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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 January 2024) | Viewed by 21531

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Special Issue Editors

Dr. Alexander A Lazarev

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Guest Editor

Institute of Control Sciences of Russian Academy of Sciences, 117997 Moscow, Russia
Interests: scheduling theory; discrete optimization; NP hardness; combinatorics; train scheduling; polynomial algorithms

Prof. Dr. Frank Werner

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Guest Editor

Faculty of Mathematics, Otto-von-Guericke-University, P.O. Box 4120, D-39016 Magdeburg, Germany
Interests: scheduling, in particular development of exact and approximate algorithms; stability investigations is discrete optimization; scheduling with interval processing times; complexity investigations for scheduling problems; train scheduling; graph theory; logistics; supply chains; packing; simulation and applications
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Bertrand M.T. Lin

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Guest Editor

Institute of Information Management, National Chiao Tung University, Taipei 100-116, Taiwan
Interests: scheduling theory; operations management; discrete optimization

Special Issue Information

Dear colleagues,

The development of software products that enable effective planning and optimization of production processes is necessary to improve the quality of the industrial sector. This Special Issue is devoted to modern approaches to solving discrete optimization problems and scheduling problems. Special attention is paid to problems with practical applications. First of all, this concerns the tasks that were updated as a result of the pandemic crisis of 2020–2021: the tasks of managing medical institutions, the tasks of cargo transportation, the tasks of production planning, and so on. NP-hard problems are the most difficult since they require significant computational resources to find a solution in general cases. Various models are studied, and their effectiveness is compared based on the study of special (pseudo-)polynomial solvable cases of problems, the measure of (pseudo-)polynomial unsolvability, the radius of stability, and the efficiency of algorithms.

Dr. Alexander A Lazarev
Prof. Dr. Frank Werner
Prof. Dr. Bertrand M.T. Lin
Guest Editors

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Keywords

scheduling theory
discrete optimization
NP hardness
combinatorics
train scheduling
plane graph
polynomial algorithm
routing
multi-agent technology
resource management

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Published Papers (11 papers)

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Editorial

Jump to: Research

3 pages, 128 KiB

Open AccessEditorial

Special Issue “Recent Advances of Discrete Optimization and Scheduling”

by Alexander A. Lazarev, Frank Werner and Bertrand M. T. Lin

Mathematics 2024, 12(6), 793; https://doi.org/10.3390/math12060793 - 8 Mar 2024

Cited by 1 | Viewed by 790

Abstract

This Special Issue of the journal Mathematics is dedicated to new results on the topic of discrete optimization and scheduling [...] Full article

(This article belongs to the Special Issue Recent Advances of Disсrete Optimization and Scheduling)

Research

Jump to: Editorial

16 pages, 541 KiB

Open AccessArticle

Approximation of the Objective Function of Single-Machine Scheduling Problem

by Alexander Lazarev, Nikolay Pravdivets and Egor Barashov

Mathematics 2024, 12(5), 699; https://doi.org/10.3390/math12050699 - 28 Feb 2024

Viewed by 749

Abstract

The problem of the approximation of the coefficients of the objective function of a scheduling problem for a single machine is considered. It is necessary to minimize the total weighted completion times of jobs with unknown weight coefficients when a set of problem instances with known optimal schedules is given. It is shown that the approximation problem can be reduced to finding a solution to a system of linear inequalities for weight coefficients. For the case of simultaneous job release times, a method for solving the corresponding system of inequalities has been developed. Based on it, a polynomial algorithm for finding values of weight coefficients that satisfy the given optimal schedules was constructed. The complexity of the algorithm is

O (n^{2} (N + n))

operations, where n is the number of jobs and N is the number of given instances with known optimal schedules. The accuracy of the algorithm is estimated by experimentally measuring the function

ε (N, n) = \frac{1}{n} \sum_{j = 1}^{n} \frac{∣ w_{j} - w_{j}^{0} ∣}{w_{j}^{0}}

, which is an indicator of the average modulus of the relative deviation of the found values

w_{j}

from the true values

w_{j}^{0}

. An analysis of the results shows a high correlation between the dependence

ε (N, n)

and a function of the form

α (n) / N

, where

α (n)

is a decreasing function of n. Full article

(This article belongs to the Special Issue Recent Advances of Disсrete Optimization and Scheduling)

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13 pages, 263 KiB

Open AccessFeature PaperArticle

Several Goethals–Seidel Sequences with Special Structures

by Shuhui Shen and Xiaojun Zhang

Mathematics 2024, 12(4), 530; https://doi.org/10.3390/math12040530 - 8 Feb 2024

Viewed by 791

Abstract

In this paper, we develop a novel method to construct Goethals–Seidel (GS) sequences with special structures. In the existing methods, utilizing Turyn sequences is an effective and convenient approach; however, this method cannot cover all GS sequences. Motivated by this, we are devoted [...] Read more.

\pm 1

polynomials can be considered a linear combination of eight polynomials with coefficients uniquely belonging to

{0, \pm 1}

. Based on this fact, we change the construction of a quad of Goethals–Seidel sequences to find eight sequences consisting of 0 and

\pm 1

. One more motivation is to obtain these sequences more efficiently. To this end, we make use of the k-block, of which some properties of (anti) symmetry are discussed. After this, we can then look for the sequences with the help of computers since the symmetry properties facilitate reducing the search range. Moreover, we find that one of the eight blocks, which we utilize to construct GS sequences directly, can also be combined with Williamson sequences to generate GS sequences with more order. Several examples are provided to verify the theoretical results. The main contribution of this work is in building a bridge linking the GS sequences and eight polynomials, and the paper also provides a novel insight through which to consider the existence of GS sequences. Full article

(This article belongs to the Special Issue Recent Advances of Disсrete Optimization and Scheduling)

17 pages, 327 KiB

Open AccessArticle

Scheduling of Software Test to Minimize the Total Completion Time

by Man-Ting Chao and Bertrand M. T. Lin

Mathematics 2023, 11(22), 4705; https://doi.org/10.3390/math11224705 - 20 Nov 2023

Cited by 1 | Viewed by 1111

Abstract

This paper investigates a single-machine scheduling problem of a software test with shared common setup operations. Each job has a corresponding set of setup operations, and the job cannot be executed unless its setups are completed. If two jobs have the same supporting setups, the common setups are performed only once. No preemption of any processing is allowed. This problem is known to be computationally intractable. In this study, we propose sequence-based and position-based integer programming models and a branch-and-bound algorithm for finding optimal solutions. We also propose an ant colony optimization algorithm for finding approximate solutions, which will be used as the initial upper bound of the branch-and-bound algorithm. The computational experiments are designed and conducted to numerically appraise all of the proposed methods. Full article

(This article belongs to the Special Issue Recent Advances of Disсrete Optimization and Scheduling)

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22 pages, 2174 KiB

Open AccessArticle

Mathematical Circuit Root Simplification Using an Ensemble Heuristic–Metaheuristic Algorithm

by Navid Behmanesh-Fard, Hossein Yazdanjouei, Mohammad Shokouhifar and Frank Werner

Mathematics 2023, 11(6), 1498; https://doi.org/10.3390/math11061498 - 19 Mar 2023

Cited by 3 | Viewed by 1614

Abstract

Symbolic pole/zero analysis is a crucial step in designing an analog operational amplifier. Generally, a simplified symbolic analysis of analog circuits suffers from NP-hardness, i.e., an exponential growth of the number of symbolic terms of the transfer function with the circuit size. This study presents a mathematical model combined with a heuristic–metaheuristic solution method for symbolic pole/zero simplification in operational transconductance amplifiers. First, the circuit is symbolically solved and an improved root splitting method is applied to extract symbolic poles/zeroes from the exact expanded transfer function. Then, a hybrid algorithm based on heuristic information and a metaheuristic technique using simulated annealing is used for the simplification of the derived symbolic roots. The developed method is tested on three operational transconductance amplifiers. The obtained results show the effectiveness of the proposed method in achieving accurate simplified symbolic pole/zero expressions with the least complexity. Full article

(This article belongs to the Special Issue Recent Advances of Disсrete Optimization and Scheduling)

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27 pages, 6141 KiB

Open AccessArticle

Autonomous Digital Twin of Enterprise: Method and Toolset for Knowledge-Based Multi-Agent Adaptive Management of Tasks and Resources in Real Time

by Vladimir Galuzin, Anastasia Galitskaya, Sergey Grachev, Vladimir Larukhin, Dmitry Novichkov, Petr Skobelev and Alexey Zhilyaev

Mathematics 2022, 10(10), 1662; https://doi.org/10.3390/math10101662 - 12 May 2022

Cited by 8 | Viewed by 2832

Abstract

Digital twins of complex technical objects are widely applied for various domains, rapidly becoming smart, cognitive and autonomous. However, the problem of digital twins for autonomous management of enterprise resources is still not fully researched. In this paper, an autonomous digital twin of enterprise is introduced to provide knowledge-based multi-agent adaptive allocation, scheduling, optimization, monitoring and control of tasks and resources in real time, synchronized with employees’ plans, preferences and competencies via mobile devices. The main requirements for adaptive resource management are analyzed. The authors propose formalized ontological and multi-agent models for developing the autonomous digital twin of enterprise. A method and software toolset for designing the autonomous digital twin of enterprise, applicable for both operational management of tasks and resources and what-if simulations, are developed. The validation of developed methods and toolsets for IT service desk has proved increase in efficiency, as well as savings in time and costs of deliveries for various applications. The paper also outlines a plan for future research, as well as a number of new potential business applications. Full article

(This article belongs to the Special Issue Recent Advances of Disсrete Optimization and Scheduling)

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10 pages, 4782 KiB

Open AccessArticle

Complexity of Solutions Combination for the Three-Index Axial Assignment Problem

by Lev G. Afraimovich and Maxim D. Emelin

Mathematics 2022, 10(7), 1062; https://doi.org/10.3390/math10071062 - 25 Mar 2022

Cited by 3 | Viewed by 1747

Abstract

In this work we consider the NP-hard three-index axial assignment problem. We formulate and investigate a problem of combining feasible solutions. Such combination can be applied in a wide range of heuristic and approximate algorithms for solving the assignment problem, instead of the commonly used strategy of selecting the best solution among the found feasible solutions. We discuss approaches to a solution of the combination problem and prove that it becomes NP-hard already in the case of combining four solutions. Full article

(This article belongs to the Special Issue Recent Advances of Disсrete Optimization and Scheduling)

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22 pages, 1175 KiB

Open AccessArticle

Special Type Routing Problems in Plane Graphs

by Tatiana Makarovskikh and Anatoly Panyukov

Mathematics 2022, 10(5), 795; https://doi.org/10.3390/math10050795 - 2 Mar 2022

Cited by 1 | Viewed by 2434

Abstract

We considered routing problems for plane graphs to solve control problems of cutting machines in the industry. According to the cutting plan, we form its homeomorphic image in the form of a plane graph G. We determine the appropriate type of route [...] Read more.

O E

-route represents an ordered sequence of chains satisfying the requirement that the part of the route that is not passed does not intersect the interior of its passed part,

A O E

-chain represents

O E

-chain consecutive edges which are incident to vertex v and they are neighbours in the cyclic order

O^{\pm} (v)

N O E

-route represents the non-intersecting

O E

-route,

P P O E

-route represents the Pierce Point

N O E

-route with allowable pierce points that are start points of

O E

-chains forming this route. We analyse the solvability of the listed routing problems in graph G. We developed the polynomial algorithms for obtaining listed routes with the minimum number of covering paths and the minimum length of transitions between the ending of the current path and the beginning of the next path. The solutions proposed in the article can improve the quality of technological preparation of cutting processes in CAD/CAM systems. Full article

(This article belongs to the Special Issue Recent Advances of Disсrete Optimization and Scheduling)

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18 pages, 423 KiB

Open AccessArticle

Decomposition of the Knapsack Problem for Increasing the Capacity of Operating Rooms

by Alexander Alekseevich Lazarev, Darya Vladimirovna Lemtyuzhnikova and Mikhail Lvovich Somov

Mathematics 2022, 10(5), 784; https://doi.org/10.3390/math10050784 - 1 Mar 2022

Cited by 2 | Viewed by 2302

Abstract

This paper is aimed at the problem of scheduling surgeries in operating rooms. To solve this problem, we suggest using some variation of the bin packing problem. The model is based on the actual operation of 10 operating rooms, each of which belongs to a specific department of the hospital. Departments are unevenly loaded, so operations can be moved to operating rooms in other departments. The main goal is to increase patient throughput. It is also necessary to measure how many operations take place in other departments with the proposed solution. The preferred solution is a solution with fewer such operations, all other things being equal. Due to the fact that the mixed-integer linear programming model turned out to be computationally complex, two approximation algorithms were also proposed. They are based on decomposition. The complexity of the proposed algorithms is estimated, and arguments are made regarding their accuracy from a theoretical point of view. To assess the practical accuracy of the algorithms, the Gurobi solver is used. Experiments were conducted on real historical data on surgeries obtained from the Burdenko Neurosurgical Center. Two decomposition algorithms were constructed and a comparative analysis was performed for 10 operating rooms based on real data. Full article

(This article belongs to the Special Issue Recent Advances of Disсrete Optimization and Scheduling)

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28 pages, 786 KiB

Open AccessArticle

JMA: Nature-Inspired Java Macaque Algorithm for Optimization Problem

by Dinesh Karunanidy, Subramanian Ramalingam, Ankur Dumka, Rajesh Singh, Mamoon Rashid, Anita Gehlot, Sultan S. Alshamrani and Ahmed Saeed AlGhamdi

Mathematics 2022, 10(5), 688; https://doi.org/10.3390/math10050688 - 23 Feb 2022

Cited by 2 | Viewed by 2664

Abstract

In recent years, optimization problems have been intriguing in the field of computation and engineering due to various conflicting objectives. The complexity of the optimization problem also dramatically increases with respect to a complex search space. Nature-Inspired Optimization Algorithms (NIOAs) are becoming dominant algorithms because of their flexibility and simplicity in solving the different kinds of optimization problems. Hence, the NIOAs may be struck with local optima due to an imbalance in selection strategy, and which is difficult when stabilizing exploration and exploitation in the search space. To tackle this problem, we propose a novel Java macaque algorithm that mimics the natural behavior of the Java macaque monkeys. The Java macaque algorithm uses a promising social hierarchy-based selection process and also achieves well-balanced exploration and exploitation by using multiple search agents with a multi-group population, male replacement, and learning processes. Then, the proposed algorithm extensively experimented with the benchmark function, including unimodal, multimodal, and fixed-dimension multimodal functions for the continuous optimization problem, and the Travelling Salesman Problem (TSP) was utilized for the discrete optimization problem. The experimental outcome depicts the efficiency of the proposed Java macaque algorithm over the existing dominant optimization algorithms. Full article

(This article belongs to the Special Issue Recent Advances of Disсrete Optimization and Scheduling)

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26 pages, 1614 KiB

Open AccessArticle

A Three-Stage ACO-Based Algorithm for Parallel Batch Loading and Scheduling Problem with Batch Deterioration and Rate-Modifying Activities

by Jae Won Jang, Yong Jae Kim and Byung Soo Kim

Mathematics 2022, 10(4), 657; https://doi.org/10.3390/math10040657 - 20 Feb 2022

Cited by 1 | Viewed by 1824

Abstract

This paper addresses a batch loading and scheduling problem of minimizing the makespan on parallel batch processing machines. For batch loading, jobs with compatible families can be assigned to the same batch process even if they differ in size; however, batches can only be formed from jobs within the same family, and the batch production time is determined by the family. During the batch scheduling, the deterioration effects are continuously added to batches processed in each parallel machine so that the batch production times become deteriorated. The deteriorated processing time of batches can be recovered to the original processing times of batches by a maintenance or cleaning process of machines. In this problem, we sequentially determine the batching of jobs and the scheduling of batches. Due to the complexity of the problem, we proposed a three-stage ant colony optimization algorithm. The proposed algorithm found an optimal solution for small-sized problems and achieved near-optimal solutions and better performance than a genetic algorithm or a particle swarm optimization for large-sized problems. Full article

(This article belongs to the Special Issue Recent Advances of Disсrete Optimization and Scheduling)

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