Valid Inequalities Based on Demand Propagation for Chemical Production Scheduling MIP Models

Transcription

1 Vald Inequalte Baed on Demand ropagaton for Chemcal roducton Schedulng MI Model Sara Velez, Arul Sundaramoorthy, And Chrto Maravela 1 Department of Chemcal and Bologcal Engneerng Unverty of Wconn Madon 1415 Engneerng Dr., Madon, WI, 5376 Abtract. The plannng of chemcal producton often nvolve the optmzaton of the ze of the ta to be performed ubect to unt capacty contrant, a well a nventory contrant for ntermedate materal. Whle everal mxed-nteger programmng (MI) model have been propoed that account for thee feature, the development of tghtenng method for thee formulaton ha receved lmted attenton. In th paper, we develop a contrant propagaton algorthm for the calculaton of lower bound on the number and ze of ta neceary to atfy gven demand. Thee bound are then ued to expre three type of tghtenng contrant whch greatly enhance the computatonal performance of the MI chedulng model. Importantly, the propoed method are applcable to a wde range of problem clae and tme-ndexed MI model for chemcal producton chedulng. 1. Introducton Chemcal producton nvolve the converon of raw materal (feedtoc) nto fnal product, both of whch are mot often flud (gae or lqud). Flud can be mxed or plt n varable proporton and are often recycled (e.g., unreacted raw materal are recycled after they are eparated from the fnal product). roducton ta convert a et of nput materal nto a et of output materal, and are carred out n equpment unt that are capable of proceng dfferent amount of (flud) materal. In other word, the ze and therefore the number of ta to be executed can be an optmzaton decon. Whle there are chemcal faclte wth a wde range of proceng charactertc and contrant, there are two maor type of chemcal proceng, equental and networ. In equental proceng, t aumed that producton ta have a ngle nput and a ngle output materal; the nput materal of a ta produced by a ngle uptream ta; and the output of a ta conumed by a ngle downtream ta. Thu, materal from dfferent ta are not mxed, and the output of a ta cannot be plt to be conumed by multple ta. Alo, t mplctly aumed that each materal produced and conumed by at mot one ta. In networ proceng, ta may conume or produce multple materal, and a materal can be produced and conumed by dfferent type of ta. Alo, materal comng from dfferent ta can be mxed n a torage veel, the output of a ta can be conumed by multple downtream ta, and materal can alo be recycled. 1 Correpondng author; emal: maravela@wc.edu; Tel:

2 Interetngly, the method that have been developed to addre chemcal producton chedulng problem follow the aforementoned clafcaton. Early wor focued on equental faclte where a batch of raw materal ha to go through multple producton tage to be converted to the fnal product. In other word, the proceng of flud repreented a the proceng of a dcrete batch that ha to go through dfferent proceng tage; amount of materal are not montored and no materal balance are ncluded. We note that th type of modelng poble becaue plttng, mxng and recyclng are not allowed. We term thee approache a batch-baed. In mot cae, t alo aumed that the ze and thu the number of batche (for nown demand) were determned pror to optmzaton [elat, 212], whle recent approache conder multaneou batchng and chedulng [raad and Maravela, 28]. In the proce ytem engneerng (SE) lterature thee faclte are termed ether a mult-tage, f the routng through the tage the ame for all batche/product, or mult-purpoe, f there are batch/product-pecfc routng [Mendez et al., 26; Maravela, 212a]. Clearly, batch-baed method cannot be ued to addre problem n networ faclte where mxng, plttng, and recyclng are allowed; materal can be produced and conumed by dfferent ta; and the ze of the ta are optmzaton decon. To addre thee problem reearcher developed what we term materal-baed model, where the amount of materal proceed by a ta, a well a the nventore of materal, are explctly modeled. Another ey modelng apect of thee approache that a tme reference grd defned, and () the tart/end tme of ta are mapped onto th grd and () nventory contrant are expreed at the tme pont of the grd. The poneerng wor of antelde and co-worer employed a dcrete-tme grd, where the tme horzon dvded nto perod of equal and nown length and proceng tme are aumed to be contant [Kondl et al., 1993; Shah et al., 1993]. Snce then, a wde range of formulaton have been propoed ncludng contnuou-tme and mxed-tme grd, a well a unt-pecfc grd. egardle of the nature of the tme repreentaton and the detal of the formulaton, there are three proceng contrant that have to be enforced [Maravela, 212b]: a) Unt utlzaton: a unt cannot perform more than one ta at a tme. b) Materal balance: the nventory of a materal n a torage veel ha to be nonnegatve and le than the capacty of the veel. c) Batch-ze: the ze of a proceng ta (batch-ze) ha to be between a nonzero lower and an upper bound (capacty). The man dfference between the varou materal-baed formulaton le n the modelng of the frt contrant. In dcrete-tme model, t enforced through the wdely ued one-machne clque contrant, whle n contnuou-tme formulaton more complex et of contrant are ued. The econd contrant alway a flow balance contrant for a materal-tme node, mlar to thoe ued n producton plannng formulaton [ochet and Woley, 26]. Fnally, the thrd contrant reult n varable lower/upper bound contrant: f a ta executed, then the amount proceed hould be wthn a lower and upper bound; otherwe t zero. The ze of a ta hould be greater than or equal to the lower bound for afety and controllablty reaon, a well a due to recpe contrant. Mot of the aforementoned type of contrant have been tuded extenvely. Frt, vald nequalte and pecalzed oluton method have been developed for tme-ndexed ngle-machne problem [Woley, 2

3 199; van den Aer et al., 1999; van den Aer et al., 2]. Alo, many contrant propagaton algorthm have been developed for th contrant [Hooer, 2; Baptte et al., 21; Van Hentenryc and Mchel, 25; Hooer, 27]. Second, lot zng problem have materal balance that are mlar to thoe n networ chedulng problem, and addng vald contrant baed on thee materal balance and bll-of-materal nformaton reult n a gnfcantly tghter formulaton [ochet and Woley, 1993; Woley, 1997; Mller and Woley, 23].Fnally, vald nequalte have been derved from varable upper bound contrant n the context of fxed-charge networ problem [Nemhauer and Woley, 1988; Woley, 1998]. However, chedulng problem wth varable lower bound contrant have not been tuded uffcently. Burard and Hatzl (25) bound the number of batche of each ta and the number of batche produced of each materal by olvng a mall MI for every ta and an L for every materal, whle Jana and Flouda (28) propoe boundng the number of batche and cumulatve producton for each ta by olvng an MI boundng problem for every ta. Accordngly, n th paper, we tudy how the varable lower bound contrant can be ued to develop trong vald nequalte for MI chemcal producton chedulng problem. Specfcally, we frt develop an algorthm that allow u to calculate tght lower bound on the number of batche of each ta and the total amount of materal proceed by each ta that are neceary to atfy the gven demand. Thee lower bound, whch are calculated from ntance data wthn econd, are then ued to wrte two clae of vald nequalte whch greatly reduce the computatonal requrement. Our algorthm applcable to all type of networ faclte, ncludng faclte wth () multple ta producng or conumng the ame materal(), () multple unt capable of performng the ame ta(), and () recycle loop. Alo, they are applcable to all type of materal-baed MI formulaton that employ tmendexed varable. It mportant to note that materal-baed tme-ndexed MI model are ued a the bacbone n a wde range of chemcal producton chedulng tool, a well a n n-houe tool developed by chemcal compane [Wac, 29]. Alo, materal-baed model have been extended to addre problem n equental producton envronment a well a faclte that combne equental and networ envronment [Sundaramoorthy and Maravela, 211]. Neverthele, one of the lmtaton of both commercal product and pecalzed tool that they reman computatonally neffcent. Thu, our method, whch are applcable to all aforementoned model, have the potental to beneft a wde range of tool and model. The paper tructured a follow. In Secton 2, we ntroduce a networ-npred repreentaton of chemcal faclte, preent a formal tatement of the problem we conder and a wdely ued MI chedulng model, and cloe wth a motvatng example. In Secton 3, we preent the contrant propagaton algorthm for the calculaton of the lower bound on the number of batche and the total materal proceed. In Secton 4, we preent two type of tghtenng contrant, a well a extenon for problem wth ntermedate due tme. In Secton 5, we perform an extenve computatonal tudy ncludng more than 7 problem ntance. We ue lowercae talc letter for ndce, uppercae bold letter for et, lowercae Gree letter for parameter, and uppercae talc for optmzaton varable. 3

4 2. Bacground 2.1. Chemcal roducton Schedulng Notaton A chemcal faclty tranform a et of raw materal nto a et of valuable product through a equence of proceng ta carred out n equpment unt. A proceng ta convert a et of nput materal (raw materal or ntermedate) nto a et of output (ntermedate or fnal product) accordng to pecfed converon coeffcent. Note that the term ta ued to decrbe a type of operaton (e.g., the converon of A to B va reacton A B a ta), whch mple that a chedule may nclude multple executon of the ame ta. A ta can n general be carred out n multple unt of unequal capacty, whch mple that the number of batche of a ta neceary to atfy gven demand not nown pror to optmzaton. Thu, the term chedulng n the SE lterature ha been ued to decrbe a problem whch nclude three level of decon: () the determnaton of the number (and ze) of batche to be performed, () the agnment of batche to proceng unt, and () the tmng of batche o that at mot one batch executed on a unt. We aume that a chemcal faclty cont of: () proceng unt, where ta are executed; () torage veel, where materal are tored; and () ppng for materal tranfer. roceng unt and torage veel have capacty lmtaton, and each proceng unt and torage veel capable of proceng/torng a ubet of ta/materal. We do not conder other hared reource uch a utlte (e.g., electrcty) and labor. Fgure 1a how the proce flow dagram (FD) of a mple chemcal faclty, where raw materal converted nto ntermedate n unt U1, and the latter converted to ether product or n unt U2 or U3. Note that the FD how the phycal equpment of the faclty, but not necearly the ta that tae place on the unt. Alo, unt can perform multple ta; e.g., unt U2 and U3 can convert to or convert to. U2 Storage for Storage for U1 Storage for U3 Storage for {U1} {U2,U3} (a) roce Flow Dagram (b) State-Ta Networ Fgure 1. roce flow dagram (phycal layout of the faclty) and tate-ta networ repreentaton of a faclty roblem Statement To tudy chedulng problem, we have to ue a repreentaton that nclude not only the phycal reource, but alo the dfferent ta that can be carred out. In th paper, we adopt the tate-ta networ repreentaton, whch baed on the followng concept: a) State (materal): nclude feed, ntermedate, and fnal product and are repreented by crcle. b) Ta: are operaton that produce and conume tate (materal) and are depcted wth rectangle. c) Unt: unary reource for the executon of ta; multple unt may be able to proce a ngle ta, and a ngle unt may be able to proce multple ta. 4

5 State (crcle) and ta (rectangle) are connected by arrow, referred to a tream, howng the flow of materal. If a ta conume (produce) a tate, an arrow pont from the tate (ta) to the ta (tate). Fgure 1b how the STN repreentaton for a chedulng problem n the faclty hown n Fgure 1a. Note that unt are hown mplctly through the ta-unt compatblty nformaton gven by the dotted lne. The tructure of the networ defned n term of the followng et (ee Fgure 2a): Indce/Set, I J, S Subet I / I J S F ta proceng unt tate ta producng/conumng tate proceng unt that can proce ta S / S tate produced/conumed by ta fnal product We are alo gven the followng parameter: ϕ mn total cutomer demand for tate / mnmum/maxmum capacty of unt / τ ζ max fracton of tate produced/conumed by ta fxed proceng tme for ta n unt ntal nventory of tate In Fgure 2a, conumed by and and produced by ; therefore, I ={, } and I ={}. In Fgure 2b, produce and conume and, o S ={} and S ={,}. In general, a plu (+) upercrpt ndcate producton, and a mnu ( ) ndcate conumpton. If and can both be proceed n ether U2 or U3, then J I I I S S (a) Set I / I (b) Set S / S Fgure 2. Illutraton of general et Dcrete-tme Model S = J ={U1, U3}. To llutrate the underlyng concept and perform computatonal tude, we ntroduce the wdely-ued MI formulaton of Shah et al. (1993). We ue th model becaue a recent computatonal tudy howed that dcrete-tme model are more effectve for large-cale problem and, mot mportantly, can be ealy extended to account for a range of proceng retrcton and charactertc (e.g., ntermedate due date, holdng and utlty cot, varable reource requrement durng ta executon) at almot no computatonal cot [Sundaramoorthy and Maravela, 211b]. However, we 5

6 hghlght that our method are applcable to all tme-ndexed MI model for chemcal producton chedulng that account for varable batch-ze. Tme pont are ndexed by t. The formulaton nclude one famly of bnary varable: X t one f unt procee ta tartng at tme t and the followng non-negatve contnuou varable: B t S t t D t batch-ze of ta proceed n unt tartng at tme t nventory of tate at tme t amount of feed tate purchaed at tme t amount of fnal product delvered to cutomer at tme t The model cont of unt utlzaton (eqn. 1), unt capacty (eqn. 2), nventory capacty (eqn. 3), and materal balance (eqn. 4) contrant. I tt 1 t X t 1, t (1) S X B X, J, t (2) mn max t t t t, t (3) max S St 1 B B D t t t t t I J I J where D t fxed to the total amount of tate due at tme t; and t only non-zero for feed tate. Varable X t are fxed to zero for the τ -1 tme pont before the end of the tme horzon Motvatng Example We conder an ntance of the faclty ntroduced n Fgure 1, wth a demand for 9g of and 25g of. For each unt, the capacty (mn-max) and proceng cot of a ta n that unt are: 25-6g and $1 for U1; 4-5g and $25 for U2; and 35-45g and $3 for U3. Our obectve to mnmze cot. We generated a model for th ntance baed on formulaton preented n 2.3. The mnmum cot for the L-relaxaton $76.7, whle the number of batche of ta,, and are 1.9, 1.8, and.5, repectvely (ee Table 1). Table 1. Effect of tghtenng on the number of batche and cot for the networ n Fgure 1. # of Batche Mn. Cot ($) Optmal oluton Mnmum # of batche Burard method ropoed method L-elaxaton No tghtenng Burard method ropoed method (4) 6

7 However, f we tae nto account the capacte of the unt, we can nfer that more batche wll be requred n any feable oluton. mut produce 9g of to atfy demand; nce the maxmum batch-ze 5g, th requre at leat two batche. Smlarly, mut produce at leat 25g of, but, nce the mnmum batch-ze 35g, mut produce at leat 35g n at leat one batch. Together, and requre a mnmum of 125g (=9+35) of, whch produce n at leat three batche. Ung th contrant propagaton procedure, we calculate lower bound on the number of batche and cumulatve producton for each ta. Table 1 how the effect of tghtenng on the example n Fgure 1. Wthout tghtenng, the mnmum cot for the L-relaxaton $76.7. Wth the propoed method, the cot of the L relaxaton $15. Interetngly, the mnmum cot and number of batche of the L-relaxaton n th mple problem, match the optmal MI oluton. Our goal to develop a method that allow u to ytematcally calculate mlar bound n general producton networ, ncludng faclte wth recycle tream. Note that ung the tghtenng method propoed by Burard and Hatzl (25) mprove the cot of the Lrelaxaton to $96.7. Burard and Hatzl (25) do not conder mnmum unt capacte, o ther method requre to produce only 115g (=9+25) of n two batche. The tghtenng method of Jana and Flouda (28) fnd a lower bound on the number of batche whch a tght a our, but requre olvng a MI for every ta whch often a tme conumng a olvng the orgnal MI model. Alo, ther approach rele on a pecfc contnuou-tme model. Our propoed method do not requre olvng any auxlary MI, are vald for any networ, and can be ued wth any model employng a tme grd. 3. Contrant ropagaton Algorthm Our contrant propagaton algorthm calculate parameter whch are then ued to formulate the tghtenng contrant. The algorthm depend on the tructure of the proce networ. We clafy networ nto four categore accordng to Fgure 3: (1) networ wth no loop (Fgure 4a); (2) networ wth loop but no recycle tate (Fgure 4b); (3) networ wth recycle tate n a ngle loop (Fgure 4c); and (4) networ wth a recycle tate and multple neted loop (Fgure 4d). A recycle loop any cloed path (movng only n the drecton of the tream arrow) wthn the networ, and a recycle tate a tate n a loop that can be produced by multple ta. Each colored box n Fgure 3 ha t own pecfc tghtenng procedure that are decrbed n the ndcated ecton. All networ ue the tghtenng procedure for general networ ( 3.1). Networ are frt dvded nto networ wth and wthout recycle loop. If there are recycle loop, 3.2 decrbe the addtonal tghtenng method. ecycle loop are further dvded nto loop wth and wthout recycle tate, and 3.3 explan more tghtenng method for networ wth recycle tate. When a loop ha a recycle tate, t clafed a havng ether a ngle loop or a neted loop, whch decrbed n 3.4. In 3.5, we preent a general algorthm for applyng the tghtenng method to any networ. In 3.6, we decrbe an alternatve approach for networ wth recycle tate that alo wor well when multple ta can produce a ngle tate. 7

8 3.1 General Networ 3.2 Loop No Loop (Fg 4a) (a) General networ wthout loop (b) Networ wth a loop and no recycle tate 3.3 ecycle State No ecycle State (4b) 3.4 Neted Loop (4d) Sngle Loop (4c) Fgure 3. Clafcaton of networ General Networ Bacward ropagaton (c) Networ wth a recycle tate (d) Networ wth neted loop Fgure 4. Example of categore of networ. ω For bacward propagaton, we ntroduce the followng parameter: lower bound on the amount of tate requred to meet fnal demand μ / ν lower bound on the producton of ta requred to meet fnal demand, μ lower bound on the amount of tate that mut be produced by ta Frt, we need to fnd the mnmum amount requred for all tate, ω, and the mnmum producton for all ta, ; we calculate thee parameter equentally by bacward propagatng demand for fnal product. We etmate ω once nown for all ta conumng tate (all I ). Smlarly, we need to now ω for all tate produced by ta (all S ) before we calculate (ee Fgure 5). We calculate ω tartng wth the fnal product: S I F F S (5) In the HS of eqn 5, the top expreon the cutomer demand for fnal product mnu any ntal nventory whle the bottom expreon the amount of ntermedate requred by all ta conumng t mnu t ntal nventory. If the ntal nventory exceed the amount requred, ω wll be negatve, and the proce doe not need to produce tate. The mnmum amount of tate that mut be produced by ta, ν, ntroduced a an ntermedate parameter. Although ω can be negatve, ν mut be at leat zero. ST max, S, I MT S \ S, I (6) S5 S6 T4 (a) STN repreentaton of example S7 S5 T4 S6 (b) Sere of calculaton n bac-propagaton and T4 mut be etmated before ω S5 ω S5 and ω S5 mut be etmated before S7 Fgure 5. Example of bacward propagaton of demand. From demand for S6 and S7, we calculate ω S6 and ω S7 ; next, we calculate and T4, followed by ω S5 and ω. We cannot fnd yet becaue ω not nown, o we calculate, followed by ω and ω. Fnally, we calculate and ω. 8

9 # of Batche where S ST the et of tate produced by a ngle ta, S MT the et of tate that can be produced by multple ta, and S the et of recycle tate. Eqn. 6 doe not conder recycle tate, whch we dcu n 3.3. If multple ta can produce a tate, we aume that any ngle ta can meet the full demand; therefore, the mnmum producton of that tate by each ta zero, and, for thee tate, an alternatve approach that gve better reult decrbed n 3.6. After calculatng ν for all tate produced by a ta, we calculate μ, max S (7) The term nde the bracet the total amount ta mut proce to meet the demand for. We tae the maxmum over all tate produced by ta to enure the ta atfe demand for all tate. At th pont, we mut fnd the mnmum attanable producton amount, μ (8) where Δμ the mnmum amount that mut be added to μ to reach an attanable producton amount and dcued n arameter provde a tghter lower bound on the requred producton of ta. We bacward propagate demand untl ω and are nown for all tate and ta Attanable roducton Amount After fndng μ, we need Δμ to calculate. When only one unt can proce a ta, t traghtforward to fnd the range of attanable producton amount for any number of batche and to chec f the requred producton n one of thoe attanable range (ee example n Fgure 6). If μ fall n an attanable regon, demand can be met exactly, and Δμ zero; otherwe, Δμ the dtance between μ and the tart of the next attanable range. However, when multple unt can proce a ta, we mut fnd and chec attanable range for every poble combnaton of batche n unt. For example, f two unt can proce a ta, we need to chec 1 batch n U1, n U2; n U1, 1 n U2; 1 n U1, 1 n U2; etc. To reduce the number of max combnaton, we fnd an upper bound on the number of batche n a partcular unt,, by dvdng μ by the larget poble batch unt can proce and roundng up, max max J (9) roducton Amount Fgure 6. Attanable range for a unt wth a capacty of 3-4g are haded. 9

10 When the number of batche for a partcular unt at t upper bound, we are guaranteed the attanable range meet or exceed demand wth ut one unt, and the number of batche n all other unt et to zero to elmnate more combnaton. In total there are K range to chec, K max J J (1) For each range, {1,2,,K }, there a unque combnaton of, where gve the number of max batche n unt for range. The frt term n eqn. 1 the number of range wth =,1,2, ( - 1) for all max J, and the econd term the number of range wth = for one J and = for all other J. We loop over all {1,2,,K } and chec f fall nto an attanable range. If, for ta, J mn max J for ome, the mnmum requred producton can be met exactly, and Δμ zero. The LHS of eqn. 11 gve the lower bound of the attanable range, and the HS gve the upper bound. If μ not attanable for any range, we perform another loop over all to fnd the range that able to meet producton requrement and whoe lower bound cloet to μ. mn mn f J J mn f J The top lne et equal to the exce amount produced (the lower bound on the range mnu μ ) f the combnaton of unt at leat able to meet demand. The econd lne et to nfnty when the combnaton capacty too mall. Once all range have been checed, Δμ calculated: mn (13) A an example, we conder a ta that can be proceed n U1 or U2 and mut proce 55 g (ee max max Fgure 7). We fnd U1 3 and U2 2 from eqn. 9; and K =8 from eqn. 1. All for {1,2,,8} are hown on the left de of Fgure 7. The attanable range are haded (eqn. 11). Snce μ = 55 doe not fall n a gray range, eqn. 11 never atfed. The exce produced by each range the dtance between μ and the tart of the next haded range (eqn. 12). For th example, the mallet exce producton 5, capacte (mn/max): U1: 2/25 U2: 45/5 α U1 α U2 μ = 55 μ = 6 Eqn. 11 I μ n attanable regon? I demand met? Eqn. 12 Exce produton Δμ 1 no no 2 1 no no 3 2 no ye no no no ye no no no ye no ye 5 roducton Amount Fgure 7. Example of calculaton of the mnmum attanable producton amount. (11) (12) 1

11 o Δμ 5, and 6. The algorthm for fndng Δμ a follow: 1 Calculate and Κ and et α. max 2 For {1,2,, Κ } 3 If eqn. 11 true 4 Set Δμ = and top 5 end 6 end 7 For {1,2,, Κ } 8 Calculate (eqn. 12) 9 end 1 Calculate Δμ (eqn. 13) The attanable producton amount, 3.2. Networ wth Loop, now calculated ung eqn. 8. When there are loop n a networ, mple bacward propagaton wll not wor. We ue tear tream to brea the loop wth one tear tream n every loop. A tear tream cont of a ta, referred to a the tear ta, and a tate produced by that ta, referred to a the tear tate. Although any tream can be choen a the tear tream, ome choce are more convenent. If a loop ha a ta that produce a tate outde the recycle loop, th ta hould be ued a the tear ta; otherwe, any ta can be choen. The et I T and S T contan all tear ta and tate, and L I l and L S l gve all ta and tate n loop l. Fgure 8 provde an example for propagatng demand for a networ wth a loop. We wrte the value of μ nde the box repreentng ta, ω nde the crcle repreentng tate, and ν next to the tream connectng ta to tate. For mplcty, all example have one unt per ta, and capacte are gven for each ta. The recycle loop hown n bold. We choe a the tear ta becaue t produce outde the recycle loop, and S5 the tear tate. We ntalze ν,s5 for the tear tream to zero (Fgure 8b). Now, we can etmate a oon a ω nown. We bacward propagate demand a follow: ω =5 =1 ω =1 =1 ω =1 =11 (μ =1 and Δμ =1) ω =11-45=65 T4=65 ω S5 =65*.8=52. We top after calculatng ω S5 =52 for the tear tate (Fgure 8c). We need a mnmum of 52 g of S5, but only produce 5g (=1*.5). Therefore, we et ν,s5 =52, and delete ω and μ for all other tate and ta (Fgure 8d). We repeat bacward propagaton ung the new value ν,s5 =52 (Fgure 8e). Now, produce the requred 52g of S5. In general, we begn by ntalzng ν for all tear tream to zero. We bacward propagate demand untl ω etmated for a tear tate. We update ν for the tear tream ung eqn. 6 and compare t to the producton of the tear ta. If producton can meet demand (ν ), there no problem, and we contnue wth the bacward propagaton. If demand greater than producton (ν > ), we reet all other ω and, and retart the bacward propagaton ung the new ntal value for the tear tream. If, on the econd pa, demand for a tear tate tll exceed producton, the problem nfeable wth the gven ntal nventore. 11

12 S6 S5 8% T4 2% 5% 5% 5 S 1 45 L1: S5 T4 T T I S S5 L IL1,,,T4 L S,,,S5 L1 (a) Example STN and data Capacte -T4 mn 55 5 max 9 15 S5 S5 52 T4 T4 S6 S6 (b) Intalze tear tream (d) Chec tear tream and tart over T4 65 S T4 65 S S6 13 S6 13 (c) ropagate demand bacward Fgure 8. Networ wth a loop and no recycle tate. (e) Fnal mnmum producton amount 3.3. Networ wth ecycle State revouly, when multple ta could produce a ngle tate, we aumed that each ta wa capable of producng the full amount, and, therefore, each ta had a mnmum producton of zero (eqn. 6). However, when a tate S MT part of a loop, we can fnd a better etmate for ν. We refer to thee tate a recycle tate, S S MT. Note that the fracton of a tate that recycled le than one; n other word, f we tart out wth ome amount of the recycle tate and run only ta n the recycle loop, we wll never end up wth more of the recycle tate than what we had ntally. We ntroduce the followng new parameter for networ wth recycle tate: upper bound on the amount of tate that can be produced when only the mnmum amount of a recycle tate, ω, avalable upper bound on the producton of ta when only the mnmum amount of a recycle tate avalable ψ upper bound on the amount of recycle tate produced by ta when only the mnmum amount of tate avalable Example For the networ n Fgure 9, we tart by electng the tear tream n exactly the ame way a n 3.2. We chooe T4 a the tear tream and ntalze ν T4, to zero. The bacward propagaton contnue untl we calculate ω =98 n Fgure 9c. There a maxmum amount of that T4 () can produce wthout ncreang the requred producton of (T4) (ee ropoton 1); th amount gven by the parameter, whch we etmate by propagatng ω forward (Fgure 9d). In the forward propagaton, we tart out wth ω =98g of, whch mean can produce at mot =14g (=98/.7) of S5 (aumng there enough ). T5 need T5=2g of S5 (from Fgure 9c) and the remanng 12g (=14-2) of S5 are avalable for T4. Therefore, T4 produce up to ψ T4, =24g 12

13 (=12*.2) of. Snce ω =98g of are needed n all (from Fgure 9c), we conclude that mut produce the remanng 74g (=98-24), and ν, =74. capable of producng all 98g of (ψ, = wth unlmted ntal nventory of feed), o T4 not requred to produce any, and ν T4, =. Snce alo a tear tate, we mut enure that the producton of by T4 exceed ν T4, ; nce t doe (.2*12), we contnue wth the bacward propagaton untl we have calculated ω and for all tate and ta (Fgure 9e). 4% 6% 7% 3% S5 2% T5 T4 S7 8% S6 (a) Example STN and data S7 2 L1,T4 T L S S I L T S6 8 SL1,S5 I T4 L1: S5 T4 Capacte - T4 T5 mn max S5 (b) Intalze tear tream T5 T4 S7 S , S5, 74 T4, 24, (d) ropagate ω forward T5 T4 12 S7 S S5 T4 1 (c) ropagate demand bacward T S7 2 S T S5 (e) Fnal mnmum producton Fgure 9. Bacward propagaton n a networ wth a ngle recycle tate n a ngle loop. T4 1 8 S7 2 S General Method for ecycle State Bacward propagaton for networ wth recycle tate tart by ntalzng ν for tear tream to zero. A wth tandard loop, when ν calculated for a tear tream, we chec f producton exceed demand. After calculatng ω for a recycle tate, we label th tate * and fnd the maxmum amount that can be recycled. The mnmum amount of the labeled recycle tate that requred to meet demand, ω *, propagated forward around the recycle loop to fnd and for all tate and ta, repectvely. For all tate and ta outde the recycle loop, we et and to nfnty; th enure no feable oluton are cutoff and aume that thee tate are produced tartng from feed wth an unlmted upply. Once nown for all ta producng a tate ( I ), we calculate. We can calculate for the labeled recycle tate mmedately. for * L L f l and max 1 S l S I I S ll * I ll * (14) The frt expreon gve for the labeled recycle tate *. The econd expreon for all other tate n the recycle loop except tate produced by multple ta (other recycle tate) wthn any loop contanng *. The frt term n the condton n the econd lne gve all tate n any loop contanng *, and the econd term the number of ta wthn the loop that produce tate. If there another recycle 13

14 tate nde the loop, there are neted loop n the networ, and dcue how to fnd. The total amount of tate avalable,, the amount produced by all ta plu the ntal nventory. Once nown for all tate conumed by ta ( S ), we fnd. I, L L l l S l L l * L* mn f I and max S S 1 (15) Eqn. 15 gve for any ta nde the recycle loop except for ta that conume multple tate wthn any loop contanng *, whch are dcued n The frt term n the condton gve all ta n any loop contanng tate * and the econd term gve the number of tate wthn the loop that are conumed by ta. In the numerator, the mnmum amount of tate requred for other ta ubtracted to gve the maxmum amount of tate avalable for ta. Dvdng by gve the total amount ta need to proce to conume all of the avalable. Tang the mnmum over all tate conumed by ta enure there enough of every tate. We do not round to an attanable producton amount; roundng up wll reult n weaer bound, and roundng down may cutoff feable pont. Once nown for all ta producng the labeled recycle tate, we fnd the maxmum amount produced by each ta. * * * I (16) Fnally, we calculate the mnmum amount of tate * produced by each ta. * max, * * I * (17) I * \ The mot a ta can produce of a recycle tate ψ and the leat zero. If ν > ψ, there not enough ntal nventory, and the problem nfeable. If the demand cannot be atfed wthout a partcular ta, eqn. 17 gve the mnmum that ta mut produce. If a ngle ta or combnaton of ta can atfy the demand, all other ta do not need to upply any materal, and the fnal term n eqn. 17 le than zero. ropoton. There a maxmum amount of each recycle tate that can be produced by each ta,, wthout ncreang the mnmum requred producton of other ta producng that tate. roof: If ω the mnmum amount of recycle tate requred to meet fnal demand, and the maxmum amount of tate that can be produced by ta when tartng wth ω, then the amount of produced by ta and max, I. I, Let be the maxmum fracton of tate that recycled by ta. If an addtonal amount of tate,, produced by ta, the total amount of now requred at leat, and the maxmum amount produced by all ta I I. The total amount of that need to be produced by ta 14

15 max, I. I I, nce 1,, and. Therefore, all ta producng tate are at ther mnmum I I, producton when all other ta recycle at mot 3.4. Networ wth Neted Loop. Not all networ wth multple loop need the addtonal procedure decrbed n th ecton. There are two cae wth multple loop where the forward propagaton wll not wor Cae 1 When multple ta n a ngle loop produce another recycle tate, the forward propagaton decrbed prevouly fal when calculatng for that tate. For the networ n Fgure 1a, produced by and, whch both belong to loop L3 contanng recycle tate. We propagate the demand a decrbed prevouly untl t tme to propagate ω forward and calculate n Fgure 1b. We need and to calculate, but needed to calculate (Fgure 1c). Durng the forward propagaton, produce at mot =12g of. revouly (Fgure 1b), we determned that mut produce a mnmum of =112.5g of, meanng can now produce an extra 7.5g (= ). If procee all addtonal 7.5g of, t wll produce another.75g (=7.5*.1) of. roceng the addtonal.75g gve.75g (=.75*.1) more of ; th a geometrc ere that eventually converge to 8.33g (=7.5/(1-.1)) more of. We add the extra 8.33g to the orgnal 125g of needed to meet the demand to gve (Fgure 1c). Demand propagaton contnue untl ω and are nown for all tate and ta (Fgure 1d). The example generalzed to any proce networ. We wll refer to the frt labeled recycle tate a * (the one for whch we are tryng to fnd ν ) and the econd recycle tate a ** (the one for whch we are tryng to fnd ). The total amount of tate ** avalable 1 L L f l and max 2 S S I I (18) NC 1 S ll * I \I ll* T T L S, S, I SL1 L L L L L S S, I = I = I, L2 L3 L1 L2 L3 L1: L2: L3: (a) Example STN and data 1% 8% 1% 1 Capacte, mn 5 6 max 75 75,, (c) ropagate ω forward, 13.3, (b) Bacward propagaton (d) Fnal mnmum producton Fgure 1. Demand propagaton wth multple recycle tate n the ame loop

16 where I NC the et of ta for whch ha not been calculated, and ξ the maxmum fracton of tate that can be recycled by the loop contanng only ta n I NC. The algorthm provde a method for eepng trac of I NC. Eqn. 18 gve for any tate excluded from eqn. 14. The frt term the addtonal amount avalable durng the forward propagaton from all ta for whch ha already been calculated. Dvdng by 1-ξ gve the total addtonal amount of ** produced durng recyclng. Fnally the orgnal ω and the ntal nventory are added to gve. arameter ξ can be etmated when there only one loop contanng ** but not *;.e., l {l:** S,* S }. L l L L I l, S Sl for L L Il, S Sl L l S (19) The numerator gve the fracton of materal remanng n the loop after ta. The denomnator the fracton of the materal conumed by ta that already n the recycle loop. Eqn. 19 only vald when there only one loop contanng ** but not *. Eqn. 18 vald for any networ, but ξ need to be etmated Cae 2 When a ta nde a loop conume multple tate n a loop, the forward propagaton fal. In Fgure 11, conume and whch both belong to recycle loop L2. In Fgure 11b, demand ha been bacward propagated to calculate the demand for recycle tate. The demand need to be propagated forward, but we cannot calculate untl and are nown, and cannot be calculated untl nown (Fgure 11c). Durng the forward propagaton we have =18g of avalable. If there enough, can proce at mot 12g. We gnore n eqn. 15, and et 12. We contnue propagatng demand (Fgure 11d). The procedure ealy generalzed to any proce networ, I, L L NC l S l S \S S ll ll * * mn f I and max S S 2 (2) L1: 1% 9% L2: L3: 1% T T L S S I SL1 L L L L L S S, I = I = I, L2 L3 L1 L2 L3 9% 9 Capacte, mn 5 6 max (c) ropagate ω forward 18 (a) Example STN and data (b) Bacward propagaton (d) Fnal mnmum producton Fgure 11. Demand propagaton n a networ wth a recycle tate and multple loop

17 where S NC the et of tate for whch ha not been calculated; thee tate are gnored when calculatng. The algorthm provde a way to eep trac of S NC. Eqn. 2 gve for any ta n the loop excluded from eqn Complete Algorthm The complete algorthm combne the tghtenng procedure for all networ type. For each of the four categore, the relevant porton of the algorthm n Fgure 12 are colored a n Fgure 3. For convenence, we defne the new et: S tate n any recycle loop contanng tate, S SL SL I ta n any recycle loop contanng tate, I SL SL S L l S ll L I l S ll I NC /I NC ta for whch μ /μ not nown I A /I A ta avalable (ω /ω ha been calculated for all S / S ) for calculatng μ /μ S NC /S NC tate for whch ω /ω not nown S A /S A tate avalable (μ /μ ha been calculated for all I / I ) for calculatng ω /ω C S S C tate for whch ν ha been calculated for ta recycle tate for whch we need to perform a forward propagaton In the algorthm, we ue the parameter t to eep trac of how many tme tear tate S T ha been updated. A mentoned n 3.2, f the tear tate updated more than once, the ntance nfeable. General networ ecycle loop ecycle tate Neted loop NO Stop: Infeable YES I t 1 S T? 7. t =t +1 S T.t. ν >ρ I T I + 1. Set ν = S T, I T I + ; t = S T ; and S C ={: S T S + } 2. Set S NC =S, I NC =I, I A =, and S A =S F 3. Calculate ω S A (eqn. 5) I S A S =? YES 4. Calculate ν S A, I + (eqn. 6) NO I ν ρ S T, I T I +? YES NO 5. Set S NC =S NC \S A, S C =S C {: S + S A }, S A =, and I A ={: I NC, S C =S + } YES I S C =? NO 11. Calculate ψ S* (eqn. 16) and ν S* (eqn. 17) I S* + and remove * from S C YES 8. Set S C =S A S 9. Chooe an S C and label t * 1. Set μ = I * SL, ω = S * SL I NC =I * SL, S NC =S * SL, I A =, and S A ={*} I S NC = & I NC =? NO 12. Calculate ω S A (eqn. 1 or 14). Set S NC =S NC \ S A, S A =, and I A ={: I NC, S - S NC = } 13. Calculate μ I A (eqn. 11 or 16). Set I NC =I NC \I A, I A =, and S A ={: S NC, I + I NC = } Stop 6. Calculate μ (eqn. 7) and (eqn. 8) I A. Set I NC =I NC \I A. I A = and S A ={: S NC, I - I NC = } YES I I A = & S A =? Fgure 12. General algorthm for propagatng demand through a networ. NO I S A or S NC =? NO YES 14. Set S A ={: S S NC, (I + I * SL )\I NC } and I A ={: I NC, (S - S * SL )\S NC, S - S * SL >1}. Calculate ξ S A (eqn. 19) 17

18 3.6. State roduced by Multple Ta When multple ta can produce a tate, eqn. 6 gve ν = for each ta, whch may mean μ zero for uptream ta that mut produce ome materal. In Fgure 13, produced from or, o each ta ha a mnmum producton of zero. However, and both requre, whch produced by. If we ued eqn. 6 to fnd ν n tep 4 of the algorthm, we fnd that μ =, whch not a tght lower bound. Intead, we can olve a mple lnear program (L ) to fnd ν. mn Q.t. Q Q I I I Q S NC (L ) where Q a potve varable for the total amount of materal ta produce n a partcular oluton of (L ). 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 nown. The obectve to mnmze the amount produced by ta. When t tme to fnd ν n the algorthm, we olve (L ) for ta and then multply the optmal obectve value by to get ν. When a ta (other than a tear ta) produce multple tate, we only need to olve (L ) once and multply by the optmal value by for each tate S. For tear ta that produce multple tate, we olve (L ) twce, frt to calculate ν for any nontear tate produced by the ta and econd to update ν for the tear tate. When we ue th method, we olve (L ) to fnd ν for all ta n the networ. Th method alo provde an alternatve to the method for recycle tate decrbed n 3.3 and 3.4. For more complcated networ wth neted loop, th method may gve tghter bound and may be much mpler to ue. Demand tll bacward propagated accordng to the algorthm n Fgure 12 wth two change: (1) we calculate ν n tep 4 of the algorthm wth (L ) ntead of wth eqn. 6, and (2) we p (or anwer ye to) all tep nvolvng recycle tate (tep 8-14). 1% Method ω for ν 9% Eqn. 6 5 (L) Fgure 13. Compare the two method for fndng ν : (1) ung eqn. 6 and (2) olvng (L). Cutomer demand 5g of and all ta are proceed n a unt wth a capacty of -5g. 18

19 4. Vald Inequalte We wrte tghtenng contrant after calculatng and ω for all ta and tate. We fnd the mnmum number of batche proceed by a ta, λ, by dvdng the requred producton by the larget poble ze of a ngle batch of ta and roundng up. max J max (21) The mnmum number of batche provde a lower bound for the um of the agnment varable. J, t X t (22) When multple ta produce a tate, eqn. 22 may not provde a tght bound. Intead, we fnd bound for the mnmum number of batche from all ta producng a tate, κ. Agan, we dvde the requred amount of each tate by the larget poble amount of that tate that can be produced n a ngle batch and round up, max I, J max (23) Now, κ provde a bound for the agnment varable for all ta producng tate. I, J, t X t MT S (24) Eqn. 24 doe not provde new nformaton for tate produced by a ngle ta. Tghtenng contrant 22 and 24 have the ame form a thoe propoed by Burard and Hatzl (25) and Jana and Flouda (28), but the bound from the dfferent method may be dfferent. When a ta can be proceed n unt wth very dfferent capacte, eqn. 22 and 24 may not provde tght bound. Intead, we ue the maxmum batch-ze and mnmum producton requrement to fnd tronger general nequalte. J, t X ˆ max t mn where μ the tghtet bound for eqn. 25. In 3.1.2, we calculate baed on. Snce eqn. 25 baed on, doe not provde a tght bound. We fnd μ by performng another loop over all {1,2,,K } max wth the ame ued to calculate the attanable producton amount. f J J (26) otherwe max max ˆ (25) 19

20 The top expreon et ˆ to the producton amount at the end of attanable range f the range large enough to meet demand. The econd expreon et ˆ to nfnty when the attanable range too mall to meet demand. After checng all range, we et μ to the mallet of all ˆ. ˆ mn ˆ (27) The LHS of eqn. 25 can only tae on a dcrete et of value when X t bnary; thee value are gven by Σα where α an nteger and are the ame a the producton amount at the end of the attanable max range hown n Fgure 7. Therefore all poble value for the LHS occur at the end of an attanable range. To fnd the value of μ that gve the tghtet bound, we ue the producton amount at the end the attanable regon that cloet to but greater than (ee Fgure 14). In general, μ the mallet max producton amount at the end of an attanable range (Σα ) that at leat able to meet demand ( ). Eqn. 25 can alo be wrtten for tate produced by multple ta: I J, t X max t MT S (28) Fgure 15 llutrate the effect of the tghtenng contrant on the feable regon of the Lrelaxaton of the model for the example n Fgure 7 and 14. The total demand 55, μ =55, =6, and μ =75. The lne how eqn. 22 and eqn. 25 wth three HS value: μ,, and μ. Eqn. 22 requre at leat two batche. Improvng (ncreang) the HS of contrant 25 mprove the L-relaxaton of the model. When bacloggng not allowed, we ue due tme to tghten the formulaton further. Now, all parameter nclude a tme ndex (ϕ t, ω t, etc.) and are zero for tme when no order are due. For every due tme, we add all earler order to get ϕ t and calculate the other parameter a before. Intead of ummng the agnment varable over the entre horzon, we um t only over the tme avalable to tart the ta. t J t X t t, t (29) For tme when no order are due, all parameter are zero, and eqn. 29 not wrtten. We can alo wrte contrant 24, 25, and 28 to conder due tme by changng the tme over whch the agnment varable are ummed. For the example n Fgure 1, uppoe cutomer C1 demand 3g of at t=6 and 25g of at t=9, and cutomer C2 demand 6g of at t=9. The total fnal demand up to t=6 and up to t=9 are calculated. Table 2 lt order ze (ϕ t ), requred producton for tate (ω t ) and ta ( t ), and agnment bound (λ t ). Table 2. arameter value when due tme are ued to tghten the formulaton. Agnment t Order (ϕ t ) ω t t Bound (λ t )

21 # of Batche n U2 capacte (mn/max): U1: 2/25 U2: 45/5 α U1 α U2 μ = 55 μ î = 75 Eqn. 26 I demand met? μ î 1 no 2 1 no 3 2 ye no ye no ye ye roducton Amount Fgure 14. Example of μ calculaton. The LHS of eqn. 25 mut belong to the et {, 25, 5, 75 }. Snce the mnmum producton = 6, the LHS of eqn. 25 mut be at leat 75 n any feable oluton Integer ont Eqn. 22 Eqn. 25: Eqn. 25: Eqn. 25: # of Batche n U1 Fgure 15. Effect of tghtenng contrant on feable regon. 5. Computatonal Study 5.1. roblem Intance To determne how well the tghtenng contrant wor, we tet 72 ntance wth dfferent obectve, networ, problem feature, and tme horzon. Each ntance run wth dfferent tghtenng formulaton. Specfcally, we compare the two type of contrant baed on the number of batche (eqn. 22 and 24) and on the mnmum producton (eqn. 25 and 28). We alo loo at the effectvene of the extenon for procee wth due tme. We conder two obectve: mnmzaton of proceng cot and maepan. The proceng cot depend on the ta and unt, but not on the batch-ze. cot t,, J I X t t, MS X t J t (31) Table 3. The even contrant et. 21 (3)

22 Mnmum # of batche (eqn. 22, 24) Mnmum producton (eqn. 25, 28) Due Tme (eqn. 29) Set F1 F2 X F3 X F4 X X F5 X X F6 X X F7 X X X We ue four networ, N1-N4, to determne the effect of recycle tream and tate produced by multple ta on the effectvene of the tghtenng contrant (Fgure A1 n Electronc Companon). roce and order data are alo gven n the Electronc Companon (Table A1-A4). Networ N1 and N4 have no recycle tream, but networ N4 ha a tate produced by multple ta. Networ N2 and N3 have a recycle tream. Networ N2 taen from Kondl et al (1993). We conder three problem clae wth dfferent feature for each networ. roblem (a) ha all order due at the end of the tme horzon. To determne the mpact of ncludng due tme n the tghtenng contrant, problem (b) ha ntermedate due tme. To compare the two type of tghtenng contrant, unt capacte are changed for problem (c) o at leat one ta can be proceed by unt wth dfferent capacte. We olve each ntance wth three tme horzon: 4, 8, and 12 hour. The order ze and due tme cale wth the tme horzon to enure all tme horzon lead to chedule that are mlarly loaded. Startng wth a 4-hour horzon, the order ze and due tme are doubled for an 8-hour horzon and trpled for a 12-hour horzon. We conder even et of tghtenng contrant (Table 3). All formulaton contan eqn Formulaton F1 ha no tghtenng contrant. F2 nclude the contrant baed on the mnmum number of batche. Set F3 contan the contrant baed on the requred producton. F4 combne F2 and F3. Fnally, et F5-F7 are the ame a et F2-F4 but wth due tme. Only problem (b) run wth formulaton F5-F7, and the reult focu on formulaton F1-F4. All problem are olved ung GAMS 23.7/CLEX 12.3 on a computer wth 6 GB of AM and a 2.67 GHz Intel Core (7-92) proceor runnng on Wndow 7. We ue a reource lmt of 18. Model and oluton tattc for all problem are gven n the Electronc Companon (Table A5-A1). For the 36 ntance, mplementng the demand propagaton algorthm tae an average of.26 wth a maxmum tme of 4.3 when ung eqn. 6 to calculate ν, and an average of 2.4 wth a maxmum tme of 11.9 when ung (L ) to calculate ν eult Table 4 how aggregate reult for formulaton F1 and F4. roblem are grouped nto four categore: (1) thoe that are olved to optmalty by both formulaton, (2) thoe that are olved to optmalty only by F4, (3) thoe that are never olved to optmalty, and (4) thoe that are olved to optmalty only by F1. The average computatonal tme or optmalty gap gven for the problem n each of the four categore. For cot mnmzaton, 14 problem are n the frt category, and tghtenng 22

23 (r<) (r<) reduce the computatonal tme from 189 to 1.4. Category 2 contan 2 problem, and the tghtened formulaton olve thee problem n an average of 2.4 whle, wthout tghtenng, there an average optmalty gap of 1.3% after 18. For the two problem n category 3, tghtenng reduce the average optmalty gap from 2.1% to 1.5%. There are no problem n the fnal category. For maepan mnmzaton, 34 problem fall n the frt category, and the average computatonal tme 61 wthout tghtenng and 32 wth tghtenng. There one problem each n categore 2 and 4. Fgure 16 a performance profle for formulaton F1-F4. For each ntance, the computatonal tme for each formulaton dvded by the fatet tme over all formulaton for that ntance to gve r. The vertcal ax the probablty a formulaton wll olve a problem wthn (on the horzontal ax) tme the fatet formulaton. For cot mnmzaton, the untghtened formulaton olve only 17% of the problem wthn 15 tme the fatet formulaton, but F4 olve 94% of the problem wthn the ame tme. Ung the tghtenng contrant baed on the mnmum producton (n ether F3 or F4) gve the bet reult. For maepan mnmzaton, F3 perform the bet, but tghtenng generally le effectve and doe not alway mprove oluton tme. Fgure 17 compare the four formulaton for the two obectve. Each pont the average computatonal tme or optmalty gap over all networ, problem clae, and tme horzon (36 run per pont). The tghtenng contrant baed on the number of batche (F2) the leat effectve, and F3 and F4 are about equvalent. For cot mnmzaton, tghtenng ncreae the fracton of problem olved to optmalty and decreae the computatonal tme, whle for maepan mnmzaton, t doe not have a large mpact on the computatonal tme. Snce maepan mnmzaton appear to be an eaer problem and tghtenng not a effectve, we wll focu on cot mnmzaton for the remander of the ecton. Table 4. Summary of tghtenng reult wth average computatonal tme or optmalty gap. Cot Mnmzaton Maepan Mnmzaton Category # of problem F % 2.1% % F % % F1 F2 F3 F (a) Cot Mnmzaton (b) Maepan Mnmzaton Fgure 16. erformance profle comparng tghtenng formulaton F1-F4. 23

24 Fracton Solved to Optmalty Computatonal Tme () Optmalty Gap (%) Fracton Solved to Optmalty Computatonal Tme () Optmalty Gap (%) Fracton Solved to Optmalty Computatonal Tme () Optmalty Gap (%) Fgure 18 compare the four formulaton for the four networ averaged over the three problem type and three tme horzon (9 run per pont). All tghtened formulaton perform better than F1, and F3 and F4 are the mot effectve. When there are recycle tream, a n N2 and N3, F2 doe not perform a well. Fgure 19 how reult for the three problem clae averaged over the four networ and three tme horzon (12 run per pont). For all problem clae, tghtenng baed on the mnmum number of batche (F2 and F5) le effectve than ung the mnmum producton requrement (F3 and F6), but ung both type of nequalte n F4 and F7 ometme better. When the problem nclude due tme n type (b), addng the due tme to the tghtenng contrant (F5-F7) ha lttle mpact on the oluton tme. Fnally, n Fgure 2 we how reult for the dfferent tme horzon, where each pont an average over all problem clae and networ (12 run per pont). A expected, ncreang the tme horzon decreae the fracton of problem olved to optmalty, but ung tghtenng ncreae the fracton olved o only 8% of the problem olved wth a 4-hour horzon are not olved wth 8- or 12-hour horzon. We alo note that the average computatonal requrement and optmalty gap decreae notably F1 F2 F3 F4 F1 F2 F3 F4 F1 F2 F3 F4 (a) Formulaton (b) Formulaton (c) Formulaton Fgure 17. eult for the two obectve. Cot MS N1 N2 N3 N4 F1 F2 F3 F4 F1 F2 F3 F4 (a) Formulaton (b) Formulaton Fgure 18. eult for four proce networ (cot mnmzaton). F1 F2 F3 F4 (c) Formulaton a 9 b 6 c a b1.5 c a b c F1 F2 F3 F4 F5 F6 F7 F1 F2 F3 F4 F5 F6 F7 (a) Formulaton (b) Formulaton Fgure 19. eult for the three problem clae (cot mnmzaton). F1 F2 F3 F4 F5 F6 F7 (c) Formulaton 24