Production Inventory Model with Different Deterioration Rates Under Linear Demand

Transcription

1 IOSR Journal of Engineering (IOSRJEN) ISSN (e): 50-0, ISSN (p): Vol. 06, Issue 0 (February. 06), V PP Production Inventory Model with Different Deterioration Rates Under Linear Demand S.R. Sheikh, Raman Patel Department of Statistics, V.B. Shah Institute of Management, Amroli, Surat, INDIA Raman Patel, Department of Statistics,Veer Narmad South Gujarat University, Surat, INDIA Abstract:- A production inventory model with linear trend in demand is developed. Different deterioration rates are considered in a cycle. Shortages not allowed. o illustrate the model numerical example is provided and sensitivity analysis is also carried out for parameters. Key Words: Production, Inventory model, Varying Deterioration, Linear demand, ime varying holding cost I. INRODUCION he basic question for making a production inventory model is how much to produce to satisfy customer demand. So, the problem is to find the optimal time to maximize the total relevant profit. Moreover, deterioration of certain items occurs due to spoilage, obsolescence, evaporation, etc. Many authors in past studied inventory model for deteriorating items. Covert and Philip [] derived an EOQ model for items with weibull distribution deterioration. Mishra [7] developed the EPQ model for deteriorating items with varying and constant rate of deterioration. Aggarwal [] discussed an order level inventory model with constant rate of deterioration. Gupta and Vrat [4] developed an inventory model with stock dependent consumption rate. Mandal and Phaujdar [6] presented an inventory model for stock dependent consumption rate. Haiping and Wang [5] studied an economic policy model for deteriorating items with time proportional demand. Other research work related to deteriorating items can be found in, for instance (Raafat [9], Goyal and Giri [], Ruxian et al. []). Roy and Chaudhary [0] studied a production inventory model for deteriorating items when demand rate is inventory level dependent and production rate depends both on stock level and demand. ripathy and Mishra [] developed an order level inventory model with time dependent Weibull deterioration and ramp type demand rate where production and demand were time dependent. Patel and Patel [8] developed a deteriorating items production inventory model with demand dependent production rate. Initially there is no deterioration in most of the products. 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 a production inventory model 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 and sensitivity analysis of the optimal solutions for major parameters is also carried out. II. ASSUMPIONS AND NOAIONS Notations: he following notations are used for the development of the model: P(t) : Production rate is a function of demand ( P(t) = ηd(t), η>0) D(t) : Demand rate is a linear function of time t (a+bt, a>0, 0<b<) 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 μ t t, 0< θ< θt : Deterioration rate during, t t, 0< θ< π : otal relevant profit per unit time. Assumptions: he following assumptions are considered for the development of the model: he demand of the product is declining as a linear function of time. Rate of production is a function of demand 7 P a g e

2 Replenishment rate is infinite and instantaneous. Lead time is zero. Shortages are not allowed. Deteriorated units neither be repaired nor replaced during the cycle time. III. HE MAHEMAICAL MODEL AND ANALYSIS Let I(t) be the inventory at time t (0 t ) as shown in figure. Figure he differential equations which describes the instantaneous states of I(t) over the period (0, ) is given by = (η - )(a + b t), + θ I(t) = (η - )(a + b t), + θ ti(t) = - (a + b t), with initial conditions I(0) = 0, I(μ ) = S, I(t ) = Q and I() = 0. Solutions of these equations are given by I(t) = (η - )(at + b t ). 0 t μ () μ t t () t t () I(t) = S + θ μ - t -(η - ) a μ -t + b μ - t + a θ μ - t + b θ μ -t -a θ t μ -t - b θ t μ -t 4 4 I(t) = a - t + b - t + a θ - t + b θ - t - a θ t - t - b θ t - t Putting t = µ in equation (4), we get S = (η - )(aμ + b μ ) Putting t = t in equations (5) and (6), we have I(t ) = S + θ μ - t (4) (5) (by neglecting higher powers of θ) - (η - ) a μ - t + b μ - t + a θ μ - t + b θ μ - t - a θ t μ - t - b θ t μ - t 4 4 I(t ) = a - t + b - t + a θ - t + b θ - t - aθ t - t - b θ t - t So, from equations (8) and (9), we have t = S ( + θ μ ) - (η - )aμ - a S θ + η aμ From equation (0), we see that t is a function of μ, and S, so t is not a decision variable. Putting value of S from equation (7) in equation (5), we have (6) (7) (8) (9) (0) 7 P a g e

3 I(t) = (η - )(a μ + b μ ) + θ μ - t - (η - ) a μ - t + b μ - t + a θ μ - t + b θ μ - t - a θ t μ - t - b θ t μ - t Putting t = t, in equation (5), we get Q = (η - )(a μ + b μ ) + θ μ - t - ( η - ) a μ - t + b μ - t + a θ μ - t + b θ μ - t - a θ t μ - t - b θ t μ - t Based on the assumptions and descriptions of the model, the total annual relevant profit (π), include the following elements: (i) Ordering cost (OC) = A () (ii) H C = 0 (x + yt)i(t)d t μ t = (x + yt)i(t)d t + (x + yt)i(t)d t + (x + yt)i(t)d t 0 μ t = -x a + b + a θ + b θ t + y η b - b μ + x η a -a μ - x - b - aθ - b θ -ya t x - b - a θ - b θ -ya -x η a θ μ + η b θ μ - a θ μ - b θ μ μ + x b θ + yaθ x a θ + y - b - a θ - b θ + -x a + y a + b a θ + b θ - x b θ + yaθ t x - η b θ + b θ + y - b + η b + a θ - η a θ t + x η a -a + y η a θ μ + η b θ μ - a θ μ - b θ μ t x η b - b + y η a -a μ 5 - yb θ t - x - η b θ + b θ + y - b + η b + a θ - η a θ μ - x η a -a + y η a θ μ + η b θ μ - a θ μ - b θ μ μ x - b + η b + a θ - η a θ + y η a -a μ + y - η b θ + b θ t - -x a + y - b - a θ - b θ t x η a θ μ + η b θ μ - a θ μ - b θ μ t x - b + η b + a θ - η a θ + y η a -a t + x a + b a θ + b θ - y - η b θ + b θ μ + yb θ (iii) t D C = c θ I(t)d t + θ ti(t)d t μ t () () (4) (by neglecting higher powers of θ) η a μ t + η b μ t -a μ t - b μ t + η a θ μ μ t - t + η b θ μ μ t - t -a θ μ μ t - t 4 - bθ μ μ t - t -η a μ t - t - η b μ t - t - η a θ μ t - t - η b θ μ t - t 4 = c θ 4 + η a θ μ t - t + η b θ μ t - t + a μ t - t + b μ t - t + a θ μ t - t bθ μ t - t -a θ μ t - t - b θ μ t - t P a g e

4 -cθ η a μ + η b μ - a μ - b μ + η a θ μ + η b θ μ - a θ μ - b θ μ cθ b θ + a θ + - b - a θ - b θ - a a + b + a θ + bθ c θ b θ t + a θ t + - b - a θ - b θ t - a t a + b + a θ + b θ t (5) (iv) S R = p (a+ b t)d t = p a + b (6) 0 he total profit (π) during a cycle, consisted of the following: π = S R - O C - H C - D C Substituting values from equations () to (6) in equation (7), we get total profit per unit. Putting µ = v and value of t from equation (0) in equation (7), we get profit in terms of. he optimal value of = * (say), which maximizes profit π can be obtained by differentiating it with respect to and equate it to zero (7) i.e. d π = 0 d provided it satisfies the condition d π < 0. d (8) (9) IV. NUMERICAL EXAMPLE Considering A= Rs.00, a = 500, b=0.05, η=, c=rs. 5, p=rs. 40, θ=0.05, x = Rs. 5, y=0.05, v =0.0, in appropriate units. he optimal value *=0.740 and Profit*= Rs and optimum order quantity Q*= he second order condition given in equation (9) is also satisfied. he graphical representation of the concavity of the profit function is also given. and Profit Graph 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 % Profit Q +0% a +0% % % θ +0% P a g e

5 x A +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 and x, 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 a production inventory model for deteriorating items with linear 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. (978): A note on an order level inventory model for a system with constant rate of deterioration; Opsearch, Vol. 5, pp []. Covert, R.P. and Philip, G.C. (97): An EOQ model for items with Weibull distribution deterioration; AIIE ransactions, Vol. 5, pp. -6. []. Goyal, S.K. and Giri, B. (00): Recent trends in modeling of deteriorating inventory; Euro. J. Oper. Res., Vol. 4, pp. -6. [4]. Gupta, R. and Vrat, P. (986): Inventory model for stock dependent consumption rate, Opsearch, pp [5]. Haiping, U. and Wang, H. (990): An economic ordering policy model for deteriorating items with time proportional demand; Euro. J. Oper.Res., Vol. 46, pp. -7. [6]. Mandal, B.N. and Phujdar, S. (989): A note on inventory model with stock dependent consumption rate; Opsearch, Vol. 6, pp [7]. Mishra, R.B. (975): Optimum production lot size model for a system with deteriorating inventory; International J. Production Research, Vol., pp [8]. Patel, R. and Patel, S. (0): Deteriorating items production inventory model with demand dependent production rate and varying holding cost; Indian J. Mathematics Research, Vol., No., pp [9]. Raafat, F. (99): Survey of literature on continuous deteriorating inventory model, J. of O.R. Soc., Vol. 4, pp [0]. Roy,. and Chaudhary, K.S. (009): A production inventory model under stock dependent demand, Weibull distribution deterioration and shortages; Intl. rans. In Oper. Res., Vol. 6, pp []. Ruxian, L., Hongjie, L. and Mawhinney, J.R. (00): A review on deteriorating inventory study; J. Service Sci. and management; Vol., pp []. ripathy, C.K. and Mishra, U. (0): An EOQ model with time dependent Weibull deterioration and ramp type demand; Inturnational J. of Industrial Engineering and Computations, Vol., pp P a g e