{{\left( {{m_{1L}}+{m_{1U}}} \right)}}{2}\,\,{\text{and}}\,\,{m_r}=\frac{{\left( {{m_{1U}} - {m_{1L}}} \right)}}{2}$are the centre and radius of ${M_1}$respectively.

Definition 2

The point $\left( {{s_p}*,g*,{t_w}*,\,T*} \right) \in \Omega ={{\mathbb{R}}^+} \times \left( {0,1} \right) \times {{\mathbb{R}}^+} \times {{\mathbb{R}}^+} \subseteq {{\mathbb{R}}^4}$ is said to be a maximizer of $\left[ {A{P_L}\left( {{s_p},g,{t_w},\,T} \right),\,A{P_U}\left( {{s_p},g,{t_w},\,T} \right)} \right]$ if the following interval inequality is satisfied:

$\left[ {A{P_L}\left( {{s_p}*,g*,{t_w}*,\,T*} \right),\,A{P_U}\left( {{s_p}*,g*,{t_w}*,\,T*} \right)} \right]{ \geqslant _{\hbox{max} }}\left[ {A{P_L}\left( {{s_p},g,{t_w},\,T} \right),\,A{P_U}\left( {{s_p},g,{t_w},\,T} \right)} \right],\,\,\forall \,\left( {{s_p},g,{t_w},\,T} \right) \in \Omega .$

Improved c-r optimization technique: ¹⁰In this technique, the following steps are used to solve interval optimization Problem 23:

Step 1. At first, to solve the following optimization problem:

$$\left\{ \begin{gathered} {\text{Maximize }}A{P_c}\left( {{s_p},g,{t_w},\,T} \right) \hfill \\ {\text{subject}}\,{\text{to}}\,\,\left( {{s_p},g,{t_w},\,T} \right) \in \Omega \hfill \\ \end{gathered} \right.$$

(24)

and let the maximizer of (24) be

$$\left( {{s_p},g,{t_w},\,T} \right)=\left( {{s_p}^{1},{g^1},{t_w}^{1},\,{T^1}} \right).$$

(25)

Step 2. To navigate all the maximizers of (24) as.

${\Omega _c}=\left\{ {\left( {{s_p},g,{t_w},\,T} \right) \in \Omega :\,\,A{P_c}\left( {{s_p},g,{t_w},\,T} \right)=A{P_c}\left( {{s_p}^{1},{g^1},{t_w}^{1},\,{T^1}} \right)} \right\}.$

Step 3. Solve the following crisp minimization problem

$$\left\{ \begin{gathered} {\text{Minimize }}A{P_r}\left( {{s_p},g,{t_w},\,T} \right) \hfill \\ {\text{subject}}\,{\text{to}}\,\,\left( {{s_p},g,{t_w},\,T} \right) \in {\Omega _c} \hfill \\ \end{gathered} \right.$$

(26)

and let the minimizer of (26) be

$$\left( {{s_p},g,{t_w},\,T} \right)=\left( {{s_p}^{2},{g^2},{t_w}^{2},\,{T^2}} \right).$$

(27)

Either (25) or (27) will be maximizer of (23) subject to the feasibility conditions of (24) and (26).

Metaheuristic algorithms

Since the objective function of the corresponding problem (23) involves the expression of the decision variables with high complexity and nonlinearity, it is difficult to solve the problem using the classical optimization techniques. Therefore, in this sub section, we have discussed the working principle of one of the recently developed metaheuristics algorithm Sparrow Search Algorithm (SSA) to avoid difficulties in solving the said problem.

Motivation for choosing SSA

In the year 2020, Xue and Shen⁸ introduced the Sparrow Search Algorithm (SSA), emerges as a prominent solution for Problem (23) when exploring literature. Among contemporary metaheuristics, SSA stands out due to its recent development and notable success in tackling diverse physical and mathematical problems. Several compelling justifications advocate for employing SSA to address Problem (23):

• SSA, a bio-inspired metaheuristic, emulates the foraging behaviors and strategies of sparrows, including responses to competitive interactions and unpredictable situations within bird flocks such as attacks by predators. This naturalistic approach sets SSA apart from other metaheuristic algorithms, enhancing its adaptability and effectiveness.

• Comparative evaluations against metaheuristic algorithms like Grey Wolf Optimization (GWO), Particle Swarm Optimization (PSO), and Gravitational Search Algorithm (GSA) demonstrate SSA’s superiority in terms of efficiency, efficacy, and feasibility across various conventional test functions. Such empirical evidence positions SSA as a preferable choice for optimization tasks compared to its counterparts.

• Notably, SSA has yet to be applied to solve production inventory and interval optimization problems, highlighting its untapped potential in addressing diverse optimization challenges, as observed from existing literature.

The details discussion about sparrow search algorithm is given below:

Sparrow search algorithm

The Sparrow Search Algorithm (SSA) draws inspiration from the behaviour of sparrows, ubiquitous and intelligent birds known for their gregarious nature and omnivorous feeding habits, primarily consuming weeds or cereal seeds. SSA’s design incorporates mathematical modelling based on sparrow behaviour, simulating key aspects such as:

• (i) The energy level of the producers is typically sufficient to give the scavengers the right instructions and directions to the food sources. The rich food sources are also identified. An individual’s reserve amount of energy is based on their fitness value.

• (ii) As soon as the sparrows spot the predator, they begin chirping to alert the others. If the alarm value exceeds the safety level, producers should direct all scavengers to take a safe area.

• (iii) Every sparrow has the ability to produce for better food sources as long as it please, but overall, the population’s distribution of scavengers and producers stays stable.

• (iv) The birds with the highest energy levels can be considered the producers. To gain more energy, many famished scavengers are more likely to fly to various locations in quest of food.

• (v) Scavengers that are in search of food follow the providers who offers the best availability of food. While attempting to obtain food, some scavengers may continue to observe the producers for accelerating their own pace of predation.

• (vi) The sparrows near the group’s edge move quickly to the safe place to take up a better location if there is a danger, whereas those in the centre of the group fly at random to become closer to their fellow sparrows.

In the experimental simulation, we need computer-generated sparrows to search food. The location of sparrows is shown using a the matrix given below:

$\chi =\left[ {\begin{array}{*{20}{c}} {{\chi _{1,1}}}&{{\chi _{1,2}}}& \cdots & \cdots &{{\chi _{1,m}}} \\ {{\chi _{2,1}}}&{{\chi _{2,2}}}& \cdots & \cdots &{{\chi _{2,m}}} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {{\chi _{n,1}}}&{{\chi _{n,2}}}& \cdots & \cdots &{{\chi _{n,m}}} \end{array}} \right]$

m be the size of the variables that need to be optimised, and n stands for the number of sparrows. Since all sparrows fall into this category, the following vector can be used to demonstrate their overall health:

${X_\chi }=\left[ {\begin{array}{*{20}{c}} {u\left( {\left[ {{\chi _{1,1}}} \right.} \right.}&{{\chi _{1,2}}}& \cdots & \cdots &{{\chi _{1,m}}\left. ] \right)} \\ {u\left( {\left[ {{\chi _{2,1}}} \right.} \right.}&{{\chi _{2,2}}}& \cdots & \cdots &{{\chi _{2,m}}\left. ] \right)} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {u\left( {[{\chi _{n,1}}} \right.}&{{\chi _{n,2}}}& \cdots & \cdots &{\left. {{\chi _{n,m}}]} \right)} \end{array}} \right]$

The value in each row indicates the individual’s level of fitness, and n is the number of sparrows. For producers in the SSA, food is prioritised throughout the search process for those with higher fitness ratings. Additionally, the producers are in charge of finding food and guiding population mobility. Therefore, the producers may search for food in a larger range of areas than the scavengers. The producer’s location changes in line with rules (i) and (ii) in the following ways:

$\chi _{{i,j}}^{{k+1}}=\left\{ {\begin{array}{*{20}{c}} {\chi _{{i,j}}^{k}.\exp \left( {\frac{{ - i}}{{\mu .{{\hbox{max} }_{it}}}}} \right)\,\,\,{\text{if}}\,{L_2}<SQ} \\ {\chi _{{i,j}}^{k}+T.M\,\,\,\,\,if\,{L_2} \geqslant SQ} \end{array}} \right.$where k denotes present iteration, $j=1,2, \ldots ,m.$$\chi _{{i,j}}^{k}$ is a representation of i-th sparrow’s j-th dimension value at kth iteration. ${\hbox{max} _{it}}$ is a constant that has had the most iterations. $\mu$be a uniformly distributed random number (R.N) in between [0, 1]. The alert value is represented by ${L_2}$ $\left( {{L_2} \in \left[ {0,1} \right]} \right)$ while the safety threshold is represented by $SQ\,\left( {SQ \in \left[ {0.5,0.1} \right]} \right)$. The R.N T follows the normal distribution. M displays a matrix of $1 \times m$ with 1 as the value of each member. When ${L_2}<SQ$, which indicates no presence of nearby predators, the producer switches to broad search mode. If ${L_2} \geqslant SQ$, some sparrows have identified the presence of predator, and they must immediately flee to another safe regions.

The rule (iv) and rule (v) must be upheld by the scroungers. As was already said, certain scrappers pay more attention to the producers. When they have found the availability of producer good food, they have immediately left the area where they are currently living to compete for it. If they win, they can immediately consume the producer’s food; otherwise, they must follow the rules (v). The scrounger’s position update formula looks like this:

$\chi _{{i,j}}^{{k+1}}=\left\{ {\begin{array}{*{20}{c}} {T.\exp \left( {\frac{{\chi _{{worst}}^{k} - \chi _{{i,j}}^{k}}}{{{i^2}}}} \right)\,\,\,{\text{if}}\,i<n/2} \\ {\chi _{P}^{{k+1}}+\left| {\chi _{{i,j}}^{k} - \chi _{P}^{{k+1}}} \right|.{A^+}.M\,\,\,\,\,{\text{otherwise}}} \end{array}} \right.$

where $\chi _{P}^{{k+1}}$ represents the best position for the producer. $\chi _{{worst}}^{k}$ stands for the current worst location in the world. A random number between 1 and − 1 is assigned to each member of matrix A, along with the letters ${A^+}={\left( {A{A_T}} \right)^{ - 1}}$. When i > n/2, the i-th scrounger with the lowest fitness score is most likely to go hungry. For the experimental simulation, we assume those sparrows, who are aware of the threat, make up 10–20% of the whole population. These sparrows are placed in their beginning locations at random. Using rule (vi), the mathematical model can be represented as follows:

$\chi _{{i,j}}^{{k+1}}=\left\{ {\begin{array}{*{20}{c}} {\chi _{{best}}^{k}+\beta .\left| {\chi _{{i,j}}^{k} - \chi _{{best}}^{k}} \right|\,\,{\text{if}}\,{h_i}>{h_g}} \\ {\chi _{{i,j}}^{k}+v.\left( {\frac{{\left| {\chi _{{i,j}}^{k} - \chi _{{worst}}^{k}} \right|}}{{\left( {{h_i} - {h_w}} \right)+\lambda }}} \right)\,\,\,\,{h_i}={h_g}} \end{array}} \right.$

where $\chi _{{best}}^{k}$ denotes the Earth’s current optimum location. The parameter controlling step size is set to $\beta$ which is a normally distributed random integers with mean 0 and variance 1. Let $v \in \left[ { - 1,1} \right]$ be a chance number. Here, ${h_i}$ denotes the fitness level of the current sparrow. The best and worst fitness values currently available are ${h_g}$ and ${h_w}$, respectively. In order to avoid a zero division, the lowest constant is chosen as$\lambda$.

The ${h_i}>{h_g}$ shows the fact that the sparrow is on the edge of the group keeps things straightforward. $\chi _{{best}}^{k}$ is secure in the vicinity of the population centre that it represents. The presence of ${h_i}={h_g}$ suggests that the sparrows situated in the centre of the population are more aware of danger and should move nearer to the other sparrows. Until the termination requirement is satisfied, v indicates both the coefficient of step size control parameter and the sparrow’s walking direction.

To compare the best-found results of the corresponding optimization problems of the model obtained by SSA, others five meta-heuristic algorithms are used which are presented below:

• Artificial Electric Field Algorithm (AEFA) (Yadav⁵⁹).

• Equilibrium Optimizer Algorithm (EOA) (Faramarzi et al.⁶³).

• Teaching Learning Based Optimization Algorithm (TLBOA) (Rao et al.⁶¹).

• Grey Wolf Optimizer Algorithm (GWOA) (Mirjalili et al.⁶²).

• The Whale Optimization Algorithm (WOA) (Mirjalili and Luis⁶⁰).

Numerical illustrations

In this section, the best-found policy of the proposed inventory control problem has been illustrated into four different cases. Depending upon the presence of the green level and warranty period in the demand rate of the present model, following cases may arise:

Case-I	Modelling without both green level $\left( g \right)$ and warranty period $\left( {{t_w}} \right)$
Case-II	Modelling with green level $\left( g \right)$ but without warranty period $\left( {{t_w}} \right)$
Case-III	Modelling without green level $\left( g \right)$ but with warranty period $\left( {{t_w}} \right)$
Case-IV	Modelling with both green level $\left( g \right)$ and warranty period $\left( {{t_w}} \right)$

To discuss the optimal policy of all previous mentioned four cases, the corresponding interval optimization problem (23) has been solved numerically under four different settings of the values of inventory parameters involved in the system. In this regard, four different numerical examples are considered for all four cases. All of these four examples are solved by modified c-r optimization technique, and meta-heuristic technique – SSA (Sparrow Search Algorithm). algorithm EOA, AEFA, GWOA, SSA, TLBOA and WOA. All of these algorithms are coded in MATLAB R2023a software.

Numerical examples and simulations

The best-found policy of the present model is justified by performing of numerical simulations into the earlier mentioned four cases as follows:

Case-I (Modelling without both green level $\left( g \right)$ and warranty period $\left( {{t_w}} \right)$):

To validate the optimal policy for this case, following real-life example has been considered. Here, value of all the system parameters is selected hypothetically.

Example 1

Suppose, a scooter production company want to launch a new model of mini scooter for kids in near future. In this regard, the manager wishes to estimate the manufacturing policy in the form interval uncertainty. It is assume that the company manufactures the product with a defective percentage lies in the interval$\left[ {{\eta _L},{\eta _U}} \right]=\left[ {5,6} \right]\%$ and company have decided to pay $\$ 8$ as a fixed reworking cost (r) per unit product. Per unit initial production cost ($\left[ {{\lambda _{0L}},{\lambda _{0U}}} \right]$) is $\$ \left[ {4.5,4.7} \right]$ and production cost increases with increase of production with production scaling parameter lying in $\left[ {{\lambda _{1L}},{\lambda _{1U}}} \right]=\left[ {1.3,1.35} \right]$. Let initial demand of the customer for the mini electric scooter lies in the interval $\left[ {{\alpha _{0L}},{\alpha _{0U}}} \right]=\left[ {355,357} \right].$ Sometimes, demand rate of a new electronic gadgets decreases due to increases of selling price and increases with the passing of time. Here, the value of selling price effective parameter is lies in $\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right]=\left[ {2.4,2.45} \right]$ and time sensitive parameter is fluctuated in between $\left[ {{\alpha _{4L}},{\alpha _{4U}}} \right]=\left[ {2.1,2.15} \right]$. The company have additional expenses in different sectors as $\$ \left[ {1.2,1.25} \right]$for per unit holding cost ($\left[ {{h_{cL}},\,{h_{cU}}} \right]$) per unit time, $\$ \left[ {505,510} \right]$for setup cost ($\left[ {{A_L},{A_U}} \right]$) and $\$ \left[ {1.2,1.25} \right]$ for carbon emission regulation tax ($\left[ {{c_{mL}},\,{c_{mU}}} \right]$). Now, based on the above set of parameters, estimate the bounds of best-found policy of the said company for new launched mini scooter.

Solution

The value of the system parameters related to the proposed model for this example is as follows.

$$\eqalign{& \left[ {{\alpha _{0L}},{\alpha _{0U}}} \right] = \left[ {355,357} \right]{\text{units}},\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right] = \left[ {2.4,2.45} \right], \cr & \left[ {{\alpha _{4L}},{\alpha _{4U}}} \right] = \left[ {2.1,2.15} \right],\,\,\,\left[ {{\gamma _{0L}},{\gamma _{0U}}} \right] = \left[ {2.2,2.3} \right], \cr & \left[ {{\gamma _{1L}},{\gamma _{1U}}} \right] = \left[ {0.001,0.002} \right],\,\,\,\left[ {{a_{mL}},\,{a_{mU}}} \right] = \left[ {0.3,0.4} \right], \cr & \left[ {{\eta _L},{\eta _U}} \right] = \left[ {0.05,0.06} \right],\,\,\left[ {{h_{cL}},\,{h_{cU}}} \right] = \$ \left[ {1.2,1.25} \right], \cr & \left[ {{c_{mL}},{c_{mU}}} \right] = \$\left[ {1.2,1.25} \right],\,\,a = 1.1,\,\,r = \$8, \cr & \left[ {{A_L},{A_U}} \right] = \$ \left[ {505,510} \right]/{\text{cycle}}. \cr}$$

Example-1 is solved by the set of previous mentioned meta-heuristic algorithms (EOA, AEFA, GWOA, SSA, TLBOA and WOA) and the computational best-found as well as worst-found results are presented in the Tables 2 and 3. Further, to check the consistency and efficiency of all adopted the algorithms, statistical simulations with respect to Example 1 have been performed and the simulated results are presented in Table 4.

Table 2 Best-found solutions for Example 1.

Table 3 Worst-found solutions for Example 1.

Table 4 Results of statistical experiment for Example 1.

Case-II (Modelling with green level $\left( g \right)$ but without warranty period $\left( {{t_w}} \right)$):

In this case, the most important concept green manufacturing is taken into the account. The optimal policy for this case has been validated by solving following real-life example:

Example 2

This example is formed with the same values of the system parameters as of Example 1. The only notable changes in this example are that the manager of the mini scooter manufacturing company wishes to manufacture green product (electric scooter). In this regard, the production cost is slightly increased due to the effect of green production factors $\left[ {{\lambda _{2L}},{\lambda _{2U}}} \right]=\left[ {0.1,0.15} \right],\,d=4.5.$ On the other side, the demand rate is slightly increased due to the green production effective parameters whose values are taken as $\left[ {{\alpha _{2L}},{\alpha _{2U}}} \right]=\left[ {3.5,3.55} \right],\,\,b=0.4.$ Now, estimate the bounds of best-found policy of the said company for new launched mini scooter.

Example-2 is solved by the set of previous mentioned meta-heuristic algorithms (EOA, AEFA, GWOA, SSA, TLBOA and WOA) and the computational best-found as well as worst-found results are presented in the Tables 5 and 6. Further, in order to check the efficiency and consistency of all adopted the algorithms, statistical simulations with respect to Example 2 have been performed and the simulated results are depicted in Table 7.

Table 5 Best-found solutions for Example 2.

Table 6 Worst-found solutions for Example 2.

Table 7 Results of statistical experiment for Example 2.

Case-III (Modelling without green level $\left( g \right)$ but with warranty period $\left( {{t_w}} \right)$):

To validate the optimal policy for this case, following real-life example has been considered. Here, value of all the system parameters is selected hypothetically.

Example 3

This example is also formed with the same values of the system parameters as of Example 1. Here, the only changes are that the manager of the mini scooter production company wishes to provide the warranty period facility in the manufactured product. For this consideration, the amount of warranty cost that has been paid is $\left[ {{\gamma _{0L}},{\gamma _{0U}}} \right]=\left[ {2.2,2.3} \right]$with the value of effective factors $\left[ {{\gamma _{1L}},{\gamma _{1U}}} \right]=\left[ {0.001,0.002} \right],\,\delta =3.1.$ On the other side, the demand rate is slightly increased due to the warranty period facility with effective parameters whose values are taken as $\left[ {{\alpha _{3L}},{\alpha _{3U}}} \right]=\left[ {2.9,2.95} \right],\,\,c=0.8.$ Now, estimate the bounds of best-found policy of the said company for new launched mini scooter.

Example-3 is tackled using the set of aforementioned meta-heuristic algorithms (EOA, AEFA, GWOA, SSA, TLBOA, and WOA), and the computational best-found and worst-found results are presented in Tables 8 and 9. To assess the efficiency and consistency of these algorithms, statistical simulations based on Example 3 are conducted, and the simulated results are outlined in Table 10.

Table 8 Best-found solutions for Example 3.

Table 9 Worst-found solutions for Example 3.

Table 10 Results of statistical experiment for Example 3.

Case-IV (Modelling with both green level $\left( g \right)$ and warranty period $\left( {{t_w}} \right)$):

To validate the optimal policy for this case, following real-life example has been considered. Here, value of all the system parameters is selected hypothetically.

Example 4

This example is formed with the values of common system parameters as of Example 1. Here, both green production and warranty period facility are taken into the account. In this regard, the production cost is slightly increased due to the effect of green production factors $\left[ {{\lambda _{2L}},{\lambda _{2U}}} \right]=\left[ {0.1,0.15} \right],\,d=4.5.$ Another cost due to warranty has to pay at the rate $\left[ {{\gamma _{0L}},{\gamma _{0U}}} \right]=\left[ {2.2,2.3} \right]$with the value of effective factors $\left[ {{\gamma _{1L}},{\gamma _{1U}}} \right]=\left[ {0.001,0.002} \right],\,\delta =3.$ On the other side, the demand rate is slightly increased due to the green production and warranty period effective parameters whose values are taken as $\left[ {{\alpha _{2L}},{\alpha _{2U}}} \right]=\left[ {3.5,3.55} \right],\,\,\left[ {{\alpha _{3L}},{\alpha _{3U}}} \right]=\left[ {2.9,2.95} \right],b=0.4,c=0.8.$ Now, estimate the bounds of best-found policy of the said company for new launched mini scooter.

Solution

The value of the system parameters related to the proposed model for this example is as follows.

$$\eqalign{& \left[ {{\alpha _{0L}},{\alpha _{0U}}} \right] = \left[ {355,357} \right],\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right] = \left[ {2.4,2.45} \right],\,\,\left[ {{\alpha _{2L}},{\alpha _{2U}}} \right] = \left[ {3.5,3.55} \right],\,\,\left[ {{\alpha _{3L}},{\alpha _{3U}}} \right] = \left[ {2.9,2.95} \right], \cr & \left[ {{\alpha _{4L}},{\alpha _{4U}}} \right] = \left[ {2.1,2.15} \right],\,\,\,\left[ {{\lambda _{0L}},{\lambda _{0U}}} \right] = \left[ {4.5,4.7} \right],\,\,\left[ {{\lambda _{1L}},{\lambda _{1U}}} \right] = \left[ {1.3,1.35} \right],\,\,\,\left[ {{\lambda _{2L}},{\lambda _{2U}}} \right] = \left[ {0.1,0.15} \right], \cr & \left[ {{\gamma _{0L}},{\gamma _{0U}}} \right] = \left[ {2.2,2.3} \right],\,\,\left[ {{\gamma _{1L}},{\gamma _{1U}}} \right] = \left[ {0.001,0.002} \right],\,\,\left[ {{a_{mL}},\,{a_{mU}}} \right] = \left[ {0.3,0.4} \right],\,\,\left[ {{\eta _L},{\eta _U}} \right] = \left[ {0.05,0.06} \right], \cr & \left[ {{h_{cL}},\,{h_{cU}}} \right] = \left[ {1.2,1.25} \right],\,\,\left[ {{c_{mL}},{c_{mU}}} \right] = \left[ {1.2,1.25} \right],\,\,a = 1.1,\,\,b = 0.4,\,\,c = 0.8,\,\,d = 4.5,\,\,\delta = 3.1,\,\,r = 8, \cr & \left[ {{A_L},{A_U}} \right] = \left[ {505,510} \right]. \cr}$$

Example-4 is addressed using the same set of meta-heuristic algorithms (EOA, AEFA, GWOA, SSA, TLBOA, and WOA), and the computed best & worst results are depicted in Tables 11 and 12. Additionally, statistical simulations were conducted to assess the efficiency and consistency of these algorithms in relation to Example 4, and the simulated outcomes are shown in Table 13.

From Tables 11, 12 and 13, it is evident that SSA exhibits superior performance compared to other algorithms up to a certain decimal place in Example 4. However, EOA, and TLBOA demonstrate equal consistency with SSA with regard to Example 4.

Table 11 Best-found solutions for Example 4.

Table 12 Worst-found solutions for Example 4.

Table 13 Results of statistical experiment for Example 4.

Comparative discussions and numerical findings for cases I-IV

From the numerical simulations and discussions of the subsection 6.1, one can observed the following findings:

• Tables 2, 3, 5, 6, 8, 9, 11 and 12 show that the best & worst found values of interval valued average profit (IVAP) corresponding to the Examples 1–4 (alternatively, Cases I-IV). From these results it is seen that the interval uncertainty of best found (worst found) value of the corresponding average profit of the present model w.r.t. Example 4 is comparatively best than that of the other examples according to the interval order relation introduced by Bhunia & Samanta.

• Furthermore, the interval uncertainty of best found (worst found) value of the same with respect to Examples 2 and 3 are comparatively better than that of the Example 1. Therefore, from these discussions one can say that the best found (worst found) policy under interval uncertainty for the proposed model is more profitable for Case-IV than other cases, and on the other side, the best found (worst found) polices of the same for Cases-II &III are more economical than the Case-I.

• From Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13, one can see the notable observations that the algorithm SSA, TLBO and EOA give the comparatively better result than others in solving Examples 1, 2 and 3 up to a certain decimal place and only SSA gives the better result than other in solving Example 4. Here, all the comparisons of interval fitness values are made by the interval order relation introduced by Bhunia & Samanta.

• Furthermore, from the statistical results one can see that EOA, SSA and TLBO are more consistent than others in solving Examples 1–4. At the conclusion from this discussions, one can say that the SSA algorithm is more efficient and consistent to analyze the best-found policy of the present model.

To show sectionally concavity of $A{P_c}$ graphically, the 3-D graphs of $A{P_c}$ w.r.t. different decision variables (taking two at a time) have been drawn using MATLAB R2023a software with regard to Example 4 and the obtained results are presented in the following Figs. 5, 6, 7, 8, 9 and 10.

Fig. 5

Concavity of $A{P_c}$w.r.t. selling price$\left( {{s_p}} \right)$ and green level of the products$\left( g \right)$of Example-4.

From Fig. 5, it is observed that the centre of average profit is sectionally concave w.r.t. the variables selling price$\left( {{s_p}} \right)$ and green level of the products$\left( g \right)$keeping other variables fixed at their optimal values.

Fig. 6

Concavity of $A{P_c}$w.r.t. selling price$\left( {{s_p}} \right)$ and warranty period $\left( {{t_w}} \right)$ of Example-4.

Figure 6 indicates that the centre of average profit is sectionally concave w.r.t. the variables selling price$\left( {{s_p}} \right)$ and warranty period $\left( {{t_w}} \right)$ keeping other variables fixed at their optimal values.

Fig. 7

Concavity of $A{P_c}$ w.r.t. selling price$\left( {{s_p}} \right)$ and cycle length $\left( T \right)$ of Example-4.

Figure 7 reveals that the centre of average profit is sectionally concave w.r.t. the variables selling price$\left( {{s_p}} \right)$ and cycle length $\left( T \right)$ keeping other variables fixed at their optimal values.

Fig. 8

Concavity of $A{P_c}$w.r.t. green level of the products$\left( g \right)$ and warranty period $\left( {{t_w}} \right)$ of Example-4.

From Fig. 8, it is seen that the centre of average profit is sectionally concave w.r.t. the variables green level of the products$\left( g \right)$ and warranty period $\left( {{t_w}} \right)$ keeping other variables fixed at their optimal values.

Fig. 9

Concavity of $A{P_c}$ w.r.t. green level of the products$\left( g \right)$ and cycle length $\left( T \right)$ of Example-4.

Figure 9 exposes that the centre of average profit is sectionally concave w.r.t. the variables green level of the products$\left( g \right)$ and cycle length $\left( T \right)$ keeping other variables fixed at their optimal values.

Fig. 10

Concavity of $A{P_c}$ w.r.t. warranty period $\left( {{t_w}} \right)$ and cycle length $\left( T \right)$ of Example-4.

It is observed from Fig. 10 that the centre of average profit is sectionally concave w.r.t. the variables warranty period $\left( {{t_w}} \right)$ and cycle length $\left( T \right)$ keeping other variables fixed at their optimal values.

To see the convergence history of all employed algorithms, a convergency graph with respect to the same example has been drawn using the values of $A{P_c}$ obtained by all algorithms in 500 generations and the graphical result has been depicted in Fig. 11.

Fig. 11

Convergence graph of different metaheuristic algorithms.

From Fig. 11, it is evident that SSA, EOA and TLBO algorithms converge rapidly where as other algorithms take more than 40 iterations to convergence in the best-found solution.

Example 5

This example is considered to justify the best-found policy of this case of the proposed model in the deterministic sense. The system parameters take the value for this example constructed from Example 4 by taking the centre values of each parameter and the obtained parameters’ values are presented in the form of degenerate intervals as follows;

$$\eqalign{& \left[ {{\alpha _{0L}},{\alpha _{0U}}} \right] = \left[ {356,356} \right],\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right] = \left[ {2.425,2.425} \right],\,\,\left[ {{\alpha _{2L}},{\alpha _{2U}}} \right] = \left[ {3.525,3.525} \right], \cr & \left[ {{\alpha _{3L}},{\alpha _{3U}}} \right] = \left[ {2.925,2.925} \right],\,\,\left[ {{\alpha _{4L}},{\alpha _{4U}}} \right] = \left[ {2.125,2.125} \right],\,\,\left[ {{\lambda _{0L}},{\lambda _{0U}}} \right] = \left[ {4.6,4.6} \right], \cr & \left[ {{\lambda _{1L}},{\lambda _{1U}}} \right] = \left[ {1.325,1.325} \right],\,\,\left[ {{\lambda _{2L}},{\lambda _{2U}}} \right] = \left[ {0.125,0.125} \right],\,\,\left[ {{\gamma _{0L}},{\gamma _{0U}}} \right] = \left[ {2.25,2.25} \right] \cr & \left[ {{\gamma _{1L}},{\gamma _{1U}}} \right] = \left[ {0.0015,0.0015} \right],\,\,\left[ {{a_{mL}},\,{a_{mU}}} \right] = \left[ {0.35,0.35} \right],\,\,\left[ {{\eta _L},{\eta _U}} \right] = \left[ {0.055,0.055} \right] \cr & \left[ {{h_{cL}},\,{h_{cU}}} \right] = \left[ {1.225,1.225} \right],\,\left[ {{c_{mL}},{c_{mU}}} \right] = \left[ {1.225,1.225} \right],\,\,a = 1.1,\,\,b = 0.4,\,\,c = 0.8 \cr & d = 4.5,\,\,\delta = 3.1,\,\,r = 8,\,\,\left[ {{A_L},{A_U}} \right] = \left[ {507.5,507.5} \right]. \cr}$$

Solution:

The computed best-found & worst-found results of Example 5 are presented in Tables 14 and 15. Statistical simulations have shown in Table 16. From Tables 14, 15 and 16, it is evident that SSA, TLBO and EOA gives the same results up to a certain decimal place in solving Example 5. Also, these algorithms demonstrate equal consistency with regard to same example.

Table 14 Best-found solutions for Example 5.

Table 15 Best-found solutions for Example 5.

Table 16 Results of statistical experiment for Example 5.

Tables 14 and 15 exhibit that the best found (worst found) values of the IVAP of Example 5 are de-generate intervals which identified by crisp values and these value lies between the bounds of the corresponding best-found (worst-found) values of the IVAP with respect to Example 4 (Tables 11 and 12). In other words, it can be stated that the best-found policy of the proposed model in crisp environment always fluctuates between the bounds of that of the proposed model in interval environment and conversely, from the policy of the model under interval uncertainty, an estimated policy for deterministic case can be obtained.

From Tables 14, 15 and 16, one can see the notable observations that the algorithm SSA, TLBO and EOA give the comparatively better result than others in solving Example 5 up to a certain decimal place. Here, all the comparisons of interval fitness values are made by the interval order relation introduced by Bhunia & Samanta.

Non-parametric statistical tests

From the numerical discussions and findings, it is evident that SSA is more competent than other algorithms in solving all cases, specially, in solving Case-IV. To test the significance changes of the solution obtained by SSA and other five algorithms (viz., GWOA, EOA, AEFA, TLBO and WOA), two non-parametric statistical tests (Friedman and Wilcoxon rank sum test) are conducted with respect to Example 4(Case-IV) and during the execution of these tests by taking SSA as the controlling algorithm. The p-values of both Friedman test and Wilcoxon rank sum test are presented in Tables 17 and 18 respectively.

Table 17 Friedman test of SSA vs. other algorithms for Example-4.

Table 18 Wilcoxon rank sum of SSA vs. other algorithms for Example-4.

Table 17 shows that p-values obtained from SSA versus AEFA, TLBOA, GWOA and WOA are less than 0.05. Thus, SSA is statistically significant in comparing with AEFA, WOA & GWOA in order to solve the both cases of Example 4 at 5% level of significance.

It can be observed from Table 18 that p-values obtained after applying Wilcoxon rank sum test from SSA versus AEFA, GWOA & WOA is less than 0.05 for Example 4. Based on the discussion, it can be remarked that SSA is statistically significant compared to AEFA, WOA, and GWOA in solving Example 4 at the 5% level of significance. Therefore, the results of the Friedman test and the Wilcoxon rank-sum test indicate that SSA outperforms AEFA, GWOA, and WOA in solving Example 4, while its performance is similar to TLBO.

Sensitivity analysis

Here, sensitivity analyses have been carried out to analyse the effect of different interval-valued inventory parameters involved in the proposed model on centre of average profit$\left( {A{P_c}} \right)$, selling price of the items$\left( {{s_p}} \right)$, green level of the products$\left( g \right)$, warranty period$\left( {{t_w}} \right)$and business cycle length$\left( T \right)$ of the manufacturer for Example 4. The computed results are shown graphically in Figs. 12, 13, 14, 15, 16, 17, 18, 19 and 20.

Fig. 12

Impact of $\left[ {{\alpha _{0L}},{\alpha _{0U}}} \right]$on the optimal policy.

Fig. 13

Impact of $\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right]$on the optimal policy.

Fig. 14

Impact of $\left[ {{\alpha _{2L}},{\alpha _{2U}}} \right]$on the optimal policy.

Fig. 15

Impact of $\left[ {{\alpha _{3L}},{\alpha _{3U}}} \right]$on the optimal policy.

Fig. 16

Impact of $\left[ {{\alpha _{4L}},{\alpha _{4U}}} \right]$on the optimal policy.

Fig. 17

Impact of $\left[ {{\lambda _{0L}},{\lambda _{0U}}} \right]$on the optimal policy.

Fig. 18

Impact of $\left[ {{\lambda _{1L}},{\lambda _{1U}}} \right]$on the optimal policy.

Fig. 19

Impact of $\left[ {{\gamma _{0L}},{\gamma _{0U}}} \right]$on the optimal policy.

Fig. 20

Impact of $\left[ {{h_{cL}},{h_{cU}}} \right]$on the optimal policy.

The following finding can be observed from the Figs. 12, 13, 14, 15, 16, 17, 18, 19 and 20.

• (i) It is observed from the previous mentioned figures that centre of IVAP of Example 4 is extremely effective w.r.t. $\left[ {{\alpha _{0L}},{\alpha _{0U}}} \right]$,$\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right]$and production cost parameter $\left[ {{\lambda _{1L}},{\lambda _{1U}}} \right]$whereas it insensitive w.r.t. remaining parameters i.e., demand parameters $\left[ {{\alpha _{2L}},{\alpha _{2U}}} \right]$,$\left[ {{\alpha _{3L}},{\alpha _{3U}}} \right]$,$\left[ {{\alpha _{4L}},{\alpha _{4U}}} \right]$, initial production cost $\left[ {{\lambda _{0L}},{\lambda _{0U}}} \right]$,warranty cost parameter$\left[ {{\gamma _{0L}},{\gamma _{0U}}} \right]$and holding cost $\left[ {{h_{cL}},{h_{cU}}} \right]$. Furthermore, $\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right]$and $\left[ {{\lambda _{1L}},{\lambda _{1U}}} \right]$have a reverse effect on the centre of IVAP.

• (ii) Selling price of the products$\left( {{s_p}} \right)$of Example 4 is extremely effective with the changes of demand parameter$\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right]$and moderately sensitive with the changes of initial demand rate c. Further, selling price of the products is not sensitive w.r.t. demand parameters $\left[ {{\alpha _{2L}},{\alpha _{2U}}} \right]$, $\left[ {{\alpha _{3L}},{\alpha _{3U}}} \right]$,$\left[ {{\alpha _{4L}},{\alpha _{4U}}} \right]$,initial production cost$\left[ {{\lambda _{0L}},{\lambda _{0U}}} \right]$,production cost parameter$\left[ {{\lambda _{1L}},{\lambda _{1U}}} \right]$, warranty cost parameter$\left[ {{\gamma _{0L}},{\gamma _{0U}}} \right]$and holding cost $\left[ {{h_{cL}},{h_{cU}}} \right]$. Also, $\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right]$has reverse effect on $\left( {{s_p}} \right)$.

• (iii) It is seen from Figs. 12, 13, 14, 15, 16, 17, 18, 19 and 20 that g is less sensitive w.r.t. demand parameters$\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right]$and$\left[ {{\alpha _{2L}},{\alpha _{2U}}} \right]$whereas it is not sensitive w.r.t. to the remaining parameters i.e., initial demand rate$\left[ {{\alpha _{0L}},{\alpha _{0U}}} \right]$,demand parameters$\left[ {{\alpha _{3L}},{\alpha _{3U}}} \right]$,$\left[ {{\alpha _{4L}},{\alpha _{4U}}} \right]$, initial production cost $\left[ {{\lambda _{0L}},{\lambda _{0U}}} \right]$, production cost parameter$\left[ {{\lambda _{1L}},{\lambda _{1U}}} \right]$, warranty cost parameter$\left[ {{\gamma _{0L}},{\gamma _{0U}}} \right]$and holding cost $\left[ {{h_{cL}},{h_{cU}}} \right]$.

• (iv) Warranty period $\left( {{t_w}} \right)$ is sensitive moderately with the changes of demand parameters$\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right]$,$\left[ {{\alpha _{3L}},{\alpha _{3U}}} \right]$. It is not sensitive with the changes of the remaining parameters i.e., initial demand rate $\left[ {{\alpha _{0L}},{\alpha _{0U}}} \right]$,demand parameters$\left[ {{\alpha _{2L}},{\alpha _{2U}}} \right]$,$\left[ {{\alpha _{4L}},{\alpha _{4U}}} \right]$, initial production cost $\left[ {{\lambda _{0L}},{\lambda _{0U}}} \right]$, production cost parameter$\left[ {{\lambda _{1L}},{\lambda _{1U}}} \right]$, warranty cost parameter$\left[ {{\gamma _{0L}},{\gamma _{0U}}} \right]$and holding cost $\left[ {{h_{cL}},{h_{cU}}} \right]$. Furthermore, $\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right]$has reverse effect on $\left( {{t_w}} \right)$.

• (v) Now, cycle length$\left( T \right)$is highly effective w.r.t. $\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right]$. It is sensitive equally w.r.t. $\left[ {{\lambda _{1L}},{\lambda _{1U}}} \right]$. Further, cycle length of the manufacturer$\left( T \right)$ is moderately sensitive with the changes of $\left[ {{\alpha _{0L}},{\alpha _{0U}}} \right]$ and holding cost $\left[ {{h_{cL}},{h_{cU}}} \right]$ whereas it is not sensitive w.r.t. the remaining of the parameters i.e., demand parameters$\left[ {{\alpha _{2L}},{\alpha _{2U}}} \right]$,$\left[ {{\alpha _{3L}},{\alpha _{3U}}} \right]$,$\left[ {{\alpha _{4L}},{\alpha _{4U}}} \right]$, initial production cost $\left[ {{\lambda _{0L}},{\lambda _{0U}}} \right]$and warranty cost parameter$\left[ {{\gamma _{0L}},{\gamma _{0U}}} \right]$.Furthermore, $\left[ {{\alpha _{1L}},{\alpha _{1U}}} \right]$and$\left[ {{\lambda _{1L}},{\lambda _{1U}}} \right]$ have reverse effect on $\left( T \right)$.

Findings and managerial implications

In this section, all the possible findings in light of the research questions posed in the introduction section and possible managerial insights are discussed.

• (i) Model development:

This study successfully develops a production inventory model by addressing the complexities of imperfect production, reworking, green investment, warranty policy and carbon emission regulation together. Then, an interval-valued optimal control problem (IVOCP) grounded in a generalized variational principle is formulated. The inclusion of interval uncertainty in the demand rate—modulated by green investments and warranty period in a highly nonlinear optimization problem captures real-world imprecision more effectively than crisp models. The model demonstrates flexibility and adaptability in scenarios with complex, uncertain parameters.

• (ii) Performance of improved hybrid optimization algorithms:

The improved c-r optimization technique and the Sparrow Search Algorithm (SSA) exhibit superior performance in terms of best-found result, convergence speed, and robustness across four distinct scenarios. Comparative analysis against other popular metaheuristic algorithms (viz. EOA, AEFA, TLBO, WOA, GWO) reveals that SSA consistently yields better or equal best-found solutions with stability. Non-parametric statistical tests and ANOVA test validate the significance of these performance differences at a 95% confidence level.

• (iii) Insights from sensitivity and statistical analyses:

Sensitivity analysis reveals that key inventory parameters—such as demand’s parameters, holding cost, production cost’s parameters and warranty cost’s parameters, significantly affect the optimal policy. For instance, higher green investment levels tend to reduce total cost in the long run, justifying sustainability-oriented strategies. Similarly, warranty periods influence not only the demand rate but also the replenishment cycle, indicating their dual role in customer satisfaction and cost structure. These findings are critical for managerial decision-making, particularly in industries aiming to balance sustainability, reliability, and profitability under uncertainty.

• (iv) Industrial Implication:

The modelling concept of this work is applicable in industries where green responsibility, product lifecycle management, and extended warranty are plays crucial role. This model is best fit for the goods like, electronics, automotive, medical devices. Specifically, the manager of a company can applies the model to make more accurate decisions on green invest management in uncertain, complex environments. Likewise, firms that provide extended warranties can apply the results to synchronize warranty longevity with investment and inventory management policies. The described method is of great utility to industries subject to regulatory compulsion to green up without losing profitability and competitiveness in uncertain markets.

Concluding remarks and future scope of research

As a brief, in this work, an imperfect production inventory model with reworking and carbon emission regulation under interval uncertainty has been studied. In the formulation, green investment as well as warranty-based interval valued demand rate has been considered. The corresponding decision-making problem is formulated as an interval-valued optimization and it has been solved by hybridization of improved centre-radius optimization method and the Sparrow Search Algorithm (SSA). The optimality of the model has been studied under four different cases based on green investment and warranty period. The findings indicate that the application of interval modelling to manage uncertainty strengthens the model significantly, particularly in dynamic production systems. Among the four cases, the fourth case, green investment and warranty period, was the most profitable and sustainable. This indicates how these elements interact to boost demand, reduce costs, and make the entire system more efficient.

The results of this study provide actionable insights for managers and strategic decision-makers operating under uncertainty. By incorporating green investment initiatives and warranty strategies into the demand structure, the proposed interval-valued optimal control model enables managers to navigate complex, uncertain environments with greater precision. The improved optimization techniques ensure more stable and reliable decision-making compared to traditional methods. Sensitivity analysis further equips managers with an understanding of how key system parameters—such as holding costs, deterioration rates, or warranty lengths—influence optimal inventory and investment strategies. Consequently, firms can dynamically adjust their policies to enhance sustainability, cost-effectiveness, and customer satisfaction while mitigating risks associated with demand uncertainty.

Although the proposed model deals with important sources of uncertainty and incorporates major sustainability aspects, there are several limitations. Firstly, the model assumes fixed bounds for all interval parameters, which might not always hold in extremely volatile markets. Secondly, the environmental impact is represented by a reduced carbon emission cost structure, which might not represent more sophisticated regulation or multi-level sustainability structures.

To address these issues, in future research, one could explore the concept of this model in a green supply chain model employing fuzzy, stochastic, Type-2 interval or hybrid approaches to respond more adequately to deep uncertainty in cases of indeterminate boundaries. Further, the idea of this work can be applied in the area of smart manufacturing supply chain system^82,83,84. Incorporating more sophisticated or hybrid optimisation algorithms could yield even better results, surpassing the capabilities of the current SSA approach. Testing the proposed model in real-world scenarios across different industries would provide practical insights and validate the theoretical outcomes.