- Email: [email protected]
Urban Freight Fleet Composition Problem
Proceedings,16th IFAC Symposium on Information Control Problems in Manufacturing Proceedings,16th IFAC Symposium on Information Control Problems in Manufacturing Bergamo, Italy, June 11-13, 2018 Proceedings,16th IFAC Symposium on Available online at www.sciencedirect.com Information Control Problems in Manufacturing Bergamo, Italy, June 11-13, 2018 Information Control Problems in Manufacturing Proceedings,16th IFAC Symposium on Bergamo, Italy, June 11-13, 2018 Bergamo, Italy, JuneProblems 11-13, 2018 Information Control in Manufacturing Bergamo, Italy, June 11-13, 2018
ScienceDirect
PapersOnLine 51-11 (2018) 582–587 UrbanIFAC Freight Fleet Composition Problem Urban Freight Fleet Composition Problem Urban Freight Fleet Composition Problem Urban Freight Composition Problem Roberto Pinto*, Fleet Alexandra Lagorio*, Ruggero Golini* Urban Freight Fleet Composition Problem Roberto Pinto*, Alexandra Lagorio*, Ruggero Golini*
Roberto Roberto Pinto*, Pinto*, Alexandra Alexandra Lagorio*, Lagorio*, Ruggero Ruggero Golini* Golini* * Department of Management, Roberto Information and Industrial Engineering, University of Bergamo, via G. Marconi, 24044, Pinto*, Alexandra Lagorio*, Ruggero Golini* * Department of Management, Information and Industrial Engineering, University of Bergamo, via G. Marconi, 24044, Dalmine (BG) – Italy (e-mail: [email protected], [email protected], [email protected]) ** Department of Management, Information and Industrial Engineering, University via Dalmine (BG) – Italy (e-mail: [email protected], [email protected], [email protected]) Department of Management, Information and Industrial Engineering, University of of Bergamo, Bergamo, via G. G. Marconi, Marconi, 24044, 24044, Dalmine (BG) – Italy (e-mail: [email protected], [email protected], [email protected]) * Department of Management, Information and Industrial Engineering, University of Bergamo, via G. Marconi, 24044, Dalmine (BG) – Italy (e-mail: [email protected], [email protected], [email protected]) Dalmine (BG) – Italy (e-mail: [email protected]) Abstract: The Urban [email protected], Fleet Composition (UFFC) [email protected], problem concerns the definition of the optimal Abstract: The Urban Freight FleetofComposition (UFFC) problemtoconcerns the definition of the delivery optimal mix in terms of types and number vehicles (fleet composition) serve the demand for parcel Abstract: The Urban Freight Fleet Composition (UFFC) problem concerns the definition of the optimal mix in terms of types and number of vehicles (fleet composition) to serve the demand for parcel delivery Abstract: The Urban Freight Fleet Composition (UFFC) problem concerns the definition of the optimal in aninurban area. Urban areas can of be vehicles subject to access restrictionstothat reduce the possibility to enter the mix terms of types and number (fleet composition) serve the demand for delivery Abstract: The Freight Fleet (UFFC) problem concerns the definition of thethe optimal in aninurban area. Urban areas cantime beComposition subject to access restrictions reduce the possibility to enter the mix terms ofUrban types and number of vehicles (fleet composition) tothat serve thevehicle demand for parcel parcel delivery area. In this paper, we consider access restriction related to specific types (i.e. timein an area. Urban areas can be subject to access restrictions reduce the enter the mix of types and number of (fleet composition) serve thevehicle demand for parcel delivery area. Interms this paper, we consider access restriction related totothat specific types (i.e.to timein aninurban urban area. Urban areas cantime be subject restrictions that reduce the possibility possibility to the enter the windows during which the access to vehicles the areato ofaccess delivery is allowed depends upon the characteristics of area. In this paper, we consider time access restriction related to specific vehicle types (i.e. the timein an urban area. Urban areas can be subject to access restrictions that reduce the possibility to enter the windows during which the access to the area of delivery is allowed depends upon the characteristics of area. In this paper, we consider time access restriction related to specific vehicle types (i.e. the timethe vehicles). Thus, we aim to discuss the impact of these restrictions on the fleet composition decision. windows during which access to the of is allowed depends upon the characteristics of area. In end, this paper, we the consider time access restriction related to model, specific vehicle types timethe vehicles). Thus, aim to discuss thearea impact of these on the fleet composition decision. windows during which the to tactical the area of delivery delivery isrestrictions allowed depends upon the characteristics of To this we introduce aaccess simple, oriented optimization which trades off(i.e. the the different the vehicles). Thus, we aim to discuss the impact of these restrictions on the fleet composition decision. windows during which the access to the area of delivery is allowed depends upon the characteristics of To this end, we introduce a simple, tactical oriented optimization model, which trades off the different the vehicles). Thus, we aim to discuss the impact of these restrictions on the fleet composition decision. costs andend, revenues in delivering goods. To we introduce simple, tactical oriented optimization model, which off different the vehicles). Thus, aimaa to discuss the impact of these restrictions on the fleettrades composition costs andend, revenues inwe delivering goods. To this this we introduce simple, tactical oriented optimization model, which trades off the the decision. different costs and revenues in delivering goods. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. To this end, we introduce a simple, tactical oriented optimization model, which trades off the different Keywords: fleet composition and sizing, urban freight delivery, logistics performance, optimization costs and revenues in delivering goods. Keywords: fleet composition andgoods. sizing, urban freight delivery, logistics performance, optimization costs and revenues in delivering Keywords: fleet fleet composition composition and and sizing, sizing, urban urban freight freight delivery, logistics logistics performance, performance, optimization optimization Keywords: delivery, Keywords: fleet composition and sizing, urban freight delivery, logistics performance, optimization their introduction is usually relatively inexpensive (although 1. INTRODUCTION their introduction isisusually relatively inexpensive (although their enforcement usually expensive). These restrictions 1. INTRODUCTION their introduction is usually relatively inexpensive (although enforcement is usually expensive). These restrictions their introduction is usually relatively inexpensive (although 1. INTRODUCTION limited traffic zones (LTZ) that must be restrictions considered Urban freight transport encompasses goods delivery and generate 1. INTRODUCTION their enforcement is usually expensive). These their introduction is usually relatively inexpensive (although generate limited traffic zones (LTZ) that must be considered enforcement is usually expensive). These restrictions Urban freight transport encompasses goods delivery and in the process of planning the urban freight transport 1. INTRODUCTION pickup performed within the boundaries of andelivery urban area. generate limited traffic zones (LTZ) that must be considered Urban freight transport encompasses goods and their enforcement is usually expensive). These restrictions in the process of planning the urban freight transport generate limited zones (LTZ) that must be considered pickup performed within the boundaries of andelivery urbanasarea. Urban freight transport encompasses goods and activities, as welltraffic as inplanning defining the set of resources needed. Such activities have gained substantial relevance the in the process of the urban freight transport pickup performed within the boundaries of an urban area. generate limited traffic zones (LTZ) that must be considered activities, as well as in defining the set of resources needed. in the process of planning urban freight transport Urban freight transport encompasses goods and Such activities have gained substantial relevance asarea. the pickup performed within the boundaries of andelivery urban market for direct-to-home deliveries of consumer goods (i.e. activities, as well as in defining the set of resources needed. Such activities have gained substantial relevance as the in the process of planning urban freight transport activities, as well as in defining the set of resources needed. pickup performed within the boundaries of an urban marketactivities for direct-to-home deliveries of consumer (i.e. Such have gained substantial relevance asarea. the e-commerce) continues to grow, especially ingoods densely 1.1 Objective and as motivation market for direct-to-home deliveries of consumer goods (i.e. activities, as well in defining the set of resources needed. Such activities have gained substantial relevance as the e-commerce) continues to grow, of especially ingoods densely market for cities direct-to-home deliveries consumer (i.e. populated (Chen and Conway, 2016). The issue of 1.1 Objective and motivation e-commerce) continues to grow, especially in densely 1.1 Objective and motivation market for direct-to-home deliveries of consumer goods (i.e. populated cities (Chen and Conway, 2016). The issue e-commerce) continues to grow, especially in densely Objective efficient and effective urban freight2016). distribution is of a 1.1 Assuming the and pointmotivation of view of a parcel delivery company or a populated cities (Chen and Conway, The issue of e-commerce) continues to grow, especially in densely efficient and effective urban freight distribution is a populated cities (Chen and Conway, 2016). The issue of 1.1 Objective and motivation Assuming the point of view of aisparcel delivery company or a contemporary issue, as it influences in positive as well as in courier, the focus of this paper on understanding the impact efficient and effective urban freight distribution is a the point of view of aaisparcel delivery company or populatedthe cities (Chen Conway, 2016). The issue contemporary issue, ascitizens’ itand influences inThe positive as well as ina Assuming efficient and effective urban freight distribution is of courier, the focus of this paper on understanding the impact Assuming the point of view of parcel delivery company orToaa negative quality of lives. increment of urban of accesstherestrictions onpaper the optimal fleet composition. contemporary issue, as it influences in positive as well as in courier, focus of this is on understanding the impact efficient and effective urban freight distribution is a negative the quality of citizens’ lives. The increment of urban contemporary issue, as it influences in positive as well as in Assuming the point view aisparcel delivery orToa of access restrictions on theof optimal fleet composition. courier, focus ofof this on operating understanding theof impact population translates into a substantial demand forof goods end,the we consider a paper company a company fleet own negative the quality of citizens’ lives. increment urban of access restrictions on the optimal fleet composition. To contemporary issue, asinto it influences inThe positive astransporting well as in this population translates a substantial demand for negative the quality of citizens’ lives. The increment of goods urban courier, the focus of this paper is on understanding the impact this end, we consider a company operating a fleet of own of access restrictions on the optimal fleet composition. To delivery, which in turn concerns several vehicles vehicles to deliver small to medium size aparcels to own the population translates into aa substantial demand for goods this end, we consider a company operating fleet of negative the quality of citizens’ lives. The increment of urban delivery, which in turn concerns several vehicles transporting population translates into substantial demand for goods of access restrictions on the optimal fleet composition. To vehicles to deliver small to medium size parcels to the this end, we consider a company operating a fleet of own goods, traveling to and from the city. E-commerce customers. The customers are located in an urban area subject delivery, which in turn concerns several vehicles transporting vehicles to deliver small to medium size parcels to the population translates into a substantial demand for goods goods, traveling to and from the city. E-commerce delivery, which in turn concerns several vehicles transporting this end, we consider a company operating a fleet of own customers. The customers are located in an urban area subject vehicles deliver small to medium size parcels to the purchasing, for example usually consist ofcity. smallE-commerce parcels that to access to restrictions; in particular, we consider the (sub)set goods, to and from the The customers are in an urban area subject delivery, traveling which in reliable turn several transporting purchasing, for example usually consist ofcity. small parcels thata customers. goods, traveling to concerns and from the vehicles E-commerce vehicles to deliver small toalocated medium size parcels tofrom the to access restrictions; in particular, we consider the (sub)set customers. The customers are located in an urban area subject requires fast and delivery. This, in turn, requires of customers located in LTZ where access purchasing, for example usually consist of small parcels that to access restrictions; in particular, we consider the (sub)set goods, traveling to and from the city. E-commerce requires fast and reliable delivery. This, in turn, requires a purchasing, for example usually consist of small parcels that customers. The customers are located in an urban area subject of customers located in a LTZ where access from to access restrictions; in particular, we consider the (sub)set fleet of vehicles circulating on the transport network, carryinga “conventional”, thermal engine vehicles is restricted to few requires fast and reliable delivery. This, turn, requires customers located in a vehicles LTZ access purchasing, usually small parcels thata of fleet offrom vehicles circulating on theconsist transport network, carrying requires fastfor andexample reliable delivery. This,ofin in turn, requires to access restrictions; in engine particular, we where consider (sub)set “conventional”, thermal is restricted tofrom few of customers located in LTZ where access from goods origin to destination. Owning (or leasing) a fleet business hours during the day a(i.e. from 8:00am tothe 10:00am). fleet of vehicles circulating on the transport network, carrying “conventional”, thermal engine vehicles is restricted to few requires fast and reliable delivery. This, in turn, requires a goods from origin to destination. Owning (or leasing) a fleet fleet of vehicles circulating on the transport network, carrying of customers located in a LTZ where access from business hours during the day (i.e. from 8:00am to 10:00am). “conventional”, thermal engine vehicles is restricted to few of vehicles is usually costly; therefore, it is relevant to The samehours LTZduring can however be from accessed by to “ecological” goods from origin to destination. Owning (or leasing) aa fleet business the day (i.e. 8:00am 10:00am). fleet of vehicles circulating on the transport network, carrying of vehicles is usually costly; therefore, it is relevant to goods from origin to destination. Owning (or leasing) fleet “conventional”, thermal engine vehicles is restricted to few The same LTZ can however be accessed by “ecological” business hours during the day (i.e. from 8:00am to 10:00am). address the optimization of the number of required vehicles vehicles (such as bicycles, cargo-bike, electric vehicles) of vehicles is costly; therefore, it is relevant to same LTZ can however be accessed by “ecological” goods from origin to destination. Owning leasing) a fleet address theand optimization of the number of(or required vehicles of vehicles is usually usually costly; therefore, it is the relevant to The business hours during the day (i.e. from 8:00am to 10:00am). vehicles (such as bicycles, cargo-bike, electric vehicles) The same LTZ can however be accessed by “ecological” (Beaujon Turnquist, 1991), trading-off capital the aswhole working day without any time address the optimization of number of required vehicles vehicles bicycles, cargo-bike, electric vehicles) of vehicles is usually costly; therefore, iscosts relevant to throughout (Beaujon Turnquist, 1991), trading-off the address the optimization of the the number of it required vehicles The same(such LTZ can however be vehicle accessed byoperate “ecological” throughout the as whole working day without any time vehicles (such bicycles, cargo-bike, electric vehicles) expenses ofand asset procurement, the operational ofcapital asset restriction. Thus, a conventional can in the (Beaujon and Turnquist, 1991), trading-off the capital throughout the working day without any address the optimization of the number of cost required expenses ofand asset procurement, the operational costs ofcapital asset vehicles (Beaujon Turnquist, 1991), trading-off the vehicles (such aswhole cargo-bike, electric vehicles) restriction. Thus, a bicycles, conventional vehicle can operate intime the throughout the whole working day without any time utilization, the customer-service level, and minimization area only for a portion of the time that an ecological vehicle expenses of asset procurement, the operational costs of asset restriction. Thus, a conventional vehicle can operate in the (Beaujon and Turnquist, 1991), trading-off the capital utilization, the customer-service level, and cost minimization expenses of asset procurement, the operational costs of asset throughout the whole working day without any time area only for a portion of the time that an ecological vehicle restriction. Thus, a conventional vehicle can operate in the goals (Wu ettheal.,customer-service 2005). can provide; ifa portion the company does not have the possibility to utilization, level, and cost minimization area only for of the time that an ecological vehicle expenses of asset procurement, the operational costs of asset goals (Wu ettheal.,customer-service 2005). utilization, level, and cost minimization restriction. Thus, a conventional vehicle can operate in the can provide; if the company does not have the possibility to area only for a portion of the time that an ecological vehicle use the conventional vehicles on other activities (i.e. goals (Wu et 2005). provide; if the company does not have the possibility to utilization, customer-service and cost Urban freight transport, also deals withminimization a vast array can goals (Wu ettheal., al., 2005). however,level, area only for a portion of the time that an ecological vehicle use the conventional vehicles on other activities (i.e. can provide; if the company does not have the possibility to in non-LTZvehicles areas) then the return on (i.e. the Urban freight transport, however, also deals to with a vast array delivering use the conventional on other activities goals (Wu et al., 2005). of restrictions and constraints. Restrictions road transport can provide; if the company does not have the possibility to delivering in non-LTZ areas) then the return on the use the conventional vehicles on other activities (i.e. Urban freight transport, however, also deals with a vast array for acquiring the fleet may bethe extremely low. On of restrictions and constraints. Restrictions to road transport Urban freightthe transport, however, also deals with a vast array investment delivering in non-LTZ areas) then return on the (at present, most relevant mode of transport in urban use the conventional vehicles on other activities (i.e. investment for acquiring the fleet may be extremely low. On delivering in non-LTZ areas) then the return on the of restrictions and constraints. Restrictions to road transport other side, some types of conventional vehicles can Urban freight transport, however, also deals with a(i.e. vast array the (at present, the most mode of transport in urban of restrictions and constraints. Restrictions to road transport investment for the fleet may be extremely low. On areas) are imposed byrelevant the urban morphology streets delivering in acquiring non-LTZ areas) then the return on than the the otherlarger side, some types of conventional vehicles can investment for acquiring the fleet may bedistance extremely low. On (at present, the most relevant mode of transport in urban provide payload or longer fuel range of restrictions and constraints. Restrictions to road transport areas) are imposed by the urban morphology (i.e. streets (at present, the most relevant mode of transport in urban the other side, some types of conventional vehicles can arrangement, street size and capacity) and by access policies investment for acquiring the fleet may be extremely low. On provide larger payload or longer fuel distance range than the other side, some types of conventional vehicles can areas) are imposed by the urban morphology (i.e. streets or evenpayload electric orvehicles (EV), thus being ablethan to (at present, the most mode of general, transport inpolicies urban bicycle arrangement, street size and and by access areas) are imposed byrelevant the capacity) urban morphology (i.e. streets provide larger longer fuel distance range meant to govern the urban traffic. In by access the other side, some types of conventional vehicles can bicycle evenquantities electric orvehicles (EV), thus being ablethan to provide or larger payload longer fuel distance range arrangement, street size and capacity) and by access policies transport larger with one trip. areas) are imposed by the urban morphology (i.e. streets meant to govern the urban traffic. In general, by access arrangement, street size and capacity) and by access policies bicycle or even electric vehicles (EV), thus being able to policies wegovern refer the to the entire set ofIn rules governing the transport provide or larger payload orvehicles longer fuel distance range than larger quantities with one trip. bicycle even electric (EV), thus being able to meant to urban traffic. general, by access arrangement, street size and capacity) and by access policies policies we refer to the entire set of rules governing the meant to govern the urban traffic. In general, by access transport larger quantities with one trip. access toweanrefer area:tosuch rules can be rules related to physical bicycle orlarger even electric vehicles (EV), thus that being ablethe to Considering alsoquantities the availability of trip. incentives reduce transport with one policies the entire set of governing the meant to govern the urban traffic. In general, by access access area: such rules can beorrules related to physical policiestoweanrefer torelated the entire setsize of governing the Considering alsoecological the availability of trip. incentives that reduce the characteristics (i.e. to the the weight of the transport larger quantities with one investment in vehicles, the company wants to access anrefer area: such rules can be related to physical also the of that reduce the policiesto the entire set of characteristics (i.e. related to the size orrules the governing weight of the Considering access towe area: such rules be related to investment in vehicles, the company wants to Considering alsoecological the availability availability of incentives incentives that the vehicles) asanwell astoto the time ofcan day (i.e. access onlyphysical during define the best fleet composition considering thereduce different characteristics (i.e. related to the size or the weight of the investment in ecological vehicles, the company wants to access to an area: such rules can be related to physical vehicles) as well as to the time of day (i.e. access only during characteristics (i.e. related to the size or the weight of the Considering also the availability of incentives that reduce the define the best fleet composition considering the different investment in ecological vehicles, the company wants to pre-specified time windows). Time-window access characteristics offleet bothcomposition conventionalconsidering and ecological vehicles vehicles) as well as to the time of day (i.e. access only during define the best the different characteristics (i.e. related to the size or the weight of the pre-specified time windows). Time-window access vehicles) as well as to the time of day (i.e. access only during investment in ecological vehicles, the company wants to characteristics of both conventional and ecological vehicles define the best fleet composition considering the different restrictions are probably one of the most common policy to such as payload, working hours, traveling time, and costs. To pre-specified time windows). Time-window access of both conventional and ecological vehicles vehicles) as are wellfreight as to the timeof ofthe daymost (i.e.etcommon access only during restrictions probably one policy to characteristics pre-specified time windows). Time-window access define the best fleet composition considering the different such as payload, working hours, traveling time, and costs. To characteristics of both conventional and ecological vehicles manage urban access (Muñuzuri al., 2005), since this end, in this paper we hours, develop a tactical model aiming to restrictions are probably one the policy to such as working traveling time, and costs. To pre-specified time access windows). Time-window access manage urban (Muñuzuri etcommon al., 2005), since restrictions are freight probably one of of the most most common policy to this characteristics both we conventional and ecological vehicles in thisofpaper develop a tactical model to suchend, as payload, payload, working hours, traveling time, and aiming costs. To manage urban freight access (Muñuzuri et al., 2005), since in we develop aa tactical model to restrictions are freight probably one of the most etcommon policy to this manage urban access (Muñuzuri al., 2005), since suchend, as payload, working traveling time, and aiming costs. To this end, in this this paper paper we hours, develop tactical model aiming to manage urban Copyright © 2018freight IFAC access (Muñuzuri et al., 2005), since 589 this end, in this paper we develop a tactical model aiming to 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 589Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 589Control. Copyright © 2018 IFAC 589 10.1016/j.ifacol.2018.08.381 Copyright © 2018 IFAC 589
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
Roberto Pinto et al. / IFAC PapersOnLine 51-11 (2018) 582–587
support the company in assessing the trade-offs that can emerge at a tactical level.
addressed by Klosterhalfen et al. (2014) developing a model to determine the optimal structure and size of a rail car fleet at a chemical company.
To this end, the paper is structured as follows: in Section 2, we provide an abridged literature review underlining the main aspects of the fleet sizing and composition problem. Such a review is then used to position the main contributions of this paper. Section 3 discusses the main elements of the fleet composition problem in an urban environment, underlining the role of the access restriction policies in the decision-making process. In this section, we also present the optimization model. Section 4 reports a simple example of application of the model, along with the main limitations of the model itself. Finally, some conclusions in section 5 close the paper, proposing further avenues of research in the urban freight fleet composition subject.
Fleet composition problems are often combined with other transportation problems: for example, Hoff et al. (2010) describe industrial aspects of combined fleet composition and routing in maritime and road-based transportation. Wu et al. (2005) address a fleet-sizing problem in the context of the truck-rental industry. The authors address both operational decisions (i.e. demand allocation and empty truck repositioning) and tactical decisions (i.e. asset procurements and sales) considering that assets of different ages carry different costs. Redmer et al. (2012) consider the problem of composing an optimal fleet of tankers in a central distribution system. The decision problem is considered as a vehicle assignment in which different types of vehicles are assigned to customers’ orders (resulting in the construction of an average / typical routes) and formulated as a single objective mathematical programming model. Alem et al., (2016) develop a model to help deciding how to rapidly supply humanitarian aid to victims of a disaster, taking into account practical characteristics such as the fleet sizing of multiple types of vehicles. To take uncertainty into account, the authors propose a scenario-based approach within a two-stage stochastic programming paradigm. One of the goals of the model is to define the overall capacity of each type of vehicle in order to perform effective emergency logistics once the disaster strikes.
2. BACKGROUND The capacity of a transportation system is directly related to the number and types of available vehicles. Determining the optimal number of vehicles requires a trade-off between the ownership and operating costs of the vehicles, and the potential costs or penalties associated with not meeting a portion of the overall demand (Beaujon and Turnquist, 1991; Wu et al., 2005). In literature, we can found at least two macro-classes of problems (Etezadi and Beasley, 1983; Du et al., 2016): 1.
Vehicle fleet size problems: given a vehicle type to be operated, the decision relates to the number of vehicles to invest into.
2.
Vehicle fleet composition problems: the decisions relate to the type of vehicles to operate (mix) and the number of each type.
583
The problem under analysis can be represented with a twostage stochastic programming model. However, the complexity of the model is relatively high, given the number of variables; the resulting model is complex and computationally demanding. Our goal is to formulate a simple yet relevant model. Thus, the model discussed in this paper addresses a fleet composition problem for a company operating according to a one-to-many transport scheme (that is, the typical case of a fleet of vehicles departing from a single depot and visiting several destinations). The model is deterministic, although it considers input (such as the demand) that comes from a statistical distribution, similarly to a deterministic equivalent of stochastic programming model with recourse. With respect to the literature, the proposed model introduces two new aspects: i) the constraint imposed by the LTZ in terms of working hours, and ii) the definition of the optimal service level, trading-off revenues and costs.
According to Żak et al. (2011), fleet composition problems can be classified according to several criteria. For example, it is possible to distinguish between one-to-one, one-to-many and many-to-many transportation schemes, or between homogeneous and heterogeneous fleet problems. Framing the time horizon of the decisions is relevant: in general, strategic decisions define markets and customer service levels, tactical decisions determine asset purchases and sales, and operational decisions include asset allocation, assignment, routing, and/or scheduling decisions (Wu et al., 2005). Under this time framework, thus, both the abovedescribed classes of problems concern tactical decisionmaking processes.
3. THE FLEET COMPOSITION PROBLEM UNDER ACCESS RESTRICTION POLICIES
Deterministic models usually stress the spatial structure of the problem neglecting the temporal dimension, and particularly the stochastic nature of demand. List et al. (2003) illustrate a solution strategy that incorporates uncertainty into optimization models for fleet sizing, with a focus on the uncertainty arising from the future demand and the productivity of the vehicles. Their focus on robust optimization allows modelling an explicit trade-off between the level of fleet investment and the level of risk in the solution. Uncertainty in demand and travel times is also
In a fleet composition problem, we are not concerned with short-term decisions, such as the specific vehicle that one should use to supply a certain customer, but with long-term decisions concerning the number and type of vehicles that the company should operate (Etezadi and Beasley, 1983). More assets allow a company to achieve a higher level of customer satisfaction at the expense of higher capital and holding costs (Wu et al., 2005). Because vehicles are generally long-lived assets, there is intrinsic uncertainty about the demands that 590
IFAC INCOM 2018 584 Bergamo, Italy, June 11-13, 2018
Roberto Pinto et al. / IFAC PapersOnLine 51-11 (2018) 582–587
they will serve over their lifetime, and about the conditions under which they will operate (List et al., 2003). Thus, in order to develop the model, we need to discuss some assumptions first.
For example, Fig. 1 depicts two different demand patterns of the daily demand d j over a 250-day horizon; these demand patterns can be determined either from historical data or via forecast. The design capacity (represented with a grey, dotted line in Fig.1) can be chosen in such a way that it covers for a given percentage of the demands, thus defining a service level measure. However, this opens the problem related to the appropriate choice of the design capacity and the related service level. In Fig. 1, the design capacity is set to a value that allows covering for at least 90% of the daily demands, that is, the capacity is enough to satisfy the demand of at least 90% of the considered period (i.e. 225 over 250 days). This also means that the company accepts to resort to another company to complete the deliveries for at most 25 days in a year. Defining the design capacity at a level lower that 100%, we should balance the trade-off between having too much and to little capacity.
We consider a company (typically, a parcel company) delivering parcels in a LTZ. The company wants to address the fleet composition problem defining the mix of vehicles that maximizes the company’s profit over a medium- to longterm planning horizon (3-5 years), considering relevant revenues and costs. We consider a revenue proportional to the quantity delivered to the customers (that is, the revenue can be calculated multiplying a fixed price by the quantity delivered). One of the first decision that should be made in this context is about how to measure the demand from the company’s perspective. Indeed, this affects the way in which we should model the capacity of the fleet. Transport activities can be measured in terms of weight of the goods transported, in terms of number of parcels, in number of trips from the departing depot, and the like. The selection of the unit of measure often depends upon the scope and goal of the model. For example, for one-to-one transportation scheme it can be relevant to measure the number of trips, whereas for parcel delivery can be better expressing the demand in terms of number of parcels or in number of required deliveries.
Daily demand
Design capacity
20000 19000
Demand
18000
15000 14000 13000 12000 11000 10000 1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211 221 231 241
Since the company wants to tackle the problem at a tactical level (as defined in Wu et al., (2005)), the details of the forecasted demand are extremely difficult to get. Thus, in order to use an aggregate measure that reduces the variability in the forecast, in this study we represent the demand and the capacity of the fleet in terms of number of deliveries rather than in terms of number of parcels or in terms of kilograms/tons. This assumption greatly affects the model that is presented further on. However, the peculiarities mainly regard the conversion from one unit of measure to another, whereas the overall structure of the model remains the same. We also assume that the demand of each customer is for small volumes, lower than the carrying capacity of each individual vehicle. In fact, the company operates a one-tomany transportation scheme.
17000 16000
Days Design capacity 20000 19000 18000
Demand
17000 16000
15000 14000 13000 12000 11000 1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211 221 231 241
10000
Days
The considered costs involve the vehicles purchasing (investment), operating and maintenance cost, as well as the cost to outsource part of the activities whenever needed (penalty cost). In fact, the size and mix of the fleet define also the overall transport capacity; however, such size and mix should be defined considering an estimated demand that can vary over the considered period. Facing a volatile demand, if we define the overall capacity based on the peak demand, we may end up with a relatively large portion of the capacity unsaturated, whereas if we use the average value of the demand we may need to resort to outsourcing way too often, with consequent cost increase. To this end, the model is developed considering the distribution of quantities to be delivered over time (i.e. on a yearly basis) and a minimum capacity that should be guaranteed in order to attain a good service level at a sustainable cost.
Fig. 1. Estimations of the design capacity based on different demand patterns If we define D * as the design capacity (also referred to as the critical demand) it means that in the days in which the demand is higher than D * we would not be able to fulfil it, unless we resort to the extra capacity provided by a third party parcel courier, of course at a higher cost. This cost represents the penalty associated with not meeting a portion of the overall demand (Beaujon and Turnquist, 1991). It is worth noticing that defining the design capacity considering the number of days that the company wants to cover with its own fleet does not necessarily provide complete information about penalty cost of resorting to a third party.
591
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
Roberto Pinto et al. / IFAC PapersOnLine 51-11 (2018) 582–587
One way to address the problem of defining the design capacity is to consider the fleet composition problem as a version of the classical newsvendor problem. In fact, the decision maker is required to make a decision that cannot be changed in the short term, trading off the risk of having too much or too little capacity during the year. However, the calculation of an equivalent fractile ratio in the discussed context can be difficult and not accurate. An alternative approach, used in this paper, is to consider the design capacity as a variable of the model that should be optimized considering all the relevant revenues and costs.
reducing the average number of delivery per hour. Thus, we assume Pe Pc e E , c C .
Finally, our proposed model does not include routing aspects since we focus on a long-term perspective. At this level, demand, costs and revenue uncertainties related to fleet operations are high, thus taking into account routing aspects on a detailed level is ineffective (Du et al., 2016; Hoff et al., 2010).
H k = working time for a vehicle k K . In the considered case, the working time is different for the two categories of vehicles. In fact, due to access restriction policies, we assume that H e H c e E , c C .
m = average unit margin for delivery (i.e. €/delivery)
Co = cost of using a third party parcel courier, expressed in the same unit of measure of the unit margin m (i.e. €/delivery)
F k = fixed cost for operating a vehicle k K ,
including ordinary maintenance cost. We assume that operating and maintaining a conventional vehicle is usually more expensive, that is Fe Fc e E , c C ). The value F k is an average value, since in reality maintenance costs can depend upon the use of the asset (i.e. the number of trips, or the distance travelled).
3.1 Notation The model uses the following notation:
T = time horizon on which the investment decision is assessed (i.e. 5 years)
J = set of days per year of the time horizon (i.e. J 1, , 250 )
K = set of vehicle types available for the fleet mix decision.
C K = subset of conventional vehicles, including pick-ups, vans, trucks, and lorries with internal combustion engines.
E K = subset of ecological vehicles, including
for
one
vehicle of
M = maximum number of vehicles composing the fleet. This parameter may reflect the necessity to limit the number of vehicles to be operated to contain cost or other constraints (such as lack of space) affecting the company’s operations.
The number of deliveries per hour Pk depends upon several factors, such as the payload of the vehicle, the distance between the customers, the speed of the vehicles, but also the traffic congestion and the morphology of the city. Therefore, this parameter may be extremely difficult to estimate; however, in this paper we assume it as deterministic and known, postponing the discussion about its determination to a future research work.
d j = daily (estimated) demand for day j J over
the year, in terms of number of deliveries. These values may come from a probability distribution (i.e. the values d j are the realization of the sampling
Considering the fleet composition problem, the main decision variables are the number of vehicles per each type, and the design capacity, that is:
from a known statistical distribution) or from historical data. We further assume that the deliveries cannot be postponed or anticipated (i.e. all the demand must be served in full either via the owned fleet or via a third party parcel delivery service). This assumption is supported by the tactical level approach of the model.
= investment cost
Ik
type k K .
electric vehicles (EV), bikes and cargo-bikes. The two sets E and C are not overlapping; that is E K \ C .
585
Y k = Number of vehicles of type k K in the fleet.
D = Optimal design capacity.
*
However, to build the model we need some further variables as follows:
Pk = average number of deliveries per hour with a
vehicle k K . We assume that conventional vehicles have usually a larger payload, so they can carry more parcels (i.e. more deliveries) and can drive faster than, for example, bicycles. Vehicles with a small payload may require going back to depot in order to load new parcels to deliver, further
q j = Number of deliveries on day j J with the
company’s own fleet.
592
Z = Total capacity (in terms of number of possible deliveries) of the company’s own fleet.
IFAC INCOM 2018 586 Bergamo, Italy, June 11-13, 2018
Roberto Pinto et al. / IFAC PapersOnLine 51-11 (2018) 582–587
constraints (7) and (8) define the types of the variables. The model thus defined is similar to a chance-constrained problem (Charnes and Cooper, 1959) where we require that the probability of not being able to satisfy the demand be less than a given threshold.
3.2 The model The objective of the model is to maximize the profit considering the margin (discounting the fleet variable operations costs from the revenues), the fleet ownership costs (i.e. the investment cost plus the fixed operations and maintenance costs) and the penalties for not having enough capacity to meet the demand.
4. APPLICATION AND LIMITATIONS OF THE MODEL The model discussed in Section 3 is a simple one; yet, it can be used to perform interesting what-if analysis. Let us assume for the sake of the example that the daily demand is normally distributed with 500 deliveries and 125 . Sampling a business year from this distribution (i.e. 250 values) we can get a demand signal similar to the one represented in the upper part of Fig. 1. Let us assume that the fleet can be composed of no more than eight vehicles chosen among those listed in Table 1. Vehicles C1 and C2 are conventional vehicles with a high payload, whereas vehicles E1 and E2 are ecological vehicles (i.e. E1 can be an EV small truck and E2 a cargo bike). With this data, the model can be solved in less than one second (Gurobi solver 6.0 on a Core i7 computer with 8GB of memory).
The resulting model is as follows: max T
max
1
m q j Co d j q j j J
kK
Fk Y k
(1)
I k Yk
kK
where
1
1 r
is the discounting factor at the r rate.
The term m q j expresses the operations margin in day j, whereas the term Co d j q j represents the penalty cost of resorting to a third party parcel delivery service in day j. Clearly, the margin and the penalty cost depend upon the decision about the design capacity D * . The constraints are as follows:
P
k
H k Yk Z
(2)
kK
Z D
*
qj d
q
j
Z
Y
k
j J
(4)
j J
(5)
M
k
0 int
(6)
Y k 0 int *
q j 0, Z 0, D 0
k K
j J
H
C1
15
C2
Fk
Ik
3
12,000
20,000
11
3
11,000
30,000
E1
15
8
8,500
35,000
E2
6
8
1,100
2,500
k
Considering an average margin per delivery m 2 . 5 € and a penalty Co = 5.0 €, the solution of one realization of the demand dj determines a design capacity D* equal to 608 deliveries per day to serve a total demand of about 124,800 deliveries during a year; to this end, the optimal fleet is composed of four vehicles E1 and four vehicles E2. The company must resort to a third party company to cover for 3,509 deliveries during the year, about 2.8 % of the total demand.
kK
X
Pk
Table 1. Data for the example
(3)
j
Vehicle
(7)
4.1 Robustness of the solution
(8)
Clearly, the results reported in the previous example are extremely partial, and can lead to a wrong conclusion: in fact, those results are related to a specific realization of the demand (i.e. the sampling from the demand distribution specified above). However, thanks to the short time required, it is possible to solve the model several times, each time with a different realization of the demand d j. In this way, we can analyse the robustness of the solution (i.e. the change of the optimal solution according to change in the demand) and the average performance.
Constraint (2) defines the overall capacity Z of the fleet in terms of possible deliveries as a function of the number of delivery per hour Pk , the daily working hours H k and the number of vehicles per each type, Y k . As required, the capacity Z is such that it covers at least the design capacity D * (but can be larger), as expressed by constraint (3). Constraints (4) and (5) together limit the number of deliveries q j to the minimum between the overall
So, let us assume to sample 100 realizations d j from the same distribution specified above, and to solve the model for each realization. The optimal solution S alternates between S1 = (E1=4, E2=4) and S2 = (E1=5, E2=3), with a design capacity
capacity Z and the actual demand d j . Constraint (6) limits the size of the fleet to M vehicles; in fact, a large number of vehicles may introduce a managerial overload. Finally, 593
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
Roberto Pinto et al. / IFAC PapersOnLine 51-11 (2018) 582–587
of 608 and 696 deliveries per day, respectively. The investment is equal to 150,000 € for S1 and 182,500 € for S2. Delving into the results, the solution S1 is the optimal one in only 31 cases out of 100, thus indicating that S2 can provide a very good coverage against the volatility of the demand. The average profit of solution S1 is about 807,000 € whereas the average profit of solution S2 is about 816,000 €.
587
off the cost of owning the vehicles with the cost to outsource part of the activities whenever needed to guarantee the delivery by the required time. The resulting model is quite simple, yet it can be used to perform a sensitivity analysis. An interesting development is the comparison of the simple procedure outlined in this paper against a stochastic programming approach. As discussed in the limitations section, this model should be considered as an initial model towards the definition of a decision support system in the context of fleet composition problems in the urban freight area.
4.2 Limitations The model discussed in the previous section has several limitations that must be brought about and addressed by further research. The model addresses the fleet composition problem at a tactical level, thus using aggregate data and information. This allows creating a relatively small model. For example, aggregating the demand on the time dimension reduces the number of variables to be considered. However, at the same time this aggregation hides the complexity of the transport problem when the vehicles have to perform multiple stops during their trips. Clearly, some levels of details may not be available at the time the decision is made (i.e. we do not know which customer should be server each day).
REFERENCES Alem, D., Clark, A., Moreno, A. (2016). Stochastic network models for logistics planning in disaster relief. European Journal of Operational Research. 255(1), 187-206. Beaujon, G.J., Turnquist, M.A. (1991). A Model for Fleet Sizing and Vehicle Allocation. Transportation Science, 25(1), 19-45. Charnes, A., Cooper, W.W. (1959). Chance-Constrained Programming. Management Science. 6(1), 73-79. Chen, Q., Conway, A. (2016). Commercial Vehicle Parking Availability and Behavior for Residential Delivery in New York City. Transportation Research Board 95th Annual Meeting. Du, J.Y., Brunner, J.O., Kolisch, R. (2016). Obtaining the optimal fleet mix: A case study about towing tractors at airports. Omega. 64, 102-114. Etezadi, T., Beasley, J.E. (1983). Vehicle Fleet Composition. The Journal of the Operational Research Society. 34(1), 87-91. Hoff, A., Andersson, H., Christiansen, M., Hasle, G., Løkketangen, A. (2010). Industrial aspects and literature survey: Fleet composition and routing. Computers & Operations Research. 37(12), 2041-2061. Klosterhalfen, S.T., Kallrath, J., Fischer, G. (2014). Rail car fleet design: Optimization of structure and size. International Journal of Production Economics. 157 (1), 112-119. List, G.F., Wood, B., Nozick, L.K., Turnquist, M.A., Jones, D.A., Kjeldgaard, E.A., Lawton, C.R. (2003). Robust optimization for fleet planning under uncertainty. Transportation research Part E. 39 (3), 209-227. Muñuzuri, J., Larrañeta, J., Onieva, L. and Cortés, P. (2005). Solutions applicable by local administrations for urban logistics improvement. Cities. 22(1), 15-28. Redmer, A., Żak, J., Sawicki, P., Maciejewski, M. (2012). Heuristic approach to fleet composition problem. Procedia - Social and Behavioral Sciences. 54, 414-427 Wu, P., Hartman, J.C., Wilson, G.R. (2005). An integrated model and solution approach for fleet sizing with heterogeneous assets. Transportation Science. 39(1), 87103. Żak, J. Redmer, A., Sawicki, P. (2011). Multiple objective optimization of the fleet sizing problem for road freight transportation. Journal of advanced transportation. 45(4), 321-347.
Further, the capacity of the fleet can be expressed in different ways, and this decision affects the resulting model. In one-toone transportation scheme, the number of required trips can be used; in such a context, the capacity of the fleet depends on the location of the customers (i.e. their distance from the depot) that determines the number of trips each vehicle can perform in a period. In one-to-many transport scheme, instead, we use the same vehicle to serve several customers with one trip. In this case, the number of trips may be not linearly correlated with the number of deliveries: one trip can serve more customers than another trip, depending upon the demand of each customer visited, and their distance from the depot and from each other. To represent the capacity of the fleet, we thus used the number of average deliveries per unit of time Pk , a measure that is also commonly reported by couriers as a performance KPI. However, we understand that assessing the values of Pk can be extremely convoluted, due to the large number of elements affecting it (i.e. payload of the vehicle, distance between the customers, the speed of the vehicles, traffic congestion during the day, morphology of the city). This requires further investigation in order to provide more robust decisions. The model considers the same demand for each year of the considered time horizon T . This has been assumed to reduce the number of variables in the model. However, the demand can vary over T, which can span over several years. Actually, the model can be easily changed to include the demand over each year in T, increasing the number of variable q j from J to T J .
5. CONCLUSIONS This paper addresses the fleet composition problem in the context of urban freight distribution, where access limitations to the delivery areas may affect the possibility of using specific vehicles. The fleet composition problem must trade594
ScienceDirect
PapersOnLine 51-11 (2018) 582–587 UrbanIFAC Freight Fleet Composition Problem Urban Freight Fleet Composition Problem Urban Freight Fleet Composition Problem Urban Freight Composition Problem Roberto Pinto*, Fleet Alexandra Lagorio*, Ruggero Golini* Urban Freight Fleet Composition Problem Roberto Pinto*, Alexandra Lagorio*, Ruggero Golini*
Roberto Roberto Pinto*, Pinto*, Alexandra Alexandra Lagorio*, Lagorio*, Ruggero Ruggero Golini* Golini* * Department of Management, Roberto Information and Industrial Engineering, University of Bergamo, via G. Marconi, 24044, Pinto*, Alexandra Lagorio*, Ruggero Golini* * Department of Management, Information and Industrial Engineering, University of Bergamo, via G. Marconi, 24044, Dalmine (BG) – Italy (e-mail: [email protected], [email protected], [email protected]) ** Department of Management, Information and Industrial Engineering, University via Dalmine (BG) – Italy (e-mail: [email protected], [email protected], [email protected]) Department of Management, Information and Industrial Engineering, University of of Bergamo, Bergamo, via G. G. Marconi, Marconi, 24044, 24044, Dalmine (BG) – Italy (e-mail: [email protected], [email protected], [email protected]) * Department of Management, Information and Industrial Engineering, University of Bergamo, via G. Marconi, 24044, Dalmine (BG) – Italy (e-mail: [email protected], [email protected], [email protected]) Dalmine (BG) – Italy (e-mail: [email protected]) Abstract: The Urban [email protected], Fleet Composition (UFFC) [email protected], problem concerns the definition of the optimal Abstract: The Urban Freight FleetofComposition (UFFC) problemtoconcerns the definition of the delivery optimal mix in terms of types and number vehicles (fleet composition) serve the demand for parcel Abstract: The Urban Freight Fleet Composition (UFFC) problem concerns the definition of the optimal mix in terms of types and number of vehicles (fleet composition) to serve the demand for parcel delivery Abstract: The Urban Freight Fleet Composition (UFFC) problem concerns the definition of the optimal in aninurban area. Urban areas can of be vehicles subject to access restrictionstothat reduce the possibility to enter the mix terms of types and number (fleet composition) serve the demand for delivery Abstract: The Freight Fleet (UFFC) problem concerns the definition of thethe optimal in aninurban area. Urban areas cantime beComposition subject to access restrictions reduce the possibility to enter the mix terms ofUrban types and number of vehicles (fleet composition) tothat serve thevehicle demand for parcel parcel delivery area. In this paper, we consider access restriction related to specific types (i.e. timein an area. Urban areas can be subject to access restrictions reduce the enter the mix of types and number of (fleet composition) serve thevehicle demand for parcel delivery area. Interms this paper, we consider access restriction related totothat specific types (i.e.to timein aninurban urban area. Urban areas cantime be subject restrictions that reduce the possibility possibility to the enter the windows during which the access to vehicles the areato ofaccess delivery is allowed depends upon the characteristics of area. In this paper, we consider time access restriction related to specific vehicle types (i.e. the timein an urban area. Urban areas can be subject to access restrictions that reduce the possibility to enter the windows during which the access to the area of delivery is allowed depends upon the characteristics of area. In this paper, we consider time access restriction related to specific vehicle types (i.e. the timethe vehicles). Thus, we aim to discuss the impact of these restrictions on the fleet composition decision. windows during which access to the of is allowed depends upon the characteristics of area. In end, this paper, we the consider time access restriction related to model, specific vehicle types timethe vehicles). Thus, aim to discuss thearea impact of these on the fleet composition decision. windows during which the to tactical the area of delivery delivery isrestrictions allowed depends upon the characteristics of To this we introduce aaccess simple, oriented optimization which trades off(i.e. the the different the vehicles). Thus, we aim to discuss the impact of these restrictions on the fleet composition decision. windows during which the access to the area of delivery is allowed depends upon the characteristics of To this end, we introduce a simple, tactical oriented optimization model, which trades off the different the vehicles). Thus, we aim to discuss the impact of these restrictions on the fleet composition decision. costs andend, revenues in delivering goods. To we introduce simple, tactical oriented optimization model, which off different the vehicles). Thus, aimaa to discuss the impact of these restrictions on the fleettrades composition costs andend, revenues inwe delivering goods. To this this we introduce simple, tactical oriented optimization model, which trades off the the decision. different costs and revenues in delivering goods. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. To this end, we introduce a simple, tactical oriented optimization model, which trades off the different Keywords: fleet composition and sizing, urban freight delivery, logistics performance, optimization costs and revenues in delivering goods. Keywords: fleet composition andgoods. sizing, urban freight delivery, logistics performance, optimization costs and revenues in delivering Keywords: fleet fleet composition composition and and sizing, sizing, urban urban freight freight delivery, logistics logistics performance, performance, optimization optimization Keywords: delivery, Keywords: fleet composition and sizing, urban freight delivery, logistics performance, optimization their introduction is usually relatively inexpensive (although 1. INTRODUCTION their introduction isisusually relatively inexpensive (although their enforcement usually expensive). These restrictions 1. INTRODUCTION their introduction is usually relatively inexpensive (although enforcement is usually expensive). These restrictions their introduction is usually relatively inexpensive (although 1. INTRODUCTION limited traffic zones (LTZ) that must be restrictions considered Urban freight transport encompasses goods delivery and generate 1. INTRODUCTION their enforcement is usually expensive). These their introduction is usually relatively inexpensive (although generate limited traffic zones (LTZ) that must be considered enforcement is usually expensive). These restrictions Urban freight transport encompasses goods delivery and in the process of planning the urban freight transport 1. INTRODUCTION pickup performed within the boundaries of andelivery urban area. generate limited traffic zones (LTZ) that must be considered Urban freight transport encompasses goods and their enforcement is usually expensive). These restrictions in the process of planning the urban freight transport generate limited zones (LTZ) that must be considered pickup performed within the boundaries of andelivery urbanasarea. Urban freight transport encompasses goods and activities, as welltraffic as inplanning defining the set of resources needed. Such activities have gained substantial relevance the in the process of the urban freight transport pickup performed within the boundaries of an urban area. generate limited traffic zones (LTZ) that must be considered activities, as well as in defining the set of resources needed. in the process of planning urban freight transport Urban freight transport encompasses goods and Such activities have gained substantial relevance asarea. the pickup performed within the boundaries of andelivery urban market for direct-to-home deliveries of consumer goods (i.e. activities, as well as in defining the set of resources needed. Such activities have gained substantial relevance as the in the process of planning urban freight transport activities, as well as in defining the set of resources needed. pickup performed within the boundaries of an urban marketactivities for direct-to-home deliveries of consumer (i.e. Such have gained substantial relevance asarea. the e-commerce) continues to grow, especially ingoods densely 1.1 Objective and as motivation market for direct-to-home deliveries of consumer goods (i.e. activities, as well in defining the set of resources needed. Such activities have gained substantial relevance as the e-commerce) continues to grow, of especially ingoods densely market for cities direct-to-home deliveries consumer (i.e. populated (Chen and Conway, 2016). The issue of 1.1 Objective and motivation e-commerce) continues to grow, especially in densely 1.1 Objective and motivation market for direct-to-home deliveries of consumer goods (i.e. populated cities (Chen and Conway, 2016). The issue e-commerce) continues to grow, especially in densely Objective efficient and effective urban freight2016). distribution is of a 1.1 Assuming the and pointmotivation of view of a parcel delivery company or a populated cities (Chen and Conway, The issue of e-commerce) continues to grow, especially in densely efficient and effective urban freight distribution is a populated cities (Chen and Conway, 2016). The issue of 1.1 Objective and motivation Assuming the point of view of aisparcel delivery company or a contemporary issue, as it influences in positive as well as in courier, the focus of this paper on understanding the impact efficient and effective urban freight distribution is a the point of view of aaisparcel delivery company or populatedthe cities (Chen Conway, 2016). The issue contemporary issue, ascitizens’ itand influences inThe positive as well as ina Assuming efficient and effective urban freight distribution is of courier, the focus of this paper on understanding the impact Assuming the point of view of parcel delivery company orToaa negative quality of lives. increment of urban of accesstherestrictions onpaper the optimal fleet composition. contemporary issue, as it influences in positive as well as in courier, focus of this is on understanding the impact efficient and effective urban freight distribution is a negative the quality of citizens’ lives. The increment of urban contemporary issue, as it influences in positive as well as in Assuming the point view aisparcel delivery orToa of access restrictions on theof optimal fleet composition. courier, focus ofof this on operating understanding theof impact population translates into a substantial demand forof goods end,the we consider a paper company a company fleet own negative the quality of citizens’ lives. increment urban of access restrictions on the optimal fleet composition. To contemporary issue, asinto it influences inThe positive astransporting well as in this population translates a substantial demand for negative the quality of citizens’ lives. The increment of goods urban courier, the focus of this paper is on understanding the impact this end, we consider a company operating a fleet of own of access restrictions on the optimal fleet composition. To delivery, which in turn concerns several vehicles vehicles to deliver small to medium size aparcels to own the population translates into aa substantial demand for goods this end, we consider a company operating fleet of negative the quality of citizens’ lives. The increment of urban delivery, which in turn concerns several vehicles transporting population translates into substantial demand for goods of access restrictions on the optimal fleet composition. To vehicles to deliver small to medium size parcels to the this end, we consider a company operating a fleet of own goods, traveling to and from the city. E-commerce customers. The customers are located in an urban area subject delivery, which in turn concerns several vehicles transporting vehicles to deliver small to medium size parcels to the population translates into a substantial demand for goods goods, traveling to and from the city. E-commerce delivery, which in turn concerns several vehicles transporting this end, we consider a company operating a fleet of own customers. The customers are located in an urban area subject vehicles deliver small to medium size parcels to the purchasing, for example usually consist ofcity. smallE-commerce parcels that to access to restrictions; in particular, we consider the (sub)set goods, to and from the The customers are in an urban area subject delivery, traveling which in reliable turn several transporting purchasing, for example usually consist ofcity. small parcels thata customers. goods, traveling to concerns and from the vehicles E-commerce vehicles to deliver small toalocated medium size parcels tofrom the to access restrictions; in particular, we consider the (sub)set customers. The customers are located in an urban area subject requires fast and delivery. This, in turn, requires of customers located in LTZ where access purchasing, for example usually consist of small parcels that to access restrictions; in particular, we consider the (sub)set goods, traveling to and from the city. E-commerce requires fast and reliable delivery. This, in turn, requires a purchasing, for example usually consist of small parcels that customers. The customers are located in an urban area subject of customers located in a LTZ where access from to access restrictions; in particular, we consider the (sub)set fleet of vehicles circulating on the transport network, carryinga “conventional”, thermal engine vehicles is restricted to few requires fast and reliable delivery. This, turn, requires customers located in a vehicles LTZ access purchasing, usually small parcels thata of fleet offrom vehicles circulating on theconsist transport network, carrying requires fastfor andexample reliable delivery. This,ofin in turn, requires to access restrictions; in engine particular, we where consider (sub)set “conventional”, thermal is restricted tofrom few of customers located in LTZ where access from goods origin to destination. Owning (or leasing) a fleet business hours during the day a(i.e. from 8:00am tothe 10:00am). fleet of vehicles circulating on the transport network, carrying “conventional”, thermal engine vehicles is restricted to few requires fast and reliable delivery. This, in turn, requires a goods from origin to destination. Owning (or leasing) a fleet fleet of vehicles circulating on the transport network, carrying of customers located in a LTZ where access from business hours during the day (i.e. from 8:00am to 10:00am). “conventional”, thermal engine vehicles is restricted to few of vehicles is usually costly; therefore, it is relevant to The samehours LTZduring can however be from accessed by to “ecological” goods from origin to destination. Owning (or leasing) aa fleet business the day (i.e. 8:00am 10:00am). fleet of vehicles circulating on the transport network, carrying of vehicles is usually costly; therefore, it is relevant to goods from origin to destination. Owning (or leasing) fleet “conventional”, thermal engine vehicles is restricted to few The same LTZ can however be accessed by “ecological” business hours during the day (i.e. from 8:00am to 10:00am). address the optimization of the number of required vehicles vehicles (such as bicycles, cargo-bike, electric vehicles) of vehicles is costly; therefore, it is relevant to same LTZ can however be accessed by “ecological” goods from origin to destination. Owning leasing) a fleet address theand optimization of the number of(or required vehicles of vehicles is usually usually costly; therefore, it is the relevant to The business hours during the day (i.e. from 8:00am to 10:00am). vehicles (such as bicycles, cargo-bike, electric vehicles) The same LTZ can however be accessed by “ecological” (Beaujon Turnquist, 1991), trading-off capital the aswhole working day without any time address the optimization of number of required vehicles vehicles bicycles, cargo-bike, electric vehicles) of vehicles is usually costly; therefore, iscosts relevant to throughout (Beaujon Turnquist, 1991), trading-off the address the optimization of the the number of it required vehicles The same(such LTZ can however be vehicle accessed byoperate “ecological” throughout the as whole working day without any time vehicles (such bicycles, cargo-bike, electric vehicles) expenses ofand asset procurement, the operational ofcapital asset restriction. Thus, a conventional can in the (Beaujon and Turnquist, 1991), trading-off the capital throughout the working day without any address the optimization of the number of cost required expenses ofand asset procurement, the operational costs ofcapital asset vehicles (Beaujon Turnquist, 1991), trading-off the vehicles (such aswhole cargo-bike, electric vehicles) restriction. Thus, a bicycles, conventional vehicle can operate intime the throughout the whole working day without any time utilization, the customer-service level, and minimization area only for a portion of the time that an ecological vehicle expenses of asset procurement, the operational costs of asset restriction. Thus, a conventional vehicle can operate in the (Beaujon and Turnquist, 1991), trading-off the capital utilization, the customer-service level, and cost minimization expenses of asset procurement, the operational costs of asset throughout the whole working day without any time area only for a portion of the time that an ecological vehicle restriction. Thus, a conventional vehicle can operate in the goals (Wu ettheal.,customer-service 2005). can provide; ifa portion the company does not have the possibility to utilization, level, and cost minimization area only for of the time that an ecological vehicle expenses of asset procurement, the operational costs of asset goals (Wu ettheal.,customer-service 2005). utilization, level, and cost minimization restriction. Thus, a conventional vehicle can operate in the can provide; if the company does not have the possibility to area only for a portion of the time that an ecological vehicle use the conventional vehicles on other activities (i.e. goals (Wu et 2005). provide; if the company does not have the possibility to utilization, customer-service and cost Urban freight transport, also deals withminimization a vast array can goals (Wu ettheal., al., 2005). however,level, area only for a portion of the time that an ecological vehicle use the conventional vehicles on other activities (i.e. can provide; if the company does not have the possibility to in non-LTZvehicles areas) then the return on (i.e. the Urban freight transport, however, also deals to with a vast array delivering use the conventional on other activities goals (Wu et al., 2005). of restrictions and constraints. Restrictions road transport can provide; if the company does not have the possibility to delivering in non-LTZ areas) then the return on the use the conventional vehicles on other activities (i.e. Urban freight transport, however, also deals with a vast array for acquiring the fleet may bethe extremely low. On of restrictions and constraints. Restrictions to road transport Urban freightthe transport, however, also deals with a vast array investment delivering in non-LTZ areas) then return on the (at present, most relevant mode of transport in urban use the conventional vehicles on other activities (i.e. investment for acquiring the fleet may be extremely low. On delivering in non-LTZ areas) then the return on the of restrictions and constraints. Restrictions to road transport other side, some types of conventional vehicles can Urban freight transport, however, also deals with a(i.e. vast array the (at present, the most mode of transport in urban of restrictions and constraints. Restrictions to road transport investment for the fleet may be extremely low. On areas) are imposed byrelevant the urban morphology streets delivering in acquiring non-LTZ areas) then the return on than the the otherlarger side, some types of conventional vehicles can investment for acquiring the fleet may bedistance extremely low. On (at present, the most relevant mode of transport in urban provide payload or longer fuel range of restrictions and constraints. Restrictions to road transport areas) are imposed by the urban morphology (i.e. streets (at present, the most relevant mode of transport in urban the other side, some types of conventional vehicles can arrangement, street size and capacity) and by access policies investment for acquiring the fleet may be extremely low. On provide larger payload or longer fuel distance range than the other side, some types of conventional vehicles can areas) are imposed by the urban morphology (i.e. streets or evenpayload electric orvehicles (EV), thus being ablethan to (at present, the most mode of general, transport inpolicies urban bicycle arrangement, street size and and by access areas) are imposed byrelevant the capacity) urban morphology (i.e. streets provide larger longer fuel distance range meant to govern the urban traffic. In by access the other side, some types of conventional vehicles can bicycle evenquantities electric orvehicles (EV), thus being ablethan to provide or larger payload longer fuel distance range arrangement, street size and capacity) and by access policies transport larger with one trip. areas) are imposed by the urban morphology (i.e. streets meant to govern the urban traffic. In general, by access arrangement, street size and capacity) and by access policies bicycle or even electric vehicles (EV), thus being able to policies wegovern refer the to the entire set ofIn rules governing the transport provide or larger payload orvehicles longer fuel distance range than larger quantities with one trip. bicycle even electric (EV), thus being able to meant to urban traffic. general, by access arrangement, street size and capacity) and by access policies policies we refer to the entire set of rules governing the meant to govern the urban traffic. In general, by access transport larger quantities with one trip. access toweanrefer area:tosuch rules can be rules related to physical bicycle orlarger even electric vehicles (EV), thus that being ablethe to Considering alsoquantities the availability of trip. incentives reduce transport with one policies the entire set of governing the meant to govern the urban traffic. In general, by access access area: such rules can beorrules related to physical policiestoweanrefer torelated the entire setsize of governing the Considering alsoecological the availability of trip. incentives that reduce the characteristics (i.e. to the the weight of the transport larger quantities with one investment in vehicles, the company wants to access anrefer area: such rules can be related to physical also the of that reduce the policiesto the entire set of characteristics (i.e. related to the size orrules the governing weight of the Considering access towe area: such rules be related to investment in vehicles, the company wants to Considering alsoecological the availability availability of incentives incentives that the vehicles) asanwell astoto the time ofcan day (i.e. access onlyphysical during define the best fleet composition considering thereduce different characteristics (i.e. related to the size or the weight of the investment in ecological vehicles, the company wants to access to an area: such rules can be related to physical vehicles) as well as to the time of day (i.e. access only during characteristics (i.e. related to the size or the weight of the Considering also the availability of incentives that reduce the define the best fleet composition considering the different investment in ecological vehicles, the company wants to pre-specified time windows). Time-window access characteristics offleet bothcomposition conventionalconsidering and ecological vehicles vehicles) as well as to the time of day (i.e. access only during define the best the different characteristics (i.e. related to the size or the weight of the pre-specified time windows). Time-window access vehicles) as well as to the time of day (i.e. access only during investment in ecological vehicles, the company wants to characteristics of both conventional and ecological vehicles define the best fleet composition considering the different restrictions are probably one of the most common policy to such as payload, working hours, traveling time, and costs. To pre-specified time windows). Time-window access of both conventional and ecological vehicles vehicles) as are wellfreight as to the timeof ofthe daymost (i.e.etcommon access only during restrictions probably one policy to characteristics pre-specified time windows). Time-window access define the best fleet composition considering the different such as payload, working hours, traveling time, and costs. To characteristics of both conventional and ecological vehicles manage urban access (Muñuzuri al., 2005), since this end, in this paper we hours, develop a tactical model aiming to restrictions are probably one the policy to such as working traveling time, and costs. To pre-specified time access windows). Time-window access manage urban (Muñuzuri etcommon al., 2005), since restrictions are freight probably one of of the most most common policy to this characteristics both we conventional and ecological vehicles in thisofpaper develop a tactical model to suchend, as payload, payload, working hours, traveling time, and aiming costs. To manage urban freight access (Muñuzuri et al., 2005), since in we develop aa tactical model to restrictions are freight probably one of the most etcommon policy to this manage urban access (Muñuzuri al., 2005), since suchend, as payload, working traveling time, and aiming costs. To this end, in this this paper paper we hours, develop tactical model aiming to manage urban Copyright © 2018freight IFAC access (Muñuzuri et al., 2005), since 589 this end, in this paper we develop a tactical model aiming to 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 589Hosting by Elsevier Ltd. All rights reserved. Peer review©under of International Federation of Automatic Copyright 2018 responsibility IFAC 589Control. Copyright © 2018 IFAC 589 10.1016/j.ifacol.2018.08.381 Copyright © 2018 IFAC 589
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
Roberto Pinto et al. / IFAC PapersOnLine 51-11 (2018) 582–587
support the company in assessing the trade-offs that can emerge at a tactical level.
addressed by Klosterhalfen et al. (2014) developing a model to determine the optimal structure and size of a rail car fleet at a chemical company.
To this end, the paper is structured as follows: in Section 2, we provide an abridged literature review underlining the main aspects of the fleet sizing and composition problem. Such a review is then used to position the main contributions of this paper. Section 3 discusses the main elements of the fleet composition problem in an urban environment, underlining the role of the access restriction policies in the decision-making process. In this section, we also present the optimization model. Section 4 reports a simple example of application of the model, along with the main limitations of the model itself. Finally, some conclusions in section 5 close the paper, proposing further avenues of research in the urban freight fleet composition subject.
Fleet composition problems are often combined with other transportation problems: for example, Hoff et al. (2010) describe industrial aspects of combined fleet composition and routing in maritime and road-based transportation. Wu et al. (2005) address a fleet-sizing problem in the context of the truck-rental industry. The authors address both operational decisions (i.e. demand allocation and empty truck repositioning) and tactical decisions (i.e. asset procurements and sales) considering that assets of different ages carry different costs. Redmer et al. (2012) consider the problem of composing an optimal fleet of tankers in a central distribution system. The decision problem is considered as a vehicle assignment in which different types of vehicles are assigned to customers’ orders (resulting in the construction of an average / typical routes) and formulated as a single objective mathematical programming model. Alem et al., (2016) develop a model to help deciding how to rapidly supply humanitarian aid to victims of a disaster, taking into account practical characteristics such as the fleet sizing of multiple types of vehicles. To take uncertainty into account, the authors propose a scenario-based approach within a two-stage stochastic programming paradigm. One of the goals of the model is to define the overall capacity of each type of vehicle in order to perform effective emergency logistics once the disaster strikes.
2. BACKGROUND The capacity of a transportation system is directly related to the number and types of available vehicles. Determining the optimal number of vehicles requires a trade-off between the ownership and operating costs of the vehicles, and the potential costs or penalties associated with not meeting a portion of the overall demand (Beaujon and Turnquist, 1991; Wu et al., 2005). In literature, we can found at least two macro-classes of problems (Etezadi and Beasley, 1983; Du et al., 2016): 1.
Vehicle fleet size problems: given a vehicle type to be operated, the decision relates to the number of vehicles to invest into.
2.
Vehicle fleet composition problems: the decisions relate to the type of vehicles to operate (mix) and the number of each type.
583
The problem under analysis can be represented with a twostage stochastic programming model. However, the complexity of the model is relatively high, given the number of variables; the resulting model is complex and computationally demanding. Our goal is to formulate a simple yet relevant model. Thus, the model discussed in this paper addresses a fleet composition problem for a company operating according to a one-to-many transport scheme (that is, the typical case of a fleet of vehicles departing from a single depot and visiting several destinations). The model is deterministic, although it considers input (such as the demand) that comes from a statistical distribution, similarly to a deterministic equivalent of stochastic programming model with recourse. With respect to the literature, the proposed model introduces two new aspects: i) the constraint imposed by the LTZ in terms of working hours, and ii) the definition of the optimal service level, trading-off revenues and costs.
According to Żak et al. (2011), fleet composition problems can be classified according to several criteria. For example, it is possible to distinguish between one-to-one, one-to-many and many-to-many transportation schemes, or between homogeneous and heterogeneous fleet problems. Framing the time horizon of the decisions is relevant: in general, strategic decisions define markets and customer service levels, tactical decisions determine asset purchases and sales, and operational decisions include asset allocation, assignment, routing, and/or scheduling decisions (Wu et al., 2005). Under this time framework, thus, both the abovedescribed classes of problems concern tactical decisionmaking processes.
3. THE FLEET COMPOSITION PROBLEM UNDER ACCESS RESTRICTION POLICIES
Deterministic models usually stress the spatial structure of the problem neglecting the temporal dimension, and particularly the stochastic nature of demand. List et al. (2003) illustrate a solution strategy that incorporates uncertainty into optimization models for fleet sizing, with a focus on the uncertainty arising from the future demand and the productivity of the vehicles. Their focus on robust optimization allows modelling an explicit trade-off between the level of fleet investment and the level of risk in the solution. Uncertainty in demand and travel times is also
In a fleet composition problem, we are not concerned with short-term decisions, such as the specific vehicle that one should use to supply a certain customer, but with long-term decisions concerning the number and type of vehicles that the company should operate (Etezadi and Beasley, 1983). More assets allow a company to achieve a higher level of customer satisfaction at the expense of higher capital and holding costs (Wu et al., 2005). Because vehicles are generally long-lived assets, there is intrinsic uncertainty about the demands that 590
IFAC INCOM 2018 584 Bergamo, Italy, June 11-13, 2018
Roberto Pinto et al. / IFAC PapersOnLine 51-11 (2018) 582–587
they will serve over their lifetime, and about the conditions under which they will operate (List et al., 2003). Thus, in order to develop the model, we need to discuss some assumptions first.
For example, Fig. 1 depicts two different demand patterns of the daily demand d j over a 250-day horizon; these demand patterns can be determined either from historical data or via forecast. The design capacity (represented with a grey, dotted line in Fig.1) can be chosen in such a way that it covers for a given percentage of the demands, thus defining a service level measure. However, this opens the problem related to the appropriate choice of the design capacity and the related service level. In Fig. 1, the design capacity is set to a value that allows covering for at least 90% of the daily demands, that is, the capacity is enough to satisfy the demand of at least 90% of the considered period (i.e. 225 over 250 days). This also means that the company accepts to resort to another company to complete the deliveries for at most 25 days in a year. Defining the design capacity at a level lower that 100%, we should balance the trade-off between having too much and to little capacity.
We consider a company (typically, a parcel company) delivering parcels in a LTZ. The company wants to address the fleet composition problem defining the mix of vehicles that maximizes the company’s profit over a medium- to longterm planning horizon (3-5 years), considering relevant revenues and costs. We consider a revenue proportional to the quantity delivered to the customers (that is, the revenue can be calculated multiplying a fixed price by the quantity delivered). One of the first decision that should be made in this context is about how to measure the demand from the company’s perspective. Indeed, this affects the way in which we should model the capacity of the fleet. Transport activities can be measured in terms of weight of the goods transported, in terms of number of parcels, in number of trips from the departing depot, and the like. The selection of the unit of measure often depends upon the scope and goal of the model. For example, for one-to-one transportation scheme it can be relevant to measure the number of trips, whereas for parcel delivery can be better expressing the demand in terms of number of parcels or in number of required deliveries.
Daily demand
Design capacity
20000 19000
Demand
18000
15000 14000 13000 12000 11000 10000 1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211 221 231 241
Since the company wants to tackle the problem at a tactical level (as defined in Wu et al., (2005)), the details of the forecasted demand are extremely difficult to get. Thus, in order to use an aggregate measure that reduces the variability in the forecast, in this study we represent the demand and the capacity of the fleet in terms of number of deliveries rather than in terms of number of parcels or in terms of kilograms/tons. This assumption greatly affects the model that is presented further on. However, the peculiarities mainly regard the conversion from one unit of measure to another, whereas the overall structure of the model remains the same. We also assume that the demand of each customer is for small volumes, lower than the carrying capacity of each individual vehicle. In fact, the company operates a one-tomany transportation scheme.
17000 16000
Days Design capacity 20000 19000 18000
Demand
17000 16000
15000 14000 13000 12000 11000 1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211 221 231 241
10000
Days
The considered costs involve the vehicles purchasing (investment), operating and maintenance cost, as well as the cost to outsource part of the activities whenever needed (penalty cost). In fact, the size and mix of the fleet define also the overall transport capacity; however, such size and mix should be defined considering an estimated demand that can vary over the considered period. Facing a volatile demand, if we define the overall capacity based on the peak demand, we may end up with a relatively large portion of the capacity unsaturated, whereas if we use the average value of the demand we may need to resort to outsourcing way too often, with consequent cost increase. To this end, the model is developed considering the distribution of quantities to be delivered over time (i.e. on a yearly basis) and a minimum capacity that should be guaranteed in order to attain a good service level at a sustainable cost.
Fig. 1. Estimations of the design capacity based on different demand patterns If we define D * as the design capacity (also referred to as the critical demand) it means that in the days in which the demand is higher than D * we would not be able to fulfil it, unless we resort to the extra capacity provided by a third party parcel courier, of course at a higher cost. This cost represents the penalty associated with not meeting a portion of the overall demand (Beaujon and Turnquist, 1991). It is worth noticing that defining the design capacity considering the number of days that the company wants to cover with its own fleet does not necessarily provide complete information about penalty cost of resorting to a third party.
591
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
Roberto Pinto et al. / IFAC PapersOnLine 51-11 (2018) 582–587
One way to address the problem of defining the design capacity is to consider the fleet composition problem as a version of the classical newsvendor problem. In fact, the decision maker is required to make a decision that cannot be changed in the short term, trading off the risk of having too much or too little capacity during the year. However, the calculation of an equivalent fractile ratio in the discussed context can be difficult and not accurate. An alternative approach, used in this paper, is to consider the design capacity as a variable of the model that should be optimized considering all the relevant revenues and costs.
reducing the average number of delivery per hour. Thus, we assume Pe Pc e E , c C .
Finally, our proposed model does not include routing aspects since we focus on a long-term perspective. At this level, demand, costs and revenue uncertainties related to fleet operations are high, thus taking into account routing aspects on a detailed level is ineffective (Du et al., 2016; Hoff et al., 2010).
H k = working time for a vehicle k K . In the considered case, the working time is different for the two categories of vehicles. In fact, due to access restriction policies, we assume that H e H c e E , c C .
m = average unit margin for delivery (i.e. €/delivery)
Co = cost of using a third party parcel courier, expressed in the same unit of measure of the unit margin m (i.e. €/delivery)
F k = fixed cost for operating a vehicle k K ,
including ordinary maintenance cost. We assume that operating and maintaining a conventional vehicle is usually more expensive, that is Fe Fc e E , c C ). The value F k is an average value, since in reality maintenance costs can depend upon the use of the asset (i.e. the number of trips, or the distance travelled).
3.1 Notation The model uses the following notation:
T = time horizon on which the investment decision is assessed (i.e. 5 years)
J = set of days per year of the time horizon (i.e. J 1, , 250 )
K = set of vehicle types available for the fleet mix decision.
C K = subset of conventional vehicles, including pick-ups, vans, trucks, and lorries with internal combustion engines.
E K = subset of ecological vehicles, including
for
one
vehicle of
M = maximum number of vehicles composing the fleet. This parameter may reflect the necessity to limit the number of vehicles to be operated to contain cost or other constraints (such as lack of space) affecting the company’s operations.
The number of deliveries per hour Pk depends upon several factors, such as the payload of the vehicle, the distance between the customers, the speed of the vehicles, but also the traffic congestion and the morphology of the city. Therefore, this parameter may be extremely difficult to estimate; however, in this paper we assume it as deterministic and known, postponing the discussion about its determination to a future research work.
d j = daily (estimated) demand for day j J over
the year, in terms of number of deliveries. These values may come from a probability distribution (i.e. the values d j are the realization of the sampling
Considering the fleet composition problem, the main decision variables are the number of vehicles per each type, and the design capacity, that is:
from a known statistical distribution) or from historical data. We further assume that the deliveries cannot be postponed or anticipated (i.e. all the demand must be served in full either via the owned fleet or via a third party parcel delivery service). This assumption is supported by the tactical level approach of the model.
= investment cost
Ik
type k K .
electric vehicles (EV), bikes and cargo-bikes. The two sets E and C are not overlapping; that is E K \ C .
585
Y k = Number of vehicles of type k K in the fleet.
D = Optimal design capacity.
*
However, to build the model we need some further variables as follows:
Pk = average number of deliveries per hour with a
vehicle k K . We assume that conventional vehicles have usually a larger payload, so they can carry more parcels (i.e. more deliveries) and can drive faster than, for example, bicycles. Vehicles with a small payload may require going back to depot in order to load new parcels to deliver, further
q j = Number of deliveries on day j J with the
company’s own fleet.
592
Z = Total capacity (in terms of number of possible deliveries) of the company’s own fleet.
IFAC INCOM 2018 586 Bergamo, Italy, June 11-13, 2018
Roberto Pinto et al. / IFAC PapersOnLine 51-11 (2018) 582–587
constraints (7) and (8) define the types of the variables. The model thus defined is similar to a chance-constrained problem (Charnes and Cooper, 1959) where we require that the probability of not being able to satisfy the demand be less than a given threshold.
3.2 The model The objective of the model is to maximize the profit considering the margin (discounting the fleet variable operations costs from the revenues), the fleet ownership costs (i.e. the investment cost plus the fixed operations and maintenance costs) and the penalties for not having enough capacity to meet the demand.
4. APPLICATION AND LIMITATIONS OF THE MODEL The model discussed in Section 3 is a simple one; yet, it can be used to perform interesting what-if analysis. Let us assume for the sake of the example that the daily demand is normally distributed with 500 deliveries and 125 . Sampling a business year from this distribution (i.e. 250 values) we can get a demand signal similar to the one represented in the upper part of Fig. 1. Let us assume that the fleet can be composed of no more than eight vehicles chosen among those listed in Table 1. Vehicles C1 and C2 are conventional vehicles with a high payload, whereas vehicles E1 and E2 are ecological vehicles (i.e. E1 can be an EV small truck and E2 a cargo bike). With this data, the model can be solved in less than one second (Gurobi solver 6.0 on a Core i7 computer with 8GB of memory).
The resulting model is as follows: max T
max
1
m q j Co d j q j j J
kK
Fk Y k
(1)
I k Yk
kK
where
1
1 r
is the discounting factor at the r rate.
The term m q j expresses the operations margin in day j, whereas the term Co d j q j represents the penalty cost of resorting to a third party parcel delivery service in day j. Clearly, the margin and the penalty cost depend upon the decision about the design capacity D * . The constraints are as follows:
P
k
H k Yk Z
(2)
kK
Z D
*
qj d
q
j
Z
Y
k
j J
(4)
j J
(5)
M
k
0 int
(6)
Y k 0 int *
q j 0, Z 0, D 0
k K
j J
H
C1
15
C2
Fk
Ik
3
12,000
20,000
11
3
11,000
30,000
E1
15
8
8,500
35,000
E2
6
8
1,100
2,500
k
Considering an average margin per delivery m 2 . 5 € and a penalty Co = 5.0 €, the solution of one realization of the demand dj determines a design capacity D* equal to 608 deliveries per day to serve a total demand of about 124,800 deliveries during a year; to this end, the optimal fleet is composed of four vehicles E1 and four vehicles E2. The company must resort to a third party company to cover for 3,509 deliveries during the year, about 2.8 % of the total demand.
kK
X
Pk
Table 1. Data for the example
(3)
j
Vehicle
(7)
4.1 Robustness of the solution
(8)
Clearly, the results reported in the previous example are extremely partial, and can lead to a wrong conclusion: in fact, those results are related to a specific realization of the demand (i.e. the sampling from the demand distribution specified above). However, thanks to the short time required, it is possible to solve the model several times, each time with a different realization of the demand d j. In this way, we can analyse the robustness of the solution (i.e. the change of the optimal solution according to change in the demand) and the average performance.
Constraint (2) defines the overall capacity Z of the fleet in terms of possible deliveries as a function of the number of delivery per hour Pk , the daily working hours H k and the number of vehicles per each type, Y k . As required, the capacity Z is such that it covers at least the design capacity D * (but can be larger), as expressed by constraint (3). Constraints (4) and (5) together limit the number of deliveries q j to the minimum between the overall
So, let us assume to sample 100 realizations d j from the same distribution specified above, and to solve the model for each realization. The optimal solution S alternates between S1 = (E1=4, E2=4) and S2 = (E1=5, E2=3), with a design capacity
capacity Z and the actual demand d j . Constraint (6) limits the size of the fleet to M vehicles; in fact, a large number of vehicles may introduce a managerial overload. Finally, 593
IFAC INCOM 2018 Bergamo, Italy, June 11-13, 2018
Roberto Pinto et al. / IFAC PapersOnLine 51-11 (2018) 582–587
of 608 and 696 deliveries per day, respectively. The investment is equal to 150,000 € for S1 and 182,500 € for S2. Delving into the results, the solution S1 is the optimal one in only 31 cases out of 100, thus indicating that S2 can provide a very good coverage against the volatility of the demand. The average profit of solution S1 is about 807,000 € whereas the average profit of solution S2 is about 816,000 €.
587
off the cost of owning the vehicles with the cost to outsource part of the activities whenever needed to guarantee the delivery by the required time. The resulting model is quite simple, yet it can be used to perform a sensitivity analysis. An interesting development is the comparison of the simple procedure outlined in this paper against a stochastic programming approach. As discussed in the limitations section, this model should be considered as an initial model towards the definition of a decision support system in the context of fleet composition problems in the urban freight area.
4.2 Limitations The model discussed in the previous section has several limitations that must be brought about and addressed by further research. The model addresses the fleet composition problem at a tactical level, thus using aggregate data and information. This allows creating a relatively small model. For example, aggregating the demand on the time dimension reduces the number of variables to be considered. However, at the same time this aggregation hides the complexity of the transport problem when the vehicles have to perform multiple stops during their trips. Clearly, some levels of details may not be available at the time the decision is made (i.e. we do not know which customer should be server each day).
REFERENCES Alem, D., Clark, A., Moreno, A. (2016). Stochastic network models for logistics planning in disaster relief. European Journal of Operational Research. 255(1), 187-206. Beaujon, G.J., Turnquist, M.A. (1991). A Model for Fleet Sizing and Vehicle Allocation. Transportation Science, 25(1), 19-45. Charnes, A., Cooper, W.W. (1959). Chance-Constrained Programming. Management Science. 6(1), 73-79. Chen, Q., Conway, A. (2016). Commercial Vehicle Parking Availability and Behavior for Residential Delivery in New York City. Transportation Research Board 95th Annual Meeting. Du, J.Y., Brunner, J.O., Kolisch, R. (2016). Obtaining the optimal fleet mix: A case study about towing tractors at airports. Omega. 64, 102-114. Etezadi, T., Beasley, J.E. (1983). Vehicle Fleet Composition. The Journal of the Operational Research Society. 34(1), 87-91. Hoff, A., Andersson, H., Christiansen, M., Hasle, G., Løkketangen, A. (2010). Industrial aspects and literature survey: Fleet composition and routing. Computers & Operations Research. 37(12), 2041-2061. Klosterhalfen, S.T., Kallrath, J., Fischer, G. (2014). Rail car fleet design: Optimization of structure and size. International Journal of Production Economics. 157 (1), 112-119. List, G.F., Wood, B., Nozick, L.K., Turnquist, M.A., Jones, D.A., Kjeldgaard, E.A., Lawton, C.R. (2003). Robust optimization for fleet planning under uncertainty. Transportation research Part E. 39 (3), 209-227. Muñuzuri, J., Larrañeta, J., Onieva, L. and Cortés, P. (2005). Solutions applicable by local administrations for urban logistics improvement. Cities. 22(1), 15-28. Redmer, A., Żak, J., Sawicki, P., Maciejewski, M. (2012). Heuristic approach to fleet composition problem. Procedia - Social and Behavioral Sciences. 54, 414-427 Wu, P., Hartman, J.C., Wilson, G.R. (2005). An integrated model and solution approach for fleet sizing with heterogeneous assets. Transportation Science. 39(1), 87103. Żak, J. Redmer, A., Sawicki, P. (2011). Multiple objective optimization of the fleet sizing problem for road freight transportation. Journal of advanced transportation. 45(4), 321-347.
Further, the capacity of the fleet can be expressed in different ways, and this decision affects the resulting model. In one-toone transportation scheme, the number of required trips can be used; in such a context, the capacity of the fleet depends on the location of the customers (i.e. their distance from the depot) that determines the number of trips each vehicle can perform in a period. In one-to-many transport scheme, instead, we use the same vehicle to serve several customers with one trip. In this case, the number of trips may be not linearly correlated with the number of deliveries: one trip can serve more customers than another trip, depending upon the demand of each customer visited, and their distance from the depot and from each other. To represent the capacity of the fleet, we thus used the number of average deliveries per unit of time Pk , a measure that is also commonly reported by couriers as a performance KPI. However, we understand that assessing the values of Pk can be extremely convoluted, due to the large number of elements affecting it (i.e. payload of the vehicle, distance between the customers, the speed of the vehicles, traffic congestion during the day, morphology of the city). This requires further investigation in order to provide more robust decisions. The model considers the same demand for each year of the considered time horizon T . This has been assumed to reduce the number of variables in the model. However, the demand can vary over T, which can span over several years. Actually, the model can be easily changed to include the demand over each year in T, increasing the number of variable q j from J to T J .
5. CONCLUSIONS This paper addresses the fleet composition problem in the context of urban freight distribution, where access limitations to the delivery areas may affect the possibility of using specific vehicles. The fleet composition problem must trade594