14.1 Facility Location Analysis: Qualitative Factor Rating and Centroid Method

Facility location decisions represent some of the most capital-intensive, strategically binding commitments an industrial enterprise can make. A suboptimally placed manufacturing plant or regional distribution center imposes recurring penalties in freight expenses, unfavorable labor rates, supply chain vulnerability, and diminished customer service that cannot be easily engineered away through operational improvements. On the FE Industrial and Systems exam, facility location problems test both qualitative multi-criteria decision frameworks and rigorous quantitative models, including center-of-gravity formulations, distance metric derivations, and cost-volume break-even analyses.

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1. Multi-Level Facility Location Decisions

Industrial engineers evaluate location alternatives through a structured, top-down spatial hierarchy that progresses from international considerations down to specific parcel boundary characteristics:

Spatial Hierarchy of Facility Location
 ├── 1. Global / International Level
 │    ├── Trade agreements & tariffs (e.g., USMCA, WTO rules)
 │    ├── Currency stability & exchange rate volatility
 │    ├── Geopolitical stability & intellectual property protection
 │    └── Cultural factors, labor laws, & environmental mandates
 ├── 2. Regional Level
 │    ├── Proximity to raw material suppliers vs. final customer markets
 │    ├── Transportation infrastructure (interstate highways, rail heads, deep-water ports)
 │    ├── Regional labor supply, wage structures, & unionization density
 │    └── Utility availability (electrical grid capacity, natural gas, industrial water)
 ├── 3. Community Level
 │    ├── Local municipal tax abatements & economic development incentives
 │    ├── Local zoning ordinances, building codes, & EPA attainment status
 │    ├── Technical workforce training programs (community colleges, trade schools)
 │    └── Community quality of life (housing affordability, schools, public safety)
 └── 4. Specific Site Selection Level
      ├── Parcel acreage, terrain topography, & expansion potential
      ├── Soil load-bearing capacity & seismic/flood-plain status (FEMA maps)
      ├── Direct utility hookups (fiber optic, high-voltage substations, sewer discharge)
      └── Traffic ingress/egress, turning radii for 53-ft trailers, & rail siding access

Raw Material vs. Market Orientation (Weber's Classification)

A foundational principle in regional location theory, first formalized by Alfred Weber, categorizes manufacturing processes by their material weight transformation:

• Weight-Losing (Raw Material-Oriented) Processes: When raw materials lose significant weight, bulk, or perishability during processing (e.g., paper mills, copper smelting, timber milling), shipping raw inputs costs substantially more than shipping finished goods. Facilities must be located near raw material sources to minimize total ton-miles.
• Weight-Gaining (Market-Oriented) Processes: When finished products become heavier, bulkier, more fragile, or more perishable than input ingredients (e.g., commercial bakeries, soft drink bottling, structural steel fabrication), shipping finished products is significantly more expensive than transporting inputs. Facilities must be located near customer consumption centers.

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2. Qualitative Factor Rating Method

Because location decisions involve conflicting qualitative attributes (e.g., labor union relations, community quality of life) alongside quantitative costs (e.g., land purchase price, utility tariffs), industrial engineers employ the Factor Rating Method. This multi-criteria decision technique normalizes disparate evaluation metrics into a single composite score.

Mathematical Formulation

1. Identify $m$ critical success factors ($j = 1, 2, \dots, m$) encompassing all relevant operational, financial, and environmental criteria.
2. Assign an importance weight $w_j$ to each factor such that the weights sum to unity: $\sum_{j=1}^m w_j = 1.0 \quad (\text{or } 100\%)$
3. Establish a standard rating scale (typically $1$ to $10$ or $1$ to $100$) where higher scores indicate superior performance.
4. Score each candidate location $i$ on factor $j$, denoted as $s_{ij}$.
5. Calculate the composite weighted score $S_i$ for each candidate location: $S_i = \sum_{j=1}^m w_j s_{ij}$
6. Select the candidate location with the maximum composite score: $\max_i {S_i}$.

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3. The Center-of-Gravity (Centroid) Method

The Center-of-Gravity (Centroid) Method is a mathematical technique used to locate a single intermediate facility (such as a central distribution center or consolidation warehouse) that minimizes total transportation cost between a set of existing geographically dispersed supply origins and customer demand destinations.

Theoretical Formulation

Let $(x_i, y_i)$ denote the Cartesian grid coordinates of existing facility $i$ (for $i = 1, 2, \dots, n$). Let $V_i$ represent the shipment volume or tonnage moving between existing facility $i$ and the proposed central facility, and let $c_i$ represent the freight transportation cost per unit volume per unit distance.

The combined weighting factor $w_i$ for facility $i$ is defined as: $w_i = V_i \times c_i$

If transportation rates per unit distance are uniform across all routes ($c_i = c$), the freight rate cancels out, and the volume $V_i$ serves directly as the weight ($w_i = V_i$).

The optimal centroid coordinates $(x^, y^)$ are determined by: $x^* = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} = \frac{\sum_{i=1}^n (V_i c_i x_i)}{\sum_{i=1}^n (V_i c_i)}$

$y^* = \frac{\sum_{i=1}^n w_i y_i}{\sum_{i=1}^n w_i} = \frac{\sum_{i=1}^n (V_i c_i y_i)}{\sum_{i=1}^n (V_i c_i)}$

Centroid Mechanics (Center-of-Gravity Analogy)
  (x3, y3) [w3 = 300 tons]
      o
       \ 
        \       (x*, y*) [Optimal Hub]
         \        *
          \      / \
           \    /   \  
            o  /     o (x2, y2) [w2 = 600 tons]
  (x1, y1) [w1 = 400 tons]

  The centroid pulls toward nodes with the highest relative volume-distance weights.

Important Mathematical Distinction: The centroid formula provides the exact spatial point that minimizes the sum of squared Euclidean distances weighted by volume: $\min \sum w_i [(x - x_i)^2 + (y - y_i)^2]$. In physical freight logistics, the true cost objective minimizes the sum of unsquared Euclidean distances: $\min \sum w_i \sqrt{(x - x_i)^2 + (y - y_i)^2}$ (the classical Weber / Fermat problem, solved via the iterative Weiszfeld algorithm). On the FE exam, NCEES specifically tests the centroid formulas shown above as the standard closed-form center-of-gravity method.

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4. Distance Metrics in Location Analysis

Facility location models depend directly on the mathematical metric used to quantify the distance $d(A, B)$ between coordinates $(x_A, y_A)$ and $(x_B, y_B)$:

Rectilinear Single-Facility Location: The Median Method

When travel occurs along orthogonal rectilinear aisles ($L_1$ norm), the objective is: $\min_{x, y} \sum_{i=1}^n w_i \left( |x - x_i| + |y - y_i| \right) = \min_x \sum_{i=1}^n w_i |x - x_i| + \min_y \sum_{i=1}^n w_i |y - y_i|$ Because the objective function decomposes into independent functions of $x$ and $y$, the optimal location $(x^, y^)$ is found by calculating the independent weighted medians along the $x$-axis and $y$-axis, respectively. The optimal coordinate $x^*$ satisfies: $\sum_{i: x_i < x^*} w_i \le \frac{1}{2} \sum_{i=1}^n w_i \quad \text{and} \quad \sum_{i: x_i \le x^*} w_i \ge \frac{1}{2} \sum_{i=1}^n w_i$ This stands in sharp contrast to the center-of-gravity method, which computes the weighted mean.

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5. Location Break-Even (Cost-Volume) Analysis

When multiple candidate locations exhibit different structures of fixed and variable operating costs, industrial engineers employ Location Break-Even Analysis to identify which site yields the lowest total cost over specified annual production volume ranges.

Cost Function Formulation

For each candidate facility location $k$: $TC_k(Q) = FC_k + VC_k \times Q$ where:

• $TC_k$ = Total annual operating cost for location $k$ ($/year)
• $FC_k$ = Fixed annual costs (land lease, facility depreciation, property taxes, base administrative overhead) ($/year)
• $VC_k$ = Variable operating cost per unit produced (direct labor, materials, utilities, per-unit outbound shipping) ($/unit)
• $Q$ = Annual production or shipment volume (units/year)

Cost-Volume Trade-Off Across Three Candidate Sites
 Total Cost ($)
     ^
     │                            / Location A (Lowest FC, Highest VC)
     │                           / 
     │                          /   / Location B (Moderate FC & VC)
     │                         /   / 
     │                        /   /   / Location C (Highest FC, Lowest VC)
     │                       /   /   /
     │                      /   x   /
     │                     /   / \ /
     │                    /   /   x
     │                   /   /   / \
  FC_C ───             /   /   /   ── Optimal: Site C
     │                /   /   /
  FC_B ───           /   x─── Optimal: Site B
     │              /   / 
  FC_A ────────────x─── Optimal: Site A
     └─────────────┼─────┼────────────> Volume Q (units/year)
                  Q1*   Q2*

Determining Crossover Points

The crossover (indifference) volume between location $A$ and location $B$ occurs where their total cost curves intersect: $TC_A(Q) = TC_B(Q) \implies FC_A + VC_A \cdot Q = FC_B + VC_B \cdot Q$

$Q_{AB}^* = \frac{FC_B - FC_A}{VC_A - VC_B}$

• If expected annual volume $Q < Q_{AB}^*$, the facility with the lower fixed cost dominates.
• If expected annual volume $Q > Q_{AB}^*$, the facility with the lower variable cost dominates.

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6. Step-by-Step Worked Engineering Calculations

Worked Example 14.1.1: Multi-Criteria Factor Rating Evaluation

Problem: An advanced electronics manufacturer must select a site for a new semiconductor packaging facility. The site selection team has identified four critical factors, established their relative percentage weights, and scored three candidate cities on a 1-to-100 scale. Evaluate the candidate sites and determine the optimal location.

Solution:

Step 1: Verify Weight Normalization $\sum w_j = 0.35 + 0.25 + 0.20 + 0.20 = 1.00 \quad \checkmark$

Step 2: Calculate Composite Weighted Score for City A $S_A = (0.35)(70) + (0.25)(80) + (0.20)(90) + (0.20)(65)$ $S_A = 24.50 + 20.00 + 18.00 + 13.00 = 75.50$

Step 3: Calculate Composite Weighted Score for City B $S_B = (0.35)(85) + (0.25)(75) + (0.20)(60) + (0.20)(80)$ $S_B = 29.75 + 18.75 + 12.00 + 16.00 = 76.50$

Step 4: Calculate Composite Weighted Score for City C $S_C = (0.35)(90) + (0.25)(60) + (0.20)(80) + (0.20)(85)$ $S_C = 31.50 + 15.00 + 16.00 + 17.00 = 79.50$

Conclusion: City C yields the highest composite score ($S_C = 79.50$) and is the recommended facility location.

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Worked Example 14.1.2: Center-of-Gravity Distribution Center Location

Problem: A nationwide retail logistics provider is planning to construct a central distribution hub to supply four major regional fulfillment centers. The grid coordinates (in kilometers) and monthly outbound shipment demands (in metric tons) for each fulfillment center are shown below. Shipping rates are uniform at $0.15 per ton-kilometer across the entire network. Determine the optimal center-of-gravity coordinates $(x^, y^)$ for the new distribution hub.

Solution:

Step 1: Compute Total Network Weight Because the shipping rate $c$ is constant across all links, $w_i = V_i$: $\sum_{i=1}^4 w_i = 400 + 600 + 300 + 700 = 2,000\text{ tons}$

Step 2: Calculate Weighted $x$-Coordinate Sum $\sum_{i=1}^4 w_i x_i = (400)(30) + (600)(80) + (300)(20) + (700)(50)$ $\sum_{i=1}^4 w_i x_i = 12,000 + 48,000 + 6,000 + 35,000 = 101,000\text{ ton-km}$

Step 3: Calculate Weighted $y$-Coordinate Sum $\sum_{i=1}^4 w_i y_i = (400)(20) + (600)(40) + (300)(60) + (700)(10)$ $\sum_{i=1}^4 w_i y_i = 8,000 + 24,000 + 18,000 + 7,000 = 57,000\text{ ton-km}$

Step 4: Determine Optimal Centroid Coordinates $x^* = \frac{101,000}{2,000} = 50.50\text{ km}$ $y^* = \frac{57,000}{2,000} = 28.50\text{ km}$

Conclusion: The optimal center-of-gravity location for the central distribution hub is $(x^* = 50.5\text{ km}, y^* = 28.5\text{ km})$.

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Worked Example 14.1.3: Three-Site Location Break-Even and Crossover Analysis

Problem: An industrial valve manufacturer is evaluating three candidate cities for a new automated foundry. The estimated fixed annual operating costs and per-unit variable manufacturing costs are:

• Location 1: Fixed Cost $FC_1 = $200,000/\text{year}$; Variable Cost $VC_1 = $60/\text{unit}$
• Location 2: Fixed Cost $FC_2 = $320,000/\text{year}$; Variable Cost $VC_2 = $40/\text{unit}$
• Location 3: Fixed Cost $FC_3 = $500,000/\text{year}$; Variable Cost $VC_3 = $25/\text{unit}$

1. Calculate the crossover production volume between Location 1 and Location 2 ($Q_{1,2}^*$).
2. Calculate the crossover production volume between Location 2 and Location 3 ($Q_{2,3}^*$).
3. Determine the optimal location range for annual production volumes ranging from $0$ to $25,000$ units.

Solution:

Step 1: Compute Crossover Volume $Q_{1,2}^*$ $FC_1 + VC_1 \cdot Q = FC_2 + VC_2 \cdot Q$ $200,000 + 60 Q = 320,000 + 40 Q$ $20 Q = 120,000 \implies Q_{1,2}^* = 6,000\text{ units/year}$

Step 2: Compute Crossover Volume $Q_{2,3}^*$ $FC_2 + VC_2 \cdot Q = FC_3 + VC_3 \cdot Q$ $320,000 + 40 Q = 500,000 + 25 Q$ $15 Q = 180,000 \implies Q_{2,3}^* = 12,000\text{ units/year}$

Step 3: Establish Volume Decision Ranges

• For $0 \le Q < 6,000$ units/year: Location 1 provides the lowest total cost. (Example at $Q = 4,000$: $TC_1 = $440,000$, $TC_2 = $480,000$, $TC_3 = $600,000$).
• For $6,000 < Q < 12,000$ units/year: Location 2 provides the lowest total cost. (Example at $Q = 10,000$: $TC_1 = $800,000$, $TC_2 = $720,000$, $TC_3 = $750,000$).
• For $Q > 12,000$ units/year: Location 3 provides the lowest total cost. (Example at $Q = 16,000$: $TC_1 = $1,160,000$, $TC_2 = $960,000$, $TC_3 = $900,000$).

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7. NCEES Reference Handbook Tips & Realistic Exam Traps

• Centroid vs. Rectilinear Median Trap: Do not use the center-of-gravity formula when the problem states that travel is confined to an orthogonal city grid or factory aisle system. For rectilinear ($L_1$) movement, the optimal single facility location is the independent median of the $x$ and $y$ coordinates, NOT the mean.
• Unit Transportation Cost Omission: In center-of-gravity calculations, do not automatically set $w_i = V_i$. Check whether shipping rates per ton-mile ($c_i$) differ across sources and markets. If City 1 costs $$0.30/\text{ton-km}$ and City 2 costs $$0.10/\text{ton-km}$, the weights are $w_1 = 0.30 V_1$ and $w_2 = 0.10 V_2$. Omitting differing rates produces an incorrect centroid.
• Crossover Feasibility Trap: When solving for multiple break-even crossovers, always plot or test the intermediate volume ranges. A site with moderate fixed and variable costs can occasionally be completely dominated (never optimal at any volume) if its crossover point with the high-fixed-cost site occurs at a lower volume than its crossover with the low-fixed-cost site.
• Distance Metric Conversion: Remember that for any two points not on the exact same horizontal or vertical coordinate, Rectilinear distance is strictly greater than Euclidean distance: $d_R \ge d_E$. Specifically, the maximum discrepancy occurs along a 45-degree diagonal where $d_R = \sqrt{2} \cdot d_E \approx 1.414 \cdot d_E$ (a 41.4% increase).