Assembly system facility design

Transcription

1 IIE Transactions (2006) 38, Copyright C IIE ISSN: X print / online DOI: / Assembly system facility design YOSSI BUKCHIN 1, RUSSELL D. MELLER 2, and QI LIU 2 1 Industrial Engineering Department, Tel Aviv University, Tel Aviv, Israel bukchin@eng.tau.ac.il 2 Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA rmeller@vt.edu; qliu@vt.edu Received September 2004 and accepted June 2005 This paper addresses the design of an assembly system facility consisting of multiple assembly lines of different shapes. In such a design problem there are two conflicting objectives: (i) to minimize the overall area of the facility; and (ii) to maximize the efficiency of the material handling transportation system. We first address the optimization problem of objective (ii) when replacing objective (i) with a constraint on the facility area. We propose a mixed-integer linear program to determine the layout of a facility with given dimensions and with given assembly line areas and shapes (that cannot be changed due to technological considerations). In the layout model, the physical placement of each line within the facility is a decision variable. The objective function of the layout model is to minimize the distances traveled by material flow. Our performance analysis provides an indication of the maximal problem size that can be solved in a reasonable amount of time and we examine the effect of the problem parameters on the solution run time. This layout model is then incorporated into an efficiency frontier approach for facility design to address both objectives. Examples are presented to illustrate the use of the proposed facility design model. 1. Introduction In designing a manufacturing facility, a facilities planner typically has competing objectives. An example of these competing objectives is apparent when one considers the size of the facility and the efficiency of the material handling transportation system. Clearly, minimizing the overall size of the facility is the least costly option in terms of investment in the building itself. However, in so doing, the design of the facility layout is severely constrained, which has an adverse impact on the efficiency of the material handling transportation system. In this paper we consider the facility design problem as a dual-objective problem and employ the well known efficiency frontier approach. In the efficiency frontier approach, a systematic search mechanism is used to consider the various combinations of weights that can be given to the various objectives. Our mechanism is discussed in detail in Section 5. Until that point in the paper, we concentrate on solving the primary sub-problem: the facility layout design problem for a fixed facility area. The traditional layout design of manufacturing facilities is based on the concept of planning departments that are either process-based or product-based entities. The departments typically are constrained in terms of area, but only Corresponding author loosely constrained in terms of shape (e.g., an aspect ratio of less than 4:1 must be maintained). Although the decisionmaker is asked to specify an interaction matrix (e.g., a quantitative from-to flow matrix or a qualitative REL chart), the detailed layout is assumed to follow the block layout design. As a result, the traditional facility layout design formulation treats the departments as malleable objects, with department shape refinement to follow based on user-based massaging. There are many examples of manufacturing layout facilities where most, if not all, of the department shape layouts are fixed due to technological constraints. One class of examples is assembly systems, where the component, subassembly and final assembly lines are located within the same facility (another class is flexible manufacturing facilities). The result is a complex system consisting of many different lines feeding one another. For example, the final assembly line may be fed by several sub-assembly lines, which may, in turn, be fed by component assembly lines, etc. The physical layout of such a system can have a large impact on both the fixed costs associated with the construction of such a system, including the automatic conveyancing systems, and the operational costs associated with the material handling cost in the facility. Moreover, the management of such a system is greatly aided by line-of-sight management, which in layout terms, means that the output point of one line is as close as possible to the input point for that X C 2006 IIE

2 54 Bukchin et al. sub-assembly or component on the next line. The assembly lines may have different shapes, such as, an I-shape, an L-shape, a U-shape, etc. The U-shaped line, for example, is currently very popular in industry and has replaced the I-shaped line in many environments. The reason for this is that the U-shaped line enables good communication among the workers and locates each worker relatively close to all the stages of the process performed in the line. The primary sub-problem in such a facility design problem is how to locate a given set of lines in a given facility such that the distances between output points and input points are minimized. As is common in layout design, rectilinear distances are applied. The main drawback of this distance measure is the fact that in some cases it assumes that material can be transported through the area of another department. Still, these cases are relatively rare and using rectilinear distances is more appropriate here than using Euclidean distances. We assume that the shape of each line is given, and the placement (i.e., location and orientation) of the line in the facility is to be determined (Savas et al., 2002). In so doing, we assume that each line has four possible orientations, determined by rotating the shape by 90, 180 and 270.In order to construct a general formulation that is not dependant on specific line shapes, we assume that each line can be divided into rectangles, as explained later. In Fig. 1 we demonstrate the problem s characteristics via an example. We can see an assembly system consisting of several lines of different shapes. Each line is divided into several rectangles, denoted as sub-lines, except for I-shaped lines, which are rectangular to begin with. The notation (i, j) refers to the rectangular sub-line j of line i. The thin arrows denote the assembly process flow for each line. The thick arrows show the flow of material within the facility. First, raw material/components are transported from the input points of the facility, denoted by FI k, to the input points of the lines (sub-lines), denoted as I i(j). After the assembly process on the line, the material is transported from the output point of the line (O i(j) ) to the input point of the next line in sequence. At the end, the sub-assemblies are moved to the final assembly line and from there to the output point of the facility (FO k ). The remainder of the paper is organized as follows. The literature review focusing on reviewing exact models for the general manufacturing layout problem is presented in the next section. In Section 3 a formal presentation of the problem is given, along with a mixed-integer programming formulation. The results from a performance analysis of the formulation in solving problems with different characteristics is given in Section 4, along with an illustrative example. In Section 5 we present our facility design model that incorporates the efficiency frontier approach described earlier. In Section 6 we conclude the paper and outline future research avenues. 2. Literature review Most of the research dealing with the design of assembly systems focuses on the operational side of the building blocks of the system, the assembly lines. A lot of research Fig. 1. An example assembly system.

3 Assembly system facility design 55 has been conducted on the line balancing problem, where a single model or several models of the same product are to be assembled on the line. When a mixed-model line is concerned, an additional problem of determining the sequence of the different models to the line arises. For recent surveys of the various types of assembly line balancing problems see Erel and Sarin (1998), Becker and Scholl (2004) and Scholl and Becker (2004). To our knowledge, there is no existing research that addresses the layout of a system of assembly lines. The layout of a flexible manufacturing system, a related problem, has been addressed in Das (1993) in which the layout is represented in a discrete fashion and it is determined heuristically. However, note that both of these problems are specialized versions of the general manufacturing facility layout problem, which has been extensively studied (see Meller and Gau (1996) for the latest survey that includes over 100 references on manufacturing layout research in the 10 years surveyed). In particular, Montreuil (1990) provided a general formulation for the layout of continuously-represented rectangular-shaped departments. In addition, Heragu and Kusiak (1991) presented a model that specifies each department s length and width assuming a rectangular-shaped department with a fixed orientation. Likewise, Sarkar et al. (2005) examined the problem of inserting a department of given area, but unknown dimensions into an existing layout via a graph-theoretic approach. We propose to adapt the general mixed-integer programming model first developed by Montreuil (1990), and later enhanced by other researchers, to our problem setting. We now review these mixed-integer programming models. The mixed-integer programming (MIP) formulation for the floor layout problem (FLP) was originally presented by Montreuil (1990). This model uses a distance-based objective, but is not based on the traditional discrete representation employed in the quadratic assignment problem. Instead, the MIP-FLP utilizes a continuous representation of a layout and considers departments with unequal areas. In this model, the locations and dimensions of departments are decision variables. A number of binary integer variables are used in the model to avoid overlapping departments. Montreuil s model is commonly referred to as FLP0. One of the problems in FLP0 is that in lieu of the exact non-linear (specifically non-convex and hyperbolic) area constraint, a bounded perimeter constraint is used to linearize the model. However, using a bounded perimeter constraint instead of an exact area constraint can lead to errors in the final area of each department. A modified MIP-FLP model based on FLP0 was presented by Meller et al. (1998) to improve the model accuracy and approach efficiency. This model is commonly referred to as FLP1. The bounded perimeter constraint from FLP0 is modified in FLP1 with a linear approximation to the area constraint, which improves area accuracy. The binary variables that ensure departments do not overlap in the facility were also redefined. FLP1 considers every allrectangular-department layout solution, but becomes difficult to solve for small instances of n = 7 (where n is the number of assembly lines) even when valid inequalities are added in order to eliminate some infeasible solutions from the solution space and to improve the algorithm s efficiency (Meller et al., 1998). A further enhanced MIP-FLP model based on FLP1 was presented by Sherali et al. (2003), which we refer to as FLP2. The first enhancement is an improved representation of the nonlinear area constraint based on a novel polyhedral outer approximation scheme. The new surrogate area constraint provides as tight a representation as desired (unlike the approximation used in FLP0 and FLP1) by using a large enough number of discretization points for the tangential supports. Another important enhancement in FLP2 concerns preventing department overlapping by two alternative formulations, DJ1 and DJ2. In addition, a new class of valid inequalities, called UB inequalities, was discovered by exploring DJ2. DJ1 and DJ2 possess certain partial convex hull properties and provide increased tightness for the MIP- FLP model, but they also lead to a substantial increase in solution time as compared with FLP1 because of their size. Thus, three different strategies were implemented in Sherali et al. (2003) to impart the tightness from DJ1 and DJ2, with any increase in the problem size, being limited. Other enhancements in FLP2 consist of a symmetrybreaking constraint and some branching priorities for the branch-and-bound search. Combing all of these enhancements leads to greatly reduced algorithm run time and improves the tractable problem size n = 9. Thus, some challenging test problems from the literature were solved for the first time by FLP2. However, the problem size is still limited and not applicable for most industrial applications, which can range from departments. As we can see from prior research on MIP-FLP models, the resulting model is difficult to solve. However, the MIP-FLP models provide a good framework for modeling manufacturing layout problems and we will specialize the MIP-FLP for assembly system layout design in the next section. 3. Problem description and model formulation An assembly system consists of multiple assembly lines of different shapes. We assume that each line can be divided into several rectangles, denoted as sub-lines. Without loss of generality, we assume that there is an orientation of the line, called the standard orientation, consisting of vertical sub-lines which are ordered from left to right. Next, we can take each pair of adjacent sub-lines as a basic block to form the constraints that connect the two sub-lines together. The connectivity constraints of all the pairs constructing a line preserve the shape of the line. The fundamental blocks of

4 56 Bukchin et al. Fig. 2. Four orientations of an adjacent sub-line pair. any line (i.e., the two adjacent rectangular sub-lines) can take one of the four possible orientations in the layout, which are shown in Fig. 2. The most left orientation is the standard orientation, and then we assume that the shape can be rotated by 90, 180 and 270. We give the general MIP formulation for a layout of the assembly system. Parameters: L s = length of the facility in direction s; n = number of assembly lines; N = assembly line set (N ={1,...,n}); m i = number of sub-lines for assembly line i; M i = sub-line set for assembly line i(m i = {1,...,m i }); i(i ) = sub-line i of assembly line i(i M i ); h i(i ) = height of sub-line i for assembly line i in the standard orientation; w i(i ) = width of sub-line i for assembly line i in the standard orientation; i(i h )(j ) = the height difference of sub-lines i and j (j = i + 1) centroids for assembly line i in the standard orientation; i(i w )(j ) = the width difference of sub-lines i and j (j = i + 1) centroids for assembly line i in the standard orientation; ai(i s ) = the relative location in direction s from the output point of sub-line i of assembly line i to the centroid of i of i; bi(i s ) = the relative location in direction s from the input point of sub-line i of assembly line i to the centroid of i of i; f i(i )j(j ) = the material flow from the output point of subline i of assembly line i to the input point of sub-line j of assembly line j; f I i(i )k f O i(i )k FI s k = the material flow from the input point k of the facility to the input point of sub-line i of assembly line i; = the material flow from the output point of subline i of assembly line i to output point k of the facility; = the location of the input point k of the facility; FO s k FI s k = the location of the output point k of the facility; = the location of the input point k of the facility. Decision variables: c s i(i ) l s i(i ) I s i(i ) O s i(i ) z s i(i )j(j ) = centroid of sub-line i of assembly line i in direction s; = side length of sub-line i of assembly line i in direction s; = input point of sub-line i of assembly line i in direction s; = output point of sub-line i of assembly line i in direction s; = Binary decision variables, which denote relative locations of sub-line i of i and j of j with respect to direction s and are used to ensure the correct connection of sub-lines in the same assembly line and prevent the overlapping of the sub-lines from different assembly lines. The definition of zi(i s )j(j ) is given as follows: Given i j or i j : zi(i s )j(j ) = 0 otherwise. 1 if sub-line i of i proceeds sub-line j of j in direction s; For the assembly lines composed of multiple sub-lines, we use the relative location variables zi(i s )i(i +1) to denote the assembly line orientation. Note that since the sub-lines of each line are ordered in one direction, as explained above, the value of the zi(i s )i(i +1) variable of any pair of sub-lines of that line also denotes the orientation of the line. For example, considering the line in Fig. 2, if zi(i x )i(j ) = 1, then we know that the standard orientation is applied. However, for the I-shaped assembly line consisting of a single rectangle, zi(i s )i(i +1) is clearly not valid. Instead, we use a new set of binary variables ri t, t {I, II, III, IV} to denote the four orientations, as show in Fig. 3, of the assembly line. The definition of ri t is given as follows: { ri t 1 if I-shaped assembly line i takes orientation t; = 0 otherwise.

5 Assembly system facility design 57 Fig. 3. Illustration of different orientations of an I-shaped assembly line. The MIP formulation of the assembly system layout problem (MIP-ASLP1) is given next. MIP-ASLP1: min ( Oi(i x ) I x j(j ) + O y i(i ) Iy ) j(j ) i j>i + i + i subject to i i k i l j f i(i )j(j ) f I i(i )k ( FI x k I x i(i ) + FI y k Iy i(i ) ) fi(i O O x )l( i(i ) FOx l + O y i(i ) FOy l ) (1) li(i s ) cs i(i ) Ls li(i s ), i, i, s, (2) ci(i x ) = cx i(i +1) + ( zi(i x +1)i(i ) zx i(i )i(i +1)) w i(i +1)i(i ) + ( z y i(i +1)i(i ) zy i(i )i(i +1)) h i(i +1)i(i ), i, i, (3) c y i(i ) = cy i(i +1) + ( zi(i x )i(i +1) zx i(i +1)i(i )) h i(i )i(i +1) + ( z y i(i +1)i(i ) zy i(i )i(i +1)) w i(i )i(i +1), i, i, (4) zi(i s )i(j ) = zs i(i )i(i +1), i, i, s; j > i, (5) zi(i s )i(j ) = zs i(i )i(i 1), i, i, s; j < i, (6) ( z s i(i )j(j ) + ) zs j(j )i(i ) = 1, i j or, i j, (7) s ri t = 1, i {j m j = 1}, (8) t ci(i s ) + ls i(i ) cs j(j ) ls j(j ) + Ls( 1 zi(i s )j(j )), i j; i, j, s, (9) li(i x ) = ( zi(i x )i(i +1) + zx i(i +1)i(i )) wi(i ) + ( z y i(i )i(i +1) + z y i(i +1)i(i )) hi(i ), i, 1 i < m i, (10) li(i x ) = ( zi(i x )i(i 1) + zx i(i 1)i(i )) wi(i ) + ( z y i(i )i(i 1) +z y i(i 1)i(i )) hi(i ), i, i = m i > 1, (11) li(i x ) = ( ri I + ri III ) wi(i ) + ( ri II + ri IV ) hi(i ) i, m i = 1, (12) li(i x ) + ly i(i ) = w i(i ) + h i(i ), i, i, (13) Oi(i x ) = cx i(i ) + ( zi(i x )i(i +1) zx i(i +1)i(i )) a x i(i ) + ( z y i(i +1)i(i ) z y ) y i(i )i(i +1) a i(i ), i, i, 1 i < m i, (14) Oi(i x ) = cx i(i ) + ( zi(i x 1)i(i ) zx i(i )i(i 1)) a x i(i ) + ( z y i(i )i(i 1) z y ) y i(i 1)i(i ) a i(i ), i, i, i = m i > 1, (15) O y i(i ) = cy i(i ) + ( zi(i x )i(i +1) ) zx y i(i +1)i(i ) a i(i ) + ( z y i(i )i(i +1) z y i(i +1)i(i )) a x i(i ), i, i, 1 i < m i, (16) O y i(i ) = cy i(i ) + ( zi(i x 1)i(i ) ) zx y i(i )i(i 1) a i(i ) + ( z y i(i 1)i(i ) z y i(i )i(i 1)) a x i(i ), i, i, i = m i > 1, (17) I x i(i ) = cx i(i ) + ( zi(i x )i(i +1) zx i(i +1)i(i )) b x i(i ) + ( z y i(i +1)i(i ) z y ) y i(i )i(i +1) b i(i ), i, i, 1 i < m i, (18) I x i(i ) = cx i(i ) + ( zi(i x 1)i(i ) zx i(i )i(i 1)) b x i(i ) + ( z y i(i )i(i 1) z y ) y i(i 1)i(i ) b i(i ), i, i, i = m i > 1, (19) I y i(i ) = cy i(i ) + ( zi(i x )i(i +1) ) zx y i(i +1)i(i ) b i(i ) + ( z y i(i )i(i +1) z y i(i +1)i(i )) b x i(i ), i, i, 1 i < m i, (20) I y i(i ) = cy i(i ) + ( zi(i x 1)i(i ) ) zx y i(i )i(i 1) b i(i ) + ( z y i(i 1)i(i ) z y i(i )i(i 1)) b x i(i ), i, i, i = m i > 1, (21) Oi(i x ) = cx i(i ) + ( ri I ri III ) a x i(i ) + ( ri II ri IV ) y a i(i ), i, i, m i = 1, (22) O y i(i ) = cy i(i ) + ( ri I ri II ) y a i(i ) + ( ri IV ri II ) a x i(i ), i, i, m i = 1, (23) Ii(i x ) = cx i(i ) + ( ri I ri III ) b x i(i ) + ( ri II ri IV ) y b i(i ), i, i, m i = 1, (24) I y i(i ) = cy i(i ) + ( ri I ri III ) y b i(i ) + ( ri IV ri II ) b x i(i ), i, i, m i = 1. (25) The objective function of problem (MIP-ASLP1), Equation (1), minimizes the total distances traveled by the internal material flows between the Input/Output (I/O) points of assembly lines and the external material flows and between the I/O points of the facility and the I/O points of assembly lines. The boundaries of sub-lines are constrained within the facility by Equation (2). The sub-lines of the same assembly line are connected by constraining the distance between centroids of adjacent sub-lines from the same assembly line in Equations (3) and (4). The relative location of non-adjacent sub-lines from the same assembly line is constrained in Equations (5) and (6). In Equation (7), we denote that for any pair of sub-lines there is only one relative location relationship. In Equation (8), we ensure that for each I-shaped line, only one of the four different orientations can be taken. We prevent the overlapping of the sub-lines from the different assembly lines through Equation (9). The side length of the sub-lines are determined in Equations (10) (13), where we use Equation (10) and (11) to determine the side length of sub-lines for multiple rectangles lines and Equation (12) to determine the side length of I-shaped assembly lines. The complementary dimension of each sub-line is determined in Equation (13). The I/O points of sub-lines of multiple rectangles lines are defined according to the relative location of the I/O points to the centroids of sub-lines in Equations (14) (21). The I/O points of the sub-lines of the I-shaped assembly lines are defined according to the relative location of the I/O points

6 58 Bukchin et al. to the centroids of the assembly lines in Equations (22) (25). The non-linear absolute value in the objective function can be easily addressed by the following replacement: a b = ab + + ab, where a, b, ab +, ab Assembly system layout formulation performance analysis The following numerical tests are based on problem (MIP- ASLP1), Equations (1) (25), given in Section 3. Throughout the testing, each problem is solved with the CPLEX 7.0 platform. Although it is clear that the run time for solving the formulation directly increases as an exponential function of the problem size, it is interesting to see what problem size can be solved in a reasonable time, and to identify empirically the factors that most affect the run time. Four main types of problems were solved. The first type consisted of only I-shaped lines, the second, L-shaped lines (two connected sub-lines in each line), the third, U-shaped lines (three connected sub-lines in a line), and the fourth, mixed-shaped lines. For each problem type, several instances were solved with different characteristics, such as, the number of lines, the size of the rectangular facility, and the number and location of the I/O points. The run time of each instance was recorded. The results are listed in Table 1. Columns 1 to 4 denote the data set, the number of assembly lines, the type of the lines (I-shaped, L-shaped, U-shaped or mixed) and the total number of sub-lines, respectively. Columns 5 to 7 refer to the facility size while denoting the dimensions of the facility and the area utilization, namely, the percentage of the facility area occupied by the assembly lines. In columns 8 to 10 we can see the number of input points, the number of output points, and the CPU run time, respectively. In order to examine how the problem parameters affect the run time, a multiple regression analysis was conducted with four independent variables: (i) the total number of lines; (ii) the total number of sub-lines; (iii) the facility area utilization; and (iv) the total number of I/O points. The best model was obtained by taking log 10 (run time) as the dependent variable with p-values of The model shows that the run time increases significantly (p-value smaller than 0.05) with the number of sub-lines as well as with the area utilization and with the number of I/O points. The former is easily understood since the number of sub-lines is directly related with the problem size. The last two effects can be explained by noting that the problem becomes more constrained as the facility area in which the lines are located decreases and as the number of I/O points increases. For a highly constrained problem, an optimal solution is more difficult to achieve. Note that the variable denoting the number of lines was not found to be significant. The reason for this is that this variable alone ignores the number of sub-lines constructing each line, and therefore, the information regarding the number of lines alone Table 1. Solution CPU run times of the MIP formulation for the assembly system layout problem Data set Number of lines Type Number of Fac Fac Area FI FO sub-lines X Y util. size size CPU time (sec) I I I I I I I I I I I L L L L L L L L L L L L L L L L L L L U U U U U U U U U U U Mixed Mixed Mixed Mixed Mixed is insufficient to indicate the problem size. Consequently, another regression model was conducted, this time without the number of lines as a dependent variable. The ANOVA table is presented in Tables 2(a) and 2(b). One can see that the p-value of the model has improved, and is now equal to

7 Assembly system facility design 59 Table 2a. ANOVA table for the run time performance df SS MS F Significance F Regression Residual Total The regression equation is then given by the following expression: log 10 (run time) = (no. of sublines) (area util.) (no. of I/O points). In a real-world situation, the number of I/O points is usually relatively low. Therefore, the most influencing and interesting factors are the number of sub-lines and the area utilization. In Fig. 4 we can see three graphs demonstrating the run time as a function of the number of sub-lines, each for a different level of area utilization. We can first see the exponential nature of the graphs with respect to the area utilization level. In addition, one can see that given a relatively low area utilization, on the average, problems up to around 50 sub-lines can be solved to optimality in a reasonable amount of time. This number goes down to around 28 sub-lines for higher values of area utilization. One should note that estimating the run time based on the average values obtained by the regression model may be quite risky due to the high variance of the run time. For example, the average run time of a configuration of 30 sublines, 50% utilization level and two I/O points is 2.23 hours. However, looking at worst-case scenarios based on the 90% confidence interval of each of the coefficients (keeping the other coefficients at their average values), the expected run time becomes much larger. Run time values of 56.3, 21.3 and 13.2 hours are obtained for the worst case of the coefficient of the number of sub-lines, area utilization and the number of I/O points, respectively. Looking at the worstcase scenario of this problem based on the 90% confidence interval of all the coefficients simultaneously, one would get a run time value of hours. This value is of course a result of the compounding of the three coefficients and is not expected to occur; however, it does give an indication of the large variability of the run time. As we realized that the model is highly affected by the area utilization, and since we assume that in a real-world environment, the facility area is well utilized, we tested the run time over several instances of very-high-facility-area uti- Table 2b. Further ANOVA data on the run time performance Standard Coefficients error t Stat P-value Intercept No.of sub-lines Area util No.of I/O points lization. The largest problem that could be solved in a reasonable amount of time (12.3 hours) was of seven (mixed) assembly lines (13 sub-lines). Such a problem represents a reasonably large assembly facility. Details on this example problem are presented next. The material flow within the example assembly system is given via the precedence diagram shown as Fig. 5. Each arrow denotes a flow of material, the circles are the I/O points of the facility and within each rectangle we can see the line number and its shape in parentheses. We located the assembly system in a facility with dimensions of and the area utilization is 80%. The optimal layout solution that minimizes the distance flow, is shown in Fig. 6, with an objective value of 42 distance units. 5. A cost model for the Assembly System Facility design problem Problem (MIP-ASLP1) locates the set of assembly lines in a given facility area and minimizes the weighted distances between the input and output points of the facility and the assembly lines. Where an automated material handling system exists to move material between the input and output points, the total cost (investment plus operational costs) is assumed to increase with the distance traveled. On the other hand, where such a system does not exist, the weighted distances are associated with the operational cost of transportation. For real-world data, the net present value of the operational costs has to be evaluated based on a planning horizon in order to be compared with the investment cost. In either case, we assume that the total material handling costs increase as the distances between the input and output points increase. Another important aspect to consider when designing (or constructing) an assembly system facility is the cost due to the size (area) of the facility. That is, since in practice a cost is associated both with material handling and with the area of the facility, a mechanism is needed to analyze the trade-off between the two. To that end, a design model is needed to examine the impact of changes in the facility area on the resulting changes in material handling cost. In developing a model for the design or construction of a facility, we may ignore the placement of some or all of the facility I/O points since such points would typically be determined after the lines have been placed (otherwise it would be very difficult to assign lines to I/O points without knowing both the location of the lines and the input points). Still, the location of one point, the output point of

8 60 Bukchin et al. Fig. 4. Run time as a function of the number of sub-lines and area utilization. the facility, can generally be determined and we use it to orient the general direction of the material flow. By considering only one facility output point (and ignoring the facility input points in the experiments) it will also make it much easier to standardize our test problems to isolate the impact of facility size versus material handling distance. In general, note that the minimal material handling transportation costs will be achieved for a facility that is large enough so as to not limit the placement of the assembly lines. For any problem, the maximum side length needed, L max, is equal to n i=1 max{ m i i =1 h i(i ), m i i =1 w i(i )}. Of course, a facility measuring L max L max would be very under-utilized. However, as the facility dimensions are reduced to decrease the area of the facility and to increase the facility utilization, the placement of the assembly lines is constrained, and as a result, the material handling transportation costs will rise in general (and certainly cannot decrease). In general, due to department shapes that have been determined a priori, it is not possible to calculate the minimum feasible facility dimensions, but clearly n i=1 mi i =1 h i(i )w i(i ) serves as a lower bound to the facility area (L x L y ). We define the bi-objective Assembly System Facility Design problem (ASFDP) as follows: (ASFDP): where w 1 and w 2 are the cost of one-unit of transportation and one-unit of area, respectively, and Transportation is defined by Equation (1), the objective to problem (MIP-ASLP1). Such a formulation has reduced a multiobjective optimization problem to a single-objective problem. Still, two problems arise here. First, the values of the weights are usually difficult to obtain, and some times other non-quantitative considerations are involved. To that end, we prefer to avoid the setting of the weights in the objective and instead adopt an efficiency frontier approach to study the different possible feasible solutions prior to selecting the preferred one. The second problem involves the non-linearity of the objective function, Equation (26), due to the facility area term. This problem is addressed below. min{w 1 (Transportation) + w 2 L x L y }, (26) Fig. 5. Material flow within the example assembly system. Fig. 6. Optimal layout of the example problem.

9 Assembly system facility design 61 Let us first look at the extreme case of the design problem, in which the material handling cost is minimized subject to constrained dimensions of the facility, while ignoring the cost of the area. Although solving problem (MIP-ASLP1) will provide the optimal solution, this solution may be somewhat inefficient, since the same transportation cost may be possible for a smaller area than the one defined by the constraint. Hence, we modify problem (MIP-ASLP1) (resulting in problem (MIP-ASLP2)) with a new objective function to incorporate the above aspect of facility size; i.e.: (MIP-ASLP2): min{m(transportation) + L x L y }, (27) Subject to Equations (2) (25), L s L s s. (28) Since M is a large number, the transportation cost becomes the primary objective and the facility area is the secondary objective, which allows us to find the minimum transportation cost for particular facility side lengths. Note that eliminating Equation (28) from problem (MIP-ASLP2), we would get the formulation for finding the minimum transportation cost along with the largest possible area beyond which there is no further reduction in the transportation cost. Similarly, if in addition we change Equation (27) to min{(transportation) + ML x L y }, the smallest possible area will be obtained, beyond which placement of the lines becomes infeasible. One can see that problem (MIP-ASLP2) has a non-linear objective function. In order to make a linear approximation, we replace the area component in Equation (27) by the perimeter of the facility, assuming that in general the two components are highly correlated (Meller et al., 1988), resulting in problem (MIP-ASLP3). MIP-ASLP3: min{m(transportation) + 2(L x + L y )} (29) Subject to Equations (2) (25) L s L s s. (30) Thus, solving problem (MIP-ASLP3) will then provide an approximation to the minimal area solution for the lowest transportation cost given the maximal values of the facility s dimensions. However, our original facility design problem is to consider the trade-off between material handling costs and facility area costs. To that end, we adopt the efficiency frontier approach (Steuer, 1986). According to this approach, all or some non-dominated (efficient) solutions have to be generated, and the preferred solution is to be selected later by the decision-maker. A solution i is defined as efficient if there is no other solution, j, which is at least as good as solution i in both objectives. The efficiency frontier approach gives a full picture of the trade-off between the two objectives as we show below. One can see that solving problem (MIP-ASLP3) with Equation (29) yields an approximation to the extreme point of the efficiency frontier, where the transportation costs are considered as a primary objective and the area cost as a secondary objective. One way to generate the efficiency frontier is by changing gradually the values of w 1 and w 2 and using Equation (26) in problem (MIP-ASLP2) (or a modified version with a perimeter component instead of the area component). Since we do not know how the efficient solutions are distributed along the efficiency frontier, this approach may consume a large number of iterations, and worse, skip efficient solutions. Instead, we suggest using the following algorithm to generate the approximated efficiency frontier. The algorithm is based on constraining one component of the objective function (the perimeter value in our case) while minimizing the other component (the transportation distance). Algorithm EF Step 1. Solve MIP-ASLP-3. Step 2. Set i, the iteration index, equal to one; i.e., i = 1. Step 3. Calculate the facility perimeter resulting from Step 1, P i. Step 4. Set P i+1 = P i, where is a resolution parameter. Step 5. Solve problem (MIP-ASLP-3) with the following additional constraint, 2(L x + L y ) P i+1. Step 6. If no feasible solution obtained, go to Step 7. Otherwise, set i = i + 1, keep the obtained solution S i, Fig. 7. Shapes of the lines in the example problem.

10 62 Bukchin et al. Table 3. The algorithm results of the example problem Solution Lx Ly Perimeter Area Utilization z Fig. 8. Material flow in the example problem. with a perimeter value of P i and a distance value of distance D i, and go to Step 4. Step 7. For each solution i: (i) calculate the facility area A i ; (ii) if there is any solution, S j, where D j D i and A j A i, eliminate solution S i as a dominated solution. The remaining set of solutions is the approximated efficiency frontier. Note that in Step 5, one could use the constraint 2(L x + L y ) < P i instead of the current constraint. In this case, each time a solution with P i was obtained, the above constraint including the strict inequality would have been used in the next iteration. This approach would guarantee finding all efficient solutions with regard to the perimeter as one of the objectives. The reason for preferring the suggested constraint is that the problem is mixed integer, the solution is partially continuous, and consequently, the efficiency frontier may contain an infinite number of solutions. In the proposed approach, we can control the number of efficient solutions in the set by changing the resolution parameter,. The following example demonstrates the generation of the approximated efficiency frontier. The considered assembly system consists of five lines and nine sub-lines, as can be seen in Fig. 7. The material flow is shown in Fig. 8. The figures in the parentheses represent the sub-lines composing each line. The facility output point was forced to the bottom of the facility (y = 0) with the x-value being left as a variable. Such an approach is flexible in terms of the lay N/A N/A N/A N/A N/A N/A out algorithm as well as the facility design algorithm since a particular facility output point may constrain the size of the facility. Without loss of generality, Algorithm EF was started without limiting the initial area and the value of was set to one-unit of distance. Twenty iterations were performed and 19 solutions were obtained, as is shown in Table 3. In columns 2 and 3 we can see the dimensions of each layout. Then, the associated perimeter and area are presented in columns 4 and 5, respectively. The area utilization is presented in column 6 and the total transportation distance presented in column 7. We can see that the first solution is the solution of problem (MIP-ASLP3) with a perimeter Fig. 9. Approximated efficiency frontiers for the example problem: (a) the approximated efficiency frontiers associated with the perimeter and transportation components; and (b) the approximated efficiency frontier associated with the total area and transportation components.

11 Assembly system facility design 63 Fig. 10. Total distance traveled against the area utilization. value of 83 distance units, an area of distance units squared, and an objective function of one distance unit. We can see that no feasible solution was obtained for a perimeter smaller than or equal to 51 distance units. Another interesting observation concerns solutions 3 and 6. These solutions were found to be dominated by other solutions, since in their iteration, the obtained area was increased although the perimeter was decreased. These solutions were eliminated as specified by Step 7 of Algorithm EF. In. Fig. 9(a) we can see the approximated efficiency frontier associated with the perimeter and transportation components. In this graph we can see that all 19 solutions are efficient. Note that the algorithm guarantees that each new solution will be an efficient one with regard to the perimeter objective. The reason for this is that each new solution is superior to the previous solutions with regard to the perimeter (due to Steps 4 and 5 of the algorithm) and inferior with regard to the transportation cost (otherwise, the value of the transportation cost would be obtained in a previous iteration while solving a less-constrained problem). In Fig. 9(b) we see the approximated efficiency frontier associated with the total area and transportation components is presented, before eliminating the two dominated solutions, which are marked by arrows. Here we can see that the facility perimeter is a relatively good surrogate measure for the facility area, since 17 out of the 19 perimeter-efficient solutions are also area-efficient solutions. Another perspective on the results can be obtained by looking at Fig. 10, and considering the relationship between the transportation distances and the area utilization. One can see that the optimal solution with regards to the transportation distances is obtained for an area utilization of 37.8%. The utilization is then increased by decreasing the facility perimeter until a maximal value of 89.7% is reached (with an associated transportation distance of 27.5 Fig. 11. Three solutions of the example problem: (a) = (z = 1); (b) = 221 (z = 6.5); and (c) = 165 (z = 27.5).

12 64 Bukchin et al. distance units). Again, we can see the two dominated solutions, marked by the arrows that should be omitted from the efficient set. Another insight can be obtained by looking at the three layouts presented in Fig. 11(a c). In Fig. 11(a), we can see the first solution obtained by Algorithm EF, which is optimal with regard to the transportation distance. In Fig. 11(c), the last solution obtained by the algorithm, with the smallest area and the highest utilization, is presented. In between, we can see a solution that was taken from the middle area of the efficiency frontier (i.e., a solution that appears to represent a good trade-off of the two objectives). In fact, this solution, S i, was selected based on the following measure, i = arg min i [ max ( Ai A min A max A min, z i z min z max z min )], namely, the solution with the smallest maximal relative distance from either of the two objectives. This solution represents a compromise between the two components of the objective function and is relatively good with respect to both components, with a total distance of 6.5 distance units and an area utilization of 67%. Note that in some cases this solution may be preferred over the other solutions if the weights, representing the costs, are relatively close. However, if there is one weight which is much higher that the other, a solution much closer to one edge of the efficiency frontier will be chosen. Such an efficiency frontier approach will provide quite a bit of information to the decision-makers as they design their assembly system facility. In addition to providing the design that represents the best trade-off between material handling cost and area utilization, the chosen design will also imply the values of w 1 and w 2 in Equation (26). Doing so may then allow the solution of the assembly system facility design problem directly in future iterations. 6. Concluding remarks In this paper we considered a facility design problem for a facility that consists of a system of assembly lines. In solving this problem, we introduced a new facility layout problem. To solve this layout problem, we proposed a MIP formulation that models each line as a set of rectangular sub-lines. Our experimental results indicate that small-to-moderatesized problems can be solved with this formulation depending on the number and types of lines considered. We also discovered that the percentage of facility area that was occupied and the number and location of I/O points also has an impact on the run time. Due to our performance analysis of the assembly system layout formulation, we believe that many assembly system facilities can be solved in a reasonable amount of computational effort with our approach. Still, there may exist problems that are larger than the ones shown here. In such a case, a heuristic will be needed. Development of a heuristic is left for future research. To solve the facility design problem, the efficiency frontier approach was applied to analyze the trade-off between the facility area and the transportation distance. An algorithm able to generate a set of efficient solutions was presented. On the practical side, the efficiency frontier can be used by designers to solve the facility design problem enabling them to see the whole picture and consider the complex trade-off between facility area costs and material handling costs. References Becker, C. and Scholl, A. (2004) A survey on problems and methods in generalized assembly line balancing. European Journal of Operational Research (forthcoming). Das, S.K. (1993) A facility layout method for flexible manufacturing systems. International Journal of Production Research, 31(2), Erel, E. and Sarin, S.C. (1998) A survey of the assembly line balancing procedures. Production Planning & Control, 9(5), Heragu, S.S. and Kusiak, A. (1991) Efficient models for the facility layout problem. European Journal of Operational Research, 53, Meller, R.D. and Gau, K.-Y. (1996) The facility layout problem: recent and emerging trends and perspectives. Journal of Manufacturing Systems, 15(5), Meller, R.D., Narayanan, V. and Vance, P.H. (1998) Optimal facility layout design. Operations Research Letters, 23(3 5), Montreuil, B. (1990) A modelling framework for integrating layout design and flow network design, in Proceedings of the Material Handling Research Colloquium, pp Sarkar, A., Batta, R. and Nagi, R. (2005) Planar area location/layout problem in the presence of generalized congested regions with the rectilinear distance metric. IIE Transactions, 37(1), Savas, S., Batta, R. and Nagi, R. (2002) Finite-size facility placement in the presence of barriers to rectilinear travel. Operations Research, 50(6), Scholl, A. and Becker, C. (2004) State-of-the-art exact and heuristic solution procedures for simple assembly line balancing. European Journal of Operational Research (forthcoming). Sherali, H.D., Fraticelli, B.M.P. and Meller, R.D. (2003) Enhanced model formulations for optimal facility layout. Operations Research, 51(4), Steuer, R.E. (1986) Multiple Criteria Optimization: Theory, Computation and Application, Wiley, New York, NY. Biographies Yossi Bukchin is a faculty member in the Department of Industrial Engineering at Tel Aviv University. He received his B.Sc., M.Sc., and D.Sc. degrees in Industrial Engineering from the Technion Israel Institute of Technology. He is a member of the Institute of Industrial Engineering (IIE) and on the Editorial Board of IIE Transactions. Dr. Bukchin has held a visiting position in the Grado Department of Industrial & Systems Engineering at Virginia Tech. His papers have been published in IIE Transactions, European Journal of Operational Research, International Journal of Production Research, Annals of the CIRP and other journals. His main research interests are in the areas of assembly systems design, assembly line balancing, facility design, operational scheduling, multi-objective optimization as well as work station design with respect to cognitive and physical aspects of the human operator.