This paper addresses the challenge faced by managers in the fast-moving consumer goods industry: the joint optimization of promotion prices and the scheduling of promotion vehicles for multiple items to boost total profit. We first propose a general multiplicative demand function that encompasses all crossperiod effects, crossitem effects, promotion vehicle effects, and crossterm effects of promotion vehicles. Then, we formulate the problem of planning sales promotions, simultaneously using price reductions and promotion vehicles, considering several business rules as constraints. To efficiently solve this mixed-integer nonlinear program, we reformulate it as a convex optimization form by using the demand function’s multiplicative structure and the concept of geometric programming. Furthermore, to reduce the running time of the large-scale instances, we develop a Lagrangian decomposition algorithm, dividing the original model into a geometric program and an integer program. The algorithm significantly improves computational efficiency as evidenced by a reduction in running time from 8,125 to 78 seconds for large-scale instances. Finally, utilizing real sales data from a meal delivery company, we demonstrate that applying the convex promotion optimization model allows the company to increase its profits by roughly 21% compared with scenarios where neither price reductions nor promotion vehicles are utilized. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0275 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0275 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Mixed-integer second-order cone programs (MISOCPs) form a novel class of mixed-integer convex programs, which can be solved very efficiently as a result of the recent advances in optimization solvers. This paper shows how various performance metrics of M/G/1 queues can be modeled by different MISOCPs. To motivate the reformulation method, it is first applied to a challenging stochastic location problem with congestion, which is broadly used to design socially optimal service systems. Three different MISOCPs are developed and compared on different sets of benchmark test problems. The new formulations efficiently solve very large-size test problems that cannot be solved by the two existing methods developed based on linear programming within reasonable time. The superiority of the conic reformulation method is next shown over a state-space decomposition method recently used to solve an assignment problem in queueing systems. Finally, the general applicability of the method is shown for similar optimization problems that use queue-theoretic performance measures to address customer satisfaction and service quality.