Determining locations and layouts for parcel lockers to support supply chain viability at the last mile

The pandemic caused by the corona virus SARS-CoV-2 raised many new challenges for humanity. For instance, governments imposed regulations such as lockdowns, resulting in supply chain shocks at different tiers. Additionally, delivery services reached their capacity limits because the demand for mail orders soared temporarily during the lockdowns. We argue that one option to support supply chain viability at the last-mile delivery tier is to use (outdoor) parcel lockers through which customers can collect their orderings 24/7 while ensuring physical distancing. The location planning of such lockers is known to be of utmost importance for their success. Another important topic to address is that the design of the compartment structure of the parcel lockers should meet the (uncertain) customer demand for different commodities. Both of the latter planning issues are combined into one optimization problem in this article. The objective is to maximize a linear function (e.g., expected profits) of the covered demand, given a budget an operator is willing to invest. An integer linear programming formulation is proposed, and a reformulation based on Benders decomposition is derived. It is shown that the Benders cuts can be separated in linear time. The developed algorithms enable solving of large-scale problem instances demonstrated by a performance analysis of computational experiments. The impact of different problem parameters on the obtained solutions is demonstrated by a sensitivity analysis. A case study based on real-world data from Austria is presented. The results show that using parcel lockers can support supply chain viability at the last-mile delivery tier. Moreover, the relatively small investment cost yields promising returns. The results further indicate that small-sized and medium-sized compartments should be preferred over large and x-large ones in the parcel locker compartment design.


Introduction
The pandemic proclaimed by the World Health Organization due to the proliferation of the corona virus SARS-CoV-2 in the year 2020 posed many new challenges for humanity. To decelerate the spread of the virus and to avoid overloading existing healthcare capacities, governments all over the world imposed regulations such as closed borders, partial lockdowns of the economy, stay-at-home orders, and physical distancing. These regulations shocked different tiers of local and global supply chains (SCs) while demand and supply ripples were observed [1] . On one hand, customers changed their buying behavior such that the demand for mail orders soared in most economies during the lockdowns [2] . Many industries were consequently affected by the bullwhip effect in which small variations in demand have large recurrent effects on SCs. On the other hand, many industries faced production disruptions that E-mail address: m.kahr@univie.ac.at propagate through SCs downstream; a phenomenon known as the ripple effect [3] . Both of the latter effects occurred simultaneously during the pandemic, which resulted in a disruption of the overall economy, and in substantial reverberations in some economies [1] . Scarpin et al. [4] , however, recently proposed that the observed SC disruptions caused by the bullwhip effect were statistically insignificant within the first 3.5 months after the pandemic was declared. The pandemic was nevertheless a strong test of the viability of SCs and demonstrated that their adaption capabilities play a crucial role in such situations [5] . SC viability is defined by Ivanov et al. [6] as the "ability to survive and exist after a disruption [...] with the re-design of the supply chain structure and re-planning economic performance with long-term impacts". Ivanov [7] also recently proposed the viable SC model suggesting that SCs should quickly adapt between agility, resilience, and sustainability.
One option to support SC viability is to consider alternative modes of transport at the last mile, because conventional door delivery is labor-intensive and time-consuming [1] . The aforementioned soaring in demand for mail orders during the lockdowns https://doi.org/10.1016/j.omega.2022.102721 0305-0483/© 2022 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ) therefore challenged courier, express and parcel (CEP) delivery services, particularly with respect to their capacity limits. Recently considered alternative last-mile delivery concepts that support relieving the delivery workforce include the use of autonomous vehicles such as delivery robots or drones; see [8,9] for comprehensive surveys. The latter concepts are, however, accompanied with legal aspects such as liability issues, which typically cannot be resolved in the short term [10] . Starting to implement last-mile delivery networks of robots or drones during SC shocks is therefore likely to fail. Also crowd-shipping concepts in which private people act as the last-mile delivery workforce are increasingly studied [11] and implemented in practice (e.g., amazon flex [12] ). Crowd-shipping is, however, similar to door delivery, hardly supporting physical distancing during pandemics. Moreover, deliverers may be affected by stay-at-home orders (or quarantine).
A promising last-mile delivery concept during pandemics is the use of automatic self-service (outdoor) parcel lockers. CEP deliverers can be relieved hereby because they can deliver several shipments at a stroke. The short-term usage of parcel lockers therefore enables expanding of delivery capacities while maintaining the existing delivery workforce. Additionally, the overall delivery (and environmental) cost can be thereby reduced [13][14][15] . Using parcel lockers for last-mile delivery further poses less legal issues than using autonomous vehicles (i.e., parcel lockers can be set up comparably fast). Further favorable properties of parcel lockers include that they (i) support physical distancing, (ii) are easy to build and maintain, (iii) enable collection (and return) of mail orders 24/7, (iv) enable location-specific (modular) design, (v) can be powered by batteries or solar panels, (vi) are operable under most geographic and climatic conditions, (vii) exist in variants suitable for (frozen) food, (viii) can be located at many places in urban areas (e.g., parking areas), and (ix) are available as mobile variants (e.g., locker trucks [16] ). Last-mile delivery via parcel lockers has therefore appealing properties compared to different modes of transport so that their usage is not limited to pandemics.
Although parcel lockers can be set up relatively fast, starting to implement them from the scratch during SC shocks might be nevertheless too slow to support SC viability at the last mile. One option to overcome this is to warehouse a backup fleet of parcel lockers, which can be set up quickly in case of a soaring demand for mail orders. The warehousing of parcel lockers may be performed, e.g., by large CEP delivery operators such as Amazon or by (local) authorities. In the latter case, one option is to provide white-label parcel lockers that are publicly (or privately) owned, and each CEP operator may use them (against a service fee) [17] . Alternatively, one can resort to parcel locker rental services; however, this option is not safe because rental services may run out of stock.
Challenges arising when planning distribution networks of parcel lockers include the decision on where to locate them. The location planning for such lockers is known to be the utmost important factor for success by reason of the customers' limited willingness to invest travel time to collect their orderings [8] . Moreover, deciding on the size and the compartment structure of parcel lockers is challenging but crucial to meet the customer demand for different commodities. For instance, small parcels are ordered more often than large ones (cf., Section 6 ). Another challenging aspect to consider is the uncertain customer demand, especially during a pandemic. The aforementioned challenges are addressed in this article from the perspective of an operator (e.g., CEP delivery operator or authority) aiming to maximize a linear function (e.g., expected profits) of the customer demand for different commodities covered by parcel lockers (within one replenishment period). The focus is on determining optimal locations for parcel lockers together with their location-specific locker configuration . A locker configuration consists of locker modules providing compartments of (possibly) different sizes and features (e.g., cooling). An illustration of such modules is given in Fig. 1 . Figure 1 a shows a base module containing a control unit, Fig. 1 b and c depict two optional modules, and Fig. 1 d illustrates a locker configuration. The locker modules contain compartments with four different sizes indicated by different shade. The base module is typically mandatory whereas optional modules can be arbitrarily combined up to a limit on the number of modules set by the manufacturer.
Although there exists (limited) research on parcel locker location planning and on the compartment layout design separately, to our knowledge, no work combines both of the latter decisions into one optimization framework. Note that the importance of pursuing this research avenue is also emphasized by Boysen et al. [8] and by Rohmer and Gendron [18] . Based on the suggestions of the latter authors, we propose and study the novel stochastic multicompartment locker location problem (SMCLLP) in which both location and layout decisions are combined. The main aims include (i) studying one option to support SC viability at the last-mile delivery tier, (ii) developing an optimization framework that enables solving large-scale problem instances, and (iii) deriving new insights for researchers and managerial implications for practitioners based on new real-world data. The contributions are as follows. In Section 2 , we summarize the related literature, focusing on parcel locker location planning and compartment layout design. In Section 3 , we define the SMCLLP in the spirit of multi-commodity multi-type facility location problems and prove its NP-hardness. In Section 4 , we propose a (to our knowledge) novel integer linear programming (ILP) formulation based on the concept of locker configurations (i.e., multi-type facilities). We further prove that a set of used integer variables can be relaxed without harming the existence of optimal integral solutions. We use the latter result to derive a reformulation based on Benders decomposition [19] in the spirit of a recent successful approach for solving location problems [20] . We further prove that the derived Benders cuts can be separated in linear time. In Section 5 , we outline the used algorithmic framework and propose a preprocessing procedure that supports avoiding solutions in which parcel lockers have excess idle capacity. In Section 6 , we demonstrate the performance of the latter framework in an extensive computational study using (artificial and) real-world data. We further derive new insights for practitioners based on a sensitivity analysis of the obtained results. In Section 7 , we present a case study in which Austrian census data and results from a representative survey we conducted are used. Finally, we draw conclusions from the presented work, discuss limitations of the proposed approach, and point to future research directions in Section 8 .

Related literature
The SMCLLP belongs to the large family of facility location problems (FLPs), which are of high practical relevance in numerous applications. Examples of application areas include (humanitarian) supply chain logistics [21,22] , (COVID-19) vaccination planning [23] , (preventive) disaster management [24] , traffic infrastructure planning [25] , agriculture [26] , and social network analysis [27] . To classify the SMCLLP into the family of FLPs, it is a stochastic, single-objective (or pseudo multi-objective), capacitated, single-source, single-period, multi-commodity, multi-type FLP defined in a discrete action space. Note that the latter combination of FLP variants has not yet been studied to the best of our knowledge. For more examples of FLP variants and application areas, we refer to comprehensive surveys [28,29] and proceed with discussing literature related to parcel lockers.
Although research on parcel lockers is growing, most articles are related to economic or environmental aspects of using parcel lockers (e.g., [17,30] ) or customer experience and adoption (e.g., [31,32] ). Research is also growing on using parcel lockers in vehicle routing problems (e.g., [13,33,34] ). The related work on parcel locker location problems is, however, limited while literature on the locker layout design is very sparse. The remainder of this section focuses on the latter two streams of research.
Deutsch and Golany [35] considered a problem of determining the optimal number, location, and size for parcel lockers to maximize the overall profit of an operator. Parcel lockers were assumed to have unlimited capacity (for one type of commodity). Customers were assumed to be willing to collect parcels from each potential location with their willingness negatively correlating with the distance to those locations. The authors showed that the considered problem can be formulated as a classical uncapacitated FLP formulation. A case study related to an operator located in Toronto, Canada, was presented in which 96 potential locations for parcel lockers were considered.
Luo et al. [36] studied a multi-objective parcel locker location problem. The objectives include maximizing the accessibility of parcel lockers to customers while minimizing the setup and procurement costs. The authors proposed a multi-objective ILP to solve the problem. Besides, the paper basically focused on the development of an active-learning Pareto evolutionary algorithm to tackle the problem. The authors demonstrated that the considered objectives are conflicting in extensive computational experiments (on artificial and real-world problem instances based on data from Shenyang, China). Che et al. [37] studied a multi-objective parcel locker location problem, aiming to identify optimal locations for homogeneous parcel lockers. The objectives include maximizing the demand coverage, minimizing the number of customers simultaneously covered by multiple parcel lockers, and minimizing the total idle capacity of lockers. The authors proposed a mixed ILP formulation, however, they used heuristics to solve their problem in the computational experiments. They focused on deriving the Pareto-frontiers of the considered multi-objectives. The authors further observed the existence of inefficient solutions involving parcel lockers with excess idle capacity. The latter issue was addressed by Rabe et al. [38] who considered a multi-period parcel locker location problem modeled as a multi-period capacitated FLP. The locker network can evolve over time depending on the estimated customer demand. Minimum locker utilization rates were enforced by constraints. The objective was to minimize the overall cost, including service costs and fixed costs (for setting up parcel lockers). The problem was solved deterministically for different demand scenarios. The reliability of the obtained solutions was subsequently cross-evaluated on demand scenarios generated by Monte Carlo simulations. Two other multi-period problems were proposed by Xu et al. [39] to minimize the cost of locating attended (e.g., convenience stores) and unattended delivery points (e.g., parcel lockers), respectively. In the "attended model", a given expected customer demand (estimated with machine learning models using real-world data) needs to be covered. In the "unattended model", the number of delivery points are fixed; however, the delivery points are allowed to move from one location to another one each period at a given relocation cost. The results of a computational study showed a negative correlation between operational cost and service distance (attended model) and a positive correlation between demand coverage and number of (movable) delivery points.
Lin et al. [40] studied a parcel locker location planning problem using a multinomial logit customer choice behavior model to maximize the overall service level. A non-linear (non-concave) programming formulation was developed, and a linearization using McCormick inequalities was derived. The presented computational results were, however, based on heuristics. The authors emphasized that their model had forced optimistic revenue estimation, which had lead to unnecessarily high investment cost in a follow-up paper [41] . Therein, the authors used the threshold Luce model, which enables relaxation of the aforementioned optimistic revenue estimation. Two profit-maximizing integer programming formulations with concave objective function, and a corresponding reformulation with linear objective function and second-order cone constraints were proposed. The results showed that using the threshold Luce model enables obtaining a more balanced use of investment cost. Peppel and Spinler [42] studied a parcel locker location problem to minimize both CO 2 equivalent emissions and economic cost. In a first step, the authors used a multinomial logit model to obtain the fraction of customers (aggregated into demand points) who prefer parcel locker delivery to home delivery. The obtained fractions are subsequently used in the linear objective function of the proposed ILP formulation that covers the aforementioned objectives. Real-world data from Europe, provided by an logistic service provider, was used to derive managerial insights. The latter include (among others) that parcel lockers should be installed at locations with a densely populated neighborhood and at frequently used routes based on both economic and environmental considerations.
Avgerinos et al. [43] proposed an FLP motivated by minimizing the cost of last-mile delivery using multi-type pick-up points (which may include parcel lockers). The setup and operating costs were assumed to increase non-linearly with the number (and type) of facilities. The latter's non-linearity was approximated with a piece-wise linear function, and a corresponding ILP formulation was proposed. The authors further used (meta)heuristics in the performed computational experiments whose focus is on the analysis of the algorithms' performance.
Lyu and Teo [44] studied a parcel locker location problem with the objective of reducing last-mile delivery traffic in the central business district of Singapore. The authors proposed a robust second-order cone formulation, aiming to minimize the proportion of parcels not delivered to parcel lockers while considering realworld (uncertain) customer demand profiles. They further showed that the customers' willingness to collect parcels drops off fast if the travel distance for collecting parcels exceeds 250 meters [m]. Yang et al. [45] proposed a bilevel parcel locker location problem. In the upper-level problem, an operator aims to identify op-timal locations for homogeneous parcel lockers such that the customer demand is met at minimum cost. In the lower-level problem, customers aim to minimize travel costs for collecting parcels. The problem was solved with a genetic algorithm which was also used in the presented case study based on data from Hunan, China.
Faugére and Montreuil [46] are to our knowledge the only authors who considered the design of the compartment layout of parcel lockers, however, neglecting location decisions. They proposed two ILP-based models for the design of parcel lockers focusing on (i) bin-packing compartments, and (ii) bin-packing predefined locker modules in the available space at the considered location. The objective was to maximize the profit of an operator while stochastic demand scenarios were considered.
As outlined in this section, the related literature addresses either parcel locker location problems or those related to the design of their compartment layout. Combining location and layout decisions into one optimization framework has not yet been addressed to the best of our knowledge. The main novelty of the proposed SMCLLP is to fill this research gap by explicitly considering the compartment layout of the parcel lockers. Moreover, all the formulations of the exact parcel locker location problem proposed by the authors mentioned in the section (except [44] ) are limited to solving problem instances up to a few hundred potential locations. In contrast, our proposed approach enables solving large-scale problem instances to proven optimality. We further use new real-world Austrian data to derive novel practical implications on parcel locker design.

Problem definition and properties
The SMCLLP is defined on a finite two-dimensional action space representing the planning area, and a finite probability space of demand scenarios. Probabilities p ω ∈ (0 , 1] , ∀ ω ∈ are given such that ω∈ p ω = 1 . The set of potential parcel locker locations is denoted by I, and the set of customers is denoted by J. The geographical coordinates of the potential locations and those of the customers are known. A locker configuration k ∈ K(M) consists of at least one locker module m ∈ M , where M and K(M ) denote the sets of all available locker modules and all possible locker configurations, respectively. For brevity, set K(M) will simply be denoted by K in the remainder of this paper. Each locker configuration k ∈ K is associated a construction cost r k ∈ R + depending on the number (and type) of locker modules it contains. Let k denote the number of locker modules contained in locker configuration k ∈ K, and let max denote an upper bound on the number of locker modules that can be combined (e.g., a technical restriction given by the manufacturer). Moreover, there exist location-specific upper bounds q i , ∀ i ∈ I on the number of locker modules that fit at a certain location. Then, set set of all locker configurations that fit at location i . Each locker module m ∈ M provides (different) compartments suitable for (different) commodities from set C. Examples for commodity c ∈ C include small parcels, large parcels, and frozen food of particular size (cf., Section 1 ). Commodities are further associated locationspecific weights v ic , ∀ i ∈ I, ∀ c ∈ C, which may correspond to profits of delivering commodity c ∈ C to location i ∈ I (cf., Remark 1 for more details). Commodities that are unsuitable for parcel locker delivery (e.g., because of size) are assumed to be shipped by different modes of transport (e.g., conventional door delivery).
The number of compartments (i.e., supply) for commodity c ∈ C provided by configuration k ∈ K is denoted by s kc ∈ Z + , ∀ k ∈ K, ∀ c ∈ C. Each compartment is assumed to contain at most one commodity. It is further assumed that only s kc := ηs kc , ∀ k ∈ K, ∀ c ∈ C, η ∈ (0 , 1] of commodities can be replenished in each replenishment period, i.e., s kc − s kc , ∀ k ∈ K, ∀ c ∈ C commodities have not been collected by customers before replenishment. Thus, the replenishment rate η determines the number of commodities that can be delivered to parcel lockers regularly (e.g., daily); cf., Remark 2 for details. The scenario-dependent customer demand for commodity c ∈ C is denoted by d ω Each customer can be assigned to at most one parcel locker within a given coverage distance δ ∈ R + in each scenario ω ∈ . Note that the latter requirement avoids multi-source pick-ups, which may cause customer dissatisfaction. Further note that distance δ may refer to any appropriate distance measure (e.g., Eucledian distance, Manhattan distance, or lengths of shortest paths in street networks). Finally, note that we do not consider transportation cost of serving customers via parcel lockers, because they are implicitly paid by the customers. The SMCLLP aims to identify a set I * ⊆ I of locations for building parcel locker configurations k * ∈ K i ⊆ K, ∀ i ∈ I * , which maximizes the expected location-specific weights of the covered customer demand for commodities from set C within one replenishment period, i.e., excluding s kc − s kc , ∀ k ∈ K, ∀ c ∈ C. Moreover, a given budget b ∈ R + for buying the parcel lockers needs to be respected.

Remark 1 (Location-specific weights)
. The objective of the SMCLLP is intentionally defined generic, i.e., location-specific weights on commodities v ic enable to maximize several objective function metrics including: (i) Profits considering, e.g., costs for transporting commodities from a depot to the locker locations, and locationspecific rents for placing lockers. (ii) Cost savings compared to conventional door delivery, which may differ per location, e.g., depending on population density. (iii) Customer satisfaction, e.g., customers may prefer collecting small parcels over collecting large parcels from lockers. (iv) Multi-objectives combined in a weightedsum fashion, e.g., considering both profit and customer satisfaction.
Remark 2 (Replenishment rate) . The number of commodities that can be replenished in parcel lockers on a regular basis is an important performance indicator for CEP delivery operators. The latter number is clearly at least the number of parcels collected within one replenishment period. For instance, Schnieder et al. [47] and Lyu and Teo [44] report that 70% and 75% of parcels are collected within 24 h, respectively. The replenishment rate η is, however, typically larger because additional parcels that reside longer than one replenishment period in a locker may be collected. Based on computational experiments, the latter rate seemingly converges is the fraction of parcels that reside longer than one replenishment period in a parcel locker. For instance, if a replenishment period is 24 h and 70% of the delivered parcels are collected within that period, we have η = 1 / 1 . 3 ≈ 77% .

Remark 3 (Number of locker configurations)
. By assumption, there exists an upper bound max on the number of locker modules that can be combined given by the manufacturer, and a finite set of (off-the-shelf) locker modules M. The number of locker configurations of size q ≤ max , q ∈ Z + is therefore equal to the binomial coefficient i.e, the number of possibilities to assign | M| locker modules to q bins while neglecting the order. The number of possible locker configurations respecting max is therefore Note that set K, and thus | K| , can be trivially adapted in case the manufacturer requires a lower bound on the size of a locker configuration (e.g., at least three locker modules).

Remark 4 (Mobile lockers)
. Parcel lockers are available in the form of parcel locker vans that could be warehoused as a backup fleet to support SC viability. The SMCLLP can be easily adapted for determining the fleet size and parking locations for locker vans (given a set of potential locations). For this purpose, only the locker configurations that fit exactly into a van need to be considered, i.e., set K needs to be adapted accordingly. To obtain a homogeneous fleet of locker vans (if desired), the corresponding constraints need to be enforced additionally.
Finally, we conclude this section by showing the NP-hardness of the SMCLLP in the following Theorem 1 .
Proof. Consider an instance of the SMCLLP in which | K| = | C| = | | = η = 1 , and s kc = | J| , k ∈ K, c ∈ C (i.e., large enough to be able to cover the overall demand). Moreover, let configuration cost be Then, this instance corresponds to an instance of the generalized maximum covering location problem (GMCLP) [48] with zero assignment costs and with potential locations I and customers J whereby I = I and J = J . The GMCLP seeks a subset of locations I * ⊆ I while respecting budget b that maximizes that covered demand. Thus, if b = b , the aforementioned instance of the SMCLLP is an instance of the GMCLP, which is known to be NP-hard.

Models for solving the SMCLLP
In this section, we propose a stochastic ILP formulation (in the form of its deterministic equivalent) for solving the SMCLLP ( Section 4.1 ). Moreover, a reformulation based on Benders decomposition is derived ( Section 4.2 ). For convenience, we use J i ⊆ J, ∀ i ∈ I to denote the set of customers within distance δ from location i ∈ I. Analogously, I j ⊆ I, ∀ j ∈ J denotes the set of locations i ∈ I within distance δ from customer j ∈ J.

Integer linear programming formulation
∀ ω ∈ correspond to the demand for commodity c ∈ C of customer j ∈ J i covered by location i ∈ I in scenario ω ∈ . Then, the SMCLLP can be formulated as x The objective function (2a) maximizes the expected weight of the covered customer demand (within one replenishment period). The number of parcel lockers that can be built is limited by budget in constraint (2b) . Constraints (2c) ensure that at most one locker configuration per location can be built. Inequalities (2d) guarantee that customers can only be covered by a location in a certain scenario if a parcel locker is installed there. Constraints (2e) ensure that each customer is assigned at most one parcel locker per scenario. Inequalities (2f) bind the customer demand covered by a certain locker to its available supply for each commodity in each scenario. Constraints (2g) ensure that customers are supplied at most their demand of the corresponding commodities in each scenario.

Reformulation based on Benders decomposition
The number of variables in formulation (2) grows rapidly with the problem instance size, which impedes to solve practically relevant real-world instances with that model. A computationally more tractable reformulation of (2) is derived in this section. First observe that variables x can be relaxed in formulation (2) as shown in Theorem 2 .

Theorem 2. There exists an optimal integral solution
Proof. Recall that each customer is assigned to at most one parcel locker in each scenario by constraints (2e) . First focus on one specific scenario ω ∈ , an arbitrary installed locker configuration y * ik = 1 at location i ∈ I * ⊆ I that supplies an arbitrary commodity c ∈ C with s kc ≥ 0 to the covered customers J ω * Suppose that the demand of the covered customers in scenario ω is at most the locker supply, i.e, j∈ J ω * i d ω i jc ≤ s kc . Then, the corresponding coordinates of variables x * are integral because the respective constraints (2g) are tight, i.e., x ω * i jc = d ω jc ∈ Z + , ∀ j ∈ J ω * i . Conversely, suppose that the demand of the covered customers exceeds the locker supply, i.e., Thus, in case of fractional values at the respective x * -coordinates, an optimal solution x * with integral values can be constructed without changing the objective value. By repeating the argument for each location, each commodity and each scenario the statement follows.
The result of Theorem 2 invites the derivation of a reformulation based on Benders decomposition. To this end, we use a similar approach as proposed in [20] . Particularly, we decompose the SM-CLLP into one Benders master problem in the (u , y , z ) -space while projecting out continuous variables x into | I| × i ∈ I | K i | × | C| × | | linear subproblems in which the scenario-specific coverage of the customer demand is performed. Variables u ω ikc ∈ R + , ∀ i ∈ I, ∀ k ∈ K i , ∀ c ∈ C, ∀ ω ∈ denote upper bounds on the demand for commodity c covered by configuration k at location i in scenario ω. For convenience, let

M. Kahr
Omega 113 (2022) 102721 The objective function (4a) maximizes the expected weight of the covered demand. Constraints (4b) are Benders optimality cuts while constraints (4c) ensure feasibility. Note that no Benders feasibility cuts are needed. Terms ∀ ω ∈ correspond to the objective values of the Benders subproblems (5) given a candidate solution ( ȳ , z ) ∈ P (y , z ) . Thus, a mixed ILP with an exponential number of constraints needs to be solved.
Note that indices i ∈ I, k ∈ K i , c ∈ C, ω ∈ are fixed in (5). The derivation of the Benders optimality cuts is therefore shown for one specific set of the latter indices. For the sake of brevity, it is useful to omit them and to define s := s kc ȳ ik , d j := d ω jc z ω i j , ∀ j ∈ J i at point ( ȳ , z ) ∈ P (y , z ) , and x j := x ω i jc , ∀ j ∈ J i . Given the latter definitions, a specific Benders subproblem (5) can be written as and its dual as where α and β j , ∀ j ∈ J i are dual variables associated with the constraints in (6) , respectively. Note that by weak duality, the following sequence of inequalities holds which is in fact a sequence of equations for each optimal pair (y * , z * ) ∈ P (y , z ) because strong duality holds by the linearity of subproblems (5). Moreover, variables u are forced to their bounds by the objective function (4a) . An optimal solution ( ᾱ * , β * ) of (7) at point ( ȳ , z ) ∈ P (y , z ) can be used to construct the Benders optimality cuts as follows.
∀ ω ∈ , and note the back-substitution to the original index space using the dual variables associated with constraints (5b) and (5c) , respectively. Then, Benders optimality cuts (4b) can be written explicitly as Note that the number of cuts (8) is exponential, however, they can be separated on-the-fly in a branch-and-cut fashion. Moreover, each cut in (8) can be separated in linear time as shown in Theorem 3 . (8)  Proof. Observe that an optimal solution of (7) is ( ᾱ, β ) ∈ { (0 , 1 ) , (1 , 0 ) } depending on whether or not s > j∈ J i d j , respectively, given ( ȳ , z ) ∈ P (y , z ) . The statement follows from J i ⊆ J, ∀ i ∈ I.

Corollary 1. Each Benders cut can be separated in linear time O(| J| )
for fractional values of ( ȳ , z ) .
Proof. The statement follows from substituting integer polyhedron P (y , z ) with its relaxed version in the proof of Theorem 3 .

Algorithmic framework
This section summarizes the algorithmic framework used in all computational experiments. Note that we restrict ourselves to present obtained results using the reformulation based on Benders decomposition (4) because the compact formulation (2) is incapable of solving large-scale problem instances as observed in preliminary tests.

Preprocessing
In a first step, the set of locker configurations K is computed. As long as there exist different locker configurations { k, k } ∈ K with equivalent supply for each compartment type, configuration k is removed. Note that this preprocessing step needs to be performed only once for a given set of locker modules M.
Sets I j , ∀ j ∈ J, and J i , ∀ i ∈ I are determined based on Euclidean distance δ, and unreachable customers are removed. We further remove inefficient locker configurations, i.e., those which obviously have excess idle capacity. More precisely, we adapt set K i , ∀ i ∈ I as follows. From all locker configurations which are able to cover all reachable demand (per compartment and per scenario), only the locker configuration with the smallest excess supply is kept. Finally, we remove inefficient potential locations, i.e., locations for which the demand in the coverage area is less than 70% of the supply of a minimum locker configuration that consists of three locker modules (cf., Section 6.3 ).

Separation of Benders cuts
To avoid unboundedness in the LP relaxation of reformulation (4), initial cuts of the form

Computational experiments and result analysis
In this section, we present the results of the performed computational experiments in which we solved artificial problem instances (generated based on real-world data). The parameter setting was mainly based on the literature, on Austrian census data, and on a survey we conducted in June 2020. The focus of the survey was on determining (i) the mail-order behavior of the respondents, and (ii) their willingness to use parcel lockers. A representative sample of the Austrian population ( n = 20 0 0 ) participated. We, however, reduced that data set such that only respondents living in an Austrian (provincial) capital were contained ( n = 801 ).

Description of the artificial planning areas
Square-shaped grids (of different size) corresponding to the planning areas were generated. Small grids are of sides { 1 km , 2 . 5 km } in kilometers [ km ], medium grids of sides { 5 km , 7 . 5 km } , and large grids of sides { 10 km , 12 . 5 km } , respectively. Note that we will use the latter categorization scheme to refer to the problem instance size. Customers were aggregated into the square-shaped grid cells of side 100 m , which is the smallest granularity for which Austrian census data is available. We will refer to the latter cells as customer cells. The geographical coordinates of the customer cells correspond to those of their centroids. The number of customers per customer cell was sampled from the Vienna census data (cf., Table 3 in Section 7.2 ) based on a uniform distribution. Note, however, that 9% of the respondents of the conducted survey denied using parcel lockers. The number of aggregated customers in each customer cell was therefore reduced by 9% (and rounded to the nearest integer).
The number of potential locations for parcel lockers was set to the number of square-shaped cells of the sides { 250 m , 500 m } that fit into the corresponding planning area. For instance, a cell of side 500 m in a planning area of side 1 km yields four potential locker locations whose coordinates were randomly generated based on a uniform distribution. Three different instance grids were generated for each of the aforementioned combinations of parameters. Details on the artificial planning areas are given in Table 1 .

Demand scenarios
To estimate the demand for mail orders during a pandemic, we used the historic data of the Austrian regulations from March 2020 until the end of February 2021. A distinction was made between "hard" and "soft" lockdowns of the economy. During hard lockdowns, only shops for basic supplies (e.g., supermarkets and pharmacies) were allowed to open. During soft lockdowns, all shops were allowed to open, however, with limitations on the number of customers allowed therein depending on the available selling space. Figure 2 illustrates the timeline of the considered period.
The base demand (i.e., in periods without lockdowns) was estimated based on the Viennese CEP delivery report 2020 [51] , stating that each Viennese citizen received on average 36 parcels in the year 2019 in the X2C market. The annual increase in e-commerce was assumed to be the average of its increase in the year 2019, which was 6 . 2% based on data from Germany [52] (due to the lack of Austrian data). Thus, each customer was assumed to have an average base demand for 0.1047 parcels per day in periods without lockdowns. The latter value was multiplied with the number of citizens in each customer cell to obtain the base demand therein. The distribution of ordered parcel sizes was obtained from the conducted survey. The respondents reported that they had ordered 37% small parcels, 35% medium parcels, 19% large parcels, and 9% x-large parcels (within a three-month period). The base demand for each different parcel size in each customer cell was adapted accordingly (and rounded to the nearest integer).
To estimate the increase in mail orders during hard and soft lockdowns, we used the same data set as before [52] . The latter data contains the monthly increase in e-commerce compared to the corresponding month of the previous year and ranges from January 2018 to March 2021. For the demand during lockdowns, the weighted averages of the corresponding periods (cf., Fig. 2 ) were computed and adjusted by the assumed annual increase in ecommerce. This yields an average increase in e-commerce of 20 . 6% and 24 . 4% during hard and soft lockdowns, respectively. Three lockdown scenarios were used in the computational experiments: (iii) | | = 3 : The scenario as shown in Fig. 2 . In total, 80 days of hard and 108 days of soft lockdowns were imposed by the Austrian government. Thus, we set p 1 = 188 / 365 ≈ 48% , p 2 = 80 / 365 ≈ 22% , and p 3 = 108 / 365 ≈ 30% .

Locker modules and locker configurations
The used locker modules are based on an Austrian manufacturer's products. A minimum locker configuration consists of one  1,3,1,1). Note that on one hand, the latter set of modules overrepresents small-and medium-sized compartments concerning the proportion of ordered parcel sizes reported by the respondents in the survey. On the other hand, the respondents' willingness to use parcel lockers was negatively correlated with the parcel sizes (cf., Section 6.4 ). The prices of the locker modules were set to € 5 0 0 0 and € 1 0 0 0 for the base module and for an optional module, respectively, based on personal communication with a responsible employee of the locker manufacturer.

Objective function metric
Like in the case study (cf., Section 7 ), we assumed that a (local) authority aims to set up a last-mile delivery network of white-label parcel lockers. We further assumed that each CEP delivery operator can use these lockers against a service fee related to their cost savings of using lockers compared to conventional door delivery (as suggested in [17] ). The authority was assumed to be concerned about customer satisfaction to some extent. We therefore used a weighted sum multi-objective function combining service fees and the willingness of customers to use parcel lockers.
For estimating the service fees, we used prices of the Austrian branch of the CEP delivery operator Direct Parcel Distribution (DPD) who charge € 4.3, € 6.3, € 8.3, and € 12.3 for delivering small, medium, large, and x-large parcels, respectively. We assumed that DPD determines these prices based on mark-ups, i.e., the last-mile delivery cost and potential cost savings obtained from locker delivery are fractions of the prices charged by DPD. In the remainder of this section, we therefore use the latter prices as service fees directly (because it affects neither the solutions of the computational experiments nor the result analysis). We detail how to obtain meaningful scale factors in the case study (cf., Section 7 ).
To assess the willingness of customers to use parcel lockers, we took recourse to the data from the conducted survey. Each respondent was shown 10 randomized exemplary scenarios for collecting parcels from lockers. The parameters parcel size, weight, and walking distances varied in each scenario. The respondents were asked to assess their willingness to use parcel lockers on a 7-Likert scale (1 = worst and 7 = best) in each scenario shown. Each of the aforementioned parameters has a significant negatively correlated impact on the respondents' willingness to collect parcels (revealed by a regression model). However, even for the largest value of the parameter walking distance of 500 m (or more), the average willingness to collect parcels is 4.1 on the Likert scale. The average values for the willingness to collect parcels within 500 m are 4.4, 4.2, 4.1, and 4 for small, medium, large, and x-large parcels, respectively.
Finally, we normalized the obtained values of the willingness to collect parcels, and the service fees, and combined them linearly by customer care rate λ ∈ [0 , 1] , i.e., if λ = 0 only the service fees are considered in the objective function.

Further parameter setting and summary of the used data
The maximum number of locker modules that can be combined to a locker configuration was set to max ∈ { 5 , 10 , 15 } . The number of possible locker configurations | K| is therefore at most 120, cf., (1) . For each potential location i ∈ I, the maximum number of modules which can be placed was drawn randomly from the integer interval [3 : max ] based on a uniform distribution. The available budget was computed via fractions of the maximum setup cost, i.e., the total cost of building the maximum number of modules (including one base module) at each location i ∈ I. The lat-  Table 2 summarizes the data and problem parameters (discussed in Sections 6.2 -6.5 ), which are used in the remainder of this section.

Performance analysis
Given the number of generated problem instance grids and the parameter setting outlined above in this section, the number of all combinations of the latter is 69 984 . We therefore restricted ourselves to a randomly chosen subset of these combinations so that the performance analysis and the sensitivity analysis are based on 35 973 computational experiments. All algorithms were implemented in Julia 1.6 together with the JuMP modeling language [53] . Moreover, IBM CPLEX 12.10 (with default settings) was used as ILP solver. Each experiment was performed on a single core of an Intel Xeon E5-2670v2 machine with 2.5 GHz. A memory limit of 16 GB RAM and a time limit of 1 0 0 0 s ( ≈ 17 min) were set. Note the we chose the latter time limit based on previous computational experiments (of the initial paper draft) in which a time limit of one hour was set. It turned out that 97% of the problem instances solved to optimality finished within 1 0 0 0 s. The shorter time limit comes, however, at the cost of larger average optimality gaps, which increased from 3.5% to 5.9%. Figure 3 shows performance profiles including runtimes and optimality gaps of the small, medium, and large problem instances. Figure 3 a shows that 86.3% of the small, 64.2% of the medium, and 53.3% of the large problem instances were solved to optimality within the set time limit. The optimality gaps of the remaining problem instances are depicted in Fig. 3 b. The latter gaps were computed by (UB -OV)/UB, where UB is the best known upper bound and OV the objective value of the best known integer solution. All problem instances with an optimality gap of 100% hit the set memory limit.
A more fine-grained analysis with focus on the impact of different problem parameters on the performance is provided in Appendix A . Figs. A .1-A .3 show a large volatility in performance with respect to the available budget b f , the maximum size of a locker configuration (measured in number of modules) max , and Table 2 Summary of the data and problem parameters discussed in Sections 6.2 -6.5 . Abbreviations S, M, L, and XL correspond to parcels or compartments of size small, medium, large, and x-large, respectively.

Data description Values
Demand   4. Runtime of separating Benders cuts relative to the total runtime of different problem instance sizes ( Fig. 4 a). Number of explored branch-and-bound (# BB) nodes of different problem instance sizes ( Fig. 4 b). The number of outliers that are not depicted is given above each boxplot in the brackets (·) .
the ratio between number of customer cells and potential locations | J| / | I| . It can be observed that the optimality gaps are correlated with b f and max , and negatively correlated with | J| / | I| , i.e., the more the potential locations in a problem instance of the same size (measured in | J| ) the larger the gaps. Parameters coverage distance δ and number of scenarios | | have a minor impact on the algorithm performance. We remark, however, that parameter δ is seemingly negatively correlated to the performance. The impact of parameters replenishment rate η and customer care rate λ on the performance is negligible and therefore unreported. Figure 4 a shows the runtimes used for the separation of Benders cuts relative to the total runtime per solved problem instance. The latter fraction can be observed to be negatively correlated with the problem instance size. The figure also shows that the most time in the solving process is used for branching, which is a bottleneck compared to the separation of the Benders cuts. Figure 4 b shows the number of explored branch-and-bound nodes clustered by different problem instance sizes.

Sensitivity analysis
In this section, we investigate the sensitivity of the solutions with respect to different problem parameters. Only results of problem instances solved to optimality are reported unless stated otherwise.   Fig. 5 b and c. The relative number of modules of type N2 and M2 used in the solutions is, however, correlated with the available budget. Note that the latter results strongly depend on the assumptions on the locker module design (cf., Section 6.3 ) and on the objective function weights (cf., Section 6.4 ). If we break down the assumed service fees per parcel to service fees per consumed height unit of a locker module, we note that the contribution to the objective function (per height unit) decreases with the compartment size. For instance, using one medium-sized compartment contributes less to the objective function than using two small-sized compartments does. The same holds true for the customers' willingness to collect parcels. The compartment structure of modules N1 and M1 , therefore, implies that using Module set 2 enables covering of more customer demand (13% on average) and obtaining larger objective function values (4% on average) compared to Module set 1 . Figure 5 d and e show the latter effect and further reveal that its impact is negatively correlated with the available budget. Note that we only compare solutions of two module sets if they are optimal and all other problem parameter values are identical. In the remaining anal-ysis, we restrict ourselves to results of problem instances using Module set 2 .
We proceed with an analysis of the locker and location utilization, i.e., the number of compartments used relative to the total number of compartments (per locker configuration) and the number of locker modules relative to the maximum number of modules that can be installed (per location), respectively. Figure 6 a shows the obtained locker utilization. It can be observed that there hardly exists any idle capacity, however, the latter is correlated with the available budget. The opposite trend can be observed in Fig. 6 b, showing the location utilization. The reason for these trends is that the model tends to increasingly choose inefficient locker configurations, which nevertheless contribute to the objective function (if budget is no restriction). The locker utilization rates are, however, quite large because (i) the demand in coverable areas is larger than the potential available supply in most cases, and (ii) obviously inefficient locker configurations were removed in the preprocessing. The locker utilization is also correlated with the optimality gap, i.e., the larger the gap, the larger the idle capacity in the obtained the solutions. This trend can be observed in Fig. 6 c and d in which results of problem instances solved with an optimality gap greater than zero are shown. A similar trend regarding the location utilization was not observable.
We proceed with an analysis of the impact of different problem parameters on the relative covered demand (cf., Fig. 7 ). Note that we again consider only results for which optimal solutions for all values of the parameter of interest exist (whereas all other problem parameter values are identical). Figure 7 a shows that the relative covered demand grows approximately linearly with the maximum number of modules that can be installed max . This again indicates that there exist more demand in coverable areas than  ( Fig. 6 a and b). Locker utilization (w.r.t. budget fraction and optimality gap) of solved problem instances with optimality gaps ( Fig. 6 c and d). the potential supply in the problem instances. Note, however, that the aforementioned linearity seemingly breaks down if budget is not restricted, which again indicates the existence of idle compartments. A similar trend can be observed in Fig. 7 b, which shows that the relative covered demand changes approximately like the relative changes in replenishment rate η. The latter trend has, however, also less impact if budget is not restricted. Figure 7 c shows a notable impact of the ratio of customers and potential locations | J| / | I| on the relative covered demand. In contrast, the impact of the customer care rate λ is negligible (cf., Fig. 7 d) and in fact not significant at a confidence level of 95% revealed by an analysis of variance. The reason is that the customers' will-ingness to collect parcels decreases with parcel size and so does the profitability per consumed locker module height unit (as discussed above in the comparison of Module set 1 and Module set 2 ). The impact of the coverage distance δ on the relative covered demand is quite low (cf., Fig. 7 e). This again indicates that there exists more demand than potential supply, i.e., most supply can be exhausted within a coverage distance of 300 m . Finally, Fig. 7 f shows the impact of the considered demand scenarios on the relative covered demand, which decreases notably in the very strict lockdown scenario ( | | = 2 ). Although not directly observable from the figure, this decrease is 17% on average. For the less strict lockdown scenario ( | | = 3 ), this decrease is only 10% on average. The latter comparison clearly shows that relying on existing parcel lockers will hardly help to cover a soaring increase in demand for mail orders as it was the case during the lockdowns. More promising options to do so include (i) augmenting the existing infrastructure with new parcel lockers, (ii) warehousing a backup fleet of parcel lockers, and (iii) ordering a parcel locker rental service.

Case study
The case study presented in this section is based on census data comprising the number of citizens (and their geographical location) of Vienna and the two largest Austrian provincial capitals Graz and Linz. Further used data was sourced from the conducted survey (cf., Section 6 ) and from the literature. The results are based on computational experiments performed on the same machine as outlined in Section 6 . The time limit was set to 24 h and the memory limit to 64 GB.

Status in Austria and mode of operation
The parcel locker supply is currently limited but emerging in Austria. Parcel lockers are mainly in use at several post offices and nearby supermarkets. There exists also a pilot project aiming to test the usage of white-label outdoor parcel lockers in the 5 th district of Vienna [17] . The results of the testing phase of the latter project are positive. Thus, there exists a large potential for building additional (white-label) parcel lockers in Austria. For the case study, we assumed that the government aims to invest in a backup fleet of such lockers that are only activated in case of a soaring demand for mail orders (e.g., during lockdowns or around Christmas). Options to warehouse that parcel lockers include: (i) to store them at some (publicly owned) area outside the city and to transport them to the determined locations if needed, and (ii) to place them directly at the determined locations and to activate them only if needed. Note that the second option gives rise to support local artists who may design individual artworks for each parcel locker. Clearly, also renting out their surface for advertisement is an option which ensures cash flows if they are idle. We used the second option in the result analysis (neglecting revenues from rents for advertising).

Problem instance description
Real-world problem instances include the cities of Vienna, Graz, and Linz. The city areas were segmented into standardized squareshaped customer cells of side 100 m , which is the smallest granularity for which census data is available. The number of residents registered in each customer cell by 2020/01/01 is known from that census data. The geographical coordinates of each cell correspond to the coordinates of its centroid. Uninhabited customer cells (e.g., covering rivers) were removed. Potential parcel locker locations were assumed to be at certain points of interest (POI) provided by OpenStreetMap [54] . Depending on the availability of POI in Table 3 Description of the real-world problem instances. The total number of inhabitants | J | , the number customer cells | J| , the average number of inhabitants per customer cell | J | / | J| , the number of potential locker locations | I| , and the number of customer cells per potential location | J| / | I| are reported. the respective cities, locations I are chosen from a subset of POI in raw data set. The following POI were chosen (based on [55] )

City name
whereby the number in the brackets (·) indicates the assumed maximum number of locker modules that can be built there: subway stations (20), train stations (20), petrol stations (10), marketplaces (10), parks (10), parking garages (10), playgrounds (10), large supermarkets (Billa Plus, Eurospar, Interspar) (10), universities (10), hospitals (10), tram stops (7), and bus stops (5). Table 3 shows details about the used real-world problem instances. Given the bounds on the number of locker modules defined for the POI, there exist | K| = 207 locker configurations (excluding those with less than three modules, which is a requirement imposed by the manufacturer). For each potential location i ∈ I, the maximum number of modules that can be installed was set to a randomly chosen integer ranging between (and including) three and the upper bound defined for the corresponding POI.
We set a coverage distance of δ = 300 m , a replenishment rate η = 77% (based on [47] ), and a customer care rate λ = 20% . We further used Module set 2 because it performed better in the sensitivity analysis (cf., Section 6.7 ). The prices for the locker modules are the same as reported in Section 6.3 , however, we assumed a 10% mark-up per year for service. As a demand scenario, we considered the Austrian lockdown scenario ( | | = 3 ) described in Section 6.2 .

Results
For the analysis of the results, we scaled the prices charged by DPD ( Section 6.4 ) as follows. We first discounted them by the value added tax (20%) and by an assumed mark-up (5%). We then took 41% of the latter values that correspond to the last-mile delivery cost (based on [41] ). The cost savings of using parcel lockers compared to that of door delivery reported in the literature deviate. For instance, Iwan et al. [55] and Turkoglu and Genevois [29] report 16%, whereas Gevaers et al. [56] and Yu et al. [57] report 58%. We therefore used these percentages to compute lower and upper bounds on the latter cost savings. Finally, we deducted 30% of these values (to incentivize CEP operators to use the parcel lockers) and obtained lower and upper bounds on the net service fees. Given the aforementioned manipulations, we effectively scaled the prices charged by DPD by factors by 3.64% (lower bound) and 13.21% (upper bound). Note that we also used an average scale factor of 8.43% (particularly in Fig. 8 c).
An overview of the results is performed in Fig. 8 . Figure 8 a shows the relative covered demand, depending on the available budget for Vienna, Graz, and Linz. We observe that the marginal relative covered demand decreases with the budget. This indicates that building large parcel lockers is the preferred option if budget is restrictive (because the price of the base locker module substantially exceeds the price of an optional module, and enough demand exists within the coverage distance). Moreover, there is a notable difference in the relative covered demand with respect to the cities. One reason for that difference is the volatility of the ratios between the number of customer cells and potential loca- Table 4 Results of the case study. The following values are reported: the budget fraction ( b f ), the runtime (rt) in hours, the optimality gap (gap) in percent, and the relative covered demand for different parcel sizes including the total covered demand ( E [ u ] / E [ D ] ) in percent, the relative compartment utilization including the total utilization (T) in percent; the relative number of locker module of type M1 (M1), the location utilization (LU), the relative number of installed parcel lockers ( | I * | / | I| ), the investment cost ( c) in million Euro, a lower bound on the expected net revenues per year ( r ) in million Euro, and an upper bound on the expected net revenues per year ( r ) in million Euro.    tions | J| / | I| (cf., Table 3 ). Figure 8 b again shows that using locker modules of type M1 is the preferred option if the budget is restrictive. Figure 8 c shows the average expected return on investment per year, i.e., the average annual net revenues r * (using a scale factor of 8.43%) relative to the investment cost c. Note that we assumed that the parcel lockers are in use 161 days per year, i.e., the number of lockdown days (except Sundays) in the considered demand scenario. It can be observed that the average return on investment decreases with the available budget for the reasons mentioned above: (i) increasingly small parcel locker configurations are constructed if there exists plenty of budget, (ii) the relative number of the less profitable locker modules of type M2 increases, and (iii) the optimality gaps increase, which is correlated with less efficient compartment utilization (cf., Table 4 ). Clearly, the return on investment will be substantially larger if the parcel lockers are in use the whole year (or rent out as advertising surface). In the latter case, the average return on investment over all problem instances is 75.7%, which yields an average amortization time of 1.4 years. This is in line with the results of Seghezzi et al. [58] , who report an amortization time of 1.3 years. Table 4 reveals further details. Note that problem instances with a runtime lower than the set time limit and an optimality gap greater than zero hit the memory limit. We observe similar trends as in the sensitivity analysis (cf., Section 6.7 ). Columns covered de-mand show that small-and medium-sized locker compartments (i.e., locker module M1 ) are preferred over large and x-large compartments (particularly if budget is limited). We further observe that the compartment utilization is acceptable, however, decreasing with the available budget. Moreover, the investment cost of roughly € 9 million (Vienna), € 1.3 million (Graz), and € 1 million (Linz) are sufficient to cover a soaring increase in demand for mail orders as observed during the lockdowns in Austria. The reported lower and upper bounds on the expected net revenues ( r and r ) are again based on 161 days of use per year.
Finally, one exemplary solution for Vienna (with b f = 50% ) is given in Fig. 9 . It can be observed that most parcel lockers cluster in the inner districts, which have the largest population density. An advantageous side-effect regards the relief of traffic in these districts, which have sparse traffic areas with many narrow alleys. Another interesting fact to remark is that most parcel lockers are located at tram stops (182), subway stations (148), parking garages (75), fuel stations (50), and train stations (45), where the number in the brackets ( ·) indicates the number of location types in Fig. 9 . The latter fact opens the possibility that customers can collect their parcels when commuting, e.g., between work and home.

Conclusions
We proposed and studied the stochastic multi-compartment locker location problem (SMCLLP), which was proven to be NPhard. An ILP formulation for the SMCLLP was proposed, and a reformulation based on Benders decomposition was derived. We showed that the Benders cuts can be separated in linear time. The performance of the developed algorithmic framework enables solving of large-scale problem instances to proven optimality, which was demonstrated by an analysis of results of computational experiments on artificial problem instances (generated based on realworld data). In a sensitivity analysis of the latter results and in a case study (based on data from Vienna, Graz, and Linz), we have derived practical considerations for setting up a (white-label) parcel locker last-mile delivery network. For instance, we showed that (given our assumptions) small-and medium-sized locker compartments should be preferred over large and x-large sized compartments when designing parcel lockers. Moreover, we showed that the expected return on investment of using parcel lockers is promising, however, decreasing with the investment cost. We further demonstrated that parcel lockers can cover a soaring increase in demand for mail orders (as observed during the COVID-19 pandemic). Thus, SC viability can be supported if operators (e.g., governments) warehouse backup parcel lockers, which are activated if needed. In contrast, relying on parcel lockers that are permanently in use will hardly support SC viability because CEP delivery operators will adapt their resources accordingly. However, the SMCLLP enables, nevertheless, to determine locations and compartment layout for such parcel lockers. Benefits of using the proposed model include that decision support for parcel locker locations and their layout can be obtained simultaneously. The proposed approach further enables solving of large-scale problem instances.
The proposed models are accompanied by some limitations. Although we were able to solve large-scale problem instances, the computational performance heavily depends on some problem parameters. One option to overcome this limitation is to solve the problem for smaller planning areas (e.g., districts) while increasing problem parameters that negatively affect the computational performance (e.g., bounds on the number of locker modules that can be combined). We observed that typically one type of locker module is preferable, which can imbalance coverage of demand for different commodities. The proposed models are not able to control a more balanced demand coverage (if desired). Also the presented case study is accompanied by some limitations. White-label par-cel lockers are not yet a well-established last-mile delivery mode. Their usage needs to be more standardized to provide a sound interaction between sellers, CEP delivery operators, and customers (which is, however, out of the scope of this work). The potential locations for parcel lockers were further chosen by assumption. Moreover, although the proposed models fit well for private profitmaximizing operators (with limited budget), a probably more intuitive approach for determining a backup fleet of parcel lockers is minimizing investment cost while ensuring a certain demand coverage. Another option is to replace the budget constraint (2b) with a constraint that bounds the expected covered demand by a targeted fraction of the overall expected demand.
The latter issues are options for future work for which we give further examples in the following. Although we considered the willingness of customers to use parcel lockers in the objective function, the probability that they do so was neglected. One option to do so is to use multinomial logit customer choice behavior objective functions. Such functions are frequently concave, i.e., a solution approach using generalized Benders decomposition for solving large-scale problem instances seems promising. Including customer choice behavior also opens an avenue to investigate the effects of different pricing strategies, e.g., the effects of discounts for customers if they opt for parcel locker delivery. Another research direction involves the design of the parcel locker compartment layout. In the SMCLLP, we choose from a set of predefined locker modules. More effort is needed on the design of such modules. Alternatively, one can consider combining location decisions while simultaneously bin-packing compartments to those locations. If the order of the compartments in a locker configuration is neglected (which is not the case in [46] ), the bin-packing procedure could be probably performed in knapsack subproblems.

Data availability
The used artificial planning areas (including population size per demand point) as well as the parcel locker module sets are available here: https://doi.org/10.13140/RG.2.2.364 4 4.26241 . The realworld instances are also included, however, the population size of each customer cell was randomly permuted to account for a license restriction. Note that the summary statistics of those instances remain unaffected.  Performance profiles of the medium problem instances with respect to problem parameters relative available budget b f , maximum number of locker modules that can be installed max , potential location density | J| / | I| , coverage distance δ, and number of scenarios | | . The left column shows the relative number of problem instances solved to optimality within the runtime limit. The right column shows the optimality gaps of the remaining problem instances.

M. Kahr
Omega 113 (2022) 102721 Fig. A.3. Performance profiles of the large problem instances with respect to problem parameters relative available budget b f , maximum number of locker modules that can be installed max , potential location density | J| / | I| , coverage distance δ, and number of scenarios | | . The left column shows the relative number of problem instances solved to optimality within the runtime limit. The right column shows the optimality gaps of the remaining problem instances.