A lagrangean relaxation algorithm for the constrained matrix problem

We present variants of a convergent Lagrangean relaxation algorithm for minimizing a strictly convex separable quadratic function over a transportation polytope. The algorithm alternately solves two “subproblems,” each of which has an objective function that is defined by using Lagrange multipliers derived from the other. Motivated by the natural separation of the subproblems into independent and very easily solved “subsubproblems,” the algorithm can be interpreted as the cyclic coordinate ascent method applied to the dual problem. We exhibit our computational results for different implementations of the algorithm applied to a set of large constrained matrix problems.     

The interactive fixed charge linear programming problem

In many decision-making situations, each activity that can be undertaken may have associated with it both a fixed and a variable cost. Recently, we have encountered serveral practical problems in which the fixed cost of undertaking an activity depends upon which other activities are also undertaken. To our knowledge, no existing optimization model can accomodate such a fixed cost structure. To do so, we have therefore developed a new model called the interactive fixed charge linear programming problem (IFCLP). In this paper we present and motivate problem (IFCLP), study some of its characteristics, and present a finite branch and bound algorithm for solving it. We also discuss the main properties of this algorithm.     

Successive approximation in separable programming: An improved procedure for convex separable programs

We implement a solution procedure for general convex separable programs where a series of relatively small piecewise linear programs are solved as opposed to a single large one, and where, based on bound calculations developed in [13] and [14], the ranges of linearization are systematically reduced for successive programs. The procedure inherits ε-convergence to the global optimum in a finite number of steps, but perhaps its most distinct feature is the rigorous way in which ranges containing an optimal solution are reduced from iteration to iteration. This paper describes the procedure, called successive approximation, discusses its convergence, tightness of the bounds, bound-calculation overhead, and its robustness. It presents a computer implementation to demonstrate its effectiveness for general problems and compares it (1) with the more standard separable programming approach and (2) with one of the recent augmented Lagrangian methods [10] included in a comprehensive study of nonlinear programming codes [12]. It seems clear from over 130 cases resulting from 80 distinct problems studied here that significant savings in terms of computational effort can be realized by a judicious use of the procedure, and the ease with which it can be used is appreciably increased by the robustness it shows. Moreover, for most of these problems, the advantage increases as the size, nonlinearity, and the degree of desired accuracy increase. Other important benefits include significantly smaller storage requirements, the ability to estimate the error in the current solution, and to terminate the algorithm as soon as the acceptable level of accuracy has been achieved. Problems requiring up to about 10,000 nonzero elements in their specification and about 45,000 nonzero elements in the generated separable programs resulting from up to 70 original nonlinear variables and 70 nonlinear constraints are included in the computations.     

The asymptotic joint distribution of the state occupancy times in an alternating renewal process

An alternating renewal process starts at time zero and visits states 1,2,…,r, 1,2, …,r 1,2, …,r, … in sucession. The time spent in state i during any cycle has cumulative distribution function F_i, and the sojourn times in each state are mutually independent, positive and nondegenerate random variables. In the fixed time interval [0,T], let U_i(T) denote the total amount of time spent in state i. In this note, a central limit theorem is proved for the random vector (U_i(T), 1 ≤ i ≤ r) (properly normed and centered) as T → ∞.     

Warranty design under buyer and seller risk aversion

This paper provides a framework in which warranty policies for non-repairable items can be evaluated according to risk preferences of both buyers and sellers. In particular, a warranty price schedule is established such that sellers are indifferent among the policies. Given this schedule, a buyer's response is expressed by selecting the price-warranty combination that minimizes disutility. Within this framework, a warranty can be viewed as an instrumet of risk management that can induce more sales and greater profitability. For given utility functions, analytical results for the development of a price schedule are developed. Numerical results illustrate the substitution effects between warranty terms, prices, and risk parameters.     

Rescheduling to minimize makespan on a changing number of identical processors

We consider the problem of rescheduling n jobs to minimize the makespan on m parallel identical processors when m changes value. We show this problem to be NP-hard in general. Call a list schedule totally optimal if it is optimal for all m = 1, …,n. When n is less than 6, there always exists a totally optimal schedule, but for n ≥ 6 this can fail. We show that an exact solution is less robust than the largest processing time first (LPT) heuristic and discuss implications for polynomial approximation schemes and hierarchical planning models.     

Algorithms for the minimax transportation problem

In this paper, we consider a variant of the classical transportation problem as well as of the bottleneck transportation problem, which we call the minimax transportation problem. The problem considered is to determine a feasible flow x_ij from a set of origins I to a set of destinations J for which max_(i,j)εIxJ{c_ijx_ij} is minimum. In this paper, we develop a parametric algorithm and a primal-dual algorithm to solve this problem. The parametric algorithm solves a transportation problem with parametric upper bounds and the primal-dual algorithm solves a sequence of related maximum flow problems. The primal-dual algorithm is shown to be polynomially bounded. Numerical investigations with both the algorithms are described in detail. The primal-dual algorithm is found to be computationally superior to the parametric algorithm and it can solve problems up to 1000 origins, 1000 destinations and 10,000 arcs in less than 1 minute on a DEC 10 computer system. The optimum solution of the minimax transportation problem may be noninteger. We also suggest a polynomial algorithm to convert this solution into an integer optimum solution.     

Optimal replacement for fault-tolerant systems

We consider the optimal replacement problem for a fault tolerant system comprised of N components. The components are distingushable, and the state of the system is given by knowing exactly which components are operationl and which have failed. The individual component failure rates depend on the state of the entire system. We assume that the rate at which the system produces income decreases as the system deteriorates and the system replacement cost rises. Individual components cannot be replaced. We give a greedy-type algorithm that produces the replacement policy that maximizes the long-run net system income per unit time.     

Optimal refueling sequence for a mixed fleet with limited refuelings

Consider a fleet of vehicles comprised of K₁ identical tankers and K₂ identical nontankers (small aircraft). Tankers are capable of refueling other tankers as well as nontankers. The problem is to find that refueling sequence of the tankers that maximizes the range simultaneously attainable by all K₂ nontankers. A recent paper established that the “unit refueling sequence,” comprised of one tanker refueling at each of K₁ refueling operations, is optimal. The same paper also proffered the following conjecture for the case that the number of refueling operations is constrained to be less than the number of tankers: A nonincreasing refueling sequence is optimal. This article proves the conjecture.     

Bounding methods for facilities location algorithms

Single- and multi-facility location problems are often solved with iterative computational procedures. Although these procedures have proven to converage, in practice it is desirable to be able to compute a lower bound on the objective function at each iteration. This enables the user to stop the iterative process when the objective function is within a prespecified tolerance of the optimum value. In this article we generalize a new bounding method to include multi-facility problems with l_p distances. A proof is given that for Euclidean distance problems the new bounding procedure is superior to two other known methods. Numerical results are given for the three methods.