Consistency Cuts for Dantzig-Wolfe Reformulations

A New Family of Valid-Inequalities for Dantzig-Wolfe Reformulation of Mixed Integer Linear Programs In “Consistency Cuts for Dantzig-Wolfe Reformulation,” Jens Vinther Clausen, Richard Lusby, and Stefan Ropke present a new family of valid inequalities to be applied to Dantzig-Wolfe reformulations with binary linking variables. They show that, for Dantzig-Wolfe reformulations of mixed integer linear programs that satisfy certain properties, it is enough to solve the linear programming relaxation of the Dantzig-Wolfe reformulation with all consistency cuts to obtain integer solutions. An example of this is the temporal knapsack problem; the effectiveness of the cuts is tested on a set of 200 instances of this problem, and the results are state-of-the-art solution times. For problems that do not satisfy these conditions, the cuts can still be used in a branch-and-cut-and-price framework. In order to show this, the cuts are applied to a set of generic mixed linear integer programs from the online library MIPLIB. These tests show the applicability of the cuts in general.

A multi-objective integrated procurement, production, and distribution problem of supply chain network under fuzziness uncertainties

In this paper, we devoted a design under uncertainty of a four-echelon supply chain network including multiple suppliers, multiple plants, multiple distributors and multiple customers. The proposed model is a bi-objective mixed integer linear programming which considers several constraints and aims to minimize the total costs including the procurement, production, storage and distribution costs as well as to maximize on-time deliveries (OTD). To bring the model closer to real-world planning problems, the objective function coefficients (e.g. procurement cost, production cost, inventory holding and transport costs) and other parameters (e.g., demand, production capacity and safety stock level), are all considered triangular fuzzy numbers. Besides, a hybrid mathematical model-based on credibility approach is constructed for the problem, i.e., expected value and chance constrained models. Moreover, to build the crisp equivalent model, we use different property of the credibility measure. The resulted crisp equivalent model is a bi-objective mixed integer linear programs (BOMILP). To transform this crisp BOMILP into a single objective mixed integer linear programs (MILP) model, we apply three different aggregation functions. Finally, numerical results are reported for a real case study to demonstrate the efficiency and applicability of the proposed model.

Stochastic optimization over the Pareto front by the augmented weighted Tchebychev program

В этой статье мы предлагаем новый алгоритм для решения многоцелевых задач стохастического целочисленного линейного программирования (MOSILP). Мы оптимизируем данную стохастическую линейную функцию φ по полному набору эффективных решений MOSILP, которые были преобразованы в эквивалентную детерминированную задачу с использованием неопределенных предположений, вводимых лицом, принимающим решения. Для этой цели мы применяем двухэтапный рекурсивный подход, при котором расширенная взвешенная программа Чебышева постепенно оптимизируется для создания эффективного решения, тем самым улучшая значение вспомогательной функции φ . Предлагаемый здесь подход определяет и решает последовательность целочисленных линейных программ с нарастающими ограничениями, так что на каждом этапе алгоритма генерируется новое эффективное решение. Для иллюстрации представлен числовой пример In this paper, we propose a novel algorithm to deal with multi-objective stochastic integer linear programming problems (MOSILP). Given a stochastic linear function φ , we will optimize it over the full set of efficient solutions of a MOSILP. We convert the latter into an equivalent deterministic problem using uncertain aspirations which are inputs specified by the decision maker. For this purpose, we adopt a 2-stage recourse approach where an augmented weighted Tchebychev program is progressively optimized to generate an efficient solution, the value of the utility function φ is improved to enumerate all efficient solutions. The approach proposed here defines and solves a sequence of progressively more constrained integer linear programs, so that a new efficient solution is generated at each step of the algorithm. A numerical example is presented for illustration

Empowering the configuration-IP: new PTAS results for scheduling with setup times

Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems where a set of items has to be placed in multiple target locations. Herein, a configuration describes a possible placement on one of the target locations, and the IP is used to choose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and, therefore, be solved efficiently. As an application, we consider scheduling problems with setup times in which a set of jobs has to be scheduled on a set of identical machines with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed, an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time $$f(1/\varepsilon )\cdot \mathrm {poly}(|I|)$$ f ( 1 / ε ) · poly ( | I | ) . Previously, only constant factor approximations of 5/3 and $$4/3 + \varepsilon $$ 4 / 3 + ε , respectively, were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine.

Cost-Minimizing System Design for Surveillance of Large, Inaccessible Agricultural Areas Using Drones of Limited Range

Drones are used increasingly for agricultural surveillance. The limited flight range of drones poses a problem for surveillance of large, inaccessible areas. One possible solution is to place autonomous, solar-powered charging stations within the area of interest, where the drone can recharge during its mission. This paper designs and implements a software system for planning low-cost drone coverage of large areas. The software produces a feasible, cost-minimizing charging station placement, as well as a drone path specification. Multiple optimizations are required, which are formulated as integer linear programs. In extensive simulations, the resulting drone paths achieved 70–90 percent of theoretical optimal performance in terms of minimizing mission time for a given number of charging stations, for a variety of field configurations.

Xeggora: Exploiting Immune-to-Evidence Symmetries with Full Aggregation in Statistical Relational Models (Extended Abstract)

We present improvements in maximum a-posteriori inference for Markov Logic, a widely used SRL formalism. Several approaches, including Cutting Plane Aggregation (CPA), perform inference through translation to Integer Linear Programs. Aggregation exploits context-specific symmetries independently of evidence and reduces the size of the program. We illustrate much more symmetries occurring in long ground clauses that are ignored by CPA and can be exploited by higher-order aggregations. We propose Full-Constraint-Aggregation, a superior algorithm to CPA which exploits the ignored symmetries via a lifted translation method and some constraint relaxations. RDBMS and heuristic techniques are involved to improve the overall performance. We introduce Xeggora as an evolutionary extension of RockIt, the query engine that uses CPA. Xeggora evaluation on real-world benchmarks shows progress in efficiency compared to RockIt especially for models with long formulas.

Representative Solutions for Bi-Objective Optimisation

Bi-objective optimisation aims to optimise two generally competing objective functions. Typically, it consists in computing the set of nondominated solutions, called the Pareto front. This raises two issues: 1) time complexity, as the Pareto front in general can be infinite for continuous problems and exponentially large for discrete problems, and 2) lack of decisiveness. This paper focusses on the computation of a small, “relevant” subset of the Pareto front called the representative set, which provides meaningful trade-offs between the two objectives. We introduce a procedure which, given a pre-computed Pareto front, computes a representative set in polynomial time, and then we show how to adapt it to the case where the Pareto front is not provided. This has three important consequences for computing the representative set: 1) does not require the whole Pareto front to be provided explicitly, 2) can be done in polynomial time for bi-objective mixed-integer linear programs, and 3) only requires a polynomial number of solver calls for bi-objective problems, as opposed to the case where a higher number of objectives is involved. We implement our algorithm and empirically illustrate the efficiency on two families of benchmarks.