batch process optimization in a multipurpose plant using tabu search with a design-space...

Computers and Chemical Engineering 29 (2005) 1770–1786

Batch process optimization in a multipurpose plant usingTabu Search with a design-space diversification

L. Cavin, U. Fischer∗, A. Mosat, K. HungerbuhlerSafety& Environmental Technology Group, Institute for Chemical- and Bioengineering Swiss Federal Institute of Technology (ETH),

CH-8093 Zurich, Switzerland

Received 10 August 2004; received in revised form 4 February 2005; accepted 21 February 2005Available online 3 June 2005

Abstract

Problem formulations for the purpose of optimization typically represent a simplification of the real physical systems that are modeled. Inorder to obtain solutions that are robust to criteria not considered in the objectives and constraints definition, we propose to include in theresults list additional solutions that are not necessarily optimal but have very different characteristics in the solution space than the optima.To this effect, we present a novel feature for optimization frameworks, namely a solution-space diversification.

ical processt t the specialr rities arec non-linearp ns with ap defined thatt

sign space.I in order tod ximizes thep©

K

1

1

taeppi

ngusedthently.ible

es ofeffortsbatchign ofthe9,,

0d

We present this feature on a backdrop of batch process design. We developed a method that optimizes the design of a new chemo be implemented in an existing multi-purpose batch plant operating in single product campaign mode and that takes into accounequirements and constraints in the corresponding production facilities. In the optimization, several objectives with different prioonsidered. A flexible meta-heuristic algorithm, Tabu Search, has been implemented to solve this multi-objective combinatorialroblem. This approach is particularly suited for the identification of the Pareto set of non-dominated solutions, to which desigronounced structural diversity are added. For this purpose an indicator of the structural difference between two designs has been

akes into account the position and nature of material transfers between equipment units.Three case studies are used to illustrate how the novel approach delivers extended Pareto-sets with a high variability in the de

t has been shown why diversification in the design space is relevant to obtain solutions representing a robust set of alternativeseal with potential constraints and limitations that are not covered in the problem formulation. Such an extended Pareto-set marobability that at least one of the proposed solutions will be suitable for industrial implementation.2005 Elsevier Ltd. All rights reserved.

eywords:Batch process design; Multi-objective combinatorial optimization (MOCO); Tabu Search (TS); Solution-space diversification

. Introduction

.1. Batch process design

Specialty chemicals and pharmaceutical products areypically produced in batch processes. Corresponding plantsre often classified as multiproduct batch plants, in whichvery product follows the same sequence through all therocess steps, or as multi-purpose batch plants, in which eachroduct follows its own distinct processing sequence by us-

ng the available equipment in a product-specific layout (e.g.

∗ Corresponding author. Tel.: +41 1 632 56 68; fax: +41 16321189.E-mail address:[email protected] (U. Fischer).

Rippin, 1983). In practice combinations of these two limitiscenarios might also arise. Multi-purpose plants can bein two main modes: either only one production runs inplant at a given time or many processes run concurreSome multi-purpose plants consist of discrete but flexproduction lines that are independent from each other.

Because of the escalating importance of these typchemical processes, in recent years increased researchhave been undertaken to develop design methods forprocesses. Many methods deal with the grassroot desmultiproduct or multi-purpose batch plants and includeequipment sizing problem (Grossmann & Sargent, 197;Papageorgaki & Reklaitis, 1990; Sparrow, Forder, & Rippin1975; Suhami & Mah, 1982; Voudouris & Grossmann

098-1354/$ – see front matter © 2005 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2005.02.039

L. Cavin et al. / Computers and Chemical Engineering 29 (2005) 1770–1786 1771

Nomenclature

A allocation (equipment class to task class) ma-trix

B block matrix containing sequences of tasks tobe conducted in the same equipment unit(s)

BR block-to-recipe one-to-many relationship ma-trix

C connectivity costs matrixD results list. Contains several designsLd single element ofD, Π or Q: a design; same

structure asLE equipment matrixF matrix containing the objective function values

of a designH Hash valueki recipe index vector: list of tasks belonging to

block ik+i recipe index vector: list of blocks following the

block ik−i recipe index vector: list of blocks preceding the

block iL allocated equipment units matrix.L is a solu-

tion to the design problemLm design resulting from the application of the

moveMm to a designLLm* best design inQM list of movesn any integer;n1 andn2 control the composition

of the results listP standard operating ranges library matrixQ list of designs resulting from the application of

the movesM◦ to the actual designLR recipe matrix consisting of a list of tasks to be

conductedS eligible equipment units vectorS* design-independent list of eligible equipment

unitsT Tabu list, contains forbidden movesU eligible equipment classes vectorX secondary recipe matrix. Subset matrix ofRfor

the tasks indicated inkZ sequence matrix allowing to handle branched

recipes

Greek letters∆ distance between two designsδ,λ algorithm internal parametersΠ non-dominated set (Pareto-optima), contains

several designsLϕ relative bounds on the objective functions

Subscriptsa counter on the objective functions (applies to

F)

i, j counters across the block matrixB

Superscriptm index onM; by extension, index onL. Lm is the

design resulting from moveMm

1992). In most cases, the authors consider the case wheremany productions run concurrently.

Relatively few publications have been presented thatdeal with the optimum design of a single batch process. Forgrassroot design,Loonkar and Robinson (1970)described aprocedure for the cost optimum design and apparatus sizingof a single batch process, whileTakamatsu, Hashimoto, andHasebe (1982)presented a similar approach that considersthe possibility of intermediate storage.Yeh and Reklaitis(1987) presented a method for the preliminary grassrootdesign of a single batch process including an approximatesizing procedure.Mauderli and Rippin (1979)developed amethod for planning and scheduling in multi-purpose batchplants. While they consider many concurrent productions,their first step consists of the heuristic generation of designalternatives for the production of single products taking intoconsideration the plant specifications and the process require-ments.Wellons and Reklaitis (1989)developed a MINLPformulation for such a design task. They generate groups ofequipment units that can handle one particular step of thechemical process. The resulting combinatorial tree is opti-mized for the maximum production. The results are typicallya list of batches (potentially of different batch size) that takedifferent paths through the plant and run in a fixed sequence.

The pre-assignment of equipment units to a given processstep is a constraint that limits the design possibilities inm sedf ocesss np athsf mentp etya uringp oftenp size.U theo ft lys ed inC

t isi ande (e.g.r erial,s nt isl con-n (e.g.

ultipurpose plants in which equipment units can be uor several tasks and can thus be assigned to different prteps.Wellons and Reklaitis (1991)revised their desigrocedure for the scheduling of different processing p

or the same product and included the equipment assignroblem into the optimization. However, for control, safnd quality reasons as well as due to good manufactractice (GMP) regulations, subsequent batches arereferred to follow the same path and to be of equalnder this perspective, the objective changes fromptimum schedule of n subsequent batchesto thedesign ohe single most efficient batch. To our knowledge, the onuch method presented so far is the one we describavin, Fischer, Glover, and Hungerbuhler (2004).In order to identify an optimal solution for this problem i

mportant to consider in the design procedure all detailsxisting constraints such as equipment specificationsange of operating temperature and pressure, lining matpecial supply pipes, the floor at which each equipmeocated), design constraints (e.g. feasible and infeasibleection of equipment units), and process requirements

1772 L. Cavin et al. / Computers and Chemical Engineering 29 (2005) 1770–1786

reaction mixture that cannot be safely transferred, thus forc-ing several operations to be conducted in the same equipmentunit). This is the approach taken in the method presented here.

The problem that has to be solved is one of combinatorialoptimization. Even when the user selects to use short-cutmodels to adapt operation durations, non-linear (e.g.stepwise) functions are evaluated to compute the objectivefunction, rendering this problem a non-linear integer system.Such systems are known to be NP-Complete and cannot besolved in polynomial time. Various algorithms and methodshave been developed to tackle similar problems that can beclassified in three main categories: heuristics, mathematicalprogramming and metaheuristic algorithms.

We chose to implement a metaheuristic method, the TabuSearch (TS) developed byGlover (1977). TS makes use ofadaptive memory to escape local minima. TS had numeroussuccessful applications in recent years (for example, thewebsite http://www.tabusearch.netlists over a thousandpresentations and articles on the method). It has been usedonly sparsely in the field of batch process design, byWang,Quan, and Xu (1999)for the problem of the grassrootdesign of multiproduct batch processes and in our earlierpublication on multipurpose plant utilization (Cavin et al.,2004). Balasubramanian and Grossmann (2003)have alsoimplemented a TS method for the scheduling of processingplants under uncertainty and compared it to a mathematicalp ctivem onsf ardM

1

ata r toc reto-oG ofm an-d ea aswL othe n-d f thes

tech-n quest le( Dy-n esso ignsa eachom tar-g only

the regions (i.e. designs) that belong to the non-dominatedset. Genetic algorithms (for instanceKasat, Kunzru, Saraf, &Gupta, 2002, in the continuous optimization of fluidized-bedcatalytic cracking units) are naturally suited for targeting atthe non-dominated set, the level of domination being chosenas fitness function. There are few multi-objective applicationsof simulated annealing (SA) yet; in the chemical engineeringfield, one such application is presented byDantus and High(1999)who use SA for studying waste minimization alterna-tives under uncertainty by targeting a decision-maker’s idealpoint and use direct comparison of designs.

Finally in TS, Hansen (2000a, 2000b)proposed theTAMOCO method that keeps a list of multiple solutionsrepresenting an estimation of the non-dominated set, anddiscriminate the search in favor of neighbors that are differentfrom the objective functions’ values of existing solutionsby manipulating weighting factors at each iteration.Yang,Suming, Ni, and Ho (2002)produce the non-dominated setefficiently by selecting the least dominated solution at eachiteration and adding a penalty for shared components in thesolution vector.

1.3. Objectives of this research

In practical optimization problems, typically not all rele-vant decision criteria can be included into the objective func-t riat tech-n fer-e tions.E re:

• may

• ent

• (e.g.ping

• de-

non-d thata reto-o lu-t fromtT hichi . itss

uc-t d bya di-v ogyd s toh

rogramming approach. They found TS to be an attraethod to obtain in a relatively short time good soluti

or large problems that might be intractable with standINLP optimization techniques.

.2. Diversification techniques

Most multi-objective optimization methods targetbroad diversification in the objective space, in orde

over the non-dominated set, i.e. the collection of Paptimal solutions, as completely as possible (seeEhrgott &andibleux, 2000, for an extended discussion and reviewulti-objective combinatorial optimization). Thus, the stard epsilon-constraint method, used e.g. bySong, Park, Lend Park (2002)in the scheduling of an oil refinery,ell as the goal-programming method (e.g.Chakraborty &inninger, 2002, for plant-wide waste management) bxplicitly target at finding an approximation of the noominated set without considering the intrinsic nature oolution.

Similar approaches were presented using differentiques, ranging from mathematical programming techni

o meta-heuristics: For instanceSteffens, Fraga, and Bog1999)use a method derived from Branch and Bound andamic Programming for solving a multi-objective bioprocptimization; they aim at obtaining many candidate desnd hence compile ranked lists of the best designs forbjective.Ciric and Huchette (1993)implement a MINLPethod relying on inner and outer approximations thatet at eliminating dominated regions and hence keep

ions or into the constraint’s formulation. Relevant critehat are not considered can range, among others, fromical constraints not dealt with in the optimization to prences outside the scope of the models and objective funcxamples of such criteria in chemical process design a

Safety considerations (e.g. some safety equipmentsbe present in some units but not in others).Technical limitations (e.g. no readily available equipmfor charging solids).Economic aspects outside the scope of the methodone unit is equipped with expensive sensors and equipanother unit would cause extra costs).Short-term availability of units (e.g. due to failures orlays in another process).

Extending the panel of designs beyond the set ofominated solutions enables the inclusion of designsre robust towards such additional criteria. Indeed, Paptimal solutions might be very similar; in particular, so

ions that are close in the objective space might sufferhe same drawbacks (Uppaluri, Linke, & Kokossis, 2004).herefore, we propose a design space diversification, w

s based on the intrinsic properties of the solution, i.etructure.

In the following, an indicator to evaluate the key strural differences between solutions is presented, followe

mathematical formulation for including design-spaceersification into optimization problems. The methodoleveloped will be applied to three industrial case studieighlight its performance and scalability.

http://www.tabusearch.net/


2. Problem description

While the concept of design-space diversification pro-posed here is of general applicability in optimization prob-lems, we present the mathematical formulation on a backdropof batch process design, thus extending earlier work (Cavinet al., 2004).

2.1. Problem definition

• Given:I. Basic heuristics

a. equipment classes suited for processing each oper-ation class;

b. design heuristics (built-in ones as well as user op-tions);

c. relationships between the processing time of eachtask class and the batch size.

II. Recipea. recipe of the production process expressed as a se-

ries of chemical/physical tasks;b. capacity requirements for each task per unit of final

product;c. base duration of each task at the input scale;d. recipe constraints (specific rules about how tasks

eci-

beon a

•hedtion

e. al-odes

2

2in-

p . Ther kst sk

belongs to a given operation class (e.g.React,Distill ) and hasa base volume requirement and duration. Further informationis provided, such as the operating pressure and temperature,as well as task-specific constraints, e.g. lining material re-quirements and design limitations embodied in flags. Mul-tiple flags are defined; the ones used in the mathematicalformulation for checking constraints are: 1:transfer requiredafter operation, 4: task cannot be conducted in parallel, 8:task cannot be conducted in series.

The sequence in which the tasks must be completed isdefined in another matrixZ, which contains a list of tasksthat must be completed before the beginning of a given task.Based on the constraints and the sequence matrixZ, thedifferent tasks are condensed into blocks of tasks (matrixB) thatmustbe conducted in the same equipment unit(s),as e.g. the reaction mixture is not transferable betweenthem. The blocks in matrixB contain an aggregation of theconstraints and requirements of all tasks in the block. Anindex-matrixBR keeps track of which task belongs to whichgroup.

The description of the production line is given in a matrixE, whose rows contain each one equipment unit, with itsclass (e.g.Reactor, Packed Column) and its properties (e.g.lining material, volume, or location in the production line). Itslimiting operating conditions (temperature and pressure) aresimilarly summarized as a pointer to a matrixP containing al

rizedi yp ractc easeo nitsc d by ac

uilti e asg t ber mored t, seeC

2fin-

i l datasm ath-e

TI

M

R , Durat entsZ rowsR inE e of de es, VolumeC ws ofE a nits

can be combined).III. Equipment

a. available equipment units and their detailed spfications;

b. connectivity costs (which equipment units canconnected with each other, and at which costsscale 0 (no costs) – 10 (impossible)).

IV. One or more objective functionsDetermine:

An extended set of Pareto-optimal solutions, enricwith designs representing a broad diversity in the soluspace. Here, a solution is a layout for the process – i.locations of equipment units to tasks, and operating m(in parallel, in series).

.2. Problem formulation

.2.1. Input and data formatAs mentioned in the problem formulation, the main

uts are the recipe and the production line’s descriptionecipe is given in a matrixRcontaining in each row the taso be completed (seeTable 1). Each physico-chemical ta

able 1nput matrices and their description

atrix Subject Description

Recipe Index, Operation Class, Maximum VolumeRecipe sequence Many-to-many relationship matrix amongEquipment ID, EquipmentClass, MasterID (in the casEquipmentconnectivity

Many-to-many relationship matrix linking ro

ist of standard conditions.Feasible interconnections between units are summa

n a many-to-many relationship matrixC, which gives for anair of equipment units (as identified by their ID) an abstost for the directional connection, that represent thef building pipeworks to connect the two units. Some uannot be connected to some others, and this is pictureut-off value of 10.

Finally, heuristics are specified. Many of them are bnto the mathematical formulation of the superstructuriven below (e.g. a built-in rule states that a unit cannoeused), while others are customizable at run-time. Foretails on data input, heuristics, and data managemenavin (2002)andCavin et al. (2004).

.2.2. SuperstructureIn the following, the different methodological steps de

ng the superstructure are summarized using a relationatructure formulation, the tuple calculus (Date, 1995). Ele-ents of tuple calculus and other operators used in the mmatical formulation are defined inTable 2.

ion, Temperature, Pressure, Design Flags, Lining Material Requiremofdicating precedence rules for taskspendant units such as a condenser), Lining Material, Operating Rangnd indicating the costs (and feasibility) of physically connecting two u


Table 2Symbols and operators used in the mathematical formulation

Symbol Meaning

a.b Columnb of matrixaa|a.c= 1 Take rows ofawhere the value in columnc is 1a.b|a.c= 1 Take only columnb (same rows as above)∈ Is contained in (single element)⊃ Contains (multiple elements)∪ Union∩ Intersection← Assign a subset ofamatrix to itself (filtering)∅ Empty (no elements)·∪ Pair-wise combination∨- XOR logical operator∨ OR logical operator

Given the input discussed above, a list of potential equip-ment units (S∗i ) that can be used for each blocki is compiledusing Eqs.(1)–(9). Let i be the main counter among all op-eration blocks in the matrixB, andj be a secondary counteron matrixB.

For each blocki, feasible equipment classes are deter-mined, retrieving the operation classes of the tasks containedin the block, and using matrixA, which contains allocationrules (many-to-many relationships between operations andequipment classes) to determine eligible equipment classesU, and finally selecting fromE the eligible units, which arestored in the matrixS* (Eqs.(1)–(4)).

ki = BR.RecipeIndex|BR.BlockID = i (1)

Xi = R|R.index∈ ki (2)

Ui = A.EquipmentClassID|A.OperationClassID

⊃ ∪Xi.OperationClassID (3)

S∗i = E|E.EquipmentClassID∈Ui (4)

All units of S∗ are then filtered according to the constraintsof the different tasks in the block (Eqs.(5)–(9)).

S∗ ∗ ∗

)

S

)

S

)

S

≥ )

S

Fig. 1. Illustration of subscripts used forL (and by extension forS); (A)single unit assigned to a task, (B) several units assigned in series (see textfor further explanation).

For each blocki, a valid designL will have one or severalelements ofS∗i assigned. However, further heuristics limitthe possibilities, restricting the superstructure from the sim-ple combinatorial arrangement. For instance, equipment unitscannot be re-used. Once a unit has been emptied, it may notbe used again for the same batch. Hence, if a given unit isassigned to a block, it cannot be used in another block – ex-cept if this would result in a continuous utilization. The unitsavailable for a blocki will depend on the units already as-signed to other blocks in the designL, whereL contains unitsassigned to each block and their arrangement: single unit, inseries units (including the order of use), or units in parallel.

To handle this, a differentiation is made between units thatcan be used (as subscripts ofS) or are used (as subscript ofL) at the beginning of the block (denoted⇐, such as E0 andE1 in Fig. 1), at the end of the block (⇒, such as E0 and E3in Fig. 1), orwithin the block (=, such as E2 inFig. 1). Thedouble arrow⇔ indicates all units used in the block (such asE0, or all three units E1, E2 and E3 inFig. 1B). Units E0 orE1 inFig. 1can also be used in the previous tasks, and E0 orE3 can also be used in the following tasks. But for instanceE1 cannot be used in subsequent tasks, as this would resultin a discontinuous utilization.

Eqs.(10a) and (10b)are used to find the blocks precedingand following the current blocki by querying the sequencematrixZ.

k

k

Eu ck(i q.( locki

o lock(

i ← (Si |(P.Pmin|P.TPrangeID= Si .TP range)

≤ min(Xi.Pressure)) (5

∗i ← (S∗i |(P.Pmax|P.TPrangeID= S∗i .TP range)

≥ max(Xi.Pressure)) (6

∗i ← (S∗i |(P.Tmin|P.TPrangeID= S∗i .TP range)

≤ min(Xi.Temperature)) (7

∗i ← (S∗i |(P.Tmax|P.TPrangeID= S∗i .TP range)

max(Xi.Temperature)) (8

∗i ← (S∗i |S∗i .LiningID ∈ ∩Xi.LiningID) (9)

+ = find next block(i, Z) (10a)

− = find previousblock(i, Z) (10b)

q. (11), based on an existing assignmentL, identifies thenits that are available for use at the beginning of bloiEq.(11a)), at the end of blocki (Eq.(11b)), or within block(Eq. (11c)). Note thatL might be empty. For instance E11a)defines that the units available for the end of the bare the ones that are either unused (S∗i /⊂

⋃j �=i

Lj,⇔), or the

nes that are used at the end of the task preceding the bi⋃j ∈ k−

∣∣Lj,⇒ ), provided no transfer is required (1/∈Bj .Flag).


Si,⇐ ←S∗i

∣∣∣∣∣∣S∗i /⊂

⋃j �=i

Lj,⇔

⋃

S∗i ⊂ ⋃j ∈ k−

∣∣∣∣∣Lj,⇒ if 1 /∈ Bj.Flag

else ∅

(11a)

Si,⇒ ←S∗i

∣∣∣∣∣∣S∗i /⊂

⋃j �=i

Lj,⇔

⋃

S∗i ⊂ ⋃j ∈ k+

∣∣∣∣∣Lj,⇐ if 1 /∈ Bi.Flag

else ∅

(11b)

Si,= ←S∗i

∣∣∣∣∣∣S∗i /⊂⋃j �=i

Lj,⇔

(11c)

C∗ = (C|C.cost �= 10) (12a)

Si,⇐ ←Si,⇐

∣∣∣∣∣∣Si,⇐.EquipmentID∈ ⋃j ∈ k−

(C∗.ToEquipmentID|C∗.FromEquipmentID∈Lj,⇒)

(12b)

Si,⇒ ←Si,⇒

∣∣∣∣∣∣Si,⇒.EquipmentID∈ ⋃j ∈ k+

(C∗.FromEquipmentID|C∗.ToEquipmentID∈Lj,⇐)

(12c)

Eq. (12) checks the connectivity feasibility if transfers arerequired, e.g. between the candidate units in the beginning ofblock i and the units present at the end of the preceding tasks.Only impossible connections are ruled out, i.e. costs of 10in the connectivity matrixC. For this purpose, the subsetC*

oc uledo stst tion.

n-mi ed ins nec-t q.( Sec-t

L

W heb tingoO tr

By taking all possible combinations of decisions in Eq.(13), all legal designs can be obtained. Thus, the combina-tion of the listsS∗i and Eqs.(10)–(13)implicitly defines thesuperstructure.1

The aim is to obtain the best possible designL, or more pre-c tlyd

2

re-g in)o

12

34 t

56 top-

amet cessp rentm hereu s (i.e.

gen-e ; suchi or in-s ry.M eneverp its ist

fC containing feasible connections is extracted (Eq.(12a));onnections absent from this list are unfeasible and rut (Eqs.(12b) and (12c)). Note that the connection co

hemselves are handled separately as an objective funcFurther constraints may limit the flexibility of the assig

ent of units to a blocki, as illustrated by Eq.(13). Fornstance, one task might not be allowed to be conducteries for safety reasons. For the sake of simplicity, conivity feasibility within units in series is not pictured in E13). B·Flag contains the task-specific constraints, seeion 2.2.1.

i = one of

available if 4/∈ Bi.Flag(parallel allowed)

Two elements ofSi,⇒⋂Si,⇐

Li.DesignType= “parallel”

available if 8/∈ Bi.Flag(series allowed)

one elements ofSi,⇐+optionally any elements ofSi,=+one elements ofSi,⇒

Li.DesignType= “series”

always available :

One single element ofSi,⇒ ∩ Si,⇐Li.DesignType= “single”

(13)

ith the help of the listsS∗i , that are computed once in teginning, it is possible to obtain any valid design by iterani through Eqs.(10)–(13), and taking any option in Eq.(13).bviously, initially,Li = Ø for all blocksi, and the differen

ows ofL are filled during the iterations oni.

isely, a listD of Pareto-optimal or structurally significanifferent designs to be proposed to the decision maker.

.3. Objective functions

The problem described above will be optimized withard to multiple objective functions to be minimized (mr maximized (max):

. Production rate [kg/h] (max)

. Production rate per reactor volume [kg/h/m3 reactor](max)

. Interconnectivity costs among units [−] (min)

. Number of key equipment units [−] (including, but nolimited to, reactors) (min)

. Total number of equipment units [−] (min)

. Floors the reaction mixture must be pumped up, i.e.down indicator [–] (min)

Objectives 1 and 2 will usually not be used at the sime, as the aim is either to use the whole plant (single-prolant) or to use efficiently a subset of the plant (concurultiproduct plant). Objective 4 is necessary for plants wnused equipment units may be used for other operation

1 It is not guaranteed that any combination of arbitrary options willrate a complete design, as no units could be left for the last blocks

ncomplete designs are deemed legal but have zero productivity. If, ftance, random options are used in Eq.(13), several trials might be necessaore sophisticated methods (e.g. using units for subsequent tasks whossible) will allow generating designs even when the number of un

ight.


usually in addition to objective 2).Key or significantunitscomprise usually reactors and centrifuges, but may be cus-tomized for each project for example according to the levelof automation of each unit or to the usage of the unit. For in-stance, if a reactor is used as a simple bottom for a distillation– where it could be replaced by a tank – it may be excludedfrom this list. Objective 5 is always necessary to control thenumber of total equipment units used and discriminate infavor of simple designs. Overall, usually 5 concurrent objec-tives are considered.

3. Tabu Search

3.1. Optimization algorithm

To solve the optimization problem presented in Section2, a Tabu Search algorithm was implemented. Tabu Search(TS) has been developed byGlover (1977)and has hadnumerous successful applications in recent years. It is ameta-heuristic method using a set of coordinated strategiesfor introducing and exploiting adaptive memory, in order togenerate a sequence of solutions that contains a subsequenceof progressively improving “best solutions found”. Repet-itively, modifications of the current solution are examined,and the one resulting in the best solution is chosen for thenext iteration, even if this successor is worse than the presents them ores xist.S opti-m sg ex-a )

ple-m tegralp hasisw tiono In afi tudyw spe-c tionw

ni -b d ind orm plexc d thet e.T od iss bseto

n ino nexti cea esign

Fig. 2. Scheme of the TS algorithm.

might be worse than the previous one. This feature allowsthe escape from local minima and makes of TS a globaloptimization algorithm.

However, at the next iteration, the search must be pre-vented from falling back to the same local minimum. This isobtained by the use of a short-term memory element, the Tabulist: when a move is chosen, i.e. its resulting design is chosenas the base solution for the next step, the reverse of the moveis deemed Tabu for a limited period of time. In the next iter-ations, Tabu moves are filtered out from the neighborhood,thus effectively limiting the risk of circling around a localminimum. This Tabu criterion will however render movesleading to previously unvisited solutions illegal, which is usu-ally positive as it encourages diversification, but might leadto the exclusion of very attractive solutions. Therefore, a cri-terion called aspiration is enabled, rendering moves leadingto previously unvisited but highly attractive solutions legal.

This iterative procedure is stopped when a given termina-tion criterion is reached. This criterion is implemented as amaximum number of iterations during which no improved so-lution has been found. The iterative modification will slowlymodify the initial design. But in the case of very large optionsfor the design, or in the case of discontinuous valid regions inthe design space, a more vigorous diversification is needed.Therefore, a third loop (loop 3 inFig. 2) allows the repetitionof this whole procedure with alternate starting points. An-o ouldb xedn thea

olution. Special memory processes continue to driveethod forward to discover regions that harbor one or m

olutions better than the current best, if such solutions eome forms of TS have a proof of finite convergence toality (Glover & Hanafi, 2002), but the most effective formenerally do not. A comprehensive description of TS withmples of applications is given inGlover and Laguna (1997.

In this study, a fundamental version of TS has been imented, which does not use all features that represent inarts of more advanced versions. However, a great empas put on multi-objective optimization and on the selecf a panel of solutions proposed to the decision maker.rst phase, the features of TS as implemented in this sill be explained and formulated; in the next section theific adaptations conducted for multi-objective optimizaill be explained.The overall TS method is shown inFig. 2. Based on a

nitial base solution, TS will iteratively examine its neighors (loop 1 inFig. 2), as defined by heuristics (discusseetails in the following section) indicating which movesodifications of a design are possible. In the case of com

ase studies, the neighborhood might be very large, anesting of all neighbors would require prohibitive CPU timherefore, often only a subset of the whole neighborhoelected for testing. In our implementation, a random suf a fixed number of neighbors is selected.

The best neighbor (see below for its determinatiour implementation) is selected as base solution for the

teration (loop 2 inFig. 2), regardless of its performans compared to the actual design – that is the new d

ther criterion is used to decide when this third loop she interrupted. In this study, the criterion is usually a fiumber of restarts. For a detailed explanation on howlgorithm parameters have been set, seeCavin et al. (2004).


3.2. Formulation of the optimization

The optimization targets at optimal layouts for the pro-cess, while operating parameters are considered fixed. Threesorts of design-modification moves are defined in the presentimplementation of the Tabu Search:Addition, where an un-used unit is added to a block (either in series, or in parallel),Removal, where a unit is removed from a block, andReplace-ment, where a unit is removed from a block and replaced byan unused unit. Replacement is therefore a combination ofan addition and a removal in one single step.

The addition is subject to the connectivity constraintsand hence the possible moves are different if the new unit isadded in the beginning, within or at the end of a block. Theremoval of a unit is possible only if it is not used both in aprevious and a subsequent block because otherwise such aremoval would provoke the reuse of the unit. It is allowedto remove all units assigned to a block. This is necessaryto have enough flexibility in situations with low degrees offreedom, i.e. different unconnected regions of the solutionspace exist that will thus be linked. Of course any incompletedesign has zero productivity.

Each element in the move listM contains a unit’s ID (ortwo units’ ID in the case of replacement), as well as the blockto which it refers and the kind of move it represents in super-script (Add,s: addition in series; Add,p addition in parallel;A ce-m yE

uniti edf igni n bec

M

So as wedi

MAdd,pi =

if Li.DesignType= “single” :

if 4 /∈ Bi.Flag :

Si,⇒ ∩ Si,⇐else :

∅

else :

∅

(15)

The sum of all possible additions is given as

MAddi = MAdd,s

i ∪MAdd,pi (16)

Eq.(17)defines the elimination of any unit, except units thatare used both before and after the current block. Eliminatingsuch units would lead to a re-use of equipment.

MRemi =

Li

∣∣∣∣∣∣Li /∈

⋃j ∈ k−

Lj,⇒

∩

⋃j ∈ k+

Lj,⇐

(17)

Replacements can target at any equipment unit, and are de-fined in Eq.(18) by pair-wise combination of any additionand any elimination.

MRepli = MRem

i·∪MAdd

i (18)

T

M

F r toEo eeni ck isa

b bu listT( d int

M

T f am fa tel iono d thec ter-m mbero

3

cti-c eria

dd: all possible additions; Rem: removal; Repl: replaent). The possible moves for a base designL are defined bqs.(14)–(20).Eq.(14)defines the possible addition of an equipment

n-series to the blocki, provided in-series designs are allowor this block (Flag 8). This option does not exist if the dess in parallel, as a built-in constraint is that no design caonducted both in series and in parallel.

Add,si =

if Li.DesignType= “parallel” :

∅

if Li.DesignType= “single” :

if 8 /∈ Bi.Flag :Si,⇐ (Add at the beginning)

Si,= (Add in the middle)

Si,⇒ (Add at the end)

else :

∅

if Li.DesignType= “series” :Si,⇐ (Add at the beginning)

Si,= (Add in the middle)

Si,⇒ (Add at the end)

(14)

imilarly additions in parallel are defined in Eq.(15). Thisption, if at all (Flag 4), is only available for blocks withingle unit assigned as not more than two units are allon parallel.

he sum of all possible moves is compiled inM by Eq.(19):

=⋃i

(MAddi ∪MRem

i ∪MRepli

)(19)

or the sake of simplicity, the connectivity checks (similaq.(12)) are ignored in Eqs.(14) and (17): if a unit is addedr removed from a design in series, the connectivity betw

ts preceding and subsequent units within the same blolso checked.

As stated above, a subset of the possible moves inMmuste evaluated. First Tabu moves that are present in the Taare eliminated, and then a random subsetM◦ ofM is chosen

Eq.(20)). Note that aspiration conditions are not depictehe formulation:

◦ ← random subset of(M|M /⊂ T ) (20)

he actual designL is transformed by the application ooveMm fromM◦ into a new designLm. The application oll moves inM◦ to the actual designL creates the candida

ist Q. All designs inQ are then evaluated. The evaluatf the design comprises a simulation of the process anomputation of its resulting productivity, as well as the deination of characteristics of the design, such as the nuf units used or the total connectivity cost.

.3. Design-space diversification

Our objective is to develop a method for optimizing praal problems where typically not all relevant decision crit


are contained in the optimization formulation. As discussedabove, these criteria can range, among others, from technicalconstraints not dealt with in the optimization to economicaspects outside the scope of the method.

Considering these criteria, there is no guarantee thatPareto-optimal designs are feasible or satisfy all preferencesof the decision-maker. Therefore, we consider that it isimportant to offer a panel of solutions that is not limitedto the set of non-dominated solutions. Moreover, someobjectives may be more important than others or someobjectives may be bounded, e.g. if a minimum productivityis not obtained, the process is not profitable, and it will not beimplemented. We developed a method taking these elementsinto consideration. One key element of this approach issolution-space, i.e. design-space diversification.

Solution-space diversification is not equivalent to theobjective-space diversification, as similar designs might havedifferent objective functions’ values, while different designsmight have similar objective functions’ values (seeFig. 3).

Interestingly, diversification in the design space is at thecore of the Tabu Search method – most of the implementa-tions for memory target at such a diversification. However, inmultiple-objective implementations reported to date, regard-less of the method used, even if such a diversification wasused in the search procedure, it was not used for the finalselection of solutions.

re oft ightb intot kingu enceo oned linge largew

oft usedf factt t re-a nces

F threec -set ofs veryd r thed

Fig. 4. Three candidate designs, and their distance according to several in-dicators.

among the reactors (e.g. lining material, or related equip-ments) are not significant for this recipe. The structure ofthose two designs is very similar: the first operation is treatedin series in two reactors, the second operation is treated inanother reactor and the final operation is treated in a reactorwith a condenser. Therefore, we deem Designs #1 and #3 asbeing similar, even if they are separated by five moves (re-moval of one unit in the first operation, three replacementsand an addition) and if all their attributes are different (inevery operation, the set of used equipment is different).

Design #2, however, even if it is separated by only twomoves from Design #1, and if only one of its attributes is dif-ferent from Design #1, shows a more significant difference:the first operation is conducted in parallel instead of beingconducted in series.

The aim is to create an indicator that captures the essenceof the design, i.e. its structural composition. We chose to usean indicator taking into consideration the position and natureof the transfers. In particular, we set three Boolean indicatorsfor each operation:

• does a transfer happenduring the operation (i.e. design inseries) [H1a];• does a transfer happenafter the operation [H1b];• Is the operation conducted in parallel (which implies two

transfers each at the beginning and at the end of the oper-

gns,w st att firstis bero

mbero nce,b sa

••

ed bys

Intuitively, as the TS method is move-based, a measuhe distance, i.e. the difference between two designs, me the number of moves required to transform one design

he other. However, many moves might be spent in manimportant modifications that do not impact on the essf the design. Similarly, if we consider the attributes ofesign (which equipment is used in which way for handach operation), unimportant differences may have aeight.For example, inFig. 4, three designs are shown. In all

hem, the two first reactors, which are highlighted, areor the same operation, either in parallel, or in series. Thehat Designs #1 and #3 are both legal while using differenctors for handling the operations shows that the differe

ig. 3. Diversity in the design space and in the objective space, withandidate designs taken as example. If the aim is to obtain the Paretoolutions, Design #3 should be eliminated. However, it is structurallyifferent from the other optimal designs and might be of interest foecision maker.

ations conducted in those units) [H2].

Moreover, to discriminate less between similar desie consider that transferring during an operation, or ju

he end of the operation is rather similar, hence the twondicators are combined, to obtainH1 = H1a ∨H1b; this re-ults in having two bit-fields of length equal to the numf operations in the recipe that characterize a design.

The effective distance between two designs is the nuf differences between their characterizations. For instaetween Designs #1 and #2 inFig. 4, the characterizationre:

Design #1,H11: 1 1 0 (= 1 0 0∨1 1 0),H1

2: 0 0 0

Design #2,H21 : 1 1 0 (= 0 0 0∨1 1 0),H2

2 : 1 0 0

The resulting distance between the designs is obtainumming all the different bits with theexclusive-oroperation


as indicated in Eq.(21).

∆(L1, L2) =∑

all bits

(H11∨-H2

1)+∑

all bits

(H12∨-H2

2) (21)

This results in a distance of zero between Designs #1 and #3(as they have the same characterizations), and in a distanceof 1 between Designs #1 and #2 (only one bit is different, inH2). Larger distances will be found (in larger case studies)mostly when different groups of operations are conducted inthe same units (H1b will then show a large variability).

By extension, ifL2 in Eq. (21) is replaced by a set ofdesigns (e.g.Q), the distance obtained is the sum of the dis-tances fromL1 to each designs in the set, as defined by Eq.(22).

∆(L1,Q) =∑d ∈Q

(∆ (L1, d)) (22)

3.4. Compilation of the results list

In this section, the selection of the best design at eachiteration, and the compilation of results to obtain the ex-tended Pareto-set are presented. First, the decision makerprioritizes the objectives so that higher priority objectivestake precedence over lower priority ones. The prioritizing isu veraln s ad culari es tob . Fi-n akeri per-cr asingd

ntet thepF

alitya si en

L

T e pri-o oft -p ed.T anL

Lm < L′ iff ∃n∣∣∣∣∣∀a

{Fma<n = F ′a<nFma=n < F ′a=n

(24)

Eq. (24) can be used by analogy to determine if one designLm has the best objective function values in a collection ofdesignsQ (denotedLm≤Q).

During the optimization, at each iteration a designLm*

will be chosen according to the following criteria (Eq.(25)):

• if one single neighbor is Pareto-optimal with regard to theprimary objectives, select this neighbor;• if no or several neighbors are Pareto-optimal, select the

highest ranked (over all neighbors or over all Pareto-optimal neighbors) according to the priority defined bythe user.

Lm∗=Lm∣∣∣∣∣ if(Q ∩Π) �= ∅ : (Lm ∈Π) ∧ (Lm ≤ (Q ∩Π))

else : Lm ≤ Q(25)

To compile the results listD, we deem that a solution be-longs toD if it belongs to the non-dominated setΠ, if it isstructurally significantly different from other solutions (whileremaining in the boundsϕ for the objective function definedby the decision maker) or if it has a very high performancein the highest ranked objective functions. These criteria ared c-tD ea am-e iza-t theo

D

D

D

A ri-o dated mec w bed undsϕ anceh

heP so elyt rento then s ofpw

nsf uar-a ost

sed to discriminate between several dominated or seon-dominated designs during the optimization. It giveirection to the search towards regions that are of parti

nterest to the decision-maker. Furthermore, the objective included in the Pareto analysis have to be specifiedally, the objectives can be bounded if the decision-m

s not interested in solutions being worse than a certainentage of the actual optimum, i.e. the bounds (denotedϕ) areelative and the absolute acceptance threshold is increuring the optimization.

All designsLm in the candidate listQ are evaluated ohe basis of a process simulation and the performanceFm ofach candidate design is obtained. The vectorFm contains

he list of objective functions’ values sorted according torioritization,Fma being the value of theath objective. Withd the performance of a designd is denoted.

Each candidate design is checked for Pareto-optims indicated in Eq.(23), wherea is limited to the objective

ncluded in the Pareto-analysis. The variableΠ denotes thon-dominated set and initiallyΠ is empty:

m ∈Π iff �n

∣∣∣∣∣∀a,∀d ∈Π{Fma �=n ≥ Fda �=nFma=n > Fda=n

(23)

he candidate designs are also ranked with regard to thritized objective functions’ values. The relative ranking

wo designsLmandL′ is given in Eq.(24). For the sake of simlicity, it is considered that all objectives must be minimizhus,Lm<L′ indicates thatLm has a better performance th′.

efined in Eqs.(26)–(28). Hereδ represents a minimal struural difference that is required to include a design inD, while

1. . .λ are the topmostλ designs inD ranked by performancccording to the prioritized objectives. The algorithm partersδ andλ are adapted automatically during the optim

ion in order to have a fitting acceptation threshold duringptimisation:

← D ∪ Lm if Lm ∈Π (26)

← D ∪ Lm if(∆(Lm,D) ≥ δ) ∧ (Fm ≤ ϕ) (27)

← D ∪ Lm if(Lm ≤ d,∀d ∈D1...λ) (28)

fter the addition of a new design (or, for efficiency, pedically when the list contains a given number of candiesigns), the listDmust be maintained, as for instance soandidates that were previously Pareto-optimal may noominated after inclusion of new candidates, or as the bomust be updated as solutions with a higher perform

ave been found.The list D is refreshed by first choosing only t

areto-optimal designs inΠ, sorted by their rank in termf prioritized objectives, secondly by adding iterativ

he n1 candidates that are structurally the most diffef all designs already in the list, and finally by adding2 candidates that have the highest ranking in termrioritized objectives and are not yet inD. The order inhich designs are included intoD is also theirrank in D.The resulting listD contains the Pareto-optimal desig

ound during the optimization. These designs are not gnteed to cover the whole non-dominated set, but will m


likely contain a good approximation of the Pareto-set in thedirection defined by the prioritization. Indeed, the methoduses a prioritization of the objectives to resolve ties amongseveral non-dominated candidates or among several Pareto-optimal candidates. Therefore, a bias is generated toward theregion of the non-dominated set that maximizes the highestranked objective. This leads to potential gaps in the other re-gions but only solutions of low relevance for the user mightpossibly be missing. Longer computation times eventuallyeliminate these gaps.

4. Case studies

In this section, three case studies based on industrial pro-cesses will be discussed. A first case study will be used tohighlight the quality of the designs found. A second, rathersmall recipe will be discussed in order to highlight the mainadvantages of the design-space diversification method. Fi-nally, a third larger process will be shortly discussed in orderto show the scalability of the method. All the following casestudies deal with finding an optimal design for a batch pro-cess in a given production line. By repeating this procedurefor several production lines, the optimal line can be identified.

In all case studies, the composition of the results list is reg-ulated as follows: Include first all Pareto-optima; afterwardsa dt ives,p g oft lutioni

4

4ding

f ed int skss hilet haveb esesR eateda ual-i ion.O own( on isc ov-e 4).T

, andd ived astet om-p ed,a re isc 9).

The intermediate is separated by a multi-drop centrifugation,and the mother liquor is sent to waste-treatment (Block 10).The solid intermediate is dissolved in the solution obtainedin Block 5, and a final reaction step occurs (Block 11). Twodecantation steps follow (Blocks 12 and 13), and the finalproduct crystallizes when the resulting mixture is cooleddown (Block 14). The crystals are separated by a multi-dropcentrifugation (Block 15) and are dried (Block 16).

The longest operation is the reactive-distillation (about12 h), followed by the multi-drop centrifugations (5 respec-tively 6 drops and a duration of about 2 h per drop). Thehighest volume requirements occur during the reactive distil-lation, which requires a volume of about 13 m3, followed bythe reaction step of Block 8 and the subsequent crystallizationthat require a volume of 11 m3.

The initial design, implemented in a rather small produc-tion line with four reactors, two centrifuges and one dryer,has a production per batch of 1.59 t of product and a cycletime of about 36 h, hence a production rate of 44 kg/h.

4.1.2. Production lineThe recipe will be implemented in a large production line

containing five reactors of 10 m3, equipped with condensers(R1, R3, R4, R7), or without condenser (R8). It contains one16 m3 reactor with a condenser (R6) and two 16 m3 reac-tors equipped with a condenser and serving as bottoms ford pedt s area ffert thed an bep e dry-i ithv

er-c s ona andh

4tudy

a um-b gns.O signst timaa d int

4re-

s al hree-d hindo ex int tionsg ons

dd up to 50 (n1) structurally different designs; finally adhe 20 (n2) best designs according to the ranked objectrovided they are not already in the list. The numberin

he designs in the figures represents the rank of the son the list, as defined by the order of addition.

.1. Case Study #1

.1.1. Recipe descriptionCase Study #1 is a rather complex production inclu

our reaction steps and three solvents, that is describhe following. Note that for readability, low importance tauch as purging or cleaning a unit are not listed below whey are included in the optimization. The blocks as theyeen built by the pre-processing are indicated in parentheactants are charged in solvent S1. The mixture is hnd the solids dissolve. A reaction is conducted, and a Q

ty Control (QC) Test is carried out to determine completnce the reaction is finished, the mixture is cooled d

Block 1). A second solvent S2 is added, and an extractionducted (Block 2). The bottom layer is distilled for recry of S1 (Block 3). The upper layer is neutralized (Blockhe mixture is then heated to reflux (Block 5).

In another unit, reactants are charged in a solventissolution occurs while heating (Block 6). A reactistillation is conducted, the overhead is sent to the w

reatment; quality control is conducted to determine cletion of the reaction (Block 7). A further reactant is addnd another reaction step occurs (Block 8). The mixtuooled down, and an intermediate crystallizes (Block

.

istillation columns (R2 and R5). All reactors are equipo conduct phase separations. Four identical centrifugevailable (Z1, Z2, Z3 and Z4). All centrifuges have buanks, thus freeing the preceding equipment unit duringrops. One dryer is also available. Several batches crocessed at the same time in the dryer, and hence th

ng operation cannot be time-limiting. Numerous tanks wolumes ranging from 8 to 25 m3 are available.

While the production line is quite flexible, some intonnections between units are difficult. Each reactor haverage low cost connections with five other reactors,igh connectivity costs to the other three.

.1.3. Objective functions and selection criteriaThe prioritized objectives considered in this case s

re: (1) production rate, (2) connectivity cost, (3) total ner of units, and (4) an indicator favoring top-down desibjectives 1–3 are included in the Pareto analysis. De

hat have the most different structure than the Pareto opnd at least 50% of the optimum productivity are include

he results list.

.1.4. Results and discussionThe characteristics of the designs included in the

ults list are displayed inFig. 5. The squares linked byine are Pareto-optima, and the line approximates the timensional Pareto-front. All dominated designs are ber above this front. Note that this Pareto-surface is conv

hree dimensions even though two dimensional projecive the impression of non-convexity. If only two dimensi


Fig. 5. Case Study #1: characteristics of the 70 designs included in the resultslist. The Pareto-optimal designs are marked with a square. The dotted lineis the Pareto-front if only two objectives (Prod. RateandNumb. Eq. Units)are taken into consideration.

are considered, some points are no longer Pareto-optima asshown with the dotted line inFig. 5.

The results (seeFig. 5) reveal that it is possible to triplethe production rate with only 10 units as compared to the 7units of the initial design, and that using 14 units leads tothe optimal production rate (Design #1 inFig. 5). Designsusing less than 10 units are filtered out because they obtainless than 50% of the optimal productivity. There is a broadrange of non-dominated or structurally diverse designs with aproduction rate close to the boundary of 50% of the optimum(Design #1), while the designs with a higher productivity areuse 14 or more units. Using 15 units does result in any furtherimprovement.

Design #1, which is the best in terms of ranked objec-tives, is depicted inFig. 6. The details of the assignments aregiven in Table 3. The quality of the design can be verifiedby plotting the occupancy times for all units. As can be seen

Table 3Case Study #1: candidate Design #1

Operation blocks Equipment units in Design #1

1 R8 in series with R32 R33 R14 R35 R36 R27 R2 in series with R58 R69 R6

10 Z2 in parallel with Z411 R712 R713 R414 R415 Z1 in parallel with Z316 Dryer

in Fig. 7, the occupancy if roughly equivalent for all units,which indicates efficient equipment utilization. This also in-dicates that there is little flexibility for structural changes inthis design: modifying the position of the transfers will oftendirectly impact on the cycle time and hence the productivity.

Indeed, all candidate designs with a production rate largerthan 150 kg/h have a layout very similar to Design #1. In par-ticular, the reactive-distillation is always conducted in seriesand this results in a high production rate. All this informationindicates that most of the solution space consists of designswith a rather low production rate (less than 140 kg/h) andthat the high-quality designs are all located in a quite narrowrange of structural diversity. Despite these difficulties, theTS found Design #1 in less than 10 min. Structurally differ-ent designs found by the diversification method have a ratherpoor performance (see e.g. Design #9 inFig. 5). This givesan insight on the loss in production rate to be expected if forsome reason the better designs are not feasible.

Design
Fig. 6. Case Study #1: layout of #1. SeeTable 3for more information.


Fig. 7. Case Study #1: equipment utilization for each unit in Design #1.

4.2. Case Study #2

4.2.1. Recipe descriptionCase Study #2 is a short synthesis with three reaction steps

without solvent change. Below the description of the recipeis given. Operation blocks (matrixB) to be conducted in thesame equipment unit as determined by the automated pre-processing routine are indicated in parenthesis. The resultingblocks are as follows: After charging the reactants, a first re-action step is conducted, followed by a QC-test (Block 1);due to the chemicals involved, Block 1 must be processedin an enamel-lined reactor. Other reactants are charged andheated separately, and the mixture is stirred to dissolve solidcomponents (Block 2). The two mixtures are combined, anda second reaction step follows (Block 3). The mixture is sep-arated by low-pressure distillation (Block 4). The overheadgoes to waste treatment. Two more components are addedto the bottoms, and after heating, the third reaction step oc-curs (Block 5). When a positive QC-test indicates reactioncompletion, the mixture is cooled down slowly and the de-sired product crystallizes (Block 6). A multi-drop centrifugeseparates the mother liquor from crystals (Block 7) that aresubsequently dried (Block 8).

The volume limiting operations in the initial recipe areBlocks 5 and 6 each requiring about 9 m3. The time limit-ing operation is the third reaction step (Block 5) in whicht etedy h.S , theb indi-c start-i ins

4tors

( ee ocesst mel-l Two

centrifuges CE01 and CE02 are available, as well as one dryer(DY01). The dryer can be used for drying multiple batches atthe same time, and hence drying is not time limiting and thedryer is not a significant piece of equipment. Numerous tanksare available which are used for handling for instance thecontinuous output flow of the distillation and will not explic-itly be listed in the designs discussed below. The productionline is built on five floors (vertical scale inFig. 10), and allinterconnections are possible among the different equipmentunits.

4.2.3. Objective functions and selection criteriaThe prioritized objectives considered in this case study

are: (1) production rate, (2) number of significant units, (3)total number of units, and (4) an indicator favoring top-downdesigns. Only objectives (1) and (2) are included in the Paretoanalysis. Designs that have a significantly different structurethan the Pareto optima and at least 50% of the optimum pro-ductivity are also included in the final results list.

4.2.4. Results and discussionThe results show that Blocks 2 and 3 can be conducted in

the same unit without becoming time limiting and hence thefive reactors available are sufficient to process the first sixblocks without reduction of the production rate.

edureo e ares ichw tingp moreu duc-t froma d oft

i.e.l nitsa ctionr ni in

he reaction requires more than 7 h for reaching the targield. The total duration of Block 5 is slightly above 10upposing that each block is processed in its own unitase production rate is 188.4 kg/h. The flow-analysisates that Blocks 1 and 2 must be completed beforeng Block 3, while the remaining blocks must followeries.

.2.2. Plant descriptionThe relevant units of the production line are five reac

RE01-RE05), with a volume of 10 m3 each. All reactors arquipped with condensers and thus can be used to pr

he low-pressure distillation. RE01 and RE03 are enained, while the other reactors are plain stainless steel.

For 70 designs that have been selected by the procutlined above, the performances in the objective spachown inFig. 8. The third dimension is the Batch Size, whas neither an objective function nor a criterion for selecromising designs. Some designs, like Design #3, usenits (e.g. the second centrifuge) while having good pro

ion rates, some other designs are structurally differentll others with production rates within the specified boun

he optimal production rate.The optimal design in terms of prioritized objectives (

argest production rate, followed by smallest number of und best top-down design) is Design #1. It has a produate of 278.0 kg/h and is shown inFig. 9(left plot). The desigs summarized inTable 4. Block 5 is spread on two units


Fig. 8. Case Study #2: characteristics of the 70 designs included in the resultslist. The first five designs in the list are highlighted with circles while thesquares and the line linking the first two represents the Pareto-front in termsof the number of units and the production rate.

Table 4Case Study #2: description of Designs #1 and #4

Operationblocks

Equipment unitsin Design #1

Equipment units in Design #4

1 RE01 RE042 RE04 RE01 in parallel with RE033 RE04 RE01 in parallel with RE034 RE03 RE01 in parallel with RE035 RE02 in series with RE05 RE02 in parallel with RE056 RE05 RE02 in parallel with RE057 CE01 CE028 DY01 DY01

series, sharing the second one with Block 6. This results in asignificantly shorter cycle time.

Only one other design (Design #2) is Pareto optimal withregard to the production rate and the number of significantunits used. This design has a production rate of 175.0 kg/h.The optimal reuse of the units among multiple blocks needsone reactor less as compared to Design #1 but at the sametime it allows for only 62% of the optimal production rate.

Fig. 10. Case Study #3: characteristics of the 70 designs in the results list.The Pareto-optimal designs are marked with a circle and a square.

If we had not considered a cut-off of 50% of the optimalproduction rate, other Pareto optima could have been found,down to using only two reactors (lower limit as Blocks 1 and2 have to be conducted in parallel) plus one centrifuge andthe dryer.

The designs that are structurally most different from thePareto-optima are labeled #3, #4 and #5 inFig. 8. It is inter-esting to note that no design with large batch size is Pareto-optimal, but several of them have been stored in the results listD as they are structurally different from the Pareto optima.

Design #4 (Fig. 9, right plot), for instance, is structurallyfundamentally different from both Pareto optima. It has aproduction rate of 252.4 kg/h and instead of running Block 5in series, Blocks 2–6 are conducted in parallel in two units,thus allowing a doubling of the batch size. Such a dominateddesign might nevertheless be considered, for instance, it islater observed that transferring the reaction mixture duringBlock 5 causes safety problems or if final quality tests areextensive and expensive and thus the savings in QC-tests thatare conducted for each batch compensate for the reducedproductivity. Overall this dominated design might possibly

Fig. 9. Case Study #2: left: optimal Design #1;
right: Design #4. SeeTable 4for a description.


Fig. 11. Case Study #3: Design #1 which shows the highest production rate.

offer a more feasible solution for industrial practice than thePareto optimal Design #2.

4.3. Case Study #3

Case Study #3 is a long recipe containing more than 100tasks building 31 operation blocks. It contains four reactionsteps and five solvents are used. It contains extensive prepara-tion of reactants (e.g. slow dissolutions that can be conductedin tanks), and several complex purification procedures, suchas cascades of distillations and three separate crystallization-centrifugation steps. The same plant as the one described inCase Study #1 will be used for the implementation of thisrecipe.

The prioritized objectives considered in this case studyare: (1) production rate, (2) number of significant units, these

being reactors and centrifuges, (3) cost of interconnectivi-ties, (4) total number of units, and (5) an indicator favoringtop-down designs. Objectives 1–3 are included in the Paretoanalysis. Designs that have a significantly different structurethan the Pareto optima and at least 50% of the optimum pro-ductivity are also included in the final results list.

As shown inFig. 10, feasible designs are found using 8important units only, as well as a couple of tanks for the prepa-ration of reactants. The best design (Design #1) in terms ofproduction rate is sketched inFig. 11. It uses 10 significantunits (one reactor, being used exclusively as distillation bot-tom is not included in the count), as well as a dryer and threetanks.

The diversification algorithm provides additional designsextending the Pareto set. For example Design #7 (seeFig. 12)is dominated because its production rate is 1.2% lower than

F artially lume. Tr before n).A

ig. 12. Case Study #3: dominated Design #2. Reactor R8 is used pesults in the connections R3→R8 (transferring half of the content of R3fter further processing, the mixture is sent to Z2 and Z3.

in parallel with R3 for an extraction requiring temporarily a larger vohisthe extraction), and R8→R3 (joining the two halves again after the extractio


the one obtained in Design #1, it uses two significant unitsmore and it has higher interconnection costs. However, thisdesign has other advantages because it does not use any tankfor the preparation of reactants, which makes it probably saferas reactors are always stirred and have better sensors andcontrol capacities. It also does not run any operation in series,as opposed to Design #1. Moreover, its batch size is 25%larger than Design #1, which might reduce quality controlefforts as discussed for Case Study #2.

5. Conclusions

Our study has focused on a new method for optimizingthe implementation of a new chemical process in a multi-purpose batch plant, in which a well-defined set of equip-ment units is available. The optimization considers severalobjectives with different priorities and bounds. The aim isto obtain a panel of candidate designs as a basis for furtherselection by the decision-maker, including, but not limitedto, the non-dominated set. Our approach embodies a spe-cialized version of Tabu Search to solve this multi-objectiveoptimization problem.

A method for compiling the extended Pareto-set has beenpresented. This method considers a diversification in the so-lution space in form of a specific indicator that measures thes itiona ersi-fi ionsr withp d int max-i olu-t

stratt ap-p hourp

ex-p ents implyi ra-t witha

n int sefuli n then ersusr es ift withr

imiza-t ringt SAa tings ods

that have the advantage of a proof of optimality might proveinteresting and challenging.

References

Balasubramanian, J., & Grossmann, I. E. (2003). Scheduling optimizationunder uncertainty – an alternative approach.Computers and ChemicalEngineering, 27(4), 469–490.

Cavin, L. (2002). A systematic approach for multi-objective process de-sign in multi-purpose batch plants, Ph.D. Thesis ETHZ #14898, ISBN:3-906734-31-5; http://e-collection.ethbib.ethz.ch/show?type=diss&nr=14898.

Cavin, L., Fischer, U., Glover, F., & Hungerbuhler, K. (2004). Multi-objective process design in multi-purpose batch plants using a TabuSearch optimization algorithm.Computers and Chemical Engineering,28-4, 459–478.

Chakraborty, A., & Linninger, A. A. (2002). Plant-wide waste manage-ment. 1. Synthesis and multiobjective design.Industrial and Engi-neering Chemistry Research, 41, 4591–4604.

Ciric, A. R., & Huchette, S. G. (1993). Multiobjective optimization ap-proach to sensitivity analysis: Waste treatment costs in discrete pro-cess synthesis and optimization problems.Industrial and EngineeringChemistry Research, 32, 2636–2646.

Dantus, M. M., & High, K. A. (1999). Evaluation of waste mini-mization alternatives under uncertainty: A multiobjective optimiza-tion approach.Computers and Chemical Engineering, 23, 1508–1793.

Date, C. J. (1995).An Introduction to Database Systems(6th ed.). NewYork: Addison/Wesley.

E gra-

G gate

G ence.

G -

G ulti-e-

H l op-

H ulti-s

K jec-tede-

L st-sizes.elop-

M ndd

P ulti--

R ulti-

S ctualwith

tructural diversity among solutions focusing on the posnd nature of the transfers. It has been shown why divcation in the design space is relevant to obtain solutepresenting a robust set of alternatives in order to dealotential constraints and limitations that are not covere

he problem formulation. Such an extended Pareto-setmizes the probability that at least one of the proposed sions will be suitable for industrial implementation.

Three case studies have been presented to demonhe benefit of considering a design-space diversificationroach. Rather small computation times of about oneer case study were needed to obtain these results.

In the present version of the methodology, there is nolicit search for structural difference. Structurally differolutions that are encountered during the search are sncluded in the results list. A promising option is the integion of intensification and diversification strategies, e.g.long-term frequency-based memory.Moreover, some other indicators for the diversificatio

he design space might be investigated and might prove un some cases. For instance, taking into consideratioature of the equipment assigned to the tasks (e.g. tank veactor), might enhance the differentiation in some reciphe class of unit used can play an important role, e.g.egard to safety considerations.

The strategies presented here can be used in any option algorithm that visits a large number of solutions duhe global search for an optimum. In particular, GA andre likely candidates for this enhancement. Implemenimilar strategies with mathematical optimization meth

e

hrgott, M., & Gandibleux, X. (2000). A survey and annotated bibliophy of multiobjective combinatorial optimization.OR Spectrum, 22,425–460.

lover, F. (1977). Heuristics for integer programming using surroconstraints.Decision Science, 8, 156.

lover, F., & Hanafi, S. (2002). Tabu Search and Finite ConvergDiscrete Applied Mathematics, 119-1, 3–36.

lover, F., & Laguna, M. (1997).Tabu Search. Kluwer Academic Publishers.

rossmann, I. E., & Sargent, R. W. H. (1979). Optimum design of mpurpose chemical plants.Industrial and Engineering Chemistry Rsearch Development, 18, 343–348.

ansen, M. P. (2000a). Tabu Search for multiobjective combinatoriatimization: TAMOCO.Control and Cybernetics, 29-3, 799–818.

ansen, M. P. (2000b). Experiments concerning hashing in the mobjective Tabu Search method TAMOCO.Control and Cybernetic,29-3, 789–798.

asat, R. B., Kunzru, D., Saraf, D. N., & Gupta, S. K. (2002). Multiobtive optimization of industrial FCC units using Elitist NondominaSorting Genetic Algorithm.Industrial and Engineering Chemistry Rsearch, 41, 4765–4776.

oonkar, Y., & Robinson, J. D. (1970). Minimization of capital invement for batch processes. Calculation of optimum equipmentIndustrial and Engineering Chemistry Process Design and Devment, 9, 625.

auderli, A., & Rippin, D. W. T. (1979). Production planning ascheduling for multi-purpose batch chemical plants.Computers anChemical Engineering, 3, 199–206.

apageorgaki, S., & Reklaitis, G. V. (1990). Optimal design of mpurpose batch plants. 1. Problem formulation.Industrial and Engineering Chemistry Research, 29, 2054–2062.

ippin, D. W. T. (1983). Design and operation of multiproduct and mpurpose batch chemical plants. An analysis of problem structure.Com-puters and Chemical Engineering, 7, 463–481.

ong, J., Park, H., Lee, D.-Y., & Park, S. (2002). Scheduling of asize refinery processes considering environmental impacts


multiobjective optimization.Industrial and Engineering ChemistryResearch, 41, 4794–4806.

Sparrow, R. E., Forder, G. J., & Rippin, D. W. T. (1975). The choice ofequipment sizes for multiproduct batch plants. Heuristics vs. branchand bound.Industrial and Engineering Chemistry Process Design andDevelopment, 14, 197–207.

Steffens, M. A., Fraga, E. S., & Bogle, I. D. L. (1999). Multi-criteria process synthesis for generating sustainable and economicbioprocesses.Computers and Chemical Engineering, 23, 1455–1467.

Suhami, I., & Mah, R. S. H. (1982). Optimal design of multi-purposebatch plants.Industrial and Engineering Chemistry Process Designand Development, 21, 94–100.

Takamatsu, T., Hashimoto, I., & Hasebe, S. (1982). Optimal design andoperation of a batch process with intermediate storage tanks.Industrialand Engineering Chemistry Process Design and Development, 21,431–440.

Uppaluri, R. V. S., Linke, P., & Kokossis, A. C. (2004). Synthesis andoptimization of gas permeation membrane networks.Industrial andEngineering Chemistry Research, 43, 4305–4322.

Voudouris, V. T., & Grossmann, I. E. (1992). Mixed-integer linearprogramming reformulations for batch process design with discreteequipment sizes.Industrial and Engineering Chemistry Research, 31,1315–1325.

Wang, C., Quan, H., & Xu, X. (1999). Optimal design of multiproductbatch chemical processes using Tabu Search.Computers and ChemicalEngineering, 23, 427–437.

Wellons, M. C., & Reklaitis, G. V. (1989). Optimal schedule generationfor a single-product production line. I. Problem formulation.Comput-ers and Chemical Engineering, 13, 201–212.

Wellons, M. C., & Reklaitis, G. V. (1991). Scheduling of multipurposebatch chemical plants. 1. Formation of single-product campaigns.In-dustrial and Engineering Chemistry Research, 30, 671–688.

Yang, S., Suming, X., Ni, P., & Ho, S. L. (2002). A Tabu-based al-gorithm for multiobjective optimizations of electromagnetic devices.International Journal of Applied Electromagnetics and Mechanics, 16,207–214.

Yeh, N. C., & Reklaitis, G. V. (1987). Synthesis and sizing ofbatch/semicontinuous processes: single product plants.Computers andChemical Engineering, 11, 639–654.

batch process optimization in a multipurpose plant using tabu search with a design-space...

Documents