Optimal Issuing Policies for Substitutable Fresh Agricultural Products under Equal Ordering Policy

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1 06 Internatonal Academc Conference on Human Socety and Culture (HSC 06) ISBN: Optmal Issung Polces for Substtutable Fresh Agrcultural Products under Eual Orderng Polcy Qao- TENG,a, and T-jun FAN, School of Busness, East Chna Unversty of Scence and Technology, Shangha, Chna a Emal: tengao@com Correspondng author Keywords: Fresh agrcultural products, LIFO,, Consumer choce, Substtuton Abstract Small fresh agrcultural products retalers often adopt eual orderng polcy Demand for the old fresh agrcultural products and the new s descrbed by consumer choce whch s analyzed wth prce and freshness Assumng that the market capacty s known and takng substtuton of demand for the old and new fresh agrcultural products nto account, we establsh LIFO and nventory models To fnd the thresholds of freshness, prce and the substtuton probablty, we gve the suggeston about optmal ssung polcy Ths study helps small fresh agrcultural products retalers to optmze ther ssung polces Introducton Fresh agrcultural product s consumers necesstes of healthy lvng besdes gran Fresh agrcultural product typcally ncludes vegetables, fruts, meat and fsh Goyal and Gr[] defned fresh agrcultural products as the products whch must be sold wthn a fxed lfetme otherwse t wll be on sale or dscard for deteroraton and out expraton date because ts value wll be declned wth tme Accordng to reports, Amercan and European countres only n the retal lnk losses up to 0% per year and the loss of fresh agrcultural products after expraton date worth bllons of dollars[,3] So t s a meanngful problem to optmze ssung polces of fresh agrcultural product Two prmary ssung polces for selectng tems wth several ages when a demand arrves are LIFO and Under LIFO (Last In Frst Out) all demand s satsfed by the youngest tem and under (Frst In Frst Out) demand s satsfed by the oldest tem Hahn and Hwang[4], level Lee C C[5] and Lee Y M[6] establshed determnstc demand rate at prce, tme and even nventory level to study the optmal ssung polcy among normal pershable tems or partcular product Also researchers studed the optmal ssung polces n stochastc demand Perskalla W and Roach C D[7] were the frst one to assume that the nventory was grouped nto categores accordng to age and demand occurred for each categores whch may be satsfed by nventory unts from that category or from any younger category And they also were the frst to consder stochastc demand and found that LIFO s optmal for blood bank Also Cohen M A and Pekelman D[8] proved that LIFO ssung polcy was optmal for mltary ammunton stockples Whle Hajema R[9] presented a new class of stock-level dependent orderng polces (s, S,, Q) and found that ssung polcy wasn t always optmal Fresh agrcultural product s more deterorate than other pershable tems Its value decreased as tme goes by, so ts freshness affects consumer s behavor more sgnfcantly than other pershable tems, whch means consumers wll choose to

2 purchase fresh agrcultural products dependng on freshness and prce Conseuently, the market s dvded nto several parts for varous consumer choces Tallur and van Ryzn[0], Zhang and Cooper[], Dong L et al[] and Akcay Yet al[3] studed consumer choce n dfferent stuaton Substtuton phenomenon also exsts among dfferent age category of fresh agrcultural products Then some age category s out of stock, consumers may purchase substtuton Parlar M[4] establshed a Markov decson model to deal wth the optmal orderng polces for a pershable and substtutable product Huang He, Xu Hongyan and Dan Bn[5] studed the optmal orderng polces wth eual orderng polcy for fresh agrcultural product Fresh agrcultural products and pershable tems ssung polcy have been researched, but few researchers studed the optmal ssung polces dependng on the consumer choce and substtuton of fresh agrcultural product under more practcal eual orderng polcy because small retalers have to adopt eual orderng polcy for lack of barganng power Ths paper studes the optmal ssung polces for substtutable fresh agrcultural product consderng consumer choce whch small retalers under eual orderng polcy should adopt when orderng uantty s the only decson varable Model Descrptons of parameters are lsted as follows: N D u Market capacty Market demand wth category Consumer utlty when consumer purchase category > > 0 Freshness of category of fresh agrcultural products, p > p > 0 p Prce of category of fresh agrcultural products, θ Q Q a j c Random coeffcent to descrbe consumers senstvty to freshness, PDF f ( θ ), CDF F( θ ) Orderng uantty, decson varable Inventory uantty of old category whch s kept log from the last perod The probablty that consumers choose substtuton category j when category s out of stock, a j [0,] (, j =, ) [6] Cost of unt resh agrcultural product π LIFO, π Retaler s revenue We assume that small retalers adopt eual orderng polcy n the sale season and can estmate the demand for experence, whch means that the demand of multple perods n the sale season look as same [3] Fgure demonstrates the event seuences of retalers under LIFO and ssung polces We defne the market capacty as N and the demand of new and old fresh agrcultural products as D and as D Retalers decde the ssung polcy n the sale season and customers wll get the nformaton Customers wll purchase followng the ssung polcy s seuence accordng to ther need Retalers wll sell the new or old category frst dependng on the decded ssung polcy If frst sell category s demand sn t satsfed, possble substtuton demand

3 α j (D -Q ) wll emergent and customers wll purchase the other category to substtute ther demand As a result, the other category s demand become D j α j (D -Q ) Estmate the market capacty N Decde orderng uantty Q Emergent possble substtuton demand α(d Q ) Decde, consumers get nformaton,sell old products Q frst, demand s D Sell new products Q,demand s D α(d Q ) One perod Estmate the market capacty N Decde orderng uantty Q Emergent possble substtuton demand α (D Q ) Decde LIFO, consumers get nformaton LIFO,sell new products Q frst, demand s D Sell old products Q, demand s D α(d Q ) One perod Fgure The event seuences under LIFO and ssung polces Demands for the New and Old Fresh Agrcultural Products Varous categores of fresh agrcultural products are vertcal dfferentaton Consumers choose dfferent category n dependent on freshness and prce and the market demand s dvded nto two parts The followng consumer utlty functon was establshed referencng Akcay Y et al s[3] and L Mngzh and Ke Xung s [8] papers u = θ p () Formula () shows consumer utlty functon wth dfferent categores descrbes dfferent category wth dfferent age = means that fresh agrcultural products age s 0 and = means that fresh agrcultural products age s When the age s, the category wth age must be dscarded for out of expre date From formula (), we fnd that when, p = F( ) α = 0 = and D p = [ F( )]N = 0 = ;

4 when <, p p α = p = F( ) = F( ) F( ) and D p p = = p = [ F( )]N [F( ) F( )]N LIFO and Inventory Models LIFO and nventory models are establshed under eual orderng polcy LIFO π (Q ) = p mn[d,q ] cq p mn[d a (D Q ),Q ] () π (Q ) = p mn[d,q ] p mn[d a (D Q ),Q ] cq (3) Formula () and (3) reference Huang He, Xu Hongyan and Dan Bn[3] and Netessne S, Rud N[9] We assume that several perods eual the same perod for the eual orderng polcy and consumers know the ssung polcy retalers adopt and wll come wth the sales seuence Let D LIFO D = D a (D Q ) where D = α N( =, ) and Q = (Q D ) before the sales season Formula () descrbe that the retaler adopts LIFO ssung polcy The retaler frst to sell fresh agrcultural products wth age 0 and the demand s D When the nventory of fresh agrcultural products wth age 0 doesn t declne or s out of stock, the retaler wll begn to sell fresh agrcultural products wth age and the demand s a (D -Q ) whch means that part of consumers wll purchase products wth age when ther demand doesn t satsfy andd Formula (3) descrbe the retaler adopts ssung polcy The retaler frst to sell fresh agrcultural products wth age and the demand s D When the nventory of fresh agrcultural products wth age doesn t declne or s out of stock, the retaler wll begn to sell fresh agrcultural products wth age 0 and the demand s a (D -Q ) whch means that part of consumers wll purchase products wth age 0 when ther demand doesn t satsfy and D Frst case:, formula () and (3) wll be: π LIFO (Q ) = p mn[d,q ] cq (4) π (Q ) = p mn[d,q ] cq (5) Theorem : When new category s freshness prce rato s hgher than old category s that, t s same for retalers to adopt LIFO or ssung polcy

5 Second case: <, formula () and (3) wll be: π (Q ) = p mn[d,q ] cq p mn[d a (D Q ),Q ] (6) LIFO π (Q ) = p mn[d,q ] p mn[d a (D Q ),Q ] cq (7) Frst, to analyss the relatonshp between the orderng uantty and retaler s revenue By analyzng formula (6)-(7), retaler s revenue wll change wth dfferent Q Theorem : Under LIFO ssung polcy, when p > c, optmal orderng uantty p Q = D D = [ F( )]N and maxmum revenue π LIFO(Q )=(p -c)d (p -c)d ; When p c, optmal orderng uantty p p Q = D = [ F( )]N and maxmum revenue π LIFO(Q )=(p -c)d Theorem 3: Under ssung polcy, when p > a p ( a )c, optmal p orderng uantty Q = D D = [ F( )]N and maxmum revenue π (Q ) = (p c)d (p c)d ; when p a p ( a )c, optmal orderng p uantty Q = D a D = [ ( a )F( ) a F( )]N and maxmum revenue π (Q ) = (p c)d (p a c)d From theorem and 3, when p a p ( a )c, π = p D c(d D ) p D (8) LIFO π = p D c(d D ) p D (9) When c p < a p ( a )c, π = p D c (D D ) p D (0) LIFO π = (p c)(d a D ) () When p < c, π = (p c)d () LIFO π = (p c)(d a D ) (3) By analyzng formula (5)-(0), we can get theorem 4 Theorem 4: If <, when p a p ( a )c, t s same for retalers to adopt LIFO or ssung polcy; when p < a p ( a )c, t s better to adopt ssung polcy

6 We get Fgure and Fgure 3 accordng to theorem and 4 Fgure shows the optmal ssung polcy when a < Area S : When p < c, retalers wll lost c p by sellng per unt of old category, so retalers wll not keep log nventory just to satsfy the demand for new category Under LIFO ssung polcy, retalers only satsfy the demand of new category Under ssung polcy, retalers could satsfy the demand of new category and the possble substtuton demand Therefore, ssung polcy s optmal when p < c p c ( a )c S 4 & LIFO S S p = p S 3 & LIFO p = a p ( a )c p = c c p Fgure a <, the choce of optmal ssung polcy Area S : When c p < a p (-a )c The proft of old category s p -c and the proft of new category s p -c as well as p -c > p -c So under LIFO ssung polcy, retalers wll satsfy the demand of new category totally wthout possble substtuton demand and keep log n the last perod to satsfy the demand of the old to maxmum ther revenue Under ssung polcy, more possble substtuton demand could get more profts for p c > p c Therefore, ssung polcy f optmal when c p < a p (-a )c Area S 3 : When a p ( a )c p < p The gap between p and p becomes smaller Under LIFO ssung polcy, all demands wll be satsfed Under ssung polcy, more possble substtuton demand could get more profts, but p c s large enough to cause huge loss for the loss of demand (-a )D So there s a balance between the two polcy that retalers satsfy the demands totally respectvely Therefore, LIFO and ssung polces are both optmal Area S 4 : When p p, u > u As a result, consumers wll not purchase old category of fresh agrcultural products even f the new category s out of stock Under LIFO or ssung polcy, retalers satsfy the demand of new category totally and the demand of old category s 0 Therefore, LIFO and ssung polces are both optmal when p p

7 p c S 4 & LIFO S p = a p ( a )c p = p p = c ( a )c S c p Fgure 3 a, the choce of optmal ssung polcy Fgure 3 shows the optmal choce of ssung polcy when a Comparng wth Fgure, S 3 s replaced by S Because a becomes bgger, n the orgnal area S 3, under ssung polcy, the loss demand ( a) D s smaller and the proft s more than the loss Therefore, ssung polcy s optmal Concluson Ths paper studes the ssung problem wth dfferent age category of fresh agrcultural products under eual orderng polcy We consder consumer choce to analyss the demands of new and old category of fresh agrcultural products The market demand s dvded nto parts dependng on freshness and prce LIFO and nventory models consderng substtuton of fresh agrcultural products are establshed and we fnd the thresholds to choose optmal ssung polcy When the market prces for the dfferent age category change, the demands for dfferent age category wll change And the substtuton probablty wll also affect the decson Ths paper gves small fresh agrcultural products retalers suggestons to choose optmal ssung polcy wth the certan market prce and substtuton probablty References [] Goyal S K, Gr B C (00) Recent trends n modelng of deteroratng nventory European Journal of operatonal research, 34(), -6 [] Kantor L S, Lpton K, Manchester A, et al (997) Estmatng and addressng Amerca s food losses Food Revew, 0(), - [3] Kärkkänen M (003) Increasng effcency n the supply chan for short shelf lfe goods usng RFID taggng Internatonal Journal of Retal & Dstrbuton Management, 3(0), [4] Hahn K H, Hwang H, Shnn S W (004) A returns polcy for dstrbuton channel coordnaton of pershable tems European Journal of Operatonal Research, 5(3), [5] Lee C C (006) Two-warehouse nventory model wth deteroraton under dspatchng polcy European Journal of Operatonal Research, 74(),

8 [6] Lee Y M, Mu S, Shen Z, et al (04) Issung for pershable nventory management wth a mnmum nventory volume constrant Computers & Industral Engneerng, 76, 80-9 [7] Perskalla W P, Roach C D (97) Optmal ssung polces for pershable nventory Management Scence, 8(), [8] Cohen M A, Pekelman D (978) LIFO nventory systems Management Scence, 4(), 50-6 [9] Hajema R (03) A new class of stock-level dependent orderng polces for pershables wth a short maxmum shelf lfe Internatonal Journal of Producton Economcs, 43(), [0] Tallur K, Van Ryzn G (004) Revenue management under a general dscrete choce model of consumer behavor Management Scence, 50(), 5-33 [] Zhang D, Cooper W L (005) Revenue management for parallel flghts wth customer-choce behavor Operatons Research, 53(3), [] Dong L, Kouvels P, Tan Z (009) Dynamc prcng and nventory control of substtute products Manufacturng & Servce Operatons Management, (), [3] Akcay Y, Natarajan H P, Xu S H (00) Jont dynamc prcng of multple pershable products under consumer choce Management Scence, 56(8), [4] Parlar M (985) Optmal orderng polces for a pershable and substtutable product: A Markov decson model Infor, 3(), 8-95 [5] Huang He, Xu Hongyan, Dan Bn (007) Study on Substtutable Pershable Products Inventory Model Based on Eual Orderng Polcy Mathematcs n Practce an Theory, 37(7), [6] McGllvray A R, Slver E A (978) Some concepts for nventory control under substtutable demand Infor, 6(), [7] Chen X, Feng Y, Kebls M F, et al (05) Optmal nventory polcy for two substtutable products wth customer servce objectves European Journal of Operatonal Research, 46(), [8] L Mngzh, Ke Xung (004) Industral Organzaton Theory Bejng: Tsnghua Unversty Press [9] Netessne S, Rud N (003) Centralzed and compettve nventory models wth demand substtuton Operatons Research, 5(),