Incentive and Production Decisions for
Remanufacturing Operations
Onur Kaya
∗
Department of Industrial Engineering, Koc University, Istanbul
[email protected]
June 2008; Revised Feb. 2009
Abstract
We consider a manufacturer producing original products using virgin materials and remanufactured
products using returns from the market where the amount of returns depend on the incentive offered by
the manufacturer. We determine the optimal value of this incentive and the optimal production quantities
in a stochastic demand setting with partial substitution. We analyze 3 different models in centralized and
decentralized settings where the collection process of the returns is managed by a collection agency in the
decentralized setting. We also analyze contracts to coordinate the decentralized systems and determine
the optimal contract parameters. Finally, we present our computational study to observe the effect of
different parameters on the system performance.
Keywords: Supply Chain Management, Remanufacturing, Demand Substitution
1
Introduction
Closed-loop supply chain management, product recovery, recycling, remanufacturing and reverse logistics
concepts received considerable attention in the last years and are becoming especially important in the context of sustainable development (Guide et al., 2003, Thierry et al., 1995, Fleischmann et al., 1997,2000,2001
etc.). Companies engage in recycling and remanufacturing processes either because of legislations in their
countries or because of economical reasons. In Europe, recycling of materials is driven by legislations and
∗ Corresponding
author: Onur Kaya, Koc University, Eng 206, Sariyer, Istanbul, 34450, Turkey, Tel: 90-212-3381583,
[email protected]
1
original equipment manufacturers (OEMs) are legally responsible for collecting their end-of-life products.
However, there are several companies in US and other countries, who are not legally bound to collect back
their products but do so to reap economic benefits from this process through remanufacturing.
Remanufacturing is the process of collecting used items, extracting the useful parts and reusing these parts
in the production of new products. Remanufacturing has both economical and environmental consequences.
In addition to saving from direct material costs, companies also save from disposal and energy costs through
remanufacturing. In 1996, Ford avoided disposal of more than 67, 700 pounds of toner cartridges, and saved
$180, 000 in disposal costs (US EPA, 1997). Studies have shown that the unit cost of remanufacturing can be
about 40−60% of the unit manufacturing cost of an original product in some industries (Dowlatshahi, 2000,
Giuntini and Gaudette, 2003). This means that the remaining value in used products may reach significant
levels. In addition to 14 million tons of material savings per year worldwide, an estimated 120 trillion
BTUs/year of energy are saved from remanufacturing globally, accounting for about 16 million barrels of
crude oil and about $500 million in energy costs (Giuntini and Gaudette, 2003). Xerox obtained over $80
million with the implementation of a remanufacturing program in 1997 (Maslennikova and Foley, 2000) and
is a successful example of the benefits that can be achieved by remanufacturing. Ayres et al. (1997) and
Sundin et al. (2005) present several examples and potentials of remanufacturing processes and their benefits
for both the companies and the society.
One of the most important tasks and also the first task that effects all the other activities in remanufacturing is the collection of used products. Some companies collect the used products directly from end
customers, but there are also independent third parties that handle used product collection for OEMs in
some industries like the auto industry. GENCO is such a company that operates return centers totaling
more than four million square feet throughout North America and reintroduces returned items to sales or
manufacturing channels.
Financial incentives offered to product holders or buy-back campaigns as mentioned in Klausner and
Hendrickson (2000) influence the quantity of returns and thus, many companies offer financial incentives to
collect more used products. As mentioned in Guide and Jayaraman (2000), offering the correct amount of
incentive is of great importance for a company to ensure a sufficient number of used products for remanufacturing. In addition, several companies are locating more and more collection centers to be close to the
customers and collect more used products. Aras and Aksen (2008) develop a mixed-integer nonlinear facility
location-allocation model to determine the optimal locations of the collection centers as well as the optimal
incentive values to offer. They develop heuristic algorithms to solve this problem.
In this paper, we consider a manufacturer that produces both original products using virgin materials
and remanufactured products using returns from the customers where the manufacturer needs to offer an
incentive to the customers to collect the right amount of returns to be used in the remanufacturing process.
We determine the optimal value of the incentive to offer and the optimal production quantities of original
2
and remanufactured products in a stochastic demand setting. We analyze this problem in the context of the
newsboy models (see Khouja, 1999, for a review of newsboy problems). Different from the classical newsboy
models, the production quantity of the remanufactured items is constrained with the amount of collected
used products which depends on the incentives offered to the product holders. In addition, we assume that,
remanufactured products and original products are partially substitutable in the market. That is, some of
the customers demanding an original product can buy the remanufactured ones if the original product is out
of stock and vice-a-versa.
In this paper, we analyze both centralized and decentralized business settings. In the centralized setting,
the manufacturer collects the used products directly from the customers and manages all the collection
and the manufacturing processes. However, in the decentralized setting, a third party collection agency
collects the used products from the market and sells them to the manufacturer at a wholesale price. In the
decentralized setting, the collection agency decides on the incentive to offer and the manufacturer decides
on the production amounts.
For both centralized and decentralized settings, we analyze 3 different models. In Model 1, we assume
that the manufacturer only uses used products in the manufacturing process and produces remanufactured
products only. In Model 2, we assume that the manufacturer can use both virgin materials and used products
in the manufacturing process but there is a single end product and the end product is not differentiated
as remanufactured or original product. In this model, it is assumed that reusing the used products in the
manufacturing process does not cause a decrease in the value of the finished product. Remanufactured and
original products are assumed to have the same valuation by the customers. Finally, in Model 3, we assume
that the manufacturer produces both remanufactured and original products and they are sold as separate
products. These products have different customer valuations and different demand functions. In addition,
we assume that these products are partially substitutable products, that is, the customers who can not find
their first-choice product in the inventory, might buy the other product if it is available.
The problem of optimally determining the stocking amounts of substitutable products is a standard
problem in operations literature (See Mahajan and Van Ryzin, 1999, Parlar and Goyal, 1984, Rajaram and
Tang, 2001, Netessine and Rudi, 2002 etc.). However, most of these studies do not consider the issues
related to remanufacturing and the used product collection. In addition, to the best of our knowledge,
the studies analyzing the inventory problems in the remanufacturing industry, do not consider the demand
substitution between the remanufactured and original products. Van der Laan and Salomon (1997) consider
a stochastic inventory system with production, remanufacturing and disposal operations and extend the
push-pull strategies to include the option of product disposal. Van der Laan et al. (1996) compare the
performances of three different procurement and inventory control strategies and Toktay et al. (2000) aim
to find an ordering policy to minimize the total expected procurement, inventory holding and lost sales costs
in a remanufacturing system. Van der Laan and Teunter (2006) also study the inventory problem in the
3
remanufacturing industry and focus on finding closed form expressions for calculating near optimal policy
parameters. They use certain extensions of the (s,Q) policy, called push and pull remanufacturing policies
and present simple, closed form expressions to approximate the optimal policy parameters.
There is a growing amount of literature on remanufacturing and closed-loop supply chains (e.g. Dyckhoff,
2003, Dekker et al., 2004). However, only a scarce portion of this literature considers incentive determination.
Although the importance of effective incentive mechanisms on the success of remanufacturing and product
recovery businesses is acknowledged in several studies (e.g. Guide, 2000, Guide and Van Wassenhove, 2001),
there are only very few number of analytical models that consider incentive determination. Bakal and
Akcali(2006) and Karakayali et al. (2007) are two studies in this stream. However, these papers consider
remanufacturing issues in isolation from the original equipment market, without considering the effects
of remanufactured items on original products, like the case in our Model 1. In addition, they consider
deterministic demand functions and focus on pricing problems.
A number of authors have studied remanufacturing models where an OEM produces both new and
remanufactured products. Tang and Teunter (2006) analyze the lot scheduling problem with two production
sources, manufacturing and remanufacturing, where all operations are performed on the same line. Teunter et
al. (2008) study the same lot scheduling problem assuming that the operations are performed on separate,
dedicated lines and develop an exact algorithm for finding the optimal common cycle-time policy. They
also compare their results with the ones in Tang and Teunter (2006)and analyze the cost benefits of using
dedicated lines. Guide et al. (2005) compare the performances of several static scheduling rules in a basic
remanufacturing shop considering the disassembly and reassembly of products. Savaskan et al. (2004)
analyze the problem of determining the best reverse channel structure to collect the returns. They consider
three options for collecting the used products: (1) Directly from the customers, (2) Utilizing the retailers, (3)
Subcontracting the collection activity to a third party. They assume deterministic demand and return rates
as functions of prices and incentives and also analyze how the closed-loop supply chain structures influence
the incentives to invest in used-product collection and the product return rates. Inderfurth (2004) considers
downward product substitution in the manufacturing/remanufacturing industry in the determination of
the optimal inventory levels and characterizes the structure of the optimal solution that coordinates the
manufacturing and remanufacturing processes.
There are also several researchers who analyze competition in the remanufacturing industry assuming
deterministic demand and product returns. Majumder and Groenevelt (2001) study a two-period model
with one OEM and one independent remanufacturer where the companies compete in the sales market and
in the procurement process. They aim to find the optimal prices and production quantities in a deterministic
setting. Ferrer and Swaminathan (2006) extend the work of Majumder and Groenevelt (2001) to multi-period
models and investigate the effects of various parameters on the system. Heese et al. (2005) study a model of
two OEMs and characterize the conditions under which it is profitable for the OEMs to sell remanufactured
4
products along with original products considering the effect of product substitutability. Debo et al. (2005),
Ferguson and Toktay (2005), Webster and Mitra (2007) are the other studies that deal with competition in
a two-period setting.
In contrast to the above papers, we consider the optimal incentive determination problem in addition
to determining the optimal production quantities in the manufacturing/remanufacturing industry using
stochastic demand functions and considering partial substitution of demand in our models. We analyze
this problem considering 3 different business models in the remanufacturing industry in centralized and
decentralized settings. We also investigate the effect of centralization for the remanufacturing companies
and the value of elimination of the third party collection agencies by building their own collection centers
and collecting the end products directly from end users.
2
Model
In our study, we consider a manufacturer that produces both original products using virgin materials and
remanufactured products by collecting used products and reusing them in the manufacturing process. In
our centralized models, the manufacturer offers to pay a certain amount of money, denoted by t, to collect
the right amount of used materials from the market. The amount of returns collected from the market is
denoted by Q(t) which is a function of t. In our analysis, we assume that, the collection amount Q(t) is a
deterministic, increasing and convex function of t. However, in section 5, we also analyze an extension of our
model where Q(t) is a stochastic function of t. In the decentralized models, the collection agency is the one
that collects the used materials from the market and thus is the one that decides on the value of t. After
collecting the used materials from the market, the collection agency sells them to the manufacturer at a unit
wholesale price w, the value of which is also decided by the collection agency.
The cost of producing remanufactured products from the used materials is denoted by cr and the unit
production cost of the original products from virgin materials is denoted by cm . Note that cr includes all
the collection, inspection, disassembly and remanufacturing costs. If cr > cm , then there is no economical
benefit for the manufacturer in remanufacturing. Thus, we assume that cm > cr . The difference of cm − cr
can be thought as the value of the used products. We note that the values of the used products might be
different from each other since they might be in different conditions when returned, however, we think of
cm − cr as the average values of the used products and assume identical returns. We also assume that the
manufacturer uses one unit of the used products to produce one unit of the remanufactured product.
In our models, we denote the selling price for the remanufactured products with pr and the selling price
for the original products with pm . Demands for the remanufactured and original products, denoted by Dr
and Dm respectively, are stochastic functions. We denote the density functions of Dr and Dm with fr and
5
Table 1: List of symbols
t
Amount of the per unit incentive offered to product holders for used products
Q(t)
Amount of the collected used products depending on the value of the incentive
w
Unit wholesale price that the collection agency sets for the used products
cr
Unit cost of remanufacturing
cm
Unit cost of manufacturing of original products
pr
Market price of the remanufactured products
pm
Market price of the original products
qr
Number of remanufactured products produced
qm
Number of original products produced
Dr
Initial demand for the remanufactured products
Dm
Initial demand for the original products
Fr (.)
Cumulative distribution function for the initial demand of the remanufactured products
Fm (.)
Cumulative distribution function for the initial demand of the original products
αrm
Rate of substitution from the remanufactured products to original products
mr
Rate of substitution from the original products to remanufactured products
α
Rr
Effective demand for the remanufactured products
Rm
Effective demand for the original products
fm and the c.d.f’s with Fr and Fm . We model the system in a single period newsvendor setting and in this
model, the manufacturer needs to decide on the amount of remanufactured products, denoted by qr , and
the amount of original products, denoted by qm , to produce. In addition, the manufacturer also needs to
decide on how much money to offer, denoted by t, for the used materials which he needs for the production of
remanufactured products. In our model, if the demand for the original products exceed the available quantity
qm , then a portion of the excess demand customers, denoted by αmr , buys the remanufactured products, if
they are available. Similarly, if the demand for the remanufactured products exceed the available quantity
qr , then a portion of the excess demand customers, denoted by αrm , buys the original products, if they
are available. Thus, the effective demand for the remanufactured and original products are denoted by
Rr = Dr + αmr [Dm − qm ]+ and Rm = Dm + αrm [Dr − qr ]+ , respectively. A list of the symbols used in the
paper is presented in Table 1.
We start our analysis in section 3.1 with Model 1, by considering the remanufactured products in isolation
from the original equipment production. Then in section 3.2, we consider Model 2 where the remanufactured
products are valued the same as the original products and the remanufactured and original products are
viewed as perfect substitutes. Then, in section 3.3, we present our third model where the remanufactured
products and original products are valued differently in the market and the demand for these products
are partially substitutable. In all of these sections, we consider both centralized and decentralized models.
6
Then, in section 4, we focus on contracts to coordinate the decentralized models and in section 5, we consider
stochastic return functions. Finally in sections 6 and 7, we present our computational results and conclusion,
respectively. All the proofs are presented in the online supplement to this paper.
3
Analysis and Results
3.1
Model 1: Pure Remanufacturing
In recent years, in addition to OEMs who start to integrate remanufacturing into their business structure,
there are also several companies that only do remanufacturing by collecting used products from the market.
In this section, we start our analysis by considering such companies that only do remanufacturing by collecting used materials from the market and reselling them after making repairs and completing the required
modifications and processes on them. These companies do not produce any original products and can not
survive their manufacturing processes in the absence of used materials that they collect from the market.
Turner Engineering, who is the remanufacturer of Land Rover engines, cylinder heads and associated components is an example of these companies. These companies depend highly on the collection of used products
since they can not produce the end products without the used products.
3.1.1
Centralized Model
In the centralized model, the remanufacturer collects the used products directly from the market and thus
is the one that decides both on the optimal value of t, the amount of the incentive to offer in order to collect
the used materials, and qr , the amount of remanufactured products to produce. So, the remanufacturer’s
problem is:
max π
t,qr
Z
=
+
pr E[Dr ] − pr E[Dr − qr ] − cr qr − tQ(t) =
qr
pr xfr (x)dx + pr qr F̄r (qr ) − cr qr − tQ(t)
0
s.t
qr ≤ Q(t)
t≥0
Observe that the objective function is jointly concave in t and qr and the constraints are convex functions.
Thus, the problem is a concave optimization problem over a convex set and thus the optimal solution satisfies
the first-order optimality conditions.
0
Proposition 1. Let t0 be the value of the incentive that satisfies the relation: Fr (Q(t0 )) =
Then, the optimal solution is:


(t0 , Q(t0 ))
∗ ∗
(t , qr ) =

(0, min{Fr−1 ( pr −cr ), Q(0)})
pr
7
if t0 ≥ 0
otherwise
∂t
pr −cr −t0 −Q(t0 ) ∂Q(t
0)
pr
.
3.1.2
Decentralized Model
In this section we assume that the used products are collected by a collection agency and this collection
agency sells these returns to the remanufacturer at a unit wholesale price w. In this case, the profit function
for the remanufacturer is denoted by πr and the profit function for the collection agency is denoted by πl
where
πr = pr E[Dr ] − pr E[Dr − qr ]+ − cr qr − wqr
πl = wqr − tQ(t)
We solve the remanufacturer’s problem first and find the optimal production quantity as below depending
on the value of w.
qr∗ = Fr−1 (
pr − cr − w
)
pr
Then, we solve the collection agency’s problem which is:
max πl
t,w
= wqr − tQ(t)
s.t
qr ≤ Q(t)
t≥0
The optimal solution to this problem is given in the following proposition:
Proposition 2. Let qr0 be the value of the incentive that satisfies the following relation.
pr − cr − pr Fr (qr0 ) − Q−1 (qr0 ) − pr qr0 fr (qr0 ) − qr0
∂Q−1 (qr0 )
=0
∂qr0
Also, let t0 and qr00 be the values that satisfy the equations Q(t0 ) = qr0 and pr − cr − pr Fr (qr00 ) − pr qr00 fr (qr00 ) = 0,
respectively. Then, the optimal solution satisfies the following.



(t0 , qr0 )
if t0 ≥ 0



(t∗ , qr∗ ) = (0, qr00 )
if t0 < 0 and qr00 ≤ Q(0)




(0, Q(0))
if t0 < 0 and qr00 > Q(0)
In all cases, the optimal wholesale price is w∗ = pr − cr − pr Fr (qr∗ ).
3.2
Model 2: Equally Valued Remanufactured and Original Products
There are many companies that produce both original products using virgin materials and remanufactured
products using the returns from the customers. In some industries, the remanufactured products have a lower
valuation than the original products in the eyes of the customers and thus they have different demands and
are sold at different prices in the market. However, in some other industries, the remanufactured products
8
are treated the same as the original products and the end products are not labeled as remanufactured or
original. Remanufactured and original products are considered as perfect substitutes in these industries.
Single-use cameras are an example of these types of products.
These companies collect the used products from the market, extract the useful parts, make the necessary
repairs or modifications and use them in their manufacturing process as original parts. The remanufacturing
processes do not affect the performance or the expected life of the end product. Thus, these companies do
not label their products as remanufactured or original even though some or all of the parts that they use in
the manufacturing process are remanufactured items. The reused materials in these industries are generally
not the critical parts for the end product and they do not cause any significant loss in the value of the
product. Note that, these companies can survive their manufacturing processes even in the absence of used
materials by using original materials but probably at a higher cost. Recycling of paper, metal, cans, plastic
or glasses can also be considered under this model since the recycled products are considered the same as
original products in the market and the production company can produce these products either by using
original materials from scratch or through recycling.
In this section, we assume that the customers have the same valuations for the original products produced
from virgin materials and the products produced by using the returns. Thus, these two types of products
have the same price and demand in the market and customers are indifferent between them.
Although producing end products using returns is less costly than using completely original materials,
to collect the returns and extract the useful parts from them also brings an extra cost to the company. We
assume that the unit cost of producing the end products from original materials is cm , the company offers $t
per unit of the returns and the total cost of extracting the useful parts from the returns and producing the
end products using these parts in the manufacturing process is cr . We assume that cr < cm . Otherwise, the
company has no benefit in collecting returns and only uses original materials in their production process.
Since the customers are indifferent between these two types of products, we denote the market price for
both types with p and the density and cumulative distribution functions for the total demand of the original
and remanufactured products with f (x) and F (x), respectively. In this section, we analyze the decisions for
the optimal amount of incentive to offer to product holders to obtain the returns and the amount of the
end products to produce. We again consider centralized and decentralized models depending on the party
collecting the returns.
3.2.1
Centralized Model
In the centralized model, the manufacturer collects the used products directly from the market and makes
all the decisions. Let qr denote the amount of production done using the returns and qm denote the amount
9
of production using original materials. The manufacturer’s problem in this case is:
max π
t,qr ,qm
=
s.t
pE[D] − pE[D − qr − qm ]+ − cm qm − cr qr − tQ(t)
qr ≤ Q(t)
t≥0
Since the objective function is jointly concave in t, qr and qm the optimal solution satisfies the first-order
optimality conditions.
Proposition 3. Let qr0 , t0 , qr00 and t00 be the solutions of the equations pF (qr0 ) = p−cr −Q−1 (qr0 )−qr0
Q(t0 ) = qr0 , cm = cr +Q−1 (qr00 )+qr00
∂Q−1 (qr00 )
∂qr00
∂Q−1 (qr0 )
,
∂qr0
and Q(t00 ) = qr00 , respectively. Then, the optimal solution satisfies
the following:
∗
(t
∗
)
, qr∗ , qm
3.2.2
=



(t0 , qr0 , 0)





(t00 , q 00 , F −1 ( p−cm ) − q 00 )
r
r
p
00
and t0 ≥ 0
00
and t00 ≥ 0
m
if F −1 ( p−c
p ) ≤ qr
m
if F −1 ( p−c
p ) ≥ qr

r

(0, F −1 ( p−c

p ), 0)




(0, Q(0), [F −1 ( p−cm ) − Q(0)]+ )
p
r
if none of the above and F −1 ( p−c
p ) ≤ Q(0)
otherwise
Decentralized Model
In this section, since the used products are collected by a collection agency and sold at a unit wholesale price
w to the manufacturer, the profit functions for both companies are:
πr = pE[D] − pE[D − qr − qm ]+ − cr qr − wqr − cm qm
πl = wqr − tQ(t)
We solve the manufacturer’s problem first and find the optimal production quantity in terms of w and then
we solve the collection agency’s problem which is:
max πl
t,w
s.t
= wqr − tQ(t)
qr ≤ Q(t)
t≥0
Note that the manufacturer will not buy any used materials from the collection agency if w > cm − cr . Thus,
in the optimal solution w ≤ cm − cr . Then, we state the following lemma for the production quantities at
the manufacturer.
Lemma 1. For a given value of w and t, the optimal production quantities are:


(F −1 ( p−cr −w ), 0)
if w < cm − cr
p
∗ ∗
(qr , qm ) =

(min{F −1 ( p−cm ), Q(t)}, F −1 ( p−cm ) − qr∗ )
if w = cm − cr
p
p
10
(3.1)
Then, we solve the collection agency’s problem to find the optimal value of w and t.
Proposition 4. Let qr0 , t0 , qr00 , t00 and t000 be the values that satisfy the equations
p − cr − pF (qr0 ) − Q−1 (qr0 ) − pqr f (qr0 ) − qr0
∂Q−1 (qr0 )
∂qr0
= 0, Q(t0 ) = qr0 ,
00
p − cr − pF (qr00 ) − pqr00 f (qr00 ) = 0, (cm − cr − t00 ) ∂Q(t
∂t00
)
m
− Q(t00 ) = 0 and Q(t000 ) = F −1 ( p−c
p ), respectively
and let w0 = p − cr − pF (qr0 ) and w00 = p − cr − pF (qr00 ). Then, the optimal solution satisfies:
(t∗ , w∗ ) =
3.3


(t0 , w0 )





(t00 , c − c )


m
r


if w0 < cm − cr and t0 ≥ 0
m
if w0 ≥ cm − cr and t00 ≥ 0 and Q(t00 ) ≤ F −1 ( p−c
p )
m
(t000 , cm − cr )
if w0 ≥ cm − cr and t000 ≥ 0 and Q(t00 ) > F −1 ( p−c
p )




00


if none of the above and w00 ≤ cm − cr and qr00 ≤ Q(0)
(0, w )




(0, min{cm − cr , p − cr − pF (Q(0))})
otherwise
Model 3: Partially Substitutable Remanufactured and Original Products
In this section, we consider a manufacturer that produces both original products using virgin materials and
remanufactured products using the returns. Different from the previous section, in this case, the original
product and the remanufactured product has different characteristics and different valuations in the market.
Thus, the demand and the market price for these two products are different from each other. Generally,
original products are valued higher than the remanufactured ones by the customers.
In this section, we denote the demands for the remanufactured and original products by Dr and Dm
respectively, with density functions fr and fm and c.d.f’s Fr and Fm . In our model, we also consider the
overflow of demand in case of the absence of one of the products in the market. That is, if the demand
for the original products exceed the available quantity qm , then a portion of the excess demand customers,
denoted by αmr , goes to buy the remanufactured products, if they are available. Similarly, if the demand
for the remanufactured products exceed the available quantity qr , then a portion of the excess demand
customers, denoted by αrm , goes to buy the original products, if they are available. Thus, the effective
demand for the remanufactured and original products are denoted by Rr = Dr + αmr [Dm − qm ]+ and
Rm = Dm + αrm [Dr − qr ]+ with density functions hr and hm and c.d.f.s Hr and Hm , respectively. Note
that the functions hr and Hr depend on the decision variable qm and the functions hm and Hm depend on
the decision variable qr .
11
3.3.1
Centralized Model
In the centralized model, the manufacturer collects the used products directly from the market and makes
all the decisions. The manufacturer’s problem in this case is:
max π
t,qr ,qm
s.t
= pr E[Rr ] − pr E[Rr − qr ]+ + pm E[Rm ] − pm E[Rm − qm ]+ − cm qm − cr qr − tQ(t)
qr ≤ Q(t)
t≥0
If we ignore the last term in the objective function, the remaining portion is demonstrated to be jointly
concave w.r.t qr and qm in Parlar and Goyal (1984). Since Q(t) is also assumed to be a concave function
w.r.t t and using the additivity property of the concave functions, this objective function is jointly concave
w.r.t. qr , qm and t. Thus, the first order conditions will supply the optimal solution.
0
Proposition 5. Let qr0 and qm
be the solutions of the following two equations and t0 be the value that satisfies
Q(t0 ) = qr0 .
0
, Dr > qr0 ) = pr − cr − t0 − qr0
pr P (Rr < qr0 ) + pm αrm P (Rm < qm
∂t0
∂qr0
0
0
pm P (Rm < qm
) + pr αmr P (Rr < qr0 , Dm > qm
) = pm − cm
00
Also let qr00 and qm
be the solutions to the following two equations.
00
pr P (Rr < qr00 ) + pm αrm P (Rm < qm
, Dr > qr00 ) = pr − cr
00
00
pm P (Rm < qm
) + pr αmr P (Rr < qr00 , Dm > qm
) = pm − cm
000
Also let qm
be the solution to the following equation
000
000
pm P (Rm < qm
) + pr αmr P (Rr < Q(0), Dm > qm
) = pm − cm
Then, the optimal solution satisfies the following:


0

(t0 , qr0 , qm
)



∗ ∗ ∗
00
(t , qr , qm ) = (0, qr00 , qm
)




(0, Q(0), q 000 )
m
if t0 ≥ 0
if t0 < 0 and qr00 ≤ Q(0)
otherwise
Note that, finding the probabilities in the above equations and solving these equations simultaneously
are not easy tasks and are not always possible for many demand distribution functions due to the terms
[Dr − qr ]+ and [Dm − qm ]+ in Rm and Rr and the correlation between the demand functions. For normally
distributed demand functions Dr and Dm , Rajaram and Tang [31] developed a ”service-rate based heuristic”
to approximate the distributions of Rm and Rr . In this ”service-rate based heuristic”, type 2 service rate (Lee
and Nahmias [25]), which represents the proportion of expected demand satisfied from on-hand inventory,
12
is used. For an inventory level of qi units, type 2 service rate is denoted with γ(qi ) =
R qi
0
xfi (x)dx+qi F̄i (qi )
µi
E[min{qi ,Di }]
E[Di ]
=
for i ∈ {m, r}
For computational purposes, we employ the ideas in the ”service-rate based heuristic” developed by
Rajaram and Tang [31] to approximate the optimal solution for this problem. Recall that the effective demand
for the remanufactured products is denoted by Rm = Dm + αrm [Dr − qr ]+ = Dm + αrm Dr (1 −
In the ”service-rate based heuristic”, the term
min{Dr ,qr }
Dr
is replaced with
E[min{Dr ,qr }]
E[Dr ]
min{Dr ,qr }
).
Dr
= γ(qr ) which is
type 2 service rate. Rajaram and Tang [31] ran simulations to test this approximation and state that the
gap for this approximation is about 1% when qr is one standard deviation above the mean and that the gap
is negligible otherwise. We use the same approximation for the effective demand of the original products.
Then we denote the approximations for the effective demands for the remanufactured and original products
with the functions:
R̂m = Dm + αrm Dr (1 − γ(qr ))
R̂r = Dr + αmr Dm (1 − γ(qm ))
Observe that γ(qr ) and γ(qm ) are constants for given qr and qm values and thus the approximations R̂m
and R̂r are sum of normally distributed functions and thus also normally distributed with means µ̂R
m and
R
R
µ̂R
r and standard deviations σ̂m and σ̂r where:
rm
µ̂R
µr (1 − γ(qr ))
m = µm + α
mr
µ̂R
µm (1 − γ(qm ))
r = µr + α
p
R
2 + [αrm σ (1 − γ(q ))]2 + 2ρσ σ αrm (1 − γ(q ))
σ̂m
= σm
r
r
r m
r
p
σ̂rR = σr2 + [αmr σm (1 − γ(qm ))]2 + 2ρσr σm αmr (1 − γ(qm ))
Then, using these normally distributed functions for Rm and Rr , we can approximate the optimal solution
for this problem.
3.3.2
Decentralized Model
In this section, since the used products are collected by a collection agency and sold at a unit wholesale price
w to the manufacturer, the profit functions for both companies are:
max πr = pr E[Rr ] − pr E[Rr − qr ]+ − cr qr − wqr + pm E[Rm ] − pm E[Rm − qm ]+ − cm qm
qr ,qm
πl = wqr − tQ(t)
13
(3.2)
We solve the manufacturer’s problem first and find the optimal production quantity in terms of w and then
we solve the collection agency’s problem which is:
max πl
t,w
= wqr − tQ(t)
s.t
qr ≤ Q(t)
t≥0
Observe that, in the optimal solution, either Q(t) = qr or t = 0 should be satisfied. Otherwise, if t > 0
and Q(t) > qr in the optimal solution, then the collection agency can increase his profit by decreasing t
which decreases Q(t) until Q(t) = qr or t = 0.
Then, for a given value of w, the manufacturer’s problem is:
max π = pr E[Rr ] − pr E[Rr − qr ]+ − cr qr − wqr + pm E[Rm ] − pm E[Rm − qm ]+ − cm qm
qr ,qm
(3.3)
Observe that this objective function is also a jointly concave function w.r.t. qr and qm and thus, solving the
first order optimality conditions
∂π
∂qr
= 0 and
∂π
∂qm
= 0, simultaneously, we can get the optimal qr and qm
values in terms of w. We get the following two equations from the first order optimality conditions:
pr P (Rr < qr ) + pm αrm P (Rm < qm , Dr > qr ) = pr − cr − w
pm P (Rm < qm ) + pr αmr P (Rr < qr , Dm > qm ) = pm − cm
(3.4)
Then, the collection agency’s problem is to maximize:
πl = wqr − tQ(t)
Although, we can not provide a closed form solution for this problem, we outline the solution procedure
below.
Recall that, in the optimal solution, either Q(t) = qr or t = 0 should be satisfied. First, we assume that
t > 0 and Q(t) = qr in the optimal solution. In this case, we can write all the terms in this objective function
in terms of w using the equations 3.4 and find the value of w∗ that maximizes this objective value. Then we
∗
find the value of qr∗ and qm
using w∗ and equations 3.4, and also get the value of t∗ by solving the equation
Q(t∗ ) = qr∗ . If the resulting t∗ > 0, then this solution will be optimal.
Otherwise t∗ = 0 will be the optimal value of the incentive and the collection agency needs to maximize the
objective function πl = wqr . For this objective function, we also use the equations 3.4 and find the value of w∗
∗
that maximizes this objective function. Then, we find the value of qr∗ and qm
by solving the manufacturer’s
problem using this w∗ . If qr∗ ≤ Q(0), then this solution is optimal. Otherwise, t∗ = 0, qr∗ = Q(0) and
∗
w∗ = pr − cr − pr P (Rr < qr ) − pm αrm P (Rm < qm , Dr > qr ) will be the optimal values. In addition, qm
will
be the solution to the following equation: pm P (Rm < qm ) + pr αmr P (Rr < qr , Dm > qm ) = pm − cm .
14
Note that, for computational purposes, we again use the same approximation technique that we presented
in the centralized case.
4
Coordination through Contracts
We observe that in the decentralized models, a lower incentive value is offered and lower amounts of remanufacturing is done as opposed to centralized models and lower total system profits are obtained. Thus, in this
section we search for contracts to coordinate the decentralized systems to eliminate the efficiency losses and
to achieve the centralized model results. There are various contracts that are used to coordinate the supply
chains. See Cachon [4], Tsay et al. [38] and Yano and Gilbert [44] for detailed surveys of supply chain
contracts and supply chain coordination. In this section, we only focus on linear contracts with transfer
payments but similar results can be obtained with other types of contracts. In linear contracts with transfer
payments, a unit price w and a fixed transfer price A is written on the contract. In this contract, A is a one
time that the manufacturer pays to the collection agency and w is the per unit payment.
Proposition 6. For all three models described above, a linear contract with transfer payments


(tc + q c ∂tcc , πl − RPl )
if tC > 0
r ∂qr
(w, A) =

(0, RPl )
if tC = 0
(4.1)
can coordinate the supply chain with arbitrary profit division and achieve the results obtained by the centralized
model where tc and qrc are, respectively, the optimal incentive value and the quantity of remanufactured
products produced in the centralized model, πl is the profit of the collection agency with this w and RPl is
the reservation profit of the collection agency.
Note that, there might be many other contracts (e.g. revenue sharing, quantity discount etc.) that also
coordinates this supply chain but we omit the analysis for different types of contracts here for brevity.
5
Stochastic Return Functions
In this section, we analyze an extension of our models in which the amount of returns, Q(t), is a stochastic
function of t. For this case, let gt and Gt denote the density and cumulative distribution functions of Q(t)
as functions of t.
15
5.1
Centralized Models
In the centralized models, first, we assume that the manufacturer sets the incentive value t and collects the
returns from the market where the amount of returns is a stochastic function of t. Then, depending on the
amount of the collected used products, the manufacturer makes his production decisions qr and qm . In all
our three models, the manufacturer’s problem is composed of two stages. First, the manufacturer needs to
decide on t and after seeing the collection amount Q, the manufacturer decides on the production quantities.
We solve this problem going backwards and solve the second stage problem first for a given value of Q. We
can state the following proposition for the case in Model 1.
Proposition 7. For a given Q, the optimal production quantity, qr∗ satisfies:
qr∗ = min{Fr−1 (
Proof.
pr − cr
), Q}
pr
For a given Q, the manufacturer’s problem is:
Z qr
M ax
pr xfr (x)dx + pr qr F̄r (qr ) − cr qr
0
s.t. qr ≤ Q
(5.1)
Since the objective function is concave w.r.t qr , looking at the first order derivative of the objective
r
function, we get: qr0 = Fr−1 ( prp−c
). So, if Q ≥ qr0 , qr0 is the optimal solution. However, if Q < qr0 ,
r
the optimal production quantity will be equal to Q. Thus, the optimal production quantity, qr∗ satisfies
r
), Q}.
qr∗ = min{Fr−1 ( prp−c
r
We can do the same analysis for models 2 and 3 and find the optimal production quantities for given
values of Q. Then, we move to the first stage problem, where the manufacturer needs to determine the value
of t to maximize his objective function. Again, for Model 1, the manufacturer’s problem in the first stage is:
Z
max π
A
=
t
{pr E[Dr ] − pr E[Dr − y]+ − cr y}gt (y)dy
y=0
+ (pr E[Dr ] − pr E[Dr − A]+ − cr A)Ḡt (A) − tEt [Q]
s.t.
t≥0
r
where A = Fr−1 ( prp−c
). Since, the objective function is a concave function w.r.t. t, we can look at the first
r
order derivative and equating to 0 will give the optimal solution if the resulting t ≥ 0. Otherwise t∗ = 0 will
be the optimal incentive value.
∂π
∂t
Z
A
=
{pr E[Dr ] − pr E[Dr − y]+ − cr y}
y=0
−
(pr E[Dr ] − pr E[Dr − A]+ − cr A)
16
∂gt (y)
dy
∂t
∂Gt (A)
∂Et [Q]
− Et [Q] − t
=0
∂t
∂t
Also, one can find the optimal value of t using one-dimensional search algorithms.
Note that the same analysis can also be done for models 2 and 3 and the optimal value of the incentive
can be found using the same approach as above. For brevity, we omit the analysis for models 2 and 3 here.
5.2
Decentralized Models
In the decentralized models, first, the collection agency sets the incentive value t and collects the returns
from the market. Then, depending on the amount of the collected used products, the collection agency sets
the wholesale price w and the manufacturer buys the used products from the collection agency and makes
his production decisions qr and qm . In all the three models, the problem is now composed of three stages
and again, we solve this problem going backwards and solve the third stage problem first. In the third stage,
the manufacturer needs to decide on the production quantities qr and qm for given values of w and Q. For
Model 1, using the same reasoning as in proposition 7, we can state that qr∗ = min{Fr−1 ( pr −cprr −w ), Q}.
Then, we go to the second stage and solve for w for a given value of Q. In the second stage, the collection
agency aims to maximize his profit function πl = wqr assuming a given value of Q. Assume that in the
optimal solution Fr−1 ( pr −cprr −w ) > Q. However, in that case, observe that the collection agency can increase
his profit by increasing w which does not change qr∗ . Thus, in the optimal solution, Fr−1 ( pr −cprr −w ) ≤ Q and
qr∗ = Fr−1 ( pr −cprr −w ). Then, using this relation, we write the second stage problem as:
max πl = wqr
qr
s.t.
= (pr − cr − pr Fr (qr ))qr
qr ≤ Q
By looking at the first order derivative of the objective function, let qr0 be the solution of the following
equation.
∂πl
= pr − cr − pr Fr (qr ) − pr qr fr (qr ) = 0
∂qr
For the given value of Q, if qr0 ≤ Q, then qr0 will be optimal and qr∗ = qr0 . However, if qr0 > Q, then due to
the concavity of the objective function, qr∗ = Q. In both cases, w∗ = pr − cr − pr Fr (qr∗ ).
Finally, we solve the first stage problem to find the optimal value of t. We can write the collection
agency’s problem in the first stage as:
Z
max πl
t
=
qr0
Q=0
s.t.
(pr − cr − pr Fr (Q))Qgt (Q)dQ + Ḡt (qr0 )(pr − cr − pr Fr (qr0 ))qr0 − tEt [Q]
t≥0
Since, the objective function is a concave function w.r.t. t, we can look at the first order derivative and
equating to 0 will give the optimal solution if the resulting t ≥ 0. Otherwise t∗ = 0 will be the optimal
incentive value. Note that we can also find the optimal value of t using one-dimensional search algorithms.
17
The same analysis can also be done for models 2 and 3 and the optimal values can be found using the
same approach as above. For brevity, we omit the analysis for models 2 and 3 here.
6
Computational Experiments
We developed several computational experiments to compare the centralized and decentralized collection
methods for all our models and to see the effect of various parameters on the optimal decisions. In this
section, we only consider the deterministic return function models. We assume that the amount of returns
is a linear function of t where Q(t) = a + bt. For our base case in Table 2, we use a = 40 and b = 5. In
addition, for Model 1, we assume that pr = 80, cr = 30 and the demand for the remanufactured products is
normally distributed with a mean µr = 100 and a standard deviation σr = 20. For Model 2, we assume that
p = 80, cm = 60, cr = 30 and D ∼ N [100, 20]. For Model 3, we assume that pm = 100, pr = 80, cm = 60,
cr = 30, Dm ∼ N [100, 20], Dr ∼ N [100, 20], ρ = 0 and αmr = αrm = 0.5.
In Table 2, we present the optimal values of the decision variables for both centralized and decentralized cases of Models 1, 2 and 3. We observe for all three models that, less incentive is offered and less
remanufacturing is done in the decentralized models as opposed to the centralized cases. We observe in
Model 1 that there is an efficiency loss of more than 10% between the centralized and decentralized cases
and the collection agency can extract most of the profit in the decentralized model 1. Thus, centralizing
the system and collecting their own used products directly is extremely critical for remanufacturers doing
remanufacturing only.
When we look at the results of Model 2, we observe that no original product is produced in Model 2 and
the same results are obtained in the centralized cases of Models 1 and 2 since doing remanufacturing is less
costly and the end products are not differentiated as new or remanufactured in Model 2. However, when
we compare the decentralized cases of Models 1 and 2, although no original product is produced again, the
amount of remanufactured products is much higher in Model 2 than Model 1 since the manufacturer uses the
possibility of producing original products as a threat against the collection agency to decrease the wholesale
price. Thus, the collection agency charges a lower price and the manufacturer produces more remanufactured
items and the efficiency loss between centralized and decentralized cases is much lower in Model 2 as opposed
to Model 1. However, still, there is a huge difference between the profits of the manufacturer in centralized
and decentralized cases and the collection agency can extract a big portion of the profit.
For both Models 1 and 2, the amount of total production is equal to the amount of collected returns and
the system is highly sustainable. However, when we look at the results for Model 3, we observe that a high
level of original products is produced since the end products are differentiated as new or remanufactured
and both types have their own demand. Thus, the manufacturer can increase his profit by producing from
18
Table 2: Comparison of the Results for Centralized and Decentralized Cases for all Models
qrc
c
qm
tc
πc
qrd
d
qm
wd
td
πrd
πld
πd
πc
πd
Model 1
88.5
-
9.7
3285.5
66.95
-
46.1
5.4
230.85
2723
2954
1.112
Model 2
88.5
0
9.7
3285.5
86.5
0
30
9.3
1491.6
1790.6
3282.1
1.001
Model 3
81.5
104.3
8.3
6914.1
67
111.5
28.6
5.4
5292
1554
6846
1.01
Table 3: Effect of the system parameters on our results for Model 1
qrc
tc
πc
qrd
wd
td
πrd
πld
πd
πc
πd
Base Case
88.5
9.7
3285.5
66.95
46.1
5.4
230.85
2723
2954
1.112
a=0
83.6
16.7
2596.6
64.5
47
12.9
170.5
2197.1
2367.7
1.097
a = 100
100
0
4361.7
72.1
43.5
0
410.9
3134.9
3545.8
1.23
b=1
44.9
4.9
2023.6
44.2
49.8
4.18
8.03
2015
2023.1
1
b = 10
96.6
5.7
3771.7
69.6
44.9
2.96
312.6
2916.2
3228.8
1.168
µr = 50
49
1.8
1762.7
32.3
34.9
0
321.8
1129
1450.9
1.215
µr = 200
144.5
20.9
4203.6
139.3
49.9
19.85
12.75
4185.1
4197.9
1.001
σr = 10
93.5
10.7
3550.3
78.6
48.7
7.7
96.5
3221.5
3318
1.07
σr = 40
81.2
8.2
2727.8
56.2
39.1
3.24
392.6
2013
2405.6
1.134
pr = 95
93.6
10.7
4588.1
69.1
59.2
5.81
350.1
3687.8
4037.9
1.136
pr = 50
65
5
958.8
55.9
19.3
3.19
33.77
901.9
935.6
1.025
cr = 10
99
11.8
5162.7
71.9
63.6
6.4
402.9
4114.1
4517
1.143
cr = 50
74.6
6.9
1644.2
59.6
28.3
3.93
90.9
1450.9
1541.9
1.066
both items. We observe that a lower incentive is offered and lower amount of remanufacturing is done in
Model 3 as opposed to Models 1 and 2 because in Model 3, the manufacturer can satisfy some of the excess
demand for remanufactured items from original products since we assume substitutable demands. When we
look at the profits, we observe that the efficiency loss is only 1% in Model 3 and the manufacturer gets a
huge portion of the total profit in the decentralized model and centralization and collection of his own used
products directly is not as critical for the manufacturer as in the other models.
Note that the results in Table 2 depends on the values of the parameters used in the experiment. Thus, we
do additional experiments by changing the values of the parameters and observe their effects on our systems.
In Tables 3, 4 and 5, we present our results for Models 1, 2 and 3 respectively for changing parameter values
from the base case. The first column in each table shows the new value of the parameter that is changed
from the base case while all other parameters remain the same.
We observe in these tables that as a or b increases, that is as the collection of used products become
less costly, more remanufacturing is done and more used products are collected. Also, the decrease in the
collection cost is more important and has a higher impact for the manufacturer in the centralized models
than in the decentralized models and the efficiency loss due to decentralization increases as the collection of
used products becomes less costly.
19
We also observe that as the price of the remanufactured products increase or the cost of producing
remanufactured products decrease, that is as the remaining value in the used products increase, more remanufacturing is done and more used products are collected. In addition, if the government (or another
third party) decides to give subsidies to the companies for each collected return, this will decrease the cost
of collection and producing remanufactured products and increase the amount of returns. Thus, the government, who aims to increase the collection amounts, can achieve its target collection amounts by increased
subsidization. We can also find the minimum subsidy value required to achieve the target collection amounts
by including subsidy amounts into our models and using one-dimensional search algorithms.
In the decentralized cases for Models 2 and 3, we observe that a change in the parameters a, b or cr
does not effect the manufacturer’s profits very much but highly effects the collection agency’s profits. So,
it is much more important for the collection agency to find new ways to increase the collection rates at a
lower cost or to decrease the remanufacturing costs since the manufacturer only does remanufacturing if it
is profitable enough and can switch to the manufacturing of original products without a huge loss in case
of costly remanufacturing processes. On the other hand, the collection agency has no other option and is
directly effected by any change in the costs of collection or remanufacturing activities since he can not reflect
any cost increase to the manufacturer. In case of any cost increase in collection or remanufacturing activities,
the collection agency has to incur most of the loss by himself. However, the collection agency also enjoys
most of the benefits by himself in case of any cost decrease in these activities. Thus, the collection agency is
the one that bears most of the risks in the collection or remanufacturing costs. We can also state the same
result for the parameter pr in Model 3. A change in the parameter pr does not effect the manufacturer’s
profits very much but highly effects the collection agency’s profits in Model 3. Thus, we conclude that,
centralization becomes more important for the manufacturer as a, b or pr increases or cr decreases.
We also look at the effect of pm for Model 3 and cm for Models 2 and 3. We observe that as pm
decreases or cm increases, remanufacturing becomes more important and incentive value is increased. In
addition, manufacturer gets much lower profits and centralization becomes much more important for the
manufacturer as pm decreases or cm increases.
When we look at the parameters of the demand distribution, we observe that as the mean demand
for remanufactured products, µr increases, more remanufacturing is done and the efficiency loss due to
decentralization decreases. However, the mean demand of original products in Model 3, µm , does not have a
significant impact on the incentive value or on the amount of remanufactured products produced. But, as µm
increases, remanufacturing covers a lower portion of the manufacturer’s profits and centralization becomes
less important for the manufacturer. When we look at varying σ values, we observe that the amount of
remanufactured products become closer to the mean as σ values decrease.
When we look at the effect of ρ, correlation of demand between the original and remanufactured items, in
Table 5, we observe that as the demand for these products become more inversely correlated, the amount of
20
Table 4: Effect of the system parameters on our results for Model 2
qrc
c
qm
tc
πc
qrd
d
qm
wd
td
πrd
πld
πd
πc
πd
Base Case
88.5
0
9.7
3285.5
86.5
0
30
9.3
1491.6
1790.6
3282.1
1.001
a=0
75
11.5
15
2616.6
75
11.5
30
15
1491.6
1125
2616.6
1
a = 100
100
0
0
4361.7
86.5
0
30
0
1491.6
2595.3
4086.9
1.067
b=1
40
46.5
0
2691.6
40
46.5
30
0
1491.6
1200
2691.6
1
b = 10
96.6
0
5.66
3771.7
86.5
0
30
4.65
1491.6
2192.9
3684.5
1.024
µ = 50
49
0
1.8
1762.7
36.5
0
30
0
491.6
1095.3
1586.9
1.11
µ = 200
95
91.5
11
5296.6
95
91.5
30
11
3491.6
1805
5296.6
1
σ = 10
93.5
0
10.7
3550.3
93.3
0
30
10.65
1745.8
1804.4
3550.2
1
σ = 40
81.2
0
8.2
2727.8
73.02
0
30
6.6
983.1
1708.4
2691.5
1.013
p = 95
93.6
0
10.7
4588.1
93.3
0
30
10.65
2783.6
1804.4
4588
1
p = 50
65
0
5
958.8
55.9
0
19.3
3.19
33.77
901.9
935.6
1.025
cr = 10
99
0
11.8
5162.7
86.5
0
50
9.3
1491.6
3502.8
5012.3
1.03
cr = 50
45
41.5
1
1896.6
45
41.5
10
1
1491.6
405
1896.6
1
cm = 40
45
55
1
3766.7
45
55
10
1
3361.7
405
3766.7
1
cm = 70
88.49
0
9.7
3285.5
76.99
0
40
7.4
670.63
2510.1
3180.7
1.033
both original and remanufactured items increase but this increase is very little and the correlation between
the demand for these two types of products does not have a significant impact on the results.
We also analyze the effect of the substitution rate between the remanufactured and original products
and observe that as αrm decreases or αmr increases, incentive value is increased and more remanufacturing
is done. However, a change in αrm has a much larger effect on the incentive than a change in αmr . As αrm
decreases, the manufacturer’s profits decrease significantly in both centralized and decentralized models and
centralization becomes especially important for the manufacturer when αrm is small. However, a change in
αmr has only a very small effect on the profits of the manufacturer.
7
Conclusion
Either because of environmental regulations or economical benefits, more and more companies are becoming
interested in remanufacturing operations. In this study, we consider a manufacturer that produces both
original products using virgin materials and remanufactured products using returns from the customers where
the manufacturer offers an incentive to the customers to collect more returns to use in the remanufacturing
process. We determine the optimal value of the incentive to offer and the optimal production quantities
of original and remanufactured products in a stochastic demand setting. We analyze 3 different models in
centralized and decentralized settings where the collection process of the returns are managed by a different
party than the manufacturer in the decentralized settings. Model 1 assumes that the manufacturer only
produces remanufactured products, Model 2 assumes that the remanufactured and original products have
21
Table 5: Effect of the system parameters on our results for Model 3
qrc
c
qm
tc
πc
Base Case
81.5
a=0
70.1
104.3
8.3
110
14.03
a = 100
100
94.74
td
πrd
πld
πd
πc
πd
qrd
d
qm
wd
6914.1
67
111.5
28.6
5.4
5292
1554
6846
1.01
6312.7
60.72
114.5
29.3
12.14
5246.2
1041.7
6287.9
1.004
0
7924.7
77.6
106.2
26.1
0
5469.8
2026.6
7496.4
1.057
b=1
40
124.7
0
6392.9
34.9
127.2
29.9
0
5190.8
1045
6235.8
1.025
b = 10
96.5
96.6
5.65
7334.4
72.73
108.6
27.5
3.27
5367.5
1762
7129.6
1.03
1.022
µr = 50
50.28
94.57
2.05
5322.9
37.2
101.4
21.8
0
4398.6
811.06
5209.7
µr = 200
94.99
147.26
11
9006.1
94.75
147.4
29.9
10.95
7210.5
1795.5
9006
1
µm = 50
81.52
54.31
8.3
4913.6
66.9
61.5
28.6
5.38
3291.9
1553.4
4845.3
1.014
µm = 200
81.54
204.31
8.3
10914
66.9
211.5
28.6
5.38
9292
1553.9
10846
1.006
σr = σm = 10
87.4
103.83
9.48
7385.3
78.1
108.4
29.6
7.61
5640.2
1716.5
7356.6
1.004
σr = σm = 40
74.4
102.8
6.88
5905.5
58.1
110.9
25.6
3.6
4546.3
1276.9
5823.1
1.014
pm = 150
40
135.3
0
12490
32.25
139.2
4.9
0
12279
158.02
12437
1.004
pm = 85
89.3
93.7
9.85
5487.4
70.5
103.7
35.3
6.1
3285.8
2059.3
5345.1
1.027
pr = 95
93.02
97.98
10.6
8182.2
71.8
109.1
41.8
6.35
5409.6
2544
7953.6
1.028
pr = 50
0
144.3
0
5136.1
0
144.3
0
0
5136.1
0
5136.1
1
cr = 10
100.3
94.6
12.1
8724.5
76.4
106.8
46.5
7.28
5439.9
2996.7
8436.6
1.034
cr = 50
44.8
122.4
0.96
5602.5
44.75
122.4
9.9
0.95
5201.9
400.5
5602.5
1
cm = 40
66.5
122.02
5.29
9176.9
58.9
125.8
19.4
3.79
8239.8
920.1
9160
1.002
cm = 80
93.3
84.7
10.67
5014.3
74
95.9
37.6
6.8
2579.9
2279.2
4859.2
1.032
ρ = −1
82
104.9
8.4
7003.7
67.9
111.9
28.6
5.58
5429.8
1563.1
6992.9
1.001
ρ=1
81
103.8
8.2
6828.8
66.6
110.8
28.5
5.31
5176.8
1543.5
6720.3
1.016
αrm = 0
93.27
92.17
10.65
6639.8
71.24
94.64
46.5
6.25
3447.9
2867.7
6315.6
1.051
αrm
=1
45.1
149.14
1.02
7522.7
44.75
149.48
9.9
0.95
7120.9
400.5
7521.5
1
αmr = 0
78.21
106.49
7.64
6886.7
63.75
113.12
28.5
4.75
5293.6
1514
6807.6
1.012
αmr = 1
84.42
103.42
8.88
6945.5
70.41
110.03
28.6
6.08
5298.3
1585.5
6883.9
1.009
22
the same market value and Model 3 assumes that the remanufactured and original products have different
demand and customer valuations and are partially substitutable products in case of excess demand. We also
look at contracts to coordinate the decentralized systems and determine the optimal contract parameters.
We also perform a computational study to analyze the effect of parameters on the system performance
and the efficiency loss in the system due to decentralization. We observe that more remanufacturing is done
and a higher incentive is offered as a, b, σ, pm , cr or αrm decrease or µr , pr , cm or αmr increase. µm
and ρ does not have a significant impact on remanufacturing amounts. We also observe that centralization
becomes especially important for the manufacturer as a, b, cm or pr increase or cr , pm , µm or αrm decrease.
In addition, the government can increase the collection amounts by giving subsidies to the companies for
each collected return.
Of course, real world situations have many more and complex characteristics that are not captured by
the model in this paper. Different models in which the returns are not identical and have different values
for the manufacturer or models with competing manufacturers or collection agencies can be analyzed. Also,
pricing decisions can be incorporated into our models in the future in which the manufacturer has the ability
to set the sales price for his products.
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