
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39251629
70881
10.1038/s41598-024-70881-x
Article
Application of water cycle algorithm with demand follows green level and nonlinear power pattern of the product for an inventory system
Das Subhash Chandra 1
Akhtar Fleming 2
Alrasheedi Adel Fahad 3
Shaikh Ali Akbar aakbarshaikh@gmail.com

2
1 Department of Mathematics, Chandrapur College, Chandrapur, West Bengal India
2 https://ror.org/05cyd8v32 grid.411826.8 0000 0001 0559 4125 Department of Mathematics, The University of Burdwan, Burdwan, 713104 India
3 https://ror.org/02f81g417 grid.56302.32 0000 0004 1773 5396 Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, 11451 Riyadh, Saudi Arabia
9 9 2024
9 9 2024
2024
14 209957 5 2024
22 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
It is commonly known that a number of variables, including price, supply levels, time, and green level, affect how quickly certain things are in demand. Furthermore, the inventory carrying cost is considered to be a nonlinear representation of time and is subject to variation throughout time. More precisely, it rises with time since longer storage times necessitate more costly warehouse space. This study presents a fully backlogged situation inventory system for a single commodity where the product’s selling price, green level, and time are used to simultaneously compute the demand rate in accordance with a power pattern. Purchase price is determined by the product’s nonlinear green level. Complete backorders are available for shortages. The impact of the product’s selling price, green level and time power function are combined to determine the product’s demand. Moreover, the holding cost also rises as the product is stored for a longer period of time. The primary goal is to determine the best inventory policy to maximise total profit per unit of time. Though the problem is highly nonlinear in nature. Hence, we cannot solve it analytically. To overcome these difficulties, we have applied several well-known popular metaheuristic algorithms (Water Cycle Algorithm (WCA), Artificial Electric Field Algorithm (AEFA), Teaching Learning Based Optimization Algorithm (TLBOA), Grey Wolf Optimizer Algorithm (GWOA), Sparrow Search Algorithm (SSA), Whale Optimizer Algorithm (WOA), Prairie Dog Optimization Algorithm (PDOA), Gazelle Optimization Algorithm (GOA), A Sinh Cosh Optimizer Algorithm (SCHOA) and White Sherk Optimizer Algorithm (WSOA), Archimedes Optimization Paradigm Algorithm (AOPA), Marine Predator Optimization Algorithm (MPOA), Geyser Inspired Algorithm (GIA), Runge Kutta Optimization Algorithm (RKOA), Lungs Performance-based Optimization Algorithm (LPOA) and Dwarf Mongoose Optimization Algorithm (DMOA)). It is observed that WCA perform better than other algorithms with respect to the convergence rate. A numerical example is taken in order to validate the proposed model. Finally, a post optimality analysis is performed in order to make a fruitful conclusion.

Keywords

Green product
Power pattern demand
Green level dependent purchase cost
Nonlinear holding cost
Fully backlogged shortages
Application of WCA algorithm
Subject terms

Engineering
Mathematics and computing
http://dx.doi.org/10.13039/501100002383 King Saud University RSP2024R323 Alrasheedi Adel Fahad issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

In any inventory situation, there exists an optimisation problem to either maximise the total/average profit or minimise the total/average expense. These optimisation problems can be categorised as constrained or bound-constrained, and they may contain continuous, integer, or mixed-integer decision variables. The general shape of a continuous optimisation problem is as follows:Optimizefusubjecttogiu≤ai,i=1,2,3,...,n1hiu=ai,i=n1+1,n1+2,...,n1+n2whereu=u1,u2,...,ud1∈Rd1.

The price at which a product is made available for purchase to customers or clients is its selling price in an inventory system. It is the price that a company or organisation sets for a particular good or service in order to make money and pay for its outlays. When determining the selling price of an item, factors including the cost of production or acquisition, any associated costs (such packing or shipping), and the intended profit margin are all taken into consideration. In addition, considerations may be given to rivalry, market demand, and pricing strategies. In an inventory system, the selling price of an item is frequently established and stored in a database or software platform. This facilitates the company’s ability to access and modify the selling price as needed. It also aids in the system’s ability to compute earnings, track the worth of each item in stock, and generate accurate sales records. Businesses must carefully consider the price of their inventory products so that cost-coverage, competitiveness, and the accomplishment of targeted profit targets. Pricing decisions often require analysis of consumer preferences, the overall business strategy, and the state of the market.

The non-linear power demand pattern of an item in an inventory system is the variation in the quantity or rate at which an item is required over time, when the changes in demand are not proportionate to changes in other variables. Here, “power” refers to the degree of strength or intensity of the demand. Compared to a linear demand pattern, which illustrates how the demand for a good may constantly rise or fall in response to changes in factors like price or market conditions, a non-linear power demand pattern exhibits more complex and unpredictable variations. Seasonality, consumer behaviour, trends, and external events are just a few of the variables that could affect these variations. In a non-linear power demand pattern, high demand intervals could be followed by low demand intervals, or the other way around. For example, the demand for some goods might rise during specific events or seasons and fall during other times of the year. Managing non-linear power demand trends in an inventory system can pose challenges for businesses. Precise forecasting techniques, data analysis, and adaptable inventory management protocols are required. Businesses may need to modify their production or procurement schedules, time marketing and promotional activities to coincide with times of peak demand, and optimise inventory levels in order to satisfy customer requests while minimising excess stock or stock outs. Predictive analytics, market trends, and historical sales data are some of the tools that sophisticated inventory management systems can employ to identify and anticipate nonlinear power usage patterns. This helps businesses to optimise inventory replenishment, improve resource allocation, and enhance supply chain management generally.

In an inventory system, a shortfall occurs when an item is not available in the intended or necessary quantity. It indicates that there isn’t enough inventory on hand to satisfy consumer demands or meet demand. Shortages can occur for several causes and have a significant effect on an organisation. If the demand for a product is miscalculated or underestimated, shortages may result. Unexpected events, changes in the market, and insufficient data analysis can all lead to inaccurate forecasts. Low inventory levels can also result from supply chain issues such delays in production, shipment issues, or shortages of suppliers. Unexpected events such as traffic bottlenecks, labour disputes, and natural calamities might cause delays. Furthermore, order processing issues like imprecise stock counts or a delay in replenishing supplies might lead to shortages. Human mistake, a lack of real-time visibility, or inefficient inventory management strategies could be the cause of these issues.

A corporation may suffer unfavourable effects from shortages. They could lead to decreased revenue, irate clients, harm to the company’s image, and even the loss of potential new business prospects. In order to quickly replenish supply, shortages might also result in increased costs due to faster shipment or manufacturing. Businesses that implement efficient inventory management techniques can reduce shortages. To achieve this, it is essential to use precise demand forecasts, real-time inventory monitoring, safety stock level setting, strong supplier connections, and automatic reorder systems. By closely monitoring inventory levels, anticipating changes in demand, and acting quickly, businesses can lessen the likelihood and effect of shortages in their inventory system.

For every business in every economic sector, product inventories are crucial. In order to choose the best policies (such as when to order, where to place it, and how many quantities), as well as, in some situations, the maximum number of shortages that may be permitted, businesses must first develop reliable inventory models. For this reason, companies usually employ an inventory management department that makes helpful recommendations for controlling inventory so that goods are constantly available to meetup the demand of the customer. This is the justification for the various inventory models that academics have been developing to appropriately match and handle inventory and its related problem. As for example, Alshanbari et al.1, Fang et al.2, have investigated the combined effect on the pricing-inventory system and Khan et al.3.

The rate of demand is often taken to be known and constant in production and inventory models. In actuality, though, a number of factors, including price, an item’s green level, stock, and storage duration, all have an impact on the pace of demand. Consequently, managers can consider all demand dependent on price, nonlinear holding costs, and patterns of power demand. The aforementioned notion has been achieved in this study, along with fully backlogged shortages that are pertinent to real-world business situations.

Two broad categories of approaches are commonly found in the literature to address optimisation difficulties. Gradient-based approaches make up one of them, while gradient-free methods make up the other. Gradient-based approaches might reach the best result fast and precisely, but they need the objective function’s derivative information and/or limitations. Therefore, these techniques cannot be used for optimisation problems with extremely non-linear objective functions or constraints that are discontinuous or non-differentiable. Gradient-free methods are capable of solving these kinds of optimisation problems because they don’t require restrictions or the objective function’s derivative information. The literature now contains a number of metaheuristic algorithms that have been developed during the past few decades. The following categories could apply to these algorithms:

Evolutionary-based algorithm

An area of optimisation algorithms known as evolutionary-based algorithms draws inspiration from the laws of natural selection. In fields like engineering, machine learning, and biology, when conventional methods are ineffective, they are frequently employed to solve complex optimisation problems. Algorithms that are based on evolution often emulate the idea of natural evolution. Goldberg and Holland4 proposed Genetic algorithms (GA), Koza5 introduced genetic programming (GP), Storn and Price6 discussed about the idea of differential evolution (DE), Rechenberg7 introduced the concept about evolutionary strategy (ES) and other well-known evolutionary-based algorithms are a few examples.

Swarm intelligence-based algorithm

Swarm intelligence-based algorithms take cues from the collective behaviour of social organisms such as birds, bees and ants to solve optimisation problems. To obtain the best answers, these algorithms imitate the self-organized, decentralised structure of natural swarming. Algorithms based on swarm intelligence are very good at handling complicated optimisation issues with plenty of search space, several targets, and changing conditions. They frequently use the combined intellect of a population of agents to deliver strong and effective solutions. Swarm activity serves as an inspiration for these kinds of algorithms. Popular Swarm-Intelligence-Based Algorithms include Particle Swarm Optimisation (PSO) introduced by Kennedy and Eberhart8, Baccalagio (BF) proposed by Passino9, Firefly Algorithm (FA) developed by Yang10, Krill Hard (KH) described by Gandomi and Alavi11, Grey Wolf Optimisation (GWO) proposed by Mirjalili et al.12 and Whale Optimisation Algorithm (WOA) introduced by Mirjalili and Lewis13, etc.

Human-based algorithm

Human-based algorithms, often referred to as human-in-the-loop algorithms, to tackle challenging issues, use human intellect and decision-making skills. These algorithms uncover solutions that are hard for machines to find on their own by combining human intuition, creativity, and subject knowledge with the computing capacity of machines. In fields like design, innovation, and complicated decision-making, where human judgement, creativity, and knowledge are crucial, human-based algorithms are useful. These algorithms can get above the drawbacks of solely computer methods and provide more efficient and significant outcomes by incorporating human input. These kinds of algorithms are inspired by human conduct. These actions could be emotional, cultural, societal, etc. Thus several human-based algorithms have been introduced. Among them are the following: tabu search (TS) introduced by Glover14, harmony search (HS) proposed by Geem et al.15, teaching–learning-based optimisation (TLBO) investigated by Rao et al.16, social group optimisation (SGO) proposed by Satapathy and Naik17, ideology algorithm (IA) introduced by Huan et al.18 and others.

Chemical-and physical and based algorithm

Algorithms that are based on physics and chemistry draw inspiration from their concepts and processes to address computational and optimisation issues. In order to investigate and optimise solutions in a computational setting, these algorithms frequently replicate chemical or physical phenomena. For the purpose of resolving complicated optimisation issues involving dynamic systems, non-linear interactions, and stochastic processes, physical and chemical-based algorithms are especially helpful. They offer strong instruments for investigating and refining solutions in a variety of disciplines, including engineering, chemistry, biology, and physics. These kinds of algorithms are inspired by physics and chemical concepts, laws, and regulations. A few well-known physical and chemical-based algorithms are atom search optimisation (ASO) proposed by Zhao et al.19, simulated annealing (SA) introduced by Kirkpatrick et al.20, gravitational local search algorithm (GLSA) investigated by Webster and Bernhard21, ray optimisation (RO) proposed by Kaveh and Khayatazad22, curved space optimisation (CSO) introduced by Moghaddam et al.23, and so on. In the next section, we are going to discuss about the literature related to inventory models.

The premise of a constant cost of carrying is the foundation of most of the models of inventory and production. However, in practice, the holding cost is different. Future research should therefore focus on creating models of inventory that take a variable holding cost into account. Many model types are used in this subject, including ones that take into account holding costs that are based on stocks, time, multiple dependencies, or any additional variation in holding costs. Alfares and Ghaithan24 provided a wonderful and thorough state-of-the-art analysis of models of inventories that consider fluctuating holding costs. When holding costs are time-dependent, authors frequently use either nonlinear or linear time functions. Considering that the item’s cost of carrying is not linearly dependent on the length of storage, Weiss25 suggested EOQ inventory models that are both probabilistic and deterministic. Goh26 examined two different forms of holding cost variations: a nonlinear function that takes into account the amount of inventory that is kept on hand as well as the length of time that the products are kept in storage. The inventory models of Goh26 examined and expanded by Giri and Chaudhuri27, who included the perishable nature of the products. Chang28 further refined Giri and Chaudhuri’s27 inventory models by maximising profit and easing the requirement of inventory level become zero. They demonstrated that the profits are much higher than those attained by the inventory model proposed by Giri and Chaudhuri27. Ferguson et al.29, who also noted that the model approximates the ideal order quantity when it comes to perishable goods, expanded upon Weiss’s25 inventory model. They included savings as well as surcharges for infrequent orders. Regarding the variable holding cost, Alfares30 suggested two different types of step functions that discontinue, in which the cost of carrying increases continuously and the items’ time of stocking is divided into multiple time span. Goh26 to describe how the cost of carrying varied over time used a nonlinear continuous function. Furthermore, the holding cost may be applied either incrementally to the next storage cycle solely or retroactively to all storage cycles as the item’s storage time gets closer to the next time period. Not to be overlooked is the requirement set by Alfares30 that the inventory level at cycle’s conclusion equal zero. On the other hand, Urban31 expanded upon and reviewed Alfares’s30 study by allowing the final level of inventory to have a value other than zero. Mahata and Goswami32 investigated fuzzy related models for decaying items, based on the presumption that holding costs change depending on how long a product is stored and that a triangular fuzzy number determines the rate of deterioration. Mao and Xiao33 constructed and handled a non-instantaneous deteriorating commodity inventory model by accounting for the possibility of shortages and the fact that they would be completely backordered. The study’s representation of the holding cost was a generalised function of the on-hand inventory. Valliathal and Uthayakumar34 created a production inventory model that treated the holding cost as a nonlinear function of time and assumed shortages with partial backordering. When the demand and the holding cost are nonlinear functions in relation to the amount of time stored, Pando et al.35 examined about the best course of action for maximising profits in an inventory system with demand dependent on stock. When there is uncertainty in the purchaser’s lead time, Sazvar et al.36 found in a three-tier supply chain with a stochastic lead time, a centralised replenishment mechanism for degrading products. When demand is contingent on inventory level, Pando et al.37 looked into an inventory model with an economic lot size. An indefinite function was incorporated into the model to represent the quantity of units kept in stock as well as the duration of storage. According to the quantity of on hand inventory, Prasher and Pundir38 examined the holding cost’s nonlinearity. Assuming a stochastic lead time and a constant and predictable demand rate over an infinite planning horizon, Sazvar et al.39 explored with an ongoing evaluation of the inventory system. Their inventory model supports fully backordered shortages and makes use of the dependent on time nonlinear holding cost function established by Weiss25. For perishable items with a uniformly distributed lead time and demand, Sazvar et al.40 incorporated the level of service requirements into their new method for calculating inventory level. San-Jose et al.41 examined an EOQ inventory model featuring non-linear unit holding costs and partial backordering. Khalilpourazari and Pasandideh42 studied an EOQ model with multiple items, nonlinear unit holding costs and partial backordering: method for moth-flame optimisation. Paknejad et al.43 proposed a power yield distribution’s shape: implications for the EOQ model with random quality and nonlinear holding costs. Pando et al.44 presented the optimal lot-size approach that maximises profitability for degrading items with stock-dependent demand. An economic order quantity model featuring ramp-type demand, nonlinear holding costs, and a partial backlog was reported by San-Jose et al.45. Making price and inventory decisions simultaneously was recommended by Edalatpour and Al-e-Hashem46 for complementary and substitute commodities with nonlinear holding costs. Pando et al.47 explored optimising the profitability ratio in an inventory model with non-linear holding costs and inventory level dependent demand rate. San-Jose et al.48 addressed the optimal pricing and quantity for non-linear holding costs and electricity demand patterns. Tripathi49 looked at models of economic order quantity for different holding cost functions and price-dependent demand. Cardenas-Barron et al.50 examined an EOQ inventory model with trade credit, nonlinear stock dependent holding cost, and nonlinear stock dependent demand.

One branch of research in the field of inventory models looks into inventory issues related to time-dependent demand (i.e., goods that have been sold at the beginning of the cycle, removed at its conclusion, or continuously consumed during the period). We refer to these particular ways that the demand occurs throughout a time period as power patterns. Many studies have been conducted that use the power demand pattern function to model the demand. Naddor51 developed an inventory model with power demand structure based on cycle time as well as time. Subsequently, numerous additional scholars have created inventory models that use of a function of time in the power from of the representation of the demand. An inventory model, for instance, was created by Goel and Aggarwal52 utilising the power form of the demand for the scenario in which a fixed percentage of the available inventory degrades over time. Next, for objects with a variable rate of deterioration, Datta and Pal53 presented an inventory system that used a power demand pattern. Girlich54 employed a power demand pattern to solve the EOQ inventory model. Afterwards, Lee and Wu55 investigated a power demand pattern in an EOQ inventory model with authorised shortages where the products degrade at a Weibull distributed rate. Apart from providing a power demand pattern and an overall time-proportional backlogging rate, Dye56 also examined and expanded on the completed work by Lee and Wu55. Jung et al.57 made a revision to Dye’s56 inventory model due to several questionable findings. Abdul-Jalbar et al.58 created an inventory model using mixed nonlinear programming and investigated the effects of backordering and the use of a power demand pattern in a scenario with one warehouse and N retailers. Singh et al.59 created an EOQ inventory model in which a power demand pattern is observed in the demand for degraded products, allowing for shortages and the utilisation of partially backordered goods. Their inventory model states that the relationship between the backordering rate and the waiting period for the next replenishment is inverse. When the products degrade with a two-parameter Weibull distribution rate, Tripathy and Pradhan60 took into account for the demand pattern taken into a power form with partially backlogs and created an EOQ inventory model. Kumar and Singh61 when modelling of the inventory system also considered the effects of a partial backlog and a combined holding cost. The item degrades subsequent to a set amount of time, known as the life duration. Rajeswari and Vanjikkodi62 investigated a model of inventory with a constant rate of product deterioration and a power pattern in demand. There is room for shortages, as these are somewhat backordered. Sarbjit and Shivraj63 introduced predictable and stochastic EOQ models of inventory with shortfalls for products with different rates of deterioration, as well as a power demand pattern. Inflation and a suitable payment delay are also considered and assessed. Singh and Sehgal64 considered an efficient consumption pattern in which shortages are tolerable and fully backordered in order to develop an inventory model. The deterioration rate for a power demand pattern with no shortages was represented by a two-parameter Weibull distribution in Krishnaraj and Ramasamy’s65 inventory system. Rajeswari and Vanjikkodi66 studied of an item inventory model with two parameters backlog and Weibull distribution deterioration is presented. Sicilia et al.67 proposed a system for deterministic inventory that incorporate power demand patterns. Additionally, models are created for total lost sales inventory and total backordering. Sicilia et al.68 developed a production-inventory system along with demand of the product is taken into power form under completely backlogging situations. The best inventory plan was identified by San-Jose et al.69 for an inventory system with a fixed partial backlog and a pattern of power consumption. An investigation by San Jose et al.70 looked into a model of inventory with fully backlogged shortages and a demand rate derived from a power-time function multiplied by a pricing function. San-Jose et al.48 examined a single item inventory model without shortages, where the demand rate was determined by multiplying a power-time function by a linear function that took into account the unit selling price. A power demand pattern is included in various inventory models that Sicilia et al.71 have presented. Sicilia et al.72 proposed the optimal course of action for power demand pattern related inventory model with backlogs, and a production rate proportionate to demand. Sicilia et al.73 introduced the optimal inventory rules for time-dependent demand in uniform replenishment systems. For linearly fading items, Rajeswari et al.74 developed an optimisation fuzzy inventory framework with power demand, partial backlog, and linear holding cost. San-Jose et al.70 studied an inventory system that takes into account backlogs in shortages and bases demand on both time and price. Gurtu75 optimises holding costs for inventory by taking into account the price, weight, and volume of each item. San-Jose et al.76 introduced best pricing and ideal policy for an inventory system with demand and backorders that are depending on time and price. Chowdhury and Ghosh77 studied on a perishable item production-inventory model includes demand-dependent production rates, shortages, and variable holding costs. Momena et al.78 developed an inventory model that involves a trade credit policy and varying holding costs over time.

Higher pricing results in buyers buying fewer items since pricing influences the demand for different products. Conversely, a low price makes people want to buy more products. In this field of study, a variety of inventory models have been developed to assess inventory policies while accounting for demand that is price-dependent. Price-dependent probabilistic demand and price reductions are included in the joint pricing and inventory problem, which was first presented by Jadidi et al.79. A price-sensitive demand model with volume flexibility and declining inventory was presented by Panda et al.80. Rubio-Herrero and Baykal-Gursoy81 investigated the price-setting newsvendor problem with additive demand’s unimodality under risk considerations. In a service-inventory system, Marand et al.82 covered integrated pricing and inventory control. The ideal quantity and price under a non-linear holding cost and power demand pattern were determined by San-Jose et al.48. Rahman et al.83 solved an inventory model for degraded items whose demand depended on selling price using a parametric technique. In the context of pricing-sensitivity of demand, Ruidas et al.84 solved an inventory model with changing carbon emission characteristics. Palanivel and Suganya85 developed an inventory model with partial backlog that takes price sensitivity into account. Narang et al.86 recently postulated that price, stock levels, and advertising activities influence the demand for production inventory.

Demand in the business sector is influenced by a wide range of factors. To illustrate this complexity, the demand is defined as a function that simultaneously depends additively on several variables. Accordingly, Herbon and Khmelnitsky87 developed an inventory model that takes into consideration the cumulative effects of time and price on demand in order to determine the optimal pricing and ordering methods for a degrading commodity. A model akin to that developed by Herbon and Khmelnitsky87 was developed by San-Jose et al.48 for degrading goods whose demand is dependent on both price and time. The focus of the current study is on this modelling element, though, because their inventory model ignored the potential for shortages. As to Dey et al.88 optimising a smart manufacturing system requires careful consideration of variables like as demand fluctuations and controllable lead times. An inventory model comprising many items was created by San-José et al.89 that considers storage capacity limitations as well as variations in demand over time. Nurhasril et al.90 developed an inventory model with two warehouses that included rework procedures and dynamic demand.

In this study, an inventory model for a good whose demand rate is the product of the impacts of a time-dependent power function and the selling price is developed and evaluated. More precisely, the demand rate fluctuates nonlinearly with respect to time and cycle length and changes linearly with respect to the selling price. The inventory model completely permits deficits due to backorders. Moreover, the holding cost is the power function of the storage period. This implies that the holding cost is nonlinear, as determined by Weiss25. The primary goal of this research project is to maximise total profit per unit of time while also determining the amount of orders, selling price, and backordering level. The selling price, green level, and an additional time-dependent function are added up to determine the product’s demand. Furthermore, the longer the commodities are stored in storage, the higher the holding costs become. Identifying the best inventory policy to maximize overall profit per unit of time is the primary goal. Even if there are many nonlinear aspects to the problem. We can’t therefore solve it analytically. We have used a number of well-known, widely used metaheuristic algorithms, including the Sparrow Search Algorithm (SSA), Whale Optimizer Algorithm (WOA), Grey Wolf Optimizer Algorithm (GWOA), Teaching Learning Based Optimisation Algorithm (TLBOA), Artificial Electric Field Algorithm (AEFA), Prairie Dog Optimization Algorithm (PDOA), Gazelle Optimization Algorithm (GOA), A Sinh Cosh Optimizer Algorithm (SCHOA), White Sherk Optimizer Algorithm (WSOA), Water Cycle Algorithm (WCA), Archimedes Optimization Paradigm Algorithm (AOPA), Marine Predator Optimization Algorithm (MPOA), Geyser Inspired Algorithm (GIA), Runge Kutta Optimization Algorithm (RKOA), Lungs Performance-based Optimization Algorithm (LPOA) and Dwarf Mongoose Optimization Algorithm (DMOA), to get past these challenges. It has been noted that from the convergency graph (cf., Fig. 2) of all metaheuristic algorithms, WCA outperforms from other algorithms.

Research gap and contribution

From the comparison table (cf. Table 1) related to our proposed work based on selling price, green level, and power pattern time-dependent demand, some research gaps are found. It is seen from Table 1 that Dey et al.88 developed an EPQ model considering non-linear advertisement-dependent demand with fully backlogged shortages where holding costs are constant. Again, Chowdhury and Ghosh77 proposed a production model with linearly time-dependent demand and holding costs without taking green level and price into account in the demand function. Further, Momena et al.78 analysed the optimal policy of an EOQ model under the effects of advertisement frequency and price-dependent multiplicative demand, linearly time-dependent holding costs, and partially backlogged shortages. In this work, the green level of the product and a fully backlogged shortage are not considered. Ali et al.92 solved a production inventory problem considering selling price, time, warranty period, and green level-dependent non-linear demand without consideration of non-linear time-dependent holding costs or fully backlogged shortages. Table 1 An analysis of the literature in comparison with the suggested inventory problem.

Reported works	Type of the model	Demand depends on	Demand type (linear/non-linear)	Holding cost	Greenness of the item	Backlogged type	Purchasing/production cost depends on	Solution methodology	
Depends on	Linear/non-linear	
San-José et al.41	EOQ	Time	Power pattern demand	Time	Non-linear	 × 	Partially backlogged	Constant	Analytical method	
Alfares and Ghaithan24	EOQ	Price	Linear	Time	Linear	 × 	–	Lot size	Analytical method	
Rajeswari et al.74	Fuzzy inventory	Time	Power pattern demand	Time	Linear	 × 	Partially backlogged	Fuzzified	Analytical method	
San-José et al.69	EOQ	Time	Power pattern demand	Constant	Linear	 × 	Partially backlogged	Constant	Analytical method	
Paknejad et al.43	EOQ	Deterministic	–	Time	Non-linear	 × 	–	Constant	Analytical method	
Edalatpour et al.46	EOQ	Price	Linear	Time	Non-linear	 × 	–	Constant	Analytical method	
Cárdenas-Barrón et al.50	EOQ	Stock level	Non-linear	Stock level	Non-linear	 × 	Partially backlogged	Constant	Analytical method	
Dey et al.88	EPQ	Advertisement	Non-linear	Constant	Linear	 × 	–	Production rate	Analytical method	
Chowdhury and Ghosh77	EPQ	Time	Linear	Time	Linear	 × 	–	Time	Analytical method	
Momena et al.78	EOQ	Advertisement frequency and price	Multiplicative	Time	Linear	 × 	Partially backlogged	Lot size	Analytical method	
Akhtar et al.91	EPQ	Greenness and expiry time	Linear	Constant	Linear	✓	–	Greenness of the product	Metaheuristic algorithms and Tournament algorithm	
Ali et al.92	EPQ	Selling price, time, warranty period and green level	Non-linear	Constant	Linear	✓	–	Production rate and green level	Metaheuristic algorithms	
Das et al.93	EPQ	Price, payment period and green level	Non-linear	Constant	Linear	✓	–	Green level of the product	Metaheuristic algorithms and TLBOA	
This work	EOQ	Price, green level and time	Power pattern demand	Time	Non- linear	✓	Fully backlogged	Green level of the product	Metaheuristic algorithms and WCA	
EOQ economic order quantity; EPQ economic production quantity.

To fill up the research gap in the existing literature, we have proposed an EOQ model considering price, green level, and time-dependent power pattern demand, along with non-linear time-dependent holding costs. Also, the purchasing cost of the product is a non-linearly increasing function of the green level of the product. Further, in this study, fully backlogged shortages are considered. The corresponding optimization problem is highly non-linear in nature. Therefore, the analytical method can’t be applied to solve this optimization problem. Due to the highly non-linear nature of the objective function of our optimization problem, metaheuristic algorithms are employed. In this perspective, WCA, SSA, AEFA, TLBOA, WSOA, GOA, GWOA, WOA, AOPA, MPOA, GIA, LPOA, DMOA, PDOA, and SCHOA are used to find the best-found solution to the optimization problem.

Organization of the manuscript

For the remaining portion, the manuscript is organised as follows. “Notation and assumptions” section introduces the nomenclature, underlying assumptions, “Model formulation” section provides the mathematical formulation of the model. In “Solution methodology” section, an efficient method is described for solving the suggested optimization problem. “Numerical analysis” section solves one numerical example, perform statistical analysis, provide convergence graph of the used algorithms and shown the concavity of the objective function graphically. In “Sensitivity analyses” section, a sensitivity analysis is shown graphically and described some managerial insight. A few proposals for future research directions and conclusion are included in “Concluding remarks” section.

Notation and assumptions

The notation in Abbreviation section is used for the development of the proposed inventory model.

The inventory model is developed taking into account the following assumptions.(i) This model deals with single item with infinite time horizon planning and the replenishment is instantaneous.

(ii) Fully backlogged shortages are considered.

(iii) The concept that a product’s cost can be influenced by its commitment to sustainability or eco-friendliness is known as “the purchase price of products being dependent on the green level”. The cost of acquiring ecologically friendly products may be higher than that of less sustainable alternatives. Green products frequently need more environmentally friendly raw materials or production techniques, which might be more costly than using conventional techniques. For instance, employing alternative energy sources or recycled materials in manufacturing may result in higher expenses. The trade-offs between cost-effectiveness and environmental sustainability are reflected in the cost of acquiring products, which is based on their green level. Green products may be more expensive up front, but they can pay off in the long run by having a positive influence on the environment and enhancing brand recognition. Mathematically, it can be represented as CP=C1+C2gξwhereC1,C2,ξ>0.

(iv) Pricing and marketing strategies should take into account the relationship that exists between items’ greenness, demand, and selling price. Customer demand may be impacted by a product’s selling price. While a lower selling price can increase demand, a higher selling price typically results in less demand. Demand may also be impacted by a product’s green rating, which expresses how eco-friendly or sustainable it is. Because they believe that green products are better for the environment, some consumers are willing to pay more for them. Demand, selling price, and green level have a complicated relationship that varies depending on consumer preferences, market dynamics, and product positioning. To maximize demand and profitability, effective pricing strategies should consider these elements. The functional form of the demand is as Dp,g,t=α-βpa+κgb+γmtT1-mm, where α,β,κ,m>0 and pa>αβ.

(v) The term “nonlinear holding costs” in inventory management describes circumstances in which the price of keeping or storing merchandise fluctuates according to the amount maintained. Holding cost per unit time can be express mathematically as,Ch=C3tδ,whereC3>0.

Model formulation

In an inventory system, a situation where the demand for a product fluctuates over time and is impacted by nonlinear factors is referred to as a time-dependent nonlinear power demand pattern. This indicates that there is a fluctuating demand for the goods throughout time. This might be the result of cyclical patterns in consumer behaviour, trends, or seasonal variations. It is implied by nonlinear demand that there is not a linear relationship between the product’s demand and other elements (such price, promotions, or outside causes). This could imply that minor adjustments to these variables cause demand to fluctuate disproportionately. This instance of “power” most likely alludes to a mathematical idea. A power demand pattern in this context denotes that the demand exhibits a pattern that may be explained by a power function, a nonlinear function. Here it is assumed that a retailer receives S+R units of item at time t=0 and after receives immediately fulfil the backlogged R quantity of items. Then, remaining Q items carry the inventory system. At time t=t1 inventory level reaches to zero and backlogged occurs. At the highest level of backlogged items R, the inventory system repeats. Thus, the governing differential equation of the inventory system is as1 dItdt=-α-βpa+κgb+γmtT1-mm,0≤t<T,

with the boundary condition I0=SandIt1=0.

Solving above differential Eq. (1) one can reach with the using I0=S, one can get2 It=S-α-βpa+κgbt+γt1mTm-1m,0≤t<T.

Now, applying It1=0 Eq. (2) gives3 S=α-βpa+κgbt1+γt11mTm-1m.

Also, the maximum shortages can obtain byR=∫t1Tα-βpa+κgb+γmtT1-mmdt,

i.e., R=α-βpa+κgb+γT-α-βpa+κgbt1-γTm-1mt11m.

Clearly, the total order quantity is S+R=α-βpa+κgb+γT.

Now,(i) Total sales revenue of the inventory system isSR=p∫0TDp,g,tdt=p∫0Tα-βpa+κgb+γmtT1-mmdt=pα-βpa+κgb-γT.

(ii) The holding cost for the inventory system isHC=∫0t1Chα-βpa+κgc+γmtT1-mmdt=∫0t1C3tδα-βpa+κgc+γmtT1-mmdt=C3α-βpa+κgcδ+1t1δ+1+γmδ+1Tm-1mt1mδ+1m.

(iii) The shortage cost is given bySC=Cs∫t1T-Itdt=CsST-t1-α-βpa+κgb2T2-t12-mγm+1Tm-1mTm+1m-t1m+1m=CsST-t1-α-βpa+κgb2T2-t12-mγm+1T2-mγm+1Tm-1mt1m+1m.

(iv) Ordering cost =Co.

(v) Total purchase costPC=Cpα-βpa+κgb+γT=C1+C2gξα-βpa+κgb+γT.

Then the average profit in the entire cycle is given by4 AP=1TSR-PC-HC-SC-CO=1Tp-C1+C2gξα-βpa+κgb+γT-C3α-βpa+κgct1δ+1δ+1+γTm-1mt1mδ+1mmδ+1-CsST-t1-α-βpa+κgb2T2-t12-mγm+1T2-mγm+1Tm-1mt1m+1m-Co

Thus, the corresponding optimization problem is arrived using the Eq. (4)

Problem 1

This optimization problem takes the form of the average profit across the cycle is5 MaximizeAPp,t1,T,gsubject top>0,t1>0,T>0,g>0.

Solution methodology

Motivation for solving optimization Problem 1, using a metaheuristic approach

After closely examining the given inventory system’s optimization Problem 1, it becomes clear that, in relation to the decision factors, the objective function in Problem 1 exhibits a notable degree of non-linearity, namely the product selling price p, stock-in period t1, business cycle length T and the green level of the product g. Here, we have investigated the Water Cycle Algorithm (WCA) (Eskandar et al.94) to tackle this extremely non-linear optimization problem.

Water cycle algorithm (WCA)

In this section, we have discussed about water cycle algorithm (WCA) in details.

Preliminary concept

The water cycle, notably the way rivers and streams naturally flow towards the sea, is the foundation for the WCA concept, which is taken (Eskandar et al.94) from nature. For your benefit, here’s a brief explanation of how rivers are created and how water gets to the sea.

Every time water moves from one location downwards to another, rivers and streams are created. Most rivers have their source high in the mountains, where long-gone glaciers or snowmelt. All of the rivers flow south. As water flows lower and eventually reaches a sea, it is gathered from streams and rains. Figure 1a presents a simple depiction of the hydrologic cycle. Plants evaporate the water of rivers and lakes by releasing water into the environment during photosynthesis. When evaporating water condenses in a cooler environment and returns to the earth, clouds are created, resulting in precipitation, or rain. We refer to this process as the hydrologic cycle, or water cycle of David95.Fig. 1 (a) Hydrologic cycle (water cycle) simplified diagram. (Eskandar et al.94). (b) Diagram showing how rivers flow into the sea and streams flow into them Image (Eskandar et al.94). (c) The city of Arkhangelsk on the Dvina River (adopted from NASA, Image Eskandar et al.94).

In reality, precipitation and snowmelt are the main ways that water reaches the aquifer. Massive areas are covered in subterranean water reserves. The aquifer may alternatively be referred to as “groundwater” (note the percolation arrow in Fig. 1a). Then, much like it would on the ground, the water in the aquifer flows downhill beneath the surface of the earth. The underground water can be discharged into a marsh stream or lake. During this cycle, additional clouds and rain will fall as a result of water transpiring from trees and other vegetation and evaporating from rivers and streams according to David95.

Figure 1b schematically depicts the flow of streams into rivers and rivers into the sea. Figure 1b resembles tree roots or trees in general. The tree-shaped image in brilliant green in Fig. 1b represents the smallest river branches, those are the tiny streams that eventually become rivers. First-order streams are what these minuscule streams are known as (Fig. 1b, green). That is when two first-order streams combine to create a second-order stream, which is depicted in white in Fig. 1b. A hypothetical globe’s lowest point, the sea, is reached by rivers that eventually drain into it. A third-order stream, depicted in blue in Fig. 1b, originates from the confluence of two second-order streams (Strahler96). Arkhangelsk, situated on the Dvina River, is seen in Fig. 1c. Stretched along both banks of the Dvina River, close to the point where the river empties into the White Sea, is the Russian city of Arkhangelsk, sometimes known as Angel in English. The Dvina River, seen in Fig. 1c, is a stream, river, and sea formation that resembles Fig. 1b in shape and is typical of a true stream.

WCA procedure

Similar to earlier metaheuristic algorithms, the suggested approach starts with an initial population known as the raindrops. Initially, we assume that precipitation—such as rain—is present. Like in a sea, the best person gets selected to be the best raindrop. After that, some extraordinary droplets are labelled as rivers, and the remainder raindrops are considered streams that drain into rivers and the ocean. The explanation of each river’s flow intensity—which dictates how much water it receives from the streams—can be found in the following subsections. As a matter of fact, a stream’s discharge into a river or the sea differs from that of other streams. River’s flow into the sea, which has the largest slope.

Initialization

To solve an optimisation problem utilising population-based metaheuristic techniques, the values of the problem variables need to be grouped into an array. This type of array is referred to as a “chromosome” in GA terminology and as a “particle position” in PSO terminology. For a single solution, this is the reason the suggested technique calls it “Raindrop”. A raindrop is an array of one in a dimensional optimisation problem. This is the definition of this array:6 Raindrop=r1,r2,...,rN.

A candidate depicting a matrix of raindrops of size Npop, Nvar is constructed to initiate the optimisation method (i.e. population of raindrops). The random matrix W, thus, is as follows (rows denote the number of population variables, and column the number of design variables, respectively):7 PopulationofRaindrop=r11r21r31...rNvar1r12r22r32...rNvar2r13r23r33...rNvar3.....................r1Npopr2Npopr3Npop...rNvarNpop.

The values of each decision variable, r1,r2,...,rNvar can be represented as a predetermined set for discrete problems or as a floating point number (actual values) for continuous problems. The assessment of the cost function (X), which is presented as follows, yields the cost of a raindrop.8 Xi=fr1i,r2i,...,rNvari,i=1,2,...,Npop,

where Npop and Nvar stand for the initial population (number of raindrops) and the number of design variables, respectively. Npop raindrops are made in the initial stage. Several Usr from the top persons (minimum values) are chosen as rivers and the sea. A raindrop is regarded as a sea when its value is the lowest of all the others. Actually, Usr is the total of one sea as specified by Eq. (9) and the number of rivers, which is a user parameter. Equation (10) is used to compute the remainder of the population, which consists of raindrops that create streams that run to rivers or may flow straight to the sea.9 Usr=numberofriver+1(sea),

10 Yraindrop=Npop-Usr.

Here is the equation that can be used to allocate raindrops to the sea and rivers based on flow intensity:11 YSn=roundXi∑i=1UsrXi×Yraindrops,n=1,2,3,...,Usr,

where YSn is how many streams run into a given river or body of water.

In what manner does a stream empty into rivers or the sea?

The streams are formed by the raindrops and merge to produce new rivers, as was stated in “Preliminary concept” section. It’s also possible for some of the streams to empty into the ocean. The sea is the ultimate, ideal destination for all rivers and streams. The stream’s schematic view is flowing towards a particular river.

Through their connecting line, a stream flows to the river at a distance that is chosen at random and provided as follows:12 Z∈0,C×D,C>1,

where C is a value close to 2 that falls between 1 and 2. One could select 2 as the optimal value for C. The symbol for the current distance between the river and the stream is D. A randomly distributed number between 0 and (C×D), distributed either uniformly or according to any suitable distribution, is represented by the value of Z in Eq. (12). Streams can flow towards rivers in diverse directions when the value of A exceeds unity.

Rivers that flow into the sea can also be thought of using this idea. Consequently, the following could be the new location for rivers and streams:13 Zstreami+1=Zstreami+rnd×C×Zriveri-Zstreami,

14 Zriveri+1=Zriveri+rnd×C×Zseai-Zriveri,

in which the random number between 0 and 1, rnd, is uniformly distributed. Streams and rivers switch places if the solution they provide outweighs that of the connected river (stream becomes river and river becomes stream). Similarly, rivers and the sea can interchange in this way. An interchange between a stream and the river that is the best option.

Condition of evaporation

A major factor that can prevent the algorithm from converging soon is immature convergence, or evaporation. This is demonstrated by nature, where water is released into the atmosphere by plants through photosynthesis and evaporates in rivers and lakes. When water evaporates, clouds carry it into the atmosphere, where it condenses in the colder air and rains back down to the land. Rain creates new streams, which eventually connect to rivers that empty into the ocean (David95).

The water cycle is the one that was discussed in “Preliminary concept” section. The sea water in the suggested approach evaporates as rivers and streams flow into the sea due to the evaporation process. The purpose of proposing this assumption is to prevent the trapping of local optima. The Psuocode that follows demonstrates how to ascertain whether a river empties into the sea.

If Zseai-Zriveri<δmax,i=1,2,3,...,Usr-1.

Evaporation and raining process end where δmax is a negligible value that approaches 0. As a result, a river has reached or joined the sea if the distance between it and the sea is smaller than δmax. The evaporation process is used in this case, and as is observed in nature, rain (precipitation) will begin after a sufficient amount of evaporation. In the vicinity of the sea, a big number for δmax discourages search activity, whereas a small value increases it. Consequently, δmax regulates the search intensity (the best option) close to the coast. The δmax value adaptively decreases as:15 δmaxi+1=δmaxi-δmaximax_it.

Raining process

The raining method is used once the evaporation process has been satisfied. As it rains, the fresh droplets split into streams at various spots (behaving like GA’s mutation operator). The following equation is used to specify the new locations of the newly generated streams:16 Zstreamnew=Lb+rnd×Ub-Lb,

where the provided problem defines the lower and upper bounds, denoted by lb and ub, respectively.

Once more, the finest recently produced raindrop is compared to a river that empties into the ocean. It is expected that the remaining rains will create new streams that either run directly to the sea or to rivers.

To maximise the convergence rate and computing efficiency of the approach for small problems, Eq. (17) is applied only to streams that terminate directly in the sea. In order to enhance near-sea exploration (the ideal answer) in the feasible zone for limited challenges, this equation seeks to promote the creation of streams that flow directly to the sea.17 Zstreamnew=Znew+α×rnd1,Nvar.

With a coefficient called α indicates the range of the search area close to the coast. rnd is a random number that is normally distributed. Exiting the viable region is more likely when α is greater. The algorithm, however, searches a smaller area close to the sea when α is less. For the value of α, 0.1 is a suitable value for the well performance of the algorithm.

In terms of mathematics, the standard deviation is denoted by the letter α in Eq. (17), and as a result, α determines variance. These ideas are then used to disperse the created individuals with variance α around the best-obtained ideal location (sea).

Now, at the same time, we are using fifteen other popular and well-known metaheuristics to solve optimization Problem 1 numerically to see how well Water Cycle Algorithm (WCA) (Eskandar et al.94 solves them numerically. The following list of metaheuristics includes them:Whole optimizer algorithm (WOA) (Mirjalili and Lewis13).

Grey wolf optimizer algorithm (GWOA) (Mirjalili12).

Teaching–Learning Based Optimizer Algorithm (TLBOA) (Rao16).

Artificial electric field algorithm (AEFA) (Yadav97).

Sparrow Search Algorithm (SSA) (Xue and Shen98).

Prairie Dog Optimization Algorithm (PDOA) (Hu et al.99).

Gazelle Optimization Algorithm (GOA) (Agushaka et al.100).

A Sinh Cosh Optimizer Algorithm (SCHOA) (Bai et al.101).

White Sherk Optimizer Algorithm (WSOA) (Braik et al.102).

Archimedes Optimization Paradigm Algorithm (AOPA) (Mehmood et al.103).

Marine Predator Optimization Algorithm (MPOA) (Mehmood et al.104).

Geyser Inspired Algorithm (GIA) (Ghasemi et al.105).

Lungs Performance-based Optimization Algorithm (LPOA) (Ghasemi et al.106).

Dwarf Mongoose Optimization Algorithm (DMOA) (Agushaka et al.107).

Runge Kutta Optimization Algorithm (RKOA) (Ahmadianfar et al.108).

Numerical analysis

We have used the numerical example to support the model in numerical form. The hypothetical input data for various inventory parameters is acquired in the provided numerical example. We use the sixteen previously described metaheuristic algorithms: WCA, AEFA, TLBOA, GWOA, SSA, WOA, AOPA, MPOA, GIA, LPOA, DMOA, PDOA, SCHOA, GOA, RKOA and WSOA. On a desktop PC running Windows 11.1 and sporting an 11th generation, 2.40 GHz Intel Core i-5 processor, these techniques are implemented using MATLAB software. Using a variety of algorithms (WCA, TLBOA, AEFA, GWOA, SSA, WOA, AOPA, MPOA, GIA, LPOA, DMOA, PDOA, SCHOA, GOA, RKOA and WSOA), the stability of the optimization results for Problem 1 was thoroughly examined. Ultimately, a population size of 50 (pop_size) and a maximum generation of 500 (max_gen) were selected for each algorithm.

Each algorithm undergoes fifty independent iterations in order to obtain the best and worst-case scenarios’ solutions. For the inventory system of Example 1 shown below, separate tables present the best and worst results from a variety of algorithms (WCA, TLBOA, AEFA, GWOA, SSA, WOA, AOPA, MPOA, GIA, LPOA, DMOA, PDOA, SCHOA, GOA, RKOA and WSOA). Moreover, convergence graphs, statistical experiments, and Friedman test & Wilcoxon sum test are performed to evaluate the effectiveness of the previously stated metaheuristic algorithms (WCA, TLBOA, AEFA, GWOA, SSA, WOA, AOPA, MPOA, GIA, LPOA, DMOA, PDOA, SCHOA, GOA, RKOA and WSOA) in solving Example 1.

Example 1

To validate the proposed model a numerical example is considered and the parameters of the numerical example are as follows:α=120;β=1;γ=10;m=0.5;C1=$30;C2=0.5;C3=$1.05;Co=$200;δ=1.5;a=1;κ=1.5;b=0.4;ξ=1.3.

To attain the maximum average profit AP∗p∗,t1∗,T∗,g∗, it is now essential to choose the ideal price p∗, stock-in period t1∗, business period T∗, and greenness of the product g∗. Finding the system’s matching ideal value is also very important.

Solution: Here, Example 1 is considered for the Problem 1 and is solved for the respective proposed inventory model.

Table 2 shows the optimal solutions for the proposed inventory system, which correspond to Example 1 of Problem 1. On the other hand, Table 3 shows the least desirable results that were obtained. Furthermore, the results of the statistical analysis are presented in Table 4. Table 2 The best-found result from a variety of metaheuristic algorithms that corresponds to Example 1 of Problem 1.

Table 3 Worst-found result from a variety of metaheuristic algorithms that corresponds to Example 1 of Problem 1.

Table 4 Results of statistical experiment of average profit AP obtained from different metaheuristic algorithms.

Observations and discussions

The following consequences are evident from Tables 2, 3 and 4:Table 2 demonstrates that, while it deviates from the WOA, GWOA, PDOA, GIA, DMOA, AOPA and SCHOA algorithms’ findings, the best-found average profit value AP for Example 1’s agrees with the outcomes of the WCA, WSOA, MPA, LPOA, AEFA, TLBOA, GOA, RKOA and SSA algorithms.

Again, Tables 2 and 3 demonstrate that the data extracted from WCA, AEFA, MPA, LPOA, TLBOA, SSA, GOA, RKOA and WSOA exhibit a correspondence between the best- and worst-found values. Moreover, WSOA requires the least amount of processing time to execute the best-found average profit value AP.

Once more, the statistical experiment (see Table 4) makes it clear that the WCA, AEFA, MPA, LPOA, TLBOA, SSA, WSOA, RKOA and GOA standard deviations are constant and minimal throughout the computation of Example 1.

As such, WCA, AEFA, MPA, LPOA, TLBOA, SSA, GOA, RKOA and WSOA are all equally efficient when it comes to solving Problem 1’s Example 1.

Non-parametric tests

Two non-parametric statistical tests are conducted, taking into account of Example 1, to determine how effective WCA, SSA, MPA, LPOA, GIA, DMOA, AOPA, RKOA, TLBOA, AEFA, PDOA, SCHOA, WSOA, GOA, WOA, and GWOA are in solving the problem. These tests are the Friedman test and the Wilcoxon rank sum test. The parametric values (p-values) for the Friedman and Wilcoxon rank sum tests, respectively, are provided in Tables 5 and 6, with WCA being considered the controlling algorithm for each test. Table 5 Friedman test analysis for different metaheuristic algorithms of Example 1.

WCA vs	p-value	Significant/not significant (5% significant level)	
AEFA	1	Not significant	
SSA	1	Not significant	
TLBOA	1	Not significant	
GOA	1	Not significant	
WSOA	1	Not significant	
MPOA	1	Not significant	
LPOA	1	Not significant	
RKOA	1	Not significant	
GIA	1.53746 × 10–12	Significant	
DMOA	1.53746 × 10–12	Significant	
AOPA	1.53746 × 10–12	Significant	
PDOA	1.53746 × 10–12	Significant	
SCHOA	1.53746 × 10–12	Significant	
GWOA	1.53746 × 10–12	Significant	
WOA	1.53746 × 10–12	Significant	

Table 6 Wilcoxon rank sum test analysis for different metaheuristic algorithms of Example 1.

WCA vs	p-value	Significant/not significant (5% significant level)	
AEFA	0.5	Not significant	
SSA	0.5	Not significant	
TLBOA	0.5	Not significant	
GOA	0.5	Not significant	
WSOA	0.5	Not significant	
MPOA	0.5	Not significant	
LPOA	0.5	Not significant	
RKOA	0.5	Not significant	
GIA	0	Significant	
DMOA	0	Significant	
AOPA	0	Significant	
PDOA	0	Significant	
SCHOA	0	Significant	
GWOA	0	Significant	
WOA	0	Significant	

Table 5 displays the p-values 1, which is more than 0.05, for SSA, AEFA, TLBOA, WSOA, MPA, LPOA, RKOA and GOA, respectively. Again, less than 0.05 is shown by the p-values of 1.53746 × 10–12 for PDOA, SCHOA, GIA, AOPA, DMOA, WOA, and GWOA. Consequently, at the 5% level of significance, WCA outperforms from PDOA, SCHOA, GIA, AOPA, DMOA, WOA, and GWOA, whereas SSA, AEFA, TLBOA, WSOA, MPA, LPOA, RKOA and GOA perform equally according to the Friedman rank sum test.

Once more, there are comparable implications for the efficiency of WCA, SSA, AEFA, TLBOA, WSOA, GOA, MPOA, RKOA, WOA, GWOA, SCHOA, GIA, DMOA, AOPA and PDOA in solving the numerical Example 1 from both the Friedman test and the Wilcoxon rank sum test (see Table 6).

Convergence graph

The convergence of the sixteen metaheuristic algorithms (WCA, AEFA, TLBOA, SSA, RKOA, WOA, GWOA, PDOA, SCHOA, GOA, WSOA, GIA, DMOA, AOPA, MPA and LPOA) that are being considered to solve the optimization problem of the suggested model for Problem 1 of Example 1 is shown in Fig. 2.Fig. 2 Convergency graphs of different metaheuristic algorithms of Example 1.

Figure 2 shows that after a certain number of iterations, the WCA, AEFA, SSA, TLBOA, GOA, MPA, LPOA, RKOA and WSOA algorithms converge to the optimal solution 2492.013795 of average profit AP. However, WCA converges to the ideal solution more quickly as compare to others.

Concavity figures

The concavity of AP with respect to the decision variables p,t1,T,andg for the inventory system are shown graphically by Figs. 3, 4, 5, 6, 7 and 8.Fig. 3 Concavity of average profit w.r. to pandt1.

Fig. 4 Concavity of average profit w.r. to Tandt1.

Fig. 5 Concavity of average profit w.r. to pandT.

Fig. 6 Concavity of average profit w.r. to pandg.

Fig. 7 Concavity of average profit w.r. to Tandg.

Fig. 8 Concavity of average profit w.r. to gandt1.

Sensitivity analyses

A sensitivity analysis is performed on optimal policy by changing one parameter − 20% to + 20% and at the same time other parameters are keeping as fixed. The effects are shown graphically by Figs. 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 and 22 corresponding to the parameter mentioned below.Fig. 9 Effects of α on optimal policy.

Fig. 10 Effects of β on optimal policy.

Fig. 11 Effects of γ on optimal policy.

Fig. 12 Effects of m on optimal policy.

Fig. 13 Effects of C3 on optimal policy.

Fig. 14 Effects of Co on optimal policy.

Fig. 15 Effects of C1 on optimal policy.

Fig. 16 Effects of C2 on optimal policy.

Fig. 17 Effects of δ on optimal policy.

Fig. 18 Effects of Cs on optimal policy.

Fig. 19 Effects of κ on optimal policy.

Fig. 20 Effects of b optimal policy.

Fig. 21 Effects of ξ on optimal policy.

Fig. 22 Effects of a on optimal policy.

Experimental observations

Figure 9 indicates that average profit AP and stock-in period t1 demonstrate a high sensitivity, while selling price p and cycle length T exhibit moderate sensitivity to changes in demand parameter α within the range of − 20% to 20%. Conversely, the green level g appears to be insensitive to such variations.

In Fig. 10, it is evident that average profit AP and green level g display a high level of sensitivity, while stock-in period t1 and cycle length T exhibit lower sensitivity. Conversely, the selling price p demonstrates a moderate level of sensitivity to variations in the selling price controlling parameter β within the specified range.

Figure 11 suggests that all parameters exhibit lower sensitivity, whereas the green level g of the product appears to be insensitive to changes in the inventory time-controlling parameter γ within the specified percentage range.

From Fig. 12, it can be observed that the stock-in period t1 and cycle length T are highly and moderately sensitive to decreasing changes, respectively, while other parameters exhibit almost insensitivity to the same percentage change of m.

Figures 13 and 17 demonstrate that the stock-in period t1 is highly sensitive, and the cycle length T is less sensitive, while other parameters exhibit almost insensitivity to changes in their respective controlling factors C3andδ respectively.

It is evident from Figs. 14 and 18 that the stock-in period t1 and cycle length T exhibit moderate sensitivity, whereas other parameters demonstrate almost insensitivity to changes in their corresponding controlling factors C0andCs respectively.

Figure 15 indicates that average profit AP is moderately sensitive, while green level g of the product is insensitive to changes in their respective controlling factor C1. Additionally, other parameters demonstrate lower sensitivity to the same percentage change.

From Figs. 16, 19, 20 and 21, it is evident that green level g exhibits high sensitivity, while the other parameters demonstrate almost insensitivity to changes in their corresponding controlling factors, namely, C2, κ, b and ξ respectively.

Figure 22 highlights that stock-in period t1 and cycle length T are highly sensitive, while other parameters demonstrate even higher sensitivity to the same percentage change in their respective controlling factor a.

Managerial insights

In contrast to single-point values, interval-valued power demand patterns depict power demand as a range or interval. This method acknowledges the unpredictability and uncertainty that are part of estimating power demand. The following managerial observations on interval-valued power demand patterns:When the index of the green level (ξ) is very high, the impact of the green level on demand is also very high in a positive way. In this situation, the decision-maker can invest a few in green-level purposes to increase profit.

When the index of the power demand pattern (m) decreases, then the products are consumed at a lower demand rate at the beginning of the business cycle. As a result, the inventory manager should store products with higher green levels, which help to increase the demand rate and hence reduce holding costs.

If the price-sensitive demand parameter is lower, the selling price has a lower impact on the demand rate. Therefore, in this case, the manager should increase the selling price for each unit to increase profits.

By adopting a strategy about the demand patterns, managers can add robustness and flexibility to their operations. Therefore, managers should consider a range of possible demand scenarios when developing systems that can respond to fluctuating demand levels. While lowering the risks related to demand volatility, this flexibility helps guarantee that customers receive a steady supply.

Concluding remarks

The demand rate of the product is determined by selling price, green level, and a power form of time in this study’s inventory model, which has been constructed and presented for a single item. As a power form of how long the products are maintained in storage, the carrying expense is also considered. Shortages are allowed with fully backlogged situation. Due to high nonlinear of the objective function, it is unable to solve the objective function analytically. Different metaheuristic algorithms are applied for solving this inventory problem. It is observed that some algorithms are equally efficient for solving this particular problem. Among them WCA algorithms converge rapidly to the best found solution or near to the global optimal solution. In addition, According to the presumptions made in the inventory model, the outcomes may be helpful for managing the inventory of commodities where consumption is responsive to price, green level, and inventory holding period. As a result, when a commodity is maintained in stock, its value drops nonlinearly. After demonstrating the model numerically, several salient management insights are extracted for the decision manager. For instance, when the level of greenness (ξ) is very high, the impact of the level of greenness on demand is also very high in a positive way. In this situation, the decision-maker can invest a few in green-level purposes to increase profit. In addition, when the index of the power demand pattern (m) decreases, then the products are consumed at a lower demand rate at the beginning of the business cycle. As a result, the inventory manager should store products with higher green levels, which help to increase the demand rate and hence reduce holding costs.

The current research can be extend by incorporating several factors such as age-dependent deterioration, credit policy, quantity discount, stochastic demand, and many more. One may extend this model in uncertain environment such as fuzzy, stochastic, interval environment.

List of symbols

Cp Purchase cost (in $) of the item, where Cp=C1+C2gξ

δ Holding cost controlling parameter

Ch Holding cost (in $) of the item per unit time, where Ch=C3tδ

Cs Shortage cost (in $) of the item

Co Ordering cost (in $) of the inventory system

α Demand parameter

β Selling price controlling demand parameter

a Demand scale parameter associated with selling price

κ Green level controlling parameter

γ Scale parameter of time-dependency controlling of demand

m Demand pattern index

b Demand scale parameter associated with green level

S Initial stock of the inventory system

R Maximum shortages allowed

It Inventory level at time t

Dp,g,t Demand function of the item

APp,g,t1,T Average profit function ($/month)

Decision variables

p Selling price (in $/unit) of the item

t1 Time period of inventory level reaches zero (in month)

T Cycle length (in month)

g Green level of the item

Acknowledgements

All authors are thankful to editorial team and anonymous reviewers for their constructive suggestion in order to improve the manuscript. The second author expresses his gratitude for the financial support granted by the UGC IN for providing JRF (NTA Ref. No.: 211610092425). King Saud University, Riyadh, Saudi Arabia, is acknowledged by the third author for funding this work under Researchers Supporting Project number (RSP2024R323). The Ministry of Science and Technology, Government of India, is thanked by the fourth author for its support of FIST (SR/FST/MSII/2017/10 (C)).

Author contributions

Subhash Chandra Das: Conceptualization; Investigation; Formal Analysis; Writing original draft. Fleming Akhtar: Conceptualization; Investigation; Formal Analysis; Writing original draft. Adel Fahad Alrasheedi: Conceptualization; Investigation; Formal Analysis; Review editing original draft; Supervision. Ali Akbar Shaikh: Conceptualization; Investigation; Formal Analysis; Review editing original draft; Supervision, Writing—review & editing.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request. We don’t used any published data for solving the optimization problem.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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References

1. Alshanbari HM El-Bagoury AAAH Khan MAA Mondal S Shaikh AA Rashid A Economic order quantity model with weibull distributed deterioration under a mixed cash and prepayment scheme Comput. Intell. Neurosci. 2021 2021 1 9588685 10.1155/2021/9588685 34527045
Alshanbari, H. M. et al. Economic order quantity model with weibull distributed deterioration under a mixed cash and prepayment scheme. Comput. Intell. Neurosci. 2021(1), 9588685 (2021).34527045 10.1155/2021/9588685
2. Feng X Xie Y Wang S Yan H Optimal structure of joint inventory-pricing management with dual suppliers and different lead times J. Manag. Sci. Eng. 2021 6 1 1 24
Feng, X., Xie, Y., Wang, S. & Yan, H. Optimal structure of joint inventory-pricing management with dual suppliers and different lead times. J. Manag. Sci. Eng. 6(1), 1–24 (2021).
3. Khan MAA Shaikh AA Khan AR Alrasheedi AF Advertising and pricing strategies of an inventory model with product freshness-related demand and expiration date-related deterioration Alex. Eng. J. 2023 73 353 375 10.1016/j.aej.2023.04.059
Khan, M. A. A., Shaikh, A. A., Khan, A. R. & Alrasheedi, A. F. Advertising and pricing strategies of an inventory model with product freshness-related demand and expiration date-related deterioration. Alex. Eng. J. 73, 353–375 (2023).10.1016/j.aej.2023.04.059
4. Goldberg DE Holland JH Genetic algorithms and machine learning Mach. Learn. 1988 3 95 99 10.1023/A:1022602019183
Goldberg, D. E. & Holland, J. H. Genetic algorithms and machine learning. Mach. Learn. 3, 95–99 (1988).10.1023/A:1022602019183
5. Koza, J. R. Evolution of subsumption using genetic programming. In Proc. First European Conference on Artificial Life 110–119 (MIT Press, 1992).
6. Storn R Price K Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces J. Global Optim. 1997 11 341 359 10.1023/A:1008202821328
Storn, R. & Price, K. Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11, 341–359 (1997).10.1023/A:1008202821328
7. Rechenberg I Evolution strategy: Optimization of technical systems by means of biological evolution Fromman-Holzboog Stuttgart 1973 104 15 16
Rechenberg, I. Evolution strategy: Optimization of technical systems by means of biological evolution. Fromman-Holzboog Stuttgart 104, 15–16 (1973).
8. Kennedy, J. & Eberhart, R. Particle swarm optimization. In Proc. ICNN’95-International Conference on Neural Networks, Vol. 4, 1942–1948 (IEEE, 1995).
9. Passino KM Biomimicry of bacterial foraging for distributed optimization and control IEEE Control Syst. Mag. 2002 22 3 52 67 10.1109/MCS.2002.1004010
Passino, K. M. Biomimicry of bacterial foraging for distributed optimization and control. IEEE Control Syst. Mag. 22(3), 52–67 (2002).10.1109/MCS.2002.1004010
10. Yang XS Firefly algorithm, stochastic test functions and design optimisation Int. J. Bio-Inspir. Comput. 2010 2 2 78 84 10.1504/IJBIC.2010.032124
Yang, X. S. Firefly algorithm, stochastic test functions and design optimisation. Int. J. Bio-Inspir. Comput. 2(2), 78–84 (2010).10.1504/IJBIC.2010.032124
11. Gandomi AH Alavi AH Krill herd: A new bio-inspired optimization algorithm Commun. Nonlinear Sci. Numer. Simul. 2012 17 12 4831 4845 10.1016/j.cnsns.2012.05.010
Gandomi, A. H. & Alavi, A. H. Krill herd: A new bio-inspired optimization algorithm. Commun. Nonlinear Sci. Numer. Simul. 17(12), 4831–4845 (2012).10.1016/j.cnsns.2012.05.010
12. Mirjalili S Mirjalili SM Lewis A Grey wolf optimizer Adv. Eng. Softw. 2014 69 46 61 10.1016/j.advengsoft.2013.12.007
Mirjalili, S., Mirjalili, S. M. & Lewis, A. Grey wolf optimizer. Adv. Eng. Softw. 69, 46–61 (2014).10.1016/j.advengsoft.2013.12.007
13. Mirjalili S Lewis A The whale optimization algorithm Adv. Eng. Softw. 2016 95 51 67 10.1016/j.advengsoft.2016.01.008
Mirjalili, S. & Lewis, A. The whale optimization algorithm. Adv. Eng. Softw. 95, 51–67 (2016).10.1016/j.advengsoft.2016.01.008
14. Glover F Tabu search, part I ORSA J. Comput. 1989 1 190 206 10.1287/ijoc.1.3.190
Glover, F. Tabu search, part I. ORSA J. Comput. 1, 190–206 (1989).10.1287/ijoc.1.3.190
15. Geem ZW Kim JH Loganathan GV A new heuristic optimization algorithm: Harmony search Simulation 2001 76 2 60 68 10.1177/003754970107600201
Geem, Z. W., Kim, J. H. & Loganathan, G. V. A new heuristic optimization algorithm: Harmony search. Simulation 76(2), 60–68 (2001).10.1177/003754970107600201
16. Rao RV Savsani VJ Vakharia DP Teaching–learning-based optimization: A novel method for constrained mechanical design optimization problems Comput. Aided Des. 2011 43 3 303 315 10.1016/j.cad.2010.12.015
Rao, R. V., Savsani, V. J. & Vakharia, D. P. Teaching–learning-based optimization: A novel method for constrained mechanical design optimization problems. Comput. Aided Des. 43(3), 303–315 (2011).10.1016/j.cad.2010.12.015
17. Satapathy S Naik A Social group optimization (SGO): A new population evolutionary optimization technique Complex Intell. Syst. 2016 2 3 173 203 10.1007/s40747-016-0022-8
Satapathy, S. & Naik, A. Social group optimization (SGO): A new population evolutionary optimization technique. Complex Intell. Syst. 2(3), 173–203 (2016).10.1007/s40747-016-0022-8
18. Huan TT Kulkarni AJ Kanesan J Huang CJ Abraham A Ideology algorithm: A socio-inspired optimization methodology Neural Comput. Appl. 2017 28 1 845 876 10.1007/s00521-016-2379-4
Huan, T. T., Kulkarni, A. J., Kanesan, J., Huang, C. J. & Abraham, A. Ideology algorithm: A socio-inspired optimization methodology. Neural Comput. Appl. 28(1), 845–876 (2017).10.1007/s00521-016-2379-4
19. Zhao W Wang L Zhang Z A novel atom search optimization for dispersion coefficient estimation in groundwater Futur. Gener. Comput. Syst. 2019 91 601 610 10.1016/j.future.2018.05.037
Zhao, W., Wang, L. & Zhang, Z. A novel atom search optimization for dispersion coefficient estimation in groundwater. Futur. Gener. Comput. Syst. 91, 601–610 (2019).10.1016/j.future.2018.05.037
20. Kirkpatrick S Gelatt CD Vecchi MP Optimization by simulated annealing Science 1983 220 4598 671 680 10.1126/science.220.4598.671 17813860
Kirkpatrick, S., Gelatt, C. D. & Vecchi, M. P. Optimization by simulated annealing. Science 220(4598), 671–680 (1983).17813860 10.1126/science.220.4598.671
21. Webster, B. & Bernhard, P. J. A local search optimization algorithm based on natural principles of gravitation. http://hdl.handle.net/11141/117 (2003).
22. Kaveh A Khayatazad M A new meta-heuristic method: Ray optimization Comput. Struct. 2012 112 283 294 10.1016/j.compstruc.2012.09.003
Kaveh, A. & Khayatazad, M. A new meta-heuristic method: Ray optimization. Comput. Struct. 112, 283–294 (2012).10.1016/j.compstruc.2012.09.003
23. Moghaddam, F. F., Moghaddam, R. F. & Cheriet, M. Curved space optimization: A random search based on general relativity theory. Preprint at http://arXiv.org/1208.2214 (2012).
24. Alfares HK Ghaithan AM Inventory and pricing model with price-dependent demand, time-varying holding cost, and quantity discounts Comput. Ind. Eng. 2016 94 170 177 10.1016/j.cie.2016.02.009
Alfares, H. K. & Ghaithan, A. M. Inventory and pricing model with price-dependent demand, time-varying holding cost, and quantity discounts. Comput. Ind. Eng. 94, 170–177 (2016).10.1016/j.cie.2016.02.009
25. Weiss HJ Economic order quantity models with nonlinear holding costs Eur. J. Oper. Res. 1982 9 1 56 60 10.1016/0377-2217(82)90010-8
Weiss, H. J. Economic order quantity models with nonlinear holding costs. Eur. J. Oper. Res. 9(1), 56–60 (1982).10.1016/0377-2217(82)90010-8
26. Goh M EOQ models with general demand and holding cost functions Eur. J. Oper. Res. 1994 73 1 50 54 10.1016/0377-2217(94)90141-4
Goh, M. EOQ models with general demand and holding cost functions. Eur. J. Oper. Res. 73(1), 50–54 (1994).10.1016/0377-2217(94)90141-4
27. Giri BC Chaudhuri KS Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost Eur. J. Oper. Res. 1998 105 3 467 474 10.1016/S0377-2217(97)00086-6
Giri, B. C. & Chaudhuri, K. S. Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost. Eur. J. Oper. Res. 105(3), 467–474 (1998).10.1016/S0377-2217(97)00086-6
28. Chang CT Inventory models with stock-dependent demand and nonlinear holding costs for deteriorating items Asia Pac. J. Oper. Res. 2004 21 04 435 446 10.1142/S0217595904000321
Chang, C. T. Inventory models with stock-dependent demand and nonlinear holding costs for deteriorating items. Asia Pac. J. Oper. Res. 21(04), 435–446 (2004).10.1142/S0217595904000321
29. Ferguson M Jayaraman V Souza GC Note: An application of the EOQ model with nonlinear holding cost to inventory management of perishables Eur. J. Oper. Res. 2007 180 1 485 490 10.1016/j.ejor.2006.04.031
Ferguson, M., Jayaraman, V. & Souza, G. C. Note: An application of the EOQ model with nonlinear holding cost to inventory management of perishables. Eur. J. Oper. Res. 180(1), 485–490 (2007).10.1016/j.ejor.2006.04.031
30. Alfares HK Inventory model with stock-level dependent demand rate and variable holding cost Int. J. Prod. Econ. 2007 108 1–2 259 265 10.1016/j.ijpe.2006.12.013
Alfares, H. K. Inventory model with stock-level dependent demand rate and variable holding cost. Int. J. Prod. Econ. 108(1–2), 259–265 (2007).10.1016/j.ijpe.2006.12.013
31. Urban TL An extension of inventory models with discretely variable holding costs Int. J. Prod. Econ. 2008 114 1 399 403 10.1016/j.ijpe.2008.02.014
Urban, T. L. An extension of inventory models with discretely variable holding costs. Int. J. Prod. Econ. 114(1), 399–403 (2008).10.1016/j.ijpe.2008.02.014
32. Mahata GC Goswami A Fuzzy EOQ models for deteriorating items with stock dependent demand and non-linear holding costs Int. J. Appl. Math. Comput. Sci. 2009 5 2 94 98
Mahata, G. C. & Goswami, A. Fuzzy EOQ models for deteriorating items with stock dependent demand and non-linear holding costs. Int. J. Appl. Math. Comput. Sci. 5(2), 94–98 (2009).
33. Mao, X. L. & Xiao, X. P. Optimal inventory policy for non-instantaneous items with stock-dependent holding cost function and shortage. In 2009 IEEE International Conference on Grey Systems and Intelligent Services (GSIS 2009) 1772–1778 (IEEE, 2009).
34. Valliathal M Uthayakumar R Optimal pricing and replenishment policies of an EOQ model for non-instantaneous deteriorating items with shortages Int. J. Adv. Manuf. Technol. 2011 54 1 361 371 10.1007/s00170-010-2913-y
Valliathal, M. & Uthayakumar, R. Optimal pricing and replenishment policies of an EOQ model for non-instantaneous deteriorating items with shortages. Int. J. Adv. Manuf. Technol. 54(1), 361–371 (2011).10.1007/s00170-010-2913-y
35. Pando V García-Laguna J San-José LA Optimal policy for profit maximising in an EOQ model under non-linear holding cost and stock-dependent demand rate Int. J. Syst. Sci. 2012 43 11 2160 2171 10.1080/00207721.2011.565134
Pando, V., García-Laguna, J. & San-José, L. A. Optimal policy for profit maximising in an EOQ model under non-linear holding cost and stock-dependent demand rate. Int. J. Syst. Sci. 43(11), 2160–2171 (2012).10.1080/00207721.2011.565134
36. Sazvar Z Jokar MA Baboli A Campagne JP Centralized replenishment policy for deteriorating items in a three echelon supply chain under stochastic lead time IFAC Proc. Vol. 2012 45 6 493 498 10.3182/20120523-3-RO-2023.00239
Sazvar, Z., Jokar, M. A., Baboli, A. & Campagne, J. P. Centralized replenishment policy for deteriorating items in a three echelon supply chain under stochastic lead time. IFAC Proc. Vol. 45(6), 493–498 (2012).10.3182/20120523-3-RO-2023.00239
37. Pando V San-José LA García-Laguna J Sicilia J An economic lot-size model with non-linear holding cost hinging on time and quantity Int. J. Prod. Econ. 2013 145 1 294 303 10.1016/j.ijpe.2013.04.050
Pando, V., San-José, L. A., García-Laguna, J. & Sicilia, J. An economic lot-size model with non-linear holding cost hinging on time and quantity. Int. J. Prod. Econ. 145(1), 294–303 (2013).10.1016/j.ijpe.2013.04.050
38. Prasher L Pundir S Optimizing production policies for flexible manufacturing system with non-linear holding cost Prestige Int. J. Manag. IT-Sanchayan 2013 2 1 114 126 10.37922/PIJMIT.2013.V02i01.009
Prasher, L. & Pundir, S. Optimizing production policies for flexible manufacturing system with non-linear holding cost. Prestige Int. J. Manag. IT-Sanchayan 2(1), 114–126 (2013).10.37922/PIJMIT.2013.V02i01.009
39. Sazvar Z Rekik Y Jokar MA Baboli A Al-E-Hashem SM A new up-to level inventory model for deteriorating products with non-linear holding cost IFAC Proc. Vol. 2013 46 9 1702 1707 10.3182/20130619-3-RU-3018.00507
Sazvar, Z., Rekik, Y., Jokar, M. A., Baboli, A. & Al-E-Hashem, S. M. A new up-to level inventory model for deteriorating products with non-linear holding cost. IFAC Proc. Vol. 46(9), 1702–1707 (2013).10.3182/20130619-3-RU-3018.00507
40. Sazvar Z Baboli A Jokar MRA A replenishment policy for perishable products with non-linear holding cost under stochastic supply lead time Int. J. Adv. Manuf. Technol. 2013 64 5–8 1087 1098 10.1007/s00170-012-4042-2
Sazvar, Z., Baboli, A. & Jokar, M. R. A. A replenishment policy for perishable products with non-linear holding cost under stochastic supply lead time. Int. J. Adv. Manuf. Technol. 64(5–8), 1087–1098 (2013).10.1007/s00170-012-4042-2
41. San-José LA Sicilia J García-Laguna J Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost Omega 2015 54 147 157 10.1016/j.omega.2015.01.007
San-José, L. A., Sicilia, J. & García-Laguna, J. Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost. Omega 54, 147–157 (2015).10.1016/j.omega.2015.01.007
42. Khalilpourazari S Pasandideh SHR Multi-item EOQ model with nonlinear unit holding cost and partial backordering: Moth-flame optimization algorithm J . Ind. Prod. Eng. 2017 34 1 42 51
Khalilpourazari, S. & Pasandideh, S. H. R. Multi-item EOQ model with nonlinear unit holding cost and partial backordering: Moth-flame optimization algorithm J. Ind. Prod. Eng. 34(1), 42–51 (2017).
43. Paknejad J Nasri F Affisco JF Shape of power yield distribution: Impact on EOQ model with nonlinear holding cost and random quality Int. J. Manag. Sci. Eng. Manag. 2018 13 4 237 244
Paknejad, J., Nasri, F. & Affisco, J. F. Shape of power yield distribution: Impact on EOQ model with nonlinear holding cost and random quality. Int. J. Manag. Sci. Eng. Manag. 13(4), 237–244 (2018).
44. Pando V San-José LA García-Laguna J Sicilia J Optimal lot-size policy for deteriorating items with stock-dependent demand considering profit maximization Comput. Ind. Eng. 2018 117 81 93 10.1016/j.cie.2018.01.008
Pando, V., San-José, L. A., García-Laguna, J. & Sicilia, J. Optimal lot-size policy for deteriorating items with stock-dependent demand considering profit maximization. Comput. Ind. Eng. 117, 81–93 (2018).10.1016/j.cie.2018.01.008
45. San-Jose LA Sicilia J Gonzalez-de-la-Rosa M Febles-Acosta J An economic order quantity model with nonlinear holding cost, partial backlogging and ramp-type demand Eng. Optim. 2018 50 7 1164 1177 10.1080/0305215X.2017.1414205
San-Jose, L. A., Sicilia, J., Gonzalez-de-la-Rosa, M. & Febles-Acosta, J. An economic order quantity model with nonlinear holding cost, partial backlogging and ramp-type demand. Eng. Optim. 50(7), 1164–1177 (2018).10.1080/0305215X.2017.1414205
46. Edalatpour MA MirzapourAl-e-Hashem SMJ Simultaneous pricing and inventory decisions for substitute and complementary items with nonlinear holding cost Prod. Eng. 2019 13 305 315 10.1007/s11740-019-00883-6
Edalatpour, M. A. & MirzapourAl-e-Hashem, S. M. J. Simultaneous pricing and inventory decisions for substitute and complementary items with nonlinear holding cost. Prod. Eng. 13, 305–315 (2019).10.1007/s11740-019-00883-6
47. Pando V San-Jose LA Sicilia J Profitability ratio maximization in an inventory model with stock-dependent demand rate and non-linear holding cost Appl. Math. Model. 2019 66 643 661 10.1016/j.apm.2018.10.007
Pando, V., San-Jose, L. A. & Sicilia, J. Profitability ratio maximization in an inventory model with stock-dependent demand rate and non-linear holding cost. Appl. Math. Model. 66, 643–661 (2019).10.1016/j.apm.2018.10.007
48. San-José LA Sicilia J Cárdenas-Barrón LE Gutiérrez JM Optimal price and quantity under power demand pattern and non-linear holding cost Comput. Ind. Eng. 2019 129 426 434 10.1016/j.cie.2019.01.054
San-José, L. A., Sicilia, J., Cárdenas-Barrón, L. E. & Gutiérrez, J. M. Optimal price and quantity under power demand pattern and non-linear holding cost. Comput. Ind. Eng. 129, 426–434 (2019).10.1016/j.cie.2019.01.054
49. Tripathi RP Economic order quantity models for price dependent demand and different holding cost functions Jordan J. Math. Stat. 2019 12 1 15 33
Tripathi, R. P. Economic order quantity models for price dependent demand and different holding cost functions. Jordan J. Math. Stat. 12(1), 15–33 (2019).
50. Cárdenas-Barrón LE Shaikh AA Tiwari S Treviño-Garza G An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit Comput. Ind. Eng. 2020 139 105557 10.1016/j.cie.2018.12.004
Cárdenas-Barrón, L. E., Shaikh, A. A., Tiwari, S. & Treviño-Garza, G. An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit. Comput. Ind. Eng. 139, 105557 (2020).10.1016/j.cie.2018.12.004
51. Naddor E Inventory Systems 1966 Wiley
Naddor, E. Inventory Systems (Wiley, 1966).
52. Goel, V. P. & Aggarwal, S. P. Order level inventory system with power demand pattern for deteriorating items. In Proc. All India Seminar on Operational Research and Decision Making 19–34 (University of Delhi, 1981).
53. Datta TA Pal AK Order level inventory system with power demand pattern for items with variable rate of deterioration Indian J. Pure Appl. Math. 1988 19 11 1043 1053
Datta, T. A. & Pal, A. K. Order level inventory system with power demand pattern for items with variable rate of deterioration. Indian J. Pure Appl. Math. 19(11), 1043–1053 (1988).
54. Girlich HJ Naddor’s demand patterns and the economic order quantity under uncertainty Eng. Costs Prod. Econ. 1990 19 1–3 327 331 10.1016/0167-188X(90)90060-U
Girlich, H. J. Naddor’s demand patterns and the economic order quantity under uncertainty. Eng. Costs Prod. Econ. 19(1–3), 327–331 (1990).10.1016/0167-188X(90)90060-U
55. Lee WC Wu JW An EOQ model for items with Weibull distributed deterioration, shortages and power demand pattern Int. J. Inf. Manag. Sci. 2002 13 2 19 34
Lee, W. C. & Wu, J. W. An EOQ model for items with Weibull distributed deterioration, shortages and power demand pattern. Int. J. Inf. Manag. Sci. 13(2), 19–34 (2002).
56. Dye CY A note on An EOQ model for items with Weibull distributed deterioration, shortages and power demand pattern Int. J. Inf. Manag. Sci. 2004 15 2 81 84
Dye, C. Y. A note on An EOQ model for items with Weibull distributed deterioration, shortages and power demand pattern. Int. J. Inf. Manag. Sci. 15(2), 81–84 (2004).
57. Jung ST Lin JSJ Chuang JPC A note on An EOQ model for items with Weibull distributed deterioration, shortages and power demand pattern Int. J. Inf. Manag. Sci. 2008 19 4 667 672
Jung, S. T., Lin, J. S. J. & Chuang, J. P. C. A note on An EOQ model for items with Weibull distributed deterioration, shortages and power demand pattern. Int. J. Inf. Manag. Sci. 19(4), 667–672 (2008).
58. Abdul-Jalbar B Gutiérrez JM Sicilia J A two-echelon inventory/distribution system with power demand pattern and backorders Int. J. Prod. Econ. 2009 122 2 519 524 10.1016/j.ijpe.2009.04.017
Abdul-Jalbar, B., Gutiérrez, J. M. & Sicilia, J. A two-echelon inventory/distribution system with power demand pattern and backorders. Int. J. Prod. Econ. 122(2), 519–524 (2009).10.1016/j.ijpe.2009.04.017
59. Singh TJ Singh SR Dutt R An EOQ model for perishable items with power demand and partial backlogging Int. J. Oper. Quant. Manag. 2009 15 1 65 72
Singh, T. J., Singh, S. R. & Dutt, R. An EOQ model for perishable items with power demand and partial backlogging. Int. J. Oper. Quant. Manag. 15(1), 65–72 (2009).
60. Tripathy CK Pradhan LM An EOQ model for Weibull deteriorating items with power demand and partial backlogging Int. J. Contemp. Math. Sci. 2010 5 38 1895 1904
Tripathy, C. K. & Pradhan, L. M. An EOQ model for Weibull deteriorating items with power demand and partial backlogging. Int. J. Contemp. Math. Sci. 5(38), 1895–1904 (2010).
61. Kumar V Singh SR A finite horizon inventory model with life time, power demand pattern and lost sales Int. J. Math. Sci. 2011 10 3 435 446
Kumar, V. & Singh, S. R. A finite horizon inventory model with life time, power demand pattern and lost sales. Int. J. Math. Sci. 10(3), 435–446 (2011).
62. Rajeswari N Vanjikkodi T Deteriorating inventory model with power demand and partial backlogging Int. J. Math. Arch. 2011 2 9 1501 1945
Rajeswari, N. & Vanjikkodi, T. Deteriorating inventory model with power demand and partial backlogging. Int. J. Math. Arch. 2(9), 1501–1945 (2011).
63. Sarbjit, S. & Shivraj, S. Deterministic and probabilistic EOQ models for products having power demand pattern. In Proc. World Congress on Engineering, Vol. 1 (2011).
64. Singh SP Sehgal VK An EOQ inventory model for Weibull distributed deteriorating items with power demand pattern and shortages JP J. Math. Sci. 2011 1 2 99 110
Singh, S. P. & Sehgal, V. K. An EOQ inventory model for Weibull distributed deteriorating items with power demand pattern and shortages. JP J. Math. Sci. 1(2), 99–110 (2011).
65. Krishnaraj RB Ramasamy K An inventory model with power demand pattern, Weibull distribution deterioration and without shortages Bull. Soc. Math. Serv. Stand. 2012 2 33 37
Krishnaraj, R. B. & Ramasamy, K. An inventory model with power demand pattern, Weibull distribution deterioration and without shortages. Bull. Soc. Math. Serv. Stand. 2, 33–37 (2012).
66. Rajeswari N Vanjikkodi T An inventory model for items with two parameter Weibull distribution deterioration and backlogging Am. J. Oper. Res. 2012 2 02 247
Rajeswari, N. & Vanjikkodi, T. An inventory model for items with two parameter Weibull distribution deterioration and backlogging. Am. J. Oper. Res. 2(02), 247 (2012).
67. Sicilia J Febles-Acosta J Gonzalez-De La Rosa M Deterministic inventory systems with power demand pattern Asia Pac. J. Oper. Res. 2012 29 05 1250025 10.1142/S021759591250025X
Sicilia, J., Febles-Acosta, J. & Gonzalez-De La Rosa, M. Deterministic inventory systems with power demand pattern. Asia Pac. J. Oper. Res. 29(05), 1250025 (2012).10.1142/S021759591250025X
68. Sicilia J González-De-la-Rosa M Febles-Acosta J Alcaide-López-de-Pablo D Optimal policy for an inventory system with power demand, backlogged shortages and production rate proportional to demand rate Int. J. Prod. Econ. 2014 155 163 171 10.1016/j.ijpe.2013.11.020
Sicilia, J., González-De-la-Rosa, M., Febles-Acosta, J. & Alcaide-López-de-Pablo, D. Optimal policy for an inventory system with power demand, backlogged shortages and production rate proportional to demand rate. Int. J. Prod. Econ. 155, 163–171 (2014).10.1016/j.ijpe.2013.11.020
69. San-José LA Sicilia J González-De-la-Rosa M Febles-Acosta J Optimal inventory policy under power demand pattern and partial backlogging Appl. Math. Model. 2017 46 618 630 10.1016/j.apm.2017.01.082
San-José, L. A., Sicilia, J., González-De-la-Rosa, M. & Febles-Acosta, J. Optimal inventory policy under power demand pattern and partial backlogging. Appl. Math. Model. 46, 618–630 (2017).10.1016/j.apm.2017.01.082
70. San-José LA Sicilia J Alcaide-López-de-Pablo D An inventory system with demand dependent on both time and price assuming backlogged shortages Eur. J. Oper. Res. 2018 270 3 889 897 10.1016/j.ejor.2017.10.042
San-José, L. A., Sicilia, J. & Alcaide-López-de-Pablo, D. An inventory system with demand dependent on both time and price assuming backlogged shortages. Eur. J. Oper. Res. 270(3), 889–897 (2018).10.1016/j.ejor.2017.10.042
71. Sicilia J Febles-Acosta J González-De la Rosa M Economic order quantity for a power demand pattern system with deteriorating items Eur. J. Ind. Eng. 2013 7 5 577 593 10.1504/EJIE.2013.057381
Sicilia, J., Febles-Acosta, J. & González-De la Rosa, M. Economic order quantity for a power demand pattern system with deteriorating items. Eur. J. Ind. Eng. 7(5), 577–593 (2013).10.1504/EJIE.2013.057381
72. Sicilia J González-De-la-Rosa M Febles-Acosta J Alcaide-López-de-Pablo D An inventory model for deteriorating items with shortages and time-varying demand Int. J. Prod. Econ. 2014 155 155 162 10.1016/j.ijpe.2014.01.024
Sicilia, J., González-De-la-Rosa, M., Febles-Acosta, J. & Alcaide-López-de-Pablo, D. An inventory model for deteriorating items with shortages and time-varying demand. Int. J. Prod. Econ. 155, 155–162 (2014).10.1016/j.ijpe.2014.01.024
73. Sicilia J González-De-la-Rosa M Febles-Acosta J Alcaide-López-de-Pablo D Optimal inventory policies for uniform replenishment systems with time-dependent demand Int. J. Prod. Res. 2015 53 12 3603 3622 10.1080/00207543.2014.983618
Sicilia, J., González-De-la-Rosa, M., Febles-Acosta, J. & Alcaide-López-de-Pablo, D. Optimal inventory policies for uniform replenishment systems with time-dependent demand. Int. J. Prod. Res. 53(12), 3603–3622 (2015).10.1080/00207543.2014.983618
74. Rajeswari N Vanjikkodi T Sathyapriya K Optimization in fuzzy inventory model for linearly deteriorating items, with power demand, partial backlogging and linear holding cost Int. J. Comput. Appl. 2017 169 1 6 12
Rajeswari, N., Vanjikkodi, T. & Sathyapriya, K. Optimization in fuzzy inventory model for linearly deteriorating items, with power demand, partial backlogging and linear holding cost. Int. J. Comput. Appl. 169(1), 6–12 (2017).
75. Gurtu A Optimization of inventory holding cost due to price, weight, and volume of items J. Risk Financ. Manag. 2021 14 2 65 10.3390/jrfm14020065
Gurtu, A. Optimization of inventory holding cost due to price, weight, and volume of items. J. Risk Financ. Manag. 14(2), 65 (2021).10.3390/jrfm14020065
76. San-José LA Sicilia J González-De-la-Rosa M Febles-Acosta J Best pricing and optimal policy for an inventory system under time-and-price-dependent demand and backordering Ann. Oper. Res. 2020 286 351 369 10.1007/s10479-018-2953-5
San-José, L. A., Sicilia, J., González-De-la-Rosa, M. & Febles-Acosta, J. Best pricing and optimal policy for an inventory system under time-and-price-dependent demand and backordering. Ann. Oper. Res. 286, 351–369 (2020).10.1007/s10479-018-2953-5
77. Chowdhury RR Ghosh SK A production-inventory model for perishable items with demand dependent production rate, shortages and variable holding cost Int. J. Procure. Manag. 2022 15 3 424 446
Chowdhury, R. R. & Ghosh, S. K. A production-inventory model for perishable items with demand dependent production rate, shortages and variable holding cost. Int. J. Procure. Manag. 15(3), 424–446 (2022).
78. Momena AF Haque R Rahaman M Mondal SP A two-storage inventory model with trade credit policy and time-varying holding cost under quantity discounts Logistics 2023 7 4 77 10.3390/logistics7040077
Momena, A. F., Haque, R., Rahaman, M. & Mondal, S. P. A two-storage inventory model with trade credit policy and time-varying holding cost under quantity discounts. Logistics 7(4), 77 (2023).10.3390/logistics7040077
79. Jadidi O Jaber MY Zolfaghari S Joint pricing and inventory problem with price dependent stochastic demand and price discounts Comput. Ind. Eng. 2017 114 45 53 10.1016/j.cie.2017.09.038
Jadidi, O., Jaber, M. Y. & Zolfaghari, S. Joint pricing and inventory problem with price dependent stochastic demand and price discounts. Comput. Ind. Eng. 114, 45–53 (2017).10.1016/j.cie.2017.09.038
80. Panda S Saha S Modak NM Sana SS A volume flexible deteriorating inventory model with price sensitive demand Tékhne 2017 15 2 117 123 10.1016/j.tekhne.2017.09.002
Panda, S., Saha, S., Modak, N. M. & Sana, S. S. A volume flexible deteriorating inventory model with price sensitive demand. Tékhne 15(2), 117–123 (2017).10.1016/j.tekhne.2017.09.002
81. Rubio-Herrero J Baykal-Gursoy M On the unimodality of the price-setting newsvendor problem with additive demand under risk considerations Eur. J. Oper. Res. 2018 265 3 962 974 10.1016/j.ejor.2017.08.055
Rubio-Herrero, J. & Baykal-Gursoy, M. On the unimodality of the price-setting newsvendor problem with additive demand under risk considerations. Eur. J. Oper. Res. 265(3), 962–974 (2018).10.1016/j.ejor.2017.08.055
82. Marand AJ Li H Thorstenson A Joint inventory control and pricing in a service-inventory system Int. J. Prod. Econ. 2019 209 78 91 10.1016/j.ijpe.2017.07.008
Marand, A. J., Li, H. & Thorstenson, A. Joint inventory control and pricing in a service-inventory system. Int. J. Prod. Econ. 209, 78–91 (2019).10.1016/j.ijpe.2017.07.008
83. Rahman MS Duary A Shaikh AA Bhunia AK An application of parametric approach for interval differential equation in inventory model for deteriorating items with selling-price-dependent demand Neural Comput. Appl. 2020 32 14069 14085 10.1007/s00521-020-04806-w
Rahman, M. S., Duary, A., Shaikh, A. A. & Bhunia, A. K. An application of parametric approach for interval differential equation in inventory model for deteriorating items with selling-price-dependent demand. Neural Comput. Appl. 32, 14069–14085 (2020).10.1007/s00521-020-04806-w
84. Ruidas S Seikh MR Nayak PK A production inventory model with interval-valued carbon emission parameters under price-sensitive demand Comput. Ind. Eng. 2021 154 107154 10.1016/j.cie.2021.107154
Ruidas, S., Seikh, M. R. & Nayak, P. K. A production inventory model with interval-valued carbon emission parameters under price-sensitive demand. Comput. Ind. Eng. 154, 107154 (2021).10.1016/j.cie.2021.107154
85. Palanivel M Suganya M Partial backlogging inventory model with price and stock level dependent demand, time varying holding cost and quantity discounts J. Manag. Anal. 2022 9 1 32 59
Palanivel, M. & Suganya, M. Partial backlogging inventory model with price and stock level dependent demand, time varying holding cost and quantity discounts. J. Manag. Anal. 9(1), 32–59 (2022).
86. Narang, P., Kumari, M. & De, P. K. Production inventory model with three levels of production and demand for deteriorating item under price, stock and advertisement dependent demand. In Applications of Operational Research in Business and Industries: Proceedings of 54th Annual Conference of ORSI 49–68 (Springer, 2023).
87. Herbon A Khmelnitsky E Optimal dynamic pricing and ordering of a perishable product under additive effects of price and time on demand Eur. J. Oper. Res. 2017 260 2 546 556 10.1016/j.ejor.2016.12.033
Herbon, A. & Khmelnitsky, E. Optimal dynamic pricing and ordering of a perishable product under additive effects of price and time on demand. Eur. J. Oper. Res. 260(2), 546–556 (2017).10.1016/j.ejor.2016.12.033
88. Dey BK Bhuniya S Sarkar B Involvement of controllable lead time and variable demand for a smart manufacturing system under a supply chain management Expert Syst. Appl. 2021 184 115464 10.1016/j.eswa.2021.115464
Dey, B. K., Bhuniya, S. & Sarkar, B. Involvement of controllable lead time and variable demand for a smart manufacturing system under a supply chain management. Expert Syst. Appl. 184, 115464 (2021).10.1016/j.eswa.2021.115464
89. San-José LA González-De-la-Rosa M Sicilia J Febles-Acosta J An inventory model for multiple items assuming time-varying demands and limited storage Optim. Lett. 2022 16 6 1935 1961 10.1007/s11590-021-01815-z
San-José, L. A., González-De-la-Rosa, M., Sicilia, J. & Febles-Acosta, J. An inventory model for multiple items assuming time-varying demands and limited storage. Optim. Lett. 16(6), 1935–1961 (2022).10.1007/s11590-021-01815-z
90. Nurhasril N Supadi SS Omar M A two-warehouse inventory model with rework process and time-varying demand Malays. J. Sci. 2023 1 17 31 10.22452/mjs.vol42no1.3
Nurhasril, N., Supadi, S. S. & Omar, M. A two-warehouse inventory model with rework process and time-varying demand. Malays. J. Sci. 1, 17–31 (2023).10.22452/mjs.vol42no1.3
91. Akhtar M Duary A Manna AK Shaikh AA Bhunia AK An application of tournament differential evolution algorithm in production inventory model with green level and expiry time dependent demand Artif. Intell. Rev. 2023 56 5 4137 4170 10.1007/s10462-022-10268-4
Akhtar, M., Duary, A., Manna, A. K., Shaikh, A. A. & Bhunia, A. K. An application of tournament differential evolution algorithm in production inventory model with green level and expiry time dependent demand. Artif. Intell. Rev. 56(5), 4137–4170 (2023).10.1007/s10462-022-10268-4
92. Ali H Akhtar F Manna AK Alrasheedi AF Shaikh AA Impact of warranty and green level of the product with nonlinear demand via optimal control theory and Artificial Hummingbird Algorithm Sci. Rep. 2024 14 1 10809 10.1038/s41598-024-61453-0 38734734
Ali, H., Akhtar, F., Manna, A. K., Alrasheedi, A. F. & Shaikh, A. A. Impact of warranty and green level of the product with nonlinear demand via optimal control theory and Artificial Hummingbird Algorithm. Sci. Rep. 14(1), 10809 (2024).38734734 10.1038/s41598-024-61453-0
93. Das SC Ali H Khan MAA Shaikh AA Alrasheedi AF Inventory model for green products with payment strategy, selling price and green level dependent demand using teaching learning based optimization algorithm Sci. Rep. 2024 14 1 3033 10.1038/s41598-024-53109-w 38321078
Das, S. C., Ali, H., Khan, M. A. A., Shaikh, A. A. & Alrasheedi, A. F. Inventory model for green products with payment strategy, selling price and green level dependent demand using teaching learning based optimization algorithm. Sci. Rep. 14(1), 3033 (2024).38321078 10.1038/s41598-024-53109-w
94. Eskandar H Sadollah A Bahreininejad A Hamdi M Water cycle algorithm—A novel metaheuristic optimization method for solving constrained engineering optimization problems Comput. Struct. 2012 110 151 166 10.1016/j.compstruc.2012.07.010
Eskandar, H., Sadollah, A., Bahreininejad, A. & Hamdi, M. Water cycle algorithm—A novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput. Struct. 110, 151–166 (2012).10.1016/j.compstruc.2012.07.010
95. David S The Water Cycle, Illustrations by John Yates 1993 Thomson Learning
David, S. The Water Cycle, Illustrations by John Yates (Thomson Learning, 1993).
96. Strahler AN Dynamic basis of geomorphology Geol. Soc. Am. Bull. 1952 63 9 923 938 10.1130/0016-7606(1952)63[923:DBOG]2.0.CO;2
Strahler, A. N. Dynamic basis of geomorphology. Geol. Soc. Am. Bull. 63(9), 923–938 (1952).10.1130/0016-7606(1952)63[923:DBOG]2.0.CO;2
97. Yadav A AEFA: Artificial electric field algorithm for global optimization Swarm Evol. Comput. 2019 48 93 108 10.1016/j.swevo.2019.03.013
Yadav, A. AEFA: Artificial electric field algorithm for global optimization. Swarm Evol. Comput. 48, 93–108 (2019).10.1016/j.swevo.2019.03.013
98. Xue J Shen B A novel swarm intelligence optimization approach: Sparrow search algorithm Syst. Sci. Control Eng. 2020 8 1 22 34 10.1080/21642583.2019.1708830
Xue, J. & Shen, B. A novel swarm intelligence optimization approach: Sparrow search algorithm. Syst. Sci. Control Eng. 8(1), 22–34 (2020).10.1080/21642583.2019.1708830
99. Hu G Guo Y Wei G Abualigah L Genghis Khan shark optimizer: A novel nature-inspired algorithm for engineering optimization Adv. Eng. Inform. 2023 58 102210 10.1016/j.aei.2023.102210
Hu, G., Guo, Y., Wei, G. & Abualigah, L. Genghis Khan shark optimizer: A novel nature-inspired algorithm for engineering optimization. Adv. Eng. Inform. 58, 102210 (2023).10.1016/j.aei.2023.102210
100. Agushaka JO Ezugwu AE Abualigah L Gazelle optimization algorithm: A novel nature-inspired metaheuristic optimizer Neural Comput. Appl. 2023 35 5 4099 4131 10.1007/s00521-022-07854-6
Agushaka, J. O., Ezugwu, A. E. & Abualigah, L. Gazelle optimization algorithm: A novel nature-inspired metaheuristic optimizer. Neural Comput. Appl. 35(5), 4099–4131 (2023).10.1007/s00521-022-07854-6
101. Bai J Li Y Zheng M Khatir S Benaissa B Abualigah L Wahab MA A sinh cosh optimizer Knowl. Based Syst. 2023 282 111081 10.1016/j.knosys.2023.111081
Bai, J. et al. A sinh cosh optimizer. Knowl. Based Syst. 282, 111081 (2023).10.1016/j.knosys.2023.111081
102. Braik M Hammouri A Atwan J Al-Betar MA Awadallah MA White Shark Optimizer: A novel bio-inspired meta-heuristic algorithm for global optimization problems Knowl. Based Syst. 2022 243 108457 10.1016/j.knosys.2022.108457
Braik, M., Hammouri, A., Atwan, J., Al-Betar, M. A. & Awadallah, M. A. White Shark Optimizer: A novel bio-inspired meta-heuristic algorithm for global optimization problems. Knowl. Based Syst. 243, 108457 (2022).10.1016/j.knosys.2022.108457
103. Mehmood K Chaudhary NI Khan ZA Cheema KM Raja MAZ Shu CM Novel knacks of chaotic maps with Archimedes optimization paradigm for nonlinear ARX model identification with key term separation Chaos Solitons Fractals 2023 175 114028 10.1016/j.chaos.2023.114028
Mehmood, K. et al. Novel knacks of chaotic maps with Archimedes optimization paradigm for nonlinear ARX model identification with key term separation. Chaos Solitons Fractals 175, 114028 (2023).10.1016/j.chaos.2023.114028
104. Mehmood K Chaudhary NI Khan ZA Cheema KM Raja MAZ Milyani AH Azhari AA Nonlinear hammerstein system identification: A novel application of marine predator optimization using the key term separation technique Mathematics 2022 10 22 4217 10.3390/math10224217
Mehmood, K. et al. Nonlinear hammerstein system identification: A novel application of marine predator optimization using the key term separation technique. Mathematics 10(22), 4217 (2022).10.3390/math10224217
105. Ghasemi M Zare M Zahedi A Akbari MA Mirjalili S Abualigah L Geyser inspired algorithm: A new geological-inspired meta-heuristic for real-parameter and constrained engineering optimization J. Bionic Eng. 2024 21 1 374 408 10.1007/s42235-023-00437-8
Ghasemi, M. et al. Geyser inspired algorithm: A new geological-inspired meta-heuristic for real-parameter and constrained engineering optimization. J. Bionic Eng. 21(1), 374–408 (2024).10.1007/s42235-023-00437-8
106. Ghasemi M Zare M Zahedi A Trojovský P Abualigah L Trojovská E Optimization based on performance of lungs in body: Lungs performance-based optimization (LPO) Comput. Methods Appl. Mech. Eng. 2024 419 116582 10.1016/j.cma.2023.116582
Ghasemi, M. et al. Optimization based on performance of lungs in body: Lungs performance-based optimization (LPO). Comput. Methods Appl. Mech. Eng. 419, 116582 (2024).10.1016/j.cma.2023.116582
107. Agushaka JO Ezugwu AE Abualigah L Dwarf mongoose optimization algorithm Comput. Methods Appl. Mech. Eng. 2022 391 114570 10.1016/j.cma.2022.114570
Agushaka, J. O., Ezugwu, A. E. & Abualigah, L. Dwarf mongoose optimization algorithm. Comput. Methods Appl. Mech. Eng. 391, 114570 (2022).10.1016/j.cma.2022.114570
108. Ahmadianfar I Heidari AA Gandomi AH Chu X Chen H RUN beyond the metaphor: An efficient optimization algorithm based on Runge Kutta method Expert Syst. Appl. 2021 181 115079 10.1016/j.eswa.2021.115079
Ahmadianfar, I., Heidari, A. A., Gandomi, A. H., Chu, X. & Chen, H. RUN beyond the metaphor: An efficient optimization algorithm based on Runge Kutta method. Expert Syst. Appl. 181, 115079 (2021).10.1016/j.eswa.2021.115079
