A note on Snyder's simple proof that the inventory position is uniformly distributed in some (Q,R) Systems
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Viewpoint A note on Snyder's simple proof that the inventory position is uniformly distributed in some (Q,R) Systems In an order-quantity reorder-point (Q, R) inventory system with backorders, the standard textbook expression for average inventory level is: AILa 12 Q R ÿ mL ;
1
where Q order quantity, R reorder point, and mL mean lead time demand. AILa is an approximation which is suf®ciently accurate only when the stock-out probability is suf®ciently low. Hadley & Whitin1 showed that if the inventory system has `Property A' (de®ned below), then the exact average inventory level may be evaluated as:
QR
Y
Y ÿ x f L
xdx dY
2 AIL
1=Q R
be very useful pedagogically since (2) will surely become increasingly widely taught and used as computerised procedures become increasingly convenient. However, there are minor errors in an intermediate step of Snyder's proof.6 Namely, the (unnumbered) equation (A6) does not have the correct integral limits. That is, instead of:
x 1 P f
x ÿ y mQdx pn
y pnÿ1
x R
y
where fL
is the density function of the lead time demand. `Property A' is:
Hadley & Whitin1 showed that Property A holds if the periodic demand is Poisson distributed and the lead time is deterministic. Numerous papers have used (1) to evaluate AIL under more general conditions, that is, non-Poisson periodic demand and=or non-deterministic lead times.2,3 They justify this procedure by citing such works as Serfozo and Stidham4 and Zipkin,5 which showed that Property A holds under these more general conditions. However, none of these works cited Snyder's 1980 J Opl Res Soc paper,6 which contains a proof that Property A exists under very general conditions. Snyder's proof is much simpler and easier to follow than those given in, for example, Hadley and Whitin,1 Serfozo and Stidham4 and Zipkin,5 and should
x
m0
pnÿ1
x
1 P m1
f
x ÿ y mQdx
the correct expression is:
RQ 1 P pnÿ1
x f
x ÿ y mQdx pn
y
0
the `inventory position' (inventory on hand plus on order minus backorders) is uniformly distributed between R and (Q R).
RQ
y R
m0
pnÿ1
x
Oklahama State University
1 P m1
f
x ÿ y mQdx A H-L Lau and H-S Lau
References 1 Hadley G and Whitin T (1963). Analysis of Inventory Systems. Prentice-Hall: New Jersey, pp 181±185. 2 Zheng YS (1992). On properties of stochastic inventory systems. Mgmt Sci 38: 87±103. 3 Platt DE, Robinson LW and Freund RB (1997). Tractable (Q, R) heuristic models for constrained service levels. Mgmt Sci 43: 951±965. 4 Serfozo R and Stidham S (1978). Semi-stationary clearing processes. Stochastic Processes and Their Applications 6: 165± 178. 5 Zipkin P (1986). Stochastic leadtimes in continuous-time inventory models. Naval Res Logis Q 33: 763±774. 6 Snyder R (1980). The safety stock syndrome. J Opl Res Soc 31: 833±837.
A Reply to Lau and Lau In their note, Lau and Lau 1 correctly identify an error in a p
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