EXTENSION OF STOCHASTIC ANALYSIS MODEL TO THREE PRODUCTS INVENTORY SYSTEM WITH SCBZ SEASONAL PRODUCTION AND SALES

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1 International Journal of Civil Engineering and Technology (IJCIET) Volume 8, Issue 10, October 2017, pp , Article ID: IJCIET_08_10_056 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed EXTENSION OF STOCHASTIC ANALYSIS MODEL TO THREE PRODUCTS INVENTORY SYSTEM WITH SCBZ SEASONAL PRODUCTION AND SALES D. Kanagajothi Department of Mathematics, Veltech Dr. RR & Dr.SR University, Avadi Chennai, Tamil Nadu, India S. Parvathi Department of Mathematics, Veltech Multitech Dr.Rangarajan Dr.Sakuntala Engineering College Avadi, Chennai, Tamilnadu, India S. Srinivasaragavan Department of Mathematics, Veltech Dr. RR & Dr.SR University, Avadi Chennai, Tamil Nadu, India ABSTRACT A study on production and sales system is performed. Three products production inventory system where the production has the impact of seasonal demand and sales. We consider the following model in which the production and sales of k triplets of three different products are studied subject to seasonal requirements. The production of the three different products is done alternately. The production times of the products have general distribution and the season is exponential with SCBZ property. The demand season has with SCBZ property. It occurs with rate and changes to be an exponential time with parameter. We present Laplace Transform of production and sales time and their means are obtained. Numerical examples are presented with MATLAB ODE Tool. The model can be applied to civil engineering problem of construction involving inventories. AMS Subject Classification: Primary 30C45, Secondary 30C80 Key words: Inventory systems, three products, product season, SCBZ property. Cite this Article: D. Kanagajothi, S. Parvathi and S. Srinivasaragavan, Extension of Stochastic Analysis Model to Three Products Inventory System with SCBZ Seasonal Production and Sales. International Journal of Civil Engineering and Technology, 8(10), 2017, pp editor@iaeme.com

2 D. Kanagajothi, S. Parvathi and S. Srinivasaragavan 1. INTRODUCTION Single commodity inventory problem of (s, S) type was discussed by several researchers. We introduce a new concept called setting the Clock Back to Zero (SCBZ) which is used in our model. Stochastic periodic review models have been investigated by Brojeswar Pal et al and Tal Avinadav [2, 10]. Murthy and Ramanarayanan [6, 7] have considered several (s, S) inventory systems. General production and sales system with SCBZ machine Time and Manpower has been studied by Madhusoodhanan P et al [6]. General manpower and machine system with Markovian production and general sales has been analyzed by Harikumar.K [5].Parvathi S et al [8, 9] have examined general analysis of single and two products inventory system. Dinçer Konur et al and Haoran Zhang et al [3, 4] discussed different delivery Schedules among suppliers and three stage stochastic programming method. Joint inventory control and pricing in a service-inventory system treated by Ata Jalili Marand, HongyanLi, Anders Thorstenson [1]. In real life inventory models different types of triplets of products are produced for sales by the companies. For examples Television manufacturing companies produce normal, LCD and LED television. In soft drinks companies produce three different drinks Pepsi, seven up, Merinda. The Ice cream paler there is demand for chocolate, strawberry and vanilla flayer in production line. In this paper we study inventory systems and their related inventory subject to seasonal demands. The demand season starts in an exponential time with rate. If it does not start with in an exponential time with rate, the rate changes to. In section 1 the double Laplace Transform of production of k triplets with seasonal threshold and sales is presented. The expected production time and expected sale time are obtained. In section 2 Numerical examples are presented. 2. MODEL DESCRIPTION 2.1. Assumptions The following are the assumptions of the model The company produces three different products A, B and C and at a time only one is produced. The production of the products A, B and C are done in order. The production times of A, B and C are random variables X, Y and Z respectively. The production time X of product A has Cdf and pdf ; the production time Y of product B has Cdf and pdf and the production time Z of product C has Cdf and pdf. The production time X+Y+Z of one triplet has Cdf G (.) and pdf g (.) which is the Cdf and the pdf of the sum of the variables. The season for the products starts in a random time with exponential distribution whose parameteris. If the season does not start before the end of a random time with exponential distribution with parameter, then the parameter changes to. Sales time starts when k number of triplets of products are produced or when the demand season starts. The products are sold in triplets and the selling time of a triplets is random variable with cdf S (.) and the selling time of product A is (.) Analysis We note that the probability of n units produced in the interval (0, t) is for n 0 where (t) is n fold Cdf convolution of G (t) and is the Cdf of When the nth triplet is completed at time x < t, then during the period (x, t) there are editor@iaeme.com

3 Extension of Stochastic Analysis Model to Three Products Inventory System with SCBZ Seasonal Production and Sales possibilities of completion or incompletion of products A, B, and C as follows. During the period (x, t) a) The production time for product A is not over or b) The product A is produced but the product B is not produced or c) The products A and B are produced but the product C is not produced. Their respective probabilities are given below. Let (x) be n fold convolution of g(x) with itself where g(x) is the pdf of X + Y + Z and (x) is the pdf of. Then the probability of n triplet units are produced in (0, t) and the (n+1)th production of A is not completed before t is P [ =, Where (x) =1-G(x).The probability of n triplet units are produced and one production of A is over before t, but production for B is not over before t is P [ =. (2.1.1) Since the selling time starts when k triplets are produced or when the season starts, we note that the time to start sales T = min (time to produce k triplets, the time at which the season starts) The occurrence of season is exponential with SCBZ property. It occurs with rate α and α changes to β in an exponential time with parameter λ. Let the probability density function of season starting time be h(y). Considering the truncation point T 0 in which the parameter changes h(y) = { y T 0 y 0 Noting the truncation point T 0 itself is a random variable with pdf λ h(y) = + With distribution function H(y) = 1- + Where =, = The pdf of T is editor@iaeme.com

4 D. Kanagajothi, S. Parvathi and S. Srinivasaragavan { { The first term of the right side of is the part of the pdf that the time to produce k Number of pairs is t and the season has not started up to time t. The second, third And fourth terms are given by flower bracket. The second term is the part of the pdf that the season starts at time t, the time to produce i number of triplet units of Product A, B and C is x and the (i + 1)th product A is not over during t - x. The Third term is the part of the pdf that the season starts at time t, the time to produce i number of A,B and C triplet units is x, the (i+1)th A product is produced at x+u But (i+1)th B product is not produced during t-x-u for 0 i k-1.the fourth term is The part of the pdf that the season starts at time t the time to produce I number of A, B and C triplet units is x the (i+1)th product A is produced at x+u, (i + 1)th B product is produced at x + v but the product C is not produced during t - x - v for 0 i k - 1. This gives the joint pdf of time to start sales T and total sales time of products S as follows. + { { } The double Laplace transform of the pdf is given by = = + { editor@iaeme.com

5 Extension of Stochastic Analysis Model to Three Products Inventory System with SCBZ Seasonal Production and Sales { } = On simplification we get = + + [ ] [ + [ ][ ] editor@iaeme.com

6 The Laplace transform of T is D. Kanagajothi, S. Parvathi and S. Srinivasaragavan = + + [ ] [ + [ ][ ] On differentiation of equation at = -E(T) E(T) = - k - k + [ ] { } - [ ] { } - * + { + [ ] { } - [ ] { } - * + { Similarly = -E(S) E(S) = + - {[ ] { } +[ ] editor@iaeme.com

7 Extension of Stochastic Analysis Model to Three Products Inventory System with SCBZ Seasonal Production and Sales { } + ] { + + }} - {[ ] { } +[ ] { } + ] { + + }} In the above case k is fixed as a constant and sales for the system are considered when it completes the production of k pairs. As a most general case k is treated as a random variable taking positive integral values such that P(k=i)= =1 The special case in which X and Y are exponential random variables with parameters a and b respectively is considered Using E (T) and E(S) are found, when X and Y are exponentials with parameters a and b. E(T) = ] ) + ] ) editor@iaeme.com

8 Lambda Lambda D. Kanagajothi, S. Parvathi and S. Srinivasaragavan And E(S) = { }[ [E(S) - - ] + + ] + { }[ [E(S) - - ] + + ] 3. NUMERICAL EXAMPLE To illustrate the applications of the above result a=5, b=10, c=15, =0.8, =0.6 are given and E (T), E(S) are obtained in the following table, for E = 20 and E=10, E=30. Table 1 E (T) and E(S) values K E(T) E(S) E(T) 70 E(s) K K Figure 1 Observation As k increase from 1 to 9, E(T) increases and E(S) also increases. Taking a=5, b=10, c=15, =10 to 50, =0.6 and vary k=10 to 50, graphs of E (T) and E(S) are drawn editor@iaeme.com

9 Lambda Lambda Extension of Stochastic Analysis Model to Three Products Inventory System with SCBZ Seasonal Production and Sales Table 2 E (T) k =10 =20 =30 =40 = Table 3 E(S) k =10 =20 =30 =40 = E(T) 25 E(s) K K Figure 2 Observation As k increases and the parameter α is fixed, the expected values E(T) and E(S) increases. As k is fixed and the parameter α increases, the expected value E(T) and E(S) decreases. 4. CONCLUSIONS When k, the number of products increases, both the expected time to sales E(T) and sales time E(S) increases. When the exponential parameter α and β increase, both the expected time to sales E(T) and sales time E(S) decreases. As k increases and the parameter α is fixed, the expected values E(T) and E(S) increases. As k if fixed and the parameter α increases, the expected value E(T) and E(S) decreases. In this paper, we have extended the stochastic analysis model to three products inventory system with SCBS seasonal production and sales. The model presented in this paper can be applied to civil engineering problem of construction involving inventories editor@iaeme.com

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