Computers and Chemical Engineering 28 (2004) 985–992

Multi-company collaborative supply chain managementwith economical and environmental considerations

Metin Türkaya,∗, Cihan Oruça, Kaoru Fujitab, Tatsuyuki Asakurab

a College of Engineering, Koç University, Rumelifeneri Yolu, Sariyer, Istanbul 34450, Turkeyb Business Process Optimization Lab, MCC-Group Science & Technology Research Center, Mitsubishi Chemical Corporation,

3-10 Ushiodori, Kurashiki, Okayama 712-8054, Japan

Abstract

Process systems must interact with other systems for a better production performance. The interaction among process systems is usuallyestablished when these systems exchange materials such as steam and electricity. Integrated analysis of different process systems can providevaluable insight and also identify improvements in the financial and environmental performance of industrial supply chain systems. Asystematic approach to identify the synergy among different process systems has been developed. The proposed approach uses three steps;the generation of standardized models for process units, integration of process unit models for the supply chain system and solution of themodel and analysis of the results. The developed approach is illustrated with an example that is a simplified version of a real problem andtested on an industrial problem. It is shown that important improvements in the cost and release of environmentally harmful emissions can beaccomplished by integration of different process systems.© 2003 Elsevier Ltd. All rights reserved.

Keywords: Supply chain management; Energy integration; Mixed-integer programming

1. Introduction

Supply chain management has attracted a lot of attentionrecently due to its role involving all of the activities in indus-trial organizations ranging from raw material procurementto final product delivery to customers. Conceptual studieson supply chain management emphasized the importanceof the strategic relationships between companies in orderto increase financial and operational performance of thesecompanies by reductions in the total cost and inventoriesthroughout the supply chain, and increased levels of sharedinformation. The essence of this relationship is concernedwith coordination between the participants (Reyniers, 1992;Sox, Thomas, & McClain, 1997; Whang, 1995). Partneringbetween industrial organizations is an increasingly commonavenue for these organizations to find and maintain compet-itive advantage (Mentzer, 1999; Mohr and Spekman, 1994).The nature of inter-firm partnering in supply chain man-agement has been studied byMentzer, Min, and Zacharia(2000). These authors suggest a continuum of strategic and

∗ Corresponding author.E-mail address: mturkay@ku.edu.tr (M. Türkay).

operational partnering based upon the orientation of the part-ners and the degree of implementation of partnering be-tween two independent industrial organizations. This workalso argued that implementation of strategic partnering leadsto sustainable competitive advantage.Maloni and Benton(1997)mentioned that extensive literature exists about con-cepts of supply chain management, but there are very fewreports describing more rigorous quantitative approaches toaddress issues related to supply chain management.Lee andKim (2002)proposed a hybrid approach combining analyticand simulation models for planning production and distribu-tion operations of supply chain systems considering capacityconstraints and operation times.

A growing number of industrial organizations started torealize the strategic importance of planning, controlling,and designing supply chain networks as a whole rather thana collection of disconnected subsystems.Min and Zhou(2002) reviewed the efforts in modeling supply chain op-erations, and identified key challenges and opportunitiesin modeling the operations in supply chain systems. Theseauthors emphasized the importance of inter-functionaland inter-organizational integration in creating a synergyamong industrial organizations to overcome challengingissues related to supply chain systems. Integrated process

0098-1354/$ – see front matter © 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.compchemeng.2003.09.005

986 M. Türkay et al. / Computers and Chemical Engineering 28 (2004) 985–992

management was discussed bySchiefer (2002)to dealwith economic, quality and environmental aspects of sup-ply chain systems and link its subsystems to make betterdecisions in the organization and control of the processes.

Energy systems have also attracted a lot of attention dueto its role and importance in the performance of all indus-trial systems.Tari and Söderström (2002)described thedevelopment of a method for creating optimization modelsfor industrial energy systems including material stores. Theauthors also determined the influence of the existence ofmaterial stores in an industrial energy system on the totalsystem cost. Process integration for the design of utilitysystems has been addressed byMarechal and Kalitventzeff(1998) to determine the optimal configuration of utilitysystems for minimization of energy requirements.Saeed,Blakemore, and Doulah (1996)addressed fuel consump-tion reduction by applying a pinch-point implementation.These approaches targeted minimization of the total cost ofsupplying energy to process systems. Financial costs andenvironmental impact of energy generation has been studiedby Gonzales-Monroy and Cordoba (2002). A single energyproduction system for satisfying electricity demand in a citywas considered and a solution to this problem was reportedusing simulated annealing.Zhou, Cheng, and Hua (2000)integrated two multi-objective decision making methods,goal programming and analytic hierarchy process, to ad-dress sustainable supply chain optimization of continuousprocess industries.

An important issue in the industrial supply chain man-agement is the satisfaction of all production requirementsto achieve high profits while observing environmental reg-ulations. Performance improvement, both financially andenvironmentally, is a must for survival and growth of in-dustrial organizations. There is a high level of competitionin all industrial sectors that are composed of several com-panies competing to produce the same or similar productsat lower costs while trying to achieve higher levels of qual-ity. There are also some environmental protection laws andprotocols, like Kyoto Protocol that organizations have tofollow that could impose significant restrictions on individ-ual organizations’ ability to sustain production levels.

Industrial organizations carry out their industrial activi-ties usually at designated areas called industrial zones. Anindustrial zone is a collection of production systems belong-ing to different companies with distinct characteristics inthe same area. Some of the production systems in the in-dustrial zone have close interaction among each other dueto supplier–producer relationships. The industrial zone isclassified as the overall system while the individual produc-tion systems (organizations) can be considered as subsys-tems that are integral part of the overall system in this study.A strong interaction among production systems can be ob-served in the supply chain integration of the energy at anindustrial zone. Since all of the subsystems require energyfor production; there is almost always a central power pro-duction facility in the industrial zone, and also a number

of subsystems may produce their own energy in their ownutility plants. Consequently, supply chain integration of en-ergy at an industrial zone is as important as any material inthe system. A distinct feature of energy that differentiatesit from the other materials is the storage: energy cannot bestored in its most effective form, as electricity or steam. Theproduction rate at a subsystem is proportional to the supplyamount of energy; therefore, energy supply is an importantfactor that determines the capacity and efficiency. Anotherimportant characteristic of energy is the fact that energy gen-eration systems release a large quantity of environmentallyharmful chemicals such as SOx, NOx, and COx.

The objective of this paper is to develop a quantitativeassessment to the question: “Does the multi-organizationcollaborative SCM create a synergy to overcome financialand environmental obstacles and difficulties for survival andgrowth?”

In this paper, it is shown that a systematic approach canidentify the synergy among a number of process systemsand detect improvements in the financial and environmen-tal performance of the integrated system while keeping thetopology of the systems fixed. The first step in the proposedapproach is the development of process models. The pro-cess units are modeled using fundamentals of thermodynam-ics, conservation of mass and energy, and after the analysisof existing process data. The process models for the mostcommon units in the energy systems are given in the fol-lowing section. The second step is the development of anMILP model for each process system that is a collection ofprocess units. The optimization model integrates the pro-cess systems in the industrial zone that involves systems oflinear disjunctive equations. The last step of the proposedapproach is the analysis of the solution to identify finan-cial and environmental improvements if the process systemsare integrated through exchange of material and energy. Theapplication of this approach is illustrated through an exam-ple that is a simplified version of a real problem in sectionthree. It is shown that important improvements in the costand release of environmentally harmful chemicals can beaccomplished by integration of different process systems atan industrial zone. In addition, application to an industrialproblem is presented in section four. Main conclusions aresummarized in the last section of the paper.

2. Problem formulation

Energy systems utilize fuel, air and other materials to gen-erate electricity and steam demanded by the other processunits in the industrial systems (seeFig. 4). The flow of ma-terials in the system is modeled with variablesxijkl repre-senting the amount of materialk in unit j of companyi instatel. The indexl, which represents the state of the mate-rial k is included to model the mass and energy conservationequations effectively and takes one of the valuesIN, OUT,CON or GEN according to the state of the material which

M. Türkay et al. / Computers and Chemical Engineering 28 (2004) 985–992 987

Boiler HP Steam

MP Steam

Electricity

Fuel

SOxWater

Fig. 1. Schematic diagram of a boiler.

can be an input to a process or output from a process, orconsumed or generated during a process. Specific materialsand states of a material are indicated with subscripts.

The models for the most common units in the energysystems are given in the following subsections.

2.1. Boiler models

Boilers generate high pressure steam by burning fuel. Asa consequence of burning fossil fuels, boilers generate envi-ronmentally harmful chemical substances such as SOx, NOx,and COx. Boilers require electricity for operating the me-chanical equipment and medium pressure steam for heatingthe boiler feed water. Material flow around a typical boileris illustrated inFig. 1.

Boiler models include the following equations:

xijkHPlGEN= 1

ηijkfuelm

cckfuelmxijkfuelmlCON (1)

xijkMPlCON = aijkMPxijkHPlGEN (2)

xijkELlCON = aijkELxijkHPlGEN (3)

xijkSOx lGEN = sffuelxijkfuellCON (4)

xijklIN + xijklGEN = xijklOUT + xijklCON (5)

xijkl′ = 0 (6)

Cijkfuel = cffuelxijkfuellCON (7)

The amount of HP steam generation as a function of thefuel consumption is modeled inEq. (1). The amount ofsteam generation is proportional to fuel consumption andthe calorific value of the particular fuel used in the boiler.In addition, boiler efficiency,ηijk, is a function of the fueltype. Eqs. (2) and (3)model the electricity and MP steamconsumption in the boiler as a function of the amount ofHP steam generated. The generation of SOx as function ofsulfur content of the particular fuel and the amount of fuelconsumption in the boilers as given inEqs. (4) and (5)relatethe states of materials in the boiler considering conservationof mass. In order to maintain consistency in the materialbalances,Eq. (6)fixes some of the states of materials to zero(e.g., since there is no HP steam consumption and HP steaminput to the boilers, corresponding states of HP,CON andIN, are fixed to 0 in the boilers). Finally,Eq. (7)models thetotal cost of fuel as function of the unit cost of fuel and fuelconsumption in the boilers.

HP Steam

Turbine

MP Steam LP Steam

G

Condensate

Electricity

Fig. 2. Schematic diagram for a turbine.

2.2. Turbine models

Turbines expand steam at higher pressures to steam atlower pressures and generate electricity by converting themechanical energy released during expansion in to electric-ity. A typical multi-stage turbine receives HP steam andproduces electricity and MP and LP steams as shown inFig. 2.

Electricity generation in a turbine is a function of theamount of HP steam feed and the amounts of MP and LPsteam and condensate generation as shown inEq. (8). Thecoefficientsbijk andgijk in Eq. (8)can be obtained by sta-tistical analysis of the existing process data. The materialbalance around turbines is expressed inEq. (9). The upperand lower bounds on the amount of electricity generated inturbines are given inEq. (10). In addition,Eqs. (5) and (6)are also included for all materials and their correspondingstates for turbines.

xijkELlGEN = bijkHPxijkHPlIN − gijkMPxijkMPlGEN

−gijkLPxijkLPlGEN − gijkCONxijkCONlGEN (8)

xijkHPlIN = xijkMPlGEN + xijkLPlGEN + xijkCONlGEN (9)

xLijkEL lGEN

≤ xijkELlGEN ≤ xUijkEL lGEN

(10)

2.3. Fuel tank models

Fuel tanks contain different kinds of fuel which aretransferred to boilers for generation of HP steam. The fueltanks have finite capacity and contain an initial inventory,Nij. The amount of fuel in tankj of companyi is repre-sented by variableIij. Tank models include the followingequations:

xijkfuellOUT =∑

j′∈boilers

xij′kfuellIN (11)

Iij = Nij −∑

j′∈boilers

xij′kfuellIN (12)

Iij ≥ γ × Nij (13)

Eq. (11)models the material balance between a fuel tankand the boilers that use the particular fuel contained inthe fuel tank. The amount of fuel outflow from a fueltank is equal to the total amount of that fuel that flowsinto the boilers.Eq. (12) updates the amount of inven-tory in a fuel tank while Eq. (13) expresses a safety

988 M. Türkay et al. / Computers and Chemical Engineering 28 (2004) 985–992

Mixer/Splitter othercompanies

other companies

boilers

boilers

turbines

turbines

other mixers

other mixers

Fig. 3. Schematic diagram for a mixer.

stock level for each fuel tank as function of the initialinventory. It is also possible to express safety stock asa function of the total storage capacity of the fuel tank;in this caseNij in Eq. (13) is replaced by total storagecapacity of the tank. The safety stock parameter,γ, de-fines the fraction of the fuel that must remain in thetank. The amount of fuel that can be used by the boil-ers is restricted to the available fuel inventory in the fueltanks.

2.4. Mixer/splitter models

Mixers/splitters are the units which receive and send thesame type of material from and to different units. There isa mixer/splitter for each type of material in the system (HPsteam mixer, LP steam mixer, MP steam mixer, electricitymixer). Material flow around a typical steam mixer is shownin Fig. 3.

Material coming to a mixer/splitter from units in the pro-cess that produce this material, other mixers/splitters, andother companies is transferred to certain units that use thismaterial, other mixers, and to other companies as shown inEq. (14).

∑j

xijklOUT +∑

j

xeiji′j′ =∑

j

xijklIN +∑

j

xei′j′ij + dij

(14)

The variablexeiji ′j′ represents the amount of material ex-changed from unitj of companyi to unit j′ of companyi′.The amount of particular material demanded by companyiis represented by parameterdij.

2.5. Fuel selection

Boilers can utilize different fuels with minimal adjust-ments in the operating conditions of boiler equipment.There are many reasons for considering alternative fu-els. One of the most important reasons is the insufficientamounts of fuel available in the inventory forcing the util-ity system to use an alternative fuel. Other reasons includethe selection of economically and/or environmentally at-tractive fuel among the available alternatives. The use ofalternative fuels in boilers is modeled using disjunctions

(Turkay & Grossmann, 1996).

vm∈Dij

Yijkfuelm

xijkHPlGEN = 1

ηijkfuelm

cckfuelmxijkfuelmlCON

xijkMPlCON = aijkMPxijkHPlGEN

xijkMPlCON = aijkELxijkHPlGEN

xijkSOx lGEN = skfuelmxijkfuelmlCON

Cijkfuelm= ckfuelm

xijkfuelmlCON

xLijkHPlGEN

≤ xijkHPlGEN ≤ xUijkHPlGEN

(15)

The above disjunction is included in the optimization modelafter the convex hull formulation as shown byTurkay andGrossmann (1996). The derivation of convex hull formula-tion for Eq. (15) is given in Appendix A. The boilers canuse only one type of fuel at a time for generating HP steamas indicated byEq. (A.28). In addition, the HP steam gener-ation (and fuel consumption as consequence ofEq. (A.20))in the boilers is restricted to between an upper bound andlower bound withEqs. (A.25) and (A.26).

The limit on the SOx emission is expressed inEq. (16)asfollows:∑

i

∑j

xijkSOx lGEN ≤ sUkSOx

(16)

A solution satisfying this constraint (16) is considered suf-ficient without any penalty for SOx release.

The objective function is defined as the minimization ofthe cost:

minZ =∑

i

∑j

∑k

Cijk

+∑

i

∑j

∑i′

∑j′

(CELB − CELS)xeiji′j′ (17)

The first term of the objective function gives the total costof fuel used in the boilers. The second term includes theexchange of material between the unitj of companyi andthe unitj′ of companyi′.

The optimization model and the analysis of the results areillustrated in the following section with an example that isderived from an industrial system.

3. Illustrative example

The illustrative example considers two energy systemswith each system having two fuel tanks with different fuels,two boilers and two turbines as shown inFig. 4. The en-ergy systems must fulfill the electricity and steam require-ment of processes they serve. The demand for steam (HP,MP, and LP) and electricity is a function of production rateand energy requirement characteristics of the processes thatthe energy systems are serving. The parameters and energyrequirements for the example are given inTables 1 and 2,respectively.

M. Türkay et al. / Computers and Chemical Engineering 28 (2004) 985–992 989

Co mpany 1

Electricity DemandElectricity Demand

LP DemandLP Demand

MP DemandMP Demand

HP DemandHP Demand

Fuel 1

Boiler 1

Fuel 2

Boiler 2

HP

T1 T2

MP

LP

Electricity

Fuel 1

Boiler 1

Fuel 2

Boiler 2

HP

T1 T2

MP

LP

Electricity

Cogeneration Facility (Company 3)

Company 2

Fig. 4. Flowsheet of the illustrative example.

Table 1Operating characteristics for the utility system in the illustrative example

Parameter Company 1 Company 2

Fuel 1 Fuel 2 Fuel 1 Fuel 2

ccf 10.500 9.650 6.650 10.200Nijk 100 40 120 100sf 7.80 1.42 1.20 5.13cf 200 76 83 145

Boiler 1 Boiler 2 Boiler 1 Boiler 2

ηijkfuel 1 0.590 0.575 0.560 0.565ηijkfuel 2 0.600 0.595 0.605 0.600aijkMP 0.11 0.12 0.11 0.12aijkEL 0.002 0.003 0.0025 0.0028xu

ijkHPlGEN550 550 600 600

Turbine 1 Turbine 2 Turbine 1 Turbine 2

bijkHP 0.150 0.175 0.160 0.170gijkMP 0.070 0.080 0.070 0.075gijkLP 0.009 0.010 0.012 0.010xu

ijkEL lGEN70 60 70 65

Table 2Energy demand in the example problem

Company 1 Company 2

Electricity 150 140HP Steam 10 10MP Steam 620 300LP Steam 300 680

The problem is modeled in OPL Studio and solved us-ing ILOG CPLEX (Ilog, 2002). The integrated model thatincludes exchange of material between companies 1 and 2contains a very large number of variables compared to thenumber of constraints as shown inTable 3. The MILP prob-lem is solved at node 0 in 1.38 CPU seconds on computerwith Pentium 4 processor at 1.5 GHz speed and 512 MB ofRAM memory. It is important to note that the problem issolved at the root node of the branch and bound algorithmillustrating zero integrality gap of the convex hull formula-tion in this problem.

In order to asses the synergy generated by integrationof companies 1 and 2 through exchange of material, thesame problem is solved to optimality after eliminating theexchange of material between these companies. Therefore,the following analysis of the solution is made by compar-ing optimal solutions for the integrated and non-integratedoperations of companies. When the results of the integratedsolution are compared with the results of the non-integratedsolution, it is observed that steam expansion that means lostenergy decreased and exchange of steam and electricity ex-ists as shown inTables 4 and 5.

Table 3Statistics of problem size and solution for illustrative example

Number of constraints 222Number of variables 4961Number of nodes 0Number of iterations 20CPU time (sec) 1.38

990 M. Türkay et al. / Computers and Chemical Engineering 28 (2004) 985–992

Table 4Material exchange in the non-integrated solution

Companyi Unit j Companyi′ Unit j′ Amount

Company 1 HP1 Company 1 MP1 140.57Company 2 MP2 Company 2 LP2 149.50Company 3 Elec3 Company 1 Elec1 53.22Company 3 Elec3 Company 2 Elec2 7.95

For example, in the optimal solution for non-integratedsystem there is a large quantity of steam flow from the higherpressure steam mixers to the lower pressure steam mixersof the same company. More specifically, there is a flow ofHP steam from HP steam mixer of company 1 to MP steammixer of the same company. Besides, there is a flow of MPsteam from MP steam mixer of company 2 to LP steammixer of the same company. The flow of steam between twosteam mixers means steam expansion, since steam that is ata higher energy level loses its energy and becomes steam ata lower energy level without its energy being extracted inthe steam turbines.

The expansion of steam from a higher pressure mixer toa lower pressure mixer means that the energy is lost be-tween two mixers. In other words, when the exchange ofmaterials (HP, MP, and LP steam) is not allowed, it is ob-served that boilers must produce more HP steam than therequired amount for fulfilling the MP and LP steam require-ment through expansion. This is a common practice in pro-cess systems: utility systems are so tightly integrated withother processes that frequently higher pressure steam mustbe expanded to fulfill lower pressure steam requirement. Onthe other hand, the energy integrated solution fulfills theenergy requirement of both of the companies at a lowercost. Because, in the integrated solution, there exist mate-rial flows between the equivalent mixers of the two com-panies. Specifically there exist steam flows from HP steammixer of company 1 to HP steam mixer of company 2, LPsteam mixer of company 1 to LP steam mixer of company2, MP steam mixer of company 2 to MP steam mixer ofcompany 1. Therefore, in the integrated system there ex-ists steam and electricity exchange between the integratedcompanies.

Financial and environmental improvements as a result ofthe integration are illustrated inTable 6. It is possible toserve the same energy requirement at a 2.25% lower cost byintegrating two companies. It is also important to notice thata significant reduction (7.1%) in the SOx emission is possi-

Table 5Material exchange in the integrated solution

Company Unit Company Unit Value

Company 1 HP1 Company 2 HP2 8.48Company 1 LP1 Company 2 LP2 281.52Company 2 MP2 Company 1 MP1 362.00Company 3 Elec3 Company 1 Elec1 31.97Company 3 Elec3 Company 2 Elec2 8.18

Table 6Comparison of the results

Non-integrated Integrated Change (%)

Total cost 16051 15690 2.25SOx release 375.84 349.23 7.1

ble through supply chain integration. The proposed approachidentified simultaneous improvements in the financial andenvironmental performances for both companies. It is im-portant to note that the same optimization model was con-sidered for non-integrated and integrated solutions; the onlydifference between the two problems is the exchange of ma-terial between the two companies. The variables that modelthe exchange of material between two companies (xeiji ′j′ ) isfixed to 0 in the non-integrated solution.

4. Example

The proposed approach developed in this paper is appliedto this real problem consisting of three different companiesand the cogeneration facility in the industrial zone. Thereare different kinds of tanks, boilers, turbines, mixers andsplitters in the energy systems of each of these companies:three boilers, six turbines, five types of steam in company 1;seven boilers, six turbines, six types of steam in company 2;six boilers, five turbines, three types of steam in company 3;six types of steam in the cogeneration facility. There are 11types of steam that have different pressures generated andused in the overall system. The boilers are fed with differenttypes of fuels with different characteristics such as calorificvalue, cost and SOx emission. There are 10 types of fuelsused in the system, and each company has five differentfuel tanks containing different fuels. Further details of theindustrial problem are considered confidential and are notincluded in this paper.

The proposed approach is also tested on the industrialproblem. The problem is modeled in OPL Studio withdatabase integration and solved using ILOG CPLEX. Thestatistics of the model size and solution are given inTable 7.

When the optimal solutions for the integrated andnon-integrated systems are compared, it is observed that thenon-integrated solution fulfills the energy requirement ofthe companies at a higher cost than the integrated solution.The difference in the cost between the two solutions is inthe same order of magnitude as the illustrative example. In

Table 7Statistics of problem size and solution for the example

Number of constraints 1353Number of variables 143553Number of nodes 0Number of iterations 20CPU time (s) 5.17

M. Türkay et al. / Computers and Chemical Engineering 28 (2004) 985–992 991

addition it is also observed availability of more types of fuelincreases the degrees of freedom in reducing SOx emissionlevels and also. The energy-integrated solution fulfills theenergy requirement of the companies at a lower cost whilereducing SOx emission to satisfactory levels.

5. Conclusions

The multi-company collaborative supply chain manage-ment has been addressed in this paper. A systematic ap-proach is proposed and illustrated in the energy integrationof companies in the same industrial zone. The proposed ap-proach consists of modeling process units using fundamen-tals of thermodynamics, conservation of mass and energyand process data; development of MILP model for the sup-ply chain integration of different process systems; and com-parative analysis of the results. The proposed approach hasbeen illustrated with an example that is a simplified versionof a real problem. It is shown that important improvementsin the financial and environmental performance of the in-dustrial organizations can be accomplished by integrationof different process systems at an industrial zone. The sameapproach is applied to a complex real world problem. Theresults and the improvements upon integration are also il-lustrated for this real problem. Significant reductions in theamount of energy wasted due to strong constraints in theindustrial systems are eliminated. A quantitative approachcan be valuable in analyzing the synergy among multiple or-ganizations in improving their financial and environmentalperformance. The application of the proposed approach thataims to asses the synergy among a number of companiesin an illustrative example and an industrial problem showedthat significant improvements in the financial and environ-mental performances of the organizations can be realized.

Appendix A. Convex hull formulation of Eq. (15)

The first step in the convex hull formulation involves dis-aggregating all of the continuous variables for each term ofthe disjunction and expressing the original continuous vari-ables as the summation of disaggregated variables.

xijkHPlGEN =∑

m∈Dij

xdmijkHPlGEN

(A.1)

xijkfuelmlCON =∑

m∈Dij

xdmijkfuelmlCON

(A.2)

xijkMPlCON =∑

m∈Dij

xdmijkMPlCON

(A.3)

xijkELlCON =∑

m∈Dij

xdmijkELlCON

(A.4)

xijkSOx lGEN =∑

m∈Dij

xdmijkSOx lGEN

(A.5)

Cijkfuel =∑

m∈Dij

Cdmijkfuel

(A.6)

The step is the replication of constraints with the disag-gregated variables and adding upper bound constraints fordisaggregated variables:

xdmijkHPlgen

= 1

ηijkfuelm

cckfuelmxdm

ijkfuelm lCON∀m ∈ Dij (A.7)

xdmijkMPlCON

= aijkMPxdmijkHPlGEN

∀m ∈ Dij (A.8)

xdmijkELlCON

= aijkEL xdmijkHPlGEN

∀m ∈ Dij (A.9)

xdmijkSOx

lGEN= Sffuelm

xdmijkfuelm

lCON∀m ∈ Dij (A.10)

xijkSOx lGEN =∑

m∈Dij

xdmijkSOx lGEN

(A.11)

Cdmijkfuel

= cffuelmxdm

ijkfuelmlCON

∀m ∈ Dij (A.12)

xLijkHPlGEN

Yijkfuelm≤ xdm

ijkHPlGEN∀m ∈ Dij (A.13)

xdijkfuelmlmCON ≤ xU

ijkfuelm lCONYijkfuelm

∀m ∈ Dij (A.14)

xdmijkSOx lGEN

≤ SUkSOx

Yijkfuelm∀m ∈ Dij (A.15)

Summation of all binary variables is equal to 1.∑m

YijkHPlfuelm= 1 (A.16)

The last step is to make necessary simplifications to elim-inate unnecessary variables and constraints:

xijkHPlGEN =∑

m∈Dij

xdmijkHPlGEN

(A.17)

xijkfuelmlCON =∑

m∈Dij

xdmijkfuelmlCON

(A.18)

xijkSOx lGEN =∑

m∈Dij

xdmijkSOx lGEN

(A.19)

xdmijkHPlGEN

= 1

ηijkfuelm

ccffuelmxijkfuelmlCON∀m ∈ Dij (A.20)

xijkMPlCON = aijkMPxijkHPlGEN (A.21)

xijkELlCON = aijkELxijkHPlGEN (A.22)

xdmijkSOx lgen

= SffuelmxijkfuelmlCON∀m ∈ Dij (A.23)

Cijkfuel = cffuelmxijkfuelmlCON (A.24)

xLijkHPlGEN

Yijkfuelm≤ xdm

ijkHPlGEN∀m ∈ Dij (A.25)

xdmijkHPlGEN

≤ xUijkHPlGEN

Yijkfuelm∀m ∈ Dij (A.26)

992 M. Türkay et al. / Computers and Chemical Engineering 28 (2004) 985–992

xdmijkSOx lGEN

≤ SUkSOx

Yijkfuelm∀m ∈ Dij (A.27)

∑m

YijkHPlfuelm= 1 (A.28)

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