2015 International Conference on Management Science & Engineering (22th)
October 19-22, 2015
Dubai, United Arab Emirates
Optimal Policies in Remanufacturing System with Product Substitution
under the Constraint of Carbon Emissions
WEI Li，CHEN Wei-da，LIU Bi-yu
School of Economics and Management, Southeast University, Nanjing 211189, P.R.China
Abstract: Given increasing concern related to CO2
and other greenhouse gas, green manufacturing is a
choice for manufacturers to reduce their carbon emission.
This paper studies the decision problem of steel
remanufacturing with two kinds of products produced in
blast furnace and electrical furnace and a downward
substitution under the constraint of carbon cap and trade
mechanism. The profit-maximization optimal production
model with carbon cap and trade mechanism is
established. The condition satisfied by the optimal policy
is given. The numerical experiments are shown to
illustrate the impact of carbon price, carbon cap on the
strategy with product substitution, carbon emission, the
total profit and the substitution quantities. Some
managerial insights and policy implication regarding the
steel remanufacturing enterprise and the government are
obtained.
Keywords: remanufacturing, product substitution,
carbon emission, production decision
1 Introduction
Given industrial development and improved living
standards, steels play an important role in our modern
societies, while steel enterprises are supposed to be a
prime source of the carbon emission. From an economic
perspective, steel recycling is cheaper than mine virgin
ore. The recycling of scrap iron and steel plays will be a
trend in supplying precious raw material for steelmaking
[1]
. Iron and steel recycling has been considered of
important to society, almost 75million tons of steel were
recycled or exported for recycling in 2008 in the US
alone [2]. In china, the consumption of scraps rose from
3.44 million in 2001 to about 8.1 million in 2009, an
increase 2.4 fold. Depending on the grade of steel and
steelmaking process, internal steel components may be
made using either 25 percent or more than 90percent
recycled steel [3]. The chemical process industries are a
major contributor to greenhouse gas emission, especially
carbon dioxide. The iron and steel recycling is also a
Supported by the National Natural Science Foundation of
China(No.71271054, 70971022), the Scientific Research
Innovation Project for College Graduates in Jiangsu
Province(No.CXZZ12_0133)
978-1-4673-6513-0/15/$31.00 ©2015 IEEE
becoming especially important in the context of
environmental benign steelmaking.
Many countries have attempted to enact legislation
or design market-based carbon trading mechanism for
controlling carbon emission since the global warming
caused by CO2 and other greenhouse gas (GHG). Due to
the production operation of manufacturing enterprise is
the main source of carbon emission, manufacturing
enterprises are under growing pressure to reduce their
carbon emission. The carbon cap and trade mechanism
are both economical problem and technical matter and
they will become even more significant in the future for
the purpose of reducing carbon emission. Laurens G. et al.
[4]
considered introducing a remanufacturable product in
a market that consists of heterogeneous consumers who
have different choice to the retreaded tires. The low-end
and high-end consumer populations are different across
markets. They discussed the impacts of the market and
technology drivers of product manufacturability and
identified some phenomena of managerial importance.
Klausner et al. [5] considered the remanufacturing of
electrical motors. Products containing remanufactured
electrical motors have no significant differences in
quality or can be sold in a low-end market at a
discounted price. In order to satisfy the customer’s
special demand for the product with different quality and
price, a remanufacturer have to produce different quality
remanufactured products. In this paper, we investigate
the decision problem of steel remanufacturing with
returned product remanufacturing and product
substitution under the constraint of carbon emissions,
where a remanufactured product with high quality level
may be used to substitute another one with low quality
level.
The following literature discussion is intended to
provide context for the production planning with carbon
cap and carbon emission trading. Letmathe et al. [6]
presented two mathematical models that can be used by
firms to determine their optimal product mix and
production quantities in the presence of several different
types of environmental constraints, in addition to typical
production constraints. In the research [7], a
mixed-integer nonlinear programming (MINLP) model
was proposed for the production planning of refinery
processes to achieve maximum operational profit while
- 219 -
reducing CO2 emissions to a given target through the use
of different CO2 mitigation options. Hua et al. [8] studied
how firms manage carbon footprints in inventory
management under the carbon emission trading
mechanism. They used an EOQ model to obtain the
optimal order quantity, and also analyzed the impacts of
carbon trade, carbon price, and carbon cap on order
decisions, carbon emissions, and total cost. Bin et al. [9]
studied the multi-item production planning problem with
carbon cap and trade mechanism and analyzed the
impacts of carbon price on the shadow price, production
decisions, carbon emission and the total profit. Note that
the decision problem of remanufacturing with returned
product remanufacturing and product substitution under
the constraint of carbon emissions has not been
investigated. Gong et al. [10] consider the manufacturer
how to make a balance between carbon emission and
production policy. Nouira et al. [11] not only cinsider the
selection of manufacturing processes, but also input
items
(components).
A
optimization
models
formanufacturing systems considerthe environmental
impacts is showed. Wellington et al. [12] study the optimal
product mix under emission restriction, propose Interior
analysis to solve this problem. Shen et al. [13] analysis the
manufacturer’s production decision-making problem
under carbon cap and carbon emissions trading.
While many works are related to the production
planning for remanufacturing, the literature dealing with
remanufacturing and products substitution at the same
time is limited. The related literature considers the
product substitution under various situations. Regarding
one-way product substitution, Inderfurth [14] present
combines methods to analysis the optimal policies in
hybrid manufacturing/remanufacturing systems. Bayındır.
et al[15] jointly consider one-way product substitution and
a
capacity
constraint
of
the
manufacturing/remanufacturing system. Rao. et.al [16]
study the Multi-product inventory planning with
downward substitution, stochastic demand and setup
costs and present a two-stage integer stochastic program
model. Bayindir et al. [17, 18] investigated the profitability
of remanufacturing option. They constructed a single
period profit model under substitution to investigate
remanufacturing was profitable. Li et al. [19] investigated
an uncapacitated muti-product production planning
problem with returned product remanufacturing and
demand substitution, where no backlog and no disposal
are allowed. A dynamic programming approach was
derived for the optimal solution. In 2007, they analyzed a
version of the capacitated dynamic lot-sizing problem
with substitutions and return products and applied a
genetic algorithm to determine all periods requiring
setups
for
batch
manufacturing
and
batch
remanufacturing[20]. To respond to government regulatory
on carbon emission and environment concerns, same
researches in operations management (OM) and
operations research (OR) have been arisen in the efforts
optimizing operational decisions. Huang et.al[21]
congsider a multiple-product newsboy problem with
partial
product
substitution,
develop
an
iterativealgorithm to solve the model. Liu et.al[22] study
the two-way productsubstitution and present an
ewsvendor game with the impact of loss aversion. Ahiska
et.al[23] also study a product substitution strategy for a
stochastic manufacturing/remanufacturing system and
formulat it as a discrete-time Markov Decision Process.
Zhang et.al[24] concentrate on the consumer
environmental awareness and channel coordination with
two substitutable products within a one-manufacturer
and one-retailer supply chain. Most studies on product
substitution focused on the impact of substitutable,
overlooked the factor of natural environment (Carbon
emissions etc.)The objective of the study was to focus on
the green manufacturing and develop an optimization
model, to achieve the optimal expect profit, by
integrating the carbon emission scheme. Integrating the
carbon emission scheme in steel remanufacturing makes
it possible to choose their remanufacturing technology to
comply with the regulations. We optimize production
planning with carbon cap and trade mechanism. A
mathematical model is presented to formulate the
problem. We derive the optimal solution of production
and carbon trading quantities.
In practice, steel can be subdivided into different
categories which represent different chemical
composition standards. People are pursuing different
grade steel products for diverse purposes in the market,
when the demand of one product is not satisfied, the
product substitution can avoid the loss coursing by the
stock out. And the same time, the steel remanufacturing
firms face the challenge of managing product production
system with product substitution under the constraint of
carbon emission. To reduce CO2 emission, the industry is
undertaking long-range, high risk research and
development into new steelmaking processes whose
designs emit much less CO2. This is the main problem
we are going to study in this paper. Zhang et.al[25] use an
empirical analysis to verify the relationship of CO2
reduction and Chinese iron & steel (IS) companies. Tang
et.al [26] recognize the energy consumption cost as a
nonlinear function of the production quantity in a steel
production process, a mixed integer nonlinear
programming(MINLP) model is formulated. Tang [27]also
investigates the effective decisions for the batching
problems
of
steelmakingand
continuous-casting
production. [28,29] study the continuous casting
planning. In order to improve the efficiency and
feasibility of charge planning, Dong et.al[30] formulated a
multi-objective mathematical programming model and
designe Two new meta-heuristics. These papers focus on
the on the production control of a complex industrial
process which is set in the steel-making process and has
been almost no consideration for the possible influence
of carbon emission. Considering the iron & steel (IS)
companies are traditional high-emission industries, to
take same action must be necessary. In this paper, we
solve the joint carbon emission and product subsititution
problem faced by an iron & steel (IS) companies.
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The rest of the study is organized as follows.
Section 2 describes the problem and gives a
mathematical formulation. In section 3, numerical
experiments are provided to illustrate the impact of
carbon price and carbon cap on production decisions,
carbon emission and the total profit. In the last section
the findings of the study are summarized.
τ i : Per unite capacity
When the demand for remanufactured products
demand is not fulfilled, a substitution takes place. The
steel products with low quality can be substituted by
steel products with high quality. This results in the
following substitution quantity q :
if x1 − D1 ≥ D2 − x2 > 0
 D2 − x2
(1)
q=

 x1 − D1
2 The problem and mathematical model
if
D2 − x2 > x1 − D1 > 0
The expected amount of substitution is denoted by
In this paper, we consider optimal policies of a steel
remanufacturing enterprise which produce two kinds of
steel products using the recycled steel and iron products
as raw material，called product 1 and product 2. There
are two main steel production processes: blast furnace
plants and electrical furnaces. The blast furnace plants
are considered as integrated plants, while the electrical
furnace plants are called semi-integrated plants. But the
electrical furnace steel making is far less energy
intensive by comparison with the blast furnace steel
making from both resources use and CO2 emissions
perspective. The product 1 is produced in integrated
plants, which is of high quality. In semi-integrated plants,
iron scrap is put directly into electrical furnace to
produce steel products, called product 2. The steel
remanufactured products with different quality have been
undistinguishable in terms of functionalities. The two
kinds of steel remanufactured products satisfy different
types of consumers at different prices. When the demand
for products with high-level quality (product 1) is
fulfilled, the producer can offer a product (product 2)
with high-level quality instead a product with low-level
quality in case of a shortage of a product with low-level
quality. We will focus on a single-period problem with
independent stochastic demands for both product types.
The following notations are used throughout the
paper.
i : product index
xi : production quantities
Di : random demand for product i
q : quantity of product substitution
fi ( x) : probability density function of the demand for
product i
Fi ( x) : probability distribution function of the demand for
product i demand
Fi −1 ( x) : inverse distribution function of the demand
forproduct i
:
Selling
price per unit product i
pi
si : Salvage value per unit product i
ci : Production cost per unit product i
ce : Carbon price per unit
ei : Carbon emission per unit product
Eg : Carbon cap
Q( x1 , x2 , D1 , D2 )= E[min[max( D2 -x2 ,0), max( x1 D1 ,0)]]]
x1
x1 + x2 -D1
0
x2
= ∫ f1 ( D1 ) ∫
x1
∞
0
x1 +x2 - D1
+ ∫ f1 ( D1 ) ∫
( D2 - x2 ) f 2 ( D2 )dD2 dD1
(2)
( x1 - D1 ) f 2 ( D2 )dD2 dD1
In this problem, for maximizing the total expected
profit, the steel remanufacturer has to decide the optimal
production quantities and the corresponding carbon
trading quantity L . And the total carbon emission will
2
be ∑ ei xi , in the optimal carbon trading quantity L it
i =1
2
xi
holds the following relation ∑ ei =
i =1
Eg + L
[12]. If L > 0 ,
the manufacturer buys emission permits form the
carbon trading market; otherwise, he sells its unused
permits − L . The carbon trading quantity L = 0 means
the steel manufacture will not involve in the carbon
trading.
Now, we are ready to present the optimization
model with product substitution under the constraint of
carbon cap.
Max
=
p ( xi , L) E[ p1 min( x1 , D1 ) + s1 ( x1 − D2 ) + ]
+ E[ p2 min( x2 , D2 ) + s2 ( x2 − D2 ) + ]
+ ( p2 − s1 ) Q( x1 , x2 , D1 , D2 ) − x1c1 − x2 c2 − ce L
(3)
subject to
2
x
∑e =
i =1
i i
2
∑τ x
i =1
i i
Eg + L
(4)
(5)
≤σ
(6)
The objective function (3) maximizes the total
expected profit for both products. In models, we
+
max(⋅,0) . For guaranteeing a profitable
let (⋅)=
remanufacturing business, we assume the cost value of
product 1 is greater than product 2, that is c1 > c2 . And,
we also assume that p1 > p2 > c1 > c2 , p2 > c2 > s2 and
c1 > s1 > s2 . To avoid the general case, the steel
remanufacturer get more revenue from selling product 1
than product 2 and the positive profit can obtained in the
situation that using product 1 to satisfy the demand of
product 2, that is p1 − c1 > p2 − c2 > p2 − c1 > 0 . It is
reasonable in practice because the products with high
quality have a higher performance than products with
low quality.
From Eq. (4), we have
xi ≥ 0
(i =
1, 2)
2
∑e x − E
: the carbon trading quantity
σ : the total capacity
L
i =1
i i
g
=
L.
(7)
Substituting Eq. (7) into Eq. (3), this also means the
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following.
( p2 − c2 − ce e2 ) − ( p2 − s2 ) F2 ( x2 ) − ( p2 − s1 )
Max=
p ( xi , Em ) E[ p1 min( x1 , D1 ) + s1 ( x1 − D1 ) ] + E[ p2 min( x2 , D2 ) + s2 ( x2 − D2 ) ]
+
+
∫
+ ( p2 − s1 ) Q( x1 , x2 , D1 , D2 ) − x1c1 − x2c2 − ce (∑ ei xi − Eg )
2
∑µ x
0
x2
−( p2 − s2 ) ∫ F2 ( D2 )dD2
i =1
0
x1 + x2 − D1
x2
x1
∞
0
x1 + x2 − D1
+ ∫ f1 ( D1 ) ∫
(8)
Proposition 1. The expected profit function is jointly
concave in xi , i = 1, 2 .
x1
F1 ( x1 ) =
∂p
= ( p1 − c1 − ce e1 ) − ( p1 − s1 ) F1 ( x1 ) + ( p2 − s1 )
∂x1
0
( p1 − c1 − ce e1 ) + λτ 1 + µ1 +( p2 − s1 ) ∫0 f1 ( D1 )(1 − F2 ( x1 + x2 − D1 ))dD1
p1 − s1
p1 − s1
(14)
f1 ( D1 ) F2 ( x1 + x2 − D1 )dD1
x1
∂p
= ( p2 − c2 − ce e2 ) − ( p2 − s2 ) F2 ( x2 ) − ( p2 − s1 )
∂x2
,
x1
(13)
λ ∗ be the optimal shadow price. Then the x1∗ , x2∗ can be
derived from the first-order optimality condition.From
equation (9) and (10), we can derive the optimal values
of each of the decision variables which are satisfied the
following equations:
Proof. First，since
∫
(12)
=0
Let x1∗ , x2∗ be the optimal production decision, and
( x1 − D1 ) f 2 ( D2 )dD2 dD1 ]
i =1
x1
i i
λ represents the shadow price of the common capacity.
2
0
(11)
λ≥0
( D2 − x2 ) f 2 ( D2 )dD2 dD1
− x1c1 − x2 c2 − ce (∑ ei xi − Eg )
∫
(10)
f 2 ( D2 )dD2 dD1 + λτ 2 + µ 2 =
0
i =1
=p1 x1 − ( p1 − s1 ) ∫ F1 ( D1 )dD1 + p2 x2
x1
x1 + x2 − D1
0
2
x1
0
f1 ( D1 ) ∫
λ (σ − ∑τ i xi ) =
0
i =1
+( p2 − s1 )[ ∫ f1 ( D1 ) ∫
x1
0
2
F2 ( x2 ) =
We have
( p2 − c2 − ce e2 ) + λτ 2 + µ2 +( p2 − s1 ) ∫0 f1 ( D1 )( F2 ( x1 + x2 − D1 ) − F2 ( x2 ))dD1
p2 − s2
p2 − s2
(15)
f1 ( D1 )( F2 ( x1 + x2 − D1 ) − F2 ( x2 ))dD1
−
x1
∂ 2p
=
−( p1 − s1 ) f1 ( x1 ) − ( p2 − s1 )[ ∫ f1 ( D1 ) f 2 ( x1 + x2 − D1 )dD1 + f1 ( x1 ) F2 ( x2 )]
2
0
∂x1
3 Numerical experiments
x1
In this section, we analyze the effect of the carbon
price and carbon cap on the strategy with product
substitution, carbon emission and the total profit of the
steel remanufacturing system with product substitution
presented in section 2. For this purpose, we design the
parameters sets for illustrating our main results and
observe the behavior of the steel remanufacturing
system.
Demands of the product 1 and 2 are assumed as
uniformly distributed, i.e., Di ∈ (ai , bi ), i =
1, 2 , where
bi > ai , ai > 0, i =
1, 2 . We set carbon quota Eg = 800 , the
=
−[( p1 − s1 ) f1 ( x1 ) + ( p2 − s1 )[ f1 ( x1 ) F2 ( x2 ) + ∫ f1 ( D1 ) f 2 ( x1 + x2 − D1 )dD1 ]]
0
x1
∂p
=
−( p2 − s2 ) f 2 ( x2 ) + ( p2 − s1 ) ∫ f1 ( D1 )(( f 2 ( x1 + x2 − D1 ) − f 2 ( x2 ))dD1
0
∂x22
2
=
− [( s1 − s2 ) F1 ( x1 ) f 2 ( x2 ) + ( p2 − s2 ) f 2 ( x2 )
x1
[1 − F2 ( x2 )] + ( p2 − s1 ) ∫ f1 ( D1 ) f 2 ( x1 + x2 − D1 )dD1 ]
0
x1
∂ 2p
=
−( p2 − s1 ) ∫ f1 ( D1 ) f 2 ( x1 + x2 − D1 )dD1
0
∂x1 x2
using for abbreviation
x1
G ( x1 , x2 ) = ( p2 − s1 ) ∫ f1 ( D1 ) f 2 ( x1 + x2 − D1 )dD1
0
we thus find
P:
=
common capacity σ = 500 , and ce = 1 . The problem
parameters are listed in tab.1.
∂ 2ππππ
∂2
∂2
∂2
−
∂x12 ∂x22 ∂x1∂x2 ∂x2 ∂x1
=
[(p1 − s1 )f1 (x1 )+( p2 − s1 ) f1 (x1 )F2 (x2 )+G ][(s1 − s2 )F1 (x1 )f 2 (x2 )
Tab.1 Parameters of the problem product
ai
τi
ci
si
bi
pi
+(p2 − s2 )f 2 ( x2 )[1 − F2 ( x2 )]] + [(p1 − s1 )f1 (x1 )+( p2 − s1 ) f1 (x1 )F2 (x2 )]
[(s1 − s2 )F1 (x1 )f 2 (x2 )+(p2 − s2 )f 2 ( x2 )[1 − F2 ( x2 )] + G ]
due to
p1 > p2 , s1 > s2 , p2 > s2 and
∂ 2π
≤0
∂x12
,
∂ 2π
≤0
∂x22
and D ≥ 0 . Namely the Hessian matrix
of the objective function is negative semi-definite.
Since π is concave, the Karush-Kuhn-Tucker (KKT)
condition can be necessary and usefully for the optimal
solutions. Let λ , µi , i = 1, 2 be the dual variables
corresponding to the constraint (5) and (6). Then, the
optimal policies to this problem are xi , i = 1, 2 , if and only
if there exists non-negative dual variable λ , µi , i = 1, 2 .
Hereby we can have
( p1 − c1 − ce e1 ) − ( p1 − s1 ) F1 ( x1 ) + ( p2 − s1 )
∫
x1
0
f1 ( D1 ) ∫
∞
x1 + x2 − D1
1
2
p2 > s1 , It is obvious that
f 2 ( D2 )dD2 dD1 + λτ 1 + µ1 =
0
(9)
95
55
45
25
15
10
100
150
0
0
3
2
ei
7
2
In order to figure out the effect of the carbon price
on the system, we conduct experiments with varying
value of the carbon price, and keep other parameters
unchanged. Below is the impacts of carbon cap and
carbon price on the shadow price of the capacity,
production decisions, carbon emission and total profit.
Fig.1 shows that the curves of optimal decisions of
steel remanufacturer, we can see that with the carbon
price increasing, the quantity of product 1 x1 is
decreases, meanwhile the quantity of product 2 x2 is
increasing with the carbon price increases until 12.52.
After that the carbon price increases to 12.52,
- 222 -
until ce increases to 12.52. After that, the total carbon
emission keeps constant. From this curve, the carbon
emission of the remanufacturer is decreases under the
situation of carbon emission strategy. It means that
carbon price is an effective factor to controlling the
carbon emission. When the carbon price increases, the
steel remanufacturing firms will choose electrical furnace
to produce more economical products which areless
carbon inefficient. So, the government can take carbon
trade as an effective measure to make firms reduce
carbon emission.
Fig.1 The curves of optimal production quantities
Remanufacturer
Fig.3 The curves of the expected profit for different carbon
caps
Fig.2 The curve of the total carbon of steel emission
x1 and x2 keep constant. Because of as the carbon price
increases, production 2 will be more economical and low
carbon emission. It indicates the steel remanufacturer
intend to choose electrical furnace as the main steel
making process to avoid
high cost and high carbon
emission. If the carbon price is increasing, steel
remanufacturer will produce more low carbon emission
productions. When carbon price is low, to produce
production 1 takes the advantage at the low cost. But
with the increasing of the carbon price, the
remanufacturer will tend to manufacture the product with
low carbon emission. Given the production capacity, the
decision variables and expected profit will have nothing
to do with ce > 12.52 .
Fig. 2 shows the total carbon emission. It shows that
the total carbon emission decreases with the carbon price
- 223 -
Fig.4 The expect profit of combination of the carbon cap
and carbon price
adjust carbon cap properly to prevent fluctuation of the
carbon price.
4 Conclusions
Fig.5 The curve of the substitution quantities
Fig.3 illustrates the impact of the carbon cap and
carbon price on expected profit. In order to investigate
how the carbon cap and carbon price effect the expected
profit, Fig.3 plots the curves of expected costs against
carbon price when=
Eg 0,=
Eg 500,=
Eg 1500 respectively.
When Eg = 0 , the expected profit decreases as the carbon
price increases. When Eg = 500 , the curve have the same
shape. They decreases with the carbon price in the case.
When Eg = 1500 , the top curve indicates that the expected
profit is increasing first and then decreasing with the
carbon price. When Eg = 0 and Eg = 500 , the steel
remanufacturing firm has to buy carbon permits from the
carbon trade market. So the cost of steel remanufacturing
increases with carbon price. As a result, the increase of
the carbon price will reduce the firm’s expect profit.
When Eg = 1500 , the steel remanufacturing firm has extra
carbon permits, so the profit increases with carbon price
until the profit reach the peak, and then decreases with
carbon price. Fig. 4 shows the expect profit of
combinations of the carbon cap and the carbon price. The
3-D object in Fig.5 increases as the carbon price and
carbon cap increase.
Fig.5 illustrates the change of the expect
substitution quantities with respect to the carbon price
for sets of data as given in the table. The curve shows
that Q decreases with the carbon price. The products
with low carbon emission will be economical than
product with high carbon emission when carbon price
increases, thus x1 decreases and x2 increases. The expect
substitution quantities is only related to the carbon price
and is no sensitive to the carbon cap.
There is a significant finding we can get from the
analyses. The optimal quantities and the expect
substitution quantities are more sensitive to the carbon
price, while maximum expected profit is insensitive to
the carbon price and the carbon cap. It makes sense to
As the growing attention on carbon emission,
manufacturers have started to integrate carbon emission
concern into their operation decision. In this paper, we
study the decision problem of remanufacturing with
substitution with the carbon emission constraint that
carbon cap and trade. A profit-maximization model is
established. By analyzing the decision process and
optimal model, we get the equations which optimal
solutions satisfied.
There are some significant findings by analyzing the
impact of carbon cap and carbon price on production
decision, carbon emission and total expect profit. Under
the carbon emission constraint, with the carbon price
increases, the remanufacturer is prefer to produce
low-carbon production. At the same time, the total
carbon emission decreases. So the carbon cap and trade
mechanism can be an effective measure to reduce carbon
emission. If remanufacturer have surplus carbon quota,
with the carbon price increases, the expect profit will
increases. Otherwise, if remanufacturer have not enough
carbon quota, with the carbon price increases, the expect
profit will decreases. According to the expect profit of
combinations of carbon cap and carbon price, the policy
makers can design efficient car cap and trade mechanism
to balance the remanufacturer’ profit and the total carbon
emission.
In this paper, there remains several questions which
give motivation for future research. At first, we can
extend the single-period and address the general
multi-period problem case. In each period, the carbon
price may be different. Another extension is that we only
consider two kinds of carbon constraint: carbon cap and
carbon trade, it is very significant to study the
remanufacturer production optimal under the other
carbon constraint or the combination of carbon
constraints.
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