Limiting survival functions of self‐similar structures

We show the existence of a unique analytic single parameter limiting survival function arising from the repeated composition of a coherent structure as the number of components tends to infinity. Examples include the repeated composition process of the bridge structure. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Optimal outbound dispatch policies: Modeling inventory and cargo capacity

In this paper, we present an optimization model for coordinating inventory and transportation decisions at an outbound distribution warehouse that serves a group of customers located in a given market area. For the practical problems which motivated this paper, the warehouse is operated by a third party logistics provider. However, the models developed here may be applicable in a more general context where outbound distribution is managed by another supply chain member, e.g., a manufacturer. We consider the case where the aggregate demand of the market area is constant and known per period (e.g., per day). Under an immediate delivery policy, an outbound shipment is released each time a demand is realized (e.g., on a daily basis). On the other hand, if these shipments are consolidated over time, then larger (hence more economical) outbound freight quantities can be dispatched. In this case, the physical inventory requirements at the third party warehouse (TPW) are determined by the consolidated freight quantities. Thus, stock replenishment and outbound shipment release policies should be coordinated. By optimizing inventory and freight consolidation decisions simultaneously, we compute the parameters of an integrated inventory/outbound transportation policy. These parameters determine: (i) how often to dispatch a truck so that transportation scale economies are realized and timely delivery requirements are met, and (ii) how often, and in what quantities, the stock should be replenished at the TPW. We prove that the optimal shipment release timing policy is nonstationary, and we present algorithms for computing the policy parameters for both the uncapacitated and finite cargo capacity problems. The model presented in this study is considerably different from the existing inventory/transportation models in the literature. The classical inventory literature assumes that demands should be satisfied as they arrive so that outbound shipment costs are sunk costs, or else these costs are covered by the customer. Hence, the classical literature does not model outbound transportation costs. However, if a freight consolidation policy is in place then the outbound transportation costs can no longer be ignored in optimization. Relying on this observation, this paper models outbound transportation costs, freight consolidation decisions, and cargo capacity constraints explicitly. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 531–556, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10030     

The effects of model parameter deviations on the variance of a linearly filtered time series

We consider a general linear filtering operation on an autoregressive moving average (ARMA) time series. The variance of the filter output, which is an important quantity in many applications, is not known with certainty because it depends on the true ARMA parameters. We derive an expression for the sensitivity (i.e., the partial derivative) of the output variance with respect to deviations in the model parameters. The results provide insight into the robustness of many common statistical methods that are based on linear filtering and also yield approximate confidence intervals for the output variance. We discuss applications to time series forecasting, statistical process control, and automatic feedback control of industrial processes. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Adjacency‐based local top‐down search method for finding maximal efficient faces in multiple objective linear programming

It is well‐known that the efficient set of a multiobjective linear programming (MOLP) problem can be represented as a union of the maximal efficient faces of the feasible region. In this paper, we propose a method for finding all maximal efficient faces for an MOLP. The new method is based on a condition that all efficient vertices (short for the efficient extreme points and rays) for the MOLP have been found and it relies on the adjacency, affine independence and convexity results of efficient sets. The method uses a local top‐down search strategy to determine maximal efficient faces incident to every efficient vertex for finding maximal efficient faces of an MOLP problem. To our knowledge, the proposed method is the first top‐down search method that uses the adjacency property of the efficient set to find all maximal efficient faces. We discuss this and other advantages and disadvantages of the algorithm. We also discuss some computational experience we have had with our computer code for implementing the algorithm. This computational experience involved solving several MOLP problems with the code.     

Properties of a mean residual life function arising from renewal theory

The present article discusses the properties of the mean residual life function in a renewal process. We examine the relationship this function has with the failure rate function and the conventional mean, variance and coefficient of variation of residual life. We also discuss some monotonicity properties of the mean residual life function. A partial order based on the renewal mean residual function is introduced along with its interrelationship with some existing stochastic orders. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Lanchester-type models of warfare and optimal control

The optimization of the dynamics of combat (optimal distribution of fire over enemy target types) is studied through a sequence of idealized models by use of the mathematical theory of optimal control. The models are for combat over a period of time described by Lanchester-type equations with a choice of tactics available to one side and subject to change with time. The structure of optimal fire distribution policies is discussed with reference to the influence of combatant objectives, termination conditions of the conflict, type of attrition process, and variable attrition-rate coefficients. Implications for intelligence, command and control systems, and human decision making are pointed out. The use of such optimal control models for guiding extensions to differential games is discussed.     

An operator theory of parametric programming for the generalized transportation problem: I. Basic theory

This paper gives a new organization of the theoretical results of the Generalized Transportation Problem with capacity constraints. A graph-theoretic approach is utilized to define the basis as a one-forest consisting of one-trees (a tree with an extra edge). Algorithmic development of the pivot-step is presented by the representation of a two-tree (a tree with two extra edges). Constructive procedures and proofs leading to an efficient computer code are provided. The basic definition of an operator theory which leads to the discussion of various operators is also given. In later papers we will present additional results on the operator theory for the generalized transportation problem based on the results in the present paper.     

A computation procedure for the exact solution of location-allocation problems with rectangular distances

A method is presented to locate and allocate p new facilities in relation to n existing facilities. Each of the n existing facilities has a requirement flow which must be supplied by the new facilities. Rectangular distances are assumed to exist between all facilities. The algorithm proceeds in two stages. In the first stage a set of all possible optimal new facility locations is determined by a set reduction algorithm. The resultant problem is shown to be equivalent to finding the p-median of a weighted connected graph. In the second stage the optimal locations and allocations are obtained by using a technique for solving the p-median problem.     

On the treatment of force-level constraints in times-equential combat problems

The treatment of force-level constraints in time-sequential combat optimization problems is illustrated by further studying the fire-programming problem of Isbell and Marlow. By using the theory of state variable inequality constraints from modern optimal control theory, sharper results are obtained on necessary conditions of optimality for an optimal fire-distribution policy (in several cases justifying conjectures made in previous analysis). This leads to simplification of the determination of the domains of controllability for extremals leading to the various terminal states of combat. (Additionally, some new results for the determination of boundary conditions for the adjoint variables in optimal control problems with state variable inequality constraints have arisen from this work.) Some further extensions of previous analysis of the fire-programming problem are also given. These clarify some key points in the solution synthesis. Some important military principles for target selection and the valuation of combat resources are deduced from the solution. As a result of this work, more general time-sequential combat optimization problems can be handled, and a more systematic solution procedure is developed.     

An operator theory of parametric programming for the transportation problem-I

This paper investigates the effect on the optimum solution of a (capacitated) transportation problem when the data of the problem (the rim conditions-i. e., the warehouse supplies and market demands-the per unit transportation costs and the upper bounds) are continuously varied as a (linear) function of a single parameter. An operator theory is developed and algorithms provided for applying rim and cost operators that effect the transformation of optimum solution associated with changes in rim conditions and unit costs. Bound operators that effect changes in upper bounds are shown to be equivalent to rim operators. The discussion in this paper is limited to basis preserving operators for which the changes in the data are such that the optimum basis structures are preserved.