An Analysis for Multi Plant Location Problem under Concurrent Implementation
of Different Carbon Policies
Zhen Chen, Qiu-hong Zhao, Hai-tao Zheng
School of Economics and Management,Beihang University,Beijing, China
(qhzhao@buaa.edu.cn)
Abstract - This paper analyzes the influence of
concurrent implementation of three different carbon
policies: carbon tax, carbon trade, carbon offset to the
company’s multi plant location decisions, while multi
capacity of any plant is considered. We present a mixed
integer math model with the objective to minimize the total
cost within the given period. The numerical results and
sensitive analysis show that the location decision depends on
several factors including the level of carbon tax and the price
of carbon trade and carbon offset.
Keywords - carbon policies, plant location, multi
capacity
I.
INTRODUCTION
The Intergovernmental Panel on Climate Change
(IPCC) reports that global warming poses a grave threat to
the world's ecological system and the human race, and it is
very likely caused by increasing concentrations of carbon
emissions, which mainly results from such human
activities as fossil fuel burning and deforestation . In order
to alleviate global warming, the United Nations (UN), the
European Union (EU), and many countries have enacted
legislation or designed mechanisms to curb the total
amount of carbon emissions. There are three main carbon
policies: carbon tax, carbon trade, carbon offset. Carbon
tax is a policy that the government makes an emission
constraint for a company and levies tax on the company’s
exceeding carbon emission. Carbon trade is a policy that
companies can trade the amount of carbon emission in the
carbon emission exchange. Companies can buy carbon
offsets in order to comply with caps on the total amount of
carbon dioxide they are allowed to emit.
There are a few studies on the operations decisions
under carbon emission regulations. Kim et al. (2009)
examine the relationship between the freight transport
costs and CO2 emissions in given intermodal and truckonly freight networks by multi-objective optimization.
Hoen et al. (2010) examine the effects of two regulation
mechanisms (emission cost vs. emission constraint) on the
transport mode selection decision and suggest that policymakers impose a constraint on freight transportation
emissions. Pan et al. (2010) examined the environmental
impact of pooling of supply chains, and they found the
supply network pooling is an efficient approach in
reducing CO2 emissions. Hua et al. (2011) investigates
how firms manage carbon footprints in inventory
____________________
This paper is sponsored by National Natural Science Foundation
of China. (71071007)
management under the carbon emission trading
mechanism. To the best of our knowledge, there are no
papers about the effects of the concurrent implementation
of three main carbon policies on companies’ operations.
However, in the reality, several carbon policies may be
carried out by the government at the same time, like the
carbon policies in Europe Union. And, since the
environment problem is more and more severe now, this
may be a trend of other governments’ policy.
The multi plant location problem is similar to the
capacitated warehouse location problem (CWLP).
Mathematically, they can be formulated as a mixed
integer program with a certain number of potential
warehouses or plants and certain number of customers.
The objective is to choose the most appropriate plants to
minimize the total cost including the fixed cost associated
with plants and the transportation cost. The difference is
that you can choose the plant production capacity in the
former problem while in the latter situation the
warehouse’s capacity is a constant value. To our review,
there are few papers about the multi plant location
problem. However, in the real situation, managers always
have some choices of the production capacity with the
common sense that the larger capacity plant can meet
more demands of the customers but with a higher fixed
cost while the smaller plant has cheaper fixed cost but
meets less demand. On the other hand, under carbon
policies, managers should pay more attention on the
plant’s capacity because it has great effects on the carbon
footprint.
The multi plant location problem is the same with
CWLP in the computation principles, so the algorithms of
CWLP are adapted to solve it. Akinc and Khumawala
(1977) propose a branch-and-bound solution method.
Bartezzaghi et al. (1981) presented a tree search algorithm
based upon a lower bound derived from a transportation
problem and gave computational results for problems
involving up to 12 potential warehouse locations and 40
customers. Guignard-Spielberg and Kim (1983) presented
a lower bound for the problem based upon lagrangean
relaxation. Computational results are reported for
problems involving up to 20 potential warehouse locations
and 35 customers. Barcelo (1984) presented an algorithm
based upon the automatic generation of cutting planes in
an attempt to move the solution of the linear programming
relaxation of the problem closer to the optimal mixedinteger solution. Computational results are reported for
problems involving up to 30 potential warehouse locations
and 50 customers. Beasley (1988) incorporates the lower
bound and the reduction tests into a tree search procedure.
By this way, it is able to solve problems involving up to
500 potential warehouse locations and 1000 customers.
The rest of this paper is organized as follows: In
Section 2 we formulate the multi plant location model
under the concurrent implementation of three carbon
policies. In Section 3 we provide some numerical
examples and make sensitive analyses to gain practical
insights from the analytical results derived in Section 2.
Finally we conclude the paper and suggest topics for
future research in Section 4.
II.
 : carbon tax rate
L : the upper limit of the carbon emission per unit
time
et : the volume of carbon traded
eo : the volume of carbon offsetted
pt : the price of carbon in trading
po : the price of carbon in offsetting
Z ik : equals 1 when setting up plant i with a capacity
MODEL FORMULATION
The multi plant location problem is in fact a
transportation problem and the distribution network is like
the figure below:
level j, equals 0 when not
Z t : equals 1 when the company trades carbon, equals
0 when not
Zo
: equals 1 when the company offsetts carbon,
equals 0 when not
Model formulation:
min (  gik    vik cik ) Z ik  b li j xij 
iI kK
iI kK
iI jJ
 pt [ L  (  gik    vik cik ) Z ik  b  li j xij ]
iI kK
Plants
Fig.1 The structure of distribution network
In our model, we assume that the company chooses
some customer’s locations to build up plants. The
production capacity of the plant is assumed to have
several levels and the decider maker chooses one
production level for each plant. We also assume that the
cost and carbon emissions in the transportation are only
related to the distance and the volume of the goods
transported.
The following notation will be used to describe the
model:
I  {1, 2,..., n} : the sets of plants
J  {1, 2,..., m} : the sets of customers
K  {1, 2,..., s} : the sets of capacity levels of plants
gik : the fixed cost of setting up plant i with a capacity
level k per unit time
vik : the unit cost of operating in plant i
cik : the maximum capacity of plant i with a capacity
level of j
iI kK
iI jJ
+ {[(  gik    vik cik ) Z ik  b  li j xij  L  qt Z t  qo Z o ] }
Customers
iI kK
iI kK
iI jJ
 pt qt Z t  po qo Z o
s.t.
x  c
ij
jJ
x
iI
ij
Z
kK
kK
ik
(1)
Z ik  0
i  1, , n
dj  0
ik
1
xijk  0, eo  0
j  1,..., m
i  1, , n
i  1,..., n; j  1,..., m; k  1,..., s
Z ik  0,1; Z t  0,1; Z o  0,1
i  1,..., n; k  1,..., s
Formula 1 is the total cost of the company per unit
time.
Formula 2 represents that the customer demand must
be meted.
Formula 3 represents that the amounts of goods
transported from a plant could not exceed its capacity.
Formula 4 represents each plant must only have only
one certain capacity level.
In Formula 5, et doesn’t have positive or negative
et  0 , it represents that the amount of
d j : the demand of customer j per unit time
constraint; if
li j : the distance of plant i to customer j
carbon the company buys from the carbon trade exchange;
if et  0 , it represents the amounts of carbon company
b : unit transportation cost
xijk : the volume of goods transported from plant i
with a capacity level k to customer j
gik : the volume of carbon emission when setting up
plant i with a capacity level k
b : unit carbon emission in transportation
sells in the carbon trade exchange.
Formula 6 is 0-1 variables.
From the objective function, we can see that in order to
minimize total cost, the decision maker would choose one
or
more
carbon
policies
to
take
effect.
(2)
(3)
(4)
(5)
(6)
From  [
 g
iI kK
ik
Zik  b ( xijk lij  L  eo Zo  et Zt ] ,
iI jJ kK
13
3
6
7
we know that the location decision is closely related
with  , pt , po .
12
4
2
11
8
14
10
5
III.
1
NUMERICAL RESULTS
15
9
In order to analysis the effects of carbon policies on
the plant location decision; we use the data in Christofides
and Eilon (1969) as the customer’s locations. We choose
15 numbers and draw the figure below. The distance
between them are listed in Appendix 1. For simplicity and
better analysis of the influence of carbon policies to
location decisions, we assume all the customers’ demands
are the same 2500and in our model the unit time is one
year. The decision maker chose several locations from
them to build plant to meet the demands of all customers.
Figure 2. Solution 1
13
3
6
7
12
4
2
11
8
14
10
5
1
15
9
3
13
6
7
12
4
2
11
8
Figure 3. Solution 2
14
10
5
1
15
meets the demands at its location, which the transportation
cost can be ignored, also including the demands of other
locations assigned for it. The details of the two solutions
are listed below.
9
Figure 1. Distance of customers
We assume that there are two levels of the capacity of
the plant and the date is based on cement plants. The
values of some imputed parameters are listed in Table 1.
As some papers have used, our data is based on an
assumption that larger production capacity plants have
smaller unit operating cost and unit operating carbon
emission.
TABLE I.
Capacity
Level
(ton)
5,000
125,000
PARAMETER VALUES FOR TWO CAPACITY PLANTS
gik
vik
gik
(yuan)
(yuan/ton)
(ton)
16,000,000
38,000,000
250
220
4000
10000
vik
(ton
CO2/ton
cement)
0.08
0.07
b is 0.5 yuan/ton*km, b is 0.00006 ton/ton*km, the
initial carbon tax rate  is 10 yuan/ton, initial carbon
emission L is 50000 ton, carbon trading price
yuan/ton, carbon offsetting price
pt is 60
po is 80yuan/ton.
It is difficult to find the optimal solution, so we
provide two feasible solutions. Solution 1 is that we build
5 high production capacity plants; Solution 2 is that we
build 1 high capacity plant and 5 small capacity plants.
For better understanding, we draw the figures below.
In the figures, the large triangle represents the high
production capacity plant and the small one represents the
low production capacity plant. For each plant, it not only
TABLE II.
DETAILS OF COST WITHOUT CARBON POLICY
Details
fixed setting up cost
operation cost
transportation cost
total cost without
carbon related cost
TABLE III.
Solution 1
11400000
82500000
22860053
Solution 2
11800000
90000000
15922577
116760053
117722577
CARBON COST WITH L=5000,
fixed
carbon
emission
operation
carbon
emission
transportation carbon
emission
total carbon emission
exceeding
carbon
emission
total cost with carbon
offsetting
 =10, pt =15,
po
=20
30000
30000
26250
28750
27432.063
19107.092
83682.063
77857.092
33682.063
27857.092
117433694
123293995
From the above data, we can get the better solution
is Solution 1 with carbon tax, that is that the decision
maker choose to pay carbon tax and setting up 3 large
production capacity plants.
We now analysis the better location strategy can be
changed with the changing value of  , pt , po .
A.
pt  po
In this situation, carbon offsetting is profitable.
Because no matter what carbon constraint level is,
companies can always offset carbon as more as possible,
and then sell their carbon portion in the carbon trading
market. We can view the carbon constraint and the carbon
tax policy as meaningless. When pt is large enough, that
TC2 '  TC1'
'
, where TC is the total cost
Ce1  Ce 2
without carbon related cost and Ce is the exceeding
is
pt  po 
carbon emission, Solution 2 is better, otherwise, if
  po  pt 
TC2'  TC1'
Solution 1 should be the
Ce1  Ce 2
right strategy, conversely, Solution 2 is better because
Solution 1 pays too much tax for its exceeding emissions.
B.
  pt  po
In this situation, neither trading carbon nor carbon
offsetting can decrease cost and paying carbon tax is
always
the
best
strategy.
When 
 po  pt 
TC2'  TC1'
, Solution 1 is better
Ce1  Ce 2
for whatever the carbon constraint level is, like the figure
(   10, pt  15, po  20 ) below. Otherwise, Solution
2 is better. Figure 5 (
pt  180, po  200, L  50000 )
show the total cost change with the changing  .
Figure 4. Total cost with different carbon tax rate
C.
pt    po or pt    po
Pt is smallest now, so companies would trade all their
exceeding emissions and Solution 1 is always better for
whatever carbon tax rate is and whatever carbon
constraint level is.
IV.
CONCLUSIONS
This paper analysis the effects of concurrent
implement of three carbon policies to the multi production
capacity plants location decisions. We formulate a mixed
integer model and present two feasible solutions. By
adjusting carbon tax rate, carbon trading price and carbon
offsetting price, we find that they can change the better
solution. This can be a reference for managers to make
location decisions.
In future research, it is necessary to employ an
effective algorithm to get the optimal solution of this
problem. Another extension of this problem may be
conducted by considering more complex carbon policies,
for example, the fluctuation the carbon trading price.
ACKNOWLEDGMENT
The authors greatly appreciate the anonymous referees
of their valuable and helpful suggestions on an earlier
version of the paper.
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