Optimizing Area Loss in Flat Glass Cutting

Transcription

1 Optimizing Area Loss in Flat Glass Cutting F. Corno, P. Prinetto, M. Rebaudengo, M. Sonza Reorda Politecnico di Torino Dipartimento di Automatica e Informatica Torino, Italy S. Bisotto Bottero SpA Automazione Vetro Piano Cuneo, Italy Abstract This paper describes GGOAL, a Genetic Algorithm for the minimization of glass loss in cutting large sheets into several pieces. The algorithm takes into account several industrial constraints, stemming both from the glass cutting technology and from the requirement that the optimization must run in real-time, concurrently with the cutting operation. The algorithm delivers comparable or better results that optimization procedures embedded in comprehensive commercial software systems, and is now distributed with all flat glass cutting machines sold by Bottero SpA. 1. Introduction The industry of glass production [HBGM] is composed of two mainly independent branches: one producing hollow glass (bottles, glasses, and other differently shaped products) and another dealing with flat glass (for windows, mirrors, and so on). This paper deals with the flat glass production, where a large glass sheet is first produced, with an area of some tens of square meters, and individual glass pieces of various dimensions are then cut from this sheet. A glass maker usually starts from a set of desired pieces that have been commissioned to him, and a set of sheets to cut from. Loss is created when a sheet can not be exactly covered by the desired pieces. Loss is constituted by small glass pieces, called scraps, that can t be utilized in any way. For economical reasons, glass makers require a given list of pieces to be placed over the smallest number of sheets, and the total glass loss to be minimized. The cutting pattern minimizing the loss must also verify several technological constraints, deriving from how the glass sheet is actually cut by automatic or semi-automatic cutting machines. Such constraints will be better detailed in Section 2, but they essentially require the cutting pattern to be composed of a series of end-to-end cuts, in the horizontal or vertical direction, and that some minimum and maximum distance is observed between neighboring cuts. This is a non trivial optimization problem, for which deterministic or heuristic solutions are very hard to develop. We propose to solve the problem with a Genetic Algorithm, called GGOAL (Genetic Glass Optimization ALgorithm) and we will show that the results of a first implementation of the algorithm are generally better that those delivered even by commercial products. The algorithm described here has been adopted by Bottero SpA, one of the world leader manufacturers of glass productions industrial systems. Commercial software products for loss minimization are usually integrated with accounting, billing, and ordering software, such that the optimization phase is directly done on data coming from the customer orders and taking into account the sheets presents in the store. This requires the optimization to be done on an office PC, separate from the cutting machine (that contains an embedded PC). To better serve small glass makers, that do not require the administrative part of the commercial software products, Bottero SpA decided to include directly on the cutting machine some optimization program that could be used on-the-fly. With such a solution, the PC already included with the cutting machine is powerful enough to enter, edit and optimize cutting patterns. Of course, data editing and optimization must not interrupt the cutting operation, requiring some real-time constraint to be met by the algorithm. Ideally, while the machine cuts a sheet the algorithm should optimize the next one. The realtime properties of Genetic Algorithms, which always deliver some acceptable solution even if prematurely interrupted, is therefore crucial in this application and prevented commercial solution from being applied. This paper presents the GGOAL genetic algorithm and provides some results showing its effectiveness to solve the problem and take into account all the constraints. In particular, Section 2 better details the problem to be solved, by analyzing the constraints coming from the technology and those coming from the real-time implementation. Section 3 describes the GGOAL genetic algorithm and its optimizations, while Section 4 reports about its performance and comparison with other tools.

2 Figure 1: Cutting pattern example Section 5 finally gives some concluding remarks and analyzes future directions. 2. Problem description 2.1. Glass cutting technology In flat glass cutting [HBGM] [MTV], two separate steps are needed to extract a set of smaller pieces from a large sheet: first, a cutting pattern is scored on the glass sheet using a cutting wheel second, pieces are separated by using breakout bars, that hit the glass and create a mechanical stress in the neighborhood of one of the lines that were scored in the previous step. Internal glass tension breaks the glass in correspondence with the scored lines. Due to the critical breakout mechanism, several constraints must be observed when scoring the cutting pattern, the most notable of which is that cuts must always be drawn from end to end of a glass piece. In practice, a cutting pattern is composed of a series of end-to-end vertical cuts (called X-cuts), breaking the sheet in a series of large stripes. Each stripe is then cut end-to-end in the horizontal direction (Y-cuts). This process may be repeated (W- and Z- cuts) in order to obtain the desired piece sizes. Fig. 1 gives an example of such a cutting pattern, where areas in gray represent a glass loss, i.e., some glass that has to be thrown away. The rightmost area of the last sheet in the cutting pattern isn t usually regarded as loss, since it tends to be large enough to be used as a blank sheet for future needs. Due to technological constraints in the score and in the breakout of the sheet, the cutting pattern must also satisfy an additional set of requirements: a minimum distance between adjacent parallel cuts must be observed, in order for breakout bars to operate correctly a maximum distance between adjacent vertical parallel cuts must be observed, since the cutting machines have a vertical size smaller than the horizontal one and the pieces need to be rotated during breakout the extreme borders of the sheet cannot be used due to irregularities: pieces must observe a minimum distance from the sheet boundary some sheets might be damaged in some internal point: no piece should be placed over these points. Due to these constraints, it is quite difficult to develop some heuristic to efficiently devise a cutting pattern which minimizes the glass loss. Very few software companies offer optimization programs for this problem Real-time constraints The reason why Bottero SpA started the development of an in-house glass loss optimizer, is to better serve small and medium glass makers. Third party optimizers, in fact, are often bound to comprehensive software solutions, that include also administrative databases, invoicing, accounting and so on. Such procedures are not always used by smaller glass makers, but contribute to the software costs significantly. They also require a dedicated PC to run the optimizer. Since all cutting machines are already equipped with an embedded PC, that is used to provide the user interface, to give instructions to the hardware controllers of the cutting wheel, the breakout bars, the loading and unloading of the sheet, and so on, it could be used to run some optimization procedure, too. In fact, the CPU is quite unloaded, since it has just to provide coordinates to the intelligent control system of the cutting wheel. By exploiting the existing PC and using its unused CPU periods, a glass maker may optimize the cutting pattern of a glass sheet while the machine is cutting the previous one. Although the CPU is relatively unloaded, some real time constraints are forced on the optimization algorithm by the concurrent cutting activity: the PC should never delay a movement of the cutting wheel, and so increase the total cutting time. This means that, whenever the cutting wheel con-

3 troller needs some input, it must be served immediately, and the optimization must be suspended. This happens with the frequency of some hundreds of milliseconds. the optimized cutting pattern of any sheet must be provided as soon as the previous one is cut. This means that the optimization algorithm must be interruptible at any time, and still provide a good solution. the initial delay needed to optimize the first sheet before starting to cut should be reduced as much as possible. A multithreaded implementation is therefore needed. Due to compatibility constraints with existing software, no real-time operating system is used, and task switching must be explicitly programmed in C code at appropriate times during the optimization. 3. The GGOAL Genetic Algorithm The above described constraints, coming from both the technology and the real time environment, are quite easily met if a Genetic Algorithm (GA) is used [Holl75]. GAs are in fact known to handle quite easily different sets of constraints, and to provide an acceptable solution when they are prematurely stopped. Furthermore, due to the iterative and not recursive nature of the code, they are extremely easy to multithread. With more traditional algorithmic techniques it would have been extremely hard to meet all the constraints. A GA has therefore been developed for devising a good cutting patterns starting from a set of blank sheets and a set of pieces to be cut out of them. The algorithm, called GGOAL (Genetic Glass Optimization ALgorithm) is an hybridized order-based GA whose characteristics are similar to the Traveling Salesman Problem [OSHo87] [GGRV85] [ReSR96] GA implementation Each individual in the population represents a different cutting pattern. The individual is encoded an order for considering the pieces. This order is then used by a greedy placement algorithm that builds a cutting pattern. The goal of the GA is therefore to devise the best permutation of the pieces such that the placement algorithm finds an optimal cutting pattern. The optimization is performed one sheet at a time: a different GA experiment is activated for each new sheet. Each experiment operates on the set of pieces that remain to be cut after the previous sheets have been optimized. The following of this section better details the implementation of the GA. Placement algorithm The placement algorithm is a simple procedure that considers a piece at a time and tries to place a piece in one of the current unused areas (scraps) of the sheet. At the end of the procedure, all scraps inevitably become glass loss. The procedure is greedy in the sense that once a piece is placed, the choice is never re-considered and no backtracks are ever performed. It is the goal of the GA to provide the pieces in such an order that the greedy placement yields a good cutting pattern. To do this, the procedure considers all the scraps in the current sheet, and finds the one that it best contains the piece to be placed. All technological constraints are taken into account in this procedure, and the rest of the GA needs not to care about them. Individual encoding Each individual is a permutation of the pieces to be cut. It is composed of a sequence of genes, one for each piece, describing: the geometrical characteristics of the piece to be considered at that point in the permutation an orientation flag, specifying to the placement procedure that the piece is to be placed rotated by 90 a placement criterion flag which links the piece with the previous one, if possible. Fitness function computation The fitness function straightforwardly takes into account the total area of glass loss. Due to the fact that the last sheet can be partially reused in subsequent optimization, two evaluation functions are actually used: for each sheet but the last one, the evaluation function gives the ratio between the utilized area of the sheet, i.e., that occupied by pieces, and the total sheet area for the last sheet, the evaluation function gives the ratio between the area of the rightmost scrap (that will be re-used) and the total sheet area. A fitness function is derived from the relevant evaluation function via linearization. Genetic Operators Several operators were defined in order to let the population evolve. The adopted crossover operator is a uniform, orderbased one [PRSR93]: a set of genes is extracted from the first parent, and copied to the newly generated individual in the same positions. Remaining genes fill the gaps, and are taken in the order in which they appear in the second parent.

4 Three mutation operators may be applied, with a given probability, to the output of a crossover operation. The first mutation operator merely swaps two genes at random. The remaining two mutation operators modify a single randomly chosen gene, and they flip the rotation flag or the placement criterion flag, respectively Optimizations Starting from the GA described in the previous Section, several optimizations were added to improve both the run time performance and the quality of the solution. This section briefly reports on the most important ones. Several performance improvements were made in the computation of the placement algorithm, to reduce the time spent for finding the best fit scrap in which to insert a piece. First, all scraps smaller than the smallest piece are immediately discarded and need not be considered anymore. To help the algorithm converge faster, the best fit placement algorithm also takes into account the possibility of rotating the piece and prefers a low-order cut (X- or Y-) over higher-order ones. Once a sheet has been successfully optimized, GGOAL tries immediately to replicate this result on as many other sheets as possible: in those cases where there are many pieces of similar sizes if often avoids optimizing several sheets completely. To help escaping local maxima, with a small probability the mutation operator may be applied to any individual in the population, not just on the result of a crossover. Care is taken not to damage the best solution. Finally, some heuristics have been developed to make the search more effective and to faster climb easy regions of the search space: In the initial population some special individuals are always inserted, corresponding to heuristically generated solution. The currently used heuristic is based on placing larger pieces first. The rest of the initial population is selected in a random way. when the best individual does not improve for a given number or generations, an heuristic operator is activated on the best few individuals. This operator tries to rotate each individual piece in the individual until better solutions are found. when the population reaches a steady state, when no improvements are done after a larger number of generations, the GA is restarted from scratch with a newly generated initial population Real-time Implementation Concerning the real time implementation of the GGOAL algorithm, two contrasting requirements arise: the GA must be ready to deliver a good solution in a very short time, not to introduce a delay when a sheet is finished to cut. whenever the GA elaboration can be longer, for instance because many sheets were replicated from a good one, or the glass maker decides to let the optimization run for a long period before starting to cut, the CPU time must be effectively spent to improve the quality of the solution. A GGOAL run must be able to run for hours and avoid being stuck in a local maximum. The two constraints were satisfied by having GGOAL perform several independent runs on the same sheet. After the first run terminates, a solution is available. Successive runs are then activated, with increasing values of the genetic algorithm parameters (population size, number of generations, and so on). Successive runs thus explore different regions of the search space with a higher effort and usually improve the solution. The algorithm never terminates by itself: it just delivers the best solution when it is asked for. From an implementation point of view, during the execution of the GA some high priority tasks must also be executed by the processor. To achieve this, a lightweight multithreading is implemented at the source code level resorting to C-programmed co-routines. 4. Experimental results To experimentally evaluate GGOAL performance, in terms of result quality and CPU time, we selected some real benchmarks and run them on a set of commercial optimizers, as well as on GGOAL. Bench # Piece Types # Pieces Tot Area [m 2 ] # Sheets Table 1: Benchmark characteristics

5 4.1. Benchmarks The characteristics of benchmarks we used are reported in Tab. 1. For each benchmark, we report the number of different piece sized and the total number of pieces. The reported area is the sum of the areas of all the pieces. We also reported the minimum number of sheets that the different optimizers needed. GGOAL was always able to reach this minimum. The benchmarks we selected are regarded as real cases since they represent actual data used by some glass makers. Further, they represent critical data, since they are some cases for which glass makers complained about the optimization efficiency of some optimizer. The total glass area being optimized in all the benchmarks (nearly 750 m 2 ) can well require several days to be cut by a cutting machine Reference tools To provide a fair comparison, we ran the 18 benchmarks on other optimizers besides GGOAL. They are all commercial products, written by specialized software houses in cooperation with Bottero SpA, and are listed in Tab. 2. They are all based on branch-and-bound techniques and make heavy use of various heuristics. All the optimizers are sold by Bottero SpA, bundled with their cutting machines. Product Tool A Tool B Tool C Algorithm Developer SCAI, Torino, Italy Albat+Wirsam Software-Vertriesbs GmbH, Linden, Germany Quality Consult SNC, Sommariva del Bosco, Italy Table 2: Reference tools The comparison that follows should not be regarded as an absolute comparison between the tools: they all have a lot of additional features, that are not taken into account here. Instead, the comparison is made just on the algorithmic point of view, by a mere comparison of the obtained area loss. Furthermore, GGOAL is the only one intended to be run on the cutting machine itself, in real time, instead of on some different PC Obtained results GGOAL has been implemented in the C language and consists of about 4,900 source code lines for the GA and 1,500 for the integration in real time with the already running software. In the following results, it has been run with the parameters shown in Tab. 3. After running the experiments on a PC with a Pentium clocked at 75MHz, the current low-end machine being integrated in the cutting machine, we obtained the results reported in Tab. 4 and graphically displayed, for the sake of ease of interpretation, in Fig. 2. Tab. 4 reports the area loss obtained by the different optimizers. The ratio is computed by dividing the useful glass area of the pieces by the employed sheet area. The rightmost scrap in the last sheet is excluded from the computation. CPU times are not reported, since GGOAL never terminates, but are chosen to satisfy the constraint: CPU time on a sheet < time to cut a sheet. The results show that GGOAL generally performs better than the other optimization algorithms, except when compared with Tool B, where it gives comparable results, and no definite winner can be identified. However, one can infer a trend, where GGOAL tends to be more efficient than Tool B in larger optimization problems. This is particularly evident on benchmark 16, the largest one. Parameter Value Individuals in the population 50 New individuals per generation 50 Generations per run 50 Mutation probability 2% Idle generations before heuristic operator 11 Table 3: GGOAL parameters Area Loss [%] Bench Tool A Tool B Tool C GGOAL Table 4: experimental data These results are rather encouraging, since with a relatively limited effort we were able to reach results

6 highly competitive with commercial solutions, while satisfying tight real time constraints. The GGOAL optimization algorithm is included on every flat glass cutting machine currently being sold by Bottero SpA. 5. Conclusions This paper reports about a Genetic Algorithm based solution to an optimization problem arising in flat glass cutting. The algorithm, called GGOAL, aims at reducing the glass loss while keeping constant the total cutting time. The implementation of the algorithm has been chosen so that it can run in parallel with the actual cutting of a glass sheet by using the same Personal Computer already embedded in the cutting machine. It is therefore possible, at no additional hardware or software cost, to allow even small and medium glass makers, which formerly could not afford the software investment, to benefit from the cutting pattern optimization. Thanks to the versatile nature of Genetic Algorithms, GGOAL has been implemented, optimized and integrated with the run time environment in about 300 working hours. Notwithstanding this limited implementation effort, it is able to deal with all the constraints coming from the cutting technology, it takes into account real time constraints derived from the concurrent cutting activity, and is still able to compete with more complete third-party tools. The application of Genetic Algorithms in this industrial context has been judged as profitable, since with a limited effort one can reach good results with an impaired versatility. The success of the tool is demonstrated that the fact that is has been integrated on the software that ships with every flat glass cutting machine coming from Bottero Spa. 6. References [GGRV85] J. Grefenstette, R. Gopal, B. Rosmaita, D. Van Gucht: Genetic Algorithms for the Traveling Salesman Problem, Proceedings of the First International Conference on Genetic Algorithms, Pittsburgh, PA (USA), 1985, pp [HBGM] The handbook of glass manufacturing, Haslee Publishing, NY USA [Holl75] J.H. Holland, Adaption in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MC (USA), 1975 [MTV] Manuale tecnico del vetro Saint Gobain, FABBRICA PISANA S.p.A. Saint Gobain (in Italian) [OSHo87] J.M. Oliver, D.J. Smith, J.R.C. Holland, A study of Permutation Operators on the Traveling Salesman Problem, Proceedings of the Second International Conference on Genetic Algorithms, Cambridge, MA (USA), 1987, pp [PRSR93] P. Prinetto, M. Rebaudengo, M. Sonza Reorda, Hybrid Genetic Algorithms for the Traveling Salesman Problem, Proceedings of the International Conference on Artificial Neural Networks and Genetic Algorithms, Innsbruck, (A), 1993, pp [ReSR96] M. Rebaudengo, M. Sonza Reorda, GALLO: A Genetic Algorithm for Floorplan Area Optimization, IEEE Transaction on Computer-Aided Design, August 1996, pp

7 30 25 Tool A Tool B Tool C GGOAL 20 Area loss [%] Benchmark Figure 2: results comparison