Elaboration of the equipment replacement terms taking into account wear and tear and obsolescence

The optimum terms of replacement of equipment subject to wear and tear and obsolescence with more advanced equipment with similar performance are investigated. To this end, the methodology of multi-criteria evaluation of equipment performance indicators when switching to a new type of equipment is proposed. The values of EAC (Equivalent Annual Cost), calculated for several equipment replacement cycles and for an infinite number of cycles are investigated. Estimates of the dispersion degree of EAC values depending on the service life of old and new equipment under conditions when the dynamics of operating costs is subject to random fluctuations are obtained. For this, covariance functions of random processes that describe the dynamics of the operating costs of old and new equipment were used. On the basis of covariance functions, estimates of the functions of standard deviations of EAC values are obtained. Using the obtained estimate of the degree of dispersion of equipment performance indicators, the multi-criteria optimization problem was investigated. This approach is of great practical interest, because for many enterprises, not only the average expected level of equipment performance indicators, but also the dispersion degree of the values of these indicators is of great importance. As a result of the research, the technique for planning the equipment replacement terms was developed. The proposed methodology allows justifying the terms of replacement of old equipment with new equipment, taking into account both the average expected EAC values and their level of fluctuations. The studies have shown that due to choosing the equipment renewal terms, it is possible to significantly reduce the degree of dispersion of equipment performance indicators, slightly sacrificing the average expected value.


Introduction
Efficient use of equipment is a key factor of successful operation of many enterprises. Therefore, the issues of justifying the optimum terms of equipment renewal are of great practical interest. The advisability of equipment replacement arises for several reasons. First, this is due to wear and tear. Wear and tear can be caused by abrasion of parts, fatigue of materials, oxidation and other reasons. The increase in wear and tear is manifested in the increase in the number of breakdowns and growth of operating costs of equipment. Secondly, the advisability of equipment replacement is associated with obsolescence, caused by the emergence of more efficient or cheaper analogs of equipment in the market. In many cases, the advisability of equipment renewal is determined not by one type of wear, but by the combined effect of wear and tear and obsolescence on equipment.
Planning of terms of replacement of aging equipment with new equipment often should take into account the impact of various random factors. Equipment performance depends on external factors such as equipment utilization rate, the nature of the work performed, weather conditions, etc., which are subject to accidental changes. Even for similar machines used under identical conditions, the dynamics of wear and tear can be significantly different. And this should be taken into account when planning the terms of equipment renewal.
Currently, many enterprises in Ukraine, including seaports, need upgrading of equipment. This causes a significant practical interest in the development of scientifically based methods for determining the optimum terms of switching to a new type of equipment. The relevance of work in this direction is also explained by the need to develop control methods aimed at reducing the degree of dispersion of possible values of equipment performance indicators.

Literature review and problem statement
In [1] and [2], the optimum terms of equipment replacement under the conditions when operating costs of equipment are subject to random fluctuations have been investigated. Also in [1], the issues of the influence of tax rates on the determination of optimum terms of equipment utilization have been studied. In [2], the influence of random fluctuations in the equipment utilization rate on the optimum service life has been examined. However, in [1,2] the possibility of capital repairs of equipment and the obsolescence factor have not been taken into account.
In [3,4], the issues of determining the terms of replacement of equipment parts have been largely solved. In [5] and [6], the issues of determining the optimum service life of equipment, taking into account the possibility of restoration have been investigated. Thus, in [5], the possibility of a

ELABORATION OF THE EQUIPMENT REPLACEMENT TERMS TAKING INTO ACCOUNT WEAR AND TEAR AND OBSOLESCENCE I . L a p k i n a
Doctor of Economic Sciences, Professor, Head of Department* Е-mail: lapkina@ukr.net

M . M a l a k s i a n o
PhD, Associate Professor* Е-mail: malax@ukr.net *Department of management of logistics systems and projects Odessa National Maritime University Mechnikova str., 34, Odessa, Ukraine, 65029 single equipment repair with a subsequent replacement for a coal-fired power plant has been considered. In [6], handling equipment has been discussed and the possibility of multiple major repairs has been allowed. Both in [5] and [6], the random factor when changing the operating costs of equipment has been taken into account. However, these works have not addressed the issues related to justifying the service life of aging equipment in connection with the emergence of the corresponding new types.

Розв'язано задачу обґрунтування термiнiв замiни обладнання, що схильне до фiзичного i морального зносу, на бiльш досконале обладнання з аналогiчним рiвнем продуктивностi. Для цього запропоновано методику багатокритерiальної оптимiзацiї значень показникiв при переходi на обладнання нового типу. Дослiджено значення EAC (Equivalent Annual Cost), якi розраховано як для декiлькох циклiв замiн обладнання, так i для нескiн-
In [7], the issues of the influence of the number of technological operations on reliability and wear and tear of equipment have been considered. In [8], the problem of finding the optimum terms of equipment replacement taking into account taxation and the depreciation policy has been investigated. For this, the mathematical model based on a partial differential equation has been proposed. For the modeling of random fluctuations, the Brownian process has been used. Failures of complex multi-component technical systems have been studied in [9]. In this work, the influence of the technical system configuration, quality of elements and interaction between them on failures of the entire system and changes in associated costs has been analyzed. Models of cost estimation for systems with various configurations of elements have been presented and a sensitivity analysis has been carried out. At the same time, the issues related to obsolescence of equipment remained beyond the scope of these studies.
In [10] and [11], the problems of stable functioning of transport systems under conditions of uneven cargo traffic and the rationale for the choice of the optimum structure of the equipment fleet have been investigated. However, in [10,11], attention hasn't been paid to the justification of equipment replacement strategies, taking into account technological progress.
The work [12] is devoted to planning the terms of introduction of new technologies. The models and conclusions proposed in it are based on certain assumptions regarding the rates of technological progress and emergence of new technologies.
In [13], the issues of early and delayed replacement of equipment in the deterministic model with the account for technological progress have been studied. It has been assumed that the level of income and expenses associated with equipment varies over time. The optimum time of deterministic replacement of equipment has been determined using the dynamic programming model.
In [14], the issues of determining the optimum time of the introduction of advanced equipment in view of uncertainty of the time of emergence of new technologies and their efficiency have been also studied. Three strategies of introduction of technological innovations have been considered. The substantiation of the choice of a strategy at various levels of uncertainty, efficiency, and also the pace of technological innovation has been carried out.
In order to justify the time of equipment replacement taking into account available spare parts stocks and considering the technical progress, it has been proposed in [15] to use the model of the Markov decision process. It has been assumed that due to technical progress in the market, new types of equipment may appear over time. But when replacing old equipment with new equipment, spare parts for old equipment become useless.
However, in [12][13][14][15], the influence of wear and tear of equipment on the determination of optimum replacement terms has not been investigated.
In a number of works, including [1][2][3][4][5][6][7][8][9][10][11], when determining the optimum terms of equipment replacement, only the factor of wear and tear or only the factor of emergence of new technologies has been taken into account. Also, many authors emphasize the need to study the influence of the random factor in justifying the equipment replacement terms. In [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], only some of the above factors have been in a varying degree considered. At the same time, in a number of cases, it is necessary to fully take into account the aggregate of all these factors when making decisions on equipment replacement. Therefore, the issues related to justifying the equipment replacement terms on the basis of a comprehensive assessment of both wear and tear and obsolescence, as well as accounting for the random factor, are of great importance.

The aim and objectives of the study
The aim of the study is to develop methods of planning the optimum terms of replacement of wearing equipment with more advanced equipment of new type, taking into account the degree of dispersion of possible values of equipment performance indicators.
To achieve the aim, the following objectives are accomplished: -to justify the choice of equipment performance assessment criteria when switching to a new type of equipment; -to carry out a quantitative estimation of the degree of uncertainty of performance indicators, depending on the choice of the service life of old and new equipment; -to determine a multi-criteria estimate of average values and degree of dispersion of possible values of equipment performance indicators, depending on the service life of old and new equipment.

1. Justification of optimum terms of equipment replacement
The situation when the operating mode and utilization rate of equipment remain constant throughout the life, and all technical and operational characteristics are governed by the relevant regulations is considered. If technical and operational characteristics cease to meet specified standards of performance, reliability or safety over time, then the use of such equipment is considered unacceptable, and it must be repaired or replaced. For example, this applies fully to seagoing vessels: their usage modes are constant, and the technical condition is strictly controlled by the maritime register. It is economically irrational to use a seagoing vessel not with full intensity because it has rather high fixed costs, which can only be covered by intensive use. The situation is similar with railway rolling stock. The same situation is with complex handling equipment of sea ports. So, reducing the operating time of an aging harbor crane or reducing the speed of the harbor crane entails the irrational use of the berth, warehouse and dockers, increases the costly unproductive time of vessel berthing in the port. For such equipment, operating costs associated with the growth in the number of maintenance and repair works usually increase over time, while performance and reliability must remain at a certain level.
The main criterion in making decisions on replacement of such equipment, as a rule, is the change in the economic performance of its operation. It is often difficult to estimate the profit share of the entire enterprise, which falls on a particular machine, if this machine is only a part of a complex production chain. Therefore, it is natural to consider the unit cost of equipment operating time (or unit cost) or indicators that are derivatives of the unit cost as an indicator of equipment performance under such conditions. The EAC indicator reflects the redistributed share of total costs of equipment per unit operating time. At the same time, EAC considers cash flow discounting, which is a mandatory requirement in case of complex equipment having a long service life [16]. The use of the EAC indicator does not require reduction to a single planning horizon when comparing the results of utilization of equipment with different service lives. With such problem statement, NPV, IRR and many other widely used indicators for equipment performance assessment are not suitable. Therefore, the EAC indicator was chosen in the paper as the main criterion for estimating the efficiency of complex port equipment.
In some cases, it may take a long time from the moment of making a decision to purchase complex equipment prior to commissioning. This is due to the fact that, as a rule, complex equipment, including complex seaport handling equipment, is made to order. The manufacture, transportation and installation of equipment, as well as previous stages associated with the preparation and approval of feasibility studies to obtain permits and attract investments for purchase, often take years. Therefore, the issues related to replacement of such equipment cannot be solved promptly, based only on the current condition of equipment or short-term forecasts. In many cases, replacement of such equipment should be planned long before its technical condition reaches an unsatisfactory level.
Thus, the delay between the control action (in this case, the decision to replace equipment) and the response of the control object (commissioning of new equipment) is too great. At the same time, the values of equipment performance indicators are subject to random fluctuations. Moreover, the values of performance indicators of equipment at the time of replacement often correlate little with the values of performance indicators at the time of making a replacement decision (for example, graphs of covariance functions of operating costs for container cranes are shown in Fig. 6, 7). In [17], the developed apparatus for the synthesis and stability analysis of adaptive control systems is presented. However, due to the above-mentioned features, it cannot be successfully used in some cases to solve the problem of finding the optimum terms for replacing complex equipment. In this paper, we propose a different approach to the study of this problem.
In most cases, equipment repair and maintenance payments are discrete and aggregated monthly, quarterly or annually. Therefore, when evaluating the equipment effectiveness post factum, discrete models are often used. However, continuous regression curves of possible changes in indicators and continuous optimization methods are more convenient for forecasting. Therefore, attention should be given to the development of an approach based on the use of continuous models.
In [18], the definition of efficiency is given and its main function as an optimization criterion is defined. However, as noted in [18], the constraint on the use of such indicator in optimization problems is the imperfection of architectural solutions of automatic productions.
The rationale for the choice of the EAC indicator as an optimization criterion is presented in [16,19].
Consider the situation where old equipment is replaced with more advanced equipment of comparable performance. We introduce the following notation: A o -the cost of purchase and installation of old equipment, USD; A n -the cost of purchase and installation of new equipment, USD; c o (t) -the average rate of operating costs of old equipment after operation for t years, USD/year; c n (t) -the average rate of operating costs of new equipment after operation for t years, USD/year; S o (t) -the cost of dismantling and sale of old equipment after operation for t years, USD; S n (t) -the cost of dismantling and sale of new equipment after operation for t years, USD; T o -the time during which it is planned to use old equipment, years; T n -the time during which it is planned to use new equipment, years.
Since the service life of complex equipment is typically several years, discounting should be used to assess the utilization efficiency throughout the life cycle. Let r be the annual continuously compounded interest rate. Using the known continuous compounding formula (for example, [19]), the current operating costs of old equipment when used for T o years can be found: The current capital costs of old equipment, when used for T o years, are found by the formula Thus, the current total costs of old equipment when used To substantiate the optimum service life of equipment, it is possible to use the current total costs of equipment in cyclic replacements with the same type of equipment during an infinite planning horizon Summing up the terms of the geometric progression, it is easy to show that To compare the equipment performance on time intervals of different lengths, the EAC (Equivalent Annual Cost) indicator is often used (for example, [16] The equipment operation term * o T in which the expression (4) or (6) will take a minimum value can be considered optimum. Although the formulas (4) and (6)  this gives grounds to consider that new equipment, when used in the given conditions, is economically more expedient than old equipment.
Let us investigate the issue of finding the optimum terms of equipment replacement when switching from old equipment to new equipment, which is comparable in performance to old, but is more economically feasible. To this end, consider two reasoning patterns: the first pattern, based on the analysis of a finite time interval consisting of two complete cycles of equipment replacement, and the second -based on an infinite time interval.
We consider the first pattern, based on the analysis of the time interval, consisting of two complete utilization cycles of equipment. Within this pattern, it is planned to use old equipment during the first cycle of T o years. Then it is planned to use new equipment during the second cycle of T n years. It is easy to see that the current total costs during these two cycles can be calculated as Let us find the EAC value for the second reasoning pattern. Within this pattern, we will analyze the infinite planning horizon. And, during the first full cycle of work, it is planned to use old equipment. Then, during all subsequent cycles of the same duration T n , new equipment will be used. In this case, obviously,

2. Assessment of the influence of equipment replacement terms on possible fluctuations in EAC values
Due to the influence of various random factors, as equipment ages, significant fluctuations in the operating costs of equipment are possible. Therefore, it makes sense to describe the dynamics of changes in the rates of the operating costs of equipment using random processes c o (t, ω) and c n (t, ω), where ω∈Ω, (Ω, A, P) is the probability space. Moreover, the mathematical expectations of these random processes are c o (t)=E(c o (t, ω)) and c n (t)=E(c n (t, ω)), respectively. In this case, the current operating costs of equipment will be described by the corresponding random processes It is natural to assume that the dynamics of the operating costs of each subsequent machine does not depend on the costs of previous machines. Therefore, using (9) and the properties of covariance functions of random processes (for example, [20]), we obtain Similarly, from (10) for the second reasoning pattern, we obtain

Results of the study of optimum terms of equipment replacement
We investigate optimum replacement terms for harbor container cranes. The costs of equipment of the old and new type are A o =240 thousand USD and A n =195 thousand USD, respectively. The average dynamics of operating costs for container cranes is described by the functions c o (t)=75.07+0.21·t 2.11 , thousand USD/year and c n (t)= =60.52+0.22·t 2.52 , thousand USD/year. These functions were obtained on the basis of the regression analysis of changes in operating costs for a sample of 27 old-type container cranes. Fig. 1 shows the box plot, which presents the dynamics of changes in operating costs of old-type container cranes according to this sample. In Fig. 2, the solid line shows the curve of changes in average operating costs of oldtype container cranes, and the dashed lines -the borders of the band, within which, with a 0.9 probability, the values of operating costs are found.
The way the form of the law of distribution of the actual values of the operating costs of equipment changes is seen from the histograms in Fig. 3, 4, reflecting distributions of values of operating costs at specified timepoints. In these figures, red lines represent the densities of normal distributions, the mathematical expectations and standard deviations of which correspond to sample indicators, as well as the results of the Kolmogorov-Smirnov and Shapiro-Wilk normality tests. As can be seen, at the beginning of equipment service life, the distribution of average costs per unit time of equipment operation almost does not differ from normal (Fig. 3, 4), but the deviation from the normal distribution law becomes noticeable over time (Fig. 4). A similar trend is observed for new equipment.    T T = and is 119.60 thousand USD. Fig. 8 shows the graphs of changes in the EAC values for each individual piece of equipment when replacing old equipment with new equipment. Fig. 9 shows the graphs of EAC changes for each individual piece of equipment with one operation cycle of old equipment and an infinite number of subsequent operation cycles of new equipment. In Fig. 8 and Fig. 9, the blue solid line represents the curve of changes in the EAC values for old equipment, and the red dotted line shows the EAC curves for new equipment. We investigate the degree of dispersion of EAC values depending on the choice of equipment replacement terms with possible random changes in operating costs. Estimation of the operating cost function c o (t) and the covariance function K o (t 1 , t 2 ) for old equipment is usually not very difficult, since in most cases there is enough statistical data for old equipment. But for new equipment, there is sometimes not enough information for reliable statistical estimation of equipment utilization under given specific conditions. In this case, the estimate of the dynamics of the average level of operating costs and the covariance function K n (t 1 , t 2 ) can be obtained on the basis of technical documentation and expert assessments. Introduction and first experience of using innovative equipment are often associated with increased risks. And this risk should be taken into account when justifying the values of the covariance function K n (t 1 , t 2 ).
Further, when describing the covariance functions for old and new equipment, we will use the following expression where the set of constant coefficients a=(a 1 ;…; a 6 ) is determined by regression analysis methods, separate for each type of equipment. Such choice of a type of covariance function makes it easy to interpret the effect of each of the constant coefficients on the properties of the studied random process of changes in the operating costs of equipment. Thus, the coefficients a 1 , a 2 and a 3 determine the growth rate, the curve shape and the initial value of the dispersion function, respectively, for the random process under study. The coefficients a 4 , a 5 and a 6 reflect the way in which the degree of interrelation between the values of the operating costs of equipment is changed at different timepoints. The use of covariance functions of the type (19) made it possible to describe quite accurately the process of changes in the operating costs for container cranes. However, this form of covariance functions is not universal. The form of the function K(t 1 , t 2 ) should be selected and justified individually for different types of equipment and different operating conditions. For the considered old type container cranes, a set of values of constant coefficients a o =(0.3501; 2.21; 35.21; 0.002; 2.50; 15.40) in the function (19) was determined based on the regression analysis. Accordingly, for container cranes of new type, the values of coefficients a n =(0.0075; 3.50; 105.21; 0.015; 3.01; 5.61) were obtained. Fig. 10, 11 present the graphs of the covariance functions K o (t 1 , t 2 ) and K n (t 1 , t 2 ) for container cranes of the old and new type determined by the function of the form (19) with the above values of the coefficients. From Fig. 10, 11, it can be seen that dispersion of the operating cost values for new equipment is much higher than for old equipment. Moreover, for new equipment, there is also a greater correlation between the values of operating costs at different timepoints than for old.
Verification of the indicators was carried out on the basis of actual operation data of harbor container cranes. The values of all input data for calculations by the formulas (1)- (18) given in the paper, including statistical estimates of the covariance functions K o (t 1 , t 2 ) and K n (t 1 , t 2 ) are obtained using the statistical analysis of actual data.  Comparing Fig. 6, 12, and also Fig. 7, 13, we can see that the values of T o and T n , optimum from the point of view of the average EAC, will not correspond to the minimum values of standard deviations of EAC. Therefore, in order to justify such a choice of equipment replacement terms, at which the balance between the minimum and the dispersion degree of EAC values is achieved, it is necessary to investigate the multi-criteria optimization problems (17) and (18). Let us analyze the relationship between the values of E(EAC on (T o , T n , ω)) and σ(EAC on (T o , T n , ω)). Fig. 14 shows the set of points, the coordinates of which are respectively equal to E(EAC on (T o , T n , ω)) and σ(EAC on (T o , T n , ω)) and are determined by the choice of the values of T o and T n . The unimprovable points lying on the Pareto frontier of the multi-criteria optimization problem are given in red (17). The coordinates of some unimprovable solutions, as well as the corresponding values of equipment replacement terms T o and T n , are shown in Table 1.  The points lying on the Pareto frontier of the multi-criteria optimization problem (18) are given in red in Fig. 15. The coordinates of some of these points are shown in Table 2.  It is obvious that the points that did not fall on the Pareto frontier are of no practical interest. To find the optimum service life of equipment, it is advisable to consider only unimprovable solutions of the problems (17), (18). This paper does not assume the use of a single integrated criterion for solving the problems (17) and (18). On the contrary, the reduction of the problems (17) and (18) to the study of any integrated indicators or convolutions of criteria would make the analysis of the problem of choosing the optimum equipment replacement terms more formal and would cut off much of the information that could help the DM (decisionmaker) to make a more reasonable choice. Not the packing of all indicators into a single integrated indicator, but the construction and analysis of the entire Pareto frontier, makes it possible to analyze the problem as deeply as possible (for example, [21]).

Discussion of the results of the study of optimum equipment replacement terms
The shape of the surfaces ( ( , , )) ω This difference is especially pronounced in cases when T n significantly exceeds T o . This is explained by the effect of overlapping and mutual compensation of random multidirectional fluctuations of independent random variables that make up an infinite sum in the evaluation of ( ( , , )).  Table 1, 2, it should be noted that the unimprovable solutions of the multicriteria optimization problem (17) correspond to longer terms of equipment utilization than those that correspond to unimprovable solutions of the problem (18).
The studies have shown that the minimum value of E(EAC on (T o , T n , ω)) equals 121.29 thousand USD and is achieved in the case when the service lives of old and new equipment are respectively 11 and 9 years. In this case, the value of σ(EAC on (T o , T n , ω)) is 5.72 thousand USD. In Fig. 14, this solution corresponds to the point A 11;9 . Let us compare the points A 11;9 . and A 9;8.5 in Fig. 14. As can be seen, by reducing the service lives of old and new equipment to 9 and 8.5 years, respectively, it is possible to significantly decrease the value of σ(EAC on (T o , T n , ω)), while slightly worsening E(EAC on (T o , T n , ω)). A similar effect can be observed in Fig. 15, analyzing the Pareto frontier of solutions of the problem (18).
Since T as optimum service lives of equipment can be considered the most reasonable, since the maximum planning horizon is taken into account. However, on the maximum planning horizon, estimates of equipment performance fluctuations may be blurred and significantly underestimated. Therefore, when studying the dispersion degree of the values of equipment performance indicators, it may be more appropriate in some cases to consider the planning horizon consisting of two equipment replacement cycles.

Conclusions
1. The paper proposes the methodology for multicriteria assessment of equipment performance when switching to a new type of equipment. For equipment performance assessment, taking into account both wear and tear and obsolescence at different time intervals, it is proposed to use the EAC indicator. The calculations used the EAC values calculated for several equipment replacement cycles (9) and for an infinite number of cycles (10). The studies have shown that when planning the equipment renewal terms, it is advisable to shorten the life of old equipment in comparison with those terms that would be optimum when replaced with old equipment.
2. The quantitative estimate of the dispersion degree of the values of the EAC indicator depending on the choice of the service lives of old and new equipment under conditions when the dynamics of operating costs is subject to random fluctuations is obtained. To do this, covariance functions for random processes that describe the dynamics of the operating costs of old and new equipment were used. Based on these covariance functions, estimates of functions of standard deviations of the EAC values were obtained.
3. Based on the obtained estimate of the dispersion degree of the EAC values, the technique for planning the equipment replacement terms was proposed. This technique allows justifying the terms of replacement of old equipment with new equipment, taking into account both the average expected EAC indicators and the level of possible EAC fluctuations. The studies have shown that due to the choice of equipment renewal terms (Fig. 14, 15), the degree of dispersion of the EAC values can be significantly reduced, slightly sacrificing the average expected value. Thus, the optimum (from the point of view of the minimum of average values and dispersion degree of EAC) replacement terms of equipment subject to wear and tear and obsolescence with new equipment were justified in the work.