Available online at www.sciencedirect.com ScienceDirect Procedia CIRP 57 (2016) 416 – 421 49th CIRP Conference on Manufacturing Systems (CIRP-CMS 2016) Hybrid multi-objective optimization method for solving simultaneously the line balancing, equipment and buffer sizing problems for hybrid assembly systems Jonathan Oesterlea,*, Thomas Bauernhansla,b, Lionel Amodeoc a Fraunhofer Institute for Manufacturing Engineering and Automation (IPA), Nobelstrasse 12, 70569 Stuttgart, Germany Institute of Industrial Manufacturing and Management (IFF), University of Stuttgart, Nobelstrasse 12, 70569 Stuttgart, Germany c Charles Delaunay Institute (ICD-LOSI), University of Technology of Troyes, STMR, UMR CNRS 6279, 12 rue Marie Curie, 10010 Troyes, France b * Corresponding author Tel.: +49(0)711/970-1199; fax: +49(0)711/970-1009 E-mail address: jno@ipa.fhg.de Abstract In today’s dynamic and uncertain markets, companies are required to regularly renew their product and process platforms through new production technologies and factory infrastructure This results in shortening products’, processes’ and factories life cycle, engendering in return an increase of the complexity of assembly planning tasks, which are seen as increasingly uncertain and complex to control This article presents a hybrid multi-objective optimization algorithm aiming at solving simultaneously the line balancing, equipment selection and buffer sizing problem under consideration of capacity and cost-oriented objectives The proposed algorithm is compared to two classical evolutionary algorithms, the NSGA2 and SPEA2 © Published by Elsevier B.V This 2015The TheAuthors Authors Published by Elsevier B.V.is an open access article under the CC BY-NC-ND license © 2016 (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of Scientific committee of the 49th CIRP Conference on Manufacturing Systems (CIRP-CMS 2016) Peer-review under responsibility of the scientific committee of the 49th CIRP Conference on Manufacturing Systems Keywords: Assembly Line Design problem; balancing problem; equipment selection; buffer sizing; Mixed Model Line; Multiobjective Introduction Since product mass customization became a viable strategy in the mid-1990s, there has been tremendous market pressure on companies to deliver personalized products and services to customers with mass production efficiency, costs and quality [1] Assembly lines, which are the most commonly used assembly systems and allow meeting these cost and efficiency requirements permit the assembly of products by workers with limited training and dedicated machines and/or robots Due to the high investment and running costs involved, the design and re-design of such lines is highly important and a number of crucial decisions have to be made, including product design, process selection, line layout configuration, line balancing and buffer sizing Usually, due to their complexity, these problems are considered successively at one time [2,3] However, successively addressing the previous steps will most likely not lead to a global optimum of the whole system The last two crucial steps are the line balancing and buffer sizing In the former case, the tasks are assigned to workstations such that the efficiency of the line is optimized The effects of the unreliability of machines and/or robots, e.g starvation and blockage, disrupting the material flow in the assembly line, are limited through buffer inclusion Additionally, they help to smooth and balance the material flow between stations However, the inclusion of buffers requires additional capital investment, floor space of the line, in-process inventory [4,5] and also increases the lead time Changeable, dynamic and uncertain markets forces companies to regularly renew their product and process platforms through new production technologies and factory infrastructure in order to fit explicitly the requirements of individual customers This results in shortening not only products life cycles but also factories’ and processes’ This consistently affects the complexity of assembly planning projects, which are seen as increasingly uncertain, complex, dynamic and difficult to control These characteristics features are mostly related to the choice of the right level of 2212-8271 © 2016 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 49th CIRP Conference on Manufacturing Systems doi:10.1016/j.procir.2016.11.072 Jonathan Oesterle et al / Procedia CIRP 57 (2016) 416 – 421 automation (e.g fully, semi-automatic or manual) and equipment regarding technical data and engendered product costs, which in turn affects the buffer location and size in the assembly line In the views of significant uncertainty, the ability to plan the most flexible and economic assembly system by taking product and process alternatives is highly important This article presents a holistic planning method for a mixed-model line under consideration of product, processes and resources alternatives, aiming at optimizing capacity- and cost-oriented objectives This planning method addresses simultaneously the process selection, line balancing and buffer sizing State-of-the-Art The Assembly Line Balancing Problem (ALBP) consists in finding a feasible line balancing, i.e., an assignment of tasks to station such that precedence constraints and possible further restrictions are fulfilled Due to different conditions in industrial manufacturing, assembly line systems and corresponding ALBPs have been extensively studied and different classification schemes and state-of-the-art have been proposed The most recent reviews of ALBPs are those provided by Becker and Scholl [6,7] Baybars [8] proposed a common classification scheme, which distinguish between the: (i) Simple Assembly Line Balancing Problem, and the (ii) Generalized Assembly Line Balancing Problem (GALBP) In the former case, only one single product is processed, and the problem is restricted by precedence relations and cycle time constraints In the latter case, problems involving e.g parallel workstations, parallel tasks, unequally equipped workstations, problems involving sequence-dependent or stochastic processing times and problems considering mixed and multimodel lines can be found Battaia and Dolgui [2] provided a taxonomy of ALBPs ALBPs can also be classified into (i) capacity-oriented objectives and (ii) cost-oriented objectives Hazir et al [9] extended the classification of Boysen et al by incorporating cost and profit aspects Two different approaches have been proposed to incorporate processing alternatives into ALBP [10] The former one is known as the equipment selection problem and is based on the assumption that there is a fixed set of equipment (exactly one of each) that has to be selected and assigned to a station The latter consists in assigning processes to tasks In addition to line balancing, for each task exactly one processing alternative has to be chosen out of a set of possible ones These alternatives are determined through task requirements concerning either technological alternatives (e.g gluing, clinching) or resource alternatives (e.g machines or manpower) Approaches dealing with processing alternatives can be found here [11–14] Capacho and Pastor [15] considers alternative variants that an assembly process may admit Each assembly variant is represented by a subgraph and determines the tasks required to assemble a part of a particular product Up to now, the problem has been defined and modelling in a restricted version and an extended version [16,15] This problem, also known as the Alternative Subgraphs Assembly Line Balancing Problem (ASALBP) considers alternative assembly precedence subgraphs that involve either the same or different set of tasks Not only heuristic methods have been developed and tested comprehensively [17,18] but also an exact method [19] The buffer allocation problem (BAP) aims at allocating a certain amount of buffer, among intermediate buffer locations of a production line to optimize some specific objective, e.g throughput of the assembly line, minimum total buffer size The solution approaches used to solve the BAP involve both a generative and evaluative method While the former aims at searching for an optimal solution, the latter aims at evaluating various performance measures by means of analytical methods and/or simulation A comprehensive survey about the BAP has been provided by Demir et al [4] While the BAP has been widely studied in the literature, only a few works are focused on solving the multiobjective BAP [20] Example of works taking only respectively one criterion into consideration are [21,22] and several criteria [23,20,24] So far, both single and multi-objective assembly line balancing problems and buffer sizing problems have almost always been investigated separately in literature, suffering from the lack of simultaneous considerations Tiacci [25] addresses the problem of simultaneously balancing a line and allocated buffers with stochastic task times and parallel workstations Despite the huge amount of research done over the last years, there is still a vast bridge between the methods provided by the literature and the current industrial problems and market features, engendering a difficult practical use of these methods Indeed, despite the direct simplicity of the available models, some recent characteristic features of the present day situation engendered shortcomings on the current methods Indeed, the requirement of quality engenders the need to not only select and plan the most reliable system, with e.g low maintenance effort, low material waste and a low number of deficient products but also the most economic one While most of the studies consider equipment costs on a high level, other product costs elements (e.g breakdowns, quality) are not examined Furthermore, most of the studies previously listed, only optimize either one capacity-oriented or costoriented objective However, since most real-life decision and planning situations involve multiple conflicting criteria, a multi-criteria optimization model that considers both time and cost based criteria, conjugated with a robust cost model, would better reflect the current industrial needs Last but not least, in order to achieve a global optimum of the whole assembly system, the process selection, the line balancing and buffer sizing have to be addressed simultaneously, which has, to the best knowledge of the authors, not been addressed yet Problem description There are ܲ models of a product, a set of tasks ܸ comprised of a set ܵ ܩof components’ and processes’ alternatives and a set of equipment ܧ For each task ݆, there is a set of available equipment ܧ with different properties, e.g task processing times, costs, scrap rate The problem is to select tasks and equipment and assign them to workstations in order to minimize idle time between models and workstations Additionally, the buffers have to be allocated between stations 417 418 Jonathan Oesterle et al / Procedia CIRP 57 (2016) 416 – 421 such that the throughput of the line is maximized and the total unit product costs, resulting from the line balancing and buffer sizing are minimized Using the node and precedence graph representation of Scholl et al [19], following notation will be used The set of nodes ܸ ൌ ሼͳǡ ǥ ǡ ܰሽ consists of sets: (i) ܸ , set of real tasks, (ii) ܸ௦ , set of entry tasks (iii) ܸ௧ , set of terminal tasks and (iv) ܸௗ , set of dummy tasks Each task ݆ א ܸ has a set of available resources ܧ ={ܴ ܹ ሽ, where, ܴ and ܹ respectively represent the set of automatic and manual resources Each equipment ݈ ܧ א has a some tasks specific and unspecific properties, such as a task processing time for task ݆ and model , ݐ , a scrap rate ݎݏ and ்ܥ and ் ܮ respectively the initial purchasing price and useful life of tools Each automatic resource ݈ ܴ א has an average failure rate ߤ ǡ ሺߤ ൌ ͳȀܨܶܶܯ ሻ , an average repair rate ݎ ǡ ሺݎ ൌ ͳȀܴܶܶܯ ሻ , and an average planned and unplanned downtime ܦ Additionally it has an average energy consumption ݁ , a useful life ܮ , and ܥா the initial capital investment Each manual resource ݈ ܹ א has a standard wage ݓ , and one-time personal costs ܿ Each task ݆ ܸ א has a wage rate ߱ and material costs ܿ Each buffer position has an initial capital investment ܥ௨ and a useful life ܮ௨ The total task time of task ݆ performed by equipment ݈is calculated by using ߙ , which represents the probability of occurrence of model ݐ ൌ σୀଵ ߙ ݐ (1) The mathematical model requires two assignment variables, namely ݔ (for ݆ ܸ א, ݇ ൌ ͳǡ ǥ ǡ ݉ ഥ , and ݈ ܧ א ) and ܻ ( ൌ ͳǡ ǥ ǡ ܤǡand ݍൌ ͳǡ ǥ ǡ ݉ ഥ െ ͳ) ͳ݂݄݅ݐ݅ݓ݇݊݅ݐܽݐݏݐ݀݁݊݃݅ݏݏܽݏ݆݅݇ݏܽݐ ݔ ൌ ൝ ݈݁ݐ݊݁݉݅ݑݍ Ͳ݁ݏ݅ݓݎ݄݁ݐ ͳ݂ܾ݅ܽݍݎ݂݂݁ݑܾݐ݀݁݊݃݅ݏݏܽݏ݅݁ݖ݅ݏݎ݂݂݁ݑ ܻ ൌ ቄ Ͳ݁ݏ݅ݓݎ݄݁ݐ (2) In order the reduce the number of variables of ݔ , the earliest and latest station, ܧԢ and ܮᇱ to which a task ݆ can be assigned, can be computed using the relation proposed by Scholl et al [19] and extended by relation ሺͳሻ, where ܲ כሺ݆ሻ and כ ܨሺ݆ሻ represent respectively the list of all predecessors and successors of a modified graph, in which all subgraphs are replaced by a fictive task representing a lower bound on the total task time, ݏݐሺݒሻ ൌ ሼݏݐሺܸ ሺ݃ሻሻ This allows to אௌீ identify the subset ܤ ൌ ሼ݆ ܸ אȁ݇ ܫܵ א ሽ representing potential tasks assignable to station ݇, where ܵܫ ൌ ሾ ܧᇱ ǡ ܮᇱ ሿ, based on the new task time ߬ ൌ ݐ Τܶܥ ܧԢ ൌ ඃ߬ σఢ כሺሻ ߬ ඇ,ܮᇱ ൌ ݉ ഥ ͳ െ ඃ߬ σఢி כሺሻ ߬ ඇ (3) The decisions that have to be taken address four related issues: the design problem, (i) where components and processes have to be selected, (ii) and where equipment for a given task has to be selected, (iii) the balancing problem, where the tasks have to be assigned to workstations, (iv) the buffer allocation problem, where the buffer size between each stations will be determined The assumptions of the problem are listed below: x Automatic resources on each stations are unreliable and each station is separated by an intermediate finite buffer x Breakdowns only occur when resources are operating x The last station is never blocked x Work pieces move through buffers with zero transit time and follow a First-In-First-Out strategy x The operating time and repair time between failures are exponentially distributed x If a breakdown occurs, unit in process will be scrap x The setup time is included in the task processing time Mathematical Model The mathematical formulation of this problem can be stated as follows The objective function to be optimized (see (4)) depends on three objectives: (i) the minimization of the idle time in the line, (ii) the minimization of the total unit costs ܥ, and (iii) the maximization of the throughput rate ܧ ܼ ൌ ሺܼଵ ǡ ܼଶ ǡ ܼଷ ሻ ഥ ܼ݊݅ܯଵ ൌ σ ୀଵሾ ܶܥെ σאೖ σאாೕ ݔ ݐ ሿ ܼ݊݅ܯଶ ൌ ܿ ܼݔܽܯଷ ൌ ܧ ܧൌ ݂ଶ ሺݔ ǡ ܻ ) ܿ ൌ ݂ଷ ሺݔ ǡ ܻ ) σאೖ σאாೕ ݔ ݐ ܶܥ, ݇ൌ ͳǡ ǥ ǡ ݉ ഥ σאೖ ݔ ܽ ܣ, ݇ൌ ͳǡ ǥ ǡ ݉ ഥ, ܧ א ݈ σאௌூ ݇ݔ σאௌூೕ ݇ݔ , ݆ൌ ܸ െ ܸ௦ , ݆ ܨ א หܵܫ ܫܵ ת ് , ܧ א ݈ σאௌூ ݔ െ σאௌூೕ ݔ ݉ ഥሺͳ െ σאௌூೕ ݔ ሻ, ݅ൌ ܸ௦ , ݆ ܨ א ȁܵܫ ܫܵ ת ് , ܧ א ݈ σאௌூ ݔ σאௌூೕ ݔ , ݅ൌ ܲ , ݆ ܸ אെ ܸ௧ , ܧ א ݈ (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) σאி σאௌூೕ ݔ െ σאௌூ ݔ ൌ Ͳ, ܸ א ݅௦ , ܧ א ݈ (15) σאೕ σאௌூ ݔ െ σאௌூೕ ݔ ൌ Ͳ, ܸ א ݅௧ , ܧ א ݈ (16) ഥ ିଵ σୀଵ σ (17) ୀଵ ሺܻ ܾ ͳሻ ܵ௫ െ ͳ σୀଵ ܻ =1, ͳ=ݍǡ ǥ ǡ ݉ ഥ െͳ (18) ݔ אሼͲǡͳሽ, ܸ א ݆, ܫܵ א ݇ ǡ ܧ א ݈ (19) ܻ אሼͲǡͳሽǡ ͳ א ǡ ǥ ǡ ͳ=ݍܤǡ ǥ ǡ ݉ ഥ െͳ Constraints (8) and (9) indicate that the throughput rate ܧ and the product cost ܿ are function of the balancing and buffer allocation, namely the selection of tasks, resources and assignment to stations and the sizes and locations of buffers Constraint (10) aims at verifying that the assignment of tasks to station is respecting the cycle time ܶܥ Constraint (11) assures that the required material space is not exceeding the available space ܣat each station Relations (12) and (13) ensure that precedence constraints are respectively respected for all nodes Constraints (14)-(16) guarantee that all are at most assigned to one station Constraint (17) ensures that the total space of the buffers and assembly stations must not exceed the total available space for the assembly line Constraint (18) imposes that a unique size must be assigned to each buffer Constraint (19) defines the binary decision variables The objective function (9) aims at assessing the unit cost per part, given by the ratio of the total annual costs and the target net volume of defect-free parts ܸ௧ to be produced ܿ ൌ ܥ௧௧ ሺݔ ǡ ܻ ሻΤܸ௧ (20) 419 Jonathan Oesterle et al / Procedia CIRP 57 (2016) 416 – 421 The total annual costs are provided by: ܥ௧௧ ൫ݔ ǡ ܻ ൯ ൌ ܥெ ൫ݔ ൯ ܥ ൫ݔ ൯ ܥைு ൫ݔ ൯ ܥா ൫ݔ ǡ ܻ ൯ ܥ ൫ݔ ǡ ܻ ൯ ܥா ൫ݔ ǡ ܻ ൯ ்ܥሺݔ ሻ (21) Resolution Method Where, ܥெ ൫୨୩୪ ൯ǡ ܥ ൫୨୩୪ ൯ , ܥைு ൫୨୩୪ ൯ǡ ܥா ൫୨୩୪ ǡ ୮୯ ൯ǡ ܥ ൫୨୩୪ ǡ ୮୯ ൯ǡ ܥா ൫୨୩୪ ǡ ୮୯ ൯ and ்ܥሺ୨୩୪ ሻ represent respectively the total material costs, the labour costs, the overhead costs, the energy costs, the annual building costs, the equipment costs and the tooling costs Each of these terms depends on either the line balancing results, the buffer allocation results or both ഥ ܥெ ሺݔ ሻ=σ ୀଵ σאೖ ݔ ܿ ൈ ܸ௦௦ , ܧ א ݈ (22) Where, ഥ ܸ௦௦ =ܸ௧ Τς ୀଵሺͳ െ ݎ ሻ, ݎ ൌ ͳ െ ςאೖ ςאாೕ ݔ ሺͳ െ ݎݏ ሻ (23) The total labour costs, ܥ , is provided by the sum of labour costs used at each station ǡ ܿ The wage of a worker is determined according to the most difficult task to be performed in station ݇ [26] If ܯrepresents the total number of required workers, then: ഥ ܥሺ௫ೕೖ ሻ σ ୀଵ σאௐೕ ሺ݉ܽݔאೖ ሺݔ ߱ ሻ ܯൈ ܿ ሻ ൈ ߬ (24) Overhead costs ܥைு are related to indirect labour required to maintain production, which is modelled by a ratio of the number of indirect ݎௗ workers, paid at wage rate ߱ௗ , for each direct worker: ܥைு ൫ݔ ൯ ൌ ݎௗ ൈ ߱ௗ ൈ ܯൈ ߬ (25) Taking the average planned and unplanned downtime ܦ of a station ݇, and its probability of failure : ഥ ߬ ൌ ܶܥ൫ͳ σ ୀଵ ܦ ൯ ൈ ܸ௦௦ (26) The energy costs ܥா are provided by the average energy costs of each automatic equipment and buffer used at station ݇, ܿா In order to distribute building, tooling and equipment costs over time, the capital recovery factor ܨܴܥ is used, where ݆ represents either building, equipment, tooling, or buffer ܨܴܥ ൌ ݎሺͳ ݎሻೕ Τሾሺͳ ݎሻೕ െ ͳሿ (27) Where ݎis the annual discount rate and ܮ is the useful life in number of years The annual building cost is computed given one-off costs, the initial building capital investment ܥǡ of size ݈ and width ݓ and running costs, e.g cost of energy for lighting, heating, air conditioning, which can be calculated by multiplying the space occupied by the assembly line with a factor ݁, representing the annual energy cost for each m² The one-off costs of equipment and tooling, ܥா ்ܥ, are provided by multiplying ܨܴܥ by the respective initial investment ܥா and ்ܥ Additionally, the costs associated to buffer size are added to ܥா ഥ ିଵ ܥ ൫ݔ ǡ ܻ ൯ ൌ ൫σୀଵ σ ୀଵ ሺܻ ܾ ͳሻ ͳ൯ ൈ ݈ ൈ ݓൈ ሺܥǡ ൈ ܨܴܥ௨ௗ Τሺ݈ ൈ ݓ ሻ ݁ሻ Where, ݈ and ݓrepresent respectively the length and width of a given station/buffer Finally, these annual costs can be used to compute a unit cost per part (28) Many methods exist for solving multiobjective optimization problems (MOP) Two main categories can be identified: (i) classical methods which use direct or gradientbased methods following some mathematical principles and (ii) non-traditional and population-based methods following some natural or physical principles Classical methods mostly attempt to scalarize multiple objectives and perform repeated applications to find a set of Pareto-optimal solutions, whereas population-based methods attempt to find a multiple Paretooptimal solutions in a single simulation run [27] Since the ALBP and BAP are NP-hard [4,28], approximation methods are most suitable to rely on solving these problems The use of population-based optimization techniques, such as evolutionary algorithms (EA) is an appropriate approach for addressing MOP However, due to their global search nature, EAs are not so efficient as regards quickly and reliably leading the population toward the optimal front A most powerful mechanism can be obtained by combining the EA with a local search, which allows a balance between global and local search [29] Combining global and local search methods is known as memetic approach Despite a better accuracy for the final solution, memetic algorithms are also able to offer a better speed of convergence [30] The developed memetic algorithm is composed of a NSGA2 and a Simulated Annealing (SA), which will be explained in the next subchapters 5.1 Structure of the NSGA2 The solution encoding uses two different chromosomes In the first chromosome, three numbers are assigned that respectively represent the selected task, equipment, and workstation assignment To the second chromosome, two numbers are assigned, the buffer position and its size This attribution is done by taking the previously listed constraints into consideration Initially, an initial population ܲ is created following classical ALBP heuristics [31] The production rate of a station ݇ in isolation, meaning its production rate when the station is not subject to starvation or blockage is given by and can be used to initiate its buffer allocation.: ൌ ቀݎ Τሺσאೖ σאாೕ ݔ ݐ ሻቁൗሺߤ ݎ ሻ (29) Where, ୩ its average repair rate, and Ɋ୩ its average failure Indeed, in order to avoid effects of starvation and blockage, stations with a poor production rate and thus a bigger chance of breakdown should have a bigger buffer Therefore, the relative criticality ܴܥ of a buffer ݍcan be determined by: ഥ ିଵ ିଵ ܴܥ ൌ ݉݅݊ሼ ǡ ାଵ ሽିଵ ൗσ ୀଵ ݉݅݊ሼ ǡ ାଵ ሽ (30) The initial buffer allocation is provided by multiplying ܴܥ by the maximum buffer size ܤat each location ݍ, for ݍൌ ͳǡ ǥ ǡ ݉ ഥ െ ͳ 420 Jonathan Oesterle et al / Procedia CIRP 57 (2016) 416 – 421 1st Chromosome Assignation 2nd Chromosome Tasks number Workstation 11 1012 Equipment 1 3 3 1 4 4 Tasks Equipment 1 1 … 10 12 Buffer 123 Buffer size 121 3 Fig Solution encoding The first population is evaluated through a discrete event simulation, which calculates the throughput and total product costs of each solution The population is then sorted based on the Pareto non-domination concept In a second step, a child population ܳ is created from ܲ by performing tournament selection, crossover and mutation operations The two populations ܲ and ܳ are combined to form a population ܴ of size 2ܰ, where ܰ ൌ ȁܲ ȁ ൌ ȁܳ ȁ A non-dominated sorting is used to classify the entire population ܴ , which is in turn subdivided in several non-dominated front The new parent population ܲଵ is filled with individuals of the best nondominated fronts Since ܴ is of size 2ܰ, while ܲଵ of size ܰ, the last allowed front will be truncated by using niching strategy to choose individuals from the last front which reside in the least crowded region of this front These steps are repeated until reaching a stopping criterion, e.g predefined number of iterations 5.2 Simulated Annealing The memetic NSGA2 procedure, based on an adaptation of the PHC-NSGA2 [32], is explained in Fig For any generation ݐ ͳ, the local search procedure, which works on both chromosomes by either swapping and/or changing tasks, equipment assignment and buffer size, is applied for each front ܨ of ܲ௧ The solutions selected for the local search, representing the set ܵܮare the least crowded solutions in the objective space For each solution ܵܮ א ݏ, the SA procedure will return a new solution After the local search, the new set of solutions ݓ̴݁݊ܵܮis added to ܨ A crowded distance in the decision space is assigned to each solutions After applying the local search for each front ܨ of ܲ௧ , the new population ܳ௧ is generated The general SA algorithm involves two main steps: (i) selection of a proper annealing scheme consisting of decreasing temperature with increasing iterations, and (ii) a method generating a neighbor near the current search position The transition probability scheme is different in multiobjective optimization and choosing a proper transition probability is difficult [33] The following geometric cooling, which is widely employed for the annealing scheme, was used: ܶ ൌ ߙ ܶ (31) The transition probability from state ݏto ݏԢ is given by: ܲሺݏǡ ݏᇱ ሻ ൌ ݉݅݊ሼ݁ ି൫௦ǡ௦ ᇲ ൯Τ் ǡ Ͳሽ (32) Begin Rt ՚ Pt Qt F ՚ Fast_non-dominated_sort (Rt, M) Pt+1 ՚ i՚1 While | Pt+1|+|Fi|N Crowding_distance_assignment (Fi, M) Pt+1 ՚ Pt+1 Fi i՚i+1 EndWhile Sort (Fi,