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. 

 
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