Optimal Stockpile Problems with Stochastic Consumption Saturation and Solution Interval

Lloyd, Justin M.; Meyer, Gérard G. L.

doi:10.1109/acc.2014.6858652

Cited by 2 publications

(7 citation statements)

References 12 publications

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“…

H = e^{- r t} U (q) - λ q + ν (\overset{q}{true} - q) .

In contrast to the deterministic case, where the application of the PMP necessary conditions requires the direct maximization of the Hamiltonian, the stochastic problem version requires the maximization of the expected value of the Hamiltonian, as in equation . Additional details on the following derivation are available in Lloyd and Meyer [], where an analogous procedure is followed but with the assumption of a truncated normal distribution for the consumption saturation limit:

\underset{q \in U}{trueprefixmax} E [H] .

…”

Section: Minimum Consumption Models With Saturation Uncertaintymentioning

confidence: 99%

“…For clarity, the constrained minimum consumption model with an uncertain embargo length is restated in equations using the form introduced by Lloyd and Meyer []. Furthermore, to illuminate the derivations of the solutions for the shifted welfare and sigmoidal welfare cases, the solution procedure is repeated as in Lloyd and Meyer [].

\begin{matrix} true \\ right boldJ (q) & = & left \int_{0}^{T_{f}} - e^{- r t} ((), U, (q (t))) d t, \\ right \dot{x} & = & left - q, \\ left - q \leq - \overset{q,}{true} \\ right prefixPr ((), T_{f}) & = & left scriptTE (() T_{f} | μ, σ, a, b), \\ right x ((), 0) & = & left X_{0}, \\ right x ((), T_{f}) & = & left X_{f} . \end{matrix}

…”

Section: Minimum Consumption Models With Embargo Length Uncertaintymentioning

confidence: 99%

“…The details of the expectation calculation are treated in detail in Lloyd and Meyer [], but the ultimate result is shown in .

\begin{matrix} true \\ right E ([], J) = & = \int_{0}^{T_{f}} \frac{e^{- (r + θ) t} - e^{- r t}}{- 1 + e^{- θ T_{f}}} U (q (t)) d t . \end{matrix}

…”

Section: Minimum Consumption Models With Embargo Length Uncertaintymentioning

confidence: 99%

“…With the development of three alternative models for consumption saturation in optimal stockpiling problems, one research extension is to consider the impact of each form on the solutions to stochastic versions of the standard optimal consumption problem. In the previous section, deterministic models were examined, but building on prior work reported in Lloyd and Meyer [], the following sections contrast each model when the critical consumption level, either modeled as a hard or soft limit, is a random variable.…”

Section: Minimum Consumption Models With Saturation Uncertaintymentioning

confidence: 99%

“…The second part of this study combines the development of the alternative forms of a consumption saturation model to previously reported results in modeling embargo length uncertainty and minimum consumption uncertainty (Bahel [], Lloyd and Meyer []). As evidenced by existing research and known economic phenomena, uncertainty plays a significant role in the accurate treatment of stockpiling problems.…”

Section: Introductionmentioning

confidence: 99%

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Methods of Modeling Consumption Saturation With Uncertainty in Optimal Stockpile Problems

Lloyd

Meyer

2015

Natural Resource Modeling

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Abstract. Economic problems in the optimal management of strategic resource stockpiles can be rigorously studied and solved by formulating them as optimal control problems in continuous time. In these optimal control problems, the proper description of the control constraints and objective function are critical to reflecting a realistic economic model. Existing work in this area often ignores fundamental saturation effects in the economic systems under scrutiny, and the following paper introduces and compares several methods for correcting this common modeling simplification in both deterministic and stochastic contexts.

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“…

H = e^{- r t} U (q) - λ q + ν (\overset{q}{true} - q) .

\underset{q \in U}{trueprefixmax} E [H] .

…”

Section: Minimum Consumption Models With Saturation Uncertaintymentioning

confidence: 99%

\begin{matrix} true \\ right boldJ (q) & = & left \int_{0}^{T_{f}} - e^{- r t} ((), U, (q (t))) d t, \\ right \dot{x} & = & left - q, \\ left - q \leq - \overset{q,}{true} \\ right prefixPr ((), T_{f}) & = & left scriptTE (() T_{f} | μ, σ, a, b), \\ right x ((), 0) & = & left X_{0}, \\ right x ((), T_{f}) & = & left X_{f} . \end{matrix}

…”

Section: Minimum Consumption Models With Embargo Length Uncertaintymentioning

confidence: 99%

“…The details of the expectation calculation are treated in detail in Lloyd and Meyer [], but the ultimate result is shown in .

\begin{matrix} true \\ right E ([], J) = & = \int_{0}^{T_{f}} \frac{e^{- (r + θ) t} - e^{- r t}}{- 1 + e^{- θ T_{f}}} U (q (t)) d t . \end{matrix}

…”

Section: Minimum Consumption Models With Embargo Length Uncertaintymentioning

confidence: 99%

Section: Minimum Consumption Models With Saturation Uncertaintymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Methods of Modeling Consumption Saturation With Uncertainty in Optimal Stockpile Problems

Lloyd

Meyer

2015

Natural Resource Modeling

View full text Add to dashboard Cite

show abstract

A dynamic game model of optimal stockpile management with price dynamics and constraints

Lloyd

Meyer

2015

2015 23rd Mediterranean Conference on Control and Automation (MED)

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Optimal Stockpile Problems with Stochastic Consumption Saturation and Solution Interval

Cited by 2 publications

References 12 publications

Methods of Modeling Consumption Saturation With Uncertainty in Optimal Stockpile Problems

Methods of Modeling Consumption Saturation With Uncertainty in Optimal Stockpile Problems

A dynamic game model of optimal stockpile management with price dynamics and constraints

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