Optimal Order Policy for Time-Dependent. Deteriorating Items in Response to Temporary. Price Discount Linked to Order Quantity

Transcription

1 Appled Mathematcal Scence, Vol. 7, 01, no. 58, HIKARI Ltd, Optmal Order Polcy for me-dependent Deteroratng Item n Repone to emporary Prce Dcount Lnked to Order uantty R. P. rpath 1 and Shweta Sngh omar 1 Graphc Era Unverty, Dehradun (Uttarakhand), Inda Indo Global College of Engneerng, Abhpur (Punjab), Inda Copyrght 01 R. P. rpath and Shweta Sngh omar. h an open acce artcle dtrbuted under the Creatve Common Attrbuton Lcene, whch permt unretrcted ue, dtrbuton, and reproducton n any medum, provded the orgnal work properly cted. Abtract: In th paper, we dcu the poble effect of a temporary prce dcount offered by a uppler on a retaler replenhment polcy for tmedependent deteroratng tem wth contant demand rate. he optmal ordered quantty of a pecal order polcy for a elected cae obtaned by maxmzng the total cot avng between pecal and regular order for the duraton of the depleton tme. An algorthm exhbted to determne the optmal oluton. Numercal example are ued to llutrate the theoretcal reult. Keyword: Inventory, me-dependent deteroratng tem, emporary prce dcount, Contant demand. 1. INRODUCION In the pat few decade, nventory problem for deteroratng tem have been wdely tuded. Mot of the phycal good deterorate over tme. In realty, ome of the tem ether decayed or deterorated or damaged or vaporzed or affected by ome other factor and are not n a perfect condton to atfy the demand. Food tem, gran, vegetable, frut, drug, pharmaceutcal, radoactve ubtance, fahon good and electronc ubtance are a few example of uch tem n whch uffcent deteroraton can take place durng the normal torage perod of the unt and conequently th lo mut be taken nto account when analyzng the ytem. herefore, the lo due to deteroraton cannot be neglected. he deteroraton rate of nventory n tock durng the torage perod conttute an mportant factor, whch ha attracted the reearcher. Whtn (1957)

2 870 R. P. rpath and Shweta Sngh omar the frt author who tuded an nventory model for fahon good deteroratng at the end of a precrbed torage perod. An exponentally decayng nventory wa developed by Ghare and Schrader (196). Emmon (1968) etablhed a replenhment model for radoactve nuclde generator. Shah and Jawal (1977) etablhed an order-level-nventory model for perhable tem wth a contant rate of deteroraton. he deteroraton occur a oon a the retaler receve the commodte that have aumed n all nventory model for deteroratng tem. A a reult, whle determnng optmal replenhment order, the lo due to deteroraton cannot be avoded. Ghare and Schrader [19] frt etablhed an EO model for exponentally-decayng tem, for whch there contant demand. Later, Covert and Phlp [] extended Ghare and Schrader [19] model and obtaned an EO model for a varable deteroraton rate, by aumng a two-parameter Webull dtrbuton. Shah [17] extended Phlp [8] model and condered the crcumtance n whch hortage allowed. A complete note on nventory lterature for deteroratng nventory model wa gven by Goyal and Gr [0] and Raafat [7]. In addton, when the uppler offer a temporary prce dcount to tmulate demand, ncreae market hare and cah flow, t mportant for the retaler to determne whether or not t advantageou to place a pecal forward buyng order. Several reearcher have tuded temporary prce dcount and propoed varou nventory model to gan deeper nght nto the relatonhp between prce dcount and order polcy. Naddor [4], Barker [0], and erne [] were the earlet reearcher to propoe nventory model n repone to a temporary prce dcount where the pecal ale concde wth the replenhment tme. Followng th, Barker and Vlcam [1], Goyal [9], erne [4], and Slver et al. [6] tuded tuaton where the pecal ale occur durng the retaler ale perod. Ardalan [1, ] developed optmal order polce wth poble combnaton of replenhment tme and ale perod. More reearch tudyng the repone to a temporary ale prce nclude Aull-Hyde [6], Abad [18], Bhaba and Mahmood [8], Lev et al. [],erne and Gengler [5], Wee and Yu [11], Chang and Dye [10], Bhavn [1] and o on. In the above nventory lterature, all reearcher aumed that the prce dcount rate ndependent of the pecal order quantty. However, n practcal bune tuaton, the uppler uually propoe a quantty dcount to encourage larger order. A a reult, the retaler may trade off purchae prce avng agant hgher total carryng cot. A to the reearch conderng quantty dcount, Lal and Staeln [7] preented a model of buyer reacton to eller prcng cheme whch aumed pecal form of dcount prce tructure wth multple buyer and contant demand. Other recent tude related to the quantty dcount recently ncluded Shue [5], Burwell et. al. [], Wee [1], Papachrto and Skour [1], etc. In recent year, many reearcher have gven conderable attenton toward the tuaton where the demand rate dependent on the level of the on-hand nventory. he aumpton of contant demand rate not alway applcable to many nventory tem uch a fahonable clothe, electronc equpment, taty food etc. a they are fluctuated n the demand rate. Demand of a product may vary wth tme or prce or ever wth the ntantaneou level of nventory dplayed n a retal hop. Wth the progre of tme,

3 Optmal order polcy 871 reearcher developed nventory model wth deteroratng tem and tmedependent demand rate. In th area, the work done by varou author (Rtche [5], Deb and Chaudhar [16], Goel and Aggarwal [4]. Inventory model for deteroratng tem wth lnearly trended demand and no hortage were condered by Dave and Patel [], Bahar-Kahan [9], Chung and ng [14] etc. he purpoe of th tudy to develop a decon proce to at retaler n decdng whether to adopt a regular or pecal order polcy. When the pecal order polcy elected at the retaler replenhment tme, the retaler optmal pecal order quantty determned by maxmzng the total cot avng between pecal and regular order durng a pecal order perod. he theoretcal analy conducted and the reult categorzng the optmal oluton.. NOAIONS AND ASSUMPIONS he mathematcal model baed on the followng notaton and aumpton..1 Notaton : A Orderng cot per regular or pecal order c Unt purchang prce D Contant demand rate h Holdng cot rate θ me-dependent deteroraton rate. Order quantty under regular order polcy. Optmal order quantty under regular order polcy. Length of replenhment cycle tme under regular order polcy. Optmal length of replenhment cycle tme under regular order polcy. Specal order quantty at dcount prce, decon varable. Length of depleton tme for the pecal order quantty. I ( t ) Inventory level at tme t when adopt the regular order polcy, 0 t. I ( t ) Inventory level at tme t when the pecal order polcy adopted, 0 t. K ( ) he total cot per unt tme.. Aumpton : (a) he demand rate known and contant. (b) Deteroraton rate tme-dependent. (c) he lead tme zero or neglgble. (d) Shortage are not allowed and the replenhment rate nfnte. (e) here a no replacement or repar for deterorated unt durng the perod under conderaton.

4 87 R. P. rpath and Shweta Sngh omar. MAHEMAICAL FORMULAION In th paper, the deteroraton rate lnear tme varyng and the demand contant. Hence the rate of change of nventory level governed by the followng dfferental equaton: di( t ) = θ ti( t ) D, 0 t (1) dt Gven the boundary condton I ( ) = 0, the oluton of (1) may be repreented by θ θ θt I( t ) = De t, t () hu, the order quantty gven by θ = I(0 ) = D 6 () When the uppler doe not offer the temporary prce dcount, the retaler follow the regular order polcy wth a unt purchang cot c and hence the total cot per order cycle the um of the orderng cot, purchang cot and holdng cot, that A c hc 0 t I( t )dt o remove the dffculty of exponental term, takng aylor expanon of e θ, neglectng hgher order term, we have the total cot per order cycle (denoted by K) 4 θ D Dθ K = A cd hc (4) 6 1 herefore, the total cot per unt tme (denoted by K ( ) ) wthout temporary prce dcount 4 1 θ D Dθ K( ) = A cd hc (5) 6 1 A K ( ) a convex functon of, hence, there ext a unque value of (ay dk( ) ) that mnmze K( ). can be found by olvng the equaton = 0, d.e. atfe the followng equaton: A cdθ D Dθ hc = 0 4 (6) When the optmal length of replenhment cycle tme, obtaned, the optmal order quantty wthout a temporary prce dcount, obtaned a follow:

5 Optmal order polcy 87 θ = D (7) 6 When the uppler offer a temporary prce dcount, the retaler may order a quantty greater than to take advantage of the dcount prce. Alternatvely, the retaler may gnore th dcount and adopt a regular order polcy. Here, we formulate the correpondng total cot avng functon when the pecal order tme occur at the retaler replenhment tme. If the retaler decde to adopt a pecal order polcy and order nventory level at tme t θt θ θt I ( t ) = De t 6 6 and the pecal order quantty θ = I (0 ) = D 6, 0 t S unt, the A the prce dcount rate beng dependent on the pecal order quantty, for the gven prce dcount rate λ, ι the total cot of the pecal order durng the tme nterval [0, ] (denoted by K 1( )) 4 θ ( ) D Dθ K ( ) = A (1 )cd 1 hc 1 λ λ (10) 6 1 On the other hand, f the retaler adopt regular order polcy ntead of placng a large pecal order polcy, then the total cot durng the tme nterval [ 0, ] can be obtaned by ung the average cot approach whch wa aerted by erne [4], and ued by Goyal. he total cot of a regular order durng the tme nterval [ 0, ] (denoted by KN ( ) ) 1 S θ D Dθ KN 1( S) = A cd hc (11) 6 1 Comparng equaton (10) wth (11), for the fxed prce dcount rate λ, the total cot avng can be formulated a follow (denoted by G ( )) G ( ) = KN ( ) - K ( ) S θ D Dθ = A cd hc θ D Dθ A (1 λ )cd (1) 6 ( 1 λ ) hc 1 (8) (9)

6 874 R. P. rpath and Shweta Sngh omar 4. HEOREICAL RESULS In th ecton, the optmal value of that maxmze the total cot avng determned. For the fxed prce dcount rate λ, takng the frt and econd order dervatve of G 1 ( ) n (1) wth repect to, gve dg ( d ) 1 = A cd θ D hc D 1 S θ θ ( 1 λ ) cd 1 ( 1 λ ) S 6 (1) and d G 1 ( S) = (1 λ )cdθ ( 1 λ ) hc[ D Dθ ] < 0 (14) d It can be ealy hown thatg 1 ( ) a concave functon of ; hence, a unque value of (ay 1 ) ext that maxmze G 1 ( ). 1 can be found by dg1 ( ) olvng the equaton = 0. Gven by d 1 y (1 λ )cd = (15) chd(1 λ ) 4 1 θ D D Where y A cd hc > 0 6 θ = 1 o enure < 1 (.e. 1 <, where can be found by olvng equaton (6)), we ubttute (15) n to the nequalty 1 <, and t reult n If δ >, then, (16) 1 0 < 1 1 = y (1 λ )cd[1 Where δ h ] Agan, ubttutng (15) nto (1) the correpondng maxmum total cot avng can be obtaned a h G 1( 1) = ( 1 λ ) cd A (17) It worth placng a pecal order only f G 1 ( 1 )>0. Otherwe the retaler wll adopt the regular order polcy (.e. the order quantty ). Let h δ G 1( 1) = ( 1 λ ) cd A hen from the above argument, we can obtan the optmal value of 1 (denoted by ) a 1

7 Optmal order polcy 875 1,f δ 1> 0 and δ > 0, 1 =,Otherwe (18) o obtan the optmal oluton, we can develop an algorthm a follow: ALGORIHM: Step 1 : Determne from equaton (6). Step : For each λ, = 1,,..., m calculate 1 n (15), δ = y (1 λ )cd[1 h ] Step : Step 4: 1 h and δ = ( 1 λ ) cd A. If δ > 1 0 and δ > 0, then ubttute 1 n to (9) to evaluate the correpondng lot ze, 1 and check 1 under δ. () It x 1 < x 1, 1 a feable oluton. Set 1 = 1 and ubttute = nto (17) to evaluate G ( 1). () It x, we can get a larger prce dcount rate whch greater 1 than λ and thu 1 not a feable oluton. () If 1 < x et 1 = x and ubttute 1 nto (9) to fnd 1. hen, ubttute nto (17) to evaluate G 1 ( ). If G 1 ( 1)>0, go to 1 tep ; otherwe, et =, = and G 1 ( ) = 0. 1 Otherwe, et 1 =, 1 = and G 1 ( 1) = 0. Max Fnd G 1 ( Max 1). If G IK ( S1K ) = G 1 ( 1), = 1,...m = 1,...m the optmal oluton and thu the optmal order quantty alo be determned. Stop S1K S1K can 5. NUMERICAL EXAMPLES o valdate the propoed model, let u conder followng example. Example 1: Conder D=600 R, A=50 per order, c = 0 / unt, h= 1 unt/year, θ = 0.0/year. From the above data, the optmal oluton for the regular order quantty are = and = he prce dcount rate chedule offered by the uppler

8 876 R. P. rpath and Shweta Sngh omar tabulated n able 1. And ung the algorthm, the oluton produce and computatonal reult are hown n able. able 1. Prce dcount rate chedule Clafcaton Specal order quantty Dcount rate λ < < λ able. Optmal Soluton of Example Note: " " denote the maxmum total cot avng. From able, we can ee that the optmal order polcy a follow: =0.47, =61.5 and G = S1 S1 1 S1 G 1 6. CONCLUSION h tudy nvetgated the poble effect of a temporary prce dcount offered by a uppler on a retaler replenhment polcy for tme-dependent deteroratng tem wth demand rate. he purpoe of th tudy to determne the optmal pecal order quantty by maxmzng the total cot avng between pecal and regular order durng the length of depleton tme for the pecal order quantty. By analyzng the total cot avng under pecal and regular order polce, reult were developed to characterze the optmal oluton. Fnally, the reult how that a we ncreae the prce dcount, the total cot avng ncreae. h reearch etablhe an algorthm to determne the optmal oluton and utlze a numercal example to llutrate the theoretcal reult. he propoed model can be extended n everal way. REFERENCES [1] A. Ardalan, Optmal orderng polce n repone to a ale, IIE ranacton, 0 (1988), 9-94.

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