TWO-STAGE NEWSBOY MODEL WITH BACKORDERS AND INITIAL INVENTORY

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1 TWO-STAGE NEWSBOY MODEL WITH BACKORDERS AND INITIAL INVENTORY Ali Cheaitou, Christian van Delft, Yves Dallery and Zied Jemai Laboratoire Génie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, Chatenay-Malabry Cedex, France, HEC Paris, 1 Rue de la Libération, Jouy-en-Josas Cedex, France, vandelft@hec.fr Abstract: In this paper, we develop a general two-stage newsboy model. Each period decision induces specific costs. In addition to the usual decision variables for such models, we consider that, at the beginning of the decision process, an initial inventory is available and some preliminary fixed orders are to be delivered at each period. The unsatisfied demands during a period are backlogged to be satisfied in the future. The model is solved by a dynamic programming approach. We then provide insight regarding this type of two-stage inventory decision process with the help of numerical examples. Keyword: Supply Chain Planning; Dynamic Programming; Demand Management; Modelling Methodologies. 1 Introduction The single-stage newsboy model has received a lot of attention in operations research and operations management literature. Basically, the numerous versions of this model determine, under different assumptions, the orders and inventory quantities that optimally satisfy an uncertain future demand. Given the intrinsic simplicity of this class of inventory model, the optimal solutions can often be explicitly given under an analytic form. In many real-life applications, such simple single-stage models do not really apply because several correlated decisions have to be sequentially taken. It is thus quite natural to consider two-period newsboy models to analyze the structure of optimal decisions in such multiperiod decision processes. This class of models is characterized by several features: the structure of demand uncertainty, the structure of the decision process and related costs and the way unsatisfied demands at a given period are dealt with. In this paper, we consider style-goods type products with a short life cycle. For this kind of product, a two-period decision model appears quite naturally. The induced costs are purchasing costs, inventory holding costs and backorder costs. The demands at the first and second period are described by independent random variables, with known probability distributions. We assume that at the end of the season, the remaining inventory can be sold to a specific market with a given salvage value. In the literature about style-goods production and inventory problem, most of the models are a newsboy single-period problems (Khouja 1999). However, several two-stage extensions have been developed. All these papers exploited the two-period horizon in order to improve the inventory management process facing the demand uncertainty. Such two-period decision processes permit one to adapt the inventory levels to the demand variability. In other words, using a single period it is ordered only once, at the beginning of the season before information about the

2 effective demand is available. On the contrary, in a two-period model, after the first order, the realized demand of the first period can be observed and a second order is made, which clearly exploits this information. Several authors have considered such models. First, Hillier Lieberman (21) analyzed a two-period model with uniformly distributed independent demands. Via a dynamic programming approach, these authors analytically solved this model and proposed an explicit optimal order-up-to policy. Lau Lau (1997,1998) developed lost sales two-period models and proposed numerical solutions via dynamic programming. Bradford Sugrue (199) proposed another class of model in which the second period demand is correlated to the first period demand. A bayesian update for the second period s demand forecast can thus be used after having observed the value of the first period demand. These authors determined a conditional order-up-to policy for the second period and an optimal order quantity for the first period. Another important two-period model has been proposed by Fisher Raman (1996). In this paper the demand of the whole horizon and the demand of the first period are characterized via a joint probability density function. Furthermore, the order size for the second period is constrained by a limited amount. Gurnani Tang (1999) considered a two-period model with a first period demand equal to zero. In their model, the dynamic structure concerns available information for the sequential decisions: at the end of the first period, exogenous information is collected which permits one to update the forecast for the second period demand. Choi, et al. (23) proposed a quite similar two-stage newsboy model with an update of the forecast of the second-period demand via some market information. Donohue (2) applies a similar approach for developing supply contracts. The contributions of our model are the followings: First, the periodic ordering process is quite general in the sense that at each time period orders can be made for the different subsequent periods, possibly with different costs, Second, the periodic selling process is quite general, in the sense that, in addition to the classical selling process, it is possible, at the beginning of each period, to sell a part of the available inventory to a parallel market, at a given salvage value, Third, the data are dynamic : the selling prices, costs, salvage values and demand probability distributions are period-dependent, Fourth, the model includes initial inventory and initially fixed order quantities to be delivered in the different periods. The remaining part of this paper is structured as follows: the second section describes the model (namely the complete decision process, the information structure and the costs and profits structure), the third section details the objective function and the dynamic programming approach. Numerical examples are solved in section four. The last section is dedicated to the conclusion. 2 Decision process and information structure 2.1 Demand processes description Define D 1 and D 2 as the demand at the first and the second period respectively. These random variables are characterized by probability distributions F 1 ( ) and F 2 ( ) and probability density functions f 1 ( ) and f 2 ( ). 2.2 Decision process description First, the state variables of the model are the inventory level at the beginning of each period, X 1 and X 2 and the inventory level at the end of each period I 1 and I 2 (I being the given initial

3 inventory for the problem). The decision variables of the model are as follows. First, we define Q ts as the quantity ordered at the beginning of period t to be received at the beginning of period s (with t s and t, s = 1, 2, 3). Then we introduce, for each period, the variable S t, the quantity that is salvaged (to the parallel market) at the beginning of period t (with t = 1, 2, 3). We will show that the decision variables Q 33 and S 3 can be optimally chosen directly as explicit functions of the other variables. Figure 1 presents the structure of the decision process and demand realization, which is the following: the available inventory at the beginning of the first period, before current orders are decided and demand occurs, is X 1 = I + Q 1, where, in fact, I and Q 1 can be considered as data. Then decision variables Q 11, Q 12 and S 1 are fixed. Then, the demand D 1 occurs and the available inventory at the end of the first period is given by I 1 = X 1 + Q 11 S 1 D 1. The decision structure for the second period is similar, with the initial inventory given by and the final inventory given by the expression X 2 = I 1 + Q 2 + Q 12, I 2 = X 2 + Q 22 S 2 D 2. The terminal decision process is then as follows: after demand occurs in the second period, it is optimal to order Q 33 = I 2 units to satisfy backlogged demand (when I 2 < ) or to sell S 3 = I 2 units with a salvage value to eliminate the remaining inventory (when I 2 ). 2.3 Costs and profits structure and assumptions In each period, any demand is charged at a price P t, even if not immediately delivered. The unit order cost of Q ts is c ts. In the case of a positive inventory at the end of a period, an inventory holding cost h t is paid. Unsatisfied orders in period t are backlogged to the next period, with a penalty b t. We will show that under the assumptions of this paper, it is optimal that all backlogged orders in a given period be satisfied at the beginning of the next period. The unit salvage value at the beginning of period t is given by s t. It is necessary to introduce some -D (P ) I h -D (P ) I h -S (s ) I b -S (s ) I b -S (s ) I X X I I +Q +Q (c ) +Q +Q (c ) +Q (c ) +Q (c ) Figure 1: Decision process assumptions about the different periodical costs in order to guarantee the coherence and interest of our model. These assumptions could be classified in three categories:

4 2.3.1 Type 1 assumptions: c 11 <c 22 + b 1, c 11 <c 12 + b 1, c 12 <c 33 + b 2 and c 22 <c 33 + b 2. These constraints aim at avoiding situations with systematic backlogs of demands to the next period. For example, if the first constraint is not satisfied, the optimal policy will consist of backlogging the first period demand to the second period and to satisfy this demand with a second period order (with c 22 as unit order cost) Type 2 assumptions: s 2 <c 11 + h 1, s 3 <c 12 + h 2, s 3 <c 11 + h 1 + h 2 and s 3 <c 22 + h 2. These constraints aim at avoiding situations where it would be profitable to order at a given period in order to sell to the parallel market at a salvage price. For example, if the first constraint is not satisfied, the optimal policy will consist of ordering an infinite Q 11 quantity in the first period and selling it at a salvage price s 2 in the second period Type 3 assumptions: s 1 <c 11, s 2 <c 22, s 2 <c 12 and s 3 <c 33. These constraints aim at avoiding other situations where it would be profitable to order at a given period and to sell at the delivery period to the parallel market at the corresponding salvage price. For example, if the first constraint is not satisfied, the optimal policy will consist of ordering an infinite quantity in the first period and selling it at a salvage price s 1 in the same period. 2.4 Terminal conditions The optimal value of the decision variables Q 33 and S 3 can be shown to be explicit functions of the state variable I 2 as follows (Cheaitou, et al.,25): 3 The dynamic programming approach 3.1 The model if I 2 Q 33 = I 2 and S 3 =, (1) if I 2 Q 33 = and S 3 = I 2. (2) First, we recall the equilibrium equations of the system, X 1 = I + Q 1 (3) I 1 = X 1 + Q 11 D 1 S 1 (4) X 2 = I 1 + Q 2 + Q 12 = X 1 + Q 11 D 1 S 1 + Q 2 + Q 12 (5) I 2 = X 2 + Q 22 D 2 S 2 (6)

5 Introduce Π(I, Q 1, Q 2, Q 11, Q 12, Q 22, Q 33, S 1, S 2, S 3 ) as the expected profit with respect to the random variables D 1 and D 2. This expected profit Π( ) is formulated as follows, Π(I, Q 1, Q 2, Q 11, Q 12, Q 22, Q 33, S 1, S 2, S 3 ) = P 1 D 1 f 1 (D 1 ) dd 1 + s 1 S 1 c 11 Q 11 c 12 Q 12 X1 +Q 11 S 1 h 1 (X 1 + Q 11 S 1 D 1 )f 1 (D 1 ) dd 1 b 1 (D 1 X 1 Q 11 + S 1 )f 1 (D 1 ) dd 1 X 1 +Q 11 S 1 + P 2 D 2 f 2 (D 2 ) dd 2 + s 2 S 2 c 22 Q 22 X2 +Q 22 S 2 h 2 (X 2 + Q 22 S 2 D 2 )f 2 (D 2 ) dd 2 b 2 (D 2 X 2 Q 22 + S 2 )f 2 (D 2 ) dd 2 X 2 +Q 22 S 2 X2 +Q 22 S 2 + s 3 (X 2 + Q 22 S 2 D 2 )f 2 (D 2 ) dd 2 c 33 (D 2 X 2 Q 22 + S 2 )f 2 (D 2 ) dd 2 X 2 +Q 22 S 2 (7) 3.2 Problem decomposition Using a dynamic programming approach, this problem can be decomposed into two one-period subproblems that are the followings. The first subproblem is associated to the second period. The solution of this problem is optimal value of the second-period decision variables, namely Q 22 and S 2. These variables are expressed as a function of the state variable X 2 and are computed as the solution of the optimization problem max {Π 2 (X 2, ξ 2 (X 2 ))}, (8) ξ 2 (X 2 ) where we formally have ξ 2 (X 2 ) = (Q 22 (X 2 ), S 2 (X 2 )). Then, the second subproblem exploits ξ 2(X 2 ) in order to find the optimal policy for the first period, namely ξ 1(X 1 ) = (Q 11(X 1 ), Q 12(X 1 ), S 1(X 1 )). This optimal policy is obtained as the solution of the problem max {Π 1(X 1, ξ 1 (X 1 )) + E D1 {Π 2(X 2, ξ2(x 2 ))}}, (9) ξ 1 (X 1 ) where Π 1 (X 1, ξ 1 (X 1 )) is the expected first period profit function, while the second term is the expectation, with respect to D 1, of the second period profit function, under the optimal policy ξ 2(X 2 ).

6 3.3 Second-period subproblem. The objective function of the second period is defined by the following expression: Π 2 (X 2, Q 22, S 2 ) = {P 2 D 2 f 2 (D 2 ) dd 2 + s 2 S 2 c 22 Q 22 X2 +Q 22 S 2 h 2 (X 2 + Q 22 S 2 D 2 )f 2 (D 2 ) dd 2 b 2 (D 2 X 2 Q 22 + S 2 )f 2 (D 2 ) dd 2 X 2 +Q 22 S 2 X2 +Q 22 S 2 +s 3 (X 2 + Q 22 S 2 D 2 )f 2 (D 2 ) dd 2 } c 33 (D 2 X 2 Q 22 + S 2 )f 2 (D 2 ) dd 2 X 2 +Q 22 S 2 (1) This class of model has been analyzed by Cheaitou, et al. (25). These authors have shown that the objective function defined by (1) is a concave function of Q 22 and S 2. Furthermore, the following properties have been proven. Property 1: The optimal values of the two decision variables Q 22 and S 22 can not be simultaneously positive. Property 2: If I 1 < and X 2 <, the optimal quantity Q 22 satisfies X 2 + Q 22. by Optimal policy: From Cheaitou, et al. (25) the second period optimal policy is given { Q if X 2 < Y1 22 = Y1 X 2, S2 =, (11) { Q if Y1 X 2 Y2 22 =, S2 =, (12) and { Q if X 2 > Y2 22 =, S2 = X 2 Y2. (13) These conditions amount to with Q 22 = max (Y 1 X 2 ; ) and S 2 = max (X 2 Y 2 ; ) (14) ( ) Y1 = F2 1 b2 + c 33 c 22 b 2 + c 33 + h 2 s 3 ( ) and Y2 = F2 1 b2 + c 33 s 2. (15) b 2 + c 33 + h 2 s 3 Note that it is easily seen that under the assumptions of this paper, one has Y 1 < Y 2.

7 3.4 First period subproblem. Using the results of the second period subproblem, it is possible to numerically solve the first period subproblem and compute the optimal values of the decision variables of the first period, Q 11, Q 12 and S 1. The total expected profit function Π( ), under the optimal second-period policy ξ 2(X 2 ) becomes: Π(I, Q 1, Q 2, Q 11, Q 12, S 1 ) = Π 1 (X 1, Q 11, Q 12, S 1 ) + E D1 {Π 2(X 2, Q 22(X 2 ), S 2(X 2 ))} (16) The optimization problem to solve for this subproblem is then the following: Π (I, Q 1, Q 2 ) = max {Π(I, Q 1, Q 2, Q 11, Q 12, S 1 )} Q 11,Q 12,S 1 = max {P 1 D 1 f 1 (D 1 ) dd 1 + s 1 S 1 c 11 Q 11 c 12 Q 12 Q 11,Q 12,S 1 X1 +Q 11 S 1 h 1 (X 1 + Q 11 S 1 D 1 )f 1 (D 1 ) dd 1 b 1 (D 1 X 1 Q 11 + S 1 )f 1 (D 1 ) dd 1 X 1 +Q 11 S 1 +E D1 {Π 2(X 2, Q 22(X 2 ), S2(X 2 ))}} (17) By using the optimal values Q 22(X 2 ) and S 2(X 2 ) defined by (14), it is easily seen that the total expected profit function Π(I, Q 1, Q 2, Q 11, Q 12, S 1 ) is concave w.r.t. Q 11, Q 12 and S 1, in such a way that the optimization problem described in equation (17) has a unique maximum. However, no closed form formula exists for general demand probability distributions and the problem has to be numerically solved. 4 Numerical examples In this section we show, via some numerical applications, how our model gives some insight regarding this type of general two-stage inventory decision process. In the first example we show the behavior of the first period optimal policy in function of the initial inventory and of the salvage value s 1. In the second example we show the impact of the variability and of the difference between the costs on the optimal value of Q 12 and the expected optimal value of Q 22. For the two following examples we will suppose normal distributed demands for the two periods. 4.1 Example 1 Q11 S1 Q11, S Initial inventory I Figure 2: First period optimal policy - low s 1 value In the first example we consider the following parameter values: D 1 = N[1, 2], D 2 = N[1, 2], h 1 = 5, h 2 = 5, P 1 = 1, P 2 = 1, b 1 = 25, b 2 = 25, c 11 = 5,

8 c 12 = 3, c 22 = 5, c 33 = 5, s 2 = 2, s 3 = 2, Q 1 = and Q 2 =, with N[µ,σ] a normal distribution with mean of µ and standard deviation σ. In this example, the optimal policy is numerically computed as a function of the initial inventory I. In the first part (Figure 2) we have s 1 = 2 and in the second part (Figure 3) we have s 1 = 29. It is clearly seen, from this Q11 S1 Q11, S Initial inventory I Figure 3: First period optimal policy - high s 1 value numerical example, that the first period optimal policy has the same structure as the second period, for which the explicit form is known. From figures 2 and 3 we deduce that there are two thresholds. For the values of I between these two thresholds, the optimal values for the decision variables Q 11 and S 1 are both equal to zero. Below the low threshold only the optimal value of Q 11 is positive and above the high threshold only the optimal value of S 1 is positive. The higher threshold value depends on s 1 (see Figure 2 and Figure 3). When s 1 increases the higher threshold value decreases. It can be seen that the low threshold value depends on c 11, while the difference between the two thresholds is proportional to the difference c 11 s Example Q11 Q12 Q22 Q11, Q12, Q The cost c12 Figure 4: Early and late reorder modes: optimal values with low D 1 variability In the second example, we have the following parameter values : D 2 = N[1, 2], h 1 = 5, h 2 = 5, P 1 = 1, P 2 = 1, b 1 = 25, b 2 = 25, c 11 = 5, c 22 = 5, c 33 = 5, s 1 = 2, s 2 = 2, s 3 = 2, I =, Q 1 = and Q 2 =. In this example, the optimal values are computed as a function of the unit ordering cost c 12. For the first part of this example (Figure 4) we consider the first period demand with a low variability, namely we have D 1 = N[1, 2]. For the second part (Figure 5) we suppose a higher demand variability, i.e. D 1 = N[1, 4] It can be seen that the c 12 values can be divided into three intervals. The first interval corresponds to small c 12 values, for which the optimal value of Q 12 is positive and the expected optimal value of Q 22 is equal to zero.

9 Q11 Q12 Q Q11, Q12, Q The cost c12 Figure 5: Early and late reorder modes: optimal values with high D 1 variability The second region corresponds to mean c 12 for which the optimal values of Q 12 and Q 22 are both positive. The third region corresponds to the situation c 12 > c 22, where the optimal value of Q 12 is equal to zero. Another insight can be seen with respect to variability of D 1. When D 1 variability increases several phenomenons occur, see figures 4 and 5: The width of the second region (for medium c 12 values) increases toward the low c 12 values side. In fact Q 22 is decided after observing the realization of D 1 and therefore this decision benefits from the realization information. Q 12 is fixed before the realization of D 1. So when the variability of D 1 increases it is profitable to wait for realization of the first period demand and then to use this information to optimize the reorder, and this even if the cost (c 22 ) is high. For low c 12 values, the optimal value of Q 12 increases with demand variability while the optimal value of Q 11 decreases. This corresponds to the different numerical costs values that we have defined. In our case, for small c 12 values, the differences between c 11 and c 12 and between c 12 and c 22 are important, in such a way that for high D 1 variability, it becomes more profitable to order a small Q 11 value and a bigger Q 12 quantity, and if necessary reorder a Q 22 quantity in period 2. So in this case Q 12 increases to face the first period variability and to satisfy the second period demands. For medium c 12 values, the optimal Q 11 and Q 12 values decrease while the expected optimal Q 22 value increases to face the variability of the first period demand and to satisfy the second period demand. For high c 12 values, the optimal value of Q 12 is equal to zero (c 12 > c 22 ). When the variability of D 1 increases, the optimal Q 11 value increases and the expected optimal Q 22 value decreases. This is associated to the fact that c 11 is equal to c 22 in our example, and that the holding inventory cost in the first period is very small with respect to the shortage cost b 1. So in this case, Q 11 faces the high D 1 variability and satisfies a part of the second period orders. 5 Conclusion In this paper, we have analyzed a general two-period newsboy model, including initial inventory, pre-fixed orders and periodic salvage opportunities. Via a dynamic programming approach, we

10 characterized the structure of the optimal policy and provided a numerical approach in order to compute this optimal policy in a general setting. Via the numerical examples, we gave some insight into the effect of the parameters values on the solution. 6 Biography Ali Cheaitou studied mechanical engineering at the Lebanese University(Beirut). In 24 he prepared a DEA in industrial engineering at Ecole Centrale Paris(France). Since October 24, he is PhD student and teaching assistant in the Laboratoire Génie Industriel of the Ecole Centrale Paris. He is mainly interested in production planning and stochastic optimization. Christian van Delft is an associate professor in the department of Industrial Management and Logistics, at HEC-Paris School of Management (France), where he teaches and researches in operations management, operations research and quality control. He has published in IEEE Transactions on Robotics and Automation, Annals of Operations Research, European Journal on Operations Research, Optimal Control: Applications and Methods and others. Yves Dallery is a Professor of Industrial Engineering at /Ecole Centrale Paris/. His research interests are in production management, supply chain management and service operations management, with a special emphasis on modelling and optimization. Zied JEMAI is an assistant Professor in the department of Industrial Engineering, at Ecole Central Paris, where he teaches and researches in stochastic models and supply chain management. He has published in IIE Transactions and European Journal of Operational Research. References Bradford, J. W. Sugrue, P. K A bayesian approach to the two-period style-goods inventory problem with single replenishment and heterogeneous poisson demands, Journal of the operational research society 41: Cheaitou, A., van Delft, C., Dallery, Y. Jemai, Z. 25. Generalized newsboy model with initial inventory and two salvage opportunities. Working paper. Choi, T.-M., Li, D. Yan, H. 23. Optimal two-stage ordering policy with bayesian information updating, Journal of the operational research society 54: Donohue, K. L. 2. Efficient supply contracts for fashion goods with forecast updating and two production modes, Management science 46: Fisher, M. Raman, A Reducing the cost of demand uncertainty through accurate response to early sales, Operations research 44: Gurnani, H. Tang, C. S Optimal ordering decisions with uncertain cost and demand forecast updating, Management science 45: Hillier, F. S. Lieberman, G. J. 21. Introduction to operations research, 5th edn, McGraw-Hill. Khouja, M The single-period (news-vendor) problem: literature review and suggestions for future research, Omega 27: Lau, A. H.-L. Lau, H.-S Decisions models for single-period products with two ordering opportunities, International journal of production economics 55: Lau, H.-S. Lau, A. H.-L A semi analytical solution for a newsboy problem with midperiod replenishment, The journal of the operational research society 48: