The model, which is applicable to small-medium size urban areas, was applied to .... during business days of the winter period. The analysis covered an all day ...
CSCE 2011 General Conference - Congrès générale 2011 de la SCGC
Ottawa, Ontario June 14-17, 2011 / 14 au 17 juin 2011
Mathematical Optimization of Commercial Vehicle Parking Stalls in Urban Areas S.M. Easa and G. Dezi Department of Civil Engineering, Ryerson University, Toronto, ON, Canada
Abstract: Better management of commercial vehicles in urban areas can bring many benefits, including reduction of air and noise pollution and an improved traffic level of service. This paper presents a mathematical optimization model for determining the optimal locations of parking stalls (loading and unloading areas) in urban areas subject to physical and operational constraints. The optimization model focuses on single commercial vehicles. The formulated model is nonlinear and can be solved using available commercial software, such as Premium Solver. The model, which is applicable to small-medium size urban areas, was applied to the city of Bologna, Italy. The master plan of the city of Bologna includes innovative aspects related to management of access to traffic zones and proper use of parking spaces through optimization of parking areas. The results of the proposed model were verified using an exhaustive search. The model presented in this paper represents an efficient management tool for freight transportation logistics in urban areas.
1. Introduction Freight transportation in urban areas is an important element of the total transportation system. Urban freight transport and logistics operations deal with the activities of delivering and collecting goods in city centres and towns (European Commission 2011). These activities, normally referred to as city logistics, entail the processes of transportation, handling, and storage of goods, the management of inventory, waste and returns as well as home delivery services. The analysis of the transportation system usually involves conflicting objectives. On one hand, there is a need to ensure an efficient distribution of goods that is capable of responding to requests from retailers to achieve just-in-time policy. On the other hand, there is a desire to place restrictions on freight traffic to minimize environmental impacts (Maggi 2001). To protect the collective interest, trade-off measures that reconcile these conflicting objectives are required. The regulations of most cities normally identify solutions to regulate access, circulation, and the parking of commercial vehicles in urban centers, implement policies without restrictions that harm economic and social prosperity (Schäffeler and Wichser 2003). For example, in the City of Bologna, the transportation of goods represents a primary element of importance. Some technical solutions for the management of parking and access for the transportation of goods have been developed by Dezi (2010). The objective of his study was to allow on-time delivery and to mitigate traffic induced issues toward citizens. In particular, attention was given to various issues related to the loading and unloading areas, and the development of a graphical optimization method for determining the locations, number, and size of parking stalls (Fig. 1). This paper presents a mathematical optimization model for determining the locations of parking stalls in urban areas that maximizes their coverage of the existing commercial stores. The following sections
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Figure 1: Examples of parking stalls in urban area
present the model formulation and its verification. Application of the model to a portion of the City of Bologna is then presented, followed by the conclusions.
2. Mathematical Optimization Model Consider first the simple case of a single parking stall located along a linear street segment AB (Fig. 2). The coordinates of the segment ends, with respect a Cartesian coordinate axes x and y, are (xA, yA) and (xB, yB). The street segment has m commercial stores located along its side. The coordinates of store i is (xi, yi), i = 1, 2, …, m. Each store has a weight, W i, that represents the attraction/generation in terms of loading and unloading, thus reflecting the density of the commercial area. This weight is expressed in terms of the number of equivalent commercial stores (ECS). The parking stall has coordinates (x p, yp) which are the (unknown) decision variables. The parking stall is used for loading and unloading, and serves the stores that are located within a circle with a radius r, where the centre of the circle is the centre of the parking stall. The sum of the weights of the stores that are located within the circle represents the coverage of the parking stall. It is required to determine the location of the parking stall that maximizes its coverage. The objective function can be written as [1]
Maximize z
with W i = 0 for Dpi ≥ r
Wi
where Dpi is the distance between parking stall p and store i which is given by [2]
2
2 0.5
Dpi = [(xp - xi) + (yp - yi) ]
To simplify the formulation, with negligible effect on accuracy, the centre of the parking stall is assumed to lie on the linear street side. Thus, to ensure that the centre of the parking stall lies on the linear street segment, the following constraint is required [3]
SAB = (yp – yA) / (xp – xA)
where SAB = slope of segment AB which is given by [4] SAB = (yB – yA) / (xB – xA) Note that the left side is the slope of line AP. In addition, the following constraints are added, [5]
yp ≤ yB
[6]
yp ≥ yA
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Figure 2: Geometry of a single parking stall on a street segment AB
Figure 3: The case when space available for parking on street segment is discrete
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Consider the case where only discrete portions of the street segment are available as parking stalls (Fig. 3). In this figure, there are two portions available for parking, uv and fg. In this case, the location of the parking stall that maximizes the coverage could be on either portion. This case can be formulated using a binary variable λ. The respective constraints are written as [7]
xp ≥ λ xf + (1 - λ) xu
[8]
xp ≤ (1 - λ) xv + λ xg
where xu, xv, xf, xg = x-coordinates of the portions u, v, f, and g, respectively, and λ = binary variable which equals 0 if the parking stall lies on the first segment and 1 if it lies on the second segment. Similar constraints can be used to allow a parking stall to lie on different street segments. Suppose that there is another parking stall on a nearby street segment (Fig. 4). Let the centres of the parking stalls be denoted by p1 and p2. Then, the constraints for p2 will be similar to those of Eqs. 2-5 and the coverage of p2 is added to the objective function of Eq. 1. However, in this case an additional constraint is needed to ensure that the coverage of p2 does not overlap with that of p1. This is satisfied using the following constraint, [9]
Dp1 p2 ≥ 2r
where Dp1 p2 = distance between the centres of parking stalls p1 and p2 which is given by [10]
2
2 0.5
Dp1 p2 = [(xp1 – xp2) + (yp1 – yp2) ]
The preceding formulation can be easily extended for the general case where there are n parking stalls on different street segments.
Figure 4: Geometry of two parking stalls on nearby street segments
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Table 1: Data for verification of two-part parking segment Store Number 1 2 3 4 5
ECS Amount 1 3 2 5 2
Location (m) xs 20 50 110 130 160
ys 0 0 0 0 0
The proposed optimization model, which is a nonlinear and non-convex, was solved using the Premium solver software (Frontline Systems 2005). The software implements the Generalized Reduced Gradient technique that uses a multi-start strategy for global optimization. The strategy generates candidate starting points with randomly selected values within the specified bounds of the parameters. These points are then grouped into clusters that are likely to lead to the same locally optimal solution. The solver is then run from a representative point in each cluster and continues with successively smaller clusters that are likely to capture each locally optimal solution. A Bayesian test is used to determine whether the process should continue (Carlin and Louis 2008), and ultimately the software converges in probability to a globally optimal solution. Other optimization software, such as LINGO (Schrage 2006), can also be used to solve the model.
3. Model Verification Consider a street segment, AB, that has five commercial stores with the amount of ECS and the locations given in Table 1. The segment is horizontal with coordinates (xA = 0, yA = 0) and (xB = 200, yB = 0). The segment has two parts available for parking, whose x-coordinates are (xu = 10, xv = 50) and (xf = 110, xg = 160). The radius of the coverage of the parking stall is r = 50 m. Applying the proposed model, the optimal parking slot that maximizes the total ECS has a location with coordinates (xp = 144.56, yp = 0). The optimal objective function is 9 and the binary variable λ = 1, indicating that the parking slot lies on the second segment. The optimal location of the parking stall covers Stores 3, 4, and 5. The distances between the parking stall and the three stores are 34.56 m, 14.56 m, and 15.44 m, respectively, which are less than 50 m. The distances from Stores 1 and 2 are 124.56 m and 94.56 m, respectively, which exceed the maximum coverage radius. The location of the preceding optimal solution can be verified by solving the problem graphically. Note that there are other locations that generate the same value of the objective function.
4. Application to City of Bologna 4.1 Freight Characteristics The city of Bologna has a plan for the distribution and collection of goods in an urban area called MerciBo2. This plan contains innovative aspects related to the management of access to limited traffic zones (LTZ) and the proper use of parking spaces, through optimization of the areas used for loading/unloading goods (Municipality of Bologna 2005). The reorganization of the distribution of goods will reduce the high number of ownership deliveries that occupy public spaces that are already deficient. This usually moderately contributes to the distribution of goods. Another major problem is the level of the vehicle’s load: 67% of the vehicles use less than 25% of their capacity and 12% use less than 50% of their capacity. There are margins for improving the distribution of LTZ to achieve a better use of the parking spaces, a greater utilization of vehicles, and a transfer of market share from ownership to third parties. The survey focused only on the categories of freight vehicles (vans < 3.5 t < box trucks < 7.5 t < trucks).
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Figure 5: Type and reason of non-use of SCVLZ
To quantify the number of commercial vehicles that enter the historic district daily, traffic count data collected in 2004 were used. The data were collected at nine electronic gates for the traffic entering LTZ during business days of the winter period. The analysis covered an all day interval of the limited traffic zone (7:00 am – 8:00 pm). There were approximately 1,900 commercial vehicles that entered LTZ daily, of which 841 have ownership, and 941 are third-party deliveries, for a total of 29,300 deliveries. The ratio of pick-ups to deliveries is 0.17. With the same number of vehicles, the deliveries carried out by third parties (15,100 operations: 12,600 deliveries and 2,500 pick-ups) are almost twice the number of deliveries carried out by ownerships (8,800 operations: 7,700 deliveries and 1,100 pick-ups). These data show that the ownerships are less organized and less efficient compared to others (Emilia-Romagna Region 2005). The MerciBo2 plan confirms that to satisfy the demand of the loading/unloading generated from regular distribution activities, it would be necessary to have at least 1,100 MCVLZs (multiple commercial vehicle loading and unloading zones) rather than the 500 currently available. Every delivery takes about 14 minutes for loading/unloading and 27% of pick-ups per day (5,700) are carried out in an area called T (area with limited access). Figure 5 shows the practices of stopping found for commercial vehicles within LTZ and the causes of non-use of the SCVLZs (single commercial vehicle loading and unloading zones). 4.2 Data Collection The Municipality of Bologna has a Geographic Information System called CityTrekWeb (Municipality of Bologna. 2011). This is a thematic online map of the urban center, where the parking areas are reported along with road directions and addresses of both residential and commercial buildings. The SCVLZs use was assessed for a period of 30 minutes (10:30 am to 11:00 am) during the week days in March 2008, on the basis of freight transportation data relative to the vehicles at the nine electronic gates that permit entering LTZ. The time and number of operations of loading and unloading were collected. To complete the study, the research team unanimously verified and photographed the existence, the location and the dimensions of the MCVLZs, as well as the presence and position of their vertical and horizontal traffic signs, the adjacency of the SCVLZs and their location approximately within LTZ (Dezi et al. 2010). The collected data were rearranged in tabular form to facilitate the next phase of the analysis.
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4.3 Data Analysis The main survey showed that the number of SCVLZs currently in the historical center equals 415 units, organized in 175 MCVLZs in various dimensions. The average data were as follows: (a) 48 SCVLZs are occupied by hotels, bins, and sidewalk barriers, (b) 213 SCVLZs are occupied by vehicles which are not authorized, and (c) 154 SCVLZs are free or occupied by vehicles that are authorized for loading and unloading. Excluding the first category from the analysis, which is in fact unsuitable for loading and unloading, the number of SCVLZs equals 367, of which 58% are illegally occupied during the survey. These data are in agreement with the data of the City of Bologna. Using these data, a series of ratios describing the context of use of existing MCVLZs was calculated. The results compared the total number of MCVLZs with those that are free, occupied, free or partially occupied, and partially occupied or occupied. The numbers calculated confirm the current situation in the city of Bologna in the peak hour of a typical business day. A graphical procedure has been used for determining the best locations of parking stalls (Dezi et al. 2010). Considering the MCVLZs as the distribution hub to serve a portion of the historic center, the area of influence of the MCVLZ can be approximated like a circle. The estimate of the size of the circular area served by a MCVLZ is based on an iterative process designed to approximate the areal relationships. Starting from a radius of influence of 70 m - derived from interviews and questionnaires - circles centered at MCVLZs were used. The calculations continued by measuring the areal relationship corresponding to the number considering that the location of MCVLZs can involve the overlap of the corresponding circles. The optimal radius of influence was found to be 50 m.
Figure 6: Commercial stores (+) and optimal parking stalls (blue rectangles) for example network (r = 40 m)
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Table 2: Data for commercial stores Coordinates of ECS x y
ECS Number
Quantity of ECS
1
1.5
-4663.124
-13912.9558
2
4
-4663.7277
-13918.0833
3
3
-4664.3081
-13923.1119
4
4.5
-4657.6039
-13936.575
5
6
-4652.0227
-13937.2581
6
3
-4637.1685
-13939.0629
7
6
-4632.5762
-13939.5648
8
1.5
-4627.6598
-13911.2945
9
3
-4615.2573
-13905.9778
10
3
-4614.0192
-13910.4436
11
1.5
-4611.2222
-13925.8584
12
3
-4610.6593
-13930.894
13
1
-4610.1627
-13936.0297
14
15
-4598.1388
-13937.6178
15
1
-4592.2305
-13935.554
16
5.5
-4587.7312
-13934.1731
17
1.5
-4582.5409
-13932.6169
18
5.5
-4568.9736
-13925.3776
19
1.5
-4570.9838
-13920.6753
20
6
-4573.0328
-13915.9171
21
15
-4560.1172
-13910.1658
22
6
-4558.0971
-13914.8505
23
4.5
-4641.7103
-13983.2276
24
24
-4626.4998
-13965.8905
25
15
-4587.7112
-13961.6998
26 27 28
9 6 6
-4582.9185 -4577.6878 -4556.8245
-13960.5803 -13959.3825 -13954.8194
Table 3: Example of the data for the location of available parking segments Location
Location
x
y
End Point
x
y
1
-4662.2785
-13905.5412
2
-4665.8058
-13936.0883
2
-4665.8058
-13936.0883
3
-4622.8004
-13940.6226
3
-4622.8004
-13940.6226
4
-4628.1255
-13905.5412
5
-4615.6103
-13905.5412
6
-4611.9172
-13919.3684
6
-4611.9172
-13919.3684
7
-4610.1946
-13939.3511
7
-4610.1946
-13939.3511
8
-4599.3852
-13938.3237
8
-4599.3852
-13938.3237
9
-4567.5445
-13928.6283
First Point
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4.4 Results of Proposed Model The proposed optimization model was applied to the example network shown in Fig. 6. The network has 28 commercial stores. The location and amount of ECS for these stores are shown in Table 2. There are two existing parking stalls (number 26 and 27). The possible segments on which each parking stall may be located should be first identified and then the respective constraints can be written. For this example, parking stall p1 may lie on lines 7-8-9 or 18-19. Parking stall p2 may lie on lines 2-3 or 14-15. The objective function for the existing parking stalls is 79. Applying the proposed model, the optimal objective function equals 152 ECS which corresponds to the following locations of the parking stalls (blue rectangles at the centre of the coverage circles): (x p1 = -4567.0, yp1 = -13928.5) and (xp2 = -4647.0, yp2 = 13938.1). The graphical solution yields the same objective function with corresponding locations of the parking stalls. The distance between the parking stalls is 80.6 m, which is greater than 80 m, indicating that the coverage of the parking stalls does not overlap. It is clear that the existing locations of the parking stalls are not efficient, covering 79 ECS compared with the optimal value of 152.
5. Conclusions This paper has presented a mathematical optimization model for determining the optimal locations of parking stalls (loading and unloading areas) in urban areas subject to physical and operational constraints. The model focuses on single commercial vehicle loading and unloading zones, subject to safety and operational requirements. The optimization model involves binary variables to allow exploring all possible solutions. Application of the model showed that it performs as expected, and its results were verified graphically. As the network becomes larger the number of binary variables considerably increases. Therefore, the proposed model is intended for application to small and medium-size urban areas. The model can be extended to include other features such as: (a) Allowing an overlap between the coverage (circles) of adjacent parking stalls as long as the overlap does not contain any served commercial stores. (b) Determining the optimal number of parking stalls required to adequately serve a given urban area. (c) Handling different types of commercial vehicles and sizes of parking stalls. (d) Accommodating other types of constraints, such as the number of ECS that a parking stall can serve and geometry-related constraints. The proposed model represents an initial step in applying mathematical optimization of freight logistics. It should allow for better management of commercial vehicles in urban areas, and for the associated operational and environmental benefits.
6. References Carlin, B.P. and Louis, T.A. 2008. Bayesian methods for data analysis. Chapman & Hall/CRC, Boca Raton, Florida. , Dezi, G., Dondi, G., and Sangiorgi, C. 2010. Urban freight transport in Bologna: Planning of commercial vehicle loading/unloading zones. Proc. of Social and Behavioural Sciences, Internet: www.sciencedirect.com. Dezi, G. 2010. City logistics: urban freight transport in Bologna. Ph.D. dissertation, DISTART, Faculty of Engineering, University of Bologna, Italy. Emilia-Romagna Region. 2005. Logistica urbana a Bologna: elementi per un progetto. Department of Mobility and Transport, Quaderni del Servizio Pianificazione dei Trasporti e Logistica n°8. European Commission. 2011. Urban freight transport and logistics: An overview of the European research and policy. Directorate-General for Energy and Transport. Internet: http://www.transportresearch.info/Upload/Documents/200608/20060831_105348_30339_Urban_freight.pdf Frontline Systems. 2005. Premium solver platform – User guide. Frontline Systems, Inc., Incline Village, Nevada.
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Maggi, E. 2001. Un approccio innovativo per la gestione del trasporto merci in ambito urbano. Working Paper; Department of Architecture and Planning; Polytechnic of Milan Municipality of Bologna. 2005. MerciBo2 Piano per la distribuzione merci in città. Bologna. Municipality of Bologna. 2011. City Trek Web; Comune di Bologna; http://urp.comune.bologna.it. Schäffeler, U. and Wichser, J. 2003. Trasporto urbano di merci e logistica della città. Portal, Materiale didattico sui trasporti. Schrage, L. 2006. Optimization modeling with LINGO. LINDO Systems, Palo Alto, California.
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