Dynamic Facility Location

The majority of facility location models are applied to strategic long-term planning.

However, customer demands, as well as the prices for production, transportation and commodities tend to change over time. Multi-period models aim at coping with these challenges by defining independent demands and costs for each time period. Early works in the domain of dynamic facility location were initiated by authors such as Ballou (1968), Wesolowsky (1973), Wesolowsky and Truscott (1975) and Sweeney and Tatham (1976). A few authors used the term dynamic in a broader context (Arabani and Zanji-rani Farahani, 2011; ZanjiZanji-rani Farahani and Hekmatfar, 2009), also including stochastic aspects. However, the majority of the literature limited the use of the term to the con-text of multi-period problems. From the modeling viewpoint, the temporal aspect is usually captured by an additional variable index t. While a few models represent the available capacity by an additional flow variable z_jt ∈ R⁺, most of the works incorpo-rate modular capacities, using binary variables of type y<sub>j`t</sub>, where ` is either a capacity level or a facility type linked to a fixed amount of capacity. While the optimal timing of a facility construction, as well as its initial capacity are important decisions (e.g., Shulman, 1991), it has often been found beneficial to adjust capacities at later time pe-riods to better respond to changing demand and market conditions (Owen and Daskin, 1998). Mathematical models that include such features have been applied in both the private and the public sectors to determine locations and capacities for production facil-ities, entire supply chains (Melo et al., 2006), telecommunications networks (Chardaire et al., 1996), schools (Antunes and Peeters, 2001), ambulances (Brotcorne et al., 2003), emergency services (Hochbaum, 1998) and many more, responding to population shifts and other environmental factors. Several surveys (Owen and Daskin, 1998; Arabani and

Zanjirani Farahani, 2011; Zanjirani Farahani and Hekmatfar, 2009) reviewed the grow-ing literature on dynamic facility location problems, which suggested different ways to adjust capacities throughout a given planning horizon:

– The construction of a facility at a certain time period.

– The expansion or reduction of capacity at an existing facility.

– The temporary closing of a facility and reopening at a later time period.

– The relocation of capacity from one location to another.

The timing of facility construction is part of most of the multi-period facility location problems. We now review the existing literature for the other three features.

2.1.3.1 Capacity Expansion and Reduction

When customer demands of certain regions permanently change and are not likely to return to their previous levels, it may be beneficial to add or reduce (or even permanently shut down) production capacities at an existing facility to permanently adjust to the new conditions.

Luss (1982) discusses modeling techniques for capacity expansion. He points out that the total capacity available at a location may either be provided by a single facility or be composed by several coexisting facilities. The first category includes models that allow one facility at a location that increases or decreases the available capacity over time (Jacobsen, 1990; Canel et al., 2001; Antunes and Peeters, 2001; Melo et al., 2006;

Behmardi and Lee, 2008). These models typically use flow variables of type z_jt ∈ R⁺ and manage the expansion (s_jt ∈ R⁺ variables) or reduction (r_jt ∈ R⁺ variables) of ca-pacity by using flow conservation constraints similar to the following:

z_jt = z_j(t−1)+ s_jt− r_jt ∀i ∈ I , ∀t ∈ T (2.6)

Models in the second category commonly use integer variables to indicate the num-ber of existing facilities at a location. When the problem allows for capacity expansion, but not reduction, the total capacity can also be composed of several binary variables, one for each constructed facility or expanded capacity (Shulman, 1991; Troncoso and

16 Garrido, 2005). This modeling technique can also be found in other classes of loca-tion problems, such as variants of the Capacitated Concentrator Problem (Gouveia and Saldanha da Gama, 2006; Gourdin and Klopfenstein, 2008; Correia et al., 2010).

When the problem involves both capacity expansion and reduction, an alternative modeling technique (Dias et al., 2007) can be used involving binary variables of type y<sub>j`t1t2</sub> to indicate that a capacity of size ` is added for a period defined by the interval [t₁,t₂]. The total capacity available at a location and time period is then computed by the sum of all facilities (capacity blocks) available at that time period, enabling a flexible expansion and reduction of capacity along time. The two different categories, using flow variables and capacity block variables, are illustrated in Figure 4.1. We refer to Section 4.3 for a detailed discussion of these modeling techniques.

Next to classical capacity expansion and reduction, several special cases with in-dividual restrictions have been presented. In the work of Antunes and Peeters (2001), facilities may either expand or decrease their capacities throughout the planning hori-zon, but not both. We refer to the book chapter of Jacobsen (1990) for more references to works that consider capacity expansion.

2.1.3.2 Temporary Facility Closing and Reopening

In some situations, it may be beneficial to temporarily close a facility, for example to avoid high maintenance costs. This may be appropriate when demand temporarily decreases, but is likely to return to its previous level afterwards. While, in practice, it may be possible to close only parts of a facility, previous studies focused on the tem-porary closing of entire facilities. Among the suggested models, certain are limited to a single closing and reopening of each facility, whereas others allow repeated closing and reopening throughout the planning horizon. The uncapacitated facility location prob-lem presented by Van Roy and Erlenkotter (1982), as well as the supply chain model of Hinojosa et al. (2008), allow one-time opening or closing of facilities: new facilities can be opened once and existing facilities can be closed once. Chardaire et al. (1996) and Canel et al. (2001) propose formulations for opening and closing facilities more than once. The former installs and removes terminals in telecommunications networks

to adapt to changes in data traffic and costs along time. The authors of both works use binary variables of type y_jt to indicate whether a facility is open or closed during a cer-tain period. A closing or reopening is then indicated by a quadratic term y_jt(1 − y_jt) in the objective function. A linear formulation for a simplified variant of this problem with fixed capacity levels has been proposed by Dias et al. (2006).

The works cited above interpret facility closing either as temporary (i.e., the facility still exists, but its capacities are temporarily unavailable) or permanent (a facility is shut down). In most cases, maintenance costs for temporarily closed facilities are low and can therefore be ignored in the model. Most of the existing formulations therefore do not explicitly distinguish temporary and permanent facility closing. Furthermore, permanent facility closing may also be seen as a special case of capacity reduction.

2.1.3.3 Facility Relocation

In certain contexts, the relocation of existing capacity from one location to another may be a possibility to shift capacity closer to the demand points. Wesolowsky and Truscott (1975) have been one of the first to consider simple relocation of facilities.

The authors use flow conservation constraints similar to (2.6), but with binary variables (instead of flow variables) to indicate whether a facility is available or not. That is, instead of variables representing capacity expansion and reduction, the model contains binary variables to represent the relocation from or to the location.

The relocation of facilities has since been considered by several researchers. Min and Melachrinoudis (1999) document a case study for a company that relocates warehouses and Melachrinoudis (2000) provides an appropriate model. Brotcorne et al. (2003) re-view location-relocation models for ambulances for deterministic and probabilistic sce-narios. Melo et al. (2006) and Melo et al. (2009b) provide an extensive modeling frame-work for modeling generic multi-level supply chain netframe-work structures. Their model is based on flow conservation constraints and focuses on gradual relocation of existing ca-pacity. The authors also show how to link binary variables to indicate the facility type, as well as the origin and destination for the relocated facility. However, it can be noted that most of the other works ignore the distance the facilities are relocated and therefore

18 allocate equal costs to all facility relocations.

Often, the closing of a facility at one location and opening at another location has also been interpreted as a facility relocation, which has been considered under the constraints of a global budget (Wang et al., 2003) and under demand uncertainty (Lim and Sonmez, 2013).

Relocation models have also been proposed under more restricted conditions. Amiri-Aref et al. (2011) present a non-linear mixed-integer formulation to relocate emergency maintenance rooms given that the transit availability for certain regions are subject to uncertainty. Zanjirani Farahani et al. (2008) locate and relocate a single facility under the condition that costs vary according to a continuous weight function. Albareda-Sambola et al. (2009) introduce a problem in which facilities must select a certain number of customers. Once served, customers have to be served in all subsequent periods.