GRASPVNS hybrid for the Strip Packing Problem

1
GRASP/VNS hybrid for the Strip Packing Problem

• Jesús David Beltrán, José Eduardo Calderón,
  Rayco Jorge Cabrera, José A. Moreno Pérez and J.
  Marcos Moreno-Vega
• Intelligent Computation Group
• University of La Laguna.
• http//webpages.ull.es/users/gci/
• gci_at_ull.es
• TIC2002-04242-C03-01 (70 FEDER)
• HM 2004 - ECAI 2004, Valencia, August 22-23

2
Abstract

• We consider an hybrid metaheuristic (HM) to solve
  the SPP.
• SPP Strip Packing Problem.
• The hybrid algorithm to use a VNS to improve the
  quality of part of the solution obtained with a
  GRASP.
• VNS Variable Neighbourhood Search.
• GRASP Greedy Randomized Adaptive Search
  Procedure.
• The results of HM(VNS/GRASP) are compared with
  those existing in the literature for the SPP.

3
The Problem

• The Strip Packing Problem (SPP) consists of
  packing a set of rectangles in a strip
  minimizing the height of the packing
• the rectangles can be rotate at 900.
• the cuttings can be non guillotine.
• (a cutting is guillotine if it goes from a side
  of the object to the front side in a
  non-guillotine cut this can not be true)

4
Literature

• Many researches have tackled these problems.
• Lodi, Martello and Monaci (2002) discuss
  mathematical models, and survey lower bounds,
  classical approximation algorithms, recent
  heuristic and metaheuristic methods and exact
  enumerative approaches for the two-dimensional
  packing problems.
• Hopper and Turton (2001) have a review of
  metaheuristic algorithms to solve two-dimensional
  strip packing problems.
• A Genetic approach was proposed for the oriented
  two-dimensional strip packing problem in Yeung
  and Tang (2004).

5
Packing Problems

• Packing problems involve constructing an
  arrangement of items that minimizes the total
  space required by the arrangement.
• These problems are encountered in many areas of
  business and industry.
• Some applications can be found in wood, glass,
  paper, textile or leather industries.
• It must be note that small improvements can be
  very valuable for industrial applications of
  these problems.

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The Hybrid Metaheuristics

• The Hybrid
• Use VNS to improve the solution provided by
  GRASP.
• VNS operates on a part of the solution given by
  GRASP.
• VNS attempts to improve the packing of the last
  rectangles introduced into the solution.

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The Strip Packing Problem (SPP)

• The Strip Packing Problem (SPP). Consider
• a strip of fixed width w and infinite height,
• a finite set of rectangles, with at least one of
  their sides less than w.
• The SPP consists of packing the rectangles in the
  strip minimizing the total height h
• (The rectangles can be rotate at 90 degrees)

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The data of the SPP

• The elements of an instance of
• the Strip Packing Problem (SPP) are
• A strip with fixed width w and infinite height.
• A set of n rectangles
• ? R(wi,hi) i 1, ..., n
• Each rectangle R(wi,hi) has
• width wi and height hi verifying
• min wi, hi ? w.

hi
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The solutions of the SPP

• The solutions are given by placing
• (ri, ai,bi) i 1, ..., n
• of the rectangles R(wi,hi), i 1, ..., n.
• where
• ri represents if R(wi,hi) is rotated or not
• if ri 1 then the R(wi,hi) is rotated at 90
  degrees
• otherwise ri 0 and it is not rotated.
• (ai,bi) are the coordinates of the position of
  the bottom-left corner of the rectangle R(wi,hi)
  with respect to the origin of coordinates (the
  bottom-left corner of the strip).

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Representing Solutions

• Representing solutions (ri, ai,bi), i 1, ...,
  n

hi
hi
hi
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The feasible placings

• The placing of each rectangle is feasible if
• The following two conditions are verified
• the rectangle is completely included in the strip
  and
• it does not overlap with any other rectangle
  already placed.
• In other words,
• If ri 0 then 0 ? ai ? w wi , 0 ? bi and
• the position of no other rectangle R( wj, hj )
  verifies
• ai lt aj lt ai wi and bi lt bj lt bi hi
• If ri 1 then 0 ? ai ? w hi , 0 ? bi and
• the position of no other rectangle R( wj, hj )
  verifies
• ai lt aj lt ai hi and bi lt bj lt bi wi

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Unfeasible Solutions

• Unfeasible solutions

aj
hi
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The feasible placings

• The placing of each rectangle is feasible if
• The following two conditions are verified
• the rectangle is completely included in the strip
  and
• it does not overlap with any other rectangle
  already placed.
• In other words,
• If ri 0 then 0 ? ai ? w wi , 0 ? bi and
• the position of no other rectangle R( wj, hj )
  verifies
• ai lt aj lt ai wi and bi lt bj lt bi hi
• If ri 1 then 0 ? ai ? w hi , 0 ? bi and
• the position of no other rectangle R( wj, hj )
  verifies
• ai lt aj lt ai hi and bi lt bj lt bi wi

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The objective function

• The objective to be minimized is
• the maximum height h
• reached by the rectangles
• It is computed by
• h max max (bihi), max (biwi)
• ri0 ri1
• Equivalently, the objective is
• to minimize the area of the strip used
• It is computed by the product
• h w.

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The Infinite set of Solutions

• Infinite equivalent feasible solutions
• could be obtained by
• shifting horizontally or vertically
• some rectangles.
• To avoid it
• consider only solutions obtained by
• introducing successively the rectangles
• following a given placement strategy.

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The Bottom-left placing

• We use the usual bottom-left strategy
• each rectangle is placed
• at the deepest location
• and, within it,
• in the most left possible location.
• Each feasible solution of the problem
• is determined by
• the order in which the rectangles
• are introduced in the strip.

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The Solution Space

• Given the bottom-left strategy
• each rectangle is placed at the deepest location
  and,
• within it, in the most left possible location.
• The space of solutions consists of
• the permutations of the numbers 1, 2 , n
• with a binary vector ri, ..., rn
• that represents if each rectangle is rotated or
  not.

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High Quality Solutions

• Small
• waste
• areas
• Smooth
• upper
• contour

h
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The GRASP metaheuristic

• GRASP Greedy Randomized Adaptive Search
  Procedure
• The GRASP metaheuristic has two phases
• a constructive phase, that constructs a good
  initial solution x.
• an improving phase, that is applied to the
  initial solution x.
• (the improving phase consists in a restricted
  VNS)
• Both phases are repeated until the stopping
  criterion is met.
• Resende and Ribeiro, 2003

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The Constructive Method of GRASP

• A constructive method
• an items is iteratively added to an initially
  empty
• structure until a solution of the problem is
  obtained.
• A heuristic constructive procedure
• The choice of the item based heuristic
  evaluation(s)
• measure(s) of the convenience of considering
• the item as belonging to the solution.
• An adaptive heuristic constructive algorithm
• The evaluation depends on the items already in
  the solution.
• The greedy strategy
• Choose the item that optimize the evaluation
  (the best item)
• A randomized rule
• Use random numbers in the choice.

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The GRASP rule

• Choose (at random) one of the best items.
• The evaluations by
• The wasted areas
• The smoothness of the upper contour.
• The best items are those that best fit to the
  upper contour
• Construct a Restricted Candidate List of items
  that conteins the best items, then choose one
  item of the list

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The GRASP pseudocode

• The Constructive phase
• Let x0 be the initial empty partial
  solution.
• Let t ? 0.
• Repeat the steps
• Construct the restricted candidate list RCL(t).
• Choose at random an element et1 of RCL(t).
• Update the partial solution xt ? xt1 U et1 .
• Set t ? t1.
• until there is not element in RCL(t).
• The post-porcessing phase
• Aply a VNS to the last k packed rectangles

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Representing the upper contour

• The upper contour is a series of segments from
  left to right.
• It is represented by the coordinates of the
  segments
• (the origin is the bottom-left corner of the
  strip)
• C ( yi, x1i, x2i ) i 1,...,c, where, for
  the i-th segment,
• yi is the y-coordinate (height) of its points
  and
• x1i, x2i is the interval of their
  x-coordinates.
• C (y1, x11, x21), (y2, x12, x22), (yi, x1i,
  x2i), (yc, x1c, x2c)
• This sequence, verifies
• x11 0 and x2c w,
• x1i1 x2i, for each i 1,...,c?1.
• yi1 ? yi, for each i 1,...,c?1.

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Representation of upper contour
(y1,x11,x21)

• y1
• y4
• y3
• y2

(y4,x14,x24)
(y3,x13,x23)
(y2,x12,x22)
0 x11 x21 x12 x22 x13
x23 x14 x24 0
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Computing the RCL

• At any iteration t, the RCL(t) is constructedas
  follows
• Let ?1(t) be the inserted rectangles and ?2(t)
  ? \ ?1(t).
• Let C(t) (yi, x1i, x2i) i 1,...,c be the
  contour at iteration t.
• Let sj be the lowest segment of C(t) (minimum
  height)
• The rectangles in ?2(t) are evaluated by its
  adjustment to sj.
• Let yj min y1, y2, ..., yc and l x2j ?
  x1j be its length.
• Given a?0,1, RCL(t), is built as follows.
• RCL(t) R(wi,hi) ? ?2(t) l ?a wi l or
  l ?a hi l .

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Updating the upper contour
(yj?1,x1j?1,x2j?1)

• yj?1
• yj1
• yj

(yj1,x1j1,x2j1)
(yj,x1j,x2j)
x1j x2j
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Update the RCL

• If there is not rectangle to choose (with some
  side less than l) then
• The rectangle x1j,x2jyj,minyj ?1,yj1
  becomes a wasted area.
• If yj1 yj?1 (the other case is similar) then
  replace
• the segments (yj,x1j,x2j) and
  (yj1,x1j1,x2j1) by (yj1,x1j,x2j1) .
• ( the size of the contour decreases i.e.,
  C(t1) C(t) ? 1 )
• Otherwise,
• If R(wi,hi) is the selected rectangle of RCL(t)
  then
• replace (yj, x1j, x2j) by (yjhi, x1j,
  x1jwi) and (yj, x1jwi, x2j).
• ( the size of the contour increases i.e.,
  C(t1) C(t) 1 )

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The VNS Metaheuristics

• Variable Neighbourhood Search (VNS) is a recent
  metaheuristic based on systematic change of the
  neighbourhood in a search.
• Usual heuristic searches are based on
  transformations of solutions that determine a
  neighbourhood structure on the solution space.
• A neighborhood structure in a solution space S is
  a mapping
• N S ? 2S. x ? N(x).
• The solutions of y ? N(x) are the neighbor
  solutions of x
• and constitute its neighborhood N(x).
• VNS uses a series of neighborhoods Nk, k 1, ,
  kmax.
• A local search takes a better neighbor while
  possible.
• The basic idea of the VNS is to change the
  neighbourhood structure when the local search is
  trapped on a local minimum.

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VNS Variable Neighbourhood Search

• Initially it takes k ? 1 and a random initial
  solution x.
• A local search from x with the neighborhood N1
  returns x.
• These steps are repeated until the stopping
  condition is met.
• Take k ? k1 and a random neighbor y of the local
  optimum x using in the neighborhoods Nk i.e., y
  ? Nk(x).
• Apply the local search from y using the
  neighborhoods N1.
• If the local minimum found y is better than x
  then the process is iterated with x ? y and k
  1.
• Otherwise, iterate the process with the current
  solution x.
• When k kmax, generate a new initial solution x
  and iterate the process with k ? 1.
• Hansen Maldenovic 2001,2003,2004

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VND Variable Neighbourhood descent

• Given a neighbourhood structure, an improving
  local search iteratively seeks for a better
  solution in the neighbourhood of the current
  solution therefore it is trapped in a local
  optimum with respect to the current neighbourhood
  structure.
• The VND (Variable Neighborhood Descent) method
  changes the neighbourhoods Nk each time a local
  optimum is reached.
• It ends when there is not improve with the all
  neighbourhoods
• The final solution provided by the algorithm
  should be a local optimum with respect to all
  kmax neighbourhoods.

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A simple VNS

• A VNS can be implemented by the combination of
  series of random and improving (local) searches.
• When the improving local search stops at a local
  minimum, a shake procedure performs a random
  search for a new starting point for a new local
  search.
• The improving local search and the random shake
  procedure are usually based on standard moves
  that determine neighbourhood structures.
• A basic local search consists of applying an
  improving move until no such move exists.
• A simple shake procedure consists of applying a
  number of random moves.
• The stopping condition in a VNS may be maximum
  CPU time allowed, maximum number of iterations,
  or maximum number of iterations between two
  improvements.

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The General VNS

• The GVNS (General Variable Neighborhood Search)
  method applies two (possibly different) series of
  neighbourhoods
• one for the shaking and one for the descent.
• Several applications use the same set of
  neighbourhood structures for shaking and descent.
  
• The neighbourhoods are often nested and based on
  a single standard move.
• Given a base neighbourhood structure N S ? 2S,
  the series of nested neighbourhood structure is
  defined as follows.
• The first neighbourhoods are the basic ones
  N1(x) N(x).
• and the next neighbourhoods are defined
  recursively by Nk1(x) N(Nk(x)).
• Nk(x) consists of the solutions that can be
  obtained from x by k base moves.

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The Nested VNS

• Initialization
• Find an initial solution x.
• Set x ? x and k ? 1.
• Iterations
• Repeat the following until the stopping
  condition is met
• Shake
• Apply k random moves to the solution x to get
  x.
• Local Search
• Apply an improving move to the solution x
  until a local minimum x is found.
• Improve or not
• If x is better than x, do x ? x and k ? 1.
• Otherwise do k ? k1. Set x ? x.

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Further details

• For the local search we use the greedy strategy
  at step 2b an iterative procedure tests all the
  base moves, and that providing the best solution
  is made until no improving move exists.
• The shake consists of applying k of random base
  moves.
• This number of random moves is the size of the
  shake.
• The base move for problems with solutions
  represented by permutations is the interchange
  move that consists in the exchange of the
  position of two elements of the permutation.
• The random generation of solutions is performed
  by successively selecting one of the remainder
  elements with the same probability.

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The Hybrid approach

• The hybrid approach proposed is obtained by using
  a nested Variable Neighborhood Search in the
  post-processing phase of the above GRASP.
• The GRASP constructs a solution by inserting at
  random a rectangle of the RCL it gets small
  interior waste.
• The VNS gets a better packing of the last packed
  rectangles it increases the smothness of the
  upper contour

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The Hybrid approach

• The last packed rectangle(s) by GRASP

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The VNS/GRASP hybrid

• Apply the contructive phase of the GRASP.
• Let R1, R2, , Rn be the solution provided by
  the GRASP.
• Let R1, R2, , Rk be the last k rectangles
  of the solution.
• Apply the nested VNS on the solution space
  consisting of
• all the permutations of the rectangles R1, R2,
  , Rk
• to be packed in the free strip.
• Let R1, R2, , Rk be the best permutation
  obtained by the nested VNS
• Return the solution R1, R2, , Rn?k,R1, R2, ,
  Rk.

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Computational Experiments

• Tuning the parameter a for constructive phase of
  the GRASP.
• the parameter that controls the fitness of the
  rectangles to the lowest segment of the upper
  contour to construct the LCR
• Randomly generated packing instances were solved.
  
• Comparative between GRASP/VNS and SABLF
• SABLF is a Simulated Annealing with the
  Bottom-Left rule proposed by Hopper and Turton
  (2001)
• GRASP(/VNS) are very much faster.
• Set of instances from Hopper and Turton (2001)
• Comparative between GRASP and GRASP/VNS
• VNS significatively improve GRASP?
• GRASP/VNS improves GRASP for large instances
• Set of instances from Hopper and Turton (2001)

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Tunning the parameter a

• We fix several values for a. 0, 0.1, 0.5.
• Each method was run 5 times for the instances
  randomly generated.
• The output variable was the average objective
  value in the niter 40 executions of the
  constructive phase
• Therefore we considered three treatments T(0),
  T(0.1) and T(0.2).
• The previous normality and variance equality
  tests were negative.
• We applied the Friedman nonparametric (Daniel
  1990).
• When the null hypothesis of equality among
  treatment was rejected,
• we applied the Friedman multiple comparison
  tests (Daniel 1990, page 274)
• to obtain the significance of the differences
  among treatments.

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Results for tunning a
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Tunning a with big instances
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Test Instancesfor thecomparativeanalysis

• The instances used in
• Hopper and Turton
• (2002)
• Data are available on
• the OR-Library at
• http//mscmga.ms.ic.ac.uk/jeb/orlib/stripinfo.html
  

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Comparative with SABLF
44
Comparative for hybridization
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Conclusions

• GRASP and GRASP/VNS have better performance than
  SABLF
• It provides a great decreasing in the running
  time (for instances in C7 it goes from 4181
  minutes to 1.37 secs or less)
• GRASP and GRASP/VNS have comparable or better
  efficacy than SABLF.
• GRASP/VNS has slightly better minimum objective
  value than GRASP
• In the majority of the problems it gives better
  minimum value than GRASP.
• GRASP/VNS is more robust than GRASP
• the difference between the maximum and minimum
  objective values is smaller in GRASPVNS than in
  GRASP.
• GRASPVNS is more useful when the number of
  rectangles increases (the number of improvements
  increases with the size of the problem).
• (this trend must be studied with new
  computational experiments)

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GRASP/VNS hybrid for the Strip Packing Problem
THANKS

• Jesús David Beltrán, José Eduardo Calderón,
  Rayco Jorge Cabrera, José A. Moreno Pérez and J.
  Marcos Moreno-Vega
• Inteligent Computation Group
• University of La Laguna.
• http//webpages.ull.es/users/gci/
• gci_at_ull.es
• Proyecto TIC2002-04242-C03-01 (70 FEDER)