Integer Formulation Research Articles

Optimization of gas metering maintenance services: A multiobjective vehicle routing problem with a set of predefined overlapping time windows

AbstractOptimization of maintenance services of a company working in the gas metering sector is investigated. In particular, daily tasks of operators are optimized by exploiting the paradigm of vehicle routing problems with a set of predefined time windows that overlap one with the others and four competing objectives to take into account. First, an exact integer formulation is presented that can be solved only for a reduced number of customer sites to visit. Then, a heuristic approach is proposed to find approximate solutions with huge savings on the computational effort, also for high‐dimensional instances. Numerical results on both real and synthetic scenarios showcase the effectiveness of the approach.

The rank pricing problem with ties

In the Rank Pricing Problem (RPP), a firm intends to maximize its profit through the pricing of a set of products to sell. Customers are interested in purchasing at most one product among a subset of products. To do so, they are endowed with a ranked list of preferences and a budget. Their choice rule consists in purchasing the highest-ranked product in their list and whose price is below their budget. In this paper, we consider an extension of RPP, the Rank Pricing Problem with Ties (RPPT), in which we allow for indifference between products in the list of preferences of the customers. Considering the bilevel structure of the problem, this generalization differs from the RPP in that it can lead to multiple optimal solutions for the second level problems associated to the customers. In such cases, we look for pessimistic optimal solutions of the bilevel problem : the customer selects the cheapest product. We present a new three-indexed integer formulation for RPPT and introduce two resolution approaches. In the first one, we project out the customer decision variables, obtaining a reduced formulation that we then strengthen with valid inequalities from the former formulation. Alternatively, we follow a Benders decomposition approach leveraging the separability of the problem into a master problem and several subproblems. The separation problems to include the valid inequalities to the master problem dynamically are shown to reduce to min-cost flow problems. We finally carry out extensive computational experiments to assess the performance of the resolution approaches.

Optimizing wind farms layouts for maximum energy production using probabilistic inference: Benchmarking reveals superior computational efficiency and scalability

Successful development of wind farms relies on the optimal siting of wind turbines to maximize the power capacity under stochastic wind conditions and wake losses caused by neighboring turbines. This paper presents a novel method to quickly generate approximate optimal layouts to support infrastructure design decisions. We model the quadratic integer formulation of the discretized layout design problem with an undirected graph that succinctly captures the spatial dependencies of the design parameters caused by wake interactions. On the undirected graph, we apply probabilistic inference using sequential tree-reweighted message passing to approximate turbine siting. We assess the effectiveness of our method by benchmarking against a state-of-the-art branch and cut algorithm under varying wind regime complexities and wind farm discretization resolutions. For low resolutions, probabilistic inference can produce optimal or nearly optimal turbine layouts that are within 3% of the power capacity of the optimal layouts achieved by state-of-the-art formulations, at a fraction of the computational cost. As the discretization resolution (and thus the problem size) increases, probabilistic inference produces optimal layouts with up to 9% more power capacity than the best state-of-the-art solutions at a much lower computational cost.

Strong IP formulations need large coefficients

The development of practically well-behaved integer programming formulations is an important aspect of solving linear optimization problems over a set X⊆{0,1}n. In practice, one is often interested in strong integer formulations with additional properties, e.g., bounded coefficients to avoid numerical instabilities. This article presents a lower bound on the size of coefficients in any strong integer formulation of X and demonstrates that certain integer sets X require (exponentially) large coefficients in any strong integer formulation. We also show that strong integer formulations of X⊆{0,1}n may require exponentially many inequalities while linearly many inequalities may suffice in weak formulations.

Baggage Carousel Assignment at Airports: Model and Case Study

Baggage operations at an airport are split into several different sub problems. The problem of assigning baggage carousels to each arriving flight is part of the important passenger arrival process. This article presents a mixed integer formulation for the underlying decision problem which aims at balancing customer satisfaction with operational needs. We introduce a static model that fulfills the different real-world requirements and balances the different objective criteria. It is shown how the model can be used in the dynamic environment of an airport. Because of the high degree of uncertainty at an airport, the issue of stability is discussed in detail. In a case study several scenarios with different time horizons and times of decision were studied. The model was verified with data from Frankfurt Airport and is successfully running as part of the decision support system at Frankfurt Airport.

Mixed integer formulations using natural variables for single machine scheduling around a common due date

While almost all existing works which optimally solve just-in-time scheduling problems propose dedicated algorithmic approaches, we propose in this work mixed integer formulations. We consider a single machine scheduling problem that aims at minimizing the weighted sum of earliness–tardiness penalties around a common due date. Using natural variables, we provide one compact formulation for the unrestrictive case and, for the general case, a non-compact formulation based on non-overlapping inequalities. We show that the separation problem related to the latter formulation is solved polynomially. In this formulation, solutions are only encoded by extreme points. We establish a theoretical framework to show the validity of such a formulation using non-overlapping inequalities, which could be used for other scheduling problems. A Branch-and-Cut algorithm together with an experimental analysis is proposed to assess the practical relevance of this mixed integer programming based methods.

Binary integer modeling of the traffic flow optimization problem, in the case of an autonomous transportation system

Our research aimed to optimize the transportation processes through the binary integer modeling of cooperative vehicle control by linking the dynamic traffic assignment approach and controlling the autonomous transport system. Our paper’s main contribution is a model transforming the optimal vehicle control problem into binary integer formulation, optimizing transport processes at the system level, and representing safety and dynamics related constraints on the vehicle level. Two small numerical case studies have illustrated the applicability and effectiveness of the model.

Pre-auction lane selection in an integrated production–distribution planning problem

ABSTRACT Integrating production scheduling with transportation decisions is an important problem that is receiving increasing interest from the logistics industry. As an order is received, the manufacturer starts planning its production while considering the appropriate decisions for delivery. The company can adopt an auction paradigm to involve external occasional drivers, besides using its own fleet for the delivery. Ahead of an auctioning process, the company should decide which deliveries should be served by its fleet and which can be assigned to the occasional drivers. This is known as the shipper lane selection problem (SLSP). This article integrates the SLSP with production scheduling and proposes an integer formulation and heuristic methods to solve the resulting integrated problem (I-SLSPS). Experiments show the validity of the proposed methods and their capability in achieving minimum cost for the integrated problem.

DSP: Schema Design for Non-Relational Applications

The way a database schema is designed has a high impact on its performance in relational databases, which are symmetric in nature. While the problem of schema optimization is even more significant for NoSQL (“Not only SQL”) databases, existing modeling tools for relational databases are inadequate for this asymmetric setting. As a result, NoSQL modelers rely on rules of thumb to model schemas that require a high level of competence. Several studies have been conducted to address this problem; however, they are either proprietary, symmetrical, relationally dependent or post-design assessment tools. In this study, a Dynamic Schema Proposition (DSP) model for NoSQL databases is proposed to handle the asymmetric nature of today’s data. This model aims to facilitate database design and improve its performance in relation to data availability. To achieve this, data modeling styles were aggregated and classified. Existing cardinality notations were empirically extended using synthetically generated queries. A binary integer formulation was used to guide the mapping of asymmetric entities from the application’s conceptual data model to a database schema. An experiment was conducted to evaluate the impact of the DSP model on NoSQL schema production and its performance. A profound improvement was observed in read/write query performance and schema production complexities. In this regard, DSP has significant potential to produce schemas that are capable of handling big data efficiently.

Polynomial size IP formulations of knapsack may require exponentially large coefficients

A desirable property of integer formulations is to consist of few inequalities having small coefficients. We show that these targets are conflicting by proving the existence of knapsack sets that need exponentially many inequalities or exponentially large coefficients in any integer formulation. Moreover, we show that there exist undirected graphs such that (in a natural model) every integer formulation of stable sets requires exponentially large coefficients if the number of non-trivial inequalities is bounded by a constant.

A Memetic Algorithm for the Cumulative Capacitated Vehicle Routing Problem Including Priority Indexes

This paper studies the Cumulative Capacitated Vehicle Routing Problem, including Priority Indexes, a variant of the classical Capacitated Vehicle Routing Problem, which serves the customers according to a certain level of preference. This problem can be effectively implemented in commercial and public environments where customer service is essential, for instance, in the delivery of humanitarian aid or in waste collection systems. For this problem, we aim to minimize two objectives simultaneously, the total latency and the total tardiness of the system. A Mixed Integer formulation is developed and solved using the AUGMECON2 approach to obtain true efficient Pareto fronts. However, as expected, the use of commercial software was able to solve only small instances, up to 15 customers. Therefore, two versions of a Memetic Algorithm with Random Keys (MA-RK) were developed to solve the problem. The computational results show that both algorithms provided good solutions, although the second version obtained denser and higher quality Pareto fronts. Later, both algorithms were used to solve larger instances (20–100 customers). The results were mixed in terms of quality but, in general, the MA-RK v2 consistently outperforms the first version. The models and algorithms proposed in this research provide useful insights for the decision-making process and can be applied to solve a wide variety of business situations where economic, customer service, environmental, and social concerns are involved.

Building Representative Matched Samples With Multi-Valued Treatments in Large Observational Studies

ABSTRACTIn this article, we present a new way of matching in observational studies that overcomes three limitations of existing matching approaches. First, it directly balances covariates with multi-valued treatments without explicitly estimating the generalized propensity score. Second, it builds self-weighted matched samples that are representative of a target population by design. Third, it can handle large datasets, with hundreds of thousands of observations, in a couple of minutes. The key insights of this new approach to matching are balancing the treatment groups relative to a target population and positing a linear-sized mixed integer formulation of the matching problem. We formally show that this formulation is more effective than alternative quadratic-sized formulations, as its reduction in size does not affect its strength from the standpoint of its linear programming relaxation. We also show that this formulation can be used for matching with distributional covariate balance in polynomial time under certain assumptions on the covariates and that it can handle large datasets in practice even when the assumptions are not satisfied. This algorithmic characterization is key to handling large datasets. We illustrate this new approach to matching in both a simulation study and an observational study of the impact of an earthquake on educational attainment. With this approach, the results after matching can be visualized with simple and transparent graphical displays: while increasing levels of exposure to the earthquake have a negative impact on school attendance, there is no effect on college admission test scores. Supplementary materials for this article are available online.

Integrated location and inventory planning in service parts logistics with customer-based service levels

We analyze the integrated logistics network design and inventory stocking problem in service parts logistics, where each customer requires a certain response time-based service level. We introduce a non-linear mixed integer optimization model that explicitly captures the interdependency between network design (locating facilities, and allocating customers’ demands to facilities) and inventory stocking decisions (stock levels and their corresponding stochastic fill rates). We then provide two different linearized mixed integer formulations for this problem that can solve small and medium size problems. We show that it is still a challenging problem in general, for which we have a Lagrangian-relaxation based approach that provides extremely tight lower and upper bounds. Our extensive computational study shows the effectiveness of the proposed approach for a variety of problem instances, producing provably near-optimal integrated solutions.

Presenting a Multi-Start Hybrid Heuristic for Solving the Problem of Two-Echelon Location-Routing Problem with Simultaneous Pickup and Delivery (2E-LRPSPD)

This study proposes a three-index flow-based mixed integer formulation to solve a two-echelon location routing problem with simultaneous pickup and delivery. In this formulation, pickup and delivery demands can be addressed using the same vehicle in each echelon of the network to reduce costs and increase logistics efficiency. We solve such NP-hard problem by developing a multistart hybrid heuristic with path relinking (MHH-PR) which is composed of local search and a variable neighbourhood descent algorithm. In the algorithm, three constructive heuristics are applied to generate diversified initial solutions, and path relinking is introduced for intensification and postoptimisation. Results indicate that MHH-PR can reduce the gap between the near optimal and global optimal solutions by 1%-2%. The proposed algorithm significantly improves computational efficiency by reducing the computational time of more than 10 min for existing cases involving 20 nodes to less than 10 s.

A Network-Constrained Rolling Transactive Energy Model for EV Aggregators Participating in Balancing Market

The increasing adoption of renewable energy sources increases the need of balancing power. The security concerns of the distribution system operators are increasing due to fast adoption of distributed energy resources (DERs). So far, various operational models are proposed to manage the DERs for economic benefit. However, there is lack of an efficient operational model that balances the security, market, and uncertainty management. With the focus on the aggregator's operation, this paper has developed an operational model for continuous operation of aggregators interactively with the grid operators. A rolling optimization model is developed to manage the uncertainties from prediction errors. The network security is ensured continuously through an iterative price negotiation process between the distribution network operator and the aggregators. The optimality of the solution is guaranteed by convexification of the mixed integer formulation.

Modularity and Integrability-based Energy Minimization in a Reconfigurable Manufacturing Environment: A Non-linear Mixed Integer Formulation

Abstract Nowadays, manufacturing environment is characterized by the necessity of customized flexibility as well as responding rapidly and cost-effectively to changing market demands while minimizing impacts on environment and society. To reach these goals, a key paradigm called sustainable manufacturing can be coupled with reconfigurable manufacturing systems (RMSs). The coupling of RMS characteristics and sustainability concerns is a basis to develop a new generation of sustainable production systems. This paper outlines sustainability in a reconfigurable environment from an energy consumption point of view. A nonlinear mathematical model is developed to optimize the energy consumption of a RMS through a redefinition of its core characteristics—modularity and integrability. The objective is to minimize the energy consumption of the system by selecting the most suitable modular machines from a set of candidate machines. The optimization problem is addressed using an exhaustive search heuristic. Finally, the applicability of the proposed approach is illustrated through a simple numerical example and the discussion of the obtained results.

P.504 Variation in cannabinoid 1 receptor and GABRA6 genes in interaction with recent stress influences distinct suicide risk-related phenotypes

In this work, an efficient formulation for the aggregation of multiple identical generating units for the mixed integer linear programming unit commitment problem is presented. To this end, as already done in literature identical units are clustered with a single integer variable defining the number of currently committed units. To correctly model the feasible aggregated ramp speed, a shifting generation level method with low computational effort is introduced, which is the main contribution of this work. In contrast to the reference binary formulation, the number of integer and continuous variables and constraints for the integer formulation does not scale with the number of aggregated units. Thus, depending on the UC problem, the computational effort can be drastically decreased, which is shown in test cases.

Coordinating resources in Stackelberg Security Games

In this work we formulate a Stackelberg Security Game that coordinates resources in a border patrol problem. In this security domain, resources from different precincts have to be paired to conduct patrols in the border due to logistic constraints. Given this structure, models that enumerate the pure defender strategies scale poorly. We describe the set of mixed strategies using a polynomial number of variables but exponentially many constraints that come from the matching polytope. We then include this description in a mixed integer formulation to compute the Strong Stackelberg Equilibrium efficiently with a branch and cut scheme. Since the optimal patrol solution is a probability distribution over the set of exponential size, we also introduce an efficient sampling method that can be used to deploy the security resources every shift. Our computational results evaluate the efficiency of the branch and cut scheme developed and the accuracy of the sampling method.

Preemptive multi-skilled resource constrained project scheduling problem with hard/soft interval due dates

This paper considers a preemptive multi-skilled resource constrained project scheduling problem in a just-in-time environment where each activity has an interval due date to be completed. In this problem setting, resuming a preempted activity requires an extra setup cost, while each time unit violation from the given due date incurs earliness or tardiness penalty. Also, processing cost of each skill to execute any activity depends on the assigned staff member to accomplish the skill. The objective function of the model aims to minimize the total cost of allocating staff to skills, earliness–tardiness penalties and preemption costs. Two integer formulations are proposed for the model which are compared in terms of number of variables, constraints and elapsed run-time to optimality. Furthermore, an ant colony based metaheuristic is developed to tackle real life scales of the proposed model. This algorithm relies on two intelligent local search heuristics. Parameters of the algorithm are calibrated using Taguchi method. The results of the experiments for the proposed algorithm confirm that the proposed algorithm has satisfying performance.

Mixed integer formulations for a routing problem with information collection in wireless networks

We study a routing-collecting problem where a system of stations is considered. A vehicle is responsible for collecting information generated continuously in the stations and to deliver it to a base station. The objective is to determine the vehicle route and the collection operations, both physical and wireless, in order to maximize the amount of information collected during a time horizon. Three mixed integer programming models are introduced and a computational study is reported to compare the performance of a solver based on each one of the models.