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A decision support system for luggage typesetting
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Expert Systems with Applications Expert Systems with Applications 35 (2008) 1620–1627 www.elsevier.com/locate/eswa
A decision support system for luggage typesetting Shao-Lun Lee Department of Information Management, Oriental Institute of Technology, Ban-Ciao, Taipei, Taiwan, ROC
Abstract The bin packing problems play an important role in plans of production and saving cost in factories. This paper is to develop a set of intellectual automatic typesetting system (IATS) for luggage factories through bin packing algorithm and genetic algorithm. Firstly, we attain the cost of raw materials by bill of material (BOM) from orders. Secondly, the producing procedure of the luggage has been divided into two parts. The first part: using the one-dimensional typesetting algorithm (ODTA) to solve the problem of fabric cutting. The second part: using the two-dimensional packing algorithm to solve the problem of leather, wood and the plastic plates cutting. Finally, we combine the IATS with mobile phone to offer an effective quick response/efficient consumer response (QR/ECR). Hence, users can look up the minimal cost of raw materials and received the quote rapidly. It is not only more effective than traditional fabric typesetting work but also saves plenty of human resources for luggage factories. 2007 Elsevier Ltd. All rights reserved. Keywords: Bin packing; Genetic algorithms; Typesetting; Bill of material
1. Introduction In traditional factories, two to three senior tailors are often hired to start producing component parts for clients’ demand within two workdays. In order to meet the demand, they need to make a list of BOM and design the model for later producing process. Then, they have to order the raw material from the material factories according to the actual demand. This traditional way is not only time-consuming but also not easy for the senior tailors to control the adequate amount of material. The result of mistakes will often be neglected when the amount of ordered spare parts is a lot and diverse. Moreover, the consequence might cause deficiency of raw material and difficulty of meeting the demand of the ordered goods. Furthermore, the result of man-made mistakes would generate problems, for instance, overestimated quote caused clients to cancel the orders, or underestimated quote caused damage of profits. Bin packing algorithm has been generally applied to the industrial field for decades. Finding an efficient packing of E-mail address: [email protected] 0957-4174/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2007.08.096
rectangular pieces within a given rectangular area has much relevance to operating systems, operations research, and manufacturing industries. It is related to the allocation of memory, CPU scheduling, or other shareable resources to a task for a specified time (Kim & Lee-Kwang, 1997; Kim, Seong, Lee-Kwang, & Kim, 1998; Kunde, Langston, & Liu, 1988; Ullman, 1975). It can also be linked to product manufacturing such as luggage, shoes, ready-made clothes, ship and integrated circuit designing, etc., which need to be made by finding the minimal material area in order to achieve minimal cost. However, bin packing algorithm lacks the concept of multi-layers, which cannot satisfy specific requirements in different industries such as luggage. Hence, this paper focuses on modifying original one-dimensional trim algorithm with fixed width to propose a new algorithm called ‘‘one-dimensional typesetting algorithm’’. The new algorithm can enhance the effectiveness of using fabric through the concept of adding ‘‘multi-layers typesetting’’ when producing massive luggage. Furthermore, we adopt ‘‘two-dimensional bin packing algorithm’’ combines with ‘‘genetic algorithm’’ to solve the cut problem of leather with fixed length and width (Hwang, Kao, & Horng,
S.-L. Lee / Expert Systems with Applications 35 (2008) 1620–1627
Customers’ Order Is it the first time to manufacture?
No
Yes Bill of Material Print BOM Fabric
Leather
One-Dimensional Trim & Genetic Algorithm
Wood
Plastic plates
Two-Dimensional Packing & Genetic Algorithm
Zippers
Accessories
Input
Amount Amount Amount Amount Amount Amount of usage of usage of usage of usage of usage of usage Quote of luggage = (∑ unit cost * amount of usage) * total number of ordered * (1 + profit margin)
Print Cutting Result Print Cutting Result Print Quotation
Inquire Quote From Mobile Phone
Fig. 1. The framework of typesetting and quote system.
1994). According to the method above, we then proposed a new system-IATS. This system can easily and automatically calculate the amount of different required materials for luggage manufacturing by the way of one- and two-dimensional bin packing algorithms. Therefore, users can attain the cost of each row material through BOM that is the unit cost * the amount of usage of each row P material. The formula of quote of luggage is as follows: ( unit cost * amount of usage) * total number of ordered * (1 + profit margin). The quote of the luggage ordered can be used extensively via internet and mobile phone. The framework of typesetting and quote system in this paper is illustrated in Fig. 1. 2. Literature review 2.1. Bin packing problem In business, there are many researchers interested in the issue of how to save material or store goods into the least space (Coffman & Stolyar, 1999; Dyckhoff, 1990; Spillman, 1995; Vanderbeck, 1999). Bin packing problem generates while a set of items pack into a set of available bins. This involves the allocation of each item into a bin in order to minimize the total number of bins being used. It is a NPcomplete problem which consists of a finite set of items
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with different sizes and a set of bins with same capacity (Baker, Brown, & Katseff, 1981). The algorithm can solve the problem of minimizing space of bins. This approach is differently applied according to actual situations in different industries. The bin packing algorithms can be divided into three kinds: one-dimensional (Johnson, Demers, Ullman, Garey, & Graham, 1974), two-dimensional (Fenrich, Miller, & Stout, 1990; Fujita & Hada, 2002; Lodi, Martello, & Vigo, 2002a; Lodi, Martello, & Vigo, 2002b), and three-dimensional (Arthur & Roger, 1992; Chandra & Sanjeev, 1999; Lodi et al., 2002a; Lodi et al., 2002b). The aim of the one-, two- and three-dimensional bin packing algorithms are the same, which are to partition the items into empty bins so that the sum of item sizes in each bin is less than or equal to the capacity of bin (Garey & Johnson, 1979). 2.2. Genetic algorithm Darwin’s theory of evolutionary selection is applied to genetic algorithm. Darwin claimed that ‘‘On the Origin of Species by Means of Natural Selection, or the Preservation of Favored Races in the Struggle for Life’’ (Burrow, 1985). Darwin also claimed that variation within species occurs randomly and that the survival or extinction of each organism is determined by that organism’s ability to adapt to its environment (Burrow, 1985). Organisms compete with each other and to survive or extinct depends on whether to adapt to its environment or not. Survivals the younger generation of species will get superior gene to adapt changeable environment through constantly evolution and will not be eliminated. This theory is extensively applied to computer science such as ‘‘genetic algorithm’’ (Goldberg, 1989). Genetic algorithms were developed by Holland (1975). It has been extensively applied to mathematics, engineering and business. His colleagues at the University of Michigan developed heuristics search algorithms according to biological field that utilizes the concept of natural selection and survival of the fittest. It solves the problem by performing a set of population where each individual is characterized by its chromosome (a series of meaningful words or numbers). Each chromosome consists of a sequence of genes. Basically, the genetic algorithm procedure includes chromosome reproduction, chromosome crossover, gene mutation, chromosome fitness and natural selection. Genetic algorithm has been applied to solve the problems of optimization and heuristics through all disciplines. Applications of genetic algorithms in production include bin packing problem (Chan, Au, & Chan, 2006; Falkenauer, 1996; Falkenauer & Delchambre, 1992), job shop scheduling (Cheung & Zhou, 2001; Kis, 2003), machining process sequencing (Wong, Chan, & Lau, 2003), and single machine scheduling problem (Koksalan & Keha, 2003; Miller, Chen, Matson, & Liu, 1999; Ng, Cheng, Kovalyov, & Lam, 2003).
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3. A new method of solving typesetting problem for luggage manufacturing 3.1. Problem statements The appearance of a luggage is a cube. The whole luggage is made of fabric piece by piece. The original form of the material used for typesetting is one long rectangle fabric with the fixed width and unrestricted length. The amount of fabric used to make a luggage is determined by the amount of using yard in material. Namely, the way of trimming fabric will affect the usage of material. Therefore, inappropriate ways of cutting will increase the amount of using yard of material, which means the surplus of fabric will also be raised to cause unnecessary waste. Most of traditional luggage factories nowadays still engage in the way of typesetting by man-made. A stack of fabric is cut at a time by cutting machine according to its fixed typesetting size. However, incorrect typesetting by man-made will cause the invisible loss. In order to enhance the efficiency of using the raw material, one-dimensional trim algorithm was proposed by Haessler (1975), which is a very popular solution. The calculation of typesetting as Haessler’s method was usually correct, however, the procedure of machinery trimming is not finished by single piece cutting. The way of cutting should be replaced by cutting a stack of fabric at a time in order to save time and cost that could meet the requirements for luggage factories. Hence, we proposed ODTA as below (referring to Section 3.2), which is to modify the original one-dimensional trim algorithm in order to cope with the problems of ‘‘multi-layers typesetting’’. Moreover, we utilized Genetic Algorithm to solve optimal typesetting problems for luggage manufacturing.
W
X
Aij W i 6 W ;
Aij P 0;
Integer
Secondly, we prearrange the object according to the width of this object in descending order for searching better usage of pattern. After prearranging the object, we start searching the best combination of the object, which is time-consuming and complex. Therefore, we used the Heuristic algorithm to cope with the depth of searching problem. Hence, we try to prearrange the width of this object by best fit descending algorithm (Chandra & Sanjeev, 1999). The process is divided into two parts: first of all, stuff the first appearing object as many as possible according to the width of the object into the fixed width of pattern. But if it cannot be stuffed more object, we need to find space that is left to fill with the next object. Similarly, follow the same principal described above, we stuff the next object step by step and test whether the object can be added as more as possible into pattern until it cannot be filled anymore. The flowchart of the ‘‘ODTA’’ is illustrated in Fig. 2. Secondly, we need to expand the longest object among the patterns horizontally. The rest will also need to be completed until we can find utility rate, which is fitted into the requirement. The quality and the result are usually better after expanding. However, the range of expanding has to be limited due to the size of cutting machine may be shorter than the expanding size. 3.3. Enhance the utility rate by applying genetic algorithm In order to enhance the utility rate of typesetting, we have to stuff object as many as possible into pattern. Because of the result of pattern search can decide whether the outcome of typesetting is good or not, the pattern searching problem above is about NP-Hard problem, which we adopt the genetic algorithm to solve it in this paper.
3.2. One-dimensional typesetting algorithm (ODTA) Firstly, we redefine the one-dimensional trim problem with new condition that can be formulated as X Min T kX k
Begin Calculate the amount of requirement
k
s:t:
X
T k X k ¼ R;
Xk P 0
ð1Þ
Search the feasible production pattern
k
where R is a vector of order requirements quantity Rk. Ak is a cutting pattern vector with elements Aijk, i = 1, . . . , n, j = 1, . . . , n where Aik is the number of sets of width Wi to be obtained from a production set, and Ajk is the number of sets of maximum length Lk to be obtained from a production set. Xk is the number of production sets to be processed according to pattern k. Tk is the number of area of trim loss incurred by pattern k.PIf W is the maximum usable width, then T k ¼ W Lk k Aik W i Lk . In order for Aij to be a feasible cutting pattern, the objective function must be satisfied
ð2Þ
Does it meet the No requirement? Yes Calculate the production set numbers of pattern Calculate the numbers of surplus requirement No
Does it produce completely? Yes End
Fig. 2. The flowchart of the ODTA.
S.-L. Lee / Expert Systems with Applications 35 (2008) 1620–1627
3.3.1. Define chromosome and gene The only difficult part is the definition for the chromosome and gene. Here we assume the cutting pattern as the chromosome in the Genetic Algorithm and the amount of stuffed objects as the gene in the chromosome, if the value of gene is zero, it represents the object is not stuffed. In order to reduce the searching time, the model of the chromosome has to be limited because of the width of fabric is fixed. Because of the number of fabric cutting for luggage will be less than 100 patterns. In order to enhance the efficiency of genetic algorithm, we set the number of population as 100 chromosomes. 3.3.2. Define fitness function After defining the chromosome model, we now set the fitness function. In this paper, the fitness function is decided by the utility rate of the material as !, n X ffitness ¼ Oiwidth Oilength ðP length WidthÞ ð3Þ i¼1
where Oiwidth and Oilength are width and length of objects. Plength is length of pattern. Width is width of fabric. The value of the fitness function will exist between 0 and 1. The utility rate will be higher if the value of fitness function is greater. However, if the summation of the stuffed object width is more than the total sum from fabric width, we will set the value as 0 that is infeasible solution. Furthermore, the value of fitness function will determine whether the survival of chromosome is able to pass to next generation or not. In this paper, we set the stop condition of search as 0.95, which means that the value of the fitness function has to be greater than 0.95 or the iteration is greater than 30,000 times, and then the genetic algorithm will stop. 3.3.3. Initial value setting In order to meet all the possible solutions in following operation of reproduction and mutation, we set the initial value can only arrange one object in one chromosome and the amount of object is one. The concept is different from that we set a secondary best solution as the initial value in the usual genetic algorithm used before. 3.3.4. The operation of reproduction and crossover The operation of reproduction is to maintain the characteristics from the last generation. New generation compete with last generation at the same time so that we do not need to undertake reproduction. Crossover is to keep the characteristics from both father and mothers’ chromosome through integrating gene. In this paper, we set the probability of crossover as 0.7. If the result of crossover is not good, the chromosome will be eliminated in next generation. 3.3.5. The operation of mutation Gene mutation can be divided into two categories in this paper. The first category is to mutate the amount of
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objects. The operation of mutation is to select one object randomly and change its amount that has to be limited in the fabric width. Moreover, we usually decrease the amount of the objects whose ratio should be greater than the ratio of increasing the amount of objects in order to reduce the probability of infeasible solution. The second category is to mutate the type of objects, which need to maintain the characteristics of the amount of the objects and replace them to other objects if necessary. We set the probability of mutation as 0.3, which is more than 0.05 in general case. That is because the result of experiment showed that the greater probability of mutation would get better typesetting result. 3.3.6. The operation of natural selection As far as we are mentioned, in genetic algorithm, if the utility rate of chromosome is not good, the chromosome will be eliminated in next generation. In typesetting problem, it will appear the same problems. For example, we can attain a better typesetting result by stuffing one object into chromosome. However, if the utility rate of pattern of the stuffed object is very low, the chromosome would be very difficult to survive in population. The chromosome cannot mate with the other chromosome as well. In order to solve this problem, we gather all the lower utility rate of chromosome to be another group that does not need to compete with population. However, the group has to mate with all members of chromosome in population. The amount of group is about 5% in population in this paper. 3.4. Case study Next, we are about to introduce an actual example of order from a customer in a luggage factory. In this case study, the customer ordered 1500 luggages. We expand the requirements of fabric according to BOM and the amount of order. Table 1 shows the requirement of fabric from the customer’s order. Table 2 shows the typesetting result through ODTA. From Table 2, we know that the utility rate of fabric is Table 1 The requirement of fabric from customer’s order Location
Width
Length
Quantity
Requirement
1 2 3 4 5 6 7 8 9
11.5 1.625 5.625 2.25 6 17 2.125 7.125 3.5
9 16.75 14.5 12.375 14.5 14 21.25 14.5 14.5
2 2 1 1 1 1 2 1 1
3000 3000 1500 1500 1500 1500 3000 1500 1500
Luggage No.: A00001; specifications: 29 · 21 · 8.5; PCS: 1500; material: 1200 D Polyester; width: 58 in.; tolerance (width): 0.25; and tolerance (length): 0.25.
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Location
Pieces (H)
Pieces (V)
Total
Layers
Production
Pattern 1 5 6
1 1
1 3
1 3
500 500
500 1500
Pattern 2 8 4 1
2 2 3
1 1 4
2 2 12
250 250 250
500 500 3000
Pattern 3 9 3
1 1
1 9
1 9
167 167
167 1503
Pattern 4 5 8
1 1
1 7
1 7
143 143
143 1001
order to avoid occurring seam caused flaw in products. In those materials, the leather is limited in length and width and its direction is not fixed when typesetting, which are different from fabric. The price of leather is more expensive while the size is larger, in order to save the cost of material, therefore, the cut of leather in luggage factories need to typeset with minimal area. According to the description above, we can define the problem of minimization of two-dimensional pin packing as follows: the entire rectangular objects can be stuffed into the minimal two-dimensional space under the conditions that the objects can be rotated 90 but do not overlap. In this paper, we adopted the method by some researchers such as Hwang et al. (1994) proposed to solve the problem of minimization of two-dimensional pin packing.
Pattern 5 7
1
24
24
125
3000
4. The implementation results
Pattern 6 2
1
30
30
100
3000
Pattern 7 5
1
9
9
96
864
Pattern 8 9
1
15
15
89
1335
Pattern 9 4
1
23
23
44
1012
In order to solve the typesetting problem for luggage factories, we developed a set of IATS that was carried out according to the environment as followings: Implementation platform (CPU: Pentium IV 2.4 GHz; Memory: 1 GB RAM, 160 G bytes HD; Operating system: Windows XP; Database: SQL Server 2005); Developing language (Visual Basic.NET; ASP.NET; Borland C++ Builder 4.0); Implementation data source (the data of order for luggage is acquired from a famous luggage factory in Taiwan).
Table 2 The typesetting result through ODTA
Customer No.: JCP; material: 1200 D Polyester; width: 58 in.; usage: 0.96147152; yard: 749.840277; unit yard: 0.49989351; total patterns: 9; tolerance (width): 0.25; and tolerance (length): 0.25.
4.1. The function of IATS
Fig. 3. The illustration of typesetting.
96.15%. The amount of using yard is 749.84, which is the length of fabric for quantity of luggage ordered that we need to purchase. The amount of unit yard used is 0.499, which represents the needed fabric for each luggage. According to the data, we can decide the price for each luggage. Fig. 3 presents the illustration of typesetting according to Table 2. In the bottom of the Fig. 3, the numbers show the concept of ‘‘multi-layers typesetting’’, for example, ‘‘500 [1]’’ means that pattern 1 needs to be cut 500 pieces at a time, ‘‘250 [2]’’ means that pattern 2 needs to be cut 250 pieces at a time and so on.
The executive frame as we started IATS is illustrated in Fig. 4. In Fig. 4, when we receive the orders from customers. First of all, we need to build up the quotation according to the customers’ requirements by pressing the button [Add] and inputting the basic information such as quotation ID, customers’ name, name of product and specification. Secondly, we need to build up the BOM table for luggage manufacturing by pressing the button [data maintenances (M)] in Fig. 4. The executive frame is illustrated
3.5. Two-dimensional typesetting problem In IATS, the material such as leather, wood and plastic plates etc. have to be cut in a same piece of material in
Fig. 4. The executive frame of IATS.
S.-L. Lee / Expert Systems with Applications 35 (2008) 1620–1627
in Fig. 5. In Fig. 5, we need to input the components of material for manufacturing luggage such as type, name, specification and unit cost. Thirdly, we need to press button [calculation] in Fig. 4. The executive frame is illustrated in Fig. 6. In Fig. 6, we can choose [batch calculation] that when customers ordered
many different types of luggage or [single calculation] that when customers assigned any kind of luggage from the order. Next, IATS can provide three different kinds of report. The first one is the frame of one-dimensional typesetting in Fig. 7. The second one is the frame of BOM in Fig. 8. The last one is the frame of quotation in Fig. 9. Finally, in Fig. 10, the quote of any order can be inquired immediately through mobile phone.
Fig. 5. The executive frame of BOM.
Fig. 8. The frame of BOM.
Fig. 6. The executing frame of IATS.
Fig. 9. The frame of quotation.
Fig. 7. The frame of one-dimensional typesetting.
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Fig. 10. The frame for inquiring quote by mobile phone.
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S.-L. Lee / Expert Systems with Applications 35 (2008) 1620–1627
4.2. The comparison of the utility rate between IATS and manual-typesetting In the process of manufacturing one luggage, manualtypesetting work will take one day to produce paperboard model, and another two days to run optimal solution. However, IATS only takes a few minutes to key in data and then attain the result within few seconds. It saves not only time in producing paperboard model but also human resources. In the meantime, it also saves the cost of materials for making luggage. In Table 3, we compared the utility rate of one-dimensional IATS with manual-typesetting. The workers in luggage factory mentioned that the utility rate of fabric typesetting by man-made was approximately 90% in the past. The overall average utility rate of fabric typesetting was 95.32% in IATS. In other words, IATS raised over 5% utility rate of fabric typesetting as compared with manual-typesetting. If we calculate from the annual 40 million in purchasing material cost of the factory, IATS can save circa 2 million in purchasing cost annually. According to Table 3, we generated a chart in Fig. 11 to present the unit yard used between one-dimensional IATS and manual-typesetting. It is obviously to show that the Table 3 The comparison of the utility rate between one-dimensional IATS and manual-typesetting Number of piece
Number of luggage
Width of fabric (in.)
Man-made calculation (yard)
IATS calculation (yard)
Utility rate of fabric by IATS (%)
14 13 14 13 14 13 9 10 7 16 14 13 11 11
2734 2734 2735 2735 2736 2736 2622 2622 2645 2617 2532 2532 3862 3862
58 60 58 60 58 60 58 60 58 58 58 60 58 60
1.738 3.663 1.552 3.2495 1.2961 2.6124 0.5973 0.8482 0.2828 1.4796 1.4775 2.8 2.15 2.0743
1.508 3.489 1.347 3.157 1.0731 2.6124 0.503 0.829 0.203 1.24 1.206 2.61 1.9363 1.9031
95.00 93.20 96.70 93.00 96.30 95.00 95.49 94.47 92.50 97.10 96.90 98.00 96.10 94.50
Average utility rate of fabric by IATS
95.32
4 3.5 3 2.5 2 1.5 1 0.5 0 2734 2734 2735 2735 2736 2736 2622 2622 2645 2617 2532 2532 3862 3862 2903 2903 Unit yard used by manual-typesetting
Unit yard used by IATS
Fig. 11. The chart of unit yard used by one-dimensional IATS and manual-typesetting.
Fig. 12. The frame of two-dimensional typesetting.
unit yard used by IATS is higher than man-made among 16 different types of luggage. About two-dimensional IATS, the factory only provided the data of manual-typesetting. Hence, we could only show the result by IATS. Fig. 12 presents the utility rate in IATS of leather is 90%, the average utility rate in manual-typesetting is 75%. It is obviously to see that the utility rate in IATS is higher than in manual-typesetting. 5. Conclusion How to increase the effectiveness of typesetting is an important issue in luggage factories nowadays. In order to enhance the competitive advantage now and in the future, this paper develops a typesetting system – IATS, which possesses some features such as ‘‘saving cost’’, ‘‘accurate calculation of typesetting’’, ‘‘feasible cut’’ and ‘‘speedy quote’’ for luggage factories. Users can attain quote rapidly through mobile phone to raise the capability of receiving orders enormously. For the future research, the IATS can be considered to apply to irregular cut such as ready-made clothes industry. References Arthur, L. C., & Roger L. W. (1992). A genetic algorithm for packing in three dimensions. In Proceedings of the 1992 ACM/SIGAPP symposium on Applied computing: Vol. II. Technological challenges of the 1990s (pp. 1021–1030). Baker, B. S., Brown, D. J., & Katseff, H. P. (1981). A 5/4 algorithm for two-dimensional packing. Journal of Algorithms, 2, 348–368. Burrow, J. W. (1985). On the origin of species. New York: Penguin Books. Chan, F. T. S., Au, K. C., & Chan, P. L. Y. (2006). A decision support system for production scheduling in an ion plating cell. Expert Systems with Applications, 30, 727–738. Chandra, C., & Sanjeev, K. (1999). On multi-dimensional packing problems. In Proceedings of the tenth annual ACM-SIAM symposium on discrete algorithms (pp. 185–194). Cheung, W., & Zhou, H. (2001). Using genetic algorithms and heuristics for job shop scheduling with sequence-dependent setup times. Annals of Operations Research, 107, 65–81. Coffman, E. G., & Stolyar, A. L. (1999). Fluid limits, bin packing, and stochastic analysis of algorithms. In Proceedings of the tenth annual ACM-SIAM symposium on discrete algorithms (pp. 877–878). Dyckhoff, H. (1990). A typology of cutting and packing problems. European Journal of Operational Research, 44, 145–159. Falkenauer, E. (1996). A hybrid grouping genetic algorithms for bin packing. Journal of Heuristics, 2, 5–30. Falkenauer, E., & Delchambre, A. (1992). A genetic algorithm for bin packing and line balancing. In Proceedings of the IEEE 1992
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Koksalan, M., & Keha, A. B. (2003). Using genetic algorithms for single machine bicriteria scheduling problems. European Journal of Operational Research, 145, 543–556. Kunde, M., Langston, M. A., & Liu, J.-M. (1988). On a special case of uniform processor scheduling. Journal of Algorithm, 9, 287–296. Lodi, A., Martello, S., & Vigo, D. (2002a). Recent advances on twodimensional bin packing problems. Discrete Applied Mathematics, 123, 379–396. Lodi, A., Martello, S., & Vigo, D. (2002b). Heuristic algorithms for the three-dimensional bin packing problem. European Journal of Operational Research, 141, 410–420. Miller, D. M., Chen, H., Matson, J., & Liu, Q. (1999). A hybrid genetic algorithm for the single machine scheduling problem. Journal of Heuristics, 5, 437–454. Ng, C. T., Cheng, T. C., Kovalyov, M. Y., & Lam, S. S. (2003). Single machine scheduling with a variable common due date and resource dependent processing times. Computers & Operations Research, 30, 1173–1185. Spillman, R. (1995). Solving large knapsack problems with a genetic algorithm. In IEEE international conference R. Systems, man and cybernetics. Intelligent systems for the 21st century (Vol. 1, pp. 632 – 637). Ullman, J. D. (1975). Complexity of sequencing problems. In E. G. Coffman (Ed.), Computer and job-shop scheduling theory. New York: Wiley. Vanderbeck, F. (1999). Computational study of a column generation algorithm for bin packing and cutting stock problems. Francois Mathematical Programming, 86(3), 565–594. Wong, T. N., Chan, L. C. F., & Lau, H. C. W. (2003). Machining process sequencing with fuzzy expert system and genetic algorithms. Engineering with Computers (London), 19(2–3), 191–202.
Expert Systems with Applications Expert Systems with Applications 35 (2008) 1620–1627 www.elsevier.com/locate/eswa
A decision support system for luggage typesetting Shao-Lun Lee Department of Information Management, Oriental Institute of Technology, Ban-Ciao, Taipei, Taiwan, ROC
Abstract The bin packing problems play an important role in plans of production and saving cost in factories. This paper is to develop a set of intellectual automatic typesetting system (IATS) for luggage factories through bin packing algorithm and genetic algorithm. Firstly, we attain the cost of raw materials by bill of material (BOM) from orders. Secondly, the producing procedure of the luggage has been divided into two parts. The first part: using the one-dimensional typesetting algorithm (ODTA) to solve the problem of fabric cutting. The second part: using the two-dimensional packing algorithm to solve the problem of leather, wood and the plastic plates cutting. Finally, we combine the IATS with mobile phone to offer an effective quick response/efficient consumer response (QR/ECR). Hence, users can look up the minimal cost of raw materials and received the quote rapidly. It is not only more effective than traditional fabric typesetting work but also saves plenty of human resources for luggage factories. 2007 Elsevier Ltd. All rights reserved. Keywords: Bin packing; Genetic algorithms; Typesetting; Bill of material
1. Introduction In traditional factories, two to three senior tailors are often hired to start producing component parts for clients’ demand within two workdays. In order to meet the demand, they need to make a list of BOM and design the model for later producing process. Then, they have to order the raw material from the material factories according to the actual demand. This traditional way is not only time-consuming but also not easy for the senior tailors to control the adequate amount of material. The result of mistakes will often be neglected when the amount of ordered spare parts is a lot and diverse. Moreover, the consequence might cause deficiency of raw material and difficulty of meeting the demand of the ordered goods. Furthermore, the result of man-made mistakes would generate problems, for instance, overestimated quote caused clients to cancel the orders, or underestimated quote caused damage of profits. Bin packing algorithm has been generally applied to the industrial field for decades. Finding an efficient packing of E-mail address: [email protected] 0957-4174/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2007.08.096
rectangular pieces within a given rectangular area has much relevance to operating systems, operations research, and manufacturing industries. It is related to the allocation of memory, CPU scheduling, or other shareable resources to a task for a specified time (Kim & Lee-Kwang, 1997; Kim, Seong, Lee-Kwang, & Kim, 1998; Kunde, Langston, & Liu, 1988; Ullman, 1975). It can also be linked to product manufacturing such as luggage, shoes, ready-made clothes, ship and integrated circuit designing, etc., which need to be made by finding the minimal material area in order to achieve minimal cost. However, bin packing algorithm lacks the concept of multi-layers, which cannot satisfy specific requirements in different industries such as luggage. Hence, this paper focuses on modifying original one-dimensional trim algorithm with fixed width to propose a new algorithm called ‘‘one-dimensional typesetting algorithm’’. The new algorithm can enhance the effectiveness of using fabric through the concept of adding ‘‘multi-layers typesetting’’ when producing massive luggage. Furthermore, we adopt ‘‘two-dimensional bin packing algorithm’’ combines with ‘‘genetic algorithm’’ to solve the cut problem of leather with fixed length and width (Hwang, Kao, & Horng,
S.-L. Lee / Expert Systems with Applications 35 (2008) 1620–1627
Customers’ Order Is it the first time to manufacture?
No
Yes Bill of Material Print BOM Fabric
Leather
One-Dimensional Trim & Genetic Algorithm
Wood
Plastic plates
Two-Dimensional Packing & Genetic Algorithm
Zippers
Accessories
Input
Amount Amount Amount Amount Amount Amount of usage of usage of usage of usage of usage of usage Quote of luggage = (∑ unit cost * amount of usage) * total number of ordered * (1 + profit margin)
Print Cutting Result Print Cutting Result Print Quotation
Inquire Quote From Mobile Phone
Fig. 1. The framework of typesetting and quote system.
1994). According to the method above, we then proposed a new system-IATS. This system can easily and automatically calculate the amount of different required materials for luggage manufacturing by the way of one- and two-dimensional bin packing algorithms. Therefore, users can attain the cost of each row material through BOM that is the unit cost * the amount of usage of each row P material. The formula of quote of luggage is as follows: ( unit cost * amount of usage) * total number of ordered * (1 + profit margin). The quote of the luggage ordered can be used extensively via internet and mobile phone. The framework of typesetting and quote system in this paper is illustrated in Fig. 1. 2. Literature review 2.1. Bin packing problem In business, there are many researchers interested in the issue of how to save material or store goods into the least space (Coffman & Stolyar, 1999; Dyckhoff, 1990; Spillman, 1995; Vanderbeck, 1999). Bin packing problem generates while a set of items pack into a set of available bins. This involves the allocation of each item into a bin in order to minimize the total number of bins being used. It is a NPcomplete problem which consists of a finite set of items
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with different sizes and a set of bins with same capacity (Baker, Brown, & Katseff, 1981). The algorithm can solve the problem of minimizing space of bins. This approach is differently applied according to actual situations in different industries. The bin packing algorithms can be divided into three kinds: one-dimensional (Johnson, Demers, Ullman, Garey, & Graham, 1974), two-dimensional (Fenrich, Miller, & Stout, 1990; Fujita & Hada, 2002; Lodi, Martello, & Vigo, 2002a; Lodi, Martello, & Vigo, 2002b), and three-dimensional (Arthur & Roger, 1992; Chandra & Sanjeev, 1999; Lodi et al., 2002a; Lodi et al., 2002b). The aim of the one-, two- and three-dimensional bin packing algorithms are the same, which are to partition the items into empty bins so that the sum of item sizes in each bin is less than or equal to the capacity of bin (Garey & Johnson, 1979). 2.2. Genetic algorithm Darwin’s theory of evolutionary selection is applied to genetic algorithm. Darwin claimed that ‘‘On the Origin of Species by Means of Natural Selection, or the Preservation of Favored Races in the Struggle for Life’’ (Burrow, 1985). Darwin also claimed that variation within species occurs randomly and that the survival or extinction of each organism is determined by that organism’s ability to adapt to its environment (Burrow, 1985). Organisms compete with each other and to survive or extinct depends on whether to adapt to its environment or not. Survivals the younger generation of species will get superior gene to adapt changeable environment through constantly evolution and will not be eliminated. This theory is extensively applied to computer science such as ‘‘genetic algorithm’’ (Goldberg, 1989). Genetic algorithms were developed by Holland (1975). It has been extensively applied to mathematics, engineering and business. His colleagues at the University of Michigan developed heuristics search algorithms according to biological field that utilizes the concept of natural selection and survival of the fittest. It solves the problem by performing a set of population where each individual is characterized by its chromosome (a series of meaningful words or numbers). Each chromosome consists of a sequence of genes. Basically, the genetic algorithm procedure includes chromosome reproduction, chromosome crossover, gene mutation, chromosome fitness and natural selection. Genetic algorithm has been applied to solve the problems of optimization and heuristics through all disciplines. Applications of genetic algorithms in production include bin packing problem (Chan, Au, & Chan, 2006; Falkenauer, 1996; Falkenauer & Delchambre, 1992), job shop scheduling (Cheung & Zhou, 2001; Kis, 2003), machining process sequencing (Wong, Chan, & Lau, 2003), and single machine scheduling problem (Koksalan & Keha, 2003; Miller, Chen, Matson, & Liu, 1999; Ng, Cheng, Kovalyov, & Lam, 2003).
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3. A new method of solving typesetting problem for luggage manufacturing 3.1. Problem statements The appearance of a luggage is a cube. The whole luggage is made of fabric piece by piece. The original form of the material used for typesetting is one long rectangle fabric with the fixed width and unrestricted length. The amount of fabric used to make a luggage is determined by the amount of using yard in material. Namely, the way of trimming fabric will affect the usage of material. Therefore, inappropriate ways of cutting will increase the amount of using yard of material, which means the surplus of fabric will also be raised to cause unnecessary waste. Most of traditional luggage factories nowadays still engage in the way of typesetting by man-made. A stack of fabric is cut at a time by cutting machine according to its fixed typesetting size. However, incorrect typesetting by man-made will cause the invisible loss. In order to enhance the efficiency of using the raw material, one-dimensional trim algorithm was proposed by Haessler (1975), which is a very popular solution. The calculation of typesetting as Haessler’s method was usually correct, however, the procedure of machinery trimming is not finished by single piece cutting. The way of cutting should be replaced by cutting a stack of fabric at a time in order to save time and cost that could meet the requirements for luggage factories. Hence, we proposed ODTA as below (referring to Section 3.2), which is to modify the original one-dimensional trim algorithm in order to cope with the problems of ‘‘multi-layers typesetting’’. Moreover, we utilized Genetic Algorithm to solve optimal typesetting problems for luggage manufacturing.
W
X
Aij W i 6 W ;
Aij P 0;
Integer
Secondly, we prearrange the object according to the width of this object in descending order for searching better usage of pattern. After prearranging the object, we start searching the best combination of the object, which is time-consuming and complex. Therefore, we used the Heuristic algorithm to cope with the depth of searching problem. Hence, we try to prearrange the width of this object by best fit descending algorithm (Chandra & Sanjeev, 1999). The process is divided into two parts: first of all, stuff the first appearing object as many as possible according to the width of the object into the fixed width of pattern. But if it cannot be stuffed more object, we need to find space that is left to fill with the next object. Similarly, follow the same principal described above, we stuff the next object step by step and test whether the object can be added as more as possible into pattern until it cannot be filled anymore. The flowchart of the ‘‘ODTA’’ is illustrated in Fig. 2. Secondly, we need to expand the longest object among the patterns horizontally. The rest will also need to be completed until we can find utility rate, which is fitted into the requirement. The quality and the result are usually better after expanding. However, the range of expanding has to be limited due to the size of cutting machine may be shorter than the expanding size. 3.3. Enhance the utility rate by applying genetic algorithm In order to enhance the utility rate of typesetting, we have to stuff object as many as possible into pattern. Because of the result of pattern search can decide whether the outcome of typesetting is good or not, the pattern searching problem above is about NP-Hard problem, which we adopt the genetic algorithm to solve it in this paper.
3.2. One-dimensional typesetting algorithm (ODTA) Firstly, we redefine the one-dimensional trim problem with new condition that can be formulated as X Min T kX k
Begin Calculate the amount of requirement
k
s:t:
X
T k X k ¼ R;
Xk P 0
ð1Þ
Search the feasible production pattern
k
where R is a vector of order requirements quantity Rk. Ak is a cutting pattern vector with elements Aijk, i = 1, . . . , n, j = 1, . . . , n where Aik is the number of sets of width Wi to be obtained from a production set, and Ajk is the number of sets of maximum length Lk to be obtained from a production set. Xk is the number of production sets to be processed according to pattern k. Tk is the number of area of trim loss incurred by pattern k.PIf W is the maximum usable width, then T k ¼ W Lk k Aik W i Lk . In order for Aij to be a feasible cutting pattern, the objective function must be satisfied
ð2Þ
Does it meet the No requirement? Yes Calculate the production set numbers of pattern Calculate the numbers of surplus requirement No
Does it produce completely? Yes End
Fig. 2. The flowchart of the ODTA.
S.-L. Lee / Expert Systems with Applications 35 (2008) 1620–1627
3.3.1. Define chromosome and gene The only difficult part is the definition for the chromosome and gene. Here we assume the cutting pattern as the chromosome in the Genetic Algorithm and the amount of stuffed objects as the gene in the chromosome, if the value of gene is zero, it represents the object is not stuffed. In order to reduce the searching time, the model of the chromosome has to be limited because of the width of fabric is fixed. Because of the number of fabric cutting for luggage will be less than 100 patterns. In order to enhance the efficiency of genetic algorithm, we set the number of population as 100 chromosomes. 3.3.2. Define fitness function After defining the chromosome model, we now set the fitness function. In this paper, the fitness function is decided by the utility rate of the material as !, n X ffitness ¼ Oiwidth Oilength ðP length WidthÞ ð3Þ i¼1
where Oiwidth and Oilength are width and length of objects. Plength is length of pattern. Width is width of fabric. The value of the fitness function will exist between 0 and 1. The utility rate will be higher if the value of fitness function is greater. However, if the summation of the stuffed object width is more than the total sum from fabric width, we will set the value as 0 that is infeasible solution. Furthermore, the value of fitness function will determine whether the survival of chromosome is able to pass to next generation or not. In this paper, we set the stop condition of search as 0.95, which means that the value of the fitness function has to be greater than 0.95 or the iteration is greater than 30,000 times, and then the genetic algorithm will stop. 3.3.3. Initial value setting In order to meet all the possible solutions in following operation of reproduction and mutation, we set the initial value can only arrange one object in one chromosome and the amount of object is one. The concept is different from that we set a secondary best solution as the initial value in the usual genetic algorithm used before. 3.3.4. The operation of reproduction and crossover The operation of reproduction is to maintain the characteristics from the last generation. New generation compete with last generation at the same time so that we do not need to undertake reproduction. Crossover is to keep the characteristics from both father and mothers’ chromosome through integrating gene. In this paper, we set the probability of crossover as 0.7. If the result of crossover is not good, the chromosome will be eliminated in next generation. 3.3.5. The operation of mutation Gene mutation can be divided into two categories in this paper. The first category is to mutate the amount of
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objects. The operation of mutation is to select one object randomly and change its amount that has to be limited in the fabric width. Moreover, we usually decrease the amount of the objects whose ratio should be greater than the ratio of increasing the amount of objects in order to reduce the probability of infeasible solution. The second category is to mutate the type of objects, which need to maintain the characteristics of the amount of the objects and replace them to other objects if necessary. We set the probability of mutation as 0.3, which is more than 0.05 in general case. That is because the result of experiment showed that the greater probability of mutation would get better typesetting result. 3.3.6. The operation of natural selection As far as we are mentioned, in genetic algorithm, if the utility rate of chromosome is not good, the chromosome will be eliminated in next generation. In typesetting problem, it will appear the same problems. For example, we can attain a better typesetting result by stuffing one object into chromosome. However, if the utility rate of pattern of the stuffed object is very low, the chromosome would be very difficult to survive in population. The chromosome cannot mate with the other chromosome as well. In order to solve this problem, we gather all the lower utility rate of chromosome to be another group that does not need to compete with population. However, the group has to mate with all members of chromosome in population. The amount of group is about 5% in population in this paper. 3.4. Case study Next, we are about to introduce an actual example of order from a customer in a luggage factory. In this case study, the customer ordered 1500 luggages. We expand the requirements of fabric according to BOM and the amount of order. Table 1 shows the requirement of fabric from the customer’s order. Table 2 shows the typesetting result through ODTA. From Table 2, we know that the utility rate of fabric is Table 1 The requirement of fabric from customer’s order Location
Width
Length
Quantity
Requirement
1 2 3 4 5 6 7 8 9
11.5 1.625 5.625 2.25 6 17 2.125 7.125 3.5
9 16.75 14.5 12.375 14.5 14 21.25 14.5 14.5
2 2 1 1 1 1 2 1 1
3000 3000 1500 1500 1500 1500 3000 1500 1500
Luggage No.: A00001; specifications: 29 · 21 · 8.5; PCS: 1500; material: 1200 D Polyester; width: 58 in.; tolerance (width): 0.25; and tolerance (length): 0.25.
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Location
Pieces (H)
Pieces (V)
Total
Layers
Production
Pattern 1 5 6
1 1
1 3
1 3
500 500
500 1500
Pattern 2 8 4 1
2 2 3
1 1 4
2 2 12
250 250 250
500 500 3000
Pattern 3 9 3
1 1
1 9
1 9
167 167
167 1503
Pattern 4 5 8
1 1
1 7
1 7
143 143
143 1001
order to avoid occurring seam caused flaw in products. In those materials, the leather is limited in length and width and its direction is not fixed when typesetting, which are different from fabric. The price of leather is more expensive while the size is larger, in order to save the cost of material, therefore, the cut of leather in luggage factories need to typeset with minimal area. According to the description above, we can define the problem of minimization of two-dimensional pin packing as follows: the entire rectangular objects can be stuffed into the minimal two-dimensional space under the conditions that the objects can be rotated 90 but do not overlap. In this paper, we adopted the method by some researchers such as Hwang et al. (1994) proposed to solve the problem of minimization of two-dimensional pin packing.
Pattern 5 7
1
24
24
125
3000
4. The implementation results
Pattern 6 2
1
30
30
100
3000
Pattern 7 5
1
9
9
96
864
Pattern 8 9
1
15
15
89
1335
Pattern 9 4
1
23
23
44
1012
In order to solve the typesetting problem for luggage factories, we developed a set of IATS that was carried out according to the environment as followings: Implementation platform (CPU: Pentium IV 2.4 GHz; Memory: 1 GB RAM, 160 G bytes HD; Operating system: Windows XP; Database: SQL Server 2005); Developing language (Visual Basic.NET; ASP.NET; Borland C++ Builder 4.0); Implementation data source (the data of order for luggage is acquired from a famous luggage factory in Taiwan).
Table 2 The typesetting result through ODTA
Customer No.: JCP; material: 1200 D Polyester; width: 58 in.; usage: 0.96147152; yard: 749.840277; unit yard: 0.49989351; total patterns: 9; tolerance (width): 0.25; and tolerance (length): 0.25.
4.1. The function of IATS
Fig. 3. The illustration of typesetting.
96.15%. The amount of using yard is 749.84, which is the length of fabric for quantity of luggage ordered that we need to purchase. The amount of unit yard used is 0.499, which represents the needed fabric for each luggage. According to the data, we can decide the price for each luggage. Fig. 3 presents the illustration of typesetting according to Table 2. In the bottom of the Fig. 3, the numbers show the concept of ‘‘multi-layers typesetting’’, for example, ‘‘500 [1]’’ means that pattern 1 needs to be cut 500 pieces at a time, ‘‘250 [2]’’ means that pattern 2 needs to be cut 250 pieces at a time and so on.
The executive frame as we started IATS is illustrated in Fig. 4. In Fig. 4, when we receive the orders from customers. First of all, we need to build up the quotation according to the customers’ requirements by pressing the button [Add] and inputting the basic information such as quotation ID, customers’ name, name of product and specification. Secondly, we need to build up the BOM table for luggage manufacturing by pressing the button [data maintenances (M)] in Fig. 4. The executive frame is illustrated
3.5. Two-dimensional typesetting problem In IATS, the material such as leather, wood and plastic plates etc. have to be cut in a same piece of material in
Fig. 4. The executive frame of IATS.
S.-L. Lee / Expert Systems with Applications 35 (2008) 1620–1627
in Fig. 5. In Fig. 5, we need to input the components of material for manufacturing luggage such as type, name, specification and unit cost. Thirdly, we need to press button [calculation] in Fig. 4. The executive frame is illustrated in Fig. 6. In Fig. 6, we can choose [batch calculation] that when customers ordered
many different types of luggage or [single calculation] that when customers assigned any kind of luggage from the order. Next, IATS can provide three different kinds of report. The first one is the frame of one-dimensional typesetting in Fig. 7. The second one is the frame of BOM in Fig. 8. The last one is the frame of quotation in Fig. 9. Finally, in Fig. 10, the quote of any order can be inquired immediately through mobile phone.
Fig. 5. The executive frame of BOM.
Fig. 8. The frame of BOM.
Fig. 6. The executing frame of IATS.
Fig. 9. The frame of quotation.
Fig. 7. The frame of one-dimensional typesetting.
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Fig. 10. The frame for inquiring quote by mobile phone.
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4.2. The comparison of the utility rate between IATS and manual-typesetting In the process of manufacturing one luggage, manualtypesetting work will take one day to produce paperboard model, and another two days to run optimal solution. However, IATS only takes a few minutes to key in data and then attain the result within few seconds. It saves not only time in producing paperboard model but also human resources. In the meantime, it also saves the cost of materials for making luggage. In Table 3, we compared the utility rate of one-dimensional IATS with manual-typesetting. The workers in luggage factory mentioned that the utility rate of fabric typesetting by man-made was approximately 90% in the past. The overall average utility rate of fabric typesetting was 95.32% in IATS. In other words, IATS raised over 5% utility rate of fabric typesetting as compared with manual-typesetting. If we calculate from the annual 40 million in purchasing material cost of the factory, IATS can save circa 2 million in purchasing cost annually. According to Table 3, we generated a chart in Fig. 11 to present the unit yard used between one-dimensional IATS and manual-typesetting. It is obviously to show that the Table 3 The comparison of the utility rate between one-dimensional IATS and manual-typesetting Number of piece
Number of luggage
Width of fabric (in.)
Man-made calculation (yard)
IATS calculation (yard)
Utility rate of fabric by IATS (%)
14 13 14 13 14 13 9 10 7 16 14 13 11 11
2734 2734 2735 2735 2736 2736 2622 2622 2645 2617 2532 2532 3862 3862
58 60 58 60 58 60 58 60 58 58 58 60 58 60
1.738 3.663 1.552 3.2495 1.2961 2.6124 0.5973 0.8482 0.2828 1.4796 1.4775 2.8 2.15 2.0743
1.508 3.489 1.347 3.157 1.0731 2.6124 0.503 0.829 0.203 1.24 1.206 2.61 1.9363 1.9031
95.00 93.20 96.70 93.00 96.30 95.00 95.49 94.47 92.50 97.10 96.90 98.00 96.10 94.50
Average utility rate of fabric by IATS
95.32
4 3.5 3 2.5 2 1.5 1 0.5 0 2734 2734 2735 2735 2736 2736 2622 2622 2645 2617 2532 2532 3862 3862 2903 2903 Unit yard used by manual-typesetting
Unit yard used by IATS
Fig. 11. The chart of unit yard used by one-dimensional IATS and manual-typesetting.
Fig. 12. The frame of two-dimensional typesetting.
unit yard used by IATS is higher than man-made among 16 different types of luggage. About two-dimensional IATS, the factory only provided the data of manual-typesetting. Hence, we could only show the result by IATS. Fig. 12 presents the utility rate in IATS of leather is 90%, the average utility rate in manual-typesetting is 75%. It is obviously to see that the utility rate in IATS is higher than in manual-typesetting. 5. Conclusion How to increase the effectiveness of typesetting is an important issue in luggage factories nowadays. In order to enhance the competitive advantage now and in the future, this paper develops a typesetting system – IATS, which possesses some features such as ‘‘saving cost’’, ‘‘accurate calculation of typesetting’’, ‘‘feasible cut’’ and ‘‘speedy quote’’ for luggage factories. Users can attain quote rapidly through mobile phone to raise the capability of receiving orders enormously. For the future research, the IATS can be considered to apply to irregular cut such as ready-made clothes industry. References Arthur, L. C., & Roger L. W. (1992). A genetic algorithm for packing in three dimensions. In Proceedings of the 1992 ACM/SIGAPP symposium on Applied computing: Vol. II. Technological challenges of the 1990s (pp. 1021–1030). Baker, B. S., Brown, D. J., & Katseff, H. P. (1981). A 5/4 algorithm for two-dimensional packing. Journal of Algorithms, 2, 348–368. Burrow, J. W. (1985). On the origin of species. New York: Penguin Books. Chan, F. T. S., Au, K. C., & Chan, P. L. Y. (2006). A decision support system for production scheduling in an ion plating cell. Expert Systems with Applications, 30, 727–738. Chandra, C., & Sanjeev, K. (1999). On multi-dimensional packing problems. In Proceedings of the tenth annual ACM-SIAM symposium on discrete algorithms (pp. 185–194). Cheung, W., & Zhou, H. (2001). Using genetic algorithms and heuristics for job shop scheduling with sequence-dependent setup times. Annals of Operations Research, 107, 65–81. Coffman, E. G., & Stolyar, A. L. (1999). Fluid limits, bin packing, and stochastic analysis of algorithms. In Proceedings of the tenth annual ACM-SIAM symposium on discrete algorithms (pp. 877–878). Dyckhoff, H. (1990). A typology of cutting and packing problems. European Journal of Operational Research, 44, 145–159. Falkenauer, E. (1996). A hybrid grouping genetic algorithms for bin packing. Journal of Heuristics, 2, 5–30. Falkenauer, E., & Delchambre, A. (1992). A genetic algorithm for bin packing and line balancing. In Proceedings of the IEEE 1992
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