A two storage inventory model with variable demand and time dependent deterioration rate and with partial backlogging

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1 Malaya Journal of Matematik, Vol. S, No., 35-40, 08 A two storage inventory model with variable demand and time dependent deterioration rate and with partial baklogging Rihi Singh *, Ashok Kumar and Dharmendra Yadav3 Abstrat In this paper an inventory model is developed for single deteriorating items, partially baklogged shortages under two storehouses (first is owned and seond is rented warehouse). The deterioration rate has been onsidered here to be time dependent. The demand rate of the item depends on () time; () selling prie; and (3) the frequeny of advertisement. The stoks of first warehouse are filled from seond warehouse by using ontinuous release pattern. Here different ases and sub-ases are assumed aording to the stok level at initial and reorder points, with respet to the relative size of storing apaities of storehouses. The proposed inventory model is formulated for eah sub-ase as non linear onstrained mixed integer optimization problem. Here GRG tehnique is used to solve the problem. Also a numerial example is presented to demonstrate the proposed model. Keywords Two-storehouse inventory; Variable deterioration rate; Variable demand rate; Partial baklogging. Researh Sholar, Meerut College Meerut, Uttar Pradesh, India. College, Meerut, Uttar Pradesh, India. 3 Vardhman College, Bijnore, Uttar Pradesh, India. *Corresponding author: singhrihi5@gmail.om; drashokkumar@meerutollege.org; 3 dharmendrayadav3580@gmail.om 08 MJM. Artile History: Reeived 4 Deember 07 ; Aepted January 08 Meerut Contents Introdution Notations and assumptions Mathematial formulation Numerial illustration Conlusions Referenes Introdution Hartely (97) firstly worked on inventory models assuming two storehouses and onstant demand rate. Deaying items are the items that happen to spoil, damage, lost of its marginal value, and so on through time. The deterioration of produts in inventory is an atual harateristi that is neessary to study its effet on inventory. Yang (004) and Hsieh et al., (008) analysed inventory models for deaying items assuming two storehouses. Xu et al., (07) analysed two storehouses inventory model with different poliies of dis- pathing assuming deaying items. Today, advertisement is an effetive way to introdue a produt in the market. As the number of advertisements inreases the demand of the produt in the market also inreases. Demand is also affeted by the selling prie of the produt. Bhunia and Maiti (997) first reported that the demand rate of a produt in the inventory model depends on number of advertisements, selling prie and time. Goswami and Chaudhuri (99) disussed the models with or without shortages for linearly time-dependent demand. Abad (000), Goyal and Gunasekaran (995), Luo (998), Mondal et al. (007) studied the effets on the demand aused by variations in prie and number of the advertisement on deteriorating items. Chung et al., (00) developed an inventory model with prie (linearly dereasing) dependent demand that also analysis the disount prie, the frequeny of deliveries and yle time. Sarkar and Sarkar (03) presented an inventory model whih depends on demand and variable (linear) deterioration rate. Bhunia et al., (05) developed a two storehouse inventory model taking demand as variable with partial baklogging for deaying items. Wu (0) disussed two inventory systems assuming demand rate as trapezoidal under

2 baklogging 3/40 baklogging and time dependent (linear) deterioration rate. In Setion the notations and assumptions of the model are shown. In setion 3 mathematial formulations of different ases with sub-ases are disussed. In Setion 4, a numerial example and optimal solution is presented. Setion 5 gives onluding remark and sope for the future researh. m (> ): The mark up rate. p: Selling prie per unit of item, whih is taken as p = mc p. N: Rate of reurrene of advertisement. A: Advertisement ost per advertisement. T : The duration of yle.. Notations and assumptions 3. Mathematial formulation The mathematial model rest on the following assumptions: Aording to the storage apaities of both store houses and highest stok level, two ases arise: Case : H < S H + H (L system) Case : S H (L system) Aording to the reorder point of stok level four sub-ases may arise: Sub-ase 3.: H < S H + H and R < 0 Sub-ase 3.: H < S H + H and R > 0 Sub-ase 3.3: S H and R < 0 Sub-ase 3.4: S H and R > 0 The Sub-ase 3. is explained as follows Sub-ase 3.: H < S H + H and R < 0 () The rate of replenishment is taken infinite. () Lead time is taken onstant. (3) The rate of demand rate is taken as D(N, p,t) = N (a bp + t), a, b,, > 0. (4) Extent of inventory sheduling is taken infinite. (5) The entire lot size is supplied in one bath. () Both the warehouses are of finite storing apaities. (7) The time gap in selling from first storehouse (OW) and oupying the spae by the units of seond storehouse (RW) is not signifiant. (8) Baklogging rate is taken as [ + (T t)], where >0. The mathematial model rest on the following notations: I(t): Inventory level during time t. H : Storing apaity of first storehouse (OW). H : Storing apaity of seond storehouse (RW). S: Highest stok level (initially) L : An inventory system having a single storing faility. L : An inventory system having two storing failities. C : Holding ost of unit in unit of time at first storehouse (OW). C0 (C0 > C ): Holding ost of unit in unit of time at seond storehouse (RW). C : Shortage ost of unit in unit of time. C3 : Purhase ost of unit of item. C4 : Transportation ost unit of item. θow (t): (= a + bt): Rate of deay in OW, where a is sale parameter and 0 < b <. θrw (t): (= a + bt): Rate of deay in RW, where a is sale parameter and 0 < b <. R: Reorder point. A : Replenishment ost/ordering ost per replenishment for L system. A : Replenishment ost/ordering ost per replenishment for L system. Figure Inventory level Ir (t) at seond storehouse (RW) is presented by the following equations dir (t) + (a + bt)ir (t) = D(t), 0 t t () s. t. Ir (t ) = 0, () Ir (t) = S H at t = 0 (3) Using equation (), the solution of equation () is as follows Ir (t) = N (t t ) t + t (a bp) + {(a bp)a + } 3

3 baklogging 37/40 t e (a t+b ) log + (T t ) (T t ) (4) Using equation (3), t S = H + N t (a bp) + {(a bp)a + } The Inventory arrying units in seond storehouse (RW) = R t I (t) r 0 (5) = In first storehouse (OW) the inventory level Io (t) during [0, T ] is presented as + (a + bt)io (t) = 0, 0 t < t + (a + bt)io (t) = D(t), t t t D(t) =, t < t T + (T t) t {4(a bp)(3 at ) + t (8 3at )}. 4 Io (t) = 0 at t = t (8) The Inventory arrying units in first storehouse () (OW ) = Z t 0 (7) Io (t) t )+ (a bp)(t t ) {(a bp)a + }(t t ) {3 a (t + t )} {3at t (8 3at ) t (4 at )}. (9) = H (t a (8) s. t. Io (t) = H at t = 0, (7) (9) (0) The total shortage ost and Z T Io (t) = R at t = T. = C () t [ I0 (t)] a bp + T + = C log + (T t ) { + (T t )} (T t)} d (t t ) R(T t ) (0) Using the ontinuity of Io (t) at t and t and above onditions the solutions of the equations are as follows t () Io (t) = H e (a +b ), 0 t < t Io (t) = N (T t ) (a bp) + {(a bp)a + } t t + t (3) e (a t+b ), t t t Io (t) = a bp + T + log + (T t) (T t) R, t < t T The total transportation ost = Ct Z t 0 () D(N, pt) = Ct N (a bp)t + t Now total profit (P = total sales revenue total ost of the system (4) P(N,t, T ) = p N (a bp)t + t + R and, we have H = N (t t ) (a bp) + {(a bp)a + } t + t (5) C3 (S + R ) (R + R ) t C H (t a ) + (a bp)(t t ) {3 a (t + t )} {(a bp)a + }(t t ) {3at t (8 3at ) t (4 at )}] N C0 t {4(a bp)(3 at ) + t (8 3at )} 4 N +C a bp + T + log + (T t ) { + (T t )} (T t )} d The above equation redues to At + Bt C = 0, where A = N {(a bp)a +}, B = N (a bp), C = W +N +(a bp)t + N t {(a bp)a + } gives the solution B + B + 4AB t = () A Using equations (3) and (4) R= a bp + T + 37

4 baklogging 38/40 (t t ) R(T t ) C4 N (a bp)t + t NA, (a bp)(t t) {3 a (t + T )} {(a bp)a + }(t t) {3at t(8 3a T ) 0 N t (4 a T )} C t {4(a bp)(3 at ) 4 + t (8 3at )} C4 N (a bp)t + t NA (8) + () where N is an integer variable and t, T are ontinuous variables. Hene the problem of Sub-ase 3. is to P, T s.t. 0 < t < T, H < S < H + H, and R < 0 Maximize Z(N,t, T ) = Hene the average profit per unit time of sub-ase 3. is given by Z(N,t, T ) = TP. (3) Sub-ase 3.: H < S H + H and R > 0 Sub-ase 3.3: S H and R < 0 Figure Figure 3 In this sub-ase, the inventory level Ir (t) at seond warehouse (RW) is presented as follows dir (t) + (a + bt)ir (t) = D(t), 0 t t s.t.ir (t ) = 0 and Ir (t) = S H at t = 0 The inventory level Io (t) at OW is presented by the equations + (a + bt)io (t) = D(t), 0 t t D(t) =, t < t T + (T t) s.t. Io (t ) = 0, Io (0) = S and Io (T ) = R. (4) (5) The inventory level Io (t) at first storehouse (OW) in the interval [0, T ] is presented as follows + (a + bt)io (t) = 0, 0 t < t + (a + bt)io (t) = D(t), t t T s.t. Io (t) = H at t = 0 and Io (t) = R at t = T. (9) (30) (3) Hene the net profit is given by P = p N (a bp)t + t + R C3 (S + R ) N R C t {4(a bp)(3 at ) + t (8 3at )} 4 N +C a bp + T + log + (T t ) { + (T t )} (T t ) d (t t ) R(T t ) NA (3) () (7) The Total profit (P) for this sub-ase is as follows P = pn (a bp)t + T C3 (S R) (R + R ) t C H t a + R(T t ) 38

5 baklogging 39/40 Hene the average profit per unit time of sub-ase 3.3 is given by Z(N,t, T ) = TP. onavity of the funtion is presented numerially. Figure4 represents that the profit funtion is onave for the above mentioned example. Sub-ase 3.4: S H and R > 0 The inventory level Io (t) at first storehouse (OW) is expressed by the following equations + (a + bt)io (t) = D(t), 0 t T s.t. Io (0) = S and Io (T ) = R. 5. Conlusions This paper introdues an inventory model with two storehouses of limited spae apaity under time dependent deterioration rate. Here demand rate of the produt hanges aording to () number of advertisement, () selling prie and (3) time. In the model shortages ours of whih only some units are baklogged. To make model more realisti the rate of deay is taken linear in both storehouses. Also the optimal net profit is alulated by solving proposed model for different system parameters. Finally a numerial example is taken to maximize the total profit of the system. By taking more assumptions, suh as inflation, learning effets, greening effets et the model an be better fitted to real world senarios. It an also be analysed further by introduing different preservation tehnologies in storehouses. (33) (34) Here the total profit is given by P = pn (a bp)t + T C3 (S R) R ST C ( a T ) t (a bp)(3 a T ) N t 3 {(a bp)a + }(4 3a T ) NA 4 (35) The average profit per unit time is given by Z(N,t, T ) = TP. Referenes 4. Numerial illustration [] C = Re..0/unit/unit time, C = Rs. /unit/unit time, C0 = Rs..5/unit/unit time, R = Rs. 50/order, R = Rs.50/order, a = 0., a = 0.08, b =.0, b =.0, C3 = Rs. 5/unit, C4 = Rs. 0.5/unit, H = 00; H = 00, a = 50, b = 3.5, = 5, = 0.3, A = Rs. 50/advertisement, =.5. We used Mathematia 5. to solve this example for determining the optimal values of N, t, t T, S, and R along with the average profit Z of the system. We obtained the following results: t =.908, t = 3.95, N = 5, T = , R = 3.705, S = 47.97, Z = [] [3] [4] [5] [] [7] T 0.3 t 0.4 [8] Figure 4. Represents the Average profit vs. t and T [9] The profit funtion Z(N,t, T ) obtained from equation (3) is highly nonlinear so it is very tedious to solve, thus the 39 R.V. Hartely, Operations Researh A Managerial Emphasis, Goodyear Publishing Company, California, CA, 97. A. Goswami and K.S. Chaudhuri, An eonomi order quantity model for items with two levels of storage for a linear trend in demand, Journal of the Operational Researh Soiety, 43(99), H. Yang, Two-warehouse inventory models for deteriorating items with shortage under inflation, European Journal of Operational Researh, 57(004), T. Hsieh, C. Dye and L.Y. Ouyang, Determining optimal lot size for a two warehouse system with deterioration and shortages using net present value, European Journal of Operational Researh, 9(00), 8 9. S.K. Goyal and A. Gunasekaran, An integrated prodution-inventory-marketing model for deteriorating items, Computers & Industrial Engineering, 8(995), X. Xu, Q. Bai and M. Chen, A omparison of different dispathing poliies in two-warehouse inventory systems for deteriorating items over a finite time horizon, Applied Mathematial Modelling, 4(07), A.K. Bhunia, A.A. Shaikh, G. Sharma and S. Pareek, A two storage inventory model for deteriorating items with variable demand and partial baklogging, Journal of Industrial and Prodution Engineering, 3(4)(05), 3 7. A.K. Bhunia and M. Maiti, An inventory model for deaying items with selling prie, frequeny of advertisement and linearly time-dependent demand with shortages, IAPQR Transations, (997), P.L. Abad, Optimal lot size for a perishable good under onditions of finite prodution and partial bak order-

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