A quadratic programming model of the Appalachian steam coal market
Abstract The authors report on an extension of recent modelling efforts to analyse the Appalachian steam coal market using a programming approach. The background to these modelling efforts is examined first. The particular aspects of the steam coal market to be modelled are then presented and the corresponding coal model described. Empirical results are analysed for the 1973 base solution and then compared to alternative solutions embodying variations in the parametric structure. Conclusions are drawn regarding potential policy applications.
Nonlinear Integer Programming Transportation Models: An Alternative Solution
The combinatorial nature of integer programming is inevitable even after taking specific model structure into consideration. This is the root problem in implementing large-scale nonlinear integer programming models regardless of which algorithm one chooses to use. Consequently, we suggest that the size of origin-destination be moderate. In the case of large origin-destination problems, more information on the size of XijÂ is needed to drastically reduce the dimensionality problem. For instance, if XijÂ is to be greater than the threshold value to be eligible for the rate break, computation time can be noticeably reduced. In the case of large right-hand-side constraints, we suggest scaling theseÂ values to the nearest thousands or millions. The approach from Excel proposed in this paper is particularly appropriate if one can balance the sizes of origindestination and right-hand-side constraints in such a way that computation time is not excessive. For a large-scale problem, one must exploit the structure of the model and acquire more information on the bounds of discrete variables. Our approach certainly provides an alternative way to solve nonlinear integer programming models with virtually all kinds of algebraic functions even for laymen who do not feel comfortable with mathematic programming jargons.
A Short-Run Model for the Coal Industry
T HE recent development of linear programming has made possible a new type of economic analysis.1 M1ethods have been developed to allow the computation of numerical solutions for theoretical problems involving economic choice. Linear programming provides a general mathematical model which is applicable to problems that can be expressed as the maximization (or minimization) of a linear form subject to a system of linear inequalities. Many economic problems can be placed in this general format, and linear programming can be used for the analysis of a plant, a firm, an industry, a national economy, or the world economy. The general methods of linear programming are used here in the formulation and solution of a short-run model for the coal industry. The present application is formally similar to the transportation problems formulated by Hitchcock and Koopmans.2 The model for the coal industry contains two separate, but interrelated, programming problems. The data of the model are spatially distributed demands for coal, the capacities of spatially distributed deposits of coal, and the unit costs of deliveries from the deposits to the demand locations. The levels of the deliveries are the variables of the first of the programming problems; they are selected to minimize the cost of meeting the demands subject to the capacity restrictions for the coal deposits. The variables of the second programming problem are the delivered prices of coal at the demand locations and the unit royalties earned by the various deposits. The values of these variables are selected to maximize total revenue net of royalty payments subject to the condition that every possible delivery must yield a non-positive profit. The optimum solutions of these two problems provide a complete description of perfectly competitive short-run equilibrium for the coal industry. Once the model has been presented, it is implemented with factual information for the demands, capacities, and costs; and numerical solutions are computed for both problems for