An Approximate Net Profit Model for a Fixed-Life Perishable Product in a Two- Echelon Inventory System

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1 Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 2007, March 21-23, 2007, Regal Kowloo Hotel, Hog Kog, Cha A Approxmate Net Proft Model for a Fxed-Lfe Pershable Product a Two- Echelo Ivetory System Kachaa Kachaasutor*ad Aulark Techatsawad Abstracts-Ths paper further exteds the exstg cost model for a fxed-lfe pershable product a two-echelo vetory problem to the et proft model. The exteded model s subsequetly evaluated by comparg ts results wth those obtaed from OptQuest, a commercal smulatooptmzato package. Wth a lmted but reasoable tme, the comparatve results do ot show a sgfcat mprovemet of usg OptQuest, but do cofrm that the proposed model ca already provde qute good qualty of vetory cotrol parameters for ths system a much shorter computatoal tme. Wth some smple modfcatos to the proposed model, the model for slow-movg Posso demad dstrbuto for the two-echelo system s also developed ad subsequetly evaluated. The comparatve results show that for the slow-movg demad, the Posso model s more effectve tha the model wth ormalty assumpto. Idex Terms- Net Proft; Pershable Product; Two-Echelo Ivetory System I. INTRODUCTION The two echelo vetory system addressed ths paper cossts of oe cetral warehouse ad a umber of ether detcal or o-detcal retal outlets. Our assumptos are as follows. The demad at each retal outlet s statc ad ormally dstrbuted. Products are suppled by outsde supplers to the cetral warehouse wth zero product age mmedately before leavg the supplers. Trasportato lead-tmes from the outsde supplers to the cetral warehouse ad from the cetral warehouse to each retal outlet are assumed costat wth dfferet postve values. Stockout at the cetral warehouse s cosdered as backorders based o the frst-come-frst-served allocato polcy, whle the usatsfed demad at retalers s cosdered lost. Past research o the two-echelo vetory problem such as La ad Lu [3], Chu [1], ad Wllams ad Patuwo [6] maly provde the polces for o-pershable products. For papers pershable tem vetory problem for a twoechelo vetory system such as Yag ad Wee [7], Yag ad Wee [8], ad Rau et al. [5] address the problem varous forms ad perspectves ad very dfferet from the problem cosdered here. Ths paper, therefore, attempts to address a approach that ca hadle the tegrated fxed product lfetme. As the problem of the two-echelo vetory system for a fxed-lfe pershable product s too complex to be gve a exact mathematcal formulato, a approxmate cost model was developed Kachaasutor ad Techatsawad [2]. I ths paper, a approxmate model based o the et proft measure s, therefore, proposed ad evaluated based o the cost model of Kachaasutor ad Techatsawad [2]. We also exted our evaluato by comparg the results wth those of smulato-optmzato techque. Here, OptQuest, a commercal smulato-optmzato package, s appled. II. SYSTEM AND NOTATIONS I ths paper, we address the same vetory system ad cotrol polces as those Kachaasutor ad Techatsawad [2] llustrated Fgure 1. The cetral warehouse follows the revew terval R wth reorder pot s ad desred maxmum vetory level S, whle the two retalers apply the detcal revew terval R wth dfferet maxmum vetory levels (S 1 ad S 2 ). For each outlet, the lot sze equal to the dfferece betwee ts maxmum vetory level ad the curret vetory posto s ordered from the cetral warehouse every revew terval. The procuremet order from the cetral warehouse to outsde supplers s geerated whe the vetory posto at the warehouse s less tha reorder pot s, whch ca be see perod 3 Fgure 1. The fxed trasportato lead-tmes from the outsde supplers to the warehouse ad from the warehouse to the retal outlets are 2 perods ad 1 perod, respectvely. I perod 4, whe the product age reaches the ed of ts lfetme, the remag products are pershed ad, cosequetly, the o had vetory level becomes zero. The otatos used throughout the paper are summarzed the fgure 2. III. APPOXIMATE MODELS FOR A FIXED- LIFE PERISHABLE PRODUCT Ths secto begs wth the cost model of Kachaasutor ad Techatsawad [2], ad later a smple modfcato to the et proft fucto s preseted. A. The approxmate cost model - the base model I order to costruct the total cost model for pershable tem, Kachaasutor ad Techatsawad modfed the total cost model for o-pershable tem of Matta ad Sha [4] (hereafter referred to as MS model). The, the approxmate expected total cost model of Kachaasutor ad Techatsawad [2] the base model ca be wrtte as: UTCC Egeerg Research Papers

2 Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 2007, March 21-23, 2007, Regal Kowloo Hotel, Hog Kog, Cha S s Q T ( a ) Cetral warehouse S = maxmum vetory level at warehouse S = reorder pot at warehouse S 1 = maxmum vetory level at retaler 1 S 2 = maxmum vetory level at retaler 2 L = trasportato lead-tme at warehouse L 1 = trasportato lead-tme at retaler 1 L 2 = trasportato lead-tme at retaler 2 T = reorder cycle at warehouse l = product lfetme O 1 = outdatg uts at retal1 O 2 = outdatg uts at retal2 Q 1 = order quatty of retaler1 perod 1 Q 2 = order quatty of retaler2 perod 1 Q = order quatty of warehouse perod 3 S 1 Q 1 L 1 O 1 Assumptos: L 2 = L 1 = R L = 2R l = 4R ( b ) Retal outlet 1 S 2 Q 2 L 2 l ( c ) Retal outlet 2 O 2. Fgure 1 The vetory system of the proposed model 146 Idex: retal outlet, = 1,, Geeral Parameters ad Varables: c purchase cost per ut (supply ut cost) θ outdatg cost per ut k product lfetme wth zero lead-tme l actual product lfetme V the dfferece betwee maxmum vetory level ad reorder pot Warehouse Parameters ad Varables: A orderg cost at the warehouse h holdg cost per ut per perod at the warehouse T orderg cycle tme at the warehouse I vetory level at the warehouse B backlog at the warehouse Z z value of the stadard ormal dstrbuto for the probablty of o stockout at warehouse Retaler Parameters ad Varables: shortage or lost sales cost per ut at π h retal outlet holdg cost per ut per perod at retal outlet UTCC Egeerg Research Papers 2007 μ σ I B Z mea demad per revew perod at retaler stadard devato of revew perod demad at retal outlet vetory level at retal outlet umber of ut lost per perod at retal outlet z value of the stadard ormal dstrbuto for the probablty of o stockout at outlet Cost ad Proft Varables: UC purchase cost per perod OC outdatg cost per perod AC orderg cost per perod IC IC SC NP TR TRC Fgure 2 Notatos used throughout the paper vetory holdg cost per perod at the cetral warehouse vetory holdg cost per perod at retal outlet shortage cost per perod at retal outlet. system et proft system total reveue total relevat cost

3 Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 2007, March 21-23, 2007, Regal Kowloo Hotel, Hog Kog, Cha TRC = E( UC) + E( OC) + E( AC) + E( IC) + E( IC ) + E( SC ). = 1 = 1 (1) Cost fucto (1) s approxmately wrtte as: A E( O) C( Δ, Z, Z ) = + c* E( U) + θ* + h E( I) + [ h E( I ) E( B) + π ] T T = 1 (2) The terms EB ( ), E( O ), EU ( ), EI ( ), ad EI ( ) are referred to expected shortage, expected outdatg uts per order cycle, the expected umber of uts requred per perod, the expected vetory level at cetral warehouse ad the expected vetory level at retaler respectvely. The calculatos of these terms ca follow Kachaasutor ad Techatsawad [2]. B. The proposed et proft model I our model, total cost of the base model s modfed to the et proft fucto (3), whle the system total cost ca be calculated from Equato (2). NP= TR TRC (3) I addto, for the far comparso ad smplcty, the dscout factor ths study s set to 1. As a result, the probablty of o stockout, F( Z ), used to calculate the expected shortage at retaler, EB ( ), ca be wrtte as: π F( Z ) =. (4) ( π + c+ k* h + θ ) For the total reveue, t ca be drectly approxmated by: TR p[ E( D) E( B) ], (5) where E( D) ad E( B) are the expected demad ad lost sales per perod, respectvely. ED ( ) = μ ad EB ( ) = EB ( ). To determe the vetory parameters from ths proposed et proft model, the algorthm followg Kachaasutor ad Techatsawad [2] s appled, whle the oe-dmesoal search s corporate the et proft fucto. IV. COMPARATIVE PERFORMANCE EVALUATION AND ANALYSES To vestgate the effcecy of the proposed proft model, ts performaces are compared wth those obtaed from smulato-optmzato techque. Accordg to ts flexblty ad smplcty applcatos ad ablty to be corporated Area, a smulato-optmzato package, OptQuest s selected as a tool ths work. A Expermetal desg The expermet s desged wth dfferet combatos of related parameters, as specfed Table 1. The expermet s desged as a full factoral desg. For each combato, the expermet s coducted based o three dfferet algorthms. Thus, there are 256 possble combatos for each algorthm. The three algorthms are preseted as follows. Algorthm 1 the proposed proft model The approxmate optmum vetory parameters are frstly determed by solvg the approxmate et proft fucto (1). The resultg parameters are the assged the smulato model as vetory cotrol parameters. Before collectg the expermetal results, several prelmary studes are coducted to determe the durato ad mpact of tal codtos o the smulato results. The prelmary smulato records o average vetory levels at all stallatos show the covergeces of the performace wth 3,000 perods. Addtoally, as the varatos of the system total costs ad et profts obtaed from these plot rus are observed sgfcat, three replcatos are practcally chose here. I summary, for each combato, the smulated tme s of 3,000 perods, of whch the frst 300 perods are trucated. After the trucato, the statstcs are recorded ad performace dcators are calculated. The results of each combato are preseted as the average values of 3 replcatos. Algorthm 2 OptQuest Wthout ay suggested tal solutos, OptQuest s ru for optmal vetory cotrol parameters. Fgure 3 presets the smulato-optmzato process of the studed problem whose objectve fucto s to maxmze the system et proft per perod. The decso varables are the vetory cotrol parameters of the proposed model reorder pot at cetral warehouse (s), maxmum vetory level at cetral warehouse (S) ad maxmum vetory level at retal outlets (S ). I order to determe the approprate rug tme for the study, a set of plot expermets s coducted o eght scearos, ad covergece of the objectve fucto value s aalyzed. It should be oted that computer used ths smulato-optmzato expermet s Petum IV wth the speed of 2 Ghz. Table 1 Parameters the expermetal smulato model Number of retalers Supply ut cost Retaler orderg cost Warehouse vetory carryg cost Retaler vetory carryg cost Shortage cost at the warehouse Sellg prce Outdatg cost /120 20/-20 Average demad at a retaler (µ) Demad varato at a retaler (σ 2 /µ) Retaler lead-tme Warehouse lead-tme Warehouse orderg cost Shortage cost at retaler Lfetme 6 / / 9 0 / 4 4 / 8 40 / / tme of total lead-tme UTCC Egeerg Research Papers

4 Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 2007, March 21-23, 2007, Regal Kowloo Hotel, Hog Kog, Cha Fgure 4 llustrates, for example, the eght plot expermets whch four expermets are for demad equal to 6. The covergece of et proft over a umber of smulato rus wth 7 hours of the optmzato process. It ca be geerally observed that the et proft s ormally mproved sgfcatly the early stage of the optmzato process, average creasg by % of total chage or % of the maxmum value obtaed wth 16 smulato rus. Although the performace graphs do ot show the tme scale owg to the lmtato of report geerated from OptQuest, the 16 rus compared wth the maxmum umber of ru ofte costtuted a very small part of smulato ad are expected to cosume a very lttle smulato tme out of the perod of 7 hours. Thus, we ca coclude from those results that 7 hours of rug tme s reasoable. Other results also dcate that the 7 hour smulato-optmzato s suffcet to solve for good solutos. [Fgure 4] Net Proft (Best Value) Optmzato Procedure of OptQuest Output (System Net Proft) Iput (s, S, S ) Mea Demad = 6 Smulato Model of the Studed Problem Fgure 3 Smulato-optmzato process of the studed problem Smulato Ru Fgure 4 Performace graphs of the optmzato process Algorthm 3 OptQuest-WI Gve the tal soluto suggested by the approxmate model (1), OptQuest cotues a subsequet search for a mproved set of parameters values. The plot expermet s also performed order to determe the approprate lower ad upper bouds of the search area. As ths algorthm, the startg soluto s set by the values of the approxmate model. The rages are arbtrarly set, so that they are equvalet to those of Algorthm 2, ad that the covergece should become apparet wth the 7-hours optmzato tme. B. Comparatve results ad aalyses The results of the three algorthms compared wth those obtaed from the orgal model of Matta ad Sha [13] are preseted Table 2. The values the table represet the average et proft by each parameter value. Our aalyss o the vetory parameters prescrbed by dfferet methods show equal values up to a hgh degree of smlarty, especally those related to the approxmate proft models (1) ad OptQuest-WI (wth suggested tal solutos). However, those obtaed from the MS model ad OptQuest aloe are qute dfferet. From Table 2, t s evdet that OptQuest (Algorthm 2) s the least preferred method, yeldg all cases the lowest average et profts. Ths drawback may be attrbuted to approprate startg solutos, ofte leadg to trapped local optmums. O the other had, t s ot surprsg that the outcomes obtaed from OptQuest-WI are the best amog others. Due to the lack of a exact model ad comparable soluto, the results obtaed from the OptQuest-WI are used here as a bass for our comparso. Although we caot compare drectly betwee the results of the MS model ad the approxmate et proft model (1) due to ther dfferet objectve fuctos optmzed, we ote that there are 36 out of 256 cases whch the MS model geerates the slghtly hgher profts tha the approxmate model (1). All of these cases are cosstetly related to hgh demad varato. Ths declato effectveess of the proposed model as already appeared Kachaasutor ad Techatsawad [2], oce aga, reveals a weakess of ths proposed approxmato method determg the expected total purchase cost ad the expected outdatg cost. However, geeral, the approxmate model does acheve sgfcat mprovemet over the MS model. I those cases, the proposed model dd geerate sgfcatly lower total costs ad cosequetly hgher et profts, despte may cases, the slghtly lower total reveues that correspod to lower probablty of o stockout at each retal outlet. However, the sales volumes (ot reported) ad total reveues dd show a sgfcat dfferece betwee the two models. I addto, the proposed proft model ormally requres a lower umber of uts ordered from outsde supplers, a lower umber of order geeratos, ad a lower umber of outdatg uts at both cetral warehouse ad retal outlets. Although a lower servce level ad hgher shortage cost do exst the proposed model, ths dsadvatage s couter-balaced by other postve effects, leadg to hgher system et profts all cases. Uder the smlar stuatos, whe demad varato s hgh, t also reduces the attractveess usg OptQuest- WI, sce several cases OptQuest-WI caot further mprove the et proft. Accordg to the stochastc ature of smulato output, the hgh demad varato geerates the hgh varato the et proft, thus reduces the effectveess of the search algorthm behd the optmzato process. I addto, Table 2 also llustrates that the shorter the lfetme, the hgher the beeft obtaed from the proposed model. Such a favorable effect also exsts those cases of larger demad, hgher orderg cost at the cetral warehouse, shorter lead-tme at both cetral warehouse ad retal outlets, ad shorter usable lfetme (product lfetme total lead-tme). Focusg o the dfferece betwee dfferet outdatg costs, we fd that 148 UTCC Egeerg Research Papers 2007

5 Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 2007, March 21-23, 2007, Regal Kowloo Hotel, Hog Kog, Cha the case of postve outdatg cost (dsposg cost for a outdatg ut), a sgfcatly hgher mprovemet ca be ejoyed tha the case of egatve outdatg cost or postve salvage value of a outdatg ut. I Table 3, soluto qualty of the proposed model by each parameter value s measured by a ormalzed mprovemet made by OptQuest-WI: Table 2 Computatoal results of dfferet algorthms by parameter values Factor MS Model Algorthm 1 Algorthm 2 Algorthm 3 Proposed Proft OptQuest OptQuest-WI Model (1) Sellg prce , , , Outdatg cost Mea demad , , , Demad varato (σ 2 /µ) Warehouse orderg cost Shortage cost Usable lfetme Remarks: OptQuest = OptQuest wthout ay suggested tal solutos (Algorthm 2) OptQuest-WI = OptQuest wth suggested tal solutos (Algorthm 3) Table 3 Devato aalyss betwee the proposed proft model (1) ad OptQuest-WI Average Proposed Proft Rage of Factor OptQuest-WI devato (%) Model (1) devato (%) (std.dev.) (0.84) Sellg prce (1.25) Outdatg cost Mea demad Demad varato (σ 2 /µ) Warehouse orderg Cost Shortage cost Usable lfetme (0.97) (1.17) (1.22) (1.06) (1.12) (0.93) (1.08) (1.16) (1.09) (1.02) (1.24) (1.04) (0.85) Overall UTCC Egeerg Research Papers

6 Proceedgs of the Iteratoal MultCoferece of Egeers ad Computer Scetsts 2007, March 21-23, 2007, Regal Kowloo Hotel, Hog Kog, Cha The averages of percet devatos are characterzed by each parameter value, whle the devato rages are determed from the overall devatos obtaed from each test problem. As the overall average devato of 2.31% ad devato rage of 0.34% 9.02% are cosdered mmal, our approxmate model ca evdetly determe qute good qualty of vetory cotrol parameters a short computatoal tme. Table 3 also llustrates that the mprovemet ca be made the most ad the least by OptQuest the case of hgher ad lower sellg prce, respectvely. Whe the sellg prce s hgh, the approxmato error s accetuated by sale volume whch s maly approxmated from the expected demad; hece, the soluto qualty obtaed from the approxmate model s dramatcally dropped. Addtoally, the low demad varato ad low usable lfetme cases, the favorable mprovemets obtaed from QptQuest are more evdet. The characterstc of the search algorthm behd OptQuest ad the approxmato ad assumpto made the MS model cause ths kd of pheomea. As metoed the prevous aalyss whe demad varato s hgh, the performace of OptQuest s dropped ad appears to be less tha that of the proposed model. The mprovemet made by OptQuest s thus less favorable. For usable lfetme, due to the assumpto we followed Secto 3, the shorter the lfetme, the more the approxmato of our proposed models, leadg to the hgher mprovemet made by OptQuest I terms of computatoal tme, all approxmate models are solved by Matlab approxmately 2.5 mutes of elapsed tme, up to 5 mutes the worst case. It should also be oted that the et proft model (1) takes a slghtly loger ru tme tha the total cost model of Kachaasutor ad Techatsawad [2]. Compared wth 8 hours rug tme or eve the tme cosumed by the frst 16 rus whch the most mprovemet s made from OptQuest, the approxmate models usually requre a sgfcatly lower computg tme, but gve qute reasoably good qualty of solutos. The vetory cotrol parameters obtaed from OptQuest-WI were regarded as the ear-optmal solutos ths paper, owg to the lack of a exact model ad soluto to evaluate agast. The expermetal results dd cofrm that the proposed model could geerally provde qute reasoably good qualty of vetory cotrol parameters for ths vestgated system a short computatoal tme, ad dd ot show ecouragg results of usg OptQuest. However, oe should be cautous applyg the models, as ther lmtato exsted whe the demad varato was hgh. REFERENCES [1] Chu H. N., A Approxmato to The Cotuous Revew Ivetory Model wth Pershable Items ad Lead Tmes., Europea Joural of Operatoal Research, vol. 87, pp [2] Kachaasutor K. ad Techatsawad A., A Approxmate Perodc Model for Fxed-Lfe Pershable Products a Two-Echelo Ivetory-Dstrbuto System. accepted for publcato the Iteratoal Joural of Producto Ecoomcs. [3] La Z. ad Lu L., Cotuous Revew Pershable Ivetory Systems: Models ad Heurstcs. IIE Trasactos, vol. 33, pp [4] Matta K. F. ad Sha D., Polcy ad Cost Approxmatos of Two-Echelo Dstrbuto Systems wth a Procuremet Cost at the Hgher Echelo. IIE Trasactos, vol. 27, pp [5] Rau H., Wu M. Y. ad Wee H. M., Itegrated Ivetory Model for Deteroratg Items uder A Mult-Echelo Supply Cha Evromet. Iteratoal Joural of Producto Ecoomcs, vol. 86, pp [6] Wllams C. L. ad Patuwo B. E., A Pershable Ivetory Model wth Postve Order Lead Tmes. Europea Joural of Operatoal Research, vol. 116, pp [7] Yag P. C. ad Wee H. M., Ecoomc Orderg Polcy of Deterorated tem for Vedor ad Buyer: A Itegrated Approach. Producto Plag ad Cotrol, vol. 11, pp [8] Yag P. C. ad Wee H. M., A Sgle-Vedor ad Multple-Buyers Producto-vetory Polcy for A Deteroratg Item. Europea Joural of Operatoal Research, vol. 143, pp V. CONCLUSIONS AND FURTHER RESEARCH I ths paper, a perodc revew vetory-dstrbuto model for pershable product vestgated ad proposed by Kachaasutor ad Techatsawad [2] was modfed to the et proft fucto. OptQuest, a smulato-optmzato package was appled to evaluate the performace of the approxmate model ad to solve ths complex vetory-dstrbuto system. The expermet was desged order to compare the et profts obtaed from the four algorthms: 1) the orgal model of Matta ad Sha [13], 2) the proposed et proft model (1), 3) the smulato-optmzato algorthm usg OptQuest, wthout ay kowledge of approprate vetory cotrol parameters, ad 4) the smulatooptmzato algorthm usg OptQuest, but wth the suggested approprate tal solutos (OptQuest-WI). 150 UTCC Egeerg Research Papers 2007