Packing job is considered as an NP-hard job. It is an optimisation job of happening an optimum agreement of a set of points in a larger container with the aim of minimising the spread on the packing country or maximising the container use. This sort of job occurs in different industries and is important in mechanical design and industry, transit and in the development of a good layout design of an endeavor [ 9 ] .

This job has been studied and different algorithms have been applied to come up with an optimum wadding agreement. Some optimisation algorithms that were used in work outing were familial algorithm and atom drove algorithm.

Furthermore, another optimisation and swarm-based algorithm called the Artificial Bee Colony ( ABC ) algorithm is presented in this paper. It is an algorithm that is based on the scrounging behavior of bees.

In this survey, the research worker will utilize the Artificial Bee Colony Algorithm in happening an optimum agreement of jammed rectangles.

Packing jobs consider a set of points and a bin that will keep the points which aim to pack points every bit dumbly as possible with really optimum to no spreads. Packing jobs are helpful and widely used in several Fieldss such as in fabrication and in state of affairss like make fulling up containers, lading burden, ship edifice, building and ornament layout and so on [ .. ] .

Packing jobs are classified depending on the job ‘s application and aim. Some of the most of import jobs such as cutting stock, backpack and bin packing [ 4 ] are defined below.

The cutting stock job is about cutting various-sized pieces out on a given stock sheets. The job can either be a trim-loss job or an mixture job.

Trim-loss Problem

Trim-loss job is the allotment onto stock sheets of pieces of different sizes demanded by the clients with the purpose of minimising the cost of waste.

Assortment Problem

The mixture job is concerned in finding the size of the pieces or points to be kept in stock of sheets such that the best choice of points is used so waste is minimized.

Knapsack Problem

The backpack job describes the procedure of packing the most valuable points into a fixed-sized storage such as a backpack. The job consists of points with matching weights and values and a backpack of limited capacity. The aim of the backpack job is to find which points that maximize the value should be in the backpack given that the entire weight of the points is at most of the weight of the backpack.

Taken from [ 12 ]

Figure.1: Illustration on the backpack job

Bin Packing

Bin wadding job has a aggregation of points of different sizes and a figure of bins holding the same horizontal and perpendicular dimensions. There are different sorts of bin packing such as 2D wadding, 3D wadding, additive wadding, battalion by weight and more [ 14 ] . The aim of the job is to merely pack the points into the bins minimising the figure of bins used.

Taken from [ 13 ]

Figure.2: Bin wadding of different forms

Loading Problems

Loading jobs deal with rectangular boxes that are to be packed into a rectangular palette. These jobs can be classified into maker ‘s and distributer ‘s burden jobs [ 4 ] . Manufacturer ‘s and distributer ‘s burden jobs are chiefly concerned with the wadding of indistinguishable and non-identical rectangular pieces severally. The aim of lading jobs is to happen an optimum wadding form such that the figure of boxes to be placed in a certain palette or container is maximized.

Taken from [ 15 ]

Figure.3: Illustration on palette burden jobs for indistinguishable boxes

Other Discrepancies of Packing Problems

There are several sorts of packing jobs. The followers are some packing jobs that deal in “ happening the maximal figure of a certain form that can be packed into a larger, possibly different form ” [ 3 ] .

Sphere in Cuboid – a sphere wadding job that involves in happening an optimum agreement of given a set of spherical objects with diameter vitamin D be packed into a cuboid with a size of a ten B x degree Celsius

Packing Circles – are some circle packing jobs that attempt to pack a set of indistinguishable circles into a circle or another form

Taken from [ 5 ] & A ; [ 6 ]

Figure.4: ( a ) Circles in circle, ( B ) Circles in square ( degree Celsius ) Circles in equilateral trigon and ( vitamin D ) Circles in regular hexagon

Packing Squares – shows a set of indistinguishable squares packed inside a form such as the square and circle

Taken from [ 5 ] & A ; [ 6 ]

Figure.5: ( a ) Squares in square and ( B ) Squares in circle

Rectangle Packing Problems

Rectangle Packing Problem Definition

Rectangle wadding job is an optimisation job of apportioning a set of rectangle points into a larger rectangle, the container with the aim of minimising the packing country or the country wasted [ 8 ] . The set of rectangle points has different dimensions of width tungsten and tallness H and the entire country of the points should non transcend the country of the container width W and height H. The arrangement of the points is described by the undermentioned set of restraints [ 8 ] [ 10 ] [ 11 ] .

No imbrication of rectangle points.

No points must be wider or taller than the size of the container.

Edges of the points must be parallel to the container ‘s border

Taken from [ 7 ]

Figure.6: Illustration of a authoritative rectangle wadding job process

Taken from [ 1 ]

Figure.7: A sample wadding of 9 rectangular points – 90o rotary motion allowed

Taken from [ 1 ]

Figure.8: A sample packing – fixed orientation, no rotary motion allowed

Other Rectangle Packing Problems

Packing Into An Open-ended Rectangle

The job is described as arrangement, without overlapping, a set of rectangle points of different sizes in an open-ended rectangular container of width W. A rotary motion of 90A° is allowed every bit long as no points overlap. All the rectangle points should be pack in manner that the entire length L of infinite occupied is minimized [ 1 ] .

Taken from [ 1 ]

Figure.9: A possible agreement of eight points

Taken from [ 1 ]

Figure.10: Packing into an open-ended rectangles with three different breadths.

Packing Into Rectangles of Fixed Size

The thought behind this job is similar to usual bin packing jobs. It consists of rectangle points of different horizontal and perpendicular dimensions and a set of rectangular sheets of fixed length and breadth. The aim is to “ happen an agreement of the pieces that minimizes the figure of sheets needed ” [ 1 ] .

Taken from [ 1 ]

Figure.11: Packing into fixed sized rectangles

Drove Intelligence

Swarm intelligence is defined as “ any effort to plan algorithms or distributed problem-solving devices inspired by the corporate behavior of societal insect settlements and other carnal societies ” [ 19 ] . What makes drove based algorithms an interesting mechanism for work outing jobs particularly NP-complete jobs is the two cardinal constructs within, self-organisation and division of labor. Some illustrations of swarm-inspired algorithms and surveies are Particle Swarm Optimization ( PSO ) , ant settlement, bee settlement, flock of birds and more [ 19 ] .

The Artificial Bee Colony Algorithm

The Artificial Bee Colony ( ABC ) algorithm is a nature-inspired optimisation algorithm defined by Dervis Karaboga in 2005 [ 2 ] . Based on the scrounging behavior of bees, the purpose of the algorithm is to happen nutrient beginnings with high nectar sums and finally take the 1 with the highest sum.

In the ABC algorithm, bees are grouped into employed bees, looker-on bees and lookout bees. The employed bees are the 1 that exploit and keep the information of a peculiar nutrient beginning. The information on a specific nutrient beginning is so shared by each employed bees to the looker-on bees through a shake dance. Then, nutrient choice is done by the looker-on bees. These bees determine the quality of the nutrient beginnings and acquire to take the best nutrient beginning. When a nutrient beginning of an employed bee has been abandoned, this bee becomes a lookout bee. Scout bees are responsible for researching and seeking possible nutrient beginnings around the country.

The Waggle Dance

Bees need to pass on with other bees in order for them to happen and garner nutrient indispensable for the endurance of their settlement. One of the most interesting and challenging mechanisms of discoursing around the carnal land peculiarly Apis melliferas is dancing. Information on a certain nutrient beginning off from their hive is passed on to other bees through a shake dance.

A shake dance is performed by a lookout bee informing his hive mates the exact way and distance to the nutrient beginning. During the shake dance, the bee performs an eight-figure form dance where it foremost walks in a consecutive line while waggling his tail back and Forth. Then looping is done in jumping waies and travels the consecutive line over and over once more capable to the way and the distance being relayed. Walking in the consecutive line indicates the way and figure of shakes refers to the distance of the nutrient beginning [ 16 ] [ 17 ] .

Taken from [ 16 ]

Figure.12: The shake dance

The Algorithm

The chief stairss of the ABC algorithm from [ 19 ] consisting of the employed bees, the looker-on bees and the lookout bees is given below.

Figure.13: Main stairss of ABC

Pseudocode of ABC

The elaborate pseudocode of the ABC algorithm shown in [ 20 ] is given below:

Initialize the population of solutions Xi

Measure the population

cycle=1

Repeat

Produce new solutions ( nutrient beginning places ) Vi in the vicinity of Xi for the employed bees.

Apply Greedy Selection

Calculate the chance values Pi for the solutions Xi by agencies of their fittingness values utilizing the equation

Normalize Pi values into [ 0,1 ]

Produce the new solutions ( new places ) Vi for the looker-ons from the solutions Xi,

Apply Greedy Selection Process for the looker-ons between Xi and Vi

Determine abandoned Solutions, and replace it with new randomly produced solutions Xi for the lookout

Memorize the best nutrient beginning place achieved so far

rhythm = cycle+1

UNTIL ( rhythm = Maximum Cycle Number )

Initially, random executable solutions ( nutrient beginnings ) are generated and evaluated. Then, the employed bees will seek for new solutions in the vicinity out of the current solutions and the greedy choice is applied. The choice of much fitter solutions is done by the looker-on bees and is dependent on the fittingness value. The nutrient beginnings that do non better after a series of loop are abandoned and the bee associated to it eventually becomes a lookout. The lookout will seek for a new nutrient beginning once more. The whole procedure continues until the expiration standard is satisfied.

Neighbourhood Search

Neighbourhood searching is the procedure of bring forthing better solutions from the current executable solutions. A new solution is generated utilizing the equation below:

( 1 )

where XAij is the value of cell in solution XAi ( current solution ) which is indiscriminately picked,

XAkj is the value of cell J in XAk, a random solution non equal to XAi,

O is a random value in the scope of [ 0, 1 ] .

Greedy Selection

The greedy choice is responsible of doing and taking the optimum solution at each phase of the procedure.

Fitness Function

A fittingness map determines the quality of a nutrient beginning ( solution ) . The higher the fitness value of the equation below, the better the solution is.

( 2 )

where is the cost map

Probability Function

The chance map determines the chance that a peculiar nutrient beginning will be preferred by looker-on bees. The chance value is calculated utilizing the expression below:

( 3 )

where I is the current nutrient beginning and Sn is the entire figure of nutrient beginnings.

Literature Review

In the paper “ An Improved Genetic Algorithm for the Packing of Rectangles ” by Ming Le

Stefan Jakobs [ 22 ] in his paper entitled “ On familial algorithms for the wadding of polygons ” implemented a intercrossed attack to familial algorithm. Jakobs used the bottom-left-condition to cut down the figure of possible wadding forms. The initial population is composed of width-sorted sequence of agreements based on the bottom-left regulation. A rectangular piece is moved get downing from the upper right corner of the country and moving every bit far as to the bottom so allotment is done every bit far as to the left corner of the bounding rectangle. This construct of a intercrossed familial algorithm was besides used in wadding of polygons job.

Chen Zhao, et al [ 23 ] introduced the construct of Discrete Particle Swarm Optimization ( DPSO ) algorithm. In this method, a. For a elaborate treatment on the said method, see [ 23 ] .

Statement of the Problem

Packing jobs such as rectangle packing belongs to the category of NP-hard jobs since there is no easy manner to find the optimum solution for every instance. Finding the best manner of suiting a figure of rectangles into a larger rectangle is a clip devouring repeating undertaking and involves a really big solution infinite.

The ABC algorithm, a new swarm-based methodological analysis, has been proven to be an efficient attack that solves optimisation jobs in assorted countries. Some research surveies besides show that ABC outperforms other optimisation techniques such as familial algorithm. In this survey, the ABC algorithm will be implemented to work out rectangle packing jobs.

Therefore, this paper will prove the efficiency of the ABC algorithm in happening the best possible agreement of packing rectangles.

Timeline

The tabular array below shows the undertakings and their corresponding clip periods that I intend to set about to successfully finish this research.

Undertaking No.

Time Period

Undertaking Description

1

Dec 13 – Dec 19, 2010

Research and reading of bing documents and published surveies sing rectangle wadding jobs and the ABC algorithm.

Making of the first bill of exchange of the Thesis Proposal.

2

Dec 21, 2010 – Jan 04, 2011

Execution of the proposed algorithm.

Making of the Proposed Approach portion of the thesis.

3

Jan 05, 2011 – Feb 2011

Testing and debugging. Experimentation stage.

Making of the 10-page conference paper and 5-page URS paper.

4

Feb – Mar 2011

Finalizing of documents and other necessary demands

5

Mar 2011

Submission twenty-four hours

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