The National Energy Modeling System (NEMS) is a large-scale mathematical model that computes equilibrium fuel prices and quantities in the U.S. energy sector and is currently in use at the U.S. Department of Energy (DOE). At present, to generate these equilibrium values, NEMS iteratively solves a sequence of linear programs and nonlinear equations. This is a nonlinear Gauss-Seidel approach to arrive at estimates of market equilibrium fuel prices and quantities. In this paper, we present existence and uniqueness results for NEMS-type models based on a nonlinear complementarity/variational inequality problem format. Also, we document mathematically, for the first time, how the inputs and the outputs for each NEMS module link together.
This exposition on the Analytic Hierarchy Process (AHP) has the following objectives: (1) to discuss why AHP is a general methodology for a wide variety of decision and other applications, (2) to present brief descriptions of successful applications of the AHP, and (3) to elaborate on academic discourses relevant to the efficacy and applicability of the AHP vis-a-vis competing methodologies. We discuss the three primary functions of the AHP: structuring complexity, measurement on a ratio scale, and synthesis, as well as the principles and axioms underlying these functions. Two detailed applications are presented in a linked document at http://mdm.gwu.edu/FormanGass.pdf .
We analyze a single-period, stochastic inventory model (newsboy-like model) in which a sequence of heterogeneous customers dynamically substitute among product variants within a retail assortment when inventory is depleted. The customer choice decisions are based on a natural and classical utility maximization criterion. Faced with such substitution behavior, the retailer must choose initial inventory levels for the assortment to maximize expected profits. Using a sample path analysis, we analyze structural properties of the expected profit function. We show that, under very general assumptions on the demand process, total sales of each product are concave in their own inventory levels and possess the so-called decreasing differences property, meaning that the marginal value of an additional unit of the given product is decreasing in the inventory levels of all other products. For a continuous relaxation of the problem, we then show, via counterexamples, that the expected profit function is in general not even quasiconcave. Thus, global optimization may be difficult. However, we propose and analyze a stochastic gradient algorithm for the problem, and prove that it converges to a stationary point of the expected profit function under mild conditions. Finally, we apply the algorithm to a set of numerical examples and compare the resulting inventory decisions to those of some simpler, naive heuristics. The examples show that substitution effects can have a significant impact on an assortment's gross profits. The examples also illustrate some systematic distortions in inventory decisions if substitution effects are ignored. In particular, under substitution one should stock relatively more of popular variants and relatively less of unpopular variants than a traditional newsboy analysis indicates.
The paper considers the single vehicle routing problem with stochastic demands. While most of the literature has studied the a priori solution approach, this work focuses on computing a reoptimization-type routing policy. This is obtained by sequentially improving a given a priori solution by means of a rollout algorithm. The resulting rollout policy appears to be the first computationally tractable algorithm for approximately solving the problem under the reoptimization approach. After describing the solution strategy and providing properties of the rollout policy, the policy behavior is analyzed by conducting a computational investigation. Depending on the quality of the initial solution, the rollout policy obtains 1% to 4% average improvements on the a priori approach with a reasonable computational effort.
Stochastic multicriteria acceptability analysis (SMAA) is a multicriteria decision support method for multiple decision makers in discrete problems. In SMAA, the decision makers need not express their preferences explicitly or implicitly. Instead, the method is based on exploring the weight space in order to describe the valuations that would make each alternative the preferred one. Inaccurate or uncertain criteria values are represented by probability distributions from which the method computes confidence factors describing the reliability of the analysis. In this paper we introduce the SMAA-2 method, which extends the original SMAA by considering all ranks in the analysis. In situations where the “elitistic” SMAA may assess large acceptability only for extreme alternatives without sufficient majority support, the more holistic SMAA-2 analysis can be used to identify good compromise candidates. The results are presented graphically. We consider also situations where partial preference information is available. We demonstrate the new method using a real-life decision problem.