OPTIMAL ORDER LOT SIZING AND PRICING

WITH CARBON TRADE

Guowei Hua

Department of Logistics Management, School of Economics and Management, Beijing Jiaotong University

100044, Beijing, P.R. China

Han Qiao

School of Economics, Qingdao University, 266071, Qingdao, P.R. China

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, P.R. China

Jian Li

School of Economics and Management, Beijing University of Chemical Technology

100029, Beijing, P.R. China

Keywords: Carbon trade, EOQ model, Pricing, Carbon footprints.

Abstract: Carbon emission trading is one of the broadly adopted methods to curb the amount of carbon emission. This

paper examines the optimal decisions of retailers under cap-and-trade. We derive the optimal order lot size

and retail price under cap-and-trade when the demand is an additive function or multiplicative function of

retail price, and analyze the impacts of carbon trade on the order decision, pricing decision, carbon emission

and profit.

1 INTRODUCTION

In order to alleviate global warming, many

measurements such as economics, legislation were

taken to curb the total amount of carbon emissions.

Carbon emission trading is generally accepted as one

of the most eﬀective market-based mechanisms,

which has been broadly adopted by UN, EU, and

many governments. For example, the Kyoto

Protocol (UNFCCC, 1997) and the European Union

Emission Trading System (EU-ETS) implement a

mandatory “cap and trade” system in 183 countries

and the 27 EU member countries (EU, 2009),

respectively. More than 20 platforms for trading

carbon are running in the world.

Facing the cap-and-trade, firms can optimize

their strategic decisions such as supply chain design

or operations decisions in production, transportation,

and inventory to reduce carbon emissions. There are

few studies on the operations decisions under carbon

emission regulations. Cachon (2009) discusses how

the new objective of reducing carbon footprints is

likely to aﬀect supply chain operations and

structures. Hua et al. (2010) examined the optimal

order quantity under carbon trade. Benjaafar et al.

(2010) introduce a series of simply 3models to

illustrate how carbon footprint considerations could

be incorporated into operations decisions. Bonney

and Jaber (2010) examined the importance of

inventory planning to the environment and the

possibility of using models to perform analyses.

However, all the researches mentioned-above are not

incorporated pricing into them.

Although there are plentiful studies of purchase

decisions incorporating pricing (Chen and Simchi-

Levi, 2010), they did not incorporate carbon

footprints into them. To fill the gap, in this paper, we

examine the optimal order lot sizing and pricing for

retailers under carbon trade.

The rest of this paper is organized as follows: In

Sections 2 we formulate EOQ model with pricing

under carbon trade, derive the optimal order quantity

and price. In Sections 3 and 4, we analytically and

533

Hua G., Qiao H. and Li J..

OPTIMAL ORDER LOT SIZING AND PRICING WITH CARBON TRADE.

DOI: 10.5220/0003597005330536

In Proceedings of the 13th International Conference on Enterprise Information Systems (DMLSC-2011), pages 533-536

ISBN: 978-989-8425-55-3

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

numerically the impacts of carbon trading on order

decisions, pricing decision, carbon emissions, and

total cost. Finally we conclude the paper and suggest

topics for future research in Section 5.

2 THE MODEL

This section we will formulate the EOQ with pricing

under carbon trade, and derive the optimal order lot

sizing and pricing. Carbon trading is also known as

cap and trade. A ﬁrm is allocated a limit or cap on

carbon emissions. If its amount of carbon emissions

exceeds the carbon cap, it can buy the right to emit

extra carbon from the carbon trading market.

Otherwise, it can sell its surplus carbon credit. We

focus on the carbon emissions caused by logistics

and warehousing activities in this paper.

The notation used in the paper is as follows:

K= ﬁxed ordering cost;

· T = the replenishment time interval;

· h= annual holding cost per unit, expressed as a

percentage of the average inventory value;

· p= the retail price (a decision variable);

· Q = order lot size in units (a decision variable);

· D(p) = annual demand or demand rate, which is

a function of the retail price p;

· w = wholesale price per unit;

· α = carbon emission quotas per unit time;

· C = carbon price per unit (ton);

· CE=the amount of carbon emission;

·e= the amount of carbon emissions in executing

an order;

·gQ = the amount of carbon emissions in holding

Q units product, where g is the variable

emission factor in warehouse;

· X = transfer quantity of carbon emissions (a

decision variable);

(,)Qp

= total proﬁt per unit time;

Following Abad and Aggarwal (2005), we

suppose the demand function satisﬁes:

(i).

() 0Dp>

for

max

0 pp<≤

;

(ii).

()

decreases with increasing p,

i.e.,

() 0Dp<

;

(iii). the marginal revenue

(()) ()

() ()

dpDp Dp

dD p D p

is a strictly increasing

function of p;

where

max

is a large number that the retail

price does not exceed.

Notice that

()

CE e g

, based on the

classical EOQ model, we can formulate our problem

()

max ( , ) ( ) ( )

()

.. .

Dp hQ

Qp p wDp K CX

Dp Q

st e g X

=− − − +

++=

Substituting

()

Xe g

=− +

into the

objective function, we have

max ( , ) ( ) ( )

()()()

Qp p wDp

KCeDp hCgQ

−

−−+

(1)

The ﬁrst-order condition for maximization yields

the optimal retail price

()pQ

for a given Q. Let

(,)

() ( ) () 0

Qp K Ce

Dp p w D p

∂

+−− =

∂

Namely,

()

Dp K Ce

Dp Q

+=+

(2)

Differentiating (2) with respect to Q, we have

*'2'

'2 2

()(2 )

dp Q D DD K Ce

dQ D Q

−+

=−

Based on the above analysis, we have the

following theorems.

Theorem 1. For any given Q, the ﬁrst-order

condition (2) yields the unique maximum

()pQ

Proof. If

()ppQ>

, then

()

pKCe

Dp Q

+>+

(,)

() ( ) () 0

Qp K Ce

Dp p w D p

∂

+−− >

∂

(),ppQ<

the

()

pKCe

Dp Q

+<+

(,)

() ( ) () 0,

Qp K Ce

Dp p w D p

∂

+−− <

∂

which indicate that

()

pQ=

is the unique

maximum of

(,)Qp

for a given Q. □

Next, we will derive the optimal order lot size

and price.

ICEIS 2011 - 13th International Conference on Enterprise Information Systems

534

Substituting

()

pQ=

into

(,)Qp

, we have

(())( )

(, ())

(()) 2

DpQ hCgQ

Qp Q C

DpQ

=− − +

(3)

Since

*'2'

'2 2

()(2 )

dp Q D DD K Ce

dQ D Q

−+

=−

we have

*'2'

(, ())

() (2 )

()

dQpQ

dp Q D D DD h Cg

dQ D

KCehCg

−+

=− −

=−

(4)

Theorem 2.

(1) when

() ,(, 0)Dp a bp ab=− > , then

satisfies

()()()()0,

()max(),

hCgQ abwKCeQbKCe

ππ

+−−+++=

and

222

awKCe

=++

(2) when

() ,( 0, 1)

Dp ap a b

−

=>>

, then

satisfies

() ( ) 0,

()max(),

b K Ce K Ce h Cg

bQ Q

ππ

−

−+ + +

+−=

and

()

bKCe

−

From (2) and (4), we can derive Theorem 2 easily,

and we also can obtain

()

Xe g

=− +

3 THE IMPACT OF CARBON

TRADE ON DECISIONS

Due to the difficulty of the problem, in this section

we will numerically examine the impact of carbon

trade on the order quantity and price. First, we

introduce the following theorem.

Notice that when C=0, our problem is the EOQ

with pricing. In this case, the optimal order lot size

and price can be found from the following formulas.

()

(())

(, ()) ,

(()) 2

Dp K

Dp Q

pQ hQ

Qp Q

DpQ

+=+

=− −

**'2'

(, ()) () (2 )

()

dQpQ dpQDD DD h

dQ dQ D

−

−−

=−

From the above formulas and (2), we can derive the

following Theorem 3 easily.

Theorem 3. If the order quantity is the same as that

without carbon trade, then the retail price should

increase. And if the price is the same as that without

carbon trade, then the order quantity should increase.

Theorem 3 shows that carbon trade increases the

cost of retailer, if his order quantity keeps constant,

he will increase his retail price in order to offset the

increased carbon cost, in other words, the end-

customers will partially pay the cost of low-carbon.

From Theorem 2, we have the following

observations.

Theorem 4. The order quantity, retail price and the

amount of carbon emission are decided by carbon

price, and have nothing to do with carbon emission

quotas.

From the following examples, we can obtain

some new observations.

Example 1. Let D(p)=6000−30p, K=$200/order,

h=$0.4/$/year, w=50, e=500, g=2,

2000

, the

results was summarized in Table 1 and Table 2.

Table 1: The results of Example 1 with increasing C.

(,)Qp

profit

0.2

0.4

0.6

0.8

(1500,125)

(1299,125.1)

(1224.7,125.2)

(1186,125.2)

(1162,125.3)

2249.3

2163.7

2141.3

2132

2127

168450

168630

168820

169000

169190

Table 2: The results of Example 1 with increasing

(,)Qp

CE profit

4000

3000

2500

2000

1500

(1299,125.1)

(1299, 125.1)

2143

169030

168830

168730

168630

168530

Example 2. Let

( ) 4000000 ,Dp p

−

K=$200/order, and h=$0.3/$/year,w=50, e=500,

OPTIMAL ORDER LOT SIZING AND PRICING WITH CARBON TRADE

535

g=2,

3000

, the results was summarized in

Table 3 and Table 4.

Table 3: The results of Example 2 with increasing C.

(,)Qp

CE profit

0.2

0.4

0.6

0.8

(99.5,104)

(84.2,107.1)

(78.2,110.2)

(74.6,113.4)

(71.8,116.7)

3764.3

4222.6

4286.4

4245.1

4159.4

38454

37940

37489

37069

36675

Table 4: The results of Example 2 with increasing

(,)Qp

CE profit

7000

6000

5000

4000

3000

(84.2,107.1)

4222.6

38740

38540

38340

38140

37940

Tables 1-4 show that the order quantity would

decrease but retail price would increase with

increasing the carbon price. The carbon emission

would decrease in an additive demand function but

increase in a multiplicative demand function with

increasing the carbon price. The profit would

decrease with increasing the carbon price, which is

straightforward.

Tables 1-4 also show that the order quantity,

retail price and the amount of carbon emission

would keep constant with decreasing carbon

emission quotas. However, the profit would decrease

with decreasing carbon emission quotas since the

carbon constraint is becoming strict.

4 CONCLUSIONS

To respond to the regulations on carbon emissions, a

firm can optimize their operations decisions in

production, transportation, and inventory to reduce

carbon emissions. This paper examines the jointly

inventory and price decisions with carbon trade, we

derive the optimal order lot size and price based on

the EOQ model. We theoretical analyze the impact

of the carbon price and carbon emission quotas on

the order and price decisions, the carbon emission

and profit. We also present some interesting

observations from numerical tests.

In this paper, we suppose that carbon price has

nothing to do with carbon emission quotas, in fact,

carbon price is effected by carbon emission quotas,

if carbon emission quotas is small, which means the

carbon policy is strict, generally speaking, the

carbon price would increase. So, to examine the

same question in this case is a good further research

direction.

ACKNOWLEDGEMENTS

This research was supported by the NSFC under

grant number 71071015, 71003057, and 70801003,

the Ph.D. Programs Foundation of the Ministry of

Education of China under grant number

20100009120009, and the Fundamental Research

Funds for the Central Universities under grant

number 2009JBZ010-3.

REFERENCES

Benjaafar S., Li Y., Daskin M., 2010. Carbon footprint

and the management of supply chains: Insights from

simple models. http://www.ie.umn.edu/faculty/ faculty/

pdf/beyada-3-31-10.pdf.

Bonney, M., Jaber, M.Y., Environmentally responsible

inventory models: Nonclassical models for a non-

classical era. International Journal of Production

Economics (2010), doi:10.1016/j.ijpe.2009.10.033.

Cachon G. Carbon footprint and the management of

supply chains. The INFORMS Annual Meeting, San

Diego, CA, 2009.

Chen X.., Simchi-Levi D. 2011. Pricing and Inventory

Management. To appear in Handbook of Pricing, eds.

R. Philips and O. Ozalp.

EU, 2009. http://ec.europa.eu/environment/climat/

emission/ets post2012 en.htm.

Hua G.W., Cheng T.C.E., Wang S.Y., Managing carbon

footprints in inventory management. International

Journal of Production Economics (2011),

doi:10.1016/j.ijpe.2011.03.024.

UNFCCC, 2009. Kyoto Protocol. http://unfccc.int/kyoto

protocol/items/2830.php.

ICEIS 2011 - 13th International Conference on Enterprise Information Systems

536