In the first part of the article by Agnieszka Hodczak-Sekulska and Adam Redmer, Figure 1 lacked labels for specific methods in selected fields. We are now publishing the full version of this figure, while apologizing to the Authors and Readers for the situation that occurred.
Editorial Team
- 16 potential warehouse locations in different voivodeships are considered; provincial cities were adopted as symbolic locations for each warehouse: Białystok, Bydgoszcz, Gdańsk, Gorzów Wielkopolski, Katowice, Kielce, Kraków, Lublin, Łódź, Olsztyn, Opole, Poznań, Rzeszów, Szczecin, Warszawa, Wrocław
- number of decision variables:
1. in stage 1 (preliminary), 192 decision variables Xij were defined, where 192 = 12 (voivodeships where stores operate) x 16 (cities as potential warehouse locations); decision variables take binary values,
2. in stage 2 (main), 177 decision variables Xij were defined (the average number of stores owned by retail chains operating in Poland), by means of which each store was assigned to a specific warehouse; variables can take values from 1 to 16 depending on the number of the warehouse that is to supply a given store.
As a result, to make it possible to use generally available optimization tools, the problem was solved in two stages, which simplified the calculations. However, such a division is not necessary from a methodological point of view. The procedure in each of the optimization stages was as follows:
a) Preliminary stage – determining warehouse locations and assigning them to serve the voivodeships where stores operate. Provincial cities were chosen as warehouse locations. Therefore, in this stage, specific names of cities where potential warehouses are to be built were already used. Meanwhile, the demand of all stores operating in individual voivodeships was aggregated to the voivodeship level. A fragment of the decision variable table is shown in Figure 2.
Generating solutions: using the MS Excel Solver tool, calculations were performed for 12 possible distribution network variants. This consisted of entering the number of warehouses as one of the constraints, and during subsequent iterations of the optimization process, this number changed from 1 to 12. In this way, results were obtained for each potential number of warehouses in the range of 1 to 12 (this is the number of voivodeships in which a typical retail chain operates). These results included: warehouse locations and the assignment of voivodeships where stores operate to the warehouses defined in this way.
b) Main model – finding the optimal assignment of each store to a specific warehouse. For this purpose, the results obtained in the preliminary model were used. Thus, the warehouse locations for each of the distribution network variants were known (it was assumed that the service areas of each warehouse, defined preliminarily in the previous stage for the purpose of determining their locations, again constitute an element of optimization, this time at the store level, and therefore more accurately than before). The decision variable table has a different form than in the preliminary model and was presented in Figure 3.
Generating solutions: in this stage, similar to the previous stage, calculations were performed for all 12 variants (from 1 to 12 warehouses). For all analyzed variants, an input solution was introduced, which was the result of the preliminary model. This consisted of entering the warehouse number for each store into the decision variable table for a specific variant in accordance with the assignment from the preliminary model (construction of the initial solution). In all variants, Evolver made such changes to the assignment of stores to warehouses from the input solution so that the sum of transport and storage costs, expressed by formula 3, was as low as possible. In some cases, Evolver changed the initial assignment of stores to warehouses located even in the same voivodeships as the store to a warehouse from another voivodeship, because the distance of these stores to warehouses located in other voivodeships turned out to be shorter.
The article comes from the magazine “Logistyka” 4/2010.