Traveling Salesman Problem

This book is a collection of current research in the application of evolutionary algorithms and other optimal algorithms to solving the TSP problem. It brings together researchers with applications in Artificial Immune Systems, Genetic Algorithms, Neural Networks and Differential Evolution Algorithm. Hybrid systems, like Fuzzy Maps, Chaotic Maps and Parallelized TSP are also presented. Most importantly, this book presents both theoretical as well as practical applications of TSP, which will be a vital tool for researchers and graduate entry students in the field of applied Mathematics, Computing Science and Engineering

Evolutionary computation applied to combinatorial optimisation problems

This thesis addresses the issues associated with conventional genetic algorithms (GA) when applied to hard optimisation problems. In particular it examines the problem of selecting and implementing appropriate genetic operators in order to meet the validity constraints for constrained optimisation problems. The problem selected is the travelling salesman problem (TSP), a well known NP-hard problem. Following a review of conventional genetic algorithms, this thesis advocates the use of a repair technique for genetic algorithms: GeneRepair. We evaluate the effectiveness of this operator against a wide range of benchmark problems and compare these results with conventional genetic algorithm approaches. A comparison between GeneRepair and the conventional GA approaches is made in two forms: firstly a handcrafted approach compares GAs without repair against those using GeneRepair. A second automated approach is then presented. This meta-genetic algorithm examines different configurations of operators and parameters. Through the use of a cost/benefit (Quality-Time Tradeoff) function, the user can balance the computational effort against the quality of the solution and thus allow the user to specify exactly what the cost benefit point should be for the search. Results have identified the optimal configuration settings for solving selected TSP problems. These results show that GeneRepair when used consistently generates very good TSP solutions for 50, 70 and 100 city problems. GeneRepair assists in finding TSP solutions in an extremely efficient manner, in both time and number of evaluations required

Genetic algorithm for process sequencing modelled as the travelling salesman problem with precedence constraints

This thesis addresses process sequencing subject to precedence constraints which arises as a subproblem in scheduling, planning and routing problems. The process sequencing problem can be modelled as the Travelling Salesman Problem with Precedence Constraints (TSPPC). In the general Travelling Salesman Problem (TSP) scenario, the salesman must travel from city to city; visiting each city exactly once and wishes to minimize the total distance travelled during the tour of all cities. TSPPC is similar in concept to TSP, except that it has a set of precedence constraints to be followed by the salesman. The exact methods that are used to find an optimal solution of the problem are only capable of handling small and medium sizes of instances. Genetic algorithms (GA) are heuristic optimization techniques based on the principles and mechanisms of natural evolution and can be used to solve larger instances and numerous side constraints with optimal or near-optimal solutions. However, the use of a conventional genetic algorithm procedure for TSP, with an order-based representation, might generate invalid candidate solutions when precedence constraints are involved. In this thesis, a new GA procedure is developed which includes chromosome’s repairing strategy based topological sort to handle the precedence constraints and to generate only feasible solution during the evolutionary process. The procedure to select the task in sequence is based on the “earliest position” techniques. This procedure is combined with a roulette wheel selection, linear order crossover and inversion mutation. The effectiveness and the stability of the proposed GA are then evaluated against a wide range of benchmark problems and the solutions are compared with the results obtained from research results published in the relevant literature. The results from the computational experiments demonstrate that the proposed GA procedure developed in this thesis is capable to tackle large-size problem and reach for optimal solutions. The developed GA procedure improved the performance of the algorithm with less number of generations and less convergence time in achieving optimal solutions. The genetic operators used are capable to always introduce new fitter offspring and free from being trapped in a local optimum. Therefore it can be stated that the proposed genetic algorithm is efficient to solve process sequencing modelled as the travelling salesman problem with precedence constraints. This result will greatly help to solve many real world sequencing problems especially in the field of assembly line design and management

HEDCOS: High Efficiency Dynamic Combinatorial Optimization System using Ant Colony Optimization algorithm

This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University LondonDynamic combinatorial optimization is gaining popularity among industrial practitioners due to the ever-increasing scale of their optimization problems and efforts to solve them to remain competitive. Larger optimization problems are not only more computationally intense to optimize but also have more uncertainty within problem inputs. If some aspects of the problem are subject to dynamic change, it becomes a Dynamic Optimization Problem (DOP). In this thesis, a High Efficiency Dynamic Combinatorial Optimization System is built to solve challenging DOPs with high-quality solutions. The system is created using Ant Colony Optimization (ACO) baseline algorithm with three novel developments. First, introduced an extension method for ACO algorithm called Dynamic Impact. Dynamic Impact is designed to improve convergence and solution quality by solving challenging optimization problems with a non-linear relationship between resource consumption and fitness. This proposed method is tested against the real-world Microchip Manufacturing Plant Production Floor Optimization (MMPPFO) problem and the theoretical benchmark Multidimensional Knapsack Problem (MKP). Second, a non-stochastic dataset generation method was introduced to solve the dynamic optimization research replicability problem. This method uses a static benchmark dataset as a starting point and source of entropy to generate a sequence of dynamic states. Then using this method, 1405 Dynamic Multidimensional Knapsack Problem (DMKP) benchmark datasets were generated and published using famous static MKP benchmark instances as the initial state. Third, introduced a nature-inspired discrete dynamic optimization strategy for ACO by modelling real-world ants’ symbiotic relationship with aphids. ACO with Aphids strategy is designed to solve discrete domain DOPs with event-triggered discrete dynamism. The strategy improved inter-state convergence by allowing better solution recovery after dynamic environment changes. Aphids mediate the information from previous dynamic optimization states to maximize initial results performance and minimize the impact on convergence speed. This strategy is tested for DMKP and against identical ACO implementations using Full-Restart and Pheromone-Sharing strategies, with all other variables isolated. Overall, Dynamic Impact and ACO with Aphids developments are compounding. Using Dynamic Impact on single objective optimization of MMPPFO, the fitness value was improved by 33.2% over the ACO algorithm without Dynamic Impact. MKP benchmark instances of low complexity have been solved to a 100% success rate even when a high degree of solution sparseness is observed, and large complexity instances have shown the average gap improved by 4.26 times. ACO with Aphids has also demonstrated superior performance over the Pheromone-Sharing strategy in every test on average gap reduced by 29.2% for a total compounded dynamic optimization performance improvement of 6.02 times. Also, ACO with Aphids has outperformed the Full-Restart strategy for large datasets groups, and the overall average gap is reduced by 52.5% for a total compounded dynamic optimization performance improvement of 8.99 times

Quantum Algorithms for Solving Hard Constrained Optimization Problems

En aquesta investigació, s'han examinat tècniques d'optimització per resoldre problemes de restriccions i s'ha fet un estudi de l'era quàntica i de les empreses líders del mercat, com ara IBM, D-Wave, Google, Xanadu, AWS-Braket i Microsoft. S'ha après sobre la comunitat, les plataformes, l'estat de les investigacions i s'han estudiat els postulats de la mecànica quàntica que serveixen per crear els sistemes i algorismes quàntics més eficients. Per tal de saber si és possible resoldre problemes de Problema de cerca de restriccions (CSP) de manera més eficient amb la computació quàntica, es va definir un escenari perquè tant la computació clàssica com la quàntica tinguessin un bon punt de referència. En primer lloc, la prova de concepte es centra en el problema de programació dels treballadors socials i més tard en el tema de la preparació per lots i la selecció de comandes com a generalització del Problema dels treballadors socials (SWP). El problema de programació dels treballadors socials és una mena de problema d'optimització combinatòria que, en el millor dels casos, es pot resoldre en temps exponencial; veient que el SWP és NP-Hard, proposa fer servir un altre enfoc més enllà de la computació clàssica per a la seva resolució. Avui dia, el focus a la computació quàntica ja no és només per la seva enorme capacitat informàtica sinó també, per l'ús de la seva imperfecció en aquesta era Noisy Intermediate-Scale Quantum (NISQ) per crear un poderós dispositiu d'aprenentatge automàtic que utilitza el principi variacional per resoldre problemes d'optimització en reduir la classe de complexitat. A la tesi es proposa una formulació (quadràtica) per resoldre el problema de l'horari dels treballadors socials de manera eficient utilitzant Variational Quantum Eigensolver (VQE), Quantum Approximate Optimization Algorithm (QAOA), Minimal Eigen Optimizer i ADMM optimizer. La viabilitat quàntica de l'algorisme s'ha modelat en forma QUBO, amb Docplex simulat Cirq, Or-Tools i provat a ordinadors IBMQ. Després d'analitzar els resultats de l'enfocament anterior, es va dissenyar un escenari per resoldre el SWP com a raonament basat en casos (qCBR), tant quànticament com clàssicament. I així poder contribuir amb un algorisme quàntic centrat en la intel·ligència artificial i l'aprenentatge automàtic. El qCBR és una tècnica d’aprenentatge automàtic basada en la resolució de nous problemes que utilitza l’experiència, com ho fan els humans. L'experiència es representa com una memòria de casos que conté qüestions prèviament resoltes i utilitza una tècnica de síntesi per adaptar millor l'experiència al problema nou. A la definició de SWP, si en lloc de pacients es tenen lots de comandes i en lloc de treballadors socials robots mòbils, es generalitza la funció objectiu i les restriccions. Per això, s'ha proposat una prova de concepte i una nova formulació per resoldre els problemes de picking i batching anomenat qRobot. Es va fer una prova de concepte en aquesta part del projecte mitjançant una Raspberry Pi 4 i es va provar la capacitat d'integració de la computació quàntica dins de la robòtica mòbil, amb un dels problemes més demandats en aquest sector industrial: problemes de picking i batching. Es va provar en diferents tecnologies i els resultats van ser prometedors. A més, en cas de necessitat computacional, el robot paral·lelitza part de les operacions en computació híbrida (quàntica + clàssica), accedint a CPU i QPU distribuïts en un núvol públic o privat. A més, s’ha desenvolupat un entorn estable (ARM64) dins del robot (Raspberry) per executar operacions de gradient i altres algorismes quàntics a IBMQ, Amazon Braket (D-Wave) i Pennylane de forma local o remota. Per millorar el temps d’execució dels algorismes variacionals en aquesta era NISQ i la següent, s’ha proposat EVA: un algorisme d’aproximació de Valor Exponencial quàntic. Fins ara, el VQE és el vaixell insígnia de la computació quàntica. Avui dia, a les plataformes líders del mercat de computació quàntica al núvol, el cost de l'experimentació dels circuits quàntics és proporcional al nombre de circuits que s'executen en aquestes plataformes. És a dir, amb més circuits més cost. Una de les coses que aconsegueix el VQE, el vaixell insígnia d'aquesta era de pocs qubits, és la poca profunditat en dividir el Hamiltonià en una llista de molts petits circuits (matrius de Pauli). Però aquest mateix fet, fa que simular amb el VQE sigui molt car al núvol. Per aquesta mateixa raó, es va dissenyar EVA per poder calcular el valor esperat amb un únic circuit. Tot i haver respost a la hipòtesi d'aquesta tesis amb tots els estudis realitzats, encara es pot continuar investigant per proposar nous algorismes quàntics per millorar problemes d'optimització.En esta investigación, se han examinado técnicas de optimización para resolver problemas de restricciones y se ha realizado un estudio de la era cuántica y de las empresas lideres del mercado, como IBM, D-Wave, Google, Xanadu, AWS-Braket y Microsoft. Se ha aprendido sobre su comunidad, sus plataformas, el estado de sus investigaciones y se han estudiado los postulados de la mecánica cuántica que sirven para crear los sistemas y algoritmos cuánticos más eficientes. Por tal de saber si es posible resolver problemas de Problema de búsqueda de restricciones (CSP) de manera más eficiente con la computación cuántica, se definió un escenario para que tanto la computación clásica como la cuántica tuvieran un buen punto de referencia. En primer lugar, la prueba de concepto se centra en el problema de programación de los trabajadores sociales y más tarde en el tema de la preparación por lotes y la selección de pedidos como una generalización del Problema de los trabajadores sociales (SWP). El problema de programación de los trabajadores sociales es una clase de problema de optimización combinatoria que, en el mejor de los casos, puede resolverse en tiempo exponencial; viendo que el SWP es NP-Hard, propone usar otro enfoque mas allá de la computación clásica para su resolución. Hoy en día, el foco en la computación cuántica ya no es sólo por su enorme capacidad informática sino también, por el uso de su imperfección en esta era Noisy Intermediate-Scale Quantum (NISQ) para crear un poderoso dispositivo de aprendizaje automático que usa el principio variacional para resolver problemas de optimización al reducir su clase de complejidad. En la tesis se propone una formulación (cuadrática) para resolver el problema del horario de los trabajadores sociales de manera eficiente usando Variational Quantum Eigensolver (VQE), Quantum Approximate Optimization Algorithm (QAOA), Minimal Eigen Optimizer y ADMM optimizer. La viabilidad cuántica del algoritmo se ha modelado en forma QUBO, con Docplex simulado Cirq, Or-Tools y probado en computadoras IBMQ. Después de analizar los resultados del enfoque anterior, se diseñó un escenario para resolver el SWP como razonamiento basado en casos (qCBR), tanto cuántica como clásicamente. Y así, poder contribuir con un algoritmo cuántico centrado en la inteligencia artificial y el aprendizaje automático. El qCBR es una técnica de aprendizaje automático basada en la resolución de nuevos problemas que utiliza la experiencia, como lo hacen los humanos. La experiencia se representa como una memoria de casos que contiene cuestiones previamente resueltas y usa una técnica de síntesis para adaptar mejor la experiencia al nuevo problema. En la definición de SWP, si en lugar de pacientes se tienen lotes de pedidos y en lugar de trabajadores sociales robots móviles, se generaliza la función objetivo y las restricciones. Para ello, se ha propuesto una prueba de concepto y una nueva formulación para resolver los problemas de picking y batching llamado qRobot. Se hizo una prueba de concepto en esta parte del proyecto a través de una Raspberry Pi 4 y se probó la capacidad de integración de la computación cuántica dentro de la robótica móvil, con uno de los problemas más demandados en este sector industrial: problemas de picking y batching. Se probó en distintas tecnologías y los resultados fueron prometedores. Además, en caso de necesidad computacional, el robot paraleliza parte de las operaciones en computación híbrida (cuántica + clásica), accediendo a CPU y QPU distribuidos en una nube pública o privada. Además, desarrollamos un entorno estable (ARM64) dentro del robot (Raspberry) para ejecutar operaciones de gradiente y otros algoritmos cuánticos en IBMQ, Amazon Braket (D-Wave) y Pennylane de forma local o remota. Para mejorar el tiempo de ejecución de los algoritmos variacionales en esta era NISQ y la siguiente, se ha propuesto EVA: un algoritmo de Aproximación de Valor Exponencial cuántico. Hasta la fecha, el VQE es el buque insignia de la computación cuántica. Hoy en día, en las plataformas de computación cuántica en la nube líderes de mercado, el coste de la experimentación de los circuitos cuánticos es proporcional al número de circuitos que se ejecutan en dichas plataformas. Es decir, con más circuitos mayor coste. Una de las cosas que consigue el VQE, el buque insignia de esta era de pocos qubits, es la poca profundidad al dividir el Hamiltoniano en una lista de muchos pequeños circuitos (matrices de Pauli). Pero este mismo hecho, hace que simular con el VQE sea muy caro en la nube. Por esta misma razón, se diseñó EVA para poder calcular el valor esperado con un único circuito. Aún habiendo respuesto a la hipótesis de este trabajo con todos los estudios realizados, todavía se puede seguir investigando para proponer nuevos algoritmos cuánticos para mejorar problemas de optimización combinatoria.In this research, Combinatorial optimization techniques to solve constraint problems have been examined. A study of the quantum era and market leaders such as IBM, D-Wave, Google, Xanadu, AWS-Braket and Microsoft has been carried out. We have learned about their community, their platforms, the status of their research, and the postulates of quantum mechanics that create the most efficient quantum systems and algorithms. To know if it is possible to solve Constraint Search Problem (CSP) problems more efficiently with quantum computing, a scenario was defined so that both classical and quantum computing would have a good point of reference. First, the proof of concept focuses on the social worker scheduling problem and later on the issue of batch picking and order picking as a generalization of the Social Workers Problem (SWP). The social workers programming problem is a combinatorial optimization problem that can be solved exponentially at best; seeing that the SWP is NP-Hard, it claims using another approach beyond classical computation for its resolution. Today, the focus on quantum computing is no longer only on its enormous computing power but also on the use of its imperfection in this era Noisy Intermediate-Scale Quantum (NISQ) to create a powerful machine learning device that uses the variational principle to solve optimization problems by reducing their complexity class. In the thesis, a (quadratic) formulation is proposed to solve the problem of social workers' schedules efficiently using Variational Quantum Eigensolver (VQE), Quantum Approximate Optimization Algorithm (QAOA), Minimal Eigen Optimizer and ADMM optimizer. The quantum feasibility of the algorithm has been modelled in QUBO form, with Cirq simulated, Or-Tools and tested on IBMQ computers. After analyzing the results of the above approach, a scenario was designed to solve the SWP as quantum case-based reasoning (qCBR), both quantum and classically. And thus, to be able to contribute with a quantum algorithm focused on artificial intelligence and machine learning. The qCBR is a machine learning technique based on solving new problems that use experience, as humans do. The experience is represented as a memory of cases containing previously resolved questions and uses a synthesis technique to adapt the background to the new problem better. In the definition of SWP, if instead of patients there are batches of orders and instead of social workers mobile robots, the objective function and the restrictions are generalized. To do this, a proof of concept and a new formulation has been proposed to solve the problems of picking and batching called qRobot. A proof of concept was carried out in this part of the project through a Raspberry Pi 4 and the integration capacity of quantum computing within mobile robotics was tested, with one of the most demanded problems in this industrial sector: picking and batching problems. It was tested on different technologies, and the results were promising. Furthermore, in case of computational need, the robot parallelizes part of the operations in hybrid computing (quantum + classical), accessing CPU and QPU distributed in a public or private cloud. Furthermore, we developed a stable environment (ARM64) inside the robot (Raspberry) to run gradient operations and other quantum algorithms on IBMQ, Amazon Braket (D-Wave) and Pennylane locally or remotely. To improve the execution time of variational algorithms in this NISQ era and the next, EVA has been proposed: A quantum Exponential Value Approximation algorithm. To date, the VQE is the flagship of quantum computing. Today, in the market-leading quantum cloud computing platforms, the cost of experimenting with quantum circuits is proportional to the number of circuits running on those platforms. That is, with more circuits, higher cost. One of the things that the VQE, the flagship of this low-qubit era, achieves is shallow depth by dividing the Hamiltonian into a list of many small circuits (Pauli matrices). But this very fact makes simulating with VQE very expensive in the cloud. For this same reason, EVA was designed to calculate the expected value with a single circuit. Even having answered the hypothesis of this work with all the studies carried out, it is still possible to continue research to propose new quantum algorithms to improve combinatorial optimization