2742 As systematic approach to engineering matters, the performance evaluation system is proposed and examined theoretically by using mathematical model. The systematic and theoretical approach to humanity is described. In the long history of human activity, engineering, culture, tradition, customs, life style, language have been formed gradually based upon politics, economics, natural and social environments. In usual, facility (F) behaves and performs a certain interaction (I) under some environments (E). This general phenomenon (physics/chemistry) is due to nature laws and also applies to a general social phenomenon and human activity. Above F,E,I are considered to be primary elements of basic system V(F,E,I).The performance of V(F,E,I) is evaluated as a result of phenomenon. As rating index (p), five elements are defined: time(t), space(x),money(m), humanity(h), quality(q). Basic system V(F,E,I) is expressed in form of V(t,x,m,h,q) because of having rating index built-in. Performance evaluation system is formulated by mathematical model (partial differentiation form )of δV(F,E,I)/ δp. Primarily, it is revised to organize the basic system V(F,E,I) ,then build each hierarchy in detail, integrating independent phenomenon. 1)Partial cause /effect analysis : δV(F,E,I)/ δp= δV1(F)/ δp+ δV2(E)/ δp+ δV3(I)/ δp . 2)Primary evaluation: δV/δp (gradient/grade), quick/slow (t), large/small (x), tough/fragile, strong/weak (q), beautiful/dirty, bright/dark (h), expensive/cheap, rich/poor (m). 3)Secondary evaluation:δ2V/δp2 (acceleration/inertia/potential), life evaluation (t), spread characteristics, broad spectrum evaluation (x), safety, reliability evaluation (q), public opinion, reputation, use-related evaluation (h), money making characteristics, economic evaluation(m). 4)Multifarious evaluation δ2V/δp1δp2: System V is revised from different viewpoints. δ2V/δmδt: change of stock prices. δ2V/δhδt: reputational future risk in time history. 5)Sequence order of evaluation time: The decision making is handled depending on a situation to develop one by one. The conclusion highly depends on time processing. 6)System V is classified to be function separated type and function integrated type, which results in big influence on performance evaluation in decision making（δ2V/δp1δp2 type）. As the two-dimensional（X,Y）problem, the expression method of block diagram is discussed. It should be orthogonally designated by independent phenomenon each other. In X-Y axis, time(t),space(x), money(m),humanity(h),quality/quantity(q) are usually chosen as the rating index which are mutually exclusive and independent phenomenon each other. As a model, a risk diagram (occurrence probability-hazard relation) is used. In which for X-Y axis, rating index m/t are orthogonally designated. Furthermore, division of risk category A,B,C,D are made as risk matrix and used for risk management/control. The shape of this block domain highly depends on nature law (probability density function). The shape factor k has some properties: 1) ｋ>1,too active/top heavy type,2) k=1, stable/natural type,3) k
