Method for Production Planning and Inventory Control in Oil

Transcription

1 Memors of the Faculty of Engneerng, Okayama Unversty, Vol.41, pp.20-30, January, 2007 Method for Producton Plannng and Inventory Control n Ol Refnery TakujImamura,MasamKonshandJunIma Dvson of Electronc and Informaton System Engneerng Graduate School of Natural Scence and Technology Okayama Unversty 3-1-1, Tsushma-Naka Okayama, (Receved November 13, 2006) Abstract In ths paper, we propose a smultaneous optmzaton method for nventory control and producton plannng problem for a chemcal batch plant. The plant conssts of blendng process, ntermedate storage tanks and fllng process. In the proposed method, the orgnal problem s decomposed nto producton plannng sub-problem and nventory control sub-problem. Then the decson varables are optmzed by alternately solvng each sub-problem. The soluton of the proposed method s compared wth that of centralzed optmzaton method. The effectveness of the proposed method s nvestgated from numercal computatonal results. 1 Introducton Recently, hgh-mx low-volume producton has been accelerated through necessty by the dversfcaton of customer s demand. Under these crcumstances, havng excess stock causes not only the ncrease n the nventory cost but also decrease n proft because of abrogaton of the stock when specfcaton of product s changed. Especally, n lubrcant manufacturng factory producng several hundred kl or more, proper nventory control s ndspensable. Therefore, t s necessary to make producton plannng that mnmzes total cost for productons wth mnmum nventory consderng entre factory at the same tme. Many of past researches about the producton management for chemcal plants drected to optmze producton plan under the condtons of gven due date for jobs or amount of producton etc [1][2]. However, such the optmzaton only of producton plan s nsuffcent from the vewpont of optmzaton of the entre factory. Therefore, t exsts necessty for plannng that mamura@cntr.elec.okayama-u.ac.jp consders both the producton plan and the nventory control at the same tme. Heretofore, the nventory control and the producton plannng n the lubrcant manufacturng factory have been herarchcally decded [3]. That s, nventory control system that s a superor system outputs the producton request to product by whch amount of nventory fall below reasonable nventory quantty, and the producton plannng system decde producton plan that s based on the producton request. However, such a method can t necessarly optmze total plan. The smultaneous optmzaton of nventory control and producton plannng have been studed[4][5], but these researches are drectng to model that s composed only sngle stage and equpment of gven process performance. However, the lubrcant manufacturng process s mult stage producton composed of the mxture process and the fllng process, etc. and ntermedate storages between them. And, the processng performance of equpment changes by the producton planoftheblendngmachneandallocatngjobtothe ntermedate storages. Therefore, past ntegrated op- Ths work s subjected to copyrght. All rghts are reserved by ths author/authors. 20

2 Takuj IMAMURA et al. MEM.FAC.ENG.OKA.UNI. Vol.41 tmzaton method can t be appled drectly to such a lubrcant manufacturng factory. In ths paper, producton system consdered producton process bear pecular complex restrctve constrants of chemcal plant and nventory control s modeled, and optmzaton method of coordnatng nventory control producton plannng s proposed. 2 Optmzaton problem of producton plannng and nventory control 2.1 Defnton of the problem Problems are for producton plannng and nventory controlofolrefneryplantasshownnfgure1. The problem treated here s defned n the followng. Fg. 1: Chemcal Plant Producton plannng Producton process s composed of blendng process, called#1process,nwhchrawmateralgroup#1and that of #2 are blended and fllng process, called #2 process, n whch materals from blendng process s packed n predetermned products wares. Both processes requre one tme perod for ts productons regardless of ts producton amounts. The capacty of blendng machne s predetermned for one perod of tmeandtsprohbtedtoblendamountsmorethan the capacty. On the other hand, fllng capacty s also determned beforehand. It s natural that both processes can not process same product at the same tme. Between these processes, there nstalled plural storage tanks actng as buffers storng blended materals n a certan perod of tme. The capactes for storage tanks are assumed to be suffcently large. It s prohbted to move blendng materals from one tank to another durng storng. Materals stored n tanks can be freely dverged nto plural jobs n the next fllng process. It s natural that the transfer lnes from some tanktothefollowngfllngprocesscannotbeusedfor other producton. In case of change n product knd for storage tank, change over s necessary nducng change over cost. Remanng product, named remanng ol, s stored n the tank for no change over. Flled materals are stored as product stocks Inventory control It s assumed that demands from customers are known forplannngperodsbothatpresentandpasttme. It s prohbted to have the shortage of nventory. Inventorycostsnducedfromtheamountofproductstorage. In our research, mnmum amount of products storage s assured to have the safety operatons. The amount s necessary for safety stock preparng probablstc changes n demand and occurrence of demands. The amount s calculated based on safety factor relatng to servce level and past demands data. The servce level s a probablty to be able to comply wth order mmedately when there s demand from the customer and the post-process[6]. Cost penalty s added n case of shortage n storage. 2.2 Mathmatcal model When t(t=1,2,,t)s set as a producton plannng perod, the ntegrated optmzaton problem of the nventory control problem and the producton plannng problem can be formulated as the followng mxed nteger lnear programmng(milp) problem. [Notatons] Sets: : Setofproducts R: Setoftanks Decson varables: St, P : Inventory level of product at the end of tme perodt ( S0, P ( ):gven) E t, : Shortageamountofnventoryofproductfrom amountofsafetystockattheendoftmeperod t 21

3 January 2007 Method for Producton Plannng and Inventory Control n Ol Refnery St,k, I : Intermedate nventory level of product n tankkattheendoftmeperodt ) (S I0,k, =0( k, ) B t,k, : Amountofblendngofproductntankkat thetmeperodt F t,k, : Amountoffllngofproductntankkatthe tme perod t { 1 (fb t,k, >0) δ t,k, = 0 (otherwse) γ t,k, = { θ t,k, = λ t,k, = ξ t,k,,= 1 (ff t,k, >0) 0 (otherwse) 1 (Ifproductsproducedusngtankk attmeperodt) 0 (otherwse) Cost coeffcents: 1 (If ntermedate nventory or remanng olofproductexstntankkattme perod t) 0 (otherwse) 1 (Iftsswtchedfrom tontank katthefrstoftmeperodt) 0 (otherwse) µ P : Factorofproductnventorycost ω: Penalty cost coeffcent for shortage amount of nventory from amount of safety stock µ I : Costcoeffcentforntermedatenventory φ: Cost coeffcent for blendng set up χ,: Penaltycostcoeffcentforproduct toproduct Constant data: T: Numberoftmeperod l: Maxmumlotsofblendngnonetmenterval F max : Maxmumamountoffllngnonetmenterval D t, : Amount of demand of product at perod t (gven) Q t, : Amountofsafetystockofproductatperodt [Problem Descrpton] (P):mn Z (1) Z= µ P St,+ P ωe t, + t, t, t,k.µ I St,k, I + φδ t,k, + χ,ξ t,k,, (2) t,k, t,k,, subject to St, P =SP t 1, + F t,k, D t, ( t, ) (3) k 0 ( t, ) (4) S P t, E t, Q t, S P t, ( t, ) (5) E t, 0 ( t, ) (6) St,k, I =SI t 1,k, +B t,k, F t,k, ( t, k, ) (7) δ t,k, l ( t) (8) k, F t,k, F max ( t) (9) k, δ t,k, +γ t,k, 1 ( t, k, ) (10) θ t,k, 1 ( t, k) (11) λ t,k, =1 ( t, k) (12) δ t,k, λ t,k, 0 ( t, k, ) (13) δ t,k, +λ t 1,k, λ t,k, 0( t, k, ) (14) δ t,k, +λ t 1,k, λ t,k, 0( t, k, ) (15) λ t 1,k, +λ t,k, 2 ξ t,k,, 0 ( t, k,, ) (16) λ t 1,k, +λ t,k, 2 ξ t,k,, 1 ( t, k,, ) (17) 0 F t,k, S I t 1,k, ( t, k, ) (18) S I t,k,,b t,k, 0 (19) (,, k R, t=1,,t) Eq.(2) represents objectve functon, and the frst term represents the nventory cost, the second term represents the penalty cost for shortage amount nventory fromamountofsafetystock, the thrdtermrepresents the ntermedate nventory cost, the forth term represents blendng set up cost and the ffth term represents change over cost. Eq.(3) represents the nventory flow constrant. Eq.(4) represents the amount of the product stock s nonnegatve. Eq.(5) and Eq.(6) represent the restrctons concernng the dfference between theamountofthesafetystockandtheamountofthe nventory. Eq.(7) represents the ntermedate nventory flow constrant. Eq.(8) represents the blendng operaton capacty constrant. Eq.(9) represents the fllng 22

4 Takuj IMAMURA et al. MEM.FAC.ENG.OKA.UNI. Vol.41 operaton capacty constrant. Eq.(10) represents that the prohbton of the smultaneous processng n blendng and fllng. Eq.(11) represents that there s one kndofproductthatcanbemantanedneachtankat the same tme. Eq.(12),Eq.(13),Eq.(14) and Eq.(15) representtheconstrantforλ t,k,. Eq.(16)andEq.(17) represent the constrant for ξ.eq.(18) represents mnmum amount and maxmum amount of the fllng of each product n correspondng tank at each tme perod. When the safety stock s calculated by usng thedemanddataath perodofthepast,theamount of demand of product D t τ, ( t = 1,,T, τ = 1,,H, =1,,I)atτ perodbeforetperodare gven, the amount of safety stock Q t, of product s computed by usng followng expressons. ( ( σ t, = 1 H H )) 2 τ=1 D t τ, D t τ, (20) H 1 H τ=1 Q t, =m LT σ t, (21) Here, σ t, represents the root-mean-square devaton ofproductatthetmeperodt. m representssafety factorofproduct. LT representsleadtmeofproduct. 2.3 Decomposton of the problem It s dffcult to optmze all varables at the same tme. The number of dscrete varables may rapdly ncrease n the model of the nventory control problem and producton plannng problem for chemcal plant. So, n ths research, the problem s optmzed decomposng orgnal problem to some sub-problems, and applyng the decentralzed optmzaton method. Artfcal varable Ft, ICP are ntroduced to the model from Eq.(1) to Eq.(19), and the constrant Eq.(22) s added. ThsproblemsnamedproblemP 2. F ICP t, = k F t,k, ( t=1,,t, ) (22) When Eq.(22) s relaxed by usng nonnegatve Lagrangemultplerν t,,relaxatonproblemrp 2 ofproblemp 2 canbeformulatedasfollows. (RP 2 ):mn L (23) L= µ P St, P + ωe t, + t, t, t,k.µ I St,k, I + φδ t,k, + χ,ξ t,k,, t,k, t,k,, + ( ν t, Ft, ICP ) F t,k, t, k subjectto Eq.(4) (19) (24) S P t,=s P t 1,+F ICP t, D t, ( t, ) (25) ( t=1,,t, ) Eq.(25) represents the nventory flow constrant. It s obtaned by transformng Eq.(3) usng Eq.(22). Lagrangan functon L can be descrbed as follows by consoldatng the varable. L=Z ICP +Z SP (26) Z ICP = t, µ P S P t, + t, ωe t, + t, ν t, F ICP t, (27) Z SP = µ I St,k,+ I t,k, t,k. t,k,φδ + ν t, F t,k, (28) t,k,,χ,ξ t,k,, t, WhenacertanLagranganmultplerν t, aregven, the relaxaton problem of mnmzng Lagrangan functon L can be decomposed to the followng sub-problem ICP andsp. (ICP):mn Z ICP (29) subjectto Eq.(4) (6),(25) F ICP t, 0 ( t=1,,t, ) (30) (SP):mn Z SP (31) subjectto Eq.(7) (19) Problem ICP s a sub nventory control problem mnmzng weghted sum of nventory cost, penalty cost for shortage amount of nventory from amount of safety stock. Here, artfcal varable Ft, ICP means the amount of the fllng of product at tme perod t that s requred of nventory control sde from the producton sde. In the followng, the Ft, ICP s called the amount of the fllng demand. Problem SP s a sub producton plannng problem mnmzng weghted sum of ntermedate nventory cost, blendng cost and change over cost. k 23

5 January 2007 Method for Producton Plannng and Inventory Control n Ol Refnery 3 Decentralzed soluton algorthm 3.1 Outlne of the algorthm In the algorthm of Lagrangan relaxaton, soluton process of each sub problem and update of Lagrangan multpler are carred out alternatvely. Bascally there s no assurance of convergence of the computaton. To prove the problem, the penalty functon method by Nshetal[7]sused. Inthemethod,thedstancefrom the feasble soluton s forced to added to the objectve functon as a penalty cost. As the results, feasblty of the obtaned soluton can be assured after ncreasng of penalty weght. The constructon of soluton process combnng ICP, nventory control sub-system, and SP, producton plannng sub-system s shown n Fgure 2. optmzaton of product plannng, and decdes nventory control plan that mnmze weghted sum of the nventory cost, the penalty cost for shortage amount of nventory from amount of safety stock and penalty to gap from feasble soluton. Step3 Product plannng The producton plannng sub-system decdes the producton plan consderng the fllng demand {Ft, ICP } to mnmze weghted sum of the ntermedate nventory cost, the blendng cost, the change over cost and penalty to gap from feasble soluton. Then,theamountofthefllng{F t,k, } obtaned by the optmzaton of product plannng s transfered to the nventory control sub-system. Step4 Evaluaton of convergence If the tentatve plan s feasble satsfyng Eq.(22), the calculaton s ended. Otherwse, algorthm proceeds to Step5. Fg. 2: Structure of optmzaton system The algorthm for solvng each sub problem s descrbed n the followng. Step1 Readng of ntal data Each sub-system retreves necessary data of the resource constrants, each cost coeffcent, and theamountofdemandofeachperodetc. Moreover,theamountofthesafetystockofeachproduct for total tme horzon s calculated. In addton, the weghtρnthe penalty terms ntalzed. Step2 Inventory control plannng The nventory control sub-system decdes the nventory control plan to mnmze the objectve functon, and passes amount of the fllng demand {Ft, ICP } the producton plannng sub-system. Here, though t optmzes wthout consderng the producton plannng sub-system n ntal teraton, n teraton snce the second tmes, t receve the amount of fllng of each product n correspondng tank at each perod {F t,k, }, t obtaned by Step5 Update of weght ρ The weght ρ s ncreased n ρ. Thereafter, algorthmrepeatsfromstep2to5untlconvergence of evaluaton. Frst of all, each sub-system retreves data as a preparaton. Afterwards, the nventory control sub-system and the producton plannng system optmze the orgnal problem by repeatng optmzaton of the each problem accordng to each objectve functon. Here, each sub-system exchanges the amount of fllng of correspondng product at each perod to satsfy the consstency of Inventory control plan and producton plan. Each sub-system adds penalty to the dfference between fllng plan preferable for each sub-system and the fllng plan obtaned from another sub-system to each objectve functon. The soluton of orgnal problem s gradually approachs to feasble soluton by ncreasng the value of the penalty coeffcent gradually fll the teraton ends. The flow of the algorthm s shownnfgure3. In the followng, optmzaton of each sub-system wll be stated. 3.2 Inventory control sub-system Inventory control sub-system In the target chemcal plant, due to the restrctons for usable number of tanks and maxmum number of 24

6 Takuj IMAMURA et al. MEM.FAC.ENG.OKA.UNI. Vol.41 for the plan. To overcome the dffcultes modfed constrant for SP sub-system s added to ICP sub-system as follows. F1,t ICP =0 ( t=1,,t) (34) Ft, ICP F max ( t=1,,t) (35) η t, K ( t=1,,t) (36) Fg. 3: Flow chart of proposed method lots n one tme nterval are predetermned. The feasblty of nventory plan determned by nventory control plan s affected from demand of fllng operatons for each tme perod. To reflect the effect, the dfference between fllng plan by nventory and that by producton plan s added to the objectve functon of nventory plannng as the penalty factor. Optmzaton problem ICP n the nventory control sub-system can be formulatedbyaddngabnaryvarableη t, asthefollowng mxed nteger lnear programng problems. η t, = { 1 (ff ICP t, >0) 0 (atherwse) ( t, ) (ICP):mn Z ICP (32) Eq.(34) represents constrant of the amount of the fllngnatthefrsttmeperod,andtsobtanedfrom the ntal condton of St,k, I, Eq.(18), and Eq.(22). Eq.(35) represents the upper bound of the amount of the fllng, and t s obtaned from Eq.(9) and (22). Eq.(36) represents upper bounds of number of productkndthatcanbethefllngprocessngforonetme perod. Ths constrant can be obtaned from problem settngofonlyonekndofproductcanbeprocessedat thesametmeneachtank. Theoptmalsolutoncan be obtaned by usng a commercal solver because ICP s mxed nteger lnear programmng problem ncludng contnuous varable. 3.3 Producton plannng In the target chemcal plant, capacty of producton s affected by producton plan because ntermedate storage of materals n tanks and dvergence of jobs n the fllng process may be occurred. The examples are shown n Fgure 4 and Fgure 5, where only one tank s usable for ntermedate storage. Z ICP = t, µ P S P t, + t, ωe t, + t, ρ η t, Γ t, (33) subjectto Eq.(4) (6),(25),(30) The thrdterm of rght sde nequaton(33) s the artfcally added factor representng dfference between thevalueoffllngplanη t, byicpandγ t, thatbysp. Γ t, represents the presence of the fllng plan of each product at each perod n producton plan computed byproductonplannnngsub-system. If k γ t,k, 1, thenγ t, =1,otherwsezero. IncreasngthevalueofweghtρaftersolvngSP sub problem, t becomes possble to derve feasble soluton. However, f we use only the penalty method, convergenttmeapttobelargeduetotheotherconstrants Fg. 4: Change of the producng capacty Fg.4 represents a Gantt chart of producton plan that desgned fllng of product A from second terms to fourth term. On the other hand, Fg.5 represents one that desgned fllng of product A from second terms tothrdtermandtofproductbatfourthterm. In both cases, one lot fllng s desgned from second term to fourth term. However, feasble soluton s obtaned n Fg.4 and nfeasble one s obtaned n Fg.5 due to the prohbton by constrants. Thus, the 25

7 January 2007 Method for Producton Plannng and Inventory Control n Ol Refnery h j,t,k =g j ( j J) (37) t,k x j,t,k =1 ( j J) (38) t,k a j = t x j,t,k ( j J) (39) t,k Fg. 5: Change of the producng capacty r j = a j d j ( j J) (40) capacty of each tme nterval n producton vares accordng to producton plan. In the plannng for nventory control, t s mpossble to reflect such change n producton capablty. As the result, calculated fllng request made by nventory control plannng may be nfeasble for producton. So t s necessary to revse the calculated fllng request from nventory for the total feasblty of obtaned results by updatng penalty factor ρ. In the proposed method, feasble soluton s created byft, ICP sdeemedtofllngjobofproductkndu (= ),duedated t (=t),amountoffllngg j (=F ICP t, ),and the devaton from due date s allowed wth penalty cost s named devaton from due date penalty cost. The producton plan sub-system can make feasble producton plan consderng the gven fllng demand, by usng ths method. The reason to gve producton plannng sub-system the devaton from due date penalty cost s the producton capacty changes greatly by changng the processng tme perod of blendng lots and fllng lots because these processes are batch process. The followng notatons are ntroduced nto problem SP. Sets: J: Setoffllngjobs. π : Setoffllngjobsthatsatsfyu j = Decson varables: h j,t,k : Amountoffllngjobjntankkattmeperod t { 1 (fh x j,t,k = j,t,k >0) 0 (otherwse) a j : Fllngdateoffllngjobj r j : Amount of fllng of product n tank k at the tme perod t Then, the followng constrants are added to problem SP. Eq.(37) and Eq.(38) represent that all the fllng jobs are processed. Eq.(39) represents the defnton constrantofa j. Eq.(40)representstheconstrantforr j. Moreover, the exstng constrant s converted as follows. S I t,k,=s I t 1,k,+B t,k, j π h j,t,k ( t, k, ) (41) h j,t,k F max ( t) (42) j,k δ t,k, +x j,t,k 1 ( t, k,, j π ) (43) 0 j π h j,t,k S I t,k, ( t, k, ) (44) (, k R, t=1,,t) Eq.(41) represents nventory flow constrants. It s obtaned by transformng Eq.(7). Eq.(42) represents the fllng operaton capacty constrant. It s obtaned by transformng Eq.(9). Eq.(43) represent that the prohbton of the smultaneous processng n blendng and fllng. It s obtaned by transformng Eq.(10). Eq.(44) represents mnmum amount and maxmum amount of fllng of each product n correspondng tank at each tme perod. It s obtaned by transformng Eq.(18). When t occurs the devaton from due date nventory plannng sub-system can t create feasble soluton that satsfy the fllng plan s obtaned by optmzaton of product plannng because of out of nventory. It causes delay of convergence of soluton. So the followng constrant that represents the lowest amount to be flled before each perod to out of nventory s not causedsaddedtoproblemsp. Thsconstrantsobtaned from Eq.(3) and Eq.(4) S P 0,+ t t =1 k j π h j,t,k (, t=1,,t) t D t, 0 ( t, ) (45) t =1 Therefore, problem SP can be formulated as a problem to mnmze weghted sum of the ntermedate nventory cost, the blendng set up cost, the changeover 26

8 Takuj IMAMURA et al. MEM.FAC.ENG.OKA.UNI. Vol.41 cost,andthedevatonfromdue date penaltycostto the fllng demand as follows. Here, κ 1, κ 2 n fourth term of Eq.(47) are added artfcally to match the value of penalty that s added to problem SP to the valueofpenaltythatsaddedtoproblemicp because those penaltes are dfferent. (SP):mn Z SP (46) Z SP = µ I St,k,+ I t,k, t,k, t,k,φδ + t,k,,w,ξ t,k,,+(κ 1 ρ+κ 2 ) j subjectto Eq.(8),(11) (17),(19),(37) (45) r j (47) It s dffcult to attan strct optmzaton because of the objectve functon of problem SP contans the changeover cost that depends on order of operaton. Then, the producton plan s optmzed by usng the algorthm of the followng SA (Smulated annealng mothod)[8] s the followng algorthms. Frst of all, to expand the search space of the soluton, the constrants of Eq.(8) and Eq.(11) are relaxed and added to the objectve functon as a penalty lke n Eq.(49). (SP):mn Z SP (48) Z SP = µ I St,k, I + t,k, t,k, t,k,φδ + t,k,,w,ξ t,k,,+(κ 1 ρ+κ 2 ) j + ζν t + ǫα t,k (49) t t,k subjectto Eq.(12) (17),(19),(37) (45) ν t δ t,k, l ( t, k, ) (50) k, ν t 0 ( t, k, ) (51) α t,k θ t,k, 1 ( t, k, ) (52) α t,k 0 ( t, k, ) (53) (, k R, t=1,,t) Here, ζ represents the penalty cost for the volaton of blendng operaton capacty constrant. ǫ represents the penalty cost for the volaton of resource constrant about tanks. Step1 Intal allocaton of the fllng jobs To satsfy due date, the fllng jobs are allocated toatank. r j Step2 Producton plannng Problem SP s solved by usng a commercal solver, and the producton plan that s satsfy allocatons ofthe jobsthatsdecde ntheprevous steps obtaned. Step3 Evaluaton of producton plan and adopton judgment The producton plan s evaluated by usng Eq.(49). And, the adopton judgment of the producton plan s decded accordng to the rule of the SA mothod. Step4 Neghborhood operaton To satsfy the constrant of Eq.(45), the allocatonofafllngjobthatstoselectatrandoms changed at random. And a regulated frequency repeatsfromstep2tostep4. 4 Numercal experments 4.1 Centralzed method To check the valdty of the proposed method, results are compared wth the centralzed total optmzaton method. The compared method s based on SA method. In the centralzed method, once fllng plan, thetmeoffllngofeachproductkndandtanknumber are made and then the volume of fllng s determnedbysaalgorthm. Andatthesametmeblendng volumes are also determned. The procedure of the centralzed method s gven as follows. Step1 Intal allocaton of γ t,k, Allgamma t,k, thatrepresentthepresenceofthe fllngplanofeachproductntankateachperod aredecded. Here,thosearedecdedasthetamp ahead plan and the constrants from Eq.(3) to Eq.(19)andγ t,k, =1aresatsfed. Step2 Decson of nventory plan and producton plan The nventory control plan and the producton plan that mnmze Eq.(2) and satsfy γ t,k, are decded n the prevous step usng a commercal solver. Step3 Evaluaton of producton plan and adopton judgment The nventory control plan and the producton plan obtaned n Step3 are evaluated. And, the 27

9 January 2007 Method for Producton Plannng and Inventory Control n Ol Refnery adopton judgment of the producton plan s decdedaccordngtotheruleofthesamethod. Step4 Neghborhood operaton t, k, and are selected at random, and γ t,k, s reversed. Table. 2: Parameters for the makng of demand product root-mean average amount -square devaton of demand A B C D E F G H I J Table. 3: Changeover cost data Fg. 6: Flow chart of centrazed method 4.2 Numercal Examples Numercal experments are conducted for 3 cases examples shown n Table1.The demand of each product at each perod n the plannng term and H term of past mmedately before the plannng term s generated usng normal random number based on root-mean -square devaton and average amount of demand are shownntable2. ThechangeovercostsshownnTable 3. The product data s shown n Table 4. Other datasshownntable5. Table. 1: Examples example Number of Number of Number of Tme Perod Product Tank CASE CASE CASE from\to Table. 4: Product Data product m LT S P 0, A B C D E F G H I J Expermental result and consderaton The system developed uses CPLEX8.0 as a commercal solver. An ntal value of the penalty coeffcent ρ 28

10 Takuj IMAMURA et al. MEM.FAC.ENG.OKA.UNI. Vol.41 Table. 5: Other data H: 5,µ I : 1,µ P : 4,l: 4 F max 1 : 500,ω: 3,φ: 100,κ 1 : 3 κ 2 : 10,ζ: 1000,ǫ: 3000 ssettobezeroand ρ=300. Moreover,theparameter of the SA method that s used when the system solvetheproblemsp ntheproposedmethodandthe centralzed method s shown n Table 6. Fg. 7: Comparson of computaton tme Fg. 8: Comparson of evaluaton value Table. 6: Parameters for the SA parameter Proposed Centrazed Method(SP) Method Maxmum Temperature Mnmum Temperture Coolng Perod Coolng Rate The comparng of computaton tme and evaluaton value of plans that are optmzed by the centralzed method and the proposed method are shown n Fgure 7 and Fgure 8. Moreover, Gantt charts that are obtanedbythosemethodsareshownnfgure9and Fgure10. Here,thealphabetsshowthekndofproduct and numbers show the value of the processng n these fgures. It can be confrmed that the proposed method obtans the better soluton n a short computaton tme compared wth centralzd method. Ths reason s thought that the SA method used for the centralzed method that can obtan the optmal soluton n nfnte tme s can t optmze problem enough n lmted tme because t s a method of searchng for the soluton at random. On the other hand, t s thought thatthe better soluton n shorttme canbe obtaned by the proposed method because t s possble to search for solutons near optcal soluton by teratng the optmzaton of the each sub-problem and the nformaton exchange between sub-systems. The effectveness of the proposed method to sach problems s nvestgated by these results. 4.4 Concluson Fg. 9: Ganttchart(Concentrated method) Fg. 10: Ganttchart(Proposed method) nventory control plannng and producton plannng are made alternatvely. The proposed method s revealed to show the better soluton n a short computaton tme compared wth centralzed method. The extenson of the proposed method to mprove soluton optmalty and reflecton of procurement whch leads to the total supply chan solver. In ths paper, decomposed soluton method s proposed to solve the chemcal plant wth two processes and ntermedate storage between them. In the method, 29

11 January 2007 Method for Producton Plannng and Inventory Control n Ol Refnery Reference [1] Toshhko Takeshta, Fujta Kaoru, Toshya Sanaka, Masash Matsukawa: Constrant-based schedulng system for Chemcal Process, Proceedngs the Schedulng Symposum 2002(2002) [2] Yoshhro Murakam, Hronobu Uchyama, Shnj Hasebe and Ior Hashmoto : Apprcaton of Repettve SA Method to Schedulng Problems of Chemcal Process, Transactons of the Insttute of Systems, Control and Informaton Engneers, Vol.10,No.3,pp (1997) [3] Hdetsugu Furuch : Development of Schedulng System for Batch Blendng of Lubrcant Ols (In Japanese), Instrumentaton, Vol. 40, No. 12 (1997) [4] Htosh Ima, Nobuo Sannomya: A Proposton for Smultaneous Determnaton of Orderng and Schedulng n a Manufacturng Process, Transactons of the Insttute of Systems, Control and Informaton Engneers, Vol. 16, No. 7, pp ,(2003) [5] M. Ago, T. Nsh, M.Konsh, J.Ima: Integrated optmzaton of schedulng and warehouse managemwnt usng an augmented Lagrangan decomposton and coordnaton method, Proceedngs of the 4th Symposum on Cybernetc Flexble Automaton, pp (2004) [6] Takao Enkawa, Kenj Ito : Method of Producton Management (In Japanese), Asakura shoten (1996) [7] T.Nsh,M.Konsh,Y.Hattor,S.Hasebe: A Decentralzed Supply Chan Optmzaton Method for Sngle Stage Producton Systems, Transactons of the Insttute of Systems, Control and Informaton Engneers, Vol. 16, No. 12, pp (2003) [8] Mutsunor Yagura, Toshhde Ibarak : Combnatonal Optmzaton Methaheurstcs (In Japanese), Asakura shoten(2001) 30