Optimal replenishment schedule and pricing policy for hybrid retailers with continuously decaying demand in time and price

This study deals with the problem of jointly predetermining optimal replenishment schedule and selling prices toward hybrid retailers who sell products via in-stores and estores concurrently, where deteriorating item is considered with two declining demands in time and price. Under the circumstances, strict concavity of the proposed problem in selling prices is first proven, and closed-form solutions for selling prices are thereby obtained. Next, when selling prices are fixed, not only the strict pseudo-concavity in replenishment interval is shown, but also the optimal schedule uniquely exists if the criterion inequality holds, which is also a premise for positive profit generation. Afterwards, three solution-finding sequences are defined, to which the convergent property is identified, by which a simple algorithm for optimal solutions is presented. Finally, numerical examples are conducted to illustrate the proposed problems.