## Estimating Performance Measures in Stochastic Cyclic Schedules (1991)

Citations: | 1 - 1 self |

### BibTeX

@TECHREPORT{Rao91estimatingperformance,

author = {Uday S. Rao and Peter L. Jackson},

title = {Estimating Performance Measures in Stochastic Cyclic Schedules},

institution = {},

year = {1991}

}

### OpenURL

### Abstract

This paper reviews a cyclic scheduling model of repetitive manufacturing environments consisting of flexible machines and multi-stage jobs. We explicitly model two system performance measures, throughput and flow time, while incorporating variability arising from uncertainty in operation processing times. We present an iterative approximation scheme to estimate the two moment behavior of our performance measures. The basic approach relies on Clark's method for computing moments of the maximum of bivariate normals. The proposed algorithm, called the Cyclic Network Clark's Algorithm, is illustrated via a simple flow shop example and subjected to extensive computational testing. 1 Introduction This research is motivated by the stochastic version of a special cyclic scheduling problem (Roundy, [16]) pertaining to repetitive manufacturing environments with highly automated, reliable machines and synchronized material handling. We investigate an iterative approximation approach for perfor...

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Citation Context ...own that any non-negative processing time distribution can be approximately arbitrarily closely by a phase-type distribution, which in turn, can be modeled as a random sum of exponentials (see Wolff, =-=[21]-=- or Whitt, [17] for details). Since we are primarily interested in a two moment analysis, any processing time with coefficient of variation c ! 1 may be modeled as a series of exponentially distribute... |

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Citation Context ...estigate an iterative approximation approach for performance evaluation of such periodic production systems with processing time uncertainty. The solution framework is based on Clark's method (Clark, =-=[5]-=-), summarized in Appendix A below, for estimating the first two moments of the maximum of correlated, normal random variables. The goal of this line of research is to develop methods that will identif... |

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Citation Context ...n-negative processing time distribution can be approximately arbitrarily closely by a phase-type distribution, which in turn, can be modeled as a random sum of exponentials (see Wolff, [21] or Whitt, =-=[17]-=- for details). Since we are primarily interested in a two moment analysis, any processing time with coefficient of variation c ! 1 may be modeled as a series of exponentially distributed processing du... |

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Citation Context ...e of the lognormal distribution (Wilhelm, [19]), extreme value distributions (Dodin & Sirvanci, [6]), GERT queuing approximations (Neumann, [13]), diffusion (Glynn, [8]) and fluid (Chen & Mandelbaum, =-=[4]-=-) approximations may be considered instead, particularly for dense networks with heavy tailed processing time distributions. Acknowledgements: The authors would like to thank Robin Roundy for suggesti... |

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Citation Context ...e OE(ff) = 1 p 2 e \Gamma ff 2 2 : (7) 24 Then, for all i 2 I, ff 3+i = 0; (a 3+i ) 2 = 2(c + 1)` 2 : (8) The correspondingsvalues are all zero ( [i] = 0 8i 2 I), \Sigma [i; 1] = 0 8i 2 I, and \Sigma =-=[2; 2]-=- = \Sigma [3; 3] = 2\Sigma [2; 3] = 2c` 2 : (9) If Z denotes the steady state cycle length, then E(Z ) = p + q 2(c + 1)OE(0)`; and V ar(Z ) = 2(c + 1)(\Phi(0) \Gamma OE 2 (0))` 2 : Proof. Based on the... |

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Citation Context ...astic maximumoperators including use of the lognormal distribution (Wilhelm, [19]), extreme value distributions (Dodin & Sirvanci, [6]), GERT queuing approximations (Neumann, [13]), diffusion (Glynn, =-=[8]-=-) and fluid (Chen & Mandelbaum, [4]) approximations may be considered instead, particularly for dense networks with heavy tailed processing time distributions. Acknowledgements: The authors would like... |

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Citation Context ... via a simple flow shop example and subjected to extensive computational testing. 1 1 Introduction This research is motivated by the stochastic version of a special cyclic scheduling problem (Roundy, =-=[16]-=-) pertaining to repetitive manufacturing environments with highly automated, reliable machines and synchronized material handling. We investigate an iterative approximation approach for performance ev... |

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Citation Context ...eneric cyclic scheduling model has been detailed in Rao [14]. Similar deterministic periodic scheduling environments and their properties have been studied in Graves et al [9], Whybark [18], Wittrock =-=[20]-=-, Hall [10], McCormick et al [12], Matsuo [11], and Roundy [16]. For a more complete review of deterministic cyclic scheduling, including an evaluation of its advantages and disadvantages, see Rao & J... |

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Citation Context ...s been detailed in Rao [14]. Similar deterministic periodic scheduling environments and their properties have been studied in Graves et al [9], Whybark [18], Wittrock [20], Hall [10], McCormick et al =-=[12]-=-, Matsuo [11], and Roundy [16]. For a more complete review of deterministic cyclic scheduling, including an evaluation of its advantages and disadvantages, see Rao & Jackson [15]. 2 Past research (Rao... |

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Citation Context ...interest. The ideas are then extended to the stochastic case corresponding to the cyclic production of identical jobs (Roundy, [16]) with "re-entrant" flows (machine revisits \Gamma see Grav=-=es et al, [9]-=-). In Section 2, we introduce the general approach, including a description of the PERT network structure and the processing time model. After discussing existence of the steady state random variables... |

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Citation Context ...ed in Rao [14]. Similar deterministic periodic scheduling environments and their properties have been studied in Graves et al [9], Whybark [18], Wittrock [20], Hall [10], McCormick et al [12], Matsuo =-=[11]-=-, and Roundy [16]. For a more complete review of deterministic cyclic scheduling, including an evaluation of its advantages and disadvantages, see Rao & Jackson [15]. 2 Past research (Rao [14], Roundy... |

5 |
Stochastic analysis of cyclic schedules
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Citation Context ...r model, an operation may be delayed due to random delays in completion of some predecessor operations. Their goal is to minimize a weighted sum of these expected operation delays. Bowman & Muckstadt =-=[3]-=- consider a continuous time Markov chain model. In theory, this Markov chain (assuming ergodicity) could be used to compute steady state system performance measures. Both Zhang & Graves and Bowman & M... |

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Citation Context ...e control of cyclic schedules. In particular, other approximate stochastic maximumoperators including use of the lognormal distribution (Wilhelm, [19]), extreme value distributions (Dodin & Sirvanci, =-=[6]-=-), GERT queuing approximations (Neumann, [13]), diffusion (Glynn, [8]) and fluid (Chen & Mandelbaum, [4]) approximations may be considered instead, particularly for dense networks with heavy tailed pr... |

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GERT-Networks and the Time-Oriented Evaluation of Projects
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(Show Context)
Citation Context ..., other approximate stochastic maximumoperators including use of the lognormal distribution (Wilhelm, [19]), extreme value distributions (Dodin & Sirvanci, [6]), GERT queuing approximations (Neumann, =-=[13]-=-), diffusion (Glynn, [8]) and fluid (Chen & Mandelbaum, [4]) approximations may be considered instead, particularly for dense networks with heavy tailed processing time distributions. Acknowledgements... |

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Citation Context ...mpute steady state system performance measures. Both Zhang & Graves and Bowman & Muckstadt resort to simulationbased approaches for solving more complex versions of their basic model. Dubois & Stecke =-=[7] have used-=- fmax, +g path algebras of timed marked graphs and Petri net techniques to model repetitive "decision-free production processes," primarily focusing on deterministic flexible manufacturing e... |

3 |
Cyclic Scheduling for Improvement
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(Show Context)
Citation Context ...ic scheduling model has been detailed in Rao [14]. Similar deterministic periodic scheduling environments and their properties have been studied in Graves et al [9], Whybark [18], Wittrock [20], Hall =-=[10]-=-, McCormick et al [12], Matsuo [11], and Roundy [16]. For a more complete review of deterministic cyclic scheduling, including an evaluation of its advantages and disadvantages, see Rao & Jackson [15]... |

3 |
Multi-stage, Identical Job Cyclic Scheduling for Repetitive Manufacturing
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- 1992
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Citation Context ...thm. A simple example is also presented. We conclude in Section 4 with results from our computational testing. 1.1 Brief Literature Review Our generic cyclic scheduling model has been detailed in Rao =-=[14]-=-. Similar deterministic periodic scheduling environments and their properties have been studied in Graves et al [9], Whybark [18], Wittrock [20], Hall [10], McCormick et al [12], Matsuo [11], and Roun... |

3 | Subproblems in identical jobs cyclic scheduling: properties, complexity and solution approaches
- Rao, Jackson
- 1991
(Show Context)
Citation Context ...[10], McCormick et al [12], Matsuo [11], and Roundy [16]. For a more complete review of deterministic cyclic scheduling, including an evaluation of its advantages and disadvantages, see Rao & Jackson =-=[15]. 2 P-=-ast research (Rao [14], Roundy [16]) indicates that deterministic cyclic scheduling problems can be solved by decomposition into two phases. The first phase determines a "cyclic precedence struct... |

3 |
Production Planning and Control at Kumera Oy
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- 1984
(Show Context)
Citation Context ...re Review Our generic cyclic scheduling model has been detailed in Rao [14]. Similar deterministic periodic scheduling environments and their properties have been studied in Graves et al [9], Whybark =-=[18]-=-, Wittrock [20], Hall [10], McCormick et al [12], Matsuo [11], and Roundy [16]. For a more complete review of deterministic cyclic scheduling, including an evaluation of its advantages and disadvantag... |

2 |
Ergodic theory of stochastic Petri networks
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- 1992
(Show Context)
Citation Context ...ee production processes," primarily focusing on deterministic flexible manufacturing environments which correspond to the production timing subproblem. Stochastic Petri nets are considered in Bac=-=elli [1]-=-. Our approach is quite different from the above methods in that it is an iterative approxi3 mation scheme that is fast and easy to implement. Furthermore, we do not impose the Zhang and Graves restri... |

1 |
Lognormal Models of Transient Operations in the Flexible Manufacturing Environment
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(Show Context)
Citation Context ...tion strategies, along with approaches to the real-time control of cyclic schedules. In particular, other approximate stochastic maximumoperators including use of the lognormal distribution (Wilhelm, =-=[19]-=-), extreme value distributions (Dodin & Sirvanci, [6]), GERT queuing approximations (Neumann, [13]), diffusion (Glynn, [8]) and fluid (Chen & Mandelbaum, [4]) approximations may be considered instead,... |

1 |
Cyclic Scheduling in a Stochastic Environment," Working Paper
- Zhang, Graves
(Show Context)
Citation Context ... production system over time. To our knowledge, only a few papers have directly addressed the stochastic cyclic scheduling problem. A convex stochastic programming approach was used by Zhang & Graves =-=[22]-=-. They assume that an underlying deterministic cyclic schedule is given and that no operation is permitted to start before its deterministic start time. However, in their model, an operation may be de... |