Abstract
whether it is an individual decision –what house to buy, what car to buy, or it is a group decision –e.g., where to build a school,
we need to take into account many different criteria.
For example, when we buy a house, we take into account: its price, its size, how far is its location from our workplaces and from stores, how noisy is this location, how close is this location to the nearest park, how safe is this location, etc.
There are many possible functions that combine several utility values into a single value. It has been proven that if we assume that criteria are, in some reasonable sense, independent from each other, then the overall utility is equal: either to a weighted linear combination of the individual utilities or to the product of these utilities raised to some powers.
In some situations, these combination functions work well. However, in many other situations, the criteria are not independent and, as a result, the traditionally used combination rules do not describe well how we make decisions.
Whatever scale we use to describe these degrees, we can always re-scale this degree to the interval [0, 1]. For example, if we ask the use to describe his/her degree of satisfaction on a scale form 0 to 10, then we can divide this degree by 10 and get the values from the interval [0, 1].
Such [0, 1]-valued degrees are a starting point for fuzzy logic analysis. However, as we will explain shortly, the author goes beyond the traditional fuzzy logic formulas. So, to avoid possible misunderstandings, he uses a more general term “graded logic" to describe Zadeh’s idea that everything is a matter of degree –without necessarily following the usual fuzzy ways of describing “and”- and “or”-operations.
The first reason is that in fuzzy logic, we usually select one single “and”-operation and use it every time we need to estimate our degree of certainty in a statement of the type “A and B”. In contrast, in decision making, how we combine the degrees of two criteria depends on how related they are: for some pairs of criteria, there is almost no relation between them, while for other pairs of criteria, we may have a strong dependence.
This fact necessitates the use of several different “and”-operations in different situations. (By the way, this need for different “and”- and “or”-operations is in perfect agreement with the use of logical connectives like “and” and “or” in natural language: e.g., when we say “or”, we sometimes mean the usual inclusive “or” and sometimes an exclusive “or”.)
Another reason for going beyond the traditional fuzzy logic reasons is that the usual fuzzy “and”-operations are commutative. So, if we use these operations to combine the degrees to which different criteria are satisfied, this would mean that we consider all criteria to be equally important. In reality, different criteria may have different importance. For example: some criteria are absolutely necessary, so a solution in which this criterion is not satisfied is not acceptable, while other criteria are more like wishes, so it is OK if one or more of these criteria are not satisfied.
Yes another reason why we need to go beyond the usual fuzzy theory is that for fuzzy “and”-operations –just like for the 2-valued “and” –if one of the statements is false, i.e., has degree of confidence 0, then the whole “and”-combination is false. In decision making, if one of the ten criteria for, e.g., buying a house is not satisfied, it may still be a good decision –especially since sometimes, this is inevitable: there is simply no alternative in which all the criteria from our wish list are satisfied.
Also, the current “and”-operations are based –via the associativity property –on using only binary relations. In reality, there may be dependence between three criteria –e.g., that the value of the third criterion is close to e.g., the sum of the first two. In this case, we do not have any dependence between any two criteria, but we do have a strong dependence between the three of them. To reflect this, it is desirable to have “and”-operations estimating the truth values of statements of the type A1 & A2 & … & A n for n > 2.
Similarly arguments explain why we need to go beyond the traditional fuzzy logic approach when describing statements including “or”.
four different “and”-operations, four different “and”-operations, and several “neutral” (averaging) operations. some of these combinations are closer to “and”, some operations are closer to “or”, and some operations are equally distant from both and are, in this sense, neutral.
It is worth mentioning that, in effect, most of these operations are not pure “and”- and “or”-operations: they combine the usual “and”- and “or”-ideas:
All these operations can be described in terms of linear combinations, products, and raising to a given power.
The author conjectures –the book calls it Graded Logic (GL) Conjecture –that we can reasonably well describe all human decision making by using only a few such n-ary operations.
As a result, to describe how people make decisions, we do not need –as the current fuzzy methodology seems to imply –to find the most appropriate functions that would describe “and”, “or”, and “not": it is now sufficient to just estimate the values of a few corresponding real-valued parameters.
This leads to the general methodology that the author proposes under the name of Logic Scoring of Preferences (this is the mysterious LSP from the book’s title): we elicit the expert degrees, we find the parameters describing the appropriate combination operations, etc.
This methodology is illustrated, in detail, on several practical examples ranging from buying a house to making a serious medical decision.
The book contains many new interesting results about the new operations –and many challenging open questions. The main open question is, of course, to explain why these few specific “and”- and “or”-operations are used. There are also many other open questions.
It can also be recommended to theoreticians who are eager to encounter new challenging tasks –this books provides many such tasks, and the good news is that they are not just tasks of mostly mathematical interest, these are tasks motivated by applications –so theoretical breakthroughs have a good chance of leading to practical successes.
And, of course, last but not the least, this book, with its numerous examples and clear explanations, will be very good for students who want to learn state-of-the-art decision making techniques.