Electronic Companion — “ A Single - Product Inventory Model for Multiple Demand Classes ” by Hasan Arslan , Stephen C . Graves , and

Proposition 1. Consider two stocking policies s1 and s2 where for some j , s2 j = s1 j − 1, s2 j−1 = s1 j−1 + 1, and s2 i = s1 i , ∀ i = j j − 1. We have that: (i) ILi s2 i s 2 N ≥ ILi s1 i s1 N for i ∈ 1 j − 1 , and (ii) ∑j i=1 Bi i s 2 i s 2 N ≥ ∑j i=1 Bi i s 1 i s 1 N . Proof. From (6) and (7) we find that ILj s2 j s 2 N = ILj s1 j − 1 s1 j+1 s1 N = ILj s1 j s1 N − 1. Thus from (8) and (9), we obtain Bj s2 j s 2 N ≤ Bj s1 j s1 N + 1 and Bj j−1 s2 j s2 N ≤ Bj j−1 s1 j s 1 N + 1. We can now use this result in (7) to show that: ILj−1 s 2 j−1 s 2 N = s j−1 −Bj j−1 s j s N ≥ s j−1 + 1−Bj j−1 s j s N − 1= ILj−1 s j−1 s N