Modelling and planning of manpower supply and demand

MODELLING AND PLANNING OF MANPOWER SUPPLY AND DEMAND YIK JIAWEI (B.Eng (Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF INDUSTRIAL & SYSTEMS ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2010 ABSTRACT Effective manpower planning is important and beneficial to a country’s development. This project will focus on engineering manpower. Engineers are an important part of the workforce, especially in Singapore where the drive is to become a knowledge-based economy. In order to make effective manpower policies, the system structure must be known so that system behaviour can be understood. Unfortunately, such knowledge is difficult to obtain and hardly well established. For this project, a hypothesis of the system structure is proposed and validated. Potential policy levers are identified and scenario analyses are carried out to understand the system behaviour. Artificial Intelligence tools are applied to support the modelling process when parts of the system structure are unknown. Fuzzy Expert System is applied to mimic decision policies and various forecasting models. Using fuzzy logic, an attempt to find a decision policy that will enable policy makers to fulfil their objectives is conducted. i ACKNOWLEDGEMENTS I would like to thank the following people for making the thesis possible: Assistant Professor Ng Tsan Sheng Adam and Associate Professor Dipti Srinivasan, my supervisors, for their support, guidance and patience throughout the course of the study And everyone who has helped in one way or another ii CONTENTS ABSTRACT i ACKNOWLEDGMENTS ii LIST OF TABLES viii LIST OF FIGURES ix CHAPTER 1 Introduction 1 1.1 Background 1 1.2 Problem Statement 2 1.3 Objectives 3 1.4 Scope of Study 4 1.5 Organisation of Thesis 5 CHAPTER 2 Literature Review 8 2.1 Perspectives related to Manpower Supply and Demand 8 2.1.1 Economics Perspective 8 2.1.2 Attractiveness Perspective 9 2.1.3 System Dynamics Perspective 2.2 The Human Factor 10 10 2.2.1 Traditional System Dynamics Method 10 2.2.2 Artificial Intelligence Tools 11 2.2.2.1 Fuzzy Expert Systems 12 2.2.2.2 Neural Networks 13 iii 2.2.2.3 Fuzzy Inductive Reasoning 2.3 Modelling: A Synthesis of Ideas 14 15 CHAPTER 3 Prototype Model 16 3.1 Hypothesis 18 3.1.1 Manpower Adjustment 18 3.1.2 Foreign Engineers 19 3.1.3 Resident Engineers 20 3.1.4 Engineer Education 22 3.1.5 Engineering Wage 22 3.1.6 Complete Causal Loop Diagram 23 3.2 Simulation Model 24 3.2.1 Key Assumptions 25 3.2.2 Model Validation 25 3.3 Scenario Analysis and Simulations 27 3.3.1 Employment Duration of Foreign Engineers 28 3.3.2 Hiring Rate of Foreign Engineers 31 3.3.3 Engineering College Admission Rate 34 3.4 Discussion and Analysis 38 3.5 Sensitivity Analysis 39 3.5.1 One Way Sensitivity Analysis 39 3.5.2 Monte Carlo Simulation 41 3.6 Conclusion 44 iv CHAPTER 4 Application of Artificial Intelligence Techniques 4.1 Generic Supply Chain Model – Engineering College Admission Rate 46 46 4.1.1 Simulation of Data 48 4.1.2 Inputs and Outputs of Fuzzy Expert System 48 4.1.3 Fuzzy Expert System – Membership Functions 49 4.1.4 Fuzzy Expert System – Rule Base 50 4.1.5 Optimisation and Calibration of Fuzzy Expert System 50 4.1.6 Validation of Fuzzy Expert System 52 4.2 Fuzzy Expert System – Engineering College Admission Rate 52 4.2.1 Output 54 4.2.2 Inputs 55 4.2.3 Membership Functions 56 4.2.4 Rule Base 56 4.2.5 First Estimate 57 4.2.6 Optimisation of Membership Functions 58 4.3 Discussion and Analysis 60 4.4 Traditional Forecasting Models using Fuzzy Logic 61 4.4.1 First Order Exponential Smoothing 62 4.4.2 Holt‟s Method 62 4.4.3 TREND Function 63 4.4.4 Fuzzy Forecasting Model 64 4.4.5 Calibration and Optimisation of Fuzzy Forecasting Model 66 v 4.5 Conclusion 71 CHAPTER 5 Optimal Decision Policy using Fuzzy Logic 73 5.1 Decision Policy and Fuzzy Logic 73 5.2 Policy Lever and Objective 74 5.3 Fuzzy Policy Model 75 5.3.1 Output of Fuzzy Logic Controller 75 5.3.2 Input of Fuzzy Logic Controller 76 5.3.3 Membership Functions of Fuzzy Logic Controller 76 5.3.4 Rule Base of Fuzzy Logic Controller 77 5.4 Calibration and Optimisation of Decision Policy Model 5.4.1 Discussion and Analysis 5.5 Performance of Fuzzy Decision Policy Model Under Noise 78 83 84 5.5.1 Noise and Errors 85 5.5.2 Robustness of Fuzzy Decision Policy Calibrated under Deterministic Setting 85 5.6 Improving the Robustness of Fuzzy Decision Policy 89 5.6.1 Results 89 5.6.2 Discussion & Analysis 92 5.6.3 Monte Carlo Simulation under Random Noise 93 5.6.4 Linear Programming 96 5.7 Conclusion 103 CHAPTER 6 Conclusion 106 6.1 Summary 106 vi 6.2 Contributions 111 6.3 Limitations 112 6.4 Suggestions for Future Research 113 REFERENCES 114 APPENDIX A: Stock-and-Flow Diagrams 117 APPENDIX B: Equations for Engineering Manpower Supply and Demand Model (iThink) 121 APPENDIX C: One Way Sensitivity Analysis 129 APPENDIX D: Equations for Supply Chain (iThink) 132 APPENDIX E: Training Data Generated by Supply Chain Model 137 APPENDIX F: Membership Function Parameters for Fuzzy Expert System (Supply Chain Model) APPENDIX G: MATLAB Model of Engineering Manpower Supply and Demand System 139 140 APPENDIX H: MATLAB Model of Fuzzy Expert System 144 APPENDIX I: Membership Function Parameters for Fuzzy Expert System (Prototype Model) 145 APPENDIX J: Test Inputs for Fuzzy Forecasting Model 146 APPENDIX K: Engineering Manpower Demand Inputs 148 APPENDIX L: Model Out from Extreme Points (Ramp Input) 150 APPENDIX M: Simplified Model (Simulink) 151 APPENDIX N: MATLAB Code for LP Model 152 vii List of Tables Table 3.1 Key Model Parameters 25 Table 3.2 Low, Nominal and High Values for Key Model Parameters 40 Table 4.1 Rule Base for Fuzzy Expert System (Supply Chain Model) 50 Table 4.2 Levels for Input and Output (Prototype Model) 56 Table 4.3 Rule Base for Fuzzy Expert System (Prototype Model) 56 Table 4.4 Rule base for fuzzy expert system - Forecasting Model 67 Table 4.5 RMSE between Fuzzy Forecasting Model vs. Traditional Forecasting Models (Calibrated with Cyclical Input) Table 4.6 RMSE between Fuzzy Forecasting Model vs. 1st Order Information Delay (Calibrated with Different Inputs) Table 4.7 RMSE between Fuzzy Forecasting Model vs. Holt‟s Method (Calibrated with Different Inputs) Table 4.8 RMSE between Fuzzy Forecasting Model vs. TREND Function (Calibrated with Different Inputs) 69 70 70 70 Table 5.1 Rule base for fuzzy policy model 78 Table 5.2 Assumed Standard Deviation of Noise 85 Table 5.3 Maximum α for Fuzzy Decision Policy (Calibrated under Deterministic Conditions) 87 Table 5.4 Assumed Standard Deviation of Noise 89 Table F.1 Optimised Membership Functions (Supply Chain Model) Table I.1 Membership Function Parameters (1st Estimate, Prototype Model) Table I.2 Membership Function Parameters (Optimised, Prototype Model) viii 139 145 145 List of Figures Figure 3.1 Simplified Representation of Engineering Manpower System 17 Figure 3.2 Causal Loop Diagram for Manpower Adjustment 18 Figure 3.3 Causal Loop Diagram for Foreign Engineers 19 Figure 3.4 Causal Loop Diagram for Resident Engineers 21 Figure 3.5 Causal Loop Diagram for Engineer Education 22 Figure 3.6 Causal Loop Diagram for Engineering Wage 23 Figure 3.7 Complete Causal Loop Diagram 24 Figure 3.8 Comparison of Historical Number of Engineers vs. Simulated Number of Engineers Figure 3.9 Comparison of Historical Engineering Wage and Simulated Engineering Wage Figure 3.10 Average EP Engineer Employment Duration (Decrement of 0.1 years) Figure 3.11 Engineering Wage w.r.t variations in Average EP Engineer Employment Duration Figure 3.12 Leakage Fraction w.r.t variations in Average EP Engineer Employment Duration Figure 3.13 Resident Engineers w.r.t variations in Average EP Engineer Employment Duration Figure 3.14 Average Time Needed to Hire an EP Engineer (Increments of 0.1 years) Figure 3.15 Engineer Wages w.r.t variations in Average Time needed to hire an EP Engineer Figure 3.16 Leakage Fraction w.r.t variations in Average Time needed to hire an EP Engineer Figure 3.17 Number of Resident Engineers w.r.t variations in Average Time needed to hire an EP Engineer 26 27 29 29 30 30 32 32 33 33 Figure 3.18 Engineering Manpower Demand (Ramp Increase) 35 Figure 3.19 Engineering Rate with varying levels of step change 35 Figure 3.20 Engineering Wage w.r.t variations in Engineering Rate 35 Figure 3.21 Number of Resident Engineers w.r.t variations in Engineering Rate 36 ix Figure 3.22 Number of Resident Jobseekers w.r.t variations in Engineering Rate 36 Figure 3.23 Confidence bands for Resident Engineers 42 Figure 3.24 Confidence bands for EP Engineers 42 Figure 3.25 Confidence bands for Resident Jobseekers 43 Figure 3.26 Confidence bands for Engineering Wage 43 Figure 3.27 Confidence bands for Leakage Fraction 44 Figure 4.1 Supply Chain Model (Engineering College Admission Rate) 47 Figure 4.2 Percentage Changes (First 200 data points) 49 Figure 4.3 Membership Functions for Fuzzy Expert System (Supply Chain Model) Figure 4.4 Fuzzy Expert System vs. Supply Chain Model (Engineering Admission Rate) Figure 4.5 Validation of Fuzzy Expert System vs. Supply Chain Model 51 51 52 Figure 4.6 A Missing Link in Engineering Manpower Model 53 Figure 4.7 Hypothesis of inputs: Percentage change in demand 55 Figure 4.8 Membership Functions (First Estimate, Prototype Model) Figure 4.9 Simulated Engineering College Admission Rate vs. Historical Engineering College Admission Rate (Prototype Model) 57 58 Figure 4.10 Optimised Membership Functions (Prototype Model) 59 Figure 4.11 Simulated Engineering College Admission Rate vs. Historical Engineering College Admission Rate (after optimisation, prototype model) 60 Figure 4.12 Surface of Rule Base (Prototype Model) 61 Figure 4.13 Stock and Flow Diagram for First Order Exponential Smoothing 62 Figure 4.14 Stock and Flow Diagram of Holt's Method 63 Figure 4.15 Stock and Flow Diagram for TREND Function 64 x Figure 4.16 Fuzzy Forecasting Model 65 Figure 4.17 Input and Output to Fuzzy Logic Controller for Fuzzy Forecasting Model Figure 4.18 Fuzzy Forecasting Model vs. 1st Order Information Delay (Calibrated with Cyclical Input) Figure 4.19 Fuzzy Forecasting Model vs. Holt's Method (Calibrated with Cyclical Input) Figure 4.20 Fuzzy Forecasting Model vs. TREND Function (Calibrated with Cyclical Input) 66 68 68 68 Figure 5.1 Fuzzy Policy Model 75 Figure 5.2 Membership Functions for Fuzzy Policy Model 77 Figure 5.3 Fuzzy Decision Policy Model 78 Figure 5.4 Normalised Ratio after Implementation of Fuzzy Decision Policy (Ramp Input, RMSE) Figure 5.5 Normalised Ratio After Implementation of Fuzzy Decision Policy (Cyclical Input, RMSE) Figure 5.6 Normalised Ratio after Implementation of Fuzzy Decision Policy (Cyclical Ramp Input, RMSE) Figure 5.7 Normalised Ratio After Implementation of Fuzzy Decision Policy (Ramp Input, Worst Absolute Error) Figure 5.8 Normalised Ratio After Implementation of Fuzzy Decision Policy (Cyclical Input, Worst Absolute Error) Figure 5.9 Normalised Ratio After Implementation of Fuzzy Decision Policy (Cyclical Ramp Input, Worst Absolute Error) Figure 5.10 Model Outputs from Extreme Points (Cyclical) Figure 5.11 Output from Corner Points when Alpha = 0.2891 (Cyclical Input, Calibrated using RMSE) Figure 5.12 Output from Corner Points when Alpha = 0.2969 (Cyclical Input, Calibrated using RMSE) 80 80 81 82 82 83 86 88 88 Figure 5.13 Optimisation Process under noise 90 Figure 5.14 Normalised Ratio for alpha =0, cyclical input 91 Figure 5.15 Normalised Ratio for alpha = 1.3359, cyclical input 91 Figure 5.16 Normalised Ratio for alpha = 1.41, cyclical input 92 xi Figure 5.17 Monte Carlo Simulation under random noise without any decision policy 94 Figure 5.18 Monte Carlo Simulations (Cyclical Input) 95 Figure 5.19 Instance of Model Output Exceeding Acceptable Range (Cyclical Input) 95 Figure 5.20 Simplified Engineering Manpower Model 97 Figure 5.21 Monte Carlo Simulation using weights found from LP Model and under random noise (beta distribution) 101 Figure 5.22 Model Output with no noise 102 Figure 5.23 Model output under random noise (Within acceptable range) Figure 5.24 Model output under random noise (Acceptable range breached) Figure A.1 Stock and Flow Diagram for Manpower Adjustment Subsystem Figure A.2 Stock and Flow Diagram for Job Vacancies Subsystem Figure A.3 Stock and Flow Diagram for Wage Adjustment Subsystem Figure A.4 Stock and Flow Diagram for Resident Engineers Subsystem 102 103 117 118 118 119 Figure A.5 Stock and Flow Diagram for EP Engineers Subsystem 120 Figure A.6 Stock and Flow Diagram for Engineering Education Subsystem 120 Figure C.1 Average Resident Engineer Career Duration 129 Figure C.2 Average Resident Engineer Job Duration 129 Figure C.3 Average EP Engineer Employment Duration 130 Figure C.4 Average Time Needed to Hire a Resident Engineer 130 Figure C.5 Average Time needed to Hire an EP Engineer 131 Figure C.6 Percentage of EP Engineers to Resident 131 Figure E.1 Percentage Change in Resident Jobseekers 137 Figure E.2 Percentage Change in Resident Hiring Rate 137 xii Figure E.3 Percentage Change in Resident Student Cohort 138 Figure E.4 Percentage Change in Engineering College Admission Rate 138 Figure G.1 Manpower Adjustment Subsystem 140 Figure G.2 Job Vacancies Subsystem 141 Figure G.3 Wage Adjustment Subsystem 141 Figure G.4 Resident Engineers Subsystem 142 Figure G.5 EP Engineers Subsystem 143 Figure G.6 Engineering Education Subsystem 143 Figure H.1 MATLAB Model of Fuzzy Expert System 144 Figure J.1 Cyclical Demand 146 Figure J.2 Saw-tooth Demand 146 Figure J.3 Pulse Shape Demand 147 Figure J.4 Mixed Shape Demand 147 Figure K.1 Cyclical Input 148 Figure K.2 Cyclical Ramp Input 148 Figure K.3 Ramp Input 149 Figure L.1 Model Output from Extreme Points (Ramp Input) 150 Figure M.1 Simplified Model ( Simulink version) 151 xiii Chapter 1 1.1 Introduction Background With the development of science and technology and the arrival of the knowledge economy as part of an international economic structure, new challenges are abound for effective human resources planning. At a national level, it is extremely important to balance between manpower capital investment to nurture promising citizens and also the execution of policies to bring in foreign expertise. Doing this well will ensure a competitive advantage in this new economic era. This is especially so for engineering manpower. With their unique skill set and talents, engineers are an important component in the drive to become a knowledge economy. Furthermore, for many countries, they are an important part of the workforce in the traditional heavyweight industries like manufacturing. Institutions of higher education (IHE) play an important role in the training of engineers. However, given that IHE can only influence the supply side in the engineering manpower market, their capacity to solve engineering manpower shortages is very limited. Furthermore, for certain countries, recent trends indicate an increasing number of engineering graduates choosing other professions outside of their specialisation. This phenomenon may upset the best manpower planning efforts of policy makers. Moreover, this “leakage” into other professions could be due to various soft factors such as perceived wage equity between competing 1 2 professions, job prestige etc. Hence, it is clear that when looking at the engineering manpower system, it is important to not only consider the economics of demand and supply, but the social element as well. 1.2 Problem Statement While the importance of the engineering manpower supply and demand is evident, knowledge about the system structure is often lacking and insufficient. Established economic models often neglect the complex interrelationships between various factors or components. The lack of knowledge about the system structure may lead to a flawed understanding about the system. This flawed understanding can lead to fallacious manpower policies implemented by policy makers and thus resulting in unwanted consequences for the economy and citizenry. Hence, it is important to obtain a better understanding of the system structure. Systems thinking is the process of understanding how things influence each other within a whole. It is a framework for seeing interrelationships rather than individual parts and for seeing patterns of change rather than static snapshots or events (Senge, 1990). Systems thinking facilitates the understanding of complex systems. System Dynamics is one tool to apply systems thinking. The System Dynamics Society defines System Dynamics as a methodology for studying and managing complex feedback systems, such as those in business, economy and other social systems. System dynamics models are not immune from forecast inaccuracies and potential misuses in decisions. However, the main utility of such models is not precision forecasting, but for understanding and learning system structure and policy design. According to Sterman (2000) , the purpose of 3 modelling is to eliminate problems by changing the underlying structure of the system. The development of causal and simulation models can be done through systems thinking (Senge, 1989, 1990; Anderson et al., 1997) and system dynamics methodology (Forrester, 1961; Sterman 2000). While it is possible to construct a model using the system dynamics method, the method is dependent on expert knowledge for the elicitation of the system structure and data for the calibration of the model. Thus, this may lead to excessive development times as elicitation of information and sourcing for data often requires a lot of time and resources. This can be a problem if there are time constraints to the modelling project. To circumvent this problem and expedite the modelling process, it is hoped that artificial intelligence techniques can be used to either replace the part of the system where there is insufficient knowledge or to model the human decision making process. However, this application of A.I. techniques in system dynamics is relatively new and the research in the area is somewhat limited. 1.3 Objectives This project aims to build a prototype model of the engineering manpower supply and demand system using a system dynamics approach. This prototype model will take into account the market supply and demand dynamics as well as the human aspects of supply and demand. It is hoped that through this prototype model, a better understanding of the system structure can be achieved. Next, answers to pertinent policy questions can be found. Some of these questions are: 4  Does the influx of foreign engineers depress local wages?  How does College Admission Rates affect manpower supply? It is hoped that through better understanding and more knowledge about the system, possible guidelines concerning manpower policies can be learnt. Furthermore, A.I. techniques will be applied alongside the system dynamics approach to evaluate their suitability in replacing parts of the model where there is insufficient knowledge about the system or where human decision making is required. Good decision policies are difficult to formulate. Current methods of obtaining such policies are also difficult to apply. Furthermore, when building a model, especially when developing a rapid prototype, parameters and data are often rough estimates to the actual values. Thus, if policies are to be designed under such situations, the policies will have to be robust against ambiguity and uncertainty in order to be useful. The ability of A.I techniques to synthesize a good and robust decision policy and to hedge against uncertainty will be tested. 1.4 Scope of Study For this study, a dynamic hypothesis of the system structure is first proposed. This prototype model will be built using the traditional system dynamics approach. The results from this prototype will then be compared with available historical annual data. The annual data is from a certain Country X, which shall not be named due to confidentiality issues. When unavailable, data or parameters needed in the model are assumed if necessary. Simulations, sensitivity analysis and scenario 5 analyses can then be carried out to study the system behaviour and possibly answer some of the questions raised. Next, A.I. techniques will be applied to a section of the model where there is insufficient knowledge about the system structure or human decision making is required. The results will be tested and evaluated to see if A.I techniques can be incorporated with system dynamics. Lastly, A.I techniques will be applied to obtain a decision policy for the model. This decision policy will be tested for effectiveness and also robustness to uncertainty and noise. 1.5 Organisation of Thesis The thesis consists of six chapters. The outline of the chapters is as follows: Chapter 1 serves as an introductory text to the research project. The background related to the research study is first described. Next, the related problem being studied is stated. The objectives of the research project are then articulated. The scope of the research work conducted is presented. Lastly, the organisation of the thesis is outlined to inform the reader of the topics covered in the following chapters. In Chapter 2, a literature review of past related research work is conducted. Manpower supply and demand is viewed from different perspectives. Then, human behaviour within the system is studied. Lastly, prospective A.I. techniques that can be applied are explored. 6 Chapter 3 deals with the building of a prototype model using a system dynamics approach. Firstly, a hypothesis of the system structure is proposed. Then, the hypothesis will be validated by the comparison of the results generated by the prototype model and historical data. After this, scenario analyses and discussion are carried out. The prototype model is then tested for its sensitivity to its assumed model parameters. In Chapter 4, an A.I technique is applied to help the modelling process. Fuzzy expert system is used to replace the decision rule of a generic supply chain model as a form of validation of the approach. Various attributes of the fuzzy expert system are discussed. A particle swarm optimisation is carried out to obtain a best fit with respect to historical data. Next, the same fuzzy expert system approach is used to mimic a policy maker deciding the engineering college admission rate. The results of the fuzzy expert system are then discussed and analysed. Lastly, the chapter also attempts to use fuzzy logic to mimic the forecasting models traditionally used in system dynamics methodology. A fuzzy expert system will be applied to see whether it is able to replicate the behaviour and replace some of the common forecasting models. Chapter 5 builds on Chapter 4. An attempt to synthesise an optimal decision policy using fuzzy logic was conducted. This decision policy will be based on a hypothetical policy lever and should allow policy makers to achieve their policy objectives while maintaining robustness to noise and uncertainty. Chapter 6 presents a conclusion to the research project. A summary of the research objectives, the activities carried out and the results obtained was 7 provided. The limitations faced by the study were discussed. Then, contributions made by the research project were noted. Lastly, further research work pertaining to the research project was suggested. Chapter 2 Literature Review This chapter reviews relevant work pertaining to the project. It covers the different perspectives concerning manpower supply and demand, the human element within the system and some possible A.I. techniques that can be applied. 2.1 Perspectives related to Manpower Supply and Demand 2.1.1 Economics Perspective Economists seek to explain the sufficiency of manpower in different sectors and its effect on economic development and growth. Some of the common theories or models are the Theory of Markets (Toutkoushian, 2005), Cobweb model (Freeman, 1971, 1975, 1976), Growth Theories (Solow, 1956) and the Leontief Input-Output Model (Brody, 1970). The economics models on manpower supply and demand are extensively studied and well established. Thus, they provide the basic framework of our understanding and knowledge on the system. However, it should be noted that most economic models fail to address adequately the regenerative loops that make up an economic system (Forrester, 2003). Furthermore, most of the models based on mathematical theory are not sophisticated enough to describe explicit solutions to real world problems. Linearity is often assumed to model a system whose essential characteristics arise from its non-linear properties. For reasons such as 8 9 these, most economic models are in fact inadequate to address our deepest and most profound questions about the system. Also, economic theories and models focus on the supply and demand for labour with equilibrium being determined by hard facts such as wages and growth. They neglect to address the possible interactions between individual actors in the system, for example students and schools, graduating students and career choice, foreign workers and local workforce etc. These interactions can be of specific interest to policy makers and other stakeholders who wish to achieve their individual agenda. 2.1.2 Attractiveness Perspective The attractiveness of a profession plays an important role in determining the supply of manpower to the workforce. This is especially true for highly skilled labour where the opportunity cost of training is high. The supply of manpower is affected by two main factors, the retention of manpower and students joining the labour pool. Firstly, it is a widely accepted fact that higher wages result in lower worker turnover and job mobility (Barth and Dale-Olsen, 1999). Next, a student‟s choice of college major is essentially based on his perceived probability of success, the predicted earnings of graduated students in all majors and the student‟s expected earnings if he fails to complete the college program. (Montmarquette, Cannings and Mahseredjian, 2002) It was also discovered that the impact of expected earnings on choice of college major varies according to gender and race. 10 2.1.3 System Dynamics Perspective Some system dynamics models for manpower supply and demand have been built and can be used as a guide to the important parameters within the model. Park, Yeon and Kim (2008) built a manpower planning model for the information security industry of Korea. In their paper, they have built a hypothesized manpower demand-supply system using system dynamics. They then tested current manpower policies implemented in Korea. Following this, they analysed the problem of imbalance in manpower supply and demand in the information security industry. While they admitted to having neglected the quantitative aspect of the model, they have managed to identify the likely trends caused by the manpower policies. From this, they were able to provide better insight into the structure of the manpower system and thus propose some solutions to the problem. 2.2 The Human Factor Humans are a very important element in any social system. However, because humans display judgement and somewhat more sophisticated thinking, it is difficult to incorporate human behaviour into any model. 2.2.1 Traditional System Dynamics Method According to Sterman (2000), the structure of all models consists of two parts: assumptions about the physical environment on one hand and assumptions about the decision processes of the agents who operate in those structures on the other. He defines the decision processes of the agents as the decision rules that 11 determine the behaviour of the actors in the system. These assumptions about human behaviour will describe the way in which people respond to different situations. Decision rules are the policies and protocols specifying how the decision maker processes available information. They do not necessarily use all available information, but use information according to the mental models of the decision maker. The decision rules in a model embody assumptions about the degree of rationality of the decision makers and decision making process, ranging from simple-minded rules to total rationality. This approach captures human decision making in a form of a table function. However, this method is often unsatisfactory and depends a lot on the modeller‟s judgement and experience. Furthermore, this method is not feasible and inaccurate when the structure is partially or not understood. 2.2.2 Artificial Intelligence Tools Artificial Intelligence (A.I.) has been defined as the study and design of intelligent agents (Poole, Mackworth and Goebel, 1998). These intelligent agents should ideally be able to perceive its environment and carry out necessary actions which maximise its chances of success (Russell and Norvig, 2003). It may be interesting to tap into existing A.I. tools to mimic human decision making in a system dynamics model. Furthermore, it is possible to apply A.I. methods to substitute for parts of the model where there is insufficient knowledge. Modellers usually face two different types of uncertainties, namely “Parameter Uncertainty” and “Structural Uncertainty”. Parameter Uncertainty is uncertainty about the exact values of the parameters of the mathematical model describing the 12 system. This is especially so for complex systems where a modeller often have to rely on incomplete and/or inaccurate data. Structural Uncertainty is uncertainty about the structure of the system being modelled. This could be due to stakeholders‟ reluctance to share their mental models and/or the structure is just too complex that no one knows for sure how it looks. A modeller can try to overcome this by proposing various hypotheses on the structure of the system. However, this trial and error method is often time consuming and impractical. A.I. presents a myriad of tools that can be solutions to the uncertainty problem. Search and optimisation tools can be used to overcome parameter uncertainty by allowing us to choose the best option in the solution space. Similarly, tools like neural networks can be used to act as black boxes when a subsystem of the model is not fully understood. Some of the relevant A.I. tools will be presented in more detail in the subsequent subsections. 2.2.2.1 Fuzzy Expert Systems An expert system is a computer system designed to mimic the problem solving nature of a human expert. A heuristic is an educated guess based on experience that simplifies and limits a search for solutions in applications which are poorly understood. In general, a human expert uses a blend of heuristics, logic and knowledge to solve a problem. Thus an expert system which mimics a human expert will allow us to solve problems which are boggled down by uncertainty or in situations where conclusions cannot be easily predicted (Gallacher, 1989). The two main components of an expert system are the Knowledge Base and Inference Engine. While an expert system has its uses and strengths, it is often 13 limited by its lack of „common sense‟. In other words, it is unable to recognise an exceptional case or know when to bend the rules unlike a human expert. The system closely models the way people perceives and reasons when faced with a problem as people tend to think qualitatively rather than quantitatively. Ghazanfari, Alizadeh and Jafari (2002) used Fuzzy Expert Systems as an alternative method for the analysis of the causal loop in a system dynamics model. They proposed the use of a fuzzy expert system to represent parametric uncertainty in system dynamics models, especially human-related parameters which have imprecise behaviours and cannot be stated precisely. Next, Kunsch and Springael (2006) demonstrated the use of fuzzy reasoning techniques as a means to aggregate external data from different sources with various credibility levels driving the model. This was used to account for dynamic parameter uncertainty within the model. 2.2.2.2 Neural Networks Neural networks are software models inspired by biological neural networks. Neural network models are built by using inductive techniques and thus need a lot of data in order to train them correctly. Depending on the quality of the data, they are able to simulate system behaviour rather accurately but do not require any understanding from the modeller. As a result, a neural network model is most useful where it is possible to specify the inputs and outputs but difficult to define analytically a relationship between them (Nebot et al., 1995). Neural networks have been shown mathematically to be universal approximators (Cybenko, 1989). Furthermore, they are inherently non-linear (Wasserman, 1989) 14 and estimate non-linear systems well (White and Gallant, 1992). Hence neural networks are able to overcome the limitations of tradition models such as linear regression models. Thus when given sufficient hidden units, they will always find a mapping between any set of independent and dependent variables. However, this results in a significant disadvantage: Neural networks may find associations in places where there are not (Ceccatto, Navone and Waelbroeck, 1997). 2.2.2.3 Fuzzy Inductive Reasoning Fuzzy Inductive Reasoning (FIR) methodology is rather similar to neural networks. It has the ability to model systems that are not well understood or where the system‟s characteristics are not known. As it is an inductive method, FIR also requires an adequate amount of data in order to train the model correctly. As with neural networks, FIR does not allow the modeller to understand the underlying system structure and adopts a „black-box‟ approach to the system it is modelling. FIR is a qualitative technique and hence requires a data fuzzification step before the model can be built. A significant advantage that FIR methodology has over neural networks is that it does not generate models that are not justifiable from the given data (Nebot, Cellier and Linkens, 1995). FIR models contain information about the likelihood of any particular state transition. This acts like an inbuilt model validation mechanism such that forecasting by the model stops if the likelihood of a particular state drops below a level specified by the modeller. A new methodology using System Dynamics and Fuzzy Inductive Reasoning was proposed by Moorthy et al. (1998) in their model used to predict U.S. food 15 demand in the 20th century. They proposed the use of FIR because it can be easily embedded into System Dynamics models. Data for level variables are more readily available than rate variables which are needed in model building using traditional system dynamics methodology. Thus, FIR can be used to predict level variables directly instead. This is a black box approach where the model predicts each variable from past values without knowing the underlying relationships and equations between them. 2.3 Modelling: A Synthesis of Ideas Existing system dynamics and economics models on manpower planning have been reviewed. It is possible to draw inspiration from them and to calibrate and change these models to suit the purposes of the project. Humans are very much a part of real world systems and human decision making play a key role in the behaviour of such systems. Hence, modelling human decision making is of paramount importance when building a model. Traditional system dynamics method of modelling human decision making is somewhat unrealistic and modelling the system itself is infeasible when there is insufficient knowledge about the system. Artificial intelligence tools may aid us in our efforts during such situations. Depending on the problem faced, different tools can be applied for us to circumvent the problem. The application of A.I. tools shows a lot of promise and potential in helping the modelling process become smoother or more constructive. Chapter 3 Prototype Model This chapter details the building of the prototype model using system dynamics methodology. A hypothesis of the system structure for engineering manpower supply and demand is proposed and its results are compared with historical data. Following this, various scenarios are envisaged and for each scenario, the system behaviour is observed and analysed. As the world develops, countries will be looking to become a knowledge economy and engineers, as knowledge workers, will have a large role in this. Thus to achieve this goal, governments have to ensure that there is a balance between industry demand of engineers and the supply of engineers in the labour pool. This would mean training sufficient engineers in local tertiary education institutions. However, in recent years, there has been an increasing trend of engineering graduates joining other professions. This phenomenon will lead to an imbalance between supply and demand, where there are insufficient local engineers to meet industry needs and foreign engineers would have to be brought in to fill in the gap. A large influx of foreign engineers threatens the domestic social fabric and raises public discontentment. Next, the resources used to train a “leaked” engineer could have been put to more efficient use by training him/her in his final chosen profession, instead of engineering. Moreover, engineering manpower planning becomes even more delicate as it is difficult to predict the number of engineering graduates becoming engineers. If there are too few engineering graduates in the pipeline, then this increases the dependence on foreign engineers in the future and exacerbates its associated problems. On the other hand, if there are too many 16 17 engineering graduates, then this is inefficient and the resources could have been put to better use. In light of the above problem and issues, it is important to understand the system in order to arrest the potential consequences. However, the engineering manpower system is complex, involving the interaction between many subsystems. The Figure 3.1 shows a possible simplified representation of the engineering manpower system. The system entails social, labour and education aspects. Hence, systems thinking and systems dynamics will be used to make sense of this complex system. Figure 3. 1 Simplified Representation of Engineering Manpower System 18 3.1 Hypothesis The hypothesised system structure can be explained using a causal loop diagram. The diagram will be expanded incrementally as different market dynamics are considered in the system. 3.1.1 Manpower Adjustment The dynamics of engineering manpower adjustment between demand and supply can be described as shown in the Figure 3.2. Figure 3. 2 Causal Loop Diagram for Manpower Adjustment Engineering Manpower Demand can be understood as the number of engineers required by the market at a given point in time. This variable is assumed to be exogenous in the model. Correspondingly, Engineering Manpower Supply is the number of engineers which is employed at a given point in time. The Engineering Manpower Gap is the difference between Engineering Manpower Demand and Engineering Manpower Supply. This gap is interpreted as the shortfall/excess in the number of engineers. Also, it is unrealistic to expect employers in the market to be aware of the exact manpower gap at any given time. Hence, the variable Perceived Manpower Gap is to model the difference between employers‟ perceived engineering manpower gap and the actual gap. Employers will act on 19 their perceived engineering manpower gap so as to adjust for manpower accordingly to their needs. For example, when an employer thinks that he needs more engineers, he will make the necessary adjustment to his company‟s HR policies to hire more engineers. This adjustment for engineering manpower can then be translated into the opening/closures of engineering Job Vacancies available in the market. 3.1.2 Foreign Engineers In an increasingly globalised work, the presence of foreign workers is becoming more and more commonplace. This trend may have either beneficial or detrimental effects for a local economy. Thus, it is important to include this presence of Foreign Engineers in the system structure in order to study the possible effects. The causal loop diagram including foreign engineers is as shown in Figure 3.3. Figure 3. 3 Causal Loop Diagram for Foreign Engineers 20 Note that Foreign Engineers are noted as EP Engineers in the diagram. EP means employment pass, which foreign engineers are required to obtain before being allowed to enter the labour pool. The two terms, Foreign Engineers and EP Engineers, will be used interchangeably in this paper. From above, the number of job vacancies will be filled by EP Engineers. The number of EP Engineers that is hired depends on the Average Time Needed to Hire an EP Engineer. This can be interpreted as the average amount of time needed by an employer to find a foreign engineer to fill a job opening. If the average time is short, it is easy for employers to fill a job opening with foreign engineers. Next, the number of EP Engineers depends on the Average EP Employment Duration as the shorter the duration, the higher the turnover rate amongst foreign engineers is. The number of EP Engineers can be counted as part of the Engineering Manpower Supply. Also, foreign engineers can choose to be assimilated into the local resident workforce and become a resident. EP Engineers to Resident Engineer Fraction is the fraction of foreign engineers who chooses to become resident engineers and EP Engineers becoming Resident Engineers is the number of foreign engineers who chooses to do so at a given point in time. The first balancing feedback loop, the EP Engineer Hiring Loop (B1), in the system can be seen in the diagram. It is a feedback loop that seeks to close the engineering manpower gap by increasing the number of foreign engineers. 3.1.3 Resident Engineers The indigenous labour pool is an important part of any manpower system. This is no exception for engineering manpower. 21 Figure 3. 4 Causal Loop Diagram for Resident Engineers Resident Engineers is the number of resident engineers currently employed in the system. The number of resident engineers is a part of the Engineering Manpower Supply. The number of resident engineers can be increased from two sources: either Resident Jobseekers who found an Engineering Job or EP Engineers becoming Resident Engineers. Resident Jobseekers can be defined as engineering trained individuals who are looking for engineering jobs. Resident engineers can choose to leave their jobs and join the Resident Jobseekers pool or they may reach retiring age and leave the system entirely. The variables which control these are Average Resident Engineer Job Duration which is the amount of time a resident engineer stays in the same job before quitting and Average Resident Engineer Career Duration which is the career length before a resident engineer retires. The rate at which resident jobseekers become employed is dependent on the Average 22 Time Need to Hire a Resident Engineer. It has the same meaning as the case for foreign engineers. As not all Resident Jobseekers looking for an engineering job will eventually find one, they can be “leaked” out to other industries if they choose to do so and hence the variable Leakage to other Professions. It should be noted that there is a second balancing feedback loop. The Resident Engineering Hiring Loop (B2) is a feedback loop that seeks to close the engineering manpower gap by increasing the number of resident engineers. 3.1.4 Engineer Education The resident jobseekers pool is supplemented by resident engineers who had left their jobs or by fresh engineering graduates. Figure 3. 5 Causal Loop Diagram for Engineer Education The Engineering Rate is the admission rate of students per year into the Engineering Student cohort. The engineering curriculum is usually four years and thus there is a delay mark on the arrow to indicate this. The Graduating Students will then join the Resident Jobseekers pool after graduation. 3.1.5 Engineering Wage Employers may choose to adjust the wages for engineers according to their perceived manpower gap. 23 Figure 3. 6 Causal Loop Diagram for Engineering Wage The Engineering Wage is compared with Non-Engineering Wage. Depending on how attractive the Engineering Wage is relative to Non-Engineering Wage, the leakage to other professions will be affected. If Engineering Wage is very attractive as compared to Non-Engineering Wage, then the leakage will be small. The Engineering Wage Loop (B3) is the third balancing feedback loop observed in the system. The loop serves to close the Engineering Manpower Gap by increasing Engineering Wages and thus leading to less leakage. With less leakage, the number of resident jobseekers increases. 3.1.6 Causal Loop Diagram The complete causal loop diagram which incorporates all the sections above is shown in Figure 3.7. 24 Figure 3. 7 Complete Causal Loop Diagram 3.2 Simulation Model Based on the above hypothesis, a simulation model of the Engineering Manpower Supply and Demand System was built on iThink. There are six subsystems and they are:  Manpower Adjustment Subsystem (Figure A.1)  Job Vacancies Subsystem (Figure A.2)  Wage Adjustment Subsystem (Figure A.3)  Resident Engineers Subsystem (Figure A.4)  EP Engineers Subsystem (Figure A.5)  Engineer Education Subsystem (Figure A.6) 25 The detailed stock and flow diagrams for each subsystem can be found in Appendix A. The equations and relationships between each variable used in iThink can also be found in Appendix B. 3.2.1 Key Assumptions Some of the key assumptions made are:  Engineering Manpower Demand, Engineering Rate and Non-Engineering Wages are considered exogenous in the model.  Aggregation of engineers across different age groups, industries and pay scales.  Source of foreign engineers is unlimited.  Model parameters are assumed when data is not available. The key model parameters are shown in the Table 3.1. Average Resident Engineer Career Duration Average Resident Engineer Job Duration Average EP Engineer Employment Duration Average Time Needed to Hire a Resident Engineer Average Time Needed to Hire a EP Engineer Percentage of EP engineers who becomes Resident Engineer 45 years 6 years 3 years 2 months 6 months 10% Table 3. 1 Key Model Parameters 3.2.2 Model Validation Annual data for the number of Resident Engineers, EP Engineers, Engineering Rate and Engineering Wage are available from 2000-2008. Using the model and 26 parameters as described above, the results from the model output can be compared with the historical data from Country X. The number of resident engineers, foreign engineers and amount of engineering wage are compared in the Figures 3.8 and 3.9. It is possible to observe that the simulated data follows the historical data trends rather closely. Hence, the prototype model can be concluded to be somewhat accurate on an aggregate level. A finer and more detailed model may be needed to explain the smaller variations in historical data. Number of Engineers 2000 2001 2002 2003 Simulated Resident Engineers 2004 Historical Resident Engineers 2005 2006 2007 Simulated EP Engineers 2008 Historical EP Engineers Figure 3. 8 Comparison of Historical Number of Engineers vs. Simulated Number of Engineers 27 Engineering Wage 2000 2001 2002 2003 2004 Simulated Engineering Wages 2005 2006 2007 2008 Historical Engineering Wages Figure 3. 9 Comparison of Historical Engineering Wage and Simulated Engineering Wage 3.3 Scenario Analysis and Simulations Policy levers are the variables which a policy maker has control over. Some of the possible policy levers identified in the system are:  Employment duration of foreign engineers (regulation of contract duration etc.)  Hiring rate of foreign engineers (regulation of hiring of foreign engineers by making it easier or more difficult)  Engineering college admission rate (setting the number of students admitted into engineering) Different scenarios can be played out by varying these policy levers and observing their effect on system behaviour. 28 3.3.1 Employment Duration of Foreign Engineers The employment duration of foreign engineers is reflected in the model by the variable Average Employment Duration of EP Engineers and was assumed to be three years. Simulations where the Average Employment Duration of EP Engineers is shortened are carried out. The effect on key variables is shown in Figure 3.10 to Figure 3.13. From the graphs, it can be observed that with a decrease in Average Employment Duration of EP Engineers, the Engineering Wage increases, the Leakage Fraction decreases, and the number of Resident Engineers increases. Shortening the employment duration of foreign engineers decreases the number of employed foreign engineers within the system as more are forced to leave while the hiring rate of foreign engineers remain the same. This results in a larger engineering manpower gap since the overall supply of engineers decreases. To close this gap, employers will respond by increasing the wages of engineers to attract more to join. Fewer resident engineers will be leaked to other industries and will join the resident engineer stock as they are enticed by the increased pay. 29 Average EP Engineer Employment Duration 3.1 2.9 2.7 2.5 2.3 2.1 1.9 2008 2008.5 2009 2009.5 2010 2010.5 2011 Base Scenario(No Change) Simulation 1 Simulation 2 Simulation 3 2011.5 2012 Simulation 4 Figure 3. 10 Average EP Engineer Employment Duration (Decrement of 0.1 years) Engineering Wage 2008 2008.5 2009 2009.5 2010 2010.5 Base Scenario(No Change) Simulation 1 Simulation 2 Simulation 3 2011 2011.5 2012 Simulation 4 Figure 3. 11 Engineering Wage w.r.t variations in Average EP Engineer Employment Duration 30 Leakage Fraction 2008 2008.5 2009 2009.5 2010 2010.5 Base Scenario(No Change) Simulation 1 Simulation 2 Simulation 3 2011 2011.5 2012 Simulation 4 Figure 3. 12 Leakage Fraction w.r.t Variations in Average EP Engineer Employment Duration Resident Engineers 2008 2008.5 2009 2009.5 2010 2010.5 Base Scenario(No Change) Simulation 1 Simulation 2 Simulation 3 2011 2011.5 2012 Simulation 4 Figure 3. 13 Resident Engineers w.r.t Variations in Average EP Engineer Employment Duration 31 3.3.2 Hiring Rate of Foreign Engineers The hiring rate of foreign engineers is reflected in the model by the variable Average Time Needed to hire an EP Engineer and was assumed to be six months (0.5 years). Simulations where the Average Time Need to hire an EP Engineer is lengthened are carried out. The effect on key variables is shown in Figures 3.14 to 3.17. From the graphs, it can be observed that with an increase in Average Time Needed to Hire an EP Engineer, the Engineering Wage increases, the Leakage Fraction decreases, and the number of Resident Engineers increases. A slower hiring rate decreases the number of employed foreign engineers within the system as less join the stock of foreign engineers in the same period of time. This results in a larger engineering manpower gap since the overall supply of engineers decreases. To close this gap, employers will respond by increasing the wages of engineers to attract more to join. Fewer resident engineers will be leaked to other industries and will join the resident engineer stock as they are enticed by the increased pay. 32 Average Time needed to hire an EP Engineer 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 2008 2008.5 2009 2009.5 2010 2010.5 2011 Base Scenario(No Change) Simulation 1 Simulation 2 Simulation 3 Simulation 4 Simulation 5 2011.5 2012 Figure 3. 14 Average Time Needed to Hire an EP Engineer (Increments of 0.1 year) Engineering Wage 2008 2008.5 2009 2009.5 2010 2010.5 2011 Base Scenario(No Change) Simulation 1 Simulation 2 Simulation 3 Simulation 4 Simulation 5 2011.5 2012 Figure 3. 15 Engineer Wages w.r.t variations in Average Time needed to hire an EP Engineer 33 Leakage Fraction 2008 2008.5 2009 2009.5 2010 2010.5 Base Scenario(No Change) Simulation 1 Simulation 2 Simulation 3 Simulation 4 Simulation 5 2011 2011.5 2012 Figure 3. 16 Leakage Fraction w.r.t variations in Average Time needed to hire an EP Engineer Resident Engineers 2008 2008.5 2009 2009.5 2010 2010.5 Base Scenario(No Change) Simulation 1 Simulation 2 Simulation 3 Simulation 4 Simulation 5 2011 2011.5 2012 Figure 3. 17 Number of Resident Engineers w.r.t variations in Average Time needed to hire an EP Engineer 34 3.3.3 Engineering College Admission Rate The Engineering College Admission Rate is reflected in the model by the variable Engineering Rate. During the model validation phase, the Engineering Rate was assumed to be exogenous and set using annual historical data. For the scenario analysis, it is assumed that the Engineering Manpower Demand increases with a ramp increase of 5000/year for six years (from 2009 to 2015). During this period, varying levels of step changes are implemented to the Engineering Rate in order to observe the effects on system behaviour. The levels of step increase for each simulation are detailed as follows:  Simulation 1: Step Increase of 5000 in Engineering Rate to match the increase in Engineering Manpower Demand.  Simulation 2: Step Increase of 2500 in Engineering Rate to match half the increase in Engineering Manpower Demand.  Simulation 3: Step Decrease of 1250 in Engineering Rate to see the effects of decreased supply.  Simulation 4: Step Decrease of 2500 in Engineering Rate to see the effects of decreased supply. The effect on key variables is shown in Figures 3.18 to 3.22. 35 Engineering Manpower Demand 2008 2013 Base Scenario(No Change) Simulation 3 2018 2023 Simulation 1 Simulation 4 2028 Simulation 2 Figure 3. 18 Engineering Manpower Demand (Ramp Increase) Input Engineering Rate 2008 2013 Base Scenario(No Change) Simulation 3 2018 Simulation 1 Simulation 4 2023 2028 Simulation 2 Figure 3. 19 Engineering Rate with varying levels of step change Engineering Wage 2008 2013 Base Scenario(No Change) Simulation 2 Simulation 4 2018 2023 Simulation 1 Simulation 3 2028 Figure 3. 20 Engineering Wage w.r.t variations in Engineering Rate 36 Resident Engineers 2008 2013 2018 2023 Base Scenario(No Change) Simulation 1 Simulation 2 Simulation 3 2028 Simulation 4 Figure 3. 21 Number of Resident Engineers w.r.t variations in Engineering Rate Resident Jobseekers 2008 2013 Base Scenario(No Change) Simulation 2 2018 2023 Simulation 1 2028 Simulation 3 Simulation 4 Figure 3. 22 Number of Resident Jobseekers w.r.t variations in Engineering Rate From the graphs, it can be observed that the effects of changing the engineering college admission rates only occurred after approximately four years. Increasing the engineering college admission rate above the base scenario had no effect on 37 the engineering wage while decreasing the engineering college admission rate increases the engineering wage. The number of resident engineers increases as the engineering college admission rate increases. However, there is no difference between the number of resident engineers in simulations 1 and 2. This suggests that increasing the engineering college admission rate will only increase the number of resident engineers up to a certain extent. Next, the number of resident jobseekers increases as the engineering college admission rate increases. It should be noted that while simulations 1 and 2 have no difference in the number of resident engineers and engineering wage, the number of resident jobseekers is vastly more for Simulation 1 than for Simulation 2. Hence, it may be concluded that, after a certain extent, increasing the engineering college admission rate will only increase the number of resident jobseekers. Also, the number of resident jobseekers continued to increase even after the increase in engineering college admission rate is discontinued. The engineering college admission rate ultimately controls the stock of resident jobseekers looking for engineering jobs within the system. When there is a shortage of resident jobseekers looking for engineering jobs, employers will respond by increasing the engineering wages in the short term. However, if this shortage is persistent, they will have to resort to hiring foreign engineers to plug the shortage. 38 3.4 Discussion and Analysis The results from the scenario analyses can be analysed from two policy aspects, namely Manpower policies and Education policies. Firstly, by retarding the hiring rate of foreign engineers (i.e. increasing the Average Time Needed to Hire an EP Engineer) or shortening the duration a foreign engineer can work locally (i.e. decreasing the Average EP Engineer Employment Duration), the engineering wage and proportion of resident engineers in the engineering labour force increases. Leakage to other industries also decreases. This indicates the possibility that an excessive influx of foreign engineers does indeed depress the wages of local engineers and hence making the profession less attractive. This may be further accentuated if the foreign engineers draw a lesser wage. Furthermore, if this is true, the long term technical ability of the resident population may be lowered due to the decreased attractiveness of engineering. Hence, it is important to balance the need for foreign engineers to supplement the local supply of engineers and also the long term survivability and attractiveness of the engineering profession amongst residents. Secondly, it can be observed that the engineering college admission rate should not be set by simply matching the increase in engineering manpower demand as this would lead to excessive inefficiency (i.e. too many resident jobseekers). Also, there is a need to take into account the delay and pipeline effects as changes in the engineering college admission rate today will not affect the system until several years later. Thus, there is a need to strike a balance between increasing the number of resident engineers and also decreasing the number of resident 39 jobseekers. One possible action going forward could be to identify the factors that should be considered when setting the engineering college admission rate. Following this, possible policy guidelines could be created so as to decrease unemployment while as the same time matching the demand for engineers. Although it is plausible that the observations made above are true and realistic, it is also true that the prototype model suffers from a lack of historical data. With only data from 2000-2008 available, it is difficult to conclude that the model is accurate for the more distant past and that it is able to simulate possible events confidently into the future. Next, the prototype model is an aggregate model of all engineers across age groups, industries, job positions and pay scales. It is possible that the system behaviour may differ on more granular level. Similarly, results from a more in-depth model may be even more pertinent for policy makers to micromanage the different industries or types of engineers. Hence, it should be noted that this prototype model is merely a work-in-progress and should be improved upon by interested individuals. 3.5 Sensitivity Analysis Due to the number of key model parameters assumed in the model, it is essential to carry out sensitivity analysis on them so as to validate the assumed parameters and the associated system behaviour. 3.5.1 One Way Sensitivity Analysis A One Way Sensitivity Analysis is first carried out by varying the key model parameters between high and low values one at a time while keeping the rest of 40 the model constant at their nominal values. The Table 3.2 shows the low, nominal values for each of the key model parameters. Low Value Average Resident Engineer Career Duration Average Resident Engineer Job Duration Average EP Engineer Employment Duration Average Time Needed to Hire a Resident Engineer Average Time Needed to Hire a EP Engineer Percentage of EP engineers who becomes Resident Engineer 42.5 years 3 years 2 years 1 month 3 months 5% Nominal Value 45 years 6 years 3 years 2 months 6 months 10% High Value 47.5 years 9 years 4 years 4 months 9 months 20% Table 3. 2 Low, Nominal and High Values for Key Model Parameters The graphs for Resident Engineers, EP Engineers, Engineering Wage, Leakage Fraction and Resident Jobseekers are plot for all six key model parameters, using the different values in the sensitivity analysis. These graphs can be found in Appendix C. Some observations that can be made from the graphs are:- Some parameters are more sensitive than others and affect more aspects of the results 41 - Average Time to Hire a EP Engineer and Average Time to Hire a Resident Engineer affects both the composition of the labour force (i.e. the number of resident engineers and foreign engineers) and the labour market in terms of Engineering Wage and Leakage Rate. - Other parameters only affect the composition of the labour force. - Average Resident Engineer Career Duration is insensitive within the defined range. For system dynamics models, a One Way Sensitivity Analysis is not necessary the best way for sensitivity analysis as the models are usually non-linear. Hence it is entirely possible that the model may vary more strongly within the defined range of values than at the defined extremes. For this reason, a Monte Carlo sensitivity analysis is carried out to determine the confidence intervals of the prototype model. 3.5.2 Monte Carlo Simulation For the Monte Carlo simulation, the key model parameters are assumed to be uniformly distributed between the ranges shown in Table 3.2. This is because the parameters are usually bounded and their actual distribution unknown. A total of two thousand simulations were carried out. The 50%, 75% and 95% confidence bands for Resident Engineers, EP Engineers, Engineering Wage, Leakage Fraction and Resident Jobseekers are shown in Figures 3.23 and 3.24. 42 Figure 3. 23 Confidence bands for Resident Engineers Figure 3. 24 Confidence bands for EP Engineers 43 Figure 3. 25 Confidence bands for Resident Jobseekers Figure 3. 26 Confidence bands for Engineering Wage 44 Figure 3. 27 Confidence bands for Leakage Fraction From the above graphs, it can be observed that Resident Engineers and EP Engineers are more sensitive to changes in the key model parameters. Next, there is less uncertainty about the Resident Jobseekers, Leakage Rate and Engineering Wage as they have much smaller confidence bands. Despite the uncertainty in the variables, the general trends in the system behaviour for each of the variable are somewhat regular and do not change. This shows that the system behaviour is insensitive to changes to the assumed key model parameters. 3.6 Conclusion In this chapter, a hypothesis for the Engineering Manpower Supply and Demand system was proposed and described. A prototype model based on the hypothesis was built. The prototype model was validated using historical data. 45 Next, scenario analyses were carried out for some identified policy levers. The possible impacts were observed and analysed. Policy considerations concerning these impacts and policy levers were discussed. A One Way Sensitivity Analysis and Monte Carlo simulation were carried out to determine the sensitivity of the system to changes in the assumed key model parameters. It was found that the general behaviour of the prototype model is insensitive to changes in the key model parameters. Chapter 4 Systems Dynamics Modelling with A.I. techniques In this chapter, A.I. tools will be applied in order to aid the modelling process. In many situations, there is neither information nor data on parts of the model structure. For instance, in the engineering manpower planning model, the structure determining the supply of engineers is unknown. Thus, in this chapter, fuzzy logic is proposed as a way to imitate decision policy. First, the method is tested on a simple text book example. Next, the same technique is applied on the engineering manpower case study. Lastly, fuzzy expert systems will be tested for its ability to replace forecasting models. 4.1 Generic Supply Chain Model – Engineering College Admission Rate A generic system dynamics supply chain model was adapted to simulate the decision rule for the engineering college admission rate. The model is shown in Figure 4.1. The equations used in the model can be found in Appendix D. The rectangle in the figure shows the decision rule of the supply chain model. The goal is to replace this rectangle entirely with a fuzzy expert system. 47 Decision Rule Figure 4. 1 Supply Chain Model (Engineering College Admission Rate) From Figure 4.1, the inputs of the decision rule are:  Resident Hiring Rate  Resident Jobseekers  Engineering Student Cohort The output is the Engineering Admission Rate. Percentage changes will be used in order to ensure that the fuzzy expert system is robust in relative terms. The number of Job Vacancies is considered to be exogenous and is a proxy to the engineering manpower demand. 48 4.1.1 Simulation of Data Since engineering manpower demand is not prone to strong variations, its percentage change can be assumed to be between -10% and 10%. Hence, the percentage change in Job Vacancies was assumed to be uniformly distributed within this range. A hundred random disturbances was generated and applied to the system. System behaviour in response to different disturbances to the system can then be observed. 4.1.2 Inputs and Outputs of Fuzzy Expert System Out of the hundred random disturbances, the data points corresponding to the first ninety disturbances will be used for the calibration of the fuzzy expert system. The Figure 4.2 shows the percentage changes in the inputs and output for the first two hundred data points. Note that the sign of percentage change in resident jobseekers was changed so as to have the same sign for all inputs and outputs. The percentage change in Engineering Admission Rate correlates well with the three inputs. Figures for the entire training data can be found in Appendix E. 49 40 30 20 10 0 0 50 100 150 200 -10 -20 -30 negative %Change (Resident Jobseekers) %Change (Resident Hire Rate) %Change (Engineering Admission Rate) %Change (Resident Student Cohort) -40 Figure 4. 2 Percentage Changes (First 200 data points) Only the percentage change in resident jobseekers and resident hiring rate will be used as inputs to the fuzzy expert system so as not to double count certain influences on system behaviour. 4.1.3 Fuzzy Expert System - Membership Functions The inputs and output are separated into different levels as follows:  %Change in Hiring Rate: Negative, Mild, Positive  %Change in Resident Jobseekers: Negative, Mild, Positive  %Change in Engineering Admission Rate: Large Negative, Negative, Mild, Positive, Large Positive 50 There are five levels for the percentage change in engineering admission rate because it tends to swing wildly at the beginning of every disturbance. Thus, to simulate this behaviour, an extra level is most likely required. 4.1.4 Fuzzy Expert System - Rule base There are nine rules as shown in the Table 4.1. %Change in Jobseekers %Change Negative Mild Positive Negative Large Negative Mild Mild Mild Negative Mild Positive Positive Mild Mild Large Positive In Hiring Rate Table 4. 1 Rule Base for Fuzzy Expert System (Supply Chain Model) 4.1.5 Optimisation and Calibration of Fuzzy Expert System Particle Swarm Optimisation (PSO) is used to calibrate the fuzzy expert system. The following membership functions in Figure 4.3 were obtained. The membership function parameters can be found in Appendix F. The engineering college admission rate generated by the fuzzy expert system and the supply chain model are compared in Figure 4.12. The RMSE is 29.9 and the percentage RMSE is 5.6%. 51 Figure 4. 3 Membership Functions for Fuzzy Expert System (Supply Chain Model) Engineering Admission Rate 1000 900 800 700 600 500 400 300 200 0 200 400 600 800 Fuzzy Engineering Rate 1000 1200 1400 1600 Simulated Engineering Rate 1800 Figure 4. 4 Fuzzy Expert System vs. Supply Chain Model (Engineering Admission Rate) 52 4.1.6 Validation of Fuzzy Expert System The last remaining ten random disturbances will be used to validate the fuzzy expert system‟s ability to replicate the decision rule. The Figure 4.5 shows the engineering college admission rate generated by the fuzzy expert system compared with the values generated by the supply chain model. 800 Engineering Admission Rate 750 700 650 600 550 500 450 400 0 50 100 150 200 Engineering Admission Rate Fuzzy Engineering Admission Rate Figure 4. 5 Validation of Fuzzy Expert System vs. Supply Chain Model The RMSE is 25.1 and the percentage RMSE is 4.4%. Thus the fuzzy expert system is able to replicate the decision rule of the supply chain model rather accurately. 4.2 Fuzzy Expert System– Engineering College Admission Rate In the prototype model, the engineering college admission rate was assumed to be exogenous. This is because there was insufficient knowledge about the 53 relationships between engineering college admission rate and the rest of the system. ? Figure 4. 6 A Missing Link in Engineering Manpower Model Despite the lack of knowledge, it is possible to try making it endogenous through certain A.I. tools. FIR methodology and neural network models are some techniques that can be considered. However, this would result in a “black box” where the system structure behind the engineering college admission rate remains a mystery. Thus, a fuzzy expert system will be applied. A fuzzy expert system has been shown in the previous Section 4.1 to be able to replicate a decision policy well. The fuzzy expert system will have to mimic a policy maker setting the engineering college admission rate. Due to the nature of the approach, a set of rules regarding the policy maker‟s decision making process will have to be 54 proposed. If a set of rules that allows us to satisfactorily recreate the engineering college admission rate exists, it is plausible that this set of rules is actually reflective of the policy maker‟s mental model. Thus, the fuzzy expert system actually allows the possible discovery of unknown information about the system. There are a few aspects to consider when implementing the fuzzy expert system: The outputs of the fuzzy expert system – What is the output and how does it relate to the rest of the system?  The inputs into the fuzzy expert system – What are the factors to consider when setting the engineering college admission rate?  The membership functions – How to interpret the level or intensity of a given factor and what is the range for this level?  The rule base – How does the different levels for each factor interact to give an output? In order to apply A.I. tools, the model will have to be reconstructed in MATLAB. The diagrams of the model in MATLAB can be found in Appendix G. The MATLAB model is verified to have similar performance to the original model 4.2.1 Output The output of the system is the percentage change in engineering college admission rate. Percentage change is used so that the fuzzy expert system is able to compute in relative terms and not absolute numbers. This will enable it to be more robust with respect to the rest of the system. 55 4.2.2 Inputs The inputs of the fuzzy expert system are a hypothesis of the factors that a policy maker actually looks at when deciding on the engineering college admission rate. The variables available in the system will have to be observed and analysed in order to deduce a suitable hypothesis of the set of inputs. The Figure 4.7 was observed. 30 Percentage Change 20 10 0 2000 -10 2002 2004 2006 2008 Percentage Change in Demand Percentage Change in Engineering Admission Rate Figure 4. 7 Hypothesis of inputs: Percentage change in demand From Figure 4.7, the Percentage Change in Demand (more specifically, engineering manpower demand) is somewhat correlated with the Percentage Change in Engineering Admission Rate. Hence, it was deduced that the Percentage Change in Demand is one of the factors considered. However, this alone is not able to account for the variations in the engineering college admission rate. Thus another input, the Previous Percentage Change in Demand, is proposed. The Previous Percentage Change in Demand can be interpreted as the outlook or “mood” of the policy maker carried over from the previous year. Depending on his mood, the policy maker‟s reaction to a change in engineering manpower demand can be different. 56 4.2.3 Membership Functions To summarise, there are two inputs (Percentage Change in Demand and Mood) and one output (Percentage Change in Engineering Admission Rate) for the fuzzy expert system. They are differentiated into three fuzzy levels each as shown in the Table 4.2. % Change in Demand Mood (Previous % Change in Demand) % Change in Engineering Admission Rate Negative Pessimistic Negative Mild Neutral Mild Positive Optimistic Positive Table 4.2 Levels for Input and Output (Prototype Model) 4.2.4 Rule Base There are nine rules in total. Table 4.3 shows the proposed rule base. %Change in Negative Demand Mild Positive Mood Pessimistic Negative Negative Mild Neutral Negative Mild Mild Optimistic Mild Mild Positive Table 4. 3 Rule Base for fuzzy Expert System (Prototype Model) The MATLAB model of the fuzzy expert system can be found in Appendix H. 57 4.2.5 First Estimate Using the above specifications, the membership functions are calibrated manually to give an approximate performance of the fuzzy expert system. The membership functions for the inputs and output are shown in the Figure 4.8. The output is then compared with the historical data in Figure 4.9. Parameters used for the membership functions can be found in Appendix I. Figure 4. 8 Membership Functions (First Estimate, Prototype Model) 58 Engineering Admission Rate 2000 2001 2002 2003 Sim Engineering Rate 2004 2005 2006 2007 2008 Real Engineering Rate Figure 4. 9 Simulated Engineering College Admission Rate vs. Historical Engineering College Admission Rate (Prototype Model) The root mean squared error (RMSE) is 111.7. To put it in context with the data set provided, the RMSE is divided by the mean of all the historical data points. The percentage RMSE is 2.84%. From Figure 4.9, the fuzzy expert system seems to perform well and able to recreate the general trend of the historical rate. 4.2.6 Optimisation of Membership Functions Although the first estimate is able follow the trend, the membership function parameters can be calibrated so that the output is able to recreate the values and the trend of the historical engineering college admission rate. Hence, a particle swarm optimisation (PSO) is carried out to fine tune the membership function parameters. Each membership function is assumed to be triangular and can be defined by three values: the left foot, triangle peak and right foot. These three values for each membership function are allowed to vary within a defined range during the PSO 59 process. The PSO will deliver the set of membership functions which gives the smallest RMSE. Figure 4. 10 Optimised Membership Functions (Prototype Model) Engineering Admission Rate 2000 2001 2002 2003 2004 Sim Engineering Rate 2005 2006 2007 2008 Real Engineering Rate Figure 4. 11 Simulated Engineering College Admission Rate vs. Historical Engineering College Admission Rate (after optimisation, prototype model) The optimised membership functions are shown in Figure 4.6. As can be observed from Figure 4.7, the fuzzy expert system gives a closer fit after optimisation. The 60 RMSE is 41.4 and the percentage RMSE is 1.1%. Parameters of the optimised membership functions can be found in Appendix I. 4.3 Discussion and Analysis A fuzzy expert system was built to mimic a policy maker setting the engineering college admission rate. The fuzzy expert system was able to follow the trend and values of the historical data provided. Hence, the rule base that was used may be credible. Figure 4.12 shows the surface of the rule base used. A few observations about the rule base can be made:  There is a strong positive reaction when the mood and percentage change in engineering manpower demand are at their extreme positive values.  There is a swifter response to negative changes as only one of the factors needs to be negative for a strong negative reaction.  There is a natural tendency to decrease the admission rate when the demand remains stagnant. From the observations above, it is possible to conclude that policy makers adopt a cautious approach to increasing engineering college admission rate. They do not react immediately when the market demand increases but only do so when their optimistic outlook is validated. Next, they are quick to offer a response when there is negative change. 61 Figure 4. 12 Surface of Rule Base (Prototype Model) It ought to be noted that the construction, calibration and analysis of the fuzzy expert system are based on the few available data. More data can be used for calibration if available so as to build on the current analysis. 4.4 Traditional Forecasting Models using Fuzzy Logic Forecasts are often used in decision making because of their predictive ability. Hence, different kinds of forecasting models have been developed to do just that. There are some useful and established forecasting models in System Dynamics methodology, namely the First Order Exponential Smoothing and TREND function. These models are used in system dynamics models to mimic forecasting made by decision makers. Holt‟s Method is an improved forecasting model based on the first order information delay. Although it is seldom found in system dynamics models, it can also be expressed in a stock-and-flow diagram. 62 4.4.1 First Order Exponential Smoothing First order exponential smoothing is also known as first order information delay. It is often used to the fact that beliefs gradually adjusts to the actual value of a variable. The model structure is shown in Figure 4.13. Figure 4. 13 Stock and Flow Diagram for First Order Exponential Smoothing 4.4.2 Holt’s Method Holt‟s method is a case of double exponential smoothing. The slope and intercept are used to generate the forecast. The equations used are: 𝑆𝑖 = 𝛼𝐷𝑖 + 1 − 𝛼 𝑆𝑖−1 + 𝐺𝑖−1 (4.1) 𝐺𝑖 = 𝛽 𝑆𝑖 − 𝑆𝑖−1 + 1 − 𝛽 𝐺𝑖−1 (4.2) 𝐹𝑖 = 𝑆𝑖 + 𝐺𝑖 (4.3) Where Si is the current intercept, 63 Gi is the current slope, Di is the observed data, Fi is the forecasted value, α and β are the smoothing constants. After some manipulation, the following two equations can be obtained: 𝐹𝑖 = 𝐹𝑖−1 + 𝐺𝑖−1 + 𝛼 1 + 𝛽 (𝐷𝑖 − 𝐹𝑖−1) (4.4) 𝐺𝑖 = 𝐺𝑖−1 + 𝛼𝛽(𝐷𝑖 − 𝐹𝑖−1 ) (4.5) The two equations are equivalent to the stock and flow diagram, Figure 4.14. Figure 4. 14 Stock and Flow Diagram of Holt's Method 4.4.3 TREND Function The TREND function is often used to model growth expectations in system dynamics (Sterman 1987). One of important point to note is that the TREND function, unlike other forecasting models, does not directly give the forecast value. Instead, it gives the growth rate as output. The TREND function can be interpreted as a behavioural theory of how people form expectations by taking into consideration the time required for data collection and analysis, the time 64 horizon considered, and the time needed to react to the changes in growth rate. The model structure is shown in Figure 4.15. Figure 4. 15 Stock and Flow Diagram for TREND Function 4.4.4 Fuzzy Forecasting Model While these three models have their appropriate uses, having a fixed model structure means that their possibilities available to them are limited. This means that in order to find the most suitable forecasting model, the various model structures would have to be replaced and integrated with the rest of the model each time a new one is considered. From a computation perspective, this may affect the efficiency of the optimisation and design of policies. In this section, we 65 propose the use of fuzzy expert systems. There are many tuning parameters in a fuzzy logic controller and changing them may lead to different forecasting behaviours. This flexibility is especially useful in determining the forecasting model to use without switching the model structure. The fuzzy forecasting model is a simple feedback controller as shown in Figure 4.16. Figure 4. 16 Fuzzy Forecasting Model The input to the fuzzy logic controller is the difference between the current input and previous forecast. Based on this input, the fuzzy logic controller will generate an output that is the percentage change in forecast. This output will then be used to calculate the new forecast. The input and output of the controller are classified into different levels:  Difference between current input and previous forecast: Negative, Neutral, Positive  Percentage change in forecast: Negative, Neutral, Positive Figure 4.17 shows the membership functions used. The membership functions for both the input and output are kept constant. 66 Figure 4. 17 Input and Output to Fuzzy Logic Controller for Fuzzy Forecasting Model The rule base is simple as there are only one input and one output. Difference between current input and Percentage change in forecast previous forecast Negative Negative Neutral Neutral Positive Positive Table 4. 4 Rule base for fuzzy expert system - Forecasting Model 4.4.5 Calibration and Optimisation of Fuzzy Forecasting Model In order to show that the fuzzy forecasting model is flexible, it has to be able to replicate the behaviour of the traditional forecasting models. Using a cyclical Engineering Manpower Demand as input, the fuzzy forecasting model is first calibrated to the output of a traditional forecasting model. Without loss of generality, all parameters, such as adjustment times, used by the traditional forecasting model are set to 1 year. After calibration, the fuzzy forecasting model 67 output is tested with respect to the traditional forecasting model using different test inputs (Appendix J). The calibration is carried out using Particle Swarm Optimisation. Given that the membership functions and rule base are assumed to be constant and fixed, the weight can be interpreted as an expert‟s or a group of experts‟ opinion of the rule. Hence, the weights of the rule base can be tuned to reflect this knowledge. The optimisation criterion is the Root Mean Squared Error between the output of the fuzzy forecasting model and the corresponding tradition forecasting model, i.e. the optimisation process will seek to minimise the error. The results are as shown in Table 4.5. Figure 4. 18 Fuzzy Forecasting Model vs. 1st Order Information Delay (Calibrated with Cyclical Input) 68 Figure 4. 19 Fuzzy Forecasting Model vs. Holt's Method (Calibrated with Cyclical Input) Figure 4. 20 Fuzzy Forecasting Model vs. TREND Function (Calibrated with Cyclical Input) 69 Fuzzy Forecasting Model vs. Traditional Forecasting Models RMSE Test Input 1st Order Holt’s Method TREND Function Cyclical 0.0359 0.0248 0.0374 Saw-tooth 0.126 0.113 0.196 Pulse 0.145 0.108 0.395 Mixed 0.0567 0.0497 0.0381 Table 4. 5 RMSE between Fuzzy Forecasting Model vs. Traditional Forecasting Models (Calibrated with Cyclical Input) Results show that the fuzzy forecasting model is able to replicate the behaviour of the different forecasting models by simply tuning the different weights of the rules. However, it can be observed that the error is much greater when the inputs contain sudden discontinuous changes like in the Saw-tooth and Pulse inputs. Hence, this may suggest that the calibration of the fuzzy forecasting model may not be robust enough when the system is noisy or rapidly changing. In order to show that the performance of the fuzzy forecasting model is insensitive to the choice of calibration data, the fuzzy forecasting model is calibrated using different calibration inputs. 70 Inputs Calibrated Using... Cyclical Saw tooth Pulse Mixed Cyclical 0.0359 0.0329 0.0394 0.412 Saw tooth 0.126 0.133 0.128 0.128 Pulse 0.145 0.147 0.146 0.144 Mixed 0.0568 0.0537 0.0593 0.0534 Table 4. 6 RMSE between Fuzzy Forecasting Model vs. 1st Order Information Delay (Calibrated with Different Inputs) Inputs Calibrated Using... Cyclical Saw tooth Pulse Mixed Cyclical 0.0248 0.0264 0.0344 0.0318 Saw tooth 0.113 0.128 0.100 0.101 Pulse 0.108 0.112 0.107 0.105 Mixed 0.0497 0.0511 0.0480 0.0700 Table 4. 7 RMSE between Fuzzy Forecasting Model vs. Holt’s Method (Calibrated with Different Inputs) Inputs Calibrated Using... Cyclical Saw tooth Pulse Mixed Cyclical 0.0374 0.0485 0.0932 0.486 Saw tooth 0.196 0.189 0.211 0.189 Pulse 0.395 0.393 0.385 0.393 Mixed 0.0381 0.0300 0.0719 0.0300 Table 4. 8 RMSE between Fuzzy Forecasting Model vs. TREND Function (Calibrated with Different Inputs) 71 From the tables, the difference in RMSE when different calibration data are used is small. This would mean that the fuzzy forecasting model‟s performance is more or less independent of the type of calibration data used. The model‟s performance is relatively insensitive to the calibration data used. 4.5 Conclusion In this chapter, a fuzzy expert system was constructed to replicate the decision rule of a generic supply chain model so as to demonstrate its ability to replace a decision policy. The fuzzy expert system was able to replicate the decision rule of the supply chain model rather satisfactorily. Next, fuzzy expert system was applied on an unknown part of the engineering manpower supply and demand system. The fuzzy expert system was created to replicate the actions of a policy maker setting the engineering college admission rate. It was found that the fuzzy expert system was able to give generally good results and its rule base revealed a possible mental model behind the policy maker‟s decision rule for engineering college admission rate. More historical data would lend more credibility to the findings. It can also be concluded the fuzzy forecasting model is able to replace the various traditional forecasting models. The fuzzy forecasting model is flexible because it is able to approximate the various types by simply changing the weights of the rules. This flexibility is desirable when searching for an optimal decision policy. This is because the forecasting model structure will not be fixed or limited and there is no need to switch in and out the various types of model structures to be explored. The fuzzy forecasting model is also able to give similar results despite 72 being calibrated using different data. This shows that the performance of the fuzzy forecasting model is insensitive to the type of calibration data. However, while the fuzzy forecasting model is able to follow the trend of the input data, the error between the outputs of the fuzzy forecasting model and the traditional forecasting model can become significant when the input is highly noisy or rapidly changing. Chapter 5 Optimal Decision Policy Using Fuzzy Logic The previous chapter demonstrated the versatility of fuzzy logic, enabling the imitation of decision policies and forecasting models. This chapter will extend on what was done previously and attempt to synthesise an optimal decision policy using fuzzy logic. This decision policy is based on a hypothetical policy lever, the Approval Fraction of foreign engineers, to move the system towards the objective. The decision policy should also enable us to handle uncertainty within the system. 5.1 Decision Policy and Fuzzy Logic A good decision policy is of great interest to policy makers as this decision policy will enable them to pursue their objectives. However, it is difficult to obtain good policies which fulfil the objectives robustly. Formally, there are three methods to obtain decision policies for high order and non-linear models: mathematical methods, optimal algorithms and guideline methods. These methods are often difficult to apply and require a certain level of sophistication and knowledge to derive them. Thus, a method which would allow us to arrive at a good decision policy with relative ease should be sought. Fuzzy logic can be one way where this can be achieved. As seen from previous chapters, fuzzy logic, along with the correct inputs, can be used to mimic decision policies and forecasting models. Thus, it is plausible that fuzzy expert systems can be tuned with respect to a policy objective so as to give 73 74 us the corresponding optimal decision policy. The Manpower Supply and Demand model will be used to study this possibility. 5.2 Policy Lever and Objective For this study, our objective is to achieve a certain Resident Engineers: EP Engineers ratio (henceforth referred to as the Objective Ratio). This is a realistic objective because policy makers often have to balance the domestic and foreign workforce. In the model, a normalised ratio will be used. Ideally, if the Objective Ratio is met, the value of the Normalised Ratio should be equal to 1. The policy lever that will be used is the Approval Fraction. This approval fraction is the fraction of EP Engineers who are permitted to enter the system. It ranges from 0 to 2. This means that depending on the situation, a policy maker can choose to allow numbers varying from zero EP engineers or twice the number of EP Engineers as stipulated by the market. Thus, a policy maker is able to decide to bring in more foreign workforce when the demand is high. On the other hand, he is also able to cut down on the foreign influx when demand is low. Although the approval fraction may not be explicitly manipulated in real life manpower policies, it is still of relevance and importance as manpower policies are inadvertently related to it. For example, the issuance of work permits or the collection of foreign workers levy has the same effect of regulating foreign manpower. Hence, the approval fraction can be considered to be a surrogate indicator of the actual manpower policies implemented. Learning how the approval fraction affects the system can lead to insights as to how actual manpower policies should be calibrated. 75 The fuzzy decision policy model will use the Normalised Ratio as an input and mimics a decision policy which controls the Approval Fraction. The goal of this decision policy is to keep the Normalised Ratio within an acceptable interval. As mentioned before, decision makers often base their decisions on forecasts. Hence, a forecast of the Normalised Ratio is needed. This forecast will be generated using a fuzzy forecasting model due to its flexibility. In effect, the fuzzy decision policy model consists of two parts, a fuzzy forecasting model which generates a forecast of the Normalised Ratio and a fuzzy policy model which acts depending on the forecast. 5.3 Fuzzy Policy Model The fuzzy policy model will be mimicking a decision policy setting the Approval Fraction. A fuzzy feedback model structure is used to generate this approval fraction. Figure 5. 1 Fuzzy Policy Model 5.3.1 Output of Fuzzy Logic Controller The output is the percentage change in the Approval Fraction. As before, the percentage change is used so that the approval fraction will be robust and compute 76 in relative terms. The percentage change in Approval Fraction will have three levels: Negative, Neutral, and Positive. 5.3.2 Input of Fuzzy Logic Controller The input to the fuzzy expert system is the forecasted value of Normalised Ratio, i.e. Current Ratio/Objective Ratio. This means that ideally the normalised ratio should be 1. The use of the normalised ratio as an input is intuitive. This is because if the ratio is more than 1, this means we have too many resident engineers and can increase our intake of EP Engineers. The converse is also true. If the ratio is less than 1, this means that there are too many EP Engineers and must decrease the influx of EP Engineers. The input will be divided into three levels: Low, Neutral, and High. 5.3.3 Membership Functions of Fuzzy Logic Controller Figure 5.2 shows the membership functions. The input is classified into three levels. A policy maker when looking at the ratio may feel that it is “Low” if the number is less than the objective, “Neutral” if the number is close to the objective or “High” if the number is more than the objective. Similarly, a policy maker may react in three different ways, “Negative” to reduce the approval fraction, “Neutral” to maintain the approval fraction or “Positive” to increase the approval fraction. 77 Figure 5. 2 Membership Functions for Fuzzy Policy Model 5.3.4 Rule Base of Fuzzy Logic Controller The rule base is simple as there are only one input and output. Hence, for a certain input considered, there is only one logical output. For instance, when the Normalised Ratio is “Low”, the output must be “Negative” as it would not make sense to further increase the number of EP Engineers when there is already too many. This rule base may be unique due to the model considered. In cases where the rule base is larger with multiple levels of inputs and outputs, discussions with domain experts to solicit consensus about the rule base can be considered. It may also be useful to consider optimisation algorithms to find a rule base. However, 78 this may lead to illogical results which may not be in line with rational thinking like the example above. Forecasted value of Normalised Ratio Percentage Fraction Low Negative Neutral Neutral High Positive change in Approval Table 5. 1 Rule base for fuzzy policy model 5.4 Calibration and Optimisation of Decision Policy Model The decision policy is made up of two fuzzy models. First, the fuzzy forecasting model will be generating a forecast of the Normalised Ratio. Next, based on the forecast, the fuzzy policy model will then generate an appropriate approval fraction. Figure 5. 3 Fuzzy Decision Policy Model Particle Swarm Optimisation is used to find the set of weights for both fuzzy models which would keep the results close to objective (i.e. Normalised Ratio = 79 1).Three scenarios for Engineering Manpower Demand as the input were considered: the demand increases linearly, the demand follows a cyclical pattern and the demand follows a cyclical pattern with a general linear increasing trend (refer to Appendix K). A simulated timeframe of forty years was used. The first five years was for the system to reach steady state. This is so that all changes after the fifth year are due to the inputs used. The fuzzy policy model and fuzzy forecasting model were calibrated to give the best results possible under a deterministic setting using input data for the next twenty years. In order to demonstrate the capability of the fuzzy decision policy after calibration, the last fifteen years were used as validation. The root mean squared error (RMSE) with respect to 1 was used as an optimisation criterion. 𝑤𝑜𝑝𝑡 = 𝑎𝑟𝑔𝑚𝑖𝑛 𝑓(𝑤𝑥 ) 𝑓 𝑤𝑥 = 𝑡 𝑖 𝑦 𝑖 𝑤 𝑥 −1 2 𝑡 Where:    wx is the set of weights for a particle x, t is the number of data points in the calibration phase yi is the Normalised Ratio at a data point i. The results obtained for each type of input are as shown in Figure 5.4 to 5.6. (5.1) (5.2) 80 1.02 Calibration Validation 1.01 1 0.99 0.98 0.97 0.96 RMSE = 0.00074 0.95 0.94 0 5 10 15 20 25 30 35 40 Figure 5.4 Normalised Ratio after Implementation of Fuzzy Decision Policy (Ramp Input, RMSE) 1.03 Calibration Validation 1.02 1.01 1 0.99 0.98 0.97 0.96 RMSE = 0.00997 0.95 0.94 0 10 20 30 40 Figure 5.5 Normalised Ratio After Implementation of Fuzzy Decision Policy (Cyclical Input, RMSE) 81 1.03 Validation Calibration 1.02 1.01 1 0.99 0.98 0.97 0.96 RMSE = 0.00979 0.95 0.94 0 5 10 15 20 25 30 35 40 Figure 5.6 Normalised Ratio after Implementation of Fuzzy Decision Policy (Cyclical Ramp Input, RMSE) Another possible optimisation criterion is the Worst Absolute Error. The Worst Absolute Error is the maximum absolute error with respect to 1 and should be minimised. 𝑤𝑜𝑝𝑡 = 𝑎𝑟𝑔𝑚𝑖𝑛 𝑔(𝑤𝑥 ) 𝑔 𝑤𝑥 = max 𝑎𝑏𝑠 𝑦𝑖 𝑤𝑥 − 1 (5.3) ∀𝑖𝜖𝑡 (5.4) Where    wx is the set of weights for a particle x, t is the data points in the calibration phase yi is the Normalised Ratio at data point i. This is used to calibrate the fuzzy decision policy and the results are as follows: 82 1.02 Calibration Validation 1.01 1 0.99 0.98 0.97 0.96 WAE = 0.00292 0.95 0.94 0 5 10 15 20 25 30 35 40 Figure 5.7 Normalised Ratio After Implementation of Fuzzy Decision Policy (Ramp Input, Worst Absolute Error) 1.02 Calibration Validation 1.01 1 0.99 0.98 0.97 0.96 WAE = 0.0183 0.95 0.94 0 5 10 15 20 25 30 35 40 Figure 5.8 Normalised Ratio After Implementation of Fuzzy Decision Policy (Cyclical Input, Worst Absolute Error) 83 1.03 Calibration 1.02 Validation 1.01 1 0.99 0.98 0.97 0.96 WAE = 0.0195 0.95 0.94 0 5 10 15 20 25 30 35 40 Figure 5.9 Normalised Ratio After Implementation of Fuzzy Decision Policy (Cyclical Ramp Input, Worst Absolute Error) 5.4.1 Discussion and Analysis From the figures above, the fuzzy decision policy is slightly sensitive to the choice of optimisation criterion. This is understandable because different optimisation criterion gives a different set of weights. This in turn leads to different results. It can be observed that the fuzzy decision policy is able to keep the Normalised Ratio close to the objective (within ±5%) under all three different inputs when given the calibration data. In addition, the fuzzy decision policy’s ability is validated as it is able to maintain its performance through the validation phase. Hence, the fuzzy decision policy can be used as the optimal decision policy or as a reference to such a decision policy. 84 The fuzzy decision policy’s performance depends on the type of input used for calibration. The fuzzy decision policy is able to keep the normalised ratio very close to 1 when the input demand is increasing linearly. On the other hand, the performance is poorer when the input includes cyclical changes. Thus, the fuzzy model may not be suitable for systems which are rapidly changing. 5.5 Performance of Fuzzy Decision Policy Model under Noise In the previous section, the inputs and parameters are deterministic. However, in reality, inputs and parameters such as demand are often difficult to estimate and are subject to errors. As a result, if the fuzzy decision policy model is to be considered useful, it has to be robust even when there are noise and errors in the inputs and parameters. Assuming that the standard deviations of the noise and errors are known, the absolute values of noise can be formulated as α*σ. σ is the standard deviation of noise and α is a constant which changes the magnitude of noise in the system. When α increases, the noise in the system also increases. Hence, the fuzzy decision policy can be considered robust if it is able to keep the Normalised Ratio between an acceptable range for a desired α. We shall assume that the acceptable range for the Normalised Ratio is defined to be between 0.95 and 1.05 (±5%). This range indicates how policy makers are tolerant of the deviation from the objective and depends on how close the system must be kept to the objective. 85 5.5.1 Noise and Errors To study the fuzzy decision policy under noise, noise is added into three parameters: Manpower Demand, the Effect of Gap on Wages and EP Engineers to Resident Engineers Fraction. The values for standard deviation of noise are assumed and are shown in Table 5.2. Parameters Magnitude of Noise Coefficient of Variance (%) Demand 500 0.833 Effect of Gap on Wages 0.01 1 EP to Resident Engineers Fraction 0.001 1 Table 5.2 Assumed Standard Deviation of Noise 5.5.2 Robustness of Fuzzy Decision Policy Calibrated under Deterministic Conditions In the above paragraphs, it is assumed that if the fuzzy decision policy is able to handle a noise α*σ, it will be able handle any noise that is less than α*σ. This is not true if the system behaviour is non-monotonic, i.e. as noise increases, the perturbations to the system output are not of the same magnitude to the level of noise or in the same direction. Using the three parameters and varying levels of noise added to them, it was found that the Engineering Manpower Model is indeed monotonic. Since there are three parameters and two signs (positive and negative) for each parameter, there are eight possible permutations, or corner points, for a defined noise α*σ. Figure 5.10 shows the model output from these eight points, using a cyclical 86 demand input. A similar figure for ramp demand input can be found in the Appendix L. 1.3 1.25 1.2 Output 1 1.15 Output 2 1.1 Output 3 1.05 Output 4 1 Output 5 0.95 Output 6 0.9 Output 7 0.85 Output 8 0.8 0 50 100 Figure 5.10 Model Outputs from Extreme Points (Cyclical) Using Monte Carlo simulations, random noise levels that are below the defined α*σ are generated. Their model outputs are compared with those observed in Figure 5.10. It was found that none of the randomly generated noises’ outputs exceed those in Figure 5.10. This demonstrates that for any noise below α*σ, the system behaviour will not deviate more than the behaviour from the one generated with noise magnitude α*σ. The envelopes of the output trajectories are always generated by the corner points of the noisy parameters. In general, such behaviour is generated if the system state levels are monotonic with respect to these parameters. . For small enough perturbations, it is reasonable to treat the underlying dynamic system as a linear one, thus producing system state levels that are monotonic with respect to the parameters. A fuzzy decision policy that is able to handle noise α*σ will be able to handle any noise below α*σ. 87 The robustness of the calibrated fuzzy decision policy found in Section 5.4 is tested. Since the system is shown to be monotonic when the changes made to the system is small, it can be assumed that the most extreme change in system behaviour is given by one of the eight points when noise is equal to α*σ. As it is not known which of the eight points changes the system behaviour the most, all of the eight points are tested. Each set of weights is used to find the maximum α that it is able to handle all eight points for the calibrated type of input. Table 5.3 shows the results. Figure 5.11 and Figure 5.12 show the model output when the noise is at maximum α and above the maximum α for the case Cyclical Input and RMSE as optimisation criterion. Type of Input Optimisation Criterion Maximum α Cyclical RMSE 0.2891 Cyclical WE 0 Ramp Cyclical RMSE 0.5391 Ramp Cyclical WE 0.6406 Ramp RMSE 2.3672 Ramp WE 2.9063 Table 5. 3 Maximum α for Fuzzy Decision Policy (Calibrated under Deterministic Conditions) 88 1.03 1.02 Series1 1.01 Series2 1 Series3 0.99 Series4 0.98 Series5 0.97 Series6 0.96 Series7 0.95 Series8 0.94 0 10 20 30 40 Figure 5.11 Output from Corner Points when Alpha = 0.2891 (Cyclical Input, Calibrated using RMSE) 1.1 1.09 1.08 1.07 1.06 1.05 1.04 1.03 1.02 1.01 1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 Series1 Series2 Series3 Series4 Series5 Series6 Series7 Series8 0 10 20 30 40 Figure 5.12 Output from Corner Points when Alpha = 0.2969 (Cyclical Input, Calibrated using RMSE) The maximum α for cyclical and ramp cyclical demand are low. Therefore, it can be deduced that the fuzzy decision policies obtained are not robust. α for ramp demand are much higher, indicating the robustness of the decision policy. The 89 reason for this could be that there are more fluctuations, variations and less predictability in the cyclical and ramp cyclical cases. Also, even though α is high for ramp demand, this is not useful because there is no real need for a decision policy if the system inputs are known to be linear. The poor robustness of these fuzzy decision policies implies that it is insufficient to simply calibrate them to best fit a policy objective under deterministic conditions. Hence, a new way to calibrate the fuzzy decision policy should be explored in order to obtain a set of robust parameters. 5.6 Improving the Robustness of Fuzzy Decision Policy One way to ensure robustness is to include noise and errors in the calibration process. For a set of weights, the maximum possible α is found using a binary search method. The best set of weights is then found using a particle swarm optimisation. The optimisation seeks to find the set of weights that maximises α. Figure 5.13 shows the optimisation process. 5.6.1 Results Alternative sets of weights for the fuzzy decision policy were found for each type of input. The maximum α for each case is shown in Table 5.4. Type of Input Maximum α Cyclical 1.3359 Ramp Cyclical 1.4844 Ramp 2.8858 Table 5.4 Maximum α for Fuzzy Decision Policy Calibrated for Robustness 90 To test the fuzzy decision policy and see that it is truly robust for all noise levels below α, α is varied from 0 to Maximum α. The following figures show the Normalised Ratio for α = 0, α = Maximum α and α slightly above Maximum α for a cyclical input. Figure 5.13 Optimisation Process under noise 91 Figure 5.14 Normalised Ratio for alpha =0, cyclical input 1.04 1.03 1.02 Series1 1.01 Series2 1 Series3 0.99 Series4 0.98 Series5 Series6 0.97 Series7 0.96 Series8 0.95 0.94 0 10 20 30 40 Figure 5.15 Normalised Ratio for alpha = 1.3359, cyclical input 92 1.05 1.04 1.03 1.02 Series1 1.01 Series2 Series3 1 Series4 0.99 Series5 0.98 Series6 0.97 Series7 0.96 Series8 0.95 0.94 0 10 20 30 40 Figure 5.16 Normalised Ratio for alpha = 1.41, cyclical input 5.6.2 Discussion & Analysis As can be observed from the graphs, the same fuzzy decision policy was able to keep the Normalised Ratio within the defined acceptable range for varying levels of noise. This is a marked improvement from the previous case where the fuzzy decision policy was calibrated for the best performance under a deterministic setting. It can be concluded that the robustness of the fuzzy decision policy indeed improved when calibrated this way. However, the performance in the noiseless case is sacrificed as it is not able to keep the Normalised Ratio as close as it could have. It is also pertinent to note that the maximum α for a fuzzy decision policy depends on the defined acceptable range for the policy objective. The robustness is gained in exchange for extra computational time. As the optimisation process needs to calculate the maximum α for each and every set of 93 weights it considers, this may pose a problem for large models or fuzzy controllers with many fuzzy weights. This is also true if there are many noisy parameters, thus leading to many corner points to consider in the optimisation process. 5.6.3 Monte Carlo Simulation under Random Noise In the previous section, it is assumed that the system is monotonic and thus noise is added at a flat level to the system throughout the simulation. A decision policy which is able to handle the maximum amount of noise was found. In reality, noise is often random and not fixed at a certain level for sustained periods of time. Thus, it is important that the fuzzy decision policy found is also robust when the noise is varying randomly. The maximum α found in the previous section serves as a reference since the fuzzy decision policy is calibrated to handle this level of noise. The fuzzy decision policy should be able to keep the normalised ratio within the acceptable range if the noise varies randomly and remains below the level of maximum α. The noise is assumed to follow a beta (3, 3) distribution between – α*σ and α*σ, α = 1.3359 and the cyclical input case was used. The beta distribution is used because it is more realistic than a uniform distribution. Smaller values of noise are more likely to happen than larger values. Monte Carlo simulations were carried out to test the fuzzy decision policy’s performance under random noise. In order to demonstrate the fuzzy decision policy’s effectiveness, the system is also subjected to random noise without any form of decision policy affecting the approval fraction. The approval fraction is set at 1. Monte Carlo simulations were 94 carried out using the same inputs and noise distribution as above, but without a decision policy. Figure 5.17 shows that, without control over the approval fraction, the acceptable range will definitely be exceeded. For each run, the normalised output also remains outside of the acceptable range for an approximate average 20% of the simulation time. Figure 5.17 Monte Carlo Simulation under random noise without any decision policy 95 Figure 5.18 Monte Carlo Simulations (Cyclical Input) Figure 5.19 Instance of Model Output Exceeding Acceptable Range (Cyclical Input) 96 It can be seen from Figure 5.18 that the fuzzy decision policy is successful in keeping the normalised ratio within the acceptable range most of the time with a confidence of 95%. Out of the hundred simulations, ninety nine of them stepped out of the acceptable range at least once. However, it was observed that when the model output exceeds the acceptable range, it spends a mere average 7.3% of the simulation time outside. This represents a significant improvement over the case where there is no fuzzy decision policy controlling the approval fraction. Thus, it can be concluded that the fuzzy decision policy is useful in helping us achieve our policy objective and is robust even when the noise is randomly distributed. 5.6.4 Linear Programming (LP) It is hoped that linear programming can help to give a good estimate of the best set of weights. Unlike the binary search algorithm where many iterations are needed to find the maximum α, the maximum α for a set of weights can be calculated directly using the LP model, thus reducing the computational time required. Linearization of the model is justifiable when the noise is small. In cases which are complex with many noisy parameters, solving a linear programming model is extremely efficient because of the simplex method. However, for clarity of exposition, a simplified version of the model will be used in this study. The simplified version of the model is proposed as shown in the following figure. 97 Figure 5.20 Simplified Engineering Manpower Model As before, the fuzzy decision policy influences the EP Engineers Hiring Rate through an approval fraction. The fuzzy decision policy is modelled using fuzzy controllers. The MATLAB version of this model and the fuzzy decision policy can be found in the Appendix M. The state variables in simplified model are expressed by the following equations. 1 1 1 1 REt = 1 􀀀 − tthr 􀀀 − arjd − 􀀀 . REt−1 + E2Rf 􀀀 − tthr . EPt−1 + arcd af tthe EPt = 1 − 􀀀 t − 1 aejd −􀀀 E2Rf . EPt−1 􀀀 − 1 tthe . aft . REt−1 + Where:          RE: Resident Engineers EP: EP Engineers tthr: Time to hire resident engineers arjd: Average Resident Engineer Job Duration arcd: Average Resident Engineer Career Duration E2Rf: EP Engineer to Resident Engineer Fraction d: Demand af: Approval Fraction tthe: Time to Hire EP Engineer af t tthe 1 tthr . dt−1 . dt−1 (5.5) (5.6) 98  aejd: Average EP Engineer Job Duration In order to build the LP model, the above state variable equations will have to be linearised about a nominal set of parameters. A state variable Y can be expressed by the following equation. 𝑌𝑡 = 𝑌𝑡 + ∆𝑌𝑡 (5.7) Where:    Y is a state variable 𝑌 is the state variable under nominal setting ∆𝑌 is the change introduced due to noise in the parameters Using the above formulation, 𝑅𝐸 and 𝐸𝑃 can be found using the equations 5.5 and 5.6 with nominal parameters. For this study, noise is added to the demand and the EP Engineer to Resident Engineer fraction. Thus, ∆RE and ∆EP will be calculated based on the noise. However, this is not all as EP also depends on the approval fraction. The approval fraction is generated by the fuzzy decision policy which is a non-linear function. Hence, the approval fraction has to be linearised. The linear form of the Approval Fraction (af) is assumed to be: aft = 𝑎𝑓𝑡 + ∆aft ∆𝑎𝑓𝑡 = 𝑐 ∗ ∆𝑅𝐸𝑡−1 + 𝑑 ∗ ∆𝐸𝑃𝑡−1 Where:    𝑎𝑓 is the nominal approval fraction, ∆af is the change in approval fraction, c and d are constants . (5.8) (5.9) 99 Using the above formulations, ∆RE and ∆EP can be expressed by the following matrix. 𝑎11 ∆𝑅𝐸𝑡 = 𝑎 ∆𝐸𝑃𝑡 21 𝑎12 ∆𝑅𝐸𝑡−1 𝑏11 𝑎22 ∆𝐸𝑃𝑡−1 + 𝑏21 𝑏12 ∆𝑑 𝑏22 ∆𝐸2𝑅𝑓 (5.10) Where: 𝑎11 = 0 𝑎12 = 𝐸2𝑅𝑓 − 𝑎21 = 𝑎22 = 1 − 1 𝑡𝑡𝑕𝑟 𝑐 𝑎𝑓𝑡−1 . 𝑑𝑡−1 − 𝑅𝐸𝑡−1 − 𝐸𝑃𝑡−1 − 𝑡𝑡𝑕𝑒 𝑐 𝑎𝑓𝑡−1 1 𝑑 − − 𝐸2𝑅𝑓 + . 𝑑𝑡−1 − 𝑅𝐸𝑡−1 − 𝐸𝑃𝑡−1 𝑡𝑡𝑕𝑒 𝑎𝑒𝑗𝑑 𝑡𝑡𝑕𝑒 𝑏11 = 1 𝑡𝑡𝑕𝑟 𝑏12 = 𝐸2𝑅𝑓 𝑏21 = 𝑎𝑓𝑡−1 𝑡𝑡𝑕𝑒 𝑏22 = 𝐸2𝑅𝑓 A LP model can be built using the equations above and the MATLAB code can be found in Appendix N. ∆d and ∆E2Rf are the noise added into the system. As in the previous section, they are in the form of α*σ, where α is a constant and σ is one standard deviation of noise. The LP model will find the maximum α possible for a particular set of weights. A particle swarm optimisation will then be used to find the set of weights that give the maximum α. Different weights give different behaviours and thus different linear coefficients. Hence, for each set of weights, 100 coefficients c and d has to be calculated from simulations varying Resident Engineers and EP Engineers. Ideally, the results found from using the LP model and the Simulink model should be the same. However, as the LP model uses an approximation of the fuzzy decision policy, the model output is expected to differ once the system moves away from the nominal level. For this study, the demand was set to follow a cyclical pattern. The optimisation process was carried out twice, once using the Simulink model and once using the LP mode. This will allow us to compare between the two models. The maximum α found for the LP model and Simulink model are 2.1094 and 4.1875 respectively. This is an improvement from a maximum α of 1.5000 when the approval fraction is kept at 1 and is not controlled by a decision policy. From the results, it can be deduced that calibration using the LP model led to a loss in robustness. The set of weights found is only able to handle about half the noise level that the best set of weights found using the Simulink model. However, this loss can be overcome by using better approximations in the LP model. Despite the loss in robustness, the set of weights found using the LP model can be used as a quick approximation to a good decision policy. This is especially the case if the model is large and contains several noisy parameters. Finding a decision policy with the Simulink model in such cases will take very long. As was the case in the previous section, a Monte Carlo simulation was carried out to test the fuzzy decision policy found by the LP model under random noise. The noise level is constrained to be between -α*σ and α*σ, where α is equals to the 101 maximum alpha of 2.1094. The noise was assumed to follow a beta (3, 3) distribution. The Figure 5.21 shows the results of the Monte Carlo simulation. From Figure 5.21, the fuzzy decision policy is able to keep the normalised ratio within the acceptable range at all times with a confidence of 95%. Out of the hundred simulations carried out, the model output stepped outside the acceptable range thirty-six times. This implies a 64% probability that the fuzzy decision policy would succeed in its objective. Furthermore, when the acceptable range is exceeded, the model output remained outside of the range for a short 1.5 time steps on average. Thus, these can be considered as outliers and does not affect the performance in general. Figure 5.23 and 5.24 show a couple of instances of the model output. Figure 5.21 Monte Carlo Simulation using weights found from LP Model and under random noise (beta distribution) 102 Figure 5.22 Model Output with no noise Figure 5.23 Model output under random noise (Within acceptable range) 103 Figure 5.24 Model output under random noise (Acceptable range breached) Hence, it can be concluded that the fuzzy decision policy found by the LP model is useful. Furthermore, judging from the results of the Monte Carlo simulation, it is likely that the fuzzy decision policy is able to handle higher levels of random noise, or a narrower acceptable range of normalised ratio. 5.7 Conclusion Fuzzy Logic was applied in an attempt to find an optimal decision policy to satisfy a policy objective. The policy objective was defined to be the ratio between Resident Engineers and EP Engineers. It was hoped that the certain ratio between the two variables can be kept. To achieve this policy objective, a policy lever called the Approval Fraction was used. The Approval Fraction is the fraction of EP Engineers that would be allowed into the system. 104 The decision policy model consists of two parts. First, a fuzzy forecasting model is used to generate a forecast of the Normalised Ratio. Next, based on this forecast, a fuzzy policy model will set the Approval Fraction to determine the number of EP Engineers allowed into the system. Three different cases of Engineering Manpower Demand input were considered. They are Ramp input, Cyclical input and Cyclical plus Ramp input. For each case, the decision policy model is calibrated, using Particle Swarm Optimisation, to minimise a certain optimisation criterion. Two optimisation criteria were tested and they were the Root Mean Squared Error and the Worst Absolute Error. It was found that the calibrated decision policy model for each scenario is able to satisfy the policy objective in all three cases. Next, it is more realistic that the parameters and input to the model are noisy. Thus, the calibrated decision policy models were tested by adding noise to the parameters and input. It was first shown that it is reasonable to assume that the system is monotonic. Noise is then added in the form of α*σ, where α is a constant and σ represents one standard deviation of noise. As a result of the monotonic assumption, a fuzzy decision policy that is able to handle noise α*σ will be able to handle any noise below α*σ. For the Cyclical and Cyclical Ramp input, the calibrated fuzzy decision policy model is not robust and is intolerant of noise. However, for the Ramp input, the calibrated decision policy model is able to satisfy the policy objective up to a noise level of approximately 2%. Thus, it can be concluded that the fuzzy decision policy calibrated under a deterministic setting is not robust. 105 The fuzzy decision policy is then calibrated by considering noise. To do this, the maximum α for each particle in the particle swarm is found by testing the extreme points. This process is a tedious one as depending on the number of noisy parameters, the number of extreme points can grow exponentially large. In the study, there are three noisy parameters and thus eight extreme points. It was found that after calibration, the fuzzy decision policies became more robust for each type of input. They were able to tolerate higher levels of noise than previously possible. Although better results were obtained, this is in exchange for additional computational time. This may not be palatable for projects where the system is large and the number of noisy parameters is plenty. To overcome this issue, linear programming (LP) was considered. A LP model should approximate the behaviour of the original model about a nominal operating point. Moreover, the LP model can be solved quickly to find the maximum α for a set of weights. Thus, this should reduce the computational time regardless of the number of noisy parameters. It was found that the fuzzy decision policy calibrated using the LP model is less robust than the one calibrated using the original model. This is expected as the LP model is only an approximation. However, the fuzzy decision policy found is useful and can serve as a quick reference if needed. To conclude, an attempt to find an optimal decision policy using fuzzy logic was conducted. It was found that the policies found under deterministic settings are not robust to noise. Fuzzy decision policies calibrated with noise are more robust but are computationally expensive. Thus, linear programming can be one way to approximate robust fuzzy decision policies. Chapter 6 Conclusion This chapter summarises the objectives, the design, the results and analyses of the study. Contributions made by the study are then presented. The limitations of the study are then discussed. Finally, scope for future research is then suggested. 6.1 Summary Effective manpower planning on a macro level is important. Doing it well will reduce the inefficient use of human resources and hence maximise productivity and growth. Also, in an increasingly globalised world, it is essential to balance the nurture and well-being of the resident workforce and the need to bring in foreign expertise to supplement and bolster the resident workforce. The engineering manpower demand and supply system was studied because of the importance of engineers in the global context. Although institutes of higher learning produce many engineers annually, not all engineering-trained graduates eventually join the workforce as engineers. Human judgement and decision often play a huge role in social systems such as these. However, these interactions are often neglected in established economic models due to their unpredictability and sometimes, irrationality. It was proposed to model the engineering manpower demand and supply system using a system dynamics approach. System dynamics models promises to overcome certain limitations that mathematical and economic models face such as non-linear behaviour and inherent feedback loops within the system structure. However, they are dependent on available knowledge and information about the 106 107 system. Artificial Intelligence techniques may be applied to replace parts of the system which are poorly understood. These techniques can also be applied to help policy makers formulate better decision policies which are robust and effective. The objectives of the study were determined in relation to the above considerations and they are:  Propose and construct a prototype model of the engineering manpower supply and demand system using the system dynamics approach.  Obtain more knowledge of the system structure so as to understand the system behaviour.  Provide answers to some policymaking questions and give some ideas to possible policymaking guidelines concerning manpower policies.  Apply A.I. tools alongside system dynamics to replace parts of the model which are poorly understood or where there is insufficient knowledge.  Use A.I. tools to obtain a good decision policy to achieve policy objectives. To achieve the objectives, a hypothesis of the system structure was proposed. Based on this hypothesis, a prototype model was built and validated using historical annual data. Some possible policy levers in the prototype model were identified. Scenario analysis was carried out by varying the policy levers and observing their impact on the system behaviour. Firstly, it was observed that an excessive influx of foreign engineers depresses the wage of engineers and hence makes the profession less attractive. This leads to an increase in the leakage of engineering trained residents into other industries. Next, the engineering college 108 admission rate is effective in increasing the number of resident engineers up to a certain extent. Beyond this, increasing the engineering college admission rate only increases the number of resident jobseekers. Matching the demand also leads to an oversupply of resident jobseekers. Also, the effect of changing the engineering college admission rate is delayed. From these observations, some conclusions can be reached. Firstly, it is important to regulate the influx of foreign engineers into the system and also maintain the critical pool of resident engineers in the system. Secondly, setting the engineering college admission rate according to changes in the engineering manpower demand is insufficient. Other factors also have to be considered in order for the system to match demand but at the same time decrease unemployment. Sensitivity analysis of the model to the key model parameters was carried out. From the One Way Sensitivity Analysis, it was concluded that certain model parameters are able to impact more aspects of the model. A Monte Carlo simulation was also carried out. It was found that the general system behaviour of the model is robust to changes made to the key model parameters. In order to test the ability of a fuzzy expert system to replicate decision policies, it was suggested a fuzzy expert system be built to replace the decision rule in a generic system dynamics supply chain model. This was done so as to demonstrate that this approach of building a fuzzy expert system and replacing a decision rule with it is feasible and reliable. Hence, a fuzzy expert system was built with this proposition in mind. The fuzzy expert system was able to replicate the output of the decision rule rather satisfactorily. 109 The engineering college admission rate was assumed to be exogenous in the model because the knowledge of the underlying system structure is unknown. In order to make it endogenous, a fuzzy expert system was used to mimic the behaviour of a policy maker deciding on the engineering college admission rate. The method was tested previously using the generic supply chain model. It was observed that the percentage change in engineering college admission rate was correlated with the percentage change in demand. Inputs, membership functions and a rule base were proposed for the construction of the fuzzy expert system. Then, a particle swarm optimisation was carried out to optimise the fuzzy expert system so as to obtain a good fit with respect to historical data. The fuzzy expert system was able to replicate the engineering college admission rate well. A glimpse into the possible mental model of the policy maker was made possible by the rule base of the fuzzy expert system. It was observed that a cautious approach was adopted when positive changes to the engineering college admission rate were to be made. However, swift changes to the engineering college admission rate were made when negative developments arise. On top of decision policies, traditional forecasting models used in systems dynamics were also shown to be easily replicated by a simple feedback structure using a fuzzy expert system. Based on what was learnt about fuzzy expert systems and its ability to mimic decision policies and forecasting models, an attempt to synthesise a good decision policy was conducted. The policy objective was defined to be the ratio between Resident Engineers and EP Engineers. To achieve this policy objective, a policy 110 lever called the Approval Fraction was used. The Approval Fraction is the fraction of EP Engineers that would be allowed into the system. The fuzzy decision policy will control the Approval Fraction so as to achieve the policy objective. Different inputs were used to calibrate the fuzzy decision policy. For each input, the fuzzy decision policy was first calibrated under a deterministic setting where there is no noise. Two optimisation criteria were used: Root Mean Squared Error and the Worst Absolute Error. It was found that the calibrated decision policy model for each scenario is able to satisfy the policy objective. Noise was subsequently added to test for robustness. The fuzzy decision policies calibrated under a noiseless setting are not able to handle the uncertainty and noise introduced. Thus, a new way to calibrate for robustness has to be explored. The system was shown to demonstrate monotonic behaviour and thus was assumed to be so. Noise is added in the form of α*σ, where α is a constant and σ represents one standard deviation of noise. As a result of the assumption of monotonic behaviour, a fuzzy decision policy that is able to handle noise α*σ will be able to handle any noise below α*σ. The fuzzy decision policy is then calibrated by considering noise through particle swarm optimisation. The maximum α for each particle in the particle swarm is found by testing the extreme points. In the study, there are three noisy parameters and thus eight extreme points. The PSO will then find the best set of parameters which allows us to obtain the maximum α possible. It was found that after calibration, the fuzzy decision policies became more robust for each type of input. They were able to tolerate higher levels of noise than previously possible. Although better results 111 were obtained, more time was needed to compute and find the set of optimal parameters. A Linear Programming model was then used to approximate the model. This is because a LP model can be solved quickly to find the maximum α for a set of parameters. Thus, this should reduce the computational time regardless of the number of noisy parameters. It was found that the fuzzy decision policy calibrated using the LP model is less robust than the one calibrated using the original model. This is expected as the LP model is only an approximation. However, the fuzzy decision policy found is useful and can serve as a quick reference if needed. 6.2 Contributions Firstly, a hypothesis of the structure of the engineering manpower supply and demand system was proposed and validated using historical data. Scenario analysis can then be carried out to study the system behaviour. Secondly, the prototype model provides a basis and framework for the construction of models for similar systems. Thirdly, policy makers may be interested in the knowledge gained about the system structure. Manpower and Education policy makers may only be experts in their respective domains. Hence, insight about the system structure may shed light on how their separate actions often affect one another and thus weakening or over strengthening the desired effect of their policies. This can lead to more effective policies from collaboration between manpower and education policymakers to 112 create combined manpower and education policies to achieve an overall impact on the system. Next, a fuzzy expert system approach was shown to be capable of replacing unknown parts of the system structure. Fuzzy expert systems also replicate forecasting models well. Hence, the approach may be adopted in the future if similar problems arise so as to expedite the modelling process. Lastly, fuzzy expert systems allow us to obtain a good decision policy with relative ease. There is no need for complex or sophisticated methods in carrying out the method. However, if the system is complex and the noisy parameters are numerous, the computational time needed may become extremely long. Hence, the use of linear programming models to approximate large scale models may be a way to quicken the optimisation process. 6.3 Limitations As discussed in the previous chapters, a recurring limitation in this study was the lack of the historical data for calibration and validation. Although this is by no means a suggestion of a flaw or mistake in the modelling methodology, more historical data would ensure that the results and analyses of the model are more meaningful and palatable to the critical eye. Due to the sensitive nature of policy making, real knowledge or mental models of policymakers are difficult to obtain. The proposed system structure is but a hypothesis and other system structures can also be proposed. The results and analyses of the model would have been more pertinent, more revealing and more 113 useful to actual policy making if real mental models about the system had been used. The prototype model is an aggregate model of the engineering supply and demand system. It is evident that people in a varied context, for instance age group or job positions etc, may react or behave differently over time. Thus, the model would have been more complete and in-depth if a lower level of aggregation was used. 6.4 Suggestion for Future Research Future research work could be to disaggregate the prototype model further to account for different industries, age groups, pay scales etc. This would allow for more in-depth knowledge about the system behaviour at a more micro level. The disaggregation effort can be helped by the collection of more data since more information about the disaggregated levels is needed. At the same time, data further into the past should be collected so as to avoid the same limitations faced by this study. Next, engineering manpower demand, engineering college admission rate and other exogenous factors can be modelled endogenously either by finding out their underlying system structure or by attempting to replace them using A.I. techniques. This would be interesting as more knowledge can obtained and bigger picture scenario analyses can be conducted. The prototype model can be applied to a different country to test whether the system structure is country specific. If it is, the differences can be studied and analysed and a more generic hypothesis can be proposed. 114 REFERENCES Sterman, J. D. (2000) Business Dynamics: Systems Thinking and Modeling for a Complex World. (Boston: Irwin/McGraw-Hill) Senge, P.M. 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(1998) Predicting U.S. food 116 demand in the 20th century: A new look at System Dynamics, SPIE Proceeding Series, Vol. 3369, 343(1998) 117 Appendix A: Stock-and-Flow Diagrams Figure A.1 Stock and Flow Diagram for Manpower Adjustment Subsystem 118 Figure A.2 Stock and Flow Diagram for Job Vacancies Subsystem Figure A.3 Stock and Flow Diagram for Wage Adjustment Subsystem 119 Figure A.4 Stock and Flow Diagram for Resident Engineers Subsystem 120 Figure A.5 Stock and Flow Diagram for EP Engineers Subsystem Figure A.6 Stock and Flow Diagram for Engineering Education Subsystem 121 Appendix B: Equations for Engineering Manpower Supply and Demand Model (iThink) Engineering Education Sector Resident_Engineering_Student_Cohort(t) = Resident_Engineering_Student_Cohort(t dt) + (Engineering_Rate - College_Attrition_Rate - Graduation_Rate) * dt INIT Resident_Engineering_Student_Cohort = 12000 INFLOWS: Engineering_Rate = Input_Engineering_Rate*Resident_Fraction OUTFLOWS: College_Attrition_Rate = Graduation_Rate*College_Attrition_Fraction Graduation_Rate = DELAY(Engineering_Rate*(1-College_Attrition_Fraction),4,3375) College_Attrition_Fraction = 0 EP Engineers Sector EP_Engineers(t) = EP_Engineers(t - dt) + (EP_Engineers_Hire_Rate EP_Engineers_Layoff_Rate - EP_Engineers_Resignation_Rate EP_to_Resident_Conversion_Rate) * dt INIT EP_Engineers = 12460 INFLOWS: EP_Engineers_Hire_Rate = Vacancies/Average_Time_Needed_to_Hire_EP_Engineer*(1STEP(Fixed_Resident_Engineer_vs_EP_Engineer_Job_Ratio_switch,2009))+EP_Engineer_ Job_Vacancies/Average_Time_Needed_to_Hire_EP_Engineer*STEP(Fixed_Resident_Eng ineer_vs_EP_Engineer_Job_Ratio_switch,2009) OUTFLOWS: EP_Engineers_Layoff_Rate = MIN(Desired_Layoff_Rate*Willingness_to_Layoff_EP_Engineer,Maximum_EP_Engineers _Layoff_Rate) EP_Engineers_Resignation_Rate = MAX(0,EP_Engineers/Average_EP_Engineer_Employment_Duration) 122 EP_to_Resident_Conversion_Rate = EP_Engineers*EP_to_Resident_Conversion_Fraction Average_EP_Engineer_Employment_Duration = Base_EP_Engineer_Employment_DurationSTEP(Step_Decrease_in_Average_EP_Engineer_Employment_Duration,2009) Average_Time_Needed_to_Hire_EP_Engineer = Base_Time_Needed_to_Hire_EP_Engineer+STEP(Step_Increase_in_Average_Time_Need ed_to_Hire_EP_Engineer,2009) Average_Time_Needed_to_Layoff_Engineer = 1/12 Base_EP_Engineer_Employment_Duration = 3 Base_Time_Needed_to_Hire_EP_Engineer = 6/12 EP_to_Resident_Conversion_Fraction = 0.1 Maximum_EP_Engineers_Layoff_Rate = EP_Engineers/Average_Time_Needed_to_Layoff_Engineer Willingness_to_Layoff_EP_Engineer = 1 Inputs and Data Accelerated_Growth = 0 Accelerated_Growth_Simulator = RAMP(5000,2009)-RAMP(5000,2015) Big_Recession = 0 Big_Recession_Simulator = RAMP(-5000,2009)-RAMP(-5000,2015) Constant_Engineering_Rate = 3375 Constant_Engineering_Rate_Switch = 0 Engineering_Manpower_Demand = Engineering_Manpower_Target_2+ Manpower_Target_Ramp_Simulator*Manpower_Ramp_Switch+ Recession*Recession_Simulator+ Big_Recession*Big_Recession_Simulator+ Stagnant*0+ Accelerated_Growth*Accelerated_Growth_Simulator+ 123 Growth*Growth_Simulator Engineering_Rate_Step_Size = 0 Growth = 0 Growth_Simulator = RAMP(2500,2009)-RAMP(2500,2015) Input_Engineering_Rate = Real_Engineering_Rate*Real_Engineering_Rate_Switch + Constant_Engineering_Rate*Constant_Engineering_Rate_Switch + Step_Engineering_Rate*Step_Engineering_Rate_Switch+ Ramp_Engineering_Rate*Ramp_Engineering_Rate_Switch Manpower_Ramp_Gradient = 0 Manpower_Ramp_Switch = 1 Manpower_Target_Ramp_Simulator = RAMP(Manpower_Ramp_Gradient,2009)RAMP(Manpower_Ramp_Gradient,2015) Ramp_Engineering_Rate = STEP(Engineering_Rate_Ramp_Gradient,2009) STEP(Engineering_Rate_Ramp_Gradient,2015) Ramp_Engineering_Rate_Switch = 0 Real_Engineering_Rate_Switch = 1 Recession = 0 Recession_Simulator = RAMP(-2500,2009)-RAMP(-2500,2015) Stagnant = 0 Step_Decrease_in_Average_EP_Engineer_Employment_Duration = 0 Step_Engineering_Rate = STEP(Engineering_Rate_Step_Size,2009) Step_Engineering_Rate_Switch = 0 Step_Increase_in_Average_Time_Needed_to_Hire_EP_Engineer = 0 Engineering_Manpower_Target_2 = GRAPH(TIME) Real_Engineering_Rate = GRAPH(TIME) Real_Engineering_Wages = GRAPH(TIME) Real_EP_Engineers = GRAPH(TIME) 124 Real_Resident_Engineers = GRAPH(TIME) Manpower Adjustment Sector Expected_Attrition_Rate(t) = Expected_Attrition_Rate(t - dt) + (Change_in_Expected_Attrition_Rate) * dt INIT Expected_Attrition_Rate = Attrition_Rate INFLOWS: Change_in_Expected_Attrition_Rate = (Attrition_RateExpected_Attrition_Rate)/Expected_Attrition_Rate_Smoothing_Time Perceived_Gap(t) = Perceived_Gap(t - dt) + (Change_in_Perceived_Gap) * dt INIT Perceived_Gap = Engineering_Manpower_Gap INFLOWS: Change_in_Perceived_Gap = (Engineering_Manpower_GapPerceived_Gap)/Perceived_Gap_Smoothing_Time Adjustment_for_Manpower = Perceived_Gap/Desired_Manpower_Adjustment_Time Attrition_Rate = EP_Engineers_Resignation_Rate+Resident_Engineer_Resignation_Rate+Resident_Engin eer_Retirement_Rate Desired_Hiring_Rate = Adjustment_for_Manpower+Expected_Attrition_Rate Desired_Layoff_Rate = MAX(0,-Desired_Hiring_Rate) Desired_Manpower_Adjustment_Time = 2/12 Engineering_Manpower_Gap = Engineering_Manpower_DemandEngineering_Manpower_Supply Engineering_Manpower_Supply = EP_Engineers+Resident_Engineers Expected_Attrition_Rate_Smoothing_Time = 1 Normalised_Gap = Perceived_Gap/Reference_Gap Perceived_Gap_Smoothing_Time = 2/12 Reference_Gap = 1000 Resident Engineers and Education Sector 125 Resident_Engineers(t) = Resident_Engineers(t - dt) + (Resident_Engineer_Hire_Rate + Residency_Take_Up_Rate - Resident_Engineer_Layoff_Rate Resident_Engineer_Resignation_Rate - Resident_Engineer_Retirement_Rate) * dt INIT Resident_Engineers = 50185 INFLOWS: Resident_Engineer_Hire_Rate = Vacancies/Average_Time_Needed_to_Hire_Resident_Engineer*(1STEP(Fixed_Resident_Engineer_vs_EP_Engineer_Job_Ratio_switch,2009)) +Resident_Engineer_Job_Vacancies/Average_Time_Needed_to_Hire_Resident_Enginee r*STEP(Fixed_Resident_Engineer_vs_EP_Engineer_Job_Ratio_switch,2009) Residency_Take_Up_Rate = EP_to_Resident_Conversion_Rate OUTFLOWS: Resident_Engineer_Layoff_Rate = MIN(Desired_Layoff_Rate,Maximum_Layoff_Rate) Resident_Engineer_Resignation_Rate = MAX(0,Resident_Engineers/Average_Resident_Engineer_Employment_Duration) Resident_Engineer_Retirement_Rate = MAX(0,Resident_Engineers)/Average_Resident_Engineer_Career_Duration Resident_Jobseekers(t) = Resident_Jobseekers(t - dt) + (Resident_Engineer_Layoff_Rate + Resident_Engineer_Resignation_Rate + Fresh_Graduates_Entrance_Rate Leakage_Rate - Resident_Engineer_Hire_Rate) * dt INIT Resident_Jobseekers = 700 INFLOWS: Resident_Engineer_Layoff_Rate = MIN(Desired_Layoff_Rate,Maximum_Layoff_Rate) Resident_Engineer_Resignation_Rate = MAX(0,Resident_Engineers/Average_Resident_Engineer_Employment_Duration) Fresh_Graduates_Entrance_Rate = Graduation_Rate OUTFLOWS: Leakage_Rate = Resident_Jobseekers*Leakage_Fraction Resident_Engineer_Hire_Rate = Vacancies/Average_Time_Needed_to_Hire_Resident_Engineer*(1STEP(Fixed_Resident_Engineer_vs_EP_Engineer_Job_Ratio_switch,2009)) 126 +Resident_Engineer_Job_Vacancies/Average_Time_Needed_to_Hire_Resident_Enginee r*STEP(Fixed_Resident_Engineer_vs_EP_Engineer_Job_Ratio_switch,2009) Average_Resident_Engineer_Career_Duration = 45 Average_Resident_Engineer_Employment_Duration = 6 Average_Time_Needed_to_Hire_Resident_Engineer = 2/12 Base_Fractional_Leakage_Rate = 0.4 Leakage_Fraction = Base_Fractional_Leakage_Rate*Effect_of_Relative_Wage_on_Leakage_Rate Maximum_Layoff_Rate = Resident_Engineers/Average_Time_Needed_to_Layoff_Engineer Effect_of_Relative_Wage_on_Leakage_Rate = GRAPH(Relative_Wage) (0.00, 2.49), (0.5, 2.20), (1.00, 1.00), (1.50, 0.675), (2.00, 0.537), (2.50, 0.35), (3.00, 0.263), (3.50, 0.188), (4.00, 0.0875), (4.50, 0.0375), (5.00, 0.00) Vacancies Sector EP_Engineer_Job_Vacancies(t) = EP_Engineer_Job_Vacancies(t - dt) + (EP_Engineer_Job_Vacancy_Creation_Rate - EP_Engineer_Job_Vacancy_Rate EP_Engineer_Job_Vacancy_Cancellation_Rate) * dt INIT EP_Engineer_Job_Vacancies = 1895*Resident_Engineer_vs_EP_Engineer_Job_Ratio INFLOWS: EP_Engineer_Job_Vacancy_Creation_Rate = Desired_Vacancy_Creation_Rate*(1Resident_Engineer_vs_EP_Engineer_Job_Ratio)*(STEP(Fixed_Resident_Engineer_vs_EP_ Engineer_Job_Ratio_switch,2009)) OUTFLOWS: EP_Engineer_Job_Vacancy_Rate = EP_Engineers_Hire_Rate*STEP(Fixed_Resident_Engineer_vs_EP_Engineer_Job_Ratio_sw itch,2009) EP_Engineer_Job_Vacancy_Cancellation_Rate = Desired_Vacancy_Cancellation_Rate*(1Resident_Engineer_vs_EP_Engineer_Job_Ratio) Resident_Engineer_Job_Vacancies(t) = Resident_Engineer_Job_Vacancies(t - dt) + (Resident_Engineer_Job_Vacancy_Creation_Rate - 127 Resident_Engineer_Job_Vacancy_Closure_Rate Resident_Engineer_job_Vacancy_Cancellation_Rate) * dt INIT Resident_Engineer_Job_Vacancies = 1895*Resident_Engineer_vs_EP_Engineer_Job_Ratio INFLOWS: Resident_Engineer_Job_Vacancy_Creation_Rate = Desired_Vacancy_Creation_Rate*Resident_Engineer_vs_EP_Engineer_Job_Ratio*STEP(F ixed_Resident_Engineer_vs_EP_Engineer_Job_Ratio_switch,2009) OUTFLOWS: Resident_Engineer_Job_Vacancy_Closure_Rate = Resident_Engineer_Hire_Rate*STEP(Fixed_Resident_Engineer_vs_EP_Engineer_Job_Rati o_switch,2009) Resident_Engineer_job_Vacancy_Cancellation_Rate = Desired_Vacancy_Cancellation_Rate*Resident_Engineer_vs_EP_Engineer_Job_Ratio Vacancies(t) = Vacancies(t - dt) + (Vacancy_Creation_Rate - Vacancy_Closure_Rate Vacancy_Cancellation_Rate) * dt INIT Vacancies = 930 INFLOWS: Vacancy_Creation_Rate = MAX(0,Desired_Vacancy_Creation_Rate)*(1STEP(Fixed_Resident_Engineer_vs_EP_Engineer_Job_Ratio_switch,2009)) OUTFLOWS: Vacancy_Closure_Rate = (EP_Engineers_Hire_Rate+Resident_Engineer_Hire_Rate)*(1STEP(Fixed_Resident_Engineer_vs_EP_Engineer_Job_Ratio_switch,2009)) Vacancy_Cancellation_Rate = MIN(Maximum_Vacancy_Cancellation_Rate,Desired_Vacancy_Cancellation_Rate)*(1STEP(Fixed_Resident_Engineer_vs_EP_Engineer_Job_Ratio_switch,2009)) Desired_Vacancy_Cancellation_Rate = MAX(0,-Desired_Vacancy_Creation_Rate) Desired_Vacancy_Creation_Rate = Desired_Hiring_Rate Fixed_Resident_Engineer_vs_EP_Engineer_Job_Ratio_switch = 0 Maximum_Vacancy_Cancellation_Rate = Vacancies/Time_needed_to_cancel_Vacancy Resident_Engineer_vs_EP_Engineer_Job_Ratio = 0.5 128 Time_needed_to_cancel_Vacancy = 0.5/12 Wage Adjustment Sector Engineering_Wage(t) = Engineering_Wage(t - dt) + (Change_in_Engineering_Wage) * dt INIT Engineering_Wage = 3551 INFLOWS: Change_in_Engineering_Wage = (Desired_Engineering_WageEngineering_Wage)/Wage_Adjustment_Smoothing_Time Desired_Engineering_Wage = Effect_of_Gap_on_Desired_Engineering_Wage*Engineering_Wage Input_Non_Engineering_Wage = 4000 + RAMP(Non_Engineering_Wage_Ramp_Gradient,2009) - RAMP (Non_Engineering_Wage_Ramp_Gradient,2013) Non_Engineering_Wages = Input_Non_Engineering_Wage Non_Engineering_Wage_Ramp_Gradient = 0 Relative_Wage = Engineering_Wage/Non_Engineering_Wages Wage_Adjustment_Smoothing_Time = 1 Effect_of_Gap_on_Desired_Engineering_Wage = GRAPH(Normalised_Gap) (-10.0, 0.00), (-9.50, 0.00), (-9.00, 0.04), (-8.50, 0.07), (-8.00, 0.13), (-7.50, 0.19), (-7.00, 0.3), (-6.50, 0.45), (-6.00, 0.57), (-5.50, 0.67), (-5.00, 0.75), (-4.50, 0.82), (-4.00, 0.85), (3.50, 0.9), (-3.00, 0.94), (-2.50, 0.94), (-2.00, 0.96), (-1.50, 0.97), (-1.00, 0.98), (-0.5, 1.00), (0.00, 1.00), (0.5, 1.06), (1.00, 1.06), (1.50, 1.06), (2.00, 1.06), (2.50, 1.12), (3.00, 1.14), (3.50, 1.18), (4.00, 1.19), (4.50, 1.20), (5.00, 1.22), (5.50, 1.23), (6.00, 1.24), (6.50, 1.26), (7.00, 1.28), (7.50, 1.33), (8.00, 1.37), (8.50, 1.43), (9.00, 1.48), (9.50, 1.59), (10.0, 1.67) Not in a sector Engineering_Rate_Ramp_Gradient = 0 129 Appendix C: One Way Sensitivity Analysis Figure C. 1 Average Resident Engineer Career Duration Figure C. 2 Average Resident Engineer Job Duration 130 Figure C. 3 Average EP Engineer Employment Duration Figure C. 4 Average Time Needed to Hire a Resident Engineer 131 Figure C. 5 Average Time Needed to Hire an EP Engineer Figure C. 6 Percentage of EP Engineers to Resident 132 Appendix D: Equations for Supply Chain Model (iThink) Engineering_Student_Cohort(t) = Engineering_Student_Cohort(t (Engineering_Admission_Rate - Graduating_Rate) * dt - dt) + INIT Engineering_Student_Cohort = Desired_Engineering_Student_Cohort INFLOWS: Engineering_Admission_Rate = Indicated_Admission_Rate OUTFLOWS: Graduating_Rate = Engineering_Student_Cohort/Graduation_Lag Resident_Jobseekers(t) = Resident_Hiring_Rate) * dt Resident_Jobseekers(t - dt) + (Graduating_Rate - INIT Resident_Jobseekers = 100 INFLOWS: Graduating_Rate = Engineering_Student_Cohort/Graduation_Lag OUTFLOWS: Resident_Hiring_Rate = Job_Vacancies/Time_Needed_to_Hire_Resident_Engineer Adjustment_for_Stock = (Desired_Resident_JobseekersResident_Jobseekers)/Resident_Jobseekers_Adjustment_Time Adjustment_for_Student_Cohort = (Desired_Engineering_Student_CohortEngineering_Student_Cohort)/Student_Cohort_Adjustment_Time Desired_Admission_Rate = Adjustment_for_Stock+Expected_Hiring_Rate Desired_Engineering_Student_Cohort = Desired_Admission_Rate*Graduation_Lag Desired_Resident_Jobseekers = 100 Expected_Hiring_Rate = Resident_Hiring_Rate Graduation_Lag = 4 Increase_in_Job_Vacancies = STEP(-0.076659186,5)+ STEP(0.090918951,25)+ 133 STEP(0.099212266,45)+ STEP(-0.005225684,65)+ STEP(-0.021135969,85)+ STEP(-0.039995384,105)+ STEP(-0.016515175,125)+ STEP(-0.054663841,145)+ STEP(-0.089321595,165)+ STEP(-0.079112602,185)+ STEP(0.022630927,205)+ STEP(0.075149899,225)+ STEP(-0.080937321,245)+ STEP(-0.056377514,265)+ STEP(0.036312507,285)+ STEP(-0.067193769,305)+ STEP(-0.062215011,325)+ STEP(-0.031092271,345)+ STEP(0.013087397,365)+ STEP(0.030135728,385)+ STEP(0.073644824,405)+ STEP(-0.068674593,425)+ STEP(-0.071694605,445)+ STEP(0.041834657,465)+ STEP(0.033879087,485)+ STEP(-0.064414467,505)+ STEP(-0.096350565,525)+ STEP(0.076275034,545)+ 134 STEP(-0.084087628,565)+ STEP(0.079913142,585)+ STEP(-0.075238021,605)+ STEP(0.05611526,625)+ STEP(0.096816835,645)+ STEP(-0.088353501,665)+ STEP(0.064418828,685)+ STEP(-0.064572306,705)+ STEP(0.064886142,725)+ STEP(-0.062995963,745)+ STEP(-0.025082719,765)+ STEP(-0.090097429,785)+ STEP(0.029061962,805)+ STEP(0.077164833,825)+ STEP(-0.003405806,845)+ STEP(0.014790499,865)+ STEP(0.063149093,885)+ STEP(-0.090313161,905)+ STEP(0.02594251,925)+ STEP(0.024154861,945)+ STEP(0.008210307,965)+ STEP(0.08923504,985)+ STEP(0.026161529,1005)+ STEP(0.093424377,1025)+ STEP(-0.004179313,1045)+ STEP(0.09463104,1065)+ 135 STEP(0.095949241,1085)+ STEP(-0.036548126,1105)+ STEP(0.083398411,1125)+ STEP(0.047987285,1145)+ STEP(0.056397441,1165)+ STEP(0.018045889,1185)+ STEP(-0.014418319,1205)+ STEP(0.029297415,1225)+ STEP(-0.060036183,1245)+ STEP(0.01195766,1265)+ STEP(0.028926198,1285)+ STEP(0.034989258,1305)+ STEP(-0.087227185,1325)+ STEP(-0.011110701,1345)+ STEP(-0.024004974,1365)+ STEP(-0.060380625,1385)+ STEP(0.059934226,1405)+ STEP(-0.092684048,1425)+ STEP(-0.01840596,1445)+ STEP(0.099249377,1465)+ STEP(0.043254494,1485)+ STEP(0.073722501,1505)+ STEP(0.010718202,1525)+ STEP(-0.002714666,1545)+ STEP(0.098600895,1565)+ STEP(0.017060387,1585)+ 136 STEP(-0.039912786,1605)+ STEP(-0.038380148,1625)+ STEP(-0.071083586,1645)+ STEP(-0.021023337,1665)+ STEP(-0.032779994,1685)+ STEP(-0.020645386,1705)+ STEP(-0.044354614,1725)+ STEP(-0.096539182,1745)+ STEP(0.011461616,1765)+ STEP(-0.027018323,1785)+ STEP(-0.023140189,1805)+ STEP(0.038234998,1825)+ STEP(0.01267485,1845)+ STEP(-0.065939164,1865)+ STEP(-0.006187541,1885)+ STEP(0.095816689,1905)+ STEP(-0.01447732,1925)+ STEP(-0.023632264,1945)+ STEP(0.05437677,1965)+ STEP(0.058425518,1985) Indicated_Admission_Rate = Adjustment_for_Student_Cohort+Desired_Admission_Rate Job_Vacancies = (1+Increase_in_Job_Vacancies)*100 Percentage_Change_in_Demand = RANDOM(-0.1,0.1) Resident_Jobseekers_Adjustment_Time = 1 Student_Cohort_Adjustment_Time = 0.75 Time_Needed_to_Hire_Resident_Engineer = 2/12 137 Appendix E: Training Data Generated by Supply Chain Model negative %Change (Resident Jobseekers) 40 30 20 10 0 0 500 1000 1500 -10 -20 -30 Figure E. 1 Percentage Change in Resident Jobseekers %Change (Resident Hire Rate) 50 40 30 20 10 0 -10 0 200 400 600 800 1000 1200 1400 1600 -20 -30 -40 Figure E. 2 Percentage Change in Resident Hiring Rate 1800 138 %Change (Resident Student Cohort) 20 15 10 5 0 0 500 1000 1500 -5 -10 -15 -20 Figure E. 3 Percentage Change in Resident Student Cohort %Change (Engineering Admission Rate) 100 80 60 40 20 0 0 500 1000 1500 -20 -40 -60 Figure E. 4 Percentage Change for Engineering College Admission Rate 139 Appendix F: Membership Function Parameters for Fuzzy Expert System (Supply Chain Model) Membership Function Number 1 2 3 4 5 6 7 8 9 10 11 Left -60 -33.6 7.38 -30 -6.02 3.33 -95 -51.34 -45 4.22 27.62 Centre -25.0 0 21.81 -11.63 0 11 -47.02 -18.14 0 45 53.210 Right -4.54 33.6 60 -4.95 6.02 30 -38.07 -1.77 45 55 95 Table F. 1 Optimised Membership Functions (Supply Chain Model) 140 Appendix G: MATLAB Model of Engineering Manpower Supply and Demand System Figure G. 1 Manpower Adjustment Subsystem 141 Figure G. 2 Job Vacancies Subsystem Figure G. 3 Wage Adjustment Subsystem 142 Figure G. 4 Resident Engineers Subsystem 143 Figure G. 5 EP Engineers Subsystem Figure G. 6 Engineering Education Subsystem 144 Appendix H: MATLAB Model of Fuzzy Expert System Figure H. 1 MATLAB Model of Fuzzy Expert System 145 Appendix I: Membership Function Parameters for Fuzzy Expert System (Prototype Model) Membership Function Number 1 2 3 4 5 6 7 8 9 Left Centre Right -25.00 -6.40 0.00 -21.79 -12.00 0.00 -18.41 -14.71 12.00 -8.21 0.00 9.00 -10.00 0.00 12.00 -13.00 -6.58 33.18 -0.12 3.08 25.00 -1.22 9.81 25.00 -2.87 20.29 34.73 Table I. 1 Membership Function Parameters (1st Estimate, Prototype Model) Membership Function Number 1 2 3 4 5 6 7 8 9 Left Centre Right -25.00 -4.26 0.12 -25.00 -12.00 2.00 -16.15 -15.49 12.43 -6.75 0.00 10.16 -10.54 0.00 11.65 -14.00 -7.72 32.53 -2.00 3.13 25.00 -0.93 9.35 25.00 -3.70 19.97 34.23 Table I. 2 Membership Function Parameters (Optimised, Prototype Model) 146 Appendix J: Test Inputs for Fuzzy Forecasting Model Figure J. 7 Cyclical Demand Figure J. 8 Saw-tooth Demand 147 Figure J. 9 Pulse Shape Demand Figure J. 10 Mixed Shape Demand 148 Appendix K: Engineering Manpower Demand Inputs Figure K.1 Cyclical Input Figure K.2 Cyclical Ramp Input 149 Figure K.3 Ramp Input 150 Appendix L: Model Output from Extreme Points (Ramp Input) 1.15 1.1 Output 1 1.05 Output 2 Output 3 1 Output 4 0.95 Output 5 Output 6 0.9 Output 7 0.85 Output 8 0.8 0 20 40 60 80 100 Figure L. 1 Model Output from Extreme Points (Ramp Input) 151 Appendix M: Simplified Model (Simulink) Figure M. 1 Simplified Model ( Simulink version) 152 Appendix N: MATLAB Code for LP Model function [a, RE, EP] = LPModel4a( cgrad,dgrad,lowerlimit,upperlimit) % begin rome tic; rome_begin; h = rome_model ( 'Engineering Manpower'); % Create Rome Model T=361; % trial time horizon % Define decision variables d1=zeros(T,1); RE1=zeros(T+1,1); EP1=zeros(T+1,1); newvar a %alpha to be maximized newvar RE(T+1) %nominal RE plus change in RE newvar EP(T+1) %nominal EP plus change in EP newvar b1(T+1) %change in RE newvar b2(T+1) %change in EP newvar a21(T+1) %auxiliary variable for change in RE newvar a22(T+1) %auxiliary variable for change in RE rome_maximize(a); %objective maximize alpha rome_box(a,lowerlimit,upperlimit); % trial values for the possible range of alpha %nominal demand (d1(t)) and change in demand (d1(t)) %change in demand for period 1:T %creation of demand input delay = 80; for i = 1:delay d1(i)=60000; end for t=delay+1:T d1(t)=3000*sin(((t-delay)*.125/3))+60000; end %Parameters for nominal RE and EP tthe=.5; aejd=3; E2Rf=.1; tthr=.16667; arcd=45; arjd=6; af=1; %nominal values (initial for ER and EP) RE1(1)=52866; EP1(1)=7134; % constraint for the nominal values of RE and EP (denoted by RE1 and EP1) %d1 is also the nominal value of demand for t=2:T 153 RE1(t)=(RE1(t-1)*(1-(1/tthr*0.125)-(1*0.125/arjd)(1*0.125/arcd))+EP1(t-1)*(E2Rf*0.125(1*0.125/tthr))+(1*0.125/tthr)*d1(t-1)); EP1(t)=(RE1(t-1)*(-1*0.125/tthe)*af+EP1(t-1)*(1-((af)*0.125/tthe)(1*0.125/aejd)-(E2Rf*0.125))+(af*0.125/tthe)*d1(t-1)); end c=cgrad; d= dgrad; %c=1.6374e-05; %d=-1.66976e-05; %auxiliary equations for change in RE and EP for t=delay+1:T rome_constraint(a21(delay)==0); %for EP rome_constraint(a21(t)==((((-c*RE1(t-1))/tthe)+(1-((d*EP1(t1))/tthe)-(1/aejd)-(E2Rf)))^(t-delay))*((1/tthe)*a*500)+(-EP1(t1)*a*0.001)); % change in EP rome_constraint(a22(delay)==0); % change in RE rome_constraint(a22(t)==((1/tthr)*a*500)+(EP1(t-1)*a*0.001)); end % constraints for change in RE and EP rome_constraint(b1(delay)==0); rome_constraint(b2(delay)==0); for t=delay+1:T rome_constraint(b1(t)==a22(t-1)+a*500); rome_constraint(b2(t)==a21(t-1)+a*500); end %RE %EP for t=delay+1:T rome_constraint(-1=b1(t)*c+d*b2(t)+b1(t-1)*c+d*b2(t-1)); end %constraint for total RE and EP (nominal plus change) rome_constraint(RE(1)==52866); rome_constraint(EP(1)==7143); for t=2:delay rome_constraint(RE(t)==(RE1(t))); rome_constraint(EP(t)==(EP1(t))); end for t=delay+1:T rome_constraint(RE(t)==(RE1(t)+b1(t))); rome_constraint(EP(t)==(EP1(t)+b2(t))); end %constraint for range rome_constraint(0=0); rome_constraint(RE(1:T)>=0); rome_constraint(EP(1:T)>=0); % solve h.solve; a =h.objective; EP=h.eval(((EP(1:T)))); RE=h.eval(((RE(1:T)))); b1=h.eval(((b1(1:T)))); 154 b2=h.eval(((b2(1:T)))); rome_end; % % % % % % disp(a); a b1 b2 EP RE toc; [...]... application of A.I techniques in system dynamics is relatively new and the research in the area is somewhat limited 1.3 Objectives This project aims to build a prototype model of the engineering manpower supply and demand system using a system dynamics approach This prototype model will take into account the market supply and demand dynamics as well as the human aspects of supply and demand It is hoped... concerning manpower supply and demand, the human element within the system and some possible A.I techniques that can be applied 2.1 Perspectives related to Manpower Supply and Demand 2.1.1 Economics Perspective Economists seek to explain the sufficiency of manpower in different sectors and its effect on economic development and growth Some of the common theories or models are the Theory of Markets... according to gender and race 10 2.1.3 System Dynamics Perspective Some system dynamics models for manpower supply and demand have been built and can be used as a guide to the important parameters within the model Park, Yeon and Kim (2008) built a manpower planning model for the information security industry of Korea In their paper, they have built a hypothesized manpower demand -supply system using... could be due to various soft factors such as perceived wage equity between competing 1 2 professions, job prestige etc Hence, it is clear that when looking at the engineering manpower system, it is important to not only consider the economics of demand and supply, but the social element as well 1.2 Problem Statement While the importance of the engineering manpower supply and demand is evident, knowledge... dynamics of engineering manpower adjustment between demand and supply can be described as shown in the Figure 3.2 Figure 3 2 Causal Loop Diagram for Manpower Adjustment Engineering Manpower Demand can be understood as the number of engineers required by the market at a given point in time This variable is assumed to be exogenous in the model Correspondingly, Engineering Manpower Supply is the number of engineers... attractiveness of a profession plays an important role in determining the supply of manpower to the workforce This is especially true for highly skilled labour where the opportunity cost of training is high The supply of manpower is affected by two main factors, the retention of manpower and students joining the labour pool Firstly, it is a widely accepted fact that higher wages result in lower worker turnover and. .. circumvent the problem The application of A.I tools shows a lot of promise and potential in helping the modelling process become smoother or more constructive Chapter 3 Prototype Model This chapter details the building of the prototype model using system dynamics methodology A hypothesis of the system structure for engineering manpower supply and demand is proposed and its results are compared with historical... relationships and equations between them 2.3 Modelling: A Synthesis of Ideas Existing system dynamics and economics models on manpower planning have been reviewed It is possible to draw inspiration from them and to calibrate and change these models to suit the purposes of the project Humans are very much a part of real world systems and human decision making play a key role in the behaviour of such systems... Engineering Manpower Gap is the difference between Engineering Manpower Demand and Engineering Manpower Supply This gap is interpreted as the shortfall/excess in the number of engineers Also, it is unrealistic to expect employers in the market to be aware of the exact manpower gap at any given time Hence, the variable Perceived Manpower Gap is to model the difference between employers‟ perceived engineering manpower. .. theories and models focus on the supply and demand for labour with equilibrium being determined by hard facts such as wages and growth They neglect to address the possible interactions between individual actors in the system, for example students and schools, graduating students and career choice, foreign workers and local workforce etc These interactions can be of specific interest to policy makers and ... concerning manpower supply and demand, the human element within the system and some possible A.I techniques that can be applied 2.1 Perspectives related to Manpower Supply and Demand 2.1.1 Economics... a prototype model of the engineering manpower supply and demand system using a system dynamics approach This prototype model will take into account the market supply and demand dynamics as well... Figure H.1 MATLAB Model of Fuzzy Expert System 144 Figure J.1 Cyclical Demand 146 Figure J.2 Saw-tooth Demand 146 Figure J.3 Pulse Shape Demand 147 Figure J.4 Mixed Shape Demand 147 Figure K.1 Cyclical