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Optimal Design with Optimal Uncertainty in Parameters and Control Variables.

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Optimal Design with Optimal Uncertainty
in Parameters and Control Variables
A.W. Longley and P.L. Douglas*
Department of Chemical Engineering, University of Waterloo
Waterloo,Ontario, CANADAN2L 3GI
This paper presents a method for determining the optimal process design while
simultaneously choosing the optimal levels of parameter and decision variable
uncertainty. The optimal level of uncertainty is valuable in accessing steady state
control systems and justifying {or refuting) further laboratory or bench-scale
investigations of process parameters. Development of the method begins with the
classical approach to optimal design, then progresses to include parameter
uncertainty. Next, the development incorporates decision variable uncertainty into
the optimization. Finally, parameter and decision variable uncertainties are
optimized. The development is accompanied by a case study of a continuous stirred
tank reactor (CSTR).
Introduction
Several different and often competing factors must be considered in the design of a
chemical plant. The necessary decisions in plant design are often complex and
beyond the capabilities of human visualization. The design problem is frequently
Author for correspondence (Email: pdouglas@cape.uwaterloo.ca).
39
A. W.Longley and P.L. Douglas
resolved by marhematical modelling and optimization. Most design optimization is
performed in a deterministic fashion with empirical overdesign factors added to the
optimal design. In reality, the model parameters and control variables are random
values with varying degrees of uncertainty. These uncertainties may have a profound
effect on the final design and should not be excluded from the design optimization.
Furthermore, the level of uncertainty is often chosen by the designer based on
experimentation, experience, and/or rules of thumb. Deciding whether further
investigations are necessary to reduce design uncertainty is frequently the designer's
responsibility. To aid in this decision. the uncertainty levels can be included as
decision variables in a design optimization. The optimal levels of uncertainty are
then compared to the present uncertainty levels, providing insight into possible
parameter investigationsand process control systems. These design methods offa an
alternative to the blind and possibly costly use of overdesign factors.
The pioneering work in this field was undertaken by Grossman and Halemane
(1982)and Halemane and Grossman (1983). Applications of these techniques to
synthesize flexible process flowsheets have been repaed by Saboo and Morari
(1984)and Westerburg and Chen (1986). Wagler and Douglas (1988)and Douglas
et al(l991) simplified these techniques and applied them to large scale systems, in
particular to a distillation sequence with and without thermal integration. No work
has been r-ed
in the literature in the use of these techniques to study the effect a€
uncertainty on chemical reactor design.
The purpose of this paper is to illustrate the
CoIlCepts of
design optimization,
decision variable uncertainty, and optimal uncmainty as applied to a chemical
reactor. The chemical reactor is the centre of many chemical processes and the CSTR
is often the reactor of choice (Hill, 1977). In addition to the standard decision
variables in reactor design (e.g. volume, flowrate, and temperature),variables can be
40
Optimal Design with Optimal Uncertainty in Parameters and Control Vahbles
introduced to represent the level of uncertainty or tolerance of the inlet flowratesand
the reactor temperature.
The opposing forces in choosing the levels of the
temperature and flowrateuncertainties are the cost of reactor overdesign and the cost
of the control system.
The uncertainty in model parameters (e.g. reaction kinetic parameters) is
accounted for by selecting critical parameter values that ensure feasibility of the
design. The level of unertainty in these parameters can also be included in the
design optimization. The opposing forces here are the cost of overdesign and the
cost of bench scale investigation of model parameters. Reducing uncertainty will
decrease overdesign but increase the cost of parameter investigation.
CSTR Model and Objective Function
The process under consideration employs a CSTR to convert the liquids A and B to
the more valuable product C. A process flow diagram is presented in Figure 1. The
reaction mechanism is simple:
?he reaction occurs at high temperature and will require a steam-heatedjacket.
For simplicity, the design will not include a dynamic heat analysis; we assume that a
residence time of 15 minutes will allow for proper heat dissipation. The isothermal
steady-state behaviour of the system is modelled by three mass balance equations:
F~ - rV- BFo = 0
rV-
m0 =
0
41
A. W.Longley and P.L. Douglas
. Fo
Fb
-a-
Volume (V)
Temperature (T)
Figure 1. Processflow diagram.
Assuming there is no volume change upon reaction,the outlet flomte must ‘equal
the sum of the inlet flowrates:
where the coefficients are the mole-volume conversion factors. Equations (I), (2),
and (3) are combined to produce the reactor model:
FA - ye-BmTABV- A(O.~FA+ 0 . 1 F ~ )= 0
y e-p’RTABV- C(O.1FA + 0 . 1 F ~ )= 0
Assume
p =
42
that
the values of the kinetic parameters
(y = 3x10’ l/mol/tnin,
3x104 calhol) were determined from bench-scale investigations and have a
Oprimal Design wirh Optimal Uncertaintyin Parameiers and Control Variclbles
-+25%tolerance. Furthermore, assume that the decision variables (FA,FB, T) can
be regdami by a process control system to within a 25% tolerance. Assumed
market research for the fictitiousproduct C dictates that a production rate of 6OOO
kmol of C per day is desired, however, occasional deviations below this
production rate are acceptable. To satisfy the prospective customers, the product
must be 80%pure. The design objective is to determine the reactor volume and
operating conditions that meet product requirements at a minimum cost
The decision variables in this problem are reactor volume, V, inlet flowrates,
(FA,FB),and reactor temperature, T. The state variables in this problem are
functions of the decision variables and are therefore not included as decision
variables. The terms of the cost function, which includes both capital and
operating costs, are:
Costl = CReactor] + [A inlet pump] + [B inlet pump]
+
[Outlet pump] + [Heater] + [Feeds]
Costl = [200Voy] + [FA + 2F:Dm] + [2F, + 4F:61]
+
[1.5Fo + 3F,Om] + [lOT + 3Toa] + [2FA + 3FB]
Additional costs used in the objective function are the cost of reactor control
systems and the cost of further parameter investigations:
Cost2 = [ A flow control]
+
+
[B flow control]
+
[ y investigation]
[P investigation] + [Temperature control1
cost2 =
[F][F][y]
[ ; . .+ [
+
+
+ 27 x l@
60000
30s
43
A. W. Longley and P.L. Douglas
These cost terns are discussed later.
Classical Optimization Approach
The classical approach to optimal steady-state design is to represent the problem
as a nonlinear programming problem (Grossmann and Sargent, 1978):
min
Cost = f(va.0)
(7)
"lx
s.t
h(v,x,B) = 0
gwxB) Io
The problem is solved at the nominal values of the uncertain parameters,
e=@,and the over-design factors are added to the solution.
The use of over-
design factors can be detrimental to a design, possibly resulting in a process
never having an opportunity to prove its economic value (Peters and Timmerhaus,
1991). Furthermore, no rational method exists for including over-design factors
therefore the resulting design is usually not optimal and may be infeasible.
Including uncertainty in Equation 7 would provide the designer with a clearer
picture of the final design, but increases the problem size and complexity.
Applying the classical optimization approach to the CSTR design problem
detailed above yields the following optimization problem:
min Cost = f(v$A$B,T)
VJA.FB,T
SL
h(VFA&,T) = O
g, = 15Fo- v s o
g, = 6 x lo6 - 1440FoCI0
g3 = 0.80A + 0.80B - 0.2OC
0
where h is the vector of model equations (4). Problem (8) was solved in
MATLAB using the optimization toolbox routine CONSTR which uses a
sequentialquadratic programming method (Grace, 1990). Multiplying the optimal
equipment design values obtained from problem (8) by an over-design factor of
44
Optimal Design with Optimal Uncertainry in Parameters and Control Variables
1.2 yields the fmal design recorded in Table 1. Readers should note that the
reactor volume and feed flows increase by a factor of 1.2. However, the
temperature increases by a factor determined from a solution of model equations
at the new flowram and reactor volume. This design method generates a process
cost at least 10% higher than the other design methods. In addition, this
approach requires a sensitivity analysis to provide any insight into the flexibility
requirements of the design variables.
Parameter Uncertainty
Grossmann and Sargent (1978) included parameter uncertainty in problem (7) and
made an important distinction between design variables. The design variable v
is divided into fixed design variables d (e.g. volume) and control variables z (e.g.
temperature). The control variables are manipulated to have optimal performance
in light of the actual parameter values realized during process operation.
Assuming the uncertain parameters are continuous bounded variables and the
expectation of the cost function is adequately represented by a finite weighted
sum, problem (7) becomes:
45
A. W.Longley and P.L. Douglas
Table 1. Design Optimization Decision Variable Results
No Uncmainty'
optimal Plus
Panmeter
Unccpinty
PanmcterBiDccision
Uncertainty
Uncertainty
a58
Dedsion'
Values
Overdesign
0)
CQSl
(13)
(22)
cost
(25)
81,394
94,312
81.409
83,750
86,140
85,361
12.856
15,427
12,860
13,391
13,373
13,212
4204
5.165
4306
4.473
4,468
4,471
4,301
4,47 1
4,464
4,470
4,312
4,476
4,469
4,455
4,267
4.454
4.448
4,337
4,271
4.456
4,451
4,338
4260
4.45 1
4.447
4,340
1,050
1,056
1.06,
1,106
1,338
1,342
1,352
1,393
777
786
791
836
5,558
5545
10,001
5,511
5,496
1,197
310
314
1,894
6.23 10'3
6.25(10")
6.25(10*3
3.93(10'3
6.25(lb)
6.25(1@)
6.25(1@)
6.23(1@)
10.11,12
13,14,15
10.11.12
13.14.15
10,11,12
13,14,15
10,11,13,14
15.16,lS
4,266
1,051
43
i
No C a t
5.1 19
1,216
cost of unartainty not included in objective function
' cost of mcertainty, as defined in quation (6). added to objective function
46
Optimal Design with Optimal Uncertaintyin Parameters and Control Variables
N
min Cost =
b’r’
qf(d.2 ,’x
’,ek)
k.1
When the inequality constraints in Equation 9 are monotonic in the uncertain
parameters (e), the maximization subproblem can be solved apriori (Grossmann
and Sargent, 1978). The solution of this maximization provides an initial set of
N critical points in the parameter space that the design must satisfy.
Theoretically, these initial N points may not be sufficient to guarantee feasibility
for the entire parameter space (Grossmann et al, 1983). However, these N points
are sufficient to enhance insight into the final design. Given these N critical
points, problem (9) becomes:
N
min
&‘a‘
S.t.
Cost =
akf(d,z,x‘
‘,e’>
k-1
h(d,z ’.x ’,@) = 0 k = l;-,N
g(d,z k , ’,@)
~
I0
This method can easily be extended to account for uncertainty in parameters
external to the process (Grossmann et al., 1983), but cannot account for
47
A. W.Longley and P.L Douglas
Uncertainties in the decision variables themselves. For example, consider a
process with one feed stream containing multiple components. Assume that the
feed composition is fuced and the design p b l e m must determine the optimal
feed flowrate. The feed composition is an external parameter, therefore problem
(10) can account for composition uncertainty, but the feed flowrate is a decision
variable, therefore problem (10) can not account for flowrate uncertainty.
Applying this method to the CSTR design problem the uncertainty in the
kinetic parameters (y,p) is now included in problem (8). To include parameter
uncertainty assume the parameters are bounded at the minimum and maximum
values. If no information is available to identify the upper and lower bounds
assume that the given parameter tolerance equals three standard deviations
(3a=tol). Therefore:
eL
= pe
eu = Pe
- 30~
+
30,
To determine the critica points in the parameter space, examine the
constraints in problem (8).
The first constraint, g,, is independent of the
parameters and therefore is not affected by the maximization subproblem in
problem (9). The other constraints, (g2,g,), are both monotonic in the parameters
and are always maximized at the extreme value (?,pu),
Assume that this point
the only one that must be satisfied to ensure feasible design of the reactor. To
generate a more representative cost function, two more points or vertices in the
parameter space are also considered. The three points and the associated cost
function weightings are:
48
Optimal Design with Optimal Uncertainty in Parameters and Control Variables
83 = (+73"]* aj = 0.10
The decision variables (V,FA&T) are divided into design variable, d = (V),
and control variables, z = (FA,FB,T).
With these modifications, problem (8)
becomes:
- 1440F:C' I 0
g t = 0.80A ' + 0.80B ' - 0.2OC ' I 0
8," = 6 x 106
Each of the constraints must now be satisfied at the three parameter points
which increases the total number of constraints from 6 to 18. The solution to
problem (13) is recorded in Table 1. Including parameter uncertainty in the
design has revealed that the reactor temperature will require a wide operating
range, from T, = 777K to T, = 1338K. Using over-design factors, the process
would have been designed for a temperature of 1261K resulting in infeasible
operation at vertices 2 and 3.
Parameter and Decision Variable Uncertainty
Including uncertainty in the decision variables (d, 2, x') requires a probabilistic
formulation of problem (10). Assuming that the equality constraints, h, in
49
A. W.Longley and P.L. Douglas
Equation 10 represent the process model, a probabilistic formulation transforms
the inequality constraints into probabilistic constraints, resulting in a chance-
consfraind model (Wagner, 1975):
N
min
Cost =
b’s’
s.t
C
~&,z”,x~,fY)
”-1
h&,z k,x ’,0L) = 0 k = l,-*-,N
~r(g,.(d,rk,xk,8t) I01 Ipi i = I,***,i
gj(d,zk
k,P)
I0
, ~
j = I + l,-;t
4 ‘2‘,x 9: (p,o’>
If the probability distribution functions (PDF) of the random variables (8,
2,x’) and probabilistic constraints are known, then the probabilistic constraints
in problem (14) can be transformed into deterministic constraints. The new
constraints are derived from the means (psi> and standard deviation (osJof the
original constraints (gi):
pli
+
C,o,, I0
The parameter 4 is determined from the PDF of the constraint i and the
associated minimum acceptable probability, pi. Often, the PDF of the random
variables and/or the probabilistic constraints are unknown or impractical to
determine. For example, even if the PDFs of the decision variables in problem
(13) were know,determining the PDF of the quality constraints,g3,
would result
in a large and complex integration problem. Solving this integration problem
50
Optimal Design with Optimal Uncertainty in Parameters and Control Variables
would be a labourous task and probably not worth the effon The parameter 5
can be determined without the PDF using Chebyshev's inequality,providing that
the PDF of the probabilistic consnaints have finite mean and standard deviation
(Harr, 1987). Chebyshev's inequality is:
where ty -
4 is the absolute value of the deviation fiom the mean.
Applying
(16) to the probabilistic constraints in problem (14) yields:
Thus, problem (14) becomes:
N
min
baa'
s.t.
Cost =
okf(d,zk
',w
, ~
k-1
h(d,z ',x ',@) = 0 k = l;-,N
p&(dJ',x ',@?
+
o,(d,z
(18)
,'@)
k , ~
I0 j = I
+
l;-,I
Id '3 ',x '1: (p,*
51
A. W.Longley and P.L Douglas
The statistical moments of the constraints &,,a& can be determined using
first-order second-moment (FOSM)analysis, (Curi and Ponnambalam, 1992).
This method approximates the fmt two moments of a function of random
variables from the first two moments of those random variables.
This
approximation may contain a large error for highly non-linear constraints and/or
high standard deviationsof the random variables. The FOSM analysis equations
are:
where y is the subset of random variables (da,z’, xa). If the random variables
are independent, equation set (19) is reduced to:
Returning to the CSTR problem, the conEol variables {FA,F,, T) are
52
Optimal Design with Optimal Uncertainty in Parameters and Control Variables
uncertain in the sense that they may vary due to external disturbances and the
lack of an adequate control system. To include this uncertainty requires the
transformation of problem (13) into a chanceconstrainedproblem. Constraint gf
is a convenient simplificationof the heat analysis and enhancing this constraint
,(:
with a probabilistic fornulation is senseless. The constraints g
g3y are
formulated probabilistically using judgement to transform the fuzzy market
requirements into numbers:
Pr {Production 2 6.0 x lo6}2 0.90
h {Conversion 2 0.80)1 0.99
Applying Equation (17) to the above constraints, problem (14) becomes:
s.t
h(VPi,F,k,T ',@) = 0
g,L = 15Ft
- V I0
pg2. + 3 . 1 6 2 ~ ~
I .0
.,p
+
loa,. I 0
where the statistical moments of the constraints (pg,c& are estimated using
Equation 20. The solution to problem (22), shown in Table 1, reveals a slightly
modified design. The required reactor volume is larger than the design with only
parameter uncertainty, problem (13). Also,including control variable uncertainty
53
A. W.Lungley and P.L. Douglas
has increased the optimal values of the control variables and the projected cost
compared to problem (13). However, the reacm volume and expected cost is at
least 11% less than the method using overdesign factors.
Problem (22)was solved again with the additional cost function terms of
equation (6)added to the objective function. These terms are representative of
the cost of parameter and control variable uncertainty. The resulting design
changes less than 1% (see Table 1). The new projected cost increases by $2239
but is still 9% ($8172)less than the cost obtained using over-design factors. The
tolerances used and associated costs are recorded in Table 2, for example the
annual cost
Quantity
Tolerancc'
(preset)'
Cost
(preset tol.)'
TolClaQCC'
(as decision)'
Cost
(decision tol.)'
of the reactor heater control system with a temperatue tolerance of 25.0% is
!§4424/year,while the annualized cost of investigating the kinetic parameter j3 to
a tolerance of i25.0 is $14l/year.
Uncertainty as a Decision Variable
When solving problem (18) in conjunction with Equations (20), the designer
54
Optimal Design with Optimal Uncertaintyin Parameiers and Control Variables
supplies the parameter and decision variable Uncertainty. Further insight into the
design problem is achieved by determining the optimal values of uncertainty.
With uncertainty as a decision variable, problem (18) becomes:
N
min
4da 'r '
qf(d.a',z k,x k,BL)
Cost =
k-1
S.t. h(d,z ',x
'.ek) = 0 k
p, (d,d,z 'sc ',OY
= l,-*,N
o,(d,$,z ,'x k,Bk) I 0 i = 1;-J
+
g,(d,z k,x '$37 I0 j = I + l;-,t
d ',z ',x '1 : ( p d )
where d is the vector of parameter and decision variable variances. Often the
critical parameter values, ek, chosen in the maximization subproblem of problem
(9). are at the extreme values of the parameters, (eL,eu). Thus, the critical
parameter values can be represented by the parameter means and variances:
fY= p&+TO&
where
(24)
is known, O, is a decision variable, and z is chosen to achieve the
desired probability of the parameter bounds.
The choice of parameter uncertainty in the CSTR problem (22) is based on
the available parameter &ta
obtained from laboratory and bench scale
investigations. The selection of control variable uncertainty in problem (22)
arises h m the selection of process control system. Clearly, the designer has
some control over parameter and control variable uncertainty in CSTR design.
55
A. W.Longley and P.L.Douglas
Two logical questions are: (a) Is the available parameter data sufficient; and (b)
How tight should the process be controlled at steady state operation? Some
insight into these two questions is achieved by including parameter and control
variable uncertainties (8)
as decision variables. Thus, problem (22)becomes:
3
min
V,a'P.PJ
k-1
S.t. h(V,F:,F,k,T
k,@)
g,L = 15F:
+
=0
- V I0
pg2t + 3.1620,.
Po.
q f ( V , a 2 , F ~ , F ~',8L>
,T
Cost =
I0
looc3.5 0
The moments of the inequality constraints (g2, g3)are functions of the parameter
and control variable uncertainty. This functionality stems ftom the FOSM
analysis of Equation 20.
Including uncertainty as a decision variable, problem (25). yields a slightly
more economical design as shown in Table 1. The most notable change is the
increased reactor temperature. More interesting are the optimal tolerances in
Table 2. The tolerance of the control variables (FA,F,, T) represent the
performance of the control system and the uncertainty in external disturbances.
The tolerance of the kinetic parameters (y$) represents the uncertainty arising
from the kinetic investigations (i.e. laboratory and bench-scale experimentation).
The optimal control variable tolerances are useful for steady state control system
decisions. For example, the inlet flowrate of B requires tight control. The design
warrants an expenditure of $3400 to achieve this flow control. However, the
56
Optimal Design with Optimal Uncertainty in Parameters and Control Variables
reactor temperature requires little control. Operator control of the reactor
temperature may be sufficient to maintain the ~ 1 1 . 8 %optimal design tolerance.
The optimal variances of the kinetic parameters are valuable in justifying or
refuting further bench or pilot-plant investigations. Assuming the present
parameter data has tolerances of +25%, the optimal tolerance of
-+25%
p is at least
therefore no further investigation of this parameter is justified. The
optimal tolerance of y is +19.82%, which is 5% less than the present value of
-+25%.
The design justifies an annualized expenditure of $59 to improve the
parameter tolerance by 5%.
Conclusions
Accounting for decision variable uncertainty gives the designer greater insight
into the final design than traditional design optimization methods. Combined
with uncertainty in the parameters, this method can reduce the reliance on
empirical over-design factors which usually yield non-optimal solutions and may
result in an infeasible design. Including uncertainty as a decision variable yields
information about the optimal level of uncertainty in the final design. This
information, when compared to actual uncertainty, provides insight into the
necessary steady state control systems and insight into the benefit of further
kinetic studies.
Acknowledgements
The authors would like to thank MRCO (ManufacturingResearch Corporation of
Ontario) and NSERC (National Sciences and Engineering Research Council) for
financial support of this project.
57
A. W.Lnngley and P.L. Douglas
Nomenclature
A
B
C
FA
FB
F.
I
N
R
T
v
d
d'
g
gl
g2
g3
h
k
Pi
r
t
V
X
X'
Y
2
2'
P
Y
0
eN
s"4
d
2
0,
58
Concenuations of species A [molfl]
Concenavltions of species B [mol/ll
Concentrations of species C [mol/l]
Inlet flowrate of species A [mol/min]
Inlet flowrate of B [moVminl
Outlet flowrate [ lhnh]
Number of probabilisticconstraints
Number of points in the parameter space that the design must satisfy
Gas canstant [cal/mol/K]
Reactor temperature [K]
Reactor volume [ 11
Fixed design variables
Subset of design variables which contain uncertainty
Vector of inequality consuaints
Residence time constraint
Production constraint
Quality constraint
Vector of Equality constraints including the process model
Reaction rate constant [I/moVmin]
Minimum acceptableprobability that constraint i is satisfied
Reaction rate [mol/l/min]
Number of consuaints
Design variables
State variables
Subset of state variables which contain uncertainty
Subset of random variables (d', z', x']
Control variables
Subset of control variables which contain uncertainty
Arrhenius exponential factor [calhol]
Arrhenius preexponentialfactor [l/mol/min]
Uncertain parameters
Nominal value of the uncertain parameters
Mean of x
Chebyshev parameter
Standard deviation of x
Vector of parameter and decision variable variances
Parameterchosen to achieve the desired probability of the UIlCenainty
bounds
Weights representative of the likelihoodof the associated parameter point
Optimal Design with Optimal Uncertainty in Parameters and Control Variables
References
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Analysis and Design of Circuits. IEEE Trans. Educ., 36( l), 5 1-56.
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Grossmann IE., and Halemane K.P. 1982. Decomposition Strategy for Designing
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and Sargent, R.WH. 1978. Optimal Design of Chemical Plants
with Uncertain Parameters. AIChE J., 24 (6). 1021-1028.
Halemane, K.P., and Grossnann, IE. 1983. Optimal Process Design under
Uncertainty, AICE J., 29(3), 425-433.
Han, ME. 1987. "Reliability-Based Design in Civil Engineebg". McGraw-Hill,
-
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Hill, C.G. 1977. "An Innoduction to Chemical Engineering Kinetics and Reactor
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Peters, M.S., and Timmerhaus, K.D. 1991 "Rant Design and Economics for
Chemical Engineers",4th ed.,McGraw-Hill, New York.
Saboo, AX., and Morari, M., 1984, Design of Resilient Processing Plants - W,
Some new results on Heat Exchanger Network Synthesis, Chem. Eng. Sci., 39(3),
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Wagler, R.M., and Douglas, PL., 1988, A Method for the Design of Flexible
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Wagner, H.M. 1975. "Principles of Operations Research", 2nd ed.,Prentice-Hall.
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Received: 15 April 1995; Accepted afier revision:27 February 1996.
59
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