Retailer s Pricing and Ordering Strategy for Weibull Distribution Deterioration under Trade Credit in Declining Market

Transcription

1 Applid Mathmatial Sins, Vol. 4, 00, no., 0-00 Rtailr s Priing and Ordring Stratgy for Wibull Distribution Dtrioration undr Trad Crdit in Dlining Markt Nita H. Shah and Nidhi Raykundaliya Dpartmnt of Mathmatis, Gujarat Univrsity Ahmdabad , Gujarat, India nitahshah@ gmail.om Abstrat In this rsarh artil, an ordring and priing poliy is formulatd for a rtailr whn th supplir offrs a dlay in paymnts to sttl th aounts against th rtailr s du. Th problm is to study afor said stratgy whn dmand of produt is subjt to dras with tim. Th units in invntory ar loosing its ffiiny with rspt to tim. A dision poliy is skthd to dtrmin th optimal slling pri and th ordring quantity to maximiz th rtailr s profit. Th numrial xampls ar givn to support th dvlopmnt of th mathmatial modl. Th snsitivity analysis of ritial paramtr is arrid out to obsrv th hangs in th dision variabl and profit. Mathmatis Subjt Classifiation: 90B05 Kywords: Finan, Wibull dtrioration, trad rdit, Priing and ordring. Introdution To attrat mor rtailr s supplir uss a promotional tool of offring th rtailr a trad rdit (say) N days to sttls th aount against th purhas mad. Th trad rdit in finan managmnt is trmd as nt N (Brigham (995)). No intrst is hargd if th aount is sttld within this N days. Howvr, if th paymnt is not mad within this rdit priod, thn th intrst is hargd on th rmaining stok in invntory. Goyal (985) xplor th onpt of trad rdit. H omputd intrst arnd on th sals rvnu on unit purhas pri. This onpt followd by Dav (985), Shah (993a, 993b, 993), Aggarwal and Jaggi (995), Jamal t al. (997), Hwang t al (997), Liao t al (000), Chung and Dy (00) and thir itd rfrns whih ignord th fat that th rtailr s slling pri is highr than th unit purhas pri. Thy onlud that th rtailr arn intrst gnratd rvnu by dlaying th paymnt up to last dat of offrd dlay priod tim. Jamal t al. (000) and Sarkr t al. (000) inorporatd th onstant slling pri and th diffrn btwn th unit

2 0 N. H. Shah and N. Raykundaliya sal pri and unit purhas pri. Tng t al. (005) xtndd th abov modl by making unit sll pri as a dision variabl. Tng (003) stablishd that it is onomially advantagous for th rtailr to rplnish ordr of smallr siz mor frquntly and tak bnfits of th prmissibl trad rdit. Chang t l. (003) xtndd Tng s modl by onsidring rdit linkd ordr quantity whn units in invntory ar subjt to th onstant dtrioration. In abov itd rfrns, th dmand is takn to b dtrministi and onstant. Howvr, th dmand of th sasonal produt is drass with tim. In this study dmand is onsidrd to b drasing funtion of tim and sal pri. It is assumd that th unit slling pri is highr than unit ost pri. Th profit is maximizd with rspt to unit sal pri and th yl tim. Th units in invntory ar subjt to dtrioration with tim i.. dtrioration of unit follow two-paramtr Wibull distribution. Numrially it is stablishd that inras in trad rdit lowr unit sal pri and inras th profit. Th snsitivity analysis is arrid out to study th variations in th dision variabls and objtiv funtion.. Notations and Assumptions Th mathmatial modl is dvlopd undr th following notations and assumptions.. Notations: η R() t = a( bt) P ; Whr a > 0 is fixd dmand, b (0 < b < ) is rat of hang of dmand and η > is makup paramtr. C: Th unit purhas ost. P: Th unit sal pri with P > C ( a dision variabl). h: Th invntory holding ost pr unit pr annum xluding intrst hargs. A: Th ordring ost pr ordr. M: Th prmissibl trad rdit offrd by th supplir to th rtailr for sttlmnt of th aount against purhass. I : Th intrst hargd pr montary unit in stoks pr annum by th supplir. I : Th intrst arnd pr montary unit pr yar on gnratd sals rvnu. Not : I > I Q : th ordr quantity (a dision variabl) β θ ( t ) : = αβ t, th dtrioration of units in invntory follows Wibull distribution. Whr α ( 0 α <. ) dnots th sal paramtr and β (> ) dnots th shap paramtr. It is assumd that th dtrioration of units inrass with tim t (> 0). I () t : Th invntory lvl at any instant of tim t, 0 t T. T: Th yl tim (a dision variabl). Z P, T : Th total profit pr tim unit. ( )

3 Rtailr s priing and ordring stratgy 03 Th total profit pr unit tim of a rtailr ompriss of: (a) Sals rvnu: SR, (b) Purhas ost: PC, () invntory holding ost xluding intrst hargs; IHC, (d) ordring ost; OC, () intrst hargd on unsold itms in th stok aftr th prmissibl rdit priod whn M < T; I, and (f) intrst arnd on th gnratd sals rvnu during th prmissibl dlay priod: I.. Assumptions:..Th invntory systm undr onsidration dals with th singl itm only.. Th planning horizon is infinit. 3. Th dmand of th produt is drasing funtion of th tim and th sal pri. 4. Shortags ar allowd and lad-tim is zro. 5. Th units in invntory ar subjt to dtriorat with rspt to tim. Th dtrioration rat follows Wibull distribution. Th dtrioratd units an nithr b rpaird nor rplad during th yl tim. 6. Th rtailr gnrats rvnu by slling th produt. Th gnratd rvnu is dpositd in an intrst arning aount during th allowabl rdit priod. At th nd of this priod, th rtailr sttls th aount for all th units sold kping th diffrn for day-to-day xpnss, and starts paying th intrst hargs on th unsold itms in th invntory. 3. Mathmatial modl Th rtailr s invntory lvl gradually dras du to th tim dpndnt and sal pri and dtrioration of units in invntory. Th instantanous stat of invntory lvl at any instant of tim t during th yl priod [ 0, T ] an b dsribd by th diffrntial quation : di( t) +θ ( t) I() t = R( t,, 0 t T () dt with th initial ondition I ( 0 ) = Q and th boundary ondition ( T ) = 0 Consquntly, th solution of () is givn by I. β t β αt αt I() t = Q R( t, dt () 0 Undr th assumption that α ( 0 α <. ) is vary small, xpanding xponntial sris by nglting α and its highr powrs, th solution () an b writtn as T η β β I( t) = Q a( bt) P ( αt ) dt ( αt ) + (3) 0

4 04 N. H. Shah and N. Raykundaliya 3 3 β + 6T 8bT β 8T β + 56Tβ + 56Tβ 4bα T η + β β + β + β + ap 8bαT + 4α T β + α T β + 6αT β Q = ( + )( + )( + ) β + β + β + 8 β β β + 40αT 8bT 8T β + 8α T + 6αT β + β + β + β + 4bα T β 0bα T β 6bαT β 4bαT β (4) Nxt w omput diffrnt ost involvd in th total profit pr tim unit. PQ A) Sals rvnu; SR pr tim unit is SR = (5) T CQ b) Purhas ost; PC of prouring Q- units pr tim unit is PC = (6) T h T ) Invntory holding ost; IHC pr tim unit is IHC = I() t dt (7) T 0 A d) Ordring ost; OC pr ordr is OC = (8) T Rgarding intrst hargs and arnd, (i.. osts () and (f) stat in stion (.) two ass may aris basd on th lngths of M and T. Viz M T or M > T. Cas: M T. In this as, th rtailr slls R(M) M units during [ 0, M] and has CR(M)M to pay th supplir. For th unsold itms in th stok, th supplir hargs at an intrst rat I during th priod[ M, T ]. Hn, th intrst hargd, IC pr tim unit is CI T IC = I()dt t T (9) M During [0, M], th rtailr slls th units and dposits th rvnu into th intrst arning aount at th rat I pr montary unit pr annum. Hn, th intrst arnd, IE pr tim unit is η + PI M P I 3 IE = R( t, tdt = am bm T 0 T 3 (0) Hn, th rtailr s profit pr tim unit is Z ( = SR PC IHC OC IC + IE () Th nssary onditions for Z( to b optimum is Z ( = 0 () P Z ( And = 0 (3) th obtaind T, P ) = ( T, P ) (say) maximizs th profit Z ( T, ) providd Z( < 0 P ( Z( and < 0 P

5 Rtailr s priing and ordring stratgy 05 Z( Z( (, ) and > 0 Z T P P P Th omplxity of th xprssion in ()-(4) suggsts that it is not asy to gt good losd form for th nssary and suffiint onditions. On an solv () and (3) for ( by a mathmatial softwar. Cas: T M In this snario, th rtailr slls R (T)T- units in all by th nd of th yl tim and has CR(T)T to pay th supplir in full by th nd of th rdit Priod M. Hn th intrst hargs IC = 0 (5) and th intrst arnd pr tim unit is PI T IE = + R( t, tdt R( T ( M T ) T 0 + ai P 3 = T bt + ( bt ) T ( M T ) T 3 (6) Hn, th rtailr s profit pr tim unit is ; Z ( = SR PC OC IHC IC + IE (7) Th optimum valu of T, ) = (, P ) (say) an b obtaind by solving ( Z ( = 0 P (8) Z ( = 0 (9) With th hlp of suitabl numrial softwar. Th obtaind (, ) profit Z ( if (4) holds for ( T, P, Z( ) At T = M, w hav Z M, = Z ( M, ) (0) ( P 4. Computational algorithm To maximiz th profit, th rtailr an go through th following stps: Stp: Tak paramtri valus in propr units. P using q( ) and (3). If M < T as is bst poliy to hav maximum profit; othrwis go to stp3. Stp: Calulat ( ),T P by solving q. (8) and (9). If M > T as givs th maximum profit; ls go to stp 4. Stp3: Comput ( ),T Stp4: Comput P from () or (8). Z( M, or Z ( M, ) is th maximum profit.

6 06 N. H. Shah and N. Raykundaliya Stp5: stop. 5. Numrial Exampls Exampl: For a = 00000, b = 0., η =. 0, h = $.00 / unit / yar, C = $ 0.00 pr unit, A = $ 50 / ordr, I = 0. / $ / yar, I = / $ / yar, α = 0 %, β =., M = 30/ 365 yar, th optimal slling pri P = $ 4.59 pr unit and yl tim T = yar. Th maximum profit pr tim unit is $ 95.5 and optimum purhas quantity is 95 units. For P = $ 4.59/ unit and T = yars. Z Z =.454 = 64.5 and P Z Z Z = >0. P P guarants maximum profit. Th 3D-plot drawn in th rang [35, 55] for P and [0.5,.5] for T xhibits that Z (4.59,0.963) $95. 5 is maximum profit. = Exampl: Considr [ a, b, η, A, C, h, I, M, α, β ] = [ ,0.,.6,50,,,54,0%,60 / 365,0.,. ] in propr units. Th optimal solution is P = $ pr unit and yl tim T = 0.47 yars. Th maximum profit pr tim unit is Z ( P,T ) $ 379 and optimum purhas quantity Q is 460 units. For P = $ pr unit and T = 0.47 yars Z Z = = and P Z Z Z = >0. P P Guarants maximum profit. Th 3D-plot (Fig.) drawn in th rang [45, 90] for P and [0.05,.5] for T xplors that obtaind profit Z (55.78,0.47) = $ 379 is maximum. Using th data of xampl, th snsitivity analysis is arrid out by hanging valus of M, α, β, b from -40%, -0%, 0%, 40%.Th variation in yl tim, slling pri, purhas units and total profit pr tim unit ar xhibitd in tabl.

7 Rtailr s priing and ordring stratgy 07 Tabl: snsitivity analysis % hangs Paramtr T P Q K M α β b It is obsrvd that inrass in dlay priod drass yl tim and slling pri Whil inrass in prourmnt quantity not signifiantly and profit signifiantly. Inras in dtrioration rat α inrass slling pri and profit. Th variation in shap paramtr signifiantly inrass yl tim and drass profit. Th dras in profit is du to mor dtrioration unit with rspt to tim. Th inras in dlining dmand rat dras yl tim, purhas quantity and profit signifiantly. 6. Conlusions Th optimal ordring and priing poliis ar xplord for a rtailr whn units in invntory ar subjt to dtrioration with tim and dmand of a produt is dlining in th markt undr trad rdit offrd by th supplir to sttl th aounts against th purhas mad. Th profit is maximizs. It is stablishd that th rtailr should rplnish smallr ordr and tak advantag of prmissibl dlay paymnts mor frquntly. Th modl an b gnralizd for stohasti dmand, random input, allowing shortags t.

8 08 N. H. Shah and N. Raykundaliya Rfrns [] Aggarwal, S. P. and Jaggi, C. K., 995. Ordring poliis of dtriorating itms undr prmissibl dlay in paymnt. Journal of th Oprational Rsarh Soity, 46 (5), [] Bringham, E. F.995. Fundamntals of finanial Managmnt. Th Drydn Prss, Florida. [3] Chung, H. J and Dy, C.Y., 00. An invntory modl for dtriorating itms with partial baklogging and prmissibl dlay in paymnts. Intrnational Journal of systm sin, 3, [4] Chang, C Ouyang, LY and Tng, JT.003. An EOQ modl for dtriorating itms undr supplir rdits linkd to ordring quantity. Applid Mathmatial Modling; 7; [5] Dav, U., 985. On Eonomi ordr quantity undr onditions of prmissibl dlay in paymnts by Goyal, Journal of th Oprational Rsarh Soity, 36, 069. [6] Goyal, S. K., 985. Eonomi ordr quantity undr onditions of prmissibl dlay in paymnts. Journal of th Oprational Rsarh Soity, 36, [7] Hwang, H. and Shinn, S. W., 997. Rtailr s priing and lot-sizing poliy for xponntially dtriorating produts undr th ondition of prmissibl dlay in paymnts. Computrs and Opration rsarh, 4, [8] Jamal, A. M. M., Sarkr, B. R, Wang, S., 997. An ordring poliy for dtriorating itms with allowabl shortags and prmissibl dlay in paymnt. Journal of th Oprational Rsarh Soity; 48, [9] Jamal, A. M. M., Sarkr, B. R, Wang, S., 000. Optimal paymnt tim for a rtailr undr prmittd dlay of paymnt by th wholsalr. Intrnational Journal of Prodution Eonomis, 66, [0] Liao, H. C., Tsai, C. H. and Su, C.T., 000. An invntory modl with dtriorating itms undr inflation whn a dlay in paymnts is prmissibl. Intrnational Journal of Prodution Eonomis, 63, [] Sarkr, B. R., Jamal, A. M. M., Wang, S., 000. Optimal paymnt tim undr prmissibl dlay for prodution with dtrioration. Prodution planning & Control;, [] Shah, Nita H., 993a. A lot - siz modl for xponntially daying invntory whn dlay in paymnts is prmissibl. CERO, 35(-), 9. [3] Shah, Nita H., 993b. A probabilisti ordr lvl systm whn dlay in paymnts is prmissibl. Journal of th Koran Oprations Rsarh and Managmnt Sin; 8 (), [4] Shah, Nita H., 993. Probabilisti tim shduling modl for xponntially daying invntory whn dlay in paymnts is prmissibl. Intrnational Journal of Prodution Eonomis; 3, 77 8.

9 Rtailr s priing and ordring stratgy 09 [5] Tng, J. T., 00. On th onomi ordr quantity undr onditions of prmissibl dlay in paymnts. Journal of th Oprational Rsarh Soity; 53, [6] Tng, J. T., Chang, C.T and Goyal, S. K., 005. Optimal priing and ordring poliy undr prmissibl dlay in paymnts. Intrnational Journal of Prodution Eonomis; 97, 9. Fig: M T

10 00 N. H. Shah and N. Raykundaliya Fig- M > T Rivd: Novmbr, 009