A NEW REPLENISHMENT POLICY BASED ON MATHEMATICAL MODELING OF INVENTORY AND TRANSPORTATION COSTS WITH PROBABILISTIC DEMAND

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1 8 th International Conference of Modeling and Simulation - MOSIM 10 - May 10-12, Hammamet - Tunisia Evaluation and optimization of innovative production systems of goods and services A NEW REPLENISHMENT POLICY BASED ON MATHEMATICAL MODELING OF INVENTORY AND TRANSPORTATION COSTS WITH PROBABILISTIC DEMAND Khaled BAHLOUL, Armand BABOLI, Jean-Pierre CAMPAGNE Université de Lyon, INSA-Lyon, Laboratoire LIESP 19 av. Jean Capelle, F-69621, France Khaled.bahloul@insa-lyon.fr, arman.baboli@insa-lyon.fr, jean-pierre.campagne@insa-lyon.fr ABSTRACT: The implementation of supply chain has to reduce the total cost of system, but generally each component of a supply chain tries to find the best policy for its company and consequently tries to find a local optimum. Knowing that the sum of local optimum cannot constitute the global optimum, it is necessary to consider all costs of system simultaneously to find the best replenishment policy for all the components of a supply chain. This paper presents a new approach based on mathematical modeling of total costs of system (transportation and inventory costs) in a supply network (multi-echelon multi-product structure). Moreover, the demand in real case is often probabilistic and it has to be taken into account. This reality justifies the necessity to consider all costs, generated by all products, in all links and all echelons. The first part of this paper presents an overall view of the approach adopted. Then, the mathematical model of the logistic costs is developed. In the next part, a new replenishment policy based on joint optimization is detailed. Finally, the numerical experimentation and an example illustrate proposed approach. KEYWORDS: Multi product Supply chain, Joint optimization, Inventory control, Transportation organization, Probabilistic demand, replenishment policy 1 INTRODUCTION The replenishment problem has been traditionally treated from a multi-echelon and multi-product perspective (Jen- Ming and Tsung-Hui 2005). A multi-echelon replenishment problem focuses on channel coordination issues for inventory replenishment, between upstream and downstream components of a supply chain, with the objective of minimizing total system costs (Sıla et al. 2005). Moreover, multi-product replenishment problems aim to coordinate the replenishment of various items in the same family or same category in order to reduce the frequency of major setups and the related costs. This can be obtained by choosing an appropriate common replenishment frequency and lot-sizes within the family of items (Bahloul et al. 2008). Several previous works have studied the problem of multi-echelon, multiproduct Supply Chain. Chen et al. (Cheng-Liang et al. 2004) have studied a multi-item inventory and transport problem with joint setup costs, referred to a joint replenishment problem. Traditionally, synchronization of different echelons is carried out in a sequential way, in the sense that outputs of the upstream echelon are regarded as inputs of the downstream echelon. This way cannot obtain an optimal plan for a company with more than one echelon in a supply chain (Zhendong et al. 2007). This problem leads the researchers to propose the integrated SCM, in which the aim is to optimize the supply chain as a whole and consider the planning of different echelons simultaneously. This can allow providing an important source of cost savings for companies operation management, particularly for inventory and transportation, which are the two most common operations of many companies. Huang et al. (Huang et al. 2005) studied the case of a fixed transportation cost and a variable cost which is linearly proportional to the volume of cartons delivered. Since any replenishment policy implemented by the third logistics service provider will eventually be translated into cost to its client and the client s customers, it is important for everyone in the system, that the service provider should find a good replenishment policy to minimize the overall logistics costs. There are some works which documented joint optimization of transportation and inventory cost. In this way, we present here three of most important works. The first one, proposed by Speranza and Ukovich (Speranza and Ukovich 1996) considers the product shipping strategy to determine shipping frequencies in which each product has to be shipped in a way that the sum of transportation and inventory costs are minimized. The second one (Bertazzi and Speranza 1999) considers

2 a periodic shipping strategy to minimize the total cost of transportation and inventory in a network with one origin, some intermediates and one destination with given frequencies. Finally the third one (Fleischmann 1999) considers the transportation of several products on a single link when shipment is conducted only at discrete times. It aims to determine the timing and the quantities of the shipments and the inventory level on a link in a specified planning horizon to minimize total cost of transportation and inventory. Several authors use mathematical models to study how postponement reduces the total inventory required for meeting a consummator service level. A nearly paper by (Tony and Marc 2007) demonstrates that component commonality may result in lower prediction errors and therefore lower levels of safety stocks, and he proposes algorithms for grouping products in clusters that are served by a common component. (Lee and Yao 2003) Have proposed mathematical models and solution algorithms for solving a multiproduct JRP (joint replenishment problem). Nevertheless, these works focus on specific cases and fail to present a global solution especially in the case of multi-echelon, multi-product Supply Chain with several links. Moreover, most works consider the demand of final clients as a deterministic or constant demand. This paper considers the problem of a multi-echelon multi-product Supply Network as a joint optimization of transportation and inventory costs with probabilistic demand. We present a downstream supply chain structure, see figure 1, consisting of a distribution center and several consumption centers. Several types of products are managed in the supply chain and all products are replenished from the same distribution center. Physical flows Information flows Figure 1: downstream Supply Chain The main contributions of this paper are: Proposal of a method to integrate the probabilistic demand. Mathematical modeling the logistic costs functions. Proposal for a new replenishment policy numerical experimentation followed by a discussion based on simulation This paper is organized as follows: in section 2.1 we present an overall view of our contribution based on a mathematical modelization and propose a replenishment policy. Next, in section 2.2 we give a method to present a probabilistic demand. Then, in section 2.3 we model the calculation function of logistic costs. Afterwards, we present an optimization part in the form of. After than, in section 3, the proposed replenishment policy is detailed. Finally, the numerical experimentation and the discussion of results conclude this paper. 2 MATHEMATICAL MODEL OF COST CALCULATION 2.1 A general view of our approach The increase of the transport and inventory costs regarding the other logistic costs and the decrease of the periods of inventory and delivery incite companies to give more importance to the costs and constraints linked to all these activities simultaneously, and manage better the functioning of their supply network. The concept of supply network appeared with these needs. The optimization of the management of physical flows of activities has started to be carried out in a simultaneous (integrated) way, in order to minimize the total cost. With this approach, the constraints and the costs of coordinated activities are integrated in the same model so as to be optimized in one single time. The optimization linked to this approach is called integrated optimization. This approach has currently become quite common with the increasing number of relocations of companies. The integrated approach can provide a schedule of activities at a lower cost than that used in a sequential approach, where the links of the chain are independently optimized. The gains obtained in an integrated optimization with regard to a sequential optimization are especially illustrated in the case of problems of the same importance as storage and transport. On the other hand, this approach leads to consider complex, large-sized systems, with strong interactions. At the level of the optimization process, it can lead to problems which are difficult to solve within a reasonable time. This rising complexity is due to the increase of the constraints to be taken into account during a mathematical modeling and also to the configuration of the objective function which becomes more complicated to optimize. This difficulty implies that there are more and more theoretical researches lead on this problem.

3 Knowing that joint optimization of transportation and inventory costs for a multi-echelon, multi-product case with a probabilistic demand is very complex, we adopt, in the one hand, a mathematical modeling for calculation of logistic costs, and in the second hand, we present a new replenishment policy. 2.2 Demand modeling In each period t, the independent random demand is defined by a probability density function (PDF) 0, IR and by a cumulative distribution function (CDF). : 0, 0, 1. At each period any received demand is charged at a price pt, even if it is satisfied only at the next period. Given a customer service level fixed by a company at for example 98 %. We first try to find the rate value in the standard normal distribution table which corresponds to a particular (normal distribution) and conclude the resulting value (eg. r=2.06). Based on this value and the characteristics of distribution (eg. m=50 and 4) we compute the safety stock quantity ss (1) and the order point follows s (2), see Figure ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0, ,500 0,503 0,1 0,5398 1,9 2 0,9803 2,1 Figure 2: Example for probabilistic demand service level x. 2. the transportation cost between two echelons (n, n- 1) incurs by the link in the echelon (n-1) 3. the transportation time between two echelons is constant 4. the lead time for an order to arrive at link is constant 5. there is not-splitting at link 6. The replenishment at link can be calculated based on the historical consumption 7. Transportation quantity = ordering stock quantity Notations: n: number of echelons k: number of products j: number of periods i: number of links v: number of vehicles A jin : ordering cost in period j, in the link i, at the echelon n h jkin : Rate of holding cost in period j of product k in the link i, at the echelon n s jkin : shortage cost in period j, of product k, in the link i, at the echelon n T k : periodic replenishment of article k t: safety factor r: resulting value of safety factor σ k : standard deviation of errors ss r σ units s m r σ m ss (1) (2) y Tk : stock level in the period T k F cvn : transportation fixed cost of the vehicle type v to level n q v : Capacity of vehicle v We base on this principle model for calculating the level of safety stock ss and the level of replenishment afterward in the section Modeling of cost function Assumptions: 1. We define the stochastic demand D k by a normal distribution, defined by two parameters, mean and standard deviation (m,, and the rate of customer V cvn : transportation variable cost of the vehicle v to level n Y vj =1 if the vehicle number v is used at period j, 0 otherwise u k : Volume of product k ss: safety stock α vkj =1 if item k is delivered by vehicle i at period j, 0 otherwise

4 MOSIM 10 - May 10-12, Hammamet - Tunisia z kj =1 if item k is ordered by retailer in period j, 0 otherwise D k : rate of demand of product k D k *: quantity to minimize L: replenishment lead time m(): average v(): variance We define two types of costs: Inventory costs: The inventory costs can be classified in three families ((Toomey 2000) and (Zermati and Mocellin 2005)): Ordering costs, holding costs and shortage costs. When optimizing the decisions relative to the inventory, one must take into account all these costs. The ordering costs SOC: The ordering costs include the salaries of the personnel, the functioning costs (buildings, offices, etc.), reception and test costs, information systems costs and customs costs. These costs represent about 2 to 5% of the value of the ordered articles (Zermati and Mocellin 2005) ; 1,2,, 2 This holding costs SHC is calculated from the probabilistic demand, define by m(d) and v (D,) and the rate of satisfaction clientele defined beforehand. The shortage costs SSC: The inventory shortage corresponds to the case where the units available at the time of the customer's demand, are not sufficient to satisfy this demand. The related costs are classified into two categories: lost sales costs and backlogging costs. In the lost sales category, if the available units are not sufficient to fully satisfy the demand, the unsatisfied demands are then completely lost and the cost in this case is the "miss to gain". In the second case, the cost will be a penalty shortage cost. The latter includes the cost difference between satisfying the demand at the time it occurs and the time when it is satisfied. In both cases, some costs can be incurred like an increase in the cost of raw materials by the use of substitute materials, as well as the cost of buying or renting a substitute product. The ordering costs SOC: is calculated from both constraints bound to the binary variables a j and z kj. The binary variable a j is equal 1 if at least item k is ordered one time (z kj =1). The holding costs SHC: This family of costs can be divided into two subfamilies; les financial and functional costs. The financial costs represent the financial interest of the money invested in providing the stocked products. The functional costs include the rent and maintenance of the required place, the salaries of the employees, the insurance costs, the equipments costs, the inter-depot transportation costs and the obsolescence costs. This family of costs represents about 12 to 25% of the value of the held products (Zermati and Mocellin 2005). This means that 12-25% of the value of stocked products is charged per year, depending on the volume and the price of products. The shortage costs SSC appears under the shape of the difference between the holding stock level and the ordered quantity. The holding stock is based on the stock possessed for the period T-1 by adding the quantity received during the period T, decreased by the quantity asked and delivered for the same period. Transportation costs When products are delivered from the supplier to the consumer, transportation costs are incurred. However, in a practical logistic system, the transportation cost of a vehicle includes both the fixed cost TfC and the variable cost TvC. The fixed cost, which is considered to be a constant sum in each period, refers to some necessary expenses such as parking fare and rewards to the driver. As to the variable cost, it depends mainly on the gasoil consumed, which is related directly to the distance travelled. In short, considering the real conditions, it is unreasonable to assume that the transportation cost can

5 be proportional to the quantity delivered or a constant sum. With the notations in Table 1, it is assumed that: F 11 <F 21 <F i1, v 11 >v 21 >v i1, q 1 <q 2 <q 3, F 2 =F 1 +q 1 (v 1 -v 2 ), The transportation fixed cost TfC is calculated by the binary variable of y vj that it takes 1 if there is one vehicle v used for the period j, 0 otherwise. F 3 =F 2 +q 2 (v 2 -v 3 ). These equations are supposed to avoid any over- cost varies declaration. Hence, the transportation according to the order quantity as shown in Figure 3. Figure 3: Variation of cost transportation 3 PROPOSED REPLENISHMENT POLICY: The variable cost TvC: is calculated by the constraints linking the variables z kj and a vkj. These two variables determine if a product k is going to be delivered for the period j by vehicle v and also determine the quantity delivered in each period j. We use the same concept of transportation cost as defined by Baboli et al. in (Baboli et al. 2007). They assume that there are three different types of vehicles (V.T) and the delivery for each order from warehouse to retailer is made by a single vehicle without splitting; these types are defined as small (S), medium (M) and large (L) and have their own fixed costs (FC), variable costs (VC) and capacities (C). The corresponding transportation scheme is shown in table 1. V.T C Destination FC S q 1 1 n F 11, F 12,, F 1n M q 2 1 n F 21, F 22,, F 2n L q 3 1 n F 31, F 32,, F 3n VC v 11,, v 1n v 21,, v 2n v 31,, v 3n In our context, the economic order quantity model does not represent the best solutions. Indeed, in the case of probabilistic demand, it proves necessary to consider specific situations, and thus, define specific hypotheses, in order to take them into account relatively to logistic cost afterwards. In this section, we propose a replenishment policy in the form of an optimization algorithm. This method is based on the hypotheses defined in the first part of the section so as to determine the use context of the policy. The principle characteristic of the method is presented in the second part. 3.1 Hypotheses: We have defined various hypotheses so as to define the condition of applicability: 1. The means of transportt exist in three different capacities: small, mediumm and large. 2. The means of transport (trucks) are chosen accordingly to the quantities of products to deliver 3. The partial replenishment of a product is forbidden Table 1: Transportation schema

6 4. Calculating the quantities of product to replenish has been done independently of the transportation capacity. 5. The replenishment costs are calculated independently to the number of references replenished and the number of products transported. - An ordering level (s i ) : this level is calculated ac- for each i product cording to the demand distribution - A replenishment level (S i ): this level is also calcu- distribution for each I lated according to the demand product. 6. Identification of family products based on the several qualitative and quantitative criteria of products, such as mean of consumption, transportation conditions, price, etc. 3.2 Proposed method In these conditions, the classical policies will be faced to a certain problems and are likely not to be adapted if we want to provide a low logistic cost for the following reasons: - The type of demand: the considerable variations of demand can cause serious shortages, which will necessarily generate much replenishment, increasing the ordering costs and the fixed transportation costs for each demand - Safety stocks: these policies opt for high quantities of safety stocks which result in two types of problems: on the one hand, a high holding cost and, on the other hand, a risk of expiry for the products with a limited lifecycle. Figure 4: operating mode of proposed policy Operating mode (figure 4): The method consists in waitordering level s i for an i ing for the overtaking of an product. This overtaking triggers replenishment for all the prodquantities ordered q * i will ucts of the same family. The be added to the quantities of the current stock to reach the replenishment level S i. In order to take into account the hypotheses previously presented and to reduce logistic costs, we have defined a new policy which is based on the following principle: In the first time, it is necessary to identify some family of product. This can be made basing on the several characteristics of products, such as mean of consumption, transportation conditions, price, etc. The method based on continues review until a product reaches to his ordering level and then replenishment all products of the family. The ordering quantity of each product depends to its level in stock (in hand quantity) to reach the replenishment level. In order to calculate the total logistic cost, we mainly focus on equations modeled in the previous section. Two types of logistic costs are considered: the inventory cost and the transportation cost. In the inventory cost, we include an ordering cost, holding cost and the shortage cost and in the transportation cost, we include a fixed cost and the variable cost. Principle: The proposed (s, S) policy is mainly characterized by: - A level of safety stock (ss) : this quantity of stock is used to partially (temporarily) meet with probabilistic demand Figure 5: algorithm of replenishment policy First, a demand can activate the operation mode of the policy. The satisfaction of this demand may be decrease the in hand quantity of a product to its reordering level. In this case, replenishment must be considered not only for the product reached to its ordering level, but also for all other product of its family. The ordering quantity of each product is calculated based on in hand quantity and replenishment level, see figure 5.

7 4 NUMERICAL EXPERIMENTATION AND DISCUSSION We have focused on comparing the replenishment policy proposed to a reference policy used in similar conditions as ours. Classic reference policy (Edward et al. 2003): The classic policy we refer to is based on two main variables and a running logic deduced from the cases provided by the literature which deals with problematic very close to ours. These two variables are: an ordering level for each product and a replenishment level. The running logic is defined as follows: launching the ordering for the product to make sure the stock level reaches the ordering level. The replenishment for this product triggers the ordering of optimal ordered quantity for this product. A family of 10 products is used for our experimentations. For each simulation, a new random demand is generated for 472 periods, using the normal distribution with an identical mean and standard deviation (m = σ). The following table represents the results of various costs of twenty simulations. Instance of model: Varia ble Designation Value n: number of echelons 2 k: number of products 10 j: number of periods 472 i: number of links 2 v: number of vehicles 3 h jkin s jkin F cvn V cvn u k rate of holding cost in period j of product k in the link i, at the echelon n shortage cost in period j, of product k, in the link i, at the echelon n transportation fixed cost of the vehicle v to level n transportation variable cost of the vehicle v to level Volume of product k Table 2: instance of model 20% 100 {20, 90, 120 } {3, 2, 1 } x Q 1cm 3 /un it This Table (table 3) presents the various logistic costs integrating the inventory costs (ordering, holding and shortage) and transportation costs (fixed and variable) for policies, the proposed one vs. the reference one. We notice that the proposed policy provides lower costs than the reference policy. This is due to the optimality of the proposed policy in this specific context and the modeling of logistic costs, which takes into account the various logistic aspects and the different situations. [1.. 20] : number of simulation m: mean of demand σ: standard deviation p. policy: proposed policy A jin ordering cost in period j, in the link i, at the echelon n 50 r. policy: reference policy

8 inventory transportation Total cost Demand Ordering cost Holding cost Shortage cost Fixed cost Variable cost m δ p. policy r. policy p. policy r. policy n. policy r. policy n. policy r. policy n. policy r. policy n. policy r. policy 1 4,80 4, ,4 6, ,6 6, ,6 4, ,7 6, ,3 6, ,2 5, ,6 4, ,2 4, ,80 4, ,9 4, ,7 6, ,3 4, ,6 4, ,6 5, ,7 6, ,8 3, ,6 5, ,3 5, ,1 5, mean σ Table 3: many cost variations Varation of total cost for two policy Total cost Total cost n. policy Total cost r. policy Number of simulation Figure 6: variation of total cost for both policies Qj Total Cost cost Table 4: Example of calculation of holding cost for p-policy Several values of the table 3 can be explained by the following example. Focusing on the first line, third column we have define the ordering cost for p. policy and it can be calculate as: 50

9 MOSIM 10 - May 10-12, Hammamet - Tunisia 1 1, ; In column 5, the holding cost for p. policy is calculated as: Table 4 presents an example of calculation of holding cost for 10 products by proposed policy. Column 7 presents the shortage cost for 472 periods. We observe for the first line 10 shortages. By p-policy this shortage cost is calculated as: Column 9 presents the transportation fixed cost and it can be calculated as: Column 11 presents the transportation variable cost and it can be calculated as: The same costs have been calculated for the reference policy (r-policy) in the columns 4, 6, 8, 10 and 12. Simulation aiming at showing results: We study the feasibility of the proposed methods in terms of logistic cost benefit, since the general objective is to minimize these costs while keeping the best efficiency for the proposed policy. In our probabilistic context, we resort to simulation for: - Calculating and comparing the logistic costs provided by proposed policy and another reference policy - Showing the benefit achieved by our proposed policy compared to the classical policy so far considered as the most efficient in our context - evaluation of the hypotheses showing that they are all feasible The figure (figure 6) shows the variation of total logistic costs by carrying out twenty simulations with different demands randomly generated for each simulation. This variation shows the difference (gain ~ 15%) between the two policies in their way of dealing with stock management combined with transport management in totally random demand. We have to observe that the logistic costs are calculated in the same way in both policies as they are based on the equation models displayed in the first part of the article. 5 CONCLUSION This paper presents in the first time, an approach to model the cost calculation functions (transportation and inventory costs) in a multi-echelon multi-product supply network with a probabilistic demand. First of all, we present an overall view of our contribution based on a mathematical modeling and propose a new replenishment policy. Secondly, we give a method to present a probabilistic demand. Then, we model the calculation function of logistic costs such as ordering, holding and shortage for the inventory costs as well as fixed and variable transportation costs. In the next section, we develop a new replenishment policy which is based on the following principle: In the first time, it is necessary to identify some family of product. This can be made basing on the several characteristics of products, such as mean of consumption, transportation conditions, price, etc. The method based on continues review until a product reaches to his ordering level and then replenishment all products of the family. The ordering quantity of each product depends to its level in stock (in hand quantity) to reach the replenishment level. Finally, we illustrate the propose approach by a numerical experimentation and we analyze the obtained results. Our future works consist in implementing scenario in the form of optimization algorithms so as to deal with specific cases in order to find a balance between the quantities carried to fill and use at its best the transport capacities; which generates inventory holding costs or minimize the stock quantities ; which generates extra transporting costs. Reference Baboli, A., J. Fondrevelle, M. Pirayesh Neghab and A. Mehrabi (2007). Centralized and decentralized replenishment policies considering inventory and transportation in a two-echelon pharmaceutical downstream supply chain. 33rd International Conference on Operational Research Applied to Health Services (ORAHS 2007). Saint-Etienne, France. Bahloul, K., A. Baboli and J.-P. Campagne (2008). Optimization methods for inventory and transportation problem in Supply Chain: literature review. International Conference on Information Systems, Logistics and Supply Chain (ILS). Madison, WI, U.S.A.

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