In the past decade, increasing interest in equity issues resulted in new methodologies in the area of operations research. This paper deals with the concept of equitably efficient solutions to multiple criteria optimization problems. Multiple criteria optimization usually starts with an assumption that the criteria are incomparable. However, many applications arise from situations which present equitable criteria. Moreover, some aggregations of criteria are often applied to select efficient solutions in multiple criteria analysis. The latter enforces comparability of criteria (possibly rescaled). This paper presents aggregations which can be used to derive equitably efficient solutions to both linear and nonlinear multiple optimization problems. An example with equitable solutions to a capital budgeting problem is analyzed in detail. An equitable form of the reference point method is introduced and analyzed.
We present methods that are useful in solving some large scale hierarchical planning models involving 0-1 variables. These 0-1 programming problems initially could not be solved with any standard techniques. We employed several approaches to take advantage of the hierarchical structure of variables (ordered by importance) and other structures present in the models. Critical, but not sufficient for success, was a strong linear programming formulation. We describe methods for strengthening the linear programs, as well as other techniques necessary for a commercial branch-and-bound code to be successful in solving these problems.
SUMMARYThe classical problem of elasto-hydrodynamic lubrication of cylinders in line contact is formulated as a non-linear complementarity problem. A direct algorithm is applied to the approximation obtained by finite differences. Implementation considerations are emphasized. The new method provides reliable and automatic location of the previously troublesome lubricant free boundary. Numerical results reveal the qualitative behaviour of the pressure distribution and the lubricant film thickness under variation of key physical parameters.
This paper is concerned with the properties of nonlinear equations associated with the Scheweitzer-Bard (S-B) approximate mean value analysis (MVA) heuristic for closed product-form queuing networks. Three forms of nonlinear S-B approximate MVA equations in multiclass networks are distinguished: Schweitzer, minimal, and the nearly decoupled forms. The approximate MVA equations have enabled us to: (a) derive bounds on the approximate throughput; (b) prove the existence and uniqueness of the S-B throughput solution, and the convergence of the S-B approximation algorithm for a wide class of monotonic, single-class networks; (c) establish the existence of the S-B solution for multiclass, monotonic networks; and (d) prove the asymptotic (i.e., as the number of customers of each class tends to m) uniqueness of the S-B throughput solution, and (e) the convergence of the gradient projection and the primal-dual algorithms to solve the asymptotic versions of the minimal, the Schweitzer, and the nearly decoupled forms of MVA equations for multiclass networks with single server and infinite server nodes. The convergence is established by showing that the approximate MVA equations are the gradient vector of a convex function, and by using results from convex programming and the convex duality theory.
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