Transcript Water quality planning. - Computing Center of the Russian

Approximation and Visualization
of Interactive Decision Maps
Short course of lectures
Alexander V. Lotov
Dorodnicyn Computing Center of Russian
Academy of Sciences and
Lomonosov Moscow State University
Lecture 10. IDM in dynamic MOO problems
and problems with uncertainty and risk
Plan of the lecture
Part I. Dynamic multiobjective problems
1.1. Controlled linear differential equations: Moving Pareto frontier
1.2. Linear differential equations with constraints imposed on state
and controls (economic systems)
1.3. Non-linear differential equations & identification
Part II. RGM/IDM technique for problems with uncertain and
stochastic data
2.1. RGM for non-precise data
2.2. Application of the IDM/RGM technique for supporting robust
decision making
2.3. Supporting the decision making under risk
Part III. Goal-related negotiations (experiments on closing the gap
between the goals)
Part I. Dynamic multiobjective
problems
Multiobjective dynamic systems
This part of the lecture is devoted to applications of the
IDM technique for exploration of the multiobjective
dynamic systems described by differential or
difference equations.
The approach is based on
1) approximating the reachable sets for the dynamic
system under study;
2) subsequent approximating the feasible sets in the
objective space (or, their EPH), and
3) visualization of their Pareto frontiers.
Thus, the only new feature of the approach is related to
the approximating the reachable sets for the dynamic
systems.
Controlled differential
equations linear in respect to
the state
The Reach Sets for linear dynamic systems
dx  Ax  Bu  a, t  [0, T ]
dt
n
u  U , t  [0, T ], x(0)  X (0)  R
Here U and X0 are given polyhedra.
Let X(t) be the reachable set for the time
moment t  [0, T ], that is, all points of the space
n
R , to which the system can be brought
from X (0)precisely at the moment t  [0, T ].
The dynamic multiobjective problems
Let us consider the criteria z=f(x),
where x  X (t ), t  [0, T ]
Thus, we can consider the feasible sets
in criteria space Z(t)=f(X(t)), their
Pareto frontier P(Z(t)) and their
m
Edgeworth-Pareto Hulls Z p (t )  Z (t )  R
Note that the mapping f may be nonlinear.
Moving Pareto frontier
Let us split the time period [0, T] into M steps
tk  kT / M
Then, we have developed methods of ER-type
for constructing polyhedral approximations of
reachable sets X(tk) for given time moments tk,
k=1,…,M, with a required precision.
Then, the sets Z(X(tk)) or ZP(X(tk)), k=1,…,M,
are approximated.
Finally, decision maps based on slices of, say,
ZP(X(tk)) are displayed one after another
providing animation of the Pareto frontier.
Approximating the reachable sets for n>2
Since the sets X(tk) are convex and compact,
the approximation methods are based on
combination of the ER method with method for
computing the support function of the
reachable set proposed by L.S.Pontryagin.
Namely, his maximum principle is used by us.
Let, for example, tk=T. If <c,x> is maximized
over X(T), then first the system
   A * , 0  t  T



is solved.


 (T )  c
Approximating…(continued 1)
Then,
 x0 , (0)  max  x, (0) 
xX 0
and, for 0<t<T,  u * (t ), (t )  max  u , (t ) ,
uU
which results in
T
x * (T )  e AT x0*   e A(T s ) u * ( s )ds
0
By using techniques for numerical integration of
differential equations, it is possible to perform
the operations with a given precision   0 .
Approximating…(continued 2)
Since the ER method can approximate
a compact convex body with any
desired precision  2  0, an estimate is
obtained h
 ( X (T ), Xˆ (T ))  1   2
where 1 is precision of computing
the support function.
Approximating the EdgeworthPareto Hull
Approximating the sets ZP(X(tk)),
k=1,…,M, on the basis of X(tk) can
be carried out by using the ER
method in the convex case and the
technique for approximating by cones
in the non-convex case.
Then, visualization can be used.
Example
All constants and the initial state (for
t=0) are given. Let t* be the moment of
the end of the movement. The three
criteria are considered:
z1  x1 (t*)
z 2  v1 (t*)
z3  t *
The system (six state variables)
 x1  v1
v  9 x  x
1
2
 1
 x 2  v2


v
 2  0.5 x1  x2  0.5 x3
 x 3  v3

1  u  1
v3  2 x2  2 x3  u,
Reachable sets for six state variables for
about M=500 time moments of the interval
[0, 60] were approximated.
Projection of the 6-dimensional
reachable set on (x1, v1) plane
Moving Pareto frontier
Linear differential equations with
constraints imposed on state and
controls simultaneously (economic
systems)
Typical linear economic
system
x  Ax  Bu  a,
n
x R ,
Cx(t )  Du(t )  d , u  R ,
x(0)  X 0
r
0t T
Terminal criteria are usually
considered:
z  f ( x), x  X (T ).
Method for constructing
reachable sets and Edgeworth-Pareto
hulls
Time is split into M steps and linear difference
equations are used instead of the differential
equations. Thus, a linear system of equalities
and inequalities in a finite dimensional space is
obtained. Alternatively, the difference equations
can be used originally for the description of an
economic system.
The Edgeworth-Pareto hull can be approximated
by using the ER method in the convex case or
approximating by the cones in the general case.
Applications
• Real-life. Specification of national goals
for a long-time development (State
planning agency of the USSR in 19851987).
• Methodological. Search for efficient
strategies against global climate change.
Non-linear differential equations &
identification
The system
x  g (v, u, t ), x R ,
n
u (t )  U , u  R , x(0)  X 0
r
0t T
Terminal criteria are usually
considered:
z  f ( x), x  X (T ).
Method of the study
• A large, but finite number of trajectories
and associated criterion vectors are
constructed and non-dominated criterion
points are selected,
• Visualization of the non-convex
Edgeworth-Pareto hull follows
• If needed the convex hull of the
Edgeworth-Pareto hull approximated by
using the ER method is studied by using
the Interactive Decision Maps technique.
Applications
• Real-life. Exploration of marginal
pollution abatement cost in the electricity
sector of Israel (Ministry of National
Infrastructures). The software system
was used at the Ministry for about five
years.
• Methodological. Development of
strategies of steel cooling in the process
of continuous steel casting (jointly with
Finnish specialists).
Identification of the state of a
dynamic system
Part II.
RGM/IDM technique for problems
with uncertain and stochastic data
RGM for non-precise data
Uncertainty of data
Values yij are given not precise: instead of
values, the (subjective) probability density
φ(yij) that describes possible values of the i-th
attribute for the j-th alternative must be given.
In the simplest case that has been studied now,
the probability density φ(yij) is a constant
value over a given interval [aij, bij] and is zero
outside of it.
Then, each alternative (row) can be associated
to a box [aj,bj] of the m-dimensional linear
criterion space that contains possible values of
the attributes for this alternative.
Thus, instead of the m-dimensional points in
the precise case, we have to study the mdimensional boxes in the fuzzy case.
Theoretical result
It was theoretically proven that specification
of the reasonable goal at the Pareto
frontier of the envelope of the best points
of the boxes provides all the theoretical
benefits of the situations without
uncertainty.
Screening the fuzzy alternatives
(maximization case)
Application of the IDM/RGM
technique in the case of large
uncertainty
• Application of the IDM/RGM technique to
several criteria used in the case of
uncertainty (minmax, maxmax, minimization
of maximal regrets, etc.) simultaneously.
Application of the IDM/RGM
technique for supporting robust
decision making
Under robust decision making one understands
selecting of decisions, which are reasonable in the
case of all possible futures. The IDM technique was
applied recently for selecting robust decisions in the
framework of selecting the parameters of the
electronic suspension system controller to be used
in future cars (on request with STMicroelectronics).
Search for a robust strategy during Russian
financial default in 1998
Search for a robust strategy before Russian financial default in
August 1998 started in February 1998 after it was clear that
some kind of unhappy event is inevitable, but it was not
clear what kind of event it will happen and when.
The question was considered: what will happen with
US$1000. Three possible futures were studied:
1) the event will not happen at all (normal development);
2) the 150% devaluation of ruble will happen;
3) A total collapse of the banking system will happen.
The decisions were the allocations of the sum between
different banks (including Russia and abroad) and in
different currencies.
Color provides results in the case of the total
collapse of the banking system
Supporting the decision making
under risk
The model
Let us consider N alternatives,
while the i-th alternative is given by its
cumulative distribution function
Fi(x)=P{v<x}, i=1,..,N,
where v is a value to be maximized
(or minimized).
Approach proposed by Y. Haimes
(University of Virginia)
Criteria are selected by using the cumulative
distribution function F(x) =P{v<x}.
Then, the values yk=P{vk<v<v+k}, k=1,..,m,
are used as criteria, where the values vk and
v+k are specified by the decision maker.
Any multi-criteria method can be used.
In contrast to Y. Haimes, we use different
criteria and apply the IDM/RGM technique.
Criteria explored by us
A criterion, which may be constructed by using the probability
function F(x) of an indicator v, can simply have a sense of
the probability that the value of the indicator is not higher
that some value z specified by the decision maker
y=F(z)=P{v<z}.
Such values have a simple sense (in contrast to the values used
by Y.Haimes):
• if the indicator v is some kind of benefit, then the value
y=F(z) is desirable to decrease, must be minimized; in
contrast,
• if the indicator v describes some kind of losses, then the
increment of the value y=F(z) is desirable.
Let the decision maker specify m values vk, k=1,..,m. Then, we
can consider m criteria yk=F(vk)=P{v<vk} and apply the
IDM/RGM technique.
Example of the IDM/RGM application in
decision making under risk
Let us consider an example problem: choice of an
alternative variant of a dam. Let consider the
probability distribution of losses which gives rise
to three criteria:
1.expectation of losses (including known annual cost);
2. probability of high losses denoted by P_h, i.e.
P_h=1-F(h) where h is a high value of losses, and
3. probability of catastrophic losses denoted by P_c ,
i.e. P_c=1-F(c) where c is a catastrophic value.
One is interested to minimize the values of the criteria.
List of the alternatives
Decision map
If the cross is specified as in the decision
map, the following alternatives are selected
Informing lay stakeholders on
risks
One can use the Web RGDB
application server for informing lay
stakeholders on environmental risks
just in the same way as concerning
any other environmental problem.
Part III.
Goal-related negotiations
(experimental closing gap
between the goals)
Experimental conflict
Loss 2
EPH
The best
Point for
Negot. 1
Initial
point
The best
Point for
Negot. 2
Loss 1
Experiment with transferable reward
Two students who have never met before were
informed that the winner will be given a
money reward if their negotiation results
small losses. They immediately (in about 5
minutes) have found an objective point with
the maximal payment, have got the money
and immediately disappeared. It seemed
that they have found the way how to share
the reward.
Experiment with non-transferable
reward
The second experiment involved non-transferable rewards. In
this experiment, twelve students of the fourth year from the
Lomonosov Moscow State University were grouped into
six groups in accordance to their wishes.
In the framework of the experiment, the additional score (or
mark) during the examination was used for a nontransferable reward. Movements along the Pareto frontier
were related to the increment of the additional score for one
student and to the decrement for another. Clearly, in this
case sharing of the reward is not possible.
results and the other student agreed to accept very
poor results. Therefore, one can state that the
experiment with the non-transferable rewards
resulted in practically the same outcome as the
experiment involving money rewards! What is the
reasons of such behavior? The students informed
that they have used some forms of compensation.
They were not obliged to inform on the form of the
compensation they have developed.
Experimental results
It took from 15 minutes to two hours to find an agreement. One pair
decided to stay at the initial point, but all other pairs decided to move
the goal.
In other pairs, one student of each pair achieved very good results and
the other student agreed to accept very poor results. Therefore, one
can state that the experiment with the non-transferable rewards
resulted in practically the same outcome as the experiment involving
money rewards! What is the reasons of such behavior?
The students informed the teacher that they have used some forms of
compensation. They were informed in advance that they are not
obliged to inform the teacher on the form of the compensation they
have developed.
Compensational payments and
moving along the Pareto frontiers
• The results of the experiments show that people can
develop compensational payments.
• Due to it, they can move along the Pareto frontiers.
In turn, this may assume that the Pareto frontier-based
negotiation support technology might be useful in
real-life negotiations.
Our Web address
• http://www.ccas.ru/mmes/mmeda/