Optimal Ordering Policies in the EOQ (Economic Order Quantity) Model with Time-Dependent Demand Rate under Permissible Delay in Payments

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1 Article International Journal of Modern Engineering Sciences, 015, 4(1):1-13 International Journal of Modern Engineering Sciences Journal homepage: wwwmodernscientificpresscom/journals/ijmesaspx ISSN: Florida, USA Optimal Ordering Policies in the EOQ (Economic Order Quantity) Model with ime-dependent Demand Rate under Permissible Delay in Payments HSShukla 1, Vivek Shukla and Sushil Kumar Yadav 1 1 Department of Mathematics and Statistics, DDU Gorakhpur University (UP) India Department of Mathematics, MMM University of echnology, Gorakhpur (UP) India Author to whom correspondence should be addressed; tripathi_rp031@rediffmailcom Article history: Received 14 November 014, Received in revised form 8 January 015, Accepted 10 January 015, Published 17 January 015 Abstract: his paper considers the problem of determining the annual total relevant cost for three different situations It is assumed that the demand rate is linearly time dependent In this paper the model has been framed to study the items whose deterioration rate is constant he objective is to minimize the total annual relevant cost: for each case Mathematical models are presented under three different situations Second order approximation has been used for finding closed form optimal solution Numerical examples are given to validate the proposed model Finally, sensitivity analysis with the variation of different parameters is also discussed Keywords: Inventory, linearly time-dependent demand, permissible delay, deterioration 1 Introduction In past few decades mathematical ideas have been used in different areas of daily life problems of controlling inventory In classical inventory models demand rate is assumed to be either timedependent or constant However, in real life demand rate is considered as time-dependent In this paper authors considered that demand of items is linearly time-dependent and deterioration is constant

2 Int J Modern Eng Sci 015, 4(1): 1-13 Deterioration is defined as decay, damage, spoilage, evaporation, obsolesce, loss of utility or loss of marginal value of commodities that result in decreasing usefulness At present, corporations have started taking control measures to prevent and reduce deterioration by using better storage facilities, technology etc Most of the items in nature deteriorated with time, certain products such as medicine, blood, green vegetables, radioactive chemicals, volatiles, etc decrease under deterioration during their normal storage period Various types of inventory models for deteriorating items were discussed by Agrawal and Jaggi [1], Roy Chowdhury and Chaudhory [], Shah [3], Chu et al [4], Chung et al [5], Liao et al [6], Padmanabhan and Vrat [7], Balkhi and Benkherout [8], Yang [9] ripathy and Pradhan [10] developed an inventory model for Weibull deteriorating items with constant demand when delay in payments is allowed to retailer to settle the account against the purchases made Shah [11] developed inventory model for deteriorating items and time value of money for a finite time horizon under the permissible delay in payments Singh et al [1] developed an inventory model for deteriorating items Huang [13] established an EOQ model for single period perishable item with volatile demand Khanra et al [14] presented an inventory model for deteriorating items with time-dependent demand under trade credits Jaggi et al [15] developed an EOQ (Economic Order Quantity) based inventory model for imperfect quality item to determine the optimal ordering policies of a retailer under permissible delay in payments with allowable shortages Yang et al [16] proposed a partial backlogging inventory lot-size model for deteriorating item with slock- dependent demand Sarkar and Moon [17] presented an EPQ model with inflation in an imperfect production system Various models have been proposed for constant demand rate with constant holding cost In reality demand for physical goods may be time-dependent, stock-dependent and price-dependent An inventory model of ameliorating items for price dependent demand rate was considered by Mondal et al [18] ripathi [19] developed an EOQ model for deteriorating items with linear time-dependent demand rate under permissible delay in payments You [0] developed an inventory model with price and time-dependent demand ripathi and omar [1] developed a model for optimal order policy for time-dependent deteriorating items in response to temporary price discount linked to order quantity ripathi [] presented an inventory model for time varying demand and constant demand with varying holding cost Huang [3] developed an inventory model with generalized type demand, deterioration and backorder rates Khanra et al [4] presented an EOQ model for deteriorating items with time-dependent quadratic demand under permissible delay in payment Sana [5] formulated optimal selling price and lot size with time varying deterioration and partial backlogging ripathi [6] developed an inventory model with shortage and exponential demand rate under permissible delay in payments ripathi [7] studied an inventory model for stock-level dependent demand rate with

3 Int J Modern Eng Sci 015, 4(1): delayed payments allowed to the retailer to settle the account against the purchases, under inflation and time discounting Singh et al [8] presented an inventory model for deteriorating items, general demand and time-dependent holding cost Bylka [9] presented continuous deterministic model to analysis the coordination and competition issue in a two-level supply chain, having one manufacturer and one retailer and determined schedules, which minimize the individual average total costs in the production distribution cycles obtained by individual decision choosing the number and sizes of deliveries he main objective of this paper is to determine the optimal ordering policies with timedependent demand rate under trade credits In this study we develop a model to determine the minimum annual total relevant cost in three different situations Numerical examples and sensitivity analysis is given to illustrate the model he remainder of this paper is organized as follows: notations and assumptions are given in the next section his is followed by mathematical formulation in section 3 In the next section 4, numerical examples are given he sensitivity analysis is mentioned in section 5 Finally, conclusion and future research direction is given in the last section 6 Notations and Assumptions he following notations are being used throughout the manuscript: A c h I e : ordering cost per order : unit purchasing cost per item : unit stock holding cost per item per year excluding interest charges : interest earned per dollar per year I k : interest charged per dollar in stocks per year by the supplier in years M N I(t) : the retailer s credit period offered by supplier in years : the customers trade credit period offered by retailer in years : the cycle time in years : the inventory level at time t VC1() : the annual total relevant cost for case 1 VC() : the annual total relevant cost for case VC3() : the annual total relevant cost for case 3 = 1 : optimal cycle time for case 1 = : optimal cycle time for case

4 Int J Modern Eng Sci 015, 4(1): = 3 : optimal cycle time for case 3 In addition, the following assumptions are being made throughout the manuscript: (1) he demand rate D = D ( t) = a bt, is linearly time-dependent () he deterioration rate θ is constant and 0 < θ < 1 (3) Shortages are not allowed (4) he lead time is zero and time period is infinite (5) M N, I k I e (6) For M, the account is settled at = M and the retailer begins paying for the interest charges in stock with rate I k For M, the retailer does not need to pay any interest charges (7) he retailer can accumulate revenue and earn interest during the period N to M with rate I e under the condition of trade credit 3 Mathematical Formulation At any time t, the inventory I ( t ) depleted due do demand and deterioration he rate of change of inventory level is governed by the following differential equation: di ( t) + θ I ( t) = R( t) = ( a bt), 0 t (1) dt subject to the boundary condition I ( 0) = Q and I ( ) = 0 he solution of (1) with the boundary condition I ( ) = 0 is given by 1 b b I ( t) = a + e 1 e t θ θ θ θ ( t) θ ( t) () 1 b θ be Q = I (0) = a ( e 1 ) θ + θ θ he total annual relevant cost consists of the following elements: 1) Annual ordering cost A = ) Annual stock holding cost θ (3) θ θ h h 1 b e 1 b e 1 = I ( t) dt a = + θ θ θ θ θ 0 (4) 3) hree cases arise due to assumption (6) occur in costs of interest charges for the items kept in stock per year Case 1 M Annual interest payable

5 Int J Modern Eng Sci 015, 4(1): cik = I ( t) dt M θ ( M ) θ ( M ) ci k 1 b e 1 b e M = a + ( M ) θ θ θ θ θ θ Case N M In this case annual interest payable= 0 Case 3 N Similar as case, annual interest payable= 0 4) Again three cases to occur in interest earned per year: Case 1 M he annual interest earned (5) M cie cie a b = ( a bt) tdt ( M N) ( M MN N ) ( M N) = (6) N Case N M he annual interest earned ci e = ( a bt) tdt +( M ) ( a bt)dt N 0 cie a b 3 3 b = ( N ) ( N ) + ( M ) a 3 Case 3 N (7) Annual interest earned cie ( M N) cie b = ( a bt) dt ( M N) a = (8) 0 he total relevant cost for the retailer can be expressed as VC() = ordering cost + stock holding cost+ interest payable interest earned VC1 if M VC ( ) = VC if N M VC3 if 0 < N VC = θ θ θ θ θ θ θ θ θ A h a b e 1 b e 1 ( ) -- - ( M ) ( M ) ci θ θ k a b e 1 b e M + ( M ) θ θ θ θ θ θ θ cie a b ( M N ) ( M N ) (9)

6 Int J Modern Eng Sci 015, 4(1): θ θ A h a b e 1 b e VC ( ) = + + θ θ θ θ θ θ θ ci e a b 3 3 b ( N ) ( N ) + ( M ) a 3 θ θ A h a b e 1 b e 1 b VC3 ( ) = + + ci e ( M N ) a θ θ θ θ θ (10) (11) Using runcated aylor s series expansion for the terms e and (11) becomes θ ( θ ) = 1 + θ + etc equations (9), (10) A h cik a cie a b VC = + a b + M b ( M N ) ( M N ) 3 (1) 3 A h ( a N b N b VC = + a b cie + ( M ) a 3 (13) A h b VC3 ( ) = + ( a b ) cie ( M N ) a (14) Since VC1 ( M ) VC ( M ) and VC ( N ) VC3 ( N ) = = is continuous and well defined All,, and VC1 VC VC3 VC are defined for 0 and (14) two times, we get dvc 1 A h k am e a b ( ) ( ) > Differentiating equation (1), (13) ci ci 3 3 = + a b + a b + Mb + ( M N ) ( M N ) d 3 (15) d VC1 ( ) A am cie a b = bh + ci 3 k b M N M N 3 3 d ( a bm ) dvc ( ) A h + b bn an 1 = + ( a b ) + ci e + d 3 3 d VC 3 ( ) A b 1 bn an = bh + ci 3 e 3 d 3 3 (16) (17) (18) dvc A h 3 ( a b ) ci M N e b = + + ( ) (19) d d VC3 ( ) A 3 bh d = (0) he optimal (minimum) solutions are obtained by equating equations (15), (17) and (19) equal to zero and solving for, we get

7 Int J Modern Eng Sci 015, 4(1): ( + ) ( + + ) 6b h ci 3 ah aci ci bm 3 k k k 3 3 { A acikm acie ( M N ) bcie ( M N )} = ( e ) ( e e ) ( e e ) b 3h + ci 3 ah + aci + bci M + 6A bci N + 3acI N = 0 () 3 bh ah bciem bcien A = (3) 4 Numerical Examples (1) Example 1 Case 1 Let A = $570/order, M = year, N = year, c = $100/ unit, I k = $015/$/year, I e = $01/$/year, h = $10/unit/year, a = 1000 and b= 0 Substituting these values in equation (1) and solving, we get optimal quantity Q ( 1 ) = units and total relevant cost 1 ( 1 ) 1 ie M = 1 = 0865year, optimal economic order VC = $ 31543, which verified case Example Case Let A= $60/order, M = year, N = year, c = $50/unit, I k = $015/$/year, I e = $ 01/$/year, h = $10/unit/year,a = 1000 and b = 0 Substituting these values in equation () and solving, we get optimal order quantity Q ( ) case ie N M = = 01005year, optimal economic = units and total relevant cost VC () = $608307, which verified Example 3 Case 3 Let A = $40/order, M = year, N = year, c = $50/unit, I k = $015/$/year, I e = $01/$/year, h = $50/unit/year, a = 1500 and b = 0 Substituting these values in equation (3) and solving, we get optimal order quantity ( 3 ) case3 ie N 5 Sensitivity Analysis Q = units and total relevant cost ( VC ) 3 3 = 3 = 00366year, optimal economic = $ , which verified We have performed sensitivity analysis by changing A, c, h and keeping the remaining parameters at their original values for all the three cases he corresponding variations in the optimal cycle time, economic order quantity and the annual total relevant cost are exhibited in able1 (able1a, 1b, 1c) for case 1, able (able a, b, c) for case, able3 ( able 3a, 3b, 3c) for case 3 respectively Case 1 M able 1

8 Int J Modern Eng Sci 015, 4(1): able1a: Variation of A keeping all the parameters same as in Example 1 A = 1 Q() = Q(1) VC1() = VC1(1) able 1b: Variation of c keeping all the parameters same as in Example 1 c = 1 Q() = Q(1) VC1() = VC1(1) able 1c: Variation of h keeping all the parameters same as in Example 1 h = 1 Q() = Q(1) VC1() = VC1(1) Case N M able able a: Variation of A keeping all the parameters same as in Example A = Q() = Q() VC() = VC() able b: Variation of c keeping all the parameters same as in Example c = Q() = Q() VC() = VC()

9 Int J Modern Eng Sci 015, 4(1): able c: Variation of h keeping all the parameters same as in Example h = Q() = Q() VC() = VC() Case 3 N able 3 able 3a: Variation of A keeping all the parameters same as in Example 3 A = 3 Q() = Q(3) VC3() = VC3(3) able 3b: Variation of c keeping all the parameters same as in Example 3 C = 3 Q() = Q(3) VC3() = VC3(3) able 3c: Variation of h keeping all the parameters same as in Example 3 h = 3 Q() = Q(3) VC3() = VC3(3) All the above observations from able 1, able and able 3 are summarized as follows: (a) From able 1a, it can be easily seen that increase of ordering cost A results in increase in optimal cycle time VC 1 1, 1 = economic order quantity ( 1 ) Q and optimal annual total relevant cost (b) From able1b, we see that increase of unit purchasing cost per item c results in decrease in optimal cycle time VC 1 1 = economic order quantity ( 1 ), 1 Q and optimal annual total relevant cost

10 Int J Modern Eng Sci 015, 4(1): (c) From able1c, we see that increase of holding cost per item h results in decrease of optimal cycle time VC, 1 = economic order quantity ( ) Q but increase in optimal annual total relevant cost (d) From ablea, we see that increase of ordering cost A results in increase in optimal cycle time = economic order quantity ( ) Q and optimal annual total relevant cost VC ( ), (e) From able b, we see that increase of unit purchasing cost per item c results in decrease in optimal cycle time VC, = economic order quantity ( ) Q and optimal annual total relevant cost (f) From able c, we see that increase of holding cost per item h results in decrease of optimal cycle time VC, = economic order quantity ( ) Q but increase in optimal annual total relevant cost (g) From able 3a, it can be easily seen that increase of ordering cost A results in increase in optimal cycle time, 3 VC 3 3 = economic order quantity ( 3 ) Q and optimal annual total relevant cost (h) From able3b, we see that increase of unit purchasing cost per item c results in decrease in optimal cycle time VC 3 3, 3 = economic order quantity ( 3 ) Q and optimal annual total relevant cost (i) From able3c, we see that increase of holding cost per item h results in decrease of optimal cycle time VC, 3 = economic order quantity ( ) Conclusion and Future Research Q but increase in optimal annual total relevant cost In this paper we have studied economic ordering policies with time-dependent demand rate under trade credits by considering three different cases he main aim of this paper is to minimize the annual total cost Demand rate is considered linearly time-dependent and decreasing function of time Second order approximation has been used for finding closed-form optimal solutions It has been observed from the sensitivity analysis that the total annual relevant cost increases with the increase of ordering cost and also with the increase of holding cost he total annual relevant cost decreases with increase of unit purchasing cost Mathematica 91 is used for numerical solution he model proposed in this paper can be extended in several ways For instance, we may extend the model for time-dependent deterioration or Weibull deterioration rate, for shortages and for probabilistic demand, etc

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