基于非优信息分析的最大次优决策
Maximum Sub⁃optimum Decision⁃making Based on Non⁃optimum Information Analysis

Abstract ：This paper presents a novel approach to sub⁃optimum decision⁃making based on non⁃optimum analysis． Our goal is to provide support for optimization of uncertainty system． Our approach allows the practice courses and results of mankind are classified by their natures into three categories： certainty， sub⁃certainty and uncertainty． It is considered that the non⁃optimum system does not exclude the targets and the results of optimum in practice． The formation of non⁃optimum system serves as the basis for existence of optimum system in uncertainty． Besides， the various characteristics and functions of the optimum system can be measured from the nonoptimum system． Furthermore， by summing the practice， the paper provides maximum sub⁃optimum principle of the system， establishes the conception of sub⁃optimum thresholds and puts forward three theorems about sub⁃optimum parameters． Through the concept of maximum sub⁃optimum， it analyzes the actual significance of system optimization． Based on sub⁃optimum analysis， it puts out the academic idea of suboptimum decision⁃making， it discusses about the general framework of sub⁃optimum．Finally， according to the previous practice of optimization， a kind of method has been developed to approach the optimum from non⁃optimum．

Keywords ：Non⁃optimum category；self⁃organization；sub⁃optimum set；maximum sub⁃optimum； risk analysis

1． Introduction

Ever since nearly half a century， the optimization theory has undoubtedly contributed extensively every branch of science and technology． It is because of its wide use that people find out it is far from actual requirements． People wonder whether ideal model analysis can solve real problems． Furthermore， it is very hard to build up a mathematic model for many of the actual complicated problems． Especially when the system is uncertain， man can only simply build up the model， but can hardly get its solution． Although there are a lot of approximate methods and theories of solving， they are far from satisfaction． The previous system analysis committed that it was impossible to realize optimum under a limited condition of time and resources． At the same time，behind the optimum， there is definitely a series of hypotheses， middle⁃way decisions，and predigesting of data． Under most conditions， the hypotheses of optimum do not exist． Although people have generalized this method to many fields， the results obtained can be only temporary， and sometimes cannot achieve the final goals．

Theoretical researches and actual applications show that it will complicate the problems if only the system optimum is emphasized， because people can hardly confirm the yardstick of optimum． Therefore， intervenient optimum theory ［1］， satisfactory principles ［2］ and third optimum ［3］ were raised one after another， whose aims were to fetch up the shortcomings of optimum theories． However， people are still staying the frame of traditional optimum theory in actual researches， and only provide acceptable ideas and methods of the formats to solve the problems．

A Chinese scholar proposed the theory and method of non⁃optimum systems in 1989 ［4］， which studies the reasons of system non⁃optimum from an inverse angle and builds up the valid approaches from non⁃optimum to optimum． Meanwhile， He Ping（2003） also proposed non⁃optimum analysis theory of systems， which discusses system optimum from the non⁃optimum analysis angle and solves the contradictions in finding system optimum of the past． Thus the cooperative transmission of system is realized in such aspects as conditions and aims， time and resources， recognition and behavior，and hypotheses and existence．

The non⁃optimum analysis theory of systems ［2］， based on the results of the recognition and practice of mankind， establishes the most optimal， sub⁃optimum and non⁃optimal research fields， in order to satisfy the subjective requirements of people and fulfill the objective regulations． The optimum category consists of the most optimal and optimum， which refers to the processes and results of certain （such as the success is derived from certain ［3］）． The sub⁃optimum category is composed of the processes and results of sub⁃certain （ such as the processes and results of satisfaction and acceptance， where a certain degree exists to selected ［4］）． The non⁃optimum category is composed of the processes and results of uncertain（such as the failures and imperfect situations is derived from uncertain ［5］）． Unfeasibility and unreasonableness are typical non⁃optimum． Although being feasible and reasonable in a certain degree， they still belong to the non⁃optimum category． In reality， every uncertainty system belongs to the non⁃optimum category． It meets the recognition and realization of mankind to analyze the causes of non⁃optimum system and the ways to reach optimum from the viewpoint of non⁃optimum category． This way of thinking is abbreviated as non⁃optimum tracing theory， and the theory of researching and tracing non⁃optimum is called nonoptimum analysis theory of system．

In the following years， the research of systems’ non⁃optimum has been developed very fast， both in theory and in practice， which involves non⁃optimum recognition of systems ［6］， evaluation of the optimum and sub⁃optimum solutions ［7］， the nonoptimum measurement of systems ［8］ and the non⁃optimum differentiation and instruction of systems in the engineering areas ［9］． Based on the studies of the past［1-20］， these paper researches deeper the non⁃optimum analysis of systems from the angle of Incomplete Information Theories （ IIT）， build up the framework of suboptimum theory， and confirm that maximum sub⁃optimum principle is a criterion and premise of the decision⁃making of uncertainty system ．

The purpose of this paper is to develop an effective model for sub⁃optimum analysis based on non⁃optimum system ［1-20］． This model uses traditional methods to select relevant features， and classify data． Its principal contribution makes use of an innovative sub⁃optimum analyzed procedure that is activated at certain intervals during the optimization process in order to make use of information obtained during that process． Our approach describes work in progress． It has elements in common with，and may be applied in， areas such as bioinformatics， which make use of small decision making in order to create fitness functions for an objective performance measure．However， the approach has not yet been applied by others to the areas of research，information analysis， and other decision making， which we plan to explore in this paper．

This paper is organized as follows． Section 2 provides a brief overview of the nonoptimum analysis on systems， sets the background for the present research， and shows the architecture of self⁃organization in non⁃optimum category． Section 3 introduces the method of sub⁃optimum analysis． Section 4 provides an example of how non⁃optimum analysis can be applied to risk management． Finally， Section 5 draws a conclusion and further work．

2． Theory and Methods of Non⁃optimum Analysis on Systems

2.1 Background

In this section， we present a brief summary of the optimal theory we have developed for uncertainty decision． We then present an overview of the implementation of uncertainty decision⁃making and our approach for learning new optimal for real⁃world applications．

2.1.1 Optimum methods and relative optimum solution

In the research of decision analysis， the common methods of researchers are optimal theory and satisfactory principle． Although people’ s understanding of optimal concept is approaching actual life and optimal methods are improving， the research of optimal theory is getting mature， the researches of system optimum are still under rational analysis， which don’t reflect the actual process of people’s pursuit to optimum nor actually appraise optimum problem． In fact， in the research of traditional theory and method， optimum problems have premises． That is， in the series of methods solving problems， it exists in a relative way and is a concept of relative optimum． It possesses two ideas： on one hand， among all the problem⁃shooting methods， optimum can realized the pre⁃described aims with a smaller input； on the other hand， it can realize more optimal results than other methods． The former judges the good and the bad with an aim， and the latter with the aim together with the methods． Before adopting the optimum methods， a serious work of comparison， analysis， calculating， reasoning，and designing has to be done， and the process itself is method optimum of the system．This is a distinctive advantage of the optimum methods over the traditional methods．The optimum criteria of the optimum method are objective， because the value of human beings pursuit in all of the actual practices is to obtain the largest value with lowest substantial and mental input possible． In the reality of solving problems and realizing aims， the criteria of optimum are objective， which are the objective criteria of whether the methods and aims are optimal．

In real life， there are no absolute optimums， and only under certain conditions， is there differentiated relative optimum． Relative optimum can be seen as satisfactory result （sub⁃optimum）， because there are a great deal of uncertainty and non⁃linearity as explained by Simone ［5］—there are three defects in the traditional decision disciplines： to ignore the uncertainty of the economic life； to ignore the non⁃linear relationship of the real life； to ignore the subjective limitations of the decision maker．Simone held that in the complicated real world， only a minority of problems can be solved through the calculus of the maximum and minimum value． Sometimes there are not the most optimal solutions at all， and under most conditions people just try to find out a satisfactory approximate solution （ relative optimum solution）． The satisfactory criteria are composed of an upper limit and a lower limit． As long as the solution lies between the two limits， it is an acceptable one．

The satisfactory criteria are composed of an upper limit and a lower limit． As long as the solution lies between the two limits， it is an acceptable one． Thereby the optimum area takes the place of the optimum point， which is paid attention by the scholars． At the same time the goal and result out of the optimum area can be seen as non⁃optical． In reality， it is impossible to keep all the behaviors of the system within the optimal area． That is to say， the size of his non⁃optimum area decides the optimum level and the behaviors of the system in the optimum area are also decided by the control of the system to the non⁃optimum area． Therefore， from this point of view， we can tell that the non⁃optimum of the system decides the optimum of the system．

2.1.2 Uncertainty and fuzzy sets

Artificial Intelligence researchers have explored different ways to represent uncertainty （Russell， Norvig， 1995）： belief networks， default reasoning， DempsterShafer theory， fuzzy sets theory． For the problems we want to solve， the optimal problem task will require a representation that explicitly deals with uncertainty． The evolutionary optimal methods that are employed must be able to work with such a representation． Uncertainty， as well as evolution， is a part of nature． When humans describe complex environments，they use linguistic descriptors of real⁃world circumstances， which are often not precise， but rather“ fuzzy”． The theory of fuzzy sets （Zadeh， 1965） provides an effective method of describing the behavior of a system which is too complex to be handled with the classical precise mathematical analysis．The theory of rough sets （ Pawlak， 1991） emerged as another mathematical approach for dealing with uncertainty that arises from inexact， noisy or incomplete information．Fuzzy sets theory assumes that the membership of the objects in some set is defined as a degree ranging over the interval ［0，1］． Rough sets theory focuses on the ambiguity caused by the limited distinction between objects in a given domain．

Fuzzy sets have been employed to represent rules generated by evolutionary learning systems． Using fuzzy concepts， Valenzuela⁃Rendon （1997） tried to overcome the limitations of the conventional rule⁃based classifier system （Holland， 1975） when representing continuous variables． He used fuzzy logic to represent the results of the genetic⁃based search of the classifier system．

Likewise， fuzzy functions have been used to describe and update knowledge in Cultural Algorithms． First， Reynolds （1994 ） employed a fuzzy acceptance and influence function in the solution of real⁃valued constrained optimization problems．Following the same idea Zhu designed a fully fuzzy cultural algorithm （Zhu， Reynolds，1998） which included a fuzzy knowledge representation scheme in order to deal with the continuous variables （Zhu， Reynolds， 1998） in the belief space， as well as a fuzzy acceptance and influence function． All these approaches were tested on real⁃values function optimization problems． More recently， Jin （2000） used a “fuzzy” knowledge representation for normative knowledge in the belief space of cultural algorithms， to solve the real⁃valued constrained function optimization．

The design of a fuzzy representation system is not an easy job， because the membership functions should be carefully chosen， and the procedures that use these functions should be specified precisely． The problem is to optimize the fuzzy membership functions for a problem and to find optimum plans related to the fuzzy performance measures． It is natural approach to use heuristics （ i． e． evolutionary algorithms） to solve this task． The heuristic optimal comes from Greek and means“to know”，“to find ”，“to discover” or“ to guide an investigation”． Specifically，“Heuristics are techniques which seek good （ near⁃optimal） solutions at a reasonable computational cost without being able to guarantee either feasibility or optimality， or even in many cases to state how close to optimality a particular feasible solution is．”（Russell， Norvig， 1995）

Heuristic optimal refers to any technique that improves the average⁃case performance on a problem⁃solving task but does not necessarily improve the nonoptimum case performance．Heuristic techniques search the problem space“intelligently” using knowledge of previously tried solutions to guide the search into fruitful areas of the search space． Often， search spaces are so large that only heuristic search can produce a solution in reasonable time． These techniques improve the efficiency of a search process， sometimes by sacrificing the completeness or the optimality of the solution．

2.2 Basic concepts

2.2.1 Actual significances of the non⁃optimum

According to the self⁃organizing principle of the system， the development of human society is forever in the dynamic and uncertainty process of moving from the less ordered toward the more ordered larger system， which is toward its destination point cycle． However we must be aware of the hidden danger under the vigorous stream which may bring about slipped up in decision making and failures． Meanwhile we have already suffered a few slip up in decision making and failures in some areas to some extent． What’ s more failures are that some failures suffered have been repeated and what could have been avoided has not．

From the reliability of the economic reform point of view， we think the most important thing in the Chinese economic reform as a new effort is not a problem of optimizing． Rather， it is one of lessen slipped up in decision making and keeping away failures and detours as possible． Even if some model is considered optimum under the present circumstances， it is hard to be a stable one because it is in the midst of a dynamic process with quite a few hidden threats lurching and many horizontal or vertical sub⁃optimum states． So， if we try to set goals for the economic reform， make plans and take measures and advocate some optimum models simply following the optimum thinking methods out of blind subjective wish， we’ll be actually putting the economic reform on an unreliable and unrealistic basis．

So we say that the non⁃optimum thinking and the methods of non⁃optimum analysis on systems with failure⁃avoiding as its basic aim are based on the non⁃optimum facts with its special ways of thinking， information gathering， analyzing and processing， and with the setting up of non⁃optimum system， these methods seek to lessen slipped up in decision making and failures， thus providing a new way of scientifically summarizing past lesson and making them lamppost for the future． There is profound potential for putting the non⁃optimum thinking into use in Chinese economic reform， and in other country’ s practice． Take the non⁃optimum guiding system for example； it can be employed in the economic reform of the country’ s macro policies， financial system as well as decision analysis． To be sure， the establishment of this non⁃optimum guiding system with computer as its means with information processing techniques as its foundation is no easy task．

We expect that the non⁃optimum thinking and the methods of non⁃optimum analysis on systems will grow into a new theoretical branch of decision theory that is system non⁃optimum analysis．Meanwhile， applying the theory and methods of information science and system engineering， the non⁃optimum thinking and the methods of non⁃optimum analysis on systems are still in their primitive stage of development and research as a new branch of learning， thinking and theory， But we believe that it will perfect and sophisticate the course of practice and debate， and will bring about results and gain its own position in the world of science．

2.2.2 Virtues of failure

The theory of non⁃optimum analysis on systems ［He Ping， 2003］， based on the results of the recognition and practice of mankind， establishes the most optimum and non⁃optimum research fields， in order to satisfy the subjective requirements of people and fulfill the objective regulations． The optimum category consists of the most optimum and optimum， which refers to the processes and results of success． The non⁃optimum category is composed of the processes and results of failures and acceptable， imperfect situations． Unfeasibility and unreasonableness are typical non⁃optimum． Although being feasible and reasonable in a certain degree， they still belong to the non⁃optimum category． In reality， every system belongs to the non⁃optimum category． It meets the recognition and realization of mankind to analyze the causes of non⁃optimum system and the ways to reach optimum from the viewpoint of non⁃optimum category． This way of thinking is abbreviated as non⁃optimum tracing theory， and the theory of researching and tracing non⁃optimum is called non⁃optimum analysis theory of system．

We know， the non⁃optimum analysis is derived from the virtues analysis of failure．The idea of the virtues of failure is featured by a historical origin in the Chinese ancient idea history． At the end of the war countries period， in his famous book 《Runs Away Collectives》 ［6］， the famous philosopher， Han Fei probed deeply the lessons of the defeated dynasties and thoroughly addressed 47 omens that led to a country’ s perdition． Meanwhile， he proposed strategies to the governors to avoid losing the reign of the country with his acute ideas． In the chapters of “Strategies”，“Nine Changes”，and of “Terrain” of the famous Chinese ancient military work 《The Art of War by Sun Zi》［7］ and of “ Army loses”，“ By Righteousness”，“General loses” and “General defeated” of 《The Art of War by Sun Bin》［8］， the writers summed up the experiences of war， particularly the lessons of failures before the Spring and Autumn times and the Warring States． These historical literatures illustrated that ever since the ancient times，man has not only analyzed problems in the fields of optimum， but also in the fields of non⁃optimum． Thus the idea of non⁃optimum analysis has a significant position in the history of man’s recognition．

Man’s knowledge of the objective world and of himself is always in the midst of ever deepening course． In human history， there has been many social practice that was carried out partly or even wholly with blindness， thus made it impossible to avoid failures completely． Yet， each of the failures adds to the improvement of human understanding of the objective and subjective world． So those social efforts that have failed occupy important positions in the chronicle of human knowledge． The motto“failure is the mother of success” and“ the virtues of failure” （Ref． ［9］， Research Corporation Newsletter， Winter， 2005） tells that human race learns from setbacks． But it fails in theorization and quantitative analysis．

In the system analysis， there lies always two schools of thinking differences in nature： one is the regular thinking， which goes along with the existed thinking pattern；the other is the reversed thinking， which goes against the existed thinking pattern． The history of scientific research shows that the regular thinking pattern might easily cause people to be rigid and stubborn， which leads to failures of scientific research； while the reversed thinking can enlighten scientists and leads to successes of scientific research．The non⁃optimum analysis of systems is created by the reversed way of thinking．

In fact， the concept of non⁃optimum is quite comprehensive ［4］． From the viewpoint of systems’ entity， non⁃optimum means unfeasible and unreasonable； from the viewpoint of systems’ behavior， it means non⁃ideal and non⁃good； from the viewpoint of systems’ capacity， it means ineffective and abnormal； from the viewpoint of systems’ change， it means obstacles， disturbance and influence． There exists a serious of non⁃optimum problem from the entity of the system to the change of the system， which causes non⁃optimum problem． As to every kind of decision⁃making problems， there is the individual non⁃optimum category as well as the common nonoptimum attributes． The so⁃called individual non⁃optimum attribute is decided by the characters of the system， while the common non⁃optimum problem is an objective entity．

From the system analysis point of view， all problems exist both in an optimum attribute and non⁃optimum attribute． In the past， people mainly studied how system operate under optimum conditions and how they can become optimum． Yet this optimization is only relative and on many occasions the conditions for it is uncertain and unattainable． So the efforts of seeking this kind of optimization are rather blindfolded，and facts have proved that merely controlling some of the conditions cannot keep the system free of non⁃optimum attribute， because some or even all of the variables to satisfy a optimum attribute may become non⁃optimum， and on the other hand conditions made for the non⁃optimum attribute may help to optimum．

2.3 Theories and methods

2.3.1 A new self⁃organization theory

In the research of the self⁃organization theory of systems， the transmission of order and non⁃order is a core question． The Theory of Dissipation Structure （ I． Prigogine，1977）， Synergetic Theory （ H． Haken， 1979） and Chaos Theory contribute a great deal to it． According to the theory of Dissipation Structure， as long as the system is open， the non⁃equilibrium state may become the source of ordered system． So nonoptimum is the source of non⁃ordered system． Only when the system goes out of the non⁃optimum category， can it come into the ordered stage， where we are to seek the optimization （See Fig． 1）．

In fact， their individual theories include non⁃optimum theory of the system ［9］．Because the major character of the self⁃organization of the system is to perfect the running of the system， develop its goals， they have to experience from non⁃optimum to optimum， and from optimum to non⁃optimum． If the system is not featured with this non⁃optimum， it doesn’ t need self⁃organization either． Analysis shows that systems always stay on the transmission of optimum and non⁃optimum， and the aim of selforganization is to bring the system from the non⁃optimum to the optimum

There is a time limitation on the system’s stay in the optimum category． Within a certain time， because the system is stable， it stays in the optimum category． However，if the system is not stable， it will soon move from the optimum category to the new border and cause a sustained situation of the system． The sustained situation is neither a developing situation， nor an ideal situation． Of course， the actual angle of the system doesn’t have the most optimum criteria， and it is also not necessary to make sure what is most optimal． As long as the system can shorten the time of moving from the nonoptimum category to the border and from the border to the optimum category， the system is satisfactory．

The attainability of the objective of the system shows that the distance between the recognized goal of the system and the actual goal of the system is acceptable． The achievability of the function of the system refers that the actual functional resources are near to the objective⁃required resources． The controllability of the environment of the system refers to the self⁃organizing capacity or the order parameters achieving the permitted value．

The so⁃called non⁃optimum system is decided by the non⁃optimum attributes；hence the definition of the non⁃optimum system is illustrated through the following methods：

Suppose S represents a system， no matter it is an optimum system or a nonoptimum system， they are all composed of system objective O ， system characteristic F and system environment E ． As to the system， if the structure S ＝ Л （ O ， C ， E ） of the system composed of objective O ， characteristic C and environment E meets the following conditions： （1 ） The objective of the system can be selected； （2 ） The characteristic of the system can be recognized； （3） The environment of the system can be controlled． Then this system is called an optimum system， it is called a nonoptimum system， otherwise．

In fact， every system exists in a non⁃optimum category． Due to the needs of the system， certain conducts and functions of the system come into being， which are confirmed by the non⁃optimum category． So the non⁃optimum cases under different conditions are difference． Analyzing the general laws behind the system’ s movement，we can sum up three different non⁃optimum types：

（1） Systems formed from the changed states of the systems’ old self in the process of system movement． The former constraint conditions are no longer in keeping with the operating conditions of the new systems， because the systems now operate in the nonoptimum．

（2） Systems formed because of changes in constraint factors and new constraints can no longer satisfy the operation of the systems．

（3 ） Systems formed from changes in both the system’ s own states and their constraints， operating in new conditions and thus making it impossible to determine their laws． Then the systems move in the non⁃optimum category．

Suppose O _r is recognition goal of the system， O _s acts as the actual goal of system， α represents the difference of the value between O _r and O _s ， which shows the degree of acceptance of the goal of the system． G _s acts as the system’ s actual functional resources， G _r acts as the resources demanded by the system’ s objective， and β expresses the functional measurement value between G _r and G _s ； The entropy of the actual system e ≤ γ ， and γ expresses the system’s standard entropy． Thereby for the system’s sub⁃optimum structure Л （ O ， G ， E ）， if there are ε ， ζ ， η ，（ random minimal discrepancy can be accepted） causing ｜ α － α ₀ ｜ ≦ ε ，｜ β － β ₀ ︱≦ ζ ，｜ λ － λ ₀ ｜ ≦ η to hold at the same time， the system S is an optimal system． α ₀ ， β ₀ ， λ ₀ is the border value of the system’s optimum and non⁃optimum， and thereby the gathered assemble Л （ α ₀ ， β ₀ ， λ ₀ ） is the criteria of the system’s non⁃optimum analysis． In the actual system analysis， under certain selected standards （ ε ， ζ ， η is known）， for α ， β ， λ ， when man can’t obtain α ₀ ， β ₀ ， λ ₀ ， the system S is called non⁃optimum system．

From the non⁃optimum analysis theory of systems， it can be concluded that people need the controllable order of the system， and non⁃optimum can also be more orderly．From the non⁃optimum reference system， the transit of the system from non⁃order into order as well as the requisites of the transit can be estimated． The non⁃optimum theory of systems will be widely used in the decision sciences． It can often transform people’s experiences into scientific means and might set up reference models with behavior attributes in the control system． This kind of model can marry the experiences and the theories， and can make actual judges to the running path of the system．

2.3.2 Decision making based on non⁃optimum analysis

Simone indicated that the traditional optimization theory has certain defeats and proposed sub⁃optimum theory （satisfactory criteria） ［2］． However， he didn’t analyze why the uncertainty， non⁃linear and subjective condition can bring difficult to optimization problem． The uncertainty problem is classified into three research categories： fuzzy of conception description， random of event， and uncertain in subjective cognition． It is well known that the major problem in real world is uncertainty． Therefore， the optimization problems are under uncertainty． From the traditional point of view， fuzzy optimization， stochastic optimization and optimization under uncertainty are all research contents of uncertainty optimization． Although stochastic and fuzzy optimization has achieved satisfactory results in theory research and practical application， the optimization problem under uncertainty has not really been resolved ［5］．

In real life， to some extent， the features of the optimum and non⁃optimum can be judged and controlled based on experience． In fact， the category of optimum and nonoptimum features depends on recognition degree． Some features are both optimum and non⁃optimum． For instance， there is acceptable and unacceptable aspect when the decision is made． At the same time， there exist satisfactory results， as well as unsatisfactory ones， et al． The uncertainty decision arises during balancing those optimum and non⁃optimum features． Thus， if a problem has optimum and non⁃optimum feature at same time， it is called sub⁃optimum problem under non⁃optimum analysis［4］． That is to say， people always exert the optimum features in the light of different uncertainty decision condition， and knows more about how to overcome non⁃optimum features． In fact， from the integrated view of traditional epistemology and set theory，the result is either optimum or non⁃optimum． It is optimum from one point of view while is non⁃optimum from the other one． Therefore， the judgment of optimum and nonoptimum is an uncertain choice based on partition． For example， in the economic analysis， every economic behavior can be seen as a gambling activity based on the analysis of the optimum and non⁃optimum． People differ in cognition of optimum and non⁃optimum features． In traditional optimization theory， the standards of optimum attribute are expressed by mathematical models． Although theoretic researches and their applications have been deep developed， it is still difficult to solve the realistic decision problems． The primary cause is uncertainty of optimum feature and the existence of non⁃optimum ones． The decision process is based on the comparison of optimum and non⁃optimum．

3． Extension Sub⁃optimum Decision⁃Making

3.1 Concept of extension sub⁃optimum

The traditional sub⁃optimum theories discuss the optimum problems under the condition of free borders and fixed borders， which actually reflect subjective and objective restrictions． Whatever the subjective and objective restrictions are， the optimization under satisfactory principles conditions is called sub⁃optimum． Suboptimum is a conditional optimum， which is different from sub⁃optimum and satisfactory principles． In fact， in the recognition and analysis of problems， we obtain a choice through the recognition of optimum and non⁃optimum， and the meaning of non⁃optimum analysis lies in confirm of their borders． The borders of optimum and non⁃optimum reflect their co⁃existence， which is the bases of the transmission of the system， and is called extension sub⁃optimum． The states and behaviors of a system stay mostly in extension sub⁃optimum． The so⁃called optimum and non⁃optimum are only temporary，which are the recognition and selection of the system to sub⁃optimum process．

3.1.1 Self⁃learning process

If the system is able to realize the transit， it has a good self⁃learning capacity． As is known from the self⁃organization theory， profound changes will not influence the system， and only the huge changes composed of profound changes might cause the evolution of the system． This conclusion can make the sub⁃optimum control of the system effective， and the system will stay naturally in the optimum category， or on the border． People can achieve the self⁃learning function on the border， e． g． the organization is open， and exchanges energies with the outside． Thus the function and behavior of the system change and new non⁃optimum control comes into being． Then，the system goes back to the optimum category． The self⁃learning through coordination and super⁃circulation can let the system replicate and consummate itself and reach the optimum category． （It still needs be emphasized that the optimum category shows the category that can be controlled by the system’s sub⁃optimum）［12］．

3.1.2 Experience analysis

There are three attributes of the recognition to the sub⁃optimum problem：experience， intuition and knowledge．The attribute of experience reflects the recognition to the characteristics of the object’ s behavior． Here the selection of the factors of the decision⁃making is discussed from the experience attribute’ s viewpoint．System experience provides extension sub⁃optimum for the problem．When the recognitions are different， the sub⁃optimum is different as well． The tracing to the system’s conditions of the past can propose a extension sub⁃optimum．

In an artificial system， different people have different behaviors and stories， thus different experiences． Sometimes experiences are called a kind of recognitions； but as the level of recognition is different， the experience of the decision is also different． The sub⁃optimum of the decision is selected and decided by the experience of the decision，and the reasonability of the experience’ s selection is also a meaningful question for discussion． For example， the increase of the function of the system can reduce the nonoptimum category， and the changes of the system’ s behavior can cause new nonoptimum factors， which change with the system’ s behavior． Thus the sub⁃optimum of the actual system is composed of non⁃optimum attribute， the amount of non⁃optimum changes and the potential non⁃optimum factors． （See Fig． 2）

Under the prerequisites of the formation of the problem’ s experience， there is a process of recognition to the non⁃optimum， which is a self⁃organized and selfaccustomed process． Natural non⁃optimum is an objective entity， which does not change with people’s will． However， when people get hold of the basic characteristics of the non⁃optimum， they can set up certain functions to avoid the non⁃optimum， which is not the main subject of the non⁃optimum analysis theory of the system． From the creation to the death of the system， there is an overall running procedure． In fact， a whole， standard running condition does not exist， and also breaches the development regulation of things． From the viewpoint of the dialectic from recognition to entity， this also accords with the entity and recognition to the non⁃optimum． For example， as a decision⁃maker of a concern， one first needs to do a series of work related to the management of the concern and the strategic development objectives． That is to say， to find out what methods to take， what problems to solve and what difficulties to conquer，the key to finish this series of work is to correctly find out the non⁃optimum problem that exists meanwhile with the objectives． Of course， these non⁃optimum problems are formed by direct experience， indirect experience and partial hypotheses． Mentioning hypotheses， people might ask： can hypotheses be hypotheses？ Can they be replaced？These doubts are unnecessary． The actual research shows that if there is no hypothesis，there is no affirmation； to accept a hypothesis is to confirm； the acceptable effect is in the direct ratio of the affirmation． Most of the chemical systems are set up on hypotheses， whose importance is obvious． Mathematics is also the conclusions made by logistic reasoning inference discursiveness discussions based on hypotheses． There are all kinds of hypotheses in economic systems as well， but the hypotheses of economics are not repeatable． When the systems are different， the hypotheses are different． Also the hypotheses of time t _j is not valid in the time period of t _i ． For example， an investor decided his goal and hypotheses in the market of t _i ， When t _i changes to t _j ， the original goals and hypotheses may not hold． Thus， the dynamic characteristics of the goals and hypotheses cause the fact that he cannot make definite answers to this investment under any conditions． The major reasons are brought out by the non⁃optimum problems．

To understand the entity and the accountability of the hypotheses is a tracing to non⁃optimum problems of the system， where experiences are most important． The experience system works with the experience environment， which is the base of the entity and development of the experience system， which in turn works on and influences the environment．

The tracing to the phenomena of the non⁃optimum system has its own nature and regulation， which has a close relationship with the nature and regulation of the system’s experience． Experience is people ’ s conclusion， improvement and accumulation through the recognition， enhancement and control of systems． Experiences develop with human being’s entity and development， and the optimization of experience is one of the most important elements in the development of the society in the history． When experiences possess certain scientific value and form a certain system， they turn into knowledge． Thus， there is a process when experiences transform into knowledge．

3.2 Approach of sub⁃optimum analysis

In the non⁃optimum analysis ［9］， we introduced the concept of sub⁃optimum degree， through which we can describe any factor in system and tell whether it belongs to optimum， non⁃optimum or border zones． Meanwhile， the factors belonging to one zone can be put into different layers according to the sub⁃optimum degree． According to quantitative expressions， we can give the following definition：

Definition 3.1 Let U ＝ ｛ u ₁ ， u ₂ ，…， u _m ｝ be a set of objects （ or a case， or an event ）， P ＝ ｛ p ₁ ， p ₂ ，…， p _n ｝a set of properties （or attributes） and R _S a binary relation defined between O and ， where O ＝ ｛ o ₁ ， o ₂ ，…， o _n ｝ be a set of optimum attributes of u ， be a set of non⁃optimum attributes of U ， then there must be a set of where j ＝ 1，2，…， n ， then R _S be called a set of suboptimum relationship of U ．

In fact， the sub⁃optimum sets is a relationship between optimum attributes and non⁃optimum attributes. Thus， this relationship can be represented by a table where each row represents， for instance， an object， a case， or an event． Every upper column represents an optimum attribute， every below column represents a non⁃optimum attribute that can be measured for each object； it can also be supplied by a human expert or user． The table is called a sub⁃optimum information table （See Table 1）．

Definition 3.2 ． Let ， u U be a value set of optimum attribute in project sets U ， be a value set of non⁃optimum attribute of project sets U ， then V _S （ u ） be a value sets of project sets U based on optimum attribute and non⁃optimum， that is， mapping V _S （ u ）→［－1，1］， u → V _S （ u ）（ u U ） is called a value sets of sub⁃optimum of U ．

The so called value of sub⁃optimum V _S （ u ） in the objects U under a restraint is indicated to provide a real number V _S （ u ）for a u U ， by which the relationship of u and P is described． In the sub⁃optimum selection and analysis， if a sub⁃optimum subset V _S （ u _i ）was given in the objects set U ， an attribute in U ， which does not belong to V _S （ u _i ）， must belong to V _S （ u _i ＋ 1 ） in the classical mathematics． But attribute in V _S （ u _i ） consists of two kinds of attribute， between which there is the difference in innate character． In order to describe this relationship， we are establishing the concept of extension sub⁃optimum set：

In the actual analysis， under certain optimum interval of aims and results， the premise is to find out the correspondent non⁃optimum interval． Thus a sub⁃optimum interval is decided by the optimum and non⁃optimum interval and if the results of decisions fall in this interval， we can call them extension sub⁃optimum， which decide the optimum degree of the system． For example， in the strategic analysis of a system，we need to go through aim—cause and result—choice three procedures．The conventional methods of theoretical researchers are to build up strategic models， obtain analysis results through quantitative index and the aim of non⁃optimum analysis is to build up its quantitative express methods．

3.3 Maximum sub⁃optimum analysis

Definition 3.3 Let U be a nonempty project set． A sub⁃optimum degree （ SOD，for short） μ _S （ u ） is an object having the form and v _S ： U →［－1，1］ where the functions and denote respectively the degree of optimum （ namely v _o （ u ）） and the degree of non⁃optimum（namely ）．

Definition 3.4 If V _Si ⊂ V _S ， then is called the subset of sub⁃optimum V _S ．

Definition 3.5 （Sub⁃optimum topology space） Let τ is a family of subset of suboptimum in U ，we have （1）Ø； V S τ （2）； for any ；（3） If （ αI ） where I is any index set， then In this case τ is called a suboptimum topology of U ， that is （ U ， τ ） is called the topology space of the sub⁃optimum．

There are optimum attribute and non⁃optimum attribute to any decision problem，we have：

Definition 3.6 Let λ be an index set of optimum degree in V _S ， and λ ＝ ｛ λ ₁ ，…， λ _l ｝，then is called λ ⁃optimum if there is a minimum limitation．

Definition 3.7 Let η be an index set of non⁃optimum degree in V _S ， and η ＝ ｛ η ₁ ，…， η _l ｝， then or ， u U ｝ is called η － non⁃optimum if there is a maximum limitation．

Theorem 3.1 Let U be a closed set definition in metric space， effect function v _o （ x ）， of optimum and non⁃optimum is homeomorphism mapping in U for any u U （ decision making）， and there are maximum optimum degree and minimum nonoptimum degree in U ， namely max v _o （ u ）， min ， then

（1） CS ＝

（2） RS ＝ max

（3） DS ＝ min

（4） ES ＝ min

Where CS ： the sub⁃optimum of can support； RS ： the sub⁃optimum of minimum risk； DS ： the sub⁃optimum of maximum effect； ES ： the sub⁃optimum of maximum difference．

Proof Let v _o （ u ） and have maximum and minimum respectively in U ：

（1） What is called the can support range of decision⁃making， in the decisionmaking process， is required optimum attribute value of decision⁃making program is greater than or equal to the minimum limitation value λ ， and non⁃optimum attribute absolute value is less than or equal to the maximum value of sustainability．

“⇒” Suppose ， but

let be homeomorphism mappings， so

namely （ O _{λ ）. Therefore， hypothesis does not hold．}

“⇐” Suppose be homeomorphism mappings，so are one⁃one mappings， then

Therefore， the decision⁃making range of can support is as follows：

（2） What is called the minimum risk decision⁃making， in the can support range of decision⁃making， is that if the maximum of corresponding to optimum attribute values is chose from the minimum absolute value of non⁃optimum attribute values． Meanwhile if it has a corresponding value in O _λ ， there exists the minimum risk value in the corresponding decision⁃making project． According to the known condensations， it exists the minimum value to in U ， and the minimum value is in ，then we will prove that there exists the minimum risk decision⁃making in O _λ ．

According to the conditions of Theorem 3.1， we have the minimum value of ｜ v—o（u） ｜ ， and from ［Ref． ［8］］． Suppose that there exists a minimum value at least in ．Let U be a closed set， according to ［Ref． 17，19］， are all closed sets， and based on Definition 3.4 and Definition 3.5， we know O _λ ， are closed sets of O ， respectively． （ O _{λ ） are closed sets from ［Ref． 19］， so} is a closed set．

Suppose and is a minimum value in ，the optimum attribute value v _o （ u ₀ ） is existed in corresponding to the minimum from Definition 3.1． Assume that the minimum have a sequence ，（ m M ）， satisfy ，（ m →∞ ）； Because is continuous， we obtain （ v _o （ u _m ））＝ u _m → u ₀ ＝ . As V _o is continuous， and has a corresponding sequence ｛ F _o （ u _m ）｝ in O _λ ， satisfying v _o （ u _m ）→ v ₀ （ u ₀ ）， because O _λ is a closed set，we have v _o （ u ₀ ） O _λ ， at the same time O _λ is a homeomorphism mapping， we obtain u ₀ （ O _λ ）. Hence， . Therefore， the minimum risk decisin⁃making is as follows：

（3） What is called the maximum effect decision⁃making， in the can support range of decision⁃making， is that if the minimum absolute value of corresponding to nonoptimum attribute values is chosen from the maximum of optimum attribute values．Meanwhile if it has a corresponding value in ，there exists the maximum effect value in the corresponding decision⁃making project．

According to the conditions of Theorem 3.1， there exists the maximum value to v _o （ u ） in U ． We have O _λ ＝ ｛ v _o （ u ）⩾ λ ： λ ＞ 0， u U ｝ from Definition 3.6． Suppose that there is a maximum value at least in O _λ ． Similarly， we can replace minimum with maximum in light of above（2）， it is easy to rewrite in the remaining part of the proof，hence， the maximum effect decision⁃making is as follows：

（4） The maximum difference decision⁃making： thinking about many decisionmaking projects， High⁃efficiency associated with high⁃risk． Optimum effect curve increases in direct proportion with non⁃optimum effect curve． Sometimes， in the can support range of decision⁃making， the price needs to be as small as possible for the greatest possible benefits． Namely， the maximum difference is chosen between optimum attribute value and non⁃optimum attribute value． Now we will prove there must exist the maximum value according to the conditions to the Theorem 3.1

Suppose ， when u _n → u ₀ ， there exists a sequence（ v _o where

As is a closed set in light of above proof process， we can obtain u ， and then

Because O _λ ， O— _η are all closed sets， we have ，i． e． there exists max ． Therefore， the maximum difference decision⁃making is as follows：

The theorem is proved．

4． Examples of Application

Whether an enterprise proceeds with product development is based on their technical ability to support the enterprise and economic ability to support the enterprise． The technical support ability depends on the integrated power about technical personals， equipments and environment of scientific research and it is limited for any enterprise． New product development is a technical exploration working for itself， and it has immense risks． An American study found that the failure rate of new product is as follows： consumer goods occupies 40％， industrial products is 20％ and services accounts for 18％［10］. By this token， the failure rate of new product is relatively higher． The main reason is that it only analyses optimum attribute of decisionmaking， and does not analyze non⁃optimum attribute under the existing optimum attribute in the program design of new product development． Then， we can obtain new product development strategy by using of sub⁃optimum analysis method， according to the practical of new product development in an enterprise．

For the enterprise， one of the keys of enterprise management decision is how to select a reasonable project of new product development． Let P ＝ ｛ P ₁ ， P ₂ ， P ₃ ｝ be a strategy set of new product development， where P ₁ ： innovate product， P ₂ ： improve product， P ₃ ：fake product． Let P _i ＝ w （ ε ， g ） （ i ＝ 1，2，3） be each of decision⁃making projects， ε ＝ T （ R ， S ， H ） denotes cooperative level of business technology， where R ：technical personnel， M ：scientific equipment， H ：the environment of scientific research．Suppose g ＝ F （ u ） is scale of fund about business invested．

For new product development， the traditional optimum decision⁃making showed that maximize economic benefits with minimum capital investments for determinate cooperative level of business technology． Namely， P _i ＝ w （ ε ， g ） →max （ i ＝ 1，2，3） when g →min ｜ _ε ．

In fact， it is very difficult to let F →min and P _i ＝ g （ T ， F ）→max under the certain ε condition in actual decision⁃making process． This is a realizable issue of limited rational decision⁃making． For example， America has undergone a serious financial crisis， because they completely depend on rational optimum decision⁃making method in decision⁃making problem of financial product innovation．

From the sub⁃optimum analysis of new research point of view， we not only analyse optimum attributes of ε and g ， but also think about unfolded non⁃optimum attributes in the basis of the deferent select of optimum attributes. When we research that the decision⁃making project， with inclusion of technology cooperation level ε and capital investment scale g ， is reasonable and satisfactory． The so⁃called optimum attribute is beneficial to a certain decision⁃making projects， and the opposite is called non⁃optimum attribute． These two properties are interrelated． That is to say， different non⁃optimum attributes will come into being if different optimum attributes are selected． For instance， we can obtain the evaluation about optimum and non⁃optimum attributes from two sides of technology and capital in allusion to three kinds of product development strategy． This can be shown in Table 2.

For an innovative product， its technology cooperation ability attains λ ＝ 0.55 at least （ namely， it invests at least 55％ of the total in the enterprise technology resources）． Resultantly， there exists the deviation between rational analysis and practicality about technology cooperation ability． The reason of deviation is nonoptimum attribute caused by uncertainty （ for example， technology research program，personal behavior influence， etc． ）． Therefore， the non⁃optimum attribute decides the size of failure risks （ the non⁃optimum effect ability is less than η ＝-0.8 at least）．According to the research in American financial market， the innovative products occupy homeland market more than 50％， enterprises may be profitable． Namely， market share is λ ＝ 0.5. Then， afford ability of capital investment companies is less than η ＝｜-0.9 ｜ （It is less than or equal to 90％ of unforeseen capital）．

Technology sub⁃optimum decision⁃making：

According to the known conditions in Table 2， we can get μ _εO （ x ） ＝ 0.4 x ＋ 0.3， ＝-0.4 x －0.3 （Geometric meaning can be seen in Fig． 3）．

We obtain the following expressions by Theorem 3.1：

1）Can support decision⁃making

2） Minimum risk decision⁃making

3） Maximum effect decision⁃making

4） Optimum difference decision⁃making

Based on the Table 2， we can gain the capital decision⁃making function in the same way．

v _εO （ u ） ＝ 0.4 u ＋ 0.2， ＝-0.5 u －0.3

Its result is 0.65 only from the optimum difference decision⁃making point of view，so that the overall decision⁃making project is

u ＝ （0.25，0.85）∩0.65 ＝ 0.65．

As a result， the enterprise does not have completely conditions of innovative product， and it should adopt the product development strategy between innovative product and improved product．

5． Conclusions and Future Research

As we all known， the previous sub⁃optimum analysis does not get rid of the framework of optimum theories． The analysis results of sub⁃optimum are regarded as a research perspective． In fact， the keys of uncertainty in decision⁃making are how to analyze and control the uncertainty of objects and attributes． It shows that by analyzing practical problems， we can determine an optimum target or optimum result. That is to say， decision⁃making problems belong to optimum category．If decision⁃making problems have complete uncertainty， we cannot ascertain the optimum character of objects and attributes， namely， decision⁃making problems belong to non⁃optimum category． Whereas， not only completely certain decision⁃making in optimum category but also completely uncertain decision⁃making in non⁃optimum category both have no actual research significance． However， it has practical significance that researching about limited certainty and limited uncertainty． The so called the uncertain decisionmaking is a kind of sub⁃certainty decision⁃making with different certainty and uncertainty． Studies have shown that a certain decision⁃making is sub⁃optimum under the conditions of optimum and non⁃optimum． The theoretical meaning of sub⁃optimum research is that it provides a valuable methodology for the theory of system cooperation，development of dissipative structures theory and actual uncertain problems， and it has great application value in economic decision⁃making theory． In future studies， for U in the attribute domain ， the necessary and sufficient conditions for the realization of trusted sub⁃optimum is existence of additively optimum attribute and controllability non⁃optimum attribute， where O and are optimum attribute and nonoptimum attribute respectively． How to build the research framework of the largest suboptimum analysis， and how to choose the optimum and non⁃optimum of decision⁃making project both of them are preconditions for research in this area． In fact， this paper gives basic principles to realize trusted sub⁃optimum on incomplete information system． The correctness and feasibility are obvious． It is important to note that there involves some following fundamental problems．

（1） is realized optimization on its attribute domain；

（2） the construction of additively optimum attribute；

（3） the evaluation of controllability non⁃optimum attribute；

（4） the analysis of ．

The above four problems construct the research framework of trusted optimization problems． Every problem has its own research contents， at the same time， which is interdependent and interactional．

It is known that human brain， which can obtain， process and evaluate information， is a system of information processing． The trusted sub⁃optimum pattern is constructed along with the experiences of information processing and extents to use knowledge． This sub⁃optimum pattern can effectively control the decision process， and decrease the risk of decision making． A trusted sub⁃optimum system can be constructed by self⁃organization and self⁃learning function． Then what is brain⁃trusted sub⁃optimum pattern？ In fact， trust is built based on comparison of optimum and non⁃optimum． The trusted sub⁃optimum learning system （TSOLS） （trusted system for short） is constructed by all the attributions and functions of trusted optimization．

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