Study On Latticing And Group Lattice Order Decision

The subject to be studied in this dissertation is theory and methods of modern decision-making based on lattice.The famous axioms for rational behavior established by Von Neumann and Morgenstern in 1944, as a sign of birth of modern decision-making science, is a cornerstone of normative decision-making theory, and be applied extensively and developed depthly in economy, management, engineering technology. In the past half century, researching on the axioms is a hot spot all the while, and many critical results have been obtained. The (linear) expected utility theory based on the axioms is also very important in modern decision-making theory. From late fifties of last century, some scientists headed by Allais and Edwards, try to check the facticity of the rational decision-making models applied to the real decision-making behavior, but find it unsatisfactory. For example, Allais Paradox and Preference Circles challenge the axioms critically. Last eighties, Fishburn and Bell tried to revise the axioms, but end with no success.Because of the extreme importance of the order in the choice of the decision alternatives, it is important to find a way that can reflect the preference of decision maker, independent of comparative condition, and meet some of the rational behavior axiom as well. Professor Guo yaohuang and his students study the lattice order characteristic of the decision preference structure and analyze the nature of the lattice order preference structure. By using modern mathematical theory, the ordering axiom is generalized to the lattice-order axiom, and construct lattice-order decision-making behavior axioms, the unique existence theorem for (linear) utility function based on the new axioms is obtained. The concept of the minimal determining set is generalized to non-binary social choice, and the unique existence condition of the minimal determining set of non-binary complete rational choice function is also obtained, and the sufficient condition is confirmed, in this way, the lattice-order decision-making theory is framed.Even so, a lot work still need to be done on the lattice order decision theory and method. For example, the lattice order decision method of the current systemis not set up yet; some elements of the preference lattice structure are absent; the operation and compare rule of the lattice are not wholesome; the decision maker can't make out the superior or inferior due to the insufficient information. Futhurmore, it studies the weakening of independent irrelevant condition and total rational when meeting the lattice preference. Main creative results are obtained as follows:1. The study of operation of lattice dicision-makingBy applying the concept, theory, method of lattice theory and graph theory, defining and stipulating the method which can reflect the superior or inferior relation among different elements on lattice, and the variable, parameter, operation rule and the compare method, For example, the preference distance and it's working with relecting the superior or inferior relation among different elements on lattice. And the theory is applied to the group decision-making with interval numbers, the arrange method of interval numbers with lattice order preference is obtained.2. Offering the strategy and method of completing the default elements on latticeWhen lattice ordering the incomplete lattice of the default elements, it is studied in two points: lattice ordering of the incomplete lattice without preference and with preference. When the incomplete lattice without preference, the interval numbers with lattice characteristic is studied as an example. It analyzes the smallest complete lattice's structure according to the graph theory, general algebra theory, and then defines the settling regulation of key elements and the minimal decision set. It also studies the probable approaches of decision preference structure lattice ordering and lattice structure completing. It analyzes the affect of introducing default elements on project preference, and defines the completing default elements rules and set up the strategy and method to complet the default elements based on it.3. The study of the weakening of independent irrelevant conditionFirst, it studies the problem that the relative properties, conditions and revealed preference structure over subsets of path independence choice functions and proves the lattice characteristic of a revealed preference binary relation.Secondly, when the preference structure in the group decision is charactered by lattice characteristic, and independent irrelevant condition (IIA) is replaced by independent decisiveness condition IIA(Si) , it studies the unique existence condition for minimal determining set, whether Arrow's possibility theorem is tentable, and whether the unique existence of minimal determining set can be weakened. The abundant and essential condition of choice is given when the independent decisiveness condition HA{Sy) is replaced by independent decisiveness conditionIIA(S2).4. The application of lattice order decision to the multi-objective group decision-makingMulti-objective group decision is important in decision-making theory, because decisions in real life always not only have more than one object and but also more than one decision maker. First, in this dissertation, the integrating method of different forms preference information on attributes is studied. Secondly, when Multi-attribute group decision making problem without determining weights and attribute values are given using linguistic term, and the prference structure of groups with partial order preference characteristic, a compromise weights method is given. Finally, it studies the integrating method of the multi-attribute group decision-making problem with lattice order preference structure, and the distance measures of the preference structure.