The bounded interval generalized assignment model

The bounded interval generalized assignment model is a “many-for-one” assignment model. Each task must be assigned to exactly one agent; however, each agent can be assigned multiple tasks as long as the agent resource consumed by performing the assigned tasks falls within a specified interval. The bounded interval generalized assignment model is formulated, and an algorithm for its solution is developed. Algorithms for the bounded interval versions of the semiassignment model and sources-to-uses transportation model are also discussed.     

A versatile scheme for ranking the extreme points of an assignment polytope

A cutting plane scheme embedded in an implicit enumeration framework is proposed for ranking the extreme points of linear assignment problems. This method is capable of ranking any desired number of extreme points at each possible objective function value. The technique overcomes storage difficulties by being able to perform the ranking at any particular objective function value independently of other objective values. Computational experience on some test problems is provided.     

A note on first passage times in birth and death and nonnegative diffusion processes

Consider a birth and death process starting in state 0. Keilson has shown by analytical arguments that the time of first passage into state n has an increasing failure rate (IFR) distribution. We present a probabilistic proof for this. In addition, our proof shows that for a nonnegative diffusion process, the first passage time from state 0 to any state x is IFR.     

Multivariate age-dependent imperfect repair

In this article we consider models of systems whose components have dependent life lengths with specific multivariate distributions. Upon failure, components are repaired. Two types of repair are distinguished. After perfect repair, a unit has the same life distribution as a new item. After imperfect repair, a unit has the life distribution of an item which is of the same age but has never failed. We study a model in which the mechanism for determining the nature of the repair is age dependent.     

Equivalence relations in the theory of repetitive consumption

Preference for food in the function of the time since last consumption and total preference in the function of eating frequency are equivalent mathematical representations of attitudes governing food consumption. The observed and postulated properties of these functions imply a formal correspondence between preference maximizing and variety seeking behavior.     

A comparison of several estimates of availability

We compare several competing estimates of the availability of a system which alternates between two states, “up” and “down,” in accordance with an alternating renewal process. Both interval and point estimators are compared under several special but representative situations. The comparison reaffirms the validity and robustness of the log-logistic jackknifed estimates. However, when the point estimates are compared from the intrinsic criterion of probability of concentration, the uniformly minimum variance estimate obtained for the Markov model performs very well.     

Applications of renewal theory in analysis of the free-replacement warranty

Under a free-replacement warranty of duration W, the customer is provided, for an initial cost of C, as many replacement items as needed to provide service for a period W. Payments of C are not made at fixed intervals of length W, but in random cycles of length Y = W + γ(W), where γ(W) is the (random) remaining life-time of the item in service W time units after the beginning of a cycle. The expected number of payments over the life cycle, L, of the item is given by M_Y(L), the renewal function for the random variable Y. We investigate this renewal function analytically and numerically and compare the latter with known asymptotic results. The distribution of Y, and hence the renewal function, depends on the underlying failure distribution of the items. Several choices for this distribution, including the exponential, uniform, gamma and Weibull, are considered.     

Optimal set partitioning,matchings and lagrangian duality

We formulate the set partitioning problem as a matching problem with simple side constraints. As a result we obtain a Lagrangian relaxation of the set partitioning problem in which the primal problem is a matching problem. To solve the Lagrangian dual we must solve a sequence of matching problems each with different edge-weights. We use the cyclic coordinate method to iterate the multipliers, which implies that successive matching problems differ in only two edge-weights. This enables us to use sensitivity analysis to modify one optimal matching to obtain the next one. We give theoretical and empirical comparisons of these dual bounds with the conventional linear programming ones.