SELECTED ASPECTS OF MULTI-CRITERIA DECISION SUPPORT IN TRANSPORT SERVICE IN URBAN AREAS. Emilian SZCZEPAŃSKI

Transcription

1 SELECTED ASPECTS OF MULTI-CRITERIA DECISION SUPPORT IN TRANSPORT SERVICE IN URBAN AREAS Emilian SZCZEPAŃSKI Warsaw University of Technology, Faculty of Transport, Department of Logistic and Transport Systems, Koszykowa 75, Warsaw, Poland; Abstract The article presents the construction of mathematical model for the urban system including multicriteria optimization problem. The idea of multistage distribution systems based on the Cargo Consolidation Centers and transshipment points ( s) was presented. In the paper presented multi-criteria decision support problem. The various elements of the network in urban areas, which served the formulation of a mathematical model were analyzed. The article contains the concept of a model of urban system including its formal record. Vector objective function was formulated. Keywords: multi criteria optimization, in urban areas, model of system 1. INTRODUCTION The article deals with the study of constructing a mathematical model taking into account the multi-criteria optimization problem. It is a very complex process and requires the determination of the appropriate limits and criteria, as well as review of available methods for its solution. Transportation of goods in urban areas on the one hand is a major factor in the development of most economic and social activities taking place in urban areas. This is connected with the stores and workplaces supply, as well as waste management. On the other hand, freight is a major factor hampering and disturbing social life and urban residents [7]. It is important to find solutions that reduce freight traffic in urban areas. One such solution is widely understood city logistics. It aims to reduce the negative effects associated with the of goods in urban areas. At the same time it supports the economic and social development of cities. In logistics, it is very important to meet customer requirements [2]. Distribution of goods and service customers should be in accordance with the principle of 7R (right item in the right quantity at the right time at the right place for the right price in the right condition to the right customer). 2. MULTISTAGE DISTRIBUTION SYSTEM IDEA Requirements related to principle 7R in urban agglomeration service require action to improve. In the case of large cities such action can be applied multi-level distribution system based on the Cargo Consolidation Centers and transhipment s. provides services such as warehousing, picking and of goods. Charges brought to after undergoing separation and picking and loading the vehicle delivered to s. There also is separation of the load and the loading of the goods on the vehicle with a lower capacity [1]. Therefore, multi-level distribution systems in urban areas are constructed Cargo Consolidation Centres () located on the outskirts of cities and urban transshipment points (). From the point of view of the process of distribution can be distinguished: direct, indirect and mixed distribution system [2],[3]. Examples of such systems are shown in Fig. 1.

2 Direct Distribution (lowtonnage vehicles) Distribution (low-tonnage vehicles) ) Distribution (low-tonnage vehicles) ) , Jeseník, Czech Republic, EU Otoczenie a) b) c) d) Fig. 1. Graphic illustration of services: a) direct system, b) indirect single system, c) the indirect two-stage system, d) mixed system Source: own work base on [3]. 3. MULTIOBJECTIVE OPTIMIZATIONS Multiobjective optimization problem relates to the decision situation, in which there is more than one criterion sought a solution acceptable from the standpoint of each of them. The ideal solution is one that is best for each objective function [4]. Possible solutions are classified into the space dominated and non dominated solutions (pareto optimal). The presentation of the multi-criteria optimization solutions to selected areas dominated and non dominated solutions shown in Fig. 2. Fig. 2. Sample presentation of the solutions in multiobjective optimization Source: own work base on [5].

3 The solution in the sense of Pareto optimal solution is belonging to the feasible solutions and to improve the value for which one of the criteria is not without worsening the value of other [8]. There are many methods of finding solutions in multi-criteria optimization tasks such methods may include: Weighting Method or Constraint Method. These methods are classified as classical approaches. With the development of information technology are often used methods based on evolutionary algorithms such as VEGA (Vector Evaluated Genetic Algorithm), NSGA (The Nondominated Sorting Genetic Algorithm), SPEA (The Strength Pareto Evolutionary Algorithm) [8]. 4. THE STRUCTURE OF TRANSPORT NETWORK The network is composed of the major elements affecting the operation of the entire system. These elements include not only senders, customers or intermediate points, but also vehicles, drivers and road connections. Thus, for the proper representation of reality in the form of the model is also characterized by its individual components. Graph used to present the structure of the study area ation network (formula 1) [4]: where: a set of graph vertices, { }; a set of graph arcs. (1) We assume that defines a set of arches and is a relation defined on the Cartesian product. The arc ( ) is understood as a transition from node to node, represented by the formula 2: ; {( ) ( ) } (2) Path in the graph from node a to b will be sequence recorded as a (formula 3): ( ) if (3) When all the nodes are different, it is a simple path. However, the set of all paths from node a to node b marked as (formula 4): { ( ) } Cyclic path in graph is a path such that for. (4) For the study, it was assumed that the set of transforms into a set of {0,1,2} tj. (wzór 5): specified function, which elements of this collection { } (5) if ( ) then -th node is interpreted as number of, but when ( ) then -th node is interpreted as number of transshipment, when ( ) then -th node is interpreted as number of customers located within the city. This allows to define the vertices in the network (major elements): set of numbers (formula 6): where: { ( ) } (6) number of -th sender; sum of senders. should be associated with the sender, each of them has a production capacity that is able to fulfill the demand of a certain size. set of s numbers (formula 7): where: { ( ) } (7) number of -th transshipment point; sum of transshipment points. Transshipment points are points where the load is split into individual routes and loading of vehicles with lower load capacity. At this point it is an important time points, the duration of the activity related to the handling of cargo, cargo allocation for each route. set of numbers of customers (formula 8): { ( ) } (8)

4 where: number of -th customers; sum of customers. Customers are also a very important part of the network. Time window, the unloading time determine ation plan. Operating time of the vehicle at a point depends on many factors, even if the available load equipment. In arranging ation plan is an important ation lead time. Going through every element of the network requires some time. Total travel time, operating at different customers cannot exceed the time allowed for the driver. For the study, defined a set of vehicle types (formula 9), each vehicle type is characterized by a load capacity: { } (9) where: number of -th type of vehicle; sum of vehicle types. 5. MATHEMATICAL MODEL CONCEPT AND FORMALL RECORD Optimization task formulation is a complex process and requires consideration of several factors as mentioned above. The task that is formulated is a modified multiple traveling salesmen problem. The issue of a salesman in simple terms involves the determination of the way between all the customers on the route to its total length is as short as possible. Important condition is that the salesman starting the travel at some point must finish in the same place. Such an approach requires finding a Hamiltonian cycle in the graph. Hamiltonian cycle is a cycle that contains all vertices exactly once. The modification of the many multiple traveling salesmen problem, there are additional constraints of vehicle capacity or time windows in customers. This task is often referred to as VRP (Vehicle Routing Problem). The problem undertaken additional difficulty is due to the optimization of a number of objectives to be achieved. The problem is known as the MVRP (Multicriteria Vehicle Routing Problem) [6]. In order to provide a mathematical model of multi-level distribution system to serve customers located in urban areas is important to specify the input data, decision variables, constraints, and function criteria. In article formulated optimization tasks with two functions criterion. The first function relates to minimize distribution costs, presented it as a formula 10: ( ) (10) ( ) [ ( )] [ ( )] { ( [( ( ) ( ) ) ( )] ( ))} In formulating the function G1 focuses on ation costs resulting from the distance of various points on the route. In addition to the cost of a Salesman also includes the cost of supplies from the s to s. This feature also includes penalties for delays in customer service, as well as penalties for not handle them. The first case occurs when the salesman came out of a time window. Acceptable is when driver arriving early and waiting to be unloaded, but it is a waste of time, which in the case of global optimization can be justified. The second situation is when the point of view of global optimum decision is made to not handling the customer. The second criterion concerns on execution time, which is also committed to a minimum. This function is shown as a formula 11: ( ) ( ( ) ) [( ( ) ( ) )] (11)

5 The G2 function included loading time in transhipment s and unloading times in individual customers. This feature also includes travel times between individual customers, but does not include a component for the time between and. The meaning of the symbols used in G1 and G2 functions are described below. For such a optimization task it is necesary to: specify the following data: graph of the network structure; [ { } ] customer demand vector; [ ] distance matrix between n-th and h-th ; [ ] distance matrix between h-th and o-th customers; [ ] distance matrix between o-th and o -th customers [ ] = [ ] transposed matrix distance between o -th customers and h-th ; [ ] jouney time matrix between n-th and h-th ; [ ] jouney time matrix between h-th and o-th customers; [ ] jouney time matrix between o-th and o -th customers; [ ] = [ ] transposed matrix jouney time between o -th customers and h-th ; loading time of one unit load on the vehicle in ; ( ) - unloading time of one unit load from the vehicle in - th customers; ( ) - the beginning of the time window in - th customers; ( ) - the end of the time window u - th customers; - driver's daily working time; ( ) fixed cost of route from by - th vehicle type; fixed cost of driver working; c 1 (n,h) cost of cargo ation to a 1 km distance between and ; c 2 ( ) cost of 1 km travel by - th vehicle type; c 3 ( ) penalty for delay in service c 4 ( ) the penalty for not handle - th customers; - th customers; q( ) capacity - th vehicle type; { } - set of drivers and number of drivers (determined in the optimization process); [ ] - set of routes and number of routes (determined in the optimization process); set value of decision variables: - moment of the beginning of service (unloading) -th customers in -th route from -th by -th driver by -th vehicle type; - moment of the end of service (unloading) -th customers in -th route from -th by -th driver by -th vehicle type; volume of cargo between -th and -th ; volume of cargo between -th and -th customers in -th route by -th driver by -th vehicle type; volume of cargo between -th customers and -th customers in -th routeby -th driver by -th vehicle type;

6 delivery volumes to -th customers in -th route by -th driver by -th vehicle type; binary variables determining the existence of a connection between -th customers and -th, in -th route by -th driver by -th vehicle type; binary variables defining the handle or not handle of -th customers; binary variables defining the service outside the time window -th customers in -th route; with restrictions: demand, submitted by -th customers must be satisfied, volume of supply coming from to must be equal to volume of outgoing supply from, volume of cargo outgoing from must be equal to demand of customers, capacity of v-th vehicle type in -th route cannot be exceeded, vehicle leaving -th must return to it, driver's working time cannot be exceeded, each customer can be visited only once during -th route, so that the vector objective function reached a minimum (formula 12): ( ) [ ( ) ] (12) ( ) 6. CONCLUSION Construction of multi-criteria models of ation systems in multistage distribution systems is extensive. The paper presents some aspects of the problem, which occurs in such distribution systems. Issues included time windows, different vehicle capacity and multistage delivery. Optimization takes place with respect to two criteria - the time and cost of. Presented restrictions and decision variables Przedstawione w pracy ograniczenia are not all to consider when constructing a model. They are exemplary and actual model contains many more. The article does not provide mathematical record of all the elements of a model and omitted some of the elements. This is due to limitations in paper spaciousness. ACKNOWLEDGEMENTS This work is carried under research project (dean grant) Model of multi-criteria decision support in service in urban areas. LITERATURE [1] Ambroziak T. Jacyna M. Wasiak M.: The Logistic Services in a hierarchical distribution System, Transport Science and Technology. ELSEVIER. Chapter 30. ISBN-13: (ISBN-10: ) DEC [2] Bluszcz M., Jacyna M.: Koncepcja modelu obsługi logistycznej miasta, Logistyka 4/2010 [3] Jacyna M.: The role of Cargo Consolidation Center in urban logistic system, Urban Transport XII, WIT Press 2011 [4] Jacyna M.: Modelowanie i ocena systemów owych, Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa 2009 [5] Michalkiewicz Z.: Algorytmy genetyczne + struktury danych = programy ewolucyjne, WNT, Warszawa 2003 [6] Michlowcz E.: Rozwiązywanie problemów dostaw w systemach dystrybucji, Logistyka 4/2012 [7] Zomer G.R.: Optimisitaion of Urban Freight Systems by strategic co-operations, materiały konferencyjne LIFE- CEDM, Lucca 2008 [8] Zitzler E.: Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications, Zürich 1999