**Dynamic Programming:**

Dynamic Programming (DP) is a method for solving complex problems by breaking them down into simpler overlapping subproblems and solving each subproblem only once, storing the solutions to avoid redundant computations.

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The approach is particularly useful for problems exhibiting the optimal substructure and overlapping subproblems properties.

**Applications of Dynamic Programming in Decision-Making:**

**Optimal Resource Allocation:**DP is used to find the optimal allocation of resources over time, considering changing conditions and constraints.**Project Scheduling:**DP helps in scheduling tasks over time to minimize completion time or costs, considering dependencies and resource constraints.**Inventory Management:**DP is applied to determine the optimal inventory levels over time, considering demand fluctuations and costs associated with holding and ordering.**Finance:**In finance, DP is used for portfolio optimization, asset allocation, and investment decisions by considering changing market conditions and risk factors.**Robotics and Control Systems:**DP is employed in designing optimal control policies for systems that evolve over time, such as robotics or manufacturing processes.

**Difference from Linear Programming:**

**Nature of Problems:**

**Dynamic Programming:**DP is used for problems where decisions are made sequentially over time, and the optimal solution depends on previous decisions.**Linear Programming:**LP deals with problems where a linear objective function is to be optimized, subject to linear equality and inequality constraints.

**Time Dependency:**

**Dynamic Programming:**Considers the time dimension, making decisions at different time points, with solutions dependent on the state at each time.**Linear Programming:**Typically does not involve time; decisions are made at a single point in time.

**Optimal Substructure:**

**Dynamic Programming:**Utilizes optimal substructure, where the optimal solution to the overall problem can be constructed from optimal solutions to its subproblems.**Linear Programming:**Involves solving a single, comprehensive problem without breaking it down into smaller subproblems.

**Memory Usage:**

**Dynamic Programming:**Involves storing solutions to subproblems to avoid redundant computations, often requiring memory for memoization.**Linear Programming:**Solves the entire problem in one go, without the need for storing intermediate solutions.

In summary, dynamic programming is a method for solving optimization problems with a temporal dimension, breaking them into overlapping subproblems. In contrast, linear programming focuses on optimizing a linear objective function subject to linear constraints without the time-dependent aspect seen in dynamic programming.