Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 4(7), 87-92, July (2015) Res.J.Recent Sci. International Science Congress Association 87 EOQ Model with Time Dependent Holding Cost under Trade Credits Mohan R.*1 andPetricia Leema RoselineDept of Mathematics, College of Military Engineering, Pune, 411 031, MH, INDIA Department ofComputer Science, Christ College, Pune, 411 014, MH, INDIA Available online at: www.isca.in , www.isca.me Received 11th January 2014, revised 28th April 2014, accepted 3rd July 2014Abstract In this paper we have developed a deterministic inventory model with constant demand with variable holding cost under Trade Credit. Here we have considered shortages are allowed and the deterioration is assumed as function of time. Suitable numerical example and sensitivity analysis is also carried out at the end. Keywords: Constant demand, deterioration rate, shortages, time dependent deterioration. IntroductionResearchers studied deterministic inventory models under Quadratic, linear, exponentially increasing/decreasing demand rate for the products like electronic goods, food stuffs, fashionable clothes etc., and some inventory models studied under price dependent demand considering variable deterioration rate. In business buying inventory with full capital amount is not viable. In developing different types of inventory models, considering permissible under trade credit is also a worth attempt. Researchers developed trade credits in different ways. Here trade credit means that supplier offers the customer (Case 1) a permissible delay in payments, i.e., trade credits to attract customers to increase demand in (Case II) to motivate quick payment and reduce credit expenses. Though many EOQ models proposed under permissible delay in payments we have developed an inventory model with constant demand considering variable holding cost with shortages. In this model deterioration rate is function of time. Hariga developed an inventory model of deteriorating items for time-varying demand also considering shortages. Chakraborti and Choudhuri proposed an EOQ model for deteriorating products of linear trend in demand with shortages in all cycles. Giri and Chaudhuri developed an economic order quantity model for deteriorating items of time varying demand and costs considering shortages. Goyal and Giri studied complete survey of recent trend in deteriorating inventory models. Mondal et. al developed an inventory model of ameliorating products for price dependent demand rate. You studied the inventory system for the products with price and time dependent demands. Ajanta Roy proposed an inventory model for deteriorating items of time varying holding cost with and without shortages in the price dependent demand. Mishra and Singh studied an inventory model for deteriorating items with time dependent demand and partial backlogging. Mishra developed an inventory model with Weibull rate of deterioration and rate of demand is constant with variable holding cost considering shortages and salvage value. Mukesh et al, proposed a deterministic inventory model for deteriorating products with price dependent demand under trade credits10. Vikas Sharma and Rekha developed an inventory model for time dependant demand for deteriorating items with Weibull rate of deterioration and shortages11. Venkateswarlu and Mohan studied an EOQ model with 2parameter Weibull deterioration, time dependent quadratic demand and salvage value12. Venkateswarlu and Mohan proposed an EOQ model for time varying deterioration and price dependent quadratic demand with salvage value13. Mohan and Venkateswarlu developed an EOQ models with variable holding cost and salvage value14. Mohan and Venkateswarlu proposed an inventory model for, time Dependent quadratic demand with salvage considering deterioration rate is time dependent15. Recently, Mohan and Venkateswarlu developed an inventory model with Quadratic Demand, Variable Holding Cost with Salvage value16. In this paper, we developed a model with time-dependent deterioration when the demand rate is constant under trade credits. Shortages are allowed. Time horizon is infinite. The optimal total cost is obtained. The sensitivity analysis is done with numerical example. The robustness of this model by increasing /deceasing of parameters, which gives the objectives from the sensitivity analysis carried out at the end. Assumptions and notationsThe mathematical model is developed on the following assumptions and notations: The demand rate D(t) at time is assumed to be D(t) = R , Lead time is zero, Replenishment rate is infinite, ) = is the deterioration rate, 0 1. A, the ordering cost, h+t, h&#x-3.3;女0, &#x-3.3;女0, the holding cost per unit, C, the cost per unit, I(t) is the inventory level at time t., The order quantity in one cycle is q. C is the shortage cost per unit per order, NDU, The number of deteriorating units per order with one cycle time. Research Journal of Recent Sciences _____________________________________________________________ ISSN 2277-2502Vol. 4(7), 87-92, July (2015) Res.J.Recent Sci. International Science Congress Association 88 Mathematical Model The differential equation which governs describes the instantaneous inventory level at time is given by Figure-1 Time - Inventory graph dtdQ (1) dtdQ (2) With the initial condition ,0atUsing the initial condition, solution of equation (1) is given by = 40 +K 22teq- 1(1(40Using the condition = 0, we get the value of K. Substituting the value K 1 in the above equation, we get {} 1240 (3) Using )0(= Q, we obtain 40 (4) The solution of equation (2) using the boundary condition is Inventory Model with Shortages The total cost (TC) of the system consists of the following costs: Ordering cost (OC) = A Inventory holding cost per cycle (IHC) = 0 )(1)( t dt (5) 3369012hRIHC (6) The number of units that deteriorated during this cycle time is t wheredtNDU (7) 40NDUCost due to deterioration =C*D 40CRCD (8) Shortage cost (SC) = t dt SC (9) Since we are considering the inventory system under Trade credits the following two cases arises: Case (i): (Here the payment on or before when inventory reaches to zero) In case-i, the credit period ended on or before when the inventory level reaches to zero. After due date M, the time horizon for the interest payable is for the inventory which are not sold after the due date. Figure-2 Time inventory graph for case i Hence Research Journal of Recent Sciences _____________________________________________________________ ISSN 2277-2502Vol. 4(7), 87-92, July (2015) Res.J.Recent Sci. International Science Congress Association 89 CI t dtCIEarnIECI t dtCICharICTotal Average Cost ()TC) is given by TC ( ) EarnCharICSCOCCDIHC(10) 403369012CICIatCRatatathRTC(11) The necessary condition for minimizing the total cost is )))) ¶ =¶ ¶ TCandTC, i,e., () [] 3360667.0EarCICharCICRathRTCand () []()403360111.012 T EarnCICharCICRatatathRTC (12) Provided TCTCTC, where ( ) []() 33642EarnCIatTC (13) () [] ))40336400111.012EarnCICharCITC (14) () [] 33601116.0 T EarnCICharCITC(15) Research Journal of Recent Sciences _____________________________________________________________ ISSN 2277-2502Vol. 4(7), 87-92, July (2015) Res.J.Recent Sci. International Science Congress Association 90 Case (ii): � (In this case Interest payable per cycle is zero when) Figure-3 Time inventory graph for case ii Total Average Cost TC is given by TC ( ) EarnIESCOCCDIHC 403369012MtEarnCICRhRTCThe necessary condition for minimizing the total cost is 0)2),1((0 1 TCandTC, i.e, () [] 3360667.0CICRathRTCEarnand () []()40336400111.012 T MtEarnCICRhRTC(16) Provided TCTCTC, Numerical Example for case I: By putting proper units of parametric values for, A =80 a = 40, C = 4, M = 0.09 =0.1 Ie =0.09, =0.01, h = 0.9, = 0.1, C= 0.1, =0.1 Ic=.12, Using MATHCAD Software, the optimal values the inventory system are: t= 1.012, T = 6.839, TC = 23.307 Numerical Example for case II: A =80 a = 40, C=4, M = 2 =0.1 Ie =0.09, =0.01, h = 0.9, = 0.1, C= 0.1, =0.1 Ic=.12 Sensitivity Analysis We will study the effect of changes made in the values of the parameters , A, a, , C, , h and on the optimal cycle time, total cost and EOQ of these models. We have the following inferences can be made from table-1 and table–2 using case I and II Decrease in the parameters , A , a decreases the total optimal system cost. Similarly the increase in these parameters increases the total optimal cost. The effect of these parameters is more pronounced on the total cost of the system. The effect of the parameter on the total optimal cost is very significant. The total optimal cost of the system is very significant by considering all variables together decreasing or increasing. Conclusion We have developed inventory management model for deteriorating items when the demand rate is assumed to be constant. It is assumed that the deterioration rate is proportional to time. We have solved the model considering shortages .The objectives are discussed in the sensitivity analysis for the various parameters and the sensitivity analysis is calculated in percentile variation rather than actual values from the tables. This model can be extended further incorporating inflation, and using fuzzy concepts. Research Journal of Recent Sciences _____________________________________________________________ ISSN 2277-2502Vol. 4(7), 87-92, July (2015) Res.J.Recent Sci. International Science Congress Association 91 Table-1 Case 1: Parameter % change 1 T TC -25 0.6917 0.07311 -0.03862 -10 0.395257 0.029244 -0.02145 10 -0.29644 -0.02924 0.017162 25 -0.98814 -0.10235 0.051487 A -25 -12.253 -13.2768 -13.4423 -10 -4.64427 -5.07384 -5.15296 10 4.44664 4.825267 4.899816 25 10.57312 11.66837 11.85481 A -25 13.73518 15.30926 -13.3093 -10 4.841897 5.35166 -5.09718 10 -4.1502 -4.62056 4.844038 25 -9.58498 -10.4694 11.71751 -25 1.976285 0.190086 -0.11155 -10 0.790514 0.07311 -0.0472 10 -0.6917 -0.07311 0.042906 25 -1.77866 -0.17546 0.102973 C -25 0.889328 0.043866 -0.08581 -10 0.395257 0.014622 -0.03432 10 -0.29644 -0.01462 0.030034 25 -0.79051 -0.05849 0.081521 -25 -10.6719 13.46688 -11.7433 -10 -10.0791 13.53999 -11.769 10 -11.3636 13.40839 -11.7175 25 -12.253 13.32066 -11.6746 H -25 21.64032 1.535312 -1.94791 -10 7.806324 0.555637 -0.69507 10 -6.81818 -0.49715 0.600678 25 -15.6126 -1.15514 1.364397 All parameters -25 -23.5178 12.58956 -33.1445 -10 5.237154 5.395526 -14.6051 10 9.881423 -3.61164 13.79843 25 -10.3755 -10.5571 39.71339 References 1.Hariga M., An EOQ model for deteriorating items with shortages and time-varying demand, Jour. of Operational Res Socty, 46, 398-404 (1995) 2.Chakraborti T. and Chaudhuri K.S., An EOQ model for items with linear trend in demand and shortages in all cycles, Int. 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International Science Congress Association 92 for Deteriorating Items with price Dependent Demand and Time Varying Holding Cost under Trade Credits, International Journal of Soft Computing and Engineering(IJSCE), 2(1), (2012)11.Vikas Sharma and Rekha Rani Chaudhuri), An inventory Model for deteriorating items with Weibull Deterioration with Time Dependent Demand and shortages, Res Jour of Management Sci., 2(3), 28-30 (2013) 12.Venkateswarlu R and Mohan R., An Inventory Model with Weibull Deterioration, Time Dependent Quadratic Demand and Salvage Value, AIMS -10, Proceedings, Bangalore, (2013a) 13.Venkateswarlu R and Mohan R., An Inventory Model for Time Varying Deterioration and Price Dependent Quadratic Demand with salvage value, Ind. 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