Radivoj Petrovic

Supply chain modelling using fuzzy sets

Abstract This paper considers a production supply chain (SC) with all facilities in a serial connection. The SC includes inventories and production facilities between them. It is assumed that the SC operates in an uncertain environment. Uncertainty is associated with: (1) customer demand, (2) supply deliveries along the SC and (3) external or market supply. Uncertainties are described by vague and imprecise phrases that are interpreted and represented by fuzzy sets. The SC fuzzy model described in this paper is developed to determine the order quantities for each inventory in the SC in the presence of uncertainties, that give an acceptable service level of the SC at reasonable total cost. Two control concepts of the SC are treated: (1) decentralised control of each inventory and (2) partial coordination in the inventories control. A special purpose simulator has been developed for examining the dynamics and performance of all the parts of the SC and the SC as a whole. Various simulation tests have been carried out to assess particularly the effects of uncertain external supply on the SC service level. Different approaches to improve SC performance in an uncertain environment have been simulated and analysed.

Modelling and simulation of a supply chain in an uncertain environment

This paper describes fuzzy modelling and simulation of a supply chain (SC) in an uncertain environment, as the first step in developing a decision support system. An SC is viewed as a series of facilities that performs the procurement of raw material, its transformation to intermediate and end-products, and distribution and selling of the end-products to customers. All the facilities in the SC are coupled and interrelated in a way that decisions made at one facility affect the performance of others. SC fuzzy models and a simulator cover operational SC control. The objective is to determine the stock levels and order quantities for each inventory in an SC during a finite time horizon to obtain an acceptable delivery performance at a reasonable total cost for the whole SC. Two sources of uncertainty inherent in the external environment in which the SC operates were identified and modelled: customer demand and external supply of raw material. They were interpreted and represented by fuzzy sets. In addition to the fuzzy SC models, a special SC simulator was developed. The SC simulator provides a dynamic view of the SC and assesses the impact of decisions recommended by the SC fuzzy models on SC performance.

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.

Optimal preventive maintenance scheduling of thermal generating units in power systems —A survey of problem formulations and solution methods

Abstract Preventive maintenance scheduling of thermal generating units occupies a significant place in power system operation and expansion planning; at the same time it is a challenging optimization problem. The purpose of this paper is to analyse essential features of the maintenance scheduling problem, including imposed constraints and various objectives, as well as to present the results of the research work done in this field using operational research methods. Papers published during last fifteen years are discussed, with special regard for the applied optimization techniques.

Coordinated control of distribution supply chains in the presence of fuzzy customer demand

This paper considers a single product inventory control in a Distribution Supply Chain (DSC). The DSC operates in the presence of uncertainty in customer demands. The demands are described by imprecise linguistic expressions that are modelled by discrete fuzzy sets. Inventories at each facility within the DSC are replenished by applying periodic review policies with optimal order up-to-quantities. Fuzzy customer demands imply fuzziness in inventory positions at the end of review intervals and in incurred relevant costs per unit time interval. The determination of the minimum of defuzzified total cost of the DSC is a complex problem which is solved by applying decomposition; the original problem is decomposed into a number of simpler independent optimisation subproblems, where each retailer and the warehouse determine their optimum periodic reviews and order up-to-quantities. An iterative coordination mechanism is proposed for changing the review periods and order up-to-quantities for each retailer and the warehouse in such a way that all parties within the DSC are satisfied with respect to total incurred costs per unit time interval. Coordination is performed by introducing fuzzy constraints on review periods and fuzzy tolerances on retailers and warehouse costs in local optimisation subproblems.

A multiobjective optimization approach to thermal generating units maintenance scheduling

Abstract This paper presents a new approach to preventive maintenance scheduling of thermal generating units in electric power systems. The problem has been formulated as a combinatorial optimization task, with explicit and simultaneous treatment of multiple objectives: minimization of fuel costs, maximization of reliability, and minimization of constraints violations. The optimization is performed by the multiobjective branch and bound algorithm with successive approximations, embedded in an outer loop where in each iteration a new initial solution is generated in a pseudorandom manner. This procedure was the basis for development of the multiobjective maintenance scheduling software package. A realistic example of annual maintenance scheduling of 21 thermal generating units demonstrates the effectiveness of the proposed methodology.

Multicriteria ranking of inventory replenishment policies in the presence of uncertainty in customer demand

Abstract This paper considers the ranking of alternative replenishment policies for a single-location inventory when customer demand is confined to a single product. It is supposed that customer demand is uncertain and described by imprecise linguistic terms defined by fuzzy sets. For example, “demand is about d”, or “demand is more or less between dL and dU”, or more complex “demand is about d, but there is a possibility to be 2d”, etc. Four alternative replenishment policies are considered in particular: (1) fixed cycle–fixed order quantity, (2) fixed cycle–variable order quantity, (3) variable review period–fixed order quantity, and (4) variable review period–variable order quantity. With each replenishment policy a number of criteria are associated. Typical criteria examined are: (a) average fill rate, (b) inventory holding cost per item delivered, (c) regularity of the replenishment orders with regard to time and quantity of ordering. Criteria values for each replenishment policy can be obtained either analytically or by using a simulation technique, or they are linguistic subjective judgements defined by fuzzy sets, like, for example, the values of criterion (c). In order to choose the best replenishment policy, a procedure for selecting a compromised optimistic–pessimistic solution is developed and applied. It enables the comparison of a finite number of alternative replenishment policies with respect to a number of criteria, simultaneously, when the criteria values are either cardinal or linguistic terms, and the relative weights of the criteria are vaguely expressed. An illustrative example is given.

Fuzzy logic based algorithms for maximum covering location problems

This paper concerns a class of maximum covering location problems in networks in uncertain environments. It is assumed that (a) relative weights of demand nodes are either deterministic or imprecise, described by linguistic expressions and (b) potential facility site locations are limited to network nodes. The concept of coverage is extended to include a degree of node coverage which means that the borders between the subset of covered demand nodes and the subset of uncovered demand nodes are inexact. The acceptable service distance/travelling times from a facility site to demand nodes are modelled by fuzzy sets. Three new algorithms for choosing the best facility locations are developed which assume that (1) demands at all nodes are equally important, (2) relative weights of demand at nodes are deterministic and (3) weights of demand at nodes are imprecise and described by linguistic terms, respectively. The algorithms are based on searching among potential facility nodes by applying comparison operations on discrete fuzzy sets. It is shown how to extend the proposed algorithms from one-site to multi-site covering problems. Illustrative examples of selecting locations for logistics centres in a distribution company are given.

SPARTA II: Further development in an expert system for advising on stocks of spare parts

Abstract This paper is part of a continuous story of SPARTA - an expert system for advising on spare stocks for maintenance purposes. Using an item approach, SPARTA II suggests the assortment and quantities of both repairable and consumable parts of electronic systems. The domain of expertise is restricted to the parts for which the exponential distribution of the times between failures can be assumed. The output of SPARTA II includes: (1) the list of recommended stocks of spares, (2) the calculated reliability with and without the spares recommended, (3) suggestions concerning further investment in spare parts.