Fuzzy multi-criteria selection of object-oriented simulation software for production system analysis

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1 Computers & Operations Research 32 (2005) Fuzzy multi-criteria selection of object-oriented simulation software for production system analysis Jeery K. Cochran a;, Hung-Nan Chen b a Department of Industrial Engineering, Arizona State University, Tempe, AZ , USA b Arizona State University, USA Abstract Discrete event computer simulation is one of the most widely used modeling tools for production systems. Object-oriented programming is revolutionizing computer simulation software packages. A fuzzy set approach for multi-criteria selection of object-oriented simulation software for analysis of production system is developed in this paper. The approach uses fuzzy set theory and algebraic operations of fuzzy numbers to characterize simulation software so that the strength and weakness of each alternative can be compared. The linguistic input from experts and decision makers are fuzzied and the results of the fuzzy inference are translated back to linguistic explanation. Further, by aggregating decision maker s preference (weighting) to the expert s rating, a single number for each candidate software can be obtained. One can use this number as an index to choose the most suitable simulation software for the specic application. A comparison between evaluations using simple triangular fuzzy numbers and using the real fuzzy set are presented.? 2003 Elsevier Ltd. All rights reserved. Keywords: Fuzzy sets; Fuzzy logic; Multi-criteria decision making; Object-oriented simulation; Production system modeling 1. Introduction The signicant contribution of discrete-event simulation, which we will refer to simply as simulation throughout this paper, to modeling production systems has gained enormous recognition in the industry. Generally, simulation software is classied into three categories: general programming languages, simulation languages, and simulators [1]. With each, there are trade-os between ease of use, exibility, reusability, and other desired characteristics. Researchers have demonstrated that an Corresponding author. Tel.: ; fax: address: j.cochran@asu.edu (J.K. Cochran) /$ - see front matter? 2003 Elsevier Ltd. All rights reserved. doi: /s (03)

2 154 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) object-oriented approach helps solve these problems [2,3]. If a manufacturing enterprise becomes interested in using object-oriented simulation (OOS), the issue then becomes how to choose a suitable tool among available alternatives. The literature has many perspectives on software selection [4 6], but the central decision method is a weighted scoring model calculation score for each alternative as the basis for ranking selections. A ranking model can be formulated as follows: n R i = w j r ij ; i=1; 2;:::;m; j=1 where R i is the ranking for alternative i, w j is the weighting (importance) for feature j, r ij is the score of feature j for alternative i, and there are m alternatives and n features to be considered. There are several problems with this approach. A scoring system needs a precisely dened number as the input to each feature and alternative; whereas in the real world, the criteria used to evaluate the suitability of a simulation tool, such as exibility and statistical ability, are usually subjective and may not be precisely dened for the decision makers. In addition, linguistic terms, such as strong, very strong, weak, etc., instead of precise numbers, are frequently used in evaluating the features of an alternative. In order to better deal with these concerns, we will integrate the fuzzy set theory developed by Zadeh [7] into scoring models to help an enterprise select the most suitable simulation tool for its application domain. In a previous study [8], an object-oriented simulation language description model (OOSLDM) was constructed to characterize and help select simulation languages. OOSLDM classies the desired features of simulation software into four categories (object-oriented, simulation, programming, and environment features) and obtained weighted scores of each feature for each alternative. Three simulation software, namely C++, SIMAN, and SIMPLE++, were evaluated based on these features. A usual scoring model was used to assist the decision-making process. However, neither numerical uncertainty nor the role of the linguistic variables was handled. The goal of this paper is to extend OOSLDM by using fuzzy set and fuzzy logic theories to achieve a more realistic and convincing result. 2. Literature review 2.1. Fuzzy sets The theory of fuzzy sets introduced by Zadeh [7] was developed to describe vagueness and ambiguity in the real-world system. In a decision-making problem, it is usually dicult to obtain a precise number for each criterion to be evaluated. The diculties come from several aspects, such as ill-dened criteria, inherently fuzzy assessments of decision makers, and the imprecise characteristics of the linguistic terms used in decision making. Zadeh denes a fuzzy set A in X (the space of objects) as a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function f A (x) which associates with each point in X a real number in the interval [0; 1], with the value of f A (x) atx representing the grade of membership of x in A. The larger the value of f A (x), the higher the grade of membership of x in A. This fuzzy set denition can be used to eectively express the vagueness of many real-world cases.

3 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) f A 1 0 α β γ R Fig. 1. Membership function of a triangular fuzzy number A =(; ; ) Triangular fuzzy numbers A triangular fuzzy number (TFN) can be dened by (; ; ) with the membership function [9] f A (x)= (x ) ; 6 x 6 ; ( ) (x ) = ; 6 x 6 ; ( ) = 0 otherwise with, as shown in Fig. 1. A crisp value has a TFN expressed as (; ; ). The reason for using TFNs as an approximation is that decision makers nd them intuitively easy to use [10]. For example, the linguistic value medium can be represented by (0:3; 0:5; 0:7). Some algebraic operations (such as addition and subtraction) on TFNs yield another TFN, but others (such as multiplication, inversion, division, maximum, and minimum) do not necessarily give a TFN [9,11] Multi-criteria decision making (MCDM) with fuzzy numbers When fuzzy numbers are associated with a multi-criteria decision-making problem, an important issue is how to aggregate the TFNs from several criteria and how to determine the ranking among several alternatives. Many methods, such as max, min, median, addition, multiplication, and mixed operators, are available to aggregate TFNs. Related literature can be found in [9,12,13]. In this paper, the addition and multiplication operations will be used to aggregate fuzzy numbers since they are required in the weighted scoring model. One point should be noted is that since the aggregated fuzzy numbers are not necessarily a TFN, an approximating formula to produce a TFN may be used to simplify the calculation [10]. Chen [14] developed the standard approximation method for ranking fuzzy numbers using maximizing and minimizing sets as fuzzy subsets with special membership functions. These functions, together with the aggregation formula, will be described in the following section. Later in the paper we will compare TFN approximation to the true fuzzy set for our class of use.

4 156 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) Methodology In this section, a fuzzy set approach for characterization and selection of object-oriented simulation software (OOSS) is developed. The approach is an eight-step process shown in Fig. 2. Step 1: Classication of desired features. The desired features for an OOSS are classied into four categories: OO features, programming features, simulation features, and environment features. Table 1 shows the detail of these features [8]. Experts 1.Classification of desired features Decisionmakers 2.Linguistic rating of features 5.Linguistic weighting of features 3. Aggregation M ij 6. Aggregation N j 4. Characterization of software alternatives 7. Aggregate M ij & N j 8.Rank alternatives; generate suggestion Fig. 2. A fuzzy approach for characterization and selection of object-oriented simulation software. Table 1 Desired features for OOSS OO features Programming features Simulation features Environment features 1. User capability 1. User capability 1. Modeling orientation 1. User interface 2. Inheritance 2. Flexibility 2. Data acquisition 2. Cost 3. Polymorphism 3. Module, subroutine 3. Interactive debugger 3. Documentation 4. Abstraction 4. Logic structure 4. Execution 4. Tutorial 5. Encapsulation 5. Variable denition 5. Animation 5. Portability 6. Object library 6. Data type 6. Output analysis 7. Reusability 7. Statistical ability 8. Late binding 8. Manufacturing constructs 9. Quick prototyping

5 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) Step 2: Obtain linguistic rating of each feature from simulation expert for each alternative. Experts who are familiar with the OOSS are asked to evaluate each alternative (OOSS) through a designed rating scale (very weak, weak, medium, strong, very strong). The membership functions of the linguistic values may vary depending on each individual expert. A generic membership function we have found useful is very weak (VW): (0; 0; 0:2). f S (x)=1 5x; 0 6 x 6 0:2; weak (W): (0; 0:2; 0:4). f S (x)=5x; 0 6 x 6 0:2; =2 5x; 0:2 6 x 6 0:4; medium (M): (0:3; 0:5; 0:7). f S (x)=5x 3 ; 2 0:3 6 x 6 0:5; 5x; 0:5 6 x 6 0:7; = 7 2 strong (S): (0:6; 0:8; 1). f S (x)=5x 3; 0:6 6 x 6 0:8; =5 5x; 0:8 6 x 6 1; very strong (VS): (0:8; 1; 1). f S (x)=5x 4; 0:8 6 x 6 1: The expert s linguistic evaluations will be transformed into TFNs according to the membership functions. Step 3: Aggregation of linguistic ratings. In this step, TFNs are calculated by using the membership functions dened in step 2. Assume there are m alternatives (OOSS). With each alternative a set of desired features (classied in four categories, with features in each category) is considered. Let X ijk =(o ijk ;p ijk ;q ijk ); i=1; 2;:::;m; j=1; 2; 3; 4; k =1; 2;:::; be the TFNs of the kth desired feature in the jth category for the candidate OOSS i. By using the averaged aggregation rule, the TFNs of the jth set of category for the candidate OOSS i will be M ij =(o ij ;p ij ;q ij ); i=1; 2;:::;m; j=1; 2; 3; 4; where n j o ij = k=1 o ijk ; p ij = k=1 p ijk and q ij = k=1 q ijk :

6 158 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) Step 4: Characterization of OOSS. In step 3, the M ij obtained by averaging several TFNs is still a TFN. Thus, using defuzzication techniques [18], a single number for each category of features for each OOSS can be obtained to indicate the strength of a certain feature set (OO, programming, simulation, or environment) for that OOSS. This number is dened as a feature coecient (FC), as it indicates the strength of a specic category of desired features. The most popular defuzzication methods are center of gravity (CoG, or centroid) and mean of maximum (MoM) [18]. CoG is more accurate in terms of representing a fuzzy set of any shape; however, calculation can be dicult and time consuming. MoM requires less computation but in some cases may misrepresent the original fuzzy set. In our case, the fuzzy numbers to be defuzzied are TFNs, whose CoG can be easily calculated, thus we use the CoG method to nd the representative crisp values. The centroid of any fuzzy number, for example the one in Fig. 1, isat((++)=3, 1 ); therefore 3 its CoG defuzzication is ( + + )=3. Step 4.1: Inference of SLS i from the proposed fuzzy rules. Let SLS i be the strength of language style for the ith candidate, and (O i ;P i ;S i ) denote the FCs for this candidate according to its OO features, programming features, and simulation features, respectively. Then, SLS i = SLS ij Rule j (j =1; 2; 3; 4; 5); where Rule 1 :IfO i and (P i or S i ) then OOSS i is an object-oriented language, Rule 2 :IfP i and S i then OOSS i is a simulation language, Rule 3 :IfO i and P i and S i then OOSS i is an OOS language, Rule 4 :If(not P i ) and S i then OOSS i is a simulator, Rule 5 :IfO i and (not P i ) and S i then OOSS i is an object-oriented simulator. The and operation in fuzzy logic is the minimum of the operands, or is the maximum of the operands, and not is the complement of one [15]. The values of O i, P i, and S i are always between 0 and 1; hence, the result of the rules is also a number falling in [0; 1] which indicates the strength of expected results from the inference rules. For example, if O i =0:3; P i =0:85; S i =0:8; E i =0:5; then applying the ve rules yields SLS i1 Rule 1 = O i and (P i or S i ) = min(o i ; max(p i ;S i ))=0:3; SLS i2 Rule 2 = P i and S i = min(p i ;S i )=0:8;

7 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) Table 2 Inference result for the illustrative example Language style Strength of language style Object-oriented language 0.3 Simulation language 0.8 OOS language 0.3 Simulator 0.15 Object-oriented simulator 0.15 SLS i3 Rule 3 = O i and P i and S i = min(o i ;P i ;S i )=0:3; SLS i4 Rule 4 =(not P i ) and S i = min(1 P i ;S i )=0:15; SLS i5 Rule 5 = O i and (not P i ) and S i = min(o i ; 1 P i ;S i )=0:15: These results are summarized in Table 2. Step 4.2: Linguistic explanation of SLS i. The SLS i obtained from the rules can be translated into linguistic terms using the following denition [16]: Attribute value = member=membership function ;:::; member=membership function : In the situation of the inference rules, the above equation becomes OOSS i is an XX language = strongly not=[0; 0:2) ; not=[0:2; 0:4) ; somewhat=[0:4; 0:6) ; is=[0:6; 0:8) ; strongly is=[0:8; 1:0] ; where XX language is one of the ve language styles described in the ve rules. Applying this equation to the ve rules in the previous example, it becomes Rule 1 OOSS i is not an object-oriented language. Rule 2 OOSS i strongly is a simulation language. Rule 3 OOSS i is not an object-oriented simulation language. Rule 4 OOSS i is strongly not a simulator. Rule 5 OOSS i is strongly not an object-oriented simulator. Step 5: Obtain linguistic weighting of each feature from the decision maker. In this step, decision makers are asked to assign the importance (weighting value) to each desired feature through a designed weighting scale (very low, low, medium, high, very high). For simplicity, the generic membership function introduced earlier is used. Very low (VL): (0; 0; 0:2). Low (L): (0; 0:2; 0:4).

8 160 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) Medium (M): (0:3; 0:5; 0:7). High (H): (0:6; 0:8; 1). Very high (VH): (0:8; 1; 1). Step 6: Aggregation of linguistic weightings. Similar to the process in step 3, TFNs for importance of features can be calculated by using the membership functions dened in step 5. Let A jk =(a jk ;b jk ;c jk ); j =1; 2; 3; 4; k =1; 2;:::; be the importance TFNs of the kth desired feature in the jth category. By using the averaged aggregation rule, the importance of the jth set of category for the candidate OOSS i will be N j =(a j ;b j ;c j ); j =1; 2; 3; 4; where n a jk j a j = ; b j = k=1 k=1 b jk ; and c j = k=1 Step 7: Aggregate M ij and N j ; Rank alternatives. In this step, the TFN G i for the ith OOSS can be obtained by the following fuzzy operation: c jk : G i =( 1 4 ) [(M i1 N 1 ) (M i2 N 2 ) (M i3 N 3 ) (M i4 N 4 )]; where denotes a fuzzy sum and denotes fuzzy multiplication. Note that the multiplication of two TFNs does not yield a TFN and thus, to be completely accurate, the calculation has to be carried out using their -cut condence intervals [9]. Alternatively, we may use a simplied formula [10] to approximate the aggregated TFN ( 4 G i j=1 o 4 ija j j=1 = ; p 4 ijb j j=1 ; q ) ijc j : Step 8: Rank alternatives and generate suggestion. The total utility function developed by Chen [14] can be used against G i at this step to obtain a number (U i ) for each candidate OOSS. The OOSS with the highest U i will be the best alternative. Assuming that the decision make is conservative or neutral, Chen s method denes the maximizing set M = {(x; f M (x)) x R} with f M (x)= (x x m) (x M x m ) ; x m 6 x 6 x M ; = 0 otherwise and the minimizing set G = {(x; f G (x)) x R} with f G (x)= (x x M ) (x m x M ) ; x m 6 x 6 x M ; = 0 otherwise;

9 where J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) x m = inf D; x M = sup D; D = m D i ; D i = {x f Gi (x) 0}; i=1; 2;:::;m i=1 and f Gi (x) is the membership function of the aggregated TFN. Further, the right utility value, UR i, and left utility value, UL i, for candidate i are dened as UR i = sup[f M (x) f Gi (x)]; i=1; 2;:::;m; UL i = sup[f G (x) f Gi (x)]; i=1; 2;:::;m and the total utility or ordering value for candidate i is U i = (UR i +1 UL i ) : 2 There are many other ranking methods available, but Prodanovic and Simonovic [17] review several ranking methods and conclude that for non-questionable cases, most methods produce identical ranking. Note that the explanation of language style in step 4 also gives the decision maker an idea of the characteristics of each alternative. This may provide extra supporting information for decision making. 4. Example The data used in this example is obtained from a previous study of ours [8] that compared three languages: C++, SIMAN, and SIMPLE++. In the example, only one decision maker and one OOSS expert are involved. Step 1: The feature set is given as in Table 1. Basically, the OO, programming, and simulation features are those that dominate the modeling style of a language, while environment features is a category beyond the programming styles but which supporting the modeling work. Step 2: The linguistic values for ratings are listed in Table 3. As described in [8], a computerintegrated just-in-time system has been modeled in each of these simulation languages. The results generated by these languages are statistically identical. The desired features of each language have been evaluated through the modeling process. Step 3: TFNs from step 2 are calculated according to the corresponding membership function and are shown in Table 4. Step 4: By using CoG defuzzication on the features for each OOSS, the feature coecients (FC) of each candidate are obtained and shown in Table 5. Step 4.1: Inference of SLS i from the Fuzzy Rules. Applying the FCs obtained in step 4 to the ve rules, the strength of language style (SLS i ) of these three languages can be compared. Given OOSS C++ = {0:837; 0:933; 0:138; 0:613}; OOSS SIMAN = {0:263; 0:689; 0:933; 0:793} and OOSS SIMPLE++ = {0:682; 0:450; 0:571; 0:640}:

10 162 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) Table 3 Linguistic rating of features for three candidates C++ SIMAN SIMPLE++ OO features User capability VS VW M Inheritance VS VW VS Polymorphism VS VW VW Abstraction VS M VS Encapsulation VS W VS Object library VS VW S Reusability S M VS Late binding VS VW VW Quick prototyping W M VS Programming features User capability VS VS W Flexibility VS VS M Module, subroutine VS VS W Logic structure VS VS W Variable denition VS W S Data type VS W S Simulation features Modeling orientation VW VS W Data acquisition VW VS VW Interactive debugger M VS VS Execution W VS VS Animation VW VS VS Output analysis VW VS W Statistical ability VW VS S Manufacturing constructs VW VS M Environment features User interface W S VS Cost S M W Documentation VS S W Tutorial W VS VS Portability VS VS VS These inference results are shown in Table 6. Step 4.2: Linguistic explanation of SLS i. Using the membership functions dened for this step from Table 6, we have C++ strongly is an object-oriented language. C++ is strongly not a simulation language. C++ is strongly not an OOS language. C++ is strongly not a simulator. C++ is strongly not an object-oriented simulator.

11 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) Table 4 TFNs of feature ratings C++ SIMAN SIMPLE++ OO features User capability (0:8; 1; 1) (0; 0; 0:2) (0:3; 0:5; 0:7) Inheritance (0:8; 1; 1) (0; 0; 0:2) (0:8; 1; 1) Polymorphism (0:8; 1; 1) (0; 0; 0:2) (0; 0; 0:2) Abstraction (0:8; 1; 1) (0:3; 0:5; 0:7) (0:8; 1; 1) Encapsulation (0:8; 1; 1) (0; 0:2; 0:4) (0:8; 1; 1) Object library (0:8; 1; 1) (0; 0; 0:2) (0:6; 0:8; 1) Reusability (0:6; 0:8; 1) (0:3; 0:5; 0:7) (0:8; 1; 1) Late binding (0:8; 1; 1) (0; 0; 0:2) (0; 0; 0:2) Quick prototyping (0; 0:2; 0:4) (0:3; 0:5; 0:7) (0:8; 1; 1) M i1 (averaged TFN) (0:69; 0:89; 0:933) (0:1; 0:3; 0:389) (0:556; 0:7; 0:79) Programming features User capability (0:8; 1; 1) (0:8; 1; 1) (0; 0:2; 0:4) Flexibility (0:8; 1; 1) (0:8; 1; 1) (0:3; 0:5; 0:7) Module, subroutine (0:8; 1; 1) (0:8; 1; 1) (0; 0:2; 0:4) Logic structure (0:8; 1; 1) (0:8; 1; 1) (0; 0:2; 0:4) Variable denition (0:8; 1; 1) (0; 0:2; 0:4) (0:6; 0:8; 1) Data type (0:8; 1; 1) (0; 0:2; 0:4) (0:6; 0:8; 1) M i2 (averaged TFN) (0:8; 1; 1) (0:533; 0:733; 0:8) (0:25; 0:45; 0:65) Simulation features Modeling orientation (0; 0; 0:2) (0:8; 1; 1) (0; 0:2; 0:4) Data acquisition (0; 0; 0:2) (0:8; 1; 1) (0; 0; 0:2) Interactive debugger (0:3; 0:5; 0:7) (0:8; 1; 1) (0:8; 1; 1) Execution (0; 0:2; 0:4) (0:8; 1; 1) (0:8; 1; 1) Animation (0; 0; 0:2) (0:8; 1; 1) (0:8; 1; 1) Output analysis (0; 0; 0:2) (0:8; 1; 1) (0; 0:2; 0:4) Statistical ability (0; 0; 0:2) (0:8; 1; 1) (0:6; 0:8; 1) Manufacturing constructs (0; 0; 0:2) (0:8; 1; 1) (0:3; 0:5; 0:7) M i3 (averaged TFN) (0:038; 0:088; 0:288) (0:8; 1; 1) (0:413; 0:588; 0:712) Environment features User interface (0; 0:2; 0:4) (0:6; 0:8; 1) (0:8; 1; 1) Cost (0:6; 0:8; 1) (0:3; 0:5; 0:7) (0; 0:2; 0:4) Documentation (0:8; 1; 1) (0:6; 0:8; 1) (0; 0:2; 0:4) Tutorial (0; 0:2; 0:4) (0:8; 1; 1) (0:8; 1; 1) Portability (0:8; 1; 1) (0:8; 1; 1) (0:8; 1; 1) M i4 (averaged TFN) (0:44; 0:64; 0:76) (0:62; 0:82; 0:94) (0:48; 0:68; 0:76) SIMAN is not an object-oriented language. SIMAN is a simulation language. SIMAN is not an OOS language. SIMAN is not a simulator. SIMAN is not an object-oriented simulator.

12 164 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) Table 5 FCs of three candidates U 1 (C++) U 2 (SIMAN) U 3 (SIMPLE++) OO features (FCo) Programming features (FCp) Simulation features (FCs) Environment features (FCe) Table 6 SLS of the selected languages C++ SIMAN SIMPLE++ Rule 1: Object-oriented language Rule 2: Simulation language Rule 3: OOS language Rule 4: Simulator Rule 5: Object-oriented simulator FCo FCp FCo FCp FCo FCp FCe FCs FCe FCs FCe FCs (a) C++ (b) SIMAN (c) SIMPLE++ Fig. 3. Characteristic of the three candidates. SIMPLE++ is somewhat an object-oriented language. SIMPLE++ is somewhat a simulation language. SIMPLE++ is somewhat an OOS language. SIMPLE++ is somewhat a simulator. SIMPLE++ is somewhat an object-oriented simulator. The characteristic of these OOSS are illustrated graphically in Fig. 3. Fig. 3 shows that C++ is strong in OO and programming features, but weak in simulation features. SIMAN is strong in simulation and environment features but weak in OO features. SIMPLE++ s features are average in each of the four categories. Step 5: Linguistic weighting of each feature. The decision maker was asked to assign linguistic weightings to each feature. Shown in Table 7 are these weightings as well as their corresponding TFNs. Step 6: Aggregation of linguistic weightings. TFNs of these weightings are calculated according to the membership function (and also shown in Table 7).

13 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) Table 7 Linguistic weightings and TFNs for each feature Weighting TFN of weighting OO features User capability VH (0:8; 1; 1) Inheritance H (0:6; 0:8; 1) Polymorphism H (0:6; 0:8; 1) Abstraction H (0:6; 0:8; 1) Encapsulation H (0:6; 0:8; 1) Object library H (0:6; 0:8; 1) Reusability VH (0:8; 1; 1) Late binding L (0; 0:2; 0:4) Quick prototyping H (0:6; 0:8; 1) N 1 (averaged TFN) (0:6; 0:667; 0:933) Programming features User capability VH (0:8; 1; 1) Flexibility VH (0:8; 1; 1) Module, subroutine H (0:6; 0:8; 1) Logic structure H (0:6; 0:8; 1) Variable denition H (0:6; 0:8; 1) Data type H (0:6; 0:8; 1) N 2 (averaged TFN) (0:667; 0:866; 1) Simulation features Modeling orientation M (0:3; 0:5; 0:7) Data acquisition M (0:3; 0:5; 0:7) Interactive debugger H (0:6; 0:8; 1) Execution VH (0:8; 1; 1) Animation H (0:6; 0:8; 1) Output analysis H (0:6; 0:8; 1) Statistical ability VH (0:8; 1; 1) Manufacturing constructs H (0:6; 0:8; 1) N 3 (averaged TFN) (0:575; 0:775; 0:925) Environment features User interface H (0:6; 0:8; 1) Cost H (0:6; 0:8; 1) Documentation VH (0:8; 1; 1) Tutorial M (0:3; 0:5; 0:7) Portability M (0:3; 0:5; 0:7) N 4 (averaged TFN) (0:52; 0:72; 0:88) Step 7: Aggregation of M ij and N j. The aggregated fuzzy set G i for each candidate is calculated using -cut condence intervals [9]. The resulted fuzzy set is a composition of two parabolic curves shown as solid curves in Fig. 4. The aggregation can also be approximated as a TFN using the formula described earlier in Section 3 from [10]. Fig. 4 shows the approximated TFN as dotted

14 166 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) Ranking of Aggregated Fuzzy Numbers (and their TFN approximations) Solid curves (parabolas) indicate actual fuzzy numbers; dotted lines indicate approximated triangular fuzzy numbers Minimizing Set Maximizing Set degree of membership Simple++ C++ SIMAN Aggregated features (Gi ) Fig. 4. Ranking of aggregated fuzzy numbers. straight lines. From the gure, it is clear that the approximations are very close to the actual curves for our use (we calculate the dierences in the next section). The aggregated fuzzy sets share the same vertices as their corresponding TFNs and are G C++ =(0:295; 0:436; 0:695); G SIMAN =(0:308; 0:476; 0:73); G SIMPLE++ =(0:255; 0:403; 0:679): Step 8: Rank alternatives and generate suggestions. To rank the three aggregated true fuzzy sets, we apply Chen s ranking method to the parabola curves. In Fig. 4, the right utility functions are found at the right intersections of the maximizing set and the membership functions of the three aggregated fuzzy numbers; the left utility functions are found at the left intersections of the minimizing set and the membership functions of the three aggregated fuzzy numbers. The results for these three candidates are UR C++ =0:592; UL C++ =0:71; U C++ =0:441; UR SIMAN =0:646; UL SIMAN =0:661; U C++ =0:493; UR SIMPLE++ =0:553; UL SIMPLE++ =0:766; U SIMPLE++ =0:394:

15 J.K. Cochran, H.-N. Chen / Computers & Operations Research 32 (2005) If calculated using approximated TFNs, the utility functions are UR C++ =0:6; UL C++ =0:705; U C++ =0:447; UR SIMAN =0:651; UL SIMAN =0:656; U C++ =0:498; UR SIMPLE++ =0:565; UL SIMPLE++ =0:762; U SIMPLE++ =0:401: The errors of utility functions using actual fuzzy sets and approximated fuzzy sets are 1.44%, 1.03%, and 2.03% for C++, SIMAN, and Simple++, respectively. Both calculations yield the same rankings (SIMAN C++ Simple++) and the errors are insignicant. To potential practitioners of our method, we note that the calculation using an approximated fuzzy set is much simpler. From above calculation, the total utility suggests that SIMAN is the best alternative. However, the characterization in step 4 reveals interesting characteristics of each alternative. These are in linguistic terms and may also be considered by the decision maker. 5. Conclusion A fuzzy set approach for characterization and selection of OOS software have been developed in this paper. The approach employs fuzzy set theory and fuzzy operations to solve the common and important decision-making problem of choosing among alternatives using rankings based on weighted scores. The method can be used in any circumstance where features can be scored and linguistic descriptions of scoring chosen. In the motivating example wherein three simulation software packages are compared, a complicated production system simulation model was built using each of the three simulation software packages C++, SIMAN, and SIMPLE++. The experience gained provided the expert rating input to the model. Weighting of the features was done by the decision maker. The use of the fuzzy set concept improves the realism of the decision-making process by considering the fuzziness of the real-world system as well as conceptualizations about it. It also obviated the need for crisp numbers when they do not exist. We found that using triangular fuzzy numbers (TFN) made data collection and interpretation of results easier for experts and decision makers alike. Interestingly, we ultimately ranked an older-generation software package (SIMAN) as the best in the case study. Mathematical manipulation of fuzzy sets can be messy. In particular, simple operations on TFN do not always result in TFN. Comparing the simpler TFN approximations of results to the true fuzzy results of our calculations showed only small errors (1 3%) and changed no rankings. Possible future work of this paper is to consider the case of multiple experts/decision makers and the sensitivity of membership functions. References [1] Mackulak GT, Cochran JK, Savory PA. Ascertaining important features for industrial simulation environments. Simulation 1994;63: [2] Adiga S, Glassey CR. Object-oriented simulation to support research in manufacturing systems. International Journal of Production Research 1991;29:

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