Scheduling and feed quality optimization of concentrate raw materials in the copper refining industry

Transcription

1 Scheduling and feed qualiy opimizaion of concenrae raw maerials in he copper refining indusry Yingkai Song, Brenno C. Menezes, Pablo Garcia-Herreros, Ignacio E. Grossmann * Deparmen of Chemical Engineering. Carnegie Mellon Universiy. Pisburgh, PA 15213, USA. Research, Developmen & Innovaion. Aurubis AG. Hamburg, Germany. ABSTRACT The scheduling and feed qualiy opimizaion for processing solid concenraes in he copper refining indusry is formulaed as a large-scale discree ime non-convex Mixed-Ineger Non- Linear Program (MINLP) ha includes logisics operaions and ad-hoc blending consrains. For his complex problem, he full space MINLP for he blending of concenraes and he scheduling of he logisics is pariioned ino a Mixed-Ineger Linear Program (MILP) and in a Non-Linear Program (NLP). The soluion sraegy considers he relax-and-fix rolling horizon wih nearby ime window overlaps and he use of muliple MILP soluions o be applied in a wo-sep MILP- NLP procedure. Two models are proposed for he flowshee balances: a spli fracion model and a process nework model. The resuls indicae ha he spli fracion model yields near opimal soluions wih a large compuaional effor, while he process nework can generae several feasible soluions faser. We presen a moivaing example and an indusrial problem wih MILP o NLP gaps close o 0%. 1

2 1. Inroducion Copper is one of he mos imporan meals in indusry because of inheren properies: high hermal and elecrical conduciviy, good corrosion resisance, malleabiliy, among ohers. 1 Wih he widespread evoluion of he inerne of hings and he brand-new projecs on smar ciies, he demand of copper for elecommunicaions, energy supply, and raffic infrasrucures, is expeced o increase. Among oher applicaions, copper plays an imporan role in he developmen of elecric vehicles. According o a research conduced by IDTechEx 2, he baery of an elecric vehicle has 83 kg of copper, which is more han 3.5 imes he conen of an inernal combusion engine. Consequenly, copper demand for elecric vehicles is expeced o increase from 185,000 ons in 2017 o 1.74 million ons in Renewable energy generaion echnologies, such as wind, ocean and solar insallaions are also copper inensive 3. The insallaion of 1 MW of offshore wind energy requires abou 6 ons of copper; a phoovolaic plan requires abou 4 ons/mw, appreciably more han an oil-fired power plan (1.1 ons/mw) or a nuclear power plan (0.7 ons/mw). Given he curren and fuure imporance of copper, his work proposes a mehodology o opimize he scheduling and feed-mix blending of concenraes in a copper refinery, wih he purpose of improving he operaional resuls based on an economic meric. The final produc of he process is copper cahodes, a high puriy shee of copper ha can be sold as a commodiy. The ypical process of ransforming copper concenraes ino copper cahodes can be divided ino hree sages: smeling, convering and elecrorefining. These processes ransform he raw maerials ino a final produc conaining more han 99.99% copper. 1 2

3 Since copper concenraes wih diverse composiion are obained from ores widespread over he world, he procuremen of concenraes is an imporan decision-making challenge. In addiion, many qualiy resricions apply o he chemical composiion of he concenraes because of he condiions required for smeling and oher seps of he refining process. Therefore, an enerprise-wide opimizaion approach inegraing he scheduling and feed blending operaions in he processing sie can lead o high-performance producion since he smeler and downsream unis are very sensiive he composiion of he raw maerials. For his reason, a blend scheduling opimizaion model can unveil he poenial value of complex qualiy feeds. To he bes of our knowledge, here is no research work repored on scheduling and feed qualiy opimizaion in he copper processing indusry. Neverheless, we can find similariies wih he crude-oil blend scheduling problem: boh processes inegrae bach, semi-coninuous, and coninuous operaions as blending and scheduling are execued considering logisic and qualiy consrains. Therefore, similar modeling and soluion sraegies can be used o ackle he blending and scheduling problems of boh crude-oils and copper concenraes, despie he difference arising from operaing wih solid maerials insead of liquids. As described in he nex secions, several logisics consrains mus be considered o ranspor, sore, and mix copper concenraes. The blend scheduling opimizaion of copper concenraes involving is logisics and qualiy aspecs in a given opology is a challenging problem since i gives rise o a complex MINLP model. To opimize such sysems of raw maerials, Lee e al. 4 developed a firs approach for scheduling in crude-oil refineries based on a discree-ime formulaion. The MINLP model is reformulaed as an MILP problem by relaxing he bilinear erms represening he blending qualiy consrains, which can lead o sreams wih differen componen concenraion leaving he 3

4 same qualiy ank. Mendez e al. 5 proposed a sequenial MILP approach o simulaneously solve he blending and scheduling opimizaion of peroleum fuels using an approximaion ha successively updaes a dela (he difference beween he linear and nonlinear blending correlaions) o ieraively correc he MILP blending predicions. However, he main sraegy in he lieraure of blending and scheduling opimizaion o consider several sages of sorage, mixing, and movemen of raw maerials involved in he logisics and feed qualiy operaions, is o decompose he MINLP model ino he soluion of MILP and NLP programs In such MILP-NLP decomposiion approach, he qualiy informaion from he mixing of sreams is negleced in he MILP sep, which migh produce infeasibiliies in he NLP subproblems if he discree assignmens found in he MILP do no allow maching he qualiy consrains of he NLP. Alernaives in he lieraure o approximae he non-lineariies of blending in he MILP before solving he NLPs make use of: a) piecewise McCormick envelopes o linearly under- and over-esimae he bilinear erms, b) muliparameric disaggregaion, 7 and c) augmened equaliy balances as cus of qualiy maerial flows. 16 Successive subsiuion o ieraively correc he MILP predicions 5,16 have also been used o sar he NLPs. In erms of he ime represenaion, despie he advances in MILP solvers, he simulaneous blending and scheduling opimizaion in boh coninuous- and discree-ime formulaions have moved away he effors from he laer o he more compac coninuous-ime approaches. 6-10,17,18 Srenghs and weaknesses of discree- and coninuous-ime formulaions have been summarized by Joly and Pino 19 and Floudas and Lin. 20 Broadly speaking, discree-ime models are faser and easier o undersand, and ulimaely o implemen a he operaing/process level. However, he ime inervals mus be sufficienly shor o properly represen he blending and scheduling 4

5 problems. Alhough he discree-ime models presen a igher LP relaxaion, he soluion of indusrial problems may demand sraegies such as Benders 21 or Lagrangean 22 decomposiion, or some sor of rolling-horizon echniques inegraed wih a relax-and-fix mehod for he discree decisions. 15,23,24 On he oher hand, coninuous-ime formulaions can be easily solved for several ypes of problems since a smaller number of discree decision are required. Major drawbacks are in he weaker LP relaxaion and possible soluions including very small-ime inervals which canno be execued in pracice. In addiion, manpower bolenecks can be a barrier o execue he scheduling in a specific poin in ime of he coninuous-ime raher han in a ime-sep window as in he discree-ime model. Lee and Maravelias 25 proposed a hybrid modeling of ime by combining coninuous-ime in a refined discree-ime approach ha can bring new alernaive o scheduling problems. Furher o he discussion on MINLP soluion sraegies and he ime represenaion, a major opic concerning he proposed blending and scheduling opimizaion problem is he modeling of he quaniy and qualiy balances in he blending or pooling locaions. Misener and Floudas 26 presen a comprehensive survey on modeling and opimizaion of he pooling problem. The main wo formulaions are: a) he p-formulaion by Haverly 27 based on oal flows and composiion variables, which is widely used in chemical indusry and b) he q-formulaion by Ben-Tal e al. 28 ha keeps rack of he proporion of raw maerials in he flows, which ofen performs beer wih branch-and-cu algorihms. 29 Sahinidis and Tawarmalani 30 also presened a pq-formulaion producing a igher relaxaion for he pooling problem by adding wo paricular consrain ses o he q-formulaion. Recenly, Loero e al. 31 formulaed he source based (SB) model which is similar o he spli fracion (SF) model proposed by Quesada and Grossmann, 32 alhough he sources are racked 5

6 insead of he specificaions. In he (SF) model, he fracion of specificaion q in a sream is defined as he amoun of flow of specificaion q in he sream divided by he oal flow beween anks (see eq. 3b). In he (SB) model, he composiion of a sream is deermined from he composiions of each of he sources presen in he sream. More deails abou he difference beween models can be found in Loero e al. 31 We propose wo alernaive formulaions for he discree-ime muliperiod blend scheduling problem of copper concenraes: he muliperiod process nework formulaion ha derives from he sandard p-formulaion and he muliperiod spli fracion formulaion ha derives from he spli fracion model proposed by Quesada and Grossmann 32, which is similar bu no idenical o he q-formulaion. The remaining of his aricle is organized as follows. In secion 2, he problem saemen of he blend scheduling of copper concenraes is explained. In secion 3, he spli fracion and process nework formulaions for modeling he mass and qualiy balances in he problem are presened, as well as he binary relaions for specific logisics operaions and qualiy consrains in erms of composiion of elemens. In secion 4, soluion sraegies o solve he decomposed MILP are presened, namely he relax-and-fix rolling horizon scheme using ime blocks wih ime-sep overlapping and he generaion of muliple binary soluions. Finally, compuaional resuls are presened o show he effeciveness of he proposed soluion sraegy and he performance of he spli fracion and process nework models. Boh, he moivaing example and he indusrial problem are modeled using he wo differen formulaions. 6

7 2. The raw maerial blend scheduling problem The raw maerial blend scheduling problem in his aricle considers he arrivals of mariime vessels conaining solid copper concenraes. These raw maerials are iniially unloaded a he mariime por, where he differen copper concenraes are separaely piled in preparaion for he firs mixing sep (see Figure 1). The mixed concenraes are loaded in ransfer ships and dispached o he processing faciliy hrough a river ransporaion sysem. These pre-blends are received a he producion faciliy o be direcly fed ino several bins of he smeler furnace. Daily arrival of non-concenrae maerials are also added o he bins in lower amouns. Figure 1. Topology of he blend scheduling nework for he copper refining plan Similar o he final qualiy consrains in blending and scheduling of crude oils, here are upper bounds on he mass fracion of key componens in he mixures o be processed in he smeler. In 7

8 addiion, he process includes inerdependency consrains based on individual key componen flowraes; hese consrains model resricions for he subsequen sages in he refining process, such as he convering and elecro-refining sages. 1 In such linear consrains, for example, a flowrae of K1 componen would be lower han a flowrae of K2, or oher linear consrains involving flowraes and oher parameers can be modeled as will be shown in he indusrial example. In a crude oil blend scheduling problem 4,6, he objecive funcion considers he gross margin for processing he raw maerials and he cos of invenory managemen and unloading. In he proposed blending and scheduling of copper raw maerials, variaions in he concenraion of key componens in he mixure are undesirable in he smeling process due o operaional sabiliy. Therefore, penalies are also considered for variaions in he concenraion of cerain key componens in he objecive funcion, since i provides a balance beween profi and operaional sabiliy. Complex logisic operaions of inermediae unis such as ransfer ships and bins, are also needed for processing he copper conaining raw maerials. They are described in deail below. Operaion of he mariime vessels. Vessels wih differen ypes of copper concenraes arrive on he por during he ime horizon. The vessels unload all he maerials as piles a he expeced arrival ime period and i is assumed ha he ime period is long enough for he vessels o discharge all maerials. There are unlimied capaciies on he por. Invenory cos and unloading cos are negleced. Operaion of he pre-blender. The pre-blender on he por has unlimied capaciy and can simulaneously receive maerial from several piles while i is no charging a ransfer ship. Par of 8

9 a pile can be sen o he pre-blender in one ime period, no necessarily he whole pile. The preblender can also be discharged o wo or more ransfer ships simulaneously. The qualiy of mixed raw maerials in he pre-blender is no under conrol. Operaion of he ransfer ships. River cargo ships ransfer he pre-blends of copper concenraes from he mariime por o he producion faciliy. There is a cycle ime for he ransfer considering ime of loading, ransporing, unloading, and reurning, which gives rise o he ranspor delay. Unloading maerials from he river ships in he producion faciliy follows a general rule: shu down when empy. This means ha unloading operaions are execued unil he ships are empy. The ransfer ships can be discharged ino wo or more bins simulaneously. Two or more ransfer ships are allowed o unload maerials simulaneously o differen bins. Operaion of he bins. Several bins feed he smeler hrough conveyor bels simulaneously and coninuously. One bin can only be charged by one resource a one ime period, eiher from one ransfer ship or from one pile of daily-arrival maerial. If a vessel arrives when all bins are being charged by difference resources, he vessel mus wai o unload he maerials. The bins funcion similarly o he charging anks in crude oil scheduling problem bu no exacly in he same way. Firs, a charging ank canno be charged and discharged simulaneously, bu a bin is charged and discharged simulaneously and coninuously feeding he smeler hroughou he ime horizon. Second, all maerials in a charging ank are well-blended; a bin can conain more han one feed mix or blend since here is no mechanical agiaion nor blending of he solids in he bins. Therefore, a single bin can successively feed he smeler wih blends having differen concenraions. In he process, he smeler is charged by a coninuous moving bel ha ranspor he raw maerials sored a he bins. For beer conrol of he process, i is required ha he feeding process from he bins are coordinaed o sar and end a given feeding recipe (or smeler 9

10 die) simulaneously. Oherwise, if one bin sars feeding he smeler wih new mixures, here migh be he need o abruply adjus anoher conveyor bel o saisfy qualiy consrains. Operaion of he sub-bins. To model he operaions for he bins described above (wihou simulaneous charging and discharging such as a sanding-gauge ank), we model each bin by dividing i ino wo sub-bins. The sub-bins follow he assumpions as shown below. i. The sub-bins have unlimied capaciy. ii. Each sub-bin canno be charged and discharged simulaneously. I can only be charged from one source a a ime, eiher from one ransfer ship or from one pile of non-concenrae maerial, when i is empy. Blending is forbidden in he sub-bins, i.e. a sub-bin canno be charged by wo differen resources in successive ime periods. iii. The sub-bins can only sar charging he smeler a few ime periods afer he ransfer process is iniiaed because here are no iniial invenories in he inermediae unis. Based on he proposed rules of operaion and he ranspor delay for he ransfer ships, i akes ime for he raw maerials o reach he smeler from he por. In his paricular case, he smeler can work coninuously afer ime period 5 as shown in Figure 2. Figure 2. Time delay for he maerial ranspor 10

11 iv. The sub-bins are grouped ino wo pre-defined ses wih only one se charging he smeler a any ime, i.e. when a group of sub-bins are feeding he smeler, he oher group can be charged. In addiion, every sub-bin in one working group mus sar o feed he smeler simulaneously and mus sop feeding he smeler simulaneously. Figure 3 illusraes how his modeling scheme works. Figure 3. Modeling mehod: 2 sub-bins for each bin. v. The sub-bins obey he general rule, shu down when empy, which means ha a working sub-bin can be supplying maerial o he smeler over one or more ime periods unil i becomes empy. If he sub-bins of he working group become empy, a feeding blend ends and anoher blend can sar using he oher group. Operaion of he daily arrivals of non-concenrae maerials. There are daily arrivals of non-concenrae maerials in he producion faciliy, which are mainly lefovers from oher refining processes. I is required ha he daily arrival maerial is no accumulaed in he producion faciliy and i mus be consumed a he end of he scheduling horizon. Operaion of he smeler. The smeler mus be operaed coninuously a full capaciy. The goal of he opimizaion problem is o maximize he gross margin obained from processing raw maerials, which implies deermining he schedule of he operaions o be 11

12 execued as well as he amoun of concenraes o be ransferred from he por o he plan. Penalies for variaion in he composiion of key componens are also considered. Therefore, he model aims o find he mos profiable blend schedule wih consideraion of he logisic resricions and he operaional sabiliy of he smeling furnace. 3. Mahemaical model The mahemaical model for blending and scheduling of copper concenraes shown in Figure 4 involves quaniy and qualiy balances, as well as operaional relaionships considering he resources and heir connecions. Figure 4 is a schemaic represenaion of he illusraive example proposed for he scheduling and feed qualiy opimizaion problem of copper concenraes. Each bin is modeled by wo sub-bins, as we shall discuss in secion The main resources used in he model are concenraes c, marine vessels v, piles p, pre-blender b, ransfer ships s, daily arrival piles dp, sub-bins sb, and he smeler s. Figure 4. Schemaic represenaion of he mixing nework of he moivaing example 12

13 In he problem, copper concenraes wih differen concenraion of key componens k are unloaded a he mariime por, and differen piles p are formed separaely. A firs mixing sep akes place in he pre-blending faciliy b, where he blend is loaded ino he ransfer ships s ha ranspor i o he processing plan. Once he ransfer ships arrive o he plan, hey feed he subbins sb. The final blend is he resul of a second mixing sep ha is conrolled by varying he speed of he conveyor bels connecing each sub-bin o he smeler s. As menioned earlier, we developed wo alernaives o describe mahemaically he flow, sorage, and mixing of copper concenraes. For a uni n, considering upsream unis i I n and downsream unis o O n, we formulae he models based on he following approaches: a) A muliperiod Process Nework model (PN) including raw maerial concenraions xn,c, in oal flows (Fi,n, or Fn,o,) and oal amouns (Mi,n, or Mn,o,), derived from he sandard p-formulaion for coninuous process in he Harvely pooling formulaion 27 ; b) A muliperiod Spli Fracion model (SF) considering flow proporions of inles i and oules o (fi,n, or fn,o,), derived from he spli fracion model for coninuous process proposed by Quesada and Grossmann Quaniy and qualiy balances of blending and scheduling We formulae he process nework and he spli fracion models for he quaniy and qualiy balances of he blending and scheduling processes ha originae a he mariime por and end a he smeler. The process nework formulaion includes oal invenories, flow, and composiion variables of each inermediae uni and sockpile; in his model, he nonlineariies arise on he blending of flows when calculaing he blended maerial composiion. Disaggregaed invenories 13

14 and flow variables for each ype of copper concenrae are considered in he spli fracion model and he bilinear erms appear when inermediae unis are discharging Process nework model (PN) The process nework formulaion is derived from he sandard p-formulaion. The mass and qualiy balances in his model for a coninuous process are shown in eq. 1a and eq. 1b, F = F i I n i, n n, o o O n n N (1a) F x = F x n N, c C (1b) i I n i, n i, c n, o o, c o O n where Fi,n represens he oal inle from uni i o he uni n and Fn,o is he oal oule from he uni n o he uni o. xi,c is he mass fracion of concenrae c in uni i and xo,c is he mass fracion of concenrae c in he uni o. The muliperiod process nework model requires wo addiional equaions: eq. 2a models he oal mass balance of invenories; eq.2b represens invenories of individual concenraes c modeled by he oal invenories Mn, imes he composiion variables xn,c,, which is he mass fracion of concenrae c in uni n a ime. M = M + F F, n, n, 1 i, n, n, o, i I o O n n n N T (2a) M x = M x + F x F x,, n, n, c, n, 1 n, c, 1 i, n, i, c, 1 n, o, n, c, 1 i I o O n n n N c C T (2b) Spli fracion model (SF) 14

15 Quesada and Grossmann 32 proposed a spli fracion model including he flow variables and invenory of differen concenraes c and he spli fracion of discharge, as shown in eq. 3a and eq. 3b. f = f n N, c C (3a) i I n i, n, c n, o, c o O n i In i In f f i, n, c = f f noc,, i, n, c n, o, c n N, o O,( c c ) C (3b) n Eq. 3a is he mass balance for inle (fi,n,c) and oule (fn,o,c) flows of concenrae c going hrough uni n; eq. 3b forces he spli fracion of he oule flows o be idenical o he aggregaed concenrae fracion of inle flows. The spli fracion model menioned above is no he same as he sandard q-formulaion 28. The spli fracion model racks exacly he specificaions, which is he individual flow of concenrae c in he proposed blending problem insead of proporion variables. The proporion variables in he q-formulaion 29 are he fracion of flows coming from differen sources, which are he raw maerials in differen vessels and iniial invenories a he por in he proposed blending problem. We apply he idea of spli fracion model for coninuous process o he muliperiod blending problem and inroduce variables for oal invenories and flowraes in he sockpiles and unis. The mass balances aggregaing he amouns of differen concenraes are shown in eq. 4a-4d. M = m n N, T (4a) n, n, c, c C 15

16 F = f,, (4b) n, o, n, o, c, c C n N o O T n Eq. 4a correspond o he balances beween oal invenory variables and disaggregaed invenories. Eq. 4b represens he balances beween oal flows and disaggregaed flows. In paricular, Mn, models he oal invenory of inermediae uni n a ime ; mn,c, is he corresponding individual invenory of concenrae c; Fn,o, represens he oal ouflow of uni n o he downsream uni o; and fn,o,c, is he corresponding individual ouflow of concenrae c. The individual invenory of concenrae c a ime is defined by adding o he individual invenory a -1 all he inle and oule individual flowrae a ime as shown in eq. 4c. m = m + f f,, n, c, n, c, 1 i, n, c, n, o, c, i I o O n n n N c C T (4c) The composiion of he producs ransferred during a ransfer operaion a mus be idenical o he composiion a he origin ank a -1, m f = n N, c C, o On, T F n, c, 1 n, o, c, M n, 1 n, o, which can be reformulaed wih bilinear erms according o eq. 4d. m F M f = n N, c C, o O, T (4d) n, c, 1 n, o, n, 1 n, o, c, n The logic consrains modeling he ransfer of maerial from uni n o heir downsream unis o O n are shown in consrains 5a and 5b, F F D n N, o O, T (5a) n, o, n, o n, o, n 16

17 F F D n N, o O, T (5b) n, o, n, o n, o, n where [ F, F ] n, o n, o are defined as he bounds on flow from uni n o uni o, and Dn,o, are binary variables indicaing if uni n is charging uni o a ime Operaions for blending and scheduling of sock piles The operaion o handle he solid sock piles and flows of concenraes c are modeled wih he mixed ineger consrains described below Operaion of he mariime vessels The vessels ransporing copper concenraes unload all he raw maerial when hey arrive o he por, in conras o he crude oil ankers 4,6, which ypically unload in muliple ime-periods due o pipeline bolenecks. In our model, he vessels can unload all maerial on he por as a pile p wihou considering unloading cos or sorage anks. Therefore, he vessels operaion is predefined. We define he parameer Fv,p, o indicae he flowrae from vessel v o pile p a ime Operaion of he pre-blender The pre-blender b canno be charged and discharged simulaneously. Insead of defining binary variables for flow from piles o he pre-blender, we define binary variables Xb, o indicae if he pre-blender is charging any ransfer ships (see Figure 5). Then, he flow from piles o he preblender is indicaed by 1-Xb,. Db,s, is a binary variable indicaing if he pre-blender b is charging ransfer ship s a ime. The corresponding consrains are given by he inequaliies

18 Figure 5. The mehod for modeling he operaions of he pre-blender If he pre-blender is charging any ransfer ship ( D = b, s, 1 ), Xb, mus be 1. D X b B, s TS, T (6a) b, s, b, If he pre-blender is no charging any ransfer ship ( Db, s, = 0 ), Xb, mus be 0. s TS Db, s, Xb,, s TS b B T (6b) If he pre-blender is no charging any ransfer ship ( X b, = 0 ), he feeding process from piles is allowed and is flow Fp,b, is given by: F F (1 X ) b B, p P, T (7a) p, b, p, b b, F F (1 X ) b B, p P, T (7b) p, b, p, b b, Operaion of he ransfer ships The operaions of he ransfer ships beween he por and producion faciliy consider a ranspor delay, which means ha he duraion ime for loading, ransporaion, and unloading are all equal 18

19 o one ime period. The ransfer ships should always unload all he maerial in he producion faciliy, which is he shu down when empy condiion. The consrains for ranspor delay are given by inequaliies 8a-8c. The ransfer ships canno be charged (Db,s,) and discharged (Ds,sb,) simulaneously, D s, sb, Db, s, 1 + b B, s TS, sb SB, T (8a) where Ds,sb, is defined as a binary variable o indicae if ransfer ship s is charging sub-bins sb a ime. The ransfer ships canno be charged and discharged in successive ime periods. D s, sb, 1 Db, s, b B, s TS, sb SB, T (8b) Ds, sb, + D b, s, b B, s TS, sb SB, T (8c) In order o model he shu down when empy condiion for unloading operaions in a single ime period, we define he binary variable Es, o indicae if here are maerials in ransfer ship s a he end of ime ; his condiion is modeled wih consrains 9a and 9b. M M E s TS, T (9a) s, s s, M M E, s, s s, s TS T (9b) where [ M, M ] are he bounds on invenories of ransfer ship s. s s 19

20 Consrains 10a-10b ensure ha each ransfer ship s charges only one sub-bin sb a he same ime; binary variables Xs, indicae if he ransfer ship s is charging any sub-bin sb a ime ; binary variables Ds,b,, indicae if ransfer ship s is charging sub-bin sb a ime. A similar formulaion was proposed by Zyngier and Kelly. 33 D X s TS, sb SB, T (10a) s, sb, s, Ds, sb, Xs,, sb SB s TS T (10b) The shu down when empy is enforced wih consrains 11a and 11b. If ransfer ship s is unloading maerial a ime ( X s, = 1), i mus be empy a he end of ime ( s, 0 E = ). X + E s TS, T (11a) s, s, 1 Furhermore, if ransfer ship s is being charged by he pre-blender a ime ( D,, = 1), he ransfer ship s mus be iniially empy ( Es, 1 = 0). b B b s E + D s TS, T (11b) s, 1 b, s, 1 b B Operaion of he daily arrivals of non-concenrae maerials The invenory of daily arrival maerial should be below a cerain value a he end of he ime horizon. This consrain is differen for he spli fracion model and he process nework model. Since he spli fracion model has variables indicaing he invenories each raw maerial in, he condiion can be easily described wih he linear consrain 12, 20

21 m + m MD c DC, = TMH (12) dp DP dp, c, sb, c, sb SB where mdp,c, and msb,c, represen he amoun of non-concenrae maerial c (c DC) in daily arrival pile dp and in sub-bin sb a ime, respecively. MD represens he maximum amoun of his non-concenrae maerial in he sysem a he end of ime horizon, TMH. The recycling and reprocessing maerials c (c DC) boh socked in he daily arrival piles and in he sub-bin downsream conneced o hem are regarded in consrains 12 and 13. In he process nework model, bilinear erms are necessary for describing he amoun of nonconcenrae maerials boh socked in he piles and wihin he sub-bins a he end of scheduling horizon. In order o propose a MILP approximaion necessary for he soluion sraegy described in Secion 4, we impose he consrains ha he sub-bins should be empy a he end of scheduling horizon. These condiion is modeled wih consrains 13a and 13b. M dp, MD, dp DP = TMH (13a) M =, sb, 0 sb SB = TMH (13b) Even hough he formulaions of he spli fracion and process nework models are no equivalen, he difference does no affec he opimal soluion as will be shown in he compuaional resuls secion. In oher words, he sub-bins wih non-concenrae maerials (c DC) are always empy a he end of ime horizon, wheher we impose eq. 13b or no Operaion of he sub-bins The purpose of he sub-bins as a modeling sraegy is o force he bins o feed he same recipe simulaneously and coordinae he change-over o he nex recipe. We divide all sub-bins (se 21

22 FSB and se SSB) ino wo groups conaining exacly one sub-bin from every bin. For each group, we define a se of binary variables, Df and Ds, indicaing if he firs (Df) or second (Ds) group of sub-bins is charging he smeler a ime. Consrains 14a and 14b ensure ha each subbin is eiher being charged wih non-concenrae maerials dp, is being charged wih concenraes from he ransfer ships s, or is feeding he smeler. D + Df + D sb FSB, T (14a) dp DP dp, sb, s, sb, 1 s TS D + Ds + D sb SSB, T (14b) dp DP dp, sb, s, sb, 1 s TS The wo groups of sub-bins should work in cycles afer he saring of smeling process ( ET) as imposed wih eq. 15. Df + Ds = 1 ET (15) In order o model he operaion shu down when empy of he sub-bins feeding process in muliple ime periods, we define he variable Esb, o indicae if here are maerials in he sub-bin sb a ime according o consrains 16a and 16b. M M E sb SB, T (16a) sb, sb sb, M M E sb SB, T (16b) sb, sb sb, New binary variables Df ˆ and Ds ˆ are inroduced o indicae when one feeding recipe ends as shown in Figure 6. Consider he firs group of sub-bins as an example. 22

23 Figure 6. Binary variables indicaing end of recipe. Consrains 17a-17c model he end of a feeding recipe wih variables Dfˆ and Dsˆ. If one feeding recipe ends a ime ( Df ˆ = 1 ), he sub-bins mus feed he smeler a ime ( Df = 1 ). Df ˆ Df T (17a) If one feeding recipe ends a ime ( Df ˆ = 1), he group of sub-bins canno feed he smeler a ime +1 ( Df + 1 = 0 ). Dfˆ + Df T (17b) If he sub-bins feed he smeler a ime ( Df = 1), and do no feed he smeler a ime +1 ( Df + 1 = 0 ), he group of sub-bins end a feeding recipe a ime ( Df ˆ = 1). Df + Dfˆ + Df T (17c) 1 The same definiion is applied o Ds ˆ as shown in consrains 18a-18c. Ds ˆ Ds T (18a) 23

24 Dsˆ + Ds T (18b) Ds Ds Ds + ˆ + 1 T (18c) The shu down when empy condiion is enforced wih consrains 19a-19c. If he sub-bins finish charging he smeler a ime (e.g., Df ˆ = 1 for he firs bins), hen hey mus be empy a he end of ime ( E = sb, 0 ). Dfˆ + E 1 sb FSB, T (19a) sb, Dsˆ + E 1, sb, sb SSB T (19b) If he sub-bin sb is non-empy ( Esb, 1 = 1), i canno be charged by any of he ransfer ships or dp sb ). s sb dp DP s TS daily arrival piles ( D,, + D,, = 0 D + D + E sb SB, T (19c) dp DP dp, sb, s, sb, sb, 1 1 s TS 3.3. Final blending and qualiy consrains The smeler works a full capaciy coninuously saring a ET and he sum of flowrae from subbins are equal o he maximum flowrae ino he smeler, as shown in eq. 20a and eq.20b, F = F T (20a) sb SB sb, s, F = F ET s (20b) 24

25 where F s corresponds o he full capaciy of he smeler. The conversion from concenraes oal flow o he flow of individual key componens F k, is calculaed based on heir assay. The consrains necessary for he spli fracion model are linear as shown in eq. 21. The equivalen consrains are nonlinear for he process nework model as in eq. 22. F f =, k, sb, s, c, k, c sb SB c C s S k K T (21) F F x =, k, sb, s, sb, c, 1 k, c sb SB c C s S k K T (22) Here, kc, is he assay describing he composiion of key componens k in concenrae c, and xsb,c,- 1 represens he mass fracion of concenrae c a ime -1. Qualiy consrains. We specify an upper bound for he mass fracion of key componens, in k he final blending. F F, (23) k k, k K T Inerdependency consrains. The consrains model complex qualiy resricions ha relae he conen of differen key componens in he blends. Equaions 24a-24c are based on individual flows, where kk is alias of k. consrains. Ue k and Ke k are he parameers relaed o he inerdependency 0.64F k 2, F k 7, T (24a) 25

26 0.58F k 2, F k 7, T (24b) Ue ( Ke F ) Ke F k K, T (24c) k kk kk, k k, kk K 3.4. Objecive funcion The objecive is o maximize he gross margin of processing differen kinds of raw maerials, while considering penalies for he variaion in concenraion of key componens. The objecive funcion for boh spli fracion model (SF) and he process nework model (PN) have he same physical inerpreaion; however, heir formulaions are differen. Eq. 25 presens he linear objecive funcion for he spli fracion model (SF). The nonlinear objecive funcion of he process nework model is presened in eq. 26. (25) max fsb, s, c, c Zk, k T sb SB s S c C T k K (26) max Fsb, s, xsb, c, 1 c Zk, k T sb SB s S c C T k K Here, c is defined as he income for processing uni concenrae c, while k is he pre-defined weigh of flow change for key componen k change for key componen k, is defined according o eq. 27a and 27b.. The posiive variables Z k,, defining he flow Zk, Fk, Fk, 1 k K, T (27a) Z F F k K, T (27b) k, k, 1 k, 26

27 4. Soluion sraegy 4.1. Soluion sraegy for spli fracion model MILP-NLP decomposiion mehod Theoreically, he nonconvex MINLP model can be solved using any global solver such as BARON or a local opimal solver such as DICOPT. However, because of he high nonlineariy and large size of he model, BARON can be prohibiively expensive o use, while DICOPT may converge o poor subopimal soluions. Therefore, a wo-sep procedure is proposed ha leads o local opimal soluions wih an esimaion of he opimaliy gap. In he firs sep, an MILP relaxaion of he model wihou he nonlinear consrains (4d), is solved using CPLEX. In he second sep, he nonlinear consrains are enforced, and all binary variables are fixed, which means he schedule of operaions is fixed. The resuling nonlinear programming model is solved using he soluion of he MILP as a saring poin by a global NLP solver, BARON or local NLP solver CONOPT. The soluion a his sage may no correspond o he global opimum of he model in he full space or may even be infeasible. However, he opimaliy gap can be esimaed from he soluion of he MILP and he feasible NLP. If he NLP reurns an objecive value idenical o he MILP s, hen global opimaliy is proven. As shown in he compuaional resuls secion, near-opimal soluions are obained in which he MILP-NLP gap is close o zero Relax-and-fix rolling horizon wih nearby ime window overlaps The MILP model of he moivaing example in a spli fracion model can be solved by CPLEX in he full space considering all binary variables are simulaneously calculaed. However, for he indusrial problem in a spli fracion model, he firs MILP model canno be solved in reasonable 27

28 soluion ime since he size of indusrial problem is oo large. Therefore, a relax-and-fix rolling horizon wih nearby ime window overlaps mehod is proposed o solve his large scale MILP model. A rolling horizon algorihm is applied similar o he producion-disribuion coordinaion of crude-oil scheduling presened by Assis e al. 15 and indusrial gases supply chains presened by Zamarripa e al. 23 The rolling horizon mehod divides he ime horizon ino a deailed ime block where binary variables are considered and an aggregae ime block where binary variables are relaxed o 0-1 coninuous variables; he mehod successively solves a se of subproblems in which discree decisions are fixed in previous ime periods, as shown in Figure 7. Figure 7. The illusraive example of rolling horizon by Zamarripa e al (2016) However, considering ha in his raw maerial blending problem we have special operaion rules, he rolling horizon menioned above can lead o infeasible soluions afer fixing binary decisions in he previous ime block; his can lead o infeasible problem in subsequen subproblems. Therefore, a rolling horizon sraegy wih relaxaion overlaps is applied in his work. We have overlapping ime periods for he binary relaxaions beween he aggregae ime block and he nex deailed ime block o avoid infeasible soluions in he soluion process. 28

29 Figure 8. Proposed rolling horizon wih relaxaion overlapping for he illusraive example. The scheme presened in Figure 8 shows an illusraive example in which we describe a problem wih 25 ime periods. We have 9 subproblems solved sequenially. Subproblem 1: ime periods 1-5 are considered as he deailed ime block, while he res ime periods are considered as he aggregae ime block. Subproblem 2: Even hough he soluion from ime period 1-5 is known, we only fix he binary variables of ime period 1-3, which means he overlap beween he firs and second deailed ime block is 2 ime periods. Then he deailed ime block of subproblem 2 is from 4-8. This procedure is repeaed unil he las subproblem in which here is only one deailed ime block o he end of he scheduling horizon. Theoreically, he objecive value by his rolling horizon wih overlap mehod is always lower han he one obained by solving he full space. However, he suiable combinaion of he lengh of he deailed ime block and overlap can reduce he gap beween he soluion by his sraegy and he global opimal soluion. There is rade-off beween he lenghs of he deailed ime block and overlap periods, and he compuaional resul. If he deailed ime block and overlap are longer, i will require a large compuaional effor, bu he soluion can be expeced o be closer 29

30 o he opimal soluion from solving he full space model. If he deailed ime block is shorer, he compuaional effor is smaller, bu he soluion migh be furher away from he opimal soluion, and here is a higher poenial for infeasible soluions. Since he lengh of he deailed ime blocks and overlaps need o be posulaed before running he rolling horizon wih he relaxaion overlapping algorihm, we have esed differen combinaions of he lengh of deailed ime blocks and overlaps and selec he bes combinaion o obain he bes objecive value wih relaively less compuaional effor. We applied his rolling horizon wih overlap sraegy o he MILP of he indusrial problem as shown in he compuaional resuls secion Soluion sraegy for process nework model MILP-NLP decomposiion mehod The MILP-NLP decomposiion algorihm is also applied o he process nework model bu wih some differences. For he spli fracion model, he objecive funcions of he MILP and NLP are he same, which is he gross margin minus he penalies of variaion in feed concenraion. However, in he process nework model, once we drop he nonlinear eq. 2b and 26 in he MILP, here is one oal flow hroughou he flowshee and he composiion informaion is no longer available, which means ha i is no possible o esimae he gross margin in he firs MILP model. To overcome his problem, we subsiue he objecive funcion shown eq.26 by eq. 28, which maximizes he oal flowrae ino he smeler during he scheduling horizon. max F (28) T Since he objecive funcions of he MILP and NLP are differen, he MILP-NLP gap canno be esimaed for he soluions obained wih he process nework model. Therefore, his MILP-NLP 30

31 mehod canno guaranee ha a feasible soluion is wihin a cerain gap of he global opimum. The moivaion here is o generae a schedule (soluion of binary variables) quickly ha can guaranee he smeler is working a full capaciy hroughou he scheduling horizon. We hen solve he NLP model o find a soluion ha maximizes he acual objecive value, bu he schedule obained from he MILP migh resric he qualiy of he soluion in reference o he original MINLP problem Muliple soluions in firs MILP In order o obain beer NLP soluions and avoid infeasible soluions, muliple soluions are generaed in he firs MILP sage. These soluions provide muliple schedules ha can keep he smeler working a full capaciy are generaed. The MILP opimaliy gap for all soluions are se o 0. Then, a se of NLP problems are generaed from all soluions in he MILP soluion pool. Theoreically, we can collec a sufficienly large number of muliple soluions (>10000) and ry o find he bes soluion in he NLP model. Bu his would require a large compuaional effor. Therefore, we collec 15 muliple soluions wih he purpose of reducing he compuaional ime. Ineviably, he soluions migh be symmeric or similar o each oher. GAMS provides a funcion for CPLEX o creae a soluion pool 34 ha generaes and keeps muliple soluions; a descripion of how o se up he corresponding parameers is presened in he Supporing Informaion. 5. Examples 5.1. Moivaing example We presen a small moivaing example o provide some insigh on he complexiy of his raw maerial blending problem. The insance consiss of 2 vessels, 4 piles wih iniial invenories a 31

32 he por, 1 pre-blender, 2 ransfer ships, 1 non-concenrae maerial, 4 bins (8 sub-bins) and 1 smeler. Figure 2 shows a schemaic represenaion of he moivaing example. The daa for he for he moivaing example represened in Figure 4 is given in Table S1 in he Supporing Informaion. The scheduling horizon is composed of 13 days and wo marine vessels are scheduled o arrive a he beginning of day 1 (=0) and day 7 ( = 6); hey conain 11,380 and 10,800 ons of concenraes, respecively. Togeher wih 3 piles of concenraes already presen a he por, hey can be mixed in he pre-blender. Two ransfer ships ranspor maerial from he pre-blender o he producion faciliy. There is one non-concenrae arriving on a daily basis o he producion faciliy. The smeler is operaed a full capaciy, 1500 ons/ime period. The properies ha are being racked are he mass fracion of four key componens (K1-K4). There is a posiive gross margin for every concenrae, bu none for he non-concenrae maerials. The ransfer flowraes for he operaions a he por are unlimied. In addiion, he inerdependency consrains and penalies for variaion in concenraion are no considered in he moivaing example. The objecive is o maximize he gross margin for processing raw maerials during he scheduling horizon Indusrial problem We provide an indusrial problem which is significanly larger han he moivaing example. The problem consiss of 5 vessels, 6 piles wih iniial invenories a he por, 1 pre-blender, 2 ransfer ships, 3 non-concenrae maerials, 6 bins (12 sub-bins) and 1 smeler. Inerdependency consrains and penalies for variaion in concenraion of one key componen are also considered. The duraion of each ime period is 0.5 days. Figure 9 shows a schemaic 32

33 represenaion of he indusrial problem. The complee problem daa is given in Table S2 in he Supporing Informaion. Figure 9. Schemaic represenaion of he mixing nework of he indusrial problem. 6. Compuaional resuls In his secion, we address he raw maerial blend scheduling problem for he small moivaing example and he indusrial problem. For each of hem, we compare he performance of he wo formulaions, he spli fracion model and process nework model. Therefore, we have 4 models o solve: 1) moivaing example in spli fracion model; 2) moivaing example in process 33

34 nework model; 3) indusrial problem in spli fracion model; 4) indusrial problem in process nework model. The differences beween spli fracion model and process nework model, as shown in secion 3, are mainly abou: a) he mass balance equaion, eq. 2a-2b and eq.4a-4d; b) he operaions for non-concenrae maerials, eq.12 and eq.13a-13b; c) he final blending, eq.21 and eq.22; d) he objecive funcion, eq.25 and eq.26. We do no consider he inerdependency consrains and penalies for he variaion in concenraion of he key componens in he moivaing example. Therefore, eq. 24a-24c and 27a-27b are excluded in he moivaing example. The equaions considered in each model are shown in Table 1. Table 1. Equaions of he four insances Insance Equaions involved Moivaing example (SF) (4)-(12),(14)-(21),(23),(25) Moivaing example (PN) (2),(5)-(11),(13)-(20),(22),(23),(26) Indusrial problem (SF) (4)-(12),(14)-(21),(23)-(25),(27) Indusrial problem (PN) (2),(5)-(11),(13)-(20),(22),(23),(24),(26),(27) We presen he compuaional resuls of applying he soluion sraegy described in he previous secion o he moivaing example and indusrial problem. The compuaional saisics of he insances are presened in Table 2. The algorihms and models were implemened in GAMS All compuaions were performed on a ASUS UX305 compuer wih wo Inel Core i5 processors a 2.30 GHz and 2.40 GHz, 8 GB of RAM. Table 2. The problems size of moivaing and indusrial insances Model Binary var. Variables Consrains Moivaing example (SF) MILP 559 5,171 5,563 NLP 5,171 7,435 Moivaing example (PN) MILP 559 1,850 3,096 NLP 2,714 4,009 Indusrial problem (SF) MILP 2,490 45,253 47,404 34

35 NLP 45,253 62,832 Indusrial problem (PN) MILP 2,490 12,090 12,963 NLP 18,404 19, Moivaing example (SF) Table 3. Performance in he moivaing example (SF) Moivaing example (SF) MILP (CPLEX, gap = 2%) NLP (CONOPT) MILP-NLP mehod soluion CPU ime soluion CPU ime gap s s 0% MINLP (BARON, gap = 2%) solved in a full space soluion CPU ime ,007 s As shown in Table 3, afer 140 s CPU ime, he opimaliy gap is 0 beween he soluion of he MILP and he NLP, which means ha he global opimum has been found. We also solve he full space MINLP model wih BARON. A similar objecive value is obained bu wih a much longer CPU ime, namely 3007 s. The slighly differen objecive values are because he MILP-NLP mehod and BARON converge o differen MILP gap wihin 2%. Therefore, he soluion by MILP-NLP mehod is applied for he following discussion. 35

36 Operaion Unloading 1 Unloading 2 Transfer 1 Transfer 2 Transfer 3 Transfer 4 Transfer 5 Transfer 6 Transfer 7 Transfer 8 Transfer 9 Transfer 10 Transfer 11 Transfer 12 Transfer 13 Transfer 14 Transfer 15 Transfer 16 Transfer 17 Transfer 18 Transfer 19 Transfer 20 Transfer 21 Transfer 22 Transfer 23 Transfer 24 Transfer 25 Feeding 1 Feeding 2 Feeding 3 Feeding 4 Feeding 5 Feeding 6 Feeding 7 Feeding Time period Figure 10. Gan char of opimal schedule for moivaing example (profi: $8,792,740) 36

37 Invenories (ons) Invenories (ons) Invenories (ons) Invenories (ons) Invenories (ons) Invenories (ons) Invenories (ons) Invenories (ons) sub-bin Time period sub-bin Time period sub-bin 3 sub-bin Time period Time period sub-bin 5 sub-bin Time period Time period sub-bin 7 sub-bin Time period Time period Figure 11. Invenories of sub-bins of he moivaing example Figure 10 depics he Gan Char (see Table S3 for serial number of he operaions) for he opimal soluion of he moivaing example wih a profi of $8,792,740. Each ask is represened by a horizonal bar and each row corresponds o a specific operaion. The feeding operaions 1-8 represen he discharging of sub-bins 1-8; we can observe ha he firs group of sub-bins (feeding 1,3,5,7) and he second group of sub-bins (feeding 2,4,6,8) work in cycles o keep he 37

38 Invenories (ons) smeler operaed coninuously. From Figure 11, eiher he firs group or he second group of subbins finish one feed recipe and becomes empy simulaneously, which physically means ha he 8 bins use up o one mixure and sar a new feed recipe simulaneously. In his case, here are 4 feeding recipes during he scheduling horizon which cover ime periods 5-8, 9-10, 11-12, and 13. Transfer ship 1 Transfer ship Time period Time priod Figure 12. The invenories of ransfer ships From Figure 12, we can see he operaion rule modeling he ranspor delay clearly apply for he ransfer ships. The cycle ime for loading, ransporing, unloading, and reurning is 4 ime periods. Consider he firs ransfer operaion of ransfer ship 1 as an example. The ship is loading a ime 2, ransporing o he producion faciliy a ime 3, unloading maerial in a ime 4, and reurning o he por a ime 5. The ransfer ship 1 ransfers maerials hree imes during he scheduling horizon, which are a imes 2-5, 6-9, The ransfer ship 2 ransfers maerial wo imes a imes2-5 and

39 6.2. Moivaing example (PN) Table 4. Performance of he moivaing example (PN) Moivaing example (PN) MILP (CPLEX, gap = 2%) MILP-NLP decomposiion soluion CPU ime Number of alernaive soluions 13,500 * 6 s 15 NLP (CONOPT) Binary decisions NLP soluion Binary decisions NLP soluion Binary decisions NLP soluion Schedule Schedule Schedule Schedule Schedule Schedule Schedule Schedule Schedule Schedule Schedule Schedule Schedule Schedule Schedule MINLP (BARON, gap = 2%) solved in a full space lower bound upper bound CPU ime ,000 s * The objecive value, 13,500 is equal o he oal amoun of processing maerials when he smeler is operaed a full capaciy (i.e. 9 1,500). We also applied he process nework model for he moivaing example. As menioned in he previous secion, he MINLP model is decomposed ino MILP and NLP subproblems. The firs MILP whose objecive funcion is o maximize he oal flowrae ino he smeler is solved by CPLEX o generae muliple soluions. Then, he differen binary decisions are fixed o solve he second NLP. The bes objecive value of NLP is seleced. As shown in Table 2, he model size of he process nework model is smaller han he spli fracion model, which is because in a process nework model here are no variables for individual flows and invenories of componens bu heir concenraions insead. On he oher hand, he non-lineariy of he NLP in he process nework model is higher han he spli fracion model. 39