On the optimality of a general production lot size inventory model with variable parameters

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1 On th optimality of a gnral production lot siz invntory modl with variabl paramtrs ZAID.. BALKHI Dpartmnt of Statistics & Oprations Rsarch Collg of Scinc, King Saud Univrsity P.O. Box 55,Riyadh 5 SAUDI ARABIA. ztbalkhi@ksu.du.sa, Abstract: A gnral production lot siz dynamic invntory modl with dtriorating itms for which th rats of dmand, production, dtrioration as wll as th cost paramtrs ar arbitrary and known functions of tim is considrd in this papr. Shortags ar allowd but ar partially backordrd. Both inflation and tim valu of mony ar takn into account. h objctiv is to minimiz th total nt invntory cost. h rlvant modl is built, solvd and som main rsults about th uniqunss and th global optimality of this solution, with th us of rigorous mathmatical mthods, ar obtaind. An illustrativ xampl is providd Kywords: Production lot siz, Invntory, Variabl paramtrs, Optimality. Introduction According to Nahmias Book (Production and Oprations Analysis (997, th invstmnt in invntoris in th Unitd Stats hld in th manufacturing, wholsal and rtail sctors during th first quartr of 995 was stimatd to b $.5 trillion. hrfor, thr is a grat nd to prform spcial rsarch on invntory control managmnt for th giant systms, in ordr to improv thir fficincy and prformanc in such a way that th total of rlvant invntory costs is minimizd. Applying such rsarch rsults ar xpctd to sav hug amounts of mony that can b usd for dvlopmnt, as it is th cas in most first class countris. An intrsting problm, rlatd to on of th assumptions of th classical Economic Production Quantity (EPQ modls, has rcivd som attntion in th litratur. It concrns with th various unit costs involvd,which ar assumd to b known and constant. Among ths Rsh t al [6], Hong t al [], Chang [], Lan t al [5], Chui []. Sugapriya and Jyaraman [8] considrd th variability of th holding cost in a non-instantanous dtrioration undr a production invntory policy. Balkhi [9] conductd anothr study in which h tratd th variability of paramtrs for an invntory modl for dtriorating itms undr trad crdit policy with partial backordring and an infinit tim horizon. Anothr form of variability in paramtrs of an EPQ invntory modl is th larning phnomnon, has bn considrd by Balkhi[5]. Alamri and Balkhi[] considrd forgtting phnomnon along with larning.darwish [] gnralizd th classical EPQ modl by studying th rlation btwn stup cost and lngth of th production cycl. An invntory modl in which products dtriorat at a constant rat and in which dmand and production rats ar allowd to vary with tim has bn introducd by Balkhi and Bnkhrouf []. Subsquntly, Balkhi [], [], [6], and Balkhi t al [7] hav introducd svral invntory modls in ach of which, th dmand, production, and dtrioration rats ar arbitrary functions of tims,and in som of which, shortags ar allowd but ar compltly backloggd. In ach of th last mntiond paprs, closd forms of th total invntory cost wr stablishd, a solution procdur was introducd and th conditions that guarant th optimality of th solution for th considrd invntory systm wr introducd. hough som of th abov ISSN: ISBN:

2 mntiond paprs don not account for dtrioration, th importanc of itms dtriorating in invntory modling in now widly acknowldgd, as shown by th rcnt survy of Goyal and Giri [] h goal of this papr is to gnraliz th modls of ths paprs in various ways. First, th costs in our modl ar gnral functions of tim instad of bing linar functions or constant. Scond, th dmand rat is assumd to b a gnral function of tim. hird, shortags ar allowd, so that part of ths shortags is lost and th rst ar backloggd. Fourth,th on-hand invntoris dtriorat and th dtrioration rat is a gnral function of tim instad of bing a linar function or constant. Fifth, w considr th ffct inflation and th tim valu of mony to on th total cost. W dal in this papr with a vry gnral (EPQ invntory modl of which many of th modls availabl in th litratur ar spcial cass. Our analysis dpns, broadns, and nrichs th availabl thortical studis, in particular th mathmatical rsults rlatd to (EPQ invntory modls, and prsnts practical xampls to show how to implmnt th solution mthodology to ral problms. Assumptions and notations h invntory modl assumptions ar:. A singl itm is producd at th bginning of th cycl and hld in stock.. Shortags ar allowd but only a fraction of th stock out is backordrd and th rst ar lost.. All costs ar affctd by inflation rat and tim valu of mony. W shall dnot by r th inflation rat and by r th discount rat rprsnting th tim valu of mony so that r = r r is th discount rat nt of inflation h paramtrs of th modl ar gnral functions of tim and ar dnotd as follows: : Dmand rat at tim t. : Production rat at tim t. θ(: Dtrioration rat at tim t. I(: Invntory lvl at tim t. c(: Itm production cost at tim t. h(: Holding cost pr unit pr unit of tim at tim t. b(: Shortag cost pr unit pr unit of tim at tim t for backordrd itms. l(: Shortag cost pr unit pr unit of tim at tim t for lost itms. k(: Stup cost at tim t. τ β ( τ =, is th rat of backordrd itms and τ = t is th waiting tim up to th nw production whr shortags start to b backloggd. Not that, β (τ is a dcrasing function of τ, which rflcts th fact that lss waiting tim implis mor backordrd itms. h proposd invntory systm oprats as follows. h cycl starts at tim t = and th invntory accumulats at a rat -- θ(i( up to tim t= whr th production stops. Aftr that, th invntory lvl starts to dcras du to dmand and dtrioration at a rat - θ(i( up to tim t =, whr shortags start to accumulat at a rat β (τ up to tim t=.production rstarts again at tim t = and nds at tim t = with a rat - to rcovr both th prvious shortags in th priod [, ] and to satisfy dmand in th priod [, ]. h procss is rpatd. In this rspct and in ordr to rcovr th backordrd itms within th priod [, ] and to satisfis th dmand in th priod [, ] w rquir that > [+ β ( ] = [+β ( τ ] Problm Formulation h changs of th invntory lvl I( is govrnd by th following diffrntial quation = θ ( I( ; t ( = θ ( I( ; t ( = β ( τ ; t ( ISSN: ISBN:

3 = ; t ( h solutions of th abov diffrntial quations ar g ( g ( u I ( { u u} du, t (5 = t g( g ( u I ( = u du ; t (6 t t I ( = β ( τ ; t (7 I ( = { u u} du, t (8 t Rspctivly, whr g( = θ( g( = ( θ Nxt, w driv th prsnt worth for ach typ of cost Prsnt worth of holding cost (PWHC. Itms ar hld in stock in th two priods [, ] and [, ].so w hav PWHC = [ H ( H ( ]{ } [ H ( H ( ] H ( = Whr g( r ( g ( g ( t d t+ (9 h( ( Prsnt worth of shortag cost for backordrd itms (PWSCB. Shortags occur ovr two priods, [, ] and [, ]. PWSCB= [ B( B( [ B( B( ] β ( ]{ } + ( -rt B ( = b( ( Prsnt worth of storag cost for lost itms (PWSCL. In a small tim priod (d w los a fraction[ β ( τ ] D (, hnc: PWSCL = l( [ β ( ] ( Prsnt worth of itm production cost (PWPC. Sinc production occurs during th two priods [, ] and [, ], w hav: PWPC = { c( + c( } ( Not that th last cost includs both consumd and dtrioratd itms. Prsnt worth of th st-up cost (PWSUC.h st-up of nw production occurs twic during any cycl, th first is at t =, and th scond is at t =. hrfor, th prsnt worth of th stup cost r ( r PWSUC = k + k( ( r = k( k( + (5 Hnc, th total rlvant cost pr unit tim as a function of,,, which w shall dnot by CU (,,, is givn by CU = { PWHC+ PWSCB+ PWSCL+ PWPC+ PWSUC} (6 Our problm is to find th optimal valus of,,, that minimiz CU (,,, givn by (6 subjct to th following constraint: < < < < (7 ( [ ] = g ( t g ( : = C β (8 C (9 Not that constraint (7 is a natural constraint sinc othrwis our problm would hav no maning. Constraint (8 coms from th fact ISSN: ISBN:

4 that, th invntory lvls givn by (7 & (8 must b qual at t= whras constraint (9 coms from th fact that th invntory lvls givn by (5 & (6 must b qual at t=. hus, our problm (call it (P is givn by Minimiz CU (,,, subjct to (7, (8 & (9 (P Problm Solution o solv problm (P, w first ignor (7. his can b justifid by th rasons that; if (7 dos not hold, thn th whol problm has no maning. Howvr, w shall not considr any solutions that do not satisfy (7. Our nw problm (call it P is: Minimiz CU (,,, subjct to (8 & (9 (P Not that (P is an optimization problm with two quality constraints, so it can b solvd by th Lagrang chniqus. Lt L(,,,, λ,λ b our Lagrangan thn: L (,,,, λ,λ = CU(,,, + λ C+ λc ( h ncssary conditions for having optima ar: dl dl dl dl =, =, =, =, d d d d dl dl =, = ( dλ dλ Now, from (6, (8 & (9 w hav: dl r g ( = c( ( ( P + λ P = d r g ( c( λ = ( dl g ( = [ H ( H ( ] d [ B ( r l ( B ( ] β ( [ β ( ] g ( λ β ( λ = ( dl d = r rk( l( β ( β ( [ B( B( ] + c( + r + λ[ ( + ] = β ( k'( r D ( dl o facilitat computations of,lt d W CU =,thn from(6,(8&(9w hav dw w dl d dc dc = + λ + λ = d d d Which givs: w CU = = [ B( B( ][ ] + r c( λ [ ] (5 Whr w is to b takn from (6. Not that, (5 givs th minimum total cost in trms of λ, &. h quations (8,(9,(,(,(,(5, ar 6 quations with 6 variabls Namly,,,, λ,λ so that th solution of ths quations ( if xists givs th critical points of L(,,,, λ,λ from which (,,, is th corrsponding critical point of CU(,,,. 5 Optimality of Solutions In this sction, w driv conditions that guarant th xistnc, uniqunss, and global optimality of solution to problm (P For that purpos, w first stablish sufficint conditions undr which th Hssian matrix of th ISSN: ISBN:

5 Lagrangan function L (,,,, calculatd at any critical point (,,, of L, is positiv dfinit. o comput th Hssian matrix of L w considr th L L following notations L =, = L i i, j i i j i,j=,,, hn th rlatd computations showd that L (,,, has th following form L L L L L L L (,,, = L L L L L L By Balkhi and Bbkhrouf [] and Stwart [7],this symmtric matrix is positiv smi-dfinit if (6 (7 (8 (9 hus, th abov argumnts lad to th following thorm. horm. Any xisting solution of (P is a minimizing solution to (P if this solution satisfis (6 through (9. Nxt w show that any minimizing solution of (P is uniqu. o s this, w not, from (8,(9,(,(,(,(5 that ach of,, can implicitly b dtrmind as a, function of (,.say = f (, = f (, = f (, = f (.Our argumnt in showing th uniqunss of th solution is basd on th ida that th gnral valu of CU givn by (6 must coincid with th minimum valu ofcu givn by (8.hat is W( f (, f (, f ( / f -, f (, f (, f ( ( CU(, = ( Whr W(, f (, f (, f ( / f ( is takn from (5 and CU(, f (, f (, f ( is takn from (6. Not that any minimizing solution of (P (if it xists is uniqu (hnc global minimum if quation (, as an quation of, has a uniqu solution. his fact has bn shown by Balkhi([],[],[5] and [8]. Hnc, th abov argumnts lad to th following thorm horm. Any xisting solution of (P for which (6 through (9 hold, is th uniqu global optimal solution to (P. 6 Illustrativ Exampl W hav vrifid our modl by th following illustrativ xampl D ( = at+ a, θ θ ( =, k ( = k, k ( = k C ct lt l( = l, ( = c, h ht ( = h, P pt ( = p, b ( t bt ( = b, β ( = h numrical rsults ar shown in abl- which consists from two parts.h first part displays th valus of th modl paramtrs for fiv diffrnt sts h scond part shows th optimal solution that corrspond to ths valus. h obtaind rsults show that th nw invntory modl is practically applicabl. 7 Conclusion In this papr, w hav considrd a gnral production lot siz invntory modl in which ach of th dmand, production, and dtrioration as wll as all cost paramtrs ar known and gnral functions of tim. Shortags ar allowd but ar partially backordrd. h objctiv is to minimiz th ovrall total rlvant invntory cost. W hav built an xact mathmatical modl and introducd a solution procdur by which w could dtrmin th optimal stopping and ISSN: ISBN:

6 rstarting production tims in any cycl. hn, quit simpl and fasibl sufficint conditions that guarant th uniqunss and global optimality of th obtaind solution ar stablishd. An illustrativ xampl which xplains th applicability of th thortical rsults ar also introducd and numrically vrifid.most of prviously rlatd modls that hav bn introducd by prvious authors ar spcial cass of our modl. his sms to b th first tim whr such a gnral (EPQ is mathmatically tratd and numrically vrifid. Acknowldgmnt. h author would lik to xprss his thanks to th Rsarch Cntr in th Collg of Scinc in King Saud Univrsity for its financial support of this rsarch undr projct Numbr Rfrncs []A.A. Alamri, Z.. Balkhi, h Effcts of Larning and Forgtting on th Optimal Production Lot Siz for Dtriorating Itms with im Varying Dmand and dtrioration Rats, Intrnational Journal of Production Economics, Vol. 7, pp. 5-8, 7. [] Z. Balkhi, On th global optimal solution to an intgrating invntory systm with gnral tim varying dmand, production, and dtrioration rats. Europan Journal of Oprational Rsarch,, 996, pp [] Z. Balkhi, Z. and L. Bnkhrouf, A production lot siz invntory modl for dtriorating itms and arbitrary production and dmand rats, Europan Journal of Oprational Rsarch 9, 996, pp. -9. [] Z. Balkhi, On th Global Optimality of a Gnral Dtrministic Production Lot Siz Invntory Modl for Dtriorating Itms, Blgian Journal of Oprational Rsarch, Statistics and Computr Scinc, 8,, 998, pp. -. [5] Balkhi, Z. (. h ffcts of larning on th optimal production lot siz for dtriorating and partially backordrd itms with tim varying dmand and dtrioration rats. Journal o Applid Mathmatical Modling 7, [6] Z. Balkhi, Viwpoint on th optimal production stopping and rstarting tims for an EOQ with dtriorating itms. Journal of oprational Rsarch Socity, 5,, pp [7] Z. Balkhi, C. Goyal and L. Giri, Viwpoint, Som nots on th optimal production stopping and rstarting tims for an EOQ modl with dtriorating itms, Journal of Oprational Rsarch Socity, 5,, pp. -. [8] Z. Balkhi. An optimal solution of a gnral lot siz invntory modl with dtrioratd and imprfct products, taking into account inflation and tim valu of mony, Intrnational Journal of Systm Scinc, 5,, pp [9]Z.. Balkhi, On th Optimality of a Variabl Paramtrs Invntory Modl for Dtriorating itms Undr rad Crdit Policy, Procdings of th th WSEAS Intrnational Confrnc on Applid Mathmatics, pp.8-9. [] H. C. Chang, A Not on th EPQ Modl with Shortags and Variabl Lad im, Information and Managmnt Scincs, Vol.5, pp.6-67,. []Y.S. P. Chiu, h Effct of Srvic Lvl Constraint on EPQ Modl with Random Dfctiv Rat, Mathmatical Problms in Enginring, Articl ID 985, pp., 6. []M. Darwish, EPQ Modls with Varying Stup Cost, Intrnational Journal of Production Economics, Vol., pp. 97-6, 8. []S.K. Goyal and B.C. Giri, Rcnt trnds in modling of dtriorating invntory. Europan Journal of Oprational Rsarch Vol.,pp-6,. [] J.D. Hong, R.R. Standroporty, and J.C. Hayya. On production policis for linarly incrasing dmand and finit production rat, Computational and Industrial Enginring,,99, pp -5. [5]C. H. Lan, Y. C. Yu, H.J. Lin, C. ung, C. P. Yn, P. Dng.A Not on th Improvd Algbraic Mthod for th EPQ Modl with ISSN: ISBN:

7 Stochastic Lad im, Information and Managmnt Scincs, Vol.8, pp.9-96, 7. [6] M. Rsh, M. Fridman, and L.C. Barbosa. On a gnral solution of a dtrministic lot siz problm with tim proportional dmand, Oprations Rsarch,, 976, pp [7] G. W. Stwart, Introduction to matrix computations, Acadmic Prss, 97 [8]C. Sugapriya, K. Jyaraman. An EPQ Modl for Non-Instantanous Dtriorating Itms in Which Holding Cost Varis with im, Elctronic Journal of Applid Statistical Analysis, Vol., pp.9-7, 8. Modl Functions Dmand Rat Production Rat Production Cost Holding Cost Stup Cost Shortag Cost for Lost Itms abl Modl Paramtr Valus Paramtr s St St St St St 5 a..5 a p..... p c c h h k 5 k 5 5 l l..... Shortag Cost for Backordrd Itms b..... b..... Inflation Rat r..... Dtrioration Rat θ Optimal Rsults Dcision Variabls St St St St St 5 λ λ t CU ISSN: ISBN: