On continuous review stochastic (s, S) inventory systems with ordering delays

Computers ind. Engng Vol. 28, No. 4, pp. 763-771,1995

Pergamon

0360-8352(95)00015-1

ON CONTINUOUS

REVIEW

SYSTEMS

WITH

STOCHASTIC ORDERING

Copyright© 1995ElsevierScienceLtd Printed in GreatBritain.All rightsreserved 0360-8352/95 $9.50+0.00

(s, S ) I N V E N T O R Y DELAYS

H Y O - S E O N G LEE Department of Industrial Engineering, Kyung Hee University, Kiheung, Yongin-goon, Kyunggi-do, 449-900, Korea (Received 21 March 1995)

Abstract--ln this paper a continuous review (s, S) inventory system with ordering delays is considered. Demands for the item arrive accordingto a Poisson process. When the inventoryleveldrops to s, the order is triggered. However,due to an unavoidabledelay, the order is actually placed after a random amount of time. Once the order is placed, it is instantaneously delivered bringing the inventory level back to S. Under a cost structure which includes a setup cost, a holding cost and a penalty cost, an expression for the expected cost per unit time for given control values is obtained. Then some properties of the cost functions are developedto characterizethe optimal policy. Based on these properties, an efficientsearch procedure to find the optimal (s, S) policy is presented.

1. INTRODUCTION The (s, S) policy is known to be optimal in a variety of single item inventory systems. Under an (s, S) policy, an order is placed to increase the inventory position to S as soon as the inventory position drops to or below s. There has been considerable work on the (s, S) inventory policy. Sivazlian [1] considered a continuous review (s, S) policy with the assumptions of unit demand arrivals and general demand inter-arrival times. He showed that the steady state distribution on inventory position is uniform on the set (s + 1. . . . . S). Richards [2] extended this result to the case with random demand size. Stidham [3, 4] and Sahin [5, 6] demonstrated some properties of the cost functions which could be exploited to find an optimal policy. A number of algorithms have also been developed to find the optimal (s, S) policy. Archibald and Silver [7] presented a procedure to compute the optimal (s, S) policy for systems with constant lead times and compound Poisson demands. Federgruen and Zipkin [8] developed an algorithm which is based on adaptation of the general policy-iteration method for solving Markov decision problems. Recently, Zheng and Federgruen [9] devised an efficient algorithm to find an optimal (s, S) policy. In addition to the exact methods, heuristic methods have been proposed by a number of researchers, for example, Freeland and Porteus [10], Tijms and Groenevelt [11] and Srinivasan and Lee [12], to name but a few. All these models assume that the order is actually placed as soon as the inventory position drops down to or below s. There are some situations, however, where placing an order must be delayed by some unavoidable reason. Weiss [13] extended the previous (s, S) inventory models to the one with ordering delays in which the difference between the time the order should be placed and the time the order is actually placed is non-trivial. This delay could be due to bookkeeping or the setup time for a production process or to time required for vendor selection [13]. For the system with random ordering delays and instantaneous delivery lead times, Weiss obtained an expression for the expected cost per unit time for a given policy and showed through computational experiments that ordering delays could be costly, especially if they are unrecognized. In that paper, however, no special properties for the cost functions were proved, that would assist in the search for the optimum. Consequently, no search procedures specially designed for the system were developed. In this paper, we analyze the same (s, S) inventory system as the one considered by Weiss. However, the approach used to obtain an expression for the expected cost per unit time is different from that in Weiss. In addition, in this paper, we characterize the behavior of cost functions by 763

764

Hyo-Seong Lee

deriving a set of properties the cost functions possess. Then by exploiting these properties, we develop an extremely efficient search procedure to find the optimal policy. We now describe our (s, S) inventory system in detail. The demand for the item arrives according to a Poisson process with rate 2. Unfilled demand is lost. The delay time between the trigger time and the order time is a random variable D which follows an arbitrary distribution V(x) and the delivery lead time is zero. If a cycle is defined to be the time interval from the instant the inventory level is raised to S to the subsequent instant the inventory level is raised back to S, the operating characteristics of our (s, S) policy are described as follows: at the beginning epoch of a cycle, the inventory level is S. The inventory level is continuously monitored to determine whether the inventory level has reached a prespecified value s or not. At the instant the inventory level drops to s, the order is triggered. However, due to an unavoidable ordering delay, the order is actually placed after a random amount of time D. Once the order is placed (after the delay), the inventory level is instantaneously raised back to S, initiating another cycle in the (s, S) system. The costs associated with this inventory system are a setup cost K per each order, a holding cost h per item in inventory per unit time and a penalty cost p for each unfilled demand. Our objective is to find an (s, S) policy which minimizes the expected cost per unit time in the long run. In the following sections, we first derive an expression for the expected cost per unit time for given control values and following that we present an efficient procedure to find the optimal control values s and S. 2. DEVELOPMENT OF THE COST FUNCTION For analytical convenience, we set r = S - s and throughout the paper we will use (r, S) instead of (s, S) to describe a policy. Thus, the (r, S) policy represents the policy with S - r as a lower control value and S as an upper control value. If we denote by C(r, S) the expected holding and penalty costs incurred per cycle and by T(r, S) the expected length of a cycle, then from the renewal reward theorem [14], the expected cost per unit time G(r, S) is expressed as:

G(r, S) =

K + C(r, S) T(r, S)

(1)

We now show how the terms C(r, S) and T(r, S) are determined. The term C(r, S) consists of the following two costs: (i) the expected cost incurred from the beginning of the cycle until the instant the order is triggered; and (ii) the expected cost incurred during a delay time. The first cost is computed easily as h/2 Es= s-,+~J. Let Ek denote the expected cost incurred during a delay time initiated with k items in inventory. Then, since the inventory level at the beginning epoch of the delay time is S - r , the second cost is just Es-r. Thus, the term C(r, S) is expressed as:

C(r, S) = h r(2S - r + 1) + Es_ r.

(2)

The expected cycle length T(r, S) is obtained easily as r/2 + E(D). Therefore, from equations (1) and (2), the expected cost per unit time is given by:

2K + h r(2S - r + 1) + 2Es_r G(r, S) =

J.

(3)

r + 2E(D)

Now if we can compute Ek, we can obtain G(r, S) using equation (3). To obtain Ek, let d, denote the probability that n items are demanded during a delay time. Then Ek is obtained from the following lemma.

Lemma 2.1 Ek=Ek_~+h~k-p

z~ d,, n=k

for

k/>l,

(4a)

Continuous review inventorysystems

765

where ~k is computed from

~k=~k-l+'~

nmk

d,,

for

kt>l,

(4b)

with initial values

Eo = 2E(D)p, Go= O. P r o o f . Suppose the length of the delay time is t and n items are demanded during t. Let the expected cost incurred during this delay time which is initiated with k items in inventory be denoted by ~,,,,(k). B y conditioning on the number of demand arrivals and length of the delay time, Ek is expressed as:

Ek= f ; ~ ~;[.'7"=o e-;"(2t)"~b""(k)}

(5)

Note that given n demand arrivals during t, the joint distribution of these arrival epochs have the same distribution as the order statistics of n independent random variables uniformly distributed on [0, t] (see, for example, Ross [14], p. 37). Using this fact, ~b,,,(k) is obtained as n

t

all, ,(k) = ~ --'-7-7,(k -j)h, •

if n ~
j = 0 n -i.. l

k

t

j~=o-~-~(k -j)h +(n -k)p,

if n > k .

(6)

Substituting (6) into (5) and using the fact that d~ = S~ (e-;"(2t)O/n! d V(t) gives:

Ek=f;{~e-%~t)"~ t k ~, e-~t(2t)" U ~ t .=o ~ j~--'o-~-~( --J)h +,,J'ff+l .n! Lj~=o~-+Y ( k - j ) h +(n -k)pJtdV(t) = fo g )'X~ e~t(2t)~+' x~

(k-J)h +

+ fo°~ ~ -e ~'t(2tl"(n -k)p

~e-~'t(2t)"+'~ T'~. I

dV(t)

n=O JO

=

~., n=0

d.+,

e-~t(2t)"+'~ ( k - j )

} dV(t)

j_o--5 --h

dV(t)

n!

n=k+l

= ~

.=,+,

min~'n}(k --j) j=0

~

min~" n , ( k - j ) --h j=O 2

h+

+

~

n=k + l JO

--dV(t)(n n[

d.(n - k)p.

-k)p (7)

n=k+l

From (7), we can obtain: Ek = Ek_, + ~ ~d~+,=o_ min{(n + l).k} - .=,

d~p.

(8)

Let: oo

~* = 2 .=~od.+, min{(n + 1). k}. Then equation (8) is expressed as:

Ek = Ek_ l + h~k - p ~ d., n=k

where ¢, is obtained from (9) as:

~k = ~,- l +

n=k

(9)

766

Hyo-Seong Lee

The initial values of Ek and ~k can be obtained from equations (7) and (9) as:

Eo=p ~, dnn =2E(D)p,

~0=0.

n=l

3. THE OPTIMAL CONTROL VALUES To find the optimal policy (r*, S*), G(r, S) must be minimized over the 2-D integer parameter space. This search could be performed quite efficiently if some properties of the cost functions are exploited. Let us denote the optimal S-value for a given r by S*(r). Then, the following properties can be proved which enable us to devise an efficient search procedure. Property l - - F o r a given value of r, C(r, S) is convex in S. Proof. If we define

AC(r, S)= C(r, S ) - C(r, S - 1), from equations (2) and (4) we obtain: hr

AC(r, S) = ~ + h~s_r-p

4"

(10)

j=S-r

To prove that C(r, S) is convex in S, it is sufficient to show the following:

AC(r,S)-AC(r,S-1)=~

1-

~

4 +pds_~ r>~O.

•

y=0

Property 2 - - I f S*(r) = k, then S*(r + 1) ~
hr C(r,k + l ) - C ( r , k ) = ~ + h ~ k + l _ r -

p

~, j=k+l

d:.>~O. r

To prove Property 2, since C(r, k) is convex in k for a given r, it is enough to show that C(r + 1, k + 2)/> C(r + 1, k + 1), which can be done using equation (10) as follows:

C(r+l,k+2)-C(r+l,k+l)=~(r+l)+h~k+~

~-p

4 j=k+l-r

= C(r,k-I- 1 ) - C ( r , k ) + ~ > O.

•

Property 3--r + 1 ~< S*(r + 1) ~< S*(r) + 1. Proof. Property 3 is a direct result of Property 2 as well as the fact that s is always non-negative.

Property 4--S*(r) = r if and only if r/> (1 - do)(2p/h - 1). Proof. From the convexity of C(r, k) as well as the fact that S*(r) >1r, we have that S*(r) = r if and only if C(r, r + 1) i> C(r, r). From equations (2) and (4), C(r, r + 1) - C(r, r) = hr/2 + (1 - do)(h/2 - p ) >1O, thereby proving Property 4. • Property 4 has the following intuitive interpretation: if we compare the expected cost incurred per cycle of the policy (r, r) with that of the policy (r, r + 1), the policy (r, r) incurs more expected penalty cost than the policy (r, r + 1) by (1 - do)p while the policy (r, r + 1) incurs more expected holding cost than the policy (r, r) by h/2(r + 1 - do). Note that the difference of the expected penalty cost between two policies is fixed whatever the value of r is. However, the difference of the expected holding cost increases as r increases. Therefore, there exists a certain threshold value over which the (r, r) policy is more economical compared to the (r, r + 1) policy. Let the policy in which the inventory system is not operated at all be called the (0, 0) policy. Therefore, in the (0, 0) policy, the inventory level is always zero. Note that the expected cost incurred per unit time when the (0, 0) policy is used is 2p. The following property gives a sufficient condition for the (0, 0) policy to be an optimal policy.

Continuous review inventory systems Property

767

5 - - I f 2p ~< h, then the (0, 0) policy is an optimal policy.

Proof. If 2p ~< h, then (1 - do)(2p/h - l) ~< 0. Thus, from Property 4, S*(r) = r for all r >1 1. N o w we can easily check that if 2p ~< h, for all r t> 1, the following relationship holds: 2K + h (r 2 + r) + 22E(D)p a~

G[r, S*(r)] = G(r, r) =

> ).p = G(O, 0).

r + 2E(D)

•

Property 6---If 2,o > h, then: (a) G[r, S*(r)] is convex in r in the range r >/(1 - do)(2p/h - 1). (b) Let - 2 E ( O ) h + x/22E(D)2h 2 - 2h x X*--

where h Z = -~ :rE(D) - )flE(D)p - 2K. Let r ° denote the optimal value of r in the range r/> (1 - do)(2p/h - 1). Then r ° is obtained as follows: if x* ~> (1 - do)(2p/h - 1), then r ° is one of the neighboring integers of x*; if x * < ( 1 - d 0 ) ( 2 p / h - 1 ) , then r ° is the smallest integer r in the range r ~> (1 -- do)(2p/h - 1). Proof. (a) Since S*(r) = r in the range r >~ (1 - do)O.p/h - 1), we only need to show that G(r, r) is convex with respect to r in this range. To this end let us treat the control variable r as a continuous variable and call it x. We make use of the fact that if G(x, x) is convex with respect to x, then G(r, r) is also convex with respect to r. By substituting x for r in equations (3) and (4), we have h 2 2K + ~ (x + x) + 22E(O)p G(x,x) =

x + 2E(D)

Using the fact that 22E(D)2h - 2~ is positive if 2p > h, we can verify that G(x, x) is convex with respect to x by dZG(x,x) dx 2

-

22E(D)2h - 2~ >0. {x + 2E(D)} 3

(b) Let x* denote the optimal value of x which minimizes G(x, x). Then from the convexity of G(x, x), x* is a solution of the equation, dG(x, x) dx

--

h x2 + 2 E ( D ) h x + X 2 ~ {x + 2E(D)} 2

0 .

Using the fact that 22E(D)2h 2 - 2hz > 0 if 2p > h, we obtain the solution of the equation as - 2 E ( D ) h + x/2ZE(D)2h 2 - 2h• x*-

h

This fact together with the convexity of G(x, x ) leads to Property 6b. •. These properties can be used effectively in devising a search procedure. When 2p ~< h, the (0, 0) policy is optimal from Property 5. On the other hand, if 2p > h, we must find the optimal policy (r*, S*) in the range r >/1. In order to find (r% S*), we should be able to find S*(r) for a given value of r, which can be done effectively using Properties 1-4 as follows: when r~>(l-do)(2p/h-1), S*(r) is obtained directly as S * ( r ) = r from Property 4. When r < (1 - do)(2p/h - 1), Property 1 can be used in searching S*(r) for a given value of r. In this case, once S*(r) is obtained for some r, we can find S*(r) readily for a different value of r using Properties 2 and 3. Property 6 restricts considerably the range of r where the search should be performed, that is, owing to Property 6, the enumerative search needs to be done only for the range

768

Hyo-Seong Lee

r <(1-do)(2p/h1). In addition, our extensive computational experience suggests that S*(r + 1) >i S*(r) and G[r, S*(r)] is unimodal in r in the range r < (1 - do)(2p/h - 1) although we could not prove these [for r I> (1 - do)(2p/h - 1), both of these properties hold by Property 6]. Owing to this observation, the enumerative search is not needed even in the range r < (1 - d o ) O . p / h - 1) for practical use. N o w we are in a position to describe our search procedure. Suppose 2p > h. To find the optimal control values (r*, S*), our algorithm makes use of Properties 1-6 as well as our observations obtained from the computational experience. Our search procedure starts by computing x*. I f x* is equal to or greater than (1 - do)(2p/h - 1), then we can find the optimal policy (r*, S*) very easily since r* is one of the neighboring integers of x* and S* is equal to r*. On the other hand, if x* is less than (1 - do)(2p/h - 1), this implies that r* does not exceed (1 - do)(2p/h - 1). When x* < (1 - do)(2p/h - 1), in all our computational tests, r* is observed to be consistently less than x*. Hence, as an initial value for r in this case, we choose the largest integer not exceeding x* and for the next search point we choose r which is smaller than the initial search point. Once these values for r are chosen, for each of these values we compute S*(r) as well as G[r, S*(r)]. The search is continued over r using a local descent search method adjusting r in the direction of descent. The procedure stops when any local change in r will increase G(r, S). When (r*, S*) is found and G(r*, S*) is computed, we should compare G(r*, S*) with 2,o. If G(r*, S*) is less than or equal to 2p, the optimal policy is (r*, S*). Otherwise, the optimal policy is (0, 0). Most of the computational effort in this algorithm is spent on calculating Ek. However, due to the recursive nature of Ek, if some Ek has been computed, very little calculation is added to obtain a new Ek if we use the information obtained earlier. Moreover, it should be noted that once S*(r) has been found for some r, from Property 2, no new Ek is needed to be computed to find S*(n) for n ~>r. If a certain condition is satisfied, we can develop more strong properties of the cost functions as follows: Property 7 - - S u p p o s e dk_ ~>>-(1 -- h/2p)dk for all k 1> 1. I f S*(n + 1) > S*(n) for some n, then: (a) S*(r + 1) is either S*(r) or S * ( r ) + 1 for r/> n. (b) G[r, S*(r)] is unimodal in r in the range r >/n. Proof. The p r o o f is given in the Appendix. Property 7 states that if dk_ ~>1[1 -- (h/2p)]dk for all k >I 1, once S*(r) increases, then it never decreases. Furthermore, in the range where S*(r) is non-decreasing, G[r, S*(r)] is unimodal with respect to r. The function, S*(r), usually begins to increase when r is very small, namely, r = 1, 2 or 3. Therefore, if the condition dk_ ~>~ (1 --h/2p)dk, k i> 1 is satisfied, Property 6 not only reduces the search time considerably but also guarantees that the policy found by the algorithm is a global optimum. Remark. Suppose the delay time D follows an exponential distribution. Then it can be easily shown that dk l = dk/(1 -- do) > (1 - h/2p)dk. For the constant delay time, if (2 - h/p) <~ 1/E(D), the condition is also satisfied. Thus Property 7 applies to both of these cases.

4. NUMERICAL EXAMPLES In order to verify the efficiency of our algorithm and to check the unimodality of G[r, S*(r)] we made extensive numerical tests. Some of these examples are presented below. In the first example, we assume that the delay time is constant, i.e. D = 10 with probability 1. Other parameter values are given as 2 = 0.3, K = 100, h = 1 and p = 50. In the second example, we consider a case where an arrival rate is very high, i.e. 2 = 5. The delay time is assumed to follow an exponential distribution with mean 5. Other parameter values are given as K = 100, h = 1 and p = 10. For the last example, we demonstrate a case where a setup cost is very high, i.e. K = 500. The delay time is assumed to follow a uniform distribution in the range [6, 10]. Other parameter values are given as 2 = 1, h = 1 and p = 30. Note that in all these examples, 2p is greater than h. The results of the policy comparisons for these examples are presented in Tables 1-3. In each table, to show the behavior of the cost functions, the values of s*(r), S*(r) and G[r, S*(r)] are

769

C o n t i n u o u s r e v i e w i n v e n t o r y systems Table I. Result of Example 1

). = 0 . 3 , K = 100, h = I, p = 50, constant delay time (D = 10)

r

s*(r)

S*(r)

G[r, S*(r)l

0 1 2 3 4 5 6 *7 8 9 10

0 4 4 3 3 3 3 2 2 2 2

0 5 6 6 7 8 9 9 10 11 12

15.000 11.875 10.700 10.019 9.587 9.389 9.346 9.344 9.403 9.536 9.726

*Indicates the optimal policy

Table 2. Result of Example 2 2 = 5, K = 100, h = l , p = 10, exponential delay time with mean 5

r

s*(r)

S*(r)

G[r, S*(r )]

0 28 29 30 31 32 *33 34 35 36 37

0 8 8 7 7 6 6 6 5 4 3

0 36 37 37 38 38 39 40 40 40 40

50.000 39.152 39.112 39.088 39.069 39.067 39.066 39.082 39.102 39.144 39.208

*Indicates the optimal policy.

Table 3. Result of Example 3 2 = l , K = 5 0 0 , h = 1,p = 3 0 ,

uniform delay time on [6, 10]

r

s*(r)

S*(r)

G[r, S*(r)]

*0 26 27 28 29 30 31 32 33 34 35

0 3 2 1 0 0 0 0 0 0 0

0 29 29 29 29 30 31 32 33 34 35

30.000 31.933 31.862 31.806 31.757 31.711 31.692 31.700 31.732 31.786 31.861

*Indicates the optimal policy.

given for each value of r, although these values need not be computed for every r in an actual search procedure. In the first example, (1 - d o ) ( 2 p / h - 1) = 13.303 and x* = 9.490. Thus, S * ( r ) = r for r >t 14. Since x* < (1 - d o ) ( 2 p / h - 1), a search to find the optimum is performed in the range r ~< 13. As an initial point for r, from the value of x*, we choose r = 9. The optimal policy for this example obtained by the algorithm in the range r/> 1 is the policy (7, 9) and the expected cost per unit time is G(7, 9) = 9.344. Since G(7, 9) < G(0, 0) = 15.000, we can conclude that the optimal policy is actually the policy (7,9). In the second example, ( 1 - d o ) ( 2 p / h - 1)=47.115 and x* = 39.031. Therefore, a search is performed in the range r ~<47, with an initial point r = 39. The optimal policy for this example found by the algorithm in the range r/> 1 is the policy (33, 39) and the expected cost per unit time is G(33, 39) = 39.066. Since G(33, 39) < G(0, 0) = 30.000, we can conclude that the optimal policy is the policy (33, 39). Note that the policy (33, 39) is guaranteed to be a global optimum since the delay time follows an exponential distribution. The optimal policy for the third example can be obtained very easily. In this example, (1 -do)(2p/h - 1) = 28.982 and x* = 31.192. Thus, from Property 6, the optimal r-value is one of the neighboring integers of x*, which turns out to be r * = 31. In this problem, however, it is checked that G(31, 31) = 31.692 > 30.000 = G(0, 0). Therefore, the optimal policy in this example

770

Hyo-Seong Lee

is the policy (0, 0). In more than 30% of the problems we have tested, we have observed that the condition x*>~ ( 1 - d o ) ( A p / h - 1) is satisfied. This implies that Property 6 plays an extremely important role in finding the optimal policy. Besides, as shown in the tables, in all the examples we have considered, S*(r) is nondecreasing and G[r, S*(r)] is unimodal in the range r >i 1 as we have expected. Although only three distributions for the delay time are demonstrated in the examples, many other distributions can be implemented easily because d: can be expressed in closed form for many distributions of practical interest. 5. CONCLUSIONS

We have analyzed a continuous review (s, S) inventory system with ordering delays considered by Weiss. Using a different approach from that in Weiss, we obtained an expression for the expected cost per unit time for given control values. We derived a set of properties of the cost functions and characterized the behavior of these cost functions as well as the optimal policy. Based on these properties, we developed an efficient search procedure to find the optimal (s, S) policy. REFERENCES 1. B. D. Sivazlian. A continuous review (s, S) inventory system with arbitrary interarrival distribution between unit demand. Opns. Res. 22, 65-71 (1974). 2. F. R. Richards. Comments on the distribution of inventory position in a continuous-review (s, S) inventory system. Opns. Res. 23, 366-371 (1975). 3. S. Stidham. Cost models for stochastic clearing systems. Opns. Res. 25, 100-127 (1977). 4. S. Stidham. Clearing systems and (s, S) inventory systems with nonlinear costs and positive lead times. Opns. Res. 34, 276-280 (1986). 5. I. Sahin. On the objective function behavior in (s, S) inventory models. Opns. Res. 30, 709-725 (1982). 6. I. Sahin. Optimality conditions for regenerative inventory systems under batch demands. Appl. Stochastic Models Data Anal 4, 173-183 (1988). 7. B. Archibald and E. Silver. (s, S) policies under continuous review and discrete compound Poisson demands. Mgmt. Sci. 24, 899-908 (1978). 8. A. Federgruen and P. Zipkin. An efficient algorithm for computing optimal (s, S) policies. Opns. Res. 34, 1268-1285 (1984). 9. Y. S. Zheng and A. Federgruen. Finding optimal (s, S) policies is about as simple as evaluating a single policy. Opns. Res. 39, 654-665 (1991). 10. J. Frceland and E. Porteus. Evaluating the effectiveness of a new method for computing approximately optimal (s, S) inventory policies. Opns. Res. 28, 353-364 (1980). 1I. J. Tijms and H. Groenevelt. Approximations for (s, S) inventory systems with stochastic lead times and a service level constraint. Eur. J. Opnl. Res. 17, 175-190 (1984). 12. M. M. Srinivasan and H. S. Lee. Random review production/inventory systems with compound Poisson demands and arbitrary processing times. Mgmt Sci. 37, 813-833 (1991). 13. H. J. Weiss. Sensitivity of continuous review stochastic (s, S) inventory systems to ordering delays. Eur. J. Opnl. Res. 36, 174-179 (1988). 14. S. M. Ross. Stochastic Processes. Wiley, New York (1983).

APPENDIX

Proof of Property 7 Pro~rty 7--Suppose dk_ i >/(! - h/Ap)dk for k I> I. If S*(n + 1) > S*(n), then: (a) S*(r + 1) is either S*(r) or S*(r) + 1, for r >t n. (b) G[r, S*(r)] is unimodal in r for r I> n.

Proof. (a) From equation (4): h ~

( E . - E , , _ , ) - ( E , , _ , - E._2) =~j~=

(AI)

If dk_ l >/(1 --h/Ap)dk for k/> I, it can be shown using equation (A1) that:

(E._I-E._2)-(En_2-En_3)>.(E~-E._I)-(E._I-E._2), for n~>3. (a2) Suppose S*(n) ~ k and S*(n + 1) -- k + 1. Note that if S*(n + 1) > S*(n), from Property 2, S*(n + !) must be k + 1. Then from C (n, k + 1) >I C (n, k): h

~n + Ek+j_n-Ek_n>>.O. Similarly, from C(n + l, k + 1) ~
(n + I)+ Ek_~--Ek_._ I ~<0.

(A3)

(A4)

C o n t i n u o u s review inventory systems

771

Inequalities (A3) a n d (A4) imply that: h

(Ek+,_o- E,_n)- (E,_~- ~,_n_ A ~>~.

(AS)

T o prove (a), it is e n o u g h to show t h a t S*(r + I)/> S*(r) for r / > n. Suppose, on the contrary, that S*(r)= m a n d S*(r + 1 ) < m for some r > n . Then, from C(r,m)<~ C ( r , m - 1) and C(r + 1, m) >1C(r + 1, m - 1), the following inequalities hold: h

~ r + E r a _ , - E~_,_l ~<0, h

(A6)

(r + 1 ) + E , , _ , _ I --Em_,_2>~O.

(A7)

Inequalities (A6) and (A7) imply that: (Em-, -- Era-r-I)

h

--

( E m - r - I -- Era-r-2) ~ "~.

N o t e that, from Property 2, if S * ( n ) = k , then S*(r)<~ k + r - n , which is, m contradicts inequalities (A2) and (A5). Therefore, S*(r + 1) >i S*(r), for r / > n. (b) To prove (b), we first need to show that:

(g8)

r
C[r + 2, S*(r + 2)] - C[r + I, S*(r + 1)] I> C[r + 1, S*(r + 1)] - C[r, S*(r)],

for

T h u s , inequality (A8)

r / > n.

(A9)

Let S*(r) = m and we consider 4 cases: (i) S*(r + l ) = m + l, S*(r + 2 ) = m + 2; (ii) S*(r + l ) = m + l, S*(r + 2 ) = m + l; (iii) S*(r + 1) = m, S*(r + 2) = m + 1; (iv) S*(r + 1) = m, S*(r + 2) = m. Note, from Property 7a, that these 4 cases, collectively consider all possibilities. Inequality (A9) can be proved for each case as follows: (i) {C(r + 2, m + 2 ) - C ( r + l , m + l ) } - { C ( r + l , m + l ) - C ( r , m ) } = h / 2 >O. (ii) {C(r + 2, m + l ) - C ( r + l , m + l ) } - { C ( r + l , m + l ) - C ( r , m ) } = C ( r + I , m ) - C ( r + l , m + l)l>O, (iii) {C(r + 2, m + I) - C(r + 1, m ) } - {C(r + 1, m) - C(r, m ) } = C(r + 1, m + 1) - C(r + 1, m) >! O. (iv) U s i n g the fact that (E,. , - E . . . . ~) - (E,. _,_ z - E . , _ , _ 2)/> h/2, which is obtained in the p r o o f of Property 7a, we can show that:

{C(r + 2, m ) - C [ r

+ 1,m)}-

{C[r + l , m ) - C ( r , m ) } = ( E m - , - E m _ , - , ) - ( E . , - r - ~ - E , ~ _ , _ 2 ) - h / 2

>10.

T o prove the unimodality of G[r, S*(r)], we only need to show that:

K + C[r, S*(r)] K + C[r + 1, S*(r + 1)] K + C[r + 1, S*(r + 1)1 K + C[r + 2, S*(r + 2)1 < , then < . (AI0) T(r) T(r + 1) T(r + I) T(r + 2) Since T(r + 1) - T(r) = T(r + 2) -- T(r + 1) = 1/2, from inequality (A9), we have: if

C[r + 2, S*(r + 2)1 - C[r + 1, S*(r + 1)] t> C[r + 1, S*(r + 1)] - C[r, S*(r)] (AI 1) T(r + 2) - T(r + 1) T(r + 1) - T(r) For positive values of a, b, c and d with a/b > c/d it can be verified that if a > c and b > d, then (a - c)/(b - d) > c/d. If we apply this algebraic fact to the given condition {K + C[r + I, S*(r + l)]}/T(r + 1) > {K + C[r, S*(r)]}/T(r), we obtain

C[r + 1, S*(r + 1 ) 1 - C[r, S*(r)l K + C[r, S*(r)] > T(r + 1 ) - T(r) T(r)

(A12)

C o m b i n i n g (A11) and (A12), we have:

C[r + 2, S*(r + 2)] - C[r + I, S*(r + 1)] C[r + 1, S*(r + 1)] - C[r, S*(r)] K + C(r, S*(r)] /> > (AI3) T(r + 2) - T(r + 1) T(r + 1) - T(r) T(r) W e n o w use a n o t h e r algebraic fact that if a, b, c, d, e, f a r e all positive values, then (a + c + e)/(b + d + f ) > (c + e)/(d + f ) holds if a/b >~c/d > elf. A p p l y i n g this algebraic fact to inequality (A13), we finally obtain: K + C [ r + 2 , S*(r+2)] K + C [ r + I , S * ( r + I ) ] > • T(r + 2) T(r + 1)