Effects of variable prepayment installments on pricing and inventory decisions with power demand pattern and non-linear holding cost under carbon cap-and-price regulation
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Khan, Md. Al-Amin et al. Article Effects of variable prepayment installments on pricing and inventory decisions with power demand pattern and non-linear holding cost under carbon cap-and-price regulation Operations Research Perspectives Provided in Cooperation with: Elsevier Suggested Citation: Khan, Md. Al-Amin et al. (2024) : Effects of variable prepayment installments on pricing and inventory decisions with power demand pattern and non-linear holding cost under carbon cap-and-price regulation, Operations Research Perspectives, ISSN 2214-7160, Elsevier, Amsterdam, Vol. 12, pp. 1-24, https://doi.org/10.1016/j.orp.2023.100289 This Version is available at: https://hdl.handle.net/10419/325774 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/
Operations Research Perspectives 12 (2024) 100289 Available online 11 November 2023 2214-7160/© 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Effects of variable prepayment installments on pricing and inventory decisions with power demand pattern and non-linear holding cost under carbon cap-and-price regulation Md. Al-Amin Khan a , b , Leopoldo Eduardo C´ ardenas-Barr´ on a , c , * , Gerardo Trevi˜ no-Garza d , Armando C´ espedes-Mota a , Imelda de Jesús Loera-Hern´ andez a , Neale R. Smith a a Tecnol´ ogico de Monterrey, School of Engineering and Sciences, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo Le´ on, Mexico b Department of Mathematics, Jahangirnagar University, Savar, Dhaka 1342, Bangladesh c Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea d Ingram School of Engineering, Texas State University, 601 University Drive San Marcos, TX 78666, USA ARTICLE INFO Keywords: Variable prepayment installment Power demand Non-linear holding cost Carbon emissions Carbon tariff rules ABSTRACT Regulators’ increasingly stringent carbon rules to protect the environment are encouraging practitioners to modify their operational activities that are accountable for releasing emissions into the atmosphere. Thereby, practitioners dealing with product inventory planning are seeking proper management strategies not only to increase profits but also to reduce released carbons from operations. In addition, increasing uncertainty in supply operations has motivated suppliers to impose prepayment mechanisms in recent decades. This study examines the best prepayment installment policy for a practitioner for the first time, where the consumption behavior of consumers changes as a result of the combined effects of unit selling price and storage time. Moreover, to make the present inventory planning more realistic, the unit holding cost function is adopted as a power function of the inventory unit’s storage period. The goal of this study is to provide the best combined installment for advance payment, price, and replenishment strategies for a practitioner under cap-and-price, cap-and-trade, and carbon tax environmental guidelines by ensuring maximum profit. For this purpose, an algorithm is created by combining all derived theoretical results from the analytical study, whereas the efficacy of the algorithm is assessed through the examination of five illustrative numerical instances. A plethora of noteworthy management insights for the practitioner are obtained by investigating the dynamic shifts in optimal strategies resulting from fluctuations in system parameters. The results reveal that if the demand is low in the nascent phases of the business cycle, then the prudent approach for the practitioner entails procuring a comparatively smaller lot-size using a modest number of payment frequencies and then setting a relatively small unit selling price to increase profits. 1. Introduction To achieve the goals of sustainable development, one of the greatest global challenges is controlling the emission of greenhouse gasses, and therefore, many countries around the world are executing environmental regulations to lessen the negative impact of emissions on climate change. It is important to note that, due to activities like warming, freezing, lighting, and product handling, warehousing in the business industry is one of the major sources of carbon emissions. Furthermore, maintaining a company’s cash flow by selecting a suitable way of collecting payments from customers is one of the most difficult jobs. An effective way for a business to sustain cash flow and lower order cancellations or postponements is through prepayment or advance payment arrangements. The items’ demand rate during the storage process is another crucial factor that has a substantial impact on any company’s inventory management practices. Through the duration of storage, customer preferences for item consumption may vary. For instance, due to their freshness, certain commodities like packaged foods, fruits, and meat with expiration dates are in higher demand early in the storage period. On the other hand, since they tend to dry out with time, allowing * Corresponding author. E-mail addresses: [email protected] (L.E. C´ ardenas-Barr´ on), [email protected] (G. Trevi˜ no-Garza). Contents lists available at ScienceDirect Operations Research Perspectives journal homepage: www.elsevier.com/locate/orp https://doi.org/10.1016/j.orp.2023.100289 Received 20 June 2023; Received in revised form 3 November 2023; Accepted 8 November 2023
Operations Research Perspectives 12 (2024) 100289 2 customers to utilize them for longer periods, goods like potatoes, garlic, and dates may experience a rise in demand towards the end of the inventory cycle. As opposed to this, the demand for goods like furniture, electrical appliances, tiles, and draperies often remains steady throughout the cycle. To account for these varying characteristics, the demand pattern is assumed to be a mix of two factors: the impact of storage time, which is modelled as a power pattern, and the impact of pricing, which is represented in a linear form. Taking these factors into account simultaneously, the current study digs into the optimization of installment plans for practitioners by thoroughly examining the effects of variable prepayment installments on profitability. This study focuses particularly on a case in which customer demand is viewed as a composite function that combines the impact of storage time following a power pattern and pricing structure in a linear format. All of these factors are taken into account in the context of strict tax laws that regulate carbon emissions. This research addresses the environmental impact from the perspective of carbon emissions restrictions, in addition to offering practitioners useful insights into financial decision-making. It is a thorough and forwardlooking examination since it takes into account the dual objectives of maximizing profit and upholding ecologically appropriate practices. 1.1. Motivation The Environmental Protection Agency (EPA) of the US revealed on its website that industry, the third largest emission source, accounted for nearly one-quarter of the greenhouse gasses released in the year 2020 (https://www.epa.gov/ghgemissions/sources-greenhouse-gas-emission s). Furthermore, the european commission (EC) has revealed that the biggest contributor to global warming is carbon emissions from human activities (https://climate.ec.europa.eu/climate-change/causes-c limate-change_en). To curb the increasing trend of global warming, governments in developed countries have taken several initiatives, one of which is the enforcement of stringent carbon rules on businesses to reduce carbon emissions. Finnish authorities implemented a carbon tax policy for the first time in the world in 1990, and a year later, Swedish authorities also imposed a tax regulation on carbon emissions. The World Bank reports that 68 carbon pricing measures have been adopted or are expected to be employed by June 2022 in 46 different nations (https://www.weforum.org/agenda/2022/07/carbon-tax-emissions-co untries/). Fig. 1 highlights different countries’ tax rates in US dollars against each metric ton of CO 2e (carbon dioxide equivalent). At the beginning of the year 2022, Uruguayan authorities established a tax regulation against carbon emissions for the first time in their country, and the tax rate is the highest in the world at $137 per metric ton of CO 2e as of April 2022. Noticing the positive consequences in terms of reducing carbon emissions, such as in Uruguay, many countries are going to enact more strict tax regulations on companies. As a result, companies in different countries are aware of their operational activities during the course of business, which are sources of carbon emissions, and they are seeking appropriate amendments to their best business policies under various carbon regulations. Warehousing in the business sector is one of the significant sources of carbon emissions because of warming, freezing, lighting, and product handling operations [30]. According to the World Economic Forum [46], 13% of released emissions from supply chain actions are produced from storage activities in logistics facilities, including product handling operations. Fig. 2 highlights the amount of released CO 2e (carbon dioxide equivalent) annually in Sweden from industrial processing and product consumption from 2011 to 2020. Therefore, companies are now planning their inventory by taking into account the amount of tax on the released emissions under the imposed carbon tax regulations. In the meantime, with ambitions to significantly reduce CO 2e , many countries are revising their regulations by imposing higher tax rates on each unit of emissions. As an example, the tax rate in Sweden was $123 per ton of CO 2e in 2019, $126 per ton of CO 2e in 2020, and $130 per ton of CO 2e in 2022 (which is the second highest rate in the world). 1,2 Due to an increase in the tax rate, Sweden’s total CO 2e emissions from industrial processing and product consumption decreased by 16.43% (1.3 million tons) in 2020 in contrast to the previous year. Furthermore, the Portuguese government nearly doubled the tax rate in 2020 to € 24 per ton of CO 2e emissions, while the Norwegian government increased the tax rate to 8.16% in 2019 [1]. Under these strict tax regulations, companies ought to meticulously prepare their best inventory and business plans without compromising their profitability. One of the most challenging tasks for any company is to maintain its cash flow by choosing an appropriate method of collecting payments from customers. In addition, cancelation or postponement of orders can become a nightmare for any manufacturer or supplier, and hence, controlling clients’ order cancellations or postponements is another big challenge for the manufacturers or suppliers of various products. A prepayment or advance payment arrangement is an effective one for a company, not only to maintain cash flow but also to reduce order cancellations or postponements. As a result, prepayment mechanisms have become a common strategy for many companies (suppliers) of various products in the present business sector. Fig. 3 illustrates the different existing payment methods of companies in Europe in the year 2019. As seen from Fig. 3, half of the total companies in Europe offered prepayment methods (partially or fully) in 2019. However, as it is not possible to earn any revenue from the invested prepayment amount before receiving products, there is an opportunity expense or capital charge against the paid amount in advance. To reduce this opportunity or capital cost for a practitioner, companies offer the multiple-installment facility for accomplishing the prepayment [16]. Most importantly, they did not consider any transaction charge for installments, though there is always a transaction charge for all prepayment installments. For instance, when installments are completed by exchanging money internationally or one bank account to another account, a transaction cost is incurred according to the existing rules; when the installments are completed by the practitioner itself or by sending an agent, the transaction charge comprises the transportation cost, and time and labor costs. Consequently, the research issue that arises in this case is: What is the ideal prepayment installment method to minimize capital costs or increase profits? 1.2. Research questions and goals The clients’ behavior regarding items’ consumption may vary over the storage period. For instance, for the reason of their freshness, certain items (such as packaged foods, fruits, meat with an expiration date, etc.) are in more demand early on in the storage period. Again, some products, like potatoes, garlic, dates, etc., may be in more demand towards the end of the inventory cycle because of their drying conditions, which encourage consumers to consume for longer periods of time, whereas other products, like furniture, electrical appliances, tiles, and curtains, have basically consistent demand throughout the cycle. To adopt these characteristics, the clients’ demand pattern is assumed to be an additive combination of the effects of storage time in power pattern and price in linear form. Now, the following research queries appear: 1 https://taxfoundation.org/sweden-carbon-tax-revenue-greenhouse-gas-e missions/ 2 https://www.carbonpricingleadership.org/blogs/2019/10/18/shouldevery-country-on-earth-copy-swedens-carbon-tax 3 https://www.statista.com/statistics/483590/prices-of-implemented-carbon -pricing-instruments-worldwide-by-select-country/ 4 https://www.statista.com/statistics/412202/annual-greenhouse-gas -emissions-from-industrial-processing-in-sweden/ 5 https://www.statista.com/statistics/931239/share-of-payment-methods -offered-by-corporations-in-europe/ Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 3 Fig. 1. Country-wise tax rates for carbon emissions in 2022 (Statista, ID: 483,590 3 ). Fig. 2. Annual CO 2e emissions in Sweden from industrial processing and product use from 2011 to 2020 (Statista, ID: 412,202 4 ). Fig. 3. Payment approaches of businesses in Europe in 2019 (Statista, ID: 931,239 5 ). Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 4 (i) How do environmental guidelines affect a practitioner’s best inventory planning when demand follows a power pattern in the storage phase? (ii) What is the ideal prepayment installment method to minimize capital costs or increase profits under environmental guidelines? (iii) The practitioner’s profit per unit of time is impacted by carbon laws to what extent? The research goals of this study are to address the research questions raised above appropriately, which are succinctly and systematically summarized as follows: •To examine the effects of a variable prepayment installment strategy on the profitability of a practitioner. •To understand how customer demand, modeled as a composite function of storage time and pricing structure, influences profitability. •To provide insights into financial decision-making for practitioners by considering the impact of strict tax laws governing carbon emissions. •To balance the dual objectives of maximizing profit and adhering to environmentally responsible practices. 2. Literature review The purpose of this work is to integrate the prepayment installment policy into an inventory planning problem for a practitioner that simultaneously includes the pricing and replenishment strategies under carbon tariff regulations such as cap-and-price (C&P), cap-and-trade (C&T), and carbon tax (CT) guidelines. In the literature, though the pricing and replenishment strategies under carbon tariff regulations are investigated, these strategies are not investigated under carbon tariff regulations after incorporating the simultaneous effects of storage time and unit selling price on the clients’ consumption behavior, while no existing work is available dealing with the prepayment installment policy for a practitioner. This section provides an outline of the existing literature related to this research work. Due to the nature of today’s competitive business, practitioners must create efficient inventory management models to maximize profits after meeting customer demands at a fair price within a reasonable time frame in order to prevent losing market share to rivals. Clients’ consumption behavior for any product is not always the same; rather, it fluctuates over the storage period due to various realistic characteristics like age of inventory, freshness of inventory, selling price, availability of items, and others. When demand is interconnected with inventory ages and its structure is characterized by the quotient of products’ storage duration and cycle length, then the demand is referred to as “power demand,” which was invented in inventory analysis by Naddor [24]. Relaxing a fixed business cycle in power demand, Sicilia et al. [38] explored the best inventory planning for a company where its business cycle is adopted as a variable. Afterwards, Sicilia et al. [39] analyzed the best reorder point for a practitioner under power demand in a manufacturing system. Taking into consideration the imperfect production, Keshavarzfard et al. [13] further extended the manufacturing system with a power demand of Sicilia et al. [39]. These studies did not investigate the consequences of selling price along with the consequences of products’ age on the consumption behavior. Avianadav et al. [3] described two inventory problems in which one takes into account the multiplicative impacts of price and product stock age on demand, while the other takes into account the additive effects of price and product stock age. Describing the impacts of price as a logit function and of the product’s stock age as power demand, the ideal price and stock plans for a company are derived by San-Jos´ e et al. [33]. Subsequently, San-Jos´ e et al. [34] further investigated demand as a composite function comprising two distinct and separable components: one associated with the market price and the other with the duration of storage, while the effects of price and storage time are described as linear and power patterns, respectively. Then, they explored the optimal price and stock plans for a practitioner under the consequences of non-linear holding costs. Later, C´ ardenas-Barr´ on et al. [4] further expanded the study of San-Jos´ e et al. [34] in a stockout environment. In this research direction, some noteworthy works are San-Jos´ e et al. [35–37]. However, all work published related to power demand has a common feature that payment follows a cash-on-delivery process that is incompatible with many current business environments. As supply chains become more fragile, suppliers are very concerned about cancelling or postponing orders from their clients. As a result, suppliers ask for a complete or fractional advance payment from their customers, and this payment method is known as “prepayment” or “advance payment” in inventory management literature. This prepayment mechanism is widely popular in China’s rapidly rising automotive industry sector [48]. According to Zia and Taleizadeh [49], and Duary et al. [7], there are two key benefits for suppliers from the prepayment mechanism: (i) they can control order cancelation markedly, and (ii) they have the opportunity to accrue interest on the prepayment sum. On the other hand, as it is not possible to earn any revenue from the invested prepayment amount before receiving products, there is an opportunity expense or capital charge for the practitioner against the prepayment sum [29]. To reduce this opportunity or capital cost for a practitioner, suppliers offer a multiple-installment facility for accomplishing the prepayment [42]. Taleizadeh et al. [42] explored the consequences of using many payments as prepayments rather than a single payment. Later, Taleizadeh [43] extended the prepayment agreement for decay items. According to the variable prepayment legislation Khan et al. [17] suggested, merchants would be compelled to prepay a tiny portion of the acquisition cost for greater quantities acquired. However, all the above-stated published works related to prepayment did not consider any transaction charge for installments, though there is always a transaction charge for every installment of prepayment, such as money exchanging charges (nationally or internationally) from one bank to another, transportation charges, and time and labor costs. During the holding cost computation of any inventory procedure, it is generally presumed that the unit carrying expense is linear in the product’s stock duration, though this assumption may be incompatible for many products in practice. Adopting the expense of carrying each unit as a non-linear function of the product’s stock duration, Naddor [24] described an inventory procedure for a practitioner. Weiss [45] showed that when the products’ value falls non-linearly with respect to their stock duration, the inventory models with the expense for carrying each unit as a non-linear function of the product’s stock duration can be suitable. The model that Weiss [45] previously looked at was reexamined in Ferguson et al.’s work [9], which clarified the model’s function as an approximation of the best ordering strategy for perishable commodities. Taking into account the non-linearity with respect to a product’s stock duration, not only in holding cost but also in demand, Pando et al. [26] examined the best inventory planning for a practitioner to maximize profit. San-Jos´ e et al. [31] included both a variable unit carrying expenditure, which was modeled as a non-linear power relationship with storing time, and a constant unit carrying expense, which reflected the expenses of accommodating products in the warehouse. Alfares and Ghaithan [2] found that the expense of holding each unit is directly dependent on the unit acquisition cost and also exhibits a linearly increasing relationship with the product’s stock duration. Later, Shaikh et al. [40] conducted another analysis incorporating a linearly increasing unit holding expense for decay items, while Khan et al. [15] explored the consequences of a linearly increasing unit holding cost function on the profit of a practitioner dealing with products having a maximum lifetime. Some other investigated inventory models under a non-linear unit holding cost are Khalilpourazari and Pasandideh [14], Paknejad et al. [25], Pando et al. [27], Edalatpour and Mirzapour Al-e-Hashem [8] and Pando et al. [28]. The warming, freezing, lighting, and product handling procedures Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 5 involved in warehousing are among the major causes of carbon emissions within a supply chain [30]. According to the World Economic Forum [46], 13% of greenhouse gas emissions from supply chain business operations are produced from storage activities in logistics facilities, including product handling operations. Authorities in various countries (see Fig. 1) impose a number of regulations on companies with the aim of seeking the best interests of our society rather than individual interests (the interests of the companies) [12]. Considering the ordering and holding operations as carbon emission sources in an inventory practice of a practitioner, Hua et al. [11] investigated the impacts of C&T regulation on the inventory planning. The ideal ordering strategy under the carbon cap (CC) legislation was further explored by Chen et al. [5] by taking procured products into account as an additional cause of emissions in a practitioner’s stock system. Later, analyzing the investment strategy for a practitioner to shrink released carbons from several operations in an inventory system, Toptal et al. [44] investigated the optimal inventory decision under three carbon tariff regulations (C&T, CT, and CC). Hovelaque and Bironneau [10] integrated not only the carbon tariffs but also the impacts on customers’ demand of the released carbon amounts by a practitioner in an inventory system. Chen et al. [6] analyzed an adjustment strategy of a retailer under the C&T regulation and a credit period opportunity. The ideal reorder strategies for several merchants operating out of a single warehouse where a carbon tariff is applied to the emissions from purchasing and storing operations were covered by Li and Hai [22]. Afterward, the stock management plan for a practitioner with time-sensitive client demand under carbon tariff rules adopting a particular business cycle was examined by Xu et al. [47]. They did not, however, take into account how prices affect customers’ purchasing decisions. Lee [21] contrasted the best practice with C&T, CC, CT, and CO (carbon offset) rules after further examining the best purchasing strategy for a corporation under the C&P regulation. Recently, Khan et al. [18] discussed the significances of carbon tariff on the best circular indexing strategy policy for a manufacturer under the CT regulation. Furthermore, a comparison of similar studies is undertaken to highlight the research’s uniqueness, as shown in Table 1. The discussion performed above reveals that there is only one previous paper [47] that has studied the inventory strategy for a practitioner with time-sensitive client demand under carbon tariff regulations adopting a specific business cycle. They did not, however, take into account how prices affect customers’ purchasing decisions. This work attempts to investigate the best inventory planning of a practitioner under carbon tariff regulations by incorporating the simultaneous effects of storage time and unit selling price on the clients’ consumption behavior. In addition, the existing prepayment literature on inventory control focuses particularly on accomplishing the prepayment by a single installment or fixed multiple installments offered by the supplier. Khan et al. [19] examined the effects of variable advance payment installments on the optimum stocking selection and took into account transaction expenses related to the payment installments in advance payment arrangements. Using a tri-exponential function of price and a power function of time to model the demand structure, which was assumed to reflect consumer behavior, they examined the installment policy to prepay under a discount depending on the amount of purchase. It is important to note that they found all the theoretical results in the absence of a carbon regulatory policy. With the demand structure considered to be an additive combination of the influence of storage time in power pattern and price in linear form under various environmental guidelines, the current study makes the first attempt to determine the optimal stock management and market price plan as well. Moreover, Khan et al. [20] investigated pricing, inventory, and installment decisions for growing items only, a specific kind of item that increases inventory levels during its feeding time. It is noteworthy that their developed model is not suitable for many general products like cosmetic products, electrical equipment, computers, clothes, furniture, Table 1 Major literature study over the prepayment installment mechanism and power demand pattern. Author(s) Demand involves Prepayment installment Holding cost Products type Environmental regulations Objective function Time impacts as Price impacts as Others Variable Fixed Growing General (nongrowing) CM PM Taleizadeh et al. [42] Con. √ Con. √ √ Taleizadeh [43] Con. √ Con. √ √ Toptal et al. [44] Con. Con. √ C&T, CT, CC √ Zia and Taleizadeh [49] Con. √ Con. √ √ Hovelaque and Bironneau [10] Linear form Emissions level Con. √ CT √ Zhang et al. [48] Random variable √ √ √ San-Jos´ e et al. [32] PDP Con. √ √ Chen et al. [6] Con. Con. √ C&T √ San-Jos´ e et al. [33] PDP Logit form Con. √ √ San-Jos´ e et al. [34] PDP Linear form Non-linear √ √ San-Jos´ e et al. [35] PDP Exponential form Con. √ √ Khan et al. [16] Linear form √ Linear √ √ Xu et al. [47] General form Con. √ CC, C&T √ Lee [21] Con. Con. √ C&T, CC, CT, CO √ Khan et al. [17] Con. √ Con. √ √ C´ ardenas-Barr´ on et al. [4] PDP Linear form Non-linear √ √ San-Jos´ e et al. [36] PDP Power form Non-linear √ √ Manna et al. [23] Intervalvalued √ Intervalvalued √ √ Khan et al. [19] PDP Exponential form √ Con. √ √ Khan et al. [20] PDP Power form √ Non-linear √ C&P, C&T, CT √ This work PDP Linear form √ Non-linear √ C&P, C&T, CT √ PDP: power demand pattern; Con.: constant; CM: cost minimization; PM: profit maximization. Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 6 and many others. Inventory planning for non-growing items differs significantly from that of growing items. Non-growing items are typically managed with a focus on demand and storage considerations. In contrast, growing items involve unique challenges related to breeding, feeding, and care over time. For non-growing items, the emphasis is on maintaining optimal stock levels to meet customer demand efficiently, while for growing items, it involves managing the lifecycles and health of the livestock to ensure a steady supply. Thus, it is crucial to conduct distinct investigations into inventory planning for growing items and general (non-growing) items. As a result, one of the big limitations of Khan et al. [20] is that it is not suitable for many general (non-growing) products. The present study tries to cover this limitation by determining the optimal pricing, inventory, and installment decisions for general products. Therefore, this research makes the first attempt to inspect the best prepayment installment strategy for a practitioner by examining the effects of variable prepayment installments on profit with demand as an additive combination of the effects of storage time in power pattern and price in linear form under strict tax regulations on released carbons. The following is a list of the study’s remaining sections: Section 3 clarifies the inventory problem definition, including the required notation and assumptions. The corresponding mixed-integer optimization problem is formulated in Section 4, while all the necessary analytical findings to arrive at the optimal stocking and installment plans are derived in Section 5. Section 6 deals with the clarification of the operational efficiency of the established algorithm by demonstrating five numerical examples. The consequences of fluctuations in system parameters are examined in Section 7, and several management insights are extracted in Section 8. Section 9 highlights the conclusion and salient findings of the work. 3. Problem definition Suppose a practitioner creates an order for a specific kind of item from an outside supplier by prepaying a particular proportion of the cost of the purchase over the course of a variable number of installments (single or multiple identical installments) from the time the order is placed to the time the order is delivered. The rest of the cost of the purchase is paid at the moment the practitioner receives the order, and then the inventory process commences at the practitioner’s storage, and the stored items are consumed through the consumption period. The clients’ behavior regarding items’ consumption may vary over the storage period such that inventory may fall rapidly at the initial stages or at the final stages of the storage period. To adopt this characteristic, the clients’ demand pattern is assumed to be an additive combination of the effects of storage time in power pattern and price in linear form. In addition, the practitioner holds items in stock bearing the cost, which is a power form of the storing period. If the prepayment is completed within a single installment, then the practitioner has a higher opportunity or capital cost. This capital cost decreases when the prepayment installment number also increases. On the other hand, each payment installment leads to transaction expenses for the practitioner (either by money transitioning from one bank account to another bank account or by the expense of transportation, labor, or time). Consequently, there is a trade-off between the transaction charge and capital expense for the practitioner with respect to the installment frequency. Furthermore, the practitioner incurs additional costs for the amount of carbon emissions in accordance with the environmental regulations (C&P, C&T, and CT) imposed by the authority / government because the practitioner’s various business activities release significant amounts of emissions into the surroundings. The major problem of this study is to address the following question: what are the most effective prepayment installment, price, and stock strategies for practitioners so that they may make the greatest profits possible in the shortest amount of time? To clarify the above inventory problem for the practitioner mathematically, the following notation and hypothesis are adopted: 3.1. Notation Parameters: ac Unit acquisition cost θ Prepayment proportion to the total cost of purchasing Lp Allowed lead time by the supplier ic Rate of interest u Maximum client demand rate, excluding the impacts of price and time v Price sensitivity factor in demand (>0) yh The greatest selling price per unit α The time-dependent demand’s scale factor with α ∈ ( − ∞,∞) ξ Index of power demand pattern with ξ>0 O Fixed ordering expense for every order, which is independent of purchased quantity units g0 Fixed storing cost for every unit g Time-varying holding cost parameter for every unit γ Index of time-varying cumulative holding expense tc Expense of transactions for each installment of payments O Quantity of carbon released during cycle setup procedures g0 Fixed quantity of carbon emissions per unit produced by lodging g Time-sensitive emission factor from holding operations ac The quantity of carbon emissions per unit of purchased goods tc The quantity of carbon emissions released during every single payment installment π Limit of the practitioner’s released carbons per unit of time p Carbon tariff per unit of emissions when the total amount of emissions exceeds π r Reward rate per unit of emissions when the total emitted carbon is below π Decision variables: y Price for every unit (y≥ac) τ Duration of each inventory cycle If Installment number for accomplishing prepayment (If∈ {1,2,3, ...}) Dependent-decision variables: Q Purchased quantity units for every cycle Functions: q(t)Available stock amount in the storage at t∈ [0, τ ] ω (t)Cumulative holding expenses to keep each unit at any time t ρ (y,t)Demand structure δ(y)Auxiliary function λ(y, τ ,If)Total carbon footprints per unit of time throughout one inventory cycle Y(y, τ , If) Practitioner’s profit per unit of time 3.2. Hypothesis (i) This study makes reference to a single practitioner’s inventory process for an indefinite planning horizon in which the practitioner sells a single item. (ii) The lead time is fixed and based on the quantity ordered. During the lead time period Lp units of time, the practitioner has to prepay the θ proportion of the total cost of purchasing using If identical multiple installments. Finally, the rest (1−θ)proportion of the total cost of purchasing is paid cash-on-delivery to receive all the goods. (iii) Owing to money transfer or shipping or manpower or time expenditure in prepayment installments, the practitioner bears the transaction charge for every installment. (iv) As the demand is deterministic, the stockout situation is not encountered by the practitioner during the inventory process. (v) To hold each unit in stock, the cumulative holding expense consists of two parts: (a) a fixed accommodating cost independent from storage time, and (b) a variable cost as a power function of the storage time t in stock [31]. This cost structure is assumed as ω (t) = g0+gtγ, where g0≥0, g>0 and γ≥1. (vi) Carbon emissions from all operations are included in the inventory coordination. As a result, the practitioner incurs an additional cost for the amount of carbon emitted in accordance with the carbon regulations (C&P, C&T, and CT) imposed by the authority / government. (vii) Clients’ demand ρ (y,t)fluctuates over the inventory cycle due to the resultant effect of unit selling price and stock age. Similar to Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 7 San-Jos´ e et al. [34], clients’ demand structure is assumed as ρ (y,t) = ρ 1(y) + ρ 2(t), where the first term, that is, the consequence of unit selling price, is given by ρ 1(y) = u−vy,with u>0,v>0 and ac≤y≤u v, and the second term, that is, the effect of inventory age, is defined by ρ 2(t) = ( α ξ)(t τ )1 ξ−1 ,with −∞< α <∞ and ξ>0. 4. Model formulation Without limiting the generality, let us assume the practitioner orders Q units of a single item prior to Lp units of time from receiving products under a prepayment regulation imposed by the supplier. In accordance with this regulation, the practitioner prepays θacQ amount of the total acquisition price acQusing If(≥ 1)equal installments and receives the order by cash-on-delivery of the residual (1−θ)acQ amount of the total cost of purchasing. Once the products are received, the main activities of the inventory procedure (storage and sale) begin (see Fig. 4). The consumption of inventories now starts with a beginning quantity of Q units, which decreases over time τ with the customer demand rate ρ (t,y)as follows: The time-sensitive component in demand, i.e., ( α ξ)(t τ )1 ξ−1, represents the consumers’ purchase decisions based on the inventory age. Customers’ purchasing patterns are categorized in one of three ways, depending on the storage duration of the inventories: (i) because of their current condition, some items (such as processed meals, fruits, meat with an expiry date, etc.) are in higher demand in the beginning than in the end, and this perception is represented by ξ>1; (ii) some items, such as potatoes, dates, and garlic, might be in greater demand later in the storage cycle because of their drying conditions, which allow consumers to consume them for longer, and this situation is reflected for 0 <ξ<1; and (iii) there are some goods (such as household items, electrical devices, tiles, curtains, etc.) whose demand is practically consistently steady throughout the cycle, and this feature is stated for ξ=1. Again, the price-sensitive component in demand, i.e., u−vy is a declining and affine function of y such that y≤yh where yh=u v is the greatest selling price per unit. In addition, the parameter u articulates the maximum client demand, excluding the impacts of price and time, and v is price sensitivity. It is noteworthy that the price-dependent portion of the demand function becomes constant, that is, independent of the effect of price, when v=0 and yh tends to infinity. Let q(t,y) = {q(t|y):0≤t≤ τ }be the item stocking process during an inventory period τ , where q denotes the amount of available stock at time t∈ [0, τ ]. As all the stored items are consumed at the demand rate ρ (t,y)during the inventory period τ , the purchased quantity amount for every cycle is precisely identical to the total demand during the period (0, τ ]. Therefore, Q=∫ τ 0 ρ (t,y)dt =∫ τ 0{u−vy +( α ξ)(t τ )1 ξ−1}dt = (u−vy + α ) τ (2) In addition, the available stock amount at t∈ [0, τ ]is described as follows: q ′ (t|y) = − {u−vy +( α ξ)(t τ )1 ξ−1},0<t< τ (3) with the auxiliary conditions q(0|y) = Q and q( τ |y) = 0. From Eq. (3), the available stock amount at time t∈ [0, τ ]is: q(t|y) = (u−vy)( τ −t) + ατ {1−(t τ )1 ξ},0≤t≤ τ .(4) Since the practitioner purchases Q units at the unit acquisition price ac and then sells them all at the unit selling price y, the practitioner’s sales revenue per unit of time (SR) and acquisition price per unit of time (AC) are y(u−vy + α )and ac(u−vy + α ), respectively. The practitioner prepays θacQ amount of the total cost of purchasing acQ by using If(≥ 1)identical multiple installments within the provided time duration LP units of time and receives the order by paying instantly the residual (1−θ)acQ amount of the total cost of purchasing. ConseFig. 4. Graphic presentation of multiple prepayment installments in an inventory procedure with a power demand. ρ (t,y) = u−vy +( α ξ)(t τ )1 ξ−1 ,with u>0,v>0,−∞< α <∞,ξ>0,and s∈[ac,u v](1) Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 8 quently, the prepayment amount incurs an additional opportunity or capital cost for the practitioner, and this capital cost (CC) per unit time is calculated from Fig. 4(a) as follows: ic τ [(θacQ If)Lp If(1+2+3+... +If)]=(If+1 2If)θicacLp(u−vy + α )(5) Every prepaid installment results in a transaction expense tc for the practitioner (due to money proceeding from one bank account to another bank account, transport, personnel, or time cost); therefore, the total transactional expense per unit of time (TC) is tcIf τ . The expense for holding operations per unit of time (HC) for the practitioner is 1 τ ∫ τ 0 (g0+gtγ) ρ (t,y)dt =g0(u−vy + α ) + g(u−vy γ+1+ α γξ +1) τ γ(6) Now, the practitioner’s total released emissions amount in a unit time is λ(y, τ ,If)= O τ +ac(u−vy + α ) +tcIf τ +g0(u−vy + α ) +g(u−vy γ+1+ α γξ +1) τ γ(7) The first and second terms in the expression of λ(y, τ ,If)indicate the released emissions amount in a unit time from setup and purchasing operations, respectively. When the prepayment installment process involves any vehicle, then the released emissions amount in a unit time from installment activity is given by the third term of λ(y, τ ,If); otherwise, tc is zero. Additionally, the fourth and fifth components show the amount of released emissions from holding actions per unit of time. Depending on the environmental regulations established by the authority or government, the practitioner may be fined or rewarded for their carbon footprint. Due to the fact that C&T and CT rules are specific examples of the C&P regulation, the practitioner’s profit is calculated using the C&P regulation in this study and expressed as follows: In Eq. (8), the term p(λ(y, τ ,If) − π )+=max{0,p(λ(y, τ ,If) − π )} indicates the tariff amount, whereas the term r( π −λ(y, τ ,If))+= max{0,r( π −λ(y, τ ,If))} denotes the reward amount for the practitioner. For the index c=p,r, let us define Note that Y(y, τ ,If)={Yc(y, τ ,If),if λ(y, τ ,If)≥ π Yr(y, τ ,If),if λ(y, τ ,If)≤ π (10) Finally, the described inventory practice generates the following mixed-integer maximization problem max (y, τ ,If)∈ΓYc(y, τ ,If),(11) where Γ≡{(y, τ ,If):ac≤y≤yh=u v, τ >0 and If∈ {1,2,3,...}} 5. Solution technique There are three decision variables y, τ , and If in the objective function Yc(y, τ ,If), where two of them (y and τ ) are continuous and the remaining one (If)is discontinuous over the positive integer set. Evidently, because the installation number must be a positive integer, the differential optimization approach cannot be used to determine the best I∗ f. In order to address the described mixed-integer optimization issue in Eq. (11), a Y(y, τ ,If)=[y−ac{1+(If+1 2If)θicLp}−g0](u−vy + α ) − O+tcIf τ −g(u−vy γ+1+ α γξ +1) τ γ −p(λ(y, τ ,If)− π )++r( π −λ(y, τ ,If))+ (8) Yc(y, τ ,If)=[y−ac{1+(If+1 2If)θicLp}−g0](u−vy + α ) − O+tcIf τ −g(u−vy γ+1+ α γξ +1) τ γ −c(λ(y, τ ,If)− π ) =[y−ac{1+(If+1 2If)θicLp}−g0−c(ac+g0)](u−vy + α ) −(O+c O) τ −(tc+ctc)If τ − (g+cg)(u−vy γ+1+ α γξ +1) τ γ+c π (9) Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 15 I∗ f=⌈1 2{−1+ 1+2acθicLp τ ∗ (tc+ctc)(u−vy∗+ α ) √}⌉, the best business cycle is τ ∗ =[{O+c O+ (tc+ctc)I∗ f}/{(g+cg)γ(u−vy∗ γ+1+ α γξ+1)}]1 γ+1, the ideal number of products to procure isQ∗= (u−vy∗+ α )[{O+c O+ (tc +ctc)I∗ f}/{(g+cg)γ(u−vy∗ γ+1+ α γξ+1)}]1 γ+1, and the practitioner’s maximum profit is Y∗ c=[y∗−ac{1+(I∗ f+1 2I∗ f)θicLp}−g0−c(ac+g0)](u−vy∗+ α ) − (γ+1){O+c O+(tc+ctc)I∗ f γ}γ γ+1{(g+cg)(u−vy∗ γ+1+ α γξ+1)}1 γ+1 +c π . Proof. The proof is similar to that of Proposition 3 . □ Corollary 4. (a) According to the C&T regulation, when carbon emissions are priced at p per unit, let (y∗, τ ∗,I∗ f) represent the optimal solution for the practitioner determined through the established algorithm. When y∗<yh and tc<acθicLp τ ∗ α 2−ptc, then the best payment installment number is I∗ f =⌈1 2{−1+ 1+2acθicLp τ ∗ (tc+ptc)(u−vy∗+ α ) √}⌉, the best business cycle is τ ∗=[{O+c O+ (tc+ptc) I∗ f}/{(g+pg)γ(u−vy∗ γ+1+ α γξ+1)}]1 γ+1, the ideal number of products to procure is Q∗= (u−v y∗+ α ) [{O+p O+ (tc+ptc)I∗ f}/{(g+pg)γ(u−vy∗ γ+1+ α γξ+1)}]1 γ+1, and the practitioner’s maximum profit is (b) According to the CT regulation, when carbon emissions are priced at p per unit, let (y∗, τ ∗,I∗ f)represent the optimal solution for the practitioner determined through the established algorithm. When y∗<yh and tc<acθicLp τ ∗ α 2−ptc, then the optimal strategies for the practitioner are identical to what the C&T rule requires. However, the practitioner’s maximum profit is Y∗ c=[y∗−ac{1+(I∗ f+1 2I∗ f)θicLp}−g0− p(ac+g0)](u−vy∗+ α ) − (γ+1){O+p O+(tc+ptc)I∗ f γ}γ γ+1 {(g+pg)(u−vy∗ γ+1+ α γξ+1)}1 γ+1. Proof. The proof is similar to that of Corollary 1. □ 5.2. Some particular cases The necessary conditions for incorporating the inventory model suggested in this work into previously examined models by others are outlined in this subsection. (i) The inventory model of San-Jos´ e et al. [34] is derived when the prepayment technique does not exist, any emission control guideline is not enforced, and no fixed lodging expense is included for holding operations purposes (i.e., θ=0, tc=0, p= r=0, and g0=0). (ii) If If is known, tc=0, p=r=0, v=0, α =0, ξ=1, g0=0, γ= 1, and y is fixed, then it is inferred that the model of Taleizadeh et al. [42] operates in the absence of shortages. (iii) The mathematical framework of Sicilia et al. [38] in a stockout situation is deduced if θ=0, tc=0, p=r=0, u=0, v=0, g0= 0, γ=1, and y is constant in the present study. Fig. 7. A visual representation of the practitioner’s profit is shown in comparison to If under Example 1. Table 2 Best business tactics and the practitioner’s greatest profit under various environmental rules for Example 1. Regulation π p r y∗ τ ∗I∗ f Q∗λ(y∗, τ ∗,I∗ f)Y(y∗, τ ∗,I∗ f) C&P 70 3 6 $89.9768/unit 2.795 months 4 111.8631 units 67.1913 tons $1823.506 C&T 70 3 3 $88.1719/unit 2.758 months 5 115.3629 units 69.5350 tons $1818.282 CT 0 3 NA $88.1719/unit 2.758 months 5 115.3629 units 69.5350 tons $1608.282 Y∗ c=[y∗−ac{1+(I∗ f+1 2I∗ f)θicLp}−g0−p(ac+g0)](u−vy∗+ α ) − (γ+1){O+p O+ (tc+ptc)I∗ f γ}γ γ+1{(g+pg)(u−vy∗ γ+1+ α γξ +1)}1 γ+1 +p π . Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 16 (iv) The current inventory approach becomes the Smith et al. [41] model when a linear consumption structure is used by adopting θ =0, tc=0, p=r=0, α =0, ξ=1, and g0 =0. (v) The inventory framework examined by Ferguson et al. [9] is obtained by considering θ=0, tc=0, p=r =0, u =0, v =0, and g0=0 in the present study. 6. Numerical findings and remarks This section deals with the clarification of the functional efficacy of the established algorithm by investigating several numerical examples. Since the algorithm offers five potential scenarios for the best-selling price by considering various parameter values, five numerical illustrations are adopted to elucidate the working efficacy of the implemented algorithm. Example 1. The parameters used in the current illustration are taken from Example 1 of the work by San-Jos´ e et al. [34], accompanied by the inclusion of all other essential data required for alignment with the aforementioned model, which are summarized as follows: O = $200/order, u=120units/month, v=1, α =10units/month, ξ =0.5, ac=$40/unit, γ=1.5, g0=$1/unit, g=$0.6/unit/month, Lp = 0.5month, θ=0.6, ic=$0.1/month, tc=$1.6/installment, O = 50tons/order, ac=0.4 ton/unit, tc=0.6 ton/installment, g0 = 0.5ton/unit, g=0.15 ton/unit/month, p=$3/ton, r =$6/ton, and π = 70tons/month. As a result, the greatest unit selling price for the practitioner is yh=u v=$120. Now, the policies for the practitioner are attained by employing the algorithm in the following way: Set j=1 and I(j) f=1. Iteration 1. Since 10 = α <u−v[ac{1+(If+1 2If)θicLp}+g0+r(ac+ g0)]=72.4, compute the point y0=u+ α 2v+1 2[ac{1+(If+1 2If)θicLp}+ g0+r(ac+g0)]=88.8. From Eq. (16), find the value Y ′ r(yh|If) = − 59.9707. As Y ′ r(yh|If)<0, determine y(1)=arg y∈(88.8,120) { Y ′ r(y|If) = 0} = 90.1803. Exploiting Eqs. (14) and (15), one finds τ (1)=2.7663 and Y(1)=1811.152, respectively. Since acθicLp τ (1) 2(u−vy(1)+ α ) − rtc= Table 3 Numerical results of Example 2. Iteration I(j) f y0 y1 y(j) τ (j)I(j+1) f Conditions j =1 1 111.2250 118.6076 120 8.1423 ⌈6.5408⌉.= 7 α <u−v[ac{1+(If+1 2If)θicLp} +g0+r(ac+g0)]=77.55, Y ′ r(yh|If) = 15.5984 >0, Y ′ r(y1|If) = 11.79188 >0, acθicLp τ (1) 2(u−vy(1)+ α ) − rtc=252.8818 >tc and I(1) f∕= I(2) f. j =2 7 111.0000 118.5176 120 8.1865 ⌈6.5598⌉.= 7 α <u−v[ac{1+(If+1 2If)θicLp} +g0+r(ac+g0)]=78, Y ′ r(yh|If) = 15.5093 >0, Y ′ r(y1|If) = 11.4887 >0, acθicLp τ (2) 2(u−vy(2)+ α ) − rtc=254.2743 >tc and I(2) f=I(3) f=7. Table 4 Best business tactics and the practitioner’s greatest profit under various environmental rules for Example 2. Regulation π p r y∗ τ ∗I∗ f Q∗λ(y∗, τ ∗,I∗ f)Y(y∗, τ ∗,I∗ f) C&P 74 3 6 $120/unit 8.1865 months 7 491.1892 units 72.4475 tons $4769.196 C&T 74 3 3 $120/unit 8.9658 months 9 537.9475 units 74.3647 tons $4767.101 CT 0 3 NA $120/unit 8.9658 months 9 537.9475 units 74.3647 tons $4545.101 Table 5 Numerical results of Example 3. Iteration I(j) f y0 y1 y(j) τ (j)I(j+1) f Conditions j =1 1 103.7272 114.4085 109.3755 4.7532 ⌈0.5777⌉.= 1 α <u−v[ac{1+(If+1 2If)θicLp} +g0+r(ac+g0)]=58.5456, Y ′ r(yh|If) = 12.4399 >0, Y ′ r(y1|If) = − 5.229<0, y=arg y∈(103.7272,114.4085) { Y ′ r(y|If) = 0} = 109.3755, 1787.194 = Yr(y|If)> Yr(yh|If) = 1759.781, acθicLp τ (1) 2(u−vy(1)+ α ) − rtc=1.1394 <tc and I(1) f=I(2) f=1. Fig. 8. A visual representation of the practitioner’s profit is shown in comparison to y and τ under Example 3. Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 17 62.4908 >1.6=tc, set I(2) f=⌈1 2{−1+ 1+2acθicLp τ (1) (tc+rtc)(u−vy(1)+ α ) √}⌉=⌈3.1⌉=4. As I(2) f∕= I(1) f, set j =2 and proceed to Step 3. Iteration 2. As 10 = α <u−v[ac{1+(If+1 2If)θicLp}+g0+r(ac+g0) ]=72.85, the point y0=u+ α 2v+1 2[ac{1+(If+1 2If)θicLp}+g0+r(ac+ g0)]=88.575. Since Y ′ r(yh|If) = − 57.3207 <0, find the unit selling price y(2)=arg y∈(88.575,120) { Y ′ r(y|If) = 0} = 89.9768. Moreover, from Eqs. (14) and (15), one has τ (2)=2.795 and Y(2)=1823.506, respectively. As acθicLp τ (2) 2(u−vy(2)+ α ) − rtc=63.5179 >1.6=tc, set I(3) f=⌈1 2{− 1+ 1+2acθicLp τ (2) (tc+rtc)(u−vy(2)+ α ) √}⌉=⌈3.1273⌉=4. Clearly, I(3) f= I(2) f, j>1 and Y(2)>Y(1). Therefore, (y∗, τ ∗,I∗ f,Y∗ c) = (89.9768,2.795,4, 1823.506). The optimal purchased quantity for the practitioner, according to Eq. (2), is Q∗=111.8631units. Within Fig. 6, the practitioner’s optimal profit point is shown as a distinct green dot, situated upon the profit function’s surface, which is contingent upon the continuous variables y and τ . It should be noted that the payment frequency is set at 4, which corresponds to the ideal value determined for Example 1. The practitioner’s profit is also plotted in Fig. 7 when the payment installment number ranges from 1 to 15. Moreover, Fig. 7 demonstrates the validity of the deduced mathematical findings in Theorem 3 and proves that the practitioner’s profit is strictly concave versus the payment installment number If for Example 1. Now the derived ideal decisions and highest profit for the practitioner in Example 1 are compared with those found in Example 1 of SanJos´ e et al. [34]. The optimal policies of Example 1 in San-Jos´ e et al. [34] are y∗=$85.6472/unit and τ ∗=2.11779months where the practitioner’s maximum profit is Y∗=$1867.18. Both the best unit price and business period for the practitioner in this study are higher than those from San-Jos´ e et al. [34]. Since the practitioner in this study bears several additional costs, such as capital or opportunity costs on the prepayment amount, transaction costs due to prepayment installments, and carbon emissions costs, the optimal unit selling price (y∗= $89.9768)is higher than the optimal selling price in Example 1 of San-Jos´ e et al. [34] in order to encounter these costs. As the unit selling price is higher, the client’s demand in this study is lower than the demand rate in Example 1 of San-Jos´ e et al. [34] and hence, the optimal business period of this example is also higher than that of Example 1 of San-Jos´ e et al. [34]. Although both y∗and τ ∗are higher, the practitioner’s highest profit of this example is lower than that of Example 1 of San-Jos´ e et al. [34] due to several additional costs in this study. However, this study helps practitioners figure out not only how to accomplish the prepayment in order to maximize their profit but also how to conduct their business through several carbon emission regulations that have not been studied by San-Jos´ e et al. [34]. Table 2 details how various environmental rules (C&P, C&T, and CT) affect the best business Fig. 9. A visual representation of the practitioner’s profit is shown in comparison to If under Example 3. Fig. 10. Graphic illustration of the function Y ′ (y|If)when If =1 under Example 3. Table 6 Best business tactics and the practitioner’s greatest profit under various environmental rules for Example 3. Regulation π p r y∗ τ ∗I∗ f Q∗λ(y∗, τ ∗,I∗ f)Y(y∗, τ ∗,I∗ f) C&P 74 3 6 $109.3755/unit 4.7532 months 1 174.0839 units 57.3372 tons $1787.194 C&T 74 3 3 $106.3614/unit 4.7715 months 1 189.1339 units 63.5458 tons $1746.479 CT 0 3 NA $106.3614/unit 4.7715 months 1 189.1339 units 63.5458 tons $1524.479 Table 7 Numerical outcomes of Example 4. Iteration I(j) f y0 y1 y(j) τ (j)I(j+1) f Conditions j =1 1 119.3673 148.5469 120 3.7386 ⌈0.5693⌉.= 1 α <u−v[ac{1+(If+1 2If)θicLp} +g0+p(ac+g0)]=81.2654, Y ′ p(yh|If) = 7.2575 >0, yh=120 ≤148.5469 = y1, acθicLp τ (1) 2(u−vy(1)+ α ) − ptc=3.3817 <tc and I(1) f=I(2) f=1. Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 18 tactics and the practitioner’s greatest profit for Example 1. The best business practices for the practitioner under C&P, C&T, and CT rules are compared in Table 2. Because the C&P policy permits the practitioner to earn more when the overall quantity of emissions is lower than the C&T or CT guidelines, it generates the practitioner’s highest profit among the three. According to the C&P rule, the reward for every ton of emissions that is released into the atmosphere is often greater than the penalty. Therefore, as is seen in Table 2, this policy encourages practitioners to produce fewer carbon emissions. Despite the fact that the C&T policy permits a reward for each unit of carbon that is released when the overall emission is less than the cap, the total carbon emission in Example 1 is higher than the C&T since both the tariff and incentive for each unit of carbon released are the same. With the aim of protecting the environment, the authority or government might approve the C&P program with a bigger reward than the tariff for each unit of emissions. The ideal price and business period under the C&P legislation are lower than those under the C&T and CT laws, but the ideal payment installment number and procured amount are larger under the C&P program. The practitioner’s profit under the C&T scheme is lower than that of the CT regulation, despite the fact that the practitioner’s combined best business tactics under the C&T and CT laws are identical. These findings support the conclusions drawn from the analysis in Proposition 4 and Corollary 4. According to the numerical findings in Example 1, the environmental guideline C&P is the optimal program when the reward is greater than the tariff for unit emissions, both for the practitioner (in regard to profit) and for the government (in regard to emissions amount). Example 2. With the following exceptions, this instance uses the identical values for the parameters as Example 1: O =$1600/order, α = 60units/month, ξ=25, ac=$35/unit, γ=2, and π =74 tons/month. Therefore, the maximum price for the practitioner is yh =u v =$120. Now, Table 3 presents the numerical results of the algorithm-based solution approach. Therefore, the ideal price, business period, and payment installment number for the practitioner are y∗=$120/unit, τ ∗= 8.1865months, and I∗ f=7 installments, respectively, where the practitioner’s highest profit is Y∗=$4769.196. Exploiting Eq. (2), the ideal number of products to procure is Q∗=491.1892units. Additionally, Table 4 provides a numerical assessment of the effects of C&P, C&T, and CT on the best business tactics and the practitioner’s highest profit within Example 2. The statistical data in Table 4 support the findings in Proposition 2 and Corollary 2. Example 3. With the following exceptions, this instance uses the identical values for the parameters as Example 2: O=$1000/order, α = 26units/month, ac=$55/unit, Lp=0.11month, θ=0.1, and ic= $0.09/month. The computational findings of the solution technique using the algorithm are shown in Table 5. Thus, the ideal price, business period, and payment installment number for the practitioner are y∗= $109.3755/unit, τ ∗=4.7532months, and I∗ f=1 installment, respectively, where the practitioner’s highest profit is Y∗=$1787.194. Exploiting Eq. (2), the ideal number of products to procure is Q∗= 174.0839units. In Fig. 8, the practitioner’s highest profit point is shown as a distinct green dot, situated upon the profit function’s surface, which is contingent upon the continuous variables y and τ . It should be noted that the payment frequency is set at 1, which corresponds to the ideal value determined for Example 3. Additionally, Fig. 9 is created by showing the practitioner’s profit when the payment installment number ranges from 1 to 10 and ensures, by graphic display, the validity of the deduced mathematical findings in Theorem 3 that the practitioner’s profit is strictly decreasing against If for Example 3. Fig. 10 reveals the behavior of the first-order derivative Y ′ r(y|If=1)with respect to the selling price per unit. More interestingly, in this case, there is an additional solution at y=$118.7187 of the equation Y ′ r(y|If=1) = 0. However, at y=$118.7187, the practitioner has a relative minimum profit as $1753.421 = Y(118.7187|If=1)< Y(yh|If=1) = $1759.781. In addition, Table 6 reveals a numerical assessment of the effects of C&P, C&T, and CT on the best business tactics and the practitioner’s highest profit within Example 3. The statistical data in Table 6 support the findings in Proposition 2 and Corollary 2. Example 4. With the following exceptions, this instance uses the identical values for the parameters as Example 2: α =80units/month, ac=$35/unit, ξ=2, Lp=0.11month, θ=0.1, ic=$0.09/month, and tc=$4/installment. The computational findings of the solution Table 8 Best business tactics and the practitioner’s greatest profit under various environmental rules for Example 4. Regulation π p r y∗ τ ∗I∗ f Q∗λ(y∗, τ ∗,I∗ f)Y(y∗, τ ∗,I∗ f) C&P 74 3 6 $120/unit 3.7386 months 1 299.09 units 119.08 tons $6018.771 C&T 74 3 3 $120/unit 3.7386 months 1 299.09 units 119.08 tons $6018.771 CT 0 3 NA $120/unit 3.7386 months 1 299.09 units 119.08 tons $5796.771 Table 9 Numerical results of Example 5. Iteration I(j) f y(j) τ (j)I(j+1) f Conditions j =1 1 46.1539 4.3424 ⌈1.7936⌉.=2 α >u−v[ac{1+(If+1 2If)θicLp} +g0+r(ac+g0)]= − 3.76, acθicLp τ (1) 2(u−vy(1)+ α ) − rtc=22.4543 >tc and I(1) f∕= I(2) f. j =2 2 46.1539 4.3602 ⌈1.798⌉.=2 α >u−v[ac{1+(If+1 2If)θicLp} +g0+r(ac+g0)]= − 2.98, acθicLp τ (2) 2(u−vy(2)+ α ) − rtc=22.5612 >tc and I(2) f=I(3) f=2. Table 10 Best business tactics and the practitioner’s greatest profit under various environmental rules for Example 5. Regulation π p r y∗ τ ∗I∗ f Q∗λ(y∗, τ ∗,I∗ f)Y(y∗, τ ∗,I∗ f) C&P 70 3 6 $46.1539/unit 4.3602 months 2 43.602 units 28.5465 tons $213.4404 C&T 70 3 3 $46.1539/unit 4.3744 months 3 43.7442 units 28.6837 tons $89.3012 CT 0 3 NA $46.1539/unit 4.3744 months 3 43.7442 units 28.6837 tons -$120.6988 Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 19 Table 11 Influences of u, v, α and ξ on the practitioner’s ideal business tactics and highest profit. v α u=100 u=120 u=140 u=160 y∗ τ ∗I∗ f Y∗y∗ τ ∗I∗ f Y∗y∗ τ ∗I∗ f Y∗y∗ τ ∗I∗ f Y∗ ξ=0.5 0.5 5 130.0198 2.8518 4 3429.605 148.0938 2.6068 5 5257.858 168.0066 2.4323 5 7496.894 187.9391 2.2929 5 10,137.46 10 133.1524 2.7207 5 3838.282 153.0503 2.5204 5 5776.307 172.9733 2.3640 5 8116.104 192.9126 2.2370 5 10,857.27 15 138.1011 2.6210 5 4281.225 158.0121 2.4434 5 6320.136 177.9434 2.3020 5 8760.608 197.8886 2.1857 5 11,602.31 20 143.0566 2.5331 5 4749.612 162.9782 2.3740 5 6889.298 182.9165 2.2453 5 9430.374 202.8666 2.1382 5 12,372.56 0.75 5 95.1690 3.0450 4 1749.494 106.4910 2.7307 5 2787.379 119.7206 2.5276 5 4125.318 132.9759 2.3694 5 5731.720 10 98.3879 2.8974 4 1975.700 109.7721 2.6295 5 3092.778 123.0152 2.4497 5 4498.282 136.2791 2.3068 5 6172.034 15 101.6267 2.7726 4 2219.539 113.0603 2.5404 5 3415.295 126.3142 2.3796 5 4888.251 139.5853 2.2497 5 6629.280 20 103.1230 2.6346 4 2482.838 116.3541 2.4611 5 3754.870 129.6169 2.3161 5 5295.188 142.8940 2.1972 5 7103.433 1 5 77.8875 3.2866 3 990.5486 87.5753 2.9244 4 1636.091 95.6098 2.6381 5 2506.418 105.5183 2.4560 5 3595.181 10 80.2318 3.0929 3 1125.454 89.9768 2.7950 4 1823.506 98.0640 2.5478 5 2755.980 107.9836 2.3852 5 3895.540 15 82.6084 2.9350 3 1274.145 90.6175 2.6531 5 2029.809 100.5240 2.4676 5 3018.440 110.4526 2.3211 5 4208.703 20 85.0076 2.8027 3 1436.370 93.0707 2.5610 5 2254.305 102.9886 2.3956 5 3293.749 112.9247 2.2626 5 4534.639 1.25 5 67.6908 3.6486 3 602.2129 75.2608 3.1295 3 1013.158 81.1775 2.7687 5 1589.062 89.0678 2.5554 5 2366.669 10 69.4565 3.3707 3 680.9471 77.1310 2.9642 3 1135.362 83.1221 2.6621 5 1764.292 91.0273 2.4741 5 2582.846 15 71.2836 3.1580 3 771.5472 79.0257 2.8267 3 1268.650 85.0746 2.5687 5 1949.999 92.9915 2.4014 5 2809.376 20 73.1492 2.9875 3 873.5618 80.9209 2.7201 4 1412.856 87.0332 2.4860 5 2146.117 94.9596 2.3357 5 3046.221 ξ=1 0.5 5 130.0663 2.9127 4 3435.895 148.1161 2.6503 5 5261.693 168.0233 2.4662 5 7500.343 187.9523 2.3204 5 10,140.62 10 133.2076 2.8259 5 3846.591 153.0903 2.5998 5 5783.678 173.0039 2.4268 5 8122.776 192.9370 2.2884 5 10,863.39 15 138.1744 2.7628 5 4293.137 158.0664 2.5525 5 6330.787 177.9856 2.3894 5 8770.302 197.9165 2.2663 6 11,611.26 20 143.1440 2.7045 5 4764.842 163.0441 2.5080 5 6903.011 182.9685 2.3540 5 9442.918 202.9026 2.2371 6 12,384.22 0.75 5 95.2306 3.1229 4 1756.456 106.5180 2.7824 5 2791.498 119.7402 2.5668 5 4128.976 132.9910 2.4004 5 5735.035 10 98.4927 3.0326 4 1988.818 109.8200 2.7224 5 3100.654 123.0507 2.5213 5 4505.333 136.3068 2.3644 5 6178.458 15 101.7626 2.9511 4 2238.182 113.1246 2.6669 5 3426.627 126.3628 2.4786 5 4898.466 139.6236 2.3302 5 6638.632 20 105.0389 2.8769 4 2504.515 116.4314 2.6151 5 3769.404 129.6763 2.4384 5 5308.369 142.9414 2.2977 5 7115.553 1 5 77.9746 3.3925 3 998.3917 87.6276 2.9919 4 1642.631 95.6333 2.6838 5 2510.324 105.5358 2.4913 5 3598.682 10 80.3740 3.2701 3 1140.027 90.0670 2.9137 4 1835.897 98.1061 2.6308 5 2763.477 108.0155 2.4503 5 3902.305 15 82.7871 3.1624 3 1294.629 90.6953 2.8025 5 2041.958 100.5809 2.5812 5 3029.260 110.4964 2.4116 5 4218.525 20 85.1953 3.0783 4 1462.185 93.1629 2.7409 5 2269.814 103.0575 2.5349 5 3307.665 112.9786 2.3749 5 4547.338 1.25 5 67.8316 3.8106 3 611.4657 75.3320 3.2182 3 1020.426 81.2063 2.8234 5 1593.269 89.0885 2.5962 5 2370.389 10 69.6697 3.6241 3 697.7270 77.2341 3.1273 4 1148.983 83.1729 2.7599 5 1772.323 91.0645 2.5488 5 2590.007 15 71.5375 3.4680 3 794.7192 79.1615 3.0354 4 1287.997 85.1424 2.7013 5 1961.536 93.0422 2.5043 5 2819.741 20 73.4267 3.3345 3 902.3164 81.0971 2.9526 4 1437.354 87.1143 2.6469 5 2160.895 95.0215 2.4624 5 3059.582 ξ=2 0.5 5 130.1104 2.9699 4 3441.569 148.1368 2.6906 5 5265.129 168.0388 2.4973 5 7503.422 187.9643 2.3455 5 10,143.42 10 135.0953 2.9504 4 3859.284 153.1298 2.6769 5 5790.425 173.0335 2.4868 5 8128.839 192.9544 2.3456 6 10,868.95 15 138.2537 2.9123 5 4304.417 158.1229 2.6635 5 6340.730 178.0284 2.4764 5 8779.263 197.9504 2.3372 6 11,619.51 20 143.2440 2.8942 5 4779.568 163.1161 2.6503 5 6916.043 183.0233 2.4662 5 9454.693 202.9464 2.3290 6 12,395.07 0.75 5 95.2901 3.1973 4 1762.774 106.5434 2.8305 5 2795.200 119.7584 2.6027 5 4132.248 133.0048 2.4287 5 5737.989 10 98.6030 3.1718 4 2001.101 109.8680 2.8141 5 3107.905 123.0856 2.5905 5 4511.767 136.3335 2.4191 5 6184.282 15 101.9165 3.1470 4 2256.120 113.1929 2.7981 5 3437.288 126.4128 2.5785 5 4907.959 139.6622 2.4097 5 6647.247 20 105.2306 3.1229 4 2527.831 116.5180 2.7824 5 3783.348 129.7402 2.5667 5 5320.826 142.9853 2.4092 6 7126.926 1 5 78.0613 3.4963 3 1005.580 87.6776 3.0559 5 1648.546 95.6553 2.7262 5 2513.828 105.5519 2.5237 5 3601.809 10 80.5311 3.4604 3 1153.898 88.2910 2.9813 5 1834.647 98.1478 2.7118 5 2770.351 108.0465 2.5128 5 3908.460 15 82.9881 3.4389 4 1314.837 90.7803 2.9616 5 2053.503 100.6405 2.6977 5 3039.383 110.5411 2.5020 5 4227.618 20 85.4603 3.4054 4 1488.338 93.2699 2.9423 5 2284.871 103.1333 2.6838 5 3320.924 113.0358 2.4913 5 4559.282 1.25 5 67.9804 3.9782 3 620.1044 75.4016 3.3039 3 1027.049 83.0820 2.9332 4 1602.586 89.1077 2.6339 5 2373.719 10 69.9276 3.9191 3 714.1623 77.3626 3.2868 4 1161.891 85.0674 2.9142 4 1777.836 91.1011 2.6211 5 2596.551 15 71.8781 3.8634 3 818.2840 79.3393 3.2581 4 1306.796 85.2151 2.8400 5 1972.420 93.0947 2.6085 5 2829.391 20 73.8190 3.8248 4 932.4896 81.3168 3.2303 4 1461.730 87.2063 2.8234 5 2175.119 95.0885 2.5962 5 3072.239 Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 20 technique are presented in Table 7 using the solution algorithm. As a result, the ideal price, business period, and payment installment number for the practitioner are y∗=$120/unit, τ ∗=3.7386months, and I∗ f=1 installment, respectively, while the practitioner’s highest profit is Y∗= $6018.771. Exploiting Eq. (2), the ideal number of products to procure is Q∗=299.09units. Additionally, Table 8 presents a numerical assessment of the effects of C&P, C&T, and CT on the ideal business tactics and the practitioner’s highest profit for Example 4. Because the tariff for unit emissions is identical and the overall emissions exceed the cap, the best practices for the practitioner in this instance are identical under all of the aforementioned environmental guidelines. The statistical data in Table 8 support the findings in Proposition 1 and Corollary 1. Example 5. With the exception of the value of v in demand, this instance takes the identical parameter values as in Example 1 into account. The new value of the price-sensitive parameter (v)is 2.6. The computational findings of the solution technique, which makes use of the algorithm, are shown in Table 9. Therefore, the ideal price, business period, and payment installment number for the practitioner are y∗= $46.1539/unit, τ ∗=4.3602months, and I∗ f=2 installments, respectively, while the practitioner’s highest profit is Y∗=$213.4404. Exploiting Eq. (2), the ideal number of products to procure is Q∗= 43.602units. Table 10 reports that this inventory problem is unprofitable under CT regulation, while both carbon C&P and C&T policies help the practitioner to make the system profitable by allowing rewards from the remaining emission amounts of the cap. 7. Sensitivity investigation The section deals with characterizing how the ideal payment installment, price, and stock decisions for the practitioner to maximize profit change from fluctuations in system parameters under the C&P environmental guideline. 7.1. Resulting from the demand parameters (u, v, α , and ξ) An exploration is conducted in this subsection to investigate the consequences of u, v, α , and ξ on the ideal business tactics and the highest profit of the practitioner in accordance with the computational results presented in Table 11 when u∈ {100,120,140,160}, v∈ {0.5, 0.75,1,1.25}, α ∈ {5,10,15,20}and ξ∈ {0.5,1,2}in Example 1. In addition, Table 11 displays the computational outcomes of another 191 new inventory examples, and these outcomes help to bring out the following behavioral properties of the demand parameters on the practitioner’s profit and ideal business decisions as well. If the highest possible market demand (u)increases under fixed values of the price sensitive parameter(v), scale parameter ( α ), and the power index (ξ), the demand always increases, and therefore all the items are consumed sooner; that is, the optimal storage duration ( τ ∗) always decreases. Because client demand is high, the practitioner sets a high price (y∗)when the maximum potential market size (u)is high. Due to higher client demand, the inventory planner orders a higher lot-size and hence, the ideal payment installment strategy (I∗ f)exhibits an ascending trend in reducing the entire capital expense from the prepayment sum. Since a high demand helps to sell all products quickly, the entire holding cost shrinks significantly for a high potential market size (u)and therefore, the practitioner’s maximum profit (Y∗)increases precipitously. To increase the profit, an important take-home measurement from the above findings is that the practitioner should adopt some effective marketing strategies (such as advertising through different media, allowing discount facilities, among others) to raise the potential market demand. The price-sensitive coefficient (v)in demand has a negative way consequence on the ideal price and payment installment policies when all remaining demand parameters are held constant. As an increment in Table 12 Influences of carbon policy parameters p and r on the practitioner’s ideal business tactics and highest for Example 1. Cap-and-price policy Cap-and-trade policy Carbon tax policy p r y∗ τ ∗I∗ f Y∗λ(y∗, τ ∗,I∗ f)p r y∗ τ ∗I∗ f Y∗λ(y∗, τ ∗,I∗ f)p y∗ τ ∗I∗ f Y∗λ(y∗, τ ∗,I∗ f) 3 4 88.7776 2.7636 4 1819.306 68.6054 3 3 88.1719 2.7580 5 1818.282 69.5350 3 88.1719 2.7580 5 1608.282 69.5350 3 6 89.9768 2.7950 4 1823.506 67.1913 3 3 88.1719 2.7580 5 1818.282 69.5350 3 88.1719 2.7580 5 1608.282 69.5350 3 8 91.2012 2.8160 3 1830.880 65.5296 3 3 88.1719 2.7580 5 1818.282 69.5350 3 88.1719 2.7580 5 1608.282 69.5350 4 3 88.1719 2.7580 5 1818.282 69.5350 4 4 88.7776 2.7636 4 1819.306 68.6054 4 88.7776 2.7636 4 1539.306 68.6054 6 3 88.1719 2.7580 5 1818.282 69.5350 6 6 89.9768 2.7950 4 1823.506 67.1913 6 89.9768 2.7950 4 1823.506 67.1913 8 3 88.1719 2.7580 5 1818.282 69.5350 8 8 91.2012 2.8160 3 1830.880 65.5296 8 91.2012 2.8160 3 1270.880 65.5296 2 6 89.9768 2.7950 4 1823.506 67.1913 2 2 87.5773 2.7419 5 1818.166 70.2342 2 87.5773 2.7419 5 1678.166 70.2342 4 6 89.9768 2.7950 4 1823.506 67.1913 4 4 88.7776 2.7636 4 1819.306 68.6054 4 88.7776 2.7636 4 1539.306 68.6054 8 6 89.9768 2.7950 4 1823.506 67.1913 8 8 91.2012 2.8160 3 1830.880 65.5296 8 91.2012 2.8160 3 1270.880 65.5296 6 4 88.7776 2.7636 4 1819.306 68.6054 6 6 89.9768 2.7950 4 1823.506 67.1913 6 89.9768 2.7950 4 1823.506 67.1913 6 8 91.2012 2.8160 3 1830.880 65.5296 6 6 89.9768 2.7950 4 1823.506 67.1913 6 89.9768 2.7950 4 1823.506 67.1913 Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 21 the price-sensitive coefficient parameter implies a precipitous drop in demand, the optimal replenishment period ( τ ∗)increases significantly when the value of v increases. Conversely, the best purchased quantity by the practitioner shrinks, so the optimal payment installment number (I∗ f)exhibits a downward trend. Consequently, for higher values of v, the overall holding expense for the practitioner increases markedly, and therefore, the maximum profit exhibits an upward trend. Similar to the potential market demand parameter (u), the scale parameter ( α )of power demand affects the optimal pricing and installment policies in a positive way when all remaining demand parameters are held constant. If α in power demand rises, the consequences of time on demand are intensified, and hence, the stored items are consumed with a higher demand, and the optimal replenishment duration shrinks. To meet a higher client demand, the practitioner increases the purchased quantity and sets a higher unit selling price. With the ambition to lessen the capital expense, the practitioner increases the prepayment installment frequency if the value of α increases. Since a higher value of α helps to sell all products quickly, the total holding cost shrinks significantly, and therefore, the practitioner’s maximum profit (Y∗)increases noticeably. The power index (ξ)is a great influencer on the behavior of the optimal strategies of the practitioner. When ξ∈ (0,1), the stored items are consumed with low demand at the beginning of the business cycle and, therefore, a high lot-size increases the whole holding expense markedly. Thus, the practitioner purchases a relatively small lot-size so that all products are consumed within a short period of time. At the same time, for a small lot-size, the practitioner also uses a small number of installment frequencies to accomplish the prepayment condition. To diminish the total holding expense, the practitioner sets a relatively small unit price. A summary of the salient findings from the discussion above is that if the demand is low at the beginning of the business period, then the practitioner should procure relatively a small lot-size through a small number of installment frequencies, and then set a relatively small unit selling price to increase profits. On the other hand, when ξ∈ (1,∞), the stored items are consumed with a high demand at the beginning of the business period, and therefore, the practitioner orders a relatively large lot-size for each cycle, and then uses a higher prepayment installment frequency to shrink the capital expense against the advance payment sum. As the demand is high at the beginning of each business period, the practitioner increases the unit selling price to increase profits. From Table 11, it is observed that when u=100units/ month, v=1.25 and α =20units/month, then for the indices of power demand ξ=0.5, ξ=1, and ξ=2, the optimal pricing policies are: y∗= $73.1492/unit, y∗=$73.4267/unit, and y∗=$73.8190/unit, respectively; the optimal replenishment policies are: τ ∗=2.9875 months, τ ∗= 3.3345 months, and τ ∗=3.8248 months, respectively; the ideal payment installment policies are: I∗ f=3 installments, I∗ f=3 installments, and I∗ f=4 installments, respectively; the highest profits are: Y∗= $873.5618, Y∗=$902.3164, and Y∗=$932.4896, respectively. Integrating the above analysis, in order to maximize profit, the notable observation is that the practitioner ought to procure a comparatively large lot size, adopt a greater frequency of payment installments, and then establish a relatively high unit price if the demand stays high at the beginning phases of the business process. 7.2. Significances of the tariff p and the reward r against each unit emission Under the cap-and-price, cap-and-trade, and carbon tax regulations, Table 12 is constructed to explore the consequences of the penalty p and the reward r for each unit carbon emission on the practitioner’s best strategies, maximum profit, and the total discharged carbons. The practitioner does not release lower emissions using the tariff p than the amount of carbon emissions using the reward r when the tariff p is not higher than the reward r against every emission unit. On the Table 13 Influences of O, ac, g0, g, tc, ic and γ on the practitioner’s ideal business tactics and highest profit for Example 1. Parameter value γ=1 γ=1.5 γ=2 γ=2.5 y∗ τ ∗I∗ f Y∗y∗ τ ∗I∗ f Y∗y∗ τ ∗I∗ f Y∗y∗ τ ∗I∗ f Y∗ O 100 89.9232 3.5951 4 1900.117 89.8217 2.5502 4 1860.852 87.9250 1.9999 4 1841.506 87.8496 1.7693 4 1830.732 200 90.0775 4.0067 4 1873.806 89.9768 2.7950 4 1823.506 88.0944 2.2290 4 1794.256 88.0128 1.9415 4 1776.891 300 90.2098 4.3995 5 1849.971 90.1360 3.0027 4 1789.016 88.2496 2.4197 4 1751.254 88.1643 2.0830 4 1727.223 400 90.3393 4.7450 5 1828.100 90.2858 3.1919 4 1756.733 88.3945 2.5851 4 1711.306 88.3072 2.2044 4 1680.592 ac 20 77.9659 3.5368 4 2808.552 77.8930 2.5258 3 2775.308 77.8070 2.0799 3 2753.352 77.7304 1.8309 3 2737.819 30 83.0992 3.7304 5 2304.773 83.0290 2.6340 4 2270.541 82.9423 2.1529 4 2247.421 82.8773 1.8804 3 2230.912 50 95.2595 4.2919 5 1493.619 95.1796 2.9381 4 1442.253 95.0761 2.3462 3 1406.153 94.9531 2.0267 3 1379.963 60 100.4724 4.6198 5 1166.470 100.4093 3.1118 4 1114.132 100.2831 2.4663 4 1076.718 100.1828 2.1082 3 1049.129 g0 0.5 89.8231 3.9949 4 1893.831 89.7220 2.7886 4 1843.581 87.8413 2.2250 4 1815.272 87.7599 1.9386 4 1797.947 0.75 89.9503 4.0008 4 1883.803 89.8494 2.7917 4 1833.528 87.9679 2.2270 4 1804.748 87.8863 1.9400 4 1787.403 1.25 90.2047 4.0126 4 1863.841 90.1042 2.7982 4 1813.516 88.2210 2.2310 4 1783.795 88.1393 1.9430 4 1766.410 1.5 90.3319 4.0185 4 1853.908 90.2317 2.8014 4 1803.558 88.3476 2.2330 4 1773.366 88.2658 1.9445 4 1755.960 g 0.3 89.9079 4.4931 5 1901.319 89.8550 3.0525 4 1850.039 88.0012 2.4919 4 1820.225 87.9398 2.1365 4 1800.928 0.45 89.9991 4.2196 4 1887.153 89.9179 2.9137 4 1836.326 88.0504 2.3457 4 1806.513 87.9785 2.0285 4 1788.189 0.75 90.1522 3.8235 4 1861.123 90.0324 2.6918 4 1811.441 88.1345 2.1325 4 1783.120 88.0439 1.8692 4 1766.695 0.9 90.2237 3.6638 4 1849.014 90.0851 2.6009 4 1800.023 88.1715 2.0510 4 1772.880 88.0722 1.8076 4 1757.375 tc 0.8 90.0640 4.0106 5 1874.705 89.9715 2.7879 4 1824.652 88.0783 2.2275 5 1795.782 88.0078 1.9365 4 1778.541 1.2 90.0669 4.0185 5 1874.206 89.9742 2.7915 4 1824.079 88.0919 2.2256 4 1794.974 88.0103 1.9390 4 1777.715 2 90.0798 4.0129 4 1873.407 89.9957 2.7868 3 1822.939 88.0970 2.2323 4 1793.538 88.0153 1.9440 4 1776.067 2.4 90.0822 4.0192 4 1873.009 89.9977 2.7895 3 1822.508 88.0996 2.2356 4 1792.822 88.0366 1.9402 3 1775.307 ic 0.05 89.8918 3.9781 3 1889.114 89.7899 2.7791 3 1839.413 87.9118 2.2191 3 1810.483 87.8304 1.9342 3 1793.372 0.075 89.9821 4.0022 4 1881.300 89.8918 2.7816 3 1831.381 87.9995 2.2274 4 1802.122 87.9316 1.9353 3 1784.949 0.125 90.1615 4.0305 5 1866.530 90.0724 2.7974 4 1816.011 88.1759 2.2372 5 1786.453 88.1077 1.9426 4 1769.027 0.15 90.2531 4.0348 5 1859.367 90.1679 2.7998 4 1808.533 88.2670 2.2386 5 1778.933 88.1852 1.9487 5 1761.315 Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 22 contrary, when the reward r is smaller than the tariff p, the released carbon amounts under the reward r are not lower than the released emissions under the tariff p. When the reward rate r is higher than or equal to the penalty rate p, the practitioner amends the operations so that the released carbons are reduced. In this situation, the practitioner reduces the prepayment installment frequency (I∗ f)to shrink released emissions, though the capital cost increases against the prepayment amount. To manage this higher capital cost, the practitioner increases the ideal unit price (y∗), and hence, the best business period ( τ ∗)also increases when the award rate r increases. Due to a high reward rate r, the practitioner growths the total profit by reducing the total released emissions. Consequently, the C&P environmental guideline ensures a higher or equal profit for the practitioner than the C&T program when the reward rate r is greater than or equal to the tariff rate p. Furthermore, an important take-home measurement for the carbon scheme regulator (authority or government) is that a reward rate allowance greater than the tariff rate against unit released carbon helps to reduce released carbons markedly. As a result, from both economic and environmental perspectives, a C&P mechanism performs better with a greater reward rate than the penalty rate against every emission unit. The practitioner’s greatest profit according to the C&T mechanism, on the other hand, will be greater than or equal to the highest profit according to the C&P mechanism if the penalty rate p is greater than the incentive rate ragainst unit emission. In this instance, under the C&T regulation, the practitioner reduces the prepayment installment frequency and increases the unit selling price. As a result, the C&T policy provides a higher or equal profit than the C&P policy when p >r. Thus, from both economic and environmental perspectives, a carbon C&T mechanism works better when a C&P regulation is imposed with p >r. As the carbon tax mechanism does not include any cap on carbon release, the practitioner does not have any opportunity to benefit by emitting a lesser amount of carbon from the operational activities. Moreover, the practitioner pays tax for every released emission unit, and therefore, the CT mechanism provides the lowest maximum profit for the practitioner among the considered regulations. It is worth noting that the optimal strategies for the practitioner according to the CT and C&P mechanisms are identical, as the CT is a particular form of the C&T regulation with the zero cap. 7.3. Consequences of O, ac, g0, g, tc, ic, and γ Table 13 highlights that when the ordering expense (O)for a single cycle increases with fixed holding cost elasticity (γ), the practitioner enlarges the purchased quantity, and hence, the ideal payment installment number (I∗ f)and business cycle ( τ ∗)strategies reveal an upward trend. Though the best unit selling price (y∗)increases, the practitioner’s maximum profit falls when O increases under fixed γ. However, when the holding expense elasticity (γ)increases with any fixed ordering cost (O), the practitioner dwindles the purchased quantity, and hence, the ideal payment installment number (I∗ f), price (y∗), and business cycle ( τ ∗)strategies reveal a downward trend. Thus, the practitioner’s highest profit falls when γ increases under fixed O. This investigation articulates that the practitioner should purchase a relatively small lot-size for a short business cycle to increase profit when the ordering expense (O)for a single cycle is low. If the unit acquisition cost (ac)increases with fixed holding cost elasticity (γ), the practitioner sets a higher selling price for each unit (y∗), and therefore, the client demand falls markedly. This is the main reason why the optimal business cycle ( τ ∗)shrinks when the unit acquisition cost (ac)increases. At the same time, a downward trend is identified on the ideal payment installment strategy (I∗ f)and the maximum profit of the practitioner. Thus, the manufacturer should try to purchase products from the supplier by negotiating the unit acquisition cost (ac)and then set a lower unit price to increase demand. The fixed accommodation cost for a single unit (g0)does not play any significant role in the ideal payment installment tactics (I∗ f)of the practitioner. Nevertheless, the best unit price (y∗)increases gradually, and therefore, the optimal business cycle also increases slowly when the fixed accommodating cost (g0)increases. Since a high fixed accommodating cost for a single unit (g0)is a cause to reduce the demand, and hence, increases the total holding cost, the practitioner’s maximum profit falls slightly if the fixed accommodating cost for a single unit (g0) increases. Consequently, to increase profits, the practitioner should reduce the fixed accommodation cost for a single unit by amending the activities appropriately. Under a fixed holding cost elasticity (γ), the best business cycle ( τ ∗) reduces as the time-sensitive holding cost parameter (g)increases. To manage the increased holding cost due to a higher value of g, the best unit price (y∗)increases slightly. However, the maximum profit for the practitioner falls when g increases. Moreover, if the holding cost elasticity (γ)under a fixed g, the best business cycle ( τ ∗)for the practitioner shrinks significantly as the purchased lot-size decreases to reduce the whole holding expense. A slight decrease is recognized on the pricing strategy of the practitioner when the time sensitive holding cost parameter (g)increases. Interestingly, the optimal prepayment installment number almost remains stable with changes in the time-sensitive holding cost parameter (g). An insignificant consequence on the practitioner’s pricing and replenishment cycle strategies is observed when the transaction charge for every payment installment (tc)increases with fixed holding cost elasticity (γ). However, the ideal payment installment number (I∗ f) shows an upward trend when tc increases with fixed γ. In addition, if the holding expense elasticity (γ)increases with stable tc, then the best price (y∗), and business period ( τ ∗)fall significantly. Thus, the maximum profit of the practitioner dwindles when tc increases with fixed γ. Though the interest charged rate (ic)does not greatly influence the practitioner’s best pricing (y∗), and business cycle ( τ ∗)decisions, it does stimulate the practitioner to increase the frequency of optimal prepayment installments (I∗ f). Since any prepayment amount over a long period at a high interest rate incurs high capital costs, the practitioner makes an effort to make all the payments in the allotted time in a number of modest installments. An important finding from the research is that when the interest charged rate is high, the practitioner ought to raise the payment installment numbers to reduce the capital expenditure from the prepayment. Table 14 Impacts of emission parameters O, ac, tc, g0, and g on the practitioner’s ideal business tactics and highest profit for Example 1. Parameter Value y∗ τ ∗I∗ f Y∗λ(y∗, τ ∗,I∗ f) O 30 89.7855 2.4995 3 1868.773 59.4666 40 89.8758 2.6590 4 1845.507 63.6110 60 88.2189 2.8472 5 1807.578 73.0090 70 88.2646 2.9326 5 1797.197 76.3216 ac 0.2 89.3653 2.7797 4 1871.901 59.7861 0.3 89.6710 2.7873 4 1847.612 63.5195 0.5 90.2826 2.8027 4 1799.584 70.8016 0.6 90.5885 2.8105 4 1775.845 74.3502 tc 0.4 89.9689 2.7844 4 1825.227 66.9141 0.5 89.9728 2.7897 4 1824.366 67.0529 0.7 89.9967 2.7881 3 1822.723 67.0607 0.8 89.9996 2.7921 3 1822.078 67.1645 g0 0.3 89.3653 2.7797 4 1871.901 59.7861 0.4 89.6710 2.7873 4 1847.612 63.5195 0.6 88.3238 2.7617 5 1805.756 73.5238 0.7 88.4757 2.7653 5 1793.276 77.4822 g 0.11 89.8807 2.9942 4 1844.436 63.7308 0.13 89.9300 2.8885 4 1833.695 65.5270 0.17 90.0215 2.7113 4 1813.799 68.7450 0.19 88.2245 2.6326 4 1808.509 72.4551 Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 23 7.4. Consequences of carbon emission parameters O, ac, tc, g0, and g Among the carbon emission parameters, the amount of released emissions during cycle setup procedures ( O), quantity of carbon emissions from the activity to purchase a unit item (ac), and fixed quantity of released emissions from per unit accommodation activity (g0)have notable impacts on the profit. When these three carbon amounts ( O, ac, and g0) are reduced, the practitioner’s profit increases markedly. Therefore, the practitioner should try to reduce the released emissions by appropriately modifying the setup, purchasing, and accommodation activities. The practitioner’s optimal business cycle ( τ ∗)experiences a large fluctuation when the quantity of carbon released during cycle setup procedures ( O)and time-sensitive emissions factor of holding operations (g)vary. Table 14 reveals that in order to increase profits, the practitioner should increase the purchased quantity when the quantity of carbon released during cycle setup procedures ( O)increases, while the purchased quantity should be reduced when the time-sensitive emissions factor of holding operations (g)increases. 8. Managerial insights Using the key outcomes from the previous section as a foundation, the subsequent insights for the practitioner are developed: •When the demand is low at the beginning of the business period, the practitioner should procure a relatively small lot-size through a small number of installment frequencies and then set a relatively small unit selling price to increase profits. However, the practitioner ought to procure a comparatively large lot size by adopting a greater frequency of payment installments and then establish a relatively high unit price if the demand stays high in the beginning phases of the business process. •A reward rate allowance greater than the tariff rate per unit released carbon helps to reduce released carbons significantly. As a result, from both economic and environmental perspectives, a C&P mechanism performs better with a greater reward rate than the penalty rate against every emission unit. •The practitioner should purchase a relatively small lot-size for a short business cycle to increase profit when the ordering expense (O)for a single cycle is low. •The inventory manager should try to purchase products from the supplier by negotiating the unit acquisition cost (ac)and then setting a lower unit price to increase demand. •When the interest charged rate is high, the practitioner ought to raise the payment installment numbers to reduce the capital expenditure against prepayment. •The practitioner should increase the purchased quantity when the quantity of carbon released during cycle setup procedures ( O)increases, while the purchased quantity should be reduced when the time-sensitive emission factor of holding operations (g)increases. 9. Conclusions This study examines the significance of environmental regulations (cap-and-price, cap-and-trade, and carbon tax) on the profitability of a practitioner dealing with a particular item under an advance payment mechanism. Taking into account the transaction fee against payment installment(s) and the capital expense against prepayment amount, the ideal installment policy of the practitioner is determined analytically. The stored items’ consumption rate is jointly connected with unit price and time, more precisely, in the addition form of a linear-price function and a power-time function. An algorithm is designed to obtain the optimal combined payment installment, price, and stock tactics by ensuring the greatest profit for the practitioner through incorporating all the developed analytical findings. By studying five numerical illustrations, the efficacy of the algorithm is subsequently established. Observing the altering patterns of the best business tactics due to variations in the system parameters, salient management insights for the practitioner are extracted. For instance, an important take-home measurement is that if the demand is low in the beginning phases of the business period, then the practitioner should procure a relatively small lot-size through a small number of installment frequencies and then set a relatively small unit selling price to increase profits. From both economic and environmental perspectives, a C&P environmental mechanism performs better with a greater incentive rate than the tariff rate against every released emission unit. The practitioner should purchase a relatively small lot-size for a short business cycle to increase profit when the ordering expense for a single cycle is low. Another salient insight from the analysis is that the practitioner should increase the prepayment installment numbers to lessen the capital expense against prepayment when the interest charged rate is high. By appropriately modifying the setup, purchasing, and accommodation operations to reduce the released emissions, the practitioner’s profit increases markedly. The current study possesses several remarkable benefits. It helps the practitioner to make the best installment decision to accomplish the prepayment to lessen the capital expense and increase profits. Moreover, this work helps to assess the efficiency of different environmental mechanisms in terms of both economic and environmental perspectives. However, the discrete feature of the prepayment installment frequency results in a fairly complex solution approach, which is one downside of this work. The increasing uncertainty of supply sometimes leads to a stockout situation for the practitioner but is not investigated when modeling the inventory procedure. Thus, allowing shortages during modeling would be worth noting in future research. Exploring investment plans to lower carbon footprints is another way to go further into this study. Declaration of Competing Interest a. The authors declare that there is not conflict of interest. b. This manuscript is the authors’ original work and has not been published nor has it been submitted simultaneously elsewhere. c. All authors have checked the manuscript and have agreed to the submission. Data availability The data is included in the paper References [1] Alegoz M, Kaya O, Bayindir ZP. A comparison of pure manufacturing and hybrid manufacturing–remanufacturing systems under carbon tax policy. Eur J Oper Res 2021;294(1):161–73. [2] Alfares HK, Ghaithan AM. Inventory and pricing model with price-dependent demand, time-varying holding cost, and quantity discounts. Comput Ind Eng 2016; 94:170–7. [3] Avinadav T, Herbon A, Spiegel U. Optimal ordering and pricing policy for demand functions that are separable into price and inventory age. Int J Prod Econ 2014; 155:406–17. [4] C´ ardenas-Barr´ on LE, Mandal B, Sicilia J, San-Jos´ e LA, Abdul-Jalbar B. Optimizing price, order quantity, and backordering level using a nonlinear holding cost and a power demand pattern. Comput Oper Res 2021;133:105339. [5] Chen X, Benjaafar S, Elomri A. The carbon-constrained EOQ. Operat Res Lett 2013; 41(2):172–9. [6] Chen X, Gong W, Wang F. Managing carbon footprints under the trade credit. Sustainability 2017;9(7):1235. [7] Duary A, Das S, Arif MG, Abualnaja KM, Khan MAA, Zakarya M, Shaikh AA. Advance and delay in payments with the price-discount inventory model for deteriorating items under capacity constraint and partially backlogged shortages. Alexandria Eng J 2022;61(2):1735–45. [8] Edalatpour MA, Mirzapour Al-e-Hashem SMJ. Simultaneous pricing and inventory decisions for substitute and complementary items with nonlinear holding cost. Prod Eng 2019;13(3):305–15. Md.A.-A. Khan et al.
Operations Research Perspectives 12 (2024) 100289 24 [9] 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):485–90. [10] Hovelaque V, Bironneau L. The carbon-constrained EOQ model with carbon emission dependent demand. Int J Prod Econ 2015;164:285–91. [11] Hua G, Cheng TCE, Wang S. Managing carbon footprints in inventory management. Int J Prod Econ 2011;132(2):178–85. [12] Jiang ZZ, He N, Xiao L, Sheng Y. Government subsidy provision in biomass energy supply chains. Enterprise Info Sys 2019;13(10):1367–91. [13] Keshavarzfard R, Makui A, Tavakkoli-Moghaddam R, Taleizadeh AA. Optimization of imperfect economic manufacturing models with a power demand rate dependent production rate. S¯ adhan¯ a 2019;44(9):1–19. [14] Khalilpourazari S, Pasandideh SHR. Multi-item EOQ model with nonlinear unit holding cost and partial backordering: moth-flame optimization algorithm. J Indus Product Eng 2017;34(1):42–51. [15] Khan MAA, Shaikh AA, Panda GC, Konstantaras I, Taleizadeh AA. Inventory system with expiration date: pricing and replenishment decisions. Comput Ind Eng 2019; 132:232–47. [16] Khan MAA, Shaikh AA, Konstantaras I, Bhunia AK, C´ ardenas-Barr´ on LE. Inventory models for perishable items with advanced payment, linearly time-dependent holding cost and demand dependent on advertisement and selling price. Int J Prod Econ 2020;230:107804. [17] Khan MAA, Shaikh AA, C´ ardenas-Barr´ on LE. An inventory model under linked-toorder hybrid partial advance payment, partial credit policy, all-units discount and partial backlogging with capacity constraint. Omega 2021;103:102418. [18] Khan MAA, C´ ardenas-Barr´ on LE, Trevi˜ no-Garza G, C´ espedes-Mota A. Optimal circular economy index policy in a production system with carbon emissions. Expert Syst Appl 2023;212:118684. [19] Khan MAA, C´ ardenas-Barr´ on LE, Trevi˜ no-Garza G, C´ espedes-Mota A. A prepayment installment decision support framework in an inventory system with all-units discount against link-to-order prepayment under power demand pattern. Expert Syst Appl 2023;213:119247. [20] Khan MAA, C´ ardenas-Barr´ on LE, Trevi˜ no-Garza G, C´ espedes-Mota A, de Jesús, Loera-Hern´ andez I. Integrating prepayment installment, pricing and replenishment decisions for growing items with power demand pattern and non-linear holding cost under carbon regulations. Comput Oper Res 2023;156:106225. [21] Lee JY. Investing in carbon emissions reduction in the EOQ model. J Oper Res Soc 2020;71(8):1289–300. [22] Li Z, Hai J. Inventory management for one warehouse multi-retailer systems with carbon emission costs. Comput Ind Eng 2019;130:565–74. [23] Manna AK, Khan MAA, Rahman MS, Shaikh AA, Bhunia AK. Interval valued demand and prepayment-based inventory model for perishable items via parametric approach of interval and meta-heuristic algorithms. Knowl Based Syst 2022;242:108343. [24] Naddor E. Inventory systems. New York: John Wiley; 1966. [25] Paknejad J, Nasri F, Affisco JF. Shape of power yield distribution: impact on EOQ model with nonlinear holding cost and random quality. Int J Manage Sci Eng Manage 2018;13(4):237–44. [26] Pando V, García-Laguna J, San-Jos´ e 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):2160–71. [27] Pando V, San-Jos´ e 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. [28] Pando V, San-Jos´ e LA, Sicilia J. Profitability ratio maximization in an inventory model with stock-dependent demand rate and non-linear holding cost. Appl Math Model 2019;66:643–61. [29] Rahman MS, Khan MAA, Halim MA, Nofal TA, Shaikh AA, Mahmoud EE. Hybrid price and stock dependent inventory model for perishable goods with advance payment related discount facilities under preservation technology. Alexandria Eng J 2021;60(3):3455–65. [30] Ries JM, Grosse EH, Fichtinger J. Environmental impact of warehousing: a scenario analysis for the United States. Int J Prod Res 2017;55(21):6485–99. [31] San-Jos´ e LA, Sicilia J, García-Laguna J. Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost. Omega 2015;54:147–57. [32] San-Jos´ e LA, Sicilia J, Gonz´ alez-De-la-Rosa M, Febles-Acosta J. Optimal inventory policy under power demand pattern and partial backlogging. Appl Math Model 2017;46:618–30. [33] San-Jos´ e LA, Sicilia J, Alcaide-L´ opez-de-Pablo D. An inventory system with demand dependent on both time and price assuming backlogged shortages. Eur J Oper Res 2018;270(3):889–97. [34] San-Jos´ e LA, Sicilia J, C´ ardenas-Barr´ on LE, Guti´ errez JM. Optimal price and quantity under power demand pattern and non-linear holding cost. Comput Ind Eng 2019;129:426–34. [35] San-Jos´ e LA, Sicilia J, Gonz´ alez-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(1):351–69. [36] San-Jos´ e LA, Sicilia J, Abdul-Jalbar B. Optimal policy for an inventory system with demand dependent on price, time and frequency of advertisement. Comput Oper Res 2021;128:105169. [37] San-Jos´ e LA, Sicilia J, Gonz´ alez-de-la-Rosa M, Febles-Acosta J. Optimal price and lot size for an EOQ model with full backordering under power price and time dependent demand. Mathematics 2021;9(16):1848. [38] Sicilia J, Febles-Acosta J, Gonz´ alez-De La Rosa M. Deterministic inventory systems with power demand pattern. Asia-Pacific J Operation Res 2012;29(05):1250025. [39] Sicilia J, Gonz´ alez-De-la-Rosa M, Febles-Acosta J, Alcaide-L´ opez-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: 163–71. [40] Shaikh AA, Khan MAA, Panda GC, Konstantaras I. Price discount facility in an EOQ model for deteriorating items with stock-dependent demand and partial backlogging. Int Transact Operation Res 2019;26(4):1365–95. [41] Smith NR, Martinez-Flores JL, C´ ardenas-Barr´ on LE. Analysis of the benefits of joint price and order quantity optimisation using a deterministic profit maximisation model. Product Planning Control 2007;18(4):310–8. [42] Taleizadeh AA, Pentico DW, Jabalameli MS, Aryanezhad M. An economic order quantity model with multiple partial prepayments and partial backordering. Math Comput Model 2013;57(3–4):311–23. [43] Taleizadeh AA. An EOQ model with partial backordering and advance payments for an evaporating item. Int J Prod Econ 2014;155:185–93. [44] Toptal A, ¨ Ozlü H, Konur D. Joint decisions on inventory replenishment and emission reduction investment under different emission regulations. Int J Prod Res 2014;52(1):243–69. [45] Weiss HJ. Economic order quantity models with nonlinear holding costs. Eur J Oper Res 1982;9(1):56–60. [46] World Economic Forum. Supply chain decarbonization. Geneva: World Economic Forum; 2009. [47] Xu C, Liu X, Wu C, Yuan B. Optimal inventory control strategies for deteriorating items with a general time-varying demand under carbon emission regulations. Energies 2020;13(4):999. [48] Zhang Q, Zhang D, Tsao YC, Luo J. Optimal ordering policy in a two-stage supply chain with advance payment for stable supply capacity. Int J Prod Econ 2016;177: 34–43. [49] Zia NP, Taleizadeh AA. A lot-sizing model with backordering under hybrid linkedto-order multiple advance payments and delayed payment. Transport Res Part E: Logist Transport. Rev. 2015;82:19–37. Md.A.-A. Khan et al.