Facility location models

“Where should the warehouse go?” sounds like one question, but different models answer different versions of it.

Center of gravity gives a quick first guess. Weight each customer’s location by how much it ships (volume times freight rate) and take the weighted average of the coordinates. If Mumbai takes most of the volume, the answer is pulled towards Mumbai. It is fast and intuitive, but it ignores roads, real estate, labour and capacity, and the “optimal” point may land in a field or the sea. Use it to narrow the search, not to sign a lease.

p-median chooses the best $p$ sites from a list of candidates to minimize the total demand-weighted distance: the average customer gets as close as possible. It suits cost-focused networks.

p-center chooses $p$ sites to minimize the worst distance: the farthest customer gets as close as possible. It suits service promises such as “every customer within 4 hours”, or emergency services.

Fixed-charge (capacitated) location is what companies actually use for big studies. It decides which candidate sites to open, a yes-or-no binary decision with a fixed cost each, and how much flows from plants to warehouses to customers, minimizing:

fixed cost of open sites + production cost + transport cost (+ inventory, + duties)

subject to meeting all demand and staying within each site’s capacity. Because it mixes yes/no decisions with flow quantities, it’s a mixed-integer linear program (MILP), solved with tools such as Gurobi, CPLEX or open-source solvers.

Centre of gravity, worked

Weight each customer’s coordinates by its volume (times freight rate, if rates differ), and average:

The centre of gravity minimizes the weighted squared distances, not the distances themselves. The point that minimizes total weighted distance (the Weber point) is found iteratively, and when one customer carries at least half of the total weight, as A does here, the best spot is at that customer. Use the centre of gravity as a rough starting area, never as the answer.

The fixed-charge location model

For real decisions, list candidate sites and let an optimizer choose:

• $y_j$ = 1 if site $j$ is opened, 0 otherwise; $f_j$ is its fixed cost and $K_j$ its capacity.
• $x_{ij}$ = quantity shipped from site $j$ to customer zone $i$, at unit cost $c_{ij}$; $d_i$ is the zone’s demand.

subject to: every zone’s demand is met, $\sum_j x_{ij} = d_i$; nothing ships from a closed site and open sites stay within capacity, $\sum_i x_{ij} \le K_j\,y_j$; and $y_j \in \lbrace 0, 1 \rbrace$, $x_{ij} \ge 0$.

The $K_j\,y_j$ term is the clever part: a closed site has zero capacity, so the model must pay $f_j$ to use it.

Choosing between the models

More than one echelon

Real networks have plants, central warehouses, regional warehouses and customers. The model adds a layer of flows for each echelon, with flow-balance constraints (what leaves a warehouse equals what arrives, give or take stock changes). It can also choose among capacity options per site (small, medium or large modules, each with its own fixed cost) and enforce single sourcing, where each customer zone is served by exactly one warehouse, which customers and operations often prefer.

Getting the inputs right

• Demand zones: thousands of customers are grouped into zones, for example by PIN-code areas, each represented by a weighted centre. Too few zones distort distances; too many slow the model without improving decisions.
• Road distances and times: straight-line distance understates real road distance, and some routes are far slower than distance suggests. Use network distances and transit times.
• Freight rates: rates aren’t linear in distance. Full truckloads, part loads and parcels are priced differently, and lanes with good backhaul are cheaper.
• Inventory: safety stock rises with the square root of the number of sites, a non-linear effect that models approximate in steps or add after optimizing flows.

Service-driven models

When the promise to customers matters more than cost, covering models apply. The set-covering model finds the fewest sites so that every customer is within a time or distance limit; the maximal covering model (Church and ReVelle, 1974) places a fixed number of sites to cover as much demand as possible. Both are common for emergency services and same-day delivery networks.

Solving large problems

Mixed-integer solvers, commercial and open-source, handle networks with hundreds of candidate sites and thousands of zones, especially with good formulations. Very large or detailed problems use heuristics, decomposition or a two-step approach: optimize the strategic structure first, then refine flows. Whatever the method, test the answer: if a small change in one assumption flips the recommendation, the decision is not robust yet.