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Chemical Engineering and Processing 49 (2010) 1076–1083

Contents lists available at ScienceDirect

Chemical Engineering and Processing:Process Intensification

journa l homepage: www.e lsev ier .com/ locate /cep

ptimization of short-time gasoline blending scheduling problemith a DNA based hybrid genetic algorithm

iao Chena,b, Ning Wanga,∗

National Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou 310027, PR ChinaInstitute of Automation, Hangzhou Dianzi University, Hangzhou 310018, PR China

r t i c l e i n f o

rticle history:eceived 31 August 2009eceived in revised form 25 July 2010ccepted 28 July 2010vailable online 6 August 2010

a b s t r a c t

Gasoline blending is a key process in the petroleum refinery industry posed as a nonlinear optimizationproblem with heavily nonlinear constraints. This paper presents a DNA based hybrid genetic algorithm(DNA-HGA) to optimize such nonlinear optimization problems. In the proposed algorithm, potentialsolutions are represented with nucleotide bases. Based on the complementary properties of nucleotidebases, operators inspired by DNA are applied to improve the global searching ability of GA for efficiently

eywords:NA computingenetic algorithmQPybrid optimization methodonlinear optimization problems

locating the feasible domains. After the feasible region is obtained, the sequential quadratic programming(SQP) is implemented to improve the solution. The hybrid approach is tested on a set of constrainednonlinear optimization problems taken from the literature and compared with other approaches. Thecomputation results validate the effectiveness of the proposed algorithm. The recipes of a short-timegasoline blending problem are optimized by the hybrid algorithm, and the comparison results show thatthe profit of the products is largely improved while achieving more satisfactory quality indicators in both

envi

hort-time gasoline blending scheduling certainty and uncertainty

. Introduction

Gasoline blending is a crucial process for the modern oil refineryhat its product, gasoline, can yield 60–70% of a typical refinery’sotal revenue [1,2]. Since a good solution may eliminate re-blendsnd reduce 5–20% of component and product inventory, the opti-ization of gasoline blending is often considered as a key problem

or the profitability of a company [1–3]. In the gasoline blendingrocess, different gasoline products are produced by combiningnumber of feedstock together with small amounts of additives

such as antioxidants, corrosion inhibitors, metal deactivators,etergents, and dyes) while meeting certain quality specifications.ence, the blending problem is naturally represented as a nonlinearptimization problem with many linear and nonlinear constraints.

Currently, many blending applications are treated as extensionsf linear problems, and linear programming (LP) strategies [4,5]nd nonlinear programming (NLP) strategies are applied to findhe optimum solutions. However, LP approaches have shortcom-ngs in terms of robustness and convergence, and NLP methods do

ot provide a straight forward extension to handle discrete com-inatorial elements in blending [6]. In view of this, applicationf genetic algorithm (GA) to overcome the drawbacks mentionedbove has attracted interest in chemical engineering, chemistry,

∗ Corresponding author.E-mail address: nwang@iipc.zju.edu.cn (N. Wang).

255-2701/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.cep.2010.07.014

ronment.© 2010 Elsevier B.V. All rights reserved.

and other fields [7–12]. Genetic algorithm is a stochastic globalsearch technique and has been used for many combinatorial opti-mization problems as in Holland [13] and Goldberg [14]. However,it is observed that the local convergence of GA is slow especially forconstrained optimization problems.

One way of improving the local convergence of GA is to forma hybrid GA by including local search methods, like SQP method[15]. In the most of the hybrid algorithms, GA is used to find thefeasible region and provide the promising starting point for theSQP method [16–20]. Thus, the capability of efficiently locating thefeasible region is crucial for the performance of the hybrid algo-rithm. In the recent years, DNA based GAs inspired by the DNAstructure are reported [21–25]. With specific encoding method andgenetic operators, DNA based GA (DNA-GA) can improve the globalsearching speed of GA and avoid the premature convergence.

Based on the above discussion, a DNA based hybrid geneticalgorithm is presented to solve nonlinear constrained optimizationproblems. In the proposed algorithm, the solutions are representedas the sequences of nucleotide bases. Based on the complementaryproperties of the nucleotide bases, DNA inspired genetic opera-tors which is different from that in traditional GAs are used toexplore the searching space and locate the feasible region. Then,

SQP (Sequence Quadratic Programming) method is used to findmore accurate solution based on the initial point provided by DNA-GA. The remainder of the paper is organized as follows: Section2 describes the details of the DNA based hybrid genetic algo-rithm including the constraints handling, encoding method and

ring and Processing 49 (2010) 1076–1083 1077

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X. Chen, N. Wang / Chemical Enginee

enetic operators. Section 3 compares the performances of the pro-osed algorithm with other reported algorithms using nonlinearonstrained functions, and depicts the evolution process of the pro-osed algorithm. In Section 4, the recipe of a gasoline blendingroblem composed of three cases is optimized by the proposedlgorithm, and the profit and the quality of the products obtainedy different optimization methods are compared.

. The DNA based hybrid genetic algorithm (DNA-HGA)

.1. Problem description and constraints handling

Generally, the nonlinear constrained optimization problem cane described as follows:

min f (x)s.t. gi(x) ≤ 0, i = 1, 2, . . . , M

x ∈ [xmin, xmax](1)

here f(x) is the objective function, x = (x1, x2, . . ., xn) represents theolution vector, and gi(x) represents the ith nonlinear constraint.

In the previous study, there are many methods to handle theonlinear constraints, such as penalty functions [26], repair func-ions [27], and Pareto solution [28]. In this paper, the fitnessunction for constraint handling is devised as follows:

If all solutions in the current generation are infeasible

itness = deviation =M∑

i=1

�iqi(x)r(qi(x)) (2)

i(x) = max{0, gi(x)}, r(qi(x)) ={

1 qi(x) < 12 qi(x) ≥ 1

(3)

Otherwise

itness ={

f (x) feasible solutionf (x) + fmax(kg) + deviation infeasible solution

(4)

here �i is the weight of the ith constraint, fmax(cg) is the objectiveunction value of the worst feasible solution in current generationg. The value of �i can change according to the problems that thealue of �i would be larger when the ith constraints is more impor-ant than the other constraints. This constraint handling guaranteesll the infeasible solutions are inferior to the feasible solutions.

.2. DNA encoding method

To better imitate the biological operation of DNA, we useucleotide bases, adenine (A), guanine (G), cytosine (C), andhymine (T), to represent the potential solutions of the opti-

ization problem. These nucleotide bases are basic units of DNAolecule and can be attracted through hydrogen bond based on

he Watson–Crick complementary principle: A bonds with T, andbonds with C. Through this pairwise attraction, a nucleotide

equence can be bonded with another sequence called the comple-ent sequence and form the well-known double helix DNA [29].oreover, for the convenience of the computer, integers 0, 1, 2, andare adopted to encode the nucleotide bases. And the mapping

rom nucleotide bases to the digital integer is 0123/CGAT. Throughnheriting the complementary properties of the nucleotide bases,he four integers pair as (2, 3) and (0, 1). Based on the above encod-

ng method, every variable xi in problem (1) is represented as annteger string of length l. And the length of one individual is L = n × l.

The decoding method of this algorithm is similar to the binaryA. First, each individual is separated into n parts, and each partill be translated into a real number corresponding to one variable

Fig. 1. Procedure of the reconstruct operation.

of the optimized problem. For part i, the sequence will be decodedinto a integer tempxi:

tempxi =l∑

p=1

bit(p) × 4l−p (5)

where bit(p) is the digital number of the pth gene for xi. Then, tempxiwill be mapped into a real number xi:

xi = tempxi

4l−1(xmax i − xmin i) + xmin i (6)

2.3. Genetic operators

Along with the DNA encoding method, more complicatedgenetic operators are designed to improve the global search abilityof GA. The genetic operators include crossover operator, recon-struction operator and three mutation operators [24].

2.3.1. CrossoverCrossover is an operator that exchanges information between

different individuals. Before the adoption of the crossover, two indi-viduals should be chosen at random from the population as theparents. In the crossover process, an individual could be selectedmore than once as a parent. After the parents are determined,two children would be generated through two-point crossover. Thecrossover operator is adopted with probability pc.

2.3.2. ReconstructionReconstruction is an operator that reconstructs two similar indi-

viduals. In this process, three individuals are selected randomlyfrom the population as the candidates. Among the candidates, twoindividuals which are more similar to each other would be cho-sen as the parents. Then, two children are produced through thereconstruction operator as shown in Fig. 1. First, one subsequenceis cut from the rear of one parent, and then the cut subsequenceis stuck at the front of another parent. Here, the length of the cut

subsequence is randomly selected. Second, another new segmentwith the same length of the cut subsequence is randomly gener-ated and it will be stuck at the rear of the shorter parent. Third, thelonger parent will be tailored to the predefined individual length L.Through this operation, two similar individuals are not as similar

1078 X. Chen, N. Wang / Chemical Engineering and Processing 49 (2010) 1076–1083

Table 1Comparison of the solutions for test functions over 50 runs.

GA [30,31] HGA [23] DNA-HGA

ε ≤ 1% Best G ε ≤ 1% Best avekg ε ≤ 1% Best avekg

13.570424.3−1.

ap

2

(

(

(

2

t

f1 29 13.59085 50 50f2 23 7060.221 4000 50f3 41 24.3725 3500 50f4 – −0.8332 10,000 –

s before. The reconstruction operator occurs with the probabilityr.

.3.3. Mutation

a) Inverse-anticodon operator (IA operator)IA operator is an operator which replaces the codon with its

inverse anticodon. For example, suppose the original individualis 002312 where 12 are chosen as a codon, the anticodon willbe 03 and the inverse anticodon is 30. Then, the new individualafter IA operator is 002330. IA operator occurs with the proba-bility pIA. And the number of the bases and the location of codonare both assigned randomly.

b) Maximum–minimum operator (MM operator)MM operator can change the chromosome by replacing the

frequently used bases with the rarely used bases in the currentchromosome. As an example, suppose the strand is 1223000where base 0 (C) is the most frequently used base and 1 (G) isthe least frequently used base, then the new strand after MMoperator is 1223111. MM operator occurs with the probabilitypMM.

c) Normal-mutation operator (NM operator)NM operator is a background operator which produces spon-

taneous random changes in the chromosomes. In this process,every base in the individual can be replaced by one of anotherthree bases with the probability pm. As an example, suppose thestrand is 1223000, then the new strand is 1221000 where 3 isreplaced by 1.

.4. The procedure of the DNA based hybrid GA

The procedure of the DNA based hybrid GA (DNA-HGA) is illus-rated as follows:

Step 1: Initialize a population containing N individuals using theDNA encoding method.Step 2: Calculate the fitness value of each individual.Step 3: Judge whether the optimum remains same for 50 succes-sive generations. If the condition is satisfied, the SQP method isapplied to explore the neighborhood for more accurate solutionby using the current best individual as the starting point, and thesolution obtained by the SQP method is inserted into the currentpopulation; otherwise, the algorithm goes to step 4.Step 4: Select two parents randomly from the population and carryout the crossover operator over the parents to produce two newindividuals. If the crossover is not performed, reconstruction oper-ator is employed. Repeat this process until N/2 new individuals aregenerated. Then insert all the new individuals into the populationwithout deleting old individuals.Step 5: Adopt IA operator, MM operator and NM operator orderlyover each individual in the population, and generate 3/2N new

individuals. Then replace all the original individuals with the newones.Step 6: Select N − 2 individuals from the population with tourna-ment selection, along with the two best individuals of the currentgeneration, to advance into the next generation.

9084 80 50 13.59084 1269.2480 532 50 7049.24802 253062 650 50 24.3062 382

1314 – – −1.1792 –

Step 7: Repeat steps 2–6 until the stop criteria are met, andthe final solution is found. The stop criterion can be the prede-fined maximum number of generations, the predefined minimumimprovement of the best individual in the consecutive generations,or the set minimum distance between the best individual and theknown optimum.

3. Numeric simulation

To investigate the performance of the proposed method, fournonlinear constrained benchmark functions are taken from the lit-erature to test the optimization ability of the proposed algorithm.Here, f1 is a nonlinear optimization problem with nonlinear objec-tive function and nonlinear constraint. f2 is a nonlinear problemwith linear objective function and nonlinear constraints. f3 has aquadratic objective function with quadratic constraints. f4 is a prob-lem with strongly nonlinear objective function and constraints. Thedetails of the functions are listed in Appendix B. Meanwhile, theperformance of other approaches including modified GAs [30,31]and a hybrid GA [23] are compared with that of the DNA-HGA.

Since GAs are stochastic algorithms, each test method runs 50times from random initial population for each problem. The param-eters of the DNA-HGA are set as: l = 10, N = 40, pc = 0.8, pr = 0.2,pIA = 0.5, pMM = 0.5, and pm = 0.001. Here, considering neither of theconstraints in the test function is more important than the oth-ers, we choose �i = 1. And the maximum evolution generation is1000. The comparison results of the performance of all the meth-ods are listed in Table 1. Here, ε ≤ 1% denotes the number of the runsthat the test algorithm satisfies the requirement ε = |f − f(x*)|/f ≤ 1%,where f is the best-so-far fitness value in the current generationand x* is the known optimal solution. The G column of GA meansthe evolution generation of GA in the corresponding test problem.And the value of avekg of HGA and DNA-HGA represents the aver-age generation that the algorithm needs to satisfy ε ≤ 1%. Since thebest optimum of f4 is unknown, we only compare the best valueobtained by different methods, and the other columns are filledwith short line. We should also notice that the solution for the con-strained problem is evaluated not only by the function value butalso by the violation of the constraints. The less the violation is, thesuperior the solution is. Here, we use a vector vio to indicate the vio-lation of the constraints, where vio(i) is a positive number denotingthe degree that the solution violates constraint i. Obviously, whenvio(i) is zero, it means constraint i is satisfied.

f1 is a two-dimensional constrained problem whose feasibleregion is a narrow crescent-shaped region. For this function, Deb’simproved GA only gets the known optimum 29 times in 50 runs. Ascomparison, HGA [23] and our method both can reach the optimumin all runs. But the value of avekg in Table 1 shows that our methodspends a little more generations to find the optimum.

For f2, the best function value obtained by Deb is 7060.221 andTao gets a better solution with the function value of 7049.2480. And

the result of 6 inequality constraints of Tao’s best solution is vio = (0,0, 0, 0, 0.0659, 0). As comparison, the best solution obtained by ourmethod is located at x = (579.306799, 1359.970661, 5109.970560,182.017709, 295.601178, 217.982291, 286.416532, 395.601178)with function value of 7049.2480. And the deviation from the con-

ring and Processing 49 (2010) 1076–1083 1079

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X. Chen, N. Wang / Chemical Enginee

traints is vio = 10−9 (0, 0.25, 0, 0, 0, 0), which is obviously superioro the solution obtained by Tao [23]. Moreover, we also noticedhat the average required generation of the proposed hybrid GA islmost half reduced compared with that of [23].

As for f3, the best value obtained by Tao [23] is f(x*) = 24.3062,nd the deviation of 8 constraints is vio = 10−3 (0, 0.6, 0.0083,, 0.001, 0.4266, 0, 0). As comparison, we obtain the best solu-ion locating at x = (2.171939, 2.363826, 8.773934, 5.095955,.990696, 1.430604, 1.321532, 9.828634, 8.279939, 8.375918) with

(x*) = 24.3062, and vio = (0, 0, 0, 0, 0, 0, 0, 0) means no constraints violated. It means that our best solution is superior to the otherwo methods.

The DNA-HGA performs better than other tested methods inhe fourth function f4. As a heavily nonlinear and multi-modalonstrained problem, the global optimum of f4 is unknown atresent. Although the proposed method is also trapped into the

ocal optimum, we can easily find that the function value ofhe solution obtained by our method is −1.1792, which is the

inimum of all the obtained solutions. The best solution obtainedy the DNA-HGA is located at x = (3.176450, 6.280301, 6.266649,.135888, 3.122428, 3.108855, 3.095371, 3.081363, 0.39946,.053473, 0.374801, 0.364003, 0.354144, 0.344725, 0.336390,.328025, 0.320468, 0.313174, 0.306399, 0.299752). And theiolation of constraints is zero.

To clearly illustrate the function scheme of the proposedethod, we show the evolution process of the DNA-HGA in two

est functions, f1 and f2, in Figs. 2 and 3. From Fig. 2, we can find thatefore 100 generation, the DNA-HGA can reach the feasible regionnly relying on the global search ability of the hybrid GA. However,t will be trapped into the local optimum and convergence speedecomes slower until the SQP method is implemented to gain thelobal optimum. This evolution process clearly illustrates the func-ion scheme of the hybrid method. The same evolution process islso can be seen in the test for f2 from Fig. 3.

. Short-time gasoline blending scheduling

.1. Complex scheduling problem

Gasoline blending is an important example of blending schedul-ng problem in the process industry. Its product, gasoline, is one of

Fig. 3. Evolution process with the best solution on f2.

Fig. 4. Illustration of gasoline blending system.

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080 X. Chen, N. Wang / Chemical Enginee

he most important refinery products as it can yield 60–70% of totalevenue of a typical refinery’s. Thus, the optimization of the recipes important for the profitability of the refinery.

Fig. 4 shows a typical blending process which produces twoypes of gasoline, regular gasoline and premium gasoline, with 5omponents: Reformate, LSR naphtha, n-Butane, Catalytic gas andlkylate [6].

Usually, the products of the gasoline blending process shouldeet certain quality specifications while the process is subject to a

umber of operation restrictions including limits on the availabilityf blending components and product storage facilities. Thus, theptimization of the gasoline blend process can be summarized toaximize the profitability of the products while satisfying all of

he product quality and operating constraints. In view of this, thebjective of blending scheduling can be described as follows [4]:

ax f =Np∑

n=1

(Cpn × Vpn −

Ncn∑m=1

Ccm × Vcn,m

)(7)

here the former part is the revenue of the products and the latters the cost of blending components, Ncn is the number of compo-ents to blend product n, Np is the number of products, Cpn is therice of product n, Vpn is the volume of product n, Ccm is the costf component m and Vcn,m is the volume of component m to blendroduct n.

As mentioned above, the constraints of the gasoline blendingroblem can be generalized into two types: operation restrictionnd quality specification of the products. The operation restrictiononstraints are given by Eqs. (8)–(10) [6].

Material balance equation between products and components

Np

n=1

Vpn =Np∑

n=1

Ncn∑m=1

VCn,m (8)

Supply capacity of components

CLm

≤Np∑

n=1

VCn,m ≤ VCUm

(9)

here VCLm

and VCUm

is the minimum and maximum supply of com-

onent m, respectively.Market demands of the products

pn = Mpn (10)

here Mpn is the market demand of product n.As for the specification of the products, they are usually

escribed as nonlinear constraints given by Eqs. (11)–(16):

Quality specification A:

QRONn ≥ RONn,min (11)

where RONn,min is the given minimum research octane number(RON) indicator of product n, and QRONn is the value of RON indi-cator of product n computed by [3]

QRON = rT x + ˛1

(rT diag(s)x − (rT x)(sT x)

eT x

)

+ ˛2

(oT

s x − (oT x)2

eT x

)+ ˛3

(aT

s x − (aT x)2

eT x

)(12)

Quality specification B:

QMONn ≥ MONn,min (13)

where MONn,min is the given minimum motor octane number(MON) indicator of product n, and QMONn is the value of MON

nd Processing 49 (2010) 1076–1083

indicator of product n computed by [3]

QMON = mT x + ˛4

(mT diag(s)x − (mT x)(sT x)

eT x

)

+ ˛5

(oT

s x− (oT x)2

eT x

)+ ˛6

10, 000(eT x)

(aT

s x− (aT x)2

eT x

)2

(14)

and x is the flow rate of feedstock for product n, r is RON vec-tor of feedstock, m is the MON vector of feedstock, s = r − m, e isunit volume vector, o is olefin content vector of feedstock (% byvolume), os is square of olefin content vector, a is aromatic con-tent vector of feedstock (% by volume), ˛i is the parameter of themodel (˛1 = 0.03224, ˛2 = 0.00101, ˛3 = 0, ˛4 = 0.0445, ˛5 = 0.00081and ˛6 = −0.00645), and as is square of aromatics content vector.Quality specification C:

QRVPn ≤ RVPn,max (15)

where RVPn,max is the given limit of reid vapor pressure (RVP) indi-cator for product n, and QRVPn is the value of RVP indicator ofproduct n computed by [3]

PRVP =(

Ncn∑i=1

ui(RVPi)1.25

)0.8

(16)

and RVPi is the RVP of the component i and ui is the volume fractionof component i.

From the above description, we can conclude that the gasolineblending problem contains different type of constraints: productdemand and material balance are both linear equality constraints,component supply is linear inequality constraint while the specifi-cations of products are nonlinear inequality constraints. To reducethe complexity of the constraints handling, the component supplyconstraints are transformed into the decoding process of the DNA-HGA. Since the value of premium gasoline is higher than the regulargasoline, we would first guarantee the component for blending thepremium gasoline to make higher profit. Thus, for each individual,the variables corresponding to the components for the premiumgasoline are decoded before that for regular gasoline that the xmax iis VCU

min Eq. (6). And if the volume of component j for premium

gasoline is vj,p, then the xmax i will be VCUm

− vj,p in Eq. (6) when the

sequence corresponding to the volume of component j for regulargasoline is decoded. The material balance constraints are trans-ferred by modifying objective function (7) as follows:

max f =Np∑

n=1

(Cpn ×

Ncn∑m=1

Vcn,m −Ncn∑m=1

Ccm × Vcn,m

)(17)

4.2. Optimization results and discussion

In this section, the proposed DNA based hybrid GA (DNA-HGA)is applied to optimize the recipes of a short-time gasoline blendingproblem which is composed of three cases. Among the three cases,two of them provide certain circumstance while the third case con-tains uncertain parameter. The parameters of the DNA-HGA are set

as follows: G = 300, N = 90, l = 10, pc = 0.8, pr = 0.1, pIA = 0.3, pMM = 0.1,and pm = 0.001, respectively. Since the previous study does not indi-cate either constraint is more important than the others, we alsochoose �i = 1. As comparison, we also provide the results optimizedby other reported methods.

X. Chen, N. Wang / Chemical Engineering and Processing 49 (2010) 1076–1083 1081

Table 2Demand and the quality requirement of the production [1,32,34].

Value, $/bbl Demand, bbl/day Min RON Min MON Max RVP

Regular 33.0 8000 88.5 77.0 10.8Premium 37.0 10,000 91.5 80.0 10.8

Table 3Parameters of component [32,33].

Component Reformate LSR naphtha n-Butane Catalytic gas Alkylate

RON 94.1 70.7 93.8 92.9 95.0MON 80.5 68.7 90.0 80.8 91.7Olefin, % 1.0 1.8 0 48.8 0Aromatics, % 58.0 2.7 0 22.8 0RVP, psi 3.8 12.0 138.0 5.3 6.6Available, bbl/day 12,000 6500 3000 4500 7000Cost, $/bbl 34.0 26.0 10.3 31.3 37.0

Note: bbl stands for barrel (a unit of volume); 1 psi = 6894.76 Pa.

Table 4Recipe for one-day gasoline blending scheduling problem.

n-Butane Catalytic gas Alkylate

150.5 126.3 0222.3 4373.7 20.4

4p

icrbf

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4p

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TC

Table 6Production requirements for three-day [34].

Demand of 1stday, bbl/day

Demand of 2ndday, bbl/day

Demand of 3rdday, bbl/day

Regular 8000 7000 9000Premium 10,000 8000 9000

Table 7Recipe for three-day gasoline blending scheduling problem.

Reformate LSR naphtha n-Butane Catalytic gas Alkylate

1st dayRegular 5172.3 2550.9 150.5 126.3 0Premium 3817.4 1566.2 222.3 4373.7 20.4

2nd dayRegular 3716.8 2243.8 126.6 912.8 0Premium 2987.3 1247.8 177.7 3587.2 0

Reformate LSR naphtha

Regular 5172.3 2550.9Premium 3817.4 1566.2

.2.1. Case 1: recipes for one-day gasoline blending schedulingroblem

The objective of case 1 is to find an optimized recipe with max-mum product profit for a one-day gasoline blending problem. Theorresponding parameters including market demand and qualityequirement listed in Table 2 are taken from [1,32,34], while thelending component quality parameters shown in Table 3 are takenrom [32,33].

The recipe obtained by the DNA-HGA [24] is shown in Table 4hile the quality indicators and the profit of products are given

n Table 5. For comparison, we also list the results of other meth-ds in Table 5. As we can see, the DNA-GA improves the profit to75,112 compared to PSO [34] and GA [6]. But the proposed DNA-GA obtained the highest profit among the four methods, $75,862,ith least redundancy of MON indicator in regular gasoline.

.2.2. Case 2: recipes for three-day gasoline blending schedulingroblem

Compared with case 1, case 2 is more complicated that it needshe optimization method provide the recipes for three-day blend-ng. The market demand of the products for three days listed in

able 6 is taken from [34], while the other parameters remain sames in case 1. It should be noticed that the remaining componentsf current day can be added to the component supply of the fol-owing days. The optimized recipes are given in Table 7 and the

able 5omparison results between different methods.

Optimization method RON MON RVP Profit,$

DNA-HGARegular 88.4999 77.8875 10.7993

75,862Premium 91.4999 79.9999 10.7992

DNA-GA [24]Regular 88.5000 77.8960 10.7823

75,112Premium 91.5390 80.0373 10.7449

PSO [34]Regular 88.7 78.3841 10.8004

74,618Premium 91.5002 80.0406 10.8006

GA [6]Regular 88.5 78.2 10.8

62,443Premium 91.5 80.1 10.7

3rd dayRegular 5490.9 2877.4 167.2 464.4 0Premium 3360.8 1403.7 199.9 4035.6 0

results of the products are shown in Table 8. Similarly, the resultsof the recipes optimized by DNA-GA [24] and PSO [34] are, respec-tively, listed in Tables 9 and 10 as comparison. From Tables 8–10,we can see that the total profit of the products obtained by our

method is $213,256, which is higher than that of other recipes,$210,041 and $210,396. Furthermore, the indicator of products’quality also shows the advantage of our obtained recipe. For exam-ple, the sum of the redundancy of RVP indicator in DNA-GA is 1.1826

Table 8Quality and profit of products optimized with the DNA-HGA.

RON MON RVP Profit, $

1st dayRegular 88.4999 77.8875 10.7993

75,862Premium 91.4999 79.9999 10.7992

2nd dayRegular 88.5000 78.0827 10.8003

64,295Premium 91.4999 79.9999 10.7993

3rd dayRegular 88.5000 77.9472 10.8005

73,099Premium 91.5001 80.0000 10.7989

1082 X. Chen, N. Wang / Chemical Engineering a

Table 9Quality and profit of products optimized by DNA-GA [24].

RON MON RVP Profit, $

1st dayRegular 88.5000 77.8960 10.7823

75,112Premium 91.5390 80.0373 10.7449

2nd dayRegular 88.6150 78.1222 10.5736

63,273Premium 91.5729 80.0294 10.4452

3rd dayRegular 88.5835 77.9873 10.7101

71,656Premium 91.5812 80.0012 10.3613

Table 10Quality and profit of products optimized by PSO [34].

RON MON RVP Profit, $

1st dayRegular 88.7000 78.3841 10.8004

74,618Premium 91.5002 80.0406 10.8006

2nd dayRegular 88.6878 78.2545 10.8008

64,068Premium 91.5412 80 10.5477

3rd dayRegular 88.6842 78.1440 10.8010

71,710Premium 91.5 80.2967 10.8006

Table 11Recipe for one-day gasoline blending scheduling problem with uncertaintyparameters.

Reformate LSR naphtha n-Butane Catalytic gas Alkylate

Regular 5170.1 2551.4 147.3 131.2 0Premium 3821.2 1566.7 217.6 4368.8 25.6

Table 12Comparison results between different methods.

Optimization method RON MON RVP Profit, $

DNA-HGARegular 88.5000 77.8862 10.7991

75,666Premium 91.4999 79.9999 10.7991

DNA-GA [24]Regular 88.5048 77.8892 10.5400

74,931Premium 91.6043 80.0528 10.7778

wvi

4p

pdacttTo

5

ih

3 8 5

PSO [6]Regular 88.5 77.68 6.0

61,680Premium 91.5 80 10.65

hich is larger than that of PSO, 0.2557. The proposed method pro-ides more preferable recipe that the sum of the redundancy of RVPndicator is only 0.0025.

.2.3. Case 3: recipes for one-day gasoline blending schedulingroblem with uncertainty parameters

Suppose the RVP of n-Butane component is an uncertaintyarameter with normal distribution of expectation 138 and stan-ard deviation 2. Then, the nonlinear constraint of RVP turns intoconstraint with uncertain parameter. Here, the RVP of n-Butane

omponent, with probability � = 0.9. The final recipe obtained byhe proposed hybrid method is listed in Table 11. We also givehe results of the recipe obtained by PSO [6] and DNA-GA [24] inable 12 as comparison. It is obvious that the products obtained byur recipe have higher profit and superior quality.

. Conclusions

Gasoline blending is considered as the key problem for the prof-tability of refineries, which is hard to optimize because of theeavily nonlinearity constraints. In this work, a DNA based hybrid

nd Processing 49 (2010) 1076–1083

GA is proposed to optimize the recipes of the gasoline blendingproblem with the objective to maximizing the product profit. Inthe DNA-HGA, genetic operators inspired by the biological DNA areapplied to improve the global search ability of GA, and the SQPmethod is used to alleviate the weak local search capability of GA.The comparison results of several constrained problems show theefficiency of the proposed DNA-HGA, and the optimized recipesof the gasoline blending problem reflect that the proposed hybridmethod improves the profit largely with more satisfactory qualityindicators compared with previously reported methods.

Acknowledgements

This work was supported by the National Natural Science Foun-dation of China under grant Nos. 60874072 and 60721062.

Appendix A. Nomenclature

as square of the aromatics contenta aromatic content of each componentCcm the cost of component mCpn the value of product neT unit matrixl each parameter encoding lengthL individual lengthm the MON vector of each componentMpn market demand of product nN population sizeNc,n number of components to blend product nNp the number of productspc crossover probabilitypIA IA operator probabilitypm NM operator probabilitypMM MM operator probabilityr the RON of each componentVpn the volume of product nVcn,m the volume of component m to blend product nVCL

mthe minimum supply of component m

VCUm

maximum supply of component m

Appendix B.

f1 :

min f (x) = (x21 + x2 − 1)

2 + (x1 + x22 − 7)

2

s.t. g1(x) ≡ 4.84 − (x1 − 0.05)2 − (x2 − 2.5)2 ≥ 0

g2(x) ≡ x21 + (x2 − 2.5)2 − 4.84 ≥ 0

0 ≤ x1 ≤ 6, 0 ≤ x2 ≤ 6

The optimal solution is given by x* = (2.246826, 2.381865),f(x*) = 13.59085

f2 :

min f (x) = x1 + x2 + x3

s.t. g1(x) ≡ 1 − 0.0025(x4 + x6) ≥ 0

g2(x) ≡ 1 − 0.0025(x5 + x7 − x4) ≥ 0

g (x) ≡ 1 − 0.01(x − x ) ≥ 0

g4(x) ≡ x1x6 − 833.33252x4 − 100x1 + 83333.333 ≥ 0

g5(x) ≡ x2x7 − 1250x5 − x2x4 + 1250x4 ≥ 0

g6(x) ≡ x3x8 − x3x5 + 2500x5 − 1, 250, 000 ≥ 0

ring a

x

f

−

x

f

0

x

f

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

X. Chen, N. Wang / Chemical Enginee

100 ≤ x1 ≤ 10, 000, 1000 ≤ (x2, x3) ≤ 10, 00010 ≤ xi ≤ 1000, i = 4, . . . , 8

The optimal solution is given by

∗ = (579.3167, 1359.943, 5110.071, 182.0174, 295.5985,

× 217.9799, 286.4162, 395.5979)

(x∗) = 7049.330923

f3 :min f (x) = x2

1 + x22 + x1x2 − 14x1 − 16x2 + (x3 − 10)2 + 4(x4 − 5)2

+ (x5 − 3)2 + 2(x6 − 1)2 + 5x27 + 7(x8 − 11)2 + 2(x9 − 10)2

+ (x10 − 7)2 + 45s.t. g1(x) ≡ 105 − 4x1 − 5x2 + 3x7 − 9x8 ≥ 0

g2(x) ≡ −10x1 + 8x2 + 17x7 − 2x8 ≥ 0g3(x) ≡ 8x1 − 2x2 − 5x9 + 2x10 + 12 ≥ 0g4(x) ≡ −3(x1 − 2)2 − 4(x2 − 3)2 − 2x2

3 + 7x4 + 120 ≥ 0g5(x) ≡ −5x2

1 − 8x2 − (x3 − 6)2 + 2x4 + 40 ≥ 0g6(x) ≡ −x2

1 − 2(x2 − 2)2 + 2x1x2 − 14x5 + x6 ≥ 0g7(x) ≡ −0.5(x1 − 8)2 − 2(x2 − 4)2 − 3x2

5 + x6 + 30 ≥ 0g8(x) ≡ 3x1 − 6x2 − 12(x9 − 8)2 + 7x10 ≥ 0

10 ≤ xi ≤ 10, i = 1, . . . , 10

The optimal solution is given by

∗ = (2.171996, 2.363683, 8.773926, 5.095984, 0.9906548,

× 1.430574, 1.321644, 9.828726, 8.280092, 8.375927)

(x∗) = 24.3062

f4 :

min f (x) = −∣∣∑n

i=1cos4(xi) − 2∏n

i=1cos2(xi)∣∣√∑n

i=1ixi

s.t.n∏

i=1

xi ≥ 0.75

n∏i=1

xi ≤ 7.5n

≤ xi ≤ 10, i = 1, . . . , n

The optimal solution is

∗ = (3.1520, 9.4137, 6.2666, 4.6805, 3.1172, 3.1085, 3.0997,

× 3.0910, 3.0821, 3.0733, 0.2080, 3.0554, 0.2003, 0.1965,

× 0.1929, 0.1894, 0.1859, 0.1826, 0.1793, 0.1762)

(x∗) = −1.1314

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