Inventory Control of Perishable Items in a Two-Echelon Supply Chain

Transcription

1 Journal of Indusrial Engineering, Universiy of ehran, Special Issue,, PP Invenory Conrol of Perishable Iems in a wo-echelon Supply Chain Fariborz Jolai *, Elmira Gheisariha and Farnaz Nojavan Deparmen of Indusrial Engineering, College of Engineering, Universiy of ehran, ehran, Iran (Received 4 November, Acceped 4 April Absrac In his paper, we develop an invenory model for perishable iems wih random lifeime in a wo-echelon producion-disribuion sysem. here is a manufacurer a he firs sage ha produces is produc wih a consan rae. Deerioraion in his sage is modeled via a woparameer Weibull disribuion. A he second sage, he reailer places he order and receives he produc insanly. he deerioraion rae a his sage is a hree-parameer Weibull disribuion, which is iniial value depends on he ime he produc has spen in he firs sage before being ransferred. he behavior of differen key parameers of he model is analyzed using numerical sudies. Keywords: Invenory conrol, Deerioraing iems, Producion-disribuion sysems, woechelon supply chains Inroducion We develop a mahemaical model for a wo-echelon supply chain wih perishable iems. Producs are produced in he firs sage and are sen o he second sage o be socked. Differen deerioraion funcions are imposed for wo sages. Our purpose is o find he opimal policy, i.e., he opimal producion ime, producion cycle, invenory level and delivery cycle of he sysem. he wide variey of perishable iems and he enormous number of facories and manufacurers dealing wih hese producs as well as he applicaion of wo-echelon supply chains for heir producion and disribuion sysems is he main moivaion of his sudy. Inegraing producion planning and disribuion planning in a supply chain is necessary o achieve is opimal performance. hese sysems are called Explici Producion-Disribuion (EPD sysems. Among differen groups of EPD defined by Chen [, join Lo-Sizing and Finished-Produc Delivery Problem is he closes o our sudy. In his sudy we assume producion period is equal o disribuion period and iems lose heir values in boh sages. he deerioraion funcions are sochasic and follow Weibull disribuion. A he firs echelon, deerioraion is modeled via a wo-parameer Weibull disribuion similar o Cover and Philip [3 and a he second echelon deerioraion rae follows a hree -parameer Weibull disribuion as Philip [8. Sudies on inegraed invenory models wih perishable iems when he manufacurers and he reailers coordinae heir producion and ordering policies have received much aenion from researchers in recen years. Yang and Wee [ considered a wo-echelon sysem wih one manufacurer and several cusomers wih consan producion rae, deerioraion rae and demand. Wee e al. [ cied wo possible flaws in he cos funcion of Wee and Yang s model and give a proposal o eradicae he flaws. Rau e al. [9 proposed a model similar o ha of Yang and Wee [ wih he only difference ha he deerioraion rae is se o be exponenial. Yang and Wee [3 developed a muli-lo-size producion and invenory model of deerioraing iems wih consan producion and demand raes. Lo e al [7 derive an opimal soluion for an inegraed producion-invenory model wih imperfec producion processes and Weibull disribuion deerioraion under inflaion. Cheng and Wee [ sudied a producion- * Corresponding auhor: el: Fax: fjolai@u.ac.ir

2 7 Journal of Indusrial Engineering, Universiy of ehran, Special Issue, invenory deerioraion model considering pricing policy, warrany period, imperfec producion and sock dependen demand. Lee and Hsu [6 sudy he wo-warehouse invenory conrol model for deerioraing iems wih finie replenishmen rae. Wang e al [ empirically showed differen deerioraion raes in each echelon affec performances of individuals and inegraed invenory policies. o he bes of our knowledge, his is he firs sudy considering perishable iems in a wo-echelon supply chain having producion a he firs sage and warehouse a he second sage. he following assumpions are made in developing he mahemaical model: - Demand rae is consan. - he produc is produced on one producion line or producion machine. here is consan se up cos a he beginning of each producion cycle. 3- Invenory conrol is coninuous. 4- Lead-ime is consan and zero 5- A he firs sage here is only one producer. 6- A he second sage here is only one reailer. 7- Shorage is no allowed. 8- Deerioraion of iems begins a he firs sage righ afer being produced. 9- he rae of deerioraion and is parameers are known for boh sages. - Replacing or repairing he deerioraed iems is no allowed. - Our purpose is o find he opimal policy, i.e., he opimal producion ime, producion cycle, invenory level and delivery cycle of he sysem. he res of his paper is organized as follows: In secion we develop a mahemaical problem based on our assumpions. In Secion 3 we solve he proposed model using numerical analysis and invesigae he behavior of opimal soluion as differen parameers of he model change. Conclusion is presened in Secion 4. Mahemaical Model For describing he model and is soluion we need he following noaions, see figure : Firs Second P I I R Figure : An illusraion of he proposed woechelon sysem D: Demand per uni of ime P: Producion per uni of ime : Producion cycle (ime beween wo consequen seups R: Delivery cycle (ime beween wo consequen deliveries A: Delivery cos S: Se-up cos s: Se-up ime C : Cos of one uni of produc for producer C : Cos of one uni of produc for he reailer h: Cos of keeping an iem in per uni of ime p : Producion ime in a producion cycle : ime (represening he age of iems I i (: Invenory level a he i h sage (i=, f i (: Densiy funcion of iems life-ime a he i h sage (i=, F i (: Cumulaive funcion of iems life-ime a he i h sage (i=, Z i (: Deerioraion funcion of iems a he i h sage (i=, Firs, we will develop differen componens of he model separaely and a he end, he mahemaical model is inroduced. We have used he resuls from Cover and Philip [3 for deerioraion rae of produc in he firs sage when we have a Weibull disribuion wih wo parameers. P D

3 Invenory Conrol of Perishable.. 7 f (: Densiy funcion of iems life-ime in he firs sage, has a Weibull disribuion wih wo parameers; is p.d.f. is as follows: f ( exp( ( in which is he scale parameer and is he shape parameer. F (: Cumulaive funcion of iems life-ime in he firs sage may be used along wih reliabiliy heory o gives us he iniial ' f( deerioraion rae as Z ( which F ( resuls in: ' exp( Z ( ( exp( If (β>, he iniial deerioraion rae is increasing wih respec o ime. If (β<, he iniial deerioraion rae is decreasing wih respec o ime. If (β=, he iniial deerioraion rae is consan and Weibull disribuion urns ino exponenial disribuion. he behavior of he iniial deerioraion ' rae, Z (, as a funcion of ime is shown in ' he figure. Noe ha Z ( is he deerioraion rae for an iem in invenory of he firs sage a ime =. In oher words, his funcion is used when iems are in invenory from he beginning of he planning period and we canno use his funcion for a producer which produces gradually, i.e., iems deeriorae afer hey are produced and all iems are no produced a ime =. Since he producion rae is P, an iem which is produced earlier begins o deeriorae earlier, oo. Deerioraion Rae > = < ime Figure : he behavior of he iniial deerioraion rae We sugges finding a funcion for deerioraion rae by ake an average over ime, so we will have: x dx Z ( (3 According o he fac ha iems go o he invenory of he firs sage immediaely afer producion and heir deerioraion rae is zero a ime =, i implies ha β <. Comparing a siuaion in which he warehouse is relaed o a producer and he iems ener he warehouse gradually o a siuaion in which warehouse is for a reailer who receives an order alogeher, we find ou ha he deerioraion rae in he firs case is /β of he deerioraion rae in he second sage: Z ( Z (. I (: Invenory level a he firs sage he following differenial equaion shows he rae of invenory a he firs sage: di ( I ( Z ( d Pd In fac he above equaion shows ha invenory increases wih producion and decreases wih deerioraion. By replacing he value of Z ( in he above equaion we ge: di ( Pd ( I( d di( I( P d Solving he above equaion wih sandard mehod resuls in: I exp( x dx k exp( ( o find he consan value of k, noe ha he invenory is equal o zero a =: I ( k

4 7 Journal of Indusrial Engineering, Universiy of ehran, Special Issue, So we will have: I exp( x exp( ( dx f ( : Densiy funcion of iem life-ime a he second sage. As menioned before, he deerioraion rae in wo-parameer Weibull case begins a ime = eiher wih he rae of zero or infiniy (see figure. Iems afer being produced and ransferred o warehouse begin o deeriorae. While ransferring he iems o he warehouse of he second sage, deerioraion rae is no zero or infiniy. In fac i is no reasonable o use wo-parameer Weibull disribuion. his jusifies he use of hree-parameer Weibull which gives us more flexibiliy in modeling he deerioraion. We have used he resuls from Philip [8 for deerioraion rae of produc in he second sage when we have a Weibull disribuion wih hree parameers. he pdf of his disribuion is defined as follows: f ( ( exp[ ( : Scale parameer (α> : Shape parameer (β> : Locaion parameer ( λ F (: Cumulaive disribuion funcion of iems life-ime in he second sage can be used o find he deerioraion rae of he second sage. Similar o he firs sage we have: f ( Z ( F ( F ( exp ( his gives us: Z ( ( In his deerioraion funcion, is no consan. Deerioraion rae a P, when he producion in a cycle of he firs sage finishes, is equal o he deerioraion rae a = a he second sage. herefore, we will have: ( p By solving he above equaion, we will p have: I ( : Invenory level a he second sage he following differenial equaion shows he invenory level a he second sage: di( I( ( d Dd By solving he above equaion, we ge: I ( Dexp[ ( x dx k ( exp[ ( in which k is a consan and is deermined in he following way: I ( k Dexp[ ( x dx Moreover, by replacing k, we will have: I ( D exp[ ( x exp[ ( dx Objecive funcion he objecive of his model is o minimize he oal cos in boh sages which has he following componens: oal cos = Seup cos + Holding cos a he firs sage + Deerioraion cos in he firs sage + Delivery cos + Holding cos a he second sage + Deerioraion cos a he second sage. We will compue all pars of he objecive funcion separaely and a las, we will add hem up o find he objecive funcion as a whole.

5 Invenory Conrol of Perishable.. 73 Seup cos As menioned in he problem assumpions, seup cos is supposed o be consan. Since his cos is imposed a he beginning of each producion cycle, seup cos per ime uni will be: Seup cos per uni of ime= S Holding cos In order o compue he holding cos, we should analyze invenory behavior in boh sages. As i is shown in figure, seup begins a ime zero a he firs sage and finishes s unis of ime laer. A ime s, producion begins wih a consan rae P and coninues unil p. A his ime, all of he produced iems are ransferred o he second sage and wih a consan rae D hey are consumed. Noe ha in he figure, deerioraion rae is no considered. According o he figure, imum invenory a he end of producion ime a he firs sage is equal o he imum invenory a he momen of enering o he warehouse a he second sage. So imum invenory a he second sage happens o be a = and we have: D exp[ ( x dx I ( I exp[ ( And because I I, we can compue holding coss of he firs sage h p as: * I *. he raio p shows he fracion of ime in which he firs sage warehouse has invenory. Invenory cos of he second sage h can simply be saed as: * I. Delivery cos he delivery cos per uni of ime is follows: A Delivery cos per uni of ime= R Deerioraion cos for he firs sage oal producion a he firs sage= p * P Amoun of deerioraed iems a he firs sage= p * P I Cos of deerioraion a he firs sage per C( p * P I uni of ime= Deerioraion cos for he second sage A he second sage, he amoun of deerioraed iems is equal o he amoun of iems ha ener he second sage minus he iems ha are used o fulfill he demand: Amoun of deerioraed iems a he second sage= I DR. Cos of deerioraion a he second sage per C ( I DR uni of ime=. R Model Consrains As menioned before, he imum invenory a he firs sage is when producion is finished jus before he whole invenory is ransferred o he second sage. Maximum invenory a he second sage is when he whole invenory is received from he firs sage. So one of he problem consrains is he consrain which shows I I. p exp( exp[ ( P x dx D x dx exp( exp[ ( p Nex, noe ha he producion cycle is equal o, which consiss of seup ime, producion ime and idle ime. Second consrain is he one ha ensures seup ime and producion ime does no exceed he producion cycle. S p In he above equaions, we have: p Obviously, he model consiss of nonlinear non-convex objecive funcion, which has a nonlinear non-convex equaliy consrain and a nonlinear non-convex inequaliy consrain. his problem is

6 74 Journal of Indusrial Engineering, Universiy of ehran, Special Issue, difficul o solve using exac mehods and in a closed form. Numerical Analysis MALAB is used and he near-opimal soluions are obained using a coordinaed search mehod. o implemen he search in a narrower space, we can use a simple upper limi and lower limi for he producion ime: Clearly, he number of cycles canno D exceed P. On he oher hand, we have: s D number of cycles p. P From he above inequaliies, we will come o he following conclusion: Ds p ( P D For an upper limi on p, i is worhy o noe ha if p becomes very large, he deerioraion rae grows so fas ha he invenory level a he firs sage does no increase a all. In oher word, a very large producion ime may cause he deerioraion rae o be greaer han he producion rae. Finding he upper limi in his way is very complicaed hough. Insead we have used he following very simple upper limi in our numerical sudy: p <-s. In his secion, for analyzing he proposed model, we have solved a numerical example and done sensiiviy analysis so ha we can find ou wha are he effecs of changing differen parameers in he model. Example Assume a sysem which has a producer a he firs sage and a reailer a he second sage. he producer produces an iem wih he producion rae P and socks he iem a is warehouse. he deerioraion of he iem a he firs sage is following wo-parameer.5 Weibull in he form of Z(. Afer 6 finishing he producion ime and ransferring iems o he warehouse of he second sage, he deerioraion funcion changes o a hree-parameer Weibull of he form Z ( (.5(( Oher parameers values of he problem are as follows: =/6, =.5, h=., C =4, D=7, A=5, S=, s=, C =4 he opimal policy of he sysem is obained as follows: Opimal producion and delivery cycle (=R=76 Producion ime=6438 Maximum invenory=777 oal cos=96 Nex we show he impac of differen assumpions and parameers on he key elemens of he opimal policy. - he influence of he assumpion ha iems are perishable on oal cos, imum invenory level, I, and opimal producion cycle, p, is summarized in able. As i is expeced, adding he assumpion of perishabiliy o he problem assumpions will resul in increasing he oal cos. he behavior of he wo variables p and I wih and wihou perishable iems are depiced in figures 3 and 4 respecively. By adding perishabiliy assumpion o he model, increasing p resuls in I being increasing up o some poin and hen declining. his is because by increasing he invenory, deerioraion rae increases and afer some ime, i becomes larger han he producion rae. he behavior of p and are exacly he same as p and I. Iem is no perishable Iem is perishable Rae of change oal Cos Maximum Invenory Opimal Producion Cycle able : he effec of perishable iems

7 Invenory Conrol of Perishable I Cos Funcion p Figure 3: Behavior when iem is no perishable k Figure 5: he influence of α on oal cos funcion 3 8 I 6 4 * p Figure 4: Behavior when iem is perishable k Figure 6: he influence of α on p - Changing in α in he inerval (,/3 will resul in he following changes in he opimal soluion: By increasing α, he rae of deerioraion will increase oo, so i is expeced ha oal sysem cos increases oo (see figure 5. By increasing α we predic ha I decreases, because increasing α makes he rae of deerioraion o increase. As a resul, he invenory should decrease. his implies ha he producion cycle decreases oo (see figure he changes in he holding cos in he inerval of (,. will resul in he following changes in he opimal soluion: By increasing he holding cos, we predic I would decrease because increasing he holding cos, implies he invenory level o decline. he resul of comparing he opimal poins wih respec o holding cos changes is summarized in he able. he deerioraion cos and seup ime did no show any influence on C, p and I. As we can see, he model is more sensiive o and holding cos bu i is no sensiive o oher parameers. his means ha using non perishable models for he case of perishable iems may lead o he significan errors in oal cos esimaion as well as misake managerial decisions such as quaniy order. Also hese experimens showed ha having precise values of and holding cos are very imporan from poin of view of managemen. Conclusion In his paper we have developed a woechelon invenory model for perishable iems wih sochasic life-ime. A he firs sage, he deerioraion rae is following a woparameer Weibull disribuion while a he second sage, i is modeled via a hreeparameer Weibull disribuion.

8 76 Journal of Indusrial Engineering, Universiy of ehran, Special Issue, he opimal soluion is obained using numerical mehods and he influence of differen model parameers on he opimal policy is considered. he resuls reveal ha a small change in he deerioraion parameers has a grea effec in he oal cos. Our goal is o provide a general framework for hese kinds of problems and illusrae how his complicaed sysem can be modeled. o he bes of our knowledge, his is he firs paper which considers wo differen deerioraion raes for wo sages wih a high level of flexibiliy in modeling. Invenory cos oal Cos Maximum Invenory Opimal Producion Cycle % % % % ٠ % % % % able : he influence of changes in holding cos References: - Chen, Z.-L. (4. Inegraed Producion and Disribuion Operaions: axonomy, Models, and Review. Handbook of Quaniaive Supply Chain Analysis: Modeling in he E-Business Era, Simchi-Levi, D., Wu, S.D., Shen, Z.-J., (Eds, Kluwer Academic Publishers. - Chung, C.J. and Wee, H-M. (8. An inegraed producion-invenory deerioraing model for pricing policy considering imperfec producion, inspecion planning and warrany-period and sock-level-dependan demand. Inernaional Journal of Sysems Science, Vol. 38, PP Cover, R. and Philip G. (973. An EOQ Model for Iems wih Weibull Disribuion Deerioraion. AIIE rans. Vol. 5, PP Hill, R.M. (997. he Single-Vendor Single-Buyer Inegraed Producion-Invenory Model wih a Generalized Policy. European Journal of Operaional Research. Vol. 97, PP Hill, R.M. (999. he Opimal Producion and Shipmen Policy for he Single-Vendor Single-Buyer Inegraed Producion-Invenory Problem. Inernaional Journal of Producion Research. Vol. 37, PP Lee, C.C. and Hsu S-L (9. A wo-warehouse producion model for deerioraing invenory iems wih ime-dependen demands. European Journal of Operaional Research. Vol. 94, PP Loa, S-, Wee, H-M. and Huang, W-C. (7. An inegraed producion-invenory model wih imperfec producion processes and Weibull disribuion deerioraion under inflaion. In. J. Producion Economics. Vol. 6, PP Philip, G.C. (974. A Generalized EOQ Model for Iems wih Weibull Disribuion Deerioraion. AIIE rans. Vol. 6, PP Rau, H., Wu, M. and Wee, H. (3. Inegraed Invenory Model for Deerioraing Iems under a Muli- Echelon Supply Chain Environmen. In. J. Producion Economics. Vol. 86, PP Wang, K-J, Lin, Y.S. and Yu, J C.P. (. Opimizing invenory policy for producs wih ime-sensiive deerioraing raes in a muli-echelon supply chain. In. J. Producion Economics. Vol. 3, PP Wee, H.M., Jong, J.F. and Jiang, J.C. (7. A Noe on a Single-Vendor and Muliple-Buyers Producion- Invenory Policy for a Deerioraing Iem. European Journal of Operaional Research. Vol. 8, PP

9 Invenory Conrol of Perishable Yang, P.C. and Wee, H. M. (. A Single-Vendor and Muliple-Buyers Producion Invenory Policy for a Deerioraing Iem. European Journal of Operaional Research. Vol. 43, PP Yang, P-C. and Wee, H-M. (3. An inegraed muli-lo-size producion invenory model for deerioraing iem. Compuers & Operaions Research. Vol. 3, PP