N
C/
a,
HD28
.M4U
no. 3500-
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Production Allocation Modeling System:
Optimizing for Competitive Advantage
in a Mature Manufacturing Industry
R.
Sloan
Brown,
J.
Shapiro and V. Singhal
WP# 3500-92-MSA
November, 1992
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Production Allocation Modeling System:
Optimizing for Competitive Advantage
in a Mature Manufacturing Industry
R.
Sloan
Brown,
J.
Shapiro and V. Singhal
WP# 3500-92-MSA
November, 1992
JAK
I
Production Allocation Modeling System:
Optimizing for Competitive Advantage
in a Mature Manufacturing Industry
R.
Brown,
J.
by
Shapiro, V. Singhal
November, 1992
I.
INTRODUCTION
Setting production targets for geographically dispersed production sites
is
a
common problem
routine decision,
that
it
in large
manufacturing companies.
Although
a
it is
encompasses important but sometimes subtle tradeoffs
have a direct impact on corporate performance. The location of
production clearly affects the cost of production, because the cost of inputs, as
well as taxes and other financial factors,
It
may show wide geographic
also affects the cost of distributing products to customers,
which
is
variation.
largely a
function of distance. Service levels are also strongly related to the proximity
of the production point to customers,
and
to
such factors as the capacity of the
plant.
Production targets also set the stage for lower level manufacturing
decisions, such as production scheduling
and inventory management. More
generally, targets largely determine the level of utilization for manufacturing
sites.
At the most extreme, the allocation of production may
be shut
down
call for a site to
or mothballed.
This paper describes a production allocation modeling system
(PRAMSYS)
that
we have done
ArON
for the
corporation, a major producer
of industrial gases (oxygen, nitrogen, argon), with
sites
and customers throughout
the United States.
active use as a routine planning tool for
more than
numerous manufacturing
The system has been
a year in the
in
company's
Mid-Atlantic region, and installations in other regions are underway.
PRAMSYS
embedded
generation.
in a
is
a large
Mixed Integer Programming (MIP) model
menu-driven interface
The immediate purpose
for data preparation
of
PRAMSYS
is
to
and report
minimize combined
regional manufacturing and distribution costs over a planning period of one
1
to three
months.
More
allocate individual
sites.
PRAMSYS
generally, corporate planners use
customer demands
to geographically
because
it
is
of general interest
costs
dispersed production
and
demonstrates the value of optimization
reduction to be limited.
model can be used
to
it
mature industry
in a
to expect opportunities for cost
how an
demonstrates
integrating
bring the company's technical expertise in production
and process engineering
PRAMSYS
In particular,
capabilities.
beyond the immediate application
where conventional wisdom might lead one
The
each month to
also optimally allocates idle time to sites in keeping with
complex relationships between production
PRAMSYS
it
on larger
to bear
strategic issues.
project also illustrates
how
a
model and an application
evolve together over the course of a project through an interplay between
practical,
computational, and theoretical considerations.
model became both more
fortuitous
correct
outcome which
applications.
Among
difficulties that
is
and simpler -
a
In this case the
happy but perhaps
by no means the rule with complex modeling
other things, this evolution illustrates the subtle
can arise
when
practitioners focus too closely
on
their
mathematical abstractions and lose sight of the practical reality behind their
models.
II.
INDUSTRY BACKGROUND
Production of industrial gases
is
in
many ways
the quintessential
mature manufacturing industry. The process of cryogenic
which
known
air is
distillation
by
separated into gaseous and liquid elemental fractions has been
for over eighty years.
Competing producers now operate
intensive plants with similar intrinsic
thermodynamic
capital
efficiencies;
no radical
breakthroughs
material,
is
in
free
production technology are to be expected. Air, the sole raw
and does not vary appreciably
in quality.
Nor
there
is
much
scope for product differentiation - except for special applications where
extreme purity
is
essential, all liquid nitrogen is very
much
the same.
Despite this stability on the supply side, however, the markets for
industrial gases are changing, largely in response to structural changes in the
national
many
and world economy. Demand
for liquid
and gaseous oxygen was
years the driving force of the industry. In recent years this
demand
been declining as the centers of basic industries such as steelmaking
offshore.
in
On
the other hand,
demand
food preparation, enhanced
oil
for liquid nitrogen
is
recovery, and other domains
the older
to a shift in the location of
midwestern industrial
The
result has
been
in the conditions
where
for a
III.
for
to alter the prevailing
is
and cannot afford
where
is
essential.
demand, away from
premises and operating
no longer principally an adjunct
to operate as
if it
were. This shift
stability.
that can adapt to the
new
It
also
opens up new opportunities
conditions.
PROJECT BACKGROUND
The
PRAMSYS
upper management
at
project originated in a general desire
ArON
on the part of
to bolster competitive position
through better
operation of the company's production and distribution system.
cost
a
underlying competition in the industry raises hazards
decades there had been
company
shift
centers.
procedures in the industry. The company
of stable larger industries
has
increasing for use
combination of very low temperature and chemical inertness
These changes have also led
for
is
one of the primary determinants of competitive advantage
Delivered
in this
industry (the other being customer service.) The two primary elements of cost
and medium term are distribution
that are subject to control over the short
and production. Production
are
still
significant -
costs are generally larger, but distribution costs
on the order of 35%.
attention should have focused at
first
It
was therefore natural
on reducing each of these
that
costs
independently of the other, particularly since such an effort meshed with the
prevailing division of responsibilities under the company's Distribution and
Production functions. The use of formal models closely paralleled
this
division.
For the purposes of production and distribution planning,
groups
principle,
it
customers and production
its
makes
any
site in a
the product
predicted customer
ArON
sites into several large regions.
In
region can serve any customer in that region, provided
demanded by
demands
the customer.
Within
a region,
are assigned to a site through a
known
or
monthly
planning cycle. These demands can then be aggregated into production
targets for each product at each site.
In practice, the regional distribution
sites, since
customers.
it
was they who managed
In
making
department assigned customers
the physical shipment of product to
this decision, the
department made heavy use of a
model based on network algorithms which had been developed
distribution costs.
the nearest
the
site.
site,
to
to
minimize
This naturally tended to favor assignment of customers to
with out taking fully into account the cost of production at
The production process and
to be represented well in
its
such a model.
economics were simply too complex
In a
very real sense, the model served
only to formalize the standard operating procedures of the distribution
department and
interests
to
optimize the allocation of production with reference to
capabilities of that department.
and
The production function
improve the
was
also
had
in place a
efficiency of production sites.
A
major element of
and modeling software
a set of data gathering procedures
ArON
Process Optimization Protocol (SIPOP).
engineers had developed SIPOP for use at each
should be operated to minimize
very successful program to
electric
to
produce
at those rates,
configured and operated to attain
this localized
called the Site
chemical and process
site to
determine
power demand. Given
how
the site
a set of target
minimum power
production rates for each product, SIPOP calculates the
demand
program
this
and indicates how the plant should be
this
minimum. Although SIPOP performs
optimization rapidly and very accurately,
it
could not in
itself
determine what the target production rates should be, since these directly
reflect
higher level decisions about the allocation of customer
demand
to the
site.
As we reviewed ArON's procedures and planning
that cost reductions in production
haphazard,
by
its
if
and
distribution
tools
would be
it
became
at best
not illusory, unless they were achieved in concert.
clear
be
Conspicuous
absence was the ability to plan both production and distribution
activities
within a single, comprehensive framework to achieve the greatest
overall cost reductions.
This kind of coordinated planning looked to be relatively untapped
area in which the
company could
distinguish
itself
from
its
competitors.
In a
competitive industry such as industrial gases cost reductions of even one or
two percent can be extremely important,
increases in profit.
translating into
much
larger
IV.
MODEL DEFINITION
Air separation
produce gaseous and liquid
sites
distributed by pipeline to customers located near the
by truck or railroad
nitrogen, argon) are distributed
deliveries; in fact, each vehicle
that the distribution
indirectly,
through
is
site.
Gases are
Liquids (oxygen,
tanker.
There are no
system - and costs - for each product are linked only
joint
production
at the sites.
commodity
PRAMSYS
is
thus represented as a set
arc networks (one for each product) linking
production (or external supply) points to customers. In general, any
deliver to
customer
any customer, but
is
if
shipment from a given
site to a
reflects the distance
intervening geography.
historical data
The
the production sites.
costs
used
in
site
may
and
be
a
PRAMSYS
are derived from
in use for distribution planning.
of the
model emerges from the representation of
The complexity of
variety of related factors,
can
between the two points, and perhaps the
and were already
Most of the structure
site
given
undesirable or impossible, then the corresponding arc
omitted from the network. Unit transportation cost between a
customer
joint
dedicated to a single product. This means
The distribution component of
of simple, single
air fractions.
among them
this representation
stems from a
joint production, electricity contracts,
and shut-down operation.
A. joint Production
A
site
produces products
to five products at once.
A
jointly
from the same production process - up
product can be produced
at
any
rate,
within upper
and lower
limits that
depend upon
the
site,
how
the site
is
configured, the
product, and the rates at which other products are being produced.
P2
Figure
The (instantaneous) power demand
of production rates for
liquid products.
1
of the site
is
an increasing function
products, with strong cross terms, particularly for
There are compelling theoretical and empirical reasons to
believe that the surface
KW=f (Pl,...Pn)
all
mcf/hr
is
KW
is
convex, but no closed form for the function
known.
4
production
rate,
Figure 2
7
product
P-
Data that defines
this surface in
PRAMSYS
although SIPOP was not developed for
SIPOP
is
to find the
difficult to
purpose. The methodology of
minimum power
This
point.
is
standard
and process modeling, where the complexity
practice in chemical engineering
and non-linearity of
ultimately derived from SIPOP,
from that of PRAMSYS. SIPOP uses
in fact radically different
random search methods
this
is
the underlying processes
implement and cumbersome
make
to use (see
gradient search methods
Martin (1982),
Wang
(1978)).
Electricity Contracts
B.
Virtually the only variable production cost
to
run compressors and
liquefiers, so that
to the site's use of electricity.
production rates of
say,
LN
LOX
(liquid
all
Because
is
the cost of electricity used
production cost
KW
demand
is
is
very closely tied
a function of the
products, a decision to assign a customer's
oxygen)
to site
A
(liquid nitrogen) at that site,
therefore alters the cost of both
even though the production
demand
for,
LOX: and
of
rate for
LN
remains the same.
But production cost
efficiency.
power
Sites are
costs are
complex and
One
at a site
is
not strictly a matter of thermodynamic
such major consumers of
electricity that
energy and
governed by special contractual terms that are often quite
that differ,
sometimes
radically,
typical contractual feature
(KWH) consumption and
demand during some
for
is
that the site
maximum
site to site.
is
charged both for energy
(instantaneous)
contract billing period.
same magnitude, although energy
from
These costs are roughly of the
costs tend to be higher.
8
power (KW)
Under most
contracts the unit cost of energy varies discontinuously by
time of day. Figure 3 depicts a situation in which the day
is
divided into on-
peak, mid-peak, and off-peak hours. Energy charges are highest during the
on-peak hours, lowest during off-peak, and take on an intermediate value
during midpeak. The relative proportion of on-,
in a
may
weekday, weekend day, and holiday may
be absent from any day type.
$/KWH
all
off-,
and mid-peak periods
be different.
Any
period type
each type of day in order to take
conditions
C.
is
maximum
advantage of contractual
by no means easy.
Shut-down Operation
Given short- and long-term fluctuations
occasionally has excess production capacity in
in
demand,
some
regions.
ArON
Gaseous
products cannot be inventoried, and inventory capacity for liquids
Therefore,
it is
often necessary to put a site into standby
mode
for
is
limited.
some
part of
the month.
It is
of
MIP
here, in the representation of site shut-down, that the principal use
arises in
PRAMSYS. A pure
choose to shut a plant
power
down
Linear
Programming
only during on-peak hours,
are both most expensive.
(LP)
when energy and
In practice, such a solution
impractical for operational reasons (repeatedly stopping
model would
and
would be
starting
production places unacceptable stresses on equipment and requires an
excessive
was not
amount
to
of operator intervention.)
schedule
site
production day by day or hour by hour,
that the solutions be operationally feasible.
impose
shut
a kind of loose parity
down
during on,
While the purpose of
off,
It
was
it
PRAMSYS
was
vital
therefore necessary to
between the length of time the
site
would be
and mid-peak.
D. Slates: Discretizing the Decision Space
Both energy and power costs can be very significant. Since both are
directly related to
KW
demand
relationships accurately.
programming, but
this
it
was
clearly important to represent these
One approach might have been
was
rejected for several reasons.
10
to use quadratic
First,
there are
no
commercial grade
at best
QP codes
capable of handling
only an empirically derived quadratic
We
rate for each product,
produce
the site to
The
function.
Each
.
at those rates.
slates define the
function.
However, the
convex
cost of
is
KW
a vector containing a production
KW
draw
associated with operating
KW production rate
is
to
determine
surface.
This
itself
would have
(so called
demand
demand over
charge)
was operated. Representing
cost
and production
V.
MODEL FORMULATION
and only
if"
activity also required the use of
of modifications
MIP
the conclusion of this section.
approach
for
how
long that
techniques.
formulation upon which
after their application to actual
and
Thus,
relationship between
MIP
was based. Experience with MIP models drawn from
number
based on
the set of production rates used during the
this "if
present here the original
is
the entire period.
period that resulted in the greatest power draw, regardless of
and
how
quite able to represent a convex cost
power
instantaneous power
prior to
site
Slates in the set cover a "grid" of production
would be incurred only by
We
we had
slate.
problem, since LP
little
maximum
is
basic decision of the model, therefore,
presented
slate
Also,
function.
approximating the actual
long to operate each potential
The
slate
and the minimum
rates for all products, closely
this cost
KW
constructs.
chose instead to discretize the production rate space for each
into a set of production slates
the
MIP
PRAMSYS
this formulation,
both
planning problems, led to a
simplifications.
These are discussed briefly
In the following section,
we
at
discuss our
implementing the system based on these models, and experience
with the system.
11
Indices
1
i:
j:
to
I
index for plants
to
J
index for slates at each plant
k:
1
to
K
index for products
m:
1
to
M
index for customers
(slate
plant shut-down)
is
Parameters
Pjj
=
power draw
ei
=
electric
energy charge
E
=
electric
power demand charge
=
cost of transporting
at plant
i
(KW)
i
($
per
KWH)
at plant
i
per
($
KW)
J
Cj^jj,
customer
a
for jth slate at plant
jjjj
=
m
($
one unit of product k from plant
i
to
per cubic foot)
instantaneous production rate of product k by jth slate at plant
i
(cubic feet per hour)
m
=
demand
R
=
minimum run
T
=
length of planning horizon (hours)
d
i^jj,
for
product k by customer
(cubic feet)
time for any slate at any plant (hours)
Variables
tjj
=
length of time plant
Wj
=
maximal power demand
|l
=
ij^jj,
jth slate at
plant
i
uses
is
jth slate
at plant
used
i
(hours)
(KW)
at a positive level
\0 otherwise
'J
y
if
i
quantity of product k shipped from plant
feet)
12
i
to
customer
m
(cubic
Production Allocation Model (PAM)
M
I
\
I
K
minimize
(1)
i
Subject
=
i
1
,
=
\\
i=l k=l
i
to:
For
i
=
.
.
M
J
^
I
,
.
^iik'^ii
Mjk Sj
^
~
m=
for
yikm
k =
1,
.
.
.,
K
(2)
l
J
(3)
Itij
j
=
t,j
-
Rx^j
tij
-
TXij
^
m = 1,
.
,
.
.
tij
The
>
j
= l,...,J
(4b)
(4C)
P X
>
M
^
i
for
y
W;
For
(4a)
fork =
dkm
Yikm
K
l
(5)
=
0,
Wi >
0,
objective function (1) in this
energy power demand
costs,
=
Xjj
or
model
and distribution
13
is
1,
the
costs.
Yikn,
sum
Note
>
(6)
of energy costs,
that energy
and
power
from plant
costs differ
electric utilities
to plant.
This
is
because the contracts with
vary by location, and furthermore, each plant has
its
unique
design and operating characteristics. Note also that the slates available for use
at
each plant, and their
costs, are
have chosen the fixed number
J
We
uniquely associated with that plant.
of
trial slates for
each plant simply for
expositional convenience.
The
constraints (2) state that the total quantity shipped from each plant
cannot exceed the
total
production. In practice, the inequality was extended to
account for small quantities of beginning and allowable ending inventories.
The constraints
the entire planning horizon
(3) state that
plant by production time and
down
The constraints
slate.)
slate is
used
at plant
minimum R and
the
constraint in (4b)
is
it
i,
if it is
that
that
(4b) state that the time
at all,
must
lie
T.
The upper bounding
based equals the
slates selected
The
specific
fewer than
by the model
for plant
distinguish
among
electricity rates
included
power demand
i.
of the
power
The constraints
the plants.
total
We
note
number
of
KM.
models generated by
complex than (PAM)
we have
maximum
demand must be met by shipments from
(6) is far
that the jth
between the conditional
most customers demand only one product. Thus, the
constraints
t-
Constraint (4c) ensures that the
is
each
at
the plant shut-
is
in the light of constraint (3);
power charge
the
and
used
redundant
demand draws among
(5) state
(4a)
time (recall that slate
maximal allowable time
for expositional purposes.
Wj upon which
down
consumed
is
for several reasons.
PRAMSYS
First,
the
turned out to be more
model was extended
peak, mid-peak and off-peak operations
when
to
the
vary significantly. Plant shut-downs were modeled more
extensively to ensure that shut-down periods occur contiguously.
14
Moreover,
may
contracts with the electric utility
example, terms relating to differences
be more complicated, involving, for
power draws between peak and
in
off-
peak periods. These complications were modeled by straightforward
extensions of the modeling techniques used above.
manufacturing
sites
were extended so
Finally, for
complex
involving several interconnected plants, the models
that they
would choose
the plant configurations as well as
the slates for each plant.
Even without these extensions, (PAM)
fixed charge variety.
Wj behave
the
in a
In particular, the
manner
is
a large scale
MIP model
power demand charges
associated with
Tricks involving cutting
similar to fixed charges.
planes on the plant objective functions derived from an optimal
proved
relatively effective in causing the
quickly.
A
uniform reduction
energy charges
A
ej
models
in size of the
LP
solution
produce good solutions
to
demand
and bound
also caused the branch
of the
charges Ej relative to the
work more
to
efficiently.
second pass through the MIP optimization with the best solution from
heuristic as
incumbent required
far less
CPU
this
time than that required from a
cold start without an incumbent.
Feedback from users
that
of
for
allowed the models
to
at the plants led to
be
still
an important simplification
more rapidly optimized. For
monthly planning, the people running
the plant prefer to
each contract period (peak, mid-peak, off-peak.) The
an optimal solution to (PAM)
for
each contract period
is
slate
the purposes
employ one
slate
suggested from
the convex
combination of the slates where the weights are the fractions of the time that
a slate
is
used. Since the surface of the cost vs. slate function for the plants
studied thus far has empirically proven to be convex,
15
we have been
able to
relax the corresponding
MIP
constructs in optimizing the model.
MIP
constructs are
VI.
IMPLEMENTATION AND RESULTS
required to properly model shut-downs.
still
PRAMSYS was implemented
the
LOGS model
However,
IBM mainframe computer using
an
for
generation language (see
Brown
et al (1986))
and the IBM
optimization package MIP/370.
It is
PRAMSYS
important to emphasize that the
LOGS model
produces a family of models. The precise formulation of a model
for a specific region consisting of several plants
it.
generation in
depends on the data passed
For example, depending upon whether a certain contractual element
present in the data, certain structures
model.
We
was merely
reiterate that the
may
or
may
to
is
not be present in the
model (PAM) discussed
in the
previous section
the point of departure for our implementation work,
and
the
creation of a generator for a family of models.
The MIP models generated thus
tended to be quite
large.
As many
far for the
Mid-Atlantic Region have
as 1000 slates for each of several plants
were
generated by the Site Process Optimization Protocol and included in the
PRAMSYS
models.
Moreover, the models incorporate upward of 1000
customers demands over a typical monthly planning horizon.
Automatic
customer aggregation procedures were implemented, but have not yet been
extensively used.
many
The
resulting
as 10,000 columns.
models have
a
few thousand rows and
as
Using the simplifications and approximations
outlined above, the models are usually optimized, at least to a close
approximation, within a few
CPU
first
minutes on an IBM 3083 computer.
16
We
believe that the use of
PRAMSYS
lead to shifts in the prevailing production
However,
as
is
in the
Mid-Atlantic Region has
and distribution
often the case in real-world applications,
patterns.
it is
difficult to
substantiate this belief with experimental results, for the simple
PRAMSYS
reason that
demands
to
is
not run in an experimental context.
month
fluctuate from
to
month, and there
show what would have been done
A
used
to
"base case"
was run
in the
is
no
and obvious
Customer
"control" process
absence of a model.
early in the project, in
which
PRAMSYS was
second guess a recent month's allocation decisions. The model
solutions
showed an
increase in distribution costs, with a decrease in
production costs that more than compensates for
estimate
is
that
PRAMSYS
this increase.
Overall, the
produces monthly production/distribution
strategies that are several percentage points lower in total cost than solutions
that
VII.
would have been obtained without
CONCLUSIONS AND FUTURE RESEARCH
PRAMSYS
at
it.
ArON.
at last
Its
has proven
itself to
be a useful and important planning tool
success demonstrates once again that computer technology has
reached a level of development permitting mathematical
programming models
to
be implemented and effectively applied to business
planning problems. The success of
blending of
this project
and experience
scientific skills
in
was
also
due
to a felicitous
chemical engineering,
mathematical programming, and computer systems design and
programming.
radically
Finally, the
new approach
to
backing of top management
planning was crucial
17
in
supporting a
to the project's success.
PRAMSYS
currently being extended for use in other
is
national
In this regard, experimentation with the Site Process Optimization
regions.
Protocol
is
required for those sites consisting of several production plants that
can be linked in different ways.
more complex
sites
experimentation
is
new
select
A
to link the Site Process
(see Shapiro (1979).
the mathematical
An MIP model
has been developed.
PRAMSYS
directly to the
and
ArON
related area of future
Optimization Protocol more
models via price directed decomposition methods
The idea would be
programming model
slates for the
Once models
for calculating slates for these
to occasionally use
to price
PRAMSYS
prices from
out slates produced by SIPOP,
model.
have been developed, the intention
for all regions
construct a longer range, national
shadow
model
for strategic
is
to
planning purposes. The
types of problems to be addressed by such a model include contract
negotiations with customers and electric
utilities,
long term plant shut-
downs, and economic evaluations of new markets.
Moving
project
is
underway
PRAMSYS
that the
in the other direction
to a
to convert the
with respect
to
time and scope, a
production planning sub-model in
production scheduling model. The reader
model (PAM)
selects
new
an optimal combination of
may have
slates,
noted
but makes no
attempt to schedule the sequence in which they should be used. In the short-
term
when one
on the
plant,
considers distinct production periods with varying
and recognize
that inventory storage for gas
limited, the sequencing of slates
to other process
principle in perfornMng
extremely
becomes important.
Finally, generalizations of the
be applicable
is
demands
models developed
for
PRAMSYS
should
manufacturing industries. The underlying
modeling research
18
in this area is to better
understand
how
to
imbed process control optimization models, which provide an
instantaneous prescription for the plant, in one or more mathematical
programming models
planning and scheduling.
for
how
methodological problem
is
world of process control
to the
in
which sequences of
and setups are
discrete,
crucial.
We
A
central
from the essentially "instantaneous"
to pass
world of operational scheduling and control,
on /off events associated with changeovers
believe the models in
PRAMSYS
are an important
step in this research direction.
VIII.
P. S.
REFERENCES
Bender, R.
W. Brown, M. H.
Purchasing Productivity
Interfaces. 15,
P. S.
Bender,
at
Isaac,
IBM with
a
and
J.
F.
Shapiro, "Improving
Normative Decision Support System,"
May-June, 1985, pp 106-115.
W.
Northup and
D.
J.
F.
Shapiro, "Practical Modeling for
Resource Management," Harvard Business Review, 59, March-April 1981, pp
163-173.
R.
W. Brown, W.
D. Northup and
J.
F.
Shapiro "LOGS:
A
Optimization System for Business Planning," pp 227-241
Modeling and
in
Computer
Assisted Decision Making, edited by G. Mitra, North-Holland, 1986.
D.
L.
Martin and
L.
J.
Randomly Directed
J.
F.
Gaddy, "Process Optimization with the Adaptive
Search,"
AIChE Symposium
Series. Z8, 1982,
pp
79-107.
Shapiro, Mathematical Programming: Structures and Algorithms. John
Wiley and Sons, 1979.
B.
Wang and
R. Luus, "Reliability of
Global Optimum,"
AIChE
Optimization Procedures for Obtaining
Journal. 24. 1978,
19
pp
619-626.
76
8
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