**ÓÄÊ: 656.615:004.272**

**Arsenyev**** ****Yu****.****N****.**

**OPTIMIZATION APPROACHES TO THE
UTILITY VALUE AND ASSESSED VALUE OF INVESTMENT WITH THE MULTITUDE OF FEATURES
AND ALTERNATIVES**

*The territorial branch of the ARDIFE
in the town of *

*Consideration of the models of
optimization of investicial and financial objects
with an illustration then on concrete examples are considered, optimization
approaches to the utility value and assessed value of investmentinto
the intellectual capital* *of
the region with the multitude of features* *and alternatives*.

One of the most effective economic and mathematical methods of
investment optimization is the utility value theory with multitude of features
applied for the situation of uncertainty (eng. MAUT-method). It is also applied
for the situations of certainty, emerging in forecasting where it is possible
to apply the value theory with multitude of features instead of the utility
value theory [1-3].

MAUT method helps to solve any multiple goal problem having applied unit
functions of utility value (merit, priority) – UVF, assigned to separate
features and evaluated numerically by the rate of substitution between
features. Cumulative utility, or value, N is defined as UFVn,
assigned to manifestations ai (I = 1,..., k) of the
objective criteria N_{M} (a_{1}, a_{2}, ..., a_{K}) = f[n_{1}(a_{1}), n_{2}(a_{2}),...,
n_{K}(a_{K})].

Multiple goal problems optimization under determinateness conditions is
kept by the MAUT-method algorithm, including: criteria choice; criteria
independence processing; UVF determination under separate features;
equilibration factors determination for the criteria; determination of the
universal utility value of the alternatives. Let us remark its properties.

Under the choice of subjectively evaluated quantitative and qualitative
criteria the primary objective is divided into sub goals in the hierarchical
order: the lowest level of objectives incorporates features which help to
determine the degree of pursuing an objective. Under qualitative features grows
a problem of measurement without ability to apply universally accepted scale.
Under processing the independent criteria, affording to summarize separate
feature properties into a single property they should be integrated into a
summary utility value function.

Under determination UVF n_{k} of
separate features k these functions are defined by detecting essential
properties a_{k} features, then performs a
normalizing of properties n_{k} on the
interval [0, 1], as this takes place the most unsuccessful manifestation a_{k} UFV is
given the value of zero, i.e. n_{k} (a_{k}) = 0, and the most successful – a digit i.e. n_{k}(a_{k}¹) = 1. UFV form in view of
straight or curve lines is set by means of inquiries by the median method.
Under feature C_{1} on the basis of values a_{1}^{0}
and a_{1}^{1} they define average value a_{1}^{0,5}, and also utility value rising from a_{1}^{0}
till a_{1}^{0,5} is identical to rising from a_{1}^{0}
till a_{1}¹. This manifestation a_{1}^{0,5} is assigned to an ordinary property of utility
value 0,5, i.e. n_{1}(a_{1}^{0,5}) = 0,5. To define a_{1}^{0,5} they take second feature C_{2}, but its
manifestation should be modified during the progress of consecutive inquiries
as a result of a_{2} level and
defining the modification D_{a2}, which is equal as for transition from
a_{1}^{0} till a_{1}^{0,5} and for transition
from a_{1}^{0,5} till a_{1}^{1}. According to
this condition we have the following formula:

(a_{1}^{0}, a_{2}^{1})
~ (a_{1}^{0,5}, a_{2}^{1}
- Δa_{2}); (a_{1}^{0,5}, a_{2}^{1})
~ (a_{1}^{1} - Δa_{2}).

During the next phases of inquiry they define median properties (a_{1}^{0,25} and a_{1}^{0,75}) also for the
intervals [a_{1}^{0}, a_{1}^{0,5}] and [a_{1}^{0,5},
a_{1}^{1}]. Possessed properties are sufficient for the
approximate determination UVF n_{1} if its type (linear function) is
known a priori. If UVF n_{1} is - a curve, then it can be determined
according to [3]. Analogously it is
possible to determine UVF (n_{2},..., n_{k}), thereat testing the organic of conclusions.
Often under multiple goal problems the accounting of complete UVF is not
necessary and it’s enough to determine the properties of an ordinary utility
value for the manifestations of essential alternatives.

Equilibration factors of criteria can be determined by the correlation
between equilibration factors appropriately of two properties, moreover these
correlations may be interpreted like substitutions norms. If two linear
objective functions are given, then integrated function of universal utility
value N_{M} has a view of N_{M} = w_{1}n_{1} +
w_{2}n_{2}.

The altitude of the rise dn_{2}/dn_{1} of the concurrent
lines is a norm of the substitution between n_{1} and n_{2}. It
shows how many units should be lessened from n_{2} to try to get common
degree of utility value with the aid of additional unit n_{1}; i.e. dn_{2}/dn_{1}
= -w_{1}/w_{2}. For the correlation between the properties the
rule is correct:

׀Δn_{2}׀w_{2} = ׀Δn_{1}׀w_{1} or
w_{1} = ( ׀Δn_{2}׀
/׀Δn_{1}׀)w_{2}.

Assigned method may be transferred under the
independence of priorities to several objective functions, determining
correlation between w1 and other equilibration factors (w_{3},…, w_{k}). So far as the condition acts w_{k} = 1, then having applied these correlations
it’s possible to comprise an equation, the solutions of which are the
equilibration factors w_{k}, liable to calculation.

Universal utility value of alternatives is determined by recount of
alternative manifestations with the aid of UFV into single properties of
utility value and succedent their accumulation with
regard of equilibration factors (universal utility value may at the most
achieve 1). As this takes place the following rules of profit evaluation are in
effect: IO is absolutely (relatively) profitable, if its universal utility
value property (whose universal utility value property) outnumbers previously
given critical volume (similar property of any submitted for the IO choice).

Let us evaluate the relative profitability of three alternative
investment projects A_{1}, A_{2}, A_{3} by their
primary objective – region intellectual assets (IA) – accounting the following
four objective criteria of the lower level (one criterion from every criteria
group): in terms of investment volume (IV), intellectual assets (IA),
structural capital (SC) and consumer capital (CC).

In the course of choice for alternatives of projects A_{1}, A_{2},
A_{3} we will obtain the following data of the essential features of
the lower level criteria (table 1). The chosen criteria are independent from
each other, thus it’s possible to find the universal utility value function.
Let us derive minimal and maximal manifestations from the table 1 data through
assignment to them appropriately properties of ordinary utility value 0 and 1
(under minimization of the objective function the smallest manifestation
obtains utility value 1). After that let us obtain the following values:

Table 1

Main properties of essential features of the
alternatives

Alternative |
IV, thous. rub. |
IA |
SC |
CC |

A1 |
60,0 |
800 |
15 |
350 |

A2 |
42,5 |
1100 |
12 |
250 |

A3 |
35,0 |
1300 |
25 |
450 |

for IV: n_{1}(35,0)
= 0; n_{1}(60,0) = 1; for IA:
n_{2}(800) = 0; n_{2}(1300) = 1;

for SC: n_{3}(12)
= 0; n_{3}(25) = 1; for CC:
n_{4}(250) = 0; n_{4}(450) = 1.

Let us assume UFV for the feature C_{1} (IV), calculated by the
median method and corresponding indifference evaluations, takes the form:

n_{1}(35,0) = 0; n_{1}(42,5) = 0,5; n_{1}(47,75)
= 0,65; n_{1}(53,0) = 0,8; n_{1}(60,0) = 1.

Hence arises, that for SAP from point 35,0 at
first takes place rather rapid growth of utility value, but after value of 42,5
utility value growth decreases.

For UFV with respect to second feature C_{2} (IA), calculated by
median method, let us find manifestation a_{2}^{0,5},
leading to the determination UFV 0,5 through engaging first criterion IV. As a
result of value a_{1}^{1} = 42,5 let
us check, what change Δa_{1} corresponds to the transition just as
from a_{2}^{0} to the required a_{2}^{0,5}, and
from a_{2}^{0,5} to a_{2}^{1}. Inquiries adhere
to, that this action decreases Δa_{1} = 7,5
(42,5 -35,0 = 7,5), that is confirmed by indifference evaluation:

(42,5, 800) ~ (35,0, 1100)
or (a_{1}^{1}, a_{2}^{0}) ~ (a_{1}^{1}
– Δa_{1}, a_{2}^{0,5});

(42,5, 1100) ~ (35,0, 1300)
or (a_{1}^{1}, a_{2}^{0,5})
~ (a_{1}^{1}, - Δa_{1}, a_{2}^{1}).

We obtain, that n_{2}(1100) = 0,5. As
far as we know UFV for all alternative manifestations, then we enclose UFV n_{2}
analysis.

It is possible to conclude the correlation between equilibration factors
w_{1} and w_{2} from the indifference evaluations. UFV
difference for the IV criterion (between 42,5 and
35,0) Δn_{1} = 0,5.
Compensating difference in utility value Δn_{2} for the IA
criterion also sets up 0,5. By this means, we have the
following correlation:

׀Δn_{2}
׀w_{2} = ׀Δn_{1} ׀w_{1} or
0,5w_{2} = 0,5w_{1}, i.e. w_{2} = w_{1}.

Therefore arises, that IV and IA criteria have
common weighting coefficients: w_{1} = w_{2}.

To determine UFV n_{3} and equilibration
factor w_{3} let us choose IV criterion again, in addition the
following indifference evaluations for IV and CC criteria are in effect:
(53,0, 12) ~ (42,5, 17); (53,0,
17) ~ (42,5, 25), i.e. this
equivalence means, that n_{3}(17) = 0,5. For the correlation of
equilibration factors w_{1} and w_{3} operates the following
rule: ׀Δn_{3}׀w_{3}
= ׀Δn_{1}׀w_{1} or 0,5w_{3} = 0,3w_{1}, i.e. w_{3} = 0,6w_{1}.

To determine ordinary
utility value n_{3} = 15, necessary for the A_{1} alternative
evaluation, let us carry on further indifference evaluations, including this
manifestation: (47,5,
12) ~ (42,5, 15); (47,5,
15) ~ (42,5, 17). According to
it we obtain, that n_{3}(15)n =
0,25.

UFV n_{4} of the correlation between w_{4} and w_{1}
may leave unassigned because of that, ordinary utility value of the important
value a_{4} = 350 UFV n_{4}(350) is
equal to 0,5. The correlation between w_{4} and w_{1} is such,
that w_{4} = 0,4w_{1}.

Taking into account normalizing (w_{1} + w_{2} + w_{3}
+ w_{4} = 1) let us determine: equilibration factors – w_{1}
=1/3; w_{2} = 1/3; w_{3} = 1/5; w_{4} = 2/15; universal
utility value N_{M} of the alternatives N_{M} = 1/3 n_{1}(a_{1}) + 1/3 n_{2}(a_{2})
+ 1/5·n_{3}(a_{3})+2/15·n_{4}(a_{4}).

Inserting the manifestations, we obtain universal utility value of the
alternatives: A_{1} - 9/20; A_{2} - 7/15; A_{3} - 8/15.
Therefore we have that alternative A_{3} is the most profitable for the
main objective.

In the large MAUT-method is theoretically established, gives standard
ranking order with detecting the decision under the range of objective
functions in comparison to utility value analysis methods and ANR under rather
strict conditions and high expenditures on data accounting. Contrary to it
PROMETHEE-method arises from the weaker conditions [1, 2].

Bibliography

1. Arsenyev Yu.N., Davidova T.Yu. Hybrid
intellect systems. Economics. Management. Education. - M.: High
School, 2008. - 566 p.

2. Davidova T.Yu.
Intellectual and Potential potential of the market
agent: management, quality assessment and efficiency methods. - M.: High
School, 2005.