Optimal control of a finite dam using P M λΤ policies and penalty cost: total discounted and long run average cases

1990 ◽  
Vol 27 (04) ◽  
pp. 888-898
Author(s):  
M. Abdel-hameed ◽  
Y. Nakhi
Keyword(s):  
Optimal Control ◽  
Wiener Process ◽  
Average Cost ◽  
Input Process ◽  
Average Case ◽  
Discounted Cost ◽  
Penalty Cost ◽  
Drift Term ◽  
Long Run ◽  
Finite Dam

Zuckermann [10] considers the problem of optimal control of a finite dam using policies, assuming that the input process is Wiener with drift term μ ≧ 0. Lam Yeh and Lou Jiann Hua [7] treat the case where the input is a Wiener process with a reflecting boundary at zero, with drift term μ ≧ 0, using the long-run average cost and total discounted cost criteria. Attia [1] obtains results similar to those of Lam Yeh and Lou Jiann Hua for the long-run average case and extends them to include μ < 0. In this paper we look further into the results of Zuckerman [10], simplify some of the work of Attia [1], [2], offering corrections to some of his formulae and extend the results of Lam Yeh and Lou Jiann Hua [7].

Optimal control of a finite dam using PMλΤ policies and penalty cost: total discounted and long run average cases

1990 ◽  
Vol 27 (4) ◽  
pp. 888-898 ◽  
Author(s):  
M. Abdel-hameed ◽  
Y. Nakhi
Keyword(s):  
Optimal Control ◽  
Wiener Process ◽  
Average Cost ◽  
Input Process ◽  
Average Case ◽  
Discounted Cost ◽  
Penalty Cost ◽  
Drift Term ◽  
Long Run ◽  
Finite Dam

Zuckermann [10] considers the problem of optimal control of a finite dam using policies, assuming that the input process is Wiener with drift term μ ≧ 0. Lam Yeh and Lou Jiann Hua [7] treat the case where the input is a Wiener process with a reflecting boundary at zero, with drift term μ ≧ 0, using the long-run average cost and total discounted cost criteria. Attia [1] obtains results similar to those of Lam Yeh and Lou Jiann Hua for the long-run average case and extends them to include μ < 0. In this paper we look further into the results of Zuckerman [10], simplify some of the work of Attia [1], [2], offering corrections to some of his formulae and extend the results of Lam Yeh and Lou Jiann Hua [7].

Optimal control of a finite dam: average-cost case

1985 ◽  
Vol 22 (02) ◽  
pp. 480-484 ◽  
Author(s):  
Lam Yeh
Keyword(s):  
Optimal Control ◽  
Wiener Process ◽  
Release Rate ◽  
Optimal Policy ◽  
Average Cost ◽  
Output Rate ◽  
Penalty Cost ◽  
Long Run ◽  
Negative Drift ◽  
Finite Dam

We consider the problem of minimizing the long-run average cost per unit time of operating a finite dam in the class of the policies of the following type. Assume that the dam is initially empty, the release rate is kept at 0 until the dam storage increases to λ, and as soon as this occurs, water is released at rate M, then the output rate is kept at M as long as the dam storage is more than τ and it must be decreased to 0 if the dam storage becomes τ. We assume that the input of water into the finite dam is a Wiener process with non-negative drift μ and variance parameter σ 2. There is a cost in increasing the output rate from 0 to M as well as in decreasing the rate from M to 0 and whenever the dam storage is below level a, there is a penalty cost per unit time depending on the level. A reward is given for each unit of water released. In this paper, the long-run average cost per unit time is determined, and therefore the optimal policy can be found numerically.

Optimal control of a finite dam: average-cost case

1985 ◽  
Vol 22 (2) ◽  
pp. 480-484 ◽  
Author(s):  
Lam Yeh
Keyword(s):  
Optimal Control ◽  
Wiener Process ◽  
Release Rate ◽  
Optimal Policy ◽  
Average Cost ◽  
Output Rate ◽  
Penalty Cost ◽  
Long Run ◽  
Negative Drift ◽  
Finite Dam

We consider the problem of minimizing the long-run average cost per unit time of operating a finite dam in the class of the policies of the following type. Assume that the dam is initially empty, the release rate is kept at 0 until the dam storage increases to λ, and as soon as this occurs, water is released at rate M, then the output rate is kept at M as long as the dam storage is more than τ and it must be decreased to 0 if the dam storage becomes τ. We assume that the input of water into the finite dam is a Wiener process with non-negative drift μ and variance parameter σ2. There is a cost in increasing the output rate from 0 to M as well as in decreasing the rate from M to 0 and whenever the dam storage is below level a, there is a penalty cost per unit time depending on the level. A reward is given for each unit of water released. In this paper, the long-run average cost per unit time is determined, and therefore the optimal policy can be found numerically.

Optimal control of a finite dam: Wiener process input

1987 ◽  
Vol 24 (1) ◽  
pp. 186-199 ◽  
Author(s):  
Lam Yeh ◽  
Lou Jiann Hua
Keyword(s):  
Optimal Control ◽  
Wiener Process ◽  
Optimal Policy ◽  
Average Cost ◽  
Transition Density ◽  
Discounted Cost ◽  
Transition Density Function ◽  
Long Run ◽  
The Resolvent Operator ◽  
Finite Dam

The problem of optimal control of a finite dam in the class of policies has been considered by Lam Yeh [6], [7]. In this paper, by using the first Dynkin formula, the same problems of specifying an optimal policy in the class of the policies to minimize the expected total discounted cost as well as the long-run average cost are considered. Both the expected total discounted cost and long-run average cost are determined explicitly, and then the optimal policy can be found numerically, Also, we obtain the transition density function and the resolvent operator of a reflecting Wiener process.

Optimal control of a dam using Pλ,τM policies and penalty cost when the input process is a compound Poisson process with positive drift

2000 ◽  
Vol 37 (2) ◽  
pp. 408-416 ◽  
Author(s):  
Mohamed Abdel-Hameed
Keyword(s):  
Optimal Control ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Input Process ◽  
Compound Poisson ◽  
Discounted Cost ◽  
Penalty Cost ◽  
Drift Term ◽  
Long Run ◽  
Negative Drift

In this paper we consider the optimal control of an infinite dam using policies assuming that the input process is a compound Poisson process with a non-negative drift term, and using the total discounted cost and long-run average cost criteria. The results of Lee and Ahn (1998) as well as other well-known results are shown to follow from our results.

Optimal control of a dam using P λ,τ M policies and penalty cost when the input process is a compound Poisson process with positive drift

2000 ◽  
Vol 37 (02) ◽  
pp. 408-416
Author(s):  
Mohamed Abdel-Hameed
Keyword(s):  
Optimal Control ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Input Process ◽  
Compound Poisson ◽  
Discounted Cost ◽  
Penalty Cost ◽  
Drift Term ◽  
Long Run ◽  
Negative Drift

In this paper we consider the optimal control of an infinite dam using policies assuming that the input process is a compound Poisson process with a non-negative drift term, and using the total discounted cost and long-run average cost criteria. The results of Lee and Ahn (1998) as well as other well-known results are shown to follow from our results.

Two-stage output procedure of a finite dam

1977 ◽  
Vol 14 (2) ◽  
pp. 421-425 ◽  
Author(s):  
Dror Zuckerman
Keyword(s):  
Optimal Control ◽  
Wiener Process ◽  
Average Cost ◽  
List Type ◽  
Type Number ◽  
Output Rate ◽  
Discounted Cost ◽  
Long Run ◽  
Control Output ◽  
Finite Dam

The input of water into a finite dam is a Wiener process with positive drift. Water may be released at either of two possible rates 0 or M. At any time the output rate can be increased from 0 to M with cost KM, (K ≧ 0), or decreased from M to 0 with zero cost. There is a reward of A monetary units for each unit of output, (A > 0). We will consider the problem of specifying an optimal control output policy under the following optimal criteria: (a)Minimum total long-run average cost per unit time.(b)Minimum expected total discounted cost.

Two-stage output procedure of a finite dam

1977 ◽  
Vol 14 (02) ◽  
pp. 421-425 ◽  
Author(s):  
Dror Zuckerman
Keyword(s):  
Optimal Control ◽  
Wiener Process ◽  
Average Cost ◽  
List Type ◽  
Type Number ◽  
Output Rate ◽  
Discounted Cost ◽  
Long Run ◽  
Control Output ◽  
Finite Dam

The input of water into a finite dam is a Wiener process with positive drift. Water may be released at either of two possible rates 0 or M. At any time the output rate can be increased from 0 to M with cost KM, (K ≧ 0), or decreased from M to 0 with zero cost. There is a reward of A monetary units for each unit of output, (A &gt; 0). We will consider the problem of specifying an optimal control output policy under the following optimal criteria: (a) Minimum total long-run average cost per unit time. (b) Minimum expected total discounted cost.

Optimal control of a finite dam: Wiener process input

1987 ◽  
Vol 24 (01) ◽  
pp. 186-199 ◽  
Author(s):  
Lam Yeh ◽  
Lou Jiann Hua
Keyword(s):  
Optimal Control ◽  
Wiener Process ◽  
Optimal Policy ◽  
Average Cost ◽  
Transition Density ◽  
Discounted Cost ◽  
Transition Density Function ◽  
Long Run ◽  
The Resolvent Operator ◽  
Finite Dam

The problem of optimal control of a finite dam in the class of policies has been considered by Lam Yeh [6], [7]. In this paper, by using the first Dynkin formula, the same problems of specifying an optimal policy in the class of the policies to minimize the expected total discounted cost as well as the long-run average cost are considered. Both the expected total discounted cost and long-run average cost are determined explicitly, and then the optimal policy can be found numerically, Also, we obtain the transition density function and the resolvent operator of a reflecting Wiener process.

The control of a finite dam

1982 ◽  
Vol 19 (4) ◽  
pp. 815-825 ◽  
Author(s):  
F. A. Attia ◽  
P. J. Brockwell
Keyword(s):  
Markov Chain ◽  
Wiener Process ◽  
Average Cost ◽  
Type B ◽  
Input Process ◽  
Long Run ◽  
The Cost ◽  
Finite Dam

The long-run average cost per unit time of operating a finite dam controlled by a PlM policy (Faddy (1974), Zuckerman (1977)) is determined when the cumulative input process is (a) a Wiener process with drift and (b) the integral of a Markov chain. It is shown how the cost for (a) can be obtained as the limit of the costs associated with a sequence of input processes of the type (b).

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