A Study of Production Inventory Policy with Stock Dependent Demand Rate

Transcription

1 International Journal of Statistics and Systems ISSN Volume 12, Number 3 (2017), pp Research India Publications A Study of Production Inventory Policy with Stock Dependent Demand Rate Himanshu Pandey 1, Ashutosh Pandey 2 and Dileep Kumar 3 1,3 Department of Mathematics and Statistics, DDU Gorakhpur University, Gorakhpur, India. 2 Lovely Professional University, Punjab, India. Abstract In the present study we have been taken a more realistic demand rate that depends on the stock level available. The stock level in itself obviously gets depleted due to the customer s demand. This set up is very close to the reality that is observed in the market. This setup is very practical and can applied to many commodities in today s market. The model conforms to present day economic condition worldwide and is beneficial for obtaining optimal industrial output. Keyword: Realistic Demand, Stock, Inventory Policy INTRODUCTION Inventory handling is an important part of our manufacturing, retail and distribution infrastructure. Researchers were engaged to develop the inventory models considering the demand of the items to be constant, linearly increasing or decreasing, increasing or decreasing with time, stock-dependent etc. Later, it was realized that the said demand patterns do not precisely depict the demand of items such as newly introduced fashion items, garments, cosmetics, automobiles etc, for which the demand increases with time as they are introduced into the market and after few time, it becomes constant. In order to consider demand of such types, the concept of ramptype demand is introduced. Gupta and Vrat (1986) were amongst the first few researchers to deliberate the effect of stock dependent consumption rate on an EOQ model. In this study they established EOQ for two cases, one for an instantaneous replenishment and another for a finite

2 432 Himanshu Pandey, Ashutosh Pandey and Dileep Kumar rate of replenishment.mandaland Phaujdar (1989) wrote a note an inventory model with instantaneous stock replenishment and stock dependent consumption rate.roy and Choudhary (2006) studied a model with stock dependent demand and constant deterioration.singh (2010) studied an inventory model for deteriorating items with stock dependent demand and storages.singh and Singh (2012) developed an EOQ model with power form stock dependent demand.baker and Urban (1988) analyzed a continuous deterministic case of an inventory system in which the demand rate is a polynomial function of the inventory level.dattaand Pal (1990) developed an inventory model with stock dependent demand until the stock level reached a particular point, after which the demand became constant.balkhi and Benkherouf(2004) analyzed a deteriorating stock dependent model for a finite horizon. It is evident from above that the minimum total cost occurs at the point where the ordering costs and inventory carrying costs are equal. Assumptions and Notations The proposed inventory model is developed under the following assumptions and notations: Assumptions:Following assumptions are given below: Demand is taken to be stock level. Production rate is taken to be decision variable. Production cost is taken to be production dependent. Deterioration rate is constant. Shortages are not allowed. Lead time is zero.

3 A Study of Production Inventory Policy with Stock Dependent Demand Rate 433 Notations: Following notations are given below: Pr= Production rate D(t) = Demand rate, D(t) = a + bi(t), a > 0 and 0 < b < 1. Cm= Material cost CTH= Tool or Die Cs= Set-up cost per unit item Ch= Holding cost per unit item =Deterioration rate G= Energy and labor cost η0(p) = Unit production cost of an item and Mathematical Model η 0 (P) = C m + G P r + C TH P r In our study we have taken a more realistic demand rate that depends on the stock level available. The stock level in itself obviously gets depleted due to the customer s demand. This set up is very close to the reality that is observed in the market. This setup is very practical and can applied to many commodities in today s market. The model conforms to present day economic condition worldwide and is beneficial for obtaining optimal industrial output. We considered the manufacturing system in which demand in market is met by its produced items. Here timehorizon(0,t) is divided into n cycles. Production starts at t=0 and continuous up to t1 and inventory level reaches at highest level. Inventory level at t=t reaches up to zero due to combined effect of demand and deterioration.

4 434 Himanshu Pandey, Ashutosh Pandey and Dileep Kumar Governing Differential equation of the above inventory system is given as : t 1. (1) di(t) dt +. I(t) = (a + b. I(t) 0 t and di(t) dt +. I(t) = a t 1 t T. (2) Conditions are I(t 1 ) = Q, I(T) = 0. Now we use MATHEMATICA for finding the solution of above equation: For equation (1):- I(t) = aet( b )+t(b+) + e t( b ) = e t(b+) (b ae t(b+) + ) = e t( b ) aet( b )+t(b+) = [ aet( b )+t(b+) + e t( b ) ] I(t) = a + be t(b+) + e t(b+).. 0 t t 1. (3) For equation (2):- I(t) = a + be t(b+) + e t(b+) I(t) = a + e t (4) Using boundary condition:- I(t 1 ) = Q = [e t 1(b+) a ].. (5)

5 A Study of Production Inventory Policy with Stock Dependent Demand Rate 435 Set up cost: Set up cost of the system is given as: Setup Cost =Cs (6) Holding Cost: Inventory carried between time periods 0 to T. So, holding cost of the system is given as: t 1 HC = C h [ I(t) dt 0 T + I(t) dt] t 1 HC = C h [ e T + e t 1 + a( T + t 1 ) 1 + e t1(b+) + at 1 ] Production Cost: Production is being between time periods 0 to t 1 So, the production cost is given as: P. C. = (C m + G P r + C TH P r ) t 1 P r 0 dt.. (7) P. C. = (C m + G P r + C TH P r ) t 1. (8) Hence, the total average cost of the inventory system is TC = Set-up cost + Holding cost + Production cost TC = C s + C h ( e T Solution Procedure: + e t1 + (C m + G P r + C TH P r ) t 1 at + at et (b+) 1 at 1 ) The optimization technique is used to minimize TC to derivepr, t1 and T as follows:.(9) Step 1: Since the number of delivery per order k, is an integer value, start by choosing an integer value of k 1 Step 2: Take the partial derivatives of T.C. with respect to Pr, t1 and T and equate the results to zero. The necessary conditions for optimality are

6 436 Himanshu Pandey, Ashutosh Pandey and Dileep Kumar TC( Pr, t1, T) TC( Pr, t1, T) 0, 0, P t r 1 TC( Pr, t1, T) 0 T These simultaneous equation can be solved for P r, t 1 and T Where TC = C s + C h ( e T TC P r = 0 Gives + e t1 + (C m + G P r + C TH P r ) t 1 at + at et (b+) 1 at 1 ) P r = G C TH TC t 1 = 0 Gives TC T = 0 Gives a. b t 1 = log [ ()(e + e (b+)] Step 3: T = log (a ) Using Pr, t1 and T found at step 2, substitute into equation (9) and drive TC ( Pr, t 1, T) and Q Numerical Illustration The numerical solution is given below to illustrate the above solution procedure. On the basis of previous studies, let us considered the following data in proper units: C L =2500, C TH =0.001, a=0.06, b=4, C s =100, C h =8, = 0.04

7 A Study of Production Inventory Policy with Stock Dependent Demand Rate 437 Then we find * t 1 =0.43 * T =4.40 * P r = TC=12546.We use mathematical software MATHMATICA for solving the system numerically. CONCLUSION In this study, we have attempted to develop a production inventory model for deteriorating items with a very realistic and practical demand rate which depends upon the stock level available. The factor of stock level takes care of the fact that the demand is influenced by the level of stock displayed and changes with respect to it. The proposed model is very useful in the present market situation as almost every item can be identified as having a demand rate varying accordingstock available. The inventory is allowed to deteriorate during the time it is stored and during this time it undergoes constant rate of deterioration. An optimal solution of the system is obtained under the assumed conditions. REFERENCES [1] S. Bose, A. Goswami and K. S. Chaudhary, An EOQ model for deteriorating items with linear time- dependent demand rate andshortage under inflation and time discounting, Journal of operational research society, 46(7)(1995), [2] J. A. Buzacott, Economic order quantities with inflation, Operations Research Quarterly, 26(1975), [3] J.M. Chen,, an inventory model for deteriorating items with time-proportional demand and shortages under inflation and time discounting, International Journal of Production Economics,55, (1), (1998), [4] C.K.Jaggi, K.K. Aggrawal and S.K. Goel, Optimal order policy for deteriorating items with inflation induced demand, International Journal of Production Economics, 103(2)(2006),

8 438 Himanshu Pandey, Ashutosh Pandey and Dileep Kumar [5] C.K.Jaggi, K.K. Aggrawal and S.K. Goel, Optimal inventory replenishment policy for deteriorating items under inflationary conditions, International Journal of Production Economics, 13,(2007), [6] A. Mirzazadeh, M.M. Seyyed, Esfahani and Ghomi, S.M.T.F.(2009), An inventory model under uncertain inflationary conditions, finite production rate and inflation-dependent demand rate for deteriorating items with shortages, International Journal of Systems Science,40(1)(2009), [7] I. Moon., and S. Lee, the effects of inflation and time value of money on an economic order quantity model with a random product life cycle, Europian Journal of Operational Research, 125(3)(2000), [8] I. Moon, B.C. Giri, and B. Ko, Economic order quantity models for ameliorating/deteriorating items under inflation and time discounting, European Journal of Operational Research, 162(3)(2005) [9] J. Ray and K.S. Chaudhari, An EOQ model with stock dependent demand, shortage, inflation and time discounting, International Journal of Production Economics, 53(1997), [10] S.R. Singh and C. Diksha, Supply Chain Model in a multi Echelon System with inflation induced demand, International Transaction in Applied Science, 1(2009), [11] H.M. Wee and S.T. Law, Replenishment and précising policy for deteriorating items taking into account the time value of money, Int. J. Production Economics 71,(2001), [12] H. L. Yang, Two-warehouse inventory models for deteriorating items with shortages under inflation, European Journal of Operations Research, 157(2004),2, [13] H.L. Yang, T.T. Teng, M.S. Chern An inventory model under inflation for deteriorating items with stock-dependent consumption rate and partial backlogging shortages, Int. J. Production Economics. 123(2010),8-19. [14] Pandey, H.andPandey,A.(2013). An Inventory Model for Deteriorating Items with two level storage with uniform demand and shortage under Inflation and completely backlogged International Journal, Investigations in Mathematical Sciences. ISSN: Vol. 3(1), 2013, 47-57]. [15] Pandey, H.andPandey,A.(2014) An optimum inventory policy for exponentially deteriorating items, considering multi variate Consumption Rate with Partial Backlogging,Mathematical Journal of Interdisciplinary Sciences (MJIS) Print Version: ISSN Online Version: ISSN X Vol 2(2)

9 A Study of Production Inventory Policy with Stock Dependent Demand Rate 439 [16] A Study of Integrated Model with Variable Production and Demand Rate Under Inflation [Journal of Computer and Mathematical Sciences, ISSN (Print), ISSN (Online) Vol. 7(12), pp , December 2016, IF 1.92]

10 440 Himanshu Pandey, Ashutosh Pandey and Dileep Kumar