A facility location model determines the spatial configuration of a logistics network. Two decisions often occur together:
Facility location decisions involve an extended planning horizon (Owen and Daskin 1998). They play a critical role in the strategic design of supply chain networks (Melo et al. 2009).
The feasible location space determines the form of the model (Daskin 1995; Farahani and Hekmatfar 2009).
| Location space | Feasible locations | Typical model |
|---|---|---|
| Continuous | Any point in a plane or region | Weber problem |
| Network | Nodes or points on a transport network | Network median or center |
| Discrete | A finite set of candidate sites | \(p\)-median, covering, or fixed-charge model |
Facility location models combine assumptions from several dimensions (Farahani and Hekmatfar 2009; Daskin 2008).
Allocation and capacity assumptions combine in different ways across facility location models (Farahani and Hekmatfar 2009; Sridharan 1995).
| Symbol | Meaning |
|---|---|
| \(I\) | Set of demand points |
| \(J\) | Set of candidate facility locations |
| \(h_i\) | Demand at point \(i\) |
| \(d_{ij}\) | Distance from demand point \(i\) to candidate \(j\) |
| \(c_{ij}\) | Cost of serving all demand \(h_i\) at point \(i\) from candidate \(j\) |
| \(f_j\) | Fixed cost of opening a facility at candidate \(j\) |
| \(q_j\) | Capacity of a facility opened at candidate \(j\), in the same units and period as \(h_i\) |
| \(x_{ij}\) | 1 if demand point \(i\) is assigned to candidate \(j\), 0 otherwise |
| \(y_j\) | 1 if a facility opens at candidate \(j\), 0 otherwise |
Median models minimize total demand-weighted distance (Daskin 1995, 2008).
The \(p\)-median problem opens exactly \(p\) facilities and assigns every demand point to one open facility (Daskin 1995).
| Component | Mathematical statement | Interpretation |
|---|---|---|
| PM1 Objective | \(\displaystyle \min \sum_{i\in I}\sum_{j\in J} h_i d_{ij}x_{ij}\) | Minimize total demand-weighted assignment distance. |
| PM2 Assignment | \(\displaystyle \sum_{j\in J}x_{ij}=1 \quad \forall i\in I\) | Assign each demand point to exactly one facility. |
| PM3 Linking | \(\displaystyle x_{ij}\leq y_j \quad \forall i\in I,\ j\in J\) | Permit assignment to candidate \(j\) only when facility \(j\) opens. |
| PM4 Cardinality | \(\displaystyle \sum_{j\in J}y_j=p\) | Open exactly \(p\) facilities. |
| PM5 Domains | \(\displaystyle x_{ij}\in\{0,1\}\ \forall i,j;\quad y_j\in\{0,1\}\ \forall j\) | Require binary assignment and facility-opening decisions. |
Assumptions: single-source allocation and no binding facility capacity.
For a fixed number \(p\) of facilities, the \(p\)-center problem minimizes the greatest distance between any demand point and its assigned facility.
Use it when the worst service distance matters.
The Location Set Covering Problem minimizes the number of facilities needed to cover all demand (Toregas et al. 1971).
The Maximal Covering Location Problem maximizes covered demand for a fixed number of facilities (Church and ReVelle 1974; Farahani and Hekmatfar 2009).
Both covering models define coverage through a distance or travel-time threshold (Daskin 1995). Figure 1 illustrates the effect of changing that threshold.
Figure 1: Warehouse locations for humanitarian relief distribution in Nepal under 100, 200, and 300 km coverage-distance scenarios (Maharjan and Hanaoka 2017).
A fixed-charge model balances the cost of opening facilities against the cost of serving demand (Daskin 2008).
The Uncapacitated Warehouse Location Problem selects warehouses and assigns each customer to one open warehouse (Daskin 2008; Fischetti et al. 2016).
| Component | Mathematical statement | Interpretation |
|---|---|---|
| WL1 Objective | \(\displaystyle \min \sum_{j\in J}f_jy_j + \sum_{i\in I}\sum_{j\in J}c_{ij}x_{ij}\) | The first term represents facility-opening cost. The second represents customer-allocation cost. |
| WL2 Assignment | \(\displaystyle \sum_{j\in J}x_{ij}=1 \quad \forall i\in I\) | Assign each customer to exactly one warehouse. |
| WL3 Linking | \(\displaystyle x_{ij}\leq y_j \quad \forall i\in I,\ j\in J\) | Permit assignment to candidate \(j\) only when warehouse \(j\) opens. |
| WL4 Domains | \(\displaystyle x_{ij}\in\{0,1\}\ \forall i,j;\quad y_j\in\{0,1\}\ \forall j\) | Require binary assignment and warehouse-opening decisions. |
Key distinction: the trade-off between opening and allocation costs determines the number of warehouses.
The capacitated model adds an upper bound on the demand assigned to each open warehouse (Farahani and Hekmatfar 2009; Sridharan 1995).
| Component | Mathematical statement | Interpretation |
|---|---|---|
| CWL1 Capacity | \(\displaystyle \sum_{i\in I}h_i x_{ij}\leq q_jy_j \quad \forall j\in J\) | Demand assigned to warehouse \(j\) cannot exceed its available capacity. |
If \(y_j=0\), the right-hand side equals zero. If \(y_j=1\), warehouse \(j\) can serve at most \(q_j\) units of demand. The remaining objective and constraints are unchanged.
Variant shown: single-source allocation; retain binary \(x_{ij}\) and \(y_j\) from WL4. Demand and capacity use the same units and planning period.
Multi-level facility location models decide which facilities open at each level. More broadly, multi-echelon models may contain levels without facility-location decisions (Ortiz-Astorquiza et al. 2018; Farahani et al. 2014).
The echelon structure is independent of the capacity and objective assumptions.
Exact algorithms use feasible solutions and bounds to establish optimality or report a gap. Lagrangian relaxation and column generation can provide bounds within these algorithms.
Heuristics seek good feasible solutions without an optimality guarantee.
Instance size influences method choice. Small warehouse location problems can use exact integer programming, whereas larger problems often require heuristic or metaheuristic methods (Farahani and Hekmatfar 2009). Large UFLP formulations with many allocation variables remain challenging for exact solvers (Fischetti et al. 2016).
Metaheuristics seek good solutions without guaranteeing optimality (Farahani and Hekmatfar 2009).
Examples include:
Use them when exact methods cannot solve the required model at the relevant scale.
A genetic algorithm for a warehouse location model can use the following cycle:
Each chromosome represents one candidate solution (Farahani and Hekmatfar 2009).
In this illustrative course exercise, an e-grocery company plans rapid delivery service in Enschede.
Decisions: warehouse openings and customer assignments.
| Planning question | Candidate model |
|---|---|
| How should a fixed number of warehouses minimize average travel? | \(p\)-median |
| How many warehouses are required to meet a delivery-time threshold? | Location Set Covering Problem |
| Which warehouses minimize opening and distribution costs? | Warehouse Location Problem |
Treat capacity and the delivery-time target as explicit assumptions.
For an illustrative 10-minute order-to-door target, reserve time for picking, waiting, and dispatch before deriving the allowable travel time. Use that travel-time threshold in a covering model or as a service constraint in a warehouse location model.
Report more than the selected sites:
Sensitivity analysis tests whether the recommended network remains defensible when inputs change.
The Weber problem is a continuous-space model that commonly uses straight-line or Euclidean distance (Farahani and Hekmatfar 2009).
Weiszfeld’s algorithm iteratively solves the geometric-median form of this problem. It does not serve as a general heuristic for discrete facility location.
Discrete-event simulation can evaluate operational behavior that a strategic location model omits (Szczepański et al. 2019).
Simulation evaluates a chosen configuration. It does not replace a clearly defined location objective.
Multi-criteria decision analysis combines qualitative and quantitative site criteria.
Warehouse site selection combines tangible and intangible criteria and can incorporate judgments from several stakeholders (Korpela and Tuominen 1996).
Facility location models may contain a single objective or multiple objectives (Farahani and Hekmatfar 2009; Melo et al. 2009).
A cost objective and a service objective may favor different networks.
The preferred design depends on how competing objectives are evaluated.
Warehousing - M-IEM - Breno A. Beirigo
