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IIE Transactions (1998) 30, 1049±1055

Optimal policy for a periodic review returnable inventory system D.J. BUCHANAN1 and P.L. ABAD2,* 1

Hamilton, Ontario, Canada, L8T 3C1 M.G. DeGroote School of Business, McMaster University, Hamilton, Ontario, Canada L9B 2E1 E-mail: [email protected]; [email protected]

2

Received July 1995 and accepted April 1998

In this paper, we consider the inventory control problem in a periodic review returnable system. In a returnable system, containers are returned by consumers to the manufacturer for reuse. We view the returns in a given period to be a stochastic function of the number of containers out in the ®eld. Using dynamic programming, we derive the optimal inventory control policy for the system.

1. Introduction One of the important problems in inventory control is the problem of managing stocks in a returnable system. A returnable system is a system in which items (e.g., containers, bottles, parts) are returned by consumers to the manufacturer for reuse. The problem has attained signi®cance given the increased emphasis on reusing/recycling resources due to environmental concerns. The only work in the area of returnable system to date is the work by Kelle and Silver [1,2] who have presented a model of a returnable system [2]. The model assumes a non-zero order cost and a zero lead time. Returns from previously issued stock are described by a binomial distribution whereas returns from stock that will be issued in the future are described by mixed binomials. The cumulative net demand (i.e., issues-returns) in a given future period is then approximated by a normal distribution. The stochastic model is further approximated to a deterministic model using the chance-constrained approach of Bookbinder and Tan [3]. The deterministic model in turn is used to determine a procurement plan using the Wagner±Whitin algorithm thus providing a heuristic for the original problem. In a related paper, Kelle and Silver [1] have provided procedures for forecasting net demand (i.e., issues-returns) over the lead time. In this paper we ®rst formulate a periodic review model of a returnable system. There are two state variables in the model: Inventory of consumable units and the number of units in the ®eld. Returns in a given period is

*Corresponding author. 0740-817X

Ó

1998 ``IIE''

assumed to be a random fraction of the number of units in the ®eld. The planning horizon is assumed to be ®nite. We derive the structure of the optimal policy and develop a procedure for determining the optimal policy. The model can be used to determine the minimum cost plan for the ®nite horizon. In the next section we list assumptions and formulate the proposed model. Then we present an analysis of the single period problem. The single period model is then extended to the N -period problem and the structure of the optimal policy is derived using stochastic dynamic programming. Finally a two-period example is presented to illustrate the solution procedure.

2. Assumptions and notation The following notation is used throughout. c purchase cost of a new container; p penalty on each unit of shortage; p > c; v0 cost associated with an unsold unit at the end of the planning horizon. If v0 < 0, then it is the salvage value. We assume that ÿv0 < c; I net stock at the beginning of the period; S quantity of containers in the ®eld at the beginning of the period; d fraction of S which are destroyed or become permanently unavailable per period; x demand in the period. x is a random variable with pdf f x and cdf F x; r fraction of S which are returned to supplier during the period. r is a random variable with pdf h r; Q quantity of new containers ordered at the beginning of the period.

1050

Buchanan and Abad

We assume that the inventory of the containers is controlled using a periodic review system. We further make the following assumptions: (1) Order cost is zero. (2) Replenishment lead time is zero. (3) Holding cost is incurred on units carried through the period. (4) I Q 0 for each period. Hence if I < 0 (i.e., if there is a backlog), the backlog is met at the beginning of the next period. Assumption 4 is not strictly necessary in the single-period problem. However, this assumption is used later in the formulation of the N -period problem. Given that containers are used to sell another primary product, it is a reasonable assumption.

3. Single period analysis Let h represent the total cost in the ®nal period. Then: 8 cQ v0 I Q r 1 ÿ dS ÿ x > > > < if x I Q r 1 ÿ dS; h 1 > cQ px ÿ I ÿ Q ÿ r 1 ÿ dS > > : if x I Q r 1 ÿ dS: Let I Q y. Here y is the order-up-to level. Then Q y ÿ I and: 8 c y ÿ I v0 y r 1 ÿ dS ÿ x > > > < if x y r 1 ÿ dS; h 2 > c y ÿ I px ÿ y ÿ r 1 ÿ dS > > : if x > y r 1 ÿ dS: Note we have assumed I Q y 0. Moreover since ^ S; y E h be the r 0; y r 1 ÿ dS 0. Let w I; expected cost in the ®nal period. Also let notation z stand for maxfz; 0g where z is any time-varying quantity. Then:

y max 0; I or y I : It is easy to show that ^ S; y @ w I; c ÿ p v0 p @y

Z1 h rF y r 1 ÿ dSdr: 0

5 and ^ S; y @ 2 w I; v0 p @y 2

Z1 h rf y r 1 ÿ dSdr: 0

^ S; y=@y 2 0. Hence w I; ^ S; y Since f :; h : 0, @ w I; is a convex function of y. Setting (5) to zero, the ®rst order condition for optimal y is Z1 pÿc : 7 h rF y r 1 ÿ dSdr v0 p 0

Let I c S represent the solution of (7) for a given S. Since the state variable I is not present in (7), I c S depends entirely on S. Taking the derivative of (7) with respect to S, we have Z1 0

dy h rf y r 1 ÿ dS r 1 ÿ d dr 0: dS

0

v0 y r 1 ÿ dS ÿ x 0

f xdx dr Z1

Z1 h r

0

px ÿ y ÿ r 1 ÿ dS f xdx dr; yr 1ÿdS

c y ÿ I E fv0 y r 1 ÿ dS ÿ x r; x

px ÿ y ÿ r 1 ÿ dS g:

3

In addition Q y ÿ I 0. Hence the restriction on y I Q is:

8

From (8), it is easy to show that dy=dS < 0. Hence I c S is a decreasing function of S. I c S is the unconstrained value of y which minimizes ^ w I; S; y. Clearly when I > 0 and I c S < I; y I c S will violate restriction (4). Under this case, given the convexity of w^ the optimal value of y; y I. Similarly when I < 0 and I c S < 0; y I c S will violate constraint (4) and y 0. As shown in Fig. 1, y is given by:

yr 1ÿdS Z

h r

c y ÿ I

6

2

^ S; y w I; Z1

4

Fig. 1. Optimal policy for the period.

Optimal policy for a periodic review returnable inventory system

y

8 c I S > > > < > I > > : 0

if I 0; I c S I

(Region A),

c

or if I < 0; I S 0; c

if I 0; I S < I c

if I < 0; I S < 0

(Region B),

also 9

(Region C).

Since y Q I, the optimal Q is given by Q y ÿ I. Note at the boundary between regions A and B, I I c S. Hence this boundary is characterized B1 fS; I j S; I I c S; I 0g:

10

Similarly at the boundary between regions A and C, I c S 0. Let S0 denotes the value of S in (7) which returns y I c S 0 0. The boundary between regions A and C is hence characterized by B2 fS; I j S S 0 ; I < 0g:

11

We have characterized the solution to the single period problem. If the goal is to solve the single period problem for a given S, then it is not necessary that Fig. 1 be developed. For the given S, one would solve for I C S using Equation (7) and then simply apply the decision rule (9). In the next section, we will formulate the N -period problem. The single-period analysis is applicable to the last period of the planning horizon. In what follows, we present a convexity result that will be useful in the analysis of the N -period problem. Theorem 1. Let ^ S; y w I; /1 I; S Min Min E fc y ÿ I v0 y r 1 ÿ dS ÿ x yI r; x px ÿ y ÿ r 1 ÿ dS :

12

Region A In this region, y I c S. Substituting this optimal value in (12), we have @ / I; S 0 0: @I 2

13

Region B In this region Q 0 or y I. Substituting this optimal value in (12), we have @/1 I; S ÿp v0 p @I

h rf I r 1ÿ dS dr 0: 15 0

@/1 I; S ÿc: 16 @I From (13) and (16), it follows that @/1 I; S=@I is continuous on boundary B1 . Region C In this region, y 0. Substituting this optimal value in (12), we have @/1 I; S @ 2 /1 I; S 00 17 ÿc and @I @I 2 Also given conditions (13) and (17), @/1 I; S=@I is continuous on boundary B2 . At the boundary between regions C and B, @/1 I; S=@I is not continuous. Substituting I 0 in (14), at the side of this boundary in region B, Z1 @/1 I; S ÿp v0 p h rF r 1 ÿ dSdr: 18 @I Given (7), when I I c S,

0

h rF I c S r 1 ÿ dSdr ÿc: 19

0

Proof. We show that in each region in Fig. 1, /1 I; S is convex. Moreover @/1 I; S=@I is continuous at the boundaries B1 and B2 and it is an increasing function of I as we move from region C to region B.

2

Z1

Moreover from (10), (14) and (7), on boundary B1 ,

ÿp v0 p

Then /1 I; S is a convex function of I for a given S.

and

@ 2 /1 I; S v0 p @I 2

Z1

yI

@/1 I; S ÿc @I

1051

Z1 h rF I r 1 ÿ dS dr: 0

14

0

Now for S > S , I c S < 0. Hence r 1 ÿ dS > I c S r 1 ÿ dS:

20

Given (18), (19), and (20), at the side of this boundary in region B, @/1 I; S > ÿc: 21 @I From (17) and (21), it follows that @/1 I; S=@I increases as we move from region C to region B. j

4. N-period analysis Let the periods be numbered in the reverse order. Also let: v unit holding cost per period (except for the last period). This cost is incurred on units present in the system at the end of the period; In net stock at the beginning of period n; Sn quantity of containers in the ®eld at the beginning of period n; xn demand during period n with pdf fn xn ; rn fraction of Sn returned during period n with pdf h rn .

1052

Buchanan and Abad convex function of y2 . Thus, w2 is a sum of convex functions; i.e., w2 is a convex function of y2 . j

The transition equations for n 2; 3; :::; N are: Inÿ1 In Qn rn 1 ÿ dSn ÿ xn yn rn 1 ÿ dSn ÿ xn :

22a

Snÿ1 Sn ÿ dSn ÿ rn 1 ÿ dSn xn 1 ÿ dSn 1 ÿ rn xn :

22b

Note that given assumption 4 (i.e., ynÿ1 Inÿ1 Qnÿ1 0), all xn units of demand is met either during period n or at the beginning of period n ÿ 1. Therefore all xn units would be in the ®eld by the beginning of period n ÿ 1. For this reason, we have added xn in the right hand side of (22b). The decision variable Qn is therefore present only in (22a). Without assumption 4, state Equation (22b) will be dierent leading to a more dicult formulation. ^ 1 ; S1 ; y1 . The N -period Now let w1 I1 ; S1 ; y1 w I problem is: For n 2; 3; . . . N solve ^ n ; Sn ; yn E/ Inÿ1 ; Snÿ1 ; 23 Min wn In ; Sn ; yn w I nÿ1

yn In

Proposition 2. Let: y2 argminy2 I2 fw2 I2 ; S2 ; y2 g. Then y2 is characterized by a decision rule with a structure similar to (9). Proof. Substituting n 2 in (23), the ®rst order condition for the unconstrained problem is ^ 2 ; S2 ; y2 @E/ I1 ; S1 @w2 I2 ; S2 ; y2 @ w I 1 0: 29 @y2 @y2 @y2 The second term in (29) is given by (27). Substituting n 2 in (24) and taking derivative w.r.t. y2 , we have ^ 2 ; S2 ; y2 @ w I @y2 Z1 c ÿ p v p

h r2 F y2 r2 1 ÿ dS2 dr2 :

30

0

where ^ n ; Sn ; yn c yn ÿ In E fvyn rn 1 ÿ dSn ÿ xn wI rn ; xn

pxn ÿ yn ÿ rn 1 ÿ dSn g;

24

/nÿ1 Inÿ1 ; Snÿ1 Min wnÿ1 Inÿ1 ; Snÿ1 ; ynÿ1 ; ynÿ1 Inÿ1

25

and E/nÿ1 Inÿ1 ; Snÿ1 E f/nÿ1 yn rn 1 ÿ dSn ÿ xn ; rn ; xn

1 ÿ dSn 1 ÿ rn xn g:

26

^ 2 ; S2 ; y2 r E; x f/ I1 ; S1 g Proposition 1. w2 I2 ; S2 ; y2 w I 1 2 2 is a convex function of y2 . ^ 2 ; S2 ; y2 is a Proof. As shown in the previous section, w I convex function of y2 . Substituting n 2 in (26) and taking derivative w.r.t. y2 , we have @E/1 I1 ; S1 @/1 I1 ; S1 @I1 27 E r2 ; x2 @y2 @y2 @I1 where from (22a) and (22b), I1 y2 r2 1 ÿ dS2 ÿ x2

and

2

2

@ E/1 I1 ; S1 @ /1 I1 ; S1 @I1 E 2 r ; x @y2 2 2 @y2 @I12

31 j The three regions in I2 ; S2 space de®ned by (31) are very similar to the three regions shown in Fig. 1. Expression (27) involves the partial derivative @/1 I1 ; S1 =@I1 . Hence solving numerically for the value of y2 that satis®es condition (29) requires the value of @/1 I1 ; S1 =@I1 at a given point I1 ; S1 . From (9) and Theorem 1, at a given point I1 ; S1 the derivative is given by: @/1 I1 ; S1 ÿc @I1

S1 1 ÿ dS2 1 ÿ r2 x2 : Now @I1 =@y2 1. Hence

Now I2 is not present in expressions (30) and (27) and therefore not present in Equation (29). Hence the value of y2 satisfying condition (29) would depend only upon S2 . Let this value be denoted as I2c S2 . Convexity of w2 I2 ; S2 ; y2 with respect to y2 and the restriction y2 I2 imply that optimal y2 is characterized by 8 c I2 S2 if I2 0; I2c S2 I2 (RegionA), > > > > > < or if I2 < 0; I2c S2 0, y2 > if I2 0; I2c S2 < I2 (Region B), > > I2 > > : 0 if I2 < 0; I2c S2 < 0 , (Region C).

when I1 < maxfI1c S1 ; 0g

@/1 I1 ; S1 ÿp v0 p @I1

28

From Theorem 1, @ 2 /1 I1 ; S1 =@I12 0 for all I1 . Hence the right hand side in 28 0. That is Ef/1 I1 ; S1 g is a

Z1 h r1 F I1 r1 1 ÿ dS1 dr 0

when I1 > maxfI1c S1 ; 0g 32 Finally given (32) and given the de®nition of I1c S1 from (7), it is easy to show that

Optimal policy for a periodic review returnable inventory system @/1 I1 ; S1 @I1

Region C In this region y2 0: Substituting this optimal value in (34), we have

Z1

maxfÿc; ÿp v0 p

h r1 F I1 r1 1 ÿ dS1 dr1 g: 0

33 Theorem 2. Let: /2 I2 ; S2 Min w2 I2 ; S2 ; y2 : y2 I2

Then /2 I2 ; S2 is a convex function of I2 for a given S2 . Proof. From (23), /2 I2 ; S2 Min c y2 ÿ I2 E fvy2 r2 1 ÿ dS2 ÿ x2 r2 ;x2

y2 I2

px2 ÿ y2 ÿ r2 1 ÿ dS2 g E f/1 y2 r2 1 ÿ dS2 ÿ x2 ; r2 ;x2

1 ÿ dS2 1 ÿ r2 x2 g:

34

We would ®rst show that @ 2 /2 I2 ; S2 =@I22 0 in each region as de®ned in (31). Region A In this region, y2 I2c S2 Substituting this optimal value in (34), we have @/2 I2 ; S2 ÿc @I2

@ 2 /2 I2 ; S2 0 0: @I22

and

35

Region B In this region Q2 0 or y2 I2 . Substituting this optimal value in (34), we have @/2 I2 ; S2 ÿp v p @I2

Z1 h r2 F2 I2 r2 1 ÿ dS2 dr2 0

@/1 I1 ; S1 @I1 ; E r2 ;x2 @I2 @I1

36

where I1 I2 r2 1 ÿ dS2 ÿ x2 : Now @I1 =@I2 1. Hence @ 2 /2 I2 ; S2 v p @I22 E

r2 ;x2

1053

Z1 h r2 f2 I2 r2 1 ÿ dS2 dr2 0

@ 2 /1 I1 ; S1 @I1 : @I2 @I12

@/2 I2 ; S2 ÿc and @I2

@ 2 /2 I2 ; S2 00 @I22

38

Similar to Theorem 1, we can show that @/2 I2 ; S2 =@I2 is either continuous or an increasing function of I2 at the various boundaries. j Theorem 3. For n 2; 3; . . . ; N let wn In ; Sn ; yn be as de®ned in (23). Then for each n, wn In ; Sn ; yn is a convex function of yn . Proof. Proof is by induction. In Theorem 2, we showed that /2 I2 ; S2 is a convex function of I2 . Now let n 3. Then: (1) /nÿ1 Inÿ1 ; Snÿ1 is a convex function of Inÿ1 . Given Equations (22a) and (22b), Inÿ1 yn rn 1 ÿ dSn ÿ xn and Snÿ1 1 ÿ dSn 1 ÿ rn xn . Hence Ern ;xn f/nÿ1 Inÿ1 ; Snÿ1 g Ern ;xn f/nÿ1 yn rn 1 ÿ dSn ÿ xn ; 1 ÿ dSn 1 ÿ rn xn g is a con^ n ; Sn ; yn is convex with vex function of yn . Also w I respect to yn . Thus wn In ; Sn ; yn is a sum of convex functions and therefore is a convex function of yn . (2) Given the convexity of wn and given that yn In , the optimal yn will be given by a decision rule similar to condition (9). (3) Using the procedure of Theorem 2, we can show that /n In ; Sn as de®ned in (25) with n ÿ 1 replaced by n is a convex function of In for a given Sn . If n N stop. Else let n n 1 and repeat steps 1, 2, and 3 above. j Note that in order to numerically determine the critical values Inc Sn for period n, we would need to calculate and store in period n ÿ 1 values of @/nÿ1 Inÿ1 ; Snÿ1 =@Inÿ1 on an appropriate grid of Inÿ1 ; Snÿ1 . For example, to determine I3c S3 values in period 3, we need to compute @/2 I2 ; S2 =@I2 values on a suitable grid of I2 ; S2 in period 2. From Theorem 2, these partial derivatives are equal to ÿc in regions A and C of (31). In region B, the partial derivative is given by expression (36).

5. An example 37

We showed in Theorem 1 that /1 I1 ; S1 is a convex function of I1 . Hence @ 2 /1 I1 ; S1 =@I12 0 for all I1 . Hence both terms in (37) are non-negative.

Suppose c $3=unit, p $7=unit short, v0 $0:5=unit, v $0:5=unit, d 0:05, f x 1=20exp ÿx=20 and with h r 1= 0:5 ÿ 0 for r 2 0; 0:5 r1 r2 r For the ®nal period, the chart (corresponding to Fig. 1) for determining the optimal lot-size is shown in Fig. 2.

1054

Buchanan and Abad

Fig. 2. Optimal policy for the ®nal period.

The chart is developed by solving Equation (7) for different values of S using the software Mathcad. Note that S10 74. Hence for S1 > 74, the solution of condition (7), I1c S1 will be less than 0. Of course, we do not solve for these negative critical values because for such a S1 ; y max0; I. To illustrate the solution of a two period problem, suppose that I2 5 and S2 40. We use condition (29) to solve for I2c 40. Closed form expression (33) is used in (29) or (27) for @/1 I1 ; S1 =@I1 . We ®nd I2c 40 22:1. Hence Q2 22:1 ÿ 5 17:1. Now suppose that the initial period has elapsed and the realizations of random variables r2 and x2 lead to I1 ÿ10, S1 35. Then from Fig. 2, I1c 35 7:5 and hence Q1 7:5 ÿ ÿ10 17:5.

6. Conclusions We have derived the optimal inventory control policy for a periodic review returnable inventory system. There are two state variables in the model: net stock of containers in the hands of the manufacturer and the number of containers in the hands of the consumers. In the proposed model, the probability of a container being returned in the current period is assumed to be independent of the age (i.e., the time since the container was last issued) of the container. This is equivalent to assuming that the ®eld life (i.e., length of time the con-

tainer stays in the hands of the current customer) is exponentially distributed. The model can also be applied to situations where the probability of return is dependent upon the age provided that the age-mix of the containers in the ®eld from period to period is stable. The return variable r in such a case can be described by its conditional distribution given the agemix of the containers in the ®eld. In future research, models will be formulated to explicitly include the age of the container in the analysis.

Acknowledgement This research is partially funded by a grant from Arts Research Board of McMaster University and by NSERC Canada Grant OGP0001226.

References [1] Kelle, P. and Silver, E.A. (1989) Forecasting the returns of reusable containers. Journal of Operations Management, 8, 17±35. [2] Kelle, P. and Silver, E.A. (1989) Purchasing policy of new containers considering the random returns of previously issued containers. IIE Transactions, 21, 349±354. [3] Bookbinder, J.H. and Tan, Y. (1988) Strategies for the probabilistic lot-sizing problem with service-level constraints. Management Science, 34, 1096±1108.

Optimal policy for a periodic review returnable inventory system [4] Heyman, D.P. and Sobel, M.J.(1984) Stochastic Models in Operations Research 2 Vol. II: Stochastic Optimization, McGraw Hill, New York. [5] Ross, S.M. (1970) Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, CA.

Biographies David J. Buchanan holds as BSc (Physics), an MBA, and a Ph.D. (1988) from McMaster University. He has co-authored an article on a

1055

lost sales inventory model and an article on locating a noxious facility with respect to several polygonal regions using asymmetric distances. He is not currently involved in the ®eld of operations research. Prakash L. Abad is a Professor of Management Science in the DeGroote School of Business at McMaster University. Dr. Abad's research interests are in production and inventory systems, pricinginventory models, nonlinear programming, stochastic dynamic programming, and classi®cation. His work has appeared in Management Science, Decision Sciences, European Journal of Operations Research, Optimal Control and Applications, International Journal of Systems Science and Computers and Operations Research.

Lihat lebih banyak...
Optimal policy for a periodic review returnable inventory system D.J. BUCHANAN1 and P.L. ABAD2,* 1

Hamilton, Ontario, Canada, L8T 3C1 M.G. DeGroote School of Business, McMaster University, Hamilton, Ontario, Canada L9B 2E1 E-mail: [email protected]; [email protected]

2

Received July 1995 and accepted April 1998

In this paper, we consider the inventory control problem in a periodic review returnable system. In a returnable system, containers are returned by consumers to the manufacturer for reuse. We view the returns in a given period to be a stochastic function of the number of containers out in the ®eld. Using dynamic programming, we derive the optimal inventory control policy for the system.

1. Introduction One of the important problems in inventory control is the problem of managing stocks in a returnable system. A returnable system is a system in which items (e.g., containers, bottles, parts) are returned by consumers to the manufacturer for reuse. The problem has attained signi®cance given the increased emphasis on reusing/recycling resources due to environmental concerns. The only work in the area of returnable system to date is the work by Kelle and Silver [1,2] who have presented a model of a returnable system [2]. The model assumes a non-zero order cost and a zero lead time. Returns from previously issued stock are described by a binomial distribution whereas returns from stock that will be issued in the future are described by mixed binomials. The cumulative net demand (i.e., issues-returns) in a given future period is then approximated by a normal distribution. The stochastic model is further approximated to a deterministic model using the chance-constrained approach of Bookbinder and Tan [3]. The deterministic model in turn is used to determine a procurement plan using the Wagner±Whitin algorithm thus providing a heuristic for the original problem. In a related paper, Kelle and Silver [1] have provided procedures for forecasting net demand (i.e., issues-returns) over the lead time. In this paper we ®rst formulate a periodic review model of a returnable system. There are two state variables in the model: Inventory of consumable units and the number of units in the ®eld. Returns in a given period is

*Corresponding author. 0740-817X

Ó

1998 ``IIE''

assumed to be a random fraction of the number of units in the ®eld. The planning horizon is assumed to be ®nite. We derive the structure of the optimal policy and develop a procedure for determining the optimal policy. The model can be used to determine the minimum cost plan for the ®nite horizon. In the next section we list assumptions and formulate the proposed model. Then we present an analysis of the single period problem. The single period model is then extended to the N -period problem and the structure of the optimal policy is derived using stochastic dynamic programming. Finally a two-period example is presented to illustrate the solution procedure.

2. Assumptions and notation The following notation is used throughout. c purchase cost of a new container; p penalty on each unit of shortage; p > c; v0 cost associated with an unsold unit at the end of the planning horizon. If v0 < 0, then it is the salvage value. We assume that ÿv0 < c; I net stock at the beginning of the period; S quantity of containers in the ®eld at the beginning of the period; d fraction of S which are destroyed or become permanently unavailable per period; x demand in the period. x is a random variable with pdf f x and cdf F x; r fraction of S which are returned to supplier during the period. r is a random variable with pdf h r; Q quantity of new containers ordered at the beginning of the period.

1050

Buchanan and Abad

We assume that the inventory of the containers is controlled using a periodic review system. We further make the following assumptions: (1) Order cost is zero. (2) Replenishment lead time is zero. (3) Holding cost is incurred on units carried through the period. (4) I Q 0 for each period. Hence if I < 0 (i.e., if there is a backlog), the backlog is met at the beginning of the next period. Assumption 4 is not strictly necessary in the single-period problem. However, this assumption is used later in the formulation of the N -period problem. Given that containers are used to sell another primary product, it is a reasonable assumption.

3. Single period analysis Let h represent the total cost in the ®nal period. Then: 8 cQ v0 I Q r 1 ÿ dS ÿ x > > > < if x I Q r 1 ÿ dS; h 1 > cQ px ÿ I ÿ Q ÿ r 1 ÿ dS > > : if x I Q r 1 ÿ dS: Let I Q y. Here y is the order-up-to level. Then Q y ÿ I and: 8 c y ÿ I v0 y r 1 ÿ dS ÿ x > > > < if x y r 1 ÿ dS; h 2 > c y ÿ I px ÿ y ÿ r 1 ÿ dS > > : if x > y r 1 ÿ dS: Note we have assumed I Q y 0. Moreover since ^ S; y E h be the r 0; y r 1 ÿ dS 0. Let w I; expected cost in the ®nal period. Also let notation z stand for maxfz; 0g where z is any time-varying quantity. Then:

y max 0; I or y I : It is easy to show that ^ S; y @ w I; c ÿ p v0 p @y

Z1 h rF y r 1 ÿ dSdr: 0

5 and ^ S; y @ 2 w I; v0 p @y 2

Z1 h rf y r 1 ÿ dSdr: 0

^ S; y=@y 2 0. Hence w I; ^ S; y Since f :; h : 0, @ w I; is a convex function of y. Setting (5) to zero, the ®rst order condition for optimal y is Z1 pÿc : 7 h rF y r 1 ÿ dSdr v0 p 0

Let I c S represent the solution of (7) for a given S. Since the state variable I is not present in (7), I c S depends entirely on S. Taking the derivative of (7) with respect to S, we have Z1 0

dy h rf y r 1 ÿ dS r 1 ÿ d dr 0: dS

0

v0 y r 1 ÿ dS ÿ x 0

f xdx dr Z1

Z1 h r

0

px ÿ y ÿ r 1 ÿ dS f xdx dr; yr 1ÿdS

c y ÿ I E fv0 y r 1 ÿ dS ÿ x r; x

px ÿ y ÿ r 1 ÿ dS g:

3

In addition Q y ÿ I 0. Hence the restriction on y I Q is:

8

From (8), it is easy to show that dy=dS < 0. Hence I c S is a decreasing function of S. I c S is the unconstrained value of y which minimizes ^ w I; S; y. Clearly when I > 0 and I c S < I; y I c S will violate restriction (4). Under this case, given the convexity of w^ the optimal value of y; y I. Similarly when I < 0 and I c S < 0; y I c S will violate constraint (4) and y 0. As shown in Fig. 1, y is given by:

yr 1ÿdS Z

h r

c y ÿ I

6

2

^ S; y w I; Z1

4

Fig. 1. Optimal policy for the period.

Optimal policy for a periodic review returnable inventory system

y

8 c I S > > > < > I > > : 0

if I 0; I c S I

(Region A),

c

or if I < 0; I S 0; c

if I 0; I S < I c

if I < 0; I S < 0

(Region B),

also 9

(Region C).

Since y Q I, the optimal Q is given by Q y ÿ I. Note at the boundary between regions A and B, I I c S. Hence this boundary is characterized B1 fS; I j S; I I c S; I 0g:

10

Similarly at the boundary between regions A and C, I c S 0. Let S0 denotes the value of S in (7) which returns y I c S 0 0. The boundary between regions A and C is hence characterized by B2 fS; I j S S 0 ; I < 0g:

11

We have characterized the solution to the single period problem. If the goal is to solve the single period problem for a given S, then it is not necessary that Fig. 1 be developed. For the given S, one would solve for I C S using Equation (7) and then simply apply the decision rule (9). In the next section, we will formulate the N -period problem. The single-period analysis is applicable to the last period of the planning horizon. In what follows, we present a convexity result that will be useful in the analysis of the N -period problem. Theorem 1. Let ^ S; y w I; /1 I; S Min Min E fc y ÿ I v0 y r 1 ÿ dS ÿ x yI r; x px ÿ y ÿ r 1 ÿ dS :

12

Region A In this region, y I c S. Substituting this optimal value in (12), we have @ / I; S 0 0: @I 2

13

Region B In this region Q 0 or y I. Substituting this optimal value in (12), we have @/1 I; S ÿp v0 p @I

h rf I r 1ÿ dS dr 0: 15 0

@/1 I; S ÿc: 16 @I From (13) and (16), it follows that @/1 I; S=@I is continuous on boundary B1 . Region C In this region, y 0. Substituting this optimal value in (12), we have @/1 I; S @ 2 /1 I; S 00 17 ÿc and @I @I 2 Also given conditions (13) and (17), @/1 I; S=@I is continuous on boundary B2 . At the boundary between regions C and B, @/1 I; S=@I is not continuous. Substituting I 0 in (14), at the side of this boundary in region B, Z1 @/1 I; S ÿp v0 p h rF r 1 ÿ dSdr: 18 @I Given (7), when I I c S,

0

h rF I c S r 1 ÿ dSdr ÿc: 19

0

Proof. We show that in each region in Fig. 1, /1 I; S is convex. Moreover @/1 I; S=@I is continuous at the boundaries B1 and B2 and it is an increasing function of I as we move from region C to region B.

2

Z1

Moreover from (10), (14) and (7), on boundary B1 ,

ÿp v0 p

Then /1 I; S is a convex function of I for a given S.

and

@ 2 /1 I; S v0 p @I 2

Z1

yI

@/1 I; S ÿc @I

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Z1 h rF I r 1 ÿ dS dr: 0

14

0

Now for S > S , I c S < 0. Hence r 1 ÿ dS > I c S r 1 ÿ dS:

20

Given (18), (19), and (20), at the side of this boundary in region B, @/1 I; S > ÿc: 21 @I From (17) and (21), it follows that @/1 I; S=@I increases as we move from region C to region B. j

4. N-period analysis Let the periods be numbered in the reverse order. Also let: v unit holding cost per period (except for the last period). This cost is incurred on units present in the system at the end of the period; In net stock at the beginning of period n; Sn quantity of containers in the ®eld at the beginning of period n; xn demand during period n with pdf fn xn ; rn fraction of Sn returned during period n with pdf h rn .

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Buchanan and Abad convex function of y2 . Thus, w2 is a sum of convex functions; i.e., w2 is a convex function of y2 . j

The transition equations for n 2; 3; :::; N are: Inÿ1 In Qn rn 1 ÿ dSn ÿ xn yn rn 1 ÿ dSn ÿ xn :

22a

Snÿ1 Sn ÿ dSn ÿ rn 1 ÿ dSn xn 1 ÿ dSn 1 ÿ rn xn :

22b

Note that given assumption 4 (i.e., ynÿ1 Inÿ1 Qnÿ1 0), all xn units of demand is met either during period n or at the beginning of period n ÿ 1. Therefore all xn units would be in the ®eld by the beginning of period n ÿ 1. For this reason, we have added xn in the right hand side of (22b). The decision variable Qn is therefore present only in (22a). Without assumption 4, state Equation (22b) will be dierent leading to a more dicult formulation. ^ 1 ; S1 ; y1 . The N -period Now let w1 I1 ; S1 ; y1 w I problem is: For n 2; 3; . . . N solve ^ n ; Sn ; yn E/ Inÿ1 ; Snÿ1 ; 23 Min wn In ; Sn ; yn w I nÿ1

yn In

Proposition 2. Let: y2 argminy2 I2 fw2 I2 ; S2 ; y2 g. Then y2 is characterized by a decision rule with a structure similar to (9). Proof. Substituting n 2 in (23), the ®rst order condition for the unconstrained problem is ^ 2 ; S2 ; y2 @E/ I1 ; S1 @w2 I2 ; S2 ; y2 @ w I 1 0: 29 @y2 @y2 @y2 The second term in (29) is given by (27). Substituting n 2 in (24) and taking derivative w.r.t. y2 , we have ^ 2 ; S2 ; y2 @ w I @y2 Z1 c ÿ p v p

h r2 F y2 r2 1 ÿ dS2 dr2 :

30

0

where ^ n ; Sn ; yn c yn ÿ In E fvyn rn 1 ÿ dSn ÿ xn wI rn ; xn

pxn ÿ yn ÿ rn 1 ÿ dSn g;

24

/nÿ1 Inÿ1 ; Snÿ1 Min wnÿ1 Inÿ1 ; Snÿ1 ; ynÿ1 ; ynÿ1 Inÿ1

25

and E/nÿ1 Inÿ1 ; Snÿ1 E f/nÿ1 yn rn 1 ÿ dSn ÿ xn ; rn ; xn

1 ÿ dSn 1 ÿ rn xn g:

26

^ 2 ; S2 ; y2 r E; x f/ I1 ; S1 g Proposition 1. w2 I2 ; S2 ; y2 w I 1 2 2 is a convex function of y2 . ^ 2 ; S2 ; y2 is a Proof. As shown in the previous section, w I convex function of y2 . Substituting n 2 in (26) and taking derivative w.r.t. y2 , we have @E/1 I1 ; S1 @/1 I1 ; S1 @I1 27 E r2 ; x2 @y2 @y2 @I1 where from (22a) and (22b), I1 y2 r2 1 ÿ dS2 ÿ x2

and

2

2

@ E/1 I1 ; S1 @ /1 I1 ; S1 @I1 E 2 r ; x @y2 2 2 @y2 @I12

31 j The three regions in I2 ; S2 space de®ned by (31) are very similar to the three regions shown in Fig. 1. Expression (27) involves the partial derivative @/1 I1 ; S1 =@I1 . Hence solving numerically for the value of y2 that satis®es condition (29) requires the value of @/1 I1 ; S1 =@I1 at a given point I1 ; S1 . From (9) and Theorem 1, at a given point I1 ; S1 the derivative is given by: @/1 I1 ; S1 ÿc @I1

S1 1 ÿ dS2 1 ÿ r2 x2 : Now @I1 =@y2 1. Hence

Now I2 is not present in expressions (30) and (27) and therefore not present in Equation (29). Hence the value of y2 satisfying condition (29) would depend only upon S2 . Let this value be denoted as I2c S2 . Convexity of w2 I2 ; S2 ; y2 with respect to y2 and the restriction y2 I2 imply that optimal y2 is characterized by 8 c I2 S2 if I2 0; I2c S2 I2 (RegionA), > > > > > < or if I2 < 0; I2c S2 0, y2 > if I2 0; I2c S2 < I2 (Region B), > > I2 > > : 0 if I2 < 0; I2c S2 < 0 , (Region C).

when I1 < maxfI1c S1 ; 0g

@/1 I1 ; S1 ÿp v0 p @I1

28

From Theorem 1, @ 2 /1 I1 ; S1 =@I12 0 for all I1 . Hence the right hand side in 28 0. That is Ef/1 I1 ; S1 g is a

Z1 h r1 F I1 r1 1 ÿ dS1 dr 0

when I1 > maxfI1c S1 ; 0g 32 Finally given (32) and given the de®nition of I1c S1 from (7), it is easy to show that

Optimal policy for a periodic review returnable inventory system @/1 I1 ; S1 @I1

Region C In this region y2 0: Substituting this optimal value in (34), we have

Z1

maxfÿc; ÿp v0 p

h r1 F I1 r1 1 ÿ dS1 dr1 g: 0

33 Theorem 2. Let: /2 I2 ; S2 Min w2 I2 ; S2 ; y2 : y2 I2

Then /2 I2 ; S2 is a convex function of I2 for a given S2 . Proof. From (23), /2 I2 ; S2 Min c y2 ÿ I2 E fvy2 r2 1 ÿ dS2 ÿ x2 r2 ;x2

y2 I2

px2 ÿ y2 ÿ r2 1 ÿ dS2 g E f/1 y2 r2 1 ÿ dS2 ÿ x2 ; r2 ;x2

1 ÿ dS2 1 ÿ r2 x2 g:

34

We would ®rst show that @ 2 /2 I2 ; S2 =@I22 0 in each region as de®ned in (31). Region A In this region, y2 I2c S2 Substituting this optimal value in (34), we have @/2 I2 ; S2 ÿc @I2

@ 2 /2 I2 ; S2 0 0: @I22

and

35

Region B In this region Q2 0 or y2 I2 . Substituting this optimal value in (34), we have @/2 I2 ; S2 ÿp v p @I2

Z1 h r2 F2 I2 r2 1 ÿ dS2 dr2 0

@/1 I1 ; S1 @I1 ; E r2 ;x2 @I2 @I1

36

where I1 I2 r2 1 ÿ dS2 ÿ x2 : Now @I1 =@I2 1. Hence @ 2 /2 I2 ; S2 v p @I22 E

r2 ;x2

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Z1 h r2 f2 I2 r2 1 ÿ dS2 dr2 0

@ 2 /1 I1 ; S1 @I1 : @I2 @I12

@/2 I2 ; S2 ÿc and @I2

@ 2 /2 I2 ; S2 00 @I22

38

Similar to Theorem 1, we can show that @/2 I2 ; S2 =@I2 is either continuous or an increasing function of I2 at the various boundaries. j Theorem 3. For n 2; 3; . . . ; N let wn In ; Sn ; yn be as de®ned in (23). Then for each n, wn In ; Sn ; yn is a convex function of yn . Proof. Proof is by induction. In Theorem 2, we showed that /2 I2 ; S2 is a convex function of I2 . Now let n 3. Then: (1) /nÿ1 Inÿ1 ; Snÿ1 is a convex function of Inÿ1 . Given Equations (22a) and (22b), Inÿ1 yn rn 1 ÿ dSn ÿ xn and Snÿ1 1 ÿ dSn 1 ÿ rn xn . Hence Ern ;xn f/nÿ1 Inÿ1 ; Snÿ1 g Ern ;xn f/nÿ1 yn rn 1 ÿ dSn ÿ xn ; 1 ÿ dSn 1 ÿ rn xn g is a con^ n ; Sn ; yn is convex with vex function of yn . Also w I respect to yn . Thus wn In ; Sn ; yn is a sum of convex functions and therefore is a convex function of yn . (2) Given the convexity of wn and given that yn In , the optimal yn will be given by a decision rule similar to condition (9). (3) Using the procedure of Theorem 2, we can show that /n In ; Sn as de®ned in (25) with n ÿ 1 replaced by n is a convex function of In for a given Sn . If n N stop. Else let n n 1 and repeat steps 1, 2, and 3 above. j Note that in order to numerically determine the critical values Inc Sn for period n, we would need to calculate and store in period n ÿ 1 values of @/nÿ1 Inÿ1 ; Snÿ1 =@Inÿ1 on an appropriate grid of Inÿ1 ; Snÿ1 . For example, to determine I3c S3 values in period 3, we need to compute @/2 I2 ; S2 =@I2 values on a suitable grid of I2 ; S2 in period 2. From Theorem 2, these partial derivatives are equal to ÿc in regions A and C of (31). In region B, the partial derivative is given by expression (36).

5. An example 37

We showed in Theorem 1 that /1 I1 ; S1 is a convex function of I1 . Hence @ 2 /1 I1 ; S1 =@I12 0 for all I1 . Hence both terms in (37) are non-negative.

Suppose c $3=unit, p $7=unit short, v0 $0:5=unit, v $0:5=unit, d 0:05, f x 1=20exp ÿx=20 and with h r 1= 0:5 ÿ 0 for r 2 0; 0:5 r1 r2 r For the ®nal period, the chart (corresponding to Fig. 1) for determining the optimal lot-size is shown in Fig. 2.

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Buchanan and Abad

Fig. 2. Optimal policy for the ®nal period.

The chart is developed by solving Equation (7) for different values of S using the software Mathcad. Note that S10 74. Hence for S1 > 74, the solution of condition (7), I1c S1 will be less than 0. Of course, we do not solve for these negative critical values because for such a S1 ; y max0; I. To illustrate the solution of a two period problem, suppose that I2 5 and S2 40. We use condition (29) to solve for I2c 40. Closed form expression (33) is used in (29) or (27) for @/1 I1 ; S1 =@I1 . We ®nd I2c 40 22:1. Hence Q2 22:1 ÿ 5 17:1. Now suppose that the initial period has elapsed and the realizations of random variables r2 and x2 lead to I1 ÿ10, S1 35. Then from Fig. 2, I1c 35 7:5 and hence Q1 7:5 ÿ ÿ10 17:5.

6. Conclusions We have derived the optimal inventory control policy for a periodic review returnable inventory system. There are two state variables in the model: net stock of containers in the hands of the manufacturer and the number of containers in the hands of the consumers. In the proposed model, the probability of a container being returned in the current period is assumed to be independent of the age (i.e., the time since the container was last issued) of the container. This is equivalent to assuming that the ®eld life (i.e., length of time the con-

tainer stays in the hands of the current customer) is exponentially distributed. The model can also be applied to situations where the probability of return is dependent upon the age provided that the age-mix of the containers in the ®eld from period to period is stable. The return variable r in such a case can be described by its conditional distribution given the agemix of the containers in the ®eld. In future research, models will be formulated to explicitly include the age of the container in the analysis.

Acknowledgement This research is partially funded by a grant from Arts Research Board of McMaster University and by NSERC Canada Grant OGP0001226.

References [1] Kelle, P. and Silver, E.A. (1989) Forecasting the returns of reusable containers. Journal of Operations Management, 8, 17±35. [2] Kelle, P. and Silver, E.A. (1989) Purchasing policy of new containers considering the random returns of previously issued containers. IIE Transactions, 21, 349±354. [3] Bookbinder, J.H. and Tan, Y. (1988) Strategies for the probabilistic lot-sizing problem with service-level constraints. Management Science, 34, 1096±1108.

Optimal policy for a periodic review returnable inventory system [4] Heyman, D.P. and Sobel, M.J.(1984) Stochastic Models in Operations Research 2 Vol. II: Stochastic Optimization, McGraw Hill, New York. [5] Ross, S.M. (1970) Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, CA.

Biographies David J. Buchanan holds as BSc (Physics), an MBA, and a Ph.D. (1988) from McMaster University. He has co-authored an article on a

1055

lost sales inventory model and an article on locating a noxious facility with respect to several polygonal regions using asymmetric distances. He is not currently involved in the ®eld of operations research. Prakash L. Abad is a Professor of Management Science in the DeGroote School of Business at McMaster University. Dr. Abad's research interests are in production and inventory systems, pricinginventory models, nonlinear programming, stochastic dynamic programming, and classi®cation. His work has appeared in Management Science, Decision Sciences, European Journal of Operations Research, Optimal Control and Applications, International Journal of Systems Science and Computers and Operations Research.

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