1. bookAHEAD OF PRINT
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Otwarty dostęp

Game theoretic model for low carbon supply chain under carbon emissions reduction sensitive random demand

Data publikacji: 14 Oct 2022
Tom & Zeszyt: AHEAD OF PRINT
Zakres stron: -
Otrzymano: 01 Dec 2020
Przyjęty: 25 Feb 2021
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Introduction

The inconvenience caused by the harsh environment has been particularly long. How to improve social and economic benefits under the harmonious coexistence of human and nature has become a key topic of concern for the government and firms.

In order to adapt to the environmental protection and sustainable development, low-carbon products initiated by the government and implemented by firms came into being. In recent years, scholars have tried their best to provide suggestions for low-carbon supply chain management (see literature review by [1]). In particular, extensive studies are published to discuss the low-carbon strategies in various polices such as carbon cap and trade (e.g., [2,3]), carbon tax (e.g., [4,5]), low carbon subsidy (e.g., [6,7]), and hybrid low carbon policy (e.g., [8,9]).

Apart form the impact of various policies, the firms’ low-carbon strategies are also related to the composition of market. For better deal with firms’ decision problems under different power structures, decision makers are often apply Stackelberg game or Nash game models. Zhou et al. [10] construct a social welfare model with Stackelberg game considering carbon emissions, and show that an increasing block carbon tax policy can encourage low-carbon production. Du et al. [11] propose a carbon-related price-discount sharing-like scheme under manufacturer Stackelberg game, and point out that this contract can achieve low carbon supply chain coordination. Xu and Wang [12] consider a two-period closed-loop supply chain with retail price and emission reduction dependent demand, and obtain the equilibrium decision strategy and profit distribution under the Stackelberg game frame and Nash game frame, respectively. In fact, there are also other literature reviews that explore the low-carbon strategy of closed-loop supply chains under the game framework from multiple perspectives (e.g., [13, 14, 15]). The above studies are conducted under deterministic demand, while those do not take the uncertain risk of demand into account.

In actual transactions, market demand is not static. The variability of the environment, the psychological expectations of consumers, and the diversity of consumer values, all of those will cause market demand become uncertain. In order to better predict the random demand, scholars have contributed their own strength. Cohen et al. [16] study government subsidies for green technology adoption, and show that an increase in demand uncertainty leads to higher production quantities and lower prices. Rong et al. [17] consider product line design under endogenous substitution, and demonstrate that the presence of demand uncertainty can reduce the benefit of market segmentation. Feng and Viswanathan [18] discuss the impact of demand uncertainty on the effectiveness of coordinating such a supply chain, and find that coordination through common replenishment epochs may not always be beneficial when the demand variance is high. Actually, more and more researchers make decisions with random demand in supply chain management (e.g., [19, 20, 21]).

In practice, manufacturers may be a Stackelberg leader, faced a random demand, and taken the make-to-stock mode. However, the previous literature has not adequately investigated the low-carbon strategies when these three issues are considered altogether. In summary, different from [22], we focus on low-carbon strategy. Different from [23] and [24], we discuss a decentralized decision, which includes Stackelberg game and Nash game. Different from [25] concerning the make-to-order mode, we demonstrate the make-to-stock mode. Different from [26, 27, 28], we investigate the impacts of demand uncertainty. The characteristics and contributions of our study are presented in Table 1.

Summary of relevant literature

Authors Low-carbon strategy
Demand form
Type of game
Production mode
With Without Deterministic Random Stackelberg Nash Make to order Make to stock
Shi et al. [22]
Jiang et al. [23]
Zhang et al. [24]
Bai et al. [25]
Liu et al. [26]
Meng et al. [27]
Shi et al. [28]
This paper

Motivated by the research gap, this paper establishes a supply chain with one risk-neutral manufacturer and one risk-neutral retailer and considers Stackelberg modes to derive the optimal low-carbon strategies under random demand. Before that, we firstly summarize our main contribution as follows and unfold them in the remaining part of this paper.

First, we establish three Stackelberg models, including Manufacture Stackelberg (MS), Retailer Stackelberg (RS) and Nash. We demonstrate the related equilibrium stocking factor, emission reduction level, wholesale price and retail price of those three models that maximize the firm’s expected profit, respectively. Second, we illustrate effects of the power structures. We show that the MS model has the highest wholesale price, while it has the lowest stocking factor. Third, we reveal the impacts of the demand uncertainty via numerical analysis. We find out that as the market potential or the low-carbon sensitivity coefficient increases, the emission reduction level, retail price, the firms’ expected profits increase. While those decisions and expected profits are decreasing in the price sensitivity coefficient. The remainder of the paper is organized as follows. In Section 2, we introduce assumptions and notions. Section 3 analysis three Stackelberg models, including MS, RS and Nash. Section 4 discusses the effects of the power structure. Section 5 investigates the impacts of demand uncertainty. Finally, Section 6 concludes and presents future research. All technical proofs are presented in Appendix.

Assumptions and Notions

We consider a single-period supply chain consisting of a risk-neutral manufacturer (she) and a risk-neutral retailer (he). In order to reduce carbon emissions and achieve environmental protection, the manufacturer need employ low-carbon technologies. In this case, the cost of low-carbon technologies for the manufacturer is ke2, where k represents the cost coefficient of eco-friendly production and operations and e represents the eco-friendly level of the product.

After purchasing eco-friendly products from the manufacturer at the wholesale price w, the retailer sales them into an uncertain markets at the price p. In this case, the random demand D consists of two parts: expected demand and demand shock. The expected demand is price dependent and carbon emission reduction sensitive, which is denoted as abp + te, where a is the market potential, b is the price sensitivity coefficient, t is the low-carbon sensitivity coefficient. The demand shock is a random variable, denoted as x, which has the probability density function (PDF) f (x), the cumulative distribution function (CDF) F(x) and the expected function E[x] = 0. Thus, the random demand is D = abp + te + x. Related notions are summarized in Table 2.

The notions and parameters.

Symbol Description
z Stocking factor
p Retail price
w Wholesale price
e Emission reduction level
ε Retail margin, ε = pw
h Cost of emergency procurement
λ Government subsidies
E Expectation operators
i Represent the firms, i ∈ {m,r}
j Represent the models, j ∈ {M,N,R}
a Market potential
b Price sensitivity coefficient
t Low-carbon sensitivity coefficient
k Emission reduction cost coefficient
c Unit production cost
v Salvaged value
f(·) Probability density function of ·
F(·) Distribution function of ·
π Expected profit
q Order quantity

For tractability, some assumptions are provided as follows.

The shock demand x has an increasing failure rate, which means f(x)f2(x)F¯(x), f'(x) \ge - {{{f^2}(x)} \over {\bar F(x)}}, where F¯(x)=1F(x) \bar F(x) = 1 - F(x) .

p > w > c > v > 0 and h > v, which means the firms obtain a positive profit, it also in line with real transactions.

λ < cv. If λcv, the manufacturer will gain unlimited positive profit from the government subsidies, which is not reasonable.

4kb > t2, which means the both manufacturer and the retailer are willing to implement low-carbon strategy (see the results of Proposition 1 in [26]). Particularly, we set 2bk > t2.

The retailer’s order quantity q satisfies that q = abp + te + z, which helps facilitating the analysis.

Model analysis

In this section, three cases are discussed according to the different power structure of the firms. Case 1, the manufacturer is a Stackelberg-leader and the retailer is the follower. Case 2, the retailer acts as a Stackelberg-leader and the manufacturer acts as the follower. Case 3, the manufacture and the retailer have equal status. Throughout this paper, the expected profits of the manufacturer and the retailer are as following, respectively. πr(w,e,p,z)=E[pD+v(qD)+h(Dq)+wq]=(pw)(abp+te)(wv)z(hv)z+F¯(x)dx, \matrix{{{\pi _r}(w,e,p,z)} \hfill & {= E[pD + v{{(q - D)}^ +} - h{{(D - q)}^ +} - wq]} \hfill \cr {} \hfill & {= (p - w)(a - bp + te) - (w - v)z - (h - v)\int_z^{+ \infty} \bar F(x){\rm{d}}x,} \hfill \cr} πm(w,e,p,z)=E[(wc+λ)qke2]=(wc+λ)(abp+te+z)ke2. {\pi _m}(w,e,p,z) = E[(w - c + \lambda)q - k{e^2}] = (w - c + \lambda)(a - bp + te + z) - k{e^2}.

Manufacture Stackelberg (MS)

In this case, the manufacturer acts as the leader and sets the wholesale price and the emission reduction level. The retailer acts as the follower and sets the retail price and the stocking factor.

Considering the retailer’s decisions, for a given wholesale price w and emission reduction level e. The retailer’s expected profit πrM \pi _r^M (w,e,p,z) is jointly concave in (p,z), and the retail price and the stocking factor are pM(w,e)=a+te+bw2b, {p^M}(w,e) = {{a + te + bw} \over {2b}}, zM(w)=F¯1(wvhv). {z^M}(w) = {\bar F^{- 1}}({{w - v} \over {h - v}}).

Taking the Eqs. (4) and (5) into Eq. (3), the manufacturer decides on the wholesale price and the emission reduction level that maximizes its profit. We can obtain that πmM(w,e)=(wc+λ)[abw+te2+F¯1(wvhv)]ke2. \pi _m^M(w,e) = (w - c + \lambda)[{{a - bw + te} \over 2} + {\bar F^{- 1}}({{w - v} \over {h - v}})] - k{e^2}.

According to Eq. (5), we have that w=(hv)F¯(z)+v w = (h - v)\bar F(z) + v . Then we rewrite πmM(w,e) \pi _m^M(w,e) in terms of the stocking factor z and the emission reduction level e as the following πmM(z,e)=[(hv)F¯(z)+vc+λ]{ab[(hv)F¯(z)+v]+te2+z}ke2. \pi _m^M(z,e) = [(h - v)\bar F(z) + v - c + \lambda ]\{{{a - b[(h - v)\bar F(z) + v] + te} \over 2} + z\} - k{e^2}.

Lemma 1

Define G(z)=a+2bvb(cλ)2t28k[(hv)F¯(z)+vc+λ]+b(hv)F¯(z)z+F¯(z)f(z). G(z) = {{- a + 2bv - b(c - \lambda)} \over 2} - {{{t^2}} \over {8k}}[(h - v)\bar F(z) + v - c + \lambda ] + b(h - v)\bar F(z) - z + {{\bar F(z)} \over {f(z)}}.

At equilibrium, the stocking factor zM and emission reduction level eM are, respectively, f(zM)G(zM)=cvλhv, f({z^M})G({z^M}) = {{c - v - \lambda} \over {h - v}}, eM=t4k[(hv)F¯(zM)+vc+λ]. {e^M} = {t \over {4k}}[(h - v)\bar F({z^M}) + v - c + \lambda ].

According to Lemma 1, by substituting zM and and Eq. (8) into Eq. (5), we obtain the equilibrium wholesale price wM wM=(hv)F¯(zM)+v. {w^M} = (h - v)\bar F({z^M}) + v.

Similarly, by substituting zM and Eq. (8) into Eq. (4), we obtain the equilibrium retail price pM pM=a+b[(hv)F¯(zM)+v]+t24k[(hv)F¯(zM)+vc+λ]2b. {p^M} = {{a + b[(h - v)\bar F({z^M}) + v] + {{{t^2}} \over {4k}}[(h - v)\bar F({z^M}) + v - c + \lambda ]} \over {2b}}.

Retailer Stackelberg (RS)

In this case, the retailer acts as the leader and sets the retail margin and stocking factor firstly. The manufacturer responds by choosing the wholesale price and emission reduction level.

Recall ε = pw, then the order quantity q = ab(w + ε) +te + z. Thus the expected profit of the retailer is πrR(z,ε)=ε[ab(w+ε)+te](wv)z(hv)z+F¯(x)dx, \pi _r^R(z,\varepsilon) = \varepsilon [a - b(w + \varepsilon) + te] - (w - v)z - (h - v)\int_z^{+ \infty} \bar F(x){\rm{d}}x, and the stocking factor zR(w) is zR(w)=F¯1(wvhv). {z^R}(w) = {\bar F^{- 1}}({{w - v} \over {h - v}}).

The manufacturer’s expected profit is πmR(w,e)=(wc+λ)[ab(w+ε)+te+F¯1(wvhv)]ke2. \pi _m^R(w,e) = (w - c + \lambda)[a - b(w + \varepsilon) + te + {\bar F^{- 1}}({{w - v} \over {h - v}})] - k{e^2}.

From Eq. (12), we know that w=(hv)F¯(z)+v w = (h - v)\bar F(z) + v . Substituting it into Eq. (13), we obtain πmR(z,e)=[(hv)F¯(z)+vc+λ][abvbε)+te+zb(hv)F¯(z)]ke2. \pi _m^R(z,e) = [(h - v)\bar F(z) + v - c + \lambda ][a - bv - b\varepsilon) + te + z - b(h - v)\bar F(z)] - k{e^2}.

Lemma 2

Define H(z)=a2bv+b(cλ)+t22k[(hv)F¯(z)+vc+λ]2b(hv)F¯(z)+zF¯(z)f(z)+cvλ(hv)f(z). \matrix{{H(z) = a - 2bv + b(c - \lambda) + {{{t^2}} \over {2k}}[(h - v)\bar F(z) + v - c + \lambda ]} \cr {- 2b(h - v)\bar F(z) + z - {{\bar F(z)} \over {f(z)}} + {{c - v - \lambda} \over {(h - v)f(z)}}.} \cr}

The manufacturer’s response wholesale price and emission reduction level are, respectively, w=(hv)F¯(z)+v,e=t2k[(hv)F¯(z)+vc+λ], \matrix{{w = (h - v)\bar F(z) + v,} \cr {e = {t \over {2k}}[(h - v)\bar F(z) + v - c + \lambda ],} \cr} where z is the unique solution to bε=H(z). b\varepsilon = H(z).

Lemma 2 reflects that the manufacturer’s response, we now examine the retailer’s setting in terms of z and e. Applying Eqs. (12) and (15) to Eq. (11) and combing the definition of H(z), we rewrite the expected profit function of retailer as the following, πrR(z)=H(z)b{ab[(hv)F¯(z)+v]H(z)+t22k[(hv)F¯(z)+vc+λ]} (hv)F¯(z)(hv)z+F¯(x)dx. \matrix{{\pi _r^R(z) = {{H(z)} \over b}\{a - b[(h - v)\bar F(z) + v] - H(z) + {{{t^2}} \over {2k}}[(h - v)\bar F(z) + v - c + \lambda ]\}} \hfill \cr {- (h - v)\bar F(z) - (h - v)\int_z^{+ \infty} \bar F(x){\rm{d}}x.} \hfill \cr}

Thus the equilibrium stocking factor zR must satisfy πrR(z)z|z=zR=0 {{\partial \pi _r^R(z)} \over {\partial z}}{|_{z = {z^R}}} = 0 . With the solution of zR, the equilibrium retail margin, emission reduction level, wholesale price can be determined as follows: εR=1bH(zR), {\varepsilon ^R} = {1 \over b}H({z^R}), eR=t2k[(hv)F¯(zR)+vc+λ], {e^R} = {t \over {2k}}[(h - v)\bar F({z^R}) + v - c + \lambda ], wR=(hv)F¯(zR)+v. {w^R} = (h - v)\bar F({z^R}) + v.

Therefore the equilibrium retail price is pR=(hv)F¯(zR)+v+1bH(zR). {p^R} = (h - v)\bar F({z^R}) + v + {1 \over b}H({z^R}).

Nash

In this case, the manufacturer and the retailer choose the wholesale price, retail margin, and emission reduction level simultaneously. For a given wholesale price w, retail margin ε, and emission reduction level e, it follows the RS model, the retailer’s ordering is q = ab(w + ε) + te + zN(w), where the stocking factor zN(w) satisfies zN(w)=F¯1(wvhv). {z^N}(w) = {\bar F^{- 1}}({{w - v} \over {h - v}}).

The manufacturer’s expected profit is πmN(w,e,ε)=(wc+λ)[ab(w+ε)+te+zN(w)]ke2. \pi _m^N(w,e,\varepsilon) = (w - c + \lambda)[a - b(w + \varepsilon) + te + {z^N}(w)] - k{e^2}.

The retailer’s expected profit is πrN(w,e,ε)=ε[ab(w+ε)+te](wv)zN(w)(hv)zN(w)+F¯(x)dx. \pi _r^N(w,e,\varepsilon) = \varepsilon [a - b(w + \varepsilon) + te] - (w - v){z^N}(w) - (h - v)\int_{{z^N}(w)}^{+ \infty} \bar F(x){\rm{d}}x.

Lemma 3

Define P(z)=a+3bv2b(cλ)2t24k[(hv)F¯(z)+vc+λ]+3b(hv)2F¯(z)z+F¯(z)f(z). P(z) = {{- a + 3bv - 2b(c - \lambda)} \over 2} - {{{t^2}} \over {4k}}[(h - v)\bar F(z) + v - c + \lambda ] + {{3b(h - v)} \over 2}\bar F(z) - z + {{\bar F(z)} \over {f(z)}}.

The unique Nash equilibrium stocking factor zN satisfies f(zN)P(zN)=cvλhv. f({z^N})P({z^N}) = {{c - v - \lambda} \over {h - v}}.

Furthermore, the unique Nash equilibrium emission reduction level, wholesale price, and retail price are, respectively, eN=t2k[(hv)F¯(zN)+vc+λ], {e^N} = {t \over {2k}}[(h - v)\bar F({z^N}) + v - c + \lambda ], wN=(hv)F¯(zN)+v, {w^N} = (h - v)\bar F({z^N}) + v, pN=a+b[(hv)F¯(zN)+v]+t22k[(hv)F¯(zN)+vc+λ]2b. {p^N} = {{a + b[(h - v)\bar F({z^N}) + v] + {{{t^2}} \over {2k}}[(h - v)\bar F({z^N}) + v - c + \lambda ]} \over {2b}}.

Lemma 3 shows that the Nash game also exist a unique pure-strategy Nash equilibrium. What are the differences of those equilibrium strategies among the MS, RS, and Nash models? Next, we will try to discuss this issue.

Effects of power structures

In this section, we compare the equilibrium strategies of the three games, and show the effects of power structure.

Theorem 4

The equilibrium stocking factors satisfy that zM < zN < zR.

Theorem 4 reflects that the stocking factor decrease as the power shifts from the retailer to the manufacturer. It lies in the fact that in order to prevent the loss of potential customers under the unknown random demand, the more power of the retailer has, the greater his willingness to order.

Theorem 5

The equilibrium wholesale prices satisfy that wR < wN < wM.

Theorem 5 reflects that the wholesale price increase as the power shifts from the retailer to the manufacturer. It is in line with the fact that as the increase in the power of the manufacturer, she pays more attention to her own income, and at this time, appropriately increasing the wholesale price has become one of her means.

Theorem 6

The equilibrium emission reduction levels satisfy that eR < eN < 2eM. The equilibrium retail prices satisfy that pN < pR and pN < 2pM.

Theorem 6 reflects that compared to the Nash game, the emission reduction level in retailer-led model is in a weak position, while retail price is higher. The reason may be found from that he retailer’s profit from the market through retail price, and the manufacturer is the main body implementing low-carbon strategy.

According to Theorem 6, it also find that twice the emission reduction level and retail price in Manufacturer-led model are higher than that in Nash game, respectively. So one interesting question is appeared: which one is higher of the emission reduction level in MS game and Nash game? Similarly, how different of the retail price in MS game and Nash game? The next section may give the answers.

Impacts of demand uncertainty

In this section, a series of numerical investigations are considered to present new management insights. We suppose that the demand risk is uniformly distributed in the region [−5,5], and the other exogenous variables follow that h = 10, c = 5, v = 1, λ = 3 and k = 0.5. Then, we show the influence of random demand on firms in two aspects.

On the one hand, we consider the impact of random demand on the equilibrium strategies. Table 3 reflects that as the market potential increases, the stocking factor decreases, while the emission reduction level, wholesale price and retail price increase. Table 4 reflects that the stocking factor is increasing with respect to the price sensitivity coefficient, while the emission reduction level, wholesale price and retail price are decreasing with respect to the price sensitivity coefficient. Table 5 reflects that as the low-carbon sensitivity coefficient continues to increase, the stocking factor shows a gradual decreasing trend, while the emission reduction level, wholesale price and retail price show a gradually increasing trend.

Impact of a on the equilibrium strategies (where b = 5 and t = 1)

a z
e
w
p
zR zN zM eR eN eM wR wN wM pR pN pM
40 6.44 6.01 5.08 2.02 2.59 1.71 4.21 4.59 5.43 7.03 6.55 6.88
50 5.96 5.41 4.29 2.64 3.13 2.07 4.64 5.13 6.14 8.49 7.88 8.28
60 5.48 4.81 3.49 3.07 3.67 2.43 5.07 5.67 6.86 9.96 9.20 9.67
70 5.00 4.20 2.69 3.50 4.22 2.78 5.50 6.22 7.58 11.42 10.53 11.07

Impact of b on the equilibrium strategies (where a = 60 and t = 1)

b z
e
w
p
zR zN zM eR eN eM wR wN wM pR pN pM
3 3.44 2.48 0.87 4.90 5.77 3.61 6.90 7.77 9.22 16.06 14.85 15.21
4 4.67 3.87 2.40 3.79 4.52 2.92 5.79 6.52 7.84 12.27 11.32 11.79
5 5.48 4.81 3.49 3.07 3.67 2.43 5.07 5.67 6.86 9.96 9.20 9.67
6 6.05 5.48 4.31 2.56 3.07 2.06 4.56 5.07 6.13 8.41 7.79 8.23

Impact of t on the equilibrium strategies (where a = 60 and b = 5)

t z
e
w
p
zR zN zM eR eN eM wR wN wM pR pN pM
0.5 5.64 4.97 3.63 1.46 1.77 1.18 4.92 5.53 6.73 9.71 8.85 9.43
1 5.48 4.81 3.49 3.07 3.67 2.43 5.07 5.67 6.86 9.96 9.20 9.67
1.5 5.15 4.51 3.23 5.05 5.91 3.82 5.36 5.94 7.09 10.43 9.86 10.12
2 4.53 4.01 2.84 7.85 8.78 5.45 5.92 6.39 7.45 11.25 10.95 10.81

All tables from table 3 to table 5 reveal that eN > eM, eN > eR, pN < pM and pN < pR. This implies that when the power is shifted from the retailer to the manufacturer, the retail price appears as a U-shaped curve, while the emission reduction level appears as an inverted U-shaped curve.

Figure 1 and Figure 2 reflect that the firms’ expected profits are increasing in the market potential. Figure 3 and Figure 4 reflect that the firms’ expected profits are decreasing regarding the price sensitivity coefficient.

Fig. 1

Impact of a on the manufacturer’s expected profit(where b = 5 and t = 1)

Fig. 2

Impact of a on the retailer’s expected profit(where b = 5 and t = 1)

Fig. 3

Impact of b on the manufacturer’s expected profit(where a = 60 and t = 1)

Fig. 4

Impact of b on the retailer’s expected profit(where a = 60 and t = 1)

Figure 5 and Figure 6 reflect that the firms’ expected profits are increasing with respect to the low-carbon sensitivity coefficient. According to the figures from Figure 1 to Figure 6, it indicates that when the power is shifted from the retailer to the manufacturer, the manufacturer’s expected profit increase while the retailer’s expected profit decrease. This further reveals the fact that firms are self-serving.

Fig. 5

Impact of t on the manufacturer’s expected profit(where a = 60 and b = 5)

Fig. 6

Impact of t on the retailer’s expected profit(where a = 60 and b = 5)

Conclusions

In order to further cherish life, environmental protection becomes more important, and a low-carbon economy is once again mentioned as one of the main means of environmental protection. This article considers the manufacturer-retailer supply chain and discusses the manufacturer’s low-carbon strategy under the setting that market demand is uncertain. The results are as follows:

Firstly, three Stackelberg models, including MS, RS and Nash, are established. We obtain the equilibrium stocking factor, emission reduction level, wholesale price and retail price, respectively.

Secondly, the effects of power structure are discussed. We show that as the power shifts from the retailer to the manufacturer, the stocking factor and the retailer’s expected profit decrease while the wholesale price and the manufacturer’s expected profit increase. Meanwhile, the retail price appears as a U-shaped curve, while the emission reduction level appears as an inverted U-shaped curve.

Thirdly, the impacts of demand uncertainty are demonstrated. We find that a higher market potential or a higher low-carbon sensitivity, leads to a higher emission reduction level, wholesale price, retail price and firms’ expected profits, while leads to a lower stocking factor. Meanwhile, a higher price sensitivity leads to a lower emission reduction level, wholesale price, retail price and firms’ expected profits, while leads to a higher stocking factor.

For further research, an interesting direction is to consider the issue of competition when multiple manufactures are involved. Moreover, given that risk aversion can effectively influence the decisions of firms, the other possible direction may be consider this scenario.

Fig. 1

Impact of a on the manufacturer’s expected profit(where b = 5 and t = 1)
Impact of a on the manufacturer’s expected profit(where b = 5 and t = 1)

Fig. 2

Impact of a on the retailer’s expected profit(where b = 5 and t = 1)
Impact of a on the retailer’s expected profit(where b = 5 and t = 1)

Fig. 3

Impact of b on the manufacturer’s expected profit(where a = 60 and t = 1)
Impact of b on the manufacturer’s expected profit(where a = 60 and t = 1)

Fig. 4

Impact of b on the retailer’s expected profit(where a = 60 and t = 1)
Impact of b on the retailer’s expected profit(where a = 60 and t = 1)

Fig. 5

Impact of t on the manufacturer’s expected profit(where a = 60 and b = 5)
Impact of t on the manufacturer’s expected profit(where a = 60 and b = 5)

Fig. 6

Impact of t on the retailer’s expected profit(where a = 60 and b = 5)
Impact of t on the retailer’s expected profit(where a = 60 and b = 5)

Impact of b on the equilibrium strategies (where a = 60 and t = 1)

b z
e
w
p
zR zN zM eR eN eM wR wN wM pR pN pM
3 3.44 2.48 0.87 4.90 5.77 3.61 6.90 7.77 9.22 16.06 14.85 15.21
4 4.67 3.87 2.40 3.79 4.52 2.92 5.79 6.52 7.84 12.27 11.32 11.79
5 5.48 4.81 3.49 3.07 3.67 2.43 5.07 5.67 6.86 9.96 9.20 9.67
6 6.05 5.48 4.31 2.56 3.07 2.06 4.56 5.07 6.13 8.41 7.79 8.23

The notions and parameters.

Symbol Description
z Stocking factor
p Retail price
w Wholesale price
e Emission reduction level
ε Retail margin, ε = pw
h Cost of emergency procurement
λ Government subsidies
E Expectation operators
i Represent the firms, i ∈ {m,r}
j Represent the models, j ∈ {M,N,R}
a Market potential
b Price sensitivity coefficient
t Low-carbon sensitivity coefficient
k Emission reduction cost coefficient
c Unit production cost
v Salvaged value
f(·) Probability density function of ·
F(·) Distribution function of ·
π Expected profit
q Order quantity

Impact of a on the equilibrium strategies (where b = 5 and t = 1)

a z
e
w
p
zR zN zM eR eN eM wR wN wM pR pN pM
40 6.44 6.01 5.08 2.02 2.59 1.71 4.21 4.59 5.43 7.03 6.55 6.88
50 5.96 5.41 4.29 2.64 3.13 2.07 4.64 5.13 6.14 8.49 7.88 8.28
60 5.48 4.81 3.49 3.07 3.67 2.43 5.07 5.67 6.86 9.96 9.20 9.67
70 5.00 4.20 2.69 3.50 4.22 2.78 5.50 6.22 7.58 11.42 10.53 11.07

Summary of relevant literature

Authors Low-carbon strategy
Demand form
Type of game
Production mode
With Without Deterministic Random Stackelberg Nash Make to order Make to stock
Shi et al. [22]
Jiang et al. [23]
Zhang et al. [24]
Bai et al. [25]
Liu et al. [26]
Meng et al. [27]
Shi et al. [28]
This paper

Impact of t on the equilibrium strategies (where a = 60 and b = 5)

t z
e
w
p
zR zN zM eR eN eM wR wN wM pR pN pM
0.5 5.64 4.97 3.63 1.46 1.77 1.18 4.92 5.53 6.73 9.71 8.85 9.43
1 5.48 4.81 3.49 3.07 3.67 2.43 5.07 5.67 6.86 9.96 9.20 9.67
1.5 5.15 4.51 3.23 5.05 5.91 3.82 5.36 5.94 7.09 10.43 9.86 10.12
2 4.53 4.01 2.84 7.85 8.78 5.45 5.92 6.39 7.45 11.25 10.95 10.81

Shaharudin, M. S.; Fernando, Y.; Jabbour, C. J. C.; Sroufe, R.; Jasmi, M. F. A. Past, present, and future low carbon supply chain management: A content review using social network analysis. Journal of cleaner production, 2019, 218, 629–643. ShaharudinM. S. FernandoY. JabbourC. J. C. SroufeR. JasmiM. F. A. Past, present, and future low carbon supply chain management: A content review using social network analysis Journal of cleaner production 2019 218 629 643 10.1016/j.jclepro.2019.02.016 Search in Google Scholar

Du, S.; Tang, W.; Song, M. Low-carbon production with low-carbon premium in cap-and-trade regulation. Journal of cleaner production, 2016, 134, 652–662. DuS. TangW. SongM. Low-carbon production with low-carbon premium in cap-and-trade regulation Journal of cleaner production 2016 134 652 662 10.1016/j.jclepro.2016.01.012 Search in Google Scholar

Wang, Y.; Chen, W.; Liu, B. Manufacturing/remanufacturing decisions for a capital-constrained manufacturer considering carbon emission cap and trade. Journal of Cleaner Production, 2017, 140, 1118–1128. WangY. ChenW. LiuB. Manufacturing/remanufacturing decisions for a capital-constrained manufacturer considering carbon emission cap and trade Journal of Cleaner Production 2017 140 1118 1128 10.1016/j.jclepro.2016.10.058 Search in Google Scholar

Ma, X.; Ji, P.; Ho, W.; Yang, C. H. Optimal procurement decision with a carbon tax for the manufacturing industry. Computers & Operations Research, 2018, 89, 360–368. MaX. JiP. HoW. YangC. H. Optimal procurement decision with a carbon tax for the manufacturing industry Computers & Operations Research 2018 89 360 368 10.1016/j.cor.2016.02.017 Search in Google Scholar

He, S.; Yin, J.; Zhang, B.; Wang, Z. How to upgrade an enterprise's low-carbon technologies under a carbon tax: The trade-off between tax and upgrade fee. Applied Energy, 2018,227, 564–573. HeS. YinJ. ZhangB. WangZ. How to upgrade an enterprise's low-carbon technologies under a carbon tax: The trade-off between tax and upgrade fee Applied Energy 2018 227 564 573 10.1016/j.apenergy.2017.07.015 Search in Google Scholar

Yu, Y.; Han, X.; Hu, G. Optimal production for manufacturers considering consumer environmental awareness and green subsidies. International Journal of Production Economics, 2016, 182, 397–408. YuY. HanX. HuG. Optimal production for manufacturers considering consumer environmental awareness and green subsidies International Journal of Production Economics 2016 182 397 408 10.1016/j.ijpe.2016.09.014 Search in Google Scholar

Shu, T.; Peng, Z.; Chen, S.; Wang, S.; Lai, K. K.; Yang, H. Government subsidy for remanufacturing or carbon tax rebate: Which is better for firms and a low-carbon economy. Sustainability, 2017, 9(1), 156. ShuT. PengZ. ChenS. WangS. LaiK. K. YangH. Government subsidy for remanufacturing or carbon tax rebate: Which is better for firms and a low-carbon economy Sustainability 2017 9 1 156 10.3390/su9010156 Search in Google Scholar

Cao, K.; Xu, X.; Wu, Q.; Zhang, Q. Optimal production and carbon emission reduction level under cap-and-trade and low carbon subsidy policies. Journal of Cleaner Production, 2017, 167, 505–513. CaoK. XuX. WuQ. ZhangQ. Optimal production and carbon emission reduction level under cap-and-trade and low carbon subsidy policies Journal of Cleaner Production 2017 167 505 513 10.1016/j.jclepro.2017.07.251 Search in Google Scholar

Chen, W.; Hu, Z. H. Using evolutionary game theory to study governments and manufacturers’ behavioral strategies under various carbon taxes and subsidies. Journal of Cleaner Production, 2018, 201, 123–141. ChenW. HuZ. H. Using evolutionary game theory to study governments and manufacturers’ behavioral strategies under various carbon taxes and subsidies Journal of Cleaner Production 2018 201 123 141 10.1016/j.jclepro.2018.08.007 Search in Google Scholar

Zhou, D.; An, Y.; Zha, D.; Wu, F.; Wang, Q. Would an increasing block carbon tax be better? A comparative study within the Stackelberg Game framework. Journal of environmental management, 2019, 235, 328–341. ZhouD. AnY. ZhaD. WuF. WangQ. Would an increasing block carbon tax be better? A comparative study within the Stackelberg Game framework Journal of environmental management 2019 235 328 341 10.1016/j.jenvman.2019.01.08230703647 Search in Google Scholar

Du, S.; Hu, L.; Wang, L. Low-carbon supply policies and supply chain performance with carbon concerned demand. Annals of Operations Research, 2017, 255(1–2), 569–590. DuS. HuL. WangL. Low-carbon supply policies and supply chain performance with carbon concerned demand Annals of Operations Research 2017 255 1–2 569 590 10.1007/s10479-015-1988-0 Search in Google Scholar

Xu, L.; Wang, C. Sustainable manufacturing in a closed-loop supply chain considering emission reduction and remanufacturing. Resources, Conservation and Recycling, 2018, 131, 297–304. XuL. WangC. Sustainable manufacturing in a closed-loop supply chain considering emission reduction and remanufacturing Resources, Conservation and Recycling 2018 131 297 304 10.1016/j.resconrec.2017.10.012 Search in Google Scholar

Li, J.; Du, W.; Yang, F.; Hua, G. The carbon subsidy analysis in remanufacturing closed-loop supply chain. Sustainability, 2014, 6(6), 3861–3877. LiJ. DuW. YangF. HuaG. The carbon subsidy analysis in remanufacturing closed-loop supply chain Sustainability 2014 6 6 3861 3877 10.3390/su6063861 Search in Google Scholar

Li, H.; Wang, C.; Xu, L.; Ou, W. Pricing, carbon emission reduction, collection decision, and coordination in a low-carbon closed-loop supply chain. Journal of Renewable and Sustainable Energy, 2017, 9(6), 065907. LiH. WangC. XuL. OuW. Pricing, carbon emission reduction, collection decision, and coordination in a low-carbon closed-loop supply chain Journal of Renewable and Sustainable Energy 2017 9 6 065907 10.1063/1.4991668 Search in Google Scholar

Wan, N. The impacts of low carbon subsidy, collection mode, and power structure on a closed-loop supply chain. Journal of Renewable and Sustainable Energy, 2018, 10(6), 065904. WanN. The impacts of low carbon subsidy, collection mode, and power structure on a closed-loop supply chain Journal of Renewable and Sustainable Energy 2018 10 6 065904 10.1063/1.5054669 Search in Google Scholar

Cohen, M. C.; Lobel, R.; Perakis, G. The impact of demand uncertainty on consumer subsidies for green technology adoption. Management Science, 2016, 62(5), 1235–1258. CohenM. C. LobelR. PerakisG. The impact of demand uncertainty on consumer subsidies for green technology adoption Management Science 2016 62 5 1235 1258 10.1287/mnsc.2015.2173 Search in Google Scholar

Rong, Y.; Chen, Y. J.; Shen, Z. J. M. The impact of demand uncertainty on product line design under endogenous substitution. Naval Research Logistics (NRL), 2015, 62(2), 143–157. RongY. ChenY. J. ShenZ. J. M. The impact of demand uncertainty on product line design under endogenous substitution Naval Research Logistics (NRL) 2015 62 2 143 157 10.1002/nav.21619 Search in Google Scholar

Feng, Y.; Viswanathan, S. Impact of demand uncertainty on coordinating supply chain inventories through common replenishment epochs. Journal of the Operational Research Society, 2007, 58(7), 964–971. FengY. ViswanathanS. Impact of demand uncertainty on coordinating supply chain inventories through common replenishment epochs Journal of the Operational Research Society 2007 58 7 964 971 10.1057/palgrave.jors.2602219 Search in Google Scholar

Chan, H. L.; Shen, B.; Cai, Y. Quick response strategy with cleaner technology in a supply chain: Coordination and win-win situation analysis. International Journal of Production Research, 2017, 56(10), 3397–C34088. ChanH. L. ShenB. CaiY. Quick response strategy with cleaner technology in a supply chain: Coordination and win-win situation analysis International Journal of Production Research 2017 56 10 3397 C34088 10.1080/00207543.2016.1278283 Search in Google Scholar

Choi, T. M. Carbon footprint tax on fashion supply chain systems. The International Journal of Advanced Manufacturing Technology, 2013, 68(1–4), 835–847. ChoiT. M. Carbon footprint tax on fashion supply chain systems The International Journal of Advanced Manufacturing Technology 2013 68 1–4 835 847 10.1007/s00170-013-4947-4 Search in Google Scholar

Drake, D. F.; Kleindorfer, P. R.; Van Wassenhove, L. N. Technology choice and capacity portfolios under emissions regulation. Production and Operations Management, 2016, 25(6), 1006–1025. DrakeD. F. KleindorferP. R. Van WassenhoveL. N. Technology choice and capacity portfolios under emissions regulation Production and Operations Management 2016 25 6 1006 1025 10.1111/poms.12523 Search in Google Scholar

Shi, R.; Zhang, J.; Ru, J. Impacts of power structure on supply chains with uncertain demand. Production and Operations Management, 2013, 22(5), 1232–1249. ShiR. ZhangJ. RuJ. Impacts of power structure on supply chains with uncertain demand Production and Operations Management 2013 22 5 1232 1249 10.1111/poms.12002 Search in Google Scholar

Jiang, W.; Chen, X. Optimal strategies for manufacturer with strategic customer behavior under carbon emissions-sensitive random demand. Industrial Management & Data Systems, 2016, 116(4), 759–776. JiangW. ChenX. Optimal strategies for manufacturer with strategic customer behavior under carbon emissions-sensitive random demand Industrial Management & Data Systems 2016 116 4 759 776 10.1108/IMDS-08-2015-0321 Search in Google Scholar

Zhang, B.; Qu, S.; Li, P.; Huang, R. Optimal Strategies for Manufacturers with the Reference Effect under Carbon Emissions-Sensitive Random Demand. Discrete Dynamics in Nature and Society, 2018, 1–15. ZhangB. QuS. LiP. HuangR. Optimal Strategies for Manufacturers with the Reference Effect under Carbon Emissions-Sensitive Random Demand Discrete Dynamics in Nature and Society 2018 1 15 10.1155/2018/2452406 Search in Google Scholar

Bai, Q.; Xu, J.; Chauhan, S. S. Effects of sustainability investment and risk aversion on a two-stage supply chain coordination under a carbon tax policy. Computers & Industrial Engineering, 2020, 142, 106324. BaiQ. XuJ. ChauhanS. S. Effects of sustainability investment and risk aversion on a two-stage supply chain coordination under a carbon tax policy Computers & Industrial Engineering 2020 142 106324 10.1016/j.cie.2020.106324 Search in Google Scholar

Liu, B.; Li, T.; Tsai, S. B. Low carbon strategy analysis of competing supply chains with different power structures. Sustainability, 2017, 9(5), 835. LiuB. LiT. TsaiS. B. Low carbon strategy analysis of competing supply chains with different power structures Sustainability 2017 9 5 835 10.3390/su9050835 Search in Google Scholar

Meng, X.; Yao, Z.; Nie, J.; Zhao, Y.; Li, Z. Low-carbon product selection with carbon tax and competition: Effects of the power structure. International Journal of Production Economics, 2018, 200, 224–230. MengX. YaoZ. NieJ. ZhaoY. LiZ. Low-carbon product selection with carbon tax and competition: Effects of the power structure International Journal of Production Economics 2018 200 224 230 10.1016/j.ijpe.2018.03.029 Search in Google Scholar

Shi, X.; Dong, C.; Zhang, C.; Zhang, X. Who should invest in clean technologies in a supply chain with competition. Journal of Cleaner Production, 2019, 215, 689–700. ShiX. DongC. ZhangC. ZhangX. Who should invest in clean technologies in a supply chain with competition Journal of Cleaner Production 2019 215 689 700 10.1016/j.jclepro.2019.01.072 Search in Google Scholar

Polecane artykuły z Trend MD

Zaplanuj zdalną konferencję ze Sciendo