Inventory Model for Single Manufacturer - Two Competing Retailers with Carbon Tax and Product Selling Price

. This paper discusses how a manufacturer and two competing retailers interact to develop an inventory model considering a carbon tax, green investment, and product selling price under stochastic demand to maximize the profitability of the supply chain. It is assumed that the manufacturer replenishes all the retailers at the same time and period. Since there are two competing retailers, the product selling price at each retailer impacts not only its demand but also the other retailer’s demand. This paper also considers the implementation of green technology since there is a carbon tax for activities that produce carbon emissions such as production, transportation, and holding activities in the supply chain. The validity of the inventory model will be tested through sensitivity analysis and numerical examples.


Introduction
The industrial revolution that took place in England in the 18th century has brought significant changes to industrial development in the world (Ministry of Environment, 2012).The revolution has succeeded in changing the industry from slow to fast.It is undeniable that the industry that is developing today is the result of this revolution.The Industrial Revolution also brought positive impacts in the form of accelerating the pace of the world economy, but behind the good impacts, the Industrial Revolution also brought adverse impacts in the form of environmental degradation.In line with the rapid growth of industry, the amount of carbon emissions and other greenhouse gases has also increased over time (Martinez, 2005).This can occur due to two main reasons: industrial activities and the use of fossil energy (Stolyarova, 2013).
Industrial development has caused many forests to change their function from oxygen-producing and carbon dioxide-absorbing to carbon dioxide-producing land (Ministry of Environment, 2012).In addition, manufacturing companies also require a large amount of fossil energy to support their supply chain activities, such as production, transportation, and storage, which are also activities that produce carbon emissions.Examples of fossil energy commonly used by industries to run manufacturing processes include petroleum, natural gas, and coal, which are sources of air pollution (Stolyarova, 2013).Any use of fossil energy will cause an increase in the amount of carbon emissions in the atmosphere.In a report released by the Global Carbon Project in 2022, it was revealed that global carbon emissions in 2022 remained at the highest level without any signs of decline.Total global CO2 emissions are projected to reach up to 40.6 tons in 2022.This is driven by fossil CO2 emissions which are projected to rise 1% compared to 2021, slightly above pre-Covid-19 levels in 2019.
According to Abdirad et al. (2021) to meet market demands, the industry must implement the concept of supply chain management.Supply chain management is a unity of production processes and activities starting from the procurement of raw materials from suppliers, the process of adding value from raw materials to finished goods, the process of storing inventory, to the process of shipping finished goods to retailers and consumers.Activities in the supply chain produce different amounts of emissions which are influenced by several factors.One example is production activities that require tools or machines that use fossil power so the more and longer the production process, the greater the level of emissions produced.The increase in emissions has harmed the environment, so several efforts have been made to reduce carbon emissions, one of which is the implementation of an emission tax for a company or industry.By applying an emission tax, companies must pay for carbon emissions generated from their operational processes according to the tax rate imposed for each unit of CO2 produced.The costs incurred by the company will increase with the application of the emission tax, but on the other hand the company also still has to carry out its operational activities to meet consumer demand.Based on this, it is necessary to have inventory management to determine the optimal point of the elements in the supply chain so that the company can determine the optimal point by minimizing costs and also the emissions generated but consumer demand can still be met.
Research conducted by Drake et al. (2016) states that many companies have begun to invest in green technology to reduce emissions produced.Bai et al. (2019) mentioned that due to pressure from the government and consumers, many industries have begun to switch to using green technology.Even so, companies do not directly apply green technology to all existing process lines due to the large investment costs required, so it is necessary to calculate the optimal green technology investment costs for the company.Based on this, the role of inventory management is needed to determine the optimal level of green technology implementation and its impact on company profitability.Harahap (2015) explains that several factors influence product purchasing decisions, one of these factors is the selling price of the product.Product selling price is one of the interesting things in inventory management.This is because the amount of price set by the retailer greatly affects demand for both the retailer itself and other retailers who are competitors.With the development of competition in the industry, many manufacturers supply the same product to several retailers.The selling price of the product at one retailer may be different from other retailers.This certainly affects product demand at each retailer so it is necessary to calculate what is the optimal product selling price for each retailer to obtain maximum profit in a supply chain unit.Bai et al. (2019) developed a demand model influenced by product selling prices at two competing retailers.In addition, the study also discusses the effects of reducing carbon emissions and comparing centralized systems with decentralized systems, but the study has not included the optimal green investment costs for the company and the costs involved are only production and storage costs so that other costs such as shortage costs, transportation costs, and emission tax costs need to be considered.
Based on the research mentioned above, no research has developed an inventory model between one manufacturer and two competing retailers by considering emission taxes, green investment, and product selling prices at retailers.This research will consider the carbon emission policy and the amount of green investment costs and demand functions that depend on the selling price of the product.The resulting inventory model will be used to obtain maximum profit in the supply chain.

Inventory Model with Periodic Review
According to Jauhari (2016), the periodic review inventory model is inventory that is checked within a fixed period (T).This model is different from the continuous review model which requires inventory managers to monitor continuously.If in the continuous review, the ordering process is carried out when the inventory level has reached the reorder point (ROP), in the periodic review the ordering process is carried out within a fixed period.The size of the order is calculated based on the difference between the maximum inventory size and the inventory size at the time of ordering so this model is considered to have an easy way to apply it.Although the periodic review policy is easy to apply, this policy has the disadvantage of requiring a high safety stock so that the amount of inventory stored in the warehouse is also high.This of course also affects the storage costs borne by the company which are getting higher.

Inventory Model with Joint Economic Lot-Sizing (JELS)
Joint Economic Lot-Sizing (JELS) is an inventory model to determine the number of production lots and product shipments obtained from the calculation of total inventory costs.The purpose of using JELS is to integrate suppliers into their production cycle and buyers into their ordering cycle to minimize the total cost incurred in the supply chain.
The JELS model was first developed by Goyal (1976).The model developed aims to minimize the total cost of inventory with deterministic and constant demand and unlimited production levels.Then Banarjee (1986) developed the previous model by considering lot-for-lot delivery and limited production levels.Then the development was carried out again by Goyal (1988) by improving the lot-for-lot shipping assumption by shipping the same amount.
Research and development of the JELS model continue to be carried out until now with increasingly diverse variations.Such as research conducted by Rad et al. (2014) who developed an inventory model that considers the effects of product price variations on market demand.JELS research is developing towards a sustainable supply chain where companies are required to consider environmental aspects of supply chain policies such as those conducted by Jauhari et al. (2022) who developed an inventory model by considering environmental aspects in the form of carbon emission tax costs and green production investments.

System Description
This research develops a supply chain inventory model consisting of two parties, namely a manufacturer and two retailers who compete with each other with stochastic demand and depend on carbon emission taxes, green investment, and demand influenced by the selling price at each retailer.The total amount of product demand at a retailer is strongly influenced by the price of the product sold at each retailer.In this model, the demand received by retailers is normally distributed with a mean D and a standard deviation σ.This average demand depends on the selling price of the product at retailer 1 being p1 and the selling price of the product at retailer 2 being p2.The average market demand function from consumers to retailer 1 is D1 (p1, p2) = d1 -y1p1 + y3 p2, while the average market demand from consumers to retailer 2 is D2 (p1, p2) = d2 -y2 p2 + y4 p1, where y1, y2 are the sensitivity coefficients to the product selling price and y3, y4 are the sensitivity coefficients to the effect of competitors' selling prices, respectively.It is assumed that y1 > y3 and y2 > y4 represent that the selling price of the product at that retailer is more influential on changes in demand than the selling price at competing retailers.So the total market demand received by the manufacturer is D(p1, p2) which is the sum of demand at retailer 1 and retailer 2.
The retailer will order D(p1, p2) units of product to the manufacturer.The manufacturer will fulfill the order by producing D(p1, p2).In the production process carried out by the manufacturer, the manufacturer requires a cost for purchasing raw materials of MC which is the product of the number of units to be produced of D(p1, p2) with the selling price of raw materials per unit of pt.Before entering the production process, the manufacturer needs to do a setup which costs SC.When the production process takes place, the manufacturer needs a cost of PC to process raw materials into finished products that will be sent to retailers.To run production using green production facilities, manufacturers need to spend an investment cost of R, which will affect the carbon generated from the manufacturing process.Before being shipped to retailers, the products at the manufacturer will be stored in a warehouse with a storage cost of Hm.The costs mentioned are not the total costs incurred by the manufacturer, but there are still emission tax costs due to the emissions generated from the manufacturing process.In this study, it is assumed that the emission tax will be imposed on the production and storage processes so that each process will bear the emission tax costs of TCEM production and TCEM storage.After the product from the manufacturer is sold to the retailer, the manufacturer will get a revenue of RM from the payment made by the retailer.
It costs the retailer OTC to order the product from the manufacturer and as a transportation cost for the product from the manufacturer to the retailer.Once the product reaches the retailer, it is stored at a storage cost of Hb.In addition, retailers also need to prepare a shortage cost of ESC in case of product shortages due to fluctuations in demand.In addition, the retailer also bears the cost of emission tax because the process that takes place at the retailer also generates carbon emissions in the transportation and storage process with the respective costs of TCEBtransport and TCEBstorage.After the retailer's product is sold, the retailer will get a revenue of RB from the payment made by the consumer.
The product will be delivered by the manufacturer to the retailer at a predetermined period.The inventory system implemented by retailers is a periodic review system.The retailer will place an order for the product to the manufacturer if it has entered the predetermined order period.If there is a shortage of inventory, the request will continue as a backorder.

Notation
The following notation is used to formulate the proposed mathematical inventory model: Parameters for the retailer: D1 Average demand of retailer 1 (unit/year) D2 Average demand of retailer 2 (unit/year) d1 Basic demand of retailer 1 (unit/year) d2 Basic demand of retailer 2 (unit/year) y1 Selling price elasticity parameter (p1 ) at retailer 1 y2 Selling price elasticity parameter (p2) at retailer 2 y3 Parameter of cross elasticity of selling price (p2) y4 Parameter of cross elasticity of selling price (p1) σ1 Standard deviation of retailer

Mathematical Models
Average demand function at retailers 1 and 2 Average demand function at the manufacturer

Retailer's Inventory Model
The following is a formulation of all costs incurred and revenue received by the retailer.With the formulation of all retailer costs and revenues, the formulation of retailer profits can be known.

Manufacturer's Inventory Model
The following is a formulation of all costs incurred and revenue received by the manufacturer.With the formulation of all manufacturer costs and revenues, the formulation of manufacturer profits can be known.In finding the optimal solution, the above equations are entered into the calculation algorithm.The following is a description of the steps in the optimal solution search algorithm using Matlab Based on those input parameters, after the computation using the previous step, these are the results for the decision variable, demand, total profit, and total emission shown in the table below.

Sensitifity Analysis
In this section, a sensitivity analysis is performed to investigate the influence of the changes in key parameters on the model's solution.Thus, we focus on evaluating some parameters, which are maximum green reduction, production rate, and carbon tax.The following are the green reduction parameter values used in the sensitivity analysis.Based on the value changes that have been made, the following values are obtained

Conclusion
In this paper, we develop a single manufacturer -two competing retailers considering a carbon tax, green investment, and product selling price under stochastic demand to maximize the profitability of the supply chain.We analyzed the impact of some parameters on each decision variable and joined the total profit of the supply chain.
The developed model can guide inventory practitioners in determining appropriate inventory levels in a supply chain system consisting of a single manufacturer and two competing retailers.The application of the inventory model can assist in maximizing the joint total profit of the supply chain and reduce total carbon emissions generated.

Table 2 .
Design Requirements With Criteria.

Decision Variable Value Cost/Profit Value
The following table shows the scenario of changing the model parameter values

Table 1 .
Result Scenario Maximum Green Reduction

Table 2 .
Result Scenario Production Rate

Table 2 .
Result Scenario Tax Emission