1. HM 2004 – ECAI 2004VNS/GRASP for SPP 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@ull.es TIC2002-04242-C03-01 (70% FEDER) HM 2004 - ECAI 2004, Valencia, August 22-23

2. HM 2004 – ECAI 2004VNS/GRASP for SPP 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. HM 2004 – ECAI 2004VNS/GRASP for 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 90 0. 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. HM 2004 – ECAI 2004VNS/GRASP for SPP 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. HM 2004 – ECAI 2004VNS/GRASP for SPP 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.

6. HM 2004 – ECAI 2004VNS/GRASP for SPP 6 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. w h

7. HM 2004 – ECAI 2004VNS/GRASP for SPP 7 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) w h

8. HM 2004 – ECAI 2004VNS/GRASP for SPP 8 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(w i,h i ): i = 1,..., n } Each rectangle R(w i,h i ) has width w i and height h i verifying: min{ w i, h i }  w. wiwi hihi w

9. HM 2004 – ECAI 2004VNS/GRASP for SPP 9 The solutions of the SPP The solutions are given by placing [(r i, a i,b i ) : i = 1,..., n] of the rectangles R(w i,h i ), i = 1,..., n. where: r i represents if R(w i,h i ) is rotated or not: if r i = 1 then the R(w i,h i ) is rotated at 90 degrees; otherwise r i = 0 and it is not rotated. (a i,b i ) are the coordinates of the position of the bottom-left corner of the rectangle R(w i,h i ) with respect to the origin of coordinates (the bottom-left corner of the strip).

10. HM 2004 – ECAI 2004VNS/GRASP for SPP 10 Representing Solutions Representing solutions [(r i, a i,b i ), i = 1,..., n] wiwi hihi wiwi hihi bibi wiwi hihi aiai r i = 0 r i = 1 r i = 0

11. HM 2004 – ECAI 2004VNS/GRASP for SPP 11 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 r i = 0 then 0  a i  w – w i, 0  b i and the position of no other rectangle R( w j, h j ) verifies: a i < a j < a i + w i and b i < b j < b i + h i If r i = 1 then 0  a i  w – h i, 0  b i and the position of no other rectangle R( w j, h j ) verifies: a i < a j < a i + h i and b i < b j < b i + w i

12. HM 2004 – ECAI 2004VNS/GRASP for SPP 12 Unfeasible Solutions Unfeasible solutions bjbj ajaj bibi wiwi hihi aiai

13. HM 2004 – ECAI 2004VNS/GRASP for SPP 13 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 r i = 0 then 0  a i  w – w i, 0  b i and the position of no other rectangle R( w j, h j ) verifies: a i < a j < a i + w i and b i < b j < b i + h i If r i = 1 then 0  a i  w – h i, 0  b i and the position of no other rectangle R( w j, h j ) verifies: a i < a j < a i + h i and b i < b j < b i + w i

14. HM 2004 – ECAI 2004VNS/GRASP for SPP 14 The objective function The objective to be minimized is the maximum height h reached by the rectangles It is computed by: h = max {max (b i +h i ), max (b i +w i )} r i =0 r i =1 Equivalently, the objective is to minimize the area of the strip used It is computed by the product: h · w. w h bibi hihi

15. HM 2004 – ECAI 2004VNS/GRASP for SPP 15 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.

16. HM 2004 – ECAI 2004VNS/GRASP for SPP 16 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.

17. HM 2004 – ECAI 2004VNS/GRASP for SPP 17 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 [r i,..., r n ] that represents if each rectangle is rotated or not.

18. HM 2004 – ECAI 2004VNS/GRASP for SPP 18 High Quality Solutions h w Small waste areas Smooth upper contour

19. HM 2004 – ECAI 2004VNS/GRASP for SPP 19 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]

20. HM 2004 – ECAI 2004VNS/GRASP for SPP 20 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.

21. HM 2004 – ECAI 2004VNS/GRASP for SPP 21 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

22. HM 2004 – ECAI 2004VNS/GRASP for SPP 22 The GRASP pseudocode The Constructive phase: Let x 0 = {} be the initial empty partial solution. Let t ← 0. Repeat the steps: Construct the restricted candidate list RCL(t). Choose at random an element e t+1 of RCL(t). Update the partial solution x t ← x t+1 U e t+1. Set t ← t+1. until there is not element in RCL(t). The post-porcessing phase: Aply a VNS to the last k packed rectangles

23. HM 2004 – ECAI 2004VNS/GRASP for SPP 23 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 = [( y i, x 1 i, x 2 i ): i = 1,...,c], where, for the i-th segment, y i is the y-coordinate (height) of its points and [x 1 i, x 2 i ] is the interval of their x-coordinates. C = [(y 1, x 1 1, x 2 1 ), (y 2, x 1 2, x 2 2 ), … (y i, x 1 i, x 2 i ), … (y c, x 1 c, x 2 c )] This sequence, verifies: x 1 1 = 0 and x 2 c = w, x 1 i+1 = x 2 i, for each i = 1,...,c  1. y i+1 ≠ y i, for each i = 1,...,c  1.

24. HM 2004 – ECAI 2004VNS/GRASP for SPP 24 Representation of upper contour y1 y1 y4 y4 y3 y3 y2 y2 (y 1,x 1 1,x 2 1 ) (y 2,x 1 2,x 2 2 ) (y 3,x 1 3,x 2 3 ) (y 4,x 1 4,x 2 4 ) 0 = x 1 1 x 2 1 = x 1 2 x 2 2 = x 1 3 x 2 3 = x 1 4 x 2 4 = 0 C = [(y 1, x 1 1, x 2 1 ), (y 2, x 1 2, x 2 2 ), (y 3, x 1 3, x 2 3 ), (y 4, x 1 4, x 2 4 )]

25. HM 2004 – ECAI 2004VNS/GRASP for SPP 25 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) = [(y i, x 1 i, x 2 i ): i = 1,...,c] be the contour at iteration t. Let s j be the lowest segment of C(t) (minimum height) The rectangles in  2 (t) are evaluated by its adjustment to s j. Let y j = min { y 1, y 2,..., y c } and l = x 2 j  x 1 j be its length. Given α  [0,1], RCL(t), is built as follows. RCL(t) = { R(w i,h i )   2 (t): l  α ≤ w i ≤ l or l  α ≤ h i ≤ l }.

26. HM 2004 – ECAI 2004VNS/GRASP for SPP 26 Updating the upper contour yj1 yj1 y j+1 yj yj (y j  1,x 1 j  1,x 2 j  1 ) (y j,x 1 j,x 2 j ) (y j+1,x 1 j+1,x 2 j+1 ) x 1 j x 2 j

27. HM 2004 – ECAI 2004VNS/GRASP for SPP 27 Update the RCL If there is not rectangle to choose (with some side less than l) then: The rectangle [x 1 j,x 2 j ]×[y j,min{y j  1,y j+1 ] becomes a wasted area. If y j+1 ≤y j  1 (the other case is similar) then replace: the segments (y j,x 1 j,x 2 j ) and (y j+1,x 1 j+1,x 2 j+1 ) by (y j+1,x 1 j,x 2 j+1 ). ( the size of the contour decreases; i.e., |C(t+1)| = |C(t)|  1 ) Otherwise, If R(w i,h i ) is the selected rectangle of RCL(t) then replace (y j, x 1 j, x 2 j ) by (y j +h i, x 1 j, x 1 j +w i ) and (y j, x 1 j +w i, x 2 j ). ( the size of the contour increases; i.e., |C(t+1)| = |C(t)| + 1 )

28. HM 2004 – ECAI 2004VNS/GRASP for SPP 28 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 → 2 S. 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 N k, k = 1, …, k max. 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.

29. HM 2004 – ECAI 2004VNS/GRASP for SPP 29 VNS: Variable Neighbourhood Search Initially it takes k ← 1 and a random initial solution x. A local search from x with the neighborhood N 1 returns x*. These steps are repeated until the stopping condition is met. Take k ← k+1 and a random neighbor y of the local optimum x* using in the neighborhoods N k ; i.e., y  N k (x*). Apply the local search from y using the neighborhoods N 1. 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 = k max, generate a new initial solution x and iterate the process with k ← 1. [Hansen & Maldenovic 2001,2003,2004]

30. HM 2004 – ECAI 2004VNS/GRASP for SPP 30 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 N k 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 k max neighbourhoods.

31. HM 2004 – ECAI 2004VNS/GRASP for SPP 31 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.

32. HM 2004 – ECAI 2004VNS/GRASP for SPP 32 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 → 2 S, the series of nested neighbourhood structure is defined as follows. The first neighbourhoods are the basic ones; N 1 (x) = N(x). and the next neighbourhoods are defined recursively by N k+1 (x) = N(N k (x)). N k (x) consists of the solutions that can be obtained from x by k base moves.

33. HM 2004 – ECAI 2004VNS/GRASP for SPP 33 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 ← k+1. Set x ← x*.

34. HM 2004 – ECAI 2004VNS/GRASP for SPP 34 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.

35. HM 2004 – ECAI 2004VNS/GRASP for SPP 35 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

36. HM 2004 – ECAI 2004VNS/GRASP for SPP 36 The Hybrid approach The last packed rectangle(s) by GRASP

37. HM 2004 – ECAI 2004VNS/GRASP for SPP 37 The VNS/GRASP hybrid Apply the contructive phase of the GRASP. Let [R 1, R 2, …, R n ] be the solution provided by the GRASP. Let [R’ 1, R’ 2, …, R’ k ] be the last k rectangles of the solution. Apply the nested VNS on the solution space consisting of: all the permutations of the rectangles [R’ 1, R’ 2, …, R’ k ] to be packed in the free strip. Let [R* 1, R* 2, …, R* k ] be the best permutation obtained by the nested VNS Return the solution [R 1, R 2, …, R n  k,R* 1, R* 2, …, R* k ].

38. HM 2004 – ECAI 2004VNS/GRASP for SPP 38 Computational Experiments 1. Tuning the parameter α 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. 2. Comparative between GRASP/VNS and SA+BLF SA+BLF 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) 3. 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)

39. HM 2004 – ECAI 2004VNS/GRASP for SPP 39 Tunning the parameter α We fix several values for α. 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 n iter = 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.

40. HM 2004 – ECAI 2004VNS/GRASP for SPP 40 Results for tunning α

41. HM 2004 – ECAI 2004VNS/GRASP for SPP 41 Tunning α with big instances

42. HM 2004 – ECAI 2004VNS/GRASP for SPP 42 Test Instances for the comparative analysis 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 120 80 97 C6C6 90 60 73 C5C5 60 49 C4C4 30 60 28 29 28 C3C3 15 40 25 C2C2 20 16 17 16 C1C1 h opt w nCategory C7C7 196 197 160 240

43. HM 2004 – ECAI 2004VNS/GRASP for SPP 43 Comparative with SA+BLF

44. HM 2004 – ECAI 2004VNS/GRASP for SPP 44 Comparative for hybridization

45. HM 2004 – ECAI 2004VNS/GRASP for SPP 45 Conclusions 1. GRASP and GRASP/VNS have better performance than SA+BLF: It provides a great decreasing in the running time: (for instances in C 7 it goes from 4181 minutes to 1.37 secs or less) 2. GRASP and GRASP/VNS have comparable or better efficacy than SA+BLF. 3. GRASP/VNS has slightly better minimum objective value than GRASP In the majority of the problems it gives better minimum value than GRASP. 4. GRASP/VNS is more robust than GRASP: the difference between the maximum and minimum objective values is smaller in GRASP+VNS than in GRASP. 5. GRASP+VNS 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)

46. HM 2004 – ECAI 2004VNS/GRASP for SPP 46 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 Inteligent Computation Group University of La Laguna. http://webpages.ull.es/users/gci/ gci@ull.es Proyecto TIC2002-04242-C03-01 (70% FEDER) THANKS