We consider a sales effort management problem under an all-or-nothing constraint. The seller will receive no bonus/revenue if the sales volume fails to reach a predetermined target at the end of the sales horizon. Throughout the sales horizon, the sales process can be moderated by the seller through costly effort. We show that the optimal sales rate is nonmonotonic with respect to the remaining time or the outstanding sales volume required to reach the target. Generally, it has a watershed structure, such that for any needed sales volume, there exists a cutoff point on the remaining time above which the optimal sales rate decreases in the remaining time and below which it increases in the remaining time. We then study easy-to-compute heuristics that can be implemented efficiently. We start with a static heuristic derived from the deterministic analog of the stochastic problem. With an all-or-nothing constraint, we show that the performance of the static heuristic hinges on how the profit-maximizing rate fares against the target rate, which is defined as the sales target divided by the length of the sales horizon. When the profit-maximizing rate is higher than the target rate, the static heuristic adopting the optimal deterministic rate is asymptotically optimal with negligible loss. On the other hand, when the profit-maximizing rate is lower than the target rate, the performance loss of any asymptotically optimal static heuristic is of an order greater than the square root of the scale parameter. To address the poor performance of the static heuristic in the latter case, we propose a modified resolving heuristic and show that it is asymptotically optimal and achieves a logarithmic performance loss. This paper was accepted by Gabriel Weintraub, revenue management and market analytics.
Decomposition algorithm and mathematical models of hierarchical structure of control of the process of primary oil processing
