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Planning for supply water demands (drinkable and irrigation water demands) is a necessary problem. For this purpose, three subjects must be considered (optimization of water supply systems such as volume of reservoir dams, optimization of released water from reservoir and prediction of next droughts). For optimization of volume of reservoir dams, yield model is applied. Reliability of yield model is more than perfect model and cost of solution of this model is less than other methods. For optimization of released water from reservoir dams, different methods can be applied. In this research, dynamic programming method (a discrete method for optimization) and genetic algorithm (a searcher method for optimization) are considered for optimization of released water from the Karaj reservoir dam. The Karaj dam locates in west of Tehran. This research shows that reliability and resiliency of GA method is more than DP method and vulnerability of GA method is less than DP method. For improving of results of GA method, mutation rate of GA method is considered from 0.005 to 0.3 for different generations. For prediction extreme droughts in future, the Markov chain method is used. Based on generated data by Markov chain method, optimum volume of reservoir dam is determined by yield model. Then optimum released water from reservoir dam is determined by DP and GA methods for different scenarios that produced by Markov chain method. The Markov chain and yield model show that volume of reservoir Karaj dam should increase 123 MCM for overcoming to next droughts.

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Dynamic lot sizing problem is one of the significant problem in industrial units and it has been considered by  many researchers. Considering the quantity discount in  purchasing cost is one of the important and practical assumptions in the field of inventory control models and it has been less focused in terms of stochastic version of dynamic lot sizing problem. In 
this paper, stochastic dynamic lot sizing problem with considering the quantity discount is defined  and  formulated.  Since  the  considered  model  is  mixed  integer  non-linear programming,  a  piecewise  linear  approximation  is  also  presented.  In  order  to  solve  the mixed integer non-linear programming, a branch and bound algorithm are presented. Each node in the branch and bound algorithm is also MINLP which is solved based on dynamic programming framework. In each stage in this dynamic programming algorithm, there  is a sub-problem which can be solved with lagrangian relaxation method. The numeric results found in this  study indicate that the proposed algorithm solve the problem faster than the mathematical  solution  using  the  commercial  software  GAMS.  Moreover,  the  proposed algorithm for  the  two  discount  levels  are  also  compared  with  the  approximate  solution  in mentioned software. The results indicate that our algorithm up to 12 periods not only can reach to the exact solution, it consumes less time in contrast to the approximate model.