* M. Larbani and **P. L. Yu * IIUM University, Kuala Lumpur, Malaysia

Yu and Larbani 31/06/2011

ISOR11

*M. Larbani and **P. L. Yu

*IIUM University, Kuala Lumpur, Malaysia

**National Chiao Tung University Taiwan and Kansas University , USA

Decision Making and Optimization in Changeable Spaces, A Paradigm Shift

Presentation Outline

1. Traditional Models of Decision Making and their Limitations

2. Habitual Domain Theory 3. Decision Making and Optimization in

Changeable Spaces 4. Applications5. Conclusion


ISOR11

1.Traditional DM ModelsThere are two approaches to decision making:

qualitative and quantitative approaches. The quantitative approach to decision making went

through two major stages. Stage 1. Deterministic Approach. A fixed set X of

decisions or alternatives is exactly known and unchangeable, and a utility (or value) function (Von Neumann)

u(.): X R,

that represents the preferences of the DMs is exactly determined.

Po-Lung Yu 112/04/20 2009 NACA--3

To each decision x in X, a value u(x) is associated; a decision x is preferred to a decision y if

u(y) < u(x). Then the problem of finding the best or

optimal decision is formulated as an optimization problem,

Maxx X u(x) (1)

Po-Lung Yu 112/04/20 2009 NACA--4

Numerous resolution methods were developed and implemented as linear programming (Dantzig) and nonlinear programming (Kuhn-Tucker).

(1) was extended to multiple criteria decision making (Yu, Steuer)

Maxx X u1(x), u2(x),…, um(x) , (2)

and games:

< {1,2,..n}, ui(x1 , x2 ,…, xn), Xi , i =1,2,…,n > (3)

(Von Neumann, Nash, Aumann, Shaplay)

Po-Lung Yu 112/04/20 2009 NACA--5

Po-Lung Yu and Larbani 31/06/2011

ISOR 11

Stage 2. Incorporation of Structured Uncertainty

(1)-(3) were extended to models incorporating partially or completely probabilistic or fuzzy or fuzzy-probabilistic inputs, (1) became

Max ŷ Ŷ û(ŷ)

where û(ŷ) and Ŷ may be stochastic or fuzzy or fuzzy-stochastic (Zadeh, Bellman)

1.2 Limitations of Traditional ModelsStructured uncertainty. In probabilistic and/or

fuzzy models it is assumed that the uncertain parameters vary within a certain range and have well known shapes of probability distributions or fuzzy membership functions.

Whereas, often real decision making involve parameters the dimensions, ranges and shapes of which may not be easily predetermined or assumed as in probabilistic and/or fuzzy models (Unstructured Uncertainty).

Po-Lung Yu 112/04/20 2009 NACA--7

Changeable Parameters. Often, in real-life decision making problems the parameters like the set of alternatives, the criteria and the DMs, change as situations and psychological states of the DMs change.

Po-Lung Yu 112/04/20 2009 NACA--8

Example 1. Horse Race

A retiring corporate chairman invited to his ranch two finalists (A and B) from whom he would select his replacement using a horse race. A and B, equally skillful in horseback riding, were given a black and white horse, respectively. The chairman laid out the course for the horse race and said, “Starting at the same time now, whoever’s horse is slower in completing the course will be selected as the next chairman!”

Po-Lung Yu 112/04/20 2009 NACA--9

Here, we have a discovering problem; the two finalists were puzzled at the beginning, it was not immediately apparent what strategy was needed to win the race, they had to discover winning strategy themselves.

This decision making problem cannot be formulated within the traditional framework, let alone solve it, because

The set of strategies X of each candidate is a changeable space for the other player and for the candidate himself.


ISOR 11

Example 2. Game of silence

Silence fell over a young couple after a family quarrel. They did not talk to each other for two days. The situation became uneasy for husband and wife. No one of them wanted to break the silence first because of the fear of losing face.

Po-Lung Yu 112/04/20 2009 NACA--11

Game of Silence

This problem cannot be formulated in the framework of traditional mathematical programming because the utility function of the couple, u(.), and its set of alternatives, are not known. The couple has to generate solutions.

Po-Lung Yu 112/04/20 2009 NACA--12

Here, the couple faces a discovering-covering (dis/covering for short) problem. They want to return to normal life (covering a target), but they don’t know how to achieve this target (discovering solutions).

Thus, despite the tremendous theoretical and practical results achieved, some important aspects of decision making have not been incorporated in the existing models as unstructured uncertainty, idea generation and human psychology.

Po-Lung Yu 112/04/20 2009 NACA--13

The new model we propose, Decision Making and Optimization in Changeable Spaces (DMCS)

is a paradigm shift from utility function –structured uncertainty based model to a model that incorporates human psychology and creativity, and can handle unstructured uncertainty.

In this presentation, we will focus on Covering and Discovering Problems, which encompass most of decision making problem.

Po-Lung Yu 112/04/20 2009 NACA--14

Yu and Larbani ISOR11

2. Habitual Domain (P.L.Yu)

The collection of ideas and actions (including ways of perceiving, thinking, responding, acting, and memory) in our brain together with their formation, dynamics, and basis in experience and knowledge, is called our Habitual Domain (HD).

Over time, unless extraordinary event occurs or purposeful effort is exerted, our HD will become stabilized within a certain domain.


ISOR11

Elements of a Habitual Domain

Potential Domain (PD) - the collection of ideas and actions that can potentially be activated.

Actual Domain (AD) - the set of ideas and actions that are actually activated.

Activation Probabilities (AP) - the probabilities that ideas and actions in PD also belong to AD.

Reachable Domain (RD) - the set of ideas and actions that can be attained from a given set in AD.


ISOR11

e

AP(e)

AD

PDRD

HD(t) = { PD(t), AD(t), AP(t), RD(t)} (4)

e


ISOR11

Eight hypotheses H1-H8 of HDH1. Circuit Pattern Hypothesis.

• Thoughts or concepts or ideas are represented by circuit patterns of the brain.

H2. Unlimited Capacity Hypothesis. • Practically every normal brain has capacity to encode

and store all thoughts, and concepts that one intends to.

H3. Efficient Restructuring Hypothesis.• According to the dictation of attention our thoughts and

memories are continuously restructured so that relevant ones can be efficiently retrieved to release charge.


ISOR11

H4. Analogy and Association Hypothesis. • The perception of new events, or ideas can be learned

primarily by analogy and association with what is already known.

H5. Goal Setting and State Evaluation. • Each one of us has a set of goal functions and for each

goal function we have an ideal state or equilibrium to reach and maintain (goal setting). We continuously monitor, where we are relative to the ideal state or equilibrium point (state evaluation). Goal setting and state evaluation are dynamic, interactive and are subject to change….

Goal’s Ideal state


ISOR11

H6. Charge Structure and Attention Allocation Hypothesis. • When there is an unfavorable deviation of

perceived value from the ideal, each goal function will produce various levels of charge. The totality of the charges created by all goal functions is called the charge structure. At any point in time, our attention will be paid to the event which has the most influence on our charge structure.

Ideal Perceived

Fifth level ( 5). The charge is extremely intense and the allowable time for discharge is very short.

Forth level (4). The charge is intense with a short allowable time for discharge.

Third level (3). The charge is less intense, but still strong, with a little longer allowable time for discharge.

Second level (2). The charge is mild and the allowable time for discharge is fairly long. This level, again, can be produced by almost all goal functions.

First level (1). The charge is slight with no specific discharge time. Again this level can be produced by almost all goal functions.


ISOR11


ISOR11

H7. Discharge Hypothesis ： the least resistance principle. • To release charges, we tend to select the action

which yields the lowest remaining charge (the remaining charge is the resistance to the total discharge).

H8. Information Input Hypothesis. • Humans have innate needs to gather external information. Unless attention is paid, external information may not be processed.

Yu and Larbani 2011 ISOR11

Compilation

CHARGE STRUCTURE

Attention Allocation

Solution Obtained?

Unsolicited

Information

Solicited

Information

ExternInf.

State Evaluation

Internal Information Processing Center

Physiological

Monitoring

Goal Setting

Self-

suggestion

Actions/Discharges

(11) (1)

(3) (2)

(4)

(5)

(6)

(8)

(7)

(12)

(13)

(9)

(10)

Experience/ Being H7:Dischange:Least Resistance

Problem Solving

Avoidance Justification

(14)

H1 : Circuit PatternsH2 : UnlimitedH3 : RestructureH4 : Association

H5 : Goal Set ＆ States

Evaluation

H6 : Charge Structures ＆ Attention Allocation

H8 : Information Input

Po-Lung Yu and Larbani 31/06/2011

ISOR11

Competence Set.

when a DM faces a Problem E at some time t, he has a perceived competence set CS t (E) to solve it.

On the other hand there is the actual or true competence set CS (E) required to solve the problem.

3. Model Construction


ISOR11

CSt (E) CS(E)

E can be solved some at time t if CS (E)⊂CSt (E), that is, when CSt (E) covers CS (E).

CS (E)\CSt (E) is the decision blindsNote, the larger the decision blind is, the more likely it is that the DMs might make important mistakes.

Decision Blind

Decision Blinds


ISOR11

A DM is in Decision Trap if his competence set CSt (E) is trapped in some area and cannot expand to fully cover the targeted competence set CS(E).

Decision Trap

Decision Trap CSt (E)

CS(E)

Po-Lung Yu 112/04/20 2009 NACA--27

Discovering is getting out of a decision trap.

Discovering

Decision Trap

CSt(E)


ISOR11

The resolution of challenging problems generally involve covering and discovering or dis/covering.

Discovering requires a target to cover, while the covering process requires discovering when it falls in a decision trap.

The 7-8-9 principles for Dis/covering

The Seven empowering operators.M1. Everyone is a priceless living entity. We are

all unique creations who carry the spark of the divine.

M2. Clear, specific and challenging goals produce energy for our lives. I am totally committed to doing and learning with confidence. This is the only way I can reach the goals.

Po-Lung Yu 112/04/20 2009 NACA--29

M3. There are reasons for everything that occurs. One major reason is to help us grow and develop.

M4. Every task is part of my life mission. I have the enthusiasm and confidence to accomplish this mission.

M5. I am the master of my living domain. I take responsibility for everything that happens in it.

Po-Lung Yu 112/04/20 2009 NACA--30

M6. Be appreciative and grateful and don’t forget to give back to society.

M7. Our remaining lifetime is our most valuable asset. I will enjoy it fully and make a 100 percent contribution to society in each moment of my remaining life.

Po-Lung Yu 112/04/20 2009 NACA--31

Eight methods for expanding and enriching HDs.

M8. Learning Actively

M9. Projecting from a Higher Position

M10. Active Association

M11. Changing the Relevant Parameters

M12. Changing the Environment

M13. Brainstorming

M14. Retreating in Order to Advance

M15. Praying or Meditating

Po-Lung Yu 112/04/20 2009 NACA--32

Nine principles of deep knowledge.M16. The Deep and Down Principle

M17. The Alternating Principle

M18. The Contrasting and Complementing Principle

M19. The Revolving and Cycling Principle

M20. The Inner Connection Principle

M21. The Changing and Transforming Principle

M22. The Contradiction Principle

M23. The Cracking and Ripping Principle

M24. The Void Principle

Po-Lung Yu 112/04/20 2009 NACA--33

Let M= {M1, M2,…, M24 } be the set of the 7-8-9 principles and

CM= {H/ H=Ms(1)o Ms(2),…oMs(e) , Ms(i) M}

be the set of finite sequences resulting from combination of elements of M.

H: Ω Ω1

Ω= The set of all knowledge humanity has reached so far.

Ω1 = set of all knowledge humanity can reach in the future.

Po-Lung Yu 112/04/20 2009 NACA--34

Formulation of Dis/covering as OCS

Given a target competence set CS(E) and an initial competence set CS0 (E) of the DMs, the dis/covering problem is formulated as the Optimization in Changeable Space (OCS) problem

find H in CM such that (5)

CS (E)⊂H(CS0(E))

Po-Lung Yu 112/04/20 2009 NACA--35

When time is emphasized,

Min t(H) (6)

CS (E)⊂H(CS0(E)) , H CM

Where t(H)= period for finding H and implementing it

When Cost is emphasized

Min c(H) (7)


Where c(H)= cost for implementing H

Po-Lung Yu 112/04/20 2009 NACA--36

When charge (state of mind) is emphasized

Min ch(H) (8)


Where ch(H)= the resulting charge after implementing H

Po-Lung Yu 112/04/20 2009 NACA--37

Note. The operator Min in the models (6)-(8) should be understood in the sense of satisfaction not in the sense of absolute minimum.

Po-Lung Yu 112/04/20 2009 NACA--38

Application

Example 1. Horse Race.

Formulation as OCS from A’s perspective:Stage 1. The candidates were in a decision trap.

Initial competence set:

CS0(A)= traditional knowledge about the rules of horse races and the individual skills of riding a horse.

Po-Lung Yu 112/04/20 2009 NACA--39

Target competence set:

CS= set of skills and knowledge needed to make sure one’s horse crosses the finishing line last.

Candidate A faces a Dis/covering problem The OCS problem of candidate A is

Min t(H) (9)

CS ⊂H(CS0(A)) , H CM

Po-Lung Yu 112/04/20 2009 NACA--40

Resolution: After a puzzling period, A jumped on B’s horse and rode as fast as he could to the finishing line, while leaving his own horse behind. By the time B realized what was going on, it was already too late! Naturally, A became the new chairman.

Let us formulate this solution in terms of operators from CM.

Po-Lung Yu 112/04/20 2009 NACA--41

Step 1. A analyzed the situation using the “deep and down principle”, M16 and the “projection from higher position” M9 to generate ideas, he narrowed down his focus to the most important elements of the race: (A, HR1) and (B,HR2), A and his horse and B and his horse:

M9 o M16 (CS0(A))= CS1(A) (10)

where CS1(A) is the resulting competence set.

Po-Lung Yu 112/04/20 2009 NACA--42

Mathematically, assume that

CS0(A)={a1, a2,…, an } ,

and let , P0(aj), be the initial activation probabilities of aj , j =1,2,…, n, from his potential domain PD to his actual domain AD. Generally speaking, the activation probability can be seen as a time dependent function with domain as the potential domain and range [0,1], that is,

Pt(.): PD [0,1].

Po-Lung Yu 112/04/20 2009 NACA--43

Then the operation (10) reduces to the transformation of the initial activation probabilities P0(aj), j =1,2,…, n to a new set of probabilities Pt(aj), j =1,2,…, n such that Pt((A, HR1)) = Pt((B, HR2)) =1, where t is the duration of operation (10).

This transformation can be expressed as a function

S: {P0(aj) , j =1,2,…, n }[0,1], such that S(P0(aj))= Pt(aj) , j =1,2,…, n .

Po-Lung Yu 112/04/20 2009 NACA--44

Step 2. The problem now is how to change the situation so that it becomes: the one who crosses the finishing line will be the winner?

The solution is to use the “alternating principle” , M17 by “jumping on B’s horse and riding it to the finishing line as soon as possible”. This operation can be formulated as follows.

Po-Lung Yu 112/04/20 2009 NACA--45

The alternating principle changed the pairs (A, HR1) and (B,HR2) into the pairs (A, HR2) and (B,HR1),

Mathematically, denote A, HR1, B, HR2 by 1,2,3,4 respectively and d=(1,2,3,4), the alternating of horses corresponds to the following operation

Po-Lung Yu 112/04/20 2009 NACA--46

d’= F d

Where d’=(1,4,3,2) and

Po-Lung Yu 112/04/20 2009 NACA--47

0010

0100

1000

0001

F

Thus the solution of the OCS (9) can be expressed as:

H*= (M16 o M9) o M17 and

Min t(H)= t* = duration of operation H*

H* corresponds to the mathematical operation FoS.

Po-Lung Yu 112/04/20 2009 NACA--48

DMCS has a great potential of application in Artificial Intelligence, Management, conflict resolution and Scientific Discovery.

5. Conclusion.

We hope that this presentation will trigger more research on this new area.

For more details please read the paper:

Larbani M., Yu P.L., Decision Making and Optimization in Changeable Spaces, a New Paradigm, Journal of Optimization Theory and Application (JOTA), Vol.155, No.3.

Po-Lung Yu 112/04/20 2009 NACA--49

References.

Yu, P.L., 1990, Forming Winning Strategies, An Integrated Theory of Habitual Domains, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo.

Yu, P.L., 2002, Habitual Domains and Forming Winning Strategies, NCTU Press, Hsinchu, Taiwan.

Yu P.L. and Larbani M., 2009, Two-Person Second-Order Games, Part 1: Formulation and Transition Anatomy, Journal of Optimization Theory and Application (JOTA).

Larbani M. and Yu P.L., 2009, Two-Person Second-Order Games, Part 2: Restructuring Operations to Reach a Win-Win Profile.

Larbani M. and Yu P.L.,2011, Second Order Games: A Paradigm Shift in Game Theory, JOTA, Vol. 149, 447-473.


ISOR11

Po-Lung Yu 112/04/20 2009 NACA--51

The End.

Thank you very much for your attention.

* M. Larbani and **P. L. Yu * IIUM University, Kuala Lumpur, Malaysia

Documents

Transcript of * M. Larbani and **P. L. Yu * IIUM University, Kuala Lumpur, Malaysia