A review meta heuristic approaches for solving rectangle packing problem


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A review meta heuristic approaches for solving rectangle packing problem

  1. 1. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME410A REVIEW: META- HEURISTIC APPROACHES FOR SOLVINGRECTANGLE PACKING PROBLEM1Dr. Leena Jain, 2GaganDeep Singh1Associate Professor & Head-MCA, Global Institute of Mgt. &. Emerging Technologies,Amritsar (India)2Ph. D., Research Scholar, Punjab Technical University Jalandhar (India), Head of Dep’t(Computer Science & Application),CKD Institute of Management & Technology, Amritsar(India)ABSTRACTPacking problems are optimization problem encountered in many areas of businessand industries and have wide applications. These problems look for good arrangement ofmultiple items in some larger containing regions with an objective to maximize the utilizationof resource materials. 2D packing problem has wide industrial applications starting fromsmall scale industries related to leather, furniture, glass, metal, and wood to large scaleindustries dealing with textile, garments, paper, shipbuilding, automobiles and VLSI design.In this paper authors have summarized the different Metaheuristic approaches used to solvepacking problem. Accordingly, this paper is a pure academic activity that catalogues latestresearch in the area of Rectangle packing Problem using Metaheuristic approaches.Keywords: Ant colony optimization (ACO), genetic algorithm (GA), Swarm ParticlesOptimization (SPO), Variable Neighborhood Search (VNS)I. INTRODUCTIONRectangular Packing problems arise in many industries where desired rectangularitems are laid on the rectangular object, which is bigger in size in comparison to items. Thisis of particular interest to industries involved with mass production as small improvements indesign layouts can lead to considerable savings in raw material vis-à-vis reduction inproduction costs [1]. The wood, glass and paper industries are mainly concerned with thecutting of regular figures. The floor planning in VLSI (very large scale integrated design) isanother tricky application of rectangle packing. On the other hand, in shipbuilding, textile andINTERNATIONAL JOURNAL OF COMPUTER ENGINEERING& TECHNOLOGY (IJCET)ISSN 0976 – 6367(Print)ISSN 0976 – 6375(Online)Volume 4, Issue 2, March – April (2013), pp. 410-424© IAEME: www.iaeme.com/ijcet.aspJournal Impact Factor (2013): 6.1302 (Calculated by GISI)www.jifactor.comIJCET© I A E M E
  2. 2. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME411leather industries irregular and arbitrary shaped items need to be packed. These items are firstpacked into rectangle enclosures and then rectangle enclosures are in turn packed ontorectangle stock sheet using rectangle packing approach [2]. Most of the standard problemsrelated to Nesting are known to be NP-complete where computation time for an exactsolution increases with N and become rapidly prohibitive in cost as N increases. The solutionapproach to these problems lies in reducing the exhaustive search of all possiblearrangements of nesting the parts and subsequently checking upon the execution time. Thedevelopment of exact algorithms, which are faster and produce near optimal solutions, is stilla major research issue in this area. Proliferation of sophisticated desktops and faith ofresearchers in meta-heuristics have further allowed them to look beyond the traditionaloptimization techniques to solve this hard problem. Singh and Jain [3] undertaken a studythat catalogues solution methodologies popularly used to handle nesting problems. Fig. 1represents survey of solution methodologies for Nesting Problem and it is clear from thefigure now a days researchers work on Metaheuristic for solving the packing/nestingproblem. This paper reviews the application of meta-heuristic methods to two dimensionalrectangle packing problem.Commonly used Metaheuristic methods in rectangle packing problem• Simulated annealing[4,5]• Tabu search [6,7,8,9,10,11]• Iterated local search [12,13,14• Genetic Algorithm[15,16,17,18]• Ant Colony Optimization. [19]Figure 1: Survey of solution methodologies for Nesting Problem
  3. 3. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME412II. SYNOPSIS OF GENETIC ALGORITHMGenetic Algorithms (GA) were developed by Holland (1975) and since then havebeen used in various fields of engineering. GA has been used quite successfully forcombinatorial problems that are NP-complete. GAs have been used in a wide variety ofoptimisation tasks, including numerical optimisation and combinatorial optimisationproblems such as travelling salesman problem (TSP) [20], packing problem [21,22,23], jobshop scheduling[24] and video & sound quality optimisation. A genetic algorithm is arandomized parallel search method modelled on natural selection and genetics[25]. Incontrast to more standard search algorithms, GAs base their progress on the performance of apopulation of candidate solutions, rather than that of a single candidate solution. Themotivation behind this is that by simultaneously searching many areas of the design space therisk of getting stuck at local optima is greatly reduced. GAs are probabilistic in nature andstart off with a population of randomly generated candidates and evolve towards bettersolutions by applying genetic operators, modelled on the natural genetic process. For solvingany problem, Genetic Algorithm is started with a set of solutions (represented bychromosomes) called population. A member of the population is a genotype, a chromosome,a string or a permutation. Solutions from one population are taken and used to form anew population. This is motivated with a hope, that the new population will be better thanthe old one. Solutions which are selected to form new solutions (offspring) are selectedaccording to their fitness - the more suitable they are, the more chances they have toreproduce. When a genotype is decoded, a packing pattern, called a phenotype, isformed. We may calculate the fitness function even to restrict the mutating parents.Working of GA represents in fig. 2. A GA generally has five components [26]1. A representation for solutions to the problem2. A method to create initial population3. An evaluation function to determine the relative fitness of the solutions4. Genetic operators that effect the composition of the offspring duringreproduction5. Values for parameters that the GA uses (e.g. population size, probabilitiesof applying the genetic operators, etc.).III. SYNOPSIS OF ANT COLONY OPTIMIZATIONThis optimization, first introduced by Colorni et al. [27,28] imitates the way antssearch for food and find their way back to their nest. First an ant explores its neighborhoodrandomly. As soon as a source of food is found it starts to transport food to the nest leavingtraces of pheromone on the ground which will guide other ants to the source. The intensity ofthe pheromone traces depends on the quantity and quality of the food available at the sourceas well as from the distance between source and nest, as for a short distance more ants willtravel on the same trail in a given time interval. As the ants preferably travel along importanttrails their behavior is able to optimize their work. Pheromone trails evaporate and once asource of food is exhausted the trails will disappear and the ants will start to search for othersources. For the heuristic, the search area of the ant corresponds to a discrete set from whichthe elements forming the solutions are selected, the amount of food is associated with anobjective function and the pheromone trail is modeled with an adaptive memory [29].
  4. 4. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME413Aco CharacteristicsExploit a positive feedback mechanismDemonstrate a distributed computational architectureExploit a global data structure that changes dynamically as each ant transverses therouteHas an element of distributed computation to it involving the population of antsInvolves probabilistic transitions among states or rather between nodest=0Y N NFigure 2: Flowchart of working of GAIV. SYNOPSIS OF SWARM PARTICLES OPTIMIZATIONParticle swarm optimization (PSO) is a population-based stochastic approach forsolving continuous and discrete optimization problems. In particle swarm optimization,simple software agents, called particles, move in the search space of an optimization problem.The position of a particle represents a candidate solution to the optimization problem at hand.Each particle searches for better positions in the search space by changing its velocityaccording to rules originally inspired by behavioral models of bird flocking. Particle swarmInitialize populationEvaluate SolutionReproductionCrossoverMutationt=t+1StartOptimalSolution
  5. 5. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME414optimization belongs to the class of swarm intelligence techniques that are used to solveoptimization problems. It was developed in 1995 by James Kennedy (social-psychologist)and Russell Eberhart (electrical engineer). It uses a number of agents (particles) thatconstitute a swarm moving around in the search space looking for the best solution. Eachparticle is treated as a point in a N-dimensional space which adjusts its “flying” according toits own flying experience as well as the flying experience of other particles. Each particlekeeps track of its coordinates in the solution space which are associated with the best solution(fitness) that has achieved so far by that particle. This value is called personal best , pbest.Another best value that is tracked by the PSO is the best value obtained so far by any particlein the neighborhood of that particle. This value is called gbest. The basic concept of PSO liesin accelerating each particle toward its pbest and the gbest locations, with a random weightedaccelaration at each time step as shown in Fig 2.Fig 3 represents working of PSOalgorithm[30].skvkvpbestvgbestsk+1vk+1skvkvpbestvgbestsk+1vk+1Figure 2: Searching point by PSOPSO CharacteristicsPopulation-based optimization technique – originally designed for solving real-valuedfunction optimizationsApplicable for optimizations in rough, discontinuous and multimodal surfacesDoes not require any gradient information of the function to be optimizedConceptually very simpleEach candidate solution of continuous optimization problem is described (encoded)by a real vector N-dimensional search space: x = x1, …, xnEach candidate solution is called PARTICLE and represents one individual of aPopulation called SWARM.The particles change their components and FLY through the multi-dimensional searchspace.Particles calculate their FITNESS function as the quality of their actual position in thesearch space using w.r.t. the function to be optimized.Particles also compare themselves to their neighbors and imitate the best of thatneighbors.sk: current searching point.sk+1: modified searching point.vk: current velocity.vk+1: modified velocity.vpbest : velocity based on pbest.vgbest : velocity based on gbest
  6. 6. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME415Figure 3: Flow chart of PSO AlgorithmV. SYNOPSIS OF ITERATED LOCAL SEARCHIterated local search [Stiitzle (1999)] is a general meta-heuristic [31]. It has two basicoperators for generating new solutions. One is a local search and other is a perturbation operator.When the local search is trapped in local optimal solution, a perturbation operator is applied tothe local optima to generate a new starting point for its local search. Itis desirable that thegenerated starting point should be in a promising area in the search space. A commonly-usedperturbation operator is a conventional mutation, which can produce a starting point in aneighboring area of the local optimum. Another perturbation operator is guided mutation operator[32,33]. Let C be the cost function of our combinational problem, cis to be minimizes. We labelcandidate solution by S, solution in S by s, cost function value of solution by C(s),Locallyoptimal solution by s*, set of locally optimal solutions by S* and Local Search defines mappingfrom S S*.Figure 4 depict procedure of Iterated Local Search [31.]procedure Iterated Local SearchS0=Generate InitialSolutionS*=Localsearch(S0)RepeatS’= Perturbation(S*,history)S*’=LocalSearch(S’)S*=AcceptanceCriterion(S,S*’, History)until termination condition metendFigure 4: procedure of Iterated Local SearchStartInitialize particles with random position and velocity vectorsFor each particle’s position (p) evaluate fitnessIf fitness (p) better than fitness (pbest) then pbest=pSet best of pBest as gBestStop: giving gBest optimal solutionUpdate Particles Velocity and PositionLoopuntilallParticlesexhaustLoopuntilMaxIteration
  7. 7. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME416VI. A BRIEF REVIEW OF LITERATURE TO RECTANGULAR PACKINGPROBLEM USING METAHEURISTIC APPROACHThe majority of literature available is on hybrid algorithms, where a GA is combinedwith a heuristic placement routine. In this two-stage approach a GA is used to determine thesequence, in which the items are to be packed. A second algorithm is then needed, whichdescribe how this sequence is allocated onto the object. One of the first few researchers whoimplemented GAs in the domain of packing was Smith (1985) [34]. He experimented withtwo heuristic packing routines, one of which implements backtracking and produces denserlayouts. However, this method was computationally more expensive. Comparisons betweenthe two hybrid approaches showed that the combination with the more sophisticatedheuristic generated better packing patterns. The packing problem Smith studied was specialin that the orientation of the rectangles was fixed.The GA was based on a directed binary tree to encode the problem [35, 36]. Thisrepresentation fixes one dimension of the position of an item in the layout. The seconddimension was determined by the bottom left condition. Since its performance wascompared to well-known packing heuristics, a relative comparison with our work is possible.Falkenauer and Delchambre (1992) [37] have developed a genetic algorithm for theclassical two-dimensional bin packing problem, where the objective was to minimise thenumber of bins. Consequently, the fitness function has to take into account how efficientlythe bin capacity is utilised. The encoding technique uses one gene per item to represent thebin in which it is packed. This method of encoding does not perform well in combinationwith the classic crossover and mutation operators. Therefore, an alternative data structurewas proposed with a chromosome consisting of two parts, an object part and a group part.The object part identifies the items which form a bin and the group part contains one geneper bin introduced in the object part.In the hybrid approach by Hwang et. al. (1994) [38] a GA was combined with awell-known heuristic from bin packing, the so-called First-Fit-Decreasing-Height algorithm(FFDH). Although this technique produces guillotineable layouts, and is suggested for thegeneral case of the rectangle packing problem. It also used a directed binary tree,which combines two rectangles to a larger rectangle by either placing them horizontally orvertically next to each other. The position within the larger rectangle was left justified. Asmentioned before, comparisons with a hybrid GA technique showed that this method wasless efficient in terms of packing height.Herbert and Dowsland (1996) [39] developed a 2D coding technique for a pallet-loading problem of identical rectangles. The layout was represented by 2D matrix indicatingavailable positions for vertical and horizontal placement, where in the horizontal one haspriority over the vertical one. This technique works well for small problems. In order toimprove the outcome for medium sized problems additional repair and enhance operatorshave been introduced by the authors.Jakobs (1996) [40] first suggested genetic algorithms to two dimensional packingproblem and employed the bottom left (BL) strategy to pack the small rectangles.The method developed by Ratanapan and Dagli (1997)[41] is different from theother approaches described so far, since it does not make use of a data structure torepresent the problem. In their approach, the items were represented as 2D pieces with theirtrue geometric dimensions. After the initialisation process, which places all items ontonon-overlapping positions on the object, a series of genetic operators which consist ofmoving, relocation and recombination operations were applied.
  8. 8. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME417Dagli and Poshyanonda (1997) [42] also used the GA to generate an input sequencefor the placement algorithm, which was based on a sliding method combined with anartificial neural network. In this method, every incoming item was placed next to the partiallayout and all scrap areas generated were recorded. If there was a match between anincoming item and one of the scrap areas, the neural network selects the best match. Asecond category of solution approach with GAs aims at incorporating some of the layoutinformation into the data structure of the GA. However, some additional rules are stillneeded to fix the position in the layout.Leung et. al. (2003) [15] found that genetic algorithms consistently outperformsimulated annealing, and that a genetic algorithm combined with simulated annealingoutperforms a pure genetic algorithm. However, a flaw in the study of Leung et al. wasthe assumption that the orientations of the small rectangles are fixed.Goncalves (2007) [43] proposed algorithm, which hybridizes a placementprocedure with a genetic algorithm based on random keys for two-dimensional (2D)orthogonal packing problem. The approach was tested on a set of instances taken from theliterature and was compared with other approaches. The computation results validate thequality of the solutions and the effectiveness of the proposed algorithm.Ranjan et. al. (2009)[16] explored the basics of genetic programming (emergingevolutionary technique) with the possibilities of its application in interdisciplinary research.This paper illustrated the difference between genetic programming and genetic algorithm. Itstated that genetic programming created computer program in the LISP computer languageas the solution and the output of genetic programming is another computer program, whichcould suitably be applied in applications, whereas the genetic algorithm created a string ofnumbers that represents the solution. Therefore, it was concluded that geneticprogramming was more powerful than genetic algorithm.Lai and Chan (1997) [44] developed an algorithm based on evolution strategy. Thispaper provides a basis for exploring the integration of the evolutionary algorithm techniquein artificial intelligence with classic two dimensional cutting stock problems, withminimum trim loss. The algorithm developed could be used to solve non-guillotine, twodimensional cutting problems. The algorithm addresses the problem of placing differentlysized small rectangles on a larger rectangle or box, in order to minimize trim loss. Theadvantage of the proposed heuristic was its easy implementation and lesser requirement ofcomputation memory.Jain and Gea (1998) [45] presented a technique for applying the genetic algorithmfor the two dimensional packing problem. This algorithm was quite effective and couldbe used for any level of discretization. The algorithm worked for both concave and complexobjects with holes. In this work, random seeds were tried on the same problems and a newconcept of a two dimensional genetic chromosome was introduced. The total layout space isdivided into a finite number of cells for mapping it into a 2D genetic algorithmchromosome. The mutation and crossover operators have been modified and are applied inconjunction with connectivity analysis for the objects to reduce the creation of faultygeneration. A new feature was added to the genetic algorithm in the form of a new operatorcalled compaction. Several examples of GA based layout were presented. Thus it wasconcluded that the problem of compacting general shaped objects could be resolvedquite accurately and in a very small period. A limitation of this approach was that itallows rotations in steps of 90 degree.
  9. 9. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME418Hopper and Turton (1999) [46] described two GAs for rectangular packing problems.Both the GAs were hybridized with a heuristic placement algorithm, one of which is thewell known bottom left routine (BLR). A second placement method which overcamesome of the disadvantages of the BLR was developed. The two hybrid geneticalgorithms were compared with heuristic placement algorithms. The GA combined withthe improved BLF heuristic outperformed the GA, using both the BL method as well as theheuristic method. The improved heuristic was sufficient to achieve high quality layouts forapplication in industry.Faina (1999) [47] proposed two new algorithms global optimization and simulatedannealing, involving guillotine and non-guillotine constraints, to solve the two dimensionalrectangular cutting stock problem. The simulated annealing is an algorithm thatcontinuously attends to transform the current configuration into one of its neighbours. Themechanism is mathematically best described by means of a Markov chain: a sequence oftrials, where the outcome of each trial depends only on the outcome of the previous one.Several tests proved the validity of these algorithms. It was derived that, for a large numberof items, the NON-GA is beyond comparison with the GA, since it gets much better cuttingpatterns in almost the same computational time. The result of the study illustrates theadditional effort necessary for realizing computer codes based on the non-guillotineconstraints.Burke and Kendall (1999)[48] in their research work considered a simplified versionof stock cutting problem. They found that all three metaheuristic techniques geneticalgorithm (GA), tabu search (TS) and simulated annealing (SA)) were able to find theoptimum solution to a small problem. When presented with larger problems, tabu searchand simulated annealing produced good quality solution, whereas genetic algorithmproduced lower quality solution despite many test during the small problem to try andascertain the best combination of parameters. It was concluded that SA and TS algorithmsproduced similar solutions and both outperformed GA.Hopper and Turton (2001a) [49] considered 2D rectangular packing problem,where a fixed set of items have to be allocated on a single object. Two heuristics belongingto the class of packing procedure that preserve bottom left (BL) stability, are hybridizedwith three meta heuristic algorithms (genetic algorithm, simulated annealing and naïveevolution and local search heuristic). This study compares the hybrid algorithms in terms ofsolution quality and computation time, on a number of packing problems of different size.In order to show the effectiveness of the design of different algorithms, their performance iscompared to random search (RS) and heuristic packing routines.In their next study Hopper and Turton (2001b) [50] compared the performances ofgenetic algorithms and simulated annealing and found that the latter yields better results thanthe former but it requires more execution time when the size of a problem is large.Babu and Babu (2001) [51] proposed both genetic and heuristic algorithms to arrangemultiple 2-D shaped parts in multiple 2-D shaped sheets with the objective of minimizingthe wastage of the sheet material. The paper proposed a new method of representing thesheet and part of geometries in a discrete form to arrange the parts on the sheet quickly,irrespective of the complexity in the geometry of the sheets and parts. The proposedheuristic algorithm, using the bottom-left positioning strategy, arranges the parts with theirorientation, on the sheets, in an effective manner .The genetic algorithm that generates thebest sequence of the sheets and parts with their orientation can be obtained for nestingthe parts in an optimal manner. Authors considered the geometry of parts and sheet with
  10. 10. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME419line and arc features. The discrete representation scheme adopted in this approach can easilyincorporate free-form features for the parts and sheets. It concluded that thecomputational complexity of the proposed approach can be reduced with the help of parallelprocessing algorithms.In 2003, Onwubolu and Mutingi [52] proposed a genetic approach for the guillotinerectangle packing problem. The approach firstly places the initial rectangle and thensearches for a rectangle that can be attached to the first whilst maintaining the guillotine cutconstraint. The chromosome within the genetic algorithm represents the piece pairs that areto be attached and the orientation in which they should be placedLin (2006) [53] in his study proposed a genetic algorithm to solve the twodimensional assortment problems. The proposed genetic algorithm uses a problem –specific encoding scheme that incorporates a novel packing process. Numerical examplesindicated that the coding method is effective for solving the assortment problem. Using thiscoding scheme the proposed genetic algorithm appeared to be more efficient and effectivethan a state of the art integer programming approach and provided acceptable solutions tolarge problems.Huang et. al. (2007) [54] recommended a new heuristic algorithm based on twoimportant concepts namely, the corner – occupying action and caving degree. Authors usedthe following rectangle packing principle: the rectangle to be packed into the box alwaysoccupies a corner, and the caving degree of the packing action should be as large aspossible. 21 rectangle packing instances were tested by the algorithm developed and 16 ofthem had achieved optimum solution within the reasonable runtime. Experimental resultsdemonstrated that the algorithm developed was fairly efficient for solving the rectanglepacking problemGoncalves (2007) [43] in his paper addressed a hybrid meta-heuristic algorithm fortwo dimensional (2D) orthogonal packing problem, where a fixed set of small rectangles hadto be placed on a larger stock rectangle in such a way that the amount of trim loss isminimized. This algorithm combines random keys based genetic algorithm with a novelfitness function and new heuristic placement policy. The experimental resultdemonstrated the excellent quality of the proposed algorithm when compared with othermeta-heuristics in absolute terms.Alvarez-Valdes et. al. (2007)[55] have developed a new heuristic algorithm based onTabu search techniques for a non-guillotine two-dimensional cutting stock problem.Two moves have been proposed, based on the reduction and insertion of blocks of pieces.The efficiency of the moves is based on a merge and fill strategy that accommodates theempty rectangles to the pieces still to be cut. Some intensification and diversificationstrategies based on long-term memory, have also been included in this study.Hadjiconstantinou and Iori (2007) [56] in their paper presented a new greedyalgorithm and hybrid genetic algorithm with elitist theory, immigration rate, heuristics on-line and tailored crossover operators, for two dimensional knapsack problems. Thealgorithm was evaluated on a very large number of test problems of varying size andcomplexity. The results indicated that as the problem size increased the heuristics performedbetter and the results outperformed existing metaheuristics.Mohamed and Adnan (2009) [19] proposed new pretreatments for the two-dimensional bin-packing problem, the non-oriented case. The problem of bin-packing in twodimensions (2BP) consists of placing a given set of rectangular items in a minimum numberof rectangular and identical containers, called bins. This study treats the case of objects
  11. 11. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME420with a free orientation of 90degree. The authors proposed an approach of resolutioncombining optimization by colony of ants (ACO) and the heuristic method IMA to resolvethis NP-Hard problem. New heuristic method (SACO) was developed for the resolution ofthe problem. Pretreatments on benchmarks taken from literature were tested and theirefficiency to simplify the instances was shown. It was concluded that on an average betterresults were shown on average by using the proposed method.Lueng et. al. (2010) [57] presented a fitness strategy based on which a constructiveheuristic algorithm was developed. The fitness strategy evaluated the fitness of everyunpacked rectangle so that one appropriate rectangle is selected to pack into the sheet. Basedon this fitness strategy, a constructive heuristic algorithm was developed to generate asolution, i.e., a given sequence of rectangles for packing. Then, a greedy strategy was usedto search for a better solution. By combining the greedy strategy and simulated annealingalgorithm, a hybrid heuristic algorithm was proposed to solve the rectangular knapsackpacking problem. The computational results on zero-waste and non-zero-waste instanceshave shown that HSA is very fast and effective in solving the rectangular knapsackpacking problem.Ducatelle and Levine (2002) [58] adopted ACO to decipher bin packing andemployed simulated ants to assemble solutions stochastically, with heuristic informationobtained by First Fit Decreasing (FFD) and an simulated pheromone trail, which isdetermined by the favourability of having two items in the similar bin. This approach mayget sound performance and a high-quality explanation even not finding the optimum.However, it’s computing time is extensive than others. As a result, Levine and Ducatelle(2004) [60] perk up the performance by combining ACO and an iterated local searchapproach, since a local search algorithm can significantly advance the performance of anACO approach. The mixed approach they projected to the rectangle packing is relativelygeneric, and could be accomplished of being tailored to solve similar grouping problems.Thiruvady [59] engaged a cross approach by combining ACO and Bottom-LeftFill(BLF) to solve two dimensional rectangular packing problems. The planning and orientationsof items are produced by ACO and the items are owed one by one with BLF heuristicaccording to the ordering and orientations given by ACO. They evaluate and analyzed fourcombinations between ordering and orientation. The results showed that packing items withlearning ordering and orientations obtained by ACO outperformed the other combinations.Xu, Y.C. et.al (2010) [60] recommended a productive heuristic to pack weighteditems in a rounded container. They utilize an ant-based algorithm and optimize the waddingorder with the base of this heuristic. In their ant algorithm, encode of pheromone matrixconsiders the favourability of choosing an item and the product of the size and the weight ofthe next packed item. By doing so, hefty and weighty items have higher priority and this isadvantageous for the packing performance. They also evaluated two edition of the ant-basedalgorithm, AS and Min-Max AS, with existing approaches, such as the genetic algorithm, andthe hybrid PSO. However, since their study only centre on the regular objects (square andrectangle), it’s a problem for them whether their methodology can obtain fine performancefor irregular formed objects.Qi yang and wang jin-min (2011)[61] solve the rectangle packing problem using PSO.The algorithm optimizes the parameter of dynamic attractive factors by updating the positionand the velocity of the particles, and applies perturbation strategy to solve the matter that it iseasy to stick at local optima. The experimental result shows that the algorithm can get a betterpacking result by less time.
  12. 12. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME421CONCLUSIONRectangular Packing problems arise in many industries where desired rectangularitems are laid on the rectangular object, which is bigger in size in comparison to items. Mostof the standard problems related to Nesting/Recatngle Packing Problem are known to be NP-complete where computation time for an exact solution increases with N and become rapidlyprohibitive in cost as N increases. The solution approach to these problems lies in reducingthe exhaustive search of all possible arrangements of nesting the parts and subsequentlychecking upon the execution time. The development of exact algorithms, which are fasterand produce near optimal solutions, is still a major research issue in this area. Proliferation ofsophisticated desktops and faith of researchers in meta-heuristics have further allowed themto look beyond the traditional optimization techniques to solve this hard problem. The paperis a pure academic activity that catalogues latest research in the area of Rectangle packingProblem using Metaheuristic approaches.REFERENCES1. Singh, K. (2001), Designing algorithms for nesting regular and irregular shapes. Ph.D.Thesis, Thapar University, Patiala (India).2. Singh, K. and Jain, L. Associating statistical relationship among stock-sheet and pieces inrectangle packing, International Journal of Engineering and Technology. Volume 2,Number 6, pp. 608-615. ISSN: 1793-8236.3. Jain L. and Singh K. A Heuristic Based Solution for Rectangle Packing Problem: AnAnalysis and Design of Heuristic Approach, LAP LAMBERT Academic Publishing(November 11, 2011), ISBN-10: 3846538426 ,ISBN-13: 978-38465384254. Burke, E. and Kendall, G. (1999), Comparison of Meta-heuristic Algorithms forClustering Rectangles. Computers and Industrial Engineering. Volume 37, Number 1, pp.383-386.5. Soke, A. and Bingul, Z. (2006), Hybrid genetic algorithm and simulated annealing fortwo-dimensional Non-guillotine rectangular packing problems. Engineering Applicationsof Artificial Intelligence. Volume 19, pp. 557–567.6. Alvarez-Valdes, R., Parreno, F. and Tamarit, J.M. (2007), A tabu search algorithm for atwo-dimensional non-guillotine cutting problem. European Journal of OperationalResearch. Volume 183, pp. 1167–11827. Lodi, A., Martello, S. and Vigo, D. (1999a), Approximation algorithms for the orientedtwo-dimensional bin packing problem. European Journal of Operational Research.Volume 112, pp. 158-166.8. Lodi, A., Martello, S. and Vigo, D. (1999b), Neighborhood search algorithm for theguillotine non-oriented two-dimensional bin packing problem. In Meta-Heuristics.Advances and Trends in Local Search Paradigms for Optimization. Voss, S., Martello, S.,Osman I. H. and Roucairol, C. (Eds). Kluwer Academic Publishers, Boston, pp. 125-139.9. Glover, F. (1989), Tabu Search — Part I. ORSA Journal on Computing. Volume 1,Number 3, pp. 190-206.10. Glover, F. (1990), Tabu Search — Part II. ORSA Journal on Computing. Volume 2,Number 1, pp. 14-32.11. Glover, F. and Laguna, M. (1997), Tabu Search. Kluwer Academic Publishers, BostonNorwell, MA. USA, pp. 263-282.
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  15. 15. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 – 6375(Online) Volume 4, Issue 2, March – April (2013), © IAEME42448. Edmund Burke Graham Kendall, ‘’Comparison of Meta-Heuristic Algorithms for ClusteringRectangles’’ in 1999 proc. Computers & Industrial Engineering 37 pp. 383-38649. Hopper, E. and Turton, B.C.H. (2001a), A Review of the application of meta-heuristicalgorithms to 2D strip packing problems. Artificial Intelligence Review. Volume 16, pp. 257–300.50. Hopper, E. and Turton, B.C.H. (2001b), An empirical investigation of meta-heuristic andheuristic algorithms for a 2D packing problem, European Journal Operational Research.Volume 128, Number 1, pp. 34-57.51. Babu, R.A. and Babu, R.N. (2001), A generic approach for nesting of 2-D parts in 2-D sheetsusing genetics and heuristic algorithms. Computer-Aided Design. Volume 33, pp. 879-891.52. Onwubolu, G.C. and Mutingi, M. (2003), A genetic algorithm approach for the cutting stockproblem, Journal of Intelligent Manufacturing, Volume 14, pp. 209-218.53. Lin, C.C. (2006), A genetic algorithm for solving the two dimensional assortment problem.Computers and Industrial Engineering. Volume 50, pp. 175-18454. Huang, W., Chen, D. and Xu, R. (2007), A new heuristic algorithm for rectangle packing.Computers and Operation Research. Volume 34, pp. 3270-3280.55. Hadjiconstantinou, E. and Iori, M. (2007), A hybrid genetic algorithm for the twodimensional single large object placement problem. European Journal of OperationalResearch. Volume 183, pp. 1150-1166.56. Leung, S.C.H., Zhang, D., Zhou, C. and Wu, T. (2010), A hybrid simulated annealingmetaheuristic algorithm for the two-dimensional Knapsack packing problem. Computers andOperations Research. doi:10.1016/j.cor.2010.10.02257. Ducatelle, F., Levine, J., “Ant Colony Optimization for Bin Packing and Cutting StockProblems.” In 2001 proc. UK Workshop on Computational Intelligence, Edinburgh58. John Levine and Frederick Ducatelle, ‘’Ant Colony Optimisation and Local Search for BinPacking and Cutting Stock Problems‘’ in 2004 proc. Journal of the Operational ResearchSociety, pp. 705–716.59. Thiruvady, D.R., Meyer, B., Ernst, ’’ Strip packing with hybrid ACO Placement order islearnable.’’ In 2008 proc. IEEE Congress on Evolutionary Computation, Hong Kong, pp.1207–121360. Xu, Y.C., Dong, F.M., Liu, Y., Xiao, R.B.: Ant Colony Algorithm for the Weighted ItemLayout Optimization Problem. eprint arXiv:1001.4099 (2010)61. Qi yang and wang jin-min(2011), “The Particle Swarm Optimization Algorithm for SolvingRectangular Packing Problem”, Advanced Material Research vol186 pp 479-48362. Mr.Vijay Kumar, Dr.Jagdev Singh, Dr.Yaduvir Singh and Dr.Sanjay Sood, “Design &Development of Genetic Algorithms for Economic Load Dispatch of Thermal GeneratingUnits”, International Journal of Computer Engineering & Technology (IJCET), Volume 3,Issue 1, 2012, pp. 59 - 75, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375.63. Sudip Kumar Sahana, Dr. Aruna Jain and Abijit Mustafi, “A Comparative Study on MulticastRouting using Dijkstra’s, Prims and Ant Colony Systems”, International Journal of ComputerEngineering & Technology (IJCET), Volume 1, Issue 2, 2010, pp. 16 - 25, ISSN Print: 0976– 6367, ISSN Online: 0976 – 6375.64. S. Kanimozhi Suguna and Dr.S.Uma Maheswari, “Comparative Analysis of Bee-Ant ColonyOptimized Routing (Bacor) with Existing Routing Protocols for Scalable Mobile Ad HocNetworks (MANETS)”, International Journal of Computer Engineering & Technology(IJCET), Volume 3, Issue 1, 2012, pp. 232 - 240, ISSN Print: 0976 – 6367, ISSN Online:0976 – 6375.