Sensitivity of continuous review stochastic (s, S) inventory systems to ordering delays

174

European Journal of Operational Research 36 (1988) 174-179 North-Holland

Theory and Methodology

Sensitivity of continuous review stochastic (s,S) inventory systems to ordering delays H o w a r d J. W E I S S

Department of Management, School of Business and Management, Temple University, Philadelphia, PA 19122, USA

Abstract: This paper examines continuous review stochastic (s, S) inventory systems with ordering delays. That is, systems where the difference between the time the order should be placed and the time the order is actually placed is non-trivial. The traditional inventory ordering, holding and penalty costs are included and the average cost for an (s, S) policy is developed and examined. Computational results are presented for two cases. In the first case, the manager is aware of the delay and uses the policy that minimizes all costs. We present the increase in cost due to having a delay. In the second case the manager is unaware of the delay and uses the (integer) square root formula. We present the increase in cost due to using the square root formula when it is inappropriate and in fact our computational results indicate that there may very well be a large increase in cost due to being unaware of the ordering delay. Keywords: Inventory, stochastic

I. Introduction In this paper we analyze a stochastic continuous review (s, S) inventory model with ordering delays. The model essentially extends those presented by Sivazlian [7], Richards [4] and Deuermeyer [3]. In the previous models units are demanded one at a time and the inter-demand times form a renewal process. The system is operated under an (s, S ) policy. That is, when the inventory level reaches s an order for S-s units is placed and immediately delivered. Shortages are not permitted. Sivazlian [7] has shown that the steady state inventory position is distributed uniformly on the set of integers between s + 1 and S and uses this information to determine the long run average expected cost. Richards [4] offers an extension to the steady-state distribution results. Received June 1986; revised July 1987

Deuermeyer [3] computes the long run average cost using a sample path approach rather than the steady state distribution approach. Other related models have been presented. Archibald and Silver [1] compute the optimal (s, S) policy for systems with exponential inter-demand times but general (rather than unit) demands. Sahin [6] has presented more general results in that he presents both the time dependent and steady state distributions of the inventory level and the inventory position for cases with general inter-demand times, general demands and constant lead times operated under an (s, S) policy. Stidham [8] has considered cost models for general stochastic clearing systems and their applications to similar continous review inventory systems but his work is mostly directed towards queueing systems and he warns that " t h e reader should be wary of trying indiscriminately to apply our results to inventory systems".

0377-2217/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)

H.J. Weiss / Continuous review stochastic (s, S) inventory systems with ordering delays

In this paper we consider the cases where we would like to order when the inventory level falls to s (the trigger time) but unavoidably must delay placing the order until some (random) time later. This delay could be due to bookkeeping or the set-up time for a production process or to time required for vendor selection. In other words, in theory we use an (s, S) policy but in practice there may be a delay between the trigger time and the order time. At the time that the order is actually placed (after the delay), delivery is instantaneous and the stock is completely replenished. That is, there are S units in stock. Notice that the delay time is not a lead time since the inventory can be fully re-stocked. Due to the delay time it is necessary to include either shortages or lost sales as they occur with a positive probability. Using the regenerative approach of Deuermeyer [2] we find the long run average cost for the lost sales model. In Section 2 of the paper we present the model and develop the long run average cost function. In Section 3 we examine the properties of the cost and computational results are presented indicating the effects of the delay in ordering. Our computational results are highly significant. We show that basing the ordering policy on the assumption that there is no delay can be very costly if, in fact, there is a delay.

2. D e v e l o p m e n t

of the cost function

The demand process is a Poisson process as defined in Cinlar [2] (as opposed to the more general renewal demand processes in [3,4,6] and [7]). Let N(t) represent the number of demands in [0, t], let ~ denote the time of the j-th demand, T0 = 0 and define X j = T j - T j _ 1. The ~ ' s are independent identically distributed exponential random variables with mean #. All demands are for one unit. Without loss of generality the inventory level is initially S > 0. The inventory level is decreased by the unit demands until the level reaches s, 0 ~
175

lnventor~ Level S

:

:

i L t m

i

'---,

s-I-

I t

s*l 5

--

s-|-

:

:

i

i

I

%

I :

I--i

[

: •

:

i

I :

--I l

6

:

i

i

i7

YI*I)I

.-,

I

v2,07

line Figure 1. A realization of the inventory process with delays

tributed random variables with the common cumulative distribution function B(t). The costs associated with the inventory system are a set-up cost per order K > 0, a holding cost rate h > 0 per item in inventory per unit time and a penalty cost p for each unsatisfied demand. Items demanded while the inventory level is zero are lost sales. We are interested in determining the long run average expected cost of any (s, S) policy. We do this using renewal theory since the trigger points Y~, Y2.... are renewal times for the inventory process. From standard renewal results (for example, see Ross [5]) we have that the long run expected average cost is the expected cost incurred between Y~ and Y2 divided by the expected length of time between Y1 and Y2 which we express as

G(s, S ) = E [ C ( s , S ) I / E [ T ( s , S ) ] . We begin by computing the expected cost per cycle. The cost C(s, S) is the sum of the costs incurred during [Y1, YI +D1] and [Yl +D1, Y2]. We first compute the cost incurred during [Ya, Y1 + D1]. As the Dj's are independent identically distributed random variables we at times drop the subscript. At the time Ya there are exactly s units on hand. For any item j, j = 1, 2 . . . . . s, the length of time that it is held in stock during [Y~, Y~ + D] is given by min(Tj, D). Without loss of generality we assume that the demand process begins at YILet qn(t) be the probability that there are n demands in [0, t]. That is, q,(t) = e- t/~(t/~t)~/n!

H.J. Weiss / Continuous review stochastic (s, S) inventory systems with ordering delays

176

We now must compute the holding costs incurred during [Y1 + D1, II2]. From Figure 1 it is clear that S - j + 1 units are held between each of the demands. Renumbering the demands so that T1 represents the first demand after Y1 + Da we have that the expected holding cost during [Y1 + D1, Y2] is given by

Conditioning on the delay time D we have that

E[minIT,, D] : ffE[min( , D)ID:,] _ooJ-1 ~'-_oe [ t l N ( t ) = n '

=J0

D= tlq"(t) dB(t)

S--s

h E (S-j+I)E[Tj-Tj-1]" j=l

n=j

The total holding cost during the period [Y1 +

Xq,(t) dB(t)

Da, ]I2] is thus given by

__e~ J - 1

h(S - s)(S + s + 1)/*/2.

= £[Tj.I - Jo n~-'=E o [ T j - t l N ( t ) = n ' D=,]

Finally we compute

Xq.(t) dB(t)

E[ T( S, s)] = E[ Y2 - I11] = E[ D ] + E[ Ts_s] = E [ D ] + (S-s)~*. (4)

j--1

~j~ - ~_~ (j-- n)/*l~qn(t)--

dB(t).

~O

n=0

The last step follows because from the conditioning there are n < j demands through time t which means that there are j - n demands remaining to get to Tj. The expected time of each of these is/,. If we denote by P, the probability of exactly n demands during the delay time, That is, P, = f~q,(t) dB(t), then j-1

Putting (1), (2) and (3) together, including the order cost K and dividing by (4) yields the following: The expected long run average cost is given by

G(s, S) = [K + hS(S + 1)/,/2 - h# L

(s - n)(s

Thus the expected holding cost during [Y1, Y1 + D] is given by

+p °=o(s-.)Po]

h

/{E[D]

(j n=0 s

+

(5)

Notice that if there is no delay (E[D] = 0) then Po = 1 and P j = 0 for j = 1, 2, 3 . . . . . We denote the cost for this case of no delay as GO and have that

j-1

= hs(s + 1)/,/2 - h# ~, ~_, ( j - n)P, j=l

1)P./2

+p(E[D]/#-s)

n=O

j=l

- n +

n=0

E[min(Tj, D)] = j / , - ~_, ( j - n ) / , P , .

j/, - / ,

(3)

n=0

= hs(s + 1)/*/2

G0(s, S) = { K + h S ( S + 1)/,/2 - h / , s ( s + 1)/2}

s

-h/, ~ (s - n)(s - n + 1 ) P , / 2 .

(1)

~{(S-s)~,}.

n=0

The second type of cost incurred during [Y1, Y1 + D1] is the lost sales cost. By definition this is given as

p ~

(n-s)P n

n=s+l

=p(EID]//,-s) +p L (s- n)P.. n=O

(2)

This, of course, agrees with the cost presented by Sivazlian who also shows that the optimal trigger is s = 0 and the optimal order quantity, which we denote by So*, is given by an integer square root formula. That is, So* satisfies

So*( So* - 1) ,%<2K/#h and S~'(So* + 1) >~2K//*h.

(6)

H.J. Weiss / Continuous review stochastic (s, S) inventory systems with ordering delays

We will return to both G O(0, S) and So* when we perform our computational results in the next section. N o w that the cost function is known we wish to address one of two questions. In certain cases, the inventory manager will know that the delay exists. Hence the interest is in finding the values of S and s that minimize G(s, S). In other cases the inventory manager will prescribe a (0, S) policy based on a traditional (non-delay) inventory model as presented by SO*. In these cases our interest is in determining the extra costs incurred due to the unrecognized delay.

3. Cost function behavior and computational resuits In this section we will present our computational results. Prior to performing the computations it is useful to analyze the behavior of the cost function for the purpose of optimization. In order to have a better form for the general cost function G(s, S) multiply both the numerator and denominator in equation (5) by 1//~ and extract h/2 from the numerator yielding.

G(s, S) = ( h/2)]2K/l~h + S( S + 1)

- ~ P.(s-n)(s-n+l) n=O

+ 2p/l~h(E[D]/Ix - s) + (2P/Ith) ,=0 ~ P , ( s - n)] /[ E[ D ]/I~ + ( S - s ) ] . It is possible to show that the cost function is partially well behaved. That is, G(s, S) is unimodal in S for fixed s if s is at least as large as the expected number of demands during the delay time. Unfortunately, if the holding cost is low and the penalty for lost sales is high s* will be less than E[D]/I~. Hence the partial good behavior of G(s, S) will not help us to find its minimum and therefore the computational optimization must be a complete search over pairs of integers for s and S. We next consider two special cases: exponentially distributed delay times and constant delay

177

times and constant delay times. Rather than examine G(S, s) as a function of the five parameters K, h, p, D and/~ we can reduce the number of parameters to three. That is, if service times are distributed exponentially or are constant, then the optimal policy (s*, S * ) is a function of the three variables K/l~h, p/l~h and E[D]/t~. Case 1. Exponential service times. For exponential service times we have Pn = (1/( E[ D]/I~ + 1))((E[D]/I~)/(E[D] + I~ + 1))". Substitution into (7) yields the fact that G(s, S) is a function of S, s, K/l~h, p/tzh and E[D]/I~ which yields the desired result. Case 2. Constant service times. For constant service times we have that P, = e (E[D]/~) (E [D ]/tt)"/n ! and substitution into (7) yields the desired result. One more result is very useful for our computational analysis. We wish to be sure that in the analysis we consider only cases where it is actually optimal to place an order. That is, we want to avoid considering cases where the long run average cost of never ordering, P/t~, is less than the optimal ordering cost, G(s*, S*). A sufficient condition for ordering to be optimal is that 2p/l~h > 2 + 2¢1 + 2K/llh. (A proof appears in the appendix). We now can begin our computational results. We compute the effects of the delay for two different situations. In the first case we presume that the inventory manager is aware of the delay and chooses the (s, S) policy that minimize the delay cost. We show the effect of the delay by presenting the ratio of the optimal cost with delay versus the optimal cost when there is no delay. The second case is using (actually misusing) the 'optimal' (0, S * ) policy. The actual cost is G(0, So* ) (although the manager expects a cost given by G0(0, So*).) We present the ratio of G(0, SO*) versus the optimal cost, G(s*, S* ). In Table 1 our computational results are presented for both the case where the delay is constant and the case where the delay is exponential. As S* and s* are functions of 2K/l~h, E[D]/I~ and 2p/l~h we have allowed these three parameters to vary. We have set 2K/l~h = 2,29 and 99; E[D]/tt = 1, 5 and 10 and 2p/l~h = 2(1 + ¢1 + 2K/l~h ) * j, j = 1, 2 and 3. The first parameter 2K/l~h was set so that S o would vary from 2 to 5 to 10 (see (6) in Section 2) and also so that 2p/l~h would be

H.J. Weiss/ Continuousreviewstochastic(s, S) inventorysystemswithorderingdelays

178

Table 1 Optimal and suboptimal costs and policies 2K

E [D ]

~,h

~,

2p

No delay

Constant delay

~,h S¢" Go(O, a(o, s~) s~

3 3 3 3 3 3 3 3 3

1 1 1 6 6 6 11 11 11

6 12 18 6 12 18 6 12 18

2 2 2 2 2 2 2 2 2

4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.50

24 24 24 24 24 24 24 24 24

1 1 1 6 6 6 11 11 11

12 24 36 12 24 36 12 24 36

5 5 5 5 5 5 5 5 5

10.80 10.80 10.80 10.80 10.80 10.80 10.80 10.80 10.80

99 99 99 99 99 99 99 99 99

1 1 1 6 6 6 11 11 11

22 44 66 22 44 66 22 44 66

10 10 10 10 10 10 10 10 10

a Largest observation,

9.00 15.00 21.00 6.50 12.50 18.50 6.27 12.27 18.27

Exponential delay

S* s* G(S*, a(s*, S*) ~) Go(O, Sg) 2 3 3 2 5 6 2 5 8

0 5.00 1 6.23 2 6.96 1 5.57 4 8.45 5 10.09 1 5.75 4 9.76 7 12.42

G(O,Sg') a(s*,S*)

S* s* a(s*, G(s*, S*) S*) ao(O,S*)

G(O,Sg')

a(0*, S*)

1.11 1.38 1.55 1.24 1.88 2.24 1.28 2.17 2.76 a

1.80 2.41 3.02 1.17 1.48 1.83 1.09 1.26 1.47

2 3 3 2 4 6 2 5 6

0 1 1 0 2 3 0 2 4

5.00 6.67 7.67 5.63 9.35 12.26 5.77 10.23 13.97

1.11 1.48 1.70 1.25 2.08 2.72 1.28 2.27 3.11

1.80 2.25 2.74 1.16 1.34 1.51 1.09 b 1.20 1.31

36.00 5 48.00 6 60.00 6 16.00 5 28.00 7 40.00 8 14.18 5 26.18 9 38.18 11

0 1 1 3 6 7 4 8 10

11.00 12.35 13.08 11.35 14.21 15.68 11.51 15.98 18.25

1.02 1.14 1.21 1.05 1.32 1.45 1.07 1.48 1.69

3.27 3.89 4.59 1.41 1.97 2.55 1.23 1.64 2.09

5 6 6 5 8 10 5 9 11

0 1 1 0 2 4 0 3 6

11.00 12.83 13.83 11.45 16.72 20.45 11.63 18.57 23.91

1.02 1.19 1.28 1.06 1.55 1.89 1.08 1.72 2.21

3.27 3.74 4.34 1.40 1.67 1.96 1.22 1.41 1.60

20.90 121.00 10 20.90 143.00 11 20.90 165.00 11 20.90 38.50 10 20.90 60.50 12 20.90 82.50 12 20.90 31.00 10 20.90 53.00 12 20.90 75.00 13

0 1 1 2 6 7 5 11 12

21.00 22.40 23.14 21.24 24.48 25.99 21.34 25.53 27.48

1.00 1.07 1.11 1.02 1.17 1.24 1.02 1.22 1.31

5.76 6.38 7.13 * 1.81 2.47 3.17 1.45 2.08 2.73

10 11 11 10 13 16 10 15 18

0 1 1 0 3 5 0 4 8

21.00 22.91 23.91 21.31 27.79 32.09 21.48 30.80 37.41

1.00 1.10 1.14 1.02 1.33 1.54 1.03 1.47 1.79

5.76 6.24 6.90 1.81 2.18 2.57 1.44 1.72 2.00

b Smallest observation.

i n t e g e r ( f o r c o n v e n i e n c e ) . T h e l a s t p a r a m e t e r is set t o e n s u r e t h a t it is b e t t e r t o o r d e r t h a n n o t t o o r d e r (see t h e s u f f i c i e n t c o n d i t i o n a b o v e a n d i n the Appendix). The results presented in the table are a representative subset of larger scale runs that we have computed. Presented in the table for each set of parameters are the optimal policy when the d e l a y is k n o w n t o e x i s t ( s * , S * ), t h e c o s t o f t h i s p o l i c y (G(s *, S * )), t h e o p t i m a l p o l i c y w h e n t h e r e is n o d e l a y (So*), t h e c o s t o f t h i s p o l i c y ( G ( 0 , So* )), the percentage increase in cost due to the delay, G(s*, S*)/Go(O, S*) a n d t h e p e r c e n t a g e i n crease in cost due to being unaware of the delay

G(O, S~)/G(s*, S*). T h e v a l u e s f o r G(s*, S*) w e r e f o u n d b y v a r y i n g s, f i n d i n g S * f o r e a c h value of s and then choosing the best pair. Several different behavior patterns can be observed from the table. In both the constant and

exponential delay cases we observe that both s*, S * a n d G(s*, S*) a r e n o n - d e c r e a s i n g i n 2p/lth ( f o r f i x e d 2K/tth a n d E[D]/II) a n d a l s o t h a t s * , S * a n d G ( s * , S * ) a r e n o n - d e c r e a s i n g i n E[D]/# a n d a l s o t h a t s * , S * a n d G(s*, S * ) ' a r e n o n - d e c r e a s i n g i n E[D]/t~ ( f o r f i x e d 2k/l~h a n d E[D]/I~). I n t h e c a s e o f c o n s t a n t s e r v i c e t i m e s w e h a v e t h a t S * - s * is n o n i n c r e a s i n g i n b o t h 2p/t~h a n d E[D]/# a n d f u r t h e r t h a t S * - s * is b o u n d e d a b o v e b y So*. H o w e v e r , i n t h e c a s e o f e x p o n e n t i a l service times S*-s* is n o t n e c e s s a r i l y m o n o t o n e , n o r is it n e c e s s a r i l y b o u n d e d a b o v e b y So*. Comparing across the two cases of exponential and constant delays we find that S*-s* is a t l e a s t as l a r g e i n t h e e x p o n e n t i a l d e l a y c a s e as i n t h e c o n s t a n t d e l a y case. I n a d d i t i o n w e f i n d t h a t G(S*, s * ) is l a r g e r f o r t h e e x p o n e n t i a l c a s e t h a n t h e c o n s t a n t case. T h e s e l a s t t w o o b s e r v a t i o n s a r e

H.J. Weiss / Continuous review stochastic (s, S) inventory systems with ordering delays

quite reasonable as there is more variance in a system with exponential delays than a system with constant delays. Our last observations are with respect to the cost ratios and are the most critical. It is clear that there will be an increase in cost due to the delay. In this set of runs the largest increase we observed is a 176% increase. However, this increase is unavoidable. The increase in cost when using the wrong policy, (0, So* ) rather than (s*, S * ) is avoidable and can be very high. The largest in this set of runs is over 600%. Even the smallest increase in the set of runs is 9%. Clearly, when ordering delays exist it is imperative that the manager become aware of these delays and modify the ordering policy to account for these delays.

179

P/t* consider only the numerator of (A.1) and denote it by N(S). We have that N ( S + 1) - N ( S ) = 2(S + 1) - 2 p / t t h . Hence the difference is minimized at the value of S that satisfies 2 ( S + 1) >/2p/l*h,

(a.2)

2 S <~2p/l*h.

(1.3)

Substitution for S according to (A.3) into N ( S ) yields N ( S ) <~2K/l*h + ( S + 1)~-~ - SZp/l*h = 2K/l*h - Sp/l*h + p/l*h. Further substitution of S according to (A.2) yields N(S)

4. Summary We have considered continuous review inventory systems with ordering delays. Our results indicate that ordering delays can be costly, especially if they are unrecognized. The implication is that inventory systems should be designed to keep track of the difference between the trigger time (theoretical reorder time) and the actual order time. If differences exist then ordering policies should be modified accordingly.

- 1 +

= 2k/l*h - ( p / l - h ) 2 + 2p/l*h.

(1.4)

Since we want to guarantee that N ( S ) is negative we solve (A.4) for p/l*h (using the quadratic formula) and have p / l * h = 1 + 1/1 + 2 K / l * h

and the result is proved.

References

Appendix Sufficient condition. If 2p/l*h > 2 + 21/1 + 2K/l*h then it is better to order than not to order. Proof. We will show that if the condition in the corollary is met then there exists a (0, S) policy such that G(O, S ) < p/l*. G(o, s)-p/l*

= ( ( h / 2 ) ( 2 K / l * h + S ( S + 1) + (2p/l*h)E[D]/l*)} ×{e[D]/l*+s)

-1

P l*

= (2K/l*h + S ( S + 1) - S 2 p / l * h ) /(E[D]/l*+

2K/l*h -

S).

(A.1)

As our concern is only with the sign of G(0, S) -

[1] Archibald, B.C. and Silver, E.A. "(s, S) policies under continuous review and discrete compound Poisson demand", Management Science 24 (1978) 899-909. [2] Cinlar, E., Introduction to Stochastic Processes, PrenticeHall, Englewood Cliffs, N J, 1975. [3] Deuermeyer, B.L., " O n continuous review (s, S) inventory systems: An application of regenerative stochastic processes", Krannert Graduate School of Management, Paper 638, October, 1977. [4] Richards, F.R., "Comments on the distribution of inventory position in a continuous review (s, S) inventory system", Operations Research 23 (1975) 366-371. [5] Ross, S.M., Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, 1970. [6] Sahin, I., " O n the stationary analysis of continuous review (s, S) inventory systems with constant lead times", Operations Research, 27 (1979) 717-729. [7] Sivazlian, B.D., "A continuous review (s, S) inventory system with arbitrary inter-arrival distribution between unit demand", Operations Research 22 (1974) 65-71. [8] Stidham Jr., S., "Cost models for stochastic clearing systems", Operations Research 25 (1977) 100-127.