Larosa Job Shop is studying where to locate its tool bin facility on the shop floor. There are five production cells (fabrication, paint, subassembly 1, subassembly 2, and assembly) at fixed locations. The following table contains the cell locations, expressed as x and y coordinates on a shop floor grid, as well as the daily demand for tools (measured in number of trips to the tool bin) at each production cell.

One way to solve this problem is to find the tool bin location that minimizes the sum of the distances from each production cell to the tool bin location. There are a number of ways to measure distance, but Larosa has decided to use Euclidean (straight-line) distance. The Euclidean distance between two points (x, y) and (a, b) is

However, considering only distance ignores that fact that some production cells make more trips for tools than others. An approach that takes into account that some cell locations use more tools than others is to minimize the sum of the weighted distances from each station to the tool bin, where the weights are the demand from each production cell.
a. Develop a model that will find the optimal tool bin location for Larosa, where the objective is to minimize the sum of the distances from the tool bin to the production cell locations (ignoring the demand). Solve your model for the optimal location.
b. Update your model to find the tool bin location that minimizes the sum of the demand-weighted distances form the tool bin to the production cell locations. Solve your model for the optimal location.
c. Compare the results from parts (a) and (b). Explain the differences in the solutions?

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Location Cell Fabrication Paint Subassembly 1 Subassembly 2 Assembly r Coordinate yCoordinate Demand 12 2A 13 2.5 17 V(x a) + (y - b)