268 _6 = ` max 7 ∈:{W − U ∙ f}4V (1) ; < In his 1988 book Economic Definition of Ore, Lane (Lane,1988) states that the way to solve this problem is through dynamic programming where for each infinitesimal tonnage increment, the integrand selects the largest economic margin across all policies g. Integrating over R yields cumulative NPV sensitivity to incremental cut‑off changes. On the other hand, Richard Sutton and Andrew Barto integrate Bellman equations and Markov Chains to mathematically define the optimal policy in Reinforcement Learning, as shown in Equation 2. This approach aims to teach a machine to perform tasks by training it within a given environment, with specific actions and rewards. The optimal value equals expected immediate reward plus discounted best future return. P∗(j>,k>)=CL(j>?!,k>?!|j>,k>) - */! mn>?! +omax 5!"# P∗(j>?!,k>?!)p (2) Based on master’s thesis in Computer Science (Lozano,2024), the mining process was modeled as a Reinforcement Learning (RL) problem. The study demonstrated that Lane’s integral presented in Equation 1 can be addressed by reformulating the mining optimization challenge through RL framework as shown in Equations 3 and 4. P7∈: = max 7∈: rs),* +t11−+uuvr1+ C s)@!,A (1+u)A@* $ A/*?! w Δ>$ Δ1$w (3) P∗(j>,k>) ≅ P7∈: (4) 2.1. Reinforcement Learning in Non-Renewable Resources The optimization of non-renewable resources, such as mining, is a significant challenge for both industry and academia. Traditional approaches, using algorithm that implement MILP or Metaheuristics, have proven effective in finding optimal solutions under given initial conditions. However, these methods exhibit several limitations, including the generation of a single solution, lack of adaptive learning, the need for re-optimization with parameter changes, and uncertainty in the success probability of the obtained solution. In this context, we propose an abstraction of mining problems using Reinforcement Learning (RL). Mining problems can be naturally modeled as reinforcement learning games, where periods represent states, extraction decisions correspond to actions, and the cash flow of the mining model serves as the reward like we can show in Figure 2 and Figure 3. This approach addresses the limitations of classical optimization methods, providing a more flexible and adaptive framework.
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