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Stochastic inventory model for two-echelon distribution systems
Computers ind. Engng Vol. 16, No. 2, pp. 245-255, 1989 Printed in Great Britain. All rights reserved
0360-8352/89 $3.00+0.00 Copyright ~ 1989 lterganum Press pk
STOCHASTIC INVENTORY MODEL FOR TWO-ECHELON DISTRIBUTION SYSTEMS K y u s 6 S. PARK and DAE H. KIM Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, P.O. Box 150, Cheongryang, Seoul, Korea
(Received for publication 29 April 1988) Almtract--This paper presents a heuristic treatment of a stochastic inventory model for a two-echelon distribution system, in which one central warehouse stocks goods for distribution to retailers. An iterative algorithm is developed to fred an optimal or near optimal inventory policy minimizing the total variable system cost per year. Numerical experience with 15 randomly generated problems shows that the algorithm converges rapidly. A numerical example is given to illustrate the savings in an)real cost by the two-echelon inventory model compared to the situation where the suboptimal inventory policies are determined sequentially, from the retailer level to the central warehouse.
1. INTRODUCTION
Multi-echelon distribution systems are commonly found in practice and their models appear in the operations research literature frequently. However, there are few useful guidelines for managing these systems. Thus, many real world distribution systems are managed, at best, using the results of single-echelon models. Such models ignore the interactions among the demand, order quantity, and reorder point at various stocking points. Therefore, these models perform very poorly in comparison with themodels designed to take advantage of the interactive systems structure. Most of the stochastic inventory models in the literature for multi-echelon distribution systems have been limited mostly to (i) models for low demand items using (S-l, S) policies [1, 2], (ii) models for the analysis of customer service level for identical retailer distribution system [3, 4], (iii) models for estimating expected service level as a function of system's parameters rather than minimizing the total system cost [3, 5]. It is generally recommended [6] that explicit backorder cost be used whenever possible in preference to trying to guess the most economic service level. This is mainly because in many instances it is possible to reason intelligently regarding the value of the backorder cost, while the selection of economic service level cannot usually be made other than by a purely arbitrary choice. Thus, the approach that explicitly trades ordering and holding cost against backorder cost is superior to the service level approach. The cost of backorder at each retailer is assumed to be proportional to the length of time for which the backorder exists. As a minimum, it is the inconvenience of backordering and customer waiting, but it may be the loss of goodwill of retailers during the stockout period. Therefore, in establishing the cost of backorders, the seriousness of the backorder condition and its time duration must be considered (time-weighted backorders). This paper determines an optimal or near optimal inventory policy for a two-echelon distribution system, in which one central warehouse feeds retailers where customer demands for these goods originate (Fig. 1). The criterion is to minimize the total variable system (ordering, holding and backorder) cost per year incurred at the central warehouse and retailers as a system. Each retailer is assumed to follow a continuous review (Q, r) policy. When its inventory position reaches reorder point r, an order for Q units is placed upon from the central warehouse. Each retailer experiences continuous (unit-sized) random demand from its customers. In this situation, a (Q, r) policy is the most commonly used because of its good performance and ease of implementation [4]. Unsatisfied demand at each retailer is backordered. The central warehouse is assumed to follow a periodic review (R, T) policy. At each review interval T, an order is placed to bring the inventory position (on-hand plus on-order minus backorders) to a level R from the outside supplier. The ce~ral warehouse experiences lumpy cam t6/2-v
245
246
KYtmo S. PARK and DAE H.
KIM
Source
Woreh~
RetaiLers
I
? ?
?
Outside customer demand
Fig. 1. Two-echelon distribution system.
demand from retailers. In such a situation, since there is overshoot of the reorder point, it may be no longer desirable to order a fixed quantity as in a (Q, r) policy [7]. However, periodic review policy offers a regular opportunity to adjust order quantity up to level R, a desirable property for managing lumpy demand pattern. If the central warehouse has inventory to fill a retailer order, it does so without delay. If the central warehouse has no inventory to fill a retailer order, it is backordered and flied as soon as adequate inventory is on hand, thus adding a stochastic delay to the retailer's nominal lead time. Thus the effective lead time at a retailer has a fixed component plus a stochastic component. The proposed model is divided into two interactive submodels: retailer submodel and central warehouse submodel. The two submodels are linked by d(R), expected delay per order cycle each retailer experiences (in addition to his nominal lead time) before the arrival of a procurement from the central warehouse, which reflects interactions between retailers and central warehouse. An overview of the interactive algorithm developed follows: for an initial value of d(R) = 0, the retailer submodel determines the optimal operating policy variables Q~ and rt. Using these values, the central warehouse submodel determines the optimal operating policy variable R, and revises the value of d(R). This value of d(R), which in turn affects the retailers' ordering policies, is fed back to the retailer submodel for the next iteration. This process is repeated until convergence occurs. Numerical experience with 15 randomly generated problems shows that the algorithm converges rapidly, A numerical example is given to illustrate the savings in annual cost by the two-echelon inventory model compared to the situation where the suboptimal inventory policies are determined sequentially, from the retailer level to the central warehouse.
Assumptions 1. Demand at retailer i is normally distributed, independent of demands incurred at all other retailers. 2. The lead time at the central warehouse and the nominal lead time of each retailer are fixed, but not necessarily equal. 3. Demand during lead time plus review interval at the central warehouse is normally distributed [8]. 4. At the central warehouse, backorders occur only in small quantities relative to the on-hand inventory. 5. If the central warehouse is unable to fill a given retailer order in full, partial shipment of retailer order is allowed. 6. There is never more than a single order outstanding at each retailer.
Stochastic inventory model
247
7. Backorder cost at each retailer is high enough so that the reorder point r, based on the net inventory, is positive at each retailer. 8. The cost of backorder at each retailer is proportional to the length of time for which the backorder exists. 9. The central warehouse and retailers constitute a system; no penalty is directly assessed for the backorders at the central warehouse to avoid double counting.
Notation N number of retailers, Di, Si mean and standard deviation of annual demand at retailer i, Do
expected annual demand at the central warehouse, (D0 = ;~= , Di),
Q i , ri
order quantity and reorder point at retailer i,
Q*,r? optimal value of Ql and ri, lo, li
fixed lead time at the central warehouse and the nominal lead time of retailer i, expected delay per order cycle each retailer experiences (in addition to his nominal lead time) before the arrival of a procurement from the central warehouse due to the central warehouse being out of stock, expected effective lead time at retailer i, [Li--li + d(R)], Li R,T target inventory and review interval at the central warehouse, R*, T* optimal value of R and T, ~ , tro mean and standard deviation of demand during lead time plus review interval at the central warehouse, ~i, O'i mean and standard deviation of effective lead time demand at retailer i, Ao, Ai ordering cost at the central warehouse and retailer i, ho, hi inventory holding cost/year per unit at the central warehouse and retailer i, 7fi backorder cost/year per unit at retailer i.
d(R)
2. RETAILER ANALYSIS
In this section, the average annual variable cost incurred at each retailer is derived, and the optimal operating policy variable Qi, ri are determined. Figure 2 depicts the inventory level variation at retailer i resulting from a continuous review (Qi, ri) policy.
Casel
q
| C
Case2
B I m
\
2/ :~ t
~
o//o,
LI--'---'--~
~=
Fig. 2. Inventory level variation at retailer i (see text for index).
Flme
248
KYuu~ S. PARKand DAEH. KIM
The effective lead time of each retailer is Lt = I, + d(R).
(I)
This approach involves a deterministic approximation to retailers' effective lead times, which is similar to that o f Deuermeyer and Schwarz [3] and Muckstadt [I]. Suppose that the distribution of effective lead time demand at retailer i is a normal probability density functionf(x,) with mean and standard deviation: #~ = LiD, (2)
(3)
=
Retailer i orders DdQi times annually, neglecting the partial lot shipment (Assumption 4). The average annual ordering cost incurred at retailer i is A,D,/Q,.
(4)
The expected on-hand inventory at retailer i is computed in two parts. First, for the period up to the time the reorder point is reached; and second for the effective lead time L~ = I~i/D~. Let £~ be the expected net inventory at the time of arrival of a procurement at retailer i. Since the expected demand during the effective lead time is/z~, £~ is (r, -/z~). The expected net inventory immediately after the arrival of a procurement is Zi = zt + Qi = Q~ + r~- #i. The expected time taken by the inventory level to reach the reorder point, r~, after the arrival of a procurement is given by Q d D i - L~. The expected value of the area Ct in Fig. 2 is given by 1/2(Q,/D,- L,)(2, + r,) = (Q, -/~,) (Q, + 2r,.- Iz,)/(2D,).
(5)
There are two possibilities for the remaining part: whether the inventory meets all demands during the effective lead time, or the inventory will be depleted before the new order arrives. If there is no depletion (x, ~
(6)
If stocks are depleted (xi > r~), we are concerned with the area of the triangle Ca in Fig. 2. By similar triangles, the time, during the lead time, with positive inventory on-hand is th = r~Li/xi. Therefore the expected value of the area C3 is given by r, e(th )
r~ -~ Li ~ (1/x,)f(x,) dxi. (7) drI Therefore the expected on-hand inventory per cycle is found by summing the relevant areas from equations (5), (6) and (7). The average annual cost of holding inventory incurred at retailer i is found by multiplying the expected on-hand inventory per cycle by h~D~/Q~. After some manipulation, it is given by =
h ~ ( ~ + r i - ~ ; ) + hd~' fr,~° (xi~r,)2f(x,)dr,.
(8)
Since the time during which a shortage occurs per cycle is td = L ~ - th = L ~ - r~Ldxi and the demand backordered is (x~-r~), the average annual time-weighted backorder cost incurred at retailer i is ~iDifr°°(xi-rl)(Ll-rlLi~ .... 7tll~'fr~ (xl--ri)2fl(xl)dx i. 2Q, , -~i ) J l t ' x ' ) ° x ' = - ~ i , xi
(9)
All the components of the average annual variable cost Kj(Q~, rt) incurred at retailer i have been found. From equations (4), (8) and (9), it is given by K,(Q,, r,) = Q,
T + ri -- #, +
2Q,
,
x,
,1o,
Stochastic inventory model
249
For a given value of d(R), the values for the mean and standard deviation of demand during effective lead time at each retailer i can be obtained from equations (1), (2) and (3). Then, we can solve N independent subproblems. The subproblem for retailer i is to minimize the average annual variable cost KI(Q~, r~) incurred at retailer i.
Lemma: The function Ki(Q~, r~) in equation (10) is strictly convex in (Q~, r~) and has, therefore, a unique minimum. Proof. See Appendix. The Q* in equation (10) can not be zero, since K~(0, r;) = ~ . Also, from Assumption 7, r* > 0. In other words, the minimum can not occur on the boundary. Thus, a necessary and sufficient condition for Q~ and rt being optimal for the ith subproblem is that they satisfy t'~Ki AiDi OQ~= - ~ + ht/2
(hi + n,)#, ~oo (x i _ r~)Z 2Q~ .j,, ~. fj(x,) dx~ = 0
(11)
and
c3Kt = h,
(h, + n,)#, Qt
t'~ri
f;
(l - r,/x,)f~(x,) dx, = 0.
,
(12)
Here we have two equations to be solved for Q~ and ri. It is convenient to write equations (11), (12) as
Q, =
J(
2/l~Di + (h, + ~l)~,
[(x,- ri)2/x, lf~(x,) dx i
t
(13)
'
hi
and
f o~ !x, -- r,)f(x, ) dxi = ,
x,
h,Q, [(h, +
rc,)~,]"
(14)
From the lemma, the solution (Q~*, r~*)for the ith subproblern obtained from equations (13) and (14) yields a unique minimum. To find the optimal pair (Qt*, r*) for the ith subproblem that will minimize Kt(Q~, r;), we use the following iterative subprocedure of Hadley and Whitin [7]. Step Step Step Step
R1. R2. R3. R4.
The initial estimate for Q~ is Q~= x/(2AiD~/ht). Call this value QI ~). Use equation (14) with Q~ = Q~I) to find the reorder point r~. Call this value r~l). Use equation (13) with r~= r[ ') to find QI2). Repeat step R2 with Qi= Q I2), etc. Convergence occurs when at iteration j, Q~ __ Q~-i) or r~ = r~J- l).
A proof of convergence of this iterative subprocedure corresponds to that of Kim and Park [9] when fl = 1, after notational changes (Q:-,R, ri~r). Finally, the minimum average annual variable cost incurred at all retailers can be represented as a function of d(R), i.e. N
K*[d(R)] ffi ~, K,(Q*, r*).
(15)
i-I
The parameter d(R) intervenes in equation (15) through equations (I), (2) and (3). 3. CENTRAL WAREHOUSE ANALYSIS
In this section, the average annual variable cost incurred at the central warehouse is derived, and the optimal operating policy variable R is determined for a fixed T. The review interval T could be the result of conditions external to the model; e.g. T can be determined so as to coordinate the replenishment of many different items, or outside supplier (factory) may be able to accept the central warehouse order only every other month.
KztrsG S. PARK and DAE H. KIM
250
~R "<-
.....
T,--
o
i • "0 ;~
lo
~
~;~
Time
r
Fig. 3. Inventory level variation at the central warehouse (see text for index).
In the proposed model, T is predetermined from the economic order interval expressed as: T* = ~/[2Ao/(Doho)]. Figure 3 depicts the inventory level variation at the central warehouse resulting from a (R, T) policy. The target inventory R is computed based on the probability distribution of the demand on the central warehouse during the lead time plus review interval. Although it is very difficult to model the central warehouse demand process exactly, Sand [8] shows that the process is quite accurately described by a normal distribution for high demand, low variability items. Thus, the demand on the central warehouse during the lead time plus review interval is assumed to follow a normal distribution f0(x) with the mean and standard deviation as given in Sand [8] or Rosenbaum [5]: N
= ( T + 1o) ~., D,,
(16)
i=l
a0;
(r+t0)
S + 2 Q ,/12 . i=1
07)
i=l
These expressions take into account total customer demand on retailers, and the fact that the central warehouse experiences lumpy demand from retailers. Since an order is placed at every review interval T, the average annual ordering cost incurred at the central warehouse is Ao/T.
(lS)
As the demand rate has a constant mean, net inventory varies on the average linearly between a maximum (just after a procurement arrives) and a minimum (just before a procurement arrives). Since the expected demand during the lead time plus review interval is ~ , the expected net inventory just before a procurement arrives is ~0 = R - ~ . The expected net inventory just after a procurement arrives is
20=R
- p.o + D o T,
where Do T is the expected demand during the review interval.Since the portion of time the central warehouse being out of stock is small (Assumption 4), the average annual cost of holding inventory is approximately h0(g0 + ~0)/2 = ho(R - ~ + DOT~2). (19)
Stochastic inventory model
251
Then the average annual variable cost K0(R) incurred at the central warehouse is, from equations
(18) and (19)
Ao ho(R - #o+ ~-~-~.
(20)
K0(R) = -~ +
Note that the backorder cost at the central warehouse does not appear in equation (20) from Assumption 9. However, backorders at the centralwarehouse increase retailers'effectivelead times, thereby affecting retailers'performance. To see this,consider the total variable costs incurred at the central warehouse and retailersfrom equations (I0) and (20): N
TVC(R,
Q,,r,)= ~ K,(Qi,ri)+ Ko(R).
(21)
i=l
To separate the retailers'problem from the total system cost model in equation (21), the retailer submodel has not considered the effect of backorder level variation at the central warehouse. Therefore, the effectof backorders at the central warehouse on the average annual variable cost incurred at all retailers must be reflected in the central warehouse submodel. This can be accomplished by introducing the concept of imputed backorder cost (Muckstadt and Issac [I0]). It measures the effectof a unit increase in backorder level at the central warehouse at a random point in time on the average annual variable cost incurred at all retailers. Let ~0 be the imputed backorder cost having the dimensions of dollars per unit year of shortage. Let B (R) denote time-averaged backorder levelat the central warehouse. Then, dK*[d(R)]/dB (R) is an estimate of ~0, since it measures the marginal effectof a unit increase in backorder level at a random point in time at the central warehouse on the average annual variable cost incurred at all retailers. Since backorders occur whenever demand during 10+ T exceeds R (x > R), by similar triangles, tp = (10+ T)(x - R)/x. Then
=
2----T-
x
.Jo(xldx,
(22)
which can be interpreted as the expected number of units backordered throughout the cycle. The expected number of units filled without delay in a cycle is D o T - B(R). The demandweighted average time that a unit demand is delayed, d(R), is d(R) = { T . B ( R ) + 0"[DoT - B(R)]}/(DoT) = B(R)/Do.
(23)
This relation is shown to hold even for a lumpy demand pattern (Deuermeyer and Schwarz [3]). The average delay per unit can be interpreted as the expected delay that a retailer's order placed at a random point in time from the central warehouse must wait before being filled. Thus, the expected delay per retailer's order cycle is also given by equation (23). Then, it can be shown thot dK*[d(R)] = dK*[d(R)] dd(R) dd(R) dB(R)
~0-~ dB(R)
=
h,D,
,- I -- -~o +
(h, +
(x, - r~*)2 1 + ( x , - ~ , ) f~(x,)
~,)D,
4Q I*Do
r
x,
~
dx,
(24)
"
from equations (10) and (15). Actual values of Q~* and r* used in equation (24) are based on the current value of R. Let ~0(R) be the imputed cost of backorders plus the average annual variable cost incurred at the central warehouse. Then, from equations (20) and (22)
~o(R)=-~+ho(R-
~ +-D-~-T)+ 7/0(1° 2 T+ T)
ff
(x
-x R)2J°"(x) dx.
Since
d21~o/dR 2 -- f~o(lo"+ T)/T I ~ [fo(x)/x] dx > O, JR
(25)
252
KvuNo
S. PARK a n d DAE H. KIM
Table 1. Data for the example problem Demand parameters
Cost parameters
Retailer i
D~ (units/yr)
lj (yr)
S1 (units/yr)
A~ ($)
hI ($/yr/unit)
($/yr/unit)
1 2 3
77 122 60
0.12 0.17 0.13
42 29 30
37 43 27
2.2 3.5 1.3
19 35 22
4 5 6 7
132 85 61 120
0.16 0.18 O.ll 0.15
40 37 33 43
32 18 26 32
4.1 2.7 3.7 2.4
I$ 27 47 34
8 9 I0
92 69 112
0.14 0.19 0.15
32 34 28
36 25 29
4.5 3.4 2.9
39 34 46
nl
the function fro is strictly convex in R and has a unique minimum. The optimal value of R is a unique solution to
d/~OdR = ho ~o(Io +T T)f:~(x -R)fo(x)dx=0, which yields ff
.
(x - R*) , , x
,
hoT
(26)
yotX) dx = fco(lo + T)"
Although the imputed backorder cost is considered in determining the optimal value of R, it is not included in the actual assessment of the total Variable system cost per year in equation (21). Finally, from this new value of R*, a new value of d(R) from equation (23) is fed back to the retailer submodel. 4. A N I T E R A T I V E A L G O R I T H M
AND NUMERICAL
EXAMPLE
The following iterative algorithm is used to find an optimal or near optimal inventory policy minimizing the total variable system cost per year. Step 1. The initial estimate for d ( R ) = 0. Step 2. (a) For a given value of d(R), compute #i and at for each retailer i using equations (1), (2) and (3). (b) For each retailer i, determine (Qt*, r*) using the iterative subprocedure developed in Section 2. Step 3. (a) Given the (Q~*, r~*), compute a0 from equation (17), and compute ~0 from equation (24). (b) Determine R* which satisfies equation (26). Revise the ~,alue of d(R) using equation (23). Step 4. For the given value of Q~*, r~* and R* obtained in steps 2 and 3, the total variable system cost per year in equation (21) is assessed. Return to step 2 with the revised value of d(R) unless the total variable system cost per year has convereged sufficiently. To illustrate the application of the model and the iterative algorithm developed, an example with ten retailers is considered. Table 1 gives the input data for retailers. Other relevant data for the central warehouse are A0 = $50, h0 = $0.8/yr per unit, 10= 0.7 yr and T* = x/[2do/(Doho)] = 0.37 yr. The intermediate computational results following the steps of the iterative algorithm are shown in Table 2. The solution to the problem was obtained in the third iteration. The minimum total variable system coat is $1634.61/yr. Because of the complexity of multi-echelon inventory problems, a sequential approach is frequently used to determine inventory policies for multi-echelon inventory systems (Schwarz [11]). In the two-echelon distribution system, the sequential approach begins at the retailer level. Each retailer determines his optimal operating policy independently: each retailer may simply use an "economic order quantity" EOQ~ = x/(2AtD~/h3 as his Q~*, and then determine the reorder point
Stochastic inventory model
253
Table 2. Intermediate computational results 1
2
3
Iteration No, Retailer (i)
(EOQt)
QIt)
r~I)
Q~2)
r~2)
Q~3)
r~3)
1 2 3 4 5 6 7 8 9 10
(51) (55) (50) (45) (34) (29) (57) (38) (32) (47)
59 63 56 57 43 36 67 46 41 53
1 15 4 8 13 4 15 7 10 15
62 64 58 59 45 37 68 48 42 55
6 24 10 17 20 10 26 15 16 25
63 64 58 60 45 38 69 48 43 55
7 25 10 18 21 10 27 16 17 26
d(R) R (units) TVC (R, Qj, rl)
0.069 809 $1754.26
0.076 796 $1635.47
.•
0.081 789 $1634.61
TDeterministic economic order quantity = ~/(2A~Di/ht), neglecting the order quantity-reorder point interaction.
based on this EOQ,. Or, each retailer can use a refined method of determining Q* and r* jointly (given in columns 3 and 4 of Table 2, respectively), in which case he must still use the iterative subprocedure developed in Section 2 with d(R) = 0. For reference, the EOQ values are given in column 2 of Table 2. Then, for the actually used values of Q* and r*, the central warehouse must take a system's point of view and determine the system's optimal operating policy R* minimizing (possibly by searching) the total variable system cost per year. (Notice that if the central warehouse takes a myopic standpoint and tries to minimize only its own cost, the target inventory becomes R ~<0, since there is no specific backorder cost at the central warehouse.) For the example given, the minimum total variable system cost turns out to be $1674.85/yr for R* = 909 units. Thus, failure to use the interactive two-echelon inventory model has cost management $1674.85 - $1634.61 = $40.25/yr as a system for a single item. The effect of this in a multi-item inventory may be quite substantial. 5. C O N V E R G E N C E
Fifteen problems were solved to test the convergence of the iterative algorithm. In particular, five problems with N = 5, 10, and 20 retailers each were generated. The values of the parameters were selected randomly from the range given in Table 3. Although the convergence of the iterative algorithm can not be proved mathematically because of the complexity of the model, the algorithm converged for all the test problems. The number of iterations and CPU times required for the problems are shown in Table 4. The convergence is very rapid for all the problems solved, and the maximum number of iterations required for each problem was 4. 6. CONCLUSIONS
Although multi-echelon distribution systems are commonly found in practice, there are few useful guidelines for managing these systems. This is partly because the stochastic model for these Table 3. The range of values of the parameters used in the 15 test problems solved Range Parameters
Minimum
Maximum
Ao($) A,($) D~(units/yr) $,($1unit/yr) ho($/yr/unit)
20 10 50 20 0.5 1 0.5 0.1 l0
100 50 100 50 I 5 I 0.2 50
h~($1yr/unit) Io(yr) /~(yr) gi($/yr/unit)
254
KYUNG S. PARK and DAE H. KIM Table 4. Number of iterations and CPU timer required for the test problems Number of retailers 5 Problem number
Iter. No.
I 2 3 4 5 Average
3 4 3 4 4 3.6
10 CPU
Iter. No.
2.51 3.11 2.37 3.20 3.14 2.87
3 4 4 4 3 3.6
20 CPU 4.53 6.26 6.25 6.54 4.83 5.68
Iter. No. 3 4 3 3 3 3.2
CPU 9.03 12.34 8.52 8.62 8.91 9.48
"['In CPU seconds on a CDC CYBER 174-16 Computer,
systems is quite complex, and it is very difficult to find an optimal inventory policy for these systems. This paper presents a heuristic treatment of a stochastic inventory model for a two-echelon distribution system. Based on the interactions between the central warehouse and retailers, an iterative algorithm is developed to find an optimal or near optimal inventory policy minimizing the total variable system cost per year. Numerical experience with 15 randomly generated problems shows that the algorithm converges rapidly. The proposed model could be useful for the practical solution of multi-echelon inventory problems. REFERENCES 1. J. A. Muckstadt. A model for a multi-item, mutli-echelon, multi-indenture inventory system. Mgmt ScL 20, 472-481 (1973). 2. C. C. Sherbrooke. METRIC: A multi-echelon technique for recoverable item control. Opns Res. 16, 122-141 (1968). 3. B. L. Deuermeyer and L. B. Sehwarz. A model for the analysis of system service level in warehouse-retailer distribution systems: the identical retailer case. In Multi-Level Production/Inventory Control Systems: Theory and Practice. North Holland, Amsterdam 0981). 4. L. B. Sehwarz, B. L. Deuermeyer and R. D. Badinelli. Fill-rate optimization in a one-warehouse N-identical retailer distribution system. Mgmt Sci. 31, 488-498 0985). 5. B. A. Rosenbaum. Service level relationships in a multi-echelon inventory system. Mgmt Sci. 27, 926-945 0981). 6. D. P. Heron. Use of dimensionless ratios to determine minimum-cost inventory quantities. Naval Res. Logist. Q. 13, 167-179 (1967). 7. G. Hadley and T. M. Whitin. Analysis of Inventory Systems. Prentice-Hall, London (1963). 8. G. Sand. Predicting demand on the secondary echelon. Eastman Kodak TP & R Working Paper No. G-05-06 (1979). 9. D. H. Kim and K. S. Park. (Q, r) Inventory model with a mixture of lost sales and time-weighted backorders. J. opl. Res. Soc. 36, 231-238 (1985). 10. J. A. Muckstadt and M. H. Issac. An analysis of single item inventory systems with returns. Naval Res. Logist. Q. 28, 237-254 (1981). II. L. B. Sehwarz. Physical distribution: the analysis of inventory and location. AIIE Trans. 13, 138-150 (1981). 12. B. V. Beek. Modeling and analysis of a multi-echelon inventory system. Eur. J. Opl Res. 6, 380-385 (1981). 13. A. J. Clark. An informal survey of multi-echelon inventory theory. Naval Res. Logist. Q. 19, 621-650 (1972). 14. M. J. Lawrence. An integrated inventory control system. Interfaces 7, 55-62 (1977). 15. C. P. Pinkus. Optimal design of multi-product multi-echelon inventory systems. Decis. Sci. 6, 492-507 (1975). 16. N. Singh and P. Vrat. A Location-allocation model for a two-echelon repair-inventory system. IIE Trans. 16, 222-228 (1984). APPENDIX
Proof of Lemma Let us write
i,=f,;
[(x~--rl)2/xt]f(xj)dx~,
12=O1,/Or,------2f,; (1-r,/x~)f~(x~)dx,, 13 = t~I2/Or~= 2
r
(l/x,)f(xt) dx~.
d rt
It is then easily shown that
~2x,/OQ2= 12A,D,+ (h, + ,~,)u,I,IQ,~> 0, t~2Ki/t~r~-~ (hi+ ~,)lAtl3/(2Qi) > O.
Stochastic inventory model
255
The determinant of Hessian matrix, [HI, is
IH I = (a 2K~/~Q ~)(~ 2K,/~r ~) - (~ ~K~/~Q,~r,) ~ = AiD~(h , + ~i)lliI3/Q~ -'[- [#i(hi + ~)/2Qfl2(21,13 - I~).
(A1)
By applying the following Schwartz inequality
where u = ~/{[(x~ - r~)2/x,]fi(xi)} and v = ~/[(1/xi)fi(xi)], it can be shown that 21113 >~ I]. Since all other terms on the right hand side of equation (AI) are positive, we conclude that K~ is strictly convex in (Q~, r~). Hence, K~ has a unique minimum.
0360-8352/89 $3.00+0.00 Copyright ~ 1989 lterganum Press pk
STOCHASTIC INVENTORY MODEL FOR TWO-ECHELON DISTRIBUTION SYSTEMS K y u s 6 S. PARK and DAE H. KIM Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, P.O. Box 150, Cheongryang, Seoul, Korea
(Received for publication 29 April 1988) Almtract--This paper presents a heuristic treatment of a stochastic inventory model for a two-echelon distribution system, in which one central warehouse stocks goods for distribution to retailers. An iterative algorithm is developed to fred an optimal or near optimal inventory policy minimizing the total variable system cost per year. Numerical experience with 15 randomly generated problems shows that the algorithm converges rapidly. A numerical example is given to illustrate the savings in an)real cost by the two-echelon inventory model compared to the situation where the suboptimal inventory policies are determined sequentially, from the retailer level to the central warehouse.
1. INTRODUCTION
Multi-echelon distribution systems are commonly found in practice and their models appear in the operations research literature frequently. However, there are few useful guidelines for managing these systems. Thus, many real world distribution systems are managed, at best, using the results of single-echelon models. Such models ignore the interactions among the demand, order quantity, and reorder point at various stocking points. Therefore, these models perform very poorly in comparison with themodels designed to take advantage of the interactive systems structure. Most of the stochastic inventory models in the literature for multi-echelon distribution systems have been limited mostly to (i) models for low demand items using (S-l, S) policies [1, 2], (ii) models for the analysis of customer service level for identical retailer distribution system [3, 4], (iii) models for estimating expected service level as a function of system's parameters rather than minimizing the total system cost [3, 5]. It is generally recommended [6] that explicit backorder cost be used whenever possible in preference to trying to guess the most economic service level. This is mainly because in many instances it is possible to reason intelligently regarding the value of the backorder cost, while the selection of economic service level cannot usually be made other than by a purely arbitrary choice. Thus, the approach that explicitly trades ordering and holding cost against backorder cost is superior to the service level approach. The cost of backorder at each retailer is assumed to be proportional to the length of time for which the backorder exists. As a minimum, it is the inconvenience of backordering and customer waiting, but it may be the loss of goodwill of retailers during the stockout period. Therefore, in establishing the cost of backorders, the seriousness of the backorder condition and its time duration must be considered (time-weighted backorders). This paper determines an optimal or near optimal inventory policy for a two-echelon distribution system, in which one central warehouse feeds retailers where customer demands for these goods originate (Fig. 1). The criterion is to minimize the total variable system (ordering, holding and backorder) cost per year incurred at the central warehouse and retailers as a system. Each retailer is assumed to follow a continuous review (Q, r) policy. When its inventory position reaches reorder point r, an order for Q units is placed upon from the central warehouse. Each retailer experiences continuous (unit-sized) random demand from its customers. In this situation, a (Q, r) policy is the most commonly used because of its good performance and ease of implementation [4]. Unsatisfied demand at each retailer is backordered. The central warehouse is assumed to follow a periodic review (R, T) policy. At each review interval T, an order is placed to bring the inventory position (on-hand plus on-order minus backorders) to a level R from the outside supplier. The ce~ral warehouse experiences lumpy cam t6/2-v
245
246
KYtmo S. PARK and DAE H.
KIM
Source
Woreh~
RetaiLers
I
? ?
?
Outside customer demand
Fig. 1. Two-echelon distribution system.
demand from retailers. In such a situation, since there is overshoot of the reorder point, it may be no longer desirable to order a fixed quantity as in a (Q, r) policy [7]. However, periodic review policy offers a regular opportunity to adjust order quantity up to level R, a desirable property for managing lumpy demand pattern. If the central warehouse has inventory to fill a retailer order, it does so without delay. If the central warehouse has no inventory to fill a retailer order, it is backordered and flied as soon as adequate inventory is on hand, thus adding a stochastic delay to the retailer's nominal lead time. Thus the effective lead time at a retailer has a fixed component plus a stochastic component. The proposed model is divided into two interactive submodels: retailer submodel and central warehouse submodel. The two submodels are linked by d(R), expected delay per order cycle each retailer experiences (in addition to his nominal lead time) before the arrival of a procurement from the central warehouse, which reflects interactions between retailers and central warehouse. An overview of the interactive algorithm developed follows: for an initial value of d(R) = 0, the retailer submodel determines the optimal operating policy variables Q~ and rt. Using these values, the central warehouse submodel determines the optimal operating policy variable R, and revises the value of d(R). This value of d(R), which in turn affects the retailers' ordering policies, is fed back to the retailer submodel for the next iteration. This process is repeated until convergence occurs. Numerical experience with 15 randomly generated problems shows that the algorithm converges rapidly, A numerical example is given to illustrate the savings in annual cost by the two-echelon inventory model compared to the situation where the suboptimal inventory policies are determined sequentially, from the retailer level to the central warehouse.
Assumptions 1. Demand at retailer i is normally distributed, independent of demands incurred at all other retailers. 2. The lead time at the central warehouse and the nominal lead time of each retailer are fixed, but not necessarily equal. 3. Demand during lead time plus review interval at the central warehouse is normally distributed [8]. 4. At the central warehouse, backorders occur only in small quantities relative to the on-hand inventory. 5. If the central warehouse is unable to fill a given retailer order in full, partial shipment of retailer order is allowed. 6. There is never more than a single order outstanding at each retailer.
Stochastic inventory model
247
7. Backorder cost at each retailer is high enough so that the reorder point r, based on the net inventory, is positive at each retailer. 8. The cost of backorder at each retailer is proportional to the length of time for which the backorder exists. 9. The central warehouse and retailers constitute a system; no penalty is directly assessed for the backorders at the central warehouse to avoid double counting.
Notation N number of retailers, Di, Si mean and standard deviation of annual demand at retailer i, Do
expected annual demand at the central warehouse, (D0 = ;~= , Di),
Q i , ri
order quantity and reorder point at retailer i,
Q*,r? optimal value of Ql and ri, lo, li
fixed lead time at the central warehouse and the nominal lead time of retailer i, expected delay per order cycle each retailer experiences (in addition to his nominal lead time) before the arrival of a procurement from the central warehouse due to the central warehouse being out of stock, expected effective lead time at retailer i, [Li--li + d(R)], Li R,T target inventory and review interval at the central warehouse, R*, T* optimal value of R and T, ~ , tro mean and standard deviation of demand during lead time plus review interval at the central warehouse, ~i, O'i mean and standard deviation of effective lead time demand at retailer i, Ao, Ai ordering cost at the central warehouse and retailer i, ho, hi inventory holding cost/year per unit at the central warehouse and retailer i, 7fi backorder cost/year per unit at retailer i.
d(R)
2. RETAILER ANALYSIS
In this section, the average annual variable cost incurred at each retailer is derived, and the optimal operating policy variable Qi, ri are determined. Figure 2 depicts the inventory level variation at retailer i resulting from a continuous review (Qi, ri) policy.
Casel
q
| C
Case2
B I m
\
2/ :~ t
~
o//o,
LI--'---'--~
~=
Fig. 2. Inventory level variation at retailer i (see text for index).
Flme
248
KYuu~ S. PARKand DAEH. KIM
The effective lead time of each retailer is Lt = I, + d(R).
(I)
This approach involves a deterministic approximation to retailers' effective lead times, which is similar to that o f Deuermeyer and Schwarz [3] and Muckstadt [I]. Suppose that the distribution of effective lead time demand at retailer i is a normal probability density functionf(x,) with mean and standard deviation: #~ = LiD, (2)
(3)
=
Retailer i orders DdQi times annually, neglecting the partial lot shipment (Assumption 4). The average annual ordering cost incurred at retailer i is A,D,/Q,.
(4)
The expected on-hand inventory at retailer i is computed in two parts. First, for the period up to the time the reorder point is reached; and second for the effective lead time L~ = I~i/D~. Let £~ be the expected net inventory at the time of arrival of a procurement at retailer i. Since the expected demand during the effective lead time is/z~, £~ is (r, -/z~). The expected net inventory immediately after the arrival of a procurement is Zi = zt + Qi = Q~ + r~- #i. The expected time taken by the inventory level to reach the reorder point, r~, after the arrival of a procurement is given by Q d D i - L~. The expected value of the area Ct in Fig. 2 is given by 1/2(Q,/D,- L,)(2, + r,) = (Q, -/~,) (Q, + 2r,.- Iz,)/(2D,).
(5)
There are two possibilities for the remaining part: whether the inventory meets all demands during the effective lead time, or the inventory will be depleted before the new order arrives. If there is no depletion (x, ~
(6)
If stocks are depleted (xi > r~), we are concerned with the area of the triangle Ca in Fig. 2. By similar triangles, the time, during the lead time, with positive inventory on-hand is th = r~Li/xi. Therefore the expected value of the area C3 is given by r, e(th )
r~ -~ Li ~ (1/x,)f(x,) dxi. (7) drI Therefore the expected on-hand inventory per cycle is found by summing the relevant areas from equations (5), (6) and (7). The average annual cost of holding inventory incurred at retailer i is found by multiplying the expected on-hand inventory per cycle by h~D~/Q~. After some manipulation, it is given by =
h ~ ( ~ + r i - ~ ; ) + hd~' fr,~° (xi~r,)2f(x,)dr,.
(8)
Since the time during which a shortage occurs per cycle is td = L ~ - th = L ~ - r~Ldxi and the demand backordered is (x~-r~), the average annual time-weighted backorder cost incurred at retailer i is ~iDifr°°(xi-rl)(Ll-rlLi~ .... 7tll~'fr~ (xl--ri)2fl(xl)dx i. 2Q, , -~i ) J l t ' x ' ) ° x ' = - ~ i , xi
(9)
All the components of the average annual variable cost Kj(Q~, rt) incurred at retailer i have been found. From equations (4), (8) and (9), it is given by K,(Q,, r,) = Q,
T + ri -- #, +
2Q,
,
x,
,1o,
Stochastic inventory model
249
For a given value of d(R), the values for the mean and standard deviation of demand during effective lead time at each retailer i can be obtained from equations (1), (2) and (3). Then, we can solve N independent subproblems. The subproblem for retailer i is to minimize the average annual variable cost KI(Q~, r~) incurred at retailer i.
Lemma: The function Ki(Q~, r~) in equation (10) is strictly convex in (Q~, r~) and has, therefore, a unique minimum. Proof. See Appendix. The Q* in equation (10) can not be zero, since K~(0, r;) = ~ . Also, from Assumption 7, r* > 0. In other words, the minimum can not occur on the boundary. Thus, a necessary and sufficient condition for Q~ and rt being optimal for the ith subproblem is that they satisfy t'~Ki AiDi OQ~= - ~ + ht/2
(hi + n,)#, ~oo (x i _ r~)Z 2Q~ .j,, ~. fj(x,) dx~ = 0
(11)
and
c3Kt = h,
(h, + n,)#, Qt
t'~ri
f;
(l - r,/x,)f~(x,) dx, = 0.
,
(12)
Here we have two equations to be solved for Q~ and ri. It is convenient to write equations (11), (12) as
Q, =
J(
2/l~Di + (h, + ~l)~,
[(x,- ri)2/x, lf~(x,) dx i
t
(13)
'
hi
and
f o~ !x, -- r,)f(x, ) dxi = ,
x,
h,Q, [(h, +
rc,)~,]"
(14)
From the lemma, the solution (Q~*, r~*)for the ith subproblern obtained from equations (13) and (14) yields a unique minimum. To find the optimal pair (Qt*, r*) for the ith subproblem that will minimize Kt(Q~, r;), we use the following iterative subprocedure of Hadley and Whitin [7]. Step Step Step Step
R1. R2. R3. R4.
The initial estimate for Q~ is Q~= x/(2AiD~/ht). Call this value QI ~). Use equation (14) with Q~ = Q~I) to find the reorder point r~. Call this value r~l). Use equation (13) with r~= r[ ') to find QI2). Repeat step R2 with Qi= Q I2), etc. Convergence occurs when at iteration j, Q~ __ Q~-i) or r~ = r~J- l).
A proof of convergence of this iterative subprocedure corresponds to that of Kim and Park [9] when fl = 1, after notational changes (Q:-,R, ri~r). Finally, the minimum average annual variable cost incurred at all retailers can be represented as a function of d(R), i.e. N
K*[d(R)] ffi ~, K,(Q*, r*).
(15)
i-I
The parameter d(R) intervenes in equation (15) through equations (I), (2) and (3). 3. CENTRAL WAREHOUSE ANALYSIS
In this section, the average annual variable cost incurred at the central warehouse is derived, and the optimal operating policy variable R is determined for a fixed T. The review interval T could be the result of conditions external to the model; e.g. T can be determined so as to coordinate the replenishment of many different items, or outside supplier (factory) may be able to accept the central warehouse order only every other month.
KztrsG S. PARK and DAE H. KIM
250
~R "<-
.....
T,--
o
i • "0 ;~
lo
~
~;~
Time
r
Fig. 3. Inventory level variation at the central warehouse (see text for index).
In the proposed model, T is predetermined from the economic order interval expressed as: T* = ~/[2Ao/(Doho)]. Figure 3 depicts the inventory level variation at the central warehouse resulting from a (R, T) policy. The target inventory R is computed based on the probability distribution of the demand on the central warehouse during the lead time plus review interval. Although it is very difficult to model the central warehouse demand process exactly, Sand [8] shows that the process is quite accurately described by a normal distribution for high demand, low variability items. Thus, the demand on the central warehouse during the lead time plus review interval is assumed to follow a normal distribution f0(x) with the mean and standard deviation as given in Sand [8] or Rosenbaum [5]: N
= ( T + 1o) ~., D,,
(16)
i=l
a0;
(r+t0)
S + 2 Q ,/12 . i=1
07)
i=l
These expressions take into account total customer demand on retailers, and the fact that the central warehouse experiences lumpy demand from retailers. Since an order is placed at every review interval T, the average annual ordering cost incurred at the central warehouse is Ao/T.
(lS)
As the demand rate has a constant mean, net inventory varies on the average linearly between a maximum (just after a procurement arrives) and a minimum (just before a procurement arrives). Since the expected demand during the lead time plus review interval is ~ , the expected net inventory just before a procurement arrives is ~0 = R - ~ . The expected net inventory just after a procurement arrives is
20=R
- p.o + D o T,
where Do T is the expected demand during the review interval.Since the portion of time the central warehouse being out of stock is small (Assumption 4), the average annual cost of holding inventory is approximately h0(g0 + ~0)/2 = ho(R - ~ + DOT~2). (19)
Stochastic inventory model
251
Then the average annual variable cost K0(R) incurred at the central warehouse is, from equations
(18) and (19)
Ao ho(R - #o+ ~-~-~.
(20)
K0(R) = -~ +
Note that the backorder cost at the central warehouse does not appear in equation (20) from Assumption 9. However, backorders at the centralwarehouse increase retailers'effectivelead times, thereby affecting retailers'performance. To see this,consider the total variable costs incurred at the central warehouse and retailersfrom equations (I0) and (20): N
TVC(R,
Q,,r,)= ~ K,(Qi,ri)+ Ko(R).
(21)
i=l
To separate the retailers'problem from the total system cost model in equation (21), the retailer submodel has not considered the effect of backorder level variation at the central warehouse. Therefore, the effectof backorders at the central warehouse on the average annual variable cost incurred at all retailers must be reflected in the central warehouse submodel. This can be accomplished by introducing the concept of imputed backorder cost (Muckstadt and Issac [I0]). It measures the effectof a unit increase in backorder level at the central warehouse at a random point in time on the average annual variable cost incurred at all retailers. Let ~0 be the imputed backorder cost having the dimensions of dollars per unit year of shortage. Let B (R) denote time-averaged backorder levelat the central warehouse. Then, dK*[d(R)]/dB (R) is an estimate of ~0, since it measures the marginal effectof a unit increase in backorder level at a random point in time at the central warehouse on the average annual variable cost incurred at all retailers. Since backorders occur whenever demand during 10+ T exceeds R (x > R), by similar triangles, tp = (10+ T)(x - R)/x. Then
=
2----T-
x
.Jo(xldx,
(22)
which can be interpreted as the expected number of units backordered throughout the cycle. The expected number of units filled without delay in a cycle is D o T - B(R). The demandweighted average time that a unit demand is delayed, d(R), is d(R) = { T . B ( R ) + 0"[DoT - B(R)]}/(DoT) = B(R)/Do.
(23)
This relation is shown to hold even for a lumpy demand pattern (Deuermeyer and Schwarz [3]). The average delay per unit can be interpreted as the expected delay that a retailer's order placed at a random point in time from the central warehouse must wait before being filled. Thus, the expected delay per retailer's order cycle is also given by equation (23). Then, it can be shown thot dK*[d(R)] = dK*[d(R)] dd(R) dd(R) dB(R)
~0-~ dB(R)
=
h,D,
,- I -- -~o +
(h, +
(x, - r~*)2 1 + ( x , - ~ , ) f~(x,)
~,)D,
4Q I*Do
r
x,
~
dx,
(24)
"
from equations (10) and (15). Actual values of Q~* and r* used in equation (24) are based on the current value of R. Let ~0(R) be the imputed cost of backorders plus the average annual variable cost incurred at the central warehouse. Then, from equations (20) and (22)
~o(R)=-~+ho(R-
~ +-D-~-T)+ 7/0(1° 2 T+ T)
ff
(x
-x R)2J°"(x) dx.
Since
d21~o/dR 2 -- f~o(lo"+ T)/T I ~ [fo(x)/x] dx > O, JR
(25)
252
KvuNo
S. PARK a n d DAE H. KIM
Table 1. Data for the example problem Demand parameters
Cost parameters
Retailer i
D~ (units/yr)
lj (yr)
S1 (units/yr)
A~ ($)
hI ($/yr/unit)
($/yr/unit)
1 2 3
77 122 60
0.12 0.17 0.13
42 29 30
37 43 27
2.2 3.5 1.3
19 35 22
4 5 6 7
132 85 61 120
0.16 0.18 O.ll 0.15
40 37 33 43
32 18 26 32
4.1 2.7 3.7 2.4
I$ 27 47 34
8 9 I0
92 69 112
0.14 0.19 0.15
32 34 28
36 25 29
4.5 3.4 2.9
39 34 46
nl
the function fro is strictly convex in R and has a unique minimum. The optimal value of R is a unique solution to
d/~OdR = ho ~o(Io +T T)f:~(x -R)fo(x)dx=0, which yields ff
.
(x - R*) , , x
,
hoT
(26)
yotX) dx = fco(lo + T)"
Although the imputed backorder cost is considered in determining the optimal value of R, it is not included in the actual assessment of the total Variable system cost per year in equation (21). Finally, from this new value of R*, a new value of d(R) from equation (23) is fed back to the retailer submodel. 4. A N I T E R A T I V E A L G O R I T H M
AND NUMERICAL
EXAMPLE
The following iterative algorithm is used to find an optimal or near optimal inventory policy minimizing the total variable system cost per year. Step 1. The initial estimate for d ( R ) = 0. Step 2. (a) For a given value of d(R), compute #i and at for each retailer i using equations (1), (2) and (3). (b) For each retailer i, determine (Qt*, r*) using the iterative subprocedure developed in Section 2. Step 3. (a) Given the (Q~*, r~*), compute a0 from equation (17), and compute ~0 from equation (24). (b) Determine R* which satisfies equation (26). Revise the ~,alue of d(R) using equation (23). Step 4. For the given value of Q~*, r~* and R* obtained in steps 2 and 3, the total variable system cost per year in equation (21) is assessed. Return to step 2 with the revised value of d(R) unless the total variable system cost per year has convereged sufficiently. To illustrate the application of the model and the iterative algorithm developed, an example with ten retailers is considered. Table 1 gives the input data for retailers. Other relevant data for the central warehouse are A0 = $50, h0 = $0.8/yr per unit, 10= 0.7 yr and T* = x/[2do/(Doho)] = 0.37 yr. The intermediate computational results following the steps of the iterative algorithm are shown in Table 2. The solution to the problem was obtained in the third iteration. The minimum total variable system coat is $1634.61/yr. Because of the complexity of multi-echelon inventory problems, a sequential approach is frequently used to determine inventory policies for multi-echelon inventory systems (Schwarz [11]). In the two-echelon distribution system, the sequential approach begins at the retailer level. Each retailer determines his optimal operating policy independently: each retailer may simply use an "economic order quantity" EOQ~ = x/(2AtD~/h3 as his Q~*, and then determine the reorder point
Stochastic inventory model
253
Table 2. Intermediate computational results 1
2
3
Iteration No, Retailer (i)
(EOQt)
QIt)
r~I)
Q~2)
r~2)
Q~3)
r~3)
1 2 3 4 5 6 7 8 9 10
(51) (55) (50) (45) (34) (29) (57) (38) (32) (47)
59 63 56 57 43 36 67 46 41 53
1 15 4 8 13 4 15 7 10 15
62 64 58 59 45 37 68 48 42 55
6 24 10 17 20 10 26 15 16 25
63 64 58 60 45 38 69 48 43 55
7 25 10 18 21 10 27 16 17 26
d(R) R (units) TVC (R, Qj, rl)
0.069 809 $1754.26
0.076 796 $1635.47
.•
0.081 789 $1634.61
TDeterministic economic order quantity = ~/(2A~Di/ht), neglecting the order quantity-reorder point interaction.
based on this EOQ,. Or, each retailer can use a refined method of determining Q* and r* jointly (given in columns 3 and 4 of Table 2, respectively), in which case he must still use the iterative subprocedure developed in Section 2 with d(R) = 0. For reference, the EOQ values are given in column 2 of Table 2. Then, for the actually used values of Q* and r*, the central warehouse must take a system's point of view and determine the system's optimal operating policy R* minimizing (possibly by searching) the total variable system cost per year. (Notice that if the central warehouse takes a myopic standpoint and tries to minimize only its own cost, the target inventory becomes R ~<0, since there is no specific backorder cost at the central warehouse.) For the example given, the minimum total variable system cost turns out to be $1674.85/yr for R* = 909 units. Thus, failure to use the interactive two-echelon inventory model has cost management $1674.85 - $1634.61 = $40.25/yr as a system for a single item. The effect of this in a multi-item inventory may be quite substantial. 5. C O N V E R G E N C E
Fifteen problems were solved to test the convergence of the iterative algorithm. In particular, five problems with N = 5, 10, and 20 retailers each were generated. The values of the parameters were selected randomly from the range given in Table 3. Although the convergence of the iterative algorithm can not be proved mathematically because of the complexity of the model, the algorithm converged for all the test problems. The number of iterations and CPU times required for the problems are shown in Table 4. The convergence is very rapid for all the problems solved, and the maximum number of iterations required for each problem was 4. 6. CONCLUSIONS
Although multi-echelon distribution systems are commonly found in practice, there are few useful guidelines for managing these systems. This is partly because the stochastic model for these Table 3. The range of values of the parameters used in the 15 test problems solved Range Parameters
Minimum
Maximum
Ao($) A,($) D~(units/yr) $,($1unit/yr) ho($/yr/unit)
20 10 50 20 0.5 1 0.5 0.1 l0
100 50 100 50 I 5 I 0.2 50
h~($1yr/unit) Io(yr) /~(yr) gi($/yr/unit)
254
KYUNG S. PARK and DAE H. KIM Table 4. Number of iterations and CPU timer required for the test problems Number of retailers 5 Problem number
Iter. No.
I 2 3 4 5 Average
3 4 3 4 4 3.6
10 CPU
Iter. No.
2.51 3.11 2.37 3.20 3.14 2.87
3 4 4 4 3 3.6
20 CPU 4.53 6.26 6.25 6.54 4.83 5.68
Iter. No. 3 4 3 3 3 3.2
CPU 9.03 12.34 8.52 8.62 8.91 9.48
"['In CPU seconds on a CDC CYBER 174-16 Computer,
systems is quite complex, and it is very difficult to find an optimal inventory policy for these systems. This paper presents a heuristic treatment of a stochastic inventory model for a two-echelon distribution system. Based on the interactions between the central warehouse and retailers, an iterative algorithm is developed to find an optimal or near optimal inventory policy minimizing the total variable system cost per year. Numerical experience with 15 randomly generated problems shows that the algorithm converges rapidly. The proposed model could be useful for the practical solution of multi-echelon inventory problems. REFERENCES 1. J. A. Muckstadt. A model for a multi-item, mutli-echelon, multi-indenture inventory system. Mgmt ScL 20, 472-481 (1973). 2. C. C. Sherbrooke. METRIC: A multi-echelon technique for recoverable item control. Opns Res. 16, 122-141 (1968). 3. B. L. Deuermeyer and L. B. Sehwarz. A model for the analysis of system service level in warehouse-retailer distribution systems: the identical retailer case. In Multi-Level Production/Inventory Control Systems: Theory and Practice. North Holland, Amsterdam 0981). 4. L. B. Sehwarz, B. L. Deuermeyer and R. D. Badinelli. Fill-rate optimization in a one-warehouse N-identical retailer distribution system. Mgmt Sci. 31, 488-498 0985). 5. B. A. Rosenbaum. Service level relationships in a multi-echelon inventory system. Mgmt Sci. 27, 926-945 0981). 6. D. P. Heron. Use of dimensionless ratios to determine minimum-cost inventory quantities. Naval Res. Logist. Q. 13, 167-179 (1967). 7. G. Hadley and T. M. Whitin. Analysis of Inventory Systems. Prentice-Hall, London (1963). 8. G. Sand. Predicting demand on the secondary echelon. Eastman Kodak TP & R Working Paper No. G-05-06 (1979). 9. D. H. Kim and K. S. Park. (Q, r) Inventory model with a mixture of lost sales and time-weighted backorders. J. opl. Res. Soc. 36, 231-238 (1985). 10. J. A. Muckstadt and M. H. Issac. An analysis of single item inventory systems with returns. Naval Res. Logist. Q. 28, 237-254 (1981). II. L. B. Sehwarz. Physical distribution: the analysis of inventory and location. AIIE Trans. 13, 138-150 (1981). 12. B. V. Beek. Modeling and analysis of a multi-echelon inventory system. Eur. J. Opl Res. 6, 380-385 (1981). 13. A. J. Clark. An informal survey of multi-echelon inventory theory. Naval Res. Logist. Q. 19, 621-650 (1972). 14. M. J. Lawrence. An integrated inventory control system. Interfaces 7, 55-62 (1977). 15. C. P. Pinkus. Optimal design of multi-product multi-echelon inventory systems. Decis. Sci. 6, 492-507 (1975). 16. N. Singh and P. Vrat. A Location-allocation model for a two-echelon repair-inventory system. IIE Trans. 16, 222-228 (1984). APPENDIX
Proof of Lemma Let us write
i,=f,;
[(x~--rl)2/xt]f(xj)dx~,
12=O1,/Or,------2f,; (1-r,/x~)f~(x~)dx,, 13 = t~I2/Or~= 2
r
(l/x,)f(xt) dx~.
d rt
It is then easily shown that
~2x,/OQ2= 12A,D,+ (h, + ,~,)u,I,IQ,~> 0, t~2Ki/t~r~-~ (hi+ ~,)lAtl3/(2Qi) > O.
Stochastic inventory model
255
The determinant of Hessian matrix, [HI, is
IH I = (a 2K~/~Q ~)(~ 2K,/~r ~) - (~ ~K~/~Q,~r,) ~ = AiD~(h , + ~i)lliI3/Q~ -'[- [#i(hi + ~)/2Qfl2(21,13 - I~).
(A1)
By applying the following Schwartz inequality
where u = ~/{[(x~ - r~)2/x,]fi(xi)} and v = ~/[(1/xi)fi(xi)], it can be shown that 21113 >~ I]. Since all other terms on the right hand side of equation (AI) are positive, we conclude that K~ is strictly convex in (Q~, r~). Hence, K~ has a unique minimum.