# Integrated Simulation-Based Optimization of Operational Decisions at Container Terminals
## 1 Introduction
### Optimizing Objective Functions without Mathematical Optimization
The longer the total observed time range for a given simulation model, the more that typically noisier sample statistics approximate the population parameters. [PERSON] et al. [29] stated that this approach does not necessarily create optimal solutions, but solutions that are close to the optimum with a very high probability. Furthermore, in industry, such approximations often satisfy requirements [29]. Even if optimal input parameters are calculated for a given simulation model, the difference in performance might be of little practical relevance. Another option to reduce computational time is to embed the simulation into an outer loop of optimization. The simulation model itself is regarded as a black-box function. If the optimization problem (i.e., the objective values derived from the simulation model) has some known structural properties, one might prefer to derive a simpler representation that enables optimization with other tools than simulation [31]. The optimization algorithm in the outer loop tests a subset of the feasible parameter configurations to approximate the optimum of the black-box function. This concept goes by many different names, such as \"simulation optimization\" [32], \"simulation evaluation\" [33], \"simulation integrated into optimization\" [34], and \"simulation-based optimization\" [23]. For cases in which a combinatorial optimization problem is solved, the term \"simheuristic\" has been coined [35]. This concept is the only combination of simulation and optimization that does not rely on maintaining an additional mathematical model [34]. If the simulation model is sufficiently complex, then the optimization algorithm can only consist of general guidelines for searching good (but not necessarily optimal) solutions. These general guidelines, which are applicable across research domains, are also referred to as meta-heuristics [36]. Meta-heuristics often start with several randomly drawn parameter configurations, and the first objective values that are obtained direct further search. A good meta-heuristic balances exploration and exploitation. During exploration, parameter configurations that are quite different from the previous samples are tested. During exploitation, well-performing parameter configurations are slightly altered to obtain an improved parameter configuration. After several iterations, a stopping criterion is reached, and the best solution found so far is returned as an approximation of the global optimum. For a guided search, a meta-heuristicneeds to keep track of past evaluations. The large number of meta-heuristics reported in the literature stems from the fact that it is a non-trivial decision as to how to continue a search given a set of observations. To the best of the authors' knowledge [37], and the authors' previous study [24] are the only ones that have used the simulation-based optimization approach at a container terminal for scaling the amount of several utilized resources. [PERSON] et al. [37] simultaneously optimized the berth length, the number of QCs, the number of gates, the fleet size of YTs, the number of export and import rows, and the number of RTGs per row. The objective function balanced the throughput and the utilization, weighted by investment costs. While a high throughput was achieved with more resources, the weighted utilization ensured that no superfluous resources were added to the container terminal. Even though not all permissible realistic values were taken into account, the authors calculated 72,576 possible parameter combinations. They decided that these were too many to fully cover in a simulation study. They built an optimization framework that consisted of two stages: First, the interactions between resources were examined to determine the most promising sequence of resource optimization tasks. As seven different types of resources were checked, \(7!=5040\) possible permutations existed. Second, the gained sequence was utilized to optimize each resource one by one.