REVIEW PERIOD REORDER POINT PROBABILISTIC INVENTORY SYSTEM FOR DETERIORATING ITEMS WITH THE MIXTURE OF BACKORDERS AND LOST SALES

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1 T. Vasudha WARRIER, PhD Nita H. SHAH, PhD Deartment of Mathematics, Gujarat University Ahmedabad , Gujarat, INDIA nita_sha_h@rediffmail.com REVIEW PERIOD REORDER POINT PROBABILISTIC INVENTORY SYSTEM FOR DETERIORATING ITEMS WITH THE MIXTURE OF BACKORDERS AND LOST SALES Abstract. This aer is develoed to describe a review eriod reorder oint inventory system in which units are subject to deterioration at a constant rate and shortages are allowed but only artially backlogged, i.e. rest goes as lost sales. Also the demand during rescribed scheduling eriod is a random variable following suitable robability distribution. The objective is to find the otimum reorder oint which gives the minimum cost during the review eriod. Key words : Review eriod, robabilistic inventory model, deterioration, artial backlogging, lost sales. JEL Classification: 90B05 1.Introduction Since the develoment of Wilson s lot size inventory model in 1915, a number of aers has been written analyzing mathematical models of inventory under different situations. In these models, it was assumed that once units enter in inventory, they last forever. Afterwards, researchers showed considerable interest in develoing mathematical models of inventory for describing otimal olicies for deteriorating items. Raafat (1991, Shah and Shah (000, Goyal and Giri (001 gave an u-to- date review of inventory models for erishable items. Most of the authors have considered simultaneous obsolescence, i.e., all units remaining in inventory at the end of the lanning horizon become useless. Hadley and Whitin (1961, 196 derived inventory models for deteriorating items under the assumtion of finite lanning horizon which was revisited by Murray and Silver (1966. Stochastic demand was incororated in the aforesaid models by Barankin and Denny (1960. Brown et al (196 established ordering olicies for inventory which is subject to obsolescence. They modeled obsolescence by assuming lanning horizon to be a random variable. Bulinskaya (196 develoed newsboy

2 T. Vasudha Warrier, Nita H. Shah roblem where delivery erishes immediately with robability and after one eriod with robability (1. Van Zyl (196 considered the model where the roduct life time is eactly two eriods with roortional costs of ordering and shortages. The demands in successive eriods were assumed to be indeendent identically distributed random variable. He gave dynamic rogramming formulation of the said model. Nahmias and Pierskalla (1973, 1975 develoed a otimal ordering olicy to balance the eected outdating and eected shortages arises in blood bank management. Various authors viz. Nahmias (1975a, 1975b, Fries (1975 etc. have considered inventory models for items with fied life time. Jani et al (1978 develoed a eriodic review inventory model for deteriorating items in an inventory as a random variable with some secified continuous robability distribution and the distribution of demand during the review eriod is stationary over time. The urose of this aer is to develo a eriodic review inventory model for deteriorating items with stochastic demand in which artial backordering and artial lost sales is allowed, with some secified continuous robability distribution for demand and assuming that the distribution of the demand during the review eriod is stationary over time.. Assumtions and Notations : The stated mathematical model is develoed with the following assumtions and notations : The inventory osition of the system is reviewed regularly at a eriod of w time units. Whenever the on hand inventory is found to be less than or equal to the reorder oint s, a rescribed lot size of q units is scheduled for relenishment. Lead time is zero. The random demand of units during the review eriod w follows uniform distribution with the robability density function f(, 0 < and average demand, µ = f( d. 0 Shortages are allowed. The α th fraction of it is backlogged and (1 - α th fraction is lost sales at any instant of time during the review eriod. Items in inventory are subject to deterioration during the eriod (0, w and there is no reair or relacement of deteriorated items during this eriod. The time taken for deterioration of an item follows a negative eonential distribution given by g(t, θ = θ e (- θ t, θ > 0, t 0 where E(t = θ -1 = average life of an item.

3 Review Period Reorder Point Probabilistic Inventory System for Deteriorating. The cumulative density function (c.d.f. of t is G(t, θ = 1 - e (- θ t, θ > 0, t 0 So that the age secific failure rate φ (t defined by Co (1963 is given by g(t, θ φ (t = = θ, a constant. 1 G(t, θ The inventory carrying cost, h (er unit er time unit, the cost of backordered shortages, π (er unit er time unit, the cost due to lost sales, τ (er unit and the unit urchase cost C (er unit are known and remain constant during the eriod under consideration. 3. Mathematical Model : Let S denote the inventory level at the beginning of each review eriod (after relenishment, if any, then S is a random variable following the.d.f. given by 1, s S s + q h(s = q (1 0, otherwise Two cases may arise : Case 1 : When shortages do not occur. The inventory level at the beginning of the review eriod is S; units is the demand during the review eriod w ; the items in the inventory are subject to deterioration during the eriod (0, w at the constant rate θ. We assume that S units can satisfy the total demand during the scheduling eriod including deteriorated units. Let Z(t, S denote the inventory level of the system at time t, 0 t w, when a demand of units occurs during the review eriod w and the initial inventory level is S units. Then the differential equation that describes the instantaneous state of the inventory level during the review eriod w is given by dz(t,s + θ Z(t,S = -, 0 t w. dt w Since, Z(0, S = S, the solution of the above first order linear differential equation is Z(t, S = e (- θ t {S - ( e (θ t - 1}, 0 t w. ( θw From (, the number of units that remain in inventory at the end of the review eriod w is given by

4 T. Vasudha Warrier, Nita H. Shah Z(w, S = e (- θ w {S - θw ( e (θw - 1}. Now Z(w, S 0 gives S 1 where S 1 is given by Sθw S 1 = e( θw 1 (3 The number of units that deteriorate during the review eriod w is given by D 1 ( S 1 = S - Z(w, S = (S + (1 - e (- θw - ( θw The average number of units in inventory during the review eriod w is given by w 1 I 11 ( S 1 = w Z(t, S dt 0 ( + Sθw = θ (1 - e (- θw - (5 w θ The costs due to backordering and lost sales do not arise as there are no shortages occurring during the review eriod in this case. Case : When shortages do occur. When t t 1, i.e. shortages do not occur, eressions for inventory in the system, number of units deteriorated during the review eriod w is same as those derived in case 1. We assume that S units, the on hand inventory at the beginning of the review eriod, satisfies the demand only u to t 1 (t t 1 time units. Among the shortages, occurring after t 1 time units till the end of the review eriod only α th fraction is backordered at any instant and (1 - α th fraction comes under lost sales. The differential equation governing the system is given by dz(t,s + θ Z(t,S = -, 0 t t 1. dt w and dz(t,s α = -, t1 t w dt w The solution of above equations is resectively given by

5 Review Period Reorder Point Probabilistic Inventory System for Deteriorating. Z(t, S = e (- θ t {S - ( e (θ t - 1}, 0 t t 1. (6 θw and Z(t, S = α w (t 1 - t, t 1 t w (7 Since, Z(t 1, S = 0, we get (from (6, S = ( e (θ t 1-1. θw which gives t 1 = 1 Sθw log 1+ θ Note that t 1 < w gives > S 1. The number of units that deteriorate during the review eriod w is given by Sθw D ( > S 1 = S - t 1 = S - log w 1+ θ (9 w The average number of units carried during the review eriod w is given by t 1 I 1 ( > S 1 = w 1 Z(t /, S dt 0 = S Sθw log 1+ (10 θ θ w Also, total number of backordered units carried during the review eriod w is given by w α I 1 ( > S 1 = (t t1 dt w w t 1 Sθw α 1 = w log( 1+ (11 w θ Finally, the total number of units lost during time w is (8 I ( > S 1 = (1 α (w t 1 w

6 T. Vasudha Warrier, Nita H. Shah Sθw 1 = ( 1 α w log( 1+ w θ Combining both the cases, the eected number of units that deteriorate during the review eriod w for a given S (using eqs. ( and (9 S 1 D(S = D 1 ( S 1 f( d + D ( > S 1 f( d (13 0 S 1, the average eected number of units in inventory during the review eriod w is (using eqs. (5 and (10 for a given S S 1 I 1 (S = I 1 1 ( S 1 f( d + I 1 ( > S 1 f( d (1 0 S 1,the average eected number of units backordered during the review eriod w is (using eq. (11 for a given S I 1 (S = I 1 ( > S 1 f( d (15 S 1 and the eected number of units lost during the review eriod w (using eq. (1 for a given S I (S = I ( > S 1 f( d (16 S 1 For a given S, the eected cost during the review eriod w is K 1 (S = Cq + C D(S + hi 1 (S + πi 1 (S + τ I (S Hence, the eected total cost during the review eriod w is s+ q K(s = K 1 (S h(s ds (17 s To obtain the otimum value of s, we have to solve d K(s / ds = 0 for s and verify that d K(s / ds > 0. In ractice, the resulting equation is very comlicated and involves several integrals which can not be elicitly evaluated. (1

7 Review Period Reorder Point Probabilistic Inventory System for Deteriorating. We can however, find the aroimate solution by considering the first order aroimation as follows : Let us assume that θ << w so that for > S 1, Sθw / <1. Hence we can take the ower series eansion of e (θ w and log (1 + Sθw / uto first degree aroimation in θ for > S 1. With this aroimation, we can take Sθw S 1 = e( θw 1 (1 - θw / ηs (18 where η = (1 - θw /. Also note that η r = (1 - rθw /. For comutation see Aendi 1. Using above relations and following Jani et al (1978, the eected total cost of the system during the review eriod w is given by s+ q 1 K(s = q K 1 (S ds (19 s. Observations : For a given and S considering t 1 as a function of θ, we have t 1 (0 = Sw / and t 1 (θ = 1 Sθw log 1+ < Sw / = t 1 (0. θ Thus, for θ > 0, (0, t 1 (θ (0, t 1 (0 which means that when θ > 0, the shortages will sufficiently more. When α = 1, i.e. when all units are backordered then this model reduces to the model given by Jani et. al. (1978. When α = 1 and θ = 0 i.e. when all units are backordered and there is no deterioration, the model develoed reduces to that of Naddor ( An Eamle : Let the.d.f. of the demand be given by 3, 0 1 f( = 0, otherwise The average demand of this distribution is µ = Substituting this in the cost function (eq. (19 and simlifying it, we get the following eression for cost function : K(s = As + Bs 3 + Ds + Es + F (0 where A, B, D, E, and F are constants. (see Aendi.

8 T. Vasudha Warrier, Nita H. Shah The otimum reorder oint s 0 which gives minimum cost is obtained by solving d K(s / ds = 0 for s and verifying that d K(s / ds > 0. Consider an inventory model which has the following arameters : Scheduling eriod, w = 1 month. Lot size, q = 1 unit (= 1000 items. Purchase cost, C = 50,000 $ / unit = 50 $ / item. Inventory holding cost, h = 9,000 $ / unit / month. Shortage cost for backorders, π = 5, 000 $ / unit / month. Shortage cost for lost sales, τ = 0,000 $ / unit. Table : Effect of variation in α and θ on Otimal reorder oint and otimal cost α θ s K(s From the above table it can be observed that increase in the deterioration of units decreases the value of reorder oint, s, resulting in more shortages and as a result increases the total cost during the review eriod w. Keeing deterioration rate constant, increase in the backordering rate increases the value of reorder oint, s, and decreases the total cost during w. 6. Conclusions : A review eriod reorder oint inventory model is develoed in which units are subject to deterioration at a constant rate and shortages are allowed but only artially backlogged, i.e. rest goes as lost sales. Also the demand during the

9 Review Period Reorder Point Probabilistic Inventory System for Deteriorating. rescribed scheduling eriod is a random variable following suitable robability distribution. The model can be alied to suer malls where the stuff like dairy roducts, vegetables, fruits, cosmetics etc is subject to deterioration. On the other hand out dated fashion goods goes as a lost sales. The model can be etended to time deendent deterioration of units, in cash discount flow aroach i.e. otimizing the resent value of all future cash out - flows, etc. Aendi 1 Sθw Sθw Sθw S 1 = e( θw 1 = = θ w θ w 1+θw θw...! + +! Sθw = θw θw ( ! θw = S (1 + 1 θw = S (1 - = η S (All eansions are done under assumtion that θ < < w.. A = Aendi 9Cθw hw hθw παw πθw τ( α θw τ( 1 α Cθw q hw q 3hθw q παw q C w + θ + + hθw B = 3παθw ( q 1 τ( 1 α q ( 1+ 3θw +

10 T. Vasudha Warrier, Nita H. Shah Cθw q q hw q h w q q ( 3+ 6 ( 3 3 θ ( παθw ( q 3q + 1 παw ( 3 q D = + + τ( 1 α q ( 1 3θw 3τ( 1 α θw + + Cθw q q q hw q q h w q q ( ( 6 θ ( παθw q ( 1 q + q παw q ( 6 q E = + + παw 8 3 τ( 1 α( q 3τ( 1 α θw q ( 1 q Cθw q ( 10q q hw q ( 10 q hθw q ( 3q παθw q ( 6q 15q + 10 παw ( 10q q 0q + 15 F = τ( 1 α θw q ( 5 3q τ( 1 α( 5 10q q REFERENCES [1]Barankin, E. and Denny, J. (1960, Eamination of an inventory model incororating robabilities of obsolescence, Logistic Review &. Miltetary Logistic., 1, 11 5; []Brown, G., Lu, J. and Wolfson, R. (196, Dynamic modeling of inventory subject to obsolescence, Management Science, Vol. 11, 51 63; [3]Bulinskaya, E. (196, Some results concerning otimal inventory olicies, Theoretical Probability Alications, Vol. 9, ; []Co, D. R., (1963, Renewal Theory, John Wiley and Sons, Inc. [5]Fries, B. (1975, Otimal ordering olicy for a erishable commodity with fied life - time, Oerational Research, Vol. 3, 6 61; [6]Goyal, S. K. and Giri, B. C. (001, Recent trends in modeling of deteriorating inventory, Euroean Journal of Oerations Research, Vol.13,1-16; [7]Hadley, G. and Whitin, T. M. (1961, An otimal final inventory model, Management Science, Vol. 7;

11 Review Period Reorder Point Probabilistic Inventory System for Deteriorating. [8]Hadley, G. and Whitin, T. M. (196, Generalization of the otimal final inventory model, Management Science, Vol. 8; [9]Jani, B. B., Jaiswal, M. C. and Shah, Y. K. (1978, (s, q - system inventory model for deteriorating items, International Journal of Production Research, Vol. 16 (1, 1 9; [10]Murray, G. R. Jr. and Silver, E. A. (1966, A bayesian analysis of the style goods inventory roblem, Management Science, Vol. 1; [11]Naddor, E. (1966, Inventory Systems, John Wiley Publications, NY; [1]Nahmias, S. (1975a, Otimal ordering olicies for erishable inventory - II, Oerations Research, Vol. 3, ; [13]Nahmias, S. (1975b, On ordering erishable inventory under erlang demand, Naval Research Logistics (Quarterly, Vol., 15 5; [1]Nahmias, S. and Pierskalla, W. P. (1973, Otimal ordering olicies for a roduct that erishes in two eriods subject to stochastic demands, Naval Research Logistics (Quarterly Vol., 07-9; [15]Nahmias, S. and Pierskalla, W. P. (1975, Otimal ordering olicies for erishable inventory - I, Proceedings of XX International meeting of the Institute of Management Science, Vol. II, 85 93; [16]Raafat, F. (1991, Survey of literature on continuously deteriorating inventory models, Journal of Oerational Research Society, Vol. (1, 7 37; [17]Shah, Nita. H. and Shah, Y. K. (000, Literature survey on inventory model for deteriorating items, Economic Annals (Yugoslavia Vol. XLIV, 1 37; [18]Nita H. Shah (008, One time only incentives and inventory olicies for deteriorating items; Journal of Economic Comutation and Economic Cybernetics Studies and Research, vol.. no. 3-, Bucharest; [19]Van Zyl, G. J. J. (196, Inventory control for erishable commodities, Doctoral dissertation, University of North Carolina.