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The Weber Problem in Logistic and Services Networks under Congestion

We investigate a location-allocation-routing problem where trucks deliver goods from a central production facility to a set of warehouses with fixed locations and known demands. Due to limited capacities congestion occurs and results in queueing problems. The location of the center is determined to maximize the utilization of the given resources (measured in throughput), and the minimal number of trucks is determined to satisfy the overall demand generated by the warehouses. Main results for this integrated decision problem on strategic and tactical/operational level are: (i) The location decision is reduced to a standard Weber problem with weighted distances. (ii) The joint decision for location and fleet size is separable.(iii) The location of the center is robust against perturbations of several system parameters on the operational/tactical level. Additionally, we consider minimization of travel times as optimization target.


Introduction
We investigate a generalized Weber problem in a dynamic and stochastic environment. The model combines a classical planar location problem with optimizing the size of the fleet of trucks in a logistic and services network and describes interacting production and distribution facilities with logistic components.
This integrated problem is related to location-allocation problems, location-routing problems (LRPs), location-inventory problems (LIPs), and transportation-location-allocation problems under random influences. A key property of our model is to integrate strategic (facility location) and tactical/operational (allocation, scheduling) aspects of decision making. Such integrated logistics-location models occur e.g.
in supply chain planning and operation, for surveys see [MNS09] and [HN19]. Difficulties arising with similar integration procedures in combined location-routing problems are described in [MJS98][p.10].
Our model comprises a set of warehouses with known locations, a single production center which produces commodities which are demanded by the warehouses, and a set of trucks which transport the commodities from the production center to the warehouses. The model captures additionally congestion which emerges at service stations with limited capacity for loading of trucks at the center and for unloading at the warehouses. Optimality of the system is defined with respect to maximization of the warehouses' throughput (which generates revenue) and minimization of the number of trucks (which generate costs) over time and in a stationary system. Our main contributions are: For given demands from the warehouses and incorporating the consequences of congestion at the center and the warehouses (i) we find an optimal location for the central production facility, and (ii) determine the needed transportation capacity, and (iii) solve jointly both optimization problems in a unified model, to combine strategic decision making (for location of the center) with decisions for tactical and operational issues (transportation capacity and scheduling rules for routing trucks), and (iv) we demonstrate that the solution of the optimization problem reveals important robustness of that solution against changes or perturbations of several parameters on the tactical/operational level (insensitivity).
Summarizing: We introduce a model which is tailored to integrate several hierarchical levels of decision making in complex systems. As indicated in the literature, neglecting these inter-dependencies often generates sub-optimal solutions, see the comments on literature in Section 2, especially concerning integration of strategic and tactical/operational aspects of planning on p. 4.
Structure of the paper. In Section 2 we review related literature. Section 3 presents an overview of the problem setting and connections to investigations of related problem settings. Details are provided in Section 4.1. Our main findings are described in Section 4.2. In Section 4.3 we analyze the system in full detail. In Section 4.4 we complement our structural results by an algorithm to determine the minimal number of trucks needed to satisfy the overall demand at the warehouses. In Section 4.5 we discuss the assumptions used in Section 4.3 and we indicate that in many cases the results hold in more general settings. Section 4.6 is devoted to determine the optimal location of the center for minimizing the round-trip times of trucks. Section 5 contains numerical experiments to underpin the structural results obtained sofar and a discussion of robustnes properties of our results. In Section 5.3 we discuss the impact of our results on the consequences of intertwining, respectively separating strategic (location) and tactical/operational (routing and scheduling) decisions. We conclude by discussing

Literature review
Stochastic location models with congestion, i.e. location problems in the context of service processes under stochastic influences have been investigated since the 1970's by many authors. Research on location problems within the scope of queuing systems was initiated with Larson's paper [Lar74] followed by work of Larson, Berman and coauthors (e.g. [BLC85] and [BLP87]) on discrete location problems. We sketch here only two main research directions: (1) In [DSSL90] and [SJD99] the authors consider location problems where mobile servers move in a plane and demands of the clients occur as Poisson processes. The mobile servers are described as queueing systems, e.g. of type M |M |1|∞ or M |G|s|∞. The guiding principle is to incorporate travel times to and from the clients into the service times of the mobile server. A survey is provided in [BK04]. Locational analysis in a randomly changing environment with occurrence of queueing phenomena is investigated in [DM19].
(2) A class of models which is somehow dual to that described in (1): The servers are fixed and clients move to the nearest service stations. Work in this direction is by [BD07] and investigated further by [ABD08] and [ABD09], where additional references are provided. In these papers the demand is generated at specified points according to a Poisson or a general renewal process and the servicing nodes usually are modeled as M/M/k/∞ systems. For different cost functions the authors determine the optimal location of a set of servers on a set of nodes. The field is reviewed in detail in [BK19] and classified there as SLCIS ( = Stochastic Location models with Congestion and Immobile Servers).
Location-inventory problems (LIPs) aim "to integrate strategic supply chain decisions with tac-4 tical and operational inventory management decisions [FBFK15]". The basic LIP, as described in [FBFK15][Section2], encompasses a single supplier (production center), several warehouses (distribution centers), and retailers. The location of the supplier and the retailers are given and the decision problem is to determine the location of the distribution centers. [FBFK15] provides a survey of research on basic LIPs and more evolved variants.
Location of local repair facilities with a central service station and inventories at the repair stations is investigated in [OBS06]. Given the location of the center, the objective is to determine optimal locations of repair facilities, ensuring a certain inventory level at each station. The optimization problem is solved by approximations and local search. In this article queueing models are used to describe the interrelation between local repair stations and the center.
Location routing problems aim to combine location analysis and planning of vehicle routing. Fundamentals of vehicle routing problems are described in [Lap88]. A recent survey with emphasis on location decisions on networks (discrete location problems) is [ASRP19]. Closer to our problem of location in the plane are [SN09] and in a more versatile setting [MTS12].
Integration of strategic and tactical/operational aspects of planning is a common topic of almost all the mentioned work. In [SR89] for LRPs it is shown that separating decision on location and routing can lead to sub-optimal decisions. Under the heading "Why logistics matters in location modeling" this problem is discussed indepth in [HN19][Section 6.2], stating as main conclusion "that making location decisions ignoring primary logistics activities . . . may result in excessive costs."

Problem description and classification
Problem setting. We are given a set of warehouses (stations), indexed by j ∈ {2, 3, . . . , J}, whose positions a j = (a j1 , a j2 ) ∈ R 2 in the plane are known. We are also given the aggregate demand (requirement) of D j truck loads per day generated by warehouse j for a single commodity, D j > 0 .
We are to find a location x = (x 1 , x 2 ) ∈ R 2 for a production center (source of commodities), indexed henceforth by 1, where the commodities are produced and dispatched.
To deliver the commodities from the production center to the warehouses, N ≥ 1 identical trucks circulate in the system. The dispatching rules are roughly as follows: A truck loaded at the center is directed to one of the warehouses according to a given schedule or plan, is unloaded there, and returns to the central station to start another delivery cycle. The next destination, determined by the scheduling regime, might be different. The center has limitations on its capacity to ship the product, especially when loading several trucks simultaneously, and the warehouses have limited capacity to unload trucks in parallel.
To assess the system's performance we take into consideration the distances between the center and the warehouses, the number of trucks, the travel times, the scheduling rules for sending out the trucks, the loading and unloading times, and additionally the delay resulting from limitations of loading and unloading capacities, i.e., congestion at loading and unloading facilities.
The optimization problem to be solved is to determine the location of the center such that the delivered amount of commodities is maximal and all requests of the warehouses are satisfied with a minimal number of trucks.
Related problems and classification. Following [MJS98] and [NS07] this problem is related to locationallocation problems because it is assumed that there are only radial trips from the center to the warehouses. The problem is related to location-routing problems because it deals with "location planning with tour planning aspects taken into account [NS07][p.1]", because we have to decide about subsequent tours for the trucks to visit different warehouses (the scheduling regimes/plans). The relation to transportation-location problems in the sense of Cooper [Coo72], [Coo76] is clear because we consider, among others, both problems. Moreover, the center has limitations on its capacity to ship the product [Coo72][p. 94]. Additionally, our optimization criterion encompasses a time dimension as it is considered in [Tap71] for the classical setting of Cooper. Aspects of our problem setting which are not included in the above standard problem classes are: • Limited unloading and loading capacities at the warehouses and the central production facility, i.e. congestion in the transportation network and queueing problems.
• Time dimension of transportation-location-allocation problems, as discussed e.g. in [Tap71][p. 383], in the optimization criterion. In [Tap71][Section 5] questions concerning delivery time lags are sketched. In our setting sequentially varied scheduling of trucks is allowed and for feasible scheduling of trucks emerges a necessary condition which optimal plans must satisfy. We sketch a toy example which highlights the problems.
Example 3.1 Consider the case J = 3, i.e. a center indexed 1 and two warehouses, numbered 2, 3, with demands D 2 = 10 and D 3 = 20 truck loads per day for a commodity and N = 3 trucks with equal capacity. Two reasonable schedules are (i) one truck serves station 2 and two trucks serve station 3, (ii) all trucks are scheduled to serve in a cycle first station 3, thereafter station 2, and finally station 3 again; these cycles are iterated.
Both schedules generate fair service for the warehouses because they guarantee that over time warehouse 2 will obtain ρ 2 = 1/3 and warehouse 3 will obtain ρ 3 = 2/3 of the overall delivered goods.
Because we investigate location decisions we have to consider long time horizons and we will therefore assume that the system has approached its stationary state. Clearly, then a necessary condition for optimality of a scheduling regime for trucks is to partition in the long run sent-out goods according to: Our problem fits into neither of the mentioned streams on queueing-location problems because we incorporate local congestion at the warehouses explicitly. Determining a center's location in [BD07], [ABD08], and [ABD09] is somehow similar to our problem but in these papers no two-way interactions 6 (here generated by trucks) occur between sources of demands and the production center.

Details of problem statement
Modeling the warehouses-production network. Warehouse j is equipped with s j ≥ 1 service facilities for unloading trucks and there is ample waiting space for trucks that arrive while all unloading facilities are busy, j ∈ {2, . . . , J}. The time for unloading a truck is exponential with rate µ j per hour.
Center 1 has s 1 ≥ 1 service facilities for loading trucks and ample waiting space for trucks that arrive while all loading facilities are occupied. Loading times are exponential with mean µ −1 1 . The queueing regime for trucks is First-Come-First-Served (FCFS). We abbreviate the service rate functions by µ j (n) := µ j · min(n, s j ), n = 0, 1, . . . , j = 1, . . . , J. (4.1) We consider general schemes for scheduling the radial trips of trucks which must meet only the following restrictions. If D := J j=2 D j denotes the total demand per day, the portion of demand that has to be delivered to warehouse j is Then a truck loaded at the center is directed to warehouse j with (average) frequency ρ j = D j /D. In the mathematical model we realize this property of the dispatching rules by a randomized schedule which selects the next warehouse with probability ρ j for destination j in a Markovian way. We emphasize that this does not mean that scheduling of trucks should be randomized, but reflects that we are interested in gross characteristics of the system layout. We comment on this later in the discussion of "Random routing. . . " on p. 13.
Demand that is not satisfied immediately will be backordered at the respective warehouses. The distance between warehouse j and center at x is denoted by d j (x) := d(a j , x) where d : R 2 ×R 2 −→ R + is a general convex function.
Details of the optimization problem. The aim is to fulfill all demands occurring at the warehouses with a minimal number of trucks and to maximize the utilization of the given resources. The latter means that we are to maximize the overall mean number of delivered goods per hour (= time unit for servicing) which is the sum of the throughputs of the warehouses measured in truck loads. The throughput of a warehouses depends on the coordinates x = (x 1 , x 2 ) of the center and the number of trucks N . Formally: Denote for T ∈ R + by A j (N ; x)(T ) the amount of commodities which arrived at warehouse j and is unloaded within time horizon [0, T ] when the center is located at x and N trucks are cycling. It will be shown that for j = 2, . . . , J the throughput for warehouse j exists and determines by standard ergodicity arguments for Markov processes the overall mean number of truck departures per hour from j in the stationary system. By stationarity this equals the mean number of truckloads delivered to j per hour. The total throughput of interest " w " indicates that we evaluate only throughputs of warehouses. This leads to It will turn out that the main effort in solving Optimization Problem 4.1 is to solve a sequence of maximization sub-problems. These are

Main results and detailed analysis
Facility location in connection with queueing problems usually leads to complex algorithms, see [BLC85]. In view of this, our first theorem is counter-intuitive.
Theorem 4.4 Consider locations a j = (a j1 , a j2 ) ∈ R 2 , j = 2, . . . , J, in the plane and associated weights ρ j from (4.2). Let x * ∈ R 2 be a solution of the standard Weber problem with weighted distances: Find min Then x * is a solution of the Optimization Problem 4.2 for any N ≥ 1 as well.
The proof is postponed to Appendix A.2. It relies on the observation that the model for the logistic and services network from Section 3 can be described in terms of a closed queueing network of Gordon-Newell type. The proof of the next theorem will be given implicitly by proving correctness of Algorithm 4.10 below.
Theorem 4.5 If a solution of the Optimization Problem 4.1 exists for the capacities µ j (·), j = 1, . . . , J, it is uniquely determined if x * is given.
Remark 4.6 The results of Theorem 4.4 and Theorem 4.5 are striking, so comments are necessary.
(i) If the Optimization Problem 4.1 has a solution, i.e. the side constraints are satisfied with capacities µ j (·), j = 1, . . . , J, these service capacities µ j (·) (respectively the number of service channels s j ) at warehouses and center do not matter for optimizing the overall warehouse throughput with respect to the location of the center. Similarly, the absolute demands D j and the number N of trucks are not relevant for the optimal location x * . The relevant information for the location decision only comprises , which determine travel times, and • the proportions ρ j = D j /D of goods to be dispatched to warehouse j.
(ii) It is intuitive that increasing the loading capacity at the center increases throughput at any warehouse. However, less intuitive is: If we fix the capacities at the center and at all but one dedicated warehouse and increase the unloading capacity at the dedicated warehouse then the throughput at all warehouses increases. Both facts are consequences of Theorem 14.B.13 of [SS94].
When capacities of loading/unloading facilities change Theorem 4.4 guarantees that the decision for the optimal location remains optimal as long as the solution of the Optimization Problem 4.1 exists.
(iii) Theorem 4.4 does not propose that warehouse throughput is independent of local properties of the warehouses. Details about functional dependencies will be provided below. Moreover, it is not clear in advance whether a prescribed overall throughput can be met with a given set of parameter values. If the throughput can be met with the given capacities, Theorem 4.5 in connection with the main result of [vdW89] guarantees that by successively adding trucks we can increase the throughput until the total requirements can be dispatched. Otherwise, if loading/unloading capacities do not suffice, bottlenecks occur. Our proofs will show that we can increase the throughput by increasing the loading and unloading capacities at the nodes, see Section 4.4. Theorem 4.4 states that in any case the selected location remains optimal.

Analysis of the model as a Gordon-Newell network
The locations in the problem setting of Section 3 can be arranged as a star-like graph with warehouses as exterior vertices 2, . . . , J and the production unit as central vertex 1. Routes from the center to the exterior nodes, and vice versa back, correspond to edges (links, lanes). The vertices contain the loading and unloading facilities modeled as queueing systems. The circulating trucks are modeled as customers requesting for service at these queueing systems.
Because the number of trucks circulating in the network is fixed these features establish a closed queueing network structure (Gordon-Newell network). Additionally, we apply a standard feature to incorporate travel times into the model for the logistic network: For each warehouse, roads from the center to that warehouse and back are modeled as two additional infinite server nodes with random or deterministic service times ( = travel times). Necessary definitions, facts, and formulas from network theory are summarized in Appendix A.1.

The detailed model of the logistic and services network
We start with node 1 (the center) and nodes 2, . . . , J (the warehouses) which are multi-server nodes with s j service channels and exponentially distributed service times with mean µ −1 j , j = 1, 2, . . . , J, and with distances d j (x) := d(a j , x), j = 2, . . . , J. N ∈ N + customers (trucks) cycle in the network.
For simplicity of presentation we assume that trucks are traveling with unit speed, i.e., 1 km/hour.
(In examples we shall introduce realistic speeds.) Whenever a truck is served (loaded) at center 1 and is routed to warehouse j it has to travel distance d j (x) to and from. Traveling these distances is modeled as trucks being served by an infinite server station for a deterministic time d j (x) or a random time with mean d j (x). The first infinite server, from 1 to j, is denoted ja. After passing the road to j the truck will be unloaded at warehouse j. When unloaded at warehouse j the truck travels back to center 1. This is modeled as being served by another infinite server station, denoted jb, with mean service time d j (x).
Summarizing: Any radial tour from the center to node j consist of three nodes, representing: (i) traveling to j (via ja), (ii) unloading at j, and (iii) return from j (via jb). Thus we have a network with 3(J − 1) + 1 nodes and any round trip (radial tour to j) is of the form: "1 → ja → j → jb → 1".
Routing decisions at node 1 are independent of the network's previous history. For simplicity of presentation we assume that the travel times of the trucks are exponentially distributed with mean d j (x) when x is the location of the center. We discuss this in detail in Section 4.5.
With these stations, routing, and customers we have constructed a star-like Gordon-Newell network.
Local states of nodes are: For station j is n j the number of customers present (in service + waiting), for station ja is n ja the number of trucks on the way from station 1 to station j and for station jb is n jb the number of trucks on the way from station j to station 1. We abbreviate state-dependent service intensities as µ j (·), j = 1, 2, . . . , J, given in (4.1) and for lanes by µ ja (x)(n ja ) = d j (x) −1 · n ja and µ jb (x)(n jb ) = d j (x) −1 · n jb , j = 2, . . . , J. (4.5) Star-like network of transport system with added lanes The overall throughput of the network is T H(N ; The total throughput at the warehouse stations is Remark 4.8 The representation of G(N,J ; x) has a remarkable interpretation. It is the same normalization constant as that for a Gordon-Newell network with N customers, J multi-server stations with the same service rates as given in (4.1) and an attached single infinite server, which will be indexed by J + 1, with visit ratio η J+1 = 1/2 and exponentially distributed service time with mean Corollary 4.9 Consider the system of Theorem 4.7 with deterministic or general random travel times with means d j (x). Then the normalization constant is the same G(N,J; x) and the relevant throughputs are (4.7) and (4.8) as well.

Determining the number of trucks
We demonstrate the power of Theorem 4.7 by showing how to determine efficiently the minimal number of trucks to fulfill the total demand. We assume that the center's location x and capacities µ j (·) are fixed and sufficiently high to satisfy demands D i eventually, i.e. with sufficiently many trucks.
Recall that in our development we assumed up to now that trucks travel with unit speed (1 km/hour).
This implies that d j (x) is exactly the time to travel distance d j (x). For the present demonstration we allow general speed S > 0 for trucks. The mean time for traveling distance d j (x) is then the mean service time d j (x)/S at the infinite servers ja and jb, j = 2, . . . , J.  If C · 1 4 of this artificial Gordon-Newell network is non-decreasing in N . As can be seen from the proof in [vdW89] the throughput is strictly increasing in N . This guarantees that the algorithm stops after a finite number of iterations, because we assumed that the capacities are high enough to satisfy all the demands eventually.

With notation from Definition
Recall that from Theorem 4.4 the optimal location for the center is independent of loading and unloading capacities and the number of trucks cycling. So, if for N trucks and optimal location x the demand D exceeds the achievable maximal throughput T H w (N ; x) for the given µ j , one may increase the loading and/or unloading capacities (for a proof see Theorem 14.B.13 of [SS94]). short survey with more relevant details of classical bottleneck analysis is [SSB93].
Because the maximal throughput at a bottleneck node can be increased by increasing the local service rate, we have the following simple recipe. Random routing when departing from the center is a modeling assumption which adjusts the distribution of the available transportation capacity in the long run and in the stationary system according to the demands of the warehouses. As discussed in Example 3.1, transitions into realizable schedules can be found easily. A realistic schedule determines a sequence of visits for each truck to warehouses.
These sequences are cyclically iterated. A necessary condition for optimal scheduling is that (in a stationary system and in the long run) the schedule must reproduce the frequencies ρ j = D j /D: Recall Example 3.1: Two warehouses with demands D 2 = 10 and D 3 = 20 truck loads per hour and N = 3 trucks with equal capacity. Both reasonable schedules described there can be modeled with deterministic routing schemes described in [Kel79][Section 3.4] to obtain a Markovian network model using a more elaborated state space. The route of the trucks (in the sense of Kelly) for the second schedule would be: . . , iterated indefinitely over time.
If we evaluate the joint queue length distribution of stations 1, 2, 3, in these models we obtain exactly the stationary distribution of our present model with ρ 2 = 1/3, ρ 3 = 2/3. "Long range plans" in the sense of Tapiero [Tap71] justify to assume that we consider a stationary system. Note that this does not mean to consider static systems. Only the random fluctuations of the system are invariant over time.

Minimizing round-trip times
Throughput maximization is probably the most important objective in the logistic and services network. Another important objective is minimization of expected round-trip times, i.e. the mean travel time for a truck between two successive departures from the center.
The expected passage-time Z j (N ; x) from the center located at x to station j and back is the sum of all expected waiting and service times, which a truck spends at station j plus traveling to and from j and thereafter at station 1. If W i (N ; x) is the expected sojourn time at station i, then   The necessary number of trucks needed to fulfill the total demand of 81 truck loads is 28 for center with weights and 29 for center without weights. Nevertheless, the thoughput at the warehouses is greater with weights and 28 trucks than without weights and 29 trucks. The details are summarized in the following is rather homogeneous and therefore the difference between the locations of the center is insignificant. The case of demand structure D pro j is more interesting because of the great distance between the centers' location which is a consequence of the more variable demand structure.
In any case the congestion, measured as P (X1 > 0), increased with increasing throughput which follows from easy computations. Our experiments show that the increase of congestion seems to be tolerable, especially if we can reduce the number of trucks in parallel.
It is easy to see that in both scenarios the loading server at the center is the bottleneck of the network. We therefore investigated the influence of the loading capacity at the center in the above examples and reduced the loading capacity from µ1 = 4 to µ1 = 3 truck loads/hour. .
In any of 4 blocks ((I),. . . ,(IV)) we performed 10 experiments. For extreme total demands it turned out that the demand can not be satisfied with the given capacities due to occurrence of bottlenecks (the number of such samples are indicated as "Num∞Tru"). We included these cases in the tables in Section A.3 and performed additional experiments to obtain in any block 10 complete data sets. Within these we observed throughout that in approximately half of the samples for the center's location with weights less trucks (usually 1 truck less) are needed than without weights (precise numbers indicated as "Num<Tru"). We observed in both blocks (III) and (IV) a single experiment where 3 trucks less are sufficient (underlined). These high differences coincided with the maximal distance (MaxDist) between the two centers' location (bold).
In all experiments where an equal number of trucks is needed for center with weights and for center without weights, this coincides with a higher throughput at the warehouses for the center's location selected with weights. This seems to be in line with intuition. On the other side, when less trucks are needed for center with weights, in almost all cases the throughput with more trucks and location selected without weights produces more throughput, i.e. more overshot.
This demonstrates that decisions on the basis of integrated models leads to better utilization of the given resources. This is substantiated by the following observations: Exceptions of the moderate decrease of needed resources (trucks) are the extreme cases in (III) and (IV). With 3 trucks less the throughput with weights exceeds the throughput without weights. Moreover, in both blocks (III) and (IV) the second largest distance (bold) between the centers' location generated with 1 truck less (with weights) a higher throughput than without weights. In Table 1 we report for any block the minimal and maximal distance between the centers and the minimal and maximal demands. Detailed results are presented in Section A.3 of the Appendix.

Sensitivity analysis and robustness
Our main results indicated that some of the system's parameters are not relevant for the decision problems which are in the focus of our investigations.
(a) Discussing modeling assumptions in Section 4.5 we indicated that for fixed mean travel time dj (x)(< ∞) of vehicles we can allow any shape of travel time distribution. The joint stationary queue length distribution, the normalization constants, and the throughputs of the network are the same.
Consequently, our model is robust against changes of these data, e.g. against variability of travel times. This flexibility is due to so-called Insensitivity theory for queueing networks. This theory dates back to insensitivity in Verallgemeinerte Bedienungsschemata [KMN74] and in BCMP and Kelly networks [Sch78], for details see (c) On the other side, separability of Optimization Problem 4.1 implies that, when the optimal location is fixed, we can optimize for the number of trucks needed to satisfy demands D2, . . . , DJ by reallocation of capacities at the unloading service stations or by adding capacity at the loading station at the center.

Managerial insights
Theorems 4.4 and 4.5 substantiate conclusions for managerial decision making.
(1) Strategic (location of central facility) and tactical and operational (routing, scheduling) decisions are usually thought to be independent and are consequently separated. Then decision about the location of a central production facility neglects the actual and future capacities, the resulting congestion, and delays at downstream warehouses. Several authors have shown that for location-routing problems (LRPs) such a structural separation produces sub-optimal solutions for allocation problems, see e.g. [NS07]. For general supply chain analysis this is discussed in [HN19].
Contrary to this, our Theorems 4.4 and 4.5 justify in a stylized but rather general model the separation of strategic decisions for locations from several tactical and operational decisions for scheduling and placing service capacities. In our setting, in a first step the optimal location can be determined by way of a standard Weber problem where future allocation/routing decisions are incorporated only via gross information (or gross assumptions) about expected demand. Later on, in a second step the fine-tuning of the radial trips can be carried out according to the service resources at hand without the central location becoming sub-optimal. In Section 4.5 it is demonstrated that results obtained in the stylized model are valid in more realistic settings as well.
(2) Once the system is built, the quality of the delivering process measured in the standard metrics throughput or round-trip time can be increased by local enhancement of service without making the central location sub-optimal.
(3) Shifting capacities between the nodes is possible without perturbing optimality of the center's location as long as the fractions of demand at the stations remain the same. More precisely: If we can fine-tune

Conclusion and directions of further research
We have developed a methodology for determining jointly optimal solutions for location-allocation-routing problems which are usually considered to be problems on different levels of decision making: strategic versus tactical/operational level. We translated the problem into a stochastic network problem and showed that (i) determining the center of a star-like network under constraints on the demands generated by the exterior nodes, and (ii) determining the optimal number of customers in the network, can be separated.
Starting from an exponential version of the problem, which allows for simple proofs, we have shown that more realistic models are covered by the result. Probably, the most important result is that the decision upon the location of the production center can be decoupled from building the exterior stations and their equipment.
Many research problems are not yet tackled in the more general area of locating additional nodes in networks of queues. These can be easily identified by investigating the more involved location problems, like p-median or p-center problems, in the continuous as well as in the discrete setting of queueing networks.
Parts of our ongoing research related to the present paper are (i) location theory in the discrete queueing network setting with prescribed network graphs, and (ii) p-center location problems in the setting of this paper.

A Appendix
A.1 Prerequisites from queueing network theory Remark. Any non-zero solution η of (A.1) is admissible to compute π. Consequently, the next algorithm can be used with any such η. The stationary distribution of the network is π(nj , j ∈J; x) = where we utilized ηja = η jb = ηj and µja(x) = µ jb (x) = d −1 j (x) for j = 2, . . . , J. The following representation of normalization constants will be of value. hj (x) n ja +n jb nja!n jb ! = n 2a +...+n Ja + n 2b +...+n Jb =n J j=2 hj hj(x) 2h(x) The last step is a continuity argument: When a sequence of finite mixtures of Erlangian service time distributions approaches (in the sense of weak convergence) the given service time distribution at some node, then that node's queue length distributions converge weakly as well. This is equivalent to (multi-dimensional) point-wise convergence in a suitable multi-dimensional real space. Finally, the normalization constants and the throughput continuously depend on the densities of the joint queue length distributions, as visible from Lemma A.5.
Lemma A.6 For all k = 0, 1, . . . , N − 2, n = 0, 1, . . . , N − k − 2, it holds that Proof: By definition, we have From [vdW89], the throughput of a Gordon-Newell-network with service rates non-decreasing in the number of customers is a non-decreasing function in the network's population size. Thus, the lemma is proved.
Proof: (of Theorem 4.4) For all x ∈ R 2 with h(x) > h(x * ) and N ∈ N we will show By Lemma A.5 ,this is equivalent to We consider the summand for k = N So the whole summand is strictly positive and the problem is reduced to prove In the second summand we apply the following summation formula twice: for a k , b k ∈ R, k = 1, . . . , N, and the theorem is proved.

A.3 Additional numerical experiments
We report in this section details of the experiments which have been summarized in Table 1 in Section 5.1.
We performed four blocks of experiments, distinguished by demand distributions which are uniform U (A) on finite demand sets A.
In any case 12 locations are sampled uniformly from {10, . . . , 410} × {10, . . . , 270}. Service is provided by single servers with intensities µj = 2, j = 2, . . . , 13, for unloading servers at warehouses. Service intensity µ1 at the center varies with the blocks. Abbreviations: DistLoc ≡ distance between center with weights and center without weights Demand: to/∧ /∨ ≡ total demand/minimal demand/maximal demand +/-≡ quantities for: center with weights(+)/ center without weights (-) The extreme cases of large demand are highlighted by boldface numbers, the extreme differences for needed trucks are colored red. The cases where the requested demand could not be delivered due to bounds determined by bottlenecks are indicated under "Trucks" as "-/-". Bottleneck was in any case the loading server at the center.