RAIRO Rech. Opé. (vol. 33, n 1, 1999, pp. 29-45) A GOOD APPROXIMATION OF THE INVENTORY LEVEL IN A(Q ) PERISHABLE INVENTORY SYSTEM (*) by Huan Neng CHIU ( 1 ) Communicated by Shunji OSAKI Abstact. This pape deives a good appoach to appoximating the expected inventoy level pe unit time fo the continuous eview peishable inventoy system. Thee existing appoximation appoaches ae examined and compaed with the poposed appoach. Thee stockout cases, including the full backode, the patial backode, and the full lost sales cases, which customes o mateial uses geneally use to espond to a stockout condition ae consideed. This study eveals the fact that the poposed appoximation is simple yet good and suitable fo incopoation into the peishable inventoy model to detemine the best odeing policy. The esults fom numeical examples and a sensitivity analysis indicate that sevee undeestimation o oveestimation of the expected inventoy level pe unit time due to the use of an inappopiate appoximation appoach would esult in geat distotion in the detemination of the best odeing policy. Keywods: Peishable inventoy, appoximation, backode, lost sales, sensitivity analysis. Résumé. Cet aticle développe une bonne façon d appoxime le niveau moyen de stock pa unité de temps pou le système d inventaie pemanent dans le cas des denées péissables. Nous examinons tois appoches existantes et les compaons avec celle qui est poposée ici. Nous considéons tois cas de uptues de stock : celui du éappovisionnement total, celui du éappovisionnement patiel, et celui où toute demande non satisfaite est entièement pedue ; ce sont les cas les plus généalement encontés. L étude évèle que l appoximation poposée est simple, et cependant bonne et appopiée à une incopoation dans le modèle pou la détemination de la meilleue politique de éappovisionnement. Les ésultats des exemples numéiques et une étude de sensibilité indiquent que d une sous-estimation ou d une suestimation sévèe du niveau moyen du stock pa unité de temps, causée pa l utilisation d une méthode inappopiée d appoximation, ésulteait une gande distosion dans la détemination de la meilleue politique de éappovisionnement. Mots clés : Stock péissable, appoximation, éappovisionnement, ventes pedues, analyse de sensibilité. (*) Received Januay 1996. ( 1 ) Depatment of Industial Management, National Taiwan Univesity of Science and Technology, Taipei, Taiwan 106, Republic of China. Recheche opéationnelle/opeations Reseach, 0399-0559/99/01/$ 7.00 EDP Sciences 1999
30 H. N. CHIU 1. INTRODUCTION The study of the detemination of the optimal o the best odeing policies fo peishable o deteioating inventoy systems has eceived a significant amount of attention in the past thee decades. Compehensive eviews in this aea can be found in Nahmias [13] and Raafat [17]. Typically, goods having finite lifetimes ae subject to eithe peishability o decay. A peishable inventoy is one in which all the units of one mateial item emaining in stock will simultaneously lose thei utility. The emaining units must be discaded if they have not yet been used (deteministic o andom demand) afte stoage fo a fixed peiod of time. Common examples of peishable inventoies ae fashion gaments, blood, and foodstuffs. On the othe hand, a decaying o deteioating inventoy geneally has a andom lifetime. It can be defined as one in which a faction of the units of an item emaining in stock loses its utility (e.g., adioactive mateials and gasoline) o in which the utility of each unit deceases ove time (e.g., fuits and vegetables). In this pape, the pimay focus is placed on the continuous eview (ode quantity/eode point) peishable inventoy system. Nahmias [11, 12] and most of the othe pevious studies such as those of Cohen [4], Chazan and Gal [2], and Nandakuma and Moton[14] have concentated on the peiodic eview and multi-peiod lifetime poblem with zeo lead time. Thei consideable effots have been spent on the development of good appoximations of the exact expected outdating (i.e., the expected peished units of an item duing a time inteval). This is because it is extemely difficult to obtain the optimal expected outdating fo a long lifetime item. In fact, this equies solving a multi-dimensional pogam with coesponding quantities fo vaious ages at the beginning of each peiod, which involves complex ecusive computation. As fa as we know, few papes have dealt with the continuous eview peishable inventoy model, which is known to be an intactable poblem. Schmidt and Nahmias [20] commented that the peishable poblem appeas to be extemely difficult when a positive lead time is intoduced. The difficulty is that peishability can only be applied to units on hand, not on ode. Recently, this autho [3] developed a simple yet good appoximation of the expected outdating fo a fixed-life peishable inventoy model with a positive lead time. This autho used an extemely ough appoximation of the expected inventoy level pe unit time since both the expected outdating of the cuent ode size and the expected shotage quantity pe cycle ae assumed to be negligible in the calculation of the expected stock level. Recheche opéationnelle/opeations Reseach
A GOOD APPROXIMATION OF THE INVENTORY LEVEL 31 Though this stong assumption can help simplify computation of the holding cost, distotion in detemining a best odeing policy may aise. Bown et al. [1] also demonstated that the penalty associated with odeing is elated to not only the lot-size eo but also to the holding cost function. Theefoe, the deivation of a good appoximation of the stock level function to educe this odeing distotion and cost penalty to a minimum is the focus of this study. The liteatue on the inventoy system with andom demand includes the classical models pesented in Hadley and Whitin [6, Sections 4-2 and 4-3]. They discussed the backode and lost sales cases unde the assumption that peishability o decay is not allowed. The optimal policy is that when the inventoy position (on hand plus on ode stock) eaches the eode point,, an ode of size units is placed. Silve [22] classified inventoy management poblems into an enomous vaiety of eseach schemes. The inventoy system with pobabilistic lead time demand, stockout, and item shelf-life consideations is of inteest fo futue application. He pointed out that commonly used distibutions of lead time demand ae the Nomal, Gamma, and Poisson distibutions. Howeve, thee always is a small pobability that the lead time demand will be negative when a nomal distibution is used fo the lead time demand. In this case, a tuncated nomal distibution is ecommended, but this may make the computation difficult. Late, Das [5] intoduced a inventoy model with time-weighted (time-popotional) backodes. Seveal inventoy models with a mixtue of backodes and lost sales wee poposed by Posne and Yansouni [16], Montgomey et al. [10], Matthews [9], Rosenbeg [19], Pak [15] and Kin and Pak [7]. Almost all the pevious eseach woks used, a faction of the unsatisfied demand backodeed (the emaining faction 1- completely lost), to model patial backodes. Recently, Rabinowitz et al. [18] modeled a inventoy system using a contol vaiable, which limits the maximum numbe of backode allowed to accumulate duing a cycle. Obviously, these pevious eseach woks did not include the undelying peishability assumption in thei model fomulations. In geneal, the cost of a shotage can be assumed to be the timeindependent stockout cost ($/unit), the time-popotional shotage cost ($/stockout duation/unit), o the stockout cost pe outage. The time-weighted shotage cost is popotional to the duation of a stockout. On the othe hand, if the shotage cost is based on an outage, then accoding to Tesine [23, p. 218], an outage can be defined as one time of the stockout without egad to the numbe of units out of stock duing a eplenishment cycle. vol. 33, n 1, 1999
32 H. N. CHIU In this pape, thee stockout cases in which customes o mateial uses can choose to eact to a stockout condition ae consideed. As peviously stated, the thee stockout cases ae the full backode, the full lost sales, and the patial backode cases, which influence computation of the expected inventoy levels pe unit time and, then, the holding costs. Hee, we assume that both the backode and lost sales costs ae independent of the duation of the stockout. In addition, othe impotant assumptions ae the stochastic demand, fixed item lifetime, backode faction, and no quantity discounts. Theefoe, the poposed peishable inventoy model is diffeent fom Shiue s [21] model and the above mentioned models. In this pape, we will examine thee existing appoximations and deive a new one to appoximate the expected inventoy level pe unit time: (1) An extemely ough appoximation, as adopted by this autho [3, p. 97, equation (7)] and Hadley and Whitin [6, p. 156, equation (4-1)]. (2) An appoximation without consideing the stockout duation and the outdate condition, as intoduced by Wagne [24, p. 825, equation (14)]. (3) An appoximation consideing the stockout duation but excluding the outdate condition, as poposed by Kin and Pak [7, p. 233, equation (5)]. (4) A good appoximation based on ou [3, p. 96, equation (4)] appoximate expected outdating, as developed in this study. 2. PROBLEM DESCRIPTION In this pape, only one peishable item (o poduct) is consideed. Each unit of the item has a fixed lifetime equal to. The inventoy level is eviewed continuously and deceased by a satisfaction of demand o by disposal of peished units. An ode size of is placed when the inventoy level eaches the eode point,. Thee is a positive leadtime,, fo each eplenishment, and a fixed odeing cost,, is incued. All the units of a eplenishment ode aive fesh o new. Each unit does not lose o decease in utility befoe its useful lifetime ends, but it must be discaded if it has not been used befoe the expiation date. An outdate cost equal to pe unit is chaged. The demand in unit time, 1, is a nonnegative andom vaiable. Assume that it follows a specific continuous o discete distibution with density o mass function 1 1 and mean. We also assume that if is cumulative demand by time, then is a stochastic pocess with stationay, independent incements. This implies that has density o mass m m and mean. In othe wods, Recheche opéationnelle/opeations Reseach
A GOOD APPROXIMATION OF THE INVENTORY LEVEL 33 has density o mass m+l m+l and mean. Units ae always depleted accoding to an FIPO (i.e., Fist into stock ae consumed fist) issuing policy. The notation to be used thoughout this pape is defined as follows: Ode quantity. Reode point. Fixed lifetime of the peishable item. Positive ode lead time. L Demand duing lead time with pobability function L L and mean, whee L is an -fold convolution of 1. Replenishment cost pe unit. Holding cost pe unit pe unit time. Fixed odeing cost pe ode. Outdate cost pe unit. Backode cost pe unit. Lost sales cost pe unit. A faction of the excess (unsatisfied) demand pe eplenishment cycle can be backodeed, and the emaining faction is lost. Expected cycle length. Expected inventoy level pe unit time. Expected outdate quantity of the cuent ode size. Expected shotage quantity pe cycle. Additionnal notations will be intoduced late when needed. Figue 1 shows a peishable inventoy model with a mixtue of backodes and lost sales. 3. CHIU S EXPECTED OUTDATING APPROXIMATION Just as demonstated by Nahmias [13], who dealt with the peiodic eview and multi-peiod lifetime poblem with zeo ode lead time, avoidance of complex computation equies developing a good appoximation of the exact expected outdating. The continuous eview peishable inventoy poblem with positive ode lead time also involves complex computation, as stated by Schmidt and Nahmias [20] and mentioned befoe. Thus, this autho [3] pesented a simple yet good appoximation to the expected outdating fo the vol. 33, n 1, 1999
34 H. N. CHIU Figue 1. continuous eview peishable inventoy system with positive ode lead time. Ou appoximate expected outdating of the cuent ode size is given by +Q 0 m+l 0 m+l m+l u<+q m+l u< m+l m+l (1) whee m+l is the pobability function of the andom vaiable m+l (i.e., the demand duing time units). Equation (1) has been shown to be a faily acceptable appoximation of the exact expected outdating in the situation whee the continuous eview stategy is used. It should be noted hee that equation (1) is analogous to pesented in Recheche opéationnelle/opeations Reseach
A GOOD APPROXIMATION OF THE INVENTORY LEVEL 35 Nahmias [11, p. 1004, equation (2-1)] with,, and eplaced by,, and, espectively. How equation (1) can appoximate the expected outdating effectively has been discussed in moe detail elsewhee [3]. In the following thee sections, ou attention will be focused on the deivations and compaisons of the expected inventoy levels pe unit time fo the thee stockout cases. 4. FULL BACKORDER CASE With full backodes, thee is no loss of sales since customes o mateial uses ae willing to wait fo the aival of the next ode o an outstanding ode. The unsatisfied demand is then filled by the aived ode immediately. Fou appoaches can be used to appoximate the expected inventoy level pe unit time in the peishable inventoy system: (1) Extemely ough appoximation This appoach assumes that the values of and ae consideably smalle than the cuent ode size,. Hence, and can be neglected, and the expected inventoy level pe unit time is (2) Equation (2) implies that thee ae no diffeences among the thee stockout cases. As mentioned ealie, this appoach has been adopted by this autho as well as by Hadley and Whitin [6]. Howeve, stock level was not coectly accounted fo when thee was a depletion case (i.e., an out of stock condition). (2) Wagne appoximation In contast to extemely ough appoximation, Wagne [24] consideed both the depletion case and the non-depletion case duing a lead time. Suppose that is much smalle than and can be ignoed in this appoximation. Then, Wagne intoduced whee 1 x> L L w (3) L L (4) vol. 33, n 1, 1999
36 H. N. CHIU The function L in equation (4) is the pobability function of the andom vaiable L. In equation (3), is called the coection tem of the expected inventoy level pe unit time. Clealy, this appoximation is educed to the extemely ough appoach when is ignoed. Howeve, this appoach does not take into account the duation of the stockout since it assumes that when L, the inventoy level becomes zeo just befoe the eplenishment aives. The pupose of this appoximation is to make the holding cost fomulas uncomplicated. (3) Modified Wagne appoximation w in equation (3), which will be demonstated in Section 6, is an oveestimation of due to neglect of the stockout duation and the outdate condition. In this pape, the Kin and Pak appoximation [7] without the outdate condition is called the modified Wagne appoximation. Refeing to the deivation of the aveage caying inventoy in Kin and Pak, the expected inventoy level pe unit time of the modified Wagne model can be expessed by m 1 2 L (5) If the ight side of the equal sign in equation (5) is multiplied by, then the esult is equivalent to Kin and Pak s [7, p. 233, equation (5)] full backode model with. Afte futhe manipulation, equation (5) can be ewitten as m 1 L (6) whee is fom equation (4). Note that in equations (5) and (6), the integal notation should be eplaced with the summation notation if the lead time demand, L, is a discete andom vaiable. Also, it should be emphasized hee that only equations used in the continuous andom vaiable case will be pesented late. (4) Chiu appoximation It is a fact that equation (5) is deived unde the assumption that is consideably smalle than and can be neglected. Inevitably, this will Recheche opéationnelle/opeations Reseach
A GOOD APPROXIMATION OF THE INVENTORY LEVEL 37 esult in an inaccuate value of obtained by using the modified Wagne appoach. Now, let 1 expected aveage inventoy level duing a lead time; 2 expected aveage inventoy level afte ode aival until next eode. In ode to simplify the deivation of the expected inventoy level pe unit time, it is assumed that when L, the inventoy level becomes zeo just befoe the odeed units aive. Then, 1 1 L L (7) 0 In fact, equation (7) can be futhe simplified to 1 L (8) 0 In equation (8), 0 L can be easily poved to equal. On the othe hand, 2 can be appoximated pecisely by consideing a ectangle, a tiangle, and a paallelogam as shown in Figue 1. Thus, 2 (9) whee, (10) is fom equation (1), and denotes the expected numbe of eplenishment cycles that the item lifetime can ove; moeove,. In othe wods, the lifetime of time units consists of eplenishment cycles, whee denotes the geatest intege less than o equal to. Fo example, in Figue 1, we set equal to 1. In addition, m epesents an outdate point of time dopped in a given cycle, which depends on the actual demand duing an time unit inteval. Clealy, equation (8) must be weighted by (due to ). Coespondingly, equation (9) should be weighted by. Multiplying the two equations by the two weights, espectively, the expected inventoy level pe unit time has the following fom: c 0 L (11) vol. 33, n 1, 1999
38 H. N. CHIU Afte some manipulations, it is given by c (12) Note that c is educed to w if is ignoed (that is, is set to zeo). Futhemoe, c becomes when both and ae neglected. The last two tems in equation (12) ae the coection tems used to make this appoximation moe effective. In ode to educe the computational effot, it is easonable to set m to the middle point of the time length. As a esult, becomes 1/2; thus, c (13) Fom equations (2), (3), and (6), we conclude that m w (14) In geneal, we have. Thus, since. Equations (2), (3), and (13) imply that c w (15) Howeve, it is difficult to compae c with m. We find that m c when the value of is vey small in equation (13). This can be seen by compaing equation (13) with equation (6) diectly. Now, the total expected aveage cost pe unit time fo the full backode case is given by (16) whee is one of the above fou appoximations. It is noted that, as given in equation (10), is also a function of the cuent ode size,, and the eode point, 5. PARTIAL BACKORDER AND FULL LOST SALES CASES In a full lost sales situation, any unsatisfied demand is completely lost, and the custome o mateial use has pesumably filled he o his need fom othe souces. Howeve, in most pactical situations, when the item is out of stock, some customes o mateial uses ae patiently waiting fo thei demand to be satisfied upon initial eceipt of the next ode while othes ae impatient and make puchases fom othe souces to fill thei Recheche opéationnelle/opeations Reseach
A GOOD APPROXIMATION OF THE INVENTORY LEVEL 39 demand. Unde these cicumstances, it is easonable to assume that only a faction,, of the shotage quantity is backodeed, and that the emaining faction,, is lost foeve. The deivation of the expected inventoy levels pe unit time fo the two stockout cases is simila to that in the full backode case. The majo diffeence between the full backode and the patial backode cases is that in the patial backode case, on aveage, units ae equied in each eplenishment cycle, as compaed to only units in the full backode case. It should be noted that the quantity of (including units backodeed) is satisfied while that of is lost foeve. As a esult, the extemely ough appoximation emains unchanged, and the othe thee appoximations can easily be deived by simply substituting fo in the elevant equations of the full backode case. The esultant equations fo the patial backode case ae: w m and 1 L c (20) whee is fom equation (1), and is fom equation (4). At one exteme,, the patial backode case educes to the full lost sales case. At anothe exteme,, it educes to the full backode case. Analogously, the patial backode and the full lost sales cases have the same popeties as expessed in elations (14) and (15). The total expected aveage cost pe unit time fo the patial backode case is given by (21) whee (22) vol. 33, n 1, 1999
40 H. N. CHIU 6. NUMERICAL EXAMPLES AND SENSITIVITY ANALYSIS In this section, numeical examples will be given and the esults of sensitivity analysis will be pesented. Twenty-fou test poblems which appeaed in Chiu [3] wee used and ae listed in Table 1. TABLE 1 Cost paametes and elevant data of 24 test poblems. Test poblem No. Cost paamete 1 5 20 10 5 2 5 20 50 5 3 5 20 100 5 4 5 40 10 5 5 5 40 50 5 6 5 40 100 5 7 15 20 10 5 8 15 20 50 5 9 15 20 100 5 10 15 40 10 5 11 15 40 50 5 12 15 40 100 5 13 5 20 10 15 14 5 20 50 15 15 5 20 100 15 16 5 40 10 15 17 5 40 50 15 18 5 40 100 15 19 15 20 10 15 20 15 20 50 15 21 15 20 100 15 22 15 40 10 15 23 15 40 50 15 24 15 40 100 15,,,, and 1 Poisson with. Fo the pupose of illustation, Test Poblem 1 in Table 1 was chosen. Then, the elevant equations of the poposed appoach (including equations (1), (4), and (20)-(22)) wee applied fo, 0.5, and 0, espectively. Afte solving this test poblem with Gino [8], a summay of the final solution was a given in Table 2. It can be seen fom Table 2 that the values of c ae 11.4899, 11.4080, and 11.0981 fo, 0.5, and 0, espectively. We may conclude hee that the expected inventoy level pe unit time deceases as the faction Recheche opéationnelle/opeations Reseach
A GOOD APPROXIMATION OF THE INVENTORY LEVEL 41 TABLE 2 Summay of esults using the poposed appoximation and Test Poblem 1 fo, and. Final solution (Full backode) (Patial backode) (Full lost sales) 13.8417 13.9178 13.6224 14.5414 14.3792 14.1564 c 11.4899 11.4080 11.0981 0.0571 0.0549 0.0431 0.1331 0.1471 0.1693 1.3785 1.3936 1.3749 71.0898 70.8247 70.5319 deceases. A moe detailed analysis to veify this conclusion was futhe conducted in this study. Table 3 pesents the solution values of c using the poposed appoximation and Test Poblem 1 fo vaious values of. The esults indicate that fo each faction of, the expected inventoy level pe unit time conveges to a fixed value when the lifetime,, inceases to a lage value (this value of is five in this example). The longe the lifetime of a peishable item has, the geate is the tendency that the peishability assumption being eleased. This implies that as the lifetime,, inceases to a sufficently lage value, and that equation (20) then appoaches equation (18). Consequently, the peishable inventoy model educes to the no-outdating model in the extemely long lifetime situation. TABLE 3 Solution values of using the poposed appoximation and Test Poblem 1 fo vaious values of. 2 8.7777 8.5570 8.4990 3 11.4899 11.4080 11.0981 4 12.5202 12.3676 12.1288 5 12.5803 12.4032 12.1890 6 12.5803 12.4032 12.1890 7 12.5803 12.4032 12.1890 A question aises about whethe caeless appoximation of the expected inventoy level pe unit time has a significant impact on detemination of the odeing policy. Table 4 pesents a summay of the esults of sensitivity analysis in which 24 test poblems, given in Table 1, wee used. Each aveage pecentage in Table 4 is the esult of, fist, subtacting the policy paamete (e.g., ) which was obtained using the poposed vol. 33, n 1, 1999
42 H. N. CHIU TABLE 4 Aveage pecentages fo odeing policy deviations. Appoximation appoach Extemely 0.10% 0.53% + 0.10% 0.84% + 0.29% 0.92% + 0.54% 1.82% ough Modified 0.14% 0.49% 0.24% 0.37% + 0.82% 1.20% 0.52% 0.40% Wagne Wagne 0.03% 0.52% 0.35% 0.18% 0.16% 0.03% 0.07% 0.19% appoximation fom the policy paamete which was obtained using one of the othe thee appoximations, and then dividing this value by the poposed policy paamete. A positive aveage pecentage shows the extent to which the policy paamete obtained by using an appoximation appoach has been oveestimated while a negative one means that the policy paamete obtained has been undeestimated. Some impotant conclusions dawn fom Table 4 ae as follows: (1) Fo each faction of, the eode points obtained by using the extemely ough, modified Wagne, and Wagne appoaches ae consistently undeestimated. (2) In the full lost sales case, deviations on the ode quantity ae geate than those in the full backode case. This may be because ode quantity,, does not include the backodeed quantity of in the full lost sales case. (3) Most of the policy paametes obtained by using the Wagne appoach have much smalle deviations than do those obtained using the extemely ough and modified Wagne appoximations. Pesumedly, the main eason is that the solution value of is vey small (one example is shown in Table 2). Thus, equation (18) is almost identical to equation (20). Nevetheless, all policy paametes detemined by using the Wagne appoach ae undeestimated. Table 5 pesents the sums of 24 solution values of. Figue 2 gives the associated gaph which shows the elative values of fo the fou appoximations and thee stockout cases. Hee, we conclude that the expected inventoy level pe unit time deceases with the decease of the faction,. It is also evident that Relations (14) and (15) ae consistent with the esults shown in Table 5 o Figue 2. Futhemoe, just as expected, the values of obtained by using the modified Wagne appoximation ae smalle than Recheche opéationnelle/opeations Reseach
A GOOD APPROXIMATION OF THE INVENTORY LEVEL 43 those obtained using the poposed appoximation since the solution values of ae vey small in this analysis. TABLE 5 Total solution values of (24 test poblems). Appoximation appoach Extemely ough 299.4603 293.4229 283.7080 268.7160 Modified Wagne 299.5945 295.4538 290.4995 283.3413 Chiu 301.2300 297.8750 293.2553 286.8325 Wagne 302.4031 298.7430 294.8651 287.9539 Figue 2. 7. CONCLUSIONS This pape has pesented a good appoach to appoximation of the expected inventoy level pe unit time fo the peishable inventoy system. Thee stockout cases which customes o mateial uses may adopt in esponse vol. 33, n 1, 1999
44 H. N. CHIU to a stockout condition have been consideed. The study has compaed the poposed appoximation appoach with thee existing appoaches which have been used in a situation whee pesishability o decay is not allowed. Obviously, the poposed appoximation is much simple than the modified Wagne appoach. This can be obseved by compaing equation (20) with equation (19). It is not complicated, as compaed with the Wagne appoximation. Theefoe, equation (20) is a pactical fomula, suitable fo incopoation into the peishable inventoy model, which can be fomulated in the fom of equations (21), (22), (1), and (4). The best odeing policy can, theeby, be obtained coectly, and distotion in detemining and can be educed to a minimum. In addition, esults fom numeical examples and a sensitivity analysis indicate that the solution values of ae undeestimated when the extemely ough appoach and the modified Wagne appoach ae used. This esult often causes deviations in the policy paametes. Moe impotantly, sevee undeestimation of due to the use of the extemely ough appoach will esult in geat distotion when detemining the best odeing policy. It is woth noting hee, as pointed out by Bown et al. [1, p. 607], that the impotance of accuately estimating the holding cost function is eadily appaent fo decision makes whose fims opeate in an envionment of diseconomies of scale (e.g., peishability, decay, o deteioation). REFERENCES 1. R. M. BROWN, T.E.CONINE and M. TAMARKIN, A Note on Holding Costs and Lot-Size Eos, Decision Sciences, 1986, 17, pp. 603-608. 2. D. CHAZAN and S. GAL, A Makovian Model fo a Peishable Poduct Inventoy, Management Science, 1977, 23, pp. 512-521. 3. H. N. CHIU, An Appoximation to the Continuous Review Inventoy Model with Peishable Items and Lead Times, Euopean Jounal of Opeational Reseach, 1995, 87, pp. 93-108. 4. M. A. COHEN, Analysis of Single Citical Numbe Odeing Policies fo Peishable Inventoies, Opeations Reseach, 1976, 24, pp. 726-741. 5. C. DAS, Inventoy Models with Time-Weighted Backodes, Jounal of the Opeational Reseach Society, 1983, 34, pp. 401-412. 6. G. HADLEY and T. M. WHITIN, Analysis of Inventoy Systems, Pentice-Hall, Englewood Cliffs, N.J., 1963. 7. D. H. KIN and K. S. PARK, Inventoy Model with a Mixtue of Lost Sales and Time-Weighted Backodes, Jounal of the Opeational Reseach Society, 1985, 36, pp. 231-238. 8. J. LIEBMAN, L. LASDON, L. SCHRAGE and A. WAREN, Modelling and Optimization with GINO, The Scientific Pess, Palo Alto, Califonia, 1986. Recheche opéationnelle/opeations Reseach
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