Simulation-based optimization

Ranking and selection methods are designed for problems where the alternatives are fixed and known, and simulation is used to estimate the system performance. In the simulation optimization setting, applicable methods include indifference zone approaches, optimal computing budget allocation, and knowledge gradient algorithms.

In response surface methodology, the objective is to find the relationship between the input variables and the response variables. The process starts from trying to fit a linear regression model. If the P-value turns out to be low, then a higher degree polynomial regression, which is usually quadratic, will be implemented. The process of finding a good relationship between input and response variables will be done for each simulation test. In simulation optimization, response surface method can be used to find the best input variables that produce desired outcomes in terms of response variables.[7]

Heuristic methods change accuracy by speed. Their goal is to find a good solution faster than the traditional methods, when they are too slow or fail in solving the problem. Usually they find local optimal instead of the optimal value; however, the values are considered close enough of the final solution. Examples of these kinds of methods include tabu search and genetic algorithms.[4]

Metamodels enable researchers to obtain reliable approximate model outputs without running expensive and time-consuming computer simulations. Therefore, the process of model optimization can take less computation time and cost.[8]

Stochastic approximation is used when the function cannot be computed directly, only estimated via noisy observations. In this scenarios, this method (or family of methods) looks for the extrema of these function. The objective function would be:[9]

${\displaystyle {\underset {{\text{x}}\in \theta }{\min }}f{\bigl (}{\text{x}}{\bigr )}={\underset {{\text{x}}\in \theta }{\min }}\mathrm {E} [F{\bigl (}{\text{x,y}})]}$

${\displaystyle y}$ is a random variable that represents the noise.

${\displaystyle x}$ is the parameter that minimizes ${\displaystyle f{\bigl (}{\text{x}}{\bigr )}}$.

${\displaystyle \theta }$ is the domain of the parameter ${\displaystyle x}$.

Derivative-free optimization is a subject of mathematical optimization. This method is applied to a certain optimization problem when its derivatives are unavailable or unreliable. Derivative-free methods establish a model based on sample function values or directly draw a sample set of function values without exploiting a detailed model. Since it needs no derivatives, it cannot be compared to derivative-based methods.[10]

For unconstrained optimization problems, it has the form:

${\displaystyle {\underset {{\text{x}}\in \mathbb {R} ^{n}}{\min }}f{\bigl (}{\text{x}}{\bigr )}}$

The limitations of derivative-free optimization:

1. Some methods cannot handle optimization problems with more than a few variables; the results are usually not so accurate. However, there are numerous practical cases where derivative-free methods have been successful in non-trivial simulation optimization problems that include randomness manifesting as "noise" in the objective function. See, for example, the following [5] .[11]

2. When confronted with minimizing non-convex functions, it will show its limitation.

3. Derivative-free optimization methods are relatively simple and easy, but, like most optimization methods, some care is required in practical implementation (e.g., in choosing the algorithm parameters).

Dynamic programming deals with situations where decisions are made in stages. The key to this kind of problems is to trade off the present and future costs.[12]

One dynamic basic model has two features:

1) It has a discrete time dynamic system.

2) The cost function is additive over time.

For discrete features, dynamic programming has the form:

${\displaystyle x_{k+1}=f_{k}(x_{k},u_{k},w_{k}),k=0,1,...,N-1}$

${\displaystyle k}$ represents the index of discrete time.

${\displaystyle x_{k}}$ is the state of the time k, it contains the past information and prepare it for the future optimization.

${\displaystyle u_{k}}$ is the control variable.

${\displaystyle w_{k}}$ is the random parameter.

For the cost function, it has the form:

${\displaystyle g_{N}(X_{N})+\sum _{k=0}^{N-1}g_{k}(x_{k},u_{k},W_{k})}$

${\displaystyle g_{N}(X_{N})}$ is the cost at the end of the process.

As the cost cannot be optimized meaningfully, it can be used the expect value:

${\displaystyle E\{g_{N}(X_{N})+\sum _{k=0}^{N-1}g_{k}(x_{k},u_{k},W_{k})\}}$

Neuro-dynamic programming is the same as dynamic programming except that the former has the concept of approximation architectures. It combines artificial intelligence, simulation-base algorithms, and functional approach techniques. “Neuro” in this term origins from artificial intelligence community. It means learning how to make improved decisions for the future via built-in mechanism based on the current behavior. The most important part of neuro-dynamic programming is to build a trained neuro network for the optimal problem.[13]