## Determining the location of the warehouse

## Methods for determining the location of the warehouse

Determining the location of the warehouse is one of the fundamental solutions in logistics, which belongs to a group of design tasks and is solved in the design of logistics systems. If we correlate the definition of the location of the warehouse with the types of managerial decisions, then we can say that this task relates to strategic decisions that form the element of a logistics strategy, such as the configuration of the logistics system. Typically, the definition of the location of the warehouse is considered in relation to distribution warehouses.

In order to determine the location of the distribution warehouse in the served region, it is necessary to know:

• location of manufacturing firms (suppliers) and consumers (customers) of products, which is usually given by * xp y, *

• volumes of deliveries of products from producers (suppliers) to consumers ((2,);

• delivery routes that depend on the characteristics of the existing transport network;

• shipping costs (or tariffs for transportation services) (7).)

There are different methods for determining the location of the warehouse, differing in the optimization criteria and the way distances between suppliers, consumers and the warehouse are taken into account. Consider two ways to account for distances. The first way is to calculate the shortest distance between the points where the suppliers, customers, and the warehouse are located. The shortest distance is determined by the formula

where, * y, - * coordinates of the supplier, the consumer;

*warehouse coordinates.*

**xc, uy -**The second way to account for distances is the so-called Manhattan distance, which allows for the consideration of distances between suppliers and consumers on a rectangular grid, which most fully corresponds to the rectangular arrangement of city streets. The & quot; Manhattan distance & quot; using the following formula:

The criteria on the basis of which the location of the warehouse is determined are transport work, transport or logistics costs. The target function in the models for determining the location of the warehouse is minimized.

The main methods for determining the location of the warehouse in logistics systems are presented in Table. 7.5.

## Determine the number of warehouses.

The task of logistics - determining the number of warehouses in the served region - paid a lot

Table 7.5. ** Methods for locating warehouses in supply chains **

attention in the works of foreign and domestic specialists. Often in the works are graphs of individual components and total costs, reflecting their dependence on the number of warehouses. An example of such graphs is shown in Fig. 7.7. It is believed that with the increase in the number of warehouses, transportation costs and lost profits from sales are decreasing, and the costs of maintaining stocks, operating the warehouse and managing the warehouse system are increasing. The presence of these contradictory tendencies determines the parabolic nature (with a clearly expressed optimum) of the dependence of the total costs on the functioning of the distribution system on the number of warehouses.

To solve this task of determining the number of warehouses, the following options can be considered.

The first option is without the use of warehouses. For this option, it is recommended to solve the classical transport problem of attaching * n * consumers to

*suppliers.*

**t**The second option is one warehouse. The solution in this variant (warehouse coordinates) can be determined using a numerical search algorithm with minimization of transport work.

The third option is two or more warehouses. The specifics of the calculations for the third option are characterized by the fact that, firstly, a condition is introduced about the approximate equality of warehouse capacities (if the capacities of warehouses can vary, the task becomes multi-criteria); Secondly, the distance between warehouses along the axis * X * (or Y) must not be less than a certain value. If we do not introduce this artificial restriction, then the general problem of finding the required dependence of transport costs on the number of warehouses under the optimal variant may degenerate.

Let's consider the iterative algorithm for finding coordinates by the example of two warehouses (in Table 7.5 is presented as the SPPGEIU method).

At the * first stage * the coordinates of the first and second warehouses are selected, then the transport task is solved (similar to the first variant of the task of determining the number of warehouses) if

**/strong>****suppliers and two consumers (warehouses).**

* Fig. 7.7. *

**Dependence of components and total costs on the functioning of the distribution system on the number of warehouses:**

1 - total costs; * 2 - * the costs of storing stocks, operating warehouses and managing the distribution system;

*the total costs for delivering the goods;*

**3 -***losses due to warehouse remoteness from the consumer*

**4 -** At the * second stage * the transport task is solved again, but on condition of two suppliers (warehouses) and

*consumers .*

**n** * The third stage * is the summation of the results of calculations performed in the first and second stages. The obtained value of the transport work is fixed as a first approximation.

On * the fourth stage * the coordinates of the warehouses change according to the selected rule and the calculations from the first to the third stage are repeated. The search for variants of warehouse coordinates is terminated in the case when the difference in the transport operation of two consecutive iterations becomes less than the specified value.

Following the proposed algorithm, you can calculate the transport component of the total logistics costs in distribution systems with two warehouses. This algorithm can be transformed for a situation where there are more than two warehouses in the logistics system. In this case, the transport task is solved twice - first from the * t * suppliers to

*warehouses, then from*

**to***warehouses to*

**to***consumers.*

**n**In Fig. 7.8 shows the algorithm for calculating transportation costs, which is the basis for determining the number and location of warehouses.

* Fig. 7.8. *

**The algorithm for determining transportation costs in a distribution network for a different number of warehouses**

Let's consider the steps of the given algorithm in more detail.

Step 1.

* If the location of suppliers and consumers is given by the coordinates of their location on the plane, then the shortest distance between suppliers and consumers of km , can be determined by the formula (7.1). *

The objective function (transport minimum * P, * m-km) can be written in the following form:

where * i = * 1,

*suppliers ;; = 1,*

**t -***consumers; & amp; - the volume of goods supplied from the/th supplier to the 7th consumer, t; IV. - the product of the weight shares of the 7th supplier and the 7th consumer. The introduction of this component is dictated by the need to take into account additional factors affecting the plan for optimal fastening of consumers to suppliers, for example, the inability of direct transit deliveries from the/th supplier to the y'th consumer or the priority of the y-th consumer in relation to others.*

**n -**Calculation of total transportation costs 5 is calculated using the formula

where * 2 ~ - * the number of loaded riders from the 1st supplier to the customer; C0 is the transportation tariff, cu/km.

The number of laden riders * X - * is calculated as follows:

where * tsU) - * the nominal carrying capacity of the rolling stock used when transporting from the/th supplier to the customer, t; y & quot; - coefficient of utilization of the carrying capacity of the rolling stock used when transporting from the/th supplier to the consumer.

Step 2.

* Solution of the warehouse positioning task. * The objective function used at this stage is:

where * bp b- - *, respectively, the distance from the warehouse to the/th supplier and to the y-th consumer; About, - accordingly volume of the cargo transported on a warehouse from 1-st supplier and from a warehouse up to ^ -th consumer.

The distance from the warehouse to the i-th supplier or the y-th customer is given by the formula (7.1), where * X = ХрУ = уЛ - * the required coordinates of the warehouse, under which the minimum of the objective function (7.6) is reached.

Transportation costs are calculated using the following formula

where * 2.p * 2 ,. - respectively, the number of riders from the 1st supplier to the warehouse and from the warehouse to the/th consumer.

Step 3.

* Determine the coordinates of the warehouses relative to the center of gravity. * As the center of gravity & quot; The found coordinates of the warehouse are located

*and*

**X***and the rules for finding the coordinates of the warehouses relative to the center of gravity are set. To determine the distance from the warehouses to the & quot; center of gravity & quot; rules introduced in the sequence are introduced:*

**Y**• determine the distance between the coordinates of the most distant from each other:

Where ХрЗД - supplier coordinates; . ^, * y} - * customer coordinates;

• choose the minimum distance and determine the radius of the circle R, on which the warehouses are located diametrically:

• the warehouses are first placed horizontally, and then vertically with respect to the coordinate axes;

• The initially adopted radius * I = * 0.1 is increased to 0.2, then to 0.3, etc.

Step 4. * Calculation of the minimum total costs for transportation for different locations of warehouses. * If there are two or more warehouses, the objective function looks like:

where * г = * 1,

*suppliers;*

**т -***1,/- warehouses;*

**to =***1,*

**} =**

**strong>****consumers;**

*respectively the distance from the/th supplier to the & amp; warehouse and the & amp; stock of the dow consumer; 0 ^ - - respectively, the volumes of cargo transportation from the 1st supplier to the & amp; warehouse and the & amp; stock of the dou-th consumer; And W ^ - ^, respectively, the product of the weight shares of the 1st supplier and the A-th warehouse,***& pound; i, bw -***of the warehouse and the ./th consumer.***the -th**
The distances from the 1st supplier to * to-go * warehouse and from

*warehouse to ^ th consumer are calculated by the formula (7.1).*

**to**Total transportation costs are calculated using the following formula

where * 2 [k, 2w - * respectively the number of laden riders from the 1st supplier to

*warehouse and from*

**to***warehouse of the dou-th consumer.*

**to-ho**The number of loaded riders is calculated by the formula (7.5).

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