The double-assignment plant location problem with co-location

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Abstract

In this paper, a new variant of the Simple Plant Location Problem is proposed. We consider additional conditions in the classic location-allocation problem for clients and facilities. Namely, some pairs have to be served by a common plant. The resulting problem can be addressed with existing models for the case of single assignment. However, to the best of our knowledge, the proposed setting when each client must be assigned to a couple of facilities is still unexplored. We examine the implications of adding such new constraints to standard formulations of the SPLP with double assignment. We compare the resulting formulations from a theoretical point of view. After that, we focus on the study of one of the models, which turned out to be a set packing problem. All the clique facets are identified and a separation algorithm is devised. Although the separation problem is proved to be NP-hard, our computational experience shows that the separation algorithm is effective and efficient, reducing computational times and duality gaps for all the instances tested.

Introduction

Facility location has been one of the most fruitful areas within operations research (see Krarup and Pruzan, 1983, Laporte et al., 2015). One of the reasons is that decision making certainly includes strategic infrastructure layout in almost every corporation, public or private. Industrial companies have to locate their facilities and warehouses so as to reach clients in their marketplaces; government agencies usually decide on the location of schools, hospitals, emergency services, etc. Other than its wide range of applications, the area also entails theoretical challenges, see for instance Cho et al. (1983) or Labbé and Yaman (2004). In this paper we study a variant of a well-known model in facility location, which, as far as we know, has not been investigated yet.

One of the seminal problems in discrete location is to decide on sites to install a set of identical facilities and on how clients are allocated to them. The problem, which has been known as Simple Plant Location Problem (SPLP) or Uncapacitated Facility Location Problem, has been extensively studied by operational researchers (Cho et al., 1983, Cornuéjols et al., 1977, Guignard, 1980, Cornuéjols and Thizy, 1982, Cánovas et al., 2002). The simplicity of the model has been fundamental as a base for the development of locational analysis, and at the same time has given room for studying a range of variants. Extensions include those considering specialized facilities in different product types (Warszawski, 1973), customer preferences (Cánovas et al., 2007), risk of disruptions in the distribution system (Swamy and Shmoys, 2008, Cui et al., 2010) or models seeking for sustainable logistics (Xifeng et al., 2013). For more references, see surveys Krarup and Pruzan (1983) or Fernández et al. (2015), which focus on the SPLP. It is worth mentioning that Fischetti et al. (2016) presented a Benders decomposition more recently, which found optimal solutions for previously unsolved instances of up to 3000 clients and location candidates and obtained speedups of several orders of magnitude with respect to other methods.

In this work, we propose a new modification of the SPLP, which considers new requirements in relation to the allocation of clients to facilities. The new variant can be considered to belong to the same family than that introduced in Marín and Pelegrín (2019). In that previous work, some pairs of clients were supposed to be incompatible, that is, they could not be assigned to the same facility. A set packing formulation for the resulting facility location problem was proposed and its polyhedral structure was studied, deriving different types of facets and separation algorithms to manage the inequalities within a branch and cut.

In our new scenario, some pairs of clients, which we will call tied, wish to be allocated to the same facility. This situation could be easily addressed by existing models for the SPLP, since allocation of each tied pair can be decided at once by adding their costs, just as if they were a single client. However, sometimes clients must be assigned to a couple of facilities, for instance when a backup service is needed, just as in Swamy and Shmoys (2008). When double assignment is considered, tied clients are pairs that like to be served by at least one common facility. Now, the new scenario gives rise to a new combinatorial problem that is a variant of the classic SPLP. Closely related topics are hub location, where origin and destination pairs have to be connected by using a couple of hubs (see for instance Contreras et al., 2015) or warehouse location, where clients are served by a facility through a warehouse (see Kaufman et al., 1977). In facility location problems with capacities, clients are also frequently assigned to more than one facility, see for instance Wu et al. (2006).

The above setting, which we will call double-assignment plant location with co-location, finds interesting applications in telecommunication networks design. A generic telecommunication network consists of a set of terminals (users), connected to concentrators (switches or multiplexers) and a backbone network which interconnects the concentrators. A primary problem in network design is to decide how many concentrators are needed and how the terminals should be assigned to the concentrators. These two decisions can be identified with that of facility location and allocation, a fact that was already observed in Gourdin et al. (2002). In the context of telecommunication networks, the setting we propose corresponds to a configuration in which some users must share a concentrator. This is a realistic assumption, since there could be users with special communication requirements that want to have a dedicated path to avoid the backbone network.

The contributions of the paper can be summarized as follows:

  • (i) a new variant of the SPLP, which considers double assignment and clients ties is proposed;

  • (ii) two formulations, inspired by classic two and three index facility location models, are presented, and their linear relaxations are theoretically compared (something that to the best of our knowledge was not done before);

  • (iii) all the clique facets of one of the formulations, which is a set packing, are disclosed;

  • (iv) the corresponding separation problem is proved to be NP-hard and a heuristic algorithm is then proposed;

  • (v) finally, a computational study to test the developed formulations and algorithm is conducted.

The paper is organized as follows. The next section introduces the double-assignment plant location problem with co-location, together with two integer programming formulations. In the following section, the linear relaxations of both are compared theoretically. Then, in Section 4 all the clique facets of one of the new models, which is a set packing problem, are described. The problem of separating them is proven to be NP-hard in Section 5, where a heuristic separation algorithm is also proposed. Finally, Section 6 reports the computational tests of the formulations and the heuristic, which includes a comparative analysis with respect to standard clique cuts incorporated by a commercial solver. Some conclusions close the paper.

Section snippets

Problem statement and formulations

Consider as initial setting that of the SPLP, where I{1,,n} and J{1,,m} are the sets of clients and candidate facilities respectively. Facility location and clients allocation decisions have been typically modeled with the following decision variables,

  • yj=1 iff no service is installed at candidate location j,jJ,

  • yj=1 iff a facility is installed at candidate location j,jJ and.

  • xij=1 iff client i is served by facility at j, iI,jJ,

where yj and yj satisfy yj=1-yj for all j. The classic

Comparing the formulations

Variables x and z are clearly related to each other. Given a client i,zijk will be one if and only if xij and xik are. This is mathematically written as followszijk=xijxikiI,j,kJ,j<kxij=k>jzijk+k<jzikjiI,jJ.

We use the second formula, which is linear, to replace x in (DPLP2) by the corresponding z-variables. The resulting formulation, which we name (DPLP’3), will be written in terms of decision variables y and z. We will use (DPLP’3) to compare the objective values of the linear

Clique facets

In this section, we identify some facets of the integer polytope of (DPLP3),Bdplpconv{(z,y){0,1}n·m(m-1)2×{0,1}m:(7)-(9)}.

In order to study the facial arrangement of Bdplp, we will leverage the set packing structure of (DPLP3). Any set packing formulation, i.e., any linear program with a 0/1 constraint matrix and a vector of ones as right-hand side, can be uniquely identified with a graph. It is usually called conflict graph or intersection graph, and has one node per variable and edges

Clique separation

The only clique facets of Bdplp that are not included in (DPLP3) are that of family (17) with |CT|>2 or |CT|=2 and |Ji|2,iCT. This section focuses on the latter, which belong to family (15). Given a tied pair (i1,i2)ET, (15) define exponentially many inequalities, even when J1J2=J. A separation algorithm is then needed to manage these inequalities within a branch and cut scheme. We will first prove the theoretical computational complexity of this problem and then provide a heuristic

Computational study

The aim of our computational study is twofold. First, we are interested in providing an empirical comparison between the LP bounds of (DPLP2) and (DPLP3). Second, we intend to validate the separation heuristic applied on (DPLP3) and test its relative performance with respect to bare formulations (DPLP2) and (DPLP3).

Conclusions

In this work, a new variant of one of the fundamentals problems in discrete location is proposed and studied. Specifically, we propose a modification of the Simple Plant Location Problem where double assignment is considered and some pairs of clients have to share at least one common facility. A set packing three-indexed formulation is proposed and its linear relaxation is proved to provide better bounds than that of a formulation with classic variables. The facial structure of the proposed set

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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  • Cited by (2)

    • New variants of the simple plant location problem and applications

      2023, European Journal of Operational Research
      Citation Excerpt :

      Applications of the SPLP other than facility location include telecommunication networks design (Gourdin, Labbé, & Yaman, 2002), distributed systems design (Klose & Drexl, 2005), sensor networks (Furuta, Sasaki, Ishizaki, Suzuki, & Miyazawa, 2009), clustering (Meira, Miyazawa, & Pedrosa, 2017), and robotics (Karch, Noltemeier, & Wahl, 2002), just to mention some of them. This work revisits two new variants of the SPLP (Marín & Pelegrín, 2019; 2021), which take into consideration some additional requirements on clients assignment to facilities. Namely, these variants allow, respectively, pairs of clients that: i) are incompatible, i.e., they cannot be assigned to the same facility; ii) must be co-located, i.e., they have to be assigned to one common facility at least (each client is assigned to two facilities in this variant).

    • Discrete Facility Location in Machine Learning

      2021, Journal of Applied and Industrial Mathematics

    Research supported by the Ministerio de Economía y Competitividad, project MTM2015-65915-R and Ministerio de Educación, Cultura y Deporte, PhD grant FPU15/05883.

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