Sustainable Ordering Policies with Capacity Constraint Under Order-Size-Dependent Trade Credit, All-Units Discount, Carbon Emission, and Partial Backordering

In today's competitive business situation, the supplier frequently offers his or her retailers a permissible delay period to stimulate sales. In addition, the capacity of any warehouse is limited in practice; thus, the retailer needs an additional rented warehouse to store the excess units when the order quantity exceeds the capacity of the own warehouse. Furthermore, with the globalization of the marketing policy, the supplier may provide the retailer a discounted price if the quantity of purchase is large enough. Green inventory management reduces the environmental impacts of a business without lacking its profit. Considering all of the factors mentioned above, in this paper, we study a green economic order quantity model with capacity constraint under order-size-dependent trade credit and all-units discount along with minimizing carbon for a cleaner environment. The paper discusses all the potential cases, which may occur in green inventory models with carbon emission cost under different allowable delay in payments. Shortages are allowed and partially backordered. The main objective is to determine the sustainable optimal ordering strategies for retailers and decide whether a rented warehouse (RW) is to obtain ordering and replenishment policies for a retailer such that the retailer’s annual profit is maximized. First, we prove that the conditions of the objective functions have interior minimizer value and the closed-form optimal solution is found. Next, an algorithm is developed to determining the global optimal solution of the problem. Finally, some management insights are drawn by observing the applicability of the developed algorithm and also by performing sensitivity analyses on different parameters.


Introduction
Providing a reasonable and efficient way to manage green inventory management and carbon emission control is one of the most challenging activities that business organizations face, which plays a critical role for the success of an organization in a competitive market. To be a part of this competitive green trade market, it is essential to provide the end-product at economical cost following with less carbon output. Efficient and green inventory management plays a vital part in saving environment from hazardous carbon emissions and maxi-mizes the overall profit. The elevated level of global warming due to the carbon emission in industrial activities is the foremost challenge for researchers and organizations. In modern inventory management models incorporation of carbon emission cost along with trade credit policy brings a new paradigm for the success of an efficient inventory system. It has been seen that in recent years, academia and industry have not much focused on the approach that drives towards environmental protection. Only 10% of organizations actively model their inventory system on the basis of clean environmental activities (Hua et al. 2011). Additionally, to attain an environmental and economic bottom line of sustainability, business players have to collaborate for financial support and improve their processes to control carbon emissions. Also, from a profitability point of view, an elevated level of inventory may cause less profit for the company. In contrast, holding less amount of inventory makes stock-out conditions, which may result in the letdown of potential customers.
In 1913, Harris proposed the classical economic order quantity (EOQ) model, which initiated the study of modern inventory theory. However, the classical EOQ model makes many simplified assumptions about the real situation, such as the retailer must pay immediately when the items are received and shortages are not allowed and the retailer's warehouse has unlimited capacity, which makes it difficult to be applied to the real business situation. In reality, these assumptions are usually not valid, especially in today's complex market environment. Therefore, scholars have devoted a lot of efforts to relaxing these assumptions in order to develop a more realistic inventory model. A basic assumption of classical EOQ model is that retailers need to pay the full purchase amount as soon as the items are received. However, with the increasing market competition, the supplier usually offers his or her buyers a permissible delay period as an outstanding promotional to boost sales and to reduce on hand stock level. Specifically, the trade credit refers to the supplier gives the retailer a grace period to pay for the purchase amount. Usually, there is no interest charge if the retailer pays off the entire purchase amount within the grace period. However, if the payment is not paid in full by the end of the grace period, then the supplier charges the retailer interest on the outstanding amount. Obviously, trade credit reduces the retailer's inventory cost and attracts the retailer who may view this as a type of price reduction. Moreover, trade credit is regarded as an important shortterm financing method among enterprises which is widely used in various business activities. It is reported that more than 80% of firms in the UK and the USA offer their products on various credit terms to increase sales or remain competitive (Seifert et al. 2013). More recently, Seifert et al. (2017) further studied a sample of 3383 groups of public US firms and concluded that company profitability is positively associated with payment delay. Therefore, it is of great practical significance to incorporate the concept of trade credit into inventory model. Goyal (1985) was the first to study EOQ model under the condition of permissible delay in payment. Later, Aggarwal and Jaggi (1995) extended Goyal's (1985) model to consider the deterministic inventory model with a constant deterioration rate. Jamal et al. (1997) further extended Aggarwal and Jaggi's (1995) model to allow for shortages, which makes the inventory model more applicable in practice. Furthermore, Huang (2003), Teng and Goyal (2007) and Mahata and Goswami (2007) developed an EOQ model in which the supplier offers the retailer an upstream credit period M, while the retailer grants a downstream credit period N to customers. Subsequently, Teng et al. (2012) discussed an EOQ model with trade credit financing under a nondecreasing demand pattern. Chen and Teng (2015) and Mahata et al. (2020) obtained the optimal ordering and credit period decisions for time-varying deteriorating product under twolevel trade credit by using discounted cash flow analysis. Wu et al. (2018) developed the optimal inventory policies for perishable products with two-level trade credit financing under trapezoidal-type demand patterns. Mahata (2015) and Feng and Chan (2019) investigated joint pricing and production decisions for new products under two-level trade credit with learning curve effects. Lashgari et al. (2018) established an ordering policy for non-instantaneous deteriorating items under hybrid partial prepayment, partial trade credit, and partial backordering. Li et al. (2019) discussed optimal pricing, lot sizing, and backordering decisions when a seller demands an advance-cash-credit payment scheme. All the inventory models discussed above assume that the length of the trade credit period is a fixed value and independent of the retailer's order quantity.
However, in practice, the supplier usually provides the retailer a trade credit depending on the retailer's order quantity. In this regard, Chang et al. (2003) and Mahata (2012) established an EOQ model with deteriorating items where the supplier offers a trade credit to the retailer only if the order quantity is greater than or equal to a specified threshold. Huang (2007) extended the model of Chang et al. (2003) to consider partial trade credit if the order quantity is lower than a specific quantity. Ouyang et al. (2009a) proposed an EOQ model for deteriorating items under partially permissible delay in payments which was considered to be linked-to-order quantity as well. Jaggi et al. (2018) obtained credit policies for deteriorating imperfect quality items with exponentially increasing demand and partial backlogging. Chung and Liao (2009) proposed optimal ordering policy of EOQ model under trade credit depending on ordering quantity from the DCF approach. Zia and Taleizadeh (2015) developed a lot-sizing model with backordering under hybrid linked-to-order multiple advance payments and delayed payment. Shah and Cárdenas-Barrón (2015) established retailer's decision for ordering and credit policies for deteriorating items when a supplier offers order-linked credit period or cash discount. Tiwari et al. (2018a, b, c) discussed retailer's optimal ordering policy for deteriorating items under order-size-dependent trade credit and complete backlogging. Recently, Taleizadeh et al. (2019) considered partial linked-to-order delayed payment and lifetime effects on decaying items ordering policy. Clearly, there are two shortcomings in order-size-dependent trade credit terms schedule based on the only one order quantity threshold: (1) The supplier needs to fully grasp all kinds of information of the retailer (including market demand information, warehouse capacity information, cost structure information) to set an appropriate order quantity threshold. It is very difficult, if not impossible, to achieve in today's highly competitive market environment, especially when the market is the buyer's market.
(2) For the retailer, the trade credit with a single order quantity threshold forces the retailer to make two extreme choices: to enjoy delay in payment by making the order quantity greater than or equal to the predetermined quantity or to pay the full purchase amount immediately when the order quantity is less than the predetermined quantity. Therefore, in order to reduce the difficulty of the supplier decisionmaking and meanwhile increase the retailer's choice, more flexible trade credit terms based on different trade credit periods and different quantities thresholds emerge. However, to the best of our knowledge, few scholars have paid attention to this kind of trade credit terms so far except Ouyang et al. (2008Ouyang et al. ( , 2009bOuyang et al. ( , 2015 and Chang et al. (2015). They considered order-size-dependent delay in payments in their model formulation. Therefore, with more adaptation with the real world, this study would introduce this type of trade credit terms schedule in the development of specific inventory model.
Clearly, when the length of the trade credit period is linked to the order quantity rather than a given parameter, the retailer is encouraged to order more products to enjoy a longer trade credit period. But if the retailer's own warehouse (OW) capacity is insufficient to store all the purchased units, the retailer needs a rented warehouse (RW) to store the excess units. Consequently, the assumption of single warehouse in classical EOQ model is no longer valid. The development of a twowarehouse inventory model will be more in line with the real business environment. Hartley (1976) first established a twowarehouse inventory model in which shortages are not allowed. Subsequently, Sarma (1987) extended Hartley's (1976) model to consider perishable products in which shortages are allowed. Zhou and Yang (2005) obtained the optimal ordering policies for a two-warehouse inventory system with stock-level-dependent demand rate. Agrawal et al. (2013) developed a two-warehouse inventory model for deteriorating items with ramp type demand and partially backordering. Recently, Shaikh et al. (2019a) established a two-warehouse inventory model for deteriorating item with partial backlogged shortages and interval-valued inventory cost. Other researchers who have studied in this topic include Lee and Hsu (2009), Ghiami et al. (2013), Xu et al. (2017), Tiwari et al. (2018a, b, c), and Khan et al. (2019a). But the influence of trade credit policy on optimal solution is not considered in the abovementioned papers. Huang (2006) first developed a two-warehouse inventory model with two-level trade credit policy. Liang and Zhou (2011) established an inventory model with two warehouses and permissible delay in payments, which assumed that the RW has a higher deterioration rate than the OW. Yang and Chang (2013) described a twowarehouse partial backlogged inventory model for deteriorating items under inflation when a delay in payment is permissible. Liao et al. (2012Liao et al. ( , 2013 developed a deterministic inventory model for deteriorating items with two warehouses and trade credit in a supply chain system. Tiwari et al. (2016) worked on two-warehouse inventory systems for non-instantaneous deteriorating items with trade credit and inflation. Jaggi et al. (2017) explored credit financing in economic ordering policies for non-instantaneous deteriorating items with price-dependent demand and two storage facilities. Chakraborty et al. (2018) established a twowarehouse partial backlogging inventory model for deteriorating items with ramp-type demand rate, inflation, and permissible delay in payments. Recently, Panda et al. (2019) investigated a two-warehouse inventory model for deteriorating item under trade credit, where the demand rate is dependent on price, stock, and frequency of advertisement.
Also, carbon emissions are due to the outcome of storing products, transporting, and warehousing. The emission from the warehouse is subject to the entire inventory, along with the consumption of energy per unit item in the warehouse. Sustainable and green inventory framework consists of balancing regional and global efforts to fulfill buyer requirements without distressing the environment (Ahmed and Sarkar 2018). The correlation due to the environmental influence of products involves a liability designed for sustainable practice. Dye and Yang (2015) considered sustainability issues on inventory decisions and joint trade credit policy under carbon emission constraints. Kazemi et al. (2016) proposed the inventory model having the impact of carbon emissions in an imperfect process. Tayyab and Sarkar (2016) explained a sustainable multi-stage manufacturing study having unplanned imperfect rate and decided the optimum shipment amount. Ahmed and Sarkar (2018) investigated the economic policy of a supply chain having the effect of uncertain parameters and carbon emission tax. Li et al. (2018) investigated the crucial findings concerning the influence of feature control on green expansion and the governing prospect of eco-friendly guiding principle on this connection. Tiwari et al. (2018a, b, c) proposed a green production system for multi-item under trade credit and partial backordering under the framework of sustainable inventory model. Gautam et al. (2019) presented strategic defect management for a sustainable green supply chain. Khanna et al. (2020) explained the effect of carbon tax and cap-and-trade mechanism on an inventory system with price-sensitive demand and preservation technology investment. Yadav et al. (2021) proposed a sustainable Inventory model for perishable products with expiration date and price reliant demand under carbon tax policy and also Yadav et al. (2021) discussed the impact of energy and carbon emission of a supply chain management with a two-level trade credit policy. Mishra et al. (2021) discussed on optimum sustainable inventory management with backorder and deterioration under controllable carbon emissions.
Also, in the present-day competitive market dynamics, quantity discount is another important penetrations and marketing strategy for the supplier to run their business successfully expect for trade credit discount (Khan et al. 2020). Several types of single-item quantity discount approaches are used in practice and have been discussed in the literature, among which all-units discount is most widely used in the practical business environment. Specific literature on all-units discounts, we recommend that readers refer to literatures of Taleizadeh and Pentico (2014), Alfares and Ghaithan (2016), Shaikh et al. (2019b), and Khan et al. (2019b). In addition, to illustrate the contribution of this study, we make a comparison between this study and previous studies in Table 1.

Research Gap and Our Contribution
From the literature review and Table 1, it clearly shows that no inventory model developed in previous studies has simultaneously considered sustainable inventory management with capacity constraint, order-size dependent trade credit, all-units discount, and partial backordering. In addition, integration of cost of carbon emission in the inventory system will help firms to emphasize on curtailing the total inventory the warehouse, which will benefit to minimize carbon cost in holding items; also, total holding cost per unit item will reduce. Combining these factors, in this paper, we derive the retailer's optimal ordering policies in sustainable inventory system that considers the following features: (1) The retailer's OW capacity is limited, which means that if the retailer's order quantity surpasses his/ her OW capacity, an additional RW is needed to store the excess units; (2) the retailer receives an order-size-dependent trade credit and an all-units discount from the supplier that also depends on the order size; (3) shortages are allowed and partial backordering; and (4) carbon emission is considered a negative feature for inventory in the warehouse, as is the carbon that is emitted from obsolete materials. Hence, in the present paper, we have generalized many existing literatures, such as Goyal (1985), Taleizadeh and Pentico (2014), and Ouyang et al. (2015). Next, we proved the existence of the optimal solution of the objective function, and then the closed-form optimal solution was found. After that, we designed an algorithm to find the global optimal solution of the problem in an integrated manner. Finally, some numerical examples are presented to illustrate theoretical results, and managerial insights are given.
The remainder of this paper is organized as follows. In Sections 2 and 3, the notations and assumptions used throughout this paper are provided. In Section 4, the mathematical model to maximize the retailer's annual profit is formulated. In Section 5, the model is analyzed, and the closed-form optimal solutions are derived. In addition, an algorithm of the solution procedure is presented. In Section 6, some numerical examples are presented to show the applicability of the model, and in Section 7, sensitivity analysis of major parameters is performed. Finally, conclusions and future research are discussed in Section 8.

Assumptions
The following assumptions are used in formulating the model: a) The inventory system involves only one item over an infinite planning horizon. b) Lead time is zero and replenishment rate is infinite. c) It is a sustainable inventory model where carbon emission cost is incorporated and demand rate is known and constant. d) Shortages are allowed and partially backordered, and the fraction of shortages is backordered at a constant rate β. e) The OW has limited capacity of W units. When the order quantity Q > W, the retailer needs to rent an additional warehouse to hold the excess units. In addition, we assume that the RW has an unlimited capacity. Moreover, in practice, the RW usually offers better preserving facilities than the OW; thus, this paper uses the relationship h r ≥ h 0 to reflect this situation. f) Because the holding cost in RW is higher than those in OW, it is cost-effective to consume the RW first. g) In actual commercial operations, the larger the retailer's order quantity, the lower the purchase price. Here, we assume that the supplier offers an all-units quantity discount to the retailer. The purchase cost is a decreasing step function of the order size Q: : : : : : where 1 = η 1 ≤ η 2 ≤ …… ≤ η λ ≤ η λ + 1 = ∞, each of which represents a boundary quantity. c ε denotes the unit purchase cost applicable to orders whose lot size Q falls in the interval h) The supplier offers a credit period N m , m = 1, 2, …, μ, which is also related to the retailer's order quantity, and the relationship is given as follows: ϑ 2 ≤ Q ≤ϑ 3 : : : : : : where 1 = ϑ 1 ≤ ϑ 2 ≤ …. . ≤ ϑ μ ≤ ϑ μ + 1 = ∞, each of which is a boundary values at which a specific credit period is offered. N m denotes the credit period applicable to orders whose lot size Q falls in the interval ϑ m to ϑ m + 1 with From assumptions (g) and (h), the retailer is presented with an order-size dependent trade credit schedule and an all-units quantity discount schedule. For convenience, we now combine the two discount schedules into a restructured new discount schedule. Rearrange boundary values η 1 , η 2 , …. η λ and ϑ 1 , ϑ 2 , …. . , ϑ μ in the order of small to large to form a new set of q 1 ≤ q 2 ≤ … ≤ q k . Then, there only exists a unique combination purchase cost c j and credit period M j applicable to the lot size falling in the interval q j to q J + 1 . The restructured discount schedule becomes . , k, the purchase cost and the length of credit period offered by supplier are c j and M j respectively, j) To curb the amount of carbon emissions, following Benjaafar et al. (2013) and Xiang and Lawley (2018), we assume that the buyer is charged a fixed dollar amount τ for every ton of emissions produced (i.e., a carbon tax regulation) by the local government. k) During the credit period, the account is not settled, and the retailer sells the items and uses the sales revenues to earn interest at a rate of I e . At the end of the credit period, the retailer pays off all units bought and starts to pay for the interest charges on the items remaining in stock with at a rate of I c .

Model Formulation
Given the notations and assumptions above, we know that the retailer's replenishment cycle time is T and the duration of the positive inventory level is KT; shortages are allowed and partially backordered. At time t = 0, the retailer receives Q units that are utilized to fulfil the total accumulated backlogged demand and consumers' demand during the time interval [0, KT]. The inventory model is constructed to maximize the retailer's annual profit. Without loss of generality, we assume that the retailer's order quantity is q j ≤ Q < q j + 1 , the purchase cost offered by supplier is c j , and the length of credit period granted by supplier is M j . Clearly, the retailer's annual profit is composed of sales revenue, ordering cost, purchasing cost, holding cost, back ordering cost, opportunity cost due to lost sales, interest charged, and interest earned. These components are computed as follows: (1) Annual sales revenue (SR): (2) Annual ordering cost (OC): OC = A/T where A is replenishment cost for replenishing the items.
(3) Annual purchasing cost (PC): (4) Annual holding cost: If the retailer's order quantity is Q ≤ W (i.e., T ≤ T w ), then he or she does not need a RW. Otherwise, the retailer must rent an extra warehouse to hold the excess units. Hence, the annual holding cost (HC) of green inventory excluding the interest charge per unit time is as follows: where h r and h o are the stock holding cost per unit time in RW and OW respectively.
(6) Annual opportunity cost due to lost sales (OCL): Benjaafar et al. (2013) and Xiang and Lawley (2018), the total amount of carbon emissions per replenishment cycle includes the fixed carbon emissions associated with placing an order (e.g., carbon emissions due to transportation and production) A ′ , the variable amount of carbon emissions associated with each unit (e.g., carbon emissions due to the handling of each unit) c ′ multiplied by the order quantity Q, and the integration of the amount of carbon emissions associated with the storage in RW and OW of each unit held per unit of time h 0 r and h 0 o (e.g., carbon emissions involved such as refrigeration in the storage of each unit in RW and OW) multiplied by inventory level I(t) throughout the replenishment cycle. Therefore, the total amount of carbon emissions per replenishment cycle is given by (8) Annual interest charged and interest earned: Based on the values of KT and M j , there are two possible situations, which are depicted in Fig. 1. We will discuss them separately.
Situation 1: KT ≤ M j (j = 1, 2, …k) In this situation, the retailer's trade credit period M j is longer than or equal to the positive inventory level length KT (see Fig. 1a). It indicates that the retailer has sold all the stock at the time M j . Therefore, there is no interest charged. On the other hand, the retailer's interest earned per cycle contains two parts: (1) During the period [0, KT], the retailer can obtain the interest earned on the sales revenue received (including sales revenues from backlogged), and (2) the retailer can use all the sales revenue to earn interest during the period, [KT, M j ]. Therefore, the annual total interest earned (IE) is Situation 2: KT ≥ M j (j = 1, 2, …k) In this situation, the retailer's delay payment period M j is shorter than or equal to the positive inventory level length KT (see Fig. 1b); it indicates that the retailer has some inventory available after due date M j . Thus, during the period [M j , KT], the retailer must pay the interest for the items in stock, and then the annual total interest charged (IC) is Also, during the period [0, M j ], the retailer can use the sales revenue to gain interest. Hence, the annual total interest earned (IE) is Combining the above results, for given M j , j = 1, 2, …. . , k, and based on the length of KT and T w , the retailer's annual profit function including ordering cost, purchase cost, holding cost excluding interest charged, interest charged for on-hand, interest earned, and the cost of carbon emissions under various situations can be expressed as follows: where, Here, Eq. (6) represents the retailer's annual profit function when he or she does not need to rent an additional warehouse (i.e., KT ≤ T w ). More specifically, for M j ≤ KT ≤ T w , it indicates that the retailer needs to use Eqs. (3) and (4) to calculate interest charged and interest earned, and the holding cost is referred to Eq. (1a). Therefore, the retailer's annual profit function in this case can be described as Eq. (8). For 0 < KT ≤ T w ≤ M j or 0 < KT ≤ M j ≤ T w , it indicates that there is no interest charged and the retailer needs to use Eq. (2) to calculate interest earned, and the holding cost is referred to Eq. (1a). Therefore, the retailer's annual profit function in this case can be described as Eq. (9).
Similarly, Eq. (7) represents the retailer's annual profit function when he or she needs to rent an additional warehouse (i.e., KT ≥ T w ). For T w ≤ M j ≤ KT or M j ≤ T w ≤ KT, it indicates that the retailer needs to use Eqs. (3) and (4) to calculate interest charged and interest earned and the holding cost is referred to Eq. (1b). Therefore, the retailer's annual profit function in this case can be described as Eq. (10). For T w ≤ KT ≤ M j , it indicates that there is no interest charged and the retailer needs to use Eq. (2) to calculate interest earned, and the holding cost is referred to Eq. (1b). Therefore, the retailer's annual profit function in this case can be described as Eq. (11).

Deriving Closed-Form Optimal Solutions
Our problem is to determine the optimal values of K * and T * such that the retailer's annual profit function ATP(K, T) is maximized. By investigating the retailer's annual profit function, we notice that they are not concave. As a result, it is obviously not feasible to use the approach of setting two first partial derivatives to zero to obtain the optimal solution. Therefore, in order to effectively solve the model, we will employ the similar approach of the work of Pentico and Drake (2009) and Ata Allah et al (2013), which would be the simplest way to prove global optimality. A sophisticated methodology is developed to establish the conditions under which the objective functions have interior optimal solution. This approach is presented for all cases as follows.
Case 1-1: M j ≤ KT ≤ T w Maximizing Eq. (8) is equivalent to minimizing the following function: where, First, for any given K, take the first and second derivatives of ATC j ð Þ 11 K; T ð Þ with respect to T; we obtain Eqs. (19) and (20), respectively: From Eq. (20), if φ 115 > 0, then where ψ 11 (K) = K 2 φ 111 − Kφ 112 + φ 114 . The discriminant of ψ 11 (K), is always negative. Thus, ψ 11 (K) has no roots, and it is always either negative or positive. Since i:e:; T ¼ ffiffiffiffiffiffiffiffiffiffi ffi Taking the first and second derivatives of Eq. (22) with respect to K, we have From Eq. (24), ATP j ð Þ 11 K ð Þ is a strictly convex function of K. We check ð Þ reaches the global minimum at K = 0, and it indicates that the best choice is that retailers do not build inventory. Therefore, we only need to consider the situation of ATC 0 j ð Þ 11 0 ð Þ < 0: We further investigate Thus, if the inequality in Eq. (27) is established, ATC j ð Þ 11 K ð Þ has a unique minimizer in the open interval (0, 1), and the global optimum values of T 11 and K 11 can be obtained using Eqs. (28) and (29), respectively (see Appendix C, Eqs. (C3) and (C4)). Otherwise, the global minimizer will lie on the boundary point K 11 = 1 (see Appendix D): Here, for the discriminant term β 11 , it should be noted that (1) If 0 ≤ β 11 ≤ β, the optimal is that the retailer uses partial backlogging, and the optimal values of T 11 and K 11 can be obtained using Eqs. (28) and (29) respectively. (2) If 0 < β < β 11 , the optimal is that the retailer employs inventory policy with without shortages (e.g., K 11 = 1 ). (3) If β 11 < 0, the retailer needs to compare the cases of no stocking (e.g., K 11 = 0) and partial backlogging to determine which is optimal.
In addition, for the solutions K 11 and T 11 found by using Eqs. (28) and (29), if the condition M j ≤ K 11 T 11 ≤ T w is not satisfied, it implies that ATC j ð Þ 11 K; T ð Þwill obtain the optimal solution at the boundary. A logical solution is to set T ¼ M j K 11 or T ¼ T w K 11 ; then we recommend that readers refer to the detailed solution process given in Appendix E.
To sum up, based on the above analysis, the order quantity for trade credit period M j and purchase cost c j can be computed form Eq. (30), namely, From Eq. (30), if the optimal order quantity (Q j ) satisfies q j ≤ Q j < q j + 1 , the solution obtained by the analysis above is feasible. Otherwise, we need to use the solution procedure given in Appendix D to determine the optimal values of T and K.
Case 1-2: 0 < KT ≤ T w ≤ M j or 0 < KT ≤ M j ≤ T w Similar to Case 1-1, maximizing Eq. (15) is equivalent to minimizing the following function: where, Note that Eqs. (31) and (12) have similar function structures (i.e., φ 121 through φ 126 instead of φ 111 through φ 116 ). So the analysis and discussion provided for Eqs. (19), (20), (21), (22), (23), (24), and (25) of Case 1-1 is also established for Case 1-2. Next, the equivalent analysis for Case 1-2 of Eq. (26) for Case 1-1 is Consequently, if the inequality in Eq. (39) is established, ATP j ð Þ 12 K ð Þ has a unique minimizer in the open interval (0,1), and the global optimum values of T 12 and K 12 can be obtained using Eqs. (40) and (41), respectively (see Appendix C). Otherwise, the global minimizer will lie on the boundary point K 12 = 1 (see Appendix D): Similar to Case 1-1, we still need to perform the following two steps to ensure the feasibility of the solution: (1) For the solutions K 12 and T 12 found by using Eqs. (40) and (41), check whether they satisfy K 12 T 12 ≤ min {M j , T w }, and if not, we need to use Appendix E to determine the optimal values of T and K.
(2) Check whether the order quantity Q j satisfies q j ≤ Q j < qj + 1 , and if not, we need to use Appendix F to determine the optimal values of T and K.
Case 2-1: T w ≤ M j ≤ KT or M j ≤ T w ≤ KT Similarly, maximizing Eq. (15) is equivalent to minimizing the following function: where, Similar to previous cases, Consequently, if the inequality in Eq. (50) is established, ð Þ has a unique minimizer in the open interval (0, 1), and the global optimum values of T 21 and K 21 can be obtained using Eqs. (51) and (52), respectively (see Appendix C). Otherwise, the global minimizer will lie on the boundary point K 21 = 1 (see Appendix D): Similar to previous cases, the feasibility of the solution needs to be checked. If the solution is not feasible, we suggest that readers refer to the detailed solution process given in Appendix E and Appendix F.
Case 2-2:T w ≤ KT ≤ M j Similarly, maximizing Eq. (17) is equivalent to minimizing the following function: where, Similarly, Consequently, if the inequality in Eq. (61) is established, ð Þ has a unique minimizer in the open interval (0, 1), and the global optimum values of T 22 and K 22 can be obtained using Eqs. (62) and (63), respectively (see Appendix C). Otherwise, the global minimizer will lie on the boundary point K 21 = 1 (see Appendix D): Similarly, Appendix E and Appendix F are used to determine the optimal values of T and K if the solution is not feasible.
Algorithm Summarizing the above results, we can establish the following algorithm to find the optimal solution ( K * , T * ).

Numerical Examples
In this section, some examples are provided to illustrate the theoretical results and solution procedure obtained in this paper. In addition, we also carry out a sensitivity analysis of major parameters.

Cases
Possibility Optimal solution "←"denotes the optimal solution for given K ." × " denotes the problem is not feasible in this case. "← a " denotes the global optimal solution 0.4940 × 350 =K * * T * * D which means that the capacity of OW is insufficient to store the ordered quantity, so the retailer needs to rent an additional warehouse.
From Table 4, when the retailer's OW capacity is relatively small (i.e., W < 200), the retailer must rent an additional warehouse to enjoy a longer credit period and a lower purchase price. Moreover, we notice that when the credit period is relatively long (i.e., (N 1 , N 2 , N 3 ) = (0.40, 0.60, 0.80) or (0.40, 0.60, 0.80)), the retailer typically orders more goods to benefit from a longer credit period. As a result, a RW is necessary. Conversely, the retailer has no incentive to order more goods, nor to rent warehouses. Furthermore, as the retailer's OW capacity increases, the retailer's optimal order quantity Q * * and the annual profit increases. The longer credit period, the larger the annual profit of the retailer is.

Sensitivity Analysis
The sensitivity analysis studies the effect of parameters A, D, h r , h o , c b , g, p, β, I e , I c and (c 1 , c 2 , c 3 ) on the percentage of duration of period (K) in which inventory level is positive, retailer's replenishment time (T), order quantity (Q) and annual profit function (ATP) when only one parameter is changed and other parameters are kept at their original values and also studies the effect of parameters W and(N 1 , N 2 , N 3 ) on these when both are changeable.
Based on the results above, we obtain the following management insights: First, the proposed discount schedule, the purchase price, and trade credit are both linked to the order quantity, can effectively encourage the retailer to increase the order quantity, and increase the retailer's profits; second, the larger capacity of OW or the longer credit period provided by the supplier, the more benefit for the retailer will have. In a word, it is the best choice for the retailer to persuade the supplier to offer a longer trade credit period schedule or to choose the supplier with a longer trade credit period. In addition, retailers can increase their profits by expanding OW capacity appropriately.
Example 3 Using the same data as those in Example 1 except for W = 300, this example outlines the impact of changes of major parameters A, D, h r , h o , c b , g, p, β, I e , I c , and (c 1 , c 2 , c 3 ) on the optimal solutions. The results are summarized in Table 5.
Based on Table 5, the main conclusions are as follows: (1) The retailer's annual profit ATP * * increases with respect to the changes of the parameters D, p, β, and I e , whereas it decreases with respect to the changes of parameters A, h r , h o , c b , g, I c , and (c 1 , c 2 , c 3 ). Understandably, the parameters D, p, β, and I e have positive influence on the retailer's profit, so the increase of their values must bring more profits to the retailer. On the contrary, the parameters A, h r , h o , c b , g, I c , and (c 1 , c 2 , c 3 ) are all cost structure parameters of inventory system, and the increase of their values must result in the decrease of retailer's profit. Moreover, we observe that the annual  profit ATP * * is highly sensitive to the changes of parameters D, p, and (c 1 , c 2 , c 3 ).
(2) Optimal order quantity Q * * increases when we increase the values of parameters D, β, I e , and (c 1 , c 2 , c 3 ), while it decreases with respect to changes of h r , h o , c b , g, and p. Hence, if ordering cost A increases, the retailer wants to decrease order frequency by increasing order quantity; if the demand parameter D or the backlogged parameter β increases, customers' demand also increases, and consequently, the retailer needs to make a large order size. In addition, the increase of parameter I e or (c 1 , c 2 , c 3 ) will motivate retailers to order more goods to enjoy longer credit period and lower purchase price. On the other hand, if inventory holding cost h r or h o increases, the retailer will lessen order quantity to maintain a lower average inventory level. If backlogging cost c b , lost sale cost c g , or unit selling price p increases, the retailer wants to shorten the replenishment cycle and the shortage period to reduce the shortage cost and the lost sales cost. As a result, the retailer will make a small order size.
(3) Optimal fraction of no shortage K * * increases with respect to the changes of the parameters c b , g, and p, whereas it decreases with respect to the change of parameters A, β, I e , and (c 1 , c 2 , c 3 ). In fact, the increase in the value of parameters c b , g, and p means that the retailer will pay more for shortages. So the retailer will shorten the shortage period, i.e., by increasing the fraction of no shortage. However, if the value of parameters A, β, I e , and (c 1 , c 2 , c 3 ) increases, the retailer will be encouraged to make a large order size. Meanwhile, the retailer wants to lengthen the shortage period (i.e., reduce the fraction of no shortage) to avoid paying excessive inventory holding costs. In addition, we observe that when the retailer faces a discount schedule when purchase price and trade credit are both linked to the order quantity, there is no specific monotonic relationship between K * * and the value of parameters D, h r , h o , and I c . and I c . Obviously, if the value of parameters A, β, I e , and (c 1 , c 2 , c 3 ) increases, the retailer will make a large order size, and thus T * * also will be increased. Similarly, if the value of parameters c b , g, p, and I c increases, the retailer will reduce their order quantity, and eventually T * * will be decreased. Moreover, optimal replenishment cycle T * * is highly sensitive to the changes of parameters A, D, h r , h o , c b , g, p, β, I e , I c , and (c 1 , c 2 , c 3 ) . (5) As the value of A, D, β, I e , and (c 1 , c 2 , c 3 ) increases, the retailer prefers to rent an additional warehouse, while as the value of h r , h o , c b , g, p, β, I e , and I c increases, the retailer tends to choose not to rent an additional warehouse.

Managerial Insights
The result demonstrates the strategic and key insights for managers in managing green inventory with capacity constraint under order-size-dependent trade credit and all-units discounts with the incorporation of carbon emission cost. The proposed discount schedule, the purchase price, and trade credit are both linked to the order quantity, can effectively encourage the retailer to increase the order quantity, and increase the retailer's profits. The scenario in which the larger capacity of OW or the longer credit period provided by the supplier, the more benefit for the retailer will have. It is the best choice for the retailer to persuade the supplier to offer a longer trade credit period schedule or to choose the supplier with a longer trade credit period. In addition, retailers can increase their profits by expanding OW capacity appropriately. The managers have to make decision grounded on replacement cost, retailer purchasing cost, holding cost, interest to be paid, interest gained, and carbon emission cost. The manager should also primarily focus on decisions related to the purchase cost of the item, as it is highly sensitive to total profit and order quantity. In link to order, quantity not only increases the holding cost but also increases the carbon emission cost on holding these items. There must be an adjustment among these parameters to get a win-win state. In addition, this research noticeably shows a track for industrial managers to cope with different scenarios of trade credit. The green inventory model can benefit from creating verdicts to progress sustainable inventory system by optimizing replenishment cycle time and order quantity. Additionally, incorporation of carbon emission cost in the model will help organizations to focus on minimizing the total inventory in the warehouse, which will help to not only reduce carbon emission cost in holding items but also in total holding cost per unit item. To acquire more profit, industrial managers have to concentrate on their credit period tactics and cycle length. The maximum optimal entire profit is achieved when the rate of interest payable is lower than the rate of interest earned. In short, the insights of this study draw a guideline for decision-makers in order to achieve the optimum in green inventory systems.

Conclusions
In this paper, we develop a green inventory model with capacity constraint under order-size-dependent trade credit, allunits discount, and partial backordering including carbon emission cost. This green inventory model proved the benefits for organizations in creating decisions to improve a green inventory system and to protect the environment by minimizing carbon emissions through optimizing order quantity and payoff time. Incorporation of carbon emission cost helped those organizations to reduce the total inventory in warehouses. As lowering total inventory in the warehouse not only reduced the holding cost but also minimized the carbon emission cost of holding these items by consuming less energy in the warehouses. The impact of carbon emission cost also affected the total replenishing cost and ordering cost of the organization. Focusing on carbon emission cost in the inventory models would enable policymakers to reduce global warming and the hazardous effect of carbon on the environment. The study also deliberated the retailer's economic policy having carbon emission cost under different circumstances of allowable delay-in payments having all-units discount and partial backordering. This green inventory study gave all the potential circumstances of inventory models with carbon emission cost under allowable delay in payments. Four possible inventory models are formulated under various cases which may occur due to different values of parameters. In order to obtain the global optimal solution for each model, the conditions of the objective functions to have interior minimizers are established, and then a closed-form optimal solution is found for each one. Subsequently, an easy to use algorithm is proposed to find the solution of all problems in an integrated manner. Finally, some examples are given to illustrate the performance of the proposed model and the algorithm. Sensitivity analysis of the major parameters is performed, and meaningful insights are gained. For instance, it is the best choice for the retailer to convince the supplier to offer a longer trade credit period schedule or choose the supplier with a longer trade credit period. The retailer also can increase their profits by expanding OW capacity appropriately.

Limitation and Future Directions
Here, a profit-maximization model with constant demand under carbon emissions has been developed, so any variation of functional demand, e.g., time and selling price-dependent and advertisement-dependent demand, can be taken into consideration in the future models. It is an inventory model under order-size-dependent trade credit, all-units discount, and partial backordering; therefore, in the future, other different cases of permissible delays can be considered to increase the profit values of the players. Furthermore, future researchers might consider an integrated cooperative scenario for multiple players in the supply chain and consider both perishable and non-perishable products to enhance their research. In addition, the proposed model can be generalized by considering twolevel credit policy or fuzzy environments.
Data Availability My manuscript has no associated data, or the data will not be deposited.

Declarations
Ethics Approval This article does not contain any studies with human participants or animals performed by any of the authors.

Conflict of Interest
The authors declare no competing interests. T, Retailer's replenishment cycle; m, Number of shipments from the supplier to the retailer; M, Number of shipments from the supplier to the retailer per production run, a positive integer; K, Percentage of duration of period in which inventory level is positive; β, Proportion of shortage that will be backordered; T w , Length of depletion time for maximum storage capacity of OW; C g , The cost of goodwill loss for a unit of lost sale; C b , The backordering cost per unit per time due to shortages; π j , Lost sales cost per unit for price range j, including the lost profit and the goodwill loss; ATP, Retailer's annual profit; ATC, Retailer's annual cost