The Capacitated Sustainable EOQ Models: Models Considering Tax Emissions

Inventory is the primary key for a company to support its smooth production [1, 2]. Inventory has a vital role in the supply chain [3]. Besides, this has an impact on the company's operating costs [4, 5]. Several approaches have been proposed to solve inventory problems. These include Economic Order Quantity (EOQ), heuristic algorithms [6], and dynamic programming [7, 8]. Nowadays, environmental issues have become the concern of all companies and the world [9, 10]. Companies are required to pay attention to environmental aspects, including carbon emissions and waste [11]. Di In developed countries, the government provides a tax policy for carbon emissions to produce emissions and waste [12]. It is intended to raise company awareness of environmental problems [13, 14]. Recently, carbon emissions have become an essential issue in the industrial sectors. It includes transportation, inventory, and storage [15, 16]. Through inventory management, companies can control carbon emissions and economic aspects [17]. Researchers have currently shown their keen interest in Sustainable EOQ (SEOQ) [18]. In inventory, the SEOQ model is a lot size method used to determine economic orders by considering environmental aspects [19]. This model has to economically balance the financial perspective with the environmental perspective so that the industry can determine the appropriate policy that supports sustainability. There have been many studies about inventory issues that accommodate carbon emissions, such as Chen, et al. (2013), Jaber, et al. (2017), dan He, et al. (2015). In general, the studies consider issues of ARTICLE INFO ABSTRACT


Introduction
Inventory is the primary key for a company to support its smooth production [1,2]. Inventory has a vital role in the supply chain [3]. Besides, this has an impact on the company's operating costs [4,5]. Several approaches have been proposed to solve inventory problems. These include Economic Order Quantity (EOQ), heuristic algorithms [6], and dynamic programming [7,8]. Nowadays, environmental issues have become the concern of all companies and the world [9, 10]. Companies are required to pay attention to environmental aspects, including carbon emissions and waste [11]. Di In developed countries, the government provides a tax policy for carbon emissions to produce emissions and waste [12]. It is intended to raise company awareness of environmental problems [13,14]. Recently, carbon emissions have become an essential issue in the industrial sectors. It includes transportation, inventory, and storage [15,16]. Through inventory management, companies can control carbon emissions and economic aspects [17]. Researchers have currently shown their keen interest in Sustainable EOQ (SEOQ) [18].
In inventory, the SEOQ model is a lot size method used to determine economic orders by considering environmental aspects [19]. This model has to economically balance the financial perspective with the environmental perspective so that the industry can determine the appropriate policy that supports sustainability. There have been many studies about inventory issues that accommodate carbon emissions, such as Chen, et al.  He, et al. (2015). In general, the studies consider issues of carbon emission impacts, order frequency, and storage amount [23,24]. Bauer, et al. (2010) proposed the SEOQ model by considering transportation costs. Subsequently, some researchers developed the model by adding tax cost on environmental impacts [26,27]. Furthermore, some researchers also included environmental and tax costs in the procurement inventory model [28] and the production inventory model [29,30]. Up to the present day, there have been several studies investigating the relationship between inventory [31] and capital constraints [32]. Raturi and Singhal (1990) developed the multiitem EOQ model by considering shortage cost, production, and capital constraints. Asadabadi (2016) also proposed the EOQ model with capital constraints for purchasing raw materials.
Although research on SEOQ has been flourishing lately, there have not been any studies that discuss SEOQ with capital restrictions for purchase and emission taxes. It has created a considerable amount of uncertainty about the relationship between SEOQ and capital constraints. Based on the description mentioned earlier, one way to overcome the problems is to accommodate capital limitation factors into the SEOQ models. The solution to the problems is based on the capital purchase function, which is the cost for the company's operations. Therefore, a new approach is needed to investigate the effect of capital in the SEOQ models. In the current study, we developed SEOQ models that considered carbon emissions by including capital restrictions on purchasing goods. This study aimed to develop a more sophisticated method to solve the problems of determining lot size by considering environmental impacts and capital constraints. Therefore, this research generated new insights in inventory, especially the SEOQ models with capital constraints. In addition, this study was intended to contribute to research on SEOQ by exposing the effects of capital on the number of lot sizes.

Assumptions and Notations
Assumptions of the SEOQ problems consist of (1) demands, order cost, purchasing cost, holding cost, tax fees, total emissions, and total capital in a fixed period; (2) components of the assumption that are deterministic; (3) each model is used for 1 item emission product; (4) tax cost per one emission unit; and (5) capital used is the capital for a purchase or emission tax. The notations used in the models include: :

Proposed SEOQ Model without Limits
In terms of the proposed SEOQ model without limit, this study developed two SEOQ models, namely SEOQ model without tax and the SEOQ model with the tax on carbon emissions.

SEOQ without Tax
In this model, the researchers considered the costs included in the sustainable inventory, such as the fixed cost of an environmental impact for each cycle, order cost, purchasing, and holding cost. The total inventory cost is formulated as follows (1): To gain the optimal Q for the SEOQ model without tax, equation (1) is differentiated to Q. The result is presented in equation (2).

SEOQ with Tax
In the problem of SEOQ with emission tax, the study considered emission tax. The SEOQ model with tax is formulated in equation (4).
Equation (4) is differentiated to Q to produce the optimal Q for the SEOQ model with emission tax. The result is presented in equation (5).

Proposed SEOQ Model with Capital Constraint
The researchers developed another model concerning capital constraint. The capital referred to the cost used in purchasing raw materials and purchase emission tax. Equation (7) denotes the capital of raw material purchase without emission tax. Equation (8) represents the capital of raw material purchase without emission tax.
The SEOQ model with capital constraints comprised two types: capital with tax and capital without tax, which has been developed by adding a constraint function, i.e., capital constraints.

Proposed SEOQ Model without Tax and with Capital Constraint
The study employed the Lagrange function to minimize the total inventory cost against the constraint. The model of total inventory cost in equation (1) is added to the constraint function in equation (7). The Lagrange function of the proposed SEOQ model without tax and with capital constraint is displayed in equation (9): Further, equation (9) is differentiated partially to Q and . The results are formulated in equations (10) and 11.

Sustainable SEOQ Model with Tax and Capital Constrain
In the sustainable SEOQ model with tax and capital constraint, the researchers developed the Lagrange formula from equation (4) by adding the constraining factor, i.e., equation (8). As a result, it is presented in equation (13).

Numerical Experiment Procedure
This section shows the numerical experiment procedure on the proposed SEOQ models. The experiment was carried out to test the sensitivity of the proposed models. The data are presented in Table 1. In the numerical experiment, this study used 20 variants of capital purchase data and emissions tax. The experiment was also conducted to find out the effects of Q lot size on the inventory cost. Table 2 and Table 3 contain the results of the experiments using the four proposed models. Table 2 recapitulates the results of the SEOQ model without tax. The findings indicate that ordering lots of SEOQ models without capital constraint reaches 58 units for each order annually. The total inventory cost is as much as $ 1023,2. According to the SEOQ experiment without tax in Table 2, if the available capital is relatively small, the lot size of orders is comparatively small. On the other hand, if the capital is relatively big, the lot size of orders is also relatively big. As reflected in Fig. 3, if > 0, Q is optimal by using equation (2). Conversely, if < 0, equation (12) is employed. Table 3 summarizes the results of the SEOQ model with tax. The findings show that the number of ordering lot of SEOQ model without capital constraints obtains 64 units per order annually. The total inventory cost is $ 1353. According to the SEOQ experiment without tax in Table 3, if the available capital is relatively small, the lot size of orders is relatively small. However, if the capital is relatively big, the lot size is also relatively big. According to Fig. 3, if > 0 is optimal by using equation (5). Nonetheless, if < 0, equation (15) Fig. 2 present the effects of lot size (Q) on the inventory cost. The results imply that the order cost and total inventory cost of the SEOQ model are affected by . For order cost, the greater the value of Q, the smaller the cost. It is due to the number of shipping frequencies that gets smaller. Conversely, the smaller the Q value, the higher the order cost. In terms of inventory cost, the higher the Q value, the greater the total inventory cost. Conversely, the smaller the value of Q, the smaller the total inventory cost.

Conclusion
The purpose of this study was to develop a more sophisticated method for solving the problems of determining lot size by considering environmental impact and capital constraint. In this study, the researchers developed SEOQ models by considering capital constraints. Capital itself is the cost used in the process of purchasing raw materials. Capital is an essential aspect that needs to be considered in decision making. This research also presented numerical experiments and analyzed the sensitivity of the models. The experimental results show that the proposed models helped solve the problems. Further, it is suggested that future researchers develop SEOQ models for multi-item products with capital and capacity constraints. [21] M. Y. Jaber, M. Bonney, and H. Jawad, "Comparison between economic order/manufacture quantity and just-in-time models from a thermodynamics point