A note on a model for evasive vehicle movement along a fixed path

A model for a vehicle moving evasively along a fixed path is defined in terms of a two- state semi-Markov process. An important feature of this model is the continuous movement of the vehicle as a function of time. One potential application of this model is the development of a strategy for the deployment of long-range missiles on long underground tracks.     

A dynamic,nonstationary inventory problem for a price/quantity setting firm

A dynamic and nonstationary model is formulated for a firm which attempts to minimize total expected costs over a finite planning horizon. The control variables are price and production. The price p and the demand ζ are linked through the relationship ζ = g(p) + η, where g(p) is the riskless demand curve and η is a random variable. The general model allows for proportional ordering costs, convex holding and stockout costs, downward sloping riskless demand curve, backlogging, partial backlogging, lost sales, partial spoilage of inventory, and two modes of collecting revenue. Sufficient conditions are developed for this problem to have an optimal policy which resembles the single critical number policy known from stochastic inventory theory. It is also shown what set of parameters will satisfy these sufficiency conditions.     

Productivity estimates of the strategic airlift system by the use of simulation

Although the strategic airlift system is under continuous analysis, C-5A problems provided impetus to analyze the airlift system productivity function by using a large-scale simulation model. Development of the simulation model (Simulation of Airlift Resources - SOAR) was initiated by the Office of Secretary of Defense (Systems Analysis) in 1966. SOAR had barely become operational in time for the study in November 1968. Since limited verification and validation tests had been performed on the simulation model, the design of experiments was of critical importance. The experimental design had to be flexible enough to salvage the maximum amount of information possible upon the discovery of either a verification or validation error. In addition, the experimental design was required to accommodate the estimation of a large number of possibly changing independent variables. The experimental design developed for the analysis was full factorial design sets for a finite number of factors. Initial analysis began with aggregated sets of factors at two levels, and information gained from experiment execution was used to parse the sets. The process was sequential and parsing continued until the major explanatory independent variables were identified or enough information was obtained to eliminate the factor from further direct analysis. This design permitted the overlapping of simulation runs to fill out the factorial design sets. In addition to estimating the airlift productivity function, several other findings are reported which tended to disprove previous assumptions about the nature of the strategic airlift system.     

A branch and bound algorithm for allocation problems in which constraint coefficients depend upon decision variables

A branch and bound algorithm is developed for a class of allocation problems in which some constraint coefficients depend on the values of certain of the decision variables. Were it not for these dependencies, the problems could be solved by linear programming. The algorithm is developed in terms of a strategic deployment problem in which it is desired to find a least-cost transportation fleet, subject to constraints on men/materiel requirements in the event of certain hypothesized contingencies. Among the transportation vehicles available for selection are aircraft which exhibit the characteristic that the amount of goods deliverable by an aircraft on a particular route in a given time period (called aircraft productivity and measured in kilotons/aircraft/month) depends on the ratio of type 1 to type 2 aircraft used on that particular route. A model is formulated in which these relationships are first approximated by piecewise linear functions. A branch and bound algorithm for solving the resultant nonlinear problem is then presented; the algorithm solves a sequence of linear programming problems. The algorithm is illustrated by a sample problem and comments concerning its practicality are made.     

A chance-constrained distribution problem

The transportation model with supplies (S_i) and demands (D_i) treated as bounded variables developed by Charnes and Klingman is extended to the case where the S_i and D_i are independently and uniformly distributed random variables. Chance constraints which require that demand at the jth destination will be satisfied with probability at least β_i and that stockout at the ith origin will occur with probability less than α_i are imposed. Conversion of the chance constraints to their linear equivalents results in a transportation problem with one more row and column than the original with some of the new arcs capacitated. The chance-constrained formulation is extended to the transshipment problem.     

Evaluation of commonly used rules for detecting “steady state” in computer simulation

A definition of the problem of the initial transient with respect to the steady-state mean value has been formulated. A set of criteria has been set forth by which the efficaacy of any proposed rule may be assessed. Within this framework, five heuristic rules for predicting the approximate end of transiency, four of which have been quoted extensively in the simulation literature, have been evaluated in the M/M/1 situation. All performed poorly and are not suitable for their intended use.     

Multiple-attribute decision making with partial information: The comparative hypervolume criterion

A new approach is presented for analyzing multiple-attribute decision problems in which the set of actions is finite and the utility function is additive. The problem can be resolved if the decision makers (or group of decision makers) specifies a set of nonnegative weights for the various attributes or criteria, but we here assume that the decision maker(s) cannot provide a numerical value for each such weight. Ordinal information about these weights is therefore obtained from the decision maker(s), and this information is translated into a set of linear constraints which restrict the values of the weights. These constraints are then used to construct a polytope W of feasible weight vectors, and the subsets H_i (polytopes) of W over which each action a_i has the greatest utility are determined. With the Comparative Hypervolume Criterion we calculate for each action the ratio of the hypervolume of H_i to the hypervolume of W and suggest the choice of an action with the largest such ratio. Justification of this choice criterion is given, and a computational method for accurately approximating the hypervolume ratios is described. A simple example is provided to evaluate the efficiency of a computer code developed to implement the method.     

A mathematical model for generating the area of a drainage basin

This paper presents a mathematical model that yields the area drained by a naturally-occurring network of streams. The model is based on empirically derived relationships in the field of quantitative geomorphology and an assumption concerning the probabilistic nature of stream system formation. A wide range of model solutions is indicated, and the model is validated by comparing the results to statistics from actual stream systems.