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Subsidising an electric vehicle supply chain with imperfect information
Accepted Manuscript An imperfect information game in subsidising the electric vehicle supply chain Xiaoyu Gu, Li Zhou, Petros Ieromonachou PII:
S0925-5273(19)30029-5
DOI:
https://doi.org/10.1016/j.ijpe.2019.01.021
Reference:
PROECO 7272
To appear in:
International Journal of Production Economics
Received Date: 2 May 2018 Revised Date:
2 November 2018
Accepted Date: 15 January 2019
Please cite this article as: Gu, X., Zhou, L., Ieromonachou, P., An imperfect information game in subsidising the electric vehicle supply chain, International Journal of Production Economics (2019), doi: https://doi.org/10.1016/j.ijpe.2019.01.021. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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An Imperfect Information Game in Subsidising the Electric Vehicle Supply Chain
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Xiaoyu Gua,∗, Li Zhoub , Petros Ieromonachoub a School
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of Economics and Management, Nanjing University of Science and Technology, Nanjing, P.R.China b Department of Systems Management and Strategy, University of Greenwich, London, United Kingdom
Abstract
This paper studies a four-echelon vehicle supply chain consisting of government,
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electric/gasoline vehicle manufacturer, retailer and consumer. The purpose is to understand how government subsidies should be allocated in order to maximise total profit of the whole supply chain. By adopting Stackelberg game theory based on conditions of imperfect information, a mathematical model was developed. The results suggest that allocation of a subsidy in the electric vehicle supply chain should first be allotted for electric vehicle customers. Specifically,
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in the early development stage, if the subsidy budget is limited, all of them should be given to the purchasers of electric vehicle customer. With an increasing budget available for subsidies, more allocation to the electric vehicle manufacturer is expected. However, more subsidies does not necessarily lead to
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more electric vehicle purchases as there is a ceiling on the market for electric vehicles. In the later development stage, subsidies may not be important in
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promoting electric vehicle uptake. Keywords: Electric vehicle, Supply chain, Energy price, Subsidy policy, Stackelberg game, Imperfect information
∗ Corresponding
author Email addresses: [email protected] (Xiaoyu Gu), [email protected] (Li Zhou), [email protected] (Petros Ieromonachou)
Preprint submitted to International Journal of Production Economics
January 16, 2019
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1. Introduction Electric vehicles (EVs) are considered a significant option for solving envi-
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ronmental issues in transportation and the automotive industry, in particular. However, for a multitude of reasons, including limited mileage before needing
a recharge, and underdeveloped charging network, and lengthy charging times, EVs have not yet been widely accepted by the customer. Specifically, according to Cazzola et al. (2017), by the end of 2016, the EV market share was 1.41%
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in the UK, 1.46% in France, 0.71% in Germany, 0.91% in the US, and 1.37% in China. As a whole, the worldwide EV market share is only 1.10%. To overcome
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this situation and with the purpose of green and sustainable development along with EV industry promotion, governments have taken some actions, such as subsidising the electric vehicle supply chain (EVSC) (Abkemeier et al., 2012). There have been some studies focusing on EVSC incentive policies. For example, Gnann et al. (2015) used alternative automobile diffusion and infrastructure models to analyse the market evolution of EVs in Germany up to 2020. They concluded that fuel prices have a notable impact on the evolution of EV
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market. Using linear regression to analyse 30 national EV markets in 2012, Sierzchula et al. (2014) concluded that although financial subsidies and a charging infrastructure are positively correlated to EV sales, they were not enough to ensure high EV diffusion rates. What is more, in regard to the environ-
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mental impact, Shin et al. (2012) conducted a case study in South Korea and found that governmental purchase price subsidies had more benefits than tax
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incentives and were more helpful for environmental improvements. In addition to the case study, Huang et al. (2013) as well as Luo et al.
(2014) studied government subsidy and incentive schemes. Huang et al. (2013) analysed the subsidy incentive policy in a duopoly market comprising a gasoline automobile supply chain and the electric-fuel hybrid automobile supply chain. A Nash equilibrium of the wholesale price in the above two supply chains is achieved when the government provides subsidies to promote the EV market based on a information condition. The conclusion was that subsidy incentives were able to help increase EV sales; additionally, it reduced the negative impacts 2
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of the gasoline vehicles (GVs) in terms of environmental hazards. Furthermore, based on this research, Luo et al. (2014) focused on the EV supply chain using a model where the customer, manufacturer, and government were all taken
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into consideration and the best EV rate and subsidy ceiling were figured out
in determining the Nash equilibrium through a perfect information cooperative bargaining game. The result showed that in order to increase EV sales, the
government’s subsidy policy likely needed to include both a price discount and
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a subsidy ceiling.
Based on the reviewes of the literature, including the articles discussed above, several research gaps can be identified. First, models for offering EV incentives
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have been developed for the supply chain that include the involvement of governments, EV manufacturers and customers (Luo et al., 2014). However, an important part of the SC, the retailer, has been rarely considered. Second, although there is existing research on subsidy allocation in other industries (e.g., Internet infrastructure, see Chen et al. (2016)), in the case of EVSC, subsidy allocation must reflect the unique characteristics of EVSC; hence, it is neces-
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sary to develop new models. Third, the majority of the research that has been conducted on the EVSC are focused on the influence of EV purchase cost (e.g., Shin et al. (2012)), but the impact on subsidy allocation of energy costs, such as fuel for GVs, and for EVs, the costs of electricity and charging time have
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been studied less, even though these factors indeed affect optimal allocation of government subsidies. And fourth, most of the research discusses the supply chain model under perfect information conditions (for example, Cai et al. (2014)
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and Bulmus et al. (2014)). However, in reality, some information is private, for example, the vehicle manufacturing cost, which makes the information imperfectly transparent in the case of the EVSC. These gaps are what motivated our study on this topic. In order to fill the gaps just discussed, this research proposed a four-tier
model comprising government, GV/EV manufacturers, GV/EV retailers and vehicle buyer based on the game scheme of the Stackelberg imperfect information game. In particular, this study attempted to answer the following questions:
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First, with regard to a given subsidy budget, what is the best way to determine an optimal allocation across the EVSC? Second, how different subsidy budgets
and the customer’s decision to buy a car?
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affect this allocation? Third, how do energy costs affect the subsidy allocation
The rest of this paper is laid out as follows: Section 2 discusses formulation of the structure of our EVSC model, including a discussion of the using cost over the life of the vehicle. With the given six EV development stages identified
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and described here, we studies two of the stages, the early EV development
stage and the later development stage. Sections 3 and 4 presents the analysis of these two stages. Then, the purchasing probability and the optimal subsidy
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allocation rate are determined. Section 5 provides the analysis with numerical experiment. Finally, section 6 gives the conclusions of the research along with its limitations.
2. Model description and assumptions
Fig. 1 illustrates the structure of a four-echelon vehicle supply chain. It is
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common practice for vehicle sales and after-sales service to be franchised to companies such as AutoNation in the USA (Szakaly and Manzi, 2015) and ZhengtongAuto in China. Our model takes into consideration GV/EV manufacturers and retailers, where the EV retailer is franchised by the EV manufacturer, and
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the GV retailer sells GVs. Customers are able to choose to buy either a GV or an EV. For example, Chevrolet manufactures GV and its retailer sells GV only.
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Tesla produces EV and its retailer sells EV only. And customer can choose to buy either Chevrolet or Tesla. The government, as the entity responsible for setting incentive policy and overseeing the entire supply chain, is the leader, in terms of Stackelberg game theory.
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(9 0DQXIDFWXUHU
(95HWDLOHU &XVWRPHU
*9 0DQXIDFWXUHU
*95HWDLOHU
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*RYHUQPHQW
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6XEVLG\WR(9FXVWRPHU
Fig. 1. Structure for the subsidy model
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In reality, GV/EV manufacturers produces vehicles and sell them to their franchised retailers, who do not know the accurate vehicle manufacturing cost, but they know the approximate range of the cost. This information about manufacturing cost, together with the wholesale price the retailer paid are taken into consideration when the retailer sets the vehicle selling price to customers. Customers are rational, and they consider buying or not buying a vehicle that
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will bring them the maximum utility by taking into consideration their own purchase intention, the vehicle price, the subsidy, and costs. The notations used in this study to describe the model include the input, the intermediate variables and the decision variables are shown in Table 1. In
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order to simplify the calculation process, we have also defined the substitutions in table 1 as well.
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Table 1. Notations and substitutions
Input parameters k
Cenvir
Vehicle purchase intention
M
The entire mileage vehicle will be driven over life cycle (km)
The environmental cost of
pf uel
Fuel price (£/L)
Ef uel
Fuel efficiency (km/L)
the GV pelec
Electricity costs (£/kWh)
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Eelec
Electricity
efficiency
vt
Time value (£/h)
tchrg
Time cost for each electric-
tf uel
Time cost for each fuel refilling (h)
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(km/kWh)
ity recharging (h)
Fuel tank volume (L)
Vev
EV battery volume (kWh)
[pM ev ,
Lower and upper bound for
[pM gv ,
Lower and upper bound for
pM ev ]
the EV manufacturing cost
pM gv ]
the GV manufacturing cost
[pRev ,
Lower and upper bound for
[pRgv ,
Lower and upper bound for
pRev ]
the EV price to retailer
pRgv ]
the GV price to retailer
Agv
GV use benefit
Aev
EV use benefit
S
The budget for all subsidies
Cf uel
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Intermediate variables
Fuel costs over the vehicle’s life cycle (£)
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Vgv
Celec
Electricity costs over the vehicle’s life cycle (£)
Time cost for the EV (£)
Cgtime
Time cost for the GV (£)
Cgv
GV use cost for the whole
Cev
EV use cost for the whole
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Cetime
life cycle
life cycle
Ugv
Utility of the GV
Uev
Utility of the EV
pCgv
GV selling price to the cus-
pCev
EV selling price to the cus-
pRgv
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tomer
GV selling price to the re-
tomer pRev
tailer
tailer
Customer’s probability of
Pgv
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Pev
EV selling price to the re-
buying the EV
Customer’s probability of buying the GV
ΠRgv
Profit for the GV retailer
ΠRev
Profit for the EV retailer
ΠM gv
Profit for the GV manufac-
ΠM ev
Profit for the EV manufac-
turer
turer
Ucustomer Customer’s utility in using
Πentire
the vehicle
The entire profit of supply chain
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Subsidy given to the EV
sm
customer fre (x)
Probability
Subsidy given to the EV manufacturer
distribution
frg (x)
PDF of GV price charged
function (PDF) of EV price
to the retailer by GV man-
charged to the retailer by
ufacturer
EV manufacturer fme (x)
PDF of EV manufacturing
fmg (x)
PDF of GV manufacturing cost
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cost Decision variables Subsidy partition ratio
Substitutions K0
Agv + Cgv
K2
(pRgv −pRgv ) 2(pM gv −pM gv )
K4
(pRev −pRev ) 2(pM ev −pM ev )
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α
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sc
K1
Aev + Cev
K3
pRgv + pRgv
K5
pRev + pRev
2.1. EV and GV cost over the whole lifecycle
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According to Greene et al. (2004), in the span of the vehicle’s lifetime, the cost is equal to the total mileage multiplied by the unit cost of energy divided by the energy’s efficiency. Hence, the cost of fuel for the GV and the cost of
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electricity for the EV are, respectively
Cf uel =
pf uel M Ef uel
(1)
Celec =
pelec M Eelec
(2)
According to Shafiei et al. (2012), charging time is one of the main factors customers consider when buying an EV, but it is not a key factor for the GV. Therefore, as a primary difference between GVs and EVs, in this study, we took this into consideration and quantified the ratio of the cost in time to the cost
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in money using Eq. 3 and Eq. 4. M Vgv Ef uel
Cetime = vt tchrg
M Vev Eelec
(3)
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Cgtime = vt tf uel
(4)
as Cgv = Cf uel + Cetime
(5) (6)
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Cev = Celec + Cgtime
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In summary, the GV/EV use cost over the vehicle’s life cycle can be expressed
2.2. Customer’s choice probability
Different people have different intentions when buying a vehicle, and based on Shafiei et al. (2012) and Helveston et al. (2015), we assumed that customers consider a vehicle’s use benefits, the cost of use, the vehicle price, and the amount of subsidies to make a vehicle buying decisions. As different people have
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different sensitivity to buy a vehicle, we used k, k ∈ [0, 1] to denote the vehicle purchase intention. According to Morwitz et al. (2014), the utility function is defined as the use benefit multiplied by the residual purchase intention minus the use cost multiplied by the purchase intention. Thus, the utility functions
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for GV and EV users can be represented by Eq. 7 and Eq. 8:
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Ugv (k) = Agv k − Cgv (1 − k) − pCgv = K0 k − Cgv − pCgv
Uev (k) = Aev k − Cev (1 − k) − pCev + sc = K1 k − Cev − pCev + sc
(7)
(8)
There are 6 relationships between Ugv and Uev which are shown in Fig. 2 below:
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Case I
Pgv
Pnull Pev Pgv
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Pnull
Pgv
Case III
Case II
1
0 Pnull Pgv Pev
Pnull
Case IV
Pev
Pnull
Pev
Case VI
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Case V
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Pnull
Fig. 2. Relationships between UCgv and UCev A discussion of how these six relationships are mutually exclusive and collectively exhaustive can be found in appendix I.
As can be seen, in Case I and Case II, all buyers would select a GV, and in Case V and Case VI, all buyers would select an EV. This means that the market
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is dominated by GVs in Cases I and II and dominated by EVs in Cases V and VI. In Cases III and IV, different people with different purchasing intentions would purchase different vehicles, so both GVs and EVs would exist in the market. For this study, only the mixed selection cases were discused (Cases III and IV).
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In Case III, the use profit for EVs is less than that for the GVs while in Case IV, the use profit for EVs is greater than that for GVs. We defined Case III
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as the early development stage, and Case IV as the later development stage. Section 3 discusses the early stage and Section 4 discusses the later stage.
3. EV early development stage 3.1. Purchase probability and customer’s utility In this stage, the GV purchasing probability is calculated as shown in Eq. 9. Pgv = 1 −
Cgv − Cev + pCgv + sc − pCev K0 − K1
9
(9)
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The probability of buying an EV is calculated as shown in Eq. 10. Cgv − Cev + pCgv + sc − pCev Cev + pCev − sc − K0 − K1 K1
The proof of Pgv and Pev can be found in Appendix II.
(10)
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Pev =
The utility for the customer is calculated as shown below. Cgv −Cev +pCgv−pCev+sc K0−K1
Z
=
1
Cgv −Cev +pCgv−pCev+sc K0−K1
Ugv dk
SC
Cev +pCev −sc K1
Z Uev dk +
2 2 Cev K0 − 2Cev (K1 (Cgv + pCgv ) + K0 (sc − pCev )) + Cgv K1
+2Cgv K1 (−K0 + K1 − pCev + pCgv + sc ) + K02 K1 −K0 K12 − 2K0 K1 pCgv + K0 p2Cev − 2K0 pCev sc + K0 s2c +2K12 pCgv − 2K1 pCev pCgv + K1 p2Cgv + 2K1 pCgv sc
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Ucustomer =
2K1 (K0 − K1 )
(11)
3.2. Profits for the retailer and manufacturer
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Generally, with the probability distribution function (PDF) fre (x), frg (x), fme (x), fmg (x) and a given price for a vehicle sold by the manufacturer to the retailer, the probability for vehicle manufacturing cost can be found with Bayes’ Theorem:
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and
frg (pRgv |pM gv )fmg (pM gv ) P (pRgv |pM gv )P (pM gv ) = R +∞ P (pRgv ) frg (pRgv |pM gv )fmg (pM gv )dpM gv 0 (12)
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P (pM gv |pRgv ) =
P (pM ev |pRev ) =
fre (pRev |pM ev )fme (pM ev ) P (pRev |pM ev )P (pM ev ) = R +∞ P (pRev ) f re (pRev |pM ev )fme (pM ev )dpM ev 0 (13)
In this model, GV/EV vehicle manufacturers know their vehicles’ manufacturing costs (pM gv /pM ev ). Based on the manufacturing cost, they are able to decide the selling price to the retailer. Then, the GV/EV retailers, decide on a price for consumers (pCgv and pCev ) based on the price they paid (pRgv and pRev ) for the vehicle. As a matter of fact, the vehicle retailer does not know the vehicle 10
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manufacturing cost, but they can estimate the range of vehicle production costs. We assumed that GV/EV manufacturing costs are uniformly distributed within the range [pM gv , pM gv ] for GV manufacturing costs and [pM ev , pM ev ] for EV
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manufacturing costs. With the manufacturing cost range, the selling price to a
retailer is assumed to be uniformly distributed in the range of [pRgv , pRgv ] for the GV and [pRev , pRev ] for the EV. Therefore, the profits for the GV and EV retailers can be calculated using Eq. 14 and Eq. 15. +∞
Pgv (pCgv − pRgv )P (pM gv |pRgv )dpRgv
(14)
Pev (pCev − pRev )P (pM ev |pRev )dpRev
(15)
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Z ΠRgv = 0
0
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+∞
Z ΠRev =
And with the uniform distribution, the profit for a GV/EV retailer can be updated to the two equations below. Z
pRgv
Pgv (pCgv − pRgv )
ΠRgv = pRgv
(pRgv − pRgv )
=
Pgv (2pCgv − pRgv − pRgv )
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2(pM gv − pM gv )
1 dpRgv pM gv − pM gv
(16)
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= K2 Pgv (2pCgv − K3 ) 2K2 p2Cgv K2 K3 (−Cev +Cgv −K0 +K1 −pCev +sc ) + − K0 −K K0 −K1 1 = K2 pCgv (2Cev −2Cgv +2K0 −2K1 +K3 +2pCev −2sc ) + K0 −K1 Z
pRev
Pev (pCev − pRev )
ΠRev =
pRev
1 dpRev pM ev − pM ev
(pRev − pRev ) Pev (2pCev − pRev − pRev ) 2(pM ev − pM ev )
AC C =
(17)
= K4 Pev (2pCev − K5 ) 2K K p2Cev K4 K5 (Cev K0 −K1 (Cgv +pCgv )−K0 sc ) − K00K14−K + 2 K1 (K0 −K1 ) 1 = K4 pCev (K0 (−2Cev +K5 +2sc )+2K1 (Cgv +pCgv )) + K1 (K0 −K1 )
Because K0 − K1 > 0, K0 > 0, K2 and K4 > 0 and K1 > 0, the maximum profit for the GV retailer and EV retailer will be achieved when
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∂ΠRgv ∂pCgv
= 0 and
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= 0, as shown below
K2 (2Cev −2Cgv +2K0 −2K1 +K3 +2pCev −2sc ) K0 −K1 K4 (K0 (−2Cev +K5 +2sc )+2K1 (Cgv +pCgv )) K1 (K0 −K1 )
−
−
4K2 pCgv K0 −K1
4K0 K4 pCev K0 K1 −K12
=0
(18)
=0
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∂ΠRev ∂pCev
And the optimal GV/EV selling price from the retailer to the customer is shown below.
2Cev K0 +2Cgv (K1 −2K0 )+K0 (4K0 −4K1 +2K3 +K5 −2sc ) 8K0 −2K1
p∗Cgv =
p∗ = Cev
2Cev (K1 −2K0 )+2Cgv K1 +2K0 K1 +2K0 K5 +4K0 sc −2K12 +K1 K3 −2K1 sc 8K0 −2K1
SC
(19)
K −4C
2C
P∗ ev
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Hence, the optimal GV/EV purchasing probability is as shown in Eq. 20 below. K +2C
K +4K 2 −4K K −2K K +K K −2K s +K K
gv 0 gv 1 0 1 0 3 0 5 0 c 1 3 ∗ 0 = ev 0 Pgv 8K02 −10K0 K1 +2K12 K0 (2Cev (K1 −2K0 )+2Cgv K1 +2K0 K1 −2K0 K5 +4K0 sc −2K12 +K1 K3 +K1 K5 −2K1 sc ) = 2K1 (K0 −K1 )(4K0 −K1 ) (20)
The manufacturer profits are the vehicle purchase probability multiplied by the revenue from the sale of the vehicle to the retailer, as shown in Eq. 21 for the
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GV manufacturer and Eq. 22 for the EV manufacturer.
(21)
ΠM ev = Pev (pRev − pM ev + sm )
(22)
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ΠM gv = Pgv (pRgv − pM gv )
With the optimal selling price and purchase probability, when the govern-
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ment decides on a subsidy, it uses the mathematical expectation of vehicle manufacturing cost and selling price, which are pM gv = pM ev +pM ev , 2
pRgv =
pRgv +pRgv 2
and pRev =
12
pRev +pRev . 2
pM gv +pM gv , 2
pM ev =
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3.3. Total profit for the supply chain The total profit for the supply chain is described as
Πtotal = =
ΠM gv + ΠRgv + ΠM ev + ΠRev +Ucustomer − Pgv Cenvir − Pev S ∗ Pgv (pCgv
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− pM gv − Cenvir )
∗ +Pev (pCev
− pM ev + sm − S) + Ucustomer
(23)
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The complete expression can be found in Eq. 51 in Appendix IV. The target for the government is to maximise total profits for the EVSC. As K0 > K1 , 3K1 − K02 S 2 (3K1 −4K0 ) 2K1 (K0 −K1 )(K1 −4K0 )2
get the maximum profit (Πtotal ), α should satisfy
∂Πtotal ∂α
= 0. Therefore,
8Cenvir K0 K1 − 2Cenvir K12 − 8Cev K02 + 6Cev K0 K1 − 2Cev K12
+8Cgv K0 K1 − 4Cgv K12 + 4K02 K5 − 16K02 pM ev − 2K0 K12 −3K0 K1 K5 + 12K0 K1 pM ev + 8K0 K1 pM gv + 2K13 − K12 K3 −2K12 pM ev − 2K12 pM gv
2K0 S(4K0 − 3K1 )
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α∗ =
< 0, and in order to
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4K0 < 0. Therefore, it is easy to get
(24)
In addition, because α is the partition ratio, it needs to meet the condition
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0 < α < 1 as well.
4. EV later development stage
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4.1. Purchase probability and customer’s utility In this stage, the purchase probability is determined using Eq. 25 for the
GV and Eq. 26 for the EV.
Pgv =
Cgv − Cev + pCgv + sc − pCev Cgv + pCgv − K0 − K1 K0
Pev = 1 −
Cgv − Cev + pCgv + sc − pCev K0 − K1
13
(25)
(26)
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The proof of probabilities is shown in appendix III. The utility function for the customer is determined as shown below.
Ucustomer =
Cgv +pCgv K0
Z Ugv dk +
1 Cgv −Cev +pCgv +sc −pCev K0 −K1
2 C K − 2C K (C − K + K − p + p + s ) 0 ev 0 gv 0 1 Cev Cgv c ev +C 2 K1 + 2K0 sc (Cgv − K0 + K1 − pCev + pCgv ) gv −2C K p 2 2 + 2C K p − K K + 2K p gv 0 Cev gv 1 Cgv 0 1 0 Cev 2 2 +K0 K1 − 2K0 K1 pCev + K0 pCev − 2K0 pCev pCgv +K0 s2c + K1 p2Cgv
SC
−2K0 (K0 − K1 )
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=
Uev dk
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Cgv −Cev +pCgv +sc −pCev K0 −K1
Z
(27)
4.2. Profits for the retailer and manufacturer
Similar to Case I, the profits for the GV and EV retailers are as shown in Eq. 28 and Eq. 29 for the GV retailer and EV retailer, respectively. +∞
Z
Pgv (pCgv − pRgv )P (pM gv |pRgv )dpRgv
ΠRgv =
(28)
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0
Z
+∞
Pev (pCev − pRev )P (pM ev |pRev )dpRev
ΠRev =
(29)
0
EP
With uniform distribution, Z
pRgv
Pgv (pCgv − pRgv )
ΠRgv =
pRgv
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=
(pRgv − pRgv )
2(pM gv − pM gv )
1 dpRgv pM gv − pM gv
Pgv (2pCgv − pRgv − pRgv ) (30)
= K2 Pgv (2pCgv − K3 ) K p (2K (C +pCev −sc )+K1 (K3 −2Cgv )) − 2 Cgv 0 evK0 (K 0 −K1 ) = 2K1 K2 p2Cgv K2 K3 (K0 (Cev +pCev −sc )−Cgv K1 ) + + 2 K0 (K0 −K1 ) K −K0 K1 0
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Z
pRev
Pev (pCev − pRev )
ΠRev = pRev
=
1 dpRev pM ev − pM ev
(pRev − pRev ) Pev (2pCev − pRev − pRev ) 2(pM ev − pM ev )
=
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(31) = K4 Pev (2pCev − K5 )
K4 pCev (2Cev −2Cgv +2K0 −2K1 −K5 −2(pCgv +sc )) K0 −K1
K K (−Cev +Cgv −K0 +K1 +pCgv +sc ) + 4 5 K0 −K1
+
2K4 p2Cev K0 −K1
achieved when
= 0 and
4K1 K2 pCgv K02 −K0 K1
−
∂ΠRev ∂pCev
= 0, which is
K2 (2K0 (Cev +pCev −sc )+K1 (K3 −2Cgv )) K0 (K0 −K1 )
K4 (2Cev −2Cgv +2K0 −2K1 −K5 −2(pCgv +sc )) K0 −K1
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∂ΠRgv ∂pCgv
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As K0 − K1 < 0, K0 > 0, K1 > 0, K2 > 0 and K4 > 0, the optimal profit is
So,
+
4K4 pCev K0 −K1
=0
(32)
=0
p∗ = − 2Cev K0 +2Cgv K0 −4Cgv K1 −2K02 +2K0 K1 +K0 K5 −2K0 sc +2K1 K3 Cgv 2K0 −8K1 p∗ = − 2Cev (K0 −2K1 )+K1 (2Cgv −4K0 +4K1 +K3 +2K5 )−2sc (K0 −2K1 ) Cev 2(K0 −4K1 )
(33)
∗ = Pgv
K1 (2Cev K0 +2Cgv (K0 −2K1 )−2K02 +2K0 K1 +K0 K3 +K0 K5 −2K0 sc −2K1 K3 ) 2K0 (K0 −4K1 )(K0 −K1 )
2Cev K0 −4Cev K1 +2Cgv K1 −4K0 K1 +K0 K5 −2K0 sc +4K12 +K1 K3 −2K1 K5 +4K1 sc 2K02 −10K0 K1 +8K12
(34)
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P∗ = ev
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And the GV/EV purchasing probabilities are
Similar to the early stage, the profits for the GV manufacturer and EV manu-
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facturer are:
ΠM gv = Pgv (pRgv − pM gv )
(35)
ΠM ev = Pev (pRev − pM ev + sm )
(36)
4.3. Total profit for the supply chain Again, we used the average vehicle manufacturing cost and price strategy,
which is pM gv =
pM gv +pM gv , 2
pM ev =
pM ev +pM ev , 2
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pRgv =
pRgv +pRgv 2
and pRev =
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The target for the government is to maximise the EVSC profit. Πtotal = =
Pgv (ΠM gv + ΠRgv ) + Pev (ΠM ev + ΠRev ) +Ucustomer − Pgv Cenvir − Pev S ∗ Pgv (pCgv − pM gv − Cenvir ) ∗ +Pev (sm + pCev − pM ev − S) +Ucustomer
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pRev +pRev . 2
(37)
seen,
K1 S 2 (4K1 −3K0 ) 2(K0 −4K1 )2 (K0 −K1 )
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The complete expression can be refereed from Eq. 52 in Appendix IV. As can be < 0, because, in this case, K1 > K0 .. In order to get ∂Πtotal ∂α
= 0. Therefore, the optimal subsidy
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the optimal profit, α should meet
allocation ratio can be shown in Eq. 38.
+4Cgv K0 K1 − 8Cgv K12 − 6K02 K1 + 2K02 pM ev + 14K0 K12 +K0 K1 K3 + 3K0 K1 K5 − 12K0 K1 pM ev + 2K0 K1 pM gv − 8K13 −4K12 K5 + 16K12 pM ev − 8K12 pM gv
6K0 K1 S − 8K12 S
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α∗ =
2Cenvir K0 K1 − 8Cenvir K12 + 2Cev K02 − 6Cev K0 K1 + 8Cev K12
(38)
Moreover, α should satisfy 0 < α < 1 as well.
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5. Analysis and discussion
This section analyses the parameters and the optimal profit. We first intro-
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duce a numerical experiment to address the model, then discuss the relationships between the amount of the subsidy, energy prices, charging time, and the trends for subsidy allocation policies, vehicle purchase probabilities and the total profit for the EVSC.
The average mileage driven per year and the average vehicle life cycle can be
found in UK Government (2017), and the prices of energy and energy efficiencies are shown in UK Power (2017) and China Automotive Technology and Research Center et al. (2016). The time value is simplified as the hourly wage in the UK (UK Government, 2018).
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All the initial values of the parameters are shown in Table 2.
Ef uel = 12.5
Eelec = 7
vt = 8
Tref uel = 0.1
Trechr = 1
Vgv = 60
Vev = 85
Cenvir = 1000
5.1. EV early development stage
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M = 300000
Table 2. Notations pf uel = 1.6 pelec = 0.15
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In the early EV development stage, we set the initial parameters as S = 3000, Agv = 100000, Aev = 70000, and the cost ranges [pM gv , pM gv ] = [6000, 8000],
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[pRgv , pRgv ] = [7000, 9000], [pM ev , pM ev ] = [8000, 12000], and [pRev , pRev ] = [10000, 14000]. In order to maximise profits along the entire supply chain, the optimal subsidy allocated to the EV manufacturer and customer and the GV/EV purchase probabilities are as shown in Table 3 with asterisks. Table 3. Optimal values in the later development stage α = 0.49
∗ = 0.38 Pgv
∗ = 0.25 Pev
pRgv = 8000
pRev = 12000
p∗Cgv = 28375
p∗Cev = 20217
Π∗total = 30209
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∗
Fig. 3 illustrates the relationships between total subsidy and the optimal subsidy allocation ratio, GV/EV purchase probability, and the total profit for
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the EVSC.
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(a) Total subsidy VS Optimal customer's subsidy ratio
1
(b) Total subsidy VS Purchase Probability
0.4
X 1500 Y 0.978
0.38 X 1500 Y 0.381
Purchase probability
0.36 0.8
0.6
0.4
0.34 GV EV
0.32 0.3 0.28
X 1500 Y 0.256
0.26 0.2 0.24 Optimal
0
0.22 500
1000
1500 2000 Total Subsidy 10
3.0209
2500 4
3000
0
1000
(c) Total subsidy VS Total profit X 1500 Y 30208.6
3.0208
3.0207
1500 2000 Total Subsidy
2500
3000
Total profit
3.0206
3.0205
3.0204
3.0203 0
500
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Total profit
500
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0
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1.2
1000
1500 2000 Total Subsidy
2500
3000
Fig. 3. Total subsidy budget vs. relevant parameters in the early development stage
In order to obtain the optimal total profit for the entire supply chain, Fig.
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3(a)indicates that if the total subsidy budget is low, all the subsidies should be distributed to EV purchasers. Increasing subsidy budgets means that more money should be invested in the EV manufacturer. As shown in Fig. 3(b), the GV purchase probability decreases as the EV purchase probability increases
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when the total subsidy increases. There are caps and floors for the probabilities: In the case of an increasing subsidy budget, the EV purchase probability is
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capped at around 0.256 and the GV probability has a floor at 0.381. Moreover, the total profit for the supply chain is capped at 30208.6 no matter how much the total subsidy grows, as shown in Fig 3(c). Therefore, in this case, we suggest that the optimal amount of subsidy should be £1500, with 97.8% of
the subsidy budget distributed to EV purchasers and 2.2% allotted for the EV manufacturers. Therefore, we suggest that if the subsidy budget of subsidy is more than 1500, it would be better to use the remaining budget for EV infrastructure development. Fig. 4 shows the relationships between fuel prices and the parameters of 18
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optimal subsidy ratio, purchase probability, and total profit. With increasing fuel prices, more subsidy should be provided to the EV customer, and when the fuel price is more than 1.52, all subsidy should be distributed to EV customers.
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In addition, the higher the fuel price, the higher the EV purchase probability and the lower the GV purchase probability. When all the subsidies have been distributed to consumers, the trend of GV/EV purchasing probabilities slows
down. Further, as fuel prices rise, the total profits for the entire supply chain
1.2
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decrease. (a) Fuel price VS Optimal customer's subsidy ratio
(b) Fuel price VS Purchase Probability 0.4
X 1.52 Y 0.3587
1 X 1.52 Y 0.9786
Purchase probability
0.35
0.8
0.6
0.4
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Optimal
0.3
X 1.52 Y 0.2823
0.25
0.2
0 1.3
1.4
1.5
1.6 1.7 Fuel price
3.1
1.8
10
1.9
4
0.2 1.3
2
1.4
1.5
1.6 1.7 Fuel price
GV EV
1.8
1.9
2
(c) Fuel price VS Total profit
Total profit
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3.05
Total profit
3
2.95
2.9
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2.85
2.8 1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Fuel price
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Fig. 4. Fuel price vs. relevant parameters in the early development stage Fig. 5 and Fig. 6 illustrate the relationships of, respectively, electricity
prices and charging time with the optimal subsidy ratio, purchase probability, and total profit. It is easy to see that the relationships are similar. When electricity prices and/or charging time increase, more subsidy should be allotted for the EV manufacturers. In addition, consumer desire for buying a GV is even stronger. In this situation, it is likely that more people mill select GVs rather than EVs. Moreover, when EV usage costs increase, the total profits for the EVSC decrease. 19
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1.2
(a) Electricity price VS Optimal customer's subsidy ratio
(b) Electricity price VS Purchase Probability
0.4
X 0.108 Y1
0.38
1
0.6 Optimal
0.4
X 0.108 Y 0.366
0.34 GV EV
0.32 X 0.108 Y 0.290
0.3
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Purchase Probability
0.36 0.8
0.28 0.26
0.2 0.24 0.22 0.06
0.08
0.1 0.12 0.14 Electricity price
3.09
104
0.16
0.18
0.06
0.08
0.1 0.12 Electricity price
(c) Electricity price VS Total profit Total profit
3.07 3.06 3.05 3.04 3.03 3.02 3.01 3 0.06
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Total profit
0.16
0.18
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3.08
0.14
0.08
0.1 0.12 0.14 Electricity price
0.16
0.18
Fig. 5. Relationship of electricity prices with optimal customer subsidy, purchase
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probability, and total profits in the early development stage
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1.2
(a) Charging time VS Optimal customer's rsubsidy ratio
(b) Charging time VS Purchase probability
0.4
Optimal
X 0.56 Y1
0.38
1
0.6
0.4
X 0.56 Y 0.366
0.34 GV EV
0.32 X 0.55 Y 0.290
0.3 0.28 0.26
0.2 0.24 0 0.2
0.4
0.6
0.8 Charging time
1
10
3.07
1.2 4
0.22 0.2
1.4
0.4
0.6
0.8 Charging time
(c) Charging time VS Total profit Total profit
3.05
1.2
1.4
3.04 3.03 3.02 3.01 3 0.2
0.4
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Total profit
1
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3.06
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Purchase Probability
0.36 0.8
0.6
0.8 Charging time
1
1.2
1.4
Fig. 6. The relationships of charging time with optimal subsidy, purchase prob-
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ability, and total profits in the early development stage 5.2. EV later development stage
In the later EV development stage, we set the initial parameters as S = 500, Agv = 100000 and Aev = 150000, [pM gv , pM gv ] = [5000, 7000], [pRgv , pRgv ] =
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[6000, 8000], [pM ev , pM ev ] = [15000, 40000] and [pRev , pRev ] = [20000, 50000].. Hence, the optimal subsidy and purchase probability were as shown in Table 4.
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Table 4. Optimal values in the later development stage
∗
α =1
∗ Pgv = 0.1
∗ Pev = 0.56
pRgv = 7000
pRev = 35000
p∗Cgv = 88342
p∗Cev = 47167
Π∗total = 44266
The results are shown in Fig. 7 through 10 below:
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(a) Total subsidy VS Optimal customer's subsidy ratio
1.2
(b) Total subsidy VS Purchase Probability
0.7 0.6
0.8
0.6
0.4
0.2
0.5 0.4 GV EV
0.3
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Purchase probability
1
0.2 0.1
Optimal
0
0
0
500
1000
1500 2000 Total Subsidy
2500
104
4.5
3000
0
500
1000
1500 2000 Total Subsidy
(c) Total subsidy VS Total profit
4.48 4.47 4.46 4.45 4.44 4.43
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Total profit
3000
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4.49
2500
Total profit
4.42 4.41 0
500
1000
1500 2000 Total Subsidy
2500
3000
Fig. 7. Total subsidy budget vs. relevant in the later development stage
(a) Fuel price VS Optimal customer's subsidy ratio
1
1
0.8
0.6
0.2
0 1.5
EP
0.4
1.6
0.4 GV EV
0.2
0
1.9 4
2
-0.2 1.5
1.6
1.7 1.8 Fuel price
1.9
2
(c) Fuel price VS Total profit
4.9 4.8 Total profit
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10
0.6
Optimal
1.7 1.8 Fuel price 5
(b) Fuel price VS Purchase Probability
0.8 Purchase probability
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1.2
4.7 4.6 4.5 4.4 Total profit
4.3 1.5
1.6
1.7 1.8 Fuel price
1.9
2
Fig. 8. Fuel price vs. relevant parameters in the later development stage
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As can be seen from Fig. 7, with the given input parameters, all the subsidies should be allotted to EV customers. In the later stage, the more subsidy that
total EVSC profit.
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is made available, the greater the EV purchase probability and the higher the
Fig. 8 shows variations of the relationships between fuel prices and the parameters of the optimal customer subsidy ratio, purchase probability, and
total profit. With increasing fuel prices, more people will select the EV, and
1.2
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the total profit for the supply chain will increase as well. (a) Electricity price VS Optimal customer's subsidy ratio
(b) Electricity price VS Purchase Probability
0.7 0.6
0.8
0.6
0.4
0.2
0.5
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Purchase Probability
1
0.4 0.3
GV EV
0.2 0.1
Optimal
0
0
0.06
0.08
0.1 0.12 0.14 Electricity price 4.7
10
4
0.16
0.18
0.06
0.08
0.1 0.12 0.14 Electricity price
0.16
0.18
(c) Electricity price VS Total profit
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Total profit
4.65
Total profit
4.6
4.55 4.5
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4.45 4.4
4.35
0.08
0.1 0.12 0.14 Electricity price
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0.06
0.16
0.18
Fig. 9. Electricity price vs. relevant parameters in the later development stage
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(a) Charging time VS Optimal customer's rsubsidy ratio 0.7
(b) Charging time VS Purchase probability
0.6 Purchase Probability
1
0.8
0.6
0.4
0.2
0.5 0.4 GV EV
0.3 0.2 0.1
Optimal
0 0.2
0.4
0.6
0.8 1 Charging time 4.65
1.2
104
0 0.2
1.4
0.4
0.6
(c) Charging time VS Total profit Total profit
1.2
1.4
4.5 4.45 4.4 4.35
0.4
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Total profit
4.55
4.3 0.2
0.8 1 Charging time
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4.6
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1.2
0.6
0.8 1 Charging time
1.2
1.4
Fig. 10. Charging time vs. relevant in the later development stage
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Fig. 9 and Fig. 10 show the relationships of electricity cost and charging time with the parameters of the optimal customer subsidy ratio, purchase probability, and total profit. As can be seen, higher electricity costs lead to greater GV purchase desire and lower total profits for the EVSC. Therefore, in the later
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stage, we can conclude that, subsidies are not as important as in the EV early development stage because by this time, EVs have been generally accepted. In summary, in the EV early development stage, to optimise the entire profit
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for the whole EVSC, if the subsidy budget is low, all subsidies should be allocated to EV buyers, and with increasing subsidy budgets, subsidies should be allocated to EV manufacturers. Governments should bear in mind that with increasing total subsidies, a cap will be reached on EV purchase probabilities, and there will be a floor for GV purchase probabilities, This means that no matter how subsidies are allocated, the EVSC total profit and vehicle purchase probabilities are capped. In terms of energy costs, increasing fuel prices mean that more subsidies should be allocated to EV buyers. However, with increase
24
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in EV use costs, more subsidies should be allocated to EV manufacturers. In addition, the higher vehicle use costs mean lower EVSC total profit. In the EV later development stage, all subsidies should be allocated to the EV customer.
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More subsidy budget or higher GV use costs will lead to higher total profits but higher EV use costs will result in lower total profits for the EVSC.
Given these results, we are able to conclude that lower EV use costs or higher GV use costs result in greater EV purchase desire, and the government should
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prioritise customers when considering subsidies for the EVSC. In addition, increasing electricity prices and EV charging times mean that it will be necessary to allocate more subsidies to EV manufacturers. The results also suggest that
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energy costs have a significant impact on the subsidy allocation ratio. It is worth noting that our floored and capped EV diffusion rates reflect the findings of Sierzchula et al. (2014), who claimed that even though financial subsidies are correlated with the EV market, they donot ensure high EV diffusion rates. Regarding the optimal subsidy amount, Gnann et al. (2015) suggested that the effective subsidy amount is 1000, and similarly, our study suggests that the op-
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timal subsidy is 1500 in the EV early stage, while in the later stage, subsidies are not quite as important for the EV uptake, as more customers have accepted EVs by that point.
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6. Conclusion
In order to fill the gaps in the study of EVSC incentive design and sub-
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sidy policy, for this study, we designed an imperfect information Stackelberg leader/follower game model that includes government, vehicle manufacturers, retailers and EV customers. Based on the results of our study, this paper discussed the effects on the EVSC of subsidy allocation to EV manufacturers or EV buyers and analysed the impacts of energy prices and EV charging times on the EVSC. With the goal of maximising profits for the entire EVSC, the optimal subsidy allocations to EV customers and manufacturers were derived, and numerical examples proposed, for both early and later EV development stages. In the lit-
25
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erature, we found that (1) EV incentive models have been previously developed for a supply chain comprising governments, EV manufacturers, and customers, but for our model, we added the missing component of the EV retailer; (2)
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Research on subsidy allocation has been conducted for other industries, but our
research focused solely on the EV industry and its unique characteristics discussed earlier; and (3) There are a number of existing studies on the EVSC, but they assume perfect information conditions, while our model uses an imperfect
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information condition to more closely reflect reality. Numerical examples are proposed on both stages.
Based on our results, the research questions were answered as follows: First,
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with a given budget for subsidising the EVSC, how can the optimal allocation be determined? In order to maximise the total profits throughout the EVSC, the subsidy allocation ratio for both early and later stages are determined, and the subsidy policy should consider EV customers first. Second, how do different amounts available for subsidy budgets affect this allocation? In the early stage, the higher the subsidy budget, the more of it that should be allocated to EV
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manufacturers. In the later development stage, it is recommended that all the subsidy should be allocated to EV customers; Third, how do the total subsidy budget and energy costs affect subsidy allocation and customer decisions to buy a car? The results showed that in the early stage, higher fuel prices, lower
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electricity prices and less EV charging times should lead to more subsidies being allocated to EV customers, and in the later stage, with the development and maturity of EV technology, all subsidies should be allocated directly to EV
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buyers. Additionally, more subsidies, lower EV use costs, and higher GV use costs will all lead to higher EV diffusion rates. Therefore, it can be concluded that governments should pay more attention
the subsidy budgets for the EVSC, including the allocation of subsidies. In the EV early development stage, if available subsidies are limited, all of them should be allocated to EV buyers, and then, as available funds for these subsidies increase, allocations should be made to EV manufacturers. This study’s results further indicated that, there is a cap for EV purchase probabilities and total
26
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profits for the EVSC, and the optimal subsidy amount should be 1500. In the later EV development stage, subsidies are no longer as important because, by that time, the user experience will have improved and more and more consumers
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will have accepted the EVs.
The contributions of this study to the field of research in EV subsidies and
government policies have threefold. Theoretically, different from those under perfect information conditions (Cai et al., 2009; Chen et al., 2012), we used
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the Stackelberg game to analyse the EVSC based on imperfect information conditions; Politically, the results of this study can serve as guiding evidence for governments when designing subsidy schemes, particularly in ensuring that
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EV buyers are afforded greater priority than other players in the EVSC. On a practical level, given that EV buyers are assured of benefiting from such a subsidy policy, manufactures and retailers can use this information in their marketing activities to promote EV uptake.
In this study, we attempted to fill the gaps in the literature on EVSC subsidy policy research. By giving attention in this study to study the impacts of
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vehicle use costs and subsidy allocation in promoting the EVSC,we showed how vehicle driving benefits can be easily converted to tangible variables (Agv and Aev ). In order to reflect real-word circumstances even more accurately, future research will examine the details of use benefits and analyse their impacts on
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subsidy allocation and will also look at optimal investment strategy in the EV infrastructure and ancillary services.
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Abkemeier, K., Appels, D., Barnes, J., Beeton, D., Brennan, G., et al., 2012. Hybrid and electric vehicles: The electric drive captures the imagination, in: Implementing Agreement for Co-operation on Hybrid and Electric Vehicle Technologies and Programmes Annual Report (IA-HEV).
Bulmus, S.C., Zhu, S.X., Teunter, R., 2014. Competition for cores in remanufacturing. European Journal of Operational Research 233, 105–113. Cai, G.G., Zhang, Z.G., Zhang, M., 2009.
27
Game theoretical perspectives
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on dual-channel supply chain competition with price discounts and pricing schemes. International Journal of Production Economics 117, 80–96.
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Cai, X., Lai, M., Li, X., Li, Y., Wu, X., 2014. Optimal acquisition and production policy in a hybrid manufacturing/remanufacturing system with core acquisition at different quality levels. European Journal of Operational Research 233, 374–382.
2017. International Energy Agency, France .
SC
Cazzola, P., Gorner, M., Schuitmaker, R., Maroney, E., 2017. Global ev outlook
Chen, J., Zhang, H., Sun, Y., 2012. Implementing coordination contracts in a
M AN U
manufacturer stackelberg dual-channel supply chain. Omega 40, 571–583. Chen, X., Wang, X., Chan, H.K., 2016. Channel coordination through subsidy contract design in the mobile phone industry. International Journal of Production Economics 171, 97–104.
China Automotive Technology and Research Center, Nissan (China) Investment
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Co., Ltd., Dongfeng Motor Co., Ltd., 2016. Annual Report on New Energy Vehicle Industry in China. Social Science Academic Press (China). Gnann, T., Pl¨ otz, P., K¨ uhn, A., Wietschel, M., 2015. Modelling market diffusion of electric vehicles with real world driving data–german market and policy
EP
options. Transportation Research Part A: Policy and Practice 77, 95–112. Greene, D.L., Duleep, K., McManus, W., 2004. Future potential of hybrid and
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diesel powertrains in the us light-duty vehicle market .
Helveston, J.P., Liu, Y., Feit, E.M., Fuchs, E., Klampfl, E., Michalek, J.J., 2015. Will subsidies drive electric vehicle adoption? measuring consumer preferences in the us and china. Transportation Research Part A: Policy and Practice 73, 96–112. Huang, J., Leng, M., Liang, L., Liu, J., 2013. Promoting electric automobiles: Supply chain analysis under a governments subsidy incentive scheme. IIE transactions 45, 826–844. 28
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Luo, C., Leng, M., Huang, J., Liang, L., 2014. Supply chain analysis under a price-discount incentive scheme for electric vehicles. European Journal of
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Operational Research 235, 329–333. Morwitz, V., et al., 2014. Consumers’ purchase intentions and their behavior. R in Marketing 7, 181–230. Foundations and Trends
´ Shafiei, E., Thorkelsson, H., Asgeirsson, E.I., Davidsdottir, B., Raberto, M., Ste-
fansson, H., 2012. An agent-based modeling approach to predict the evolution
Forecasting and Social Change 79, 1638–1653.
SC
of market share of electric vehicles: a case study from iceland. Technological
M AN U
Shin, J., Hong, J., Jeong, G., Lee, J., 2012. Impact of electric vehicles on existing car usage: A mixed multiple discrete–continuous extreme value model approach. Transportation research part D: transport and environment 17, 138–144.
Sierzchula, W., Bakker, S., Maat, K., van Wee, B., 2014. The influence of finan-
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cial incentives and other socio-economic factors on electric vehicle adoption. Energy Policy 68, 183–194. Szakaly,
S.,
America’s
Manzi,
P.,
franchised
2015.
new-car
Annual
financial
dealerships.
profile
of
Available
at
EP
https://www.nada.org/WorkArea/DownloadAsset.aspx?id=21474839497, Accessed July 10, 2018. Government,
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UK
proval
(IVA):
2017.
model
Individual
reports
for
cars.
Vehicle Available
Apat
https://www.gov.uk/government/publications/publicivamodelreports, Accessed February 18, 2018.
UK Government, 2018. National Minimum Wage and National Living Wage rates. Available at https://www.gov.uk/national-minimum-wage-rates, Accessed February 18, 2018.
29
ACCEPTED MANUSCRIPT
UK Power, 2017.
Gas Electricity Tariff Prices per kWh.
Available at
https://www.ukpower.co.uk/homeenergy/tariffsperunitkwh, Accessed Febru-
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EP
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SC
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ary 18, 2018.
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Appendix I. The proof for the possibilities of the relationship between UCgv and
We rewrite Ugv and Uev as shown below: Ugv (k) = K0 k − Cgv − pCgv
(39)
(40)
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Uev (k) = K1 k − Cev − pCev + sc
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UCev
As 0 < k < 1 and Ugv (k), Uev (k) are straight line, we first classified the rela-
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tionship based on whether there is an intersection between them in 0 < k < 1. It is easy to be found that, if (Ugv (0) − Uev (0))(Ugv (1) − Uev (1)) > 0, there is no intersection between Ugv (k) and Uev (k). As shown in Case I and Case VI below.
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In Case I, we have UCgv > UCev in 0 < k < 1 and in Case VI, UCev > UCgv .
Pnull
Pgv
Case I
Pnull
Pev Case VI
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Then, if the intersection existed in 0 < k < 1, there are two situations, which is whether the value of the intersection is positive. It is easy to find that the intersection can be achieved when k ∗ =
Cgv +pCgv −Cev −pCev +sc . K0 −K1
when
AC C
K0 k ∗ −Cgv −pCgv ≤ 0, the value for the intersection is not positive (Case II and V) otherwise the the value is positive, as shown in Case III and IV. Furthermore, in Case II and III, K0 > K1 while in Case V and IV, we have K0 < K1 .
1
0 Pnull
Pgv
Case II
Pnull
Pnull Pev Pgv
Pev
Case III
Case V
31
Pnull Pgv Pev Case IV
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II. The proof of purchase probability in the early development stage The early stage should satisfy the conditions below:
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K0 > K1 pCgv > 0
pCev − sc > 0 (Cev + pCev − sc − Cgv − pCgv )(K0 − pCgv − Cgv − K1 + Cev + pCev − sc ) < 0 p +C +s −p −C 0 < Cgv gv c Cev ev < 1 (41)
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The schematic diagram is shown below.
SC
K0 −K1
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Pnull Pev Pgv
Early stage
EP
Fig. 11. Possibility of relationships between UCgv and CCev
AC C
The cross point between UCgv and UCev is
k=
pCgv + Cgv + sc − Cev − pCev K0 − K1
(42)
The cross point between UCev and U = 0 is
k=
pCev + Cev − sc K1
(43)
ev −sc Therefore, people with intention [0, pCev +C ] will not buy a car. People K1 ev −sc with intention [ pCev +C , K1
pCgv +Cgv +sc −Cev −pCev ] K0 −K1
32
will choose to buy an EV
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and the remaining customer will buy a GV. So, we get
Pev =
Cgv − Cev + pCgv + sc − pCev K0 − K1
(44)
RI PT
Pgv = 1 −
Cgv − Cev + pCgv + sc − pCev Cev + pCev − sc − K0 − K1 K1
(45)
III. The proof of purchase probability in the later development stage
K1 > K0 pCgv > 0
SC
The later stage should satisfy the conditions below:
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pCev − sc > 0 (Cev + pCev − sc − Cgv − pCgv )(K0 − pCgv − Cgv − K1 + Cev + pCev − sc ) < 0 p +C +s −p −C 0 < Cgv gv c Cev ev < 1 K0 −K1
(46)
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The sketch of this stage is shown below.
Later stage
AC C
EP
Pnull Pgv Pev
Fig. 12. Possibility of relationships between UCgv and CCev
The cross point between UCgv and UCev is
k=
pCgv + Cgv + sc − Cev − pCev K0 − K1
33
(47)
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The cross point between UCgv and U = 0 is
People with intention [0,
pCgv +Cgv ] K0
p +C +sc −Cev −pCev p +C ] [ CgvK0 gv , Cgv gvK0 −K 1
pCgv + Cgv K0
(48)
will not buy a car. People with intention on
will choose to buy a GV and the remaining
customer will buy an EV. Therefore, the probabilities are
Cgv − Cev + pCgv + sc − pCev Cgv + pCgv − K0 − K1 K0 Cgv − Cev + pCgv + sc − pCev K0 − K1
AC C
EP
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M AN U
Pev = 1 −
SC
Pgv =
RI PT
k=
34
(49)
(50)
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IV. The Mathematical expression for the total profit The total profit for the EVSC in the EV early development stage is expressed
K0 S(2Cenvir K1 (K1 − 4K0 ) + 2Cev 4K02 − 3K0 K1 + K12
−8Cgv K0 K1 + 4Cgv K12 − 4K02 K5 + 16K02 pM ev + 2K0 K12
+K12 K3 + 2K12 pM ev + 2K12 pM gv )
SC
+3K0 K1 K5 − 12K0 K1 pM ev − 8K0 K1 pM gv − 2K13
2K1 (K0 −K1 )(K1 −4K0 )2
−4Cev K0 (2Cenvir K1 (4K0 − K1 ) + 2Cgv K1 (8K0 − 3K1 )
− − − + 4K0 K1 K3 3 +3K0 K1 K5 + 12K0 K1 pM ev + 8K0 K1 pM gv + 4K1 2 2 2 2 −2K1 K3 − K1 K5 − 2K1 pM ev − 2K1 pM gv ) 2 −4Cenvir K1 (4K0 − K1 )(2Cgv (K1 − 2K0 ) + 4K0 2 2 +K0 (−4K1 − 2K3 + K5 ) + K1 K3 ) − 36Cgv K0 K1 2 2 2 2 2 +4Cev K0 12K0 − 9K0 K1 + 2K1 + 48Cgv K0 K1 3 3 2 2 2 +8Cgv K1 − 96Cgv K0 K1 + 136Cgv K0 K1 2 −4K1 pM gv (4K0 − K1 )(2Cgv (K1 − 2K0 ) + 4K0 +K0 (−4K1 − 2K3 + K5 ) + K1 K3 ) + 16Cgv K02 K1 K3 2 2 3 −16Cgv K0 K1 K5 − 32Cgv K0 K1 pM ev − 40Cgv K0 K1 2 2 2 −12Cgv K0 K1 K3 + 8Cgv K0 K1 K5 + 8Cgv K0 K1 pM ev 3 4 3 2 3 +4Cgv K1 K3 + 48K0 K1 − 52K0 K1 − 16K0 K1 K3 3 3 2 3 2 3 −32K0 K1 pM ev − 4K0 K5 + 32K0 K5 pM ev − 4K0 K1 2 2 2 2 2 2 2 2 +28K0 K1 K3 + 4K0 K1 K5 + 40K0 K1 pM ev − 4K0 K1 K3 −16K02 K1 K3 pM ev + 3K02 K1 K52 − 24K02 K1 K5 pM ev 3 4 3 3 +8K0 K1 − 12K0 K1 K3 − 4K0 K1 K5 − 8K0 K1 pM ev 2 2 2 2 +3K0 K1 K3 + 2K0 K1 K3 K5 + 4K0 K1 K3 pM ev 2 +4K0 K1 K5 pM ev
+8K02 K1
4K02 K5
16K02 pM ev
12K0 K12
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− = +
AC C
EP
Πtotal
K02 S 2 (3K1 −4K0 ) 2 2α 2K 1 (K0 −K1 )(K1 −4K0 )
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RI PT
as
α
8K1 (K0 −K1 )(K1 −4K0 )2
(51)
35
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And the total profit for the EVSC in the EV later development stage is expressed as
K1 S 2 (4K1 −3K0 ) α2 2 2(K 0 −4K1 ) (K0 −K1 )
SC
M AN U
AC C
EP
TE D
Πtotal
RI PT
2 2 2Cenvir K1 (K0 − 4K1 ) + 2Cev K0 − 3K0 K1 + 4K1 2 2 2 2 +4C K K − 8C K − 6K K + 2K p + 14K K gv 0 1 gv 1 1 M ev 0 1 0 0 S +K K K + 3K K K − 12K K p 0 1 3 0 1 5 0 1 M ev 3 2 2 +2K1 pM gv (K0 − 4K1 ) − 8K1 − 4K1 K5 + 16K1 pM ev + α 2(K0 −4K1 )2 (K0 −K1 ) 4Cev K0 (2Cenvir K1 (K0 − 4K1 ) + 2Cgv K1 (3K0 − 8K1 ) −10K02 K1 + K02 K5 + 2K02 pM ev + 34K0 K12 + 2K0 K1 K3 −3K0 K1 K5 − 12K0 K1 pM ev + 2K1 pM gv (K0 − 4K1 ) − 24K13 −4K 2 K3 + 4K 2 K5 + 16K 2 pM ev ) + 4Cenvir K1 (K0 − 4K1 ) 1 1 1 2 · 2Cgv (K0 − 2K1 ) − 2K0 + K0 (2K1 + K3 + K5 ) − 2K1 K3 2 2 2 2 2 2 2 = +4Cev K0 2K0 − 9K0 K1 + 12K1 + 8Cgv K0 K1 − 36Cgv K0 K1 2 +48Cgv K13 − 16Cgv K03 K1 + 48Cgv K02 K12 + 4K1 pM gv (K0 − 4K1 ) 2 · 2Cgv (K0 − 2K1 ) − 2K0 + K0 (2K1 + K3 + K5 ) − 2K1 K3 +4Cgv K02 K1 K3 + 8Cgv K02 K1 K5 + 8Cgv K02 K1 pM ev −12C K K 2 K − 16C K K 2 K − 32C K K 2 p gv 0 1 3 gv 0 1 5 gv 0 1 M ev +16C K 3 K + 8K 4 K − 4K 3 K 2 − 4K 3 K K − 12K 3 K K gv 1 3 0 1 0 1 0 1 3 0 1 5 3 3 2 3 2 2 −16K0 K1 pM ev + 4K0 K5 pM ev − 52K0 K1 + 4K0 K1 K3 +28K02 K12 K5 + 80K02 K12 pM ev + 2K02 K1 K3 K5 + 4K02 K1 K3 pM ev +3K02 K1 K52 c − 24K02 K1 K5 pM ev + 48K0 K14 − 16K0 K13 K5 −64K0 K13 pM ev + 3K0 K12 K32 − 16K0 K12 K3 pM ev − 4K0 K12 K52 +32K0 K12 K5 pM ev − 4K13 K32 − 32Cgv K0 K13 − 8K0 (K0 −4K1 )2 (K0 −K1 ) (52)
36
S0925-5273(19)30029-5
DOI:
https://doi.org/10.1016/j.ijpe.2019.01.021
Reference:
PROECO 7272
To appear in:
International Journal of Production Economics
Received Date: 2 May 2018 Revised Date:
2 November 2018
Accepted Date: 15 January 2019
Please cite this article as: Gu, X., Zhou, L., Ieromonachou, P., An imperfect information game in subsidising the electric vehicle supply chain, International Journal of Production Economics (2019), doi: https://doi.org/10.1016/j.ijpe.2019.01.021. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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An Imperfect Information Game in Subsidising the Electric Vehicle Supply Chain
RI PT
Xiaoyu Gua,∗, Li Zhoub , Petros Ieromonachoub a School
SC
of Economics and Management, Nanjing University of Science and Technology, Nanjing, P.R.China b Department of Systems Management and Strategy, University of Greenwich, London, United Kingdom
Abstract
This paper studies a four-echelon vehicle supply chain consisting of government,
M AN U
electric/gasoline vehicle manufacturer, retailer and consumer. The purpose is to understand how government subsidies should be allocated in order to maximise total profit of the whole supply chain. By adopting Stackelberg game theory based on conditions of imperfect information, a mathematical model was developed. The results suggest that allocation of a subsidy in the electric vehicle supply chain should first be allotted for electric vehicle customers. Specifically,
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in the early development stage, if the subsidy budget is limited, all of them should be given to the purchasers of electric vehicle customer. With an increasing budget available for subsidies, more allocation to the electric vehicle manufacturer is expected. However, more subsidies does not necessarily lead to
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more electric vehicle purchases as there is a ceiling on the market for electric vehicles. In the later development stage, subsidies may not be important in
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promoting electric vehicle uptake. Keywords: Electric vehicle, Supply chain, Energy price, Subsidy policy, Stackelberg game, Imperfect information
∗ Corresponding
author Email addresses: [email protected] (Xiaoyu Gu), [email protected] (Li Zhou), [email protected] (Petros Ieromonachou)
Preprint submitted to International Journal of Production Economics
January 16, 2019
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1. Introduction Electric vehicles (EVs) are considered a significant option for solving envi-
RI PT
ronmental issues in transportation and the automotive industry, in particular. However, for a multitude of reasons, including limited mileage before needing
a recharge, and underdeveloped charging network, and lengthy charging times, EVs have not yet been widely accepted by the customer. Specifically, according to Cazzola et al. (2017), by the end of 2016, the EV market share was 1.41%
SC
in the UK, 1.46% in France, 0.71% in Germany, 0.91% in the US, and 1.37% in China. As a whole, the worldwide EV market share is only 1.10%. To overcome
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this situation and with the purpose of green and sustainable development along with EV industry promotion, governments have taken some actions, such as subsidising the electric vehicle supply chain (EVSC) (Abkemeier et al., 2012). There have been some studies focusing on EVSC incentive policies. For example, Gnann et al. (2015) used alternative automobile diffusion and infrastructure models to analyse the market evolution of EVs in Germany up to 2020. They concluded that fuel prices have a notable impact on the evolution of EV
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market. Using linear regression to analyse 30 national EV markets in 2012, Sierzchula et al. (2014) concluded that although financial subsidies and a charging infrastructure are positively correlated to EV sales, they were not enough to ensure high EV diffusion rates. What is more, in regard to the environ-
EP
mental impact, Shin et al. (2012) conducted a case study in South Korea and found that governmental purchase price subsidies had more benefits than tax
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incentives and were more helpful for environmental improvements. In addition to the case study, Huang et al. (2013) as well as Luo et al.
(2014) studied government subsidy and incentive schemes. Huang et al. (2013) analysed the subsidy incentive policy in a duopoly market comprising a gasoline automobile supply chain and the electric-fuel hybrid automobile supply chain. A Nash equilibrium of the wholesale price in the above two supply chains is achieved when the government provides subsidies to promote the EV market based on a information condition. The conclusion was that subsidy incentives were able to help increase EV sales; additionally, it reduced the negative impacts 2
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of the gasoline vehicles (GVs) in terms of environmental hazards. Furthermore, based on this research, Luo et al. (2014) focused on the EV supply chain using a model where the customer, manufacturer, and government were all taken
RI PT
into consideration and the best EV rate and subsidy ceiling were figured out
in determining the Nash equilibrium through a perfect information cooperative bargaining game. The result showed that in order to increase EV sales, the
government’s subsidy policy likely needed to include both a price discount and
SC
a subsidy ceiling.
Based on the reviewes of the literature, including the articles discussed above, several research gaps can be identified. First, models for offering EV incentives
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have been developed for the supply chain that include the involvement of governments, EV manufacturers and customers (Luo et al., 2014). However, an important part of the SC, the retailer, has been rarely considered. Second, although there is existing research on subsidy allocation in other industries (e.g., Internet infrastructure, see Chen et al. (2016)), in the case of EVSC, subsidy allocation must reflect the unique characteristics of EVSC; hence, it is neces-
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sary to develop new models. Third, the majority of the research that has been conducted on the EVSC are focused on the influence of EV purchase cost (e.g., Shin et al. (2012)), but the impact on subsidy allocation of energy costs, such as fuel for GVs, and for EVs, the costs of electricity and charging time have
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been studied less, even though these factors indeed affect optimal allocation of government subsidies. And fourth, most of the research discusses the supply chain model under perfect information conditions (for example, Cai et al. (2014)
AC C
and Bulmus et al. (2014)). However, in reality, some information is private, for example, the vehicle manufacturing cost, which makes the information imperfectly transparent in the case of the EVSC. These gaps are what motivated our study on this topic. In order to fill the gaps just discussed, this research proposed a four-tier
model comprising government, GV/EV manufacturers, GV/EV retailers and vehicle buyer based on the game scheme of the Stackelberg imperfect information game. In particular, this study attempted to answer the following questions:
3
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First, with regard to a given subsidy budget, what is the best way to determine an optimal allocation across the EVSC? Second, how different subsidy budgets
and the customer’s decision to buy a car?
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affect this allocation? Third, how do energy costs affect the subsidy allocation
The rest of this paper is laid out as follows: Section 2 discusses formulation of the structure of our EVSC model, including a discussion of the using cost over the life of the vehicle. With the given six EV development stages identified
SC
and described here, we studies two of the stages, the early EV development
stage and the later development stage. Sections 3 and 4 presents the analysis of these two stages. Then, the purchasing probability and the optimal subsidy
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allocation rate are determined. Section 5 provides the analysis with numerical experiment. Finally, section 6 gives the conclusions of the research along with its limitations.
2. Model description and assumptions
Fig. 1 illustrates the structure of a four-echelon vehicle supply chain. It is
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common practice for vehicle sales and after-sales service to be franchised to companies such as AutoNation in the USA (Szakaly and Manzi, 2015) and ZhengtongAuto in China. Our model takes into consideration GV/EV manufacturers and retailers, where the EV retailer is franchised by the EV manufacturer, and
EP
the GV retailer sells GVs. Customers are able to choose to buy either a GV or an EV. For example, Chevrolet manufactures GV and its retailer sells GV only.
AC C
Tesla produces EV and its retailer sells EV only. And customer can choose to buy either Chevrolet or Tesla. The government, as the entity responsible for setting incentive policy and overseeing the entire supply chain, is the leader, in terms of Stackelberg game theory.
4
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6XEVLG\WR(9PDQXIDFWXUHU
(9 0DQXIDFWXUHU
(95HWDLOHU &XVWRPHU
*9 0DQXIDFWXUHU
*95HWDLOHU
RI PT
*RYHUQPHQW
SC
6XEVLG\WR(9FXVWRPHU
Fig. 1. Structure for the subsidy model
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In reality, GV/EV manufacturers produces vehicles and sell them to their franchised retailers, who do not know the accurate vehicle manufacturing cost, but they know the approximate range of the cost. This information about manufacturing cost, together with the wholesale price the retailer paid are taken into consideration when the retailer sets the vehicle selling price to customers. Customers are rational, and they consider buying or not buying a vehicle that
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will bring them the maximum utility by taking into consideration their own purchase intention, the vehicle price, the subsidy, and costs. The notations used in this study to describe the model include the input, the intermediate variables and the decision variables are shown in Table 1. In
EP
order to simplify the calculation process, we have also defined the substitutions in table 1 as well.
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Table 1. Notations and substitutions
Input parameters k
Cenvir
Vehicle purchase intention
M
The entire mileage vehicle will be driven over life cycle (km)
The environmental cost of
pf uel
Fuel price (£/L)
Ef uel
Fuel efficiency (km/L)
the GV pelec
Electricity costs (£/kWh)
5
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Eelec
Electricity
efficiency
vt
Time value (£/h)
tchrg
Time cost for each electric-
tf uel
Time cost for each fuel refilling (h)
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(km/kWh)
ity recharging (h)
Fuel tank volume (L)
Vev
EV battery volume (kWh)
[pM ev ,
Lower and upper bound for
[pM gv ,
Lower and upper bound for
pM ev ]
the EV manufacturing cost
pM gv ]
the GV manufacturing cost
[pRev ,
Lower and upper bound for
[pRgv ,
Lower and upper bound for
pRev ]
the EV price to retailer
pRgv ]
the GV price to retailer
Agv
GV use benefit
Aev
EV use benefit
S
The budget for all subsidies
Cf uel
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Intermediate variables
Fuel costs over the vehicle’s life cycle (£)
SC
Vgv
Celec
Electricity costs over the vehicle’s life cycle (£)
Time cost for the EV (£)
Cgtime
Time cost for the GV (£)
Cgv
GV use cost for the whole
Cev
EV use cost for the whole
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Cetime
life cycle
life cycle
Ugv
Utility of the GV
Uev
Utility of the EV
pCgv
GV selling price to the cus-
pCev
EV selling price to the cus-
pRgv
EP
tomer
GV selling price to the re-
tomer pRev
tailer
tailer
Customer’s probability of
Pgv
AC C
Pev
EV selling price to the re-
buying the EV
Customer’s probability of buying the GV
ΠRgv
Profit for the GV retailer
ΠRev
Profit for the EV retailer
ΠM gv
Profit for the GV manufac-
ΠM ev
Profit for the EV manufac-
turer
turer
Ucustomer Customer’s utility in using
Πentire
the vehicle
The entire profit of supply chain
6
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Subsidy given to the EV
sm
customer fre (x)
Probability
Subsidy given to the EV manufacturer
distribution
frg (x)
PDF of GV price charged
function (PDF) of EV price
to the retailer by GV man-
charged to the retailer by
ufacturer
EV manufacturer fme (x)
PDF of EV manufacturing
fmg (x)
PDF of GV manufacturing cost
SC
cost Decision variables Subsidy partition ratio
Substitutions K0
Agv + Cgv
K2
(pRgv −pRgv ) 2(pM gv −pM gv )
K4
(pRev −pRev ) 2(pM ev −pM ev )
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α
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sc
K1
Aev + Cev
K3
pRgv + pRgv
K5
pRev + pRev
2.1. EV and GV cost over the whole lifecycle
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According to Greene et al. (2004), in the span of the vehicle’s lifetime, the cost is equal to the total mileage multiplied by the unit cost of energy divided by the energy’s efficiency. Hence, the cost of fuel for the GV and the cost of
AC C
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electricity for the EV are, respectively
Cf uel =
pf uel M Ef uel
(1)
Celec =
pelec M Eelec
(2)
According to Shafiei et al. (2012), charging time is one of the main factors customers consider when buying an EV, but it is not a key factor for the GV. Therefore, as a primary difference between GVs and EVs, in this study, we took this into consideration and quantified the ratio of the cost in time to the cost
7
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in money using Eq. 3 and Eq. 4. M Vgv Ef uel
Cetime = vt tchrg
M Vev Eelec
(3)
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Cgtime = vt tf uel
(4)
as Cgv = Cf uel + Cetime
(5) (6)
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Cev = Celec + Cgtime
SC
In summary, the GV/EV use cost over the vehicle’s life cycle can be expressed
2.2. Customer’s choice probability
Different people have different intentions when buying a vehicle, and based on Shafiei et al. (2012) and Helveston et al. (2015), we assumed that customers consider a vehicle’s use benefits, the cost of use, the vehicle price, and the amount of subsidies to make a vehicle buying decisions. As different people have
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different sensitivity to buy a vehicle, we used k, k ∈ [0, 1] to denote the vehicle purchase intention. According to Morwitz et al. (2014), the utility function is defined as the use benefit multiplied by the residual purchase intention minus the use cost multiplied by the purchase intention. Thus, the utility functions
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for GV and EV users can be represented by Eq. 7 and Eq. 8:
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Ugv (k) = Agv k − Cgv (1 − k) − pCgv = K0 k − Cgv − pCgv
Uev (k) = Aev k − Cev (1 − k) − pCev + sc = K1 k − Cev − pCev + sc
(7)
(8)
There are 6 relationships between Ugv and Uev which are shown in Fig. 2 below:
8
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Case I
Pgv
Pnull Pev Pgv
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Pnull
Pgv
Case III
Case II
1
0 Pnull Pgv Pev
Pnull
Case IV
Pev
Pnull
Pev
Case VI
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Case V
SC
Pnull
Fig. 2. Relationships between UCgv and UCev A discussion of how these six relationships are mutually exclusive and collectively exhaustive can be found in appendix I.
As can be seen, in Case I and Case II, all buyers would select a GV, and in Case V and Case VI, all buyers would select an EV. This means that the market
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is dominated by GVs in Cases I and II and dominated by EVs in Cases V and VI. In Cases III and IV, different people with different purchasing intentions would purchase different vehicles, so both GVs and EVs would exist in the market. For this study, only the mixed selection cases were discused (Cases III and IV).
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In Case III, the use profit for EVs is less than that for the GVs while in Case IV, the use profit for EVs is greater than that for GVs. We defined Case III
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as the early development stage, and Case IV as the later development stage. Section 3 discusses the early stage and Section 4 discusses the later stage.
3. EV early development stage 3.1. Purchase probability and customer’s utility In this stage, the GV purchasing probability is calculated as shown in Eq. 9. Pgv = 1 −
Cgv − Cev + pCgv + sc − pCev K0 − K1
9
(9)
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The probability of buying an EV is calculated as shown in Eq. 10. Cgv − Cev + pCgv + sc − pCev Cev + pCev − sc − K0 − K1 K1
The proof of Pgv and Pev can be found in Appendix II.
(10)
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Pev =
The utility for the customer is calculated as shown below. Cgv −Cev +pCgv−pCev+sc K0−K1
Z
=
1
Cgv −Cev +pCgv−pCev+sc K0−K1
Ugv dk
SC
Cev +pCev −sc K1
Z Uev dk +
2 2 Cev K0 − 2Cev (K1 (Cgv + pCgv ) + K0 (sc − pCev )) + Cgv K1
+2Cgv K1 (−K0 + K1 − pCev + pCgv + sc ) + K02 K1 −K0 K12 − 2K0 K1 pCgv + K0 p2Cev − 2K0 pCev sc + K0 s2c +2K12 pCgv − 2K1 pCev pCgv + K1 p2Cgv + 2K1 pCgv sc
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Ucustomer =
2K1 (K0 − K1 )
(11)
3.2. Profits for the retailer and manufacturer
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Generally, with the probability distribution function (PDF) fre (x), frg (x), fme (x), fmg (x) and a given price for a vehicle sold by the manufacturer to the retailer, the probability for vehicle manufacturing cost can be found with Bayes’ Theorem:
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and
frg (pRgv |pM gv )fmg (pM gv ) P (pRgv |pM gv )P (pM gv ) = R +∞ P (pRgv ) frg (pRgv |pM gv )fmg (pM gv )dpM gv 0 (12)
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P (pM gv |pRgv ) =
P (pM ev |pRev ) =
fre (pRev |pM ev )fme (pM ev ) P (pRev |pM ev )P (pM ev ) = R +∞ P (pRev ) f re (pRev |pM ev )fme (pM ev )dpM ev 0 (13)
In this model, GV/EV vehicle manufacturers know their vehicles’ manufacturing costs (pM gv /pM ev ). Based on the manufacturing cost, they are able to decide the selling price to the retailer. Then, the GV/EV retailers, decide on a price for consumers (pCgv and pCev ) based on the price they paid (pRgv and pRev ) for the vehicle. As a matter of fact, the vehicle retailer does not know the vehicle 10
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manufacturing cost, but they can estimate the range of vehicle production costs. We assumed that GV/EV manufacturing costs are uniformly distributed within the range [pM gv , pM gv ] for GV manufacturing costs and [pM ev , pM ev ] for EV
RI PT
manufacturing costs. With the manufacturing cost range, the selling price to a
retailer is assumed to be uniformly distributed in the range of [pRgv , pRgv ] for the GV and [pRev , pRev ] for the EV. Therefore, the profits for the GV and EV retailers can be calculated using Eq. 14 and Eq. 15. +∞
Pgv (pCgv − pRgv )P (pM gv |pRgv )dpRgv
(14)
Pev (pCev − pRev )P (pM ev |pRev )dpRev
(15)
SC
Z ΠRgv = 0
0
M AN U
+∞
Z ΠRev =
And with the uniform distribution, the profit for a GV/EV retailer can be updated to the two equations below. Z
pRgv
Pgv (pCgv − pRgv )
ΠRgv = pRgv
(pRgv − pRgv )
=
Pgv (2pCgv − pRgv − pRgv )
TE D
2(pM gv − pM gv )
1 dpRgv pM gv − pM gv
(16)
EP
= K2 Pgv (2pCgv − K3 ) 2K2 p2Cgv K2 K3 (−Cev +Cgv −K0 +K1 −pCev +sc ) + − K0 −K K0 −K1 1 = K2 pCgv (2Cev −2Cgv +2K0 −2K1 +K3 +2pCev −2sc ) + K0 −K1 Z
pRev
Pev (pCev − pRev )
ΠRev =
pRev
1 dpRev pM ev − pM ev
(pRev − pRev ) Pev (2pCev − pRev − pRev ) 2(pM ev − pM ev )
AC C =
(17)
= K4 Pev (2pCev − K5 ) 2K K p2Cev K4 K5 (Cev K0 −K1 (Cgv +pCgv )−K0 sc ) − K00K14−K + 2 K1 (K0 −K1 ) 1 = K4 pCev (K0 (−2Cev +K5 +2sc )+2K1 (Cgv +pCgv )) + K1 (K0 −K1 )
Because K0 − K1 > 0, K0 > 0, K2 and K4 > 0 and K1 > 0, the maximum profit for the GV retailer and EV retailer will be achieved when
11
∂ΠRgv ∂pCgv
= 0 and
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= 0, as shown below
K2 (2Cev −2Cgv +2K0 −2K1 +K3 +2pCev −2sc ) K0 −K1 K4 (K0 (−2Cev +K5 +2sc )+2K1 (Cgv +pCgv )) K1 (K0 −K1 )
−
−
4K2 pCgv K0 −K1
4K0 K4 pCev K0 K1 −K12
=0
(18)
=0
RI PT
∂ΠRev ∂pCev
And the optimal GV/EV selling price from the retailer to the customer is shown below.
2Cev K0 +2Cgv (K1 −2K0 )+K0 (4K0 −4K1 +2K3 +K5 −2sc ) 8K0 −2K1
p∗Cgv =
p∗ = Cev
2Cev (K1 −2K0 )+2Cgv K1 +2K0 K1 +2K0 K5 +4K0 sc −2K12 +K1 K3 −2K1 sc 8K0 −2K1
SC
(19)
K −4C
2C
P∗ ev
M AN U
Hence, the optimal GV/EV purchasing probability is as shown in Eq. 20 below. K +2C
K +4K 2 −4K K −2K K +K K −2K s +K K
gv 0 gv 1 0 1 0 3 0 5 0 c 1 3 ∗ 0 = ev 0 Pgv 8K02 −10K0 K1 +2K12 K0 (2Cev (K1 −2K0 )+2Cgv K1 +2K0 K1 −2K0 K5 +4K0 sc −2K12 +K1 K3 +K1 K5 −2K1 sc ) = 2K1 (K0 −K1 )(4K0 −K1 ) (20)
The manufacturer profits are the vehicle purchase probability multiplied by the revenue from the sale of the vehicle to the retailer, as shown in Eq. 21 for the
TE D
GV manufacturer and Eq. 22 for the EV manufacturer.
(21)
ΠM ev = Pev (pRev − pM ev + sm )
(22)
EP
ΠM gv = Pgv (pRgv − pM gv )
With the optimal selling price and purchase probability, when the govern-
AC C
ment decides on a subsidy, it uses the mathematical expectation of vehicle manufacturing cost and selling price, which are pM gv = pM ev +pM ev , 2
pRgv =
pRgv +pRgv 2
and pRev =
12
pRev +pRev . 2
pM gv +pM gv , 2
pM ev =
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3.3. Total profit for the supply chain The total profit for the supply chain is described as
Πtotal = =
ΠM gv + ΠRgv + ΠM ev + ΠRev +Ucustomer − Pgv Cenvir − Pev S ∗ Pgv (pCgv
RI PT
− pM gv − Cenvir )
∗ +Pev (pCev
− pM ev + sm − S) + Ucustomer
(23)
SC
The complete expression can be found in Eq. 51 in Appendix IV. The target for the government is to maximise total profits for the EVSC. As K0 > K1 , 3K1 − K02 S 2 (3K1 −4K0 ) 2K1 (K0 −K1 )(K1 −4K0 )2
get the maximum profit (Πtotal ), α should satisfy
∂Πtotal ∂α
= 0. Therefore,
8Cenvir K0 K1 − 2Cenvir K12 − 8Cev K02 + 6Cev K0 K1 − 2Cev K12
+8Cgv K0 K1 − 4Cgv K12 + 4K02 K5 − 16K02 pM ev − 2K0 K12 −3K0 K1 K5 + 12K0 K1 pM ev + 8K0 K1 pM gv + 2K13 − K12 K3 −2K12 pM ev − 2K12 pM gv
2K0 S(4K0 − 3K1 )
TE D
α∗ =
< 0, and in order to
M AN U
4K0 < 0. Therefore, it is easy to get
(24)
In addition, because α is the partition ratio, it needs to meet the condition
EP
0 < α < 1 as well.
4. EV later development stage
AC C
4.1. Purchase probability and customer’s utility In this stage, the purchase probability is determined using Eq. 25 for the
GV and Eq. 26 for the EV.
Pgv =
Cgv − Cev + pCgv + sc − pCev Cgv + pCgv − K0 − K1 K0
Pev = 1 −
Cgv − Cev + pCgv + sc − pCev K0 − K1
13
(25)
(26)
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The proof of probabilities is shown in appendix III. The utility function for the customer is determined as shown below.
Ucustomer =
Cgv +pCgv K0
Z Ugv dk +
1 Cgv −Cev +pCgv +sc −pCev K0 −K1
2 C K − 2C K (C − K + K − p + p + s ) 0 ev 0 gv 0 1 Cev Cgv c ev +C 2 K1 + 2K0 sc (Cgv − K0 + K1 − pCev + pCgv ) gv −2C K p 2 2 + 2C K p − K K + 2K p gv 0 Cev gv 1 Cgv 0 1 0 Cev 2 2 +K0 K1 − 2K0 K1 pCev + K0 pCev − 2K0 pCev pCgv +K0 s2c + K1 p2Cgv
SC
−2K0 (K0 − K1 )
M AN U
=
Uev dk
RI PT
Cgv −Cev +pCgv +sc −pCev K0 −K1
Z
(27)
4.2. Profits for the retailer and manufacturer
Similar to Case I, the profits for the GV and EV retailers are as shown in Eq. 28 and Eq. 29 for the GV retailer and EV retailer, respectively. +∞
Z
Pgv (pCgv − pRgv )P (pM gv |pRgv )dpRgv
ΠRgv =
(28)
TE D
0
Z
+∞
Pev (pCev − pRev )P (pM ev |pRev )dpRev
ΠRev =
(29)
0
EP
With uniform distribution, Z
pRgv
Pgv (pCgv − pRgv )
ΠRgv =
pRgv
AC C
=
(pRgv − pRgv )
2(pM gv − pM gv )
1 dpRgv pM gv − pM gv
Pgv (2pCgv − pRgv − pRgv ) (30)
= K2 Pgv (2pCgv − K3 ) K p (2K (C +pCev −sc )+K1 (K3 −2Cgv )) − 2 Cgv 0 evK0 (K 0 −K1 ) = 2K1 K2 p2Cgv K2 K3 (K0 (Cev +pCev −sc )−Cgv K1 ) + + 2 K0 (K0 −K1 ) K −K0 K1 0
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Z
pRev
Pev (pCev − pRev )
ΠRev = pRev
=
1 dpRev pM ev − pM ev
(pRev − pRev ) Pev (2pCev − pRev − pRev ) 2(pM ev − pM ev )
=
RI PT
(31) = K4 Pev (2pCev − K5 )
K4 pCev (2Cev −2Cgv +2K0 −2K1 −K5 −2(pCgv +sc )) K0 −K1
K K (−Cev +Cgv −K0 +K1 +pCgv +sc ) + 4 5 K0 −K1
+
2K4 p2Cev K0 −K1
achieved when
= 0 and
4K1 K2 pCgv K02 −K0 K1
−
∂ΠRev ∂pCev
= 0, which is
K2 (2K0 (Cev +pCev −sc )+K1 (K3 −2Cgv )) K0 (K0 −K1 )
K4 (2Cev −2Cgv +2K0 −2K1 −K5 −2(pCgv +sc )) K0 −K1
M AN U
∂ΠRgv ∂pCgv
SC
As K0 − K1 < 0, K0 > 0, K1 > 0, K2 > 0 and K4 > 0, the optimal profit is
So,
+
4K4 pCev K0 −K1
=0
(32)
=0
p∗ = − 2Cev K0 +2Cgv K0 −4Cgv K1 −2K02 +2K0 K1 +K0 K5 −2K0 sc +2K1 K3 Cgv 2K0 −8K1 p∗ = − 2Cev (K0 −2K1 )+K1 (2Cgv −4K0 +4K1 +K3 +2K5 )−2sc (K0 −2K1 ) Cev 2(K0 −4K1 )
(33)
∗ = Pgv
K1 (2Cev K0 +2Cgv (K0 −2K1 )−2K02 +2K0 K1 +K0 K3 +K0 K5 −2K0 sc −2K1 K3 ) 2K0 (K0 −4K1 )(K0 −K1 )
2Cev K0 −4Cev K1 +2Cgv K1 −4K0 K1 +K0 K5 −2K0 sc +4K12 +K1 K3 −2K1 K5 +4K1 sc 2K02 −10K0 K1 +8K12
(34)
EP
P∗ = ev
TE D
And the GV/EV purchasing probabilities are
Similar to the early stage, the profits for the GV manufacturer and EV manu-
AC C
facturer are:
ΠM gv = Pgv (pRgv − pM gv )
(35)
ΠM ev = Pev (pRev − pM ev + sm )
(36)
4.3. Total profit for the supply chain Again, we used the average vehicle manufacturing cost and price strategy,
which is pM gv =
pM gv +pM gv , 2
pM ev =
pM ev +pM ev , 2
15
pRgv =
pRgv +pRgv 2
and pRev =
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The target for the government is to maximise the EVSC profit. Πtotal = =
Pgv (ΠM gv + ΠRgv ) + Pev (ΠM ev + ΠRev ) +Ucustomer − Pgv Cenvir − Pev S ∗ Pgv (pCgv − pM gv − Cenvir ) ∗ +Pev (sm + pCev − pM ev − S) +Ucustomer
RI PT
pRev +pRev . 2
(37)
seen,
K1 S 2 (4K1 −3K0 ) 2(K0 −4K1 )2 (K0 −K1 )
SC
The complete expression can be refereed from Eq. 52 in Appendix IV. As can be < 0, because, in this case, K1 > K0 .. In order to get ∂Πtotal ∂α
= 0. Therefore, the optimal subsidy
M AN U
the optimal profit, α should meet
allocation ratio can be shown in Eq. 38.
+4Cgv K0 K1 − 8Cgv K12 − 6K02 K1 + 2K02 pM ev + 14K0 K12 +K0 K1 K3 + 3K0 K1 K5 − 12K0 K1 pM ev + 2K0 K1 pM gv − 8K13 −4K12 K5 + 16K12 pM ev − 8K12 pM gv
6K0 K1 S − 8K12 S
TE D
α∗ =
2Cenvir K0 K1 − 8Cenvir K12 + 2Cev K02 − 6Cev K0 K1 + 8Cev K12
(38)
Moreover, α should satisfy 0 < α < 1 as well.
EP
5. Analysis and discussion
This section analyses the parameters and the optimal profit. We first intro-
AC C
duce a numerical experiment to address the model, then discuss the relationships between the amount of the subsidy, energy prices, charging time, and the trends for subsidy allocation policies, vehicle purchase probabilities and the total profit for the EVSC.
The average mileage driven per year and the average vehicle life cycle can be
found in UK Government (2017), and the prices of energy and energy efficiencies are shown in UK Power (2017) and China Automotive Technology and Research Center et al. (2016). The time value is simplified as the hourly wage in the UK (UK Government, 2018).
16
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All the initial values of the parameters are shown in Table 2.
Ef uel = 12.5
Eelec = 7
vt = 8
Tref uel = 0.1
Trechr = 1
Vgv = 60
Vev = 85
Cenvir = 1000
5.1. EV early development stage
RI PT
M = 300000
Table 2. Notations pf uel = 1.6 pelec = 0.15
SC
In the early EV development stage, we set the initial parameters as S = 3000, Agv = 100000, Aev = 70000, and the cost ranges [pM gv , pM gv ] = [6000, 8000],
M AN U
[pRgv , pRgv ] = [7000, 9000], [pM ev , pM ev ] = [8000, 12000], and [pRev , pRev ] = [10000, 14000]. In order to maximise profits along the entire supply chain, the optimal subsidy allocated to the EV manufacturer and customer and the GV/EV purchase probabilities are as shown in Table 3 with asterisks. Table 3. Optimal values in the later development stage α = 0.49
∗ = 0.38 Pgv
∗ = 0.25 Pev
pRgv = 8000
pRev = 12000
p∗Cgv = 28375
p∗Cev = 20217
Π∗total = 30209
TE D
∗
Fig. 3 illustrates the relationships between total subsidy and the optimal subsidy allocation ratio, GV/EV purchase probability, and the total profit for
AC C
EP
the EVSC.
17
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(a) Total subsidy VS Optimal customer's subsidy ratio
1
(b) Total subsidy VS Purchase Probability
0.4
X 1500 Y 0.978
0.38 X 1500 Y 0.381
Purchase probability
0.36 0.8
0.6
0.4
0.34 GV EV
0.32 0.3 0.28
X 1500 Y 0.256
0.26 0.2 0.24 Optimal
0
0.22 500
1000
1500 2000 Total Subsidy 10
3.0209
2500 4
3000
0
1000
(c) Total subsidy VS Total profit X 1500 Y 30208.6
3.0208
3.0207
1500 2000 Total Subsidy
2500
3000
Total profit
3.0206
3.0205
3.0204
3.0203 0
500
M AN U
Total profit
500
SC
0
RI PT
1.2
1000
1500 2000 Total Subsidy
2500
3000
Fig. 3. Total subsidy budget vs. relevant parameters in the early development stage
In order to obtain the optimal total profit for the entire supply chain, Fig.
TE D
3(a)indicates that if the total subsidy budget is low, all the subsidies should be distributed to EV purchasers. Increasing subsidy budgets means that more money should be invested in the EV manufacturer. As shown in Fig. 3(b), the GV purchase probability decreases as the EV purchase probability increases
EP
when the total subsidy increases. There are caps and floors for the probabilities: In the case of an increasing subsidy budget, the EV purchase probability is
AC C
capped at around 0.256 and the GV probability has a floor at 0.381. Moreover, the total profit for the supply chain is capped at 30208.6 no matter how much the total subsidy grows, as shown in Fig 3(c). Therefore, in this case, we suggest that the optimal amount of subsidy should be £1500, with 97.8% of
the subsidy budget distributed to EV purchasers and 2.2% allotted for the EV manufacturers. Therefore, we suggest that if the subsidy budget of subsidy is more than 1500, it would be better to use the remaining budget for EV infrastructure development. Fig. 4 shows the relationships between fuel prices and the parameters of 18
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optimal subsidy ratio, purchase probability, and total profit. With increasing fuel prices, more subsidy should be provided to the EV customer, and when the fuel price is more than 1.52, all subsidy should be distributed to EV customers.
RI PT
In addition, the higher the fuel price, the higher the EV purchase probability and the lower the GV purchase probability. When all the subsidies have been distributed to consumers, the trend of GV/EV purchasing probabilities slows
down. Further, as fuel prices rise, the total profits for the entire supply chain
1.2
SC
decrease. (a) Fuel price VS Optimal customer's subsidy ratio
(b) Fuel price VS Purchase Probability 0.4
X 1.52 Y 0.3587
1 X 1.52 Y 0.9786
Purchase probability
0.35
0.8
0.6
0.4
M AN U
Optimal
0.3
X 1.52 Y 0.2823
0.25
0.2
0 1.3
1.4
1.5
1.6 1.7 Fuel price
3.1
1.8
10
1.9
4
0.2 1.3
2
1.4
1.5
1.6 1.7 Fuel price
GV EV
1.8
1.9
2
(c) Fuel price VS Total profit
Total profit
TE D
3.05
Total profit
3
2.95
2.9
EP
2.85
2.8 1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Fuel price
AC C
Fig. 4. Fuel price vs. relevant parameters in the early development stage Fig. 5 and Fig. 6 illustrate the relationships of, respectively, electricity
prices and charging time with the optimal subsidy ratio, purchase probability, and total profit. It is easy to see that the relationships are similar. When electricity prices and/or charging time increase, more subsidy should be allotted for the EV manufacturers. In addition, consumer desire for buying a GV is even stronger. In this situation, it is likely that more people mill select GVs rather than EVs. Moreover, when EV usage costs increase, the total profits for the EVSC decrease. 19
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1.2
(a) Electricity price VS Optimal customer's subsidy ratio
(b) Electricity price VS Purchase Probability
0.4
X 0.108 Y1
0.38
1
0.6 Optimal
0.4
X 0.108 Y 0.366
0.34 GV EV
0.32 X 0.108 Y 0.290
0.3
RI PT
Purchase Probability
0.36 0.8
0.28 0.26
0.2 0.24 0.22 0.06
0.08
0.1 0.12 0.14 Electricity price
3.09
104
0.16
0.18
0.06
0.08
0.1 0.12 Electricity price
(c) Electricity price VS Total profit Total profit
3.07 3.06 3.05 3.04 3.03 3.02 3.01 3 0.06
M AN U
Total profit
0.16
0.18
SC
3.08
0.14
0.08
0.1 0.12 0.14 Electricity price
0.16
0.18
Fig. 5. Relationship of electricity prices with optimal customer subsidy, purchase
AC C
EP
TE D
probability, and total profits in the early development stage
20
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1.2
(a) Charging time VS Optimal customer's rsubsidy ratio
(b) Charging time VS Purchase probability
0.4
Optimal
X 0.56 Y1
0.38
1
0.6
0.4
X 0.56 Y 0.366
0.34 GV EV
0.32 X 0.55 Y 0.290
0.3 0.28 0.26
0.2 0.24 0 0.2
0.4
0.6
0.8 Charging time
1
10
3.07
1.2 4
0.22 0.2
1.4
0.4
0.6
0.8 Charging time
(c) Charging time VS Total profit Total profit
3.05
1.2
1.4
3.04 3.03 3.02 3.01 3 0.2
0.4
M AN U
Total profit
1
SC
3.06
RI PT
Purchase Probability
0.36 0.8
0.6
0.8 Charging time
1
1.2
1.4
Fig. 6. The relationships of charging time with optimal subsidy, purchase prob-
TE D
ability, and total profits in the early development stage 5.2. EV later development stage
In the later EV development stage, we set the initial parameters as S = 500, Agv = 100000 and Aev = 150000, [pM gv , pM gv ] = [5000, 7000], [pRgv , pRgv ] =
EP
[6000, 8000], [pM ev , pM ev ] = [15000, 40000] and [pRev , pRev ] = [20000, 50000].. Hence, the optimal subsidy and purchase probability were as shown in Table 4.
AC C
Table 4. Optimal values in the later development stage
∗
α =1
∗ Pgv = 0.1
∗ Pev = 0.56
pRgv = 7000
pRev = 35000
p∗Cgv = 88342
p∗Cev = 47167
Π∗total = 44266
The results are shown in Fig. 7 through 10 below:
21
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(a) Total subsidy VS Optimal customer's subsidy ratio
1.2
(b) Total subsidy VS Purchase Probability
0.7 0.6
0.8
0.6
0.4
0.2
0.5 0.4 GV EV
0.3
RI PT
Purchase probability
1
0.2 0.1
Optimal
0
0
0
500
1000
1500 2000 Total Subsidy
2500
104
4.5
3000
0
500
1000
1500 2000 Total Subsidy
(c) Total subsidy VS Total profit
4.48 4.47 4.46 4.45 4.44 4.43
M AN U
Total profit
3000
SC
4.49
2500
Total profit
4.42 4.41 0
500
1000
1500 2000 Total Subsidy
2500
3000
Fig. 7. Total subsidy budget vs. relevant in the later development stage
(a) Fuel price VS Optimal customer's subsidy ratio
1
1
0.8
0.6
0.2
0 1.5
EP
0.4
1.6
0.4 GV EV
0.2
0
1.9 4
2
-0.2 1.5
1.6
1.7 1.8 Fuel price
1.9
2
(c) Fuel price VS Total profit
4.9 4.8 Total profit
AC C
10
0.6
Optimal
1.7 1.8 Fuel price 5
(b) Fuel price VS Purchase Probability
0.8 Purchase probability
TE D
1.2
4.7 4.6 4.5 4.4 Total profit
4.3 1.5
1.6
1.7 1.8 Fuel price
1.9
2
Fig. 8. Fuel price vs. relevant parameters in the later development stage
22
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As can be seen from Fig. 7, with the given input parameters, all the subsidies should be allotted to EV customers. In the later stage, the more subsidy that
total EVSC profit.
RI PT
is made available, the greater the EV purchase probability and the higher the
Fig. 8 shows variations of the relationships between fuel prices and the parameters of the optimal customer subsidy ratio, purchase probability, and
total profit. With increasing fuel prices, more people will select the EV, and
1.2
SC
the total profit for the supply chain will increase as well. (a) Electricity price VS Optimal customer's subsidy ratio
(b) Electricity price VS Purchase Probability
0.7 0.6
0.8
0.6
0.4
0.2
0.5
M AN U
Purchase Probability
1
0.4 0.3
GV EV
0.2 0.1
Optimal
0
0
0.06
0.08
0.1 0.12 0.14 Electricity price 4.7
10
4
0.16
0.18
0.06
0.08
0.1 0.12 0.14 Electricity price
0.16
0.18
(c) Electricity price VS Total profit
TE D
Total profit
4.65
Total profit
4.6
4.55 4.5
EP
4.45 4.4
4.35
0.08
0.1 0.12 0.14 Electricity price
AC C
0.06
0.16
0.18
Fig. 9. Electricity price vs. relevant parameters in the later development stage
23
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(a) Charging time VS Optimal customer's rsubsidy ratio 0.7
(b) Charging time VS Purchase probability
0.6 Purchase Probability
1
0.8
0.6
0.4
0.2
0.5 0.4 GV EV
0.3 0.2 0.1
Optimal
0 0.2
0.4
0.6
0.8 1 Charging time 4.65
1.2
104
0 0.2
1.4
0.4
0.6
(c) Charging time VS Total profit Total profit
1.2
1.4
4.5 4.45 4.4 4.35
0.4
M AN U
Total profit
4.55
4.3 0.2
0.8 1 Charging time
SC
4.6
RI PT
1.2
0.6
0.8 1 Charging time
1.2
1.4
Fig. 10. Charging time vs. relevant in the later development stage
TE D
Fig. 9 and Fig. 10 show the relationships of electricity cost and charging time with the parameters of the optimal customer subsidy ratio, purchase probability, and total profit. As can be seen, higher electricity costs lead to greater GV purchase desire and lower total profits for the EVSC. Therefore, in the later
EP
stage, we can conclude that, subsidies are not as important as in the EV early development stage because by this time, EVs have been generally accepted. In summary, in the EV early development stage, to optimise the entire profit
AC C
for the whole EVSC, if the subsidy budget is low, all subsidies should be allocated to EV buyers, and with increasing subsidy budgets, subsidies should be allocated to EV manufacturers. Governments should bear in mind that with increasing total subsidies, a cap will be reached on EV purchase probabilities, and there will be a floor for GV purchase probabilities, This means that no matter how subsidies are allocated, the EVSC total profit and vehicle purchase probabilities are capped. In terms of energy costs, increasing fuel prices mean that more subsidies should be allocated to EV buyers. However, with increase
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in EV use costs, more subsidies should be allocated to EV manufacturers. In addition, the higher vehicle use costs mean lower EVSC total profit. In the EV later development stage, all subsidies should be allocated to the EV customer.
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More subsidy budget or higher GV use costs will lead to higher total profits but higher EV use costs will result in lower total profits for the EVSC.
Given these results, we are able to conclude that lower EV use costs or higher GV use costs result in greater EV purchase desire, and the government should
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prioritise customers when considering subsidies for the EVSC. In addition, increasing electricity prices and EV charging times mean that it will be necessary to allocate more subsidies to EV manufacturers. The results also suggest that
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energy costs have a significant impact on the subsidy allocation ratio. It is worth noting that our floored and capped EV diffusion rates reflect the findings of Sierzchula et al. (2014), who claimed that even though financial subsidies are correlated with the EV market, they donot ensure high EV diffusion rates. Regarding the optimal subsidy amount, Gnann et al. (2015) suggested that the effective subsidy amount is 1000, and similarly, our study suggests that the op-
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timal subsidy is 1500 in the EV early stage, while in the later stage, subsidies are not quite as important for the EV uptake, as more customers have accepted EVs by that point.
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6. Conclusion
In order to fill the gaps in the study of EVSC incentive design and sub-
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sidy policy, for this study, we designed an imperfect information Stackelberg leader/follower game model that includes government, vehicle manufacturers, retailers and EV customers. Based on the results of our study, this paper discussed the effects on the EVSC of subsidy allocation to EV manufacturers or EV buyers and analysed the impacts of energy prices and EV charging times on the EVSC. With the goal of maximising profits for the entire EVSC, the optimal subsidy allocations to EV customers and manufacturers were derived, and numerical examples proposed, for both early and later EV development stages. In the lit-
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erature, we found that (1) EV incentive models have been previously developed for a supply chain comprising governments, EV manufacturers, and customers, but for our model, we added the missing component of the EV retailer; (2)
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Research on subsidy allocation has been conducted for other industries, but our
research focused solely on the EV industry and its unique characteristics discussed earlier; and (3) There are a number of existing studies on the EVSC, but they assume perfect information conditions, while our model uses an imperfect
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information condition to more closely reflect reality. Numerical examples are proposed on both stages.
Based on our results, the research questions were answered as follows: First,
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with a given budget for subsidising the EVSC, how can the optimal allocation be determined? In order to maximise the total profits throughout the EVSC, the subsidy allocation ratio for both early and later stages are determined, and the subsidy policy should consider EV customers first. Second, how do different amounts available for subsidy budgets affect this allocation? In the early stage, the higher the subsidy budget, the more of it that should be allocated to EV
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manufacturers. In the later development stage, it is recommended that all the subsidy should be allocated to EV customers; Third, how do the total subsidy budget and energy costs affect subsidy allocation and customer decisions to buy a car? The results showed that in the early stage, higher fuel prices, lower
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electricity prices and less EV charging times should lead to more subsidies being allocated to EV customers, and in the later stage, with the development and maturity of EV technology, all subsidies should be allocated directly to EV
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buyers. Additionally, more subsidies, lower EV use costs, and higher GV use costs will all lead to higher EV diffusion rates. Therefore, it can be concluded that governments should pay more attention
the subsidy budgets for the EVSC, including the allocation of subsidies. In the EV early development stage, if available subsidies are limited, all of them should be allocated to EV buyers, and then, as available funds for these subsidies increase, allocations should be made to EV manufacturers. This study’s results further indicated that, there is a cap for EV purchase probabilities and total
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profits for the EVSC, and the optimal subsidy amount should be 1500. In the later EV development stage, subsidies are no longer as important because, by that time, the user experience will have improved and more and more consumers
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will have accepted the EVs.
The contributions of this study to the field of research in EV subsidies and
government policies have threefold. Theoretically, different from those under perfect information conditions (Cai et al., 2009; Chen et al., 2012), we used
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the Stackelberg game to analyse the EVSC based on imperfect information conditions; Politically, the results of this study can serve as guiding evidence for governments when designing subsidy schemes, particularly in ensuring that
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EV buyers are afforded greater priority than other players in the EVSC. On a practical level, given that EV buyers are assured of benefiting from such a subsidy policy, manufactures and retailers can use this information in their marketing activities to promote EV uptake.
In this study, we attempted to fill the gaps in the literature on EVSC subsidy policy research. By giving attention in this study to study the impacts of
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vehicle use costs and subsidy allocation in promoting the EVSC,we showed how vehicle driving benefits can be easily converted to tangible variables (Agv and Aev ). In order to reflect real-word circumstances even more accurately, future research will examine the details of use benefits and analyse their impacts on
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subsidy allocation and will also look at optimal investment strategy in the EV infrastructure and ancillary services.
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Abkemeier, K., Appels, D., Barnes, J., Beeton, D., Brennan, G., et al., 2012. Hybrid and electric vehicles: The electric drive captures the imagination, in: Implementing Agreement for Co-operation on Hybrid and Electric Vehicle Technologies and Programmes Annual Report (IA-HEV).
Bulmus, S.C., Zhu, S.X., Teunter, R., 2014. Competition for cores in remanufacturing. European Journal of Operational Research 233, 105–113. Cai, G.G., Zhang, Z.G., Zhang, M., 2009.
27
Game theoretical perspectives
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on dual-channel supply chain competition with price discounts and pricing schemes. International Journal of Production Economics 117, 80–96.
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Cai, X., Lai, M., Li, X., Li, Y., Wu, X., 2014. Optimal acquisition and production policy in a hybrid manufacturing/remanufacturing system with core acquisition at different quality levels. European Journal of Operational Research 233, 374–382.
2017. International Energy Agency, France .
SC
Cazzola, P., Gorner, M., Schuitmaker, R., Maroney, E., 2017. Global ev outlook
Chen, J., Zhang, H., Sun, Y., 2012. Implementing coordination contracts in a
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manufacturer stackelberg dual-channel supply chain. Omega 40, 571–583. Chen, X., Wang, X., Chan, H.K., 2016. Channel coordination through subsidy contract design in the mobile phone industry. International Journal of Production Economics 171, 97–104.
China Automotive Technology and Research Center, Nissan (China) Investment
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Co., Ltd., Dongfeng Motor Co., Ltd., 2016. Annual Report on New Energy Vehicle Industry in China. Social Science Academic Press (China). Gnann, T., Pl¨ otz, P., K¨ uhn, A., Wietschel, M., 2015. Modelling market diffusion of electric vehicles with real world driving data–german market and policy
EP
options. Transportation Research Part A: Policy and Practice 77, 95–112. Greene, D.L., Duleep, K., McManus, W., 2004. Future potential of hybrid and
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diesel powertrains in the us light-duty vehicle market .
Helveston, J.P., Liu, Y., Feit, E.M., Fuchs, E., Klampfl, E., Michalek, J.J., 2015. Will subsidies drive electric vehicle adoption? measuring consumer preferences in the us and china. Transportation Research Part A: Policy and Practice 73, 96–112. Huang, J., Leng, M., Liang, L., Liu, J., 2013. Promoting electric automobiles: Supply chain analysis under a governments subsidy incentive scheme. IIE transactions 45, 826–844. 28
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Luo, C., Leng, M., Huang, J., Liang, L., 2014. Supply chain analysis under a price-discount incentive scheme for electric vehicles. European Journal of
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Operational Research 235, 329–333. Morwitz, V., et al., 2014. Consumers’ purchase intentions and their behavior. R in Marketing 7, 181–230. Foundations and Trends
´ Shafiei, E., Thorkelsson, H., Asgeirsson, E.I., Davidsdottir, B., Raberto, M., Ste-
fansson, H., 2012. An agent-based modeling approach to predict the evolution
Forecasting and Social Change 79, 1638–1653.
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of market share of electric vehicles: a case study from iceland. Technological
M AN U
Shin, J., Hong, J., Jeong, G., Lee, J., 2012. Impact of electric vehicles on existing car usage: A mixed multiple discrete–continuous extreme value model approach. Transportation research part D: transport and environment 17, 138–144.
Sierzchula, W., Bakker, S., Maat, K., van Wee, B., 2014. The influence of finan-
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cial incentives and other socio-economic factors on electric vehicle adoption. Energy Policy 68, 183–194. Szakaly,
S.,
America’s
Manzi,
P.,
franchised
2015.
new-car
Annual
financial
dealerships.
profile
of
Available
at
EP
https://www.nada.org/WorkArea/DownloadAsset.aspx?id=21474839497, Accessed July 10, 2018. Government,
AC C
UK
proval
(IVA):
2017.
model
Individual
reports
for
cars.
Vehicle Available
Apat
https://www.gov.uk/government/publications/publicivamodelreports, Accessed February 18, 2018.
UK Government, 2018. National Minimum Wage and National Living Wage rates. Available at https://www.gov.uk/national-minimum-wage-rates, Accessed February 18, 2018.
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UK Power, 2017.
Gas Electricity Tariff Prices per kWh.
Available at
https://www.ukpower.co.uk/homeenergy/tariffsperunitkwh, Accessed Febru-
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ary 18, 2018.
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Appendix I. The proof for the possibilities of the relationship between UCgv and
We rewrite Ugv and Uev as shown below: Ugv (k) = K0 k − Cgv − pCgv
(39)
(40)
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Uev (k) = K1 k − Cev − pCev + sc
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UCev
As 0 < k < 1 and Ugv (k), Uev (k) are straight line, we first classified the rela-
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tionship based on whether there is an intersection between them in 0 < k < 1. It is easy to be found that, if (Ugv (0) − Uev (0))(Ugv (1) − Uev (1)) > 0, there is no intersection between Ugv (k) and Uev (k). As shown in Case I and Case VI below.
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In Case I, we have UCgv > UCev in 0 < k < 1 and in Case VI, UCev > UCgv .
Pnull
Pgv
Case I
Pnull
Pev Case VI
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Then, if the intersection existed in 0 < k < 1, there are two situations, which is whether the value of the intersection is positive. It is easy to find that the intersection can be achieved when k ∗ =
Cgv +pCgv −Cev −pCev +sc . K0 −K1
when
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K0 k ∗ −Cgv −pCgv ≤ 0, the value for the intersection is not positive (Case II and V) otherwise the the value is positive, as shown in Case III and IV. Furthermore, in Case II and III, K0 > K1 while in Case V and IV, we have K0 < K1 .
1
0 Pnull
Pgv
Case II
Pnull
Pnull Pev Pgv
Pev
Case III
Case V
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Pnull Pgv Pev Case IV
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II. The proof of purchase probability in the early development stage The early stage should satisfy the conditions below:
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K0 > K1 pCgv > 0
pCev − sc > 0 (Cev + pCev − sc − Cgv − pCgv )(K0 − pCgv − Cgv − K1 + Cev + pCev − sc ) < 0 p +C +s −p −C 0 < Cgv gv c Cev ev < 1 (41)
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The schematic diagram is shown below.
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K0 −K1
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Pnull Pev Pgv
Early stage
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Fig. 11. Possibility of relationships between UCgv and CCev
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The cross point between UCgv and UCev is
k=
pCgv + Cgv + sc − Cev − pCev K0 − K1
(42)
The cross point between UCev and U = 0 is
k=
pCev + Cev − sc K1
(43)
ev −sc Therefore, people with intention [0, pCev +C ] will not buy a car. People K1 ev −sc with intention [ pCev +C , K1
pCgv +Cgv +sc −Cev −pCev ] K0 −K1
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will choose to buy an EV
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and the remaining customer will buy a GV. So, we get
Pev =
Cgv − Cev + pCgv + sc − pCev K0 − K1
(44)
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Pgv = 1 −
Cgv − Cev + pCgv + sc − pCev Cev + pCev − sc − K0 − K1 K1
(45)
III. The proof of purchase probability in the later development stage
K1 > K0 pCgv > 0
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The later stage should satisfy the conditions below:
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pCev − sc > 0 (Cev + pCev − sc − Cgv − pCgv )(K0 − pCgv − Cgv − K1 + Cev + pCev − sc ) < 0 p +C +s −p −C 0 < Cgv gv c Cev ev < 1 K0 −K1
(46)
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The sketch of this stage is shown below.
Later stage
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Pnull Pgv Pev
Fig. 12. Possibility of relationships between UCgv and CCev
The cross point between UCgv and UCev is
k=
pCgv + Cgv + sc − Cev − pCev K0 − K1
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(47)
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The cross point between UCgv and U = 0 is
People with intention [0,
pCgv +Cgv ] K0
p +C +sc −Cev −pCev p +C ] [ CgvK0 gv , Cgv gvK0 −K 1
pCgv + Cgv K0
(48)
will not buy a car. People with intention on
will choose to buy a GV and the remaining
customer will buy an EV. Therefore, the probabilities are
Cgv − Cev + pCgv + sc − pCev Cgv + pCgv − K0 − K1 K0 Cgv − Cev + pCgv + sc − pCev K0 − K1
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Pev = 1 −
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Pgv =
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k=
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(49)
(50)
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IV. The Mathematical expression for the total profit The total profit for the EVSC in the EV early development stage is expressed
K0 S(2Cenvir K1 (K1 − 4K0 ) + 2Cev 4K02 − 3K0 K1 + K12
−8Cgv K0 K1 + 4Cgv K12 − 4K02 K5 + 16K02 pM ev + 2K0 K12
+K12 K3 + 2K12 pM ev + 2K12 pM gv )
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+3K0 K1 K5 − 12K0 K1 pM ev − 8K0 K1 pM gv − 2K13
2K1 (K0 −K1 )(K1 −4K0 )2
−4Cev K0 (2Cenvir K1 (4K0 − K1 ) + 2Cgv K1 (8K0 − 3K1 )
− − − + 4K0 K1 K3 3 +3K0 K1 K5 + 12K0 K1 pM ev + 8K0 K1 pM gv + 4K1 2 2 2 2 −2K1 K3 − K1 K5 − 2K1 pM ev − 2K1 pM gv ) 2 −4Cenvir K1 (4K0 − K1 )(2Cgv (K1 − 2K0 ) + 4K0 2 2 +K0 (−4K1 − 2K3 + K5 ) + K1 K3 ) − 36Cgv K0 K1 2 2 2 2 2 +4Cev K0 12K0 − 9K0 K1 + 2K1 + 48Cgv K0 K1 3 3 2 2 2 +8Cgv K1 − 96Cgv K0 K1 + 136Cgv K0 K1 2 −4K1 pM gv (4K0 − K1 )(2Cgv (K1 − 2K0 ) + 4K0 +K0 (−4K1 − 2K3 + K5 ) + K1 K3 ) + 16Cgv K02 K1 K3 2 2 3 −16Cgv K0 K1 K5 − 32Cgv K0 K1 pM ev − 40Cgv K0 K1 2 2 2 −12Cgv K0 K1 K3 + 8Cgv K0 K1 K5 + 8Cgv K0 K1 pM ev 3 4 3 2 3 +4Cgv K1 K3 + 48K0 K1 − 52K0 K1 − 16K0 K1 K3 3 3 2 3 2 3 −32K0 K1 pM ev − 4K0 K5 + 32K0 K5 pM ev − 4K0 K1 2 2 2 2 2 2 2 2 +28K0 K1 K3 + 4K0 K1 K5 + 40K0 K1 pM ev − 4K0 K1 K3 −16K02 K1 K3 pM ev + 3K02 K1 K52 − 24K02 K1 K5 pM ev 3 4 3 3 +8K0 K1 − 12K0 K1 K3 − 4K0 K1 K5 − 8K0 K1 pM ev 2 2 2 2 +3K0 K1 K3 + 2K0 K1 K3 K5 + 4K0 K1 K3 pM ev 2 +4K0 K1 K5 pM ev
+8K02 K1
4K02 K5
16K02 pM ev
12K0 K12
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− = +
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Πtotal
K02 S 2 (3K1 −4K0 ) 2 2α 2K 1 (K0 −K1 )(K1 −4K0 )
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as
α
8K1 (K0 −K1 )(K1 −4K0 )2
(51)
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And the total profit for the EVSC in the EV later development stage is expressed as
K1 S 2 (4K1 −3K0 ) α2 2 2(K 0 −4K1 ) (K0 −K1 )
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Πtotal
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2 2 2Cenvir K1 (K0 − 4K1 ) + 2Cev K0 − 3K0 K1 + 4K1 2 2 2 2 +4C K K − 8C K − 6K K + 2K p + 14K K gv 0 1 gv 1 1 M ev 0 1 0 0 S +K K K + 3K K K − 12K K p 0 1 3 0 1 5 0 1 M ev 3 2 2 +2K1 pM gv (K0 − 4K1 ) − 8K1 − 4K1 K5 + 16K1 pM ev + α 2(K0 −4K1 )2 (K0 −K1 ) 4Cev K0 (2Cenvir K1 (K0 − 4K1 ) + 2Cgv K1 (3K0 − 8K1 ) −10K02 K1 + K02 K5 + 2K02 pM ev + 34K0 K12 + 2K0 K1 K3 −3K0 K1 K5 − 12K0 K1 pM ev + 2K1 pM gv (K0 − 4K1 ) − 24K13 −4K 2 K3 + 4K 2 K5 + 16K 2 pM ev ) + 4Cenvir K1 (K0 − 4K1 ) 1 1 1 2 · 2Cgv (K0 − 2K1 ) − 2K0 + K0 (2K1 + K3 + K5 ) − 2K1 K3 2 2 2 2 2 2 2 = +4Cev K0 2K0 − 9K0 K1 + 12K1 + 8Cgv K0 K1 − 36Cgv K0 K1 2 +48Cgv K13 − 16Cgv K03 K1 + 48Cgv K02 K12 + 4K1 pM gv (K0 − 4K1 ) 2 · 2Cgv (K0 − 2K1 ) − 2K0 + K0 (2K1 + K3 + K5 ) − 2K1 K3 +4Cgv K02 K1 K3 + 8Cgv K02 K1 K5 + 8Cgv K02 K1 pM ev −12C K K 2 K − 16C K K 2 K − 32C K K 2 p gv 0 1 3 gv 0 1 5 gv 0 1 M ev +16C K 3 K + 8K 4 K − 4K 3 K 2 − 4K 3 K K − 12K 3 K K gv 1 3 0 1 0 1 0 1 3 0 1 5 3 3 2 3 2 2 −16K0 K1 pM ev + 4K0 K5 pM ev − 52K0 K1 + 4K0 K1 K3 +28K02 K12 K5 + 80K02 K12 pM ev + 2K02 K1 K3 K5 + 4K02 K1 K3 pM ev +3K02 K1 K52 c − 24K02 K1 K5 pM ev + 48K0 K14 − 16K0 K13 K5 −64K0 K13 pM ev + 3K0 K12 K32 − 16K0 K12 K3 pM ev − 4K0 K12 K52 +32K0 K12 K5 pM ev − 4K13 K32 − 32Cgv K0 K13 − 8K0 (K0 −4K1 )2 (K0 −K1 ) (52)
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