Optimal Replenishment Strategy for Inventory Mechanism with a Known Price Increase and Backordering in Finite Horizon

For the finite horizon inventory mechanism with a known price increase and backordering, based on minimizing the inventory cost, we establish two mixed integer optimization models. By buyer’s cost analysis, we present the closed-form solutions to the models, and by comparing the minimum cost of the two strategies, we provide an optimal ordering policy to the buyer. Numerical examples are presented to illustrate the validity of the model, and sensitivity analysis on major parameters is also made to show some insights to the inventory model.

Inverse Mixed Integer Optimization: Polyhedral Insights and Trust Region Methods

Inverse optimization—determining parameters of an optimization problem that render a given solution optimal—has received increasing attention in recent years. Although significant inverse optimization literature exists for convex optimization problems, there have been few advances for discrete problems, despite the ubiquity of applications that fundamentally rely on discrete decision making. In this paper, we present a new set of theoretical insights and algorithms for the general class of inverse mixed integer linear optimization problems. Specifically, a general characterization of optimality conditions is established and leveraged to design new cutting plane solution algorithms. Through an extensive set of computational experiments, we show that our methods provide substantial improvements over existing methods in solving the largest and most difficult instances to date.

Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

Many central problems throughout optimization, machine learning, and statistics are equivalent to optimizing a low-rank matrix over a convex set. However, although rank constraints offer unparalleled modeling flexibility, no generic code currently solves these problems to certifiable optimality at even moderate sizes. Instead, low-rank optimization problems are solved via convex relaxations or heuristics that do not enjoy optimality guarantees. In “Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints,” Bertsimas, Cory-Wright, and Pauphilet propose a new approach for modeling and optimizing over rank constraints. They generalize mixed-integer optimization by replacing binary variables z that satisfy z2 =z with orthogonal projection matrices Y that satisfy Y2 = Y. This approach offers the following contributions: First, it supplies certificates of (near) optimality for low-rank problems. Second, it demonstrates that some of the best ideas in mixed-integer optimization, such as decomposition methods, cutting planes, relaxations, and random rounding schemes, admit straightforward extensions to mixed-projection optimization.

Solution of Mixed-Integer Optimization Problems in Bioinformatics with Differential Evolution Method

The solution of the so-called mixed-integer optimization problem is an important challenge for modern life sciences. A wide range of methods has been developed for its solution, including metaheuristics approaches. Here, a modification is proposed of the differential evolution entirely parallel (DEEP) method introduced recently that was successfully applied to mixed-integer optimization problems. The triangulation recombination rule was implemented and the recombination coefficients were included in the evolution process in order to increase the robustness of the optimization. The deduplication step included in the procedure ensures the uniqueness of individual integer-valued parameters in the solution vectors. The developed algorithms were implemented in the DEEP software package and applied to three bioinformatic problems. The application of the method to the optimization of predictors set in the genomic selection model in wheat resulted in dimensionality reduction such that the phenotype can be predicted with acceptable accuracy using a selected subset of SNP markers. The method was also successfully used to optimize the training set of samples for such a genomic selection model. According to the obtained results, the developed algorithm was capable of constructing a non-linear phenomenological regression model of gene expression in developing a Drosophila eye with almost the same average accuracy but significantly less standard deviation than the linear models obtained earlier.

Crew Assignment with Duty Time Limits for Transport Services: Tight Multicommodity Models

Modeling Crew Assignments for Urban Transport Services Using Differentiated Flows Public transit agencies need to judiciously deploy their limited crew members to operate numerous daily scheduled services, while meeting duty and working time regulations for each crew member. Since crew costs account for a large portion of the organizations’ operating expenses, minimizing the total crew and transfer costs is very important. But, with hundreds of daily trips and millions of possible crew itineraries, optimizing trip-to-crew assignment decisions is challenging. In “Crew Assignment with Duty Time Limits for Transport Services: Tight Multicommodity Models,” Balakrishnan, Mirchandani, and Lin propose a novel integer optimization model that represents itineraries as multicommodity flows, differentiated by first trip and depot, to capture the duty time limits and incorporate additional requirements such as selecting equitable schedules. The authors show that this compact model can be tighter than previous formulations, further strengthen the model, and propose a restricted optimization approach combined with an optimality test to generate near-optimal solutions quickly. Extensive computational tests using well-known and real-life problem instances show that the proposed model and solution approach can be very effective in practice.

Parameter Estimation for Interrupted Sampling Repeater Jamming Based on ADMM

By repeatedly sampling, storing, and retransmitting parts of the radar signal, interrupted sampling repeater jamming (ISRJ) based on digital radio frequency memory (DRFM) can produce a train of secondary false targets symmetrical to the main false target, threatening to mislead or deceive the victim radar system. This paper proposes a computationally-effective method to estimating the parameters for ISRJ by resorting to the framework of alternating direction method of multipliers (ADMM). Firstly, the analytical form of pulse compression is derived. Then, for the purpose of estimating the parameters of ISRJ, the original problem is transformed into a nonlinear integer optimization model with respect to a window vector. On this basis, the ADMM is introduced to decompose the nonlinear integer optimization model into a series of sub-problems to estimate the width and number of ISRJ’s sample slices. Finally, the numerical simulation results show that, compared with the traditional time-frequency (TF) method, the proposed method exhibits much better performance in accuracy and stability.

The cost of not knowing enough: mixed-integer optimization with implicit Lipschitz nonlinearities

It is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit constraint functions that cannot be solved for any parameter analytically. For this class of mixed-integer problems we propose a novel algorithm based on an approximation of the feasible set in the domain of the nonlinear function—in contrast to an approximation of the graph of the function considered in prior work. This method is shown to compute approximate global optimal solutions in finite time and we also provide a worst-case iteration bound. In some first numerical experiments we show that the “cost of not knowing enough” is rather high by comparing our approach with the open-source global solver . This reveals that a lot of work is still to be done for this highly challenging class of problems and we thus finally propose some possible directions of future research.

Course Scheduling Under Sudden Scarcity: Applications to Pandemic Planning

Problem definition: Physical distancing requirements during the COVID-19 pandemic have dramatically reduced the effective capacity of university campuses. Under these conditions, we examine how to make the most of newly scarce resources in the related problems of curriculum planning and course timetabling. Academic/practical relevance: We propose a unified model for university course scheduling problems under a two-stage framework and draw parallels between component problems while showing how to accommodate individual specifics. During the pandemic, our models were critical to measuring the impact of several innovative proposals, including expanding the academic calendar, teaching across multiple rooms, and rotating student attendance through the week and school year. Methodology: We use integer optimization combined with enrollment data from thousands of past students. Our models scale to thousands of individual students enrolled in hundreds of courses. Results: We projected that if Massachusetts Institute of Technology moved from its usual two-semester calendar to a three-semester calendar, with each student attending two semesters in person, more than 90% of student course demand could be satisfied on campus without increasing faculty workloads. For the Sloan School of Management, we produced a new schedule that was implemented in fall 2020. The schedule allowed half of Sloan courses to include an in-person component while adhering to safety guidelines. Despite a fourfold reduction in classroom capacity, it afforded two thirds of Sloan students the opportunity for in-person learning in at least half their courses. Managerial implications: Integer optimization can enable decision making at a large scale in a domain that is usually managed manually by university administrators. Our models, although inspired by the pandemic, are generic and could apply to any scheduling problem under severe capacity constraints.