Nur Faizatus Sa’idah, Andi Cakravastia, Udjianna Sekteria Pasaribu , Bermawi Priyatna Iskandar

Corresponding email: nurfaizatuss@gmail.com

Corresponding email: nurfaizatuss@gmail.com

**Published at : ** 04 Apr 2023

**Volume :** **IJtech**
Vol 14, No 2 (2023)

**DOI :** https://doi.org/10.14716/ijtech.v14i2.5105

Sa’idah, N.F., Cakravastia, A., Pasaribu, U.S., Iskandar, B.P., 2023. A Two-dimensional Maintenance Service Contract Considering Availability and Maintenance Cost.

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Nur Faizatus Sa’idah | Doctoral Program in Industrial Engineering and Management, Faculty of Industrial Engineering, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia |

Andi Cakravastia | Industrial Engineering, Faculty of Industrial Engineering, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia |

Udjianna Sekteria Pasaribu | Statistics Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia |

Bermawi Priyatna Iskandar | Industrial Engineering, Faculty of Industrial Engineering, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia |

Abstract

In this paper, we study a two- dimensional Maintenance Service Contract (MSC) characterized by two limits (dimensions) of age and usage. It is considered that an agent offers a two-dimensional MSC by guaranteeing a certain level of equipment available to consumers. The agent needs to reduce the total maintenance cost to offer competitive MSC prices. Preventive maintenance actions (PM) are periodically carried out, and each PM action is considered to improve reliability modeled by reducing the failure rate function. Two decision variables (the PM interval (T) and the reduction in the intensity function are obtained by considering two performance measures that are relevant to agents and consumers (i.e., availability and total maintenance cost). A numerical example is presented by considering three types of equipment usage rates: low, medium, and high. The optimization of the two performance measures can ensure availability targets and, at the same time, minimize total maintenance costs.

Availability; Imperfect preventive maintenance; Maintenance service contract Total cost; Two-dimensional

Introduction

Maintenance Service Contract (MSC) is defined as
an equipment maintenance contract that agents offer to
consumers within a certain period. Maintenance contracts, generally, are characterized
by a time limit (e.g., 1 year), called an MSC with one dimension. However, for
equipment such as dump trucks whose failure patterns are affected by age and
use, it is necessary to include a usage limit (e.g., 100,000 km) in
addition to the age limit. Different types of equipment may have different ways
of measuring usage. For example, a photocopier's usage can be measured by the
number of copies made, while a machine tool's usage can be measured by the
hours it is used. This MSC is referred to as an MSC with two dimensions. To
appeal to consumers, the MSC may include promised equipment performance (e.g.,
Target 94% equipment availability) in addition to the price charged. A
comprehensive review of MSC can be found in Murthy
and Jack (2014), and MSC is studied

From
the consumer side, equipment used to support business processes will
deteriorate with age and use, and eventually, failure will occur (Jiang and Murthy, 2008). Maintenance is an effective way to slow down
the deterioration of equipment so that failure can be minimized and
availability can be kept high (Pariaman et
al., 2017; Suthep and Kullawong, 2015; Jackson and Pascual, 2008). Consumers who own equipment,
generally need MSC because maintenance activities are not their main business
(core business), so they do not need to be done in-house.

Using an agent to maintain equipment through a
Maintenance Service Contract (MSC) is more cost-effective and reliable than
in-house maintenance, which requires expensive investments in human resources,
equipment and technology. MSC offers consumers the benefits of saving on
maintenance costs and improving equipment performance. Many studies have
explored MSC from the consumer's perspective, including research conducted by Huber and Spinler (2012) and Jensen and
Stonecash (2009). In general,
research from the consumer’s point of view always wants (i)
minimum cost or maximum profit, with additional considerations of other
performance measures such as (ii) service quality (eg. delivery time) (De-Almeida, 2007) (iii) product reliability (Laksana and
Hartman, 2010), or (iv) availability (Datta and Roy, 2010). Due to the important role of maintenance in
maintaining the condition of equipment, the agents respond proactively to the
consumers’ needs by offering MSCs that promise high availability. In addition to getting regular income, agents
can also build good relationships with consumers, which positively impacts the agent’s image (brand building).

From the agent’s point of view, research was
carried out by Tarakci *et al.* (2006), where the agent offers specific incentives
based on the combination of target uptime and bonuses in the contract to
attract consumers. In addition, reliability improvement is also considered.
Generally, the agent looks for the optimal PM time interval to fulfill the
performance promise to the customer, which maximizes the profit. MSC can be
grouped into two categories: one-dimension MSC and two-dimension MSC.
One-dimensional MSC is characterized by a one-time limit (age). For example,
the MSC of a piece of equipment is for one year, while the two-dimension MSC is
characterized by an area of a two-dimensional plane where one dimension
describes a time limit and the other a usage limit (Iskandar *et al.*, 2014). One-dimensional MSC research has considered several important
performance measures for agents (i.e., total cost and benefit) as well as
consumers (i.e., reliability and availability). First, many MSC studies
consider profit performance measures (Darghouth, Ait-kadi,
and Chelbi, 2017; Hamidi, Liao, and Szidarovszky, 2013; Chang and Lin, 2012). Second, the reliability performance measure is
considered in addition to the benefits (Laksana and
Hartman, 2010). Third, the
availability performance measure is also involved (Su and Cheng, 2018; Iskandar *et al.*, 2014). The effect of PM's actions on increasing
reliability can be represented through a reduction in (i) failure rate or (ii)
virtual age. Some of them are researched by Darghouth, Ait-kadi, and Chelbi (2017) and Pasaribu, Husniah, and
Iskandar (2012), who specifically examined one-dimensional MSC with periodic PM
policies and the impact of PM reducing a virtual age. Other works are the
research of Husniah *et al.* (2019) and Yeh and
Chang (2007), which examine
one-dimensional MSC with a periodic PM policy model using an intensity
reduction function. There is also research by Iskandar
and Husniah (2017) and Yeh, Kao, and Chang (2009), who investigate one-dimensional contracts
using a PM policy that ensures equipment reliability.

Meanwhile, two-dimensional MSC
studies have not received much attention. As in one-dimensional MSC, the most
commonly used performance measure in two-dimensional MSC is profit (Huang, Gau, and Ho, 2015; Husniah *et al.,* 2014). No studies have considered the availability
of two-dimensional MSCs. At the same time, consumers want high equipment
availability to be guaranteed. The motivation to prioritize availability is
because the losses (costs) due to equipment downtime are very large. For
example, the breakdown of equipment such as draglines (in the mining business)
and airplanes (in the transportation business) will result in huge losses for
the company operating the equipment. Performance in two-dimensional MSC can be
achieved by implementing various preventive maintenance (PM) policies, which
can be divided into two categories, namely periodic and non-periodic (Jiang and Murthy, 2008).

In this paper, we propose a two-dimensional MSC that implements the imperfect PM policy carried out periodically. The effect of the PM action on the two-dimensional MSC will be developed by extending the formulation of the one-dimensional MSC. As in the one-dimensional MSC studied by Yeh and Chang (2007), It is assumed that the effect of PM reduces the intensity function. Furthermore, this paper focuses on two-dimensional MSC, The study is carried out from the agent's point of view, considering two performance measures, namely: (i) equipment availability and (ii) total maintenance costs, and involving two decision variables, i.e., the PM time interval (T), and the failure rate reduction value for each PM. As a result, the main contributions of this paper are (i) developing a PM policy that ensures the equipment availability target with minimum costs and (ii) obtaining optimal solutions of PM policies for a two-dimensional MSC with two objective functions, namely maximizing availability and minimizing total cost. This is in accordance with the goals desired by consumers for MSC, which are to ensure high availability (for example, 94%) and simultaneously meet the agency's goal of minimizing total maintenance costs. This paper is organized as follows: The model formulation is given in Section 2, which includes a discussion of the two-dimensional MSC formulation, failure modeling, and PM impact modeling. Section 3 describes an optimization to find the optimal solution that guarantees target availability with the minimum total cost. Section 4 provides numerical examples and a discussion of the results. Finally, we present conclusions and further research topics in Section 5.

Experimental Methods

**Model
Formulation**

This section will detail the two-dimensional MSC under study, failure
modeling, and PM effect modeling.

*2**.**1**. **Two-dimensional
MSC*

Consider
a two-dimensional MSC for equipment (e.g. dump trucks), which is characterized
by two parameters, L (time limit) and U (usage limit). For example, a
maintenance contract for a dump truck with a time limit of 1 year and a usage
limit of 100,000 km. Thus, these two contract limits form a rectangular area (Iskandar *et al.*,
2014). Suppose that the equipment with the rate of use is said to be moderate (normal) usage rate, low usage rate, and high usage rate . The contract will end at

*2.2. **Failure Modeling*

Equipment failures that occur randomly during the
contract period are considered random points that fall in the maintenance
contract area (which is rectangular). There are three approaches to modeling
random points (equipment failure) on a two-dimensional plane (Murthy and Jack, 2014). In this paper, we shall use the formulation of a one-dimensional point process as has been done by
Iskandar, Murthy, and Jack* *(2005). Let

Equipment is repairable, and the
equipment failure over time is modeled with an intensity function (Iskandar and Murthy, 2003).
Suppose*et al.*, 2014), and
this formulation will be used in this paper.

Suppose is a random variable representing the time of the
first failure for a given usage. The distribution function is given by the Weibull distribution function.
Next, the effect of usage patterns on equipment failure modeling will be
explained. It is considered that equipment with a high usage rate will deteriorate faster than equipment with a low
usage rate . The AFT formulation can be used to model the effect
of this usage pattern as follows. Let be the time to first failure for usage rate The relationship
of and is given
by, If the distribution function for is given by , with is the scale
parameter, then the distribution function for is the same as the distribution function for but with the scale parameter given by: Thus, we get . The intensity and cumulative functions with
respect to are given by and is a density
function with respect to

*2.3. **PM Policy*

The agent requires a proper PM
policy to reduce failures and downtime, and this will increase availability. In
this paper, two PM policies are considered.

__2.3.1. PM Policy 1__

PM
actions are carried out periodically or at the time of *L *

__2.4.1. PM Policy 1__

As mentioned previously, the effect
of PM action can be modeled through (a) virtual age or (b) failure intensity
function. Here, the impact of PM is modeled by reducing the failure intensity
function and is described as follows. For the use of ,
the PM action performed at and after
PM, the failure rate will be reduced bysuch that the failure rate in the interval
becomes becomes (See Fig. 2). The decision variable depends on y. One can group consumers by usage rate
y – e.g., low usage rate moderate usage rate and high usage rate;
hence each usage group has a unique .

__2.4.2. PM Policy 2__

In this PM policy, whenever PM (if the failure rate reaches ), failure rate will be reduced by . For a given Y = y, number of PM actions
throughout the period *L*

*2.5. **Availability and Total Maintenance Cost*

__2.5.1.
A____vailability__

The contract period is divided into intervals with the length of the first and last intervals Suppose that shows the availability value at the interval for is given by:

where are
the length of time PM and CM action. The
formula (availability in interval )is
used to get the optimal value that is at least the same as the availability
target. Availability
during the contract *L *or
the entire interval is described as follows.
Suppose is
the expected availability over
conditional on Y = y and
it is given by:

In practice, equipment owners want
high availability of equipment to achieve their production targets . In this, we propose an approach to guarantee an
availability target of in the entire MSC period by
determining the failure rate
reduction, such that the availability of
each interval is at least equal to the availability target.

__2.5.2. Total Maintenance Cost__

PM Policy 1: Total maintenance costs include minimal repair costs (including expected downtime penalties) and PM action costs. The total expected cost of a two-dimensional MSC over the period is given by:

PM
Policy 2: The formula for total
maintenance costs consists of minimal repair costs (including expected
penalties due to downtime) and PM action costs. The expected total cost for the
period is given by

**Optimal Decision **

This section discusses the optimization scheme
(Optimization Schema) of the two-dimensional MSCs under consideration.

*3**.**1**. **Two-dimensional MSC with PM 1 Policy
(Proposed)*

Three optimization schemes are considered: Optimization Schemes 1, 2, and
3.

__Optimization
Scheme 3:__
Find the
value of by
considering availability and total cost together. This is done using
optimization with two objective functions. Using the Weighted Sum method from
the study of Grodzevich and Romanko (2006), the structure of the
optimization function with two objectives is given below.

Subject to constraints:

Hence,** **the value of** ** obtained
will maximize the availability and minimize the total cost simultaneously (not
done sequentially as in the previous two schemes).

*3**.**2**. **Two-dimensional MSC with PM Policy 2*

Again, the three optimization schemes are also used to find the optimal solution.

__Optimization
Scheme 1:__
(i) First, using the algorithm proposed by Yeh and Chang (2007) finding the optimal value of which
minimizes the expected total cost. (ii) Second, based on the optimal
value of (i), then calculate the availability with the equation in (2).

Step
1. Set

Step
2. Solve the system of equations with the objective function:

Results and Discussion

Consider that is Weibull distribution
with , where is the scale parameter and is the shape parameter. Suppose that each
failure incurs costs (including the minimal repair cost and possible penalty
cost) and the cost for performing a PM action with maintenance degree . A similar form of maintenance cost is used by Supriatna *et al.*
(2020). These parameter values were used in Yeh and Chang (2007) and will be considered in this section given
in Table 1.

In this
section, we shall evaluate the performance of two-dimensional MSC with PM Policy 1
(proposed PM) and two-dimensional with PM Policy 2.

*4**.**1**. **Discussion of MSC 2D results with PM 1*

Tables 2 and 3 show the results for low usage (y=0.9), comparing results with Schemes 1 and 2 and then results with Schemes 2 and 3. From Tables 2 and 3, we have the following findings. Increasing the parameter value of gives the same or increased availability for Schemes 1-3. This is expected because larger means higher reliability (this agrees with the result of Yeh and Chang (2007) for a 1D MSC case). The results with Scheme 2 always give higher availability values than Scheme 1. In addition, Scheme 2 also provides a lower total cost of up to 71.70% (see Table 2, for the parameter ) than the results of Scheme 1. The total cost is smaller because the PM Policy 1 improves reliability by optimizing (is not constant). Furthermore, Scheme 3 also provides an availability value that is as large as the results of Scheme 2. The advantage of Scheme 3 is that the total cost obtained is lower than the total cost with Schemes 1 and 2. Total maintenance costs can be reduced by up to 71.70% (see Table 2 on parameters ). Thus, the best optimization scheme for PM 1 policy is Scheme 3, and this pattern also holds for medium and high usage rates (Note: the results cannot be included due to the number of page limitations of a paper).

We have the following findings from Table 4. The
results of Schemes 1 and 2 are not different. Scheme 3, which optimizes
two performance measures simultaneously, gives the best results. Availability using Scheme 3 increases up to 0.10%. In addition, the total cost of
maintenance can also be reduced by up to 43.81% (see Table 4, in parameter . The Scheme 3 is also the best for availability
and total cost measures for medium and high usages.

In
each of the 2D MSCs considered, Scheme 3 gives the best results. MSC 2D with PM
1 is superior both in terms of performance measures of availability and total
cost (see Table 5). Regarding availability, MSC 2D with PM 1 is up to 0.40%
superior (see Table 5, in parameter ) to MSC 2D with PM Policy 2. Meanwhile, regarding
total costs, MSC 2D with PM 1 provides a smaller total cost of almost 1/4 times
the 2D MSC with PM 2 - see Table 8 with the 2D MSC parameter with PM 2 up to 396.38%.

The advantage of the PM 1 policy is the
optimization of (the reduction in intensity function) for each PM action. Whereas in the PM 2 policy, the value for
each PM action is the same, even though the longer the age of the equipment or
the higher the usage rate. This advantage makes PM Policy 1 (proposed PM)
superior to PM Policy 2. For an agent who offers a 2D MSC with an availability
target, the results of this study are helpful in determining maintenance
policies that can meet availability targets with minimum total maintenance
cost.

Conclusion

In this paper, a two-dimensional
Maintenance Service Contract (MSC) study is conducted with (1) periodic and (2)
non-periodic PM policies by considering availability performance measures and
total maintenance costs. The two-dimensional MSC with periodic PM policy
provides the best performance – in terms of availability (can guarantee an
availability target) and low total costs at any rate of usage considered. Here, a two-dimensional MSC study was
conducted from the agent's point of view. One of the further research topics is
a two-dimensional MSC study that takes into account the two parties- i.e., the
agent and the consumer, who are very concerned about the price of MSC. This
topic can be modeled by a game theory formulation and provides optimal results for
both parties. In addition, from an equipment maintenance perspective, MSC can
also consider condition-based preventive maintenance options so equipment
performance can be even better. Furthermore, as the maintenance services can be
provided by the Original Equipment Manufacturer (OEM) or the agent, another
interesting topic is to study a 2D MSC involving three parties– i.e., the
Original Equipment Manufacturer (OEM), the agent, and the consumer. The
integration of three-party decision problems is an interesting topic because it
can mathematically describe the relationship of competition or cooperation
between agents. The formulation can be done using two-level game theory and it
is expected to choose the optimal option for consumers as well as the optimal three-party
coordinated option. The findings will be very beneficial for the parties
involved with the following details: (i) the agent can achieve the availability
target with minimum total maintenance costs, and (ii) the consumer benefits
from the optimal target of equipment availability and MSC price to maximize
profit. Research on these topics is on the way.

Acknowledgement

The research and study were supported by the
Ministry of Research, Technology, and Higher Education (Kementrian Ristek-Dikti)
Indonesia through the Program Magister menuju Doktor untuk Sarjana Unggul (PMDSU).

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