Plane Thermoelastic Waves in Infinite Half-Space Caused Decision Making: Applications in Management and Engineering ISSN: 2560-6018 eISSN: 2620-0104 DOI:_https://doi.org/10.31181/dmame0319102022d * Corresponding author. E-mail address: anamika1994.dash@gmail.com (A. Dash), bcgiri.jumath@gmail.com (B. C. Giri), aksarkar.jumath@gmail.com (A. K. Sarkar) COORDINATION OF A SINGLE-MANUFACTURER MULTI- RETAILER SUPPLY CHAIN WITH PRICE AND GREEN SENSITIVE DEMAND UNDER STOCHASTIC LEAD TIME Anamika Dash1*, Bibhas C. Giri1 and Ashis Kumar Sarkar1 1 Department of Mathematics, Jadavpur University, Kolkata 700032, India Received: 5 December 2021; Accepted: 1 September 2022; Available online: 17 October 2022. Original scientific paper Abstract: When dealing with uncertainties in supply chain and ensuring customer satisfaction, efficient management of lead time plays a significant role. Likewise, besides managing inventory and pricing strategies adeptly in multi-retailer supply chain, it has become inevitable to the firms to embrace green and sustainable business practices. In this context, this paper considers a two-level supply chain consisting of a single manufacturer and multiple retailers in which the manufacturer produces a single product and delivers it to the retailers in some equal-sized batches. Each retailer faces a price and green sensitive market demand. The lead time is assumed to be a random variable which follows a normal distribution. Shortages for retailer inventory are allowed to occur and are completely backlogged. The centralized model and a decentralized model based on leader-follower Stackelberg gaming approach are developed. A price discount mechanism between the manufacturer and retailers is proposed. For the acceptance of this contract, the upper and lower limits of the price discount rate are established. Numerical outcomes exhibit that the price discount mechanism effectively coordinates the supply chain and enhances both environmental and economical performances. A sensitivity analysis with respect to some key parameters is performed, and certain managerial insights are emphasized. Key words: Two-level supply chain, multiple retailers, stochastic lead time, price and green sensitive demand, price discount mechanism. 1. Introduction The growing importance of environmental protection and pollution reduction has been felt all over the world in recent years. Green supply chain management aims to prevent pollution while also producing environmentally friendly products. It involves many activities including green manufacturing, green packaging, green distribution, mailto:anamika1994.dash@gmail.com mailto:bcgiri.jumath@gmail.com mailto:aksarkar.jumath@gmail.com Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 2 remanufacturing and waste management. Many industries (Walmart, Coca-Cola, Nike, Adidas, and others) are showing great interest in environmentally friendly supply chains. They are successfully influencing consumers' attitudes toward green products by emphasizing the benefits and necessity of a green supply chain. LG India has pioneered the creation of eco-friendly electronic gadgets. They have strictly used halogen or mercury, trying to reduce the use of dangerous substances in their products. TCS has already earned the title of Newsweek’s Top Greenest Company in the world, with a global green score of 80.4% due to its worldwide recognized sustainability practices. Dell has promoted an efficient and effective safe disposal system by allowing their customers to return their product to the company for free. As consumer awareness grows, more people are willing to buy environmentally friendly products and are willing to pay more for those products. The government is also trying to make people aware of eco-friendly products through various guidelines and legislation. Researchers and practitioners are focusing on integrating environ-mental concerns into supply chain management. Lead time plays a vital role in supply chain management. The assumption of deterministic lead time is not valid in most real world situations because of various reasons such as delays in production process, transit time, inspection, loading and unloading, and so on. Therefore, dealing with stochastic lead time is very fascinating and challenging. To avoid a planned shortage at the buyer’s end and to efficiently manage the phenomena of early arrival, researchers are developing supply chain models with stochastic lead time (He et al., 2005; Lieckens and Vandaele, 2007; Barman et al., 2021b). Price is another important factor that influences the customer demand. In this context, a good quality product with relatively lower price always attracts customers. In traditional supply chain management, the manufacturer determines the quality of the product and the retailers set their selling prices independently. Therefore, it has become an important managerial concern to implement an effective coordination between the manufacturer and the retailers for balancing the social and economical issues equitably. Suitable coordination schemes can improve the efficiency of the entire supply chain by creating incentives for all members to adopt it. Through such coordination mechanisms, the members of the supply chain develop a collaborative relationship between themselves. Researchers have performed a significant amount of work to coordinate the supply chain with an appropriate contract such as revenue sharing contract (Zhang and Feng, 2014; Mondal and Giri, 2021), cost sharing contract (Saha and Goyal, 2015; Zhu et al., 2018), delay in payments (Ebrahimi et al., 2019; Duary et al., 2022), etc. In today’s competitive market, manufacturers do not rely on a single retailer to sell their produced goods; instead, they deal with multiple retailers. In this study, we consider a two-level supply chain which is comprised of a single manufacturer and multiple retailers trading for a single product. The manufacturer delivers the retailers' order quantities in equal-sized batches and invests in green technologies to produce eco-friendly products. The product's greening level and selling price influence customer demand at each retailer. Replenishment lead time is assumed to be random. Both the centralized and decentralized models are considered. We demonstrate cooperation between the manufacturer and the retailers by a price discount mechanism. Our primary goal is to fulfill the research gap and find answers to the following research questions: • What will be the optimal strategies of the manufacturer and retailers when the market demand is price and green sensitive? Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 3 • What is the impact of a price discount contract on the optimal decisions of the supply chain? • Is the price discount mechanism capable of coordinating the supply chain? • What is the effect of greening investment on the profitability of the supply chain? The contributions of this study are as follows: Firstly, we incorporate a price discount mechanism with green initiatives in a single-manufacturer multi-retailer supply chain model under stochastic lead time. Secondly, we examine whether the proposed price discount contract is able to coordinate the supply chain or not. Finally, we look at the influence of the price discount contract on supply chain members' profitability and determine the conditions under which they accept the price discount contract. The rest of this paper is structured as follows: Section 2 contains a brief review of the existing literature relevant to this work. Section 3 introduces notations and assumptions that are used throughout the paper. The problem description is given in Section 4. In section 5, mathematical models are formulated. Section 6 is devoted to numerical analysis. A sensitivity analysis of some key parameters is performed in Section 7. Section 8 discusses some managerial implications of this study. Finally, Section 9 concludes the paper with some limitations and future research directions. 2. Literature Review In this section, we review some of the existing literatures which are related to our current work across four research streams: price- and green-sensitive demand, stochastic lead time, single-manufacturer multi-retailer supply chain model and price discount contract. 2.1 Price- and green-sensitive demand Price is one of the important factors that influence market demand. A preliminary work focusing on price dependent demand was carried out by Whitin and Thomson (1955). Later, many researchers and practitioners (Ho et al., 2008; Yang et al., 2009; Lin and Ho, 2011; Atamer et al., 2013; Rad et al., 2014; Jaggi et al.,2015; Alfares and Ghaithan, 2016) have done numerous works on price dependent demand. Researchers and practitioners are currently focused on issues including the reduction of harmful effects of production on the environment. Swami and Shah (2013) studied a vertical supply chain consisting of a single manufacturer and a single retailer where the members put an effort for greening their operations, and the customer demand at the retailer’s end is price and green sensitive. Zanoni et al. (2014) investigated a two-level joint economic lot size model with customer demand sensitive to price and environmental quality, and concluded that investing in improving a product's environmental performance is more beneficial, and implementing an integrated policy can increase both environmental and economic performance. Ghosh and Shah (2015) explored the positive impact of a cost sharing contract on the optimal decisions of a green supply chain to enhance the profit level and produce items with higher greening quality. Li et al. (2016) initiated e-commerce in green supply chain management and proposed a coordination mechanism for decentralized dual channel green supply chain. Basiri and Heydari (2017) investigated coordination issues in a green supply chain with a non-green traditional product and a substitutable green product under price, greening level and sales effort dependent demand. Giri et al. (2018) analyzed a two-level closed-loop supply chain model where the customer demand is affected by Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 4 selling price, warranty period and greening level of the product. They proposed a revenue sharing contract in order to develop both social and economic performances. Heydari et al. (2019) developed a three-tier dual channel supply chain model with price and green sensitive demand that is not only economically beneficial but also reduces the selling price in both channels. Heydari et al. (2021) proposed a hybrid coordination scheme of cost sharing contract and revenue sharing contract in a two- level green supply chain with price and green sensitive demand. In a two-level supply chain model with imperfect production system, price, advertisement, and green sensitive customer demand, Giri and Dash (2022) established a cost-sharing contract between the manufacturer and the retailer. Sepehri and Gholamian (2022) investigated the impacts of shortages in a sustainable inventory model with price and emission sensitive demand considering quality improvement and inspection process concurrently. 2.2 Stochastic lead time To address the shortcomings of deterministic lead time, researchers devised supply chain models that take into account the stochastic nature of lead time. Sajadieh et al. (2009) developed a single vendor single buyer supply chain model with stochastic lead time following exponential distribution and deterministic demand, and exhibited a significant cost reduction in integrated system than decentralized ones. Hoque (2013) presented an integrated inventory model with stochastic lead time following normal distribution under combined equal and unequal batch shipment policy. Lin (2016) considered an integrated vendor-buyer model with stochastic lead time, and demonstrated that further investment can reduce lead time variability and achieve enough savings for the entire system. Giri and Masanta (2019) derived optimal production and shipment policy for a closed-loop supply chain model with stochastic lead time, and observed that learning in production and remanufacturing leads to a significant cost reduction for the supply chain. Giri and Masanta (2020) developed a closed-loop supply chain model with learning in production, price and quality sensitive demand under stochastic lead time, and elaborated the positive impact of learning in production process on the optimal decisions. Sarkar et al. (2020a) investigated an integrated vendor-buyer model considering time value of money with partially backlogged shortage under stochastic lead time where the lead time is variable but dependent on the order size of the buyer and production rate at the vendor. Safarnezhad et al. (2021) derived optimal ordering, pricing and inspection policies in a vendor-buyer supply chain model with price dependent demand and stochastic lead time. Hoque (2021) developed a single-manufacturer multi-retailer supply chain model under stochastic lead time where the manufacturer delivers the lots to the retailers either only with equal batch sizes or only with unequal batch shipments. 2.3 Single-manufacturer multi-retailer supply chain model To come closer to the reality, focusing on multi-retailer models has become a great topic of interest for the researchers. Recently, Giri and Roy (2016) considered a supply chain model consisting of a single manufacturer and multiple retailers with price sensitive customer demand. They found that lead time reduction by paying extra crashing cost does not affect the retail price significantly but enhances the entire system profit. Chen and Sarker (2017) investigated a single-manufacturer multi- retailer production-inventory model for deteriorating items with price sensitive demand under just-in-time delivery environment. They solved the model using Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 5 particle swarm optimization (PSO) and quantum-behaved PSO (QBPSO) techniques. Majumder et al. (2018) studied a single-vendor multi-buyer supply chain model with variable production rate and controllable lead time reduction where the production cost at the vendor is a function of the production rate. Chan et al. (2018) proposed a coordination mechanism in a single-vendor multi-buyer supply chain model with stochastic demand, and synchronized the manufacturer’s production cycle and retailers’ ordering cycle. Ben-Daya et al. (2019) developed a single manufacturer multi-retailer closed-loop supply chain model with an environment-friendly approach of remanufacturing the used products under consignment stock policy. Giri et al. (2020b) developed a single-manufacturer multi-retailer inventory model with stochastic lead time and price sensitive demand. Esmaeili and Nasrabadi (2021) presented a single-vendor multi-retailer supply chain model for deteriorating items with trade credit and inflationary conditions, where the demand is price sensitive. Najafnejhad et al. (2021) used an imperialist competitive algorithm to solve a single- vendor multi-retailer inventory model under vendor managed inventory policy considering upper limits of inventories as decision variables. Nandra et al. (2021b) studied a single-vendor multi-buyer model that took into account variable production cost, imperfect items and environmental factors. Malleeswaran and Uthayakumar (2022) introduced a discrete investment for ordering cost reduction in a single- manufacturer multi-retailer EPQ model with green and environmental sensitive consumer demand and reworking system under carbon emissions policies. 2.4 Price discount contract Coordination between manufacturers and retailers has received a lot of attention as a means of improving inventory control, and researchers have done a lot of work to coordinate the supply chain with the appropriate contract. As we consider a price discount coordination scheme in our study, we cover some literatures which address similar issues. Viswanathan and Piplani (2001) analyzed a single-vendor multi-buyer model with a coordination mechanism in which the vendor specifies the replenishment period and all the buyers agree to order at the same time in exchange for a price discount. Li et al. (2011) investigated the impact of a price discount mechanism in a single-vendor single-buyer supply chain model with service level constraint and controllable lead time. Aljazzar et al. (2017) dealt with a three-level supply chain with two types of trade credit mechanism, and concluded that implementing both delay in payment and price discount coordination mechanisms at a time lead more profit for the entire supply chain rather adopting these contracts individually. Nouri et al. (2018) proposed a compensation-based wholesale price contract between the manufacturer and the retailer where the customer demand is stochastic and dependent on innovation and promotional efforts. Furthermore, they devised a profit-sharing strategy on the basis of bargaining power of the members. Xu et al. (2018) investigated the role of a price discount contract in coordinating a dual- channel supply chain under carbon emission capacity regulation, with consumer demand in both online and offline channels influenced by the product's selling price. They provided the necessary conditions for which the price discount contract coordinates the dual-supply chain in both online and offline modes. Sarkar et al. (2020b) suggested a price discount coordination mechanism in a two-level supply chain with price sensitive customer demand to encourage the supply chain players to take part in joint decision-making strategy. Yang et al. (2021) explored the optimal cooperation strategy between an upstream supplier and two competing manufacturers considering a wholesale price contract and manufacturers' technology investment. In order to reduce products’ carbon emissions. Zu et al. (2021) analyzed Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 6 a single-manufacturer single-retailer supply chain model under two different mechanisms viz. wholesale price contract and consignment contract. Zhang et al. (2022) performed a comparative analysis between wholesale price contract and cost- sharing contract in a two-level green supply chain model. They looked at which contract is more effective in improving the product's greenness and promoting demand, taking into account the consumer reference pricing effect. 2.5 Research gaps in the existing literature Table 1 summarizes the research gaps in the existing literature as follows:  Although there are numerous research papers available that explore stochastic lead time and single-manufacturer multi-retailer supply chain models, no attempt has been made to maximize individual profits of supply chain stakeholders. The majority of these research focused on maximizing (or minimizing) overall supply chain profit (or cost).  Most of these studies considered deterministic customer demand. They overlooked some crucial factors such as the selling price, greening level, promotional effort, advertising and product quality, all of which have an impact on market demand.  No one has incorporated environmental awareness into a single-manufacturer multi-retailer supply chain model with stochastic lead time, and none of these studies looked at the influence of greening investment on both the supply chain's economic and environmental performance.  Almost no study has ever suggested a channel coordination mechanism. The above literature review reveals a significant research gap and indicates that no attempt has been made in implementing price discount coordination mechanism in a single-manufacturer multi-retailer supply chain model with price and green sensitive demand under stochastic lead time. It would be interesting and contributory to consider all the genuine issues like the stochastic nature of lead time, the impact of retail price and environmental awareness on market demand, single-manufacturer multi-retailer business situations and so on under one umbrella. Although, Hoque (2021) extended the model of Hoque (2013) in multi-retailer scenario, but he considered the demand of each retailer as deterministic and minimizes the total cost of the supply chain. In this paper, our aim is to fulfill this research gap and implement an appropriate coordination scheme which efficiently improves each supply chain member’s profitability as well as environmental performance. A comparison of the present work with the relevant existing literature is presented in Table 1. Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 7 Table 1. A comparison of the present model with some existing literature Authors Retailer Batch shipment Demand Lead time Coordination Sajadieh et al. (2009) Single Equal Deterministic Stochastic No Li et al. (2011) Single Equal Deterministic Controllable Price discount Hoque (2013) Single Equal & unequal Deterministic Stochastic No Sarkar et al. (2017) Single Equal Deterministic No No Giri et al. (2018) Single No Price, green and warranty period sensitive No Revenue sharing Sarkar et al. (2018) Multiple Equal Deterministic Variable No Giri and Masanta (2019) Single Equal Deterministic Stochastic No Giri et al. (2020a) Single Equal & unequal Price and green dependent No Cost sharing Sarkar et al. (2020b) Single Equal Price dependent No Price discount Agrawal and Yadav (2020) Multiple Equal Price dependent constant Profit sharing Esmaeili and Nasrabadi (2021). Multiple No Price dependent No No Nandra et al. (2021a) Multiple Equal Deterministic Controllable No Sarkar et al. (2021) Single Equal Online & offline price dependent Distribution free approach & normal No Safarnezhad et al. (2021) Single No Price dependent Stochastic No Hoque (2021) Multiple Equal or unequal Deterministic Stochastic No This paper Multiple Equal Price and green dependent Stochastic Price discount 3. Notations and Assumptions The following notations are used for developing the proposed model: Parameters: 𝑅 production rate (units/ year) 𝐴𝑣 set-up cost per set-up ($/set-up) ℎ𝑣 manufacturer’s holding cost per item per unit time ($/unit /year) 𝐹 transportation cost per batch shipment($/shipment) 𝑤 unit wholesale price($/unit) 𝐼 greening investment parameter ($) 𝑁 number of retailers (positive integer) Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 8 𝑄 total order quantity [= ∑ 𝑄𝑖 𝑁 𝑖=1 ](units) 𝐷 total market demand [= ∑ 𝐷𝑖 𝑁 𝑖=1 ](units /year) 𝐿 lead time, a random variable with p.d.f. 𝑓𝐿(.) 𝑖-th retailer: 𝐴𝑖 ordering cost per order ($/order) ℎ𝑖 holding cost per item per unit time ($/unit /year) 𝐷𝑖 demand rate [𝑅 > ∑ 𝐷𝑖 𝑁 𝑖=1 ](units /year) 𝑎𝑖 basic market demand (units /year) 𝛼𝑖 consumer sensitivity coefficient to greening level 𝛽𝑖 consumer sensitivity coefficient to retail price 𝑄𝑖 order quantity (units) 𝑐𝑖 shortage cost per item per unit time ($/unit /year) 𝑟𝑖 reorder point(units) 𝜎𝑖 standard deviation of the lead time Decision variables: 𝑛 number of batches delivered to each retailer (positive integer) 𝜃 greening improvement level 𝑞𝑖 batch size of the 𝑖-th retailer (units) 𝑝𝑖 unit retail price of the 𝑖-th retailer ($/unit) 𝜙 price discount ratio, 𝜙 ∈ [0,1] (.)^𝑑 decision variable in decentralized policy (.)^𝑐 decision variable in centralized policy (.)^𝑐𝑜 decision variable in coordinated mechanism Profit functions: 𝐴𝐸𝑃𝑚 average expected profit of the manufacturer($/year) 𝐴𝐸𝑃𝑖 average expected profit of the 𝑖-th retailer($/year) 𝐴𝐸𝑃𝑠 average expected profit of the supply chain ($/year) The basic assumptions for developing the proposed model are as follows: 1. A single manufacturer produces a single item and meets the demand of multiple retailers (Sarkar et al., 2018). 2. The manufacturer transfers the products to the retailers in a number of equal sized batches (Sarkar et al., 2020b). 3. The retailers face a consumer demand dependent on the selling price and greenness of the product (Ghosh and Shah, 2015). We assume that the demand rate of the 𝑖-th retailer is a linear function of retail price and greening level of the product given by 𝐷𝑖(𝑝𝑖,𝜃) = 𝑎𝑖 − 𝛽𝑖𝑝𝑖 + 𝛼𝑖𝜃, where 𝑎𝑖 is the basic market demand, 𝛽𝑖 and 𝛼𝑖 are positive integers such that 𝑎𝑖 + 𝛼𝑖𝜃 > 𝛽𝑖𝑝𝑖 for all 𝑖 = 1,2, . . . . ,𝑁. 4. The manufacturer produces the product at a constant production rate 𝑅 in one set-up and the production rate is greater than the sum of demands of all retailers i.e., R > ∑ Di N i=1 (Hoque, 2021). 5. Shortages are allowed and are assumed to be completely backlogged (Sarkar et al., 2018). Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 9 6. The 𝑖-th retailer places his next order when his inventory stock level reaches to a certain reorder level 𝑟𝑖 (Hoque, 2013). 7. The lead time to meet the retailer’s demand is a random variable which follows a normal distribution and the lead time for each shipment is independent of the others (Hoque, 2013). 8. Annual greening investment for the product is taken as 𝐼𝜃2, which is increasing and convex in the greening improvement level 𝜃 (Ghosh and Shah, 2015). 4. Problem Definition This study develops a green supply chain model where the single manufacturer deals with multiple retailers for a single product. Figure 1 exhibits the schematic diagram of the proposed model. Figure 1. Logistics diagram of the proposed single-manufacturer multi- retailer green supply chain model The manufacturer produces the items at a fixed production rate in a single set-up and delivers the order quantities of the retailers with an equal sized batch shipment policy. Due to various unavoidable circumstances such as late start in production, varying transportation time, loading, unloading, etc., the batches may arrive early or late at the retailers. To deal with this type of delivery uncertainty, lead time is treated as a stochastic random variable which follows a normal distribution. Customer demand is assumed to be affected by the retail price and environmental performance of the product. The manufacturer adopts a green investment strategy to maintain his environmental responsibility as well as stimulate the customer demand in an eco- conscious market. In both decentralized and centralized settings, the manufacturer's and all retailers' optimal pricing and inventory strategies are derived. Following that, a wholesale price discount contract is implemented between the manufacturer and the retailers to coordinate the supply chain. 5. Model Formulation We suppose that the manufacturer sells the produced items to 𝑁 retailers. The manufacturer transfers the ordering quantity 𝑄𝑖 of the 𝑖-th retailer in 𝑛 equal batches of size 𝑞𝑖. Total order quantity of 𝑁 retailers is 𝑄. Therefore, 𝑄𝑖 = 𝑛𝑞𝑖 and 𝑄 = ∑ 𝑄𝑖 𝑁 𝑖=1 . Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 10 The 𝑖-th retailer places the next order when the inventory stock reaches to a level 𝑟𝑖. The shipment is expected to arrive to the retailer’s end at or before the time of selling this 𝑟𝑖 quantity. The mean lead time is 𝑟𝑖 𝐷𝑖 . Due to various reasons, the batches may reach early or late. We assume that the lead time follows a normal distribution. Depending on the length of the lead time, three cases may arise: Case (i) When the batch 𝑞𝑖 reaches to the retailer earlier i.e., 0 < 𝑙𝑖 < 𝑟𝑖 𝐷𝑖 . In this case, similar to Hoque (2013), the inventory holding area of the 𝑖-th retailer can be determined from Figure 2(a) as = 𝐴𝑟𝑒𝑎 ( ABCD + EFG + GHDE) = 1 2 (𝑟𝑖 − 𝐷𝑖𝑙𝑖 +𝑟𝑖)𝑙𝑖 + 1 2 (𝑞𝑖 − 𝐷𝑖𝑙𝑖) (𝑞𝑖 −𝐷𝑖𝑙𝑖) 𝐷𝑖 + 𝑟𝑖(𝑞𝑖 − 𝐷𝑖𝑙𝑖) 𝐷𝑖 = 1 2 [ 𝑞𝑖 2 𝐷𝑖 + 2𝑞𝑖 ( 𝑟𝑖 𝐷𝑖 − 𝑙𝑖)] where 𝑟𝑖 = 𝑞𝑖𝐷𝑖 𝑅 Then the order quantity 𝑄𝑖 of the 𝑖-th retailer is given by 𝑛 2 [ 𝑞𝑖 2 𝐷𝑖 + 2𝑞𝑖 ( 𝑟𝑖 𝐷𝑖 −𝑙𝑖)] The holding cost refers to the investment in storing the unsold products. The expected inventory holding cost for the order quantity 𝑄𝑖 of the 𝑖-th retailer is ℎ𝑖 ∫ 𝑛 2 [ 𝑞𝑖 2 𝐷𝑖 + 2𝑞𝑖 ( 𝑟𝑖 𝐷𝑖 −𝑙𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑟𝑖 𝐷𝑖 0 Figure 2. Inventory of 𝑖- th retailer under stochastic lead time Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 11 Case (ii) When the batch 𝑞𝑖 reaches late to the 𝑖-th retailer and the lead time 𝑙𝑖 lies in the range 𝑟𝑖 𝐷𝑖 ≤ 𝑙𝑖 ≤ 𝑟𝑖+𝑞𝑖 𝐷𝑖 . In this case, shortages occur at the retailer’s end. From Figure 2(b), the shortage area at the 𝑖-th retailer is obtained as = 𝐴𝑟𝑒𝑎 ( CDE) = 1 2𝐷𝑖 (𝐷𝑖𝑙𝑖 − 𝑟𝑖) 2. So, the expected shortage cost of the 𝑖-th retailer for 𝑛 batches is given by 𝑛𝑐𝑖 2 ∫ (𝐷𝑖𝑙𝑖 − 𝑟𝑖) 2 𝐷𝑖 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 Inventory holding area of the 𝑖-th retailer for the batch qi is = 𝐴𝑟𝑒𝑎 ( ABC + FGH+ EFHJ) = ri 2 2Di + (qi-Dili) 2 2Di + ri(qi-Dili) Di = (𝑞𝑖 − 𝐷𝑖𝑙𝑖+𝑟𝑖) 2 2𝐷𝑖 Hence the expected inventory holding cost of the 𝑖-th retailer for 𝑛 shipments is obtained as 𝑛ℎ𝑖 ∫ (𝑞𝑖 − 𝐷𝑖𝑙𝑖 + 𝑟𝑖) 2 2𝐷𝑖 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 It is assumed that, during this delay period, the batches remain in the manufacturer’s stockhouse. So, it causes an extra holding cost to the manufacturer. The extra inventory for this delayed delivery is ∑𝑁𝑖=1 𝑛𝑞𝑖(𝐷𝑖𝑙𝑖−𝑟𝑖) 𝐷𝑖 . So, in this case, the additional inventory holding cost for the manufacturer is ℎ𝑣 ∑∫ 𝑛𝑞𝑖(𝐷𝑖𝑙𝑖 −𝑟𝑖) 𝐷𝑖 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑁 𝑖=1 . Case(iii) When the batch 𝑞𝑖 arrives late to the retailer with lead time in the range 𝑟𝑖+𝑞𝑖 𝐷𝑖 ≤ 𝑙𝑖 < ∞. In this case, shortages occur at the retailer’s end and from Figure 2(c), the shortage area for the batch 𝑞𝑖 is obtained as = 𝐴𝑟𝑒𝑎 ( CDEF ) = 𝑞𝑖 2 2𝐷𝑖 + 𝐴𝑟𝑒𝑎( DEFG ) So, the expected shortage cost of the 𝑖-th retailer for all batch shipments is 𝑛𝑐𝑖 ∫ [ 𝑞𝑖 2 2𝐷𝑖 + 𝑞𝑖 ( 𝐷𝑖𝑙𝑖 − 𝑞𝑖 − 𝑟𝑖 𝐷𝑖 )] ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 Similar to case(ii), the additional expected inventory holding cost for the manufacturer is ℎ𝑣 ∑∫ 𝑛𝑞𝑖(𝐷𝑖𝑙𝑖 − 𝑟𝑖) 𝐷𝑖 ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑁 𝑖=1 . Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 12 Combining all three cases, the expected holding cost of the 𝑖-th retailer for all batches is given by 𝑛ℎ𝑖 ∫ 1 2 [ 𝑞𝑖 2 𝐷𝑖 + 2𝑞𝑖 ( 𝑟𝑖 𝐷𝑖 − 𝑙𝑖)] 𝑟𝑖 𝐷𝑖 0 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑛ℎ𝑖 ∫ (𝑞𝑖 − 𝐷𝑖𝑙𝑖 + 𝑟𝑖) 2 2𝐷𝑖 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 and the expected shortage cost for all batch shipments is 𝑛𝑐𝑖 ∫ (𝐷𝑖𝑙𝑖 − 𝑟𝑖) 2 2𝐷𝑖 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑛𝑐𝑖 ∫ [ 𝑞𝑖 2 2𝐷𝑖 + 𝑞𝑖 ( 𝐷𝑖𝑙𝑖 −𝑞𝑖 − 𝑟𝑖 𝐷𝑖 )] ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 5.1. Decentralized Model (DM) In the decentralized model, the manufacturer and the retailers independently take their decisions in order to maximize their own profits. Here we consider the retailers to be the Stackelberg leader and the manufacturer as the follower. The manufacturer sets the number of shipments and greening level of the products. Then taking these responses into consideration, the retailers decide their optimal retail price and batch sizes. Average expected profit of the manufacturer The manufacturer's total extra holding cost from cases(ii) and (iii) is ℎ𝑣 ∑∫ 𝑛𝑞𝑖(𝐷𝑖𝑙𝑖 − 𝑟𝑖) 𝐷𝑖 ∞ 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑁 𝑖=1 Figure 3. Joint inventory of the manufacturer and the retailers In Figure 3, the trapezium ABCD represents the joint inventory of the manufacturer and retailers. The average inventory of the manufacturer-retailer system is = 𝐴𝑟𝑒𝑎 ( ABCD)× 𝐷 𝑄 Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 13 = 1 2 ×(𝐴𝐵 + 𝐶𝐷) × 𝑄 × 𝐷 𝑄 = 1 2 [ ∑ 𝑞𝑖 𝑁 𝑖=1 𝑅 + ( 𝑄 𝐷 + ∑ 𝑞𝑖 𝑁 𝑖=1 𝑅 − 𝑄 𝑅 )]𝐷 = 𝐷∑ 𝑞𝑖 𝑁 𝑖=1 𝑅 + 𝑄 2 (1− 𝐷 𝑅 ) Average inventory holding of 𝑁 retailers is ∑ ( 𝑞𝑖 2 2𝐷𝑖 )( 𝐷𝑖 𝑄𝑖 )𝑁𝑖=1 = ∑ 𝑞𝑖 2 2𝑄𝑖 𝑁 𝑖=1 Therefore, the average inventory holding of the manufacturer is 𝐷∑ 𝑞𝑖 𝑁 𝑖=1 𝑅 + 𝑄 2 (1 − 𝐷 𝑅 ) −∑ 𝑞𝑖 2 2𝑄𝑖 𝑁 𝑖=1 The set-up cost incorporates the costs of materials and labours to get ready the machinery system for processing the new production lot of goods. It plays an important role in start-up of a new business and smooth running of it. The 𝑖-th retailer places an order of quantity 𝑄𝑖. The manufacturer produces the total order quantity 𝑄 = ∑ 𝑄𝑖 𝑁 𝑖=1 . The cycle length of the manufacturer is 𝑄 𝐷 . Therefore, the average set up cost is 𝐴𝑣𝐷 Q . Investment for greening supports the environmentally-conscious business practices. In this case, the manufacturer's average greening investment is 𝐼𝜃2. The average expected profit of the manufacturer is 𝐴𝐸𝑃𝑚(𝑛,𝜃) = 𝑤𝐷 − ℎ𝑣 [ 𝐷∑  𝑁𝑖=1𝑞𝑖 𝑅 + 𝑄 2 (1− 𝐷 𝑅 ) − ∑  𝑁𝑖=1 𝑞𝑖 2 2𝑄𝑖 ]− ℎ𝑣 ∑   𝑁 𝑖=1 ∫   ∞ 𝑟𝑖 𝐷𝑖 (𝐷𝑖𝑙𝑖 −𝑟𝑖)𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 − 𝐴𝑣𝐷 𝑄 − 𝐼𝜃2 = 𝑤𝐷 − 𝐴𝑣𝐷 𝑄 − ℎ𝑣 [ 𝐷∑  𝑁𝑖=1𝑞𝑖 𝑅 + 𝑄 2 (1− 𝐷 𝑅 ) −∑  𝑁𝑖=1 𝑞𝑖 2 2𝑄𝑖 ]− ∑  𝑁𝑖=1 ℎ𝑣𝐷𝑖𝜎𝑖 √2𝜋 − 𝐼𝜃2 (1) From (1), we have 𝜕𝐴𝐸𝑃𝑚 𝜕𝑛 = 𝐴𝑣𝐷 𝑛2𝑠 − ℎ𝑣𝑠 2 + ℎ𝑣𝑠𝐷 2𝑅 − ℎ𝑣𝑠 2𝑛2 (2) 𝜕𝐴𝐸𝑃𝑚 𝜕𝜃 = [𝑤 − 𝐴𝑣 𝑄 − ℎ𝑣𝑠 𝑅 + ℎ𝑣𝑄 2𝑅 − (∑𝑁𝑖=1 ℎ𝑣𝜎𝑖 √2𝜋 )]𝑢 − 2𝐼𝜃 (3) 𝜕2𝐴𝐸𝑃𝑚 𝜕𝑛2 = − 2𝐴𝑣𝐷 𝑛3𝑠 + ℎ𝑣𝑠 𝑛3 (4) 𝜕2𝐴𝐸𝑃𝑚 𝜕𝜃2 = −2𝐼 (5) 𝜕2𝐴𝐸𝑃𝑚 𝜕𝑛𝜕𝜃 = 𝐴𝑣𝑢 𝑛2𝑠 + ℎ𝑣𝑠𝑢 2𝑅 (6) 𝜕2𝐴𝐸𝑃𝑚 𝜕𝜃𝜕𝑛 = 𝐴𝑣𝑢 𝑛2𝑠 + ℎ𝑣𝑠𝑢 2𝑅 , where 𝑠 = ∑𝑁𝑖=1 𝑞𝑖 and 𝑢 = ∑ 𝑁 𝑖=1 𝛼𝑖 (7) Proposition 1. The average expected profit function of the manufacturer is jointly concave in 𝑛 and 𝜃 if 8𝐼𝑅2𝑛𝑠(2𝐴𝑣𝐷𝑠 − ℎ𝑣𝑠 2) > (2𝐴𝑣𝑅𝑢 + ℎ𝑣𝑢𝑛 2𝑠2)2. Proof. Considering 𝑛 as real, the Hessian matrix is 𝐻 = ( 𝜕2𝐴𝐸𝑃𝑚 𝜕𝜃2 𝜕2𝐴𝐸𝑃𝑚 𝜕𝜃𝜕𝑛 𝜕2𝐴𝐸𝑃𝑚 𝜕𝑛𝜕𝜃 𝜕2𝐴𝐸𝑃𝑚 𝜕𝑛2 ) = ( −2𝐼 𝐴𝑣𝑢 𝑛2𝑠 + ℎ𝑣𝑠𝑢 2𝑅 𝐴𝑣𝑢 𝑛2𝑠 + ℎ𝑣𝑠𝑢 2𝑅 − 2𝐴𝑣𝐷 𝑛3𝑠 + ℎ𝑣𝑠 𝑛3 ) (8) (1) Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 14 Here, 𝜕2𝐴𝐸𝑃𝑚 𝜕𝜃2 = −2𝐼 < 0 . So, the expected average profit function of the manufacturer will be concave in 𝜃 and 𝑛 if |𝐻| > 0. Substituting the values of the partial derivatives from the above and using the condition |𝐻| > 0, we get after simplification, 8𝐼𝑅2𝑛𝑠(2𝐴𝑣𝐷𝑠 − ℎ𝑣𝑠 2) > (2𝐴𝑣𝑅𝑢 + ℎ𝑣𝑢𝑛 2𝑠2)2. Proposition 2. At the equilibrium, the optimal number of shipments to each retailer, and the optimal greening level of the product are as follows: 𝑛∗ = √ 𝑅(2𝐴𝑣𝐷−ℎ𝑣𝑠 2) ℎ𝑣𝑠 2(𝑅−𝐷) (9) 𝜃∗ = [𝑤− 𝐴𝑣 𝑄 − ℎ𝑣𝑠 𝑅 + ℎ𝑣𝑄 2𝑅 −(∑𝑁𝑖=1 ℎ𝑣𝜎𝑖 √2𝜋 )]𝑢 2𝐼 (10) Proof. At the equilibrium, we have 𝜕𝐴𝐸𝑃𝑚 𝜕𝑛 = 𝐴𝑣𝐷 𝑛2𝑠 − ℎ𝑣𝑠 2 + ℎ𝑣𝑠𝐷 2𝑅 − ℎ𝑣𝑠 2𝑛2 = 0 (11) and 𝜕𝐴𝐸𝑃𝑚 𝜕𝜃 = [𝑤 − 𝐴𝑣 𝑄 − ℎ𝑣𝑠 𝑅 + ℎ𝑣𝑄 2𝑅 − (∑𝑁𝑖=1 ℎ𝑣𝜎𝑖 √2𝜋 )]𝑢 − 2𝐼𝜃 = 0 (12) Solving equations (11) and (12), we get the optimal values of 𝑛 and 𝜃 as given in equations (9) and (10) above. For integer optimal value of 𝑛, 𝑛𝑜𝑝𝑡 = { ⌊𝑛∗⌋, 𝑖𝑓 𝐴𝐸𝑃𝑚(⌊𝑛 ∗⌋,𝜃) ≥ 𝐴𝐸𝑃𝑚(⌈𝑛 ∗⌉,𝜃) ⌈𝑛∗⌉, 𝑖𝑓 𝐴𝐸𝑃𝑚(⌊𝑛 ∗⌋,𝜃) ≤ 𝐴𝐸𝑃𝑚(⌈𝑛 ∗⌉,𝜃) Taking these response functions of the manufacturer, the retailers then set their batch sizes and retail prices. Average expected profit of the 𝑖-th retailer Since the expected cycle length for the 𝑖-th retailer is Qi Di , therefore, the average ordering cost of the 𝑖-th retailer is given by 𝐴𝑖𝐷𝑖 𝑄𝑖 . From manufacturing to delivery to the end customer and even returns, transportation is essential to the entire production process. It is practically impossible for a logistics firm to conduct business efficiently without transportation. As the number of shipments increases, the transportation cost increases. Since the manufacturer delivers order quantity to the 𝑖-th retailer in 𝑛 shipments and the expected cycle length for the 𝑖-th retailer is 𝑄𝑖 𝐷𝑖 , therefore, the average variable transportation cost is 𝑛𝐹𝐷𝑖 𝑄𝑖 . The expected total profit of the 𝑖-th retailer is 𝑝𝑖𝑄𝑖 − 𝑤𝑄𝑖 − 𝐴𝑖 − 𝑛𝐹 − 𝑛ℎ𝑖 2 [∫   𝑟𝑖 𝐷𝑖 0 [ 𝑞𝑖 2 𝐷𝑖 +2𝑞𝑖 ( 𝑟𝑖 𝐷𝑖 − 𝑙𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫   𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 (𝑞𝑖 − 𝐷𝑖𝑙𝑖 + 𝑟𝑖) 2 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] − 𝑛𝑐𝑖 2 [∫   𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 (𝐷𝑖𝑙𝑖 − 𝑟𝑖) 2 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫   ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 [ 𝑞𝑖 2 𝐷𝑖 + 2𝑞𝑖 ( 𝐷𝑖𝑙𝑖 −𝑞𝑖 − 𝑟𝑖 𝐷𝑖 )]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 15 Therefore, the average expected profit of the 𝑖-th retailer is obtained as 𝐴𝐸𝑃𝑖(𝑞𝑖,𝑝𝑖) = 𝑝𝑖𝐷𝑖 − 𝑤𝐷𝑖 − (𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑄𝑖 − ℎ𝑖 2 [∫   𝑟𝑖 𝐷𝑖 0 [𝑞𝑖 + 2(𝑟𝑖 − 𝐷𝑖𝑙𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫   𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 (𝑞𝑖−𝐷𝑖𝑙𝑖+𝑟𝑖) 2 𝑞𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] − 𝑐𝑖 2 [∫   𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 (𝐷𝑖𝑙𝑖−𝑟𝑖) 2 𝑞𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫   ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 [𝑞𝑖 + 2(𝐷𝑖𝑙𝑖 − 𝑞𝑖 −𝑟𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] (13) Proposition 3. The average expected profit of the 𝑖-th retailer is concave in 𝑞𝑖 for given 𝑝𝑖 if 2(𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑛 + (ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝐷𝑖 2𝑙𝑖 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. Proof. Differentiating (13) twice with respect to 𝑞𝑖, we obtain ∂𝐴𝐸𝑃𝑖 ∂𝑞𝑖 = (𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑛𝑞𝑖 2 − ℎ𝑖 2 ∫   𝑞𝑖 𝑅 0 ( 𝑅+2𝐷𝑖 𝑅 )𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 − ℎ𝑖 2 ∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 ( 𝑞𝑖 2(𝑅+𝐷𝑖) 2−𝑅2𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 2𝑅2 )𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑐𝑖 2 ∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 ( 𝑅2𝐷𝑖 2𝑙𝑖 2−𝑞𝑖 2𝐷𝑖 2 𝑞𝑖 2𝑅2 )𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑐𝑖 2 ∫   ∞ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 ( 𝑅+2𝐷𝑖 𝑅 )𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 (14) ∂2𝐴𝐸𝑃𝑖 ∂𝑞𝑖 2 = − 2(𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑛𝑞𝑖 3 − ℎ𝑖 ∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 3 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 − 𝑐𝑖 ∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 3 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 (15) For given 𝑝𝑖, the average expected profit function of the 𝑖-th retailer is concave in 𝑞𝑖 if 𝜕2𝐴𝐸𝑃𝑖 𝜕𝑞𝑖 2 is negative. This implies the condition 2(𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑛 + (ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝐷𝑖 2𝑙𝑖 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. Proposition 4. The average expected profit of the 𝑖-th retailer is concave in 𝑝𝑖 for given 𝑞𝑖 if 2𝛽𝑖 + (ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛽𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. Proof. Differentiating (13) with respect to 𝑝𝑖, we obtain ∂𝐴𝐸𝑃𝑖 ∂𝑝𝑖 = 𝐷𝑖 − 𝛽𝑖𝑝𝑖 + ℎ𝑖 ∫   𝑞𝑖 𝑅 0 𝛽𝑖(𝑞𝑖 − 𝑅𝑙𝑖) 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + ℎ𝑖 ∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛽𝑖(𝑞𝑖 − 𝑅𝑙𝑖)[𝑞𝑖𝑅 + 𝐷𝑖(𝑞𝑖 −𝑅𝑙𝑖)] 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +𝑤𝛽𝑖 + (𝐴𝑖 + 𝑛𝐹)𝛽𝑖 𝑄𝑖 + 𝑐𝑖 ∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛽𝑖𝐷𝑖(𝑞𝑖 −𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 − 𝑐𝑖 ∫   ∞ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝛽𝑖(𝑞𝑖 − 𝑅𝑙𝑖) 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 (16) (12) (13) (14) (15) (16) (16) Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 16 ∂2𝐴𝐸𝑃𝑖 ∂𝑝𝑖 2 = −2𝛽𝑖 − (ℎ𝑖 + 𝑐𝑖)∫   𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛽𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 (17) Since 𝛽𝑖,ℎ𝑖 and 𝑐𝑖 all are positive, therefore, it implies that 2βi + (hi +ci)∫ qi(R+Di) RDi qi R βi 2(qi-Rli) 2 qiR 2 fL(li)dli > 0. Therefore, the average expected profit function 𝐴𝐸𝑃𝑖 is concave in 𝑝𝑖 for given 𝑞𝑖 if the above condition satisfies. Solution Algorithm Taking the best response from the manufacturer, the average expected profit of the 𝑖-th retailer can be optimized using the following solution algorithm. To optimize the expected average profit of the 𝑖-th retailer, we consider initial guess values to the decision variables of the remaining (𝑁 − 1) retailers. Step 1: Set 𝑘 = 1. Step 2: Set 𝑖 = 1 and 𝑞𝑗 = 𝑞𝑗 (𝑘−1) , 𝑝𝑗 = 𝑝𝑗 (𝑘−1) for all 𝑗 = 𝑖 + 1,𝑖 + 2, . . . . . ,𝑁. Step 3: Optimize 𝐴𝐸𝑃𝑖 taking 𝑛 and 𝜃 from the response functions of the manufacturer and 𝑞𝑗 = 𝑞𝑗 (𝑘−1) ,𝑝𝑗 = 𝑝𝑗 (𝑘−1) for all 𝑗 = 𝑖 +1,𝑖 + 2, . . . . . ,𝑁. Set the optimal results as 𝑞𝑖 = 𝑞𝑖 (𝑘) and 𝑝𝑖 = 𝑝𝑖 (𝑘) . Step 4: Set 𝑖 = 𝑖 +1. Step 5: Optimize 𝐴𝐸𝑃𝑖 taking 𝑛 and 𝜃 from the manufacturer’s response functions and 𝑞𝑗 = 𝑞𝑗 (𝑘) , 𝑝𝑗 = 𝑝𝑗 (𝑘) for 𝑗 = 1,2, . . . , 𝑖 − 1 and 𝑞𝑗 = 𝑞𝑗 (𝑘−1) ,𝑝𝑗 = 𝑝𝑗 (𝑘−1) for 𝑗 = 𝑖 + 1, 𝑖 + 2, . . . . ,𝑁. Set the optimal results as 𝑞𝑖 = 𝑞𝑖 (𝑘) and 𝑝𝑖 = 𝑝𝑖 (𝑘) . Step 6: Repeat steps 4 and 5 until 𝑖 = 𝑁. Step 7: Stop if 𝑞𝑗 (𝑘) = 𝑞𝑗 (𝑘−1) and 𝑝𝑗 (𝑘) = 𝑝𝑗 (𝑘−1) for all 𝑗 = 2,3, . . . . . ,𝑁 and consider 𝑞𝑗 (∗) = 𝑞𝑗 (𝑘) and 𝑝𝑗 (∗) = 𝑝𝑗 (𝑘) for all 𝑗 = 1,2,3, . . . . . ,𝑁. Otherwise, set 𝑘 = 𝑘 +1 and repeat steps 2 to 6. Step 8: Evaluate the optimal values of 𝑛∗ and 𝜃∗ taking 𝑞𝑗 ∗ and 𝑝𝑗 ∗ for all 𝑗 = 1,2,3, . . . . . ,𝑁 . Step 9: Using these results, calculate optimal values of 𝐴𝐸𝑃𝑚 and 𝐴𝐸𝑃𝑠. 5.2. Centralized Model (CM) In this scenario, the manufacturer and all the retailers of the supply chain act jointly as a single decision maker. They determine the optimal selling prices of the product, greening improvement level, number of shipments and batch sizes in order to maximize the entire system profit rather than focusing on their individual profits. The average expected profit of the supply chain is 𝐴𝐸𝑃𝑠(𝑛,𝜃, 𝑞𝑖, 𝑝𝑖) = ∑ 𝑝𝑖𝐷𝑖 − 𝐴𝑣𝐷 𝑄 − ℎ𝑣 [ 𝐷∑  𝑁𝑖=1 𝑞𝑖 𝑅 + 𝑄 2 (1 − 𝐷 𝑅 )− ∑  𝑁𝑖=1 𝑞𝑖 2 2𝑄𝑖 ]𝑁𝑖=1 −∑  𝑁𝑖=1 ℎ𝑣𝐷𝑖𝜎𝑖 √2𝜋 − 𝐼𝜃2 − ∑ [ (𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑄𝑖 + ℎ𝑖 2 [∫   𝑟𝑖 𝐷𝑖 0 [𝑞𝑖 + 2(𝑟𝑖 − 𝑁 𝑖=1 Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 17 𝐷𝑖𝑙𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫   𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 (𝑞𝑖−𝐷𝑖𝑙𝑖+𝑟𝑖) 2 𝑞𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] + 𝑐𝑖 2 [∫   𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 (𝐷𝑖𝑙𝑖−𝑟𝑖) 2 𝑞𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫   ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 [𝑞𝑖 + 2(𝐷𝑖𝑙𝑖 − 𝑞𝑖 −𝑟𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖]] (18) Proposition 5. The average expected profit of the system is concave in 𝑛 for given θ, 𝑞𝑖 and 𝑝𝑖 if ℎ𝑣𝑠 2 < 2(𝐴𝑣𝐷 +𝑠𝑔), where 𝑔 = ∑ 𝑁 𝑖=1 𝐴𝑖𝐷𝑖 𝑞𝑖 and the optimal number of shipments is given by 𝑛∗ = √ 𝑅(2𝐴𝑉𝐷−ℎ𝑣𝑠 2+2𝑠𝑔) ℎ𝑣𝑠 2(𝑅−𝐷) (19) Proof. Considering 𝑛 as real, from equation (18), we derive the following partial derivatives: 𝜕𝐴𝐸𝑃𝑠 𝜕𝑛 = 𝐴𝑣𝐷 𝑛2𝑠 − ℎ𝑣𝑠 2 + ℎ𝑣𝑠𝐷 2𝑅 − ℎ𝑣𝑠 2𝑛2 + 𝑔 𝑛2 (20) 𝜕2𝐴𝐸𝑃𝑠 𝜕𝑛2 = − 2𝐴𝑣𝐷 𝑛3𝑠 + ℎ𝑣𝑠 𝑛3 − 2𝑔 𝑛3 where 𝑠 = ∑ 𝑞𝑖 𝑁 𝑖=1 (21) The average expected profit of the system will be concave in 𝑛, for given θ, 𝑞𝑖 and 𝑝𝑖, if ∂2AEPs ∂n2 < 0, which implies that ℎ𝑣𝑠 2 < 2(𝐴𝑣𝐷 + 𝑠𝑔). If the above condition holds then the system profit function attains the maximum value with respect to 𝑛, and the optimal value of 𝑛 can be obtained by using the first order optimality condition i.e., 𝜕𝐴𝐸𝑃𝑠 𝜕𝑛 = 0. Solving it for 𝑛, one can get the optimal number of shipments as 𝑛∗ = √ 𝑅(2𝐴𝑉𝐷−ℎ𝑣𝑠 2+2𝑠𝑔) ℎ𝑣𝑠 2(𝑅−𝐷) . For integer optimal value of 𝑛, 𝑛𝑜𝑝𝑡 = { ⌊𝑛∗⌋, 𝑖𝑓 𝐴𝐸𝑃𝑠(⌊𝑛 ∗⌋,𝜃,𝑞𝑖,𝑝𝑖 ) ≥ 𝐴𝐸𝑃𝑠(⌈𝑛 ∗⌉,𝜃,𝑞𝑖,𝑝𝑖) ⌈𝑛∗⌉, 𝑖𝑓 𝐴𝐸𝑃𝑠(⌊𝑛 ∗⌋,𝜃,𝑞𝑖,𝑝𝑖) ≤ 𝐴𝐸𝑃𝑠(⌈𝑛 ∗⌉,𝜃,𝑞𝑖,𝑝𝑖) Proposition 6. For given values of 𝑛, 𝑞𝑖 and 𝑝𝑖, the average expected profit function of the supply chain is concave in 𝜃 if 2𝐼 + ∑𝑁𝑖=1 (ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛼𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. Proof. From equation (18), we get 𝜕𝐴𝐸𝑃𝑠 𝜕𝜃 = −[ 𝐴𝑣 𝑄 + ℎ𝑣𝑠 𝑅 − ℎ𝑣𝑄 2𝑅 + (∑ ℎ𝑣𝜎𝑖 √2𝜋 𝑁 𝑖=1 )]𝑢 − 2𝐼𝜃 − ∑[−𝑝𝑖𝛼𝑖 + (𝐴𝑖 + 𝑛𝐹)𝛼𝑖 𝑄𝑖 𝑁 𝑖=1 +ℎ𝑖 ∫ 𝛼𝑖(𝑞𝑖−𝑅𝑙𝑖) 𝑅 𝑞𝑖 𝑅 0 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +ℎ𝑖 ∫ 𝛼𝑖(𝑞𝑖−𝑅𝑙𝑖)[𝑞𝑖𝑅+𝐷𝑖(𝑞𝑖−𝑅𝑙𝑖)] 𝑞𝑖𝑅 2 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +𝑐𝑖 ∫ 𝛼𝑖𝐷𝑖(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 − 𝑐𝑖 ∫ 𝛼𝑖(𝑞𝑖−𝑅𝑙𝑖) 𝑅 ∞ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] (22) (21) (22) Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 18 𝜕2𝐴𝐸𝑃𝑠 𝜕𝜃2 = −2𝐼 − ∑ (ℎ𝑖 + 𝑐𝑖)∫ 𝛼𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝑁 𝑖=1 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 (23) It can be easily seen that 𝜕2𝐴𝐸𝑃𝑠 𝜕𝜃2 < 0 if 2𝐼 + ∑𝑁𝑖=1 (ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛼𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. Therefore, we can conclude that the average expected profit function of the supply chain system is concave in θ for given 𝑛, 𝑞𝑖 and 𝑝𝑖, if this condition holds. Proposition 7. The average expected profit function of the supply chain system is concave in 𝑞𝑖 for given 𝑛, θ and 𝑝𝑖 if 2𝐴𝑣𝐷 𝑠3 + 2(𝐴𝑖 + 𝑛𝐹)𝐷𝑖 𝑞𝑖 3 + 𝑛(ℎ𝑖 + 𝑐𝑖) ∫ 𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 3 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 Proof. Differentiating (18) with respect to 𝑞𝑖, we get 𝜕𝐴𝐸𝑃𝑠 𝜕𝑞𝑖 = 𝐴𝑣𝐷 𝑛𝑠2 − ℎ𝑣𝐷 𝑅 − 𝑛ℎ𝑣 2 (1 − 𝐷 𝑅 )+ ℎ𝑣 2𝑛 + (𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑛𝑞𝑖 2 − ℎ𝑖 2 ∫ ( 𝑅+2𝐷𝑖 𝑅 ) 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑞𝑖 𝑅 0 − ℎ𝑖 2 ∫ ( 𝑞𝑖 2(𝑅 + 𝐷𝑖) 2 − 𝑅2𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 2𝑅2 ) 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑐𝑖 2 ∫ ( 𝑅2𝐷𝑖 2𝑙𝑖 2 − 𝑞𝑖 2𝐷𝑖 2 𝑞𝑖 2𝑅2 ) 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 + 𝑐𝑖 2 ∫ ( 𝑅+2𝐷𝑖 𝑅 ) 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 ∞ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 (24) 𝜕2𝐴𝐸𝑃𝑠 𝜕𝑞𝑖 2 = − 2𝐴𝑣𝐷 𝑛𝑠3 − 2(𝐴𝑖 + 𝑛𝐹)𝐷𝑖 𝑛𝑞𝑖 3 − ℎ𝑖 ∫ 𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 3 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 −𝑐𝑖 ∫ 𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 3 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 (25) Now, the average expected profit of the entire supply chain is concave in 𝑞𝑖 for given 𝑛, θ and 𝑝𝑖, if 𝜕2𝐴𝐸𝑃𝑠 𝜕𝑞𝑖 2 < 0, which gives 2𝐴𝑣𝐷 𝑠3 + 2(𝐴𝑖+𝑛𝐹)𝐷𝑖 𝑞𝑖 3 + 𝑛(ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝐷𝑖 2𝑙𝑖 2 𝑞𝑖 3 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. Proposition 8. The average expected profit function of the supply chain system is concave in 𝑝𝑖 for given 𝑛, 𝜃 and 𝑞𝑖 if 2𝛽𝑖 +(ℎ𝑖 +𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛽𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. Proof. Differentiating (18) with respect to 𝑝𝑖, we get (23) (24) (25) (26) Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 19 𝜕𝐴𝐸𝑃𝑠 𝜕𝑝𝑖 = 𝐷𝑖 −𝛽𝑖𝑝𝑖 + 𝐴𝑣𝛽𝑖 𝑄 + ℎ𝑣𝛽𝑖𝑠 𝑅 − ℎ𝑣𝑄𝛽𝑖 2𝑅 + ℎ𝑣𝛽𝑖𝜎𝑖 √2𝜋 + (𝐴𝑖 + 𝑛𝐹)𝛽𝑖 𝑄𝑖 + ℎ𝑖 ∫ 𝛽𝑖(𝑞𝑖 −𝑅𝑙𝑖) 𝑅 𝑞𝑖 𝑅 0 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +ℎ𝑖 ∫ 𝛽𝑖(𝑞𝑖 − 𝑅𝑙𝑖)[𝑞𝑖𝑅 + 𝐷𝑖(𝑞𝑖 − 𝑅𝑙𝑖)] 𝑞𝑖𝑅 2 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑐𝑖 ∫ 𝛽𝑖𝐷𝑖(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 (26) −𝑐𝑖 ∫ 𝛽𝑖(𝑞𝑖−𝑅𝑙𝑖) 𝑅 ∞ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝜕2𝐴𝐸𝑃𝑠 𝜕𝑝𝑖 2 = −2𝛽𝑖 − (ℎ𝑖 + 𝑐𝑖)∫ 𝛽𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 (27) The average expected profit function of the entire supply chain is concave in 𝑝𝑖 for given 𝑛, 𝜃 and 𝑞𝑖, if 𝜕2𝐴𝐸𝑃𝑠 𝜕𝑝𝑖 2 < 0 i.e., if 2𝛽𝑖 + (ℎ𝑖 + 𝑐𝑖)∫ 𝑞𝑖(𝑅+𝐷𝑖) 𝑅𝐷𝑖 𝑞𝑖 𝑅 𝛽𝑖 2(𝑞𝑖−𝑅𝑙𝑖) 2 𝑞𝑖𝑅 2 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 > 0. 5.3. Coordinated Model (COM) Supply chain players agree to accept the joint-decision making policy only if it provides a better profit than the decentralized model scenario. To motivate supply chain members to make integrated decisions, an incentive strategy is required. In this section, we propose a coordination mechanism between the manufacturer and the retailers, which motivates the members to accept the integrated decision-making policy. In this coordination mechanism, the manufacturer requests the retailers to decide their optimal batch sizes (𝑞𝑖) and retail prices (𝑝𝑖) according to the centralized policy and, in return, the manufacturer decreases his wholesale price (𝑤). Suppose that the manufacturer offers the price discount scheme to the 𝑖-th retailer as follows: 𝑤 = { 𝑤, 𝑖𝑓 𝑞𝑖 < 𝑞𝑖 𝑐∗ 𝑤(1 −𝜙𝑖), 𝑖𝑓 𝑞𝑖 ≥ 𝑞𝑖 𝑐∗ (28) For this price discount scheme, the average expected profit of the 𝑖-th retailer becomes 𝐴𝐸𝑃𝑖 𝑐𝑜(𝑞𝑖 𝑐,𝑝𝑖 𝑐,𝜙𝑖) = 𝑝𝑖 𝑐𝐷𝑖 𝑐 −(1 −𝜙𝑖)𝑤𝐷𝑖 𝑐 − (𝐴𝑖+𝑛 𝑐𝐹)𝐷𝑖 𝑐 𝑄𝑖 𝑐 − [ ℎ𝑖 2 ∫ [𝑞𝑖 𝑐 + 2(𝑟𝑖 − 𝑟𝑖 𝐷 𝑖 𝑐 0 𝐷𝑖 𝑐𝑙𝑖)]𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +ℎ𝑖 ∫ (𝑞𝑖 𝑐 −𝐷𝑖 𝑐 𝑙𝑖+𝑟𝑖) 2 2𝑞𝑖 𝑐 𝑟𝑖+𝑞𝑖 𝑐 𝐷 𝑖 𝑐 𝑟𝑖 𝐷 𝑖 𝑐 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖]− [ 𝑐𝑖 2 ∫ (𝐷𝑖 𝑐 𝑙𝑖−𝑟𝑖) 2 𝑞𝑖 𝑐 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑟𝑖+𝑞𝑖 𝑐 𝐷 𝑖 𝑐 𝑟𝑖 𝐷 𝑖 𝑐 𝑐𝑖 2 ∫ [𝑞𝑖 𝑐 + 2(𝐷𝑖 𝑐𝑙𝑖 − 𝑞𝑖 𝑐 − 𝑟𝑖)] ∞ 𝑟𝑖+𝑞𝑖 𝑐 𝐷 𝑖 𝑐 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] (29) and the average expected profit of the manufacturer becomes (27) (28) (29) Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 20 𝐴𝐸𝑃𝑚 𝑐𝑜(𝑛𝑐,𝜃𝑐,𝜙𝑖) = ∑(1 −𝜙𝑖)𝑤𝐷𝑖 𝑐 − 𝐴𝑣𝐷 𝑐 𝑄𝑐 − ℎ𝑣[ 𝐷𝑐 ∑ 𝑞𝑖 𝑐𝑁 𝑖=1 𝑅 + 𝑄𝑐 2 (1− 𝐷𝑐 𝑅 ) 𝑁 𝑖=1 −∑ 𝑞𝑖 𝑐 2𝑛𝑐 ] −∑ ℎ𝑣𝐷𝑖 𝑐𝜎𝑖 √2𝜋 𝑁 𝑖=1 𝑁 𝑖=1 − 𝐼𝜃2 (30) Proposition 9. The minimum value of 𝜙𝑖 for which the 𝑖-th retailer accepts the coordination mechanism is 𝜙𝑖 𝑚𝑖𝑛 = (𝑝𝑖 𝑑 𝐷𝑖 𝑑 −𝑝𝑖 𝑐 𝐷𝑖 𝑐 )−𝑤(𝐷𝑖 𝑑 −𝐷𝑖 𝑐 )−∆𝑑+∆𝑐 𝑤𝐷𝑖 𝑐 where, ∆= (𝐴𝑖 +𝑛𝐹)𝐷𝑖 𝑄𝑖 + ℎ𝑖 2 [∫ [𝑞𝑖 + 2(𝑟𝑖 − 𝐷𝑖𝑙𝑖)] 𝑟𝑖 𝐷𝑖 0 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 +∫ (𝑞𝑖 − 𝐷𝑖𝑙𝑖 + 𝑟𝑖) 2 𝑞𝑖 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖] +[ 𝑐𝑖 2 ∫ (𝐷𝑖𝑙𝑖 − 𝑟𝑖) 2 𝑞𝑖 𝑓𝐿(𝑙𝑖)𝑑𝑙𝑖 + 𝑟𝑖+𝑞𝑖 𝐷𝑖 𝑟𝑖 𝐷𝑖 𝑐𝑖 2 ∫ [𝑞𝑖 + 2(𝐷𝑖𝑙𝑖 − 𝑞𝑖 − 𝑟𝑖)]𝑓𝐿 ∞ 𝑟𝑖+𝑞𝑖 𝐷𝑖 (𝑙𝑖)𝑑𝑙𝑖] Proof. The retailer's goal in engaging in the coordination is to find the minimum discount level so that his profit is more or equal to the profit in the decentralized situation. So, 𝐴𝐸𝑃𝑖 𝑐𝑜(𝑞𝑖 𝑐,𝑝𝑖 𝑐,𝜙𝑖) ≥ 𝐴𝐸𝑃𝑖 𝑑(𝑞𝑖 𝑑,𝑝𝑖 𝑑) (31) Solving the inequality (31), we get 𝜙𝑖 ≥ (𝑝𝑖 𝑑 𝐷𝑖 𝑑 −𝑝𝑖 𝑐 𝐷𝑖 𝑐 )−𝑤(𝐷𝑖 𝑑 −𝐷𝑖 𝑐 )−∆𝑑+∆𝑐 𝑤𝐷𝑖 𝑐 (32) Therefore, if the wholesale price discount offered by the manufacturer does not satisfy the above condition, the 𝑖-th retailer will not accept the contract. So, to motivate the 𝑖-th retailer, the manufacturer should give at least 𝜙𝑖 discount level given by 𝜙𝑖 𝑚𝑖𝑛 = (𝑝𝑖 𝑑 𝐷𝑖 𝑑 −𝑝𝑖 𝑐 𝐷𝑖 𝑐 )−𝑤(𝐷𝑖 𝑑 −𝐷𝑖 𝑐 )−∆𝑑+∆𝑐 𝑤𝐷𝑖 𝑐 (33) Proposition 10. The maximum discount level offered by the manufacturer to the 𝑖-th retailer is given by 𝜙𝑖 𝑚𝑎𝑥 = 𝑤(𝐷𝑖 𝑐 −𝐷𝑑)−∇𝑐+∇𝑑+∑ (1−𝜙𝑗)𝑤𝐷𝑗 𝑐𝑁 𝑗=1 𝑗≠𝑖 𝑤𝐷𝑖 𝑐 (34) Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 21 where,∇= 𝐴𝑣𝐷 𝑄 + ℎ𝑣 [ 𝐷∑ 𝑞𝑖 𝑁 𝑖=1 𝑅 + 𝑄 2 (1 − 𝐷 𝑅 ) − ∑ 𝑞𝑖 2𝑛 𝑁 𝑖=1 ]+ ∑ ℎ𝑣𝐷𝑖𝜎𝑖 √2𝜋 + 𝐼𝜃2 𝑁 𝑖=1 Proof. The manufacturer will offer the price discount scheme if he/she gains more profit after giving price discount to all the retailers in this coordination than the decentralized scenario. So, if the manufacturer provides a 𝜙𝑖 discount level to the 𝑖-th retailer, then 𝐴𝐸𝑃𝑚 𝑐𝑜(𝑛𝑐,𝜃𝑐,𝜙𝑖) ≥ 𝐴𝐸𝑃𝑚 𝑑(𝑛𝑑,𝜃𝑑) (35) Simplifying (35), we get, 𝜙𝑖 ≤ 𝑤(𝐷𝑖 𝑐 −𝐷𝑑)−∇𝑐+∇𝑑+∑ (1−𝜙𝑗)𝑤𝐷𝑗 𝑐𝑁 𝑗=1 𝑗≠𝑖 𝑤 𝐷𝑖 𝑐 (36) So, if the manufacturer gives 𝜙𝑖% price discount to the 𝑖-th retailer, then the maximum allowable discount level for the manufacturer will be 𝜙𝑖 𝑚𝑎𝑥 = 𝑤(𝐷𝑖 𝑐 −𝐷𝑑)−∇𝑐+∇𝑑+∑ (1−𝜙𝑗)𝑤𝐷𝑗 𝑐𝑁 𝑗=1 𝑗≠𝑖 𝑤 𝐷𝑖 𝑐 From Propositions (9) and (10), it can be observed that, the 𝑖-th retailer will accept the discount offer for all 𝜙 ≥ 𝜙𝑖 𝑚𝑖𝑛 . Therefore, all the 𝑁 retailers will accept the discount scheme (28) if 𝜙 ≥ 𝜙𝑚𝑖𝑛, where 𝜙𝑚𝑖𝑛 = 𝑚𝑎𝑥{𝜙1 𝑚𝑖𝑛 ,𝜙2 𝑚𝑖𝑛 ,𝜙3 𝑚𝑖𝑛 , . . . . . . . . . . ,𝜙𝑁 𝑚𝑖𝑛 } and the manufacturer will provide this price discount only if 𝜙 ≤ 𝜙𝑚𝑎𝑥 = min {𝜙1 𝑚𝑎𝑥,𝜙2 𝑚𝑎𝑥,𝜙3 𝑚𝑎𝑥,……. ,𝜙𝑁 𝑚𝑎𝑥} Hence, for all 𝜙 in [𝜙𝑚𝑖𝑛,𝜙𝑚𝑎𝑥] the coordination through the price discount scheme (28) will result better profit level for both the manufacturer and the retailers than the decentralized scenario. Since the manufacturer sells the product to all the retailers at the same wholesale price, therefore, we assume that he/she offers the same price discount ratio 𝜙 to each retailer. 6. Numerical Analysis In this section, we consider three numerical examples to analyze the behaviour of our proposed model and its applicability. Here we focus on the scenario where one manufacturer is trading with two retailers. Example 1: The following set of parameter-values presented in Table 2 are considered to demonstrate the proposed model numerically. As it is difficult to get access to the actual industrial data, some of the parameter-values are taken from Hoque (2013) and the rest are hypothetical. The p.d.f. of lead time (𝑙𝑖) of the 𝑖-th retailer is 𝑓𝐿(𝑙𝑖) = 1 √2Π𝜎𝑖 𝑒 − 1 2𝜎 𝑖 2(𝑙𝑖− 𝑟𝑖 𝐷𝑖 ) 2 . Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 22 Table 2. Set of parameter-values for Example 1 Parameter Value Parameter Value 𝑅 3000 units/ year 𝑎1 1500 units / year 𝐴𝑣 $400 /set up 𝑎2 1500 units / year 𝐴1 $40 /order 𝛽1 4 𝐴2 $45 / order 𝛽2 4.5 𝑤 $100 / unit 𝛼1 2 𝐹 $10 /shipment 𝛼2 1.5 ℎ𝑣 $3.5/ unit / year 𝜎1 0.12 ℎ1 $5.8 / unit / year 𝜎2 0.13 ℎ2 $5 / unit / year 𝐼 $40 𝑐1 $7 / unit / year 𝑁 2 𝑐2 $7 / unit / year Figure 4. Concavity of average expected profit function of the first retailer Figure 5. Concavity of average expected profit function of the second retailer As shown in Figures 4 and 5, for given parameter-values, the average expected profit functions of both the retailers are found to be concave with respect to the batch sizes and retail prices of the product. The optimal results are obtained using the computational software Mathematica 10.0 with the command FindMaximum. From the numerical results given in Table 3, we observe that the optimal order quantity, retail price, greening level of the product and number of shipments decided in the centralized scenario gives more system profit than that obtained in the decentralized scenario. In the centralized scenario, both the retailers can sell the product to the end customers at a cheaper price than the decentralized case. Since the customer demand is assumed to be price sensitive, the lower priced product attracts more customers. Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 23 Table 3. Optimal results for different models Models 𝑛∗ θ* 𝑞1 ∗ (unit) 𝑞2 ∗ (unit) 𝑝1 ∗ ($/unit) 𝑝2 ∗ ($/unit) ϕmin 𝜙𝑚𝑎𝑥 ϕ 𝐴𝐸𝑃1 ∗ ($/year) AEP2 * ($/year) 𝐴𝐸𝑃𝑚 ∗ ($/year) 𝐴𝐸𝑃𝑠 ∗ ($/year) DM COM CM 4 6 6 4.36 7.87 7.87 91.98 71.77 71.77 111.79 79.96 79.96 238.74 189.82 189.82 217.53 168.34 168.34 - 0.14 - - 0.27 - - 0.205 - 76577 83132 - 61799 66707 - 105584 115684 - 243960 265523 265523 Also, the manufacturer can produce more greener product by making the optimal decisions jointly. As the product's greening level has a positive impact on customer demand, customer demand in the centralized case is considerably higher than in the decentralized case, and all the retailers increase their order quantities. As a result, joint-decision making generates a higher system profit than the separate profit optimization. Figure 6. Price discount rate vs. average expected profit It is also observed from Table 3 that the channel coordination can be achieved through the price discount mechanism between the manufacturer and the retailers. The optimal results of the models reflect that embracing the price discount coordination mechanism boosts not only the total system profit but also the profits of individual supply chain members. For the first retailer, the minimum discount ratio to undertake the coordination mechanism is obtained as 0.12 and for the second retailer, it is 0.14. It is clear that if the manufacturer offers 12% discount, the first retailer will accept the offer but the second retailer will not, as it will cause a loss to him. Therefore, to motivate both the retailers for participating in the coordination, the manufacturer has to give at least 14% price discount. Again, the maximum discount ratio for which the manufacturer does not face any loss is obtained as 0.27. So, the manufacturer can Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 24 offer each retailer a maximum discount of 27%. Therefore, the win-win situation which occurs in the interval [𝜙𝑚𝑖𝑛,𝜙𝑚𝑎𝑥] is appeared as [0.14,0.27]. For any value of 𝜙 in this interval, the price discount mechanism becomes profitable for the manufacturer and both the retailers than the decentralized scenario. For 𝜙 lying in the interval [0.14,0.27], the average expected profits of the first and second retailers vary within the intervals [$78215,$88049] and [$61799,$71610], respectively and the average expected profit of the manufacturer lies within the interval [$105584,$125503]. In all cases, the average expected system profit remains $265523 i.e., our suggested coordination method effectively achieves channel coordination and results in the supply chain members sharing extra profit that occurs in the centralized scenario. Naturally, whenever the value of 𝜙 increases from 𝜙𝑚𝑖𝑛, the retailers profitability increases gradually and attains their maximum profits at 𝜙𝑚𝑎𝑥 while the manufacturer’s profit decreases, and at 𝜙𝑚𝑎𝑥, the manufacturer attains the same profitability as that of the decentralized case. This fact is plotted in Figure 6. The supply chain members can fix the value of 𝜙 through bargaining. Here we take the value of 𝜙 as the mean of the feasible interval [0.14,0.27] i.e., 0.205. Example 2: We consider the set of parameter values given in Table 4 to demonstrate the model, and the optimal results thus obtained are provided in Table 5. Table 4. Set of parameter-values for Example 2 Parameter Value Parameter Value R 4000 units/ year 𝑎1 2000 units / year 𝐴𝑣 $500 /set up 𝑎2 1800units / year 𝐴1 $50 /order 𝛽1 4.2 𝐴2 $50/ order 𝛽2 5 𝑤 $90/ unit 𝛼1 3 𝐹 $15/shipment 𝛼2 2.5 ℎ𝑣 $3/ unit / year 𝜎1 0.12 ℎ1 $6/ unit / year 𝜎2 0.13 ℎ2 $5.5/ unit / year 𝐼 $30 𝑐1 $7.4 / unit / year 𝑁 2 𝑐2 $7.4 / unit / year Table 5 shows that, when compared to a decentralized system, integrated decision making provides higher supply chain profit. Both the order quantity of each retailer as well as the product's greening improvement level increase in the centralized scenario compared to the decentralized scenario. In addition, the product's retail price falls at both the retailers. As a consequence, customers are enticed by a greener product at a lesser cost, which significantly increases market demand. In the coordinated model, the minimum wholesale price discount ratios for the two retailers are obtained as 2% and 7%, respectively, while the maximum allowable price discount ratio for the manufacturer is 17%. Therefore, for any price discount lying in the interval [7%,17%], a win-win situation arises, i.e., the wholesale price contract benefits every member of the supply chain. The value of 𝜙 is taken as the mean of this feasible interval [7%,17%] i.e., 12%. Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 25 Table 5. Optimal results for different models Models 𝑛∗ 𝜃∗ 𝑞1 ∗ (unit) 𝑞2 ∗ (unit) 𝑝1 ∗ ($/unit) 𝑝2 ∗ ($/unit) ϕmin 𝜙𝑚𝑎𝑥 ϕ 𝐴𝐸𝑃1 ∗ ($/year) 𝐴𝐸𝑃2 ∗ ($/year) 𝐴𝐸𝑃𝑚 ∗ ($/year) 𝐴𝐸𝑃𝑠 ∗ ($/year) DM COM CM 4 5 5 8.22 19.94 19.94 132.42 112.12 112.12 161.82 124.16 124.16 286.18 245.58 245.58 227.19 185.35 185.35 - 0.07 - - 0.17 - - 0.12 - 161024 170783 - 93638 97731 - 131443 140097 - 386105 408611 408611 From Table 5, it can be noticed that, by accepting the wholesale price discount contract, the profits of the two retailers are increased by 6% and 4%, respectively. Furthermore, the manufacturer earns about 7% more profit from this contract. 6.1. A comparative study with existing literature In this section, we attempt to compare the findings of our study to some previous research. Sarkar et al. (2020b) developed a single-vendor single-buyer model with equal-sized batch shipment policy and price-dependent demand in this direction. They did, however, take into account variable backorder and the inspection process, that are not considered in this study. Furthermore, their model didn’t take into account for stochastic lead time and greening investment. To compare the proposed model to Sarkar et al. (2020b), common parameter values from Sarkar et al. (2020b) are used, while the remaining parameter values are chosen at random. The proposed model is compared to Sarkar et al.'s (2020b) model in two different situations: without greening investment and with greening investment. The parameter values considered are given in Table 6. Table 6. Set of parameter-values for comparative study Parameter Value Parameter Value 𝑎1 11,000 units / year 𝛽1 320 𝐴𝑣 $200/set up 𝛼1 3 𝐴1 $20/order 𝜎1 0.02 𝑤 $10/ unit 𝐼 $80 𝐹 $5/shipment 𝑁 1 ℎ𝑣 $2/ unit / year 𝐷 𝑅 0.4 ℎ1 $5/ unit / year 𝑐1 $7.5/ unit / year For the case of without green sensitivity of the customer demand, we set 𝛼1 = 0, 𝐼 = 0. Figure 7 shows a comparative graphical representation of the average expected supply chain profit. Their centralized model obtains optimal batch size as 279 units, optimal number of shipments as 5, optimal retail price as $17 and optimal profit of the entire supply chain as $92021. Whereas our proposed model without green investment results the optimal batch size as 155.2 units, optimal number of shipments as 9, optimal retail price as $17.30, and the average expected supply chain profit as $92287. Furthermore, the proposed model with stochastic lead time and greening investment provides the optimal batch size as 223.26 units, optimal number of shipments as 8, optimal retail price as $28.24, optimal green level as 35.07 and the Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 26 average expected supply chain profit as $152186. As a conclusion of the above numerical results, it is apparent that adding the stochastic lead time and greening investment strategy makes the supply chain significantly more profitable. Figure 7. Comparison with existing literature 7. Sensitivity analysis In order to explore the impact of model parameters on the optimal decisions as well as the average expected system profit, in this section we vary one parameter- value at a time while keeping other parameter-values unchanged in Example 1. The results are shown in Table 7 from which the following conclusions can be drawn: From Table 7 and Figure 8, a significant change in overall profit of the system under the price discount coordination mechanism is observed for higher basic market demand. The first retailer can charge a higher price for the product whenever the customer demand increases at his side. This is because the first retailer compensates the effect of higher price by the higher market demand. He places order for more quantity from the manufacturer. Consequently, the profit of the first retailer as well as the manufacturer increases significantly. The changes in the order quantity and retail price of the product for the second retailer are almost negligible. As a result, the overall profit of the system increases. Similar scenario occurs whenever the market demand increases at the second retailer’s side. Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 27 Figure 8. Average expected profit vs. 𝑎2 Table 7. Sensitivity analysis of the parameters 𝑎1, 𝛽1, 𝛼1 and 𝐹 Parame- ters Values n* θ* 𝑞1 ∗ (unit) q2 * (unit) 𝑝1 ∗ ($/unit) 𝑝2 ∗ ($/unit) 𝐴𝐸𝑃1 𝑐𝑜 ($/year) 𝐴𝐸𝑃2 𝑐𝑜 ($/year) 𝐴𝐸𝑃𝑚 𝑐𝑜 ($/year) 𝐴𝐸𝑃𝑠 𝑐𝑜 ($/year) 𝑎1 1300 1400 1500 1600 1700 6 6 6 6 6 7.24 7.55 7.87 8.19 8.50 65.20 68.50 71.77 75.10 78.20 78.88 79.41 79.96 80.55 81.17 164.687 177.253 189.82 202.387 214.955 168.26 168.301 168.34 168.383 168.424 55569 68719 83132 98812 115761 66592 66649 66707 66765 66823 108057 111879 115684 119486 123285 230219 247242 265523 285065 305865 𝛽1 3 3.5 4 4.5 5 6 6 6 6 6 9.47 8.55 7.87 7.34 6.92 71.90 71.83 71.77 71.58 71.48 80.07 80.01 79.96 79.57 79.90 253.508 217.082 189.82 168.652 151.736 168.607 168.455 168.34 168.253 168.183 131649 103873 83132 67049 54229 67014 66838 66707 66605 66524 114825 115343 115684 115917 116088 313488 286055 265523 249571 236841 α1 1.5 2 2.5 3 3.5 6 6 6 6 6 6.67 7.87 9.08 10.31 11.56 71.58 71.77 72.01 72.27 72.58 79.87 79.96 80.07 80.18 80.30 189.104 189.82 190.69 191.718 192.908 168.143 168.34 168.543 168.747 168.954 82276 83132 84177 85419 86868 66478 66707 66940 67177 67418 116083 115684 115210 114654 114012 264837 265523 266327 267250 268298 F 0 10 20 30 40 11 6 4 4 3 7.87 7.87 7.87 7.87 7.87 38.99 71.77 97.95 103.46 124.43 43.67 79.96 111.89 116.70 144.46 189.752 189.82 189.89 189.911 189.98 168.282 168.34 168.41 168.421 168.48 83199 83132 83084 83022 82983 66751 66707 66680 66626 66605 115816 115684 115588 115563 115496 265766 265523 265352 265211 265084 Figure 9 depicts how the price sensitivity of the consumer demand affects the decision variables and the system profit. The figure shows that when the price sensitivity of consumer demand increases, the greater price of the product influences the customers' choice of alternatives. As a result, if customer demand becomes more price sensitive, the corresponding retailers lower their product prices to meet market demand, reducing the product's greenness. Figure 9 and Table 7 show that, under the coordination scheme, the average expected profit of both retailers and the total system profit decrease at a decreasing rate as price elasticity increases. Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 28 Figure 9. Average expected profit vs. 𝛽2 Table 7 shows the effect of customers' environmental awareness on optimal decisions and supply chain members’ profitability. When the values of 𝛼1 and 𝛼2 increase, customers are more concerned about the environmental performance of the product and they are willing to spend more for environmentally friendly products. In such a scenario, to satisfy the customers requirement, the manufacturer increases the greening level of the product. This fact is presented in Figure 10. Figure 10. Product’s greening level vs. 𝛼2 However, the higher greening level increases the expense of the manufacturer. So, the average expected profit of the manufacturer gradually decreases. On the other hand, the retailers can enhance the retail price of the product and achieve higher profitability with higher greener product. It is further observed that the average expected system profit increases for greater values of 𝛼1 and 𝛼2. Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 29 Figure 11. Product’s greening level vs. 𝐼 Table 7 illustrates the effect of the greening investment on the coordinated profit of the supply chain members and the profit of the entire supply chain. Figure 11 shows that the product's greening level falls rapidly while 𝐼 increases. When 𝐼 increases, the manufacturer produces lower greener product in order to curb his expenditure but it makes a negative impact on customer demand. So, for higher 𝐼, the profitability of the retailers decreases and the average expected profit of the entire supply chain also decreases gradually. The effect of the transportation cost is found to be negligible on the supply chain’s profitability. If we ignore the transportation cost then the optimal number of shipments is obtained as 11. As the transportation cost increases, the optimal number of shipments declines from 11. Figure 12 reflects that the system profit decreases at a diminishing rate as F increases. Figure 12. Average expected profit vs. 𝐹 Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 30 8. Managerial Insights From the numerical study and sensitivity analysis of our proposed model, the following key managerial insights are derived: (1) Business managers can improve the sales volume and economic efficiency by adopting green manufacturing technologies and suitable coordination scheme. Though, from the point of view of social welfare, it is always desirable to produce green products, the firms should estimate the profitable growth before adopting green manufacturing. From the outcomes of sensitivity analysis, it is evident that environmental awareness of the consumers and greening investment play a crucial role in the profitability of the supply chain members. By participating in the proposed price discount coordination mechanism, the business managers can improve the greening quality of the product to a remarkable higher level. It not only increases their profits but also maintains their social responsibility and increases their reputation in the business market for adopting such green initiative. (2) The proposed price discount scheme is capable of coordinating the supply chain. Under this mechanism, the manufacturer reduces his wholesale price and increases greening level of the product and encourages the retailers to set their prices and ordering quantities according to the centralized model. This improves the economic level of all members. Moreover, by participating in such coordination, the end customers get more eco-friendly product at a cheaper price than if they used an individual optimization strategy. Retailers should also remember that when consumers' price sensitivity is too high, they should lower their sales price to retain profitability. (3) The business managers may not always agree to adopt joint decision-making process even though it yields higher profit for the entire supply chain but it may not be profitable for all the chain members. To convince the members to make coordination, such price discount scheme is very effective as in this scheme increment of each member’s profitability is guaranteed. All the members could enjoy the coordination agreement as it is beneficial both socially and economically. (4) The delivery of the order quantity may not reach to the retailer’s end in time due to various reasons such as variation in transportation time, inspection time, loading and unloading times, etc. Therefore, the business managers should understand the stochastic nature of the lead time and account for all possibilities of early arrival, on time arrival, and late arrival to conduct the business efficiently. 9. Concluding Remarks In this study, we have designed a two-level supply chain model consisting of a single manufacturer and multiple retailers. To develop a realistic model, the lead time between placing an order and receiving its delivery is taken to be stochastic in nature. The retailers face a price sensitive demand from the end customers. The customer demand is also affected by the greening improvement level of the product as determined by the manufacturer. We have studied the decentralized model where supply chain members optimize their own profits without worrying about the profit of others. Stackelberg gaming approach is used where the retailers are assumed to act as the leader and the manufacturer as the follower. A solution algorithm is suggested to find the optimal solution of the proposed model. The performance of the whole supply chain is also investigated under integrated decision-making model. Though the entire supply chain experiences a better economical and environmental performances in the centralized scenario but it may not be beneficial for all the members Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 31 individually. Since the retail price of the product is decided by the retailers and the manufacturer determines the greening level of the product, and both these factors influence the customers demand, it is therefore essential to make these decisions in an efficient and coordinated manner which enriches the profit levels of each members. A price discount mechanism has been proposed to convince the supply chain members to make decisions in a coordinated manner. The maximum and minimum satisfactory discount rates are found so that all the members become interested for participating in this price discount coordination. This coordination mechanism is effective in both cases whether the market demand is high or low. There are some limitations of this study and the present model can be extended in many directions to further enhance the scope of our study. It is widely adopted in the literature but the policy of equal sized batch shipment is very limited in nature, and it may not be always possible to supply the order quantities of all the retailers in some integer number of equal sized batches. So, it would be more realistic to consider a combined equal and unequal sized batch shipment policy (Hoque, 2013). Another limitation of this study is that it is based on a single product being traded between the manufacturer and the retailers. To simulate a real-world scenario, it can be expanded to include many items (Barman et al., 2021a) and multiple manufacturers. Another shortcoming of our study is the consideration of complete backlogging strategy. It is desirable to consider partially backlogging of shortages for a more realistic approach (Duary et al., 2022). In our study, we have considered constant production rate, perfect production system at the manufacturer. One can enrich the study by taking into account variable production rate (Sarkar et al., 2018) and/or imperfect production system (Sepehri and Gholamian, 2022). The competition between the retailers will be another interesting research idea (Mondal and Giri, 2020). Our developed model can be modified by considering bargaining between manufacturer and retailers to share the profits among all the members (Nouri et al., 2018). In our study, we have proposed a price discount coordination scheme. It would be interesting to employ other contracts such as greening cost sharing contract between the manufacturer and the retailers (Giri and Dash, 2022). Consideration of set up cost reduction investment (Sarkar et al., 2017), and promotional effort (Ebrahimi et al., 2019) would also be fruitful extensions of this model. Author Contributions: Anamika Dash - Conceptualization, methodology, software, validation, formal analysis, investigation, resources, data curation, writing—original draft preparation, funding acquisition. Bibhas C. Giri - Conceptualization, methodology, formal analysis, investigation, writing—review and editing, visualization, supervision. Ashis Kumar Sarkar - Conceptualization, investigation, supervision. All authors have read and agreed to this version of the manuscript. Funding: This research was funded by the Department of Science and Technology, Government of India (grant number - IF170698). Data Availability Statement: The authors confirm that the data supporting the findings of this study are available within the article. Acknowledgments: The authors would like to thank the editor and the reviewers for their comments which led to considerable improvement in this article. Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 32 Conflicts of Interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References Agrawal, A.K., & Yadav, S. (2020). Price and profit structuring for single manufacturer multi-buyer integrated inventory supply chain under price-sensitive demand condition. Computers & Industrial Engineering, 139, 106208. https://doi.org/10.1016/j.cie.2019.106208 Alfares, H.K., & Ghaithan, A.M. (2016). Inventory and pricing model with price- dependent demand, time-varying holding cost, and quantity discounts. Computers & Industrial Engineering, 94, 170–177. https://doi.org/10.1016/j.cie.2016.02.009 Aljazzar, S.M., Jaber, M.Y., & Moussawi-Haidar, L. (2017). Coordination of a three-level supply chain (supplier–manufacturer–retailer) with permissible delay in payments and price discounts. Applied Mathematical Modelling, 48, 289–302. https://doi.org/10.1016/j.apm.2017.04.011 Atamer, B., Bakal, ˙I.S., & Bayındır, Z.P. (2013). Optimal pricing and production decisions in utilizing reusable containers. International Journal of Production Economics, 143(2), 222–232. https://doi.org/10.1016/j.ijpe.2011.08.007 Barman, A., Das, R., & De, P. K. (2021a). Optimal pricing, replenishment scheduling, and preservation technology investment policy for multi-item deteriorating inventory model under shortages. International Journal of Modeling, Simulation, and Scientific Computing, 12(05), 2150039. https://doi.org/10.1142/S1793962321500392 Barman, D., Mahata, G. C., & Das, B. (2021b). Advance payment based vendor–buyer production inventory model with stochastic lead time and continuous review policy. Opsearch, 58(4), 1217-1237. https://doi.org/10.1007/s12597-021-00521-9 Basiri, Z., & Heydari, J. (2017). A mathematical model for green supply chain coordination with substitutable products. Journal of cleaner production, 145, 232– 249. https://doi.org/10.1016/j.jclepro.2017.01.060 Ben-Daya, M., As’ ad, R., & Nabi, K.A. (2019). A single-vendor multi-buyer production remanufacturing inventory system under a centralized consignment arrangement. Computers & Industrial Engineering, 135, 10–27. https://doi.org/10.1016/j.cie.2019.05.032 Chan, C.K., Fang, F., & Langevin, A. (2018). Single-vendor multi-buyer supply chain coordination with stochastic demand. International Journal of Production Economics, 206, 110–133. https://doi.org/10.1016/j.ijpe.2018.09.024 Chen, Z., & Sarker, B.R. (2017). Integrated production-inventory and pricing decisions for a single-manufacturer multi-retailer system of deteriorating items under jit delivery policy. The International Journal of Advanced Manufacturing Technology, 89( 5), 2099–2117. https://doi.org/10.1007/s00170-016-9169-0 Duary, A., Das, S., Arif, M. G., Abualnaja, K. M., Khan, M. A. A., Zakarya, M., & Shaikh, A. A. (2022). Advance and delay in payments with the price-discount inventory model for deteriorating items under capacity constraint and partially backlogged shortages. Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 33 Alexandria Engineering Journal, 61(2), 1735-1745. https://doi.org/10.1016/j.aej.2021.06.070 Ebrahimi, S., Hosseini-Motlagh, S.M., & Nematollahi, M. (2019). Proposing a delay in payment contract for coordinating a two-echelon periodic review supply chain with stochastic promotional effort dependent demand. International Journal of Machine Learning and Cybernetics,10(5), 1037–1050. https://doi.org/10.1007/s13042-017- 0781-6 Esmaeili, M., & Nasrabadi, M. (2021). An inventory model for single-vendor multi- retailer supply chain under inflationary conditions and trade credit. Journal of Industrial and Production Engineering, 38(2), 75-88. https://doi.org/10.1080/21681015.2020.1845248 Ghosh, D., & Shah, J. (2015). Supply chain analysis under green sensitive consumer demand and cost sharing contract. International Journal of Production Economics, 164, 319–329. https://doi.org/10.1016/j.ijpe.2014.11.005 Giri, B. C., & Dash, A. (2022). Optimal batch shipment policy for an imperfect production system under price-, advertisement-and green-sensitive demand. Journal of Management Analytics, 9(1), 86-119. https://doi.org/10.1080/23270012.2021.1931495 Giri, B., & Masanta, M. (2019). Optimal production policy for a closed-loop supply chain with stochastic lead time and learning in production. Scientia Iranica, 26(5), 2936– 2951. 10.24200/SCI.2019.21537 Giri, B., & Masanta, M. (2020). Developing a closed-loop supply chain model with price and quality dependent demand and learning in production in a stochastic environment. International Journal of Systems Science: Operations & Logistics, 7(2), 147–163. https://doi.org/10.1080/23302674.2018.1542042 Giri, B., & Roy, B. (2016). Modelling supply chain inventory system with controllable lead time under price-dependent demand. The International Journal of Advanced Manufacturing Technology, 84(9), 1861–1871. https://doi.org/10.1007/s00170- 015-7829-0 Giri, B., Dash, A., & Sarkar, A. (2020a). A single-vendor single-buyer supply chain model with price and green sensitive demand under batch shipment policy and planned backorder. International Journal of Procurement Management, 13(3), 299–321. https://doi.org/10.1504/IJPM.2020.107478 Giri, B., Mondal, C., & Maiti, T. (2018). Analysing a closed-loop supply chain with selling price, warranty period and green sensitive consumer demand under revenue sharing contract. Journal of cleaner production, 190, 822–837. https://doi.org/10.1016/j.jclepro.2018.04.092 Giri, B.C., Dash, A., & Sarkar, A. (2020b). A single- manufacturer multi-retailer integrated inventory model with price dependent demand and stochastic lead time. International Journal of Supply and Operations Management, 7(4), 384–409. 10.22034/IJSOM.2020.4.7 He, X.J., Kim, J.G., & Hayya, J.C. (2005). The cost of lead-time variability: The case of the exponential distribution. International Journal of Production Economics, 97(2), 130– 142. https://doi.org/10.1016/j.ijpe.2004.05.007 Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 34 Heydari, J., Govindan, K., & Aslani, A. (2019). Pricing and greening decisions in a three- tier dual channel supply chain. International Journal of Production Economics, 217, 185–196. https://doi.org/10.1016/j.ijpe.2018.11.01 Heydari, J., Govindan, K., & Basiri, Z. (2021). Balancing price and green quality in presence of consumer environmental awareness: a green supply chain coordination approach. International Journal of Production Research, 59(7), 1957-1975. https://doi.org/10.1080/00207543.2020.1771457 Ho, C.H., Ouyang, L.Y., & Su, C.H. (2008). Optimal pricing, shipment and payment policy for an integrated supplier–buyer inventory model with two-part trade credit. European Journal of Operational Research, 187(2), 496–510. https://doi.org/10.1016/j.ejor.2007.04.015 Hoque, M. (2013). A manufacturer–buyer integrated inventory model with stochastic lead times for delivering equal-and/or unequal-sized batches of a lot. Computers & operations research, 40(11), 2740–2751. https://doi.org/10.1016/j.cor.2013.05.008 Hoque, M. A. (2021). An optimal solution policy to an integrated manufacturer- retailers problem with normal distribution of lead times of delivering equal and unequal-sized batches. Opsearch, 58(2), 483-512. https://doi.org/10.1007/s12597- 020-00485-2 Jaggi, C., Pareek, S., Khanna, A., & Sharma, R. (2015). Two-warehouse inventory model for deteriorating items with price-sensitive demand and partially backlogged shortages under inflationary conditions. International Journal of Industrial Engineering Computations, 6(1), 59-80. 10.5267/j.ijiec.2014.9.001 Li, B., Zhu, M., Jiang, Y., & Li, Z. (2016). Pricing policies of a competitive dual-channel green supply chain. Journal of Cleaner Production, 112, 2029–2042. https://doi.org/10.1016/j.jclepro.2015.05.017 Li, Y., Xu, X., & Ye, F. (2011). Supply chain coordination model with controllable lead time and service level constraint. Computers & Industrial Engineering, 61(3), 858– 864. https://doi.org/10.1016/j.cie.2011.05.019 Lieckens, K., & Vandaele, N. (2007). Reverse logistics network design with stochastic lead times. Computers & Operations Research, 34(2), 395–416. https://doi.org/10.1016/j.cor.2005.03.006 Lin, H.J. (2016). Investing in lead-time variability reduction in a collaborative vendor– buyer supply chain model with stochastic lead time. Computers & Operations Research, 72, 43–49. https://doi.org/10.1016/j.cor.2016.02.002 Lin, Y.J., & Ho, C.H. (2011). Integrated inventory model with quantity discount and price-sensitive demand. Top, 19(1), 177–188. https://doi.org/10.1007/s11750-009- 0132-1 Majumder, A., Jaggi, C.K., & Sarkar, B. (2018). A multi-retailer supply chain model with backorder and variable production cost. RAIRO-Operations Research, 52(3), 943–954. https://doi.org/10.1051/ro/2017013 Malleeswaran, B., & Uthayakumar, R. (2022). A single-manufacturer multi-retailer sustainable reworking model for green and environmental sensitive demand under discrete ordering cost reduction. Journal of Management Analytics, 1-20. https://doi.org/10.1080/23270012.2022.2030255 Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 35 Mondal, C., & Giri, B. C. (2020). Retailers’ competition and cooperation in a closed-loop green supply chain under governmental intervention and cap-and-trade policy. Operational Research, 1-36. https://doi.org/10.1007/s12351-020-00596-0 Mondal, C., & Giri, B. C. (2021). Analyzing a manufacturer-retailer sustainable supply chain under cap-and-trade policy and revenue sharing contract. Operational Research, 1-36. https://doi.org/10.1007/s12351-021-00669-8 Najafnejhad, E., Tavassoli Roodsari, M., Sepahrom, S., & Jenabzadeh, M. (2021). A mathematical inventory model for a single-vendor multi-retailer supply chain based on the Vendor Management Inventory Policy. International Journal of System Assurance Engineering and Management, 12(3), 579-586. https://doi.org/10.1007/s13198-021-01120-z Nandra, R., Majumder, A., & Mishra, M. (2021a). A multi-retailer sustainable supply chain model with information sharing and quality deterioration. RAIRO-Operations Research, 55, S2773-S2794. https://doi.org/10.1051/ro/2020113 Nandra, R., Marak, K. R., Kaur, R., Dey, B. K., & Majumder, A. (2021b). Establishing relation between production rate and product quality in a single-vendor multi-buyer supply chain model. International Journal of Services Operations and Informatics, 11(2-3), 315-331. https://doi.org/10.1504/IJSOI.2021.117256 Nouri, M., Hosseini-Motlagh, S.M., Nematollahi, M., & Sarker, B.R. (2018). Coordinating manufacturer’s innovation and retailer’s promotion and replenishment using a compensation-based wholesale price contract. International Journal of Production Economics, 198, 11–24. https://doi.org/10.1016/j.ijpe.2018.01.023 Rad, M.A., Khoshalhan, F., & Glock, C.H. (2014). Optimizing inventory and sales decisions in a two-stage supply chain with imperfect production and backorders. Computers & Industrial Engineering, 74, 219–227. https://doi.org/10.1016/j.cie.2014.05.004 Safarnezhad, M., Aminnayeri, M., & Ghasemy Yaghin, R. (2021). Joint pricing and lot sizing model with statistical inspection and stochastic lead time. INFOR: Information Systems and Operational Research, 59(1), 111-144. https://doi.org/10.1080/03155986.2020.1794227 Saha, S., & Goyal, S. (2015). Supply chain coordination contracts with inventory level and retail price dependent demand. International Journal of Production Economics, 161, 140–152. https://doi.org/10.1016/j.ijpe.2014.12.025 Sajadieh, M.S., Jokar, M.R.A., & Modarres, M. (2009). Developing a coordinated vendor– buyer model in two-stage supply chains with stochastic lead-times. Computers & Operations Research, 36(8), 2484–2489. https://doi.org/10.1016/j.cor.2008.10.001 Sarkar, B., Dey, B. K., Sarkar, M., & AlArjani, A. (2021). A sustainable online-to-offline (O2O) retailing strategy for a supply chain management under controllable lead time and variable demand. Sustainability, 13(4), 1756. https://doi.org/10.3390/su13041756 Sarkar, B., Majumder, A., Sarkar, M., Dey, B. K., & Roy, G. (2017). Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction. Journal of Industrial & Management Optimization, 13(2), 1085. Dash et al./Decis. Mak. Appl. Manag. Eng. (2022) 36 Sarkar, B., Majumder, A., Sarkar, M., Kim, N., & Ullah, M. (2018). Effects of variable production rate on quality of products in a single-vendor multi-buyer supply chain management. The International Journal of Advanced Manufacturing Technology, 99(1), 567-581. https://doi.org/10.1007/s00170-018-2527-3 Sarkar, S., Giri, B.C., & Sarkar, A.K. (2020a). A vendor–buyer inventory model with lot- size and production rate dependent lead time under time value of money. RAIRO- Operations Research, 54(4), 961–979. https://doi.org/10.1051/ro/2019030 Sarkar, S., Tiwari, S., Wee, H.M., & Giri, B. (2020b). Channel coordination with price discount mechanism under price-sensitive market demand. International Transactions in Operational Research, 27(5), 2509–2533. https://doi.org/10.1111/itor.12678 Sepehri, A., & Gholamian, M. R. (2022). A green inventory model with imperfect items considering inspection process and quality improvement under different shortages scenarios. Environment, Development and Sustainability, 1-29. https://doi.org/10.1007/s10668-022-02187-9 Swami, S., & Shah, J. (2013). Channel coordination in green supply chain management. Journal of the operational research society, 64(3), 336–351. https://doi.org/10.1057/jors.2012.44 Viswanathan, S., & Piplani, R. (2001). Coordinating supply chain inventories through common replenishment epochs. European Journal of Operational Research, 129(2), 277–286. https://doi.org/10.1016/S0377-2217(00)00225-3 Whitin, T.M. (1955). Inventory control and price theory. Management science, 2(1), 61–68. https://doi.org/10.1287/mnsc.2.1.61 Xu, J., Qi, Q., & Bai, Q. (2018). Coordinating a dual-channel supply chain with price discount contracts under carbon emission capacity regulation. Applied Mathematical Modelling, 56, 449–468. https://doi.org/10.1016/j.apm.2017.12.018 Yang, C.T., Ouyang, L.Y., & Wu, H. H. (2009). Retailer’s optimal pricing and ordering policies for non-instantaneous deteriorating items with price-dependent demand and partial backlogging. Mathematical Problems in Engineering, 2009, Article ID 198305. https://doi.org/10.1155/2009/198305. Yang, S., Liu, H., Wang, G., & Hao, Y. (2021). Supplier’s cooperation strategy with two competing manufacturers under wholesale price discount contract considering technology investment. Soft Computing, 1-18. https://doi.org/10.1007/s00500-021- 05904-0 Zanoni, S., Mazzoldi, L., Zavanella, L.E., & Jaber, M.Y. (2014). A joint economic lot size model with price and environmentally sensitive demand. Production & Manufacturing Research, 2(1), 341–354. https://doi.org/10.1080/21693277.2014.913125 Zhang, K., & Feng, S. (2014). Research on revenue sharing coordination contract in automobile closed-loop supply chain. In Proceedings of 2014 IEEE International Conference on Service Operations and Logistics, and Informatics, IEEE, 298–302. https://doi.org/10.1108/IMDS-05-2017-0181 Zhang, R., Liu, J., & Qian, Y. (2022). Wholesale-price vs cost-sharing contracts in a green supply chain with reference price effect under different power structures. Kybernetes, https://doi.org/10.1108/K-11-2021-1096. Coordınatıon of a sıngle-manufacturer multı-retaıler supply chaın wıth prıce and green…. 37 Zhu, Q., Li, X., & Zhao, S. (2018). Cost-sharing models for green product production and marketing in a food supply chain. Industrial Management & Data Systems, 118, 654- 682. https://doi.org/10.1108/IMDS-05-2017-0181 Zu, Y., Deng, D., & Chen, L. (2021). Optimal control of carbon emission reduction strategies in supply chain with wholesale price and consignment contract. Environmental Science and Pollution Research, 28(43), 61707-61722. https://doi.org/10.1007/s11356-021-15080-1 © 2022 by the authors. 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