A SINGLE PERIOD INVENTORY MODEL OF A DETERIORATING ITEM SOLD FROM TWO SHOPS WITH SHORTAGE VIA GENETIC ALGORITHM

Yugoslav Journal of Operaons Research 7 (7), Number, 7594 DOI:.98/YUJOR775M A SINGLE PERIOD INVENTORY MODEL OF A DETERIORATING ITEM SOLD FROM TWO SHOPS WITH SHORTAGE VIA GENETIC ALGORITHM S. K. MONDAL, J.K. DEY, M. MAITI Deparmen of Appled Mahemacs wh Oceanology and Compuer Programmng, Vdyasagar Unversy, Paschm Mdnapore, Inda Deparmen of Mahemacs, Mahshadal Raj College, Mahshadal, Eas Mdnapur, Inda Receved: November 3 / Acceped: May 6 Absrac: Invenory of dfferenal uns of a deerorang em purchased n a lo and sold separaely from wo shops under a sngle managemen s consdered. Here deeroraon ncreases wh me and demands are me and prcedependen for fresh and deeroraed uns respecvely. For he fresh uns, shorages are allowed and laer parallybacklogged. For he deeroraed uns, here are wo scenaros dependng upon wheher nal rae of replenshmen of deeroraed uns s less or more han he demand of hese ems. Under each scenaro, fve subscenaros are depced dependng upon he me perods of he woshops. For each sub scenaros, prof maxmzaon problem has been formulaed and solved for opmum order quany and correspondng me perod usng genec Algorhm (GA) wh Roulee wheel selecon, arhmec crossover and unform muaon and Generalzed Reduced Graden mehod (GRG). All subscenaros are llusraed numercally and resuls from wo mehods are compared. Keywords: Deerorang em, wo shops problem, me dependen demand, sngle perod nvenory model, genec algorhm.. INTRODUCTION Snce he developmen of EOQ model by Harrs (95), he researchers have formulaed and solved he dfferen ypes of nvenory models. Dealed revews on he developmen n hs area can be obaned n Hadley and Whn (963), Naddor (966), ec. In realy, here are many suaons where he demand rae depends on me. The demand of some ems especally seasonable producs lke garmens, shoes, mangoes,

76 S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model omaoes ec., s low a he begnnng of he season bu ncreases as he season progresses.e., changes wh me. Donaldson (99), Wee and Wang (999), Chang and Dye (999), Bhuna and Ma (998) and ohers developed her nvenory models wh me varyng demand. One of he mos mporan assumpons n classcal nvenory models s ha he lfeme of an em s nfne whle s n sorage. Bu, n nvenory managemen, he decay of he ems plays an mporan role. In realy, some of he ems are eher damaged or decayed or vaporzed or affeced by some oher facors,.e., hey do no reman n a perfec condon o sasfy he demand. The rae of deeroraon of an em may be consan, me dependen or sock dependen. Some ems, whch are made of glass, chna clay or ceramc, are ofen broken durng her sorage perod and n hs case, he deeroraon rae depends upon he sze of he oal nvenory. The decayng ems such as phoographc flm, elecronc goods, frus and vegeables ec. gradually lose her uly wh me. In he exsng models, s generally assumed ha he deeroraed uns are complee loss o he nvenory managemen. Bu, n realy, s no always rue. There are some pershable ems (e.g., frus, vegeables, food grans, ec), whch have a demand o some parcular cusomers even afer beng parally deeroraed. Ths phenomenon s very common n he developng counres where majory of people lve under povery lne. In busness, he parally affeced ems are beng mmedaely and connuously separaed from he lo o save he fresh ones, oherwse he good ones wll be affeced by geng n conac wh he spoled ones. These damaged uns are sold from he adjacen secondary shop. Here, he fresh/good uns may be sold wh a prof whle he deeroraed ones are usually sold a a lower prce, even ncurrng a loss, n such a way ha he managemen makes a prof ou of he oal sales from he wo shops. For he soluon of decsonmakng problems, here are some nheren dffcules n he radonal drec and gradenbased opmzaon echnques used for hs purpose. Normally, hese mehods () are nal soluon dependen, () ge suck o a sub opmal soluon, () are no effcen n handlng problems havng dscree varables, (v) can no be effcenly used on parallel machnes and (v) are no unversal, raher problem dependen. To overcome hese dffcules, recenly genec algorhms (GAs) are used as opmzaon echnques for decson makng problems. GAs[Goldberg(989), Davs(Ed)(99), Mchalewcz(99)] are adapve compuaonal procedures modeled on he mechancs of naural genec sysems. They explo he hsorcal nformaon o speculae on new offsprng wh expeced mproved performance [Goldberg(989), Pal e al(997)].these are execued eravely on a se of coded soluons (called populaon) wh hree operaors: selecon/reproducon, crossover and muaon. An eraon of hese hree operaors s known as a generaon n he parlance of GAs. Snce a GA works smulaneously on a se of coded soluons, has very lle chance o ge suck a local opma. Here, he resoluon of he possble search space s ncreased by operang on poenal soluons and no on he soluons hemselves. Furher, hs search space needs no o be connuous. Recenly, GAs have been appled n dfferen areas lke neural nework [Pal e al (997)], ravellng salesman [Forres (993)], schedulng [Davs (Ed) (99)], numercal opmzaon [Mchalewcz (99)], paern recognon [Gelsema (995)], ec. In hs paper, an nvenory model for a deerorang em, especally frus, vegeables, ec., comprsng boh good and damaged producs and purchased n a lo s formulaed under he assumpon ha he demand of he good uns s me dependen

S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model 77 whereas he deeroraed ones havng only sellng prce dependen demand. Demand of he good un lnearly ncreases wh me ll he shorages occur and afer ha, gradually decreases durng he shorage perod. I s also assumed ha a he begnnng, a lo of he em ncludng fresh and damaged uns are receved a he prmary shop and he damaged ones are spoed, separaed and ransferred o oher place known as secondary shop. Only good producs are sold from he prmary shop. Also durng he sale a he prmary shop, as he me progresses, some fresh uns are damaged and hese spoled ones are spoed and ransferred o he secondary shop connuously. These damaged uns are sold a he reduced prce. Shorages are allowed and fully backlogged a he prmary shop bu no n he secondary shop. In he prmary shop, shorages are me by he fresh uns specally purchased a hgher prce a he end of he cycle. There may be wo cases for he presen model dependng upon he rae of nal replenshmen of deeroraed uns o he secondary shop beng greaer han or less han s demand rae. Agan under each case, here may be fve scenaros n he secondary shop dependng upon he me perods of he wo shops. The me perod of he secondary shop may be equal, less han and greaer han he me perod of he prmary shop. When s less, may occur before, afer or exacly a he me of occurrence of he shorages a he prmary shop. The me perod of he secondary shop may be equal, less han or greaer han ha of he prmary shop. Dependng upon all hese crera, fve dfferen scenaros are observed for each case. For each scenaro, nvenory model has been formulaed akng boh prmary and secondary shops no accoun. To acheve he maxmum prof ou of he oal proceeds from wo shops, he problem has been solved for opmum order quanes and he correspondng me perods usng Genec Algorhm and a graden based opmzaon mehod (GRG). All he subscenaros of he model have been numercally llusraed and resuls are compared.. NOTATIONS AND ASSUMPTIONS To develop he nvenory model of a deerorang em for boh prmary and secondary shops under a sngle managemen, he followng noaons and assumpons are used: For he prmary shop: () Lead me s consdered o be neglgble. () Replenshmen rae s nfne. () c s he purchasng cos per em. (v) Shorages are allowed a he prmary shop bu backlogged by he specally purchased goods a a hgher prce, c per un em a he end of he cycle, where c = mc ( m > ). (v) s he me of shorage pon. (v) The demand rae, D() for good uns s lnear funcon of me.e., d + d, durng no shorage perod D () = D ( ) δ ( ), durng shorage perod where d, d, δ >. (v) The rae of deeroraon, θ ( ) s lnearly dependen on me,.e., θ ( ) = a, a >.

78 S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model (v) The nvenory holdng and shorage coss per un per un me are C p and C p respecvely and he replenshmen cos s C 3 p per perod. (x) s he me perod of he prmary shop. (x) p s he sellng prce per un em of he prmary shop. For he Secondary shop: () Lead me s consdered o be neglgble. () Shorages are no allowed. () The demand rae, λ of deerorang uns s dependen on he sellng prce, p = rc, < r <, where p s he sellng prce of he deerorang uns and λ = α β p, α > and β >. (v) Cs, Cs are he holdng cos per un em per un me and he se up cos per replenshmen perod respecvely. (v) The rae of deeroraon θ s assumed o be consan. (v) 3 s he me perod of he secondary shop. 3. MODEL FORMULATION I s assumed ha nally afer he arrval of a lo of S uns, deeroraed uns ha are a ceran fracon (say μ ) of he nal lo sze are separaed and ransferred o he secondary shop. Therefore, he onhand nvenory level n he prmary shop s ( μ )S a = and up o =, gradually declnes manly o mee up he demand of fresh uns and parally due o deeroraon of he uns whch are connuously ransferred o secondary shop for sale. The sock level reaches zero a me = and hen shorages are allowed and connue up o he me = when nex lo arrves. A =, he maxmum shorage level le s S. These S fresh uns are specally purchased a a hgher prce a he end of he cycle. The geomercal represenaon of he model s gven n Fgure. A he secondary shop, he deerorang uns are sold and shorages are no allowed. In hs shop, nally he amoun of sock s μ S. Dependng upon he rae of deeroraon, here wo cases may arse. In he frs case, rae of replenshmen of deerorang uns s nally less han he demand per un me and he nvenory level gradually declnes up o he me = wh he sock level S. Durng hs perod, demand s me up parly from he curren deerorang uns receved from he prmary shop and parly from he sock. Afer =, demand s me up fully from he sock as he prmary shop goes o shorages a = and nvenory level gradually declnes o zero a = 3. In hs suaon, hree separae subscenaros may arse n he secondary shop dependng upon he cases when he me perod 3 of he secondary shop s equal, less han, or greaer han. When 3 >, he sock a n he secondary shop s S 3. When

S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model 79 3 <, here may be anoher hree dfferen subscenaros dependng upon 3 <, 3 > and 3 =. The geomercal represenaons of he subscenaros are gven n Fgures 6. In he second case, replenshmen rae s nally greaer han he demand rae and gradually ncreases up o he me = when he sock aans a level S 4. As he amoun of replenshmen gradually declnes, we assume ha afer me =, he demand rae s greaer han he rae of replenshmen and demand s me up parly from currenly deeroraed uns receved from he prmary shop and parly from he sock. Ths process connues up o = when sock aans a level S. Afer =, supply of he deeroraed ems sops as shorages sar by ha me a he prmary shop. The nvenory level S gradually declnes o zero a me = 3 (say). As before, n hs case also, here may be fve subscenaros ha are depced n he Fgures 7. 3.. Prmary Shop: The dfferenal equaons governng he nsananeous sae of nvenory q() (Fgure ) a he prmary shop are: dq() { ( q ) ( ) D ( )} f d = θ + { D( ) δ ( )} f wh boundary condons: ( μ) S f = q () = f = S f = () () (μ)s S Fgure. The soluon of he equaon () s:

8 S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model ( d + d){ F( ) F( )} f f() q () = d ( ) δ ( ) f where f(), F() and d are gven by (3) a f () = ( d + d ) e (4) F() = f( u) du d = d + ( d + δ ) (5) From he boundary condons, we ge: F( ) S = (6) μ δ S = d( ) ( ) (7) Toal number of deerorang ems durng (, ) s d d θ() () ( μ) ( ) (8) S = q d = S d + The holdng cos over he perod (, ) s C = C q() d (9) The shorage cos durng he perod (, ) s p Csp = C q() d hp δ d δ 3 3 = C {( d )( ) ( ) + ( )} () p 6 Hence, he oal prof of he nvenory sysem for he perod (, ) s gven by p Z (, ) = p ( μ ) S + p S cs cs C C C p S () p hp sp 3p d 3.. Secondary Shop: Scenaro: Inally, θ( q ) ( ) < λ. Subscenaro a: In hs case, he me perod of he secondary shop s equal o he me perod of he prmary shop,.e., 3 =. The dfferenal equaons descrbng he nvenory level I() (Fg. ) are gven by

S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model 8 di() θ() q () λ f, θ() q () < λ + θ I () = d λ f wh boundary condons: μ S f = I() = S f = f = () (3) μs S The soluon of he equaon () s: θ λ θ θ ar() e + ( e ) + μse f θ I () = λ θ ( ) { e } f θ where R() s gven by (= 3 ) Fgure : (subscenaroa) (4) θ u R() = uq( u) e du (5) From he boundary condons (3), we ge θ S ar( ) e λ θ θ ( e = + ) + μse θ Toal number of deerorang ems durng (, ) s (6) where d θ () θ ( ) (7) S = I d = I + I μs θ λ θ = = + + θ θ θ I Id () ( e ) { ( e )} av ( ) (8)

8 S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model λ θ ( ) = = θ θ I Id () { e } ( ) (9) Here V() s gven by The holdng cos over he perod (, ) s θ u V () = Rue ( ) du () hs = s = s + C C I() d C ( I I ) () θ S The cycle lengh s gven by = + log( + ) θ λ Hence, he reurn from he secondary shop durng he perod (, ) s gven by s d d hs 3s () Z (, ) = ( μ S + S ) p S p C C (3) where he expressons of S, respecvely. S and d S are subsued from (6), (8) and (7) d Subscenarob: In hs case, he me perod of he secondary shop s less han he me perod of he prmary shop,.e., 3 <. Dfferenal equaons and expressons can be obaned by replacng by 3 n subscenaroa. The nsananeous sae of nvenory s shown n Fgure 3. μs S 3 Fgure 3. (subscenarob) Subscenaroc: In hs case, he me perod of he secondary shop s greaer han he me perod of he prmary shop,.e., 3 >. Here he dfferenal equaons descrbng he nvenory level I() (Fg. 4) are gven by equaon () and he boundary condons are he same as (3) excep a =. A =, I() = S3. The soluons of he dfferenal equaons are he same as equaons (4) when and anoher soluon for s as follows

S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model 83 λ I () = { e } + Se θ θ ( ) θ ( ) 3 (4) μs S S 3 Here S can be obaned by equaon (6). Toal number of deerorang ems durng (, ) s S = θ ( I + I ) where I d s obaned by equaon (8) and I s as follows λ ( ) 3 ( ) { θ S } ( ) { θ I = e e } θ + θ (5) θ The holdng cos over he perod (, ) s Chs = C s ( I + I). The cycle lengh s gven by + = + + θ λ θ λ θ S log( ) + S3 Hence, he reurn from he secondary shop durng he perod (, ) s gven by s d 3 d hs 3s 3 (6) Z (, ) = ( μ S + S S ) p S p C C + p S (7) where p = mp,< m <. (< 3 ) Fgure 4: (subscenaroc) Subscenarod: In hs case, he me perod of he secondary shop s equal o he me of shorage pon of he prmary shop,.e., 3 =. The dfferenal equaons descrbng he nvenory level I() (Fg. 5) are gven by di() + θ I () = θ() q )) λ f (8) d wh boundary condons: μ S a = I () = (9) a =

84 S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model μs (= 3 ) Fgure 5: (subscenarod) The soluon of he equaon (8) s he same as he frs equaon of (4). From he boundary condons, we ge λ ar e e Se θ θ ( ) θ θ ( + ) + μ = Toal number of deerorang ems durng (, ) s S d = θ I where I s same as (8) and he holdng cos over he perod (, ) s Chs = C si. Hence, he reurn from he secondary shop durng he perod (, ) s gven by equaon (3). Subscenaroe: In hs case, he me perod of he secondary shop s less han he me of shorage pon of he prmary shop,.e., 3 <. The dfferenal equaon descrbng he nvenory level I() (Fg. 6) and boundary condons are he same as equaons (8) and (9) respecvely only replacng by 3. The soluon of he dfferenal equaon, he boundary equaon, number of deerorang ems and holdng cos are he same as he subscenarod only replacng by 3. Hence, he reurn from he secondary shop durng he perod (, 3) s gven by (3). (3) μs 3 Fgure 6: (subscenaroe)

S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model 85 Scenaro: Inally, θ( q ) ( ) > λ Subscenaroa: In hs case, he me perod of he secondary shop s equal o he me perod of he prmary shop,.e., 3 =. The dfferenal equaons descrbng he nvenory level I() (Fg.7) are gven by θ() q () λ f, θ() q () > λ di() + θ I () = θ() q () λ f, θ() q () < λ d λ f wh boundary condons : μ S f = S4 f = I () = S f = f = (3) (3) θ()q()>λ θ()q()<λ μs S 3 S / (= 3 ) Fgure 7: (subscenaroa) The soluons of he equaons (3) are θ λ θ θ ar() e + ( e ) + μse f θ θ λ θ ( ) θ ( ) I() = ar { () R ( )} e + { e } + Se 4 f θ λ θ ( ) { e } f θ From he boundary condons, we ge (33)

86 S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model λ S ar R e e Se θ θ θ ( ) θ ( ) = { ( ) ( )} + { } + 3 λ S4 = ar( ) e + ( e ) + μse θ θ θ θ (34) (35) λ = a ( d + d ){ F( ) F( )}/ f ( ) (36) Toal number of deerorang ems durng (, ) s S = θ Id () = Id () + Id () + Id () = θ ( I + I + I ) d 3 where I s obaned from equaon (8) only replacng by and I have same 3 expresson as n (9) and I can be obaned by a θ θ I = av { ( ) V ( )} + R ( )( e e ) θ λ θ ( ) S 4 θ ( ) + [ { e } ( )] + { e } θ θ θ The holdng cos over he perod (, ) s Chs = C s ( I + I + I3) and he cycle lengh s gven by (). Hence, he reurn from he secondary shop durng he perod (, ) s gven by (3). (37) Subscenarob: In hs case, he me perod of he secondary shop s less han he me perod of he prmary shop,.e., 3 <. Dfferenal equaons and expressons can be obaned from Subscenaroa only replacng by 3. The nsananeous sae of nvenory s shown n Fgure 8. θ()q()>λ θ()q()<λ μs S 4 S / 3 Fgure 8: (subscenarob)

S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model 87 Subscenaroc: In hs case, he me perod of he secondary shop s greaer han he me perod of he prmary shop,.e., 3 >. Here he dfferenal equaons descrbng he nvenory level I() (Fg.9) are gven by equaons (3) and he boundary condons are same as (3) excep a =. A =, I () = S. The soluons of he dfferenal equaons are same 3 as equaons (33) when, and oher soluon for s as follows λ θ ( ) θ ( ) I () = { e } + Se 3 θ All boundary condons are same as equaons (34), (35) and (36). The oal number of deerorang ems durng (, ) s S = θ ( I + I + I ) where I d 3 s obaned from equaon (8) only replacng by and I have same expresson as n (37) and I s 3 same as he expresson of (5) only changng S by S. The holdng cos over he perod 4 3 (, ) s C = C ( I + I + I ). The cycle lengh hs s 3 s obaned from equaon (6). Hence, he reurn from he secondary shop durng he perod (, ) s gven by he equaon (7). θ()q()>λ θ()q()<λ μs S 4 S S 3 / (< 3 ) Fgure 9: (subscenaroc) Subscenarod: In hs case, he me perod of he secondary shop s equal o he me of shorage pon of he prmary shop,.e., 3 =. The dfferenal equaons descrbng he nvenory level I() (Fg.) are gven by he frs wo equaons of (3) wh boundary equaons μ S f = I () = S f = 4 f = (38)

88 S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model The soluons of hese equaons are same as he frs wo equaons of (33). The boundary equaons are same as (34)(36). The number of deerorang ems and oal holdng cos are S = θ ( I + I ) and C = C ( I + I ) where I s same as (8) only d hs s replacng by and I s same as (37). The expresson for reurn s same as (3). θ()q()>λ μs S 4 θ()q()<λ / (= 3 ) Fgure : (subscenarod) Subscenaroe: In hs case, he me perod of he secondary shop s less han he me shorage pon of he prmary shop,.e., 3 <. Dfferenal equaons and expressons can be obaned from Subscenarond only replacng by 3. The nsananeous sae of nvenory s shown n Fgure. θ()q()>λ μs S 4 θ()q()<λ / 3 Fgure : (subscenaroe) Toal Average Prof: Therefore, he oal average prof (Z) of he sysem from wo shops for Subscenaros (a)(e) s gven by

S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model 89 where Z + Z p s Z = (39) Z p and Z s have dfferen expressons for dfferen Subscenaros, Z p s a funcon of, and Z s a funcon of () s, (),, 3 (), (v), (v),, (v),, (v),,,, (v) 3 3,, (x),, and (x),,, for he 3 subscenarosae respecvely. 4a. Implemenng GA Toal average prof Z gven by (39) s maxmzed hrough he mplemenaon of GA n he followng way. 4a. Parameers: Frsly, we se he dfferen parameers on whch hs GA depends. All hese are he number of generaon (MAXGEN), populaon sze (POPSIZE), probably of crossover (PXOVER), probably of muaon (PMU). There s no clear ndcaon as o how large should a populaon be. If he populaon s oo large, here may be dffculy n sorng he daa, bu f he populaon s oo small, here may no be enough srng for good crossovers. In our expermen, POPSIZE = 5, PXOVER =., PMU =., MAXGEN = 5. 4a. Chromosome represenaon: An mporan ssue n applyng a GA s o desgn an approprae chromosome represenaon of soluons of he problem ogeher wh genec operaors. Tradonal bnary vecors used o represen he chromosome are no effecve n many physcal nonlnear problems. Snce he proposed problem s nonlnear, hence o overcome he dffculy, a real number represenaon s used. In hs represenaon, each chromosome V s a srng of n number of genes G j (=,,..,POPSIZE and j=,,..,n ) where hese n number of genes respecvely denoe he number of decson varables among,, and 3, he value of n depends on he subscenaros(ae)gven n secon 3.. 4a.3 Inal populaon producon: To nalze he populaon, we frs deermne he dependen and ndependen varables and hen her boundares. Here, he dependency and ndependency of he varables are dfferen for dfferen subscenaros. Snce all varables are relaed o me, here he boundares of all ndependen varables are assumed o be (,.). For each chromosome V, every gene G, whch represens he ndependen varable, s randomly generaed j beween s boundary ( LB, UB ) where LB and k k k of ha varable and he gene UB are he lower and upper bounds k G j whch s he dependen varable, are generaed from

9 S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model dfferen condons for dfferen subscenaros gven n secon 3., unl s feasble, =,,..,POPSIZE. 4a.4 Evaluaon: Evaluaon funcon plays he same role n GA as ha whch he envronmen plays n naural evoluon. The evaluaon funcon EVAL for each chromosome V s defned as EVAL( V )=objecve funcon value for V. 4a.5 Selecon: Ths selecon process s based on spnnng he roulee wheel POPSIZE mes, each me we selec a sngle chromosome for he new populaon n he followng way: (a) Calculae he fness value EVAL( V ) for each chromosome V (=,,..,POPSIZE). (b) Fnd he oal fness of he populaon as (c) Calculae he probably of selecon b p = EVAL( V )/ f. POPSIZE f = EVAL( V ). = b p for each chromosome V as (d) Calculae he cumulave probably q for each chromosome V as q = p. (e) Generae a random real number r n (, ). (f) If r < q hen he frs chromosome s V oherwse selec he h chromosome V ( POPSIZE) such ha q < r q. (g) Repea seps (e) and (f) POPSIZE mes and oban POPSIZE copes of chromosomes. 4a.6 Crossover operaon: The exploraon and exploaon of he soluon space s made possble by exchangng genec nformaon of he curren chromosomes. Crossover operaes on wo paren soluons a a me and generaes offsprng soluons by recombnng boh paren soluon feaures. Afer selecon of chromosomes for new populaon, he crossover operaon s appled. Here, he whole arhmec crossover operaon s used. I s done n he followng way: (a) Frsly, we generae a random real number r n (, ). (b) Secondly, we selec wo chromosomes V k and V l randomly among populaon for crossover f r < PXOVER. (c) Then wo offsprng V and V are produced as follows: k l j= b

S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model 9 V = c V + ( c) V k k l V = ( c) V + c V l k l where c [,]. (d) Repea he seps (a),(b) and (c) POPSIZE/ mes. 4a.7 Muaon operaon: Muaon operaon s used o preven he search process from convergng o local opma rapdly. Unlke crossover, s appled o a sngle chromosome V. Here, he unform muaon operaon s used, whch s defned as follows: mu G j =random number from he range (, UPB) where UPB s upper boundary o he correspondng gene. 4a.8 Termnaon: If number of eraon s less han or equal o MAXGEN hen he process connues, oherwse ermnaes. The basc srucure of GA s descrbed as follows: Genec Alg orhm() begn nalze Populaon() evaluae Populaon() whle( no er mn aon condon) begn + selec Populaon() from Populaon( ) aler ( by crossover and muaon) Populaon( ) evaluae Populaon() end wre he opmum resul end

9 S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model 4b. Advanages of GA The advanages of GA [Goldberg (989)] nclude he followngs: () Smple: The algorhm s easy o develop and valdae. () Effcen: The algorhm s parallel, usng he resource of a whole populaon nsead of a sngle ndvdual. Durng he evoluon, he dfferen ndvduals can exchange nformaon by crossover. Exernal nformaon s nroduced by muaon. Hence he algorhm s effcen. Even f begns wh a very poor orgnal populaon, wll progress rapdly owards sasfacory soluons. () Global opmum: Use of populaon, crossover and muaon leads he resuls oward he global opmum nsead of rappng no local peaks. (v) Doman ndependen: The algorhm s a paramerc mehod, suable n a wde range of applcaons. I does no requre preknowledge abou daa dsrbuon, connuy or he exsence of dervave. 5. NUMERICAL ILLUSTRATION To llusrae he model, we consder: r =.8, Cs =.5, C3s = 4, θ =.6, α = 6, β =., μ =., c = 5., p = 9., C =.85, C = 4.5, C =, a =., d = 75, d = 4, δ =.8, m =.4, m =.8. p p 3p The opmal values of,,, 3, S, S, S, S3 and S4 along wh he average maxmum average prof have been calculaed for dfferen subscenaros by GA and GRG (usng sandard sofware package Suden LINGO/PC release 3. verson ) and resuls are dsplayed n Table. 6. DISCUSSION Table gves he opmum values usng genec algorhm. From he able, s observed ha n he frs case, when replenshmen rae of deeroraed ems s less han he demand rae, he scenaroe gves more prof han he oher four scenaros. The nex preferable scenaros are d, b, a and c respecvely. In he second case, when he replenshmen rae s nally more and hen gradually reduces o an amoun less han he demand rae, he scenaroe s beer han he ohers. The nex preferable scenaros are d, b, a and c respecvely.

S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model 93 Table : Resul for he model Subscenaros Mehods a b c d e a b c d e GA.84 GRG.83 GA.75 GRG.75 GA.9 GRG.9 GA.74 GRG.74 GA.9 GRG.9 GA.54.86 GRG.45.8 GA.45.8 GRG.54.86 GA.54.86 GRG.58.9 GA.36.74 GRG.37.74 GA.3.7 GRG.9.75 3.5.5...3.77.4.7..9.98.74.98.74.6.74.6.74.5.5.93.9.93.5.5..7.98.74.99.74.95.65.97.66 S S 38.8 3.39 36.9 9.4.4 4.89.7 4.36 5.7 44.5 53.5 4.64 8.74 34.5 8.74 35. 5.44 38.73 5.44 39.9 43.63 4.4 3.44 8.95 3.44 43. 43.63 4.56 43.63.9 5.7 54.97 8.74 35. 8.74 36.6 3. 34.5.7 3.73 S S 3 3.35 3..3.3 6.3.53 6..77.... 4.36..97..97 4.36 4.36.97 6..44....6 S 4 Z ($) 59.88 59.8 6.45 6.39 57. 56.93 6.75 6.75 63.4 63.4 7.3 59. 4.93 58.48 4.93 6.68 7.3 59.9 7.3 56.7 9. 56.6 3. 6.75.99 6.73.3 6.7 3.9 6.6

94 S.K. Mondal, J.K. Dey, M. Ma / A Sngle Perod Invenory Model 7. CONCLUSION Here, for a realer, afer purchasng n a lo, he sale of boh good and deeroraed ems from wo shops under a sngle managemen has been consdered and solved va genec algorhm. Ths phenomenon s very common n he developng counres lke INDIA, BANGLADESH, and NEPAL ec. In hese counres, here s a marke for boh fresh and deeroraed uns. Hence, a realsc and common problem faced by he realers has been nvesgaed and opmum decsons are presened. These resuls are applcable for he producs lke frus, vegeables ec. whch are sold o he realers n a lo. Presen mehodologes can be exended o oher nvenory models wh All Un Dscoun (AUD), Incremenal Quany Dscoun (IQD), fxed me horzon ec. These models can also be formulaed and solved n probablsc, fuzzy and fuzzysochasc envronmens and solved va genec algorhm. REFERENCES [] Bhuna, A.K., and Ma, M., A wo warehouse nvenory model for deerorang ems wh a lnear rend n demand and shorages, J. Oper. Res. Soc., 49 (998) 879. [] Chang, H.J., and Dye, C.Y., An EOQ model for deerorang ems wh mevaryng demand and paral backloggng, J. Oper. Res. Soc., 5 (999) 768. [3] Davs, L. (ed.), Genec Algorhms and Smulaed Annealng, Pman Publshng, London, 987. [4] Davs, L. (ed.), Handbook of Genec Algorhms, Van Nosrand Renhold, New York, 99. [5] Donaldson, W.A., Invenory replenshmen polcy for a lnear rend n demand an analycal soluon, Operaonal Research Quarerly, 8(99) 66367. [6] Forres, S. (ed.), Proceedngs of 5h Inernaonal Conference on Genec Algorhms, Morgan Kaufnann, Calforna, 993. [7] Gelsema, E. (ed.), Specal ssue on genec algorhms, Paern Recognon Leers, 6(8) (995). [8] Goldberg, D.E., Genec Algorhms: Search, Opmzaon and Machne Learnng, Addson Wesley, Massachuses, 989. [9] Hadley, G., and Whn, T. M., Analyss of Invenory Sysem, PrenceHall, England, Clffs, 963. [] Harrs, F., Operaons and Coss (Facory Managemen Seres), A.W. Shaw Co, Chcago, 95, 485. [] Mchalewcz, Z., Genec Algorhms + Daa Srucure = Evoluon Programs, Sprnger Verlag, Berln, 99. [] Naddor, E., Invenory Sysem, John Wley, New York, 966. [3] Pal, S.K., De, S., and Ghosh, A., Desgnng hopfeld ype neworks usng genec algorhms and s comparson wh smulaed annealng, In. J. Pa. Prog. Ar. Inell, (997) 447 46. [4] Wee, H.M., and Wang, W.T., A varable producon schedulng polcy for deerorang em wh mevaryng demand, Compuers and Operaons Research, 6 (999) 3744.