Facility Location and Network Structure Models

Introduction

Facility location is the strategic decision of determining where warehouses, distribution centres, cross-docks, fulfilment centres, plants, service depots and other logistics assets should be established. Network structure adds a second question: how many facilities are required, what role should each perform, which customers should each serve, and how should inventory and transportation flows be organised?

These decisions are difficult because they are long-lived, capital-intensive and interdependent. A location that minimises transport distance may have high labour costs, weak infrastructure or regulatory disadvantages. Conversely, a low-cost site may increase delivery distance and safety stock. Facility-location analysis therefore combines demand geography, transportation, capacity, customer service, fixed costs, variable costs, risk and strategic flexibility.

At postgraduate level, facility location should be understood not as a search for a single mathematically optimal point, but as a structured decision process. Quantitative models clarify trade-offs; managerial judgement validates assumptions and addresses risks that are difficult to quantify.

Learning Objectives

After studying this chapter, the reader should be able to:

• distinguish facility roles and network structures;
• apply factor-rating, centre-of-gravity, load-distance and break-even methods;
• formulate and interpret p-median, covering and fixed-charge location models;
• determine facility numbers, customer territories and hierarchical roles;
• evaluate assumptions, data quality, uncertainty and model limitations;
• translate model results into implementable network decisions.

1. Facility Roles and Network Structure

A distribution network normally contains some combination of the following facilities:

A single-echelon network serves customers directly from one facility layer. A multi-echelon network includes two or more layers, such as a national centre supplying regional centres that serve customers. A hub-and-spoke network consolidates flows through hubs, while a point-to-point network connects origins and destinations more directly.

The appropriate structure depends on demand density, product value, delivery promises, transport economics and uncertainty. Centralisation generally reduces duplicated inventory and facility cost but increases average delivery distance and exposure to disruption. Decentralisation improves responsiveness but usually increases fixed cost, safety stock and coordination complexity.

A useful design sequence is:

1. define customer service requirements and product flows;
2. forecast demand by geographic zone and time period;
3. identify feasible sites and facility capacities;
4. estimate fixed, variable, handling, inventory and transportation costs;
5. select a suitable analytical model;
6. test scenarios and risks;
7. assign facility roles and customer territories;
8. validate the result operationally, financially and legally.

Figure: The centralisation–responsiveness trade-off in network structure.

2. Qualitative and Semi-Quantitative Models

Factor-rating method

The factor-rating method compares candidate sites against weighted criteria. It is useful when strategic, qualitative and quantitative considerations must be combined.

Here, S_j is the score of site j, w_i is the weight of criterion i, and r_ij is the rating of site j on criterion i. Usually, weights sum to 1 and ratings use a common scale, such as 1–10. The method assumes that criteria can reasonably be represented on a common scale and that weighted additive compensation is acceptable.

For example, suppose transport cost, labour availability, infrastructure and regulatory climate receive weights 0.35, 0.25, 0.20 and 0.20. Site A ratings are 8, 6, 9 and 7; Site B ratings are 7, 9, 8 and 8. Site A scores 0.35×8 + 0.25×6 + 0.20×9 + 0.20×7 = 7.60. Site B scores 0.35×7 + 0.25×9 + 0.20×8 + 0.20×8 = 8.00. Site B ranks higher. The result should be tested by changing weights and excluding any site that fails a non-compensatory requirement, such as legal prohibition or inadequate flood protection.

Centre-of-gravity method

The centre-of-gravity method estimates a location that minimises a weighted distance measure in a continuous plane. For demand points i with coordinates (x_i,y_i) and weights w_i, the commonly used weighted-average estimate is:

Weights may represent shipment volume, demand, revenue or number of trips. Coordinates must use a consistent distance system, and the method assumes transportation cost is approximately proportional to weighted distance and that the plane is unconstrained. It does not automatically account for roads, congestion, land availability, capacity or nonlinear freight rates.

Consider three demand zones: A (2, 8), 100 units; B (8, 6), 200 units; and C (6, 2), 300 units. Total demand is 600. Thus x^*=(100×2+200×8+300×6)/600=5.67 and y^*=(100×8+200×6+300×2)/600=4.67. The point (5.67, 4.67) is a starting estimate, not necessarily a feasible site. A road-network or GIS analysis should identify the nearest practical alternatives.

Load-distance method

The load-distance method evaluates specific candidate sites using loads and distances:

L_i is the load shipped between candidate site j and demand point i, and d_ij is the relevant distance. If freight rates differ, use freight-weighted distance, ∑ L_i c_i d_ij, where c_i is cost per load-distance unit. The preferred site has the lowest measure, subject to feasibility. Distances should preferably be road-network distances or travel times rather than straight-line distances when geography is irregular.

Break-even analysis

Break-even analysis compares fixed and variable costs for alternative facility configurations. For configuration j:

TC_j is total annual cost, F_j is annual fixed cost, v_j is variable cost per unit, and Q is annual throughput. The crossover quantity between alternatives A and B is:

This expression applies when v_A ≠ v_B and both alternatives serve the same unit of output. If A has lower fixed cost but higher variable cost, A is generally preferred below the crossover and B above it. The method excludes step costs, service-level penalties, inventory costs and uncertainty unless these are explicitly incorporated.

3. Mathematical Location Models

p-Median model

The p-median model selects exactly p facilities and assigns demand nodes to facilities to minimise weighted service distance or cost. A standard formulation is:

subject to:

where x_ij=1 if demand node i is assigned to facility j, y_j=1 if site j is opened, h_i is demand or shipment weight, and d_ij is distance or service cost. Variables are binary. The first constraint assigns every demand node once; the second prevents assignment to a closed facility; the third fixes the number of facilities.

A capacitated version adds:

where K_j is facility capacity. The p-median objective emphasises average weighted accessibility, not necessarily maximum service distance or opening cost.

Covering models

Covering models emphasise service reach. In the set-covering model, all demand must be covered by at least one facility within an acceptable distance R, while the number or cost of facilities is minimised:

subject to:

Here f_j is facility cost. The model is appropriate where every customer must receive a specified access standard, such as emergency parts coverage.

The maximal-covering model fixes a facility budget and maximises covered demand:

where z_i indicates whether demand node i is covered. Constraints include z_i ≤ ∑_{j:d_ij≤R} y_j and ∑_j y_j ≤ P, where P is the available number of facilities. Covering models use threshold service standards; they do not distinguish between a facility 1 kilometre and 9 kilometres away when both are within the threshold.

Fixed-charge facility-location model

The uncapacitated fixed-charge model jointly decides facility openings and customer assignments:

subject to assignment and linking constraints above. F_j is the fixed opening cost and c_ij is the variable cost of serving demand node i from site j. If c_ij=h_i t_ij, then t_ij is unit cost per unit of demand. Capacity, minimum throughput, inventory, emissions and multi-product constraints may be added.

This is a mixed-integer programming problem. Exact optimisation may become computationally difficult as the number of sites, customers, periods and products increases. Practical approaches include commercial solvers, decomposition, valid inequalities, heuristics, local search and scenario-based optimisation.

4. Worked Integrated Example

An organisation is considering three candidate sites—North, Central and South—to serve four demand zones. Annual demand is 100, 150, 200 and 250 pallets. Transportation cost is assumed proportional to pallet-distance. The distance matrix is:

The load-distance totals are:

• North: 100×10 + 150×30 + 200×50 + 250×70 = 28,500 pallet-distance units.
• Central: 100×20 + 150×10 + 200×15 + 250×35 = 14,750 pallet-distance units.
• South: 100×50 + 150×30 + 200×10 + 250×15 = 12,250 pallet-distance units.

South is best under the load-distance criterion. However, assume annual fixed costs are North 0.8 million, Central 1.0 million and South 1.3 million monetary units, and variable transportation costs are respectively 14, 9 and 7 per pallet-distance unit. For 850 pallets, the simplified annual costs are:

Central is preferred under this cost structure despite South having the lowest distance. This illustrates why distance minimisation alone is insufficient. A final decision would also test capacity, service time, labour, land, taxes, resilience and future demand growth.

5. Determining Facility Number, Territories and Hierarchy

The number of facilities is determined by the trade-off among fixed facility cost, transport cost, inventory duplication, handling cost and service requirements. Adding facilities typically reduces outbound distance but creates diminishing returns. The optimal number is the point at which the incremental benefit of another facility no longer exceeds its incremental cost and risk.

Customer territories can be assigned by nearest facility, lowest delivered cost, maximum service feasibility or optimisation. Nearest-facility assignment is simple but may be wrong when capacities, tolls, congestion or facility-specific costs differ. Territory boundaries should therefore be reviewed for workload balance, route feasibility and customer continuity.

Hierarchy should reflect product and service logic. A national centre may hold slow-moving inventory, regional centres may hold high-volume stock, and local depots may handle urgent parts or returns. Cross-docks should be positioned where shipment synchronisation and consolidation benefits exceed the cost of additional handling.

A robust design distinguishes:

• strategic flows: long-term inventory positioning and sourcing;
• tactical flows: seasonal allocation and capacity planning;
• operational flows: daily order assignment, routing and dispatch.

6. Case Study: Illustrative E-Commerce Network Redesign

Background

A fictional online retailer serves a country from one fulfilment centre. Order volume has grown rapidly, and next-day delivery is available in only 62% of postal zones. Management is considering one national centre, two regional centres or three smaller facilities.

Problem

The decision must balance fixed operating cost, parcel transportation, inventory duplication, labour availability, delivery promise and disruption resilience. Demand is highly concentrated in three metropolitan regions but includes substantial rural demand.

Analysis

The team first segmented products. Fast-moving standard products were candidates for regional stocking; slow-moving and long-tail products remained centralised. A centre-of-gravity calculation suggested a location near the largest demand cluster, but road-network analysis showed that a site near a motorway interchange produced better actual travel times. A p-median model evaluated average weighted delivery distance, while a fixed-charge model included facility opening cost and variable fulfilment cost. A covering model tested the percentage of population reachable within next-day parcel standards.

Scenario analysis considered base demand, 20% growth, a fuel-price increase and the temporary closure of one facility. The two-regional-centre design produced the lowest expected total cost under growth and improved next-day coverage to 91%. Three facilities produced slightly better coverage but excessive inventory duplication and labour risk. The one-centre design remained cheapest under low demand but failed the service requirement.

Approach and solution

The recommended structure was one national centre for long-tail products and two regional fulfilment centres for high-volume stock. Customer orders were assigned by promised service and total delivered cost rather than geographic proximity alone. A contingency rule allowed temporary reassignment during disruption.

Outcome

The illustrative solution increased service coverage, reduced average parcel distance and preserved centralised inventory for low-demand items. Its success depended on phased implementation, accurate stock classification, systems integration and periodic review rather than on the optimisation model alone.

Learning points

1. Different products may require different network structures.
2. A mathematically attractive location may be infeasible on the road network.
3. Service constraints can change the preferred number of facilities.
4. Resilience should be evaluated through scenarios, not assumed from redundancy alone.
5. Network design is a governance process as well as a technical exercise.

7. Limitations, Risk and Contemporary Practice

Location models are sensitive to forecast error, aggregation, cost assumptions and distance measurement. Annual averages can conceal seasonal peaks; geographic centroids can conceal barriers; and average delivery distance can conceal unacceptable service failures in remote zones. Fixed costs may be uncertain because leases, labour, taxes and energy prices change.

Best practice includes sensitivity analysis, stress testing, alternative demand scenarios, capacity validation and explicit documentation of assumptions. Robust or stochastic models can represent uncertain demand and disruptions, although they require probability estimates or scenario sets. Multi-objective models may include cost, service, carbon emissions and employment, but decision-makers should make trade-offs transparent rather than hide them in arbitrary weights.

Technology supports, but does not replace, judgement. GIS platforms improve road-distance and accessibility analysis; warehouse-management and transport-management systems provide actual flow data; digital twins can simulate network changes; and AI can assist with demand forecasting, scenario generation and anomaly detection. AI outputs require controls for data quality, explainability, bias, cybersecurity and human approval. A model trained on historical demand may reproduce past geographic inequities or fail under structural change.

Environmental and social considerations are increasingly material. Network design should consider emissions from transport and facilities, land-use impacts, labour conditions, community effects, noise, traffic and climate exposure. Regulatory requirements may include zoning, planning permission, building standards, hazardous-material rules, customs controls, data protection and occupational safety. Ethical analysis is especially important when facility placement affects communities unevenly or when service differentiation disadvantages remote customers.

Best-Practice Decision Framework

A defensible facility-location study should:

1. establish the decision scope, horizon and service promises;
2. use disaggregated demand and realistic time periods;
3. separate hard constraints from preference criteria;
4. compare at least one qualitative, one continuous-location and one discrete-location method;
5. include facility, transport, inventory, handling, technology and transition costs;
6. validate distances, capacities and lead times with operating experts;
7. test demand, cost, disruption and regulatory scenarios;
8. document assumptions, model version, data lineage and approval rights;
9. design implementation phases and exit or expansion options;
10. monitor actual performance and periodically re-optimise.

Chapter Summary

Facility location and network structure determine the physical architecture through which products, information and service commitments move. Facility roles range from central distribution and regional fulfilment to cross-docking and service depots. Centralisation reduces duplication but can weaken responsiveness; decentralisation improves proximity but increases fixed cost and inventory complexity.

Factor-rating supports structured qualitative comparison. Centre-of-gravity provides a weighted geographic starting point, while load-distance compares feasible candidate sites. Break-even analysis evaluates volume-dependent cost trade-offs. The p-median model selects a specified number of facilities to minimise weighted distance or cost; covering models enforce reach requirements; and fixed-charge models jointly decide facility openings and assignments while accounting for fixed and variable costs.

No model is universally best. The choice depends on whether the priority is average accessibility, guaranteed coverage, facility cost, capacity, resilience or a combination. Sound network design integrates optimisation with GIS, operational validation, scenario analysis, governance, environmental assessment and managerial judgement.

Key Terms

Facility location: The decision of where logistics facilities should be established.

Network structure: The number, hierarchy, roles and connections among facilities.

Distribution centre: A facility that receives, stores, processes and dispatches products.

Cross-dock: A facility that transfers shipments with minimal storage.

Centralisation: Concentrating inventory or operations in relatively few facilities.

Decentralisation: Spreading facilities closer to demand points.

Factor rating: A weighted scoring method for comparing candidate sites.

Centre of gravity: A weighted-average geographic estimate of a location.

Load-distance: The sum of shipment load multiplied by distance.

Break-even quantity: The output level at which alternative cost functions are equal.

p-Median model: A model selecting exactly p facilities to minimise weighted service cost.

Covering model: A model imposing a specified service-radius requirement.

Fixed-charge model: A model including the cost of opening facilities.

Demand node: A geographic point or zone representing customer demand.

Facility capacity: The maximum workload or throughput a facility can handle.

Customer territory: The demand area assigned to a facility.

Multi-echelon network: A network containing multiple facility layers.

Sensitivity analysis: Testing how results change when assumptions vary.

Resilience: The ability of a network to withstand and recover from disruption.

Delivered cost: The complete cost of supplying a customer, including relevant logistics costs.

Review Questions

Short Answer Questions

1. What is the centralisation–responsiveness trade-off?
2. Define the load-distance method.
3. What decision does the p-Median model make?
4. Why may a centre-of-gravity result be infeasible?
5. What is the purpose of a covering model?

Long Answer Questions

1. Explain the roles of central distribution centres, regional distribution centres and cross-docks in a multi-echelon network.
2. Compare factor-rating, centre-of-gravity and load-distance methods, including their assumptions and limitations.
3. Explain the formulation and managerial interpretation of a fixed-charge facility-location model.
4. Discuss how capacity, service requirements and inventory duplication influence the number of facilities.
5. Evaluate the contribution and limitations of AI, GIS and digital twins in facility-location decisions.

Analytical / Application Questions

1. A company has three candidate sites and four demand zones. Design a spreadsheet model that calculates load-distance, fixed cost, variable cost and total cost for each site. State which assumptions require sensitivity testing.
2. Formulate a capacitated p-Median model for two facilities, five demand zones and maximum delivery distance constraints. Define all decision variables and constraints.
3. A retailer must guarantee next-day access for at least 90% of demand while minimising facility cost. Recommend whether a set-covering, maximal-covering or fixed-charge model is most appropriate, and justify your choice.

Multiple Choice Questions with Answers

Question 1

Which method combines weighted criteria such as labour availability, infrastructure and transport cost into a site score?

A. Factor-rating method

B. Centre-of-gravity method

C. p-Median model

D. Set-covering model

Correct Answer: A. Factor-rating method

Question 2

In the centre-of-gravity method, the weights commonly represent what?

A. Building height

B. Demand or shipment volume

C. Number of candidate sites

D. Fixed facility cost only

Correct Answer: B. Demand or shipment volume

Question 3

A p-Median model normally selects:

A. At least one facility for every customer

B. Exactly p facilities to minimise weighted service cost

C. The facility with the highest fixed cost

D. Only facilities within a circular boundary

Correct Answer: B. Exactly p facilities to minimise weighted service cost

Question 4

What does a set-covering model primarily require?

A. Every demand node to be within an acceptable service distance

B. Every facility to have identical capacity

C. Exactly one facility to be opened

D. Transport cost to be zero

Correct Answer: A. Every demand node to be within an acceptable service distance

Question 5

If fixed cost is 400,000 and variable cost is 12 per unit, what is total cost at 10,000 units?

A. 412,000

B. 520,000

C. 1,200,000

D. 4,120,000

Correct Answer: B. 520,000

Question 6

Which constraint prevents demand from being assigned to a closed facility in a fixed-charge model?

A. ∑j yj = p

B. ∑j xij = 1

C. xij ≤ yj

D. ∑i hi ≤ Kj

Correct Answer: C. xij ≤ yj

Question 7

A load-distance calculation uses which basic relationship?

A. Fixed cost divided by demand

B. Load multiplied by distance and summed

C. Distance divided by facility count

D. Demand minus capacity

Correct Answer: B. Load multiplied by distance and summed

Question 8

Which network structure generally duplicates more inventory across locations?

A. Highly centralised network

B. Decentralised network

C. Single cross-dock

D. Direct shipping with no facilities

Correct Answer: B. Decentralised network

Question 9

What is a major limitation of a simple centre-of-gravity calculation?

A. It cannot use demand weights

B. It automatically includes congestion

C. It may identify a point where no feasible site exists

D. It requires exactly p facilities

Correct Answer: C. It may identify a point where no feasible site exists

Question 10

Which technology is especially useful for calculating realistic road distances and visualising candidate sites?

A. GIS

B. Barcode printing only

C. Payroll software

D. Word processing software

Correct Answer: A. GIS