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Citation: Zhu, X.; Song, Y.; Lin, G.;
Xu, W. Pricing Decisions and
Coordination in E-Commerce Supply
Chain with Wholesale Price Contract
Considering Focus Preferences. J.
Theor. Appl. Electron. Commer. Res.
2023,18, 1041–1068. https://doi.org/
10.3390/jtaer18020053
Academic Editor: Danny C. K. Ho
Received: 5 April 2023
Revised: 7 May 2023
Accepted: 30 May 2023
Published: 3 June 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
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distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
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4.0/).
Article
Pricing Decisions and Coordination in E-Commerce Supply
Chain with Wholesale Price Contract Considering
Focus Preferences
Xide Zhu 1, Yao Song 1, Guihua Lin 1and Weina Xu 2,*
1School of Management, Shanghai University, Shanghai 200444, China
2School of Economics and Management, Shanghai University of Political Science and Law,
Shanghai 201701, China
*Correspondence: xuweina@shupl.edu.cn
Abstract:
Decision makers’ behavioral preferences have always been important in coordinating
the supply chain. Decision makers need to choose a partner wisely to increase the profitability of
the entire supply chain, especially in the competitive e-commerce environment. In this paper, we
examine a two-echelon e-commerce supply chain with one retailer and one supplier using the most
popular wholesale price contract to facilitate collaboration. Traditional research has shown that
the classical expectation model cannot coordinate the supply chain. We apply the focus theory of
choice to describe the retailer’s behavior as a follower, and we examine the impact of the retailer’s
pricing decisions on the supplier under different focus preferences and the coordination for the
entire supply chain. The lower the parameter
ϕ
, which represents the degree of positivity, and the
higher the parameter
κ
, which represents the level of confidence, the closer the profit of the whole
supply chain is to the coordination result—both are visualized through numerical experiments and
images. In the case of
ϕ
determination, the lower the
κ
, the better the supply chain coordination.
The finding implies that the retailer may be able to coordinate the supply chain and produce better
results than the expectation model when he or she makes choices using a positive evaluation system
that includes both higher levels of optimism and lower levels of confidence. The findings of the FTC
model can simultaneously offer a theoretical foundation for expanding collaboration among supply
chain participants and management insights for decision makers to choose cooperation partners.
Keywords:
supply chain coordination; behavioral preference; focus theory of choice; e-commerce
supply chain; supplier-led supply chain
1. Introduction
In the era of e-commerce, the speed of product replacement and life cycle has become
shorter, and there are more competing products and substitutes for similar products.
Upstream suppliers in the supply chain can increase their competitiveness by setting up
direct online sales, while downstream retailers are constantly improving their decision-
making flexibility in response to changing market needs. Now many retailers are relying
on e-commerce platforms to change their sales model to a pre-order system, retaining
operational flexibility to respond to demand information by promising consumers to
order in advance while being able to reduce their costs [
1
]. The supply chain has been
transformed from the traditional form based on internal enterprise information to a modern
e-commerce supply chain that relies on electronization, intellectualization, and digitization.
Current market transactions are based on many e-commerce platforms, and according
to official statistics from Taobao, as of January 2023, 26,160,060 shops had been entered.
E-commerce platforms include a variety of sales models and can meet the sales needs of
multiple categories of merchants and have full applicability to a wide range of supply
chains. However, e-commerce supply chains still suffer from the efficiency problems
J. Theor. Appl. Electron. Commer. Res. 2023,18, 1041–1068. https://doi.org/10.3390/jtaer18020053 https://www.mdpi.com/journal/jtaer
J. Theor. Appl. Electron. Commer. Res. 2023,18 1042
found in traditional supply chains, with decentralized decision-making between subjects
leading to double marginalization of utility and reduced system efficiency [
2
]. Multiple
coordination mechanisms in supply chain management remain a very important element
in e-commerce supply chain coordination, facilitating supply chain members to coordinate
with each other as a whole and maintain the unity of purpose [
3
,
4
], which can reduce costs,
improve the economic stability of the supply chain, and reduce channel conflicts when
the supply chain makes decisions as a whole [
5
]. The formation of supply chain contracts
can further improve the efficient payoff distribution among decision makers as well as
the sustainability of the supply chain, indicating that overall coordination is still a key
component of business cooperation for e-commerce supply chains.
The decentralized supply chain is coordinated when the order volume of the decen-
tralized supply chain equals the order volume of the centralized supply chain, resulting
in the total expected payoff of the decentralized supply chain equaling the total expected
payoff of the centralized supply chain [
6
–
8
]. The most widely applied wholesale price
contract provides insights on adjusting decision makers’ target utility to improve overall
efficiency. Of course, there are other complex contracts that can coordinate the supply
chain, and given the importance of the contract in the coordination mechanism, the very
common question is which contract to adopt [
9
]. The design of cooperative contracts among
supply chain members takes into account issues such as supply chain coordination, profit
sharing, and the cost of maintaining the contract [
10
]. Therefore, considering the many
influencing factors, the designer of the contract may prefer a wholesale price contract to
other complex contracts.
We used the focus theory of choice (FTC) to emphasize the importance of salient infor-
mation in decisions of uncertainty, and the limited rationality characteristics of individuals
based on FTC are more consistent with human behavior patterns in the process of making
decisions, consistent with the results of psychological experiments conducted by Stew-
art [
11
]. We use FTC to promote coordination, aiming to be similar to the real situation and
involve the impact of individuals with different personality traits on coordination. Based
on the e-commerce environment and a one-time decision problem, the product market
demand is more stochastic, and the uncertainty of demand is an important cause of the
pull-to-center effect, which makes the decision deviate from the optimal [
12
]. Establishing
a supply chain contract under a stochastic demand setting considers numerous influencing
factors. A number of scholars are currently studying issues such as a demand-dependent
two-echelon supply chain with retail price and the impact of variable demand on produc-
tion, further emphasizing the problem of uncertainty in product demand and proposing
some solutions [
13
–
15
]. Scholars provided supply chain operation insights on the one hand
by building new supply chain contracts or applying new behavioral preferences, and on
the other, they applied new management strategies to improve management efficiency
and production cost-effectiveness [
16
]. The limitations of human judgment can exacerbate
supply chain challenges and even potentially undermine strategic initiatives, highlighting
the need to redesign and validate “human factors” to better inform relevant decisions [
17
].
Below we summarize our contributions. We consider a two-echelon e-commerce
supply chain consisting of a single supplier and a single retailer that adopts a wholesale
price contract. In the FTC framework, a new supply chain coordination model is developed
to analyze the changes in the decision-making of retailers with positive focus preference and
the impact on the whole supply chain coordination under the wholesale price contract. Our
results show that FTC can coordinate the supply chain and solve the double marginalization
effect under certain conditions. The model also provides a theoretical basis for suppliers and
retailers to establish further cooperation and offers new theoretical ideas for coordinating
the supply chain. The retailer’s FTC-based decision-making behavior allows the overall
supply chain profitability to achieve results better than the classical expectation model
under many conditions.
The remainder of this paper is organized as follows. Section 2reviews the literature.
In Section 3, we review the classical expectation-based wholesale price contract model in
J. Theor. Appl. Electron. Commer. Res. 2023,18 1043
the two-echelon supply chain and the double marginalization problem of the channel as
the basis for the focus model. In Section 4, based on the expected value model, we develop
a focus model of the retailer’s decision process to find the optimal retail price of the retailer
and infer the optimal wholesale price of the supplier’s decision as the dominant player
in the face of different retailers’ personality characteristics. In Section 5, give numerical
experiments and results. In Section 6, we compare another behavioral model to analyze
the supply chain coordination and overall channel payoff change under the positive focus
model. Section 7concludes this paper. Some of the proofs are given in Appendix A.
2. Literature Review
Decision makers’ personalities can have a big impact on how supply chain transactions
turn out. The focus theory of choice, which outperforms traditional models and is more
in line with the decision-making process of realistic decision makers, is used in this study
to assess retailers’ responses. Hence, the related literature review is classified into three
areas—(i) supply chain contracts and coordination, (ii) theoretical study on behavioral
supply chains, and (iii) study on the focus theory of choice.
2.1. Supply Chain Contracts and Coordination
When establishing cooperation among supply chain members, coordination issues
are an important factor to consider. As an example, due to centralized or decentralized
supply chain operations, the distribution of finished products for the final process may
require different coordination mechanisms. Scholars have proposed a variety of cooperative
contract models, such as the most studied wholesale price contract [
18
], and Holmstrom
and Milgrom have long suggested that, in reality, channel members often adopt a simpler
collaborative contract [
19
], so the adoption of wholesale price contracts in the supply chain
has many aspects of applicability. Scholars Cachon et al., observed that revenue-sharing
contracts have some coordination effect in the video leasing industry [
20
], and Pasternack
demonstrated the coordination of buyback contracts in the newsvendor model [
21
]. Katok
and Wu show that revenue-sharing contracts and buyback contracts are mathematically
equivalent in certain settings but do not usually lead to equivalent supply chain perfor-
mance in practice [
22
]. Zhang et al., again demonstrate this equivalence from the supply
chain perspective [
23
]. This part of scholars’ research on supply chains focuses on coor-
dination mechanisms, mostly comparing the performance differences between different
coordination mechanisms and the issue of equivalence, without considering the influence
of behavioral preferences on the final decision outcome.
For e-commerce businesses, the e-commerce clothes industry, the online food fresh
category, and many other industries, wholesale price contracts are still commonly em-
ployed [
24
]. Given the complexity of the supply chain, the wholesale price contract is
the most popular form of contract [
25
,
26
]. Early scholars Bernstein et al., studied supply
chain performance through a simple wholesale price contract where suppliers were able to
coordinate the supply chain under a specific discount setting [
27
]. With the development of
e-commerce, coordinating the supply chain remains a key issue. Qiu et al., investigated the
coordination problem for a two-tier decentralized supply chain consisting of a supplier and
a retailer who sells the product both offline and online to consumers with reference quality
effect, considering the factors influencing market demand and the impact of supply chain
contracts on coordination [
28
]. Shu et al., consider a supplier selling substituted products
to an e-retailer through wholesale selling mode or mixed use of wholesale and agency
selling mode (hybrid contracts) [
29
]. However, the above-mentioned papers continue to
concentrate on the issue of creating coordination mechanisms without considering the
influence of behavioral preferences on coordination.
2.2. Behavioral Supply Chain
The classical expected utility theory has been continuously applied in operations
research-based supply chain models, where fully rational decision makers can coordinate
J. Theor. Appl. Electron. Commer. Res. 2023,18 1044
the supply chain through some cooperative contracts [
30
,
31
]. With the development of be-
havioral economics, the Allais paradox and Ellsberg paradox challenged the expected utility
theory and the subjective expected utility theory, respectively [
32
,
33
]. A growing number of
experimental behavioral studies have challenged the “economic man” hypothesis. Experi-
mental results show that a variety of factors, such as market conditions and information
differences, influence upstream and downstream decision makers and their actual economic
decisions often deviate from predictions based on expected utility maximization [
12
,
34
–
36
].
Schweitzer and Cachon found that these decision biases were characterized by a “pull-to-
center effect” in the experiment, as the average number of orders was low when it should
have been high, and vice versa [
12
]. Later, Bostian et al., replicated this effect in a laboratory
study, and this feature remained very evident in multiple rounds of experiments [
37
]. The
classical expected utility theory has proved to be unable to solve the decision bias in the
supply chain, and the behavioral factors are more closely and importantly linked to the
supply chain [38].
For better coordination, the role of behavior in the supply chain has become increas-
ingly important [
39
]. Research has greatly enriched behavioral theories, such as prospect
theory and heuristic decision bias, which have corrective and developmental implications
for expected utility theory, as well as the study of irrational behavior with perspectives such
as fairness concern, psychological accounts, and overconfidence [
40
–
45
]. Some of these
theories have ruled out explanations for the “pull-to-center effect”, such as risk aversion
or risk-seeking, loss aversion, and other behavioral theories [
12
]. Majeed et al., think that
behavioral preferences are distinct from economic motivation and will influence behaviors
in the supply chain. Decision makers will make decisions based on not only self-interests
but also the interests of others, reciprocity, and fairness [46].
Many irrational behavior theories are more biased toward behavioral economics re-
search that ignores the procedural nature of behavioral decisions and have more restrictions
in their application to supply chains. The fairness concern theory is only applicable to two-
or multi-person settings. Additionally, if the decision maker is independent or does not
know the profit composition of the cooperating party, the decision maker with fairness
concern has no reference to utility and does not consider the pull-to-center effect in the
decision bias phenomenon. Overconfidence is defined as a cognitive bias that cannot fully
account for the decision-making process and can only explain a single personality trait.
2.3. The Focus Theory of Choice
FTC was originally proposed by Guo, who believed that human attention is limited
and the decision-making process has a certain order, similar to buying a lottery ticket. The
decision maker will consider the probability of winning and the prize amount, and people
with different personality traits will focus on different probabilities and benefits to obtain
the final decision results [
47
]. This procedure involves two steps: in the first step, for each
action, some specific event that can bring about a relatively high payoff with a relatively
high probability or a relatively low payoff with a relatively high probability is selected as the
positive or negative focus, respectively; in the second step, based on the foci of all actions,
a decision maker chooses a most-preferred action [
48
,
49
]. FTC handles decision-making
with risk or when facing ambiguity or ignorance within a unified framework. Additionally,
FTC explains the St. Petersburg paradox, the Allais paradox, and the Ellsberg paradox with
the central idea that finite rational decision makers select the salient information (focus) of
greatest concern to evaluate various options in events that are more consistent with human
decision-making processes in real life. Additionally, salient information plays a crucial role
in the decision-making process. Based on FTC, scholars proposed some new approaches to
solve stochastic optimization problems [48,50].
Some scholars have used FTC to incorporate decision makers’ personality traits into
mathematical models and analyzed the impact of FTC on decisions, supporting the impor-
tance of decision makers with different traits for strategic choices [
51
]. Zhu et al., used FTC
to solve the behavioral decision problem in the newsvendor problem and solved for the
J. Theor. Appl. Electron. Commer. Res. 2023,18 1045
retailer’s optimal order quantity in the framework of FTC, showing that when a retailer is
more optimistic about uncertain demand and more confident in his or her decision, he or
she tends to aim high and act more aggressively by ordering a higher quantity [
52
]. This
paper only explores the retailer’s decision in the newsvendor model and does not analyze
the impact on the supplier when applied to the supply chain, but it can serve as the basis
for this paper on supply chain coordination. Later, a study by Zhu et al., who studied the
optimal replenishment of retailers under the vendor-managed inventory model based on
FTC, provided a new perspective to analyze the behavior of individual suppliers in a VMI
program with revenue-sharing contracts [
53
]. Zhu et al., studied a two-echelon supply
chain under retailer-led buyback contracts in the presence of uncertain market demand,
and this paper used FTC to describe the behavioral tendencies of the supplier and to theo-
retically obtain the optimal wholesale price based on the supplier’s focus preference [
54
].
These two papers examine revenue-sharing contracts and buyback contracts, respectively,
and the emphasis of the papers is on emphasizing differences in decision-making among
decision makers with different personality characteristics and the effect of optimism on
outcomes, with no following comparison descriptions. The FTC-related papers do not eval-
uate the impact on supply chain performance or the function of supply chain coordination;
instead, they primarily discuss the varied decision outcomes of decision makers with focus
preferences under distinct types of personalities. In addition to this, we use the wholesale
price contracts to analyze decision outcomes, firstly because the real-life generalizability
is a bit stronger, and secondly because the literature studying wholesale price contracts is
more extensive and easier to compare with the results of other behavioral preferences.
3. The Classical Model of the Wholesale Price
The classical model studied in this paper consists of a fully rational two-echelon supply
chain consisting of a single supplier and a single retailer. Both make decisions according
to their expected payoff maximization, and the decentralized decision is made first by the
supplier as the dominant player, who sets a wholesale price
(w)
, and then the retailer sets
the retail price
(r)
under the given wholesale price. The list of notations used in this study
is given in Table 1.
Table 1. Variable Definition.
Notation Definition
r retail price/unit
w wholesale price/unit
x the market potential demand
b the sensitivity of product market demand concerning the retail price
c the supplier’s cost of production/unit
vs/vr/v
the payoff function of the supplier/retailer/supply chain in the classical model
ϕthe retailer’s degree of optimism
κthe retailer’s level of confidence
u(·)the retailer’s satisfaction function
π(·)the relative likelihood function
l/h/m the minimum/maximum/average values of market potential demand
xP(·)the positive demand focus of the retailer concerns
rP(·)the optimal retail price of the retailer concerns
H(·)the payoff function of the supplier in the classical model
Assume that the market potential demand faced by the product is a random variable,
denoted by the capital letter
X
, has a probability density function
f(·)
and obeys a cumula-
tive distribution function
F(·)
. The range belongs to the interval
[l, h]
;
l
denotes the lowest
market potential demand,
h
denotes the highest market potential demand, 0
≤l≤h
, and
µ
denotes the expected value of the market potential demand
X
. The supplier produces
at a fixed cost
c
per unit of production, satisfying the size relationship of 0
<c≤w≤r
.
Additionally, the retailer faces linear uncertainty product demand;
D(X, r)=X−br
, where
J. Theor. Appl. Electron. Commer. Res. 2023,18 1046
b(>
0
)
is the sensitivity of product market demand concerning the retail price. The supplier
offers a cooperation contract to the retailer, and the retailer either rejects this offer, in which
case neither partner makes any payoffs, or orders
D(X, r)
units. Since retailers adopt a
reservation system in e-commerce platforms, this paper assumes that the retailer’s orders
are matched with retail price, only the supplier has unit production cost, and the retailer’s
wholesale products can be sold fully without out-of-stock cost (opportunity cost), inventory
cost, and salvage value.
The payoff functions of the supplier and retailer are given as follows:
vs(r, w, X)=(w−c)(X−br),
vr(r, w, X)=(r−w)(X−br).
The whole potential payoff of the supply chain is given as follows:
v(r, X)=(r−c)(X−br),
where assumes
D(l, r)≥
0 to ensure that the product market demand is non-negative when
the potential market demand takes the lowest value
l
, i.e.,
r≤l
b
; assuming
w≤2l−h
b
, it is
guaranteed that payoff loss may occur when market demand is too high or low.
The expected payoff of the whole supply chain (denoted as V(r)) is given by
V(r) = −br2+(µ+bc)r−cµ.
whereas
b>
0, it can be judged that the second-order derivative of
V(r)
is negative, i.e.,
−2b <0. The function V(r)is a concave function about r. The maximum point r∗=µ+bc
2b
is within the valid interval, and the maximum value of
V(r)
is obtained as
V(r∗)=(µ−bc)2
4b
.
In decentralized decision-making, the retailer’s decision is first analyzed according to
the standard inverse induction of the Stackelberg game. Denote the expected payoffs of
the retailer and supplier as
Vr(r
,
w)
and
Vs(w)
. In this case, the retailer’s expected payoff
function is given below:
Vr(r, w) = −br2+(µ+bw)r−wµ.
Since
r1=µ+bw
2b
is the symmetry axis of
Vr(r
,
w)
must be within the valid interval,
the optimal expected payoff of the retailer is obtained as
Vr(r1
,
w) = (µ−bw)2
4b
. The optimal
wholesale price of the supplier and the optimal expected payoff is obtained from
r
as
w1=µ+bc
2b
and
Vs(w1)=(µ−bc)2
8b
. The final outcomes of the retailer are
r1=3µ+bc
4b
and
Vr(r1, w1)=(µ−bc)2
16b
. At this point, the whole payoff of the supply chain (denoted as
V∗(r)
)
is given by
V∗(r) = Vr(r1, w1) + Vs(w1)=3
4·(µ−bc)2
4b .
The sum of the payoffs generated by the supplier and the retailer under the decen-
tralized decision is lower than the optimal payoff under the centralized channel when
considering the expected payoff, leading to double marginalization.
4. The Wholesale Price Model with the Positive Focus Theory of Choice
4.1. FTC Model
We analyze the decision-making model of a retailer under a positive evaluation system,
assuming that the retailer’s decision-making process is a single-cycle decision that requires
a one-time decision on the retail price. When the supplier dominates, the wholesale price is
provided first, and the retailer with a positive focus preference decides on the e-commerce
platform through a reservation system. Based on the given wholesale price and all potential
J. Theor. Appl. Electron. Commer. Res. 2023,18 1047
retail prices, the positive demand focus is determined by comparing the satisfaction and
likelihood of potential market demand. Finally, the optimal retail price focus is selected by
comparing all possible retail prices under the positive demand focus.
Whereas there is often only one identified demand occurring for all potential market
demands, the retailer needs to determine which part of the demand to focus on. According
to FTC, the retailer’s decision process is divided into two steps: the first step is for each
possible retail price, and the retailer determines the positive demand focus by comparing
the payoff and corresponding probability of all possible demands; the second step is for the
retailer to choose the optimal retail price by comparing the focus of potential retail prices.
The Stackelberg game is established to analyze the optimal decision problem following
the transaction model of wholesale price contracts, but the decision makers are only
partially rational in the actual decision process. Instead of focusing solely on the expected
payoff, their focus is limited, or there is a certain personal preference [
55
–
58
]. Because
personal characteristics have a significant impact on decision-making, we examine the
retailer’s decisions with positive focus preferences. As shown in Definitions 1 and 2, this
paper converts the retailer’s payoff function into a satisfaction function and the demand
probability density function into a relative likelihood function [50].
Definition 1.
Let
V
be the payoff
u:V→[0, 1]
is called a range of the retailer’s payoff. The
satisfaction function if
u(v1)>u(v2)⇔v1>v2
,
∀v1
,
v2∈V
, and
∃vc∈V
such that
u(vc) = max
v∈Vu(v) = 1.
To facilitate comparative analysis, we denote the retailer’s composite function
u(vr(r, w, x))
as
u(r, w, x)
. For any given wholesale price
w∈hc, 2l−h
bi
and retail price
r∈hw, l
bi
,
u(r, w, x)
indicate the retailer’s satisfaction level with the final payoff as the potential
market demand
x
changes. With the observed potential market demand
x
, the payoff
function of the retailer is expressed as follows:
vr(r, w, x)=(r−w)(x−br). (1)
According to the range of variables therein, the maximum payoff of the retailer can be
obtained as
vmax
r=(h−bw)2
4b
and the minimum payoff as 0. According to Definition 1, this
paper sets the retailer’s satisfaction function as follows:
u(r, w, x)=vr(r, w, x)−0
vmax
r−0=4b(r−w)(x−br)
(h−bw)2. (2)
Definition 2.
Let
f:[l,h]→R+
be the density function of stochastic demand. The function
π:[l,h]→[0, 1]is called the relative likelihood function if it satisfies that
π(x1)>π(x2)⇔f(x1)>f(x2),∀x1,x2∈[l,h],
and ∃xc∈[l,h],such that π(xc)=max
x∈[l,h]π(x)=1.
For any
x∈[l, h]
, we call
π(x)
as the relative likelihood degree of
x
. A normalized
probability density function is used to indicate the relative likelihood degree of various
events and is the basis for the relative likelihood function expressed in Definition 2. The
relative probability function of the normal distribution, which matches the distribution of
the potential market demand, is used in this study:
π(x)=1−a(m−x)2
(h−m)2. (3)
J. Theor. Appl. Electron. Commer. Res. 2023,18 1048
where
a∈(0, 1]
.
π(x)
is strictly growing in the interval
[l, m]
and strictly decreasing in the
interval [m, h], as m represents the mean value of demand.
The relative likelihood function and satisfaction function can be used as the primary
inputs for decision-making without involving magnitude, which is the first reason for not
using the original probability density function and payoff function in FTC. The second
reason is that there is mounting evidence that relative values are easier to understand and
accept than absolute ones. According to Definitions 1 and 2, the analysis will use the relative
likelihood function and the satisfaction function as its primary decision-making inputs.
FTC postulates that a decision maker inherently owns two opposite evaluation sys-
tems: positive and negative. In the positive evaluation system, events with relatively high
probabilities and relatively high payoffs will be more salient. Conversely, in the negative
evaluation system, events that produce relatively low payoffs with relatively high prob-
abilities will be more salient. There will always be a system that responds to a decision
condition, and the two systems typically correspond to different thinking systems. Which
system is active depends heavily on the personality trait of the decision maker: the positive
evaluation system is more active for optimistic decision makers, and the negative evalu-
ation system is more active for pessimistic decision makers. Thus, FTC can illustrate the
behavioral characteristics of a decision maker’s performance when facing risks. Addition-
ally, by asking the decision maker some simple questions, it is simple and straightforward
to determine which system is active. Specifically applied to the two-echelon supply chain
model in this paper, if a high probability of high gain is more significant than a high
probability of high loss (or low payoff) for a retailer, then the retailer is optimistic, and the
positive evaluation system is more appropriate. On the other hand, if a retailer perceives
a high probability of a high loss (or low payoff) event to be more significant than a high
probability of a high payoff, then the retailer is pessimistic, and a negative evaluation
system is more appropriate to describe its decision behavior.
4.2. Decision-Making Model for a Retailer under the Positive Evaluation System
4.2.1. Model Construction
Based on this process presented in Section 4.1, the retailer’s decision model under the
positive evaluation system is specified below. The first step is given by Equation (4), and
the second step is given by Equation (5).
Under the positive evaluation system, the retailer determines the most significant
demand with a relatively high satisfaction level and a relatively high likelihood. For any
given wholesale price
w
and retail price
r
, denote
Xp(r, w)
as the optimal set of solutions to
the following optimization problem:
max
x∈[l,h]{ϕπ(x)+u(r, w, x)}, (4)
where parameter
ϕ
is a positive real number that acts as a scaling factor and can express the
weight of the relative likelihood when deriving internally for Equation (4). The optimization
problem Equation (4) is formed by adding the functions
ϕπ(x)
and
u(r, w, x)
, which are,
respectively, quadratic and linear functions on
x
. The Pareto optimal solution set is made up
of all the Pareto optimal solutions to this problem, all of which are Pareto optimal solutions.
The Pareto optimal frontier surface is the objective function value that corresponds to
the Pareto optimal solution. The parameter
ϕ
in Equation (4) can control the slope of
the Pareto front surface. Increasing the
ϕ
value causes the
ϕπ(x)
to account for a larger
proportion and
u(r, w, x)
for a smaller proportion, and the existence of the optimal
x
causes
the optimization problem Equation (4) to appear with a higher likelihood. In contrast,
decreasing the
ϕ
value causes the
ϕπ(x)
to shrink, resulting in the optimal
x
order shown
in Equation (4), which has a considerably lower likelihood and higher satisfaction. The
ϕ
value can be used in the model as a weight to indicate the relative importance that the
retailer gives to satisfaction and likelihood, and decreasing the
ϕ
value means that the
retailer aims to pursue higher payoffs by sacrificing the probability. In the personality trait,
J. Theor. Appl. Electron. Commer. Res. 2023,18 1049
optimism is represented by the
ϕ
value, so the lower the
ϕ
value, the more optimistic the
retailer is.
Equation (2) and Definition 2 lead to the conclusion that for any market potential
demand
x∈[l, h]
and retail price
r∈hw, l
bi
, there exists only one element in
Xp(r, w)
indicated as
xP(r, w)
, which is referred to as the positive focus of the market potential
demand
x
. Based on the selection of the positive demand focus in the first step, the second
step requires determining the optimal retail price for the retailer in the positive demand
focus. In the face of all possible retail price focuses, we will obtain the optimal retail price
through the optimization problem Equation (5):
max
r∈[w, l
b]
{κπ(xP(r, w)) + u(r, w, xP(r, w))}, (5)
where
κ
is a positive real number and is a scaling factor, and the optimal solution set of
the optimization problem Equation (5) is denoted by
RP(w)
. If there is only one element in
RP(w)
, it is said to be the optimal retail price under the positive evaluation system, denoted
as
rP(w)
. Similar to the interpretation of parameter
ϕ
, raising
κ
will lead to a higher relative
likelihood of retail prices as well as a relatively lower satisfaction level. Therefore, the
parameter
κ
can be used to measure the retailer’s level of confidence, and the higher the
value of κ, the lower the retailer’s confidence level tends to be.
4.2.2. Retailer Decision Outcome Analysis
Based on the underlying definitions, this section determines the retailer’s active
demand focus and optimal retail price using the focus theory of choice. The range of
parameters for the analysis process is based on any given wholesale price
w∈hc, 2l−h
bi
,
parameter
b∈(0, 2l−h
ci
, and retail price
r∈hw, l
bi
. First, for the lower-level problem
Equation (4), let
f(x)=ϕ"1−a(m−x)2
(h−m)2#+4b(r−w)(x−br)
(h−bw)2.
It is simple to verify that
f(x)
is a quadratic concave function, and as a result, the
maximal point xϕof this function is as follows:
xϕ=m+1
ϕ∗2b(r−w)(h−m)2
a(h−bw)2.
According to the FTC analysis process, first, Lemma 1 represents the positive focus
of the retailer’s choice of demand in a positive evaluation system for any given wholesale
price and every potential retail price.
Lemma 1. The positive demand focus of retailer concerns is described as follows:
xP(r,w) = minxϕ,h. (6)
Further obtained by
(i)
When r ∈[w,r0],satisfying xϕ<h,then xP(r,w) = xϕ;
(ii)
When r ∈hr0,l
bi,satisfying xϕ≥h,then xP(r,w) = h.
where r0=minw+aϕ(h−bw)2
2b(h−m),l
b.
Assume that
rϕ=w+aϕ(h−bw)2
2b(h−m)
. For brevity, we leave the detailed derivations in
Appendix A. Lemma 1 demonstrates the significance of
ϕ
in deciding the active demand
J. Theor. Appl. Electron. Commer. Res. 2023,18 1050
focus of retailers. In the first case, the focus is judged as
xP(r, w)=xϕ
; the
ϕ
value reaches
one of the higher ranges when it has a lower satisfaction level at a higher relative likelihood.
When the
ϕ
value is sufficiently small in the second case, the relative likelihood function
largely determines the focus
xP(r
,
w) = h
since the focus has a lower relative likelihood at
a higher satisfaction level.
The range of the positive focus of retailer interest under the positive evaluation system
is
[m, h]
. The
ϕ
value steadily decreases, and
xP(r, w)
gradually moves from
xϕ
to the
maximum possible market demand
h
. The conclusions of Theorems 1 and 2 can be found
by applying Lemma 1.
Theorem 1.
When
r∈[w,r0]
,
xP(r
,
w)
is continuously monotonically increasing with respect to
the retail price r;when r ∈hr0,l
bi,xP(r,w) = h.
Theorem 1 proves the relationship between the positive demand focus
xP(r
,
w)
, which
is of interest to retailers, and the retail price
r
: when
r
is in the higher range,
xP(r, w)
is at
the maximum value
h
, indicating that the retailer will focus on the maximum potential
demand with the higher retail price set; when the value of
r
varies from high to low, for
any given retail price
ri∈[w, r0](i=
1, 2
)
, if
r1<r2
, then the positive focus of
r1
is smaller
than the positive focus of
r2
, indicating that the higher the retail price
r
is priced, the retailer
under the positive evaluation system will prefer to focus on higher demand, gradually
converging to the maximum value h.
Theorem 2.
When
r∈[w,r0]
,
π(xP(r,w))
is continuously monotonically decreasing with respect
to the retail price r;when r ∈hr0,l
bi,π(xP(r,w)) =π(h)=1−a.
The proof of Theorem 2 is given in Appendix A. Theorem 2 proves the relationship
between the value of the relative likelihood function
π(xP(r, w))
of the active demand
focus and the retail price
r
.
π(xP(r, w))
is continuous with respect to
r
: as the retail price
r
gradually increases, the value of the relative likelihood function of the positive focus
gradually decreases from the maximum value
π(m)
to
π(h)
; as
r
continues to increase, the
relative likelihood value of the demand of the retailer’s concern remains at
π(h)
. Based on
the above Lemma and theorems, the optimal retail price for retailer decision is deduced
under the positive evaluation system.
Theorem 3.
The optimal retail price of retailer concerns under positive evaluation system
rP(w)
denotes the following three classifications:
(i)
if ϕ<h−m
a(h−bw),
rP(w)=rc,i f κ≤κ3,
rκ,i f κ>κ3.
(ii)
if h−m
a(h−bw)≤ϕ≤2(l−bw)(h−m)
a(h−bw)2,
rP(w)=rϕ,i f κ≤κ1,
rκ,i f κ>κ1.
(iii)
if ϕ>2(l−bw)(h−m)
a(h−bw)2,
rP(w)=rh,i f κ≤κ2,
rκ,i f κ>κ2.
The relevant parameters are expressed as:
J. Theor. Appl. Electron. Commer. Res. 2023,18 1051
κ1=
2
ϕ+ϕ(m−bw)(h−m)−aϕ2(h−bw)2
(h−m)2
,
κ2=
2
ϕ+aϕ2(h−bw)2(m+bw−2l)
2(l−bw)(h−m)2
and
κ3=ϕ−aϕ2(h−bw)2
2(h−m)2+1
2a+s1
2a−ϕ−aϕ2(h−bw)2
2(h−m)22
−ϕ2(m−bw)2
(h−m)2
,
rκ=w+aϕ2(h−bw)2(m−bw)
2b(h−m)2(κ−2ϕ)+2ab ϕ2(h−bw)2,rc=h+bw
2b,rϕ=w+aϕ(h−bw)2
2b(h−m),rh=l
b.
The detailed analysis and proof are in Appendix A.
Theorem 3 suggests that under the positive evaluation system, the optimal retail
price that retailers focus on is strongly associated with their optimism degree
ϕ
and their
confidence level
κ
. Based on the above, the
κ
value reflects the retailer’s level of confidence
in the decision after the retailer has determined the positive focus of demand. We can
interpret the behavioral patterns in the results of Theorem 3. When the retailer is highly
confident (κtakes a small value), the very optimistic (ϕtakes a large value) will choose rc
as the optimal retail price. The generally optimistic (
ϕ
takes value in the middle range) and
the very unoptimistic (
ϕ
takes a small value) will choose
rϕ
and
rh
, respectively. When
retailers are not confident enough (
κ
takes a large value), they will choose
rκ
as the retail
price regardless of the degree of optimism (regardless of the value of ϕ).
Under the positive evaluation system, the retailer does not concentrate on demand
below
m
. Instead, he or she bases the optimal retail pricing decision on the demand that
has his or her attention. Lemma 1 to Theorem 3 provides a comprehensive analysis of the
demand focus categories that retailers concentrate on and the optimum retail price they
select under the positive evaluation system. Based on this result, it will then be compared
to the classical expectation value model to analyze the impact of retailer personality traits
on the coordinated supply chain under FTC.
4.3. Supplier’s Decision under the Wholesale Price Contract
The supplier needs to fully understand the personality characteristics of the retailer
when setting the wholesale price. Therefore, we assume that for the supplier, the retailer’s
personality characteristics are full information, and the supplier decides the wholesale
price by maximizing its expected payoff.
All The most typical B2B (Business to Business) e-commerce model is a supply chain
that is led by suppliers. Suppliers can create an online platform to provide their products
or services to retailers online. Suppliers only collect a fee from the retailer during the
transaction. According to the decision sequence, this section analyzes the result of the
upper-level supplier in the Stackelberg game. As a dominant player, the supplier can
effectively predict the
rP(w)
and determine the optimal wholesale price
w∗
p
by maximizing
its expected payoff. The retailer determines the optimal retail price
rPw∗
p
after observing
the w∗
p. The supplier optimization problem is expressed as
max
c≤w≤2l−h
b
(w−c)E{X−b∗rP(w)}(7)
Assuming that the supplier knows about the personality characteristics of the retailer
well, write the optimization problem (7) with respect to
w
in the following functional form:
H(w) = (w−c)[m−b∗rP(w)]. (8)
Analyzing the optimal wholesale price decision of the supplier requires calculating the
maximum value point of the function (8) in the range
hc, 2l−h
bi
of the wholesale price. The
first-order derivative
H0(w)
and second-order derivative
H00 (w)
of the function
H(w)
are
H0(w)=m−brP(w)−b(w−c)(rP(w))0
and
H00 (w)=−
2
b(rP(w))0−b(w−c)(rP(w))00
.
Based on the analysis of the first-order derivative function of
H(w)
as well as the second-
order derivative function, this section considers the following three possible situations
to analyze the existence of supplier optimal wholesale price in the face of retailers with
positive focus preference.
J. Theor. Appl. Electron. Commer. Res. 2023,18 1052
(i) If
H(w)
is strictly increasing concerning the wholesale price
w
, i.e., when
H0(w)>
0,
it indicates that the supplier’s expected payoff increases as the wholesale price
w
increases
in the face of an aggressive retailer. In this case, the supplier needs to offer a maximum
wholesale price to the retailer within the effective range of the wholesale price to maximize
its expected payoff, while incentivizing the retailer to place more orders.
(ii) If
H(w)
is strictly decreasing concerning the wholesale price
w
, i.e., when
H0(w)<0
,
it indicates that the supplier’s expected payoff decreases as the wholesale price
w
increases
in the face of an aggressive retailer. In this case, the supplier needs to offer a minimum
wholesale price to the retailer within the effective range of the wholesale price to maximize
its expected payoff.
(iii) If
H(w)
is strictly concave with respect to
w
in the valid range, i.e.,
H00 (w)<
0
and there exists a wholesale price
w0
such that
H0(w0) =
0, it shows that there exists an
optimal wholesale price for the supplier to maximize its own expected payoff.
The supplier faces a defined retailer personality profile, and the above three trends
of variation may exist segmentally in the final
H(w)
function. We need to analyze the
existence of optimal wholesale prices specifically for specific values. According to Theorem
3, substituting
rP(w)
can obtain the supplier’s payoff function (8) given in the following
four forms:
(1) When rP(w)=rκ,
H(w)=(w−c)"m
2−b
2w+(h−m)2(κ−2ϕ)(m−bw)
2(h−m)2(κ−2ϕ)+2aϕ2(h−bw)2#. (9)
(2) When rP(w)=rϕ,
H(w) = (w−c)"m− bw +aϕ(h−bw)2
2(h−m)!#. (10)
(3) When rP(w)=rc,
H(w) = (w−c)m−h+bw
2. (11)
(4) When rP(w)=rh=l
b,
H(w) = (w−c)[m−l]. (12)
The optimal value of the parametric model is hard to solve. The following is a
brief description of the trend change characteristics of the supplier by parameter settings
a=
0.8,
b=
50,
c=
15,
l=
2100,
m=
2300,
h=
2500,
ϕ=
0.3,
κ=
0.1. The range
of wholesale prices
w
is
[15, 34]
, and the supplier predicts that retailer will have different
decisions in the face of the various
w
. Figure 1shows three segments of the retailer’s
decisions for different
w
. The first part when
w∈[15, 31]
, satisfying Theorem 3(iii), at this
time, the retailer’s decision is
rκ
and the supplier’s payoff function is Equation (9), showing
an increasing and then decreasing trend. The second part indicates that when
w∈[31, 33.3]
,
satisfying conditions in Theorem 3(ii), the retailer’s decision is
rϕ
; at this time, the supplier’s
payoff function is Equation (10), which shows a monotonically increasing trend on this
interval, and the supplier will choose the upper limit of the wholesale price of 33.3($). The
third part indicates that when
w∈[33.3, 34]
, satisfying Theorem 3(i), the retailer’s decision
is
rc
, when the supplier’s payoff function is Equation (11), which shows a monotonically
decreasing trend, and the supplier will choose the lower bound of the wholesale price of
33.3($).
J. Theor. Appl. Electron. Commer. Res. 2023,18 1053
JTAER 2023, 18, FOR PEER REVIEW 13
retailer’s decision is rκ and the supplier’s payo function is Equation (9), showing an in-
creasing and then decreasing trend. The second part indicates that when w ∈ [31,33.3] ,
satisfying conditions in Theorem 3(ii), the retailer’s decision is rφ; at this time, the sup-
plier’s payo function is Equation (10), which shows a monotonically increasing trend on
this interval, and the supplier will choose the upper limit of the wholesale price of 33.3($).
The third part indicates that when w∈ [33.3,34], satisfying Theorem 3(i), the retailer’s
decision is rc, when the supplier’s payo function is Equation (11), which shows a mon-
otonically decreasing trend, and the supplier will choose the lower bound of the wholesale
price of 33.3($).
Comprehensive at the three intervals, the supplier will eventually choose the rst
part of the maximum point as the overall payo optimal decision, and it can achieve the
supplier optimal payo of 4194.2($). Figure 2 comprehensively demonstrates the existence
of the supplier optimal wholesale price, containing the three cases of monotonically in-
creasing, monotonically decreasing, and rst increasing and then decreasing, as described
in the previous section. Figure 2 illustrates that in the face of retailers with dierent per-
sonality characteristics, the payo function of the supplier may be a combination of seg-
ments of Equation (9) to Equation (12). By comparing the optimal payo of each interval
and then determining the global optimal payo, the actual solution of the supplier’s opti-
mal wholesale price requires specic analysis. In this paper, we assume that the supplier
will maximize its expected payo by seing the wholesale price w in calculating the opti-
mal wholesale price. This assumption simplies the process of solving the optimal solu-
tion for the supplier, which can directly calculate the optimal decision and payo, and
more intuitively observe the interaction between the upper and lower levels in the supply
chain.
Analyzing the coordination of the supply chain under the active focus model requires
comparing the nal rP(w) with the optimal retail price r∗=μ+bc
2b under the centralized
channel in the classical expectation model. If rP(w)=r∗ can be achieved, it means that
the retailer under positive focus preference can coordinate the supply chain and achieve
the optimal overall supply chain performance under certain conditions. To facilitate com-
parison with the classical expectation model, we will further set specic parameters to
solve for the optimal wholesale price and retail price and specically analyze the inuence
of the retailer’s optimism and condence level on the decision to study the supply chain
coordination.
Figure 1. Reference model of the supplier’s payo function.
Figure 1. Reference model of the supplier’s payoff function.
Comprehensive at the three intervals, the supplier will eventually choose the first part
of the maximum point as the overall payoff optimal decision, and it can achieve the supplier
optimal payoff of 4194.2($). Figure 2comprehensively demonstrates the existence of the
supplier optimal wholesale price, containing the three cases of monotonically increasing,
monotonically decreasing, and first increasing and then decreasing, as described in the
previous section. Figure 2illustrates that in the face of retailers with different personality
characteristics, the payoff function of the supplier may be a combination of segments
of Equation (9) to Equation (12). By comparing the optimal payoff of each interval and
then determining the global optimal payoff, the actual solution of the supplier’s optimal
wholesale price requires specific analysis. In this paper, we assume that the supplier will
maximize its expected payoff by setting the wholesale price
w
in calculating the optimal
wholesale price. This assumption simplifies the process of solving the optimal solution
for the supplier, which can directly calculate the optimal decision and payoff, and more
intuitively observe the interaction between the upper and lower levels in the supply chain.
JTAER 2023, 18, FOR PEER REVIEW 18
Figure 2. Image change (1) of H(w) with φ = 0.1.
Figure 3. Image changes (2) of H(w) with φ = 0.1.
Figure 2. Image change (1) of H(w)with ϕ=0.1.
Analyzing the coordination of the supply chain under the active focus model requires
comparing the final
rP(w)
with the optimal retail price
r∗=µ+bc
2b
under the centralized
channel in the classical expectation model. If
rP(w)=r∗
can be achieved, it means that
the retailer under positive focus preference can coordinate the supply chain and achieve
the optimal overall supply chain performance under certain conditions. To facilitate com-
parison with the classical expectation model, we will further set specific parameters to
J. Theor. Appl. Electron. Commer. Res. 2023,18 1054
solve for the optimal wholesale price and retail price and specifically analyze the influ-
ence of the retailer’s optimism and confidence level on the decision to study the supply
chain coordination.
5. Numerical Examples and Result Analyses
This section analyzes the model results through numerical experiments. The supplier
is a company with a dominant market position in the apparel category, and the retailer
adopts a reservation sales system in facing the consumers, thus acquiring the product
market demand by determining the pre-sale quantity. Because the clothing product is
updated frequently and has a short sales season, and frequently requires only one purchase
during a single sales season, the retailer’s choice is a one-time, single-cycle choice. With
an effectively forecasted sales volume, neither side must bear the risk of inventory and
out-of-stock risk.
5.1. Parameter Settings and Numerical Results
By referring to the assignment of the relevant literature [
12
,
59
], the following relevant
parameters are set: the unit production cost of the supplier
c=
10 ($) and the sensitivity
coefficient of product demand to the retail price
b=
50. Since the demand is assumed to
be in the form of a normal distribution,
a=
0.8 is set. The range of the random variable
market demand is
[1600, 2000]
and the mean value of demand is
m=
1800. Assume that the
supplier has complete knowledge of the retailer’s personality traits. Based on the parameter
settings of the above model, the satisfaction function (2) of the retailer is obtained as
u(r, w, x)=2(r−w)(x−50r)
25(40 −w)2. (13)
The relative likelihood function of the potential market demand is given by
π(x)=1−(1800 −x)2
50, 000 . (14)
According to Section 3, the reaction function
rP(w)
and the positive focus of potential
demand
xP(rP(w)
,
w)
of the retailer under the positive evaluation system are first deter-
mined. At different levels of optimism and self-confidence, Theorem 3 shows that the
optimal reaction function, as well as the positive demand focus, will be different. The
solution yields the following classification of the optimal retail price in the face of different
ranges of wholesale prices as follows:
(1) When w >40 −5
ϕ,
rP(w)=
40+w
2, if(4
5κ−1)ϕ2(40−w)2
16 +ϕ2(36−w)2
16 + (κ−2ϕ)(κ−5
4)≤0,
w+ϕ2(40−w)2(36−w)
40(κ−2ϕ)+2ϕ2(40−w)2, if(4
5κ−1)ϕ2(40−w)2
16 +ϕ2(36−w)2
16 + (κ−2ϕ)(κ−5
4)>0.
(2) When w ≤40 −5
ϕand ϕ(40 −w)2−10(32 −w)≤0,
rP(w)=
w+ϕ(40−w)2
10 , ifϕ2(40 −w)2−5ϕ(36 −w)+20(κ−2ϕ)≤0,
w+ϕ2(40−w)2(36−w)
40(κ−2ϕ)+2ϕ2(40−w)2, ifϕ2(40 −w)2−5ϕ(36 −w)+20(κ−2ϕ)>0.
(3) When ϕ2(40 −w)2−10(32 −w)>0,
rP(w)=
32, ifϕ2(40 −w)2(w−28)−40(κ−2ϕ)(32 −w)≥0,
w+ϕ2(40−w)2(36−w)
40(κ−2ϕ)+2ϕ2(40−w)2, ifϕ2(40 −w)2(w−28)−40(κ−2ϕ)(32 −w)<0.
J. Theor. Appl. Electron. Commer. Res. 2023,18 1055
The two thresholds for
ϕ
are
ϕ1=5
40−w
and
ϕ2=10(32−w)
(40−w)2
. According to the
range of
w∈[10, 24]
, we can obtain
ϕ1∈h1
6,5
16 i
and
ϕ2∈h11
45 ,5
16 i
. Additionally, the
three thresholds for
κ
is
κ1=
2
ϕ+5ϕ(36−w)−ϕ2(40−w)2
20
,
κ2=
2
ϕ+ϕ2(40−w)2(w−28)
40(32−w)
and
κ3=5
8+ϕ−ϕ2(40−w)2
40 +s5
8−ϕ+ϕ2(40−w)2
40 2
−ϕ2(36−w)2
16 . Furthermore, r ∈[w, 32].
The retailer’s positive demand focus and optimal retail price are subject to changes in
their own optimism and confidence level. It is necessary to reasonably set the values of
ϕ
and
κ
, and then calculate the optimal wholesale price
w∗
p
, the optimal retail price
rPw∗
p
,
the supplier’s payoff
Hw∗
p
, the retailer’s satisfaction level
uw∗
p, xP(rPw∗
p, w∗
p)
, and
the relative likelihood
πxP(rPw∗
p, w∗
p)
. The equilibrium solutions with different
ϕ
and
κ
values are presented in tables to visualize the effect of retailer’s optimism and confidence
level on the equilibrium results.
5.2. Results Analysis
The data in Tables 2–7present the effects of different levels of optimism
ϕ
and confi-
dence level
κ
of retailers on the final decision. The analysis of the results can be obtained
for two specific aspects:
Table 2. Solutions of the proposed model with ϕ=0.1.
κw*
prPw*
pHw*
puw*
p,xP(rPw*
p,w*
p)πxP(rPw*
p,w*
p)
0.1 21 30.5 3025 1 0.2
0.5 21 30.5 3025 1 0.2
1 14.8 17.8 4365 0.4 0.8
2 23.3 23.8 8141.9 0.1 1
10 23.1 23.2 8388.8 0.01 1
Table 3. Solutions of the proposed model with ϕ=0.3.
κw*
prPw*
pHw*
puw*
p,xP(rPw*
p,w*
p)πxP(rPw*
p,w*
p)
0.1 23.3 31.7 2881.6 1 0.2
0.5 22.2 29.6 3891.5 0.9 0.5
1 24 28.5 5282.5 0.7 0.7
2 24 28.5 6504.1 0.7 0.7
10 23.6 24.3 7951.1 0.1 1
Table 4. Solutions of the proposed model with ϕ=0.5.
κw*
prPw*
pHw*
puw*
p,xP(rPw*
p,w*
p)πxP(rPw*
p,w*
p)
0.1 20.1 29.8 3122.6 0.8 0.8
0.5 21.5 29.7 3617.1 0.8 0.8
1 23 29.5 4225 0.7 0.8
2 24 28.6 5200 0.6 0.9
10 24 25.6 7298.4 0.3 1
J. Theor. Appl. Electron. Commer. Res. 2023,18 1056
Table 5. Solutions of the proposed model with ϕ=1.
κw*
prPw*
pHw*
puw*
p,xP(rPw*
p,w*
p)πxP(rPw*
p,w*
p)
0.1 21.5 29.7 3648.1 0.7 1
0.5 21.9 29.7 3771.9 0.7 1
1 22.3 29.6 3925.5 0.7 1
2 23 29.5 4225 0.7 1
10 24 27.7 5815.4 0.5 1
Table 6. Solutions of the proposed model with ϕ=2.
κw*
prPw*
pHw*
puw*
p,xP(rPw*
p,w*
p)πxP(rPw*
p,w*
p)
0.1 22.2 29.6 3933 0.6 1
0.5 22.4 29.6 3963.6 0.6 1
1 22.5 29.6 4001.5 0.6 1
2 22.6 29.5 4076.8 0.6 1
10 23.9 29.3 4640.9 0.6 1
Table 7. Solutions of the proposed model with ϕ=10.
κw*
prPw*
pHw*
puw*
p,xP(rPw*
p,w*
p)πxP(rPw*
p,w*
p)
0.1 22.9 29.5 4166 0.6 1
0.5 22.9 29.5 4167.6 0.6 1
1 22.9 29.5 4169.1 0.6 1
2 22.9 29.5 4172.1 0.6 1
10 22.9 29.5 4195.7 0.6 1
(1) Under the positive evaluation system, for a supplier, the retailer’s optimism level
ϕ
and confidence level
κ
have a more intuitive effect on the supplier’s wholesale price
decision and payoff. The supplier’s decision is based on the premise that the information
about the retailer’s personality characteristics is better known. When the retailer has a high
level of optimism and confidence, i.e., a low
ϕ
value (
ϕ=
0.1, 0.2, 0.3) and a low
κ
value
(
κ=
0.1), the supplier will choose the optimal wholesale price of 21, while the retailer
will choose the optimal retail price of 30.5. This result will be maintained in the range of
ϕ<
0.3125. As the retailer’s confidence level decreases, i.e., when
κ
increases to a larger
range (
κ=
1, 2, 10), the wholesale price chosen by the supplier will plummet, followed
by an increasing and then decreasing trend. After the retailer’s optimism level decreases,
i.e., after the
ϕ
value becomes larger (
ϕ=
0.5, 1, 2, 10), the optimal wholesale price chosen
by the supplier slowly increases and the expected payoff slowly increases as the retailer’s
confidence level gradually decreases (
κ=
0.1, 0.5, 1, 2, 10) for
ϕ>
0.3125 and the value is
determined. Overall, the larger the
κ
, the higher the supplier’s payoff will be for a fixed
value of ϕ.
(2) There is also a significant trend in the effect of the retailer’s optimism level
ϕ
and
confidence level
κ
on his or her satisfaction level and positive demand focus. When the
retailer has high levels of optimism and confidence (
ϕ=
0.1,
κ=
0.1, 0.5), the retailer’s sat-
isfaction level is close to 1 and focuses on the market potential demand with low probability.
When there is a significant decrease in the level of confidence (
κ=
1, 2, 10), the retailer’s
satisfaction level decreases rapidly, and the focus on the market demand is infinitely close
to 1. The retailer will pay more attention to the occurrence of potential demand in the
market with a higher probability. After the level of the retailer’s optimism decreases to
a certain level, as the
ϕ
increases (
ϕ=
0.5, 1, 10), the level of the retailer’s satisfaction
decreases as the retailer’s confidence level decreases gradually (
κ=
0.1, 0.5, 1, 2, 10) and
the concerned demand increases close to 1. Overall, the lower the
ϕ
value, the higher the
J. Theor. Appl. Electron. Commer. Res. 2023,18 1057
satisfaction level of the retailer, and the lower the probability of occurrence of the concerned
potential demand in the market. The lower κhas the same trend of change.
The table data can observe changes under different personality traits to a certain
extent. However, there are still some shortcomings in the amount of data, resulting in a
less intuitive presentation of specific trend variations. The cases of
ϕ
are divided into three
ranges. The table data determine that
ϕ<1
6
belongs to Theorem 3(i), and
ϕ>5
16
belongs
to Theorem 3(iii). The conclusions are more complex since
ϕ
in the range
h1
6,5
16 i
spans the
presence of Theorem 3(i) and 3(ii). The following images of the supplier’s payoff function
show more clearly the trend variation under the different ϕand κ.
Firstly, in the range of
ϕ<1
6
, taking
ϕ=
0.1 as an exemplar can indicate a certain
regular variation. As shown in Figure 2, the supplier believes that the retailer will choose
the retail price
rc
, when the wholesale price and the optimal retail price will remain constant
within a smaller range of
κ
. In Figure 3, there is a bound of
κ
between 0.87 and 0.88, which
causes the supplier’s payoff function to change from a single peak to a double peak as the
retailer’s confidence level declines.
JTAER 2023, 18, FOR PEER REVIEW 18
Figure 2. Image change (1) of H(w) with φ = 0.1.
Figure 3. Image changes (2) of H(w) with φ = 0.1.
Figure 3. Image changes (2) of H(w)with ϕ= 0.1.
Where the left peak in Figure 3indicates that the supplier believes that this range of
the retailer will lead them to choose the retail price
rκ
, the right peak remains constant.
As
κ
rises, the left peak climbs, and the supplier ’s optimal wholesale price stays the
same up until it equals the right peak, taking the right peak point—a range where the
retailer’s level of confidence has no bearing on the supplier ’s decision. In Figure 4, as the
retailer’s confidence level decreases, the black circle indicates the maximum value of the
function. Additionally,
κ
exists between 0.95 and 0.96 with bimodal equality, indicating
that the supplier’s wholesale price decision will produce a sudden drop when the retailer’s
confidence level decreases. As
κ
continues to increase, as shown in Figure 5, the range
of retailer’s choice of
rc
gradually decreases and is occupied by
rκ
. When
κ
exceeds the
threshold value of 1.1679, the supplier only considers the situation in which retailers accept
rκ
. At this point, both the wholesale price and the optimum retail price show a trend of first
increasing and then decreasing, though the variations in both the wholesale and retail prices
are very subtle. After the retailer’s confidence level falls to a certain bound, the supplier’s
pricing decision roughly exhibits a monotonically increasing situation, converging to the
highest wholesale price 24.
J. Theor. Appl. Electron. Commer. Res. 2023,18 1058
JTAER 2023, 18, FOR PEER REVIEW 18
Figure 2. Image change (1) of H(w) with φ = 0.1.
Figure 3. Image changes (2) of H(w) with φ = 0.1.
Figure 4. Image changes (3) of H(w)with ϕ= 0.1.
JTAER 2023, 18, FOR PEER REVIEW 19
Figure 4. Image changes (3) of H(w) with φ = 0.1.
The inuence of the retailer’s optimism and self-condence level on the decision can
be obtained as follows:
(1) When the level of optimism and condence of the retailer are both high, the pric-
ing decision of the supplier and retailer do not produce uctuations and are relatively
stable. The best wholesale price will produce a trend from high to low as the retailer’s
degree of condence gradually declines while it is high.
(2) When the retailer’s optimism is determined, the lower the condence level, the
higher the supplier’s payo.
The image comparison shows that it is more benecial for the supplier to partner
with a retailer who has a higher level of optimism and a lower level of condence to expect
payo growth.
Figure 5. Image changes (4) of H(w) with φ = 0.1.
6. Comparison
6.1. Comparison with Classical Expectation Model
To compare and analyze whether supply chain coordination is possible under spe-
cic circumstances with decentralized decision-making by suppliers and retailers under
positive focus decision-making, numerical experiments are substituted into the classical
expectation value model in this section. According to the parameter seings in Section 5.1,
the results of the classical model under considering only the expected payo are shown
in Table 8.
Table 8. Results of the classical expectation model.
𝐫
𝐕𝐫
𝐰∗
𝐕𝐬
𝐕
Centralization
23
/
/
/
8450
Decentralization
29.5
2112.5
23
4225
6337.5
It is because the wholesale price is an endogenous variable that only the overall op-
timal retail price, as well as the payo, are available in the results of the focus decision in
Table 8. To present more clearly the coordination of the supply chain, the representative
φ and κ are taken in combination with the data in the table and according to the range
Figure 5. Image changes (4) of H(w)with ϕ= 0.1.
After that, we discuss the situation in the
ϕ∈h1
6,5
16 i
. The overall change is more
similar to the trend at
ϕ=
0.1, and the trend also changes from single-peak to double-peak
and then to single-peak again. When
ϕ=
0.3 the bimodal peaks already appear at smaller
values of
κ
. The maximum payoff of the supplier increases as
κ
increases, with the change
in
κ
between 0.2 and 0.3 appearing as the left peak enclosing the right peak. This trend
indicates that for the supplier, the lack of confidence is the more favorable situation in the
case of the degree of optimism of the retailer and the existence of opportunities to enable
the supplier to reach a higher payoff.
Finally, we discuss the case when
ϕ>5
16
, when we can guarantee
ϕ>ϕ2
and satisfy
the Theorem 3(iii). When
ϕ=
0.4 and
κ
is small, the right peak in the double peak has
been almost covered by the left, and the process of changing from a single peak to a double
peak disappears. As the
κ
value increases currently, the image changes to a single-peak at a
faster rate. When the ϕvalue increases, it is almost a complete S-peak state.
When the
ϕ
value reaches a higher range, after the retailer optimism decreases sig-
nificantly, the supplier believes that the retailer’s decision to choose the retail price is
J. Theor. Appl. Electron. Commer. Res. 2023,18 1059
only
rκ
. When
ϕ=
2, the supplier’s payoff image that the image of the function (12),
the change of the image is very insignificant; only when there is a large change in
κ
will
it cause a subtle increase in the optimal wholesale price; as
κ
increases to higher range,
the optimal wholesale price will be infinitely close to the upper limit of 24. The trend
illustrates that when the value of
ϕ
is small, the change floats more and produces larger
fluctuations according to the change in
κ
. As the value of
ϕ
increases, the payoff function
of the supplier is gradually decreased by
κ
, and the trend becomes smaller and smaller. It
indicates that when the retailer’s optimism is low, the supplier’s payoff is almost negligible
by the retailer’s confidence level, and the supplier ’s payoff fluctuates more only when the
retailer is more optimistic.
The influence of the retailer’s optimism and self-confidence level on the decision can
be obtained as follows:
(1) When the level of optimism and confidence of the retailer are both high, the pricing
decision of the supplier and retailer do not produce fluctuations and are relatively stable.
The best wholesale price will produce a trend from high to low as the retailer’s degree of
confidence gradually declines while it is high.
(2) When the retailer’s optimism is determined, the lower the confidence level, the
higher the supplier’s payoff.
The image comparison shows that it is more beneficial for the supplier to partner with
a retailer who has a higher level of optimism and a lower level of confidence to expect
payoff growth.
6. Comparison
6.1. Comparison with Classical Expectation Model
To compare and analyze whether supply chain coordination is possible under specific
circumstances with decentralized decision-making by suppliers and retailers under positive
focus decision-making, numerical experiments are substituted into the classical expectation
value model in this section. According to the parameter settings in Section 5.1, the results
of the classical model under considering only the expected payoff are shown in Table 8.
Table 8. Results of the classical expectation model.
r Vrw*VsV
Centralization
23 / / / 8450
Decentralization
29.5 2112.5 23 4225 6337.5
It is because the wholesale price is an endogenous variable that only the overall
optimal retail price, as well as the payoff, are available in the results of the focus decision in
Table 8. To present more clearly the coordination of the supply chain, the representative
ϕ
and
κ
are taken in combination with the data in the table and according to the range of the
three
ϕ
values analyzed for the supplier’s payoff function to show the change curves of the
expected payoff of the whole channel.
Figure 6shows the change of the expected payoff image for each decision subject
when
ϕ=
0.1,
κ=
0.5. The supplier will choose the wholesale price, 21. However, it can be
seen in Figure 6that the wholesale price cannot coordinate with the channel at this time.
According to the influence of optimism degree and confidence level on decision-making
analyzed in Section 5.1, with constant optimism of the retailer (
ϕ=
0.1) and increasing the
value of
κ
, Figure 7shows that the supply chain is close to coordination at
ϕ=
0.1,
κ=
10.
At this point, the payoff of the whole channel corresponding to the point of the maximum
payoff value of the supplier is infinitely close to the result under coordination. The image
indicates that the supplier payoff and the channel payoff are close to overlap, while the
retailer payoff is close to 0. It shows that when the retailer’s confidence level decreases
infinitely, the channel payoff increases while leading to the loss of the retailer payoff.
J. Theor. Appl. Electron. Commer. Res. 2023,18 1060
JTAER 2023, 18, FOR PEER REVIEW 20
of the three φ values analyzed for the supplier’s payo function to show the change
curves of the expected payo of the whole channel.
Figure 6 shows the change of the expected payo image for each decision subject
when φ = 0.1, κ =0.5. The supplier will choose the wholesale price, 21. However, it can
be seen in Figure 6 that the wholesale price cannot coordinate with the channel at this
time. According to the inuence of optimism degree and condence level on decision-
making analyzed in Section 5.1, with constant optimism of the retailer (φ=0.1) and in-
creasing the value of κ, Figure 7 shows that the supply chain is close to coordination at
φ =0.1, κ = 10. At this point, the payo of the whole channel corresponding to the point
of the maximum payo value of the supplier is innitely close to the result under coordi-
nation. The image indicates that the supplier payo and the channel payo are close to
overlap, while the retailer payo is close to 0. It shows that when the retailer’s condence
level decreases innitely, the channel payo increases while leading to the loss of the re-
tailer payo.
On the other hand, the level of optimism of the retailer is adjusted. Increasing the
value of φ with a constant level of retailer condence, in contrast to Figure 7, which is
nearly coordinated, Figure 8 depicts the payo curve for the parameter conguration of
φ =0.4, κ = 10. The parameter seings in Figure 8 cannot coordinate the supply chain,
but the existence of a certain range of wholesale prices makes the channel payo innitely
close to the maximum value. This indicates that under certain circumstances, the whole-
sale price may change if the supplier can predict the form of the payo distribution curve
or if the supplier’s decision is not based exclusively on expected payo maximization. The
premise of this theory is to ensure that the overall payo increase compensates for the
supplier’s loss.
Figure 6. Payo image changes with φ = 0.1, κ =0.5.
Figure 6. Payoff image changes with ϕ=0.1, κ=0.5.
JTAER 2023, 18, FOR PEER REVIEW 21
Figure 7. Payo image changes with φ = 0.1, κ =10.
Figure 8. Payo image changes with φ = 0.4, κ =10.
6.2. Comparison with Fairness Concern
Referring to the literature that rst introduced fairness concern into the supply chain,
the decision results are obtained by seing the retailer’s sensitive parameters α and β
for advantageous inequality and disadvantageous inequality [42].
Table 9 shows that the fairness concern with α = 0.5, β =0.5 or α =0.7, β = 0.6 is
able to coordinate the supply chain, in line with the results of Cui et al [42]. In the fairness
concern model, the supplier sets a wholesale price that aligns the incentives of all channel
members with the interests of the entire channel so that the double-marginalization prob-
lem does not occur. When both advantageous inequality and disadvantageous inequality
are high, the retailer with fairness concern will accomplish coordination in line with the
ndings of our outcomes. Additionally, it is further shown in conjunction with FTC that
xed wholesale price contracts can coordinate the supply chain as long as the retailer has
certain fairness concern preference. There are good reasons for suppliers to choose this
simpler and more feasible pricing mechanism.
Figure 7. Payoff image changes with ϕ=0.1, κ=10.
On the other hand, the level of optimism of the retailer is adjusted. Increasing the
value of
ϕ
with a constant level of retailer confidence, in contrast to Figure 7, which is
nearly coordinated, Figure 8depicts the payoff curve for the parameter configuration of
ϕ=
0.4,
κ=
10. The parameter settings in Figure 8cannot coordinate the supply chain,
but the existence of a certain range of wholesale prices makes the channel payoff infinitely
close to the maximum value. This indicates that under certain circumstances, the wholesale
price may change if the supplier can predict the form of the payoff distribution curve or
if the supplier’s decision is not based exclusively on expected payoff maximization. The
premise of this theory is to ensure that the overall payoff increase compensates for the
supplier’s loss.
J. Theor. Appl. Electron. Commer. Res. 2023,18 1061
JTAER 2023, 18, FOR PEER REVIEW 21
Figure 7. Payo image changes with φ = 0.1, κ =10.
Figure 8. Payo image changes with φ = 0.4, κ =10.
6.2. Comparison with Fairness Concern
Referring to the literature that rst introduced fairness concern into the supply chain,
the decision results are obtained by seing the retailer’s sensitive parameters α and β
for advantageous inequality and disadvantageous inequality [42].
Table 9 shows that the fairness concern with α = 0.5, β =0.5 or α =0.7, β = 0.6 is
able to coordinate the supply chain, in line with the results of Cui et al [42]. In the fairness
concern model, the supplier sets a wholesale price that aligns the incentives of all channel
members with the interests of the entire channel so that the double-marginalization prob-
lem does not occur. When both advantageous inequality and disadvantageous inequality
are high, the retailer with fairness concern will accomplish coordination in line with the
ndings of our outcomes. Additionally, it is further shown in conjunction with FTC that
xed wholesale price contracts can coordinate the supply chain as long as the retailer has
certain fairness concern preference. There are good reasons for suppliers to choose this
simpler and more feasible pricing mechanism.
Figure 8. Payoff image changes with ϕ=0.4, κ=10.
6.2. Comparison with Fairness Concern
Referring to the literature that first introduced fairness concern into the supply chain,
the decision results are obtained by setting the retailer’s sensitive parameters
α
and
β
for
advantageous inequality and disadvantageous inequality [42].
Table 9shows that the fairness concern with
α=
0.5,
β=
0.5 or
α=
0.7,
β=
0.6 is
able to coordinate the supply chain, in line with the results of Cui et al. [
42
]. In the fairness
concern model, the supplier sets a wholesale price that aligns the incentives of all channel
members with the interests of the entire channel so that the double-marginalization problem
does not occur. When both advantageous inequality and disadvantageous inequality are
high, the retailer with fairness concern will accomplish coordination in line with the
findings of our outcomes. Additionally, it is further shown in conjunction with FTC that
fixed wholesale price contracts can coordinate the supply chain as long as the retailer has
certain fairness concern preference. There are good reasons for suppliers to choose this
simpler and more feasible pricing mechanism.
Table 9. Results of the fairness concern model.
θInequality Parameters w*rw*Supplier Profit Channel Profit
θ=1α=0.5, β=0.2 18 26 4000 8000
θ=1α=0.5, β=0.4 17.1 24.2 4190.1 8380.2
θ=1α=0.5, β=0.5 16.5 23 4225 8450
θ=1α=0.6, β=0.4 17.1 24.2 4190.1 8380.2
θ=1α=0.7, β=0.6 16.5 23 4225 8450
Note:
θ
represents the retailer’s equitable payoff parameter.
α
and
β
represent the retailer’s disadvantageous and
advantageous inequality parameters, respectively.
6.3. Discussion on the Results
Combining the values of the classical expectation model in Figure 7comparing
Tables 2and 8,
we find that channel coordination is likely to be achieved when the
ϕ
is small. As
κ
increases, the retail price of the final decision will keep approaching the
optimal retail price under the classical model; that is, the wholesale price contract is likely
to achieve channel coordination when the retailer is relatively optimistic and has a low
level of confidence.
This finding further solidifies the importance of the retailer’s optimism and confidence
levels for supply chain coordination under the FTC. Decision makers need to ensure
J. Theor. Appl. Electron. Commer. Res. 2023,18 1062
cooperation and mutual benefits between supplier and retailer if they want to coordinate
and achieve optimal channel payoffs. If channel coordination is premised on sacrificing the
payoffs of one of the channel members, cooperation between the two decision makers is
difficult to sustain in the long run, so this paper provides a theoretical basis for establishing
further cooperation between supplier and retailer. To achieve channel coordination, the
supplier can provide a certain allowance to the retailer to ensure that the retailer is not
making zero payoffs, or a cooperation contract can be established between the supplier and
the retailer to redistribute channel payoffs.
We introduce personality characteristics into the supply chain for modeling, and the
result shows the impact of personality characteristics on supply chain coordination, leading
to certain management insights on cooperation among decision makers as follows: The
dominant supplier has the right to preferentially choose retailers. The above supplier can
do this by recognizing the personality characteristic of the retailer through tests and by
choosing retailers who are more optimistic and less confident. The retailer, as a follower,
can modify his or her own optimism and self-confidence levels to promote supply chain
coordination, improve the results of the classical expectation value model, and solve the
double marginalization effect. In short, in any partnership, it is important to effectively
consider the personality traits of the partner. Additionally, FTC can consider the influence
of different personality characteristics in the decision-making process of decision makers,
making the partnership in the supply chain more transparent and achieving profitable
both sides.
7. Conclusions
The classical expectation value model using wholesale price contracts cannot coordi-
nate the supply chain if the decision maker is fully rational, considering only the expected
payoff. In this paper, the model considers the behavioral model under e-commerce and
introduces the positive focus decision model to the retailer to study the coordination of
the e-commerce supply chain under a wholesale price contract. The retailer’s decision is
divided into two processes: the first process is to decide on the positive focus of potential
demand in the product market of concern, and the second process is to select the retail price
focus of concern among the positive demand focuses and determine the optimal retail price.
The analysis shows that the wholesale price contract in the context of e-commerce may
achieve channel coordination in the case of a more optimistic and low level of confidence of
the retailer. Additionally, in many conditions, the realized payoffs are already better than
the results of the classical expectation value model. At the same time, if the supplier is im-
perfectly rational, the conditions for coordinating the channel may become somewhat more
lenient. Additionally, the supplier’s decisions may change depending on the degree of the
retailer’s focus preference; that is, a retailer with a positive focus preference, different levels
of optimism, and confidence can have a significant impact on channel coordination. With
the supplier as the dominant player, it is a matter for further discussion in the supply chain
to fully understand the personality traits and preferences of the retailer and how to make
decisions that are more conducive to channel coordination without losing their interests.
The results of the model under focus preference corroborate that the positive focus
preference of the study channel member is beneficial for coordinating the channel. The
focus theory of choice contains two different evaluation systems. In the negative evalu-
ation system, it is possible that an event that yields relatively lower returns with higher
probability has higher salience for the decision maker, and if the retailer’s personality
profile is biased toward the negative evaluation system, the conditions under the final
channel coordination may be opposite to the results under the positive evaluation system.
Additionally, in many cases, both evaluation systems may be present in the personality
profile of the decision maker, and the retailer ’s decision range may be somewhat broader
and the conditions under channel coordination somewhat more lenient.
In business management, competition among retailers can be intense in the face of
a complex e-commerce environment. The dominant supplier needs to comprehensively
J. Theor. Appl. Electron. Commer. Res. 2023,18 1063
assess the retailer’s sales level, personality characteristics, and other factors. This paper
shows that to promote supply chain coordination and improve overall channel performance,
the supplier should choose retailers with a higher degree of positivity and a lower level of
self-confidence to cooperate. The downstream retailers in the follower position, facing the
competitive situation, need to improve their sales ability while reasonably adjusting their
personality preference in the direction required by the supplier to improve the efficiency of
cooperation. At the same time, there are various sales models in e-commerce, and retailers
can improve the advantage of strategic market leadership to a certain extent through the
reservation system, and they need to effectively protect the benefits of the consumers.
There is space for further expansion of the research. First, the model in this paper only
assumes the existence of positive focus preference of the retailer, the supplier is assumed
to be completely rational, and the subsequent research can study only the existence of
focus preference of the supplier or both. Second, this paper does not have other costs,
such as inventory costs; if the product can be sold by the retailer under a certain sales
model, such as the reservation system, and can avoid a lot of cost output, many risk costs
will be transferred to the consumer. However, if consumers are not satisfied with the
product or the reservation time is too long, the return rate at a later stage will lead to
higher maintenance costs, and the article will be further extended if the after-sale costs
can be considered subsequently. Finally, the potential demand considered in this paper
is linear and not general, and the potential demand of the market can be considered
non-linear subsequently.
Author Contributions:
Conceptualization, X.Z. and Y.S.; methodology, X.Z.; software, Y.S.; validation,
X.Z., G.L. and W.X.; formal analysis, X.Z.; investigation, Y.S.; resources, W.X.; data curation, G.L.;
writing—original draft preparation, Y.S.; writing—review and editing, X.Z.; visualization, G.L.;
supervision, W.X.; project administration, G.L.; funding acquisition, W.X. All authors have read and
agreed to the published version of the manuscript.
Funding:
This research was funded by the National Natural Science Foundation of China, grant
number Nos. 11901380, 12071280.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
Proof of the Lemma 1. π(x)
is calculated according to the quadratic function of the normal
distribution, and the optimization problem (4) is set up as a problem to solve the optimal
solution of
f(x)
. First, because of
r≥w
,
h>m
,
bw ≤
2
l−h<h
, we can obtain
xϕ≥m
,
so Equation (6) holds. According to the range of
x
, we can obtain the size relationships as
follows: if
ϕ>2b(r−w)(h−m)
a(h−bw)2and r <w+aϕ(h−bw)2
2b(h−m)
, satisfying
xϕ<h
, the active focus
of demand takes
xϕ
at this point; if
ϕ≤2b(r−w)(h−m)
a(h−bw)2and r ≥w+aϕ(h−bw)2
2b(h−m)
, satisfying
xϕ≥h
, the active focus of demand takes
h
. Combining the range of values of retail price
r
,
Lemma 1 is proved.
Proof of the Lemma 2. xP(r, w)=xϕ
is monotonically increasing with respect to r when
r∈[w, r0]
; when
r∈[r0, l/b]
and
xP(r
,
w) = h
all the time. Since
π(·)
is monotonically
increasing on the interval
[l, m]
and monotonically decreasing on the interval
[m, h]
, and
xP(r, w)∈[m, h]
,
π(·)
is monotonically decreasing about
xP(r, w)
. Therefore, according to
the principle of composite functions, we know that
π(xP(r, w))
is continuously monoton-
J. Theor. Appl. Electron. Commer. Res. 2023,18 1064
ically decreasing about
r
on the interval
[w, r0]
and constant 1
−a
on the interval
hr0,l
bi
.
Proof of the Lemma 3.
For model (5), for the sake of simplicity, let
g(r)=κπ(xP(r, w)) +
u(r
,
w
,
xP(r, w))
. Substituting the positive demand focus
xP
of the first step into this func-
tion,
g(r)
can be expressed as
g(r)=κ1−a(m−xP(r,w))2
(h−m)2+4b(r−w)(xP(r,w)−br)
(h−bw)2
. The prov-
ing procedure consists of two steps.
Step 1: Solve the optimal solution of the function g(r)by interval.
When on the interval [w, r0], xP(r, w) = xϕ,
g1(r) = 4b2
(h−bw)2"(h−m)2(2ϕ−κ)
ϕ2a(h−bw)2−1#(r−w)2+4b(m−bw)
(h−bw)2(r−w)+κ.
The function
g1(r)
denotes the retailer’s optimization function on the interval
[w, r0]
,
which is a quadratic function on
r
with symmetry axis
rκ=w+aϕ2(h−bw)2(m−bw)
2b(h−m)2(κ−2ϕ)+2abϕ2(h−bw)2
and its quadratic term coefficient
A=4b2
(h−bw)2(h−m)2(2ϕ−κ)
ϕ2a(h−bw)2−1
. When
κ<
2
ϕ−
aϕ2(h−bw)2
(h−m)2
, i.e., when
A>
0, it is easy to prove that
rκ<w
, when the optimal retail price
is
r∗
1=r0
; when
κ≥
2
ϕ−aϕ2(h−bw)2
(h−m)2
, i.e., when
A≤
0, it is easy to prove that
rκ≥w
.
In this case, the position of the symmetry axis is classified on the interval
[w, r0]
, and the
optimal retail price under this condition is obtained as r∗
1=min{r0, rκ}.
According to the two classifications of
A
, the optimal retail price on the interval
[w, r0]is
r∗
1=r0, ifA>0,
min{r0, rκ}, if A≤0.
when on the interval hr0,l
bi, xP(r, w)=h, g2(r) = κ(1−a) + 4b(r−w)(h−br)
(h−bw)2.
The function
g2(r)
is a quadratic function with a downward opening, and its axis of
symmetry is
rc=h+bw
2b
, according to the condition
rc≤l
b
. According to the position of the
axis of symmetry, rccan obtain the optimal retail price in the range of hr0,l
bias
r∗
2=max{rc, r0}.
Step 2:
Solve for the global optimal retail price
rP(w)
and compare the magnitude of
g1(r∗
1)
and g2(r∗
2).
When
A>
0
or A ≤
0
and r0≤rκ
,
g1r∗
1=g1(r0)
holds on
[w, r0]
;
g2(r0)≤g2(r∗
2)
holds on the interval
[r0, l/b]
. The function
g(r)
is continuous with respect to
r
and satisfies
the equation
g1(r0) = g2(r0)
, which leads to the inequality relation
g1(r∗
1) = g1(r0) =
g2(r0)≤g2(r∗
2). The optimal retail price at this point is rP(w)=r∗
2=max{rc, r0}.
When
A≤
0,
r0>rκ
and
r0=l/b
,
g1(r∗
1) = g1(rκ)>g2(r∗
2)
is satisfied by taking
rP(w)=r∗
1=rκ.
When
A≤
0,
r0>rκ
and
r0=rϕ
, in the case
r∗
2=rϕ
,
g1(r∗
1) = g1(rκ)
holds on the
interval
[w, r0]
. Satisfying the relation
g1(rκ)≥g1(r0) = g2(r∗
2)
, we can obtain the optimal
retail price at this time as rP(w)=rκ.
J. Theor. Appl. Electron. Commer. Res. 2023,18 1065
When
A≤
0,
r0>rκ
and
r0=rϕ
, in the case
r∗
2=rc
, the function
g(r)
is continuous
with respect to
r
and satisfies the inequality relation
g1(rκ)≥g1(r0) = g2(r0)≤g2(rc)
.
Where the maximum values on the intervals [w, rϕ]and [r0, l/b], respectively, are
g1(rκ) = aϕ2(m−bw)2
(h−m)2(κ−2ϕ) + aϕ2(h−bw)2+κ,
g2(rc) = κ(1−a) + 1.
Taking the verification approach of judging the positive and negative of the difference
between the two values, comparing the magnitude between
g1(rκ)
and
g2(rc)
such that
G(κ)=g1(rκ)−g2(rc), we obtain
G(κ)=aϕ2(m−bw)2
(h−m)2(κ−2ϕ)+aϕ2(h−bw)2+aκ−1.
The first- and second-order derivatives of G(κ)are obtained as
G0(κ)=−aϕ2(m−bw)2(h−m)2
h(h−m)2(κ−2ϕ)+aϕ2(h−bw)2i2+a,
G00 (κ)=2aϕ2(m−bw)2(h−m)4h(h−m)2(κ−2ϕ)+aϕ2(h−bw)2i
h(h−m)2(κ−2ϕ)+aϕ2(h−bw)2i4.
Because of
A≤0⇒κ≥κ0
, where
κ0=
2
ϕ−aϕ2(h−bw)2
(h−m)2
, we can prove that
G00 (κ)≥0
,
while
G0(κ)
is monotonically increasing. Letting
G0(κ)=
0 yields the zero of the first-order
derivative
G0(κ)
as
κ1=
2
ϕ−aϕ2(h−bw)2
(h−m)2+ϕ(m−bw)
h−m
. According to the preconditioner
rϕ>rκ
of the fourth case,
κ>κ1
can be inferred, so
G(κ)
is monotonically increasing in
the range of
κ>κ1
. The existence of the zero point of
G(κ)
for
κ>κ1
is further determined
by making G(κ)=0 have
a(h−m)2κ2−h(1+2aϕ)(h−m)2−a2ϕ2(h−bw)2iκ+aϕ2(m−bw)2−aϕ2(h−bw)2+2ϕ(h−m)2=0. (A1)
The left and right roots of Equation (A1) are
κ3=κ0
2+1
2a +v
u
u
t1
2a −κ0
22
−ϕ2(m−bw)2
(h−m)2,
κ4=κ0
2+1
2a −v
u
u
t1
2a −κ0
22
−ϕ2(m−bw)2
(h−m)2.
Comparing the magnitude of the right root
κ3
with the zero
κ1
of the first-order
derivative G0(κ), the difference can be obtained as follows:
κ3−κ1=κ0
2+1
2a +r1
2a −κ0
22−ϕ2(m−bw)2
(h−m)2−κ0+ϕ(m−bw)
h−m
=1
2a −κ0
2−ϕ(m−bw)
h−m+r1
2a −κ0
22−ϕ2(m−bw)2
(h−m)2.
J. Theor. Appl. Electron. Commer. Res. 2023,18 1066
According to the inequality relation
aϕ(h−bw)
h−m2−
2
aϕ(h−bw)
h−m+
1
≥
0, we can launch
the relation of
1
2a −κ0
2≥ϕ(m−bw)
h−m
, which satisfies the size relationship
κ3≥κ1
. Simi-
larly, it can be proved that
κ4≤κ1
. Based on the above comparison, it is shown that
κ1
ranges between the two zeros of
G(κ)
, which eventually leads to the following classification:
(g1(rκ)−g2(rc)<0, ifκ1<κ<κ3,
g1(rκ)−g2(rc)≥0, ifκ≥κ3.
Under the conditions that
ϕ≤(h−m)
a(h−bw)
,
κ>κ1
are satisfied, the optimal retail price is
chosen as
rP(w)=(rc, ifκ1<κ<κ3,
rκ, ifκ≥κ3.
The categorical proofs in the four cases eventually sum up to the form of Theorem 3,
which is proved.
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