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Towards a General Theory of Arbitrary Decisions
Matteo Cristani [email protected]
Francesco Bellomi [email protected]
Dipartimento di Informatica, Universita di Verona,
Ca Vignal 2, strada Le Grazie, I-37134 Verona (Italy)
Abstract
Many recent investigations of Computational Decision Theory focused on the problemof making decisions in presence of more than one single criterion. The so called multiplecriteria decision theory, in particular, deals with such problems, in depth. In general theseinvestigations tend to be specifically quantitative, and withal in those approaches that lookat purely qualitative models, the solution of such problems is a procedure that looks foroptimal tradeoff among criteria.
In this paper we argue that multiplicity of criteria really exists only when the criteriaare strictly incompatible, namely when they provide methods to evaluate decisions thatcannot be merged together. We study the problem of making decisions when at least twoadmissible evaluation criteria are strictly incompatible. Decisions made in such contextare named arbitrary. The notion of arbitrary is examined as opposite to the notion ofmechanical; we nevertheless argue that such decisions are not irrational.
An arbitrary decision is good, generally speaking, whenever it makes sense to provide ajustification for it, namely an explanation of the reasoning process that brought us to sucha decision. The notion of justification is the core notion of all the theory developed in thispaper. A justification should be the explanation of the reasoning process that has broughtto the decision we indeed justify. We model justification by means of the notion of perfectdecision. A decision is perfect whenever all the applicable criteria select that decision asoptimal. A justification is, in the model we present here, a subproblem of the originallyposed one for which the decision is perfect. In other words, a justification of a decision isa reduction of a decision problem to a subproblem for which the decision is the best one,namely such that all the criteria agree on it.
We analyze the notion of arbitrariness as foundational for a theory of purely qualitativedecisions, provide a formal model of such decisions, and study the computational complexityof the problems of exhibiting a justification.
1. Introduction
A significant effort has been spent in recent investigations of Artificial Intelligence, Com-puter Science, Database, Operation Research and Management Science for obtaining anaffordable model of decision making as a computational problem. These efforts have lookedfor methods that essentially reduce multiple criteria decision, namely decision in presenceof potentially contradictory criteria, to single criterion decisions. We argue that this is avery limited approach for at least three reasons:
• There exist problems for which criteria simply cannot be considered as factors con-tributing to the individuation of best element. In these cases two criteria are intrin-
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sically independent, like price and quality. The definition of best tradeoff is possiblebut not necessarily desirable, in many cases.
• There exist many applications for which the major interest is not in the definition ofoptimal decision, which is considered to be utopian, but the definition of a reasonabledecision.
• There are cases in which we would like to retain the responsibility of making deci-sions and want as well be able to interpret a decision already made (which we aregoing to consider as arbitrary in this paper) at best. We look, indeed, to opportunejustifications of decisions.
In a context like the one we have sketched above there are many possible applicable resultswe can derive from the techniques we have studied and are presented in this paper. Inbelow we propose three application cases for which the notion of arbitrary decision and themethods we document here result effective in solving open applicative problems.
Example 1 (Deliberative reasoning)A company has established internal procedures for the deliberation of assignment of new
tasks to the employees. This procedure is, roughly, a trial-model one, in which we have fourbasic phases:
1. Preliminary collection. In this phase the office devoted to the assignment of tasksestablishes which skills are relevant for the current decision procedure and obtain skillclaims by the employees;
2. Evaluation Criteria Establishment. In this phase the office decides which criteriaare relevant for the evaluation in progress;
3. Application of Criteria to the Skills. The employees and the tasks are crossedover to evaluate their skills as applied to the tasks by means of the criteria;
4. Final Decision. The tasks are assigned.
The company desires to use an automated support for the entire process, but strictly wantsthe following basic behaviors:
• The system has to make recommendations, but has not to be autonomous in decisionmaking, being the final decision something that the humans have the right and theneed to be in charge of;
• The system has to be flexible about criteria, but should not make mixup of the criteriathemselves.
The two above mentioned characteristics are not possessed by traditional Decision SupportSystems, and we specifically need to extend those system in the direction we shall indicatein this paper.
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Preliminary collection
Evaluation criteria
establishment
Application of criteria to
the skills
Final decision
EmployeesClaims
Decision
Support
System
Figure 1: A Decision Support System that is employed in task assignments of the companyin Example 1.
In Figure 1 we show a schema of the behavior of the Decision Support System sketchedin this example.
Example 2 (User Modelling in On-Line Auctions and E-Business applications)A web site manages on-line auctions. The system is based on a method of offer thatcommunicates the availability of a piece to those users that are specifically interested in thetype of product. For instance, if I am interested in buying guitars and material regardingguitars, when a electric guitar is available for selling, the system automatically sends me anemail, and I can make an offer that competes in the auction process.
The criteria that establish the interest of users of the web site in specific products are,in various cases, incompatible, as in the case of price, general interest in the product type,brand affection and others. The users are modelled by the system in order to be effectivein proposing opportunities, since an excess of pressure is dangerous for the probability ofacceptance, but abandoning the customer is dangerous as well.
In Figure 2 we schematize the behavior of the sketched application.
Example 3 (Authoring tools for Knowledge Management) In a network oftechnicians employed by a big company distributed overall the world, the management hasdecided to downsize the expenses due to the huge number of trips involving managers.There are two uses of information technology that are useful in this context:
• Teleconferencing;
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On-line auction
Participants
system
Profiling System
Profile Manager
Merchandiser
Figure 2: A support system for automated profiling in on-line auction.
• Decision responsibility distribution.
The second option consists in providing a system where decisions are made by single per-sons and the confirmation process of the other people involved in the decision procedure isa second step, whose conclusion is the decision adoption by the management. The manage-ment needs, in order to confirm decisions, reasonable justifications of the decision makingperformed by the person in charge of the decision, and the company wants to employ acomputer program that supports the decision process.
In particular every technician should be able to provide explanation of the process thathas brought him or her to the claimed decision, and conceptual definition of the frameworkin which the decision has been made.
In Figure 3 we show a schema of the above sketched application use.
This paper documents a traditional artificial intelligence investigation. We look for aconceptual model of the logical procedure we want to define and then obtain a formalizationof this model; we then study the computational properties of the obtained representation.
The model we have obtained is general and can be applied to many different situations.The computational results are practically significant, though, in principle, negative for theapplicability to the worst case, which we argue not to be frequent in practice.
In this paper we obtain:
• The definition of the notion of qualitative decision criterion;
• The definition of the notion of arbitrariness in the context of decision making;
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Proposals
Figure 3: A schema of the use of a Knowledge Management tool that utilizes arbitrarydecision support.
• The definition of the notion of justification and, in particular, of the notion of optimaljustification for arbitrary decisions;
• The analysis of the problem of computing optimal justifications for arbitrary decisionmaking.
The paper is organized as follows. In Section 2 we provide references to the currentliterature of artificial intelligence about decision theory and in Section 3 we introduce ter-minology and basic definitions to be employed in the rest of the paper. Section 4 is the coresection of the paper and is organized in three subsections: Subsection 4.1 is devoted to theanalysis of some preliminary algorithmic problems to be solved in order to provide justifica-tion computation, and is used also to provide some basic combinatorial analysis; Subsection4.2 analyzes the problem of finding justifications based on the intuitive optimality criterionof minimal exclusion of decisions and Subsection 4.3 performs the same analysis when theintuitive optimality criterion is maximal agreement amongst criteria. Section 5 analyzessome framework conditions that constrain the practical case of applicability for the algo-rithms introduced in Section 4 and provides a practical computational analysis. FinallySection 6 takes some conclusions and sketches further work.
2. Related work
The philosophical quest for Decision Theory dates back to Aristotle, whose NicomacheanEthics (350 B.C.) introduced for the first time the concept of rationally justifiable action.Antoine Arnauld and Pierre Nicole, followers of Rene’ Descartes and disciples of BlaisePascal, described in the Port-Royal Logics (1662) the first quantitative formula for choosingthe most suitable action between several available alternatives. Jeremy Bentham (whoconied the terms ”maximize” and ”minimize”) and his student John Stuart Mill, built a
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whole ethical system (Utilitarianism, 1863) upon the Principle of Utility: ”Do those actswhich will result in the greatest good for the greatest number of people”.
The first formal framework for Decision Theory has been introduced by John Von Neu-mann and Oskar Mergenstern (1944), and combines Probability Theory, which describeswhat an agent should believe on the basis of empirical evidence, with Utility Theory, whichdescribes consistent valuations of possible outcomes: a rational agent is bound to follow theMaximum Expected Utility principle, evaluating all possible actions and choosing the onethat leads to the best expected outcome.
Whereas Decision Theory only describes a set of consistency constraints that choicesmust satisfy in order to be considered optimal, Decision Analysis is an engineering disci-pline that addresses the search for effective ways to represent decision problems, and com-putationally efficient methods for finding the best decisions. Starting from the ’60, manyscholars of Computer Science, Artificial Intelligence, Information Retrieval, OperationalResearch and Management Science have regarded the decision making as a computationalproblem.
Prior and conditional probabilities are basic statements about the beliefs of an agent.Using Bayes’ rule and joint distribution, basic probability statements can be combined,and unknown probabilities can be computed from known ones. Conditional independencedefines direct causal relationships. Belief Networks are a way to represent (both qualita-tive and quantitative) conditional independence information, using a directed graph wherenodes represent random variables and arrows represent direct causal relationships. Deci-sion Networks are an extension of Belief Networks, containing decision and utility nodes inaddition to random variable nodes. Given the probability distribution of a set of knownvariables, Belief Networks can be used to efficiently compute the probability distribution ofa set of other variables, whereas Decision Networks can be used to efficiently determine theaction with the highest utility.
For a general reference to the computational aspects of Decision making see (Raiffa,1968), and for a more comprehensive analysis of the technical implementation aspects in-cluding the computational complexity and computability of problems of making decisionssee (Holtzman, 1989).
The majority of the investigations referred to in the above mentioned manuscripts tendto consider decision as a probability computation problem. For a more complete and focusedapproach you can refer to (Boutilier, 1994) and to (Tan & Pearl, 1994), where, conversely,decision making is regarded as a purely qualitative method.
In practice the analysis of single criterion decision making is nothing difficult, since asingle criterion simply provides the optimal value, or at least, a minimal acceptability limit.The cases in which something more complicated arises are those in which we have morethan one single criterion, the so called Multiple Criteria Decision Problems. For a generaldescription of the framework and of the approaches that brought to the development of atheory of multiple criteria decision in computer science and operation research see (Keeney& Raiffa, 1976). The general approach is based on the concept of utility independence,that consists in reducing a multiple criteria decision to a single criterion decision basedon (possibly variable) method for assigning weights to the single involved criteria. For atechnical reference to the notion of utility as applied to multiple criteria decision makingsee (Koksalan & Sagala, 1995). For utility independence in general see (Bacchus & Grove,
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1996, 1997). The major technique is named elicitation and the reader can refer to (Basu,Hirsh, & Cohen, 1998; Chajewska, Getoor, Norman, & Shahar, 1998; Ha & Haddawy, 1997,1998).
For the sake of completeness we recall here two other disciplines that have investigatedthe problems of decision making: economic sciences and psychology.
Investigations in the economic science area have dealt mainly with the problem of defin-ing methods to improve quality of decisions in economic terms; a decision would be consid-ered better than another when the economic value of the former is higher than the one ofthe latter. For a general reference see (Kahneman & Tversky, 1979; Shafir, Simonson, &Tversky, 1993).
Psychologists, on the other hand, looked at decisions as a model of human behavior insocial interactions. These investigations have mainly be concerned with the definition ofcharacters of decisions, in terms of motivation and consequence. For a general reference see(Kahneman & Tversky, 1979; March, 1991; Simon, 1954; Thaler, 1980).
3. Terminology and definitions
We introduce here the basic notions to be employed in the rest of the paper. In particularwe define the notions:
• Decision;
• Decision Criterion;
• Decision Problem;
• Arbitrariness;
• Justification of Arbitrary Decisions;
• Optimality of justifications.
We generically name decision the choice of a single value in a set of at least two possiblevalues we call decision domain. Formally, we can introduce Definition 1
Definition 1 [Decision]Given a set ∆ such that |∆| ≥ 2 an element of ∆ is called a decision.
For any strong partial order C defined on a domain ∆ the maximal elements of ∆ by C arethe x in ∆ such that for any y in ∆ y E x.
Definition 2 [Decision criterion]A partial order C whose set of maximal elements is a singleton is said to be a decisioncriterion.
Note that, in particular, a total order defined on a finite set is always a decision criterion.Usually the representation employed for partial orders are Directed Acyclic Graphs (DAG)whose vertices represent element of the ordered domain, and edges represent the induced
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Figure 4: Three different DAGs and their representation power with respect to partial or-ders.
Player name Goals Crosses Defences
Diego 11 15 24
Massimo 14 11 25
Alex 5 19 26
John 13 24 13
Table 1: Fours players and their scores
relation of sup (or inf). In figure 4 we show an example of a DAG that does not representa partial order, a forest that represents a partial order with a non-singleton set of maxi-mal elements, and finally a tree, that represents a particular partial order with maximumelement.
A decision problem is, in general terms, a finite set of possible decisions, for which wehave a finite set of applicable decision criteria. Clearly we are interested only in thosedomains for which elements are decisions, namely domains with at least two elements. Wemoreover assume to have at least two criteria, being otherwise the problem a single criterionone, for which no natural arbitrariness exists indeed.
Definition 3 [Decision problem]Given a finite set ∆ , and a finite set Ω with Ω with Ω =C1, C2, . . . , Cn of decisioncriteria defined on ∆ the pair 〈∆, Ω〉 is said to be a decision problem iff :
• |∆| ≥ 2;
• |Ω| ≥ 2.
To make the idea simpler to understand we provide an example in below.
Example 4 [Football comparison]Consider the case in which we have to decide which one among four football players namedDiego, Massimo, Alex, John is the best one, and we have three evaluation criteria: numberof goals, number of crosses serving for goal, number of defence action that avoid a goal.Table 1 reports the results of the players in the last year.
Clearly, by means of criterion of goals the best player is Massimo, for the Crossescriterion is John and for Defence criterion is Alex.
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D M
J A
C
G, D
D
C, G
D G, C CG, D
Figure 5: The representation of the Football example by a graph.
Obviously we can try to mix up the criteria, for example by summing the values, namelyby considering Goals, Crosses and Defences as equivalent. However, the sums result all tovalue 50, and thus do not establish any difference among the players. This is an extremecase, since the three admissible criteria are all incompatible to each other, therefore if weemploy the notion of justification, any justification only admits one single criterion.
An interesting representation of Decision Problems can be obtained by means of graphs.Vertices represent decisions, an edge exists between a vertex a and a vertex b when b is thesup of a by at least one criterion. An edge between a and b is labelled by all the criteria thatestablish that b is sup of a. For instance, the decision problem of Example 3 is representedin Figure 5.
In general, a decision problem is simple when all the applicable criteria establish thesame maximum.
Definition 4 [Simple Decision Problem]A decision problem P = 〈∆, Ω〉 is said to be simple iff for any two criteria Ci and Cj in Ωwith Ω =C1, C2, . . .Cn the maximum of Ci in ∆ is the same of Cj:
∀i, j[[1 ≤ i 6= j ≤ n]→ [maxCi
(∆) = maxCj
(∆)]]
Given a simple decision problem P = 〈∆, Ω〉, the maximum assigned by any relation in Ωis said to be the optimal decision. Any decision which is not optimal in P is said to bearbitrary. Formally we have Definition 5.
Definition 5 [Arbitrary Decisions]Given a decision problem P = 〈∆, Ω〉 with Ω =C1, C2, . . .Cn the decision δ such that
∀i[δ = maxCi
(∆)]
is said to be optimal. The non-optimal decisions are called arbitrary.
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Product Price Quality measure
X1 −259 99
X2 −248 98
Table 2: Price and quality of two different products. Prices are indicated by negative num-bers to preserve the notion of optimal as maximality.
If a decision problem P is not simple, then we say that it is ambiguous. If a decision isarbitrary then we cannot exhibit any perfect decision procedure, namely a computation ofagreeing criteria that establish the optimal decision.
Note that in an ambiguous problem the decisions are all arbitrary. Given a partial orderC defined on ∆, for any subset ∆′ the relation C/∆′ denotes C ∩ (∆′ ×∆′).
Definition 6 [Justification]Given a decision problem P = 〈∆, Ω〉 and an arbitrary decision δ ∈ ∆, a simple decisionproblem P ′ = 〈∆′, Ω′〉, is said to be a justification of δ, iff :
1. ∆′ ⊆ ∆;
2. δ ∈ ∆′;
3. Ω′ = Ck1/∆′, Ck2
/∆′, . . . , Ckm/∆′
where ∀i[1 ≤ ki ≤ n] and ∀i[[1 ≤ i ≤ m]→ [Cki∈ Ω]];
4. ∀i[[1 ≤ i ≤ m]→ [δ = maxCki(∆′)]].
Let us now consider a novel example.
Example 5 [Choice of a product]We have to buy a product X, and have two possible choices X1 and X2, two differentbrands for the product in word, and two criteria: price and quality. Table 2 describes thecombinations.
We have only three possible subsets of both decisions and criteria, that cross to 9 casesoverall. The cases are reported in Table 3. If we look at this table, we note that everydecision has more than one justification, and that certain justifications are better thanothers from the point of view of number of applicable criteria, and others are better fromthe point of view of number of dominated decisions.
We are in particular interested in optimal justifications, where optimality is based onthe maximality of the justification with respect to the other possible justifications.
Definition 7 [Optimal Justification]Given a decision problem P = 〈∆, Ω〉, an arbitrary decision δ in ∆ and a justificationdecision problem P ′ = 〈∆′, Ω′〉 we say that P ′ is:
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Decisions Criteria Justified decision
X1 Price X1
X2 Price X2
X1, X2 Price X2
X1 Quality X1
X2 Quality X2
X1, X2 Quality X1
X1 Price, Quality X1
X2 Price, Quality X2
X1, X2 Price, Quality −
Table 3: Price and quality of two different products. Decisions and criteria as a base forjustification.
A
B
P
K
B
PK
A
K
P
A B
Figure 6: Price, Taste, Healthfulness of four fruits.
• domain-optimal iff for any other justification P ′′ = 〈∆′′, Ω′′〉 of δ hold:
∆′′ ⊆ ∆′;
[∆′′ = ∆′]→ [Ω′′ ⊆ Ω′];
• criteria-optimal iff for any other justification P ′′ = 〈∆′′, Ω′′〉 of δ, hold:
Ω′′ ⊆ Ω′;
[Ω′′ = Ω′]→ [∆′′ ⊆ ∆′];
Let us explain the above defined concept by a simple example.
Example 6 [Justifying a purchase: optimal justifications]We have a list of possible decisions to be made amongst products to be purchased: Apple,Kiwi, Banana, Pear. We then have three criteria to be applied: Price, Taste, Healthfulness.The partial orderings established by the criteria appear in Figure 6.
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Criteria Dominated fruits
Decision Price Taste Health Apple Pear Kiwi Banana Opt.
Apple • • • • X
Apple • • •
Apple • • •
Apple • • •
Apple • •
Apple • • • • • X
Apple • •
Pear • • • • X
Pear • • •
Pear • • •
Pear • • •
Pear • •
Pear • • • X
Pear • • • X
Kiwi • • • • • X
Kiwi • • • •
Kiwi • • •
Kiwi • • •
Kiwi • • • • • X
Kiwi • •
Kiwi • • •
Banana • • • • • X
Banana • • • • •
Banana • • • •
Banana • • • • •
Banana • • •
Banana • • • •
Banana • • • • • X
Table 4: The admissible justifications for the choices of one of the fruits of Example 3.
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There are several ways of justifying the four choices. In particular, in Table 4 we list, forall the four possible choices, the admissible justifications and note their optimality, if any.
We employ the term MaxSet as referred to a decision problem 〈∆, Ω〉 . In particular, byMaxSet(〈∆, Ω〉 ) we denote the set of maximum elements of ∆ with respect to the orderrelations in Ω.We now formulate two problems we will deal with in the rest of the paper.
Problem 1 (Domain-Optimal Justification Problem - DOJP) Given a decisionproblem P = 〈∆, Ω〉, and an arbitrary decision δ in ∆, find a domain-optimal justificationfor δ.
Problem 2 (Criteria-Optimal Justification Problem - COJP) Given a decisionproblem P = 〈∆, Ω〉, and an arbitrary decision δ in ∆, find a criteria-optimal justificationfor δ.
What do these problems correspond to, in practice? A Domain-Optimal Justification isa justification that provides maximal elimination, namely this kind of justifications recuron the idea that a decision is good because it is better than many other decisions. ACriteria-Optimal justification provides instead maximum agreement, namely a decision isgood because many criteria agree on the maximum.
4. Justifying arbitrary decisions
In this section we shall provide four basic results. In Subsection 4.1 we evaluate the sizeof the problems we are interested to and solve the basic problems related to the notion ofarbitrary decision: establishing whether a given partial order is a decision criterion or not,and recognizing whether a decision problem 〈∆′, Ω′〉 is a justification of an arbitrary decisionfor a decision problem 〈∆, Ω〉 . In Subsection 4.2 we study the problem of finding Domain-optimal justifications, and in Subsection 4.3 we solve the problem of finding Criteria-optimaljustifications.
4.1 Combinatorial analysis and solution of preliminary problems
First of all let us evaluate the size of the problems we deal with from a purely combinatorialpoint of view. Given a set ∆ with n elements, the number of decision criteria we can providefor that set are defined by the number of posets on ∆ with a single maximal element. Now,since the possible ways of being related for maximal elements are computed for posetsaltogether, this number is simply given by the number of posets (ωπ) with (n−1) elements.Let us name this number ν(n). The recursive formula which provides the computation ofν(n) is
ν(1) = 1
ν(2) = 2
ν(3) = ωπ(2) = 3
ν(n) = ωπ(n− 1) = ωπ(n− 2)× 2× (n− 2)
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A rough but intuitively simpler analysis is obtained by considering that any decision criteriais undoubtedly a relation on ∆ and therefore is a subset of ∆ × ∆ , and that a total orderon ∆ is a decision criteria. Therefore
n! ≤ ν(n) ≤ 2n2
In general we determine the size of a decision problem by means of two values, n shallindicate henceforth the number of possible decisions and m the number of admissible criteria.By definition: 2 ≤ m ≤ ν(n).
We have a few problems to be dealt with preliminarily. In particular, in order to provideoptimal justifications, we shall be able to solve the following problems.
Problem 3 (Decision Criterion Recognition Problem - DCRP) Given a relationR on a domain ∆ establish whether R is a Decision Criterion.
Problem 4 (Justification Recognition Problem - JRP) Given two decision prob-lems decide whether the second one is a justification for an arbitrary decision of the firstone;
In order to solve Problem DCRP we need to be able to:
• Decide whether a given relation is a partial order;
• Determine whether a given partial order relation has a unique maximal element (inother terms whether it has a maximum).
The first problem is well-known in the current literature about algorithms and can be easilysolved in O(n2) being n the number of elements in the domain where the partial orderrelation is defined. For a general reference see (Aho, Hopcroft, & Ullman, 1974; Cormen,Leiserson, Rivest, & Stein, 2001). For the second problem, though elementary, we did notfound a reference to the method we propose here, that is optimal. Other optimal methodsexist in the current literature on algorithms, but the method employed here is novel.
We employ an algorithm for partial order detection based on the elimination of transitiveedges, known as maximal chain. This method has been successfully employed for totalorders, in particular for total dense orders (Gerevini & Schubert, 1995). The idea is toperform an acyclicity test. This test can be performed in O(n2) by computing the socalled Strongly Connected Components of the graph representing the relation. This isnot enough, since we need to test also transitivity. However, for our purpose, we canassume, without loss of generality, that each potential criterion of decision is specified inform of a minimized DAG (directed acyclic graph) obtained by eliminating transitive pairsin the relation employing maximal chain method. In this kind of representation each edgerepresents the inf relation; such a graph will henceforth called a inf-DAG. A specificationin terms of a graph whose edges are the pairs of the order relation is called a c-DAG. As amatter of fact, such a representation, since the inf of an element of a domain where a partialorder relation is defined is unique, the obtained DAG is a forest, by definition. In case wetest both acyclicity and transitivity the second part of this test obviously takes O(n3).
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input A partial order relation specified as an inf-DAG D = 〈V, E〉
output Yes, if the relation is a decision criteria, No otherwise
D′ ← Maximal chain(D)D′ is a forest iff D is acyclicS ← Vfor all x, y ∈ V do
if (x, y) ∈ E thenS ← S\x
end ifend forif |S| = 1 then
Return Yesend if
Figure 7: An algorithm to solve the DCRP-I problem
The DAG obtained by maximal chain (transitive edge elimination) can be tested topossess one single maximal element as follows: for a relation C defined in ∆ :
• start with candidate set S = P
• for any pair x E y, remove x form S;
• at the end of this process, if S has cardinality 1, the only element of S is the maximum.
This is equivalent to looking at the graph and counting the vertices without exiting edges.Assuming that removing x from S takes O(1) (which can be obtained by an appropriatedata structure) this method is quadratic in the number of elements of P . If the graph is aDAG, the above algorithm can take O(n) (always with appropriate data structure).
We distinguish two versions of DCRP. In one version we specify a partial order relation interms of an intransitive DAG (DCRP-I). In the second version we specify the relation simplyas a list of pairs (DCRP-N). We can now formulate the following Proposition whose proofis a trivial consequence of the above reasoning and is omitted for the sake of conciseness.
Proposition 1 For any relation defined on a domain ∆ with n elements,
• DCRP-I can be solved in O(n2);
• DCRP-N can be solved in O(n3).
A synthesis of the above described procedures is exhibited in Figure 7 and 8.For the second preliminary problem JRP, consider two decision problems 〈∆, Ω〉 and
〈∆′, Ω′〉 . We name Ω′′ the set ωi ∩ (∆′ ×∆′)|ωi ∈ Ω. We set n = |∆|, and m = |Ω|. Thetest consists in establishing whether:
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input A partial order relation specified as a c-DAG D = 〈V, E〉
output Yes, if the relation is a decision criteria, No otherwise
D′ ← Maximal chain(D)D′′ ← Transitivity elimination(D′)D′ is a forest iff D is acyclicD′′ is a forest iff D′ is transitiveS ← Vfor all x, y ∈ V do
if (x, y) ∈ E thenS ← S\x
end ifend forif |S| = 1 then
Return Yesend if
Figure 8: An algorithm to solve the DCRP-N problem
1. ∆′ ⊆ ∆ ;
2. Ω′ ⊆ Ω′′;
3. Elements of Ω′ are decision criteria;
4. The maximum of each element of Ω′ is the same of any other element.
Evidently, testing condition 1 is O(n), testing condition 2 is O(m), testing condition 3 isO(m ·n2) and testing condition 4 is O(m). Overall, we have the following result. We exhibitan algorithm for solving JRP in Figure 9.
Proposition 2 For any pair of decision problems P and P’, with n number of elements inthe domain of P and m number of decision criteria in P
• JRP can be solved O(m · n2).
4.2 Finding Domain-optimal justifications
Let us take into consideration the problem of finding a domain-optimal justification for adecision problem P = 〈∆, Ω〉, and an arbitrary decision δ in ∆ .
We denote with p(δ, C), with C ∈ Ω, the order relation C′ = C/∆m, where ∆m is thelargest subset of ∆ such that δ = max(C′). Such C′ is unique, and can be determined inO(n2).
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input Two Decision Problems 〈∆, Ω〉 and 〈∆′, Ω′〉
output Yes, if 〈∆′, Ω′〉 is a justification of 〈∆, Ω〉 , No otherwise
Ω′′ ← ωi ∩ (∆′ ×∆′)|ωi ∈ ΩB1 ← ∆′ ⊆ ∆B2 ← Ω′ ⊆ Ω′′
for all c ∈ Ω′ doEstablish if c is a decision criterion
end forEstablish if B1 ∧B2 and MaxSet(〈∆′, Ω′〉 ) is a singleton
Figure 9: An algorithm to solve the JRP problem
input A decision problem 〈Ω, ∆〉 and an arbitrary decision δ
output A domain-optimal justification 〈Ω′, ∆′〉 of δ in 〈Ω, ∆〉
Ω′′ ← p(δ, C)|C ∈ ΩΩm ← C
′|C′ ∈ Ω′′, ∀C′′ ∈ Ω′′, |dom(C′)| ≥ |dom(C′′)|Ω′ ← ∅for all C′ ∈ Ωm do
J ← C′for all C′′ ∈ Ω′′\C′ do
if max(C′′/dom(C′)) = δ thenJ ← J ∪ C′′
end ifend forif |J | > |Ω′| then
Ω′ ← J∆′ ← dom(C′)
end ifend for
Figure 10: An algorithm for finding a domain-optimal justification.
17
A domain-optimal justification J = (∆′, Ω′) can be found by the algorithm of Figure10, which we name DOJ-Algorithm.
Finding a domain-optimal justification with the algorithm above takes O(nm2), sincethe two nested loops have O(m) iterations each, and both computing C = C′′/dom(C′) andtesting whether max(C) = δ take O(n).Based on the above reasoning we can claim the following theorem
Theorem 1 DOJ-Algorithm correctly solves DOJP in O(nm2).
More interestingly we can claim that this algorithm is optimal.
Theorem 2 DOJ-Algorithm is optimal.
ProofLet J(δ′), with δ′ ∈ ∆, denote the set of justifications of d; given a justification J = (∆j , Ωj)we denote ∆j with ∆(J ) and Ωj with Ω(J ).
We define dp(δ, Ωp), with δ ∈ ∆ and Ωp non-empty subset of Ω composed by orders thatcontain δ in their domains, as the largest subset ∆′ of ∆ such that, for each C ∈ Ωp, wehave that max(dom(C/∆′)) = δ.
We define J∆max(δ) as the subset of J(δ) such that for each J ′ ∈ J∆max and for each J ′′ ∈J(δ), we have |∆(J ′)| ≥ |∆(J ′′)|, and we define ∆max(δ) as the set ∆′|∆′ = ∆(J ′′), J ′′ ∈J∆max(δ).
We have that:
1. (dp(δ, Ωp), Ωp) is a justification of δ;
2. Ω′
p ⊂ Ωp ⇒ dp(δ, Ωp) ⊂ dp(δ, Ω′
p);
3. thus, ∆max(δ) = dp(δ, Ω′
p)|Ω′
p = C′, C′ ∈ Ω, ∀Ω′′
p ⊂ Ω, |dp(δ, Ω′
p)| ≥ |dp(δ, Ω′′
p)|,and |∆max(δ)| is O(m);
4. domain-optimal justifications are (∆′, Ω′)|(∆′, Ω′) ∈ J(δ), ∆′ ∈ ∆max(δ), ∀J ′′ ∈J(δ), |Ω′| ≥ |Ω(J ′′)|.
In order to find a domain optimal justification, it is necessary to count, for each∆′′ ∈ ∆max(d) how many of the m orders in Ω are compatible with it, that is, to checkfor each C′ ∈ Ω whether max(dom(C′/∆′′)) = δ. This check can be performed in O(n) foreach order, that amounts to O(m) for each ∆′′ ∈ ∆max(d), that is O(nm2) in total.
4.3 Finding Criteria-optimal justifications
Let us take into consideration the problem of finding a criteria-optimal justification for adecision problem P = 〈∆, Ω〉, and an arbitrary decision δ in ∆ .
In order to find the criteria-optimal justification, it is necessary to find the largestprojection of the chosen orders, such that δ is the maximum in each of them. Such ordersare generally independent, so the O(m) projections must be computed independently. LetJ be the criteria-optimal justification. We have:
18
input A decision problem 〈Ω, ∆〉 and an arbitrary decision δ
output A criteria-optimal justification (Ω′, ∆′) of δ in 〈Ω, ∆〉
Ω′ ← C|C ∈ Ω, δ ∈ dom(C)∆′ ← δfor all δ′ ∈ ∆ do
p← true
for all C′ ∈ Ω′ doif (δ, δ′) 6∈ C′ then
p← false
end ifend forif p then
∆′ ← ∆′ ∪ δ′end if
end for
Figure 11: An algorithm to solve the COJP problem
• Ω(J ) = C|C ∈ Ω, δ ∈ dom(C);
• ∆(J ) = δ ∪ δ1|δ1 ∈ ∆, ∀C ∈ Ω|(δ1, δ) ∈ C.
Ω(J ) is computed in O(m), ∆(J ) takes O(m) tests for each one of the n elements in ∆. Sothe whole computation is dominated by the computation of ∆(J ), which takes O(n ·m).
The above defined work schema is synthesized in the algorithm COJ-Algorithm showedin Figure 11.The consequence of the above reasoning is in the below theorem.
Theorem 3 COJ-Algorithm correctly solves the COJP in O(nm).
We can claim that this performance is optimal.
Theorem 4 COJ-Algorithm is optimal.
5. Some empirical observations
The computational analysis performed in Section 4 about the problems DCRP, JRP, DOJPand COJP have proved that, as it is posed, each of these problems is polynomially solvableboth in the size n of the domain and in the number m of admissible criteria.
However, in Subsection 4.1 we have proved that the relation between n and m definesm as an exponential number in n, in the worst case, so that the cost above strictly dependon the actual relation that is established in practice.
19
First of all, note that, obviously, if we fix m, then n results, in the worst case, linearin m, so that the above mentioned limit is not significant. Clearly, the number n is morerelevant in practice, and, as a theoretical limit, we tend to fix m not to be too big withrespect to n. In general, it makes sense to assume that m is reasonably close to a polynomialin n.
The kind of criteria we are interested in, that are significant in practice are two:
• Quasi-Total orders, namely those orders that compare objects in such a way thatwhen x is incomparable to y and x is comparable to z then y is comparable to z withmaximum;
• Random Partial orders with maximum.
The former is interesting since it establishes, in particular, the case in which the ordering isobtained by means of an evaluation function that attributes a numeric value to each elementof the domain. The latter is the general case.
This assumptions, however, do not limit in a very strong way the combinatorial numericrelation established in Subsection 4.1, since the number of total orders is the minimum ofthe constraints to ν(n) but still n!, whilst the case of partial orders represented by trees cansimply be seen as total orderings for partitions of a set of n elements, that is much biggerthan n!. However, in practice, the number of orderings employed in a realistic decision shalldepend on the following facts:
• The more we introduce criteria, the less is the probability that a given arbitraryrelation establishes significant dominance amongst the elements of the domain. Thisholds, in particular, when the number of elements in the domain is low; a number ofcriteria m linear in n is a reasonable constraint.
• The more we introduce criteria, the more these criteria tend to be incompatible, andtherefore difficult to group. This implies that the criteria should not exceed again thenumber of elements.
Note that the relation m = O(n) suggested by the above observations, still preserve thepossibility of two criteria to be incompatible in terms of the maximum they select.
We analyzed the performance of implementation of the algorithms COJ and DOJ. Werealized the algorithms in Java and tested them on a random benchmark generated by arandom generation algorithm that constrained the m criteria to the maximum number n.The code employed in this experiments appear in Appendix C. The results of the testsare summarized in the Table of Appendix A. In Appendix B we show the graphics of theexperimental results.
Let us now describe some technicalities about the experiments.
• The Sun Java Virtual Machine employs a dynamic profiling approach to perform arun-time optimization of the compilation of the byte-code into machine code. Thiscompilation is indeed performed at run-time, so that the measured execution timeof algorithm codes is potentially affected by compilation steps. To avoid significantincidence of this phenomenon in the evaluation of performances we execute the test
20
suite twice within the same run-time VM session, and consider only the second testresult. In fact the optimization process is typically performed in reason of repetitionsof code part executions, thus the compilation of these parts will be performed atrun-time, for our test suite, during the first execution;
• We only measure the elapsed time, not the actual machine time employed in the exe-cution of the algorithms and the elapsed time includes the operating system activities.Obviously we left the execution of the tests alone with respect to other applications,intentionally avoiding to have any other program running during the tests;
• We intentionally avoided dynamic heap allocation during the elapsed time for eachtest, because it is possible to provide an admissible measure of the time spent forallocation and therefore we cannot evaluate the contribution to the total elapsedtime;
• The precision of the employed timer is about a millisecond. Many tests can performbelow the declared precision, especially for rather small sizes of the test itself. Toavoid consequences on the performance measure we execute many different tests ateach size and consider average time. Statistically this reduces the risk of measuring 0time for classes of small-size tests.
6. Conclusions and further work
We have investigated the problem of representing decision making in presence of multiplecriteria that are strictly incompatible, and simply cannot be merged together. In thiscontext we have provided basic concepts for a theory of arbitrariness in decision making,and have defined basic problems of decision criterion evaluation. We thus provide the notionof justification, and in particular the notion of optimal justification. Based on these notionswe have described the problems of finding an optimal justification for the single cases.
We have obtained an optimal algorithm for finding optimal justifications to arbitrarydecisions, both in the case in which we desire to have maximal elimination and in the casein which we desire to have maximal agreement.
In Table 5 we summarize the results of this paper.Some basic problems have remained open. In particular we did not deal with the problem
of finding optimal justifications in real-time. We think that, based on the analysis of theproblems we have been able to provide here, an anytime algorithm would be worthy. Asecond aspect that has been left to further investigations is the provision of valid andpractically significant heuristics, for the reduction of time consumption when computingoptimal justifications, based on the relevance of further descriptors like tenability of thedecision, or economic evaluation.
Let us now introduce a quasi-serious remark. We have in fact to refer a small anecdoteregarding the investigation documented in this paper that is rather fun.
During the extension of an early version of the paper we were thinking at a motivationallevel and were looking at the examples provided in the introduction. Suddenly Francescorecalled that there Douglas Adams, the well-known Science Fiction novelist provided an
21
Problem Complexity
DCRP-I O(n2)
DCRP-N O(n3)
JRP O(m · n2)
DOJP O(m2 · n2)
COJP O(m · n2)
Table 5: A summary of paper results. The symbol n represents the number of elements inthe domain, whilst m represents the number admissible criteria.
example of use for the very same ideas we were looking at. We looked at the novel itselfand this is the quotation, in our mind, very impressive and visionary.
“Well,” he said, “it’s to do with the project which first made the software in-carnation of the company profitable. It was called Reason, and in its own wayit was sensational.”“What was it?”“Well, it was a kind of back-to-front program. It’s funny how many of the bestideas are just an old idea back-to-front. You see there have already been severalprograms written that help you to arrive at decisions by properly ordering andanalysing all the relevant facts so that they then point naturally towards theright decision. The drawback with these is that the decision which all the prop-erly ordered and analysed facts point to is not necessarily the one you want.”“Yeeeess ...” said Reg’s voice from the kitchen.“Well, Gordon’s great insight was to design a program which allowed you tospecify in advance what decision you wished it to reach, and only then to give itall the facts. The program’s task, which it was able to accomplish with consum-mate ease, was simply to construct a plausible series of logical-sounding stepsto connect the premises with the conclusion.”
Douglas AdamsDirk Gently’s Holistic Detective Agency
Pan Books, London 1987
22
A criticism that can raise to Adams’ propositions above is that the actual problem of findingan optimal justification cannot be solved with consummate ease being in the best case acomplex polynomial problem. Moreover, an optimal justification is not a series of logical-sounding steps, but a series of actual logical steps. But our paper was not ready when hewrote the above mentioned novel, so we cannot blame him for this errors...
There are several ways in which this research can be taken further. Clearly we have tolook at open problems as exposed above. However, we should also refine the classificationwe have provided here based on finer notions of decision criterion, and different ways ofthinking at optimality for justifications.
In particular we are interested in the definition of sequences of decisions, where the orderestablishes priority of decision making. This kind of arbitrariness can be considered in thecontext of collective decision making, and for modelling of vote as well.
Acknowledgments
Authors gratefully thank ACP s.r.l. for funding. This work has taken place within theproject D.A.V.A. “Tecniche di Decisione Automatica con criteri multipli e Valutazioni Arbi-trarie” (Techniques of automated decision with multiple criteria and arbitrary evaluation).
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Chajewska, U., Getoor, L., Norman, J., & Shahar, Y. (1998). Utility elicitation as aclassification problem. In Proceedings of the Fourteenth Conference on Uncertainty inArtificial Intelligence.
Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2001). Introduction to algo-rithms. MIT Press. Second Edition.
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Gerevini, A., & Schubert, L. K. (1995). On computing the minimal label in time pointalgebra networks. Computational Intelligence, 11 (3), 443–448.
Ha, V., & Haddawy, P. (1997). Problem-focused incremental elicitation of multi-attributeutility models. In Proceedings of the Thirteenth Conference on Uncertainty in ArtificialIntelligence, pp. 215–222.
Ha, V., & Haddawy, P. (1998). Towards case-based preference elicitation: Similarity mea-sures on preference structures. In Proceedings of the Fourteenth Conference on Un-certainty in Artificial Intelligence, pp. 193–201.
Holtzman, S. (1989). Intelligent Decision Systems. Addison-Wesley, Reading,Mass.
Kahneman, D., & Tversky, A. (1979). Prospect Theory: an analisys of decision under Risk.Econometrica, 47, 263–291.
Keeney, R., & Raiffa, H. (1976). Decisions with Multiple Objectives: Preferences and ValueTradeoffs. Wiley and Sons, New York.
Koksalan, M., & Sagala, P. (1995). Interactive approaches for discrete alternative multiplecriteria decision making with monotone utility functions. Management Science, 7 (41),1158–1171.
March, J. G. (1991). How decisions happen in organizations. Human Computer Interaction,6, 95–117.
Raiffa, H. (1968). Decision Analysis. Addison-Wesley, Reading,MA.
Shafir, E., Simonson, I., & Tversky, A. (1993). Reason-based choice. Cognition, 49.
Simon, H. A. (1954). A behavioral model of rational choice. The Quarterly Journal ofEconomics, 69.
Tan, S., & Pearl, J. (1994). Qualitative decision theory. In Proceedings of the TwelfthNational Conference on Artificial Intelligence, pp. 928–932.
Thaler, R. (1980). Toward a theory of consumer choice. Journal of Economic Behaviourand organization, 1, 39–60.
24
Appendix A. Experimental data
2 4 6 8 10 12 14 16
90 0,10 0,40 1,20 1,31 1,70 2,00 2,20 2,20
120 0,50 0,70 1,20 1,61 1,81 2,91 3,20 2,90
150 0,60 1,20 1,31 1,81 1,80 2,51 3,20 3,91
180 0,40 1,21 1,91 2,20 3,00 3,31 4,11 4,31
210 0,61 1,70 1,50 2,71 3,61 3,60 4,80 5,21
240 0,90 1,40 1,70 3,21 3,70 4,71 4,91 5,61
270 0,90 1,70 2,30 2,50 4,31 4,71 6,31 7,02
300 0,90 2,10 2,80 3,60 4,40 4,91 6,91 7,62
330 0,80 1,80 2,90 4,30 5,10 6,91 7,91 8,92
360 1,10 2,41 3,61 4,61 5,40 7,31 7,91 8,92
390 1,10 2,40 3,90 4,71 6,01 7,21 7,51 9,92
420 1,10 2,30 3,70 4,20 6,91 7,92 9,21 10,32
450 1,40 2,71 3,71 5,81 7,82 8,41 10,42 10,82
m (number of criteria)
n (
num
ber
of
decis
ions)
Table 6: Data of the experiments: the case of DOJ with quasi-total orders
2 4 6 8 10 12 14 16
90 0,20 0,30 0,00 0,30 0,20 0,30 0,50 0,80
120 0,00 0,30 0,30 0,50 0,70 0,20 0,40 1,31
150 0,00 0,10 0,60 0,70 1,30 1,30 1,21 1,20
180 0,20 0,30 0,30 0,80 0,80 1,50 1,20 1,80
210 0,20 0,30 1,00 0,80 1,10 1,71 1,41 1,80
240 0,00 0,50 1,11 0,80 1,31 1,40 2,30 2,90
270 0,20 0,40 1,00 2,01 1,40 2,10 1,70 2,10
300 0,30 0,30 0,90 1,31 1,81 3,10 1,90 2,70
330 0,81 0,80 1,10 1,21 1,71 1,30 2,40 2,30
360 0,20 0,40 0,90 1,40 2,01 2,40 2,70 3,40
390 0,51 0,60 1,01 1,80 2,00 2,51 4,10 3,10
420 0,40 1,00 1,51 2,61 2,00 2,90 3,31 4,00
450 0,20 0,90 1,90 1,60 1,70 3,00 3,00 5,00
m (number of criteria)
n (
num
ber
of
decis
ions)
Table 7: Data of the experiments: the case of COJ with quasi-total orders
2 4 6 8 10 12 14 16
90 0,30 0,40 0,80 0,90 1,10 1,90 1,91 2,80
120 0,10 0,81 0,80 1,70 2,00 2,00 2,30 2,90
150 0,50 1,00 1,30 1,50 2,40 2,91 3,31 4,81
180 0,60 1,10 1,70 2,31 3,11 3,80 5,00 6,12
210 0,80 1,21 1,70 3,21 4,00 4,50 6,11 7,72
240 0,91 1,40 2,51 4,10 5,01 7,52 9,82 10,01
270 0,90 1,70 3,42 5,10 6,90 6,90 9,81 13,12
300 0,80 2,01 3,90 4,40 6,22 9,13 10,80 13,61
330 1,30 1,81 3,11 5,90 9,00 10,34 11,20 15,03
360 0,80 2,00 4,51 7,41 8,51 10,43 15,92 19,62
390 0,70 3,10 5,11 8,22 10,42 12,32 17,12 19,04
420 1,20 3,21 5,51 7,60 11,62 15,96 17,53 22,34
450 1,31 3,71 5,90 9,24 11,41 16,42 20,92 23,93
m (number of criteria)
n (
num
ber
of
decis
ions)
Table 8: Data of the experiments: the case of DOJ with randomly generated partial orders
2 4 6 8 10 12 14 16
90 0,10 0,00 0,30 0,60 0,70 0,30 0,90 0,90
120 0,10 0,20 0,20 0,30 0,50 1,40 1,81 2,00
150 0,00 0,20 0,60 1,11 1,30 2,10 3,11 3,20
180 0,20 0,50 0,90 1,70 2,00 3,11 3,91 5,10
210 0,00 0,80 1,90 1,90 3,10 4,90 5,61 6,33
240 0,10 0,70 2,60 3,11 5,01 5,31 6,50 9,43
270 0,30 1,20 1,80 3,11 3,80 6,81 7,41 8,20
300 0,50 1,60 2,32 4,50 6,51 8,01 10,10 11,31
330 0,20 2,30 3,90 4,70 5,62 8,10 12,13 12,81
360 1,00 2,40 3,50 4,31 7,62 11,31 10,72 12,52
390 1,30 2,50 3,72 5,84 8,61 12,51 13,11 16,62
420 1,50 2,30 4,21 7,71 9,22 11,52 15,61 17,02
450 1,31 2,70 5,21 7,81 11,72 14,12 16,13 20,23
m (number of criteria)
n (
num
ber
of
decis
ions)
Table 9: Data of the experiments: the case of COJ with randomly generated partial orders
25
Appendix B. Graphics of the experimental results
90120
150180
210240
270300
330360
390420
450
24
68
1012
1416
0
2
4
6
8
10
12
time (m
s)
nm
Figure 12: The Quasi-Total order case of the experiment for DOJ.
90
150
210
270
330
390
450
24
68
1012
1416
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
time (m
s)
n
m
Figure 13: The Quasi-Total order case of the experiment for COJ.
26
90120
150180
210240
270300
330360
390420
450
24
68
1012
1416
0
5
10
15
20
25
time (m
s)
nm
Figure 14: The Random case of the experiment for DOJ.
90120
150180
210240
270300
330360
390420
450
24
68
1012
1416
0
5
10
15
20
25
time (m
s)
nm
Figure 15: The Random case of the experiment for COJ.
27
Appendix C. Java code of the experimental test suite
package univr.adp;
import java.util.Random;
import java.text.DecimalFormat;
import java.text.DecimalFormatSymbols;
/** Test for arbitrary decision problems
*
* @author Francesco Bellomi - [email protected]
* @version 1.0 (created: 23-mar-2003)
*/
public class ADPTestSuite
public static final Random rand = new Random();
public static final boolean[][] omega2 = new boolean[100][1000];
public static final int[] domainCard = new int[100];
public static final boolean[] omega1 = new boolean[100];
public static final int[] m2 = new int[100];
static public void main (String args[])
for (int i = 1; i <= 20; i++)
final int card = 30 * i;
for (int j = 1; j <= 10 + i; j+= 2)
final int cri = 4 + j;
System.out.print(pad(Integer.toString(card))+","+
pad(Integer.toString(j))+",");
testQuasiTotal(cri, card);
testPartial(cri, card);
System.out.println();
static private String pad(String s)
28
return " ".substring(s.length()) + s;
private static void COJP (int decision, int card, Relation[] r)
final int m = r.length;
int count = 0;
for (int i = 0; i < m; i++)
final boolean b = r[i].inDomain(decision);
omega1[i] = b;
if (b)
m2[count++] = i;
for (int i = 0; i < card; i++ )
boolean inDelta = true;
for (int j = 0; j < count; j++)
if (!r[m2[j]].r(i, decision))
inDelta = false;
private static void DOJP (int decision, int card, Relation[] r)
final int m = r.length;
int maxDomainCard = 0;
for (int i = 0; i < m; i++)
projectMax(r[i], decision, omega2[i], card);
domainCard[i] = domainCard(omega2[i], card);
maxDomainCard = Math.max(maxDomainCard, domainCard[i]);
for (int i = 0; i < m; i++)
if (domainCard[i] == maxDomainCard)
for (int j = 0; j < m; j++)
if (j != i)
boolean deltaMax = true;
for (int k = 0; k < card; k++)
if (omega2[j][k] && r[j].r(decision, k))
29
deltaMax = false;
break;
if (deltaMax)
;
private static void testPartial(final int cri, final int card)
int zz = 10;
double time1 = 0, time2 = 0;
for (int z = 0; z < zz; z++)
final Relation r[] = new Relation[cri];
for (int k = 0; k < cri; k++)
r[k] = new PartialOrder(card);
for (int i = 0; i < 100; i++)
final int decision = rand.nextInt(card);
final long start1 = System.currentTimeMillis();
COJP(decision, card, r);
final long end1 = System.currentTimeMillis();
time1 += (end1-start1);
final long start2 = System.currentTimeMillis();
DOJP(decision, card, r);
final long end2 = System.currentTimeMillis();
time2 += (end2-start2);
time1 /= 100;
time2 /= 100;
System.out.print(pad(ft.format(time1)) +","+ pad(ft.format(time2)));
private static final DecimalFormat ft = new DecimalFormat("0.000");
static
final DecimalFormatSymbols dfl = new DecimalFormatSymbols();
30
dfl.setDecimalSeparator(’.’);
ft.setDecimalFormatSymbols(dfl);
private static void projectMax(Relation r, int max,
boolean[] result, int card)
for (int i = 0; i < card; i++)
result[i] = r.inDomain(i) && (r.r(i, max) || i == max);
private static int domainCard(boolean[] d, int card)
int c = 0;
for (int i = 0; i < card; i++)
if (d[i])
c++;
return c;
private static void testQuasiTotal(final int cri, final int card)
int zz = 10;
double time1 = 0, time2 = 0;
for (int z = 0; z < zz; z++)
final Relation r[] = new Relation[cri];
for (int k = 0; k < cri; k++)
r[k] = new QuasiTotalOrder(card);
for (int i = 0; i < 100; i++)
final int decision = rand.nextInt(card);
final long start1 = System.currentTimeMillis();
COJP(decision, card, r);
final long end1 = System.currentTimeMillis();
time1 += (end1-start1);
final long start2 = System.currentTimeMillis();
DOJP(decision, card, r);
final long end2 = System.currentTimeMillis();
time2 += (end2-start2);
time1 /= 100;
time2 /= 100;
31
System.out.print(pad(ft.format(time1)) +","+
pad(ft.format(time2))+",");
interface Relation
boolean r(int a, int b);
boolean inDomain(int a);
class QuasiTotalOrder
implements Relation
private final int[] map;
public QuasiTotalOrder(int cardinality)
map = new int[cardinality];
final int range = cardinality / 2;
for (int i = 0; i < cardinality; i++)
map[i] = ADPTestSuite.rand.nextInt(range);
public boolean r(int a, int b)
return map[a] < map[b];
public boolean inDomain(int a)
return true;
class PartialOrder
implements Relation
private final int[] map;
private final boolean[] domain;
private final boolean[][] relation;
32
public PartialOrder(int cardinality)
map = new int[cardinality];
relation = new boolean[cardinality][cardinality];
domain = new boolean[cardinality];
// forces antisymmetry
for (int i = 0; i < cardinality-1; i++)
final int j = ADPTestSuite.rand.nextInt(cardinality-1);
final int k = j+1+ADPTestSuite.rand.nextInt(cardinality-j-1);
relation[j][k] = true;
domain[k] = true;
domain[j] = true;
// forces transitivity
for (int i = 0; i < cardinality; i++)
for (int j = i; j < cardinality; j++)
for (int k = j; k < cardinality; k++)
if (relation[i][j] && relation[j][k])
relation[i][k] = true;
// map to mix up elements
for (int i = 0; i < cardinality; i++)
map[i] = i;
for (int i = 0; i < cardinality; i++)
final int a = ADPTestSuite.rand.nextInt(cardinality);
final int b = ADPTestSuite.rand.nextInt(cardinality);
final int z = map[a];
map[a] = map[b];
map[b] = z;
public boolean r(int a, int b)
return relation[map[a]][map[b]];
public boolean inDomain(int a)
return domain[a];
33
