Inventory Model with Different Deterioration Rates with Stock and Price Dependent Demand under Time Varying Holding Cost

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1 International Refereed Journal of Engineering and Science (IRJES) ISSN (Online) 39-83X, (Print) 39-8 Volume 4, Issue (November 05), PP.0- Inventory Model with Different Deterioration Rates with Stock and Price Dependent Demand under ime Varying Holding Cost Raman Patel, S.R. Sheikh Department of Statistics, Veer Narmad South Gujarat University, Surat, INDIA Department of Statistics, V.B. Shah Institute of Management, Amroli, Surat, INDIA Abstract:- An inventory model for deteriorating items with stock and price dependent demand is developed. Holding cost is considered as function of time. Shortages are not allowed. Numerical examples are provided to illustrate the model and sensitivity analysis is also carried out for parameters. Keywords:- Inventory model, Deterioration, Price dependent demand, Stock dependent demand, ime varying holding cost I. INRODUCION Deteriorating items inventory model have been studies by many authors in past. Certain products such as medicine, volatile liquids, food stuff decrease under deterioration during their normal storage period. Ghare and Schrader [963] first developed an EOQ model with constant rate of deterioration. he model was extended by Covert and Philip [973] by considering variable rate of deterioration. Shah and Jaiswal [977] further extended the model by considering shortages. he related work are found in (Nahmias [98], Raffat [99], Goyal and Giri [00], Ouyang et al. [006], Wu et al. [00]). Aggarwal and Goel [984] discussed an inventory model with weibull rate of decay with selling price dependent demand. Patra et al. [00] developed a deterministic inventory model when deterioration rate was time proportional. Demand rate was taken as a nonlinear function of selling price, deterioration rate, inventory holding cost and ordering cost were all functions of time. ripathy and Mishra [00] dealt with development of an inventory model when the deterioration rate follows Weibull two parameter distribution, demand rate is a function of selling price and holding cost is time dependent. Patel and Parekh [04] developed an inventory model with stock dependent demand under shortages and variable selling price. Inventory models for non-instantaneous deteriorating items have been an object of study for a long time. Generally the products are such that there is no deterioration initially. After certain time deterioration starts and again after certain time the rate of deterioration increases with time. Here we have used such a concept and developed the deteriorating items inventory models. In this paper we have developed an inventory model with stock and price dependent demand with different deterioration rates for the cycle time under time varying holding cost. Shortages are not allowed. Numerical example is provided to illustrate the model. Sensitivity analysis of the optimal solutions for major parameters is also carried out. II. ASSUMPIONS AND NOAIONS NOAIONS: he following notations are used for the development of the model: D(t) : Demand rate is a linear function of price and inventory level (a bi(t) - ρp, a>0, 0<b<, ρ>0) A : Replenishment cost per order c : Purchasing cost per unit p : Selling price per unit : Length of inventory cycle I(t) : Inventory level at any instant of time t, 0 t Q : Order quantity θ : Deterioration rate during t, 0< θ < θt : Deterioration rate during, t, 0< θ < π : otal relevant profit per unit time. ASSUMPIONS: he following assumptions are considered for the development of two warehouse model. he demand of the product is declining as a function of price and inventory level. Replenishment rate is infinite and instantaneous. Page

2 Lead time is zero. Shortages are not allowed. III. HE MAHEMAICAL MODEL AND ANALYSIS Let I(t) be the inventory at time t (0 t ) as shown in figure. Q S 0 Figure he differential equations which describes the instantaneous states of I(t) over the period (0, ) is given by di(t) = - (a bi(t) - ρp), dt di(t) θi(t) = - (a bi(t) - ρp), dt 0t () t () di(t) θti(t) = - (a bi(t) - ρp), t (3) dt with initial conditions I(0) = Q, I( ) = S and I() = 0. Solutions of these equations are given by I(t) = Q( - bt) - (at bt -ρpt - ρbpt - abt ρbpt ), (4) a - t - ρp - t a θb - t - ρp θb - t I(t) = S θb - t (5) - a θb t - t ρp θb t - t - abθt - t ρpbθt - t a - t - ρp - t ab - t - ρpb - t aθ - t - ρpθ - t I(t) = - abt - t bρpt - t - abθt - t ρbpθt - t - aθt - t. (6) 6 6 ρpθt - t ρbpθt - t - abθt - t (by neglecting higher powers of θ) From equation (4), putting t =, we have S Q = a b - bρp - ab ρbp. (7) - b - b From equations (5) and (6), putting t =, we have a - ρp a θb - ρp θb I( ) = - a θb ρp θb - abθ bρθp (8) S θb Page

3 a - ρp ab - - ρbp aθ - - ρθp I( ) = - ab bρp - abθ ρbθp. (9) aθ ρθp ρbθp - abθ So from equations (8) and (9), we get S = θb a - ρp ab - ρbp aθ - ρθp ab bρp - abθ ρbθp - aθ 6 6 (0) ρθp ρbθp - abθ - a ρp. - a θb ρp θb a θb - ρp θb abθ - bρθp Putting value of S from equation (0) into equation (7), we have Q = - b θb a - ρp ab - ρbp aθ - ρθp - ab bρp - abθ ρbθp - aθ ρθp 6 6 ρbθp - abθ - a ρp - a θb ρp θb a θb - ρp θb abθ - bρθp a b bρp - ab. () - b Using () in (4), we have (-bt) I(t) = - b θb a - ρp ab - ρbp aθ - ρθp - ab bρp - abθ ρbθp - aθ ρθp 6 6 ρbθp - abθ - a ρp - a θb ρp θb a θb - ρp θb abθ - bρθp (-bt) a b bρp - ab - at bt - ρpt ρbpt abt () - b 3 Page

4 Based on the assumptions and descriptions of the model, the total annual relevant profit include the following elements: (i) Ordering cost (OC) = A (3) 0 (ii) HC = (xyt)i(t)dt = (xyt)i(t)dt (xyt)i(t)dt (xyt)i(t)dt = x a - ρp ab - ρbp aθ - ρθp y abθ - ρbp a (θb) a - ρp ab - ρbp aθ - ρθp ab bρp - abθ ρbθp - aθ 6 6 x ρθp ρbθp - abθ - a ρp - a θb ρp θb a θb - ρp θb abθ - bρθp (θb) a(θb) - ρp(θb) a θb - x a - ρp ab - ρbp aθ - ρθp 6 6 θb a(θb) - ρp(θb) a θb bρp - ab - abθ ρbθp - aθ ρθp x ρbθp - abθ - a ρp - a θb ρp θb a θb - ρp θb abθ - bρθp θb a(θb) - ρp(θb) 4 Page

5 x - ρbp ρθp ρbθp ab - aθ - abθ y -a ρp ρbθp - ab ρbp - abθ x aθ - ρθp y - ρbp ρθp ρbθp ab - aθ - abθ x abθ - ρbθp y aθ - ρθp x - a ρp ρbθp - ab ρbp - abθ y a - ρp ab - ρbp aθ - ρθ x abθ - ρbθp y a θb - ρp θb 4 4 x aθb - ρp θb -a θb a - ρp ab - ρbp aθ - ρθp ab bρp - abθ ρbθp - aθ y ρθp ρbθp - abθ - a ρp - a θb ρp θb a θb - ρp θb abθ - bρθp -θ - b - a θb ρp θb - abθ ρbθp ρp a θb a - ρp ab - ρbp aθ - ρθp ab bρp - abθ ρbθp - aθ x ρθp ρbθp - abθ - a ρp - a θb ρp θb a θb - ρp θb abθ - bρθp -θ -b - a θb ρp θb - abθ ρbθp ρp Page

6 a θb a - ρp ab - ρbp aθ - ρθp ab bρp - abθ ρbθp - aθ y ρθp ρbθp - abθ - a ρp - a θb ρp θb a θb - ρp θb abθ - bρθp θb a θb - ρp θb x - ρbp ρθp ρbθp ab - aθ - abθ x abθ - ρbθp y aθ - ρθp y -a ρp ρbθp - ab ρbp - abθ x aθ - ρθp y - ρbp ρθp ρbθp ab - aθ - abθ b θb a -ρp ab - ρbp aθ - ρθp -ab bρp x - abθ ρbθp - aθ ρθp ρbθp - abθ -a ρp - a θb ρp θb a θb -ρp θb abθ -bρθp a b ρbp - ab - b y- b- ρbp ab yabθ - ρbθp x abθ - ρbθp y a θb - ρpθb x a θb - ρp θb a θb a - ρp ab - ρbp aθ - ρθp - ab bρp abθ ρbθp - aθ ρθp ρbθp - abθ y a ρp - a θ b ρpθb a θb -ρpθb abθ - bρθp - θ -b - a θb ρpθb - abθ ρbθp ρp 3 6 Page

7 -a θb a - ρp ab - ρbp aθ - ρθp - ab bρp abθ ρbθp - aθ ρθp ρbθp - abθ x a ρp - a θb ρpθb a θb - ρpθb abθ - bρθp -θ -b - a θb ρpθb - abθ ρbθp ρp a θb a - ρp ab - ρbp aθ - ρθp - ab bρp abθ ρbθp - aθ ρθp ρbθp - abθ y a ρp - a θb ρpθb a θb - ρpθb abθ - bρθp a θb a θb - ρpθb x - b - ρbpab - -b θb a - ρp ab - ρbp aθ - ρθp - ab bρp y b - abθ ρbθp - aθ ρθp ρbθp - abθ - a 6 6 ρp - a θb ρp θb a θb -ρpθb abθ -bρθp b a b ρbp - ab - - a ρp -b - -b θb a - ρp ab - ρbp aθ - ρθp - ab bρp x b - abθ ρbθp - aθ ρθp ρbθp - abθ - a 6 6 ρp - a θb ρp θb a θb -ρpθb abθ -bρθp b a b ρbp - ab - - a ρp -b -b θb a - ρp ab - ρbp aθ - ρθp - ab bρp a - abθ ρbθp - aθ ρθp ρbθp y - abθ - a 6 6 ρp - a θb ρp θb a θb -ρp θb abθ -bρθp a b ρbp - ab -b 3 7 Page

8 5 5 - y abθ - ρbθp - xa - ρp ab - ρbp aθ - ρθ x -aρp ρbθp -ab ρbp- abθ y a-ρp ab - ρbp aθ - ρθ - yabθ - ρbθp (by neglecting higher powers of θ) (iii) DC = c θi(t)dt θti(t)dt a - ρp a θb - ρp θb a θb ρp θb - abθ ρpbθ θb ( - ) = cθ a - - ρp - ab - ρbp aθ - ρθp - ab (- ) bρp (- ) - abθ ρbθp - aθ - ρθp ρbθp - abθ - a(- ) ρp(- ) - a θb ρp θb a θb (- ) - ρp θb (- ) abθ - ρbθp θb (4) a a θb - ρp θb - abθ ρbθp 6 6 θb (- ) a - -ρp - ab - ρbp aθ - ρθp -ab ( - ) 6 6 -cθ bρp ( - ) - abθ ρbθp - aθ - ρθp ρbθp - abθ - a( - ) ρp( - )- a θb ρp θb a θb ( - ) - ρp θb (- ) abθ - ρbθp θb abθ- ρbθp aθ- ρθp - ρbp ρθp ρbθp ab- aθ- abθ cθ aρp ρbθp -abρbp- abθ a-ρp ab - ρbp aθ - ρθp abθ- ρbθp aθ- ρθp - ρbp ρθp ρbθp ab - aθ - abθ cθ aρp ρbθp -abρbp- abθ a- ρp ab - ρbp aθ - ρθp (5) 8 Page

9 (iv) Inventory Model with Different Deterioration Rates With Stock and Price Dependent Demand under SR = p (abi(t) - p)dt 0 = p (abi(t) - p)dt (abi(t) - p)dt (abi(t) - p)dt 0 -b θb (- ) a - -ρp - ab - - ρbp - aθ - - ρθp ab (- )bρp (- )- abθ - ρbθp - - aθ ρθp - ρbθp - - abθ - -a( - ) ρp( - ) = p a b - aθb - ρp θb - a θb (- )-ρp θb (- ) abθ - ρbθp a b ρbp - ab - b (-b ) a - b ρp - ρbp ab a - ρp a θb - ρpθb a θb ρpθb -abθ ρpbθ θb ( - ) a - - ρp- ab - ρbp aθ - ρθp p a b 3 3 -ab (- )bρp (- )- abθ - ρbθp - - aθ ρθp - ρbθp - - abθ - -a(- )ρp(- ) - aθb - ρpθb a θb (- )-ρpθb (- ) abθ - ρbθp θb -ρp 9 Page

10 a a θb - ρpθb - abθ ρbθp 6 6 θb ( - ) a - -ρp- ab - - ρbp - aθ - ρθp - -ab (- ) 6 6 -p a b 3 3 bρp (- ) - abθ ρbθp - aθ - ρθp - -ρp 6 6 ρbθp - - abθ - -a(- )ρp(- )- a θb - ρpθb - a θb (- ) - ρpθb (- ) abθ - ρbθp θb pa b a - ρp ab - ρbp aθ ρθp - abθ ρbθp - ρp a -ρp ab - ρbp aθ (6) ρθp - ab bρp - abθ p a b -ρp ρbθp - aθ ρθp ρbθp abθ he total profit during a cycle, π consisted of the following: π = SR - OC - HC - DC (7) Substituting values from equations (3) to (6) in equation (7), we get total profit per unit. Putting µ = v and µ =v in equation (7), we get profit in terms of and p. Differentiating equation (7) with respect to and p and equate it to zero, we have π(,p) π(,p) i.e. = 0, = 0 (8) p provided it satisfies the condition π(,p) π(,p) π(,p) π(,p) π(,p) < 0, < 0 and - >0. (9) p p p IV. NUMERICAL EXAMPLE Considering A= Rs.00, a = 500, b=0.05, c=rs. 5, ρ= 5, θ=0.05, x = Rs. 5, y=0.05, v = 0.30, v = 0.50, in appropriate units. he optimal values of * =0.556, p* = , Profit*= Rs and optimum order quantity Q* = he second order conditions given in equation (9) are also satisfied. he graphical representation of the concavity of the profit function is also given. and Profit p and Profit, p and Profit Graph Graph Graph 3 0 Page

11 V. SENSIIVIY ANALYSIS On the basis of the data given in example above we have studied the sensitivity analysis by changing the following parameters one at a time and keeping the rest fixed. able Sensitivity Analysis Parameter % p Profit Q a 0% % % % θ 0% % % % x 0% % % % A 0% % % % ρ 0% % % % From the table we observe that as parameter a increases/ decreases average total profit and optimum order quantity also increases/ decreases. Also, we observe that with increase and decrease in the value of θ, x and ρ, there is corresponding decrease/ increase in total profit and optimum order quantity. From the table we observe that as parameter A increases/ decreases average total profit decreases/ increases and optimum order quantity increases/ decreases. VI. CONCLUSION In this paper, we have developed an inventory model for deteriorating items with price and inventory dependent demand with different deterioration rates. Sensitivity with respect to parameters have been carried out. he results show that with the increase/ decrease in the parameter values there is corresponding increase/ decrease in the value of profit. REFERENCES []. Aggarwal, S.P. and Goel, V.P. (984): Order level inventory system with demand pattern for deteriorating items; Eco. Comp. Econ. Cybernet, Stud. Res., Vol. 3, pp []. Covert, R.P. and Philip, G.C. (973): An EOQ model for items with Weibull distribution deterioration; American Institute of Industrial Engineering ransactions, Vol. 5, pp [3]. Ghare, P.N. and Schrader, G.F. (963): A model for exponentially decaying inventories; J. Indus. Engg., Vol. 5, pp [4]. Goyal, S.K. and Giri, B. (00): Recent trends in modeling of deteriorating inventory; Euro. J. Oper. Res., Vol. 34, pp. -6. [5]. Nahmias, S. (98): Perishable inventory theory: a review; Operations Research, Vol. 30, pp [6]. Ouyang, L. Y., Wu, K.S. and Yang, C.. (006): A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments; Computers and Industrial Engineering, Vol. 5, pp [7]. Patel, R. and Parekh, R. (04): Deteriorating items inventory model with stock dependent demand under shortages and variable selling price, International J. Latest echnology in Engg. Mgt. Applied Sci., Vol. 3, No. 9, pp Page

12 [8]. Patra, S.K., Lenka,.K. and Ratha, P.C. (00): An order level EOQ model for deteriorating items in a single warehouse system with price depended demand in nonlinear (quadratic) form; International J. of Computational and Applied Mathematics, Vol. 5, pp [9]. Raafat, F. (99): Survey of literature on continuously deteriorating inventory model, Euro. J. of O.R. Soc., Vol. 4, pp [0]. Shah, Y.K. and Jaiswal, M.C. (977): An order level inventory model for a system with constant rate of deterioration; Opsearch; Vol. 4, pp []. ripathy, C.K. and Mishra, U. (00): An inventory model for Weibull deteriorating items with price dependent demand and time varying holding cost; Applied Mathematical Sciences, Vol. 4, pp []. Wu, K.S., Ouyang, L. Y. and Yang, C.. (006): An optimal replenishment policy for noninstantaneous deteriorating items with stock dependent demand and partial backlogging; International J. of Production Economics, Vol. 0, pp Page