REVISTA INVESTIGACIÓN OPERACIONAL VOL., 33, NO. 3, , 2012.

Transcription

1 REVISA INVESIGACIÓN OPERACIONAL VOL., 33, NO. 3, 33-44,. ORDERING POLICY FOR INVENORY MANAGEMEN WHEN DEMAND IS SOCK- DEPENDEN AND A EMPORARY PRICE DISCOUN IS LINKED O ORDER UANIY Nta H. Shah Department of Mathematc, Gujarat Unverty Ahmedabad 389, Gujarat, Inda ABSRAC In th artcle, the effect of the ale promotonal cheme vz. prce dcount offered by the uppler on the retaler orderng polcy tuded when demand tock-dependent. It aumed that the prce dcount rate depend on the order uantty of the retaler. h tudy wll help the retaler to take the decon whether to adopt a regular or pecal order polcy. he optmum pecal uantty decded by maxmzng the total cot avng between the pecal and regular order for the cycle tme. he algorthm propoed to take the favorable decon. he numercal example are gven to valdate the derved reult. he entvty analy carred out to determne the crtcal nventory parameter. KEYWORDS : Inventory model, tock-dependent demand, a temporary prce dcount, pecal order MSC : 9B5 RESUMEN En el preente artículo, el efecto del plan de promocón de venta, a aber decuento obre el preco ofrecdo por el proveedor, con la polítca de peddo del mnorta, e etuda cuando la demanda e extenca-dependente. Se upone ue la taa de decuento obre el preco de la orden depende de la cantdad del peddo del mnorta. Ete etudo ayudará al mnorta a aceptar la decón de adoptar un período ordnaro o extraordnaro como polítca para comprar. La cantdad epecal óptma e fjada maxmzando el coto total del ahorro entre la orden epecal y la regular para un cclo de tempo. El algortmo propueto determna la decón favorable. Ejemplo numérco on dado para valdar lo reultado. El anál de enbldad e lleva a cabo para determnar lo parámetro crítco óptmo del nventaro.. INRODUCION he offer of prce dcount by the uppler boot the demand, attract more retaler, ncreae the cah-flow to reduce h nventory. But then the ueton I t alway advantageou to aval of dcount for a pecal order? at the retaler end. Dxt and Shah (5) gave a revew artcle on nventory model when a temporary prce dcount offered by the uppler to the retaler to tudy the relatonhp between prce dcount and order polcy. he all-unt uantty dcount orderng polcy dcued by Arcelu et al. (3), Shah et al. (5), Bhaba and Mahmood (6), Abad (7), Dye et al. (7), Shah et al. (8), Mhra and Shah (9). hey aumed that the prce dcount rate ndependent of the pecal order uantty. However, n market, t oberved that the uppler offer a uantty dcount to encourage larger order. For the larger order, the hgher prce dcount rate gven by the uppler. A a reult, the retaler ha to ettle the trade-off for purchae prce avng agant hgher total holdng cot. ntahhah@gmal.com 33

2 Ouyang et al. (9) dcued the effect of a temporary prce dcount on a retaler orderng polcy by aumng that the prce dcount rate lnked to pecal order uantty. hey aumed that the demand contant and determntc. In th paper, we tudy the mpact of a temporary prce dcount on a retaler orderng polcy when demand tock-dependent and prce dcount rate lnked to pecal order uantty. h tudy wll help the retaler to take decon about adoptng or declnng the ale promoton tool. he retaler optmal pecal order uantty obtaned by maxmzng the total cot avng between the pecal and regular order durng a pecal order cycle tme. wo cenaro are dcued : () the pecal order tme occur at the retaler replenhment tme, and () the pecal order tme occur durng the retaler regular cycle tme. An algorthm derved to compute the optmal oluton. he numercal example are gven to valdate the theoretcal reult. he entvty analy of the optmal oluton carred out wth repect to the model parameter. he manageral ue are derved.. NOAIONS AND ASSUMPIONS he followng notaton and aumpton are ued n th artcle : Notaton R(I(t)) : (= α + βi(t)), tock-dependent demand rate where α > cale demand and < β < tock-dependent parameter C : Unt purchang cot A : Orderng cot per regular or pecal order : Holdng cot rate per annum : Order uantty under regular polcy : Optmal order uantty when regular order polcy adopted : Cycle tme when regular order polcy adopted : Optmal cycle tme for ung a regular order polcy : Specal order uantty at dcounted prce (a decon varable) : Cycle tme for the pecal order uantty : Inventory level before the arrval of the pecal order uantty; t : Cycle tme when unt deplete to zero W : Cycle tme for depleton of the nventory level W = + I(t) : Inventory level at tme t when the regular order polcy adopted; t I (t) : Inventory level at tme t when the pecal order polcy adopted; t I W (t) : Inventory level at tme t when the pecal order polcy adopted; t W where W = + Aumpton (a) he nventory ytem under conderaton deal wth a ngle tem. (b) he uppler offer the retaler a temporary prce dcount f the order uantty larger than the regular order uantty. he dcount rate depend on the uantty ordered and the dcount chedule a follow : Cla Specal order uantty Dcount rate d d 3 n d n n n 34

3 where k the k-th dcount rate breakng pont, k =,,, n+ and n ; d k the prce dcount rate offered by the uppler when the retaler order uantty belong to the nterval [ k, k+ ) and n d d d. (c ) he prce dcount not paed on to the cutomer. Only one tme prce dcount offered. (d) he replenhment rate nfnte. (e) he lead-tme zero and hortage are not allowed. Mathematcal Model he am of the tudy to decde the advantage of temporary prce dcount for larger order than the regular order, under the aumpton of the tock-dependent demand. If the retaler adopt to follow regular order polcy wthout a temporary prce dcount, then the nventory deplete n the retaler nventory ytem due to the tock-dependent demand. he change n nventory level governed by the dfferental euaton di() t ( I( t)), t () dt Wth boundary condton I() =, the oluton of () I( t) exp( ( t )), t () Hence, the order uantty I() exp( ) (3) In th cae, the retaler total cot per order cycle A C C I() t dt..e. C C A exp( ) exp( ) (4) herefore, the total cot per unt tme wthout temporary prce dcount C C C( ) A exp( ) exp( ) (5) he convexty of C() can be proved a gven n Dye et al. (7). It guarantee that there ext unue value of (ay) that mnmze C(). can be obtaned by ettng dc( ) C( ) A exp( ) exp( ) d (6) Knowng the regular order cycle tme, the optmal order uantty wthout a temporary prce dcount, can be obtaned a exp( ) (7) 35

4 he followng two cenaro may are when the uppler offer a temporary prce dcount and the retaler take advantage of th by orderng uantty greater than : () when the pecal order tme occur at the retaler cycle tme; and () when the pecal order tme occur durng the retaler cycle tme. Next, we formulate the correpondng total cot avng for thee two cenaro. Scenaro : When the pecal order tme occur at the retaler cycle tme (Fgure ) Fgure Specal order tme occur at the retaler cycle tme Here, f the retaler decde to gve order of pecal uantty unt, argung a above, the nventory level at tme t I( t) exp( ( t )), t (8) and the pecal order uantty I () exp( ) (9) For each prce dcount rate d, the total cot CS ( ) of the pecal order durng the tme nterval [, ] C( d ) C( d ) CS ( ) A exp( ) exp( ), =,,,n () On the other hand, f the retaler decde to follow regular order polcy of - unt ntead of puttng a large pecal order, the total cot durng [, ] 36

5 C C CN ( ) A exp( ) exp( ) () Obvouly, CN ( ) > CS ( ) for gven d. Hence, the total cot avng, G ( ) becaue of the offer of a temporary prce dcount G ( ) CN ( ) CS ( ) () Scenaro : When the pecal order tme occur durng the retaler cycle tme (Fgure ) Fgure Specal order tme occur durng the retaler cycle tme We want to analyze the tuaton when the pecal order tme occur durng the retaler cycle tme. Here, the retaler ha unt and order for unt whch rae h nventory to W = +. When the pecal order of - unt placed, the total cot durng the nterval [, W ] compre of orderng cot; A, purchae cot a C( d ) exp( ) and the holdng cot whch calculated a follow: Wth the arrval of pecal order uantte, the tock on hand ncreae ntantaneouly from to W, where W exp( ) exp( t) exp( ) exp( t ) (3) he nventory level at any ntant of tme t gven by IW( t) exp( ( W t )), t W (4) and W I () exp( ) (5) W W 37

6 From (3) and (5), we get W ln exp( ) exp( t ) (6) C he holdng cot of unt purchaed at $ C per unt exp( t ) t and that of the pecal order uantty avalable at $ C(-d ) per unt W C( d ) I ( t) dt exp( t ) t C( d ) = W exp( ) exp( t ) ln(exp( ) exp( t ) ) (exp( t ) t ) (7) Hence, the total holdng cot of W unt durng the tme nterval [, W ] C( d ) Cd exp( ) exp( t ) ln(exp( ) exp( t ) ) (exp( t ) t ) herefore, for the fxed prce dcount rate d, the total cot CS ( ) of the pecal order durng the tme nterval [, W ] (8) C( d ) Cd CS ( ) A exp( ) (exp( t ) t ) C( d ) + exp( ) exp( t ) ln(exp( ) exp( t ) ) (9) If the retaler doe not opt for the prce dcount and contnue to follow regular order polcy, the total cot durng the nterval [, W ] wll be computed for two perod. In the frt perod, he ncur the holdng cot for unt a t C C I( t) dt exp( t ) t and n the next perod total cot a ( W t) C C A exp( ) exp( ) (ln(exp( ) exp( t ) ) t ) C C A exp( ) exp( ) Hence, the total cot of the nventory ytem 38

7 C CN ( ) exp( t ) t (ln(exp( ) exp( t ) ) t ) C C + A exp( ) exp( ) () From (9) and (), for a fxed dcount rate d, the total cot avng; G ( ) G ( ) CN ( ) CS ( ) () WLOG, we aume that the total cot avng n () and () are both potve for pecal order polcy. 3. ANALYIC RESULS In th ecton, we wll determne the optmal value of that maxmze the total cot avng. Scenaro : When the pecal order tme occur at the retaler cycle tme For the fxed dcount rate d, the dervatve of G ( ) n () wth repect to gve dg ( ) C C A exp( ) exp( ) d C( d ) C( d ) exp( ) exp( ) () and d G d ( ) C( d )( ) exp( ) (3) E. (3) prove that G ( ) a concave functon of. Hence, a unue value of maxmze G ( ). Euatng () to be zero gve value of a (ay) ext that C( d ) ln x C( d ) ( ) (4) where C C x A exp( ) exp( ). Clearly, f and only f..e. (5) where C( d ) x C( d ) exp( ) exp( ) Ung (4) n () gve the correpondng maxmum total cot avng a C( d) ( ) G ( ) exp( ) exp A (6) 39

8 G ( ). he retaler opt for pecal order only f Denote. Otherwe, he wll contnue wth the regular order polcy of - unt. Hence, the optmal value of (denoted by for cenaro, f and (7), otherwe Scenaro : When the pecal order tme occur durng the retaler cycle tme For the fxed prce dcount rate d, ettng the frt order dervatve of G ( ) n () wth repect to to be zero gve dg ( ) ( ) exp( ) ( )( ) C d C d x exp( ) (8) d exp( ) exp( t ) (ay) x C( d ) ( )( ) exp( ) ln C d C( d )( ) he econd order dervatve (9) d G ( ) C( d)( ) d t exp( ) exp( ) exp( ) G ( ) maxmum. Next to enure that guarantee that.e., ubttute (9) nto th neualty whch reult n f and only f 3 (3) where x C( d)( ) exp( ) exp( t ) C( d) 3 Ung (9) nto () gve the correpondng maxmum total cot avng a G ( ) C( d )( ) ln(exp( ) exp( t ) ) t (exp( ) exp( t ) ) Clearly, 4 - C( d )( ) (exp( ) ) A G ( ) to ualfy for pecal order otherwe retaler hould follow the regular order polcy. Hence, the optmal value of (denoted by, f and 3 4, otherwe ) for cenaro Next we outlne computatonal procedure to obtan the optmal cycle tme for the two cenaro. (3) (3) and the optmal pecal order uantty 4

9 Computatonal Procedure Step. If =, then compute and go to tep. Otherwe calculate t from t ln, and go to tep 4. Step. For each d, =,,, n obtan from (4), from (5) and from (6). If and then ubttute nto (9) and obtan. Check under d. If a feable oluton. Set and compute G ( )., then, then larger prce dcount rate poble and thu G ( ). () () not a feable oluton. Set () then et. Subttute nto (9) and fnd and hence computeg ( ). If G ( ) >, go to tep 3; otherwe et, and G ( ) =. Step 3. Fnd max,,..., n G ( ). Go to tep 6. Step 4. For each d, =,,, n obtan from (9), 3 and 4. If 3 nto (3) and obtan, then () () G ( ). and 4 then ubttute. Check under d. If a feable oluton. Set and computeg ( )., then larger prce dcount rate poble and thu not a feable oluton. Set () then et. Subttute nto (9) and fnd and hence computeg ( ). If G ( ) >, go to tep 5; otherwe et, and G ( ) =. Step 5. Fnd max,,..., n G ( ). Go to tep 6. Step 6. Stop. In the next ecton, numercal example are preented to valdate the propoed problem. 4. NUMERICAL EXAMPLES Example Conder the followng parametrc value for the retaler nventory ytem when the pecal order due at the regular order cycle tme : C = $ / unt, α = unt / year, A = $ 5 / order, = 3 % per annum, β = %. Ung tep of computatonal procedure, the optmum cycle tme =.7 year and regular order uantty 75 unt per order. he prce dcount rate offered by the uppler gven n able. Cla I able. Prce dcount rate chedule Specal order uantty 5 % % 8 % Ung tep and 3, the oluton obtaned a gven n able. Dcount rate d 4

10 Shaded oluton the optmal oluton. able. Optmal oluton for Example G d % % % From able, t oberved that the retaler ave $ 7.4 by orderng 4 unt avalable at the dcount rate 8 %. Example. Conder the data a gven n example except for. Here, we want to valdate cenaro when the pecal order tme durng the retaler cycle tme. he optmal orderng polce for = 5, and are gven n able 3. able 3. Optmal oluton for Example for dfferent value of G From able 3, t can be een that the total cot avng negatvely very entve to the remnant nventory. It drect the logtc manager to keep remnant nventory a low a poble when the pecal order tme occur durng the cycle tme. Example 3. In able 4, we tudy the effect of change n the nventory parameter C, α, A, and β on the optmal prce dcount rate, pecal order uantty and total cot avng. he data taken a that of Example and = 5. able 4 Sentvty analy d Parameter Value G C A β he cloe look on able 4 gve followng manageral nght : 4

11 () he retaler wll fnd the optmal pecal order uantty by determnng the advantage of the prce dcount compared to addtonal holdng cot, he wll have to ncur. For example, for = 5 or β = 5 %, the retalor adopt the regular order polcy. Alo, for the optmal order uantty, the retaler not only maxmze h total cot avng but alo the prce dcount rate. () Increae n cale parameter of demand; and orderng cot: A, ncreae total cot avng. h ugget that when demand and orderng cot are lkely to ncreae, t advantageou for the retaler to take the advantage of pecal order uantty at the dcounted rate. (3) Increae n holdng charge fracton decreae pecal order uantty and total cot avng. he oppote effect oberved when the tock-dependent parameter condered. 4. CONCLUSIONS he retaler orderng polcy analyzed when a uppler offer a temporary prce dcount lnked to order uantty and demand tock-dependent. he optmal polcy of pecal order determned to maxmze the total cot avng. A decon makng algorthm propoed to fnd the optmal oluton. he theoretcal reult are valdated by numercal example. A entvty analy carred out to determne mot crtcal nventory parameter. Eventually, the reult dvulge that () the pecal order uantty hould be calculated by takng the dfference between the total cot when the retaler aval / not aval of a temporary prce dcount, () retaler hould keep remnant nventory a low a poble, (3) to aval of the offer of a temporary prce dcount advantageou when the unt prce, market demand and orderng cot are lkely to ncreae. hu decon polcy provde a buldng block to the retaler n order to urvve n the compettve market. he developed model can be tuded to compare varou promotonal cheme offered by the uppler. he model can be analyzed when retaler pae the part od prce dcount to h cutomer. he model can alo be tuded for dfferent demand functon. RECEIVED APRIL, REVISED MAY, REFERENCES ABAD, P. L. (7): Buyer repone to a temporary prce reducton ncorporatng freght cot. European Journal of Operatonal Reearch, 8, ARCELUS, F. SHAH, NIA H. and SRINIVASAN, G., (3):. Retaler prcng and credt and nventory polce for deteroratng tem n repone to temporary prce/credt ncentve. Internatonal Journal of Producton Economc, 8 8, BHABA, R. S. and MAHMOOD, A. K., (6): Optmal orderng polcy n repone to a dcount offer. Internatonal Journal of Producton Economc,, DIXI, V. N. and SHAH, NIA, H. (5): Prce dcount tratege : A revew. Revta Invetgacón Operaconal, 6, DYE, C. Y., CHANG, H. J. and WU, C. H., (7): Purchae-nventory decon model for deteroratng tem wth a temporary ale prce. Internatonal Jouranl of Informaton and Management Scence, 8, OUYANG, L. Y., YANG, C.. and YEN, H. F., (9); Optmal order polcy for deteroratng tem n repone to temporary prce dcount lnked to order uantty. amkang Journal of Mathematc, 4, PANDEY P. and SHAH, NIA H., (9): Webull dtrbuted deteroraton of unt wth alvage value nventory model for a temporary prce dcount. ASOR Bulletn, 8, 9. 8 SHAH, B. J., SHAH, NIA H. and SHAH, Y. K., (5):. EO model for tme-dependent deteroraton rate wth a temporary prce dcount. Aa-Pacfc Journal of Operatonal Reearch,,

12 9 SHAH, NIA H., SHAH, B. J. and WEE, H. M., (9): A lot-ze model for webull dtrbuted deteroraton rate wth dcounted ellng prce and tock-dependent demand. Internatonal Journal of Data Analy echnue and Stratege,,