1 Introduction

The availability has become a concept of dominant importance in the complex and high investment-oriented scenario of modern industrial units. There is a series–parallel or hybrid combinations of the subsystems in these complex modern industries. The complexity of today's industrial units has made reliability a vital aspect of the overall design of these units (Al Salamah et al. 2006). Overall performance of an industry is highly affected by ineffective utilization of the operating units due to the non-availability of the units. So, the high-level availability of the units and subsystems of any industry leads to enhanced productivity. The high monetary investment is very much necessary due to the need for automation in the modern production and processing plants like paper plants, thermal plants, fertilizer plants, sugar plants, etc. (Corvaro et al. 2017). For the endurance of these plants, it becomes vital for these plants to provide maximum productivity and profits. This can only be achieved if all systems and subsystems of these plants are highly consistent regarding reliability and availability (Kajal et al. 2010). Productivity enhancement is another outcome of the increased availability of industrial systems. To emphasize operational management is the most important task for achieving the best outputs in quality and quantity. Another important term is available for the design and development of the maintenance phase. Several methods are there to improve the system availability. These methods include redundancy, reducing the system complexity, and making a proper schedule for maintenance (Rao and Naikon 2015). Recently, numerous authors described the various methods for solving the problems related to reliability, availability, and maintainability for several systems.

Aggarwal et al. (2017) developed the regression modeling for a crystallization system of the sugar plant with the help of a mnemonic rule to obtain the partial differential equations. The reliability of the concerned system has been analyzed using fuzzy reliability. Aman et al. (2019) and Garg et al. (2012) developed the regression models to analyze the availability of the different repairable subsystems of a sugar plant using the Markov process. Gupta et al. (2013) reviewed the extensive study on inventory and its control by utilizing a specified ordering policy method. It is concluded that productivity is enhanced by optimizing the values of availability and other factors (Gupta et al. 2004, 2011). Kajal and Tewari (2014) developed the performance model to determine the system's availability and optimize its performance. Khanduja et al. (2009) discussed the behavior of the bleaching system and the planning of maintenance of a paper plant under a steady state. Kumar and Garg (2016) computed the best possible availability of diverse systems of a working brewery plant with the Particle Swarm Optimization technique. Structural and regression modeling of the system uses the Markov method to examine the criticality level of different components and devise maintenance strategies. Kumar et al. (2016) determined the maintenance priority level of a thermal plant's repairable flue gas and air system. Kumar et al. (2014) studied the performance model to find out the availability of a pump with the help of the Markov approach in a coal-based thermal power station.

Kumar and Tewari (2011) applied a genetic algorithm technique to CO2 cooling system in the fertilizer plant to optimize the system performance. Malik and Tewari (2020) employed the PSO technique for optimizing the performability of the coal handling unit of a thermal power plant. The highest performability level of the coal handling unit is observed as 93.33 percent at a generation size of 100 and a population size of 40. Panwar and Kumar (2021) evaluated the performance of the feeding unit of the paper plant using the probabilistic approach. The feeding unit consisted of three repairable subsystems. The transition diagram of the feeding unit is used to obtain the differential equations with the help of a mnemonic rule. The availability model is developed based on obtained differential equations using normalization conditions. Rauzy( 2004) presented the study for computing the transient state probabilities of the Markov models with six different methods. The complicated unit consists of other systems and subsystems arranged in series, parallel, and a hybrid combination. Ramirez et al. (2005) discussed a Monte-Carlo simulation method to estimate the reliability of the multi-state network. Rauzy (2004) and Saini et al. (2021) analyzed the reliability and maintainability of the different subsystems in a sugar manufacturing plant. Sharma et al. (2017) discussed performance modeling and analyzed the availability of a Leaf Spring Manufacturing Industry. The developed model was beneficial for defining different preventive maintenance decisions and actions. Wan et al. (2016) and Wang et al. (2018) introduced a stochastic model based on the Markov technique for predicting the thermal reliability of electronic systems. Yu et al. (2007) studied the design of a system with redundancy, considering a type of failure dependency called a redundant dependency.

In today's scenarios, automation in the industries is a vital problem owing to huge capital investment. For example, a sugar plant industry requires a multifaceted system with tremendous capital investment and planning. Besides this, the failure of components is also another problem related to this industry. To overcome all these problems, a proper arrangement for preventive maintenance must be necessary. Thus, a mannered, maintained schedule is required for all the components of the sugar industry. In addition, the sugar plant industries equipment is always working in harsh conditions; owing to this; it requires repairing and replacing component-time to time. It means the condition of a machine depends upon the operating conditions. These conditions may be different for different units. The sugar plant industries priority is to retain all the units available during the process.

The availability of the plant can be measured on the basis of operational time without failure. In current research work, the main focus is to preserve the plant under working conditions without any failure by maintaining the different plant units (i.e., evaporation and crystallization) in a failure-free state. It is a vital part of the sugar plant industry and consists of five subsystems organized in sequences. The unit's outcome is interconnected with the reliability and maintainability of the equipment. It depends upon the number of failures. For this, an optimal maintenance strategy is required by taking the highest maintenance priority to the most critical subsystem of the unit. In this current research, a birth–death Markov method is utilized to solve the problems related to sugar plant industries. The transition diagram for different sugar plant industry units is drawn according to the working condition of the unit.

Moreover, the differential equations are generated for each unit by utilizing the transition diagram. In addition, the performance model is designed as per the transition diagram. Finally, the performance analysis of various units is measured as per the decision matrix obtained from the performance model (Fig. 1).

Fig. 1
figure 1

Schematic flow diagram of evaporation and crystallization unit

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2 Unit description

For carrying out the performance modeling, five main subsystems of the Evaporation and Crystallization unit are identified:

  1. i.

    Subsystem E1: The first subsystem combines four evaporators arranged in parallel. If one evaporator fails, the unit goes to reduced capacity. The failure of more than one evaporator would result in the complete failure of the unit.

  2. ii.

    Subsystem E2: The second subsystem comprises three pans in parallel. If any of the pan's subsystems fails, the unit goes to reduced capacity. If more than one subsystem fails, it results in the complete failure of the unit.

  3. iii.

    Subsystem E3: It consists of eight crystallizers in parallel. If one crystallizer fails, the unit starts working at a reduced capacity. If two or more two crystallizers fail, the complete unit fails.

  4. iv.

    Subsystem E4: This subsystem has fourteen centrifuges in parallel. This unit is also failing, utterly similar to the units mentioned above.

  5. v.

    Subsystem E5: It is composed of four sugar graders in series. If one of the sugar graders fails, the complete unit gets unable.

3 Assumptions and notations

Assumptions

The assumptions for developing the performance model for the evaporation and crystallization unit of a sugar plant as given below:

  1. i.

    The failure and repair rates are taken constantly with respect to time.

  2. ii.

    The performance of components/equipment after maintenance will be similar to that of new for a specified period.

  3. iii.

    The maintenance of failed components/equipment starts at a time by providing the appropriate maintenance facilities.

  4. iv.

    The maintenance facilities follow the repair or replacement of components/equipment.

  5. v.

    The subsystems of the concerned unit can work with reduced capacity.

  6. vi.

    The failure of various subsystems of the concerned unit cannot occur simultaneously.

  7. vii.

    In last, the simultaneous failure of subsystems cannot be considered.

Notations.

The various notations are employed to make the transition diagram of the concerned unit and then obtain the first-order differential equations. These notations are given below:

Represents the system with a full capacity state

Describes the system working with a reduced capacity state

Illustrates the system in a failed state

\({\mathrm{P}}_{\mathrm{S}0}\left(\mathrm{t}\right)\)

Probability of full capacity state

\({\mathrm{P}}_{\mathrm{S}1}\left(\mathrm{t}\right)\) to \({\mathrm{P}}_{\mathrm{S}5}\left(\mathrm{t}\right)\)

Probability of reduced capacity state

\({\mathrm{P}}_{\mathrm{S}6}\left(\mathrm{t}\right)\) to \({\mathrm{P}}_{\mathrm{S}29}\left(\mathrm{t}\right)\)

Probability of failed state

\({{\mathrm{P}}^{\mathrm{^{\prime}}}}_{\mathrm{Si}}\left(\mathrm{t}\right)\)

Derivative concerning time (t) for the ith state

E1, E2, E3, E4, E5

Subsystems are operating

e1, e2, e3, e4, e5

Subsystems are in failed states

Фi, i = 17 to 21

Mean failure rates (MFR) of the subsystems E1, E2, E3, E4 & E5 respectively

µi, i = 17 to 21

Mean repair rates (MRR) of the subsystem E1, E2, E3, E4 & E5 respectively

Figure 2 depicts the transition diagram for the Evaporation and Crystallization Unit (different states). 0 state means a system working with full capacity, whereas states 1, 2, 3, 4, and 5 represent the system working with reduced capacity. Conversely, states 6 to 29 represent the failure states.

Fig. 2
figure 2

Transition diagram of evaporation and crystallization unit

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4 Results and discussions

4.1 Performance modeling

Markov Process is a powerful stochastic process used to develop the performance model of systems that exhibit probabilistic behaviour. It has many important applications in time-based reliability as well as availability analysis. Here in Movkov state transition diagrams are used for modelling of stochastic behaviour of system. It has a set of discrete states that the system can be in due course of time. The speed at which transitions fired between these states take place can be specified. The transition of system from one state to another purely depends on the previous one state irrespective of series of states that system reached to this state. The Markovian Chain models are of two types i.e. Continuous-Time-Markov-Chain (CTMC) and Discrete-Time-Markov-Chain (DTMC) model. Explosion of a number states is the main problem with the markovian chain models which makes difficult to deal with the tedious mathematical calculations.

The work of Gupta et al. (2004) and Kumar et al. (2016, 2014, 2011) on the availability analysis of on various process plant is an example of the application of Markov modeling in a process system design. More recently, Malik and Tewari (2020), applied Markov Chains for modeling and evaluated availability for different power plants.

Keeping in view of this, in the present study we have considered a Sugar plant facing several challenges as mentioned earlier. The system description and performance modeling is described in subsequent sections here after.

The Markov Birth Death technique (probabilistic approach) is utilized to develop the concerned unit performance model. Moreover, it is also used to derive the differential equation on the basis of a transition diagram using the mnemonic rule. According to this rule, each state probability is equal to the sum of all incoming probabilities from other states to the current state minus the sum of all outgoing probabilities from the current state to other states.The general form of the differential equation is given below:

$${{\mathrm{P}}^{\mathrm{^{\prime}}}}_{\mathrm{S}0}(\dot{t)}+\sum_{i=17}^{21}{\Phi }_{\mathrm{i}}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}=\sum_{j=17,n=1,\dots ,5}^{21}{\upmu }_{\mathrm{j}}{\mathrm{P}}_{\mathrm{Sn}}(\dot{t)}$$
(a)
$${{\mathrm{P}}^{\mathrm{^{\prime}}}}_{\mathrm{S}0}(\dot{t)}+{\Phi }_{17}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}+{\Phi }_{18}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}+{\Phi }_{19}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}+{\Phi }_{20}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}+{\Phi }_{21}{\mathrm{P}}_{\mathrm{S}0}(\dot{t)}={\upmu }_{17}{\mathrm{P}}_{\mathrm{S}1}(\dot{t)}+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}2}(\dot{t)}+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}3}(\dot{t)}+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}4}(\dot{t)}+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}5}(\dot{t)}$$
(1)

The complete set of equations is presented in the appendix of the manuscript from Eqs. (1)-(30).

5 Using initial conditions, at t = 0

Pi \((\dot{t)}\)= 1 for i = 0,

Pi \((\dot{t)}\)= 0 for i ≠ 0.

6 Solution of equations

6.1 Steady state behaviour

All the process plant units, like the sugar plant, must be accessible for elongated periods. Therefore, the behavior of the Evaporation and Crystallization unit under steady-state can be analyzed by setting the time derivative (d/dt) as zero and taking all state probabilities independent of time t. The linear equation of state probabilities is given below (Equation b):

$$\sum_{i=17}^{21}{\Phi }_{\mathrm{i}}{\mathrm{P}}_{\mathrm{S}0}\left(\mathrm{t}\right)=\sum_{j=17, n=1,\dots ,5}^{21}{\upmu }_{\mathrm{j}}{\mathrm{P}}_{\mathrm{Sn}}\left(\mathrm{t}\right) $$
(b)
$$\left[{\Phi }_{17}+{\Phi }_{18}+{\Phi }_{19}+{\Phi }_{20}+{\Phi }_{21}\right]{\mathrm{P}}_{\mathrm{S}0}\left(\mathrm{t}\right)={\upmu }_{17}{\mathrm{P}}_{\mathrm{S}1}\left(\mathrm{t}\right)+{\upmu }_{18}{\mathrm{P}}_{\mathrm{S}2}\left(\mathrm{t}\right)+{\upmu }_{19}{\mathrm{P}}_{\mathrm{S}3}\left(\mathrm{t}\right)+{\upmu }_{20}{\mathrm{P}}_{\mathrm{S}4}\left(\mathrm{t}\right)+{\upmu }_{21}{\mathrm{P}}_{\mathrm{S}5}\left(\mathrm{t}\right)$$
(31)

The set of equations is presented in the appendix of the manuscript from Eqs. 31–60.

After solving the Eqs. (31-60) recursively, the values obtained are as:

\({\mathrm{P}}_{\mathrm{S}1}={\mathrm{K}}_{17}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}11}={\mathrm{K}}_{17}{\mathrm{K}}_{18}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}21}= {\mathrm{K}}_{17}{\mathrm{K}}_{20}{\mathrm{P}}_{\mathrm{So}}\)

\({\mathrm{P}}_{\mathrm{S}2}={\mathrm{K}}_{18}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}12}= {\mathrm{K}}_{18}{\mathrm{K}}_{18}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}22}= {\mathrm{K}}_{18}{\mathrm{K}}_{20}{\mathrm{P}}_{\mathrm{So}}\)

\({\mathrm{P}}_{\mathrm{S}3}={\mathrm{K}}_{19}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}13}={\mathrm{K}}_{19}{\mathrm{K}}_{18}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}23}= {\mathrm{K}}_{19}{\mathrm{K}}_{20}{\mathrm{P}}_{\mathrm{So}}\)

\({\mathrm{P}}_{\mathrm{S}4}={\mathrm{K}}_{20}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}14}= {\mathrm{K}}_{20}{\mathrm{K}}_{18}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}24}= {\mathrm{K}}_{21}{\mathrm{K}}_{20}{\mathrm{P}}_{\mathrm{So}}\)

\({\mathrm{P}}_{\mathrm{S}5}={\mathrm{K}}_{20}{\mathrm{K}}_{20}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}15}={\mathrm{K}}_{21}{\mathrm{K}}_{18}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}25}= {\mathrm{K}}_{17}{\mathrm{K}}_{20}{{\mathrm{K}}_{20}\mathrm{P}}_{\mathrm{S}0}\)

\({\mathrm{P}}_{\mathrm{S}6}={\mathrm{K}}_{17}{\mathrm{K}}_{17}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}16}= {\mathrm{K}}_{17}{\mathrm{K}}_{19}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}26}= {\mathrm{K}}_{18}{\mathrm{K}}_{20}{{\mathrm{K}}_{20}\mathrm{P}}_{\mathrm{S}0}\)

\({\mathrm{P}}_{\mathrm{S}7}= {\mathrm{K}}_{18}{\mathrm{K}}_{17}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}17}={\mathrm{K}}_{18}{\mathrm{K}}_{19}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}27}= {\mathrm{K}}_{19}{\mathrm{K}}_{20}{{\mathrm{K}}_{20}\mathrm{P}}_{\mathrm{S}0}\)

\({\mathrm{P}}_{\mathrm{S}8}={\mathrm{K}}_{19}{\mathrm{K}}_{17}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}18}={\mathrm{K}}_{19}{\mathrm{K}}_{19}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}28}= {\mathrm{K}}_{20}{\mathrm{K}}_{20}{{\mathrm{K}}_{20}\mathrm{P}}_{\mathrm{S}0}\)

\({\mathrm{P}}_{\mathrm{S}9}={\mathrm{K}}_{20}{{\mathrm{K}}_{17}\mathrm{P}}_{\mathrm{S}0}\) \({\mathrm{P}}_{\mathrm{S}19}={\mathrm{K}}_{20}{\mathrm{K}}_{19}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}29}= {\mathrm{K}}_{21}{\mathrm{K}}_{20}{{\mathrm{K}}_{20}\mathrm{P}}_{\mathrm{S}0}\)

\({\mathrm{P}}_{\mathrm{S}10}={\mathrm{K}}_{21}{\mathrm{K}}_{17}{\mathrm{P}}_{\mathrm{So}}\) \({\mathrm{P}}_{\mathrm{S}20}={\mathrm{K}}_{21}{\mathrm{K}}_{19}{\mathrm{P}}_{\mathrm{So}}\)where, \({\mathrm{K}}_{\mathrm{i}}={\Phi }_{\mathrm{i}}/{\upmu }_{\mathrm{i}}\) i = 17, 18, 19, 20, 21.

Application of Normalizing condition or equating the addition of all the state probabilities to one, we get:

$${\mathrm{P}}_{\mathrm{S}0}+{\mathrm{P}}_{\mathrm{S}1}+{\mathrm{P}}_{\mathrm{S}2}+{\mathrm{P}}_{\mathrm{S}3}+{\mathrm{P}}_{\mathrm{S}4}+{\mathrm{P}}_{\mathrm{S}5}+{\mathrm{P}}_{\mathrm{S}6}+{\mathrm{P}}_{\mathrm{S}7}+{\mathrm{P}}_{\mathrm{S}8}+{\mathrm{P}}_{\mathrm{S}9}+{\mathrm{P}}_{\mathrm{S}10}+{\mathrm{P}}_{\mathrm{S}11}+{\mathrm{P}}_{\mathrm{S}12}+{\mathrm{P}}_{\mathrm{S}13}+{\mathrm{P}}_{\mathrm{S}14}+{\mathrm{P}}_{\mathrm{S}15}+{\mathrm{P}}_{\mathrm{S}16}+{\mathrm{P}}_{\mathrm{S}17}+{\mathrm{P}}_{\mathrm{S}18}+{\mathrm{P}}_{\mathrm{S}19}+{\mathrm{P}}_{\mathrm{S}20}+{\mathrm{P}}_{\mathrm{S}21}+{\mathrm{P}}_{\mathrm{S}22}+{\mathrm{P}}_{\mathrm{S}23}+{\mathrm{P}}_{\mathrm{S}24}+{\mathrm{P}}_{\mathrm{S}25}+{\mathrm{P}}_{\mathrm{S}26}+{\mathrm{P}}_{\mathrm{S}27}+{\mathrm{P}}_{\mathrm{S}28}+{\mathrm{P}}_{\mathrm{S}29} =1$$
$${\mathrm{P}}_{\mathrm{S}0}\left[1+{\mathrm{K}}_{17}+{\mathrm{K}}_{18}+{\mathrm{K}}_{17}+{\mathrm{K}}_{20}+{{(\mathrm{K}}_{20})}^{2}+{{(\mathrm{K}}_{17})}^{2}+{\mathrm{K}}_{18}{\mathrm{K}}_{17}+{\mathrm{K}}_{19}{\mathrm{K}}_{17}+{\mathrm{K}}_{20}{\mathrm{K}}_{17}+{\mathrm{K}}_{21}{\mathrm{K}}_{17}+{\mathrm{K}}_{17}{\mathrm{K}}_{18}+{{(\mathrm{K}}_{18})}^{2}+{\mathrm{K}}_{19}{\mathrm{K}}_{18}+{\mathrm{K}}_{20}{\mathrm{K}}_{18}+{\mathrm{K}}_{21}{\mathrm{K}}_{18}+{\mathrm{K}}_{17}{\mathrm{K}}_{19}+{\mathrm{K}}_{18}{\mathrm{K}}_{19}+{{(\mathrm{K}}_{19})}^{2}+{\mathrm{K}}_{20}{\mathrm{K}}_{19}+{\mathrm{K}}_{21}{\mathrm{K}}_{19}+{\mathrm{K}}_{17}{\mathrm{K}}_{20}+{\mathrm{K}}_{18}{\mathrm{K}}_{20}+{\mathrm{K}}_{19}{\mathrm{K}}_{20}+{\mathrm{K}}_{21}{\mathrm{K}}_{20}+{\mathrm{K}}_{17}{{(\mathrm{K}}_{20})}^{2}+{\mathrm{K}}_{18}{{(\mathrm{K}}_{20})}^{2}+{\mathrm{K}}_{19}{{(\mathrm{K}}_{20})}^{2}+{{(\mathrm{K}}_{20})}^{3}+{\mathrm{K}}_{21}{{(\mathrm{K}}_{20})}^{2}\right]=1$$
$${\mathrm{P}}_{\mathrm{S}0}=1/[1+{\mathrm{K}}_{17}\left(2+{\mathrm{K}}_{17}+{\mathrm{K}}_{18}+{\mathrm{K}}_{19}+{\mathrm{K}}_{20}+{\mathrm{K}}_{21}+{\mathrm{K}}_{18}+{\mathrm{K}}_{19} +{{\mathrm{K}}_{20})+ {\mathrm{K}}_{18}{\left(1+{\mathrm{K}}_{18}+{\mathrm{K}}_{19}+{\mathrm{ K}}_{20}+{\mathrm{K}}_{21}+{\mathrm{K}}_{19}+{\mathrm{K}}_{20}\right) +{\mathrm{K}}_{20} (1+{\mathrm{K}}_{19}+{\mathrm{K}}_{19}+{\mathrm{K}}_{21}) +(\mathrm{K}}_{20})}^{2}\left(1+{\mathrm{K}}_{17}+{\mathrm{K}}_{18}+{\mathrm{K}}_{19}+{\mathrm{K}}_{21}\right)+{\mathrm{K}}_{19}+\left({\mathrm{K}}_{19}+{\mathrm{K}}_{21}\right)+{{(\mathrm{K}}_{20})}^{3}\right]$$

The availability (AV) of the Evaporation and Crystallization Unit for all state conditions is shown below. It is calculated by using the normalizing condition, i.e., Ʃ (states 0 (full capacity) and states 1–5 (reduced capacity)).

$$\mathrm{AV}={\mathrm{P}}_{\mathrm{S}0}+{\mathrm{P}}_{\mathrm{S}1}{ +\mathrm{ P}}_{\mathrm{S}2}{ +\mathrm{ P}}_{\mathrm{S}3}{ +\mathrm{ P}}_{\mathrm{S}4}+{\mathrm{P}}_{\mathrm{S}5}$$
$$\mathrm{AV}={\mathrm{P}}_{\mathrm{S}0}[(1+{\mathrm{K}}_{17}+{\mathrm{K}}_{18}+{\mathrm{K}}_{19}+{\mathrm{K}}_{20}+{{(\mathrm{K}}_{20})}^{2}]$$
$$\mathrm{AV}=[\left(1+{\mathrm{K}}_{17}+{\mathrm{K}}_{18}+{\mathrm{K}}_{19}+{\mathrm{K}}_{20}+{{(\mathrm{K}}_{20})}^{2}\right] / [1+{\mathrm{K}}_{17}\left(2+{\mathrm{K}}_{17}+{\mathrm{K}}_{18}+{\mathrm{K}}_{19}+{\mathrm{K}}_{20}+{\mathrm{K}}_{21}+{\mathrm{ K}}_{18}+{\mathrm{K}}_{19} +{{\mathrm{K}}_{20}) {\mathrm{K}}_{18}{\left(1+{\mathrm{K}}_{18}+{\mathrm{K}}_{19}+{\mathrm{ K}}_{20}+{\mathrm{K}}_{21}+{\mathrm{K}}_{19}+{\mathrm{K}}_{20}\right) +{\mathrm{K}}_{20} (1+{\mathrm{K}}_{19}+{\mathrm{K}}_{21}) +(\mathrm{K}}_{20})}^{2}\left(1+{\mathrm{K}}_{17}+{\mathrm{K}}_{18}+{\mathrm{K}}_{19}+{\mathrm{K}}_{21}\right)+{\mathrm{K}}_{19}+\left({\mathrm{K}}_{19}+{\mathrm{K}}_{21}\right)+{{(\mathrm{K}}_{20})}^{3}\right]$$

6.2 Performance analysis of the unit

The availability is considered a performance parameter for current research work. The maintenance history sheets obtain the values of failure and repair rates. For carrying out the availability analysis, decision matrices are developed and shown in Tables 1, 2, 3, 4, 5. Subsequently, decision-making regarding maintenance planning and priorities are considered for diverse subsystems. The best possible combination (Φ, µ) is selected for the current study as per the decision matrix Tables 1, 2, 3, 4, 5.

Table 1 Decision Matrix for 'Evaporator' Subsystem
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Table 2 Decision Matrix for 'Pan' Subsystem
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Table 3 Decision Matrix for 'Crystallizer' Subsystem
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Table 4 Decision Matrix for 'Centrifuge' Subsystem
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Table 5 Decision Matrix for 'Sugar Grader' Subsystem
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Table 1 and Fig. 3 reveal that the maximum availability is observed at the failure rate of 0.01 and repair rate of 0.30, i.e., 0.962. In contrast, the minimum availability is obtained at a failure rate of 0.030 and a repair rate of 0.10 (i.e., 0.726). Figure 3 depicts the 3D plots for the evaporator unit. It is evident that the availability is maximum, the repair rate is maximum, and the failure rate is observed as a minimum. It is similar to the previous findings of Kumar et al. ( 2016) that determined the maintenance priority level of a thermal plant repairable flue gas and air system. Moreover, a drastic change is also observed in the availability from 72.6 to 88.8%. On the other hand, if a booster change is observed for repairing the unit is decreased from 10 to 3.5 h.

Fig. 3
figure 3

3-D map between the failure and repair rates for evaporator

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Table 2 and Fig. 4 reveals the effect of failure and repair rates of the Pan Subsystem on the availability of evaporation and crystallization unit. It is observed that the performance of subsystems decreases with an increase in the failure rate up to a certain point, and after that, it is increased significantly with an increase in repair rate. Pan subsystem failure rate increases from 0.010to 0.030, and as a result, the unit availability decreases sharply from 0.895 to 0.829, i.e., 6.6%. In contrast, the repair rate of the Pan Subsystem increases from 0.15 to 0.35 and attribute to a significant increase in the availability from 0.829 to 0.8854, i.e., 5.64%. On the other hand, in case hours a rooter change is observed for repairing the unit is decreased from 6.6 to 3 h.

Fig. 4
figure 4

3-Dmap between the failure and repair rates for pan

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The decision matrix described in Table 3 and respective Fig. 5 for the crystallizer reflects the unit availability, which shifted with the trend of failure rate and repair rate of the crystallizer. A radical fall of 12% in the level of unit availability occurs with the rise in the failure rate (Φ19) from 0.004 to 0.008 (i.e., once in 250 h to once in 125 h). Also, an appreciable enhancement of 5% in unit availability can be observed with the rise of repair rate (µ19) from 0.020 to 0.040 (i.e., once in 50 h to once in 25 h).

Fig. 5
figure 5

3-D map between the failure and repair rates for crystallizer

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Table 4 and Fig. 6 illustrate the effect of input parameters on the availability of the centrifuge subsystem. A massive fall of 10% in the level of unit availability occurs with the rise in the failure rate (Φ20) from 0.002 to 0.006 (i.e., once in 500 h to once in 166 h). Also, a substantial improvement of 3% in unit availability can be observed with the rise of repair rate (µ20) from 0.015 to 0.035 (i.e., once in 66 h to once in 28 h).

Fig. 6
figure 6

3-D map between the failure and repair rates for centrifuge

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In the case of sugar grader, the availability of the subsystem is decreased with a marginal fall of 1% with an increase in the failure rate from 0.004 to 0.008(i.e., once in 250 h to once in 125 h) as shown in Fig. 7. Besides this, a small increase of 1% is observed in unit availability with the rise in repair rate (µ21) from 0.10 to 0.30 (i.e., once in 10 h to once in 3 h) (Table 6).

Fig. 7
figure 7

3-D map between the failure and repair rates for sugar grader

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Table 6 Priorities of Subsystems as per their maintenance schedule
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6.3 Determination of maintenance priorities

The Performance analysis of a sugar plant Evaporation and Crystallization unit is accomplished with the help of the birth–death–birth process. The outcomes of the analysis are described in the decision matrix. These decision matrices will assist the maintenance department in decision-making regarding the maintenance priorities of the concerned unit. The Evaporator subsystem is found to be the highest critical subsystem for the current study. The increase in the repair rate of this subsystem resulted in 8% of enhancement in the availability of the concerned unit. Owing to this, this subsystem is on top maintenance priority. Aforementioned performance analysis shows the order of maintenance of different subsystems as given below:

7 Conclusion

The current research uses the Markov Birth death technique to find the best economical maintenance schedule for sugar plant industry equipment. The following conclusion is drawn listed below:

  1. 1.

    The most critical unit is the Evaporator unit, in which the availability increased from 0.7260 to 0.8888, i.e. (16.20%). In contrast, Sugar Grader is the least critical unit with an availability enhanced from 0.8740 to 0.8893 (i.e., 1.53%).

  2. 2.

    A drastic fall of 16.20% in the level of unit availability occurs with the rise in the failure rate of the evaporator subsystem (Φ17) from 0.010 to 0.030 (i.e., once in 100 h to once in 33 h). Also, a significant boost of 8% in unit availability can be observed with the rise in repair rate evaporator subsystem (µ17) from 0.10 to 0.30 (once in 10 h to once in 3.5 h).

  3. 3.

    A huge fall of 7% in the level of pan unit availability occurs with the rise in the failure rate of the pan subsystem (Φ18) from 0.010 to 0.030 (i.e., once in 100 h to once in 33 h). Also, a noticeable increase of 2% in unit availability can be observed with the rise in repair rate of the pan subsystem (µ18) from 0.15 to 0.35 (i.e., once in 6.6 h to once in 3 h).

  4. 4.

    A radical fall of 12% in the level of unit availability occurs with the rise in the failure rate of the crystallizer subsystem (Φ19) from 0.004 to 0.008 (i.e., once in 250 h to once in 125 h). Also, an appreciable enhancement of 5% in unit availability can be observed with the rise of repair rate of the crystallizer subsystem(µ19) from 0.020 to 0.040 (i.e., once in 50 h to once in 25 h).

  5. 5.

    A massive fall of 10% in the level of unit availability occurs with the rise in the centrifuge subsystem failure rate (Φ20) from 0.002 to 0.006 (i.e., once in 500 h to once in 166 h). Also, a substantial improvement of 3% in unit availability can be observed with the rise of the centrifuge (µ20) repair rate from 0.015 to 0.035 (i.e., once in 66 h to once in 28 h).

  6. 6.

    A marginal fall of 1% in the level of unit availability occurs with the rise in the failure rate of sugar graders (Φ21 from 0.004 to 0.008 (i.e., once in 250 h to once in 125 h). Also, a mere step-up of 1% can be observed in unit availability with the rise in repair rate of sugar grader (µ21) from 0.10 to 0.30 (i.e., once in 10 h to once in 3 h).

  7. 7.

    For the complete analysis of sugar plant industries, the different units of sugar plant industries can be arranged according to maintenance priorities as follows: Evaporator, Crystallizer, Centrifuge, Pan, and Sugar Grader.

  8. 8.

    The current research work can be implemented in a sugar plant industry to increase the availability of machines during working hours. The maintenance priority can be utilized to make a repair schedule for the different units of sugar plant industries.