- Email: [email protected]
Parallel cooperative optimization through hyperheuristics
Mario R. Eden, Marianthi Ierapetritou and Gavin P. Towler (Editors) Proceedings of the 13th International Symposium on Process Systems Engineering – PSE 2018 July 1-5, 2018, San Diego, California, USA © 2018 Elsevier B.V. All rights reserved. https://doi.org/10.1016/B978-0-444-64241-7.50129-4
Parallel cooperative optimization through hyperheuristics Paola P. Oteizaa, Diego A. Rodríguezb, c , Nélida B. Brignolea, b* a
Planta Piloto de Ingeniería Química (Universidad Nacional del Sur - CONICET) Camino La Carrindanga km. 7 - 8000 Bahía Blanca – Argentina b Laboratorio de Investigación y Desarrollo en Computación Científica (LIDECC)Departamento de Ciencias e Ingeniería de la Computación (DCIC), Universidad Nacional del Sur (UNS), Bahía Blanca, Argentina. c Facultad de Ciencias Exactas, Universidad Nacional de Salta (UNSa), Salta, Argentina. [email protected]
Abstract A hyperheuristics that coordinates the interaction between various metaheuristic techniques is presented. The proposed algorithm, which we called Parallel Optimizer With Hyperheuristics (POWH), includes a Genetic Algorithm, Simulated Annealing, and Ant Colony Optimization. In view of the need to escape from local optima, information exchanges take place between these metaheuristics. In this way, it is possible to take advantage of each metaheuristics’ particular strengths during the search process. Testing related to the hyperheuristic approach was carried out by using the following real-life case studies: I. the optimal design of a subsea pipeline network and II. the urban bus-transit optimal planning. In both cases, a satisfactory reduction of the computational time was achieved due to the parallel implementation that allowed several metaheuristics to run simultaneously. Moreover, better results were also obtained thanks to the parallel cooperative combination of metaheuristics compared with serial executions. Keywords: parallel programming, metaheuristics, pipelines, LRP, hyperheuristics.
1. Introduction Nowadays, metaheuristic optimization algorithms have turned out to be quite attractive because of their distinct advantages over traditional algorithms (Gupta and Ramteke, 2014). Since metaheuristics can solve multiple-objective multiple-solution and nonlinear formulations, they are employed to find high-quality solutions to an ever-growing number of complex real-world problems, such as the combinatorial ones. Evolutionary techniques, like Genetic Algorithms (GA) and Ant Colony Optimization (ACO), have successfully been employed in solving many general optimal problems, such as the optimization of fed-batch fermentation. Optimization frameworks combining GA and Simulated Annealing (SA) have also been applied for the design and synthesis of heat exchanger networks (Shelokar et al., 2014), as well as for model predictive control (Venkateswarlu and Reddy, 2008). In particular, some variants of hybrid-GA techniques have been reported as novel prediction techniques (Sukumar et al., 2014). In the present study we have proposed a hyperheuristic approach by incorporating GA, SA and ACO. A hyperheuristics (Burke et al., 2013) is a general search method that has a set of solvers (low-level methods) and manages the execution of the most convenient
806
P. P. Oteiza et al.
technique at a given time during the search process. When solving difficult computational search problems, hyperheuristics have proved to be advantageous. In operational research, there are some hyperheuristic-based algorithms that have been developed for single kinds of classical metaheuristics (Koulinas et al, 2014), but the combination of several types of metaheuristics has not been thoroughly analysed. Branke et al. (2016) summarized the state-of-the-art alternatives for hyperheuristic design choices and identified the main issues that future work should focus on. They pinpointed that their high computational requirements constitute a major disadvantage of hyperheuristics. In this context, parallelism emerges as an advanced strategy to reduce computational times significantly. Both the demand for fast solutions of optimization problems, together with the advent of novel concurrent computing architectures constitute incentives for the development of parallel algorithms (Alba, 2005) in order to solve challenging problems efficiently. Regarding computational efficiency in urban planning strategies, Agrawal and Mathew (2004) have implemented a parallel algorithm based on a GA model for a large-scale problem, reporting satisfactory results in computational time, speedup and efficiency. In view of the widespread need to solve challenging large-scale PSE problems, a parallel implementation was adopted and the proposed algorithm was called Parallel Optimizer With Hyperheuristics (POWH). The parallelization allows several metaheuristics to run simultaneously in threads, thus achieving a satisfactory decrease of the computational time. POWH design is briefly described in Section 2. Then, in Section 3 both pipelining and bus planning are presented as test problems, whose main features are pointed out before the discussion of some computational results, which are given in Section 4. Finally, some conclusions and hints for future work are summarized in Section 5.
2. Methodology A cooperative strategy applicable to optimization problems is presented. The proposed algorithm is a hyper-heuristics, which comprises 3 well-known canonical metaheuristics with various strengths. The algorithm for the Parallel Optimizer With Hyperheuristics (POWH) is based on the master–slave paradigm (Fig. 1) and has multiple metaheuristics that cooperate periodically and work concurrently on distributed computing environments. Its organizational framework is an A-team architecture (Talukdar et al., 1998), where the autonomous agents are the following metaheuristics: Simulated Annealing (SA), Genetic Algorithm (GA) and Ant Colony Optimization (ACO). The partnership is useful because it has been conceived by combining different primary procedures: SA is a trajectory-based technique, GA is a population-based method and ACO is a constructive approach.SA's strategy avoids local minima by allowing with a certain probability to choose a solution whose fitness value is worse than the current solution. Therefore, the SA algorithm converges relatively slowly towards the final solution. This drawback can be remediated by both implementing parallel programming and hybridizing the algorithm with ACO, which is particularly advantageous due to the remarkable ability of artificial ants to construct solutions guided by the pheromone trails. Moreover, SA tends to find local optima, thus delaying convergence. This weakness can also be surpassed thanks to a combination of SA with other metaheuristics that may contribute to diversify the search by enlarging the search space by means of efficient explorations. For this purpose, it was necessary to incorporate another exploratory method, such as GA.
Parallel cooperative optimization through hyperheuristics
807
In GA the initial population is chosen randomly, while the best solution designated by the master is always incorporated as an elite solution. In general, the termination criteria are all based on the amount of executions. These limits are: 50 generations for GA, 100 iterations for SA and 50 colonies for ACO.
Figure 1. Hyperheuristic Model.
As to the control system, the choice of a suitable metaheuristics is guided by a ranking index (Ind) that assesses each metaheuristics’ performance through Eq. (1-6), whenever a processor becomes idle. The technique with the highest rank will thus be eligible for the next execution. Besides, the particular problem is not only incorporated in modules for the model equations and the objective function, but also in the filter module because filtering is necessary to detect unfeasible solutions. The filter is a problem-dependent selector that indicates which solutions should be erased. ݀݊ܫൌ ݏݏ݁݊ݐ݅ܨ ݊݅ݐܽݑ݈ܽݒܧ ܶ݅݉݁ ܱ ݕݐ݈݅ܽ݉݅ݐ ܴ݁݊݅ݐ݅ݐ݁
(1)
ൌͳ െ ൫ݏݏ݁݊ݐ݅ܨݐݏ݁ܤൗܵݏݏ݁݊ݐ݅ܨ݉ݑ൯
(2)
ൌͳ െ ݈ܽݒܧൗܵ ݈ܽݒܧ݉ݑ
(3)
ൌͳ െ ݁݉݅ܶݐݏ݁ܤൗܵ݁݉݅ܶ݉ݑ
(4)
ൌͳ െ ቆݏݏ݁݊ݐ݅ܨݐݏ݁ܤ൘
(5)
ቇ ൫ܵܣ ܣܩ ܱܥܣ ൯
ൌͳ െ ቀ݁
ோ
ቁ ൗܳ ௨௦
(6)
3. Test Cases A couple of PSE-representative case studies were solved. Aiming at algorithmic generalization, Case II was chosen since it also serves to create very large challenging instances that may demand long computational times. CASE I: The optimal design of a subsea pipeline network. It was modelled based on the problem formulation in Oteiza et al. (2015). In contrast, the well platforms were represented by concentrating nodes, which are located by the optimizer in the adapted model. Besides, the objective function (Eq. (7)) was adapted in this work so as to comprise offshore pipelining by contemplating specific terms related to installation and operating costs. The Net Present Value (NPV) of the project was calculated from revenue
P. P. Oteiza et al.
808
flows and terms related to the construction and operating costs for each well platform ܹܲ . In Eq (7), the building costs were taken into account by considering both pipeline construction CCij,t and the installation costs of every well platform δk,t. The operating costs include the maintenance cost MCij,t and the operative labor cost LCt. The transport tariff Pt is useful to determine whether the project is economically viable. Since fitness values were measured in terms of NPV, this is a maximization problem.
ܸܰܲ ൌ ݐൌͳ
݁
ܲݐܳ ݐ
ሺͳ ݎሻݐ݀݁
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݆݅ܥܥǡݐ
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ሺͳ ݎሻݐ݁െͳ
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݆݅ܲܨെ ݐൌͳ ݇ൌͳ
ܹܲ݇ െ ݐൌͳ
ߜ݇ǡݐ
ሺͳ ݎሻݐ݁െͳ
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ሺͳ ݎሻݐ݀݁
ܹܲ݇
(7)
CASE II: Bus planning. It was modelled by means of an LRP (Location Routing Problem), which constitutes an NP-complete problem (Wolsey, 2000). The stops and routes were determined in order to meet the demand with minimal cost Z. The location and routing model for the urban bus transport was based on the formulation of classical LRPs (Ceder, 2016) with suitable modifications in fitness evaluation, whose function was formulated in terms of network costs Z. It includes three terms to cover the offer, while the fourth one contemplates the demand in more detail through a matrix, whose elements ݉ represent walking-distance penalties. As to the offer, the following terms were included: the costs of traversed edges, the fixed costs of opening bus stops and the fixed vehicular costs associated with employed buses. Then, the lower costs become, the better.
4. Performance Results All the algorithms were implemented in Java and run on a PC with 8 GB of RAM and Intel Core i5-3330. For all instances, each test was repeated independently 30 times. As to the “best average fitness”, it was calculated as the average across the 30 runs of the best value identified in each run. In the first place, serial algorithms were programmed to evaluate the behaviour of the individual metaheuristics whose computational times (costs) are employed as the upper bounds for this analysis. An incremental variation of the amount of nodes was considered on Table 1. It shows the best average fitness and the corresponding average time required for each metaheuristics running sequentially for different sample problems. The solution quality (fitness) was always satisfactory, but computational time also increased as problem size grew up. It can be concluded that POWH´s organization (GA-SA-ACO) is effective because the addition of various agents has provided benefits in terms of solution quality (fitness) for large instances. Table 1 The effect of metaheuristics on the computational time (ms) Algorithms Case
GA-SA-ACO GA SA ACO Size Fitness Time Fitness Time Fitness Time Fitness Time (Nodes) (U$S) (ms) (U$S) (ms) (U$S) (ms) (U$S) (ms)
I Small
5
9.69E+07
9.69E+07
328
8.60E+07
527
9.69E+07
I Large
18
3.19E+08 3,233 1.97E+08
928
2.00E+08
923
1.14E+11 1,370
217
677
Since substantially smaller times are required, a parallel combination of metaheuristics (POWH) was implemented. Parallelization by threading allows several metaheuristics to run simultaneously in threads. On Table 2 the impact of parallelism was corroborated by
Parallel cooperative optimization through hyperheuristics
809
comparing serial and parallel times.The average deviation between the best lower bounds (POWH) and the upper bound (SOWH) is around 50% on smaller instances and around 30% on some large instances, whenever solutions of the same quality are reached. Table 2. Contrasting Parallel (POWH) and Serial (SOWH) algorithms for Case I Algorithm Case I Small I Large
Size (Nodes)
SOWH Fitness (U$S) Time (ms)
5
9.69E+07
18
1.14E+11
POWH Fitness (U$S) Time (ms)
1,458
9.69E+07
677
2,097
1.14E+11
1,370
As to solution quality and for Case II, Table 3 shows that as the number of processors increases, POWH finds solutions of better quality (i.e., lower Z values) in shorter times. Table 3. Contrasting Parallel (POWH) and Serial (SOWH) cooperative algorithms. Algorithm
# Processes
Average Z
Average Time
(U$S)
(ms)
Speed-up
Efficiency (%)
SOWH
1
6,354
2,764
POWH
4
5,804
1,897
1.46
36.5
POWH
7
2,473
1,793
1.54
22.0
5. Conclusions and Future Work Some real-life PSE optimization problems proved to be benefited from quick solutions that could be achieved with the help of parallel programming by means of the proposed hyperheuristic method (POWH) presented here. The solution method coordinates interactions among metaheuristics to create a fast strategy that is able to escape from local optima, thus performing an effective search of a solution space. Three metaheuristics have been combined: Simulated Annealing, Genetic Algorithm, and Ant Colony Optimization. The solution strategy worked out satisfactorily in practical problems. For different sizes, POWH showed promising performance in the search for solutions since it was able to obtain high-quality results in reasonably shorter computing times. For the continuation of this research work, it is necessary to verify in general terms that POWH is always both rigorous and efficient. Then, further testing that involves a wide choice of sample problems oriented to parallel computing running in threads is vital to complete thoroughly the computational design.
Notation ACObf= The best solution yielded by the ACO technique BestFitness = The best fitness yielded for the best attained solution BestTime = The shortest time demanded by an execution CCij,t: construction cost from the i-th to the j-th point, in the t-th period δk,t: installation cost of the k-th Well Platform in the t-th period d: total number of time periods related to WP construction e: total number of pipeline construction time periods Eval = The minimum number of evaluations f: number of Wells
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P. P. Oteiza et al.
FPij: feasible path from the i-th to the j-th point GAbf = The best solution yielded by the GA technique Ind = Ranking index LCt: operative labor cost in the t-th period MCk,t: maintenance cost of the WP structure o: project lifetime p: number of plants onshore Pt: sale price of gas in the t-th period q: number of Well Platforms that have been activated Qt: amount of gas transported in the t-th period Qruns = Number of performed executions r: discount rate Rep = Counter SAbf = The best solution yielded by the SA technique SumEval = The sum of all evaluations SumFitness = The sum of all BestFitness values SumTime= Runtimes required by the past Ne executions WPk: for the k-th Well Platform: WPk=1 if active, otherwise WPk=0 Z = Total system cost per trip
References E. Alba, 2005, Parallel Metaheuristics: A New Class of Algorithms, Wiley. J. Agrawal, T.V. Mathew, 2004, Transit route network design using parallel genetic algorithm, Journal of Computing in Civil Engineering, 18, 3, 248-256. E. K. Burke, M. Gendreau, M. Hyde, G. Kendall, G. Ochoa, E. Özcan, R. Qu, 2013, Hyperheuristics: A survey of the state of the art, J. Operational Research Society, 64 ,12, 1695-1724. J. Branke, S. Nguyen, C.W. Pickardt, M. Zhang, 2016, Automated design of production scheduling heuristics: A review, IEEE Transactions on Evolutionary Computation, 20, 1, 110-124. A. Ceder, 2016, Public Transit Planning and Operation: Modeling, Practice and Behavior, 2nd Ed., Boca Raton, FL, USA, CRC Press. S. K. Gupta, M. Ramteke, 2014, Applications of genetic algorithms in chemical engineering I: , Springer, 39-59. Methodology, In: Applications of Metaheuristics in Process Eng., G. Koulinas, L. Kotsikas, K. Anagnostopoulos, 2014, A PSO based hyper-heuristic algorithm for the classic resource constrained project scheduling problem, Inf. Sci., 277, 680-693. P.P. Oteiza, M. De Meio, D.A. Rodriguez, V. Viego, N.B. Brignole, 2015, Metaheuristic Techniques for the Optimal Design of NGL Pipelining, Computer-Aided Chem. Eng., 37, 785-790. P. Shelokar, A. Kulkarni, V.K. Jayaraman, P. Siarry, 2014, Metaheuristics in Process Eng.: A Historical Perspective, In: Applications of Metaheuristics in Process Eng., , 1-38. N. Sukumar, G. Prabhu, P. Saha, 2014, Applications of genetic algorithms in QSAR/QSPR modeling, In: Applications of Metaheuristics in Process Eng., Switzerland, Springer, 315-324. S. Talukdar, L. Baerentzen, A. Gove, P. De Souza, P, 1998, Asynchronous teams: Cooperation schemes for autonomous agents, Journal of Heuristics, 4, 4, 295-321. C. Venkateswarlu, A.D. Reddy, 2008, Nonlinear model predictive control of reactive distillation based on stochastic optimization, Ind. Eng. Chem. Res., 47(18), 6949–6960 L.A. Wolsey, 2000, Integer programming, IIE Transactions, 32, 273-285, 2-58.
Parallel cooperative optimization through hyperheuristics Paola P. Oteizaa, Diego A. Rodríguezb, c , Nélida B. Brignolea, b* a
Planta Piloto de Ingeniería Química (Universidad Nacional del Sur - CONICET) Camino La Carrindanga km. 7 - 8000 Bahía Blanca – Argentina b Laboratorio de Investigación y Desarrollo en Computación Científica (LIDECC)Departamento de Ciencias e Ingeniería de la Computación (DCIC), Universidad Nacional del Sur (UNS), Bahía Blanca, Argentina. c Facultad de Ciencias Exactas, Universidad Nacional de Salta (UNSa), Salta, Argentina. [email protected]
Abstract A hyperheuristics that coordinates the interaction between various metaheuristic techniques is presented. The proposed algorithm, which we called Parallel Optimizer With Hyperheuristics (POWH), includes a Genetic Algorithm, Simulated Annealing, and Ant Colony Optimization. In view of the need to escape from local optima, information exchanges take place between these metaheuristics. In this way, it is possible to take advantage of each metaheuristics’ particular strengths during the search process. Testing related to the hyperheuristic approach was carried out by using the following real-life case studies: I. the optimal design of a subsea pipeline network and II. the urban bus-transit optimal planning. In both cases, a satisfactory reduction of the computational time was achieved due to the parallel implementation that allowed several metaheuristics to run simultaneously. Moreover, better results were also obtained thanks to the parallel cooperative combination of metaheuristics compared with serial executions. Keywords: parallel programming, metaheuristics, pipelines, LRP, hyperheuristics.
1. Introduction Nowadays, metaheuristic optimization algorithms have turned out to be quite attractive because of their distinct advantages over traditional algorithms (Gupta and Ramteke, 2014). Since metaheuristics can solve multiple-objective multiple-solution and nonlinear formulations, they are employed to find high-quality solutions to an ever-growing number of complex real-world problems, such as the combinatorial ones. Evolutionary techniques, like Genetic Algorithms (GA) and Ant Colony Optimization (ACO), have successfully been employed in solving many general optimal problems, such as the optimization of fed-batch fermentation. Optimization frameworks combining GA and Simulated Annealing (SA) have also been applied for the design and synthesis of heat exchanger networks (Shelokar et al., 2014), as well as for model predictive control (Venkateswarlu and Reddy, 2008). In particular, some variants of hybrid-GA techniques have been reported as novel prediction techniques (Sukumar et al., 2014). In the present study we have proposed a hyperheuristic approach by incorporating GA, SA and ACO. A hyperheuristics (Burke et al., 2013) is a general search method that has a set of solvers (low-level methods) and manages the execution of the most convenient
806
P. P. Oteiza et al.
technique at a given time during the search process. When solving difficult computational search problems, hyperheuristics have proved to be advantageous. In operational research, there are some hyperheuristic-based algorithms that have been developed for single kinds of classical metaheuristics (Koulinas et al, 2014), but the combination of several types of metaheuristics has not been thoroughly analysed. Branke et al. (2016) summarized the state-of-the-art alternatives for hyperheuristic design choices and identified the main issues that future work should focus on. They pinpointed that their high computational requirements constitute a major disadvantage of hyperheuristics. In this context, parallelism emerges as an advanced strategy to reduce computational times significantly. Both the demand for fast solutions of optimization problems, together with the advent of novel concurrent computing architectures constitute incentives for the development of parallel algorithms (Alba, 2005) in order to solve challenging problems efficiently. Regarding computational efficiency in urban planning strategies, Agrawal and Mathew (2004) have implemented a parallel algorithm based on a GA model for a large-scale problem, reporting satisfactory results in computational time, speedup and efficiency. In view of the widespread need to solve challenging large-scale PSE problems, a parallel implementation was adopted and the proposed algorithm was called Parallel Optimizer With Hyperheuristics (POWH). The parallelization allows several metaheuristics to run simultaneously in threads, thus achieving a satisfactory decrease of the computational time. POWH design is briefly described in Section 2. Then, in Section 3 both pipelining and bus planning are presented as test problems, whose main features are pointed out before the discussion of some computational results, which are given in Section 4. Finally, some conclusions and hints for future work are summarized in Section 5.
2. Methodology A cooperative strategy applicable to optimization problems is presented. The proposed algorithm is a hyper-heuristics, which comprises 3 well-known canonical metaheuristics with various strengths. The algorithm for the Parallel Optimizer With Hyperheuristics (POWH) is based on the master–slave paradigm (Fig. 1) and has multiple metaheuristics that cooperate periodically and work concurrently on distributed computing environments. Its organizational framework is an A-team architecture (Talukdar et al., 1998), where the autonomous agents are the following metaheuristics: Simulated Annealing (SA), Genetic Algorithm (GA) and Ant Colony Optimization (ACO). The partnership is useful because it has been conceived by combining different primary procedures: SA is a trajectory-based technique, GA is a population-based method and ACO is a constructive approach.SA's strategy avoids local minima by allowing with a certain probability to choose a solution whose fitness value is worse than the current solution. Therefore, the SA algorithm converges relatively slowly towards the final solution. This drawback can be remediated by both implementing parallel programming and hybridizing the algorithm with ACO, which is particularly advantageous due to the remarkable ability of artificial ants to construct solutions guided by the pheromone trails. Moreover, SA tends to find local optima, thus delaying convergence. This weakness can also be surpassed thanks to a combination of SA with other metaheuristics that may contribute to diversify the search by enlarging the search space by means of efficient explorations. For this purpose, it was necessary to incorporate another exploratory method, such as GA.
Parallel cooperative optimization through hyperheuristics
807
In GA the initial population is chosen randomly, while the best solution designated by the master is always incorporated as an elite solution. In general, the termination criteria are all based on the amount of executions. These limits are: 50 generations for GA, 100 iterations for SA and 50 colonies for ACO.
Figure 1. Hyperheuristic Model.
As to the control system, the choice of a suitable metaheuristics is guided by a ranking index (Ind) that assesses each metaheuristics’ performance through Eq. (1-6), whenever a processor becomes idle. The technique with the highest rank will thus be eligible for the next execution. Besides, the particular problem is not only incorporated in modules for the model equations and the objective function, but also in the filter module because filtering is necessary to detect unfeasible solutions. The filter is a problem-dependent selector that indicates which solutions should be erased. ݀݊ܫൌ ݏݏ݁݊ݐ݅ܨ ݊݅ݐܽݑ݈ܽݒܧ ܶ݅݉݁ ܱ ݕݐ݈݅ܽ݉݅ݐ ܴ݁݊݅ݐ݅ݐ݁
(1)
ൌͳ െ ൫ݏݏ݁݊ݐ݅ܨݐݏ݁ܤൗܵݏݏ݁݊ݐ݅ܨ݉ݑ൯
(2)
ൌͳ െ ݈ܽݒܧൗܵ ݈ܽݒܧ݉ݑ
(3)
ൌͳ െ ݁݉݅ܶݐݏ݁ܤൗܵ݁݉݅ܶ݉ݑ
(4)
ൌͳ െ ቆݏݏ݁݊ݐ݅ܨݐݏ݁ܤ൘
(5)
ቇ ൫ܵܣ ܣܩ ܱܥܣ ൯
ൌͳ െ ቀ݁
ோ
ቁ ൗܳ ௨௦
(6)
3. Test Cases A couple of PSE-representative case studies were solved. Aiming at algorithmic generalization, Case II was chosen since it also serves to create very large challenging instances that may demand long computational times. CASE I: The optimal design of a subsea pipeline network. It was modelled based on the problem formulation in Oteiza et al. (2015). In contrast, the well platforms were represented by concentrating nodes, which are located by the optimizer in the adapted model. Besides, the objective function (Eq. (7)) was adapted in this work so as to comprise offshore pipelining by contemplating specific terms related to installation and operating costs. The Net Present Value (NPV) of the project was calculated from revenue
P. P. Oteiza et al.
808
flows and terms related to the construction and operating costs for each well platform ܹܲ . In Eq (7), the building costs were taken into account by considering both pipeline construction CCij,t and the installation costs of every well platform δk,t. The operating costs include the maintenance cost MCij,t and the operative labor cost LCt. The transport tariff Pt is useful to determine whether the project is economically viable. Since fitness values were measured in terms of NPV, this is a maximization problem.
ܸܰܲ ൌ ݐൌͳ
݁
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ሺͳ ݎሻݐ݀݁
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െ ݐൌͳ ݇ൌͳ
݆݅ܥܥǡݐ
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ݍ
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ܹܲ݇ െ ݐൌͳ
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ሺͳ ݎሻݐ݁െͳ
ݐܥܮ
ሺͳ ݎሻݐ݀݁
ܹܲ݇
(7)
CASE II: Bus planning. It was modelled by means of an LRP (Location Routing Problem), which constitutes an NP-complete problem (Wolsey, 2000). The stops and routes were determined in order to meet the demand with minimal cost Z. The location and routing model for the urban bus transport was based on the formulation of classical LRPs (Ceder, 2016) with suitable modifications in fitness evaluation, whose function was formulated in terms of network costs Z. It includes three terms to cover the offer, while the fourth one contemplates the demand in more detail through a matrix, whose elements ݉ represent walking-distance penalties. As to the offer, the following terms were included: the costs of traversed edges, the fixed costs of opening bus stops and the fixed vehicular costs associated with employed buses. Then, the lower costs become, the better.
4. Performance Results All the algorithms were implemented in Java and run on a PC with 8 GB of RAM and Intel Core i5-3330. For all instances, each test was repeated independently 30 times. As to the “best average fitness”, it was calculated as the average across the 30 runs of the best value identified in each run. In the first place, serial algorithms were programmed to evaluate the behaviour of the individual metaheuristics whose computational times (costs) are employed as the upper bounds for this analysis. An incremental variation of the amount of nodes was considered on Table 1. It shows the best average fitness and the corresponding average time required for each metaheuristics running sequentially for different sample problems. The solution quality (fitness) was always satisfactory, but computational time also increased as problem size grew up. It can be concluded that POWH´s organization (GA-SA-ACO) is effective because the addition of various agents has provided benefits in terms of solution quality (fitness) for large instances. Table 1 The effect of metaheuristics on the computational time (ms) Algorithms Case
GA-SA-ACO GA SA ACO Size Fitness Time Fitness Time Fitness Time Fitness Time (Nodes) (U$S) (ms) (U$S) (ms) (U$S) (ms) (U$S) (ms)
I Small
5
9.69E+07
9.69E+07
328
8.60E+07
527
9.69E+07
I Large
18
3.19E+08 3,233 1.97E+08
928
2.00E+08
923
1.14E+11 1,370
217
677
Since substantially smaller times are required, a parallel combination of metaheuristics (POWH) was implemented. Parallelization by threading allows several metaheuristics to run simultaneously in threads. On Table 2 the impact of parallelism was corroborated by
Parallel cooperative optimization through hyperheuristics
809
comparing serial and parallel times.The average deviation between the best lower bounds (POWH) and the upper bound (SOWH) is around 50% on smaller instances and around 30% on some large instances, whenever solutions of the same quality are reached. Table 2. Contrasting Parallel (POWH) and Serial (SOWH) algorithms for Case I Algorithm Case I Small I Large
Size (Nodes)
SOWH Fitness (U$S) Time (ms)
5
9.69E+07
18
1.14E+11
POWH Fitness (U$S) Time (ms)
1,458
9.69E+07
677
2,097
1.14E+11
1,370
As to solution quality and for Case II, Table 3 shows that as the number of processors increases, POWH finds solutions of better quality (i.e., lower Z values) in shorter times. Table 3. Contrasting Parallel (POWH) and Serial (SOWH) cooperative algorithms. Algorithm
# Processes
Average Z
Average Time
(U$S)
(ms)
Speed-up
Efficiency (%)
SOWH
1
6,354
2,764
POWH
4
5,804
1,897
1.46
36.5
POWH
7
2,473
1,793
1.54
22.0
5. Conclusions and Future Work Some real-life PSE optimization problems proved to be benefited from quick solutions that could be achieved with the help of parallel programming by means of the proposed hyperheuristic method (POWH) presented here. The solution method coordinates interactions among metaheuristics to create a fast strategy that is able to escape from local optima, thus performing an effective search of a solution space. Three metaheuristics have been combined: Simulated Annealing, Genetic Algorithm, and Ant Colony Optimization. The solution strategy worked out satisfactorily in practical problems. For different sizes, POWH showed promising performance in the search for solutions since it was able to obtain high-quality results in reasonably shorter computing times. For the continuation of this research work, it is necessary to verify in general terms that POWH is always both rigorous and efficient. Then, further testing that involves a wide choice of sample problems oriented to parallel computing running in threads is vital to complete thoroughly the computational design.
Notation ACObf= The best solution yielded by the ACO technique BestFitness = The best fitness yielded for the best attained solution BestTime = The shortest time demanded by an execution CCij,t: construction cost from the i-th to the j-th point, in the t-th period δk,t: installation cost of the k-th Well Platform in the t-th period d: total number of time periods related to WP construction e: total number of pipeline construction time periods Eval = The minimum number of evaluations f: number of Wells
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P. P. Oteiza et al.
FPij: feasible path from the i-th to the j-th point GAbf = The best solution yielded by the GA technique Ind = Ranking index LCt: operative labor cost in the t-th period MCk,t: maintenance cost of the WP structure o: project lifetime p: number of plants onshore Pt: sale price of gas in the t-th period q: number of Well Platforms that have been activated Qt: amount of gas transported in the t-th period Qruns = Number of performed executions r: discount rate Rep = Counter SAbf = The best solution yielded by the SA technique SumEval = The sum of all evaluations SumFitness = The sum of all BestFitness values SumTime= Runtimes required by the past Ne executions WPk: for the k-th Well Platform: WPk=1 if active, otherwise WPk=0 Z = Total system cost per trip
References E. Alba, 2005, Parallel Metaheuristics: A New Class of Algorithms, Wiley. J. Agrawal, T.V. Mathew, 2004, Transit route network design using parallel genetic algorithm, Journal of Computing in Civil Engineering, 18, 3, 248-256. E. K. Burke, M. Gendreau, M. Hyde, G. Kendall, G. Ochoa, E. Özcan, R. Qu, 2013, Hyperheuristics: A survey of the state of the art, J. Operational Research Society, 64 ,12, 1695-1724. J. Branke, S. Nguyen, C.W. Pickardt, M. Zhang, 2016, Automated design of production scheduling heuristics: A review, IEEE Transactions on Evolutionary Computation, 20, 1, 110-124. A. Ceder, 2016, Public Transit Planning and Operation: Modeling, Practice and Behavior, 2nd Ed., Boca Raton, FL, USA, CRC Press. S. K. Gupta, M. Ramteke, 2014, Applications of genetic algorithms in chemical engineering I: , Springer, 39-59. Methodology, In: Applications of Metaheuristics in Process Eng., G. Koulinas, L. Kotsikas, K. Anagnostopoulos, 2014, A PSO based hyper-heuristic algorithm for the classic resource constrained project scheduling problem, Inf. Sci., 277, 680-693. P.P. Oteiza, M. De Meio, D.A. Rodriguez, V. Viego, N.B. Brignole, 2015, Metaheuristic Techniques for the Optimal Design of NGL Pipelining, Computer-Aided Chem. Eng., 37, 785-790. P. Shelokar, A. Kulkarni, V.K. Jayaraman, P. Siarry, 2014, Metaheuristics in Process Eng.: A Historical Perspective, In: Applications of Metaheuristics in Process Eng., , 1-38. N. Sukumar, G. Prabhu, P. Saha, 2014, Applications of genetic algorithms in QSAR/QSPR modeling, In: Applications of Metaheuristics in Process Eng., Switzerland, Springer, 315-324. S. Talukdar, L. Baerentzen, A. Gove, P. De Souza, P, 1998, Asynchronous teams: Cooperation schemes for autonomous agents, Journal of Heuristics, 4, 4, 295-321. C. Venkateswarlu, A.D. Reddy, 2008, Nonlinear model predictive control of reactive distillation based on stochastic optimization, Ind. Eng. Chem. Res., 47(18), 6949–6960 L.A. Wolsey, 2000, Integer programming, IIE Transactions, 32, 273-285, 2-58.