The Replenishment Policy for an Inventory System with a Fixed Ordering Cost and a Proportional Penalty Cost under Poisson Arrival Demands

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1 Scientiae Mathematicae Japonicae Online, e-211, The Replenishment Policy for an Inventory System with a Fixed Ordering Cost and a Proportional Penalty Cost nder Poisson Arrival Demands Hitoshi Hohjo Received April 1, 211; revised Agst 21, 211 Abstract. A store with a bonded tank sells a random resorce to cstomers arriving according to the Poisson process. It reasonably has to be managed in a balance between a fixed ordering cost and a proportional penalty cost. The store shold lay down the safety stock level so as to keep these losses as minimal as possible. Then he adopts an ordering policy in which the tank is filled with resorce when the stock level falls to the safety stock level. We decide the optimal safety stock level so as to minimize the expected cost per nit time in the infinite periods. 1 Introdction Inventory control problems have been discssed in the academic literatre since the 195s. There are several policies sch as order-p-to-level policy, (Q, r) policy, (s, S) policy, continos review reorder policy, and so on. In recent stdies Hariga [6] examined a single item continos review inventory problem with stochastic demand and a bonded capacity. Babai, Jemai, Dallery [3] investigated a continos oder-p-to level policy for a single echelon inventory system where the demand is modelled as a compond Poisson process. On the stdy of reorder level Hohjo, Teraoka [8] sggested a single item continos review inventory model, assming that consmers arrive at a store according to the Poisson process and by a resorce expressed by continos qantity, with the constant ordering cost and the constant penalty cost for stocking-ot and analyzed on the optimal safety stock level nder minimizing of the expected cost per nit time in the infinite period. However the penalty cost is often sitable to be given by a fnction related to cmlative qantity of shortages in some sitations. This paper extends it to a model with the constant ordering cost and a proportional penalty cost. Or model doesn t deal with the holding cost bt the ordering cost and the penalty one. This paper is constrcted as follows. Section 2 describes an inventory control model with a fixed ordering cost and a proportional penalty cost nder Poisson arrival demands. Section 3 provides the calclations of the expected cost per nit time in the infinite period by sing a renewal reward theorem [14], which leads to the optimal replenishment policy. Nmerical examples illstrate some examinations for or reslt in Section 4. Section 5 gives some conclsions. 2 Model We consider an inventory control model on a resorce expressed by continos qantity. The model is described as follows: A store with an existing tank of maximm permissible qantity U deals in a liqescent resorce. Cstomers arrive at the store according to the Poisson process with intensity fnction. The j-th cstomer prchases Y j nits of resorces, where Y j (j 1, 2,... ) is a seqence of identical and independent nonnegative random variable having a general distribtion fnction G(x) P r{y j x}. For sch cstomers behavior, the store shold lay down the safety stock level so as to keep these losses 2 Mathematics Sbject Classification. 9B5. Key words and phrases. Inventory replenishment, Compond Poisson process, Efficient management, Renewal reward process.

2 162 H. Hohjo as minimal as possible. Then he adopts an ordering policy in which the tank is filled with resorce when the stock level falls to the safety stock level ( U). The ordered resorce arrives instantaneosly and the ordering cost is charged by C r regardless of the order qantity. In being ot of stock, the penalty cost is charged by p for one nit. Under these assmptions, the objective is to decide the optimal safety stock level so as to minimize the expected cost, denoted by C(), per nit time in the infinite period. 3 Formlation and Analysis We begin or analysis by calclating the expected cycle cost R() and the expected cycle length L(). Let Z j (j, 1, 2, ) denote the cmlative qantity when the j cstomers have prchased resorces, where Z. Then the cmlative process Z j can be expressed as (1) j Z j Y i for j 1, 2, 3, i1 with (2) P r{z j x} G (j) (x) for j, 1, 2,, where G (j) (x) is the j-fold Stieltjes convoltion of G(x) with itself, and { G () 1, x (x), x <. The probability α(), of replenishing resorces when the stock level reaches between and, is given by (3) α() P r{z j 1 U < Z j U} P r{u Z j 1 < Y j U Z j 1 Z j 1 x} dg (j 1) (x) [G(U x) G(U x)] dg (j 1) (x) G(U x) dg (j 1) (x) M(U ), where (4) M() G (j) (). Using Eq.(3), the expected cycle cost R() can be written as R() {C r + p(z j U)}P r{z j 1 U, Z j > U} + C r α() {C r + p(x + Y j U)}P r{y j > U x}dg (j 1) (x) + C r α()

3 THE REPLENISHMENT POLICY FOR AN INVENTORY SYSTEM 163 (5) C r +p C r + p [1 G(U x)]dg (j 1) (x) (x + Y j U)P r{y j > U x}dg (j 1) (x) (x + y U)dG(y)dG (j 1) (x). (6) The probability density fnction that the j-th cstomer arrives at time t is given by (t) j 1 (j 1)! e t (j 1, 2, ), which is known as the Erlang distribtion. Letting Ḡ(x) denote the srvivor fnction for a continos distribtion fnction G(x), i.e. Ḡ(x) 1 G(x) P r{y j > x}, the expected cycle length L() can be written as (7) L() + + t P r{z j 1 U, Z j > U} (t)j 1 t P r{z j 1 U < Z j U} (t)j 1 t (t)j 1 t (t)j 1 t (t)j 1 j [G(j 1) (U ) G (j) (U )] 1 + M(U ). Ḡ(U x)dg (j 1) (x) [G(U x) G(U x)]dg (j 1) (x) [1 G(U x)]dg (j 1) (x) The stock level is renewed by replenishing resorces. When we define an interval of two adjacent renewal time as a cycle for the renewal process, the renewal reward theorem leads s to the expected cost C() per nit time in the infinite period. From Eqs.(5) and (7) it can be expressed as (8) C() R() L() 1 + M(U ) C r + p (x + y U)dG(y)dG (j 1) (x). To find the safety stock level of minimizing the expected cost per nit time, we differentiate C() with respect to and set it to be eqal to. Then we obtain { } (9) (y )dg(y) (x + y U)dG(y) dg (j 1) (x) C r p.

4 164 H. Hohjo See Appendix A for the detailed calclation to derive Eq.(9). Ptting the left side of Eq.(9) by V () and differentiating it with respect to, we have (1) dv () d dv () { Ḡ()[1 + M(U )]. } dg(y) dg (j 1) (x) Then the derivative fnction d is negative for all U. Hence, V () is a monotone decreasing fnction in U, and it satisfies that (11) V () U { } ydg(y) + (U x)ḡ(u x) dg (j 1) (x) > and (12) V (U). As a reslt, if V () > C r /p, there exists a niqe root satisfying Eq.(9). It holds the ineqalities V () > C r /p for < and V () < C r /p for < U, which are implied C() is a decreasing fnction in over < and an increasing one over < U. On the other hand, if V () C r /p, it holds the ineqality V () < C r /p for all U, which is implied the fnction C() increases in. Therefore the sfficient condition of the minimizing problem is garanteed. From these argments, we obtain the following theorem. Theorem. 1 The optimal replenishment policy is given as follows: (i) If V () > C r /p, then there exists the niqe root satisfying Eq.(9). The optimal replenishment policy is to order p to U as soon as the stock level falls below the safety level. And the corresponding expected cost per nit time is given by C( ) p (y )dg(y). (ii) If V () C r /p, then minimizes the vale of the objective fnction C(). It means that the optimal replenishment policy is to order p to U after stocking ot. The optimal expected cost per nit time is given by C( U ) C() C r + p (x + y U)dG(y)dG (j 1) (x). 1 + M(U) 4 Nmerical Examples We give a sensitive analysis in this section. Sppose that the fnction G(x) is exponentially distribted with its mean 1/θ, i.e. G(x) 1 e θx, and let C r 1, 1. Then Eq.(9) can be rewritten as (13) 1 θ [ ] j 1 e θ {θ(u )} i e θu i! i C r p. We give the optimal safety stock level and its corresponding expected cost C( ) for each p and U on fixed vale θ.1 in Figre 1. The optimal safety stock level takes vale in p.1 and p.2 on U 5, which reslts from a small tank. As

5 THE REPLENISHMENT POLICY FOR AN INVENTORY SYSTEM 165 Figre 1: the optimal safety stock level and the corresponding total cost for p Figre 2: the optimal safety stock level and the corresponding total cost for θ the nit penalty cost p increases, the optimal safety stock level also increases. If the maximm permissible qantity U has large enogh, the optimal expected cost can keep cheap. For more than 5 of the maximm permissible qantity U, the corresponding expected costs almost takes the same vale. There is no meaning that enlarges the limited capacity becase it can sppress the optimal expectation cost to few with some size of the maximm permissible qantity. Figre 2 represents the optimal safety stock level and its coresponding expected cost C( ) for each θ and U on fixed vale p.1. When θ is small, the optimal total cost is affected by vale of U. Then establishing the optimal safety stock level is valid for ctting down the total cost. 5 Conclding Remarks This paper considered the optimal replenishment policy in an inventory control problem with cstomers arriving according to the Poisson process. Under assmptions with a fixed ordering cost and a proportional penalty cost, we showed an effective reslt on the setting of the safety stock level. Acknowledgments The athor thanks the editor and reviewers for their comments and sggestions that helped improve the paper. Appendix A Let g (i) () denote the probability density fnction corresponding to the cmlative distribtion fnction G (i) (). The first derivative of fnction C() R() L() is obtained by dc() d R ()L() R()L () {L()} 2.

6 166 H. Hohjo The nmerator is calclated as R ()L() R()L () p g (i) (U ) i1 C r + p 1 g (i) (U ) p i1 +C r + p (y )dg(y) 1 g (i) (U ) C r p i1 ] dg (j 1) (x). 1 + M(U ) (x + y U)dG(y)dG (j 1) (x) (y )dg(y) (x + y U)dG(y)dG (j 1) (x) ( 1 dg (j 1) (x) { (y )dg(y) By setting the nmerator to be eqal to, we can obtain Eq.(9). References ) g (i) (U ) i1 } (x + y U)dG(y) [1] B.C.Archibald, E.A.Silver, (s, S) policies nder continos review and discrete compond Poisson demand, Management Science 24 (1978), [2] K.J.Arrow, T.Harris and J.Marschak, Optimal inventory policy, Econometrica 19 (1951), [3] M.Z.Babai, Z.Jemai, Y.Dallery, Analysis of order-p-to-level inventory systems with compond Poisson demand, Eropean Jornal of Operational Research 21 (211), [4] M.Beckmann, An inventory model for arbitrary interval and qantity distribtions of demands, Management Science 8 (1961), [5] D.Beyer, SP.Sethi, R.Sridhar, Stochastic mlti-prodct inventory models with limited storage, Jornal of Optimization Theory and Applications 111 (21), [6] M.A.Hariga, A single-item continos review inventory problem with space restriction, International Jornal of Prodction Economics 128 (21), [7] D.P.Heyman and M.J.Sobel, Handbooks in Operations Research and Management Science 2, Elsevier Science Pblishers, North-Holland, Amsterdam, 199. [8] H.Hohjo and Y.Teraoka, The replenishment policy for an inventory system with Poisson arrival demands, Scientiae Mathematicae Japonicae 58 (23), [9] S.G.Johansen, A.Thorstenson, An inventory model with Poisson demand and emergency orders, International Jornal of Prodction Economics 56/57 (1998), [1] C.Larsen, A.Thorstenson, A comparison between the order and the volme fill rate for a basestock inventory control system nder a compond renewal demand process, Jornal of the Operational Research Society 59 (28), [11] D.N.P.Mrthy and D.G.Ngyen, Stdy of two-component system with failre interaction, Naval Research Logistics Qarterly 32 (1985), [12] S.Osaki, Introdction to Probability Models (in Japanese), Asakra Press, Tokyo, [13] S.M.Ross, Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, 197.

7 THE REPLENISHMENT POLICY FOR AN INVENTORY SYSTEM 167 [14] S.M.Ross, Stochastic processes, 2nd ed., Wiley, New York, [15] A.F.Veinott,Jr. and H.M.Wagner, Compting optimal (s, S) inventory policies, Management Science 11 (1965), [16] Y.Zheng, On properties for stochastic inventory systems, Management Science 33 (1992), commnicated by Yoshinob Teraoka ; DEPARTMENT OF MATHEMATICS AND INFORMATION SCIENCES, OSAKA PREFECTURE UNIVERSITY, 1-1 GAKUEN-CHO, NAKA-KU, SAKAI, OSAKA , JAPAN hojo@mi.s.osakaf-.ac.jp