liên tục cho một khoảng thời gian t, chứ không phải là giống hệt nhau chỉ ởNhư thể hiện trong các đề xuất sau đây, nếu X (0) = 0 yêu cầu này ngụ ý rằng các hệ thống năng động trưng bày một hành vi tĩnh đặc trưng bởi giá bán lẻ không đổi và tỷ lệ xử lý cũng như số không mức hàng tồn khotương ứng. | 200 4 MODELING IN AN INTERTEMPORAL FRAMEWORK constant for a period of time t GT rather than identical only at t 0 and t T as imposed by . As shown in the following proposition if X 0 0 this requirement implies that the dynamic system exhibits a static behavior characterized by constant retailer pricing and processing rates as well as zero inventory levels. Proposition . If b f b1 a t a1 for t GT T0 0 T X t 0 and 0 a1-b1 cr cs 2U then X t 0 for t GT and the system-wide optimal processing and pricing policies are u t ai - - c and p t ai - b- - cc for t GT. 2 2b1 respectively. Proof Consider the following solution for the state co-state and decision variables X t 0 t -- o. - c p t ai - b c - cc u t ai- - -c for t GT. 2b1 2 It is easy to observe that this solution satisfies the optimality conditions - . Furthermore this solution is always feasible if conditions and hold which is ensured by 0 a1-b1 cr cs 2U as stated in the proposition. Finally the centralized objective function involves only concave and piece-wise linear terms which implies that the maximumprinciple based optimality conditions are not only necessary but also sufficient. System-wide optimal solution transient-state conditions Transient-state conditions do not introduce much sophistication into the centralized supply chain. Indeed it is easy to verify that if the change in demand parameters is such that 0 a2-b2 cr cs 2U holds then instantaneous change in customer sensitivity does not affect the form of the solution presented in Proposition . The price and the processing rate are simply adjusted to the changes as stated in the following proposition. Proposition . If b t b1 a t a1 for t ts t tf X t 0 0 a1-b1 cr cs 2U and b t b2 a t a2 for t ts t tf 0 a1-b1 cr cs 2U then X t 0 and the system-wide optimal processing and pricing policies are u t d t a2 - b- - and p t a- - fr - cc for t t 2 2f s t tf respectively. INTERTEMPORAL INVENTORY GAMES 201 Proof The proof is .