A Stochastic Methodology for the Optimal Management of Infrequent Demand Spare Parts in the Automotive Industry C. Ronzoni ⋆ A. Ferrara A. Grassi Dipartimento di Scienze e Metodi dell’Ingegneria, Universit` a degli Studi di Modena e Reggio Emilia, Via Amendola 2, 42122 Reggio Emilia, Italy Abstract: The paper deals with the management of the spare parts in the automotive sector introducing a first definition of business model aimed to reduce holding costs of stocks, focusing on spare parts distributors point of view, and adopting a probabilistic mathematical approach. The management of spare parts is not trivial due to the infrequent profile that characterizes market demand, in contrast with the necessity of the distributors to keep in stock a large number of items (part numbers) in order to avoid lost sales (back ordering is not allowed). Moreover, given that different distributors can deal the same products among the national market, the total amount of stocks of the same products is multiplied along the supply chain. A mathematical approach adopting probabilistic dynamic programming is presented aimed to optimize the management of spare parts, considering the creation of a platform in which different distributors can participate to share the management of items. The main purpose of the model is to find the optimal allocation of products among distributors and, at the same time, ensuring a balancing in the costs that each player has to bear period by period, providing the optimal reorder policy for each product. A numerical example is also reported. Keywords: automotive, spare parts, probabilistic dynamic programming, pooling, supply chain. 1. INTRODUCTION The paper deals with the management of spare parts in the automotive business sector, in particular focusing on the role of spare parts distributors. Despite the current economic crisis that characterized last years of global stage, spare parts business stands in opposite trend compared to other businesses. For this reason, spare parts actors have to face with not trivial issues, such as the growth of demand, the reduction of supply lead time imposed by the market and the efficiency of logistics and warehouse activities as a consequence of the elimination of geographic barriers thanks to internet and e-commerce platforms. Generally, the supply chain of spare parts is structured as follow: at the first level there are producers and manufacturers, represented by two types of actors: on one hand, very large multinational companies who provided automotive manufactures directly for the most part of original equipment (OE) components, on the other hand smaller producers or importers providing high quality substitutive components (after market components). The latter can offers their products just only after the end of warranty period. On the second level of the supply chain, there are spare parts distributors and then final resellers (shops, car dealerships, etc.). Depending on the brand they deal with, the number of distributors can range by less than 5 units for the national market to more than one distributor per national region. The main task of distributor is to ensure ⋆ Corresponding author:
[email protected] (C. Ronzoni)
the fulfilment of urgent requests, being able to ensure the supply of also just one components the morning for the evening of the same day to final clients. This task can not be provided directly by producers for obvious costs and management reasons. That is why manufacturers grant an higher discount to distributors in order to allow them to keep stock of spare parts and to make revenues for urgent service. On the other hand, from the distributors point of view, this task is very challenging and it imposes keeping stock of a large number of different items (part numbers) with small quantity, mostly characterized by infrequent demand, for immediate disposition whenever needed (Braglia et al., 2004). Nowadays, distributors deal with different OE and After Market brands suffering management and warehouse issues because, typically, in order to avoid lost sales (backorder is not allowed) they are forced to keep in stock a very large number of items with erratic demand, incurring in high inventory costs and lost sales. Moreover, from the supply chain point of view, stocks of same products are generally multiplied among the distributors, increasing total costs of the supply chain. Recently, there is a trend that is spreading nationwide, that is the creation of buying groups of limited number of distributors who shares their warehouses, thanks to informatic platforms, in order to aggregate demand of the same products they deal with, then reducing the total amount of stock. Given that, informatic platforms can only represent a pool of information shared among the
actors, since they do not allow to define an optimized method for the spare parts management. In the following sections, the paper will present a first development of a method intended for the optimized management of this particular spare parts, supporting the creation of network of distributors. The sequel of the paper is organized as follows. In Section 2 the addressed problem is clarified. The model of the system is then developed in Section 3. Finally, some concluding remarks are reported in Section 4. 2. PROBLEM STATEMENT As said in the intorduction, the aim of this paper is to define a model able to optimize the management of aftermarket spare parts in the automotive industry. Aftermarket spare parts are produced by companies that are independent from car manufacturer, and typically they are bought by customers when car warranty expires, since they are less expensive that the original ones. Hence they mainly refer to old cars, or cars that are aging. This latter fact explains the reason why the number of spare parts to be managed by the aftermarket channel is very large, and the most of them are characterized by small and infrequent demand. From this derives the typical configuration of the supply chain, formed by distributors that put large orders to the producers to obtain strong price discounts, but resulting in increased stock costs and obsolescence risks. This effect is even more relevant when spare parts characterized by small and infrequent demand are considered (in the sequel of the paper, they will be simply referred as spare parts). Hence, the new model proposed in this paper aims to redefine the supply chain for the management of spare parts in the automotive aftermarket distribution channel. The model considers the creation of a platform in which different distributors, covering different market areas of regional scale (in relation with European country size, but potentially extensible to other situation), can participate to share the management of the spare parts. In particular, to lower inventory costs and obsolescence risks, the specific item (i.e., the specific spare part) is kept in stock by only one distributor, while it is seen as available by all of the distributors adhering to the platform. Hence, each distributor owns the stock, and then bears the cost, of a subset of the whole spare parts, while every distributor can sell every item to their customers. The key aspect here is then to allocate the ownership of the items among the distributors in such a way as to the costs are balanced. The profit is then determined by the capability of the distributor to sell the item at the highest possible price, may exploiting contingent situations (e.g. the fact that customers frequently need expedited delivery). In this way, the distributor is focused on what it is really important for his business, that is, marketing and selling, while the optimization of the logistic system and its related costs is left to the platform. Morevoer, the platform could also act as a third party logistic provider for the distributors, directly managing material handling, stocking and shipment, since they can be simply the owner of the stocks.
The model here developed adopts a multi-period approach. It considers an inital group of distributors and a set of spare parts to be managed. The model then attempts to compute the probabilistic cost of ownership of every spare part, considering that customer demand vary over years, typically in a decreasing fashion. The result of this first step (referred as “first layer” in the sequel) is the optimal order policy and the inventory level over the years, and the related holding cost year by year. Then, a “second layer” takes into account the problem of reassigning the items among the distributors to guarantee both cost balancing year by year, and the reduction of the number of changes of items ownership. Moreover, each ownership change has an associated cost. The proposed model can be considered as a part of the literature related to the “pooling” strategy for inventory management. Specifically, studies related to the application of pooling strategy to spare parts management are rather recent (Wong et al., 2007; Kranenburg and van Houtum, 2009; Karsten et al., 2012; Guajardo and R¨ onnqvist, 2014; Mo et al., 2014), given also the need to adopt complex modeling techniques to produce suitable models. Hence, new contributions are particularly wellcome in this field. In our approach, the first layer of the model is deveoped by using the probabilistic dynamic programming (Hillier and Liberman, 2009) which is useful mathematical technique for making a sequence of interrelated decisions. Given that we need to work with a multi-period perspective and we have also to deal with probabilistic demand, such a technique appears to be the best fitting one. In spite of the wide adoption of dynamic programming to deal with deterministic inventory management problems, branch that was started by Wagner and Whitin (2004), applications to probabilistic dynamic programming to inventory modeling are very hard to find and rather recent (see e.g. Li et al., 2009; Arshad Naeem et al., 2013). The reason behind this is due to the behavior of the technique that needs to enumerate all of the possible branches (from a probabilistic point of view) before identifying those that do not bring to a feasible solution. So, at each stage, the number of possible states and transitions has to be limited to allow the problem to be handled. In fact, the papers from Li et al. (2009) and Arshad Naeem et al. (2013) deal with remanufacturing where, of course, remanufacturing events are rare if the manufacturing process is well controlled. Likely, in our case we are in a similar situation since the spare parts we refer to are subject to infrequent demand that brings to small demand in each stage, and so are also inventory levels. For what concerns situations in which the possible states at each stage are in numbers, Gavirneni and Tayur (2001) give an alternative procedure to the dynamic programming in the non-stationary inventory control. The spare parts demand can be suitably modeled with a Bernoulli distribution (see e.g. Eaves and Kingsman, 2004), since it can be related to the number of vehicles that are active and the probability that one of them needs for a specific spare part. Nowadays, national agencies can provide statistics about the number of active vehicles distinguishing the specific models, and trends can also be forecasted considering historical data. Moreover, parts producers provide the distributors with complete digital
catalogs able to put in relation every spare parts with the vehicles that can be equipped with. Another characteristics of such a type of demand is that it changes the parameters of its probability distribution over time, and that is the reason why we can consider it also as non-stationary. (For works related to non-stationarity see Graves and Willems, 2008; Bollapragada and Rao, 2006). Once the optimal policy is identified for every spare part, the “second layer” takes care of reallocating the ownership stage by stage (i.e. year by year in our case) among the distributors who participate to the platform. This approach has not been widely investigated in literature, where service level and cost optimization is typically addressed by other approaches, among them the one that mainly attracted the interest of scientists in the last decade has been “lateral transshipments” (see Paterson et al., 2011, Section 4.2.1). Kranenburg and van Houtum (2009) is an example of approach using pooling and lateral transshipments for optimizing inventory management of spare parts in a case in which service level is critical. In our case, characterized by the fact that each spare part has to be managed by only one distributor at a time, the key aspect is to reallocate the ownership of the different spare parts among the distributors in such a way as to costs are balanced. So, in the second layer, a deterministic dynamic programming model is proposed to reach this goal. The model is conceived in a way as to keep the distributors cost as much as possible closest to the average cost in the network, while reducing the number of requested changes of ownership. 3. MODEL Objective of this model is to allocate a set of products to a player of the network minimizing the total costs during all periods n = 1, . . . , N and, at the same time, ensuring a balancing in the costs that each player a = 1, . . . , A has to face period by period. In order to reach this purpose it is fundamental to detect the optimal reorder policy of each product j = 1, . . . , J for each period n. The proposed model is composed of two different layers: the first one wants to identify the optimal reorder policy for the products, the second one aims to identify the optimal set of products that has to be allocated to each player. 3.1 Notation as the amount z of product j demanded in period n; p(dz,j,n ) as the probability of meeting a demand equal to z of product j in period n; Lj as the reorder quantity of the product j; Ij as the maximum value of inventory for the product j; ij,n as the inventory quantity of product j in period n; as the handling cost per unit incurred by a player ch j in order to satisfy the demand dz,j,n ; as the reordering cost per unit for product j; cr j as the opportunity cost per unit for product j; co j as the stockout cost per unit for product j; csoutj xz,j,n as the first layer decision variable. It is a binary variable;
dz,j,n
c∗hj,n c∗rj,n c∗oj,n c∗trj,n c∗trn,a Ctotj,n Cnet,n A Cnet,n,a Cn,a α yj,a,n
as the handling total cost in the period n coming from the optimal inventory policy of the product j; as the reorder total cost in the period n coming from the optimal inventory policy of the product j; as the opportunity total cost in the period n coming from the optimal inventory policy of the product j. as the transfer cost of a surplus units of product j at the end of the period n, as the total transfer cost of the surplus units at the end of period n related to the player a; as the total cost of a product j in period n; as the total cost of all products managed in the network in period n; as the number of players in the network; as the part of total cost of the network that each player a should face in a uniform distributed costs allocation in period n; as the cost faced by the player a in period n; as a parameter representing the weight with which the network can manage the objective function of the second layer; as the second layer decision variable. It is a binary variable.
3.2 The first layer In this first layer, the objective is to identify the optimal inventory policy for each product j during N . Outcome of this layer is the total expected cost for the product in the period n, that will be used in the second layer of the model. Considering a specific product j (from here on the subscript j will be omitted) we assume that: (1) the demand probability function p(dz,j,n ) is based on past sales and adapted to the car fleet of the period n by using a Bernoulli distribution; (2) I is the maximum value of a range R = {0, . . . , I} representing a good inventory policy. Values of I out of the range R mean bad inventory policy (i.e. an error in the order policy), and thus considered not feasible; (3) the reorder quantity Lj is fixed and it can’t exceed the maximum quantity I; (4) the amount of the costs chj , crj , coj per unit is equal for all players and known; (5) no backlog is allowed; (6) the lead time θ for the product j is the same for each player in the network; (7) the reorder policy does not depend on the player managing the product j in the period n; (8) the reorder point is at the beginning of each period. By adopting a dynamic programming approach, we denote a state as (1). sn (in , dz,n ) (1) The purpose of the reorder policy is to end the N period with an inventory level ≥ 0 and ≤ I. With this aim, we consider as winning states, the ones characterized by
Fig. 1. States and transitions for each product j. 0 ≤ iN +1 ≥ I. This states have an equally distributed probability of winning in the period N + 1, and we computed it as: P (sN ) =
1 I +1
mL ≥ i0
Therefore, the objective function of the first layer of the model results in (8).
(2) obj = min
By referring to Figure 1, the probability transition function from a generic state n to n − 1 is expressed as (3),
P (sn ) =
K X
· p(dk,n−1 )
(3)
where k = (1, . . . , K) is the set of states converging in the state sn in the period n − 1 and dk the demand associated to the state skn−1 . The cost incurred in the period n depends on the values of the state variable and the decision variable in period n. The cost function in period n can be expressed as: Z X z=1
where xz,n = and
(
∗ fn+1
N X
fn (sn ) + C0 (i0 )
(8)
n=1
3.3 Numerical example P (skn−1 )
k=1
fn (sn ) =
(7)
P (dz,n ) (co (in + xz,n L − dz,n )+
(4)
+ cr xz,n L + ch dz,n )+ ∗ max(0, csout (dz,n − in − xz,n L)) + fn+1
xz,n = 1 if it is needed a reorder in period n because of demand z (5) xz,n = 0 otherwise
represents the optimal solution in the state n+1.
The suboptimal policy may start in the period n = 1 with an inventory level not equal to 0, thus (6) is needed. C0 (i0 ) = mL(co + cr )
(6)
where m is the minimum value that guarantees an inventory level at the beginning of period 1 satisfying (7).
Purpose of this example is to make the first layer approach clearer to the reader. We consider, under the same assumptions in section 3.2 : N =3 p(dz,n )
periods; a demand probability function as represented in table 1, for the first (p(dz,1 )), second (p(dz,2 )) and third (p(dz,3 )) year; I=6 as the maximum value of inventory for each period. Being z = 6 the maximum value of demand allowed in the system, a stock level bigger than 6 would lead at a bad inventory policy as what concerns erratic products; L=2 as the reorder quantity; ch = 2 e as the handling cost per unit; cr = 1 e as the reordering cost per unit; co = 3 e as the opportunity cost per unit; csout = 50 e as the stockout cost per unit; ∗ fN . +1 = 0 Table 1. Demand probability function z 0 1 2 3 4 5 6
p(dz,1 ) 4% 10% 17% 23% 25% 15% 6%
p(dz,2 ) 10% 15% 20% 25% 20% 8% 2%
p(dz,3 ) 30% 39% 15% 8% 8% 0% 0%
Applying the approach presented for the first level of our model, by means of (2) we are able to indentify the
probability of the winning states (i.e. allowed states) in period N + 1 as in table 2.
The focus in this part of the model is on the products in their whole.
The total costs for each period are calculated starting from the period N and going back to n = 1 (i.e. adopting a repetitive procedure) by means of (4). The costs for n = 3 are expressed in table 3.
The assumptions for the second layer of the model are:
For each value of inventory, we calculate the costs that would be met with a demand probability function p(dz,1 ) expressed in table 1. According to (8) the optimal cost for each state results in fn∗ (sn ). The reorder policy is given by x∗n . In table 4 and table 5 we stored the total costs for n = 2 and n = 1 periods, respectively. The costs for starting period n = 1, c0 are illustrated in table 6 and computed by means of (6). The minimum expected total cost for managing product j is 77.22 e, and in this example, the optimal reorder policy results in starting at the first period with an inventory of 4 items and make an order (L). A the beginning of the second period, in order to have the minimum expected total cost in N , an order (L) should be put only if the inventory level is ≤ 3. For what concerns the beginning of the third period, the order (L) should be put only if the inventory level is ≤ 2. This optimal policy may be different with a variation in the stockout cost. In fact, the more we increase this cost, the more the reorder policy will be conservative.
(1) total costs c∗hj,n , c∗rj,n , c∗oj,n for the product j in the period n are not related to the player managing the product. (2) the transfer cost c∗trj,n is met by a player when he still has inventory of product j at the end of period n. This is a cost related to the transer of property (in case of the product is stored in a centralized warehouse by the network) or a physical transfer of the product (in case of each player stores the product he is managing in the period in its own warehouse); (3) the geographical distance between the players of the network is not considered; (4) the only relevant cost in this part of the model is c∗trj,n ; (5) the number of player A in the network does not vary all along n = 1, . . . , N . In a generic period n, the total cost of the system results in (9),
Cnet,n =
Ctotj,n = c∗hj,n + c∗rj,n + c∗oj,n
Objective of the second layer, is to allocate the products to the players of the network so that each player has to face costs in each period aligned with the average cost of the system, minimizing total expected costs of the network. Table 2. Probability of allowed states in N + 1 iN+1 0 1 2 3 4 5 6
p(iN+1 ) ∼ 0.14 ∼ 0.14 ∼ 0.14 ∼ 0.14 ∼ 0.14 ∼ 0.14 ∼ 0.14
x3 = 0 65 30.9 17.47 11.99 10.75 13.75 16.75
x3 = 1 33.97 20.99 15.75 14.75 17.75 20.75 23.75
f3∗ (s3 ) 33.97 20.99 15.75 11.99 10.75 13.75 16.75
Table 4. Costs [e] for the second period x2 = 0 152.52 107.82 71.07 44.92 32.02 29.72 31.66
x2 = 1 109.57 73.42 48.02 35.72 33.66 35.66 38.66
f2∗ (s2 ) 109.57 73.42 48.02 35.72 32.02 29.72 31.66
Considering A the number of players in the network, the total cost that each one has to face in the period n is computed as in (11). Cnet,n A
(11)
The transition function results in:
x∗3 1 1 1 0 0 0 0
x∗2 1 1 1 1 0 0 0
(10)
where (10) gives the total cost of the product j in the period n.
Cnet,n,a =
Table 3. Costs [e] for the third period
i2 0 1 2 3 4 5 6
(9)
Ctotj,n
j=1
3.4 The second layer
i3 0 1 2 3 4 5 6
J X
Table 5. Costs [e] for the first period i1 0 1 2 3 4 5 6
x1 = 0 210.94 163.06 120.48 86.91 65.53 57.4 57.22
x1 = 1 164.48 122.41 89.53 68.9 61.22 61.22 64.22
f1∗ (s1 ) 164.48 122.41 89.53 68.9 61.22 57.4 57.22
x∗1 1 1 1 1 0 0 0
Table 6. Costs [e] considering C0 i0 0 1 2 3 4 5 6
C0 164.48 130.41 97.53 84.9 77.22 81.4 81.22
fn (sn ) =
A X
∗ [α | Cnet,n,a − Cn,a | +(1 − α)c∗trn,a ] + fn+1
a=1
(12)
∗ where fn+1 represents the optimal solution in the state n + 1, J X (13) yj,a,n Ctotj,n Cn,a = j=1
is the cost for each player related to the products he manages, and α is a parameter with which the network can manage the model, that is to give more importance to the costs balancing rather than the transfer costs or viceversa. Moreover,
c∗trn,a =
J X
yj,a,n c∗trj,n
(14)
j=1
represents the total transfer cost of the surplus units of each product j at the end of period n related to the player a with the yj,a,n variable:
yj,n,a =
(
yj,a,n = 1 if product j is managed by player a in period n yj,a,n = 0 otherwise
(15)
Therefore, the ojective function for the second layer of the model results in (16) obj = min fn (sn )
(16)
4. CONCLUSIONS A mathematical model has been proposed to address the optimal management of infrequent demand spare parts in the automotive industry. The main goal is to support the creation of a new business model in which spare parts are managed by only one distributor in the network but are made available also to the other players. In this way, the same service level can be guaranteed with lower inventory levels in the network. The main purpose of the model is then to find the optimal allocation of products among distributors, ensuring a balancing in the costs that each player has to face period by period, providing the optimal reorder policy for each product. Probabilistic behavior of demand is also considered, in a multi-period perspective. A numerical example is also provided. The approach proposed in this study represents, to the authors knowledge, a novelty in the field, and can be considered as a starting point for further extensions. REFERENCES Arshad Naeem, M., Dias, D.J., Tiberwal, R., Chang, P.C., and Tiwari, M.K. (2013). Production planning optimization for manufacturing and remanufacturing system in stochastic environment. Journal of Intelligent Manufacturing, 24(4), 717–728.
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