Please note the following correction to the article “The Effect of Discreteness in Vendor-Buyer Relationships” which appeared in the previous issue of IIE Trans. 2004, 36(6):583–589.
Unfortunately, an error has fallen in the proof of Lemma 1. Luckily, the lemma still holds due to the following:
The proof considers a production cycle of k+1 deliveries and its first part, up to Equation 3, is correct. Then it says: “This production cycle ends at (kq+r)/D−t time after the first delivery, with a joint inventory level of I 0 + Q−(k+1)q = I 0 + kq + r−(k+1)q = I 0−(q−r). The first delivery in the next production cycle takes place at (k+1)q/D time after the first delivery of the cycle in reference; i.e., t+(q−r)/D time after the cycles' change. Therefore, I 0−(q−r) ≥ [t+(q−r)/D]D or I 0 ≥ Dt+2(q−r) must be satisfied to prevent shortages upon the first delivery of the next cycle. Certainly, the weaker condition:
Indeed, this production cycle ends at (kq+r)/D − t time after the first delivery. However, the joint inventory level at this point of time is, of course, not I 0−(q−r), but I 0, again. Yet, this cycle ends at (kq+r)/D − t − kq/D = r/D − t time after the last, k+1st delivery, and hence, I 0+r−tD ≥ q must be satisfied to prevent shortages upon the last delivery. In addition, as above, the first delivery in the next production cycle takes place t+(q−r)/D time after the cycles' change, therefore, I 0 ≥ [t+(q−r)/D]D or I 0 ≥ Dt + (q−r) must be satisfied to prevent shortages upon this delivery. This way or the other
Consequently, the remaining of the proof is correct and the Lemma holds.