Optimizing a Production-Inventory Systems with Deteriorating and Remanufacturing Products

6th IFAC Conference on Management and Control of Production and Logistics The International Federation of Automatic Control September 11-13, 2013. Fortaleza, Brazil

Optimizing a Production-Inventory Systems with Deteriorating and Remanufacturing Products O. S. Silva Filho I. R. Salviano Center for Information Technology Renato Archer - CTI Campinas – SP – Brazil (e-mail: [email protected]) Abstract: The environment is now a short-term priority of all projects that involves a supply chains sustainable. The concept of green supply chain can be measured by the ability in which companies can recycle or remanufacture products or items. Indeed, such a concern can reduce the use of extraction of natural resources, and this can surely reduce the environmental impact. Reverse processes is part of the process to reduce this drawback. A linear quadratic stochastic problem subject to production-inventory systems is formulated. The objective of this problem is to meet the demand for a product that can be remanufactured after used. This product can be deteriorated over a period of time and also be remanufactured to be reused. Assuming the demand as stationary and normally distributed, an associated equivalent deterministic problem is introduced. From this last problem, optimal inventory-production scenarios can be created by variation of parameters such as: return rate of used products, delay of return, or even both. A simple hypothetical example illustrates the use of these scenarios for decision-making. for remanufacturing or recycling, which allows used products returning to the serviceable inventory of the company. At last, the third problem is related to the planning of the reuse of items, parts and products without any additional process of remanufacturing. There are many different approaches to deal with each one of these problems.

1. INTRODUCTION In reason of the great importance given for the rational use of resources extracted from environment, reverse logistics became an essential part of an integrated supply chain. To realize such a fact, it is enough to note how typical activities of the forward channel of a supply chain (e.g. planning, scheduling, controlling, etc.) has being replicated to the reverse channel. The main objective of a reverse system is to move products from their final destination to a place where they can get value or can be properly disposal. In this way, some products can capture values along the reverse chain by mean of recycling or remanufacturing processes. In short, someone can understand the above objective as a way that all companies have to contribute with the environment reducing the waste and, in parallel, improving the profit of the supply chain by mean of remanufacturing or recycling.

In this paper, the focus is on the second kind of problems that is problems, where products are entirely recovered by remanufacturing and then replaced to the market to be consumed. These problems are usually formulated by a criterion of minimum-cost subject to a dynamic productioninventory system with a special structure for recovering or disposing used-products that return from marketplace. Additionally, the random nature of the demand affects this dynamic system and it becomes a stochastic process. As a result, these problems belong to the class of stochastic mathematical programming, which means that they are more complex to be solved.

Any industrial company belonging to a particular supply chain can be identified by its production-inventory system. Such a system can be mathematically formulated through two different, but integrated parts: the forward (direct) and backward (reverse) channels. In the forward channel, products are manufactured, stored in serviceable inventory units, and then moved to marketplace in order to meet demand. On the other hands, in the backward channel, usedproducts are recovered or discarded. Authors, like Fleischmann et al (1997), have provided a typology of quantitative models for reverse logistics. Usually, in the literature, three kinds of problems involving such models are formulated. The first one considers the collection and transportation of used products and packages. The distribution process along of the return channel starts from the marketing, particularly where customers leave their outof-use products. The second considers the schedule scheme 978-3-902823-50-2/2013 © IFAC

It is worth mentioning that stochastic production-inventory systems are usually found in reverse logistics problems, particularly in reason of the return rate not known precisely over the future periods. Fleischmann et al. (1997) have shown that the traditional classification of production and inventory stochastic problems, based on discrete or continuous-time models, can be also applied to model and solve problems of products recovering. In this paper, a discrete-time linear and quadratic stochastic model with constraints is considered to represent a recovery problem. In such a formulation, demand’s fluctuations and return’s rates over periods are stationeries and normally distributed random variables. Based on these assumptions, the stochastic problem can be easily converted to an equivalent deterministic problem. Sensibility analysis, provided from simple variation of some parameters as return rate of used products, delay of return, or 217

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both, allows creating production scenarios that help managers to make prospered decisions. In order to illustrate the applicability of this model, a simple numerical example is introduced.

remanufacturing activities; and the second has a financial justification, in which remanufacturing all products can significantly raise the inventories, and, as a result, such strategy can increase the overall production cost.

This paper brings up a discussion about recovery usedproducts via the following interrelated sections: the section 2 discusses about a discrete-time Linear Quadratic Gaussian (LQG) model with constraints for reverse logistics problem, which is, particularly, formulated to represent an usedproduct recovery system; the section 3 introduces an equivalent, but deterministic, problem that preserves some properties of the original stochastic model; in sequence, the section 4 comments, briefly, about a method of solution for the deterministic problem; and the section 5 presents a simple example to illustrate the application of this equivalent model. Basically, it shows how scenarios of production, provided by the model, can help managerial decisions related to recovery used-products in a reverse channel of a supply chain.

2.1. Inventory-production system The inventory-production process, illustrated in figure 1, can mathematically be modeled by a discrete-time stochastic control system with two state variables, which are described by the inventory levels of the store 1 and 2, and three control variables that are related to manufacturing, remanufacturing, and discard rates. Next, the main aspects of this formulation are presented. The discrete-time stochastic control system is described by the following two difference equations, which represent, respectively, the inventory balance systems related to forward and reverse channel of the supply chain:

2. THE STOCHASTIC MODEL

x1(k+1) = (1-)x1(k)+(1-)u1(k)+u2(k)-d(k) x2(k+1) = (1-)x2(k)+u1(k)-u2(k)-u3(k)+d(k-τ) x3(k+1) = x1(k)+x2(k)

Figure 1 illustrates the forward and backward channels of a unique product supply chain. Note that there are two stores in this figure: the first one (Store 1) stocks manufactured and remanufactured products to meet the demand, and the second store (Store 2) collects the returned products. These returned products can be remanufactured or disposal properly.

(1) (2) (3)

where, for each period k, the notation is given as follows: x1(k) = inventory level of serviceable products (store 1);

It is worth emphasizing that the demand for products must be fulfilled by the combination between new products (manufactured) and remanufactured products (i.e., used products that are collected from marketplace and, if possible, overhauled). Other features and properties considered for the system exhibited by figure 1 are: a) the demand is a random variable that follows a stationary stochastic process, while the return process is assumed essentially deterministic; b) both manufacturing and remanufacturing process has infinite capacity; c) similarly, the maximum physical storage capacity of inventory for warehouses 1 and 2 are assumed unlimited; d) there is a constant time-delay associated with return products from the market; and e) used-products may be disposed after collection. There are two main reasons to discard used-products: the first has a technical justification, which is related to inappropriate returned products for

x2(k) = inventory level of returned and defective products (store 2); x3(k) = cumulative level of depreciate products that is disposal in industrial landfill ; u1(k) = production rate of manufacturing of new products; u2(k) = production rate of remanufacturing of used-products; u3(k) = discard rate of unserviceable products; d(k) = demand level of serviceable products; r(k) = return level of used-products from the marketplace The dynamic system (1) depends on the demand d(k) that is a Gaussian random variable. Thus, the inventory level x1(k) is a normal random variable. Graves (1999) discusses the importance of normal distribution in manufacturing issues.

1

Manufacturing

Inventory unit Sum operator

Remanufacturing

Quality control Final discard

2 Fig. 1. Diagram of a closed-loop system with uncertain demand and return

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relatively small penalties for small deviation; see Bertesekas (2000).

Additionally, it is worth observing that due to the linearity of the system (1), the inventory variable x1(k) follows similar distribution to the demand d(k); see Silva Filho (2001).

It is interesting to observe that the criterion (4) is not deterministic. In fact, because of the stochastic nature of the system, the variable x1(k) is a random variable, and so its real value can only be estimated. This explains the reason of applying expectation operator to this variable, that is, E{x1(k)2}.

Note yet that the dynamic system (1) represents the flow of serviceable products through the forward channel of the supply chain. In this channel, after being submitted to quality inspection, some manufactured products can be returned to be remanufacturing in a rate of  percent. Also, some serviceable products can be deteriorate during their storage process in the inventory unit 1 (see figure 1). It is assumed that such a deterioration occurs in a percentage rate of something close to . These unserviceable products are promptly discarded.

2.3. Discrete-time stochastic optimal control problem Based on above, a Linear Quadratic Gaussian (LQG) problem with constraints can be formulated, and an optimal production-inventory sequential policy {u1(k), u2(k) and u3(k) with k = 0,1,2, …, T-1} can provided from this problem. The constrained LQG problem can be formulated as follows:

System (2) denotes the reverse channel of the supply chain. It starts with an input of used-products that returns from marketplace in a rate described mathematically by r(k)=d(k-) where 01 denoting the percentage of returning unites after a delay of  periods. Since d(k-) is knew previously, the inventory level x2(k) is a deterministic variable). Note that the store 2 receives products that do not pass in quality control of forward channel, described by system (1). A quality control is also applied in the backward channel and some used products are disposed to industrial landfill. It is assumed that the amount of deteriorate products in store 2 is close to the percentage of .

Min h 1 E x 12 (T) 2  h 2 x 22 (T) 2  h 3 x 32 (T)  u1 ,u 2 ,u 3

{



x 2 (k  1)  (1  ) x 2 (k )  u 1 (k )  u 2 (k )  u 3 (k )  r (k ) x 3 (k  1)  x 1 (k )  x 2 (k )

Pr ob. x 1 (k )  x 1   

x 2 (k )  x 2 ; x 3 (k )  0; u 1 (k )  0; u 2 (k )  0; u 3 (k )  0 where E{.} and Prob.[.] denote the expectation and probability operators, respectively. The index  is a probabilistic index that denotes the level of customer satisfaction; see Silva Filho (2009). Here, it denotes the chance of satisfying customers with manufactured or remanufactured products. Sometimes this index is called of level of customer satisfaction

Zk(x1,x2,u1,u2,u3) = h E x 2 (k )  h x 2 (k )  h x 2 (k)  2

2

3 3

c 1 u (k )  c 2 u (k )  c 3 u (k ) 2 1

2 2

2 3

(4)

3. THE DETERMINISTIC MODEL

where h1, h2, and h3 denote respectively the inventory holding coefficients (prices) in the first and second store, and h3 denotes the cost for disposing defective products in the landfill. The coefficient c1, c2 and c3 denote prices related to manufacture, remanufacture and discard operations, respectively. The symbol E{.} denotes the expectation operator; which is here related to random variable x1(k). At last, it is important to add that the total production cost is given by Z T 

T  zk k 1

(5)

x 1 (k  1)  (1  ) x 1 (k )  (1   )u 1 (k )  u 2 (k )  d(k )

For each period k of planning horizon T, the function that models the production cost for running the process (1)-(3) is given as follows: 1

}

s.t.

2.2. The overall functional criterion



h 1 E x 12 (k ) h 2 x 2k (k )  h 3 x 32 (k ) 

c 1 u 12 (k )  c 2 u 22 (k )  c 3 u 32 (k )}

At last, the dependent system (3) denotes the industrial landfill. It is a storage point where depreciated and spoiled (i.e. returnable but impossible for remanufacturing) products are sent at their end life. It is a cumulative variable that shows the amount of products disposed and the cost to discard them

1

T 1  k 0

Due to some characteristics of the problem (4) such as linearity of system (1) and convexity of quadratic criterion (3), the classical certainty equivalence principle (Bertesekas, 2000) can be considered as procedure to simply the original problem. This procedure consists in applying the mathematical expectation operator in the random variables of the problem (4) (i.e. x1(k) and d(k)), and, in sequel, solving an equivalent deterministic model as described below.

. 3.1. Basic statistics of the system (1)

Some comments about (3):

As discussed before, the demand is a random variable that has its first and second statistic moments perfectly known for each period k, that is, E{d(k)}= dˆ (k ) and

The use of quadratics functions to represent production costs has some advantages when compared with other functions. For instance, quadratic cost induces high penalties for large deviations of the decision variables from the origin but

V (k)  

 0; k. Thus, since the inventory variable x1(k) is contaminated by demand, it is possible to determine the d

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3.2. An equivalent model for (4)

first xˆ 1 (k ) and second Vx (k ) statistic moments for this variable from the balance equation given in (1): 1

From the statistics given by (6), (9) and (10), it is possible to transform the problem (5) into a deterministic equivalent one, that is formulated as follows:

First statistic moment: Applying the expectation operator directly into the equation (1) results that:

Min h 1 xˆ 12 (T)  h 2 x 22 (T)  h 3 x 32 (T)  k 0 { h 1 xˆ 12 (k )  T 1

u1 ,u 2 ,u 3

Ex1 (k  1)  E(1 - ) x1 (k )  u1 (k )  u 2 (k )  d(k ) 

h 2 x 22 (k )  h 2 x 22 (k )  c 1 u 12 (k )  c 2 u 22 (k )  c 3 u 32 (k )}  K

(1 - )Ex1 (k )  u1 (k )  u 2 (k )  Ed(k )

s.t.

xˆ 1 (k  1)  (1  )xˆ 1 (k)  (1  )u1 (k)  u 2 (k)  dˆ (k)

(6)

Note that the balance equation (6) represents the mean evolution of inventory variable, for each period k of the planning horizon T, and it is a deterministic version of the system (1).

(11) ˆ xˆ 1 (k  1)  (1  ) xˆ 1 (k )  (1   )u 1 (k )  u 2 (k )  d(k ) xˆ 2 (k  1)  (1  ) xˆ 2 (k )  u 1 (k )  u 2 (k )  u 3 (k )  r (k ) xˆ 3 (k  1)  xˆ 1 (k )  xˆ 2 (k ) xˆ 1 (k )  x 1 ,  d ; xˆ 2 (k ); xˆ 3 (k ), u 1 (k ), u 2 (k ), u 3 (k )  0

Note that the problem (11) can be solved by any applicable mathematical programming technique. Both constant term K and the lower boundary of inventory store 1 (i.e. x 1 ) depend

Second statistic moment: Taking the difference between the equations (1) and (6), follows that

on the variance of inventory variable Vx (k ) and are 1

calculated, from (9)-(10), as follows (see Silva Filho (2011)):

(x1 (k  1)  xˆ 1 (k  1))  (x1 (k)  xˆ 1 (k))  (d(k)  dˆ (k))

T

T

k 0

k 0

K   Vx1 (k )   k   d2  T   d2 Assuming that x 1 (k ) and d(k ) are independent random variables, and applying the mathematical expectation operator to the square of the differences in the expression above, it follows, as a consequence, that:





2 2 E x1 (k  1)  xˆ 1 (k  1)   E x1 (k )  xˆ 1 (k )   d(k )  dˆ (k )

1

E(a k  aˆ k ) 2   Va (k)  E{a 2k }  aˆ 2k then from (7) that

follows

Vx (k  1)  Vx (k)  Vd (k)  Vx (k)  d2

(8)

Considering

k

1

Pr ob.(x 1 (k )  0)  Pr ob.(xˆ 1 (k )  k   d   k  0)   xˆ 1 (k )    Pr ob.  k    k  d  

2

2

2

1

1

1

 xˆ (k ) Pr ob.  k   1  k  d 

1

be exactly defined by:

 xˆ (k )   1  k  d 

(9)

1

(13)

where k denotes a white noise (i.e. a variable with mean 0 and standard deviation 1). Follows from (13) that:

Assuming that Vx (0)  0 , the evolution of equation (8) can

Vx (k)  k  d2

x 1 (k)  xˆ 1 (k)  Vx1 (k)   k 

xˆ 1 (k)  k   d   k ; and x 1 = 0, follows that

  (7) ˆ  E  x (k )  xˆ (k )   E  d(k )  d(k )  





and, considering that

(12)

   

 ˆ   1     x 1 (k )   1 (1  )  k  d 

xˆ 1 (k) 

As a consequence, from (6) and (9), it is possible to write that:

  ˆ     1    x 1 ( k )   k  d   

k  d   1 ()  x1 (, d )

(14)

4. SOLVING THE PROBLEM (11)

E{x (k  1)}  Vx (k)  xˆ (k)  k.  xˆ (k) 2 1

1

2 1

2 d

2 1

(10)

For reasons of simplicity, the problem (11) can be represented in a matrix format as follows:

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5. NUMERICAL EXAMPLE

T 1

Min ZT  K V  x TT Qx T  k 0 (x Tk Qx k  u Tk Ru k )

A company of a particular supply chain, which follows the scheme showed in the figure 1, makes products to inventory, and, simultaneously, pushes them to the market. It is not only responsible to manufacture the products but also to distribute them to the market and also to return them to the company after they have being used by customers. One of these products has a special feature; it deteriorates after staying long periods in the serviceable inventory of the company. If such deterioration is detected during quality inspection, the product is separated for remanufacturing; otherwise it is discarded almost immediately.

s.t. x k 1  Ax k  Bu k  w k xk  x

(15)

k min

0  u k  u kmax where xk, uk and wk denote vectors of state, control and demand of the system (2) and (5). These vectors and matrices are described below:

x Tk  xˆ 1 (k ) x 2 (k ) x 3 (k )  u Tk  u 1 (k ) u 2 (k ) u 3 (k )



w Tk  dˆ (k ) r (k  )

Looking now at the reverse channel of this supply chain, it is possible to say that all returned products are stored in the returnable inventory unit of this company. During the period in which these products stay in the returnable unit, if some products appear in certain state of deterioration, they are immediately discarded. The rest of products will be submitted to a quality control before being sent to remanufacturing.



h 1 0 0  c 1 0 0  1 0 0  Q  0 h 2 0  R  0 c 2 0  A  0 1 0  0 0 h 3  0 0 c 3  1 1 0      

Statistics show that over a monthly period such deterioration occurs in a rate base of 10%. Similarly, used-products return after having concluded its life cycle in a rate of 90%. These products are then checked in order to decide if they can be remanufactured; usually, 40% of them are discarded. Next, follows other information: (a) The cost of remanufacturing used-products is assumed lower than the cost of manufacturing new products. This means that used-products can be easily replaced to marketplace; and (b) the demand for this product is stationary, which means that the fluctuation of sales can be estimated with a good accuracy over periods of the planning horizon. The demand is described by the equation: d(t) = 600+(0,25t)*(1+sin(*t/2))+50*rand.

 x 1 (,  d )  u 1  1 1 0 k   u kmax   u 2  B x  0 min     0 1 1 0     In order to provide an optimal control policy for manufacturing, remanufacturing and discarding (i.e, T u k  u 1* (k) u *2 (k) u *3 (k) ), an Open-Loop No-updating (OLN) approach is applied to problem (10) (Bertesekas, 2000). The figure 2 illustrates how this approach works.

In the way of developing an annual production-inventory policy, based on manufacturing new products and remanufacturing used-products, a LQG problem with restrictions, formulated in (11), is considered. The data of this problem are listed in Table 1.

Table 1. Problem’s data Fig.2. OLN policy for reverse logistics

T=12 months (planning horizon) N=6 (number of decision variables) α=0.9 (customer satisfaction level) =0.1 (rate of defective products) γ=0.4 (rate of discarded products) =0.3 (rate of deteriorated used-products) =50 (standard deviation of demand)  = 2 months (time delay for returning) x01=300; x02=200; x03 =0 (initial states) x1  0 ; x 2  0 ; x 3  0 (inventory lower bounds)

The optimal control policy provided by OLN approach depends on the initial state of the system (i.e., the state of the system in the period k=0), which is denoted by x0. As a result, if further information becomes available for k>0, they are completely neglected, which means that the system will be run in an open-loop pattern, see Pekelman & Rausser (1978). In fact, the OLN is a suboptimal deterministic approach that provides an optimal open-loop control sequence {u *1 u *2 u *T-1 } , which takes into account only the initial state x0 that is usually a random variable with mean and variance known. Thus, the OLN is an approach that does not take into account any kind of feedback scheme when it provides an optimal policy for (1-3).

u 1  750 ; u 2  500

(upper production bounds)

h1=2; h2=1; h3=5; c1=3; c2=2; c3=2 221

(all costs)

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5.1. Scenario without a reverse policy

800 700 600 500 400 300 200 100 0

Initially, let’s assume that the company does not use a reverse system as the one exhibited by Figure 1. The operation of the company in terms of inventory and production is shown in Figure 3, given bellow. It is worth observing that company is running in its maximum load, which means that u1(k)= u 1  750 . With regard to the inventory levels, every product manufactured by company is practically used to meet the demand at once. However, a small part of products are kept in stock to ready-delivery, because the company uses some statistics about the customer satisfaction; and it knows that it should guarantee in 95% stock for ready-delivery. The inequality (14) shows how to keep a save stock for readydelivery with 95% of chances of customer satisfaction.

Inventory level Production rate

0

1

2

3

4

5

6

7

8

9 10 11 12 months

Fig. 3. Inventory-production levels without reverse policy

Table 2 provides all costs of company to run its production operation without a reverse policy. Such costs includes costs to hold serviceable inventory, discard deteriorated products, and remanufacture defective products that are immediately identified after being manufactured.

Inventory Levels

800 700 600 500 400 300 200 100 0

5.2. Evaluating two scenarios with reverse logistics policy Two scenarios are considered here: the first one considers a return rate of used-products around 90% (i.e., μ=0.9) to the store 2 (see Figure 1), while the second scenario considers this rate close to 50% (=0,4). In sequel, these scenarios are briefly compared with respect to the cost incurred by the same company, but without using a reverse logistics policy.

Store 1

0

1

2

3

4

5

Store 2

6

7

8

9 10 11 12 months

Fig. 4. Inventory level of manufactured

Scenario (1): return rate of used-product  = 40% 800 700 600 500 400 300 200 100 0

Figures 4 and 5 illustrate optimal inventory and production trajectories for forward and reverse channels. In this scenario, 40% of used-products are brought from the marketplace. These products are checked and, in following, part of them can be remanufactured or properly disposed. An interesting aspect to be highlight through this scenario is the significant reduction of the manufacturing of new products. Practically, new products production rate was reduced to 40% compared to previous situation (i.e., without reverse policy; figure 3). Note, from figure 5, that remanufacturing level of usedproducts was close to maximum capacity. Surely, this new feature reduces costs, particularly due to purchase of components to produce new products. Note that this feature explains the reason of the cost for remanufacturing usedproducts to be less than the cost for manufacturing new products.

Production and disposal rates

0

1

2

M anufacture

Remanufature

Dispose defective

Dispose deteriorated

3

4

5

6

7

8

9

10 months 11

Fig. 5. Optimal production and disposal rates

Inventory Levels

800 700 600 500 400 300 200 100 0

Scenario (2): return rate of used-product  = 90% This scenario is exhibited in figures 6 and 7. It is equivalent to the previous one, the difference is that there is a significant number of returnable. As a consequence, a significant increase number of discarded products, including products that could be remanufactured, but for capacity constraint they cannot. See, in figure 7, the level of used product discarded over periods of planning. It is possible to conclude from this scenario that any investment to increase production capacity for remanufacturing will be improve productivity and reduce business costs.

Store 1

0

1

2

3

4

5

6

Store 2

7

8

9 10 11 12 months

Fig. 6. Optimal inventory levels of manufactured and remanufactured 222

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products, it incurs in more costs. Strategically, the company can try to adopt a cheaper scheme of disposal, and so making more attractive a reverse policy with a high rate of return.

Production and disposal rates

800 700 600 500 400 300 200 100 0

Manufacture

Remanufature

Dispose defective

Dispose deteriorate

6. CONCLUSION

0

1

2

3

4

5

6

7

8

9

In this paper, a reverse logistics channel is added to a forward channel in order to reduce inventory and production costs. A stochastic dynamic problem is introduced to this proposal. The problem is, therefore, defined by two productioninventory systems; one is a forward stochastic system, and the other a reverse system. During the operation of these systems, manufacturing and remanufacturing products are available in a serviceable inventory to meet the demand. After a period, products already used return from the market to the company using a backward channel, which includes a returnable inventory and a remanufacturing process. A stochastic linear quadratic Gaussian (LQG) model with constraints has been formulated to provide optimal annual plans for manufacturing, remanufacturing and disposal variables. Through scenarios analyses, it is possible to compare these plans for different return rates, helping the manager to make important decisions about an appropriate policy of returning for the company. From a simple example of a make-to-stock company, it is possible to reach to an important conclusion regards to the use of returned products to reduce costs for company.

10 11 months

Fig. 7. Optimal production and disposal rates 5.3. Comparing scenario’s costs Table 2 provides the costs related to scenarios. Next a brief discussion about these costs are given:

Table 2. Costs of each scenario ($)

Acknowledgment: The paper has been supported by CNPq under Process Number: 310606/2010-1. REFERENCES

5.2. Final comments about scenarios

Bertesekas, D. P. (2000). Dynamic programming and stochastic control, Athena scientific, Vol. 1. USA. Fleischmann, M., J. M. Bloemhof-Ruwaard, R. Dekker, E. Van der Laan, J. A. E. E. Van Nunen and L. N. Van Wassenhove (1997). Quantitative models for reverse logistics: A review, European Journal of Operational Research, 103. Graves, S. C. (1999). A single-item inventory model for a non-stationary demand process, Manufacturing & Service Operations Management, Vol. 1, No 1. Pekelman, D. & Rausser, G. C. (1978), Adaptive Control: Survey of Methods and Applications, Applied Optimal Control, In: TIMS Studies in the Management Science, Vol. 9, 89-120, North-Holland. Silva Filho, O. S. (2001), Linear quadratic gaussian problem with constraints applied to aggregated production planning, Proceeding of the American Control Conference, Arlington, VA. Silva Filho, O. S. (2009), Optimal linear quadratic gaussian problem applied to reverse logistics system, Proceedings of the 13th IFAC Symposium on Information Control Problems in Manufacturing Moscow, Russia, June 3-5. Silva Filho, O. S. (2011), An open-loop solution for stochastic production remanufacturing planning problem, Proceeding of 8th ICINCO, Netherlands.

A comparative cost analysis based on data from Table 2, together with a visual analysis of the behavior of inventory, production and discard trajectories provided by Figures 3-7, allow extracting interesting elements for managerial decision making process regard to use or not use a reverse logistics policy. Two points are very crucial in such analysis: Point 1: the use of a reverse logistics scheme joint to a good return policy for used products can reduce the holding and production costs of the company. However, it is required that the cost for remanufacturing is at least slightly less than the cost for manufacturing new products. This can be verified by observing the production policies in three scenarios: in scenario 1, the manufacturing process of new products is intense and uses all its capacity that is expensive. Already, comparatively, scenarios 2 and 3 consider returnable products in the remanufacturing process, and thus the manufacturing process of new products is highly reduced, which means less expensive cost for the company. This characteristic can be seen in the Table 2. Point 2: Comparing only scenarios 2 and 3, it is possible to see that it is more advantageous to adopt a return rate of 50% for used-product than one close to 100%. The main reason is that the cost for used-product disposal is relatively high. Thus, since scenario 2 provides 90% of returns for used 223