Sales Effort Management Under All-or-Nothing Constraint

Abstract

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.

Contingent stimulus in crowdfunding

“…(2018) and Du et al. (2022). Moreover, in our crowdfunding setting, the cost of a stimulus is paid out only if the hard constraint is met; however, the cost associated with an action in those two papers is sunk regardless of whether the hard constraint is satisfied.…”

“…The slight difference is that there the firm would only collect the residual revenues after paying the debt, whereas crowdfunding creators collect all revenues if the project is successful. Du et al (2022) study a setting similar to Besbes et al (2018) in the sense that the firm can continuously update its decisions (prices in Besbes et al, 2018, andsales rates in Du et al, 2022) under an all-or-nothing constraint. In contrast, our focus is on whether and when to apply a one-shot stimulus policy in a setting where the pledges exhibit a cascade effect that is absent in Besbes et al (2018) and Du et al (2022).…”

“…(2022) study a setting similar to Besbes et al. (2018) in the sense that the firm can continuously update its decisions (prices in Besbes et al., 2018, and sales rates in Du et al., 2022) under an all‐or‐nothing constraint. In contrast, our focus is on whether and when to apply a one‐shot stimulus policy in a setting where the pledges exhibit a cascade effect that is absent in Besbes et al.…”

“…Du et al (2022) study a setting similar to Besbes et al (2018) in the sense that the firm can continuously update its decisions (prices in Besbes et al, 2018, andsales rates in Du et al, 2022) under an all-or-nothing constraint. In contrast, our focus is on whether and when to apply a one-shot stimulus policy in a setting where the pledges exhibit a cascade effect that is absent in Besbes et al (2018) and Du et al (2022). Moreover, in our crowdfunding setting, the cost of a stimulus is paid out only if the hard constraint is met; however, the cost associated with an action in those two papers is sunk regardless of whether the hard constraint is satisfied.…”

“…Du et al. (2022) study a setting similar to Besbes et al. (2018) in the sense that the firm can continuously update its decisions (prices in Besbes et al., 2018, and sales rates in Du et al., 2022) under an all‐or‐nothing constraint.…”