A finite-source inventory system with service facility and postponed demands

Abstract

In this article, a continuous review finite-source inventory system with single-server service facility is studied. The arrival of customers for unit item follows quasi-random process. The service time to process the item follows phase-type distribution. (s, S) policy is adopted for replenishing an order. The lead time follows phase-type distribution. An arriving customer who finds waiting hall full, (s)he either enters into the pool or leaves the system immediately according to a Bernoulli trial. A pooled customer is selected according to a prefixed selection policy. The joint probability distribution of the inventory level, number of customers in the pool and number of customers in the waiting hall is obtained in the steady-state case. Various stationary system performance measures are derived and total expected cost rate is calculated. Some numerical examples including optimality of the total expected cost rate are also presented.

Appendices

Appendix 1

In this appendix, we provide the general structure of the sub matrices of the infinitesimal generator P and their dimensions.

$$\begin{aligned} {[B_0]}_{kl}= & {} \left\{ \begin{array}{lll} {\tilde{B}}_{00}, &{} l = k, &{} k = 0,\\ {B_{00}}, &{} l = k, &{} k = 1,2,\ldots , M,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[\tilde{B_{00}}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0\otimes I_{m_2}, &{} n = m-1, &{} m = 1,2,\ldots , K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[B_{00}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0\otimes I_{m_2}, &{} n = m-1, &{} m = 1,2,\ldots ,K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \end{aligned}$$

Except the dimension of matrix, $\tilde{B_{00}}$ is same as $B_{00}$.

$$\begin{aligned} {[B_1]}_{kl}= & {} \left\{ \begin{array}{lll} {\tilde{B}}_{10}, &{} l = k-1, &{} k = 1,\\ {B_{10}}, &{} l = k-1, &{} k = 2,3,\ldots , M,\\ B_{11}, &{} l = k, &{} k = 0,\\ B_{12}, &{} l = k, &{} k = 1,2, \ldots M,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[{\tilde{B}}_{10}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0 \alpha _1\otimes I_{m_2}, &{} n = m, &{} m = 1,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[B_{10}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0 \alpha _1\otimes I_{m_2}, &{} n = m, &{} m = 1,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \end{aligned}$$

Except the dimension of matrix, ${\tilde{B}}_{10}$ is same as $B_{10}.$

$$\begin{aligned} {[B_{11}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0 \otimes I_{m_2}, &{} n = m-1, &{} m = 1,\\ T_1^0 \alpha _1\otimes I_{m_2}, &{} n = m-1, &{} m = 2,3,\ldots ,K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[B_{12}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0 \alpha _1\otimes I_{m_2}, &{} n = m-1, &{} m = 2,3,\ldots ,K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[B_2]}_{kl}= & {} \left\{ \begin{array}{lll} {\tilde{B}}_{20}, &{} l = k-1, &{} k = 1,\\ {B_{20}}, &{} l = k-1, &{} k = 2,3,\ldots , M,\\ B_{21}, &{} l = k, &{} k = 0,\\ B_{22}, &{} l = k, &{} k = 1,2, \ldots M,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[{\tilde{B}}_{20}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0 \alpha _1\otimes \alpha _2, &{} n = m, &{} m = 1\\ {\textbf {0}}, &{} \text{ otherwise }. \end{array} \right. \\ {[B_{20}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0 \alpha _1\otimes \alpha _2, &{} n = m, &{} m = 1\\ {\textbf {0}}, &{} \text{ otherwise }. \end{array} \right. \end{aligned}$$

Except the dimension of matrix, ${\tilde{B}}_{20}$ is same as $B_{20}.$

where

$$\begin{aligned} {[M_0]}_{kl}= & {} \left\{ \begin{array}{lll} {\tilde{B}}_{30}, &{} l = k-1, &{} k = 1,\\ {B_{30}}, &{} l = k-1, &{} k = 2,3,\ldots , L,\\ B_{32}, &{} l = k, &{} k = 0,\\ B_{33}, &{} l = k, &{} k = 1,2, \ldots L,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[M_1]}_{kl}= & {} \left\{ \begin{array}{lll} {B_{31}}, &{} l = k-1, &{} k = L+1,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[M_2]}_{kl}= & {} \left\{ \begin{array}{lll} {B_{34}}, &{} l = k, &{} k = L+1,L+2,\ldots , M,\\ B_{31}, &{} l = k-1, &{} k = L+2,L+3, \ldots M,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[{\tilde{B}}_{30}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0 \alpha _1, &{} n = m, &{} m = 1,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[B_{30}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0 \alpha _1, &{} n = m, &{} m = 1,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \end{aligned}$$

Except the dimension of matrix, $B_{30}$ is same as $\tilde{B_{30}}.$

$$\begin{aligned} {[B_{31}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0 \alpha _1, &{} n = m, &{} m = 1,\\ pT_1^0\alpha _1, &{} n = m, &{} m =2,3,\ldots , K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[B_{32}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0, &{} n = m-1, &{} m = 1,\\ T_1^0\alpha _1, &{} n = m-1, &{} m =2,3,\ldots , K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[B_{33}]}_{mn}= & {} \left\{ \begin{array}{lll} T_1^0\alpha _1, &{} n = m-1, &{} m =2,3,\ldots , K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[B_{34}]}_{mn}= & {} \left\{ \begin{array}{lll} (1-p)T_1^0\alpha _1, &{} n = m-1, &{} m =2,3,\ldots , K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[{\tilde{C}}]}_{kl}= & {} \left\{ \begin{array}{lll} {\tilde{C}}_{01}, &{} l = k-1, &{} k = 1,\\ {{\tilde{C}}_{02}}, &{} l = k-1, &{} k = 2,3,\ldots , M,\\ {\tilde{C}}_{00}, &{} l = k, &{} k = 0,\\ {\tilde{C}}_1, &{} l = k, &{} k = 1,2, \ldots M,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[{\tilde{C}}_{01}]}_{mn}= & {} \left\{ \begin{array}{lll} T_2^0\alpha _1, &{} n = m+1, &{} m = 0,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[{\tilde{C}}_{02}]}_{mn}= & {} \left\{ \begin{array}{lll} T_2^0\alpha _1, &{} n = m+1, &{} m = 0\\ {\textbf {0}}, &{} \text{ otherwise }. \end{array} \right. \end{aligned}$$

Except the dimension of matrix, ${\tilde{C}}_{01}$ is same as ${\tilde{C}}_{02}.$

where

$$\begin{aligned} {[A_{10}]}_{mn}= & {} \left\{ \begin{array}{lll} qM\gamma I_{m_1}\otimes I_{m_2}, &{} n = m, &{} m = K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ \end{aligned}$$

$\text{ for } \ k=1,2,\ldots ,M-1$

$$\begin{aligned} {[A_{1k}]}_{mn}= & {} \left\{ \begin{array}{lll} q(M-k)\gamma I_{m_1}\otimes I_{m_2}, &{} n = m, &{} m = K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[{\tilde{A}}_{10}]}_{mn}= & {} \left\{ \begin{array}{lll} N\gamma \alpha _1\otimes I_{m_2}, &{} n = m+1, &{} m= 0,\\ (N-m)\gamma I_{m_1}\otimes I_{m_2}, &{} n = m+1, &{} m = 1,2,\ldots ,K-1,\\ T_2-N\gamma I_{m_2}, &{} n = m, &{} m = 0,\\ T_1\oplus T_2-(N-m)\gamma I_{m_1}\otimes I_{m_2}, &{} n = m, &{} m = 1,2\ldots ,K-1,\\ T_1 \oplus T_2-qM\gamma I_{m_1}\otimes I_{m_2}, &{} m = n, &{} m = K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ \end{aligned}$$

$\text{ for } \ k=1,2,\ldots ,M-1$

Table 10 Dimension of sub-matrices of P

where

$$\begin{aligned} {[A_{20}]}_{mn}= & {} \left\{ \begin{array}{lll} qM\gamma I_{m_1}, &{} n = m, &{} m = K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ \end{aligned}$$

$\text{ for } \ k=1,2,\ldots ,M-1,$

$$\begin{aligned} {[A_{2k}]}_{mn}= & {} \left\{ \begin{array}{lll} q(M-k)\gamma I_{m_1}, &{} n = m, &{} m = K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[{\tilde{A}}_{20}]}_{mn}= & {} \left\{ \begin{array}{lll} N\gamma \alpha _1, &{} n = m+1, &{} m = 0,\\ (N-m)\gamma I_{m_1}, &{} n = m+1, &{} m = 1,2,\ldots ,K-1,\\ -N\gamma , &{} n = m, &{} m = 0,\\ T_1-(N-m)\gamma I_{m_1}, &{} n = m, &{} m = 1,2\ldots ,K-1,\\ T_1-qM\gamma I_{m_1}, &{} m = n, &{} m = K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ \end{aligned}$$

$\text{ for } \ k=1,2,\ldots ,M-1,$

$$\begin{aligned} {[{\tilde{A}}_{2k}]}_{mn}= & {} \left\{ \begin{array}{lll} (N-m-k)\gamma I_{m_1}, &{} n = m+1, &{} m = 1,2,\ldots ,K-1,\\ T_1-(N-m-k)\gamma I_{m_1}, &{} n = m, &{} m = 1,2\ldots ,K-1,\\ T_1 -q(M-k)\gamma I_{m_1}, &{} m = n, &{} m = K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \\ {[{\tilde{A}}_{2M}]}_{mn}= & {} \left\{ \begin{array}{lll} (N-m-M)\gamma I_{m_1}, &{} n = m+1, &{} m = 1,2,\ldots ,K-1,\\ T_1-(N-m-M)\gamma I_{m_1}, &{} n = m, &{} m = 1,2\ldots ,K-1,\\ T_1, &{} m = n, &{} m = K,\\ {\textbf {0}}, &{} \text{ otherwise. } \end{array} \right. \end{aligned}$$

The dimensions of the above matrices are given in the Table 10.

Appendix 2

In this appendix, we have provided a detailed study of the proposed model with a simple example given below. For this, we fix $N = 7, K = 3, L=2, M=4, S=5, s=2.$ We obtain the matrix P with following sub-matrices:

Table 11 $\Pi$ Values

From the structure of P that the homogeneous continuous time Markov chain $Z(t)=\left\{ (L(t),X(t),Y(t), t\ge 0\right\}$ on the finite state space E is irreducible. The steady state probability distribution of this process, denoted by $\pi _{<0>}, \pi _{<1>},\pi _{<2>},\pi _{<3>},\pi _{<4>}$ and $\pi _{<5>}$. We can compute $\Pi$ values by solving the equation $\Pi P = 0$ and $\Pi e= 1$. Table 11 gives the values of the vector $\Pi$ which are computed using Julia software.

In order to compute system parameter measures, we fix parameter values as $p=\frac{3}{4}; q=\frac{3}{4}; \gamma =\frac{4}{5}; \mu =\frac{16}{5}; \beta =\frac{1}{2};$ and hence the system parameter measures are calculated as $\eta _{I} = 0.9741195, \eta _{R} =0.4336778, \eta _{TR} = 0.2265795, \eta _{P} = 3.05270895, \eta _{W} = 2.1897712, \eta _{L} = 0.0812148.$ After computing the system parameter measures, we have obtained the total expected cost $TC= 4.979400146$ by fixing the cost parameters as $c_{h} = 0.0015; c_{s}=1.8; c_{tr}=0.5; c_{wp} = 0.4; c_{wb} = 1.3; c_{l} = 0.2.$