Optimization with Multiple Performance Criteria. Application to

OPTIMIZATION WITH MULTIPLE PERFORMANCE CRITERIA Application to Minimization o f Parameter Sensitivities in a Refinery Model J O H N

H .

SEINFELD

A N D

W A R R E N

1. M C B R I D E '

Downloaded by FLORIDA STATE UNIV on September 9, 2015 | http://pubs.acs.org Publication Date: January 1, 1970 | doi: 10.1021/i260033a010

Chemical Engineering Laboratory, California Institute of Technology, Pasadena, Calif. 91 109

In complex optimization problems there often exist several noncomparable performance criteria which a designer must consider. This situation is formulated as a vector maximum problem, for which the available theory on proper efficient solutions is outlined. A practical scheme is proposed for problems in which the criteria can be ranked in order of importance. A simplified refinery model is formulated in which it is desired to maximize total yearly profit as well as minimize the sensitivity of the profit to variations in refinery conditions. Detailed nonlinear programming results are presented.

T HE mathematical theories of steady-state and dynamic optimization require that the performance of a system be represented by a single scalar-valued criterion (Saaty and Bram, 1964). I n complex systems, in general, many considerations enter into an evaluation of performance, and the choice of a single performance index to represent all or some of these considerations is often arbitrary or subjective. The traditional approach is to add together each of the individual criteria, using different weighting factors for each. However, since the result of the optimization problem can vary significantly as the weighting coefficients change, and since very little is usually known about how to choose the weighting factors, a necessary approach is to solve the same problem for many different values of the coefficients. Still, confronted with all these solutions, the system designer must then choose among them, presumably on the basis of some other criterion. Obviously, if each individual criterion can be expressed in equivalent terms-e.g., dollars and cents-there is no problem. I t is when different criteria are not comparable that the real difficulty arises. Zadeh (1963) was the first to consider the optimal control problem for a class of systems in which not all the criteria are comparable. He suggested t h a t a vector performance index be defined to order the possible systems partially. The ordering would include the subclasses of systems: (i) better than some system S, (ii) inferior to or equal t o S, and (iii) not comparable to S. Once these classes of systems have been determined, one must choose a single system from the class of systems which are either not comparable or equivalent. Nelson (1964) suggested an alternative approach to the 1

Present address, American Oil Co., Whiting, Ind.

Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 1, January 1970

optimal control problem when two noncomparable cost factors are involved. As an example he considered a satellite attitude control problem in which both fuel expenditure and time to reach a desired goal are important. For a specified fuel consumption the maximum time can be determined. A curve of fuel consumption us. time can be constructed from solution of the minimum time problem for different levels of fuel consumption. A point is then chosen on the hyperbola-type curve representing a compromise between the two requirements. This graphical trade-off analysis gives the designer knowledge of relationships between changes in each criterion. However, for more than two criteria the method becomes unwieldy. A similar analysis of the minimum rime-minimum fuel problem has been presented by Kalinin (1965). Waltz (1967) has considered the optimal control problem with multiple performance criteria in the case when the criteria can be ordered in terms of importance. He proposes that first optimization be carried out with respect to the primary criterion. Then a second optimal control problem is solved using the secondary criterion, taking as the class of admissible controls only those from the original admissible set which are within some engineering tolerance of the primary criterion. This restriction is incorporated as an inequality constraint on the second problem. The principal aim of this work is to consider the analogous problem in steady-state optimization, and to formulate and solve a realistic steady-state optimization problem with multiple performance criteria. At the same time it appeared that the question of parameter sensitivities in nonlinear programming warranted study. If, in addition to the usual objective function, it is desired to minimize the sensitivity of the objective function to variations in parameters, the two problems can be formulated within 53

the same framework. The determination of the optimal steady-state operating point for a petroleum refinery was chosen. I n addition to the customary objective function of profit per year, the sensitivity of the yearly profit to changes in refinery conditions, which appear as parameters in the refinery model, represents additional performance criteria. First we outline the nature of the parameter sensitivity problem in nonlinear programming. Then the theory behind vector maximization is sketched, and finally the results are applied to the refinery problem. Parameter Sensitivities in Nonlinear Programming

Consider the standard nonlinear programming problem: Maximize the objective function (performance index) f(x,w)with respect to x subject to the constraints

" a,x,

- b, I 0

i = 1,2,. . .,m


1=1

(1)

where x is an n-vector of adjustable variables and w is a p-vector of constant parameters. a , and b, are assumed to be known constants. If ut, i = 1,2,. . .,p, are known exactly, the solution of this problem is straightforward. Often, however, many parameters in the process model are not known exactly; rather only nominal or most likely values are known. The parameters may reflect the effect of environmental conditions, raw material quality, product specifications, etc. Let us denote the nominal values as wO.The optimization problem can be carried out to maximize f(x,w0)subject to Constraint 1. The results can be denoted f*(x*,w')and x* = x*(wo)because the solution certainly depends on the parameter values wo. The actual value of the process parameters may be w,not necessarily equal to the nominal value wo.Although the process has been set to operate a t x*(wo),in reality it is operating with w and a performance index f[x*(wo),w]. Of course, this value of f may actually be larger than f" because w is different from wo.I n any event, it may be desirable to select the operating point, x, such that the performance index-e.g., total profit-is in some manner insensitive to changes in the process parameters, w. This would most certainly be the case if variations of w away from wo cause f to decrease significantly. For small variations in w relative to wo,f[x*(wo),w] can be expanded in a Taylor series about wo,

Since (w - w") is unknown, f[x*(wo),w] can be kept by minimizing thep-vector I [ d f dw] 1, close to f[x*(wo),wo] termed the parameter sensitivity vector of f. We n%v have an optimization problem with multiple performance criteria:

The parameter sensitivity coefficients, ( a f / a ~ , ) have ~~, units of dollars per parameter unit, whereas f has units of dollars only. Thus, the criteria are not comparable. The combined maximization of an objective function and its partial derivatives with respect to parameters represents only one possible multiple criterion optimization problem. We will subsequently consider these ideas in relation to refinery optimization. One further point deserves mention here. I n general, the constraint parameters, a , and b,, in Constraint 1 may depend on w. We have not attempted to include this aspect in the optimization problem because of the complexity this addition introduces. However, especially for equality constraints, knowledge of the effect of changes in w upon constraints in which elements of w appear is necessary. In the refinery example which follows, parameters wlo and wll, which appear in the seventh, eighth, and tenth constraints of Table IV, as well as in the objective function in Table V, are allowed to vary. I n the computations these parameters were assumed constant and equal to iupo and wh in the constraints. Although this approximation did not appear to have a significant effect on the results, this aspect requires further study. Recent results in stochastic programming might be applied in this regard. Theory of Vector Maximization

A fundamental approach to the multiple criterion optimization problem lies in the concept of efficiency (Kuhn and Tucker, 1950). Roughly speaking, a process is said to be operating a t an efficient point if no criterion can be improved without worsening a t least one other criterion. This concept has been an important part of economic analysis and game theory, where conflicts arise among noncomparable criteria. Given a q-dimensional performance vector f(x), we can pose the problem of maximizing f(x) as one of finding all efficient points 2 , where 2 is efficient if there exists no other feasible point x such that fb(x) L f , ( 2 ) , but f,(x) # f , ( 2 ) for all i. Using this definition of efficiency, no limit is placed on the marginal gains that can be obtained in certain of the elements off by reduction of other elements off. Thus, the quantity [f,(x) - f d ( 2 ) ] / [ f ! ( 2-) f,(x)]could be arbitrarily large. This implies that the marginal gains in can be made arbitrarily large relative to each of the marginal losses incurred by other criteria. If this were the case, one might select x rather than 2 because of the large relative increase in f , obtained. Geoffrion (1967) has recently proposed a slightly restricted definition of efficiency, called proper efficiency, which eliminates this problem. 2 is said to be a proper efficient solution if it is efficient and there exists a scalar M > 0 such that, for each i, fh(x) > f E ( 2 )and ft

1. Maximize f(x,wo) subject t o Constraint 1 2.

p

Maximize

+ 1. Maximize

54

-

subject t o

for some J such that f(x) < f ( 2 ) . The problem facing the system designer is the determination of the proper efficient solutions to a given vector maximization problem. Consider the associated scalar maximum problem

subject t o

Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 1 , January 1970


for which a computational solution can readily be obtained. Geoffrion (1967) has proved the following two theorems: THEOREM 1. Let A, > 0, i = 1,2,. . .,q, be fixed. If f is the optimal solution of the scalar maximum problem, then f is a properly efficient solution of the vector maximum problem. THEOREM 2. Let xeX, a set of feasible points. Let X be a convex set and let the f abe concave on X . Then f is a properly efficient solution of the vector maximum problem if and only if f is the optimal solution of the scalar maximum problem for A, > 0, i = 1,2,. . .,q. Additional necessary conditions for a proper efficient solution of the vector maximum problem in the absence of concavity are provided by Geoffrion. T h e principal result is that the determination of proper efficient solutions is equivalent to the solution of the scalar maximum problem with parameters A,, i = 1,2,. . .,q. If the f zare concave, this will yield all properly efficient solutions. However, the choice of which solution to use, corresponding to a particular choice of the A,, is still arbitrary. For large q this represents a difficult decision. If the elements off can be ranked in order of importance, the concept of hierarchical optimization can be used. If the numbering 1 to q reflects this ordering, we first solve the problem of maximizing fi (x), the primary criterion. The next problem is to maximize f Z ( x ) , the secondary criterion, but subject to the additional constraint that a t this optimal x, called x ', the first criterion f ~ ( x )should not be less than e% below its value a t the first, x",

(3) Since fl(x) is nonlinear, it must be linearized to conform with 1. An alternative constraint might be that each element of x differ by no more than e3% from x ' ,

(4) the implementation of which requires two constraints per variable. Thus, it is possible to obtain solutions to the vector maximum problem by reduction to particular scalar maximum problems. Each of the two techniques described in this section is used for the refinery problem to follow.

Refinery Model

A flowsheet of the refinery model is presented in Figure 1. I t consists of the following units: an atmospheric crude

distillation tower, A , a vacuum distillation tower, B , a reformer, C, a hydrocracker for producing gasoline from blends of light gas oil and light catalytic cycle oil, D , a fluid catalytic cracker, E , a hydrocracker for upgrading vacuum tower bottoms, F , and a hydrogen plant, G. The model is based on 200,000 barrels per stream day of 32" A.P.I. crude. I t was assumed for convenience that each unit was capable of producing at design specifications and costs for any input conditions. Hence, no product specifications are included. Gas recovery, special treating, blending, and storage facilities are not included. The refinery was designed with the goal of producing three grades of gasoline: premium, high octane, and low octane, with smaller quantities of jet fuel, kerosine, and fuel oil. Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 1, January 1970

F-I P+I I

IX9

Figure 1. Refinery flowsheet

-

A Atmospheric distillation tower

B -Vacuum distillation tower C - Reformer D - Hydrocracker E - Fluid catalytic cracker

'

F Hydrocracker ~

Table I lists all the refinery stream variables that are to be determined. The symbols for capital and operating costs of the units are C A , d a (dollars per year), respectively, for the atmospheric distillation tower, cE, d s for the vacuum distillation tower, etc. Table I1 presents the syftlbols for product values and raw material costs. Table I11 lists the refinery parameters. I n Table IV the material balance constraints are presented. The hydrocracker ( F ) split shown in Figure 1 was chosen as reasonable. I n addition, the material balance constraints in Table IV contain the

Table 1. Refinery Stream Variables Light virgin gas oil, bbl./day Heavy virgin gas oil, bbl./day Vacuum tower bottoms, bbl./day Vacuum tower bottoms, bbl./day Reformer premium gasoline product, bbl./day Reformer hydrogen product, SCF/day Hydrogen feed to hydrocracker, SCF/ day Hydrocracker high octane gasoline product, bb1.i day Light catalytic cycle oil, bbl./day

Table II. Refinery Product Values and Raw Material Costs Products

Dollars/ Bbl.

Premium gasoline High octane gasoline Low octane gasoline Jet fuel Kerosine Fuel oil Raw Materials Crude Methane and steam

h l , dollars/ bbl. h2,dollars/SCF HZ

55

Table Ill. Refinery Parameters Distillation tower fraction, boiling point -18O'F. Distillation tower fraction, boiling point 180-400" F. Kerosine product Atmospheric distillation tower bottoms Heavy gas oil from vacuum distillation Jet fuel product Fuel oil to residual ratio Crude input Total high octane gasoline product High octane gasoline fraction of FCC product Low octane to high octane ratio of hydrocracker product Minimum low octane gasoline product

u l , bbl. day fip,bbl./day LL?, bbl./day

L L ~ ,bbl./day

u., bbl. /day fig,

bbl./ day

fi-

u R ,bbl. I day fig,

bbl. / d a y

maximize f2(x) = - (afl/dw1~),:. The particular costs and nominal parameter values used are presented in Table VI. Computations were carried out with the IBM Share nonlinear programming routine based on Zoutendijk's method of feasible directions (Zoutendijk, 1960). The two methods presented above were used: METHOD1. Maximizef(x) = hfl(x) + (1 - X)f2(x),Ac(0,l). METHOD2. Maximize fl(x) as a primary criterion, then f ? ( x ) as a secondary criterion subject to 3 or 4. Method 1 was used with f2(x) = ( a f / a w ~ ) with , ~ the following results:

u Ill

x

+ X? + + a', + i'U U'i = + X: + XI I;= o.g(rCj + 0.1X3) = XI

+u r


f?(X)

x 10~-

5.13 5.13 15.81 15.81 15.94

Wl

lLj

xi; = 1300(Ui + 0.1~7) (1 W l l ) X " + WI> =

+

1.2(*1 + 0.15x1 + Xy) X; = 2000(~1 + 0.15~7+ XY) l.l(~ + ?u'i + 0.6~:) =

+ I . ~ L c ~ ,+~ (0x. 6? ~ 7+ G ) + W ( X ~+ 0.15 xa)

Atmospheric tower Vacuum tower Reformer (over-all balance) Reformer (Hp balance) Hydrocracker (over-all balance) balance) Hydrocracker (H,

XQ

l . l ~ m+( 0~. 6~~ 3+ W S )+ x s 2

ws

x 10-

0.94 0.94 1.0527 1.0527 1.0533

0.0 0.7 0.8 0.9 1.0

Table IV. Refinery Material Balance Constraints U'r

fl(Xl

(1 + u-)(x.t+ 0.15X3) _I 15,000 u1 + W ,x.;2 u11

Fluid catalytic cracker High octane gasoline Fuel oil Low octane gasoline

Method 2 was used with xl0,w l l , and hr with Constraints 3 and 4, the results of which are presented in Table VII. The primary f , values in Table VI1 differ because of different constraint values employed. For the LCIC case, wI2 was used as 90,000 and for the hr case as 100,000. For the wll case the low octane gasoline constraint was omitted altogether. I t is expected that the total profit, fl(x), will be decreased a t the expense of decreased sensitivity to changes in parameter values. The advantage of operating a t such

Table VI. Numerical Values

selected upper and lower limits of 15,000 and ~ 1 1 ~barrels 2 per day on fuel oil and low octane gasoline, respectively. The volume balances on hydrogen contain the appropriate conversion factors.

d.4 = 1.6 x 10 ' d g = 1.6 X 10 d, = 1.4 x 10' d!J = 2.02 x 10' d i = 2.22 x 10' dF = 2.656 x 10' dc = 1.12 x IO-'

= 4.70 X 10' CH = 4.73 x 10' cc = 1.84 X 10' CA

C I ~=

3.38 x 10'

Ct

= 1.15 x 10"

CF

= 2.16

Results

CG

= 8.75 x

The usual refinery design problem is to find that value of x which maximizes total profit (Table V). Even the simplified refinery model presented here has 9 design variables, 10 constraints, and 12 parameters. I t is possible that the values of some of the parameters may vary during the operation of the refinery. As outlined earlier, it might be desirable to minimize the partial derivatives of the total profit with respect to these parameters, in addition to maximizing total profit. The following parameters were selected as ones whose values may be subject to change: wlo, wll, h?. In each case, a two-criterion problem was solved-e.g., maximize fl(x) = total profit and

el = 5.36 e? = 4.62 e : = 4.41 er = 4.00 =2 x 4x U ? = 10' ur = 4 x u, = 2 x ~ ) = b 8 x ui

W? =

10'

X

lo-'

ej = 3.78 e6 = 2.1 hi = 2.8 h? = 1.14 x 10.' U';=

10' 10'

W8

1

= 2 x lo5

= 3 x 10' I = 5 x 10 L c I l = 3.5 W I ? = 90,000, 100,000 Wy

10' 10' loJ

Lc!O

Table VII. Two-step Optimization Results

Primary Secondary

(c

= 4)

1.0533 0.96 fI(X)

Primary Secondary ( I = 4) Secondary ( e , = 15)

15.94 12.5

x

1.025 0.901 0.703

3.85 2.07 3.27

f l i x / x Io-& Primary Secondary ( e = 4) Secondary ( e t = 15)

56

1.036 0.948 0.779

5.12 4.99 4.60

Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 1, January 1970


a point is that the total profit after a change in a parameter will be greater than the total profit after the same change in the parameter if the system were operating a t the maximum of f ~ ( x ) alone. However, if the parameter changes do not occur, the total profit will be less than the maximum if the design including sensitivity considerations is used. The designer is thus faced with a choice; each time a parameter changes, the new maximum can be computed and the refinery moved from the old to the new steady state, or the one design based on sensitivity minimization can be used. The key factor in this decision is the desirability of inducing large transients in the whole refinery and the ability to move the entire refinery quickly to the new desired steady state. If such transients are undesirable, the methods proposed here would be most useful. Suppose a 25% increase in w11 occurs when the plant is operating at the maximum fl = 1.025 x lo8,corresponding to wpl = 3.5. The increase in W I will ~ cause the yearly profit f l to decrease to 0.68 x lo8, using Method 2, if the design variables are kept a t their prior values. If, however, the plant were operating a t the secondary optimum of 0.901 x 10’ and the same change in WII takes place, the yearly profit will decrease to only 0.72 x lo’, and the ultimate profit after the change in wI1 would be greater than in the first case. However, in the cases of wl0 and h Z ,approximately 50 and 100 % changes, respectively, in these values are necessary for the total profit to be greater at the secondary point. Thus, the sensitivity minimizing procedure may not be successful in every case. Computationally, Method 1 is simpler, since only one optimization problem need be solved and no additional constraints need be imposed. I n Method 2 , the choice of c or t t has significance in terms of dollars or barrels per day, whereas represents the relative value placed on the two objectives.

Acknowledgment

Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this work.

Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 1, January 1970

Nomenclature

alJ

b,

CA,

.

CB,.

da, ds,. . . e, fi

h, X

W

M

constants in constraints constants in constraints capital costs of units A,B,. . . operating costs of units A,B,. . . product values performance criteria raw material costs n-dimensional vector of process variables p-dimensional vector of process parameters constant

GREEK = maximum percentage deviations A, = weighting coefficients

C , C,

SUPERSCRIPTS = nominal * = optimal = efficient (1) = primary solution (2) = secondary solution literature Cited

Geoffrion, A. M., “Proper Efficiency and the Theory of Vector Maximization,” Rand Corp. Memorandum RM-5454-PR (November 1967). Kalinin, V. N.,Automation Remote Control 26 (2), 365 (1965). Kuhn, H . W., Tucker, A. W., “Nonlinear Programming,” Proceedings of Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, p. 481, 1950. Nelson, W. L., I.E.E.E. Trans. Automatic Control AC-9, 469 (1964). Saaty, T. L., Bram, J., “Nonlinear Mathematics,” McGraw-Hill, New York, 1964. Waltz, F. M., I.E.E.E. Trans. Automatic Control AC-12, 179 (1967). Zadeh, L. A., I.E.E.E. Trans. Automatic Control AC-8, 59 (1963). Zoutendijk, G., “Methods of Feasible Directions,” Elsevier, Amsterdam, 1960. RECEIVED for review June 24, 1968 ACCEPTED August 27, 1969

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Optimization with Multiple Performance Criteria. Application to

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