Physical Asset Life Cycle Evaluation Models—A Comparative Analysis towards Sustainability

Abstract

In order to reach a sustainable circular economy, it is important to maximise the life cycle of a Physical Asset. An evaluation of a Physical Asset Life Cycle can be conducted via several approaches, and these may provide different results. The differences may be insignificant, but they must be taken into consideration because they have consequences for a manager’s decisions. This allows for a wider time interval to decide when to withdraw a Physical Asset or renew it and/or if it ought to continue functioning when profits are higher than expenses, thus allowing for a reduction in waste and increase in sustainability. These are some of the aspects that are discussed in this paper; it presents several models for the evaluation of the Physical Asset Life Cycle, considering the market value, devaluation methods and a more generalised use of Fisher’s Equation, which can include the Risk Tax, among others. The results are discussed based on data that support evaluations obtained with the models, and these are used for each Life Cycle model with the aim of evaluating the differences among them. Not only do all of the models consider expenses, namely those in Investment and Functioning, but also profits, which allows for a more holistic evaluation of the Physical Asset Life Cycle. The models are significantly versatile, allowing for a quantitative evaluation of changes in maintenance policies, energy price variations, risks, variations of profits according to the real market and so on. The results demonstrate the robustness of the approach described and indicate that it maximises the Physical Asset Life Cycle, allowing for the consumption of world resources to be minimised and, as a result, contributing to a more sustainable world.

1. Introduction

According to ISO 55000 [1], “An asset is an item, thing or entity that has potential or actual value to an organization. The value will vary between different organizations and their stakeholders, and can be tangible or intangible, financial or non-financial. The period from the creation of an asset to the end of its life is the asset life. An asset’s life does not necessarily coincide with the period over which any one organization holds responsibility for it; Instead, an asset can provide potential or actual value to one or more organizations over its asset life, and the value of the asset to an organization can change over its asset life”.

A sustainable world implies the maximisation of a Physical Asset’s Life Cycle, aiming to fully utilise its profitability. Knowing the time when a Physical Asset reaches its end of life contributes to its maximum use, thus contributing to sustainability because of the use of new Physical Assets being minimised. Maletic et al. [2] present a relationship between physical asset management and sustainability, reinforcing the need to cover the assets’ entire life cycle, highlighting the significance of sustainability when performing asset management.

On the other hand, Weerasekara et al. [3] link life cycle management to industry 4.0 in order to achieve sustainability goals. Ratnayake [4] emphasises business strategy and management shifts in favour of sustainability, linking it to PAS 55. However, because of technological upgrades and environmental evaluation, the manager can decide the best time to replace assets through the knowledge and use of the Life Cycle models presented in this paper.

The circular economy is the key to the future of the planet, and to bring it to fruition, it is necessary to respect the environment and achieve sustainability.

Peña et al. [5] recommend consistent use of a life cycle assessment in the global development, adoption and implementation of the circular economy in order to move more swiftly and effectively towards environmental sustainability.

According to ISO/TC 251 [6], “the Circular Economy in Asset Management should consider assets as material resources, maintaining, for as long as possible, a maximum number of materials/pieces/ components and, in any case, renewing them, prioritizing their repair, reuse, and recycling. Minimizing the use of resources, closing resource loops, and improving durability, performance and lifetime are all measures that should be promoted under a Life Cycle management perspective that strongly contributes to a circular economy”, and “the alignment of Asset Management with circularity is oriented to preserving environmental quality, contribute to a resilient and regenerative society, economic prosperity, and social equity, for the benefit of current and future generations, while also offering the opportunity to generate better financial and competitive returns”.

Additionally, and from a more global approach, we may consider the 17 Sustainable Development Goals of the United Nations, from which Goal 11 can be emphasised: Sustainable cities and communities. To help reach this objective, optimization of the Physical Asset Life Cycle may contribute significantly, and the models presented in this paper can help accomplish this goal.

The Life Cycle models presented in this paper can also be used to evaluate the impact of Environmental Norms, as well as Energy and Risk Norms. For example, the introduction of Environmental Norms can be evaluated economically, because, beyond its intrinsic importance, there are more and more economical incentives to implement them, aiming to contribute to the planet’s sustainability.

This paper aims to present and discuss several Physical Asset Life Cycle models and to make a comparative analysis, aiming to help support a manager’s decision. It is important to have good and robust quantitative support when deliberating on the most appropriate time to withdraw or renew a Physical Asset. It is with this objective that several Life Cycle models are discussed, including a reference to the Useful Life Cycle. This type of analysis is also very important for defining the most adequate value for a product or service provided to a customer, as well as the best maintenance policy, among other approaches. In fact, possibilities for combining model variables are immense, allowing for their adaptation to any type of business and for any type of analysis.

Based on the preceding, the following Research Questions (RQ) are formulated:

• RQ1—What type of variables can be used to assess Physical Asset Evaluation Models and in which situations?
• RQ2—What is the right time to withdraw or renew a Physical Asset to minimise waste and increase sustainability?
• RQ3—What is the importance of Apparent Rate in a Physical Asset Life Cycle?
• RQ4—What is the most adequate Physical Asset Life Cycle that best supports the manager’s decision?

This paper has the following structure:

• Section 2 presents a Literature Review;
• Section 3 presents Physical Asset Life Cycle Evaluation Models;
• Section 4 presents a Comparative Analysis of the Models;
• Section 5 discusses the results reached in the preceding section;
• Section 6 presents the Conclusions and Future Developments.

2. State of the Art

The problem of Life Cycle Cost (LCC) has received very little attention throughout time. LCC is also known as Life Cycle Assessment (LCA), or it is more rarely called Life Cycle Investment (LCI). As a result, the perspectives and activity sectors from which Physical Assets are seen may provide a specific approach. Liu et al. (2023) present a state-of-the-art review of Levelized Cost of Energy (LCOE) modelling and an economic analysis of wind power generators, namely after 2011, from the perspective of wind farm type, single unit capacity, output, time horizon and application region. Concerning this subject, and before this year, the authors refer to the work of Hoffman (2011), which complements the information about LCC, and presents 49 models that address parts of the life cycle or the entire life cycle of an offshore wind farm [7,8].

Lu et al. refer to the assertion that “Life Cycle Cost Analysis (LCCA) plays an essential role in the economic sustainability assessment of buildings, and Building Information Modelling (BIM) offers a potentially valuable approach to fulfil its requirement”. The paper reviews 45 relevant peer-reviewed articles through a systematic literature search, selection and assessment. The results show that three data exchange methods integrate BIM and LCCA through data input, calculation and output, emphasizing the importance of LCCA [9].

Mondello et al. present a comprehensive literature review focused on the “analysis of scientific articles in which the environmental and economic impacts of maritime means of transport have been assessed using the Life Cycle Assessment (LCA) and Life Cycle Cost (LCC) methods”. This study analyses 77 scientific papers, as well as integrated approaches. “The analysis is performed using three review methods: bibliometric, network, and systematic analysis” [10].

Hussien et al. present a literature review related to environmental Life Cycle Assessment (LCA) for buildings and buildings’ refurbishment from 1994 to 2022. Their paper concerns the following subjects: Life Cycle Assessment; LCA and Life Cycle Cost Assessment; LCA and life cycle cost analysis; LCA and social life cycle assessment [11].

Borroto Pentón et al. present “the state of the art of the application of optimization tools such as Genetic Algorithms, Simulation, Neural Networks, Markov Chains and Bayesian Networks in the Physical Asset maintenance management. The bibliographic references used were extracted from a detailed search that allowed the selection of the empirical studies presented, in the time horizon from 2010 to 2021” [12].

Khasreen, Banfill and Menzies present a review of LCA from the perspective of buildings. They highlight the need for its use within the building sector and the importance of LCA as a decision-making support tool [13].

Wittmanová et al. focus on the possibilities of using Life Cycle analysis to design specific integrated systems for stormwater management facilities, aiming to apply them in the future as a basis for the design of efficient and sustainable drainage systems in urban areas [14].

Seyedabadi and Eicker try “to determine whether the current studies have adequately addressed the gap in knowledge by effectively implementing Life Cycle Assessment (LCA) methodologies throughout the entire lifecycle of buildings while ensuring a sufficient level of detail and reliability in their findings”. However, the authors state that the papers they analysed do not identify the most adequate models to manage the assets under study. They also state the need “to have models to evaluate the physical assets’ life cycles aiming to support the manager’s decision, namely, to identify the most adequate time to withdraw or renew the assets” [15].

Hill et al. assert that, despite a notable increase in the literature on sustainable Construction and Demolition Waste management, there continues to be a lack of studies that comprehensively address the full Life Cycle Analysis of the building environment [16].

Hellweg et al. discuss the role of LCA in evaluating and shaping strategies on the decarbonization of energy systems, the circular economy, sustainable consumption and sustainable finance. The authors mention that cross-comparisons between LCA applications for various mitigation strategies reveal differences in maturity levels, methodological choices and the way that environmental assessment tools have been combined with LCA. They also state that economy-wide LCAs on the decarbonization of energy systems and sustainable consumption are already common, whereas economy-wide applications to the circular economy and prospective LCAs for sustainable finance are still in their infancy. Future research should develop a systematic classification of decision support problems, harmonised data and comprehensive guidance to improve the robustness and credibility of prospective economy-wide LCAs [17].

Acaroğlu and García Márquez present two very significant studies on “a life-cycle cost analysis of High Voltage Direct Current utilization for solar energy systems” where a “27-year Life-cycle cost projection has shown that the Voltage Source Converter-High Voltage Direct Current-Overhead Transmission Line outperformed the best with a 448.61 MEUR net present value, and a break-even 9 years after the beginning of the project pertained to a 1500 MW power rating option” [18]. In this study, an LCC is employed for economic analysis, involving some results like Net Present Value (NPV), net savings, internal rate of return, payback period and savings-to-investment ratio; the LCC considers all the costs that are incurred for the duration of a life cycle of work, a service or product.

Acaroğlu and García Márquez also present another interesting study on a life-cycle cost analysis with voltage source converters for bulk power transmission; this work examines high voltage direct current systems through submarine cables for offshore wind farms [19]. In this study, the authors present a cash flow analysis on the Life Cycle Cost (LCC) of the utilization of a High Voltage Direct Current (HVDC) in offshore wind farms through a Voltage Source Converter (VSC). Also, in this paper, the authors use the LCC analysis based on two main variables: cash flows and NPV.

In both studies, the preceding authors are aligned with some of the models presented in this paper.

As is stated in the preceding references, there is a gap in the economic models available to support Physical Assets’ LCAs, namely from a sustainability perspective. Aiming to fill this gap, the next section presents several models that are evaluated based on data supporting the models’ assessment in Section 4.

3. Physical Asset Life Cycle Evaluation Models

Aiming to maximise the use of Physical Assets with the objective of diminishing waste and increasing sustainability, the following models were developed:

• Uniform Annual Income Method (UAI)

  ○

  UAI with a reduction in Present Value;

  ○

  UAIV without a reduction in Present Value.

• Minimization of Total Average Cost Method (MTAC)

  ○

  MTAC with a reduction in Present Value;

  ○

  MTAC without a reduction in Present Value.
• Life Cycle Investment (LCI);
• Life Cycle of Physical Assets with Recovery (LCR).

3.1. Uniform Annual Income Method

The Uniform Annual Income Method is used to estimate the annual income needed to recover a given investment. It takes various factors into account, such as expenses, inflation rate and other taxes, to determine the amount of annual income required.

It is important to define the way the Apparent Rate (i_A) is calculated and influences the results of the models, because it is used in all of the models. In this paper, not only was the Real Interest Rate, Inflation Rate and Profit Rate introduced but also the Risk Rate. The relationship among these rates is described using Fisher’s equation (Equation (1)):

$$\left(1+i_A\right)=\left(1+r\right)\times \left(1+h\right)\times \left(1+p\right)\times \left(1+R\right)$$

(1)

where

i_A = Apparent Rate, r = Interest Rate, h = Inflation rate, p = Profit Rate, R = Risk Rate.

It is important to emphasise the integration of the Risk Tax in Equation (1), thus allowing for a measurement of the influence of this variable in the models under discussion. This approach is innovative, and it allows for many situations to be solved, including the impact of risk standards in organizations.

The Uniform Annual Income Method, including the Reduction to Present Value, is given in Equation (2):

$${UAI}_n=\frac{i_A{\left(1+i_A\right)}^n}{{\left(1+i_A\right)}^n-1}\ast {PNV}_n$$

(2)

Being the Present Net Value (PNV_n), given in Equation (3):

$${PNV}_n=II+{\sum }_{j=0}^n\frac{C{M_j+CO}_j}{{\left(1+i_A\right)}^j}-\frac{V_n}{{\left(1+i_A\right)}^n}$$

(3)

where

II = Initial Investment, CM_j = Cost of Maintenance in year j = 1, 2, 3, …n, CO_j = Cost of Operation in year j = 1, 2, 3, …n, i_A = Apparent rate, V_n = Value of the equipment over a period n = 1, 2, 3… N.

3.2. Minimization of Total Average Cost Method

The Minimization of Total Average Cost Method allows for a determining of the lowest average cost of ownership of the equipment and the respective year in which it occurs, which corresponds to its optimal replacement time.

The Minimization of Total Average Cost Method, including a reduction in Present Value, is given in Equation (4):

$$C_{n(MMTAC-RPV)}^{ }{=C_n^{\prime }+C_n^{″}=\mathrm{m}\mathrm{i}\mathrm{n}}_{n \in \left\{1, 2, \dots , \left.N\right\}\right.} \frac{1}{n}\left(II-\frac{V_n}{{\left(1+i_A\right)}^n}+{\sum }_{j=1}^n(\frac{C{M}_j+C{O}_j}{{(1+i_A)}^j})\right)$$

(4)

where

II = Initial Investment, CM_j = Cost of Maintenance in year j = 1, 2, 3, … N, CO_j = Cost of Operation in year j = 1, 2, 3, …n, V_n = Value of the equipment over a period n = 1, 2, 3…N, i_A = Apparent rate, n = Number of years n = 1, 2, 3, …n, C_n(MMTAC-RPV) = Average total cost.

3.3. Life Cycle Investment

The Life Cycle Investment Method not only considers expenses and taxes but also profits. This method allows for an evaluation of the right time as to when an investment does not return a positive value. Equation (5) defines this approach:

$${GR}_n={\sum }_{j=0}^n\frac{B_j}{{\left(1+i_A\right)}^j}-{\sum }_{j=0}^n\frac{F_j}{{\left(1+i_A\right)}^j}-{\sum }_{j=0}^n\frac{M_j}{{\left(1+i_A\right)}^j}$$

(5)

where

GR = Global Result from expenses and benefits in year n, i_A = Apparent Rate, B_j = Value of Benefits in year j, F_j = Cost of Functioning in year j, M_j = Cost of Maintenance in year j.

3.4. Life Cycle of Physical Assets with Recovery

The value of the Physical Asset over time may be seen from two perspectives: Devaluation, taking into account the Apparent Rate; and the value of a new asset, taking into account the Apparent Rate.

For many companies, it is important to create financial reserves that aim to be financially available to renew a physical asset or buy a new one at the end of its economic life cycle. Because of this, it is important to know when these two lifecycle times occur.

The preceding perspectives are opposites and are evaluated as follows:

The devaluation of the Physical Asset, taking into account the Apparent Rate (Equation (6))

$$C_{0D}^t=C_{0D}^{t-1}\times i_A$$

(6)

where

$C_{0D}^t$ = Value of the Physical Asset at time t, $C_{0D}^{t-1}$ = Value of the Physical Asset at time t − 1, i_A = Apparent Rate.

The value of a new Physical Asset, taking into account the Apparent Rate (Equation (7))

$$C_{0N}^t=C_0^{t-1}/i_A$$

(7)

where

$C_{0N}^t$ = Value of the new Physical Asset at time t, $C_{0N}^{t-1}$ = Value of the new Physical Asset at time t − 1, i_A = Apparent Rate.

The Total NPV, considering the devaluation of the Physical Asset and taking into account the Apparent Rate, is given in Equation (8)

$${NPV}_{TD}={NPV}_{FpD}+{NPV}_{Fe}$$

(8)

where

NPV_TD = Total Net Present Value considering Physical Asset Devaluation, NPV_FpD = Net Present Value of the Financial production movement, NPV_Fe = Net Present Value of the Financial Expenses movement.

The Total NPV, considering the value of a new Physical Asset and taking into account the Apparent Rate, is given in Equation (9)

$${NPV}_{TN}={NPV}_{FpN}+{NPV}_{Fe}$$

(9)

where

NPV_TN = Total Net Present Value considering a New Physical Asset, NPV_FpN = Net Present Value of the Financial Production movement, NPV_Fe = Net Present Value of the Financial Expenses movement.

3.5. Complementary Approaches

In all of the models, maintenance Key Performance Indicators (KPI) are used, namely the Mean Time Between failures (MTBF), the Mean Time To Repair (MTTR), the Mean Waiting Time (MWT) and the Availability (A)—Equations (10)–(13):

$$MTBF=\frac{{\sum }_{i=1}^{n}TB{F}_i}{n}$$

(10)

where

TBF = Time of Good Functioning (Time Between Failures), n = Number of intervals of Good Functioning;

$$MTTR=\frac{{\sum }_{i=1}^{n}TT{R}_i}{n}$$

(11)

where

TTR = Time To Repair, n = Number of intervals of interventions;

$$MWT=\frac{{\sum }_{i=1}^{n}W{T}_i}{n}$$

(12)

where

WT = Waiting Time to Repair the Physical Asset, n = Number of interventions.

$$A=\frac{MTBF}{MTBF+MWT+MTTR}$$

(13)

Relations among the preceding KPI may also be given in Equation (14):

$$MTTR=\frac{MTBF\ast \left(1-A\right)}{A}$$

(14)

These are the following devaluation methods necessary for the Uniform Annual Income Method and the Minimization of Total Average Cost Method:

3.5.1. Linear Depreciation Method

This method considers the decay rate of the value of the Physical Asset as constant over the years and is calculated as follows (Equation (15)):

$$d_l=\frac{II-{VC}_n}{N}$$

(15)

where

d_l = Annual depreciation quota; II = Initial Investment; VC_n = Residual value of the equipment at the end of N time periods; N = Time of life corresponding to VC_n, l—l = 1, 2, 3…n; V_n = Equipment value in period n = 1, 2, 3 … N.

3.5.2. Sum of the Digits Method

In this case, the annual devaluation is non-linear, being calculated as follows (Equation (16)):

$$d_l=\frac{\frac{2\ast N-\left(l-1\right)}{N+1}\ast II-V{C}_n}{N}$$

(16)

where

d_l = Annual depreciation quota; II = Initial Investment; N = Time of life corresponding to VC_n; VC_n = Residual value of the equipment at the end of N time periods; l—l = 1, 2, 3…n; V_n = Equipment value in period n = 1, 2, 3 … N.

3.5.3. Exponential Method

The exponential method incurs an annual declining depreciation charge over the life of the equipment, being calculated as follows (Equation (17)):

$$d_l=V{C}_{l-1}\times \left(1-\sqrt[N]{\frac{V{C}_n}{II}}\right)$$

(17)

where

d_l = Annual depreciation quota; II = Initial Investment; N = Time of life corresponding to VC_n; VC_n = Residual value of the equipment at the end of N time periods; l—l = 1, 2, 3…n; V_n = Equipment value in period n = 1, 2, 3 … N.

The value of the Physical Asset, V_n in a period n less than N, is given in Equation (18):

$$V_n=II-l\times d_l$$

(18)

4. Comparative Analysis of the Models

The first Research Question (RQ1—What type of variables can be used to assess Physical Asset Evaluation Models and in which situations?) are the following ones: Initial Investment (II); Residual Value of equipment (RV); Functioning costs (F); Energy costs (E); Maintenance costs (M); Benefits (B); Internal Rate of Return (r); Risk (R); Apparent Rate; MTBF; MTTR; MWT.

To evaluate the comparative analysis of the models, the same data are used. Some values from

Table 1. Data to support the models’ assessments.

Time (t)	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
r—real interest rate (1 + r)		1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05
h—inflation rate (1 + h)		1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02	1.02
p—profit rate		1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05	1.05
i—money rate (nominal interest rate)		0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12	0.12
i—Apparent Rate with Risk		0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14	0.14
1/((1+i)^t))		0.88	0.78	0.68	0.6	0.53	0.47	0.41	0.36	0.32	0.28	0.25	0.22	0.19	0.17	0.15	0.13	0.11	0.1	0.09	0.08	0.07	0.06	0.05	0.05	0.04
Initial Investment (II) (C0)	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000	−12,000
Initial Investment (II) (C0) with Devaluation (NPV)	−12,000	−10,565	−9302	−8190	−7211	−6349	−5590	−4921	−4333	−3815	−3359	−2957	−2604	−2292	−2018	−1777	−1565	−1377	−1213	−1068	−940	−828	−729	−642	−565	−497
Initial Investment (II) (C0) considering NEW Asset (NPV)	−12,000	−13,630	−15,480	−17,583	−19,970	−22,682	−25,762	−29,261	−33,234	−37,747	−42,873	−48,695	−55,307	−62,818	−71,348	−81,037	−92,041	−104,540	−118,736	−134,860	−153,174	−173,974	−197,599	−224,432	−254,909	−289,524
Fp Financial Movement (Production)	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000	50,000
Fp Financial Movement (NPV)	50,000	40,020	35,235	31,023	27,313	24,048	21,173	18,641	16,413	14,450	12,723	11,201	9862	8683	7645	6731	5926	5218	4594	4045	3561	3135	2760	2430	2140	1884
Fp Financial Movement (NPV) Accumulated	50,000	40,020	75,255	106,278	133,591	157,639	178,812	197,453	213,866	228,316	241,039	252,240	262,102	270,785	278,430	285,161	291,087	296,305	300,899	304,943	308,505	311,640	314,400	316,831	318,970	320,854
Functioning Costs	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500	−4500
Functioning Costs (NPV) Accumulated	−4500	−3962	−7450	−10,521	−13,226	−15,606	−17,702	−19,548	−21,173	−22,603	−23,863	−24,972	−25,948	−26,808	−27,565	−28,231	−28,818	−29,334	−29,789	−30,189	−30,542	−30,852	−31,126	−31,366	−31,578	−31,765
Energy	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000	−1000
Energy Accumulated (NPV) Accumulated	−1000	−880	−1656	−2338	−2939	−3468	−3934	−4344	−4705	−5023	−5303	−5549	−5766	−5957	−6125	−6274	−6404	−6519	−6620	−6709	−6787	−6856	−6917	−6970	−7017	−7059
Maintenance Costs	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300	−300
Maintenance Costs (NPV) Accumulated	−300	−267	−502	−708	−891	−1051	−1192	−1316	−1426	−1522	−1607	−1681	−1747	−1805	−1856	−1901	−1940	−1975	−2006	−2033	−2056	−2077	−2096	−2112	−2126	−2139
Total Costs (NPV) Accumulated	−5800	−5229	−14,181	−26,411	−41,527	−59,184	−79,078	−100,942	−124,540	−149,666	−176,135	−203,788	−232,484	−262,097	−292,517	−323,649	−355,407	−387,716	−420,511	−453,733	−487,332	−521,262	−555,483	−589,961	−624,666	−659,569
FT Net Present Value (NPV) (With II Devaluated)		24,226	51,772	71,677	84,854	92,107	94,144	91,590	84,992	74,835	61,544	45,494	27,015	6396	−16,105	−40,265	−65,884	−92,789	−120,825	−149,858	−179,767	−210,450	−241,812	−273,772	−306,260	−339,212
FT Net Present Value (With II New Asset)		21,162	45,594	62,285	72,094	75,773	73,972	67,250	56,091	40,903	22,030	−243	−25,689	−54,129	−85,435	−119,525	−156,361	−195,952	−238,349	−283,650	−332,001	−383,596	−438,682	−497,563	−560,604	−628,239

stand out, namely the following:

• Initial Investment (II) = −12,000
• Life Time simulated (years) = 25
• Residual value of equipment (RV) = 10
• Functioning costs? (F) = −4500
• Energy costs = −200
• Maintenance costs (M) = −300
• Benefits (B) = 50,000
• Internal Rate of Return (r) = 0.12
• Risk (R) = 0.11
• Apparent Rate = 0.2432
• Maintenance Increase Rate = 0.01
• MTBF = 500
• MTTR = 50
• MWT = 10
• Availability (%) = 0.8929

Figure 1 shows the results of the three devaluation methods without a reduction in Present Value. These results are relevant because of the next discussion of the Life Cycle models. A time interval of 25 years was considered. However, the real value of the market ought to be considered instead of using the Physical Asset Devaluation model.

It is also important to evaluate the Physical Asset Useful Life Cycle. This analysis is relevant because it allows for the Physical Asset Useful Life to be compared with the Economic Life, as discussed in the next subsections. The Useful Life of an asset ends when its maintenance costs exceed maintenance costs plus the capital amortization of an equivalent new Physical Asset. Figure 2 shows the Useful Life that results from the data above without a reduction in Present Value; the image shows that the useful life is 15 years, after which the equipment should be replaced or undergo renewal intervention.

4.1. Uniform Annual Income Method

4.1.1. The First Evaluation Is with a Reduction in Present Value

Figure 3 shows the Global Life Cycle analysis based on the Uniform Annual Income Method with a reduction in Present Value.

Figure 4 shows the Uniform Annual Income based on the three devaluation methods with a reduction in Present Value. As can be seen, Uniform Annual Income (UAI) is reached near the year 20. It is relevant to emphasise that this UAI approach only considers expenses.

Figure 5 shows the Global Result of the Uniform Annual Income Method, considering expenses and profits with a reduction in Present Value.

Comparing Figure 4 and Figure 5, it can be seen that the positive returns of a Physical Asset’s Life end at year 13. According to Figure 4, the Physical Asset is near the minimum value at year 13, which means that these two approaches converge. This result helps answer RQ2 (What is the right time to withdraw or renew a Physical Asset to minimise waste and increase sustainability?).

4.1.2. The Second Evaluation Is without a Reduction in Present Value

Figure 6 shows the Global Life Cycle analysis based on Uniform Annual Income Method without a reduction in Present Value.

Figure 7 shows the Uniform Annual Income based on the three devaluation methods without a reduction in Present Value. As can be seen, the Uniform Annual Income (UAI) is reached at year 4. It is relevant to emphasise that this UAI approach only considers expenses.

Figure 8 shows the Global Result of the Uniform Annual Income Method, considering expenses and profits without a reduction in Present Value.

As can be stated and according to what was expected, if we did not have any Taxes and Risks, and with an investment of 12,000 Monetary Units (MU) and an annual profit of 50,000 MU, it is obvious that the Global Result would be positive from the second year onwards. This result helps answer RQ3 (What is the importance of Apparent Rate in a Physical Asset Life Cycle?) because, in this case, the importance of the Apparent Rate is strongly demonstrated in the Life Cycle.

4.2. Minimization of Total Average Cost Method (MTAC)

4.2.1. The First Evaluation Is with a Reduction in Present Value

Figure 9 shows a Global Life Cycle analysis based on the Minimization of Total Average Cost Method (MTAC) with a reduction in Present Value.

Figure 10 shows the Minimization of Total Average Cost Method (MTAC) based on the three devaluation methods with a reduction in Present Value. As can be seen, the minimum value is reached at year 5. It is relevant to emphasise that this MTAC approach only considers expenses.

Figure 11 shows the Global Result of the Minimization of Total Average Cost Method (MTAC), considering expenses and profits with a reduction in Present Value.

Comparing Figure 10 and Figure 11, it can be seen that the positive returns of a Physical Asset’s Life end at year 13, which is higher than the one given by the Minimization of Total Average Cost Method (MTAC), as calculated by one of the devaluation methods. This result helps answer RQ2 (What is the right time to withdraw or renew a Physical Asset to minimise waste and increase sustainability?).

4.2.2. The Second Evaluation Is without a Reduction in Present Value

Figure 12 shows a Global Life Cycle analysis based on the Minimization of Total Average Cost Method (MTAC) without a reduction in Present Value.

Figure 13 shows the Minimization of Total Average Cost Method (MTAC) based on the three devaluation methods without a reduction in Present Value. As can be seen, the minimum value is reached at year 13. It is relevant to emphasise that this MTAC approach only considers expenses.

Figure 14 shows the Global Result of the Minimization of Total Average Cost Method (MTAC), considering expenses and profits without a reduction in Present Value.

As can be seen, if there were no Taxes or Risks, in a utopic economy with an investment of 12,000 Monetary Units (MU) and an annual profit of 50,000 MU, it is obvious that the Global Result would be positive from the second year onwards. This result helps answer RQ3 (What is the importance of Apparent Rate in a Physical Asset Life Cycle?) because, in this case, the importance of the Apparent Rate in the Life Cycle is strongly demonstrated.

4.3. Life Cycle Investment Method

Figure 15 shows a Global Life Cycle analysis based on the Life Cycle Investment Method.

Figure 16 shows the Global Result of Life Cycle Investment (LCI), considering expenses and profits with a reduction in Present Value.

As can be seen, with the Life Cycle Investment (LCI) Method, the positive Physical Asset Life again lasts until year 13, as was the case with the previous method of the preceding section. This result helps answer RQ2 (What is the right time to withdraw or renew a Physical Asset to minimise waste and increase sustainability?).

4.4. Life Cycle of the Physical Asset Method with the Recovery Method

Figure 17 shows a Global Life Cycle analysis based on the Recovery Method.

Figure 18 shows the Global Result of the Life Cycle of the Physical Asset Method with the Recovery Method, considering expenses and profits.

As can be seen, based on the method under discussion and considering normal devaluation, a positive Life Cycle ends at year 13. However, if we want to reinvest in a new Physical Asset to replace the old one, the right time is year 11.

When comparing the results of all the evaluations, it can be concluded that year 12 is the most adequate time to withdraw or renew a Physical Asset. Even in Figure 18, the interval of decision-making is between years 11 and 13, with year 12 being suggested as the most preferable option. This result helps answer RQ2 (What is the right time to withdraw or renew a Physical Asset to minimise waste and increase sustainability?).

5. Discussion

The preceding section presented an evaluation for each model of the Physical Asset Life Cycles, and many differences among the models can be noticed. The preceding model assessments allow for clearer answers to the Research Questions initially placed.

As for RQ1 (What type of variables can be used to assess Physical Asset Evaluation Models and in which situations?), the answer is the following:

• In this paper, four Life Cycle models, considering several variables, were compared. With the exception of two situations in two models where both the use and disuse of the Apparent Rate were compared, all of the remaining values in the variables were maintained constant. The objective was to make the comparative analysis as rigorous as possible.
• However, the objective of the models was that they remained as versatile as possible. All of the variables in the models could be changed, making it possible to simulate a wide range of situations that aimed to respond to all real-life situations.
• The main independent variables are the following: Initial Investment (II); Residual Value of Physical Asset (RV); Functioning Expenses (F); Energy Expenses (E); Maintenance Expenses (M); Benefits (B); Internal Rate of Return (r); Risk (R); Apparent Rate (I_A); Maintenance Increase Rate; MTBF; MTTR; MWT.

As for RQ2 (What is the right time to withdraw or renew a Physical Asset to minimise waste and increase sustainability?), the answer is the following:

• As can be stated by the preceding discussion, the interval between years 13–15 was the most adequate for the example data. This is the time interval that maximises the use of Physical Assets and, by consequence, diminishes waste and increases sustainability.
• Based on the preceding information, the importance of using several evaluation methods when aiming to reach a time interval for decision-making is emphasised, due to the fact that the results are slightly different among all the methods. This reinforces the importance of the models under discussion, as well as accurate data to apply to the models.

As for RQ3 (What is the importance of Apparent Rate in a Physical Asset Life Cycle?), the answer is the following:

• As was discussed in the Uniform Annual Income Method without a reduction in Present Value and in the Minimization of Total Average Cost Method without a reduction in Present Value, we stated that, if profits are considered (since they are higher than investments), the Global Results are always positive, which is not the case in a real economy. Therefore, the Apparent Rate is structural in the evaluation of the Physical Asset Life Cycle.
• The Apparent Rate can also incorporate the Risk Tax, which enlarges its importance and extends its potential for application.

As for RQ4 (What is the most adequate Physical Asset Life Cycle that best supports the manager’s decision?), the answer is the following:

• According to the Uniform Annual Income Method (UAI) with a reduction in Present Value, the Physical Asset Life Cycle provides positive returns that end at year 13. According to devaluation methods, a Physical Asset reaches its minimum value in year 13, which means that these two approaches converge to show that year 13 is a tipping point year for an asset.
• According to the Uniform Annual Income Method (UAI) without a reduction in Present Value and according to devaluation methods, a Physical Asset reaches its minimum value at year 4. However, if there are no Taxes and Risks, the Global Result is always positive. Therefore, this last approach is useless in supporting decision-making due to the fact that it is not realistic.
• According to the Minimization of Total Average Cost Method (MTAC) with a reduction in Present Value, it can be seen that the Physical Asset Life provides positive returns at year 13, which is higher than the one given by MTAC and calculated by one of the devaluation methods (year 5).
• According to the Minimization of Total Average Cost Method (MTAC) without a reduction in Present Value and according to devaluation methods, the Physical Asset reaches the minimum value at year 13. However, if we do not have any Taxes and Risks, the Global Result is always positive. Then, this last approach, again, must not be considered to support decision.
• According to the Life Cycle Investment Method, the Physical Asset Life is positive until year 13.
• According to the Life Cycle of Physical Assets with the Recovery Method and considering normal devaluation, a positive Life Cycle ends at year 15. However, if we want to reinvest in a new Physical Asset to replace the old one, the right time is year 13.
• Additionally, it may be emphasised that the preceding information is consistent with the value reached for the Useful Physical Asset Life Cycle.

It can be concluded that all the proposed methods have some advantages to support management decisions. However, the most adequate Physical Asset Life Cycle method that best supports a manager’s decision is the Life Cycle Investment or the Life Cycle of Physical Assets with the Recovery Method, since a Physical Asset provides profits. The Uniform Annual Income Method (UAI) and the Minimization of Total Average Cost Method (MTAC) are methods that may be used alone if profits are not considered. Additionally, it is desirable to use all the Taxes involved.

Based on the Life Cycle models presented and validated in this paper, a manager can know the optimal time to renew or withdraw a Physical Asset, allowing for an increase in sustainability due to the following:

• Physical Assets are used for the maximum time possible while they return positive results;
• New Physical Assets are only purchased when the older assets reach their maximum advantageous use;
• Instead of buying new Physical Assets, the manager can choose to renew them, allowing for an increase in waste reduction;
• This approach allows for the maximisation of Physical Asset use and, as a result, increases responsibility for the environment and, obviously, contributes to sustainability.

6. Conclusions

This paper presents and demonstrates the potential of Life Cycle models for Life Cycle Physical Asset assessments. Four research questions were formulated, and four models demonstrated their robustness through model assessments and with variations.

The approach presented in this paper maintains the innovative idea that a circular economy is the key to a better future for this planet. To implement it, it is necessary to respect the environment and create environmental sustainability, bringing greater social responsibility on the part of organizations and greater return on assets. Optimal management decisions create greater economic return, thus representing social and economic sustainability.

Based on this perspective and using data from model assessments, the most adequate time to withdraw or renew a Physical Asset was identified. This allows for maximisation of the Economical Life Cycle, aiming to minimise world resources and reduce waste.

Situations with and without an Apparent Rate, and with the aim of emphasizing their importance in a real economy, were considered. The importance of profits in a Physical Asset Life Cycle was also referred to, which may allow for the selection of the most adequate time to withdraw or renew a Physical Asset.

The results are strongly robust, which demonstrate that the models presented may help support managers in their decisions about the Physical Assets that they manage.

The results also demonstrated that maximising the Physical Asset Life Cycle allows for minimisation of the consumption of world resources and, by consequence, contributes to a more sustainable planet.

The results of the model assessments are mainly represented graphically and for generic situations. However, they have the capacity for adaptation to any other type of Physical Asset.

In the near future, the models presented will be used in several case studies on real economies, using many variations, and this will progressively increase the robustness and versatility of the models presented.