Analysis of the sensitivity of solutions of multi-criteria problems based on parametric partial preference relations
We give a survey of approaches for analyzing the sensitivity of non-dominated alternatives
to changes in the parameters of partial quasi-orderings that define preferences. Such
parameters can include values of importance coefficients for different criteria or boundaries of
interval estimates of the degrees of superiority in the importance of some criteria over others,
boundaries of intervals of criteria value tradeoffs uncertainty, and others.
Exact efficient numerical methods are proposed for solving bilinear optimization problems that arise when various solution variants are compared based on their preferability using an additive value function in the case of interval estimates of the degrees of superiority of certain criteria over others and in the case of interval restrictions on the growth of preferences along the criteria range.
This abstract offers a method for ranking alternatives in a decison making problem. It determines importance of the criteria with help of factor analysis. Though the alternatives are evaluated by each of the criteria by a group of experts, the weights for the criteria are to be found with the help of factor analysis.
The algorithm of the method is as follows:
1. Under the constraint that the problem handles several evaluation criteria, several items to compare (alternatives) and several experts to give their evaluation.
2. Find the principal components that replace the input criteria implicitly.
3. To find the final mark for each of the alternatives the marks given by experts are multiplied with the regression coefficients, found in the step 2.
4. The final marks are represented in axes „crieria“ and „mark“ so that each alternative is described with a curve (trajectory). These curves represent the map of graded alternatives. Depending on the problem to be solved (min or max,) a record for each main criteria is to be found.
5. With help of special deviation measure procedures (Minkowski, Chebyshev e.t.s) a matrix of deviations from ideal solution is to be built.
6. The alternatives are to be rated in accordance to the deviation from the ideal trajectory.
To prove the effectiveness of the method it was applied to a problem for 5 alternatives, 3 experts and 38 evaluation criteria. The problem was also solved with the help of most popular method of Weighted Sum Model (WSM) and TOPSIS method. The problem was also being solved by finding the geometric mean for each alternative. The results for approaches were compared and the method, offered in this abstrat, proved itself as a feasible one.
In this paper we consider choice problems under the assumption that the preferences of the decision maker are expressed in the form of a parametric partial weak order without assuming the existence of any value function. We investigate both the sensitivity (stability) of each non-dominated solution with respect to the changes of parameters of this order, and the sensitivity of the set of non-dominated solutions as a whole to similar changes. We show that this type of sensitivity analysis can be performed by employing techniques of linear programming.
The influence of the assumption about the existence of cardinal coefficients of criteria importance, consistent with the importance ordering of criteria (according to the definitions in the criteria importance theory), on the preference relation generated by this information, is investigated.
Influence of the assumption about the existence of quantitative coefficients of criteria importance, consistent with the criteria importance order (according to the definitions in the criteria importance theory), on the preference relation, generated by this information, is investigated.
WE present a general approach to the solution of the multicriteria choice problem by methods of the Criteria importance theory. The overview of methods of vector estimates comparison by preference using various types of information about the preferences of the decision-maker is given. These methods are implemented in the computer system DASS.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.