# Wasting the Percentage of On-hand Inventory in an Instantaneous Economic Order Quantity Model Due to Deterioration

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1 Jounal of mathematics and compute Science 7 (2013) Wasting the Pecentage of On-hand Inventoy in an Instantaneous Economic Ode Quantity Model Due to Deteioation Aticle histoy: Received Mach 2013 Accepted Api 2013 Available online May 2013 M. Pattnaik Dept. of Business Administation, Utkal Univesity, Bhubaneswa, India Abstact In this pape the taditional instantaneous economic ode quantity model is extended to allow the pecentage of on-hand inventoy is wasted due to deteioation. That is, the wasted/ no wasted pecentage of on-hand inventoies in instantaneous EOQ models due to deteioation ae consideed simultaneously. Fo any given numbe of eplenishment cycles the existence of a unique optimal eplenishment schedule is poved and futhe the concavity of the total pofit function of the inventoy system in the numbe of eplenishments is established. A new mathematical model is developed and consideed as an enhancement to the EOQ model so that the net pofit is maximized and the numeical analysis show that an appopiate policy can benefit the etaile and that policy is impotant, especially fo wasting of deteioating items. Finally, sensitivity analysis of the optimal solution with espect to the majo paamete, pecentage of onhand inventoy which is lost due to deteioation is also caied out. Keywods: EOQ, Deteioation, Units lost,, Pofit 1. Intoduction Inventoy management plays a cucial ole in businesses since it can help companies each the goal of ensuing pompt delivey, avoiding shotages, helping sales at competitive pices and so foth. The mathematical modeling of eal-wold inventoy poblems necessitates the simplification of assumptions to make the mathematics flexible. Howeve, excessive simplification of assumptions esults in mathematical models that do not epesent the inventoy situation to be analyzed. Many models have been poposed to deal with a vaiety of inventoy poblems. The classical analysis of inventoy contol consides thee costs fo holding inventoies. These costs ae the pocuement cost, caying cost and shotage cost. The classical analysis builds a model of an inventoy system and calculates the EOQ which minimize these thee costs so that thei sum is satisfying minimization citeion. One of the unealistic assumptions is that items stocked peseve thei physical chaacteistics duing thei stay in inventoy. Items in stock ae subject to many possible isks, e.g. damage, spoilage, dyness; vapoization etc., those esults decease of usefulness of the oiginal one and a cost is incued to account fo such isks. 154

2 The EOQ inventoy contol model was intoduced in the ealiest decades of this centuy and is still widely accepted by many industies today. Compehensive eviews of inventoy models can be found in [9], [14] and [17]. In pevious deteministic inventoy models, many ae developed unde the assumption that demand is eithe constant o stock dependent fo deteioated items. [7] developed a stochastic dynamic pogamming model pesented fo detemining the optimal odeing policy fo a peishable o potentially obsolete poduct so as to satisfy known time-vaying demand ove a specified planning hoizon. They assumed a andom lifetime peishability, whee, at the end of each discete peiod, the total emaining inventoy eithe becomes wothless o emains usable fo at least the next peiod. [8] exploed the inventoy model fo time dependent holding cost and deteioation with salvage value whee shotages ae allowed. [3] examined the simultaneous selection poduct duability and ode quantity fo items that deteioate ove time. Thei choice of poduct duability is modeled as the values of a single design paamete that effects the distibution of the time-to-onset of deteioation (TOD) and analyzed two scenaios; the fist consides TOD as a constant and the stoe manage may choose an appopiate value, while the second assumes that TOD is a andom vaiable. [2] consides the effect of diffeent maketing policies, e.g. the pice pe unit poduct and the advetisement fequency on the demand of a peishable item. [1] consideed an economic ode quantity inventoy model fo deteioating goods developed with a linea, positive tend in demand allowing inventoy shotages and backlogging. [1] and [5] consideed the effects of inflation and the time-value of money with the assumption of two inflation ates athe than one, i.e. the intenal (company) inflation ate and the extenal (geneal economy) inflation ate. [4] agued that the analysis of [1] contained mathematical eos fo which he poposed the coect theoy fo the poblem supplied with numeical examples. [11] explained a single item EOQ model with demand dependent unit cost and vaiable setup cost. [10] pesented an EOQ inventoy model fo peishable items with a stock dependent selling ate. They assumed that the selling ate is a function of the cuent inventoy level and the ate of deteioation is taken to be constant. Unlike the wok of [18] who studied the case of patial backlogging. [10] consideed the cases of no backlogging, and of patial and full backlogging. The most ecent wok found in the liteatue is that of [6] who extended his ealie wok by assuming a time-vaying demand ove a finite planning hoizon. [11] who assumes instant deteioation of peishable items with constant demand whee discounts ae allowed. [12] pesented an entopic ode quantity (EnOQ) model unde instant deteioation fo peishable items with constant demand whee discounts ae allowed. [15] studied an EOQ inventoy model in which it assumes that the pecentage of on-hand inventoy wasted due to deteioation is a key featue of the inventoy conditions which goven the item stocked. [16] studied pofit maximization economic ode quantity model fo deteioated items with time and pice dependent linea and deceasing demand fo finite planning hoizon unde pomotion facto fo acquiing moe pofit. In this pape, eplenishment decision unde wasting the pecentage of on-hand inventoy due to deteioation ae adjusted abitaily upwad o downwad fo pofit maximization model in esponse to the change in maket demand within the finite planning hoizon. The objective of this pape is to detemine optimal eplenishment quantities in an instantaneous eplenishment pofit maximization model. All mentioned above inventoy liteatues with deteioation o no wasting the pecentage of onhand inventoy due to deteioation have the basic assumption that the etaile owns a stoage oom with optimal ode quantity. In ecent yeas, companies have stated to ecognize that a tadeoff exists between poduct vaieties in tems of quality of the poduct fo unning in the maket smoothly. In the absence of a pope quantitative model to measue the effect of poduct quality of the poduct, these companies have mainly elied on qualitative judgment. This pape postulates that measuing the behavio of poduction systems may be achievable by incopoating the idea of etaile in making optimum decision on eplenishment with wasting the pecentage of on-hand inventoy due to deteioation and then compaes the optimal esults with no wasting the pecentage of on-hand inventoy due to deteioation taditional model. The majo assumptions used in the above eseach aticles ae summaized in Table1. 155

3 Table - 1 Summay of the Related Reseaches Autho(s) and published Yea Haiga (1994) Tsao et al. (2008) Pattnaik (2009) Pattnaik (2011) Pesent Pape (2012) Stuctue of the model (EnOQ) Demand Time Time and Pice (Deteministic) (Deteministic) (Deteministic) Demand pattens Nonstationay Linea and Deceasing Deteioation Planning Units Lost due Deteioation Model Yes Finite No Cost Yes Finite No Pofit Yes (Instant) Yes (Instant) Yes (Wasting) Finite No Pofit Finite No Pofit Finite Yes Pofit The emainde of the pape is oganized as follows. In section 2 assumptions and notations ae povided fo the development of the model. The mathematical fomulation is developed in section 3. The solution pocedue is given in section 4. In section 5, numeical example is pesented to illustate the development of the model. The sensitivity analysis is caied out in section 6 to obseve the changes in the optimal solution. Finally section 7 deals with the summay and the concluding emaks. 2. Notations and Assumptions Consumption ate t c Cycle length h Holding cost of one unit fo one unit of time. HC (q) Holding cost pe cycle K Setup cost pe cycle c Puchasing cost pe unit P s Selling Pice pe unit α Pecentage of on-hand inventoy that is lost due to deteioation q Ode quantity q** Modified economic odeing / poduction quantity (EOQ/EPQ) q* Taditional economic odeing quantity (t) On-hand inventoy level at time t (q) Net pofit pe unit of poducing q units pe cycle in cisp stategy π (q) Aveage pofit pe unit of poducing q units pe cycle in cisp stategy 3. Mathematical Model Denote (t) as the on-hand inventoy level at time t. Duing a change in time fom point t to t+dt, whee t + dt> t, the on-hand inventoy dops fom (t) to (t+dt). Then (t+dt) is given as: (t+dt) = (t) dt α (t) dt (1) Equation (1) can be e-witten as: (t+dt) (t) dt = α (t) (2) and dt 0, equation (2) educes to: d (t) + α (t) + = 0 (3) dt Equation (3) is a diffeential equation, solution is (t) = + (q + ) α α e αt (4) Whee q is the ode quantity which is instantaneously eplenished at the beginning of each cycle of length tc units of time. The stock is eplenished by q units each time these units ae totally depleted as a esult of outside demand and deteioation. The cycle length, tc, is 156

4 detemined by fist substituting tc into equation (4) and then setting it equal to zeo to get: t c = 1 + ln ( q ) (5) α Equation (4) and (5) ae used to develop the mathematical model. It is wothy to mention that as α appoaches to zeo, t c appoaches to q. Then the total numbe of units lost pe cycle, L, is given as: L = [ q 1 ln (αq+)] (6) α The total cost pe cycle, TC(q), is the sum of the pocuement cost pe cycle, K+cq, and the holding cost pe cycle, HC(q). HC (q) is obtained fom equation (4) as : HC(q) = tc 0 =h [ q α 1 ln( q + ) 0 hφ(t)dt =h [ + (q + ) α α e αt ] dt 2 ln (αq+ )] (7) TC(q) = K + cq + h [ q α2 ln (αq+)] (8) The total cost pe unit of time, TCU (q), is given by dividing equation (8) by equation (5) to give: TCU(q) = [K + cq + h [ q ln (αq+)] ] [ 1 + ln ( q )] 1 α α2 α = K +(c +h)q ln(1+ q ) h α As α appoaches zeo, then equation (9) educes to TCU (q) = K Whose solution is given by the taditional EOQ fomula, q = 2K 1(q). 1(q) = (q-l) P s TC (q) q + c + hq 2 h (9). The total pofit pe cycle is = (q-l) P s K- cq h [ q α α2 ln (αq+)] (10) Whee L, the numbe of units lost pe cycle due to deteioation, and TC (q) the total cost pe cycle, ae calculated fom equations (6) and (8), espectively. The aveage pofit (q) pe unit time is obtained by dividing tc in 1(q). Hence the pofit maximization poblem is Maximize 1 (q) q > 0 (11) 4. Solution Pocedue (Optimization) The optimal odeing quantity q pe cycle can be detemined by diffeentiating equation (10) with espect to q, then setting these to zeo. In ode to show the uniqueness of the solution in, it is sufficient to show that the net pofit function thoughout the cycle is concave in tems of odeing quantity q. The second ode deivates of equation (10) with espect to q ae stictly negative. Conside the following popositions. Poposition 1.The net pofit (q) pe cycle is concave in q. Conditions fo optimal q The second ode deivative of the net pofit pe cycle with espect to q can be expessed as (12) 157

5 (13) Since > 0 and equation (13) is negative. Popositions 1 shows that the second ode deivative of equation (11) with espect to q ae stictly negative. The objective is to detemine the optimal values of q to maximize the unit pofit function of equation (11). It is vey difficult to deive the optimal values of q, hence unit pofit function. Thee ae seveal methods to cope with constaints optimization poblem numeically. But hee LINGO 13.0 softwae is used to deive the optimal values of the decision vaiables. 5. Numeical Example Conside an inventoy situation whee K is Rs. 200 pe ode, h is Rs. 5 pe unit pe unit of time, is 1200 units pe unit of time, c is Rs. 100 pe unit, the selling pice pe unit P s is Rs. 125 and is 5%,. The optimal solution that maximizes equation (10) and and ae detemined by using LINGO 13.0 vesion softwae and the esults ae tabulated in Table 2. Table-2 Optimal Values of the Poposed Model Model Iteation q % Change 6. Sensitivity Analysis It is inteesting to investigate the influence of on etaile behaviou. The computational esults shown in Table 3 indicates the following manageial phenomena: when the pecentage of on-hand inventoy that is lost due to deteioation inceases, the eplenishment cycle length, the optimal eplenishment quantity, optimal net pofit pe unit pe cycle and optimal aveage pofit pe unit pe cycle decease espectively. The optimal total numbe of units lost pe cycle inceases with inceases in the pecentage value of the majo paamete α. Table-3 Sensitivity Analysis of α Iteation % Change in Conclusions In this pape, a modified EOQ model is intoduced which investigates the optimal ode quantity assumes that a pecentage of the on-hand inventoy is wasted due to deteioation as a chaacteistic featue and the inventoy conditions goven the item stocked. This pape povides a useful popety fo finding the optimal pofit and odeing quantity with deteioated units of lost sales. A new mathematical model is developed and compaed to the taditional EOQ model numeically. The economic ode quantity, and the net pofit fo the modified model, wee found to be less than that of the taditional, i.e. and the net pofit espectively. But the modified aveage pofit pe unit pe cycle is moe than that of the taditional pofit pe unit pe cycle. Finally, wasting the pecentage of on-hand inventoy 158

6 due to deteioation effect was demonstated numeically to have an advese effect on the aveage pofit pe unit pe cycle. Hence the utilization of units lost due to deteioation makes the scope of the application boade. Futhe, a numeical example is pesented to illustate the theoetical esults, and some obsevations ae obtained fom sensitivity analysis with espect to the majo paamete α. The model in this study is a geneal famewok that consides wasting/ no wasting the pecentage of on-hand inventoy due to deteioation simultaneously. In the futue study, it is hoped to futhe incopoate the poposed model into seveal situations such as shotages ae allowed and the consideation of multi-item poblem. Futhemoe, it may also take patial backlogging into account when detemining the optimal eplenishment policy. Refeences [1] S. Bose, A. Goswami, K.S. Chaudhui, An EOQ model fo deteioating items with linea timedependent demand ate and shotages unde inflation and time discounting. Jounal of Opeational Reseach Society, (1995) 46: [2] S.K Goyal, A. Gunasekaan, An integated poduction-inventoy-maketing model fo deteioating items. Computes and Industial Engineeing, (1995) 28: [3] D. Gupta, Y. Gechak, Joint poduct duability and lot sizing models. Euopean Jounal of Opeational Reseach, (1995) 84: [4] M. Haiga, An EOQ model fo deteioating items with shotages and time-vaying demand. Jounal of Opeational Reseach Society, (1995) 46: [5] M. Haiga, An EOQ model fo deteioating items with time-vaying demand. Jounal of Opeational Reseach Society, (1996) 47: [6] M. Haiga, Economic analysis of dynamic inventoy models with non-stationay costs and demand. Intenational Jounal of Poduction Economics, (1994) 36: [7] K. Jain, E. Silve, A lot sizing fo a poduct subject to obsolescence o peishability. Euopean Jounal of Opeational Reseach, (1994) 75: [8] V.K. Misha, Inventoy model fo time dependent holding cost and deteioation with salvage value and shotages. The Jounal of Mathematics and Compute Science, (2012) 4(1): [9] J.S. Osteyoung, D.E. Mc Caty, W.L Reinhat., Use of EOQ models fo inventoy analysis. Poduction and Inventoy Management, 3 d Qt., (1986) [10] G. Padmanabhan, P. Vat, EOQ models fo peishable items unde stock dependent selling ate. Euopean Jounal of Opeational Reseach, (1995) 86: [11] M. Pattnaik, A note on nonlinea optimal inventoy policy involving instant deteioation of peishable items with pice discounts. The Jounal of Mathematics and Compute Science, (2011) 3(2): [12] M. Pattnaik, An entopic ode quantity (EnOQ) model unde instant deteioation of deishable items with pice discounts. Intenational Mathematical Foum, (2010) 5(52): [13] M. Pattnaik, Decision-making fo a single item EOQ model with demand-dependent unit cost and dynamic setup cost. The Jounal of Mathematics and Compute Science, (2011) 3(4), [14] F. Raafat, "Suvey of liteatue on continuously deteioating inventoy models. Jounal of Opeational Reseach Society, (1991) 42: [15] M.K. Salameh, M.Y. Jabe, N. Noueihed, Effect of deteioating items on the instantaneous eplenishment model. Poduction Planning and Contol, (1993) 10(2): [16] Y.C. Tsao, G.J. Sheen, Dynamic picing, pomotion and eplenishment policies fo a deteioating item unde pemissible delay in payment. Computes and Opeations Reseach, (2008) 35: [17] C.D.J. Wates, Inventoy Contol and Management. (Chicheste: Wiley), [18] H.M. Wee, Economic Poduction lot size model fo deteioating items with patial backodeing. Computes and Industial Engineeing, (1993) 24:

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