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Operations management for urban snow removal and disposal
Transpn. Res.-A. Vol.29A,No. 5,pp. 359-370, 1995 Copyright 0 1995 Elscvis Science Ltd Printed in Great Britain. All rights reserved 0965-8564/9J S9.50 + .OO
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OPERATIONS MANAGEMENT FOR URBAN SNOW REMOVAL AND DISPOSAL JAMES F. CAMPBELL ’ School of Business Administration, University of Missouri-St. Louis, 8001 Natural Bridge Road. St. Louis, MO 63121499. U.S.A. AND& LANGEVIN GERAD and &ole Polytechnique de Montreal, 5255, Avenue Decelles, Montreal, Canada H3C 3J7 (Received 1 April 1994) Abstract- Urban snow removal and disposal operations include a number of challenging strategic and tactical problems that are relatively unstudied. Those operations include spreading de-icers and abrasives, plowing snow, loading snow into trucks, and transporting snow to disposalsites. We describe urbansnowremovalanddisposal operations in detail, identify relevant management problems, review previous studies, and highlight opportunities for future research. We also describe a snow removal and disposal decision support system, currently under development, that addresses several major problems in an integrated fashion.
1. INTRODUCTION
Many cities face a winter problem of clearing large quantities of snow from streets and sidewalks. Plowing, along with warmer temperatures, may be sufficient to control moderate snowfalls. However, in urban areas that experience heavy snowfall and persistent low temperatures, snow plowed to the sides of streets and sidewalks accumulates and impedes pedestrian and vehicular traffic. In many cities, snow is loaded into trucks and hauled to snow disposal sites, where it remains until melting, perhaps several months later. Urban snow removal and disposal operations include several challenging strategic and tactical engineering and managerial problems that are relatively unstudied. This article describes urban snow removal and disposal operations, identifies relevant management problems and reviews analytically-based approaches, and discusses directions for future research. The remainder of the article is organized as follows: Section 2 describes snow removal and disposal activities in Montreal; Section 3 presents operations management problems and relevant literature; Section 4 describes a decision support system for managing urban snow removal and disposal; the conclusion section follows. 2. SNOW REMOVAL AND DISPOSAL IN MONTREAL
Heavy winter snowfall hinders transportation in many cities around the world. Table I lists average snowfall amounts for several North American cities. The city of Montreal is regarded as a world leader in snow removal and disposal operations. Montreal covers 187 square kilometers and is home to over 1 million citizens, which is 55% of the population of the Montreal Urban Community (MUC). Montreal receives about 10 major snow storms each winter and the average annual snowfall is 223 cm with a standard deviation of 63 cm (Ville de Montreal, 1989). For each snowfall exceeding 2.5 cm, snow must be cleared from approximately 2,000 km of streets and 3,200 km of sidewalks. Special attention is paid to pedestrian crossings, fire hydrants, bus stops, and subway entrances. Montreal is divided into 60 sectors for snow removal and the typical sector contains ‘Requests for reprints should be addressed to James F. Campbell. School of Business Administration, University of Missouri, St. Louis, 8001 Natural bridge Road, St. Louis, MO 63121-4499. 359
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Table 1. Average snowfall in selected cities Average Snowfall (cm/year)
City Quebec, Canada Syracuse, USA Rochester, USA Mont&l, Canada Ottawa, Canada Buffalo, USA Toronto, Canada
330 300 230 223 220 170 130
approximately 30 km of roads and 50-60 km of sidewalks. Figure 1 shows a portion of the street network and several sectors as well as disposal sites. Snow removal and disposal operations are conducted simultaneously by 60 crews using up to 1,500 vehicles, including spreader trucks (to spread salt and abrasives), graders, snowplows, sidewalk snowplows, front-end loaders, snowblowers, service vehicles, and trucks for hauling snow to disposal sites. Snow removal equipment is described in detail in Gray and Male ( 1981) and Highway Research Board (1970). Depending on conditions, manpower ranges from 700 to 3,080 workers. The 1991budget for snow removal operations in the city of Montreal was about US $50 million. There are four major steps in snow removal and disposal operations: ( 1) spreading de-icers and abrasives, (2) snow clearing (plowing), (3) snow loading, and (4) snow disposal. The data for Montreal given are from Ville de Montreal (1992). When snow begins to fall, salt and abrasives (sand and/or crushed stone) are spread first on major arteries and then on secondary streets by 100 spreader trucks. Sidewalks are treated by 106 small snowplows/spreaders to ensure pedestrian safety. On average, almost 120,000 metric tons of salt and 28,000 metric tons of abrasives (approximately half sand and half crushed stone) have been used each year. However, in recent years, Montreal has experimented with using less salt and more abrasives: in 1991, approximately 60,000 metric tons of salt, 15,000 metric tons of sand and 53,000 metric tons of crushed stone were used. When the snow accumulation reaches 2.5 cm, snow clearing begins. The equipment used includes 325 street plows, 208 sidewalk plows, and 123 front-end loaders (used
n
Sewer chute
Fig. 1. Street network and snow removal sectors.
Urban snow removal mainly in narrow lanes and dead end streets). In each sector,
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main arteries are cleared before secondary streets. The snow is plowed as close as possible to the side of the road or the sidewalk. Salt and abrasive spreading continues while plows operate. Once the precipitation ends, a decision is made, based on the size of the snow banks and weather reports whether or not to load the snow into trucks and haul it to disposal sites. If the decision is made to load the snow, parking restrictions must be put in effect to clear the way for the snowblowers. Montreal uses 30,000 moveable signs and 10,000 fixed (flashing) signs. Movable signs are placed 8-12 hours before the snow is loaded to allow citizens time to move their cars. The restrictions prohibit parking for 12 hours (either 7 a.m.-7 p.m. or 7 p.m.-7 a.m.) on one side of a street at a time (to minimize inconvenience ) . Once one side of the street is cleared of cars, snow loading can begin, starting with the main arteries. Street plows and sidewalk plows work together in every sector to create a long row of snow in the middle of the street. The snow is then removed by a snowblower which moves slowly along the row of snow and loads the snow into trucks moving alongside. This is nearly a continuous process: as soon as a truck is filled, it departs for the assigned snow disposal site and another truck takes its place. Each sector has an hourly removal rate, which is the rate in m3 of snow per hour at which snow is sent out of the sector to the disposal site. This depends on the type and amount of equipment operating in the sector and is equal to either 400 m3/hr or 700 m3/ hr. Each sector also generates an annual volume of snow that is estimated, based on historical data, as four times the length of streets in the sector in meters (i.e., each linear meter of street generates an average of 4 m3 of snow over an entire year). The snow-loading operation must be completed within a specific time period, based on the total snow accumulation. In sectors that contain less than 25 km of streets, the deadlines are 72 hours for up to 20 cm of snow, 84 hours for 20 to 25 cm of snow, and % hours for more than 25 cm of snow. For larger sectors, the deadlines are 96 hours, 108 hours, and 132 hours, respectively. Snow loading involves over 350 street and sidewalk plows, 90 snowblowers, 84 front-end loaders, almost 300 service vehicles, and 660 trucks. Each year, an average volume of 7 millions cubic meters of snow is carried to disposal sites in approximately 300,000 truck loads. All the snow from a particular sector is transported to the same snow disposal site. In winter 1991-1992 there were 20 disposal sites of four types: 3 river sites, 1 quarry site, 10 sewer chutes, and 6 surface sites. Figure 2 shows the 60 sectors and 20 disposal sites. Table 2 shows the costs and percentage usage
= Sewer chutes A Surfacesites l Rlversites 0 Francon quarfy
Fig. 2. Sectors and disposal sites.
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Table 2. Snow disposal methods and costs Method St Lawrence River Francon Quarry Sewer Chutes Surface Sites
Volume
Cost (S/m’)
31% 25% 23% 21%
0.121 0.181 0.149 0.506
for each type of disposal site in 1991-1992. The average cost is about $0.24 per cubic meter of snow. Each disposal site has an hourly receiving rate capacity (in m30f snow per hour), which depends on the logistics and configuration of unloading facilities at the disposal site. A disposal site can also have an annual capacity corresponding to the finite space available for storing snow. Table 3 presents the hourly and annual rate capacities for the 20 disposal sites. The quarry and river disposal sites have relatively high hourly capacities due to multiple unloading stations (e.g., there are 13 piers for unloading in the quarry). The sewer chute sites have effectively unlimited annual capacities but are restricted by the relatively small hourly capacities. The river sites allow snow to be dumped directly into the St. Lawrence River. This accounts for the largest fraction of the snow and is the most economical disposal method. However, snow plowed from the streets and sidewalks is contaminated with a variety of urban pollutants (e.g., lead, cadmium, copper, zinc, etc.) that pose a threat to the environment. Therefore, the Province of Quebec has enacted legislation that will phase out river disposal by 1996 and replace it with disposal that allows melted snow to be treated in waste water treatment facilities. (Pollution from contaminated snow is also a concern in many other cities, e.g., in the Neva River at Leningrad; see Tsvetkov, 1989.) This is forcing a redesign of the snow disposal system and provides added impetus for research to find economical and environmentally sound snow disposal solutions. When the river disposal sites are closed, new sites may be added. Due to the limited size of the Francon quarry and the constraints on sewer chute disposal, the new sites may be relatively expensive surface sites involving greater travel distances. The quarry, sewer chute and surface disposal sites allow melted snow to be treated
Table 3. Disposal site capacities
Disposal Site
Hourly Capacity (m’/hr)
Annual Capacity (m’/ycar)
River
Pont de la Concorde Quai 130 Quai IS2
10,000 10,000 5,ooo
00 QD 0
QuarrY
Francon Quarry St-Pierre Anbar Dickson nord Iberville Poincart Benuharnois De 1*p&e SauvC Millen Brousseau
10,000 2,800 700 600 700 z
3,000,000 (x, 00 00 Qo 00 03 OD (x, QJ cn
Sewer Chutes
Surface Sites
Part Neuman Royalmount Contrccocur Mont& St-Ltonard M. A. Fortin Armand-Chaput
400 600 2,000 l.ooO 1.000 2,000 t*E 2:Oo0 6W3
154,000 250,000 700,000 z% 1,oso:OOO
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before release to the environment. The snow dumped in the Francon Quarry (approximately 25% of the total) is pumped out of the quarry after melting and treated in the MUC Waste Water Plant before being flushed into waterways. Sewers have been an important disposal method in many cities since the early part of this century (Highway Research Board, 1970). The sewer chute sites in Montreal are special hoppers built over major sewer structures. This is a relatively inexpensive disposal method and it offers the advantage of treating the snow before it is released into waterways. However, the unloading capabilities are somewhat limited by logistics at the site and by the condition that the water temperature in the sewer system should not fall below 2oc. Surface sites are empty lots on which the snow is piled. When the thaw comes, the melted snow runs off into the drainage system and is treated before release into waterways. (Montreal has experimented with snow melting machines, but this is currently too expensive for substantial usage.) The operation of snow disposal sites and the spreading of de-icers and abrasives are conducted by the city of Montreal. The snow clearing, parking restriction set-up, and loading activities are completed by private contractors in some sectors, city employees in other sectors and by a joint partnership between contracts and the city of Montreal in still other sectors. However, all snow removal work is performed in accordance with the same specific action and management plan. In 1991-1992, forty sectors were contracted to private firms, six sectors ivere completed entirely by the city of Montreal, and fourteen sectors were cleared jointly, with private contractors hauling the snow to disposal sites and the city performing the other tasks. Contracts are for a Cmonth period from November 15 to March 15 and the pay is based on the amount of snow during this period and the distance between the sector and its assigned disposal site.
3. OPERATIONS MANAGEMENT PROBLEMS
As the preceding section illustrates, urban snow removal and disposal operations involve a host of engineering and managerial problems. Most of these problems have received little attention from researchers. This section describes the major problems and focuses on analytical approaches that have been used. Table 4 summarizes the problems and the related works. Our purposes are to briefly review previous work and to highlight opportunities for future research.
Table 4. Snow removal problems and related works Problems Level of service (deadlines) Site location
Literature
Leclerc, 1985
Sector design Sector assignment
Campbell & Langevin, 1995 Leclerc. 1985, 1989
Fleet mix
Campbell & Langevin, 1992 Savas, 1973
Routing
Cook & Alp&, 1976 E&se, 1994 Eglese & Li, 1992 Evans, 1990 GClinas. 1992 Gilbert. 1990 Haslam & Wright, 1991 Lemieux & Campagna, 1984 Liebling, 1973 Marks & Stricker, 1971 Tucker & Clohan, 1973
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Level of Service The main objective in managing snow removal and disposal operations is to provide a specified level of service at minimum cost. The level of service is given by the deadlines for completing snow removal and disposal. These deadlines are set based on political and economic considerations. To allow a large city to be cleared of snow in a timely fashion, it is divided into many sectors that are cleared concurrently. For political reasons (equity for instance), every sector should probably have the same deadline. To minimize the inconvenience and cost to the public and to business (e.g., lost productivity), snow should be cleared promptly. However, the cost for snow removal and disposal typically decreases as the deadline increases. The tradeoff between the added inconvenience and cost of extending the deadline and the resulting cost savings is an important area for sensitivity analyses. Minsk ( 1973) identified social, environmental, and equipment characteristics to be considered in selecting a level of service, within the framework of a system-wide approach to snow removal and disposal.
Disposal site location A fundamental strategic problem that provides the foundation for snow removal and disposal operations is to locate snow disposal sites. Locating snow disposal sites can be viewed as a facility location problem in discrete space, because a set of candidate disposal sites can usually be identified based on infrastructure (e.g., se\?rers), geography and land use (e.g., vacant parcels). Disposal in rivers or lakes has often been preferred, although disposal sites that allow melted snow to be processed in waste water treatment facilities provide environmental benefits. Snow disposal sites may be viewed as seasonally obnoxious facilities because they generate a large volume of truck traffic and around-the-clock activities during snow disposal operations. There is a vast literature on facility location models (e.g., Brandeau & Chiu, 1989; Mirchandani & Francis, 1990), and a growing literature on obnoxious facility location (Erkut & Neuman, 1989), but the only analytical work on locating snow disposal sites, that we have found, is a brief post-optimal analysis for a transportation model (Leclerc, 1985).
Sector design Another fundamental strategic problem is to partition the city into snow removal sectors. The partitioning problem is similar to districting problems for distribution systems, emergency services, schools, and political jurisdictions, which have been addressed by a variety of approaches (e.g., Daganzo, 1991; Ferland & Guenette, 1990; Franklin & Koenigsberg, 1973). Techniques used in continuous approximation modelling of distribution systems (Daganzo, 1991; Langevin et al., 1995; Langevin & Saint Mleux, 1992) might be useful to partition a city into snow removal sectors. Sectors should be small enough to allow clearing to be completed by a single crew by the specified deadline. To minimize many of the fixed costs (e.g., equipment, crews, etc.), the number of sectors should be as small as possible. This implies that snowblowers and snowplows should operate continuously or as much as possible to maximize the size of the sectors. Two other strategic problems involving sectors and disposal sites are to decide: whether all sectors should be cleared by the city or if some should be cleared by private firms; and, whether the city should allow other municipalities to use the city’s disposal sites and the appropriate fee. Both of these problems have significant political components.
Sector assignment Once the sectors and disposal sites have been defined, the snow disposal assignment problem is to assign each sector to a disposal site. The assignments may have to satisfy an annual capacity for each site (m3 of snow per year) and an hourly receiving capacity for each site (m3 of snow per hour). Leclerc ( 1985; 1989) presented a linear programming model for the snow disposal assignment problem that considers annual but not hourly capacity constraints at the disposal sites. This model allows each sector to be assigned to
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(i.e., send snow to) several disposal sites. However, for operational reasons, the assignment of each sector may be restricted to a single site, as in Montreal. Campbell and Langevin ( 1995) developed a more comprehensive model that includes both annual and hourly capacity constraints as well as assignment of each sector to a single site. This model is formulated as a multi-resource generalized assignment problem and solved using a new heuristic. Results and sensitivity analyses are presented for the City of Montreal. We present next a generalization of the model of Campbell and Langevin ( 1995) for the combined problems of snow disposal site location and sector assignment. According to operating rules in the City of Montreal, each sector should be assigned to a single snow disposal site (i.e., all the snow removed from a given sector must be sent to the same disposal site). The snow disposal site location and assignment problem (SLAP) can be formulated as an integer program as follows: SLAP: Min
C C
i j
+ C
aijdijvixij
i
bj
C ViXu + C fiYj i i
(1)
subject to
C
ViXij I
V;.
for all sites j
(2)
C
riXij I
Rj
for all sites j
(3)
for all sectors i
(4)
i
i
C
Xij
i
xij
s
=1
for all sectors i and sites j
Yj xijsYj
E
(091)
for all i,j
(5) (6)
where 1 if sector i is assigned to site j and 0 otherwise, Y/ = 1 if site j is established and 0 otherwise, dij = distance from the centroid of sector i to site j, = annual volume of snow in sector i (m3/yr), vi = snow removal rate in sector i ( m3/hr), ri V, = annual capacity of site j (m ‘/hr), Rj = maximum snow receiving rate of site j (m3/hr), transportation cost per unit from the centroid of sector i to site j ( %/km/m3), aij = = disposal cost for site j ( %/m3) and = fixed cost to establish disposal site j ($/yr). Xii =
The objective is to minimize the sum of three costs. The first is the transportation cost for hauling snow from the sectors to the disposal sites, the second is the variable cost to operate the disposal sites; and the third is the fixed cost to establish disposal sites. Constraints (2) and (3) limit the assignments of sectors to disposal sites according to the annual and hourly receiving capacity of the disposal sites. Constraints (4) assure that each sector is assigned to a disposal site, Constraints (5) ensure that a disposal site is established if any sectors are assigned to it. Integrality constraints (6) prevent sectors from being assigned to more than one site. This model extends that in Campbell and Langevin (1995) by incorporating the site location decision and fixed and variable disposal costs. The resulting SLAP integer programming formulation is very difficult to solve exactly. It is a capacitated facility location problem in which each facility has two capacities: an hourly capacity for receiving snow and an annual capacity for storing snow. For any fiied set of disposal site
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locations (i.e., values of rj), the problem “reduces” to a multiple resource generalized assignment problem (GAP), which is still very difficult (e.g., NP-hard). Problem SLAP could be solved using a heuristic that incorporates the heuristic presented in Campbell and Langevin (1995) for the multiple resource GAP. The snow disposal assignment problem and the partitioning problem to define sectors are complicated by interdependencies: the size and shape of the sector should probably depend on the assignment. Separating these decisions simplifies the analysis, but will likely produce a suboptimal system.
Fleet mix Snow removal and disposal involves a number of fleet mix, equipment selection, and crew scheduling problems. A fleet of snow removal equipment (e.g., street plows, sidewalk plows, snowblowers, etc.) operates in each sector. The size and composition of the fleet in each sector depends on the configuration of the streets and sidewalks, land use (e.g., residential or commercial) and density of development, and the deadline. Savas ( 1973) discussed the geographical deployment of equipment and manpower for snow removal in New York City, especially the political aspects. Trucks must also be allocated to each sector to haul snow to the assigned disposal sites. The minimum number of trucks is constrained by the deadline for snow removal and the desire that the snowblower is never idle. Campbell and Langevin (1992) described preliminary models for allocating trucks to sectors and show how numbers and sizes of trucks depend on the distances from sectors to disposal sites, determined by the assignment of sectors to sites. Analytical research on crew scheduling is surveyed by Bodin et al. (1983).
Routing Snow removal and disposal involves a number of routing problems for spreaders, graders, snowplows, snowblowers, and trucks. These problems are complicated by precedence constraints (e.g., clearing main arteries first) and by time-window constraints (e.g., maximal time interval, parking restrictions, and other traffic issues). Arc routing problems are the most studied of any snow removal and disposal problems. These are important practical examples of the Chinese Postman Problem (CPP) and more general capacitated arc routing problems (Golden C Wong, 1981). See Eiselt et al. ( 1992) for a recent survey of arc routing. Because of the inherent difficulties of the problems (e.g., they are NP-hard), almost all algorithms developed for routing snowplows and spreader trucks are heuristics. Cook and Alprin (1976) present a salt spreader truck routing heuristic based on closest street selection to minimize the time to cover all branches in a network. The objective is to balance the work load between the vehicles. The authors validate the results of the heuristic by simulation experiments. Eglese (1994) presented a heuristic algorithm to minimize the distance travelled by gritting vehicles. The heuristic allows multiple depots, limited vehicle capacities, and roads with different priorities. The heuristic involves solving optimally an unconstrained CPP, followed by the use of Simulated Annealing for the constrained problem. Haslam and Wright (1991) discussed the strengths and weaknesses of several mathematical programming approaches and present a multiple objective heuristic methodology for the design of routes for intrastate highway snow and ice control. Liebling (1973) presented a study done for the city of Zurich for snow removal routing, based on a CPP procedure and a heuristic to partition the city between the vehicles. Marks and Stricker (1971) addressed the problem of urban snow removal with a cluster first-route second heuristic that uses a CPP model for routing. Evans ( 1990) described a decision support system for spreader truck routing and fleet mix decisions. Lemieux and Campagna ( 1984) addressed rural snow plowing with precedence relations. Their algorithm traces an Eulerian circuit on a directed graph and would have to be implemented with a heuristic method to take into account the “rural” component. Eglese and Li (1992) discussed differences between routing spreader trucks in sparse “rural” networks and more grid-like city networks. Gilbert (1990) modelled snowblower routing as an asymmetric CPP with duration, precedence, and time windows constraints and develops an insertion heuristic. GClinas (1992) described an optimal
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dynamic programming solution procedure for the CPP with precedence relations and includes an application to snow plowing in Montreal. Finally, simulation studies (Tucker & Clohan, 1973; England, 1982) have been conducted to evaluate routes produced by different procedures. Another type of routing problem is for the trucks loaded with snow that travel from sectors to disposal sites and back to the sectors. This is a difficult shortest path problem, because of the dynamic nature of traffic and the movement of the snowblower while a truck travels to and from the disposal site. Truck routing also involves political and equity issues due to the disruptive nature of large numbers of trucks converging on disposal sites. There is an enormous literature on node routing and shortest path problems (e.g., Bodin et al., 1983; Laporte, 1992) that may be applicable to snow disposal problems. However, none to our knowledge, has specifically addressed snow disposal. There has been very little analytical research on aspects of snow removal and disposal other than arc routing. The focus of previous research on arc routing problems is not surprising, since in most regions, adequate snow removal can be accomplished by plowing and spreading de-icers without hauling snow to disposal sites. Lower levels of urban concentration also make snow disposal less important because there may be sufficient room along streets to store the plowed snow until it melts. However, higher annual snowfall may contribute to greater levels of urban concentration by restricting travel (Guterbock, 1990). 4. A DSS FOR SNOW DISPOSAL
This section briefly describes a decision support system (DSS) currently under development to address several of the important strategic problems in urban snow removal and disposal. The DSS will be an interactive and “user-friendly” personal computer-based system that encourages planners to modify designs and explore alternate solutions. It will provide color maps, graphical displays, design guidelines, and real-time computation of performance measures. The DSS uses a detailed street map of the city being modeled with the MapInfoTM desktop mapping software and the MapBasicTM programming language. User input is via the keyboard and mouse and output is displayed on the screen and can be printed to various output devices. The DSS is being tested using data for the city of Montreal, but it could be applied to any urban region, given appropriate data. The DSS is being developed in two primary phases of increasing complexity and scope. The first phase, which forms the core of the DSS, addresses the snow disposal assignment problem (SDAP). The SDAP heuristic algorithm described in Campbell and Langevin ( 1995) has been linked to the desktop mapping software to automate the procedure of assigning sectors to disposal sites. The DSS displays the assignments on a map of the city as shown in Figure 3. Planners can modify the assignments and the DSS calculates and displays the revised costs and other parameters. This phase has been completed and the DSS has been demonstrated to the city of Montreal. The SDAP algorithm has produced very encouraging results for the city of Montreal. The assignment of sectors to sites produced by the SDAP algorithm reduced costs by C%236,000 or 3.2070, compared to the cost for the assignment currently used by the city. This cost reduction includes savings of C$226,090 in transportation costs and $10,000 in disposal site operating costs. The SDAP solution differed from the City’s solution in the assignment of 13 sectors. Other results (from Campbell & Langevin, 1995) showed that closing one of the river disposal sites as has been requested by the Casino de Montreal to improve traffic flow, would increase costs by only C$2,400 compared to the assignment currently used by the city. These results are based only on reassigning the existing sectors. Larger cost savings will likely result from designing new sectors and assigning them with the SDAP algorithm. The second phase of development is to incorporate the sector design problem into the DSS. Sectors will be created by aggregating more fundamental geographic units (e.g., based on administrative, political, geographic, and transportation boundaries). This is an interactive procedure in which the DSS will provide a template of the optimal sector size
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Fig. 3. Assignment of sectors to sites.
and shape for an idealized setting. The planner then sequentially selects (using the mouse) a series of the fundamental geographic units that in aggregate approximate the size and shape of the template. Planners are provided with editing capabilities to adjust sector size and shape by reassigning fundamental geographic units among sectors. The DSS helps guide the planner by providing templates and by providing updated performance measures (e.g., volume of snow per sector, length of streets in a sector, etc.). Once the sectors are defined, the assignment of sectors to disposal sites is handled as described in the first phase. This phase is currently being undertaken. This DSS will be useful not only for strategic design of a system for snow removal and disposal but also for responding to contingencies, such as equipment breakdowns, traffic incidents, or reduced snow receiving rates at disposal sites. The snow disposal DSS can also be used to address the problem of selecting disposal site locations. The DSS can be employed repeatedly to evaluate opening and closing disposal sites, as an alternative to solving a large integer programming problem for locating disposal sites. This will be most useful when considering opening or closing a few sites, as opposed to designing a completely new system. Different levels of service (i.e., deadlines) could also be analyzed by iterative use of the DSS. Future developments to the DSS may include routing components for snow removal and disposal vehicles, and additional functionality to better model real world operations. 5. CONCLUSION Snow removal and disposal is an important and expensive winter operation in many cities. This article described urban snow removal and disposal operations and identified a number of challenging engineering and managerial problems that await future research. Although there has been some work on arc routing problems in the context of snow removal, there is almost no analytical research on many of the issues specific to snow disposal, including the design of sectors for snow removal, the assignment of sectors to disposal sites and the allocation of equipment to sectors. Analysis of individual problems will be useful, but research that treats snow removal and disposal from a global, or system-wide, perspective would be most beneficial. This article also describes a snow removal and disposal DSS, currently under development that addresses several major problems in an integrated fashion. Because of the inherent complexity of the problems, an interactive DSS would be most useful to allow solutions to be adjusted to incorporate political and difficult to quantify factors. Hence, fast heuristic algorithms that produce good approximate solutions will be preferred over optimal algorithms that require excessive computer time or power.
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Two important uses for the DSS would be to help find solutions to the fundamental problems of partitioning the city into sectors and to evaluate potential disposal site locations. It would also be useful for real-time response to contingencies, including equipment breakdowns. Such a DSS will be especially important to redesign and re-engineer snow removal and disposal systems to reduce costs and decrease environmental impacts. Acknowledgements-The first author was supported by a Quebec Studies Grant from the Quebec Ministry of International Affairs. The second author was supported by a NSERC grant and a Quebec FCAR Program grant.
REFERENCES Bodin, L.. Golden, B., Assad, A., & Ball, M. (1983). Routing and scheduling of vehicles and crews: The state of the art. Computers and Operations Rmrch X0,63-21 1. Brandeau, M. L., & Chiu, S. S. (1989). An overview of representative problems in location research. Management Sci. 35.645-674. Campbell, J. F., & Langevin, A. (1992). Mode& for urban snow removal operations. Presented at the 34th ORSAITIMS Joint National Meeting, San Francisco, CA. Campbell,J. F. & Langevin, A. (1995). The snow disposal assignment problem. J. Operational Res. Sot., (in press). Cook, T. M., & Alprin, B. S. (1976). Snow and ice removal in an urban enviromnent. Management Sci. 23, 227-234. Daganzo, C. F. (1991). Logirticssystems anafysis. Berlin: Springer-Verlag. E&se, R. W. ( 1994). Routing winter gritting vehicles. Discrete Applied Mathematics 48,231~244. Eglese, R. W., & Li, L. Y. 0. (1992). Efficient routing for winter gritting. J. Operational Res. Sot. 43, 10311034. Eiseh. H. A.. Gendreau, M., & Laporte. G. (1992). Arc routing: Problems, applications and algorithms. Publication CRT-872, Centre for Research on Transportation, University of Montreal. England. R. (1982). Computer analysis ensures a clean sweep. Surveyor 6, May, IS. Erkut, E. R., & Neuman, S. (1989). Analytical models for locating undesirable facilities. European Journal of Operationai Research 40.275-291. Evans, J. R. (1990). Design and implementation of a vehicle routing and planning system for snow and ice Control. Working Paper QA-1990-002, College of Business Administration, University of Cincinnati, Cincinnati. OH. Ferland, J. A., & Guenette, G. (1996). Decision support system for the school districting problem. Opns. Res. 38.15-21. Franklin, A. D., 8r Koenigsberg, E. (1973). Computed school assignments in a large district. Opns. Res. 20, 413-426. Gthrms, E. ( 1992). Le probleme du pastier chinois avec contraintes g&&ales de pr&ance. M.Sc.A. Dissertation, E&e Polytechnique de Montreal. Gilbert, J. ( 1990). GtWrateur d’itindraiirerd’enlPvement de la neige. Publication CRT-648. Centre for Research on Transportation, University of Montreal. Golden, B.. t Wong, R. (1981). Capacitated arc routing problems. Networks II. 305-315. Gray, D. M., Bi Male, D. H. (eds.), (1981). Handbook of snow. Willowdale. Ontario, Canada: Pergamon Press. Guterbock, T. M. ( 1990). The effect of snow on urban density patterns in the United States. Environment and Behavior 22,358-386. Haslam, E., & Wright, J. R. (1991). Application of routing technologies to rural snow and ice control. Transpn. Rer. Record 1304,202-211. Highway Research Board. (1970). Snow removal and ice control research. Highway Research Board Special Report 115, Washington, DC. Langcvin, A., Mbaraga, P., & Campbell, J. F. (1995). Continuous approximation models in physical dktribution: An overview. Publication CRT-992, Centre for Research on Transportation, University of Montreal. Langevin, A., & Saint-Mleux, Y. (1992). A decision support system for physical distribution planning. Journal Of De&On Systems I, 273-256. Laporte, G. ( 1992). The vehicle routing problem: An overview of exact and approximate algorithms. European Journal of Operational Research, 59, 345-358. Leclerc, G. (1985). Least cost allocation of snow zones to elimination sites: Formulation and post-optimal analysis. Civil Engineering Systems 2, 217-222. Leclen& 0. ( 1989). Post-optimal analysis of a degenerate optimal solution to the Hitchcock formulation. Civil Engineering System 6.92-101. Lemieux, P. F., & Campagna, L. (1984). The snow ploughing problem solved by a graph theory algorithm. Civil Engineering Systems I, 337-341. Liebling, T. M. (1973). Routing problems for street cleaning and snow removal. In R. Deininger (ed.), Mode/s for environmental pollution control (pp. 363-374). Ann Arbor, MI: Ann Arbor Publishers. Marks, H. D., B Snicker, R. (1971). Routing for public service vehicles. ASCE Journal of the Urban P/arming andDeveiopment Division 97, 165-178. Minsk, D. L. (1973). A systems study of snow removal. Transportation Research Board Special Research Report Number 185.220-225. Mirchandani, P. B., & Francis, R. L. (1990). Discrete location theory. New York: Wiley. Sava~. E. S. ( 1973). The political properties of crystalline HP: Planning for snow emergencies in New York. Management Sci. 20, 137-145.
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Tsvetkov, S. (1989). Neva solution begets neva pollution. WorldPaper, November 1989, S4. Tucker, W. B.. & Clohan. 0. M. (1973). Computer simulation of urban snow removal. In Transportation Research Board Special Research Report Number 185,293-302. Ville de Montreal. (1992). Taming Montreal’s snow. Ville de Montreal Service des Travaux Publics, Montreal. Ville de Montreal. (1989). Plan directeur de la gestion intt!gr&edes neiges us&s-Plan d’action. Ville de Montreal Service des Travaux Publics, Montreal.
Pergnmon
OPERATIONS MANAGEMENT FOR URBAN SNOW REMOVAL AND DISPOSAL JAMES F. CAMPBELL ’ School of Business Administration, University of Missouri-St. Louis, 8001 Natural Bridge Road. St. Louis, MO 63121499. U.S.A. AND& LANGEVIN GERAD and &ole Polytechnique de Montreal, 5255, Avenue Decelles, Montreal, Canada H3C 3J7 (Received 1 April 1994) Abstract- Urban snow removal and disposal operations include a number of challenging strategic and tactical problems that are relatively unstudied. Those operations include spreading de-icers and abrasives, plowing snow, loading snow into trucks, and transporting snow to disposalsites. We describe urbansnowremovalanddisposal operations in detail, identify relevant management problems, review previous studies, and highlight opportunities for future research. We also describe a snow removal and disposal decision support system, currently under development, that addresses several major problems in an integrated fashion.
1. INTRODUCTION
Many cities face a winter problem of clearing large quantities of snow from streets and sidewalks. Plowing, along with warmer temperatures, may be sufficient to control moderate snowfalls. However, in urban areas that experience heavy snowfall and persistent low temperatures, snow plowed to the sides of streets and sidewalks accumulates and impedes pedestrian and vehicular traffic. In many cities, snow is loaded into trucks and hauled to snow disposal sites, where it remains until melting, perhaps several months later. Urban snow removal and disposal operations include several challenging strategic and tactical engineering and managerial problems that are relatively unstudied. This article describes urban snow removal and disposal operations, identifies relevant management problems and reviews analytically-based approaches, and discusses directions for future research. The remainder of the article is organized as follows: Section 2 describes snow removal and disposal activities in Montreal; Section 3 presents operations management problems and relevant literature; Section 4 describes a decision support system for managing urban snow removal and disposal; the conclusion section follows. 2. SNOW REMOVAL AND DISPOSAL IN MONTREAL
Heavy winter snowfall hinders transportation in many cities around the world. Table I lists average snowfall amounts for several North American cities. The city of Montreal is regarded as a world leader in snow removal and disposal operations. Montreal covers 187 square kilometers and is home to over 1 million citizens, which is 55% of the population of the Montreal Urban Community (MUC). Montreal receives about 10 major snow storms each winter and the average annual snowfall is 223 cm with a standard deviation of 63 cm (Ville de Montreal, 1989). For each snowfall exceeding 2.5 cm, snow must be cleared from approximately 2,000 km of streets and 3,200 km of sidewalks. Special attention is paid to pedestrian crossings, fire hydrants, bus stops, and subway entrances. Montreal is divided into 60 sectors for snow removal and the typical sector contains ‘Requests for reprints should be addressed to James F. Campbell. School of Business Administration, University of Missouri, St. Louis, 8001 Natural bridge Road, St. Louis, MO 63121-4499. 359
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Table 1. Average snowfall in selected cities Average Snowfall (cm/year)
City Quebec, Canada Syracuse, USA Rochester, USA Mont&l, Canada Ottawa, Canada Buffalo, USA Toronto, Canada
330 300 230 223 220 170 130
approximately 30 km of roads and 50-60 km of sidewalks. Figure 1 shows a portion of the street network and several sectors as well as disposal sites. Snow removal and disposal operations are conducted simultaneously by 60 crews using up to 1,500 vehicles, including spreader trucks (to spread salt and abrasives), graders, snowplows, sidewalk snowplows, front-end loaders, snowblowers, service vehicles, and trucks for hauling snow to disposal sites. Snow removal equipment is described in detail in Gray and Male ( 1981) and Highway Research Board (1970). Depending on conditions, manpower ranges from 700 to 3,080 workers. The 1991budget for snow removal operations in the city of Montreal was about US $50 million. There are four major steps in snow removal and disposal operations: ( 1) spreading de-icers and abrasives, (2) snow clearing (plowing), (3) snow loading, and (4) snow disposal. The data for Montreal given are from Ville de Montreal (1992). When snow begins to fall, salt and abrasives (sand and/or crushed stone) are spread first on major arteries and then on secondary streets by 100 spreader trucks. Sidewalks are treated by 106 small snowplows/spreaders to ensure pedestrian safety. On average, almost 120,000 metric tons of salt and 28,000 metric tons of abrasives (approximately half sand and half crushed stone) have been used each year. However, in recent years, Montreal has experimented with using less salt and more abrasives: in 1991, approximately 60,000 metric tons of salt, 15,000 metric tons of sand and 53,000 metric tons of crushed stone were used. When the snow accumulation reaches 2.5 cm, snow clearing begins. The equipment used includes 325 street plows, 208 sidewalk plows, and 123 front-end loaders (used
n
Sewer chute
Fig. 1. Street network and snow removal sectors.
Urban snow removal mainly in narrow lanes and dead end streets). In each sector,
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main arteries are cleared before secondary streets. The snow is plowed as close as possible to the side of the road or the sidewalk. Salt and abrasive spreading continues while plows operate. Once the precipitation ends, a decision is made, based on the size of the snow banks and weather reports whether or not to load the snow into trucks and haul it to disposal sites. If the decision is made to load the snow, parking restrictions must be put in effect to clear the way for the snowblowers. Montreal uses 30,000 moveable signs and 10,000 fixed (flashing) signs. Movable signs are placed 8-12 hours before the snow is loaded to allow citizens time to move their cars. The restrictions prohibit parking for 12 hours (either 7 a.m.-7 p.m. or 7 p.m.-7 a.m.) on one side of a street at a time (to minimize inconvenience ) . Once one side of the street is cleared of cars, snow loading can begin, starting with the main arteries. Street plows and sidewalk plows work together in every sector to create a long row of snow in the middle of the street. The snow is then removed by a snowblower which moves slowly along the row of snow and loads the snow into trucks moving alongside. This is nearly a continuous process: as soon as a truck is filled, it departs for the assigned snow disposal site and another truck takes its place. Each sector has an hourly removal rate, which is the rate in m3 of snow per hour at which snow is sent out of the sector to the disposal site. This depends on the type and amount of equipment operating in the sector and is equal to either 400 m3/hr or 700 m3/ hr. Each sector also generates an annual volume of snow that is estimated, based on historical data, as four times the length of streets in the sector in meters (i.e., each linear meter of street generates an average of 4 m3 of snow over an entire year). The snow-loading operation must be completed within a specific time period, based on the total snow accumulation. In sectors that contain less than 25 km of streets, the deadlines are 72 hours for up to 20 cm of snow, 84 hours for 20 to 25 cm of snow, and % hours for more than 25 cm of snow. For larger sectors, the deadlines are 96 hours, 108 hours, and 132 hours, respectively. Snow loading involves over 350 street and sidewalk plows, 90 snowblowers, 84 front-end loaders, almost 300 service vehicles, and 660 trucks. Each year, an average volume of 7 millions cubic meters of snow is carried to disposal sites in approximately 300,000 truck loads. All the snow from a particular sector is transported to the same snow disposal site. In winter 1991-1992 there were 20 disposal sites of four types: 3 river sites, 1 quarry site, 10 sewer chutes, and 6 surface sites. Figure 2 shows the 60 sectors and 20 disposal sites. Table 2 shows the costs and percentage usage
= Sewer chutes A Surfacesites l Rlversites 0 Francon quarfy
Fig. 2. Sectors and disposal sites.
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Table 2. Snow disposal methods and costs Method St Lawrence River Francon Quarry Sewer Chutes Surface Sites
Volume
Cost (S/m’)
31% 25% 23% 21%
0.121 0.181 0.149 0.506
for each type of disposal site in 1991-1992. The average cost is about $0.24 per cubic meter of snow. Each disposal site has an hourly receiving rate capacity (in m30f snow per hour), which depends on the logistics and configuration of unloading facilities at the disposal site. A disposal site can also have an annual capacity corresponding to the finite space available for storing snow. Table 3 presents the hourly and annual rate capacities for the 20 disposal sites. The quarry and river disposal sites have relatively high hourly capacities due to multiple unloading stations (e.g., there are 13 piers for unloading in the quarry). The sewer chute sites have effectively unlimited annual capacities but are restricted by the relatively small hourly capacities. The river sites allow snow to be dumped directly into the St. Lawrence River. This accounts for the largest fraction of the snow and is the most economical disposal method. However, snow plowed from the streets and sidewalks is contaminated with a variety of urban pollutants (e.g., lead, cadmium, copper, zinc, etc.) that pose a threat to the environment. Therefore, the Province of Quebec has enacted legislation that will phase out river disposal by 1996 and replace it with disposal that allows melted snow to be treated in waste water treatment facilities. (Pollution from contaminated snow is also a concern in many other cities, e.g., in the Neva River at Leningrad; see Tsvetkov, 1989.) This is forcing a redesign of the snow disposal system and provides added impetus for research to find economical and environmentally sound snow disposal solutions. When the river disposal sites are closed, new sites may be added. Due to the limited size of the Francon quarry and the constraints on sewer chute disposal, the new sites may be relatively expensive surface sites involving greater travel distances. The quarry, sewer chute and surface disposal sites allow melted snow to be treated
Table 3. Disposal site capacities
Disposal Site
Hourly Capacity (m’/hr)
Annual Capacity (m’/ycar)
River
Pont de la Concorde Quai 130 Quai IS2
10,000 10,000 5,ooo
00 QD 0
QuarrY
Francon Quarry St-Pierre Anbar Dickson nord Iberville Poincart Benuharnois De 1*p&e SauvC Millen Brousseau
10,000 2,800 700 600 700 z
3,000,000 (x, 00 00 Qo 00 03 OD (x, QJ cn
Sewer Chutes
Surface Sites
Part Neuman Royalmount Contrccocur Mont& St-Ltonard M. A. Fortin Armand-Chaput
400 600 2,000 l.ooO 1.000 2,000 t*E 2:Oo0 6W3
154,000 250,000 700,000 z% 1,oso:OOO
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before release to the environment. The snow dumped in the Francon Quarry (approximately 25% of the total) is pumped out of the quarry after melting and treated in the MUC Waste Water Plant before being flushed into waterways. Sewers have been an important disposal method in many cities since the early part of this century (Highway Research Board, 1970). The sewer chute sites in Montreal are special hoppers built over major sewer structures. This is a relatively inexpensive disposal method and it offers the advantage of treating the snow before it is released into waterways. However, the unloading capabilities are somewhat limited by logistics at the site and by the condition that the water temperature in the sewer system should not fall below 2oc. Surface sites are empty lots on which the snow is piled. When the thaw comes, the melted snow runs off into the drainage system and is treated before release into waterways. (Montreal has experimented with snow melting machines, but this is currently too expensive for substantial usage.) The operation of snow disposal sites and the spreading of de-icers and abrasives are conducted by the city of Montreal. The snow clearing, parking restriction set-up, and loading activities are completed by private contractors in some sectors, city employees in other sectors and by a joint partnership between contracts and the city of Montreal in still other sectors. However, all snow removal work is performed in accordance with the same specific action and management plan. In 1991-1992, forty sectors were contracted to private firms, six sectors ivere completed entirely by the city of Montreal, and fourteen sectors were cleared jointly, with private contractors hauling the snow to disposal sites and the city performing the other tasks. Contracts are for a Cmonth period from November 15 to March 15 and the pay is based on the amount of snow during this period and the distance between the sector and its assigned disposal site.
3. OPERATIONS MANAGEMENT PROBLEMS
As the preceding section illustrates, urban snow removal and disposal operations involve a host of engineering and managerial problems. Most of these problems have received little attention from researchers. This section describes the major problems and focuses on analytical approaches that have been used. Table 4 summarizes the problems and the related works. Our purposes are to briefly review previous work and to highlight opportunities for future research.
Table 4. Snow removal problems and related works Problems Level of service (deadlines) Site location
Literature
Leclerc, 1985
Sector design Sector assignment
Campbell & Langevin, 1995 Leclerc. 1985, 1989
Fleet mix
Campbell & Langevin, 1992 Savas, 1973
Routing
Cook & Alp&, 1976 E&se, 1994 Eglese & Li, 1992 Evans, 1990 GClinas. 1992 Gilbert. 1990 Haslam & Wright, 1991 Lemieux & Campagna, 1984 Liebling, 1973 Marks & Stricker, 1971 Tucker & Clohan, 1973
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Level of Service The main objective in managing snow removal and disposal operations is to provide a specified level of service at minimum cost. The level of service is given by the deadlines for completing snow removal and disposal. These deadlines are set based on political and economic considerations. To allow a large city to be cleared of snow in a timely fashion, it is divided into many sectors that are cleared concurrently. For political reasons (equity for instance), every sector should probably have the same deadline. To minimize the inconvenience and cost to the public and to business (e.g., lost productivity), snow should be cleared promptly. However, the cost for snow removal and disposal typically decreases as the deadline increases. The tradeoff between the added inconvenience and cost of extending the deadline and the resulting cost savings is an important area for sensitivity analyses. Minsk ( 1973) identified social, environmental, and equipment characteristics to be considered in selecting a level of service, within the framework of a system-wide approach to snow removal and disposal.
Disposal site location A fundamental strategic problem that provides the foundation for snow removal and disposal operations is to locate snow disposal sites. Locating snow disposal sites can be viewed as a facility location problem in discrete space, because a set of candidate disposal sites can usually be identified based on infrastructure (e.g., se\?rers), geography and land use (e.g., vacant parcels). Disposal in rivers or lakes has often been preferred, although disposal sites that allow melted snow to be processed in waste water treatment facilities provide environmental benefits. Snow disposal sites may be viewed as seasonally obnoxious facilities because they generate a large volume of truck traffic and around-the-clock activities during snow disposal operations. There is a vast literature on facility location models (e.g., Brandeau & Chiu, 1989; Mirchandani & Francis, 1990), and a growing literature on obnoxious facility location (Erkut & Neuman, 1989), but the only analytical work on locating snow disposal sites, that we have found, is a brief post-optimal analysis for a transportation model (Leclerc, 1985).
Sector design Another fundamental strategic problem is to partition the city into snow removal sectors. The partitioning problem is similar to districting problems for distribution systems, emergency services, schools, and political jurisdictions, which have been addressed by a variety of approaches (e.g., Daganzo, 1991; Ferland & Guenette, 1990; Franklin & Koenigsberg, 1973). Techniques used in continuous approximation modelling of distribution systems (Daganzo, 1991; Langevin et al., 1995; Langevin & Saint Mleux, 1992) might be useful to partition a city into snow removal sectors. Sectors should be small enough to allow clearing to be completed by a single crew by the specified deadline. To minimize many of the fixed costs (e.g., equipment, crews, etc.), the number of sectors should be as small as possible. This implies that snowblowers and snowplows should operate continuously or as much as possible to maximize the size of the sectors. Two other strategic problems involving sectors and disposal sites are to decide: whether all sectors should be cleared by the city or if some should be cleared by private firms; and, whether the city should allow other municipalities to use the city’s disposal sites and the appropriate fee. Both of these problems have significant political components.
Sector assignment Once the sectors and disposal sites have been defined, the snow disposal assignment problem is to assign each sector to a disposal site. The assignments may have to satisfy an annual capacity for each site (m3 of snow per year) and an hourly receiving capacity for each site (m3 of snow per hour). Leclerc ( 1985; 1989) presented a linear programming model for the snow disposal assignment problem that considers annual but not hourly capacity constraints at the disposal sites. This model allows each sector to be assigned to
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(i.e., send snow to) several disposal sites. However, for operational reasons, the assignment of each sector may be restricted to a single site, as in Montreal. Campbell and Langevin ( 1995) developed a more comprehensive model that includes both annual and hourly capacity constraints as well as assignment of each sector to a single site. This model is formulated as a multi-resource generalized assignment problem and solved using a new heuristic. Results and sensitivity analyses are presented for the City of Montreal. We present next a generalization of the model of Campbell and Langevin ( 1995) for the combined problems of snow disposal site location and sector assignment. According to operating rules in the City of Montreal, each sector should be assigned to a single snow disposal site (i.e., all the snow removed from a given sector must be sent to the same disposal site). The snow disposal site location and assignment problem (SLAP) can be formulated as an integer program as follows: SLAP: Min
C C
i j
+ C
aijdijvixij
i
bj
C ViXu + C fiYj i i
(1)
subject to
C
ViXij I
V;.
for all sites j
(2)
C
riXij I
Rj
for all sites j
(3)
for all sectors i
(4)
i
i
C
Xij
i
xij
s
=1
for all sectors i and sites j
Yj xijsYj
E
(091)
for all i,j
(5) (6)
where 1 if sector i is assigned to site j and 0 otherwise, Y/ = 1 if site j is established and 0 otherwise, dij = distance from the centroid of sector i to site j, = annual volume of snow in sector i (m3/yr), vi = snow removal rate in sector i ( m3/hr), ri V, = annual capacity of site j (m ‘/hr), Rj = maximum snow receiving rate of site j (m3/hr), transportation cost per unit from the centroid of sector i to site j ( %/km/m3), aij = = disposal cost for site j ( %/m3) and = fixed cost to establish disposal site j ($/yr). Xii =
The objective is to minimize the sum of three costs. The first is the transportation cost for hauling snow from the sectors to the disposal sites, the second is the variable cost to operate the disposal sites; and the third is the fixed cost to establish disposal sites. Constraints (2) and (3) limit the assignments of sectors to disposal sites according to the annual and hourly receiving capacity of the disposal sites. Constraints (4) assure that each sector is assigned to a disposal site, Constraints (5) ensure that a disposal site is established if any sectors are assigned to it. Integrality constraints (6) prevent sectors from being assigned to more than one site. This model extends that in Campbell and Langevin (1995) by incorporating the site location decision and fixed and variable disposal costs. The resulting SLAP integer programming formulation is very difficult to solve exactly. It is a capacitated facility location problem in which each facility has two capacities: an hourly capacity for receiving snow and an annual capacity for storing snow. For any fiied set of disposal site
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locations (i.e., values of rj), the problem “reduces” to a multiple resource generalized assignment problem (GAP), which is still very difficult (e.g., NP-hard). Problem SLAP could be solved using a heuristic that incorporates the heuristic presented in Campbell and Langevin (1995) for the multiple resource GAP. The snow disposal assignment problem and the partitioning problem to define sectors are complicated by interdependencies: the size and shape of the sector should probably depend on the assignment. Separating these decisions simplifies the analysis, but will likely produce a suboptimal system.
Fleet mix Snow removal and disposal involves a number of fleet mix, equipment selection, and crew scheduling problems. A fleet of snow removal equipment (e.g., street plows, sidewalk plows, snowblowers, etc.) operates in each sector. The size and composition of the fleet in each sector depends on the configuration of the streets and sidewalks, land use (e.g., residential or commercial) and density of development, and the deadline. Savas ( 1973) discussed the geographical deployment of equipment and manpower for snow removal in New York City, especially the political aspects. Trucks must also be allocated to each sector to haul snow to the assigned disposal sites. The minimum number of trucks is constrained by the deadline for snow removal and the desire that the snowblower is never idle. Campbell and Langevin (1992) described preliminary models for allocating trucks to sectors and show how numbers and sizes of trucks depend on the distances from sectors to disposal sites, determined by the assignment of sectors to sites. Analytical research on crew scheduling is surveyed by Bodin et al. (1983).
Routing Snow removal and disposal involves a number of routing problems for spreaders, graders, snowplows, snowblowers, and trucks. These problems are complicated by precedence constraints (e.g., clearing main arteries first) and by time-window constraints (e.g., maximal time interval, parking restrictions, and other traffic issues). Arc routing problems are the most studied of any snow removal and disposal problems. These are important practical examples of the Chinese Postman Problem (CPP) and more general capacitated arc routing problems (Golden C Wong, 1981). See Eiselt et al. ( 1992) for a recent survey of arc routing. Because of the inherent difficulties of the problems (e.g., they are NP-hard), almost all algorithms developed for routing snowplows and spreader trucks are heuristics. Cook and Alprin (1976) present a salt spreader truck routing heuristic based on closest street selection to minimize the time to cover all branches in a network. The objective is to balance the work load between the vehicles. The authors validate the results of the heuristic by simulation experiments. Eglese (1994) presented a heuristic algorithm to minimize the distance travelled by gritting vehicles. The heuristic allows multiple depots, limited vehicle capacities, and roads with different priorities. The heuristic involves solving optimally an unconstrained CPP, followed by the use of Simulated Annealing for the constrained problem. Haslam and Wright (1991) discussed the strengths and weaknesses of several mathematical programming approaches and present a multiple objective heuristic methodology for the design of routes for intrastate highway snow and ice control. Liebling (1973) presented a study done for the city of Zurich for snow removal routing, based on a CPP procedure and a heuristic to partition the city between the vehicles. Marks and Stricker (1971) addressed the problem of urban snow removal with a cluster first-route second heuristic that uses a CPP model for routing. Evans ( 1990) described a decision support system for spreader truck routing and fleet mix decisions. Lemieux and Campagna ( 1984) addressed rural snow plowing with precedence relations. Their algorithm traces an Eulerian circuit on a directed graph and would have to be implemented with a heuristic method to take into account the “rural” component. Eglese and Li (1992) discussed differences between routing spreader trucks in sparse “rural” networks and more grid-like city networks. Gilbert (1990) modelled snowblower routing as an asymmetric CPP with duration, precedence, and time windows constraints and develops an insertion heuristic. GClinas (1992) described an optimal
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dynamic programming solution procedure for the CPP with precedence relations and includes an application to snow plowing in Montreal. Finally, simulation studies (Tucker & Clohan, 1973; England, 1982) have been conducted to evaluate routes produced by different procedures. Another type of routing problem is for the trucks loaded with snow that travel from sectors to disposal sites and back to the sectors. This is a difficult shortest path problem, because of the dynamic nature of traffic and the movement of the snowblower while a truck travels to and from the disposal site. Truck routing also involves political and equity issues due to the disruptive nature of large numbers of trucks converging on disposal sites. There is an enormous literature on node routing and shortest path problems (e.g., Bodin et al., 1983; Laporte, 1992) that may be applicable to snow disposal problems. However, none to our knowledge, has specifically addressed snow disposal. There has been very little analytical research on aspects of snow removal and disposal other than arc routing. The focus of previous research on arc routing problems is not surprising, since in most regions, adequate snow removal can be accomplished by plowing and spreading de-icers without hauling snow to disposal sites. Lower levels of urban concentration also make snow disposal less important because there may be sufficient room along streets to store the plowed snow until it melts. However, higher annual snowfall may contribute to greater levels of urban concentration by restricting travel (Guterbock, 1990). 4. A DSS FOR SNOW DISPOSAL
This section briefly describes a decision support system (DSS) currently under development to address several of the important strategic problems in urban snow removal and disposal. The DSS will be an interactive and “user-friendly” personal computer-based system that encourages planners to modify designs and explore alternate solutions. It will provide color maps, graphical displays, design guidelines, and real-time computation of performance measures. The DSS uses a detailed street map of the city being modeled with the MapInfoTM desktop mapping software and the MapBasicTM programming language. User input is via the keyboard and mouse and output is displayed on the screen and can be printed to various output devices. The DSS is being tested using data for the city of Montreal, but it could be applied to any urban region, given appropriate data. The DSS is being developed in two primary phases of increasing complexity and scope. The first phase, which forms the core of the DSS, addresses the snow disposal assignment problem (SDAP). The SDAP heuristic algorithm described in Campbell and Langevin ( 1995) has been linked to the desktop mapping software to automate the procedure of assigning sectors to disposal sites. The DSS displays the assignments on a map of the city as shown in Figure 3. Planners can modify the assignments and the DSS calculates and displays the revised costs and other parameters. This phase has been completed and the DSS has been demonstrated to the city of Montreal. The SDAP algorithm has produced very encouraging results for the city of Montreal. The assignment of sectors to sites produced by the SDAP algorithm reduced costs by C%236,000 or 3.2070, compared to the cost for the assignment currently used by the city. This cost reduction includes savings of C$226,090 in transportation costs and $10,000 in disposal site operating costs. The SDAP solution differed from the City’s solution in the assignment of 13 sectors. Other results (from Campbell & Langevin, 1995) showed that closing one of the river disposal sites as has been requested by the Casino de Montreal to improve traffic flow, would increase costs by only C$2,400 compared to the assignment currently used by the city. These results are based only on reassigning the existing sectors. Larger cost savings will likely result from designing new sectors and assigning them with the SDAP algorithm. The second phase of development is to incorporate the sector design problem into the DSS. Sectors will be created by aggregating more fundamental geographic units (e.g., based on administrative, political, geographic, and transportation boundaries). This is an interactive procedure in which the DSS will provide a template of the optimal sector size
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Fig. 3. Assignment of sectors to sites.
and shape for an idealized setting. The planner then sequentially selects (using the mouse) a series of the fundamental geographic units that in aggregate approximate the size and shape of the template. Planners are provided with editing capabilities to adjust sector size and shape by reassigning fundamental geographic units among sectors. The DSS helps guide the planner by providing templates and by providing updated performance measures (e.g., volume of snow per sector, length of streets in a sector, etc.). Once the sectors are defined, the assignment of sectors to disposal sites is handled as described in the first phase. This phase is currently being undertaken. This DSS will be useful not only for strategic design of a system for snow removal and disposal but also for responding to contingencies, such as equipment breakdowns, traffic incidents, or reduced snow receiving rates at disposal sites. The snow disposal DSS can also be used to address the problem of selecting disposal site locations. The DSS can be employed repeatedly to evaluate opening and closing disposal sites, as an alternative to solving a large integer programming problem for locating disposal sites. This will be most useful when considering opening or closing a few sites, as opposed to designing a completely new system. Different levels of service (i.e., deadlines) could also be analyzed by iterative use of the DSS. Future developments to the DSS may include routing components for snow removal and disposal vehicles, and additional functionality to better model real world operations. 5. CONCLUSION Snow removal and disposal is an important and expensive winter operation in many cities. This article described urban snow removal and disposal operations and identified a number of challenging engineering and managerial problems that await future research. Although there has been some work on arc routing problems in the context of snow removal, there is almost no analytical research on many of the issues specific to snow disposal, including the design of sectors for snow removal, the assignment of sectors to disposal sites and the allocation of equipment to sectors. Analysis of individual problems will be useful, but research that treats snow removal and disposal from a global, or system-wide, perspective would be most beneficial. This article also describes a snow removal and disposal DSS, currently under development that addresses several major problems in an integrated fashion. Because of the inherent complexity of the problems, an interactive DSS would be most useful to allow solutions to be adjusted to incorporate political and difficult to quantify factors. Hence, fast heuristic algorithms that produce good approximate solutions will be preferred over optimal algorithms that require excessive computer time or power.
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Two important uses for the DSS would be to help find solutions to the fundamental problems of partitioning the city into sectors and to evaluate potential disposal site locations. It would also be useful for real-time response to contingencies, including equipment breakdowns. Such a DSS will be especially important to redesign and re-engineer snow removal and disposal systems to reduce costs and decrease environmental impacts. Acknowledgements-The first author was supported by a Quebec Studies Grant from the Quebec Ministry of International Affairs. The second author was supported by a NSERC grant and a Quebec FCAR Program grant.
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