xv Preface Decision making is an important task of any industry. Operations research is a discipline that helps to solve decision making problems to make viable decision one needs exact and reliable information related to the decision variables involved in the problem. A stochastic process is a random process evolving with time. A stochastic process is a family of random variables {X(t), t T} where T is a parameter or index set. Stochastic processes have acquired importance mainly due to lot of applications in various fields such as physics, biology, statistics, economic, management, social and information sciences. After the advent of computers, Science and Technology had changed its face due to vast computational potential. Industrial Mathematics and Operations Research Management Science (OR/MS) techniques lead to efficient mathematical tools to generate new processes and innovative ideas to solve industrial and management decision problems. Most of the real life models are not deterministic. So the study of Operations research problems in stochastic nature is important today. This edited book Stochastic Processes and Models in Operations Research aims to assist scholars and researchers by providing comprehensive exposure to the concepts, trends and technologies relevant to current work in stochastic process modeling. This book aims to provide a concise exploration of inventory, queueing and other operations research problems. An overview of the chapters is as follows: Chapter 1, A Method for Analysing Queues in Fuzzy Environment by Thillaigovindan considers a new method for analysing queues in fuzzy environment is presented. This method combines the Zadeh s extension principle, the a-cut approach and the parametric non-linear programming technique. After explaining the new technique, it is applied to a fuzzy bulk queue with modified Bernoulli vacation and restricted admissible customers. Some special cases are discussed and a numerical study is also carried out. The new method can be applied to any queueing system in fuzzy environment. Chapter 2, M/C k /1 Queue With Impatient Customers by Umay Uzunoglu Kocer studies a single server queuing system with impatient customers and phase-type service is examined. We assume that arrivals are Poisson with a constant rate. When the server is busy upon an arrival, customer joins the queue and there is an infinite capacity of the queue. Closed-form solutions and performance measures like the mean queue length and the mean waiting time are derived by using probability generating function. Chapter 3, Performance Analysis of an M/G/1 Feedback Retrial queue with Two types of Service and Bernoulli Vacation by Arivudainambi and Gowsalya analyzes a single server retrial queue with two types of service, Bernoulli vacation and feedback. For this queueing model using supplementary variable technique, the steady state distributions of the server state and the number of customers in the orbit are obtained. Finally the average number of customers in the system and average number of customers in the orbit are also obtained.
Chapter 4, A Single Server Retrial queueing System with Two types of Batch arrivals by Kalyanaraman, provides a retrial queueing system with two types of customers called type I customer and type II customer. The type I customers arrive in batches of size k with probability c k and type II customers arrive in batches of size k with probability d k according to two independent Poisson processes. If type II customers upon arrival finds the server busy, they enter in to an orbit of infinite capacity in order to seek service again after random amount of time. All the customers in the retrial group behave independent of each other. The retrial time is exponentially distributed. The type I customers are queued in a priority queue of infinite capacity after blocking, that is, if the server is busy. As soon as the server is free, the customers in the priority queue are served using FCFS rule and the customers in the retrial group are served, if there are no customers in the priority queue. The service time distribution for both type of customers are identically independently distributed random variables and have different distributions. Supplementary variable technique is used for the analysis and the variable is the residual service time of a customer in service. Chapter 5, Analysis of Feedback Retrial Queue with Starting Failure and Server Vacation by Sathya and Ayyappan, discusses a batch arrival feedback retrial queue with Bernoulli vacation, where the server is subjected to starting failure. Any arriving batch finding the server busy, breakdown or on vacation enters an orbit. Otherwise one customer from the arriving batch enters a service immediately while the rest join the orbit. Repair times, service times and vacation times are assumed to be arbitrarily distributed. The time dependent probability generating functions have been obtained in terms of their Laplace transforms. Chapter 6, Performance Analysis of a Markovian Queuing System with Reneging and Retention of Reneged Customers by Rakesh kumar considers a finite capacity single server Markovian queuing system with reneging and retention of reneged customers. It is envisaged that a reneging customer may be convinced to stay for his service if some customer retention mechanism is employed. Thus, there is a probability that a reneging customer may be retained. Steady-state balance equations of the model are derived using Markov chain theory. The steady-state probabilities of system size are obtained explictly by using iterative method. The performance measures like expected system size, expected rate of reneging, and expected rate of retention are obtained. The effect of probability of retaining a reneging customer on the performance measures is studied. The economic analysis of the model is performed by developing a cost model. Chapter 7, A Stochastic Inventory Model with Multiple Vacations and N Policy by Jeganathan, considers a continuous review perishable inventory system at a service facility with the maximum capacity for S units. The waiting hall space is limited to accommodate a maximum number M of customers including the one at the service point. The arrival of customers is assumed to form a Poisson process. The demand is for single item per customer. The service starts only when the customer level reaches a prefixed level N(< M), starting from the epoch at which no customer is left behind in the system. The demanded item is delivered to the customer after a random time of service. The service times of items are assumed to be independent of each other and distributed as negative exponential. The life time of the commodity is assumed to be distributed as negative exponential. We have assumed that an item of inventory that makes it into the service process cannot perish while in service. An (0, S) ordering policy is adopted with zero lead time. When the waiting hall size is zero, the server leaves for a vacation whose duration is exponentially distributed. If the server find the customer level is less than N, at the end of a vacation, he takes another vacation immediately (multiple vacations). If the server returns from xvi
the vacation and finds at least N customers in the waiting hall then he immediately starts to serve the waiting customers. While the server serves a customer, the service may get interrupted with the interruption process governed by a Poisson process. It is assumed that while the server is under interruption, no further interruption can befall the server. On completion of an interruption the service restarts, with the duration of an interruption exponentially distributed. An impatient customer leaves the system independently after a random time which is distributed as negative exponential. Note that in this model we have assumed that the servicing customer can t impatient. We also assume that the inter arrival times, service times, the lead times, server vacation times, repair times and life time of each items are mutually independent random variables. Chapter 8, Analysis of Stochastic Inventory System in Supply Chain by Bakthavachalam studies a continuous review inventory control system to multi-echelon system (tandem network), which is a building block for supply chain. This can also be viewed as a serial supply chain that is modeled as a central warehouse, with one distribution centre and one retailer system handling a single non-perishable product. We assume that a finished product is supplied from warehouse (upper echelon) to Distribution Centre (middle echelon) which adopts one-for-one replenishment policy. The replenishment of items is take place in terms of packets from warehouse to distribution centre with exponential lead time. It is also assume that a batch of Q = S-s, items supplied to retailer (lower echelon) who adopts (s, S) policy. The lead time for retailer order of Q item is administrated with exponential lead time having parameter μ 0 > 0. The demand at retailer node follows a Poisson process. Demands occurring during the stock out periods are assumed to be lost sales and the maximum inventory level at retailers node S is fixed and reorder level s is varying such that S-s = Q and Q > s. The maximum inventory level at distribution centre is M where M = nq; Q = S-s, n N 0. Here we assume Q > S to avoid perpetual shortage at retailer node. Chapter 9, Stochastic Inventory System with Compliment item and Retrial Customers by Anbazhagan deals a two commodity stochastic inventory system under continuous review with maximum capacity of i-th commodity is S ( i = 1,2). In this two commodity one is main item and the other is i compliment item. It is assumed that demand for the i-th commodity is of unit size and demand time points form a Poisson process with parameter λ i ( i = 1,2). The compliment item is supplied as a gift whenever the demand occurs for the main item, but no main item is provided as a gift for demanding a compliment item. Reordering for supply is initiated as soon as the on-hand inventory level of the main item reaches a certain level s 1, and there is a lead time until the reorder arrives but instantaneous replenishment for the compliment item. The arriving any primary demands enter into an orbit, when the inventory level of main item is zero. Also the time between consecutive retrials is exponential. The limiting probability distribution for both commodities and the number of demands in the orbit, is computed and various operational characteristics are derived. The results are illustrated with numerical examples. Chapter 10, Analysis of Two-Echelon Inventory System with Direct and Retrial demands by Krishnan deals with a simple supply chain that is modeled as a single warehouse and multiple retailer system handling a single product. In order to avoid the complexity, at the same time without loss of generality, we assumed identical demand pattern at each node., we consider a two level supply chain inventory system. It consists of one warehousing facility and one retailer. We assumed that the demands to the Distribution Centre follow Poisson process. The direct demand gets Q units at a time. The demands initiated at retailer node follow Poisson process. The demand to the retailer node requires single item at a time. The lead times are exponentially distributed. The retailer follows (s, S) policy to maintain inventory and the distributor follow (0, nq) policy for maintaining inventory. The items are perishable& Non- perishable xvii
in nature, and it if it is perishable, it is assumed that the items are perishes only at the retailer node. The life time of an item is exponentially distributed. The unsatisfied customers are treated as retrial customers and they are waiting in the orbit with finite capacity N. The arriving demands finds the empty stock and the orbit is full are considered to be lost. Chapter 11, Continuous Review Substitutable Inventory System with Partial Backlogging by Vigneswaran studies A two commodity stochastic inventory system with the maximum capacity units for -th commodity is considered. The demand for -th commodity is of unit size and the time points of demand occurrences form independent Poisson processes. The reorder level for the -th commodity is fixed at, and ordering quantity for -th commodity is items when both inventory levels are less than or equal to their respective reorder levels. The lead time is assumed to be distributed as negative exponential distribution. The two commodities are assumed to be substitutable. That is, if the inventory level of one commodity reaches zero, then any demand for this commodity will be satisfied by the item of the other commodity. If no substitute is available, then this demand is backlogged. The backlog is allowed upto the level, for the -th commodity. Whenever the inventory level reaches or, both inventory levels are pulled back to their maximum levels and instantaneously and the previous order gets cancelled. Chapter 12, Perishable Inventory System with Bi-level Service System by Gomathi studies a perishable inventory system under continuous review at a bi-level service system with finite waiting hall of size N. The maximum storage capacity of inventory system is S units. We assumed that a demand for the commodity is of unit size. The arrival time points of customers form a Poisson process. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The effect of the two modes of operation on the system performance measures is also discussed. It is also assumed that lead time for the reorders is distributed as exponential and is independent of the service time distribution. The items are perishable in nature and the life time of each item is assumed to be exponentially distributed. The demands that occur during stock out periods are lost. The joint probability distribution of the the inventory level, number of customers (waiting and being served) in the system and status of the Mode in the steady-state case is derived. Various system performance measures in the steady state are derived. The results are illustrated with numerically. Chapter 13, On Sequential Decision Problems with Constant Costs of Observation by Sören Christensen and Albrecht Irle, presents a solution technique for optimal stopping problems with constant costs of observation in a diffusion setting. Such problems arise naturally, e.g., in Wald s type sequential decision problems and the Portfolio optimization model by Morton and Pliska. The main result is that the treatment of such problem boils down to the determination of the maximum points of a class of explicitly given functions. The findings are illustrated by a variety of examples and generalized to random costs of observation. Chapter 14, Building Fuzzy Models of Stochastic Processes to Determine Probabilistic and Linguistic Types of Uncertainty by Anna Walaszek-Babiszewska discusses the objective of stochastic data analysis is to interpret the past, present or future states of the observed system. In the topic literature of stochastic processes theory and applications there are many models and approaches to modeling and analysis real dynamic systems. We can distinguish such models as: Markov processes, random walk processes, processes of independent increments, diffusion processes and many other. On the other hand, time series analysis plays a pivotal role in analyzing data of financial markets and control systems. The values and changes of stochastic processes observed by experts are often described in linguistic categories, as e.g. strong course, increasing trend, high level, etc. The fuzzy set theory with its basic notions: fuzzy sets, linguistic variables, fuzzy events offers approximations of imprecisely formulated, linguistic categories. xviii
In the chapter we intend to show the advanced fuzzy modeling method which joints fuzzy and probabilistic approaches. We formulate a linguistic random variable, a stochastic process with fuzzy states and their probability distributions. Fuzzy models in the form of the rule knowledge representation, as well as, the inference procedure use the adequate probability distributions. We built such models for chosen type of stochastic processes, based on real data or using assumed theoretical probability distributions. The structure of the rule knowledge representation shows the empirical probability (frequency) of fuzzy states in past moments. The fuzzy conclusion (prediction) represents a fuzzy mean value of the process. We also discuss the effectiveness of proposed models. Chapter 15, Utility Maximization and Optimal Portfolio Selection by Swaminathan Udayabaskaran analyzes a portfolio selection problem by two different approaches. The first approach is the meanvariance approach where the optimal portfolio is selected by maximizing the expected terminal return and minimizing the variance of the terminal wealth. In the second approach, the optimal portfolio is selected by maximizing the expected utility of the terminal wealth. Although the first approach is quite straight-forward and simple to implement, it has the limitation of not being robust to more sophisticated models of securities. On the other hand, the second approach is very much robust and can be applied for all models of securities. In the second approach, we mainly employ two different methods. In one method, the stochastic control methodology is used which leads often to intractable partial differential equations. In the other method, the martingale methods are used which are quite different both in actual computations and in general structural results. The objective of the present chapter is to discuss various stochastic models of prices of securities subject to stochastic volatility and provide analytical/numerical solutions for optimal portfolios for investors trading in different financial markets. Chapter 16, Operations Research in Healthcare Supply Management under Fuzzy-Stochastic Environment by Priyan and Uthayakumar, studies a helathcare supply chain problem under fuzzy-stochastic approach. Operations research is for mankind in almost all aspects of our life. Applying the scientific method to the management of organizations, industry, government and other enterprises play a vital role in OR. It is used to increase productivity, to improve customer service, to improve quality and to reduce costs. Healthcare has attracted a great attention of governments in order to provide sufficient health services to the people. The provision of healthcare is very complicated and very responsible, that the right drug to the right people at the right time and in good condition to fight the disease. Today, the importance and significance of planning in healthcare can hardly be over emphasized when providing proper and adequate service continues to be a key concern of most countries. Operations research provides a wide range of methodologies that can help health care systems to significantly improve their operations. It helps to solve approximately all the problems involved in healthcare with its useful modeling techniques. The main objective of this book is to expand the knowledge of stochastic modeling and in turn help researchers and practitioners to develop suitable strategies, methods and models for decision making problems. This book facilitate the researchers for initiate their research in stochastic modeling and provide them the channels required to do research in stochastic models in operations research meticulously to help the decision making problems improve their performance through the successful implementation of stochastic models. This book intends to be a forum for exchanging new ideas and developments in the field of stochastic processes and modeling. xix