Bennett L. Fox

Searching for the Multiplier in One-Constraint Optimization Problems

One-constraint optimization problems are approached via Lagrange multipliers. Sequential search schemes for generating suitable trial multiplier values are compared, and it is shown that, in general, the minimax sequential search is bisection. For certain applications, it pays to design search procedures that take advantage of special structure, such as recursively defined functions. An efficient, one-pass search procedure based on bisection for a multi-item, multi-echelon inventory problem is also presented.

Letter to the Editor-Age Replacement with Discounting

We consider age replacement with discounting. For known failure distribution, our principal results are for the case when the failure rate is continuous and strictly increasing to ∞. We also show what happens for small discount rate. For unknown failure distribution, we give the mini-max strategy.

Scientific Applications: An algorithm for identifying the ergodic subchains and transient states of a stochastic matrix

An algorithm for identifying the ergodic subchains and transient states of a stochastic matrix is presented. Applications in Markov renewal programming and in the construction of variable length codes are reviewed, and an updating procedure for dealing with certain sequences of stochastic matrices is discussed. Computation times are investigated experimentally and compared with those of another recently proposed method.

Finite-state approximations to denumerable-state dynamic programs

Abstract We show how to find a sequence of policies for essentially finite-state dynamic programs such that the corresponding vector of optimal returns converges pointwise to that of a denumerable-state dynamic program. The corresponding result for stochastic games is also given.

Adaptive age replacement

Loss structure models concerning age replacement policies, discussing mathematical aspects of maintenance with dynamic programming formulation

Letter to the Editor-Finding Minimal Cost-Time Ratio Circuits

In the treatment of routing problems, other authors have used column generators to introduce, into the basis of the master problem, the solution that corresponds to a cycle in a graph with minimal cost-to-time ratio. This subproblem is of independent interest and corresponds to deterministic Markov renewal programming; this note presents an efficient method for its solution, intuitive basis for which is a search for a way to route flow so that the cost-time trade-off is optimal. This flow-circulation problem is solved parametrically by the out-of-kilter algorithm.

Accelerating List Processing in Discrete Programming

I t is pointed out how two (existing) l is t-handling algorithms, heapsort and balanced tree search, can accelerate incremental allocation and branch and bound a lgor i thms-in particular, certain integer programming algorithms.

Letter to the Editor-Nonoptimality of Planned Replacement in Intervals of Decreasing Failure Rate

The planned replacement problem assumes a known failure distribution Ft and that replacement occurs at the earlier of a planned replacement interval at cost c1 and the time of failure at cost c2. We show that it cannot be optimal to select a planned replacement interval at a point for which the failure rate ft/[1-Ft] is decreasing.

Accelerating LP algorithms

It is shown how a novel method for computing (related) inner products can accelerate the pricing phase of LP algorithms. Other LP applications are indicated.