scholarly journals Generalized multi-peg tower of hanoi problem

1996 ◽  
Vol 38 (2) ◽  
pp. 201-208 ◽  
Author(s):  
A. A. K. Majumdar
Keyword(s):  
Dynamic Programming ◽  
Explicit Expression ◽  
Programming Technique ◽  
Optimality Equation ◽  
Tower Of Hanoi ◽  
Minimum Number ◽  
Explicit Expressions

AbstractThis paper treats the multi-peg generalization of the Tower of Hanoi problem with n(≥ 1) disks and p(≥ 3) pegs, P1, P2,…, Pp. Denoting by M(n, p) the presumed minimum number of moves required to transfer the tower of n disks from the source peg, P1, to the destination peg, Pp, under the condition that each move transfers the topmost disk from one peg to another such that no disk is ever placed on top of a smaller one, the Dynamic Programming technique has been employed to find the optimality equation satisfied by M(n, p). Though an explicit expression for M(n, p) is given, no explicit expressions for the partition numbers (at which M(n, p) is attained) are available in the literature for p ≥ 5. The values of the partition numbers have been given in this paper.

scholarly journals The Tower of Hanoi Problem with Evildoer Discs

2020 ◽  
Vol 43 (2) ◽  
pp. 205-209
Author(s):  
AAK Majumdar
Keyword(s):  
Explicit Form ◽  
Optimality Equation ◽  
Tower Of Hanoi ◽  
New Variant ◽  
Minimum Number ◽  
Academy Of Sciences

This paper deals with a variant of the classical Tower of Hanoi problem with n ( ≥ 1) discs, of which r discs are evildoers, each of which can be placed directly on top of a smaller disc any number of times. Denoting by E(n, r) the minimum number of moves required to solve the new variant, is given a scheme find the optimality equation satisfied by E(n, r). An explicit form of E(n, r) is then obtained. Journal of Bangladesh Academy of Sciences, Vol. 43, No. 2, 205-209, 2019

scholarly journals Dynamic Operation Management of a Renewable Microgrid including Battery Energy Storage

2018 ◽  
Vol 2018 ◽  
pp. 1-19 ◽  
Author(s):  
Shaozhen Jin ◽  
Zhizhong Mao ◽  
Hongru Li ◽  
Wenhai Qi
Keyword(s):  
Dynamic Programming ◽  
Energy Storage ◽  
Renewable Energy Sources ◽  
Main Idea ◽  
Optimal Operation ◽  
Optimal Control Strategy ◽  
Programming Technique ◽  
Battery Energy Storage ◽  
Comparison Results ◽  
Battery Energy

In this paper, a novel dynamic programming technique is presented for optimal operation of a typical renewable microgrid including battery energy storage. The main idea is to use the scenarios analysis technique to proceed the uncertainties related to the available output power of wind and photovoltaic units and dynamic programming technique to obtain the optimal control strategy for a renewable microgrid system in a finite time period. First, to properly model the system, a mathematical model including power losses of the renewable microgrid is established, where the uncertainties due to the fluctuating generation from renewable energy sources are considered. Next, considering the dynamic power constraints of the battery, a new performance index function is established, where the Lagrange multipliers and interior point method will be presented for the equality and inequality operation constraints. Then, a feedback control scheme based on the dynamic programming is proposed to solve the model and obtain the optimal solution. Finally, simulation and comparison results are given to illustrate the performance of the presented method.

scholarly journals On the Star Puzzle

2019 ◽  
Vol 39 ◽  
pp. 1-14
Author(s):  
AAK Majumdar
Keyword(s):  
Recurrence Relation ◽  
Additional Condition ◽  
Tower Of Hanoi ◽  
Minimum Number

In the star puzzle, there are four pegs, the usual three pegs, S, P and D, and a fourth one at 0. Starting with a tower of n discs on the peg P, the objective is to transfer it to the peg D, in minimum number of moves, under the conditions of the classical Tower of Hanoi problem and the additional condition that all disc movements are either to or from the fourth peg. Denoting by MS(n) the minimum number of moves required to solve this variant, MS(n) satisfies the recurrence relation . This paper studies rigorously and extensively the above recurrence relation, and gives a solution of it. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 1-14

Optimality conditions for a Markov decision chain with unbounded costs

1980 ◽  
Vol 17 (04) ◽  
pp. 996-1003
Author(s):  
D. R. Robinson
Keyword(s):  
Dynamic Programming ◽  
Optimality Conditions ◽  
Average Cost ◽  
Optimality Equation ◽  
Dynamic Programming Equation ◽  
Potential Condition ◽  
Markov Decision ◽  
General Conditions

It is known that when costs are unbounded satisfaction of the appropriate dynamic programming ‘optimality' equation by a policy is not sufficient to guarantee its average optimality. A ‘lowest-order potential' condition is introduced which, along with the dynamic programming equation, is sufficient to establish the optimality of the policy. Also, it is shown that under fairly general conditions, if the lowest-order potential condition is not satisfied there exists a non-memoryless policy with smaller average cost than the policy satisfying the dynamic programming equation.

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