# 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

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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