The maximum collection problem with time-dependent rewards

We consider a routing problem where the objective is to maximize the sum of the rewards collected at the nodes visited. Node rewards are decreasing linear functions of time. Time is spent when traveling between pairs of nodes, and while visiting the nodes. We propose a penalty-based greedy (heuristic) algorithm and a branch-and-bound (optimal) algorithm for this problem. The heuristic is very effective in obtaining good solutions. We can solve problems with up to 20 nodes optimally on a microcomputer using the branch-and-bound algorithm. We report our computational experience with this problem. © 1996 John Wiley & Sons, Inc.     

Requirements of an “optimizer” for computer simulations

Computer simulation has many advantages. However, one major disadvantage is that, in all too many cases, the attempt to use computer simulation to find an optimum solution to a problem rapidly degenerates into a trial-and-error process. Techniques for overcoming this disadvantage, i. e., for making optimization and computer simulation more compatible, are applicable at two points in the development of the overall computer simulation. Techniques which are used within actual construction of the mathematical models comprising the simulation will be labeled as internal methods, while those which are used after the simulation has been completely developed will be termed external methods Because external methods appear to offer the largest potential payoff, discussion is restricted to these methods, which are essentially search techniques. In addition, the development of an “Optimizer” computer program based on these techniques is suggested Although drawbacks to the use of search techniques in the computer simulation framework exist, these techniques do offer potential for “optimization.” The modification of these techniques to satisfy the requirements of an “Optimizer” is discussed.     

A general axiomatization of additive measurement with applications

Necessary and sufficient conditions are specified for a general theory of additive measurement that presumes very little set-theoretic structure. The theory is illustrated for numerical representations in extensive, conjoint, difference, threshold, expected utility, probability, ambiguity, and subset measurement. © 1992 John Wiley & Sons, Inc.     

Computing equilibria via nonconvex programming

The problem of determining a vector that places a system in a state of equilibrium is studied with the aid of mathematical programming. The approach derives from the logical equivalence between the general equilibrium problem and the complementarity problem, the latter being explicitly concerned with finding a point in the set S = {x: < x, g(x)> = 0, g(x) ≦ 0, x ≧ 0}. An associated nonconvex program, min{? < x, g(x) > : g(x) ≦ 0, x ≧ 0}, is proposed whose solution set coincides with S. When the excess demand function g(x) meets certain separability conditions, equilibrium solutions are obtained by using an established branch and bound algorithm. Because the best upper bound is known at the outset, an independent check for convergence can be made at each iteration of the algorithm, thereby greatly increasing its efficiency. A number of examples drawn from economic and network theory are presented in order to demonstrate the computational aspects of the approach. The results appear promising for a wide range of problem sizes and types, with solutions occurring in a relatively small number of iterations.     

Simulation of multi-echelon macro-inventory policies

Alternative repair part inventory policies are examined for a multi-echelon logistics system. The policies assessed pertain to the design of multi-echelon systems rather than the evaluation of stock level decisions for individual repair parts. Although the context is one of a military inventory system, the policies examined are of importance in nonmilitary systems where item failure rates are low, and only periodic access to resupply is possible.     

A backward recursive technique for optimal sequential sampling plans

This paper presents a procedure akin to dynamic programming for designing optimal acceptance sampling plans for item-by-item inspection. Using a Bayesian procedure, a prior distribution is specified, and a suitable cost model is employed depicting the cost of sampling, accepting or rejecting the lot. An algorithm is supplied which is digital computer oriented.     

The discrete max-min problem

The historic max-min problem is examined as a discrete process rather than in its more usual continuous mode. Since the practical application of the max-min model usually involves discrete objects such as ballistic missiles, the discrete formulation of the problem seems quite appropriate. This paper uses an illegal modification to the dynamic programming process to obtain an upper bound to the max-min value. Then a second but legal application of dynamic programming to the minimization part of the problem for a fixed maximizing vector will give a lower bound to the max-min value. Concepts of optimal stopping rules may be applied to indicate when sufficiently near optimal solutions have been obtained.