Download New approaches for heuristic search: linkage with artificial

European Journal of Operational Research 39 (1989) 119-130
North-Holland
119
Invited Review
New approaches for heuristic search: A bilateral
linkage with artificial intelligence
Fred GLOVER
Center for Applied Artificial Intelligence, Graduate School of Business, University of Colorado, Boulder,
CO 80309, USA
H a r v e y J. G R E E N B E R G
Mathematics Department, University of Colorado at Denver, Denver, CO 80204, USA
Abstract: This survey considers emerging approaches of heuristic search for solutions to combinatorially
complex problems. Such problems arise in business applications, of traditional interest to operations
research, such as in manufacturing operations, financial investment, capital budgeting and resource
management. Artificial intelligence is a revived approach to problem-solving that requires heuristic search
intrinsically in knowledge-base operations, especially for logical and analogical reasoning mechanisms.
Thus, one bilateral linkage between operations research and artificial intelligence is their common interest
in solving hard problems with heuristic search. That is the focus here. But longstanding methods of
directed tree search with classical problem heuristics, such as for the Traveling Salesman P r o b l e m - - a
paradigm for combinatorially difficult p r o b l e m s - - a r e not wholly satisfactory. Thus, new approaches are
needed, and it is at least stimulating that some of these are inspired by natural phenomena.
Keywords: Artificial intelligence, operations research, heuristic search, combinatorial optimization, decision support, genetic algorithms, neural networks, simulated annealing, tabu search
Introduction
One of the most challenging problems in artificial intelligence is to deal effectively with the
combinatorial explosion. As pointed out in recent
survey articles in Science and Scientific American,
practical applications of artificial intelligence must
wrestle with the combinatorial explosion in nearly
every area of major concern, including those as
diverse as knowledge based systems, VLSI design,
robotics, scheduling and pattern recognition.
Received May 1988
Broadly speaking, the combinatorial explosion
is encountered in those situations where choices
are sequentially compounded, leading to a vast
mushrooming of possibilities, as examplified by
the compounded possibilities for choosing alternative routes in a maze. Notable situations where
this type of compounding occurs in the business
domain include financial investment, manufacturing operations, inventory management, resource
allocation, and capital budgeting. Common examples from engineering include minerals exploration, meteorological data analysis, water resources
management, integrated circuit design, satellite
monitoring operations, and systems maintenance.
0377-2217/89/$3.50 '~'~1989, ElsevierScience Publishers B.V. (North-Holland)
120
F. Glover, H.J. Greenberg / New approaches for heuristic search
Similar examples come from economics, psychology and biology, typically occurring in those
settings that involve prediction, attribution, classification, monitoring and control of complex
processes.
The attempt to deal with these important problems has encountered many obstacles. It is not
enough to have 'expert knowledge' in order to
handle them effectively. Even the most skilled
analysis can make poor (and sometimes disastrous) decisions in the face of the combinatorial
explosion. Nor is it enough to rely on the computing power of high speed mainframe computers.
Classical examples of problems where the combinatorial explosion prevails--problems far simpler
than encountered in most practical settings--show
that an attempt to generate all relevant alternatives by computer would be hopeless.
The key for dealing with such problems is to go
a step beyond the direct application of expert skill
and knowledge, and make recourse to a special
procedure (or framework) which monitors and
directs the use of this skill and knowledge. Lacking such a procedure, the rules of expertise can
become bogged down, leading to a point where no
improvement can be perceived, although far superior alternatives exist (unreachable because they
require a more complex chain of analysis than the
best current expertise can provide).
Recently, five approaches have emerged for
handling complex decision problems: genetic
algorithms, neural networks, simulated annealing,
tabu search, and target analysis. The first
two--genetic algorithms and neual networks--are
inspired by principles derived from biological sciences; and, simulated annealing derives from
physical science, notably the second law of thermodynamics. Tabu search and target analysis stem
from the general tenets of intelligent problemsolving. These methods need not be viewed competitively, as we shall see, and they comprise the
emergence of promise for conquering the combinatorial explosion in a variety of decision-making
arenas.
As these approaches are described in the next
section, the Traveling Salesman Problem (TSP) is
used as a paradigm for a wide class of problems
having complexity due to the combinatorial explosion. The TSP is defined over n cities, where n is
some positive integer, and a cost matrix C = [cij ],
which describes the one-step cost of going from
city i to city j. A salesman must visit every city
exactly once, then return to his home city (i.e.,
from where he started). Each possible solution,
called a tour, may be represented by a permutation of the n integers. For example, ~1, 2 . . . . . n)
is the sequence whereby the salesman starts in city
1, then visits city 2, then city 3, and so on until he
visits city n and returns to city 1. The total cost of
this tour is the sum of the one-step costs: c12 + c23
+ . . - +c~1. The TSP is to find a tour with the
least total cost. There are n! tours, which is a
combinatorial explosion. For a 10 city problem
(i.e., n = 10) the number of tours is more than 3
million. For only 100 cities (still a fairly small
problem in practice) it exceeds the postulated age
of the universe in microseconds.
New approaches
Genetic algorithms, introduced by Holland
(1975), are based on the notion of propagating
new solutions from parent solutions, employing
mechanisms modeled after those currently believed to apply in genetics. The best offspring of
the parent solutions are retained for a next generation of mating, thereby proceeding in an evolutionary fashion that encourages the survival of the
fittest. As the quality of the fittest (best offspring)
eventually builds to the highest level compatible
with the environment (the governing problem contraints), the best overall solution is recorded and
becomes the candidate proposed by the method
for an optimal solution.
The component processes of genetic algorithms
may be described under three headings of reproduction, crossover and mutation. Reproduction is
the random pairing of individuals (trial solutions)
from a population to create one or more offsprings from them. Crossover defines the outcome,
as gene exchange, whose specific values are called
alleles--that is, alleles may be conceived as instantiations in an expert system sense. The exchange of genes (information types and their attributes) follows positional rules traditionally
modeled after biological reproduction. They must
be modified and particularized to different types
of combinatorial problems, however, in order to
make sense (by permitting certain constraining
relations to hold) and to afford an opportunity for
progeny that do in fact improve upon their parents.
F. Glover, H.J. Greenberg / New approaches for heuristic search
Finally, mutation is simply the introduction of a
random element, often used to amend the result of
a gene exchange when the outcome does not successfully meet appropriate restrictions. In this respect, mutation is more strongly biased to be
helpful in the application of genetic algorithms
than in biological genetics.
To illustrate, consider a particular crossover
operator for the TSP, based on that of Oliver,
Smith and Holland (1987). Let two parent tours
be given by
M = m l m 2 ... m~
and
D=dld2...d~.
To create an offspring pick either m i or d, for
each i = 1 . . . . . n. If m, is picked, however, city d,
must be picked from M to ensure each city is
visited. This contraint may be described by ¢ycle
labels for M and D, as in the following example.
M: 1 2 3 4 5 6 7 8 9 10
D: 1 7 2 5 4 9 3 6 10 8
Cycle label: U 1 1 2 2 3 1 3 3 3
Whenever m, = d~ the cycle label is U to indicate
a unary cycle. The first cycle (labeled 1) is for
positions 2, 3 and 7. This was obtained because if
m 2 is picked, city 7 ( = d2) must also come from
M, which is in position 7. This then constrains city
3 ( = dr) to be picked from M. The cycle ends
because city 3 is in position 3, where d 3 = 2 has
already been picked from M.
These parents, M and D, have 8 possible offsprings, by picking M or D for each of the 3
nonunary cycles. Here is one of them:
Offspring: I 7 2 4 5 6 3 8 9 10
Parent:
M D D M M M D M M M
Genetic algorithms have found their major applications in the optimization of functions over
bounded, but otherwise unrestricted, domains. A
worthwhile avenue for research relative to combinatorial problems is accordingly to characterize
types of penalty functions for which the genetic
algorithms perform effectively. An attempt to treat
constraints in more explicit form is less likely to
yield a fertile avenue for investigation in this
approach, chiefly because of the difficulty of identifying a crossover rule that assures the progeny of
two given parents will be able to satisfy the constraints already satisfied by the parent solutions
(such as extreme points visited in relative order
when the TSP cost matrix is Euclidean). For corn-
121
plex problems, where such a rule may lack a
constant form (or require an i m m a c u l a t e conception by the intervention of some form of super
parent), conceivably none of the offspring produced by the normal operation of the process
would survive, let alone improve on their progenitors.
Another area for research is to determine
whether penalty functions for more complex problems possess features that differ substantially from
those of the functions on which genetic algorithms
are tested. A critical philosophical issue is why
solutions should be limited to a form that permits
their attributes to be derived only from parents,
rather than more broadly making use of all available elements of a solution space. We conjecture
that by allowing the contribution of each parent
to extend to the elements within a neighborhood
of the parent, strategically identified, it is possible
to broaden significantly the framework of genetic
algorithms, opening the door to more flexible and
interesting ways of generating new solutions. Research into such issues is, we feel, a potential
source for useful advances in these evolutionary
methods.
Neural networks, which stem from the works of
McCulloch and Pitts (1943) and Rosenblatt (1962),
are founded on a generalized stimulus/response
(or r e i n f o r c e m e n t / d e c a y ) paradigm that emphasizes the importance of structural links, and
rules for transmitting signals across these links,
allowing the signals to be amplified or attenuated
in various combinations. Broadly speaking, the
neural network model postulates a collection of
functions at nodes (synapses of neurons) of a
highly connected system, which operate independently of each other within small time intervals.
More accurately, neural networks are inherently a n a l o g - - t h e y change state continuously over
time, Also, they are inherently massitJely parallel,
making their potential for computer architectures
of the future. Signals propagate by functions that
gate, or otherwise modify, inputs to a node across
modifiable connections, which combine with the
node's present state to determine its new state. At
the same time the node's inputs are being received, it sends out its signal to its neighbor nodes.
Neural networks vary by their topology (that is,
their interconnectedness), the states that can be
realized and the f u n c t i o n s that govern the system's
122
F. Glover, H.J. Greenberg / New approaches for heuristic search
dynamics. In addition, unlike conventional computers, neural networks form a neural computer
that is trained rather than programmed to perform certain tasks, like heuristic search.
A neural network can be organized, by appropriate choice of topology, states and functions,
to behave as an optimizing system (at least locally)
for a combinatorial problem. The function to be
optimized over a specified set is replaced by an
energy function to be minimized, typically over a
larger (relaxed) domain, constructed to incorporate constraints into the objective by means of
penalty functions--a c o m m o n a p p r o a c h in
mathematical o p t i m i z a t i o n - - s o that the set of optimal solutions for the energy function is the same
as that of the original function. The domain for
the energy function in this approach is represented
as a set of neural interconnections. This is generally accomplished for combinatorial problems by
reference to a z e r o - o n e integer programming formulation, where each variable is conceived as a
neuron that can fire with a voltage output in the
continuous range from 0 to 1. Inhibitory connections are introduced so that the energy function
can communicate among the neurons to discourage values that are inadmissible for the original
problem constraint set. The connections between
the neurons are called synapses, and the magnitude of the signal, inhibitory or excitatory, transmitted from one neuron to another is called the
synapse value. (This value is analogous to the flow
across an arc in network flow optimization, except
that the nodes of a neural network may perform
operations, like thresholding, that are more complex than conserving flow.)
The goal of the process is to induce the state of
the system to evolve over time to an equilibrium
state--that is, when there is no further change in
state values--which corresponds to a locally optimal solution. (More will be said on global optima
in the next section.) The energy function serves as
a Liapunov function to ensure stability of the
equilibrium.
Consider now the H o p f i e l d - T a n k (1985) neural
network model for the TSP. The topology is n 2
nodes arranged like a matrix with columns corresponding to position numbers and rows to cities.
Then, a tour corresponds to a permutation matrix
- - t h a t is, each node level is 0 or 1, and every row
and column has exactly one 1. More generally, the
neural states are contained in the interval [0,1]
with a sigmoid function that mimics spin glass
problems. The dynamics are those of gradient
descent:
x,.j ( t + 1) = S( xij( t ) - cOe( x ( t ) )/Oxij ),
where
x~y(t) is the strate of node (i, j ) at time t;
S(-)
c
E
is a sigmoid function to keep x in [0,1];
is a small positive constant;
is the energy function.
The energy function in this case is a multimodal quadratic with two parts: E ( x ) = D ( x ) +
P ( x ) , where D and P are non-negative functions
that describe the total cost and penalty, respectively. When x is a permutation matrix, it corresponds to a tour (and conversely). In that case
D ( x ) equals the cost of the tour, and P ( x ) = O.
There is an important point that distinguishes
the neural network model from standard application of penalty functions: the neural model must be
realizable with circuitry. In the above TSP network,
the node states are actual voltages, and the dynamics are realizable as an analog computer. This
distinction has some disadvantages: not just any
penalty function can be chosen, and projection
m a y not be realizable (in this case to remain in the
assignment polytope).
It is crucial to realize that a neural network is
not merely amenable to parallel computation, it is
inherently parallel and massively so. Unlike the
genetic algorithm approach, the neural network
approach cannot be evaluated independently of
computer hardware; it is not designed for serial
computation. For the same reason, plus some related reasons, it is improper to evaluate algorithms
derived from neural network models in the same
way as we would evaluate to penalty function on a
conventional computer. They key is that we are
dealing with a different form of automata, one
that is presently only simulated on present computers.
The neural network approach has had most
success in pattern recognition problems that arise
in vision. Although its roots are in seminal works
of 1943 and 1960, it is only recently that it has
been considered as an approach to heuristic search.
The experiments so far just show that it can be
done, and it does look promising on small problems. Although Hopfield and Tank did not em-
E Glouer, H.J. Greenberg / New approaches for heuristic search
phasize the role of setting the parameters in the
dynamical equations, this role is crucial to successful implementation. Despite its relative infancy, the study of neural networks has attracted
much attention and is growing sharply as a viable
approach to solve complex problems. Its particular strength in pattern-matching is being explored
by Greenberg (1988) in the development of an
Intelligent Mathematical Programming System.
Simulated annealing, introduced by Cerny
(1985) and by Kirkpatrick, Gelatt and Vecchi
(1983), has been heralded as a new and powerful
methodology for combinatorial problems, with
implications for the field of artificial intelligence.
More broadly applied to optimization than neural
network methods, simulated annealing is nevertheless compatible with them, and has been proposed
as one possible adjunct to these methods as a way
of enhancing their performance.
The name "simulated annealing" derives from
the intent to pattern the approach after the physical process of annealing, which is a means for
reducing the temperature of a material to its
minimum energy or ground state. Such a state,
called thermal equilibrium, may be viewed as the
value of a minimand, such as the cost function in
the TSP.
The annealing process begins with a material in
a melted state and then gradually lowers its temperature, analogous to decreasing an objective
function value by a series of improving moves. In
the physical setting, however, the temperature must
not be lowered too rapidly, particularly in its early
stages. Otherwise certain locally suboptimal configurations will be frozen into the material and the
ideal low energy state will not be reached. To
allow a temperature to move slowly through a
particular region corresponds to permitting nonimproving moves to be selected with a certain
p r o b a b i l i t y - - a probability which diminishes as the
energy level (objective function value) of the system diminishes. Thus, in the analogy to combinatorial problem solving, it is postulated that the
path to an optimal state likewise begins from one
of diffuse randomization, somewhat removed from
optimality, where nonimprovtime incrementing
moves are initially accepted with a relatively high
probability which is gradually decreased over time.
The form of the process for the purpose of
simulation may be specified as follows. Potential
123
moves of the system, viewed as small displacements from its present state, are examined one at
a time, and the change 8 in the objective function
value is calculated. If 8 < 0, indicating an improvement, the move is automatically accepted.
Otherwise the move is accepted with probability
P(8) = e ~/kT, where T is the current objective
function value (temperature) and k is a constant
adapted to the application (Boltzmann's constant
in the physical setting). The heuristic adaptation
of k at different levels of T is referred to as
creating an annealing schedule.
Intuition suggests that a successful adaptation
must be done with latitude to depart from the
framework derived from the physical setting. For
example, in the implementations that are highly
faithful to the simulated annealing framework, the
solutions used for a basis of comparison are those
obtained after a certain elapsed time, rather than
the best obtained through all stages of the process.
While a material may stabilize at an optimum
ground state after a sufficient duration, an analogous stabilized condition for general combinatorial systems seems quite unlikely. In particular,
fluctuations away from an optimal solution are
apt to be significant even in the combinatorial
vicinity of such a solution. If this supposition is
true, simulated annealing would undoubtedly benefit by a modification that does not rely on a
strong stabilizing effect over time. (Under assumptions that imply an appropriate stabilizing effect
exists, simulated annealing and Markov generalizations can be shown to yield an optimal solution
with probability 1 after a sufficient number of
iterations at each level. The implicit required effort has not, however, been shown better than that
of complete enumeration.) In one approach, for
example, an attempt to reduce reliance on the
stability assumption occurs by introducing a final
phase to see if any improved solutions appear
during some additional time increment after
reaching the supposedly stable area.
There are additional ways that simulated annealing stands to be improved through artificial
intelligence, by reference to the type of reasoning
humans employ. It is reasonable to postulate that
human behavior resembles the simulated annealing process in one respect: a human may take
nongoal directed moves with greater probability at
greater distances from a perceived destination.
(Similarly, an animal on the hunt, when con-
124
F. Glover, H.J. Greenberg / New approaches for heuristic search
fronted with a relatively weak scent, is inclined to
wander to pick up additional scents rather than to
zero in immediately on a particular target.) Yet
the human fashion of converging upon a target is
to proceed not so much by continuity as by
thresholds. Upon reaching a destination that provides a potential home base (local optimum), a
human maintains a certain threshold--not a progressively vanishing probability--for wandering
in the vicinity of that base. Consequently, a higher
chance is maintained of intersecting a path that
leads in a new improving direction.
Moreover, if time passes and no improvement
is encountered, the human threshold for wandering is likely to be increased, the reverse of what
happens to the probability of accepting a nonimproving move in simulated annealing over time.
On the chance that humans may be better equipped
for dealing with combinatorial complexity than
particles wandering about in a material, it appears
worth investigating the usefulness of an adaptive
threshoM strategy as an alternative, or supplement,
to the strategy of simulated annealing. Research
into such possibilities are expected to point toward useful improvements in the simulated annealing approaches.
Tabu search, introduced by Glover (1977, 1986),
constitutes a meta-procedure that organizes and
directs the operations of a subordinate method.
Thus it shares with simulated annealing the ability
to employ as a subroutine any procedure that
might be used to obtain local optima, including
neural network methods and genetic algorithms.
In contrast to simulated annealing, tabu search
neither makes recourse to randomization nor undertakes to proceed slowly to a local optimum on
the supposition that the proper rate of descent will
make the local optimum, when finally reached,
closer to a global optimum. Instead of terminating
upon reaching a point of local optimality, tabu
search structures the operation of its embedded
heuristic in a manner that permits it to continue.
This is accomplished by forbidding moves with
certain attributes (making them tabu), and choosing moves from those remaining that the embedded heuristic assigns to a highest evaluation.
In this respect, the method is a constrained search
procedure, where each step consists of solving a
secondary optimization problem (simple enough
to be accomplished by inspection within the
framework of most evaluation rules), admitting
only those solutions, i.e., moves, that are not excluded by the currently reigning tabu conditions.
The tabu conditions have the goal of preventing
cycling and inducing the exploration of new regions. The need for a means to avoid cycling arises
because, upon moving away from a local optimum, an unconstrained choice of moves (pursuing
those with high evaluations) likely will lead back
to the same local optimum. The tabu restrictions
are designed to prevent this, and more generally to
prevent any series of moves dominated by those
forbidden at the primary level.
The philosophy of tabu search contrasts with
that of branch and bound, another means of preventing cycling, by seeking to impose its constraining conditions to inhibit subsequent moves
in a flexible and adaptive manner--whereas
branch and bound inhibits subsequent moves by
accepting a high degree of rigidity. The greater
flexibility of tabu search serves the purpose of
giving the evaluation rules increased latitude to
drive the method toward attractive regions. This
latitude is counterbalanced by an integration of
considerations that allow both regional intensification and global diversification of the search.
Specifically, the process of tabu search achieves
this flexibility and balance in the following principal ways: (1) by allowing moves that would
violate the implicit tree structure (partial ordering)
of branch and bound; (2) by incorporating strategic forgetting, based on short term memory function; (3) by allowing the tabu status of a move to
be overruled if certain aspiration level conditions
are met; (4) by respecting dominance and deficiency conditions which do not arise in local
optimality searches; (5) by inducing strategic
oscillation of key parameters or structural elements; (6) by intensifying the focus on promising
regional features based on an intermediate term
memory function; and, (7) by diversifying the
search to encompass contrasting non-regional features based on long term memory function. A
recent proposal by Hansen (1986), applied in an
interesting study by Hansen and Jaumard (1987),
also incorporates the first two elements, which
appear by themselves to provide a highly effective
solution strategy.
It may be noted that the short term strategic
forgetting, in conjunction with the intermediate
and long-term concentration and diffusion of
F. Glover, H.J. Greenberg / New approachesfor heuristic search
search, provides an interplay between learning
and unlearning. The short term strategic forgetting
element is not random but systematically governed,
based on the premise that the likelihood of returning to a previous point of departure (or of repeating a prior escape route from that point) is inversely related to the distance from that point. A
simple measure of such distance, which has proved
quite effective in practice, is the number of moves
taken since reaching a given point, under the
condition that no intervening move is allowed to
backtrack toward the point. The non-backtracking
stipulation on intervening moves is given the interpretation that no move is allowed that is equivalent to, or dominated by, the reversal of one of
the moves in the intervening subset.
A simple way to impose tabu conditions with
memory decay is as follows. In m a n y applications
of interest, a move may be viewed as a transition
from a trial solution x ' to another trial solution
x " . (If a trial solution is partial, symbolic values
rather than numeric values are given to components not yet assigned.) A simple attribute of x '
(or sometimes of the pair x ' , x " ) may be recorded
that is sufficient to identify a class of moves that
includes the reverse transition from x " to x ' . For
example, if x~-~X~', then recording the pair
t
Pt
(x j, xj ), is sufficient to identify a class of moves
that includes the one from x " to x ' . This recorded attribute is used to give a tabu status to all
moves that possess it (for example, all moves that
would give xj the value xj ).
In general, the goal is to record an attribute
that is easily checked while imparting a tabu status
to as few moves as possible. However, making the
class of tabu moves larger than necessary for
blocking simple reversals can have desirable
aspects. In the preceding example, by making
x / = x~ tabu, it is clearly impossible to get back to
the solution x ' by any sequence of steps as long
as the tabu condition is in effect. Merely preventing the transition from x " to x ' does not
have this result. Typically, this type of stronger
blocking property is an automatic outcome of
attributes that are easy to record.
A list of such attributes, simple or complex,
which are recorded in the same sequence as their
corresponding solutions are generated, is called a
tabu list. Broadly speaking, this list defines a set
of constraints that must be satisfied by all admissible moves, or by all solutions that are considered
tt
125
legitimate to reach, while the elements of the tabu
list are in effect. Once the last position, say r, on
the list is filled, the list cycles back to the first
position, so that each new attribute replaces the
attribute recorded r moves in the past. For example, in a problem of partitioning elements into
multiple sets, when an element e is moved from
set A to some other set, the attribute represented
by the ordered pair (A, e) could be recorded on
the tabu list as a shorthand for the constraint of
preventing e (and any element equivalent to e)
from being transferred to set A for r succeeding
moves. Implementation of such a tabu list is often
facilitated by the use of a tabu status array, e.g., a
matrix M ( A , e), where M ( A , e ) = 1 if (A, e) is
on the tabu list and M ( A , e) = 0 otherwise.
Experimentation indicates the appropriate
length of the tabu list to be a very stable and
robust parameter, which has been easily identified
in all classes of problems thus far studied. In some
variants, tabu list entries are not created for all
moves, but only for moves with particular properties, such as that cause specified variables (or the
objective function) to change in specified directions. Likewise, tabu status itself is sometimes
graduated, by penalizing less heavily those tabu
moves that occurred farther in the past.
Several applications of tabu search that have
employed tabu conditions designed to block repetition of solutions have found the magic number 7
(+_ 2) to be a remarkably good choice for tabu list
size. This outcome has invited speculation as to
whether the role of such a number in human short
term memory is the result of a natural selection
process related to human problem solving ability.
Evidently m a n y attributes of the solution space
can affect the ideal tabu list size, but it may also
be that problems typically encountered in practice
are structured to favor a range of values that
cluster around 7. An application of tabu search to
traveling salesman problems, however, has invited
the use of tabu conditions that are designed to
prevent cycling of moves rather than solutions,
and it has been found in this setting that ideal
tabu list sizes grow as a function of problem size.
Equivalence and dominance operate somewhat
differently in tabu search than in usual optimization contexts, and require reference to the nature
of the move. We will not undertake to describe
these elements here, but we point out that a tabu
condition prohibiting a given move should prefer-
126
;E Glover, H.J. Greenberg / New approaches for heuristic search
ably be applied in a manner that prevents all
moves dominated by that move. In addition, situations can arise where certain moves can offer no
possibility of an improved solution even though
they receive a higher evaluation than alternatives,
and these nonproductive moves should be avoided.
The aspiration level component of tabu search
plays a particularly role, allowing tabu status to be
overcome if a move is sufficiently attractive. This
component is organized to be compatible with the
goal of avoiding cycling, while providing the ability to find an improved best solution if a move
exists than can reach it. To do this, select some
attribute of the pair x ' , x " associated with the
objective function values, c(x') and c(x"), of
these solutions. For example, an aspiration level
may be established by recording the best objective
function value that has been achieved by moving
away from the value c(x"), or that could have
been achieved by reversing a past move. The value
c(x") is a candidate for this best value by the
latter criterion. The aspiration level is to do better
than reaching this historical value. Consequently,
whenever such an aspiration level can be achieved
by a move that is classed as tabu, the tabu status
m a y be legitimately disregarded. Recent experience suggests that it can be valuable to give the
aspiration level a tenure that parallels its tabu
tenure, rather than allowing this level to be
governed by the outcomes of remote history. This
parallelism of aspiration level and tabu conditions
creates a partnership whereby tabu conditions m a y
be viewed as a component of dynamic aspiration
criteria, and conversely.
It is worthwhile in certain applications, as where
constraints may confine feasible solutions to a
fairly narrow region, to use additional tabu lists to
induce strategic oscillation of selected parameters.
To illustrate, one problem successfully approached in this matter was a lock box application
modeled as a p-median problem with p fixed. A
second tabu list was created for this problem that
compelled successive moves to climb or descend
to alternating depths on each side of the fixed p
value, keeping track of the best candidate solution
each time the value p was reached at the point of
crossing.
The use of strategic oscillation of this type has
several advantages. First, it permits the execution
of moves that are less complex than might otherwise be required. Second, moving outside of a
boundary and returning from different directions
uncovers opportunities for improvement that are
not readily attainable when the search is more
narrowly confined.
Such an observation was one of the original
motivations behind the development of tabu
search. F r o m a more general standpoint, m a n y
types of heuristic search procedures have mirror
opposites, as exemplified by the dichotomous classifications of methods: constructive or destructive,
interior or exterior, feasible or infeasible, primal or
dual, and so forth. Rather than selecting only one
type of search from such a classification, a wider
range of opportunity is opened by alternating
between them. In fact, opposing strategies typically can be organized so that each point of crossing from one to the other represents a point of
local optimality (relative to a given search orientation). T a b u lists that do not simply prohibit
certain move reversals, but compel such crossings
and returns, offer an effective way to avoid the
suboptimal entrapment of standard searches.
We finally comment briefly on the intermediate
and tong term m e m o r y functions of tabu search.
The intermediate term memory operates by recording and comparing features of a selected
number of best trial solutions generated during a
particular period of search. Features that are comm o n to all or a compelling majority of these
solutions (such as values received by particular
variables) are taken to be a regional attribute of
good solutions. The method then seeks new solutions that exhibit these features, by correspondingly restricting or penalizing available moves during a subsequent period of regional search intensification. Instances of this strategy have been
applied in early heuristic approaches to the traveling salesman problem.
The long term m e m o r y function, whose goal is
to diversify the search, employs principles that are
roughly the reverse of those for intermediate term
memory. In contrast also to those methods that
seek diversity by generating a series of random
starting points, and hence which afford no opportunity to learn from the past, long term memory is
used to provide evaluation criteria used by a heuristic designed to produce a new starting point.
These criteria penahze the features that long term
m e m o r y finds to be prevalent in previous executions of the search process. A traveling salesman
application provides a convenient illustration. A
F. Glover, H.J. Greenberg / New approaches for heuristic search
simple form of long term memory in this setting is
a count of the number of times each edge appears
in the tours generated. Penalizing each edge on the
basis of this count favors the generation of initial
'good' tours (according to the heuristic) that tend
to avoid those edges most commonly used in the
past. The diversity achieved by this criterion can
be increased by retaining the penalties for a period
after transferring to the (possibly different) heuristic incorporated within tabu search, and using the
tabu search procedure to seek improved solutions.
Afterward, the penalties are dropped and tabu
search proceeds according to its normal evaluation
criteria. This same type of procedure can be used
to continue directly from the present point of
search to a new region without going back to
generate a new solution from scratch.
Tabu search has been applied across a wide
range of problem settings in which it consistently
has found better solutions than methods previously applied to these problems. As with the
methods of neural networks, simulated annealing
and genetic algorithms, tabu search has been applied to traveling salesman problems as a test of
its ability to perform effectively on a well studied
classical problem. It is important to emphasize
that these general methods, which embody broadly
applicable solution principles, are not to be expected to compete with specialized solution approaches evolved over a significant span of time
for high performance on a given problem class.
Nevertheless, in contrast to the other general
methods, tabu search in several instances has provided solutions for traveling salesman problems
superior to the best known in the literature. These
outcomes were obtained, moreover, for simple implementations based on an unsophisticated component heuristic. Similar results showing tabu
search to be superior in solution quality and speed
to simulate annealing for graph coloring problems
has been reported by Hertz and de Werra (1987),
and more extensive applications and comparisons
are reported in Hertz, de Werra and Widmer
(1988). The fact that the solution framework of
tabu search has proved powerful enough to compete favorably with both general and specialized
strategies provides encouraging motivation for research in applying this framework to additional
classes of combinatorial problems.
127
Target analysis, developed by the authors (see
Glover, 1986), is an integration of artificial intelligence with operations research that gives a new
strategy for solving combinatorial optimization
problems. One may use any conventional strategy,
such as implicit enumeration, and subordinate its
control parameters to a learning model patterned
after classification problems. One early application of learning algorithm controls was by Crowston, Glover, Thompson and Trawick (1963) for
jop shop scheduling. Here target analysis represents a generalization of that strategy.
For example, let an implicit enumeration
strategy have a forward branching strategy of
choosing a best branch of the form x = b, where x
is a choice of variable and b is a choice of fixing
its value (where b is 0 or 1). Suppose k criteria
were used in making this branching choice with
linear weight values w 1, w 2. . . . . w k. For simplicity,
suppose this branch (x, b) was chosen because
E wiz(x, b)
= min[Y'~wiF,( X, /3)" (X, h ) ~ B],
where B is the set of admissible branches. After
the problem is solved, hindsight is used to examine branches to be curtailed thereby avoiding
dead paths. The idea is to adjust the weights w to
ensure the best winner (known with hindsight) for
the training problems.
For this case linear inequalities for w are generated as follows. Let (u, v) denote the branch we
should have taken (with hindsight). Then, adjust
the weights to make this win:
Eiw,F,(u,
_<E,w,F,(x,
for all (X, fl) ~ B.
As long as there is a solution to these inequalities,
a new weight is chosen. By construction, the chosen weight ensures a shortest path to a solution for
all training problems (with no dead paths). Eventually, there may not exist a weight vector to
ensure this. One may, of course, define a reasonable proxy, like least squares, but a better alternative is to begin to classify problems and have the
learning mechanism deal simultaneously with
problem classification and optimal algorithm
parameter settings within each class. Still other
possibilities exist.
128
F Glover, H.J. Greenberg / New approaches for heuristic search
One successful application of this idea, but
with more structure derived from the problem
attributes, was described by Glover, Klingman
and Philips [1988].
Future directions
These new approaches are not mutually exclusive. Genetic algorithms have been proposed to
train neural networks. Simulated annealing may
be viewed, like a Boltzmann machine, as one way
to find a global minimum in the presence of many
local minima. (Combinatorially complex problems
are computationally equivalent to minimization of
a nonconvex quadratic function.) As made explicit
in Tabu Search and Target Analysis, one may
choose any algorithmic framework and superimpose these techniques as an intelligent adapter.
Theoretical limits do exist. A neural network is
not likely to find an exact solution to a hard
problem like TSP with only a polynomial number
of nodes. Simulated annealing is not certain to
end successfully within polynomial time (the cooling rate may need to be exponentially slow). The
practical question is in the realm of heuristic
search: Can these new approaches do much better at
finding good solutions to hard problems?
The powers and limitations of genetic algorithms, neural networks, simulated annealing,
tabu search, and target analysis remain to be
charted fully. In spite of many published expositions, including mathematical convergence proofs
under special assumptions, the fundamental scope
and supporting rationale for these approaches are
less than completely known. Current research is
unable to explain adequately why or when such
approaches are likely to succeed or fail. There is
no taxonomy of combinatorial strategies for linking these methods into a common framework
and no theory enabling them to be taken apart
and reassembled into new and possibly more effective methods. The significance and pervasiveness of the combinatorial explosion makes it highly
important to gauge the reliability and effectiveness
of alternative strategies for different problem
classes. It is also important to determine whether
variants or extensions of these procedures will
lead to improve solution of specific real world
problems. To be certain, there exists a sufficient
number of dramatic practical successes to show
that the potential of these methods is more than
conjectural.
The challenge of evolving a unifying framework
for these strategies is complicated by their intimate reliance on procedures outside themselves.
Such general approaches must nevertheless be tied
to the specific, i.e., they must be carefully tailored
to take advantage of domain specific representations and expertise that apply to a particular
setting. Under such circumstances, the essential
character and contribution of a general strategy
and its component methods are not always easy to
differentiate. Each problem class entails its own
adaptation, and the form of this adaptation is far
from unique. Thus, a given general strategy can be
applied in a variety of ways to the solution of a
particular type of problem, and its performance
will naturally depend on which of these way is
chosen.
Advances in the development and refinement
of general strategies therefore depend in part on
identifying the type of adaptation to a specific
problem domain that will prove most effective.
Such identification can come about only by means
of empirical study. As 'good marriages' of general
strategies and component procedures become
known, broader themes may be expected to emerge
that leads to more rapid and effective exploitation
problems newly encountered.
It should be noted that most of the strategies
previously described are founded on analogies that
link them to processes that presumably occur in
nature. There is a danger in this insofar as analogy
has the power to obscure as well as elucidate.
Following the path dictated by an analogy can
discourage the exploration of alternative paths
whose features do not conform to the patterns the
analogy endorses. It is particularly worthwhile,
therefore, to re-express the principles and procedures of different general strategies in alternative
frameworks that are free of the references that
originally inspired these approaches. This opens
the way to comparisons and useful integrations
not possible when the methods are expressed in
terms of competing and (not entirely commensurable) perspectives.
The development of general frameworks for
different combinatorial search methods further
stimulates an interdisciplinary synthesis of concepts and contributions. Those who variously approach AI from the perspectives of computer sci-
F. Glover. H.J. Greenberg / N e w approaches for heuristic search
ence, operations research, psychology and biology,
for example, often fail to appreciate or understand
parallel developments in allied domains that could
significantly augment their own explorations. The
potential for accelerated advances is greatly improved by cross fertilization among groups pursuing divergent approaches to related problems, and
this in turn depends on the development of frameworks that make the contributions of each field
accessible to the others. Even within a given field,
major advances often hinge on evolving a conceptual unification of diverse elements, thereby opening the door to an enriched realm of possibilities
that transcends the components on which it is
based.
References
Ackley, D.H. (1987), "A connectionist algorithm for genetic
search", Technical Report, Department of Computer Science, Carnegie-Mellon University, Pittsburgh, PA.
Bonomi, E., and Lutton, J.-L. (1984), "The N-city travelling
salesman problem: Statistical mechanics and the metropolis
algorithm", S I A M Review 26, 551-568.
Cerny, V. (1985), "Thermodynamical approach to the traveling
salesman problem: An efficient simulation algorithm",
Journal of Optimization Theory and Applications 45, 41-52.
Crowston, W.B., Glover, F., Thompson, G.L., and Trawick,
J.D. (1963), "Probabilistic and parametric learning combinations of local job shop scheduling rules", ONR Research
Memorandum No. 117, GSIA, Carnegie-Mellon University,
Pittsburg, PA.
Farlow, S. (1984), Self-Organizing Methods in Modelling."
GMDH Type Algorithms, Marcel Dekker, New York.
Galliant, S.I. (1988), "Connectionist expert systems", Communicatins of A C M 31, 152-169.
Geman. S., and Hwang, C.-R. (1983), "Diffusions for global
optimization", Technical Report, Brown University, Providence, RI.
Glover, F. (1977), "Heuristics for integer programming using
surrogate constraints", Decision Sciences 8, 156-166.
Glover, F. (1986), "Future paths for integer prograrnmang and
links to artificial intelligence", Computers and Operations
Research 13, 533-549.
Glover, F. (1987), "Tabu search", Technical Report, Center for
Applied Artificial Intelligance, University of Colorado,
Boulder, CO.
Glover, F., Klingman, D., and Phillips, N.V. (1988), "A network-related nuclear power plant model with an intelligent
branch-and-bound solution approach", Technical Report
CBDA 139, University of Texas, Austin, TX.
Goldberg, D.E. (1988), Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley.
Goldberg, D.E., and Lingle, Jr., R. (1987), "Alleles, loci, and
the traveling salesman problem", Technical Report, Department of Engineering Mechanics, The University of
Alabama, University, AL.
129
Greenberg, H.J. (1988a), "Foundations for an intelligent
mathematical programming system", Technical Report,
University of Colorado at Denver, Denver, CO.
Greenberg, H.J. (1988b), "Learning networks", Final Report
of Mathematics Clinic for Spring 1988, University of Colorado at Denver, Denver, CO.
Grefenstette, J. (1986), "Optimization of control parameters
for genetic algorithms", 1EEE Transactions" on ,~vstems,
Man, and Cybernetics 16, 122-128.
Grefenstette, J., Gopal, R., Rosmaita, B., and Van Gucht, D.
(1985), "Genetic algorithms for the traveling salesman
problem", Proceedings" of the International Conference on
Genetic Algorithms and Their Applications.
Gutzmann, K.M. (1987), "Combinatorial optimization using a
continuous state Bohzmann machine", Proceedings of 1EEE
International Conference on Neural Networks, Vol. Ill (June),
721-733.
Hansen, P. (1986), "The steepest ascent mildest descent heuristic for combinatorial programming", presented at the Congress on Numerical Methods in Combinatorial Optimization, Capri, Italy.
Hansen, P., and Jaumard, B. (1987), "Algorithms for the
maximum satisfiability problem", RUTCOR Research Report RR 4- 43-87, Rutgers, New Brunswick, NJ.
Hertz, A., and Werra, D. de (1987), "'Using tabu search
techniques for graph coloring", Computing 39, 345-351.
Hertz, A., Werra, D. de, and Widmer, M. (1988), "Some new
applications of tabu search", presented at the 13th International Symposium on Mathematical Programming, Tokyo,
Japan.
Hicklin, J. (1987), "Application of the genetic algorithm to
neural network design", Technical Report, Laguna Research Laboratory, Fallbrook, CA.
Holland, J.H. (1975), "Adaptation in natural and artificial
systems", University of Michigan Press, Ann Arbor, MI.
Hopfield, J.J., and Tank, D.W. (1985), "Neural computation
of decisions in optimization problems", Biological
Cybernetics 52, 141-152.
Hansen, P., and Jaumard, B. (1987), "Algorithms for the
maximum satisfiability problem", RUTCOR Research Report RR:~43-87, Rutgers University, New Brunswick, NJ.
Kirkpatrick, S., Gelatt, Jr., C.D., and Vecchi, M.P. (1983),
"Optimization by simulated annealing", Science 220,
671-680.
Kohonen, T. (1987), Self-Organization and Associatit;e Memory',
Springer, Berlin.
McCulloch, W.S., and Pitts, W. (1943), "A logical calculus of
the ideas immanent in nervous activity", Bulletin of
Mathematical Biophysics 5, 115-133.
McEliece, R.J., Posner, E.C., Rodemich, E.R., and Venkatesh,
S.S. (1987), "The capacity of the Hopfield associative memory", IEEE Transactions on Information Theory 33.
461 482.
Oliver, I.M., Smith, D.J., and Holland, J.R.C. (1987), "A study
of permutation crossover operators on the traveling salesman problem", Technical Report, Texas Instruments Ltd.,
Dallas, TX.
Omohundro, S.M. (1987), "Efficient algorithms with neural
network behavior", Technical Report, Department of Computer Science, University of Illinois at Urbana-Champaign,
IL.
130
E Glover, H.J. Greenberg / New approaches for heuristic search
Rosenblatt, F. (1962), Principles of Neurodynamics, Spartan,
Washington, DC.
Rumelhart, D.E., and McClelland, J.L. (1986), Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1." Foundations, MIT Press, Cambridge, MA.
Tank, D.W., and Hopfield, J.J. (1986), "Simple optimization
networks: An A / D converter and a linear programming
circuit", 1EEE Transactions on Circuits And Systems 33,
533-541.