Protecting vulnerable people during pandemics through home delivery of essential supplies: a distribution logistics model

Pandemics and epidemics are far-reaching threat scenarios that are increasing in frequency (Nandy and Basak, 2017). The resulting crises can have serious consequences, especially from a medical, social and economic perspective, as demonstrated by the COVID-19 pandemic in 2020.

In pandemics, measures such as reducing social interaction and self-isolating aim to contain the disease and soften negative impacts of the crisis. Since grocery stores are places of close personal contact, they can cause infections. If public isolation recommendations are followed, visits should be limited (Ekici et al., 2013; Haug et al., 2007). However, the population's supply of essential goods, especially food, must be maintained at all times. In case commercial supply chains are not able to ensure smooth distribution, public authorities must intervene as part of their general duties. The long-term nature of pandemics requires a stable food supply for several weeks or months. In this way, pandemics differ from other disasters such as earthquakes that require an immediate supply of relief goods with short response times and durations (Ekici et al., 2013; Maghfiroh and Hanaoka, 2018; Toth and Vigo, 2014). Preparatory planning of a coordinated humanitarian supply chain can lead to greater efficiency and save human lives (Balcik and Beamon, 2008).

Especially vulnerable groups should limit social interactions to avoid the high risk of an infection's adverse effects. A general definition of “vulnerable groups” with regard to pandemics does not exist and should be determined on a case-by-case basis (Vaughan and Tinker, 2009). Generally, vulnerable groups are subdivided based on medical, social and economic vulnerability (Brimmer et al., 2020). From an economic perspective, entrepreneurs and employees are financially affected by shutdowns or reduced demand. Workers without formal working contracts are economically vulnerable due to the lack of furlough schemes, for instance. Social vulnerability refers to people with a high likelihood of infection due to the high rate of interpersonal contact in their living and/or working conditions (e.g. urban dwellers). Medically vulnerable population groups are very susceptible to the severe consequences of infection. Studies during the COVID-19 pandemic have shown that an individual's age and underlying health conditions affect the progression of the disease (Jordan et al., 2020; Yang et al., 2020). COVID-19 cases in Germany have shown that 86% of the deceased are over the age of 69 with median age of 82 years. In contrast, the median age of infected individuals is only 40 years old (Robert Koch Institute, 2021, as of 2021–01–19). Therefore, the urban population above 69 years old is very vulnerable from a social and medical perspective. Since the food supply is essential, solutions must be developed to circumvent supermarkets as a potential source of infection for vulnerable populations. In the future, other pandemics or epidemics may reveal different vulnerable population groups.

Nowadays, home delivery is an established concept for food distribution in primarily urban areas and provides an option to avoid contacts during grocery shopping. As a reaction to the COVID-19 pandemic, several grocery delivery concepts evolved to offer special protection to the vulnerable population. Some of the existing services assigned a high priority to orders of differently defined vulnerable groups. The Swiss company Farmy reserved delivery time slots for people with the verified status of medical vulnerability (Farmy, 2020). Another initiative by Mat.se, a Swedish food e-commerce company, provided pre-picked weekly food packages for vulnerable persons in the urban areas of Stockholm and Gothenburg. Due to capacity restrictions, they were only able to serve around 500 persons per week (Gunnilstam, 2020).

A first collaborative approach across companies without public authorities was established by the German food retailer Rewe and the parcel service DHL. Within the district Heinsberg with large numbers of affected people, the delivery concept regularly supplied around 37,000 households with uncooled essentials from supermarkets (Klasen, 2020). In the United Kingdom, a large-scale delivery concept was established between the government, retail companies and groups of registered inhabitants who are “clinically extremely vulnerable” and do not have personal contact to people serving them with essentials (Department for Environment, Food and Rural Affairs, 2020). The government identified all adults aged over 69 and those with previous illness as high risk, which included around 25% of the population (Jordan et al., 2020). Eligible persons were provided with a weekly parcel consisting of dry essential supplies.

These delivery concepts stress the practical relevance of home delivery for vulnerable groups during pandemics. However, while the small-scale delivery concepts in Switzerland and Sweden supplied both fresh and dry goods, the large-scale applications only offered dry goods. This brings up the question of resource requirements and operational limitations for a large-scale home delivery concept of fresh and dry goods.

The purpose of this paper is to investigate the feasibility of home deliveries for an urban vulnerable population during pandemics with a collaborative delivery concept. Based on a mathematical model, we quantify the required transportation capacities. For a developed case study of a large-scale urban area and dry as well as fresh goods, we investigate resource-efficient alternatives to the regular brick-and-mortar point of sale during pandemics.

Transport logistics is essential for a resilient food supply (Dasaklis et al., 2012). Huff et al. (2015) highlight that transports along the food supply chain are crucial due to long transport distances and small inventories. While food reaches the population via multiple channels, the traditional distribution via grocery stores is currently paramount. In Germany's case, online grocery in combination with home deliveries accounts for less than 2% of total grocery revenues (HDE, 2020). However, home delivery is a logistics solution, which reduces social interactions and is therefore suitable for pandemic conditions (Ekici et al., 2013; Haug et al., 2007).

Disturbances in the logistics chain can threaten supply (Dalton, 2006). Last-mile relief logistics is the final stage in relief supply chains and aims to distribute goods to affected people (Balcik et al., 2008). Balcik et al. (2008) as well as Maghfiroh and Hanaoka (2018) stress the relevance of limited transport resources for emergency supplies in last-mile logistics.

In a related study on food supply during pandemics, Ekici et al. (2013) assume a mixed supply of the population, with points of distribution (PoD) within walking distance and home delivery. The authors dynamically model the spread of a disease and solve a facility location problem to minimize the travel distance of the population. By doing so, the authors focus on the question of facility location and resource allocation, and not on the required fleet size for home delivery.

The task of determining the required fleet size can be modeled as a vehicle routing problem (VRP) (Kim et al., 2015; Renaud and Boctor, 2002), which is a well-established optimization model in literature, with many variations (Gansterer and Hartl, 2017).

VRPs can be motivated by different commercial or humanitarian objectives (Holguín-Veras et al., 2012). De la Torre et al. (2012) classify relief routing based on the objectives of cost minimization, unsatisfied demand, latest arrival, total response time and travel reliability and provide a broad overview of further VRP literature in relief logistics. While the humanitarian context requires minimizing the population's suffering, commercial settings often aim for minimal costs (Holguín-Veras et al., 2012; Toth and Vigo, 2014). In a pandemic, however, not only the immediate response but also the efficient resource utilization is crucial to cope with the longevity of such a crisis. For example, with their minimization of the total time needed to serve all customers, Wang et al. (2018) show that results differ from those that have a cost focus in routing problems.

In order to identify the required number of vehicles, fleet size minimization in VRPs can be applied. Generally, minimizing fleet size results in the smallest number of vehicles to satisfy demand while considering case-specific restrictions (Escuín et al., 2012; Figliozzi, 2011). A commercial example is the fleet size minimization of logistics service providers for economic reasons (Kim et al., 2015). When transporting food, products with different temperature requirements make distribution more complex and require different delivery vehicle types (Kuo and Chen, 2010; Renaud and Boctor, 2002). The fleet size and mix vehicle routing problem (FSMVRP) determines the required fleet size and plans the routes for different types of vehicles. Due to accessibility issues, this is a common problem in crisis situations when vehicles of different owners are used jointly (Maghfiroh and Hanaoka, 2018). Fleet size minimization can be integrated in the objective function of a VRP by minimizing total costs consisting of vehicle fixed costs and variable transportation costs (Renaud and Boctor, 2002). Another approach by Balseiro et al. (2011) optimizes the fleet size with an objective function minimizing the number of vehicles and total travel time. In Figliozzi (2011), the VRP first minimizes the number of required vehicles, and then the distance-dependent travel costs. All of the variants require vehicles with capacities and are therefore related to the capacitated vehicle routing problem (CVRP), introduced by Dantzig and Ramser (1959). Besides the use of CVRPs in operational contexts, it is also applicable for a wide range of tactical or strategic investigations, for example, determination of resource use (vehicle fleet or drivers) and network planning (Toth and Vigo, 2014). A broad overview of the variety of CVRPs was provided by Toth and Vigo (2002).

Two studies particularly relate to our research within the literature on VRPs during pandemics. Herrmann et al. (2009) model medical distribution during a pandemic with a single-commodity time-minimizing VRP and a fixed number of vehicles. However, the model distributes to PoDs and not households, which means fewer destinations and higher transport consolidation. Shen et al. (2009) minimize the unmet demand and assume a fixed number of capacitated vehicles. Hence, both studies model distribution logistics in pandemics without addressing variable fleet sizes.

The complexity of urban last-mile relief logistics leads to large-scale problems (Barzinpour and Esmaeili, 2014). Large amounts of demand points, representing, for example, households, make such VRPs complex to solve. Therefore, the development of suitable solution techniques plays a major role in reducing computation time (Arnold et al., 2019).

Resources for transportation, for example, vehicles, are usually in operation by numerous different owners, business as well as public authorities or NGOs. Collaboration between owners can increase the available logistics capacities, provide access to logistical resources and increase the provided service level. In relief logistics, collaboration between multiple actors like private or humanitarian organizations is often investigated (Gansterer and Hartl, 2017; Nurmala et al., 2018; Tomasini and Wassenhove, 2009). In crises, emergency legislation can allow public disaster agencies to access commercial information and to control commercial resources, for example, coordinating commercial transports (centralized collaborative planning) (Gansterer and Hartl, 2017; Wiens et al., 2018). Potentially differentiating objectives between public and private actors might complicate such a joint supply. Therefore, the concept of a public-private emergency collaboration (PPEC) can mitigate conflicting objectives between public and private actors by providing incentives for both parties (Wiens et al., 2018).

To conclude, we contribute to the existing literature by applying a CVRP with variable fleet size for a large-scale urban home delivery with food during pandemics. To our knowledge, these areas of research have not yet been combined.

Existing VRPs differ in terms of variables, objective function, restrictions and suitable calculation approach selection and implementation. In most cases, the VRP can be solved by methods of linear programming. Since the problem formulation does not only deal with binary decision variables and values, it can be characterized as a mixed-integer linear program (MILP).

The literature distinguishes between exact and heuristic solution approaches (Laporte, 1992; Toth and Vigo, 2002). A literature consensus apparently exists about the solvability of VRPs, which belongs to the category of NP-complete problems. Toth and Vigo (2014) prove that exact algorithms are able to solve CVRPs with 100, and, in some cases, up to 200 customers. This outcome is verified by Pecin et al. (2017) as well as by Borcinova (2017). Further improvements in terms of algorithms, software and hardware can increase the number of customers. A comparison of exact CVRP approaches with its performance in practical applications by Toth and Vigo (2014) demonstrates that algorithms based on branch-and-cut can be regarded as the most efficient method for large-scale CVRP. Nevertheless, exact solutions can claim a high computing effort of several days (Pecin et al., 2017).

To achieve results for such large-scale problems in a reasonable time, a variety of heuristics and metaheuristics, for example, tabu search, simulated annealing or genetic algorithms, can be used (Cordeau et al., 2002; Laporte, 1992; Liong et al., 2008).

The application of different heuristic approaches to benchmark data sets has shown that, currently, heuristics are able to find a good solution for CVRPs with up to 500 customers (Laporte et al., 2014, p. 109 f.).

Another solution technique of the CVRP is continuous approximation models which replace numerical with analytical methods. This can be beneficial in case of strategic investigations and inaccurate data, but comes with disadvantages for cases where demand is not uniformly distributed and high solution accuracy is desired (Franceschetti et al., 2017; Saberi and Verbas, 2012).

Commercial MIP solvers also work with hybridizations of heuristic and exact optimization methods to combine the advantages of both solution approaches. The results are easily assessed regarding their optimization quality (gap to the optimal solution) (Laporte et al., 2014). Comparisons of different mathematical solvers show that CPLEX and Gurobi achieve the best optimization efficiency for most test instances, with Gurobi holding a small lead for memory usage (Le Bodic and Nemhauser, 2015; Mittelmann, 2018).

If the number of customers is large, the connection of the CVRP with a clustering algorithm, for example, the k-means algorithm, for partitioning areas can be suggested. With this approach, a defined number of smaller CVRP will replace the single large-scale CVRP. With a k-means-algorithm, all demand points are assigned based on the shortest Euclidean distances to a desired number of spatially distributed points in the model area. Within certain iteration steps, the centroids are moved until the sum of all assigned demand points is minimized (Faber, 1994). A two-stage procedure with clustering and CVRP optimization enables problems to be iteratively solvable within an affordable running time. A higher number of clustered regions lowers the final computation time but leads to less accuracy due to the higher probability of suboptimal routings in the overall context (Geetha et al., 2009; He et al., 2009).

The proposed methodology for generating a feasible and resource-efficient home delivery concept for the vulnerable population is presented in the flow chart in Figure 1. To adopt the specific application characteristics, a CVRP model is built, including suitable preprocessing and postprocessing steps. Since the vehicle fleet is investigated from a tactical perspective, the focus of the calculations lies on the quality of the results, and computation time improvements are not extensively investigated.

If the quantity of defined demand points exceeds the vehicle capacity, the delivery will be divided into multiple shipments with a certain number of full truck load (FTL) shipments and the residual amount as input for the CVRP. This preprocessing step ensures the feasibility of the CVRP regarding the vehicle capacity restriction. In a postprocessing step, bin packing tactically assigns trips to delivery vehicles (Martello and Toth, 1990). In case large-scale application with limited computability is handled, an additional step aims to divide the model area based on a k-means clustering algorithm, as discussed in 2.2. In the case of a spatially limited but representative model area, some applications appear to be predestined for extrapolating results to a larger sector.

We introduce an MIP formulation designed for a calculation with exact algorithms inspired by Borcinova (2017), Laporte (1992) and Toth and Vigo (2002).

The defined CVRP is designed to serve demand points from a single origin within a limited time and with multiple capacitated vehicles assigned to numerous routes. In contrast to the basic CVRP, which aims to minimize total costs, our model purpose calculates a route plan that optimizes the total number of delivery vehicles by minimizing the time needed to serve all customers.

The main characteristics of the developed CVRP are listed in Table 1.

The frequently used CVRP formulation of Laporte (1992) is based on a two-index vehicle flow formulation with one single decision variable, which indicates if an arc or edge $(i, j)$ is passed by any vehicle or not. This kind of formulation neglects the assignment of specific routes or vehicles traversing $(i, j)$ and is therefore “generally […] inadequate for more complex versions of vehicle routing problems” (Toth and Vigo, 2002). To easily include the assignment of vehicles, usually a second binary variable or a three-index vehicle-flow formulation is suggested (Borcinova, 2017; Laporte, 1992).

In the present model, a binary three-index vehicle-flow formulation with the decision variable $xrij$ is defined to indicate if the route $r$ traverses arc $(i, j)$ in an optimal solution. This approach allows a higher degree of flexibility to incorporate additional constraints, like time or capacity restrictions (Borcinova, 2017; Rieck and Zimmermann, 2010). The binary decision variable is defined as follows:

The mathematical model uses the following value ranges and parameters:

The objective function of the CVRP is formulated as follows:

Subject to:

With the objective function presented in (1), the model aims to minimize the total time to serve all destination points $j$⁠. It consists of the loading time $Lr$ at the depot, which is considered to be constant for every route $r$⁠; the service time $sj$⁠, which depends on the number of households assigned to the routes' destination points; and the travel times $tij$ of all active arcs $(i, j)$ in the routing. The so-called degree constraints (2) impose that each destination is visited in exactly one route. The capacity constraints (3) ensure that the sum of the assigned demands per route $dj$ does not exceed the vehicle capacity $C$⁠. Constraints (4) represent the time constraints, which ensure that the total time per route (including loading, service and travel times) does not exceed a given time limitation $T$⁠, for example, the duration of a working day. The flow constraints (5) guarantee that all routes begin at the origin point $i=0$⁠. The set of routes is defined with $ℛ$⁠, which describes a freely chosen limit in the maximum number of routes. Because of the fixed loading time per route, it is beneficial to use as few routes as possible in the time minimization. Constraints (5) allow the definition of arc $(0, 0)$ to indicate deactivated routes in the routing plan. The artificial constraints (7) ensure that no other arcs are assigned if the route is deactivated. With the flow constraints (6), it is guaranteed that the number of vehicles arriving at every customer and entering the depot is equal to the number of the vehicles leaving that point. Constraints (8) introduce a calculation simplification and ensure that all active routes consist of at least two destinations.

Equations (6)–(8) constitute the so-called sub-tour elimination. This work presents a model with indicator constraints designed for the use in a mathematical optimization solver. Since these constraints represent a significant challenge in VRP and traveling salesman problems, there is a wide range of literature describing different ways to handle the sub-tour elimination (e.g. Desrochers and Laporte, 1991 or Pferschy and Staněk, 2017).

Based on the time minimization, this model is suitable for calculating the number and length of required tours and vehicles for home delivery to vulnerable groups during a pandemic. To achieve resource efficiency, other model characteristics like selectable time windows are neglected.

The practical feasibility of our model is analyzed with a case study of the urban area of Berlin, Germany. This area serves as an appropriate modeling exercise, because urban areas incur social vulnerability due to high population density, and medical vulnerability is prevalent since all age groups share neighborhoods. In Berlin, protecting vulnerable groups from infections is a particular concern because hospital beds are relatively scarce (Federal Health Monitoring System, 2020). Additionally, an extensive base of open secondary data is available for Berlin.

The data basis for this work builds on the depicted understanding of medically vulnerable groups in section 1.1 based on pandemics like COVID-19. Because an exact geographic determination of all vulnerable persons is not available, the age of the inhabitants is used as the statistically most evident identification parameter. COVID-19 data from Germany show that protecting all people over 69 years would prevent 85% of fatalities (Robert Koch Institute, 2021). For investigating the effect of an enlarged vulnerable group, this case study also includes the Berlin population over 64 years of age. No open access to data representing the remaining quantity of younger individuals with underlying medical conditions is available, and therefore excluded from this work.

For solving the introduced model, open secondary data are used to derive discrete georeferenced points representing the vulnerable population's demand.

Berlin has a population of around 3.77 million registered inhabitants and a population density of more than 4,000 inhabitants per square kilometer (as of 2019–12–31) (Statistical Office for Berlin-Brandenburg, 2020). Based on public data, Berlin has about 715,000 senior citizens over the age of 64 and 527,000 over the age of 69 (Senate Departement for Urban Development and Housing, 2019). These population groups are the vulnerable groups for the home delivery concept.

The city is divided into 12 boroughs with a population range between 245,000 and 409,000. The highly different areas of the districts submit to a wide population density range between 1,600 and 13,600 inhabitants per square kilometer. The city administration of Berlin provides an additional subdivision into 447 so-called LOR planning areas with a detailed age distribution of the population, which are created based on uniform building and living structures (Senate Departement for Urban Development and Housing, 2019). The most accurate open population data for Berlin are provided by the Statistical Office for Berlin-Brandenburg (2019), which divides the model region into 14,759 districts, each representing residential blocks and the associated number of inhabitants. In order to define an age distribution for these districts, the data of the 447 LOR areas are interpolated depending on the georeference by assuming the same age structure for all subordinated areas of one LOR area. The geometric center points of the districts are used for the demand representation of the model.

The contained data of the senior population (over 64 and 69 years of age, respectively) have been further enriched with Germany-wide statistical information about the proportion of pure senior households and seniors living alone or in shared accommodations (Destatis, 2014). This allows us to specify the number of seniors and senior households for every demand point, which is the basis for the quantity of food and the number of stops per demand point, respectively. It is also assumed that all registered seniors who are living in pure senior household will use the home delivery service. About 70% of the seniors live alone or in accommodations with only seniors. On average, about two seniors live in every purely senior household.

The demand quantity per person includes the average consumption of food and beverages for the German population. Other essential goods, for example, medicines, are not considered in this model. Open sales data forecasts for food and beverages in Germany in 2020 were used to calculate an average daily demand per person (German Wine Institute, 2018; Statista Market Analytics, 2020; wafg, 2020). In this calculation, weight statistics for the packaging of food and beverages are also included (Destatis, 2020; pwc, 2011; Stoll, 2018). This results in 1.71 kg of dry goods (0.44 kg dry food and 1.27 kg beverages) and 0.65 kg of fresh goods per person per day. This leads to a total daily demand by the in Berlin registered seniors aged over 64 of about 1.7 million kg and over 69 of 1.2 million kg, respectively.

For our case study, we consider established food delivery service providers in Berlin, such as Amazon Fresh or Rewe delivery service, as the most suitable delivery actors. They are already familiar with the last-mile food distribution and would be able to provide convenient urban storage areas with loading zones, trained staff and delivery vans with cooling units. Therefore, the geographic location of a real urban food delivery hub is the origin point for the CVRP. Food delivery hub capacity is neglected due to the lack of information and in order to focus on the fleet size calculation.

To define the vehicle capacities of the CVRP, we assume a typical delivery van with a payload of 1,200 kg, which is used by parcel delivery services and by food delivery services equipped with a cooling unit (Kress, 2020).

Additionally, we assume an average driving speed in Berlin of 24 km/h (Forbes, 2008) and a loading time per vehicle per trip of 15 min. This necessitates prepared food packages and is therefore small compared to usual loading times of normal parcel delivery vans (Clarke and Leonardi, 2017). Travel distances between demand points and between hub and demand points are calculated based on the road distances with the help of the street layer of OpenStreetMap and the open geographic information system QGIS.

An asymmetric cost matrix helps to include the special travel characteristics in urban areas, for example, one-way directions (Toth and Vigo, 2002). To lower the combinatory complexity of the problem, only the 25 closest relations to other demand points are included for every demand point rather than the whole distance matrix. Tests have shown that computation time can be significantly reduced with negligible effects on the optimal solution.

As outlined in section 2.2, CVRPs with thousands of customers cannot be solved with the current state of technology. Therefore, one representative district has been selected to run the calculations. For deciding for one district the criteria vulnerable population density, demand point density and average distance to the closest hub are considered. Berlin-Lichtenberg has only a very small deviation from the mean values in all these categories (5,600 inhabitants/km², 17 points/km², 8 kilometers road distance to the closest hub) and can therefore be evaluated as suitable for extrapolation to all of Berlin (Statistical Office for Berlin-Brandenburg, 2020). This district has 39,834 inhabitants aged over 64 (29,424 over 69), who live in approximately 28,000 households (21,000 over 69) and must be provided with around 94,000 kg of food (70,000 for those over 69) per day. A well-located hub of an established food delivery service is selected as the origin for the home deliveries. It is not located directly within the model district, but is the closest option with an average road distance to the customers of around 7.8 km. The resulting 826 demand points of the model area represent a median number of 14 households with inhabitants aged over 64 years (10 households aged over 69). Due to very different demand areas, the number of households per demand point considerably vary with a first quartile of 5 and a third quartile of 37 households (aged over 69 years: 4 and 27). In practical implementation, all households per demand point need to be served successively by the deliverymen. Assuming geographically evenly distributed households within the demand areas, the average euclidean distances between households is less than 20 m but could reach 400 m in the worst case. This case study excludes a detailed routing within demand points and considers a fixed service time per household.

Based on the population data collection described in section 4.1.1, the number of vulnerable people per demand point is heterogenous. The variance leads to high weight differences to be delivered per demand point, as shown in Figure 2, in case all people over the age of 64 are served once per week. Because a significant number of demand points amount to a quantity higher than the vehicle capacity of 1,200 kg, FTL trips are defined in a preprocessing step. All points with a demand of 1,200 kg and more are assigned to the weight-based required number of FTL shipments. The remaining residual demand quantities must be served with LTL shipments and are still included in the CVRP.

On that basis, a large-scale CVRP, as explained in section 3, is implemented. The model with more than 800 customers in Berlin-Lichtenberg is not solvable in a reasonable time. The discussed two-stage solution approach is adopted to first cluster the demand points using a k-means algorithm, and then solve smaller-scale CVRPs. In this case study, two different clustering specifications will be compared. Because the literature in section 2.2 showed that problem sizes with 100 demand points are frequently solvable, a clustering in eight areas is applied. It results in a mean of 103 demand points per area, including two larger statistical outliers with 57 and 187 points per area. The clustered model region is illustrated with different point colors in Figure 2. Additionally, the same process is performed with 15 clustered regions, which is the least number that results in less than 100 points in all areas. The purpose of this variation is to investigate the effects of the clustering on the total computing time and the quality of results.

The individual trips that form the solution of the CVRP are then assigned to delivery vehicles with a standard bin packing algorithm (Martello and Toth, 1990). Thus, the minimal number of vehicles that are required to fulfill the transportation task per period is derived.

Within this work, the two following delivery modes are tested:

1. Complete delivery by food delivery services in actively cooled vans.

2. Delivery of fresh goods by food delivery services in actively cooled vans and of dry goods by parcel delivery services.

The second collaborative mode attempts to decrease the need for actively cooled vans by using uncooled vans for non-perishable goods. The first mode represents the classical food delivery model.

Different delivery scenarios are formed by varying the parameters number of weekly deliveries per household (once, twice), service time per household (1.5 min, 3 min), daily delivery service working hours (8 h, 10 h) and minimum age of inhabitants to deliver (64, 69). A sensitivity analysis evaluates the consequences of different measures during crises on the resulting number of required vehicles and other delivery characteristics. For delivery mode A, all 16 parameter combinations, and for delivery mode B, only selected scenarios are calculated.

All calculations are performed with the mathematical MIP solver Gurobi 9.0.1 in combination with the language Python 3.6.1 on a standard notebook (Intel Core i5 2.5 GHz processor, 16 GB RAM). A maximum computation time of three hours or a maximum gap of 10% between the lower and upper objective bound per clustered area is chosen to terminate the calculations. As presumed, the results of the smaller areas are significantly better bounded than the larger-scale calculations. While in most cases the gap between the lower and upper bound is just below the desired 10% limit, smaller cases quickly achieved gaps under 1%, while larger ones only reached 30%.

The presented results refer to the clustering in eight differently sized areas. The effects of performing the calculation with 15 areas are presented at the end of this chapter. All results correspond to the extrapolated values for the entire Berlin senior population based on results from Berlin-Lichtenberg.

The results of the sensitivity analysis for the number of required delivery vehicles are graphically presented in Figure 3 and illustrate a huge impact of the different parameter settings.

The baseline scenario A1 assumes a moderate service time per household of three minutes and a delivery frequency of twice per week, with a daily working time of eight hours. The required number of vehicles amounts to more than 900 to provide the +64 age group for entire Berlin with food and almost 800 for the +69 age group.

Figure 3 points out appropriate strategies to reduce vehicle demand in the given setting. The relative savings per strategy are almost identical between the examined age groups. Scenarios A2 to A4 show that changing only one of the parameters in the given extent can result in up to 44% vehicle savings. The parameter changes represent different measures in the crisis. Providing the vulnerable groups with food only once per week (A4) enlarges the routing efficiency because of consolidation effects, but leads to more challenges for the households, especially regarding the shelf life of fresh produce. A second measure (A3) significantly decreases the service time, assuming that the food was not delivered right to the household's door but instead to a central pick up location, for example, the court of a condominium. Increasing the working hours of the staff to ten hours (A2) leads to a fleet size reduction of 20%.

Two measures combined in the scenarios A5 to A7 help to raise the savings. The combination of a service time and delivery frequency reduction could eventually reduce the number of vehicles by up to two-thirds. Taking increased working hours into account leads to maximum savings of 72% in delivery mode A. Therefore, in the optimistic scenario A8, 250 cooling vans would be required to directly supply all Berlin seniors over 64 years of age with food, and 226 vehicles for those over 69 years of age.

Collaborative delivery with food delivery services for fresh food and parcel services for dry food has been calculated for two selected scenarios B1 (one delivery per week, three minutes service time and eight working hours) and B2 (1.5 min service time and ten working hours) to compare with delivery mode A. The breakdown in two product categories has a negative overall effect on the efficiency of the last-mile distribution system, because all customers need to be supplied twice as much as with the collective delivery. Increase in travel, service and loading times causes an increase in the totally required vehicles of around 80% in scenario B1 compared to the reference scenario A4, and 70% in B2 compared to A8.

Assuming the cooling vehicles as a limited resource and regarding the usual delivery vans as sufficiently available, however, Figure 3 illustrates the positive effect of the collaborative concept. The collaboration helps to achieve smaller vehicle savings between 10% for B1 and 25% for B2 compared to the reference scenario without collaboration. This results in less than 200 required cooling vans, which results in 210 vulnerable persons served per van. Section 5 provides a discussion of the estimated supply of cooling vans and helps to better interpret the vehicle demand.

The overview of results and additional performance indicators, given in  Appendix, Table A1, helps to gain deeper insights about the different delivery modes and scenarios.

In principle, the observed cases have shown that the vehicle capacity is the critical factor for calculating routes, which becomes evident by means of the high average vehicle utilization over all scenarios of 88%. This fact also explains the huge differences in the supplied seniors per tour in Table A1. The scenarios A1, A2, A3 and A5 with doubled delivery frequency lead to about twice as many stops per tour and to a higher proportion of the travel time. Figure 4 also illustrates this observation and provides the distribution of total time in categories for selected scenarios. The share of service time in the baseline scenario can be considerably decreased, not only with reduced time per household but also with less delivery frequency. The combination of both measures enhances this effect and cuts the service time per route by three-quarters. Age groups and working times do not influence time distribution significantly.

As shown in scenario A1 and particularly in B1, with lower food quantities per person, the time limit of the daily working hours rather than vehicle capacity presents the critical factor. This causes inefficient routings for the fresh food delivery in scenario B1 with less than half-loaded vans.

Performing the same calculations with 15 clustered districts in the model area leads to 1.4% increased target value of 914 cooled delivery vans in the baseline scenario. Computation time savings of around 5% justify the more detailed clustering if solution quality is less important.

To implement the proposed approach in practice, some remarks and challenges must be solved.

We assume the number of cooling delivery vans as the limiting factor for food distribution. With our model, we calculate a demand of about 900 vehicles, which can be lowered to around 200 vehicles under optimal conditions and with a lower service quality for the vulnerable population. The service time per household has a high influence on the required vehicles. Decreasing the service time is usually not only within the power of the deliverer, but is also customer-dependent. In practice, the service time could strongly vary due to different distances for deliverymen between households within demand points. A detailed routing between defined households could enhance the model, for instance with a traveling salesman problem or an additional VRP like implemented by Pureza et al. (2012).

To evaluate the resulting vehicle demands, we estimate the existing actively cooled vehicles in Berlin. Available statistics of the German Federal Motor Transport Authority (2019) suggest that around 200 vehicles in the category of vans with chilled loading bays in the suitable capacity category are registered in Berlin. This number considers the amount of cooling vehicles across all payload categories and a general proportion of appropriate delivery vans. Hence, the home delivery concept could be feasible with strong service quality restrictions. However, practical limitations can complicate the implementation and increase the number of required vehicles. The existing vehicles belong to different owners and companies, so the available number of food delivery services could be much lower in practice. Notably, these existing resources are in high demand during a pandemic, as the long waiting times for orders in the online food sector during the COVID-19 crisis showed (de Prez, 2020).

The presented concept requires vehicle-efficient route optimization focusing on vehicle utilization instead of customer-oriented delivery, with freely selectable time windows. Such an optimized distribution plan with a large number of customers can lead to vehicle utilization over 80% compared to primarily less than half-loaded vehicles, as observed in existing business models of food delivery services (Allen et al., 2018).

Forward-looking grocery delivery concepts often include an enhanced sustainable last-mile consolidation structure with tiny micro hubs in residential areas, only a few hundred assigned customers per day and often in connection with cargo bike delivery (e.g. Melkonyan et al., 2020; Morganti and Gonzalez-Feliu, 2015). Only short route durations could make a passive cooling of perishable goods applicable, for example, in portable coolers. Thereby, the shortage of special cooling vans could be partially compensated. The micro hubs could be restocked at nighttime in order to reserve more vehicles for last-mile deliveries throughout the day (Bertazzo et al., 2016).

In practice, when public disaster response agencies manage relief supply, a collaborative home delivery would require transparency about demand, supply and especially available vehicles. To implement such a collaboration, formal cooperation agreements between companies and the government are conceivable (Holguín-Veras et al., 2014). In addition, “latent private-sector partnerships” (Gabler et al., 2017) are suggested, which can be intensified during crises. Informal cooperation can also work if partners are intrinsically motivated and economically benefit from the cooperation (Wiens et al., 2018). For that, a profound understanding of each of the participating partners' objectives and motivation is required (Bealt et al., 2016; Gabler et al., 2017). Incentives for companies to engage must be provided, since food delivery services might aim to maintain their business processes and therefore try to not compromise the delivery of existing clients. Public authorities should therefore find ways to compensate the involved companies' additional expenses and take the leading role in overall process coordination (Izumi and Shaw, 2015; Wiens et al., 2018).

An appropriate way must also be found to generate incentives for the vulnerable population to use home delivery services. Finding a good medium of communication with seniors can be a practical challenge. Smartphone apps and websites of the food delivery services mainly address a younger target group, but telephone or postal communication can lead to high staff and time expenditures.

In this paper, we present a modeling and solution approach for protecting vulnerable populations during pandemics with home delivery of essential supplies. We model the last-mile distribution concept through a CVRP, which aims to minimize the total time for serving customers and implicitly optimizes fleet size. A developed case study based on the city of Berlin shows the model's feasibility and applicability. A two-stage approach with a k-means clustering and an MIP calculation based on the solver Gurobi is applied. Thereby, this paper proves that good solutions can be achieved within a reasonable runtime.

In our baseline scenario, we estimate that around 900 cooling vans are required to serve the medically vulnerable group of Berlin inhabitants over 64 years of age with food. With a sensitivity analysis, we investigate different operational measures of the delivery concept during pandemics and show how a fleet size of less than 200 vehicles could be reached. Among other things, this requires collaboration between food and parcel delivery services and a division into fresh and dry food. Our findings reveal that the available cooling vehicles could be sufficient to implement the home delivery concept in Berlin, albeit with restricted service quality. The analysis also demonstrates that lowering the service time per household or the delivery frequency has a similarly high resource-saving impact.

The success of this distribution concept for the vulnerable population depends on the coordination of the collaboration. To achieve a better response time, we suggest already elaborating a comprehensive concept in a pre-disaster context (preparedness phase) with all involved public and private actors and stakeholders.

Besides the presented fleet size optimization, alternative scenarios that require different optimization objectives, for example, a shortage of drivers, are conceivable during pandemics. As stated by Huff et al. (2015), diseases or quarantines are likely to cause worker absenteeism in the food and logistics sector due to interpersonal contacts and potential infection chains within and across companies. For a home delivery concept that requires a large number of delivery staff, this scenario could shift the focus to staff as the most critical resource.

Additionally, we suggest research taking into account multiple hub locations to include the decision about which hubs to use in order to serve all vulnerable groups and to decrease the travel times of the last-mile delivery. Enhancing the model with a heterogeneous fleet formulation and considering larger trucks, especially for delivering demand points with high demand, could facilitate a more efficient last-mile delivery.

Applying the CVRP for all other Berlin districts besides Lichtenberg would lead to more accurate results and supersede the extrapolation step. A data base with non-aggregated households could avoid the high variance per demand point and make the results even more reliable. Lastly, this model should be further applied to other geographical regions, other products (e.g. medical supply) or differently defined vulnerable groups, such as persons with preexisting conditions or those in quarantine. Other actors like the fire brigade, army or non-governmental disaster relief organizations could improve the concept with their vehicle support.

Another opportunity to generate higher cooling vehicle savings would be a last-mile distribution system with a higher number of decentralized micro hubs and a delivery of passively cooled food with unspecific vehicles. Proving this concept needs further investigation.

To comprehensively view this problem, we seek advice from other fields of research, especially regarding legal restrictions, the epidemiological effects of home delivery strategies and their acceptance among vulnerable populations.

We conclude that the benefits of our research are valuable to retailers and logistics companies operating a home delivery service for vulnerable populations during pandemics. Moreover, our research serves policymakers who are devising plans to improve infection prevention while ensuring the population's supply security.

This work has partly been funded by the German Federal Ministry of Education and Research (BMBF) in the NOLAN (grant number: 13N14459) project.