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Leakage Inventory Models In Fuzzy Environment |
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| Paper Id :
18108 Submission Date :
2023-07-07 Acceptance Date :
2023-07-18 Publication Date :
2023-07-25
This is an open-access research paper/article distributed under the terms of the Creative Commons Attribution 4.0 International, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. DOI:10.5281/zenodo.10068326 For verification of this paper, please visit on
http://www.socialresearchfoundation.com/anthology.php#8
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| Abstract |
After the introduction of the first economic order quantity
(EOQ) model, a wide range of inventory models have been set up and extended to
suit the real world situations. Of the various problems, leakage is a common
phenomenon in an inventory system. By incorporating fuzzy set theory, the
inventory models have been given an improved outlook to be more practical. So,
in this paper, two leakage inventory models are discussed in fuzzy approach and
are compared with the respective models in the classical environment. Some
uncertain inventory parameters are represented by fuzzy numbers and the optimum
result is defuzzified by signed distance method. This paper will serve to
encourage researchers towards the emerging trends of fuzzy inventory models. |
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| Keywords | Leakage, Fuzzy Number, Signed Distance Method, Fuzzy Inventory Model. | ||||||||||||||||||||||||||||||||||||||||
| Introduction | In any business, inventory represents the stock of items needed for the production or sales to run the business smoothly. Inventory includes the final product manufactured, the raw materials involved, the spare parts and the partly-finished goods which are either in the storehouse or in the factory. Variables in any inventory problems are either controlled variables or uncontrolled variables. The quantity acquired, the frequency of timing of acquisition, the stage of completion of stocked items etc. where the management has control over are the controlled variables. The different cost variables like holding cost, set up cost, shortage cost, etc. are uncontrolled variables. An inventory system is the whole of the business, i.e., from policies and controls to the hardware used in monitoring and maintaining the various levels of inventory to achieve a smooth and profitable operation of the organization, i.e., to minimize the total inventory cost. An efficient inventory control policy requires making appropriate decisions at each level of the system to meet the various goals like keeping the various inventory costs under control so that the cost of production and overall cost are reduced, to minimize losses through deterioration, leakage, shortage, etc. and maximize the profit. In any inventory system concerning liquid stock, there may be leakages. Leakages in the inventory system may be considered to be of very small quantity and hence are not easily detectable by the management immediately. Such small leakages will indirectly increase the expected operational cost of the firm and hence affect the expected profit. The leakages are generally temporary in nature as measures can be taken to overcome it once it is detected. With the occurrence of leakage in the stock, an idle time which is the time during which no goods are available for supply or sales is created and hence a penalty cost, which is the cost supposed to be incurred for having lost on the quantity in the stock, arises. In other sense, leakages can also be considered as a special kind of shortages. Tomba and Geeta [1] considered the leakage occurred in a particular stock while developing an inventory model. |
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| Objective of study | A mathematical model for the management of inventory within
the framework of fuzzy logic is suggested. Such a model is meaningful as there
are parameters that optimize the inventory with the dependence of the inventory
on parameters being vary with time or spatial evolution. On the other hand,
when storage of raw materials is involved in the inventory, leakage or wastage
of resources should be taken care of. The present model will be helpful in
understanding the management of inventory of such establishments. |
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| Review of Literature | The review of literature is already in the section
"Introduction". |
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| Main Text |
In real life situations, there arise problems or conditions which are very complex to be understood quantitatively or imprecise which are due to certain uncertainties. These cannot be explained by the classical set theory. In this context, fuzzy set theory has been playing an important role to describe such imprecise or vague situations or data by attributing a degree to which a certain object belongs to a set. This concept of uncertainty became a paradigmatic change in Mathematics and Science. This evolution of modern concept of uncertainties and vagueness was explained by Zadeh [2] in his paper Fuzzy Sets", which is a totally different concept from the classical set theory. Since then, fuzzy set theory has been playing important roles in various branches of Mathematics and Science. Zimmermann [3] discussed and pointed out the usefulness of fuzzy set in various applications involved in research and development of different fields. An inventory model considered with imprecise parameters or fuzzy parameters is called a fuzzy inventory model. The inventory model in this environment provides better and acceptable inferences than the crisp inventory model since the interpretation of the parameters are more realistic. Hence, fuzzy model is well suited than the traditional model. Lee and Yao [4] investigated an economic order quantity without backorder by taking fuzzy values and after defuzzification compare the total cost in the two models. Dhivya and Pandian [5] gave a review on some of the different inventory models in fuzzy environment. Jayjayanti [6] also discussed on fuzziness in inventory management problems. Brindhavanam and Rosario [7] studied an inventory model without shortages in fuzzy sense by considering different fuzzy numbers for the ordering and holding cost and find the optimal order quantity and optimal cost. The results obtained were then compared. Nabendu and Malakar [8] developed an inventory model with shortages considering different fuzzy numbers. Chandrasiri [9], in his inventory model, considered triangular fuzzy number as one of the parameters to draw the conclusion. Dutta and Kumar [10] considered trapezoidal fuzzy number in developing their model. Jaggi et al. [11], in their inventory model, considered deteriorating items with time-varying demand having shortages by taking some of the parameters involved as triangular fuzzy numbers. Three defuzzification methods are used too compare the results. In this paper, we have analyzed the model set up of Tomba and Geeta [1] and a fuzzy leakage inventory model with uniform demand rate having no shortages is discussed in two ways different ways: (i) production rate is instantaneous, and (ii) production rate is finite, by considering the holding cost and ordering cost as triangular fuzzy numbers. The optimal results obtained are defuzzified using signed distance method and are compared with the result of the crisp model. Definitions and Preliminaries 1.1 Fuzzy Set:
1.2 Fuzzy Number: A
fuzzy set
1.3 Triangular fuzzy number
Fig.1 : Triangular fuzzy number. 1.4 Some operations of triangular fuzzy number (TFN):
1.5 Fuzzy point :
1.6 Signed Distance Method of defuzzification :
2. Notations and Assumptions: 2.1 Notations: We define the following symbols: h = holding cost per unit quantity per unit time s = set up cost or ordering cost per order d = demand rate per unit time q = order quantity per run l = leakage rate per unit time TC = average total cost per unit time
t = time interval between runs K = finite production rate per unit time
2.2 Assumptions: The following assumptions are considered: 1. Demand is constant. 2. There is no shortage. 3. Holding cost and set up cost are fuzzy in nature. 4. Lead time is zero. 5. Production is instantaneous. 3. Proposed Inventory Models in two environments: Case 1: Leakage inventory model without shortages having uniform demand rate with instantaneous production rate
Figure 2: Instantaneous production leakage inventory
In actual operation of factories, uncertainties and fluctuations in the stock of inventory over a long period of time happen many a time and it is difficult to maintain a constant cost of procuring raw materials and operation. The present model for the optimization based on a triangular fuzzy leakage model leads to the optimization of operation and production of a factory.
Case 2: Leakage inventory model with uniform rate of demand, finite production rate and without shortages
Since
the ordering and holding cost are imprecise in nature, they are represented by
triangular fuzzy numbers to discuss the model in fuzzy sense. Let
4. Numerical Example : Case 1: Crisp Model Consider the inventory system with the following data in usual notation as d = 600 units per unit per year, h = Rs.10, s = Rs.100 per order, l = 10 units per unit per year Then, optimal order quantity q* = 108 and the minimum cost per year Cmin= Rs.1104 when there is no leakage, the minimum cost per year Cmin = Rs.1095 Case 2. Fuzzy model We
take d= units per unit per year, Table 1: Optimum solution for the model in fuzzy sense
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| Conclusion |
Two inventory leakage EOQ models without shortages, one with instantaneous production and the second with finite production rate are discussed both in traditional and fuzzy environment separately. Here, the holding cost and ordering cost are assumed to be uncertain and hence expressed as triangular fuzzy numbers. The minimum costs are obtained by the method of defuzzification using signed distance. The EOQ from the two environments are somewhat equivalent and the total cost from the fuzzy environment is greater than the crisp total cost. So, the fuzzy model is considered to be more reliable than that of classical model. |
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| References | 1. I. Tomba & O. Geeta, Some deterministic
leakage inventory models, Bull. Pure Appl. Sci., 27(2)(2008), 267-276. |
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