This paper is concerned with the development of an algorithm to solve continuous polynomial programming problems for which the objective function and the constraints are specified polynomials. A linear programming relaxation is derived for the problem based on a Reformulation Linearization Technique (RLT), which generates nonlinear (polynomial) implied constraints to be included in the original problem, and subsequently linearizes the resulting problem by defining new variables, one for each distinct polynomial term. This construct is then used to obtain lower bounds in the context of a proposed branch and bound scheme, which is proven to converge to a global optimal solution.A numerical example is presented to illustrate the proposed algorithm.

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This paper deals with the design of dynamic time-space and calibrated static strategic planning models, along with solution algorithms, for the multilevel rail-car fleet management problem faced by RELOAD ® , a branch of the Association of American Railroads (AAR). We discuss a prevalent fleet sizing management model that is static in nature, and propose an alternative dynamic model based on a time-space network representation. This model accurately represents the problem, and also provides information regarding the issue of storing and retrieving empty cars. A suitable decomposition heuristic, that is based on solving subproblems defined for overlapping time segments, is developed to solve this model. This heuristic is shown to recover an optimal solution for all the test problems with a reasonable effort. We also investigate a procedure for calibrating the static model based on this improved time-space representation. Our results show that for the static model, a calibrated use of available data can yield near-optimal total fleet size requirements. This enables the use of such a simple, calibrated static model for accurately conducting fleet sizing, the determination of fleet size allocations among railroads, as well as for analyzing various "what-if" scenarios. The proposed methodology is being currently implemented at the AAR, and the status of this process as well as some test results are presented.fleet sizing, dynamic time-space network, strategic planning models, rail-car management