Optimal inventory policy for items with power demand, backlogging and discrete scheduling period

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1 Optimal ivetory policy for items with power demad, bacloggig ad discrete schedulig period Joaquí Sicilia Departmet of Mathematics, Statistics ad Operatios Research Uiversity of La Lagua, Teerife, Caary Islads, Spai Tel: (+34) , Mauel Gozález-de-la-Rosa Departmet of Busiess Admiistratio ad Ecoomic History Uiversity of La Lagua, Teerife, Caary Islads, Spai Tel: (+34) , Luis A. Sa-José IMUVA, Departmet of Applied Mathematics Uiversity of Valladolid, Valladolid, Spai Tel: (+34) , Jaime Febles-Acosta Departmet of Busiess Admiistratio ad Ecoomic History Uiversity of La Lagua, Teerife, Caary Islads, Spai Tel: (+34) , Abstract. I this paper a ivetory system for items with a power demad patter ad baclogged shortages is developed. A ow fixed time period is assumed ad, moreover, the ivetory cycle must be a multiple of that period. The holdig cost, the bacloggig cost ad the orderig cost are the relevat costs cosidered i the ivetory system maagemet. The formulatio of the ivetory problem leads to a o-liear iteger mathematical programmig problem. A algorithmic procedure to calculate the ecoomic order quatity ad the optimal schedulig period that miimize the total cost per ivetory cycle is proposed. Some umerical examples are solved to illustrate the theoretical results. Keywords: EOQ ivetory models, Baclogged shortages, Power demad patter, Discrete schedulig period. 1. INTRODUCTION The basic ivetory models suppose that demad rate is costat. However, geerally demad for items varies with time ad ivetory models must cosider that situatio. Thus, it is more realistic to aalyze ivetory systems where demad chages with time. It would allow to model appropriately the behavior ad evolutio of the ivetory of products. I the literature of ivetory theory, models ivolvig time variable demad have received attetio from several researchers. Thus, Silver ad Meal (1973) established a approximate approach for a determiistic ivetory system with time-depedet demad. Doaldso (1977) preseted a algorithm for solvig the classical oshortage ivetory policy for a liear tred i demad over a fixed time period. Ritchie (1984) aalyzed the ecoomic order quatity (EOQ) model with liear icreasig demad. Dave (1981) studied a lot-size ivetory model with a liear tred i demad ad allowig shortages. Bahari- Kashai (1989) studied the optimal repleishmet for deterioratig items with time-proportioal demad. Goswami ad Chaudhuri (1991) developed a ivetory model, assumig shortages ad liear demad. Hariga ad Goyal (1995) aalyzed a ivetory model with timevaryig demad rate, shortages ad cosiderig effects of iflatio. Bose et al. (1995) aalyzed a EOQ model for

2 deterioratig items with a liear positive tred i demad ad shortages baclogged. Bhuia ad Maiti (1998) studied a ivetory system for deterioratig items cosiderig repleishmet cost depedet o lot-size ad liear tred i demad. Chag ad Dye (1999) discussed a EOQ model for deterioratig items with time varyig demad ad partial bacloggig. Wu (00) discussed also a EOQ model with time-varyig demad, cosiderig deterioratio ad shortages. Yag et al. (004) preseted a approach to aalyze a ivetory system with o-liear decreasig demad. Saaguchi (009) developed a ivetory policy for a system with time-varyig demad. Omar ad Yeo (009) studied a ivetory system that satisfied a cotiuous time-varyig demad. Mishra ad Sigh (010) developed a ivetory model with costat rate of deterioratio ad time depedet demad. The demad patters are referred to as differet ways by which products are tae out of ivetory to supply customer demad. If the demad rate is the same durig all the ivetory cycle, the demad patter is ow as uiform demad patter. However, there are other ways by which the uits may be withdraw throughout the period. The power demad patter allows to suit demad to more practical situatios. Thus, this patter ca ot oly represet the behavior of demad whe is uiformly distributed throughout the period, but also models situatios where a high percetage of uits may be withdraw at the begiig or at the ed of the period. Several papers o ivetory systems cosider that the demad follows a power patter. Thus, Goel ad Aggarwal (1981) studied a order-level ivetory system with power demad patter for deterioratig items. Datta ad Pal (1988) aalyzed a ivetory model with power demad patter ad variable rate of deterioratio. Lee ad Wu (00) preseted a ivetory model for items with deterioratio, shortages ad power demad patter. Dye (004) exteded this last model to a geeral class with time-proportioal bacloggig rate. Sigh et al. (009) aalyzed a EOQ model for perishable items with power demad patter ad partial bacloggig. Rajeswari ad Vajiodi (011) studied a ivetory model for deterioratig items with partial bacloggig ad power demad patter. Recetly, Mishra ad Sigh (013) preseted a ecoomic order quatity model for deterioratig items with power demad patter ad shortages partially baclogged. I all the above papers the schedulig period is a fixed period. Thus, the ivetory total cost depeds o a sigle decisio variable. If the ivetory system is without shortages, the decisio variable is either the lot size or the iitial stoc level. O the other had, whe the ivetory system allows shortages, the decisio variable is time where the et stoc level is zero. However, i the model here preseted the ivetory total cost will deped o two decisio variables. Thus, we develop a ivetory model for items with power demad o a ow basic period. The ivetory cycle must be a multiple of that period ad the lot size must be also a multiple of the total quatity demaded alog that basic period. Shortages are allowed ad completely baclogged. The holdig cost, the bacloggig cost ad the orderig cost are the three costs cosidered i the system maagemet. The objective of the ivetory problem cosists of determiig the optimal schedulig period ad the ecoomic lot size such that the total ivetory cost per uit time is miimized. The formulatio of this objective leads to a o-liear iteger mathematical programmig problem. A algorithmic procedure to calculate the ecoomic order quatity ad the optimal schedulig period that miimize the total cost per ivetory cycle is proposed. The orgaizatio of the paper is as follows. I Sectio we itroduce the otatio used throughout of the paper ad the basic assumptios cosidered for the ivetory system. I Sectio 3, the mathematical model that describes the ivetory problem is formulated. A algorithm to solve the ivetory problem ad determie the optimal policy is preseted i Sectio 4. Some umerical examples are provided i Sectio 5 to illustrate the solutio procedure. Fially, coclusios ad future research are commeted.. HYPOTHESIS AND NOTATION We will use throughout the paper the followig otatio. Basic time period. That period is fixed ad ow. D Demaded total quatity alog the basic period τ. That quatity is ow. r Average demad per basic period (r = D/τ). f(t) Demad up to time t (0 t τ) alog each basic period. Note that f(τ) = D. T Demad patter idex (>0). Legth of ivetory cycle or schedulig period. That period T must be a multiple of the basic period τ. Number of basic periods that cotais the ivetory cycle (T = ). S Iitial stoc level or order level. That ivetory level S must be a multiple of the demad D. m Iteger positive such that S = md. s Reorder poit. The ivetory should be repleished whe s = (m-)d. Q Lot size or order quatity. The lot size must be equal to Q = D. I(t) Net ivetory (o had - bacorders) level at time t (0 t T). Time at which the ivetory level reaches zero. t 0

3 A Orderig cost. H Holdig cost per uit ad per time uit. W Bacloggig cost per uit ad per time uit. I 1(,m) Average amout carried i ivetory. I (,m) Average shortage i ivetory. I 3() Number of repleishmets per uit of time. C 1(,m) Holdig cost per uit of time. C (,m) Bacloggig cost per uit of time. C 3() Orderig cost per uit of time. C(,m) Total cost per uit of time of the ivetory system. The ivetory system is based o the followig assumptios: A sigle item is cosidered i the ivetory system. The plaig horizo is ifiite. The fluctuatios of the et stoc level are repeated o time alog each ivetory cycle. The repleishmet rate is ifiite. Lead time is zero or egligible. Shortages are allowed ad completely baclogged. Demad durig the basic period τ follows a power demad patter. The average demad r (i.e., r = D/τ) per basic period is determiistic, but the maer i which quatities are extracted of ivetory depeds o the time whe they are removed. This way by which demad occurs durig the period τ is ow as the demad patter. Thus the demad f(t) up to time t (0 t τ) varies with time ad is assumed to be f(t) = rt 1/ /τ (1-)/, where r is the average demad ad is the patter idex, with 0<<. The demad rate at time t (0 t τ) is f (t) = rt (1-)/ /τ (1-)/. This patter i the demad is called the power demad patter (see Naddor, 1966; Datta ad Pal, 1988; Lee ad Wu, 00; Rajeswari ad Vajiodi, 011; Sicilia et al., 01; ad Sicilia et al., 013). The decisio variables are the iteger values ad m, with 1 m. If we determie those values, the we obtai the iitial stoc level S = md, the schedulig period T =, the reorder poit s = (m-)d ad the lot size Q = D. The iput parameters of the ivetory system are D, τ,, h, w ad A. The aim cosists of determiig the optimal schedulig period, the ecoomic orderig quatity (EOQ) ad the optimal reorder poit that optimize the maagemet of the ivetory system described by the previous hypotheses. I the followig sectio we preset a mathematical model to study the ivetory policy. 3. MATHEMATICAL FORMULATION OF THE INVENTORY PROBLEM Let us cosider a ivetory system i which there is a basic period τ durig which the behavior of customers demad follows a power patter with idex. A example of such a situatio occurs for products as coffee, mil, breads, croissats, fruit, ewspapers, jourals, etc. I these cases the demads of these goods toward the begiig of the day (basic period) are greater that o the afteroo or eveig. Those demads are modeled by usig a power demad patter with idex > 1. Aother situatio occurs whe the demad of a product toward the begiig of the day is smaller tha the demad at the ed of the day. That situatio occurs, for example, for items as pizzas, hamburgers, hot dogs, ciema ticets, popcors, etc. I these cases demads are modeled by a power demad patter with idex < 1. The repleishmet of the ivetory is made every T time uits. That ivetory cycle or schedulig period T must be a multiple of the basic period τ. Hece, T = τ, beig a positive atural umber. As the ow demad over the basic period is D, the alog every ivetory cycle the total demad will be D uits. Therefore, the repleishmet size or lot size must be Q = D uits, because the orderig quatity to repleishmet the ivetory must be equal to the total demad alog the ivetory cycle. Let I(t) be the et stoc (o had - bacorders) level at time t (0 t T). At the begiig of the ivetory cycle, the iitial et stoc level is S uits. That level must be a multiple of the D uits because that quatity represets the demad alog the basic period τ. Hece, we have that S = md, beig m a iteger umber. Thus, at time t 0 = mτ, the ivetory level attais a level zero. Durig the schedulig period [0,T] the stoc level declies due to demad of items. At t = T, the ivetory attais a level s (reorder poit). That reorder poit is determied by the differece S Q. Hece, the reorder poit is a value give by s = (m)d uits. Next, the ivetory is filled with a repleishmet quatity or lot size Q = D uits, which raises the ivetory stoc up to level S ad a ew ivetory cycle starts agai. Note that i each ivetory cycle the lot size Q is equal to the total demad rt because = T/ ad r = D/. The ivetory cycle, or time period, T ca be partitioed i time itervals of equal legth τ. Thus, we have [0, T] [0, ] [, ]... [( 1), T] Uder the assumptios previously commeted i Sectio, the differetial equatio that describe the fluctuatios of the et stoc level I(t) over every time iterval ((i-1)τ, iτ), with i = 1,,...,, is (1 )/ di( t) r[ t ( i 1) ] (1) (1 )/ dt The boudary coditios are I(0) = S ad I(iτ)=(m-i)D, for i=1,,,-1. Note that these coditios imply that I(t 0) = 0 ad I(T) = s. The solutio of the above differetial equatio over

4 the time iterval [0,τ] leads to r I t S t t () ( ) 1/, if 0 1 (1 )/ For i =,...,, the equatio of the ivetory level over the time period [(i-1)τ, iτ] is r 1/ Ii ( t) I(( i 1) ) [ t ( i 1) ], (1 )/ if ( i 1) t i Thus, the ivetory level fuctio I(t) over the schedulig period [0,T=] is give by r I t I t m i D t i if ( i 1) t i, for i 1,,...,. 1/ ( ) i ( ) ( 1) [ ( 1) ], (1 )/ Note that the et stoc level I(t) is always a cotiuous ad decreasig fuctio o the time period [0,T). I additio, the decreasig evolutio of the stoc level is the same over every time iterval [(i-1)τ, iτ], for i = 1,,,. Both the average amout carried i ivetory I 1(,m) ad the average shortage I (,m) deped o the positive iteger variables ad m. As 1 m, the some ivetories are carried ad also shortages ca occur. At time t = t 0 = mτ the ivetory stoc attais a level zero. Thus, alog the iterval [t 0,T), shortages occur i the ivetory system ad they are baclogged at the ed of period. The total amout carried i ivetory durig the period [0,t 0] is the sum of the quatities carried i ivetory alog of periods [(i-1)τ, iτ], for i = 1,,,m. That total amout is m i i i1 ( i1) m( 1) 1 I () t dt md ( 1) (3) (4) (5) Thus, the average amout carried i ivetory durig the period [0,T] is give by the expressio 1 m( 1) 1 I1(, m) ( md ) T ( 1) md m( 1) 1 ( 1) Now, we have to calculate the shortages durig the ivetory cycle. It is determied by the sum of the shortages obtaied i every time iterval [(i-1),i], for i =m+1,...,. Thus, we have i im1 ( i1) ( m)( 1) 1 [ Ii ( t)] dt ( m) D ( 1) (7) Hece, the average shortage i the ivetory durig the period [0,T] is (6) 1 ( m)( 1) 1 I(, m) ( m) D T ( 1) ( m) D ( m)( 1) 1 ( 1) Fially, the average umber of repleishmets per uit of time is always I 3() = 1/T=1/(). I the followig paragraphs, we determie the costs subject to cotrol i the ivetory system. The ivetory total cost per uit time cosists of the followig cost compoets: holdig cost, bacloggig cost, ad orderig cost. The holdig cost per uit of time is md m( 1) 1 ( 1) C 1(, m ) h The bacloggig cost per uit of time is ( m) D ( m)( 1) 1 C (, m) w ( 1) The orderig cost per uit of time is (8) (9) (10) ( ) A A C3 T (11) The total ivetory cost per uit of time is the sum of the above costs. Hece, that total cost is give by md m( 1) 1 C(, m) ( h w) ( 1) ( m)( 1) 1 A wd ( 1) (1) Thus, we have to solve the followig o-liear iteger mathematical programmig problem: md m( 1) 1 Mi C(, m) ( h w) ( 1) ( m)( 1) 1 A wd ( 1) subject to ad m are iteger variables with 1 m 4. SOLVING THE INVENTORY PROBLEM (13) To fid the solutio of the ivetory problem, we have to see the miimum of the fuctio C(,m) for the rage 1 m, with ad m iteger variables. Relaxig the iteger restrictios o the variables, ad assumig that ow ad m are cotiuous variables, the we could use the differetial calculus to obtai the optimal

5 solutio for the cotiuous ivetory problem. Differetiatig with respect to m, we have C ( h w) md ( h w) D(1 ) wd m ( 1) ad differetiatig with respect to, we obtai (14) C ( h w) md m( 1) 1 wd A (15) ( 1) To fid the miimum poit ( 0,m 0), we have to solve the equatios ( h w) md ( h w) D(1 ) wd 0 ( 1) ( h w) md m( 1) 1 wd A 0 ( 1) From (16), we obtai m as a fuctio of, that is, w (1 ) m m( ) hw ( 1) (16) (17) (18) Substitutig m = m() i equatio (17), ad solvig that equatio with respect to variable, we have From (18), we obtai m A( h w) ( h w) (1 ) hw D 4 hw( 1) 0 (1 ) 1 0 Aw w h ( h w) D 4 h ( 1) ( 1) (19) (0) Substitutig 0 ad m 0 give i (19) ad (0) i the cost fuctio (1), we have the cost hwd A ( h w) D(1 ) C0 C( 0, m0 ) (1) h w 4( 1) Note that the terms iside the roots i (19), (0) ad (1) should be positive values. Thus, the coditio 8 A( 1) ( h w) D (1 ) () must be satisfied. I additio, a easy computatio shows that the Hessia at poit ( 0,m 0) is positive whe () is true. Therefore, if the coditio () is satisfied, the ( 0,m 0) is a miimum poit of the fuctio C(,m) give i (1) whe the variables ad m are cotiuous variables. If the variable m is fixed, the followig result shows that the cost fuctio C(,m) has a miimum poit o the straight lie determied by m. Theorem 1. Cosider m ad as cotiuous variables o the regio = {(,m)/ >0, m 1}. Fixed m, with m 1, the cost fuctio C m() = C(,m) is a covex fuctio with respect to the variable ad has a miimum poit at the value m give by m ( h w) md [( m 1) m 1] A( 1) wd ( 1) Proof. The proof is immediate. (3) If we substitute m i the fuctio C(,m) give i (1), the we obtai a cost fuctio that depeds of m oly, that is md m( 1) 1 C( m) C( m, m) ( h w) m ( 1) (4) ( m m)( 1) 1 A wd ( 1) m I the followig result we study how is the behavior of the fuctio C(m) give by (4). Theorem. Let us assume that m is a cotiuous variable. (a) If the coditio () is true, the C(m) = C( m,m) is a covex fuctio o the regio characterized by m 1. (b) If () is false, the C(m) = C( m,m) is a cocave ad icreasig fuctio o the regio determied by m 1. Proof. Substitutig (3) ito (4) ad differetiatig C(m) with respect to m, we have Cm ( ) m h w wd m 3 ( 1) 1 ( ) ( 1) ( 1) ( hw) md [( m1) m1] A( 1) wd (5) Note that the above derivative is positive if, ad oly if, the equatio m +bm+c > 0, beig 1 ( 1) ( ) c h w Aw 1 Dh ( h w) b ad. 4( 1) h As the discrimiate of the above equatio is b 4c [8 ( 1) ( )( 1) ] A h w D w Dh ( h w)( 1) (6) We have the followig cases: - If 8A(+1) = (h+w)dτ(1-), the = 0. Thus, for all m 1, we have 1 m bm c m ( ) 1 0 Hece, i this case, the derivative C(m)/ m >0 ad C(m) is a icreasig fuctio o the iterval [1,). - If 8A(+1) < (h+w)dτ(1-), the < 0 ad m +bm+c > 0 always. Therefore, i this case, C(m)/ m >0 ad C(m) is icreasig for all m. - If 8A(+1) > (h+w)dτ(1-), the > 0 ad the equatio m +bm+c = 0 has two real roots. Thus, i this case, the fuctio C(m) is decreasig for m m 0 ad is

6 icreasig for m m 0, beig m 0 the value give by (0). I additio, the secod derivative of C(m) is Cm ( ) D w h w A h w D m ( )[8 ( 1) ( )( 1) ] 4[( hw) md [ m( 1) 1 ] A( 1)] m (7) As m > 0 for all m 1, the the sig of the secod derivative depeds o whether the coditio () is or ot is true. Thus, if 8A(+1) >(h+w)dτ(1-), the that secod derivative is positive ad the fuctio C(m) is covex for all m. Otherwise, the fuctio C(m) is icreasig ad cocave for m 1. Corollary 1. Let us cosider that m is a cotiuous variable. (a) If the coditio () is true, the the miimum poit of the fuctio C(m) = C( m,m) o the regio characterized by m 1 is give by the formula (0). (b) If () is false, the the miimum poit of the fuctio C(m) = C( m,m) o the regio is m=1. Proof. It follows easily from Theorem. Taig ito accout the above results, we ca develop a procedure to obtai the optimal iteger solutio for the problem show i (13), which is exposed i the followig sectio. 4.1 Optimal approach A simple procedure to give a optimal iteger solutio for the ivetory problem is below preseted Algorithm Step 1 Set m = 1. Usig the formula (3), calculate the poit 1 ( h w) D A( 1) wd ( 1) (8) - If 1 1, the choose as iitial solutio (*,m*) = (1,1). Compute C(*,m*) = C(1,1) by usig the formula (1). Go to step 4. - Otherwise, go to step. Step Determie the largest iteger ot greater tha 1 ad the smallest iteger ot less tha 1, that is, the itegers 11= 1 ad 1= 1. Compute the costs C( 11,1) ad C( 1,1). - If C( 11,1) C( 1,1), the choose the value ' 1 = 11. Go to step 3. - Otherwise, cosider ' 1= 1. Go to step 3. Step 3 Set as iitial solutio (*,m*)=(' 1,1) with cost C(*,m*) = C(' 1,1). Go to step 4. Step 4 Set m =. Calculate the poit by usig the formula (3), that is ( h w) D ( 3) A( 1) wd ( 1) (9) Compute the cost C(,m=) by usig the formula (4). Go to step 5. Step 5 If 8A(+1) > (h+w)d(1-), go to step 6. Otherwise, go to step 9. Step 6 Obtai the optimal solutio ( 0,m 0) give by (19) ad (0) for the ivetory problem with cotiuous variables. Calculate the miimum cost C 0 give by (1). Go to step 7. Step 7 If m < m 0, the go to step 10. Otherwise, go to step 9. Step 8 Put m = m+1. Calculate the poit m by usig the formula (3). Determie the cost C( m,m) by usig the formula (4). Go to step 9. Step 9 If C(*,m*) C( m,m), the the optimal solutio is the poit (*,m*) ad the miimal iteger cost is C(*,m*). Stop. Otherwise, go to step 10. Step 10 Determie the largest iteger ot greater tha m ad the smallest iteger ot less tha m, that is, the itegers m1= m ad m= m. Compute the costs C( m1,m) ad C( m,m). - If C( m1,m) C( m,m), the choose the value ' m = m1. Save the poit (' m,m) ad the cost C(' m,m). Go to step Otherwise, choose ' m = m. Keep the poit (' m,m) ad the cost C(' m,m). Go to step 11. Step 11 Compare the costs of the poits (*,m*) ad (' m,m). - If C(*,m*) C(' m,m), the eep the poit (*,m*) ad remove the poit (' m,m). Go to Step 8. - Otherwise, save the poit (' m,m). Set (*,m*) = (' m,m) ad C(*,m*) = C( 'm,m). Bac to Step 8. It is iterestig to compare the solutio (*,m*) obtaied by the above algorithm with the solutio ( 0,m 0) i which o iteger costrait is imposed o the m ad values. Note that C* C 0 always, ad the gap betwee C* ad C 0 gives a measure of the goodess of the iteger solutio. Thus, the ratio = (C*- C 0)/C 0 ca be used as a measure of the approximatio betwee the iteger ad cotiuous solutios. 5. NUMERICAL EXAMPLES I this sectio we preset several umerical examples to illustrate the theoretical results.

7 Example 1. Let us cosider the followig parametric values for a ivetory system with appropriate uits: the basic period is = 1 wee, demad is D = 40 uits, average demad is r = 40 uits per wee, A = $ 600 per order, h = $ 4 per uit per wee, w = $ per uit per wee, ad the demad patter idex = 1/. Firstly, from (8), we calculate the poit 1= The, we cosider the itegers 11 = 1 = 4 ad 1 = 1 = 5. Now, we compute the costs C(4,1) = $ ad C(5,1) = $ As C(4,1) < C(5,1), the iitial solutio is (*,m*) = (4,1). Next, from (9), we calculate = ad compute the cost C(,) = $ The, we chec that the coditio () is satisfied. Followig the algorithm give i Sectio 4, we calculate the values m 0 ad 0 give i (19) ad (0). Thus, we obtai 0 = ad m 0 = As m = is greater tha m 0, i Step 9, we compare C(*,m*) with C(,). Thus, the optimal solutio is C(*,m*) = (4,1) with cost C* = C(4,1) = $ Hece, the schedulig period is T* = 4 wees ad the lot size is Q* = 160 uits. From (1), the miimum cost for the cotiuous ivetory problem is C 0 = $ Hece, the measure of the goodess of the optimal iteger solutio with respect to the cotiuous oe is = It meas that the iteger solutio is very close to the miimum cost of the cotiuous optimal policy. Example. We cosider ow the same iput parameters tha the above example, but oly chagig the idex of the power demad patter to = 3. From (8), we have 1 = Next, we calculate C(4,1) = $ 65; ad C(5,1) = $ 7. As the lower cost correspods to (4,1), the iitial solutio is (*,m*) = (4,1). The, from (9), we obtai = with cost C(,) = $ The coditio () is agai satisfied. Thus, from (19) ad (0), we have 0 = ad m 0 = I this case, the miimum cost of the cotiuous ivetory problem is C 0 = $ As m = > m 0, we compare C(*,m*) = $ 65 with C(,) = $ As C(*,m*) > C(,), the we calculate the costs C(4,) = $ 60 ad C(5,) = $ 5. Keep the poit (5,) with cost C(5,) = $ 5. Now, the ew solutio is (*,m*) = (5,). Next, we set m = 3, calculate 3 = ad the cost C( 3,3) = As C(5,) = $ 5 < C( 3,3), the the optimal iteger solutio is the poit (5,) with cost C(5,) = $ 5. The ivetory cycle is T* = 5 wees ad the lot size is Q* = 00 uits. As ca be see, the cost of the optimal iteger policy is very close to the cotiuous oe. Example 3. Cosider the same iput parameters tha the Example 1, but oly chagig demad to D = 8000 uits. We start calculatig the poit 1 = Next, we calculate the costs C(,1) = $ ad C(3,1) = $ As the lower cost correspods to (,1), that poit is the iitial solutio, that is, (*,m*) = (,1). We set m = ad calculate = with cost C(,) = $ I this case, the coditio () is false. As C(,1) < C(,), the the optimal solutio is the poit (*,m*) = (,1) ad the miimal iteger cost is C(,1) = $ Hece, the ivetory cycle is T* = wees ad the lot size is Q* = uits. 6. CONCLUSIONS AND FUTURE RESEARCH I this paper we develop a ivetory model for a sigle product where the demad over a basic time period follows a power demad patter. The shortages are allowed ad will be filled with the arrival of a ew lot of products. The repleishmet of the ivetory is made every schedulig period, beig that period a multiple of the basic period. Also, the lot size or orderig quatity to repleishmet the ivetory must be equal to a multiple of the demad alog the basic period. We have proposed a approximate approach to calculate the ivetory policy. Thus, we have preseted a algorithmic procedure to determie easily the ivetory policy, that is, the schedulig period, the ecoomic lot size ad the ivetory cost. A future study will icorporate i the proposed model ew assumptios such as deterioratio of the products, repleishmet o-istataeous, ad/or lost sales cost. ACKNOWLEDGMENTS This wor is partially supported by the Spaish Miistry of Sciece ad Iovatio, through the research project MTM P icluded i the Spaish Natioal Pla of Scietific Research, Techological Developmet ad Iovatio. REFERENCES Bahari-Kashai, H. (1989) Repleishmet schedule for deterioratig items with time-proportioal demad. Joural of the Operatioal Research Society, 40, No.1, Bhuia, A.K. ad Maiti, M. (1998) A two-warehouse ivetory model for deterioratig items with liear tred i demad ad shortages. Joural of the Operatioal Research Society, 49, No. 3, Bose, S., Goswami, A. ad Chaudhuri, K.S. (1995) A EOQ Model for Deterioratig Items with Liear Timedepedet Demad Rate ad Shortages Uder Iflatio ad Time Discoutig. Joural of the Operatioal Research Society, 46, No. 6, Chag, H.J. ad Dye, C.Y. (1999) A EOQ model for deterioratig items with time varyig demad ad

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