Computers and Chemical Engineering 68 (2014) 128139

Contents lists available at ScienceDirect

Computers and Chemical Engineering

j our na l ho me pa g e: www.elsev ier .com/ locate /compchemeng

Robust biointegra cocost ob

Kailiang a State Key Lab g, Zheb Department o SA

a r t i c l

Article history:Received 13 MReceived in reAccepted 3 MaAvailable onlin

Keywords:Biofuel supplyPetroleum reneryRobust optimizationMixed-integer linear fractionalprogramming

gn and froe ex

mixedesigradeo

More-inte

optimization algorithm. County level cases in Illinois are analyzed and compared to show the advantage ofthe proposed optimization framework. The results show that the preconversion to petroleum-upgradingpathway is more economical when applying the unit cost objective.

2014 Elsevier Ltd. All rights reserved.

1. Introdu

Biofuelsthe future. ety of biom(DOE, 2012are lower tet al., 2014afuels targetof bioenerg(RFS), part of 2007, esgallons of be advanceadverse impopment of biofuels cacrop residuet al., 2014

CorresponE-mail add

grong@iipc.zju

http://dx.doi.o0098-1354/ ction

have been shown to be a promising fuel source ofThey can be produced domestically from a wide vari-ass sources and reduce the dependence on fossil fuels). Moreover, greenhouse-gas emissions from biofuelshan those from their petroleum counterparts (Tong). Consequently, many countries have set national bio-s and provided incentives to accelerate the growthy industry. In the U.S., the Renewable Fuels Standardof the Energy Independence and Security Act (EISA)tablishes an annual production target of 36 billionbiofuels by 2022, of which 16 billion gallons shouldd biofuels made from non-starch feedstocks to avoidacts on the food market (EISA, 2007). With the devel-the third generation biofuel technologies, advancedn now be produced from cellulosic biomass such ases, wood residues or dedicated energy crops (Yueb). Moreover, advanced hydrocarbon biofuel products

ding author. Tel.: +1 847 467 2943.resses: you@northwestern.edu (F. You),.edu.cn (G. Rong).

(e.g. cellulosic-biomass-derived gasoline, diesel and aviation fuel)are functionally equivalent to the petroleum derivatives. Consid-ering all these promising properties, the expansion of thehydrocarbon biofuel industry is foreseeable in the next fewdecades, thus requiring the design and development of cost-effective biomass-to-biofuel supply chains (DOE, 2012).

Many studies have been conducted on the design and plan-ning of biofuel supply chains from economic and environmentalaspects (Akgul et al., 2012; Bowling et al., 2011; Dunnett et al.,2008; Elia et al., 2011; Giarola et al., 2011; Sokhansanj et al., 2006;You et al., 2012; Yue and You, 2014; Yue et al., 2014a; Zamboniet al., 2009). However, the drop-in property of advanced hydrocar-bon biofuel is not well explored. The U.S. Department of Energy,DOE (2012) has pointed out the opportunities for the integra-tion of emerging hydrocarbon biofuel supply chains with existingpetroleum production and distribution infrastructures. Althoughminimum retrotting costs would be required in petroleum ren-ery for compatibility reasons, the integration indicates considerablecapital savings on the construction of biofuel production facilities,which would help the biofuel products to be cost-competitive andbring in extra environmental benets to the petroleum reneries(DOE, 2012). Researchers are investigating the possibility of con-verting biomass into biofuel in the traditional renery. Huber andCorma (2007) summarized that catalytic cracking, hydrotreating,

rg/10.1016/j.compchemeng.2014.05.0032014 Elsevier Ltd. All rights reserved. design and operations of hydrocarbon ting with existing petroleum reneriesjective

Tonga, Fengqi Youb,, Gang Ronga

oratory of Industrial Control Technology, Department of Control Science and Engineerinf Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208, U

e i n f o

arch 2014vised form 30 April 2014y 2014e 14 May 2014

chain

a b s t r a c t

This paper addresses the optimal desiwith the unit cost objective. Benetethe supply chain takes advantage of thcapital and transportation savings. Ataneously consider the supply chain robust optimization approach which tthe demand and supply uncertainty.cost-competitive. The resulting mixedfuel supply chainnsidering unit

jiang University, Hangzhou 310027, China

d planning of the advanced hydrocarbon biofuel supply chainm the drop-in properties of advanced hydrocarbon biofuels,isting petroleum infrastructure, which may lead to signicantd-integer linear programming model is proposed to simul-n, integration strategy selection, and production planning. Affs the performance and conservatism is adopted to deal withover, the unit cost objective makes the nal products moreger linear fractional programming model is solved by tailored

K. Tong et al. / Computers and Chemical Engineering 68 (2014) 128139 129

and hydrocracking are the three main techniques for convertingbiomass into biofuel in existing petroleum reneries. DOE (2012)is investigating three possible insertion points into the petroleumrenery. They note that after converting the biomass into liquidbio-intermeCrude Distito produce economic echain point2014b). In tmodel intewith unit co

Most of tformance, stotal cost. Htional unit improving tbe reectedunit (Yue etmodels withusually giveproduce anall cost. Thicost might tion can guaproducts mas the total unit. The prfractional pmixed-inteMINLP solvtailored solYou, 2014) 2013a) areproblems asear programproblems. A(You et al., objective iscomparisoncost minimthe unit cos

The uncebe carefullygeographica2012; Tongdue to unset al., 2011Mas et al., data due to(Tong et almay lead tomal performunder unceming (Birgconstraint p2009), and et al., 2011;ming is to and maximdom variabHowever itparametersrepresentedtribution an1959). In ro

are considered. Robust optimization aims to nd the solution thatis feasible under all the uncertainties (Ben-Tal et al., 2009). Gen-erally speaking, a solution that is feasible for all the uncertaintiesusually does not leads to the best objective value, so the trade-

ween optihis wed h

unceon (B

trads in bork

mod dropi.e. es. M

and, arece anallst m

Illin the iofueximacases

and restuel s

in tent. modmened sompae con

kgro

supes illu

chaicilitieivatess feeect pt et nd uermecien

o-intslassh DOthat istingssiblrade ies inich ptionfuelrtionude and jdiates, they can be mixed with crude oil that feeds thellation Units (CDU), or sent directly to upgrading unitsgasoline and diesel. However, the studies on explicitvaluation of the integration possibility from the supply

of view are limited (Elia et al., 2012; Tong et al., 2014a,his work, we propose and analyze a biofuel supply chaingrating with existing petroleum reneries, combinedst objective and the robust optimization framework.he existing works consider the absolute economic per-uch as maximizing the total prot or minimizing theowever, the economic objective along with per func-

of the nal product provides further opportunities forhe economic performance, as the cost or prot would

in the functioning outputs of the system: functional al., 2013b). For instance, in the traditional supply chain

cost minimization objective, the product demands aren with lower bounds. The optimal solution tends tod sell biofuels as less as possible to reduce the over-s is obviously not an economical solution, as the unitbe high. By using the unit objective, the optimal solu-rantee the lowest cost per unit, hence making the nalore cost-competitive. The unit objective can be denedcost or prot divided by the total amount of functionaloblem can then be formulated as a mixed-integer linearrogram (MILFP), which is a special type of non-convexger nonlinear programs (MINLPs). Although the generalers can be used, there might not be so efcient as someution algorithms. The parametric algorithm (Zhong andand reformulation-linearization approach (Yue et al.,

two efcient tailored solution algorithms for MILFP they take advantage of the efcient mixed-integer lin-ming (MILP) methods to globally optimize the MILFPlthough unit objective has been chosen in many studies2009; Yue et al., 2013a, 2013b), the advantage of unit

rarely discussed. In this work, we present a detailed between total cost minimization objective and unitization objective, and try to nd out advantages of usingt objective.rtainties in biofuel supply chain are critical and should

considered. These uncertainties include seasonal andl uctuation of biomass supply (Gebreslassie et al.,

et al., 2014a, 2014b), variability of biofuel demandtable economic situations (Giarola et al., 2013; Kim; Marvin et al., 2012), uctuating market price (Dal-2011; Kim et al., 2011), and imprecise processing of

the process uctuation and immature technologies., 2014a). The inability to handle these uncertainties

either an infeasible supply chain design or subopti-ance. The widely used approaches for optimization

rtainty (Sahinidis, 2004) include stochastic program-e and Louveaux, 1997; Tong et al., 2011), chancerogramming (Charnes and Cooper, 1959; Yang et al.,

robust optimization (Ben-Tal and Nemirovski, 2002; Li Li and Floudas, 2012). The goal of stochastic program-nd the decision that is feasible for all the instancesizes the expectation of objective function over the ran-les (Birge and Louveaux, 1997; McLean and Li, 2013).

relies on the probability distributions of uncertain. In the chance-constrained approach, uncertainties are

through random variables with known probability dis-d included in the constraints (Charnes and Cooper,

bust optimization, only bounds of uncertain parameters

off betrobust

In tadvancsupplymizatiused totaintiemany wMILFP for thechain, tructur(Zhong2013a)formanAdditiounit coon theprefersduce bthe maworst vatism

Theof bioflightedstatemMILFP and nospecisive coand th

2. Bac

Thechain isupplysion fais cultbiomafor dir(Wrighsion abio-intand efthe bi(Gebre

Botnoted the exit is poto upgrenerity, whintegrathe bioIn insewith crdiesel robustness and performance is the main issue in themization.ork, we address the optimal design and planning ofydrocarbon biofuel supply chain under demand andrtainties. Bertsimas and Sims robust counterpart opti-ertsimas and Sim, 2003; Li and Ierapetritou, 2008) iseoff the robustness and performance. Although uncer-iofuel supply chain optimization have been studied ins, robust optimization is rarely used. A spatially explicitel with the unit cost objective is proposed to account-in properties of advance hydrocarbon biofuel supply

integrating with existing petroleum renery infras-oreover, two efcient algorithms, parametric method

You, 2014) and reformulation-linearization (Yue et al., adopted in this model, and their computational per-re compared with general purpose MINLP solvers.y, the difference between total cost minimization andinimization are carefully analyzed and discussed basedois cases. The results show that the unit cost objectivepreconversion to petroleum-upgrading pathway to pro-ls. And the budget parameter , which is dened asl number of uncertain parameters that can reach their, provides a mechanism to tradeoff between conser-

economic performance. of this article is organized as follows. The backgroundupply chain and its major drop-in feature are high-he next section. This is followed by a formal problemA brief introduction of the mathematical formulation ofel is presented in Section 4. Detailed model formulationclature can be found in the supporting materials. Thelution approaches are given in Section 5. A comprehen-rison and analysis is presented in the case study section,cluding remarks are given at the end of the paper.

und

rstructure of the advanced hydrocarbon biofuel supplystrated in Fig. 1. A typical advanced hydrocarbon biofueln consists of harvesting sites, bioreneries, preconver-s, upgrading facilities, and demand zones. The biomassd and harvested in harvesting sites. The harvesteddstocks can either be sent to integrated bioreneriesroduction, or undergo a two-stage conversion processal., 2008; You and Wang, 2011), namely preconver-pgrading. Preconversion stage converts biomass intodiates (e.g. bio-oil and bio-slurry) that is economicalt for transportation, whereas upgrading stage upgradesermediates into nal products (e.g. gasoline, diesel)ie et al., 2013; Wang et al., 2013; Zhang et al., 2014).E (DOE, 2012; NABC, 2012) and UOP (Marker, 2005)advanced hydrocarbon biofuel can take advantage of

petroleum renery infrastructure, which means thate to use the upgrading facilities in the existing renerybio-intermediates into nal products. Moreover, most

the United States do not operates at their full capac-rovides a great opportunity for biofuel supply chains

. DOE proposed three insertion points for integrating supply chain with a petroleum renery (DOE, 2012).

point 1 (blue dashed lines), bio-slurry is rst mixedoil, then sent to CDU, and nally converted to gasoline,et fuel through a series of upgrading units. In insertion

130 K. Tong et al. / Computers and Chemical Engineering 68 (2014) 128139

chain

point 2 (redunits to proan additioncontaminanunits in relines), biofurenery andbution netw

3. Problem

The probThe sup

petroleum biomass hapossible uptions, existiof biomass energy croptent, harveare provideat each demFor each proity, and biorcapacity levcosts, and cin petroleurates, retrotransportating unit traknown. Wecost is annuproject life

The objenomic perfo

sionsecisiocatiopgraon p

othvestiine Fig. 1. Superstructure of advanced hydrocarbon biofuel supply

dashed lines), bio-oil is directly sent to the upgradingduce biofuels. Note that in both insertion points 1 and 2,al pretreatment is needed to remove oxygen and otherts before the intermediates are sent to the operationnery (Marker, 2005). For insertion point 3 (green dashedels can be blended with conventional fuels in petroleum

then sent to customers using existing pipeline distri-ork.

of decining dsize, loities, uinsertiOn theto hardeterm statement

lem we addressed in this work is stated as below.erstructure of biofuel supply chain integrating withreneries are shown in Fig. 1. We are given a set ofrvesting sites, potential preconversion facility locations,grading facility locations, potential biorenery loca-ng petroleum reneries, and the demand zones. A setfeedstocks (namely crop residues, wood residues, ands) with their major properties, including moisture con-sting cost, and the availability at each harvesting sited. The demand of fuels (gasoline, diesel, and jet fuel)and zone is given with the upper and lower bounds.duction facility (preconversion facility, upgrading facil-enery), we are given a set of conversion technologies,els, and their corresponding conversion rates, operationapital costs. The data associated with insertion pointsm renery is also given. These data include conversiontting capacity bounds, and capital costs. The available

ion modes, transportation capacities, and correspond-nsportation costs and transportation distances are all

consider the annual case for our model, and the capitalalized by a constant discount rate throughout the giventime.ctive is to minimize the functional-unit-based eco-rmance in terms of unit cost, by determining two types

tion planniin preconvpetroleum transportat

In our adopted to harvesting are regardeassumed torepresents represents tdenes theworst case,of the solut

4. Mathem

We deveplanning ofing with exThis model by Tong et replaced bymodel is remodel. Conduction of tassociated integrating petroleum reneries.

: supply chain design decisions and supply chain plan-ns. The supply chain design decisions include number,n, and technology selection of the preconversion facil-ding facilities and the integrated bioreneries, and theoint determination for existing petroleum reneries.er hand, supply chain planning decisions are relatedng, production, and distribution. Harvesting decisionsthe biomass feedstock harvesting amount. Produc-

ng includes feedstock consumption and product yieldersion facilities, upgrading facilities, bioreneries andreneries. The distribution decisions determine theion amount for each transportation link and mode.model, robust counterpart optimization approach isdeal with uncertainties. The biomass availability at eachsite and the product demand at each demand zonesd as uncertain parameters. All the uncertainties are

be symmetrically bounded, [a a, a + a], where the uncertainty, represents the nominal value, and he variation amplitude. The budget parameter , which

maximal number of uncertainties that can reach their is introduced to control the degree of the conservatismion.

atical model formulation

lop an MILFP model addressing the optimal design and advanced hydrocarbon biofuel supply chain integrat-isting petroleum reneries with the unit cost objective.is a modication and simplication of the previous workal. (2014a). The overall cost minimization objective is

the unit cost minimization objective. The multiperiodplaced by the single period spatially explicit designsidering the length of the article, we give a brief intro-his model in this section. A detailed description and thenomenclature are given in the supporting materials.

K. Tong et al. / Computers and Chemical Engineering 68 (2014) 128139 131

The objective is to minimize the functional-unit-based eco-nomic performance, which is dened as the annualized overall costdivided by the total biofuel in terms of Gasoline Gallon Equivalent(GGE) sold to the customers. The annualized cost involves the cap-ital cost forand govern

min p,s

where Cannuone unit ofGGE, and sozone sf. Beconvexity oproblem.

The capitcapkk, upgand retrotthe discoun

Ccapital =IR

(1{k

tcap

The opertion and matransportat

Coperation = The gov

incentives ausage (You biofuel p

Cincentive =({

k

inck

The modconstraintsdene the mply chain. Trelationshipstraints limand the train advancedexisting peusing the bpetroleum rpoint 2. Thedetailed de

5. Solution

5.1. Robust

Robust handle uncdistributioncounterpartcope best w

and Ierapetritou, 2008). There are several well-known robust coun-terpart optimization formulations, namely Soysters formulation(Soyster, 1973), Ben-Tal and Nemirovskis formulation (Ben-Tal andNemirovski, 1999) and Bertsimas and Sims formulation (Bertsimas

m, 2vativalizatl ana trarodu. In onot oes a wvatis

robucede a vonstunceobjentroase.

genjectiv

min

m

alm

assualuesents ude.ertsiue d to rme

xm +

Sl repand eter lue ced

xm +

xm e tha)(1

es atmMl

m

zlm facility installation, operation cost for fuel production,ment incentives.

Cannual

f p soldp,sf= Ccapital + Coperation Cincentive

p,sf p soldp,sf(1)

al is the annualized overall cost, p is the coefcient for product p in volume to its energy content in terms ofldp,sf is the amount of product p delivered to demandcause both Cannual and soldp,sf are variables, the non-f the objective function leads to a nonconvex MILFP

tal cost consists the cost from preconversion facilitiesrading facilities tcapnn, integrated bioreneries tcapll,

in petroleum reneries tcapsrsr, and is annualized byt rate IR. NY is the project life time.

(1 + IR)NY + IR)NY 1

kk +

n

tcapnn +

l

tcapll +

sr

tcapsrsr

}(2)

ation cost comes from biomass harvesting cost, opera-intenance (O & M) cost, variable production cost, and

ion cost.

Charvest + COM + Cproduction + Ctransport (3)ernment incentives include the facility constructionnd the volumetric incentives for biofuel production andet al., 2012). INCVOp is the unit volumetric incentive for

IR(1 + IR)NY1 + IR)NY 1

k +

n

incnn +

l

incll

}+p,sf

INCVOp soldp,sf (4)

el satises the mass balance constraints, production, and capacity constraints. The mass balance constraintsass conservation of materials at each node of the sup-he production constraints describe the inputoutput

at the material processing facilities. Capacity con-it the raw material purchases, production amounts,nsportation ows. In our model, the drop-in property

hydrocarbon biofuel supply chain (integrating withtroleum reneries) can be appropriately modeled byinary variables isr1sr and isr2sr, which denote whetherenery sr is retrotted by insertion point 1 or insertion

readers can refer to the supporting information for thescription of the proposed model formulation.

approach

optimization

optimization usually outperforms other methods toertainties due to its independence on the probability

of uncertain parameters. The objective of the robust optimization is to choose a solution that is able toith the various realizations of the uncertainty data (Li

and Siconsertial reBen-Taallow the intmancewhich providconserIn thisintroducan takin the call the so the to coworst cbe.

Thethe ob

P1 :

s.t.

Wetake vrepresamplit

In Bget valallowetransfo

m

alm

whereeters, paramlute vaintrodu

m

alm

um Not

lem (10

variabl

max

s.t.003), etc. However, the Soysters formulation is tooe as it ensures the feasibility against all the poten-ions, which sacrices the optimal performance. Thed Nemirovskis formulation provides a mechanism todeoff between robustness and performance. However,ced nonlinearity complicates the computational perfor-ur work, the Bertsimas and Sims approach is adopted,nly avoids complicated nonlinear optimization, but alsoay to consider the tradeoff between performance and

m (Bertsimas and Sim, 2003; Li and Ierapetritou, 2008).st counterpart optimization, a budget parameter is

to control the degree of conservatism of the solution. Italue between 0 to the number of uncertain parametersraints and not necessarily to be integer. It is unlikely thatrtain parameters get their worst values simultaneously,ctive of this formulation is to use a user-dened valuel up to

parameters that are allowed to reach theirThe larger is, the less likely constraint violation would

eral optimization problem can be denoted as P1 withe function (5) and constraint (6).

cx (5)

xm pl, l (6)

me that some of the parameter alm are uncertain, and within the range of [alm alm, alm + alm], where almthe nominal value and alm represents the variation

mas and Sims robust formulation, a user-dened bud-l is introduced to control up to

parameters that are

reach their worst case. Then the constraint (6) is rstd into constraint (7)

maxSltl

mSl

alm|xm| +(l

l)

altlxtl

pl, l

(7)

resents the subset that containsluncertain param-

tl is an index to describe an additional uncertainif l is not an integer. In order to remove the abso-|xm|, an additional nonnegative variable um = |xm| is. Then the constraint (7) can be reformulated as

maxSltl

mSl

almum +(l

l)

altl utl

pl, l

(8)

um, m (9)t the maximization problem in (8) is equivalent to prob-2), because the optimal solution z

lmmust consist of

l

1 and one variable at l l.

almumzlm (10)

l, l (11)

132 K. Tong et al. / Computers and Chemical Engineering 68 (2014) 128139

0 zlm 1, l, m (12)The dual problem of (10)(12) is described as follows

min lzl +mMl

qlm (13)

s.t. zl + qlm almum, l, m (14)qlm 0, zl 0, l where qlm czl correspon

The robuing counterwith the eq

P2 : min

s.t.

m

alm

zl + qlm a

um xm

qlm 0, zl 0, l,

For thisstraints vioparametersthis probaband Ierapet

In our considered as DEMUP/

p,sf

of DEMUP/LBp,sf

original dem

soldp,sf D

soldp,sf D

The uncstraints. To (23), an addconstraints

soldp,sf +

Thus, thbudget paralowing conrobust optim

soldp,sf +

zLBp,sf + qLBp,sfSimilarly

the followin

soldp,sf DE

zUPp,sf + qUPp,sf

Similarly, we assume the biomass availability BAb,i takes valuein the range of [BAb,i BAb,i, BAb,i + BAb,i]. Then after introducingthe budget parameter b,i with the range of [0,1], the original con-straint (30) can be reformulated into (31) and (32).

harvb,i BAb,i, i, b (30)harvb,i BA

b,i

ixed-

e thfract

reg MINms a

Outconvhm optim

metoptimzatio(Sahil-puvane ap

nameulatie pe

min

s.t.

MILrodusolvems a

in

(

t. C

bproprobinearor th

. By

. Sol

. If t, sto

1 =

thert al.,m innal cl, m (15)

(16)

orresponds to the dual variable of the inequality zlm 1,ds to the dual variable of inequality

mjzlm l.

st formulation can then be transformed to the follow-part P2 by substituting the inner optimization problemuivalent optimization problem.

cx (17)

xm + lzl +mMl

qlm pl, l (18)

lmum, l, m (19) um, m (20)l, m (21)

m (22)

robust counterpart, the probability bounds of con-lation can be calculated by the number of uncertain

n, and the budget parameter l. For more details aboutility bounds calculation, please refer to the work by Liritou (2008) and Bertsimas and Sim (2003).work, fuel demands and biomass availabilities areuncertain. The demand uncertainties are denoted

LB, which take value in the symmetric region

[DEMUP/LB

p,sf DEMUP/LB

p,sf, DEMUP/LB

p,sf+ DEMUP/LB

p,sf

]. The

and constraints are listed by Eqs. (23) and (24).

EMLBp,sf p, sf (23)

EMUPp,sf p, sf (24)ertainties appears in the right hand side of the con-apply the robust formulation to the demand constraintitional variable with xed value of 1 is added and the

is formulated as follows:

DEMLBp,sf 1 0, p, sf (25)is constraint has only one uncertain coefcient, and themeter LB

p,sftakes a value in the range of [0,1]. The fol-

straints (26) and (27) are then incorporated into theization.

DEMLBp,sf + zLBp,sf LBp,sf + qLBp,sf 0, p, sf (26)

DEMLBp,sf , p, sf (27), for the demand upper bound constraints, we can getg constraints

MUPp,sf + UPp,sf zUPp,sf + qUPp,sf 0, p, sf /= (28)

DEMUPp,sf , p, sf (29)

zb,i + q

5.2. M

Notlinear can beconvexprobleon theto be algoritto subboundglobal optimirithm generatake adwork, wlems, reformpare th

P3 :

Thewe inttively proble

P4 : m

s.

The suMILFP a nonlsteps f

Step 1Step 2Step 3ance

let qr+

Ano(Yue eprobleadditiob,i + b,i zb,i + qb,i 0, b, i (31)

BAb,i, b, i (32)

integer linear fractional programming

at in our model, only the objective function has theional form, whereas other constraints are all linear. Itarded as an MILFP problem, which is a class of non-LP problems. The general-purpose solvers for MINLPre DICOPT, SBB, and BARON, etc. DICOPT is baseder-Approximation algorithm, but assumes the modelex (Duran and Grossmann, 1986). Applying DICOPTin the non-convex optimization problems may leadal solutions. SBB implements the simple branch and

hod (Bussieck and Drud, 2001). Also, it only guaranteesality for convex MINLP problems. BARON is a global

n algorithm and based on the branch-and-reduce algo-nidis, 1996). The computational performances of theserpose solvers are not satisfactory, because they do nottage of the special structure of MILFP problems. In thisplied two tailored solution approaches for MILFP prob-ly parametric approach (Zhong and You, 2014) andon-linearization approach (Yue et al., 2013a), and com-rformance with the MINLP solvers.

A0 +

iA1ixi +

jA2jyj

B0 +

iB1ixi +

jB2jyj

C0k +

i

C1ikxi +

j

C2jkyj = 0, k(33)

FP model in this paper can be generalized by P3. Firstce the parametric approach. The main idea is to itera-

a series of equivalent MILP subproblems. These MILPre given by problem P4.

A0 +

i

A1ixi +

j

A2jyj

) qr

(B0 +

i

B1ixi +

j

B2jyj

)

0k +

i

C1ikxi +

j

C2jkyj = 0, k

(34)

blem P4 has exactly the same constraints as the originallem P3, but with a linear objective function, instead of

one. Following the idea of Newtons method, the maine algorithm can be summarized as following.

setting r = 1, initialize qr to 0.ve P1 and denote the optimal solution as xr and yr.he optimal value in P1 is less than the optimality toler-p and output (xr, yr) as the optimal solution; otherwise,A0+

iA1ix

ri+

jA2jy

rj

B0+

iB1ix

ri+

jB2jy

rj

, go to step 2 and replace qr with qr+1.

approach is the reformulation-linearization method 2013a). The main idea is to reformulate the MILFPto an exactly equivalent MILP problem by introducingontinuous variables and constraints.

K. Tong et al. / Computers and Chemical Engineering 68 (2014) 128139 133

Firstly, nonnegative variables u and Xi, such thatu = 1

B0+

iB1ixi+

jB2jyj

, and Xi = xiB0+iB1ixi+

jB2jyj

= u xi areintroduced. Then the objective function in P3 can be reformulatedas min A0 u +

iA1iXi +

jA2jYj, where the additional variables

Yj = u yj is introduced to remove nonlinearity. By multiplying bothsides of the constraints in P3 by u, the resulting constraints can bereformulated as C0k u +

iC1ik Xi +

jC2jk Yj = 0, k. Moreover,

the only nonlinear term Yj = u yj can be linearized by using theGlovers linearization scheme (Glover, 1975), where the Big-Mis introduced. Therefore, the MILFP problem P3 can be exactlyreformulated to an MILP problem P5.

P5 : min A0 u +

i

A1iXi +

j

A2jYj

s.t. C0k u +

i

C1ik Xi +

j

C2jk Yj = 0, k

There is ables xi, yj axi do not exbe solved oextra variabreected an

5.3. Integraand MILFP

In this stion are intstrategy ofconsideringply chain mminimizatioThen robusMILFP modties. Note tonly the cosome additing robust be solved bapproachesFig. 2.

Parametric

Fig. 2. Steps objective.

6. Case study

6.1. Input data

To illustsolved a serthe data reas the one obtained frliteratures (

The statis recognizesite, a potena potential types of biocorn stover(e.g. forest ucts, namel

roposideneryroceversed be

and , bioroce

al., 2oducrenre ary theocatetrolsing gradinsertrderobleprevieduc9 possiblethe c-930MS 2ver CRON

mit o

ixed-

his sn thizative.

rstinist

paraize th

annu

detee con. In c

objeB0 u +i

B1i Xi +j

B2j Yj = 1

Yj u, jYj M yj, jYj u M (1 yj), j

(35)

exactly one to one mapping between the original vari-nd the introduced variables Xi, Yj. Continuous variableist in the new formulation. Problem P5 only needs tonce, but it is more computationally expensive as someles and constraints are introduced. These properties ared carefully analyzed in the case study section.

ted optimization framework for robust optimization

ection, the MILFP and robust counterpart optimiza-egrated in our optimization framework. The solution

this integrated model is depicted as follows. When the unit cost objective, we transform the original sup-odel into MILFP problem by changing the total costn objective into the unit cost minimization one (1).t counterpart programming can be applied into theel to deal with the demand and supply uncertain-hat when applying the robust optimization approach,nstraints with uncertain parameters are modied andional constraints are introduced. Therefore, the result-counterpart formulation is still MILFP model, and cany either general-purpose MINLP solvers or the tailored

presented in this paper. These steps are summarized in

Basic supply chain model

Sup ply chain mod el with unit cost objective

Robust counterpart with unit cost objective

MINLP solver appro ach Refor mulation -linea riza tion appro ach

Opti mal d esign

for solving robust model for biofuel supply chain with unit cost

to be pwe conbiorehydroppreconuidiz2008),cationhydropUslu etfuel prand bio

ThenamelTheir leach pprocesfor upthree i

In otion prin our After rsites, 120 pos

All Core i7the GAthe solSBB, BAtime li

6.2. M

In tbetweeminimobjecti

Wedetermall theminim

min C

Theand thmationas therate the application of the proposed framework, weies of county level cases for the state of Illinois. Exceptlated to time periods, all the other data are the samein the previous work (Tong et al., 2014a), which areom technical reports (DOE, 2011, 2012) and existingYou and Wang, 2011).e of Illinois is comprised of 102 counties, each of whichd as a node in our model that represents a harvestingtial preconversion facility, a possible upgrading center,integrated biorenery, or a demand zone. Three majormass are considered in our cases: crop residues (e.g.), energy crops (e.g. switchgrass), and wood residuesresidues). We consider three types of liquid fuel prod-y gasoline, diesel and jet fuel. Their demand are assumedrtional to the population in each county. In our cases,r two integrated conversion pathways for the integrated

(gasication + FT synthesis and pyrolysis followed byssing) (Swanson et al., 2010; Wright et al., 2010), twoion technologies (rotating cone reactor pyrolysis andd reactor pyrolysis) (Magalhaes et al., 2009; Uslu et al.,three upgrading technologies (bio oil to fuel using gasi-slurry to fuel using gasication, and bio oil to fuel usingssing) (Gebreslassie et al., 2013; Magalhaes et al., 2009;008). Three capacity levels are considered for all the bio-tion facilities (preconversion facility, upgrading facility,ery).e four existing petroleum reneries located in Illinois,

Lemont, Joliet, Robinson, and Wood River reneries.ions and capacity levels are known. We assume thateum renery operates at 95% of its capacity level forcrude oil, while the remaining 5% capacity could be usedng bio-intermediate (Tong et al., 2014a). In addition,ion points are analyzed in our case.

to reduce the complexity of the supply chain optimiza-m, we use the same heuristic model reduction rules asous work (Tong et al., 2014a) to reduce candidate sites.tion, the supply chain network contains 40 harvestingsible preconversion facilities, 20 potential bioreneries,

upgrading facilities, and 102 demand zones.ases were performed on a Dell desktop with an Intel

2.80 GHz and 6 GB RAM. All the models are coded in4.1.3 (Rosenthal, 2012). The MILP model is solved withPLEX 12.5, and the MINLP model is solved by DICOPT,

12.7 solvers. The optimal gap in all cases is set to 0. Af 10 h is applied to all the models.

integer fractional programming

ubsection, we present a comprehensive comparisone supply chain optimization problems with total coston objective and those with unit cost minimization

dene two models considered in this subsection, theic model and the MILFP model. In deterministic model,meters are deterministic. Moreover, the objective is toe annualized cost.

al = Ccapital + Coperation Cincentive (36)rministic model consists of the objective function (36)straints (S1)(S52), (S54)(S80) in the supporting infor-ontract to the deterministic model, we use the unit costctive function in the MILFP model. The MILFP model

134 K. Tong et al. / Computers and Chemical Engineering 68 (2014) 128139

consists of those in deMINLP solvreformulati

First, wmance betwcomputatio

Table 1Computationa

Model and a

Determinist

MILFP mode

a With the lb Reformulac Out of med ComputatFig. 3. Optimal design for the deterministic model.

the objective function (1) and the constraints same asterministic model. This model can be solved by eitherers, such as DICOPT, SBB, BARON, or the parametric andon-linearization approaches.e compare the economic and computational perfor-een the deterministic model and MILFP model. The

nal result is shown in Table 1. The total annualized cost

is $2229 mthan the onapproach). higher thanmizing theextent. Wetailored alg

l performance in deterministic model and MILFP model.

lgorithm Total cost (MM $) Unit cost ($/GGE)a Time (s) G

ic 2229 3.180 169

l

Parametric 2421 2.979 889 Reformulationb 2424 2.981 (2.873) 17,052c

DICOPT 2421 2.979 3577 SBB 2573 3.180 (2.654) 36,000d 1BARON 12.7 3461 4.223 (2.699) 36,000d 3

ower bound in the parentheses.tion means reformulation-linearization method.mory error.ional time limit of 10 h is applied to all the algorithms.Fig. 4. Optimal design for the MILFP model.

illion in the deterministic model, which is 8% lowere in the MILFP models ($2421 million by parametric

However, the unit cost is $3.180 per GGE, which is much the one in the MILFP models. This shows that mini-

total annualized cost may increase unit cost to some also compare the computational results between theorithms with the general-purpose MINLP solvers. The

ap No. of binary No. of continuous No. of constraints

0% 422 17,890 32,731

0% 422 17,893 32,7343.65% 560 18,313 40,6870% 422 17,891 32,7326.53% 422 17,891 32,7326.09% 422 17,891 32,732

K. Tong et al. / Computers and Chemical Engineering 68 (2014) 128139 135

mode

parametric 889 s, whicThe reformcost of $2.9of memory a large optioptimal resthe same oalthough itMINLP probSBB and BAlimit with tTheir optimvalue. Notelinearizatioreturn the zusing BAROnd more incases do noincreases frdecreases frfrom $2.6516.54% to 1does not imacceptable and 100 h cinformation

The optFig. 3, wherparametric icant differemodel, 10 pging from 3in Iroquois pyrolysis teers use the

. Eigh1 MMroprostru

ar. Foilt in

rotapacit pying fFig. 5. Cost breakdown in four

method gets the optimal solution of $2.979/GGE withinh is demonstrated to be the most efcient approach.ulation-linearization approach terminates at the unit81/GGE at around 4.74 h when it encounters the outerror. Although it stopped before time limit and it hasmality gap (3.65%), the unit cost is quite close to theult ($2.981/GGE vs. $2.979/GGE). DICOPT solver returnsptimal solution as parametric approach within 3577 s,

can only guarantee the optimal solution for convexlems. By contrast, the computational performances of

bio-oil17 to 9by hydare conper yeare buuse thewith cause fasupgradRON 12.7 are not satisfactory. They stop at 10 h timehe optimality gap of 16.53% and 36.09%, respectively.al values are also far away from the global optimal

that in MILFP model, the ones using reformulation-n approach, SBB solver, and BARON 12.7 solver do notero gap at the 10 h time limit. We tested MILFP modelN and SBB solvers with 100 h time limit, and try toformation. Unfortunately, the optimal solutions in botht increase in the last 90 h. The lower bound in BARONom $2.692/GGE to $2.728/GGE, with the optimality gapom 36.25% to 35.40%. The lower bound in SBB increases4/GGE to $2.668/GGE, with the gap decreasing from6.10%. This shows that increasing computational timeprove the solution too much, and 10 h time limit is

for these cases. The detailed comparison between 10 homputational time limit is provided in the supporting.

imal design for the deterministic model is shown ineas the optimal design for MILFP model obtained fromapproach is shown in Fig. 4. We can see several signif-nces between these two models. In the deterministicreconversion facilities are built with the capacities ran-30 to 894 kton per year. The preconversion facilitiesand Livingston Counties use the uidized bed reactorchnology, which produces the bio-slurry, while oth-rotating cone reactor pyrolysis technology to produce

bio-oil to liin the MILFwhereas thefacilities andeterministespecially inistic modemakers tenminimize thetc. Howeveper GGE solsome extenamount of pconversionwhich leads

The costin Fig. 5 (a fWe can seedecreases fthe meantimcost, and Ocan make uthe expense

As can bies are retrJoliet renels.

t bioreneries are built with the capacities ranging from GGE per year, all of which use fast pyrolysis followedcessing. In the MILFP model, 12 preconversion facilitiescted with the capacity ranging from 235 to 1331 ktonur preconversion facilities using uidized bed reactorChristian, Pike, Vermilion, and Wayne Counties. Othersting cone reactor technology. Six bioreneries are builtties ranging from 16 to 30 MM GGE per year, all of whichrolysis followed by hydroprocessing. Additionally, oneacility with technology of hydroprocessing converting

quids is built in Kane County. Apparently, the unit costP model is lower than the one in deterministic model,

overall cost is higher. And we have more preconversiond fewer bioreneries in the MILFP model than those inic model. In the MILFP model, more fuels are produced,n petroleum renery. This is because, in the determi-l with overall cost minimization objective, the decisionds to meet the minimal demand requirement in order toe production cost, transportation cost, harvesting cost,r, in the MILFP model, we tend to minimize the unit costd to the customers. It is more related to protability tot. And it is a tradeoff between the overall cost and totalroduct sold. From the unit cost point of view, the pre-

to petroleum upgrading pathway is more economical, to more production in renery.

breakdown information in these two models is shownor the deterministic model and b for the MILFP model).

in the MILFP model, the proportion of capital cost,rom 16% to 13% compared with deterministic model. In

e, other costs, such as transportation cost, harvesting & M cost increase slightly. It can be concluded that wenit cost lower by changing the supply chain structure at

of increasing transportation cost and production cost.e seen from Fig. 6 and Table 2, all these four rener-otted. However, in the deterministic model, only thery used all the remaining capacity in renery (5%). The

136 K. Tong et al. / Computers and Chemical Engineering 68 (2014) 128139

Table 2Renery retrot information in four models.

Renery Deterministic MILFP Robust Robust-MILFP

1a 2 %b 1 2 % 1 2 % 1 2 %

Lemont 70 80% 87 100% 76 86% 26 61 100%Joliet 59 95 100% 154 100% 49 106 100% 154 100%Robinson 53 47% 108 96% 51 46% 70 43 100%Wood River 70 42% 21 147 100% 75 45% 29 139 100%Total 129 219 67% 129 388 99% 124 232 68% 124 397 100%

a 1 and 2 stands for the retrotted capacity (MM GGE per year) in renery by insertion 1 and 2, respectively.b % stands for the percentage of unused capacity retrotted in reneries.

0 20 40 60 80

100 120 140 160 180

dete

rmin

istic

CA

PAC

ITY

(MM

GG

E/Y

)

insertion point 1 insertion point 2

overall unuconguratiofuel in petrobest way toall the petrbiofuels. Mbiorenerie

Fig. 7 shMarket shaels produceand petrolethe determof all, the pmodel is grGGE per yethe determof total bio

0

100

200

300

400

500

600

700

800

900

de

Mar

ket s

hare

(MM

GG

E/Y

)

Fig

While in the MILFP model, petroleum reneries produce 66.1% bio-fuels, bioreneries 16.8%, and upgrading facilities 17.1%. This againshows that the pathway of preconversion to petroleum upgrad-ing is more economic from the unit cost point of view. Moreover,in the MILFP model, all the unused capacity in petroleum rener-ies are utilized for upgrading biofuels, and an additional upgradingfacility is built to upgrade biofuels instead of producing them in theintegrated

bust

this modobje32), ag info

rstdget

modiabilhis petershe mnualitatio8, thof coMIL

FP

robu

st

robu

st-M

ILFP

dete

rmin

istic

MIL

FP

robu

st

robu

st-M

ILFP

dete

rmin

istic

MIL

FP

robu

st

robu

st-M

ILFP

dete

rmin

istic

MIL

FP

robu

st

robu

st-M

ILFP

Lemont Joliet Robinson Wood River

Fig. 6. Reenry retrot information in four models.

sed renery capacity utilization is 67%. This is the bestn in the cost minimization problem. Upgrading bio-leum renery is a good alternative, but not always the

produce biofuel. However, in the MILFP model, almostoleum reneries use their unused capacity to upgradeoreover, one upgrading facility is built instead of usings. It is a more economical way in terms of unit cost.ows the important market share in these two models.re in this paper is dened as the percentage of biofu-

6.3. Ro

In robustof the (31)(portin

Weent burobust5% varsolve tparam is, tthe ancompuIn Fig. lation d in different facilities (biorenery, upgrading facility,um renery). The difference of market share betweeninistic model and the MILFP model is signicant. Firstroducts upgraded in petroleum reneries in the MILFPeater than those in the deterministic model (538 MMar compared to 364 MM GGE per year). Moreover, ininistic model, the petroleum reneries produce 52.5%fuels, whereas bioreneries produces the other 47.5%.

47.5%

16.8%

48.2% 31.1%

52.5%

66.1%

51.8% 68.9%

17.1%

terministic MILFP robust robust-MILFP

biorefinery petroleum upgrading

. 7. Market share of production facilities in four models.

cost, harvesand capital

We cholater compabetween thputational annualizedGGE are a lithe conservis 109 CPUsincrease th

0

200

400

600

800

1000

tran

Cos

t (M

M $

)

Fig. 8. Cbioreneries.

optimization

subsection, another two models are introduced, theel and the robust-MILFP model. Robust model consistsctive function (36), the robust counterpart (26)(29),nd other constraints (S2)(S51), (S54)(S80) in the sup-rmation.

compare the results for the robust model with differ- value in order to choose a good budget value forel in comparison. We set the demand uncertainty withity, biomass availability with 10% variability. Then weroblem by robust optimization with 5 different budget

ranging from 0 to 1. As noted previously, the largerore conservative the result would be. Therefore, bothzed cost and unit cost increase as the increases. Thenal results listed in Table 3 also support this conclusion.e increase of budget parameter, which means less vio-nstraints, contributes to the increase of transportationting cost, and production cost. However, the O & M cost

cost almost remain constant.ose a budget value of 0.75 in our robust model forrison with the deterministic model, which is a tradeoffe conservatism and economic performance. The com-results are shown in Table 4. In this robust model, the

cost of $2354 million and the unit cost of $3.244 perttle higher than those in the deterministic model due toatism of the robust formulation. The computation time. It shows that Bertsimas and Sims formulation do note computational time.sportation O & M production harvest capital

=0 =0.25 =0.5 =0.75 =1

ost breakdown in robust model with different budget value .

K. Tong et al. / Computers and Chemical Engineering 68 (2014) 128139 137

Table 3Computational result in robust model with different budget value .

Transportation (MM $) O & M (MM $) Production (MM $) Harvest (MM $) Capital (MM $) Annual (MM $) Unit cost ($/GGE) Time (s)

0 402.9 205.3 856.5 977.0 470.6 2229 3.180 196.30.25 431 464.5 2269 3.200 257.90.5 441 476.6 2311 3.222 262.00.75 455 480.2 2354 3.244 352.41 486 477.7 2399 3.268 38.4

Table 4Computationa

Model and a (s) Gap No. of binary No. of continuous No. of constraints

Robust 9 0% 422 19,726 33,835

Robust-MILFmodel

3 0% 422 19,729 33,8380c 2.85% 560 20,149 41,6073 0% 422 19,727 33,836

0c 16.27% 422 19,727 33,8360c 422 19,727 33,836

a With the lb Reformulac Computati

In this roties are builyear. The otechnologyreactor pyrwith the caof which ufour petrolethe conservtighter demwith the decost and thfacilities anrenery uti

6.4. Integra

The lastintegrates objective Mthe robust c(S2)(S51), solved by eMINLP solveputational rreturns an The reformsolution at remaining wis very closestill outperfmal solutionWithin 10 hgap (lower solutions wthe best chcan return tnot guarant

The opti12 preconvging from 2located in Lnology of reactor pyr.8 204.5 866.6 993.4

.4 209.9 876.7 1006.4

.5 211.7 886.8 1027.3

.9 209.3 896.9 1043.1

l performance in robust model and robust-MILFP model.

lgorithm Total cost (MM $) Unit cost ($/GGE)a Time

2354 3.244 10

P

Parametric 2413 3.072 289Reformulationb 2414 3.073 (2.984) 36,00DICOPT 2413 3.072 303SBB 2580 3.269 (2.737) 36,00BARON 12.7 (2.786) 36,00

ower bound in the parentheses.tion means reformulation-linearization method.onal time limit of 10 h is applied to all the algorithms.

bust model, as shown in Fig. 9, 11 preconversion facili-t, with the capacities ranging from 352 to 648 kton pernes in Cook, LaSalle, and Whiteside Counties use the

of uidized bed reactor, while others use rotating coneolysis technology. Eight bioreneries are constructedpacity ranging from 16 to 93 MM GGE per year, all

se fast pyrolysis followed by hydroprocessing. All theum reneries are retrotted (Table 2). Note that due toatism manner in robust optimization, it tends to haveand bounds and lower biomass availability comparedterministic model. Therefore, it has higher annualizede unit cost. In the robust model, more preconversiond biofuel reneries are built, and it has higher petroleumlization.

ted optimization framework

model considered is the robust-MILFP model, whichthe robust counterpart optimization with unit costIFLP model. It consists of the objective function (1),ounterpart (26)(29), (31)(32), and other constraints(S54)(S80) in the supporting information. It can beither the tailored algorithms, or the general-purposers. The solution strategy is presented in Fig. 2. The com-esults are shown in Table 4. The parametric approachoptimal solution of $3.072 per GGE within 2893 CPUs.

ulation-linearization approach gets the suboptimalthe 10 h time limit. Although there is still 2.85% gapith $2.984 lower bound, the result of $3.073 per GGE

to the optimal one. For general MINLP solvers, DICOPTorms other solvers. It takes 3033 CPUs to get the opti-, which is slightly longer than the parametric approach.

time limit, SBB returns the result of $3.269 with 16.27%bound of $2.737). BARON does not return any feasibleithin 10 h. These show that the parametric approach isoice to this problem. DICOPT is a good alternative andhe optimal solution for this problem although it couldee the global optimality for all the models.mal design of robust-MILFP model is shown in Fig. 10.ersion facilities are constructed with the capacity ran-97 to 1052 kton per year. Four preconversion facilitiesee, Pike, Vermilion, and Wayne Counties use the tech-uidized bed reactor, while others use rotating coneolysis technology. Eight bioreneries are constructed Fig. 9. Optimal design for the robust model.

138 K. Tong et al. / Computers and Chemical Engineering 68 (2014) 128139

with the caof which ufour petrolcapacity to this paper els shown iall the remalized. This uunit cost obFrom the coand producare the mospercentageferent objecdown will is shown inbiofuels, wh

7. Conclus

This papadvanced h

existing petroleum infrastructure considering the unit cost objec-tive. Two tailored algorithms are adopted in solving the resultingMILFP problems. Demand and supply uncertainties are incor-porated in the robust optimization framework. Comprehensive

risons between overall cost minimization objective andst objective, and between deterministic model and robustzatio

simucussversst saramehanis

ic p

wled

authsion acknility aal Ba

dix Acompaunit cooptimimodelare dispreconunit coget paa mececonom

Ackno

Thediscuslike to tainabNation

AppenFig. 10. Optimal design for the robust-MILFP model.

pacity ranging from 14 to 89 MM GGE per year, allse fast pyrolysis followed by hydroprocessing. All theeum reneries are retrotted with their full unusedupgrade biofuels. Among the four models presented in(deterministic, MILFP, robust, and robust-MILFP mod-n Fig. 6 and Table 2), only in the robust-MILFP models,ining capacity in four petroleum reneries are fully uti-tilization percentage is much higher than those withoutjective (the deterministic model and the robust model).st breakdown information in Fig. 5, the harvesting costtion cost account for 34% and 30% of the total cost, andt expensive components. The O & M account for the least

of 7% of the total cost. Although, the models using dif-tives could result in different solutions, the cost break-not change too much. The market share information

Fig. 7. The petroleum reneries produce 68.9% of theich is the largest proportion among all the four models.

ions

er addresses the robust design and planning of theydrocarbon biofuel supply chain integrating with

Supplemin the onlin2014.05.00

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You F, Tao L, Graziano DJ, Snyder SW. Optimal design of sustainable cellulosic biofuelsupply chains: multiobjective optimization coupled with life cycle assessmentand inputoutput analysis. AIChE J 2012;58:115780.

You F, Wang B. Life cycle optimization of biomass-to-liquid supply chainswith distributed-centralized processing networks. Ind Eng Chem Res2011;50:1010227.

Yue D, You F. Fair prot allocation in supply chain optimization with transfer priceand revenue sharing: MINLP model and algorithm for cellulosic biofuel supplychains. AIChE J 2014., http://dx.doi.org/10.1002/aic.14511, Accepted.

Yue D, Guilln-Goslbez G, You F. Global optimization of large-scale mixed-integerlinear fractional programming problems: a reformulation-linearization methodand process scheduling applications. AIChE J 2013a;59:425572.

Yue D, Kim MA, You F. Design of sustainable product systems and sup-ply chains with life cycle optimization based on functional unit: generalmodeling framework, mixed-integer nonlinear programming algorithms andcase study on hydrocarbon biofuels. ACS Sustain Chem Eng 2013b;1:

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Robust design and operations of hydrocarbon biofuel supply chain integrating with existing petroleum refineries considerin...1 Introduction2 Background3 Problem statement4 Mathematical model formulation5 Solution approach5.1 Robust optimization5.2 Mixed-integer linear fractional programming5.3 Integrated optimization framework for robust optimization and MILFP

6 Case study6.1 Input data6.2 Mixed-integer fractional programming6.3 Robust optimization6.4 Integrated optimization framework

7 ConclusionsAcknowledgementsAppendix A Supplementary dataAppendix A Supplementary data