Exact Solution Method Research Articles

Identification and Analysis of Supply Chain and Supplier Risks Using Frog-Leaping and Genetic Algorithms (A Case Study in the Automotive Industry)

Objective: The increasing complexity of the production process, the use of machines with advanced and modern capabilities, and the growing demand for products compel industry owners to maximize their capabilities at minimal costs towards production, reduce the risk of product manufacturing, and also improve the quality of their manufactured products to access broader markets. One of the most important factors in achieving this goal is the assessment of risk in product manufacturing. Given the financial difficulties in the automotive industry, supply chain management is of high importance because employing high-risk suppliers increases the possibility of rework and is economically detrimental to the automotive industry. Methodology: To examine the risks in the automotive supply chain, mathematical modeling was used, and the model was executed with two metaheuristic algorithms. Findings: Comparing the results, it is evident that the research model was analyzed with both exact solution methods and a metaheuristic approach, where in the metaheuristic method, it was solved using genetic and frog-leaping algorithms. By comparing the research outcomes, it can be said that the exact solution method shows better and more optimal results. Conclusion: However, due to the NP-HARD nature of the problem, this method does not reach a solution in a logical time frame for larger dimensions of the problem. Therefore, the problem was solved with metaheuristic algorithms. Comparing the algorithms, it can be seen that the frog-leaping algorithm has provided more optimal results, but the solving time was less in the genetic algorithm compared to the frog-leaping algorithm.

An exact solution method for <i>seru</i> scheduling problems considering past-sequence-dependent setup time and adjustment activities

An exact solution method for <i>seru</i> scheduling problems considering past-sequence-dependent setup time and adjustment activities

An exact solution method for seru scheduling problems considering past-sequence-dependent setup time and adjustment activities

An exact solution method for seru scheduling problems considering past-sequence-dependent setup time and adjustment activities

An exact solution method for seru scheduling problems with multiple rate-modifying activities and learning effect

An exact solution method for seru scheduling problems with multiple rate-modifying activities and learning effect

An exact solution method for <i>seru</i> scheduling problems with multiple rate-modifying activities and learning effect

An exact solution method for <i>seru</i> scheduling problems with multiple rate-modifying activities and learning effect

An Exact Method for (Constrained) Assortment Optimization Problems with Product Costs

We study the problem of optimizing assortment decisions in the presence of product-specific costs when customers choose according to a multinomial logit model. This problem is NP-hard, and approximate solutions methods have been proposed in the literature to obtain both lower and upper bounds in a tractable manner. We propose the first exact solution method for this problem and show that provably optimal assortments of instances with up to 1,000 products can be found, on average, in about 2/10 of a second. In particular, we propose a bounding procedure to enhance an approximation method originally proposed by Feldman and Topaloglu and provide tight lower and upper bounds at a fraction of a second. We show how these bounds can be used to effectively identify an optimal assortment. We also describe how to adapt our approach to handle cardinality or space/resource capacity constraints on the assortment as well as assortment optimization under a mixed-multinomial logit model. In both cases, our solution method provides significant computational boosts compared with exact methods from the literature.

Solving nonlinear partial differential equations using a novel Cham method

Nonlinear partial differential equations (NLPDEs) have been of great interest in recent years due to their numerous applications. While there are several methods for finding exact solutions to various NLPDEs, more solutions are still required. This paper first proposes the Cham method, a new method for solving NLPDEs that can generate eight families of solutions. The method is then successfully employed to solve the (2+1)-dimensional Bogoyavlenskii's breaking soliton equations. The dynamic behaviour of these equations and the bifurcation of traveling waves are also discussed. Finally, we graphically depict some solutions corresponding to some discovered solutions with different coefficient values. The Cham method is general, effective, and adaptable to many NLPDEs.

Scatter search for capacitated minimum spanning tree

We develop a scatter search algorithm for the classical capacitated minimum spanning tree (CMST) problem. We address both the homogeneous and the more difficult heterogeneous variant of the problem. The CMST is central in a wide range of network design applications in industrial engineering, routing and logistics, and communication networks. Since the problem is NP-Complete, heuristic solution methods are often required to find high-quality solutions within practical time limits. Research efforts have been focused on complex algorithm designs combining advanced neighborhood search techniques with exact solution methods or hybrids of genetic algorithms with various alternative search techniques and likewise supplemented with optimization procedures. In this study we go back to the roots and offer an alternative algorithm that competes with the best algorithms in the literature and both avoids complicated artifacts and is simpler to implement. Key features of the proposed algorithm include an oscillation neighborhood search method characterized by two classes of neighborhoods and an evolutionary search procedure based on a scatter search framework.

Solving the park-and-loop routing problem by branch-price-and-cut

The park-and-loop routing problem is a variation of the vehicle routing problem in which routes include a main tour that is completed using a vehicle and subtours that are carried out on foot after parking the vehicle. Additionally, the route duration and total walking distance are bounded. To solve the problem, we propose an exact solution method based on the branch-price-and-cut framework. In particular, our method uses problem-specific components to solve the pricing problem. We report on computational experiments carried out on a standard set of 40 instances with up to 50 customers. The results show that our method delivers solutions that compare favorably to existing metaheuristic algorithms, matching all previously best-known solutions and improving 11 of them in reasonable computational times. Moreover, our method provides optimality certificates for 39 out of the 40 instances.

Solitary waves of the complex Ginzburg-Landau equation with anti-cubic nonlinearity

The complex Ginzburg-Landau equation with anti-cubic nonlinearity is considered. Using a modification of the simplest equation method for finding exact solutions of nonlinear partial differential equations we get new solitary waves with arbitrary amplitude and speed of this special case of the complex Ginzburg -Landau equation. The first integral of nonlinear ordinary differential equation is found. The general solution of equation is presented taking into account the first integral of equation. Conserved density of the solitary wave solution is calculated.

Multi-attribute two-echelon location routing: Formulation and dynamic discretization discovery approach

We study the two-echelon location-routing system under tight synchronization constraints, in addition to several other interacting attributes. Prompted, in particular, by city-logistics applications, the system we address concerns a two-echelon distribution layout composed of a set of platform facilities and a set of intermediate satellite facilities to deliver freight from supply zones outside the city to customers within. The problem setting includes time-dependent multicommodity demand, time windows, lack of storage capacity at intermediate facilities, and synchronization at these facilities of the fleets operating on different echelons. The problem requires the selection of facilities at both levels, the allocation of suppliers to platforms and of customers to satellites, and the routing and scheduling of vehicles at each echelon, in order to deliver the freight from platforms to customers, through the satellites. The lack of storage capacity of the shared facilities, the satellites, requires tight scheduling of the vehicle routes and demand itineraries, i.e., departure times from the platforms and satellites, and the synchronization of vehicle routes at satellites for efficient transshipment operations. We introduce the problem setting, present a mixed-integer programming formulation, and a dynamic discretization discovery-based exact solution method for the problem. We perform thorough analyses to assess the impact of the problem attributes and requirements on the system behaviour and algorithm performance.

Energy consumption optimization for sustainable flexible robotic cells: Proposing exact and metaheuristic methods

Many manufacturing companies are always looking for a way to reduce energy consumption by utilizing energy-efficient production methods. These methods can be different depending on the type of products and production technology. For instance, one of the ways to increase energy efficiency and keep the precision of production is to use robots for the transportation of the parts among the machines and loading/unloading the machines. This technology is affordable compared to the technologies used in manufacturing companies. Manufacturing companies that rely on robotics technology must have a strategy to reduce energy costs and at the same time increase production by adjusting the intensity of processing or controlling the production rate. This study presents an exact solution method for flexible robotic cells to control the production rate and minimize energy consumption, which aims to both reduce electricity prices and minimize greenhouse gas (GHG) emissions under a lead time of production. Then, considering the NP-hardens nature of the problem, a heuristic solution method based on the genetic algorithm (GA) is proposed. Using the proposed approach, manufacturing companies will be able to make more accurate decisions about processing intensity and process scheduling while ensuring sustainability.

Dynamic Project Expediting: A Stochastic Shortest-Path Approach

We deal with the problem of managing a project or a complex operational process by controlling the execution pace of the activities it comprises. We consider a setting in which these activities are clearly defined, are subject to precedence constraints, and progress randomly. We formulate a discrete-time, infinite-horizon Markov decision process in which the manager reviews progress in each period and decides which activities to expedite to balance expediting costs with delay costs. We derive structural properties for this dynamic project expediting problem. These enable us then to devise exact solution methods that we show to reduce computational burden significantly. We illustrate how our method generalizes and can be used to tackle a wide range of so-called stochastic shortest-path problems that are characterized by an intuitive property and can capture other applications, including medical decision-making and disease-modeling problems. Moreover, we also deal with the state identification issue for our problem, which is a challenging task in and of itself, owing to precedence constraints. We complement our analytical results with numerical experiments, demonstrating that both our solution and state identification methods significantly outperform extant methods for a supply chain example and for various randomly generated instances. This paper was accepted by Chung Piaw Teo, optimization. Funding: R. Mogre acknowledges support from the U.S.-UK Fulbright Commission and the Lloyd’s Tercentenary Research Foundation through the Fulbright-Lloyd’s Scholar Award, which allowed him to spend an extended period of time at Massachusetts Institute of Technology. Supplemental Material: The data files are available at https://doi.org/10.1287/mnsc.2023.4876 .

Lagrangian matheuristics for the Quadratic Multiple Knapsack Problem

The Quadratic Multiple Knapsack Problem (QMKP) is a challenging combinatorial optimization problem combining the well-known Quadratic Knapsack Problem with the Multiple Knapsack Problem. After a long stream of research devoted to metaheuristic approaches for large-scale instances, only recently some authors started to study the mathematical properties of the QMKP and proposed exact solution methods for benchmark instances of smaller size. In this paper, we propose the first matheuristic approach for solving the QMKP approximately. The proposed method exploits the strength of a Lagrangian relaxation for the natural quadratic formulation of the QMKP to derive heuristic solutions. Postoptimization local search procedures are embedded in the final framework. Experimental studies show that the resulting deterministic matheuristic approach consistently delivers solutions of very good quality in short computing times. • First matheuristic for the Quadratic Multiple Knapsack Problem. • Competitive with currently best performing exact method. • Tighter Lagrangian bounds on medium-size instances.

Capacity allocation in service systems with preferred delivery times and multiple customer classes

Motivated by operational problems in click-and-collect systems, such as curbside pickup programs, we study a joint admission control and capacity allocation problem. We consider systems where customers have preferred service delivery times and can be of different priority classes. The service provider can reject customers upon arrival or serve jobs via overtime when service capacity is insufficient. The service provider’s goal is to find the minimum-cost admission and capacity allocation policy to dynamically decide when to serve and whom to serve. We model this problem as a Markov Decision Process and present structural results to partially characterize suboptimal solutions. We then develop a linear programming-based exact solution method using these results. We also present a problem-specific approximation method using a new state aggregation rule to address computational challenges faced due to large state and action spaces. Finally, we develop heuristic policies for large instances based on the behavior of optimal policies in small problems. We evaluate our methods through extensive computational experiments where we vary the service capacity, arrivals, associated service costs, customer segmentation, and order patterns. Our solution methods perform significantly better than several benchmarks in managing the tradeoff between the computation time and solution quality.

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

In this paper, we introduce a special system of linear equations with a symmetric, tridiagonal matrix, whose solution vector contains the values of the analytical solution of the original ordinary differential equation (ODE) in grid points. Further, we present the derivation of an exact scheme for an arbitrary mesh grid and prove that its application can completely avoid other errors in discretization and numerical methods. The presented method is constructed on the basis of special local green functions, whose special properties provide the possibility to invert the differential operator of the ODE. Thus, the newly obtained results provide a general, exact solution method for the second-order ODE, which is also effective for obtaining the arbitrary grid, Dirichlet, and/or Neumann boundary conditions. Both the results obtained and the short case study confirm that the use of the exact scheme is efficient and straightforward even for ODEs with discontinuity functions.

Exact Solution of Interacting Particle Systems Related to Random Matrices

We consider one-dimensional diffusions, with polynomial drift and diffusion coefficients, so that in particular the motion can be space-inhomogeneous, interacting via one-sided reflections. The prototypical example is the well-known model of Brownian motions with one-sided collisions, also known as Brownian TASEP, which is equivalent to Brownian last passage percolation. We obtain a formula for the finite dimensional distributions of these particle systems, starting from arbitrary initial condition, in terms of a Fredholm determinant of an explicit kernel. As far as we can tell, in the space-inhomogeneous setting and for general initial condition this is the first time such a result has been proven. We moreover consider the model of non-colliding diffusions, again with polynomial drift and diffusion coefficients, which includes the ones associated to all the classical ensembles of random matrices. We prove that starting from arbitrary initial condition the induced point process has determinantal correlation functions in space and time with an explicit correlation kernel. A key ingredient in our general method of exact solution for both models is the application of the backward in time diffusion flow on certain families of polynomials constructed from the initial condition.

Identification of crack location in metallic biomaterial cantilever beam subjected to moving load base on central difference approximation

Abstract If not detected early, the cracks in structural components may ultimately result in the failure of the structure. This issue becomes even more critical when the component under investigation is a prosthesis placed in the human body. This study presents a crack location identification method based on the time domain in a cantilever beam of metallic biomaterials (CBMB). The absolute difference between the central difference approximation of the root mean square (RMS) of displacement of points on the cracked and uncracked beams was applied as a cracked location indicator. Captured time-domain data (displacement) at each node of the cracked and uncracked beams were processed into a central difference approximation of the RMS of displacement. Then, the crack could be detected by a sudden change of the cracked location indicator. The feasibility and effectiveness of the proposed method were validated by numerical simulations. The finite-element simulation of a CBMB with a transverse notch was analyzed in the numerical study. The notch or crack was detected along the beam under a moving load at various locations. A set of simulation experiments and numerical calculations was performed to determine whether the proposed identification method would accurately detect the location of a crack in a cantilever beam under a moving load compared to the location found by an exact solution method. The results showed that the proposed method was not only as able as the analytical method but also robust against noise: it was able to detect a crack precisely under 5% noise.

Trajectory Planning for an Autonomous Vehicle with Conflicting Moving Objects Along a Fixed Path – An Exact Solution Method

Trajectory planning for autonomous vehicles (AVs) by considering conflicting moving objects (CMOs) is a challenging problem to AV operations. This paper investigates an AV trajectory planning problem where the AV follows a given spatial path, and the trajectories of CMOs are predictable. With the spatial path fixed, the two-dimensional trajectory planning problem is reduced to a one-dimensional speed planning problem that decides the optimal speeds and accelerations of the AV along the spatial path. This paper first analyzes the conflict area caused by a single CMO in the space-time diagram, which reveals upper and lower bounds to the conflict area. Then a multi-area fusion algorithm is proposed to extend the upper and lower bound analyses to a relatively complex traffic scenario with multiple CMOs. To facilitate the computation of the investigated problem, a customized dynamic programming (DP) algorithm is developed, which employs the revealed upper and lower bounds and arriving time constraints to cut invalid trajectories at each stage. With this, the number of stages and states, as well as the computational time for solving the proposed problem, are largely reduced. A set of numerical experiments are conducted to evaluate the performance of the customized DP-based algorithm. The results show that the proposed customized DP-based algorithm can solve the investigated problem within milliseconds, which enables applications to real-time AV control. This much outperforms a state-of-the-art commercial solver, Gurobi, especially for complex traffic scenarios. We further implemented the proposed method in a real-world case study, and the results show that the trajectory generated by the proposed model has a great potential to enhance the future traffic system.

An exact cutting plane method for the Euclidean max-sum diversity problem

This paper aims to answer an open question recently posed in the literature, that is to find a fast exact method for solving the max-sum diversity problem, a nonconcave quadratic binary maximization problem. We show that, for Euclidean max-sum diversity problems (EMSDP), the distance matrix defining the quadratic term is always conditionally negative definite. This interesting property ensures that the cutting plane method is exact for (EMSDP), even in the absence of concavity. As such, the cutting plane method, which is primarily designed for concave maximisation problems, converges to the optimal solution of (EMDSP). The method was evaluated on several standard benchmark test sets, where it was shown to outperform other exact solution methods for (EMSDP), and is capable of solving two-coordinate problems of up to eighty-five thousand variables.