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A retrial inventory system with single and modified multiple vacation for server

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Abstract

This paper considers a continuous review stochastic (s,S) inventory system with Poisson demand and exponentially distributed delivery time. The demands that occur during the stock out period or during the server vacation period enter into an orbit of infinite size. These orbiting demands retry to get satisfied by sending out signals so that the time durations are exponentially distributed. We consider two models which differ in the way that server go for vacation. The joint probability distribution of the inventory level, the number of demands in the orbit and the server status is obtained in the steady state case. Various system performance measures in the steady state is derived and the long-run total expected cost rate is calculated. Several numerical examples, which provide insight into the behaviour of the cost function, are presented.

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Abbreviations

e :

a column vector of appropriate dimension containing all ones.

\({\bf 0}\) :

a zero matrix of appropriate dimension.

\({\bf 0}_{(i,j)}\) :

a zero matrix with i rows and j columns.

I (n,n) :

Identity matrix of order n

H(x):

Heaviside function

δ ij :

Kronecker delta

\(\bar{\delta}_{ij}\) :

=1−δ ij

e s+1(1):

\(=\left (\begin{array}{c} 1 \\ 0 \\ \vdots\\ 0 \end{array} \right )_{(s+1, 1)}\)

\(\mathbf{e}^{T}_{s+1}(1)\) :

Transpose of e s+1(1)

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Acknowledgements

The authors wish to thank the anonymous referees for their comments, which significantly improved the presentation of this paper.

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Authors and Affiliations

  1. School of Mathematics, Madurai Kamaraj University, Madurai, INDIA

    I. Padmavathi, B. Sivakumar & G. Arivarignan

Authors
  1. I. Padmavathi
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  2. B. Sivakumar
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  3. G. Arivarignan
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Corresponding author

Correspondence to B. Sivakumar.

Additional information

The research of I. Padmavathi is supported by UGC-Rajiv Gandhi National Fellowship awarded No. F. 14-2(SC)/2010(SA-III).

The research of B. Sivakumar is supported by the SERB-Fast Track Young Scientist Scheme No. SR/FTP/MS-051/2011.

Appendices

Appendix A

To compute the R matrix, we use the following set of non-linear equations. This can be solved by using Gauss–Seidel iterative process. These equations are derived by exploiting the coefficient matrices appearing in (3),

$$\begin{aligned} r_{00'}\alpha-r_{00}(\lambda+\mu+\beta)+\lambda =&0, \\ r_{Q0'}\alpha-r_{Q0}(\lambda+\mu+\beta) =&0, \\ r_{0'0'}\alpha-r_{0'0}(\lambda+\mu+\beta) =&0, \end{aligned}$$
$$\begin{aligned} r_{00}\mu-r_{0Q}(\lambda+\beta) =&0, \\ r_{Q0}\mu-r_{QQ}(\lambda+\beta)+\lambda =&0, \\ r_{0'0}\mu-r_{0'Q}(\lambda+\beta) =&0, \end{aligned}$$
$$\begin{aligned} r_{00}r_{11}\theta+r_{0Q}r_{21} \theta+r_{00'}r_{31}\theta+r_{00} \beta-d_2r_{00'}+r_{11}\lambda =&0, \\ r_{Q0}r_{11}\theta+r_{QQ}r_{21} \theta+r_{Q0'}r_{31}\theta+r_{Q0} \beta-d_2r_{Q0'}+r_{21}\lambda =&0, \\ r_{0'0}r_{11}\theta+r_{0'Q}r_{21} \theta+r_{0'0'}r_{31}\theta+r_{0'0} \beta-d_2r_{0'0'}+r_{31}\lambda+\lambda =&0, \end{aligned}$$

For i=1,2,…,s,

$$\begin{aligned} r_{1(i+1)}\lambda+r_{00}r_{1(i+1)}\theta+r_{0Q}r_{2(i+1)} \theta+r_{00'}r_{3(i+1)}\theta =&r_{1i}d_3, \\ r_{2(i+1)}\lambda+r_{Q0}r_{1(i+1)}\theta+r_{QQ}r_{2(i+1)} \theta+r_{Q0'}r_{3(i+1)}\theta =&r_{2i}d_3, \\ r_{3(i+1)}\lambda+r_{0'0}r_{1(i+1)}\theta+r_{0'Q}r_{2(i+1)} \theta+r_{0'0'}r_{3(i+1)}\theta =&r_{3i}d_3, \end{aligned}$$

For i=s+1,s+2,…,Q−1,

$$\begin{aligned} r_{1(i+1)}\lambda+r_{00}r_{1(i+1)}\theta+r_{0Q}r_{2(i+1)} \theta+r_{00'}r_{3(i+1)}\theta =&r_{1i}d_4, \\ r_{2(i+1)}\lambda+r_{Q0}r_{1(i+1)}\theta+r_{QQ}r_{2(i+1)} \theta+r_{Q0'}r_{3(i+1)}\theta =&r_{2i}d_4, \\ r_{3(i+1)}\lambda+r_{0'0}r_{1(i+1)}\theta+r_{0'Q}r_{2(i+1)} \theta+r_{0'0'}r_{3(i+1)}\theta =&r_{3i}d_4, \end{aligned}$$

For i=Q,

$$\begin{aligned} r_{1(i+1)}\lambda+r_{00}r_{1(i+1)}\theta+r_{0Q}r_{2(i+1)} \theta+r_{00'}r_{3(i+1)}\theta+ r_{0Q} \beta+r_{00'}\mu =&r_{1i}d_4, \\ r_{2(i+1)}\lambda+r_{Q0}r_{1(i+1)}\theta+r_{QQ}r_{2(i+1)} \theta+r_{Q0'}r_{3(i+1)}\theta+ r_{QQ} \beta+r_{Q0'}\mu =&r_{2i}d_4, \\ r_{3(i+1)}\lambda+r_{0'0}r_{1(i+1)}\theta+r_{0'Q}r_{2(i+1)} \theta+r_{0'0'}r_{3(i+1)}\theta+ r_{0'Q} \beta+r_{0'0'}\mu =&r_{3i}d_4, \end{aligned}$$

For i=Q+1,Q+2,…,S−1,

$$\begin{aligned} r_{1(i+1)}\lambda+r_{00}r_{1(i+1)}\theta+r_{0Q}r_{2(i+1)} \theta+r_{00'}r_{3(i+1)}\theta+ r_{1(i-Q)} \mu =&r_{1i}d_4, \\ r_{2(i+1)}\lambda+r_{Q0}r_{1(i+1)}\theta+r_{QQ}r_{2(i+1)} \theta+r_{Q0'}r_{3(i+1)}\theta+ r_{2(i-Q)} \mu =&r_{2i}d_4, \\ r_{3(i+1)}\lambda+r_{0'0}r_{1(i+1)}\theta+r_{0'Q}r_{2(i+1)} \theta+r_{0'0'}r_{3(i+1)}\theta+ r_{3(i-Q)} \mu =&r_{3i}d_4, \end{aligned}$$

For i=S,

$$\begin{aligned} r_{1(i-Q)}\mu =&r_{1i}d_4, \\ r_{2(i-Q)}\mu =&r_{2i}d_4, \\ r_{3(i-Q)}\mu =&r_{3i}d_4, \end{aligned}$$

Appendix B

To compute the ϕ (0) vector, we use the following set of non-linear equations.

$$\begin{aligned} -\phi^{(0,0,0)}(\lambda+\mu+\beta)+\phi^{(0,1,0)}\alpha =&0, \\ \phi^{(0,0,0)}\mu-\phi^{(0,0,Q)}(\lambda+\beta) =&0, \\ \phi^{(0,0,0)}(\beta+r_{11}\theta)+\phi^{(0,0,Q)}r_{21} \theta+\phi^{(0,1,0)}(-d_2+r_{31}\theta)+ \phi^{(0,1,1)}\lambda =&0, \end{aligned}$$

For i=1,2,…,s,

$$\begin{aligned} \phi^{(0,0,0)}r_{1(i+1)}\theta+\phi^{(0,0,Q)}r_{2(i+1)} \theta+\phi^{(0,1,0)}r_{3(i+1)}\theta-\phi^{(0,1,i)}d_1 +\phi^{(0,1,i+1)}\lambda =&0, \end{aligned}$$

For i=s+1,s+2,…,Q−1,

$$\begin{aligned} \phi^{(0,0,0)}r_{1(i+1)}\theta+\phi^{(0,0,Q)}r_{2(i+1)} \theta+\phi^{(0,1,0)}r_{3(i+1)}\theta-\phi^{(0,1,i)} \lambda+ \phi^{(0,1,i+1)}\lambda =&0, \end{aligned}$$

For i=Q,

$$\begin{aligned} \phi^{(0,0,0)}r_{1(i+1)}\theta+\phi^{(0,0,Q)}(r_{2(i+1)} \theta+\beta)+\phi^{(0,1,0)}(r_{3(i+1)}\theta+\mu) - \phi^{(0,1,i)}\lambda \\ +\phi^{(0,1,i+1)}\lambda =&0, \end{aligned}$$

For i=Q+1,Q+2,…,S−1,

$$\begin{aligned} \phi^{(0,0,0)}r_{1(i+1)}\theta+\phi^{(0,0,Q)}r_{2(i+1)} \theta+\phi^{(0,1,0)}r_{3(i+1)}\theta-\phi^{(0,1,i)} \lambda+ \phi^{(0,1,i+1)}\lambda \\ +\phi^{(0,1,i-Q)}\mu =&0, \end{aligned}$$

For i=S,

$$\begin{aligned} -\phi^{(0,1,i)}\lambda+\phi^{(0,1,i-Q)}\mu =&0. \end{aligned}$$

Appendix C

To compute the \(\tilde{R}\) matrix, we use the following set of non-linear equations. This can be solved by using Gauss–Seidel iterative process. These equations are derived by exploiting the coefficient matrices appearing in (14),

$$\begin{aligned} \tilde{r}_{00}\tilde{r}_{11}\theta+\tilde{r}_{0Q} \tilde{r}_{21}\theta+\tilde{r}_{00'}\tilde{r}_{31} \theta+\tilde{r}_{11}\lambda-\tilde{r}_{00}(\lambda+\mu+ \beta)+\lambda =&0, \\ \tilde{r}_{Q0}\tilde{r}_{11}\theta+\tilde{r}_{QQ} \tilde{r}_{21}\theta+\tilde{r}_{Q0'}\tilde{r}_{31} \theta+\tilde{r}_{21}\lambda-\tilde{r}_{Q0}(\lambda+\mu+ \beta) =&0, \\ \tilde{r}_{0'0}\tilde{r}_{11}\theta+\tilde{r}_{0'Q} \tilde{r}_{21}\theta+\tilde{r}_{0'0'}\tilde{r}_{31} \theta+\tilde{r}_{11}\lambda-\tilde{r}_{0'0}(\lambda+\mu+ \beta) =&0, \end{aligned}$$
$$\begin{aligned} \tilde{r}_{00}\mu-\tilde{r}_{0Q}(\lambda+\beta) =&0, \\ \tilde{r}_{Q0}\mu-\tilde{r}_{QQ}(\lambda+\beta)+ \lambda =&0, \\ \tilde{r}_{0'0}\mu-\tilde{r}_{0'Q}(\lambda+\beta) =&0, \end{aligned}$$
$$\begin{aligned} \tilde{r}_{00}\beta-f_1\tilde{r}_{00'} =&0, \\ \tilde{r}_{Q0}\beta-f_1\tilde{r}_{Q0'} =&0, \\ \tilde{r}_{0'0}\beta-f_1\tilde{r}_{0'0'} \lambda =&0, \end{aligned}$$

For i=1,2,…,s,

$$\begin{aligned} \tilde{r}_{1(i+1)}\lambda+\tilde{r}_{00}\tilde{r}_{1(i+1)} \theta+\tilde{r}_{0Q}\tilde{r}_{2(i+1)}\theta+ \tilde{r}_{00'}\tilde{r}_{3(i+1)}\theta =&\tilde{r}_{1i}f_2, \\ \tilde{r}_{2(i+1)}\lambda+\tilde{r}_{Q0}\tilde{r}_{1(i+1)} \theta+\tilde{r}_{QQ}\tilde{r}_{2(i+1)}\theta+ \tilde{r}_{Q0'}\tilde{r}_{3(i+1)}\theta =&\tilde{r}_{2i}f_2, \\ \tilde{r}_{3(i+1)}\lambda+\tilde{r}_{0'0}\tilde{r}_{1(i+1)} \theta+\tilde{r}_{0'Q}\tilde{r}_{2(i+1)}\theta+ \tilde{r}_{0'0'}\tilde{r}_{3(i+1)}\theta =&\tilde{r}_{3i}f_2, \end{aligned}$$

For i=s+1,s+2,…,Q−1,

$$\begin{aligned} \tilde{r}_{1(i+1)}\lambda+\tilde{r}_{00}\tilde{r}_{1(i+1)} \theta+\tilde{r}_{0Q}\tilde{r}_{2(i+1)}\theta+ \tilde{r}_{00'}\tilde{r}_{3(i+1)}\theta =&\tilde{r}_{1i}f_3, \\ \tilde{r}_{2(i+1)}\lambda+\tilde{r}_{Q0}\tilde{r}_{1(i+1)} \theta+\tilde{r}_{QQ}\tilde{r}_{2(i+1)}\theta+ \tilde{r}_{Q0'}\tilde{r}_{3(i+1)}\theta =&\tilde{r}_{2i}f_3, \\ \tilde{r}_{3(i+1)}\lambda+\tilde{r}_{0'0}\tilde{r}_{1(i+1)} \theta+\tilde{r}_{0'Q}\tilde{r}_{2(i+1)}\theta+ \tilde{r}_{0'0'}\tilde{r}_{3(i+1)}\theta =&\tilde{r}_{3i}f_3, \end{aligned}$$

For i=Q,

$$\begin{aligned} \tilde{r}_{1(i+1)}\lambda+\tilde{r}_{00}\tilde{r}_{1(i+1)} \theta+\tilde{r}_{0Q}\tilde{r}_{2(i+1)}\theta+ \tilde{r}_{00'}\tilde{r}_{3(i+1)}\theta+ \tilde{r}_{0Q} \beta+\tilde{r}_{00'}\mu =&\tilde{r}_{1i}f_3, \\ \tilde{r}_{2(i+1)}\lambda+\tilde{r}_{Q0}\tilde{r}_{1(i+1)} \theta+\tilde{r}_{QQ}\tilde{r}_{2(i+1)}\theta+ \tilde{r}_{Q0'}\tilde{r}_{3(i+1)}\theta+ \tilde{r}_{QQ} \beta+\tilde{r}_{Q0'}\mu =&\tilde{r}_{2i}f_3, \\ \tilde{r}_{3(i+1)}\lambda+\tilde{r}_{0'0}\tilde{r}_{1(i+1)} \theta+\tilde{r}_{0'Q}\tilde{r}_{2(i+1)}\theta+ \tilde{r}_{0'0'}\tilde{r}_{3(i+1)}\theta+ \tilde{r}_{0'Q} \beta+\tilde{r}_{0'0'}\mu =&\tilde{r}_{3i}f_3, \end{aligned}$$

For i=Q+1,Q+2,…,S−1,

$$\begin{aligned} \tilde{r}_{1(i+1)}\lambda+\tilde{r}_{00}\tilde{r}_{1(i+1)} \theta+\tilde{r}_{0Q}\tilde{r}_{2(i+1)}\theta+ \tilde{r}_{00'}\tilde{r}_{3(i+1)}\theta+ \tilde{r}_{1(i-Q)} \mu =&\tilde{r}_{1i}f_3, \\ \tilde{r}_{2(i+1)}\lambda+\tilde{r}_{Q0}\tilde{r}_{1(i+1)} \theta+\tilde{r}_{QQ}\tilde{r}_{2(i+1)}\theta+ \tilde{r}_{Q0'}\tilde{r}_{3(i+1)}\theta+ \tilde{r}_{2(i-Q)} \mu =&\tilde{r}_{2i}f_3, \\ \tilde{r}_{3(i+1)}\lambda+\tilde{r}_{0'0}\tilde{r}_{1(i+1)} \theta+\tilde{r}_{0'Q}\tilde{r}_{2(i+1)}\theta+ \tilde{r}_{0'0'}\tilde{r}_{3(i+1)}\theta+ \tilde{r}_{3(i-Q)} \mu =&\tilde{r}_{3i}f_3, \end{aligned}$$

For i=S,

$$\begin{aligned} \tilde{r}_{1(i-Q)}\mu =&\tilde{r}_{1i}f_3, \\ \tilde{r}_{2(i-Q)}\mu =&\tilde{r}_{2i}f_3, \\ \tilde{r}_{3(i-Q)}\mu =&\tilde{r}_{3i}f_3, \end{aligned}$$

Appendix D

To compute the ψ (0) vector, we use the following set of non-linear equations.

$$\begin{aligned} \psi^{(0,0,0)}\bigl(\tilde{r}_{11}\theta-(\lambda+\mu+\beta) \bigr)+\psi^{(0,0,Q)}\tilde{r}_{21}\theta+\psi^{(0,1,0)} \tilde{r}_{31}\theta+\psi^{(0,1,1)}\lambda =&0, \\ \psi^{(0,0,0)}\mu-\psi^{(0,0,Q)}(\lambda+\beta) =&0, \\ \psi^{(0,0,0)}\beta-\psi^{(0,1,0)}f_1 =&0, \end{aligned}$$

For i=1,2,…,s,

$$\begin{aligned} \psi^{(0,0,0)}\tilde{r}_{1(i+1)}\theta+\psi^{(0,0,Q)} \tilde{r}_{2(i+1)}\theta+\psi^{(0,1,0)}\tilde{r}_{3(i+1)} \theta-\psi^{(0,1,i)}f_1 +\psi^{(0,1,i+1)}\lambda =&0, \end{aligned}$$

For i=s+1,s+2,…,Q−1,

$$\begin{aligned} \psi^{(0,0,0)}\tilde{r}_{1(i+1)}\theta+\psi^{(0,0,Q)} \tilde{r}_{2(i+1)}\theta+\psi^{(0,1,0)}\tilde{r}_{3(i+1)} \theta-\psi^{(0,1,i)}\lambda+\psi^{(0,1,i+1)}\lambda =&0, \end{aligned}$$

For i=Q,

$$\begin{aligned} \psi^{(0,0,0)}\tilde{r}_{1(i+1)}\theta+\psi^{(0,0,Q)}( \tilde{r}_{2(i+1)}\theta+\beta)+\psi^{(0,1,0)}(\tilde{r}_{3(i+1)} \theta+\mu)-\psi^{(0,1,i)}\lambda \\ +\psi^{(0,1,i+1)}\lambda =&0, \end{aligned}$$

For i=Q+1,Q+2,…,S−1,

$$\begin{aligned} \psi^{(0,0,0)}\tilde{r}_{1(i+1)}\theta+\psi^{(0,0,Q)} \tilde{r}_{2(i+1)}\theta+\psi^{(0,1,0)}\tilde{r}_{3(i+1)} \theta-\psi^{(0,1,i)}\lambda+\psi^{(0,1,i+1)}\lambda \\ +\psi^{(0,1,i-Q)}\mu =&0, \end{aligned}$$

For i=S,

$$-\psi^{(0,1,i)}\lambda+\psi^{(0,1,i-Q)}\mu=0. $$

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Padmavathi, I., Sivakumar, B. & Arivarignan, G. A retrial inventory system with single and modified multiple vacation for server. Ann Oper Res 233, 335–364 (2015). https://doi.org/10.1007/s10479-013-1417-1

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