Integer Programming Techniques Research Articles

A mathematical programming method for constructing the shortest interconnection VLSI arrays

Mesh-connected processor array is an extensively investigated architecture in parallel processing. Massive studies have addressed the problem of using reconfiguration algorithms to solve the fault tolerance of faulty mesh-connected processor arrays. However, the subarrays generated by the previous studies still contain large interconnection length, which will lead to the increase of capacitance, power dissipation and dynamic communication cost. First, a mathematical model is established for the array reconfiguration. Then, the proposed method treats the interconnections between each PEs as a function with different integer variables, which can be solved by using effective integer programming techniques. Finally, an effective solver is called to find the optimal solution. Simulation results show that the proposed method can reduce the interconnection length of the array in the row and column directions simultaneously, thereby generating a subarray with the shortest interconnection length. On a 32 × 32 host array with fault density of 30%, the total interconnection length of the subarray can be reduced by 8.36% compared with state-of-the-art, and the average interconnection length can be reduced by 39.30%, which is more closer to the lower bound.

Improving emergency services efficiency during Islamic pilgrimage through optimal allocation of facilities

AbstractThe proper allocation of facilities within Islamic holy places is barely studied. These places annually witness millions of pilgrims and guests. The number of people during pilgrimage has been growing recently and is expected to grow further in the future. Different facilities should be optimally allocated to properly serve this large number of people and efficiently respond to their requests. In this paper, we target the problem of optimally allocating facilities within the largest Islamic holy place, Arafat. We evaluate the current allocation with respect to distance, coverage, and cover inequality metrics. Average‐case and worst‐case values of the three metrics are considered for evaluation. Results show that the current allocation strategy is far from being optimal. For the three considered metrics, we use crowdedness‐based techniques to allocate facilities within the area of Arafat. Optimal allocations are first obtained by solving integer programming (IP) models. Thereafter, two widely used metaheuristics, genetic algorithms (GA) and simulated annealing, are experimented and evaluated. Results show that the optimal solution could be easily obtained for coverage and cover inequality metrics. For the distance metric, the computation time of the IP technique is large and GA appears as a good candidate to balance between computation time and solution quality. Finally, we study allocating facilities from a multiobjective perspective. Both scalar‐weighted formulation and nondominated sorting genetic algorithm II techniques are considered. Results show that the latter technique outperforms the former technique in the number of generated Pareto‐optimal allocations as well as the quality of these allocations.

An analytics-based heuristic decomposition of a bilevel multiple-follower cutting stock problem

This paper presents a new class of multiple-follower bilevel problems and a heuristic approach to solving them. In this new class of problems, the followers may be nonlinear, do not share constraints or variables, and are at most weakly constrained. This allows the leader variables to be partitioned among the followers. We show that current approaches for solving multiple-follower problems are unsuitable for our new class of problems and instead we propose a novel analytics-based heuristic decomposition approach. This approach uses Monte Carlo simulation and k-medoids clustering to reduce the bilevel problem to a single level, which can then be solved using integer programming techniques. The examples presented show that our approach produces better solutions and scales up better than the other approaches in the literature. Furthermore, for large problems, we combine our approach with the use of self-organising maps in place of k-medoids clustering, which significantly reduces the clustering times. Finally, we apply our approach to a real-life cutting stock problem. Here a forest harvesting problem is reformulated as a multiple-follower bilevel problem and solved using our approach.

Utilization of the Karnaugh Map in Exploring Cause-effect Relations Modeled by Partially-defined Boolean Functions

This paper utilizes a modern regular and modular eight-variable Karnaugh map in a systematic investigation of cause-effect relationships modeled by partially-defined Boolean functions (PDBF) (known also as incompletely specified switching functions). First, we present a Karnaugh-map test that can decide whether a certain variable must be included in a set of supporting variables of the function, and, otherwise, might enforce the exclusion of this variable from such a set. This exclusion is attained via certain don’t-care assignments that ensure the equivalence of the Boolean quotient w.r.t. the variable, and that w.r.t. its complement, i.e., the exact matching of the half map representing the internal region of the variable, and the remaining half map representing the external region of the variable, in which case any of these two half maps replaces the original full map as a representation of the function. Such a variable exclusion might be continued w.r.t. other variables till a minimal set of supporting variables is reached. The paper addresses a dominantly-unspecified PDBF to obtain all its minimal sets of supporting variables without resort to integer programming techniques. For each of the minimal sets obtained, standard map methods for extracting prime implicants allow the construction of all irredundant disjunctive forms (IDFs). According to this scheme of first identifying a minimal set of supporting variables, we avoid the task of drawing prime-implicant loops on the initial eight-variable map, and postpone this task till the map is dramatically reduced in size. The procedure outlined herein has important ramifications for the newly-established discipline of Qualitative Comparative Analysis (QCA). These ramifications are not expected to be welcomed by the QCA community, since they clearly indicate that the too-often strong results claimed by QCA adherents need to be checked and scrutinized.

Constrained App Data Caching Over Edge Server Graphs in Edge Computing Environment

In recent years, edge computing, as an extension of cloud computing, has emerged as a promising paradigm for powering a variety of applications demanding low latency, e.g., virtual or augmented reality, interactive gaming, real-time navigation, etc. In the edge computing environment, edge servers are deployed at base stations to offer highly-accessible computing capacities to nearby end-users, e.g., CPU, RAM, storage, etc. From a service provider’s perspective, caching app data on edge servers can ensure low latency in its users’ data retrieval. Given constrained cache spaces on edge servers due to their physical sizes, the optimal data caching strategy must minimize overall user latency. In this article, we formulate this Constrained Edge Data Caching (CEDC) problem as a constrained optimization problem from the service provider’s perspective and prove its <inline-formula><tex-math notation="LaTeX">$\mathcal {NP}$</tex-math></inline-formula> -hardness. We propose an optimal approach named CEDC-IP to solve this CEDC problem with the Integer Programming technique. We also provide an approximation algorithm named CEDC-A for finding approximate solutions to large-scale CEDC problems efficiently and prove its approximation ratio. CEDC-IP and CEDC-A are evaluated on a real-world data set. The results demonstrate that they significantly outperform four representative approaches.

Exact Facetial Odd-Cycle Separation for Maximum Cut and Binary Quadratic Optimization

The exact solution of the NP-hard (nondeterministic polynomial-time hard) maximum cut problem is important in many applications across, for example, physics, chemistry, neuroscience, and circuit layout—which is also due to its equivalence to the unconstrained binary quadratic optimization problem. Leading solution methods are based on linear or semidefinite programming and require the separation of the so-called odd-cycle inequalities. In their groundbreaking research, F. Barahona and A. R. Mahjoub have given an informal description of a polynomial-time algorithm for this problem. As pointed out recently, however, additional effort is necessary to guarantee that the inequalities obtained correspond to facets of the cut polytope. In this paper, we shed more light on a so enhanced separation procedure and investigate experimentally how it performs in comparison with an ideal setting where one could even employ the sparsest, most violated, or geometrically most promising facet-defining odd-cycle inequalities. Summary of Contribution: This paper aims at a better capability to solve binary quadratic optimization or maximum cut problems and their various applications using integer programming techniques. To this end, the paper describes enhancements to a well-known algorithm for the central separation problem arising in this context; it is demonstrated experimentally that these enhancements are worthwhile from a computational point of view. The linear relaxations of the aforementioned problems are typically solved using fewer iterations and cutting planes than with a nonenhanced approach. It is also shown that the enhanced procedure is only slightly inferior to an ideal, enumerative, and, in practice, intractable global cutting-plane selection.

WITHDRAWN: Performance analysis of a single scheduling machine with cluster supply system, retardation, makespan and deterrent protection using genetic algorithm

In this paper, a Genetic algorithm (GA) approach for single machine is to diminish the total makespan time and deterrent protections are considered. The planning with delivery dates are used commonly in production. The integer programming technique is utilized in order to reduce the makespan in this paper. For avoiding the machine deteriorating, a deterrent protection are presented. An algorithm approach such as Genetic algorithm (GA) for solved the problem effectively and the performance measure has been analysed numerically.

Managing aquatic invasions: Optimal locations and operating times for watercraft inspection stations

Aquatic invasive species (AIS) cause significant ecological and economic damages around the world. A major spread mechanism for AIS is traffic of boaters transporting their watercraft from invaded to uninvaded waterbodies. To inhibit the spread of AIS, Canadian provinces and American states often set up watercraft inspection stations at roadsides, where potentially infested boats are screened for AIS and, if necessary, decontaminated. However, since budgets for AIS control are limited, watercraft inspection stations can only be operated at specific locations and daytimes. Though theoretical studies provide managers with general guidelines for AIS management, more specific results are needed to determine when and where watercraft inspections would be most effective. This is the subject of this paper. We show how linear integer programming techniques can be used to optimize watercraft inspection policies under budget constraints. We introduce our approach as a general framework and apply it to the prevention of the spread of zebra and quagga mussels (Dreissena spp.) to the Canadian province of British Columbia. We consider multiple scenarios and show how variations in budget constraints, propagule sources, and model uncertainty affect the optimal policy. Based on these results, we identify simple, generally applicable principles for optimal AIS management.

Integer and constraint programming approaches for providing optimality to the bandwidth multicoloring problem

In this paper, constraint and integer programming techniques are applied to solving bandwidth coloring problems, in the sense of proving optimality or finding better feasible solutions for benchmark instances from the literature. The Bandwidth Coloring Problem (BCP) is a generalization of the classic vertex coloring problem (VCP), where, given a graph, its vertices must be colored such that not only adjacent ones do not share the same color, but also their colors must be separated by a minimum given value. BCP is further generalized to the Bandwidth Multicoloring Problem (BMCP), where each vertex can receive more than one different color, also subject to separation constraints. BMCP is used to model the Minimum Span Channel Assignment Problem (MS-CAP), which arises in the planning of telecommunication networks. Research on algorithmic strategies to solve these problems focus mainly on heuristic approaches and the performance of such methods is tested on artificial and real scenarios benchmarks, such as GEOM, Philadelphia and Helsinki sets. We achieve optimal solutions or provide better upper bounds for these well-known instances, We also compare the effects of multicoloring demands on the performance of each exact solution approach, based on empirical analysis.

Criticality-Awareness Edge User Allocation for Public Safety

Edge computing provides a novel computing paradigm by deploying services on edge servers to serve nearby end-users with low latency. In this regard, a suitable allocation strategy is crucial that maximizes the number of users served at the minimum overall cost, which is referred to as the Edge User Allocation (EUA) problem. However, when edge computing meets public safety, some critical issues have not been fully considered by existing EUA approaches. Among these issues, the levels of danger to individuals quantitatively indicate whether individuals are in danger in an emergency. Hence, the inclusion of these levels impacts the priority for allocating resources in the EUA problem. In this paper, these levels are defined as individual criticalities formally. Then, we take them into account to formulate the novel CRiticality-EUA (CR-EUA) problem, and prove its NP-hardness. To solve this problem, an optimal approach, named CR-EUA-O, is proposed by utilizing the Integer Programming technique. Furthermore, we propose an approach with a proven approximation ratio, named CR-EUA-H, as an effective and efficient solution. Experiments are conducted on a real-world dataset to evaluate our approaches against four representative approaches. The results show the superior performance of our approaches in the overall criticality and execution time.

Mathematical Models for a Social Partitioning Problem

In this paper we develop modeling techniques for a social partitioning problem. Different social interaction regulations are imposed during pandemics to prevent the spread of diseases. We suggest partitioning a set of company employees as an effective way to curb the spread, and use integer programming techniques to model it. The goal of the model is to maximize the number of direct interactions between employees who are essential for company’s work subject to the constraint that all employees should be partitioned into components of no more than a certain size implied by the regulations. Then we further develop the basic model to take into account different restrictions and provisions. We also give heuristics for solving the problem. Our computational results include sensitivity analysis on some of the models and analysis of the heuristic performance.

Refocusing on Operational Harvest Planning Model for State-Owned Forestry in Turkey

The growing concerns on forest ecosystem services and sustainable management of the resources with workforce, material, and products require effective planning of forestry operations in a hierarchical level. Operational planning as a component of the hierarchy generates short-term harvest planning decisions to minimize total costs by making production and distribution decisions during all seasons. Operational harvest planning of wood harvesting has been not used in Turkish conditions. Many developments and changes in managerial and operational processes in Turkish state forestry require the right product in the right place at the right time. This indicates that it is time to use operational planning to solve the wood harvesting problem with respect to specific conditions of Turkish forestry. This study introduces a model for annual planning of harvest operations/operational harvest planning (OHARP) from stand to storage for a one-year time horizon. The article presents how the operational decisions can be optimized for selection of the most appropriate harvesting blocks, time, system, landing location, and transportation mode to provide the best balance between time and cost. The mathematical model of the planning problem was formulated with linear and mixed integer programming techniques. The data for the model comes from the forest planning units and operation systems which is combined to minimize total supply costs subject to technical, environmental and socio-economic constraints. The model was tested with the real harvesting data from a forest district in the Mediterranean Region for a one year planning horizon. The test results demonstrated that when the OHARP model was implemented in the test area and compared with the actual cost of the harvest operations realized in this area, a savings of at least 4% could be achieved by better matching appropriate harvesting systems and methods to the terrain using the OHARP methodology. When operational decisions including resource constraints were optimized, up to a 30% cost reduction could be achieved in terms of average harvesting and transportation cost.

Optimized Landing of Drones in the Context of Congested Air Traffic and Limited Vertiports

Drone fleet operators must be able to land the whole fleet in short notice. In practical operations, landing spots are usually much fewer than airborne drones. When many drones gravitate toward the limited landing spots simultaneously, congestion management becomes a challenge. This paper characterizes the fleet landing problem using mixed integer programming techniques and proposes a series of computational enhancements to reduce the solution time from hours to seconds. The solution algorithms are implemented in a software prototype for traffic management, and are thoroughly validated via extensive numerical examples and field simulations. For a fleet of 18 drones navigating at the same altitude layer within a 4-square kilometer area, all routing and trajectory computations can be completed in less than 5 seconds, and the entire fleet is able to complete landing at three pre-planned landing spots within about 3 minutes. Therefore, the models and algorithms are suitable for practical use.

A Decision Support System for Optimal Selection of Enterprise Information Security Preventative Actions

Types and complexity of information security related vulnerabilities are growing rapidly and present numerous challenges to the enterprises. One of the key challenges is to identify the optimal set of precautions with limited budget. Despite the fact that majority of enterprises have a budget constraint for installing and maintaining the protection systems, the majority of the previous work only focus on prioritization of security targets and do not consider the preventative actions and budget constraints. This article presents a decision support system (DSS) based on analytical hierarchical process and mixed integer programming techniques for optimal selection of enterprise information security preventative actions. The proposed approach enables maximizing the amount of risk prevented for a fixed amount of budget by identifying the optimal set of precautions. The new DSS also assists enterprise decision-makers in determining the minimum enterprise information security budget for a given level of risk. The main contribution of the paper is that it provides a risk management method to identify a multi-level threat model and the corresponding optimal combination of preventative actions for an enterprise while considering the budget constraints. The treemap information visualization technique is also integrated into the proposed method to improve information security related management decisions.

Solving integrated operating room planning and scheduling: Logic-based Benders decomposition versus Branch-Price-and-Cut

Integrated operating room planning and scheduling (IORPS) allocates patients optimally to different days in a planning horizon, assigns the allocated set of patients to ORs, and sequences/schedules these patients within the list of ORs and surgeons to maximize the total scheduled surgical time. The state-of-the-art model in the IORPS literature is a hybrid constraint programming (CP) and integer programming (IP) technique that is efficiently solved by a multi-featured Branch-Price-&Cut (BP&C) algorithm. We extend the IORPS literature in two ways: (i) we develop new mixed-integer programming (MIP) and CP models that improve the existing CP-IP model and (ii) we develop various combinatorial Benders decomposition algorithms that outperform the existing BP&C algorithm. Using the same dataset as used for the existing methods, we show that our MIP model achieves an average optimality gap of 3.84%, outperforming the existing CP-IP model that achieves an average optimality gap of 11.84%. Furthermore, our MIP model is 54–92 times faster than the CP-IP model in some of the optimally solved instances of the problem. We demonstrate that our best Benders decomposition approach achieves an average optimality gap of 0.88%, whereas the existing BP&C algorithm achieves an average optimality gap of 2.81%.

Cost-Effective App Data Distribution in Edge Computing

Edge computing, as an extension of cloud computing, distributes computing and storage resources from centralized cloud to distributed edge servers, to power a variety of applications demanding low latency, e.g., IoT services, virtual reality, real-time navigation, etc. From an app vendor's perspective, app data needs to be transferred from the cloud to specific edge servers in an area to serve the app users in the area. However, according to the pay-as-you-go business model, distributing a large amount of data from the cloud to edge servers can be expensive. The optimal data distribution strategy must minimize the cost incurred, which includes two major components, the cost of data transmission between the cloud to edge servers and the cost of data transmission between edge servers. In the meantime, the delay constraint must be fulfilled - the data distribution must not take too long. In this article, we make the first attempt to formulate this Edge Data Distribution (EDD) problem as a constrained optimization problem from the app vendor's perspective and prove its NP-hardness. We propose an optimal approach named EDD-IP to solve this problem exactly with the Integer Programming technique. Then, we propose an O(k)-approximation algorithm named EDD-A for finding approximate solutions to largescale EDD problems efficiently. EDD-IP and EDD-A are evaluated on a real-world dataset and the results demonstrate that they significantly outperform three representative approaches.

Closing the Gap for Makespan Scheduling via Sparsification Techniques

Makespan scheduling on identical machines is one of the most basic and fundamental packing problems studied in the discrete optimization literature. It asks for an assignment of n jobs to a set of m identical machines that minimizes the makespan. The problem is strongly NP-hard, and thus we do not expect a ([Formula: see text])-approximation algorithm with a running time that depends polynomially on [Formula: see text]. It has been recently shown that a subexponential running time on [Formula: see text] would imply that the Exponential Time Hypothesis (ETH) fails. A long sequence of algorithms have been developed that try to obtain low dependencies on [Formula: see text], the better of which achieves a quadratic running time on the exponent. In this paper we obtain an algorithm with an almost-linear dependency on [Formula: see text] in the exponent, which is tight under ETH up to logarithmic factors. Our main technical contribution is a new structural result on the configuration-IP integer linear program. More precisely, we show the existence of a highly symmetric and sparse optimal solution, in which all but a constant number of machines are assigned a configuration with small support. This structure can then be exploited by integer programming techniques and enumeration. We believe that our structural result is of independent interest and should find applications to other settings. We exemplify this by applying our structural results to the minimum makespan problem on related machines and to a larger class of objective functions on parallel machines. For all these cases, we obtain an efficient PTAS with running time with an almost-linear dependency on [Formula: see text] and polynomial in n.

Mixed Integer Programming Technique according to Traffic Light for Autonomous Collision Free Ground based on Traffic System

Abstract Human-computer interaction (HCI) is the study of how people design, implement, and use interactive computer systems and how computers affect individuals, organizations and society. This encompasses not only ease of use but also new interaction techniques for supporting user tasks, providing better access to information, and creating more powerful forms of communication. It involves input and output devices and the interaction techniques that use them; how information is presented and requested; how the computer’s actions are controlled and monitored; all forms of help, documentation, and training; the tools used to design, build, test, and evaluate user interfaces; and the processes that developers follow when creating Interfaces. This paper is an attempt to highlights the study of interaction between people (users) and computers.

Reactive planar non-prehensile manipulation with hybrid model predictive control

This article presents an offline solution and online approximation to the hybrid control problem of planar non-prehensile manipulation. Hybrid dynamics and underactuation are key characteristics of this task that complicate the design of feedback controllers. We show that a model predictive control approach used in tandem with integer programming offers a powerful solution to capture the dynamic constraints associated with the friction cone as well as the hybrid nature of contact. We introduce the Model Predictive Controller with Learned Mode Scheduling (MPC-LMS), which leverages integer programming and machine learning techniques to effectively deal with the combinatorial complexity associated with determining sequences of contact modes. We validate the controller design through a numerical simulation study and with experiments on a planar manipulation setup using an industrial ABB IRB 120 robotic arm. Results show that the proposed algorithm achieves closed-loop tracking of a nominal trajectory by reasoning in real-time across multiple contact modalities.

Designing Price Incentives in a Network with Social Interactions

Problem definition: The recent ubiquity of social networks allows firms to collect vast amount of data on their customers and on their social interactions. We consider a setting in which a monopolist sells an indivisible good to consumers who are embedded in a social network. Academic/practical relevance: This is an important problem as sellers can use available data to design and send targeted promotions that account for social externality effects and ultimately increase their profits. Methodology: We capture the interactions among consumers using a broad class of nonlinear utility models. This class extends the existing models by explicitly capturing externalities from subsets of agents (communities or groups) and includes several existing models as special cases (e.g., full information version of the triggering model). Assuming complete information about the interactions, we model the optimal pricing problem as a two-stage game. First, the firm designs prices to maximize profits and then consumers choose whether to purchase the item. Results: Under positive network externalities, we show the existence of a pure Nash equilibrium that is preferred by both the seller and the buyers. Using duality theory and integer-programming techniques, we reformulate the problem into a linear mixed-integer program (MIP). We derive efficient ways to optimally solve the MIP using its linear-programming relaxation for two pricing strategies: discriminative and uniform. Finally, we propose two intuitive heuristic algorithms to solve the problem for which we derive worst-case parametric performance bounds. Managerial implications: We draw interesting insights on the structure of the optimal prices and the seller’s profit. In particular, we quantify the effect on prices when using a nonlinear utility model relative to a linear model and identify settings with which it is beneficial to offer a price below cost to influential agents. Finally, we extend our model and results to the case in which the seller offers incentives (in addition to prices) to solicit actions so as to ensure network externality effects.