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

 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

332211

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

Remanufacture

ρ·u

Serviceable unit

Remanufacturable unit

Inspection unit

(1-ρ)u

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:

- Serviceable inventory level at period k.

- Remanufacturable inventory level at period k.

- Production rate at period k.

- Remanufacturing rate at period k;

- Disposal rate at period k;

- Demand level at period k.

- Return rate at period k.

- Cumulative level of demand at period k

- Cumulative return rate at period k

- Serviceable holding cost

- Remanufacturable holding cost

- Production cost

- Remanufacturing cost

- Disposal cost

∈

[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

∑

and

∑

are random

variable that follow cumulative normal distributions

of probability. The first variable is related to each

demand level d

(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=

∑

k;r

k;d

(2)

where

and

denote mean values of demand level

and return rate during period i. The pair

),(

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:

[

]

()

uuux

uu)1(x

+−−ρ+=

−+ρ−+=

∑

(3)

where

{

}

xEx

and

{

}

xEx

are the expected

levels of serviceable and remanufacturable units.

Note also that

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:

{} {} {} {}

{

}

12 12

11 2 2 3 3

ZhE hE hE hE

NN kk

xxx

cu c u cu

−

=++ +

++ +

∑

(4)

is transformed int o anequivalente to

{

12 12

12 111122

ˆˆ ˆˆ

Zh h h h

NN kk

xxxcucu

−

=++ +++

∑

(5)

where

and

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

(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.

(7)

or, also, in the form:

α≥

−+

≤ε )

(Prob.

(8)

where

[]

∑

+ρ−=

uu)1(y

and

is a normal

random variable related to the variable D

. Thus, the

variable

have the same cumulative distribution

function (c.d.f) of D

, denoted here as Φ

Using basic knowledge of statistical theory, the

constraint (8) can be easily handled, and rewritten as

shown below:

)(D

yx)(

kkk

αΦ⋅Θ≥−+⇒αΦ≥

−+

−−

Taking into account the deterministic system (3)

and remembering that

, it is possible to show

that

yx x

−+=

. As a result, the determinist

equivalent constraint related to the chance constraint

(1.d) is given by:

)(kx)(x

αΦ⋅σ⋅≥⇒αΦ⋅Θ≥

−−

(9)

where

−

(.) is the inverse distribution of

probability of variable D

Proceeding in a similar way for the chance-

constraints (1.e), it is possible to find that

)(kx)(x

βΦ⋅σ⋅≥⇒βΦ⋅Θ≥

−−

(10)

where

−

(.) is the inverse distribution of

probability of variable R

As a last observation, note that the inventory

levels

xandx

are random variables that, due to

the linearity of the system, follow similar

distribution of probability of D

and R

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

12 12

12 1211

u,u ,u

22 33

11 12

22 123

k1 k 2

ˆˆ ˆˆ

Min Z h h h h

s.t.

x(1) (11)

x()

x ( )

0u ;0u ;

NN kk

ii k

ii i k

xx xxcu

cu cu

xuuD

xuuuR

σα

σβ

−

=+ + +++

⎡⎤

=+ − + −

⎣⎦

=+ −− +

≥⋅⋅Φ

≤≤ ≤≤

∑

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.

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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 1a 2a 3a 4a 5a 6a 7a 8a

600 595 610 615 605 585 600 610

Standard-deviations:

≈ 20, ∀k

From the Table 1, it is important to calculate the

absolute mean value (amv) of demand variable d

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,

, ...,

). 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:

∑

≅=

.602d

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

∀

= .

Table 2: Other data of the problem.

Initial inventories

300x

and

225x

Inventory costs

2$h

and

1$h

Production costs

1$c

and

20,1$c

Disposal cost

14,0$c

Probabilistic indexes

%95=

and

%80=β

Rate of defective

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

∀

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

∀=σ

In the sequel, the two situations are implemented

in the format given by (11), and their solutions are

compared.

6.1 1

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

= 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.,

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

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

= 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.

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weeks

Inventory levels

Serviceable

Remanufacturable

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500

600

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weeks

Production rates

Manufacture

Remanufature

Disposal

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150

200

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weeks

Inventory levels

Serviceable

Remanufacturable

100

200

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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

) is the double of the

remanufacturing holding cost (h

). It is considered

here that serviceable products need special care with

storage, packing, etc.

b) Cost of the remanufacturing process (c

) is 20%

more expensive than the cost for manufacturing new

products (c

). 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

situation 2

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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