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Academic Journal of Management and Social Sciences
ISSN: 2958-4396 | Vol. 3, No. 2, 2023
65
ResearchontheDynamicBehaviorofSupplyChain
SystemsConsideringProductGreenness
Jing Wang*, Fengshan Si, Tongtong Ge
School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China
* Corresponding author: Jing Wang (Email: crystalstella@126.com)
Abstract: In order to meet consumers' demand for green products, it is necessary for manufacturers to improve the greenness
of their products during the production process. On the basis of considering the greenness of products, the strategy evolution
behavior of multi-period game model was studied, and the stability and complexity of multi-period game system was explored.
The stability domain of the system was given, and the effects of product market demand and strategy adjustment speed on the
stability domain were analyzed. Research has shown that there are scope limitations on the adjustment speed of game strategies,
and the stability domain of the system is positively correlated with the price sensitivity coefficient, while negatively correlated
with the sensitivity coefficient of product greenness and product demand. A stable system helps the strategy reach equilibrium
after multi-period of adjustment, while an unstable system leads to drastic fluctuations in prices and profits.
Keywords: Product Greenness; Supply Chain System; Complexity; Bifurcation; Chaos.
1. Introduction
In order to reduce carbon emissions and meet consumers'
low-carbon preferences, enterprises need to take measures to
improve the greenness of their products, but they also have to
pay a certain cost for this. Research on green products mainly
focuses on the level of product greening and optimal pricing
decisions. Liu et al. [1] studied the impact of consumer
expected regret behavior on the optimal pricing and
greenization level of manufacturers producing green products.
Hu et al. [2] constructed four different supply chain models
and studied the impact of fairness concern behavior on two-
level green supply chains. Liang et al. [3] analyzed the impact
of free riding behavior and price differences on the optimal
decision-making of supply chain members in a dual channel
supply chain, and designed a two-part tariff contract to
coordinate the supply chain. Cai et al. [4] compared and
analyzed the impact of cost sharing contracts and revenue
sharing contracts on the greenness and sales of supply chain
products, and achieved Pareto improvement of the supply
chain by designing the parameters of the two contracts
reasonably. Feng et al. [5] explored the optimal selection
strategy for green product research and development in
competitive supply chains, analyzed the impact of price
competition and green product research and development
costs on product pricing, green level, and enterprise profits,
and applied a two-part pricing contract to coordinate the
optimal strategy. Shen et al. [6] designed a consumer selection
model driven by price and quality in a two-level supply chain,
and determined the optimal design of green and non-green
products from the perspective of quality differences. Dey et
al. [7] analyzed the impact of three power structures and
inventory on the development of cost intensive green products.
Fadavi et al. [8] studied the impact of different green product
structures in the supply chain on environmental performance,
as well as the decision-making of green product price
competition and marketing cost allocation among members in
vertical supply chains. Liu et al. [9] analyzed the impact of
consumer green sensitivity and cost coefficient on the optimal
pricing of green products. Shan et al. [10] constructed three
pricing models based on two manufacturers and one retailer,
and analyzed the changes in the optimal profits of supply
chain members and the optimal greenness of complementary
products. Xue et al. [11] studied the impact of government
subsidies on the energy saving level of green products, market
demand, supply chain profits and social welfare in the case of
centralized decision-making, decentralized decision-making
and the existence of revenue sharing contracts. Shao et al. [12]
studied the impact of government subsidies on the greenness
of products and supporting products under four contract
models, and achieved Pareto improvement of the supply chain
and its members through revenue sharing and cost sharing
contracts. Xue et al. [13] studied multi product pricing and
green product design strategies under different supply chain
structures and government subsidy strategies, and obtained
the optimal equilibrium pricing, green product design, and
government subsidy strategies in different supply chain
structures.
In summary, existing literature has studied the greenness of
products, pricing of green products, and profit coordination in
green supply chains, laying the foundation for further
research in this paper. On this basis, this paper studies the
complex characteristics of the supply chain system from the
perspective of system stability. By constructing a dynamic
game model, it analyzes the process of the strategy from the
initial state to the optimal state through multi-period, repeated
adjustments, and gives the evolution behavior under bounded
rationality, which is also an effective supplement to previous
research.
2. Structure Model
This paper studies a secondary supply chain system
consisting of a manufacturer and a retailer. Among them, the
manufacturer is responsible for product production, with a
unit production cost is c, and a green degree of production
is
g
. The manufacturer wholesales the product to retailers by
price w, with a retail price is p, and the market demand for
the product is q.
This paper assumes the following:
66
(1) The higher the greenness of the product, the higher the
degree of low-carbon of the product. In markets where
consumers have low-carbon preferences, product greenness is
positively correlated with product demand, 0g.
(2) This paper only considers the one-time cost invested by
manufacturers and is not related to the production quantity of
green products. The cost of achieving green degree
g
i s
2/2g
, and
is the cost coefficient, 0
.
(3) To ensure that manufacturers and retailers are profitable,
it is necessary to meet 0pwc
.
(4) This paper studies the Stackelberg game model. The
manufacturer, as the leader, first decides the wholesale price
wand the product green degree
g
. Retailers, as followers,
decide the retail price p last.
Based on the above description, the market demand
function of the product is:
qa p g
(1)
where ais the potential maximum demand for the product,
is the sensitivity coefficient of demand to retail price, and
is the sensitivity coefficient of demand to the greenness of
the product.
The profit functions of manufacturers and retailers are:
2
1
(,) ( ) 2
mwg w cq g
(2)
() ( )
r
p
pwq
(3)
In the real market, it is difficult for manufacturers and
retailers to obtain all decision-making information, and they
can only refer to historical data and experience to make the
most favorable decisions. Therefore, this paper conducts
model analysis based on their Bounded rationality as decision
makers. In addition, the stability of the market has a direct
impact on the profits of decision-makers. Therefore, this
paper will study the decision-making dynamics behavior of
manufacturers and retailers from the perspective of system
stability.
The decision of manufacturers and retailers is based on
marginal profit. If the marginal profit is positive, they will
increase the price, otherwise they will reduce the price. The
manufacturer's marginal profit on wholesale price and
product greenness, and the retailer's marginal profit on retail
price are:
(,)
(,)
(
11
22
1
)
2
m
m
r
wg
w
wg
g
g
awc aw g
wc g
ap pw g
p
p
(4)
Manufacturers and retailers make multi period decisions,
and they make the next decision according to the current price
and product greenness. The decision game system is [14]:
1
2
3
(,)
(1) () ()
(,)
(1) () ()
()
(1) () ()
m
m
r
wg
wt wt kwt w
wg
gt gt kgt g
p
pt pt kpt p
(5)
where 1
k , 2
k and 3
k are the adjustment speeds of
wholesale price, product greenness, and retail price,
respectively. For example, the speed of price adjustment can
be understood as the frequency of price adjustment per unit
time, and the faster the speed of price adjustment, the more
frequent the price changes. System (5) can be divided into two
parts: the manufacturer's decision system and the retailer's
decision system [14]. The manufacturer's decision system is:
1
2
(,g)
(1) () ()
(,)
(1) () ()
m
m
w
wt wt kwt w
wg
gt gt kgt g
(6)
The retailer's decision system is:
3
()
(1) () () r
p
pt pt kpt
p
(7)
3. Analysis of Manufacturer's Decision
System
Based on Eq. (4) and Eq. (6), the manufacturer's decision
system is obtained as follows:
1
2
(1) () ()
(1) (
1
)
1
2
2
1
)2
(
w
wt
ac
w
a
c
ttkw
wg g
gt gt kgt w g
(8)
Let (1) ()wt wt
, (1) ()gt gt , obtain the
following equilibrium points for system (8):
1(),0
2
Eac
,2()0, 0E,3()0, 2
Ec
,
2
422
22
,
44
()Eac
cac
According to the research background and practical
economic significance of this paper, the situation where the
wholesale price or product greenness is zero does not meet
the requirements, so this paper only studies the stability of
system (8) at equilibrium point 4
E. The Jacobian matrix of
the manufacturer's system at equilibrium point is:
11 12
21 22
JJ
JJJ
(9)
where
2
1
2
11
2
14
Jacck
,
1
12
2
2
2
24
Jcca k
,
2
21
2
2
24
Jac k
,
2
22 2
14
ac k
J
The characteristic equation of Eq.(9) is:
2
210
()
f
zazaza
(10)
67
where
21
1aa
,
2
012
22
2
(2 4
44 )/(44)
ac ack ack
ac k
According to the Julie criterion, the system (8) must meet
the following conditions to ensure stability:
210
210
02
0
0
0
(1)
(1)
faaa
faaa
aa
(11)
3.1. The Stability Domain of System Regarding
1
k and
2
k
According to Eq. (11), the stability domain of system (8)
regarding wholesale price adjustment speed
1
k
and product
greenness adjustment speed
2
k
is shown in Figure 1.
Figure 1. The stability domain of system (8) regarding
1
k
and
2
k
Figure.1 shows that when
12
(, )kk
is in a stable region,
system (8) is stable. At this point, the wholesale price and
product greenness converge to an equilibrium state after
multiple period games, ensuring that manufacturers can
obtain maximum profits. When exceeding this area, the
system will lose stability.
To demonstrate the impact of the system's state on the game
strategy, let
12
2, 1kk
, then
12
(, )kk
is within the
stable domain of Figure 1. The time series of wholesale price
and product greenness are shown in Figure.2(a). Assuming
12
3, 1kk
,
12
(, )kk
is within the instability domain of
Figure1, and their time series are shown in Figure.2(b).
Comparing Figure 1 and Figure 2, it can be clear that the
system is stable, and the wholesale price and product
greenness will reach an equilibrium state after multiple
rounds of adjustment. The system is unstable, with periodic
fluctuations in wholesale prices and product gr eenness, whic h
will harm manufacturers, retailers, and consumers.
(a)system (8) is stable (
12
2, 1kk
)
(b)system (8) is unstable (
12
3, 1kk
)
Figure 2. Time series diagram of wholesale price and product
greenness
3.2. The Influence of Sensitivity Coefficient on
the Stability Domain of System (8)
The impact of retail price sensitivity coefficient on the
stable domain of system is shown in Figure 3(a), and the
impact of product greenness sensitivity coefficient on the
stable domain of system is shown in Figure 3(b).
From Figure 3, it can be seen that the larger the retail price
sensitivity coefficient, the larger the system stability domain,
and the larger the product greenness sensitivity coefficient,
the smaller the system stability domain. This indicates that
high sensitivity to price or low sensitivity to product
greenness helps manufacturers have sufficient wholesale
price and product greenness adjustment space.
3.3. The Impact of Wholesale Price
Adjustment Speed on System Complexity
The impact of wholesale price adjustment speed
1
k
on the
stability of system (8) and the profits of manufacturers and
retailers is shown in Figure 4.
Figure 4 (a) shows that system transitions from a stable
state to a chaotic state through period doubling bifurcation as
1
k
increases, and the stability domain of the system with
respect to
1
k
is
1
(0,2.69)k
.
0 50 100 150 200 250 300
time
0
0.2
0.4
0.6
0.8
1
1.2
w
g
0 50 100 150 200 250 300
time
0
0.2
0.4
0.6
0.8
1
1.2
g
w
68
(a)
(b)
Figure 3. The impact of sensitivity coefficient on the stability
domain of system (8). (a) the impact of
on the stability
domain, (b) the impact of
on the stability domain.
When
1
2.69k
, the system undergoes 2-cycle
bifurcation, then goes through 4, 8 cycles, and finally falls
into chaos. At this time, wholesale prices and product
greenness show significant fluctuations. Figure 4 (b) uses the
Lyapunov exponent to describe the stability, bifurcation, and
chaos of the system. When the exponent value is less than
zero, it indicates that the system is in a stable state. When the
exponent value is equal to zero, it indicates that the system is
in a bifurcation state. When the exponent value is greater than
zero, it indicates that the system is in a chaotic state. From the
figure, it can be seen that 2.69 is the first bifurcation point,
which is consistent with Figure 4(a).
Figure 4(c) uses an entropy graph to characterize the
complexity of the system. When the entropy value is equal to
zero, it indicates that the system is in a stable state. When the
entropy value is greater than zero, it indicates that the system
complexity increases. The larger the entropy value, the more
co mplex the s ystem i s. Comp ari ng Figu re 4(a) and Figure 4(c),
it can be seen that it is precisely because the system undergoes
bifurcation that it begins to become more complex, so the
value of
1
k
cannot exceed 2.69. Figure 4(d) is a trend
chart of profits for manufacturers and retailers affected by
1
k
.
(a) bifurcation diagram
(b) the largest Lyapunov exponent
(c) entropy diagram
(d) profits trend
Figure 4. The impact of wholesale price adjustment speed on
complexity of system (8)
0 0.5 1 1.5 2 2.5 3 3.5 4 4 .5
k1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
m
r
X: 2.69
Y: 0.184
69
It shows that system instability will lead to irregular
fluctuations in their profits, and it is difficult for them to
stabilize their profits. So, ensuring the stability of the system
can reduce the complexity of the system and contribute more
to the healthy development of the game subjects.
3.4. Bifurcation Diagram of the System with
Two Parameters
Figure 5. Double parameter bifurcation diagram of system (8)
regarding 1
k and 2
k
The different colors in Figure 5 represent different states of
the system. The blue region represents stable states, the red
region represents 2-cycle bifurcation, the green region
represents 4-cycle bifurcation, the yellow region represents
non convergent states, and the black region represents
divergent states. Comparing Figure 5 and Figure 1, it is found
that the stability domains of system (8) regarding 1
k and
2
k are the same.
4. Analysis of Retailer’s Decision
System
Based on Eq. (4) and Eq. (7), the retailer’s decision system
is obtained as follows:
3
(1) () ()( )pt pt kpt pawgp
(12)
Let (1) ()pt pt can obtain the equilibrium price of
system (12) as
2
2
3
4
cac
.
Bringing the equilibrium price into system (12) yields the
Jacobian matrix as follows:
2
3
2
23
14
cack
(13)
According to Eq. (13), the characteristic equation is:
10
0ax a
(14)
Where
11a ,
2
0
3
2
23
14
aacck
Furthermore, the characteristic root is obtained as follows:
2
3
2
23
14
cack
x
(15)
According to the stability condition of the system, the
system is stable when the eigenvalue lies within the unit circle.
So, the necessary and sufficient condition for the stability of
system (12) is 1x
. The parameter value remains
unchanged, and it is calculated that it is 1x
when
3
00.9615k
, the system (12) is stable.
4.1. The Impact of Retail Price Adjustment
Speed 3
k on System Stability
(a) bifurcation diagram
(b) the largest Lyapunov exponent
Figure 6. The impact of 3
k on the stability of system (12)
Similar to Figure 6(a) and Figure 4(a), the speed of retail
price adjustment 3
k can also cause system (12) instability.
The first bifurcation point is 30.9615k , and thereafter
the system undergoes a period doubling bifurcation and enters
a chaotic state. This indicates that there are constraints on the
speed of price and product greenness adjustments, and
frequent adjustments can disrupt the stability of the system.
70
4.2. The Impact of Market Demand a on the
Stability Domain of System (5)
Let a take values of 0.8, 1, and 1.2 respectively, and the
trend of stability domain changes regarding 1
k , 2
k and 3
k
is shown in Figure 7.
(a) 0.8a
(b) 1a
(c) 1.2a
Figure 7. The impact of aon the stability domain of
system (5)
Figure 7 shows that as the maximum potential market
demand for the product increases, the stability domain of the
system gradually narrows, meaning that the value range of
i
k decreases, 1, 2, 3i. This is because the market has a
high demand for products, making it easier for manufacturers
and retailers to profit. There is no need for frequent strategic
adjustments to wholesale prices, retail prices, and product
greenness, which means the adjustment speed is reduced.
4.3. Phase Diagram of Wholesale Price, Retail
Price, and Product Greenness
For Figure 7 (b), with 12 3
(, , )kk k located in different
regions, the phase diagram of system (5) is shown in Figure
8.
Figure 8(a) is a phase diagram of the system in a stable state,
that is, 123
(, , )kk k is within the stable domain of Figure
7(b). In this state, after multiple cycles and repeated strategic
adjustments, prices and product greenness eventually
converge to an equilibrium state, with equilibrium values
(,g,)wp
is (0.9, 0.6, 1.3). At this point, wholesale prices,
product greenness, and retail prices do not need to be adjusted
further, and manufacturers and retailers have already
benefited the most. Compared with Figure 8(a), the
adjustment speed 1
kof wholesale prices in Figure 8(b) has
been adjusted from 2 to 4, the adjustment speed 2
kof product
greenness in Figure 8(c) has been adjusted from 1 to 9. Based
on Figure 1, it can be seen that the system has lost stability in
both cases. After continuous strategic adjustments, prices and
product greenness cannot reach an equilibrium state, but
instead fall into chaotic fluctuations. Compared with Figure
8(a), the adjustment speed 3
k of retail prices in Figure 8(d)
has been adjusted from 0.5 to 1.2. Based on Figure 6(a), it can
be seen that the system is unstable at this time, and the game
results are still difficult to achieve an ideal equilibrium state.
In addition, it can be seen from Figure 8(a) that the
decision-making subject, as a bounded rationality decision-
maker, will eventually converge to the equilibrium state after
long-term and repeated adjustment of the strategy from the
initial state in a stable system, which indicates that a stable
system is conducive to the equilibrium of the game strategy
and the maximization of profits for manufacturers and
retailers.
5. Conclusion
This paper adopts the theory of system complexity to study
the decision-making problem of multi cycle supply chain
systems, explores the influence of decision parameters on the
optimal strategy, clarifies the important role of system
stability in the strategy adjustment process, and uses tools
such as bifurcation diagrams, phase diagrams, stability
domain diagrams, and time series diagrams to depict the
evolutionary behavior of the system during the strategy
adjustment process, striving to provide reference for the
decision-making of manufacturers and retailers.
Research has shown that there are scope limitations in
adjusting decision variables, otherwise it can easily lead to
system instability. In a stable system, the game strategy can
converge to an equilibrium state after multiple cycles and
repeated adjustments, maximizing the profits of
manufacturers and retailers. In an unstable system, the
adjustment of strategies will undergo period doubling
bifurcation and fall into a chaotic state, unable to achieve
equilibrium. At this time, the profits of manufacturers and
retailers will exhibit disorderly fluctuations.
71
(a)
1
2k
,
2
1k
,
3
0.5k
(b)
1
4k
,
2
1k
,
3
0.5k
(c)
1
2k
,
2
9k
,
3
0.5k
(d)
1
2k
,
2
1k
,
3
1.2k
Figure 8. Phase diagram of system (5)
Acknowledgments
The authors thank the reviewers for their careful reading
and pertinent suggestions. This work was supported by the
Natural Science Research Project of Education Department in
Anhui Province (2022AH050589); the school-level Science
Research Project of Anhui University of Finance and
Economics (ACKYB22015).
Conflicts of Interest
The authors declare no conflict of interest.
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