Reliability Measures and Profit Exploration of Windmill Water-Pumping Systems Incorporating Warranty and Two Types of Repair

Abstract

Wind energy is a kind of renewable energy that plays a significant role in remote areas for pumping water. The windmill is also used to generate electricity. The windmill is also known as a wind pump when it is used for pumping water. In this work, the authors proposed a complex hybrid model of an example of combined system (windmill, rechargeable battery and pumping system) to evaluate the system’s performance. System performance was affected by system degradation due to system failure. These factors also affected the profit of the user. Two types of repair facilities for continuous and satisfactory performance of the system were assumed. To illustrate the system modeling using the Gumbel–Hougaard family of the copula, numerical examples were used for the exploration of Markov results of the reliability measures and the profit of the system with the warranty period, with this also being demonstrated graphically.

1. Introduction

Worldwide, two billion people suffer from water shortage problems. Water is necessary for fulfilling human daily requirements such as drinking, bathing, washing, to protecting health, safeguarding food production, harvesting energy, and refurbishing environments, in addition to facilitating social and economic enhancement. A water-pumping system needs a source of power to operate, but in remote areas and farms in rural areas, the power supply is a challenging issue [1,2]. Renewable energy sources perform a key role in reducing the consumption of conventional energy sources and their impacts on the environment. There are five main groups of renewable energy sources, including solar photovoltaic (PV) water pumping systems, solar thermal water-pumping systems, wind energy water-pumping systems, biogas water-pumping systems, and hybrid renewable energy systems [3,4].

A very effective combined system has been widely used since the late 19th century in many countries. People use windmill water-pumping systems on farms and ranches worldwide. It is of great significance to accomplish the prophecy of water pumping. However, machine learning, as an auspicious method for many realistic applications, has been infrequently used in this field due to difficulties in determining the analytical configuration with the best performance [5]. The study on machine learning for reliability and safety demands done in past is very extensive but patchy that can be used to overcome severe problems and creating rational solutions [6,7].

Supply, especially in electrical and fuel-driven pumps-free areas, is not universally available. Wind characteristics are significant at all levels of wind-energy system development, including the design, performance evaluation, and operation [8,9]. Maximum windmills, which are used for water-pumping, consist of horizontal-axis multi-bladed rotors that can supply a high torque to harvest the mechanical energy required to initially operate the water pump. However, the windmills that are used to generate electricity consist of vertical-axis rotors or high-speed propeller rotors because generating electricity entails low preparatory torques [10]. Water-pumping systems can convert wind into energy with the help of useful factors attached to the windmill. Additionally, from an economical point of view, utilizing a combined system is the most essential factor in choosing the appropriate technology to be used in any assignment and locality. Combined systems are highly reliable, economical, and easily operable. They have major advantages including no fuel requirements, no fume production, they are potentially long-lasting, and they work well in windy spots [11,12].

Vick and Neal [13] discussed the important advantages and disadvantages of off-grid wind turbines and solar photovoltaic water-pumping systems. The authors described the different types of solar arrays and improvements in the pumping system using various useful techniques and methods. Bouzidi [14] compared and proposed two types of solar and water-pumping energy systems based on the case studied. The authors determined the best results and improved the quality of windmill solar power systems due to the availability of water and wind. Odeh et al. [15] analyzed the system sensitivity and performance availability compared to other systems’ terms and conditions and also discussed the mismatched supply of water and wind at the time of the operation of the system. Mohsen and Akash [16] explained the performance of water pumping system using wind energy based on available data and the potential energy of the wind pumping system. Awad [17] studied water windmill energy power plants, which are more reliable, renewable energy sources, and low-cost, thus, explaining their useful applications in real-life. Poompavai and Kowsalya [18] evaluated renewable energy research based on control strategies and a water windmill system and also discussed energy-management strategies with subsystems within the system.

Argaw [19] studied the various technologies and methods to prove the quality of water windmills, such as the safety, design, maintenance, and installation of the system. The authors used various tools and techniques to improve the results of the water-pumping system. Rehman and Sahin [20] considered the idea of a wind–photovoltaic-battery hybrid power system based on renewable sources to be utilized and established in remote areas for domestic requirements, in particular, in five locations in Saudi Arabia. A financial evaluation of technologies based on renewable energy has been discussed by researchers in India [2]. The authors provided a detailed analysis of the economic effects on farmers as a consequence of various systems, such as photovoltaic systems, windmills, and biogas.

Today, the main issue is to increase the system reliability performance of network multi-state systems; many researchers have used various binary and multi-state systems and applied different methods to improve the quality of network systems in the fields of communication, information sciences, and real-life engineering areas. Zio and Sansavini [21] studied multi-state networks with nodes and arcs with different performance states. The authors calculated the reliability and improved the efficiency of the network system with the help of Monte Carlo simulation methods. Liu and Kapur [22] evaluated the reliability of various multi-state systems usingMarkov process, including structure function and availability in the upper and lower structures. Natvig [23] dealt with various complex systems, i.e., binary and multi-state systems, with either working or failed performance states. The author used various methods to evaluate different measures such as reliability, availability, sensitivity, and mean time to failure using Bayesian and other techniques. Lin [24] discussed the network reliability, availability, and maintenance of a multi-state system with various nodes and arcs using routing policy. The network system has a minimal path set to increase the performance of the network multistate system. Lin and Chang [25] considered a multi-state network system with a working and multiple failure rate for evaluating the performance of the system. The authors also discussed the algorithm base method for computing probability or reliability of the system networking minimal path set, which satisfied the demand and cost.

Lin et al. [26] calculated the system reliability of the multi-state network system, proposed the algorithm with a minimal path, and evaluated the performance probability of the system. The system was based on three parameters: route capacity, delivery time, and time window. Wenbin et al. [27] considered a reliability cost system of two failure modes with maintenance activities. For cost analysis, the authors used Markov techniques and calculated the system availability using the Genetic algorithm in case of optimal repair. Lin and Pham [28] studied a voting system, which had two stages cut in the decision-making system and calculated the reliability and cost optimization of the system. The authors also discussed a new approach for solving decision-making problems. Rushdi [29] evaluated the reliability of a multi-state complex system with multiple elements of a symmetric switching system using algebraic techniques. Deng et al. [30] studied the chatter stability for a milling process multi-state neural network system. The network system performance was calculated using the second-order fourth moments method and Monte Carlo Method. Sarsour and Sabri [31] determined the availability and performance of a real-life system based on forecasting the long-term behavior. Additionally, the analysis assessed the time to failure of the stock market decision by the Markov chain process.

The above studies focused on producing electricity using various systems including windmills, wind turbines, and photovoltaic systems. All of these types of systems are used to generate electricity. Here, the authors study a combined system comprising of a windmill, rechargeable battery (with limited use), and a pumping system.

The article is organized as follows: the required notations and assumptions are presented in Section 2 named as the Materials and Methods. In this section, the system is also described in detail. The mathematical model and its solution are given in Section 3. Section 4 explores the reliability measures and expected profit of the user, with reference to the different warranty periods. The numerical study is also presented for a better understanding of the results. Finally, the obtained results are displayed and conclusions are drawn in Section 5.

2. Materials and Methods

2.1. Notation

Notations associated with this work are described in

Table 1. Notations.

Symbol	Description
t/s	Time scale/Laplace transform variable.
$\overline{P}(s)$	Laplace transformation of $P(t)$.
${\lambda }_B/{\lambda }_S/{\lambda }_{GB}/{\lambda }_{PS}/{\lambda }_{mi}$	Failure rates of battery/shaft/gearbox/pumping cylinder/minor error in pumping system.
${\lambda }_K/{\lambda }_{K+1}$	Failure rate for k blades not to two adjacent/k + 1 blades.
g	Average failure rate after warranty.
w	Warranty period.
l	Total expected life of the system.
$\alpha$	Rate of completion of warranty.
$\beta$	Failure rate during the warranty period.
$\mu$	Repair rates for the degraded states S2 and S3.
$e^y$, $\psi (y)$	Repair rate for the degraded state S4 and y is the elapse repair time for this state.
q	The joint probability (failed state S4, to normal state S0) according to the Gumbel–Hougaard family is given as: $\exp {[y^{\theta }+{\{\log \psi (y)\}}^{\theta }]}^{\frac{1}{\theta }}$.
${\varphi }_i(x_i)$	Repair rate for completely failed state Si+4; i = 1 … 5.
$P_i(t)$	The probability that the system is in state Si, where i = 0 … 4.
$P_{i+4}(x_i,t)$	The probability that the system is the state Sj+4, and xi is the elapse repair time where i = 1, 2, …5.
$P_{up}(t)$	Probability of working state of the system.
$P_{down}(t)$	Probability of failed state of the system.
Ep(t)	Expected profit during the interval [0, t).
K1, K2	Revenue and service cost per unit time.

.

2.2. System Description

India has a wide-ranging area available for agriculture, however, in many remote areas the water level is situated in a deep well and electricity is not available. Therefore, in this case, a combined system is necessary. In this research study, a combined system, modeled mathematically using the Markov process, is considered. The combined system consists of many components, such as the tail, multi-bladed rotor, shaft, tower, pump rod, discharge pipe, well seal, storage tank, tower footing, drop pipe, pumping cylinder, and screen. Its configuration diagram is shown in Figure 1.

The combined system is a repairable system, so, two types of repair policy have been considered. One follows general distribution and the other one follows an exponential distribution. When people invest in a machine, then they think about its reliability, maintenance cost, and profit. Therefore, the following assumptions have been taken:

• Initially, the system is in good working condition.
• The system has a fixed warranty period.
• The gearbox is repaired by two types of repair policy via the Gumbel–Hougaard family of Copula and the other components are repaired generally.
• The system is partially working when k blades are in a failed state, but two of them should not be adjacent.

The combined system has three types of states: good, degraded, and failed, depending on its function as a consequence of the effect of the failure of its components, battery, pumping system, and a minor error in the pumping system. The designed system has a total of ten states, in which two are good states, three are degraded states and five are failed states. The state transition diagram of the designed system is revealed in Figure 2 and the states are described below:

S0	The initial state in which the system is in good condition as per the assumptions.
S1	The good working state of the system when the warranty period is over.
S2	The degraded state of the system when k-out-of-n blades have been failed but in which two blades should not be adjacent.
S3	The degraded state of the system due to minor failure in the pumping system.
S4	Partially failed state of the combined system due to failure in the gearbox.
S5	Failed state of the system due to k+1 faulty blades.
S6	Failed state of the system due to break or error in the shaft.
S7	Failed state of the system caused by battery discharge.
S8	Failed state of the system when the system can fail due to any failure under the warranty period and in this system is repaired by the company free of charge.
S9	Failed state of the system because of any major failure in the pumping system.

S₁ and S₈ are the states of the system that can occur during warranty, other states occur after completion of the warranty period. When the system fails due to any reason or factor, either the system is repaired in each state—either under the warranty or out of the warranty—bringing the system back to good condition. If the combined system has any type of minor failure that decreases the system efficiency, then the system goes to the state S₃.

3. Mathematical Model

3.1. Formulation of the Model

The following differential equations present the mathematical model, which was derived using the Markov process, supplementary variable technique, and Gumbel–Hougaard family of the copula.

$$\left[\frac{\partial }{\partial t}+\alpha +\beta \right]P_0(t)={\displaystyle \underset{0}{\overset{\infty }{\int }}{\varphi }_4(x_4)}{P}_8(x_4,t)d{x}_4$$

(1)

$$\begin{gathered} \left[\frac{\partial }{\partial t}+{\lambda }_B+{\lambda }_{GB}+{\lambda }_S+{\lambda }_K+{\lambda }_{PS}+{\lambda }_{mi}\right]P_1(t)=\exp {\left\{y^{\theta }+{(\log \psi (y))}^{\theta }\right\}}^{1/\theta }{P}_4(t)+\alpha P_0(t) \\ +{\displaystyle \sum _{i=1,2,3,5}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\varphi }_i(x_i)}{P}_{i+4}(x_i,t)d{x}_i}+\mu {\displaystyle \sum _{i=2}^3{P}_i(t)} \end{gathered}$$

(2)

$$\left[\frac{\partial }{\partial t}+{\lambda }_{K+1}+\mu \right]P_2(t)={\lambda }_K{\displaystyle \sum _{i=1,4}{P}_i(t)}$$

(3)

$$\left[\frac{\partial }{\partial t}+{\lambda }_{PS}+\mu \right]P_3(t)={\lambda }_{mi}{P}_1(t)$$

(4)

$$\left[\frac{\partial }{\partial t}+{\lambda }_B+{\lambda }_S+{\lambda }_K+\psi (y)\right]P_4(t)={\lambda }_{GB}{P}_1(t)$$

(5)

$$\left[\frac{\partial }{\partial t}+\frac{\partial }{\partial x_i}+{\varphi }_i(x_i)\right]P_{i+4}(x_i,t)=0;i=1\dots \dots 5$$

(6)

Boundary conditions

$$P_5(0,t)={\lambda }_{K+1}{P}_2(t)$$

(7)

$$P_i(0,t)={\lambda }_j{\displaystyle \sum _{i=1,4}{P}_i(t)};i=6, 7;j=S, B$$

(8)

$$P_8(0,t)=\beta P_0(t)$$

(9)

$$P_9(0,t)={\lambda }_{PS}{\displaystyle \sum _{i=1,3}{P}_i(t)}$$

(10)

Initial condition

$$P_i(0)=\{\begin{array}{ll}0,& i=0\\ 1,& i\ge 1\end{array}$$

(11)

3.2. Solution of the Model

The equations derived in Section 3.1 are solved by Laplace transformation and provide the probability of each state.

$${\overline{P}}_0(s)=\frac{1}{s+\alpha +\beta (1-{\overline{S}}_{{\varphi }_4}(s))},$$

$${\overline{P}}_1(s)=\frac{\alpha }{s+C_1-K_2(s)}{\overline{P}}_0(s),$$

$${\overline{P}}_2(s)=(1+C_3)C_4{\overline{P}}_1(s),$$

$${\overline{P}}_3(s)=\frac{{\lambda }_{mi}}{s+{\lambda }_{PC}+\mu }{\overline{P}}_1(s),$$

$${\overline{P}}_4(s)=\frac{{\lambda }_{GB}}{s+C_2}{\overline{P}}_1(s),$$

$${\overline{P}}_{i+4}(s)=\left[\frac{1-{\overline{S}}_{{\varphi }_i}(s)}{s}\right]{\overline{P}}_{i+4}(0,s);i=1\dots \dots 5,$$

where,

$$C_1={\lambda }_B+{\lambda }_{GB}+{\lambda }_S+{\lambda }_K+{\lambda }_{PS}+{\lambda }_{mi},$$

$$C_2={\lambda }_B+{\lambda }_S+{\lambda }_K+\psi (y),$$

$$C_3=\frac{{\lambda }_{GB}}{s+C_2},$$

$$C_4=\frac{{\lambda }_K}{s+{\lambda }_{K+1}+\mu }.$$

3.3. Working State and Failed State Probability of the System

The working state of the system is the sum of working states either with full capacity or with degradation. If the system is not working, then it is a failed state of the system. The probability of the system in the working state and failed state are:

$$\begin{array}{cc}\hfill {\overline{P}}_{up}(s)& ={\displaystyle \sum _{i=0}^4{\overline{P}}_i(s)}\hfill \\ & =\left[1+\left\{1+(1+C_3)C_4+\frac{{\lambda }_{mi}}{s+{\lambda }_{PS}+\mu }+C_3\right\}\frac{\alpha }{s+C_1-K_2(s)}\right]{\overline{P}}_0(s)\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\overline{P}}_{down}(s)& ={\displaystyle \sum _{i=5}^9{\overline{P}}_i(s)}\hfill \\ & =\left[(1+C_3){\displaystyle \sum _{i=1}^3\xi \left(\frac{1-{\overline{S}}_{{\varphi }_i}(s)}{s}\right)}+\beta \left(\frac{1-{\overline{S}}_{{\varphi }_4}(s)}{s}\right)+{\lambda }_{PS}\left(1+\frac{{\lambda }_{mi}}{s+{\lambda }_{PS}+\mu }\right)\left(\frac{1-{\overline{S}}_{{\varphi }_i}(s)}{s}\right)\right]{\overline{P}}_0(s)\hfill \end{array}$$

(13)

4. Numerical Example

4.1. Availability Analysis

The availability of the combined system is calculated by taking the inverse Laplace transform of Equation (12). Setting the value of failure rates as $\alpha =0.03$, $\beta =0.01$, ${\lambda }_{GB}=0.05$, ${\lambda }_B=0.04$, ${\lambda }_K=0.03$, ${\lambda }_{K+1}=0.01$, ${\lambda }_S=0.01$, ${\lambda }_{PS}=0.02$, ${\lambda }_{mi}=0.05$, $\mu =0.9$, $\underset{i=1\dots 4}{{\varphi }_i(x_i)}=1$, ${\varphi }_i(x_i)=0.9$, $q=2.71828$ and obtaining the reliability as a function of time t.

$$\begin{array}{cc}\hfill P_{up}(t)=& e^{(-0.52 t)}\left[3.147679\cosh (0.490306t)+3.126634\sinh (0.490306t)\right]\hfill \\ & -0.000005e^{(-2.849770 t)}-0.001399e^{(-1.070243 t)}-0.0001e^{(-0.907169 t)}\hfill \\ & +e^{(-0.928960 t)}\left[0.000492\cos (0.018284t)-0.001778\sin (0.018284t)\right]\hfill \\ & -2.146671e^{(-0.043176 t)}\hfill \end{array}$$

(14)

Now, varying the time from 0 to 50 months (in 5 months interval time) in Equation (14) allows us to obtain the availability of the system, as shown in

Table 2. Availability of the combined system as a function of time.

Time (t in Months)	Availability Pup(t)
0	1.00000
5	0.97452
10	0.93721
15	0.88624
20	0.82708
25	0.76383
30	0.69943
35	0.63596
40	0.57483
45	0.51697
50	0.46293

. The behavior of the system’s availability during the first 50 months is also revealed by the graph in Figure 3.

From the above graph, one can observe that the availability of the combined system decreases uniformly as time increases. The graph of availability is a straight line. Between 30 to 35 months. We can see that the value of the availability varies between 0.6 to 0.7, which means that we have a 60 to 70 percent chance of system availability. After 50 months, the system will be around 40 to 50% available.

4.2. Reliability Analysis

The reliability of any structure or system is the probability of the system working without failure, which means the repair facility does not affect the system’s reliability. Therefore, a repair facility is not available. Assuming the values of failure rates are $\alpha =0.03$, $\beta =0.01$, ${\lambda }_{GB}=0.05$, ${\lambda }_B=0.04$, ${\lambda }_K=0.03$, ${\lambda }_{K+1}=0.01$, ${\lambda }_S=0.01$, ${\lambda }_{PS}=0.02$, ${\lambda }_{mi}=0.05$ in Equation (12), we can obtain the reliability of the combined system as expressed in Equation (15).

$$\begin{array}{cc}\hfill Rl=& -0.040022e^{(-0.2 t)}+0.270677e^{(-0.01 t)}-0.178571e^{(-0.08 t)}+0.416667e^{(-0.02 t)}\hfill \\ & +0.53125e^{(-0.04t)}\hfill \end{array}$$

(15)

By varying the time from 0 to 50 (in 5 months interval time) months in Equation (15), we calculated the reliability as given in

Table 3. Reliability of the combined system as a function of time.

Time (t in Months)	Reliability Rl
0	1.00000
5	0.93502
10	0.85651
15	0.77743
20	0.70283
25	0.63452
30	0.57290
35	0.51776
40	0.46862
45	0.42492
50	0.38608

and demonstrated it graphically in Figure 4.

The graph shows that the reliability of the combined system decreases in a curvilinear manner as time increases. It approaches zero with an increment in time. From the critical examination of the reliability graph, we can see that after 50 months, the reliability of the system is 0.38 which means that after 50 months the system is only 38% reliable.

4.3. Mean Time to Failure (MTTF)

Mean Time to Failure (MTTF) is the mean time of the system until the first failure occurs. It is defined as

$$\begin{array}{cc}\hfill MTTF& =\underset{s\to 0}{\lim }{\overline{P}}_{up}(s)\hfill \\ & =\frac{1}{\alpha +\beta }\left[1+\frac{\alpha L}{{\lambda }_B+{\lambda }_G+{\lambda }_S+{\lambda }_K+{\lambda }_{PS}+{\lambda }_{mi}}\right]\hfill \end{array}$$

(16)

where,

$$L=1+\frac{{\lambda }_K}{{\lambda }_{K+1}}\left[1+\frac{{\lambda }_G}{{\lambda }_B+{\lambda }_S+{\lambda }_K}\right]+\frac{{\lambda }_G}{{\lambda }_B+{\lambda }_S+{\lambda }_K}+\frac{{\lambda }_{mi}}{{\lambda }_{PS}}.$$

Now, varying the value of failure rates from 0.01 to 0.09 one by one and setting other failure rates are fixed as $\alpha =0.03$, $\beta =0.01$, ${\lambda }_{GB}=0.05$, ${\lambda }_B=0.04$, ${\lambda }_K=0.03$, ${\lambda }_{K+1}=0.01$, ${\lambda }_S=0.01$, ${\lambda }_{PS}=0.02$, ${\lambda }_{mi}=0.05$ in Equation (16). The MTTF, concerning each failure rate of the considered system, is computed as shown in

Table 4. MTTF of the combined system.

Failure Rates	MTTF
	α	β	λGB	λB	λK	λK+1	λS	λPS	λmi
0.01	72.50000	58.75000	57.81250	71.32353	50.69445	58.75000	58.75000	70.39474	57.81250
0.02	63.33333	47.00000	58.08823	65.97222	55.16917	49.60937	56.15079	58.75000	58.08823
0.03	58.75000	39.16667	58.33334	61.93609	58.75000	46.56250	53.97727	54.16667	58.33334
0.04	56.00000	33.57143	58.55263	58.75000	61.70635	45.03906	52.12450	51.42045	58.55263
0.05	54.16667	29.37500	58.75000	56.15079	64.20454	44.12500	50.52083	49.45652	58.75000
0.06	52.85714	26.11111	58.92857	53.97727	66.35375	43.51562	49.11538	47.91667	58.92857
0.07	51.87500	23.50000	59.09091	52.12450	68.22917	43.08036	47.87088	46.64286	59.09091
0.08	51.11111	21.36364	59.23913	50.52083	69.88461	42.75391	46.75926	45.55288	59.23913
0.09	50.50000	19.58333	59.37500	49.11538	71.35989	42.50000	45.75893	44.59876	59.37500

. The variation in the MTTF of each failure rate is graphically explained in Figure 5.

From the graphs presented in Figure 5, it can be seen that the MTTF of the combined system decreases as the component failure rates increase, except in the case of the gearbox, a minor failure in the pumping system, and the failure rate of k-out-of-n blades. In the case of variation in the failure rate of minor failures in the pumping system and gearbox, the MTTF of the system increases very slightly and it is almost constant. While with an increased failure rate of the k-out-of-n blades, the MTTF of the system increases. From the critical investigation of the graph, one can see that the MTTF of the system is most critical in the case of failure under warranty and least critical in the case of blades failure.

4.4. Expected Profit

Profit plays an important role when users invest or use any machine. Before investing, they are concerned about factors such as profit, maintenance costs, and the warranty period. The expected profit is defined as

$$E_p(t)=K_1{\displaystyle \underset{0}{\overset{t}{\int }}{P}_{up}(t)} dt-(l-w)g{K}_2$$

(17)

Setting the value of parameters as $\alpha =0.03$, $\beta =0.01$, ${\lambda }_{GB}=0.05$, ${\lambda }_B=0.04$, ${\lambda }_K=0.03$, ${\lambda }_{K+1}=0.01$, ${\lambda }_S=0.01$, ${\lambda }_{PS}=0.02$, ${\lambda }_{mi}=0.05$, $g=0.03$ $\mu =0.9$, $\underset{i=1\dots 4}{{\varphi }_i(x_i)}=1$, ${\varphi }_i(x_i)=0.9$, $q=2.71828$, l = 60 months and K₁ = 1 in Equation (17) and examine the expected profit for the user in two cases.

Case 1: When the combined system has a 12 month warranty then the expected profit is analyzed as:

$$\begin{array}{cc}\hfill E_p(t)=& -0.010415\left[\cosh (1.010306)-\sinh (1.010306)\right]+0.000002e^{(-2.84977t)}\hfill \\ & -105.649606\left[\cosh (0.029694)-\sinh (0.029694)\right]+0.001307e^{(-1.070243t)}\hfill \\ & -e^{(-0.92896t)}\left[0.000492\cos (0.018284t)-0.001923\sin (0.018284t)\right]\hfill \\ & +0.000111e^{(-0.907169t)}+49.718867e^{(-0.043176t)}+55.940228-1.44K_2\hfill \end{array}$$

Case 2: When the combined system has a 24 month warranty then the expected profit is analyzed as:

$$\begin{array}{cc}\hfill E_p(t)=& -0.010415\left[\cosh (1.010306)-\sinh (1.010306)\right]+0.000002e^{(-2.84977t)}\hfill \\ & -105.649606\left[\cosh (0.029694)-\sinh (0.029694)\right]+0.001307e^{(-1.070243t)}\hfill \\ & -e^{(-0.92896t)}\left[0.000492\cos (0.018284t)-0.001923\sin (0.018284t)\right]\hfill \\ & +0.000111e^{(-0.907169t)}+49.718867e^{(-0.043176t)}+55.940228-1.08K_2\hfill \end{array}$$

(18)

By varying the service costs by 1%, 5%, 10%, and 50% at different times in both cases, the estimated profit for the user is presented in

Table 5. Expected profit of the combined system as a function of time.

Time (t) (In Months)	Expected Profit
	K2 = 0.01		K2 = 0.05		K2 = 0.1		K2 = 0.5
	w = 12	w = 24	w = 12	w = 24	w = 12	w = 24	w = 12	w = 24
0	−0.01440	−0.0108	−0.07200	−0.05400	−0.14400	−0.108	−0.72000	−0.540
5	4.91808	4.92168	4.86048	4.87847	4.78848	4.82448	4.21248	4.39247
10	9.70443	9.70803	9.64683	9.66483	9.57483	9.61083	8.99883	9.17883
15	14.26750	14.27110	14.20990	14.22790	14.13790	14.17390	13.56190	13.74190
20	18.55286	18.55646	18.49526	18.51326	18.42326	18.45926	17.84726	18.02726
25	22.55368	22.55728	22.49608	22.51408	22.42408	22.46008	21.84808	22.02808
30	26.60545	26.60905	26.54785	26.56585	26.47585	26.51185	25.89985	26.07985
35	197.55858	197.56218	197.50098	197.51898	197.42898	197.46498	196.85298	197.03298
40	221.91050	221.91410	221.85290	221.87090	221.78090	221.81690	221.20490	221.38490
45	250.14638	250.14998	250.08878	250.10678	250.01678	250.05278	249.44078	249.62078
50	282.84736	282.85096	282.78976	282.80775	282.71776	282.75376	282.14176	282.32176

and demonstrated in Figure 6. The comparative study of both cases is explained by the graphs revealed in Figure 7.

From the analysis of Table 5 and the corresponding Figure 6, one can see that the user’s profit increases as time increases, but as the service cost increases, the profit is almost stable during some periods and slightly decreases within other intervals, such as at 0–5 months, 20–25 months, 45–50 months for both cases. This means that under both warranty periods, profit increases in the same manner. However, from the investigation of the graphs presented in Figure 7, we find that if the warranty period rises to 24 months then the user’s expected profit also escalates.

5. Conclusions

The repair policy has a great impact on the system’s availability, maintenance costs, and expected profits. This is one of the ways by which reducing the mean time between failures and increasing the system’s availability provides high expected profits. Using the Markov process, the authors propose a mathematical model for a combined system by employing the supplementary variable technique for elapse repair time and the Gumbel–Hougaard family of copula for two types of repair.

From critical examination of the model, the authors conclude that the combined system is highly functional due to the two types of repair facilities. One can see that the MTTF of the system very slightly increases with the increment in the failure rate of the gearbox, while it has two types of repair policy. The MTTF of the combined system is most sensitive under warranty. It was also determined that the user’s profit is almost constant during the system life, due to the warranty and maintenance policy. Therefore, the analyzed results are beneficial for the engineers to make this system highly reliable and more profitable in remote areas.