Dynamic Closed-Loop Supply Chain Model with Product

Remanufacturing

Zong-sheng HUANG1, and Jia-jia NIE

2

12

School of Economics & Management, Southwest Jiaotong University, Chengdu

610031, China; email: chris163@yeah.net; nie_jia@126.com

ABSTRACT

A dynamic closed-loop supply chain model with product remanufacturing is

proposed. In the closed-loop supply chain system, used-product return rate has

dynamic characteristics. The collecting efforts in this period would directly affect the

return rate to increase or decrease compared to the last period. By build the

differential equation about product return rate, whose variation velocity is determined

by investment in the reverse channel, dynamic model of manufacturer collecting used

product is developed. The optimal control strategies of both manufacturer and retailer

are found by the differential game approach. The optimal strategies and optimal profit

of both manufacturer and retailer are analyzed.

1. INTRODUCTIONS

Product remanufacturing has received increased attention in recent days.

Remanufacturing could not only reduce the natural resources needed and waste

produced, but also lower the firm’s product cost (Guide et al., 2002). As a result,

remanufacturing is a kind of production pattern which is not only beneficial to

environment protection and realize low carbon production, but also improve the profit

level of the enterprises. The supply chain contains both forward channel and reverse

channel is called Closed-loop supply chain (Blackburn et al., 2004). There are three

kinds of product collect channel in practice, called manufacturer collect model,

retailer collect model and third-party collect model (Savaskan et al., 2004). This paper

attempts to study the dynamic optimal control strategy in the closed-loop supply

chain composed by a single manufacturer and a single retailer.

As the importance of closed-loop supply chain in environment protection

and sustainable development, there are lots of the research literatures concerning

closed-loop supply chain and remanufacturing. Guide and Wassenhove (2001) studied

the used-product collect problem when facing quality uncertainty. Nakashima et al.

(2004) studied the optimal control problem in remanufacturing system. Savaskan et al.

(2004) studied the three used-product collecting models in the closed-loop supply

chain, which is found that the retailer collect model is the best model for the entire

supply chain member. Savaskan et al. (2006) further studied the reverse channel

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design problem of closed-loop supply chain when retailers competing. Guide et al.

(2006) studied the product value attenuation problem in reverse supply chain.

Subramanian et al. (2009) studied the effect of extended producer responsibility (EPR)

on remanufacturing supply chain.

At present, the research about remanufacturing closed-loop supply chain

adopts a time static models most of the time. The real life changes every second, such

as price, market capacity and collect effect are functions of time, and thus the optimal

solution under the static circumstance is only the firm’s optimal strategy for a short

time. The static model does not consider the manufacturer’s and the retailer’s long run

profit, the optimal solution is just local optimal but not the global optimal. In the

closed-loop supply chain system, used-product return rate has dynamic characteristics.

When the enterprise’s investment in product collecting is large enough, the product

return rate increases with time; and when the enterprise’s investment in product

collecting becomes zero, the product return rate will not become zero for instant, but

decrease with time. The reason of the return rate decrease may come from the

consumer turns to the competitor collect firm. This paper will examine the optimal

return strategy of manufacturer in the dynamic environment and differential game

approach is adopted to solve the open loop control strategy of both manufacturer and

retailer.

2. Model Description and Key Assumptions

In the researches about remanufacturing closed-loop supply chain, it is normal to

set the return rate as time static function of collecting efforts (Savaskan et al. 2004,

2006). Such a model can get the optimal value of the collecting efforts, but could not

get an optimal path over time, thus limited guidance to enterprises in reality. If a

dynamic collecting model could be established, then the manufacturer and retailer

would have more specific guidance in practice. In fact, instantaneous product return

rate has a "cumulative effect" feature, that is, one period’s product return rate is a

function of the cumulative collecting efforts in the past. The collecting efforts in this

period would directly affect the return rate to increase or decrease compared to the

last period. Dividing the collecting efforts cycle 0,T into n small periods, the first

period is 1t and the nth period is nt . The collecting effort in every period

is 1jA j n , and product return rate in each period is 1j j n . According to

the previous description, the product return rate in every single period is the function

of the total collecting efforts of all the previous periods. For the sake of simplicity, we

assume the function f to be linear function. So1

1

1

j

j k

k

f A

,

1

j

j k

k

f A

，…. The

variation of return rate is 1

1

1 1

j j

j j k k j

k k

f A f A f A

. That is to say the

collecting efforts in j period jA would directly impact the variation of the return

rate j compared to the last period return rate 1j . Dividing the time period into

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infinite subdivision and consider about the return rate and collecting effort in every

moment, then the variation velocity of return rate should be the function of the

collecting effort in this moment. As the properties of return rate is similar to the

goodwill model, here ideas were borrowed from the advertising goodwill model

proposed by Nerlove and Arrow (1962), set return rate as state variable, whose

variation velocity is the function of used-product collecting effort and return rate,

therefore

0, 0 0

d tA t t

dt

(1)

Where A is collecting efforts and is return rate. represents the impact of collecting

efforts for return rate. represents the attenuation factor of return rate. It can be seen

from (1), the return rate would increase faster as the firm input larger collecting effort.

However, the firm needs to invest more in order to ensure that the return rate will not

decreased when the attenuation factor is large. In dynamic environment, collecting

effort is time varied function, and so is the return rate.

The key assumptions of the dynamic collecting model:

(1) The instant demand of product is the function of instant price, i.e.

D p t p t (2)

Where 0 represents the capacity of the market, 0, mc represents the

demand parameter. It is consistent with reality that the price and demand here both

changes over time.

(2)The unit cost of the product which manufacturer uses used-product is rc , and

the unit cost of product using new materials is mc . And r mc c , which means the

remanufacturing is profitable. Let m rc c , which represents the unit cost saving

by remanufacturing.

(3)The collecting cost function of manufacturer is 2 2C t kA t , where k

represents manufacturer’s collecting cost coefficient.

(4)To collect the used-product the manufacturer gives the consumer a fix

payment , without loss of generality, assume 0 , which could be seen from the

model analysis that even if 0 , it would not change the result of this paper.

(5)The discount rate is r , and the decision period is 0, .

In the manufacturing collecting model, manufacturer is responsible for the

collecting of used-product as well as the manufacturing of product, determine the

wholesale price w t and the collecting efforts A t in the same time. The retailer is

responsible for the sale of product, determine the retail price p t . The objective

function of manufacturer

2

,0

1max

2

rt

m mw t A t

J e w t c t p t kA t dt

(3)

The objective function of retailer

0

max rt

rp t

J e p t w t p t dt

(4)

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The constraint condition is (1). Section 2 will adopt differential game theory to

resolve the optimal control strategy of both manufacturer and retailer.

3. OPEN-LOOP CONTROL STRATEGIES

Differential game theory is applied to solve the stackelberg game between

manufacturer and retailer, where manufacturer act as a leader. Because of the

complexity of the proposed model, open-loop control strategy was utilized to solve

the differential game; the solution process is referred to Nair and Narasimhan (2006).

The current value Hamiltonian for retailer is given by

, , , ,r r rH p w A p w p A (5)

The necessary conditions for equilibrium are given by

2 0, , r r rr r r

r

H H Hp w A r r

p

(6)

From (6) we obtain the optimal reaction function of retailer

as * 4p w . Taken the retailer’s reaction into consideration, the

Hamiltonian of manufacturer

* * 21, , ,

2m m m mH w A w c p kA A (7)

The necessary conditions of manufacturer for equilibrium are given by

* *

* *

*

1 10, 0

2 2

,

m m

m m

m m

m m m

m

H Hw c kA

w A

H HA r r p

(8)

,r m represent the costate variables for retailer and manufacturer’s optimal

problem. Proposition 1 concludes the optimal control strategy of manufacturer and

retailer.

Proposition 1 The optimal retail price strategy of retailer is given by

2

20

3( )

4 4

tm k

cp t e

(9)

The optimal wholesale price strategy and collecting efforts strategy are given by

2

20( )

2 2

tm k

cw t e

(10)

2

2 202

2( )

2m

tk

kA t e

k

(11)

0 represents the initial value of return rate. Among which,

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2 2 2

1 2

2 2 2

2 2

2 , ,

4

4

m

m m

k k r rk rk

c k r

k c k r

(12)

Proof From (8) it could be derived

* *, 2

mm

cw A

k

(13)

Substituting for * 4p w into (13)

3

4

mcp

(14)

Substitute (13) and (14) into (8)

2

21 1

4 4

m

m

m m m

k

r c

(15)

By solving the differential equations constituted in(15), the optimal control

strategy of manufacturer and retailer could be obtained which are stated in

proposition 1. Assume that the initial return rate 0 , proposition 2 gives the

optimal control strategy properties of manufacturer and retailer.

Proposition 2 The return rate increases over time; retailer’s optimal retail price

and manufacturer’s optimal wholesale price decrease over time, the optimal collecting

efforts of manufacturer increases over time.

Proposition 2 concludes the characteristic of manufacturer and retailer’s optimal

price strategy and collecting strategy. The price would decrease with time in the

dynamic closed-loop supply chain system, which benefit the consumer as well.

Meanwhile, the manufacturer would invest more in the collecting system to improve

the used-product return rate to further save the product cost and improve the profit

level.

The stable value and m of return rate and costate variable m could be

obtained by let 0m in (15), which are given in (12). Substitute and m into

(13) and (14) we could obtain the closed-loop supply chain’s stable retail

price 3 4 4mp c , wholesale price 2 2mw c ,

collecting efforts 2 24mc k rA , and the manufacturer’s

stable instant profit 2 2 22 4 2M

m m k , the retailer’s stable instant

profit 2

4M

r , where 4mc .

As can be seen from the above expression, the collecting efforts, return rate and

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instant profit have positive relationship with the potential market demand mc .

The larger the market demand as well as the lower the product cost, the higher the

collecting efforts level and the return rate.

In dynamic environment, return rate would increase with time to the stable value,

and the corresponding price would decrease with time, the collecting efforts would

increase with time to the stable value. The dynamic collecting model has a greater

significance for decision makers compared to the static model, as it gives guidance

for every moment.

4. CONCLUSIONS

Dynamic closed-loop supply chain model with product remanufacturing is

proposed. By building the differential equation about return rate, which is function of

investment in the reverse channel, dynamic models of manufacturer collecting used

product is developed. The optimal open-loop control strategies are found by the

differential game approach. The optimal strategies and optimal profit of both

manufacturer and retailer are analyzed. It is found that the return rate is increasing

with time, the retail price and wholesale price are decreasing with time, and the

collecting effort is increasing with time.

Due to the complexity of the dynamic environment, only the optimal strategy was

analyzed in the supply chain when manufacturer collecting. In fact, the retailer

collecting model and third-party collecting model are also common in practice, and

thus the further research directions include the optimal control strategy when retailer

collecting and third-party collecting.

REFERENCES

Blackburn J. D., Guide V. D., Souza G. C., et al. (2004). Reverse supply chains for

commercial returns. California Management Review, 46 (2): 6-22.

Guide V. D., Wassenhove L. N. (2001). Managing product returns for

remanufacturing. Production and Operations Management, 10 (2): 142-155.

Guide V. D., Wassenhove L. N. (2002). The reverse supply chain. Harvard Business

Review, 80 (2): 25-26.

Guide Jr., Souza, Wassenhove, Blackburn. (2006). Time value of commercial product

returns. Management Science, 52 (8): 1200-1214.

Nerlove M., Arrow K. J. (1962). Optimal advertising policy under dynamic

conditions. Economica, 39: 129-142.

Nakashima K., Arimitsu H., Nose T., et al. (2004). Optimal control of a

remanufacturing system. International Journal of Production Research, 42(7):

3619-3625.

Nair A., Narasimhan R. (2006). Dynamics of competing with quality and

advertising-based goodwill. European Journal of Operational Research, 175(12):

462-474.

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