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The design of a logistics network, including sets of suppliers, customers, hubs, and capacities of vehicles traveling between them. Constants such as distance, capacity, and request are provided, along with the use of sets p1, p2, and p3 for vehicles traveling among real suppliers and hubs, supplier hubs and customer hubs, and customer hubs and real customers, respectively. The document also mentions the cost of building hubs and the cost of transportation per km for various vehicles.
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7.2 Possibilities to continue in the future on basis of this dissertation thesis ....................................................................................................................................... 8 CONCLUSION........................................................................................................ 39
LIST of the publication done by the student and further activities in the field of the doctoral STUDIES................. 44 Curriculum vitae................................................................................................ 46 Curriculum vitae................................................................................................ 47
This dissertation thesis with title „Mathematical modelling for hub and spoke technology“ has analyzed the possibilities of the mathematical modelling methods for hub and spoke technology.
In the chapter 1 of the dissertation thesis we describe the hub and spoke technology in the distribution logistics. In the chapter 2 of the dissertation thesis we formulate 4 targets:
In the chapter 3 of the dissertation thesis we have started analysing the state of art about the mathematical modelling for hub and spoke technology.
Targets of the dissertation thesis were got in the chapters 4,6 and 7 of the dissertation thesis. In the proposed models two criteria were used:
The functionality of the proposed models was validated in the model examples.
In the chapter 8 of the dissertation thesis we analyzed the benefits of this dissertation thesis for the science and for the practise. The dissertation thesis includes 5 new models that differ from the previous ones in fact these models were completed by:
cost of transportation for the vehicle per 1 km (both ways) [CZK/km]
cost of transportation for the vehicle per 1 km (both ways) [CZK/km]
m number of real suppliers [ - ] n number of real customers [ - ] p number of hubs built up [ - ] binary variable that defines the existence of the customer hub k (for both ways); it has the following values: or. If it means that place k is a hub otherwise if it means that place k is not a hub [ - ] binary variable that defines the existence of the supplier hub (for both ways); it has the following values: or. If it means that place is a hub otherwise if it means that place is not a hub [ - ] number of trips moving between the real supplier and the supplier hub [ - ]
number of trips for the vehicle travelling between real supplier and supplier hubfor the first way [ - ] number of trips for the vehicle travelling between customer hub and real customer for the second way [ - ] number of trips for the vehicle with capacity travelling between real supplier and supplier hubfor the first way [ - ] number of trips for the vehicle with capacity travelling between customer hub and real customer for the second way [ - ] number of trips for the vehicle travelling between the customer hub k and the real customer for the first way [ - ] number of trips for the vehicle travelling between the real supplier and the supplier hub k for the second way [ - ] number of trips for the vehicle with capacity travelling between the customer hub k and the real customer for the first way [ - ] number of trips for the vehicle with capacity travelling between the real supplier and the supplier hub k for the second way [ - ] binary variable that represents the connection between the supplier huband the customer hub k for the first way; it can have these values: or , if the supplier hubis connected to the customer hub k ; for the supplier hubis not connected to the customer hub k [ - ] number of trips for the vehicle with capacitytravelling between the supplier hub and the customer hub k for the first way [ - ] number of trips for the vehicle with capacitytravelling between the supplier hub k and the customer hub for the second way [ - ] flow of goods defined from the real supplier to supplier hub [unit of goods]
flow of goods defined from the supplier to supplier hub for the first way [unit of goods]
flow of goods defined between customer hub and real customer for the second way [unit of goods] flow of goods defined between the customer hub k and the real customer [unit of goods]
flow of goods defined between the real customer and customer hub k for the second way [unit of goods] flow of goods defined between the customer hub k and the real customer for the first way [unit of goods]
In this chapter, we want to describe the hub and spoke technology in the distribution logistics. According to the distribution logistics the flow of goods can be organized with many different ways by us, one of them is the hub and spoke technology, which is a logistics technology where every single transport among the customers will be organized through a logistics center (it can not be the direct transportation to customers). The hub and spoke technology is commonly used in industry, in transport and the telecommunications.
We can see the main principle of the hub and spoke technology in figure 1.1.
Fig 1.: Hub and spoke principle
Let’s go to describe the hub and spoke technology according to the figure 1.1.
When we propose the hub and spoke technology, we must define the area in which we operate and in this area, there are customers who want to have some transportation among themselves. The flow of goods is not so big, so direct transportation among customers is not efficient. The basic idea of this technology is the maximation of the transportation utility, the using of ecologically advantageous transportation vehicles (train transportation and waterway transportation) on long distances. In order to be able to use possible advantages of this technology, it is important to concentrate the most of the flow of goods to the certain places. The transportation in the first and the last phase can be realized thanks to the road transportation.
This technology is also ecological and it has a very competitive price and these are some of the advantages of this technology. In hub and spoke technology we have that the cost for single unit transported will be smaller if these units are transported in longer distances inside vehicle with higher capacity.
In the hub and spoke technology, we usually want to optimize two factors:
After describing this technology we are going to explain the main principles and factors of this technology. The main principles of hub and spoke technology are:
(in the hubs, in the transship ports, in the airports).
Others logistics technologies are described in the literature [11] and [15].
After introducing the hub and spoke technology in this dissertation thesis we are going to introduce the state of art of the hub and spoke technology, so we analyze the classic method of solution of the models.
The second group describes the multiple-assignment hub and spoke technology models and an example of this configuration will be shown in the figure 3.2.
Fig 3.: Multiple-assignment hub and spoke technology [2]
For the multiple-assignment hub and spoke technology there is a characteristic at least one customer is assigned to more than one hub. For example the following models are described in the bibliography:
problems [6],
models [9]. From the figures 3.1 and 3.2, we can see that single-assignment hub and spoke technology is the special event of the multiple-assignment network.
In the dissertation thesis we are going to propose models with single-assignment hub and ske technology.
The fruition of the targets will be executed with the help of linear programming methods. The linear programming methods will be used because we can find the optimal solution. The known models are sometimes not useful for solving our targets, they do not consider to divide the model into parts in order to build up a customer hub and a supplier hub. So in order to solve these problems we have to divide the models into parts. With these models the hub can be build up in such places that they will minimize the total covered distance. The first target contains the requests for the creation of a model through distances and not through cost, because we do not know the cost, but we know the distance. We know the capacity of the vehicles, the customer’s requests and the supplier’s capacities. In all models we will build up one supplier hub and customer hub. We build up the supplier hub in one of real suppliers and we build up the customer hub in one of real customers. In the proposed models, we are going to decide where to build hubs and minimize the covered distances between suppliers, hubs and customers. In chapter 4 we will present 5 mathematical models:
We have a set of real suppliers and a set of real customers. We know the distances among all real suppliers and we know the distances among all real customers and we know the distances among suppliers and customers. We know the capacity of the suppliers and the request of the customers, the capacities of the vehicles moving only among the real suppliers, only among the real customers and among real suppliers and real customers.
In the model we will use the following constants: is the capacity of goods for the real supplier ,
is the request of goods for the real customer ,
is the distance between the real supplier and the supplier hub ,
is the distance between the supplier huband the customer hub k ,
is the distance between the customer hub k and the real customer ,
is the capacity of the vehicle travelling among real suppliers and supplier hubs,
is the capacity of the vehicle travelling among customer hubs and real customers,
T it is a prohibitive parameter we decide its value as: T=100000.
We think that among set of supplier hubs and set of customer hubs only one trip will be.
In the model we will use the following variables: is the number of trips for the vehicle travelling between real supplier and supplier hub.
is a binary variable that represents the connection between the supplier huband the customer hub k; it can have these values: or. If the supplier hub is connected to the customer hub k ; for the supplier hub is not connected to the customer hub k. is the number of trips for the vehicle travelling between the customer hub k and the real customer.
is the flow of goods defined from the real supplier to supplier hub.
is the flow of goods defined between the customer hub k and the real customer.
is a binary variable that defines the existence of the customer hub k ; it has the following values: or. If it means that place k is a customer hub otherwise if it means that place k is not a customer hub. is a binary variable that defines the existence of the supplier hub ; it has the following values: or. If it means that place is a supplier hub otherwise if it means that place is not a supplier hub.
For the situation: (4.1.1)
we have the following model:
For the situation:
(4.1.21)
The model will change in the group of constraints (4.1.3). This group of constraints will have the form:
for (4.1.22)
the group of constraints 4.1.22 ensures that we do not exceed the capacity of the vehicle.
For the situation: < (4.1.23)
This model will change in the group of constraints (4.1.4). This group of constraints will have the form:
for (4.1.24)
The group of constraints (4.1.24) ensures that we consume maximally the request of the customer.
In this chapter we consider the same situation like in the chapter 4.1 but the transportation will be organized in two ways. The first way will be from the real suppliers to the real customers and the second way will be back. This means that for the second way the real customers will be the places in which the flow of goods will begin. For the two direction ways we want to organize these transportations like this: at first we want that all goods will be transported to one place – the supplier hub. Then we want that all the goods will be transported to one place – to the customer hub and from this place will be transported to the real customers.
The flow of goods will be shown in the figure 4.7.
Fig 4.: Two ways flow of goods
Our network has the following situation: we have the distance cij among the real suppliers and the supplier hub (for the first way) and among customer hub and real customers (for the second way), djk between supplier hub and the customer hub (for the first way) and between the supplier hub and the customer hub (for the second way), and ekl among customer hub and the real customers (for the first way) among the real suppliers and the supplier hub (for the second way), as we can see in the figure 4.8.
Fig 4.: The network analyzed
We want that the supplier hub and customer hub will be the same for both ways.
By analogy like for the previous problem the capacities of the vehicles are different for:
In the model we will use the following constants: is the capacity of goods for the real supplierit is the request of goods for the real customer for the first way flow of goods, for the first way flow of goods,
it is the capacity of goods for the real supplier for the second way flow of goods,
it is the request of goods for the real customer for the second way flow of goods,
is the distance between the real supplier and the supplier hub for the first way,
is the distance between the customer hub and real customer for the second way,
is the distance between the supplier hub and the customer hub for the first way,
is the distance between the supplier hub and the customer hubfor the second way,
is the distance between the customer hub and the real customer for the first way,
is the distance between the real supplier and the supplier hub for the second way,
is the capacity of the vehicle travelling between real suppliers and supplier hubs,
is the capacity of the vehicle travelling between customer hubs and real customers,
T is a prohibitive parameter we decide its value as: T=100000.
In the model we will use the following variables: is the number of trips for the vehicle travelling between real supplier and supplier hubfor the first way. is the number of trips for the vehicle travelling between customer hub and real customer for the second way. is a binary variable that represents the connection between the supplier huband the customer hub for the first way; it can have these values: or , if the supplier hubis connected to the customer hub ; for the supplier hubis not connected to the customer hub. is the number of trips for the vehicle travelling between the customer hub and the real customer for the first way. is the number of trips for the vehicle travelling between the real supplier and the supplier hub for the second way. is the flow of goods defined from the supplier to supplier hub for the first way.
is the flow of goods defined between the real customer and customer hub for the second way.
is the flow of goods defined between the customer hub and the real customer for the first way.
is the flow of goods defined between customer hub and real customer for the second way.
is a binary variable that defines the existence of the customer hub (for both ways); it has the following values: or. If it means that place is a hub otherwise if it means that place is not a hub. is a binary variable that defines the existence of the supplier hub (for both ways); it has the following values: or. If it means that place is a hub otherwise if it means that place is not a hub.
For the situation:
for (4.2.29)
for (4.2.30)
for (4.2.31)
Comments: (4.2.3) is the optimization criterion that represents the total covered distance for both ways flow of goods. We want to minimize the optimization criterion value.
The group of constraints: (4.2.4) we exhaust the capacity of all real suppliers for the first way, (4.2.5) the requests of the real all customers are satisfied for the first way, (4.2.6) we exhaust the capacity of the all real suppliers for the second way, (4.2.7) the request of the all real customers are satisfied for the second way, (4.2.8) calculates the number of trips between the real supplier and supplier hub for the first way, (4.2.9) calculates the number of trips between the customer hub to the real customer for the first way, (4.2.10) calculates the number of trips between the customer hub to the real customer for the second way, (4.2.11) calculates the number of trips between the real supplier to the supplier hub for the second way, (4.2.12) describes that we build just one hub in the set for both ways flow of goods, (4.2.13) describes that we build just one hub in the set K for both ways flow of goods, (4.2.14) describes that just one connection is possible between the hub in the set J and the hub in the set K , (4.2.15) describes that if we go at least once to then in the hub will be built for the first way, (4.2.16) describes that if we departure from then in the hub will be built for the second way, (4.2.17) describes that if we go at least once to then in the hub will be built for the first way, (4.2.18) describes that if we departure from then in the hub will be built for the second way, (4.2.19) describes that after we built up a supplier hub in there will be trips departing from the hub, (4.2.20) describes that if there are incoming trips to the customer hub in the hub K will be built. From (4.2.21) to (4.2.31) we have the obligatory constraints.
We assume the same problem like the chapter 4.1. In the chapter 4.1 we knew the type of vehicles that realized the transportation among real suppliers and supplier hub, between the supplier hub and customer hub and among the customer hub and real customers. In the model number 3 we will not know the concrete type of vehicles. In the model it will be decided where to build up the supplier hub and customer hub and the fleet selection too. Let’s go to present the mathematical model in which for example we can rent different type of vehicle.
The symbols I, J, K, L have the same meaning like the chapter 4.1. We add the following symbols:
P1P2 set of vehicles that travel among real suppliers and possible supplier hub locations,set of vehicles that travel among possible supplier hub locations and possible customer hub locations, P3 set of vehicles that travel among possible customer hub locations and real customers,
In the model we will use the following constants:
is the capacity of goods for the real supplier ,it is the request of goods for the real customer ,
is the distance between the real supplier and the supplier hub ,
is the distance between the supplier hub and the customer hub k,
is the distance between the customer hub kand the real customer ,
is the capacity of the vehicle travelling between real suppliers and supplier hub,
is the capacity of the vehicle travelling between supplier hub and customer hub,
is the capacity of the vehicletravelling between customer hub and real customers,
T is a prohibitive parameter we decide its value as: T=100000.
In the model we will use the following variables: is the number of trips for the vehicle with capacity travelling between real supplier and supplier hub. is the number of trips for the vehicle with capacitytravelling between the supplier hub and the customer hub k for the first way. is the number of trips for the vehicle with capacity travelling between the customer hub k and the real customer l. is the flow of goods defined from the supplier to supplier hub.
is the flow of goods defined between the customer hub k and the real customer.
is a binary variable that defines the existence of the customer hub k (for both ways); it has the following values: or. If it means that place k is a hub otherwise if it means that place k is not a hub. is a binary variable that defines the existence of the supplier hub (for both ways); it has the following values: or. If it means that place is a hub otherwise if it means that place is not a hub.
Let’s now define the balanced system thanks to the following expression:
we have the following model: (4.3.2)
subject to:
for (4.3.3)
for (4.3.4)
for (4.3.5)
for (4.3.6)
hub and customer hub and the fleet selection too. Let’s go to present the mathematical model in which for example we can rent different type of vehicle such as in the chapter 4.2.
The symbols I, J, K, L have the same meaning like the chapter 4.2. We add the following symbols:
P1 set of vehicles that travel among real suppliers and possible supplier hub locations and back for the second way,
P2 set of vehicles that travel among possible real supplier hub locations and possible customer hub locations and back for the second way,
P3 set of vehicles that travel among possible customer hub locations and real customers and back for the second way.
In the model we will use the following constants: is the capacity of goods for the real supplier for the first way flow of goods,is the request of goods for the real customer for the first way flow of goods,
is the capacity of goods for the real supplier for the second way flow of goods,
is the request of goods for the real customer for the second way flow of goods,
is the distance between the real supplier and the supplier hub for the first way,
is the distance between the customer hub and the real customer for the second way,
is the distance between the supplier hub and the customer hub kfor the first way,
is the distance between the supplier hub k and the customer hubfor the second way,
is the distance between the customer hub k and the real customer for the first way,
is the distance between the real supplier and the supplier hub k for the second way,
is the capacity of the vehicle travelling between real suppliers and supplier hub for the first way and between the customer hub and real customer for both ways, is the capacity of the vehicle travelling between supplier hub and customer hub for both ways,
is the capacity of the vehicle travelling between customer hub and real customers for the first way and between real supplier and the supplier hub for both ways,
T is a prohibitive parameter we decide its value as: T=100000.
In the model we will use the following variables: is the number of trips for the vehicle with capacitytravelling between real supplier and supplier hub for the first way. is the number of trips for the vehicle with capacitytravelling between customer hub and real customer for the second way. is the number of trips for the vehicle with capacitytravelling between the supplier hub and the customer hub k for the first way. is the number of trips for the vehicle with capacitytravelling between the supplier hub k and the customer hub for the second way. is the number of trips for the vehicle with capacitytravelling between the customer hub k and the real customer for the first way. is the number of trips for the vehicle with capacitytravelling between the real supplier and the supplier hub k for the second way. is the flow of goods defined from the supplier to supplier hub for the first way.
is the flow of goods defined between the real customer and customer hub k for the second way.
is the flow of goods defined between the customer hub k and the real customer for the first way.
is the flow of goods defined between customer hub and real customer for the second way.
is a binary variable that defines the existence of the customer hub k (for both ways); it has the following values: or. If it means that place k is a hub otherwise if it means that place k is not a hub. is a binary variable that defines the existence of the supplier hub (for both ways); it has the following values: or. If it means that place is a hub otherwise if it means that place is not a hub.
We have a balanced transportation problem; it means that the sum of all the requests of the customers is equal to the sum of all the capacity of goods of the suppliers.
(4.4.1)
(4.4.2)
Let’s go to present the mathematical model:
subject to:
for (4.4.4)
for (4.4.5)
for (4.4.6)
for (4.4.7) J
for , (4.4.8)
for (4.4.9)
for , (4.4.10)
for (4.4.11)
(4.4.12)
(4.4.13)
(4.4.14)
for , (4.4.15)
for , (4.4.16)
for (4.4.17)
for (4.4.18)
