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## Inventory Control for Supply Chains with Service Level Constraints: A Synergy between Large Deviations and Perturbation Analysis (2002)

### Citations

5411 | Convex Analysis
- Rockafellar
- 1970
(Show Context)
Citation Context ... and Λ ∗− ⎧ ⎨ Λ (a) � ⎩ ∗ (a) if a < m 0 if a ≥ m. Notice that Λ ∗+ (a) is a non-decreasing and Λ ∗− (a) is a non-increasing function of a, respectively. The convex duals of these functions are (see (=-=Rockafellar, 1970-=-) on convex duality) Λ + ⎧ ⎨ Λ(θ) if θ ≥ 0 (θ) � ⎩ +∞ if θ < 0 and Λ − ⎧ ⎨ Λ(θ) if θ ≤ 0 (θ) � ⎩ +∞ if θ > 0 respectively. In particular, Λ ∗− (a) = sup θ(θa − Λ − (θ)) and Λ ∗+ (a) = sup θ(θa − Λ + (... |

1224 |
Nonlinear Programming. Athena Scientific,
- Bertsekas
- 1999
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Citation Context ...are the sample functions corresponding to the workload and stockout probability defined in (21) and (23), respectively. The algorithm we proposed is based on the so called method of multipliers (see (=-=Bertsekas, 1999-=-)). The natural question that arises is whether this algorithm converges to a stationary point (i.e., a point where optimality conditions are satisfied) w.p. 1. This can be shown in two steps. First s... |

1090 |
Large Deviations Techniques and Applications.
- Dembo, Zeitouni
- 1992
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Citation Context ... X) is a rare event. Specifically, its probability behaves as e −nr(a) , as n → ∞, where the function r(·) determines the rate at which the probability of this event is diminishing. Cramér’s theorem (=-=Dembo and Zeitouni, 1993-=-) determines r(·), and is considered the first Large Deviations statement. In particular, r(a) = sup(θa − log E[e θ θX1 ]). Consider next a sequence {S1, S2, . . .} of random variables, with values in... |

769 |
Introduction to Discrete Event Systems
- Cassandras, Lafortune
- 1999
(Show Context)
Citation Context ...chalidis and Liu, 2000). In this paper we will combine Large Deviations (LD) (employed in (Bertsimas and Paschalidis, 2001) and (Paschalidis and Liu, 2000)) and Perturbation Analysis (PA) techniques (=-=Cassandras and Lafortune, 1999-=-). We will demonstrate that there is a natural synergy between these rather distinct approaches in tackling the complex stochastic optimization problem we are considering. LD analysis is an off-line a... |

265 |
Optimal policies for a multi echelon inventory problem
- Clarck, Scarf
- 1970
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Citation Context ...for linear and the other for quadratic inventory cost assumptions). In a multi-stage single-class system without capacity limits, and for an expected linear inventory and 1backorder cost objective, (=-=Clark and Scarf, 1960-=-) in their seminal paper have shown the optimality of a production policy where each facility follows a base-stock policy based on the total inventory available locally and in the downstream facilitie... |

87 |
An inventory model with limited production capacity and uncertain demands.
- Federgruen, Zipkin
- 1986
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Citation Context ...blem is significantly simpler. It has been shown in a variety of settings (all without set-up costs) that a so called base-stock policy minimizes expected linear inventory and backorder costs (e.g., (=-=Federgruen and Zipkin, 1986-=-)). Such a policy produces when inventory falls below a prescribed level and idles otherwise; this threshold level is usually referred to as hedging point or safety stock. In multiclass single-stage s... |

70 |
Nested Partitions Method for Global Optimization”.
- Shi, Olafsson
- 2000
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Citation Context ...ill, though, left with the challenge of a constrained stochastic integer optimization problem. Such problems are notoriously hard, despite some recent advances (e.g., (Gokbayrak and Cassandras, 2001; =-=Shi and Olafsson, 2000-=-)). Moreover, the fact that there are now multiple stages increases the computational burden of updating FPA-based estimates with every event in the system. Thus, we choose to convert the queueing mod... |

65 |
Stochastic Approximation for Constrained and Unconstrained Systems.
- Kushner, Clark
- 1978
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Citation Context ...ent Infinitesimal Perturbation Analysis (IPA) methods to estimate a cost gradient, and employ a gradient-based optimization algorithm based on fairly standard stochastic approximation schemes (e.g., (=-=Kushner and Clark, 1978-=-)). Such SFMs have been recently used with success in communication network applications (Liu and Gong, 1999; Cassandras et al., 2002). We emphasize that an SFM is used to determine only the form of c... |

55 | Exponential approximations for tail probabilities in queues I: Waiting times”,
- Abate, Choudhury, et al.
- 1995
(Show Context)
Citation Context ... using the expression (7) P[Xt(w) ≤ 0] ≈ αe −wθ∗ . 8Thus, based on this approximation, (6) is modified according to (8) w ∗ = − log(η/α) θ ∗ . An estimate of α can be obtained by using an idea from (=-=Abate et al., 1995-=-) and assuming that (7) is based on the exact distribution of the inventory process. Using (4), the right hand side of (7) also approximates the tail of Lt. Hence, integrating the right hand side of (... |

55 |
Sensitivity analysis for base stock levels in multi-echelon production-inventory systems.
- Glasserman, Tayur
- 1995
(Show Context)
Citation Context ...lities (echelon inventory). In the more general case where capacity limits exist and demand and service processes are autocorrelated, such a policy is not necessarily optimal. Under a similar policy (=-=Glasserman and Tayur, 1995-=-) proposed a perturbation analysis approach to compute the hedging points in a capacitated single-class multi-stage system, and (Glasserman, 1997) has developed large deviations asymptotics to approxi... |

28 |
Perturbation analysis for on-line control and optimization of stochastic fluid models.
- Cassandras, Wardi, et al.
- 2002
(Show Context)
Citation Context ... based on fairly standard stochastic approximation schemes (e.g., (Kushner and Clark, 1978)). Such SFMs have been recently used with success in communication network applications (Liu and Gong, 1999; =-=Cassandras et al., 2002-=-). We emphasize that an SFM is used to determine only the form of cost gradient estimators, while their actual values are still based on the original queueing model. Note that while a SFM may be too “... |

27 |
Bounds and asymptotics for planning critical safety stocks,
- Glasserman
- 1997
(Show Context)
Citation Context ...necessarily optimal. Under a similar policy (Glasserman and Tayur, 1995) proposed a perturbation analysis approach to compute the hedging points in a capacitated single-class multi-stage system, and (=-=Glasserman, 1997-=-) has developed large deviations asymptotics to approximate stockout probabilities under renewal demand and constant production capacities. For a similar capacitated single-class multi-stage supply ch... |

13 |
On a Class of Stochastic Recursive Sequences Arising in Queueing Theory
- Baccelli, Liu
- 1992
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Citation Context ...[D 1 t ] < min i=1,...,n E[Bi t], which, by stationarity, carries over to all time slots t. Stability can be shown under the base-stock policy we will consider in this paper by using techniques from (=-=Baccelli and Liu, 1992-=-). 4We propose a base-stock policy that maintains a safety stock equal to wi for the (local) inventory of every stage i, i = 1, . . . , n. The inventory process becomes a function of w = (w1, . . . ,... |

13 | Probabilistic service level guarantees in make-to-stock manufacturing systems
- Bertsimas, Paschalidis
- 2001
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Citation Context ...rred to as hedging point or safety stock. In multiclass single-stage systems the optimal policy is not known in general. Only special cases or asymptotic results have appeared in the literature (see (=-=Bertsimas and Paschalidis, 2001-=-) for a detailed literature review). Combining fluid and large deviations techniques, the single-stage multiclass problem was analyzed in (Bertsimas and Paschalidis, 2001) and two families of producti... |

13 | On the large deviations behaviour of acyclic networks of G/G/1 queues
- Bertsimas, Paschalidis, et al.
- 1998
(Show Context)
Citation Context .... . . , n, are arbitrary, stationary, and mutually independent stochastic processes that satisfy certain mild technical conditions (a large deviations principle, see (Bertsimas and Paschalidis, 2001; =-=Bertsimas et al., 1998-=-) for details). These assumptions are satisfied by a fairly large class of stochastic processes, which includes renewal processes, Markov-modulated processes (where D 1 t for example is a function of ... |

12 | Optimal policies and simulation based optimization for capacitated production inventory systems. - Kapuscinski, Tayur - 1999 |

12 | Large deviations-based asymptotics for inventory control in supply chains
- Paschalidis, Liu
- 2003
(Show Context)
Citation Context ...kout probabilities under renewal demand and constant production capacities. For a similar capacitated single-class multi-stage supply chain, and under autocorrelated demand and production processes, (=-=Paschalidis and Liu, 2000-=-) explores the full power of large deviations asymptotics to compute hedging points under two base-stock policies: one that operates based on local inventory information and an echelon base-stock poli... |

7 |
Cassandras, Optimization of kanban-based manufacturing systems
- Panayiotou, G
- 1999
(Show Context)
Citation Context ...me driven by cost sensitivity estimates obtained through PA. In (Paschalidis et al., 2001) we relied on a Finite Perturbation Analysis (FPA) approach, which may be extended to multiple stages, as in (=-=Panayiotou and Cassandras, 1999-=-). We are still, though, left with the challenge of a constrained stochastic integer optimization problem. Such problems are notoriously hard, despite some recent advances (e.g., (Gokbayrak and Cassan... |

5 | Online Surrogate Problem Methodology for Stochastic Discrete Resource Allocation Problems,”
- Gokbayrak, Cassandras
- 2001
(Show Context)
Citation Context ...and Cassandras, 1999). We are still, though, left with the challenge of a constrained stochastic integer optimization problem. Such problems are notoriously hard, despite some recent advances (e.g., (=-=Gokbayrak and Cassandras, 2001-=-; Shi and Olafsson, 2000)). Moreover, the fact that there are now multiple stages increases the computational burden of updating FPA-based estimates with every event in the system. Thus, we choose to ... |

5 | Perturbation analysis for stochastic fluid queueing systems
- Liu, Gong
- 1999
(Show Context)
Citation Context ...timization algorithm based on fairly standard stochastic approximation schemes (e.g., (Kushner and Clark, 1978)). Such SFMs have been recently used with success in communication network applications (=-=Liu and Gong, 1999-=-; Cassandras et al., 2002). We emphasize that an SFM is used to determine only the form of cost gradient estimators, while their actual values are still based on the original queueing model. Note that... |

4 |
On-line inventory cost minimization for make-to-stock manufacturing systems
- Panayiotou, Cassandras, et al.
- 2002
(Show Context)
Citation Context ...ous inflow at that time. Moreover, the estimators are independent of any assumptions on demand or production processes. Finally, it can be formally shown that IPA estimates in (32) are unbiased; see (=-=Panayiotou et al., 2002-=-) for details. Two-stage system Following the main ideas used in the single-stage system, we can derive the required derivatives with respect to w1 and w2 in a two-stage system, though the derivation ... |

2 | Threshold-based Control for Maketo-Stock Models: A Synergy Between Large Deviations and Perturbation Analysis - Paschalidis, Liu, et al. - 2001 |