Approximating the Nonlinear Newsvendor and Single-Item Stochastic Lot-Sizing Problems When Data Is Given by an Oracle

The single-item stochastic lot-sizing problem is to find an inventory replenishment policy in the presence of discrete stochastic demands under periodic review and finite time horizon. A closely related problem is the single-period newsvendor model. It is well known that the newsvendor problem admits a closed formula for the optimal order quantity whenever the revenue and salvage values are linear increasing functions and the procurement (ordering) cost is fixed plus linear. The optimal policy for the single-item lot-sizing model is also well known under similar assumptions. In this paper we show that the classical (single-period) newsvendor model with fixed plus linear ordering cost cannot be approximated to any degree of accuracy when either the demand distribution or the cost functions are given by an oracle. We provide a fully polynomial time approximation scheme for the nonlinear single-item stochastic lot-sizing problem, when demand distribution is given by an oracle, procurement costs are provided as nondecreasing oracles, holding/backlogging/disposal costs are linear, and lead time is positive. Similar results exist for the nonlinear newsvendor problem. These approximation schemes are designed by extending the technique of K-approximation sets and functions.National Science Foundation (U.S.) (Contract CMMI-0758069)United States. Office of Naval Research (Grant N000141110056

An algorithm for the serial capacitated economic lot-sizing problem with non-speculative costs and stationary capacities

We address the serial capacitated economic lot-sizing problem under particular assumptions on the costs and the capacity pattern. We prove that when the involved costs are non-speculative with respect to the transfer to future periods and the capacity pattern is stationary for all levels, the optimal plan for each level can be obtained independently in O(T 3) time. This leads to an O(T 3L) algorithm for the problem with L levels. and the capacity pattern is stationary for all levels, the optimal plan for each level can be obtained independently in O(T 3) time. This leads to an O(T 3L) algorithm for the problem with L levels

Approximation schemes for non-separable non-linear Boolean programming problems under nested knapsack constraints

We consider a fairly general model of “take-or-leave”decision-making. Given a number of items of a particular weight, the decision-maker either takes (accepts) an item or leaves (rejects) it. We design fully polynomial-time approximation schemes (FPTASs) for optimization of a non-separable non-linear function which depends on which items are taken and which are left. The weights of the taken items are subject to nested constraints. There is a noticeable lack of approximation results on integer programming problems with non-separable functions. Most of the known positive results address special forms of quadratic functions, and in order to obtain the corresponding approximation algorithms and schemes considerable technical difficulties have to be overcome. We demonstrate how for the problem under consideration and its modifications FPTASs can be designed by using (i) the geometric rounding techniques, and (ii) methods of K -approximation sets and functions. While the latter approach leads to a faster scheme, the running times of the of both algorithms compare favorably with known analogues for less general problems

ОПТИМИЗАЦИЯ ВЫПУСКА КОМПЛЕКТОВ ИЗДЕЛИЙ И ИНТЕНСИВНОСТЕЙ ИХ ИЗГОТОВЛЕНИЯ В УСЛОВИЯХ СЛУЧАЙНОГО СПРОСА

The problem of optimizing the output of multi-product batch and the intensities of its items manufacture in the production line over a number of time intervals is considered. The line has a linearly ordered multiple positions without buffers. Workpieces of the input sequence composed of cyclically repeated identical subsequences (batches) are processed consecutively one by one in each working position in the order of their location in the line. Only a single workpiece is disposed in each position at each time point. The operation of the line consists of takts of simultaneous processing of all workpieces located in respective positions by the sets of tools corresponding to workpieces and positions. The composition of a batch does not vary from interval to interval. The ranges of possible demand quantities for each product and the probability distribution of the demand in these ranges are assumed known for each time interval. The sum of manufacturing cost, costs of storage and/or penalties for unmet demand on products is used as objective function. Manufacturing cost depends on processing intensities to be defined and increases with an increase in the number of batches produced in the current interval. Storage cost of unclaimed product units as well as penalty for product units not supplied to the customer do not decrease with the increase of number of such units. A two-level decomposition method for solving the problem based on the ideas of multi-step optimization is proposed.Рассматривается задача оптимизации на ряде временных интервалов программы выпуска производственной линией комплектов изделий нескольких наименований и интенсивностей их изготовления. Линия состоит из ряда линейно упорядоченных рабочих позиций без буферов. Заготовки из входной последовательности, состоящей из циклически повторяющихся идентичных подпоследовательностей (комплектов), обрабатываются последовательно одна за другой на каждой рабочей позиции линии в порядке их расположения, и в каждый момент времени на каждой позиции обрабатывается только одна заготовка. Работа линии состоит из тактов одновременной обработки на всех позициях всех расположенных на них заготовок соответствующими позициям и заготовкам наборами инструментов. Состав комплекта не изменяется от интервала к интервалу. Диапазоны возможных величин спроса на каждое изделие комплекта и распределение вероятностей спроса в этих диапазонах считаются известными для каждого временного интервала. В качестве целевой функции используется сумма производственных затрат, затрат на хранение невостребованных изделий и/или штрафов за неудовлетворенный спрос на них. Производственные затраты зависят от принимаемой интенсивности обработки и возрастают с увеличением количества комплектов, выпускаемых в текущем интервале. Затраты на хранение невостребованных изделий каждого наименования, а также штрафы за недопоставленные заказчикам изделия не убывают с ростом числа таких изделий. Предложен двухуровневый декомпозиционный метод решения задачи, основанный на идеях многошаговой оптимизации.

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

We present a framework for obtaining fully polynomial time approximation schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone single-period cost functions. This framework is developed through the establishment of two sets of computational rules, namely, the calculus of K-approximation functions and the calculus of K-approximation sets. Using our framework, we provide the first FPTASs for several NP-hard problems in various fields of research such as knapsack models, logistics, operations management, economics, and mathematical finance. Extensions of our framework via the use of the newly established computational rules are also discussed

ОПТИМИЗАЦИЯ ВЫПУСКА И ИНТЕНСИВНОСТЕЙ ОБРАБОТКИ ГРУППЫ ДЕТАЛЕЙ ПРИ НЕСТАЦИОНАРНОМ СПРОСЕ

We consider a problem of optimizing the output of a batch of parts and intensities of its processing with tool blocks on a multiposition equipment under non-stationary demand and predetermined time intervals. The batch content does not vary from one interval to another. The objective function is the sum of production cost, storage cost of excess parts, and penalties for unmet demand. The production cost depends on processing intensities. A decomposition method for solving the problem is proposed.Рассматривается задача оптимизации размеров выпуска группы деталей и интенсивностей их обработки на многопозиционном оборудовании блоками инструментов при нестационарном спросе на заданных временных интервалах. Состав группы не меняется от интервала к интервалу. В качестве целевой функции принята сумма затрат на производство, хранение избытков деталей и штрафы за неудовлетворенный спрос на них. Затраты на выпуск группы деталей зависят от интенсивностей их обработки. Предложен декомпозиционный метод решения задачи

Dynamic lot sizing for a warm/cold process: Heuristics and insights

Cataloged from PDF version of article.We consider the dynamic lot sizing problem for a warm/cold process where the process can be kept warm at a unit variable cost for the next period if more than a prespecified quantity has been produced. Exploiting the optimal production plan structures, we develop nine rule-based forward solution heuristics. Proposed heuristics are modified counterparts of the heuristics developed previously for the classical dynamic lot sizing problem. In a numerical study, we investigate the performance of the proposed heuristics and identify operating environment characteristics where each particular heuristic is the best or among the best. Moreover, for a warm/cold process setting, our numerical studies indicate that, when used on a rolling horizon basis, a heuristic may also perform better costwise than a solution obtained using a dynamic programming approach. & 2012 Elsevier B.V. All rights reserved

LP-based Approximation Algorithms for the Capacitated Facility Location Problem

The capacitated facility location problem is a well known problem in combinatorial optimization and operations research. In it, we are given a set of clients and a set of possible facility locations. Each client has a certain demand that needs to be satisfied from open facilities, without exceeding their capacity. Whenever we open a facility we incur in a corresponding opening cost. Whenever demand is served, we incur in an assignment cost; depending on the distance the demand travels. The goal is to open a set of facilities that satisfy all demands while minimizing the total opening and assignment costs. In this thesis, we present two novel LP-based approximation algorithms for the capacitated facility location problem. The first algorithm is based on LP-rounding techniques, and is designed for the special case of the capacitated facility location problem where capacities are uniform and assignment costs are given by a tree metric. The second algorithm follows a primal-dual approach, and works for the general case. For both algorithms, we obtain an approximation guarantee that is linear on the size of the problem. To the best of our knowledge, there are no LP-based algorithms known, for the type of instances that we focus on, that achieve a better performance