Abstract
This paper examines a discrete material two echelon supply chain system. Multiple reliable non identical merging suppliers send products to an intermediate storage area which can in turn feed a distribution centre with parallel identical reliable distribution channels. It is assumed that each merging supplier may have parallel identical reliable supply channels. The service rate of each supplier and each identical channel at the distribution centre is assumed to be exponentially distributed. The examined model is analyzed as a continuous time Markov process with discrete states. An algorithm that can create the system’s transition probabilities matrix for any value of its parameters is presented and various performance measures are calculated. The comparison of the proposed method with simulation showed that the proposed algorithm provides very accurate estimations of the system’s performance measures. Additionally the optimal values of the system’s parameters to optimize its various performance measures are also explored thoroughly.
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Appendices
Appendix A
Furthermore:
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The matrix of the transition probabilities of the sub matrix \({}_{{( - S_{1} *\mu_{1} )}}D_{1}^{3(2,3,4),1,2}\) is the same with the sub matrix \({}_{{( - 1*\mu_{1} )}}D_{1}^{3(2,3,4),1,2}\), with the only difference that \(- \mu_{1}\) has been added to all diagonal cell values except from the last one.
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The matrix of the transition probabilities of the sub matrix \(S_{1} *U_{0}^{3(2,3,4),1,2}\) is a diagonal matrix with all diagonal cell values equal to \(2*\mu_{1}\) (since \(S_{1} = 2\)).
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The matrix of the transition probabilities of the sub matrix \((S_{1} - 1)*U_{0}^{3(2,3,4),1,2}\) is a diagonal matrix with all diagonal cell values equal to \(1*\mu_{1}\) (since \(S_{1} - 1 = 1\)).
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The matrix of the transition probabilities of the sub matrix \(N*L_{1}^{3(2,3,4),1,2}\) is a diagonal matrix with all diagonal cell values equal to \(2*\mu_{1}\) (since \(N = 2\)) (see Tables 19, 20).
Table 19 Matrix of the transition probabilities of sub matrix \(D_{0}^{3(2,3,4),1,2}\) Table 20 Matrix of the transition probabilities of sub matrix \({}_{{( - 1*\mu_{1} )}}D_{1}^{3(2,3,4),1,2}\)
Appendix B
Table 21 presents the form of sub matrix (2(3,4),1,2).
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Vidalis, M., Koukoumialos, S., Diamantidis, A. et al. Analysis of a two echelon supply chain with merging suppliers, a storage area and a distribution center with parallel channels. Oper Res Int J 22, 703–740 (2022). https://doi.org/10.1007/s12351-019-00540-x
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DOI: https://doi.org/10.1007/s12351-019-00540-x