FACILITIES' CLOSENESS RELATIONSHIPS. The objective function in site layout planning is to minimize the total travel distance

Transcription

1 FACILITIES' CLOSENESS RELATIONSHIPS The objective function in site layout planning is to minimize the total travel distance within the site, which is a direct function of the desired closeness relationship among facilities Often, the inter-relationships among facilities are decided by the user and involve some degree of fuzziness and ambiguity Using the direct closeness relationship and the fuzzy logic theory to quantify the facilities relationships will be discussed in the following two subsections Direct Closeness Weights The closeness relationships represent the project manager's preference in having the facilities close or apart from each other, and can be determined using quantitative or qualitative methods Setting of the proximity or closeness relationships (weights) is mainly problem-dependent and is done through a pair-wise assessment of the desired relationship between each two facilities to be placed on the site If the two facilities are required to be close to each other, in the user's judgment, a high weight value is specified between them to force the two facilities to be close to each other in the optimization process, and vice versa Using Qualitative Measure Six closeness relationships are usually set in advance and the user can give desired weight values associated each category In the present study, the weight values used are shown in Table 1, expressing an exponential relationship with desired closeness A high proximity weight between two facilities means that they 1

2 share a high level of interaction and accordingly the distance between them should be small A weight value of unity also means that the two facilities have no interaction between them and the distance separating them is irrelevant Once all pair-wise assessments are made, a half-symmetrical preference matrix is constructed (Table 2) and used as basis for placing of facilities, as described later It is noted that the values in Table 1 are presented for illustration only and the project manager can set other values based on his/her own judgment or use a quantitative measure Table 1: Closeness Relationship Values Desired relationship between facilities (1) Proximity Weight (2) Absolutely necessary (A) Especially important (E) Important (I) Ordinary closeness (O) Unimportant (U) Undesirable (X) Table 2: Facilities Closeness Relationship Matrix Facility Number n-2 n-1 n* n * n: number of facilities 2

3 Using Quantitative Measure Instead of using a qualitative measure, a quantitative measure can be used to assess the facilities closeness values The actual flow of material, equipment, personnel, or/and information can be used directly as a closeness weight among facilities Also, the actual transportation cost per unit distance between facilities can be used as a measure for the closeness weight Knowing the closeness weights among facilities, constructing Table 2 will be the same as described above Fuzzy Logic Approach In large complicated sites, due to the fuzziness and ambiguity inherent in the facilities closeness relationships, the project manager may be unable to assess a value for the closeness weights among facilities However, the project manager may be able to specify facilities' relationships only in the form of linguistic expression such as "as far as possible" or "very close" It is difficult, therefore, to accurately quantify a closeness value (weight) between each two facilities To address this problem, a fuzzy quantifier module has been developed The developments made in this module use the concept of fuzzy set theory originated by Zadeh (1965) and the concepts of fuzzy control by Mamdani (1974) Fuzzy decision making has been applied successfully in many civil and construction engineering applications The main benefit of fuzzy sets over conventional sets is that fuzzy sets provide a representation of the degree by which elements belong to a set 3

4 In site layout planning, for example, a fuzzy linguistic variable "Closeness Weight" (R) is a fuzzy variable that can be represented by a family of linguistic terms (fuzzy sets A, E, I, O, U, and X as shown in Fig 1) These six fuzzy sets cover the space of closeness weight solutions, ranging from "Absolutely Important" for A to "Undesirable" for X Each of these sets (eg, set A, which signifies that the closeness between two facilities is "Absolutely Important") has a membership function with a triangular shape The six membership functions have some overlaps as shown in Fig 1 It is noted that, the ranges shown in Fig 1 for the different membership functions (eg, set A ranges from 25-75) were designed to exhibit an exponential increase in the closeness values when facilities are required to be close to each other This gives high weight values when facilities are required to be close to each other, thus enforcing this relationship later during layout optimization These membership functions are used to quantify a crisp value for the closeness relationship between each two facilities as discussed in the following sub-sections Degree of Membership (y) 1 X U O I E A Linguistic Sets A E I O U X Absolutely Important Especially Important Important Ordinary Unimportant Undesirable Closeness Values Figure 1: Fuzzy Sets for the Output Variable "Closeness Weight" 4

5 Membership Functions and Fuzzy Decision Rules In plant layout planning, various factors have been considered by researchers in determining the closeness weight among departments, including equipment flow, material flow, personnel flow, and information flow In construction sites, besides the flow of material, equipment, personnel, and information, there are still other important factors, which can affect the location of temporary facilities used by experts in site layout planning For example, safety/environmental concerns, quality assurance, supervisory requirements, building code restrictions, and user preference If the construction activities require strict quality control, then the pollutant facilities should be isolated from the construction activities even there must be a long travel distance between them To help prevent construction equipment from being stolen, it is better to locate the equipment storage yard in areas that have appropriate illumination Furthermore, some temporary facilities are located in certain areas simply because of convenience For example, planners select entrance that is close, easy to access, to the local roads or highways Developing the Membership Functions In the present development, therefore, three factors have been considered in determining the closeness weights between each two facilities: 1) the level of work flow between the two facilities; 2) the level of safety/environmental hazard; and 3) the user s preference The work flow between two facilities greatly affects site productivity and encompasses the total flow of material, equipment, personnel, and information between the two facilities The level of safety/environmental hazard also 5

6 represents any concerns that may arise when the two facilities are close to each other, which may affect site workers by increasing the likelihood of accidents, noise, uncomfortable temperature, and/or pollution The third factor user s preference represents the project manager s desirability of having the facilities close or apart from each other This factor becomes important when the project manager desires to have the two facilities close to each other, despite the little or no work flow between them Based on the above discussion, the problem at hand involves three fuzzy input variables: Work Flow (WF); Safety/Environmental concerns (SE); and User s Preference (UP) These three variables affect the Closeness Weight fuzzy variable identified earlier Determining the membership value for a given quantity in a linguistic term is a problem-dependent and required an experiment to define it properly Also, the membership functions can be determined subjectively; the closer an element to satisfying the requirements of a set, the closer its grade of membership is to 1, and vice versa To determine an appropriate representation of the membership functions of the input variables, Fig 2 shows three different formulations of the membership functions of any of the three input linguistic variables Each variable is represented by three fuzzy sets, each of them cover a range of values in either triangular or trapezoidal shape 6

7 Degree of Membership (y) 1 Low Medium High SD min (min+max)/2 max Variable values (a) Degree of Membership (y) 1 Low Medium High SD min (min+max)/2 max Variable values (b) Degree of Membership (y) 1 Low Medium High min (min+max)/2 max Variable values (c) Figure 2: Different Shapes of Fuzzy Sets for the Same Linguistic Variable 7

8 Experimenting using the three representations showed in Fig 2, Fig 2b selected to represent the input variables, because all the fuzzy sets are represented by trapezoidal shapes which gives a smooth transit not as sharp as triangular shapes Also, as shown in Table 3, the closeness weights obtained using the membership functions of type b shows a clear distinguish between the values of the output variables Accordingly, the membership function shapes in Fig 2b will be used through out this study Table 3: Closeness Weights Obtained from Different Membership Function Shapes Membership function shape (1) Input variable rank (variable1, variable2, variable3) high, high, high (2) medium, medium, medium (3) low, low, low (4) a b c A family of fuzzy sets has been formulated for the three fuzzy variables and, for simplicity, each variable was limited to three-membership function Low (L), Medium (M), and High (H) (Fig 3) The shape and range of values of the three membership functions (L, M, and H) were determined through experimentation Accordingly, triangular and trapezoidal shapes were adopted (Figs 1, and 3) These two shapes are also the most frequently used in the literature 8

9 Degree of Membership (y) 1 Low Medium High Work Flow (trips / day) (a) Work flow Degree of Membership (y) 1 Low Medium High Safety/Environmental Concerns (b) Safety/Environmental Degree of Membership (y) 1 Low Medium High User s Preference (c) User s Preference Figure 3: Fuzzy Sets of the Input Variables 9

10 Work flow is assumed to vary from to 2 daily trips of material, equipment, personnel, and information The shape of the WF membership functions is symmetrical and centers around 1 trips/day Modifying the membership function values requires an intensive survey among practitioners The SE membership functions, on the other hand, are biased towards the "High" safety/environmental concerns Therefore, while the SE indicator (x-axis) has a range from to 1 (where means no concerns and 1 means completely unsafe), the High membership function has values starting from 4 to 1 The third membership function UP is similar to that of the WF, however, with a different scale, ranging from to 1 (where means far apart, 1 means close, and 5 means indifferent) Developing Fuzzy Decision Rules So far, the "Closeness Rating" desired to be determined is governed by three fuzzy variables, WF, SE, and UP Since each of these fuzzy variables has three membership functions, L, M, and H, there could be a total of 3 3 (27) different combinations of preconditions that affect closeness rating These preconditions have to be stored in the form of rules (called fuzzy rules) along with the decisionmaker's preference in their associated closeness rating An example rule is: Rule 6: IF Work Flow (WF) is AND Degree of Safety (SE) is AND User s Preference (UP) is (1) THEN Closeness Weight (R) is Important (I) 1

11 As shown in this rule, the THEN part refers to one of the six membership functions associated with the fuzzy output variable "Closeness Weight" Table 4: Fuzzy Decision Rules Rule no (1) Work Flow (2) Safety/Environmental Concerns (3) User's Preference (4) Closeness Rating (5) Medium(M) Ordinary (O) Important (I) Especially Important (E) Unimportant (U) Ordinary (O) Important (I) Undesirable (X) Unimportant (U) Ordinary (O) Important (I) Especially Important (E) Absolutely Important (A) Ordinary (O) Important (I) Especially Important (E) Unimportant (U) Ordinary (O) Important (I) Especially Important (E) Absolutely Important (A) Absolutely Important (A) Important (I) Especially Important (E) Absolutely Important (A) Ordinary (O) Important (I) Especially Important (E) In developing the fuzzy rules for the problem at hand, a systematic approach was used to determine the appropriate membership function (A, E, I, O, U, or X) to associate with the three preconditions of each rule For each input variable, a score of 3, 2, and 1 was given to the High, Medium, and Low linguistic terms, 11

12 respectively, of the WF and UP variables On the other hand, a score of 1, 2, and 3 was given to the High, Medium, and Low linguistic terms, respectively, of the SE variable Considering the fuzzy rule in Eq 1, for example, the three preconditions of the rule have a total score of 6 (1 for WF + 2 for SE + 3 for UP) Once the total score is calculated, it was compared to a preset values of 3, 4, 5, 6, 7, and 8 that relate to the use of membership functions X, U, O, I, E, and A, respectively Following this process, the fuzzy rules were formulated as shown in Table 4 Step1: Input the numeric values of expected WF, SE, and UP between each two facilities Step2: Fuzzify inputs by applying the input values to the rules to get their membership values Step3: Calculate the firing strength of each rule using the minimum operator Step4: Proportion the consequence of each rule to its firing strength Step5: Aggregate the consequences of all rules to form the overall membership function of the output Step6: Defuzzify the output by getting the center of area to obtain the facility closeness Figure 4: Steps of Fuzzy Rule-Based Module 12

13 Determining Facility Closeness Using Fuzzy Rule-Based System With the membership functions and fuzzy rules formulated, it is possible to use them with specific values of the input variables (numeric not linguistic) to compute a numeric value of the output variable This process is known as the fuzzy rule-based inferencing Fig 4 shows the typical steps used in a fuzzy rule-based system which are applied to define the facilities' closeness relationships As shown in Fig 4, the process first requires the user to input numeric values for the WF, SE, and UP between each two facilities The process then fuzzifies these values through the membership functions of the input variables For example, assume the user inputs WF, SE, and UP values 9, 3, 6 respectively, between two specified facilities These values are applied on the 27 rules, one-by-one, to determine the firing strength of each rule and how much it contributes to the output value Fig 5 shows the calculations in an intermediate rule (rule 6), which was described in Eq 1 earlier According to the rule, the WF value of 9 was applied to its "L" membership function, the SE value of 3 was applied to its "M" membership function, and the UP value of 6 was applied to its "H" membership function The intersection of these values with the membership functions provided membership values ω1, ω2, and ω3 of 2, 67, and 33, respectively (Fig 5) The firing strength of that rule is then calculated using the minimum operator, which is the smallest of ω1, ω2, and ω3 membership values (o2) of the rule 13

14 User input Value of variable 1 WF= Value of variable 2 SE = Value of variable 3 UP = Value of output R =? Rule 1 Area1 Rule 6 y y L 1 1 ω2=67 y M 1 y H 1 y I Area6 at min of ω1, ω2, and ω3 ω1=2 ω3= x Rule 27 Area Overall membership function based on the 27 rules MF Values Area6 Envelop enclosing all areas (area1 to area27) Output Values Overall output = Center of Area=21 Figure 5: Calculating the Overall Membership Function for the Output Variable 14

15 The determined firing strength (2) was used to truncate the membership function for the output, thus forming the shaded area (Area6) that define the contribution of this rule to the overall output Once these calculations are completed for all rules, the union operator is used to aggregate the consequences (Area1 to Area27) of the 27 rules to form an overall membership function (Fig 5) Finally, this overall membership function is converted into a crisp (non-fuzzy) value through a defuzzification process Various methods can be used to defuzzify the overall membership function, among which the center of area method is most common Using the center of area, a crisp value for the closeness rating between two the facilities is 21 In a similar fashion, a closeness rating between any other two facilities can be calculated from the user input of related WF, SE, UP values 15