Soft Computing Evaluation Logic
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A novel approach to decision engineering, with a verified framework for modeling human reasoning

Soft Computing Evaluation Logic provides an in-depth examination of evaluation decision problems and presents comprehensive guidance toward the use of the Logic Scoring of Preference (LSP) method in modeling complex decision criteria. Fully aligned with current developments in computational intelligence, the discussion covers the design and use of LSP criteria for evaluation and comparison in diverse areas, such as search engines, medical conditions, real estate, space management, habitat mitigation projects in ecology, and land use and residential development suitability maps, with versatile transfer to other similar decision-modeling contexts.

Human decision making is rife with fuzziness, imprecision, uncertainty, and half-truths—yet humans make evaluation decisions every day. In this book, such decision processes are observed, analyzed, and modeled. The result is graded logic, a soft computing mathematical infrastructure that provides both formal logic and semantic generalizations of classical Boolean logic. Graded logic is used for logic aggregation in the context of evaluation models consistent with observable properties of human reasoning.  The LSP method, based on graded logic and logic aggregation, is a vital component of an industrial-strength decision engineering framework. Thus, the book:

  • Provides detailed theoretical background for graded logic
  • Provides a theory of logic aggregators
  • Explains the LSP method for designing complex evaluation criteria and their use
  • Shows techniques for evaluation, comparison, and selection of complex systems, as well as the cost/suitability analysis, optimization, sensitivity analysis, tradeoff analysis, and missingness-tolerant aggregation
  • Includes a survey of available LSP software tools, including ISEE, ANSY and LSP.NT.

With quantitative modeling of human reasoning, novel approaches to modeling decision criteria, and a verified decision engineering framework applicable to a broad array of applications, this book is an invaluable resource for graduate students, researchers, and practitioners working within the decision engineering realm.


JOZO DUJMOVIĆ, ScD, is a professor of computer science and former chair of the Computer Science Department at San Francisco State University, where he teaches and researches soft computing, software metrics, and computer performance evaluation. He is the author of more than 170 refereed publications and the founder and principal of SEAS, a company specializing in soft computing decision models and software support for the LSP method.


Preface xvii

About the Companion Website xxiii

Previous Publications xxiv

Acknowledgments xxv

List of Symbols and Abbreviations xxvii


1.1 Intuitive Evaluation as a Logic Decision Process 5

1.1.1 Main Observable Steps of the Intuitive Evaluation Process 6

1.1.2 Subjective and Objective Components in Evaluation 18

1.2 Quantitative Evaluation—An Introductory Example 21

1.2.1 Stakeholders and Their Goals 21

1.2.2 Attributes 22

1.2.3 Attribute Criteria 23

1.2.4 Simple Direct Ranking 27

1.2.5 Aggregation of Attribute Suitability Degrees 29

1.2.6 Using Cost and Suitability to Compute the Overall Value 32

1.3 Drawbacks of Simple Additive and Multiplicative Scoring and Utility Models 35

1.3.1 Simple Additive Scoring: The Irresistible Attractiveness of Simplicity 36

1.3.2 Simple Multiplicative Scoring 45

1.3.3 Logic Unsuitability of Scoring and Utility Theory Models in Professional Evaluation 47

1.4 Introduction to Professional Quantitative Evaluation 51

1.4.1 Five Fundamental Types of Professional Evaluation Problems 51

1.4.2 A Survey of Typical Professional Evaluation Problems 54

1.4.3 Components of Methodology for Professional Quantitative Evaluation 58


2.1 Graded Logic as a Generalization of Classical Boolean Logic 69

2.1.1 Aggregators and Their Classification 70 Means 71 General Aggregation Functions 71 Logic Aggregators 73 Triangular Norms and Conorms 73

2.1.2 How Do Human Beings Aggregate Subjective Categories? 75

2.1.3 Definition and Classification of Logic Aggregators 85

2.1.4 Logic Bisection, Trisection, and Quadrisection of the Unit Hypercube 92

2.1.5 Propositions, Value Statements, Graded Logic, and Fuzzy Logic 95

2.1.6 Classical Bivalent Boolean Logic 100

2.1.7 Six Generalizations of Bivalent Boolean Logic 108 Expansion of Function Domain 109 Expansion of Logic Domain 111 Expansion of Annihilator Adjustability 112 Expansion of Semantic Domain 115 Expansion of Compensative Logic Functions 117 Expansion of the Range of Andness/Orness from Drastic Conjunction to Drastic Disjunction 118

2.1.8 GL Conjecture: Ten Necessary and Sufficient GL Functions 123

2.1.9 Basic Idempotent GL Aggregators 127

2.1.10 A Summary of Differences between Graded Logic and Bivalent Boolean Logic 134

2.1.11 Relationships between Graded Logic, Perceptual Computing, and Fuzzy Logic 136

2.1.12 A Brief History of Graded Logic 142

2.2 Observable Properties of Human Evaluation Logic 147

2.2.1 Perceptual Computer and Its Basic Properties 152

2.2.2 Simultaneity and Substitutability in Evaluation Models 177

2.2.3 Basic Semantic Aspects of Evaluation Logic Reasoning 190

2.2.4 Multipolarity: Grouping and Aggregation of Semantically Heterogeneous Inputs 212

2.2.5 Grouping and Aggregation of Semantically Homogeneous Inputs 218

2.2.6 Imprecision, Incompleteness, Logic Inconsistency, and Errors 222

2.3 Andness and Orness 237

2.3.1 A General Definition of Andness/Orness 237

2.3.2 Local Andness and Orness in the Simplest Case of Two Variables 239

2.3.3 Variability of Local Andness 242

2.3.4 Mean Local Andness and Orness in the Case of Two Variables 248

2.3.5 Local and Mean Local Andness and Orness in the Case of n Variables 251

2.3.6 Global Andness and Orness 253

2.3.7 Mean Global Andness/Orness Theorems and Their Applications 272

2.3.8 Geometric Interpretations of Andness and Orness 275

2.4 Graded Conjunction/Disjunction and Logic Modeling of Simultaneity and Substitutability 283

2.4.1 Definitions and Basic Mathematical Properties of Logic Aggregators 284

2.4.2 Classification of Conjunctive and Disjunctive Logic Aggregators 295

2.4.3 Properties of Means Used in Logic Aggregation 298

2.4.4 Algebraic Properties of Aggregators Based on Weighted Power Means 304

2.4.5 Logic Aggregators Based on Weighted Means with Adjustable Andness/Orness 313

2.4.6 Selection and Use of the Threshold Andness Aggregator 318

2.4.7 Andness-Directed Interpolative GCD Aggregators 327

2.4.8 Uniform and Nonuniform Interpolative GCD Aggregators 334 The Uniform Interpolative GCD Aggregator (UGCD) 334 An Extremely Soft Interpolative Aggregator 338 An Extremely Hard Interpolative Aggregator 338

2.4.9 Extending GCD to Include Hyperconjunction and Hyperdisjunction 342

2.4.10 From Drastic Conjunction to Drastic Disjunction: A General GCD Aggregator 347

2.4.11 Gamma Aggregators versus Extended GCD Aggregators 348 Multiplicative and Additive Gamma Aggregators 351 Comparison of Gamma Aggregators and GCD 355

2.4.12 Four Main Families of GCD Aggregators and Sixteen Conditions They Must Satisfy 361

2.5 The Percept of Importance and the Use of Weights 367

2.5.1 Multiplicative, Implicative, and Exponential Weights as Importance Quantifiers 369 Multiplicative Weights 370 Implicative Weights and the Weighted Conjunction/Disjunction 374 Exponential Weights 390

2.5.2 Impact of Weights on Aggregation Results 393

2.5.3 Semantic Components in Logic Aggregation Models 398

2.5.4 Seven Techniques for Weight Adjustment 402 Importance Decomposition Method 402 Direct Weight Assessment 405 Weights Based on Ranking 405 Weights Based on Menu 407 Collective Weight Determination 409 Weights Obtained from Pairwise Comparisons 411 Weights Based on Preferential Neuron Training 414

2.5.5 Multivariate Weighted Aggregation Based on Binary Aggregation Trees 417

2.6 Partial Absorption: A Fundamental Asymmetric Aggregator 429

2.6.1 Conjunctive Partial Absorption 430

2.6.2 Disjunctive Partial Absorption 436

2.6.3 Visualizing the Partial Absorption Function, Penalty, and Reward 439

2.6.4 Mathematical Models of Penalty and Reward 442

2.6.5 Selecting Parameters of Partial Absorption 449

2.7 Logic Functions That Use Negation 453

2.7.1 Negation and De Morgan’s Duality 453

2.7.2 De Morgan’s Laws for Weighted Aggregators and Dualized Weighted Aggregators 455

2.7.3 De Morgan’s Duals of Compound Functions 458

2.7.4 Nonidempotent Logic Functions 460

2.8 Penalty-Controlled Missingness-Tolerant Aggregation 463

2.8.1 Missing Data in Evaluation Problems 463

2.8.2 Penalty-Controlled Numerical Coding of Missing Data 465

2.8.3 A Penalty-Controlled Missingness-Tolerant Aggregation Algorithm 467

2.8.4 The Impact of Penalty on Missingness-Tolerant Aggregation 472

2.9 Rating Scales and Verbalization 475

2.9.1 Design of Rating Scales 476 Strict Monotonicity of Linguistic Labeling 477 Linearity of Rating Scales 483 Balance of Rating Scales 486 Cardinality of Rating Scales 488 Hybrid Rating Scales 489

2.9.2 Stepwise Refinement of Rating Scales for Andness and Orness 491

2.9.3 Scaling and Verbalizing Degrees of Importance 496

2.9.4 Scaling and Verbalizing Degrees of Suitability/Preference 497

Part Three LSP METHOD 499

3.1 An Overview of the LSP Method 501

3.1.1 Characterization of Stakeholder and Organization of an Evaluation Project 503

3.1.2 Development of the Suitability Attribute Tree 506

3.1.3 Elementary Attribute Criteria 514

3.1.4 Logic Aggregation of Suitability 519 Logic Aggregation Using Graded Conjunction/ Disjunction 523 Logic Aggregation Using Partial Absorption 526

3.1.5 Cost/Suitability Analysis and Comparison of Evaluated Objects Using Their Overall Value 536

3.1.6 Summary of Properties of the LSP Method 540

3.2 LSP Decision Engineering Framework for Professional Evaluation Projects 543

3.2.1 Participants in a Professional Evaluation Process Based on LSP DEF 544

3.2.2 Relationships between Evaluators and Domain Experts 546

3.2.3 The Structure of LSP DEF and the Corresponding Professional Evaluation Process 547

3.2.4 Predictive Nature of Evaluation Models 551

3.2.5 Interpretation of Evaluation Results 552

3.2.6 Complexity, Completeness, and Accuracy of Evaluation Models 553

3.2.7 Combining Opinions of n Experts 555 The Maximum Likelihood Estimate 555 The Expert Competence Estimate 557

3.3 Elementary Attribute Criteria 561

3.3.1 Notation of Elementary Criteria 561

3.3.2 Verbalization of Elementary Criteria 565

3.3.3 Continuous Nonlinear Elementary Criteria 566

3.3.4 Classification of Twelve Characteristic Types of Elementary Criteria 569

3.4 Aggregation Techniques and Tools 579

3.4.1 Selecting GCD Aggregators for an LSP Project 579

3.4.2 Selecting GCD Aggregators by Training Preferential Neurons 581

3.4.3 Analytic Techniques for Selecting Partial Absorption Aggregators 589 AH Version of the Conjunctive Partial Absorption Aggregator 589 AH Version of the Disjunctive Partial Absorption Aggregator 594

3.4.4 Boundary Penalty/Reward Tables for Selecting Partial Absorption Aggregators 595

3.4.5 Selecting Partial Absorption Aggregators by Training Preferential Neurons 597

3.4.6 Nonstationary LSP Criteria 602

3.4.7 Graphic Notation of Aggregation Structures 606

3.5 Canonical Aggregation Structures 611

3.5.1 Conjunctive CAS with Increasing Andness 611

3.5.2 Disjunctive CAS with Increasing Orness 614

3.5.3 Aggregated Mandatory/Optional and Sufficient/Optional CAS 616

3.5.4 Design of a Simple LSP Evaluator Tool 617

3.5.5 Distributed Mandatory/Optional and Sufficient/Optional CAS 619

3.5.6 Nested Mandatory/Desired/Optional and Sufficient/Desired/Optional CAS 621

3.5.7 Decreasing Andness and Decreasing Orness CAS 622

3.6 Cost/Suitability Analysis as a Graded Logic Problem 623

3.6.1 Cost Analysis 623

3.6.2 Cost/Suitability Analysis Based on Linear Equi-Value Model 626

3.6.3 Using Cost/Suitability Analysis in Competitive Bidding 627

3.6.4 Conjunctive Suitability-Affordability Method 630

3.7 Sensitivity Analysis and Tradeoff Analysis 635

3.7.1 Sensitivity Analysis 635 Sensitivity with Respect to Input Suitability Scores 637 Sensitivity Properties of Basic Aggregators 641 Sensitivity with Respect to Input Attributes 643

3.7.2 Tradeoff Analysis 644 Compensatory Properties of LSP Criteria and Graded Logic Aggregators 647 The Concept of Compensation Ratio 651

3.8 Reliability Analysis 655

3.8.1 Sources of Errors in LSP Criteria and Their Empirical Analysis 655

3.8.2 The Problem of Confidence in Evaluation Results 660

3.8.3 Case Study of Reliability Analysis for a Computer Evaluation Project 664

3.9 System Optimization 671

3.9.1 Three Fundamental Constrained Optimization Problems 671

3.9.2 The Cloud Diagram and the Set of Optimum Configurations 673

3.9.3 A Case Study of Computer Configuration Optimization 675

3.10 LSP Software Technology 683


4.1 Job Selection 693

4.1.1 Job Selection Attribute Tree 694

4.1.2 Elementary Attribute Criteria for Job Selection 697

4.1.3 Logic Aggregation of Suitability for the Job Selection Criterion 701

4.1.4 A Job Selection Example 705

4.2 Home Selection 711

4.2.1 Home Selection Using ORE Websites and LSPhome 711

4.2.2 Home Attribute Tree and Elementary Criteria 716

4.2.3 Home Suitability Aggregation Structure as a Shade Diagram 717

4.2.4 Using Missingness-Tolerant LSP Criteria 725

4.2.5 The Optimum Home Pricing Problem 728

4.2.6 A Personalized Home Selection Criterion 731

4.3 Evaluation of Medical Conditions 737

4.3.1 Evaluation of Disease Severity and Patient Disability 738

4.3.2 Limitations of Medical Rating Scales 740

4.3.3 LSP Models for Computing OSD, ODD, and PDD 743

4.3.4 Evaluation of PDD for Peripheral Neuropathy 745

4.3.5 The Risky Therapy Decision Problem 752

4.3.6 A Case Study of Anti-MAG Neuropathy 755

4.3.7 LSPmed—An Internet Tool for Medical Evaluation 758 LSPmed User Types and Their Functions 758 The Use of LSPmed 760 Serving a Patient 762

4.4 LSP Criteria in Ecology: Selecting Multi-Species Habitat Mitigation Projects 769

4.4.1 Multi-Species Compensatory Mitigation Projects 769

4.4.2 A Generic LSP Attribute Tree for Evaluation of Habitat Mitigation Projects 771

4.4.3 Attribute Criteria and the Logic Aggregation Structure 772

4.4.4 Sensitivity Analysis 777

4.4.5 Logic Refining of the Aggregation Structure 779

4.4.6 Cost/Suitability Analysis 781

4.4.7 MSHCP Software Support 783

4.5 Space Management Decision Problems 785

4.5.1 A Decision Model for School Location 785 Statement of the Problem 785 School Locations Attribute Tree 786 Elementary Criteria 786 Aggregation of Suitability Degrees 792 Cost Analysis 794 Competitive Locations 795 Cost/Suitability Analysis 796

4.5.2 Suitability of Locations for Residential Development 798

4.6 LSP Suitability Maps 803

4.6.1 The Concept of Map Logic and LSP Suitability Maps 803

4.6.2 Suitability Maps Based on Points of Interest 806

4.6.3 The Problem of Optimum Location of City Objects 810

4.6.4 Suitability Analysis of Urban Locations Using the LSPmap Tool 816

4.6.5 GIS-LSP Suitability Maps Based on TerrSet/Idrisi 821

4.6.6 GIS-LSP Suitability Maps Based on ArcGIS 823

4.7 Evaluation and Comparison of Search Engines 833

4.7.1 Search Engine User and Workload Models 834

4.7.2 SEben—A Search Engine Benchmarking Tool 837

4.7.3 LSP Criterion for Evaluation of Search Engines 838

4.7.4 Search Engine Evaluation Results 843

References 847

Index 871