23. Marine company economic optimality in case of uncertainty of the investment objective or data

The economic efficiency and optimality of the characteristics for the commercial order of the commercial vessel or other real asset as a part of the property complex of the marine company when working out the investment project and in case of comparative estimation of the options is connected with the concept of economic multicriteriality when there is indeterminate of the purpose or data.
           So the generalized (complex) criteria of optimality which unite particular criteria should be formulated and used. And each of particular criteria, in turn, is put in conformity of one of the possible purposes of the investor under condition of objective uncertainty.
           The same applies to the data certainty – to a so–called situational uncertainty: the criterion is put in conformity to each combination of external conditions (data).
           Uncertainty is considered as formally irremovable factor, and formalization of the generalized criteria requires participation of the expert. And uncertainty, as an ineradicable factor, also appears in process of coordinating the results of evaluation of the commercial vessel, other asset or the property complex of the marine company by independent approaches.
           In spite of the fact that there is a set of expert recommendations how to generalize independent approaches results of value estimation, at the same time there is no and cannot be any materialistically strict demonstrative formal methods of coordination results, and difference of results is a consequence of difference of the purposes and data sets for each of independent approaches for value estimation.

23.1. Causes of the objective and situational uncertainty resulting in the economic multicriteriality

The cause of the purpose or situational uncertainty which usually leads to multicriteriality is considered to be a breakdown of connections between economic models of various levels within the limits of the systematic approach. In applied tasks the broken connections usually are replaced with expert opinions.
           To apply the generalized criteria of optimality it is required, in particular, that the generalized particular economic criteria should have methodologically similar sense coinciding with the resultant generalized criterion, for example the general time scale, similar dimension and so on.
           Besides, all local economic efficiency criteria as a part of the generalized criteria should have a conterminous order of preference; they should be simultaneously maximized or minimized.
           At the same time when there is an uncertainty of data there is no need to transform local economic criteria to a similar form for use as a part of the generalized criterion, since particular criteria are homogeneous in itself, and difference of their values is connected with ambiguity of data.
           Conformity of the methodological sense of local economic criteria as a part of the generalized criterion in case of objective uncertainty certainly is fulfilled, if particular criteria are presented in the form of investments (value) as a part of the generalized criterion expressed as value.
           It is necessary to pay attention that all economic criteria which can be applied to optimization of characteristics for the commercial vessel order, other asset or property complex of the marine company, contain in the structure the index of investments MV (or RC) which therefore can mathematically be derived on the basis of any economic criterion.
           As the basic economical indexes: the income I, operation costs C and investments MV (or RC), – do not reflect critical judgments, suitable for choosing optimum solutions, and are not the criteria of choice in optimization, but serve to formulate the criteria which mathematically are determined by correlation of the basic indexes, it should be noted that concepts of efficiency, expediency and utility are introduced by mathematical operations on the basic indexes for formalization of one criterion or another in the form of criterion function. And if mathematically to derive investments MV (or RC) from any criterion the unequivocal conformity of mathematical actions which in one case leads to generalization of the basic indexes to criterion of efficiency will remain, and in other case – to generalization of the basic indexes and criterion to the value index (investments), thus the criterion is derived into restriction.
           On the other hand, as critical judgments about efficiency of commercial vessel, other asset or property complex of the marine company are formed by mathematical operations on the basic indexes (I, C and MV or RC), the determination of investments (value index) depending on other indexes and criterion by the same mathematical operations contains not less data about mathematical structure of economic criterion, appropriate to fine a compromise in process of optimization.
           For the multiobjective optimization and formulation of the generalized criteria of efficiency of marine company real asset the preliminary mathematical transformation of each of particular criteria to the form of value (investments) eliminates possible subject discrepancies of generalized particular criteria, in particular possible discrepancies of dimension, time scale and order of preference.
           Besides uncertainty of data or purpose, the reasons of two–criterial property of optimality of the commercial vessel, other asset or property complex of the marine company can be in the methodological duality of optimality criterion from the economic point of view, since there should be a preference to optimize the choice and comparison of options with both an economic environment, and with comparable options.
           For example, in case of indeterminate of investor’s motives to maximize of internal rate of return, which is the criterion of general efficiency of commercial vessel, other asset or property complex of the marine company in relation to a macroeconomic environment, or to minimize the costs (resulted or discounted) which characterize the comparative efficiency, the specified criteria should be generalized in a compound criterion.
           But first, it is required to transform the particular criteria to a homogeneous kind, let us assume, to express the value (investments) on the basis of particular criteria of the internal rate of return and the discounted costs, and then to combine these particular criteria expressed in the value, as parts of the generalized complex criterion which also will be expressed as value.
           For the subsequent generalization as a part of the complex criterion the economic criterion of the discounted (resulted) costs determined with the account the introduced assumptions should be transformed , to the form of the value (investments) as .

           Economic determination of the value index by transformation of the criterion of the discounted costs means the evaluation of investments which are justified in relation to average market discounted costs  for comparable investments into the commercial vessel, other asset or the property complex of the marine company on condition that the efficiency and the risks level of investment correspond to the rate of discounting i.

           Transformation of the efficiency criterion in the form of the internal rate of return to the value index of investments looks like: , or for the discounting by the exponential function of The Euler’ number.

           The evaluation by transformation of the criterion in the form of the internal rate of return means the estimation of investments which are justified under condition that the level of profitableness of the commercial vessel (other asset or the property complex of the marine company), estimated by the indexes  of net operating income (I-C) and the risks of investment correspond to the rate of discounting i, in relation to the average market value of internal rate of return f₂ for comparable investments.

,

(23.13)

where α_j – the expert weight factor which is characterizes the methodological importance of the particular criterion f_j  and simultaneously is a scale multiplier (in case when the particular criteria in the formula are non–uniform and have different scale); J – the total of the particular criteria in compound criterion.
           Strictly speaking, any criteria of efficiency are compound, since they represent the sum of two or several indexes, for example, they represent the sum of investments MV and the flow of the operation costs C, divided by the factor i, as a part of discounted (resulted) costs criterion of  or other criterion.
           The compound criterion in the form of the criterion function for the optimization of the marine company real asset, constructed on the basis of the discounted (resulted) costs criterion in the form of the value  and internal rate of return in the form of the value , provided that the «weight» of the discounted costs criterion is equal to α, in the compound criterion and it is formulated as

f=α×MV1+(1-α)×MV2.

(23.14)

           The compound economic criterion defined by the transformation of the value indexes of the commercial vessel, other asset or the property complex of the marine company on the basis of the criteria of the discounted costs and the internal rate of return, with share weight α, means the estimation of investments which are justified in relation to the average market discounted costs f₁ upon condition that the level of risks of the investment corresponds to the rate of discounting i, and with share weight (1–α) – it means the estimation of investments which are justified upon condition that the level of profitableness estimated according to an index of net operating income (I-C) in relation to the average market value of the internal rate of return f₂  for comparable investments.
           After the substitution of the particular criterion of the discounted costs in the form of the value MV₁ and particular criterion of the internal rate of return in the form of the value MV₂ the compound criterion looks like:

,

(23.15)

or for the discounting by the exponential function of the Euler’ number
           .
           It’s worth mentioning, that if in case of objective uncertainty the generalized criterion (the compound criterion considered above or the generalized criterion of other type) is formalized on the basis of methodologically similar particular criteria preliminary transformed to the form of the value (investments), then the generalized criterion also is the value of the marine company real asset with the account that combination of conditions which expertly or mathematically are taken as a principle of particular criteria generalizations.
           And mathematical and expert principles of the particular criteria generalization as a part of the complex criteria also are a source of judgments about efficiency, optimality and utility as well as mathematical operations on the basic indexes: the income I, the prime cost C and investments MV (or RC), – when formalizing particular criteria.
           The Laplas’ criterion, which is defined formally almost the same as the compound criterion can be used as a form of the compound criterion for economic optimization of the characteristics of the vessel or other real asset of the marine company.
           The difference is that in the structure of the Laplas’ criterion the weight factor α_j characterizes the probability of j–th situation (a combination of data) in case of the situational uncertainty or the probability of j–th objective of the investor in case of the objective uncertainty.
           Thus the sum of probabilities of all J situations is equal to one, that is, the sum of the expert weight factors as a part of generalized Laplas’ criterion is equal to .
           The Wilde’s criterion, or the principle of «the greatest care» concerning judgments about efficiency and optimality of the commercial vessel, other asset or property complex of the marine company is formulated as follows: if the particular criteria f_j, as a part of the complex Wilde’s criterion, are minimized the complex criterion is minimized too and is defined as .
           If the particular criteria f_jas a part of the complex Wilde’s criterion are maximized, the criterion formulated as:

,

(23.16)

is maximized.
           The preference order by the particular criteria which are a parts of Wilde’s criterion and by the generalized Wilde’s criterion coincides.
           The methodological sense of the generalized Wilde’s criterion as a principle of «the greatest care» of investor to the judgments about the efficiency and optimality of the commercial vessel, other asset or the property complex of the marine company can be explained as follows: the best combination of characteristics of the object by the Wilde’s criterion is the combination of characteristics of the object for which the criterion has the best value in the worst circumstances.
           Figuratively speaking, by Wilde’s criterion the best is the combination of the marine company real asset characteristics for which the maximum losses are minimum. And losses are considered not in terms applied in accounting, but from the methodological point of view, that is, as estimations of deviations from optimality by each of particular criteria.
           For generalization of particular criteria of the discounted costs and internal rate of return the maximized Wilde’s criterion is defined as

,

(23.17)

or for the discounting by the exponential function of the Euler’ number
            .
           Wilde’s criterion estimated by transformation of value indexes on the basis of the criteria of discounted costs and internal rate of return means the least value of the evaluation which in any case is justified as in relation to average market discounted costs  upon condition that the investment risks level corresponds to the rate of discounting i, and in relation to profitableness of the commercial vessel, other asset or the property complex of the marine company, the internal rate of return  estimated in relation to the average market value and to the index of net operating income (I-C)  for comparable investments.
           By Horvitz’ criterion (unlike Wilde’s criterion) for judgments about vessel characteristics optimality, other asset or property complex of the marine company there are considered not only the worst situations (a combination of data) with share weight (1-γ) , but also the best with share weight γ.
           The preference order by Horvitz’ criterion corresponds to the preference order by the particular criteria which are a part Horvitz’ of generalized criterion. If the particular criteria f_j are minimized the Horvitz’ criterion is formulated as , where  – the expert factor of «optimism» when determining the criterion of Horvitz.
           Accordingly if the particular criteria f_jare maximized the Horvitz’ criterion for use as a part of an optimality concept of the commercial vessel, other asset or the property complex of the marine company is defined as

.

(23.18)

           For generalization of the particular criteria: the discounted costs and the internal rate of return, transformed to the value, Horvitz’ complex maximized criterion is defined in the form of the criterion function as

(23.19)

or for the discounting by the exponential function of The Euler’ number
            .
           Horvitz’ criterion defined by the transformation of value indexes on the basis of criteria of discounted costs and internal rate of return, with share weight (1–γ) means the least value which in any case is justified as in relation to average market discounted costs  by the requirement of the investment risks level corresponding to the rate of discounting i, and in relation to profitableness of the commercial vessel (or other asset of the marine company), internal rate of return  estimated by the index of the net operating income (I-C) for comparable investments, and with share weight γ– the greatest the market value which can be justified by one of entering particular criteria upon condition that the successful circumstances are reflected by this criterion.
           According to Savage’s method for generalization of the particular criteria as a part of the complex efficiency criterion of commercial vessel, other asset or the property complex of the marine company it is supposed that one of combinations of external conditions (or one of the purposes of the investor) for each of the comparable by optimization options is the best. In comparison with the best combination of the data other combinations are «unprofitable» in sense of deviation from optimality.
           Savage’s criterion in the considered formulation is minimized irrespectively of the order of preference of the particular criteria entered into it, as it is based on minimization of «losses». Certainly, concepts «losses» and «advantages» as a part of Savage’s criterion are applied in methodological sense, instead of in terms, applied in accounting.
           The best, by Savage’s criterion, is the combination of characteristics of the vessel, other asset or the property complex of the marine company for which the greatest possible «losses» are the least.
           For the minimized particular criteria Savage’s complex criterion (minimized) is formulated as , accordingly, for maximized particular criteria Savage’s complex criterion is formulated as .
           To generalize the local efficiency criteria of the discounted costs and the internal rate of return the minimized Savage’s complex criterion in the form of the criterion function is determined as

,

(23.20)

or for the discounting by the exponential function of The Euler’ number as
           .
           To use Savage’s criterion the algorithm should contain determination of optimum combinations of characteristics by each of particular criteria, and also values of particular criteria corresponding to them in the form of the investments value ( and ).
           The criterion function for an example of fishing vessel optimization by minimized Savage’s criterion is shown on the chart (fig. 23.1) .

           The maximized Savage’s criterion (or the principle of «maximization of the minimum advantages») unlike minimized Savage’s criterion (that is the principle of «minimization of the maximum losses») can be formulated as follows.
           One of the external combinations of the conditions (one of the investor’s purposes) for each of the compared options of characteristics of the commercial vessel, other asset or the property complex of the marine company for the commercial order is the worst. In comparison with this combination other combinations are favorable. The best combination of characteristics by maximized Savage’s criterion, for which the least possible advantages will be the greatest, is the best one.
           For minimized particular criteria the maximized generalized Savage’s criterion is defined as , for maximized particular criteria the maximized Savage’s criterion is formulated .
           For maximized Savage’s criterion as a principle of «maximization of the minimum advantages», this order of preference remains irrespective of entering into it particular criteria order of preference.
           The criterion representing convolution in the form of functions Kobb–Douglas, is formulated as , where – is a symbol of particular criteria multiplication. A special example of the similar generalized criterion is multiplicative convolution .
           Assuming, that the particular criteria have similar dimension or are methodologically similar, for example, when they are expressed in the form of the investments (value), the generalized criterion representing convolution, can be improved:
           ,
for example, it is possible to represent the multiplicative convolution as .
           For particular criteria: the discounted costs and the internal rate of return, expressed in the form of investments (value), the criterion in the form of multiplicative convolution is determined as

,

(23.21)

or for the discounting by the exponential function of The Euler’ number
            .
           The multiplicative convolution of local economic criteria means the average quadratic evaluation of investments into commercial vessel, other asset or the property complex of the marine company which are justified in relation to the average market discounted costs  upon the condition that the investment risks level corresponds to the rate of discounting i, and investments which are justified providing the profitableness level estimated by index of net operating income (I-C)  in relation to the average market value of internal rate of return  for comparable investments.
           The expert way of particular criteria generalization is the method of concessions, and the sequence of concessions application can be considered as thecriterion function of characteristics optimization for the commercial order of the commercial vessel, other asset or the property complex of the marine company.
           To use the concessions method for multiobjective optimality determination of particular criteria should be arranged according to their importance for the optimization of characteristics of the commercial vessel or other asset of the marine company for commercial order.
           The sequence of the multiobjective optimization by concessions method is as follows. The combinations of the marine company real asset characteristics compared by criteria are estimated by that particular criterion which is considered the most important, for example .
           The permissible deviation from the optimality by the first (most important) criterion  is set and among the set of compared combinations of characteristics there should be chosen the optimum one by the criterion , as well as those combinations which differ from the optimum characteristics by the value not exceeding the value of the permissible deviation from the optimality by the criterion .
           Operations are repeated among the set of the chosen combinations of characteristics by the second criterion  (the next in order of importance) etc. The use of concessions method supposes that the sequence of such operations leads to an optimum option.
           If particular criteria are maximized the concessions method is formulated as

,

(23.22)

where 1, 2, …, j–1, j – the sequence of particular criteria ranged according to the order of importance decrease.
           The criterion function by concessions method is presented for the example of fishing vessel optimization in the chart (fig. 23.2).

           The methodological principles of compromises in the economic optimization of characteristics of commercial vessel or other real asset of the marine company for the commercial order in the presented definition need to be referred to a number of the formal approaches directed to determination of area of optimum solutions.
           Following the formal principles of compromises, the area of optimum combinations of vessel characteristics or other marine company asset is in so–called Pareto’s area , outside of which all local economic criteria simultaneously worsen, and Pareto’s area  is defined in space of variables X.
           The optimization method is to minimize the Pareto’s area , which is «shrinked» to the optimum combination  of characteristics as the combination is located in Pareto’s area .
           If particular criteria  are maximized the definition of Pareto’s area  is formulated as

(23.23)

           For economic optimization of the fishing vessel intended for the autonomous use of project characteristics the test calculations of the criterion function and the Pareto’s area are presented in the figure (fig. 23.3).
           And the criterion function is defined as a particular criteria combination of the discounted costs and the internal rate of return transformed to the form of the investments, and the Pareto’s area is shown by boundaries.
           The minimization of Pareto’s area  (that is «shrinking» to an optimum combination of characteristics ) of the vessel or other real asset of the marine company) is made generally by one of two or by two methods: the expert participation and, hence, by increasing of subjectivity of the result or by increasing the number of used particular criteria which are algorithmically used to determine the Pareto’s area, where the criterion function is defined as well.