# Criteria for warehouse location

It's important to determine rational logistics channels and the location of the commercial network (retailers, wholesalers, warehouses). The optimal location of the warehouse helps reduce cost and to create a well-organized logistics system.

## Warehouse location

Using the Pythagorean theorem, the distance (not known) to newly established commercial centers of the area served can be determined using the following formula\[d_i = \sqrt{(x-x_i)^2+(y-y_i)^2}\]

where:

- x, y - the coordinates of the desired point of warehouse location
- xi. yi - coordinates of the area that will be served by this new facility

This equation is sometimes expanded by the number of inhabitants (mi) and the area. You can determine the minimum of the function\[Q = \sum_{i=1}^N m_i \sqrt{(x-x_i)^2+(y-y_i)^2}\]

Appointment of decision variables x and y is followed by differentiation with respect to their functions Q and equating the derivatives to zero. Then, after appropriate transformations, formulas to determine the coordinates of the warehouse can be obtained. Alternatively, instead of a linear minimum distance we could use minimum square distance (ie. to find the minimum of the square of the distance, so that the function Q does not contain a square root). Using the minimum square distance we can calculate the optimal location of the warehouse, taking into account the position of suppliers and customers.

We denote the coordinates on the xi and suppliers yi and s customers by uj and vj deliveries from individual suppliers to wholesalers by pi, and wholesalers to individual recipients by qj. Function for Q, expressing the squares of the distance, which is the minimum that we seek has the form\[Q = \sum_{i=1}^r p_i [(x-x_i)^2+(y-y_i)^2]+ \sum_{j=1}^s q_j [(x-u_j)^2+(y-v_j)^2]\]

After differentiating this function with respect to x and y and aligning the derivative to zero yields the following formulas designating the warehouse coordinates\[x = \frac{\sum{p_i x_i} + \sum{q_j u_j}}{\sum p_i + \sum q_j}, y = \frac{\sum{p_i y_i} + \sum{q_j v_j}}{\sum p_i + \sum q_j}\]

## References

- Feldman, E. F. A. T. L., Lehrer, F. A., & Ray, T. L. (1966). Warehouse location under continuous economies of scale. Management Science, 12(9), 670-684.
- Perl, J., & Daskin, M. S. (1985). A warehouse location-routing problem. Transportation Research Part B: Methodological, 19(5), 381-396.