: 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 Tula

 

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 NM (a1, a2, ..., aK) = f[n1(a1), n2(a2),..., nK(aK)].

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 nk of separate features k these functions are defined by detecting essential properties ak features, then performs a normalizing of properties nk on the interval [0, 1], as this takes place the most unsuccessful manifestation ak UFV is given the value of zero, i.e. nk (ak) = 0, and the most successful a digit i.e. nk(ak¹) = 1. UFV form in view of straight or curve lines is set by means of inquiries by the median method. Under feature C1 on the basis of values a10 and a11 they define average value a10,5, and also utility value rising from a10 till a10,5 is identical to rising from a10 till a1¹. This manifestation a10,5 is assigned to an ordinary property of utility value 0,5, i.e. n1(a10,5) = 0,5. To define a10,5 they take second feature C2, but its manifestation should be modified during the progress of consecutive inquiries as a result of a2 level and defining the modification Da2, which is equal as for transition from a10 till a10,5 and for transition from a10,5 till a11. According to this condition we have the following formula:

(a10, a21) ~ (a10,5, a21 - Δa2); (a10,5, a21) ~ (a11 - Δa2).

During the next phases of inquiry they define median properties (a10,25 and a10,75) also for the intervals [a10, a10,5] and [a10,5, a11]. Possessed properties are sufficient for the approximate determination UVF n1 if its type (linear function) is known a priori. If UVF n1 is - a curve, then it can be determined according to [3]. Analogously it is possible to determine UVF (n2,..., nk), thereat testing the organic of conclusions. Often under multiple goal problems the accounting of complete UVF is not necessary and its 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 NM has a view of NM = w1n1 + w2n2.

The altitude of the rise dn2/dn1 of the concurrent lines is a norm of the substitution between n1 and n2. It shows how many units should be lessened from n2 to try to get common degree of utility value with the aid of additional unit n1; i.e. dn2/dn1 = -w1/w2. For the correlation between the properties the rule is correct:

׀Δn2׀w2 = ׀Δn1׀w1 or w1 = ( ׀Δn2׀ /׀Δn1׀)w2.

Assigned method may be transferred under the independence of priorities to several objective functions, determining correlation between w1 and other equilibration factors (w3,, wk). So far as the condition acts wk = 1, then having applied these correlations its possible to comprise an equation, the solutions of which are the equilibration factors wk, 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 A1, A2, A3 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 A1, A2, A3 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 its 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: n1(35,0) = 0; n1(60,0) = 1; for IA: n2(800) = 0; n2(1300) = 1;

for SC: n3(12) = 0; n3(25) = 1; for CC: n4(250) = 0; n4(450) = 1.

Let us assume UFV for the feature C1 (IV), calculated by the median method and corresponding indifference evaluations, takes the form:

n1(35,0) = 0; n1(42,5) = 0,5; n1(47,75) = 0,65; n1(53,0) = 0,8; n1(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 C2 (IA), calculated by median method, let us find manifestation a20,5, leading to the determination UFV 0,5 through engaging first criterion IV. As a result of value a11 = 42,5 let us check, what change Δa1 corresponds to the transition just as from a20 to the required a20,5, and from a20,5 to a21. Inquiries adhere to, that this action decreases Δa1 = 7,5 (42,5 -35,0 = 7,5), that is confirmed by indifference evaluation:

(42,5, 800) ~ (35,0, 1100) or (a11, a20) ~ (a11 Δa1, a20,5);

(42,5, 1100) ~ (35,0, 1300) or (a11, a20,5) ~ (a11, - Δa1, a21).

We obtain, that n2(1100) = 0,5. As far as we know UFV for all alternative manifestations, then we enclose UFV n2 analysis.

It is possible to conclude the correlation between equilibration factors w1 and w2 from the indifference evaluations. UFV difference for the IV criterion (between 42,5 and 35,0) Δn1 = 0,5. Compensating difference in utility value Δn2 for the IA criterion also sets up 0,5. By this means, we have the following correlation:

׀Δn2 ׀w2 = ׀Δn1 ׀w1 or 0,5w2 = 0,5w1, i.e. w2 = w1.

Therefore arises, that IV and IA criteria have common weighting coefficients: w1 = w2.

To determine UFV n3 and equilibration factor w3 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 n3(17) = 0,5. For the correlation of equilibration factors w1 and w3 operates the following rule: ׀Δn3׀w3 = ׀Δn1׀w1 or 0,5w3 = 0,3w1, i.e. w3 = 0,6w1.

To determine ordinary utility value n3 = 15, necessary for the A1 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 n3(15)n = 0,25.

UFV n4 of the correlation between w4 and w1 may leave unassigned because of that, ordinary utility value of the important value a4 = 350 UFV n4(350) is equal to 0,5. The correlation between w4 and w1 is such, that w4 = 0,4w1.

Taking into account normalizing (w1 + w2 + w3 + w4 = 1) let us determine: equilibration factors w1 =1/3; w2 = 1/3; w3 = 1/5; w4 = 2/15; universal utility value NM of the alternatives NM = 1/3 n1(a1) + 1/3 n2(a2) + 1/5n3(a3)+2/15n4(a4).

Inserting the manifestations, we obtain universal utility value of the alternatives: A1 - 9/20; A2 - 7/15; A3 - 8/15. Therefore we have that alternative A3 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.