Distribution of Aggregate Utility Using Stochastic Elements of Additive Multiattribute Utility Models

Conventionally, elements of a multiattribute utility model characterizing a decision maker's preferences, such as attribute weights and attribute utilities, are treated as deterministic, which may be unrealistic because assessment of such elements can be imprecise and erroneous, or differ among a group of individuals. Moreover, attempting to make precise assessments can be time consuming and cognitively demanding. We propose to treat such elements as stochastic variables to account for inconsistency and imprecision in such assessments. Under these assumptions, we develop procedures for computing the probability distribution of aggregate utility for an additive multiattribute utility function (MAUF), based on the Edgeworth expansion. When the distributions of aggregate utility for all alternatives in a decision problem are known, stochastic dominance can then be invoked to filter inferior alternatives. We show that, under certain mild conditions, the aggregate utility distribution approaches normality as the number of attributes increases. Thus, only a few terms from the Edgeworth expansion with a standard normal density as the base function will be sufficient for approximating an aggregate utility distribution in practice. Moreover, the more symmetric the attribute utility distributions, the fewer the attributes to achieve normality. The Edgeworth expansion thus can provide a basis for a computationally viable approach for representing an aggregate utility distribution with imprecisely specified attribute weights and utilities assessments (or differing weights and utilities across individuals). Practical guidelines for using the Edgeworth approximation are given. The proposed methodology is illustrated using a vendor selection problem.     

L(j,k)- and circular L(j,k)-labellings for the products of complete graphs

Let j and k be two positive integers with j≥k. An L(j,k)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between labels of any two adjacent vertices is at least j, and the difference between labels of any two vertices that are at distance two apart is at least k. The minimum range of labels over all L(j,k)-labellings of a graph G is called the λ _j,k -number of G, denoted by λ _j,k (G). A σ(j,k)-circular labelling with span m of a graph G is a function f:V(G)→{0,1,…,m−1} such that |f(u)−f(v)| _m ≥j if u and v are adjacent; and |f(u)−f(v)| _m ≥k if u and v are at distance two apart, where |x| _m =min {|x|,m−|x|}. The minimum m such that there exists a σ(j,k)-circular labelling with span m for G is called the σ _j,k -number of G and denoted by σ _j,k (G). The λ _j,k -numbers of Cartesian products of two complete graphs were determined by Georges, Mauro and Stein ((2000) SIAM J Discret Math 14:28–35). This paper determines the λ _j,k -numbers of direct products of two complete graphs and the σ _j,k -numbers of direct products and Cartesian products of two complete graphs. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. This work is partially supported by FRG, Hong Kong Baptist University, Hong Kong; NSFC, China, grant 10171013; and Southeast University Science Foundation grant XJ0607230.