Research ArticleA Novel Algorithm Combining Finite State Method and GeneticAlgorithm for Solving Crude Oil Scheduling Problem

Qian-Qian Duan Gen-Ke Yang and Chang-Chun Pan

Department of Automation and Key Laboratory of System Control and Information Processing Shanghai Jiao Tong UniversityMinistry of Education of China Shanghai 200240 China

Correspondence should be addressed to Gen-Ke Yang sjtu1019163com and Chang-Chun Pan 390635304qqcom

Received 4 October 2013 Accepted 4 December 2013 Published 18 February 2014

Academic Editors Q Cheng and J Yang

Copyright copy 2014 Qian-Qian Duan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A hybrid optimization algorithm combining finite state method (FSM) and genetic algorithm (GA) is proposed to solve the crudeoil scheduling problem The FSM and GA are combined to take the advantage of each method and compensate deficiencies ofindividual methods In the proposed algorithm the finite state method makes up for the weakness of GA which is poor at localsearching ability The heuristic returned by the FSM can guide the GA algorithm towards good solutions The idea behind thisis that we can generate promising substructure or partial solution by using FSM Furthermore the FSM can guarantee that theentire solution space is uniformly covered Therefore the combination of the two algorithms has better global performance thanthe existing GA or FSM which is operated individually Finally a real-life crude oil scheduling problem from the literature is usedfor conducting simulation The experimental results validate that the proposed method outperforms the state-of-art GA method

1 Introduction

In recent years refineries have to explore all potential cost-saving strategies due to intense competition arising fromfluctuating product demands and ever-changing crude pricesScheduling of crude oil operations is a critical task in theoverall refinery operations [1ndash3] Basically the optimiza-tion of crude oil scheduling operations consists of threeparts [4] The first part involves the crude oil unloadingmixing transferring and multilevel crude oil inventorycontrol process The second part deals with fractionationreaction scheduling and a variety of intermediate producttanks control The third part involves the finished productblending and distributing process In this paper we focuson the first part as it is a critical component for refineryscheduling operations Scheduling of crude oil problem isoften formulated as mixed integer nonlinear programming(MINLP)models [2 5 6]The solution approaches for solvingMINLP can be roughly divided into two categories [7]deterministic approaches and stochastic approaches Somedeterministic methods have been available for many years[8] These methods require the prior step of identification

and elimination of nonconvexity and decompose the MINLPmodels into relevant nonlinear programming (NLP) andmixed integer linear programming (MILP) and then thesesubproblems have to be iteratively solvedThemost commonalgorithms are branch and bound [9] outer-approximation[10] generalized benders decomposition [11] and so forthAlso some commercial MINLP solvers have been developedfor solving the problem at hand optimally [12] Howeverthe commercial solver can only handle MINLPs with specialproperties The other stream of global optimization is thestochastic algorithms for example simulated annealing (SA)GA and their variants [7] GA proposed by Holland [13]because of their simple concept easy scheme and the globalsearch capability independent of gradient information havebeen developed rapidly Much other attention is given tothe development of GA for MINLP For instance Yokotaet al developed a penalty function that is suitable for solvingMINLP problems [14] Costa and Oliveira also implementedanother type of penalty function to solve various MINLPproblems including industrial-scale problems [15]They alsonoted that the evolutionary approach is efficient in termsof the number of function evaluations and is very suitable

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 748141 11 pageshttpdxdoiorg1011552014748141

2 The Scientific World Journal

to handle the difficulties of the nonconvexity Going onestep further some mixed coding methods were proposedwhich include mixed-coding genetic algorithm [15] andinformation-guided genetic algorithm (IGA) Ponce-Ortegaet al [16] proposed a two-level approach based on GA tooptimize the heat exchanger networks (HENs) The outerlevel is used to perform the structural optimization for whicha binary GA is used Bjork and Nordman [17] showed thatthe GA is very suitable to solve a large-scale heat exchangernetwork

Obviously the two different approaches previously dis-cussed have their own advantages and disadvantages Onthe one hand a deterministic approach usually involvesconsiderable algebra and undeviating analysis to the problemitself whereas the evolutionary approach does not have thisproperty On the other hand some deterministic approachessuch as mathematical programming usually cannot providepractical solutions in reasonable time whereas the evolution-ary approach can generate satisfying solutions In this worka novel genetic algorithm which combined the finite statemethod and GA is proposed to solve crude oil schedulingproblem A MINLP model is formulated based on thesingle-operation sequencing (SOS) time representation Adeterministic finite automation (DFA) model which capturesvalid possible schedule sequences is constructed based on thesequencing rules The initialization and mutation operationof GA is based on the model which builds legal schedulescomplying with sequencing rules and operation conditionThus the search space of the algorithm is substantiallyreduced as only legal sequence is explored The rest of thepaper is organized as follows the MINLP model is specifiedin Section 2 Section 3 reviews the background of finitestate theory In Section 4 a novel genetic algorithm whichcombined the finite statemethod andGA is proposed to solvethe MINLP model A test problem is studied to verify ourapproach in Section 5 In the last section conclusive remarksare given

2 Mathematic Model

In this section the MINLP model of refinery crude oilscheduling problem is described [18] This problem has beenwidely studied from the optimization viewpoint since thework of Lee et al [19] It consists of crude oil unloadingfrom marine vessels to storage tanks transfer and blendingbetween tanks and distillation of crude mixtures The goalis to maximize profit and meet distillation demands for eachtype of crude blend (eg low sulfur or high sulfur blends)while satisfying unloading and transfer logistics constraintsinventory capacity limitations and property specifications foreach blend The logistics constraints involve nonoverlappingconstraints between crude oil transfer operations

21 Sets The following sets will be used in the model

(i) 119879 = 1 119899 is the set of priority-slots(ii) 119882 is the set of all operations119882 ≜ 119882

119880cup119882119879cup119882119863

(iii) 119882119880sub 119882 is the set of unloading operations

(iv) 119882119879sub 119882 is the set of tank-to-tank transfer opera-

tions

(v) 119882119863sub 119882 is the set of distillation operations

(vi) 119877 is the set of all operations 119877 = 119877119881cup 119877119878cup 119877119862cup 119877119863

(vii) 119877119881sub 119877 is the set of vessels

(viii) 119877119878sub 119877 is the set of storage tanks

(ix) 119877119862sub 119877 is the set of charging tanks

(x) 119877119863sub 119877 is the set of distillation units

(xi) 119868119903sub 119882 is the set of inlet transfer operations on

resource 119903

(xii) 119874119903sub 119882 is the set of outlet transfer operations on

resource 119903

(xiii) 119862 is the set of products (ie crudes)

(xiv) 119870 is the set of product properties (eg crude sulfurconcentration)

22 Parameters Parameters used in the paper are definedbelow

(i) 119867 is the scheduling horizon

(ii) [119881119905V 119881119905V ] are bounds on the total volume transferredduring transfer operation 119881 in all instances 119881119905V = 0

for all operations except unloading for which119881119905V = 119881119905Vis the volume of crude in the marine vessel

(iii) [119873119863 119873119863] are the bounds on the number of distilla-

tions

(iv) [119865119877V 119865119877V] are flow rate limitations for transfer oper-ation V

(v) 119878119903is the arrival time of vessel 119903

(vi) [119909V119896 119909V119896] are the limits of property 119896 of the blendedproducts transferred during operation V

(vii) 119909119888119896is the value of the property 119896 of crude 119888

(viii) [119871119905119903 119871119905119903] are the capacity limits of tank 119903

(ix) [119863119903 119863119903] are the bounds of the demand on products

to be transferred out of the charging tank 119903 during thescheduling horizon

(x) 119866119888is the gross margin of crude 119888

23 Variables

231 Assignment Variables

119885119894V isin 0 1 119894 isin 119879 V isin 119882 (1)

119885119894V = 1 if operation V is assigned to priority-slot 119894 119885

119894V = 0

otherwise

The Scientific World Journal 3

232 Time Variables

119878119894V ge 0 119863

119894V ge 0 119894 isin 119879 V isin 119882 (2)

119878119894V is the start time of operation V if it is assigned to priorityslot 119894 119878

119894V = 0 otherwise119863119894V is the duration of operation V if it is assigned to

priority slot 119894119863119894V = 0 otherwise

233 Operation Variables

119881119905

119894V ge 0 119881119894V119888 ge 0 119894 isin 119879 V isin 119882 119888 isin 119862 (3)

119881119905

119894V is the total volume of crude transferred during operationV if it is assigned to priority slot 119894 119881119905

119894V = 0 otherwise119881119894V119888 is the volume of crude 119888 transferred during operation

V if it is assigned to priority slot 119894 119881119894V119888 = 0 otherwise

234 Resource Variables

119871119905

119894119903 119871119894119903119888 119894 isin 119879 119903 isin 119877 119888 isin 119862 (4)

119871119905

119894119903is the total accumulated level of crude in tank 119903 isin 119877

119878cup119877119862

before the operation was assigned to priority-slot 119894119871119894119903119888

is the accumulated level of crude 119888 in tank 119903 isin 119877119878cup119877119862

before the operation was assigned to priority-slot 119894

24 Objective Function The objective is to maximize thegross margins of the distilled crude blends Let 119866

119888be the

individual gross margin of crude 119888

maxsum119894isin119879

sum

119903isin119877119863

sum

Visin119868119903

sum

119888isin119862

119866119888sdot 119881119894V119888 (5)

25 General Constraints It should be noted that the crudecomposition of blends in tanks is tracked instead of theirpropertiesThe distillation specifications are later enforced bycalculating a posteriori the properties of the blend in termsof its composition For instance in the problem a blendcomposed of 50 of crude A and 50 of crude B has a sulfurconcentration of 0035 which does not meet the specificationfor crude mix X nor for crude mix Y

251 Assignment Constraints In the SOS model exactly oneoperation has to be assigned to each priority slot

sum

Visin119882119885119894V = 1 119894 isin 119879 (6)

252 Variable Constraints Variable constraints are given bytheir definitions Start time duration and global volumevariables are defined with big-119872 constraints

119878119894V + 119863119894V le 119867 sdot 119885

119894V 119894 isin 119879 V isin 119882

119881119905

119894V le 119881119905

V sdot 119885119894V 119894 isin 119879 V isin 119882

119881119905

119894V ge 119881119905

V sdot 119885119894V 119894 isin 119879 V isin 119882

(7)

Crude volume variables are positive variables whose sumequals the corresponding total volume variable

sum

119888isin119862

119881119894V119888 = 119881

119905

119894V (8)

Total and crude level variables are defined by adding tothe initial level in the tank all inlet and outlet transfer volumesof operations of higher priority than the considered priorityslot

119871119905

119894119903= 119871119905

0119903+ sum

119895isin119879119895lt119894

sum

Visin119868119903

119881119905

119894V minus sum

119895isin119879119895lt119894

sum

Visin119874119903

119881119905

119894V

119894 isin 119879 119903 isin 119877

(9)

119871119894119903119888= 119871119900119903119888

+ sum

119895isin119879119895lt119894

sum

Visin119868119903

119881119894V119888 minus sum

119895isin119879119895lt119894

sum

Visin119874119903

119881119894V119888

119894 isin 119879 119903 isin 119877 119888 isin 119862

(10)

253 Sequencing Constraints Sequencing constraintsrestrict the set of possible sequences of operationsCardinality and unloading sequence constraints are specificcases of sequencing constraints More complex sequencingconstraints will also be discussed later

254 Cardinality Constraint Each crude oil marine vesselhas to unload its content exactly oncesum

119894isin119879sumVisin119874119903 119885119894V = 1 119903 isin

119877119881The total number of distillation operations is bounded by

119873119863and119873

119863in order to reduce the cost of CDU switches

119873119863le sum

119894isin119879

sum

Visin119882119863

119885119894V le 119873119863 (11)

255 Unloading Sequence Constraint Marine vessels have tounload in order of arrival to the refinery Considering twovessels 119903

1 1199032isin 1198771198811199031lt 1199032signifies that 119903

1unloads before 119903

2

sum

119895isin119879119895lt119894

sum

Visin1198741199032

119885119895V + sum

119895isin119879119895ge119894

sum

Visin1198741199031

119885119895V le 1 (12)

256 Scheduling Constraints Scheduling constraints restrictthe values taken by time variables according to logistics rules

257 Nonoverlapping Constraint A nonoverlapping con-straint between two sets of operations119882

1sub 119882 and119882

2sub 119882

states that any pair of operations (V1 V2) sub 119882

1times1198822must not

be executed simultaneouslyUnloading operations must not overlap

sum

Visin119882119880

(119878119894V + 119863119894V) le sum

Visin119882119880

119878119895V + 119867 sdot (1 minus sum

Visin119882119880

119885119895V)

119894 119895 isin 119879 119894 lt 119895

(13)

4 The Scientific World Journal

Inlet and outlet transfer operations on a tank must notoverlap

sum

Visin119868119903

(119878119894V + 119863119894V) le sum

Visin119874119903

119878119895V + 119867 sdot (1 minus sum

Visin119874119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119878cup 119877119862

sum

Visin119874119903

(119878119894V + 119863119894V) le sum

Visin119868119903

119878119895V + 119867 sdot (1 minus sum

Visin119868119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119878cup 119877119862

(14)

Although we do not consider crude settling in storagetanks after vessel unloading it could be included in themodelwith amodified version of constraint (14) taking into accounttransition times We define TR

119881as the transition time after

unloading operation V isin 119882119880

and TR as the maximumtransition time TR = maxVisin119882119880TR119881

sum

Visin119868119903

(119878119894V + 119863119894V + TRV sdot 119885119894V)

le sum

Visin119874119903

119878119895V + (119867 + TR) sdot (1 minus sum

Visin119874119903

119885119895V)

(15)

Constraint (15) is valid in the four possible cases

(existV1isin 119868119903 119885119894V1 = 1)

and (existV2isin 119874119903 119885119895V2 = 1) 997904rArr 119878

119894V + 119863119894V1 + TRV1 le 119878119895V2

(existV1isin 119868119903 119885119894V1 = 1)

and (⋁ V2isin 119874119903 119885119895V2 = 1) 997904rArr 119878

119894V + 119863119894V1 le 119867 + TR minus TRV1

(⋁ V1isin 119868119903 119885119894V1 = 0)

and (existV2isin 119874119903 119885119895V2 = 1) 997904rArr 0 le 119878

119895V2

(⋁ V1isin 119868119903 119885119894V1 = 0)

and (⋁ V2isin 119874119903 119885119895V2 = 0) 997904rArr 0 le 119867 + TR

(16)

A tank may charge only one CDU at a time

sum

Visin119874119903

(119878119894V + 119863119894V) le sum

Visin119874119903

119878119895V + 119867 sdot (1 minus sum

Visin119874119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119862

(17)

A CDUmay be charged by only one tank at a time

sum

Visin119868119903

(119878119894V + 119863119894V) le sum

Visin119868119903

119878119895V + 119867 sdot (1 minus sum

Visin119868119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119863

(18)

To avoid schedules in which a transfer is being performedtwice at a time thus possibly violating the flow rate limita-tions constraint (19) is included in the model

119878119894V + 119863119894V le 119878119895V + 119867 sdot (1 minus 119885

119895V) 119894 119895 isin 119879 119894 lt 119895 V isin 119882(19)

258 Continuous Distillation Constraint It is required thatCDUs operate without interruption As CDUs perform onlyone operation at a time the continuous operation constraintis defined by equating the sum of the duration of distillationsto the time horizon

sum

119894isin119879

sum

Visin119868119903

119863119894V = 119867 119903 isin 119877

119863 (20)

259 Resource Availability Constraint Unloading of crudeoil vessels may start only after arrival to the refinery Let 119878

119903

be the arrival time of vessel 119903

119878119894V ge 119878119903 sdot 119885119894V 119894 isin 119879 119903 isin 119877V V isin 119874119903 (21)

2510 Operation Constraints Operation constraints restrictthe values taken by operation and time variables according tooperational rules

2511 Flow Rate Constraint The flow rate of transfer opera-tion V is bounded by FRV and FRV

FRV sdot 119863119894V le 119881119905

119894V le FRV sdot 119863119894V 119894 isin 119879 V isin 119882 (22)

2512 Property Constraint The property 119896 of the blendedproducts transferred during operation V is bounded by119909V119896 and 119909V119896 The property 119896 of the blend is calculated fromthe property 119909

119888119896of crude 119888 assuming that the mixing rule is

linear

119909V119896 sdot 119881119905

119894V le sum

119888isin119862

119909119888119896119881119894V119888 le 119909V119896 sdot 119881

119905

119894V 119894 isin 119879 V isin 119882 119896 isin 119870

(23)

2513 Composition Constraint It has been shown that pro-cesses including both mixing and splitting of streams cannotbe expressed as a linear model Mixing occurs when twostreams are used to fill a tank and is expressed linearly inconstraint (10) Splitting occurs when partially discharging atank resulting in two parts the remaining content of the tankand the transferred products This constraint is nonlinearThe composition of the products transferred during a transferoperation must be identical to the composition of the origintank

119871119894119903119888

119871119905

119894119903

=119881119894V119888

119881119905

119894V 119894 isin 119879 119903 isin 119877 V isin 119874

119903 119888 isin 119862 (24)

Constraint (24) is reformulated as an equation involvingbilinear terms

119881119894V119888 sdot 119871119905

119894119903= 119871119894119903119888sdot 119881119905

119894V 119894 isin 119879 119903 isin 119877 V isin 119874119903 119888 isin 119862 (25)

The Scientific World Journal 5

6

37

8

1

2

4

5

Vessels Storage tanksCharging tanks

CDUs

Figure 1 Crude oil operations system for the problem

Note that constraint (25) is correct even when operationV is not assigned to priority-slot 119894 as then

119881119905

119894V = 119881119894V119888 = 0 (26)

2514 Resource Constraints Resource constraints restrict theuse of resources throughout the scheduling horizon

2515 Tank Capacity Constraint The level of materials inthe tank 119903 must remain between minimum and maximumcapacity limits 119871119905

119903and 119871119905

119903 respectively Let 119871119905

0119903be the initial

total level and let 1198710119903119888

be the initial level of crude 119888 in thetank 119903 As simultaneous charging and discharging of tanks isforbidden the following constraints are sufficient

119871119905

119903le 119871119905

119894119903le 119871119905119903 119894 isin 119879 119903 isin 119877

119878cup 119877119862

0 le 119871119894119903119888le 119871119905119903 119894 isin 119879 119903 isin 119877

119878cup 119877119862 119888 isin 119862

119871119905

119903le 119871119905

0119903+ sum

119894isin119879

sum

Visin119868119903

119881119905

119894V minus sum

119894isin119879

sum

Visin119874119903

119881119905

119894V le 119871119905

119903

119903 isin 119877119878cup 119877119862

0 le 1198710119903119888

+ sum

119894isin119879

sum

Visin119868119903

119881119894V119888 minus sum

119894isin119879

sum

Visin119874119903

119881119894V119888 le 119871

119905

119903

119903 isin 119877119878cup 119877119862 119888 isin 119862

(27)

2516 DemandConstraint Demand constraints define lowerand upper limits 119863

119903and 119863

119903 on total volume of products

transferred out of each charging tank 119903 during the schedulinghorizon

119863119903le sum

119894isin119879

sum

Visin119874119903

119881119905

119894V le 119863119903 119903 isin 119877119862 (28)

3 Finite State Theory

This section presents in a somewhat informal way those basicnotions and definitions from formal language and finite statetheories which are relevant for the sections to follow Relateddefinitions are taken from literature [20 21] Readers whoare unfamiliar with formal language theory are advised toconsult the sources whenever necessary

0 1

2

3

44a

b b

bc

Figure 2 A deterministic finite state automaton (DFA)

0 1

2

3

4

o a

d c

o o o o

g t

Figure 3 Finite state transducer encoding the relation (dog cat)(dog cow)

31 Finite State Automata A DFA is a 5-tuple (119876 Σ 120575 119894 119865)where119876 is a set of states Σ is an alphabet 119894 is the initial state119865 sube 119876 is a set of final states and 120575 is a transition functionmapping 119876 times Σ to 119876 That is for each state 119906 and symbol119886 there is at most one state that can be reached from 119906 byldquofollowingrdquo 119886 (Figure 2)

32 Finite State Transducers A finite state transducer (FST)is a 6-tuple (Σ

1 Σ2 119876 120575 119894 119865) where 119876 119894 and 119865 are the same

as for DFA Σ1is input alphabet Σ

2is output alphabet and 120575

is a function mapping119876times (Σ1cup 120576) times (Σ

2cup 120576) to a subset of

the power set of119876 (Figure 3) Intuitively an FST is much likean NFA except that transitions are made on strings instead ofsymbols and in addition they have outputs

33 Finite State Calculus As argued in Karttunen [22ndash25]many of the rules used can be analyzed as special cases ofregular expressionsThey extend the basic regular expressionwith new operators These extensions make the finite stateautomation and finite state transducer become more suit-able for particular applications The system described belowwas implemented using FSA Utilities [26] a package forimplementing and manipulating finite state automata whichprovides possibilities for defining new regular expressionoperatorsThe part of FSAs built in regular expression syntaxrelevant to this paper is listed in Table 4

One particular useful extension of the basic syntax ofregular expressions is the replace-operator Karttunen [22ndash25] argues that many phonological and morphological rulescan be interpreted as rules which replace a certain portionof the input string Although several implementations of thereplace-operator are proposed the most relevant case for ourpurposes is the so-called ldquoleftmost longest-matchrdquo replace-ment In case of overlapping rule targets in the input thisoperator will replace the leftmost target and in cases where

6 The Scientific World Journal

Schedule 7 - 6 - 8

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

- 3 - 5 - 1 - 3 - 7 - -6 2

Binary

Z61 = 1

Z102

Z43 = 1 Z73 = 1

Z55 = 1

Z26 = 1 Z96 = 1

Z17 = 1 Z87 = 1

= 1

Z38 =

0

0

1

Operation

1

2

3

4

5

6

7

8

6

4 7

2

5

10

9

1 8

3

t0 t1 t2 t3 t4 t5 t6 t7 t8

variable

executions

S =

Z =

Sequence 7-6-8-3-5-1-3-7-6-2

Figure 4 An example to indicate the relationship between binary variable and schedule

a rule target contains a prefix which is also a potential targetthe longer sequence will be replaced Gerdemann and vanNoord [27] implement leftmost longest-match replacementin FSA as the operator

replace (Target LeftContextRightContext) (29)where Target is a transducer defining the actual replacementand LeftContext and RightContext are regular expressionsdefining the left and right context of the rule respectivelyThe segmentation task discussed in the mutation proceduremakes crucial use of longest-match replacement

4 The Hybrid Algorithm

From the point view of optimization efficiency and robust-ness a novel two-level optimization framework based onfinite statemethod andGA is proposed for theMINLPmodelin this section

41 Two-Level Optimization Structure As the foundationof the framework a two-level optimization structure isintroduced Once all binary variables are fixed the originalproblem becomes a relatively simpler model with only con-tinuous variable Following this deal we rewrite (5) as follows

max (119869 (120585 119911)) lArrrArr max119911

[max120585

119869 (120585 119911)] (30)

where 120585 and 119911 represent continuous and binary variablesrespectively Equation (30) shows when 119911 is fixed as 119911 thesubmodel 119869(120585 119911) can be solved optimally by continuous-optimization solvers in the inner level then we update 119911towards the best binary solution 119911lowast in the outer level

We used an example in Figure 4 to show how binarysolution can be mapped to a scheduling sequence Theschedule 119878 = [7683513762] where 7 stands for the specificoperation 7 to assign to position 1 corresponding to the binarydecisions 119885

17= 1

42 Initial Population Based on the sequencing rules [18]and the extension to the regular expression calculus [22ndash25]a DFA model which builds legal schedules complying withsequencing rules and operation condition is constructedThewhole set of possible schedules is too huge to be processed atonceTheDFAmodel of the schedule constitutes a reasonableframework capturing all possible schedules and removingmany redundant sequences of operations Initial values ofdecision variables must satisfy the equality constraints andoperation condition and therefore represent a feasible oper-ating point

Here we still use the instance with 8 operations fromMouret et al [18] to describe an efficient sequencing rule by

The Scientific World Journal 7

macro (proceduresegmentation segmentation of the input sequence into a set of sub-sequencemutation apply mutation rulesclean up) remove markers

Algorithm 1

using a regular expression A feasible sequence V1sdot sdot sdot V119894sdot sdot sdot V119899

can be described by the following

sequence = (120576 + 119871119886) (119871119887sdot 119871119886)lowast(120576 + 119871

119887)

119871119886= 7 (120576 + 4) (120576 + 6) (120576 + 1 + 14) (120576 + 2 + 26)

119871119887= 8 (120576 + 3) (120576 + 5) (120576 + 1 + 13) (120576 + 2 + 25)

(31)

However this automation suffers from a serious problemof overgeneration For example the short length of thesequence may lead to infeasibility while the long length ofthe sequence may result in an unsolvable model It is aninteresting challenge for finite state syntactic description tospecify a sublanguage that contains all and only the sequencesof valid length

Our solution is to construct a suitable constraint for thesequences of valid length The constraint expressions denotea language that admits sequences of valid length but excludesall others We obtain the desired effect by intersecting theconstraint language with the original language of sequenceexpressions The intersection of the two languages containsall and only the valid dates

ValidSequence = Sequence cap ValidLength (32)

The ValidLength constraint is a language that includes allsequences of length 119899

ValidLength = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)119899 (33)

We have now completed the task of describing thelanguage of valid sequences from the set of possible sequenceexpressions It is also possible to create an automation onthe basis of the regular expression and ValidSequence andthen generate all possible sequences V

1sdot sdot sdot V119894sdot sdot sdot V119899accepted

by the automaton The processes are implemented usingFSA Utilities [26] that is a package for implementing andmanipulating DFA and finite state transducer In order togenerate all possible sequences When all possible sequencesV1sdot sdot sdot V119894sdot sdot sdot V119899accepted by the automaton are generated and

the population of the according possible binary decisions isgenerated In the initial population stage of GA the popu-lation size is the number of individuals When the numberof individuals is given a population of candidate solutions isgenerated by randomly selecting from the population of theall possible binary decisions

43 Rule-Based Mutation Approach In the mutation stagewe use a finite state transducer for this rule-based muta-tion process The rule-based mutation strategy must obey

Input7681325712

Segmentationtransducer

76-81325-712 74-83132-741

Mutationtransducer transducer

Output

748313274

Cleanup

Figure 5 An example of mutation

replace ([identity(SSequence)[]x-][]])

Algorithm 2

the sequencing rule and the nonoverlapping constraint suchthat all involved solutions in GA are feasible

The proposedmutation approach is a two-step procedure

Step 1 Segmentation of the input sequence into a set ofsubsequences (ie the subsequence which belongs to theregular language L7 or L8)

Step 2 Mutation of the subsequences into others

Formally the rule-based mutation procedure is imple-mented as the composition of three transducers (seeAlgorithm 1)

An example of mutation including the intermediate stepsis given for the sequence ldquo7681325712rdquo as shown in Figure 5

431 Segmentation Transducer Segmentation transducersplits an input sequence into subsequences The goal ofsegmentation is to provide a convenient representation levelfor the next mutation step

Segmentation is defined as shown in Algorithm 2The macro ldquoSSequencerdquo defines the set of subsequences

The subsequences which belong to the regular languageL7 and L8 are displayed in Tables 1 and 2 Segmentationattaches the marker ldquondashrdquo to each subsequence The Targetsare identified using leftmost longest-match and thus at eachpoint in the input only the longest valid segment is marked

432 The Mutation Rules In the GA process the mutationrules are made by carefully considering nonoverlapping con-straint between operations A concrete instance for partiallyillustrating the mutation rules is given in Algorithm 3 Notethat the final element of the left-contextmust be amarker and

8 The Scientific World Journal

marco (conversion 1198717subsequence rules)

∘ 1198718subsequence rules)

marco (1198717subsequence rules

replace (2 4 6 times 1 7 mdash)∘ replace (14 26 41 42 46 61 62 times 12 7 mdash)∘ replace (142 412 414 426 461 462 612 614 626 times 126 7 mdash)marco (119871

8subsequence rules

replace (2 3 5 times 1 8 mdash)∘ replace (13 25 31 32 35 51 52 times 12 8mdash)∘ replace (132 312 313 325 351 352 512 513 525 times 125 8 mdash)

Algorithm 3 An example to demonstrate the mutation rules

Table 1 Subsequence belonging to 1198717

Length Sequences belonging to 1198717

1 72 71 72 74 763 712 714 726 741 742 746 761 7624 7126 7142 7412 7414 7426 7461 7462 7612 7614 76265 71426 74126 74142 74612 74614 74626 76126 761426 741426 746126 746142 7614267 7461426

the target itself ends in ldquondashrdquo This ensures that mutation rulescannot apply to the same subsequence

5 Experimental Study

In this section the same problem from the literature [18] isused for computational experiments The proposed method-ology is compared with existing promising algorithmsmixed-coding GA [15 28] Figure 1 depicts the refineryconfiguration for problem The data involved in the problemare given in Table 3 The performance comparison withdifferent computing times such as 350 s 500 s 2400 s isconducted The objective value is used to statistically analyzethe optimization results

The performance comparison between the two method-ologies used is illustrated in Figure 6 which shows that thehybrid optimization algorithm which combined the finitestate method and GAwill statistically outperform the mixed-coding counterpart The genetic algorithm which combinedthe finite state method and GA finds feasible solutions veryfast and is able to find better solutions in reasonable time

In Figure 7 we compare the objective variance of eachiteration in the two evolution processes of these two kindsof methodology By tracking the evolution process we findthat the mixed-coding GA is easy to stick in a local minimalsequence solution This situation only can be improvedthrough increasing the mutation scaling factor Howeverthis may result in a hard convergence unless sufficientiterations are implemented As for the hybrid optimizationalgorithm the optimization processes of binary variableand continuous variable are separated The performanceof the whole methodology mainly depends on the FSM

350 500 650 800 950 1200 1500 1800 240020

30

40

50

60

70

80

Computer time

Obj

ectiv

e val

ue

Mixed-code GAOur algorithm

Figure 6 Average objective values of two methodologies

which captures most promising schedules and removes manyredundant sequences of operations so that the user can usea small population size of corresponding discrete variables toobtain suboptimal solutions From Figure 7 we see that theproposed method has converged at 350 iterations as opposedto 2400 iterations for the mixed-coding GA

The success of the proposed algorithm lies in a compre-hensive analysis of the region of the search space and itscapacity to focus the search on the regions with the partialsolution One of the good merits of the hybrid algorithm is

The Scientific World Journal 9

Table 2 Subsequence belonging to 1198718

Length Sequences belonging to 1198718

1 82 81 82 83 853 812 813 825 831 832 835 851 8524 8125 8132 8312 8313 8325 8351 8352 8512 8513 85255 81325 83125 83132 83512 83513 83525 85132 851256 831325 835125 835132 8513257 8351325

Table 3 Problem data

Scheduling horizon 8 daysVessels Arrival time Composition Amount of crudeVessel 1 0 100 A 1000Vessel 2 4 100 B 1000Storage tanks Capacity Initial composition Initial amountTank 1 [0 1000] 100 A 250Tank 2 [0 1000] 100 B 750Charging tanks Capacity Initial composition Initial amountTank 1 (mix X) [0 1000] 100 C 500Tank 1 (mix X) [0 1000] 100 D 500Crudes 1 Gross margin Crude mixtures Property1 DemandCrude A 001 9 Crude mix X [0015 0025] [1000 1000]

Crude B 006 4 Crude mix Y [0045 0055] [1000 1000]

Crude C 002 8 Unloading flow rate [0 500]

Crude D 005 5 transfer flow rate [0 500]

Table 4 A fragment of FSA regular expression syntax and 119880

transducers and 119877 can be either

[] The empty string[1198771 119877

119899] Concatenation

1198771 119877

119899 Disjunction

119877Λ Optionality

Identity (119860) Identity the transducer which maps eachelement in 119860 onto itself

119879 ∘ 119880 Composition of the transducers 119879 and 119880macro (Term 119877) Use term as an abbreviation for 119877

that each solution involved in theGA algorithm is guaranteedto be feasible by using the mutation rules generated by DFMmethod while in existing GA algorithms the procedure togenerate feasible solution under complex process constraintsis very time costive The deterministic finite automata (DFA)can easily represent this kind of structure Furthermorethe complex process constraints can be very difficult toexpress with mixed integer programming Consequently itis unfeasible to solve the industrial problem by using MIPsolver

6 Conclusion

In this paper a novel hybrid optimization algorithm whichcombined the finite state method and GA is proposed

350 500 650 800 950 1200 1500 1800 24000

02

04

06

08

1

12

14

Computing time (s)

Varia

nce v

alue

s

Mixed-code GAOur algorithm

Figure 7 Variance values of two methodologies

The proposed algorithm constitutes a reasonable frameworkcapturing both the operating condition and sequencing ruleof the schedule The solution captures all possible sched-ules and removes many redundant sequences of operationsThe algorithm is equivalent to introducing new structure

10 The Scientific World Journal

information into the optimization process which will helpreduce the risk of trapping in a local minimal sequencesolution The hybrid optimization algorithm is an effectiveand robust tool to solve the crude oil scheduling problem interms of efficiency and reliability Algorithms only with thetwo properties are suitable for solving practical engineeringapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partially supported by the China NationalNatural Science Foundation under Grant 61203178 Grant61304214 and Grant 61290323 The authors thank the finan-cial funds fromShanghai Science andTechnologyCommitteeunder Grant 12511501002 and Grant 13511501302

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] C A Mendez I E Grossmann I Harjunkoski and P KaboreldquoA simultaneous optimization approach for off-line blendingand scheduling of oil-refinery operationsrdquo Computers andChemical Engineering vol 30 no 4 pp 614ndash634 2006

[6] M Pan X Li and Y Qian ldquoNew approach for scheduling crudeoil operationsrdquo Chemical Engineering Science vol 64 no 5 pp965ndash983 2009

[7] M F Cardoso R L Salcedo S F de Azevedo and D BarbosaldquoA simulated annealing approach to the solution of minlpproblemsrdquo Computers and Chemical Engineering vol 21 no 12pp 1349ndash1364 1997

[8] I E Grossmann ldquoReview of nonlinear mixed-integer anddisjunctive programming techniquesrdquo Optimization and Engi-neering vol 3 no 3 pp 227ndash252 2002

[9] E L Lawler and D E Wood ldquoBranch-and-bound methods asurveyrdquo Operations Research vol 14 no 4 pp 699ndash719 1966

[10] M A Duran and I E Grossmann ldquoAn outer-approximationalgorithm for a class of mixed-integer nonlinear programsrdquoMathematical Programming vol 36 no 3 pp 307ndash339 1986

[11] AMGeoffrion ldquoGeneralized Benders decompositionrdquo Journalof Optimization Theory and Applications vol 10 no 4 pp 237ndash260 1972

[12] C DrsquoAmbrosio andA Lodi ldquoMixed integer nonlinear program-ming tools a practical overviewrdquo 4OR vol 9 no 4 pp 329ndash3492011

[13] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence The University of Michigan Press AnnArbor Mich USA 1975

[14] T Yokota M Gen and Y-X Li ldquoGenetic algorithm for non-linear mixed integer programming problems and its applica-tionsrdquo Computers and Industrial Engineering vol 30 no 4 pp905ndash917 1996

[15] L Costa and P Oliveira ldquoEvolutionary algorithms approachto the solution of mixed integer non-linear programmingproblemsrdquo Computers and Chemical Engineering vol 25 no 2-3 pp 257ndash266 2001

[16] J M Ponce-Ortega M Serna-Gonzalez and A Jimenez-Gutierrez ldquoHeat exchanger network synthesis includingdetailed heat exchanger design using genetic algorithmsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8767ndash8780 2007

[17] K-M Bjork and R Nordman ldquoSolving large-scale retrofitheat exchanger network synthesis problems with mathematicaloptimization methodsrdquo Chemical Engineering and ProcessingProcess Intensification vol 44 no 8 pp 869ndash876 2005

[18] S Mouret I E Grossmann and P Pestiaux ldquoA novel priority-slot based continuous-time formulation for crude-oil schedul-ing problemsrdquo Industrial and Engineering Chemistry Researchvol 48 no 18 pp 8515ndash8528 2009

[19] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[20] J E Hopcroft Introduction to AutomataTheory Languages andComputation Pearson Education India New Delhi India 3rdedition 2008

[21] E Roche and Y Schabes Finite-State Language Processing TheMIT Press Cambridge Mass USA 1997

[22] L Karttunen ldquoConstructing lexical transducersrdquo in Proceedingsof the 15th conference on Computational Linguistics vol 1Association for Computational Linguistics 1994

[23] L Karttunen ldquoThe replace operatorrdquo in Proceedings of the 33rdAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1995

[24] L Karttunen ldquoDirected replacementrdquo in Proceedings of the 34thAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1996

[25] L Karttunen and K R Beesley Two-Level Rule Compiler XeroxCorporation Palo Alto Research Center 1992

[26] G vanNoord ldquoFSAutilities a toolbox tomanipulate finite-stateautomatardquo in Automata Implementation pp 87ndash108 SpringerNew York NY USA 1997

The Scientific World Journal 11

[27] D Gerdemann and G van Noord ldquoTransducers from rewriterules with backreferencesrdquo in Proceedings of the 9th Conferenceon European Chapter of the Association for ComputationalLinguistics Association for Computational Linguistics 1999

[28] Y-C Lin K-S Hwang and F-S Wang ldquoA mixed-codingscheme of evolutionary algorithms to solve mixed-integer non-linear programming problemsrdquo Computers and Mathematicswith Applications vol 47 no 8-9 pp 1295ndash1307 2004

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

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Electrical and Computer Engineering

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RoboticsJournal of

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Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 The Scientific World Journal

to handle the difficulties of the nonconvexity Going onestep further some mixed coding methods were proposedwhich include mixed-coding genetic algorithm [15] andinformation-guided genetic algorithm (IGA) Ponce-Ortegaet al [16] proposed a two-level approach based on GA tooptimize the heat exchanger networks (HENs) The outerlevel is used to perform the structural optimization for whicha binary GA is used Bjork and Nordman [17] showed thatthe GA is very suitable to solve a large-scale heat exchangernetwork

Obviously the two different approaches previously dis-cussed have their own advantages and disadvantages Onthe one hand a deterministic approach usually involvesconsiderable algebra and undeviating analysis to the problemitself whereas the evolutionary approach does not have thisproperty On the other hand some deterministic approachessuch as mathematical programming usually cannot providepractical solutions in reasonable time whereas the evolution-ary approach can generate satisfying solutions In this worka novel genetic algorithm which combined the finite statemethod and GA is proposed to solve crude oil schedulingproblem A MINLP model is formulated based on thesingle-operation sequencing (SOS) time representation Adeterministic finite automation (DFA) model which capturesvalid possible schedule sequences is constructed based on thesequencing rules The initialization and mutation operationof GA is based on the model which builds legal schedulescomplying with sequencing rules and operation conditionThus the search space of the algorithm is substantiallyreduced as only legal sequence is explored The rest of thepaper is organized as follows the MINLP model is specifiedin Section 2 Section 3 reviews the background of finitestate theory In Section 4 a novel genetic algorithm whichcombined the finite statemethod andGA is proposed to solvethe MINLP model A test problem is studied to verify ourapproach in Section 5 In the last section conclusive remarksare given

2 Mathematic Model

In this section the MINLP model of refinery crude oilscheduling problem is described [18] This problem has beenwidely studied from the optimization viewpoint since thework of Lee et al [19] It consists of crude oil unloadingfrom marine vessels to storage tanks transfer and blendingbetween tanks and distillation of crude mixtures The goalis to maximize profit and meet distillation demands for eachtype of crude blend (eg low sulfur or high sulfur blends)while satisfying unloading and transfer logistics constraintsinventory capacity limitations and property specifications foreach blend The logistics constraints involve nonoverlappingconstraints between crude oil transfer operations

21 Sets The following sets will be used in the model

(i) 119879 = 1 119899 is the set of priority-slots(ii) 119882 is the set of all operations119882 ≜ 119882

119880cup119882119879cup119882119863

(iii) 119882119880sub 119882 is the set of unloading operations

(iv) 119882119879sub 119882 is the set of tank-to-tank transfer opera-

tions

(v) 119882119863sub 119882 is the set of distillation operations

(vi) 119877 is the set of all operations 119877 = 119877119881cup 119877119878cup 119877119862cup 119877119863

(vii) 119877119881sub 119877 is the set of vessels

(viii) 119877119878sub 119877 is the set of storage tanks

(ix) 119877119862sub 119877 is the set of charging tanks

(x) 119877119863sub 119877 is the set of distillation units

(xi) 119868119903sub 119882 is the set of inlet transfer operations on

resource 119903

(xii) 119874119903sub 119882 is the set of outlet transfer operations on

resource 119903

(xiii) 119862 is the set of products (ie crudes)

(xiv) 119870 is the set of product properties (eg crude sulfurconcentration)

22 Parameters Parameters used in the paper are definedbelow

(i) 119867 is the scheduling horizon

(ii) [119881119905V 119881119905V ] are bounds on the total volume transferredduring transfer operation 119881 in all instances 119881119905V = 0

for all operations except unloading for which119881119905V = 119881119905Vis the volume of crude in the marine vessel

(iii) [119873119863 119873119863] are the bounds on the number of distilla-

tions

(iv) [119865119877V 119865119877V] are flow rate limitations for transfer oper-ation V

(v) 119878119903is the arrival time of vessel 119903

(vi) [119909V119896 119909V119896] are the limits of property 119896 of the blendedproducts transferred during operation V

(vii) 119909119888119896is the value of the property 119896 of crude 119888

(viii) [119871119905119903 119871119905119903] are the capacity limits of tank 119903

(ix) [119863119903 119863119903] are the bounds of the demand on products

to be transferred out of the charging tank 119903 during thescheduling horizon

(x) 119866119888is the gross margin of crude 119888

23 Variables

231 Assignment Variables

119885119894V isin 0 1 119894 isin 119879 V isin 119882 (1)

119885119894V = 1 if operation V is assigned to priority-slot 119894 119885

119894V = 0

otherwise

The Scientific World Journal 3

232 Time Variables

119878119894V ge 0 119863

119894V ge 0 119894 isin 119879 V isin 119882 (2)

119878119894V is the start time of operation V if it is assigned to priorityslot 119894 119878

119894V = 0 otherwise119863119894V is the duration of operation V if it is assigned to

priority slot 119894119863119894V = 0 otherwise

233 Operation Variables

119881119905

119894V ge 0 119881119894V119888 ge 0 119894 isin 119879 V isin 119882 119888 isin 119862 (3)

119881119905

119894V is the total volume of crude transferred during operationV if it is assigned to priority slot 119894 119881119905

119894V = 0 otherwise119881119894V119888 is the volume of crude 119888 transferred during operation

V if it is assigned to priority slot 119894 119881119894V119888 = 0 otherwise

234 Resource Variables

119871119905

119894119903 119871119894119903119888 119894 isin 119879 119903 isin 119877 119888 isin 119862 (4)

119871119905

119894119903is the total accumulated level of crude in tank 119903 isin 119877

119878cup119877119862

before the operation was assigned to priority-slot 119894119871119894119903119888

is the accumulated level of crude 119888 in tank 119903 isin 119877119878cup119877119862

before the operation was assigned to priority-slot 119894

24 Objective Function The objective is to maximize thegross margins of the distilled crude blends Let 119866

119888be the

individual gross margin of crude 119888

maxsum119894isin119879

sum

119903isin119877119863

sum

Visin119868119903

sum

119888isin119862

119866119888sdot 119881119894V119888 (5)

25 General Constraints It should be noted that the crudecomposition of blends in tanks is tracked instead of theirpropertiesThe distillation specifications are later enforced bycalculating a posteriori the properties of the blend in termsof its composition For instance in the problem a blendcomposed of 50 of crude A and 50 of crude B has a sulfurconcentration of 0035 which does not meet the specificationfor crude mix X nor for crude mix Y

251 Assignment Constraints In the SOS model exactly oneoperation has to be assigned to each priority slot

sum

Visin119882119885119894V = 1 119894 isin 119879 (6)

252 Variable Constraints Variable constraints are given bytheir definitions Start time duration and global volumevariables are defined with big-119872 constraints

119878119894V + 119863119894V le 119867 sdot 119885

119894V 119894 isin 119879 V isin 119882

119881119905

119894V le 119881119905

V sdot 119885119894V 119894 isin 119879 V isin 119882

119881119905

119894V ge 119881119905

V sdot 119885119894V 119894 isin 119879 V isin 119882

(7)

Crude volume variables are positive variables whose sumequals the corresponding total volume variable

sum

119888isin119862

119881119894V119888 = 119881

119905

119894V (8)

Total and crude level variables are defined by adding tothe initial level in the tank all inlet and outlet transfer volumesof operations of higher priority than the considered priorityslot

119871119905

119894119903= 119871119905

0119903+ sum

119895isin119879119895lt119894

sum

Visin119868119903

119881119905

119894V minus sum

119895isin119879119895lt119894

sum

Visin119874119903

119881119905

119894V

119894 isin 119879 119903 isin 119877

(9)

119871119894119903119888= 119871119900119903119888

+ sum

119895isin119879119895lt119894

sum

Visin119868119903

119881119894V119888 minus sum

119895isin119879119895lt119894

sum

Visin119874119903

119881119894V119888

119894 isin 119879 119903 isin 119877 119888 isin 119862

(10)

253 Sequencing Constraints Sequencing constraintsrestrict the set of possible sequences of operationsCardinality and unloading sequence constraints are specificcases of sequencing constraints More complex sequencingconstraints will also be discussed later

254 Cardinality Constraint Each crude oil marine vesselhas to unload its content exactly oncesum

119894isin119879sumVisin119874119903 119885119894V = 1 119903 isin

119877119881The total number of distillation operations is bounded by

119873119863and119873

119863in order to reduce the cost of CDU switches

119873119863le sum

119894isin119879

sum

Visin119882119863

119885119894V le 119873119863 (11)

255 Unloading Sequence Constraint Marine vessels have tounload in order of arrival to the refinery Considering twovessels 119903

1 1199032isin 1198771198811199031lt 1199032signifies that 119903

1unloads before 119903

2

sum

119895isin119879119895lt119894

sum

Visin1198741199032

119885119895V + sum

119895isin119879119895ge119894

sum

Visin1198741199031

119885119895V le 1 (12)

256 Scheduling Constraints Scheduling constraints restrictthe values taken by time variables according to logistics rules

257 Nonoverlapping Constraint A nonoverlapping con-straint between two sets of operations119882

1sub 119882 and119882

2sub 119882

states that any pair of operations (V1 V2) sub 119882

1times1198822must not

be executed simultaneouslyUnloading operations must not overlap

sum

Visin119882119880

(119878119894V + 119863119894V) le sum

Visin119882119880

119878119895V + 119867 sdot (1 minus sum

Visin119882119880

119885119895V)

119894 119895 isin 119879 119894 lt 119895

(13)

4 The Scientific World Journal

Inlet and outlet transfer operations on a tank must notoverlap

sum

Visin119868119903

(119878119894V + 119863119894V) le sum

Visin119874119903

119878119895V + 119867 sdot (1 minus sum

Visin119874119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119878cup 119877119862

sum

Visin119874119903

(119878119894V + 119863119894V) le sum

Visin119868119903

119878119895V + 119867 sdot (1 minus sum

Visin119868119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119878cup 119877119862

(14)

Although we do not consider crude settling in storagetanks after vessel unloading it could be included in themodelwith amodified version of constraint (14) taking into accounttransition times We define TR

119881as the transition time after

unloading operation V isin 119882119880

and TR as the maximumtransition time TR = maxVisin119882119880TR119881

sum

Visin119868119903

(119878119894V + 119863119894V + TRV sdot 119885119894V)

le sum

Visin119874119903

119878119895V + (119867 + TR) sdot (1 minus sum

Visin119874119903

119885119895V)

(15)

Constraint (15) is valid in the four possible cases

(existV1isin 119868119903 119885119894V1 = 1)

and (existV2isin 119874119903 119885119895V2 = 1) 997904rArr 119878

119894V + 119863119894V1 + TRV1 le 119878119895V2

(existV1isin 119868119903 119885119894V1 = 1)

and (⋁ V2isin 119874119903 119885119895V2 = 1) 997904rArr 119878

119894V + 119863119894V1 le 119867 + TR minus TRV1

(⋁ V1isin 119868119903 119885119894V1 = 0)

and (existV2isin 119874119903 119885119895V2 = 1) 997904rArr 0 le 119878

119895V2

(⋁ V1isin 119868119903 119885119894V1 = 0)

and (⋁ V2isin 119874119903 119885119895V2 = 0) 997904rArr 0 le 119867 + TR

(16)

A tank may charge only one CDU at a time

sum

Visin119874119903

(119878119894V + 119863119894V) le sum

Visin119874119903

119878119895V + 119867 sdot (1 minus sum

Visin119874119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119862

(17)

A CDUmay be charged by only one tank at a time

sum

Visin119868119903

(119878119894V + 119863119894V) le sum

Visin119868119903

119878119895V + 119867 sdot (1 minus sum

Visin119868119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119863

(18)

To avoid schedules in which a transfer is being performedtwice at a time thus possibly violating the flow rate limita-tions constraint (19) is included in the model

119878119894V + 119863119894V le 119878119895V + 119867 sdot (1 minus 119885

119895V) 119894 119895 isin 119879 119894 lt 119895 V isin 119882(19)

258 Continuous Distillation Constraint It is required thatCDUs operate without interruption As CDUs perform onlyone operation at a time the continuous operation constraintis defined by equating the sum of the duration of distillationsto the time horizon

sum

119894isin119879

sum

Visin119868119903

119863119894V = 119867 119903 isin 119877

119863 (20)

259 Resource Availability Constraint Unloading of crudeoil vessels may start only after arrival to the refinery Let 119878

119903

be the arrival time of vessel 119903

119878119894V ge 119878119903 sdot 119885119894V 119894 isin 119879 119903 isin 119877V V isin 119874119903 (21)

2510 Operation Constraints Operation constraints restrictthe values taken by operation and time variables according tooperational rules

2511 Flow Rate Constraint The flow rate of transfer opera-tion V is bounded by FRV and FRV

FRV sdot 119863119894V le 119881119905

119894V le FRV sdot 119863119894V 119894 isin 119879 V isin 119882 (22)

2512 Property Constraint The property 119896 of the blendedproducts transferred during operation V is bounded by119909V119896 and 119909V119896 The property 119896 of the blend is calculated fromthe property 119909

119888119896of crude 119888 assuming that the mixing rule is

linear

119909V119896 sdot 119881119905

119894V le sum

119888isin119862

119909119888119896119881119894V119888 le 119909V119896 sdot 119881

119905

119894V 119894 isin 119879 V isin 119882 119896 isin 119870

(23)

2513 Composition Constraint It has been shown that pro-cesses including both mixing and splitting of streams cannotbe expressed as a linear model Mixing occurs when twostreams are used to fill a tank and is expressed linearly inconstraint (10) Splitting occurs when partially discharging atank resulting in two parts the remaining content of the tankand the transferred products This constraint is nonlinearThe composition of the products transferred during a transferoperation must be identical to the composition of the origintank

119871119894119903119888

119871119905

119894119903

=119881119894V119888

119881119905

119894V 119894 isin 119879 119903 isin 119877 V isin 119874

119903 119888 isin 119862 (24)

Constraint (24) is reformulated as an equation involvingbilinear terms

119881119894V119888 sdot 119871119905

119894119903= 119871119894119903119888sdot 119881119905

119894V 119894 isin 119879 119903 isin 119877 V isin 119874119903 119888 isin 119862 (25)

The Scientific World Journal 5

6

37

8

1

2

4

5

Vessels Storage tanksCharging tanks

CDUs

Figure 1 Crude oil operations system for the problem

Note that constraint (25) is correct even when operationV is not assigned to priority-slot 119894 as then

119881119905

119894V = 119881119894V119888 = 0 (26)

2514 Resource Constraints Resource constraints restrict theuse of resources throughout the scheduling horizon

2515 Tank Capacity Constraint The level of materials inthe tank 119903 must remain between minimum and maximumcapacity limits 119871119905

119903and 119871119905

119903 respectively Let 119871119905

0119903be the initial

total level and let 1198710119903119888

be the initial level of crude 119888 in thetank 119903 As simultaneous charging and discharging of tanks isforbidden the following constraints are sufficient

119871119905

119903le 119871119905

119894119903le 119871119905119903 119894 isin 119879 119903 isin 119877

119878cup 119877119862

0 le 119871119894119903119888le 119871119905119903 119894 isin 119879 119903 isin 119877

119878cup 119877119862 119888 isin 119862

119871119905

119903le 119871119905

0119903+ sum

119894isin119879

sum

Visin119868119903

119881119905

119894V minus sum

119894isin119879

sum

Visin119874119903

119881119905

119894V le 119871119905

119903

119903 isin 119877119878cup 119877119862

0 le 1198710119903119888

+ sum

119894isin119879

sum

Visin119868119903

119881119894V119888 minus sum

119894isin119879

sum

Visin119874119903

119881119894V119888 le 119871

119905

119903

119903 isin 119877119878cup 119877119862 119888 isin 119862

(27)

2516 DemandConstraint Demand constraints define lowerand upper limits 119863

119903and 119863

119903 on total volume of products

transferred out of each charging tank 119903 during the schedulinghorizon

119863119903le sum

119894isin119879

sum

Visin119874119903

119881119905

119894V le 119863119903 119903 isin 119877119862 (28)

3 Finite State Theory

This section presents in a somewhat informal way those basicnotions and definitions from formal language and finite statetheories which are relevant for the sections to follow Relateddefinitions are taken from literature [20 21] Readers whoare unfamiliar with formal language theory are advised toconsult the sources whenever necessary

0 1

2

3

44a

b b

bc

Figure 2 A deterministic finite state automaton (DFA)

0 1

2

3

4

o a

d c

o o o o

g t

Figure 3 Finite state transducer encoding the relation (dog cat)(dog cow)

31 Finite State Automata A DFA is a 5-tuple (119876 Σ 120575 119894 119865)where119876 is a set of states Σ is an alphabet 119894 is the initial state119865 sube 119876 is a set of final states and 120575 is a transition functionmapping 119876 times Σ to 119876 That is for each state 119906 and symbol119886 there is at most one state that can be reached from 119906 byldquofollowingrdquo 119886 (Figure 2)

32 Finite State Transducers A finite state transducer (FST)is a 6-tuple (Σ

1 Σ2 119876 120575 119894 119865) where 119876 119894 and 119865 are the same

as for DFA Σ1is input alphabet Σ

2is output alphabet and 120575

is a function mapping119876times (Σ1cup 120576) times (Σ

2cup 120576) to a subset of

the power set of119876 (Figure 3) Intuitively an FST is much likean NFA except that transitions are made on strings instead ofsymbols and in addition they have outputs

33 Finite State Calculus As argued in Karttunen [22ndash25]many of the rules used can be analyzed as special cases ofregular expressionsThey extend the basic regular expressionwith new operators These extensions make the finite stateautomation and finite state transducer become more suit-able for particular applications The system described belowwas implemented using FSA Utilities [26] a package forimplementing and manipulating finite state automata whichprovides possibilities for defining new regular expressionoperatorsThe part of FSAs built in regular expression syntaxrelevant to this paper is listed in Table 4

One particular useful extension of the basic syntax ofregular expressions is the replace-operator Karttunen [22ndash25] argues that many phonological and morphological rulescan be interpreted as rules which replace a certain portionof the input string Although several implementations of thereplace-operator are proposed the most relevant case for ourpurposes is the so-called ldquoleftmost longest-matchrdquo replace-ment In case of overlapping rule targets in the input thisoperator will replace the leftmost target and in cases where

6 The Scientific World Journal

Schedule 7 - 6 - 8

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

- 3 - 5 - 1 - 3 - 7 - -6 2

Binary

Z61 = 1

Z102

Z43 = 1 Z73 = 1

Z55 = 1

Z26 = 1 Z96 = 1

Z17 = 1 Z87 = 1

= 1

Z38 =

0

0

1

Operation

1

2

3

4

5

6

7

8

6

4 7

2

5

10

9

1 8

3

t0 t1 t2 t3 t4 t5 t6 t7 t8

variable

executions

S =

Z =

Sequence 7-6-8-3-5-1-3-7-6-2

Figure 4 An example to indicate the relationship between binary variable and schedule

a rule target contains a prefix which is also a potential targetthe longer sequence will be replaced Gerdemann and vanNoord [27] implement leftmost longest-match replacementin FSA as the operator

replace (Target LeftContextRightContext) (29)where Target is a transducer defining the actual replacementand LeftContext and RightContext are regular expressionsdefining the left and right context of the rule respectivelyThe segmentation task discussed in the mutation proceduremakes crucial use of longest-match replacement

4 The Hybrid Algorithm

From the point view of optimization efficiency and robust-ness a novel two-level optimization framework based onfinite statemethod andGA is proposed for theMINLPmodelin this section

41 Two-Level Optimization Structure As the foundationof the framework a two-level optimization structure isintroduced Once all binary variables are fixed the originalproblem becomes a relatively simpler model with only con-tinuous variable Following this deal we rewrite (5) as follows

max (119869 (120585 119911)) lArrrArr max119911

[max120585

119869 (120585 119911)] (30)

where 120585 and 119911 represent continuous and binary variablesrespectively Equation (30) shows when 119911 is fixed as 119911 thesubmodel 119869(120585 119911) can be solved optimally by continuous-optimization solvers in the inner level then we update 119911towards the best binary solution 119911lowast in the outer level

We used an example in Figure 4 to show how binarysolution can be mapped to a scheduling sequence Theschedule 119878 = [7683513762] where 7 stands for the specificoperation 7 to assign to position 1 corresponding to the binarydecisions 119885

17= 1

42 Initial Population Based on the sequencing rules [18]and the extension to the regular expression calculus [22ndash25]a DFA model which builds legal schedules complying withsequencing rules and operation condition is constructedThewhole set of possible schedules is too huge to be processed atonceTheDFAmodel of the schedule constitutes a reasonableframework capturing all possible schedules and removingmany redundant sequences of operations Initial values ofdecision variables must satisfy the equality constraints andoperation condition and therefore represent a feasible oper-ating point

Here we still use the instance with 8 operations fromMouret et al [18] to describe an efficient sequencing rule by

The Scientific World Journal 7

macro (proceduresegmentation segmentation of the input sequence into a set of sub-sequencemutation apply mutation rulesclean up) remove markers

Algorithm 1

using a regular expression A feasible sequence V1sdot sdot sdot V119894sdot sdot sdot V119899

can be described by the following

sequence = (120576 + 119871119886) (119871119887sdot 119871119886)lowast(120576 + 119871

119887)

119871119886= 7 (120576 + 4) (120576 + 6) (120576 + 1 + 14) (120576 + 2 + 26)

119871119887= 8 (120576 + 3) (120576 + 5) (120576 + 1 + 13) (120576 + 2 + 25)

(31)

However this automation suffers from a serious problemof overgeneration For example the short length of thesequence may lead to infeasibility while the long length ofthe sequence may result in an unsolvable model It is aninteresting challenge for finite state syntactic description tospecify a sublanguage that contains all and only the sequencesof valid length

Our solution is to construct a suitable constraint for thesequences of valid length The constraint expressions denotea language that admits sequences of valid length but excludesall others We obtain the desired effect by intersecting theconstraint language with the original language of sequenceexpressions The intersection of the two languages containsall and only the valid dates

ValidSequence = Sequence cap ValidLength (32)

The ValidLength constraint is a language that includes allsequences of length 119899

ValidLength = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)119899 (33)

We have now completed the task of describing thelanguage of valid sequences from the set of possible sequenceexpressions It is also possible to create an automation onthe basis of the regular expression and ValidSequence andthen generate all possible sequences V

1sdot sdot sdot V119894sdot sdot sdot V119899accepted

by the automaton The processes are implemented usingFSA Utilities [26] that is a package for implementing andmanipulating DFA and finite state transducer In order togenerate all possible sequences When all possible sequencesV1sdot sdot sdot V119894sdot sdot sdot V119899accepted by the automaton are generated and

the population of the according possible binary decisions isgenerated In the initial population stage of GA the popu-lation size is the number of individuals When the numberof individuals is given a population of candidate solutions isgenerated by randomly selecting from the population of theall possible binary decisions

43 Rule-Based Mutation Approach In the mutation stagewe use a finite state transducer for this rule-based muta-tion process The rule-based mutation strategy must obey

Input7681325712

Segmentationtransducer

76-81325-712 74-83132-741

Mutationtransducer transducer

Output

748313274

Cleanup

Figure 5 An example of mutation

replace ([identity(SSequence)[]x-][]])

Algorithm 2

the sequencing rule and the nonoverlapping constraint suchthat all involved solutions in GA are feasible

The proposedmutation approach is a two-step procedure

Step 1 Segmentation of the input sequence into a set ofsubsequences (ie the subsequence which belongs to theregular language L7 or L8)

Step 2 Mutation of the subsequences into others

Formally the rule-based mutation procedure is imple-mented as the composition of three transducers (seeAlgorithm 1)

An example of mutation including the intermediate stepsis given for the sequence ldquo7681325712rdquo as shown in Figure 5

431 Segmentation Transducer Segmentation transducersplits an input sequence into subsequences The goal ofsegmentation is to provide a convenient representation levelfor the next mutation step

Segmentation is defined as shown in Algorithm 2The macro ldquoSSequencerdquo defines the set of subsequences

The subsequences which belong to the regular languageL7 and L8 are displayed in Tables 1 and 2 Segmentationattaches the marker ldquondashrdquo to each subsequence The Targetsare identified using leftmost longest-match and thus at eachpoint in the input only the longest valid segment is marked

432 The Mutation Rules In the GA process the mutationrules are made by carefully considering nonoverlapping con-straint between operations A concrete instance for partiallyillustrating the mutation rules is given in Algorithm 3 Notethat the final element of the left-contextmust be amarker and

8 The Scientific World Journal

marco (conversion 1198717subsequence rules)

∘ 1198718subsequence rules)

marco (1198717subsequence rules

replace (2 4 6 times 1 7 mdash)∘ replace (14 26 41 42 46 61 62 times 12 7 mdash)∘ replace (142 412 414 426 461 462 612 614 626 times 126 7 mdash)marco (119871

8subsequence rules

replace (2 3 5 times 1 8 mdash)∘ replace (13 25 31 32 35 51 52 times 12 8mdash)∘ replace (132 312 313 325 351 352 512 513 525 times 125 8 mdash)

Algorithm 3 An example to demonstrate the mutation rules

Table 1 Subsequence belonging to 1198717

Length Sequences belonging to 1198717

1 72 71 72 74 763 712 714 726 741 742 746 761 7624 7126 7142 7412 7414 7426 7461 7462 7612 7614 76265 71426 74126 74142 74612 74614 74626 76126 761426 741426 746126 746142 7614267 7461426

the target itself ends in ldquondashrdquo This ensures that mutation rulescannot apply to the same subsequence

5 Experimental Study

In this section the same problem from the literature [18] isused for computational experiments The proposed method-ology is compared with existing promising algorithmsmixed-coding GA [15 28] Figure 1 depicts the refineryconfiguration for problem The data involved in the problemare given in Table 3 The performance comparison withdifferent computing times such as 350 s 500 s 2400 s isconducted The objective value is used to statistically analyzethe optimization results

The performance comparison between the two method-ologies used is illustrated in Figure 6 which shows that thehybrid optimization algorithm which combined the finitestate method and GAwill statistically outperform the mixed-coding counterpart The genetic algorithm which combinedthe finite state method and GA finds feasible solutions veryfast and is able to find better solutions in reasonable time

In Figure 7 we compare the objective variance of eachiteration in the two evolution processes of these two kindsof methodology By tracking the evolution process we findthat the mixed-coding GA is easy to stick in a local minimalsequence solution This situation only can be improvedthrough increasing the mutation scaling factor Howeverthis may result in a hard convergence unless sufficientiterations are implemented As for the hybrid optimizationalgorithm the optimization processes of binary variableand continuous variable are separated The performanceof the whole methodology mainly depends on the FSM

350 500 650 800 950 1200 1500 1800 240020

30

40

50

60

70

80

Computer time

Obj

ectiv

e val

ue

Mixed-code GAOur algorithm

Figure 6 Average objective values of two methodologies

which captures most promising schedules and removes manyredundant sequences of operations so that the user can usea small population size of corresponding discrete variables toobtain suboptimal solutions From Figure 7 we see that theproposed method has converged at 350 iterations as opposedto 2400 iterations for the mixed-coding GA

The success of the proposed algorithm lies in a compre-hensive analysis of the region of the search space and itscapacity to focus the search on the regions with the partialsolution One of the good merits of the hybrid algorithm is

The Scientific World Journal 9

Table 2 Subsequence belonging to 1198718

Length Sequences belonging to 1198718

1 82 81 82 83 853 812 813 825 831 832 835 851 8524 8125 8132 8312 8313 8325 8351 8352 8512 8513 85255 81325 83125 83132 83512 83513 83525 85132 851256 831325 835125 835132 8513257 8351325

Table 3 Problem data

Scheduling horizon 8 daysVessels Arrival time Composition Amount of crudeVessel 1 0 100 A 1000Vessel 2 4 100 B 1000Storage tanks Capacity Initial composition Initial amountTank 1 [0 1000] 100 A 250Tank 2 [0 1000] 100 B 750Charging tanks Capacity Initial composition Initial amountTank 1 (mix X) [0 1000] 100 C 500Tank 1 (mix X) [0 1000] 100 D 500Crudes 1 Gross margin Crude mixtures Property1 DemandCrude A 001 9 Crude mix X [0015 0025] [1000 1000]

Crude B 006 4 Crude mix Y [0045 0055] [1000 1000]

Crude C 002 8 Unloading flow rate [0 500]

Crude D 005 5 transfer flow rate [0 500]

Table 4 A fragment of FSA regular expression syntax and 119880

transducers and 119877 can be either

[] The empty string[1198771 119877

119899] Concatenation

1198771 119877

119899 Disjunction

119877Λ Optionality

Identity (119860) Identity the transducer which maps eachelement in 119860 onto itself

119879 ∘ 119880 Composition of the transducers 119879 and 119880macro (Term 119877) Use term as an abbreviation for 119877

that each solution involved in theGA algorithm is guaranteedto be feasible by using the mutation rules generated by DFMmethod while in existing GA algorithms the procedure togenerate feasible solution under complex process constraintsis very time costive The deterministic finite automata (DFA)can easily represent this kind of structure Furthermorethe complex process constraints can be very difficult toexpress with mixed integer programming Consequently itis unfeasible to solve the industrial problem by using MIPsolver

6 Conclusion

In this paper a novel hybrid optimization algorithm whichcombined the finite state method and GA is proposed

350 500 650 800 950 1200 1500 1800 24000

02

04

06

08

1

12

14

Computing time (s)

Varia

nce v

alue

s

Mixed-code GAOur algorithm

Figure 7 Variance values of two methodologies

The proposed algorithm constitutes a reasonable frameworkcapturing both the operating condition and sequencing ruleof the schedule The solution captures all possible sched-ules and removes many redundant sequences of operationsThe algorithm is equivalent to introducing new structure

10 The Scientific World Journal

information into the optimization process which will helpreduce the risk of trapping in a local minimal sequencesolution The hybrid optimization algorithm is an effectiveand robust tool to solve the crude oil scheduling problem interms of efficiency and reliability Algorithms only with thetwo properties are suitable for solving practical engineeringapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partially supported by the China NationalNatural Science Foundation under Grant 61203178 Grant61304214 and Grant 61290323 The authors thank the finan-cial funds fromShanghai Science andTechnologyCommitteeunder Grant 12511501002 and Grant 13511501302

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] C A Mendez I E Grossmann I Harjunkoski and P KaboreldquoA simultaneous optimization approach for off-line blendingand scheduling of oil-refinery operationsrdquo Computers andChemical Engineering vol 30 no 4 pp 614ndash634 2006

[6] M Pan X Li and Y Qian ldquoNew approach for scheduling crudeoil operationsrdquo Chemical Engineering Science vol 64 no 5 pp965ndash983 2009

[7] M F Cardoso R L Salcedo S F de Azevedo and D BarbosaldquoA simulated annealing approach to the solution of minlpproblemsrdquo Computers and Chemical Engineering vol 21 no 12pp 1349ndash1364 1997

[8] I E Grossmann ldquoReview of nonlinear mixed-integer anddisjunctive programming techniquesrdquo Optimization and Engi-neering vol 3 no 3 pp 227ndash252 2002

[9] E L Lawler and D E Wood ldquoBranch-and-bound methods asurveyrdquo Operations Research vol 14 no 4 pp 699ndash719 1966

[10] M A Duran and I E Grossmann ldquoAn outer-approximationalgorithm for a class of mixed-integer nonlinear programsrdquoMathematical Programming vol 36 no 3 pp 307ndash339 1986

[11] AMGeoffrion ldquoGeneralized Benders decompositionrdquo Journalof Optimization Theory and Applications vol 10 no 4 pp 237ndash260 1972

[12] C DrsquoAmbrosio andA Lodi ldquoMixed integer nonlinear program-ming tools a practical overviewrdquo 4OR vol 9 no 4 pp 329ndash3492011

[13] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence The University of Michigan Press AnnArbor Mich USA 1975

[14] T Yokota M Gen and Y-X Li ldquoGenetic algorithm for non-linear mixed integer programming problems and its applica-tionsrdquo Computers and Industrial Engineering vol 30 no 4 pp905ndash917 1996

[15] L Costa and P Oliveira ldquoEvolutionary algorithms approachto the solution of mixed integer non-linear programmingproblemsrdquo Computers and Chemical Engineering vol 25 no 2-3 pp 257ndash266 2001

[16] J M Ponce-Ortega M Serna-Gonzalez and A Jimenez-Gutierrez ldquoHeat exchanger network synthesis includingdetailed heat exchanger design using genetic algorithmsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8767ndash8780 2007

[17] K-M Bjork and R Nordman ldquoSolving large-scale retrofitheat exchanger network synthesis problems with mathematicaloptimization methodsrdquo Chemical Engineering and ProcessingProcess Intensification vol 44 no 8 pp 869ndash876 2005

[18] S Mouret I E Grossmann and P Pestiaux ldquoA novel priority-slot based continuous-time formulation for crude-oil schedul-ing problemsrdquo Industrial and Engineering Chemistry Researchvol 48 no 18 pp 8515ndash8528 2009

[19] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[20] J E Hopcroft Introduction to AutomataTheory Languages andComputation Pearson Education India New Delhi India 3rdedition 2008

[21] E Roche and Y Schabes Finite-State Language Processing TheMIT Press Cambridge Mass USA 1997

[22] L Karttunen ldquoConstructing lexical transducersrdquo in Proceedingsof the 15th conference on Computational Linguistics vol 1Association for Computational Linguistics 1994

[23] L Karttunen ldquoThe replace operatorrdquo in Proceedings of the 33rdAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1995

[24] L Karttunen ldquoDirected replacementrdquo in Proceedings of the 34thAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1996

[25] L Karttunen and K R Beesley Two-Level Rule Compiler XeroxCorporation Palo Alto Research Center 1992

[26] G vanNoord ldquoFSAutilities a toolbox tomanipulate finite-stateautomatardquo in Automata Implementation pp 87ndash108 SpringerNew York NY USA 1997

The Scientific World Journal 11

[27] D Gerdemann and G van Noord ldquoTransducers from rewriterules with backreferencesrdquo in Proceedings of the 9th Conferenceon European Chapter of the Association for ComputationalLinguistics Association for Computational Linguistics 1999

[28] Y-C Lin K-S Hwang and F-S Wang ldquoA mixed-codingscheme of evolutionary algorithms to solve mixed-integer non-linear programming problemsrdquo Computers and Mathematicswith Applications vol 47 no 8-9 pp 1295ndash1307 2004

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

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Electrical and Computer Engineering

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Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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ArtificialNeural Systems

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RoboticsJournal of

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Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 3

232 Time Variables

119878119894V ge 0 119863

119894V ge 0 119894 isin 119879 V isin 119882 (2)

119878119894V is the start time of operation V if it is assigned to priorityslot 119894 119878

119894V = 0 otherwise119863119894V is the duration of operation V if it is assigned to

priority slot 119894119863119894V = 0 otherwise

233 Operation Variables

119881119905

119894V ge 0 119881119894V119888 ge 0 119894 isin 119879 V isin 119882 119888 isin 119862 (3)

119881119905

119894V is the total volume of crude transferred during operationV if it is assigned to priority slot 119894 119881119905

119894V = 0 otherwise119881119894V119888 is the volume of crude 119888 transferred during operation

V if it is assigned to priority slot 119894 119881119894V119888 = 0 otherwise

234 Resource Variables

119871119905

119894119903 119871119894119903119888 119894 isin 119879 119903 isin 119877 119888 isin 119862 (4)

119871119905

119894119903is the total accumulated level of crude in tank 119903 isin 119877

119878cup119877119862

before the operation was assigned to priority-slot 119894119871119894119903119888

is the accumulated level of crude 119888 in tank 119903 isin 119877119878cup119877119862

before the operation was assigned to priority-slot 119894

24 Objective Function The objective is to maximize thegross margins of the distilled crude blends Let 119866

119888be the

individual gross margin of crude 119888

maxsum119894isin119879

sum

119903isin119877119863

sum

Visin119868119903

sum

119888isin119862

119866119888sdot 119881119894V119888 (5)

25 General Constraints It should be noted that the crudecomposition of blends in tanks is tracked instead of theirpropertiesThe distillation specifications are later enforced bycalculating a posteriori the properties of the blend in termsof its composition For instance in the problem a blendcomposed of 50 of crude A and 50 of crude B has a sulfurconcentration of 0035 which does not meet the specificationfor crude mix X nor for crude mix Y

251 Assignment Constraints In the SOS model exactly oneoperation has to be assigned to each priority slot

sum

Visin119882119885119894V = 1 119894 isin 119879 (6)

252 Variable Constraints Variable constraints are given bytheir definitions Start time duration and global volumevariables are defined with big-119872 constraints

119878119894V + 119863119894V le 119867 sdot 119885

119894V 119894 isin 119879 V isin 119882

119881119905

119894V le 119881119905

V sdot 119885119894V 119894 isin 119879 V isin 119882

119881119905

119894V ge 119881119905

V sdot 119885119894V 119894 isin 119879 V isin 119882

(7)

Crude volume variables are positive variables whose sumequals the corresponding total volume variable

sum

119888isin119862

119881119894V119888 = 119881

119905

119894V (8)

Total and crude level variables are defined by adding tothe initial level in the tank all inlet and outlet transfer volumesof operations of higher priority than the considered priorityslot

119871119905

119894119903= 119871119905

0119903+ sum

119895isin119879119895lt119894

sum

Visin119868119903

119881119905

119894V minus sum

119895isin119879119895lt119894

sum

Visin119874119903

119881119905

119894V

119894 isin 119879 119903 isin 119877

(9)

119871119894119903119888= 119871119900119903119888

+ sum

119895isin119879119895lt119894

sum

Visin119868119903

119881119894V119888 minus sum

119895isin119879119895lt119894

sum

Visin119874119903

119881119894V119888

119894 isin 119879 119903 isin 119877 119888 isin 119862

(10)

253 Sequencing Constraints Sequencing constraintsrestrict the set of possible sequences of operationsCardinality and unloading sequence constraints are specificcases of sequencing constraints More complex sequencingconstraints will also be discussed later

254 Cardinality Constraint Each crude oil marine vesselhas to unload its content exactly oncesum

119894isin119879sumVisin119874119903 119885119894V = 1 119903 isin

119877119881The total number of distillation operations is bounded by

119873119863and119873

119863in order to reduce the cost of CDU switches

119873119863le sum

119894isin119879

sum

Visin119882119863

119885119894V le 119873119863 (11)

255 Unloading Sequence Constraint Marine vessels have tounload in order of arrival to the refinery Considering twovessels 119903

1 1199032isin 1198771198811199031lt 1199032signifies that 119903

1unloads before 119903

2

sum

119895isin119879119895lt119894

sum

Visin1198741199032

119885119895V + sum

119895isin119879119895ge119894

sum

Visin1198741199031

119885119895V le 1 (12)

256 Scheduling Constraints Scheduling constraints restrictthe values taken by time variables according to logistics rules

257 Nonoverlapping Constraint A nonoverlapping con-straint between two sets of operations119882

1sub 119882 and119882

2sub 119882

states that any pair of operations (V1 V2) sub 119882

1times1198822must not

be executed simultaneouslyUnloading operations must not overlap

sum

Visin119882119880

(119878119894V + 119863119894V) le sum

Visin119882119880

119878119895V + 119867 sdot (1 minus sum

Visin119882119880

119885119895V)

119894 119895 isin 119879 119894 lt 119895

(13)

4 The Scientific World Journal

Inlet and outlet transfer operations on a tank must notoverlap

sum

Visin119868119903

(119878119894V + 119863119894V) le sum

Visin119874119903

119878119895V + 119867 sdot (1 minus sum

Visin119874119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119878cup 119877119862

sum

Visin119874119903

(119878119894V + 119863119894V) le sum

Visin119868119903

119878119895V + 119867 sdot (1 minus sum

Visin119868119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119878cup 119877119862

(14)

Although we do not consider crude settling in storagetanks after vessel unloading it could be included in themodelwith amodified version of constraint (14) taking into accounttransition times We define TR

119881as the transition time after

unloading operation V isin 119882119880

and TR as the maximumtransition time TR = maxVisin119882119880TR119881

sum

Visin119868119903

(119878119894V + 119863119894V + TRV sdot 119885119894V)

le sum

Visin119874119903

119878119895V + (119867 + TR) sdot (1 minus sum

Visin119874119903

119885119895V)

(15)

Constraint (15) is valid in the four possible cases

(existV1isin 119868119903 119885119894V1 = 1)

and (existV2isin 119874119903 119885119895V2 = 1) 997904rArr 119878

119894V + 119863119894V1 + TRV1 le 119878119895V2

(existV1isin 119868119903 119885119894V1 = 1)

and (⋁ V2isin 119874119903 119885119895V2 = 1) 997904rArr 119878

119894V + 119863119894V1 le 119867 + TR minus TRV1

(⋁ V1isin 119868119903 119885119894V1 = 0)

and (existV2isin 119874119903 119885119895V2 = 1) 997904rArr 0 le 119878

119895V2

(⋁ V1isin 119868119903 119885119894V1 = 0)

and (⋁ V2isin 119874119903 119885119895V2 = 0) 997904rArr 0 le 119867 + TR

(16)

A tank may charge only one CDU at a time

sum

Visin119874119903

(119878119894V + 119863119894V) le sum

Visin119874119903

119878119895V + 119867 sdot (1 minus sum

Visin119874119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119862

(17)

A CDUmay be charged by only one tank at a time

sum

Visin119868119903

(119878119894V + 119863119894V) le sum

Visin119868119903

119878119895V + 119867 sdot (1 minus sum

Visin119868119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119863

(18)

To avoid schedules in which a transfer is being performedtwice at a time thus possibly violating the flow rate limita-tions constraint (19) is included in the model

119878119894V + 119863119894V le 119878119895V + 119867 sdot (1 minus 119885

119895V) 119894 119895 isin 119879 119894 lt 119895 V isin 119882(19)

258 Continuous Distillation Constraint It is required thatCDUs operate without interruption As CDUs perform onlyone operation at a time the continuous operation constraintis defined by equating the sum of the duration of distillationsto the time horizon

sum

119894isin119879

sum

Visin119868119903

119863119894V = 119867 119903 isin 119877

119863 (20)

259 Resource Availability Constraint Unloading of crudeoil vessels may start only after arrival to the refinery Let 119878

119903

be the arrival time of vessel 119903

119878119894V ge 119878119903 sdot 119885119894V 119894 isin 119879 119903 isin 119877V V isin 119874119903 (21)

2510 Operation Constraints Operation constraints restrictthe values taken by operation and time variables according tooperational rules

2511 Flow Rate Constraint The flow rate of transfer opera-tion V is bounded by FRV and FRV

FRV sdot 119863119894V le 119881119905

119894V le FRV sdot 119863119894V 119894 isin 119879 V isin 119882 (22)

2512 Property Constraint The property 119896 of the blendedproducts transferred during operation V is bounded by119909V119896 and 119909V119896 The property 119896 of the blend is calculated fromthe property 119909

119888119896of crude 119888 assuming that the mixing rule is

linear

119909V119896 sdot 119881119905

119894V le sum

119888isin119862

119909119888119896119881119894V119888 le 119909V119896 sdot 119881

119905

119894V 119894 isin 119879 V isin 119882 119896 isin 119870

(23)

2513 Composition Constraint It has been shown that pro-cesses including both mixing and splitting of streams cannotbe expressed as a linear model Mixing occurs when twostreams are used to fill a tank and is expressed linearly inconstraint (10) Splitting occurs when partially discharging atank resulting in two parts the remaining content of the tankand the transferred products This constraint is nonlinearThe composition of the products transferred during a transferoperation must be identical to the composition of the origintank

119871119894119903119888

119871119905

119894119903

=119881119894V119888

119881119905

119894V 119894 isin 119879 119903 isin 119877 V isin 119874

119903 119888 isin 119862 (24)

Constraint (24) is reformulated as an equation involvingbilinear terms

119881119894V119888 sdot 119871119905

119894119903= 119871119894119903119888sdot 119881119905

119894V 119894 isin 119879 119903 isin 119877 V isin 119874119903 119888 isin 119862 (25)

The Scientific World Journal 5

6

37

8

1

2

4

5

Vessels Storage tanksCharging tanks

CDUs

Figure 1 Crude oil operations system for the problem

Note that constraint (25) is correct even when operationV is not assigned to priority-slot 119894 as then

119881119905

119894V = 119881119894V119888 = 0 (26)

2514 Resource Constraints Resource constraints restrict theuse of resources throughout the scheduling horizon

2515 Tank Capacity Constraint The level of materials inthe tank 119903 must remain between minimum and maximumcapacity limits 119871119905

119903and 119871119905

119903 respectively Let 119871119905

0119903be the initial

total level and let 1198710119903119888

be the initial level of crude 119888 in thetank 119903 As simultaneous charging and discharging of tanks isforbidden the following constraints are sufficient

119871119905

119903le 119871119905

119894119903le 119871119905119903 119894 isin 119879 119903 isin 119877

119878cup 119877119862

0 le 119871119894119903119888le 119871119905119903 119894 isin 119879 119903 isin 119877

119878cup 119877119862 119888 isin 119862

119871119905

119903le 119871119905

0119903+ sum

119894isin119879

sum

Visin119868119903

119881119905

119894V minus sum

119894isin119879

sum

Visin119874119903

119881119905

119894V le 119871119905

119903

119903 isin 119877119878cup 119877119862

0 le 1198710119903119888

+ sum

119894isin119879

sum

Visin119868119903

119881119894V119888 minus sum

119894isin119879

sum

Visin119874119903

119881119894V119888 le 119871

119905

119903

119903 isin 119877119878cup 119877119862 119888 isin 119862

(27)

2516 DemandConstraint Demand constraints define lowerand upper limits 119863

119903and 119863

119903 on total volume of products

transferred out of each charging tank 119903 during the schedulinghorizon

119863119903le sum

119894isin119879

sum

Visin119874119903

119881119905

119894V le 119863119903 119903 isin 119877119862 (28)

3 Finite State Theory

This section presents in a somewhat informal way those basicnotions and definitions from formal language and finite statetheories which are relevant for the sections to follow Relateddefinitions are taken from literature [20 21] Readers whoare unfamiliar with formal language theory are advised toconsult the sources whenever necessary

0 1

2

3

44a

b b

bc

Figure 2 A deterministic finite state automaton (DFA)

0 1

2

3

4

o a

d c

o o o o

g t

Figure 3 Finite state transducer encoding the relation (dog cat)(dog cow)

31 Finite State Automata A DFA is a 5-tuple (119876 Σ 120575 119894 119865)where119876 is a set of states Σ is an alphabet 119894 is the initial state119865 sube 119876 is a set of final states and 120575 is a transition functionmapping 119876 times Σ to 119876 That is for each state 119906 and symbol119886 there is at most one state that can be reached from 119906 byldquofollowingrdquo 119886 (Figure 2)

32 Finite State Transducers A finite state transducer (FST)is a 6-tuple (Σ

1 Σ2 119876 120575 119894 119865) where 119876 119894 and 119865 are the same

as for DFA Σ1is input alphabet Σ

2is output alphabet and 120575

is a function mapping119876times (Σ1cup 120576) times (Σ

2cup 120576) to a subset of

the power set of119876 (Figure 3) Intuitively an FST is much likean NFA except that transitions are made on strings instead ofsymbols and in addition they have outputs

33 Finite State Calculus As argued in Karttunen [22ndash25]many of the rules used can be analyzed as special cases ofregular expressionsThey extend the basic regular expressionwith new operators These extensions make the finite stateautomation and finite state transducer become more suit-able for particular applications The system described belowwas implemented using FSA Utilities [26] a package forimplementing and manipulating finite state automata whichprovides possibilities for defining new regular expressionoperatorsThe part of FSAs built in regular expression syntaxrelevant to this paper is listed in Table 4

One particular useful extension of the basic syntax ofregular expressions is the replace-operator Karttunen [22ndash25] argues that many phonological and morphological rulescan be interpreted as rules which replace a certain portionof the input string Although several implementations of thereplace-operator are proposed the most relevant case for ourpurposes is the so-called ldquoleftmost longest-matchrdquo replace-ment In case of overlapping rule targets in the input thisoperator will replace the leftmost target and in cases where

6 The Scientific World Journal

Schedule 7 - 6 - 8

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

- 3 - 5 - 1 - 3 - 7 - -6 2

Binary

Z61 = 1

Z102

Z43 = 1 Z73 = 1

Z55 = 1

Z26 = 1 Z96 = 1

Z17 = 1 Z87 = 1

= 1

Z38 =

0

0

1

Operation

1

2

3

4

5

6

7

8

6

4 7

2

5

10

9

1 8

3

t0 t1 t2 t3 t4 t5 t6 t7 t8

variable

executions

S =

Z =

Sequence 7-6-8-3-5-1-3-7-6-2

Figure 4 An example to indicate the relationship between binary variable and schedule

a rule target contains a prefix which is also a potential targetthe longer sequence will be replaced Gerdemann and vanNoord [27] implement leftmost longest-match replacementin FSA as the operator

replace (Target LeftContextRightContext) (29)where Target is a transducer defining the actual replacementand LeftContext and RightContext are regular expressionsdefining the left and right context of the rule respectivelyThe segmentation task discussed in the mutation proceduremakes crucial use of longest-match replacement

4 The Hybrid Algorithm

From the point view of optimization efficiency and robust-ness a novel two-level optimization framework based onfinite statemethod andGA is proposed for theMINLPmodelin this section

41 Two-Level Optimization Structure As the foundationof the framework a two-level optimization structure isintroduced Once all binary variables are fixed the originalproblem becomes a relatively simpler model with only con-tinuous variable Following this deal we rewrite (5) as follows

max (119869 (120585 119911)) lArrrArr max119911

[max120585

119869 (120585 119911)] (30)

where 120585 and 119911 represent continuous and binary variablesrespectively Equation (30) shows when 119911 is fixed as 119911 thesubmodel 119869(120585 119911) can be solved optimally by continuous-optimization solvers in the inner level then we update 119911towards the best binary solution 119911lowast in the outer level

We used an example in Figure 4 to show how binarysolution can be mapped to a scheduling sequence Theschedule 119878 = [7683513762] where 7 stands for the specificoperation 7 to assign to position 1 corresponding to the binarydecisions 119885

17= 1

42 Initial Population Based on the sequencing rules [18]and the extension to the regular expression calculus [22ndash25]a DFA model which builds legal schedules complying withsequencing rules and operation condition is constructedThewhole set of possible schedules is too huge to be processed atonceTheDFAmodel of the schedule constitutes a reasonableframework capturing all possible schedules and removingmany redundant sequences of operations Initial values ofdecision variables must satisfy the equality constraints andoperation condition and therefore represent a feasible oper-ating point

Here we still use the instance with 8 operations fromMouret et al [18] to describe an efficient sequencing rule by

The Scientific World Journal 7

macro (proceduresegmentation segmentation of the input sequence into a set of sub-sequencemutation apply mutation rulesclean up) remove markers

Algorithm 1

using a regular expression A feasible sequence V1sdot sdot sdot V119894sdot sdot sdot V119899

can be described by the following

sequence = (120576 + 119871119886) (119871119887sdot 119871119886)lowast(120576 + 119871

119887)

119871119886= 7 (120576 + 4) (120576 + 6) (120576 + 1 + 14) (120576 + 2 + 26)

119871119887= 8 (120576 + 3) (120576 + 5) (120576 + 1 + 13) (120576 + 2 + 25)

(31)

However this automation suffers from a serious problemof overgeneration For example the short length of thesequence may lead to infeasibility while the long length ofthe sequence may result in an unsolvable model It is aninteresting challenge for finite state syntactic description tospecify a sublanguage that contains all and only the sequencesof valid length

Our solution is to construct a suitable constraint for thesequences of valid length The constraint expressions denotea language that admits sequences of valid length but excludesall others We obtain the desired effect by intersecting theconstraint language with the original language of sequenceexpressions The intersection of the two languages containsall and only the valid dates

ValidSequence = Sequence cap ValidLength (32)

The ValidLength constraint is a language that includes allsequences of length 119899

ValidLength = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)119899 (33)

We have now completed the task of describing thelanguage of valid sequences from the set of possible sequenceexpressions It is also possible to create an automation onthe basis of the regular expression and ValidSequence andthen generate all possible sequences V

1sdot sdot sdot V119894sdot sdot sdot V119899accepted

by the automaton The processes are implemented usingFSA Utilities [26] that is a package for implementing andmanipulating DFA and finite state transducer In order togenerate all possible sequences When all possible sequencesV1sdot sdot sdot V119894sdot sdot sdot V119899accepted by the automaton are generated and

the population of the according possible binary decisions isgenerated In the initial population stage of GA the popu-lation size is the number of individuals When the numberof individuals is given a population of candidate solutions isgenerated by randomly selecting from the population of theall possible binary decisions

43 Rule-Based Mutation Approach In the mutation stagewe use a finite state transducer for this rule-based muta-tion process The rule-based mutation strategy must obey

Input7681325712

Segmentationtransducer

76-81325-712 74-83132-741

Mutationtransducer transducer

Output

748313274

Cleanup

Figure 5 An example of mutation

replace ([identity(SSequence)[]x-][]])

Algorithm 2

the sequencing rule and the nonoverlapping constraint suchthat all involved solutions in GA are feasible

The proposedmutation approach is a two-step procedure

Step 1 Segmentation of the input sequence into a set ofsubsequences (ie the subsequence which belongs to theregular language L7 or L8)

Step 2 Mutation of the subsequences into others

Formally the rule-based mutation procedure is imple-mented as the composition of three transducers (seeAlgorithm 1)

An example of mutation including the intermediate stepsis given for the sequence ldquo7681325712rdquo as shown in Figure 5

431 Segmentation Transducer Segmentation transducersplits an input sequence into subsequences The goal ofsegmentation is to provide a convenient representation levelfor the next mutation step

Segmentation is defined as shown in Algorithm 2The macro ldquoSSequencerdquo defines the set of subsequences

The subsequences which belong to the regular languageL7 and L8 are displayed in Tables 1 and 2 Segmentationattaches the marker ldquondashrdquo to each subsequence The Targetsare identified using leftmost longest-match and thus at eachpoint in the input only the longest valid segment is marked

432 The Mutation Rules In the GA process the mutationrules are made by carefully considering nonoverlapping con-straint between operations A concrete instance for partiallyillustrating the mutation rules is given in Algorithm 3 Notethat the final element of the left-contextmust be amarker and

8 The Scientific World Journal

marco (conversion 1198717subsequence rules)

∘ 1198718subsequence rules)

marco (1198717subsequence rules

replace (2 4 6 times 1 7 mdash)∘ replace (14 26 41 42 46 61 62 times 12 7 mdash)∘ replace (142 412 414 426 461 462 612 614 626 times 126 7 mdash)marco (119871

8subsequence rules

replace (2 3 5 times 1 8 mdash)∘ replace (13 25 31 32 35 51 52 times 12 8mdash)∘ replace (132 312 313 325 351 352 512 513 525 times 125 8 mdash)

Algorithm 3 An example to demonstrate the mutation rules

Table 1 Subsequence belonging to 1198717

Length Sequences belonging to 1198717

1 72 71 72 74 763 712 714 726 741 742 746 761 7624 7126 7142 7412 7414 7426 7461 7462 7612 7614 76265 71426 74126 74142 74612 74614 74626 76126 761426 741426 746126 746142 7614267 7461426

the target itself ends in ldquondashrdquo This ensures that mutation rulescannot apply to the same subsequence

5 Experimental Study

In this section the same problem from the literature [18] isused for computational experiments The proposed method-ology is compared with existing promising algorithmsmixed-coding GA [15 28] Figure 1 depicts the refineryconfiguration for problem The data involved in the problemare given in Table 3 The performance comparison withdifferent computing times such as 350 s 500 s 2400 s isconducted The objective value is used to statistically analyzethe optimization results

The performance comparison between the two method-ologies used is illustrated in Figure 6 which shows that thehybrid optimization algorithm which combined the finitestate method and GAwill statistically outperform the mixed-coding counterpart The genetic algorithm which combinedthe finite state method and GA finds feasible solutions veryfast and is able to find better solutions in reasonable time

In Figure 7 we compare the objective variance of eachiteration in the two evolution processes of these two kindsof methodology By tracking the evolution process we findthat the mixed-coding GA is easy to stick in a local minimalsequence solution This situation only can be improvedthrough increasing the mutation scaling factor Howeverthis may result in a hard convergence unless sufficientiterations are implemented As for the hybrid optimizationalgorithm the optimization processes of binary variableand continuous variable are separated The performanceof the whole methodology mainly depends on the FSM

350 500 650 800 950 1200 1500 1800 240020

30

40

50

60

70

80

Computer time

Obj

ectiv

e val

ue

Mixed-code GAOur algorithm

Figure 6 Average objective values of two methodologies

which captures most promising schedules and removes manyredundant sequences of operations so that the user can usea small population size of corresponding discrete variables toobtain suboptimal solutions From Figure 7 we see that theproposed method has converged at 350 iterations as opposedto 2400 iterations for the mixed-coding GA

The success of the proposed algorithm lies in a compre-hensive analysis of the region of the search space and itscapacity to focus the search on the regions with the partialsolution One of the good merits of the hybrid algorithm is

The Scientific World Journal 9

Table 2 Subsequence belonging to 1198718

Length Sequences belonging to 1198718

1 82 81 82 83 853 812 813 825 831 832 835 851 8524 8125 8132 8312 8313 8325 8351 8352 8512 8513 85255 81325 83125 83132 83512 83513 83525 85132 851256 831325 835125 835132 8513257 8351325

Table 3 Problem data

Scheduling horizon 8 daysVessels Arrival time Composition Amount of crudeVessel 1 0 100 A 1000Vessel 2 4 100 B 1000Storage tanks Capacity Initial composition Initial amountTank 1 [0 1000] 100 A 250Tank 2 [0 1000] 100 B 750Charging tanks Capacity Initial composition Initial amountTank 1 (mix X) [0 1000] 100 C 500Tank 1 (mix X) [0 1000] 100 D 500Crudes 1 Gross margin Crude mixtures Property1 DemandCrude A 001 9 Crude mix X [0015 0025] [1000 1000]

Crude B 006 4 Crude mix Y [0045 0055] [1000 1000]

Crude C 002 8 Unloading flow rate [0 500]

Crude D 005 5 transfer flow rate [0 500]

Table 4 A fragment of FSA regular expression syntax and 119880

transducers and 119877 can be either

[] The empty string[1198771 119877

119899] Concatenation

1198771 119877

119899 Disjunction

119877Λ Optionality

Identity (119860) Identity the transducer which maps eachelement in 119860 onto itself

119879 ∘ 119880 Composition of the transducers 119879 and 119880macro (Term 119877) Use term as an abbreviation for 119877

that each solution involved in theGA algorithm is guaranteedto be feasible by using the mutation rules generated by DFMmethod while in existing GA algorithms the procedure togenerate feasible solution under complex process constraintsis very time costive The deterministic finite automata (DFA)can easily represent this kind of structure Furthermorethe complex process constraints can be very difficult toexpress with mixed integer programming Consequently itis unfeasible to solve the industrial problem by using MIPsolver

6 Conclusion

In this paper a novel hybrid optimization algorithm whichcombined the finite state method and GA is proposed

350 500 650 800 950 1200 1500 1800 24000

02

04

06

08

1

12

14

Computing time (s)

Varia

nce v

alue

s

Mixed-code GAOur algorithm

Figure 7 Variance values of two methodologies

The proposed algorithm constitutes a reasonable frameworkcapturing both the operating condition and sequencing ruleof the schedule The solution captures all possible sched-ules and removes many redundant sequences of operationsThe algorithm is equivalent to introducing new structure

10 The Scientific World Journal

information into the optimization process which will helpreduce the risk of trapping in a local minimal sequencesolution The hybrid optimization algorithm is an effectiveand robust tool to solve the crude oil scheduling problem interms of efficiency and reliability Algorithms only with thetwo properties are suitable for solving practical engineeringapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partially supported by the China NationalNatural Science Foundation under Grant 61203178 Grant61304214 and Grant 61290323 The authors thank the finan-cial funds fromShanghai Science andTechnologyCommitteeunder Grant 12511501002 and Grant 13511501302

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] C A Mendez I E Grossmann I Harjunkoski and P KaboreldquoA simultaneous optimization approach for off-line blendingand scheduling of oil-refinery operationsrdquo Computers andChemical Engineering vol 30 no 4 pp 614ndash634 2006

[6] M Pan X Li and Y Qian ldquoNew approach for scheduling crudeoil operationsrdquo Chemical Engineering Science vol 64 no 5 pp965ndash983 2009

[7] M F Cardoso R L Salcedo S F de Azevedo and D BarbosaldquoA simulated annealing approach to the solution of minlpproblemsrdquo Computers and Chemical Engineering vol 21 no 12pp 1349ndash1364 1997

[8] I E Grossmann ldquoReview of nonlinear mixed-integer anddisjunctive programming techniquesrdquo Optimization and Engi-neering vol 3 no 3 pp 227ndash252 2002

[9] E L Lawler and D E Wood ldquoBranch-and-bound methods asurveyrdquo Operations Research vol 14 no 4 pp 699ndash719 1966

[10] M A Duran and I E Grossmann ldquoAn outer-approximationalgorithm for a class of mixed-integer nonlinear programsrdquoMathematical Programming vol 36 no 3 pp 307ndash339 1986

[11] AMGeoffrion ldquoGeneralized Benders decompositionrdquo Journalof Optimization Theory and Applications vol 10 no 4 pp 237ndash260 1972

[12] C DrsquoAmbrosio andA Lodi ldquoMixed integer nonlinear program-ming tools a practical overviewrdquo 4OR vol 9 no 4 pp 329ndash3492011

[13] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence The University of Michigan Press AnnArbor Mich USA 1975

[14] T Yokota M Gen and Y-X Li ldquoGenetic algorithm for non-linear mixed integer programming problems and its applica-tionsrdquo Computers and Industrial Engineering vol 30 no 4 pp905ndash917 1996

[15] L Costa and P Oliveira ldquoEvolutionary algorithms approachto the solution of mixed integer non-linear programmingproblemsrdquo Computers and Chemical Engineering vol 25 no 2-3 pp 257ndash266 2001

[16] J M Ponce-Ortega M Serna-Gonzalez and A Jimenez-Gutierrez ldquoHeat exchanger network synthesis includingdetailed heat exchanger design using genetic algorithmsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8767ndash8780 2007

[17] K-M Bjork and R Nordman ldquoSolving large-scale retrofitheat exchanger network synthesis problems with mathematicaloptimization methodsrdquo Chemical Engineering and ProcessingProcess Intensification vol 44 no 8 pp 869ndash876 2005

[18] S Mouret I E Grossmann and P Pestiaux ldquoA novel priority-slot based continuous-time formulation for crude-oil schedul-ing problemsrdquo Industrial and Engineering Chemistry Researchvol 48 no 18 pp 8515ndash8528 2009

[19] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[20] J E Hopcroft Introduction to AutomataTheory Languages andComputation Pearson Education India New Delhi India 3rdedition 2008

[21] E Roche and Y Schabes Finite-State Language Processing TheMIT Press Cambridge Mass USA 1997

[22] L Karttunen ldquoConstructing lexical transducersrdquo in Proceedingsof the 15th conference on Computational Linguistics vol 1Association for Computational Linguistics 1994

[23] L Karttunen ldquoThe replace operatorrdquo in Proceedings of the 33rdAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1995

[24] L Karttunen ldquoDirected replacementrdquo in Proceedings of the 34thAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1996

[25] L Karttunen and K R Beesley Two-Level Rule Compiler XeroxCorporation Palo Alto Research Center 1992

[26] G vanNoord ldquoFSAutilities a toolbox tomanipulate finite-stateautomatardquo in Automata Implementation pp 87ndash108 SpringerNew York NY USA 1997

The Scientific World Journal 11

[27] D Gerdemann and G van Noord ldquoTransducers from rewriterules with backreferencesrdquo in Proceedings of the 9th Conferenceon European Chapter of the Association for ComputationalLinguistics Association for Computational Linguistics 1999

[28] Y-C Lin K-S Hwang and F-S Wang ldquoA mixed-codingscheme of evolutionary algorithms to solve mixed-integer non-linear programming problemsrdquo Computers and Mathematicswith Applications vol 47 no 8-9 pp 1295ndash1307 2004

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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Advances in

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4 The Scientific World Journal

Inlet and outlet transfer operations on a tank must notoverlap

sum

Visin119868119903

(119878119894V + 119863119894V) le sum

Visin119874119903

119878119895V + 119867 sdot (1 minus sum

Visin119874119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119878cup 119877119862

sum

Visin119874119903

(119878119894V + 119863119894V) le sum

Visin119868119903

119878119895V + 119867 sdot (1 minus sum

Visin119868119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119878cup 119877119862

(14)

Although we do not consider crude settling in storagetanks after vessel unloading it could be included in themodelwith amodified version of constraint (14) taking into accounttransition times We define TR

119881as the transition time after

unloading operation V isin 119882119880

and TR as the maximumtransition time TR = maxVisin119882119880TR119881

sum

Visin119868119903

(119878119894V + 119863119894V + TRV sdot 119885119894V)

le sum

Visin119874119903

119878119895V + (119867 + TR) sdot (1 minus sum

Visin119874119903

119885119895V)

(15)

Constraint (15) is valid in the four possible cases

(existV1isin 119868119903 119885119894V1 = 1)

and (existV2isin 119874119903 119885119895V2 = 1) 997904rArr 119878

119894V + 119863119894V1 + TRV1 le 119878119895V2

(existV1isin 119868119903 119885119894V1 = 1)

and (⋁ V2isin 119874119903 119885119895V2 = 1) 997904rArr 119878

119894V + 119863119894V1 le 119867 + TR minus TRV1

(⋁ V1isin 119868119903 119885119894V1 = 0)

and (existV2isin 119874119903 119885119895V2 = 1) 997904rArr 0 le 119878

119895V2

(⋁ V1isin 119868119903 119885119894V1 = 0)

and (⋁ V2isin 119874119903 119885119895V2 = 0) 997904rArr 0 le 119867 + TR

(16)

A tank may charge only one CDU at a time

sum

Visin119874119903

(119878119894V + 119863119894V) le sum

Visin119874119903

119878119895V + 119867 sdot (1 minus sum

Visin119874119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119862

(17)

A CDUmay be charged by only one tank at a time

sum

Visin119868119903

(119878119894V + 119863119894V) le sum

Visin119868119903

119878119895V + 119867 sdot (1 minus sum

Visin119868119903

119885119895V)

119894 119895 isin 119879 119894 lt 119895 119903 isin 119877119863

(18)

To avoid schedules in which a transfer is being performedtwice at a time thus possibly violating the flow rate limita-tions constraint (19) is included in the model

119878119894V + 119863119894V le 119878119895V + 119867 sdot (1 minus 119885

119895V) 119894 119895 isin 119879 119894 lt 119895 V isin 119882(19)

258 Continuous Distillation Constraint It is required thatCDUs operate without interruption As CDUs perform onlyone operation at a time the continuous operation constraintis defined by equating the sum of the duration of distillationsto the time horizon

sum

119894isin119879

sum

Visin119868119903

119863119894V = 119867 119903 isin 119877

119863 (20)

259 Resource Availability Constraint Unloading of crudeoil vessels may start only after arrival to the refinery Let 119878

119903

be the arrival time of vessel 119903

119878119894V ge 119878119903 sdot 119885119894V 119894 isin 119879 119903 isin 119877V V isin 119874119903 (21)

2510 Operation Constraints Operation constraints restrictthe values taken by operation and time variables according tooperational rules

2511 Flow Rate Constraint The flow rate of transfer opera-tion V is bounded by FRV and FRV

FRV sdot 119863119894V le 119881119905

119894V le FRV sdot 119863119894V 119894 isin 119879 V isin 119882 (22)

2512 Property Constraint The property 119896 of the blendedproducts transferred during operation V is bounded by119909V119896 and 119909V119896 The property 119896 of the blend is calculated fromthe property 119909

119888119896of crude 119888 assuming that the mixing rule is

linear

119909V119896 sdot 119881119905

119894V le sum

119888isin119862

119909119888119896119881119894V119888 le 119909V119896 sdot 119881

119905

119894V 119894 isin 119879 V isin 119882 119896 isin 119870

(23)

2513 Composition Constraint It has been shown that pro-cesses including both mixing and splitting of streams cannotbe expressed as a linear model Mixing occurs when twostreams are used to fill a tank and is expressed linearly inconstraint (10) Splitting occurs when partially discharging atank resulting in two parts the remaining content of the tankand the transferred products This constraint is nonlinearThe composition of the products transferred during a transferoperation must be identical to the composition of the origintank

119871119894119903119888

119871119905

119894119903

=119881119894V119888

119881119905

119894V 119894 isin 119879 119903 isin 119877 V isin 119874

119903 119888 isin 119862 (24)

Constraint (24) is reformulated as an equation involvingbilinear terms

119881119894V119888 sdot 119871119905

119894119903= 119871119894119903119888sdot 119881119905

119894V 119894 isin 119879 119903 isin 119877 V isin 119874119903 119888 isin 119862 (25)

The Scientific World Journal 5

6

37

8

1

2

4

5

Vessels Storage tanksCharging tanks

CDUs

Figure 1 Crude oil operations system for the problem

Note that constraint (25) is correct even when operationV is not assigned to priority-slot 119894 as then

119881119905

119894V = 119881119894V119888 = 0 (26)

2514 Resource Constraints Resource constraints restrict theuse of resources throughout the scheduling horizon

2515 Tank Capacity Constraint The level of materials inthe tank 119903 must remain between minimum and maximumcapacity limits 119871119905

119903and 119871119905

119903 respectively Let 119871119905

0119903be the initial

total level and let 1198710119903119888

be the initial level of crude 119888 in thetank 119903 As simultaneous charging and discharging of tanks isforbidden the following constraints are sufficient

119871119905

119903le 119871119905

119894119903le 119871119905119903 119894 isin 119879 119903 isin 119877

119878cup 119877119862

0 le 119871119894119903119888le 119871119905119903 119894 isin 119879 119903 isin 119877

119878cup 119877119862 119888 isin 119862

119871119905

119903le 119871119905

0119903+ sum

119894isin119879

sum

Visin119868119903

119881119905

119894V minus sum

119894isin119879

sum

Visin119874119903

119881119905

119894V le 119871119905

119903

119903 isin 119877119878cup 119877119862

0 le 1198710119903119888

+ sum

119894isin119879

sum

Visin119868119903

119881119894V119888 minus sum

119894isin119879

sum

Visin119874119903

119881119894V119888 le 119871

119905

119903

119903 isin 119877119878cup 119877119862 119888 isin 119862

(27)

2516 DemandConstraint Demand constraints define lowerand upper limits 119863

119903and 119863

119903 on total volume of products

transferred out of each charging tank 119903 during the schedulinghorizon

119863119903le sum

119894isin119879

sum

Visin119874119903

119881119905

119894V le 119863119903 119903 isin 119877119862 (28)

3 Finite State Theory

This section presents in a somewhat informal way those basicnotions and definitions from formal language and finite statetheories which are relevant for the sections to follow Relateddefinitions are taken from literature [20 21] Readers whoare unfamiliar with formal language theory are advised toconsult the sources whenever necessary

0 1

2

3

44a

b b

bc

Figure 2 A deterministic finite state automaton (DFA)

0 1

2

3

4

o a

d c

o o o o

g t

Figure 3 Finite state transducer encoding the relation (dog cat)(dog cow)

31 Finite State Automata A DFA is a 5-tuple (119876 Σ 120575 119894 119865)where119876 is a set of states Σ is an alphabet 119894 is the initial state119865 sube 119876 is a set of final states and 120575 is a transition functionmapping 119876 times Σ to 119876 That is for each state 119906 and symbol119886 there is at most one state that can be reached from 119906 byldquofollowingrdquo 119886 (Figure 2)

32 Finite State Transducers A finite state transducer (FST)is a 6-tuple (Σ

1 Σ2 119876 120575 119894 119865) where 119876 119894 and 119865 are the same

as for DFA Σ1is input alphabet Σ

2is output alphabet and 120575

is a function mapping119876times (Σ1cup 120576) times (Σ

2cup 120576) to a subset of

the power set of119876 (Figure 3) Intuitively an FST is much likean NFA except that transitions are made on strings instead ofsymbols and in addition they have outputs

33 Finite State Calculus As argued in Karttunen [22ndash25]many of the rules used can be analyzed as special cases ofregular expressionsThey extend the basic regular expressionwith new operators These extensions make the finite stateautomation and finite state transducer become more suit-able for particular applications The system described belowwas implemented using FSA Utilities [26] a package forimplementing and manipulating finite state automata whichprovides possibilities for defining new regular expressionoperatorsThe part of FSAs built in regular expression syntaxrelevant to this paper is listed in Table 4

One particular useful extension of the basic syntax ofregular expressions is the replace-operator Karttunen [22ndash25] argues that many phonological and morphological rulescan be interpreted as rules which replace a certain portionof the input string Although several implementations of thereplace-operator are proposed the most relevant case for ourpurposes is the so-called ldquoleftmost longest-matchrdquo replace-ment In case of overlapping rule targets in the input thisoperator will replace the leftmost target and in cases where

6 The Scientific World Journal

Schedule 7 - 6 - 8

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

- 3 - 5 - 1 - 3 - 7 - -6 2

Binary

Z61 = 1

Z102

Z43 = 1 Z73 = 1

Z55 = 1

Z26 = 1 Z96 = 1

Z17 = 1 Z87 = 1

= 1

Z38 =

0

0

1

Operation

1

2

3

4

5

6

7

8

6

4 7

2

5

10

9

1 8

3

t0 t1 t2 t3 t4 t5 t6 t7 t8

variable

executions

S =

Z =

Sequence 7-6-8-3-5-1-3-7-6-2

Figure 4 An example to indicate the relationship between binary variable and schedule

a rule target contains a prefix which is also a potential targetthe longer sequence will be replaced Gerdemann and vanNoord [27] implement leftmost longest-match replacementin FSA as the operator

replace (Target LeftContextRightContext) (29)where Target is a transducer defining the actual replacementand LeftContext and RightContext are regular expressionsdefining the left and right context of the rule respectivelyThe segmentation task discussed in the mutation proceduremakes crucial use of longest-match replacement

4 The Hybrid Algorithm

From the point view of optimization efficiency and robust-ness a novel two-level optimization framework based onfinite statemethod andGA is proposed for theMINLPmodelin this section

41 Two-Level Optimization Structure As the foundationof the framework a two-level optimization structure isintroduced Once all binary variables are fixed the originalproblem becomes a relatively simpler model with only con-tinuous variable Following this deal we rewrite (5) as follows

max (119869 (120585 119911)) lArrrArr max119911

[max120585

119869 (120585 119911)] (30)

where 120585 and 119911 represent continuous and binary variablesrespectively Equation (30) shows when 119911 is fixed as 119911 thesubmodel 119869(120585 119911) can be solved optimally by continuous-optimization solvers in the inner level then we update 119911towards the best binary solution 119911lowast in the outer level

We used an example in Figure 4 to show how binarysolution can be mapped to a scheduling sequence Theschedule 119878 = [7683513762] where 7 stands for the specificoperation 7 to assign to position 1 corresponding to the binarydecisions 119885

17= 1

42 Initial Population Based on the sequencing rules [18]and the extension to the regular expression calculus [22ndash25]a DFA model which builds legal schedules complying withsequencing rules and operation condition is constructedThewhole set of possible schedules is too huge to be processed atonceTheDFAmodel of the schedule constitutes a reasonableframework capturing all possible schedules and removingmany redundant sequences of operations Initial values ofdecision variables must satisfy the equality constraints andoperation condition and therefore represent a feasible oper-ating point

Here we still use the instance with 8 operations fromMouret et al [18] to describe an efficient sequencing rule by

The Scientific World Journal 7

macro (proceduresegmentation segmentation of the input sequence into a set of sub-sequencemutation apply mutation rulesclean up) remove markers

Algorithm 1

using a regular expression A feasible sequence V1sdot sdot sdot V119894sdot sdot sdot V119899

can be described by the following

sequence = (120576 + 119871119886) (119871119887sdot 119871119886)lowast(120576 + 119871

119887)

119871119886= 7 (120576 + 4) (120576 + 6) (120576 + 1 + 14) (120576 + 2 + 26)

119871119887= 8 (120576 + 3) (120576 + 5) (120576 + 1 + 13) (120576 + 2 + 25)

(31)

However this automation suffers from a serious problemof overgeneration For example the short length of thesequence may lead to infeasibility while the long length ofthe sequence may result in an unsolvable model It is aninteresting challenge for finite state syntactic description tospecify a sublanguage that contains all and only the sequencesof valid length

Our solution is to construct a suitable constraint for thesequences of valid length The constraint expressions denotea language that admits sequences of valid length but excludesall others We obtain the desired effect by intersecting theconstraint language with the original language of sequenceexpressions The intersection of the two languages containsall and only the valid dates

ValidSequence = Sequence cap ValidLength (32)

The ValidLength constraint is a language that includes allsequences of length 119899

ValidLength = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)119899 (33)

We have now completed the task of describing thelanguage of valid sequences from the set of possible sequenceexpressions It is also possible to create an automation onthe basis of the regular expression and ValidSequence andthen generate all possible sequences V

1sdot sdot sdot V119894sdot sdot sdot V119899accepted

by the automaton The processes are implemented usingFSA Utilities [26] that is a package for implementing andmanipulating DFA and finite state transducer In order togenerate all possible sequences When all possible sequencesV1sdot sdot sdot V119894sdot sdot sdot V119899accepted by the automaton are generated and

the population of the according possible binary decisions isgenerated In the initial population stage of GA the popu-lation size is the number of individuals When the numberof individuals is given a population of candidate solutions isgenerated by randomly selecting from the population of theall possible binary decisions

43 Rule-Based Mutation Approach In the mutation stagewe use a finite state transducer for this rule-based muta-tion process The rule-based mutation strategy must obey

Input7681325712

Segmentationtransducer

76-81325-712 74-83132-741

Mutationtransducer transducer

Output

748313274

Cleanup

Figure 5 An example of mutation

replace ([identity(SSequence)[]x-][]])

Algorithm 2

the sequencing rule and the nonoverlapping constraint suchthat all involved solutions in GA are feasible

The proposedmutation approach is a two-step procedure

Step 1 Segmentation of the input sequence into a set ofsubsequences (ie the subsequence which belongs to theregular language L7 or L8)

Step 2 Mutation of the subsequences into others

Formally the rule-based mutation procedure is imple-mented as the composition of three transducers (seeAlgorithm 1)

An example of mutation including the intermediate stepsis given for the sequence ldquo7681325712rdquo as shown in Figure 5

431 Segmentation Transducer Segmentation transducersplits an input sequence into subsequences The goal ofsegmentation is to provide a convenient representation levelfor the next mutation step

Segmentation is defined as shown in Algorithm 2The macro ldquoSSequencerdquo defines the set of subsequences

The subsequences which belong to the regular languageL7 and L8 are displayed in Tables 1 and 2 Segmentationattaches the marker ldquondashrdquo to each subsequence The Targetsare identified using leftmost longest-match and thus at eachpoint in the input only the longest valid segment is marked

432 The Mutation Rules In the GA process the mutationrules are made by carefully considering nonoverlapping con-straint between operations A concrete instance for partiallyillustrating the mutation rules is given in Algorithm 3 Notethat the final element of the left-contextmust be amarker and

8 The Scientific World Journal

marco (conversion 1198717subsequence rules)

∘ 1198718subsequence rules)

marco (1198717subsequence rules

replace (2 4 6 times 1 7 mdash)∘ replace (14 26 41 42 46 61 62 times 12 7 mdash)∘ replace (142 412 414 426 461 462 612 614 626 times 126 7 mdash)marco (119871

8subsequence rules

replace (2 3 5 times 1 8 mdash)∘ replace (13 25 31 32 35 51 52 times 12 8mdash)∘ replace (132 312 313 325 351 352 512 513 525 times 125 8 mdash)

Algorithm 3 An example to demonstrate the mutation rules

Table 1 Subsequence belonging to 1198717

Length Sequences belonging to 1198717

1 72 71 72 74 763 712 714 726 741 742 746 761 7624 7126 7142 7412 7414 7426 7461 7462 7612 7614 76265 71426 74126 74142 74612 74614 74626 76126 761426 741426 746126 746142 7614267 7461426

the target itself ends in ldquondashrdquo This ensures that mutation rulescannot apply to the same subsequence

5 Experimental Study

In this section the same problem from the literature [18] isused for computational experiments The proposed method-ology is compared with existing promising algorithmsmixed-coding GA [15 28] Figure 1 depicts the refineryconfiguration for problem The data involved in the problemare given in Table 3 The performance comparison withdifferent computing times such as 350 s 500 s 2400 s isconducted The objective value is used to statistically analyzethe optimization results

The performance comparison between the two method-ologies used is illustrated in Figure 6 which shows that thehybrid optimization algorithm which combined the finitestate method and GAwill statistically outperform the mixed-coding counterpart The genetic algorithm which combinedthe finite state method and GA finds feasible solutions veryfast and is able to find better solutions in reasonable time

In Figure 7 we compare the objective variance of eachiteration in the two evolution processes of these two kindsof methodology By tracking the evolution process we findthat the mixed-coding GA is easy to stick in a local minimalsequence solution This situation only can be improvedthrough increasing the mutation scaling factor Howeverthis may result in a hard convergence unless sufficientiterations are implemented As for the hybrid optimizationalgorithm the optimization processes of binary variableand continuous variable are separated The performanceof the whole methodology mainly depends on the FSM

350 500 650 800 950 1200 1500 1800 240020

30

40

50

60

70

80

Computer time

Obj

ectiv

e val

ue

Mixed-code GAOur algorithm

Figure 6 Average objective values of two methodologies

which captures most promising schedules and removes manyredundant sequences of operations so that the user can usea small population size of corresponding discrete variables toobtain suboptimal solutions From Figure 7 we see that theproposed method has converged at 350 iterations as opposedto 2400 iterations for the mixed-coding GA

The success of the proposed algorithm lies in a compre-hensive analysis of the region of the search space and itscapacity to focus the search on the regions with the partialsolution One of the good merits of the hybrid algorithm is

The Scientific World Journal 9

Table 2 Subsequence belonging to 1198718

Length Sequences belonging to 1198718

1 82 81 82 83 853 812 813 825 831 832 835 851 8524 8125 8132 8312 8313 8325 8351 8352 8512 8513 85255 81325 83125 83132 83512 83513 83525 85132 851256 831325 835125 835132 8513257 8351325

Table 3 Problem data

Scheduling horizon 8 daysVessels Arrival time Composition Amount of crudeVessel 1 0 100 A 1000Vessel 2 4 100 B 1000Storage tanks Capacity Initial composition Initial amountTank 1 [0 1000] 100 A 250Tank 2 [0 1000] 100 B 750Charging tanks Capacity Initial composition Initial amountTank 1 (mix X) [0 1000] 100 C 500Tank 1 (mix X) [0 1000] 100 D 500Crudes 1 Gross margin Crude mixtures Property1 DemandCrude A 001 9 Crude mix X [0015 0025] [1000 1000]

Crude B 006 4 Crude mix Y [0045 0055] [1000 1000]

Crude C 002 8 Unloading flow rate [0 500]

Crude D 005 5 transfer flow rate [0 500]

Table 4 A fragment of FSA regular expression syntax and 119880

transducers and 119877 can be either

[] The empty string[1198771 119877

119899] Concatenation

1198771 119877

119899 Disjunction

119877Λ Optionality

Identity (119860) Identity the transducer which maps eachelement in 119860 onto itself

119879 ∘ 119880 Composition of the transducers 119879 and 119880macro (Term 119877) Use term as an abbreviation for 119877

that each solution involved in theGA algorithm is guaranteedto be feasible by using the mutation rules generated by DFMmethod while in existing GA algorithms the procedure togenerate feasible solution under complex process constraintsis very time costive The deterministic finite automata (DFA)can easily represent this kind of structure Furthermorethe complex process constraints can be very difficult toexpress with mixed integer programming Consequently itis unfeasible to solve the industrial problem by using MIPsolver

6 Conclusion

In this paper a novel hybrid optimization algorithm whichcombined the finite state method and GA is proposed

350 500 650 800 950 1200 1500 1800 24000

02

04

06

08

1

12

14

Computing time (s)

Varia

nce v

alue

s

Mixed-code GAOur algorithm

Figure 7 Variance values of two methodologies

The proposed algorithm constitutes a reasonable frameworkcapturing both the operating condition and sequencing ruleof the schedule The solution captures all possible sched-ules and removes many redundant sequences of operationsThe algorithm is equivalent to introducing new structure

10 The Scientific World Journal

information into the optimization process which will helpreduce the risk of trapping in a local minimal sequencesolution The hybrid optimization algorithm is an effectiveand robust tool to solve the crude oil scheduling problem interms of efficiency and reliability Algorithms only with thetwo properties are suitable for solving practical engineeringapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partially supported by the China NationalNatural Science Foundation under Grant 61203178 Grant61304214 and Grant 61290323 The authors thank the finan-cial funds fromShanghai Science andTechnologyCommitteeunder Grant 12511501002 and Grant 13511501302

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] C A Mendez I E Grossmann I Harjunkoski and P KaboreldquoA simultaneous optimization approach for off-line blendingand scheduling of oil-refinery operationsrdquo Computers andChemical Engineering vol 30 no 4 pp 614ndash634 2006

[6] M Pan X Li and Y Qian ldquoNew approach for scheduling crudeoil operationsrdquo Chemical Engineering Science vol 64 no 5 pp965ndash983 2009

[7] M F Cardoso R L Salcedo S F de Azevedo and D BarbosaldquoA simulated annealing approach to the solution of minlpproblemsrdquo Computers and Chemical Engineering vol 21 no 12pp 1349ndash1364 1997

[8] I E Grossmann ldquoReview of nonlinear mixed-integer anddisjunctive programming techniquesrdquo Optimization and Engi-neering vol 3 no 3 pp 227ndash252 2002

[9] E L Lawler and D E Wood ldquoBranch-and-bound methods asurveyrdquo Operations Research vol 14 no 4 pp 699ndash719 1966

[10] M A Duran and I E Grossmann ldquoAn outer-approximationalgorithm for a class of mixed-integer nonlinear programsrdquoMathematical Programming vol 36 no 3 pp 307ndash339 1986

[11] AMGeoffrion ldquoGeneralized Benders decompositionrdquo Journalof Optimization Theory and Applications vol 10 no 4 pp 237ndash260 1972

[12] C DrsquoAmbrosio andA Lodi ldquoMixed integer nonlinear program-ming tools a practical overviewrdquo 4OR vol 9 no 4 pp 329ndash3492011

[13] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence The University of Michigan Press AnnArbor Mich USA 1975

[14] T Yokota M Gen and Y-X Li ldquoGenetic algorithm for non-linear mixed integer programming problems and its applica-tionsrdquo Computers and Industrial Engineering vol 30 no 4 pp905ndash917 1996

[15] L Costa and P Oliveira ldquoEvolutionary algorithms approachto the solution of mixed integer non-linear programmingproblemsrdquo Computers and Chemical Engineering vol 25 no 2-3 pp 257ndash266 2001

[16] J M Ponce-Ortega M Serna-Gonzalez and A Jimenez-Gutierrez ldquoHeat exchanger network synthesis includingdetailed heat exchanger design using genetic algorithmsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8767ndash8780 2007

[17] K-M Bjork and R Nordman ldquoSolving large-scale retrofitheat exchanger network synthesis problems with mathematicaloptimization methodsrdquo Chemical Engineering and ProcessingProcess Intensification vol 44 no 8 pp 869ndash876 2005

[18] S Mouret I E Grossmann and P Pestiaux ldquoA novel priority-slot based continuous-time formulation for crude-oil schedul-ing problemsrdquo Industrial and Engineering Chemistry Researchvol 48 no 18 pp 8515ndash8528 2009

[19] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[20] J E Hopcroft Introduction to AutomataTheory Languages andComputation Pearson Education India New Delhi India 3rdedition 2008

[21] E Roche and Y Schabes Finite-State Language Processing TheMIT Press Cambridge Mass USA 1997

[22] L Karttunen ldquoConstructing lexical transducersrdquo in Proceedingsof the 15th conference on Computational Linguistics vol 1Association for Computational Linguistics 1994

[23] L Karttunen ldquoThe replace operatorrdquo in Proceedings of the 33rdAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1995

[24] L Karttunen ldquoDirected replacementrdquo in Proceedings of the 34thAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1996

[25] L Karttunen and K R Beesley Two-Level Rule Compiler XeroxCorporation Palo Alto Research Center 1992

[26] G vanNoord ldquoFSAutilities a toolbox tomanipulate finite-stateautomatardquo in Automata Implementation pp 87ndash108 SpringerNew York NY USA 1997

The Scientific World Journal 11

[27] D Gerdemann and G van Noord ldquoTransducers from rewriterules with backreferencesrdquo in Proceedings of the 9th Conferenceon European Chapter of the Association for ComputationalLinguistics Association for Computational Linguistics 1999

[28] Y-C Lin K-S Hwang and F-S Wang ldquoA mixed-codingscheme of evolutionary algorithms to solve mixed-integer non-linear programming problemsrdquo Computers and Mathematicswith Applications vol 47 no 8-9 pp 1295ndash1307 2004

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 5

6

37

8

1

2

4

5

Vessels Storage tanksCharging tanks

CDUs

Figure 1 Crude oil operations system for the problem

Note that constraint (25) is correct even when operationV is not assigned to priority-slot 119894 as then

119881119905

119894V = 119881119894V119888 = 0 (26)

2514 Resource Constraints Resource constraints restrict theuse of resources throughout the scheduling horizon

2515 Tank Capacity Constraint The level of materials inthe tank 119903 must remain between minimum and maximumcapacity limits 119871119905

119903and 119871119905

119903 respectively Let 119871119905

0119903be the initial

total level and let 1198710119903119888

be the initial level of crude 119888 in thetank 119903 As simultaneous charging and discharging of tanks isforbidden the following constraints are sufficient

119871119905

119903le 119871119905

119894119903le 119871119905119903 119894 isin 119879 119903 isin 119877

119878cup 119877119862

0 le 119871119894119903119888le 119871119905119903 119894 isin 119879 119903 isin 119877

119878cup 119877119862 119888 isin 119862

119871119905

119903le 119871119905

0119903+ sum

119894isin119879

sum

Visin119868119903

119881119905

119894V minus sum

119894isin119879

sum

Visin119874119903

119881119905

119894V le 119871119905

119903

119903 isin 119877119878cup 119877119862

0 le 1198710119903119888

+ sum

119894isin119879

sum

Visin119868119903

119881119894V119888 minus sum

119894isin119879

sum

Visin119874119903

119881119894V119888 le 119871

119905

119903

119903 isin 119877119878cup 119877119862 119888 isin 119862

(27)

2516 DemandConstraint Demand constraints define lowerand upper limits 119863

119903and 119863

119903 on total volume of products

transferred out of each charging tank 119903 during the schedulinghorizon

119863119903le sum

119894isin119879

sum

Visin119874119903

119881119905

119894V le 119863119903 119903 isin 119877119862 (28)

3 Finite State Theory

This section presents in a somewhat informal way those basicnotions and definitions from formal language and finite statetheories which are relevant for the sections to follow Relateddefinitions are taken from literature [20 21] Readers whoare unfamiliar with formal language theory are advised toconsult the sources whenever necessary

0 1

2

3

44a

b b

bc

Figure 2 A deterministic finite state automaton (DFA)

0 1

2

3

4

o a

d c

o o o o

g t

Figure 3 Finite state transducer encoding the relation (dog cat)(dog cow)

31 Finite State Automata A DFA is a 5-tuple (119876 Σ 120575 119894 119865)where119876 is a set of states Σ is an alphabet 119894 is the initial state119865 sube 119876 is a set of final states and 120575 is a transition functionmapping 119876 times Σ to 119876 That is for each state 119906 and symbol119886 there is at most one state that can be reached from 119906 byldquofollowingrdquo 119886 (Figure 2)

32 Finite State Transducers A finite state transducer (FST)is a 6-tuple (Σ

1 Σ2 119876 120575 119894 119865) where 119876 119894 and 119865 are the same

as for DFA Σ1is input alphabet Σ

2is output alphabet and 120575

is a function mapping119876times (Σ1cup 120576) times (Σ

2cup 120576) to a subset of

the power set of119876 (Figure 3) Intuitively an FST is much likean NFA except that transitions are made on strings instead ofsymbols and in addition they have outputs

33 Finite State Calculus As argued in Karttunen [22ndash25]many of the rules used can be analyzed as special cases ofregular expressionsThey extend the basic regular expressionwith new operators These extensions make the finite stateautomation and finite state transducer become more suit-able for particular applications The system described belowwas implemented using FSA Utilities [26] a package forimplementing and manipulating finite state automata whichprovides possibilities for defining new regular expressionoperatorsThe part of FSAs built in regular expression syntaxrelevant to this paper is listed in Table 4

One particular useful extension of the basic syntax ofregular expressions is the replace-operator Karttunen [22ndash25] argues that many phonological and morphological rulescan be interpreted as rules which replace a certain portionof the input string Although several implementations of thereplace-operator are proposed the most relevant case for ourpurposes is the so-called ldquoleftmost longest-matchrdquo replace-ment In case of overlapping rule targets in the input thisoperator will replace the leftmost target and in cases where

6 The Scientific World Journal

Schedule 7 - 6 - 8

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

- 3 - 5 - 1 - 3 - 7 - -6 2

Binary

Z61 = 1

Z102

Z43 = 1 Z73 = 1

Z55 = 1

Z26 = 1 Z96 = 1

Z17 = 1 Z87 = 1

= 1

Z38 =

0

0

1

Operation

1

2

3

4

5

6

7

8

6

4 7

2

5

10

9

1 8

3

t0 t1 t2 t3 t4 t5 t6 t7 t8

variable

executions

S =

Z =

Sequence 7-6-8-3-5-1-3-7-6-2

Figure 4 An example to indicate the relationship between binary variable and schedule

a rule target contains a prefix which is also a potential targetthe longer sequence will be replaced Gerdemann and vanNoord [27] implement leftmost longest-match replacementin FSA as the operator

replace (Target LeftContextRightContext) (29)where Target is a transducer defining the actual replacementand LeftContext and RightContext are regular expressionsdefining the left and right context of the rule respectivelyThe segmentation task discussed in the mutation proceduremakes crucial use of longest-match replacement

4 The Hybrid Algorithm

From the point view of optimization efficiency and robust-ness a novel two-level optimization framework based onfinite statemethod andGA is proposed for theMINLPmodelin this section

41 Two-Level Optimization Structure As the foundationof the framework a two-level optimization structure isintroduced Once all binary variables are fixed the originalproblem becomes a relatively simpler model with only con-tinuous variable Following this deal we rewrite (5) as follows

max (119869 (120585 119911)) lArrrArr max119911

[max120585

119869 (120585 119911)] (30)

where 120585 and 119911 represent continuous and binary variablesrespectively Equation (30) shows when 119911 is fixed as 119911 thesubmodel 119869(120585 119911) can be solved optimally by continuous-optimization solvers in the inner level then we update 119911towards the best binary solution 119911lowast in the outer level

We used an example in Figure 4 to show how binarysolution can be mapped to a scheduling sequence Theschedule 119878 = [7683513762] where 7 stands for the specificoperation 7 to assign to position 1 corresponding to the binarydecisions 119885

17= 1

42 Initial Population Based on the sequencing rules [18]and the extension to the regular expression calculus [22ndash25]a DFA model which builds legal schedules complying withsequencing rules and operation condition is constructedThewhole set of possible schedules is too huge to be processed atonceTheDFAmodel of the schedule constitutes a reasonableframework capturing all possible schedules and removingmany redundant sequences of operations Initial values ofdecision variables must satisfy the equality constraints andoperation condition and therefore represent a feasible oper-ating point

Here we still use the instance with 8 operations fromMouret et al [18] to describe an efficient sequencing rule by

The Scientific World Journal 7

macro (proceduresegmentation segmentation of the input sequence into a set of sub-sequencemutation apply mutation rulesclean up) remove markers

Algorithm 1

using a regular expression A feasible sequence V1sdot sdot sdot V119894sdot sdot sdot V119899

can be described by the following

sequence = (120576 + 119871119886) (119871119887sdot 119871119886)lowast(120576 + 119871

119887)

119871119886= 7 (120576 + 4) (120576 + 6) (120576 + 1 + 14) (120576 + 2 + 26)

119871119887= 8 (120576 + 3) (120576 + 5) (120576 + 1 + 13) (120576 + 2 + 25)

(31)

However this automation suffers from a serious problemof overgeneration For example the short length of thesequence may lead to infeasibility while the long length ofthe sequence may result in an unsolvable model It is aninteresting challenge for finite state syntactic description tospecify a sublanguage that contains all and only the sequencesof valid length

Our solution is to construct a suitable constraint for thesequences of valid length The constraint expressions denotea language that admits sequences of valid length but excludesall others We obtain the desired effect by intersecting theconstraint language with the original language of sequenceexpressions The intersection of the two languages containsall and only the valid dates

ValidSequence = Sequence cap ValidLength (32)

The ValidLength constraint is a language that includes allsequences of length 119899

ValidLength = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)119899 (33)

We have now completed the task of describing thelanguage of valid sequences from the set of possible sequenceexpressions It is also possible to create an automation onthe basis of the regular expression and ValidSequence andthen generate all possible sequences V

1sdot sdot sdot V119894sdot sdot sdot V119899accepted

by the automaton The processes are implemented usingFSA Utilities [26] that is a package for implementing andmanipulating DFA and finite state transducer In order togenerate all possible sequences When all possible sequencesV1sdot sdot sdot V119894sdot sdot sdot V119899accepted by the automaton are generated and

the population of the according possible binary decisions isgenerated In the initial population stage of GA the popu-lation size is the number of individuals When the numberof individuals is given a population of candidate solutions isgenerated by randomly selecting from the population of theall possible binary decisions

43 Rule-Based Mutation Approach In the mutation stagewe use a finite state transducer for this rule-based muta-tion process The rule-based mutation strategy must obey

Input7681325712

Segmentationtransducer

76-81325-712 74-83132-741

Mutationtransducer transducer

Output

748313274

Cleanup

Figure 5 An example of mutation

replace ([identity(SSequence)[]x-][]])

Algorithm 2

the sequencing rule and the nonoverlapping constraint suchthat all involved solutions in GA are feasible

The proposedmutation approach is a two-step procedure

Step 1 Segmentation of the input sequence into a set ofsubsequences (ie the subsequence which belongs to theregular language L7 or L8)

Step 2 Mutation of the subsequences into others

Formally the rule-based mutation procedure is imple-mented as the composition of three transducers (seeAlgorithm 1)

An example of mutation including the intermediate stepsis given for the sequence ldquo7681325712rdquo as shown in Figure 5

431 Segmentation Transducer Segmentation transducersplits an input sequence into subsequences The goal ofsegmentation is to provide a convenient representation levelfor the next mutation step

Segmentation is defined as shown in Algorithm 2The macro ldquoSSequencerdquo defines the set of subsequences

The subsequences which belong to the regular languageL7 and L8 are displayed in Tables 1 and 2 Segmentationattaches the marker ldquondashrdquo to each subsequence The Targetsare identified using leftmost longest-match and thus at eachpoint in the input only the longest valid segment is marked

432 The Mutation Rules In the GA process the mutationrules are made by carefully considering nonoverlapping con-straint between operations A concrete instance for partiallyillustrating the mutation rules is given in Algorithm 3 Notethat the final element of the left-contextmust be amarker and

8 The Scientific World Journal

marco (conversion 1198717subsequence rules)

∘ 1198718subsequence rules)

marco (1198717subsequence rules

replace (2 4 6 times 1 7 mdash)∘ replace (14 26 41 42 46 61 62 times 12 7 mdash)∘ replace (142 412 414 426 461 462 612 614 626 times 126 7 mdash)marco (119871

8subsequence rules

replace (2 3 5 times 1 8 mdash)∘ replace (13 25 31 32 35 51 52 times 12 8mdash)∘ replace (132 312 313 325 351 352 512 513 525 times 125 8 mdash)

Algorithm 3 An example to demonstrate the mutation rules

Table 1 Subsequence belonging to 1198717

Length Sequences belonging to 1198717

1 72 71 72 74 763 712 714 726 741 742 746 761 7624 7126 7142 7412 7414 7426 7461 7462 7612 7614 76265 71426 74126 74142 74612 74614 74626 76126 761426 741426 746126 746142 7614267 7461426

the target itself ends in ldquondashrdquo This ensures that mutation rulescannot apply to the same subsequence

5 Experimental Study

In this section the same problem from the literature [18] isused for computational experiments The proposed method-ology is compared with existing promising algorithmsmixed-coding GA [15 28] Figure 1 depicts the refineryconfiguration for problem The data involved in the problemare given in Table 3 The performance comparison withdifferent computing times such as 350 s 500 s 2400 s isconducted The objective value is used to statistically analyzethe optimization results

The performance comparison between the two method-ologies used is illustrated in Figure 6 which shows that thehybrid optimization algorithm which combined the finitestate method and GAwill statistically outperform the mixed-coding counterpart The genetic algorithm which combinedthe finite state method and GA finds feasible solutions veryfast and is able to find better solutions in reasonable time

In Figure 7 we compare the objective variance of eachiteration in the two evolution processes of these two kindsof methodology By tracking the evolution process we findthat the mixed-coding GA is easy to stick in a local minimalsequence solution This situation only can be improvedthrough increasing the mutation scaling factor Howeverthis may result in a hard convergence unless sufficientiterations are implemented As for the hybrid optimizationalgorithm the optimization processes of binary variableand continuous variable are separated The performanceof the whole methodology mainly depends on the FSM

350 500 650 800 950 1200 1500 1800 240020

30

40

50

60

70

80

Computer time

Obj

ectiv

e val

ue

Mixed-code GAOur algorithm

Figure 6 Average objective values of two methodologies

which captures most promising schedules and removes manyredundant sequences of operations so that the user can usea small population size of corresponding discrete variables toobtain suboptimal solutions From Figure 7 we see that theproposed method has converged at 350 iterations as opposedto 2400 iterations for the mixed-coding GA

The success of the proposed algorithm lies in a compre-hensive analysis of the region of the search space and itscapacity to focus the search on the regions with the partialsolution One of the good merits of the hybrid algorithm is

The Scientific World Journal 9

Table 2 Subsequence belonging to 1198718

Length Sequences belonging to 1198718

1 82 81 82 83 853 812 813 825 831 832 835 851 8524 8125 8132 8312 8313 8325 8351 8352 8512 8513 85255 81325 83125 83132 83512 83513 83525 85132 851256 831325 835125 835132 8513257 8351325

Table 3 Problem data

Scheduling horizon 8 daysVessels Arrival time Composition Amount of crudeVessel 1 0 100 A 1000Vessel 2 4 100 B 1000Storage tanks Capacity Initial composition Initial amountTank 1 [0 1000] 100 A 250Tank 2 [0 1000] 100 B 750Charging tanks Capacity Initial composition Initial amountTank 1 (mix X) [0 1000] 100 C 500Tank 1 (mix X) [0 1000] 100 D 500Crudes 1 Gross margin Crude mixtures Property1 DemandCrude A 001 9 Crude mix X [0015 0025] [1000 1000]

Crude B 006 4 Crude mix Y [0045 0055] [1000 1000]

Crude C 002 8 Unloading flow rate [0 500]

Crude D 005 5 transfer flow rate [0 500]

Table 4 A fragment of FSA regular expression syntax and 119880

transducers and 119877 can be either

[] The empty string[1198771 119877

119899] Concatenation

1198771 119877

119899 Disjunction

119877Λ Optionality

Identity (119860) Identity the transducer which maps eachelement in 119860 onto itself

119879 ∘ 119880 Composition of the transducers 119879 and 119880macro (Term 119877) Use term as an abbreviation for 119877

that each solution involved in theGA algorithm is guaranteedto be feasible by using the mutation rules generated by DFMmethod while in existing GA algorithms the procedure togenerate feasible solution under complex process constraintsis very time costive The deterministic finite automata (DFA)can easily represent this kind of structure Furthermorethe complex process constraints can be very difficult toexpress with mixed integer programming Consequently itis unfeasible to solve the industrial problem by using MIPsolver

6 Conclusion

In this paper a novel hybrid optimization algorithm whichcombined the finite state method and GA is proposed

350 500 650 800 950 1200 1500 1800 24000

02

04

06

08

1

12

14

Computing time (s)

Varia

nce v

alue

s

Mixed-code GAOur algorithm

Figure 7 Variance values of two methodologies

The proposed algorithm constitutes a reasonable frameworkcapturing both the operating condition and sequencing ruleof the schedule The solution captures all possible sched-ules and removes many redundant sequences of operationsThe algorithm is equivalent to introducing new structure

10 The Scientific World Journal

information into the optimization process which will helpreduce the risk of trapping in a local minimal sequencesolution The hybrid optimization algorithm is an effectiveand robust tool to solve the crude oil scheduling problem interms of efficiency and reliability Algorithms only with thetwo properties are suitable for solving practical engineeringapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partially supported by the China NationalNatural Science Foundation under Grant 61203178 Grant61304214 and Grant 61290323 The authors thank the finan-cial funds fromShanghai Science andTechnologyCommitteeunder Grant 12511501002 and Grant 13511501302

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] C A Mendez I E Grossmann I Harjunkoski and P KaboreldquoA simultaneous optimization approach for off-line blendingand scheduling of oil-refinery operationsrdquo Computers andChemical Engineering vol 30 no 4 pp 614ndash634 2006

[6] M Pan X Li and Y Qian ldquoNew approach for scheduling crudeoil operationsrdquo Chemical Engineering Science vol 64 no 5 pp965ndash983 2009

[7] M F Cardoso R L Salcedo S F de Azevedo and D BarbosaldquoA simulated annealing approach to the solution of minlpproblemsrdquo Computers and Chemical Engineering vol 21 no 12pp 1349ndash1364 1997

[8] I E Grossmann ldquoReview of nonlinear mixed-integer anddisjunctive programming techniquesrdquo Optimization and Engi-neering vol 3 no 3 pp 227ndash252 2002

[9] E L Lawler and D E Wood ldquoBranch-and-bound methods asurveyrdquo Operations Research vol 14 no 4 pp 699ndash719 1966

[10] M A Duran and I E Grossmann ldquoAn outer-approximationalgorithm for a class of mixed-integer nonlinear programsrdquoMathematical Programming vol 36 no 3 pp 307ndash339 1986

[11] AMGeoffrion ldquoGeneralized Benders decompositionrdquo Journalof Optimization Theory and Applications vol 10 no 4 pp 237ndash260 1972

[12] C DrsquoAmbrosio andA Lodi ldquoMixed integer nonlinear program-ming tools a practical overviewrdquo 4OR vol 9 no 4 pp 329ndash3492011

[13] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence The University of Michigan Press AnnArbor Mich USA 1975

[14] T Yokota M Gen and Y-X Li ldquoGenetic algorithm for non-linear mixed integer programming problems and its applica-tionsrdquo Computers and Industrial Engineering vol 30 no 4 pp905ndash917 1996

[15] L Costa and P Oliveira ldquoEvolutionary algorithms approachto the solution of mixed integer non-linear programmingproblemsrdquo Computers and Chemical Engineering vol 25 no 2-3 pp 257ndash266 2001

[16] J M Ponce-Ortega M Serna-Gonzalez and A Jimenez-Gutierrez ldquoHeat exchanger network synthesis includingdetailed heat exchanger design using genetic algorithmsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8767ndash8780 2007

[17] K-M Bjork and R Nordman ldquoSolving large-scale retrofitheat exchanger network synthesis problems with mathematicaloptimization methodsrdquo Chemical Engineering and ProcessingProcess Intensification vol 44 no 8 pp 869ndash876 2005

[18] S Mouret I E Grossmann and P Pestiaux ldquoA novel priority-slot based continuous-time formulation for crude-oil schedul-ing problemsrdquo Industrial and Engineering Chemistry Researchvol 48 no 18 pp 8515ndash8528 2009

[19] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[20] J E Hopcroft Introduction to AutomataTheory Languages andComputation Pearson Education India New Delhi India 3rdedition 2008

[21] E Roche and Y Schabes Finite-State Language Processing TheMIT Press Cambridge Mass USA 1997

[22] L Karttunen ldquoConstructing lexical transducersrdquo in Proceedingsof the 15th conference on Computational Linguistics vol 1Association for Computational Linguistics 1994

[23] L Karttunen ldquoThe replace operatorrdquo in Proceedings of the 33rdAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1995

[24] L Karttunen ldquoDirected replacementrdquo in Proceedings of the 34thAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1996

[25] L Karttunen and K R Beesley Two-Level Rule Compiler XeroxCorporation Palo Alto Research Center 1992

[26] G vanNoord ldquoFSAutilities a toolbox tomanipulate finite-stateautomatardquo in Automata Implementation pp 87ndash108 SpringerNew York NY USA 1997

The Scientific World Journal 11

[27] D Gerdemann and G van Noord ldquoTransducers from rewriterules with backreferencesrdquo in Proceedings of the 9th Conferenceon European Chapter of the Association for ComputationalLinguistics Association for Computational Linguistics 1999

[28] Y-C Lin K-S Hwang and F-S Wang ldquoA mixed-codingscheme of evolutionary algorithms to solve mixed-integer non-linear programming problemsrdquo Computers and Mathematicswith Applications vol 47 no 8-9 pp 1295ndash1307 2004

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 The Scientific World Journal

Schedule 7 - 6 - 8

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

- 3 - 5 - 1 - 3 - 7 - -6 2

Binary

Z61 = 1

Z102

Z43 = 1 Z73 = 1

Z55 = 1

Z26 = 1 Z96 = 1

Z17 = 1 Z87 = 1

= 1

Z38 =

0

0

1

Operation

1

2

3

4

5

6

7

8

6

4 7

2

5

10

9

1 8

3

t0 t1 t2 t3 t4 t5 t6 t7 t8

variable

executions

S =

Z =

Sequence 7-6-8-3-5-1-3-7-6-2

Figure 4 An example to indicate the relationship between binary variable and schedule

a rule target contains a prefix which is also a potential targetthe longer sequence will be replaced Gerdemann and vanNoord [27] implement leftmost longest-match replacementin FSA as the operator

replace (Target LeftContextRightContext) (29)where Target is a transducer defining the actual replacementand LeftContext and RightContext are regular expressionsdefining the left and right context of the rule respectivelyThe segmentation task discussed in the mutation proceduremakes crucial use of longest-match replacement

4 The Hybrid Algorithm

From the point view of optimization efficiency and robust-ness a novel two-level optimization framework based onfinite statemethod andGA is proposed for theMINLPmodelin this section

41 Two-Level Optimization Structure As the foundationof the framework a two-level optimization structure isintroduced Once all binary variables are fixed the originalproblem becomes a relatively simpler model with only con-tinuous variable Following this deal we rewrite (5) as follows

max (119869 (120585 119911)) lArrrArr max119911

[max120585

119869 (120585 119911)] (30)

where 120585 and 119911 represent continuous and binary variablesrespectively Equation (30) shows when 119911 is fixed as 119911 thesubmodel 119869(120585 119911) can be solved optimally by continuous-optimization solvers in the inner level then we update 119911towards the best binary solution 119911lowast in the outer level

We used an example in Figure 4 to show how binarysolution can be mapped to a scheduling sequence Theschedule 119878 = [7683513762] where 7 stands for the specificoperation 7 to assign to position 1 corresponding to the binarydecisions 119885

17= 1

42 Initial Population Based on the sequencing rules [18]and the extension to the regular expression calculus [22ndash25]a DFA model which builds legal schedules complying withsequencing rules and operation condition is constructedThewhole set of possible schedules is too huge to be processed atonceTheDFAmodel of the schedule constitutes a reasonableframework capturing all possible schedules and removingmany redundant sequences of operations Initial values ofdecision variables must satisfy the equality constraints andoperation condition and therefore represent a feasible oper-ating point

Here we still use the instance with 8 operations fromMouret et al [18] to describe an efficient sequencing rule by

The Scientific World Journal 7

macro (proceduresegmentation segmentation of the input sequence into a set of sub-sequencemutation apply mutation rulesclean up) remove markers

Algorithm 1

using a regular expression A feasible sequence V1sdot sdot sdot V119894sdot sdot sdot V119899

can be described by the following

sequence = (120576 + 119871119886) (119871119887sdot 119871119886)lowast(120576 + 119871

119887)

119871119886= 7 (120576 + 4) (120576 + 6) (120576 + 1 + 14) (120576 + 2 + 26)

119871119887= 8 (120576 + 3) (120576 + 5) (120576 + 1 + 13) (120576 + 2 + 25)

(31)

However this automation suffers from a serious problemof overgeneration For example the short length of thesequence may lead to infeasibility while the long length ofthe sequence may result in an unsolvable model It is aninteresting challenge for finite state syntactic description tospecify a sublanguage that contains all and only the sequencesof valid length

Our solution is to construct a suitable constraint for thesequences of valid length The constraint expressions denotea language that admits sequences of valid length but excludesall others We obtain the desired effect by intersecting theconstraint language with the original language of sequenceexpressions The intersection of the two languages containsall and only the valid dates

ValidSequence = Sequence cap ValidLength (32)

The ValidLength constraint is a language that includes allsequences of length 119899

ValidLength = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)119899 (33)

We have now completed the task of describing thelanguage of valid sequences from the set of possible sequenceexpressions It is also possible to create an automation onthe basis of the regular expression and ValidSequence andthen generate all possible sequences V

1sdot sdot sdot V119894sdot sdot sdot V119899accepted

by the automaton The processes are implemented usingFSA Utilities [26] that is a package for implementing andmanipulating DFA and finite state transducer In order togenerate all possible sequences When all possible sequencesV1sdot sdot sdot V119894sdot sdot sdot V119899accepted by the automaton are generated and

the population of the according possible binary decisions isgenerated In the initial population stage of GA the popu-lation size is the number of individuals When the numberof individuals is given a population of candidate solutions isgenerated by randomly selecting from the population of theall possible binary decisions

43 Rule-Based Mutation Approach In the mutation stagewe use a finite state transducer for this rule-based muta-tion process The rule-based mutation strategy must obey

Input7681325712

Segmentationtransducer

76-81325-712 74-83132-741

Mutationtransducer transducer

Output

748313274

Cleanup

Figure 5 An example of mutation

replace ([identity(SSequence)[]x-][]])

Algorithm 2

the sequencing rule and the nonoverlapping constraint suchthat all involved solutions in GA are feasible

The proposedmutation approach is a two-step procedure

Step 1 Segmentation of the input sequence into a set ofsubsequences (ie the subsequence which belongs to theregular language L7 or L8)

Step 2 Mutation of the subsequences into others

Formally the rule-based mutation procedure is imple-mented as the composition of three transducers (seeAlgorithm 1)

An example of mutation including the intermediate stepsis given for the sequence ldquo7681325712rdquo as shown in Figure 5

431 Segmentation Transducer Segmentation transducersplits an input sequence into subsequences The goal ofsegmentation is to provide a convenient representation levelfor the next mutation step

Segmentation is defined as shown in Algorithm 2The macro ldquoSSequencerdquo defines the set of subsequences

The subsequences which belong to the regular languageL7 and L8 are displayed in Tables 1 and 2 Segmentationattaches the marker ldquondashrdquo to each subsequence The Targetsare identified using leftmost longest-match and thus at eachpoint in the input only the longest valid segment is marked

432 The Mutation Rules In the GA process the mutationrules are made by carefully considering nonoverlapping con-straint between operations A concrete instance for partiallyillustrating the mutation rules is given in Algorithm 3 Notethat the final element of the left-contextmust be amarker and

8 The Scientific World Journal

marco (conversion 1198717subsequence rules)

∘ 1198718subsequence rules)

marco (1198717subsequence rules

replace (2 4 6 times 1 7 mdash)∘ replace (14 26 41 42 46 61 62 times 12 7 mdash)∘ replace (142 412 414 426 461 462 612 614 626 times 126 7 mdash)marco (119871

8subsequence rules

replace (2 3 5 times 1 8 mdash)∘ replace (13 25 31 32 35 51 52 times 12 8mdash)∘ replace (132 312 313 325 351 352 512 513 525 times 125 8 mdash)

Algorithm 3 An example to demonstrate the mutation rules

Table 1 Subsequence belonging to 1198717

Length Sequences belonging to 1198717

1 72 71 72 74 763 712 714 726 741 742 746 761 7624 7126 7142 7412 7414 7426 7461 7462 7612 7614 76265 71426 74126 74142 74612 74614 74626 76126 761426 741426 746126 746142 7614267 7461426

the target itself ends in ldquondashrdquo This ensures that mutation rulescannot apply to the same subsequence

5 Experimental Study

In this section the same problem from the literature [18] isused for computational experiments The proposed method-ology is compared with existing promising algorithmsmixed-coding GA [15 28] Figure 1 depicts the refineryconfiguration for problem The data involved in the problemare given in Table 3 The performance comparison withdifferent computing times such as 350 s 500 s 2400 s isconducted The objective value is used to statistically analyzethe optimization results

The performance comparison between the two method-ologies used is illustrated in Figure 6 which shows that thehybrid optimization algorithm which combined the finitestate method and GAwill statistically outperform the mixed-coding counterpart The genetic algorithm which combinedthe finite state method and GA finds feasible solutions veryfast and is able to find better solutions in reasonable time

In Figure 7 we compare the objective variance of eachiteration in the two evolution processes of these two kindsof methodology By tracking the evolution process we findthat the mixed-coding GA is easy to stick in a local minimalsequence solution This situation only can be improvedthrough increasing the mutation scaling factor Howeverthis may result in a hard convergence unless sufficientiterations are implemented As for the hybrid optimizationalgorithm the optimization processes of binary variableand continuous variable are separated The performanceof the whole methodology mainly depends on the FSM

350 500 650 800 950 1200 1500 1800 240020

30

40

50

60

70

80

Computer time

Obj

ectiv

e val

ue

Mixed-code GAOur algorithm

Figure 6 Average objective values of two methodologies

which captures most promising schedules and removes manyredundant sequences of operations so that the user can usea small population size of corresponding discrete variables toobtain suboptimal solutions From Figure 7 we see that theproposed method has converged at 350 iterations as opposedto 2400 iterations for the mixed-coding GA

The success of the proposed algorithm lies in a compre-hensive analysis of the region of the search space and itscapacity to focus the search on the regions with the partialsolution One of the good merits of the hybrid algorithm is

The Scientific World Journal 9

Table 2 Subsequence belonging to 1198718

Length Sequences belonging to 1198718

1 82 81 82 83 853 812 813 825 831 832 835 851 8524 8125 8132 8312 8313 8325 8351 8352 8512 8513 85255 81325 83125 83132 83512 83513 83525 85132 851256 831325 835125 835132 8513257 8351325

Table 3 Problem data

Scheduling horizon 8 daysVessels Arrival time Composition Amount of crudeVessel 1 0 100 A 1000Vessel 2 4 100 B 1000Storage tanks Capacity Initial composition Initial amountTank 1 [0 1000] 100 A 250Tank 2 [0 1000] 100 B 750Charging tanks Capacity Initial composition Initial amountTank 1 (mix X) [0 1000] 100 C 500Tank 1 (mix X) [0 1000] 100 D 500Crudes 1 Gross margin Crude mixtures Property1 DemandCrude A 001 9 Crude mix X [0015 0025] [1000 1000]

Crude B 006 4 Crude mix Y [0045 0055] [1000 1000]

Crude C 002 8 Unloading flow rate [0 500]

Crude D 005 5 transfer flow rate [0 500]

Table 4 A fragment of FSA regular expression syntax and 119880

transducers and 119877 can be either

[] The empty string[1198771 119877

119899] Concatenation

1198771 119877

119899 Disjunction

119877Λ Optionality

Identity (119860) Identity the transducer which maps eachelement in 119860 onto itself

119879 ∘ 119880 Composition of the transducers 119879 and 119880macro (Term 119877) Use term as an abbreviation for 119877

that each solution involved in theGA algorithm is guaranteedto be feasible by using the mutation rules generated by DFMmethod while in existing GA algorithms the procedure togenerate feasible solution under complex process constraintsis very time costive The deterministic finite automata (DFA)can easily represent this kind of structure Furthermorethe complex process constraints can be very difficult toexpress with mixed integer programming Consequently itis unfeasible to solve the industrial problem by using MIPsolver

6 Conclusion

In this paper a novel hybrid optimization algorithm whichcombined the finite state method and GA is proposed

350 500 650 800 950 1200 1500 1800 24000

02

04

06

08

1

12

14

Computing time (s)

Varia

nce v

alue

s

Mixed-code GAOur algorithm

Figure 7 Variance values of two methodologies

The proposed algorithm constitutes a reasonable frameworkcapturing both the operating condition and sequencing ruleof the schedule The solution captures all possible sched-ules and removes many redundant sequences of operationsThe algorithm is equivalent to introducing new structure

10 The Scientific World Journal

information into the optimization process which will helpreduce the risk of trapping in a local minimal sequencesolution The hybrid optimization algorithm is an effectiveand robust tool to solve the crude oil scheduling problem interms of efficiency and reliability Algorithms only with thetwo properties are suitable for solving practical engineeringapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partially supported by the China NationalNatural Science Foundation under Grant 61203178 Grant61304214 and Grant 61290323 The authors thank the finan-cial funds fromShanghai Science andTechnologyCommitteeunder Grant 12511501002 and Grant 13511501302

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] C A Mendez I E Grossmann I Harjunkoski and P KaboreldquoA simultaneous optimization approach for off-line blendingand scheduling of oil-refinery operationsrdquo Computers andChemical Engineering vol 30 no 4 pp 614ndash634 2006

[6] M Pan X Li and Y Qian ldquoNew approach for scheduling crudeoil operationsrdquo Chemical Engineering Science vol 64 no 5 pp965ndash983 2009

[7] M F Cardoso R L Salcedo S F de Azevedo and D BarbosaldquoA simulated annealing approach to the solution of minlpproblemsrdquo Computers and Chemical Engineering vol 21 no 12pp 1349ndash1364 1997

[8] I E Grossmann ldquoReview of nonlinear mixed-integer anddisjunctive programming techniquesrdquo Optimization and Engi-neering vol 3 no 3 pp 227ndash252 2002

[9] E L Lawler and D E Wood ldquoBranch-and-bound methods asurveyrdquo Operations Research vol 14 no 4 pp 699ndash719 1966

[10] M A Duran and I E Grossmann ldquoAn outer-approximationalgorithm for a class of mixed-integer nonlinear programsrdquoMathematical Programming vol 36 no 3 pp 307ndash339 1986

[11] AMGeoffrion ldquoGeneralized Benders decompositionrdquo Journalof Optimization Theory and Applications vol 10 no 4 pp 237ndash260 1972

[12] C DrsquoAmbrosio andA Lodi ldquoMixed integer nonlinear program-ming tools a practical overviewrdquo 4OR vol 9 no 4 pp 329ndash3492011

[13] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence The University of Michigan Press AnnArbor Mich USA 1975

[14] T Yokota M Gen and Y-X Li ldquoGenetic algorithm for non-linear mixed integer programming problems and its applica-tionsrdquo Computers and Industrial Engineering vol 30 no 4 pp905ndash917 1996

[15] L Costa and P Oliveira ldquoEvolutionary algorithms approachto the solution of mixed integer non-linear programmingproblemsrdquo Computers and Chemical Engineering vol 25 no 2-3 pp 257ndash266 2001

[16] J M Ponce-Ortega M Serna-Gonzalez and A Jimenez-Gutierrez ldquoHeat exchanger network synthesis includingdetailed heat exchanger design using genetic algorithmsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8767ndash8780 2007

[17] K-M Bjork and R Nordman ldquoSolving large-scale retrofitheat exchanger network synthesis problems with mathematicaloptimization methodsrdquo Chemical Engineering and ProcessingProcess Intensification vol 44 no 8 pp 869ndash876 2005

[18] S Mouret I E Grossmann and P Pestiaux ldquoA novel priority-slot based continuous-time formulation for crude-oil schedul-ing problemsrdquo Industrial and Engineering Chemistry Researchvol 48 no 18 pp 8515ndash8528 2009

[19] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[20] J E Hopcroft Introduction to AutomataTheory Languages andComputation Pearson Education India New Delhi India 3rdedition 2008

[21] E Roche and Y Schabes Finite-State Language Processing TheMIT Press Cambridge Mass USA 1997

[22] L Karttunen ldquoConstructing lexical transducersrdquo in Proceedingsof the 15th conference on Computational Linguistics vol 1Association for Computational Linguistics 1994

[23] L Karttunen ldquoThe replace operatorrdquo in Proceedings of the 33rdAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1995

[24] L Karttunen ldquoDirected replacementrdquo in Proceedings of the 34thAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1996

[25] L Karttunen and K R Beesley Two-Level Rule Compiler XeroxCorporation Palo Alto Research Center 1992

[26] G vanNoord ldquoFSAutilities a toolbox tomanipulate finite-stateautomatardquo in Automata Implementation pp 87ndash108 SpringerNew York NY USA 1997

The Scientific World Journal 11

[27] D Gerdemann and G van Noord ldquoTransducers from rewriterules with backreferencesrdquo in Proceedings of the 9th Conferenceon European Chapter of the Association for ComputationalLinguistics Association for Computational Linguistics 1999

[28] Y-C Lin K-S Hwang and F-S Wang ldquoA mixed-codingscheme of evolutionary algorithms to solve mixed-integer non-linear programming problemsrdquo Computers and Mathematicswith Applications vol 47 no 8-9 pp 1295ndash1307 2004

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 7

macro (proceduresegmentation segmentation of the input sequence into a set of sub-sequencemutation apply mutation rulesclean up) remove markers

Algorithm 1

using a regular expression A feasible sequence V1sdot sdot sdot V119894sdot sdot sdot V119899

can be described by the following

sequence = (120576 + 119871119886) (119871119887sdot 119871119886)lowast(120576 + 119871

119887)

119871119886= 7 (120576 + 4) (120576 + 6) (120576 + 1 + 14) (120576 + 2 + 26)

119871119887= 8 (120576 + 3) (120576 + 5) (120576 + 1 + 13) (120576 + 2 + 25)

(31)

However this automation suffers from a serious problemof overgeneration For example the short length of thesequence may lead to infeasibility while the long length ofthe sequence may result in an unsolvable model It is aninteresting challenge for finite state syntactic description tospecify a sublanguage that contains all and only the sequencesof valid length

Our solution is to construct a suitable constraint for thesequences of valid length The constraint expressions denotea language that admits sequences of valid length but excludesall others We obtain the desired effect by intersecting theconstraint language with the original language of sequenceexpressions The intersection of the two languages containsall and only the valid dates

ValidSequence = Sequence cap ValidLength (32)

The ValidLength constraint is a language that includes allsequences of length 119899

ValidLength = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)119899 (33)

We have now completed the task of describing thelanguage of valid sequences from the set of possible sequenceexpressions It is also possible to create an automation onthe basis of the regular expression and ValidSequence andthen generate all possible sequences V

1sdot sdot sdot V119894sdot sdot sdot V119899accepted

by the automaton The processes are implemented usingFSA Utilities [26] that is a package for implementing andmanipulating DFA and finite state transducer In order togenerate all possible sequences When all possible sequencesV1sdot sdot sdot V119894sdot sdot sdot V119899accepted by the automaton are generated and

the population of the according possible binary decisions isgenerated In the initial population stage of GA the popu-lation size is the number of individuals When the numberof individuals is given a population of candidate solutions isgenerated by randomly selecting from the population of theall possible binary decisions

43 Rule-Based Mutation Approach In the mutation stagewe use a finite state transducer for this rule-based muta-tion process The rule-based mutation strategy must obey

Input7681325712

Segmentationtransducer

76-81325-712 74-83132-741

Mutationtransducer transducer

Output

748313274

Cleanup

Figure 5 An example of mutation

replace ([identity(SSequence)[]x-][]])

Algorithm 2

the sequencing rule and the nonoverlapping constraint suchthat all involved solutions in GA are feasible

The proposedmutation approach is a two-step procedure

Step 1 Segmentation of the input sequence into a set ofsubsequences (ie the subsequence which belongs to theregular language L7 or L8)

Step 2 Mutation of the subsequences into others

Formally the rule-based mutation procedure is imple-mented as the composition of three transducers (seeAlgorithm 1)

An example of mutation including the intermediate stepsis given for the sequence ldquo7681325712rdquo as shown in Figure 5

431 Segmentation Transducer Segmentation transducersplits an input sequence into subsequences The goal ofsegmentation is to provide a convenient representation levelfor the next mutation step

Segmentation is defined as shown in Algorithm 2The macro ldquoSSequencerdquo defines the set of subsequences

The subsequences which belong to the regular languageL7 and L8 are displayed in Tables 1 and 2 Segmentationattaches the marker ldquondashrdquo to each subsequence The Targetsare identified using leftmost longest-match and thus at eachpoint in the input only the longest valid segment is marked

432 The Mutation Rules In the GA process the mutationrules are made by carefully considering nonoverlapping con-straint between operations A concrete instance for partiallyillustrating the mutation rules is given in Algorithm 3 Notethat the final element of the left-contextmust be amarker and

8 The Scientific World Journal

marco (conversion 1198717subsequence rules)

∘ 1198718subsequence rules)

marco (1198717subsequence rules

replace (2 4 6 times 1 7 mdash)∘ replace (14 26 41 42 46 61 62 times 12 7 mdash)∘ replace (142 412 414 426 461 462 612 614 626 times 126 7 mdash)marco (119871

8subsequence rules

replace (2 3 5 times 1 8 mdash)∘ replace (13 25 31 32 35 51 52 times 12 8mdash)∘ replace (132 312 313 325 351 352 512 513 525 times 125 8 mdash)

Algorithm 3 An example to demonstrate the mutation rules

Table 1 Subsequence belonging to 1198717

Length Sequences belonging to 1198717

1 72 71 72 74 763 712 714 726 741 742 746 761 7624 7126 7142 7412 7414 7426 7461 7462 7612 7614 76265 71426 74126 74142 74612 74614 74626 76126 761426 741426 746126 746142 7614267 7461426

the target itself ends in ldquondashrdquo This ensures that mutation rulescannot apply to the same subsequence

5 Experimental Study

In this section the same problem from the literature [18] isused for computational experiments The proposed method-ology is compared with existing promising algorithmsmixed-coding GA [15 28] Figure 1 depicts the refineryconfiguration for problem The data involved in the problemare given in Table 3 The performance comparison withdifferent computing times such as 350 s 500 s 2400 s isconducted The objective value is used to statistically analyzethe optimization results

The performance comparison between the two method-ologies used is illustrated in Figure 6 which shows that thehybrid optimization algorithm which combined the finitestate method and GAwill statistically outperform the mixed-coding counterpart The genetic algorithm which combinedthe finite state method and GA finds feasible solutions veryfast and is able to find better solutions in reasonable time

In Figure 7 we compare the objective variance of eachiteration in the two evolution processes of these two kindsof methodology By tracking the evolution process we findthat the mixed-coding GA is easy to stick in a local minimalsequence solution This situation only can be improvedthrough increasing the mutation scaling factor Howeverthis may result in a hard convergence unless sufficientiterations are implemented As for the hybrid optimizationalgorithm the optimization processes of binary variableand continuous variable are separated The performanceof the whole methodology mainly depends on the FSM

350 500 650 800 950 1200 1500 1800 240020

30

40

50

60

70

80

Computer time

Obj

ectiv

e val

ue

Mixed-code GAOur algorithm

Figure 6 Average objective values of two methodologies

which captures most promising schedules and removes manyredundant sequences of operations so that the user can usea small population size of corresponding discrete variables toobtain suboptimal solutions From Figure 7 we see that theproposed method has converged at 350 iterations as opposedto 2400 iterations for the mixed-coding GA

The success of the proposed algorithm lies in a compre-hensive analysis of the region of the search space and itscapacity to focus the search on the regions with the partialsolution One of the good merits of the hybrid algorithm is

The Scientific World Journal 9

Table 2 Subsequence belonging to 1198718

Length Sequences belonging to 1198718

1 82 81 82 83 853 812 813 825 831 832 835 851 8524 8125 8132 8312 8313 8325 8351 8352 8512 8513 85255 81325 83125 83132 83512 83513 83525 85132 851256 831325 835125 835132 8513257 8351325

Table 3 Problem data

Scheduling horizon 8 daysVessels Arrival time Composition Amount of crudeVessel 1 0 100 A 1000Vessel 2 4 100 B 1000Storage tanks Capacity Initial composition Initial amountTank 1 [0 1000] 100 A 250Tank 2 [0 1000] 100 B 750Charging tanks Capacity Initial composition Initial amountTank 1 (mix X) [0 1000] 100 C 500Tank 1 (mix X) [0 1000] 100 D 500Crudes 1 Gross margin Crude mixtures Property1 DemandCrude A 001 9 Crude mix X [0015 0025] [1000 1000]

Crude B 006 4 Crude mix Y [0045 0055] [1000 1000]

Crude C 002 8 Unloading flow rate [0 500]

Crude D 005 5 transfer flow rate [0 500]

Table 4 A fragment of FSA regular expression syntax and 119880

transducers and 119877 can be either

[] The empty string[1198771 119877

119899] Concatenation

1198771 119877

119899 Disjunction

119877Λ Optionality

Identity (119860) Identity the transducer which maps eachelement in 119860 onto itself

119879 ∘ 119880 Composition of the transducers 119879 and 119880macro (Term 119877) Use term as an abbreviation for 119877

that each solution involved in theGA algorithm is guaranteedto be feasible by using the mutation rules generated by DFMmethod while in existing GA algorithms the procedure togenerate feasible solution under complex process constraintsis very time costive The deterministic finite automata (DFA)can easily represent this kind of structure Furthermorethe complex process constraints can be very difficult toexpress with mixed integer programming Consequently itis unfeasible to solve the industrial problem by using MIPsolver

6 Conclusion

In this paper a novel hybrid optimization algorithm whichcombined the finite state method and GA is proposed

350 500 650 800 950 1200 1500 1800 24000

02

04

06

08

1

12

14

Computing time (s)

Varia

nce v

alue

s

Mixed-code GAOur algorithm

Figure 7 Variance values of two methodologies

The proposed algorithm constitutes a reasonable frameworkcapturing both the operating condition and sequencing ruleof the schedule The solution captures all possible sched-ules and removes many redundant sequences of operationsThe algorithm is equivalent to introducing new structure

10 The Scientific World Journal

information into the optimization process which will helpreduce the risk of trapping in a local minimal sequencesolution The hybrid optimization algorithm is an effectiveand robust tool to solve the crude oil scheduling problem interms of efficiency and reliability Algorithms only with thetwo properties are suitable for solving practical engineeringapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partially supported by the China NationalNatural Science Foundation under Grant 61203178 Grant61304214 and Grant 61290323 The authors thank the finan-cial funds fromShanghai Science andTechnologyCommitteeunder Grant 12511501002 and Grant 13511501302

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] C A Mendez I E Grossmann I Harjunkoski and P KaboreldquoA simultaneous optimization approach for off-line blendingand scheduling of oil-refinery operationsrdquo Computers andChemical Engineering vol 30 no 4 pp 614ndash634 2006

[6] M Pan X Li and Y Qian ldquoNew approach for scheduling crudeoil operationsrdquo Chemical Engineering Science vol 64 no 5 pp965ndash983 2009

[7] M F Cardoso R L Salcedo S F de Azevedo and D BarbosaldquoA simulated annealing approach to the solution of minlpproblemsrdquo Computers and Chemical Engineering vol 21 no 12pp 1349ndash1364 1997

[8] I E Grossmann ldquoReview of nonlinear mixed-integer anddisjunctive programming techniquesrdquo Optimization and Engi-neering vol 3 no 3 pp 227ndash252 2002

[9] E L Lawler and D E Wood ldquoBranch-and-bound methods asurveyrdquo Operations Research vol 14 no 4 pp 699ndash719 1966

[10] M A Duran and I E Grossmann ldquoAn outer-approximationalgorithm for a class of mixed-integer nonlinear programsrdquoMathematical Programming vol 36 no 3 pp 307ndash339 1986

[11] AMGeoffrion ldquoGeneralized Benders decompositionrdquo Journalof Optimization Theory and Applications vol 10 no 4 pp 237ndash260 1972

[12] C DrsquoAmbrosio andA Lodi ldquoMixed integer nonlinear program-ming tools a practical overviewrdquo 4OR vol 9 no 4 pp 329ndash3492011

[13] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence The University of Michigan Press AnnArbor Mich USA 1975

[14] T Yokota M Gen and Y-X Li ldquoGenetic algorithm for non-linear mixed integer programming problems and its applica-tionsrdquo Computers and Industrial Engineering vol 30 no 4 pp905ndash917 1996

[15] L Costa and P Oliveira ldquoEvolutionary algorithms approachto the solution of mixed integer non-linear programmingproblemsrdquo Computers and Chemical Engineering vol 25 no 2-3 pp 257ndash266 2001

[16] J M Ponce-Ortega M Serna-Gonzalez and A Jimenez-Gutierrez ldquoHeat exchanger network synthesis includingdetailed heat exchanger design using genetic algorithmsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8767ndash8780 2007

[17] K-M Bjork and R Nordman ldquoSolving large-scale retrofitheat exchanger network synthesis problems with mathematicaloptimization methodsrdquo Chemical Engineering and ProcessingProcess Intensification vol 44 no 8 pp 869ndash876 2005

[18] S Mouret I E Grossmann and P Pestiaux ldquoA novel priority-slot based continuous-time formulation for crude-oil schedul-ing problemsrdquo Industrial and Engineering Chemistry Researchvol 48 no 18 pp 8515ndash8528 2009

[19] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[20] J E Hopcroft Introduction to AutomataTheory Languages andComputation Pearson Education India New Delhi India 3rdedition 2008

[21] E Roche and Y Schabes Finite-State Language Processing TheMIT Press Cambridge Mass USA 1997

[22] L Karttunen ldquoConstructing lexical transducersrdquo in Proceedingsof the 15th conference on Computational Linguistics vol 1Association for Computational Linguistics 1994

[23] L Karttunen ldquoThe replace operatorrdquo in Proceedings of the 33rdAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1995

[24] L Karttunen ldquoDirected replacementrdquo in Proceedings of the 34thAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1996

[25] L Karttunen and K R Beesley Two-Level Rule Compiler XeroxCorporation Palo Alto Research Center 1992

[26] G vanNoord ldquoFSAutilities a toolbox tomanipulate finite-stateautomatardquo in Automata Implementation pp 87ndash108 SpringerNew York NY USA 1997

The Scientific World Journal 11

[27] D Gerdemann and G van Noord ldquoTransducers from rewriterules with backreferencesrdquo in Proceedings of the 9th Conferenceon European Chapter of the Association for ComputationalLinguistics Association for Computational Linguistics 1999

[28] Y-C Lin K-S Hwang and F-S Wang ldquoA mixed-codingscheme of evolutionary algorithms to solve mixed-integer non-linear programming problemsrdquo Computers and Mathematicswith Applications vol 47 no 8-9 pp 1295ndash1307 2004

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 The Scientific World Journal

marco (conversion 1198717subsequence rules)

∘ 1198718subsequence rules)

marco (1198717subsequence rules

replace (2 4 6 times 1 7 mdash)∘ replace (14 26 41 42 46 61 62 times 12 7 mdash)∘ replace (142 412 414 426 461 462 612 614 626 times 126 7 mdash)marco (119871

8subsequence rules

replace (2 3 5 times 1 8 mdash)∘ replace (13 25 31 32 35 51 52 times 12 8mdash)∘ replace (132 312 313 325 351 352 512 513 525 times 125 8 mdash)

Algorithm 3 An example to demonstrate the mutation rules

Table 1 Subsequence belonging to 1198717

Length Sequences belonging to 1198717

1 72 71 72 74 763 712 714 726 741 742 746 761 7624 7126 7142 7412 7414 7426 7461 7462 7612 7614 76265 71426 74126 74142 74612 74614 74626 76126 761426 741426 746126 746142 7614267 7461426

the target itself ends in ldquondashrdquo This ensures that mutation rulescannot apply to the same subsequence

5 Experimental Study

In this section the same problem from the literature [18] isused for computational experiments The proposed method-ology is compared with existing promising algorithmsmixed-coding GA [15 28] Figure 1 depicts the refineryconfiguration for problem The data involved in the problemare given in Table 3 The performance comparison withdifferent computing times such as 350 s 500 s 2400 s isconducted The objective value is used to statistically analyzethe optimization results

The performance comparison between the two method-ologies used is illustrated in Figure 6 which shows that thehybrid optimization algorithm which combined the finitestate method and GAwill statistically outperform the mixed-coding counterpart The genetic algorithm which combinedthe finite state method and GA finds feasible solutions veryfast and is able to find better solutions in reasonable time

In Figure 7 we compare the objective variance of eachiteration in the two evolution processes of these two kindsof methodology By tracking the evolution process we findthat the mixed-coding GA is easy to stick in a local minimalsequence solution This situation only can be improvedthrough increasing the mutation scaling factor Howeverthis may result in a hard convergence unless sufficientiterations are implemented As for the hybrid optimizationalgorithm the optimization processes of binary variableand continuous variable are separated The performanceof the whole methodology mainly depends on the FSM

350 500 650 800 950 1200 1500 1800 240020

30

40

50

60

70

80

Computer time

Obj

ectiv

e val

ue

Mixed-code GAOur algorithm

Figure 6 Average objective values of two methodologies

which captures most promising schedules and removes manyredundant sequences of operations so that the user can usea small population size of corresponding discrete variables toobtain suboptimal solutions From Figure 7 we see that theproposed method has converged at 350 iterations as opposedto 2400 iterations for the mixed-coding GA

The success of the proposed algorithm lies in a compre-hensive analysis of the region of the search space and itscapacity to focus the search on the regions with the partialsolution One of the good merits of the hybrid algorithm is

The Scientific World Journal 9

Table 2 Subsequence belonging to 1198718

Length Sequences belonging to 1198718

1 82 81 82 83 853 812 813 825 831 832 835 851 8524 8125 8132 8312 8313 8325 8351 8352 8512 8513 85255 81325 83125 83132 83512 83513 83525 85132 851256 831325 835125 835132 8513257 8351325

Table 3 Problem data

Scheduling horizon 8 daysVessels Arrival time Composition Amount of crudeVessel 1 0 100 A 1000Vessel 2 4 100 B 1000Storage tanks Capacity Initial composition Initial amountTank 1 [0 1000] 100 A 250Tank 2 [0 1000] 100 B 750Charging tanks Capacity Initial composition Initial amountTank 1 (mix X) [0 1000] 100 C 500Tank 1 (mix X) [0 1000] 100 D 500Crudes 1 Gross margin Crude mixtures Property1 DemandCrude A 001 9 Crude mix X [0015 0025] [1000 1000]

Crude B 006 4 Crude mix Y [0045 0055] [1000 1000]

Crude C 002 8 Unloading flow rate [0 500]

Crude D 005 5 transfer flow rate [0 500]

Table 4 A fragment of FSA regular expression syntax and 119880

transducers and 119877 can be either

[] The empty string[1198771 119877

119899] Concatenation

1198771 119877

119899 Disjunction

119877Λ Optionality

Identity (119860) Identity the transducer which maps eachelement in 119860 onto itself

119879 ∘ 119880 Composition of the transducers 119879 and 119880macro (Term 119877) Use term as an abbreviation for 119877

that each solution involved in theGA algorithm is guaranteedto be feasible by using the mutation rules generated by DFMmethod while in existing GA algorithms the procedure togenerate feasible solution under complex process constraintsis very time costive The deterministic finite automata (DFA)can easily represent this kind of structure Furthermorethe complex process constraints can be very difficult toexpress with mixed integer programming Consequently itis unfeasible to solve the industrial problem by using MIPsolver

6 Conclusion

In this paper a novel hybrid optimization algorithm whichcombined the finite state method and GA is proposed

350 500 650 800 950 1200 1500 1800 24000

02

04

06

08

1

12

14

Computing time (s)

Varia

nce v

alue

s

Mixed-code GAOur algorithm

Figure 7 Variance values of two methodologies

The proposed algorithm constitutes a reasonable frameworkcapturing both the operating condition and sequencing ruleof the schedule The solution captures all possible sched-ules and removes many redundant sequences of operationsThe algorithm is equivalent to introducing new structure

10 The Scientific World Journal

information into the optimization process which will helpreduce the risk of trapping in a local minimal sequencesolution The hybrid optimization algorithm is an effectiveand robust tool to solve the crude oil scheduling problem interms of efficiency and reliability Algorithms only with thetwo properties are suitable for solving practical engineeringapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partially supported by the China NationalNatural Science Foundation under Grant 61203178 Grant61304214 and Grant 61290323 The authors thank the finan-cial funds fromShanghai Science andTechnologyCommitteeunder Grant 12511501002 and Grant 13511501302

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] C A Mendez I E Grossmann I Harjunkoski and P KaboreldquoA simultaneous optimization approach for off-line blendingand scheduling of oil-refinery operationsrdquo Computers andChemical Engineering vol 30 no 4 pp 614ndash634 2006

[6] M Pan X Li and Y Qian ldquoNew approach for scheduling crudeoil operationsrdquo Chemical Engineering Science vol 64 no 5 pp965ndash983 2009

[7] M F Cardoso R L Salcedo S F de Azevedo and D BarbosaldquoA simulated annealing approach to the solution of minlpproblemsrdquo Computers and Chemical Engineering vol 21 no 12pp 1349ndash1364 1997

[8] I E Grossmann ldquoReview of nonlinear mixed-integer anddisjunctive programming techniquesrdquo Optimization and Engi-neering vol 3 no 3 pp 227ndash252 2002

[9] E L Lawler and D E Wood ldquoBranch-and-bound methods asurveyrdquo Operations Research vol 14 no 4 pp 699ndash719 1966

[10] M A Duran and I E Grossmann ldquoAn outer-approximationalgorithm for a class of mixed-integer nonlinear programsrdquoMathematical Programming vol 36 no 3 pp 307ndash339 1986

[11] AMGeoffrion ldquoGeneralized Benders decompositionrdquo Journalof Optimization Theory and Applications vol 10 no 4 pp 237ndash260 1972

[12] C DrsquoAmbrosio andA Lodi ldquoMixed integer nonlinear program-ming tools a practical overviewrdquo 4OR vol 9 no 4 pp 329ndash3492011

[13] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence The University of Michigan Press AnnArbor Mich USA 1975

[14] T Yokota M Gen and Y-X Li ldquoGenetic algorithm for non-linear mixed integer programming problems and its applica-tionsrdquo Computers and Industrial Engineering vol 30 no 4 pp905ndash917 1996

[15] L Costa and P Oliveira ldquoEvolutionary algorithms approachto the solution of mixed integer non-linear programmingproblemsrdquo Computers and Chemical Engineering vol 25 no 2-3 pp 257ndash266 2001

[16] J M Ponce-Ortega M Serna-Gonzalez and A Jimenez-Gutierrez ldquoHeat exchanger network synthesis includingdetailed heat exchanger design using genetic algorithmsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8767ndash8780 2007

[17] K-M Bjork and R Nordman ldquoSolving large-scale retrofitheat exchanger network synthesis problems with mathematicaloptimization methodsrdquo Chemical Engineering and ProcessingProcess Intensification vol 44 no 8 pp 869ndash876 2005

[18] S Mouret I E Grossmann and P Pestiaux ldquoA novel priority-slot based continuous-time formulation for crude-oil schedul-ing problemsrdquo Industrial and Engineering Chemistry Researchvol 48 no 18 pp 8515ndash8528 2009

[19] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[20] J E Hopcroft Introduction to AutomataTheory Languages andComputation Pearson Education India New Delhi India 3rdedition 2008

[21] E Roche and Y Schabes Finite-State Language Processing TheMIT Press Cambridge Mass USA 1997

[22] L Karttunen ldquoConstructing lexical transducersrdquo in Proceedingsof the 15th conference on Computational Linguistics vol 1Association for Computational Linguistics 1994

[23] L Karttunen ldquoThe replace operatorrdquo in Proceedings of the 33rdAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1995

[24] L Karttunen ldquoDirected replacementrdquo in Proceedings of the 34thAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1996

[25] L Karttunen and K R Beesley Two-Level Rule Compiler XeroxCorporation Palo Alto Research Center 1992

[26] G vanNoord ldquoFSAutilities a toolbox tomanipulate finite-stateautomatardquo in Automata Implementation pp 87ndash108 SpringerNew York NY USA 1997

The Scientific World Journal 11

[27] D Gerdemann and G van Noord ldquoTransducers from rewriterules with backreferencesrdquo in Proceedings of the 9th Conferenceon European Chapter of the Association for ComputationalLinguistics Association for Computational Linguistics 1999

[28] Y-C Lin K-S Hwang and F-S Wang ldquoA mixed-codingscheme of evolutionary algorithms to solve mixed-integer non-linear programming problemsrdquo Computers and Mathematicswith Applications vol 47 no 8-9 pp 1295ndash1307 2004

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 9

Table 2 Subsequence belonging to 1198718

Length Sequences belonging to 1198718

1 82 81 82 83 853 812 813 825 831 832 835 851 8524 8125 8132 8312 8313 8325 8351 8352 8512 8513 85255 81325 83125 83132 83512 83513 83525 85132 851256 831325 835125 835132 8513257 8351325

Table 3 Problem data

Scheduling horizon 8 daysVessels Arrival time Composition Amount of crudeVessel 1 0 100 A 1000Vessel 2 4 100 B 1000Storage tanks Capacity Initial composition Initial amountTank 1 [0 1000] 100 A 250Tank 2 [0 1000] 100 B 750Charging tanks Capacity Initial composition Initial amountTank 1 (mix X) [0 1000] 100 C 500Tank 1 (mix X) [0 1000] 100 D 500Crudes 1 Gross margin Crude mixtures Property1 DemandCrude A 001 9 Crude mix X [0015 0025] [1000 1000]

Crude B 006 4 Crude mix Y [0045 0055] [1000 1000]

Crude C 002 8 Unloading flow rate [0 500]

Crude D 005 5 transfer flow rate [0 500]

Table 4 A fragment of FSA regular expression syntax and 119880

transducers and 119877 can be either

[] The empty string[1198771 119877

119899] Concatenation

1198771 119877

119899 Disjunction

119877Λ Optionality

Identity (119860) Identity the transducer which maps eachelement in 119860 onto itself

119879 ∘ 119880 Composition of the transducers 119879 and 119880macro (Term 119877) Use term as an abbreviation for 119877

that each solution involved in theGA algorithm is guaranteedto be feasible by using the mutation rules generated by DFMmethod while in existing GA algorithms the procedure togenerate feasible solution under complex process constraintsis very time costive The deterministic finite automata (DFA)can easily represent this kind of structure Furthermorethe complex process constraints can be very difficult toexpress with mixed integer programming Consequently itis unfeasible to solve the industrial problem by using MIPsolver

6 Conclusion

In this paper a novel hybrid optimization algorithm whichcombined the finite state method and GA is proposed

350 500 650 800 950 1200 1500 1800 24000

02

04

06

08

1

12

14

Computing time (s)

Varia

nce v

alue

s

Mixed-code GAOur algorithm

Figure 7 Variance values of two methodologies

The proposed algorithm constitutes a reasonable frameworkcapturing both the operating condition and sequencing ruleof the schedule The solution captures all possible sched-ules and removes many redundant sequences of operationsThe algorithm is equivalent to introducing new structure

10 The Scientific World Journal

information into the optimization process which will helpreduce the risk of trapping in a local minimal sequencesolution The hybrid optimization algorithm is an effectiveand robust tool to solve the crude oil scheduling problem interms of efficiency and reliability Algorithms only with thetwo properties are suitable for solving practical engineeringapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partially supported by the China NationalNatural Science Foundation under Grant 61203178 Grant61304214 and Grant 61290323 The authors thank the finan-cial funds fromShanghai Science andTechnologyCommitteeunder Grant 12511501002 and Grant 13511501302

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] C A Mendez I E Grossmann I Harjunkoski and P KaboreldquoA simultaneous optimization approach for off-line blendingand scheduling of oil-refinery operationsrdquo Computers andChemical Engineering vol 30 no 4 pp 614ndash634 2006

[6] M Pan X Li and Y Qian ldquoNew approach for scheduling crudeoil operationsrdquo Chemical Engineering Science vol 64 no 5 pp965ndash983 2009

[7] M F Cardoso R L Salcedo S F de Azevedo and D BarbosaldquoA simulated annealing approach to the solution of minlpproblemsrdquo Computers and Chemical Engineering vol 21 no 12pp 1349ndash1364 1997

[8] I E Grossmann ldquoReview of nonlinear mixed-integer anddisjunctive programming techniquesrdquo Optimization and Engi-neering vol 3 no 3 pp 227ndash252 2002

[9] E L Lawler and D E Wood ldquoBranch-and-bound methods asurveyrdquo Operations Research vol 14 no 4 pp 699ndash719 1966

[10] M A Duran and I E Grossmann ldquoAn outer-approximationalgorithm for a class of mixed-integer nonlinear programsrdquoMathematical Programming vol 36 no 3 pp 307ndash339 1986

[11] AMGeoffrion ldquoGeneralized Benders decompositionrdquo Journalof Optimization Theory and Applications vol 10 no 4 pp 237ndash260 1972

[12] C DrsquoAmbrosio andA Lodi ldquoMixed integer nonlinear program-ming tools a practical overviewrdquo 4OR vol 9 no 4 pp 329ndash3492011

[13] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence The University of Michigan Press AnnArbor Mich USA 1975

[14] T Yokota M Gen and Y-X Li ldquoGenetic algorithm for non-linear mixed integer programming problems and its applica-tionsrdquo Computers and Industrial Engineering vol 30 no 4 pp905ndash917 1996

[15] L Costa and P Oliveira ldquoEvolutionary algorithms approachto the solution of mixed integer non-linear programmingproblemsrdquo Computers and Chemical Engineering vol 25 no 2-3 pp 257ndash266 2001

[16] J M Ponce-Ortega M Serna-Gonzalez and A Jimenez-Gutierrez ldquoHeat exchanger network synthesis includingdetailed heat exchanger design using genetic algorithmsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8767ndash8780 2007

[17] K-M Bjork and R Nordman ldquoSolving large-scale retrofitheat exchanger network synthesis problems with mathematicaloptimization methodsrdquo Chemical Engineering and ProcessingProcess Intensification vol 44 no 8 pp 869ndash876 2005

[18] S Mouret I E Grossmann and P Pestiaux ldquoA novel priority-slot based continuous-time formulation for crude-oil schedul-ing problemsrdquo Industrial and Engineering Chemistry Researchvol 48 no 18 pp 8515ndash8528 2009

[19] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[20] J E Hopcroft Introduction to AutomataTheory Languages andComputation Pearson Education India New Delhi India 3rdedition 2008

[21] E Roche and Y Schabes Finite-State Language Processing TheMIT Press Cambridge Mass USA 1997

[22] L Karttunen ldquoConstructing lexical transducersrdquo in Proceedingsof the 15th conference on Computational Linguistics vol 1Association for Computational Linguistics 1994

[23] L Karttunen ldquoThe replace operatorrdquo in Proceedings of the 33rdAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1995

[24] L Karttunen ldquoDirected replacementrdquo in Proceedings of the 34thAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1996

[25] L Karttunen and K R Beesley Two-Level Rule Compiler XeroxCorporation Palo Alto Research Center 1992

[26] G vanNoord ldquoFSAutilities a toolbox tomanipulate finite-stateautomatardquo in Automata Implementation pp 87ndash108 SpringerNew York NY USA 1997

The Scientific World Journal 11

[27] D Gerdemann and G van Noord ldquoTransducers from rewriterules with backreferencesrdquo in Proceedings of the 9th Conferenceon European Chapter of the Association for ComputationalLinguistics Association for Computational Linguistics 1999

[28] Y-C Lin K-S Hwang and F-S Wang ldquoA mixed-codingscheme of evolutionary algorithms to solve mixed-integer non-linear programming problemsrdquo Computers and Mathematicswith Applications vol 47 no 8-9 pp 1295ndash1307 2004

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 The Scientific World Journal

information into the optimization process which will helpreduce the risk of trapping in a local minimal sequencesolution The hybrid optimization algorithm is an effectiveand robust tool to solve the crude oil scheduling problem interms of efficiency and reliability Algorithms only with thetwo properties are suitable for solving practical engineeringapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partially supported by the China NationalNatural Science Foundation under Grant 61203178 Grant61304214 and Grant 61290323 The authors thank the finan-cial funds fromShanghai Science andTechnologyCommitteeunder Grant 12511501002 and Grant 13511501302

References

[1] J M Pinto M Joly and L F L Moro ldquoPlanning and schedulingmodels for refinery operationsrdquo Computers and Chemical Engi-neering vol 24 no 9-10 pp 2259ndash2276 2000

[2] J Li I A Karimi and R Srinivasan ldquoRecipe determination andscheduling of gasoline blending operationsrdquoAIChE Journal vol56 no 2 pp 441ndash465 2010

[3] J Li RMisener andCA Floudas ldquoContinuous-timemodelingand global optimization approach for scheduling of crude oiloperationsrdquo AIChE Journal vol 58 no 1 pp 205ndash226 2012

[4] Z Jia M Ierapetritou and J D Kelly ldquoRefinery short-termscheduling using continuous time formulation crude-oil oper-ationsrdquo Industrial and Engineering Chemistry Research vol 42no 13 pp 3085ndash3097 2003

[5] C A Mendez I E Grossmann I Harjunkoski and P KaboreldquoA simultaneous optimization approach for off-line blendingand scheduling of oil-refinery operationsrdquo Computers andChemical Engineering vol 30 no 4 pp 614ndash634 2006

[6] M Pan X Li and Y Qian ldquoNew approach for scheduling crudeoil operationsrdquo Chemical Engineering Science vol 64 no 5 pp965ndash983 2009

[7] M F Cardoso R L Salcedo S F de Azevedo and D BarbosaldquoA simulated annealing approach to the solution of minlpproblemsrdquo Computers and Chemical Engineering vol 21 no 12pp 1349ndash1364 1997

[8] I E Grossmann ldquoReview of nonlinear mixed-integer anddisjunctive programming techniquesrdquo Optimization and Engi-neering vol 3 no 3 pp 227ndash252 2002

[9] E L Lawler and D E Wood ldquoBranch-and-bound methods asurveyrdquo Operations Research vol 14 no 4 pp 699ndash719 1966

[10] M A Duran and I E Grossmann ldquoAn outer-approximationalgorithm for a class of mixed-integer nonlinear programsrdquoMathematical Programming vol 36 no 3 pp 307ndash339 1986

[11] AMGeoffrion ldquoGeneralized Benders decompositionrdquo Journalof Optimization Theory and Applications vol 10 no 4 pp 237ndash260 1972

[12] C DrsquoAmbrosio andA Lodi ldquoMixed integer nonlinear program-ming tools a practical overviewrdquo 4OR vol 9 no 4 pp 329ndash3492011

[13] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence The University of Michigan Press AnnArbor Mich USA 1975

[14] T Yokota M Gen and Y-X Li ldquoGenetic algorithm for non-linear mixed integer programming problems and its applica-tionsrdquo Computers and Industrial Engineering vol 30 no 4 pp905ndash917 1996

[15] L Costa and P Oliveira ldquoEvolutionary algorithms approachto the solution of mixed integer non-linear programmingproblemsrdquo Computers and Chemical Engineering vol 25 no 2-3 pp 257ndash266 2001

[16] J M Ponce-Ortega M Serna-Gonzalez and A Jimenez-Gutierrez ldquoHeat exchanger network synthesis includingdetailed heat exchanger design using genetic algorithmsrdquoIndustrial and Engineering Chemistry Research vol 46 no 25pp 8767ndash8780 2007

[17] K-M Bjork and R Nordman ldquoSolving large-scale retrofitheat exchanger network synthesis problems with mathematicaloptimization methodsrdquo Chemical Engineering and ProcessingProcess Intensification vol 44 no 8 pp 869ndash876 2005

[18] S Mouret I E Grossmann and P Pestiaux ldquoA novel priority-slot based continuous-time formulation for crude-oil schedul-ing problemsrdquo Industrial and Engineering Chemistry Researchvol 48 no 18 pp 8515ndash8528 2009

[19] H Lee JM Pinto I E Grossmann and S Park ldquoMixed-integerlinear programming model for refinery short-term schedulingof crude oil unloading with inventory managementrdquo Industrialand Engineering Chemistry Research vol 35 no 5 pp 1630ndash1641 1996

[20] J E Hopcroft Introduction to AutomataTheory Languages andComputation Pearson Education India New Delhi India 3rdedition 2008

[21] E Roche and Y Schabes Finite-State Language Processing TheMIT Press Cambridge Mass USA 1997

[22] L Karttunen ldquoConstructing lexical transducersrdquo in Proceedingsof the 15th conference on Computational Linguistics vol 1Association for Computational Linguistics 1994

[23] L Karttunen ldquoThe replace operatorrdquo in Proceedings of the 33rdAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1995

[24] L Karttunen ldquoDirected replacementrdquo in Proceedings of the 34thAnnual Meeting on Association for Computational LinguisticsAssociation for Computational Linguistics 1996

[25] L Karttunen and K R Beesley Two-Level Rule Compiler XeroxCorporation Palo Alto Research Center 1992

[26] G vanNoord ldquoFSAutilities a toolbox tomanipulate finite-stateautomatardquo in Automata Implementation pp 87ndash108 SpringerNew York NY USA 1997

The Scientific World Journal 11

[27] D Gerdemann and G van Noord ldquoTransducers from rewriterules with backreferencesrdquo in Proceedings of the 9th Conferenceon European Chapter of the Association for ComputationalLinguistics Association for Computational Linguistics 1999

[28] Y-C Lin K-S Hwang and F-S Wang ldquoA mixed-codingscheme of evolutionary algorithms to solve mixed-integer non-linear programming problemsrdquo Computers and Mathematicswith Applications vol 47 no 8-9 pp 1295ndash1307 2004

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 11

[27] D Gerdemann and G van Noord ldquoTransducers from rewriterules with backreferencesrdquo in Proceedings of the 9th Conferenceon European Chapter of the Association for ComputationalLinguistics Association for Computational Linguistics 1999

[28] Y-C Lin K-S Hwang and F-S Wang ldquoA mixed-codingscheme of evolutionary algorithms to solve mixed-integer non-linear programming problemsrdquo Computers and Mathematicswith Applications vol 47 no 8-9 pp 1295ndash1307 2004

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014