Iopel D., Bilan S., Goncharova Y.

Oles Honchar Dnipropetrovsk National University


Topicality of the problem. The task of determining the locations of storage capacities on the criterion of transport costs is one of the fundamental problems of logistics. The founders of this research were A. Weber and J. Peak in the “Theory of the Location of Industries”, (1909). But nowadays the problem of determining the structure of the warehouse network remains urgent.

Goal of Research – the algorithm design of determining the effect of the number of warehouses for logistics transportation costs based on real transport communications and possiblelocations of distribution centers.

The presentation of the basic material of the research. Suppose that there are M suppliers (формула) and J consumers (формула). Within the precincts of a city there are Koptions for placement of storage capacities (формула). Crossroads of the city are markedформула. Graph G, corresponding to the transport network, is defined by the table, in which a path length from the vertex формулаtoформулаis situated at the intersection of i-th row and j-th column, if these vertices are adjacent.

  At the preliminary stage it is necessary to find all shortest paths between all vertices. This problem can be solved, for example, using the Dantzig's algorithm [4, p.58]. Received distances between each pair of vertices are entered into the data set and hereafter a chunk of received array is used at each stage.

In addition the database is formed:   

формула – the amount of products supplied by m-th enterprise (формула);

формула – the volume of production required by j-th customer (формула);

С0 – the transport tariff, cu. / km;

формула the rated capacity of rolling stock used for the transport from the m-th supplier to the j-th customer, T;

 the coefficient of capacity of the rolling stock used for transportation from the m-th supplier to the j-th customer.

To account the additional factors that influence on the plan of optimal consolidation of customers around suppliers (the impossibility of direct deliveries, consumer preferences in the selection of suppliers, etc.), weight fractions m-th supplier формула and j-th consumer формула are entered.

The first stage. Traditionally, the task of fixing the transport suppliers for consumers in the presence of two or more warehouses is solved twice – from M suppliers to K warehouses and from K warehouses to J consumers. If the warehouse is only one then for minimization of transportation costs it is enough to select warehouse corresponding to the vertex, for which:


where формулаформула – the shortest paths from suppliers формула to warehouse формула and from warehouse формула to consumers формула. In this case, the total cost for transportation S can be calculated by the formula:


where формула the number of loaded trips between places and that is calculated by the next formula:

формула. (1)

The second stage. Warehouse positioning. The vertex, selected in the first stage, is determined as “basic”.

Then the next magnitude формула is determined. Next vertices from the vertex формула are taken into consideration no more than distanceформула. First, two peaks in oppositedirections are taken, then two, etc. If in circle of the radius формула there is only one vertex that is different fromформула then it is viewed as the second, solving the problem of fixing suppliers forconsumers and determining the logistics transportation costs.

Thereafter, vertices, situated from "basic" at a distance over формула and no more thanформула, then at a distanceформула, etc., are taken into consideration.

The third stage. Calculation of minimum total costs of transportation at different locations of warehouses.

In the presence of two or more warehouses it is necessary to determine the volume of transportations from suppliers to warehouses and from warehouses to customers, minimizingtransportation work P (m × km). Thus, the objective function will be:


where m=1, 2,…, M – suppliers; k=1, 2,…,K* (K*≥2) – warehouses, selected in the second stage; j=1, 2,…, J – consumers;

формула and формула – the distances from the m-th supplier to the k-th warehouse and from the k-th warehouse to the j-th consumer;

формула and формула – unknown volumes of cargo transportation from the the m-th supplier to the k-th warehouse and from the k-th warehouse to the j-th consumer, respectively;

формула and формула – weight fractions from the m-th supplier to the k-th warehouse and from the k-th warehouse to the j-th consumer, respectively.

After determining the volume of traffic from suppliers to warehouses and from warehouses to consumers the logistics transportation costs are calculated as follows:


where the number of loaded trips формула is defined by (1).

The resulting database of the traced analysis of the amount of transportation costs and the location of the warehouse can be used to make a decision on the formation of the optimal warehouse network with the available options for the location of distribution centers. As well, the represented algorithm can be used to select the locations for additional warehouses in the expansion of the distribution system.

The solution to the above problem is quite time-consuming, requires the use of computer systems and assistance of an operator during the choosing of opposite located vertices.

Conclusion. The represented algorithm can be applied when the objects are set not by Cartesian coordinates, but the actual distance between them, and when you need to consider reallypossible location of the storage capacity.

The list of references:

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