Online scheduling with immediate and reliable lead-time quotation

This paper studies an online scheduling problem with immediate and reliable lead-time quotation. A manufacturer either accepts an order by quoting a reliable lead-time on its arrival or rejects it immediately. The objective is to maximize the total revenue of completed orders. Keskinocak et al. (2001) studied a linear revenue function in a discrete model with integer release time of order, and proposed a competitive strategy Q-FRAC. This paper investigates a relaxed revenue function in both discrete and continuous models where orders are released at integer and real time points, respectively. For the discrete model, we present a revised Q-FRAC strategy that is optimal in competitiveness for concave and linear revenue functions with unit length and uniform weight of order, improving the previous results in Keskinocak et al. (2001). For the scenario with uniform length $p$ and nonuniform weight of order, we prove an optimal strategy for the case with $p=1$ and the nonexistence of competitive strategies for the case with $p>1$ . For the continuous model, we present an optimal strategy in competitiveness for the case with uniform weight of order and linear revenue functions, and prove the nonexistence of competitive strategies for the other case with nonuniform weight of order.