The Impact of Lean Production on the Cycle Time: A Case Study of a Welding As...
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1. Journal of Quality in Maintenance Engineering
Model for economic replacement time of mining production rigs including
redundant rig costs
Hussan Saed Al-Chalabi Jan Lundberg Majid Al-Gburi Alireza Ahmadi Behzad Ghodrati
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Hussan Saed Al-Chalabi Jan Lundberg Majid Al-Gburi Alireza Ahmadi Behzad Ghodrati ,
(2015),"Model for economic replacement time of mining production rigs including redundant rig costs",
Journal of Quality in Maintenance Engineering, Vol. 21 Iss 2 pp. 207 - 226
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4. DMC Decreasing maintenance cost (%)
nl Number of labours
DRC Decreasing redundant rig cost (%)
cl Man hour cost (cu/h)
RT Replacement time (month)
SPPi Spare part cost for preventive
maintenance (cu)
ERTs Scaled economical replacement time
LCPi Labour cost for preventive
maintenance (cu)
TC Total cost (cu)
RCi Redundant rig cost (cu)
cu Currency unit
PTi Using time of redundant rig (h)
AC Acquisition cost (cu)
CRi Redundant rig cost per hour (cu/h)
i Time period (month)
TRi Logistic time for redundant rig (h)
MCi Maintenance cost (cu)
TFi Restoring time of faulty rig
to operation (h)
OCi Operating cost (cu)
T1i Moving time of redundant rig from
its location to production point (h)
COi Compensation cost (cu)
T2i Moving time of redundant rig from
production point to its original
location (h)
Si Resale value (cu)
TMi Moving time of faulty rig from
production point to workshop (h)
r Discount rate (%)
TWi Time in workshop of faulty rig (h)
N Number of replacement cycles
TLi Moving time of repaired rig from
workshop to production point (h)
CMi Corrective maintenance cost (cu)
tdi Delay time in workshop of faulty rig
before repair (h)
PMi Preventive maintenance cost (cu)
tri Actual repair time of faulty rig (h)
SPCi Spare part cost for corrective
maintenance (cu)
tIi Idle time in workshop of faulty rig
after repair (h)
1. Introduction
Industrial companies, or more specifically, mining companies put huge funds, often
millions of dollars into their annual budgets to purchase heavy mobile equipment
(HME) such as drilling rigs, scaling rigs, wheel dozers, wheel loaders, dump trucks, etc.
Given the enormous costs of acquiring, operating and maintaining their HME, it is
important for companies to optimise their replacement and procurement strategies
(Richardson et al., 2013). As the HME operating hour’s rise, so too do the maintenance
and operating costs. At some point in the equipment’s life span, these costs will be too
high; it will no longer be economically viable to continue using the old equipment, so it
should be replaced (Verheyen, 1979). An essential economic consideration in industrial
companies is to find a model that can discriminate this point (i.e. the point at which
the equipment replacement time is expected to yield minimal life-cycle cost (LCC)).
Obviously, for mining companies, one of the most important decisions is determining
the economic replacement time (ERT) of capital equipment; this can be done with the
help of LCC analysis. The main reason for the increasing use of the life cycle costing
concept for HME is that at some point the operating and maintenance (O&M) costs will
exceed their acquisition costs.
In general, LCC is determined by summing up all potential costs associated with
equipment over its life time (i.e. the total of ownership and acquisition costs). It is well
known that the value of expenditure today costs more than the same expenditure next
year because of the decreasing “time value of money”. In this study we use a discount
rate to account for the time value of money. To compare costs incurred at different
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5. times, we must shift expenditure to a reference point in time. Thus, we calculate the
present equivalent value of the costs by considering the discount rate factor.
1.1 Literature review
Standard models for ERT decisions contain an estimation of the discounted costs by
minimising the total cost of the equipment. The assumption of these models is that
equipment will be replaced at the end of its economic lifetime by a continuous sequence
of identical equipment (Hartman and Tan, 2014). Bellman (1955) developed the first
optimal asset replacement model for variable lifetime of assets. Wagner (1975) offered
dynamic programming formulation for the equipment replacement problem in which
the state of the system is the time period and the decision at each period is to keep the
equipment for N periods. His formulation has been extended by researchers to deal
with realities of technological changes (e.g. see (Oakford et al., 1984; Bean et al., 1985;
Hartman and Rogers, 2006; Hritonenko and Yatsenko, 2008)). These authors assumed
a finite horizon in their approaches to the problem of equipment replacement under
non-stationary costs. Elton and Gruber (1976) showed that an equal life policy was
optimal on an infinite horizon under technological changes. In contrast, Hartman
and Murphy (2006) studied an asset replacement problem for a stationary finite
horizon; they illustrated how a bound on the number of times an asset is retained at its
economic life can be obtained, thus suggesting it is optimal to replace the asset at
its economic lifetime.
Dynamic programming models have been utilised in real cases of calculating
equipment replacement time because of the important uncertainties associated with
LCC (Richardson et al., 2013). The net present value of all LCC associated with an
infinite sequence of equipment life cycles has also been used to make equipment
replacement decisions (Bethuyne, 1998; Scarf and Bouamra, 1999; Hartman, 2005;
Yatsenko and Hritonenko, 2005). Other researchers have used different equipment
replacement models to analyse a variety of equipment, such as forklifts, buses and
aircraft (Eilon et al., 1966; Keles and Hartman, 2004; Bazargan and Hartman, 2012).
Although Tanchoco and Leung (1987) found replacement decisions could be influenced
by capacity considerations, others have noted that technological changes can encourage
decision makers to utilise equipment beyond its economic life (Cheevaprawatdomrong
and Smith, 2003). Still other researchers have considered reliability, maintainability
and optimum replacement decisions; readers are referred to, e.g., Wijaya et al. (2012);
Dandotiya (2012), Golmakani and Pouresmaeeli (2014) and Al-Chalabi et al. (2014a,b) for
further discussion of the recent literature.
1.2 Aim of the study
Blanchard et al. (1995) mentioned that the costs associated with equipment support,
operation and maintenance can account for more than 75 per cent of the equipment
LCC. Thus, careful consideration must be given to estimating the capital equipment
ownership costs. Given the importance of O&M costs and in order to assure a specific
level of availability, industries must consider using redundant equipment to overcome
production loss when a failure occurs. Another way to ensure production performance
is to make a pooling agreement with other companies, renting commonly owned
equipment to ensure that failed equipment will be replaced by serviceable machines.
But any of these compensation strategies cost money for the operator. In the
assessment of equipment replacement time, the compensation cost (i.e. redundant rig
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6. cost in this case study) associated with a machine should be taken into account.
Thus, the aim of this paper is to develop a model to determine the ERT of production
equipment, in this case, a drilling rig, considering redundant rig cost. In this paper, we
also consider the relative importance of other cost factors on the rig’s ERT; these
factors are the equipment acquisition, operating, maintenance and redundant rig costs.
Finally, in the model, we consider the time value of money by using a discount rate.
2. Case study
In this study, our model of ERT for production machines is implemented in a case
study of a drilling rig used in mining industry. Our case study was selected from the
mining sector, as maintenance costs in this sector account for 20-30 per cent of the total
cost of production (Kumar, 1994). Kumar (1994) also maintained that to achieve optimal
performance from the capital intensive mining equipment and systems, mine operators
must ensure world-class maintenance in line with other advanced industries. The
drilling rig is selected as a case study for several reasons: drilling is the first step in a
typical mining cycle and thus is extremely important (see Figure 1); the drilling rigs are
heavily loaded; the rigs’ acquisition and maintenance costs are high; finally, drilling
represents a critical bottleneck for production.
3. Model formulation
The ERT of capital equipment is the age that minimises its total cost. In this study, the
total cost is represented by acquisition (initial or investment) cost and ownership cost.
The ownership cost includes O&M costs and compensation cost. All repairable systems
wear over time; consequently, the ownership cost increases and the resale value
decreases.
In this study, the ERT is defined as the value of the replacement time (RT ) which
minimises the total discounted cost, calculated on a monthly basis as follows:
MinTC ¼ Min AC þ
XRT
i¼1
MCi þOCi þCOið Þ
" #
ÀSi
!
Â
1
1þrð Þ
i
12
( )
 N
" #
(1)
The objective of the proposed model (i.e. Equation (1)) is to determine the ERT which
minimises the total discounted cost over the rig’s planned lifetime. We assume the
1. Drilling
2. Charging
3. Blasting
4. Loading
5. Scaling
6. Bolting
Figure 1.
Typical underground
mining cycle
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7. replacement rig (i.e. the new rig) has the same performance and cost as the existing rig
(i.e. identical rigs). The number of replacement cycles during the planned lifetime is
represented as:
N ¼
T
RT
(2)
where T and RT represent the planned lifetime and the replacement time (in months),
respectively.
3.1 Maintenance cost
Maintenance can be defined as any work used to keep something in an appropriate
condition. Maintenance can corrective or preventive. Corrective maintenance refers
to actions which take place after an unscheduled breakdown to return an item to
a specified condition. Preventive maintenance refers to regularly scheduled actions
planned to keep an item in the desired condition. The maintenance cost can be labelled
as a summation of the materials and labour expense required to keep an item in suitable
working condition. In this study, due to the company’s regulations, all costs data are
encoded and expressed as currency unit (cu). The maintenance cost is represented as
follows:
MCi ¼ CMi þPMi (3)
where CMi and PMi represent corrective and preventive maintenance cost (cu),
respectively:
CMi ¼ SPCi þLCCi (4)
where SPCi and LCCi represent spare part and labour costs for corrective maintenance
(cu), respectively:
SPCi ¼ SPVi þSPLi (5)
where SPVi and SPLi are spare part value and spare part logistic costs (cu), respectively:
LCCi ¼ rt  nl  cl (6)
where rt represents repair time (h), ni is number of repairs and cl is man hour cost
(cu /h):
PMi ¼ SPPi þLCPi (7)
where SPPi and LCPi represent spare part and labour costs for preventive maintenance
(cu), respectively:
SPPi ¼ SPVi þSPLi (8)
LCPi ¼ rt  nl  cl (9)
3.2 Operating cost
Data on O&M costs were collected over four years and stored in the MAXIMO
computerised maintenance management system (CMMS). Operating cost can be
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8. defined as recurring costs for efficiently operating the equipment, in our case study,
a drilling rig. The operating costs include administration, energy, fuel, indirect
overhead costs, consumables like steel rods, operator’s salary, a figure given to us
by experts at the collaborating company. In CMMS, the cost data are recorded based on
calendar time. Since drilling is not a continuous process, the operating cost is estimated
by considering the utilisation of the drilling rig. The company plans to use the machine
for 120 months. Therefore, extrapolation for the O&M cost data was done. Figures 2
and 3 illustrate the maintenance and operating costs determined by the data
extrapolation.
In Figures 2 and 3, the dots represent the real historical data for maintenance and
operating costs. Curve fitting is done by using Table curve 2D software to show the
behaviour of these costs before and after the time when data were collected. Note that
the fitting would be better if more data were available for a time period of more than
four years. This software uses the least squares method to find a robust (maximum
likelihood) optimisation for nonlinear fitting. It is worth mentioning that the drilling
machine in this case study has no multi-level preventive maintenance programme.
In addition, it was new at the start of utilisation. This is the main reason why the
maintenance cost is quite low in earlier months. The history shows that when
the maintenance costs started growing, the user company began to keep track of cost
400
300
200
100
0
0 50 100
Time (month)
Maintenance cost
Maintenancecost(cu)
150 200
−100
−200Figure 2.
Maintenance cost
150
100
50
0
0 50 100 150
Time (month)
Operation cost
Operationcost(cu)
200
−50Figure 3.
Operating cost
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9. data by using CMMS. The equation “Lorentzian Cumulative” of extrapolation for
expected maintenance cost obtained by the software is expressed as:
Y ¼
a
p
arctan
xÀb
c
þ
p
2
(10)
where Y represents the expected maintenance cost, a ¼ 217.42, b ¼ 112.37, c ¼ 13.63,
r2
(adj.)¼ 0.97 and X represents the time (1, 2, 3, 4, […], n months). Similarly, the equation
“Lorentzian Cumulative” of extrapolation for expected operating cost is expressed as:
Y ¼
a
p
arctan
xÀb
c
þ
p
2
(11)
where Y represents the expected operating cost, a ¼ 79.89, b ¼ 109.2, c ¼ 13.85,
r2
(adj.)¼ 0.91 and X represents the time (1, 2, 3, 4, […], n months).
As the figures show, the OM costs increase over time. In fact, the number of
failures increases with time and/or the machine consumes more energy due to machine
degradation.
3.3 Compensation cost
In the present study, we focus on the compensation cost by using a redundant rig cost
as one of the critical factors affecting the ERT. Mining companies, for example, lose
a large amount of money each year from lost production which, in turn, is due to the
production equipment’s downtime. In fact, this may be the most important factor
affecting the ERT of production machines. In this study, we assume when a drilling rig
fails and is sent to the workshop for maintenance, to continue production without stops,
the company uses a redundant rig which has the same performance as the existing
faulty rig. Since in the mining industry, downtime in production is almost zero, the
compensation cost in this case represents the cost of using a redundant rig. In this
study, the experts at the collaborating mine classified the rig’s failures in three
categories as[1]:
(1) failures fixed by maintenance team at the workshop;
(2) failures fixed by maintenance team at the production point (mining room); and
(3) failures fixed by operators at the production point (mining room).
Detailed information, such as experience in years and work position of the experts,
appears in Table I.
The compensation cost based on a Category 1 failure of the rig is modelled as
follows:
COi ¼ RCi (12)
where RCi represents redundant rig cost (cu):
RCi ¼ PTi  CRi (13)
where PTi and CRi represent the used time of the redundant rig (h) and redundant rig
cost per hour (cu/h), respectively:
PTi ¼ TRi þTFi (14)
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10. where TRi and TFi represent the logistic time of the redundant rig (h) and the time to
restore the faulty rig to operation (h), respectively:
TRi ¼ T1i þT2i (15)
where T1i and T2i represent the time to move the redundant rig from its location to the
production point and the return time from the production point to its original location
(h), respectively:
TFi ¼ TMi þTWi þTLi (16)
where TMi, TWi and TLi represent for the time to move the faulty rig from the
production point to the workshop (h), time in workshop (h) and return time after repair
(from workshop to production point) (h), respectively:
TWi ¼ tdi þtri þtIi (17)
where tdi, tri and tIi represent delay time in workshop before repair (h), actual repair time
(h) and idle time in workshop after repair (h), respectively.
Figure 4 illustrates the time the redundant rig is used due to a category 1 failure in
the existing rig.
Table II represents the clarifications of symbols A, B, C, D, E and G of Figure 2.
Since the moving speed inside the underground mine is limited to low speed, we
assume the moving time of the maintenance team from the workshop to the production
point is almost equal to the moving time of the faulty rig from a production point to the
same workshop. Thus, the using time a redundant rig is used after a category 2 rig
failure is modelled as[2]:
PTi ¼ TRi þTMi þtri (18)
Table III illustrates the minimum and maximum time values given by the maintenance
expert in the collaborating mine; these are used in the model.
Current position at companies (U) and (M) Expert field and experience (no. of years)
Maintenance engineer for open pit and
underground mines (U)
Maintenance of mobile and fixed equipment’s (23)
Mine production foreman (U) Underground drill machines (30)
Mine production manager (U) Mine drilling and production (15)
Mine production planner (U) Mine production planning (22)
Maintenance supervisor (U) Maintenance of mobile equipment’s (30)
Maintenance manager (U) Maintenance of mobile equipment’s (26)
Mine production manager (U) Mine drilling and production (32)
Maintenance foreman (U) Maintenance of mobile equipment’s (25)
Maintenance engineer for fixed equipment (U) Maintenance of fixed equipment’s (10)
Global service operations manager (M) Maintenance of equipment (20)
Design engineer–underground drill rigs (M) Designing underground equipment (10)
Global fleet manager Marketing and business management (8)
Vice president service operations (M) Parts and Service Business management and
Maintenance of mobile equipment (18)
Regional business-Europe and product line
manager-rental (M)
Project management and business management (10)
Table I.
Description of
expertise of the
experts used in the
present study
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11. We assume the moving time of the redundant rig T1i is equal to the moving time of the
faulty rig TMi. It is worth mentioning that the time values tdi, tIi, TMi, TLi, T1i and T2i
are randomly generated by using MATLAB code, since this type of data is not
available from the collaborating mine. We use a discount rate of 10 per cent to
consider the “time value of money” following the suggestions of the collaborating
mining company.
3.4 Resale value
A Matheson formula (declining balance depreciation model) was used to estimate the
resale value of the rig after each month of operation. In this method, a fixed percentage
of the book value at the beginning of the month represents the monthly depreciation of
the rig. The rig resale value is its value if/when the firm wants to sell it at any time
Symbol Clarification
A Production stops and a redundant rig starts moving from its location
B Production starts with a redundant rig and a faulty rig starts moving to the workshop
C Faulty rig enters the workshop
D Faulty rig exits the workshop
E Faulty rig starts work after repair and redundant rig starts moving to its original location
G Redundant rig arrives at the original location
Table II.
Clarifications of
symbols A, B, C, D,
E and G of Figure 2
C
T1i TM i
tdi
tri tIi
D
TLi
TW i
TFi
PTi
A B
T2i
E
G
Time
Figure 4.
Time the redundant
rig is used due to
a category
one rig failure
Time Minimum Maximum
Moving time of faulty rig from production point to workshop (TMi) 30 60
Delay time in workshop before repair (tdi) 30 90
Idle time in workshop after repair (tIi) 30 60
Table III.
Minimum and
maximum time
values (minute) used
in the model
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12. during its lifetime. The resale value denoted by Si is calculated by using the following
model (Luderer et al., 2010; Eschenbach, 2010; Dhillon, 2010):
Si ¼ BV1 Â 1ÀDrð Þi
(19)
where “i” represents a time (number of months), i ¼ 1, 2, 3, […]120 planed lifetime, BV1
and Dr represent the rig value on the first day of operation and depreciation rate,
respectively. In addition:
BV1 ¼ AC Â A (20)
where A represents the percentage of decrease multiplied by the rig acquisition cost to
represent the rig value on the first day of use. During discussions with us, company
experts agreed that the rig acquisition cost decreases by 10 per cent on the first day of
use (i.e. A ¼ 0.9). In this study, the rig acquisition cost is 6,000 (cu). Hence, the rig value
on the first day of use is 5,400 (cu).
The depreciation rate that allows for full depreciation by the end of the planned
lifetime of the rig is modelled by the following formula (Luderer et al., 2010; Dhillon,
2010):
Dr ¼ 1À
SV
BV1
1
T
(21)
where T and SV represent the planned lifetime of the rig, 120 months, and rig scrap
value, respectively. The rig is assumed to reach scrap value after 10 years. The rig
resale value is calculated by:
Si ¼ AC Â A Â 1ÀDrð Þi
(22)
The declining balance depreciation model is suitable in our case study because this
model writes off the cost of the rig early in its lifespan at an accelerated rate and at
correspondingly lower monthly charges close to the end of its lifespan. It also considers
the rig to be more productive when it is new, and its productivity declines continuously
due to rig aging. Therefore, in the early years of its lifespan, a rig will generate more
revenue than in later years. In accountancy, depreciation refers to two aspects of the
same concept. The first is the decrease in the rig value. The second is the systematic
allocation of the capital cost of the rig over its lifespan. The scrap value is an estimate
of the value of the equipment at the time it is sold or disposed of. In our case study, 50
(cu) is assumed to be the scrap value of the rig at the end of its planned lifetime, a figure
given to us by company experts.
4. Results and discussion
We tested the model for ERT on a case study of a drilling rig. This rig is manufactured
by Atlas Copco Company and used by Boliden mineral AB Company in Sweden.
MATLABTM
software is used to enable a variation of the replacement time (RT ) of
Equation (1) which minimises the total cost.
Figure 5 shows the optimisation curve and the ERT of our case study at a redundant
rig cost per hour equal to 1 (cu/h).
The results show the lowest possible total cost can be achieved by replacing the rig
at 104 months of its planned lifetime. A decision to replace the rig before or after its
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13. ERT incurs greater costs for the user company. The use of a lower replacement age
(i.e. less than 104 months) incurs higher costs due to the high investment cost.
Meanwhile, if the lifetime of the rig exceeds its ERT (i.e. more than 104 months), losses
will increase for two reasons:
(1) the OM and redundant rig costs increase when the operating hours increase
due to rig degradation; and
(2) the rig resale value will decrease for each month of operation until it reaches its
scrap value by the end of its planned lifetime.
As Figure 5 also shows, there is a range 97-109 (months) when the minimum total cost
can be still achieved in practice. In this study, we call it the economic replacement
range. Finding the economic replacement range is an important result of our study, as it
can help decision makers in their planning. To show the effect of the redundant rig cost
per hour (CRi) in the ERT of our case study, we change the values of the redundant rig
cost per hour from 1 to 6 (cu/h). Figure 6 shows the result.
It is clear from Figure 6 that increasing the CRi (cu/h) has a negative effect on the
ERT of the drilling rig. To determine the effect of other factors on the ERT, we perform
a sensitivity analysis on rig acquisition, operating, maintenance and redundant rig
costs (cu) using the ANN technique. Four MATLAB codes for six cases of CRi (1-6 cu/h)
are used to identify the effect of increased acquisition cost (IAC), decreased operating
cost (DOC), decreased maintenance cost (DMC) and decreased redundant rig cost
(DRC). The resulting ERT from these four codes is fed as input to the ANN and the
results translated into a relatively simple equation to estimate the ERT of the drilling
rig. The method of partitioning weights, proposed by Garson (1991) and adopted
by Goh (1995), is used to determine the relative importance of the various input factors
(see Figure 7).
As evident in Figure 7, the most important factor is the redundant rig cost, followed
by the acquisition, maintenance and operating costs. Therefore, a design for reliability
and maintainability should be adapted to reduce the downtime and maintenance costs
of the drilling rig. As mentioned earlier, four MATLAB codes are used to identify the
effect of IAC, DOC, DMC and DRC on the ERT of drilling rig. We choose the case where
0 20 40 60 80 100 120 140 160 180 200 220 240
0
5
10
15
18
x 104 Economic replacement time of a drilling rig
Replacement time RT (month)
Totalcost(cu)
Redundant rig cost = 1 (cu/h)
ERT=104 months
Figure 5.
Economic
replacement time of
the drilling rig
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14. CRi ¼ 1 (cu/h) to demonstrate the effect of these factors on the ERT. Figure 8 shows
the correlation of DRC and IAC for a given 25 per cent DOC and DMC. Figure 9 shows the
correlation of DRC and DMC for a given 25 per cent IAC and DOC. Figure 10
shows the correlation of DRC and DOC for a given 25 per cent IAC and DMC.
As Figures 8-10 show, DRC, IAC, DMC and DOC have a positive effect on the ERT
of the drilling rig, but it is also evident that DRC has a more positive effect, followed
by IAC, DMC and ROC.
0
30,000
60,000
90,000
120,000
150,000
180,000
0 20 40 60 80 100 120 140 160 180 200 220 240
Totalcost(cu)
Replacement time (month)
Effect of redundant cost per hour
CRi=1 (cu/h). ERT=104 month CRi=2 (cu/h). ERT 94 month
CRi=3 (cu/h). ERT 87 month CRi=4 (cu/h). ERT 82 month
CRi=5 (cu/h). ERT 79 month CRi=6 (cu/h). ERT 76 month
CRi = Redundant rig cost per hour
Figure 6.
Effect of the
redundant rig cost
per hour on the ERT
of the drilling rig
0
10
20
30
40
50
60
1 2 3 4 5 6
IAC (cu) 33.1 33.1 31.7 34.6 30.5 32.8
DOC (cu) 14.1 11.9 8.0 9.4 10.3 3.2
DMC (cu) 20.4 15.0 17.7 10.1 12.3 6.2
DRC (cu) 32.2 39.8 42.3 45.7 46.6 57.5
Relative1mportance(%)
Figure 7.
Relative importance
of input factors on
ERT of drilling rig
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15. 0 5 10 15 20 25 30 35 40 45 50
110
112
114
116
118
120
122
124
126
ERT(month)
Decreasing redundant rig cost (%)
IAC=10%
IAC=20%
IAC=30%
IAC=40%
IAC=50%
DOC=25%
DMC=25%
Figure 8.
Correlation of DRC
and IAC for a given
25 per cent DOC
and DMC
0 5 10 15 20 25 30 35 40 45 50
112
114
116
118
120
122
124
126
ERT(month)
Decreasing redundant rig cost (%)
DMC=10%
DMC=20%
DMC=30%
DMC=40%
DMC=50%
IAC=25%
DOC=25%
Figure 9.
Correlation of DRC
and DMC for a given
25 per cent IAC
and DOC
0 5 10 15 20 25 30 35 40 45 50
114
116
118
120
122
124
ERT(month)
Decreasing redundant rig cost (%)
DOC=10%
DOC=20%
DOC=30%
DOC=40%
DOC=50%
IAC=25%
DMC=25%
Figure 10.
Correlation of DRC
and DOC for a given
25 per cent IAC
and DMC
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16. 4.1 Training and testing the proposed ANN model
ANNs can perform nonlinear modelling without prior information and are able to
learn complex relationships between inputs and outputs; the process is also fast
(Ahmadzadeh and Lundberg, 2013). Our ANN analyses are based on the results
obtained from the four cases represented by four MATLAB codes, as explained above.
The resulting ERTs from these codes are fed as inputs to ANN and the results
translated into a relatively simple equation which can be used to estimate the overall
ERT of the drilling rig. The equation is transformed to an Excel spread-sheet to make
ERT estimation quick and easy for any engineer to apply. As mentioned earlier, the
proposed model has four inputs: IAC, DOC, DMC and DRC. A hidden layer with three
neurons and a nonlinear transfer function allows the network to learn nonlinear and
linear relationships between input and output variables. The number of neurons in the
output layer is constrained to one, as the output only requires one parameter, in this
case, the ERT of the drilling rig. 90 per cent of the data are used in training and
10 per cent in testing the neural network; see Figures 11 and 12. The model shown
in Figures 11 and 12 have very high values of R ¼ 99 per cent for ANN. However, as
also shown in the figures, the neural network model yields outputs very close to the
desired targets with a high level of accuracy.
The proposed ANN model is used to construct a formula to calculate the ERT of our
case study. The formula is transformed to an Excel spread-sheet to make ERT
estimation quick and easy for any engineer to apply. The structure of the optimal ANN
model is shown in Figure 13; its connection weights and threshold levels are
summarised in Table IV.
Pre-processing data by scaling improves the training of the neural network. To
avoid a slow rate of learning, specifically near the end points of the output range (due to
the property of the sigmoid function, which is asymptotic to values 0 and 1), the input
105 110 115 120 125 130 135
105
110
115
120
125
130
135
Targets T
OutputsY,LinearFit:Y=(1)T+(−0.44)
Outputs vs Targets, R = 0.99819
Data Points
Best Linear Fit
Y = T
Figure 11.
Training capability
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17. and output data are scaled in the interval between 0.1 and 0.9 (Oreta, 2004). It should be
noted that any new input data should be scaled before being presented to the network
and the corresponding predicted values should be un-scaled before use (Yousif, 2007).
The linear scaling equation is expressed by:
Xs ¼
0:8
D
X þ 0:9À
0:8Xmax
D
(23)
105 110 115 120 125 130 135
105
110
115
120
125
130
135
Targets T
OutputsY,LinearFit:Y=(1)T+(−0.81)
Outputs vs Targets, R =0.99829
Data Points
Best Linear Fit
Y = T
Figure 12.
Testing capability
IAC (%)
DOC (%)
DRC (%)
DMC (%)
Output
(ERT)
Inputfactors
1
2
3
4
5
6 8
7
Figure 13.
Optimal structure
of artificial
neural network
(ANN) model
Hidden layer nodes
wij weights from node ith input layer to node jth
hidden layer
Hidden threshold (θj)i ¼ 1 i ¼ 2 i ¼ 3 i ¼ 4
j ¼ 5 0.085 0.089 −0.034 0.108 −8.776
j ¼ 6 3.006 0.379 1.441 1.784 −1.120
j ¼ 7 1.416 0.452 1.509 1.876 −5.001
Output layer nodes wij weights from node ith hidden layer to
node jth output layer
Output threshold (θj)
i ¼ 5 i ¼ 6 i ¼ 7
j ¼ 8 4.679 3.410 5.376 −3.581
Table IV.
Weights and
threshold levels of
proposed ANN
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18. where Xs represents the scaled value of input factors and X represents the un-scaled
value of input factors. As it used in MATLAB code for neural networks, Equation (23)
is used here for an IAC, DOC, DMC and DRC between a minimum increasing or
decreasing percentage of 1 per cent (Xmin) and a maximum increasing or decreasing
percentage of 50 per cent (Xmax). This results in:
D ¼ XmaxÀXmin (24)
The equation length depends on the number of nodes in the hidden layer. Adopting
three nodes gives an accuracy of 99 per cent. The small number of connection weights
of the neural network enables the ANN model to be translated into a relatively simple
formula, in which the predicted ERT can be expressed as follows:
ERTs ¼
1
1þexp
À y8 þ w5:8
1
1 þ eÀx1
þ w6:8
1
1 þ eÀx2
þ w7:8
1
1 þ eÀx3
n o (25)
where ERTs represents the scaled ERT derived from the ANN model, θj represents the
output threshold and wij represents the weight from node i in the hidden layer to node j
in the output layer. Hence:
x1 ¼ y5 þw5:1 Â IAC þw5:2 Â DOC þw5:3 Â DMC þw5:4 Â DRC (26)
x2 ¼ y6 þw6:1 Â IAC þw6:2 Â DOC þw6:3 Â DMC þw6:4 Â DRC (27)
x3 ¼ y7 þw7:1 Â IAC þw7:2 Â DOC þw7:3 Â DMC þw7:4 Â DRC (28)
To obtain the actual value of ERT, the predicted ERTs must be re-un-scaled using the
following formula:
X ¼ Xs
D
0:8
À0:9
D
0:8
þXmax (29)
Equation (29) is obtained by solving Equation (23), considering the input variable (X) as
unknown and the output variable (Xs) as known. An Excel spread-sheet can be used as
a substitute for fast and accurate calculation of the ERT of the drilling rig. Equation
(30) is applied to estimate the actual value of the ERT of the drilling rig as follows:
ERT ¼ ERTs
ERTmaxÀERTmin
0:8
À0:9
ERTmaxÀERTmin
0:8
þERTmax (30)
where ERTmax and ERTmin represent the maximum and minimum values of ERT,
respectively, derived from the optimisation model.
In this paper, it is worth to mention that the ANN techniques were used for the
following two main reasons:
(1) One aim was to help the engineers and decision makers in the user company to
estimate the ERT of new drilling rigs without needing to use complicated
software.
(2) Another aim was to determine the relative importance of factors which were
used in the optimisation model and which would affect the ERT of new drilling
rigs. The factors which have the highest impact on the ERT of new rigs should
be prioritised in the development process of new drilling rigs.
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19. 4.2 Regression analysis
The regression analysis of the results obtained from the above four MATLAB codes
uses Minitab software and the least squares method. ERT is modelled as a linear
function of IAC, DOC, DMC and DRC. The regression analysis results in the following
mathematical model:
ERT ¼ 104þ0:19 Â IAC þ0:04 Â DOC þ0:14 Â DMC þ0:17 Â DRC (31)
It is evident from the regression analysis for this particular case (i.e. CRi ¼ 1 cu/h) that
the IAC has the greatest effect on the ERT of the drilling rig, followed by DRC, DMC
and DOC, but at the same time, CRi (cu/h) increases from two to six for other cases.
Ultimately, the DRC has the largest effect on ERT in our case study; see Figure 7. The
R2
value obtained from the regression analysis, R2
(adj) ¼ 99 per cent, indicates that the
ERT of the drilling rig depends linearly on the factors of IAC, DOC, DMC and DRC,
supporting the results obtained in the sensitivity analysis.
5. Conclusions
This paper presents a model for the economical replacement time of production
machines. Although the problem has been solved previously by other researchers
using different models, our model can more readily examine the relationship between
the factors affecting the ERT of production machines, especially the cost of using
a redundant rig. The model is found to be a good choice for estimating the ERT in a case
study of the drilling rig used in underground mines in Sweden, and it can be extended
to other production capital assets in other industries. In our case study, the results of
the sensitivity analysis show that the redundant equipment cost has the highest impact
on the ERT followed by equipment acquisition, maintenance and operating costs. The
results of the sensitivity analysis also indicate that decreasing the operating,
maintenance and redundant rig costs have a positive effect on increasing the ERT. The
results obtained from the optimisation curves show that increasing the redundant rig cost
per hour has a negative effect on the ERT. Therefore, improving the reliability and
maintainability of production equipment is essential to reduce their downtime and
maintenance costs.
The absolute ERT of the drilling rig when CRi ¼ 1 cu/h is 104 months. However, the
ERT has a range of 97-109 months, during which period the total cost remains almost
constant. This means the user company has the flexibility of making replacements
within the optimum replacement age range (12 months). The results of the regression
analysis show that the ERT of the new equipment depends linearly on its acquisition,
operating, maintenance and redundant rig costs. These results confirm the results
of the sensitivity analysis. In summation, this study presents a comprehensive and
very practical approach which can determine the ERT of any mobile equipment with
higher levels of certainty by using ANN analysis.
Notes
1. Note: we obtained information on the drilling process and maintenance of drilling rigs by
talking with experts at the user company (U) and manufacturing company (M).
2. Note: as the failures fixed by operators are classified as small failures and take only a short
time, the mining company does not use a redundant rig in the third category of failures.
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About the authors
Hussan Saed Al-Chalabi received Bsc Eng Degree in Mechanical Engineering from the Mosul
University, Iraq in 1994 and MSc Degree in Mechanical Engineering in Thermal Power from the
Mosul University, Iraq in 2008. Then he joined the Department of Mechanical Engineering at
the Mosul University as a Lecturer. Since 2011, he joined the Division of Operation, Maintenance
and Acoustics at LTU as a Doctoral Student. Hussan Saed Al-Chalabi is the corresponding
author and can be contacted at: hussan.hamodi@ltu.se
Jan Lundberg is a Professor of Machine Elements at the Luleå University of Technology and
also a Professor in Operation and Maintenance with focus on product development. During the
years 1983-2000, his research concerned mainly about engineering design in the field of machine
elements in industrial environments. During the years 2000-2006, his research concerned mainly
about industrial design, ergonomic and related problems as cultural aspects of design and
modern tools for effective industrial design in industrial environments. From 2006 and forward,
his research is completely focused on maintenance issues like methods for measuring failure
sources, how to do design out maintenance and how to design for easy maintenance.
225
Model for ERT
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22. Majid Al-Gburi received BSc Eng Degree in Civil Engineering from the Mosul University, Iraq
in 1998 and MSc Degree in Structural Engineering in time dependent of concrete behaviour from
the Mosul University, Iraq in 2001. Then he joined the Department of Dam and Water Resources
Engineering at the Mosul University as a Lecturer. Since 2011, he joined the Division of
Structural Engineering and Production at the LTU as a Doctoral Student.
Alireza Ahmadi is an Assistant Professor at the Division of Operation and Maintenance
Engineering, Luleå University of Technology (LTU), Sweden. He has received his PhD Degree in
Operation and Maintenance Engineering in 2010. Alireza has more than ten years of experience
in civil aviation maintenance as a Licensed Engineer, and Production Planning Manager.
His research topic is related to the application of RAMS and Maintenance optimisation.
Behzad Ghodrati is an Associate Professor of Maintenance and Reliability Engineering at the
Lulea˚ University of Technology. He obtained his PhD Degree on “Reliability based spare parts
planning” from the Lulea˚ University of Technology and he was awarded the Postdoctoral
Research Fellowship from the University of Toronto in 2008.
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Model for economic replacement time of mining production rigs including
redundant rig costs
Hussan Saed Al-Chalabi Jan Lundberg Majid Al-Gburi Alireza Ahmadi Behzad Ghodrati
Article information:
To cite this document:
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(2015),"Model for economic replacement time of mining production rigs including redundant rig costs",
Journal of Quality in Maintenance Engineering, Vol. 21 Iss 2 pp. 207 - 226
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