Abstract
We analyze a Markov model of a two-stage production system capable of producing two part types. Each stage consists of an unreliable machine and the different stages are decoupled by two intermediate buffers of finite capacity, one for each part type. Unlike previous work, we specifically consider non-negligible machine setup times during changeovers and also assume that machine failure probabilities are dependent on the part type being produced. We assume that machine processing times, repair/failure times and setup times are exponentially distributed and may have different mean rates for each machine and for each part-type. We describe a solution method to evaluate the system performance that reduces the total number of equations to be solved from a multiplicative function to an additive function of buffer sizes. This model may then be integrated with a new decomposition method for analyzing longer lines. The results show the relative influence of different factors on system performance and thus provide guidance to the optimal choice of system parameters such as buffer sizes.
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The authors would like to thank the three anonymous reviewers for their valuable suggestions which have considerably enhanced the quality of the paper.
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Appendices
Appendix 1: Balance equations
The balance equations (BEs) for the 2M2B model are provided below. These BEs are listed according to the nine BE groups defined in Table 1 of Sect. 3.4.
BE Group I. n a = 0 and n b = 0
BE Group II. \( 1 \le n_{A} \le N_{A} - 1{\text{ and }}n_{B} = 0 \)
For \( n_{A} = 1{\text{ and }}n_{B} = 0 \) only Eq. 16 is as follows:
BE Group III. \( n_{A} = N_{A} {\text{ and }}n_{B} = 0 \)
BE Group IV. \( n_{A} = 0{\text{ and }}1 \le n_{B} \le N_{B} - 1 \)
For \( n_{A} = 0 \,{\text{and}}\, n_{B} = 1\) only Eq. 38 is as follows:
BE Group V. \(n_{A} = 0 \,{\text{and}}\, n_{B} = N_{B} \)
BE Group VI. \( 1 \le n_{A} \le N_{A} - 1{\text{ and }}1 \le n_{B} \le N_{B} - 1 \)
BE Group VII. \( n_{A} = N_{A} {\text{ and }}1 \le n_{B} \le N_{B} - 1 \)
For \( n_{A} = N_{A} {\text{ and }}n_{B} = N_{B} - 1 \) only Eq. 72 is as follows:
BE Group VIII. \( 1 \le n_{A} \le N_{A} {\text{ - 1 and }}n_{B} = N_{B} \)
For \( n_{A} = N_{A} - 1{\text{ and }}n_{B} = N_{B} \) only Eq. 85 is as follows:
BE Group IX. \( n_{A} = N_{A} {\text{ and }}n_{B} = N_{B} \)
Appendix 2: Solution methodology
The solution method can be described in the following three steps:
2.1 Step 1: Set the basic variables
The steady state probabilities of the lower boundary states (i.e., states where both buffer levels are zero) are all set as basic variables. In addition, the following steady state probabilities are also set as basic variables:
-
1.
\( p(n_{A},0,W_{A},S_{A}),\quad {\text{for}}\;(1 \le n_{A} \le N_{A} - 1) \)
-
2.
\( p(n_{A},0,\Updelta_{A},S_{A}),\quad {\text{for}}\;(1 \le n_{A} \le N_{A} - 1) \)
-
3.
\( p(0,n_{B},W_{B},S_{B}),\quad {\text{for}}\;(1 \le n_{B} \le N_{B} - 1) \)
-
4.
\( p(0,n_{B},\Updelta_{B},S_{B}),\quad {\text{for}}\;(1 \le n_{B} \le N_{B} - 1) \)
The reason for selecting the above steady state probabilities as basic variables is as follows: In the 2M2B model, there are no direct transitions from the internal states (defined in Table 1) to the lower boundary states (i.e. in balance Eqs. 5–14 of Appendix 1). Therefore, expressing the internal states in terms of only the lower boundary states proves impossible. We therefore check the balance equations (BEs) of the states where one buffer level is zero and the other is internal (i.e., states where \( n_{A} = 0, \) \( 1 \le n_{B} \le N_{B} - 1 \) and \( n_{B} = 0, \) \( 1 \le n_{A} \le N_{A} - 1 \)). We find that the BEs for the particular states listed above are the only BEs in these groups (BE groups II and IV of Appendix 1) that include internal states. Therefore we select these steady state probabilities also as basic variables.
2.2 Step 2: Solve for the steady state probabilities of all other states in terms of the basic variables
The non-basic variables are then expressed in terms of the basic variables using the BEs of Appendix 1 in the following sequence. The BEs are categorized into various groups as defined in Table 1 of Sect. 3.4.
BE Group I:
Express the following steady state probabilities in terms of the basic variables using BEs Eqs. 5–14 sequentially:
BE Group II:
FOR \( (n_{A} = 1;n_{A} \le N_{A} - 1;n_{A} + + ) \) |
IF \( (n_{A} = = 1) \) |
Express \( p(1,1,W_{A},W_{B}),p(1,0,\Updelta_{A},\Updelta_{A}),p(1,0,W_{A},\Updelta_{A}) \) in terms of the basic variables using BEs Eqs. 27, 19 and 17, sequentially |
ENDIF |
Express \( p(n_{A},0,\Updelta_{A},\Updelta_{A}) \) in terms of the basic variables using BE Eq. 19 |
IF \( (n_{A} \le N_{A} - 2) \) |
Express \( p(n_{A} + 1,0,W_{A},W_{A}),p(n_{A} + 1,0,\Updelta_{A},W_{A}) \) in terms of the basic variables using BEs Eqs. 15 and 18 sequentially |
ENDIF |
IF \( (n_{A} \ge 2) \) |
Express \( p(n_{A},0,W_{A},\Updelta_{A}),p(n_{A},1,W_{A},W_{B}) \) in terms of the basic variables using BEs Eqs. 16 and 17, sequentially |
ENDIF |
Express the following steady state probabilities in terms of the basic variables using BEs Eqs. 20–26 sequentially: \( p(n_{A},1,\Updelta_{A},W_{B}),p(n_{A} + 1,0,S_{B},W_{A}),p(n_{A},0,S_{B},\Updelta_{A}),p(n_{A} + 1,0,W_{B},W_{A}),p(n_{A},0,W_{B},\Updelta_{A}),p(n_{A} + 1,0,\Updelta_{B},W_{A}),p(n_{A},0,\Updelta_{B},\Updelta_{A}) \) |
END |
BE Group III:
Express the following steady state probabilities in terms of the basic variables using BEs Eqs. 28–36 sequentially:
BE Group IV:
FOR \( (n_{B} = 1;n_{B} \le N_{B} - 1;n_{B} + + ) \) |
IF \( (n_{B} = = 1) \) |
Express \( p(1,1,W_{B},W_{A}),p(0,1,\Updelta_{B},\Updelta_{B}),p(0,1,W_{B},\Updelta_{B}) \) in terms of the basic variables using BEs Eqs. 39, 41 and 49, sequentially |
ENDIF |
Express \( p(0,n_{B},\Updelta_{B},\Updelta_{B}) \) in terms of the basic variables using BE Eq. 41 |
IF \( (n_{B} \le N_{B} - 2) \) |
Express \( p(0,n_{B} + 1,W_{B},W_{B}),p(0,n_{B} + 1,\Updelta_{B},W_{B}) \) in terms of the basic variables using BEs Eqs. 37 and 41, sequentially |
ENDIF |
IF \( (n_{B} \ge 2) \) |
Express \( p(0,n_{B},W_{B},\Updelta_{B}) \) in terms of the basic variables using BE Eq. 39 |
Express \( p(1,n_{B},W_{B},W_{A}) \) in terms of the basic variables using BE Eq. 35 |
ENDIF |
Express the following steady state probabilities in terms of the basic variables using BEs Eqs. 42–48 sequentially: \( p(1,n_{B},\Updelta_{B},W_{A}),p(0,n_{B} + 1,S_{A},W_{B}),p(0,n_{B},S_{A},\Updelta_{B}),p(0,n_{B} + 1,W_{A},W_{B}),p(0,n_{B},W_{A},\Updelta_{B}),p(0,n_{B} + 1,\Updelta_{A},W_{B}),p(0,n_{B},\Updelta_{A},\Updelta_{B}) \) |
END |
BE Group V:
Express the following steady state probabilities in terms of the basic variables using BEs Eqs. 50–58 sequentially:
BE Group VI:
FOR \( (n_{A} = 1;n_{A} \le N_{A} - 1;n_{A} + + ) \) |
FOR \( (n_{B} = 1;n_{B} \le N_{B} - 1;n_{B} + + ) \) |
Express the following steady state probabilities in terms of the basic variables using BEs Eqs. 59–66 sequentially: |
\( p(n_{A},n_{B},W_{A},\Updelta_{B}),p(n_{A},n_{B},\Updelta_{A},\Updelta_{B}),p(n_{A},n_{B} + 1,W_{A},W_{B}),p(n_{A},n_{B} + 1,\Updelta_{A},W_{B}),p(n_{A},n_{B},W_{B},\Updelta_{A}),p(n_{A},n_{B},\Updelta_{B},\Updelta_{A}),p(n_{A} + 1,n_{B},W_{B},W_{A}),p(n_{A} + 1,n_{B},\Updelta_{B},W_{A}) \) |
END |
END |
BE Group VII:
FOR \( (n_{B} = 1;n_{B} \le N_{B} - 1;n_{B} + + ) \) |
Express the following steady state probabilities in terms of the basic variables using BEs Eqs. 67–70 sequentially: \( p(N_{A},n_{B},W_{B},S_{A}),p(N_{A},n_{B},\Updelta_{B},S_{A}),p(N_{A},n_{B},W_{B},\Updelta_{A}),p(N_{A},n_{B},\Updelta_{B},\Updelta_{A}) \) |
IF \( (n_{B} \le N_{B} - 2) \) |
Express \( p(N_{A},n_{B},S_{B},\Updelta_{B}) \) in terms of the basic variables using BE Eq. 71 |
Express \( p(N_{A},n_{B} + 1,S_{B},W_{B}) \) in terms of the basic variables using BE Eq. 72 |
ELSE |
Express\( p(N_{A},N_{B},W_{A},W_{B}) \) in terms of the basic variables using BE Eq. 79 |
ENDIF |
Express the following steady state probabilities in terms of the basic variables using BEs Eqs. 73–76 sequentially: \( p(N_{A},n_{B},W_{B},W_{B}),p(N_{A},n_{B},W_{B},\Updelta_{B}),p(N_{A},n_{B},\Updelta_{B},\Updelta_{B}),p(N_{A},n_{B} + 1,\Updelta_{B},W_{B}) \) |
END |
BE Group VIII:
FOR \( (n_{A} = 1;n_{A} \le N_{A} - 1;n_{A} + + ) \) |
Express the following steady state probabilities in terms of the basic variables using BEs Eqs. 80–83 sequentially: \( p(n_{A},N_{B},W_{A},S_{B}),p(n_{A},N_{B},\Updelta_{A},S_{B}),p(n_{A},N_{B},W_{A},\Updelta_{B}),p(n_{A},N_{B},\Updelta_{A},\Updelta_{B}) \) |
IF \( (n_{A} \le N_{A} - 2) \) |
Express \( p(n_{A},N_{B},S_{A},\Updelta_{A}) \) in terms of the basic variables using BE Eq. 84 |
Express \( p(n_{A} + 1,N_{B},S_{A},W_{A}) \) in terms of the basic variables using BE Eq. 85 |
ELSE |
Express \( p(N_{A},N_{B},W_{B},W_{A}) \)in terms of the basic variables using BE Eq. 92 |
ENDIF |
Express the following steady state probabilities in terms of the basic variables using BEs Eqs. 86–89 sequentially \( p(n_{A},N_{B},W_{A},W_{A}),p(n_{A},N_{B},W_{A},\Updelta_{A}),p(n_{A},N_{B},\Updelta_{A},\Updelta_{A}),p(n_{A},N_{B},\Updelta_{A},W_{A}) \) |
END |
BE Group IX:
Express the following steady state probabilities in terms of the basic variables using BEs Eqs. 93–98 sequentially:
2.3 Step 3: Using the normalization equation and remaining equations from Step 2 to solve for the basic variables
In Step 2, the non-basic variables were expressed in terms of the basic variables using the BEs. The following BEs were however not used in Step 2:
BEs Eqs. 15 and 18 from BE group II for \( n_{A} = N_{A} - 1 \)
BEs Eqs. 16 and 17 from BE group II for \( n_{A} = 1 \)
BEs Eqs. 37 and 40 from BE group IV for \( n_{B} = N_{B} - 1 \)
BEs Eqs. 38 and 39 from BE group IV for \( n_{B} = 1 \)
BEs Eqs. 77 and 78 from BE group VII for \( (n_{B} = 1;n_{B} \le N_{B} - 1;n_{B} + + ) \)
BEs Eqs. 90 and 91 from BE group VIII for \( (n_{A} = 1;n_{A} \le N_{A} - 1;n_{A} + + ) \)
BEs Eqs. 99–102 from BE group IX
Choosing any of these \( 2*(N_{A} + N_{B}) + 5 \) BEs and the normalization equation, the \( 2*(N_{A} + N_{B}) + 6 \) equations are solved for the basic variables set in Step 1. Once the basic variables are calculated, since the non-basic variables were expressed in terms of the basic variables, all the steady state probabilities can be determined.
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Senanayake, C.D., Subramaniam, V. Analysis of a two-stage, flexible production system with unreliable machines, finite buffers and non-negligible setups. Flex Serv Manuf J 25, 414–442 (2013). https://doi.org/10.1007/s10696-011-9115-2
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DOI: https://doi.org/10.1007/s10696-011-9115-2