Global Journal of Pure and Applied Mathematics. ISSN 0973-768 Volume 3, Number 6 (07), pp. 537-55 Research India Publications http://www.ripublication.com Inventory Model with Different Deterioration Rates for Imperfect Quality Items and Linear Demand B.T. Naik Department of Statistics V.S. Patel College of Arts and Science Bilimora, INDIA R.D. Patel Department of Statistics Veer Narmad South Gujarat University Surat, INDIA Abstract Many times it happens that units produced or ordered are not of 00% good quality. A deterministic inventory model with imperfect quality is developed when deterioration rate is different during a cycle. Here it is assumed that holding cost is time dependent. Demand is considered as linear function of time. Numerical example is taken to support the model. Keywords: Inventory model, Varying Deterioration, Linear demand, Time varying holding cost, defective items. INTRODUCTION: Deterioration of items is a general phenomenon for many inventory systems and therefore deterioration effect cannot be ignored in real life. Ghare and Schrader [3] considered inventory model with constant rate of deterioration. Covert and Philip [] extended the model by considering variable rate of deterioration. The related work are found in (Nahmias [5], Raffat [6], Ruxian, et al [7]).Many times it happens that units produced or ordered are not of 00% good quality.
538 B.T. Naik and R.D. Patel Cheng [] developed a model of imperfect production quantity by establishing relationship between demand dependent unit production cost and imperfect production process. Salman and Jaber [8] developed an inventory model in which items received are of defective quality and after 00% screening, imperfect items are withdrawn from the inventory and sold at a discounted price. Jaggi et al. [4] developed an inventory model for deteriorating items with imperfect quality under permissible delay in payment. Patel and Patel [0] developed an EOQ model for deteriorating items with imperfect quantity items. Patel and Sheikh [9] developed an inventory model with different deterioration rates and time varying holding cost. Generally the products are such that initially there is no deterioration. Deterioration starts after certain time 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 for imperfect quality items with different deterioration rates for the cycle time. Holding cost is taken as function of time. Shortages are not allowed. To illustrate the model, numerical example is provided. Sensitivity analysis for major parameters is also carried out.. ASSUMPTIONS AND NOTATIONS: NOTATIONS: The following notations are used for the development of the model: D(t) : Demand rate is a linear function of time t (a+bt, a>0, 0<b<) c : Purchasing cost per unit p : Selling price per unit d : defective items (%) -d : good items (%) λ : Screening rate SR : Sales revenue A : Replenishment cost per order z : Screening cost per unit pd : Price of defective items per unit h(t) : Variable Holding cost (x + yt) t : Screening time T : Length of inventory cycle I(t) : Inventory level at any instant of time t, 0 t T Q : Order quantity θ : Deterioration rate during t, 0< θ< θt : Deterioration rate during, t T, 0< θ< π : Total relevant profit per unit time.
Inventory Model with Different Deterioration Rates for Imperfect Quality Items 539 ASSUMPTIONS: The following assumptions are considered for the development of the model. The demand of the product is declining as a linear function of time. Replenishment rate is infinite and instantaneous. Lead time is zero. Shortages are not allowed. The screening process and demand proceeds simultaneously but screening rate (λ) is greater than the demand rate i.e. λ > (a+bt). The defective items are independent of deterioration. Deteriorated units can neither be repaired nor replaced during the cycle time. A single product is considered. Holding cost is time dependent. The screening rate (λ) is sufficiently large. In general, this assumption should be acceptable since the automatic screening machine usually takes only little time to inspect all items produced or purchased. 3. THE MATHEMATICAL MODEL AND ANALYSIS: In the following situation, Q items are received at the beginning of the period. Each lot having a d % defective items. The nature of the inventory level is shown in the given figure, where screening process is done for all the received items at the rate of λ units per unit time which is greater than demand rate for the time period 0 to t. During the screening process the demand occurs parallel to the screening process and is fulfilled from the goods which are found to be of perfect quality by screening process. The defective items are sold immediately after the screening process at time t as a single batch at a discounted price. After the screening process at time t the inventory level will be I(t) and at time T, inventory level will become zero due to demand and partially due to deterioration. Q Also here t= () λ (x+yt) and defective percentage (d) is restricted to d - () λ Let I(t) be the inventory at time t (0 t T) as shown in figure.
540 B.T. Naik and R.D. Patel Figure The differential equations which describes the instantaneous states of I(t) over the period (0, T) is given by di(t) = - (a + bt), dt di(t) + θi(t) = - (a + bt), dt 0t (3) t (4) di(t) + θti(t) = - (a + bt), dt t T (5) with initial conditions I(0) = Q, I() = S and I(T) = 0. Solutions of these equations are given by I(t) = Q - (at + bt ), a - t + b - t + aθ - t I(t) = + S + θ - t 3 3 + bθ - t - aθt - t - bθt - t 3 3 3 a T - t + bt - t + aθt - t 6 I(t) =. (8) 4 4 + bθt - t - aθt T - t - bθt T - t 8 4 (by neglecting higher powers of θ) After screening process, the number of defective items at time t is dq. So effective inventory level during t t T is given by I(t) = - (at + bt ) + Q(-d). (9) (6) (7)
Inventory Model with Different Deterioration Rates for Imperfect Quality Items 54 From equation (6), putting t =, we have Q = S + a + b. (0) From equations (7) and (8), putting t =, we have a + b + aθ I( ) = + S + θ 3 3 + bθ - aθ - bθ 3 () 3 3 a T + bt + aθ T 6 I( ) =. () 4 4 + bθt - aθ T - bθ T 8 4 So from equations () and (), we get S = 3 3 a T + bt + aθ T 6 + bθ T - aθ T - bθ T 8 4 + θ - a - b - aθ - bθ + aθ + bθ 3 4 4 3 3. (3) Putting value of S from equation (3) into equation (0), we have 3 3 a T + bt + aθ T 6 4 4 + bθt - - aθ T - - bθ T - 8 4 Q = + a + b. +θ- - a - b - aθ 3 3 - bθ - +aθ - + bθ - 3 (4)
54 B.T. Naik and R.D. Patel Using (4) in (6), we have 3 3 a T + bt + aθt 6 + bθ T - aθ T - bθ T 8 4 I(t) = + θ - a - b - aθ - bθ + aθ + bθ 3 + a - t b - t. 4 4 3 3 Similarly, using (4) in (9), we have 3 3 a T + bt + aθ T 6 + bθ T - - aθ T - - bθ T - (-d) 8 4 I(t ) = +θ - - a - b - aθ - bθ - +aθ - + bθ - 3 + (-d) a + b - (at + bt ) 4 4 3 3 Similarly putting value of S from equation (3) in equation (7), we have (5) (6) 3 3 a T + bt + aθ T 6 + θ - t + bθ T - aθ T - bθ T 8 4 I(t) = + θ - a - b - aθ 3 3 - bθ + aθ + bθ 3 a - t + b - t + aθ - t + 3 3 + bθ - t - aθt - t - bθt - t 3 4 4 (7)
Inventory Model with Different Deterioration Rates for Imperfect Quality Items 543 Based on the assumptions and descriptions of the model, the total annual relevant profit (), include the following elements: (i) Ordering cost (OC) = A (8) (ii) Purchasing cost (PC) = cq (9) (iii) Screening cost (SrC) = zq (0) (iv) T HC = (x+yt)i(t)dt 0 t T = (x+yt)i(t)dt + (x+yt)i(t)dt + (x+yt)i(t)dt + (x+yt)i(t)dt 0 t 3 3 a T + bt + aθ T 6 + bθ T - aθ T - bθ T 8 4 + θ - bθ + aθ + bθ 3 +a + b 4 4 = x - a - b - aθ 3 3 t 3 3 a T + bt + aθ T 6 + bθ T - aθ T - bθ T 8 4 -d + θ - x - a - b - aθ - bθ + aθ + bθ 3 + -da + b 4 4 3 3 t
544 B.T. Naik and R.D. Patel 6 8 3 4 + x at + bt + aθt + bθt T 3 3 a T + bt + aθ T 6 + bθ T - aθ T - bθ T 8 4 +θ 4 4 + θ +x - a - b - aθ 3 3 - bθ + aθ + bθ 3 3 +a + b + aθ + bθ 3 - xbθ + y- b + aθ 4 6 4 - + θ 3 3 a T + bt + aθt 6 + bθ T - aθ T - bθ T 8 4 - x - b + aθ +y θ 3 - a - b - aθ 3 3 - bθ + aθ + bθ 3 -a-aθ- bθ 4 4 3
Inventory Model with Different Deterioration Rates for Imperfect Quality Items 545 5 6 5 - ybθ + ybθt + ybθ 30 48 30 3 3 a T + bt + aθt 6 4 4 + bθt - aθ T - bθ T 8 4 - θ + θ x - a - b - aθ 3 3 - bθ + aθ + bθ 3 -a-aθ- bθ + 3 3 a T + bt + aθt 6 4 4 + bθt - aθ T - bθ T 8 4 +θ + θ +y - a - b - aθ 3 3 - bθ + aθ + bθ 3 3 +a + b + aθ + bθ 3 + xbθ + yaθ T + xaθ + y- b - aθt - bθt T 5 8 3 4 3 4 5 4 + x- b - aθt - bθt - ya T + -xa + yat + bt + aθt + bθt T 3 4 6 8 3 3 4 3 3 a T + bt + aθt 6 4 4 + bθt - aθ T - bθ T 8 4 -d + θ + -xa+y - a - b - aθ 3 3 - bθ + aθ + bθ 3 + -da + b
546 B.T. Naik and R.D. Patel 3 3 a T + bt + aθt 6 4 4 + bθt - aθ T - bθ T 8 4 - θ + θ x - a - b - aθ 3 3 - bθ + aθ + bθ 3 -a-aθ- bθ - 3 3 a T + bt + aθt 6 + bθ T - aθ T - bθ T 8 4 +θ + θ +y - a - b - aθ 3 3 - bθ + aθ + bθ 3 3 +a + b + aθ + bθ 3 4 4 3 3 a T + bt + aθt 6 + bθ T - aθ T - bθ T 8 4 -d + θ - -xa+y - a - b - aθ 3 3 - bθ + aθ + bθ 3 + -da + b 4 4 t
Inventory Model with Different Deterioration Rates for Imperfect Quality Items 547 - xat+ bt + aθt + bθt - ybθ 6 8 48 3 4 6 3 3 a T + bt + aθt 6 + bθ T - aθ T - bθ T 8 4 + θ 3 3 - bθ + aθ + bθ 3 +a b 4 4 + -xa + y - a - b - aθ + xbθ + y- b + aθ 4 6 4 - + θ 3 3 a T + bt + aθt 6 + bθ T - aθ T - bθ T 8 4 + x - b + aθ +y θ 3 - a - b - aθ 3 3 - bθ + aθ + bθ 3 -a-aθ- bθ 4 4 3 3 a T + bt + aθt 6 + bθ T - aθ T - bθ T 8 4 -d + θ + x - a - b - aθ - bθ + aθ + bθ 3 + -da + b 4 4 3 3 t 3
548 B.T. Naik and R.D. Patel - yb 8 4 3 3 a T + bt + aθ T 6 + bθ T - aθ T - bθ T 8 4 +θ 4 4 + θ -x - a - b - aθ - bθ + aθ + bθ 3 3 +a + b + aθ + bθ 3 3 3 - xbθ + yaθ - xaθ + y- b - aθt - bθt 5 8 3 4 3 4 5 4 3 3 4 - x- b - aθt - bθt - ya - -xa + yat + bt + aθt + bθt () 3 4 6 8 3 + - xb -yb 3 (by neglecting higher powers of θ) T (v) DC = c θi(t)dt + θti(t)dt + θ(- ) 3 3 4 4 a T- + bt + aθ T + bθ T - aθ T- 6 8 3 3 - bθ T - a(- ) - b - aθ - bθ 4 3 = cθ +aθ (- ) + bθ +θ 3 3 3 4 +a + b + aθ + bθ 3 3 3 4 3 4 -aθ - bθ 3 4
Inventory Model with Different Deterioration Rates for Imperfect Quality Items 549 3 3 4 4 a T- + bt + aθ T + bθt 6 8 - aθ T- - bθ T - a(- ) - b 4 + θ( - ) - cθ - aθ - bθ +aθ ( - ) + bθ 3 + θ 3 3 4 + a + b + aθ + bθ 3 6 8 3 3 bθt + aθt + - b - aθt- bθt T 48 5 4 4 +cθ 3 3 4 - at at + bt + aθt + bθt T 3 6 8 6 5 4 bθ + aθ + - b - aθt- bθt 48 5 4 4 - cθ 3 3 4 - a at + bt + aθt + bθt 3 6 8 6 5 4 T (vi) SR = p (a+bt)dt + pddq = pat + bt + pddq 0 (3) The total profit (π) during a cycle consisted of the following: π = SR - OC - PC - SrC - HC - DC (4) T Substituting values from equations (8) to (3) in equation (4), we get total profit per unit. Putting µ= vt and µ= vt and value of t and Q in equation (4), we get profit in terms of T. Differentiating equation (4) with respect to T and equate it to zero, we have () i.e. dπ = 0 dt provided it satisfies the condition d π < 0. dt (5) (6) 4. NUMERICAL EXAMPLE: Considering A= Rs.00, a = 500, b=0.05, c=rs. 5, p= Rs. 40, pd = 5, d= 0.0, z = 0.40, λ= 0000, θ=0.05, x = Rs. 5, y=0.05, v=0.30, v = 0.50, in appropriate units. The optimal value of T* =0.74, Profit*= Rs. 90.7443 and optimum order quantity Q* = 36.4673.
550 B.T. Naik and R.D. Patel The second order conditions given in equation (6) are also satisfied. The graphical representation of the concavity of the profit function is also given. T and Profit Graph 5. SENSITIVITY 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. Table Sensitivity Analysis Parameter % T Profit Q +0% 0.489 34.459 49.6038 a +0% 0.598 80.957 43.553-0% 0.869 763.0045 9.3743-0% 0.3043 5307.3889.99 +0% 0.505 956.036 5.473 x +0% 0.608 987.73 30.6433-0% 0.857 955.375 43.463-0% 0.30 99.6680 50.88 +0% 0.70 95.7560 35.450 θ +0% 0.73 98.447 35.943-0% 0.735 93.550 36.998-0% 0.746 95.7770 37.578 +0% 0.980 950.695 49.347 A +0% 0.855 984.8953 43.0459-0% 0.586 958.4097 9.5389-0% 0.440 998.06.09 +0% 0.74 90.8583 36.4673 λ +0% 0.74 90.8064 36.4673-0% 0.73 90.6683 36.47-0% 0.73 90.5734 36.47
Inventory Model with Different Deterioration Rates for Imperfect Quality Items 55 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. 6. CONCLUSION: In this paper, we have developed an inventory model for deteriorating items with linear demand with different deterioration rates. Sensitivity with respect to parameters have been carried out. The results show that with the increase/ decrease in the parameter values there is corresponding increase/ decrease in the value of profit. REFERENCES [] Cheng, T.C.E. (99): An economic quantity model with demand dependent unit production cost and imperfect production process; IIE Transactions, Vol. 3, pp. 3-8. [] Covert, R.P. and Philip, G.C. (973): An EOQ model for items with Weibull distribution deterioration; American Institute of Industrial Engineering Transactions, Vol. 5, pp. 33-38. [3] Ghare, P.N. and Schrader, G.F. (963): A model for exponentially decaying inventories; J. Indus. Engg., Vol. 5, pp. 38-43. [4] Jaggi, C.K., Goyal, S.K. and Mittal, M. (0) : Economic order quantity model for deteriorating items with imperfect quality and permissible delay in payment; Int. J. Indus. Engg. Computations, Vol., pp. 37-48. [5] Nahmias, S. (98): Perishable inventory theory: a review; Operations Research, Vol. 30, pp. 680-708. [6] Raafat, F. (99): Survey of literature on continuous deteriorating inventory model, J. of O.R. Soc., Vol. 4, pp. 7-37. [7] Ruxian, L., Hongjie, L. and Mawhinney, J.R. (00): A review on deteriorating inventory study; J. Service Sci. and management; Vol. 3, pp. 7-9. [8] Salameh, M.K. and Jaber, M.Y. (000): Economic production quantity model for items with imperfect quality; J. Production Eco., Vol. 64, pp. 59-64. [9] Patel, R. and Sheikh, S.R. (05): Inventory Model with Different Deterioration Rates under Linear Demand and Time Varying Holding Cost; International J. Mathematics and Statistics Invention, Vol. 3, No. 6, pp. 36-4.
55 B.T. Naik and R.D. Patel [0] Patel, S. S. and Patel, R.D. (0) : EOQ Model for Weibull Deteriorating Items with Imperfect Quality and Time Varying Holding Cost under Permissible Delay in Payments; Global J. Mathematical Science: Theory and Practical, Vol. 4, No. 3, pp. 9-30.