SIMPLE ASSEMBLY LINE BALANCING

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1 A R C H I V E S O F M E C H A N I C A L T E C H N O L O G Y A N D A U T O M A T I O N Vol. 33 no JAN UREK, MAREK PASTWA, MARCIN WINIEWSKI SIMPLE ASSEMBLY LINE BALANCING Assembly line balancing is one of the challenges that appear when technological process of an assembly is being developed. It is a particularly difficult, computably complex task, belonging to the NP-complete class. Apart from the task mentioned above, this group includes a number of key computational problems. Due to the practical value of these complexities, as well as great cost reductions related to their optimisation, precise algorithms are being constantly investigated for them. These tasks include scheduling, compaction, permutation, sequencing, etc. What is vital, effectively developed algorithms for resolving one of the NP problems are easy to be implemented into another ones, which makes the assembly line balancing universal. The article describes balancing of an assembly line pictured as graphs. Basic concepts and important assumptions were discussed, including the restrictions and types of simple straight assembly line balancing. Key words: straight assembly line balancing, types of simple assembly line balancing 1. ASSEMBLY LINE BALANCING IN THE FORM OF A GRAPH Assembly line balancing is, to some extent, mapped out in the graph of sequence restrictions G = {V, A, t} known as the graph of precedence [1, 8]. It is a directed graph (digraph). It consists of a nonempty V = {1,..., n} set of vertices representing the operations and a finite family of arcs (edges) A = {(i, ) iv and F i }, where F i represents a set of operations directly following the I one. Each of the vertices is assigned with a t value standing for the operation time. An i operation preceding a operation is called the operation predecessor, while an i operation that can be performed subsequently to a one is known as its successor. The relationship between i and operations is called direct when there is an (i, ) arc or a (, i) one. The graph vertices relating to the operations without predecessors are called source vertices, while the ones referring to the Prof. dr hab. in. Institute of Mechanical Technology, Poznan University of Technology. Mgr in. Dr in. Sparta sp. z o.o.

2 62 J. urek, M. Pastwa, M. Winiewski operations without successors are known as sink vertices. If each of the (i, ) arcs is replaced by the (, i) in graph G, a G r reversed precedence graph is obtained. In the latter graph, preceding operations become successors, successors turn into predecessors, sources come to be sinks and sinks become sources. A sequence restrictions graph is an acyclic one, which means it does not contain any closed paths, as well as a simple graph, which makes it a digraph without any repetitive edges or loops, that is multiple (i, ) edges and (i, i) arcs [2]. A G * = (V, A *, t) subgraph can be distinguished within the G graph. The subgraph mentioned contains (i, ) arcs for each i and operation connected by a path within the G graph, that is A * = {(i, ) iv and F * i}, where F * i represents all operations (graph s nodes) following the i operation, the successors of i. The A * set contains all the arcs representing the relationships following the i operation, and the A A * set subtraction is a set of arcs corresponding to the relationships preceding the i operation. Framework for assembly line balancing is consisted in dividing the V = {1,..., n} set which assigns every operation to a S k set for k = 1,..., m, where m is the number of stations. This assignment needs to meet certain requirements. The first condition is that each operation has to be assigned to exactly one station, while the second one dictates keeping the sequence that stems from the sequence restrictions graph, which makes it indispensable to meet the following condition: if is o and S p, then for each arc (i, ) A po. The division mentioned should also meet the condition of the total t(s k ) time needed for an operation assigned to a given station to be less than or equal to given c line takt time. 2. AN EXAMPLE OF ASSEMBLY LINE BALANCING A basic assembly line balancing, known as SALB, is simplified in a number of ways, which makes it easy during the tests, yet most often different from the methods used in industrial practice. Basic conditions of this method are: assemblage of homogenous products, known, unvariable duration time of particular operations, restrictions to operation grouping expressed only in a form of a precedence graph, determined and constant line production rate, serial configuration of the assembly line, with the stations located on one side, independent and equally equipped stations. In its basic form, that is when no additional conditions are formulated, assembly line balancing narrows down to a decision problem [5, 9], concerning the

3 Simple assembly line balancing 63 question whether the operations are assigned to a given number of stations and pointing out that assignment. Figure 1 maps out a sequence restrictions graph for an exemplary Jackson s task. Particular operations are indicated by nodes, while the arcs reflect the precedence relationships among them. The task consists of 11 operations, whose total production time equals 46 units, the maximum time needed for performing one operation is 7 units and the minimum is 1 unit. The value of a given line s production rate determines its theoretical efficiency (a product leaves the line every cycle). If the production rate is stated to be 14 units and the unit equals one minute, the production rate is 60/14 products per hour, which means there are over 34 products per an 8-hour shift, while the minimum number of stations is 4 {{1, 2, 3, 5}, {6, 8, 10}, {4, 7}, {9, 11}}. It is easy to notice that the load of Station 2 is 13 units, of Station 3 10 units and of Station 4 only 9 units, in the operation division proposed. 2/2 6/2 8/6 10/5 1/6 3/5 7/3 9/5 11/4 4/7 4/10 5/1 number of operations working time Figure 1. Sequence restrictions graph for the Jackson s task If the demand for products raises to 40 per shift, the line production rate has to be decreased to 12 per unit, however, increasing the number of stations is not necessary. The division of operations among 4 stations, with a maximum load on one of them being 12 units is: {{1, 2, 5, 6},{3, 4},{8, 10},{7, 9, 11}}. The line s takt time, equalling 12 units of time, is at the same time the smallest one that allows for dividing the operations among 4 stations. A takt time that consists of 11 units requires increasing the number of stations.

4 64 J. urek, M. Pastwa, M. Winiewski 3. RESTRICTIONS IN ASSEMBLY LINE BALANCING Restrictions referring to the sequence of operations may be pictured by means of a sequence restrictions graph. As it can be observed, these restrictions may be used to determine the earliest station: E and the last one: L on which a given operation can be performed (it can be performed as early as the operations directly and indirectly preceding it are assigned), when the line production rate is determined and the number of stations is maximum. Therefore, the earliest station where a given operation can be performed is, in other words, the quotient of the total of the operation times (of the given operation, as well as its direct and indirect predecessors) and the line production rate determined. The value obtained should be rounded to the smallest integer greater or equalling it (the socalled ceiling). If the total number of the line stations is known, the last station where a given operation has to be performed can be easily determined. To indicate it, one makes an assumption that all the operations directly and indirectly following the given one have to be performed on the station where this operation takes place, or on one of the next ones. The value obtained needs to be rounded in the same way the previous one is. These relationships can be expressed by the following formulas: L E t h c * P t h for = 1,..., n (1) t th * hf m 1 c for = 1,..., n (2) where: E the earliest permissible station for a operation, L the last permissible station for a operation, m number of stations, c line takt time, t, t h and h operation times, respectively, P * operation s direct and indirect predecessors, F * operation s direct and indirect successors.

5 Simple assembly line balancing 65 The earliest and latest stations for the Jackson s task Table 1 E L SI The sequence restrictions adopted allow for stating the station interval where a certain operation has to be performed. This interval, written SI, is defined by the earliest and the last station a particular operation might be performed on: SI = = [E, L ]. By this, B k operation sets are obtained and subsequently may be potentially assigned to particular stations: B k k SI for k = 1,..., m (3) Table 1 presents the results of the Jackson s task discussed above, with the takt time equalling 14 units and the division consisting 4 stations. Basing on the data provided, sets of operations that can be assigned to particular stations are: B 1 ={1, 2, 3, 4, 5, 6}, B 2 ={2, 3, 4, 5, 6, 7, 8, 9, 10}, B 3 ={2, 3, 4, 5, 6, 7, 8, 9, 10}, B 4 ={7, 9, 10, 11}. What is vital, the station intervals stem only from the sequence restrictions, without other indispensable conditions for the solution to be acceptable. 4. TYPES OF SIMPLE ASSEMBLY LINE BALANCING Depending on the product, assembly line balancing can be in formof one of the three types of tasks described below: 1) SALBP-1 [4] minimisation of the m number of stations at a given production rate c: min{m (m, c) is feasible for a given c production rate}, 2) SALBP-2 [3] minimisation of the c line production rate with a given number of stations: {c (m, c) is feasible for the m number of stations}, 3) SALBP-E maximisation of the E line effectiveness, that is minimisation of the product of m c, max{ t sum /(m c) (m, c) is feasible for possible m and c vaues}.

6 66 J. urek, M. Pastwa, M. Winiewski The SALBP-1 and SALBP-2 tasks are similar. It can be observed that the constant c line production rate parameter in SALBP-1 undergoes minimisation in SALBP-2 and vice versa. The invariable m number of stations in SALBP-2 is, in turn, minimised in SALBP-1, which makes these two tasks mutually complementary. A suboptimal solution (that is, dividing into the lowest possible number of stations), found under the criterion of line stations minimisation, can be improved by minimising the line production rate. In this case the number of stations is invariable. Due to such steps taken, the effectiveness of an assembly line may be improved, which makes it resemble SALBP-E. The criterion is the best assembly line effectiveness when m and c numbers of stations are determined at particular intervals and a given c production rate. 5. CONCLUSION As literature [19] shows, assembly line production rate minimisation for a given number of stations always leads to a decreased total of idle times. Reducing the production rate, and at the same time minimising the number of stations, makes the idle times increase. Therefore, stating which changes of the stations number and the line production rate (m and c) are acceptable is crucial for the quality of the derived solutions. However, it is uneasy, as no guidelines are known before the solution appears. SALBP-1 and SALBP-E are mostly often used in the design of new assembly lines in the industrial practice, and their structure and required productivity value can be fully determined. SALBP-2 is, in turn, used in order to improve the effectiveness of existing assembly lines, as well as when there is a need for redesigning them to a limited extent. What is vital, though seemingly similar, the assembly line balancing SALBP-1 and SALBP-2 types require different methods and algorithms. REFERENCES [1] Ciszak O., urek J., Wyznaczanie kolenoci montau czci i zespoów maszyn, Archiwum Technologii Maszyn i Automatyzaci, 1998, vol. 18, nr 2. [2] Cormen T.H., Leiserson C.E., Rivest R.L., Introduction to algorithms, Warszawa, WNT [3] Nearchou A.C., Balancing large assembly lines by a new heuristic based on differential evolution method, International Journal of Advanced Manufacturing Technology, 2007, vol. 34, issue 9 10, p [4] Pastor R., Ferrer L., An improved mathematical program to solve the simple assembly line balancing problem, International Journal of Production Research, 2009, vol. 47, issue 11, p

7 Simple assembly line balancing 67 [5] Pastwa M., Balansowanie linii montaowe za pomoc algorytmów ewolucynych, Ph.D. thesis, Poznan University of Technology [6] urek J., Algorytmizaca balansowania linii montaowe, Archiwum Technologii Budowy Maszyn, 1992, vol. 10. [7] urek J., Technologia montau, Zeszyty Naukowe Politechniki Poznaskie, Mechanika, 1994, nr 39. [8] urek J., Ciszak O., Modelowanie oraz symulaca kolenoci montau czci i zespoów maszyn za pomoc teorii grafów, Pozna, Wydawnictwo Politechniki Poznaskie [9] urek J., Pastwa M., Próba zastosowania algorytmu genetycznego do balansowania linii montaowe, in: Technika i technologia montau maszyn. IV International Science and Technology Conference Materials, Rzeszów ZADANIE BALANSOWANIA PROSTEJ LINII MONTAOWEJ S t r e s z c z e n i e W artykule opisano zagadnienia dotyczce balansowania linii montaowe przedstawione za pomoc grafów. Okrelono podstawowe pocia, omówiono niezbdne zaoenia, w tym ograniczenia oraz odmiany prostego zadania balansowania proste linii montaowe. Sowa kluczowe: balansowanie linii montaowe proste, odmiany prostego zadania balansowania linii montaowe

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