AN OPEN-LOOP SOLUTION FOR A STOCHASTIC
PRODUCTION-REMANUFACTURING PLANNING PROBLEM
Oscar Salviano Silva Filho
Center for Information Technology Renato Archer, Rod. D. Pedro I (SP 65), Campinas, SP, Brazil
Keywords: Reverse logistics, Supply chain, Remanufacturing, Optimization.
Abstract: A stochastic linear production planning problem with chance-constraints is introduced in order to provide a
production plan that optimizes a reverse logistics system. Such a system is composed of two channels: in the
forward channel, new and remanufactured products are produced and stored into a serviceable inventory
unit. On the other hands, in the reverse channel, used and defected products are stored into the
remanufacturable inventory unit. The uncertainties about the fluctuation of demand and the amount of
returnable are the reason of the stochastic nature of the problem. Since global optimal solution is not easy to
be achieved, an equivalent-deterministic suboptimal problem is proposed. An example of applicability of
this deterministic problem is presented. In this example, two situations are proposed: the first considers that
50% of used-products are returnable; while the second considers 100%. It is assumed that 5% of new
products are rejected during the quality inspection process. As a result, the example shows that, under
certain circumstances, it is possible to improve the performance of the system by increasing the return rate
for used-products.
1 INTRODUCTION
Planning and control the flow of used-products
throughout the reverse channel has become
increasingly crucial for the success of any supply
chain. A major reason that explains this fact is the
pressure that governments and society have imposed
to companies for that they preserve the environment.
In this sense, collect and recovery used products can
reduce the necessity for extraction of raw materials,
and, as a consequence, helps the preservation of the
environment. Thus, reverse logistic issue has
become an important topic of the supply chain area.
Conceptually, reverse logistics term can be
understood as a process of dealing with different
activities that involve, for instance: the collecting of
used-products from the market for remanufacturing,
recycling or disposing them. More generically, the
objective of reverse logistics is to move used-
products from the market to their final destination
with the aim of capturing value, or proper disposal.
In the literature, there are several papers related
to logistic reverses issues. Most of them are based
on quantitative models that are used to represent
remanufacturing and recycling activities in the
reverse channel. Fleischmann et al (1997) provide a
typology of quantitative models for reverse logistics,
which is based on three classes of problems, namely:
(i) reverse distribution problems; (ii) inventory
control problems in systems with return flows; and
(iii) production planning problem with reuse of parts
and materials. In short, the first class of this
typology is concerned with the collection and
transportation of used-products and packages.
According the authors: “the reverse distribution can
take place through the original forward channel,
through a separate reverse channel, or through
combinations of the forward and the reverse
channel”. The second class is related to appropriate
control mechanism that allows returning the used
products into the market; and, the third class is
associated with the planning of the reuse of items,
parts and products without any additional process of
remanufacturing. At last, it is worth mentioning that
there are many different approaches to deal with
each one of these problems.
The paper considers a sequential stochastic
production planning problem with chance
constraints on decision variables. Such a problem is
formulated in order to deal with a stochastic
production-inventory system with a special structure
for collecting used-products from the market, and
369
Silva Filho O..
AN OPEN-LOOP SOLUTION FOR A STOCHASTIC PRODUCTION-REMANUFACTURING PLANNING PROBLEM.
DOI: 10.5220/0003542103690378
In Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2011), pages 369-378
ISBN: 978-989-8425-75-1
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
remanufacturing or disposing them. The stochastic
nature of this system is due to the fact of demand
fluctuation for serviceable products and return rate
of used-products are random variables.
Stochastic production-remanufacturable systems
are very commons in the reverse logistics field. In
fact, return rates are usually estimated with base on
demand level for serviceable products. This means
that they are directly dependent on the stochastic
fluctuation of demand over periods of the planning
horizon. Fleischmann et al. (1997) consider that
traditional techniques used to deal with stochastic
inventory balance systems can be reapplied for
reverse systems. Based on this, the stochastic linear
programming technique is taking into account here
to model and solve a combined stochastic
production/remanufacturing problem with chance
constraints on inventory variables.
Without loss of generality, it is assumed here that
the demand fluctuation for serviceable products and
the return rate of used-products are described by
uncorrelated normal stationary processes; see Graves
(1999). Thus, the production/remanufacturing
problem with constraints is stochastically well-
defined. This means that its random variables have
probability distributions well-known along periods
of the planning-horizon.
Since a global optimal solution for a sequential,
constrained stochastic problem is not easy to obtain,
nearly-optimal (heuristics) should be considered. In
this paper, the certainty-equivalence principle is
used to transform the stochastic problem into an
equivalent-deterministic, see Bertsekas (2007). An
open-loop solution is provided for this equivalent
problem. This sub-optimal solution can be used by
managers to improve their decision-making skills.
For instance, considering that the object of this paper
is to solve a production/remanufacturing problem,
the open-loop solution provided enables managers to
decide about a proper return rate of used-products
that improve the productivity of the company. In
fact, production/remanufacturing plans provided by
the equivalent problem can be used to create
production/remanufacturing scenarios. These
scenarios are built from the variation of some
parameters of the original stochastic problem, as for
instance: percentage of the return rate of used
products, time-delays for collecting used-products,
customer satisfaction level due to safety-stock of
serviceable products, and so on.
An example is introduced to show the
applicability of the proposed problem. It compares
two distinct production/remanufacturing policies
that are provided by solving the equivalent problem.
The first policy is the result of a production situation
where only 50% of used-products return from the
market. On the other hands, the second policy
considers the situation where 100% is assumed to
return. It is also considered for both situations that
5% of new products present some kind of defect,
which is detected during quality inspection. At last,
from analyzes of the two situations, it is possible to
conclude that 100% of the return rate can improves
the productivity of the company. However, it is
important to mention that such a conclusion depends
on production/inventory costs and operational
conditions specified to the problem. These aspects
will be discussed ahead.
The paper is stated as follows: initially, section 2
introduces comments on stochastic remanufacturing
literature (a brief review). In the section 3, the
reverse system is described. In sequel, the section 4
formulates the stochastic problem. Section 5
presents the transformation process form the original
stochastic problem to equivalent-deterministic one.
Each step of this conversion process is brief
considered. Section 6 illustrates the applicability of
this equivalent-problem in a company that deal with
a single product production/remanufacturing system.
2 LITERATURE REVIEW
In this section, it is introduced a brief review of the
literature. It highlights aspects of planning and
inventory control of dynamic models that include
remanufacturing processes.
In recent years, companies have begun to seek
for solutions that increase the life cycle of their
products. Many reasons can explain such an interest,
but a special is the scarcity of material resources
(i.e., the environment is finite). A way considered by
companies is to recovery returnable products by, for
instance, using remanufacturing systems. Thus, the
planning and inventory control of remanufacturing
systems have become an essential task. All modeling
and solving techniques used to plan and control the
production process in the forward channel can be
reapplied to reverse channel.
Glancing at the literature, we find that the
remanufacturing problems can be classified as
deterministic and stochastic. This classification will
depend on the hierarchical decision level where the
problem is being formulated. For example, at the
operational level (short term decisions) a good part
of the problem is deterministic, while at the strategic
level (long term decisions), they are stochastic. The
planning problem proposed in this paper belongs to
ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics
370
a class of stochastic problems.
Some papers that are directly or indirectly related
to the problem proposed here are due to: Shi et al.
(2011) that discuss a stochastic production planning
problem for a multi-product closed-loop system and
solve it using a Lagrangian relaxation based
approach; Wei et al. (2010) that propose an
inventory control model for the return and
remanufacturing processes under uncertains of the
demand and returns. They propose a linear
programming approach to deal with the uncertainty
of the problem; Ahiska and King (2010) that use a
Markov decision process to model and solve an
inventory problem of single product recoverable
manufacturing system; Inderfurth (2005) that
considers a multi-period stochastic closed-loop
supply chain with remanufacturing. A heuristic is
proposed that allows evaluating this environment;
Nakashima et al. (2004) that study a stochastic
control problem of a remanufacturing system, which
is modeled and solved via Markov decision process
approach; and Dobos (2003) that formulates a
quadratic production planning problem to deal with
a reverse logistic system, and uses control theory to
solve it. However, it is important to add that the
processes of modeling and solution, adopted in this
article, are different from those found in the papers
mentioned above.
3 THE REVERSE SYSTEM
Figure 1 illustrates a schematic view of the reverse
logistics system. Two channels define such a
system: the forward channel pushes serviceable
products (i.e. new and remanufacturable) to the
market. In the backward (or reverse) channel, used-
products are collected from the market. Part of them
will be remanufactured, and others disposed.
Figure 1: The production/remanufacturing system.
Note also that new products are previously
inspected and all that fails, during this process, are
sent to remanufacturable inventory unit (see dashed
lines in the Figure 1).
Typical examples of organizations that follow
the scheme of Figure 1 are companies that
remanufacture bottles, cans, containers, etc.
Some features and properties of the system
exhibited by Figure 1 are the following: (a) demand
for serviceable products should be met by the
combination of both manufactured and
remanufactured products; (b) the fluctuation of
demand and return rate are no correlate stationary
stochastic processes that follow the normal
distribution functions, whose first and second
statistics moments are perfectly known; (c) there are
upper limits for both manufacturing and
remanufacturing capacities, but not for inventory
units; and (d) used-products may be discarded after
being collected from the market. There are two main
reasons to discard used-products: the first has a
technical justification, which is related to the quality
of used-product, which may be inappropriate for
remanufacturing. The second reason has a financial
justification. It is motivated by the idea in which
remanufacturing all products that return, can
increase unnecessarily holding inventory costs of
serviceable products. In short, the last strategy
brings the possibility of increasing the overall cost
for running the system.
4 THE STOCHASTIC PROBLEM
The combination of production/remanufacturing
stochastic planning problem with chance-constraints
can be formulated by a classical linear programming
model as follows:
{} {} {} {}{
}
[]
()
(f) 0u;uu0;uu0
)e(0)(xProb.
)d(0)(xProb.
)c(Ruuuxx
)1()b(Duu)1(xx
s.t.
)a(ucucuc
xEhxEhxEhxEhZMin
3
k2
2
k1
1
k
2
k
1
k
k
k
1i
3
i
2
i
1
i
2
0
2
k
k
k
1i
2
i
1
i
1
0
1
k
332211
1N
0k
2
k2
1
k1
2
N2
1
N1
u,u,u
321
β
α
+ρ+=
+ρ+=
+++
+++=
=
=
=
Problem (1) minimizes the sum of linear of expected
holding costs, at each store units, production costs
for manufacturing and remanufacturing products,
Productio
n
Remanufacture
u
3
u
2
r
ρ·u
1
Serviceable unit
Remanufacturable unit
Inspection unit
u
2
u
1
(1-ρ)u
1
d
Supply
AN OPEN-LOOP SOLUTION FOR A STOCHASTIC PRODUCTION-REMANUFACTURING PLANNING PROBLEM
371
and costs with disposal. As a result, it provides an
optimal global sequential plan, which encompasses
optimal inventory goals for serviceable and
remanufacturable units and, optimal production rates
goals for manufacturing and remanufacturing
processing units, as well.
For each period k of the planning horizon N,
variables and parameters that compose (1) have the
following notation:
1
k
x
- Serviceable inventory level at period k.
2
k
x
- Remanufacturable inventory level at period k.
1
k
u
- Production rate at period k.
2
k
u
- Remanufacturing rate at period k;
3
k
u
- Disposal rate at period k;
d
k
- Demand level at period k.
r
k
- Return rate at period k.
D
k
- Cumulative level of demand at period k
R
k
- Cumulative return rate at period k
h
1
- Serviceable holding cost
h2
- Remanufacturable holding cost
c
1
- Production cost
c
2
- Remanufacturing cost
c
3
- Disposal cost
ρ
[0, 1] – Historical index of new product that are
rejected during the control quality process.
α and β – Probabilistic indexes that assure the
validity of the probabilistic constraints (1.b)-(1.c).
Both
=
=
k
1i
ik
dD
and
=
=
k
1i
ik
rR
are random
variable that follow cumulative normal distributions
of probability. The first variable is related to each
demand level d
i
(i.e. orders placed at period i) that is
accumulated until reaching the period k. The second
variable refers to the cumulative number of returns
of products collected during each period i. They are
assumed stationary and not correlated variables with
first and second statistics moments given,
respectively, by:
σ=Θ=
σ=Θ=
=
=
r
R
k
k
1i
ik
d
D
k
k
1i
ik
k;r
ˆ
R
ˆ
k;d
ˆ
D
ˆ
(2)
where
i
d
ˆ
and
i
r
ˆ
denote mean values of demand level
and return rate during period i. The pair
),(
rd
σ
σ
denotes their finite standard deviations, respectively.
It is important to detach that the normal
distribution is usually a good approximation for a
variety of other types of distributions. For example,
the Poisson distribution is commonly used to
represent events that involve arrivals. Thus, in a
production environment where products are moved
from inventory as soon as orders are placed, an
interesting representation for these orders is the
Poisson distribution. However, whenever the arrival
flow becomes intense, the normal distribution can be
a good approximation for the Poisson Process, see
(DasGupta, 2010). Additionally, Graves (1999)
justifies the use of normality hypothesis to describe
demand fluctuation in a manufacturing environment.
It is important to emphasize here that the
problem formulated in (1) reflects the economic
aspects of the productive environment that intends to
represent. This means that processes of collecting,
storing and remanufacturing used-products are
economically viable, when transport and inventory
costs, in reverse channel, consume only a small
portion of the budget of the organization.
Additionally, if the cost of remanufacturing is less
than the cost of manufacturing new-products, the
chances of reverse logistics model to be
economically viable are very high. Also, it is
important to have in mind that costs with raw-
material and components to manufacture new
products can increase significantly the production
cost of new products. In short, the production/
manufacturing policy, provided by problem (1),
must be interpreted from cost viewpoint, in order to
conclude something about its economic viability.
4.1 Solving Problem (1)
Find an optimal global solution to the problem (1) is
not an easy task. The reason of this is the stochastic
nature of the inventory balance system and
constraints on main decision variables. As a
consequence, heuristic techniques are viable
alternatives that allow finding a near-optimal
solution to problem (1). The normal distributions,
assumed to random variables of problem (1), makes
possible to use the certainty-equivalence principle to
transform problem (1) into an equivalent, but
deterministic, problem. This deterministic problem
is entirely based on the first and second statistic
moments of the random variables (Bertsekas, 2007).
5 AN EQUIVALENT PROBLEM
The certainty-equivalence principle is commonly
ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics
372
employed to reduce the complexity of a stochastic
programming problem. In fact, it allows
transforming problems like (1) to a deterministic-
equivalent form, which is easier to solve.
The deterministic-equivalent problem is derived
by using the first and second statistic moment of the
serviceable and remanufacturable inventory
variables. Because of the normal assumption, the
process of conversion is very simple. The main steps
of this process are given in the sequel.
5.1 Deterministic Inventory-production
Balance System
The stochastic inventory-production system, descry-
bed in the Figure 1, is mathematically modeled by
discrete-time balance equations given in (1.b) and
(1.c). Note that these equations have five decision
variables. Two of them are state variables, which
describe inventory units for serviceable and
returnable (i.e. remanufacturable) products, while
three others are control variables describe rates of
manufacturing, remanufacturing, and discarding.
Thus, taking into account directly the expected value
on (1.b) and (1.c), results that they can be rewritten
as follows:
[
]
()
k
k
1i
3
i
2
i
1
i
2
0
2
k
k
k
1i
2
i
1
i
1
0
1
k
R
ˆ
uuux
ˆ
x
ˆ
D
ˆ
uu)1(x
ˆ
x
ˆ
+ρ+=
+ρ+=
=
=
(3)
where
{
}
1
k
1
k
xEx
ˆ
=
and
{
2
k
2
k
xEx
ˆ
=
are the expected
levels of serviceable and remanufacturable units.
Note also that
1
0
x
ˆ
=
1
0
x
is provided a prior by
manager.
5.2 Deterministic Criterion
The linearity of the criterion (1.a) associated to the
normal random nature of decision variables allow
that this criterion can be quickly transformed to an
equivalent, but deterministic linear criterion, which
preservers the first and second statistical moments of
random variables.
Based on above, it is only necessary to compute
the expected value of (1.a). Thus proceeding, the
original criterion, given by:
{} {} {} {}
{
}
N1
12 12
12 12
k0
11 2 2 3 3
ZhE hE hE hE
NN kk
x
xxx
cu c u cu
=
=++ +
++ +
(4)
is transformed int o anequivalente to
{
N1
12 12
12 111122
k0
33
ˆ
ˆˆ ˆˆ
Zh h h h
NN kk
x
xxxcucu
cu
=
=++ +++
+
(5)
where
1
k
x
ˆ
and
2
k
x
ˆ
denote the mean values of inventory
variables, for each period k [0, N]. Note that these
values are perfectly computed from (3).
5.3 Deterministic Constraints
The serviceable and remanufacturable inventory
variables, that is, states of (1.b) and (1.c) systems,
are assumed non-negative variables. It is also
assumed that there is no upper limit of storage for
these systems. These characteristics can be
represented as follows:
<
<
Infx0
Infx0
2
k
1
k
(6)
Note that the above characteristic of considering
no upper bound to inventory variables is not unusual
it seems. In fact, this occurs because many
companies produce lots of products that are made to
stock, and these lots are rapidly absorbed by the
market. This means that these lots remain for a short
period of time stored in the serviceable unit. As a
consequence, in such a case, it is not necessary to set
upper limits into the model. Analogously, products
collected from the market remains short periods in
the inventory remanufacturing unit. Usually, after a
quick inspection, they are sent to remanufacture or
to dispose.
Due to the randomness of inventory variables,
constraints (6) cannot be used directly into the
stochastic problem (1). There are, however, two
ways of including (6) into (1): the first one is a
classical procedure from mathematical programming
theory that consists in penalizing (6) directly into the
criterion of the problem (1). The second format
consists in taking the constraints in (6) on
probability operator. As a result, they become
chance-constraints. This last format is usually more
realist, particularly because it allows explicitly
preserving constraints (6) into the stochastic
problem (1); see (1.d) and (1.e). The transformation
of these constraints for deterministic equivalent
forms is performed in the sequel.
AN OPEN-LOOP SOLUTION FOR A STOCHASTIC PRODUCTION-REMANUFACTURING PLANNING PROBLEM
373
Firstly, using the inventory balance equation
(1.b), the chance-constraint (1.d) can be rewritten as
follows:
α+ )Dy(xProb.
kk
1
0
(7)
or, also, in the form:
α
Θ
+
ε )
D
ˆ
yx
(Prob.
D
k
kk
1
0
d
(8)
where
[]
=
+ρ=
k
1i
2
i
1
ik
uu)1(y
and
d
ε
is a normal
random variable related to the variable D
k
. Thus, the
variable
d
ε
have the same cumulative distribution
function (c.d.f) of D
k
, denoted here as Φ
D
.
Using basic knowledge of statistical theory, the
constraint (8) can be easily handled, and rewritten as
shown below:
)(D
ˆ
yx)(
D
ˆ
yx
1
D
D
kkk
1
0
1
D
D
k
kk
1
0
αΦΘ+αΦ
Θ
+
Taking into account the deterministic system (3)
and remembering that
1
0
x
=
1
0
x
ˆ
, it is possible to show
that
kk
1
0
1
k
D
ˆ
yx x
ˆ
+=
. As a result, the determinist
equivalent constraint related to the chance constraint
(1.d) is given by:
)(kx)(x
1
Dd
1
k
1
D
D
k
1
k
αΦσαΦΘ
(9)
where
1
D
Φ
(.) is the inverse distribution of
probability of variable D
k
.
Proceeding in a similar way for the chance-
constraints (1.e), it is possible to find that
)(kx)(x
1
Rr
2
k
1
R
R
k
2
k
βΦσβΦΘ
(10)
where
1
R
Φ
(.) is the inverse distribution of
probability of variable R
k
.
As a last observation, note that the inventory
levels
2
k
1
k
xandx
are random variables that, due to
the linearity of the system, follow similar
distribution of probability of D
k
and R
k
.
5.4 The Equivalent Problem
Gathering together all the transformed parts as
exhibited above, an equivalent, but deterministic,
problem can be stated to represent the stochastic
problem (1). It is formulated as follows:
{
}
()
123
N1
12 12
12 1211
u,u ,u
k0
22 33
11 12
k0
1
22 123
k0
1
11
k
21
k
12
k1 k 2
ˆ
ˆˆ ˆˆ
Min Z h h h h
s.t.
ˆ
ˆ
x(1) (11)
ˆ
ˆ
x
ˆ
x()
ˆ
x ( )
0u ;0u ;
NN kk
k
ii k
i
k
ii i k
i
dD
rR
xx xxcu
cu cu
xuuD
xuuuR
k
k
uu
ρ
ρ
σα
σβ
=
=
=
=+ + +++
+
⎡⎤
=+ +
⎣⎦
=+ +
≥⋅Φ
≥⋅Φ
≤≤ ≤≤
3
k
u0
Some advantages of problem (11) are:
- Linearity of criterion (1.a) and inventory balance
equations, given by (1.b) and (1.c), are totally
preserved.
- In the same way, chance-constraints (1.d) and (1.e)
are converted to equivalent determinist constraints,
in which are maintained original statistical
characteristics of the problem (1); see that first and
second statistical moments, given in (2), are
explicitly into the constraints (9) and (10).
- The equivalent problem (11) is very simple to be
solved. Classical techniques of linear programming,
available in the literature, can be applied.
6 AN EXAMPLE
A company sells a kind of product that can be
remanufactured. Structurally speaking, the
production/remanufacturing process of this company
can be seen as a closed-loop system, as that one
shown by Figure 1. In the forward direction of this
system, new and remanufactured products are stored
into a serviceable inventory unit. These products are
previously stored in serviceable unit in order to meet
the demand that fluctuates in an uncertain way over
weekly periods. On the other hand, in the reverse
(backward) direction, used-products are collected
weekly from the market, with base on an uncertain
return rate. They are stored into the
remanufacturable inventory unit and, after being
inspected, they are sent to remanufacture or to
discard. Besides, a very small part of products stored
in the remanufacturable inventory are defective
products. They are new products that fail during the
quality inspection process.
ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics
374
The demand fluctuation for serviceable products
and the rate of return products are considered
uncorrelated random variables that follow stationary
normal processes. Such processes are well-defined
by their first and second statistics moments. They
follow the same equations given in (2) to describe
mean and standard-deviation.
The objective of the company is to analyze if,
under certain conditions related to inventory and
production costs, it is advantageous to increase the
return rate of used products. More specifically, the
question to be answered is: can the increase of the
weekly return rate improve the performance of
processes (1.b) and (1.c)?; and what is the
implication on the total production cost?
To carry out such an analysis, the company
decided to develop production plans for the next
eight weeks (N=8), which are based on the solution
of deterministic problem (11).
Before proceeding with the deployment of this
study, it is necessary to introduce general data of the
problem, which are mainly available in the Tables 1
and 2.
Table 1 presents the first and second statistic
moments related to the normal distribution function
of demand variable.
Table 1: Weekly statistics of demand d
k
.
k 1a 2a 3a 4a 5a 6a 7a 8a
k
d
ˆ
600 595 610 615 605 585 600 610
Standard-deviations:
d
σ
20, k
From the Table 1, it is important to calculate the
absolute mean value (amv) of demand variable d
k
.
This absolute value is based on the arithmetic mean
of first statistical moments of demand, known a prior
for each period k of the planning horizon N (i.e,
1
d
ˆ
,
2
d
ˆ
,
3
d
ˆ
, ...,
N
d
ˆ
). Thus, considering the mean values
provided in Table 1, for 8 weekly periods (N=8), the
absolute mean value for this example is computed as
follows:
=
=
8
1i
i
.602d
ˆ
amv
(12)
The amv value (12) will be used ahead to determine
the mean value of the return rate of used-products.
Table 2 introduces the current information about the
current state of the production/remanufacturing
system, and about parameters and costs of the
problem (9).
Based on these data, two situations will be
analyzed by managers of the company: in the first
one, managers consider that the mean value of the
return rate variable is set equal to 50% of absolute
mean value (amv) of demand, that is,
k 301,r
ˆ
k
= .
Table 2: Other data of the problem.
Initial inventories
300x
1
0
=
and
225x
2
0
=
Inventory costs
2$h
1
=
and
1$h
2
=
Production costs
1$c
1
=
and
20,1$c
2
=
Disposal cost
14,0$c
3
=
Probabilistic indexes
%95=
α
and
%80=β
Rate of defective
%5
=
ρ
This means that on an average flow of returned
products is 50% smaller than the flow of orders
placed (i.e. demand for serviceable products).
The second situation considers that on average
100% of used-products are weekly collected. This
means that the return rate is close to
k 602,r
ˆ
k
=
.
Additionally, it pertinent to mention that, for
both situations, the second statistical moment will be
set exactly equal to 10 units, that is,
k ,01
r
=σ
.
In the sequel, the two situations are implemented
in the format given by (11), and their solutions are
compared.
6.1 1
st
Situation: 50% Returnable
In this situation, used-products that are weekly
collected from the market have the same weekly
average return rate of
301,r
ˆ
k
= with standard-
deviation of 15%.
Figures 2 and 3 illustrate optimal inventory-
production trajectories for forward and reverse
channels of the system illustrated generically in the
Figure 1. It is interesting to note that serviceable and
remanufacturable inventory levels show a similar
sharp in their optimal trajectories. In fact, except for
the first week, when inventory-production process
(1.b) and (1.c) are still being adjusted to the initial
conditions of the problem
,)xandx(i.e.,
2
0
1
0
the
remaining weeks show the continuous growing of
serviceable and remanufacturable inventory levels
over the weekly periods. As a result, safety-stocks
are providing. In the case of serviceable inventory,
the idea of a safety-stock is to guarantee ready
delivery of products to meet demand; and, in the
case of remanufacturable inventory, the idea is to
provide a buffer of used products that can be used in
remanufacturing process, during future periods, for
avoiding backlogging occurrences.
AN OPEN-LOOP SOLUTION FOR A STOCHASTIC PRODUCTION-REMANUFACTURING PLANNING PROBLEM
375
It is worth mentioning that these safety-stocks
are due to the original stochastic nature of the
problem (1). During the transformation process,
from (1) to the equivalent problem (11), safety-
stocks are created to preserve the feasibility of (11);
to realize this, see constraints (9) and (10).
The figure 2 shows optimal rates of
manufacturing and remanufacturing, which evolve
over weekly periods of the planning horizon. It is
possible to observe that the manufacture of new
products has met quite completely the weekly
fluctuation of demand. The remanufactured products
only complement a small part of serviceable
products. Practically, the weekly amount of disposal
products is very close to the amount of products that
are weekly remanufactured. It is important to
remember that 5% of products, which are processed
in the remanufacturing unit, are defective products.
Production and inventory costs associated to the
operation of this system are presented in Table 3,
given ahead. The idea is to compare total production
costs for the two situations analyzed.
Figure 2: Serviceable and remanufacturable levels.
Figure 3: Production rates.
6.2 2
nd
Situation: 100% Returnable
In this situation, used-products that are weekly
collected from the market have the same weekly
average return rate of
602,r
ˆ
k
= with standard-
deviation of 15%.
The optimal inventory and production
trajectories for forward and reverse channels are
respectively exhibited in Figures 4 and 5. Note that
the trajectories, depicted in the Figure 4, are exactly
equal to trajectories exhibited in Figure 2. Note also
that the characteristic of inventory levels increase
over the week periods of the planning horizon is
typical behavior imposed by chance constraints (9)
and (10). Once again it is important to emphasize
that such a behavior creates safety-stock to protect
inventory units against stockout occurrences.
Contrasting with the behavior of inventories
trajectories that do not change with the increase of
the return rate (as seen in Figures 2 and 4), the
behavior of manufacturing, remanufacturing, and
disposal variables, over weekly periods, change
completely (when compared with last situation). In
fact, in the Figure 3, it is observed that the level of
remanufactured products only complements the level
of new manufactured products. This means that the
amount of “new” products predominates in the
weekly production of serviceable products. On the
other hands, the Figure 5 shows that the weekly
remanufacturing rate is greater than the weekly rate
of manufacturing. Therefore, in such a situation, the
amount of remanufactured products is that
effectively predominate. This characteristic reveals
the importance of increasing return rate percentage.
Figure 4: Serviceable and remanufacturable levels.
Figure 5: Production and disposal rates.
0
50
100
150
200
250
300
350
012345678
weeks
Inventory levels
Serviceable
Remanufacturable
0
100
200
300
400
500
600
01234567
weeks
Production rates
Manufacture
Remanufature
Disposal
0
50
100
150
200
250
300
350
012345678
weeks
Inventory levels
Serviceable
Remanufacturable
0
100
200
300
400
500
600
01234567
weeks
Production rates
Manufacture
Remanufature
Disposal
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6.3 Assumptions, Costs and Final
Comments, and Future
Perspectives
With the objective of developing the situations 1 and
2, the problem (11) took into account some
particular features of the company’s production
environment. These features had strong implications
on optimal costs provided by this problem, and
allowed some reflections about other perspectives
for using the current model. In the sequel this
features are discusses as problem’s assumptions.
6.3.1 Some Assumptions
a) Serviceable holding cost (h
1
) is the double of the
remanufacturing holding cost (h
2
). It is considered
here that serviceable products need special care with
storage, packing, etc.
b) Cost of the remanufacturing process (c
2
) is 20%
more expensive than the cost for manufacturing new
products (c
1
). In practice, remanufacturing process is
usually an expensive activity because it involves a
series of sub-activities like dismantling, replacing
parts, reprocessing of parts, testing, etc.
c) It is assumed that the cost of disposing is quite
insignificant when compared with the cost of
remanufacturing. This means that can be
advantageous to discard used products.
d) It was not taken into account raw material
purchase costs for manufacturing new products.
e) Transport costs are indirectly included in holding
inventory and production costs.
6.3.2 Costs Evaluations
Features, listed previously, explain the optimal plans
provide by situation 1 and 2, whose costs generated
are given in Table 3.
Table 3: Total and individual costs.
Costs
1
st
situation 2
nd
situation
Serviceable holding 1.672,80 1.672,80
Remanufacturing holding 430,85 430,85
Manufactured rate 3.191,90 1.928,70
Remanufactured rate 1.829,10 3.332,10
Disposal rate 165,21 325,49
Total cost
7.289,86 7.689,94
The costs of holding inventory for serviceable
and remanufacturable units are exactly the same for
both situations. This means that the optimal
inventory trajectories, as shown in Figures 2 and 4
did not vary with increasing rate of return used. The
justification for such a result is that the weekly
demand is sufficient to absorb all products available
in the serviceable inventory unit (i.e. “new” and
remanufactured products that are processed
concurrently; see Figures 3 and 5). Other interesting
point is that the cost for disposing is very cheap, so
all used products that return or defective products
are immediately remanufactured or discarded.
6.3.3 Final Comments
It is important to note that although the total cost of
the second situation has been 5.5% higher than the
total cost verified by the first situation, the resulting
optimal production plan shows a balanced
distribution on the amount of “new” products (i.e.
manufactured products) and remanufactured
products. This characteristic can be observed
through Figures 3 and 5. In fact, serviceable
products are now composed of approximately 35%
of manufactured (i.e. new products), 5% of
recovered products (they were identified as
defectives) and around 60% of remanufactured
products; Figure 5 illustrates partially this
characteristic. Particularly, the main advantage of
this type of operation is on the cost reduction for
raw-material purchasing. Thus, it is sure that a large
amount of remanufactured products implies in lower
raw-material acquisition. As a result, total
production costs can be minimized. Unfortunately,
Table 2 does not reflect such a reduction in
production costs.
6.3.4 Further Extensions of the Model
As prospects for future study, it might be considered
an extended version of the model here proposed.
Such a version should be idealized to allow a greater
realism in the formulation of the original stochastic
planning problem (1). Some improvements into the
model are, for instance:
a) to adopt more realistic functions to describe the
production and inventory costs. Thus, one can
consider nonlinear functions, but, in particular, it is
desirable that they have convexity and concavity
properties in order that deterministic transformations
can be easily performed;
b) to include upper storage limits for serviceable and
remanufacturable stores and consider them into
chance-constraints formulation;
c) to consider multi-products and include new
constraints on the remanufacturing and
manufacturing processes that allow sharing
production operations among these products in their
respective processing units; and
AN OPEN-LOOP SOLUTION FOR A STOCHASTIC PRODUCTION-REMANUFACTURING PLANNING PROBLEM
377
d) to consider sequential solutions based on rolling
horizon techniques, which allow optimal adjustment
of the generated plans in order to follow actual
demand fluctuations during each period of the
planning horizon, see Pereira and De Sousa (1997).
With these improvements, the model will be
closer to reality, and, as a result, it will be possible
to develop more accurate studies, and to provide
more efficient plans for management purposes.
7 CONCLUSIONS
From a linear structure that represents an inventory-
production dynamic system with forward and
reverse flows, a production planning problem based
on a stochastic linear programming model with
chance constraints was proposed. The solution of
this optimization problem can be very useful for
those companies that deal with operations involving
both used-products return, and defective products, as
well.
The main difficulty here is that a global solution
to this stochastic problem cannot be trivial. Thus, it
was assumed without lost of generality that the
statistical behavior of the demand could be
approximated by a normal process. As an immediate
consequence, an easy-to-solve equivalent determi-
nistic problem was proposed. This provides an open-
loop sub-optimal solution that would be optimal
global if current demand observed at each period of
the planning horizon was exactly equal to mean
demand. However, this sub-optimal solution is a
good estimate to the original stochastic problem,
once demand levels and return rates are both
stationary processes.
A simple example was proposed to illustrate the
applicability of the equivalent deterministic
problem, and, at the same time, to compare the
effect of increasing the return rate of used-products
on the optimal production policies provided by
solving the problem (11). In order to make the
analysis interesting, it was considered that the cost
of remanufacture would be 20% higher than the cost
to manufacture new products. In this case, it was
expected that the manufacture of new products
would be preferred instead of the remanufacture of
used-products, but this was not effectively observed.
Indeed, the results showed that as the return rate of
used-products becomes close to the average absolute
value (amv), the remanufactured products becomes
more attractive to the company. This kind of
operating system (1) is illustrated by Figure (5).
The model considered here is, therefore, an
interesting management tool that allows not only
develop a production plan for implementation within
the hierarchy of business decisions, but also allows
helping in the process of decision making on new
strategies, as discussed in the example of section 5.
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