Solution Methods for Time-indexed MIP Models for Chemical Production Scheduling

Transcription

1 Ian Davd Lockhart Bogle and Mchael Farweather (Edtor), Proceedng of the 22nd European Sympoum on Computer Aded Proce Engneerng, 17-2 June 212, London. 212 Elever B.V. All rght reerved. Soluton Method for Tme-ndexed MIP Model for Chemcal Producton Schedulng Sara Zenner and Chrto T. Maravela Department of Chemcal and Bologcal Engneerng, Unverty of Wconn 1415 Engneerng Drve, Madon, WI 5376, USA Abtract Chemcal producton chedulng problem are often formulated a mxed nteger programmng (MIP) model whch, depte advance n computer hardware and optmzaton technologe, reman hard to olve. Mot prevou effort to obtan oluton fater have gone nto formulatng maller MIP model. In th paper, we take an alternatve approach: we ue problem pecfc nformaton to generate addtonal tghtenng contrant that can gnfcantly reduce computatonal tme. We backpropagate the demand for fnal product to fnd the mnmum amount of materal and number of batche each tak mut proce, provdng a lower bound on the number of tme each tak mut be run. We tudy procee wth and wthout recycle tream. Keyword: contrant propagaton, vald nequalte, nteger programmng. 1. Introducton Snce Kondl et al. (1993) ntroduced the tate-tak network (STN) model for chedulng batch chemcal procee, a lot of effort ha gone nto wrtng better model to fnd near optmal oluton fater. For example, contnuou-tme model wth a varety of tme grd were ntroduced to reduce formulaton ze (Mendez et al., 26). However, problem of practcal mportance are tll dffcult to olve n a reaonable tme (Sundaramoorthy and Maravela, 211a). We propoe generatng tghtenng contrant, baed on the total demand for fnal product, whch mprove the LPrelaxaton of the chedulng model and gnfcantly reduce computatonal tme. One of the major advantage of our approach that t doe not requre olvng auxlary MIP model a prevouly propoed (Burkard and Hatzl, 25; Janak and Flouda, 28). Furthermore, we ntroduce a new type of tghtenng contrant. In 2, we preent contrant propagaton method that allow u to calculate tght lower bound on () the number of batche and () the cumulatve producton needed to meet demand. In 3, we ue thee bound to generate trengthenng contrant, and n 4 we preent computatonal reult. Due to pace lmtaton, we preent only an outlne of the contrant propagaton algorthm followed by llutratve example. We ue lowercae talc for ndce, uppercae bold font for et, uppercae talc for varable, and lowercae Greek letter for parameter. 2. Tghtenng Method 2.1. Backward Propagaton We conder a general faclty contng of proceng unt, j J, wth a et of proceng tak, I, and tate (materal) S. Frt, we mut fnd the mnmum producton of a tak, μ ; and the mnmum amount requred for a tate, ω, where, for

2 2 S. Zenner and C.T. Maravela fnal product, ω the cutomer demand. Parameter μ and ω are calculated equentally by back-propagatng the demand. When we know ω (μ ) for all tate (tak) produced by tak (conumng tate ), we calculate μ (ω ) ung eqn. 1 (2): μ = max ω ρ (1) { } S { } I ω = max, ρμ ζ (2) where S (I - ) the et of tate (tak) produced by (conumng) tak (tate ), ζ the ntal nventory of tate, and ρ (ρ - ) the fracton of tate produced (conumed) by tak. In eqn. 1, the term nde the bracket the total amount tak mut proce to meet the demand for tate. We take the maxmum over all tate produced by tak to enure the tak atfe demand for all tate. In eqn. 2, ω the amount of ntermedate requred by all tak conumng t mnu any ntal nventory and mut be at leat zero. After calculatng μ, we fnd the mnmum attanable producton amount, μ (a decrbed n 2.4), whch provde a tghter lower bound on the requred producton of tak and hould be ued n place of μ n eqn. 2. For example, f the only unt for a tak ha a capacty of 3-4kg and the tak mut produce at leat 5kg, the mnmum attanable producton 6kg. We back-propagate demand untl ω and μ are known for all tate and tak. An llutratve example hown n Fgure 1. S1 S5 S6 T3 8% 2% 6% 4% S4 4% 6% T4 S7 1 ω S6 = 75 Cutomer demand ω S7 = 5 Cutomer demand 2 μ T3 = 75 To produce S6 μ T4 = 5 To produce S7 3 ω S4 =.4*5 = 2 Requred by T4 ω S5 = 75.6*5 = 15 Requred by T3 and T4 4 μ = 15 To produce S5 5 ω =.2*15 = 21 Requred by 6 μ = max{21/.6, 2/.4} = 5 To produce uffcent and S4 Fgure 1. Product demand back-propagated through the network. Each group n the table repreent one equence of calculaton. Decrpton of each calculaton are gven on the rght Recycle Stream For network wth recycle loop, mple back-propagaton doe not work, and we mut tear all loop. Frt, we chooe a tate n each loop, called the tear tate, and ntalze ω =. Next, we back-propagate demand a before and eventually calculate an updated value for ω for the tear tate. Ung the updated value, we back-propagate demand through the network a econd tme; th repeated for each tear tate. For any feable ntance, we only need to update ω for each tear tate once. Fgure 2 provde an algorthm for back-propagaton, where I k (S k ) the et of tak (tate) for whch μ (ω ) known, and S T the et of tear tate. Note that ω recalculated for tear tate n tep 3, and the updated value ued n the next teraton. Step Step 1 Step 2 Step 3 Set ω = for tear tate, calculate ω for product, and et n = Set I k = and S k = {product & tear tate} Calculate μ and then μ I k.t. S \ S k =. Add thee tak to I k Calculate ω (S k \S T ).t. I - \ I k =. Add thee tate to S k Doe I k =I & S k =S? I n S T? Set n = n1 Fgure 2. Algorthm for back-propagatng demand wth recycle tream. No Ye No Ye Stop

3 Soluton Method for Tme-ndexed MIP Model for Chemcal Producton Schedulng 3 An example provded n Fgure 3. Each row n the table repreent tep and 1 or one executon of tep 2 followed by tep 3. Tak (tate) wth value n black belong to I k (S k ). Value n gray are from the prevou teraton, and the correpondng tak (tate) do not belong to I k (S k ). Baed on the gven unt capacte, μ mut be a multple of 3 for and equal to μ for. S1 2% Tear State: ζ S1=48 Cutomer demand: 44kg U1 for, U2 for U1 capacty: 3 kg exactly U2 capacty: -35 kg 8% µ /μ ω Step S1 n = n = Tear tate and cutomer demand / Step 2: μ. Step 3: ω /6 55/ Step 2: μ. Step 3: update ω S1 = /6 55/ Retart wth new value for ω S /6 6/ Fnd better value for μ and ω 2-3 6/6 6/ ω S1 doe not change th tme Fgure 3. The algorthm appled to a network wth a loop (n bold) State Produced by Multple Tak When multple tak can produce a tate, eqn. 1 nvald a there no way of knowng beforehand how much of a tate each tak mut produce. Often t poble for each tak to meet the entre demand of the tate, o each tak wll have a mnmum producton of zero. Intead, we olve a mple LP to fnd μ. μ = mn P.t. ζ ρ P ρ P (LP) I I k ρ P ω S I where P a nonnegatve varable denotng the total amount of materal tak produce for a partcular oluton of (LP). The frt contrant requre that, for each tate, the amount produced plu any ntal nventory greater than the amount conumed. The econd contrant enforce that the amount produced of a tate mut exceed ω and only wrtten for tate for whch ω known. The objectve to mnmze the producton for tak and gve the value for μ. Demand tll back-propagated accordng to the algorthm n Fgure 2, and μ tll calculated after μ ; the only change that we calculate μ wth (LP) ntead of wth eqn. 1. Th method hould be ued for all tak for any network wth a tate that can be produced by multple tak and work even for network wth loop. S1 1% T3 μ =μ ω Step T3 S1 S4 9% S Cutomer demand or T3 can meet full demand Fnal mnmum value Fgure 4. Back-propagaton for a network where S4 produced by multple tak where μ found by olvng (LP). All unt have a capacty of -2kg, o any producton amount attanable and μ = μ. We aume that feed S1 and have unlmted ntal nventory when olvng (LP). Tak (tate) wth value n black belong to I k (S k ).

4 4 S. Zenner and C.T. Maravela 2.4. Attanable Producton Amount Once μ known, we mut fnd the attanable producton amount, μ. When only one unt can proce a tak, t traghtforward to fnd the range of attanable producton for any number of batche and to check f the requred producton n one of thoe attanable range. When multple unt can proce a tak, we fnd and check attanable range, ndexed by k, for every poble combnaton of batche. For example, f two unt, U1 and U2, can proce a tak, then we check 1 batch n U1, n U2; n U1, 1 n U2; 1 n U1, 1 n U2; etc. The attanable range for a partcular et of unt from Σα j k β j mn to Σα j k β j max, where α j k the number of batche n unt j for range k, and β j max (β j mn ) the maxmum (mnmum) capacty of unt j. For example, conder a tak whch mut proce 55kg and can be carred out n unt U1 and U2 (Fgure 5). The attanable range for each unt combnaton haded n Fgure 5. Snce μ =55 doe not fall n an attanable range, μ the producton amount at the tart of the next range and, for th example, 6. k αu1 αu2 mn max β / β : U1: 2/25 U2: 45/5 1 μ = 55 ˆ μ = % μ = Producton Amount Fgure 5. Attanable producton amount. # of Batche n U Integer Pont Eqn. 3 Eqn. 4: µ Eqn. 4: μ Eqn. 4: µ # of Batche n U1 Fgure 6. Effect of tghtenng contrant on feable regon. 3. Vald Inequalte Let X P jt be the agnment varable n a tme-ndexed MIP chedulng model;.e., X P jt = 1 f tak tart n unt j at tme t. We wrte two type of tghtenng contrant once μ known for all tak (J P the et of unt that can proce tak ): P % μ Xjt max P j, max t { β (3) J } P j j J max P β X ˆ μ (4) P j J, t j jt The rght-hand-de (RHS) of eqn. 3 the mnmum number of batche a tak mut proce, whch we fnd by dvdng μ by the larget poble ze of a ngle batch of tak and roundng up. The mnmum number of batche gve a lower bound on the um of X P jt. However, when a tak can be proceed n unt wth very dfferent capacte, eqn. 3 may not provde a tght bound; ntead, eqn 4 ue the maxmum capacty and mnmum producton requrement. In eqn 4, μˆ the mallet value (greater than ~ μ ) that the left-hand-de can be when all X P jt are bnary. In Fgure 5, μˆ the producton amount at the end the attanable regon that cloet to but greater than μ ĩ. Fgure 6 how how the ue of μˆ on the RHS of eqn. 4 provde a tghter formulaton than ung μ or μ.

5 Soluton Method for Tme-ndexed MIP Model for Chemcal Producton Schedulng 5 4. Reult To llutrate the effectvene of the tghtenng method, we preent reult for the STN model from Shah et al. (1993). Our method can be appled to other dcrete-tme model wth mlar reult (Sundaramoorthy and Maravela, 211b). We run 36 problem wth varou network, tme horzon, and problem feature wth the objectve of mnmzng proceng cot. We ue GAMS 23.7/CPLEX 12 on a computer wth 6 GB of RAM and a 2.67 GHz Intel Core (7-92) proceor runnng on Wndow 7 wth an 18 reource lmt. Fgure 7a a performance profle for dfferent combnaton of tghtenng contrant. For each problem, r the rato of the computatonal tme for a formulaton to the fatet tme over all formulaton. The vertcal ax the probablty that r le than (on the horzontal ax). The orgnal formulaton olve 17% of the problem wthn 15 tme the fatet formulaton, but the formulaton wth eqn. 3 and 4 olve 81% of the problem wthn the ame tme. Fgure 7b compare the orgnal formulaton to the formulaton wth eqn 3 and 4. Fourteen problem are olved to optmalty wth both formulaton, and tghtenng reduce the average computatonal tme from 189 to 1.4, more than two order of magntude. Twenty problem are olved to optmalty only wth tghtenng; the average optmalty gap after 18 wthout tghtenng 1.3%, whle the average computatonal tme wth tghtenng only 2.4. Two problem are never olved to optmalty wthn 18. In concluon, the addton of the trengthenng nequalte baed on the bound calculated by contrant propagaton method gnfcantly reduce the computatonal tme and ncreae the proporton of problem that are olved to optmalty wthn Eqn. 3 & 4.8 No tghtenng 2.1% Eqn %.6 Eqn. 3 Tghtenng (Eqn. 3 & 4) 1.5% P(r<).2 No tghtenng Problem Optmal oluton Sub-optmal oluton (a) Performance Profle (b) Comparon of Reult Fgure 7. (a) Performance profle and (b) comparon of reult wth and wthout tghtenng. Reference C.A. Mendez; J. Cerda; I.E. Gromann; I. Harjunkok; M. Fahl, 26, State-of-the-art revew of optmzaton method for hort-term chedulng of batch procee, Comput. Chem. Eng., 3, E. Kondl; C.C. Pantelde; R. Sargent, 1993, A general algorthm for hort-term chedulng of batch operaton I. MILP Formulaton, Comput. Chem. Eng., 17, A. Sundaramoorthy; C.T. Maravela, 211a, Computatonal Study of Schedulng Approache for Batch Proce Network, Ind. & Eng. Chem. Re., 5(9), N. Shah; C.C. Pantelde; R.W.H. Sargent. 1993, A general algorthm for hort-term chedulng of batch operaton II. Computatonal Iue, Comput. Chem. Eng., 17, R.E. Burkard; J. Hatzl, 25, Revew, extenon and computatonal comparon of MILP formulaton for chedulng of batch procee, Comput. Chem. Eng., 29, S.L. Janak; C.A. Flouda, 28, Improvng unt-pecfc event baed contnuou-tme approache for batch procee: Integralty gap and tak plttng, Comput. Chem. Eng., 32, A. Sundaramoorthy; C.T. Maravela, 211b, A General Framework for Proce Schedulng, AIChE J., 57(3),