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A framework for resource assignments in skill-based environments

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Title:
A framework for resource assignments in skill-based environments
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English
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Otero, Luis Daniel
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University of South Florida
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Subjects / Keywords:
Personnel assignments
Resource allocation
Data envelopment analysis
Tobit regression
Capability assessments
Dissertations, Academic -- Industrial and Management Systems -- Doctoral -- USF   ( lcsh )
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non-fiction   ( marcgt )

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Summary:
ABSTRACT: The development of effective personnel assignment methodologies has been the focus of research to academicians and practitioners for many years. The common theory among researchers is that improvements to the effectiveness of personnel assignment decisions are directly associated with favorable outcomes to organizations. Today, companies continue to struggle to develop high quality products in a timely fashion. This elevates the necessity to further explore and improve the decision-making science of personnel assignments. The central goal of this research is to develop a novel framework for human resource assignments in skill-based environments. An extensive literature review resulted in the identification of the following three areas of the general personnel assignment problem as potential improvement opportunities: determining assignment criteria, properly evaluating personnel capabilities, and effectively assigning resources to tasks.Thus, developing new approaches to improve each of these areas constitute the objectives of this dissertation work. The main contributions of this research are threefold. First, this research presents an effective two-stage methodology to determine assignment criteria based on data envelopment analysis (DEA) and Tobit regression. Second, this research develops a novel fuzzy expert system for resource capability assessments in skill-based scenarios. The expert system properly evaluates the capabilities of resources in particular skills as a function of imprecise relationships that may exist between different skills. Third, this research develops an assignment model based on the fuzzy goal programming (FGP) technique. The model defines capabilities of resources, tasks requirements, and other important parameters as imprecise/fuzzy variables.The novelty of the research presented in this dissertation stems from the fact that it advances the science of personnel assignments by combining concepts from the fields of statistics, economics, artificial intelligence, and mathematical programming to develop a solution approach with an expected high practical value.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2009.
Bibliography:
Includes bibliographical references.
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Statement of Responsibility:
by Luis Daniel Otero.
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Title from PDF of title page.
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Document formatted into pages; contains 133 pages.
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Includes vita.

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oclc - 436936185
usfldc doi - E14-SFE0002921
usfldc handle - e14.2921
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A Framework for Resource Assignments in Skill-Based Environments by Luis Daniel Otero A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Industrial and Management Systems Eng ineering College of Engineering University of South Florida Major Professor: Grisselle Centeno, Ph.D. Kingsley Reeves, Ph.D. Jos L. Zayas-Castro, Ph.D. Miguel Labrador, Ph.D. Alex J. Ruiz-Torres, Ph.D. Date of Approval: March 14, 2009 Keywords: Personnel assignments, Resource allocatio n, Data envelopment analysis, Tobit regression, Capability assessments, Fuzzy exp ert systems, and Fuzzy goal programming Copyright 2009, Luis Daniel Otero

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DEDICATION I dedicate this dissertation to my family. I am ex tremely grateful to God for such an exceptional family. You guys are truly blessing s in my life! To my parents Angel L. Otero (Pa) and Lydia E. Rive ra (Ma) : Thank you for always being there for me. Among the many great things that you taught me, today I thank you for teaching me to set high goals in life and to work hard to achieve these goals. Most importantly, I thank you for teaching me to have a strong faith in God and put Him first in every aspect of my life This dissertation is mainly dedicated to you! To my brothers Angel R. (Rafi) Otero and Carlos E. (Quique) Otero : As you know, you two are tremendous role models in my life. Your outstanding qualities as professionals and human beings motivat e me to reach the highest possible goals. I know that you are as proud of me as I am of you two. To my beautiful wife Aymara and my son Christian Da niel (and the others to come) : You are my inspiration in everything that I do! I always carry you in my mind and in my heart.

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ACKNOWLEDGEMENTS I would like to extend my deepest gratitude to my m ajor advisor, Dr. Grisselle Centeno, for her support and encouragement througho ut the course of this research. Dr. Centeno, I sincerely thank you for giving me the op portunity to reach my greatest academic achievement. It has been a privilege work ing with you! Also, thanks to my committee members Dr. Jos L. Zayas-Castro, Dr. Kin gsley Reeves, Dr. Miguel Labrador, and Dr. Alex J. Ruiz-Torres for been inte gral part of my academic journey. I want to gratefully acknowledge the following spec ial faculty at Florida Institute of Technology for their positive influence and guid ance which further motivated me to pursue a doctoral degree: Dr. Wade Shaw, Dr. Muzaf far Shaikh, and Dr. Paul Cosentino. In particular, I want to thank Dr. Wade Shaw for hi s continuous and effective effort to positively influence students’ lives. Dr. Shaw, yo ur extraordinary dedication to students inside and outside a classroom provides the standar d that I will strive to follow as a professor. Thank you! I would like to specially acknowledge my family for all their love and support. In particular, I would like to thank my beautiful wife Aymara for selflessly sharing me with school and work over the last few years. Aymara, I know that my journey as a doctoral student represented a big sacrifice for you. Never theless, you were always by my side showing me unconditional love. You made possible t he completion of this dissertation! I sincerely thank you for that.

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i TABLE OF CONTENTS LIST OF TABLES..................................... ................................................... .....................iv LIST OF FIGURES.................................... ................................................... ....................vi ABSTRACT .......................................... ................................................... .....................viii CHAPTER 1 INTRODUCTION........................... ................................................... ........1 1.1 Motivation..................................... ................................................... ...................2 1.2 Research Objectives............................ ................................................... .............3 1.3 Solution Approach and Contributions............ ................................................... .3 1.4 Organization of Dissertation................... ................................................... .........5 CHAPTER 2 A DEA-TOBIT ANALYSIS TO IDENTIFY KEY ASSI GNMENT CRITERIA IN SKILL-BASED ENVIRONMENTS............... ...................6 2.1 Abstract....................................... ................................................... .....................6 2.2 Introduction and Overview...................... ................................................... ........7 2.3 Related Literature............................. ................................................... ................9 2.4 Solution Approach and Methodology.............. .................................................12 2.4.1 First Stage – DEA Analysis................... ................................................... 13 2.4.1.1 DEA Characteristics........................ ................................................... .14 2.4.1.2 Undesirable Variables and Isotonicity...... ..........................................16 2.4.1.3 DEA Input/Output Parameters and Number of D MUs.......................17 2.4.1.4 Orientation of DEA Model................... ..............................................18 2.4.1.5 DEA Model Selection........................ .................................................19 2.4.2 Second Stage Tobit Regression Analysis..... ..........................................20 2.4.2.1 Independent Variables...................... ..................................................2 1 2.5 Example – Software Development Setting......... ..............................................22 2.5.1 Previous Studies............................. ................................................... ........23 2.5.2 Data for Analysis............................ ................................................... .......28 2.5.3 First Stage – DEA Analysis................... ................................................... 30 2.5.4 Second Stage Tobit Regression Model........ ...........................................33 2.5.5 Discussion................................... ................................................... ...........37 2.6 Summary and Contributions...................... ................................................... ....39 CHAPTER 3 A FUZZY EXPERT SYSTEM ARCHITECTURE FOR CAPABILITY ASSESSMENTS IN SKILL-BASED ENVIRONMENTS....................................... ............................................42 3.1 Abstract....................................... ................................................... ...................42

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ii 3.2 Introduction................................... ................................................... .................43 3.3 Fuzzy Expert System Architecture............... ................................................... .45 3.3.1 Presentation, Data, and Global Layers........ ..............................................46 3.3.2 Fuzzy Logic System........................... ................................................... ....48 3.3.2.1 Fuzzy Sets................................. ................................................... .......48 3.3.2.2 Fuzzy Propositions......................... ................................................... ..50 3.3.2.3 Fuzzy Logical Operators.................... .................................................51 3.3.2.4 Fuzzy Reasoning............................ ................................................... ..52 3.3.2.4.1 Generalized Modus Ponens and the Composit ional Rule of Inference.......................................... ............................................52 3.3.2.5 Mamdani Max-Min Inference Approach......... ...................................54 3.3.2.6 Defuzzification............................ ................................................... .....56 3.3.3 Expert System Data Flow...................... ................................................... 56 3.3.3.1 Pre-conditions............................. ................................................... .....57 3.3.3.2 Step 1: User Inputs........................ ................................................... ...57 3.3.3.3 Step 2: Fuzzification...................... ................................................... ..57 3.3.3.4 Step 3: Inference Engine and Fuzzy Rules.. ......................................58 3.3.3.5 Step 4: Defuzzification.................... ................................................... 58 3.3.3.6 Step 5: Display Results.................... ................................................... 58 3.4 Example Software Development Setting......... ...............................................58 3.4.1 Problem Statement and Pre-conditions......... ............................................59 3.4.2 User Inputs.................................. ................................................... ...........61 3.4.3 Fuzzification................................ ................................................... ..........62 3.4.4 Inference Engine............................. ................................................... .......63 3.5 Summary and Contributions...................... ................................................... ....67 3.5.1 Research Extensions.......................... ................................................... ....68 CHAPTER 4 A FUZZY GOAL PROGRAMMING MODEL FOR SKILLBASED RESOURCE ASSIGNMENT PROBLEMS................. ..............70 4.1 Abstract....................................... ................................................... ...................70 4.2 Introduction................................... ................................................... .................71 4.3 Related Literature............................. ................................................... ..............75 4.3.1 Approaches for Personnel Assignments......... ..........................................75 4.3.1.1 Mathematical Programming Approaches........ ...................................75 4.3.1.2 Artificial Intelligence Approaches......... .............................................78 4.3.1.2.1 Fuzzy Set Theory Approaches.............. .......................................79 4.3.1.2.2 Global Optimization Approaches........... .....................................81 4.3.1.3 Other Approaches........................... ................................................... .82 4.3.2 Modeled Parameters........................... ................................................... ....84 4.3.3 Summary of Findings.......................... ................................................... ...85 4.4 Justification for FGP as a Solution Method..... .................................................86 4.4.1 Goal Programming............................. ................................................... ....86 4.4.2 Fuzzy Set Theory............................. ................................................... ......88 4.4.2.1 Membership Functions....................... .................................................88 4.4.3 FGP for the Skill-Based Assignment Problem... ......................................89

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iii 4.5 Solution Approach and Methodology.............. .................................................91 4.5.1 Membership Functions......................... ................................................... ..92 4.5.2 Priorities................................... ................................................... ..............95 4.5.2.1 Priorities with Fuzzy Weighted Average..... .......................................95 4.5.2.2 Priorities with Desirable Achievement Degre es.................................97 4.5.2.3 Membership Function Relaxation............. ..........................................97 4.5.3 FGP-MFR Model................................ ................................................... .100 4.6 Example Software Development Setting......... .............................................102 4.6.1 Problem Statement and Pre-conditions......... ..........................................103 4.6.2 Establishing Imprecise Parameters............ .............................................104 4.6.3 Identifying Traits of Resources.............. .................................................10 6 4.6.4 Develop Membership Functions for Goals....... ......................................107 4.6.5 MFR Process.................................. ................................................... ......110 4.6.6 FGP Model Results............................ ................................................... ..114 4.7 Summary and Contributions...................... ................................................... ..115 4.7.1 Research Extensions.......................... ................................................... ..117 CHAPTER 5 CONCLUSIONS AND FUTURE RESEARCH.......... ..........................119 5.1 Conclusions.................................... ................................................... ..............119 5.1.1 Research Extensions.......................... ................................................... ..122 REFERENCES......................................... ................................................... ...................124 ABOUT THE AUTHOR................................... ................................................... .End Page

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iv LIST OF TABLES Table 2.1 Inputs/Outputs for DEA Model........ ................................................... ........18 Table 2.2 Examples of Parameters for Tobit Mod els in Particular Disciplines..........22 Table 2.3 Selected Literature on Team Factors Affecting Quality and/or Productivity....................................... ................................................... .......25 Table 2.4 Correlation Analysis for DEA Paramet ers (Average-Complexity Tasks)............................................. ................................................... ..........31 Table 2.5 Correlation Analysis for DEA Paramet ers (High-Complexity Tasks)........31 Table 2.6 DEA Results Efficiency of Personne l Assignments.................................33 Table 2.7 Independent Variables............... ................................................... ...............34 Table 2.8 Correlation of Independent Variables in Tobit (AverageComplexity)........................................ ................................................... .....35 Table 2.9 Correlation of Independent Variables in Tobit (High-Complexity)............36 Table 2.10 Tobit Regression Results............. ................................................... .............37 Table 3.1 Classical Modus Ponens Form......... ................................................... ........52 Table 3.2 Generalized Modus Ponens Form....... ................................................... .....54 Table 3.3 Multiconditional Reasoning Structure ................................................... ......54 Table 3.4 Crisp Evaluation Ratings in Various Programming Languages..................62 Table 3.5 Fuzzy Evaluation Ratings............ ................................................... .............62 Table 3.6 Fuzzy Rules for C++................. ................................................... ...............63 Table 3.7 Initial and Modified C++ Ratings.... ................................................... ........67 Table 4.1 Characteristics of the General Skill -Based Personnel Assignment Problem............................................ ................................................... ........72 Table 4.2 Selected Recent Literature on Skillbased Resource Assignment...............84 Table 4.3 Notation for the FGP-MFR Model...... ................................................... ...101

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v Table 4.4 Desired Expertise Levels for Skills (Priorities of Skills in Parentheses)....................................... ................................................... ....105 Table 4.5 Tasks' Priorities................... ................................................... ...................105 Table 4.6 Skill Matrix of Available Candidates Based on a 0-5 Rating Scale..........106 Table 4.7 Motivation Levels of Resources with Tasks Based on a 0-5 Rating Scale.............................................. ................................................... .........106 Table 4.8 Set of General Rules................ ................................................... ...............110 Table 4.9 Solution to the Personnel Assignment Problem........................................115

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vi LIST OF FIGURES Figure 2.1 Conceptual Diagram of Solution Appro ach................................................ 13 Figure 2.2 Software Defect Introduction and Rem oval Process..................................24 Figure 2.3 Modular Project Structure........... ................................................... .............28 Figure 3.1 Conceptual Fuzzy Expert System...... ................................................... ......45 Figure 3.2 Layered Software Architecture....... ................................................... .........46 Figure 3.3 Example of Triangular Fuzzy Set..... ................................................... .......49 Figure 3.4 Mamdani Max-Min Inference........... ................................................... .......55 Figure 3.5 Fuzzy Sets of Skill Levels.......... ................................................... ..............60 Figure 3.6 Membership Functions for Fuzzy Sets of Skill Levels...............................61 Figure 3.7 Capacity Assessment for Engineer_6.. ................................................... .....65 Figure 3.8 Defuzzified Rating in C++ (Modified) ................................................... ....66 Figure 4.1 Sample Scenario..................... ................................................... ..................77 Figure 4.2 Solution Approach: Steps and Activi ties............................................... ....92 Figure 4.3 Sample Membership Function to Minimi ze Deviations from a Target Value....................................... ................................................... ......94 Figure 4.4 Sample Membership Function to Minimi ze Deficiencies from a Target Value....................................... ................................................... ......94 Figure 4.5 Triangular Membership Functions for Priority........................................... 96 Figure 4.6 Fuzzy Sets for Novice and Expert.... ................................................... ........99 Figure 4.7 Sample MFR to Minimize Deviations fr om Target Goals..........................99 Figure 4.8 Sample MFR to Minimize Deficiencies from Target Goals.....................100 Figure 4.9 Fuzzy Sets to Minimize Deviations fr om Target Values..........................108 Figure 4.10 Fuzzy Sets to Minimize Deficiencies f rom Target Values.......................109 Figure 4.11 Fuzzy Set for Highly Motivated....... ................................................... ......110 Figure 4.12 Fuzzy Set of Novice Expertise After M FR...............................................11 2

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vii Figure 4.13 Fuzzy Set of Proficient Expertise Aft er MFR...........................................11 2 Figure 4.14 Fuzzy Set of Highly Proficient Expert ise After MFR...............................113 Figure 4.15 Piecewise Linear Membership Functions for Tasks’ Priorities................114

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viii A Framework for Resource Assignments in Skill-Based Environments Luis Daniel Otero ABSTRACT The development of effective personnel assignment m ethodologies has been the focus of research to academicians and practitioners for many years. The common theory among researchers is that improvements to the effec tiveness of personnel assignment decisions are directly associated with favorable ou tcomes to organizations. Today, companies continue to struggle to develop high qual ity products in a timely fashion. This elevates the necessity to further explore and impro ve the decision-making science of personnel assignments. The central goal of this research is to develop a n ovel framework for human resource assignments in skill-based environments. An extensive literature review resulted in the identification of the following thr ee areas of the general personnel assignment problem as potential improvement opportu nities: determining assignment criteria, properly evaluating personnel capabilitie s, and effectively assigning resources to tasks. Thus, developing new approaches to improve each of these areas constitute the objectives of this dissertation work. The main contributions of this research are threefo ld. First, this research presents an effective two-stage methodology to determine ass ignment criteria based on data

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ix envelopment analysis (DEA) and Tobit regression. S econd, this research develops a novel fuzzy expert system for resource capability a ssessments in skill-based scenarios. The expert system properly evaluates the capabiliti es of resources in particular skills as a function of imprecise relationships that may exist between different skills. Third, this research develops an assignment model based on the fuzzy goal programming (FGP) technique. The model defines capabilities of resou rces, tasks requirements, and other important parameters as imprecise/fuzzy variables. The novelty of the research presented in this disse rtation stems from the fact that it advances the science of personnel assignments by combining concepts from the fields of statistics, economics, artificial intelligence, and mathematical programming to develop a solution approach with an expected high practical value.

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1 CHAPTER 1 INTRODUCTION The development of effective personnel assignment m ethodologies have been the focus of research to academicians and practitioners for many years. The common theory among researchers is that improvements to the effec tiveness of personnel assignment decisions are directly associated with favorable ou tcomes to organizations [1]. These outcomes may include enhanced quality of products, increased employee productivity, lower turnover rates, increased market shares, and competitive advantage. The continued struggle of companies to develop high quality products in a timely fashion elevates the necessity to further explore a nd improve the decision-making science of personnel assignments. For example, the U.S. Go vernment recently spent nearly 8 billion dollars in the software development industr y to rework software due to qualityrelated issues [2]. In the accounting field, audit quality problems are currently a major concern given “the cascade of audit failures in the concluding years of the last century and the first few years of the new century” [3]. I n fact, “developing [quality] products faster has become critical to success in many indus tries, whether the product is an office building, software package, or computer chip” [1]. From a personnel assignment point of view, a common denominator in the types of industries mentioned above is the presence of hi ghly imprecise parameters. For instance, expertise levels of personnel in various specialized areas are more adequately

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2 described with imprecise parameters (e.g., high, av erage, low) rather than using precise values (e.g., 12 units per hour). These parameters are typically defined by decision makers. Some examples include describing the exper tise of an auditor in a particular accounting software tool, the expertise of a progra mmer with a programming language, or the expertise of a statistician with stochastic processes. Similarly, tasks’ requirements are more adequately defined with imprecise paramete rs. The type of assignment problem characterized by imp recise personnel capabilities and tasks requirements is denoted in this research as the skill-based resource assignment problem (SBRAP). The focus of this research is to develop a new solution approach to the SBRAP. Although there is extensive literature related to personnel assignment approaches, most of these approaches deal with prec ise parameters. Moreover, relatively minor research has been conducted on the topic of c ompetence-based assignment of employees to workplaces [4]. 1.1 Motivation The motivation for conducting this research grew fr om the particular industry experience of the author as a software engineer in major software projects for the defense industry. Experiencing first-hand the absence of p roper processes for assigning software developers to software tasks provided the initial p ush to pursue this research. A thorough review of the current literature, as well as discus sions with software managers regarding the problem statement, demonstrate an evident oppor tunity and confirm that this study has the potential to make significant contributions to the general personnel assignment literature.

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3 1.2 Research Objectives The central goal of this research is to develop a n ovel framework for human resource assignments in skill-based environments. To this end, a literature review was conducted to investigate current resource assignmen t methodologies applicable to skillbased environments in order to develop new approach es that address the major weaknesses found in current methods. Through the r eview of the literature, three areas of the general personnel assignment problem were ident ified as opportunities for improvement. They include: determining assignment criteria, properly evaluating personnel capabilities, and effectively assigning r esources to tasks. Thus, developing new approaches to improve each of these areas const itute the objectives (or subproblems) of this dissertation work. 1.3 Solution Approach and Contributions The main contributions of this research are threefo ld. The first one focuses on the development of an effective two-stage methodology, based on data envelopment analysis (DEA) and Tobit regression, to determine assignment criteria. DEA analyzes data from previously completed tasks to determine relative ef ficiencies of personnel assignments. Then, Tobit regression analysis models DEA scores a gainst factors believed to affect efficiency. The model incorporates capabilities of resources and task factors as independent variables. The capability of the metho dology was demonstrated with data collected from a major software development organiz ation. The results obtained were compared to results from existing approaches.

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4 Secondly, this research presents a methodology for resource capability assessments in skill-based scenarios. This methodo logy is an extension to an exploratory approach developed by the author in [5]. The metho dology suggests that capability levels in particular skills are influenced by resources’ k nowledge in other related skills. To properly evaluate the capabilities of resources in particular skills, the methodology employs concepts from fuzzy logic and fuzzy set the ory to account for the imprecise relationships that may exist between different skil ls. Thirdly, this research develops an assignment model based on the fuzzy goal programming (FGP) technique. The approach defines capabilities of resources, tasks requirements (i.e., goals), and other important par ameters as imprecise variables. Thus, it develops fuzzy sets for these parameters, which are then meticulously manipulated to incorporate fuzzy priorities of goals and tasks. T he resulting fuzzy values are then fed to the FGP model to develop a solution that maximizes the suitability of resources with tasks. An important aspect of the FGP approach is that the author developed a software application to determine the fuzzy suitability of r esources with tasks. This lays the foundation for the future development of a complete software package to serve as a decision support system, including the solution met hodologies to determine assignment criteria and assess resources’ capabilities. This presents a significant opportunity to further extend this research, given that “the compe tence-based assignment of employees to workplaces is not supported by any commercially available software system” [4]. The novelty of the research presented in this disse rtation stems from the fact that it advances the science of personnel assignments by combining concepts from the fields

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5 of statistics, economics, artificial intelligence, and mathematical programming to develop a solution approach with an expected high practical value. 1.4 Organization of Dissertation The rest of this dissertation is organized into fiv e chapters. Chapter 2 through Chapter 4 are independent sections structured as jo urnal articles to address each of the three major objectives of this dissertation. Chapt er 2 focuses on the DEA-Tobit methodology to determine relative priorities for as signment criteria in skill-based environments. Chapter 3 presents a methodology for fuzzy resource capability assessments in skill-based scenarios. In Chapter 4 a fuzzy goal programming model for resource assignment in skill-based environments is presented. Finally, Chapter 5 concludes with a global summary of the contribution s to the literature and recommendations for future research.

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6 CHAPTER 2 A DEA-TOBIT ANALYSIS TO IDENTIFY KEY ASSIGNMENT CRI TERIA IN SKILL-BASED ENVIRONMENTS 2.1 Abstract This research presents a two-stage methodology to i dentify important assignment criteria in skill-based environments. These enviro nments are characterized by the need to assess the ability of available resources to successfully complete a set of tasks. The first stage uses data envelopment analysis (DEA) to estab lish relative efficiencies of personnel assignments in previous tasks. Efficiency is defin ed as a ratio of weighted outputs (i.e., quality and productivity measures) over weighted in puts (i.e., effort and overall industry experience). The second stage uses Tobit regressio n analysis to model DEA scores against factors believed to affect efficiency. The se factors include experience of resources on specific skills and particular charact eristics of working environments. A software development industrial setting is explor ed to validate the practical value of the methodology. Data related to tasks fr om a leading software development organization are analyzed and key assignment criter ia are determined. The contribution of this research to the literature is two-fold. First, it presents an innovative methodology to prioritize assignment cri teria in skill-based environments. Second, it develops an efficiency model for personn el assignments using real industrial

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7 software development data. To the best of our know ledge, an efficiency model of this type is non-existent in the literature regarding pe rsonnel assignments. 2.2 Introduction and Overview Research regarding methodologies to identify and pr ioritize assignment criteria in human resource assignment problems is very limited. This is particularly true for skillbased resource assignment problems (SBRAPs), which are characterized by the need to assess the ability of candidates to successfully complete specific tasks. Examples of environments where decision-makers encounter SBRAPs are software engineering, healthcare, and research and development (R&D) orga nizations among others. In SBRAPs, assignment criteria and their associated priorities are key parameters to determine the suitability of resources to execut e certain tasks. Nevertheless, assignment criteria are usually determined subjecti vely [6], or based on the effect of particular factors to a single performance measure. Furthermore, priorities for assignment criteria are usually not included in per sonnel assignment approaches, and are mostly determined intuitively by project leaders or supervisors. Consequently, the effectiveness and practical value of current method ologies suffer significantly. According to Acua et al. [6], this presents an ope n area for conducting research that incorporates a diversity of factors of individual e mployees in the assignment decision such as personal preferences and technical knowledg e and skills. The objective of this research is to develop an app roach to effectively select assignment criteria in skill-based resource allocat ion scenarios. The result is a two-stage methodology composed of data envelopment analysis ( DEA) and Tobit regression. The

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8 first stage applies DEA to analyze data from comple ted tasks to determine efficiencies of personnel assignments based on quality and producti vity measures. DEA first constructs an empirical production frontier composed of the mo st efficient assignments, which are the ones that produced the most outputs with the le ast amount of inputs. DEA determines the efficiencies of the assignments that are not in the production frontier based on the distance to their closest point (i.e., assignment) in the production frontier [7]. There are several benefits from using DEA over othe r methods. One of these benefits is that DEA considers multiple outputs sim ultaneously. This produces more thorough efficiency evaluations. Another benefit i s that DEA enables the comparison of personnel assignments with best performers (i.e., a ssignments in the efficient production frontier), which results in more rigorous efficienc y assessments. The second stage employs Tobit regression analysis to model DEA scores against parameters assumed to affect efficiency. These par ameters include capabilities of resources and task factors. Tobit regression was s elected over ordinary least squares methods because the dependent variable (i.e., DEA s core) always falls between two corner solutions (i.e., zero and one), and Tobit re gression is more robust in such situations [8], [9]. To demonstrate its practical value, the methodology was used to identify key assignment criteria with data from a leading softwa re development organization. The company specializes in the development of software applications for the defense industry and is rated a capability maturity model integratio n (CMMI) level 5 organization. A level 5 ranking means that the company has the high est standards for quantitative process monitoring and improvement. The organization provi ded data under nondisclosure

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9 agreements, as has been the case in prior studies [ 10], [11]. The data provided information about software tasks such as the number and types of software defects, the size in terms of number of software lines of code ( SLOC), and programming language and domain experience of resources. This paper is organized into five sections of which this introduction is the first one. Section 2.2 describes literature related to m ethodologies for identifying assignment criteria in skill-based environments. Section 2.3 explains the proposed DEA-Tobit regression solution approach. Section 2.4 describe s the application of the proposed methodology with data from a software development c ompany. Finally, Section 2.5 concludes with contributions to the literature and recommendations for future research. 2.3 Related Literature The literature in SBRAPs shows a limited number of methods used to determine and prioritize assignment criteria. Holness [12] m entions the lack of analyses to explain the selection of factors included in personnel assi gnment models. That is, most studies incorporate assignment criteria without explaining the rationale behind the selection of such criteria. Other studies determine assignment criteria using methods such as standard personality tests, interviews and surveys, the anal ytical hierarchy process (AHP), regression analysis, and case studies. Relevant li terature associated with these methods is discussed next. Standard personality tests are commonly used to det ermine assignment criteria. These tests usually rely on the Myers-Briggs scale to determine personality characteristics of available candidates, and classi fy candidates in four personality areas:

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10 extrovert versus introvert (E/I), sensing versus in tuitive (S/N), thinking versus feeling (T/F), and judgment versus perception (J/P) [13]. These personality characteristics are used as criteria in assignment processes to create heterogeneous teams. Examples of studies that used the Myers-Briggs scale for assign ment criteria are [13], [14], and [15]. Other studies such as [6] and [16] used the 16 pers onality factors (16PF) and the “assessment center method” standard tests to determ ine assignment criteria. Interviews and survey analyses are also used to determine assignment criteria. For example, Ng and Skitmore [17] conducted a surve y and analyzed responses with a discriminant analysis to identify similarities and differences between responses. Peslak [18] conducted a survey among university students a nd included personality factors using the Myers-Briggs scale. The author used principal component analysis and multiple linear regression to analyze survey responses and d etermine assignment criteria. Wong et al. [19] statistically analyzed survey responses wi th the Spearman rank correlation test and a two-way analysis of variance (ANOVA). Hausch ildt et al. [20] presented an interview and survey study that asked respondents t o rate employees based on a list of traits, and conducted a factor analysis to reduce t he list. Banaitiene and Banaitis [21], Zhang and Pham [22], and Cheney et al. [23] also co nducted interviews and surveys to determine assignment criteria. The literature on assignment criteria also shows st udies that used AHP. Most recently, El-Sawalhi [24] presented a model that pr ioritizes assignment criteria using AHP. The authors used a three-step screening proce ss to determine assignment criteria. First, they conducted a literature review to create a general criteria list. Second, they refined the list by including only criteria that we re recommended by more than three

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11 authors in the literature. Third, they conducted a n e-mail questionnaire to refine the list one more time, and establish a final criteria set. Al-Harbi [25] also presented a method that uses AHP to establish priorities for assignmen t criteria. In the study by Cheung et al. [26], the authors developed a multi-criteria approa ch to describe subjective judgment in a structured manner. The authors gathered data using a questionnaire survey, and applied AHP as a second stage analysis. Empirical tests that include regression analysis as a tool to determine assignment criteria are common in the literature regarding tea m formation and team performance analysis. Agrawal and Chari [10] developed a regre ssion model to determine criteria that affects quality and performance. Other similar stu dies that use regression analyses are [11], [27], [28], and [29]. Case study analyses and the Delphi technique have a lso been used to determine assignment criteria. Pieterse et al. [30] conducte d a case study analysis using students as subjects, and analyzed data with the non-parametric Spearman rank correlation test. Karn and Cowling [31] used a similar approach. Wynekoop and Walz [32] used the Delphi method to determine characteristics of top performe rs, and conducted a case study to support the results obtained from the Delphi method The Delphi method involves several rounds of data gathering from experts in th e field until a consensus is reached [33]. Patanakul et al. [34], Patanakul and Milosev ic [35], and Milosevic and Patanakul [36] also used case studies in conjunction with the Delphi method to determine assignment criteria. The literature shows two interesting insights relat ed to the use of priorities for assignment criteria. First, most personnel assignm ent methodologies do not consider

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12 relative priorities of criteria (e.g., [4], [37], [ 38], [39], [40], [41], [42], [43], [44], [45], [46], and [47]). Second, the methodologies that in corporate priorities do not explain their rationale for determining the priorities. That is, there is no process to help decisionmakers establish these priorities. Examples of suc h methodologies are found in [48], [49], [50], [51], [52], and [53]. This research as sumes that prioritizing assignment criteria helps to develop more accurate assessments of the suitability of candidates with tasks, hence leading to assignments that are more e fficient. In summary, the current literature shows that there are opportunities to improve areas regarding assignment criteria in SBRAPs. The following list highlights the major gaps found in the literature: Most assignment methodologies incorporate assignmen t criteria without explaining the rationale behind the selection of su ch criteria. Methods to determine assignment criteria are based on the effect of parameters to a single performance measure. There is a lack of m ethodologies to select assignment criteria based on data analysis that con sider multiple performance measures. Priorities for assignment criteria are seldom inclu ded in personnel assignment approaches, and are mostly determined subjectively. 2.4 Solution Approach and Methodology A conceptual diagram of the solution approach is sh own in Figure 2.1. The goal is to develop a generalized approach that can be ea sily transferred and customized to various industrial settings. The following subsect ions provide a detailed explanation of

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13 the two-stage methodology proposed to identify key assignment criteria in skill based environments. The methodology will be further expl ained in Section 2.4 through an example. Figure 2.1 Conceptual Diagram of Solution Approac h 2.4.1 First Stage – DEA Analysis DEA is a non-parametric methodology based on linear programming to evaluate the relative efficiencies of a group of entities ca lled decision making units (DMUs). n r n n nnn nn r nn nr n n r nrrn nn n n rn rn r nn n nrn n rrn # nrn n $% & $% $% $% & & &

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14 DMUs “must complete similar types of activities, pr oduce similar types of products and service, consume similar types of resources, and pe rform under similar environmental constraints” [54]. The DMUs in this research are p ersonnel assignments to tasks. These are basically assignments of expertise (i.e., years of experience) to tasks, which result in significant impact to quality and productivity meas ures. This way, expertise is treated as a discretionary variable since decision-makers may control the amount of expertise assigned to tasks. DEA estimates an empirical production frontier comp osed of the most efficient DMUs (i.e. those DMUs that are 100% efficient). Th e efficiency/inefficiency of a DMU not in the production frontier is calculated as the distance from the DMU to its corresponding reference point on the frontier. DMUs are classified as efficient/inefficient based on the “Pareto improvement” and “Pareto efficient” concepts. A Pareto improvem ent is an allocation that results in an improvement of at least one entity without worsenin g other entities. For example, a Pareto improvement occurs if reallocation of an emp loyee from project X to project Y improves the productivity of project X and does not affect the productivity of project Y. A Pareto efficient allocation (a.k.a. Pareto optimu m) occurs when there is no possibility for a Pareto improvement. Therefore, DMUs consider ed efficient cannot improve their position without worsening the position of other DM Us. 2.4.1.1 DEA Characteristics There are several characteristics of DEA that are r elevant and appealing to this study. First, DEA allows multiple outputs to be si multaneously considered, whereas

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15 other tools such as stochastic production frontier are limited to one output. This is very important in studies where multiple output paramete rs are necessary to properly determine efficiencies of DMUs. Second, being a no n-parametric approach, DEA does not assume functional relationships between paramet ers nor assumes the distribution of efficiency scores. Third, DEA evaluates the effici ency of a DMU relative to the efficiencies of other DMUs. This way, a DMU is alw ays compared to the best performer instead of being compared with an average performan ce as in regression analyses. Fourth, DEA assumes the responsibility of assigning weights to parameters. This characteristic makes DEA very suitable in situation s where differences in the production practices of DMUs are difficult to comprehend and t he level of importance of parameters may not be the same across DMUs. DEA assigns weigh ts in order to show a DMU in its “best possible way”, and then compares the efficien cy of the DMUs considered. If the “best possible way” scenario results in another DMU being more efficient than the DMU in question, then there is strong evidence for inef ficiency of the DMU. As such, DEA can focus on finding evidence of inefficiency for a DMU compared to a set of DMUs. Furthermore, DEA gives important insights into ways to increase the efficiency of DMUs by determining which input and output parameters ne ed to be improved. There are some limitations to DEA when using it to evaluate efficiencies. First, being a non-parametric approach, outliers and stati stical noise may significantly affect efficiency calculations. Therefore, decision-maker s must try to eliminate outliers from data samples. Second, a relatively small number of DMUs may lead to underestimated efficiency calculations. This can be overcome by s electing a small number of relevant inputs and outputs.

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16 2.4.1.2 Undesirable Variables and Isotonicity In DEA, an efficient DMU is one that can produce th e most outputs consuming the least amount of inputs. There are two fundamen tal rules about input/output parameters that must be followed to properly determ ine efficiency scores. First, DEA expects increases in output values and decreases in input values to be beneficial. Therefore, output parameters such as project durati on and inputs parameters such as workload per employee must be transformed so that t hey become beneficial. Input and output variables that require transformation to com ply with this rule are called undesirable parameters. There are several methods discussed in the DEA lite rature to model undesirable variables. One of the most common methods is calle d the [TR] transformation. In the [TR] transformation, an undesirable output is subtract ed from a larger scalar value such that all transformed values are positive and increa sing values are desirable. “The large scalar value is usually selected as a value just sl ightly larger than the maximum value of the undesirable output observed in the data set, si nce choosing a value that is much greater than this maximum value can distort model r esults” [54]. The second fundamental DEA rule is that an increase in an input variable must improve each of the outputs. This is called the is otonicity property of DEA parameters. Correlation analyses must be conducted to ensure po sitive relations between inputs and outputs. Negative correlation results indicate tha t one or more parameters may need to be excluded from the model. Testing for isotonicity o f parameters is essential to validate DEA models.

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17 2.4.1.3 DEA Input/Output Parameters and Number of DMUs The minimum number of DMUs for a DEA analysis needs to be carefully selected, given that DEA could identify a large por tion, if not all, of the DMUs as efficient. This can occur due to an inadequate num ber of degrees of freedom. Dyson et al. [55] recommends having at least twice as many D MUs as the total number of inputs and outputs. However, as a rule of thumb stated by one of the creators of the DEA technique in [7], the number of DMUs should be at l east equal to ( ) ) ( 3 max s m s m + * where m and s are the number of inputs and outputs respectively. Obtaining data for analysis in skill-based environm ents is often very difficult [10], which results in limited number of DMUs to conduct DEA studies. Since the minimum required number of DMUs is a function of the number of inputs and outputs, it is advisable to keep the number of inputs and outputs as small as possible. This helps to improve the efficiency estimation capability of DEA One way to minimize the number of parameters is to include those that serve as pro xies to other parameters. For example, overall years of experience of an employee can be u sed to represent salary, organizational experience, and exposure to company processes. Oth er types of parameters, such as specific knowledge in particular skills, will be in cluded in the Tobit regression analysis during the second stage. The generalized DEA model consists of two inputs an d two outputs. These parameters are shown in Table 2.1, as well as their definition in particular disciplines. Overall experience is defined as the number of year s of experience of resources that were assigned to a task. Effort, quality, and performan ce are application-specific measures that must be determined by decision-makers. Correl ation tests need to be performed to

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18 ensure that the parameters adhere to the isotonicit y property of DEA. Again, negative correlation between parameters may cause the exclus ion of a parameter from the model. Table 2.1 Inputs/Outputs for DEA Model Software EngineeringR&D Projects Overall experience Input Years of industry experience Years of experience of a resource as a Ph.D. EffortInput Number of engineers assigned per KSLOC Hours per Project QualityOutput KSLOC per software defect Number of publications per project PerformanceOutput Cycle time density (i.e., number of SLOC per hour) Adherence to Schedule Input/ Output Examples Parameter 2.4.1.4 Orientation of DEA Model DEA provides two basic model orientations: output m aximizing and input minimizing. The selection of model orientation dep ends on the objectives of the study. An output maximizing oriented model determines the maximum proportional increase in outputs relative to the actual input values, which is adequate to establish a set of target output values. Output maximizing models are also u sed when output levels are discretionary but input levels are relatively fixed (i.e. non-discretionary) [54]. An input minimizing oriented model determines the amount by which the input values can be decreased while still producing the same outputs, w hich is adequate to evaluate the efficiencies of internal processes. For this resea rch, an input-oriented model is used

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19 given that the main objective is to allocate resour ces more efficiently based on input parameters rather than to improve the outputs. 2.4.1.5 DEA Model Selection Returns to scale is an important concept in the fie ld of Economics that needs to be well understood since it is used by DEA models to f orm efficient frontiers. There are three types of returns to scale: increasing, decrea sing, and constant. Constant returns to scale describe the case where an increase of input by a constant amount results in an increase in output by the same constant amount. If the output increases by more than the constant amount, then it is called increasing retur ns to scale, or economies of scale. If the output increases by less than the constant amount, then it is called decreasing returns to scale or diseconomies of scale [56]. Employees in skill-based environments are more like ly to operate under both economies and diseconomies of scale. Skirbekk [57] mentions that “job experience improves productivity for several years, but there does come a point at which further experience no longer has an effect.” That is, more experience does not necessarily equate to increased productivity. Therefore, it will be a ppropriate to select a DEA model that allows resources in the efficient frontier to opera te under diseconomies of scale. The DEA model selected is the input-oriented BCC mo del, named after its inventors Banker, Charnes, and Cooper in 1984 [7]. The model assumes variable returns to scale frontiers, which means that efficient DMUs may operate under increasing, decreasing, or constant returns to scale. Hence, t he model allows DMUs operating under diseconomies of scale to be classified as efficient (i.e. be part of the efficient frontier).

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20 Complexity of tasks must be considered when determi ning efficiencies of past personnel assignments. Decision-makers have two op tions to deal with complexity. The first option is to compare performances of personne l assignments in tasks with similar complexity levels. That is, in the case of low and high-complexity tasks, develop an input-oriented BCC model for low-complexity tasks a nd another for high-complexity tasks. The second option is to compare performance s among tasks with different complexity levels. More specifically, performances of personnel assignments to lower level complexity tasks may be compared to those wit h higher level complexity tasks, but not vice versa. This option requires a hierarchica l categorical model, which is easily incorporated into the BCC model. Cooper et al. [7] call this model the categorical variable DEA model. 2.4.2 Second Stage Tobit Regression Analysis The DEA analysis from the first stage provides effi ciency scores for personnel assignments. After focusing on the level of effici ency of the assignments, the main challenge is to understand the impact of personnel skills on efficiency scores. This can be achieved through regression analysis. Efficiency scores are considered censored variables because they are continuous and distributed over a limited interval, in this ca se between 0-1. The common regression analysis using the ordinary least squares approach provides bias results in the presence of censored variables [58]. The preferred choice amon g researchers is Tobit regression, which is based on maximum likelihood procedures. A recent study comparing

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21 approaches for modeling DEA scores indicates that T obit regression is an effective tool that provides reliable results [8]. Although Tobit regression analysis has been previou sly used to model DEA efficiency scores, a DEA-Tobit regression approach for personnel assignments in skillbased settings has not been addressed in the litera ture. Equation (2.1) shows the Tobit model specification, where iqis the DEA efficiency score for personnel assignmen t i ijxare independent variables (j = 1 to k) for personnel assignment i, and ie is the disturbance term. Standard linear regression assum ptions for the disturbance term must be met [59]. That is, appropriate tests for normal distribution and constant variances of the error terms must be conducted. i k j ij i ixe b b q+ + == 1 0 (2 .1) 2.4.2.1 Independent Variables The most important independent variables to conside r are skills/expertise of personnel. However, other factors (e.g., task fact ors or team factors) can be included if necessary to improve the performance of the model. Table 2.2 shows examples of parameters that can be used to develop Tobit models for particular disciplines. These parameters can b e modeled using either quantitative or categorical variables.

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22 Table 2.2 Examples of Parameters for Tobit Models in Particular Disciplines Examples of Independent Variables Type of Factor Software Engineering R&D Projects Programming language experience Domain expertise Domain expertise Statistical software experience Personnel Application expertise Expertise in non-parametric approaches Task Size (i.e., number of SLOC) Scope 2.5 Example – Software Development Setting Data from a software development organization was u sed to test the capability of the solution approach. Task assignment in software development environments is considered one of the most critical decisions since it influences the performance and quality of projects [6]. Quality, as evidenced in the U.S. General Accounting Office Report in [2], continues to be a major struggle to software companies. This report states that in 2004 the U.S. Department of Defense spent n early 8 billion dollars to rework software because of quality-related issues. Even m ore important than huge monetary costs is the fact that software failures in safetycritical systems may result in lifethreatening situations. Tsai et al. [43] stated th at “evidence reveals that the failure of software development projects is often a result of inadequate human resource project planning”. Despite its importance, the literature reveals majo r gaps related to the assignment criteria and methodology in software development pr ojects. To close these gaps, it is necessary to determine factors that significantly a ffect the efficiency of assignments of software developers to software tasks. Efficiency is measured in terms of how the overall

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23 experience of developers, considering different lev els of task complexities, affect the number of software defects and cycle time (i.e., th e time it takes to complete tasks) simultaneously. The research questions addressed t hrough this example are the following: What are the relative impacts of various personnel and task factors on the technical efficiency of software tasks? How do these relative impacts compare with the conc lusions of studies in the literature regarding factors affecting the quality at the project level? These questions are of importance from both the pra ctical and theoretical perspective. The purpose of applying the DEA-Tobit methodology i n a software development setting is two-fold. First, this example serves to demonstrate the capability of the methodology using real industry data. Second, the results significantly contribute to the software engineering literature by identifying and prioritizing assignment criteria based on the effects of particular factors to the quality and duration of tasks. This type of analysis, which considers multiple performance meas ures simultaneously, has not been conducted in the software engineering field. 2.5.1 Previous Studies The software development literature shows that soft ware defects increase repair costs [60]. The common peer review technique for d efect-detection catches from 31 to 93 percent defects, with a median of approximately 60 percent [61]. However, “very few research efforts have been conducted with respect t o factors influencing defect injection” [60]. Figure 2.2 shows defect introduction and rem oval pipes similar to [60]. A

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24 percentage of residual defects from the earlier pha ses of software development will continue into subsequent phases, increasing the pro bability of more costly defects at the later phases, and eventually becoming field defects Despite the fact that minimizing faults in code is the responsibility of individual programmers, most methods ignore causal effects of programmers [62]. !nn rn !'r n n !n nr nr Figure 2.2 Software Defect Introduction and Remov al Process Table 2.3 shows a selection of studies on team fact ors affecting the quality and productivity of software projects. Factors such as project size, team capabilities, team average domain experience, communication among team members, and task complexity

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25 have been found to influence the quality and produc tivity of software projects. Other studies provided contradictory results, concluding that project size and complexity [63], and professional experience [64] do not affect the outcome of projects. These contradictions elevate the necessity to conduct a m ore detailed investigation regarding the factors affecting important performance measures of software tasks. Table 2.3 Selected Literature on Team Factors Aff ecting Quality and/or Productivity Study Selected Dependent Variables Selected Independent Variables Industry Findings Agrawal and Chari (2007) [10] Effort, Quality, Cycle Time Product size, Complexity, Team size, Team capability CMMI level 5 organization (mainly business applications) Product size was the only significant driver of effort, cycle time, and quality. Jacobs et al. (2007) [60] N/A N/A Various This was a literature survey to determine factors that affect defect injection. Capability, domain knowledge, team parameters, complexity, process maturity, and communication affect quality. Tiwana (2004) [65] Design effectiveness and efficiency, and design density Knowledge integration (business domain and technical knowledge) Unknown Knowledge integration affects development effectiveness and defect density. Nan et al. (2003) [66] Effort, Quality, Cycle Time Schedule pressure Unknown Schedule pressure may reduce effort and cycle time without impacting quality.

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26 Table 2.3 (Continued) Krishnan et al. (2000) [11] Productivity, Quality Product size, Team capability, Usage of tools, Process factors, Proportion of front-end investments Commercial software systems applications Product size, team capability, front-end investment, and software process affect quality. The usage of tools was not a significant factor affecting the quality. Faraj (2000) [64] Team performance (based on expert judgment of quality, goals met, and team operations) Technical expertise (subjective average of technical, design, and domain expertise), Professional expertise (years of experience), Administration measures (number of status meetings, etc.) Large software company developing software for commercial clients Technical expertise coordination affects team performance more than the actual presence of team expertise and administrative coordination. Professional experience had no impact on team effectiveness. Social integration contributes to performance more than technical integration. Fenton and Ohlsson (2000) [63] Quality Product size, Complexity Ericsson Telecom AB Quality is not affected by product size or complexity. Krishnan and Kellner (1999) [28] Quality CMMI software process practices, Product size, Team capability Commercial software systems applications Consistent adoption of CMMI practices reduces field defects. Team capability affects the number of field defects. Krishnan (1998) [29] Quality, Cost Product size, Team capability, Programming language experience, Domain experience Commercial packaged software projects Team capability, domain experience, and product size affect the quality. Team capability and product size affect the development cost. Domain experience has no effect on the development costs. Programming language experience has no effect on either quality or development costs. Gaffney (1984) [67] Quality Product size Unknown Product size is a good estimator of quality.

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27 A major drawback from previous studies is that data samples, most of the time, come from students and not professional employees [ 29]. The reason for this is that obtaining software development data from corporatio ns is very complicated in the best of circumstances [10]. Therefore, it is necessary to conduct more research studies with industry data in order to significantly contribute to the literature on software quality. Another limitation of previous studies is that most are based on multiple-inputsingle-output analyses (e.g., [10], [68], and [69]) To the best of our knowledge, a study that considers the multiple-input and multiple-outp ut case has not been addressed in the literature regarding software quality and productiv ity. The literature also shows studies that investigate important factors of individual team members. In [70], the authors conducted a con trolled experiment and found that years of experience in specific software domains wa s a significant factor affecting the time it took programmers to find planted bugs. Acu a et al. [6] described capabilities of individuals based on standard tests for behavioral assessments. Other studies such as [14], [15], and [71] examined individual characteri stics for software development team success with different standard personality tests. Examples of additional studies that have considered personality traits of top performin g software developers can be found in [72], [73], [74], and [75]. Personal characteristi cs that have been identified as common traits of top performing engineers include creative problem solving skills, leadership skills, and communication skills, among others. Re searchers have also looked at technical skills of top performing developers by co llecting data from interviews and surveys and using subjective performance measures [ 22], [23]. In [76], the authors studied the ability of teams to work together based on the working style of individual

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28 members. A methodology to add personnel to the tea m with the objective of reducing conflict was developed. 2.5.2 Data for Analysis Data for this research was collected from a leading CMMI level 5 organization specializing in the development of software applica tions for the defense industry. The data included information from two projects. Each project was divided into smaller software components called computer software config uration items (CSCIs), where each CSCI was divided into computer software components (CSCs). Figure 2.3 shows this modular project structure which is necessary to imp rove the management of software products. On average, four engineers were assigned to each CSC. The data collected contained information on 76 CSCs. For simplicity, the rest of this paper uses the term “task” instead of CSC. Therefore, as mentioned in Section 0, the DMUs in this research are personnel assignments to tasks. Figure 2.3 Modular Project Structure

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29 The data provided a categorical parameter to descri be the complexity of each task: high and average-complexity. Levels of complexity were assigned based on types of applications. For example, creating operating syst ems or real-time embedded software applications were considered of high complexity. D eveloping graphical user interface applications or client-server applications were con sidered of average complexity. In addition, meetings with software analysts were cond ucted to ensure the validity of the data. There were 36 average-complexity tasks and 40 of hi gh complexity. According to [10], a sample size of 30 or higher is an adequa te size for the analysis. It is also comparable with related studies [77]. Moreover, th is sample size is especially significant for this study since there are only 141 CMMI level 5 organizations worldwide [78]. The input parameters considered for the DEA model a re overall experience and effort. For each task, overall experience is defin ed as the average number of years of industry experience of its resources working with s oftware architectures, specifications, and requirements. This input serves as a proxy to parameters such as salary, leadership, and organizational experience. On the other hand, effort is defined as the number of engineers assigned for a thousand software lines of code (KSLOC). That is, effort is normalized by the size of software tasks to allow f air comparisons between assignments. For example, two engineers that completed two KSLOC and one engineer that completed one KSLOC results in the same effort value (i.e., o ne engineer per KSLOC). Effort may also be explained in terms of workload (i.e., KSLOC per engineer). As effort values increase, workloads per engineer decrease. Less wo rkload per engineer should result in better performance since debugging software applica tions becomes more complex as the

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30 number of SLOC increases. These inputs are good in dicators of the overall knowledge and costs invested to complete software tasks. The output parameters considered are quality and pr oductivity. In [10], the authors define quality as the “total number of defe cts that escaped to the customer”. Studies such as [28] and [29] also define quality a s number of defects. Instead of defect counts, this research defines quality as the number of KSLOC per post-release defects. This measure of quality has been used in previous s tudies such as [11] and [68]. KSLOC per defect is selected over defect counts because i t controls the effect of varying SLOC sizes among tasks. The measurement for productivity is cycle time dens ity which is the number of SLOC written per hour. This allows cycle time to b e modeled as a desired output variable since higher values of this parameter are preferred. This definition is slightly different than the usual one found throughout the l iterature, which is the number of days that elapsed from starting the requirements or desi gn phases to completing the development phase [10], [66]. 2.5.3 First Stage – DEA Analysis The goal of this stage was to develop DEA models to determine relative efficiencies of personnel assignments to average an d high-complexity tasks. First, correlation analyses were conducted to verify the p resence of isotonicity between inputs and outputs. Table 2.4 and Table 2.5 show the corr elation results.

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31 Table 2.4 Correlation Analysis for DEA Parameters (Average-Complexity Tasks) KSLOC per Defect Productivity Experience 0.85 0.73 Effort (Staff per KSLOC) -0.61 -0.42 Table 2.5 Correlation Analysis for DEA Parameters (High-Complexity Tasks) KSLOC per Defect Productivity Experience 0.63 0.59 Effort (Staff per KSLOC) -0.63 0.10 The results from the correlation analyses showed a strong positive correlation between experience and both output parameters. How ever, there was negative correlation between effort and KSLOC per defect in both analyses, and between effort and productivity in one of the analyses. Therefore the effort parameter was removed from the DEA analyses due to lack of isotonicity. Increasing the effort assigned to tasks was expecte d to improve both KSLOC per defect and productivity. The rationale was that in creasing the number of staff per KSLOC would have decreased workloads per staff, the refore resulting in improvement of outputs. Correlation results clearly showed that t his was not the case. A possible explanation for this behavior is that increasing th e number of staff may have also increased communication overhead. As in [79], incr eased communication overhead could have led to non-productive results. Other input parameters such as average cost per KSL OC or average cost per staff would have been adequate if data were available. H owever, research data was limited in

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32 this regards. As mentioned before, experience enco mpasses different important parameters such as salary, leadership, and organiza tional experience; therefore, experience is the only input parameter considered i n the DEA analyses. Z-tests were conducted to determine if the means of the output parameters, normalized by years of experience, from average com plexity tasks were statistically equal to those from high complexity tasks. In other word s, the goal of these tests was to determine if productivity (and quality) per years o f experience was different between the high and average tasks. The results from the z-tes ts provided evidence, at an alpha of 0.05, that the normalized means were statistically different between both types of tasks. This justifies conducting separate DEA analyses for high and average complexity tasks to allow fair comparisons between DMUs. Table 2.6 shows the results of the DEA analyses. R ecall that input-oriented BCC models were used.

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33 Table 2.6 DEA Results Efficiency of Personnel A ssignments 2.5.4 Second Stage Tobit Regression Model Tobit regression analyses were conducted to investi gate the factors that significantly affect the efficiency of personnel as signments to average and high DMUDEA ScoreDMUDEA Score Hi_1 1.000Nom_11.000 Hi_2 1.000Nom_21.000 Hi_3 1.000Nom_31.000 Hi_4 0.975Nom_40.714 Hi_5 0.941Nom_50.500 Hi_6 1.000Nom_60.667 Hi_7 1.000Nom_70.500 Hi_8 1.000Nom_80.621 Hi_9 0.662Nom_91.000 Hi_10 0.500Nom_101.000 Hi_11 1.000Nom_110.555 Hi_12 1.000Nom_121.000 Hi_13 1.000Nom_130.625 Hi_14 1.000Nom_140.759 Hi_15 0.500Nom_150.640 Hi_16 0.730Nom_160.526 Hi_17 0.668Nom_170.624 Hi_18 0.659Nom_181.000 Hi_19 0.802Nom_190.564 Hi_20 0.668Nom_201.000 Hi_21 0.629Nom_210.742 Hi_22 0.602Nom_220.705 Hi_23 0.500Nom_230.785 Hi_24 0.823Nom_240.735 Hi_25 0.629Nom_250.756 Hi_26 0.250Nom_260.960 Hi_27 0.530Nom_270.750 Hi_28 0.618Nom_280.480 Hi_29 0.333Nom_291.000 Hi_30 0.382Nom_301.000 Hi_31 0.795Nom_310.703 Hi_32 0.375Nom_320.667 Hi_33 0.566Nom_330.882 Hi_34 0.558Nom_340.782 Hi_35 0.987Nom_350.587 Hi_36 0.475Nom_360.882 Hi_37 0.301 Hi_38 1.000 Hi_39 0.389 Hi_40 0.916ComplexityHighAverage Avg. = 0.719Avg. = 0.770

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34 complexity tasks. The idea is to identify potentia l assignment criteria based on the factors that significantly increase efficiency. Th e dependent variable in the Tobit models is the DEA score. Independent variables include th e personnel and task factors shown in Table 2.7. Table 2.7 Independent Variables Type of Factor Factor Name Variable (abbreviation) Description Measurement Type Programming language experience PL Experience with the programming language required by the task Categorical variable with two levels: High = 1 Low = 0 Development system experience DSE Experience with the software and hardware tools to complete the task Categorical variable with two levels: High = 1 Low = 0 Practices and methods experience PME Experience with the software processes and methods particular to the task, such as design reviews and other QA activities Categorical variable with two levels: High = 1 Low = 0 Personnel Factors Programmer Capabilities PC Subjective measure of ability, including motivation and communication skills Categorical variable with two levels: High = 1 Low = 0 Size SIZE SLOC count Quantitative Task Factors Requirements volatility REQ Frequency and scope of requirement changes after being approved. Categorical variable with two levels: High = 1 Low = 0 Personnel factors are modeled as dichotomous catego rical variables with high and low levels. High levels of experience are defined as more than two years of experience. It is important to not confuse years of experience with programmer capabilities (PC). Instead, capability subjectively measures the abili ties of resources based on their perceived potential, including motivation and commu nication skills.

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35 The size of tasks (SIZE) is measured using number o f functional SLOC. The number of SLOC has been shown in the literature to affect both the quality and cycle time of software tasks [10], [29], [28]. Requireme nts volatility (REQ) captures the frequency and scope of requirement changes. These changes may be caused by the inability of the customers to define requirements d uring the initial stages of projects, inability to properly characterize and document req uirements, and other unexpected constraints imposed by software/hardware tools. Correlation analyses between independent variables were conducted to test for multicollinearity. Correlation between dichotomous variables is usually computed with the phi-coefficient or point biserial methods. Com rey and Lee [80] explained that the Pearson correlation coefficient yields the same res ults if dichotomous variables are scored 1 for the higher category and 0 for the lowe r one. Therefore, the Pearson coefficient method was used to calculate the correl ation coefficients (see Table 2.8 and Table 2.9). Table 2.8 Correlation of Independent Variables in Tobit (Average-Complexity) PL DSE PME PC PL 1 DSE 0.478 1 PME 0.181 0.076 1 PC 0.331 0.277 -0.021 1

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36 Table 2.9 Correlation of Independent Variables in Tobit (High-Complexity) PL DSE PME PC PL 1 DSE 0.498 1 PME -0.020 -0.108 1 PC 0.332 0.175 0.233 1 The results show mostly weak correlations between p arameters. However, there is a weak-to-moderate correlation between programmi ng language and development system experience in both cases, which is expected. The lack of strong correlations between the parameters satisfies the multicollinear ity assumption in multiple regression analysis. Equation (2.2) specifies the empirical model for th e DEA efficiency scores. Equation (2.3) shows the Tobit regression model, wh ere *q is the vector of DEA efficiency scores. Efficiency = Function (PL, DSE, PME, PC, SIZE, REQ) (2.2) ) ( ) ( ) ( ) ( ) ( ) (6 5 4 3 2 1 0 *REQ SLOC PC PME DSE PLb b b b b b b q+ + + + + + = (2.3) The Tobit regression analyses were developed using the R statistical software tool. Residual analyses and normal probability plo ts showed that the assumptions of constant variance and normal distribution of the er ror terms were met. Table 2.10 shows the results of the Tobit regressio ns. The goodness-of-fit measure for the models was the square of the correl ations between actual and expected DEA scores [9]. This measure, denoted pseudo-R2, represents the variability of the DEA scores that is explained by the independent variabl es. The Wald Chi-Square statistic

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37 result rejects the null hypothesis that the regress ion coefficients, except for the intercept term, are not significantly different from zero [81 ]. Table 2.10 Tobit Regression Results Explanatory variable Personnel assignments Average-complexity __________________ High-complexity __________________ Estimated b coefficient Estimated b coefficient INTERCEPT 0.434* 0.968** Personnel Factors PL 0.239 0.095 DSE 0.178 0.992** PME 0.082 0.140 PC 0.302** -0.013 Task Factors SIZE -2.789E-06 -1.851E-05** REQ -0.135 -0.201* Pseudo-R2 0.400 0.530 Wald Chi-Square statistic 16.06 on 6 DF (p = 0.0134 ) 27.54 on 6 DF (p = 0.0001) = significant at 5% ** = significant at 1% 2.5.5 Discussion The results from the Tobit analyses show important differences between high and average-complexity tasks. For personnel assignment s to high-complexity tasks, the results show that both task factors are statistical ly significant and negatively affect the efficiency scores. These results are compatible wi th other studies in the literature which

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38 concluded that the number of SLOC and changes in re quirements significantly affect the quality and productivity of software projects [10], [11]. However, both tasks factors were not significant in average complexity tasks, which suggests that resources working these tasks are able recuperate from requirement changes without a significant effect to quality and productivity. This also suggests that increase d values of SLOC and changes in requirements result in additional complications tha t significantly affect the outcome of high-complexity tasks. Regarding high SLOC values, managers must ensure that objectoriented (i.e., software modularity) standards are strictly followed by developers. Regarding changes in requirements, there is a vast amount of literature on methods for creating and managing software requirements [82], [ 83], [84]. The effect of programming language experience on ef ficiency was not statistically significant for either average or high-complexity t asks. These results are compatible with the study of Krishnan [29], where it was concluded that programming language experience had no effect on software quality. This is a critical finding since often programming language is used as the main criteria f or resource assignments [5]. The experience of resources in software practices a nd methods was not a significant contributor to efficiency for both type s of tasks. Studies such as [11] and [28] analyzed the effects of implementing consistent sof tware practices and processes and concluded that they significantly affect quality. However, the literature lacks a study that incorporates the knowledge of resources in software practices as a potential driver for quality and productivity. To the best of the autho rs’ knowledge, this study is the first one to incorporate and analyze the effect of such facto r.

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39 Of the four personnel factors in the model, develop ment system experience was found to be the only significant contributor to eff iciency in high-complexity tasks. In average-complexity tasks, only programmer capabilit y was found to be significant. This suggests that in-depth knowledge of software techni ques and hardware tools are drivers of efficiency in challenging tasks, whereas motivat ion and communication skills are the efficiency drivers for the less challenging ones. Consequently, development system experience should be given higher priority as an as signment criterion for high-complexity tasks, and programmer capability for average-comple xity ones. 2.6 Summary and Contributions This study presented a methodology based on DEA and Tobit regression to analyze the impact of factors believed to affect th e efficiency of personnel assignments in skill-based tasks. The methodology was used to ana lyze data regarding software tasks from a leading software development company. The d ata were divided into two categories: average and high-complexity tasks. Usi ng DEA, efficiency scores were computed for each of the two categories. Input and output parameters for the DEA analyses were validated by conducting correlation t ests to verify that the models followed the isotonicity assumption of DEA. Tobit regression models were developed to regress t he DEA scores against personnel and task factors believed to affect effic iency. Task factors included number of SLOC and frequency of changes in requirements. Per sonnel factors included programmer capability, programming language experie nce, practices and methods experience, and development system experience. The results showed that both task

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40 factors were significant in high-complexity tasks o nly. Furthermore, programming language experience was not a significant factor af fecting efficiency. The results indicated that development system experience was th e only significant personnel factor for high-complexity tasks, and programmer capabilit y for average-complexity tasks. This work contributes to personnel assignment resea rch by presenting an analytical approach that considers multiple outputs simultaneously and eliminates subjectivity when determining relative priorities f or assignment criteria in skill-based environments. This is of significant use and relev ance to decision makers since most personnel assignment decisions in industry settings involve the evaluation of several performance measures and a struggle for decision ma kers to subjectively determine important parameters. The methodology presented in this research provides a new mechanism for decision makers to objectively identify assignment criteria based on the factors that significantly affect efficiency. The methodology r educes subjectivity in two ways. First, it eliminates the need for decision makers to estab lish subjective weights for parameters when determining efficiencies, as the best possible weights for each parameter are determined by DEA. Second, assignment criteria are identified as a result of regression analyses from actual data. An important aspect of the methodology is that it d etermines efficiencies of previous personnel assignments as a function of the efficiency of best performers. This results in more rigorous evaluation of relative eff iciencies than other methodologies which determine efficiencies as a function of avera ge performances.

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41 Demonstrating the capability of the methodology usi ng software development data from a major corporation resulted in the ident ification of drivers of efficiency (i.e., assignment criteria) of personnel assignments per t ask complexity. The resulting assignment criteria are readily available for decis ion makers in software development settings, which is another key contribution of this research. To further confirm the capability of the research p resented, future work is needed to apply the methodology in different industrial se ttings. Furthermore, it is necessary to determine the acceptance of the results by decision -makers from other environments. Doing so will help to further establish the real pr actical value of the solution approach. Another future research opportunity for software en gineering researchers is to confirm and expand the results of this study. That is, the data provided for this study were limited to four personnel factors. It will be beneficial to conduct research with additional personnel and task factors to increase o ur understanding of drivers of efficiency of software applications. This research was motivated by a notable gap in the literature regarding a lack of adequate methodologies to assign resources to tasks in skill-based scenarios. The outcome of this research fills this gap by providin g a process that can be measured and improved, therefore promoting a mentality of contin uous improvement.

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42 CHAPTER 3 A FUZZY EXPERT SYSTEM ARCHITECTURE FOR CAPABILITY ASSESSMENTS IN SKILL-BASED ENVIRONMENTS 3.1 Abstract The fast pace at which new technologies and techniq ues are being developed to improve the design and development of products incr eases the demand for specialized individual skills in the workforce. As a result of higher demands, candidates with exact required skills to work tasks are usually unavailab le. Due to the lack of proper methods to assess personnel capabilities, decision makers a re forced to assign resources to tasks based on shallow assessments. To tackle this issue this research presents a layered expert architecture where subcomponents can be cust omized to specific industrial settings. A fuzzy logic scheme is described to mod el personnel capabilities as imprecise parameters, and to consider complete skill sets of resources when evaluating their levels of expertise in a skill. The proposed approach lea ds to thorough capability assessments, as well as an increased number of capable candidate s.

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43 3.2 Introduction Despite all the research and advances in the projec t management field, managing human resources remains a very complicated endeavor A major contributor to this complexity is the increased demand for specialized individual skills in the workforce, which results from high turnover rates and the fast pace at which new technologies and techniques are being developed. As a result of hig her demands, candidates with exact required skills to work tasks are usually unavailab le. Due to the lack of proper methods to evaluate personnel capabilities, decision makers struggle to efficiently assign resources to tasks. This results in excess training times th at significantly affect the cycle time for product development, as well as overall quality mea sures. Therefore, further studies of processes and techniques for personnel capability a ssessments are necessary to provide better solutions in terms of quality, cost, and sch edule. This research proposes a fuzzy expert system archit ecture as a solution to the personnel capability assessment problem. The propo sed architecture is divided into four layers: user interface, fuzzy logic system, data re pository, and global layers. The scope of this research is to provide a detailed descripti on of the fuzzy logic inference system (a.k.a. approximate reasoning), and briefly describ e the rest of the layers to give a clear idea of the expected flow of data throughout the sy stem. As such, this research lays out the foundation for the development of fuzzy expert systems for personnel capability assessments in industrial environments. The fuzzy logic scheme described in this research i s an extension to an exploratory approach developed by Otero et al. [5]. Their methodology, denoted by the

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44 authors as the best-fitted resource (BFR) methodolo gy, suggests that capability levels in particular skills are influenced by resources’ know ledge in related skills. That is, resources without proper experience in required ski lls perhaps are proficient in similar skills which can accelerate the learning process. For example, knowledge in the C++ programming language can decrease, to some extent, the training time of a programmer to become proficient in the C# programming language because they are both objectoriented languages and have a somewhat similar synt ax. This approach of considering relationships between skills leads to more thorough capability assessments and increases the set of possible candidates to work tasks that r equire specific skills. This research extends the BFR methodology in two wa ys based on the assumption that capability ratings and skill relationships are essentially imprecise factors. First, this study employs fuzzy set theory to describe the capa bility ratings of resources in particular skills as degrees of membership in various fuzzy se ts. The BFR methodology, on the other hand, describes capability ratings as crisp v alues based on classical set theory. Second, this research describes skill-relationships using fuzzy rules, whereas the BFR method uses crisp values for the development of the ir skill-relationship tables. Although fuzzy expert systems for personnel assignments have already been introduced to the literature (e.g., [41] and [49]) to the best of our knowledge the use of a fuzzy logic approach to determine personnel capabilities is a n ew contribution to the literature. This chapter is organized as follows. Section 3.2 describes the proposed fuzzy expert system architecture. It provides a review o f important fuzzy logic concepts that are necessary for understanding the functionality o f the expert system. The section concludes with a description of the step-by-step fl ow of data throughout the expert

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45 system. Section 3.3 formulates a personnel capabil ity assessment problem in a software development setting to demonstrate the implementati on of the solution approach. The last section provides conclusion remarks, contribut ions to the literature, and ideas for future research. 3.3 Fuzzy Expert System Architecture An expert system is a “computer-based system that e mulates the reasoning process of human experts within a specific domain o f knowledge” [85]. An expert system generally consists of three components: a us er interface, usually a graphical user interface (GUI), that receives user inputs and show s final results; a logic system to make inferences about data; and a data repository used t o store/receive information. Figure 3.1 shows the general components of an expert system an d the bidirectional relationship that often exists among them. Figure 3.1 Conceptual Fuzzy Expert System User Interface Data Repository Fuzzy Logic System User Interface User Interface Data Repository Data Repository Fuzzy Logic System Fuzzy Logic System

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46 Figure 3.2 shows the proposed high-level software a rchitecture developed from the conceptual expert system shown in Figure 3.1. It corresponds to a layered architecture that minimizes dependencies between co mponents. This type of architecture allows the system to be flexible to accommodate fut ure expansions such as different subcomponents in the data layer (e.g., data files, Oracle database), or various types of presentation subcomponents (e.g., command line, Jav a GUI, C# GUI). The following subsections describe each of the architecture layer s in mode detail. Figure 3.2 Layered Software Architecture 3.3.1 Presentation, Data, and Global Layers The presentation layer corresponds to any type of i nterface used to gather inputs and show information to users. The two commonly us ed interfaces are command lines and GUIs. Usually GUIs are preferred due to their user-friendly interfaces that facilitate the data retrieving/displaying activities. Data Layer Fuzzy Logic System Fuzzy Logic System Fuzzification Module Defuzzification Module Fuzzy Inference Engine Fuzzy Rules Presentation Layer Global Layer Employee_Rep Knowledge_Rep Data Layer Data Layer Fuzzy Logic System Fuzzy Logic System Fuzzification Module Defuzzification Module Fuzzy Inference Engine Fuzzy Rules Fuzzy Logic System Fuzzy Logic System Fuzzy Logic System Fuzzification Module Defuzzification Module Fuzzy Inference Engine Fuzzy Rules Presentation Layer Presentation Layer Global Layer Global Layer Employee_Rep Knowledge_Rep

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47 The data layer is composed of two repositories: Kno wledge_Rep and Employee_Rep. The Knowledge_Rep repository contain s a set of fuzzy rules to be used by the logic system to make inferences. In additio n, this repository manages the set of membership functions used to model levels of expert ise of employees in various skills, and those that are used to establish fuzzy implicat ions between skill levels. The Employee_Rep respository manages crisp rating v alues representing the capabilities of resources in various skills. For e xample, consider a rating scale from 0-5 and let { s rt } denote the crisp rating rt of a resource in skill s where the number of skills in the resource’s skill set is three. Then, values like {1, 2.5}, {2, 4}, and {3, 1} would indicate that the crisp capability rating of the re source in the first skill is 2.5, in the second skill is 4, and in the third skill is 1. The global layer acts as a mediator for the rest of the layers to communicate with each other. This is possible because the global la yer is equipped with information regarding the subcomponents responsible for any req uest. For instance, whenever the presentation layer wants to retrieve information fr om the data layer, the presentation layer makes a request using an interface provided by the global layer. This interface guarantees that the request is forwarded to the app ropriate subcomponent in the data layer. This means that the presentation layer requ ests information without worrying about the type of data repository subcomponent used in the data layer to hold such information. When the required information is gath ered, the data layer provides the desired information to the presentation layer throu gh the global layer. This type of architecture minimizes dependencies between layers by making them communicate with each other only through the global layer. Therefor e, new subcomponents added to the

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48 data layer to handle requests from the presentation layer, for example, will not require any modifications to the presentation layer. This type of architecture follows the objectoriented paradigm by being reusable, robust, and ea sy to maintain. 3.3.2 Fuzzy Logic System Logic is the study of methods for reasoning [85]. Classical logic relies on the assumption that propositions are either true or fal se. Fuzzy logic, on the other hand, relies on the assumption that propositions are true to some degree. This way, fuzzy logic allows logical reasoning with partially true imprec ise statements. The following subsections describe the type of fuzz y reasoning employed in the proposed expert system. First, a description of fu zzy sets, fuzzy propositions, and fuzzy logical operators are presented. The understanding of these concepts is fundamental to comprehend the description of the fuzzy logic syste m. 3.3.2.1 Fuzzy Sets Fuzzy set theory allows parameters to be represente d with simple linguistic terms. The functions used to develop fuzzy sets are called membership functions, and their job is to map elements from any universal set into real numbers within the range 0-1. The resulting values represent the degrees of membershi p of elements to particular fuzzy sets, where values closer to 1 represent higher degrees o f membership. Figure 3.3 shows an example of a triangular fuzzy set to denote LOW_CAP ABILITY of employees in a particular skill as a function of years of experien ce. Here, a resource with one year of experience fully belongs to the fuzzy set; therefor e the degree of membership is 1.0. Employees with one and a half years of experience h ave a 0.5 degree of membership to

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49 the fuzzy set, and any employee with more than two years of experience does not belong to the fuzzy set at all. Fuzzy set theory provides various forms of membersh ip functions. The capability to determine appropriate membership functions in th e context of each particular application is crucial for making fuzzy set theory practically useful [85]. Triangular, trapezoidal, and linear shapes of membership functi ons are most commonly used to represent fuzzy numbers. Triangular membership fun ctions are usually preferred due to their combination of solid theoretical basis and si mplicity [86]. However, there are situations that require more complex functions to m ore accurately represent the degrees of membership of elements to fuzzy sets. There are several methods for constructing membersh ip functions. Klir and Yuan [85] discussed direct and indirect methods that inv olve single or multiple experts. These Figure 3.3 Example of Triangular Fuzzy Set 0.0 0.2 0.4 0.6 0.8 1.0 0.00.51.01.52.02.53.0 Years of ExperienceDegree of Memberhsip LOW_CAPABILITY

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50 methods involve gathering and processing responses from experts in particular fields or from extensive literature reviews. 3.3.2.2 Fuzzy Propositions A fuzzy proposition is a statement that has a truth value associated with it. For example, the statement “element x belongs to set A ” has a truth value in the range of [0,1]. A truth value of zero means that x does not belong to set A Similarly, a truth value of one means that x completely belongs to set A. Truth values between zero and one, also known as partial truth, imply that x belongs to set A to some degree. The partial truth of a fuzzy proposition is represented by a de gree of truth similar to degrees of membership of elements to fuzzy sets. A common type of proposition used in fuzzy logic is the conditional and unqualified proposition. The objective of this pro position is to denote a relationship between elements from either similar or different s ets. This type of proposition is expressed with an “if-then” statement such as “if x belongs to set A then y belongs to set B ”. The first part of the proposition (i.e., the “i f” part), is called the antecedent; the second part is called the consequence. Uncondition al and unqualified propositions are used for imprecise reasoning to describe the decisi on process that human beings undergo to express cause and effect relationships. A proposition with an antecedent composed of only o ne statement is called a singleton. When the antecedent contains more than one statement (i.e., non-singleton proposition), fuzzy logical operators are used to r esolve the antecedent into a single truth

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51 value. An example of a non-singleton proposition i s “if x belongs to set A AND x belongs to set B then x belongs to set C ”. 3.3.2.3 Fuzzy Logical Operators Similar to classical set theory, there are three lo gical operators that are commonly used with fuzzy sets. These are the intersection, union, and complement, which correspond to AND, OR, and NOT operators, respectiv ely. For fuzzy sets A and B the intersection corresponds to all the elements that a re included simultaneously in both sets, and is represented as A B Equations (3.1) and (3.2) show the commonly used formulas for calculating the intersection between two fuzzy sets. The union of both sets, represented as A B corresponds to elements that are in either set. Equations (3.3) and (3.4) show the commonly used formulas to determine the union between two sets. The complement of a set, denoted as A for set A corresponds to all elements that are not in the set. Equation (3.5) shows the formula for calc ulating the complement of a set. )] ( ) ( min[ ) ( x x xB A B A m m m = (3.1 ) ) ( ) ( ) ( x x xB A B A m m m = (3.2) )] ( ) ( max[ ) ( x x xB A B A m m m = (3.3 ) ) ( ) ( ) ( ) ( ) ( x x x x xB A B A B A m m m m m + = (3.4) ) ( 1 ) ( x xA A m m = (3.5)

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52 3.3.2.4 Fuzzy Reasoning Fuzzy reasoning is the process of developing logica l inferences from imprecise premises. One way to develop fuzzy inferences is v ia the compositional rule of inference, which was introduced by Zadeh in 1975 [8 7]. This inference rule has been the foundation for various fuzzy reasoning methods pres ented in the literature [88]. One of such methods, namely the Mamdani Max-Min approach [ 89], is the selected inference method in this research. The following subsections provide a description of the compositional rule of inference and the Mamdani Max -Min approach. 3.3.2.4.1 Generalized Modus Ponens and the Compositional Rule of Inference A widely used inference rule in classical logic is the modus ponens, also known as forward chaining. It states that a conclusion c an be inferred given a conditional proposition and a fact. For example, a modus ponen s type of inference using the relationship between the levels of expertise of an employee in two skills can be expressed as shown in Table 3.1. Table 3.1 Classical Modus Ponens Form Type of Statement Statement Proposition Knowledge_Skill_1 = x Proposition Knowledge_Skill_1 Knowledge_Skill_2 Conclusion Knowledge_Skill_2 = x This simply says that if an employee has expertise x in Skill_1, and knowledge in Skill_1 implies expertise in Skill_2, then it can be inferr ed that the employee has expertise x in Skill_2. Notice that this type of inference struct ure deals with binary-valued

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53 propositions. That is, the solution set to describ e the expertise of an employee in a skill is {0,1} when using the classical modus ponens. To be used for fuzzy reasoning purposes, the classi cal modus ponens is customized through a process called the generalized modus ponens. Generalization of the classical modus ponens is achieved in three way s. First, the generalized version considers degrees of membership of elements to fuzz y sets. From the previous example, this means that the solution set to describe the ex pertise of an employee in a skill is expanded from {0,1} to [0,1]. Second, propositions showing completely true implications via the ‘’ symbol are replaced with fuzzy rules. Recall tha t a fuzzy rule is basically a conditional and unqualified proposition that implies a fuzzy relationship between an antecedent and a consequence. This rela tionship, also known as a fuzzy implication, is not explicit but rather embedded wi thin the proposition and determined for all values of antecedents and consequences [90]. T he literature presents various approaches to determine fuzzy implications (see [85 ]). The third way to generalize the classical modus pon ens is by using the compositional rule of inference shown in equation ( 3.6) for reasoning. Assuming that R is a fuzzy relation on X x Y and A and B are fuzzy sets on X and Y respectively, equation (3.6) can obtain degree of membership ) ( yB m for all Y y given a fuzzy implication R and a degree of membership ) ( xA m [85]. )] ( ) ( min[ sup ) (X xy,x R, x yA B m m = (3.6)

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54 This means that using the compositional rule of inf erence, a fuzzy conclusion can be obtained given a fuzzy rule and a fuzzy fact. This generalized modus ponens form of inference is shown in Table 3.2. Table 3.2 Generalized Modus Ponens Form Type of Statement Statement Fuzzy Rule If x is A Then y is B Fact ) ( xA m Fuzzy Conclusion ) ( yB m 3.3.2.5 Mamdani Max-Min Inference Approach The inference approach used in this research is the Mamdani Max-Min method, which employs the generalized modus ponens process for each fuzzy rule in the system. This approach follows the multiconditional reasonin g structure shown in Table 3.3. Table 3.3 Multiconditional Reasoning Structure Type of Statement Statement Rule 1 If x is A1, then y is B1 Rule 2 If x is A2, then y is B2 Rule 3 If x is A3, then y is B3 …. …. Rule n If x is An, then y is Bn Fact ) ( xA m Conclusion ) ( yB m

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55 The Mamdani method specifies that the fuzzy implica tion R for each rule, which is required by the compositional rule of inference, equals the truth value of the antecedent. More specifically, the fuzzy relation R for singleton fuzzy rules (i.e., antecedents composed of only one statement) equals the degree of membership of the only statement in the antecedent (see Figure 3.4a). For non-singleton fuzzy rules (see Figure 3.4b), the relation R is computed as the intersection of the statements in the antecedent via the minimum logical operation using equation (3.1). Figure 3.4 Mamdani Max-Min Inference An antecedent with a truth value greater than zero automatically implies that its consequence also has a truth value greater than zer o. In fuzzy reasoning terms, a true 0.0 0.2 0.4 0.6 0.8 1.0 Crisp ValueDegree of Memberhsip 0.0 0.2 0.4 0.6 0.8 1.0 Crisp ValueDegree of Memberhsip (B) (C) (A) 0.0 0.2 0.4 0.6 0.8 1.0 Crisp ValueDegree of Memberhsip 0.0 0.2 0.4 0.6 0.8 1.0 Crisp ValueDegree of Memberhsip 0.0 0.2 0.4 0.6 0.8 1.0 Crisp ValueDegree of Memberhsip 0.0 0.2 0.4 0.6 0.8 1.0 Crisp ValueDegree of Memberhsip Centroid

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56 antecedent causes a rule to fire. The fired rules are then combined into a new fuzzy set which will be used to make final inferences (see Fi gure 3.4c). 3.3.2.6 Defuzzification Defuzzification is the process of converting a set of fuzzy conclusions into a single crisp value. Several methods are available for defuzzification. One of such methods is the center of gravity approach, which ca lculates the area of a combination of fuzzy sets using integrals. A more commonly used m ethod which is reliable, less complicated, and less time consuming is the weighte d average method shown in equation (3.7) to approximate the center of gravity [91]. F igure 3.4c shows an example of the estimated center of gravity of a fuzzy set composed of two fired fuzzy rules. = ==r j j r j j js y1 1*m m (3.7) In equation (3.7), j m is the degree of membership of the fuzzy set resul ting from fuzzy rule r and sj is the center of gravity of the fuzzy set resultin g from fuzzy rule r 3.3.3 Expert System Data Flow This section describes the stepwise flow of data wi thin the expert system architecture as shown in Figure 3.2. Following is a concise description of each step. Implementation details are later described through an example in Section 3.3.

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57 3.3.3.1 Pre-conditions The solution approach requires three pre-conditions to be satisfied. First, decision-makers must agree on a crisp rating scale to evaluate employees’ capabilities. Second, linguistic terms (e.g., High, Low) must be established to denote the levels of expertise of employees in skills. Third, fuzzy set s must be created for each linguistic term to determine the degrees of membership of cris p evaluation ratings in each fuzzy set. 3.3.3.2 Step 1: User Inputs In the first step, a subcomponent in the presentati on layer (e.g., GUI) gathers user information to define three critical problem parame ters. The first parameter involves the selection of skills that are of interest to decisio n makers. The second parameter involves a decision regarding the personnel to be evaluated (i.e., either all available resources or a selected group). The third parameter is the select ion of the membership functions (e.g., triangular, trapezoidal, sigmoidal) to be used in t he fuzzy logic system to fuzzify employees’ expertise ratings. 3.3.3.3 Step 2: Fuzzification In the second step, the presentation layer subcompo nent forwards user data to the fuzzy logic system to begin the capability assessme nt process. Then, the logic system interacts with the Employee_Rep subcomponent to col lect the crisp personnel capability evaluation ratings representing the expertise of em ployees in various skills. Subsequently, the logic system interacts with the K nowledge_Rep subcomponent to

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58 convert the crisp evaluation ratings into fuzzy one s based on the types of membership functions selected by the user through the presenta tion layer. 3.3.3.4 Step 3: Inference Engine and Fuzzy Rules Based on a set of pre-determined fuzzy rules and ac tual expertise ratings, the system evaluates the complete capability set of a r esource to make inferences about his/her fuzzy expertise in a skill that is required for a task. 3.3.3.5 Step 4: Defuzzification The system employs the weighted average defuzzifica tion method to convert the capability of the resource in the required skill fr om a fuzzy value to a crisp one. 3.3.3.6 Step 5: Display Results The fuzzy logic system forwards its data inference conclusions to the presentation layer. Finally, the presentation layer displays th e results to the user. 3.4 Example Software Development Setting A capability assessment problem in a software devel opment setting was formulated to illustrate the implementation of the solution approach. This particular setting is relevant given that personnel assignment s are considered one of the most critical decisions that affect the performance and quality of software projects [6]. This is confirmed by Tsai et al. [43] with the following qu ote: “evidence reveals that the failure of software development projects is often a result of inadequate human resource project planning”. Considering that effective capability a ssessments are critical for efficient

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59 personnel assignments, efforts to improve capabilit y evaluations are necessary to significantly upgrade the outcome of personnel assi gnments decisions. Quality, as evidenced in the U.S. General Accountin g Office Report in [2], continues to be a major struggle to software compan ies. The report states that in 2004 the U.S. Department of Defense spent nearly 8 billion d ollars to rework software because of quality-related issues. Even more important than h uge monetary costs is the fact that software failures, especially in safety-critical sy stems, may result in life-threatening situations. Another reason that makes this example relevant is that it directly addresses areas of future research from the current software develo pment literature. Recently, Otero et al. [5] presented an approach for resource allocati on in software projects. Their methodology used precise parameters to determine ca pabilities of resources. The authors acknowledged the limitations of using precise param eters and encouraged researchers to develop methodologies that incorporate fuzzy parame ters instead. 3.4.1 Problem Statement and Pre-conditions The problem formulated to implement the solution ap proach involves evaluating the capabilities of various software engineers in t he C++ programming language. For this example, two experts from leading software eng ineering companies agreed to act as decision-makers for developing the required fuzzy r ules. Using real industry experts adds value to this example and helps to properly ex ecute the solution approach. Both decision makers have an average of 16 years of expe rience working for top U.S.A.

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60 organizations that specialize in the development of software applications for the defense industry. Following the solution approach described in the pr evious section, decision makers must ensure that pre-conditions are satisfie d. The first pre-condition is to establish a crisp rating scale to evaluate skill le vels. The decision-makers agreed on a rating scale from 0 to 5, where higher ratings repr esent higher evaluations. This rating scale is commonly used for yearly evaluations of th e performance of engineers. The second and third pre-conditions involve establi shing fuzzy sets to associate crisp evaluation ratings with degrees of membership The selected linear and triangular fuzzy sets, shown in Figure 3.5, correspond to the following levels of expertise: None, Novice, Proficient, Highly Proficient, and Expert. Figure 3.5 Fuzzy Sets of Skill Levels 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00.51.01.52.02.53.03.54.04.55.0 Crisp Rating in Skill jDegree of Memberhsip None Novice Highly Proficient Expert Proficient

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61 The membership functions for each fuzzy set are sho wn in Figure 3.6. Figure 3.6 Membership Functions for Fuzzy Sets of Skill Levels 3.4.2 User Inputs The definition of the problem parameters are as fol lows. First, the skill that is of interest to decision makers is the level of experti se of personnel in the C++ programming language. Second, seven software engineers are sel ected as the personnel to be evaluated. Third, the membership functions to be u sed to fuzzify crisp evaluation ratings are those shown the previous section. £ £ > = 1 x 0 for x 1 1 x for 0 )x (Nonem £ < £ £ > < = 5.2 x 5.1 for x 5.2 5.1 x 5.0 for 5.0 x 5.2 x and 5.0 x for 0 )x (Novicem £ < £ £ > < = 5.3 x 5.2 for x 5.3 5.2 x 5.1 for 5.1 x 5.3 x and 5.1 x for 0 )x(roficient Pm £ < £ £ > < = 5.4 x 5.3 for x 5.4 5.3 x 5.2 for 5.2 x 5.4 x and 5.2 x for 0 )x (oficient Pr Highlym < = 5.3 x for 3 ) 5.3 x (2 5.3 x for 0 )x (Expertm

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62 3.4.3 Fuzzification The crisp evaluation ratings for the software engineers in the C++ and Java programming languages are shown in Table 3.4. Evaluation ratings in the Java skill were included because they can potentially improve the skill ratings of en gineers in the C++ language. The decision makers explained that in practice, a cri sp evaluation rating of an engineer in a particular skill is heavily based on the number of y ears of industry experience with the skill. Therefore, it is common in industry to e ncounter situations were an engineer would have significantly different ratings i n two similar skills (e.g., Java and C++). Table 3.4 Crisp Evaluation Ratings in Various Pro gramming Languages Using the membership functions from the previous section, the fuzzifi ed evaluation ratings obtained for each engineer are shown in Table 3.5. Table 3.5 Fuzzy Evaluation Ratings C++JavaC++JavaC++JavaC++JavaC++JavaC++JavaC++JavaNo----------------------------Novice0.5----------0.5--0.5--1.0------Proficient--0.51.0--1.0------0.5--------1.0Highly Proficient--0.5--1.0----------1.0--0.51.0--Expert----------0.67--1.0------0.33----Engineer_7Degrees of MembershipEngineer_4Engineer_5Engineer_6 Engineer_1Engineer_2Engineer_3Fuzzy Set C++ Java Engineer_11.03.0Engineer_22.53.5Engineer_32.54.5Engineer_40.55.0Engineer_52.03.5Engineer_61.54.0Engineer_73.52.5 Resources Crisp Evaluation Ratings

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63 3.4.4 Inference Engine Table 3.6 shows the set of fuzzy rules that was developed by the dec ision makers. Using rule number 5 as an example, the table reads as follows: I f the initial C++ rating is Novice, and the Java rating is Highly Proficient, then the Modified C ++ rating is Proficient. Table 3.6 Fuzzy Rules for C++ These rules were developed for cases were particular levels of knowledge in the Java language result in improved skill ratings in the C++ language Therefore, in cases were none of the rules apply, the initial skill rating in C++ is used. For exam ple, consider the case where a software engineer possesses a 2.5 crisp rat ing in C++ and no experience in Java. This means that the fuzzy rating in C++ is Proficie nt and in Java is None, which causes none of the rules from Table 3.6 to fire. In this case, the i nitial crisp rating in C++ cannot be improved based on the actual Java knowledge of the engineer. T herefore, the capability assessment of the engineer in C++ remains at the initial cr isp rating of 2.5. As an example, Figure 3.7 shows the fuzzy inference process for Engineer_6. Based on the initial crisp evaluation ratings of this engineer, only Rules #5 and #6 are fired. For C++ Java 1NoneProficientNovice2NoneHighly ProficientProficient3NoneExpertProficient4NoviceProficientNovice5NoviceHighly ProficientProficient6NoviceExpertHighly Proficient7ProficientHighly ProficientHighly Proficient8ProficientExpertHighly Proficient9Highly ProficientExpertExpert Skills Modified C++ Rating Rule No.

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64 each of these two rules, equation (3.1) is used to resolve the AND log ical operator of the antecedent into a single degree of membership m antecedent(x). This value also represents the degree of truth of the antecedent. Recall that in the Mamdani process, the truth value of the antecedent equals the fuzzy relation R that is embedded within the rule. Hence, for Rule #5 the fuzzy relationship R between the Novice C++ and Highly Proficient Java fuzzy sets is calculated as R = min ( m + +C Novice= 1.0, m Java oficient Pr Highly= 0.5) = 0.5 = ) ( xantecedent m For Rule #6, the fuzzy relationship R between the Novice C++ and Expert Java fuzzy sets is calculated as R = min ( m + +C Novice= 1.0, m Java Expert= 0.33) = 0.33 = ) ( xantecedent m Subsequently, the compositional rule of inference is invoked using equation (3.6) to develop a modified fuzzy set for each rule. Theref ore, the fuzzy inference for Rule #5 is m + +ModifiedC(x) = X xsup min[ m antecedent(x), R(Novice_C++, Highly_Proficient_Java)] = X xsup min[0.5, 0.5] = 0.5. Since the Modified C++ rating for Rule #5 corresponds to a Proficient fuzzy set, the inferred conclusion ba sed on this rule is that m + +C roficient P = 0.5. Similarly for Rule #6, the inferred conclusion is that m + +C oficient Pr Highly = 0.33.

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65 Figure 3.7 Capacity Assessment for Engineer_6 0.0 0.2 0.4 0.6 0.8 1.0 0.01.02.03.04.05.0Crisp Rating C++Degree of Memberhsip Novice 0.0 0.2 0.4 0.6 0.8 1.0 0.01.02.03.04.05.0Crisp Rating in JavaDegree of Memberhsip Highly Proficient RULE #5 0.0 0.2 0.4 0.6 0.8 1.0 0.01.02.03.04.05.0Modified Crisp Rating in C++Degree of Memberhsip Proficient 0.0 0.2 0.4 0.6 0.8 1.0 0.01.02.03.04.05.0Crisp Rating C++Degree of Memberhsip No 0.0 0.2 0.4 0.6 0.8 1.0 0.01.02.03.04.05.0Crisp Rating in JavaDegree of Memberhsip Expert RULE #6 0.0 0.2 0.4 0.6 0.8 1.0 0.01.02.03.04.05.0Modified Crisp Rating in C++Degree of Memberhsip Highly Proficient

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66 Figure 3.8 shows the combination of the inferred fuzzy sets into a single set to begin the defuzzification process via the weighted average center of gravity. Using equation (3.7), the defuzzified rating is computed as 0.33 0.5 0.33(3.5) 0.5(2.5) + + = 2.9. This means that the evaluation crisp rating in C++ of En gineer_6 is improved from 1.5 to almost 3.0 due to the engineer’s level of expertise in Java. Figure 3.8 Defuzzified Rating in C++ (Modified) Table 3.7 shows the modified C++ ratings for each o f the engineers. Notice that the initial and modified ratings for Engineer_7 are equal since none of the fuzzy rules were fired based on the engineer’s initial C++ and Java ratings. 0.0 0.2 0.4 0.6 0.8 1.0 0.01.02.03.04.05.0Modified Crisp Rating in C++Degree of Memberhsip Highly Proficient Proficient centroid

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67 Table 3.7 Initial and Modified C++ Ratings 3.5 Summary and Contributions This research presents a four-layered fuzzy expert system architecture for evaluating personnel capabilities. Although a desc ription of each of the layers is presented, the main emphasis of this research is on the development of the fuzzy logic system layer. A personnel capability assessment pr oblem in a software development setting was formulated to demonstrate the implement ation of the solution approach. There are two major contributions that this researc h study makes to the personnel capability assessment body of knowledge. The first significant contribution is the introduction of a high-level layered architecture w here each layer is adaptable to contextspecific subcomponents. That is, each layer can be customized with different subcomponents without major changes to the architectur e. This is accomplished through a global layer that is used as the only channel of in teraction between any two layers. Therefore, implementation details of any layer are hidden from the others. This way, a layer is not susceptible to changes due to modifica tions in other layers. This provides decision makers the flexibility to add/delete/modif y subcomponents in any layer based on Engineer_11.02.0Engineer_22.53.5Engineer_32.53.5Engineer_40.52.5Engineer_52.03.0Engineer_61.53.0Engineer_73.53.5 Initial C++ Rating Modified C++ Rating Resources

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68 their particular needs without having to incur in e xpensive architectural system modifications. The second significant contribution from this work is the approach taken to resolve the following three main areas of the perso nnel capability assessment problem: modeling personnel levels of expertise, establishin g relationships between skills, and making inferences about the capabilities of personn el. These critical areas are considered to be naturally imprecise; therefore, they are esta blished using fuzzy concepts. Personnel levels of expertise are modeled with fuzzy sets ins tead of using the common classical set theory. Relationships between skills are described with fuzzy rules, and capability assessments are performed via approximate reasoning based on the compositional rule of inference. This realistic representation of imprec ise parameters and activities with fuzzy concepts has the potential to provide a high practi cal value to the expert system proposed in this research. 3.5.1 Research Extensions A major challenge for any researcher is to develop new methodologies that become widely accepted by practitioners. To achiev e this, it is important for researchers to properly market their solution approaches by bri nging these novel methodologies into industry scenarios to show field experts the capabi lities of such new approaches. With this in mind, the approach developed in this resear ch needs to be applied to different industry settings to validate its applicability and acceptability. For this, it is necessary to complete the design phase of the expert system and move into the coding phase. Since this research provides the high-level software desi gn architecture, the next step would be

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69 to divide the architecture into components and deve lop detailed designs for each component using object-oriented tools such as class diagrams. The final product must include proper software engineering documentation, such as: software requirements specification, software design document, software m anual, and test description document. Another potential research extension is to conduct a survey analysis to investigate if it is reasonable to develop baselines of members hip functions for general/common skills in particular environments. For example, it may be possible to interview experts from different software development organizations t o come up with fuzzy sets for technical capability assessments that can be used a s standards across companies. A similar survey analysis can be conducted to examine the possibility of establishing fuzzy rules’ baselines to describe the relationship betwe en various skills.

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70 CHAPTER 4 A FUZZY GOAL PROGRAMMING MODEL FOR SKILL-BASED RESO URCE ASSIGNMENT PROBLEMS 4.1 Abstract This research presents a fuzzy goal programming (FG P) model for personnel assignments in skill-based environments. The prior itized goals for each resource assignment are to meet desired target values for te chnical expertise, team parameters, and personnel preferences. These target values are rep resented with fuzzy sets which are developed with the help of decision makers. A pers onnel assignment problem in a software development industrial setting is formulat ed to demonstrate the proper implementation of the solution approach. Two softw are engineering field experts acted as decision-makers and participated in the developm ent of the fuzzy sets for the goals. The contribution of this research to the literature is two-fold. First, it develops a new FGP model for personnel assignments that consid ers imprecise parameters such as personnel capabilities and tasks’ requirements. Se cond, it presents an innovative methodology that is capable of representing relativ e priorities of skills and tasks. This methodology, denoted as membership function relaxat ion, is incorporated into the FGP specification. To the best of our knowledge, this study presents the first multi-objective optimization model that simultaneously considers th e following fuzzy parameters:

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71 competence levels of resources in various skills, m otivation levels of resources with tasks, priorities of tasks and skills, and required levels of skills. 4.2 Introduction Effective personnel assignment approaches in skillbased environments are essential to achieve high-quality products in a tim ely manner and within budget constraints. Skill-based environments are characte rized by the need to assess the ability of candidates to successfully complete specific tasks. Examples of such environments are: software engineering, research and development (R&D), and healthcare organizations. The review of current literature highlights researc h opportunities to improve the effectiveness of personnel assignment decisions. O ne of these opportunities which represents a significant contribution to the litera ture involves the development of enhanced assignment models that consider critical p arameters which are typical of skillbased resource assignment situations. Table 4.1 sh ows some of these parameters and provides possible definitions for these factors in various industrial settings. A major challenge is to effectively model these essential p arameters, given their highly imprecise nature. Moreover, complexity in the decision-proce ss increases when there are several levels of these parameters. Due to the lack of ade quate methodologies to undertake these complexities, decision-makers would typically appro ach the problem as a non-skill-based assignment. That is, human resources are considere d as uniform entities. This results in ignoring important characteristics such as specific capability levels and motivation factors [38].

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72 Table 4.1 Characteristics of the General Skill-Ba sed Personnel Assignment Problem Another common and challenging situation in skill-b ased environments is that candidates with the exact required skills to work o n a task are seldom available [5]. This CharacteristicSoftware EngineeringHealth CareR&D Pr ojects Imprecise competence level The level of knowledge of a software developer in C++ is described as average The level of fluency of a registered nurse (RN) in the Italian language is described as poor The level of knowledge of a researcher in Data Envelopment Analysis is excellent Skill Preference: A software developer prefers a task that involves developing code in C++. Skill Preference: An RN enjoys and has experience working with elderly patients. Skill Preference: A researcher prefers working with projects that involve the use of non-parametric analyses. Workplace Preference: A software developer prefers a task that does not require overtime. Workplace Preference: An RN prefers to assist Dr. Jones, instead of assisting Dr. Smith. Workplace Preference: A researcher prefers working with projects related to advancing the quality of education of young students. Imprecise priorities of tasks A safety critical task (i.e. involves human safety) is more important than any other task. Assisting a patient that is recovering from a heart attack is much more important than attending another patient with minor cuts. Research studies that are expected to have major impacts to society are more important than studies with lower expected impacts to society. Imprecise priorities of skills required Programming language (PL) experience is more important than domain experience for task X, but domain experience is more important than PL experience for task Y To assist patient X, an RN's fluency in foreign languages is more important than the RN's knowledge on cancer treatments. For research study X, knowledge of Markov processes is much more important than knowledge in a particular statistical software package. Imprecise level of skill required The development of a particular Windows application for Project X requires an expert level of skill in Visual Basic programming. To attend patient X, the required level of fluency in the Italian language is expressed as very fluent Research study X requires a researcher with a high level of knowledge in Markov processes. Imprecise task complexity and duration The time that will take to complete the development of a software application is described as long The time that will take to diagnose and treat a patient's condition is described as short (depending on the stage). The time that will take to complete research project X cannot be accurately estimated. Fixed limited resources A software manager must assign readily available software engineers to software tasks. A hospital manager must assign readily available registered nurses to patients. The manager of a R&D division must assign available researchers to a set of research studies. Limited or no training time A project that is running late needs a software developer to design and develop a Windows application. An RN attending a patient with several cuts does not have time to learn how to sanitize cuts. A proposal for a funded research study related to stochastic processes did not bid for a researcher to be trained in Markov processes. Task Environment Types of parameters Examples Resources Motivation to work in particular tasks

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73 is mainly due to the continuous and rapid introduct ion of new technologies to improve the development of products. This limitation often results in inefficient allocation of resources that increase costs and the probability o f developing unreliable products [5], [62]. Challenges such as the ones mentioned above drive r esearchers to advocate for improved personnel assignment models. For example, Acua et al. [6] mentioned the need to incorporate a diverse set of factors relate d to employees such as personal preferences, psychological tests, technical knowled ge and skills, career goals, promotion records, and job leveling. Baykasoglu et al. [37] also discussed future research needs in the area of team formation and assignment of tasks based on individual skills. The authors stated that “there is a need to develop ana lytic models and software systems that can incorporate important factors and multiple obje ctives”. Furthermore, there are other studies such as [43] where the authors acknowledged critical limitations in their model, including the absence of quality and performance pa rameters. In the study by Faraj and Sproull [64], the authors concluded that “while expertise is a necessary input, its mere presence on the team is not sufficient to affect performance effectiveness if team members cannot coordinate the ir expertise”. In other words, successful expertise coordination requires that eac h team member knows the expertise areas of each other in order to seek help when need ed. The point that can be made here is that it seems far more efficient to correctly match individual skills of team members with the skills required by tasks in order to minimize t he number of times that team members encounter difficulties completing their tasks.

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74 Patanakul et al. [34] stated that “the methodologie s proposed in the literature for assigning projects are based solely on project requ irements and skills of project managers”. This statement could be generalized for current personnel assignment approaches where many important parameters are omit ted, thus limiting the applicability of most assignment methods in diverse industrial se ttings. Enhanced assignment models may represent benefits s uch as increased employee and customer satisfaction, as well as higher profit s for companies. Moreover, efficient employee assignments can significantly improve the reliability of products, resulting in a positive impact to important social aspects such as public safety (e.g., software for airplanes). Therefore, it is imperative to follow the “continuous improvement” paradigm and pursue further research to improve the outcome of personnel assignment decisions. The principal research question that guided this st udy is the following: How can a novel approach for the assignment of resources to t asks in skill-based environments be developed? An extensive review of the literature h as been conducted to address this important research inquiry. As a result, this rese arch develops a personnel assignment fuzzy goal programming (FGP) model for skill-based environments. The model considers employees with various skills and prefere nces, as well as tasks with imprecise requirements. This research study is organized as follows. Secti on 4.2 presents a summary of relevant literature. Section 4.3 discusses the jus tification for using FGP as a solution approach. Section 4.4 provides the solution approa ch and model development. Section 4.5 demonstrates the capability of the model with a n example of a personnel assignment

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75 scenario in a software development setting. Finall y, Section 4.6 provides conclusions and future research directions. 4.3 Related Literature The purpose of the literature review effort for thi s research was twofold. The first objective was to identify the methods used for pers onnel assignments in skill-based environments. The second objective was to identify the parameters that were considered for assignments and how were these parameters model ed (e.g., index values or fuzzy variables). The following sections describe the fi ndings corresponding to both objectives. 4.3.1 Approaches for Personnel Assignments The literature shows various methodologies for assi gning employees to tasks. These approaches include the use of tools such as m athematical programming models and artificial intelligence techniques. Other approach es such as Taguchi’s parameter design and subjective measures have also been used by rese archers. The following subsections discuss these approaches in more detail. 4.3.1.1 Mathematical Programming Approaches The approaches based on mathematical programming te chniques include integer and goal programming (GP). Patanakul et al. [34] d eveloped an integer programming model to optimize the assignments of projects to pr oject managers. The objective function considered the suitability between project s and managers, and the strategic importance of projects to an organization. Boon an d Sierksma [42] presented a linear

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76 programming model to create teams based on the aggr egated value that each team member adds to the team. Subjective precise weight s were used to represent the value of a team member to a specific position. Karsak [47] introduced a multi-objective linear program to minimize cost and maximize the number of required skills that are fulfilled for a single task. Bassett [46] presented a mixed integer linear progr amming approach for personnel assignments. First, an initial list of a vailable resources and their suitability with tasks is constructed based on subjective opini ons. Then, assignments are made as a function of the candidates’ available time and the estimated effort required to complete the tasks. Therefore, this approach relies heavily on estimated durations of tasks and will cause problems to managers if tasks take more time to complete than their expected completion time. Majozi and Zhu [39] also used mix ed integer linear programming as a solution approach. Very recently, Peters and Zelewski [4] developed a GP model for personnel assignments in a software development setting. The model considers goals that include meeting technical requirements and preferences of e mployees regarding general workplace conditions. Team parameters such as team cohesiveness and communication skills are not considered. The objective function is to minimize the deficiencies of resources with the goals required by tasks. The an alytical hierarchy process (AHP) method is used to assign weights to goal deficienci es to determine their relative importance to the decision maker. This approach is based on the assumption that the experience levels of resources are defined by crisp values. For example, consider the situation depicted in Figure 4.1 in which a decisio n maker has to determine the

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77 compatibility between two resources and a task. Th e decision maker gathers the required information, goes through a decision process, and f inally comes up with a solution. In this case, the task requires four years of experien ce in a particular skill. Although both resources have four years of experience in this req uired skill, the actual experience of each resource with the skill in previous tasks will most likely be different. This will make the experience level of one resource more at p ar with the task than the experience level of the other resource, even if both resources have equal years of experience. Particular characteristics of this problem, like th e one just mentioned, create an important opportunity for significant research in skill-based resource allocation environments by incorporating fuzzy set theory to determine degrees of membership of resources to tasks. In fact, Peters and Zelewski [4] emphasized the nee d to develop FGP models for the skillbased assignment problem. Figure 4.1 Sample Scenario Which resource should be assigned to the task? Decision Maker Required expertise in skill_1 = 4 years Task Expertise in skill_1 = 4 years Resource_2 Expertise in skill_1 = 4 years Resource_1 Decision process … Resource assignment solution Which resource should be assigned to the task? Decision Maker Which resource should be assigned to the task? Decision Maker Required expertise in skill_1 = 4 years Task Required expertise in skill_1 = 4 years Task Required expertise in skill_1 = 4 years Task Expertise in skill_1 = 4 years Resource_2 Expertise in skill_1 = 4 years Resource_1 Expertise in skill_1 = 4 years Resource_2 Expertise in skill_1 = 4 years Resource_2 Expertise in skill_1 = 4 years Resource_1 Expertise in skill_1 = 4 years Resource_1 Decision process … Resource assignment solution

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78 In [92], the authors developed a nurse-scheduling G P model. The authors mentioned that nurses posses various levels of capa bilities due to their training, allowing them to work as registered nurses, practical nurses or aids. The authors proposed the creation of subgroups of nurses in order to assign nurses to shifts. No distinction is made between the capabilities of nurses within subgroups This means that nurses within subgroups are assumed to be equally capable so that performance is not affected by the selection of nurses. The authors included preferen ces of nurses as an assignment criterion. These preferences were not modeled base d on the preferences of available nurses. Instead, they were modeled based on survey results and therefore they represented the preferences of the majority and not of the individual nurses that correspond to a particular assignment problem. Another GP assignment model was developed by [93]. Here, the objective was to assign multiple projects to managers. The model us es estimated times for resources to complete projects as a proxy for resource capabilit y. 4.3.1.2 Artificial Intelligence Approaches There are two main artificial intelligence approach es that are used for personnel assignments methodologies. The first one deals wit h fuzzy set theory to represent the imprecise nature of particular parameters. The sec ond one corresponds to global optimization methods. The following subsections sh ow studies that have implemented methodologies using these artificial intelligence c oncepts.

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79 4.3.1.2.1 Fuzzy Set Theory Approaches There are several methods in the literature that in volve fuzzy parameters. An example is the study by Drigas et al. [41], where f uzzy variables are used to determine the suitability of candidates with tasks. This stu dy only considered the skills of candidates as parameters for assignments. Motivati on and other important factors were not taken into account. Petrovic-Lazarevic [45] al so developed a personnel selection fuzzy model that considered only imprecise competen ce levels of resources. In [49], the authors developed a methodology for th e personnel assignment problem based on fuzzy set theory and fuzzy rules. The authors used fuzzy variables to describe competence levels of resources and priorit ies for assignment parameters. For example, one of the factors considered was communic ations, which had the following measure indicators to determine the level of compet ency of a resource in this skill: listening, oral communication, oral presentation, a nd written communication. The “listening” measure indicator was given the highest priority, meaning that it will be the most important factor considered when evaluating th e level of competence of a resource. This consideration of imprecise priorities of the r equired skills is one of the strengths of this study. However, this study considered only th e single-task-multiple-resources case, making it not suitable for multiple-tasks-multipleresources situations. The authors used fuzzy rules for the selection of the best resource for a task. Part of the results from the research conducted by Liang and Wang [94] was a methodology to adequately pair candidates with jobs The authors used fuzzy variables to describe the subjective importance of skills req uired for a job and the expertise of a candidate on each skill. Incorporating the extensi on principle for fuzzy sets [85],

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80 assessments made by a panel of decision makers were aggregated into a fuzzy suitability index between candidates and jobs. Liang and Wang [53] presented a very similar methodology with the main distinction being that th e authors incorporated objective criteria to their methodology. The methodology con siders priorities of individual skills but excludes required levels of skills. Methodolog ies that consider required levels of skills are more complete and therefore more valuabl e to decision makers in the field. In the study by Yaakob and Kawata [52], the authors developed a methodology for the personnel assignment problem similar to the one developed by Liang and Wang [53]. The distinction in this study is that the au thors incorporated an evaluation of the fuzzy relationships between team members to avoid c onflicts. This parameter was defined as an average fuzzy value of the relationsh ips of every pair of workers. Shen et al. [51] developed a multi-criteria decision model that used the pair comparison method described by Yaakob and Kawata [52] to denote a soc ial relationship factor between team members. This methodology considers the case where employees are responsible for multiple tasks at any given time. Furthermore, the methodology considers capabilities of candidates with respect to the skills required to p erform a task, and whether tasks are conflicting or complementary with the current workl oad of candidates. Fuzzy variables are used to evaluate a candidate’s suitability with each task. Kozanoglu and Ozok [50] provided an approach to sol ve the single-task skillbased personnel assignment problem. Their approach relates customer requirements to engineering solutions using the Quality Function De ployment technique. The authors defined customer requirements as the characteristic s, or subtasks, of a task that need to be completed, and engineering solutions as the require d skills to successfully complete

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81 subtasks. Fuzzy parameters described the importanc e of subtasks, the priorities of the skills required, and the capability of candidates. Although no particular assignment method was specified, the authors recommended the s election of the most appropriate candidates using ranking fuzzy methods. Although t he study presented a significant contribution to the literature, its value could be significantly enlarged by extending their approach to consider parameters such as preferences of candidates, required levels of skills, multiple tasks, and task priorities. In [44], the authors used fuzzy set theory to compu te an index representing the relation between required skills and actual skills of candidates. A particular aspect of their methodology is that it inflates the suitabili ty level of a resource with a task if the resource exceeds the required levels of skills. A different and arguably more appropriate approach would have been to maximize the number of times that required skill-levels are met. In addition, priorities for required skills s hould be considered. 4.3.1.2.2 Global Optimization Approaches Recent studies show the use of artificial intellige nce search and optimization methods, such as simulated annealing and genetic al gorithms, for personnel assignments. The goal of these methods is to find a reasonable a pproximation to the global optimum solution of a function in a large search space. In [37], the authors presented a multiobjective assignment approach based on simulated an nealing. The objectives were to maximize the minimum suitability of each candidate to a team and the minimum team sizes. In [38], the authors adopted genetic algori thms for their multi-objective assignment approach. The objectives were to meet c areer path satisfaction levels of

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82 resources, levels of skills required by projects, a nd resources’ motivation levels. Duggan et al. [95] also developed an optimization model fo r task allocation based on genetic algorithms. The competencies of employees were mod eled using a categorical variable with five levels. Each of these competency levels was associated with an expected productivity per day, as well as an expected number of defects per unit of productivity. 4.3.1.3 Other Approaches Methodologies for personnel assignment in skill-bas ed scenarios also include techniques such as cluster analyses, assessment of behavioral competences, subjective assessments, and AHP. Furthermore, the goal of som e team assignment methods is simply to create heterogeneous groups, since resear ch has shown that these groups are usually more creative, innovative, and cooperative [13]. Examples of such methodologies are provided by [13], [14], and [15]. The method proposed by Hauschildt et al. [20] uses cluster analysis to classify candidates into five categories based on pre-define d assignment criteria. Then, a discriminant analysis determines the types of tasks that are more suitable with each of the five categories. The assignment policy is to assig n the candidate that is most suitable with a task based on the results from the discrimin ant analysis. Other studies such as [6] and [16] developed procedures for allocating person nel to tasks based on the assessment of behavioral competencies. The AHP and Taguchi’s parameter design techniques a re also used in the literature for resource assignments. Al-Harbi [25] presented an assignment method that uses AHP for the prequalification contractor proble m. The method relies on assignment

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83 criteria such as experience, quality performance, a nd workload. In [43], the authors proposed a methodology for assigning employees to t asks based on a critical resource diagram and the Taguchi’s parameter design approach The performance measures of the assignments were cost and cycle time. The critical resource diagram focused on resource scheduling rather than activity scheduling to repre sent human-resource workflow and tasks’ precedence. The Taguchi’s parameter design was used to obtain a scheme that would optimize the selection of resources for tasks under dynamic and stochastic conditions such as task complexity. The authors in [48] developed a multiple objectives methodology for personnel assignment in an R&D environment. The objective fu nctions were to maximize the satisfaction of skills required by each project, ma ximize the skills available throughout the project’s duration based on a learning curve fa ctor for each candidate in each skill, and maximize the average preference of each pair of resources to work together. The skill levels of candidates and the preferences of p airs of candidates to work together were expressed using fuzzy variables. The methodology f irst approximates a Pareto-optimal frontier of solutions using the lexicographic goal programming, weighted sum, and constraint methods. This way, the number of soluti ons to be analyzed is reduced significantly. The methodology then uses the ELECT RE III multi-criteria decisionmaking procedure to select the best solution among the ones in the Pareto-optimal frontier.

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84 4.3.2 Modeled Parameters The second objective of the literature review was t o identify parameters that were taken into account for personnel assignments and ho w were these parameters modeled. Table 4.2 contains selected literature on personnel assignment methodologies and describes the parameters considered. Table 4.2 Selected Recent Literature on Skill-bas ed Resource Assignment Resources Tasks Research Study Competence level of resources Motivation with tasks Priorities of tasks Priorities of required skills Level of skill required [40], [42] Precise (Index) Not considered Not considered Not considered Not considered [45], [41] Imprecise (Fuzzy) Not considered Not considered Not considered Not considered [94], [53], [50] Imprecise (Fuzzy) Not considered Not considered Imprecise (Fuzzy) Not considered [16], [6], [96] Index Not considered Not considered Not considered Weight (High, Medium) [44], [39], [47] Imprecise (Fuzzy) Not considered Not considered Not considered Imprecise (Fuzzy) [43] Probabilistic Not considered Not considered Not considered Not considered [46] Precise Not considered Not considered Not considered Not considered [20] Precise Not considered Precise (Index) Not considered Not considered [49], [52] Imprecise (Fuzzy) Imprecise (Fuzzy) Not considered Imprecise (Fuzzy) Not considered [4], [38] Precise (Index) Precise (index) Not considered Not considered Precise (index) [93] Precise Not considered Precise (Index) Not considered Not considered [37] Imprecise (Fuzzy) Precise (index) Not considered Not considered Imprecise (Fuzzy) [51] Imprecise (Fuzzy) Imprecise (Fuzzy) Not considered Imprecise (Fuzzy) Not considered [48] Imprecise (Fuzzy) Imprecise (Fuzzy) Precise (I ndex) Precise (Index) Not considered [96] Precise Not considered Precise (Index) Precise (Index) Precise (index)

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85 Table 4.2 shows that most personnel assignment meth odologies exclude critical parameters related to resources and tasks. Undoubt edly, levels of competences of resources in required skills are key parameters for successful assignments. However, the literature shows that other factors such as motivat ion levels and priorities of tasks are also critical factors that must be considered in the dec ision process [4], [38]. This is evidenced by various studies in the literature. Fo r example, Matsuodani [97] stated that the outcome of complex tasks that depend on the com petences and other individual characteristics of people is strongly related to th e motivation of personnel to engage in specific tasks. In addition, Hendriks et al. [98] indicated that the dedication of a candidate to a particular task increases efficiency Furthermore, it is very important to decide how to properly model these parameters. The values of these parameters are mor e imprecise than random or crisp, which represent a good opportunity for the applicat ion of fuzzy set theory techniques [45]. 4.3.3 Summary of Findings In summary, the current literature shows that there are opportunities to improve the effectiveness of personnel assignment decisions The following list highlights the major gaps found in the literature: Critical parameters such as levels of motivation of employees with tasks, priorities of required skills, and priorities of ta sks are seldom included in personnel assignment approaches.

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86 Most approaches model parameters that are imprecise in nature (e.g., capability levels of employees) as crisp values. A FGP model for personnel assignments in skill-base d scenarios is non-existent in the current literature. 4.4 Justification for FGP as a Solution Method Before discussing the use of FGP as part of the sol ution approach, it is important to justify it as an appropriate modeling tool for p ersonnel assignments in skill-based environments. To this end, it is necessary to brie fly discuss and justify GP and fuzzy set theory separately, followed by the combination of t hese approaches into FGP. 4.4.1 Goal Programming Personnel assignment decisions in skill-based scena rios typically involve multiple objectives. These objectives are associated with e xpectations from decision-makers and employees. That is, for a set of tasks, decision m akers expect personnel assignments to meet the tasks’ required levels of technical skills At the same time, employees expect assignments to agree with their personal preference s such as working with particular skills or in small teams. Consequently, personnel assignment policies formulated with single objectives can produce results that fall ver y short from meeting expectations that are essential to decision makers and employees. Lo gically, the best-case scenario would be to make assignments that fulfill the complete se t of requirements from managers and workers. However, many times it is impossible to m ake such assignments, resulting in unfeasible solutions to accomplish these multiple o bjectives. An alternative approach to problems with various objectives is to find a solut ion that satisfies a set of constraints

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87 and, at the same time, is close to meeting each of the objectives. Such an approach is called GP. GP is a multi-objective optimization mathematical m odel based on linear programming techniques. GP minimizes unwanted devi ations from target values (i.e., goals) subject to a set of constraints [99]. A sta ndard GP formulation requires precise target values and priorities for each goal. The cl assic GP simple additive model is the following [100]: Minimize = +=m i i id d Z1) ( (4.1 ) i i i ig d d x AG = ++ -) ( (4.2) l BX u AX £ (4.3) Equation (4.1) shows that the objective function is to minimize the overall sum of deviations from targets. Equation (4.2) adds a +id or subtracts a -id amount to the value achieved in goal “ i ” () ( x AGi) in order to reach the target value ofig Incorporating deviations in equation (4.2) guarantees that the mo del finds a feasible solution. Equation (4.3) ensures that any upper and lower value constr aints are met. There are a vast amount of studies that have used GP for solving decision p roblems with multiple criteria [101]. GP models are either preemptive or non-preemptive. In preemptive GP, each goal is assigned a priority level, where higher priority levels are infinitely more important than any lower priority level. This means that a “series of mathematical programming problems are solved sequentially, first considering highest priority goals only, and then continuing with lower priority ones, under the cons traints imposed by the alternative optimal solutions of the problems that included the higher priority goals” [101]. In non-

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88 preemptive (a.k.a. “weighted”) GP, a weight is assi gned to each goal to quantify their relative importance. The objective is to minimize the weighted sum of the deviations. 4.4.2 Fuzzy Set Theory In classical set theory, the decision to determine if an individual meets the skill levels demanded by a task is a crisp one (i.e., yes or no). Considering the case depicted in Figure 4.1, a resource with two years of experie nce in the required skill would not meet the required skill level of four years. In ot her words, this resource does not belong to the set of resources that meet the skill level d emanded by the task. A different approach to the classical set theory is the fuzzy s et theory, which utilizes degrees of membership of elements to sets [85]. In the exampl e just mentioned, the individual with two years of experience possesses a degree of membe rship to the set of resources that meet the skill level demanded by the task. Further more, an individual with four years of experience in the specialized skill may still not c ompletely meet the demanded skill level of the task, depending on the prior experience and the environment in which the individual utilized the skill. Using the degrees o f membership concept provides a more realistic way to describe the fit of resources with tasks. 4.4.2.1 Membership Functions Fuzzy set theory allows parameters to be defined us ing simple linguistic terms (e.g., high, low). These factors are then translat ed into quantitative values using membership functions. More specifically, the job o f membership functions is to map elements from any universal set into real numbers w ithin the range 0-1. The resulting

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89 values represent the degrees of membership of eleme nts to particular sets. Values closer to 1 represent higher degrees of membership. Fuzzy set theory provides various forms of membersh ip functions. The capability to determine appropriate membership functions in th e context of each particular application is crucial for making fuzzy set theory practically useful [85]. Triangular, trapezoidal, and linear shapes of membership functi ons are most commonly used to represent fuzzy numbers. Triangular membership fun ctions are usually preferred due to their combination of solid theoretical basis and si mplicity [86]. However, there are situations that require more complex functions to m ore accurately represent the degrees of membership of elements to fuzzy sets. There are several methods for constructing membersh ip functions. Klir and Yuan [85] discussed direct/indirect methods that involve single/multiple experts. These methods involve gathering and processing responses from experts in particular fields or from extensive literature reviews. 4.4.3 FGP for the Skill-Based Assignment Problem As previously mentioned, personnel assignment probl ems involve imprecise parameters and multiple objectives. In order to de velop feasible solutions to such imprecise multi-objective problems, fuzzy set theor y has been used since the early 1980s in combination with GP to form what is known as FGP [101]. The main difference between FGP and GP is that the latter requires crisp values for each objective to be achieved, whereas in FGP t hese values are specified in an

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90 imprecise manner [100]. Basically, instead of mini mizing deviations from targets as GP does in equation (4.1), FGP maximizes the degrees o f membership to each of the goals. The simple weighted additive FGP model is shown in equation (4.4) [100]. Parameters im and iw represent the degrees of membership (from a linear membership function) and relative weight of the ith goal, respectively. Zimmermann [102] defines the degrees of membership for the ith fuzzy goal i ig x AG) ( and i ig x AG) (with equations (4.5) and (4.6), respectively [100]. The operator means approximately greater than, whereas means approximately less than. Maximize ==m i i iw Z1m (4. 4) £ £ < =i i i i i i i i i i i iL x AG if g x AG L if L g L x AG g x AG if ) ( 0 ) ( ) ( ) ( 1m (4.5) < £ £ =i i i i i i i i i i i iU x AG if U x AG g if g U x AG U g x AG if ) ( 0 ) ( ) ( ) ( 1m (4.6) Equations (4.5) and (4.6) state that it is acceptab le to come short of meeting goal ig up to a specified lower (iL ) or upper (iU ) boundary. A FGP model for skill-based personnel assignments can be obtained as an extensi on to the simple additive model

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91 presented in equations (4.4) (4.6). This extensi on includes modifications to the objective and membership functions which will be de scribed in Section 4.5. Specifying precise target values and priorities for each goal can be a difficult task for decision makers [103]. Consequently, FGP has b een the modeling tool of choice for researchers to solve a variety of problems in diffe rent applications. However, FGP has not been applied to the specific area of skill-base d resource assignments. This is evidenced by statements from very recent research s tudies, stating that “future research should be directed towards developing fuzzy goal pr ogramming models for the competence and preference-based workplace assignmen t” [4]. Furthermore, Baykasoglu et al. [37] mentioned that there is an unfortunate lack of adequate approaches and procedures for assigning workers to teams. 4.5 Solution Approach and Methodology This section presents the proposed stepwise solutio n approach to the personnel assignment problem. Figure 4.2 provides a diagram showing each of the steps and their associated activities. Satisfying necessary pre-co nditions, defining imprecise parameters, and identifying traits of resources constitute the first three steps of the methodology. The fourth step is to properly develop fuzzy sets for t he goals. In the fifth step, membership functions are adequately manipulated to represent f uzzy priorities. The final step is to set up and run the assignment model to obtain a feasibl e solution that considers several goals corresponding to technical capabilities, team param eters, and personnel preferences. The following subsections explain the procedure to prop erly execute the last three steps and ensure a successful implementation of the solution approach.

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92 Figure 4.2 Solution Approach: Steps and Activiti es 4.5.1 Membership Functions Developing membership functions for the target valu es of goals constitute a very important step in the solution approach. Careful e valuation of the membership function shown in equation (4.5) reveals that this function must be modified for the skill-based personnel assignment problem. This equation states that beyond a lower limitiL every element in the solution set has a zero degree of me mbership. This means, for example, that for a task that requires a high level of exper tise in a particular skill, resources with Activities Pre-conditions Define imprecise parameters Identify traits of resources Develop membership functions for the goals Conduct MFR process Run FGP-MFR model -Establish a rating scale to evaluate candidates.-Establish a rating scale for resources to grade th eir level of motivation to work particular tasks. -Construct fuzzy sets for the target values of the goals. -Manipulate the existing fuzzy sets for the goals t o incorporate the imprecise priorities of the goals. -Construct fuzzy sets that incorporate priorities o f tasks. Steps -Calculate priority-based degrees of membership of resources with tasks. -Run the FGP model to obtain a solution. Using linguistic terms:-Assign target values for the goals of each task.-Assign priority levels to each goal.-Assign priority levels to each task. -Develop a skill-matrix for each candidate.-Develop a table with the motivation levels of the resources with each task. Activities Steps Activities Pre-conditions Define imprecise parameters Identify traits of resources Develop membership functions for the goals Conduct MFR process Run FGP-MFR model -Establish a rating scale to evaluate candidates.-Establish a rating scale for resources to grade th eir level of motivation to work particular tasks. -Construct fuzzy sets for the target values of the goals. -Manipulate the existing fuzzy sets for the goals t o incorporate the imprecise priorities of the goals. -Construct fuzzy sets that incorporate priorities o f tasks. Steps -Calculate priority-based degrees of membership of resources with tasks. -Run the FGP model to obtain a solution. Using linguistic terms:-Assign target values for the goals of each task.-Assign priority levels to each goal.-Assign priority levels to each task. -Develop a skill-matrix for each candidate.-Develop a table with the motivation levels of the resources with each task. Activities Steps

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93 medium levels of knowledge in the skill may be trea ted the same (i.e., have a degree of membership of zero) as those with lower levels of k nowledge. To avoid this situation, this iL parameter needs to be eliminated (i.e., set to zer o). This way, each lower level of expertise results in some value added. Outcomes of decisions based on fuzzy approaches dep end heavily on the appropriateness of the membership functions used. Consequently, careful selection of membership functions is vital for effective decisio n making processes [104]. One way to improve the development of membership functions is to work directly with decision makers to model these functions based on their expe rtise. However, most FGP formulations assume linear membership functions whi ch are established without the involvement of decision makers [105]. Membership functions corresponding to fuzzy sets of imprecise capabilities depend on whether the main objective of an assignme nt policy is to minimize deficiencies from target values, or minimize deviat ions (i.e., deficiencies plus surplus). A reason for selecting to minimize deviations is that studies have shown that assigning over-qualified personnel to tasks decreases product ivity due to a lack of motivation given that tasks might not be challenging enough [106]. Figure 4.3 shows an example of a linear interval membership function to minimize dev iations. Here, deviations to either side of the target value reduce the degrees of memb ership of an element in that particular fuzzy set. On the other hand, decision-makers may rather prefer to meet minimum requirements as much as possible, even if that mean s assigning an expert in a particular skill to a task that requires a low expertise level In this case, an assignment policy to minimize deficiencies is appropriate. Figure 4.4 s hows an example of a linear interval

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94 membership function to minimize deficiencies. Here deviations to the left side of the target value reduce the degrees of membership, wher eas any deviations to the right side results in a degree of membership of one. Figure 4.3 Sample Membership Function to Minimize Deviations from a Target Value Figure 4.4 Sample Membership Function to Minimize Deficiencies from a Target Value 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Degree of membership Target Overqualified Underqualified 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Degree of membership Target Overqualified Underqualified

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95 4.5.2 Priorities Personnel assignment methodologies should consider priorities of goals and tasks to develop more thorough assessments of the alterna tives. The literature shows two common approaches to consider fuzzy priorities. Th e first approach uses a method known as fuzzy weighted average (FWA), and the seco nd uses desirable achievement degrees. The following sections explain these appr oaches in more detail, as well as the method that will be used in this research to model priorities. 4.5.2.1 Priorities with Fuzzy Weighted Average The FWA method is perhaps the simplest and most com mon approach to incorporate fuzzy priorities. It is used in decisi on problems that require assessments of alternatives with respect to some assignment criter ia and the corresponding importance of such criteria. With FWA, these assessments involve three basic operations, namely scoring, weighting, and aggregating the criteria [1 07]. The general specification for the weighted average is shown in equation (4.7). Here, ]1,0[ iw and ==n i iw11 Therefore, iw must be normalized to 'iw as shown in equation (4.8). () ()n n n n n n nx w x w w w x w x w w w x x f y' 1 1 1 1 1 1 1... ... ... ,..., ,..., + + = n n r r + + + + = = (4.7) n i i iw w w w + + =...' (4.8) The study in [37] provides an example that uses FWA The authors categorized priorities into four linguistic terms: poor, fair, good, and very good. Figure 4.5 shows the triangular membership functions used for each of th e four terms.

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96 Figure 4.5 Triangular Membership Functions for Pr iority For each of the fuzzy sets, the ratings correspondi ng to a degree of membership of 1.0 are used as defuzzified ratings. That is, a poor prior ity has a defuzzified rating equal to 0.3, fair equals 0.5, good equals 0.8, and very good equ als 1.0. To develop relative priorities, each of the defuzzified ratings must be normalized using equation (4.8), where]0.1,8.0 ,5.0,3.0[ iw This way ensures that the sum of the priorities equal to one, which is a requirement for fuzzy weighted aver age operations. For example, consider three goals with priorities medium, high, and very high. That is, the first goal has a defuzzified priority of iw = 0.5, the second goal has iw = 0.8, and the third goal has iw = 1.0. The degrees of membership for the prioriti es of the three goals are the following: For goal #1: 0.5 / 2.3 = 0.217 For goal #2: 0.8 / 2.3 = 0.348 For goal #3: 1.0 / 2.3 = 0.435 PoorFairGoodVery GoodDegree of MembershipWeights (defuzzified crisp values)0 1 0.30.50.70.91.0 PoorFairGoodVery GoodDegree of MembershipWeights (defuzzified crisp values)0 1 0.30.50.70.91.0

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97 Chen and Tsai [108] stated that using FWA operation s to determine priorities of goals can produce undesirable results. To prove th eir point, the authors modified the weights in the example provided by [100]. The resu lts showed a decreased achievement degree of a fuzzy goal after significantly increasi ng the goal’s weight, which is an undesirable outcome. Therefore, Baykasoglu et al. [37] implies that the use of FWA priorities is justified in situations that impede m ore structure decision approaches. Such situations are distinguished by a strong lack of ob jective and reliable information [109], such as scenarios that prohibit inputs from field e xperts. 4.5.2.2 Priorities with Desirable Achievement Degrees Studies such as [108] and [110] use desirable achie vement degrees to represent priorities of goals. In other words, high priority goals would denote higher desirable achievement degrees. The authors use linguistic te rms to denote fuzzy priorities. Afterwards, these linguistic terms are mapped to th eir corresponding defuzzified values, which are used as crisp constraints in a linear pro gramming model. For example, the constraint for a “good” priority goal (using Figure 4.5) would be represented as8.0im. The evident drawback from this approach is that it may produce unfeasible results [108]. 4.5.2.3 Membership Function Relaxation This research presents a new method, denoted as mem bership function relaxation (MFR), to incorporate fuzzy priorities for goals an d tasks in personnel assignment problems. The purpose of the MFR method is to modi fy membership functions as a result of the flexibility of decision makers to mee t fuzzy goals. Such flexibility is driven by decision makers’ perceived imprecise priorities of the goals.

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98 Flexibility is represented as a manipulation to an existing membership function. More specifically, flexibility corresponds to an al lowable expansion of a membership function, as well as a reduction of the maximum att ainable degrees of membership for lower priority goals and goals that have lower targ et values. For example, assume that the goals of a decision maker are to select a resou rce that is an expert in skill-x and a novice in skill-y. Notice that a novice might be p referred over an expert in some tasks in order to avoid having overqualified employees. Mor eover, assume that the decision maker agrees to use the membership functions from F igure 4.6 to describe the fuzzy sets for an expert and a novice. Figure 4.7 shows a pos sible expansion policy to minimize deviations from the target goals based on different priority levels. It can be seen that lower priority levels increase the flexibility of a decision maker to meet a goal, causing the membership function to widen around its middle value. In addition, the highest achievable degrees of membership for lower priority goals are smaller, which results in higher degrees of membership for higher priority go als. Similarly, the highest achievable degrees of membership for the novice fuzzy set are smaller than for the expert fuzzy set. This follows the rationale that resources with high er levels of expertise are usually in shorter supply than those with lower expertise leve ls. This way, for example, assigning an expert to a task that requires expert capability is valued more than assigning a novice to a task that requires novice capability. Degrees of membership resulting from the MFR process are called priority-based degrees of member ship. Figure 4.8 shows an expansion policy to minimize deficiencies from the target goa l. Here, any rating higher than the target rating has a degree of membership of one.

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99 Figure 4.6 Fuzzy Sets for Novice and Expert Figure 4.7 Sample MFR to Minimize Deviations from Target Goals 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 RatingPriority-based degrees of membership Novice (High Priority) Novice (Low Priority) Expert (High Priority) Expert (Low Priority) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 RatingDegrees of membership Novice Expert

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100 Figure 4.8 Sample MFR to Minimize Deficiencies fr om Target Goals The success of any methodology that considers impre cise parameters using fuzzy set theory depends on the use of membership functio ns that make sense to experts and decision makers in the particular personnel assignm ent scenario. Consequently, it is critical to involve decision makers throughout the MFR process. This will help to more accurately resemble the perceived priority levels o f decision makers and avoid undesirable results as much as possible. 4.5.3 FGP-MFR Model This section presents the skill-based resource assi gnment optimization model called the FGP-MFR model. More specifically, the m odel has a FGP specification where the priorities of goals are established with the MF R process. The notation for the FGPMFR model is shown in Table 4.3. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 RatingPriority-based degrees of membership Novice (High Priority) Novice (Low Priority) Expert (High Priority) Expert (Low Priority)

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101 Table 4.3 Notation for the FGP-MFR Model Variable Name Description Gt Number of fuzzy goals for task t Target(gt) Fuzzy target value for goal g of task t where g = 1 to Gt and Target( gt ) [Low, Medium, High, Very High] Proximity(rt) Measure of the proximity of resource r to the aggregated target values of task t PDOM_Task(rt) Priority-based degree of membership of resource r in task t Xrt 1 if resource r is assigned to task t ; 0 otherwise PDOM_Goal(rgt) Priority-based degree of membership of resource r in fuzzy goal g of task t MAX[* gtm] Maximum priority-based degree of membership that ca n be achieved in goal g of task t The general model specification is shown in equatio n (4.9). Here, a goal g of task t is that a resource r closely meets the target value Target(gt), as shown in equation (4.10). Maximize Z = ( ) == T 1 t R 1 r ) rt( ) rt(X Task PDOM (4.9) Goal(gt): @ Goal PDOM) rgt (Target(gt) (4.10) PDOM_Task(rt) @ m(Proximity(rt)) (4.11) Proximity( rt ) @ t r G 1 g gt G 1 g ) rgt, MAX Goal PDOMt t" = =] [(m (4.12) r T 1 t rt(, X" £=1) (4. 13) t R 1 r rt(, X" ==1) (4.14)

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102 The objective function (4.9) states that the best a ssignment is the one that maximizes the sum of the priority-based degrees of membership of resources with tasks. Equation (4.12) defines the proximity of a resource to fulfill the set of goals of a task. This measure is defined as the ratio of the aggrega ted priority-based degrees of membership attained by the resource to the aggregat ed maximum priority-based degree of membership that can be achieved in the goals. E quation (4.11) considers priorities of tasks by mapping Proximity( rt ) to a priority-based degree of membership from the membership function m(Proximity( rt )). Next section provides an example that further explains the model. The constraint in (4.13) states that a resource may be assigned to at most one task. Constraint (4.14) states that each task must be ass igned to a resource. This constraint is valid only if the number of available resources is greater than or equal to the number of tasks; otherwise it must be removed. 4.6 Example Software Development Setting A personnel assignment problem in a software develo pment setting was formulated to illustrate the implementation of the solution approach. This particular setting is relevant given that personnel assignment s are considered one of the most critical decisions that affect the performance and quality of software projects [6]. This is confirmed by Tsai et al. [43] with the following qu ote: “evidence reveals that the failure of software development projects is often a result of inadequate human resource project planning”.

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103 Quality, as evidenced in the U.S. General Accountin g Office Report in [2], continues to be a major struggle to software compan ies. This report states that in 2004 the U.S. Department of Defense spent nearly 8 billi on dollars to rework software because of quality-related issues. Even more important tha n huge monetary costs is the fact that software failures, especially in safety-critical sy stems, may result in life-threatening situations. Another reason that makes this example relevant is that it directly addresses areas of future research from the current software develo pment literature. Recently, Otero et al. [5] presented an approach for resource allocati on in software projects. Their methodology used precise parameters to determine ca pabilities of resources. The authors acknowledged the limitations of using precise param eters and encouraged researchers to develop methodologies that incorporate fuzzy parame ters instead. In addition, the authors emphasized the need to extend their methodo logy to incorporate priorities of tasks. Incorporating tasks’ priorities into resour ce allocation processes “will provide more effective staffing decisions to high-priority projects, which will result in better returns of investment for companies” [5]. 4.6.1 Problem Statement and Pre-conditions The personnel assignment problem formulated to impl ement the solution approach consisted of ten available software engine ers and ten tasks. For this example, two experts from leading software engineering compa nies agreed to act as decisionmakers. Involving real industry experts adds value to this example and facilitates the proper execution of the solution approach. Both de cision makers have an average of 16

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104 years of experience working for top U.S.A. organiza tions that specialize in the development of software applications for the defens e industry. Following the solution approach depicted in Figure 4.2, the first step is to make sure that pre-conditions are satisfied. The first pre-condition is to establish a rating scale to evaluate candidates. The decision-makers agreed on a rating scale from 0 to 5, where higher ratings represent higher evaluations. This rating scale is commonly used for yearly evaluations of the performance of engineers. The second pre-condition is to determine a rating scale for resources to grade the ir level of motivation to work particular tasks. Similarly, a 0 to 5 rating scale was select ed where higher ratings represent higher motivation levels. 4.6.2 Establishing Imprecise Parameters After establishing pre-conditions, the next step is to establish fuzzy target values for the goals of each task. The goals are associat ed with three main types of assignment criteria: technical expertise, personnel preference s, and team parameters. Desired target values are generated with the following linguistic terms: None, Novice, Proficient, Highly Proficient, and Expert. Similarly, the prio rity of each goal needs to be established with the following linguistic terms: Low, Medium, a nd High. For this particular example, different priority levels were assigned on ly to Proficient and Highly Proficient target goals. The rationale for having various pri ority levels for selective goals will be explained during the implementation of the MFR proc ess.

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105 Table 4.4 shows the capability levels desired for e ach of the tasks’ required skills, as well as the priority of each skill to their corr esponding task. Table 4.5 presents the priorities assigned to the tasks. Table 4.4 Desired Expertise Levels for Skills (Pr iorities of Skills in Parentheses) Table 4.5 Tasks' Priorities Task NamePriority Task_1MediumTask_2MediumTask_3HighTask_4HighTask_5MediumTask_6LowTask_7LowTask_8MediumTask_9MediumTask_10High Team FactorsCC++C# Satellite communications Command & Control Avionics Embedded Programming GUI Programming CommunicationTask_10HP (H)NHP (H)0HP (H)0HP (H)P (L) Task_2 0HP (H)P (L)HP (M)0HP (L)0HP (H)HP (M) Task_3 0EHP (H)P (L)0HP (M)0HP (H)P (L) Task_4 E000HP (H)0E0HP (M) Task_5 00HP (M)P (L)000HP (H)P (L) Task_6 0HP (H)0HP (M)0HP (M)0P (M)N Task_7 00HP (M)P (H)0HP (M)0HP (H)HP (M) Task_8 00E00P (L)0EP (L) Task_9 00HP (H)0NP (L)0EHP (M) Task_10 HP (M)0HP (H)HP (L)0N00P (M) 0 = No expertise N = Novice level of expertise P (x) = Proficient level of expertise; skill priority lev el is x, where x ( {L=Low, M=Medium, H=High} HP (x) = Highly proficient level of expertise; skill prior ity level is x, where x ( {L = Low, M = Medium, H = High} E = Expert level of expertiseTask Name PLDomainApplication

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106 4.6.3 Identifying Traits of Resources The next step is to develop a skill-matrix for each candidate, as well as a tabular representation of the motivation levels of the reso urces for each task. Data for the skillmatrix are usually readily available to decision ma kers from databases that store employees’ self-evaluations as well as assessments made by lead personnel [5]. Table 4.6 shows the skill matrix for the available candid ates in this sample case. Table 4.7 presents the motivation levels of the resources wit h the tasks. Table 4.6 Skill Matrix of Available Candidates Ba sed on a 0-5 Rating Scale Table 4.7 Motivation Levels of Resources with Tas ks Based on a 0-5 Rating Scale Resource Task_1 Task_2 Task_3 Task_4 Task_5 Task_6 Task_7 Task_8 Task_9 Task_10 R_15.04.02.55.04.01.01.04.00.02.5 R_2 5.01.04.00.05.01.00.01.00.00.0 R_3 1.02.52.50.04.02.54.05.00.00.0 R_4 0.01.01.05.00.02.50.01.01.02.5 R_5 4.02.52.50.04.00.00.00.01.00.0 R_6 4.02.52.50.04.04.05.00.05.04.0 R_7 4.01.04.00.02.54.00.00.00.04.0 R_8 2.52.52.50.02.52.52.52.52.52.5 R_9 4.04.04.00.04.02.52.54.05.02.5 R_10 4.04.04.01.04.05.02.54.04.05.0 Team Factors CC++C# Satellite communications Command & Control Avionics Embedded Programming GUI Programming CommunicationR_12.553.543.53.53.553R_2143414043.5R_31453.51.53052.5R_4540.552553.53.5R_501.51102022.5R_6233.530403.52.5R_7152.5503.5055R_80.5333.51213.51R_9044.5312.5042.5R_1042.53.53.51323.54Resource PLDomainApplication

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107 4.6.4 Develop Membership Functions for Goals The objective of this step is to construct fuzzy se ts for the target values of the goals. To this end, the direct method with multipl e experts approach presented by Klir and Yuan [85] was used. To facilitate this process the decision makers were initially presented with various shapes of possible membershi p functions. These included linear, triangular, trapezoidal, normally distributed, and sinusoidal shapes [111]. Both experts preferred the sinusoidal shapes because these funct ions provided smooth non-linear transitions between fuzzy sets that span across the entire x-axis (i.e., rating scale). Figure 4.9 shows the sinusoidal membership function s for an assignment policy to minimize deviations from target values. That is, t hese membership functions are used when there is a penalty associated with assigning o verqualified resources to tasks. Furthermore, this figure shows that deviations to t he right of the target values (i.e., overqualified rating), for all but the Novice set, resu lt in higher degrees of membership than similar deviations to the left (i.e., under-qualifi ed rating). The reason for having these non-symmetrical shapes is that the decision makers preferred overachievement to underachievement.

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108 Figure 4.9 Fuzzy Sets to Minimize Deviations from Target Values Figure 4.10 shows the sinusoidal membership functio ns for an assignment policy whose objective is to minimize deficiencies from ta rget values. These membership functions are used when there is no penalty associa ted with assigning overqualified resources to tasks. However, the sample scenario p resented in this section assumes penalties for over-qualification; therefore, only t he membership functions from Figure 4.9 will be used for the MFR process. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00.51.01.52.02.53.03.54.04.55.0 RatingDegree of membership Novice Proficient Highly proficient Expert

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109 Figure 4.10 Fuzzy Sets to Minimize Deficiencies f rom Target Values To consider fuzzy goals representing the motivation of employees with each task, a single fuzzy set was identified with the followin g linguistic term: Highly Motivated. This means that one of the goals in every resource assignment is to match an employee with a task that he/she is Highly Motivated to tack le. The membership function for this fuzzy set is depicted in Figure 4.11. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00.51.01.52.02.53.03.54.04.55.0 RatingDegree of membership Novice Proficient Highly proficient Expert

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110 Figure 4.11 Fuzzy Set for Highly Motivated 4.6.5 MFR Process The next step is to consider the three possible pri ority levels for the goals (i.e., Low, Medium, and High) by developing expansion poli cies for each fuzzy set. With this in mind, the decision makers produced the set of ge neral rules shown in Table 4.8. Table 4.8 Set of General Rules Rules Description Rule 1 For high-priority goals over-qualification is highl y preferred to under-qualification. Rule 2 For medium-priority goals over-qualification is som ewhat preferred to under-qualification. Rule 3 For low-priority goals under-qualification is somew hat preferred to over-qualification. Rule 4 For goals with Novice or Expert target values there is no distinction between priority levels. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00.51.01.52.02.53.03.54.04.55.0 RatingDegree of membership Highly Motivated

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111 The rationale for Rule 4 is that the decision maker s associate novice and expert target values with a single priority level each. A goal that prefers a novice expertise is often logically viewed as low priority since they a re relatively of low complexity. On the contrary, a goal that prefers an expert is usually perceived as high priority given that the number of resources with expert capabilities is oft en limited in industry. The decision makers explained that there are cases where a highly proficient level of knowledge in a skill is desired for a low priori ty goal. For example, consider a task in which a high level of embedded programming expertis e is desired to develop a software component. That is, the goal is to assign a resour ce that is highly knowledgeable in embedded programming. Now, assume that there is mu ch legacy code from previous completed tasks that can be reused for this new com ponent, in addition to detailed documentation that clearly explains this legacy cod e. This may cause decision makers to be more flexible and treat the desired level of ski ll as a low priority goal, hence expanding the set of possible solutions. After several iterations to incorporate the prefere nces of the decision makers, the Expert fuzzy set remained unchanged. The resulting membership functions for the Novice, Proficient, and Highly Proficient fuzzy set s are shown in Figure 4.12, Figure 4.13 and Figure 4.14, respectively. Each fuzzy set shows reductions to their maximum attainable degrees of membership. For instance, th e maximum attainable degree of membership for the Highly Proficient fuzzy set is s maller (i.e., 0.85) than for the Expert set (i.e., 1.0). This provides a higher incentive to match an expert with a task that requires an expert capability, rather than to match a highly proficient resource with a task that requires highly proficient expertise. This sa me rationale is applied to the remaining

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112 sets. Moreover, these reductions avoid overcompens ating higher achievements in lower priority goals. Figure 4.12 Fuzzy Set of Novice Expertise After M FR Figure 4.13 Fuzzy Set of Proficient Expertise Aft er MFR 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00.51.01.52.02.53.03.54.04.55.0 RatingPriority-based degree of membership High priority Medium priority Low priority 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00.51.01.52.02.53.03.54.04.55.0 RatingPriority-based degree of membership

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113 Figure 4.14 Fuzzy Set of Highly Proficient Expert ise After MFR Finally, to consider priorities of tasks, it is nec essary to construct a new fuzzy set to represent the following linguistic term: Excelle nt Assignment. First, a fuzzy set is developed based on the decision makers’ preference of the Proximity( rt ) values that constitute an excellent resource assignment to a hi gh priority task. Then, this fuzzy set is relaxed and the maximum attainable degrees of membe rship for lower priority tasks are reduced through the MFR process. The decision make rs decided that piecewise linear membership functions were adequate to model these f uzzy sets (see Figure 4.15). 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00.51.01.52.02.53.03.54.04.55.0 RatingPriority-based degree of membership High priority Medium priority Low priority

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114 Figure 4.15 Piecewise Linear Membership Functions for Tasks’ Priorities 4.6.6 FGP Model Results A software application was developed to compute the parameters necessary for the FGP solution. That is, the software implemente d the solution approach up to the last step, which is to run the FGP model. This simplifi ed the process of determining prioritybased degrees of membership of resources with tasks (i.e., PDOM_Task( rt )), given the various combinations of factors involved in such ca lculations. The software was implemented with the object-orient ed C++ programming language. The output of the program is a DOS windo w with the PDOM_Task( rt ) values associated with all the resources and tasks. These PDOM_Task( rt ) values were then used as inputs to the FGP model, which produced the assi gnments presented in Table 4.9. Table 4.9 shows the proximity values for the resou rces with the goals of the tasks, as well as the resulting priority-based degrees of membership of each assignment. These values provide a measure of the level of satisfacti on of the decision makers with each allocation after carefully considering priorities o f goals and tasks. For each of the three 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00.10.20.30.40.50.60.70.80.91.0 Proximity(rt)Priority-based degree of membership High priority Medium priority Low priority

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115 high-priority tasks, the resulting priority-based d egrees of membership were over 0.9. This means that based on the membership functions u sed for the FGP-MFR model, these three resource assignments are very close to fully belonging to the Excellent Assignment fuzzy set. The same conclusions can be made about the medium-priority tasks, since the obtained degrees of membership were close to the ma ximum allowable value of 0.7 that resulted from the MFR process. Notice that the ass ignment of resource R_5 to Task_7 resulted in a zero priority-based degree of members hip. This is due to the overall limited expertise of R_5 with the set of skills required by any of the tasks. Therefore, in this case it resulted more efficient to assign R_5 to a low-p riority task. Table 4.9 Solution to the Personnel Assignment Pr oblem 4.7 Summary and Contributions This study presented a new FGP personnel assignment approach in scenarios characterized by imprecise tasks’ requirements and resources’ capabilities. The goals for each resource assignment are to meet desired target values for technical expertise, team ResourceTask Proximity(rt)Task Priority PDOM_Task(rt)R_2Task_10.87031Medium0.66701R_8Task_20.60866Medium0.60000R_9Task_30.79067High0.90018R_4Task_40.82923High0.93080R_6Task_50.94001Medium0.70000R_7Task_60.65400Low0.27444R_5Task_70.15043Low0.00000R_3Task_80.96847Medium0.70000R_1Task_90.62524Medium0.60000 R_10Task_100.86186High0.96270

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116 parameters, and personnel preferences. These targe t values are represented with fuzzy sets which are developed with the help of decision makers. Then, priorities of goals are considered by adequately manipulating membership fu nctions of target values. Priorities of tasks are considered in a similar way. A fuzzy set is constructed to model the decision makers perceived definition of an excellent resourc e assignment based on the suitability of a resource with the set of goals of a task. Thi s fuzzy set is then modified for lower priority tasks to promote better assignments to hig her priority tasks. There are two major contributions that this researc h study makes to the personnel assignment body of knowledge. The first significan t contribution is the introduction of FGP to the specific area of skill-based resource as signments. The second significant contribution is related to the types of parameters considered in current skill-based personnel assignment methodologies, as well as the approaches for modeling them. An extensive review of relevant literature highlighted several significant limitations in this area. The solution approach developed in this rese arch addresses these limitations in three main ways. First, it includes several critic al parameters associated with resources and tasks that must be considered in the decision p rocess and are omitted in current methodologies. Some of these parameters include pr iorities of skills and tasks, as well as the motivation levels of employees to work particul ar jobs. Taking into account these parameters in the decision process leads to more th orough evaluations of alternative solutions. Second, the solution approach considers the definition of these parameters to be naturally imprecise. Thus, these parameters are modeled using fuzzy sets instead of using the common classical set theory. This realis tic representation of imprecise parameters with fuzzy concepts has the potential to provide a high practical value to the

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117 methodology. Third, the solution approach directly involves decision makers in the process of defining imprecise parameters through th e construction of the fuzzy sets. This may also provide higher practical value to the solu tion approach, given that current fuzzy approaches to personnel assignments provide simple membership functions (e.g., triangular or trapezoidal) that were created withou t consulting decision makers. 4.7.1 Research Extensions A major challenge for any researcher is to develop new methodologies that become widely accepted by practitioners. To achiev e this, it is important for researchers to properly market their solution approaches by bri nging these novel methodologies into industry scenarios to show field experts the capabi lities of such new approaches. With this in mind, the approach developed in this resear ch needs to be applied to different industry settings to validate its applicability and determine its acceptability. Furthermore, a user-friendly software implementation of the solu tion approach is necessary. This effort was initiated with the software developed in this research to determine the degrees of membership of resources with tasks. However, pr oper software engineering processes must be followed to develop a complete decision sup port system to meet the expectations/requirements of decision makers. Another research extension is to conduct experiment al control group analyses to determine the impact of applying the personnel assi gnment methodology developed in this research versus using the conventional subject ive approach. This would provide evidence to support (or not) the existence of signi ficant gains from using the new methodology.

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118 Finally, this research could be expanded by conduct ing a survey analysis to investigate if it is reasonable to develop a baseli ne of membership functions for general/common skills in particular environments. For example, it may be possible to interview experts from different software developme nt organizations to come up with fuzzy sets for technical capability assessments tha t can be used as standards across companies. This same approach can be followed and adopted in other fields.

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119 CHAPTER 5 CONCLUSIONS AND FUTURE RESEARCH This chapter presents the conclusions of this disse rtation and summarizes extensions for further research. 5.1 Conclusions Personnel assignment methodologies have been the fo cus of active research for various decades. Nevertheless, companies continue to struggle to deliver quality products on schedule and within budget constraints. This research presents a systematic analysis approa ch to develop a robust solution to the personnel assignment problem in skill-based environments. First, the problem was decomposed into three main activities: identifying assignment criteria, evaluating personnel capabilities, and assigning personnel to tasks. Second, an extensive literature review was conducted to determine specific opportun ities for improvement in each of the three areas. Based on the literature findings, thi s work presents a framework for resource allocation composed of enhanced methodologies to ef ficiently identify assignment criteria, conduct thorough assessments of personnel capabilities, and effectively assign resources to tasks. The general methodology developed in this research to identify assignment criteria is based on a two-stage DEA-Tobit regressi on approach that determines the

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120 impact of personnel factors to quality and producti vity measures. This methodology contributes to the personnel assignment body of kno wledge by presenting an analytical approach that considers multiple outputs simultaneo usly and eliminates subjectivity when determining relative priorities for assignment crit eria in skill-based environments. This tool is of significant use and relevance to decisio n makers since most personnel assignment decisions in industry involve the evalua tion of several performance measures and pose a challenge for decision makers to subject ively determine important parameters. The methodology was validated by analyzing data fro m a software development corporation, which resulted in the identification o f drivers of efficiency of personnel assignments per task complexity. The resulting ass ignment criteria can be used by decision makers in software development settings, w hich is another key contribution of this research. For evaluating personnel capabilities, this work pr esents an expert system architecture capable of making fuzzy inferences. T his approach uses fuzzy theory to represent personnel levels of expertise, establish relationships between skills, and make inferences about the qualifications of personnel. This realistic representation of imprecise parameters and activities using fuzzy con cepts has the potential to provide a high practical value to the expert system. The mai n contribution of the proposed approach is the introduction of a high-level layere d architecture where each layer is adaptable to context-specific subcomponents. More specifically, each layer can be customized with different subcomponents without imp acting the code implementation of the other layers. This is accomplished by introduc ing a global layer that is used as the only channel of interaction between any two layers. Therefore, implementation details of

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121 any layer are hidden from the others, making change s concealed and imperceptible to other layers. This provides decision makers the fl exibility to add/delete/modify subcomponents in any layer based on their particula r needs without having to incur into expensive architectural system modifications. Finally, this research introduces a new FGP model f or personnel assignments in scenarios characterized by imprecise tasks’ require ments and resources’ capabilities. The goals for each resource assignment are to meet desi red target values for technical expertise, team parameters, and personnel preferenc es. These target values are represented with fuzzy sets which are developed wit h the assistance of decision makers. Priorities of goals and tasks are considered by ade quately manipulating membership functions of target values. The FGP approach addresses three significant limita tions from the existing literature. First, it includes several critical pa rameters associated with resources and tasks that must be considered in the decision process and are omitted in current methodologies. Some of these parameters include priorities of skil ls and tasks, as well as the motivation levels of employees to work particular jobs. Takin g into account these parameters in the decision process leads to more thorough evaluations of alternative solutions. Second, the solution approach considers the definition of these parameters to be naturally imprecise. Thus, these parameters are modeled using fuzzy sets instead of using the common classical set theory. This realistic representatio n of imprecise parameters with fuzzy concepts has the potential to provide a high practi cal value to the methodology. Third, the solution approach directly involves decision ma kers in the process of defining imprecise parameters through the construction of th e fuzzy sets. This may also provide

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122 higher practical value to the solution approach, gi ven that current fuzzy approaches to personnel assignments provide simple membership fun ctions (e.g., triangular or trapezoidal) that were developed independently from decision makers. 5.1.1 Research Extensions There are various areas for future research associa ted with each of the methodologies developed in this dissertation. This presents opportunities for researchers to continue investigating and enhancing the science of personnel assignments. The DEA-Tobit approach developed to identify assign ment criteria was evaluated using data from a software development organization Given that the data were limited to four personnel factors, an apparent expansion is th e necessity to confirm and extend the results with additional personnel and task factors to increase the understanding of drivers of efficiency in software applications. A necessary research extension for the expert syste m presented for capability assessments is to complete its design phase and pro ceed to the coding phase. Since this research provides the high-level software design ar chitecture, the next step would be to divide the architecture into components and develop detailed designs for each component using object-oriented tools such as class diagrams. The final product must include proper software engineering documentation such as software requirements specification, software design document, software manual, and test description document. Another potential research extension to the propose d expert system is to conduct a survey analysis to investigate if it is reasonable to develop baselines of membership functions for general/common skills in particular e nvironments. For example, it may be

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123 possible to interview experts from different softwa re development organizations to establish fuzzy sets for technical capability asses sments that can be used as standards across companies. A similar survey analysis can be conducted to examine the possibility of establishing fuzzy rules’ baselines to describe the relationship between various skills. The FGP approach for personnel assignments presents a preliminary effort to develop a user-friendly software implementation of the solution approach. This initial step includes C++ code to determine the degrees of membership of resources with tasks. However, proper software engineering processes must be followed to develop a complete decision support system that meets the expectations /requirements of decision makers. A major challenge prompted by the development of ne w methodologies is the validation of these novel approaches. Although the implementation of the FGP approach was demonstrated through an example, further resear ch is necessary to validate the methodology. One suggestion is to conduct experime ntal control group analyses to determine the impact of applying the FGP personnel assignment methodology developed in this research versus using the conventional subj ective approach. This would provide evidence to support (or not) the existence of signi ficant gains from using the FGP approach. Another major challenge of new methodologies is the ir acceptance by practitioners. To achieve this, it is important fo r researchers to properly market their solution approaches by bringing these novel methodo logies into industry scenarios to show field experts the capabilities of such new app roaches. With this in mind, each of the methodologies developed in this research must b e applied to different industry settings to validate their applicability and accept ability.

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ABOUT THE AUTHOR Luis Daniel Otero received his Ph.D. in Industrial and Management Systems Engineering in 2009 from the University of South Fl orida, Tampa, FL. He received a M.S. in Computer Information Systems in 2000 and a M.S. in Engineering Management in 2002 from Florida Institute of Technology (FIT), Melbourne, FL. He is currently an Adjunct Professor in the departments of Computer In formation Systems and Engineering Systems at FIT. He was a recipient of a prestigiou s National Science Foundation fellowship from 2002-2005. He has worked in the defense industry as a software engineer for more than eight years. He won a Next Level Award in 2001 from Harr is Corporation Government Communications Systems Division, and a Timely Award in 2007 from Northrop Grumman Corporation Integrated Systems (NGIS). I n 2007, he was nominated by NGIS for the Society of Hispanic and Professional E ngineers Outstanding Young Hispanic Engineering award.


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ABSTRACT: The development of effective personnel assignment methodologies has been the focus of research to academicians and practitioners for many years. The common theory among researchers is that improvements to the effectiveness of personnel assignment decisions are directly associated with favorable outcomes to organizations. Today, companies continue to struggle to develop high quality products in a timely fashion. This elevates the necessity to further explore and improve the decision-making science of personnel assignments. The central goal of this research is to develop a novel framework for human resource assignments in skill-based environments. An extensive literature review resulted in the identification of the following three areas of the general personnel assignment problem as potential improvement opportunities: determining assignment criteria, properly evaluating personnel capabilities, and effectively assigning resources to tasks.Thus, developing new approaches to improve each of these areas constitute the objectives of this dissertation work. The main contributions of this research are threefold. First, this research presents an effective two-stage methodology to determine assignment criteria based on data envelopment analysis (DEA) and Tobit regression. Second, this research develops a novel fuzzy expert system for resource capability assessments in skill-based scenarios. The expert system properly evaluates the capabilities of resources in particular skills as a function of imprecise relationships that may exist between different skills. Third, this research develops an assignment model based on the fuzzy goal programming (FGP) technique. The model defines capabilities of resources, tasks requirements, and other important parameters as imprecise/fuzzy variables.The novelty of the research presented in this dissertation stems from the fact that it advances the science of personnel assignments by combining concepts from the fields of statistics, economics, artificial intelligence, and mathematical programming to develop a solution approach with an expected high practical value.
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