Lead time control in multi-server multi-stage assembly system

Abstract

In this paper, two multi-objective models to optimally control the lead time in multi-server multi-stage assembly system by considering server allocation problem (SAP) and also service rate control problem (SCP) are presented, where new product orders, including all their operations, are entered to the system according to a Poisson process, only one type of products also is produced by the production system, and multi-servers can be settled in each service station. Each operation of any work is operated at a devoted service station with only one of the servers located at a node of the network based on a first-come-first-serve discipline, while the processing times are independent random variables with exponential distributions. Furthermore, it is also assumed that the transport times between each pair of service stations are independent random variable with generalized Erlang distributions. Such system can be modeled as a queueing network, where the system is in the steady state and the lead time is controllable. For modeling of multi-server multi-stage assembly system, initially the network of queues is transformed into an appropriate stochastic network with exponentially distributed arc lengths. A differential equations system is organized to solve and obtain approximate lead time distribution for any particular wok by applying a proper finite-state continuous-time Markov model. Then, two multi-objective models including four conflicted objectives are presented to optimally control the servers allocated to the service stations in SAP and also service rate of service stations in SCP. For solving a discrete-time approximation of the primary multi-objective problems, the goal attainment technique is employed. In this research, reactive controlling in a multi-server multi-stage assembly system also is discussed.

Appendix

If there is m _a server in the service station settled in the ath node, then the queueing system is M/M/m _a , and therefore, the density function of sojourn time (processing time plus waiting time in queue) is calculated as follow [45]:

$$ {w_a}(t) = \left( {\frac{{\lambda - {m_a}{\mu_a} + \mu w_a^q(0)}}{{\lambda - \left( {{m_a} - 1} \right){\mu_a}}}} \right){\mu_a}{e^{{ - {\mu_a}t}}} + \left( {1 - \frac{{\lambda - {m_a}{\mu_a} + \mu w_a^q(0)}}{{\lambda - \left( {{m_a} - 1} \right){\mu_a}}}} \right)\left( {{m_a}{\mu_a} - \lambda } \right){e^{{ - \left( {{m_a}{\mu_a} - \lambda } \right)t}}}\quad t \succ 0 $$

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where λ and μ _a are, respectively, the arrival rate of new product order and the service rate of each server settled in node a. Also, $w_a^q(0)$, the probability of being zero queue length, and P ₀ are obtained as follow:

$$ w_a^q(0) = 1 - \frac{{{m_a}{{\left( {\frac{\lambda }{{{\mu_a}}}} \right)}^{{{m_a}}}}}}{{{m_a}!\left( {{m_a} - \frac{\lambda }{{{\mu_a}}}} \right)}} \times {P_0} $$

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$$ {\text{and: }}{P_0} = {\left( {\sum\limits_{{n = 1}}^{{{m_a} - 1}} {\frac{1}{{n!}}{{\left( {\frac{\lambda }{{{\mu_a}}}} \right)}^n} + \frac{1}{{{m_a}!}}{{\left( {\frac{\lambda }{{{\mu_a}}}} \right)}^{{{m_a}}}}\left( {\frac{{{m_a}{\mu_a}}}{{{m_a}{\mu_a} - \lambda }}} \right)} } \right)^{{ - 1}}} $$

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As is observed, obtaining of w _a (t) in an M/M/m _a is very hard. We can rewrite the w _a (t) as follow that is similar to two series of exponential distribution with parameters (m _a μ _a  − λ) and μ _a :

$$ {w_a}(t) = \left( {1 - w_a^q(0)} \right)\left( {\frac{{{\mu_a}}}{{{\mu_a} - \left( {{m_a}{\mu_a} - \lambda } \right)}}} \right)\left( {{m_a}{\mu_a} - \lambda } \right){e^{{ - \left( {{m_a}{\mu_a} - \lambda } \right)t}}} - \left( {1 - \frac{{{\mu_a}w_a^q(0)}}{{{m_a}{\mu_a} - \lambda }}} \right)\frac{{{m_a}{\mu_a} - \lambda }}{{{\mu_a} - \left( {{m_a}{\mu_a} - \lambda } \right)}}{\mu_a}{e^{{ - {\mu_a}t}}}\quad t \succ 0 $$

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It is seemed that we can approximate density function of time spent in service station a with two series exponential. Therefore, our approximate for density function of time spent in service station a would be two series exponential with parameters $\left( {\frac{{{m_a}{\mu_a} - \lambda }}{{{\rho_a}}}} \right)$ and $\left( {\frac{{{\mu_a}}}{{1 - \frac{{\left( {1 - {\rho_a}} \right){\mu_a}}}{{{m_a}{\mu_a} - \lambda }}}} = \frac{{{m_a}{\mu_a}}}{{{m_a} - 1}}} \right)$, where ${\rho_a} = \frac{\lambda }{{{m_a}{\mu_a}}}$. This approximation is very simple and easy and also no need to calculating P ₀ and $w_a^q(0)$ that are boring, especially when m _a is large. Furthermore, our approximation can be used in mathematic programming problem and Markov chain conveniently.

We want to evaluate the mean of sojourn time and, therefore, the expected number of works and also the cumulative distribution function of sojourn time. In Fig. 7, the expected number of works in service station a as ρ _a changes is presented. Also, some sections of figures are highlighted to enhance their readability. Note that, in Fig. 7, when ρ _a  → 1, our approximation is very good.

In the literature, some approximations for the sojourn time in M/M/m _a have been introduced. Sakasegawa [46] proposed closed-form approximation for the expected waiting time in queue, and therefore, the approximation of expected sojourn time, W, in service station a was calculated as follow:

$$ W \approx \frac{1}{\mu }\left( {1 + \frac{{\rho_a^{{\sqrt {{2({m_a} + 1)}} - 1}}}}{{{m_a}\left( {1 - {\rho_a}} \right)}}} \right) $$

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Also, Halfin and Whitt [47] developed a closed-form approximation for the $w_a^q(0)$, and therefore, the approximation of expected sojourn time in service station a was obtained as follow:

$$ W \approx \frac{1}{\mu }\left( {1 + \frac{1}{{{m_a}\left( {1 - {\rho_a}} \right)\left( {1 + \sqrt {{2\pi }} \beta \cdot \Phi \left( \beta \right) \cdot {e^{{ \frac{{{\beta^2}}}{2} }}}} \right)}}} \right) $$

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where $\beta = \left( {1 - {\rho_a}} \right)\sqrt {{{m_a}}}$ and Φ(t) are the cumulative distribution function of standard normal distribution having mean 0 and variance 1. On the other hand, our approximation for the expected sojourn time in service station a would be:

$$ W \approx \left( {\frac{{{\rho_a}}}{{{m_a}{\mu_a} - \lambda }}} \right) + \left( {\frac{{{m_a} - 1}}{{{m_a}{\mu_a}}}} \right) $$

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Obviously, our approximation is very simple and its components easily, without boring computations, are controllable, i.e., the complexity of our approximation is less than other approximations. Furthermore, as mentioned before, our approximation can be used in mathematic programming problem and Markov chain conveniently.

In Table 8, the expected number of works in service station a, L _a , and the error of our approximation are presented.

As m _a  ≤ 5 and ρ _a  ≤ 0.3, our approximation for expected number of works in service stations is very poor; therefore, for coping this shortage, we consider $0.{3} \leqslant {\rho_a} \prec {1}$. Also, in Figs. 8 and 9, the exact cumulative distribution function and our approximate cumulative distribution function, respectively, for m _a  = 5 and m _a  = 10 with various utilization factors, is shown. Also some sections of figures are highlighted to enhance their readability. Note that, in Figs. 8 and 9, when ρ _a  = 0.98, our approximation for cumulative distribution function of sojourn time is very good.

Moreover, in Table 9, the maximum difference (MD) between the exact cumulative distribution function and our approximate cumulative distribution function for various number of server and utilization factors is shown.

Consequently, we have:

• 1. If m _a  = 1, then the queueing system would be an M/M/1 queue, and the density function of time spent in the service station a, w _a (t), would be exponentially with parameter μ _a  − λ; therefore, w _a (t) is calculated as follows:

  $$ {w_a}(t) = \left( {{\mu_a} - \lambda } \right){e^{{ - \left( {{\mu_a} - \lambda } \right)t}}}\quad t \succ 0,\;{\text{if}}\;{m_a} = 1 $$

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• 2. If m _a  = ∞, then the queueing system is M/M/∞, and the density function of time spent in the service station a would be exponentially expressed with parameter μ _a ; therefore, w _a (t) is calculated as follows:

  $$ {w_a}(t) = {\mu_a} \cdot {e^{{ - {\mu_a}t}}}\quad t \succ 0,\;{\text{if}}\;{m_a} = \infty $$

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• 3. If ${1} \prec {m_a} \prec \infty$, then the queueing system is M/M/m _a , and the density function of time spent in the service station a approximately would be two series exponential with parameters $\left( {\frac{{{m_a}{\mu_a} - \lambda }}{{{\rho_a}}}} \right)$ and $\left( {\frac{{{m_a}{\mu_a}}}{{{m_a} - 1}}} \right)$. Therefore, w _a (t) approximately is calculated as follows:

  $$ {w_a}(t) \approx \left( {\frac{{\left( {\frac{{{m_a}{\mu_a}}}{{{m_a} - 1}}} \right)}}{{\left( {\frac{{{m_a}{\mu_a}}}{{{m_a} - 1}}} \right) - \left( {\frac{{{m_a}{\mu_a} - \lambda }}{{{\rho_a}}}} \right)}}} \right)\left( {\frac{{{m_a}{\mu_a} - \lambda }}{{{\rho_a}}}} \right){e^{{ - \left( {\frac{{{m_a}{\mu_a} - \lambda }}{{{\rho_a}}}} \right)t}}} - \left( {\frac{{\left( {\frac{{{m_a}{\mu_a} - \lambda }}{{{\rho_a}}}} \right)}}{{\left( {\frac{{{m_a}{\mu_a}}}{{{m_a} - 1}}} \right) - \left( {\frac{{{m_a}{\mu_a} - \lambda }}{{{\rho_a}}}} \right)}}} \right)\left( {\frac{{{m_a}{\mu_a}}}{{{m_a} - 1}}} \right){e^{{ - \left( {\frac{{{m_a}{\mu_a}}}{{{m_a} - 1}}} \right)t}}}\quad t \succ 0 $$

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Table 8 Numerical value of exact and approximation expected number of works in service station

Table 9 MD between the exact and our approximate distribution function

Fig. 7

A comparison of our approximation with exact values for the expected number of works

Fig. 8

Cumulative distribution function of sojourn time for m _a  = 5

Fig. 9

Cumulative distribution function of sojourn time for m _a  = 10