Representation of general and polyhedral subsemilattices and sublattices of product spaces

AbstractIt is shown that each element of the lattice of meet (resp., join) sublattices of a product S of n chains has a representation as the intersection of n subsets of S, the ith of which is decreasing (resp., increasing) for each fixed value of the ith coordinate for each i. This result is applied to show that an arbitrary element of the lattice of sublattices of S has a representation as the intersection of n2 subsets, the ijth of which is decreasing for each fixed value of the ith and increasing for each fixed value of the jth coordinate for each i, j. Irreducible representations are given in each case, providing an alternative proof of an instance of Hashimoto's (1952) representation of sublattices of a distributive lattice. Moreover, irreducible representations are given for the polyhedral members of the lattice of closed convex subsets of n-dimensional Euclidean space that are at once subsemilattices or sublattices. It is alsoshown that the polyhedral subsemilattices and sublattices can be represented as duals respectively of pre-Leontief substitution systems and generalized network-flow problems. Finally, the problems of checking whether a polyhedral set is a subsemilattice or sublattice are reduced to that of solving a system of linear inequalities, thereby showing that these recognition problems can be solved in polynomial time

Subextremal functions and lattice programming

Let M and N be the set of minimizers of a function f over respective subsets K and L of a lattice, with K being lower than L. This paper characterizes the class of functions f for which M is lower (resp., weakly lower, meet lower, join lower, chain lower) than N for all K lower than L. The resulting five classes of functions, called subextremal variants, have alternate characterizations by variants of the downcrossing-differences property, i.e., their first differences change sign at most once from plus to minus along complementary chains.Comparative statics, supermodular functions

The Status of Mathematical Inventory Theory

This paper surveys the current status of mathematical inventory theory. The review is limited to studies which seek optimal policies for dynamic inventory models. Models with certain and uncertain demands are discussed. Particular attention is focused on multi-item and/or multi-echelon inventory systems.

Least d-Majorized Network Flows with Inventory and Statistical Applications

It is shown that for any feasible network flow model, there is a flow which simultaneously minimizes every d-Schur convex function of the flows emanating from a single distinguished node called the source. The vector of flows emanating from the source in the minimizing flow is unique and is the "least d-majorized" flow. This flow can be found by solving the problem for the special case where the d-Sehur convex function is separable and quadratic. Once this flow is found, the solution of the dual problem is reduced to evaluating the conjugate of a function appearing in the dual objective function at the above flow. This computation is extremely simple when the function is separable. These results are extended to situations in which the variables must be integers. An important special case of the problem can be solved geometrically by choosing, from among all paths joining two points in the plane and lying between two given nonintersecting paths, the path with minimum euclidian length. Applications of the results are given, to deterministic production-distribution models (e.g., the Modigliani-Hohn [30] production smoothing model), certain of the stochastic inventory-redistribution models examined by Ignall and Veinott [27], a deterministic price speculation and storage model (including Cahn's warehouse problem [11]), and a zero lead time case of the Clark-Scarf series multi-echelon model [13]. In addition, applications are given to several maximum likelihood estimation problems in which the parameters satisfy certain linear inequalities, e.g., those surveyed in Brunk [8], [9, pp. 1347-1349], and a few others.

Production Planning with Convex Costs: A Parametric Study

We consider the problem of choosing the amounts of a single product to produce in each of a finite number of time periods so as to minimize the (convex) production and (convex) inventory carrying costs over the periods while satisfying known requirements. The effect of changes in the various parameters, viz., the requirements and the production and inventory capacity limits, upon the optimal production levels is studied. The results obtained are exploited to provide simple and intuitive computational procedures for finding optimal production schedules for a range of parameter values.

Constrained Markov Decision Chains

We consider finite state and action discrete time parameter Markov decision chains. The objective is to provide an algorithm for finding a policy that minimizes the long-run expected average cost when there are linear side conditions on the limit points of the expected state-action frequencies. This problem has been solved previously only for the case where every deterministic stationary policy has at most one ergodic class. This note removes that restriction by applying the Dantzig-Wolfe decomposition principle.

Optimality of Myopic Inventory Policies for Several Substitute Products

Multiproduct inventory systems with proportional ordering costs and stochastic demands are studied. New conditions are obtained under which a myopic ordering policy (a policy of minimizing expected cost in the current period alone) is optimal for a sequence of periods for all initial inventory levels. An important one of these, the substitute property, holds when the myopic policy is such that increasing the initial inventory of one product does not increase the quantity ordered of any product. Conditions on the one period expected holding and shortage cost function, which are of independent interest in nonlinear programming, are shown to imply the substitute property. Applications of these conditions to models with storage or investment limitations and to a multiechelon model are given. Under backlogging the usual extension to a fixed delivery lag is obtained. Some non-stationary cases are also treated.

Computing Optimal (s, S) Inventory Policies

A complete computational approach for finding optimal (s, S) inventory policies is developed. The method is an efficient and unified approach for all values of the model parameters, including a non-negative set-up cost, a discount factor 0 \leqq \alpha \leqq 1, and a lead time. The method is derived from renewal theory and stationary analysis, generalized to permit the unit interval range of values for \alpha . Careful attention is given to the problem associated with specifying a starting condition (when \alpha