# Complementing Solutions to Optimization Problems via Crowdsourcing on Video Game Plays
## Implementation
### Game 2: Facility Location Problem
#### 3.3.3 Game 2 Description, Facility Location  
The FLP game is modeled after previous work on the robust facility location problem (RFLP) using genetic algorithms [43], where the RFLP is formulated as three different optimization problems. At a high level, these sub-problems are as follows:  
* Minimizing both the cost of opening facilities and the cost of traveling between the facilities and demand centers; this is a multi-objective formulation of the FLP. * Finding the worst possible failure combination, namely, the set of open facilities that result in the highest total distance should they fail, determining how each solution proposed in the previous step will perform as a solution to the RFLP. * Finding optimal solutions to the RFLP, which involves a trade-off between the optimal solution when no facilities fail and its optimal value after facility failures. This sub-problem provides a human decision-maker with the ability to trade-off for a given domain. The game is designed to address the FLP in two stages, played in consecutive order. On each stage, there is a category of levels, where each level corresponds to a single instance of the problem. The details are as follows:  
* The player proposes solutions for instances of the FLP. A solution consists of a set of facilities that the player chooses to open. * The player is presented with solutions provided by other players, and the player is asked to improve the total cost. Players must choose a set of open facilities; however, in this case, the game gives them a starting point. In a match, opening a facility has an associated cost; nevertheless, to compute each candidate solution, we use a single score for total score of the players, incorporating both the total distance and opening facilities. This implicit assumption that the cost of opening a facility is equivalent to the cost of some distance may bias the players, making them more likely to focus their search with a fixed number of open facilities. The number of open facilities minimizes the total cost, leaving much of the solution space unexplored, and the game designer determines the cost for each open facility placed. Some levels include a limit on the number of facilities that the player can open, which is chosen at random. Limiting the number of facilities that a player can open encourages the player to explore every possible number of open facilities and to provide reasonable solutions along the entire Pareto boundary. Making this distinction among separate stages enables us to complement the genetic algorithm because the crowd-computing-based stage can complement those output solutions not found by the genetic algorithm. User Interface Design  
To keep the game independent of different domains in which instances of the problem may arise, we chose to employ abstract representations of the demand centers, facilities, weights, and assignments, which were free of any references to cities, vehicles, factories, or any other real-life element of theseinstances, as shown in Figure 2. The visualization for the FLP consists of a 2D white canvas that displays demand centers as circular spots in the coordinates specified by the dataset. Each demand center may have a facility on it, displayed as a black square inscribed in the circle. All demand centers are shown in blue; however, their saturation varies with their weight--centers with low demand are a lighter blue while centers with a high demand are dark blue, as depicted in Figure 2.  
A straight line connecting two circles represents an assignment between a demand center and a facility. The line color saturation reflects the cost of the assignment, based on a function of the distance between the center and the facility, and the weight of the demand centers. Unassigned demand centers have blinking borders to attract the attention of the players. For the game stages, the player needs to choose where to open a facility within a demand center. To open a facility, the player needs to tap an empty demand center.