Dr.Ram Manohar Lohia Avadh University, Faizabad , (Uttar Pradesh) INDIA 1 Department of Computer Science & Engineering,

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1 Vnod Kumar et. al. / Internatonal Journal of Engneerng Scence and Technology Vol. 2(4) Generalzaton of cost optmzaton n (S-1 S) lost sales nventory model Vnod Kumar Mshra 1 Lal Sahab Sngh Department of Mathematcs & Statstcs Dr.Ram Manohar Loha Avadh Unversty Fazabad-2241 (Uttar Pradesh) INDIA 1 Department of Computer Scence & Engneerng Kumaon Engneerng College Dwarahat Almora (Uttarakhand) INDIA vkmshra25@gmal.com 1 snghdrlalsahab7@gmal.com 2 Abstract Ths paper consder the problem of fndng optmal crtcal levels for an nventory system for multtem wth multple demand classes Posson demand process lost sales and ample supply. The orderng polcy s assumed to be the ( S 1 S) type.e. replacement tem s ordered as soon as a unt of stock s used and represent three accurate and effcent heurstc algorthms at a gven base stock level. Keywords: Inventory System Mult-tem multple demand and optmal crtcal level. 1. Introducton In ths paper we study the ( S 1 S) lost sales nventory model for multtem that s demanded by dfferent demand classes that have dfferent penalty cost values. A penalty s ncurred f a demand s not fulflled from stock. Wthn ths settng we am to mnmze the total nventory holdng and penalty cost and we dstngush between multtem and dfferent classes by ntroducng crtcal levels. A demand for any value for a certan class s only fulflled f the physcal stock s above the base stock level for that class. The problem of multple demand classes has been ntroduced by Venott n1965 [1]. He also ntroduced the concept of crtcal level polces. After that the problem has been studed n a number of mathematcan and researcher and these papers can be dstngushed n two dfferent stream of research. The frst stream studes the structure of the optmal polcy wthn ths stream there are nterestng studes that derves the optmalty of crtcal level polcy of sngle tem wth multple customer classes. Topks n 1968 [2] consder a perodc revew model wth generally dstrbuted demand and zero lead tme. In that stuaton the optmal crtcal level are dependent on the remanng tme perod. In 1997 A.Y. Ha [4] has studed a contnuous revew model wth posson demand process a sngle exponental server for replenshment and lost sales. He derves the optmalty of crtcal level polcy and ths stuaton both the base stock level and crtcal levels are tme ndependent. In 22 Devercourt et.al. [9] studed the same model as Ha but wth back orderng of unsatsfed demand. The second stream conssts of studes that consder evaluaton and optmzaton wthn a gven class of polcy wthn ths stream there are nterestng contrbuton n 2 gven by Melchors et.al. [8] In 22 by Dekker et.al. [5] n 23 Deshpande et.al. [6] and Dekker et. al. [5] derved exact and heurstcs procedures for the generaton of an optmal crtcal level polcy for a contnuous revew model wth multple customer classes posson demands ample supply and lost sales. For the case wth two customer classes Melchors et.al. [8] extend ths work for fxed quantty orderng. In ths model fxed orderng cost the base stock level and sngle crtcal level are optmzed n order to mnmze the sum of fxed orderng nventory holdng and lost sales cost. Deshpande et. al. [6] consder the smlar model but wth back orderng of unsatsfed demand. For detal see the papers gven n the reference. In ths paper we study a mult tem contnuous revew model wth multple demand classes posson demand process lost sales and ample supply. The ample supply represents that the suppler can delver as much as desred wthn a gven replenshment lead tme. We lmt ourselves to class of crtcal level polces wth tme ndependent crtcal and base stock levels. Crtcal levels are easy to explan n practce and the results on optmal polcy n the frst stream of research suggest that ths s at least close to optmal. Under the gven values and class of crtcal level polces our problem s to optmze ( I J ) crtcal levels and one base stock level smultaneously where I s the set of tems and J s the set of demand classes. By a proecton of ( I + 1 J + 1) dmensonal total cost functon on the dmenson of base stock level and the defnton of approprate convex lower and upper bound functon: an exact soluton method s obtaned for the full problem. Enumeraton however s expensve from computatonal ponts of vew especally when number of tems and demand classes greater than two. The man contrbuton of ths paper s generalzaton of cost optmzaton n ( S 1 S) lost sale nventory model and formulaton of three effcent heurstcs algorthms. The computatonal tmes that take the heurstcs algorthms are small and far less than of explct enumeraton. So these heurstcs are accurate and ISSN: Electronc copy avalable at:

2 Vnod Kumar et. al. / Internatonal Journal of Engneerng Scence and Technology Vol. 2(4) effcent. 2. Mathematcal Model Let I J denote the set of tem and demand classes respectvely wth I 1 J 1.Consder mult tem { : m } that s demanded by a number of demand classes ( : n ) or customer groups. For each class J demands are assumed to occur accordng to a Posson process wth constant rate m ( > ). If any tem s not delvered to class J on request the demand s lost and a penalty cost p ( > ) s to be pad n such a way that p 1 / p / p PJP ( : m). The tems are controlled by a contnuous -revew crtcal level polcy ths means that total stock s controlled by a base stock polcy wth base stock level. [ N = N {}] and that there s a crtcal level S c ( N) per class J wth c 1 c2... c J S. The orderng for the crtcal levels s assumed because of the opposte orderng n the penalty cost parameters we call ths orderng of the crtcal level the monotoncty constrant. A crtcal level polcy s denoted by vector ( c S ) wth c : = ( c c... c ) If a class demand arrve at a moment that the physcal stock s larger than 1 2 then ths demand s satsfed otherwse the demand s lost. At and below level than c physcal stock can be seen as stock that s reserved for more mportant class. For easy of notaton we defne c 1 S = + replenshment lead tme are d wth mean tme t and holdng cast per unt tme s h. Let β ( c S ) denotes the fll rate for class J and I under the crtcal level polcy( c S ) an expresson for β ( c S ) can be derved as follows: If the number of parts of sku n the ppe lne s K ( S ) and thus the number of parts n the physcal stock s K then the demand rate S µ k = / k< ( s ) (1... 1) c m K S Our nventory model can be descrbed by a closed queung network wth S customers and two statons: () An ample server wth mean servce tme t whch represents the ppe lne stocks. () A load dependent exponental sngle server wth frst come frst serve servce dscplne whch represent the physcal stock. The servce rates of the load dependent server are gven by the µ k. by applyng the theory of (Baskett et.al 1975[12]). We fnd that the steady state probabltes q ( K 1... S ) for havng K tems n the ppe lnes s gven by q K 1 k t k = Π µ h K! q h= q µ Wth the conventon that K 1 Π µ = 1 for k = h= k K (1... S ) S K 1 t { } k = Π h K! K = h= 1 (Ths result also follows from Gendenko and Kovalenko [13] p.p ). By ths steady state probablty we obtan the fll rate β S C 1 ( c S ) = q I and J k K = wth the conventon that ths sum s empty f S c 1< ( e.. β ( c S ) = f c = S ) c ISSN: Electronc copy avalable at:

3 Vnod Kumar et. al. / Internatonal Journal of Engneerng Scence and Technology Vol. 2(4) Notce that 1 β 1( c S ) β 2( c S )... β ( c S ) Our obectve s to mnmze the average nventory holdng and penalty cost per unt tme. The average cost of a polcy ( c S ) s C( c S ) = hs + p m {1 β ( c S )} I and J. J Our optmzaton problem s a non lnear nteger programmng problem and s stated as follows: ( P ) Mn ( C S ). subect to c c... c S 1 2 J c N and S Mn h S + p m {1 β ( c S )} e. I I J N ( I and J) An optmal polcy for problem P s denoted by ( c S ) and the correspondng cost C ( c S ) For the stuaton wth a fxed base stock level S N let the problem ( PS ( )) denote the problem of fndng the crtcal levels such that average cost ( C( c S )) s mnmzed. Problem ( PS ( )) s stated as ( PS ( )) Mn ( c S ). subect to c c... c S 1 2 J c N and ( I and J) Mn h S + p m {1 β ( c S )} e. I I J An optmal polcy for problem ( PS ( )) s denoted by { c ( S ) S} and the correspondng optmal cost s C{ c ( S ) S}. Obvously { c ( S ) S } = ( c S ). Note that n problem ( PS ( )) the holdng cost term hs consttutes a constant factor. It s ncluded n the formulaton of problem ( PS ( )). 4. Analytcal Soluton 4.1. Exact Method for Problem ( P) A method to solve problem ( P ) exactly has been descrbed by Dekker et al [5]. Ths method explots convex lower bound and upper bound functons for the functon C{ c ( S ) S}. Frst a lower bound S PS s solved s determned for the optmal base stock level. Next startng at ths lower bound Problem ( ( )) by enumeraton for ncreasng values of S untl a stoppng crteron s met that mples that an optmal polcy has been found. An upper bound for the functon C{ c ( S ) S} s obtaned by takng all crtcal levels equal to for each S. Ths gves the upper bound functon c ( S ) = C ( S ). Ths functon s convex whch u follows from the convex behavor of the Erlang loss probablty for all S N {for proof see smth [3]} A lower bound functon for C{ c ( S ) S} s obtaned by replacng all penalty cost parameter p by the lowest penalty cost parameter p n problem ( PS ( )). Under an optmal polcy for ths modfed problem all crtcal levels are zero the resultng cost are denoted by C ( S ) and also for ths functon the convexty follows from the convexty of the Erlang loss probablty. The exact algorthm for problem P s as follows: Frst defne S as a mnmzng pont for the upper bound functon C u( S ). Next C u( S ). Next S l s defned as the lowest S N for whch Cl ( S) Cu ( Sl ).. Ths S l s a lower bound for the optmal base stock level S. Then for S = S l l ISSN:

4 Vnod Kumar et. al. / Internatonal Journal of Engneerng Scence and Technology Vol. 2(4) Problem ( PS ( )) s solved usng explct enumeraton and the current best soluton s set equal to { c ( S ) S}. Here after current best soluton s adapted f { ( ) }. l l S s ncreased by one ( PS ( )) s solved usng explct enumeraton and the c S S provde a better one. Ths s done repeatedly unt C ( S + 1) > C ( S ) and C ( S + 1) s larger than or equal to the current best soluton. (These two l condtons mply that for any base stock level greater than the current base stock level no better soluton can be found). At ths pont the current best soluton consttutes an optmal soluton for problem P Algorthm for problem ( PS ( )) The exact method uses enumeraton to solve multple values of S and thus s tme consumng n partcular for problem wth two or more demand classes. Therefore n ths secton we descrbed and fast heurstcs for problem( PS ( )).. The heurstcs that we consder are local search algorthms. We test the accuracy n an extensve computatonal experment and we fnd that the heurstcs produce an optmal soluton n all nstances. Before we formulate the heurstcs we show the typcal behavor of the functon C( c S ) for a fxed S. The behavor of cost functon are shown n the table for the example wth I = 2 tems and J = 3 demand classes. In ths example the crtcal level for class 1 s fxed at that s known to be optmal We see n the followng table that C( c S ) s unmodel n c 13 and n c. 23 for any fxed c12 and c 22 vce versa. Ths means that the sgn of the frst order of dfference of the cost term changes at most once. If t changes from mnus nto plus ntutvely the observed unmodalty ncreases the chance for local search type algorthm to fnd an optmal soluton but a guarantee cannot be gven. the property that a local mnmum s a global mnmum s not obtaned for drect extensons of the concept of convexty to dscrete spaces but t s obtaned for multmodular functons as ntroduced see by Altman et. al [11] for a whole theory used on multmodularty ISSN:

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6 Vnod Kumar et. al. / Internatonal Journal of Engneerng Scence and Technology Vol. 2(4) Smlarly we can prepare for all possble values of (c 11 c 12 ) & (c 22 c 23 ) Parameter settng for cost functon C( C S ) as functon of C12 C 13 and C22 C23 and S1 = S2 = 5 at nput parameters J = 3 M11 = M12 = M13 = M21 = M22 = M23 = 1 J = (1 2 3) I = (1 2) t1 = t2 = 1 C11 = C21 = h1 = h2 = 1 p 11 = 1 p 12 = 1 p 13 = Descrpton of Heurstcs In ths secton we formulate the heurstcs algorthms for problem ( PS ( )) whch was frst ntroduced by Dekker et. al [5] for problem P after that A.A.Kraneburg G.J. Van Houtum [14] for sngle tem. The dfference n ther heurstcs s that the Dekker ncorporates optmzaton of base stock level whle Kranenburg assume constant base stock level. Both are them explan the heurstc for sngle tem wth multple demand classes and here we explan the heurstc algorthm for mult tem wth multple demand classes at constant base stock level. Algorthm 1. Start wth an arbtrary choce c I J J > k J J where k = max{ p = p11} I then defne the neghborhood of ths current polcy ( c S ) as all polces that stll satsfy the monotoncty constrant and that have crtcal levels that dffer at most one from the correspondng crtcal level n the orgnal polcy. If the cost of the cheapest neghbor s smaller than the cost of current soluton then select ths neghbour and set ths polcy as current soluton and repeat the process of evaluatng all neghbors for ths new polcy. Otherwse stop and take the current soluton as the soluton found by the algorthm. Algorthm 2.Start wth an arbtrary choce for c I J J > k that satsfy the monotoncty constrant for M = J fnd cm ( cm 1... cm + 1) wth the lowest cost at fxed values of the other crtcal levels and change c M. accordngly. Repeat ths optmzaton for one crtcal level at a tme for M = J 1 down to K + 1. After that optmze agan for M = J. Contnue ths teratve process untl for of the M values an mprovement s found. Ths s the soluton found by the algorthm. Algorthm3. Start wth all crtcal levels equal to zero. We frst consder ncreasng c by one and accept ths ncrease f t has lower cost than current soluton. then the crtcal levels c J 1 down to c K+ 1 are ncreased wth 1 (one) crtcal level at a tme and each tme an ncrease of a crtcal level s accepted f lower cost are obtaned f c was ncreased wth one n ths frst teraton then we execute another teraton and so one. The process stops as soon as c J has not been ncreased n teraton. The polcy found at the end of last teraton s the soluton found by the algorthm Algorthm for problem P Problem P an exact method was presented n prevous secton. In the exact method problem ( PS ( )) has to be solved multple tmes and ths s to done by explct enumeraton. In prevous secton we solve the problem ( PS ( )) effcently by one of the proposed heurstcs algorthms. In ths secton the algorthms (1) (2) and (3) for problem PS ( ) can be plugged nto the exact method for problem P thus replacng the explct enumeraton. The resultng algorthms are called algorthm (4) (5) and (6). For algorthm (4) (5) and (6) the choce for the startng pont s fxed at c = (...). 5. Concluson In ths paper ( S 1 S) lost sales nventory model for multtem wth multple demand classes havng Posson demand process and controlled by crtcal level polces has been consdered. In problem P the base stock level and crtcal level are optmze n order to obtan mnmal nventory holdng and penalty cost. In a sub ISSN:

7 Vnod Kumar et. al. / Internatonal Journal of Engneerng Scence and Technology Vol. 2(4) problem problem PS ( ) the base stock level s fxed and crtcal levels are optmzed. The man contrbuton of ths paper s generalzaton of cost optmzaton n ( S 1 S) lost sales nventory model and formulaton of three heurstcs algorthm for problem PS ( ) and these algorthm can be used to solve the problem P n whch problem PS ( ) has to solved many tmes as sub problem. References [1] A.A.KranenburgG.J.van HoutumCost Optmzaton n the ( S 1 S) lost sales nventory model wth multple demand classes Technsche Unverstet Endhoven (25). [2] A.F.Venott Optmal polcy n a dynamc sngle product non statonary nventory model wth several demand classes Operaton research13 (1965) pp [3] A.Y.Ha Inventory ratonng n a make to stock producton system wth several demand classes and lost sales Management Scence 43 (1997) pp [4] B.V.GnedenkoI.N.KovalenkoIntroducton of queueng theoryisrael Programe of scentfc Translaton1968. [5] D.M.Topks Optmal orderng and ratonng polcy n a non statonary dynamc nventory model wth n demand classes Management scence 15 (1968) pp [6] E.Altman B.Gaual A.Hordk Dscrete event control of stochastc networks:multmodularty and Regularty sprnger 23. [7] F.Baskett K.M.ChandyR.R.MuntzF.G.PalacosOpenClosed and mxed networks of queues wth dfferent class of customerjournal of the assocaton eor computng machnery 22(1975) pp [8] F.De.Vercourt F.Karaesment Y.Dallery Optmal stock allocaton for a capacted supply systemmanagement Scence 48 (22) pp [9] Hadley G. Whtn T. (1963). Analyss of Inventory Systems. Prentce Hall Englewood Clffs. [1] L.Dowdy D.Eager K.Gourdon L.Saxton Throughput concavty and response tme convexty Informaton processng Letter 19(1984) pp [11] M.A.Cohen P.R.Klendorfera and H.L.Lee Servce constraned ( ss ) nventory system wth prorty demand classes and lost salesmanagement Scence 34 (4) (1988) pp [12] Naddor E. (1966) Inventory System Wlley New York. [13] P.Melchors R.Dekker M.J.Klen Inventory ratonng n an ( SQ ) nventory model wth lost sales and to demand classes Journal of the operaton research socety 51 (2) pp [14] R.Dekker R.M.Hll M.J.Klen R.H.Teunter On the ( S 1 S) lost sales nventory model wth prorty demand classes Naval Research Logstc 49 (22) pp [15] S.A. Smth Optmal nventory for an ( S 1 S) system wth no back orders Management Scence 23 (1977) pp [16] V.Deshpande M.A.Cohen K.Donohue A threshold nventory ratonng polcy for servce dfferentated demand classes Mangement Scence 49 (23) pp ISSN: