Municipal routing problems: a challenge for researchers and policy makers?

Municipal routing problems: a challenge for researchers and policy makers? Olli Bräysy, University of Jyväskylä (Agora Innoroad Laboratory) Wout Dullaert, University of Antwerp (Institute of Transport and Maritime Management Antwerp) Pentti Nakari, University of Jyväskylä (Agora Innoroad Laboratory) Abstract In many European countries, municipalities offer their inhabitants a wide variety of social services. In this paper we will focus on efficiently scheduling home care, transportation of the elderly, and home meal delivery. These so-called municipal or communal routing problems can be modelled as different variants of the vehicle routing problem, a well-known optimization problem from the literature. We present a focussed literature review and report on case studies using Finnish data. The computational results show that there is a significant potential for cost savings for all applications considered. Keywords: vehicle routing, city logistics, case study, optimization 1. Introduction The municipal sector offers a variety of services that are related to moving people or materials to the benefit of the community. Most of these problems can be modeled as variants of routing problems. Routing problems are amongst the most studied optimization problems in the operations research literature. In its most basic form, the Vehicle Routing Problem (VRP) consists of designing the cheapest distribution pattern to service a number of geographically scattered customers with known demand from a single depot, using identical, capacitated vehicles traveling at a constant speed (see e.g. Toth and Vigo, 2001). In this paper we focus on the organization of home care, transportation of the elderly, and home meal delivery in Finland. Other important areas of application are e.g. waste collection and school bus rouing. The rationale for using the Finnish situation as an example is that Finland, similar to other Nordic countries, has traditionally been a country where the public sector plays a significant role in society. As most OECD countries, Finland is facing the challenge of offering home (care) services an aging population with limited public budget. This paper aims at defining the underlying routing problems to explore the cost savings potential of routing optimization for improving the efficiency of municipal routing problems in the future. The following sections present a literature review, discuss model formulation and the results of real-life cases for routing home nurses (Section 2), 1

transportation of the elderly (Section 3), and home meal delivery (Section 4). Section 5 presents conclusions and directions for further research. 2. Routing home care nurses Cities offer different types of home care services and service housing to support higher quality living at home as long as possible and support social interaction of the elderly. Home care refers to home health care and home service, involving services such as making breakfast and helping with dressing. The pillars of these main services are the various support services such as catering, cleaning, transport, laundry, massage and sauna services, security services and library and shopping services. The large number of home care customers, nurses and daily visits creates large-scale logistical problems which must be solved on a day-to-day basis. 2.1. Literature review We start our review of recent literature on routing home care nurses with Begur et al. (1997) who develop software to minimize total travelling time, balance nurse workloads en visualize the patients locations. Cheng and Rich (1998) present the routing home care nurses problem as a vehicle routing problem with time windows and multiple depots. The problem is to determine optimal routes minimizing the total distance, the overtime worked by regular nurses, and the number of hours worked by part time nurses. Hindle et al. (2000) develop estimates for the travelling distances of health and social service professionals visiting homes in given areas. Results are demonstrated with real data from Northern Ireland. De Causmaecker et al. (2001) present an agent-system for mobile nursing with time windows. The presented method is quite comprehensive and can for example handle a number of immeasurable and incomparable preferences of personnel. Eveborn and Rönnqvist (2004) describe a general scheduling software called SCHEDULER for staff planning and give numerical results to illustrate its performance. The system is based on elastic set-partitioning models and uses a branch-and-price algorithm for the solution. Eveborn et al. (2004) present a decision support system called LAPS CARE, specifically designed for home care staff planning. Case study results from Sweden demonstrate 20% and 7% savings potential in transportation and total costs respectively. Bertels and Fahle (2006) present an approach (PARPAP) combining linear programming, constraint programming and meta-heuristics for the optimization of home health care routes. The model is solved by a construction heuristic that generates an initial solution and two improvement heuristics. Melachrinoudis et al. (to appear) propose a double request dial-a-ride (DARP) model with soft time windows and demonstrate the results with a case from Boston Metropolitan area. The solution methods are Branch & Bound and the Tabu Search Heuristic. Campbell and Savelsbergh 2

(2006) examine the use of incentives to influence consumer behavior to reduce delivery costs in delivery services. Haughton (2007) studies a routing problem consisting of random day-to-day customer demands to develop assignment rules to achieve maximum customer-driver familiarity. Although the research is not aimed at routing home nurses to maximize patient-nurse familiarity, retaining continuity in terms of having the same nurse visit a specific customer is an important aspect of customer service in home care. The suggested strategies are simulated by varying the number of customers, customer demands and truck capacities according to a known probability distribution and using real customer addresses from Ontario, Canada. Moreover, the different scenarios were evaluated over a period of 250 simulated days, each day presenting a separate set of customer demands. 2.2. Problem description Currently in Finland, each home care nurse makes typically 4-10 customer visits per day. Visits range from a few times per week to up to 6 times per day. The visiting order to the customers varies depending on the schedules of the patients (e.g. visits to day hospitals or doctors may affect the schedule). One visit can last from 10 minutes to 3 hours and sometimes two nurses are needed to treat one patient. The nurses operate in groups (each group taking care of a specific part of the city) and some planning can be done in advance, e.g for customers having personal treatment plans. However, not all tasks are known beforehand due to the fact that new customer requests are received on a daily basis. Home care and home health care services are currently separate. Home care is often organized in two-shifts, home health care is limited to day work. Currently, there is a single base or depot within each service area where personnel can for example have lunch breaks and pick up different equipment according to the service required by a particular customer. Currently customers in a given service area are allocated to a given team and workers typically only work in their team only. Personnel have a maximum number of working hours per day and different levels of education and different skills (such as nurses and practical nurses), often aimed at performing specific tasks. The service requirement by the customer (patient) and the skill level of the nurse should also be taken into account. Typically, each customer requires several visits within given intervals that can vary from a few hours to days or even weeks. A personal plan is made for each customer, based on his/her needs. The customers specify a time interval in which they want to receive service. These time windows are not 3

absolutely restrictive as a certain amount of flexibility is allowed. Time window violations are actually quite common, but customers are informed by phone. Work schedules are typically planned three weeks ahead and customer visits are scheduled one week ahead, requiring a periodic optimization model. Regulations for breaks and maximum working hours must be taken into account, and if possible workers preferences should also be taken into account. 2.3. Case study The city of Jyväskylä provided data on home nursing services for one week in March 2006. During that week 38 full time nurses made 765 home visits. The average length of a visit was about half an hour. Working shifts were divided into two periods with a lunch break in between, and thus in practice there were 76 work periods in one day. Given the limited size of the sample and the fact that not all of the nurses tasks were monitored by the current management system, only preliminary conclusions can be drawn on the redesign on the Jyväskylä home care nursing services. Four different scenarios are tested for optimization. The optimization is done using the SPIDER commercial VRP optimization solver (see www.spidersolutions.no). In each scenario, the typical duration of customer visits is based on real-life data. The speed for the vehicles is set to 20 km/h because different nurses use different means of transportation (walking, personal cars, bicycles and buses). Nurses start and end their shifts at their own team headquarters, except in the fourth scenario. In the first scenario the visiting dates and times are taken from the real life data and adjusting them is not allowed. In addition, there is a constraint that each customer can only be served by nurses from the customer's local team headquarters. The results from Table 1 show that in this scenario the nursing work requires 48 work shifts (with an average length of 2.3 hours) and 36 nurses, implying that the majority of the nurses work partial days. It should be noted that the study deals with home visits and lunch break only, whereas in real life the nurses perform other tasks as well (such as escorting patients to the doctor, planning, ordering materials, distributing medication etc.) The other three scenarios, however, clearly show that there is optimization potential from the point view of the actual nursing as well. Also, the preliminary results from Table 1 show that the nurses in Scenario 1 travel in total 277.2 km per day. The structure of the routes is illustrated by Figures 1 and 2. 4

Table 1. Scenarios for Jyväskylä home nurse services Current Scenario 1 Scenario 2 Scenario 3 Scenario 4 Travel distance/ km 277.2 238.8 123.6 60.3 Shifts per day 76 48 32 23 22 Savings in number of 36.8% 57.9% 69.7% 71.1% shifts Fig. 1. Considering the geographical team boundaries. 5

Fig. 2. Optimization without geographical team boundaries. In the second scenario the team headquarter constraint is removed and nurses are allowed to serve any customer regardless of the customer's address. Otherwise the constraints are the same as in the first scenario. It should be noted that although at the moment customers have their own "personal nurses" and it is not recommended that nurses should constantly change, the optimization does not significantly affect the assignment of nurses to customers. Firstly the personal nursed assigned in the optimization can be targeted to correspond reasonably well to the current situation and secondly, if a change is required after all, this should be carried out once, and only once. After the change, the allocation of nurses to customers will again prove stable. The results from the second scenario show that 27 nurses and 32 work shifts (with a mean length of 3.4 hours) will be sufficient to service all customers, thus reducing the number of work shifts by almost 58% compared to the current situation.??also distance travelled is to reduced to 238.8 km despite the fact that the nurses now travel across the geographical team boundaries. Although based on a limited data set, the results show that the traditional geographic team boundaries defined for the headquarters are not optimal and the impact of an optimization is significant. 6

In the third scenario the time windows for the home visits are also optimized to replace the current fixed dates and times for customer visits. In addition the team boundary constraint is removed as in the second scenario. The time windows are optimized so that each customer is visited as many times as in the real life data and the visits were evenly spaced (e.g. not on consecutive days if the customer requires visiting two or three times a week). The duration of the visits was based on the collected data and the visiting times were defined to correspond to the work shifts of the nurses. The computational results are quite significant: only 23 work shifts (with a mean length of 3.8 hours) and 12 nurses were needed. Compared to the first scenario the distance travelled is by 55% to only 123.6 km. To make the most of the optimization of the service time windows, the procedure should be repeated on a regular basis taking into account the effect of additions or removals of customers over time. The fourth and last scenario has the same constraints as the third scenario, except that nurses are allowed to start their shifts at the first customer's home instead of at team headquarters. The test results indicate that this would save one nurse and reduce the distance travelled by another by 52.2% compared to the third scenario. 3. Transportation of the elderly In most Western countries, governments, either local or national, offer different kinds of transportation services to the elderly. Typical transportation service types include transportation of the elderly to health care and other services (scheduled service bus lines and taxis) and transportation to day care. Transportation of the elderly is an important application type with significant and constantly increasing need as cities have a significant share of elderly citizens. In Finland these services are organized at municipal level. 3.1. Literature review Previous research on the topic can be found e.g. in Horn (2002) who describes a software system designed for managing the deployment of a fleet of demand-responsive passenger vehicles such as variably routed buses and taxis. Fu and Ishkhanov (2004) address the fleet size and mix problems related to paratransit services whilst presenting a practical heuristic procedure for determining the optimal fleet mix and a real-life example. Renard et al. (2004) present a pick-up and delivery transportation problem with time windows and heterogenous vehicles for an on-request urban transport system. The suggested solution method combines an agent-based parallel computing with a cheapest insertion heuristic and constraint programming. Crainic et al. (2005) present a meta-heuristic for demand-responsive transit systems in which active transportation induce detours in the given 7

sequence of compulsory stops always to be serviced. Various solution approaches based on memoryenhanced greedy randomized multi-trial constructive heuristics and a tabu search heuristic are presented. Potvin et al. (2006) describe a model for the dynamic vehicle routing and scheduling problem with time windows and define different dispatching strategies based on an insertion heuristic used to dispatch requests to vehicle routes, followed by a local improvement procedure and solution update mechanism discrete-event simulation scheme used to handle dynamic events. Rekiek et al. (2006) implement a Grouping Genetic Algorithm to find high quality routing solutions for handicapped people in terms of service quality and number of vehicles required. 3.2. Problem description In general, basically two different models exist for transporting the elderly. The most typical model (model 1) is to have fixed routes through stop points located near customers homes to service locations (such as the hospital, shop, pharmacy or local administration offices). In this case the same route is often driven several times per day. In practice about half of the customers request detours and, where possible, they are picked up from and delivered back to their home address. The customers can call in beforehand and request transport, or they can negotiate the next day's transport with the driver during a trip. The other model (model 2) is to operate in accordance with dynamic customers requests only. The main task is to pick up the elderly and transport them to specific services (e.g. hospitals) and back home. In model 2 the vehicles are divided according to service areas. Each service area has one vehicle, and one checkpoint that all routes include and where the vehicle can wait should it be idle. In both cases, the time needed to drop off a customer may also vary significantly as customers are sometimes accompanied to their destination by the driver and/or the assistant. In addition it may become necessary to define priorities to customers in order to ensure that the oldest or most disabled have a shorter waiting time. 3.3. Case study Customers are transported from their home to the City of Jyväskylä's day hospital at Kyllö health centre and back home after the visit. The day hospital can handle up to 12 patients at a time. The visiting times and dates vary depending on the nature of the customer care (e.g. physical, activity or speech therapy). Each customer typically visits the day hospital 1-5 times a week and the care period can last from 1 week to several months or in some cases even up to one year for rehabilitation patients. Usually the visiting days for each customer are fixed but these can be adjusted as and when required. 8

The day hospital has about 50-60 active customers every month. During year 2006 there were a total of 241 customers who made 1264 visits to the day hospital. Transportation of the customers is outsourced to a local company operating a fleet of small buses. The transportation should not last longer than half an hour and in an ideal case, about five patients can (?) be transported in one vehicle. Equipment such as wheel chairs requires extra space so in some cases the number of passengers is lower. The driver picks up each customer at home. Helping a passenger from the house to the bus can take up to 15 minutes. Disembarking the passengers at the day hospital takes about 5-10 minutes. Not all customers use the transportation service since some are driven to the hospital and back by family members. The budget for day hospital transportation was 23,759 euros in 2006. The schedules of the customers cannot be changed without changing hospital schedules since out-patients share a number of resources with in-patients (e.g. catering, laboratory services, x-rays, doctors and nurses and different kinds of therapy). At present only one vehicle has been handling transportation and making at least two pick-up trips every morning. This has caused customers to arrive late thus missing scheduled treatment, laboratory tests and therapy at the day hospital. A two vehicle approach was tested during spring 2007 and this has helped to get all the customers to the day hospital before 9.00 in the morning. The optimization was done using the SPIDER commercial solver. Based on the obtained results, the daily kilometres dropped from 316 to 181, i.e., a 42.7% savings in distance traveled. We consider this to be very significant, considering the small-scale of the operation which favours manual route design. It appears that optimization is necessary and helpful in this kind of small scale transportation as well. 4. Home meal delivery routing In catering operations the problem is to divide the customers to routes and then order the customers in each route so that all meals are delivered within a given time span (i.e. by lunch time, currently restricted to a 3.5 hour period). There is also a government recommendation that a meal should spend at most two hours in the vehicle (to guarantee a warm meal), though the trend in is towards distributing meals cold and warming them up afterwards. Due to the time limit, vehicles sometimes have to make several visits to kitchen to service their customers. The meals are delivered directly to customers' home addresses. When making the delivery, the driver may also assist the customer in starting the meal, possibly helping in taking medication etc which may significantly increase customer service time. 9

All routes start, staggered, from the same central kitchen, and sometimes (depending on the packaging used) vehicles need to return there at the end of the route to return the empty meal packages. Generally speaking, the duration constraint is more limiting than vehicle capacity (a vehicle can carry more meals than it can deliver in the given time limit), so often there is no need to consider the vehicle capacity constraint in the model. In conclusion, the catering problem can be modelled as open or closed vehicle routing problem without capacity constraints and with maximum route duration constraint. To the best of our knowledge, there is no dedicated literature on home meal delivery routing except for Bräysy et al. (2007a) from which we derive the following case study. Jyväskylä is a medium-sized city in Central Finland, having about 85,000 inhabitants. In 2005 the meal service had 827 customers and a total number of 95,625 meals were delivered. On average, 262 meals were delivered per day and 7,900 per month. In 2006 the total transportation costs for the city to provide the service, organised in nine separate lines, was about 174,000 euros. For the casein question, nine vehicles are used. The SPIDER software and Navteq digital maps are used to build a realistic model of the meal service, taking into account speed and distance information and driving directions. Seven different optimization scenarios are tested. The scenarios vary according to the maximum duration of delivery tours, the allowed delivery time window per customer, the number of tours per vehicle, the loading time window at the depot (central kitchen), possible return to the depot at the end of the day, and whether only the given tour sequences or the whole VRP is optimized. The objective function was to minimize the number of vehicles and total distance. Six different closed tour scenarios are considered with varying maximum tour duration and time window specifications. Because the current practice (best approximated by scenario 3) is to have the vehicles return to the depot to return the reusable food containers, only a single open routing scenario is considered. Given that current practice only allows for loading one single vehicle at a time, only a single scenario for loading two vehicles simultaneously is considered (Scenario 5). The scenarios summarized in Table 2 were executed on an AMD Athlon 64 X2 3800+ computer with 1 GB memory. 10

Table 2. The tested scenarios. Scen. TSP/ VRP Open/Closed VRP Max tour duration Time windows Number Tours Simultaneous Loading 1 TSP Closed 10-13.30 1-2 No 2 VRP Closed 2 h 10-13.30 2 No 3 VRP Closed 3.5 h 10-13.30 1 No 4 VRP Closed 2.5 h 10-13.30 2 No 5 VRP Closed 3.5 h 10-13.30 1 Yes 6 VRP Closed 5 h 8.30-13.30 1 No 7 VRP Open 3.5 h 10-13.30 1 No The savings obtained through optimizing each scenario are presented in Table 3. The optimization was stopped after 5 seconds in Scenario 1 and after 20 seconds in other scenarios as little additional improvement (0.5-1%) was found for longer computation times (up to 500 seconds) and different available search diversifiers. We believe this is mainly because the solutions obtained with the standard setting are already of very good quality. Table 3. The savings obtained compared to the current situation Savings Scen. 1 Scen. 2 Scen. 3 Scen. 4 Scen. 5 Scen. 6 Scen. 7 Distance 22.68 % 12.41 % 33.35% 20.50% 33.52% 36.67% 51.24% Vehicles 0 % 26.98 % 30.16% 26.98% 34.92% 50.79% 30.16% Just by optimizing the order in which the deliveries are made in the existing routes (i.e., a TSP optimization), total distance travelled can be reduced by 22.68%. By allowing also the optimization of the allocation of customers to tours, (i.e., a regular VRP optimization), the savings can only be larger. Probably the best benchmark for current practice is scenario 3 where one has a 3.5 h time window available to make deliveries. According to Table 3, the use of routing software would generate 33.35% savings in distance and a 30.16% saving in the number of vehicles needed compared to the current practice, while keeping all service conditions the same. Scenario 2 best matches the current recommendations on route length with its 2-hour time limit for routes. Scenario 2 and Scenario 4 (having a 2.5h time limit on route duration) are more constrained than the current routes, nevertheless they would both generate 10 to 20% savings in both distance and number of vehicles. 11

Currently only one vehicle can be loaded at a time, causing most vehicles to depart from the central kitchen between 10 and 11 when deliveries are already allowed, thus reducing the possible delivery time window. This could be improved by constructing a larger loading platform that allows loading of multiple vehicles at the same time. Results of scenario 5 report on small additional savings (about 4 percentage points) compared to Scenario 3 when two vehicles can be loaded simultaneously. For this particular case, it is unlikely to be worthwhile making the investment. Scenario 6, on the other hand, demonstrates the results of a 5-hour time window for the distribution of cooled meals to be heated at destination. As one can see, this new distribution approach provides substantial savings, up to 50.79% in the number of vehicles and 36.67% in total distance, compared to current practice and involves little to no additional investments. Currently the meals are packed in special containers that need to be returned to the central kitchen at the end of the day. There the packages are cleaned ready for use the next day. In many cases, returning to the central kitchen at the end of the day requires substantial input in terms of driving and time, compared to a situation where there is no need to return. The latter would be possible by either using disposable containers, or by having a greater number of containers thus allowing these to be returned to the kitchen the next morning after delivery the next day s meal. As can be seen from Table 3, Scenario 7 would generate 51% savings compared to the current practice. For this particular scenario, one could balance the environmental impact of the additional mileage required to collect and clean the packaging versus the social cost of having the packaging collected and recycled by regular waste collection services. 5. Conclusions Many of the services that municipalities offer to citizens involve the routing and scheduling of scarce resources. Although experienced manual planners can often manage to satisfy all constraints involved, the complexity of the routing problems makes it difficult to (efficiently) meet increased demands within limited budgets. The case studies presented in this paper illustrate the potential of applying optimization software for home care, transportation of the elderly, and home meal delivery. Even when off-the-shelf professional routing software is used with little to no adaptation to the problem in question, savings of tens of percents appear to be within reach for policy makers willing to take on the challenge of having them implemented. 12

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