Linear Programming Formulation Research Articles

Drone‐enabled material handling in smart manufacturing

AbstractAmidst labor shortages, increasing land prices, and other aspects of the post‐pandemic world, our logistics chain must consider new technological alternatives. Automated material handling equipment often requires valuable floor space and specialized layouts. Uncrewed aerial vehicles, popularly called drones, offer a flexible and cost‐effective alternative. To explore the potential benefits and challenges of such systems, we used a version of the multi‐trip vehicle routing problem with time windows to model two realistic manufacturing environments as use cases. We provide the mixed integer linear programming formulation of this problem and compare results with those of a human‐only based system in a detailed simulated environment. Furthermore, as a physical proof‐of‐concept, we outfitted a commercially available drone with pick‐up and carry capabilities. We found that, even with limited drone carrying capacity, there are economic benefits realized from time savings, compared to ground‐based material handling systems. Besides these time savings, drones are sustainable as they operate on battery power and do not impair the air quality or the traffic experienced on the ground.

Linear programming formulations for the hydro production function in a water value calculation

Linear programming formulations for the hydro production function in a water value calculation

Regeneration sites selection in survivable translucent elastic optical networks

Quality of transmission of optical signals, which are propagated beyond their transparent reach, is ensured by re-amplification, re-timing and re-shaping (3R-regeneration) at certain node(s) designated as regeneration site(s). Inclusion of regeneration sites increases capital expenditure (CAPEX) of Translucent Elastic Optical Networks (T-EONs) and so Regeneration Sites Selection (RSS) is an important problem to address. Survivability of T-EONs is a vital issue as a fiber-cut (link failure) can cause disruption of huge traffic. In this paper, we address the RSS problem for survivable T-EONs (ST-EONs). We formulate RSS as an offline problem for ST-EON design. We propose an Integer Linear Program (ILP) formulation for small networks and also heuristic algorithms for large networks. Results from simulation experiments performed on three network topologies suggest that the proposed algorithms can minimize CAPEX for ST-EON design.

Boole’s Probability Bounding Problem, Linear Programming Aggregations, and Nonnegative Quadratic Pseudo-Boolean Functions

Hailperin (1965) introduced a linear programming formulation to a difficult family of problems, originally proposed by Boole (1854, 1868). Hailperin’s model is computationally still difficult and involves an exponential number of variables (in terms of a typical input size for Boole’s problem). Numerous papers provided efficiently computable bounds for the minimum and maximum values of Hailperin’s model by using aggregation that is a monotone linear mapping to a lower dimensional space. In many cases the image of the positive orthant is a subcone of the positive orthant in the lower dimensional space, and thus including some of the defining inequalities of this subcone can tighten up such an aggregation model, and lead to better bounds. Improving on some recent results, we propose a hierarchy of aggregations for Hailperin’s model and a generic approach for the analysis of these aggregations. We obtain complete polyhedral descriptions of the above mentioned subcones and obtain significant improvements in the quality of the bounds.

Quadratic descriptors and reduction methods in a two-layered model for compound inference

Compound inference models are crucial for discovering novel drugs in bioinformatics and chemo-informatics. These models rely heavily on useful descriptors of chemical compounds that effectively capture important information about the underlying compounds for constructing accurate prediction functions. In this article, we introduce quadratic descriptors, the products of two graph-theoretic descriptors, to enhance the learning performance of a novel two-layered compound inference model. A mixed-integer linear programming formulation is designed to approximate these quadratic descriptors for inferring desired compounds with the two-layered model. Furthermore, we introduce different methods to reduce descriptors, aiming to avoid computational complexity and overfitting issues during the learning process caused by the large number of quadratic descriptors. Experimental results show that for 32 chemical properties of monomers and 10 chemical properties of polymers, the prediction functions constructed by the proposed method achieved high test coefficients of determination. Furthermore, our method inferred chemical compounds in a time ranging from a few seconds to approximately 60 s. These results indicate a strong correlation between the properties of chemical graphs and their quadratic graph-theoretic descriptors.

Sub‐Tree Scheduling for Wireless Sensor Networks With Partial Coverage: Complexity and Polynomial‐Size Formulations

ABSTRACTGiven an undirected graph whose edge weights change over time slots, the sub‐tree scheduling for wireless sensor networks with partial coverage asks to partition the vertices of in non‐empty trees such that the total weight of the trees is minimized. In this article, we show that the problem is NP‐hard in both the cases where is part of the input and is a fixed instance parameter. In both our proofs we reduce the cardinality of the Steiner tree problem. Then, in order to provide easy‐to‐implement and effective computational tools to benchmark heuristics, we introduce new polynomial‐size integer linear programming formulations for the problem. Being defined by a polynomial number of variables and constraints, the formulations can be used to address instances of the problem by means of off‐the‐shelf mixed‐integer linear programming solvers with minimum implementation efforts. All proposed formulations share the property that the integrality requirement can be relaxed on subsets of variables. This enables several algorithmic choices to solve the models, including Benders' decomposition approaches and branch‐and‐bound with specific branching priorities. We experimentally identify the best resolution method for each formulation. Moreover, the experimental results obtained on benchmark instances of the problem, compared against those reported in the literature, support the effectiveness arising from using the new proposed formulations.

Regional consensus of switched positive multi-agent systems with multiple equilibria

This paper investigates regional proportional-integral-derivative consensus of switched positive multi-agent systems with multiple equilibria. A distributed proportional-integral-derivative control protocol is developed by integrating the communication protocol, agent state, and consensus error. A novel switched positive consensus error system is established and analyzed using copositive Lyapunov functions. Subsequently, a Luenberger observer with multiple equilibria is constructed to facilitate the design of an observer-based distributed proportional-integral-derivative control protocol. The regional consensus protocol is designed in the form of linear programming. By employing the proposed protocol, all states of agents are nonnegative and driven into a specific region. The main contributions of the work are summarized as: (i) Construction of a regional proportional-integral-derivative consensus protocol framework for switched positive multi-agent systems with multiple equilibria, (ii) Proposal of an observer-based distributed proportional-integral-derivative control strategy, and (iii) Analysis of the consensus using a matrix decomposition technique, copositive Lyapunov functions, and linear programming. Finally, numerical examples are provided to illustrate the effectiveness of the obtained results.

Multi-Agent Grouping and Routing with Cooperative Tasks

This paper introduces a grouping and routing problem of multiple agents for cooperative missions. The introduced problem, referred to as the Vehicle Grouping and Routing Problem with Profits, aims to maximize the total reward obtained by conducting a multi-agent mission (e.g., cooperative reconnaissance) while reducing its makespan by appropriately grouping the agents and determining their routes under operational constraints (e.g., fuel, endurance). A mixed-integer linear programming formulation and a conservative column generation-based solution procedure for the problem are proposed. A case study with homogeneous and heterogeneous agents and numerical experiments involving a cooperative reconnaissance mission with multiple unmanned aerial vehicles demonstrate the validity of the proposed formulation and solution procedure.

Evaluating and extending speedup techniques for optimal crossing minimization in layered graph drawings.

A layered graph is an important category of graph in which every node is assigned to a layer, and layers are drawn as parallel or radial lines. They are commonly used to display temporal data or hierarchical graphs. Previous research has demonstrated that minimizing edge crossings is the most important criterion to consider when looking to improve the readability of such graphs. While heuristic approaches exist for crossing minimization, we are interested in optimal approaches to the problem that prioritize human readability over computational scalability. We aim to improve the usefulness and applicability of such optimal methods by understanding and improving their scalability to larger graphs. This paper categorizes and evaluates the state-of-the-art linear programming formulations for exact crossing minimization and describes nine new and existing techniques that could plausibly accelerate the optimization algorithm. Through a computational evaluation, we explore each technique's effect on calculation time and how the techniques assist or inhibit one another, allowing researchers and practitioners to adapt them to the characteristics of their graphs. Our best-performing techniques yielded a median improvement of 2.5-17× depending on the solver used, giving us the capability to create optimal layouts faster and for larger graphs. We provide an open-source implementation of our methodology in Python, where users can pick which combination of techniques to enable according to their use case. A free copy of this paper and all supplemental materials, datasets used, and source code are available at https://osf.io/5vq79.

An Improved Freight Transportation Planning System for Less‐Than‐Truckload Operations of a Third‐Party Logistics Carrier

ABSTRACTLess‐than‐truckload (LTL) transportation is a widely used shipping modality within the logistics industry, facilitating the movement of loads insufficient in size to occupy the entirety of a truck's capacity. Drawing on the real‐life problem of a third‐party logistics (3PL) carrier, which navigates an expansive transportation network and unpredictable demand patterns, we propose a centralized planning approach for LTL transportation, taking advantage of consolidation opportunities at cross‐dock hubs to minimize long‐term freight costs while ensuring on‐time delivery. We present an integer linear programming (ILP) formulation to generate a multi‐day plan with known demands and a daily planning problem (DPP) model to be solved daily, emulating the available order information in a real planning scenario. DPP optimizes truck routes and loading plans, incorporating a penalty cost for postponed transportation requests. We devise a novel multi‐stage matheuristic integrating geographical decomposition with tractable integer programming models and column generation‐based heuristics. With this approach, the DPP is solved in fewer than 2 h of run‐time for real problem instances with as high as 1400 transportation orders. We benchmark our results against a baseline heuristic that mirrors the firm's existing planning approach through a multi‐period simulation. Our comparative evaluations reveal a noteworthy 7.8% improvement in long‐term freight costs, concurrently maintaining adherence to the 98% on‐time delivery target.

Collaborative logistics: optimizing fixed-scheduled container trains

We propose a new logistics concept for rail-based container transportation where, instead of booking individual slots on specific trains (which is the current state of the art), customers only specify from where to where they want their containers to be delivered, as well as appropriate time windows for pickup and delivery. A containers actual transportation route is then chosen by a freight forwarder based on the availability and compatibility of freight trains. We provide an optimization algorithm based on an integer linear programming formulation that can be used to find globally optimal transportation schedules for a given set of containers. Our computational experiments show that this new paradigm can significantly reduce the number of trains required for serving a given container transportation demand, improving the logistic systems’ overall environmental and economic efficiency.

A Branch-and-Price-and-Cut Algorithm for the Inland Container Transportation Problem with Limited Depot Capacity

As an effective solution to the first- and last-mile logistics of door-to-door intermodal container transportation, inland container transportation involves transporting containers by truck between terminals, depots, and customers within a local area. This paper is the first to focus specifically on the inland container transportation problem with limited depot capacity, where the storage of empty containers is constrained by physical space limitations. To reflect a more realistic scenario, we also consider the initial stock levels of empty containers at the depot. The objective of this problem is to schedule trucks to fulfill inland container transportation orders such that the overall cost is minimum and the depot is neither out of stock or over stocked at any time. A novel graphical representation is introduced to model the constraints of empty containers and depot capacity in a linear form. This problem is then mathematically modeled as a mixed-integer linear programming formulation. To avoid discretizing the time horizon and effectively achieve the optimal solution, we design a tailored branch-and-price-and-cut algorithm where violated empty container constraints for critical times are dynamically integrated into the restricted master problem. The efficiency of the proposed algorithm is enhanced through the implementation of several techniques, such as a heuristic label-setting method, decremental state-space relaxation, and the utilization of high-quality upper bounds. Extensive computational studies are performed to assess the performance of the proposed algorithm and justify the introduction of enhancement strategies. Sensitivity analysis is additionally conducted to investigate the implications of significant influential factors, offering meaningful managerial guidance for decision-makers.

Combined Barrier-Target Coverage for Directional Sensor Network.

Over the past twenty years, camera networks have become increasingly popular. In response to various demands imposed on these networks, several coverage models have been developed in the scientific literature, such as area, trap, barrier, and target coverage. In this paper, a new type of coverage task, the Maximum Target Coverage with k-Barrier Coverage (MTCBC-k) problem, is defined. Here, the goal is to cover as many moving targets as possible from time step to time step while continuously maintaining k-barrier coverage over the region of interest (ROI). This approach is different from independently solving the two tasks and then merging the results. An Integer Linear Programming (ILP) formulation for the MTCBC-k problem is presented. Additionally, two types of camera clustering methods have been developed. This approach allows for solving smaller ILPs within clusters, and combining their solutions. Furthermore, a polynomial-time greedy algorithm has been introduced as an alternative to solve the MTCBC-k problem. An example was also provided of how the aforementioned methods can be modified to handle a more realistic scenario, where only the targets detected by the cameras are known, rather than all the targets within the ROI. The simulations were run with both dense and sparse camera placements, convincingly supporting the usefulness of the clustering and greedy methods.

A 0,1 Linear Programming Approach to Deadlock Detection and Management in Railways

In railway systems, a deadlock occurs when trains accidentally occupy positions that prevent each other from moving forward. Although deadlocks are rare events, they do occur from time to time, requiring costly recourse actions and generating significant knock-on delays. In this paper, we present a noncompact 0,1 linear programming formulation and a methodology for discovering (possibly future) deadlocks and the subsequent implementation of optimal recovery measures. The approach is implemented in a tool to dispatch trains in real time developed in cooperation with Union Pacific (UP) and currently in operations on the entire UP network. Funding: This work was partially funded by Europe’s Rail, Flagship Project MOTIONAL [Action Horizon JU Innovation, Project 101101973]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2024.0521 .

Dynamic open time‐dependent traveling salesman problem with speed optimization

AbstractIncreased awareness of people of the problems caused by CO2 emissions brings companies to consider environmental issues in their distribution systems. The rapid advance in technology allows logistics companies to tackle with dynamic nature of distribution networks (e.g., a change in the vehicle speed due to unexpected events). The planned routes at the beginning of the time horizon could be subject to modification at any point in time to account for the recent traffic information. This study addresses a dynamic open time‐dependent traveling salesman problem. The problem also involves speed optimization that aims to find optimal vehicle speed in a dynamic setting by respecting real‐time traffic conditions. We develop a mixed integer linear programming (MILP) formulation for the addressed problem to determine routing and vehicle speed decisions. Furthermore, a MILP‐based myopic‐clustering decomposition heuristic algorithm has been introduced to solve large‐sized instances within reasonable solution times. The use of the heuristic algorithm provides decision‐makers with a responsiveness capacity by enabling fast incorporation of dynamically observed data during operations. The numerical analyses demonstrate the potential benefits of employing the proposed tools.

Last train schedule optimization for metro systems considering minimum adjustment cost under elastic passenger demand

ABSTRACT Optimizing the last train schedule is challenged by the elastic nature of passenger flow and the immutability of the schedule once announced to the public. The elastic passenger demand is first mathematically described, and an optimization method is proposed to minimize the adjustment to the last trains while maximizing the accessible passengers within the metro network. Linearization techniques are then employed to transform it into a linear programming form. The proposed method is verified through a real-world case study, which indicate an 8.4% increase in accessible passenger flow on a weekday with adjustments required for only 3 lines and 10 stations’ last train times. In a special event scenario, a 9.8% is achieved with adjustments to 2 lines and 8 stations’ last train times. The results suggest that for existing metro systems, judiciously adjusting the last train times can effectively enhance the overall level of nighttime operational service.

Stability and [formula omitted]-gain analysis of switched positive systems with unstable subsystems and sector nonlinearities

In this paper, we are concerned with exponential stability and L1-gain performance of a type of switched nonlinear positive systems (SNPSs) with time-varying delay made up of partially unstable subsystems. What sets it apart from other articles is that nonlinear parts of this article are confined in a certain region. Besides, the adequacy criteria are obtained in a linear programming format via designing the novel nonlinear Lyapunov-Krasovskii (L-K) functionals and a set of special switching sequences under average dwell time (ADT) switching rule. Eventually, with the help of comparison principle, the theoretical results can be further expanded to time-varying switched systems. As practical applications, we apply the theoretical achievements to stability of switched neural networks to confirm the feasibility of the conclusions.

Complexity issues concerning the quadruple Roman domination problem in graphs

Given a graph G with vertex set V(G), a mapping h:V(G)→{0,1,2,3,4,5} is called a quadruple Roman dominating function (4RDF) for G if it holds the following. Every vertex x such that h(x)∈{0,1,2,3} satisfies that h(N[x])=∑v∈N[x]h(v)≥|{y:y∈N(x)andh(y)≠0}|+4, where N(x) and N[x] stands for the open and closed neighborhood of x, respectively. The smallest possible weight ∑x∈V(G)h(x) among all possible 4RDFs h for G is the quadruple Roman domination number of G, denoted by γ[4R](G).This work is focused on complexity aspects for the problem of computing the value of this parameter for several graph classes. Specifically, it is shown that the decision problem concerning γ[4R](G) is NP-complete when restricted to star convex bipartite, comb convex bipartite, split and planar graphs. In contrast, it is also proved that such problem can be efficiently solved for threshold graphs where an exact solution is demonstrated, while for graphs having an efficient dominating set, tight upper and lower bounds in terms of the classical domination number are given. In addition, some approximation results to the problem are given. That is, we show that the problem cannot be approximated within (1−ϵ)ln⁡|V| for any ϵ>0 unless P=NP. An approximation algorithm for it is proposed, and its APX-completeness proved, whether graphs of maximum degree four are considered. Finally, an integer linear programming formulation for our problem is presented.

DeLuxing: Deep Lagrangian Underestimate Fixing for Column-Generation-Based Exact Methods

Advancing Column Generation by a Novel Variable Fixing Method In the paper titled “DeLuxing: Deep Lagrangian Underestimate Fixing for Column-Generation-Based Exact Methods,” Dr. Yu Yang introduces DeLuxing—an innovative variable-fixing technique that significantly advances column-generation-based exact methods for solving large-scale optimization problems, particularly vehicle routing problems (VRPs). DeLuxing leverages a novel linear programming formulation with a small subset of the enumerated variables, which is theoretically guaranteed to yield qualified dual solutions for computing Lagrangian underestimates. By eliminating over 75% of the unnecessary variables, DeLuxing drastically boosts computational efficiency, doubling the size of CMTVRPTW (capacitated multitrip vehicle routing problem with time windows) instances that can be solved optimally. Additionally, this breakthrough accelerates top VRP solvers like RouteOpt by up to 71%. The core concept underpinning DeLuxing extends to broader contexts such as variable type relaxation and cutting plane addition, achieving an additional 25% speedup for difficult instances.

Learning the Graph Structure of Regular Vine-Copulas from Dependence Lists

Regular vine copulas (R-vines) provide a comprehensive framework for modeling high-dimensional dependencies using a hierarchy of trees and conditional pair-copulas. While the graphical structure of R-vines is traditionally derived from data, this work introduces a novel approach by utilizing a (conditional) pairwise dependence list. Our primary goal is to construct R-vine graphs that include the maximum possible number of dependence relationships specified in such lists. To tackle this optimization challenge, characterized by exponential growth in the search space and the structural constraints of R-vines, we propose two distinct methodologies: A 0-1 linear programming formulation and a Genetic Algorithm (GA). Additionally, the Randomized Constructive Technique (RCT) is employed to generate the initial population of the GA, serving as a baseline for our comparison. Experimental results reveal the superior performance of the GA over the RCT in terms of success rate, incorporating more relationships than RCT into the constructed R-vine graphs and achieving near-optimal or optimal graph structures.