Exact Optimization Research Articles

An Inventory Ordering Model for Deteriorating Items with Compounding and Backordering

We consider the optimal order quantity problem for exponentially deteriorating items where the opportunity cost is based on compound interest and backorders are allowed. Our objectives in this research are to develop a model that accurately models deterioration, compound interest and backordering, and determine a near-optimal and intuitive closed-form solution for the proposed model. Deteriorating items include various chemicals, gasoline and petroleum products, fresh produce, bulk and liquid food products, batteries, and some electronic components. These items incur losses over time due to spoilage, evaporation, chemical decomposition, breakdown, or deterioration in general. Exponential deterioration is commonly used to model this phenomenon, which results in a negative exponential inventory level function, which is asymmetric in the sense that the rate of depletion is highest at the beginning of an ordering cycle, and lowest at the end. On the other hand, the rate of deterioration for individual items is the same at both ends of the cycle, which means it is symmetric. Compounding also leads to exponential terms in the opportunity cost function. Both of these factors result in a total cost function that does not have a closed-form optimal solution. We therefore approximate the total cost function using a Taylor series expansion approximation of the exponential function and derive a closed-form solution that is simple and logical, and very close to the exact optimum, which makes it attractive to the practitioners as a quick and accurate calculation. Our closed form solutions for both the basic and the planned backorders models are very close to the exact optimum, as shown by extensive numerical experiments.

City-scale optimal location planning of Green Infrastructure using piece-wise linear interpolation and exact optimization methods

Green Infrastructure (GI) location planning involves multi-objective spatial analysis considering simultaneous benefits. Fast optimization results are often required (e.g., collaborative modeling exercises and stakeholder involvement workshops), while the simulation-optimization approach is impractical for these cases if it relies on time-consuming simulations of complex drainage system models. This paper proposes a new method using a surrogated model based upon targeted executions of a more complex urban drainage model. This method uses a piece-wise linear interpolation to predict the hydrological impact of individual impervious-to-pervious area changes and the Mixed Integer Linear Optimization to find the optimal location of GI practices. We applied the methodology to the capital city of Colombia, Bogotá, with the objectives of reducing runoff, combined sewer overflows (CSOs), and separate sanitary overflows (SSOs). Two multi-objective optimization approaches (i.e., a lexicographic and a weighted-sum model) were explored, showing the high efficiency of this approach in building Pareto Fronts and rapidly suggesting alternative solutions for different budgets. It was found that by implementing GIs on one-third of the city’s GI-feasible public areas (2.2% of city area), under a 10-year 6-h rainfall event, the total reduction on runoff, CSO, and SSO are 0.9%, 1.8%, and 2.4%, respectively. These results show the importance of promoting GI among private-land owners, especially in cities with land predominantly private-owned. This new approach is particularly useful for applications under data scarcity and high-uncertainty scenarios.

On the relationship between lean scheduling and economic performance in shipbuilding: A proposed model and comparative evaluation

Few studies have examined shipbuilding production processes from a lean scheduling perspective, a production scheduling approach where the primary focus is on minimizing key sources of production waste while ensuring production continuity and on-time delivery. Specifically, this research develops a lean scheduling approach and analytical model for shipbuilding production processes that minimize two specific types of production waste - excessive overproduction and waiting time - critical to the economic viability and management of projects in this industry. From a lean practice-based view (PBV) perspective of the production scheduling nuances in shipbuilding, we develop and numerically apply a lean scheduling model that targets these specific sources of waste and predicts the performance improvement gains. The applicability of the proposed scheduling approach for shipbuilding production is demonstrated through a corresponding numerical and sensitivity analysis where we compare the proposed model's performance with two alternative Makespan scheduling approaches commonly used in shipbuilding practice. Exact optimization results show a significant reduction in both forms of wastes using our proposed lean scheduling approach, leading to a remarkable boost in shipbuilding cost performance over the more traditional Makespan-minimizing scheduling approaches commonly used in this industry.

Reduction of Annual Operational Costs in Power Systems through the Optimal Siting and Sizing of STATCOMs

The problem of the optimal siting and placement of static compensates (STATCOMs) in power systems is addressed in this paper from an exact mathematical optimization point of view. A mixed-integer nonlinear programming model to present the problem was developed with the aim of minimizing the annual operating costs of the power system, which is the sum of the costs of the energy losses and of the installation of the STATCOMs. The optimization model has constraints regarding the active and reactive power balance equations and those associated with the devices’ capabilities, among others. To characterize the electrical behavior of the power system, different load profiles such as residential, industrial, and commercial are considered for a period of 24 h of operation. The solution of the proposed model is reached with the general algebraic modeling system optimization package. The numerical results indicate the positive effect of the dynamic reactive power injections in the power systems on annual operating cost reduction. A Pareto front was built to present the multi-objective behavior of the studied problem when compared to investment and operative costs. The complete numerical validations are made in the IEEE 24-, IEEE 33-, and IEEE 69-bus systems, respectively.

Dynamic Reactive Power Compensation in Power Systems through the Optimal Siting and Sizing of Photovoltaic Sources

The problem of the optimal placement and sizing of photovoltaic power plants in electrical power systems from high- to medium-voltage levels is addressed in this research from the point of view of the exact mathematical optimization. To represent this problem, a mixed-integer nonlinear programming model considering the daily demand and solar radiation curves was developed. The main advantage of the proposed optimization model corresponds to the usage of the reactive power capabilities of the power electronic converter that interfaces the photovoltaic sources with the power systems, which can work with lagging or leading power factors. To model the dynamic reactive power compensation, the η-coefficient was used as a function of the nominal apparent power converter transference rate. The General Algebraic Modeling System software with the BONMIN optimization package was used as a computational tool to solve the proposed optimization model. Two simulation cases composed of 14 and 27 nodes in transmission and distribution levels were considered to validate the proposed optimization model, taking into account the possibility of installing from one to four photovoltaic sources in each system. The results show that energy losses are reduced between 13% and 56% as photovoltaic generators are added with direct effects on the voltage profile improvement.

Virtual Savant as a generic learning approach applied to the basic independent Next Release Problem

This article presents how Virtual Savant (VS) can be used to automatically learn, from an exact algorithm, how to solve the basic independent Next Release Problem in a quick and accurate way. This variant of the Next Release Problem (NRP) is in essence a 0/1 Knapsack Problem and VS is applied to solve the underlying optimization problem. VS is a generic problem-solving approach based on machine learning and heuristics, that works by mimicking how a reference program produces solutions to problem instances. Essentially, VS learns how to generate solutions to a given problem from a reference algorithm and a training set of instances, and it is able to efficiently solve new problem instances by using the acquired knowledge. In this paper, an exact optimizer is used as a reference algorithm. Hence, we are using VS to learn from optimal solutions, which helps to reduce the approximation error inherent to the learning process. We compare five versions of VS (differing in the heuristics they implement) on a large benchmark, composed of problems with different sizes and difficulties. For the best VS configuration, which also has the lowest computational complexity, computed solutions differ less than 1% from the optima in the worst case. Therefore, VS succeeds in learning how to solve the problem under study, and it does so in a highly efficient way, exempting the programmer from having a deep knowledge of the problem domain or highly specialized parallel programming skills.

Global exact optimization for covering a rectangle with 6 circles

We address the problem of covering a rectangle with six identical circles, whose radius is to be minimized. We focus on open cases from Melissen and Schuur (Discrete Appl Math 99:149–156, 2000). Depending on the rectangle side lengths, different configurations of the circles, corresponding to the different ways they are placed, yield the optimal covering. We prove the optimality of the two configurations corresponding to open cases. For the first one, we propose a mathematical mixed-integer nonlinear optimization formulation, that allows one to compute global optimal solutions. For the second one, we provide an analytical expression of the optimal radius as a function of one of the rectangle side lengths. All open cases are thus closed for the optimal covering of a rectangle with six circles.

Mechanical and Durability Analysis of Recycled Materials

The mechanical and durability properties were best at 45% GGBS and 5% Waste Glass with 0.4 water/cement ratio. The recycled materials implemented for mix proportion were waste glass provided considerably to enhance its properties when added with GGBS. In most of the research work, the effect of WG and GGBS in concrete as a partial substitution of fine aggregate and cement individually is analyzed. Previous studies only show the individual impact of these concrete recycled materials on mechanical and durability properties. In the present study, an exact optimum substitution level of cement by GGBS (15 – 60% at an increment of 15%) and fine aggregate by the waste glass (5 – 20% at an increase of 5%) combined for OPC concrete mix. Mechanical (compressive strength, split tensile strength and flexural strength) and microstructural properties (FESEM) were observed on the combination of waste glass and GGBS concrete mix.

Solving the nuclear dismantling project scheduling problem by combining mixed-integer and constraint programming techniques and metaheuristics

Scheduling of megaprojects is very challenging because of typical characteristics, such as expected long project durations, many activities with multiple modes, scarce resources, and investment decisions. Furthermore, each megaproject has additional specific characteristics to be considered. Since the number of nuclear dismantling projects is expected to increase considerably worldwide in the coming decades, we use this type of megaproject as an application case in this paper. Therefore, we consider the specific characteristics of constrained renewable and non-renewable resources, multiple modes, precedence relations with and without no-wait condition, and a cost minimisation objective. To reliably plan at minimum costs considering all relevant characteristics, scheduling methods can be applied. But the extensive literature review conducted did not reveal a scheduling method considering the special characteristics of nuclear dismantling projects. Consequently, we introduce a novel scheduling problem referred to as the nuclear dismantling project scheduling problem. Furthermore, we developed and implemented an effective metaheuristic to obtain feasible schedules for projects with about 300 activities. We tested our approach with real-life data of three different nuclear dismantling projects in Germany. On average, it took less than a second to find an initial feasible solution for our samples. This solution could be further improved using metaheuristic procedures and exact optimisation techniques such as mixed-integer programming and constraint programming. The computational study shows that utilising exact optimisation techniques is beneficial compared to standard metaheuristics. The main result is the development of an initial solution finding procedure and an adaptive large neighbourhood search with iterative destroy and recreate operations that is competitive with state-of-the-art methods of related problems. The described problem and findings can be transferred to other megaprojects.

Star-based learning correlation clustering

Correlation clustering (CC) is a clustering method using a signed graph as input without specifying the number of clusters a priori. It had been widely used in real applications, such as social network and text mining. However, its exact optimization or approximate algorithms often give unsatisfactory results, especially for large-scale signed graphs. This paper tackles this problem and proposes a novel CC algorithm, termed star-based learning correlation clustering (SL-CC). The proposed SL-CC contains two phases. The first is a scale reduction for signed graphs. We propose a special motif, called a star structure, for reducing the scale of signed graphs. We assign the vertices within a star structure to have the same cluster label and then merge these vertices as a new vertex in the graph so we can shrink a large-scale graph to a much small-scale one. The second is to give a learning schema for the local search on the reduced graphs. It can discover some important stars as seeds of clusters according to the graph structure, and then justify whether the other stars need to be merged with seeds or not. We also construct a new integer linear programing (ILP) model based on cycle inequalities to perform the local search with final clustering results. The experiments and comparisons of the proposed SL-CC with some existing CC methods on synthetic and real data sets with variant scale structures of signed graphs demonstrate the efficiency and usefulness of the SL-CC algorithm.

Mexican Axolotl Optimization: A Novel Bioinspired Heuristic

When facing certain problems in science, engineering or technology, it is not enough to find a solution, but it is essential to seek and find the best possible solution through optimization. In many cases the exact optimization procedures are not applicable due to the great computational complexity of the problems. As an alternative to exact optimization, there are approximate optimization algorithms, whose purpose is to reduce computational complexity by pruning some areas of the problem search space. To achieve this, researchers have been inspired by nature, because animals and plants tend to optimize many of their life processes. The purpose of this research is to design a novel bioinspired algorithm for numeric optimization: the Mexican Axolotl Optimization algorithm. The effectiveness of our proposal was compared against nine optimization algorithms (artificial bee colony, cuckoo search, dragonfly algorithm, differential evolution, firefly algorithm, fitness dependent optimizer, whale optimization algorithm, monarch butterfly optimization, and slime mould algorithm) when applied over four sets of benchmark functions (unimodal, multimodal, composite and competition functions). The statistical analysis shows the ability of Mexican Axolotl Optimization algorithm of obtained very good optimization results in all experiments, except for composite functions, where the Mexican Axolotl Optimization algorithm exhibits an average performance.

A hybrid multi-objective approach for real-time flexible production scheduling and rescheduling under dynamic environment in Industry 4.0 context

With the advent of industry-4.0 era, industrial production are evolving towards high flexibility, diversity, customisation, and dynamism. We address a realistic scenario of a smart manufacturing system, which concerns the production scheduling of complex multi-level products under a dynamic flexible job shop environment with shop floor disruptions incorporated. The products are assembled from multiple basic parts, whose fabrication processes are highly flexible, involving alternative process plans, alternative machines and alternative processing sequences of operations. We aim at providing Pareto solutions, with consideration of three typical optimisation objectives, including makespan, maximum machine workload, and total tardiness. A hybrid MPGA-CP approach is designed for the problem. To the best our knowledge, this is the first attempt to embed an exact optimisation technique into a meta-heuristic algorithm in the domain of production scheduling. Compared with other alternative approaches, its efficiency and performance are proven to be outstanding in solving medium-to-large scale problems, covering the largest proportion of Pareto solutions among all tested approaches. Furthermore, we constructed a simulation model of a real-time production scheduling control system, in which our approach is embedded as the kernel algorithm, to study the impacts of some uncertainties that are concerned in practice. Based on the results of simulation experiments and sensitivity analysis, meaningful managerial insights have been provided.

Bi-Objective Optimization of Data-Parallel Applications on Heterogeneous HPC Platforms for Performance and Energy Through Workload Distribution

Performance and energy are the two most important objectives for optimization on modern parallel platforms. In this article, we show that moving from single-objective optimization for performance or energy to their bi-objective optimization on heterogeneous processors results in a tremendous increase in the number of optimal solutions (workload distributions) even for the simple case of linear performance and energy profiles. We then study full performance and energy profiles of two real-life data-parallel applications and find that they exhibit shapes that are non-linear and complex enough to prevent good approximation of them as analytical functions for input to exact algorithms or optimization software for determining the Pareto front. We, therefore, propose a solution method solving the bi-objective optimization problem on heterogeneous processors. The method's novel component is an efficient and exact global optimization algorithm that takes as an input performance and energy profiles as arbitrary discrete functions of workload size, which accurately and realistically take into account resource contention and NUMA inherent in modern parallel platforms, and returns the Pareto-optimal solutions (generally speaking, load imbalanced). To construct the input discrete energy functions, the method employs a methodology that accurately models the energy consumption by a hybrid data-parallel application executing on a heterogeneous HPC platform containing different computing devices using system-level power measurements provided by power meters. We experimentally analyse the proposed solution method using three data-parallel applications, matrix multiplication, 2D fast Fourier transform (2D-FFT), and gene sequencing, on two connected heterogeneous servers consisting of multicore CPUs, GPUs, and Intel Xeon Phi. We show that it determines a superior Pareto front containing the best load balanced solutions and all the load imbalanced solutions that are ignored by load balancing methods.

Alternating Pruned Dynamic Programming for Multiple Epidemic Change-Point Estimation

In this article, we study the problem of multiple change-point detection for a univariate sequence under the epidemic setting, where the behavior of the sequence alternates between a common normal state and different epidemic states. This is a nontrivial generalization of the classical (single) epidemic change-point testing problem. To explicitly incorporate the alternating structure of the problem, we propose a novel model selection based approach for simultaneous inference on both change-points and alternating states. Using the same spirit as profile likelihood, we develop a two-stage alternating pruned dynamic programming algorithm, which conducts efficient and exact optimization of the model selection criteria and has as the worst case computational cost. As demonstrated by extensive numerical experiments, compared to classical general-purpose multiple change-point detection procedures, the proposed method improves accuracy for both change-point estimation and model parameter estimation. We further show promising applications of the proposed algorithm to multiple testing with locally clustered signals, and demonstrate its advantages over existing methods in large scale multiple testing, in DNA copy number variation detection, and in oceanographic study. Supplementary material for this article is available online.

Stochastic joint homecare service and capacity planning with nested decomposition approaches

In this paper, we study a joint service authorization and capacity planning problem for the home health care (HHC) system with demand uncertainty. We formulate the problem as a two-stage stochastic programming with recourse which maximizes the expected total revenue. The service authorization and capacity planning are the mixed binary-and-continuous first stage decisions, and the service hour (resource) allocation is modeled as the second-stage decision which adapts to the demand realizations. The resulting sample average approximation problem (of the stochastic program) could be a large scaled mixed integer linear program. To solve the model in a more scalable fashion, we propose a supergradient-based nested decomposition algorithm that exploits the nice decomposable structure of the problem to cope with the binary and continuous variables separately. The proposed nested decomposition scheme consists of two-layer of decomposition, where the outer decomposition solves for the authorization decision (binaries) with the supergradient cuts and each supergradient can be computed iteratively in the inner decomposition scheme. The proposed nested decomposition algorithm is feasibility-cut-free and is guaranteed to reach the exact optimality in finite steps. Furthermore, we extend our stochastic HHC planning model to a more general framework of Conditional Value-at-Risk (CV@R), and by model transformation with variable change we show that the CV@R model can actually be reformulated in a structure that the proposed nested decomposition scheme can be applied almost directly. Finally, a comprehensive computational study is performed which demonstrates the effectiveness of our model and the performance of the algorithm.

A heuristic scheduling approach for fog-cloud computing environment with stationary IoT devices

The emergence of the Internet of Things (IoT) paradigm has led to the rise of a variety of applications with different characteristics and Quality of Service (QoS) requirements. Those applications require computational power and have time sensitive requirements. Cloud computing paradigm provides an illusion to consumers with unlimited computation resource power. However, cloud computing fails to deliver on the time-sensitive requirements of applications. The main challenge in the cloud computing paradigm is the associated delays from the edge IoT device to the cloud data center and from the cloud data center back to the edge device. Fog computing extends limited computational services closer to the edge device to achieve the time sensitive requirement of applications. This work proposes a scheduling solution which adopts the three-tier fog computing architecture in order to satisfy the maximum number of requests given their deadline requirements. In this work, an optimization model using mixed integer programming is introduced to minimize deadline misses. The model is validated with an exact solution technique. The scheduling problem is known to be an NP-hard, and hence, exact optimization solutions are inadequate for a typical size problem in fog computing. Given the complex nature of the problem, a heuristic approach using the genetic algorithm (GA) is presented. The performance of the proposed GA was evaluated and compared against round robin and priority scheduling. The results show that the deadline misses of the proposed approach is 20%–55% better than the other techniques.

Safe Approximation—An Efficient Solution for a Hard Routing Problem

The Disjoint Connecting Paths problem and its capacitated generalization, called Unsplittable Flow problem, play an important role in practical applications such as communication network design and routing. These tasks are NP-hard in general, but various polynomial-time approximations are known. Nevertheless, the approximations tend to be either too loose (allowing large deviation from the optimum), or too complicated, often rendering them impractical in large, complex networks. Therefore, our goal is to present a solution that provides a relatively simple, efficient algorithm for the unsplittable flow problem in large directed graphs, where the task is NP-hard, and is known to remain NP-hard even to approximate up to a large factor. The efficiency of our algorithm is achieved by sacrificing a small part of the solution space. This also represents a novel paradigm for approximation. Rather than giving up the search for an exact solution, we restrict the solution space to a subset that is the most important for applications, and excludes only a small part that is marginal in some well-defined sense. Specifically, the sacrificed part only contains scenarios where some edges are very close to saturation. Since nearly saturated links are undesirable in practical applications, therefore, excluding near saturation is quite reasonable from the practical point of view. We refer the solutions that contain no nearly saturated edges as safe solutions, and call the approach safe approximation. We prove that this safe approximation can be carried out efficiently. That is, once we restrict ourselves to safe solutions, we can find the exact optimum by a randomized polynomial time algorithm.

Modified range equation for exact modeling and design optimization of active laser remote sensing systems

An exact modeling for laser target illumination and electro-optic detection parameters used in most common laser remote sensing systems, such as laser microphones, LiDARs, laser range finders, laser spot tracking, laser designation, stand-off laser spectroscopy (Raman, LIBS, LIF,…), laser imaging of remote targets and all laser transceiver system having a laser source and an electro-optical detection measure in one assembly. The model makes possible to embed most basic parameters in one equation to ease the design of the main system components from engineering point of view, instead of the currently used general models. The resulting equation corrects calculations for short standoff target detection parameters.

A Branch-and-Cut-and-Price Algorithm for the Electric Vehicle Routing Problem with Multiple Technologies

We provide an exact optimization algorithm for the electric vehicle routing problem with multiple recharge technologies. Our branch-and-cut-and-price algorithm relies upon a path-based formulation, where each column in the master problem represents a sequence of customer visits between two recharge stations instead of a whole route. This allows for massive decomposition, and parallel implementation of the pricing phase, exploiting the large number of independent pricing sub-problems. The algorithm could solve instances with up to thirty customers, nine recharge stations, five vehicles and three technologies to proven optimality. Near-optimal heuristic solutions were obtained with a general-purpose MIP solver from the columns generated at the root node.

Lévy-flight moth-flame optimisation algorithm-based micro-grid equipment sizing: An integrated investment and operational planning approach

Bridging the gap between simulation and reality for successful micro-grid (MG) implementation requires accurate mathematical modelling of the underlying energy infrastructure and extensive optimisation of the design space defined by all possible combinations of the size of the equipment. While exact mathematical optimisation approaches to the MG capacity planning are highly computationally efficient, they often fail to preserve the associated problem nonlinearities and non-convexities. This translates into the fact that the available MG sizing tools potentially return a sub-optimal (inferior) MG design. This brings to light the importance of nature-inspired, swarm-based meta-heuristic optimisation algorithms that are able to effectively handle the nonlinear and non-convex nature of the MG design optimisation problem – and better approximate the globally optimum solution – though at the expense of increased computational complexity. Accordingly, this paper introduces a robust MG capacity planning optimisation framework based on a state-of-the-art meta-heuristic, namely the Lévy-flight moth-flame optimisation algorithm (MFOA). An intelligent linear programming-based day-ahead energy scheduling design is, additionally, integrated into the proposed model. A case study is presented for a real grid-tied community MG in rural New Zealand. A comparison of the modelling results with those of the most popular tool in the literature and industry, HOMER Pro, verifies the superiority of the proposed meta-heuristic-based MG sizing model. Additionally, the efficiency of the Lévy-flight MFOA is compared to nine well-established meta-heuristics in the MG capacity planning literature. The comparative analyses have revealed the statistically significant outperformance of the Lévy-flight MFOA to the examined meta-heuristics. Notably, its superiority to the original MFOA, the hybrid genetic algorithm-particle swarm optimisation, and the ant colony optimiser, by at least ~6.5%, ~8.4%, and ~12.8%, is demonstrated. Moreover, comprehensive capital budgeting analyses have confirmed the financial viability of the test-case system optimised by the proposed model.