Smarajit Barik Registration No

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1 SYNOPSIS ON A STUDY ON INVENTORY MANAGEMENT SYSTEM Thesis submitted in partial fulfillment of the requirements of the degree of Doctor of Philosophy By Smarajit Barik Registration No Under the Supervision of Prof. (Retd.) Umakanta Misra Post Graduate Department of Mathematics Berhampur University Bhanja Bihar Berhampur(Odisha), India & Dr. Susanta Kumar Paikray Post Graduate Department of Mathematics VSS University of Technology Burla, Sambalpur (Odisha), India Department of Mathematics Institute of Technical Education and Research SIKSHA O ANUSANDHAN UNIVERSITY Bhubaneswar, Odisha, India 2016

2 1. Introduction: The study of Operations Research is the gift of the Armed Forces to the modern society. This is a humble effort of an army officer who has spent years in teaching. In 1940, during World war-ii, military management of United Kingdom called on scientists from various disciplines and organized them into teams to assist in solving strategic and tactical problems associated with military context. Soon after the war many of the scientists who had been active in the military or groups turned their attention to the possibilities of applying a similar approach to civilian problems like business, industry, agriculture, aeronautical engineering, management science, economics, genetic engineering etc. to investigate complicated real word systems with an aim of improving optimum solution. The most important thing is the stocking of items from a smaller retailer to a larger production firm or shop to meet up the customer s demand as far as possible. Again, stocking of items depends on different factors such as deterioration, amelioration, demand, replenishment of order. The handling of such type of problems is known as inventory management. Inventory is the physical stock of goods that a business keeps on hand in order to promote the smooth and sufficient running of its affairs. It may be held before the production cycle, in the form of raw-material inventory; at an intermediate stage in the production cycle as in process inventory; or at the end of production cycle, as finished-goods inventory. The theory on inventory was developed in 1920s. At first, it had very simple models that used only a few parameters to capture the key factors. Later these models were embellished to include more details by adding more parameters but ignored variability and uncertainty. Gradually probabilistic models were developed in 1950s to capture the affects of unpredictable demand and lead times. All of these models suffered from one limitation they dealt with only one product at a time. The real life inventory facing many people was to manage on enormous variety of items, with enough interrelations among them to pose a management problem. Consequently, a separate data processing oriented subject called inventory control or inventory management evolved. Here the major concern was for organizing and maintaining records. Gradually the optimizing performance has been enhanced. 2

3 2. Literature Review on Inventory Models: There are different inventory models those played a vital role in the field of productions system which are assumed to have great importance because of their applicability and usefulness in the business world. Some literatures of particular models which are relevant to our work are surveyed and briefly described below. Most of the inventory models in the literature assume that items cannot be stored indefinitely to fulfill the future requirements. However some particular type of items deteriorate / ameliorate or become obsolete in the course of time and hence they are unstable. The commonly used goods like vegetables, fruits, foodstuffs, meat, fishes, mushroom, broiler (duck or pig), medicine, perfumes, alcohol, chemical, gasoline, radioactive substances, electronic components, photographic film etc, where deterioration / amelioration is usually observed during their normal storage period or at their farm houses. Therefore, if the rate of deterioration is not sufficiently low, its effect in the modeling of such an inventory system cannot be neglected. Inventory goods can be generally classified into mainly four categories such as: (a) (b) (c) (d) Obsolescence Deterioration / amelioration Deterioration Neither obsolescence nor deterioration Due to rapid changes of technology or entering of a new product by a competitor, the items those loss their value in course of time is known as obsolescence. The price of style goods is substantially reduced after the season is over or otherwise the goods are disposed off. For instance, after replacement model is introduced, the inventories like style goods, military air craft etc. become obsolete. In recent days businessmen are quite aware of the requirement for exactness in the field of stock control [29] of deteriorating items. Deterioration means the damage, spoilage, dryness and vaporization etc. of the products. The products such as green vegetables, food stuffs, human blood, photographic film etc. having maximum usable lifetime are known as perishable products and product like alcohol, gasoline, radioactive substances etc. having no shelf life at all are 3

4 known as decaying products. On the other hand, the shelf life of some products can be indefinite and hence they belong to neither obsolescence nor deterioration category. In the present proposed study we are not entertaining the replenishment polices for inventory of items under obsolescence because once the items become obsolete they are not required to record. A very few researchers have been attracted to study the inventory of obsolescence items. K. Cobbart and Oudheusden [7] developed inventory models for fast moving items subjected to sudden death obsolescence. In the various models, different cases of obsolescence risk have been studied allowing both shortages and without shortages. Many researchers have developed inventory models for deteriorating items from time to time. In this area Within [36] started his research work considering fashionable goods which deteriorate at the end of the prescribed storage period. Ghare and Schrader [14] initially developed an exponentially decaying inventory model. Buzacott [4] was one of the pioneer developers of inventory model with inflation. Shah et. al. [24] set up a request level-stock model for perishable items where the rate of deterioration is constant. Also different works in this direction have been carried out by Benkherouf [3] and Giri et.al. [15]. These models are all created for a infinite replenishment rate. Chu et. al. [5] have studied economic order quantity of deteriorating items under permissible delay in payments. Covert et. al. [8] have studied an EOQ model for items with Weibull distribution deterioration. Further many researchers like Dave [10], Bahari-kashani [1], Teng. et. al [31], Wee [35], Lin et. al. [20], Jaggi et. al. [19], Sarkar [25], Datta et. al. [9], Hardik [16] and many other have studied inventory models of deteriorating items. Time-varying demands were first considered by Silver and Meal [28]. In this model, they established the Heuristic solution procedure. Also Silver and Meal [28] developed a model for deterministic time-varying demand, which also gives an approximate solution procedure termed as Silver Meal Heuristic. Many research articles by Donaldson [13], Silver [27], Dave and Patel [11], Dave [10], Bahari-Kashani [1], Chung and Ting [6], Giri, Goswami and Choudhuri [15], Lin, Tan and Lee [20] and many more have analyzed liner time-varying demand. Roy and Choudhuri [23] established a non-static single-choice acquisition model for weakening things in which time-changing ramp-type demand, consistent deterioration and deficiency were 4

5 considered. Recently, Begum et al. [2] added to a stock model with time-shifting quadratic demand. The variability in the holding cost was set up in the stock models by Van der Veen [33], Muhlemann and Valtis-Spanopoulous [21], Giri, Goswami and Chaudhuri [15], Wee [35] and many more researchers. Neves [22] analyzed an optimal inventory policy with ordering costs that decrease gradually as ordering increases. Amelioration is a natural phenomenon observing in much life stock models. A few researchers have focused on ameliorating system. The vast majority of the traditional stock models depend on the rule that the estimation of stock stays consistent after some time. This is an uncommon sort of stock model where scientists have concentrated on for both ameliorating and deteriorating things, where items ameliorate when stay at breeding yard and deteriorates when in the distribution systems. Hwang [17] developed an inventory model for ameliorating items only. Again Hwang [18] added to a stock model for both ameliorating and deteriorating things independently. Tadj et. al. [30] has considered a creation stock model for both ameliorating and deteriorating items. Professionals did not give much attention for fast growing animals like broiler, ducks, pigs etc. in the poultry farm, highbred fishes in berry (pond) which are known as ameliorating items. When these items are in storage, the stock increases (in weight) due to growth of the items and also decrease due to death, various diseases or some other factors. At the point when these things are away, the stock increments (in weight) because of development of the things. Furthermore the stock diminishes because of death, different illnesses or due to some different components. Sana et. al. have studied [24] an EOQ model with time-dependent demand under inflation and time money value, Tripathi [32] has studied inventory model with different demand rate and different holding cost. In the literature, Fuzzy set theory has been applied to inventory problems to handle the uncertainties related to the demand or cost coefficients. The upside of utilizing the fuzzy set hypothesis as a part of demonstrating the stock issues is its capacity to measure vagueness and imprecision. In specific circumstances, instabilities are because of fuzziness and are basically presented by Zadeh [37]. In 1970, Zadeh et. al. [38] proposed a few methodologies for choice 5

6 making in fuzzy environment. Wide uses of fuzzy set hypothesis can be found in Zimmerman [39]. In essential EOQ model we distinguish the request measure that minimizes the aggregate of yearly expenses of stock holding and settled setup to place orders. Vujosevic et. al. [34] established an EOQ model when inventory cost is fuzzy. De and Rawat [12] has studied a fuzzy inventory model without shortages using triangular fuzzy number. 3. Objectives of the present work: One of the most important problems faced in inventory management is how to control and maintain the inventories of deteriorating items. The effect of deterioration of physical goods cannot be neglected in any inventory system because almost all the physical goods deteriorate over time i.e., the quality and quantity of goods decreases in course of time due to deterioration. The main objective of the thesis is to discuss the inventory models of deteriorating items. Further consideration of exponential and weibull distribution under time varying and price dependent demand of actual degree of deterioration under the influence of inflation is one of the significant works in the present study. Also amelioration being a natural phenomenon in life stock inventory so inventory model of ameliorating items is incorporated in the present study. To avoid inadequacy in demand, inventory models under fuzzy environment has also studied at the end. 4. Organization of Thesis: The thesis is embodied with eight chapters. The first chapter is the introduction and literature survey on the related work where the background literature has been described briefly in the light of which the investigation of the author has been presented in the subsequent chapters. Chapter 2 to Chapter 7 deals with the contribution of the author in the field of Inventory Management System. The last chapter is a brief conclusion and outlines of some unsolved problems for further investigation. The bibliography section and the information about the research publications of the papers are presented at the end. The chapter wise summary of the proposed work is given below. 6

7 Chapter 1: It provides an overview of the conceptual theoretical aspects inventory management system. The literature survey and the motivation for the present work have been discussed. Chapter 2: This chapter includes an inventory model of deteriorating items under time varying demand condition. The objective of the model is to discuss an inventory model for deteriorating perishable items and obtain the total optimal cost of the model with time varying type of demand rate with time proportional deterioration without shortage. The time varying holding cost is assumed to be a linear function of time. The result is illustrated with numerical example and the sensitivity analysis of various parameters is carried out. Chapter 3 presents inventory model of deteriorating items for nonlinear holding cost with time dependent demand has been discussed. The objective of the model is to investigate the inventory system for perishable items where time proportional deterioration rate is considered. The Economic order quantity is determined for minimizing the average total cost per unit time. Time dependent demand rate is used with finite time horizon. Nonlinear holding cost with shortage is considered. The result is illustrated with numerical example. Chapter 4 reports the inventory model for weibull ameliorating, deteriorating items under the influence of inflation has been analyzed. The objective of this model is to discuss the development of an inventory model for ameliorating items. Generally fast growing animals like duck, pigs, broiler etc. in poultry farm, high-bred fishes are these types of items. This paper investigates an instantaneous replenishment model for the above type of items under cost minimization in the influence of inflation and time value of money. A time varying type of demand rate with infinite time horizon, constant deterioration and without shortage is considered. The result is illustrated with numerical example. Chapter 5 deals with a deteriorating inventory model with shortages under price dependent demand and inflation. The objective of this model is to discuss the inventory system for deteriorating items with price dependent demand pattern where weibull rate of deterioration is considered. Here shortages are assumed and are 7

8 partially backlogged with inflation. The Economic order quantity is determined for maximizing the profit per unit time. The result is illustrated with numerical example. Chapter 6 provides an optimal inventory model with single item under various demand conditions. The objective of this model is to discuss the inventory model for time varying demand and constant demand; and time dependent holding cost and constant holding cost for case 1 and case 2 respectively. Mathematical model has been developed for determining the optimal order quantity, the optimal cycle time and optimal total inventory cost for both cases. Numerical examples are given for both cases to validate the proposed model. Sensitivity analysis is carried out to analyze the effect of changes in the optimal solution with respect to change in various parameters. Chapter 7 presents an Inventory control model of deteriorating items in Fuzzy environment has been discussed. The objective of this model is to investigate the inventory system for perishable items where fixed deterioration rate and stock dependent demand rate is used with finite time horizon is considered. The analytical development is provided to obtain the fuzzy optimal solution to minimize the total cost per time unit and for defuzzification, graded unit-preference integration method is used. Chapter 8 presents concluding remark with a brief summary of the present work. Also it provides the scope of extension of the work in future. 5. Methodology: The following systematic procedure to be followed during our proposed study (i) (ii) (iii) (iv) (v) (vi) (vii) Imagination of the physical problem Basic assumptions and notations Formation of a Mathematical model Solution of the differential equation under boundary conditions. Testing of the optimality conditions. Properties through sensitivity analysis tables. Theory developed. 8

9 6. Conclusions: Inventory management is one of the important aspects in production system as well as in business affairs. At present, the applications of inventory management covers almost all the branches of scientific studies like oil and gas plant, atomic power plant, chemical power plant, pharmaceutical power plant, ocean and space research, study of anthropology, architectures, ancient human dynasties etc. The early period of inventory management system was mainly focused towards the study of simple inventory models. But with the progress of time the model becomes more and more complicated due to various factors. In the present study different inventory models under various factors like time varying demand condition, non linear holding cost etc. have been studied under the influence of inflation. The ameliorating life stock items should be purchased when they are young and sold when they become grow up. It has also been reflected in the present thesis. The customer s impatience has also been realized allowing shortages and their painfulness to purchase the materials due to inflation has also been presented in the present work. In real life situations, the exact data are inadequate in many inventory models. Human judgments including inclinations are frequently obscure and in this way can't be assessed with precise numerical information. To avoid such ambiguity the vital decision regarding inventory models under fuzzy environment has been presented in the thesis. Further we have provided an update of inventory management system literature made by notable contributors to the analysis of both fixed life, perishable inventory and inventory subject to continuous exponential decay provided a comprehensive survey on continuously deteriorating inventory models where the deterioration was considered as function of the on hand level of inventory. Inventory models have so far been developed and analyzed extensively either from deterministic approach or stochastic approach. In deterministic approach, the parameters are assumed to be known, the objective is formulated under fixed constraints. In real world inventory of deteriorating items, the information available is not always well defined or precise, rather it is vague imprecise or insufficient. 9

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