# Theoretical Foundations for Preference Representation in Systems Engineering

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## Abstract

**:**

## 1. Introduction

^{2}matrices, functional flow diagrams, and modeling and simulations [9]. The system level requirements are decomposed into subsystem and component level requirements which are flowed down the organizational hierarchy to aid in decision-making [11,12]. These requirements only serve as constraints in the solution space, not informing about the differences between feasible solutions. Multiple US National Defense Industry Association (NDIA) reports have identified the requirements definition, development and management processes currently being practiced, as one of the top five issues in systems and software engineering [13,14]. Value-based approaches communicate preferences through special-case objective functions but assume that a preference is understood in order to form the value models [15,16,17,18,19,20,21,22,23].

- To provide formal definitions for the different types of stakeholder preferences that may be encountered in a systems engineering context.
- To prove theorems that improve understanding of how stakeholder preferences affect the solution space:
- To formally define inconsistencies in stakeholder preferences and study the effect of inconsistent preferences on the solution space;
- To understand the effect of changes in stakeholder preferences on the solution space.

## 2. Background

## 3. Preference Representation—Formalism

#### 3.1. Syntax for Modal Preference Logic

**Definition 1**

**(Modal preference language).**

#### 3.2. Semantics for Modal Preference Logic

#### 3.2.1. Preference Structure

**Definition 2**

**(Preference structure).**

**Definition 3**

**(Valuation function).**

**Definition 4**

**(Partial order/partially ordered set or poset).**

**Definition 5**

**(Total order).**

**Definition 6**

**(Betterness relation).**

#### 3.2.2. Types of Preferences

**Definition 7**

**(Attributes, propositions, and preference statements).**

**Definition 8**

**(Maximal/minimal element in a poset).**

**Definition 9**

**(Greatest/least element in a poset).**

**Definition 10**

**(Solution space).**

**Definition 11**

**(Acceptable solutions).**

**Definition 12**

**(Optimal solutions).**

_{S}) are the set of greatest elements (definition 9), i.e., highest-ranked elements, based on the betterness relation in the solution space (S) that satisfy all the preference statements that are elicited from the stakeholder.

**Definition 13**

**(Comparative preference).**

**Definition 14**

**(Absolute preference).**

**Example 1**

**(Absolute preference).**

_{1}(swept wing), which would thus be the preferred choice of the decision-maker.

**Example 2**

**(Absolute and comparative preferences).**

_{1}= swept wing, w

_{2}= rectangular wing, and w

_{3}= elliptical wing. In order to make a decision, the decision-maker has to imagine multiple worlds with each element in S as a choice by taking the propositions p and q into consideration. Let us say that the decision-maker has the following preferences.

**Definition 15**

**(Conditional preference).**

_{1}.

**Definition 16**

**(Target-oriented preferences).**

**Definition 17**

**(Design-dependent preferences).**

**Definition 18**

**(Objective-oriented preferences).**

_{1}, which is the greatest element in the solution space (S) based on the betterness relation that satisfies the preference statement ${p}_{0,1}\left[Pref\right]{p}_{1,2}\wedge {p}_{1,2}\left[Pref\right]{p}_{2,3}\wedge {p}_{2,3}\left[Pref\right]{p}_{3,4}$. Such an extension using conjunction can be done automatically to any finite set of masses or mass ranges. One will seldom have a need to fully unpack the expression because we need only find acceptable solutions. While a stakeholder’s statements can be compactly represented by terms like $\downarrow \left({M}_{S}\right)$, the point is that these can be syntactically defined (or axiomatically connected) to finite conjunctions of claims using only comparative preference.

#### 3.2.3. Relationship between Stakeholder Preferences and Solution Space

- The mathematical structure (betterness relation) of preferences;
- Types of preferences;
- Inconsistency in preferences;
- Changes in preferences.

**Definition 19**

**(Preference base).**

**Theorem**

**1.**

**Proof.**

**Theorem 2.**

**Proof.**

**Example 3.**

_{1}where both p and q are true.

**Theorem 3.**

**Proof.**

**Example**

**4.**

_{1}= System A, w

_{2}= System B, w

_{3}= System C, and w

_{4}= System D. Let us assume that the following propositions are in consideration.

_{1}and w

_{2}and worlds w

_{3}and w

_{2}are comparable based on the truth values of propositions p, whereas worlds w

_{3}and w

_{4}and worlds w

_{1}and w

_{4}are comparable based on the truth value of proposition q. However, worlds w

_{1}and w

_{3}are incomparable with each other. For a case like this, the betterness relation is a partial order that allows for incomparability, i.e., no arrows exist between these worlds as shown in Figure 3. In this case, worlds w

_{1}and w

_{3}are preferred over worlds w

_{2}and w

_{4}, respectively, as shown by the arrows in Figure 3, but the decision-maker cannot compare worlds w

_{1}and w

_{3}, which leads to no decision.

**Theorem 4.**

**Proof.**

**Example**

**5.**

**Theorem**

**5.**

**Proof.**

**Example**

**6.**

**Definition 20**

**(Satisfiability).**

**Definition 21**

**(Consistency in preferences).**

**Theorem 6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

## 4. Discussion

## 5. Conclusions and Future Work

- How can we represent domain knowledge of engineers in a formal manner?
- What is the impact of the knowledge structure on the solution space?
- How can one formally accommodate for changes in stakeholder preferences?
- How does a change in preference base affect the knowledge of engineers?
- Issue of consistency in the knowledge base.
- Issue of consistency between preference and knowledge bases.
- A mathematical framework that can aid in resolving incomparability.
- How can we leverage modal preference logic in formulating value functions?
- Another future direction is a study involving multiple stakeholders in a game theoretic context.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- United States Department of Defense. Technology and Logistics. In Performance of the Defense Acquisition Systems: 2016 Annual Report; Defense Pentagon: Washington, DC, USA, 2016. [Google Scholar]
- Simpson, T.W.; Martins, J.R. Multidisciplinary design optimization for complex engineered systems: Report from a national science foundation workshop. J. Mech. Des.
**2011**, 133, 101002. [Google Scholar] [CrossRef][Green Version] - Paul, D.C. Report on the Science of Systems Engineering Workshop. In Proceedings of the 53rd AIAA Aerospace Sciences Meeting, Kissimmee, FL, USA, 5–9 January 2015. [Google Scholar]
- Bloebaum, C.L.; Collopy, P.; Hazelrigg, G.A. NSF/NASA Workshop on the Design of Large-Scale Complex Engineered Systems—From Research to Product Realization. In Proceedings of the 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Indianapolis, IN, USA, 17–19 September 2012. [Google Scholar]
- DARPA/NSF. DARPA/NSF Systems Engineering and Design of Complex Aerospace Systems Workshop; DARPA/NSF: Arlington, VA, USA, 2009.
- Collopy, P. Final Report: National Science Foundation Workshop on the Design of Large Scale Complex Systems; National Science Foundation: Alexandria, VA, USA, 2011. [Google Scholar]
- Hazelrigg, G.A. Fundamentals of Decision Making for Engineering Design and Systems Engineering; Pearson Education, Inc.: New York, NY, USA, 2012. [Google Scholar]
- Abbas, A.E.; Howard, R.A. Foundations of Decision Analysis; Pearson Higher Education: New York, NY, USA, 2015. [Google Scholar]
- Blanchard, B.S.; Fabrycky, W.J.; Fabrycky, W.J. Systems Engineering and Analysis; Prentice Hall: Englewood Cliffs, NJ, USA, 1990; Volume 4. [Google Scholar]
- Wasson, C.S. System Engineering Analysis, Design and Development: Concepts, Principles and Practices; John Wiley & Sons: New York, NY, USA, 2015. [Google Scholar]
- NASA. NASA Systems Engineering Handbook; Volume NASA/SP-2007-6105, Rev1; NASA: Washington, DC, USA, 2007.
- Buede, D.M. The Engineering Design of Systems: Models and Methods; John Wiley & Sons: New York, NY, USA, 2009; Volume 55. [Google Scholar]
- NDIA Systems Engineering Division and Software Committee. Top Software Engineering Issues in the Defense Industry. 2006. Available online: https://ndiastorage.blob.core.usgovcloudapi.net/ndia/2006/systems/Wednesday/rassa6.pdf (accessed on 12 December 2019).
- National Defense Industrial Association Systems Engineering Division. Top Five Systems Engineering Issues in Defense Industry. 2003. Available online: https://www.ndia.org/-/media/sites/ndia/divisions/systems-engineering/studies-and-reports/ndia-top-se-issues-2016-report-v7c.ashx?la=en. (accessed on 12 December 2019).
- Collopy, P.D.; Hollingsworth, P.M. Value-Driven Design. J. Aircr.
**2011**, 48, 749–759. [Google Scholar] [CrossRef] - Kannan, H.; Mesmer, B.; Bloebaum, C.L. Increased System Consistency through Incorporation of Coupling in Value-Based Systems Engineering; Systems Engineering (INCOSE): Hoboken, NJ, USA, 2015; Under Review. [Google Scholar]
- Mesmer, B.L.; Bloebaum, C.L.; Kannan, H. Incorporation of Value-Driven Design in Multidisciplinary Design Optimization. In Proceedings of the 10th World Congress of Structural and Multidisciplinary Optimization (WCSMO), Orlando, FL, USA, 19–24 May 2013. [Google Scholar]
- Cheung, J.; Scanlan, J.; Wong, J.; Forrester, J.; Eres, H.; Collopy, P.; Hollingsworth, P.; Wiseall, S.; Briceno, S. Application of Value-Driven Design to Commercial Aero-Engine Systems. In Proceedings of the 10th AIAA Aviation Technology, Integration, and Operations (ATIO) Conference, Fort Worth, TX, USA, 13–15 September 2010. [Google Scholar]
- Claudia, M.; Price, M.; Soban, D.; Butterfield, J.; Murphy, A. An Analytical Study of Surplus Value using a Value Driven Design Methodology. In Proceedings of the 11th AIAA Aviation Technology, Integration, and Operations (ATIO) Conference, Virginia Beach, VA, USA, 20–22 September 2011. [Google Scholar]
- Collopy, P.; Poleacovschi, C. Validating Value-Driven Design. In Proceedings of the Third International Air Transport and Operations Symposium, Delft, The Netherlands, 18–20 June 2012; IOS Press: Amsterdam, The Netherlands, 2012. [Google Scholar]
- Hollingsworth, P. An Investigation of Value Modelling for Commercial Aircraft. In Proceedings of the Second International Air Transport and Operations Symposium, Delft, The Netherlands, 28–29 March 2011; IOS Press Inc.: Amsterdam, The Netherlands, 2011. [Google Scholar]
- Bhatia, G.; Mesmer, B. Integrating Model-Based Systems Engineering and Value-Based Design with an NEA Scout Small Satellite Example. In Proceedings of the AIAA SPACE and Astronautics Forum and Exposition, Orlando, FL, USA, 12–14 September 2017. [Google Scholar]
- Miller, S.W.; Simpson, T.W.; Yukish, M.A.; Stump, G.; Mesmer, B.L.; Tibor, E.B.; Bloebaum, C.L.; Winer, E.H. Toward a Value-Driven Design Approach for Complex Engineered Systems Using Trade Space Exploration Tools. In Proceedings of the ASME 2014 International Design Engineering Technical Conference & Computers and Information in Engineering Conference, Buffalo, NY, USA, 17–20 August 2014. [Google Scholar]
- Neumann, L.J.; Morgenstern, O. Theory of Games and Economic Behavior; Princeton University Press: Princeton, NJ, USA, 1947. [Google Scholar]
- Von Wright, G.H. The Logic of Preference Reconsidered. Theory Decis.
**1972**, 3, 1401–1469. [Google Scholar] [CrossRef] - Von Wright, G.H. The Logic of Preference; Edinburgh University Press: Edinburgh, UK, 1963. [Google Scholar]
- Moutafakis, N.J. The Logics of Preference: A Study of Prohairetic Logics in Twentieth Century Philosophy; Springer Science & Business Media: New York, NY, USA, 2012; Volume 14. [Google Scholar]
- Hansson, S. Preference Logic in Handbook of Philosophical Logic; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2001. [Google Scholar]
- Hansson, S.O. Preference-based deontic logic (PDL). J. Philos. Log.
**1990**, 19, 75–93. [Google Scholar] [CrossRef] - Van Benthem, J.; Liu, F. Dynamic logic of preference upgrade. J. Appl. Non Class. Log.
**2007**, 17, 157–182. [Google Scholar] [CrossRef][Green Version] - Liu, F. Von wright’s “the logic of preference” revisited. Synthese
**2010**, 175, 69–88. [Google Scholar] [CrossRef] - Van Benthem, J.; van Otterloo, S.; Roy, O. Preference Logic, Conditionals and Solution Concepts in Games; University of Amsterdam: Amsterdam, The Netherlands, 2005. [Google Scholar]
- Lang, J. Logical Representation of Preference: A Brief Survey. In Decision Theory and Multi-Agent Planning; Springer: Berlin/Heidelberg, Germany, 2006; pp. 65–88. [Google Scholar]
- Hansson, B. Fundamental axioms for preference relations. Synthese
**1968**, 18, 423–442. [Google Scholar] [CrossRef] - Hansson, S.O. A new semantical approach to the logic of preference. Erkenntnis
**1989**, 31, 1–42. [Google Scholar] [CrossRef] - Chisholm, R.M. The intrinsic value in disjunctive states of affairs. Noûs
**1975**, 295–308. [Google Scholar] [CrossRef] - Chisholm, R.M.; Sosa, E. On the Logic of “Intrinsically Better”. Am. Philos. Q.
**1966**, 3, 244–249. [Google Scholar] - Quinn, P.L. Improved foundations for a logic of intrinsic value. Philos. Stud.
**1977**, 32, 73–81. [Google Scholar] [CrossRef] - Pigozzi, G.; Tsoukias, A.; Viappiani, P. Preferences in artificial intelligence. Ann. Math. Artif. Intell.
**2016**, 77, 361–401. [Google Scholar] [CrossRef] - Wilson, N. Extending CP-Nets with Stronger Conditional Preference Statements. In Proceedings of the national conferenec on Artificial Intelligence, San Jose, CA, USA, 25–29 July 2004; pp. 735–741. [Google Scholar]
- Boutilier, C.; Brafman, R.I.; Domshlak, C.; Hoos, H.H.; Poole, D. CP-nets: A tool for representing and reasoning withconditional ceteris paribus preference statements. J. Artif. Intell. Res.
**2004**, 21, 135–191. [Google Scholar] [CrossRef] - Allen, T.E. CP-nets: From theory to practice. In Proceedings of the International Conference on Algorithmic Decision Theory, Lexington, KY, USA, 27–30 September 2015. [Google Scholar]
- Van Benthem, J.; Girard, P.; Roy, O. Everything else being equal: A modal logic for ceteris paribus preferences. J. Philos. Log.
**2009**, 38, 83–125. [Google Scholar] [CrossRef][Green Version] - Divers, J. Possible Worlds; Routledge: Abingdon, UK, 2006. [Google Scholar]
- Ditmarsch, H.; Halpern, J.Y.; van der Hoek, W.; Kooi, B.P. Handbook of Epistemic Logic; College Publications: London, UK, 2015. [Google Scholar]
- Fagin, R.; Halpern, J.Y.; Moses, Y.; Vardi, M. Reasoning about Knowledge; MIT Press: Cambridge, MA, USA, 2004. [Google Scholar]
- Keeney, R.L. Value-Focused Thinking: A Path to Creative Decisionmaking; Harvard University Press: Cambridge, MA, USA; London, UK, 1992. [Google Scholar]
- Abbas, A.E. Foundations of Multiattribute Utility; Cambridge University Press: Cambridge, UK, 2018. [Google Scholar]
- Murugaiyan, S.; Kannan, H.; Mesmer, B.L.; Abbas, A.; Bloebaum, C. A comprehensive study on modeling requirements into value formulation in a satellite system application. In Proceedings of the 14th Annual Conference on Systems Engineering Research (CSER 2016), Huntsville, AL, USA, 22–24 March 2016. [Google Scholar]
- Bhatia, G.V.; Kannan, H.; Bloebaum, C.L. A Game Theory approach to Bargaining over Attributes of Complex Systems in the context of Value-Driven Design: An Aircraft system case study. In Proceedings of the 54th AIAA Aerospace Sciences Meeting, San Diego, CA, USA, 4–8 January 2016. [Google Scholar]
- Kannan, H. An MDO Augmented Value-Based Systems Engineering Approach to Holistic Design Decision-Making: A Satellite System Case Study. Ph.D. Thesis, Iowa State University, Ames, IA, USA, 2015. [Google Scholar]
- Kannan, H.; Shihab, S.; Zellner, M.; Salimi, E.; Abbas, A.; Bloebaum, C.L. Preference Modeling for Government-Owned Large-Scale Complex Engineered Systems: A Satellite Case Study. In Disciplinary Convergence in Systems Engineering Research; Springer: Berlin/Heidelberg, Germany, 2018; pp. 513–529. [Google Scholar]
- Kwasa, B.; Kannan, H.; Bloebaum, C.L. Impact of Organization Structure in a Value-based Systems Engineering Framework. In Proceedings of the 2015 ASEM International Annual Conference, Indianapolis, IN, USA, 7–10 October 2015. [Google Scholar]
- Bhatia, G.; Mesmer, B.; Weger, K. Mathematical Representation of Stakeholder Preferences for the SPORT Small Satellite Project. In Proceedings of the 2018 AIAA Aerospace Sciences Meeting, Kissimmee, FL, USA, 8–12 January 2018; p. 0708. [Google Scholar]
- Clerkin, J.H.; Mesmer, B.L. Representation of knowledge for a NASA stakeholder value model. Syst. Eng.
**2019**, 22, 422–432. [Google Scholar] [CrossRef] - Goetzke, E.D.; Bloebaum, C.L.; Mesmer, B. Value-driven design of non-commercial systems using bargain modeling. In Proceedings of the 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Kissimmee, FL, USA, 5–9 January 2015; p. 0134. [Google Scholar]
- Jung, S.; Simpson, T.W.; Bloebaum, C.; Kannan, H.; Winer, E.; Mesmer, B. A value-driven design approach to optimize a family of front-loading washing machines. In Proceedings of the ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Charlotte, NC, USA, 21–24 August 2016. [Google Scholar]
- Keeney, R.L. Value-Focused Thinking; Harvard University Press: Cambridge, MA, USA, 1996. [Google Scholar]
- Malone, P.; Apgar, H.; Stukes, S.; Sterk, S. Unmanned aerial vehicles unique cost estimating requirements. In Proceedings of the 2013 IEEE Aerospace Conference, Big Sky, MT, USA, 2–9 March 2013; pp. 1–8. [Google Scholar]
- Malone, P.; Wolfarth, L. Measuring system complexity to support development cost estimates. In Proceedings of the 2013 IEEE Aerospace Conference, Big Sky, MT, USA, 2–9 March 2013; pp. 1–13. [Google Scholar]
- Dwyer, M.; Selva, D.; Cameron, B.; Crawley, E.; Szajnfarber, Z. The impact of technical complexity on the decision to collaborate and combine. In Proceedings of the 2013 IEEE Aerospace Conference, Big Sky, MT, USA, 2–9 March 2013; pp. 1–11. [Google Scholar]
- Bearden, D.A. A complexity-based risk assessment of low-cost planetary missions: When is a mission too fast and too cheap? Acta Astronaut.
**2003**, 52, 371–379. [Google Scholar] [CrossRef] - Leising, C.J.; Wessen, R.; Ellyin, R.; Rosenberg, L.; Leising, A. Spacecraft complexity subfactors and implications on future cost growth. In Proceedings of the 2013 IEEE Aerospace Conference, Big Sky, MT, USA, 2–9 March 2013; pp. 1–11. [Google Scholar]
- Salado, A.; Nilchiani, R. The Tension Matrix and the concept of elemental decomposition: Improving identification of conflicting requirements. IEEE Syst. J.
**2015**, 11, 2128–2139. [Google Scholar] [CrossRef] - Salado, A.; Nilchiani, R. The concept of order of conflict in requirements engineering. IEEE Syst. J.
**2014**, 10, 25–35. [Google Scholar] [CrossRef] - Carson, R.S. 1.6. 4 Requirements Completeness: A Deterministic Approach. In INCOSE International Symposium; Wiley Online Library: New York, NY, USA, 1998. [Google Scholar]
- Robertson, S.; Robertson, J. Mastering the Requirements Process: Getting Requirements Right; Addison-Wesley: New York, NY, USA, 2012. [Google Scholar]
- Liu, X.F.; Yen, J. An analytic framework for specifying and analyzing imprecise requirements. In Proceedings of the 18th International Conference on Software Engineering, London, UK, 13–14 May 2014; pp. 60–69. [Google Scholar]
- Salado, A.; Nilchiani, R.; Verma, D. Aspects of a Formal Theory of Requirements Engineering: StaNeholder Needs, System Requirements, Solution Spaces, and RequirementsГ Qualities. Syst. Eng.
**2013**. submitted. [Google Scholar] - Van Lamsweerde, A.; Darimont, R.; Letier, E. Managing conflicts in goal-driven requirements engineering. IEEE Trans. Softw. Eng.
**1998**, 24, 908–926. [Google Scholar] [CrossRef][Green Version] - Gervasi, V.; Zowghi, D. Reasoning about inconsistencies in natural language requirements. ACM Trans. Softw. Eng. Methodol.
**2005**, 14, 277–330. [Google Scholar] [CrossRef] - Ali, R.; Dalpiaz, F.; Giorgini, P. Reasoning with contextual requirements: Detecting inconsistency and conflicts. Inf. Softw. Technol.
**2013**, 55, 35–57. [Google Scholar] [CrossRef][Green Version] - Van Dalen, D. Logic and Structure; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Gensler, H.J. Introduction to Logic; Routledge: London, UK, 2012. [Google Scholar]
- Pospesel, H. Propositional Logic; Prentice-Hall: Englewood Cliffs, NJ, USA, 1974. [Google Scholar]
- Smullyan, R.R. First-Order Logic; Springer Science & Business Media: New York, NY, USA, 2012; Volume 43. [Google Scholar]
- Blackburn, P.; van Benthem, J.F.; Wolter, F. Handbook of Modal Logic; Elsevier: Amsterdam, The Netherlands, 2006; Volume 3. [Google Scholar]
- McCracken, D.D.; Reilly, E.D. Backus-naur form (BNF). In Encyclopedia of Computer Science, 4th ed.; John Wiley and Sons Ltd.: Chichester, UK, 2003. [Google Scholar]
- Kripke, S. Semantical Considerations of the Modal Logic. 2007. Available online: https://philpapers.org/rec/KRISCO (accessed on 11 December 2019).
- NASA. Systems Engineering Postulates, Principles, Hypotheses. Available online: https://www.nasa.gov/consortium/postulates-principles-hypotheses (accessed on 11 December 2019).
- Salado, A.; Wach, P. Constructing True Model-Based Requirements in SysML. Systems
**2019**, 7, 19. [Google Scholar] [CrossRef][Green Version]

Type of Preferences | Example (Stakeholder X) | |
---|---|---|

Absolute | Unconditional | Target-oriented: X prefers uninterrupted communication; Design-dependent: X prefers Solar arrays for power generation; Objective-oriented: X prefers low total satellite mass; |

Conditional | Target-oriented: If the satellite is parked in LEO, then X prefers uninterrupted communication; Design-dependent: If transponder ‘y’ is used, then X prefers solar arrays for power generation; Objective-oriented: If the satellite weighs more than 1000 kg, then X prefers high signal quality; | |

Comparative | Unconditional | Target-oriented: X prefers a system mass less than 1000 kg to uninterrupted communications; Design-dependent: X prefers Solar arrays to Nuclear reactor; Objective-oriented: X prefers low total cost to high signal quality; |

Conditional | Target-oriented: If it is a multi-satellite system, X prefers uninterrupted communications to a system mass less than 1000 kg; Design-dependent: If it is a multi-satellite system, then X prefers solar arrays over nuclear reactors; Objective-oriented: If the satellite weighs more than 1000 kg, then X prefers high signal quality to low total cost; |

Design | Mass (kg) | SNR (dB) |
---|---|---|

w_{1} | 0.5 | 1 |

w_{2} | 2.5 | 4 |

w_{3} | 4 | 7 |

w_{4} | 3.5 | 3 |

Definitions | Description |
---|---|

Solution space | Set of all possible worlds considered by the decision-maker |

Optimal solutions | Set of greatest elements based on betterness relation in the solution space |

Comparative preference | An agent prefers $\phi $ to $\psi $ if and only if all the states where $\phi $ holds is better than all the states where $\psi $ holds |

Absolute preference | An agent can be said to prefer $\phi $ simpliciter if the agent prefers $\phi $ to $\neg \phi $ |

Conditional preference | A conditional preference is defined in a preference statement as a ceteris paribus preference, where in this context “ceteris paribus” means “all other things being normal” |

Target-oriented preference | A target-oriented preference is specified on targets. The targets may be satisfied or not satisfied. |

Design-dependent preference | A design-dependent preference is one in which the stakeholder directly specifies preferences over propositions on solution alternatives. |

Objective-oriented preference | An objective-oriented preference is one in which the stakeholder indicates the direction (high-$\uparrow $ or low-$\downarrow $) without encroaching on the solution space. |

Preference base | The union of all preference statements elicited from the stakeholder |

Consistency | An agent has a consistent preference base $PB$ (Definition 19) if and only if there exists a structure $M=\left(S,\succcurlyeq ,\pi \right)$ and a world $w\in S$ such that $\left(M,w\right)\models PB.$ |

Theoretical Contributions | Description |
---|---|

How do elicited preferences impact the solution space? | Theorem 2:A betterness relation with a total order always results in an optimal solution, given a finite non-empty set of possible worlds/states. |

Theorem 3:If some of the attributes are incomparable for the stakeholder, then optimal solutions may not exist. | |

Relationship between types of preferences and solution space | Theorem 4:Target-oriented preferences may constrain the solution space. |

Theorem 5:Design-dependent preferences will always constrain the solution space | |

Effect of inconsistent preference base on solution space | Theorem 6:An inconsistent preference base results in no acceptable solutions |

Theorem 7:A change (update, addition, or deletion of preference statements) in the stakeholder preference base requires a new consistency check |

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**MDPI and ACS Style**

Kannan, H.; Bhatia, G.V.; Mesmer, B.L.; Jantzen, B.
Theoretical Foundations for Preference Representation in Systems Engineering. *Systems* **2019**, *7*, 55.
https://doi.org/10.3390/systems7040055

**AMA Style**

Kannan H, Bhatia GV, Mesmer BL, Jantzen B.
Theoretical Foundations for Preference Representation in Systems Engineering. *Systems*. 2019; 7(4):55.
https://doi.org/10.3390/systems7040055

**Chicago/Turabian Style**

Kannan, Hanumanthrao, Garima V. Bhatia, Bryan L. Mesmer, and Benjamin Jantzen.
2019. "Theoretical Foundations for Preference Representation in Systems Engineering" *Systems* 7, no. 4: 55.
https://doi.org/10.3390/systems7040055