A Novel Mathematical Formulation for the Strategic Planning of a
Reverse Supply Chain Network
Theoretical and Computational Results
Ernesto D. R. Santibanez-Gonzalez
1
and Nelson Maculan
2
1
DECOM, Universidade Federal de Ouro Preto, Ouro Preto, MG, Brazil
2
COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil
Keywords: Reverse Supply Chain, Supply Chain, Integer Programming.
Abstract: In the last decade, literature on strategic planning of a supply chain network grew rapidly. In this paper we
address a classical three-layer remanufacturing supply chain network design problem that covers sourcing,
reprocessing and remanufacturing activities, in which strategic decisions regarding the number, location of
reprocessing units and the flow of returns through the logistics network are made. First, we propose an
alternative mixed-integer mathematical programming (MILP) formulation for this problem and provide
theoretical proof of equivalence between the classical and the proposed mathematical formulation. Second,
the goodness of both formulations is compared by means of a computational study, and the results for large
instances of the problem are discussed. We empirically prove that the proposed formulation provides tighter
linear relaxation lower bounds and obtains the integer solutions several times faster than the classical
formulation.
1 INTRODUCTION
Many authors have studied the relationship that
exists between Supply Chain Management (SCM)
and Advanced Planning and Scheduling (APS)
systems (see for example Stadtler and Kilger, 2005).
For some authors, one of the advantages of an APS
approach is that it makes possible to include
suppliers and customers in the planning process and
thereby optimise the entire supply chain on a real-
time basis. As a consequence it enables “to extract
real-time information from that chain, with which to
calculate a feasible schedule, resulting in a fast,
reliable response to the customer” (Amstel, 1998).
The planning of supply chain is considered a
strategic issue. On this regards, “the strategic level
deals with decisions that have a long-lasting effect
on the firm. These include decisions regarding the
number, location and capacities of warehouses and
manufacturing plants, or the flow of material
through the logistics network.” (Simchi-Levi et al.,
2007).
This paper addresses a class of planning
problems that arise in the design of a
remanufacturing supply chain network. We study a
classical three-layer facility location model for
designing a reverse supply chain that covers the
sourcing, processing and remanufacturing activities.
The management of return products and waste
stewardship has become major concerns for
companies and organisations that are interested in
sustainable practices. In this context,
remanufacturing activities are recognised as a main
option of recovery in terms of their feasibility and
benefits.
The problem is an NP-hard combinatorial
optimisation problem and it has been previously
modeled as a mixed integer linear programming
(MILP) problem (Jayaraman et al., 2003).
This paper makes three main contributions. First,
it is proposed a novel MILP formulation for the
problem. Second, we provide theoretical proof that
the proposed MILP formulation is stronger than the
existing classical weak and strong formulation of the
problem. Third, we conclude that the proposed
MILP formulation outperforms the classical
formulation in terms of the quality of the linear
relaxation lower bound and the computing times.
The remainder of this paper is organized as
follows: In the second section, we provide a
literature review with a brief introduction to
570
D. R. Santibanez-Gonzalez E. and Maculan N..
A Novel Mathematical Formulation for the Strategic Planning of a Reverse Supply Chain Network - Theoretical and Computational Results.
DOI: 10.5220/0004432205700577
In Proceedings of the 15th International Conference on Enterprise Information Systems (SSOS-2013), pages 570-577
ISBN: 978-989-8565-59-4
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
sustainable and reverse supply chains and with a
special emphasis on the remanufacturing case. In the
third section, we propose a new mathematical model
and prove that the convex-hull associated with the
linear relaxation of this formulation is contained in
the convex-hull of the classical formulation. In the
fourth section, we present experimental results for
large sets of data that were generated randomly. The
last section contains conclusions and directions for
future research.
2 LITERATURE REVIEW
In the last few years, mathematical modeling and
solution methods for the efficient management of
return flows (and/or integrated with forward flows)
has been studied in the context of reverse logistics,
closed-loop supply chains, and sustainable supply
chains. The problem of locating facilities and
allocating customers is not new to the operations
research community and covers the key aspects of
supply chain design (Daskin et al., 2005). This
problem is one of “the most comprehensive strategic
decision problems that need to be optimized for
long-term efficient operation of the whole supply
chain” (Altiparmak et al., 2006).
Mathematical models proposed in the literature
for planning reverse logistics networks have been
reviewed by Fleischmann et al., (2000). Fleischmann
et al., (2001) proposed a generic recovery network
model based on the elementary characteristics of
return networks identified in Fleischmann et al.
(2000). Zhou and Wang (2008) proposed a generic
mixed integer model for the design of a reverse
distribution network including repairing and
remanufacturing options simultaneously.
Models for reverse logistics networks in
connection with location problems have been
discussed by Bloemhof-Ruwaard, Salomon, and Van
Wassenhove (1996) and Barros, Dekker, and
Scholten (1998). For example, the last authors
described a network for recycling sand from
construction waste and proposed a two-level
location model to solve the location problem of two
types of intermediate facility.
Regarding remanufacturing location models,
Krikke et al., (1999) described a small reverse
logistics network for the returns, processing, and
recovery of discarded copiers. They presented a
MILP model based on a multi-level uncapacitated
warehouse location model. The model was used to
determine the locations and capacities of the
recovery facilities as well as the transportation links
connecting various locations. In Jayaraman et al.,
(2003), a 0-1 MILP model for a product recall
distribution problem is proposed. They analysed a
particular case in which the customer returns the
product to a retail store and the product is sent to a
refurbishing site which will rework the product or
dispose it properly. The reverse supply chain is
composed of origination, collection, and
refurbishing sites. With the objective to minimize
fixed and distribution costs, the model has to decide
which collection sites and which refurbishing sites to
open, subject to a limit on the number of collection
sites and refurbishing sites that can be opened.
Several authors have studied different aspects of
closed-loop supply chain design problems. See, for
example, Jayaraman et al., (1999), Fleischmann
(2003), Barbosa-povoa et al., (2007); Guide and Van
Wassenhove (2009), and Neto et al., (2010). For
example, Sahyouni et al., (2007) presented three
generic facility location MIP models for the
integrated decision making in the design of forward
and reverse logistics networks. The formulations are
based on the well-known uncapacitated fixed-charge
location model, and they include the location of used
product collection centers and the assignment of
product return flows to these centers. Lu and Bostel
(2007) presented a two-level location problem with
three types of facilities to be located in a reverse
logistics system. They proposed a 0–1 MILP model
which simultaneously considers “forward” and
“reverse” flows and their mutual interactions. The
model has to decide the number and locations of
three different types of facilities: producers,
remanufacturing centers, and intermediate centers.
Reverse logistics models are recently discussed
by Salema et al., (2010); Gomes et al., (2011) and
Alumur et al., (2012). Almost all this research
proposed MILP models. The majority of solution
methods are based on standard commercial
packages.
3 MATHEMATICAL MODEL
In this section a new MILP model for the problem of
designing a remanufacturing and sustainable supply
chain network is proposed. This problem is a single
product, static, three-layer, capacitated location
model with known demands. The remanufacturing
supply chain network consists of three types of
members: sourcing facilities (origination sites such
as a retail store), collection sites (reprocessing
facilities) and remanufacturing facilities. At the
customer layer, there are product demands and used
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products that are ready to be recovered. It is
assumed that customers return the products to
origination sites such as a retail store. In the second
layer of the supply chain network, there are
reprocessing sites that are used only in the reverse
channel, and they are responsible for activities such
as cleaning, disassembly, checking and sorting
before the returned products are sent back to the
remanufacturing facilities. In the third layer,
remanufacturing facilities accept the checked returns
from reprocessing facilities, and they are responsible
for the process of remanufacturing. A classical
MILP model was proposed by Jayaraman et al.,
(2003). In such a supply chain network, the reverse
flow, from customers through collection sites to
remanufacturing facilities, is formed by used
products, while the other flow (the “forward” flow)
is from remanufacturing facilities directly to the
point of sales for the “new” products. The number
and location of reprocessing and remanufacturing
facilities must be decided to minimise the
transportation, distribution and fixed costs. As noted
by Jayaraman et al. (2003), the type of
remanufacturing problem that is addressed in this
paper is more like the type called product recalls,
whereby customers return products that have
reached the end their useful life or are defective.
3.1 Classical Model
This is a MILP model proposed by Jayaraman et al.,
(2003). It is assumed that the product demands (new
products) and the available quantities of used
products from the customers are known and
deterministic. All of the returned products are first
shipped back to collection facilities, where some of
them will be disposed of for various reasons, such as
poor quality. The checked return-products will then
be sent back to remanufacturing facilities, where
some of them could still be disposed of. The product
demands from the customers can be met by point of
sale facilities, which receive products from the
remanufacturing facilities. In this problem,
remanufactured products are considered to be the
same as the new products coming from “traditional”
producers in terms of satisfying the customer
demands.
The model proposed by Jayaraman et al., (2003)
introduces the triply subscripted flow variable
to
represent a fraction of the unit demand at location j
that is shipped to l through a reprocessing facility
located at k. We introduce a constant M, which
represents the cardinality of set J.
We introduce the following inputs and sets:
J = the set of sourcing facilities in the first layer,
indexed by j
L = the set of candidate remanufacturing facility
locations in the third layer, indexed by l
K = the set of candidate reprocessing facility
locations in the middle layer, indexed by k
a
j
= the supply quantity at the source location j ϵ J
b
l
= the demand quantity at the remanufacturing
location l ϵ L
f
k
= the fixed cost of locating a mid-layer
reprocessing facility at candidate site k ϵ K
g
l
= the fixed cost of locating a remanufacturing
facility at candidate site l ϵ L
c
ikl
= the unit cost of delivering products at l ϵ L
from a source facility located in j ϵ J through facility
k ϵ K
m
k
= the capacity at reprocessing facility location k ϵ
K
We consider the following decision variables:
w
k
= 1 if we locate a reprocessing facility at
candidate site k ϵ K; 0 otherwise
y
l
= 1 if we locate a remanufacturing facility at
candidate site l ϵ L; 0 otherwise
X
jkl
= fraction of the unit flow from the source
facility j ϵ J to the remanufacturing facility located at
l ϵ L through facility k ϵ K
RSCLP:
v(RSCLP) =
(1)
1 ∀
(2)
∀
(3)
∀
(4)
∀
(5)
∀
(6)
0,
,
0,1
∀
,∀,∀
(7)
In this formulation, constraint (2) ensures that all of
the products from the sourcing facility j are
transported to remanufacturing facilities l through
collection sites k. Constraint (3) ensures that all of
the products that arrive at site l must be less than its
demand. Constraint (4) ensures that all of the
products that arrive to and ship from collection site k
must be less than its capacity. Constraint (5) ensures
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that the products that arrive to and ship from a
collection site have already been opened at site k.
Constraint (6) warrants that all of the products that
arrive at the remanufacturing facilities have already
been opened at site l. Constraint (7) is a positive and
binary constraint.
This model has O(n
3
) positive variables, where
n=max{|J|, |K|, |L|} and {|K|+|L|} are binary
variables. The number of constraints is O(n). In fact,
this model has (nqm+q+m) variables and
(n+2q+2m) constraints, where |J| = n, |K| = q and
|L| = m.
Following the results from the well-known
uncapacitated facility location problem (UFLP), this
is a weak formulation for the problem because of
constraints (5) and (6). A stronger formulation
(RSCLP-T) is obtained by replacing constraints (5)
and (6), respectively, by the following constraints:
∀
,∀,∀
(8)
∀
,∀,∀
(9)
By limiting the number of reprocessing and
remanufacturing facilities to open, the following two
sets of constraints must be added to the RSCLP
model:
(10)
(11)
Parameters p
min
and p
max
limit the minimum and
maximum quantity of reprocessing facilities to open.
The same reasoning applies for parameters q
min
and
q
max
on the remanufacturing facilities.
Jayaraman et al., (2003) solved model RSCLP
and RSCLP-T using AMPL and CPLEX to optimally
solve for instances of networks of up to 100 sourcing
facilities, 40 candidate sites for locating reprocessing
facilities (p
max
=8) and 30 candidate sites for locating
remanufacturing facilities (q
max
= 6), using a
maximum of 50,028.6 seconds of computing time.
They also developed some heuristics for solving the
problem, but it is not the objective of this paper to
discuss them.
3.2 Proposed Model
In this proposed model, we introduce two new
variables, x
jk
and z
kj
, to break down the flows of
return products from the sourcing site j to the facility
l into two parts, i.e., the flow that goes from j to k
and the flow that goes from k to l. The remaining
variables and parameters retain the same values as in
the above RSCLP model. We also eliminated some
constraints, and other constraints are included, as
described in the following:
We consider the following parameters and
decision variables:
c
jk
= the unit cost of delivering products at k
∈
K
from a source facility located at j
∈
J
d
kl
= the unit cost of delivering products at l
∈
L
from a reprocessing facility located at k
∈
K
x
jk
= the flow from source facility j
∈
J to the
reprocessing facility located at k
∈
K
z
kl
= the flow from the reprocessing facility located
at k
∈
K to the remanufacturing facility l
∈
L
Note that models such as the RSCLP-P do not
account for the origin of the products that arrive at
the remanufacturing facilities; we lose track of the
origins of those products. In some real applications,
for example, biomedical waste, we would like to
control the origin of the products that arrive at the
remanufacturing facilities.
RSCLP-P:
v(RSCLP-P)
(1a)
∀
(2a)
∀
(3a)
∀
(4a)
∀
(5a)
,
0,
,
0,1
∀,∀
,∀
(7)
In this formulation, the first two terms of the
objective function (1a) sum up the installation costs,
and its two second terms sum up the transportation
and delivery costs. Constraint (2a) guarantees that
all of the return products available at the origination
site j must be shipped. Constraint (3a) ensures that
all of the products that arrive at reprocessing site k
must be lesser than its capacity and that this facility
must be opened. Constraint (4a) guarantees that all
of the products that are delivered at the
remanufacturing facility l must be less than its
capacity and that this facility must be opened.
Constraint (5a) is a type of flow conservation, which
ensures that all of the products that arrive at
reprocessing facility k must also leave it. Note that
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in this model, the number of positive variables is
O(n
2
), in contrast to O(n
3
) of the classical model.
The number of integer variables remains the same
for both models. Compared with the weak
formulation, both of the models have O(n) number
of constraints. However, this RSCLP-P model has
O(n) compared with O(n
3
) constraints of the strong
formulation of RSCLP. In fact, this model has
(2nq+q+m) variables and (n+2q+m) constraints
versus (nqm+q+m) variables and (n+2q+2m)
constraints in the weak formulation of RSCLP.
3.3 Proof of Equivalency and Strength
between the MILP Models
For space reasons, theoretical proof of equivalency
between the proposed RSCLP-P model and the weak
and strong classical models are not presented here.
For the same reason, it is only summarized the proof
of strength between the proposed RSCLP-P model
and weak classical model.
Lemma 1: RSCP-P Model is stronger than RSCP
Model. Let S
RSCP
= {(X,w,y) ϵ R
|J|x|K|x|L|+|k|+|L|
| X
jkl
,
w
k
and y
l
satisfy (2)-(7), (10)-(11)} be the set of
feasible solutions of RSCLP, S
RSCP-P
= {(x,z,w,y) ϵ
R
|J|x|K|+|K|x|L|+|k|+|L|
| x
jk
, z
kl
, w
k
and y
l
satisfy (2a)-(5a),
(7), (10)-(11)} the set of feasible solutions of
RSCLP-P,
RSCP
= {(X,w,y) ϵ R
|J|x|K|x|L|+|k|+|L|
| X
jkl
≥0,
0≤w
k
≤1, 0≤y
l
≤1 and satisfying (2)-(6), (10)-(11)}
the set of solutions to the LP relaxation of RSCLP,
and
RSCP-P
= {(x,z,w,y) ϵ R
|J|x|K|+|K|x|L|+|k|+|L|
|x
jk
≥0, z
kl
≥0, 0≤w
k
≤1, 0≤y
l
≤1 and satisfying (2a)-(5a), (10)-
(11)} the set of solutions to the LP relaxation of
RSCLP-P. Then,
RSCP-P
⊆
RSCP.
Proof: Consider an instance of RSCP with
J={1,2,3}, K={1,2}, L={1,2}; C
jkl
=1,
∀
j ϵ J, k ϵ K, l
ϵ L; f
k
=1,
∀
k ϵ K; g
l
=1,
∀
l ϵ L; a
j
=1,
∀
j ϵ J; u
k
=2,
∀
k ϵ K; b
l
=3,
∀
l ϵ L; p
min
=q
min
=0; p
max
=q
max
=2.
Let (X,w,y) ϵ
RSCP
with
X
111
= X
121
= 0.5 and X
211
= X
321
= 1; w
1
= w
2
=
0.5; y
1
= 1 and y
2
= 0, and the remaining values
equal to zero.
We can verify that (X,w,y) satisfy (1)-(7), (10)
and (11).
From (12) and (13), we obtain for (x,z)
x
11
= x
12
= 0.5; x
21
= x
32
= 1; z
11
= z
21
= 1.5 and,
w
1
= w
2
= 0.5; y
1
= 1 and y
2
= 0. All of the other
values are equal to zero.
However, constraint (3a) is violated
∀
j ϵ {1,2,3}
and
∀
k ϵ {1,2}.
Thus model, RSCLP-P provides a tighter
formulation than model RSCLP for the same
problem.
4 COMPUTATIONAL
EXPERIMENTS
The goodness of both formulations is compared by
means of a computational study, and the results for
large instances of the problem are discussed here.
Our interest was to benchmark the computation
times and the quality of the lower bound provided by
the linear programming relaxation. The models were
implemented in GAMS, and it was used CPLEX
(version 12.2) with default settings to solve all of the
test instances that are presented in this section. All
of the computational tests were performed on a PC
with 1 GB of RAM memory and a 2.3 GHz
processor.
We randomly generated 10 test problems
following similar methodologies used for well-
known related supply chain problems (for example:
(Fleischmann et al., 2001); (Lu and Bostel, 2007)).
These test problems correspond to networks of up to
600 origination sites, 100 candidate sites for locating
reprocessing facilities and 40 candidate sites for
locating remanufacturing facilities. The data sets for
the test problems are given in Table 1, and they are
available from the authors; because of the restriction
on the size of papers, the authors have not provided
the full data in this paper. All of the transportation
costs were generated randomly using a uniform
distribution with parameters [1,40]. The fixed costs
for the remanufacturing facilities were obtained by
multiplying by 5 the fixed costs of the reprocessing
facilities, following Jayaraman et al., (2003).
Sourcing units (a
j
), the capacity of reprocessing
facilities (m
k
) and the capacity of remanufacturing
facilities (b
l
) are shown in Table 1.
Table 1: Data set.
# J K L
Fixed
costs f
k
Fixed
costs g
l
a
j
m
k
b
l
1 40 20 15 3000 15000 150 400 2000
2 70 30 20 5000 25000 150 500 2500
3 100 40 20 5000 25000 150 500 2500
4 150 40 20 10000 50000 200 800 4000
5 200 80 20 10000 50000 300 800 4000
6 300 80 40 20000 100000 200 800 4000
7 350 100 40 20000 100000 200 800 4000
8 400 100 40 20000 100000 200 1500 7500
9 500 100 40 20000 100000 200 1500 7500
10 600 100 40 25000 125000 300 3000 15000
Table 2 illustrates for some instances, the size
differences between the RSCLP model and proposed
formulation. For example, for instance #7 the total
number of continuous variables is 1,400,000 and
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39,000 for the weak classical and proposed model
respectively. For instance #10 the number of
continuous variables is 2,400,000 and 64,000 for the
weak classical and proposed model respectively.
Although they share the same number of integer
variables, the classical model has up to 50 times the
number of continuous variables.
Table 2: Size differences between the classical model and
the proposed formulation.
Classical (weak) model Proposed formulation
# X
jkl
y
l
;
w
k
No.
cstr
x
jk
; z
kl
y
l;
w
k
No.
cstr
7 1,400,000 140 632 39,000 140 592
8 1,600,000 140 682 44,000 140 642
9 2,000,000 140 782 54,000 140 742
10 2,400,000 140 882 64,000 140 842
4.1 Results Analysis
Table 3 displays the linear programming (LP)
relaxation lower bound values (v
L
) and the gap
between the integer optimal values of the objective
function (v*) and the LP values for all of the models.
They also show the computing times in seconds
(secs). Remember that v
L
is obtained by solving the
LP relaxation of the respective model. The
computing time was limited to 5,200 seconds.
Columns 2 to 5 give the results that correspond to
the classical weak RSCLP model; columns 6 to 9
display the results that correspond to the classical
strong RSCLP-T model, and columns 10 to 13 show
the results that correspond to the RSCLP-P model
(the proposed formulation). To illustrate, for
problem instance 4, the LP relaxation lower bounds
(v
L
) are 168,200.0, 282,916.0, and 858,200.0 for the
RSCLP, RSCLP-T, and RSCLP-P models,
respectively; while for instance 6, these values are
280,800.0 and 3,160,800.0 for the RSCLP and
RSCLP-P models, respectively. For this instance, the
RSCLP-T model was unable to provide the LP value
because it exceeded the computing time limit. The
results indicate that, for all of the test instances, the
proposed RSCLP-P model outperforms the classical
weak (RSCLP) and strong formulation (RSCLP-T) in
terms of the quality of the lower bound and the
computing times. In summary, we observed the
following:
4.1.1 Lower Bounds (LP)
For instances with 800 and 1000 sourcing nodes,
the weak classical formulation is unable to provide
the LP relaxation solution because it runs out of
memory.
For the strong classical formulation, this scenario
occurs with instances that have more than 300
sourcing sites
The lower bound provided by the proposed
RSCLP-P model is significantly better than that
provided by the weak (RSCLP) and strong (RSCLP-
T) classical models
4.1.2 Optimal Integer Solutions
For test instances with more than 300 origination
sites (marked by *), the weak (RSCLP) classical
formulation is unable to provide the optimal integer
solution because it runs out of memory or the
defined time limit for execution is reached;
For the case of RSCLP-T model, this scenario
occurs for networks with more than 100 origination
sites;
4.1.3 Computing Time
The same performance can be observed in terms
of the computing times. To illustrate, for instance 4,
the computing times for obtaining the lower bounds
are 5.16 and 647.0 for the weak and strong classical
models, respectively, while the corresponding time
is 0.67 for the proposed model. For instance 5, those
times are 15.45, 4013.0 and 0.84 seconds for the
weak and strong classical models and for the
proposed formulation, respectively.
4.1.4 Gaps (%)
Table 3: Lower bound (v
L
) and computing times (seconds).
#
RSCLP (weak
classical)
RSCLP-T
(strong
classical)
RSCLP-P
(proposed)
v
L
Secs v
L
secs v
L
secs
1 37200 0.70 50938 3.97 109200 0.59
2 75300 1.63 119025 45.55 255300 0.41
3 74650 2.88 122070 317.61 419650 0.50
4 168200 5.16 282916 647.19 858200 0.67
5 254000 15.45 426674 4013.3 1694000 0.84
6 280800 49.41 * - 3160800 1.19
7 290400 125.55 * - 3670400 7.5
8 305700 90.00 * - 2319033 1.52
9 341200 137.28 * - 2887867 6.38
10 594900 290.05 * - 3444900 5.27
It was observed that gaps (%) obtained from the
weak (RSCLP) and the strong (RSCLP-T) classical
models are significantly worse than the gaps
obtained by the proposed RSCLP-P model.
To illustrate this point, for problem instance 1,
the gaps (%) are 213.84, 129.20 and 6.91 for the
RSCLP, RSCLP-T and RSCLP-P model,
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respectively, while for problem instance 5, the gaps
(%) are 572.32, 300.24 and 0.81 for the RSCLP,
RSCLP-T and RSCLP-P model, respectively;
The average gap for the proposed model is
3.69%, while this number is 210.25% and 625.19%
for the strong and weak classical formulations,
respectively.
4.2 Model Sensitivity
We investigated the sensitivity of the models
generating several scenarios to provide further
information regarding the goodness of both
formulations. For example, we increased the
capacity of the reprocessing facilities by 200% and
the capacity of the remanufacturing facilities by
100%. The results are summarized in Table 4.
Table 4: Lower bound (v
L
), integer optimal solution (v*),
and Gap [100(v*- v
L
)/v
L
].
RSCLP (weak classical) RSCLP-P(proposed)
v
L
Gap%
v
L
Gap%
1
36300 101.24 55800 30.91
2
73800 138.08 131300 33.82
3
67950 219.28 194200 11.71
4
153600 196.35 406100 12.09
5
216300 287.61
b
781300 7.19
6
262400 467.00
b
1392400 6.06
7
276400 511.72
c,d
1614733 4.71
d
8
305000 304.10
c,d
1073889 14.77
d
9
341200 328.02
c,d
1332311 9.61
d
10
595200 226.75
c,d
1695200 14.72
d
Min
101.24 4.71
Max
511.72 33.82
Ave
277.77 14.41
a
Algorithm terminated
at 5,200 CPU seconds without reaching
an optimal solution
b
Gap is calculated using the integer optimal solution provided by
the RSCLP-P model
c
Algorithm ran out of memory
d
Gap is calculated using the best integer solution provided by the
RSCLP-P model
In Table 4, for the classical model and for
problem instances 1-10, the maximum and minimum
gaps are 511.72% and 101.24%, respectively, with
an average gap of 277.77%. The maximum and
minimum gaps for the proposed model are 33.82%
and 4.71%, respectively, with an average gap of
14.41%. The average gap of the classical model is
more than 20 times larger than the proposed model
average gap. Observe that, for problem instances 7-
10, the classical models ran out of memory and for
problem instances 5-6 the same model hit the
computational limit. For those instances, the gap was
obtained using the integer optimal solution provided
by the proposed model.
5 CONCLUSIONS
In this paper, we proposed a new formulation for the
problem of planning a reverse supply chain network,
and it is provided theoretical and empirical proofs
that this model is stronger than the classical (weak
and strong) formulations of the problem. We
analysed the performance of the proposed MILP
formulation in terms of the computing times and the
quality of the lower bounds provided by the linear
relaxation. We showed, for the large-scale instances
with up to 600 sourcing sites, 100 candidate sites for
locating reprocessing facilities and 40 candidate
sites for locating remanufacturing facilities, that the
proposed RSCLP-P model outperforms the classical
weak and strong formulation with a gap that is
several times lower than the gap provided by the
weak formulation and with significantly less
computing time. Furthermore, the weak formulation
cannot provide integer optimal solutions for some
instance cases, and it is also unable to obtain the
linear optimal solution in a reasonable amount of
computing time.
ACKNOWLEDGEMENTS
This research was partially supported by FAPEMIG,
CNPq, COPPE/UFRJ and UFOP. The support is
gratefully acknowledged.
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