MSEC PLANT LAYOUT OPTIMIZATION CONSIDERING THE EFFECT OF MAINTENANCE

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1 Proceedings of Proceedings of the 211 ASME International Manufacturing Science and Engineering Conference MSEC211 June 13-17, 211, Corvallis, Oregon, USA MSEC PLANT LAYOUT OPTIMIZATION CONSIDERING THE EFFECT OF MAINTENANCE Seungchul Lee University of Michigan Ann Arbor, MI, USA Adam Brzezinski University of Michigan Ann Arbor, MI, USA Jun Ni University of Michigan Ann Arbor, MI, USA ABSTRACT With increasing production costs and constraints, demand has increased for manufacturers to minimize maintenance cost and product transport time. We address some aspects of this problem by examining how to choose the optimal layout of stations (machines or buffers) in a production facility based on how the station layout affects the maintenance and product transport times. Specifically, we consider how the location of the stations relative to the maintenance facility affects the overall maintenance time as well as how the location of the final station affects the product transport time. Hence, we can address maintenance cost during the design-phase of a production facility. By employing discrete-design optimization techniques, we generate and evaluate various station layouts to choose an optimal layout which satisfies all geometric and adjacency constraints. We focus on a single, serial production line including a set of n stations 1 INTRODUCTION Station layout is a key aspect of the design of a manufacturing system. The facility layout problem focuses on finding the most efficient arrangement of n machines within a facility [1 3]. Mathematical programming [4 6], graph theory [7 9], meta-heuristic search [1 12], and simulation [13] have all been applied to optimize the layout of a manufacturing system. However, a few researchers have included an aspect of maintenance when choosing an optimal station layout; instead, most researchers have focused on minimizing material handling cost. In an increasingly competitive market, facility designers can gain a competitive advantage by including additional aspects which affect product cycle time in their manufacturing facility layout optimization. Hence, we discuss how maintenance downtime and product transport time can be included into a model for facility design optimization. We use discrete-design techniques including graph theory and graph search algorithms to generate and evaluate various station layouts. We consider a two-dimensional production facility in which the order of machines in a line is given and all production lines are serial. Then we search for the optimal layout which satisfies the following requirements: 1) The layout must be contained inside the production facility (a 4 4 grid) and none of the machines may overlap any other machines. Furthermore, 2 machines may share grid points if and only if the end point of one machine corresponds to the start point of the other machine. 2) The last machine in a production line must be as close to the exit of the production facility as possible. 3) Each machine in line must be as close to the maintenance facility as possible. Figure 1 shows the layout of our production facility. Since the maintenance facility is far from the exit of the production facility, we expect that trade-offs will occur between satisfying items 1) and 2) above. Note that the corners of a machine must correspond to points on the production facility grid. The remainder of this paper is organized as follows. Our modeling approach is described in Section 2. Section 3 discusses the optimization method (Dijkstra s algorithm) we use to obtain an optimal plant layout. In Section 4, we provide numerical examples to illustrate the proposed technique. Finally, Section 1 Copyright c 211 by ASME

2 4 3 3 A = B = C = Drawings of each type of machine FIGURE 1: Overview of Maintenance Facility Corners of machine A FIGURE 2: Machine types and corner conventions provides conclusions and future work. 2 MATHEMATICAL MODEL Here we discuss our notation conventions and define our design variables, constraints, and objective function. 2.1 Notation Conventions 1) The machine types (see Figure 2) are given by MT {A,B,C}. 2) The corners associated with machine i (see Figure 2) are given by {(x i j,y i j ), j = 1,,m}, (1) where m is the total number of corners of machine i. 3) The probability of failure of machine type H is given by P H = p : p [,1], (2) 2.2 Design Variables and Constraints 1) The sum of the distances from each machine to the maintenance facility (see Figure 3) is given by d maint = S i=1 d i. () 2) The distance from the final machine to the production facility exit (see Figure 3) is given by d exit = (4 x S ) 2 + (4 y S ) 2. (6) 3) The entire layout is contained in a 4 4 grids: 4) No two machines can overlap. {(x i,y i ) Z 2 : x i,y i 4} (7) where H {A,B,C}. We assume without loss of generality that all machines are equally likely to fail. 4) The distance from machine i to the maintenance facility is d i. ) The sequence of machines in a production line is given by where S i MT. 6) The total number of machines is S = {S 1,,S n }, (3) n = S. (4) 2.3 Model Development The objective function J is given by J ω M d maint + ω E d exit, (8) where ω M and ω E are the relative weights of maintenance and product transport times, respectively. We define the homogeneous transformation matrices [14,1] T = 1 1,T a = 1 1 2,T b = ,T c = (9) 2 Copyright c 211 by ASME

3 T b T a O 3 O 3 d 2 2 d 3 d exit 1 T a O O FIGURE 4: Modeling of station A FIGURE 3: Illustration of linear distances from maintenance facility and production facility exit to each machine. These matrices allow us to translate and rotate the origin of our coordinate system, which we use to generate all possible station layouts. First we define T A1 T a T b T a = (1) 1 o o o' A1 A2 o' o B o' o o' o o' o o' C1 C2 C3 to illustrate how Equation 9 can be applied to move the origin based on the insertion of station A into the production line. Assuming the origin begins at (,), T A1 translates and rotates the origin from its initial position and orientation to its final position and orientation. This procedure is illustrated in Figure 4. Next, we define 1 3 T A2 T a T b T c = 1 3, T B T b T b = 1 1 4, (11) 1 1 which translate and rotate the origin as station B or a mirror image of station A, respectively. Finally, we note that station C translates and rotates the origin through either T a, T b, or T c. An illustration of how stations A, B, and C can manipulate the origin is shown in Figure. Once we have defined a sequence of stations, we use T A1, T A2, T B, T C1, T C2, and T C3 to map the origin of the coordinate system from the starting point and orientation of the production line to the end point and final orientation of the production line. Each machine in the production line is inserted into the line by mapping the origin through a coordinate transformation from its current position to its new position. In essence, the production line is represented as a mapping of the origin from the starting point through n transformation matrices to the end of the production line. FIGURE : How the addition of a station changes the orientation and position of the origin Before the origin is mapped by a transformation matrix, the corner locations associated with each machine are calculated. These locations are used to assess the feasibility of each layout generated. If any corner of any machine in the production line lies outside the production facility or overlaps a corner of another machine, the corresponding production line is infeasible and therefore rejected as a possible optimal solution. 3 OPTIMIZATION METHODS For a station sequence S = {A,B,A,B, }, where S = 18, we first generate all 2 9 = 12 possible station layouts. Next, we check all of the corner locations associated with each of the stations in each given layout. If all corners associated with all stations in a given production line satisfy the design constraints, we store the associated production line as a feasible layout. Figures 6 and 6 show infeasible and feasible station layouts, respectively. 3 Copyright c 211 by ASME

4 d 6,2 1 2 d 6, An infeasible layout FIGURE 7: Illustration of how d i,1 and d i,2 are calculated A feasible layout Realistic Distance Calculation In a production facility, some stations may block the maintenance crew from walking in a straight line. Also, the location on the station where maintenance must be performed may not be the closest corner of the station to the maintenance facility. These concepts are illustrated in Figure 8. Here we discuss a more realistic distance measurement d i,2 which accounts for station interference effects and fixed maintenance locations. Note that d i,2 is calculated using the solid lines in Figure 7. FIGURE 6: Various layouts evaluated by the proposed algorithm. 3.1 Distance Calculation Naive Distance Calculation First we calculate the distance from the maintenance facility to each station by assuming that the maintenance worker could travel in a straight line from the maintenance facility to the station. This distance estimate is given by d 1,2 d 3,2 d 2,2 Maintenance points d i,1 xi 2 + (4 y i) 2. (12) In this case, a maintenance worker could pass through a station. Although this distance calculation is unrealistic, it provides a rough estimate of the relative distances of each station from the maintenance facility. The distance from the final station to the exit of the facility is calculated in the same way, ignoring potential interference effects from other stations in the production facility. The dotted lines in Figure 7 illustrate this type of distance measurement. FIGURE 8: Illustration of how more realistic distance is determined To calculate d i,2, we use Dijkstra s algorithm [16] to find the shortest path from the maintenance facility to the maintenance point on each station. Dijkstra s algorithm can be used to find the shortest path in a graph [17]. We also apply Dijktra s algorithm to find the shortest path from the material handling point on the last station to the production facility exit. The length of the path 4 Copyright c 211 by ASME

5 from the maintenance facility to station i is denoted by d i,2. As the left drawing in Figure 8 shows, d i,1 and d i,2 are the same for some machines but not others. The right side of Figure 8 shows how we define fixed maintenance points for each type of station. Node 1 added 3.2 Removal of Infeasible Layouts Instead of generating all possible layouts before evaluating the feasibility of each machine in each possible layout, we use a graph theory [18] to facilitate efficient layout generation. A graph G = (V,E) consists of a non-empty finite set V, whose members are called nodes, together with a set E of unordered pairs (u,v) of nodes, where u v. The elements of E are called edges. In this paper we are concerned with graphs whose nodes represent machines such that the edges (u,v) imply that u and v are adjacent. We initialize a tree with the first node corresponding to the first station. Then each subsequent station is added to the tree as a set of child nodes of the previous station. For example, if the machine sequence begins with machine A facing to the right (as shown in Figure 9), the first level of the tree is composed of a single node. If the next station in the sequence is station B, then only one child of the first node exists (since station B can only manipulate the origin in a single way) and the second level of the tree also comprises a single node. However, if the third station of the sequence is station A, the third level of the tree comprises of 2 nodes, because station A can manipulate the origin in 2 possible ways. When this procedure is extended to a sequence of 18 stations in the {A,B,A,B, } configuration, the tree structure in Figure 9 results. In this paper, a node corresponds to a specific station type and orientation. For example, while both nodes 3 and 4 in Figure 9 correspond to station A, node 3 corresponds to one orientation of station A while node 4 corresponds to another. By using a tree structure to generate feasible layouts, we are able to stop generating stations for a layout once we know that at least one station in that layout is infeasible. By pruning infeasible nodes, we obtain a tree structure where only the branches which have not been pruned correspond to feasible layouts. 3.3 Optimal Solution Calculation If d i = d i,1, we can convert the tree generated in Section 3.2 into a graph. To do so, we assign the cost of edge j to be d j+1,1. Since the location and orientation of the first machine in the sequence is fixed, d 1,1 does not affect which layout is optimum. Once the tree has n levels, we add d exit to the edge weight of the final edge. This procedure for assigning edge weights is illustrated in Figure 1. In this case, the edge weight assignment can be done dynamically as the tree grows. Once we have assigned costs to each of the edges in the tree, we generate a graph. Then, we use Dijkstra s algorithm again node 4 infeasible(in) IN IN Node 3 added IN IN Node 2 added node 1 node 2 node 3 IN IN IN IN IN IN Node 4 added IN IN IN IN IN FIGURE 9: Illustration of how nodes are added to a tree structure d 1,1 d 2, IN node 1 c 1 =d 1,1 node 3 node 2 c 2 =d 2,1 FIGURE 1: Way in which edge weights are assigned to tree structure Copyright c 211 by ASME

6 to search through the graph and assign node weights to each of the nodes in the graph. Once the node weights are assigned, we find the optimal solution by comparing the node weights of each node in the n th generation. We report the optimal layout by returning the set of nodes required to generate the n th node with the smallest node weight. Dijkstra s algorithm [16] can briefly be explained as follows: d 1,2 d 2,2 d 1,2 d 2,2 Input: graph G = V,E with edge weights c uv R, (u,v) E and start node s V Output: array ρ[v] containing shortest path length from s to each node v V 1: for each v V [G]\{s} do 2: ρ[v] 3: end for 4: ρ[s] : Q {s} 6: while Q / do 7: select an element v Q s.t. ρ[v] = min u Q ρ[u] 8: Q Q\{v} 9: mark v 1: for each ω Adj(v) do 11: if ω is not marked then 12: ρ[v] min{ρ[ω],ρ[v] + c vω } 13: Q Q {ω} 14: end if 1: end for 16: end while 17: return ρ[v] FIGURE 11: Illustration of how d i,2 can change as stations are added to a potential layout 4 RESULTS 4.1 Naive Distance Calculation We let S = {A,B,A,B, }, S = 18, ω M = 1, and ω E = 2 and evaluate the objective functions for each feasible station layout; we draw the optimal layout we obtain in Figure 12. Next, we let ω M = 1, and ω E = 21 and draw the associated optimal layout in Figure If d i = d i,2, we also generate a tree to evaluate the feasibility of each possible layout. In this case, we cannot assign edge weights because, as shown in Figure 11, as the tree corresponding to one layout grows, the straight-line access to some machines is impeded by the addition of new machines to the layout. Hence, one edge can have multiple weights and we cannot generate a graph for the tree because we cannot uniquely assign edge weights to all the edges in the tree. Since we cannot express this problem as a graph, we compare the objective function values associated with each of the feasible layouts and choose the feasible layout associated with the minimum objective function value as the optimal layout. FIGURE 12: weights Optimal station layouts for different relative Then we let S = {A,B,C,A,B,C, }, S = 1, ω M = 1, and ω E = 3. In this case, we obtain the optimal layout shown in Figure 13. If we let ω M = 1 and ω E =, we obtain the optimal layout shown in Figure 13. We summarize the results of these case studies in Table 1. 6 Copyright c 211 by ASME

7 higher objective function value than the optimal layout recommended by the realistic distance calculation when the realistic distance calculation is used to evaluate the cost function FIGURE 13: Comparison of optimal layouts for different relative weights J = J = TABLE 1: Summary of two case studies Sequence S # of feasible ω M ω E Opt. J layouts {A,B,A,B, } {A,B,C,A,B,C, } FIGURE 14: Effect of type of distance calculation on optimal layout We note that the optimal station layout obtained using the realistic distance calculation is less intuitive than when the simple linear distance is used. In Figure 14, we see the effect that the orientation of a station has on the distance the maintenance personnel must travel to service the station. For an optimal layout, it is more important that the maintenance points of as many stations as possible face the maintenance facility than that the station locations be clustered near the facility. We see from the above figures that the optimal configuration clusters the most stations as close as possible to the maintenance facility but also ends the layout as close to the exit of the facility as possible. When the relative weight of the exit distance increases (in Figure 12 and Figure 13), we see that the end of the production line is very close to the exit; when the relative weight of the maintenance distance increases, the stations are clustered around the maintenance facility (in Figure 12 and Figure 13). These results match our expectations. 4.2 Realistic Distance Calculation We let S = {A,B,A,B, }, S = 18, ω M = 1, and ω E = 2. For these relative weights, we obtain the optimal layout for the realistic distance calculation shown in Figure 14. The layout shown in Figure 14 is very different from the layout recommended by the linear distance calculation (see Figure 12). We use the realistic distance calculation to evaluate the cost of the layout in Figure 12 and plot the result in Figure 14. Comparing Figures 14 and 14, we see that the optimal layout recommended by the linear distance calculation has a significantly CONCLUSIONS In this paper, we provide a plant layout optimization technique which considers product transport and maintenance times. The proposed technique uses graph search theory to obtain an optimal layout for a set of n stations in a serial production line. We employ two metrics to measure the distance between the maintenance facility and each station. The metrics result in significantly different optimal layouts; the first distance metric requires less computational time than the second, but the second distance metric more accurately reflects the actual distance. In the future, we plan to investigate parallel and mixed production lines to address more general production facilities. We also plan to consider multiple production lines in a single facility and further refine our distance metric to reduce computational time. The complexity of this problem should be also carefully investigated to see if it is feasible for more complicated manufacturing facility. We plan to modify our proposed technique to deal with dynamically-changing probabilities of failure (for example, due to machine degradation). Finally, we will take the material handling cost into account to make our problem more realistic. 7 Copyright c 211 by ASME

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