Stochastic modelling and meta heuristic optimization of a batch production system

1.1 Literature survey

To comprehend the research gaps present, a thorough review of the prior studies is presented as follows:

Several experts have investigated client arrivals in bulks. Moreover, due to its multidisciplinary nature, most research focuses on batch service queues operating under various regulations. In their groundbreaking work, Chaudhry and Templeton (1983) devised a QS called quorum queue, in which a server prioritizes and serves requests in batches of fixed size r $\le$ 1. By employing embedded Markov chain methods, Tadj and Ke (2008) addressed a hysteretic bulk quorum queue. Tadj et al. (2012) also scrutinized a dual-phase unreliable bulk quorum queue. Taking into account a variety of arrival scenarios, Kashifalghitaa and Al-Obaidi (2023) recently examined a quorum queue with an $N$ -policy.

If a client is dissatisfied with the service they received, they can request that the service be redone until they are satisfied. Lately, an asymptotic analysis of a RQ with feedback was carried out by Nazarov et al. (2023). In addition, Laxmi et al. (2023) scrutinized a feedback batch arrival QS with second discretionary service.

Further, distinct systems face varying kinds of vacations based on the circumstances. Servi and Finn (2002) were among the first to introduce an $M/M/1$ QS subjected under WV. An batch arrival unreliable RQ with negative consumers under Bernoulli working vacation (BWV) was addressed by Rajadurai et al. (2016). Further, an analysis of a bulk arrival pre-emptive priority RQ subjected to delayed repairs under a modified Bernoulli vacation schedule was done by Yuvarani and Saravanarajan (2017). Lately, a transient analysis of a two classes of bulk arrivals, retrial clients with non-pre-emptive priority service and WVs was studied by Ayyappan and Thamizhselvi (2018)

While developing any system, preventative steps should be incorporated. Therefore, one of the most crucial topics in this study is server breakdowns and their repair. Thus, many of the explored systems have thought about this issue, including Bagyam et al. (2013) etc,. Recently, Bharathidass et al. (2018) analysed a batch service queue with server breakdowns and repairs. A finite Markovian QS subjected to breakdown was also recently analysed by Das et al. (2022).

There is not much of work on RQ that addresses cost optimization and evaluates appropriate control parameters for the queueing model using many optimization techniques. Lately, Vaishnawi et al. (2022) analysed the accuracy of a $Ge{o}^{\left[X\right]}/Geo/1$ recurrent model in discrete time to induce the lowest possible cost for the system. Moreover, refer to articles (Kumar and Jain, 2023; Deora et al., 2021; Jain and Kumar, 2022) for a review of published works on cost optimization.

1.2 Research novelty

Based on the literature review section (Section 1.1), no work on quorum RQ with BWV has been conducted. Thus, with the objective of filling this research gap, this study addresses a BPS via bulk quorum feedback RQ with BWV, random breakdown, and a stand-by server, which is evaluated by the supplementary variable technique (SVT). Further, the proposed framework has been analysed using six distinct optimization methods.

1.3 Rationale for choosing a BPS and its benefits

• Flexibility: Allows production of varied products without extensive changeovers.

• Efficiency: Optimizes resource use by grouping similar tasks.

• Cost Efficiency: Leads to economies of scale and efficient resource use.

• Customization: Easily accommodates custom orders and product variations.

• Risk Mitigation: Smaller batches reduce the risk of large-scale production defects.

• Resource Allocation: Improves resource allocation by dedicating specific resources to particular batches.

1.4 Structure of the work

The rest of the work is outlined as follows: Section 2 presents an analysis of the proposed framework. The system’s stability criterion, associated equations, and steady state (SS) outcomes are determined in Section 3. Several essential system metrics are covered in Section 4. The system’s behaviour is addressed via a few numerical examples in Section 5. Section 6 delves into the detailed assessment of cost optimization via distinct approaches. In Section 7, the suggested model is discussed briefly. Overall study is summarized in Section 8.

2.1 Interpretation of the proposed model into BPS

The proposed framework introduces a novel application for batch production systems, enhancing production facility operations.

Products arrive in batches, allowing efficient processing of multiple items simultaneously, reducing setup times, and increasing throughput. If the processing system is busy, batches wait in an orbit or may balk to avoid delays. Batches in orbit retry processing later if the server remains busy or the queue limit is reached, managing workload efficiently. To handle random equipment breakdowns, a standby server activates during failures. The secondary stand-by server acts as a backup, remaining inactive to conserve resources but swiftly activating during server failures or peak demand. Its role improves system reliability by minimizing downtime and disruptions, while also optimizing resource utilization for consistent and efficient production. During periods of low demand or when the system is not operating at full capacity, servers may enter a working vacation mode where they operate at a reduced capacity. Finally, the system undergoes feedback mechanisms where the queue lengths and processing times, as well as adjusting system parameters such as batch sizes, processing rates, server availability, and system reliability, are continuously monitored. Thus, the production process significantly influences the modelling, analysis, and outcomes of the study leading to more effective and resilient production management strategies.

3.1 The steady state equations

In SS, we take into consideration that $\mathcal{G}\left(0\right)=0$ , $\mathcal{G}\left(\infty \right)=1$ , ${\mathcal{Q}}_b\left(0\right)=0$ , ${\mathcal{Q}}_b\left(\infty \right)=1$ , and ${\mathcal{Q}}_v\left(0\right)=0$ , ${\mathcal{Q}}_v\left(\infty \right)=1$ , are continuous at $\tilde{x}=0$ and $\mathcal{S}\left(0\right)=0$ , $\mathcal{S}\left(\infty \right)=1$ , are continuous at $\hat{y}=0$ . Hence, the functions $\theta \left(\tilde{x}\right)$ , ${\upsilon }_b\left(\tilde{x}\right)$ , ${\upsilon }_v\left(\tilde{x}\right)$ , and $\nu \left(\hat{y}\right)$ are the “conditional completion rates (hazard rates)” for retrial, regular service, on BWV, and under repair, respectively. $$\theta \left(\tilde{x}\right)d\tilde{x}=\frac{d\mathcal{G}\left(\tilde{x}\right)}{1-\mathcal{G}\left(\tilde{x}\right)};{\upsilon }_b\left(\tilde{x}\right)d\tilde{x}=\frac{d{\mathcal{Q}}_b\left(\tilde{x}\right)}{1-{\mathcal{Q}}_b\left(\tilde{x}\right)};{\upsilon }_v\left(\tilde{x}\right)d\tilde{x}=\frac{d{\mathcal{Q}}_v\left(\tilde{x}\right)}{1-{\mathcal{Q}}_v\left(\tilde{x}\right)};\nu \left(\hat{y}\right)d\hat{y}=\frac{d\mathcal{S}\left(\hat{y}\right)}{1-\mathcal{S}\left(\hat{y}\right)}$$ ${\mathcal{G}}^0\left(\check{\varepsilon }\right)$ , ${\mathcal{Q}}_b^0\left(\check{\varepsilon }\right)$ , ${\mathcal{Q}}_v^0\left(\check{\varepsilon }\right)$ and ${\mathcal{S}}^0\left(\check{\varepsilon }\right)$ indicate the elapsed retrial period, service times, BWV, and repair period during time $\check{\varepsilon }$ , correspondingly. Furthermore, create the random variable, $$\Theta \left(\check{\varepsilon }\right)=\left\{\begin{array}{cc}0,\hfill & \text{the server being available and in WV period}\hfill \\ 1,\hfill & \text{the availability and normal operation of the server}\hfill \\ 2,\hfill & \text{the server being occupied and in normal service period at time}\check{\varepsilon }\hfill \\ 3,\hfill & \text{the server being occupied and in WV period at time}\check{\varepsilon }\hfill \\ 4,\hfill & \text{the server is under repair at time}\check{\varepsilon }.\hfill \end{array}\right.$$

Furthermore, we emphasize how the bivariate Markov process can be utilized to describe the system’s state at a given time $\check{\varepsilon }$ { $\Theta \left(\check{\varepsilon }\right)$ , $\Xi \left(\check{\varepsilon }\right)$ ; $\check{\varepsilon }$ $\ge$ 0 }, where $\Theta \left(\check{\varepsilon }\right)$ indicates the server state $(0,1,2,3,4)$ based on whether the server is inactive, engaged, on BWV or under repair. For all products in the orbit at time $\check{\varepsilon }$ , $\Xi \left(\check{\varepsilon }\right)$ is the number that represents them. If $\Theta \left(\check{\varepsilon }\right)=1$ and $\Xi \left(\check{\varepsilon }\right)>0$ , then ${\mathcal{G}}^0\left(\check{\varepsilon }\right)$ corresponds to the elapsed retrial time. If $\Theta \left(\check{\varepsilon }\right)=2$ and $\Xi \left(\check{\varepsilon }\right)\ge 0$ , then ${\mathcal{Q}}_b^0\left(\check{\varepsilon }\right)$ is equivalent to the elapsed time of the product being manufactured regularly. If $\Theta \left(\check{\varepsilon }\right)=3$ and $\Xi \left(\check{\varepsilon }\right)\ge 0$ , then ${\mathcal{Q}}_v^0\left(\check{\varepsilon }\right)$ proportional to the elapsed time of the product spent manufacturing in slow service period. The elapsed time of the server getting fixed corresponds to ${\mathcal{S}}^0\left(\check{\varepsilon }\right)$ if $\Theta \left(\check{\varepsilon }\right)=4$ and $\Xi \left(\check{\varepsilon }\right)\ge 0$ .

SVT, a technique for analysing non-Markovian processes { $\Xi \left(\check{\varepsilon }\right)$ ; $\ge$ 0 }, is crucial for studying complex systems like BPS. By incorporating supplementary variables, SVT transforms the BPS into a Markovian process, enabling analysis using generating function methods. This approach enhances precision in calculating performance metrics and offers a deeper understanding of system behaviour. SVT’s flexibility allows for comprehensive analysis of the BPS under different scenarios, facilitating informed decision-making.

Let { ${\check{\varepsilon }}_n;n=1,2,\dots$ } be the set of epochs during which a regular service period, BWV period, or repair period ends. As a result, an MC is generated and added in the retry QS utilizing a series of random vectors $D_n=\{\Theta ({\check{\varepsilon }}_n+),\Xi ({\check{\varepsilon }}_n+)\}$ . Theorem 3.1 asserts that $D_n;n\in N$ is ergodic iff $\left[s+{\mathcal{X}}_1(1-{\mathcal{G}}^{\ast }\left(ϵ\right))-\left(ϵb{\mathcal{X}}_1(1+\delta {\mu }^{\left(1\right)})+\delta \alpha {\mu }^{\left(1\right)}\right){\beta }_b^{\left(1\right)}\right]<1$ , consequently our system must be stable. The probabilities, ${\Lambda }_0\left(\check{\varepsilon }\right)=P\{\Theta \left(\check{\varepsilon }\right)=0,\Xi \left(\check{\varepsilon }\right)=0\}$ and ${\Psi }_0\left(\check{\varepsilon }\right)=P\{\Theta \left(\check{\varepsilon }\right)=1,\Xi \left(\check{\varepsilon }\right)=0\};$ and probability densities for the method $\{\Xi \left(\check{\varepsilon }\right),\check{\varepsilon }\ge 0\}$ , are stated below, $${\Psi }_n(\tilde{x},\check{\varepsilon })d\tilde{x}=P\{\Theta \left(\check{\varepsilon }\right)=0,\Xi \left(\check{\varepsilon }\right)=n,\tilde{x}\le {\mathcal{G}}_i^0\left(\check{\varepsilon }\right)<\tilde{x}+d\tilde{x}\},\text{for}\check{\varepsilon },\tilde{x}\ge 0\text{and}n\ge 1$$ $${\Gamma }_{b,n}(\tilde{x},\check{\varepsilon })d\tilde{x}=P\{\Theta \left(\check{\varepsilon }\right)=1,\Xi \left(\check{\varepsilon }\right)=n,\tilde{x}\le {\mathcal{Q}}_b^0\left(\check{\varepsilon }\right)<\tilde{x}+d\tilde{x}\},\text{for}\check{\varepsilon },\tilde{x}\ge 0,n\ge 0$$ $${\Lambda }_{v,n}(\tilde{x},\check{\varepsilon })d\tilde{x}=P\{\Theta \left(\check{\varepsilon }\right)=2,\Xi \left(\check{\varepsilon }\right)=n,\tilde{x}\le {\mathcal{Q}}_v^0\left(\check{\varepsilon }\right)<\tilde{x}+d\tilde{x}\},\text{for}\check{\varepsilon },\tilde{x}\ge 0,n\ge 0$$ $${\Omega }_n(\tilde{x},\hat{y},\check{\varepsilon })d\tilde{x}=P\{\Theta \left(\check{\varepsilon }\right)=3,\Xi \left(\check{\varepsilon }\right)=n,\hat{y}\le {\mathcal{S}}^0\left(\check{\varepsilon }\right)<\hat{y}+d\hat{y}/{\mathcal{Q}}_b^0\left(\check{\varepsilon }\right)=\tilde{x}\},$$ $$\text{for}\check{\varepsilon }\ge 0,(\tilde{x},\hat{y})\ge 0,n\ge 0$$

We believe the follow-up meets the stability criteria, and as a result we get ${\Psi }_0={lim}_{\check{\varepsilon }\to \infty }{\Psi }_0\left(\check{\varepsilon }\right)$ and ${\Lambda }_0={lim}_{\check{\varepsilon }\to \infty }{\Lambda }_0\left(\check{\varepsilon }\right)$ . We can therefore designate limiting probabilities for $\tilde{x},\hat{y}>0$ , $n\ge 0$ $${\Psi }_n\left(\tilde{x}\right)=\underset{\check{\varepsilon }\to \infty }{lim}{\Psi }_n(\tilde{x},\check{\varepsilon });{\Gamma }_{b,n}\left(\tilde{x}\right)=\underset{\check{\varepsilon }\to \infty }{lim}{\Gamma }_{b,n}(\tilde{x},\check{\varepsilon });$$ $${\Lambda }_{v,n}\left(\tilde{x}\right)=\underset{\check{\varepsilon }\to \infty }{lim}{\Lambda }_{v,n}(\tilde{x},\check{\varepsilon });{\Omega }_n(\tilde{x},\hat{y})=\underset{\check{\varepsilon }\to \infty }{lim}{\Omega }_n(\tilde{x},\hat{y},\check{\varepsilon }).$$ SVT yields the subsequent equations, that regulate the behaviour of the system dynamics.

3.2 The steady state solution

The generating function technique assists to generate the SS solution to fit the RQ system. Furthermore, the generating functions for $\mid \breve{z}\mid <1$ are stated beneath in order to solve the equations above: $$b\left(\breve{z}\right)=\sum _{n=1}^{\infty }{\mathcal{B}}_n{\breve{z}}^n;\Psi (\tilde{x},\breve{z})=\sum _{n=1}^{\infty }{\Psi }_n\left(\tilde{x}\right){\breve{z}}^n;\Psi (0,\breve{z})=\sum _{n=1}^{\infty }{\Psi }_n\left(0\right){\breve{z}}^n;$$ $${\Gamma }_b(\tilde{x},\breve{z})=\sum _{n=0}^{\infty }{\Gamma }_{b,n}\left(\tilde{x}\right){\breve{z}}^n;{\Gamma }_b(0,\breve{z})=\sum _{n=0}^{\infty }{\Gamma }_{b,0}\left(0\right){\breve{z}}^n;{\Lambda }_v(\tilde{x},\breve{z})=\sum _{n=0}^{\infty }{\Lambda }_{v,n}\left(\tilde{x}\right){\breve{z}}^n;$$ $${\Lambda }_v(0,\breve{z})=\sum _{n=0}^{\infty }{\Lambda }_{v,0}\left(0\right){\breve{z}}^n;\Omega (\tilde{x},\hat{y},\breve{z})=\sum _{n=0}^{\infty }{\Omega }_n(\tilde{x},\hat{y}){\breve{z}}^n;\Omega (\tilde{x},0,\breve{z})=\sum _{n=0}^{\infty }{\Omega }_0(\tilde{x},0){\breve{z}}^n$$

Multiply the SS equations and boundary conditions from (3.3) to (3.10) by ${\breve{z}}^n$ and sum on $n$ where $(n=0,1,2,\dots )$ .