Optimal design of exchange networks with blind inputs and its application to Eco-industrial parks

OATAO is an open access repository that collects the work of Toulouse

researchers and makes it freely available over the web where possible

Any correspondence concerning this service should be sent

to the repository administrator:

tech-oatao@listes-diff.inp-toulouse.fr

This is an author’s version published in:

https://oatao.univ-toulouse.fr/27139

To cite this version:

Salas, David and Van, Kien Cao and Aussel, Didier and Montastruc,

Ludovic Optimal design of exchange networks with blind inputs and

its application to Eco-industrial parks. (2020) Computers &

Chemical Engineering, 143. 107053. ISSN 0098-1354

Open Archive Toulouse Archive Ouverte

Official URL :

Optimal

design

of

exchange

networks

with

blind

inputs

and

its

application

to

Eco-industrial

parks

David

Salas

a,b,c,∗

_,

_Kien

_Cao

_Van

a

_,

_Didier

_Aussel

a

_,

_Ludovic

_Montastruc

b

a Laboratoire PROMES, UPR CNRS 8521, Université de Perpignan Via Domitia, Perpignan 66100, France b Laboratoire de Génie Chimique, UMR 5503 CNRS/INP/UPS, Université de Toulouse, Toulouse 31432, France c Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Libertador Bernardo O’Higgins 611, Rancagua, Chile

a

r

t

i

c

l

e

i

n

f

o

Keywords:

Optimization Eco Industrial Park Game theory

Single-Leader-Multi-Follower

a

b

s

t

r

a

c

t

MotivatedbythedesignandoptimizationofthewaterexchangenetworksinEco-IndustrialParks(EIP), weinvestigatetheabstractBlind-Inputmodelforgeneralexchangenetworks.Thisabstractmodelisbased onaGameTheoryapproach,formulatingitasaSingle-Leader-Multi-Follower(SLMF)game:attheupper level,thereisanauthority(leader)thataimstominimizetheconsumptionofnaturalresources,while, atthelowerlevel,agents(followers)trytominimizetheiroperatingcosts.Weintroducethenotionof Blind-Inputcontract,whichisaneconomiccontractbetweentheauthorityand theagentsinorderto ensuretheparticipationofthelatteronesintheexchangenetworks.Moreprecisely,whenparticipating intheexchangenetwork,eachagentacceptstohaveablindinputinthesensethatshecontrolsonlyher outputfluxes,and theauthoritycommits toguaranteeaminimalrelative improvementincomparison withtheagent’sstand-aloneoperation.TheSLMFgameisequivalentlytransformedintoasingle mixed-integeroptimization problem. Thankstothisreformulation,examples ofEIP ofrealistic sizearethen studiednumerically.

1. Introduction

Inthe last fewdecades,the developmentoftheindustrialized countries hasled to an increasing depletion ofnatural resources suchasfreshwaterandenergy(see,e.g.,UNEP,2000;Scientificand Organization),2009).Theconservationandsustainableuseofsuch resources play an important role in both, environmental impact andbusinesssuccess withintheindustry.Inresponse topreserve the environment while increasing the utilities of the enterprises, theconceptofindustrialecologyhasemerged(Boixetal.,2015).

Industrial ecology (IE) wasfirstintroduced in Froschand Gal-lopoulos(1989).Theywrote“theconsumption ofenergyand ma-terials isoptimized,wastegenerationis minimizedandthe efflu-entsofone process... serveastherawmaterialforanother pro-cess”.Thisisanapproachtotheindustrialdesignofproductsand processes and the implementation of sustainable manufacturing strategies. The idea is directly relatedto another concept, indus-trial symbiosis, whichinvolves “separate industriesin a collective

∗ _{Corresponding author at: Laboratoire de Génie Chimique, UMR 5503}

CNRS/INP/UPS, Université de Toulouse, 31432 Toulouse, France.

E-mail addresses: david.salas@uoh.cl (D. Salas), kien.van@promes.cnrs.fr (K.C. Van), aussel@univ-perp.fr (D. Aussel), ludovic.montastruc@ensiacet.fr (L. Montas- truc).

approachtocompetitiveadvantage involvingphysicalexchangeof materials,energy,waterand/or by-products” (seeChertow,2000). Onekeyconcept ofindustrialsymbiosisis thentheexchange net-works.

A perfect example of an exchange network which illustrates thenotionofindustrialsymbiosisistheconceptofEco-Industrial Parks(EIP).Thisnotionhasseveraldefinitions,butonewidely ac-cepted is “an industrial system of planned materials and energy exchangesthat seeks to minimize energy andraw materials use, minimize waste, and build sustainable economic, ecological and social relationships” Alexander et al. (2000); Boix et al. (2015);

Montastrucetal.(2013).

Recently,inworksofBoixetal.(2015)andKastneretal.(2015), ithasbeenpointedoutthatthereisstillalackofsystematic meth-odsfordesigning theoptimalconfigurationofan EIP.Inprevious studies (Boix et al., 2011; 2015; Montastruc et al., 2013), water integration networks (which is a classical example of EIP) were modeled as a cooperative economy, in the framework of multi-objectiveoptimization(MOO).Thisapproach consistincreatinga vectorfunctionofn+1coordinatesgivenby

C

(

F

)

=



Cost1

(

F

)

,...,Costn

(

F

)

,Z

(

F

)



where Costi( · ) is the cost function of the enterprise i, Z( · )

isthe global consumption ofnatural resources, andF is the flux

Nomenclature

Latinsymbols

n numberofindependentagents

m numberofregulatedagents

P setofindependentagents

R setofregulatedagents

IP indexsetofindependentagents IR indexsetofregulatedagents I assemblyofindexsetsIPandIR I0 assemblyofindexsetIandsinknode0

E networktopology

Emax setofalladmissibleconnectionsofthenetwork

Ec _{setofconnectionsthatarenotinE} Ei,act setofactivearcsofagenti

Est stand-alonetopology E setofallvalidtopologies Cost_i(_{· )} operatingcostofagenti

STCi stand-alonecostofagenti C(i,j) arcclassof(i,j)

Ci familyofallarcclassesexitingfromagenti D setofallarcclassesofactiveagents

y booleanvariable

x_i,j fluxthroughtheconnection(i,j)

xi outletfluxvectorofagenti

x_−i vectorofallfluxesnotexitingfromagenti xP

−i vector ofall fluxesexiting froman independent

agentotherthani

x completevectoroffluxesthroughthenetwork

zi consumptionofnaturalresourceoftheithagent Z(_{· )} totalconsumptionofnaturalresources

gi(· ) inputvalidationfunctionofagenti

Fi vector of fluxesexiting from enterprise i (water

exchangenetwork)

F_−i vector ofall fluxes not exiting fromenterprise i

(waterexchangenetwork)

FP _{vector of fluxes exiting from enterprises (water}

exchangenetwork)

FP

−i vector of all fluxes exiting from an enterprise

otherthani(waterexchangenetwork)

FR _{vector of fluxes exiting from regeneration units}

(waterexchangenetwork)

F fluxvectordescribingthedistributioninthe wa-terexchangenetwork

Mi contaminantloadofenterprisei[g/h]

Ci,in,Ci,out maximumcontaminant concentration allowedin

inlet/outletofprocesses[ppm]

Cr,in minimum inlet concentration allowed of reg.

units[ppm]

C_r,in exact outlet contaminant concentration of reg. units[ppm]

A thelifetimeofthepark[h]

Coef Penalizationcoefficientofstand-aloneagents

Acronyms

EIP Eco-IndustrialPark

GNEP generalizedNashequilibriumproblem Eq thesetofequilibriafortheinducedGNEP KKT Karush-Kuhn-Tucker

MILP Mixed-integerlinearprogramming

MPEC mathematicalprogramswithequilibriumconstraint SLMF Single-Leader-Multi-Follower

STC stand-alonecost

Greeksymbol

α

the minimal relative gain that each agent ask for participatinginthenetwork

c themarginalcostoffreshwaterconsumption[$/T]

β

i,0 thedischargecostofpollutedwaterofenterprise i

[$/T]

δ

i,j thecostsendingpollutedwaterfromenterpriseito j[$/T]



r themarginalcostofregeneratingwater[$/T]

ψ

powerassociatedto



r

vector describingthe distributioninthe exchangenetwork. Then, the aim is to solve the problem of “minimizing” C with respect to F, satisfying the physical constraints of the model. The result of such minimization is called a Pareto front, which consists in all vectors F for which noneof the coordinates of C can be im-provedwithoutworsenanotherone(McCain,2010;Emmerichand Deutz,2018).Usuallyanauthority,representingtheEIP’sdesigner, selectsoneofthissolutionsconsideringascriteriathedistanceto anutopiapoint.

Themainproblemwithsuchan approachisthatpointsofthe Paretofrontarenotnecessarilyeconomicallystable:first,aPareto pointrequirestheenterprisestocooperateandshareinformation, whichisrarelythe caseofan EIP.Second, dueto the noncooper-ativeeconomy,thedifferententerprisesmaydeviate fromthe se-lectionoftheauthoritysincetheymayimprovetheircostfunction byunilaterallychanging theiroperation.Intermsofgametheory, asolutionoftheMOOapproachisasocialoptimizationwhichmay failtorespectincentives(seeNisanetal.,2007,Chapter1).

Tosolvethisincompatibility,againinthe contextofwater in-tegration networks, in the seminal work of Ramos et al. (2016), further developed in Ramos et al. (2018b), a novel game theory approach has been proposed, by modeling the EIP design prob-lem as a Single-Leader-Multi-Follower (SLMF) game (see Aussel and Svensson,2020; Hu and Fukushima, 2015): since the agents donotwanttoexchangeinformation,aconfidentialcentralization through an authority of the parkis introduced. Then, at the up-per level,there istheEIPauthority whichwantstominimize the consumption of natural resources Z(F), while at the lower level, each enterprise tries to minimize her cost function Cost_i(F), re-latedtoherprocesses,consumptionofnaturalresourcesand activ-itywithintheEIP.Theauthorityoftheparkmustchoosethe con-nectionsoftheexchangenetworkandtheoperationofthe regen-erationunits,whileeachenterprise controlstheir consumptionof naturalresourcesandtheir output fluxdistribution.Based onthe EIP authority decisions, all enterprises compete with each other ina parametric non-cooperative generalizedNash game withthe strategies of the EIP authority as exogenous parameters. Fig. 1.1

showsthegeneralschemeofsuchamodel,wheretheenterprises are considered the economic agents of the game. We refer the readertoNisanetal.(2007);Ichiishi(1983)fora primerin non-cooperativegames, toPangandFukushima (2005); Facchineiand Kanzow(2010)forasurveyofGeneralizedNashEquilibrium prob-lems,andDempeetal.(2015);DempeandZemkoho(2020)forthe theory of bilevel optimization. For Single-Leader-Multi-Follower games, we refer to Hu andFukushima (2015) andthe references therein.

The main implicitassumption done in Ramos et al. (2016) is thateachenterprisecanonlycontrolheroutletdistributionandher ownfreshwaterconsumption,buttheyare forcedtoaccept what-everissenttothem throughtheexchangenetwork.Furthermore, they haveno knowledgeabouttheparticularactionsoftheother agentsof thenetwork, excepting onlythe amountandquality of the final inlet flux. In practice, this situation corresponds to the

Fig. 1.1. General scheme of SLMF game.

Fig. 1.2. Blind-Input Schema. z i , F k,i and F r,i are freshwater consumption, wastewater sent from agent k to i , and regenerated water sent from regeneration unit r to i , respectively.

casewhen,attheentrance,eachagentofthenetworkhasamixer, and so she is only aware of the total input she is receiving, as

Fig. 1.2 illustrates. In other words, when participating in the ex-changenetwork,eachagentacceptstohaveablindinput.

While this model respects incentive consistency, it has two main drawbacks:thefirstone isthat therulethat thepark’s au-thorityimposes, that is,theblind input,istoorestrictive. Indeed, under thisparadigm, an enterprise may be forcedto receive too much pollutedwaterwhichcouldturnintohighercosts thanthe stand-aloneoperationoutsidethepark(examplesareeasyto con-structwithtwoenterprises).Thisviolatestheeconomicalprinciple (wellknownincontracttheoryandmechanismdesign)of individ-ual rationality: an enterprise will participatein the EIP only ifit isconvenienttoher(see Jackson,2014;Salanié,2005;Boltonand Dewatripont, 2005);thesecond one is thestrategy tocompute a solution.InRamosetal.(2016),theauthorsimplementedthe clas-sic general approach to solve bilevel games, that is, to reformu-late it as a mathematical programming with complementarity con-straints (MPCC): looselyspeaking,fora givennetwork,they write the Karush-Kuhn-Tucker(KKT) conditionsof each problemofthe lower level game, andput themas constraintsin the authority’s problem.ThentheyimplementedaBranch-and-Boundheuristicto obtainanapproximatedoptimalexchangenetwork,solvingateach iteration theproblemdescribedabove.However, itis knownthat the MPCC problems, which is a particular class of mathematical programmingwithequilibriumconstraints(MPEC), arehardtosolve (see, e.g., Baumruckeret al., 2008; Tseveendorj, 2013;Luo et al., 1996) andtheheuristic itself doesn’tguaranteea realsolutionof theproblem(AusselandSvensson,2019;DempeandDutta,2012). Theliterature ontheoreticalandalgorithmicaspects ofMPCC and MPEC problems is large and still an active field of research in mathematics.

In this work, we further investigate the model proposed in

Ramosetal.(2016)forwaterexchangenetworks,brieflydescribed in Section 2 and fully exposed in Section 5, but considering its abstract formfor generalexchange networksin Section 3.2. This abstractmodel is calledBlind-Input model, since we consider the constraintoffullacceptanceforeachenterprise.Tosolvethe draw-backgivenbytheIndividualRationalityconstraint,weintroducein

Section3.3thenotionofBlind-Inputcontract,whichisan econom-icalcontractbetweentheauthorityandeachenterpriseinorderto participatein theBlind-Input model.We prove that, undersome linearstructureofthecostsfunctionsCost_i( _{· )}ofeachenterprise, theBlind-Inputmodelcanbereducedfroma Single-Leader-Multi-Followerproblemtoasinglemixed-integer optimizationproblem. This reduction, which is our main contribution, is presented in

Section4.

TheproposedreformulationoftheBlind-Inputmodelopensthe doortoalotofnewdevelopments,fromthenumericaltreatment ofhugesizeproblemsthanks toclassical MILPsolvers to exhaus-tivesearchofequilibriaforsmall/mediumsizeapplications.Thisis illustratedinthesecondpartofthearticleforwaterexchange net-worksinEco-IndustrialParks:Section6illustratesacaseofstudy and the obtained results which are then discussed in Section 7. ConclusionsandperspectivesarepresentedinSection8.

Itisworthtomentionthat,eventhoughthisworkismotivated by the design problem of water exchange networks, its abstract formulationpresentedinSection3allowstoapplyittoothertype of networks, asfor example energy networks (Boix et al., 2015; Nevesetal.,2020). InSection8,we willcommentwhichare the main elements needed to apply the Blind-Input model to other contexts.

Tosurvey our contributions, a comparison betweenthiswork andRamosetal.(2016)isgiveninTable1.Itisimportantto men-tionthat the nooncoperative approachusing SLMFgames inEIPs isvery recentand, up to ourknowledge, thereis noother refer-enceintheliteraturedifferentfromRamosetal.(2016,2018b)to compareourresultswith.

2. Motivation:EIPmodelforwaterexchange

Inthissection,webrieflydescribethemodelofwaterexchange networkused to describeEco-Industrial Parks. The model canbe found inRamos etal.(2016); Boixet al.(2015) among others.A detailedversionisfurtherexposedinSection5.

Table 1

Comparison between Ramos et al. (2016) and the present work. The first two rows are related to the numerical exam- ples used in each article.

Comparison criteria Ramos et al. (2016) . This work

Number of enterprises 3 15

Number of processes per enterprise

5 1

Regeneration units Yes Yes

Admits multiple processes per enterprise

Yes No

Tools to model the EIPs SLMF game SLMF game

Presence of Blind-Input model Implicitly used. Not formalized. Economic drawbacks.

Explicit formalization. Introduction of Blind-Input contract as economic instrument.

Solution Method MPCC reformulation + Branch-

and-Bound Heuristic

Mixed-Integer Linear programming (MILP) reduction. Properties of the solution MPCC is hard to solve and existing

algorithms are not robust. The solution of the MPCC may fail to be a solution of the SLMF game.

MILP alogirthms are robust. Commercial solvers are available. Any global solution of the MILP problem is a global solution of the SLMF game.

The operating cost of each participating enterprise in the EIP is lower than that of stand-alone.

No Yes

In an Eco-Industrial Park (EIP), several enterprises exchange wastestoreducetheglobalconsumptionofnaturalresources.Each timeanenterpriseusesthenaturalresourceinherindustrial pro-cess, it comes out degraded, but still can be used as input for otherenterprisesin thepark.Oneofthe mostclassicalexamples ofEIP (see,e.g.,Boix etal., 2015;Boixetal.,2012corresponds to themodelingof waterexchangenetworks: eachenterprise needs toconsumewaterforher industrialprocessesandtheoutcoming waterispartiallypolluted.Other examplesusingdifferentnatural resources likeenergy orheat can be found in Boix etal. (2011);

Ramosetal.(2018a).

InRamosetal.(2016),thedesignofawaterexchangenetwork istreated accordingto the following assumptions:first, the park hasafixednumberofnenterprises,eachenterpriseihastodilute anamountMiofcontaminant,andtheoutletconcentrationof

con-taminantmustbelessthanafixedconcentrationCi,out.Itisusually

assumedthateachenterpriseihasalwaysanoptimaloperation,in thesense that theoutlet concentration ofcontaminant is always equaltoC_i,out.

Second, each enterprise i can accept partially polluted water, butwithamaximalconcentrationCi,in.Thisconcentrationis

mea-suredafteramixer(seeFig.1.2)insuchawaythatnoenterprise canreally knowthe operation ofthe other enterprises. However, thismeasurement,that we willdenote gi andwhichdependson

theactionsoftheotherenterprises,allowsenterpriseitoperform twofundamentalactions:(1)reportinfeasibilitiestotheauthority ofthe park, wheneverthe income water afterthe mixer doesn’t fulfilltheconstraints;and(2)computehowmuchfreshwatershe needs to complete its process attaining the outlet concentration

Ci,out.

Third, each enterprise has a cost function that depends on four factors: (1) the marginal cost of fresh water that she con-sumes, that we denote ci; (2) the marginal cost ofpolluted

wa-ter that she dischargesto the environment, that we denote

γ

i,0;

(3) the cost of sending polluted water through a connection of the park; and (4) the cost of receiving water from other agents of the park (other enterprises but also regeneration units con-trolledby the authority). The authority transfers the investment costofthe EIP tothe enterprisesvia thelast two costs:the first one, via a marginal cost

γ

which depends on the connections that enterprise i uses to send water; and the second one via a costfunction Costin_i that will depend onthe actions ofthe other enterprises.

Moreover,themainassumptionsforthepricinginstrumentsare thatthepricesoffreshwateranddischargedwaterareexogenous, andthat theauthority hasno interest of making anyprofit, and thereforeshe willfix the pricesofusing theconnections only to recoverthe investmentandmaintenance costs. Thisyields to the following scenario:each enterprise wants to minimizeits cost of theuseofwaterwhiletheauthorityisinchargeoftheecological concernsbyminimizingthefreshwaterconsumption.

Finally,aswementionedbefore,theauthoritymayhave regen-erationunits.Eachregenerationunitrreceivespollutedwaterand reducesitscontaminantconcentration uptoacertain valueCr,out.

Then,itsendsthewatertotheenterprisesforreuse.Thecosts as-sociated to the regeneration units are chargedto the enterprises throughtheinletcostfunctionCostin_i .

3. Blind-inputmodel

Taking inspiration from the water management model de-scribedinSection2,ouraiminthissectionistodefinetheconcept ofabstract Blind-Inputmodel forgeneralexchange networks. We divided themodel intwo parts: the physicalmodel, whichgives theconstraintsthatthenetworkmustsatisfy;andtheeconomical model,whichgivestheincentivesofeachagentofthenetwork,as well as theBlind-Input contract betweenthe agents andthe au-thority,whichwillensuretheparticipationoftheagents.

3.1. Networkmodel

Wefirstconsidertwomainactors:asetofagentsparticipating to an exchangenetwork, andan authoritythat aims to minimize theconsumption ofnaturalresources. Among the agents,we dif-ferentiate aset _P:₌

{

P1,...,Pn

}

ofindependent agents,andaset R=

{

R1,...,Rm

}

ofregulatedagents(controlledbytheauthority).

Regulatedagentsdon’t haveeconomicalmotivations,butthey act ontheexchange networkfollowing theindicationsofthe author-ity.InthecontextofwaterexchangeinEIP,theindependentagents aretheenterprises,andtheregulatedonesmodeltheregeneration units(Ramosetal.,2016).

We identify the independent agents with the index set IP=

{

1,...,n

}

andtheregulatedagentswithIR=

{

n+1,...,n+m

}

.We

setI=IP∪IR andI0=

{

0

}

∪I,where0representsthesinknode.

Wedefine an exchangenetworkasa simpledirectedgraph(I0,

part of her output to the agent j. The extra node 0 is identified asasinknode,whichrepresentthepossibilityofdischargeofthe output.Avalidnetwork(I0,E)mustsatisfythefollowingfive

con-ditions:

I. E⊆Emax,whereEmax isthesetofall admissibleconnections of

thenetwork.

II. (I0,E)isasimplegraph,thatis,thereisnomultipleedges nor

graphloopsinE.

III. Eachindependentagenti_∈IPisconnectedwiththesinknode,

thatis,(i,0)∈E.

IV. Eachregulatedagentr∈IRisnotconnectedwiththesinknode,

thatis,(r,0)∈E.

V. The sinknode has not exitedges inE that is(0, i)∈Emax, for

anyi∈I.

Inwhatfollows,wewillcallEthetopology ofthenetwork(I0,

E),andwewilldenotebyE the setofallvalidtopologies. Never-theless, inorderto simplifynotations,thenetwork(I0, E) willbe

onlyrepresentedbyitstopologyE.Observethatthisrepresentation mayleadtoambiguity,sincethesetEdoesn’tallowtodistinguish possible isolated regulated agents (independent agents are never isolated,givenhypothesisIII).However,thisisnotaproblem,since anyisolatedregulatedagentwillbesimplyremovedfromthe net-work.

Foreach edge (i, j) ∈Emax, weset thevariable xi,j which

rep-resents the flux through the connection (i, j). For each i _∈ I, we setxi:=(xi,j:(i,j)∈Emax),beingthustheoutcomevectorofagent

i.Finally, weset x=

(

xi, j :

(

i,j

)

∈Emax

)

, thecomplete vector of

fluxesthroughthenetwork.

To simplify the mathematical model we use, let us introduce some notation. Weput xR _{:= (x}

r:r∈IR)andxP := (xi:i ∈IP). In

whatfollows,foranagenti_∈I,wewillwrite

x_−i:=



xk, j :

(

k,j

)

∈Emax, k∈I

\

{

i

}



, xP_−i:=



xk, j :

(

k,j

)

∈Emax, k∈IP

\

{

i

}



.

ForatopologysubsetA⊆Emax,wewrite

x



_A:=

(

xi, j :

(

i,j

)

∈A

)

.

Similarly,wedefinex_i|A,x−i



A,x P_|

A,xP_−i



_AandxR|A.Itwillbeuseful

alsotodenoteAc_:=E

maxࢨA.

3.2. Physicalmodel

Letusfixa networktopology E_∈E. IfE isimplemented,then foreachagenti∈I,thephysicalmodelofthenetworkisgivenby thefollowingsixoperationalconstraints:

1. Nullfluxesoutsidethenetwork:each agentcanuseonlythe connectionsinthetopologyE.Thus,weset

xi



_Ec=0, (3.1)

thatis,foreveryedge(i,j)_∈E,thefluxx_i,j iszero.

2. Consumptionofnaturalresource:theconsumptionofnatural resource ofthe ith agentis givenby the output fluxesofthe otherplayers,thatis,

zi=zi

(

x−i

)

. (3.2)

Thisassumption is derived froman optimal response hypoth-esis: we assume that, fora given value ofx_−i, the agenti is capableofcomputeexactlytheminimalamountofnatural re-sourcezithatshehastoconsumeinordertoperformherinner

processes.

3. Balance constraint:the fluxes must satisfy the Kirchoff’slaw fortheagenti∈I,thatis,

zi

(

x−i

)

+  (k,i)∈E xk,i=  (i, j)∈E x_{i, j}. (3.3)

Since 0isthe sinknode,it isnotsubjectto thisbalance con-straint.

4. Inputconsistency:thereexistsareal-valuedfunctiongiwhich

allows the agenti _∈I to validate the input coming from the other agents.Wewritethisvalidationasan abstractinequality constraint

gi

(

x−i

)

≤ 0. (3.4)

Thisconstraintmayrepresentmaximalinletfluxes,maximal in-letcontaminantconcentration,minimalinlettemperature,etc. 5. Positivityoffluxes:weassumethatthefluxesonthegraph,as

well astheconsumednaturalresource areallpositive, thatis,

xi≥ 0 and zi

(

x−i

)

≥ 0. (3.5)

6. Extra authority constraints: the exchange network may re-quireadditionalconstraints.Wewillmodelthemherethrough anabstractinclusion

x∈X,

where X⊂ R|Emax| _{represents the abstract additional feasible} set.

Remark3.1. Here,weassume thatthedegradationofthenatural resource is implicitin the connections of the topology E.In this generalmodel,wesupposethatagenticancomputethe degrada-tionofitsinletfluxthroughthefunctionsgiandzi.

Animportantelement ofthismodel isthe totallack ofdirect information among the agents. We suppose that agent i cannot know the actions of other agents, that is, she doesn’t have ac-cessto theexact value ofx_−i.However, she countswithindirect observations: even though x_−i is unknown, the values of zi

(

x−i

)

,

g_i

(

x_−i

)

andthe total inlet flux



_(k,i)∈Ex_k,i are available. For wa-ter exchange, thiscould be interpreted asa measurement of the amountofwaterandcontaminantconcentrationafterthemixerof

Fig.1.2.Thisisaveryimportantfeatureofourmodel,since enter-priseswanttokeep asmuch private informationaspossible.The onlyagentthathasallinformationistheauthority,whohasaccess tothefullvectorx.

3.3.Economicalmodel

Inthissetting,thenetwork authorityhastwo vectorsof deci-sionvariables:shemustchoosethetopologyofthenetworkE_∈E

andshecontrolstheoperationoftheregulatedagents,thatis,the outputvectorsxr,forevery r∈IR.Each independentagenti ∈IP

controlsheroutputvectorx_i.

Weassume that the authoritydoesn’t payanycost associated totheimplementationandoperationofthenetwork.Instead,she transfersall thesecosts througha function

γ

:Emax→R+,where

γ

((

i,j

))

=

γ

i, jrepresentsthemarginalcostforsendingoneunitof

fluxthroughtheconnection(i,j). Usingthispricing, the indepen-dentagentswillpaytheinvestmentcostofthenetworkandalso theoperationoftheregulatedagents.Thus,ifthereisaconnection (r1,r2)∈Emaxbetweentworegulatedagentsr1,r2∈IR,weassume

that

γ

r₁,r2=0.

Sinceall the investmentcost is transferedto the independent agents,theauthorityisonlyconcernedaboutminimizingthe con-sumptionof the naturalresources, andso she aims to minimize thefunction

Z

(

x

)

:= i∈I

zi

(

x−i

)

. (3.6)

Remark 3.2. It could be argued that the authority must be also concernedaboutefficiencyofthenetwork,byconsideringthetotal investmentcostofthepark.However, weassumethatthepricing

instrument

γ

isgivenexclusivelytopaytheinvestmentand main-tenancecostofthepark,andthatit willbeimplementedas effi-cientlyaspossible.Thediscussionoverefficiencyandrightpricing instruments,isoutofthescopeofthiswork.

Ontheotherhandanyindependentagenti∈IPwantsto

mini-mizeherglobalcostCosti,whichcanbeseparatedintothree

com-ponents:theconsumptionofthenaturalresourcezi

(

x−i

)

,thecost

ofdischarging(usingtheconnection(i,0)),andtheuseofthe ex-changenetwork.ThereforehercostfunctionCostiisgivenas: Costi



xi,xP_−i,xR,E



=ci· zi

(

x−i

)

+Costini



xP −i,xR



+  (i, j)∈E

γi

, j· xi, j. (3.7) whereCostin_i



xP −i,xR



isthe inletoperatingcostofanagenti,and itsatisfiesthat  (k,i)∈Emax xk,i=0⇒Costini



xP −i,xR



=0.

Observe that, the cost concerning the exit connections is linear, andso,thecostfunctionislinearinthefirstcomponentx_i.

Remark3.3. Again, in termsofcosts, agent i doesn’thave direct accessto theactions ofthe otheragents.However, shemust pay an operating cost Costin_i

(

xP

−i,xR

)

that is communicated to her by

the authority. The choice of this function as pricing instrument could be studied, but this is out of the scope of the work. For now,wewillsupposethatagentihasenoughindirectinformation (throughmeasurementsafterthemixerofFig.1.2)toconsiderthe costCostin_i



xP

−i,xR



ascorrectandthereforetoacceptit.

Withthismodel,theminimizationproblemoftheith indepen-dentagent(parametrized by the topology E, the actions of regu-latedagents xR _{andthe actions ofthe other independent agents} xP

−i)leadstoproblemPi



xP −i,xR,E



: minxi Costi



xi,xP_−i,xR,E



s.t.

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

zi

(

x−i

)

+  (k,i)∈E xk,i=  (i, j)∈E xi, j gi

(

x−i

)

≤ 0 zi

(

x−i

)

≥ 0 xi≥ 0 xi



_Ec=0. (3.8)

Wedenoteby Eq(xR_{,E)the setofequilibriafortheinduced}

gen-eralizedNashequilibriumproblem(GNEP,forshort) givenbythe vectorxR _{andthetopologyE,thatis}

xP_∈Eq

₍

_xR_,E

₎

_⇐⇒

_∀

_i∈I P,xisolvesPi



xP −i,xR,E



. (3.9)

AswealreadydiscussedinSection1,themainproblemofthis model is that each independent agent only controls her output vectorxi,whichis not realistic.She isforcedby the authority to

fullyacceptanyinletfluxes,whichmaybeharmful.Thus,without anyextraconstraint, agent i maynot be willing toparticipate in thenetwork.

Thus, tosolve thisproblem, theauthority must“buy” the par-ticipationof agenti. Thisis modeled by the Blind-Input contract: agentiacceptstocontrolonlyheroutputfluxes,andtheauthority commitstoguaranteeaminimalrelativeimprovementofhercost, withrespecttothestand-aloneoperationofagenti.

Toformalizethisrequirementinthecontract,letusdenotethe stand-alonetopologybyEst∈E,thatis,

Est:=

{

(

i,0

)

: i∈IP

}

.

Foreach independentagenti_∈IP wedefine thestand-alonecost

STCi,astheoptimalvalueoftheproblemPi(0,0,Est),thatis,

STCi=

(

ci+

γi

,0

)

· zi

(

0

)

Inotherwords,STCi isthecost oftheithagentassumingthat all

other agents (independent andregulated) are inactive, i.e. when agentionlysend fluxesto thesinknodeanddoesn’treceiveany complementaryfluxesfromother agents.Then, foreach indepen-dentagentPi,we canformulate thecommitmentofminimal

im-provementintheBlind-Inputcontractasthefollowingconstraint:

Costi

(

xi,xP_−i,xR,E

)

≤

α

· STCi, (3.10)

where

α

_∈ ]0,1[istheminimalrelativegain thateachagentask forparticipatinginthenetwork.Weassumethat

α

>0since,itis impossibletoeliminateallcosts,andthat

α

<1sincenoagentis indifferentconcerningher participationinthenetwork.Indeed,if Costi

(

xi,xP_−i,xR,E

)

=STCi,then the agenti willprefer not to

par-ticipate, since she has no gain, entering an exchange network is complicatedandsheknowsshemaybe“helpingthecompetition”.

Finally,wecanwritetheauthority’sproblemas

min_E∈E,x∈R|Emax| Z

(

x

)

s.t.

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

x∈X, zr

(

x−r

)

+  (k,r)∈E xk,r=  (r, j)∈E x_{r, j},

∀

r∈IR, zr

(

x−r

)

≥ 0,

∀

r∈IR, gr

(

x−r

)

≤ 0,

∀

r∈IR, xR_{≥ 0,} xR



Ec=0, xP_∈Eq

₍

_xR_,E

₎

_, Costi

(

xi,xP_−i,xR,E

)

≤

α

· STCi,

∀

i∈IP. (3.11)

Theoptimizationproblem(3.11)canbe interpretedasfollows: theauthoritywillproposetotheagentsatopologyEandan oper-ationx∈R|Emax| _{whichsatisfyallthephysicalconstraintsandalso,} such thattheoperation xrespects:1) theincentiveconsistency,in the sense that no agentwillhave incentivesto unilaterally devi-atefromtheproposalduetotheconstraintxP_∈Eq(xR_,E);and2)

theindividualrationalityofeachagent,inthesensethatallagents will participateinthe networksince their participation hasbeen bought through the constraint (3.10). The first criteria solves the economicalinconsistencyofMOO approach,andthesecond crite-riasolves theparticipation problemof theSingle-Leader-Follower approach.

Remark 3.4. In this work, we do not claim novelty in the

constraint xP _{∈ Eq(x}R_{, E). This is the main contribution of}

Ramosetal.(2016).However,theconstraint(3.10)isnew.Interms ofmodelingandinthiscontext,thefactto“attract” the indepen-dent agentstowards a participation inthe generalexchange net-workconstitutesoneoftheimportantnoveltiesofthiswork.

Remark 3.5. After reading the forthcoming Section 4, thereader will observe that all proofs and reductions could be made con-sideringdifferentvaluesof

α

foreach independentagent, thatis, puttingavalue

α

_i∈ ]0,1[ foreach i _∈IP.Thevalue of

α

i

repre-sentsthe“cost” ofbuyingtheparticipationoftheithindependent agent,whichisexactly

(

1−

α

i

)

STCi.However,allowingtohave

dif-ferentcostsdependingontheenterpriserisesthenaturalquestion ofhow todecide thesevalues.This problemliesincontract the-ory(foranintroductiontothefield,werefertoBoltonand Dewa-tripont(2005); Salanié (2005))anditisout thescope ofthe arti-cle.Thus,wewillconsideronlyuniformvaluesof

α

,whichcanbe interpreted asa publiccall forparticipationin thenetwork. Uni-formvaluesof

α

,however,implythat thecostofbuyingthe par-ticipationofanagentisproportionaltohersize,duetothefactor STCi.

Remark3.6. Animportantfactorwedonotconsiderinthiswork isthereboundeffectthatcostsreductionsmayhaveonthe opera-tionofagents.Forexample,ittermsofwaterexchange,a

diminu-tionofcosts ofagentiwithrespecttoSTCimayinducean

incre-mentofwastesproduction,thatis,avariationinMi.Thus,this

re-boundeffectmaychangethevalueofzi

(

x−i

)

.Eventhoughthisisa

very interestingproblem, wesuppose thatthedemand ofnatural resourceisgivenbyafixedprocess,onwhichthecostswithinthe networkhavenoeffect.Inotherwords,theconsumptionofnatural resourceofeachagentisinelastic.

4. Mixed-integerprogrammingreduction

Theformulationoftheauthority’sproblem(3.11)hastheform of a general MPEC problem (see, e.g., Baumrucker et al., 2008; Tseveendorj, 2013; Luo et al., 1996). This section is devoted to prove that thisMPECformulation,which isknown tobe hard to solve,canbereformulatedasasingleMixed-Integerprogramming problem.

Thisreductioncanbe interpretedasfollows:Blind-Input mod-els are a social optimizationproblemwhere,through Blind-Input contracts, the cooperation of each independent agent has been bought.This socialoptimizationis alsoeconomicallystable, since implicitlyitrespectanequilibriumconstraint(xP_∈Eq(xR_,E)).This

reduction/reformulationwillbepresentedinthreesteps.

4.1. Characterizationofequilibria

The followingtheoremcharacterizestheequilibriumsetEq(xR_, E) as a system of equations.This allows to reduce the MPEC of problem (3.11) to a single optimization problem. The reduction we do here is based on the observation that, once every agent hascommittedtoaBlind-Inputcontract,her actionsbecome pre-dictablethroughthecostfunctions.Thus,theauthoritycanchoose the network E such that each action ofan independent agentis inducedtoreachthesocialoptimum.

Toformalizethisidea,letusintroducethenotionofactivearcs. Given a topology E, foreach independent agenti _∈IP we define

the set ofactive arcs ofi,denoted by Ei,act, asall the arcse ∈E

havingminimumcost,thatis,

Ei,act:=

(

i,j

)

∈E :

γ

i, j=

γ

i∗:=₍min_i ,k)∈E

γi,k



. (4.1)

As convention, for any regulated agent r _∈ IR, we set Er,act=

{

(

r,j

)

:

(

r,j

)

∈E

}

.

Theorem4.1. ForE_{∈E and}xR _{≥ 0fixed,theequilibriumsetEq(x}R_, E)isgivenby Eq

(

xR_,E

₎

₌

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

xP _:

_∀

_i∈I P, zi

(

x−i

)

+  (k,i)∈E x_k,i=  (i, j)∈E xi, j gi

(

x−i

)

≤ 0 zi

(

x−i

)

≥ 0 xi



_Ec i,act =0 xi≥ 0

⎫

⎪

⎪

⎪

⎪

⎪

⎬

⎪

⎪

⎪

⎪

⎪

⎭

(4.2)

Thus, the authority’s problem (3.11) is equivalent to the following Mixted-IntegerProgrammingproblem:

min_x∈R|Emax|_,E∈E Z

(

x

)

s.t.

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

x∈X, zi

(

x−i

)

+  (k,i)∈E x_k,i=  (i, j)∈E x_{i, j},

∀

i∈I xi



_Ec i,act =0,

∀

i∈I gi

(

x−i

)

≤ 0,

∀

i∈I zi

(

x−i

)

≥ 0,

∀

i∈I

Costi

(

xi,xP_−i,xR,E

)

≤

αi

· STCi,

∀

i∈IP

x≥ 0.

(4.3)

Proof. Thesecond partoftheproof iseasilyverifiedbyreplacing theconstraintxP_∈Eq(xR_{,E)bythesystemofequationsintheright}

handof equality(4.2),and thenjust reorganizing. Thus, we only needtoprove(4.2).

Tosimplify notation, let us denote by S(xR_{, E) the right-hand}

setof(4.2).First,letusprovethatS(xR_,E)⊆Eq(xR_,E).FixxP_∈S(xR_, E). SinceEi,act⊂ Eforeach i ∈IP,itis nothard toseethat xi isa

feasiblesetofPi

(

xP_−i,xR,E

)

.

Now, fix i _∈ IP and let x_i be another feasible point of Pi

(

xP_−i,xR,E

)

. Then, xi≥ 0 and it satisfies the balance constraint

(3.3),whichyieldsthat

Costi=  (i, j)∈E

γi, j

x_{i, j}−

γ

∗ i



 (i, j)∈Ei,act x_{i, j}



≥

γ

∗ i



 (i, j)∈E x_{i, j}−  (i, j)∈Ei,act xi, j



≥ 0,

where

Costi:=Costi

(

xi,xP−i,xR

)

− Costi

(

xi,x_−iP,xR

)

andthelast

in-equalityisduetothefactthat

 (i, j)∈E x_{i, j}=zi

(

x−i

)

+  (k,i)∈E xk,i =  (i, j)∈E x_{i, j}=  (i, j)∈Ei,act x_{i, j}.

Thus,xi solves Pi

(

xP_−i,xR,E

)

, andsincethis holdsforevery i ∈IP,

wededucethatxP_∈Eq(xR_,E).

Now,letusprovethatEq(xR_,E)⊆S(xR_,E).LetxP_∈Eq(xR_,E),and

suppose that xP_∈S(xR_{, E). Since for each i ∈I}

P the vector xi is a

feasiblepointofP

(

xP

−i,xR,E

)

,theonlywayforxPnottobelongto

S(xR_{,E)isthatthereexisti}

0∈IPsuchthatxi₀



_Ec i₀,act=

0.Thus,there is

(

i0,j0

)

∈E

\

Ei₀,act such thatxi₀, j0 >0.Let(i0,j1)∈Ei,act (which isnonemptybydefinition)andletusconsiderthevectorx_i

0 given by x_i0,k=



_x i0,k ifk∈I

\

{

j0,j1

}

, 0 ifk=j0, xi0, j1+xi0, j0 ifk=j1. Wehavethatx_i

0≥ 0(sincexi0≥ 0)andalso

zi

(

x−i0

)

+  (k,i0)∈E xk,i0=  (i0, j)∈E xi0, j=  (i0, j)∈E x_i 0, j. Thus, since x_−i0 remains the same, x_i

0 is a feasible point of Pi

(

xP_−i

0,x

R_,E

₎

_{. Furthermore, denoting by}

_Cost i₀= Costi₀

(

xi₀,xP−i0,x R_,E

₎

_{− Cost} i₀

(

xi₀,xP_−i 0,x R_,E

₎

_,wehavethat

Costi0=  (i0, j)∈E

γ

i0, jxi0, j−  (i0, j)∈E

γ

i0, jxi0, j =



γi

0, j1−

γi

0, j0



xi0, j0 =



γ

∗₋

_γ

i0, j0



xi0, j0<0, since,byconstruction,

γ

_i

0, j0>

γ

∗.Thisyieldsthatxi0 doesn’tsolve

Pi

(

xP_−i

0,x

R_,E

₎

_{,whichisacontradiction.Thus,x}P_∈S(xR_{,E),finishing}

theproof. _

Intuitively, the above theorem says that, given a topology E, each independent agent i _∈ IP will only use the connections of

minimalcost tosendtheexcessofflux,that is,shewilluseonly heractivearcs.Furthermore,eachindependentagentisindifferent tothedistributionoffluxesamongtheactivearcs,soanyfeasible vectorxP_{satisfyingtheconstraintx}

i



_Ec i,act=

0foreveryi ∈IPmust

beanequilibrium.Thissimplificationisstronglybasedonthe lin-earity ofthe costs functions withrespect to the agent’s variable

4.2.Mixed-integerformulation

Theorem4.1establishestheremarkablefactthattheMPEC for-mulationoftheauthority’sproblemcanbereformulatedasa “clas-sical” programmingproblem.Butactually,apartofthevariablesof thisprogrammingproblemliesinthesetoftopologiesofthe ex-changenetworkandso,itcanbe consideredasdifficultto imple-mentnumerically.Thisiswhy,in thissection, we willshowhow onecanfinallyworkwithamoreclassicalmixed-integer program-mingproblem.

Let us first introduce the key notion that we will use to ar-riveto thefinal formulation,that is,what we callarc classes:let (i,j)∈Emax.Wedefinethearcclassof(i,j)astheset

C

(

i,j

)

:=

{

(

i,k

)

∈Emax

γi,k

=

γi, j

}

ifi∈IP

{

(

i,k

)

∈Emax

}

ifi∈IR. (4.4)

Wedenote by Ci the familyof all arcclassesexiting fromi,that

is,Ci=

{

C

(

i,j

)

:

(

i,j

)

∈Emax

}

.Finally,forC∈Ciwedefinethe

uti-lizationcostoftheclassby

γ

(

C

)

:=

γ

i, j,

where(i,j)isanyrepresentativeofC.

Observethat,fortwoarcs(i,j),(i,k)∈Emaxsuchthat

γ

i, j=

γ

i,k,

one hasthat C

(

i,j

)

=C

(

i,k

)

. Thus, a classC∈Ci may havemany

representationsof theform C(i, j). Furthermore, thefamily Ci

in-ducesapartitionofthesetofarcs“exitingfrom” agenti,thatis

• C∈CiC=

{

e∈Emax : e=

(

i,j

)

forsome j∈I0

}

.

• ForanytwoclassesC,C∈Ci,eitherC=C orC∩C=∅.

Moreover, it isnot hard toverifythat foreach topology E∈E

andforeachagenti∈IP,thereexistsoneclassC∈Cisuchthat

E_i,act⊆ C, (4.5)

andthisclassmustsatisfythat

γ

(

C

)

≤

γ

(

C

(

i,0

))

. (4.6)

ThisclassisthengivenbyC=C

(

i,j

)

where(i,j)isanyelementof

Ei,act.WewillcallittheactiveclassofEoftheagenti,andwewill

denoteitbyCi(E).

Without loss of generality, we will assume that every class

C∈Ci satisfies (4.6). If not, any connection in a class violating

(4.6)would neverbeen used,andtherefore,inpractice,they can beerasedfromEmaxwithoutchangingtheproblem.

Now, let D=i∈IPCi, the setof allarc classesof independent

agents.Weintroducethebooleanvariabley=

(

yC

)

C∈D∈

{

0,1

}

|D|in

thefollowingway:foreachindependentagenti ∈IP andeacharc

classC∈Ci,weset

yC=

1 ifC istheactiveclassofi,

0 otherwise.

Fromy_∈{0,1}|D|_{,wewillbuildthegraphassociatedtoyas}

E

(

y

)

=

(



{

: yC=1

}

)

∪

{

,0

)

: i∈IP

}

∪

{

(

r,j

)

∈Emax : r∈IR

}

. (4.7)

Weconsider thenthefollowing Mixed-Integeroptimization prob-lem: min_x∈RN_,y∈{0,1}|D| Z

(

x

)

s.t.

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

x∈X, zi

(

x−i

)

+  (k,i)∈Emax xk,i=  (i, j)∈Emax xi, j,

∀

i∈I,  C∈Ci yC=1,

∀

i∈IP,  (i, j)∈C xi, j≤ B· yC,

∀

C∈D, gi

(

x−i

)

≤ 0,

∀

i∈IP, zi

(

x−i

)

≥ 0,

∀

i∈I, Costi

(

xi,xP_−i,xR,E

(

y

))

≤

α

i· STCi,

∀

i∈IP, x≥ 0, (4.8)

where Bis a realnumber chosen arbitrarily, butbiggerthan the maximumofthetotalentering fluxover allenterprises. Asimple option to set Bis the value Z(0),which corresponds to the total consumption of the natural resource when there is no exchange network.

Here, the constraint _C∈C

iyC=1 says that, for the ith agent,

onlyoneclassisactive.Also,theconstraint

 (i, j)∈C

xi, j≤ B· yC,

∀

C∈D

ensures that, whenever(i, j) doesn’t belong tothe active classof theithagent,thenxi, j=0.

Theorem4.2. Foreveryfeasiblepoint(x,y)of(4.8),thepair(x,E(y))

isafeasiblepointof(4.3).Conversely,foreveryfeasiblepoint(x,E)of

(4.3),thepair(x,yE_{)isa feasiblepointof(4.8),wherey}E _∈{0,1}|D|

isgivenby yE C=

1 ifC=Ci

(

E

)

forsomei∈IP, 0 otherwise.

Finally,onehasthat

1. if(x,E) isanoptimalsolutionof(4.3),then(x,yE_{)isan optimal} solutionof(4.8).

2. if(x,y)isanoptimalsolutionof(4.8),then(x,E(y))isanoptimal solutionof(4.3).

Proof. Let(x, y) be a feasible point of(4.8). Letus fix an agent

i _∈IP andlet Cibe theuniqueclass inCi suchthat yC_i=1.Then,

byconstruction,weknowthat

E

(

y

)

_i,act=Ciand  (i, j)∈Emax\Ci x_{i, j}≤ B·  C∈Ci\{Ci} yC=0.

Wededucethenthat

xi



_E(y)c i,act

=0.

Sincethisconstraintisvalidforeveryactiveagenti∈IP,andsince E(y) containsall exitingarcs forevery regulated agentr_∈IR,we

canrewritethebalanceconstraintinproblem(4.8)as

z

(

x_−i

)

+  (k,i)∈E(y) xk,i=  (i, j)∈E(y) x_{i, j},

∀

i∈I.

Wededucethenthat(x,E(y))isafeasiblepointofproblem(4.3). Now,let(x,E)beafeasiblepointofproblem(4.3).Byinclusion

(4.5),foreachindependentagenti∈IP,thereexistsauniqueactive

class Ci(E). LetusdefineyE ∈{0,1}|D|asin thestatementofthe

theorem.

Then,foreveryi∈IP,C∈Ciy

E

C=1.Now,fixaclassC∈D,and

leti∈IPsuchthatC∈Ci.Wehavethat  (i, j)∈C x_{i, j}≤

B=B· yE C ifC=Ci

(

E

)

, 0=B· yE C ifC=Ci

(

E

)

,

where thesecond inequality comes fromthe fact that,whenever

C=Ci(E),thenC⊆ Eic,act andsoxi



_C=0.

Foranagenti ∈IP,thefactthatEi,act⊆E(yE)leadustothefact

that

Costi

(

xi,x_−iP,xR,E

(

yE

))

=Costi

(

xi,xP_−i,xR,E

)

,

andso,theconstraint(3.10)issatisfied.We deducethat (x,yE_)is

afeasiblepointof(4.8),sinceallotherconstraintsaredirectly sat-isfiedgiventhat(x,E)isfeasibleforproblem(4.3).

Now,letusassumethat(x,E)isalsooptimalforproblem(4.8). Fromthedevelopmentabove,foreveryotherfeasiblepoint(x,y) of(4.8),weknowthat(x,E(y))isalsoafeasiblepointofproblem

(4.3),andso,Z(x)≤ Z(x).Thus,(x,yE_{) isoptimalfortheproblem}

(4.8).

Let nowassume that (x, y) is an optimal solutionof problem

(4.8)andsuppose,byabsurd,that(x,E(y))isnotoptimalfor prob-lem (4.3). Then, there exists a feasible point (x, E) of problem

(4.3) such that Z(x) _< Z(x). But,asproved above,

(

x_,yE

₎

_{is also}

feasible for problem (4.8), showing that (x, y) is not optimal for

(4.8),whichisacontradiction.Theproofisthencompleted.  Thereadercouldobservethat,apriori,themixed-integer prob-lem(4.8)issmallerthanproblem(4.3)insomesense,sinceit ad-mitsonlycertaintopologies(thoseonesoftheformE(y)forsome feasiblepointy_∈{0,1}|D|_{).Howevertheabovetheoremshowsthat}

thesetoffluxdistributionsxforwhich(x,E)isanoptimalsolution of(4.3) forat leastonetopology E coincides withtheset offlux distributionsxforwhich(x,y)isanoptimalsolutionof(4.8)forat leastoney.

4.3. Nullclassasexitoption

Physically, we know that the network has always a feasible point, which is the stand-alone configuration, that is, the topol-ogy Est and the fluxesgiven by the individual operationsof the

independentagentsandinactivityoftheregulatedones.However, when we include the individual rationality constraint (3.10), the problemmaybecomeinfeasible.

Infeasibilityofproblem(4.3)meansthattheauthorityisnot ca-pabletofindasolutionthatrespecttheBlind-Inputcontractswith alltheagents.Thus,weneedtoincludethepossibilityofexcluding someagentsfromthenetwork.

Formally, for each independent agent i ∈ IP, we include a

booleanvariabley_i,null∈{0,1}suchthat

y_i,null=

1 ifibreaks theBlind-Inputcontract,

0 otherwise.

Withthisnewvariable,wemodifyproblem(4.8)asfollows: 1. Foreachagenti_∈IP,weput

yi,null+

 C∈Ci

yC=1,

meaningthat,eitheronearcclassisactiveortheagentis out-sidethenetwork.

2. Foreachagenti_∈IP,weput  (i, j)∈C(i,0) x_{i, j}≤ B·

(

y_C(i,0)+yi,null

)

 (i, j)∈Emax, j=0 x_{i, j}≤ B·

(

1− yi,null

)

Thisistoensurethat,iftheagentbreakstheBlind-Input con-tract,thenshewillusethedischargearc(i,0).

3. Foreachagenti_∈IP,weput 

(k,i)∈Emax

xk,i≤ B·

(

1− yi,null

)

.

This constraintestablishes that, iftheagent breaksthe Blind-Inputcontract,thennobodycansendheranyflux.

4. Foreachagenti∈IP,weput

Costi

(

xi,xP_−i,xR,E

(

y

))

≤

αi

STCi·

(

1− yi,null

)

+STCi· yi,null.

(4.9)

Here, the individual rationality constraint is active only when

yi,null=0. Otherwise, since the agent isnot connected to the

network,hercostwillcoincidewithSTC_i.

We set D=D∪

{

Nulli : i∈IP

}

, where Nulli is the null class,

associatedtoy_i,null,andD0=D

\

{

C

(

i,0

)

: i∈IP

}

.Denoting STCi

(

yi,null

)

:=

αi

STCi·

(

1− yi,null

)

+STCi· yi,null,

thenewoptimizationproblembecomes

min_x∈RN_,y∈{0,1}|D|Z

(

x

)

s.t.

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

x∈X, zi

(

x−i

)

+  (k,i)∈Emax x_k,i=  (i, j)∈Emax x_{i, j},

∀

i∈I, y_i,null+  C∈Ci yC=1,

∀

i∈IP,  (i, j)∈C x_{i, j}≤ B· yC,

∀

C∈D0,  (i, j)∈C(i,0) x_{i, j}≤ B·

(

y_C(i,0)+yi,null

)

,

∀

i∈IP,  (i, j)∈Emax, j=0 xi, j≤ B·

(

1− yi,null

)

,

∀

i∈IP,  (k,i)∈Emax xk,i≤ B·

(

1− yi,null

)

,

∀

i∈IP, gi

(

x−i

)

≤ 0,

∀

i∈IP, zi

(

x−i

)

≥ 0,

∀

i∈I,

Costi

(

xi,xP_−i,xR,E

(

y

))

≤ STCi

(

yi,null

)

,

∀

i∈IP,

x≥ 0.

(4.10)

Observethat, whenevery_i,null=0_, then all constraintsfor the

ith agent are the same that those established in problem (4.8). Also,ifyi,null=1,theonlyfeasiblesolutionforiisthestand-alone

operation.Thus,inthisnewproblem,theauthorityfirstchooseall theagentsthatwillparticipateinthenetwork,representedbythe set

I_P =

{

i∈IP : yi,null=0

}

,

and then it solves problem (4.8) replacing I by I=I_P∪IR. Of

course,as itis formulated, the authority takesboth decisions si-multaneously, by solving problem (4.10). It is not hard to verify thatanyoptimalsolutionofproblem(4.10)isanoptimalsolution ofProblem(4.8)forthereducedsetofagentsI.Weleavethis ver-ificationtothereader.

5. Blind-Inputmodelforwaterexchangenetworks

In thissection we come back to our original motivation pre-sentedinSection2,thewaterexchangenetworksinEco-Industrial Parks.Wearenowreadytodescribeindetailthemodel,andhow itfitsintotheBlind-Inputmodeldevelopedsofar.

First, an EIP consists in a set of enterprises P:=

{

P1,...,Pn

}

,

thatareconnectedinanexchangenetwork.EachenterpriseP_ican beconnectedeithertoother enterprises, ortosome regeneration units,which we denoteR:=

{

R1,...,Rm

}

.The regeneration units

arecontrolledby acentralauthority. Thisauthorityplaystherole ofdesigner (when deciding the connections within the network) and regulator of the park’s operation. Finally, we include a sink node0,thatrepresentsawastes’pittodischargeuseless polluted water.WeidentifyP withtheindexsetIP:=

{

1,...,n

}

andRwith IR:=

{

n+1,...,n+m

}

. We put I=IP∪IR and I0=

{

0

}

∪I. Finally,