Book: Facilities Planning 4th Edition

You are given the following layout consisting of 6 departments. The total layout is 60x30 in size (so each square is 10x10). Interior lines are just to indicate the 40x40 squares; these are not walls. The cost of moving a load from one department to another department (1) is 1 for all combinations of departments. 5 1 1 1 1 5 244 2 4 4 5 3 3 6 6 5 I The from-to chart is given in the table below. 1 2 4 5 6 from to 1 3 No 3 2 2 5 8 2 3 1 6 w w 3 2 5 7 2 4 1 3 2 5 4 2 N 0 3 2. 6 4 6 a) Calculate the distance based objective (book formula 6.1 on page 304) measuring distances rectilinear between department centroids. In your answer clearly document the steps you take and the intermediate results obtained. b) For the same layout determine the normalized adjacency score (formula 6.3), again using the data of the from-to chart. In your answer clearly document the steps you take and the intermediate results obtained. 304 Part Two DEVELOPING ALTERNATIVES: CONCEPTS AND TECHNIQUES to be used for the pair. Given the significant increase in the number of all possible department pairs as a function of the number of departments, relationship charts are often not practical for problems with 20 or more departments. The same increase in the number of department pairs also applies to from-to charts. For medium-to large-sized problems (say, 2030 or more departments), fill- ing out each entry in the from-to chart would not be practical. In such cases, how- ever, the from-to chart can be constructed in reasonable time by using the production route data for each product (or product family). For example, if product type A is processed through departments 1-2-5-7, and is moved at a rate of, say, 20 loads per hour, then we set fiz = fs = 152 = 20 in the from-to chart. Repeating this process for each product (or product type) completes the from-to chart, which is often a "sparse matrix" (i.e., it contains many blank entries). In fact, in many cases, it is useful to first construct a separate and complete from-to chart for each product (or product type) so product-level flow data remain available to the analyst at all times. Subsequently, individual product-level from-to charts can be combined into one cumulative chart, using appropriate weights for individual product types if necessary. Of course, the construction of the from-to chart in the above manner is done by computers in most cases. Also, if the unit of flow for the product changes as it moves from one process to the next, then appropriate multipliers can be inserted into the routing data to scale the flow intensity up or down based on the unit of flow. Layout algorithms can also be classified according to their objective functions. There are two basic objectives: one aims at minimizing the sum of flows times dis- tances, while the other aims at maximizing an adjacency score. Generally speaking, the former, that is, the distance-based" objectivewhich is similar to the classical quadratic assignment problem (QAP) objectiveis more suitable when the input data is expressed as a from-to chart, and the latter, that is, the adjacency-based" objective, is more suitable for a relationship chart. Consider first the distance-based objective. Let m denote the number of depart- ments, f, denote the flow from department i to department j (expressed in number of unit loads moved per unit time), and C, denote the cost of moving a unit load one distance unit from department i to department j. The objective is to minimize the cost per unit time for movement among the departments. Expressed mathemat- ically, the objective can be written as min z = fcd (6.1) i-1-1 where d; is the distance from department i to j. In many layout algorithms, d, is measured rectilinearly between department centroids, however, it can also be mea- sured according to a particular aisle structure (if one is specified). Note that the c;; values in Equation 6.1 are implicitly assumed to be independ- ent of the utilization of the handling equipment, and they are linearly related to the length of the move. In those cases where the C, values do not satisfy the above as- sumptions, one may set c;= 1 for all i and jand focus only on total unit load travel in the facility (i.e., the product of the fi, and the d,, values). In some cases, it may also be possible to use the cy; values as relative "weights" (based on unit load attri- butes such as size, weight, bulkiness, etc.) and minimize the weighted sum of unit load travel in the facility. 305 6 LAYOUT PLANNING MODELS AND DESIGN ALGORITHMS Consider next the adjacency-based objective, where the adjacency score is computed as the sum of all the flow values (or relationship values) between those departments that are adjacent in the layout. Letting Xi = 1 if departments i and jare adjacent (that is, if they share a border) in the layout, and 0 otherwise, the objective is to maximize the adjacency score; that is, max z = fij Bip (6.2) i-1 j-1 Although the adjacency score obtained from Equation 6.2 is helpful in comparing two or more alternative layouts, it is often desirable to evaluate the relative efficiency of a particular layout with respect to a certain lower or upper bound. For this pur- pose, the layout planner may use the following normalized" adjacency score: * 1 fijxy (6.3) , Note that the normalized adjacency score (which is also known as the efficiency rating) is obtained simply by dividing the adjacency score obtained from Equation 6.2 by the total flow in the facility. As a result, the normalized adjacency score is always between zero and 1. If the normalized adjacency score is equal to 1, it implies that all department pairs with positive flow between them are adjacent in the layout. You are given the following layout consisting of 6 departments. The total layout is 60x30 in size (so each square is 10x10). Interior lines are just to indicate the 40x40 squares; these are not walls. The cost of moving a load from one department to another department (1) is 1 for all combinations of departments. 5 1 1 1 1 5 244 2 4 4 5 3 3 6 6 5 I The from-to chart is given in the table below. 1 2 4 5 6 from to 1 3 No 3 2 2 5 8 2 3 1 6 w w 3 2 5 7 2 4 1 3 2 5 4 2 N 0 3 2. 6 4 6 a) Calculate the distance based objective (book formula 6.1 on page 304) measuring distances rectilinear between department centroids. In your answer clearly document the steps you take and the intermediate results obtained. b) For the same layout determine the normalized adjacency score (formula 6.3), again using the data of the from-to chart. In your answer clearly document the steps you take and the intermediate results obtained. 304 Part Two DEVELOPING ALTERNATIVES: CONCEPTS AND TECHNIQUES to be used for the pair. Given the significant increase in the number of all possible department pairs as a function of the number of departments, relationship charts are often not practical for problems with 20 or more departments. The same increase in the number of department pairs also applies to from-to charts. For medium-to large-sized problems (say, 2030 or more departments), fill- ing out each entry in the from-to chart would not be practical. In such cases, how- ever, the from-to chart can be constructed in reasonable time by using the production route data for each product (or product family). For example, if product type A is processed through departments 1-2-5-7, and is moved at a rate of, say, 20 loads per hour, then we set fiz = fs = 152 = 20 in the from-to chart. Repeating this process for each product (or product type) completes the from-to chart, which is often a "sparse matrix" (i.e., it contains many blank entries). In fact, in many cases, it is useful to first construct a separate and complete from-to chart for each product (or product type) so product-level flow data remain available to the analyst at all times. Subsequently, individual product-level from-to charts can be combined into one cumulative chart, using appropriate weights for individual product types if necessary. Of course, the construction of the from-to chart in the above manner is done by computers in most cases. Also, if the unit of flow for the product changes as it moves from one process to the next, then appropriate multipliers can be inserted into the routing data to scale the flow intensity up or down based on the unit of flow. Layout algorithms can also be classified according to their objective functions. There are two basic objectives: one aims at minimizing the sum of flows times dis- tances, while the other aims at maximizing an adjacency score. Generally speaking, the former, that is, the distance-based" objectivewhich is similar to the classical quadratic assignment problem (QAP) objectiveis more suitable when the input data is expressed as a from-to chart, and the latter, that is, the adjacency-based" objective, is more suitable for a relationship chart. Consider first the distance-based objective. Let m denote the number of depart- ments, f, denote the flow from department i to department j (expressed in number of unit loads moved per unit time), and C, denote the cost of moving a unit load one distance unit from department i to department j. The objective is to minimize the cost per unit time for movement among the departments. Expressed mathemat- ically, the objective can be written as min z = fcd (6.1) i-1-1 where d; is the distance from department i to j. In many layout algorithms, d, is measured rectilinearly between department centroids, however, it can also be mea- sured according to a particular aisle structure (if one is specified). Note that the c;; values in Equation 6.1 are implicitly assumed to be independ- ent of the utilization of the handling equipment, and they are linearly related to the length of the move. In those cases where the C, values do not satisfy the above as- sumptions, one may set c;= 1 for all i and jand focus only on total unit load travel in the facility (i.e., the product of the fi, and the d,, values). In some cases, it may also be possible to use the cy; values as relative "weights" (based on unit load attri- butes such as size, weight, bulkiness, etc.) and minimize the weighted sum of unit load travel in the facility. 305 6 LAYOUT PLANNING MODELS AND DESIGN ALGORITHMS Consider next the adjacency-based objective, where the adjacency score is computed as the sum of all the flow values (or relationship values) between those departments that are adjacent in the layout. Letting Xi = 1 if departments i and jare adjacent (that is, if they share a border) in the layout, and 0 otherwise, the objective is to maximize the adjacency score; that is, max z = fij Bip (6.2) i-1 j-1 Although the adjacency score obtained from Equation 6.2 is helpful in comparing two or more alternative layouts, it is often desirable to evaluate the relative efficiency of a particular layout with respect to a certain lower or upper bound. For this pur- pose, the layout planner may use the following normalized" adjacency score: * 1 fijxy (6.3) , Note that the normalized adjacency score (which is also known as the efficiency rating) is obtained simply by dividing the adjacency score obtained from Equation 6.2 by the total flow in the facility. As a result, the normalized adjacency score is always between zero and 1. If the normalized adjacency score is equal to 1, it implies that all department pairs with positive flow between them are adjacent in the layout