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

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

David Salas Videla, Kien Cao Van, Didier Aussel, Ludovic Montastruc

To cite this version:

David Salas Videla, Kien Cao Van, Didier Aussel, Ludovic Montastruc. Optimal design of exchange

networks with blind inputs and its application to Eco-industrial parks. Computers & Chemical Engi-

neering, Elsevier, 2020, 143, pp.107053. �10.1016/j.compchemeng.2020.107053�. �hal-03080152�

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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

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https://doi.org/10.1016/j.compchemeng.2020.107053

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

aLaboratoire PROMES, UPR CNRS 8521, Université de Perpignan Via Domitia, Perpignan 66100, France

bLaboratoire de Génie Chimique, UMR 5503 CNRS/INP/UPS, Université de Toulouse, Toulouse 31432, France

cInstituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Libertador Bernardo O’Higgins 611, Rancagua, Chile

a rt 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.TheSLMFgameisequivalentlytransformedintoasinglemixed- 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 FroschandGal- lopoulos(1989).Theywrote“theconsumption ofenergyandma- terials isoptimized,wastegenerationis minimizedandthe efflu- entsofone process... serveastherawmaterialforanotherpro- 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: [email protected] (D. Salas), [email protected] (K.C.

Van), [email protected] (D. Aussel), [email protected] (L. Montas- truc).

approachtocompetitiveadvantage involvingphysicalexchangeof materials,energy,waterand/or by-products” (seeChertow,2000).

Onekeyconcept ofindustrialsymbiosisis thentheexchangenet- works.

A perfect example of an exchange network which illustrates thenotionofindustrialsymbiosisistheconceptofEco-Industrial Parks(EIP).Thisnotionhasseveraldefinitions,butonewidelyac- 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), ithasbeenpointedoutthatthereisstillalackofsystematicmeth- 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 Cost_i( · ) is the cost function of the enterprise i, Z( · ) isthe global consumption ofnatural resources, andF is the flux https://doi.org/10.1016/j.compchemeng.2020.107053

Nomenclature

Latinsymbols

n numberofindependentagents m numberofregulatedagents P setofindependentagents R setofregulatedagents

I_P indexsetofindependentagents I_R indexsetofregulatedagents I assemblyofindexsetsI_PandI_R I₀ assemblyofindexsetIandsinknode0 E networktopology

Emax setofalladmissibleconnectionsofthenetwork E^c setofconnectionsthatarenotinE

E_i,act setofactivearcsofagenti Est stand-alonetopology E setofallvalidtopologies Cost_i(·) operatingcostofagenti STC_i stand-alonecostofagenti C(i,j) arcclassof(i,j)

Ci familyofallarcclassesexitingfromagenti D setofallarcclassesofactiveagents y booleanvariable

x_i,j fluxthroughtheconnection(i,j) x_i outletfluxvectorofagenti

x_−i vectorofallfluxesnotexitingfromagenti x^P_−i vector ofall fluxesexiting froman independent

agentotherthani

x completevectoroffluxesthroughthenetwork z_i consumptionofnaturalresourceoftheithagent Z(·) totalconsumptionofnaturalresources

g_i(·) inputvalidationfunctionofagenti

F_i vector of fluxesexiting from enterprise i (water exchangenetwork)

F_−i vector ofall fluxes not exiting fromenterprise i (waterexchangenetwork)

F^P vector of fluxes exiting from enterprises (water exchangenetwork)

F₋^P_i vector of all fluxes exiting from an enterprise otherthani(waterexchangenetwork)

F^R vector of fluxes exiting from regeneration units (waterexchangenetwork)

F fluxvectordescribingthedistributioninthewa- terexchangenetwork

M_i contaminantloadofenterprisei[g/h]

C_i,in,C_i,out maximumcontaminant concentration allowedin inlet/outletofprocesses[ppm]

C_r,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 thenoncooper- ativeeconomy,thedifferententerprisesmaydeviate fromthese- lectionoftheauthoritysincetheymayimprovetheircostfunction byunilaterallychanging theiroperation.Intermsofgametheory, asolutionoftheMOOapproachisasocialoptimizationwhichmay failtorespectincentives(seeNisanetal.,2007,Chapter1).

Tosolvethisincompatibility,againinthe contextofwaterin- 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,consumptionofnaturalresourcesandactiv- itywithintheEIP.Theauthorityoftheparkmustchoosethecon- nectionsoftheexchangenetworkandtheoperationoftheregen- 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 primerinnon- cooperativegames, toPangandFukushima (2005); Facchineiand Kanzow(2010)forasurveyofGeneralizedNashEquilibriumprob- 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,iand F r,iare 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’sau- thorityimposes, that is,theblind input,istoorestrictive. Indeed, under thisparadigm, an enterprise may be forcedto receive too much pollutedwaterwhichcouldturnintohighercosts thanthe stand-aloneoperationoutsidethepark(examplesareeasytocon- structwithtwoenterprises).Thisviolatestheeconomicalprinciple (wellknownincontracttheoryandmechanismdesign)ofindivid- 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),theauthorsimplementedtheclas- 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.Tosolvethedraw- backgivenbytheIndividualRationalityconstraint,weintroducein Section3.3thenotionofBlind-Inputcontract,whichisaneconom- icalcontractbetweentheauthorityandeachenterpriseinorderto participatein theBlind-Input model.We prove that, undersome linearstructureofthecostsfunctionsCost_i( ·)ofeachenterprise, theBlind-InputmodelcanbereducedfromaSingle-Leader-Multi- Followerproblemtoasinglemixed-integer optimizationproblem.

This reduction, which is our main contribution, is presented in Section4.

TheproposedreformulationoftheBlind-Inputmodelopensthe doortoalotofnewdevelopments,fromthenumericaltreatment ofhugesizeproblemsthanks toclassical MILPsolvers toexhaus- tivesearchofequilibriaforsmall/mediumsizeapplications.Thisis illustratedinthesecondpartofthearticleforwaterexchangenet- 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.Itisimportanttomen- 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 timeanenterpriseusesthenaturalresourceinherindustrialpro- 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 anamountM_iofcontaminant,andtheoutletconcentrationofcon- taminantmustbelessthanafixedconcentrationC_i,out.Itisusually assumedthateachenterpriseihasalwaysanoptimaloperation,in thesense that theoutlet concentration ofcontaminant is always equaltoC_i,out.

Second, each enterprise i can accept partially polluted water, butwithamaximalconcentrationC_i,in.Thisconcentrationismea- suredafteramixer(seeFig.1.2)insuchawaythatnoenterprise canreally knowthe operation ofthe other enterprises. However, thismeasurement,that we willdenote g_i 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 C_i,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 c_i; (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,theauthoritymayhaveregen- erationunits.Eachregenerationunitrreceivespollutedwaterand reducesitscontaminantconcentration uptoacertain valueCr,out. Then,itsendsthewatertotheenterprisesforreuse.Thecostsas- 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,wedif- ferentiate aset P:=

{

^P1,...,P_n

}

^ofindependent agents,andaset R=

{

^R1,...,Rm

}

^{ofregulatedagents}(controlledbytheauthority).

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

We identify the independent agents with the index set I_P=

{

¹,...,n

}

^{andtheregulatedagentswithI}R=

{

ⁿ+1,...,n+m

}

^.We

setI=IP∪IR andI0=

{

⁰

}

∪I,where0representsthesinknode.

Wedefine an exchangenetworkasa simpledirectedgraph(I₀, E), wherethe edgee=(ⁱ,j)∈E means thatthe agenti cansend

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

I. E⊆Emax,whereEmax isthesetofall admissibleconnections of thenetwork.

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

III. Eachindependentagenti∈I_Pisconnectedwiththesinknode, thatis,(i,0)∈E.

IV. Eachregulatedagentr∈I_Risnotconnectedwiththesinknode, 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 thesetofallvalidtopologies.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 anyisolatedregulatedagentwillbesimplyremovedfromthenet- work.

Foreach edge (i, j) ∈Emax, weset thevariable x_i,j whichrep- resents the flux through the connection (i, j). For each i ∈ I, we setx_i:=(x_i,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 x^R := (xr:r∈I_R)andx^P := (x_i:i ∈I_P). In whatfollows,foranagenti∈I,wewillwrite

x_−i:=

x_k,j :

(

^k,j

)

^{∈Emax, k∈I}

\ {

ⁱ

}

, x^P_−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,x^P_−i

Aandx^R|_A.Itwillbeuseful alsotodenoteA^c:=EmaxࢨA.

3.2. Physicalmodel

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

1. Nullfluxesoutsidethenetwork:each agentcanuseonlythe connectionsinthetopologyE.Thus,weset

xi

E^c=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,

z_i=z_i

(

^x−i

)

. (3.2)

Thisassumption is derived froman optimal responsehypoth- esis: we assume that, fora given value ofx_−i, the agenti is capableofcomputeexactlytheminimalamountofnaturalre- sourcez_ithatshehastoconsumeinordertoperformherinner processes.

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

z_i

(

^x−i

)

⁺

(k,i)∈E

x_k,i= (i,j)∈E

x_i,j. (3.3)

Since 0isthe sinknode,it isnotsubjectto thisbalancecon- straint.

4.Inputconsistency:thereexistsareal-valuedfunctiong_iwhich allows the agenti ∈I to validate the input coming from the other agents.Wewritethisvalidationasan abstractinequality constraint

g_i

(

^x−i

)

≤0. (3.4)

Thisconstraintmayrepresentmaximalinletfluxes,maximalin- letcontaminantconcentration,minimalinlettemperature,etc.

5.Positivityoffluxes:weassumethatthefluxesonthegraph,as well astheconsumednaturalresource areallpositive, thatis,

x_i≥0 and z_i

(

^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,wesupposethatagenticancomputethedegrada- tionofitsinletfluxthroughthefunctionsg_iandz_i.

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 z_i(^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,sinceenter- priseswanttokeep asmuch private informationaspossible.The onlyagentthathasallinformationistheauthority,whohasaccess tothefullvectorx.

3.3.Economicalmodel

Inthissetting,thenetwork authorityhastwo vectorsofdeci- sionvariables:shemustchoosethetopologyofthenetworkE∈E andshecontrolstheoperationoftheregulatedagents,thatis,the outputvectorsxr,forevery r∈I_R.Each independentagenti ∈I_P 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, theindepen- dentagentswillpaytheinvestmentcostofthenetworkandalso theoperationoftheregulatedagents.Thus,ifthereisaconnection (r₁,r₂)∈Emaxbetweentworegulatedagentsr₁,r₂∈I_R,weassume that

γ

^r1,r₂=0.

Sinceall the investmentcost is transferedto the independent agents,theauthorityisonlyconcernedaboutminimizingthecon- 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

^isgivenexclusivelytopaytheinvestmentandmain- tenancecostofthepark,andthatit willbeimplementedaseffi- cientlyaspossible.Thediscussionoverefficiencyandrightpricing instruments,isoutofthescopeofthiswork.

Ontheotherhandanyindependentagenti∈I_Pwantstomini- mizeherglobalcostCost_i,whichcanbeseparatedintothreecom- ponents:theconsumptionofthenaturalresourcez_i(^x−i),thecost ofdischarging(usingtheconnection(i,0)),andtheuseoftheex- changenetwork.ThereforehercostfunctionCost_iisgivenas:

Cost_i

x_i,x^P_−i,x^R,E

=c_i·z_i

(

^x−i

)

+Costⁱⁿ_i

x^P_−i,x^R

+

(i,j)∈E

γ

i,j·x_i,j.

(3.7) whereCostⁱⁿ_i

x^P_−i,x^R

isthe inletoperatingcostofanagenti,and itsatisfiesthat

(k,i)∈Emax

x_k,i=0⇒Costⁱⁿ_i

x^P_−i,x^R

=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,x^R) ^{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

x^P_−i,x^R

ascorrectandthereforetoacceptit.

Withthismodel,theminimizationproblemoftheithindepen- dentagent(parametrized by the topology E, the actions ofregu- latedagents x^R andthe actions ofthe other independent agents x^P_−i)leadstoproblemP_i

x^P_−i,x^R,E

: minxi Cost_i

x_i,x^P_−i,x^R,E

s.t.

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

zi

(

^x−i

)

+ (k,i)∈E

xk,i= (i,j)∈E

xi,j

g_i

(

^x−i

)

≤0 zi

(

^x−i

)

≥0 x_i≥0 x_i

E^c=0.

(3.8)

Wedenoteby Eq(x^R,E)the setofequilibriafortheinduced gen- eralizedNashequilibriumproblem(GNEP,forshort) givenbythe vectorx^R andthetopologyE,thatis

x^P∈Eq

(

^xR,E

)

⇐⇒

∀

^{i∈IP,xisolvesPi}

x^P_−i,x^R,E

. (3.9)

AswealreadydiscussedinSection1,themainproblemofthis model is that each independent agent only controls her output vectorx_i,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” thepar- 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∈I_P wedefine thestand-alonecost STC_i,astheoptimalvalueoftheproblemP_i(0,0,Est),thatis, STC_i=

(

^ci+

γ

i,0

)

^·zi

(

⁰

)

Inotherwords,STC_i isthecost oftheithagentassumingthat all other agents (independent andregulated) are inactive, i.e. when agentionlysend fluxesto thesinknodeanddoesn’treceiveany complementaryfluxesfromother agents.Then, foreachindepen- dentagentP_i,we canformulate thecommitmentofminimalim- provementintheBlind-Inputcontractasthefollowingconstraint:

Costi

(

^xi,x^P_−i,x^R,E

)

≤

α

^{·STCi, (3.10)}

where

α

∈ ]0,1[istheminimalrelativegain thateachagentask forparticipatinginthenetwork.Weassumethat

α

>0since,itis impossibletoeliminateallcosts,andthat

α

<1sincenoagentis indifferentconcerningher participationinthenetwork.Indeed,if Cost_i(^xi,x^P_−i,x^R,E)=STC_i,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

x_k,r= (^r,j)∈E

x_r,j,

∀

^r∈IR,

zr

(

^x−r

)

≥0,

∀

^r∈IR, gr

(

^x−r

)

^≤0,

∀

^r∈IR, x^R≥0,

x^R

E^c=0, x^P∈Eq

(

^xR,E

)

,

Costi

(

^xi,x^P_−i,x^R,E

)

≤

α

·STCi,

∀

ⁱ∈IP.

(3.11)

Theoptimizationproblem(3.11)canbe interpretedasfollows:

theauthoritywillproposetotheagentsatopologyEandanoper- ationx∈R^|Emax| whichsatisfyallthephysicalconstraintsandalso, such thattheoperation xrespects:1) theincentiveconsistency,in the sense that no agentwillhave incentivesto unilaterally devi- atefromtheproposalduetotheconstraintx^P∈Eq(x^R,E);and2) theindividualrationalityofeachagent,inthesensethatallagents will participateinthe networksince their participation hasbeen bought through the constraint (3.10). The first criteria solves the economicalinconsistencyofMOO approach,andthesecondcrite- riasolves theparticipation problemof theSingle-Leader-Follower approach.

Remark 3.4. In this work, we do not claim novelty in the constraint x^P ∈ Eq(x^R, E). This is the main contribution of Ramosetal.(2016).However,theconstraint(3.10)isnew.Interms ofmodelingandinthiscontext,thefactto“attract” theindepen- 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 ∈I_P.Thevalue of

α

irepre- sentsthe“cost” ofbuyingtheparticipationoftheithindependent agent,whichisexactly(¹−

α

i)^STCi.However,allowingtohavedif- ferentcostsdependingontheenterpriserisesthenaturalquestion ofhow todecide thesevalues.This problemliesincontract the- ory(foranintroductiontothefield,werefertoBoltonandDewa- tripont(2005); Salanié (2005))anditisout thescope ofthearti- cle.Thus,wewillconsideronlyuniformvaluesof

α

^,whichcanbe

interpreted asa publiccall forparticipationin thenetwork. Uni- formvaluesof

α

^{,however,implythat thecostofbuyingthepar-}

ticipationofanagentisproportionaltohersize,duetothefactor STC_i.

Remark3.6. Animportantfactorwedonotconsiderinthiswork isthereboundeffectthatcostsreductionsmayhaveontheopera- tionofagents.Forexample,ittermsofwaterexchange,adiminu-

tionofcosts ofagentiwithrespecttoSTC_imayinducean incre- mentofwastesproduction,thatis,avariationinM_i.Thus,thisre- boundeffectmaychangethevalueofz_i(^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-Inputmod- els are a social optimizationproblemwhere,through Blind-Input contracts, the cooperation of each independent agent has been bought.This socialoptimizationis alsoeconomicallystable, since implicitlyitrespectanequilibriumconstraint(x^P∈Eq(x^R,E)).This reduction/reformulationwillbepresentedinthreesteps.

4.1. Characterizationofequilibria

The followingtheoremcharacterizestheequilibriumsetEq(x^R, 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 actionsbecomepre- dictablethroughthecostfunctions.Thus,theauthoritycanchoose the network E such that each action ofan independent agentis inducedtoreachthesocialoptimum.

Toformalizethisidea,letusintroducethenotionofactivearcs. Given a topology E, foreach independent agenti ∈I_P we define the set ofactive arcs ofi,denoted by E_i,act, asall the arcse ∈E havingminimumcost,thatis,

Ei,act:=

(

i,j

)

∈E :

γ

ⁱ,j=

γ

i^∗:= min (ⁱ,k)∈E

γ

i,k

. (4.1)

As convention, for any regulated agent r ∈ I_R, we set E_r,act=

{

(^r,j)^: (^r,j)∈E

}

^.

Theorem4.1. ForE∈E andx^R ≥0fixed,theequilibriumsetEq(x^R, E)isgivenby

Eq

(

^xR,E

)

⁼

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎩

x^P :

∀

^i∈IP,

zi

(

^x−i

)

+ (^k,i)∈E

x_k,i= (ⁱ,j)∈E

xi,j

g_i

(

^x−i

)

≤0 z_i

(

^x−i

)

^≥0 x_i

E_i,^c_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, z_i

(

^x−i

)

+

(^k,i)∈E

x_k,i= (ⁱ,j)∈E

x_i,j,

∀

^i∈I

xi

E_i,^c_act=0,

∀

^i∈I

gi

(

^x−i

)

≤0,

∀

ⁱ∈I z_i

(

^x−i

)

^≥0,

∀

^i∈I

Cost_i

(

^xi,x^P_−i,x^R,E

)

≤

α

i·STC_i,

∀

^i∈IP

x≥0.

(4.3)

Proof. Thesecond partoftheproof iseasilyverifiedbyreplacing theconstraintx^P∈Eq(x^R,E)bythesystemofequationsintheright

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

Tosimplify notation, let us denote by S(x^R, E) the right-hand setof(4.2).First,letusprovethatS(x^R,E)⊆Eq(x^R,E).Fixx^P∈S(x^R, E). SinceE_i,act⊂Eforeach i ∈I_P,itis nothard toseethat x_i isa feasiblesetofP_i(^xP_−i,x^R,E)^.

Now, fix i ∈ I_P and let x_i be another feasible point of P_i(^xP_−i,x^R,E)^{. Then, x}_i≥0 and it satisfies the balance constraint (3.3),whichyieldsthat

Cost_i= (i,j)∈E

γ

i,jx_i,j−

γ

i^∗

(i,j)∈Ei,act

x_i,j

≥

γ

i^∗

(i,j)∈E

x_i,j− (i,j)∈E_i,act

xi,j

≥0,

where Cost_i:=Cost_i(^x_i,x^P_−i,x^R)−Cost_i(^xi,x^P_−i,x^R)^{andthelastin-} equalityisduetothefactthat

(i,j)∈E

x_i,j=z_i

(

^x−i

)

+ (k,i)∈E

x_k,i

=

(i,j)∈E

x_i,j= (i,j)∈Ei,act

x_i,j.

Thus,x_i solves P_i(^xP₋i,x^R,E), andsincethis holdsforevery i ∈I_P, wededucethatx^P∈Eq(x^R,E).

Now,letusprovethatEq(x^R,E)⊆S(x^R,E).Letx^P∈Eq(x^R,E),and suppose that x^P∈S(x^R, E). Since for each i ∈IP the vector x_i is a feasiblepointofP(^xP₋i,x^R,E),theonlywayforx^Pnottobelongto S(x^R,E)isthatthereexisti₀∈I_Psuchthatx_i

0

E^c

i0,act=0.Thus,there is(ⁱ0,j₀)∈E

\

^Ei₀,act such thatx_i

0,j₀ >0.Let(i₀,j₁)∈E_i,act (which isnonemptybydefinition)andletusconsiderthevectorx_i

0 given by

x_i0,k=

_x

i0,k ifk∈I

\ {

^j0,j1

}

^,

0 ifk=j0,

x_i0,j1+x_i0,j0 ifk=j1. Wehavethatx_i

0≥0(sincex_i0≥0)andalso z_i

(

^x−i0

)

+

(k,i0)∈E

x_k,i0= (i0,j)∈E

x_i0,j= (i0,j)∈E

x_i

0,j. Thus, since x_−i0 remains the same, x_i

0 is a feasible point of P_i(^xP₋i₀,x^R,E)^. Furthermore, denoting by Cost_i

0= Cost_i

0(^x_i

0,x^P_−i

0,x^R,E)−Cost_i

0(^xi₀,x^P_−i

0,x^R,E),wehavethat Costi0=

(ⁱ0,j)∈E

γ

ⁱ0,jx_i0,j− (ⁱ0,j)∈E

γ

ⁱ0,jxi0,j

=

γ

i0,j1−

γ

i0,j0

xi0,j0

=

γ

^∗−

γ

ⁱ0,j0

xi0,j0<0,

since,byconstruction,

γ

i₀,j0>

γ

^{∗.Thisyieldsthatx}i₀ doesn’tsolve P_i(^xP₋i₀,x^R,E),whichisacontradiction.Thus,x^P∈S(x^R,E),finishing theproof.

Intuitively, the above theorem says that, given a topology E, each independent agent i ∈ I_P will only use the connections of minimalcost tosendtheexcessofflux,that is,shewilluseonly heractivearcs.Furthermore,eachindependentagentisindifferent tothedistributionoffluxesamongtheactivearcs,soanyfeasible vectorx^Psatisfyingtheconstraintx_i

E^c_i,act=0foreveryi ∈I_Pmust beanequilibrium.Thissimplificationisstronglybasedonthelin- earity ofthe costs functions withrespect to the agent’s variable x_i.