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A Heuristic for the 3-Staged 2D Cutting Stock Problem with Usable Leftover

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A Heuristic for the 3-Staged 2D Cutting Stock Problem with Usable Leftover International Conference of Electrical, Automation and Mechanical Engineering (EAME 2015) A Heuristic for the 3-staged 2D Cutting Stock Problem with Usable Leftover Q.L. Chen L.P. Li School of Business Administration College of Computer and Electronic Information South China University of Technology Guangxi University Guangzhou, China Nanning, China College of Computer and Electronic Information Guangxi University Nanning, China Y. Chen School of Business Administration, South China University of Technology, Y.D. Cui Guangzhou, China College of Computer and Electronic Information College of Computer and Electronic Information Guangxi University Guangxi University Nanning, China Nanning, China X.Y. Lu College of Computer and Electronic Information Guangxi University Nanning, China Abstract—A heuristic approach is proposed for the 3-staged only homogenous strips and is referred to as the homogenous two-dimensional cutting stock problem with usable leftover, 3-staged cutting pattern with usable leftover (3HL pattern). where the unused rectangular segments can be returned to stock for future use. The approach is based on the recursive technique II. PROBLEM DESCRIPTION combined with beam search. A forward recursion is used to The paper proposes a heuristic algorithm to determine the generate normal segments that consist of homogenous strips, and cutting plan that consists of 3HL patterns, where a set of m a beam search heuristic is used to obtain the 3-staged cutting types of rectangular items are cut from stock plates of patterns considering usable leftover. The computational results show that the algorithm is effective in reducing material cost. dimensions L W , each item type mIi },,1{ , has a length li , a width wi , a demand di and a value vi . The Keywords-3-staged cutting; beam search; recursive technique; objective is to minimize the difference between the cost of the usable leftover plate used and the value of usable leftovers. I. INTRODUCTION If not specified, we assume that the cuts on the plate are The two-dimensional (2D) cutting stock problem consists infinitely thin, i.e. the saw thickness is zero. Otherwise the of seeking an optimal cutting plan that consists of different saw thickness is added to the dimensions of the plate and cutting patterns. We are particularly concerned with the items. special case where the non-used material (leftover) of the A. Usable Leftover cutting pattern can be stocked for use in the subsequent cutting process, if large enough. With the usability of leftovers, A 3-staged pattern contains three stages of orthogonal a considerable amount of material can be saved. guillotine cuts. If the first-stage cuts are vertical, it is an X-pattern, otherwise a Y-pattern. Only the unused segment The cutting stock problem with usable leftovers (CSPUL) along the pattern direction is considered as leftover. In other was first introduced in the early 1970s by Brown [1]. Authors words, only the unused rectangular segment on the right of an made extensive studies in the 1D CSPUL in the past two X-pattern or on the top of a Y-pattern can be considered as the decades [2]. For the 2D CSPUL, only Andrade et al. [3, 4] potentially usable leftover (see Figure 1). As for that, the deal with non-exact 2-staged (2NE) patterns. To the best of usable leftover has the same width, which is useful to simplify our knowledge, no paper deals with the 3-staged cutting the cutting process and the inventory management. pattern with usable leftovers. This paper considers the 2D CSPUL with 3-staged patterns, where the unused rectangular segments of the cutting patterns can be stored as the usable leftovers. Each pattern contains © 2015. The authors - Published by Atlantis Press 776 Furthermore, this paper considers the adjacent segments jointly to get all the normal segments within a short time. Forward recursion of dynamic programming is used to obtain the segment x W through building strips x wi , i I x on the top of sub-segment x y , to obtain a new sub-segment x() y wi . The number of items in the sub-segment should not exceed demand, So , where is the number of ei min[ x / li , d i n ( x , y , i )] n(,,) x y i type-i items included in the sub-segment x y . Update v(,) x i , and the value of the sub-segment x() y w is: (a) X-pattern (b) Y-pattern i FIGURE I. 3-STAGED PATTERNS WITH USABLE LEFTOVER. (A) X-PATTERN. F(x , y wi ) max{F(x , y wi ),F( x , y ) v ( x , i )}, (B) Y-PATTERN. y W wi, i I x (4) B. The Model for 3HL Pattern Where F(,) x y is the value of sub-segment x y . To simplify the description without loss of generality, the Initially F( x ,0) 0 . approach is described for the X-pattern with oriented items. It Suppose that the solution of normal segment is can be easily adapted for the Y-pattern and non-oriented items. qk W known as F(q ,y ) . The length of its adjacent segment is q . Let l be the length of the usable leftover, c be the unit k k 1 If q// l q l , ignore item type i ; Otherwise, add it to the value (value of unit area), lmin mini I {l i , w i }. The value Vl k i k1 i of the leftover is defined as: new item set I N . The initial value of the adjacent segment is F(qk 1 ,y ) F(qk , y ) . Replace I x with I N , and obtain the clW, if l lmin (1) Vl adjacent segment. The computation time of adjacent segments ,0 otherwise is reduced, because |IIN | |x | . Let z be the value of the 3HL pattern, ai be the number of type-i items included, N be the set of non-negative C. Pattern-Procedure numbers. The following model determines the optimal 3HL Beam search (BS) is used to generate 3HL patterns. BS is pattern, where the objective is to maximize the pattern value incomplete derivative of branch and bound algorithm where (sum of the values of items and leftovers). only elite nodes that have high potential are investigated. It is m used to solve combinatorial optimization problems including z max a v V ; a d,, a N i I (2) i1 i i l i i i two-staged cutting problems [6-7] and circular packing problem [8]. III. ALGORITHM Three basic elements are required for a BS heuristic: node Both the X-pattern and Y-pattern are generated, and the definition, branching strategy and evaluation operator. The better one is chosen as the optimal pattern. Assume that V * , node is defined by a pair of sub-rectangles X Y (( ), )). Where ( ) is the partial pattern V and V are respectively the optimal pattern value, the x W ((L x ) W x W that contains a set of normal segments with sum length equal optimal X-pattern value, and the optimal Y-pattern value, then to x . ((L x ) W ) is the blank part remains to be filled. * Y V =max {V X , V } (3) The root node is ((0WLW ),( )) , where no segment has Items in the 3-staged pattern are cut in three stages, been packed. subsequently the problem of finding the optimal 3HL X-pattern can be divided into three procedures: Branching on a node ((x W ), ((L x ) W )) is strip-procedure, segment-procedure and pattern-procedure. equivalent to packing normal segments into the blank part, which creates a set of child nodes (( ()x q W ), A. Strip-Procedure k ( ()L x q W )), (L x q ) 0 , q Q . Only the top A batch of strips x w , i I { i | l x } are generated k k k i x i potential child nodes are kept for further branching. The for segment x W . Initially the number of type-i items in potential of a node is z VP U() L x , where VP is the strip x w is e min( x / l , d ) , and the strip value is i i i i partial pattern value known as the sum value of the normal v(,) x i ei v i . segments included. U()L x is the upper bound of ()L x W . B. Segment-Procedure U can be obtained by solving the following knapsack Normal segments , are used to improve the x qk W qk Q problem: algorithm, where Q is the set of normal lengths [5] . 777 B. Compares with GGL Patterns (5) U x max{FqWt (k , ) k qtk k xt,,} k NkK k K k K Two instances are used in this subsection, the saw thickness of Instance 1 is zero, and that of Instance 2 is 2. 7 Where t is the frequency of segment ()q W , and k k types of non-oriented items are cut from stock plate of . When is obtained by solving the K{ k | qk x , qk Q } U L dimensions 510 200 .The computational results of 3HL problem using the dynamic programming, all U x patterns are compared with the GGL patterns respectively corresponding to sub-rectangles x W ( x 0,1, , L ) are also from Vassiliadis [9] and Ge [10]. obtained. TABLE II. AREA OF USABLE LEFTOVERS. D. Generation of Cutting Plan ID LA3HL LAVa LAGe The Generation of the cutting plan contains three phases: 1 282000 21000 - (1) Generate the current 3HL pattern using the 2 8000 0 4200 pattern-procedure, (2) update the remaining demands of the All patterns use one plate in two instances. Table 2 shows item types by deleting the items included in the current pattern; the area of usable leftovers, where LAVa denotes the usable (3) Repeat Phases (1) and (2) until all the remaining demands leftover area of Vassiliadis, LAGe denotes that of Ge.
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