# Manufacturing and Fractional Cell Formation Using the Modified Binary Digit Grouping Algorithm

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1 Proceedings o the 2014 International Conerence on Industrial Engineering and Operations Management Bali, Indonesia, January 7 9, 2014 Manuacturing and Fractional Cell Formation Using the Modiied Binary Digit Grouping Algorithm Mroue Hassan and Dao Thiên-My Department o Mechanical Engineering École de Technologie Supérieure (University o Quebec) Montreal, Quebec H3C 1K3, Canada Abstract A modiied version o the new binary digit grouping algorithm is presented in order to search or the optimal solution to the manuacturing and ractional cell ormation problem in the context o the design o cellular manuacturing systems. This algorithm leads to a coniguration which groups the machines having common production characteristics as well as the parts having common processing requirements into part amilies according to the concept o group technology. As a inal step, manuacturing cells can be ormed each o which involves a set o machines and parts. The nonzero entries which remain outside the manuacturing cells are called exceptional elements. When a lot o exceptional elements are obtained, an additional cell called ractional (or remainder) cell can be ormed. The aim o the remainder cell is to reduce the number o exceptional elements since it must contain all o the machines as well as the greatest possible number o these elements. The nonzero entries which are included within the manuacturing or ractional cells will be no longer considered as exceptional elements. This algorithm was tested through a MATLAB code by using illustrative examples rom the literature and succeeded to give optimal inal solutions. Keywords Binary digit grouping algorithm, Cell ormation, Manuacturing cell, Fractional cell, Exceptional elements 1. Introduction A new algorithm entitled "Binary Digit Grouping Algorithm" used or the ormation o manuacturing cells according to the concept o group technology was presented in a previous conerence (Mroue and Dao 2012). In this paper, a modiied version o the same algorithm is provided in order to demonstrate its capability to orm not only manuacturing cells, but also an additional ractional cell when applicable. The cellular manuacturing system (CMS) results rom applying the concept o group technology (GT) (Asokan et al. 2001). This concept is an industrial philosophy which aims to group the machines having common production capabilities into manuacturing cells, as well as the parts having common geometric shapes or processing requirements into part amilies in order to beneit rom these similarities (Xiaodan et al. 2007). The manuacturing cell ormation problematic is considered as a non-polynomial hard (NP-hard) problem and it was classiied or a long time as being the most challenging one because the processing time required to solve it increases exponentially with the size o the problem (Ben Mosbah and Dao 2010). On the other hand, the inter-cell movement occurs when a part is treated by the relevant machine outside the manuacturing cells. In such cases, the allocated elements will be called exceptional elements. In the problems where we ind that a lot o exceptional elements a ractional cell (also called remainder cell) may be ormed. The remainder cell must contain all o the machines o the system as well as the greatest possible number o exceptional elements. By this way, the movements o parts between the manuacturing cells on one hand, and the remainder cell on the other hand are not considered inter-cell movements (Chandrasekharan and Rajagopalan 1989). Numerous researchers worked on this problem and provided methodologies which succeeded to give optimal or near-optimal solutions. Mark et al. (2000) proposed an adaptive genetic algorithm in order to solve the cell ormation problem. Liang and Zolaghari (1999) provided a new neural network approach to solve the comprehensive grouping problem. Solimanpur et al. (2010) worked on this problem through an ant colony optimization (ACO) method. Lei and Wu (2006) approached the problem by presenting a tabu search method etc. Concerning the ormation problematic o the additional ractional cell, the authors who addressed it are very ew, Venkumar and Noorul Haq (2006) applied a modiied ART1 neural networks algorithm in order to treat it whereas Chandrasekharan and Rajagopalan (1989) used a simulated annealing approach or the same purpose. 378

2 2. The Modiied Binary Digit Grouping Algorithm The irst step o orming manuacturing cells consists o using a matrix which is called incidence matrix. The size o an incidence matrix is M N where M represents the machines and N the parts. The matrix can be presented in the ollowing orm: A = [a mn ] where a mn is the workload (production volume multiplied by the unit processing time) o the part number n when being processed on the machine number m (Mak et al. 2000). Let us take the ollowing 5 5 incidence matrix as an example: Table 1: A 5 5 incidence matrix m \ n A nonzero entry (i.e. a 1 digit) means that the relevant part will be processed by the concerned machine: i we take the nonzero entry in the upper let corner as an example, it means that the part number 1 will be processed by the machine number 1; whereas, a zero entry means the inverse. We can divide the elements o the incidence matrix into 3 categories: Elements in the corner o the matrix (the 4 elements highlighted in pink in the ollowing matrix): Table 2: The corner elements (highlighted in pink) o the incidence matrix m \ n Elements in the borders (but not the corners) o the matrix (highlighted in bright green): Table 3: The border elements (highlighted in bright green) o the incidence matrix m \ n

3 Finally, elements in the heart (i.e. not in the borders) o the matrix (highlighted in turquoise): Table 4: The elements in the heart (highlighted in turquoise) o the incidence matrix m \ n I we begin with the corner elements and consider only (isolate) the nonzero entries as well as their surrounding elements (in this example, we have a single nonzero entry in the 4 corners which is located in the upper let one is shown in the ollowing table): Table 5: A nonzero entry in the upper let corner o the incidence matrix with its surrounding elements For each non zero entry, we calculate the nonzero entry neighboring actor ( N ) as being the sum o all the surrounding elements: N

4 Now let us consider the second type o the nonzero entries which are those located in the borders (but not corners) o the matrix and we take the nonzero entry located in the irst row and third column as an example by isolating it together with its surrounding elements: Table 6: A nonzero entry in the upper border o the incidence matrix with its surrounding elements We proceed as previously: N Finally, let us consider the third and last type o nonzero entries which are those located in the heart o the matrix by taking one o them (the element located in the ourth row and ourth column in the next igure) as an example: Table 7: A nonzero entry in the heart o the incidence matrix with its surrounding elements N Ater calculating the individual nonzero entry neighboring actor ( N ) o all the nonzero entries o the incidence matrix, we multiply them by 2 and sum up the resulting values all together in order to get the "nonzero entries neighborhood actor o the whole matrix" ( N M ). As a next step, we swap randomly either 2 rows and/or 2 columns (swapping 2 rows as well as 2 columns once every ew iterations helps to avoid getting trapped in local optima) in order to get a new coniguration o the incidence matrix. The maximum number o dierent combinations (i.e. o 381

5 permutations) which may result rom applying this process on a certain matrix will be equal to the multiplication o the actorial o the number o rows by the actorial o that o the columns (in this example, we have 5! 5!=14400 dierent possible conigurations o the incidence matrix). Ater each swapping process, we re-calculate everything rom the beginning or the new coniguration in order to get the new value o ( N M ) called ( N M *) and compare it with the previous one: i the new one is greater than the previous one, the new matrix will be considered as the new solution or the problem; otherwise, the previous coniguration will be kept. The computational process continues in this manner until testing all (or at least a great number) o the possible conigurations and inally, the one with the greatest value o ( N M ) will be selected as the inal solution. In the case where we have big size problems (i.e. when we have numerous rows and/or columns in the incidence matrix), the computational time may become very long; that is why, it will be limited to a reasonable duration (such as 30 minutes) and the best ound solution will be adopted as the inal one (even in such cases, the best ound solution will be mostly the best possible one because a lot o apparently dierent conigurations will lead to the ormation o the same cells). The binary digit grouping algorithm can be illustrated as ollows: Figure 1: The binary digit grouping algorithm As can be easily deduced, the binary digit grouping algorithm aims to group the nonzero entries as much as possible together within the incidence matrix in order to come up with manuacturing cells which have to include the greatest possible numbers o nonzero entries. The eectiveness o the resulting solution can be tested and compared to those resulting rom the application o other algorithms using some ormulas which were used or the evaluation o other algorithms such as the genetic algorithm (Mark et al. 2000). I we let: K be the number o manuacturing cells which will be ormed within the incidence matrix n1 represents the number o nonzero entries existing in the manuacturing cells M kand N k where k = (1,2,...,K) be consecutively the numbers o machines and parts which are assigned to the manuacturing cell number k; we can get: 382

6 e 1 K k 1 n 1 M N k k Where ( e 1 ) is a measure o the cell density which means that the more the machines and parts in a manuacturing cell are similar, the greatest is the value o ( e 1 ) and vice-versa. On the other hand, i we let ( n 2 ) be the number o the exceptional elements (which are the nonzero entries that are not located within the manuacturing cells), we can get: n1 e2 1- (2) n1 n2 Where ( e 2 ) is a measure o the intercellular material low and it increases with the increase in the number o exceptional elements and vice-versa. Finally, the grouping eiciency (e) can be calculated as ollows: e e1- e2 (3) The numerical value o (e) always belongs to the interval [-1, 1]. 3. Cell ormation Ater applying the algorithm, a inal coniguration o the incidence matrix will be gotten. The nonzero entries are expected to be grouped as close to each other as possible. The next step consists o creating only manuacturing cells by keeping in mind three rules: The greatest possible number o nonzero entries must be contained in the ormed cells. Every machine and/or part must be involved in a cell. A machine and/or a part cannot be assigned to more than one cell. According to the distribution o the nonzero entries in the matrix coniguration obtained and in cases where we obtain a lot o exceptional elements, a re-distribution o the cells may take place according to what ollows: Creating manuacturing cells where each contains a certain number o machines and parts. Not all o the machines but all o the parts must be assigned to the manuacturing cells. A machine and/or a part cannot be assigned to more than one manuacturing cell. Creating one additional ractional cell which contains all o the parts in addition to only the machines which are not assigned to manuacturing cells. The nonzero entries which are included in the manuacturing cells or the ractional one are not considered as exceptional elements; that is why, the addition o the ractional cell may play a major role in reducing the number o these elements. The ollowing section gives urther illustrations through two illustrative examples. 4. Illustrative Examples We are going in what ollows to test this algorithm through two illustrative examples taken rom the literature. The irst one results only in the ormation o manuacturing cells and the second one shows the ormation o manuacturing cells as well as an additional ractional cell. Consider the ollowing incidence matrix is taken rom SrinlVasan et al. (1990). Table 8: The incidence matrix or the irst illustrative example m \ n (1) Ater running the MATLAB code o the binary digit grouping algorithm, the ollowing solution was gotten: 383

7 Table 9: The cell design or the irst illustrative example m \ n In this example, 4 manuacturing cells were created; the number o nonzero entries existing in the manuacturing cells ( n 1 ) is equal to 49, the number o exceptional elements ( n 2 ) is equal to 0 which means that there is no intercellular material low. The calculations led to a value o the cell density measure ( e 1 ) equal to 1 and to a value o 0 or the intercellular material low ( e 2 ). The value o the grouping eiciency: e e1-e2 1. There is no need or a ractional cell in this example since there are no resulting exceptional elements. Let us consider now the second illustrative example which is represented by the ollowing matrix which is taken rom King and Nakornchai (1982). Table 10: The incidence matrix or the second illustrative example m \ n Ater applying the modiied binary digit grouping algorithm, the ollowing solution which includes only manuacturing cells was obtained, with a value o (N m ) equal to 650: Manuacturing cell 1: o Machines 10, 7 o Parts 26, 25, 13, 1, 12, 39, 31 Manuacturing cell 2: o Machines 15, 2, 9, 16, 14, 3, 12, 11, 13 o Parts 3, 24, 27, 30, 22, 36, 35, 17, 7, 40, 42, 2, 37, 32, 38, 10, 18, 28, 4, 16, 20, 11, 34, 6 Manuacturing cell 3: o Machines 1,4,5,8,6 o Parts 15, 8, 23, 43, 19, 21, 14, 5, 29, 41, 33, 9, Number o exceptional elements: o 34 Table 11: The design o the manuacturing cells or the second illustrative example 384

8 m\n As a result, a great number o exceptional elements were obtained (which is not good). In such cases, the problem may be solved by constructing manuacturing cells as well as one additional ractional (remainder) cell. The nonzero entries will be distributed between the manuacturing cells and a remainder cell in order to get a number o exceptional elements as low as possible. The reormation o the cells is illustrated as ollows: Manuacturing cell 1: o Machines 10, 7 o Parts 26,, 25, 13, 1, 12, 39, 31 Manuacturing cell 2: o Machines 15 o Parts 15, 8, 23, 43, 19, 21, 14, 5, 29, 41, 33 Manuacturing cell 3: o Machines 2, 9, 16, 14, 3, 12, 11, 13 o Parts 9, 3, 24, 27, 30, 22, 36, 35, 17, 7, 40, 42, 2, 37, 32, 38, 10, 18, 28, 4, 16, 20, 11, 34, 6 Remainder (ractional) cell: o Machines 1, 4, 5, 8, 6 Number o exceptional elements: o 0 Table 12: The design o the manuacturing cells in addition to one ractional cell or the second illustrative example m\n The solution demonstrates that the problem o exceptional elements problem may be completely solved by implementing a remainder cell. In addition, it demonstrates that the binary digit grouping algorithm may be able not only to solve the manuacturing cell ormation problems, but also the ractional (remainder) cell ormation problematic as well. 4. Conclusion A modiied version o the binary digit grouping algorithm is presented and explained in order to solve the machinepart grouping problem or manuacturing and ractional cell-ormation within the lexible manuacturing systems. The simple steps and computational procedures make it particularly powerul in quickly creating and conserving the 385

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