Systems defense games: Colonel blotto,command and control

The classical “Colonel Blotto” games of force allocation are generalized to include situations in which there are complementarities among the targets being defended. The complementarities are represented by means of a system “characteristic function,” and a valuation technique from the theory of cooperative games is seen to indicate the optimal allocations of defense and attack forces. Cost trade-offs between systems defense and alternative measures, such as the hardening of targets, are discussed, and a simple example is analyzed in order to indicate the potential of this approach.     

A squared-variable transformation approach to nonlinear programming optimality conditions

We show that the well-known necessary and sufficient conditions for a relative maximum of a nonlinear differentiable objective function with nonnegative variables constrained by nonlinear differentiable inequalities may be derived using the classical theory of equality constrained optimization problems with unrestricted variables. To do this we transform the original inequality-constrained problem to an equivalent equality-constrained problem by means of a well-known squared-variable transformation. Our major result is to show that second order conditions must be used to obtain the Kuhn-Tucker conditions by this approach. Our nonlinear programming results are motivated by the development of some well-known linear programming results by this approach.     

Some experiments in global optimization

When applied to a problem which has more than one local optimal solution, most nonlinear programming algorithms will terminate with the first local solution found. Several methods have been suggested for extending the search to find the global optimum of such a nonlinear program. In this report we present the results of some numerical experiments designed to compare the performance of various strategies for finding the global solution.     

Analysis of the turbulent regime of the progress curve when new learning additions have variable slopes

Learning curves have been used extensively for predictive purposes in the airframe and other industries. In many instances this has led to erroneous results because analysts failed to extend learning curve theory and develop adequate analytical techniques in the turbulent regime of the cost history characterizing these industries. It is this area where a series of design changes induces a series of perturbations whose turbulence intensity is a function of the frequency of occurrence and magnitudes of the design changes under consideration. In Ref. 1, a series of formulations amenable to machine programming was developed for the accurate determination of perturbed unit costs. This development was based on additions of new learning having a constant slope. In this discussion, the development of Ref. 1 will be generalized by developing formulas for the addition of new effort having variable slopes. Consideration will also be given to the expressions involving elementary unit cost expressions so that cumulative average and cumulative total values can readily be obtained from existing experience curve tables. Conversely, the problem of determining the magnitudes of design changes and the slopes of new effort from graphical data will also be considered.     

Theoretical analysis of lanchester-type combat between two homogeneous forces with supporting fires

This paper studies combat between two homogeneous forces modelled with variable-coefficient Lanchester-type equations of modern warfare with supporting fires not subject to attrition. It shows that this linear differential-equation model for combat with supporting fires may be transformed into one without the supporting fires so that all the previous results for variable-coefficient Lanchester-type equations of modern warfare (without supporting fires) may be invoked. Consequently, new important results for representing the solution (i.e. force levels as functions of time) in terms of canonical Lanchester functions and also for predicting force annihilation are developed for this model with supporting fires. Important insights into the dynamics of combat between two homogeneous forces with such supporting fires are discussed.     

On convexity proofs in location theory

It is often assumed in the facility location literature that functions of the type ø_i(x_i, y) = β_i[(x_i-x)²+(y_i-y)²]^K/2 are twice differentiable. Here we point out that this is true only for certain values of K. Convexity proofs that are independent of the value of K are given.     

“simple-approximate” battle-outcome-prediction conditions for variable-coefficient lanchester-type equations of modern warfare

This article considers combat between two homogeneous forces modeled by variable- coefficient Lanchester-type equations of modern warfare and develops new “simple-approximate” battle-outcome-prediction conditions for military engagements terminated by two different types of prescribed conditions being met (fixed-force-level-breakpoint battles and fixed-force-ratio-breakpoint battles). These battle-outcome-prediction conditions are sufficient (but not necessary) to determine the outcome of battle without having to explicitly compute the force-level trajectories, and they are characterized by their simplicity, requiring no advanced mathematical knowledge or tabulations of “special functions” for their application. Integrability properties of the Lanchester attrition-rate coefficients figure prominently in their results, and involved in their development is a generalization of Lanchester's famous square law to variable-coefficient Lanchester-type combat and several other novel mathematical developments for the analysis of ordinary differential equations. Examples are given, with the attack of a mobile force against a static defensive position (both sides armed with weapons whose firepower is range dependent) being examined in detail.     

Toward a study of bidding processes part IV - games with unknown costs

This paper represents a continuation of three previous papers [1-.3] in the study of competitive bidding processes. It treats the case where a bidder's knowledge of his competitor's cost i s given by a probability distribution over a certain interval. The results obtained extend the work of Vickrey [4] to the case where the cost intervals a r e not necessarily symmetric.