Modeling transportation routes of the pick-up system using location problem: a case study

Ondrej Stopka; Vladimír Ľupták; Pawel Droździel; Iwona Rybicka

doi:10.1515/eng-2020-0089

Article Open Access

Modeling transportation routes of the pick-up system using location problem: a case study

Ondrej Stopka , Vladimír Ľupták , Pawel Droździel and Iwona Rybicka

Published/Copyright: September 2, 2020

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From the journal Open Engineering Volume 10 Issue 1

Abstract

In today’s increasingly globalizing world, the demand for the transport of more and more specific kinds of goods is constantly growing. Greater diversity of the objects of transport extends the range of consignees and consignors, which implies the reduction of the number and volume of consignments. Modern enterprises are no longer interested in mass supplies of large quantities of the same material but strive for flexible supplies of smaller quantities of specific materials out of more suppliers. This manuscript deals with the streamlining of specific transport-logistics processes, including a detailed analysis of the existing pick-up system in relation to transport serviceability of the certain territory starting in the city of České Budějovice (Czech Republic) and a determination of more effective variants for supply. The primary objective will be to find and analyze specific options for streamlining the processes being optimized using the selected method of the graph theory, namely the location problem. In the context of optimization, a specific case study is elaborated with the aim to increase the efficiency of the current pickup system, i.e. serviceability of a particular territory.

Keywords: Logistics; forwarding; graph theory; location problem; pick-up system; supply system

1 Introduction

These days, customers expect the lowest price of postal services but still look forward to their wide availability, security and delivery of mail items on time. The most important matter in these services while generating the total costs is to deal with transport cost of postal items [5]. First of all, it is necessary to rebut the generally accepted assumption that logistics equals the transportation of goods from the place of dispatch to the destination. Logistics is a combination of a wide range of business processes addressing the flow of material and goods, energy, human resources, and information, thus creating the entire value-creating chain. Companies no longer purchase tens of pallets of material or goods but rather smaller quantities of more specific materials from a wider range of suppliers, which increases the demand for single-piece shipments. Single-piece shipments are mostly offered by various pick-up systems [20]. Pick-up system refers to a kind of transport where single-piece shipments are consolidated into one common transportation unit, thus sharing the transportation cost. Transportation cost depends on many factors; i.e. purchase price of the vehicle, labor cost of the drivers, cost of maintenance and repairs, fuel cost etc. Hence, it is so important to find organizational solutions that can reduce the cost associated with the transportation of postal shipments, thereby reducing the total cost of services. An excellent way for doing it is to minimize the distance of transportation routes, which means to define the shortest possible connections between post offices in the area being investigated [10].

Thanks to the shared costs and the related low purchasing prices of this service, this form of transportation is a highly sought alternative. From the provider’s perspective, this service has its drawbacks and risks. Therefore, it is necessary to use and manage the whole system properly and effectively. Freight forwarders operating at domestic or international level are the most common pick-up service providers. Planning of logistics centers deployment aimed at the optimization of the serviced area is one of the important decision-making problems. The choice of an appropriate location in the real network may affect a number of

other factors. Improper location in the given network can result in counterproductive serviceability of the rest of the network. Therefore, it is necessary to pay great attention to such a decision [21].

The paper presented is focused on the specific process in the context of supplying a given transport territory. These processes will be analyzed and the findings obtained will be further interpreted. If potential for increasing effectiveness in some of the processes examines is found by means of the analysis, a proposal of the process measures will be drawn in order to increase the effectiveness. The research problem of this contribution is to analyze the selected logistics process regarding the territory serviceability. Logistics processes, specifically the pick-up system, will be analyzed and evaluated, and a recommendation for streamlining the given process will be proposed on the basis of the acquired research results [6].

2 Literature review

Multi-criteria analysis is presented in detail in the studies dealing with the assessment of independent components when addressing decision-making problems [24], while comparative analysis is focused on the application of the TOPSIS method in the context of sustainable supplier selection for sourcing strategies [25, 34]. The application part of our manuscript was prepared on the basis of the international journal on safety and ergonomics and the article discussing the optimization of operation performance and safety [1]. The scientific article [35] deals with the examples of group evaluation that decide on a sustainable approach to supplier selection using the modified method TOPSIS at a Pythagorean-valued interval. Our paper is also based on the specific study addressing the application of automatic identification systems [29] and the application of RFID technology in logistics companies operating in the EU countries [3]. Finally, for the purposes of topic being examined, it was necessary to utilize the analysis of vehicle repair costs in a selected transport company, which is addressed in [17]. The scientific publications [9, 16] describe in detail research findings related to model supplier evaluation in terms of the factors in a real company, and the selection of places suitable for the allocation of a particular company.

The case studies focused on maritime transport [12, 14] deal with the rationalization of logistics processes in transport and in the view of deployment of transshipments and terminals in a certain territory. The article [18] is focused on the identification of objects in intermodal transport units’ transshipment. Logistics network in relation to resources harmonization is discussed in the manuscript [28]. The case study [32] describes in detail the optimization of transport routes, where the authors, by quantifying the transit service abilities of particular railway stations, forecast the overall transport time when applying specific Monte Carlo simulations, and determine two best-possible transport routes. The allocation of depots using the Operations Research techniques is a subject of the paper [2]. In [13], a similar problem of vehicle routing is outlined, which is directly related to the research analyzed in our contribution. The literatures [19, 22] point out the characteristics of transportation of goods in batches, which is an important aspect of flexibility in the logistics supply process, as also addressed in this case study.

The problem of routing shipments between two depots is described in the study [27]. Perishable goods in terms of vehicle routing problem represents another issue pointing to distribution (serviceability) [30], which has to be taken into account as another variable in time aspects of the research. The allocation of time window in the distribution and supply [33], as well as specification of vehicles’ daily circulation in regard to passengers’ transportation [15] are another relevant aspects being emphasized, and thus objectives which need to be rendered. As far as our research is concerned, even the technologies “Just-in-time” (JIT) and “Just-in-sequence” (JIS) constitute significant concepts associated with a fluent pick-up or distribution system; hence, the original transportation system being implemented could be replaced by an innovative and progressive type of supply system, specifically using the technology of automated logistics trains [8].

3 Materials and Methods

Processing large amounts of data requires the application of certain mathematical methods. In the contribution, they are used for the purpose of streamlining the current state of transport and logistics disciplines examined; i.e. the results of the method, specifically the location problem, served as a foundation for an own proposal of a solution. To this end, the mathematical method (location problem) was applied to the research issue. The procedure and methodology are described in detail in the following parts of the manuscript.

To streamline the selected sets of vertices, the authors implemented the specific location-allocation problem. Allocation refers to the initial depot determination in order to optimize the material flows in a given network. The output of the method will constitute the value of transport performance depending on the number of operators’ needed in individual vertices. For each set of vertices, a location problem will be elaborated considering the number and weight of shipments; i.e. for each set, two problems will be addressed, which will subsequently be evaluated properly. First of all, distance matrices of individual graph vertices will be compiled. As for the calculations, Formula 1 (see below) based on the graph theory – location-allocation problem will be used [26]:

(1)fD′k=∑∑2∗du,v×w(u)

Where: D_k – is a set of depots, v – is a vertex, u – is a node, d(u, v) – is a minimum distance between two graph vertices, w(u) – is a weight of the vertex served (number of requirements for service), in this contribution it refers to the number or weight of shipments.

In the event of supplying the network of pick-up system, it is not assumed that vehicles returns to the service depot, which means that, in the calculations, the constant “2” representing return journeys is omitted [20]. It will be either replaced by constant “1” or can be completely omitted from the formula. Subsequently, the optimal solution will be identified, which will require the lowest transport performance of all possible location combinations, in other words, it will represent the minimum according to Formula 2 [21]:

(2)fDk=minf(D′k)

In the case of two or more service depots in the network, iterative algorithm can be used. Here, it is necessary to define the concept of catchment area in the graph. Catchment area in the graph refers to the area in which the selected service depot center operates; i.e. it is prime, which means it includes the closest vertices from the center and the allocated catchment area. If the central points are situated at the same distance from one vertex, the decision-maker will consider to which vertex the catchment area will be allocated. A design of optimal allocation of logistics centers in a given network represents an important discipline of decision-making problems. The choice of locations is influenced by a number of criteria, such as the distances and capacities of the centers, the amount of costs or density settlement, etc. The most frequently used criterion is distance; the actual problems, however, are much more complex and all the necessary criteria shall be taken into consideration in order to achieve the optimal design. The objective of location problems is to find a place of service central points so that the value of the given function is minimal [7].

4 Streamlining the supply activities – case study

First of all, the values of graph vertices will be determined. The values will be defined based on the number and weight of shipments to be delivered to the vertices.

4.1 Selecting a graph set

Another step consists in dividing the graph into relevant subsets according to the criteria described above. We will choose sets of vertices coming from the branches in Hof, Brno, and Kladno. If the graph was divided according to the current lines and applied location problems, the current state would not be changed by streamlining. On the basis of current dispositions and capacities of the examined branch in České Budějovice (starting point), a new line could be taken into account. To this end, a set of graph vertices, which is partly an intersection of the set from Brno and Hof directions, will be selected. The capacity of the Austrian branch Hörsching is comparable to the capacity of Brno and Hof branches. The defined set will be represented by a network graph (see Figure 1A below). The broken line marks the route České Budějovice – Hörsching planned by the authors.

Figure 1

Selected (A) and modified (B) set for streamlining

The graph shows real edges (lines). The authors decided that the vertex Hof, which replaced a large number of graph vertices due to the fact that both from Hof in Germany and Hörsching in Austria have the same transport routes, will remain in the graph. This correction will not affect the results of this manuscript. The distances will be averaged and the value of the vertex will be quantified by summing the values of vertices replaced by vertex Hof. Since the branch České Budějovice is not deemed to be a graph vertex but only a source of shipments (starting supply point), it was removed from the graph. Figure 1B shows the graph modified by the authors, including its valued edges.

4.2 Determining values of examined set vertices

The basic data shall be used to compile a table of values of selected graph set vertices according to defined set of vertices. Table 1 shows the modified values of the selected set.

Table 1

Values of vertices of the selected graph set. Source: authors

Customer state code	Depot	Number of shipments (pcs)	Weight of shipment (kg)
AUT	Himberg	57	9,088
AUT	Hörsching	68	7,869
AUT	Graz	21	1,977
AUT	Innsbruck	11	2,244
DEU	Hof	268	24,027
DEU	Hof’	1,105	152,446
CZ	Brno	1,486	207,130

4.3 Minimum distance matrix

For the application of the location problem and calculation of transport work, it is imperative to identify individual values of the graph edges [23]. As aforementioned, the values will represent the individual real routes with the distances (in km) among the relevant branches according to the graph in Figure 1B (see Table 2 above).

Table 2

The minimum distance matrix of the selected set of vertices

km	Himberg	Hörsching	Graz	Innsbruck	Hof	Hof’	Brno
Himberg	0	204	198	468	609	985	140
Hörsching	204	0	229	281	405	781	327
Graz	198	229	0	510	634	1,010	338
Innsbruck	468	281	510	0	686	1,062	608
Hof	609	405	634	686	0	665	455
Hof’	985	781	1,010	1,062	665	0	1,108
Brno	140	327	338	608	455	1,108	0

4.4 Application of location problem

The very application of location problem in the manuscript defines the third service vertex in the graph network in Figure 1B. Two current service centers are situated at the vertices “HOF” and “BRNO”,which, in this case, indicates the existence of actual regular routes from the branch in České Budějovice. These are very important service vertices.

The quality of final solution is influenced by selected combination of service vertices. In this problem, two service vertices are specified. When selecting service vertices, it is inevitable to realize which vertices being indicated represent the right choice by their nature. An expert estimate is that the vertices with the highest values within the pickup system network being the right choice. In the following part, the authors shall try to confirm this expert estimate [11].

Table 1 indicates that, according to the authors, it is possible to place the third service vertex in the vertex “HIMBERG” based on the weight requirements, and, in the vertex “HÖRSCHING”, on the basis of the requirement for the number of shipments. The particular question arises why the “HOF”’ central point is not taken into consideration. The reason is that this vertex represents a subset of vertices, which means it cannot be a single service vertex. This vertex is included for the purposes of this paper, since in the real network, the given subset of vertices can be supplied from the vertex “HÖRSCHING”, which was taken into account in the text.

The location problem will be applied for four assignments. The first one will consist in the selection of the third service depot at the vertex “HIMBERG”, the second one will represent the selection of a service depot at the vertex “HÖRSCHING”. Both problems will be addressed with the values corresponding to the number and, subsequently, the weight of shipments.

4.4.1 Third service depot in vertex “HIMBERG” – criterion: number of shipments

A set of three service depots is given as follows:

D3pHI={HOF,BRNO,HIMBERG},

where: p – is an attribute of the set where the criterion is the number of shipments, HI – is an attribute of a set where the third service depot is HIMBERG,

and a set of vertices to be served N = {HOF’, HÖRSCHING, INNSBRUCK, GRAZ}; a value of transport work is to be calculated f(D3pHI).

Allocated catchment areas:

AHOFHI=HOF,HOF′ABRNOHI=BRNOAHIMBERGHI=HIMBERG,HÖRSCHING,INNSBRUCK,GRAZ}fD3pHI=fDHOFpHI+fDBRNOpHI+fDHIMBERGpHI

After substituting concrete values into the formula for calculating transport performance of a given subset, we get outcomes as follows:

f(DHOFpHI)=2⋆0⋆268+2⋆665⋆1,105=734,825 kmf(DBRNOpHI)=2⋆0⋆1,486=0 km

f(DHIMBERGpHI)=2⋆0⋆57 + 2⋆204⋆68 + 2⋆468⋆11+2⋆198⋆21=23,178 kmf(D3pHI)=734,825+0+23,178=758,003 km

The transport performance for the service vertices HOF, BRNO and HIMBERG when taking into account the criterion of the number of consignments is of 758,003 km.

4.4.2 Third service depot at the vertex “HIMBERG”– criterion: weight of shipments

A set of three service depots is given as follows:

D3mHI={HOF, BRNO, HIMBERG},

where: m – is an attribute of the set where the criterion is the weight of shipments, HI – is an attribute of a set where the third service depot is HIMBERG and a set of vertices to be served N = {HOF’, HÖRSCHING, INNSBRUCK, GRAZ}; the value of transport work is to be calculated f(D3mHI).

Allocated catchment areas:

AHOFHI={HOF, HOF'}ABRNOHI={BRNO}AHMBERGHI={HIMBERG, HÖRSCHING, INNSBRUCK,GRAZ}f(D3mHI)=f(DHOFmHI)+f(DBRNOmHI)+f(DHIMBERGmHI)

After substituting particular values into the formula for calculating transport performance of a given subset,we get results as follows:

f(DHOFmHI)=2⋆0⋆24,027+2⋆665⋆152,446=101,376,590kmf(DBRNOmHI)=2⋆0⋆207,130=0 kmf(DHIMBERGmHI)= 2⋆0⋆9,088 + 2⋆204⋆7,869 +2⋆468⋆2,244+2⋆198⋆1,977=3,046,914 kmf(D3mHI)=101,376,590+0+3,046,914=104,423,504km

The transport performance for the service vertices HOF, BRNO and HIMBERG when taking into account the criterion of the weight of consignments is of 104,423,504 km.

4.4.3 Third service depot at the vertex “HÖRSCHING” – criterion: number of shipments

A set of three service depots is given as follows:

D3pHO=HOF,BRNO,HÖRSCHING,

where: p – is an attribute of the set where the criterion is the number of shipments, HO – is an attribute of the set where the third service depot is HÖRSCHING and a set of vertices to be serviced N = {HOF’, HIMBERG, INNSBRUCK, GRAZ}; the value of transport work is to be calculated f(D3pHO).

Allocated catchment area:

AHOFHO=HOF,HOF'ABRNOHO=BRNO,HIMBERGAHRSCHINGHO=HÖRSCHING,INNSBRUCK,GRAZfD3pHO=fDHOFpHO+fDBRNOpHO+fDHRSCHINGpHO

After substituting concrete values into the formula for calculating transport performance of a given subset, we get outcomes as follows:

f(DHOFpHO)=2⋆0⋆268+2⋆665⋆1,105=734,825 kmf(DBRNOpHO)=2⋆0⋆1,486+2⋆140⋆57=7,980 kmf(DHRSCHINGpHO)=2⋆0⋆68+2⋆281⋆11+2⋆229⋆21=7,900kmf(D3pHO)=734,825+7,980+7,900=750,750 km

The transport performance for the service vertices HOF,BRNOand HÖRSCHING when taking into account the criterion of the number of consignments is of 750,705 km.

4.4.4 Third service depot in vertex “HÖRSCHING” – criterion: weight of shipments

A set of three service depots is given as follows:

D3mHO=HOF,BRNO,HÖRSCHING,

where: m – is an attribute of the set where the criterion is the weight of shipments, HO – is an attribute of the set where the third service depot is HÖRSCHING and a set of vertices to be served N = {HOF’, HIMBERG, INNSBRUCK, GRAZ}; the value of transport work is to be calculated f(D3mHO).

Allocated catchment areas:

AHOFHO=HOF,HOF'ABRNOHO=BRNO,HIMBERGAHRSCHINGHO=HÖRSCHING,INNSBRUCK,GRAZfD3mHO=fDHOFmHO+fDBRNOmHO+fDHRSCHINGmHO

After substituting particular values into the formula for calculating transport performance of a given subset, we get results as follows:

f(DHOFmHO)=2⋆0⋆24,027+2⋆665⋆152,446=101,376,590kmf(DBRNOmHO)=2⋆0⋆207,130+2⋆140⋆9,088=1,272,320 kmf(DHRSCHINGmHO) = 2⋆0⋆7,869 + 2⋆281⋆2,244 +2⋆229⋆1,977=1,083,297 kmf(D3mHO)=101,376,590 + 1,272,320 + 1,083,297 =103,732,207 km

The transport performance for the service vertices HOF,BRNO and HÖRSCHING when taking into account the criterion of the weight of consignments is of 103,732,207 km.

Figure 2B shows the routes being examined in the context of the location problems above. The graphs are modified for the visualization of a sub-graph of serviceability in the event of selecting a service center (newly generated route is represented by a broken line). Route ČB – HÖRSCHING is shown in Figure 2A, second route is ČB – HIMBERG.

Figure 2

Graphical comparison of the results of third service depot location problem

5 Evaluation of results achieved

The location problem was used to determine the transport performance of the branches Himberg and Hörsching given that these branches would serve as the third service vertex from the graph in Figure 1B while applying the criteria of the serviceability required; i.e. the number and weight of shipments [31]. The specific findings are summarized in Table 3.

Table 3

Outcome of the location problem

Criterion – number of shipments (pcs)	Graz (km)	Innsbruck (km)	Hof (km)	Hof’ (km)	Brno (km)	Transport performance (km)
Himberg	198	468	609	985	140	758,003
Hörsching	229	281	405	781	327	750,705


Criterion – weight of shipments (kg)	Graz (km)	Innsbruck (km)	Hof (km)	Hof’ (km)	Brno (km)	Transport performance (km)

Himberg	198	468	609	985	140	104,423,504
Hörsching	229	281	405	781	327	103,732,207

Table 3 compares the outputs of all tasks performed regarding the location problem. Furthermore, the table shows that when choosing the third service depot for the given graph in Figure 1B, the graph vertices (branches) will be supplied at lower transport performance for both criteria. The selection of the third service depot is described using the formula below [4]:

(3)f(Okg)=min{f(Dk)}

Where: f(Okg)is the (edge) optimal solution for deployment k of the depots in the networks when applying g criterion. For the two depots considered, the deployment is as follows (see below). if:

f(D3pHO)=750,705 km< f(D3pHI)=758,003 kmf(D3mHO)=103,732,207 km<f(D3mHI)=104,423,504 km

then:

f(O3p)=f(D3pHO)

f(O3m)=f(D3mHO)

The service depots were placed at the same location for both criteria, namely, in vertex Hörsching. The visual representation of the location problems solution is shown in Table 4 below.

Table 4

Visual representation of the application of location problem

Criterion – number of shipments (pcs)	Transport performance (km)
Himberg	758,003	→MIN→750,705
Hörsching	750,705


Criterion – weight of shipments (kg)	Transport performance (km)

Himberg	104,423,504	→MIN→103,732,207
Hörsching	103,732,207

6 Conclusion

Based on examining the branch in České Budějovice and by application of the mathematical calculations using the location problem, the authors of the paper presented propose the introduction of a new supply (serviceability) route that will extend the existing networks of the pick-up system. The route České Budějovice (CR) – Hörsching (Austria) shall ensure the optimal serviceability of the catchment area of the depot in České Budějovice. Implementing the new route shall also positively affect the current utilization of the present České Budějovice routes and thus the entire supply system. The new route should relieve the existing load of the routes and extend the pick-up system network. Adding new supply route (graph edge) between the vertices ČB – HÖRSCHING enabled to omit other routes from the graph, more precisely the routes that are not imperative for ensuring the serviceability of the individual branches. The proposed solution would result in higher efficiency of the depot in the city of České Budějovice within the pick-up system as well as a more diversified distribution of shipments into a complex network of transport-logistics processes, which means a lower load of the network.

Application of the location problem on the investigated section actually enabled to determine the direction of the new route from České Budějovice (Czech Republic) to Hörsching (Austria). By adding a new edge into the selected graph, it was possible to optimize the performance of transport work and the number of transshipments compared to the original state. Both optimized quantities were assessed and confirmed for both criteria, namely for the number and weight of shipments (indicated in pieces and kilograms, respectively) that had to be delivered in the monitored period. Specifically, as far as service depots HOF, BRNO and HÖRSCHING are concerned when taking into account criterions of the number of consignments (in pcs) and the weight of consignments (in kg), the total transport performances of 750,705 pcs*km and 103,732,207 kg*km, respectively were quantified. And, in regard to service vertices HOF, BRNO and HIMBERG when taking into consideration the same criterions, the transport performances of 758,003 pcs*km and 104,423,504 kg*km, respectively were calculated. Hence, the quantification procedure has confirmed that it is reasonable to place the additional service point into the location of Hörsching.

As for the future efforts in the subject being discussed, it is recommended to address the actual rationalization of the current state, which could result in the optimization of the examined technical indicators, as well as the economic (financial) impact of the measures proposed.

Acknowledgement

This manuscript was supported within solving the research project entitled “Autonomous mobility in the context of regional development LTC19009” of the INTER-EXCELLENCE program, the VES 19 INTER-COST subprogram.

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Received: 2020-04-24

Accepted: 2020-05-07

Published Online: 2020-09-02

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/eng-2020-0089

Keywords for this article

Logistics; forwarding; graph theory; location problem; pick-up system; supply system

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BY 4.0