Development of an economic replacement time model for mining equipment: a case study

In mining operation equipment replacement represents a strategic decision problem. This paper presents an economic replacement time model for mining drill rigs. A total ownership cost minimization model was developed to optimize the lifetime of a drill rig used in Tara underground mine in Ireland. The developed methodology allows an innovative practical evaluation of the replacement process by applying sensitivity and regression analysis to rank the factors affecting the replacement time of existing and new models of the production drill rig. Compared to previous studies presented in the literature, the present study represents a further development in this field as it has resulted in a practical optimization model that can be used to estimate the economic replacement time of repairable equipment used in the mining and other production industries. The proposed model shows that the absolute economic replacement time of the drill rig investigated in this case study is 81 months and the mining company operating the rig can replace it with an identical one within an optimal replacement range of 6 months (i.e. from month 79–84) when the minimum total cost can still be achieved in practice. Sensitivity and regression analyses show that the maintenance cost has the largest impact on the economic replacement time of the drill rig. The study finds that decreasing the operating and maintenance costs of the drill rig will have the positive effect of increasing the economic replacement time linearly for a new model of the drill rig. The proposed model helps decision-makers to plan the replacement of old rigs and purchase new ones from an economic view point. Thus, this new model can be extended and used for more general applications in the mining industry.


Introduction
Rational production machinery replacement is a very important issue for most companies in different branches of industry, and includes the replacement of mining rigs, aircraft, trucks, cars, buses for local transport, and production machinery, indeed almost all types of machinery (Grinyer 1973;Elton and Gruber 1976;Bean et al. 1994;Scarf and Hashem 1997;Bethuyne 1998;Regnier et al. 2004;Rogers and Hartman 2005;Hritonenko and Yatsenko 2007;Mercier 2008;Richardson et al. 2013;Nguyen et al. 2013;Yatsenko and Hritonenko 2017;Huang et al. 2021). In modern industrial economics, the need for scientifically justified replacement decisions is increasing at an extremely rapid pace. Practical and effective replacement decisions are required for different assets, for example personal computers in higher education (Babbitt et al. 2009), public infrastructure assets (Al-Douri et al. 2020;van den Boomen et al. 2019), machines and equipment in steel production (Šebo et al. 2013), machines and equipment in forestry (Diniz and Sessions 2020), navy aircraft (Keating et al. 2014) and expensive mining machinery (Al-Chalabi et al. 2015a, b).
Decisions on the replacement of old machinery with new depend on many factors, for example the degradation rate of the machinery, the market prices, technological changes, etc. (Hartman and Tan 2014). This study focuses on a replacement policy for production drill rigs used in the mining industry, more specifically, the drill rigs used underground in Tara Mine in Ireland. In this study, a practical optimization model for the economic replacement time of the drill rigs was developed. In the context of of mining in Tara mine, drilling is the process of making holes in the faces and walls of underground mine rooms, to prepare those rooms for the subsequent operation, which is the charging process. From the mining point of view, the drilling process represents the bottleneck of the mining production cycle, since drilling is the first process of this cycle (Hamodi 2014). The drill rigs used in underground mines are subject to degradation due to their harsh working environment throughout their operating life. This reduces the rate of production and increases the maintenance and operating costs, causing a negative economic effect. Given all these factors, it is important for mining companies to perform life cycle cost (LCC) analysis to estimate the economic replacement time (ERT) of their production rigs, even prior to purchasing them (Markeset and Kumar 2018;Galar et al. 2017).
The life cycle cost of equipment is determined by summing up all the potential costs associated with the equipment over its lifetime (i.e. the acquisition costs and the total cost of ownership). It is known that the value of expenditure today costs more than the same expenditure next year because of the decreasing money value concept in the "time value of money". In this study, a discount rate was used to account for the time value of money. To compare costs incurred at different times, we must shift different costs to a reference point in time. Therefore, the present value of the costs for the case study was calculated by considering the discount rate factor.
Standard models for equipment replacement decisions contain an estimation of the discounted costs by minimizing the total ownership cost for the equipment. The assumption of these models is that equipment will be replaced at the end of its optimal lifetime by a continuous sequence of identical equipment (Hartman and Tan 2014). Recently, a number of researchers have studied the optimal replacement time of different repairable and non-repairable assets considering different cost parameters and maintenance issues (Yun and Choi 2000;Moghaddam and Usher 2010;Wijaya et al. 2012;Al-Chalabi et al. 2015a, b;Adkins and Paxson 2017;Yatsenko and Hritonenko 2020a, b;Petroutsatou et al. 2021). Yatsenko and Hritonenko (2020a, b) analysed a profit-maximizing serial replacement problem, taking into account variable asset productivity, the operating cost, and a replacement cost that depended on the asset age and installation time. These authors examined and highlighted essential differences between profit maximizing and cost minimizing replacement strategies in the infinite horizon framework for industrial asset replacement. Their study focused on the complex dynamics of sequential asset replacements. Some researchers have applied the theory of dynamic programming to consider technological changes in their study of the optimal replacement time, and have used different mathematical frameworks to solve different asset replacement problems in connection with general technological changes. For example, Sadeghpour et al. (2019) presented a novel approximate dynamic programming (ADP) approach for solving large-scale nonlinear constrained equipment replacement problems. Their ADP approach used a rollout algorithm to formulate the problem in a rolling horizon, and their model was solved using a genetic algorithm. They implemented their approach for a case study of 497 transformers belonging to a power distribution company and found that their framework possessed favourable features, such as minimizing the effect of uncertainties in the state variables and measurement inaccuracies. Other researchers have considered continuous technological changes in their approaches to calculating the equipment replacement time (Rogers and Hartman 2005;Hritonenko and Yatsenko 2008;Roy et al. 2016). Furthermore, several researchers have studied the optimal replacement time of assets by considering discontinuous technological changes (Goldstein et al. 1988;Mahrez and Berman 1994;Yatsenko and Hritonenko 2009). In addition, the uncertainty involved in determining the optimal replacement policy has been considered by different researchers in various studies (Apeland and Scarf 2003;Richardson et al. 2013;Zheng and Chen 2018).
Although many researchers have studied the optimal replacement time of different assets by applying standard models and theories of dynamic programming to consider general, discontinuous and continuous technological changes and uncertainty in connection with determining the optimal replacement policy, more practical models for economic replacement time estimation are required for easier application in industry. Thus, the aim of the present study was to develop a practical optimization model using available cost data from the mining industry. This developed model was used to estimate the ERT of a drill rig used in Tara Mine in Ireland as a case study. The available cost data were the rig's purchase price and operating and maintenance costs. The production loss due to the downtime of the drill rig was not considered due to the availability of a redundant drill rig. In this study, the equivalent present value of these costs was considered by using a discount rate. Figure 1 shows the flow chart for the method used in this study. The study started with the collection of the maintenance cost (MC) data, the acquisition cost and the information regarding the operating cost (OC) of the drill rig investigated in the case study. The second step involved investigating the data collected from a data quality point of view. After filtering and sorting the data based on calendar time, the maintenance cost were calculated and the operating cost were estimated based on the information from Tara Mine for a period of 42 months. The planned operating time for the drill rig was determined by Tara Mine to be 10 years, and, therefore, data prediction (i.e. extrapolation) for the period stretching from month 43-120 was performed for the operating and maintenance cost. The optimization model for estimating the ERT of the drill rig was created in the next step of the LCC analysis. To identify the effect of the various cost factors and their correlation with the ERT of the drill rig, single and multi-variable sensitivity analyses were carried out, followed by regression analysis, to provide the company operating the rig with a mathematical model that could be used to estimate the ERT of a new model of the drill rig before purchasing it from the manufacturing company.

Method
In this study, a practical optimization model was developed based on the total discounted cost. The MATLAB™ software was used to vary the replacement time for an optimization time horizon of 10 years (120 months) and to permit sensitivity analysis. The TableCurve 2D software was used to show the behaviour of the operating and maintenance cost data after the period covered by the data collected (i.e. to predict the operating and maintenance cost for the period after that covered by the data collected). Regression analysis was carried out for the results obtained from the MATLAB codes by using the Minitab software.
In summary, the research described herein proposes an approach to life cycle cost optimization of drill rigs used in the mining industry. The proposed approach is based on estimation of the operating and maintenance cost, as well as the purchase price, for a case study on the Epiroc Simba. A discount rate of 10% was used to account for the time value of money.

Data collection
The data used in this study were collected from the MAX-IMO database of the computerized maintenance management system (CMMS) for a drill rig used by a mining company. A case study was conducted on the Epiroc Simba drill rig used underground in Tara Mine in Ireland, utilising cost data collected for a period of around 3.5 years (April 2016 to September 2019) for the analysis.
The cost data in the MAXIMO database include corrective and preventive maintenance costs. In the CMMS, the cost data are recorded based on work orders for the maintenance of the drill rig. Every work order contains a work order number, the reporting date, the repair time in hours, the labour, material and service costs, a failure description, and a description of the maintenance actions performed. Table 1 presents a sample of the raw data for this study. The operating costs, equipment purchase price, installation cost and other costs relevant to the case study were collected as well. The maintenance instructions from the company manufacturing the drill rig were also collected. Because drilling is not a continuous process, the operating cost was estimated by considering the utilization of the drill rig. For this study, all the cost data were encoded and expressed as a number of currency units (cu) to comply with the regulations of the company operating the rig. It is important to mention here that all the cost data used in this study are real costs unadjusted for inflation.
The data in MAXIMO are gathered from a variety of sources and, if they are incorrect, incomplete or inaccurate, the decisions based on them will be incorrect as well. The issue of data quality problems has been investigated in a study conducted by Castaño Arranz et al. (2020). MAX-IMO includes basic modules for the identification and codification of assets, work orders, preventive maintenance, and actions taken by equipment purchasing managers and warehouse managers, as well as tools for analysing information. These basic modules provide the foundation for an effective system of maintenance management. Figure 2 shows the calculated maintenance cost per calendar month for the period covered by the data collected.

Estimation of the operating cost for the drill rig
The operating cost can be defined as the sum of the recurring costs for efficiently operating the equipment in question. The operating cost data were collected from the mining company for the same period as that covered by the maintenance cost data collected (i.e. April 2016 to September 2019). The information and data regarding the Epiroc Simba drill rig operations include the following details: To estimate the operating cost of the electric motor of the Epiroc Simba, the following factors were considered: • the motor power = 16 kw, • the electricity price = 0.09 cu/kwh, • the actual electric power time (in hours).
where (cu) represents the currency unit. The data for the actual electric power time (in hours) were collected for the same period as that covered by the maintenance data collected. The data were sorted based on calendar time for each month of operation and the actual electric power time was calculated.
The operating cost for the electric power motor was estimated for each calendar month of operation using Eq. 1: Three different operating time values were calculated: • the electric power time, • the drilling time, • the diesel engine working time.
The electric power time is recorded when the electric motor is running and represents the time spent at the mine face. The drilling time is only recorded during the drilling process. The diesel engine working time (1) OC el =Motor power × Electricity price × Actual electric power time To estimate the operating cost for the diesel engine of the Epiroc Simba, the following factors were considered: • the diesel fuel consumption (in litres/hour), • the diesel fuel price = 0.6 cu/litre,x • the actual diesel engine working time (in hours).
The engine fuel consumption data concerned the same period as that covered by the maintenance data collected and were generated as random numbers between 15 and 25 L/hour based on information received from the experts at Tara Mine.
The data for the actual diesel engine working time (in hours) were collected for the same period as that covered by the maintenance data collected. The data were sorted based on calendar time for each month of operation, and the actual diesel engine working time was estimated.
The operating cost for the diesel engine was estimated for each calendar month of operation using Eq. 2: To estimate the operating cost for the drilling consumables (bits, rods, etc.), the following factors were considered: • the number of metres drilled (metres/month), • the cost of the consumables used for drilling one metre (cu/metre).
The data on the number of metres drilled concerned the same period as that covered by the maintenance data collected and were generated as random numbers between 3000 and 8000 m/month based on the information received from the experts at Tara mine.
The operating cost for the drilling consumables was estimated for each calendar month of operation using Eq. 3: To estimate the operating cost for operator training, the following factors were considered: The operating cost for the operator training was estimated for each calendar month of operation using Eq. 4: The total operating cost for the Epiroc Simba was estimated for each calendar month of operation using Eq. 5: Figure 3 shows the estimated operating cost data per month for the same period as that covered by the maintenance data collected.
In general the operating cost of the drill rig should increase with time due to the degradation of it. However in this case the operating cost showing slightly decrease with the time. This is due to the unavailability of the exact data for drilled meters per month for this particular drill rig. Therefore, a random numbers of drilling metres per month is used to estimate the operating cost for the drilling consumables based on the information received from the experts at Tara mine.

Present value estimation of the maintenance and operating costs
The maintenance costs (the sum of the corrective and preventive maintenance costs) for each month of operation were calculated. The present value (PV) of the maintenance and operating costs, as well as the rig purchase price, was estimated using the following equations: (4) OC tr = Training period × Shift duration × Cost for training 12 where PV MC , PV OC and PV PP are the present value of the maintenance cost, operating cost and drill rig purchase price, respectively; MC i and OC i are the maintenance and operating costs, respectively, for the ith months; r is the interest rate, equal to 10%, and RT is the replacement time of the drill rig.
When developing the ERT optimization model, a period of 10 years (120 months) was selected as the planned lifetime of the drill rig based on information received from the mining company, and this length of time represents the optimization time horizon. Therefore, the MC and OC data were extrapolated (i.e. for data prediction) to generate data for the unavailable period (after September 2019). Figures 4 and 5 illustrate the expected present value (EPV) of the cumulative maintenance and operating costs for a period of 10 years. Moreover, Fig. 6 illustrates the EPV of the rig purchase price for a period of 10 years as well.
In Figs. 4, 5 and 6, the dots represent the present value of the real historical data for the maintenance and operating costs and the drill rig purchase price. Curve fitting was performed using the TableCurve 2D software to show the behaviour of these costs after the period covered by the data collected. It is important to mention that the fitting would have been better if more data had been available for a longer period than 42 months. TableCurve 2D uses the least squares method to find a robust (maximum likelihood) optimization for nonlinear fitting.
The next step in the calculations was to estimate the present value of the total ownership cost over each operating period. The expected present value of the total ownership cost was estimated using the following equation: where EPV TC is the expected present value of the total cost.

Optimization model development
A life cycle cost optimization model was developed to estimate the ERT of the Epiroc Simba drill rig. The objective was to minimize the total ownership cost over all the periods of the planned lifetime for the drill rig. The replacement rig was assumed to have the same performance (9)  Fig. 6 Expected present value of the purchase price of the Epiroc Simba and cost as the old one (i.e. to be identical to the old rig). Therefore, the number of replacement cycles during the optimization time horizon was formulated as follows: where RC represents the number of replacement cycles and RT represents the replacement time, 1, 2, 3,…120. The economic replacement time is the value of replacement time that minimizes the expected present value of the total ownership cost for the drill rig, as shown in the following equation: Figure 7 shows the results obtained when the MATLAB™ software was used to enable a variation of the replacement time parameter in Eq. (11) for a period of 120 months. In this figure one can identify the ERT of the drill rig that minimizes the expected present value of the total ownership cost. The figure shows that the lowest possible EPV TC can be achieved by replacing the drill rig almost every 7 years (81 months) if it is replaced with an identical rig working in the same operating conditions and working environment.

Results and discussion
The total operating time of the drill rig (i.e. the sum of the actual electric power time and the actual diesel engine working time) is 12,121 h based on the data received from the underground mine. The average operating time of the drill rig during 42 months of operation is 289 h per month. Since the estimated ERT of the drill rig is 81 months, the company operating the rig can replace it after 23,376 operating hours from an economic point of view. Figure 7 also shows that there is a range of 6 months (from month 79-84) when the minimum EPV TC can still be achieved in practice. In this study, this was called the economic replacement range (ERR). Finding the ERR is an important result of this study as it can help the company operating the rig in their replacement planning. Nevertheless, although 23,376 h of drill rig operation generate the absolute minimum cost, the company operating the rig can replace it within an ERR of 22,800-24,250 operating hours.

Single-variable sensitivity analysis
In single-variable sensitivity analysis, one factor is varied while the others are kept constant. The factors that were considered in the sensitivity analysis for the present study included the drill rig purchase price and the operating and maintenance costs. Figure 8 illustrates the effect of arising increase factor of the purchase price (IFPP) on the ERT of the drill rig. Figure 8 shows that the ERT is an increasing step function of the purchase price (PP) (based on percentage increases in the purchase price); i.e. the ERT remains constant for a specific range of IFPP increments and then increases stepwise. For example, if the purchase price factor increases from 1 to 2%, the ERT remains constant. This means that the ERT increases stepwise with specific PP percentage increments of 3,6,10,13,17,21,25,30,34,39,44 Fig. 7 Economic replacement time of the drill rig Fig. 8 Effect of increasing the purchase price of the drill rig on the ERT also show that the drill rig ERT will increase linearly for 12 out of 50 percentage increments of the IFPP. Figure 9 shows the effect of the decrease factor of the operating cost (DFOC) on the ERT of the drill rig (based on percentage decreases in the operating cost). It can be seen that when the drill rig operating cost decreases, the ERT will increase linearly, although it remains constant within a specific range of OC decrease. This means that the ERT is sensitive to a specific range of operating cost reductions, i.e. 7, 22, 36 and 50%. The results also show that the ERT of the drill rig will increase linearly for 4 out of 50 percentage increments of the DFOC. Thus, the drill rig's operating cost has a lower effect than its purchase price on the ERT. Figure 10 shows the effect of the decrease factor of the maintenance cost (DFMC) on the ERT of the drill rig. It can be seen that when the drill rig's maintenance cost decreases, the ERT will increase linearly, although it remains constant within a specific range of MC decrease. This means that the ERT is sensitive to a specific range of maintenance cost reductions (i.e. 3, 7, 11, 15, 19, 22, 26, 29, 33, 36, 39, 42, 45, 48 and 50%). The results also show that the drill rig's ERT will increase linearly for 15 out of 50 percentage increments of the DFMC. Thus, the drill rig's maintenance cost has a larger impact on the ERT than the purchase price and operating cost of the rig.

Multi-variable sensitivity analysis
To increase our understanding of the correlation of the input and output variables in the optimization model, a multi-variable sensitivity analysis was performed, considering three different cases. More specifically, MATLAB™ was used to vary in conjunction the three variables of the IFPP, DFMC and DFOC, to show their effects on the ERT of the drill rig. In all three cases, the purchase price increased while the operating and maintenance costs decreased.
As Fig. 11 shows, increasing the purchase price by 1%-50%, while decreasing the maintenance cost by the above-mentioned percentages and decreasing the operating  cost by 7% has the positive effect of increasing the ERT of the drill rig. Additional figures for decreasing the operating cost by different percentages (i.e. 22, 36 and 50%) are presented in Appendix 1.
As Fig. 12 shows, decreasing the operating cost by 1%-50% while increasing the purchase price by the abovementioned percentages and decreasing the maintenance cost by 3% has the positive effect of increasing the ERT of the drill rig. Additional figures for decreasing the maintenance cost by different percentages (7, 11, 15, 19, 22, 26, 29, 33, 36, 39, 42, 45, 48 and 50%) are presented in Appendix 2.
Case three represents the effect of the DFMC with percentage of 1%-50%, with different percentage of the IFPP, again according to the results obtained from the single-variable sensitivity analysis (i.e. 3, 6, 10, 13, 17, 21, 25, 30, 34, 39, 44 and 50%) and with different given percentage of the DFOC (i.e. 7, 22, 36 and 50%). Figure 13 shows the correlation of decreasing the maintenance cost by 1-50% and increasing the purchase price by the above-mentioned percentages with a given 7% decrease in the operating cost. As the figure shows, there is a positive effect on the ERT of the drill rig when the maintenance cost is decreased, the purchase price is increased and the operating cost decreased. Additional figures for decreasing the operating cost by different percentages (22, 36 and 50%) are presented in Appendix 3.
In conclusion, it can be seen from Figs. 11-13 that decreasing the maintenance cost has a larger impact on the ERT of a new model of the drill rig than increasing the purchase price and decreasing the operating cost.

Regression analysis
A regression analysis was performed on the results obtained from the sensitivity analysis for the three cases using the Minitab software and the least squares method. The ERT was modelled as a linear function of an IFPP, a DFOC and a DFMC. The regression analysis resulted in the following  1 3 mathematical model, which can be used by the mining company operating the drill rig to estimate the ERT of a new model of the rig before purchasing it. The assumption is that the new model of the drill rig will be used in the same mine under the same working conditions. Example of estimating the ERT of new manufactured model of drill rig based on assumption that the new model of drill rig have higher reliability, easy to maintain, consume less energy and more productive (i.e. drill more number of hols for the same period of time compared with the existing drill rig) than the existing case study is illustrated as shown: where IFPP = 10%, DFOC = 7% and DFMC = 10%. The regression model calculated the ERT of a new model of the drill rig as follows: The ERT obtained from the regression model is compatible with the values shown in Fig. 13.
The high R-squared adjusted value obtained from the regression analysis, 99.05, indicates that the ERT of the new drill rig depends linearly on the variables of the IFPP, DFOC and DFMC.
Following the results of the sensitivity and regression analyses, the factors affecting the increase in the ERT of a new model of the drill rig were ranked as follows: 1. the maintenance cost, 2. the purchase price, 3. the operating cost.

Conclusions
Based on the results obtained from this study the following conclusions have been reached.
• This study proposes a practical approach to determine the economic replacement time of a mining drill rig, the Epiroc Simba, used in Tara Mine in Ireland. The approach presented herein is based on financial data on the purchase price and the operating and maintenance costs col- lected from the mine, and it maybe useful for companies operating or manufacturing drill rigs. • According to the results obtained from the optimization curve, the absolute ERT of the Epiroc Simba is approximately 7 years (81 months). However, the ERT has a range of 6 months (month 79 to 84) during which the total ownership cost remains almost constant. This means that the company operating the rig has the flexibility of making replacements within an optimum replacement age range of 6 months. Therefore, there is no fixed date or age at which the total cost is minimum. In general, a range of 6 months provides the minimum total ownership cost. Finding the economic replacement range is an important result for the company operating the rig, as this can help them in their planning for a drill rig replacement. • The results of the sensitivity analysis performed to identify the effect of various factors on the ERT of the Epiroc Simba and the correlation of these factors with the ERT indicate that the IFPP, DFOC and DFMC have the positive effect of increasing the ERT of a new model of the drill rig. • The results of the regression analysis using three factors, the IFPP, DFOC and DFMC, show that the ERT of a new drill rig depends linearly on these factors. These results confirm the computation and the results of the sensitivity analysis. • The results from the sensitivity and regression analyses show that the maintenance cost has the largest impact on the ERT of the existing and the new model of the drill rig, followed by the purchase price and operating cost. Thus, the company manufacturing new drill rigs has to focus on decreasing the maintenance cost by increasing the reliability and maintainability of the new drill rig. This will help the manufacturing company to reduce the number of failures and decrease the operating and maintenance costs and finally increase the ERT of new drill rigs produced by the company.

Multi-variable sensitivity analysis
Effect of the IFPP, DFMC and DFOC on the ERT of the Epiroc Simba drill rig. See Figs. 14, 15, 16.