article

Conditions for the discovery of solution horizons

Authors:

Robert L. SmithAuthors Info & Claims

Mathematical Programming: Series A and B, Volume 59, Issue 1-3

Pages 215 - 229

Published: 01 March 1993 Publication History

Abstract

We present necessary and sufficient conditions for discrete infinite horizon optimization problems with unique solutions to be solvable. These problems can be equivalently viewed as the task of finding a shortest path in an infinite directed network. We provide general forward algorithms with stopping rules for their solution. The key condition required is that of weak reachability, which roughly requires that for any sequence of nodes or states, it must be possible from optimal states to reach states close in cost to states along this sequence. Moreover the costs to reach these states must converge to zero. Applications are considered in optimal search, undiscounted Markov decision processes, and deterministic infinite horizon optimization.

References

[1]

J. Bean, J. Birge and R. Smith, "Aggregation in dynamic programming," Operations Research 35 (1987) 215-220.

Digital Library

[2]

J. Bean, J. Lohmann and R. Smith, "A dynamic infinite horizon replacement economy decision model," The Engineering Economist 30 (1985) 99-120.

[3]

J. Bean and R. Smith, "Conditions for the existence of planning horizons," Mathematics of Operations Research 9 (1984) 391-401.

Digital Library

[4]

J. Bean and R. Smith, "Optimal capacity expansion over an infinite horizon," Management Science 31 (1985) 1523-1532.

Digital Library

[5]

J. Bean, R. Smith and C. Yano, "Forecast horizons for the discounted dynamic lot size model allowing speculative motive," Naval Research Logistics 34 (1987) 761-774.

[6]

C. Bes and S. Sethi, "Concepts of forecast and decision horizons: applications to dynamic stochastic optimization problems," Mathematics of Operations Research 13 (1988) 295-310.

Digital Library

[7]

S. Bhaskaran and S. Sethi, "Conditions for the existence of decision horizons for discounted problems in a stochastic environment: A note," Operations Research Letters 4 (1985) 61-65.

Digital Library

[8]

S. Chand and T. Morton, "Minimal forecast horizon procedures for dynamic lost size models," Naval Research Logistics Quarterly 33 (1986) 111-122.

[9]

W. Hopp, "Identifying forecast horizons in nonhomogeneous Markov decision processes," Operations Research 37 (1989) 339-343.

Digital Library

[10]

W. Hopp, J. Bean and R. Smith, "A new optimality criterion for nonhomogeneous Markov decision processes," Operations Research 35 (1987) 875-883.

[11]

J. Lasserre, C. Bes and F. Roubellat, "Detection of planning horizons for the general inventory problem with discounted concave costs," IEEE Transactions on Automatic Control 29 (1984) 562-564.

[12]

C. Lee and E. Denardo, "Rolling planning horizons: Error bounds for the dynamic lot size model," Mathematics of Operations Research 11 (1986) 423-432.

Digital Library

[13]

L. McKenzie, "Turnpike theory," Econometrica 44 (1976) 841-864.

[14]

S. Ross, "Arbitrary state Markov decision processes," The Annals of Mathematical Statistics 39 (1968) 2118-2122.

[15]

S. Ryan, J. Bean and R. Smith, "A tie-breaking rule for discrete infinite horizon optimization," Operations Research 40 (1992) $117-S126.

Digital Library

[16]

E. Seneta, Non-negative Matrices and Markov Chains (Springer, New York, 1981).

[17]

J. F. Shapiro and H.M. Wagner, "A finite renewal algorithm for the knapsack and turnpike models," Operations Research 15 (1967) 318-341.

Digital Library

[18]

D.K. Skilton, "Imbedding posets in the integers," Order 1 (1985) 229-233.

[19]

H.M. Wagner and T.M. Whitin, "Dynamic version of the economic lost size model," Management Science 5 (1958) 89-96.

Digital Library

[20]

J. Wilson, "Approximating an infinite stage search problem with a finite horizon model," Mathematics of Operations Research 14 (1989) 433-447.

Digital Library

Cited By

Kiatsupaibul SSmith RZabinsky Z(2016)Solving infinite horizon optimization problems through analysis of a one-dimensional global optimization problemJournal of Global Optimization10.1007/s10898-016-0423-766:4(711-727)Online publication date: 1-Dec-2016
https://dl.acm.org/doi/10.1007/s10898-016-0423-7
Lortz TDolinskaya IGhate ASmith R(2015)Solvability in infinite horizon optimizationOperations Research Letters10.1016/j.orl.2015.07.00343:5(498-503)Online publication date: 1-Sep-2015
https://dl.acm.org/doi/10.1016/j.orl.2015.07.003
Ghate ASmith R(2009)Optimal Backlogging Over an Infinite Horizon Under Time-Varying Convex Production and Inventory CostsManufacturing & Service Operations Management10.1287/msom.1080.021811:2(362-368)Online publication date: 1-Apr-2009
https://dl.acm.org/doi/10.1287/msom.1080.0218
Show More Cited By

Index Terms

Conditions for the discovery of solution horizons
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
2. Theory of computation
  1. Design and analysis of algorithms
    1. Algorithm design techniques
      1. Dynamic programming

Recommendations

Technical Note: Identifying Forecast Horizons in Nonhomogeneous Markov Decision Processes

<P>A procedure for identifying forecast horizons in nonhomogeneous Markov decision processes, based on convergence results for relative value functions, is developed. Two different algorithmic implementations of this procedure are discussed, and a ...
Mean-Variance Tradeoffs in an Undiscounted MDP

A stationary policy and an initial state in an MDP Markov decision process induce a stationary probability distribution of the reward. The problem analyzed here is generating the Pareto optima in the sense of high mean and low variance of the stationary ...
A Stopping Rule for Forecasting Horizons in Nonhomogeneous Markov Decision Processes

We formulate a mixed integer program to determine whether a finite time horizon is a forecast horizon in a nonhomogeneous Markov decision process. We give a Bender's decomposition approach to solving this problem that evaluates the stopping rule, ...

Comments

Information & Contributors

Information

Published In

cover image Mathematical Programming: Series A and B

Mathematical Programming: Series A and B Volume 59, Issue 1-3

March 1993

408 pages

ISSN:0025-5610

Issue’s Table of Contents

Copyright © Copyright © 1993 The Mathematical Programming Society, Inc.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 March 1993

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

7
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 22 Sep 2024

Other Metrics

View Author Metrics

Citations

Cited By

Kiatsupaibul SSmith RZabinsky Z(2016)Solving infinite horizon optimization problems through analysis of a one-dimensional global optimization problemJournal of Global Optimization10.1007/s10898-016-0423-766:4(711-727)Online publication date: 1-Dec-2016
https://dl.acm.org/doi/10.1007/s10898-016-0423-7
Lortz TDolinskaya IGhate ASmith R(2015)Solvability in infinite horizon optimizationOperations Research Letters10.1016/j.orl.2015.07.00343:5(498-503)Online publication date: 1-Sep-2015
https://dl.acm.org/doi/10.1016/j.orl.2015.07.003
Ghate ASmith R(2009)Optimal Backlogging Over an Infinite Horizon Under Time-Varying Convex Production and Inventory CostsManufacturing & Service Operations Management10.1287/msom.1080.021811:2(362-368)Online publication date: 1-Apr-2009
https://dl.acm.org/doi/10.1287/msom.1080.0218
Cheevaprawatdomrong TSchochetman ISmith RGarcia A(2007)Solution and Forecast Horizons for Infinite-Horizon Nonhomogeneous Markov Decision ProcessesMathematics of Operations Research10.1287/moor.1060.022432:1(51-72)Online publication date: 1-Feb-2007
https://dl.acm.org/doi/10.1287/moor.1060.0224
Federgruen ATzur M(1995)Fast Solution and Detection of Minimal Forecast Horizons in Dynamic Programs with a Single Indicator of the FutureManagement Science10.5555/2909477.290947841:5(874-893)Online publication date: 1-May-1995
https://dl.acm.org/doi/10.5555/2909477.2909478
Federgruen ATzur M(1995)Fast Solution and Detection of Minimal Forecast Horizons in Dynamic Programs with a Single Indicator of the FutureManagement Science10.5555/2896314.289631541:5(874-893)Online publication date: 1-May-1995
https://dl.acm.org/doi/10.5555/2896314.2896315
Federgruen ATzur M(1995)Fast Solution and Detection of Minimal Forecast Horizons in Dynamic Programs with a Single Indicator of the FutureManagement Science10.5555/2784079.278408841:5(874-893)Online publication date: 1-May-1995
https://dl.acm.org/doi/10.5555/2784079.2784088

View Options

View options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents