A study of manufacturing and inventory strategies in close-loop supply chain: a two-phase news-vendor model with an extended warranty

Closed-Loop Supply Chain procedures provide manufacturers with a competitive advantage over their competitors. Also, the supply chain involves a manufacturer and customers, in which the manufacturer produces and sells products from the manufacturing market. Interestingly, combining forward and reversal strategies presents, challenges on both the on-demand and supply sides of the market. When customers purchase products, they are uncertain about the quality of the product, and the manufacturer offers a non-renewing extended warranty (EW) to assure the customers about the reliability and quality of the products. This paper presents a two-phase News-Vendor framework for warranted and non-warranted products with inventory carryover to describe the best manufacturing and re(manufacturing) procedures. The demand is modeled in stochastic nature with one-dimensional EW for both new and reman products. This study aims to demonstrate the importance of providing an EW with a relationship between manufacturing and inventory strategy. Also, the numerical analysis carried out on the expected total profit affects the manufacturer’s influence when providing an EW, and graphical results reveal that offering an EW and lowering holding expenditure enhances the manufacturer’s expected total profit over two phases.


Introduction
In recent years, the development of re(manufacturing) and closed-loop supply chain processes has gained prominence in the broad spectrum of industries.The phrase Closed-Loop Supply Chain (CLSC) is attributed to the idea that forward and reversal manufacturing processes are interconnected.It is also possible to incorporate the following • Product demand is uncertain (Used in two-phase News-Vendor framework) • Straightforward connection between previous sales and the re(manufacturing) of product returns • There is no market disposal A manufacturer who wants to promote their re(manufacturing) industry should start by lowering the negative effects of their new and reman products' competitiveness in the supply chain market (the cannibalization of the reman product's sales).As a result, new and reman products need to use different product pricing strategies to attract distinct categories of customers.The product manufacturer can offer the reman product at a price that's less than the price for the new product.This will help the manufacturer to compensate for lost sales for the new product by capturing sales from customers.Furthermore, the manufacturer needs to deliberate carefully over the pricing of the new and reman product in light of the demand as well as the availability of product returns.
Abbey et al. [2] determine that discounting the price for the reman product compared to the new product can enhance demand and profit despite the availability of product cannibalization.Yan et al. [36] considered the reman product's stock level when determining the best price for either product.Karimi et al. [18] investigated the issue of traveling employees within a sustainable supply network, considering greenhouse gas emissions, storage options, and the choice of suppliers.Tiwari et al. [33] analyzed a two-phase supply chain between suppliers and retailers for stochastic demands.According to Das et al. [11], the optimum values for the restock point, safeguard storage size, the number of orders, and customer service level at every inventory point are determined.Further, Jackson et al. [17] reviewed the relevant literature indicating that simulation frameworks are the most suitable methodology to deploy when comparing and contrasting various inventory management strategies.Recently, the warranty was included in such studies.Keshavarz-Ghorbani and Arshadi Khamseh [19] suggested a multiple-phase optimization framework for a CLSC system along with the warranty in light of considering both quality control and price strategies.Most studies have examined the manufacturer's production and inventory strategies for new and reman products when a warranty is offered, while EW was not considered in such studies.Because of this, and to address a few questions: • How do re(manufacturing) and inventory affect the manufacturer's expected profits during EW? • How does the EW length affect the profit for both scenarios without and with an EW strategy?• How should the product manufacturer establish the optimal prices for both products under the EW strategy when inventory value changes?• Which variables (factors) will affect the EW?
The paper aims to study the manufacturing and inventory strategies in a CLSC using a two-phase News-Vendor model with a non-renewing EW for the new and reman products.A two-phase mathematical framework is constructed to derive the expected total profit without and with an EW for both products.The framework is used to evaluate how the EW scenarios and inventory holding expenditure affect the expected total profit in the manufacturer's decision.Furthermore, the numerical analysis carried out gives insight into EW scenarios when analyzing the essential factors such as the re(manufacturing) saving expenditure, the product quantity, and expenditure of holding inventory.
Sections of the study are as follows: Section (2) provides the essential related work, Section (3) presents the model description, table of notations, and assumptions.Sections (4) and (5) follow the mathematical framework expressions and analysis about the suggested model.Section (6) concludes with graphical demonstrations and managerial implications followed by conclusions in Section (7).

Related work
The section discusses the literature on CLSC with pricing, the News-Vendor model, re(manufacturing), and product warranties.The study of the related work is threefold.Firstly, warranty and inventory management are complicated concern areas in CLSC management.Most research works are carried out to focus only on either inventory or remanufacturing without warranty.If a warranty is used, it is only a basic warranty.Secondly, the manufacturer is ready to participate in remanufacturing and inventory strategy, by considering a few important factors like inventory holding expenditure, warranty, and saving expenditure due to remanufacturing.Finally, offering warranties for new and reman products in supply chains has piqued the curiosity among many researchers.Hence, the following literature is analyzed to study on warranty in inventory and remanufacturing.
Reimann and Lechner [26] studied CLSC production and re(manufacturing) strategies using the News-Vendor concept and concluded that new and reman products are perfect alternatives.Dwicahyani et al. [9] introduced a distributor-depot stocking (inventory) framework that includes pollution reduction, energy consumption, and re(manufacturing) production.Jauhari et al. [15] also looked into a multi-level inventory system that allowed for refunds.When considering CLSC models, Dwicahyani et al. [10] suggested a threephase framework for two product recovery processes and imperfect production.Tang et al. [31] investigated the pricing and warranty consideration in a two-phase CLSC framework between a manufacturer and a retailer by using Stackelberg game analysis.Tao et al. [32] consider a monopoly manufacturer who provides new product return services at the outset, then sells refurbished products to find the best possible pricing and manufacturing techniques.Many studies have shown a strong relationship between performance and customer pleasure.Ali and Anwar [1] examined the pricing strategies to determine the impact of consumer behaviors.To adhere to regulations, manufacturers must engage in retrieval attempts to boost used product gathering refer into Jauhari et al. [16].
Taleizadeh and Mokhtarzadeh [30] investigated online and offline marketing sales routes to study the manufacturer's two-dimensional warranty policy.Ma et al. [23] considered the EW given by retailers in two competing supply chains, and they glimpse at the equilibrium in both decentralized and co-ordinate frameworks.Sun et al. [29] examined the influence of the warranty length on the competitiveness between new and reman products in the market with the manufacturer, re(manufacturer), and retailer.They showed repair expenditures for reman products increase, the re(manufacturer) obligated to limit the warranty duration, whereas the manufacturer will be able to increase the wholesale price of the new product to generate more revenue.
Re(manufactured) and reconditioned products are two examples of quality that can be viewed from different perspectives refer to Christy et al. [7].Wang et al. [34] investigated the customer's willingness to purchase an EW after the end of the standard (base) warranty.Dai et al. [8] examined the potential of various preventive maintenance techniques for used products protected by a warranty from the aspect of both merchants and consumers.Zhang et al. [38] investigated the impact of consumer choices and discovered that, in the dualchannel model, more customers purchase extended warranties from the retailer leading to higher total supply chain profitability.
Liu et al. [22] examined how the warranty period affects both new and reman products and discovered that both ratios of the unit costs and length of the warranty of new and reman products are essential factors in determining the manufacturer's best production strategy.Chen et al. [6] established a revamped News-Vendor framework with a riskaverse retailer concerned about conditional risk measurement.Chen et al. [5] proposed the News-Vendor model to explore an uncertain retailers procurement strategy under repurchase guarantee sources of finance and stochastic demand for both single-period and multiperiod frameworks.Qu et al. [25] investigate the effects of carbon emissions through each level of the News-Vendor model problem while taking cap and trade policy investment into action for the reduction.
Zhang et al. [39] studied a Waste Electrical and Electronic Equipment (WEEE) CLSC that involves a retailer, a manufacturer, and a third-party re-converter, where the manufacturer either re(manufactured) return WEEE goods or permits the retailer to do so.By merging the re(manufacturing) mode and government fund policy, four dynamic game frameworks were developed and investigated the influence of government fund policy on CLSC behaviors.Guo et al. [12] explored the competitiveness between domestically created new apparel and overseas second-hand apparel.They developed a two-stage Stackelberg supply chain game model.Wen et al. [35] examined the operational frameworks on fashion retail supply chain management within the standard functional research literature.Shu et al. [28] investigated the impact of fairness concerns on the supply chain, deciding to use the Nash Equilibrium solution as a fairness reference point.
The related work addressed above was found through an extensive examination of scholarly databases such as Google Scholar and Web of Science using the keywords optimal production, re(manufacturing), extended warranty, decision-making, and inventory management.A comprehensive investigation of the prior information concerning a study of manufacturing and inventory strategies in CLSC with warranty was made possible to examine, evaluate, and summarize the relevant published material.The significant findings, methodology, and conceptual frameworks used previously were studied to enhance and develope this research work.In this paper, the demand is modeled in stochastic nature with one-dimensional non-renewing EW for both new and reman products.The suggested model includes an EW under uncertain demand when determining the manufacturer's expected total profit.The profits of the manufacturer are analyzed using two scenarios.Finally, the illustrated data reveals how expected profit affects the influence of EW in twophase modeling.

Model description
A mathematical model with two phases is regarded, in which the manufacturer produces a new product q 1 in the first phase and offers an option of both new and reman products q 2 , q 3 in the second phase to the end customers as shown in Fig. 1.The customer returns from the first phase are used to produce reman products.Generally, it's known considering both new and reman products are suitable alternatives.Additionally, an EW is offered for the first and second phases.The supply of second-phase products comes from two different sources: manufacturing new products and re(manufacturing) the returned products.Figure 2 shows the work process for re(manufacturing) and inventory process.Furthermore, because of demand uncertainty, inventory carryover is used to meet demand in the second phase.
In both phases, the customer demand D i is undetermined with known probability den- sity and cumulative distribution functions f x i and F x i , respectively.In the entire paper, demand distribution is made continuous.Further, two scenarios are considered in this framework: the first scenario is to determine the manufacturer's expected total profit for both new and reman products without EW, and the second scenario covers the expenditure of offering an EW in determining the manufacturer's expected total profits.

Assumptions
The following are the additional assumptions for the model.

Mathematical framework and analysis
The main objective of this study is to determine the manufacturer's expected total profit in two scenarios: without and with an EW.While analyzing the EW, the manufacturer assumes that the standard warranty has been normalized to neutral.With this presumption, let us narrow our focus to the investigation of the EW one can refer to Li et al. [20].Also, the unmet demand treats a lost sale, with the manufacturer incurring an expected shortage expenditure of s m per unit.At the end of the selling season, the remaining unsold inventory is sold at an expected salvage price of l m per unit.
Further, the expected sales revenue for both products using the standard News-Vendor model is given The product selling price in both phases j = I, II is given by P i (j) , while the product manufacturing expenditure with- out an EW for the both product is c a i < P i (j) and with an EW is c w e a i < P i (j) .The saving expenditure due to re(manufacturing) is and the expenditure of − c a i (where i = r, r 1 ) will be incurred in the second phase due to the re(manufacturing).Under the subsections, the expected total profit of the manufacturer in two scenarios is discussed.

Manufacturer expected total profit without EW
The following is the manufacturer's individual expected profit over the two phases without an EW.

Case 1:
The expected first-phase sales quantity is determined by the first-phase production choice of new products q 1 and its denoted by E S q 1 .
In the first phase, the new product's expected profit E 1 is given by,

It is clearly known,
The following is the result of using Eq.(2) in Eq. (1), Equation (3) becomes, (1) The value of the quantity of new products is obtained by differentiating Eq. ( 4) with regard to q 1 and equating the result to zero.

Case 2:
The expected second-phase sales quantity is determined by the second-phase production choice of new products q 2 and its denoted by E S q 2 .
In the second phase, the new product's expected profit E 2 is given by, The following is the result of using the Eq. ( 2) in Eq. ( 5), Equation ( 6) becomes, The value of the quantity of new products is obtained by differentiating Eq. ( 7) with regard to q 2 and equating the result to zero.

Case 3:
The expected second-phase sales quantity is determined by the second-phase production choice of reman products q 3 and its denoted by E S q 3 .
In the second phase the reman product's expected profit E 3 is given by, ) The following is the result of using the Eq. ( 2) in Eq. ( 8): Equation ( 9) becomes, The value of the quantity of reman products is obtained by differentiating Eq. ( 10) with regard to q 3 and equating the result to zero.
The expected total profit of the manufacturer without EW over two phases is given by,

Manufacturer Expected Total Profit With an EW
The following is the manufacturer individual expected profit over the two phases offered with an EW.Before estimating individual expected profits, the manufacturer's total EW expense for new and reman products must be determined: where the unit expenditure of expected EW for a new product in the first phase a n 1 is acquired by, When the aforementioned Eq. ( 13) is differentiated, the expected unit EW expenditure of the new product in first-phase is calculated as follows: (9) Substituting Eq. ( 14) in Eq. ( 12) yields the following expected EW expenditure: Similarly, the expenditure of expected EW a for new products in the second phase E c a n 1 w e is acquired by, where the unit expenditure of expected EW for reman products in the second-phase a r 1 is acquired by, When the aforementioned Eq. ( 17) is differentiated, the expected unit EW expenditure of reman products in second-phase is calculated as follows: Substituting Eq. ( 18) in Eq. ( 12) yields the following expected EW expenditure: Case 4: The expected first-phase sales profit and quantity are determined by the firstphase production choice of a new product q 1 with an EW E c a n 1 w e .
In the first phase, the new product's expected profit E 4 is given by, Equation (20) The value of the quantity of new products is obtained by differentiating Eq. ( 21) with regard to q 1 and equating the result to zero.

Case 5:
The expected second-phase sales profit and quantity are determined by the second-phase production choice of a new product q 2 with an EW E c a n 1 w e .
In second-phase, the new product expected profit E 5 is given by, Equation ( 22) becomes, The value of the quantity of new products is obtained by differentiating Eq. ( 23) with regard to q 2 and equating the result to zero.

Case 6:
The expected second-phase sales profit and quantity are determined by the second-phase production choice of reman product q 3 with an EW E c a r 1 w e .
In the second phase, the new product's expected profit E 6 is given by, e )F(we)−we∫ w e 0 F(t 1 )dt1} P n 1 (I) Equation ( 24) becomes, The value of the quantity of reman products is obtained by differentiating Eq. ( 25) with regard to q 3 and equating the result to zero.
The expected total profit of the manufacturer with an EW over two phases is given by,

Particular Cases
In this section, the demand for new and reman products is modeled using three distinct probability distributions named: Uniform, Exponential, and Normal, while the failure time of both products is sold under an EW using the Weibull distribution.The new and reman products of the manufacturer's expected total profit for both models under the three distinct demand distributions are calculated in the following subsections:

Manufacturer Expected Total Profit Without an EW
For different demand distributions, this subsection shows the manufacturer expected total profit without an EW.Substituting the above values in Eq. ( 11) yields the manufacturer estimated total profit under uniform demand becomes, Case 8: When the demand for new and reman products is exponentially distributed with Substituting the above values in Eq. ( 11) yields the manufacturer estimated total profit as: Case 9: Finally, assume that the demand for new and reman products follows a Normal distribution with Substituting the above values in Eq. ( 11) yields the manufacturer estimated total profit under normal demand becomes, ( 27)

Manufacturer Expected Total Profit With an EW
For different demand distributions, this subsection shows the manufacturer expected total profit with an EW.When the failure time of the new and reman products follows the Weibull distribution with shape parameter and scale parameter with , t b ≥ 0 .The expected EW expenditure of new and reman products can be determined by using Eqs.( 15), (16), and (18).Equation ( 15) becomes, By using a lower incomplete gamma function definition, the Eq. ( 30) becomes, Similarly, the new product expected EW expenditure in second-phase is acquired by using Eq. ( 31) in Eq. ( 16), The reman product expected EW expenditure in second-phase by using ( 18), ( 30) Case 10: The manufacturer expected total profit for a uniform demand with an EW and a Weibull failure rate is given by,

Case 11:
The manufacturer expected total profit for an exponential demand with an EW and a Weibull failure rate is given by, ( Case 12: The manufacturer expected total profit for a normal demand with an EW and a Weibull failure rate is given by,
The effects of parameters q b , , , k and on the expected total expenditure under the different demands are shown in the Table 1

below:
The manufacturer increases the quantities for both products in both scenarios-with an EW and without an EW-the expected total profit for both new and reman products generally increases.If a manufacturer re(manufactures) all returns, the expected profit for the new product in the first phase decreases as the quantity of reman products increases. ( Furthermore, increasing the new product production in the first phase results in an inventory situation.The inventory of leftover products from the first phase is carried to the second phase to enable the manufacturer to acquire beneficial higher profit.On the other hand, the expected profit for the second phase increases as the quantity of reman products increases. Table (1) reveals some fascinating results.The increasing gaps between expected total profits without and with an EW show the ability to hold inventory which enhances the advantage of increasing re(manufacturing) saving expenditure .As increases, the expected profits for both products increase steadily.The total profit strategies over two phases are independent.In the first phase, the profit for the new product is unaffected if the reman product is not sufficiently profitable.The expected total profit maximized over the two phases of the reman product is viable.
To illustrate, the impact of holding expenditure on expected total profit k is varied.Reduced holding expenditure enhances the expected total profit without and with an EW under various demands.It noticed that maintaining inventory is reasonable and profitable to the manufacturer.
The following figures are the discoveries derived from the proposed work.Figures 3, 4 and 5 demonstrate the expected profit scenarios without and with an EW under uniform, normal, and exponential demands.The figures show that offering an EW increases the manufacturer's profit.
Figure 3 shows that the expected total profit for the uniform demand increases as the quantity of the new and reman products increases over the two phases.The above result reveals that providing an EW for both products increases the manufacturer's expected total profit over the two phases when compared to the cases in which no EW is offered.
Figure 4 reveals the expected total profit for the exponential demand increases as the quantity of both products increases while providing an EW.On the other hand, the expected total profit for the exponential demand decreases during the no EW.As a result, the manufacturer faces less profit compared to the EW.
Similarly, as shown in Fig. 5, the expected profit increases as the quantity of new and reman products increases, and the manufacturer would face a higher profit under the normal demand when compared to the uniform and exponential demands.
Figures 3, 4 and 5 concluded that an EW stimulates the quantity, especially if the manufacturer offers an EW: it's beneficial for the manufacturer to enhance sales.The figures revealed that the quantity has positively impacted the expected total profit of the new and reman products, and the customers are willing to purchase an extended warranted product from the manufacturer.Offering an EW by the manufacturer is always acceptable and enhances the expected total profits for both products.The manufacturer would always 1 3 benefit from providing an EW policy as part of the product manufacturing process, and it is more effective due to the EW policy offered, which boosts the demand and increases the expected total profit.As a result, the manufacturer is keen on providing an EW to entice customers to buy the products and enhance the expected profit across two phases.The expected total profits are due to q 1 , (q 2 and q 3 ) show similar results.Furthermore, the expected total profits over the three different demands increase when the holding expenditure is reduced.In particular, Table (2) shows that with an EW, the expected total profit varies from 2.4787 when k = 0 to 2.4373 when k = 0.115 .Additionally, when no EW is offered, the expected total profit varies from 1.7406 when k = 0.115 to 1.7820 when k = 0 .It helps to discover a few intriguing results regarding holding expenditure and inventories.The scenarios help the manufacturer to decide on the production of both products as inventory and re(manufacturing) are the primary second phase supply.Further, the manufacturer can boost its efficiency whenever holding inventories is feasible and profitable.
On the other hand, if there is excessive production in the first phase, the level of re(manufacturing) and utilized inventory carryover continuously increases with lowering holding expenditure k .While examining the second phase supply, inventory carryover and re(manufacturing) work together to offset the second phase expenditure of new product production.Table 2 revealed some further fascinating information.
The capability to maintain inventory magnifies the benefits when is increased and observed by the increasing differences between expected total profits with and without an EW.Furthermore, when increases, expected total profits under uniform, normal, and exponential demand increase simultaneously.The second phase profit has increased relative to the first phase profit.
Finally, manufacturers conclude that by reducing the holding expenditure k , both quan- tities of inventory and re(manufacturing) increase consistently.As a result, the manufacturer can use his increasing first-phase profit in exchange for a decrease in second-phase expenditures on new and reman products.Exclusively, most of the reman products are purchased by low-end consumers, but the results initiate high-end customers to be interested in buying the reman products.The decision to purchase an EW depends on the customer's willingness.The EW expenditure and time should ideally be within the manufacturer's discretion.By doing this, manufacturers can attract more customers to purchase extended warranties, which boosts the manufacturer's revenue and is also beneficial to the customers regarding the reliability of the reman products.

Result Comparison
Warranty and inventory management are complicated concern areas in CLSC management.However, the model addresses this by developing a two-phase News-Vendor framework including EW.This section compares the expected profit with an EW discussed in subsection (5.2) to those found in subsection (5.2) developed by Yedida and Sekar [36].The comparisons are depicted in Fig. 9 to analyze the expected profit Eq. ( 35) when providing the EW during exponential demand.The results point out the importance of adopting the EW along with the inventory strategy as it shows a positive impact on the manufacturer's expected profit when compared to Yedida and Sekar [36].

Managerial Implications
This subsection provides an overview of the managerial repercussions discussed in this paper.
How do re(manufacturing) and inventory affect the manufacturer's expected profits during EW? and which variables (factors) will affect the EW?
When a manufacturer offers an EW, the manufacturer's expected total profit relies on the expenditure of re(manufacturing) ( ) and the expenditure of inventory holding (k) .The increase in value of enhances the manufacturer's expected total profits under distinct demand distributions.Also, the expected total profit is influenced by holding expenditure due to the excess production, because of this the manufacturer is unable to acquire a sufficient number of product returns for the re(manufacturability).To balance this, the manufacturer needs to minimize the holding expenditure.Also, re(manufacturing) helps the environment in many waysby preventing the waste of raw materials, cutting down on energy consumption, and lowering emissions of greenhouse gases-it's important to show encouragement in re(manufacturing) process.Further, used product returns endure a significant effect on the expected total profit earned by the manufacturer as well as the production of the first phase of new products.Interestingly, if there is excessive production in the first phase, the level of re(manufacturing) and utilized inventory carryover continuously increases with lowering the holding expenditure k .Alternatively, offering an EW for reman products is viable as it promotes the business strategy and safeguards the customers against the product quality.

Conclusion
Reman product inventory management is a very complex business.Manufacturers in all sectors regularly struggle with the burden of having too much product on hand, which can have serious financial repercussions.Therefore, effectively handling surplus inventory is essential for safeguarding profitability, ensuring customer's satisfaction, and making the most efficient use of storage space.The firms who frequently make use of excess production inventories for new and reman products include consumer electronics: such as Apple, Sony, and Samsung; fashion and apparel: such as H&M, and Nike; automotive parts suppliers: such as Ford, General Motors, and Bosch, etc., and heavy equipment re(manufacturer) like Caterpillar.The above industries handle their excess production in inventory to accommodate customer demands despite a product announcement and respond immediately to supply chain breakdowns.The model can have a significant effect on a firm's bottom line to make wise choices about how to optimize the activities and increase the profitability when to have a solid understanding of models involving inventory management, remanufacturing, and EW.In this study, the authors presented a two-phase stochastic mathematical framework for a single product adopting the News-Vendor model approach for estimating the manufacturer's optimal production with expected total profit and re(manufacturing) strategies under unpredictable and uncertain demands.The risk of stocking first-phase product production and the availability of returned components is dependent on the previous manufacturing and supply options for brand-new products.The analytical expression for the optimal production strategies like excess first phase new product production and using inventory carryover for the new product with production and reman products in the second phase are derived.

Assumption 1 :
The manufacturer validates both new and reman products with the same EW length w e in the product market.Assumption 2: The expected EW expenditure is given by c w e a i (t b ) = c w e a i (1 + t b w e ) where c w e a i = (c a i + a i ) , b = 1, 2, 3. Assumption 3: An inventory-maintaining expenditure k is used.Assumption 4: For each unsold (lost sale) unit of product demand, the manufacturer is incurring a per-unit shortage expenditure s m , m = 1, 2. Assumption 5: Unsold inventory is offered (expected salvage price) for a unit salvage price of l m , where m = 1, 2.

Fig. 2
Fig. 2 Inventory Management and Work Process for Re(manufacturing)

Case 7 :
Let the new and reman products demand x b be uniformly distributed between [ , ] with F x b = x b − − , ≤ x b ≤ .

Table of notation
The following table summarizes the mathematical framework parameters.

Table 1
Effects of q b , , , k and on expected total profits ( ↑ for increasing; ↓ for decreasing; − for no effect)

Table 2
Expected profits without and with an EW under varying re(manufacturing) saving expenditure δ