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,
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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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