Mirko Vujošević

EOQ formula when inventory cost is fuzzy

Abstract Various types of uncertainties and imprecision are inherent in real inventory problems. They are classically modeled using the approaches from the probability theory. However, there are uncertainties that cannot be appropriately treated by usual probabilistic models. The questions how to define inventory optimization tasks in such environment and how to interpret optimal solutions arise. This paper considers the modification of EOQ formula in the presence of imprecisely estimated parameters. For example, holding and ordering costs are often not precisely known and are usually expressed by linguistic terms such as: ”Holding cost is approximately of value c h ″, or: “Ordering cost is about value c o or more”. These imprecise parameters are presented by fuzzy numbers, defined on a bounded interval on the axis of real numbers. Alternative approaches to determining the optimal order quantity in a fuzzy environment are developed, illustrated by a selection of examples, and discussed.

Fuzzy models for the newsboy problem

Abstract This paper presents two fuzzy models for the newboy problem in an uncertain environment. It is assumed that uncertainties may appear in demand and in inventory costs. Fuzzy demand is used to describe a subjective estimate, linguistically expressed by the phrase “demand is about d ”. Also, fuzzy demand could be derived from evidences about demand recorded in the past. Imprecise inventory costs, such as overage and shortage costs, are represented by fuzzy sets, too. The quantity that should be ordered for a fixed time period minimizes the possible total cost. The computational aspects of the fuzzy models and their interpretations are illustrated by examples.

Reliability Evaluation and Optimization of Redundant Dynamic Systems

This paper analyzes the reliability of a simple dynamic system with redundant element and demonstrates an optimization technique for system reliability. A computer program was implemented to calculate the true reliability and the near optimal redundancy allocation for a dynamic subsystem. The solutions for the examples were always truly Optimum which is illustrated by an example.

Spares allocation in the presence of uncertainty

In this paper we present a new combined optimization-simulation approach to determining the spares of an inventory for the maintenance of a system when some parameters are not completely known. The system under consideration consists of a number of randomly failing replaceable components, in series. It is supposed that the failure rates and unit prices of components are not precisely known. The basic idea is to classify all system components into a given number of subcategories. Each subcategory is defined by ranges for the failure rate and the representative unit price. It is assumed that the real failure rate of a given component is randomly distributed in the corresponding subcategory range. Initially, the number of spares can be obtained if we choose a point-valued failure rate- unit price representative pair for each spare subcaterory, (λ, c ) and then apply some routine optimization technique for calculating the optimum number of spares. After that, the simulation procedure is applied in order to check that the chosen pairs are acceptable representatives of the subcategory, with regard to the required probability that the total reliability of the system with spares is satisfied. If the simulation shows that too many or too few spares are obtained, which means unacceptable costs or unacceptable system operation, ‘better’ (λ, c ) representatives are chosen and new spares allocations with succeeding simulation are iteratively repeated. Actual programming results indicate that the iterative algorithm, with optimization and simulation phases following each other, can efficiently handle problems in systems with hundreds of different components.

Reliability analyses for a tree-structured hierarchic control system

Mathematical models for reliability calculations of a three-level centralized control system are given. The highest level is a central control unit which sends control messages to peripheral control units that process them and transmit appropriate instructions to the lowest level, the terminal units. The following probabilities are defined, modeled, and calculated: operating terminal reliability, branch reliability, group reliability, and system reliability. >

Spare parts inventory planning for a redundant system subject to a phased mission

Abstract The problem considered is a priori planning of spare parts inventory intended to keep operable a redundant system consisting of n identical and statistically independent elements. The system is subject to a certain mission proceeding in a number of successive time periods, called phases, during which environmental conditions and, thus, element reliabilities vary. Maintenance actions, intended to determine the system state and replace failed elements by new ones, may be performed only during overhauls between two successive phases. Spares inventories for replacement purposes are planned in advance for each overhaul assuming that spare parts remaining unused from previous overhauls can be used in succeeding ones. The mathematical model of the described system is developed in the paper and expressions for calculating relevant performances are derived. An optimization problem is stated in which the total purchase and holding costs are the criterion function and the stockout probabilities represent constraints.

Uncertainty in reliability evaluation processes and a simulation approach to treating it

Abstract This paper considers the problem of uncertainty in reliability data that are used in many decision making processes and describes a simulation approach to dealing explicitly with this uncertainty. The causes of uncertainty are discussed for three different situations: (a) in the process of designing new systems, when failure data are not yet available, (b) after performing reliability test and gathering failure data, and (c) in mission reliability prediction. It is concluded that, in performing reliability prediction or failure rate prediction, one should use interval estimates rather than point estimates. These intervals can be used to perform component classification and then, employing simulation, to obtain tables or scattergrams for the mean time between failures of a system or for system reliability.

A recurrence relation for calculating a reliability increment in a series-parallel system

A recurrence relation is given for the easy calculation of an increment in the reliability of a series-parallel system if the number of redundant elements is increased by one. This relation is used for cold or standby redundancy when the number of failures obeys the Poisson distribution. The formulas have been implemented in various computer programs for system reliability calculation and optimization. The experience with many real and hypothetical examples has shown no computational problems. There is no need to care either about the magnitude of the number of redundancies or about the accuracy of computation. This shows the advantage of this algorithm over the approximation proposed by M. Messinger and M. Shooman (1970). >