# Supply Chain Network Design Concept Card: [Your course or network]

> A concept card for the supply chain elective: how many facilities a company needs, where to put them, which customers
> each one serves, and how the answer trades off fixed facility cost, inbound and outbound transport, inventory and
> delivery time. It covers the facility location model, the centre of gravity as a first screen, risk pooling and the
> square-root law, pipeline inventory and landed cost for near-shore versus far-shore sourcing. It builds on the core
> inventory card (safety stock, Little's Law): read mba/mba-inventory-card first, because this card uses its notation
> and does not re-teach it; the linear programming and Solver basics are in mba/mba-decision-analytics-guide. It is for
> MBA students in a supply chain elective and for managers in retail, distribution, manufacturing and service parts who
> face a network decision. One example is fully solved, one is half solved and one is left for you, and the self-test
> answers sit in a separate key. Source basis: Weber (1909) for location theory, Hakimi (1964) for the median problem,
> Geoffrion and Graves (1974) for the optimisation model, Maister (1976) and Eppen (1979) for inventory pooling, all
> written up in original words with fictional data (Northfield Outdoor Co. and Corvane Robotics are invented companies).
> Replace every [bracketed] value and every row marked *Example*.

<!--
Guidance: How to use this template
- Start with "When to use it" and the cost trade-off chart. A network question is almost never "what is the cheapest
  network?" alone: it is "what does a given delivery promise cost, and is it worth it?"
- Rebuild Worked example A in a spreadsheet: distances, cost per unit, assignment, then the 15 configurations. Set up
  the Solver model shown and check that it finds the same answer as the enumeration.
- Keep units straight: demand in units per year, transport in currency per unit, fixed cost per year, safety stock from
  weekly standard deviations.
- Core cards this builds on: mba/mba-inventory-card (safety stock, pooling, Little's Law), mba/mba-decision-analytics-guide
  (linear programming and Solver), mba/mba-npv-irr-card (when a new facility is an investment decision).
- Related templates: mba/mba-supply-chain-guide, mba/mba-bullwhip-effect-card, mba/mba-sourcing-procurement-card,
  mba/mba-supply-chain-resilience-card, mba/mba-inventory-card, mba/mba-operations-guide,
  mba/mba-decision-analytics-guide, mba/mba-npv-irr-card, mba/mba-relevant-costs-card, mba/mba-formula-sheet,
  mba/mba-error-log, mba/mba-concept-flashcards, finance/capex-request, finance/working-capital-review,
  business/vendor-evaluation-scorecard.
-->

## Document control

<!-- Guidance: Network studies depend on a demand snapshot and a cost snapshot. Record both dates; a freight rate change can reorder the ranking of options. -->

| Field | Value |
|---|---|
| Course or project | [Supply chain elective, network design sessions / client project] |
| Learner or analyst | [Your name] |
| Network analysed | [Northfield distribution network, five regions, four candidate sites (fictional) / your network] |
| Demand data and year | [Units by region, YYYY, forecast version] |
| Cost data | [Freight rate cards dated YYYY-MM-DD, site cost quotes, holding cost rate approved by finance] |
| Service promise | [Next-day within 450 road km / your promise] |
| Tool | [Excel with Solver / Python with scipy / a network design package] |
| Reviewed by | [Study group / tutor / mentor] |
| Last updated | YYYY-MM-DD |
| Version | [0.1] |

## When to use it

<!-- Guidance: Network design decisions are long-lived and expensive to reverse. Use the card when the decision changes the number, place or role of facilities, not for day-to-day routing. -->

**One-sentence rule:** use this card whenever you must decide how many stocking or production points to run, where,
and which customers each serves, and the answer changes fixed cost, transport, inventory and delivery time together.

| Situation | Model on this card | Key output |
|---|---|---|
| First screen for a new warehouse or plant region | Centre of gravity | A starting point on the map, not an answer |
| Choose among candidate sites with known costs | Facility location model (enumeration or mixed-integer programme) | Open sites, assignments, total cost |
| Consolidate or split stocking points | Risk pooling and the square-root law | Safety stock change |
| Near-shore versus far-shore supplier or plant | Landed cost with pipeline and safety stock | Cost per unit delivered |
| A new delivery promise (next-day, same-day) | Location model with a service constraint | The cost of the promise |
| An acquisition with an overlapping network | Location model on the combined sites | Sites to close, synergy estimate |

| Use when | Do not use alone when |
|---|---|
| Demand by region and cost by lane can be estimated | Demand is shifting fast between channels (stores to online) without a forecast |
| The decision horizon is several years | The question is next week's truck routes (vehicle routing) |
| Candidate sites are a manageable list | Real estate, labour availability or incentives dominate: add them as constraints or scores |
| Costs are roughly linear in volume | Strong economies of scale in a site (automation) need step costs |

## The idea in plain language

<!-- Guidance: Read this aloud before the formulas. If you can explain why more warehouses lower outbound freight but raise inventory and fixed cost, the rest of the card follows. -->

A distribution network is a set of trade-offs. **Adding a facility** brings stock closer to customers, so outbound
freight per unit falls and delivery gets faster. But every facility has a **fixed cost** (rent, staff, systems), and
every facility holds its own **safety stock**. Splitting one pool of stock into several separate pools raises the
total buffer, because each pool must cover its own demand swings; the swings no longer cancel out (Eppen 1979). Inbound
transport (from the port or factory to the facilities) usually rises a little with more sites, because full trucks
must go to more places.

Total cost across the number of facilities is often **flat near the optimum**: in Worked example A the best one-, two-,
three- and four-site networks differ by about 2% in cost (USD 7.537m to 7.690m). That is good news and a warning. Good news, because
other criteria (delivery promise, risk, labour markets, expansion room) can decide among near-equal options. A
warning, because a small error in a freight rate can flip the ranking, so the robustness check matters more than the
optimum.

Location theory goes back to Weber (1909), who placed a plant to minimise weighted transport distance. Hakimi (1964)
proved that, on a network, a location minimising total weighted distance can always be found among the nodes, a
result he later extended to several facilities; that turned location into a combinatorial choice. Geoffrion and Graves (1974) built the mixed-integer model that
modern network design software still descends from: open sites, assign flows, minimise fixed plus variable cost,
subject to capacity and service.

The **centre of gravity** is the demand-weighted average location. It is quick and visual, but it minimises squared
distance, not distance, and ignores roads, costs and real estate, so treat it as a screening tool. **Landed cost**
extends the logic to sourcing: the cheapest factory price is not the cheapest delivered unit once freight, duties,
pipeline inventory and safety stock for a long lead time are added.

## Formulas and notation

<!-- Guidance: Write units next to every input. Most network errors come from mixing per-unit and per-year numbers or weekly and annual standard deviations. -->

**Facility location model** (uncapacitated form, after Geoffrion and Graves 1974). Open site $j$ ($y_j = 1$) at fixed
cost $f_j$; serve region $i$ from site $j$ ($x_{ij} = 1$) at cost $c_{ij}$ per unit:

$$
\min \sum_{j} f_j\,y_j + \sum_{i}\sum_{j} d_i\,c_{ij}\,x_{ij}
\quad \text{s.t.} \quad \sum_{j} x_{ij} = 1 \ \forall i, \qquad x_{ij} \le y_j \ \forall i, j, \qquad y_j \in \{0, 1\}, \ x_{ij} \ge 0
$$

With capacities $K_j$, add $\sum_i d_i\,x_{ij} \le K_j\,y_j$. With a service promise, set $x_{ij} = 0$ for every lane
longer than the promised distance or time.

**Centre of gravity** (demand-weighted mean location):

$$
\bar{x} = \frac{\sum_i w_i x_i}{\sum_i w_i}, \qquad \bar{y} = \frac{\sum_i w_i y_i}{\sum_i w_i}
$$

The point that minimises total weighted straight-line distance $\sum_i w_i \lVert (x, y) - (x_i, y_i) \rVert$ is
found by the iterative method of Weiszfeld (1937); it can differ from the centre of gravity and can sit on a heavy
demand point.

**Risk pooling.** Serving regions with weekly demand standard deviations $\sigma_i$ and pairwise correlation
$\rho_{ik}$ from one facility, and the square-root law for $n$ equal, independent markets (Maister 1976):

$$
\sigma_{\text{pooled}} = \sqrt{\sum_i \sigma_i^2 + 2\sum_{i<k}\rho_{ik}\,\sigma_i\,\sigma_k}, \qquad
SS = z\,\sigma_{\text{pooled}}\sqrt{L}, \qquad SS_n = SS_1\sqrt{n}
$$

**Pipeline inventory** (Little's Law applied to goods in transit) and **landed cost**:

$$
I_{\text{pipeline}} = D \times T, \qquad
\text{Landed cost per unit} = p\,(1 + \tau) + F + \frac{h\,(p\,I_{\text{pipeline}} + c_L\,SS)}{D}
$$

| Symbol | Meaning | Unit |
|---|---|---|
| $i$, $j$ | Demand region, candidate site | index |
| $d_i$, $D$ | Annual demand of region $i$, total annual demand | units per year |
| $f_j$, $K_j$ | Fixed annual cost and capacity of site $j$ | currency per year, units per year |
| $c_{ij}$ | Inbound plus outbound transport and handling cost per unit from site $j$ to region $i$ | currency per unit |
| $y_j$, $x_{ij}$ | Open site (0 or 1), share of region $i$ served from $j$ | none |
| $w_i$, $(x_i, y_i)$ | Weight (usually volume, or volume x rate) and coordinates of region $i$ | units, km |
| $\sigma_i$, $\rho_{ik}$ | Weekly demand standard deviation of region $i$, correlation between regions | units per week, none |
| $z$, $L$ | Standard normal quantile of the cycle service level, replenishment lead time | none, weeks |
| $T$ | Transit time | years (or weeks with weekly demand) |
| $p$, $\tau$, $F$ | Purchase price, duty rate, freight per unit | currency per unit, %, currency per unit |
| $h$, $c_L$ | Holding cost rate, landed unit cost used to value safety stock | % per year, currency per unit |

**Excel equivalents.** Road distance: `=1.2*SQRT((x1-x2)^2+(y1-y2)^2)`. Total cost:
`=SUMPRODUCT(fixed,open)+SUMPRODUCT(demand_matrix,cost_matrix,assign_matrix)`. Pooled SD:
`=SQRT(SUMSQ(sigmas))` for independent regions. Safety stock: `=NORM.S.INV(CSL)*pooled_SD*SQRT(L)`. Centre of
gravity: `=SUMPRODUCT(w,x)/SUM(w)`.

**Solver set-up** (Excel, Simplex LP): objective = total cost cell, minimise; variable cells = the 4 open flags and the
20 assignment cells; constraints: open flags binary, assignment cells binary, each region's assignment row sums to 1,
each assignment cell $\le$ its site's open flag. Add a capacity row per site, and fix lanes beyond the service distance
at 0, when those apply.

## Assumptions and limits

<!-- Guidance: Name the assumption that matters most in your network before you present a recommendation. Usually it is the demand forecast by region five years out. -->

| Assumption | What breaks it | What to do |
|---|---|---|
| Demand by region is known for the planning horizon | Channel shift, new markets, growth | Run scenarios (low, base, high) and keep options that are good in all |
| Transport cost is linear in distance and volume | Rate tables with breaks, backhaul, fuel surcharges | Use lane rates from rate cards; test fuel scenarios |
| Inventory follows the square-root law | Correlated demand across regions, different service levels, different product mixes by region | Pool with the correlation formula; size stock by item, not by site |
| Fixed cost is fixed | Leases with break clauses, step costs for automation | Model steps as extra binary variables |
| Each region is served from one site | Split deliveries, store-to-online fulfilment | Allow fractional $x_{ij}$ or add a second echelon |
| Costs are today's | Labour inflation, tariffs, carbon prices | Sensitivity tables on the two or three largest cost lines |
| Static, one-period model | Seasonality, ramp-up | Model peak weeks for capacity and the average for cost |

Tariffs and duty rates change by product and origin and are policy decisions; take them from the official tariff
schedule of the importing country on the date of the study, and check them again before a decision. The trade terms
that decide who pays freight, insurance and duties are usually written with the ICC Incoterms rules: as of October 2026
the current set is Incoterms 2020, eleven rules in force since 1 January 2020 (check
https://iccwbo.org/business-solutions/incoterms-rules/ before you rely on it).

## How managers use it

<!-- Guidance: Connect each output to a decision someone signs. Network changes touch real estate, labour, IT, customers and finance, so the model is the start of the conversation. -->

- **Price the delivery promise.** Marketing wants next-day delivery everywhere; the model says what it costs (Worked
  example A: USD 0.153m a year, 2.0% of network cost). The decision becomes a margin question, not a debate.
- **Consolidation programmes.** Closing stocking points releases fixed cost and safety stock (Worked example B); the
  model shows what happens to freight and service.
- **Merger integration.** Overlapping networks after an acquisition are a classic synergy source; the location model on
  the combined sites gives a defensible estimate (see mba/mba-supply-chain-guide).
- **Near-shoring and dual sourcing.** Landed cost with pipeline and safety stock often narrows or reverses a far-shore
  price advantage (Exercise C); resilience adds a further argument (mba/mba-supply-chain-resilience-card).
- **Capital budgeting.** A new facility is an investment: the annual saving from the model is the cash flow line in
  the NPV (mba/mba-npv-irr-card and finance/capex-request).
- **Emissions.** Transport emissions grow with tonne-km; network models can carry an emissions line beside cost.
  Logistics emissions accounting commonly follows ISO 14083:2023 (published March 2023) and the GLEC Framework (version
  3.0, September 2023, aligned with ISO 14083; later versions are reported, so check smartfreightcentre.org for the
  current one) before you report figures.

## Choosing the method

<!-- Guidance: Use this flow at the start of a network question to pick the right level of model for the decision. -->

Pick the lightest method that answers the question: a centre of gravity to frame a search, enumeration when candidates
are few, a mixed-integer model when they are many, and a landed-cost comparison when the question is about sources
rather than warehouses.

```mermaid
flowchart TD
  A["Network question"] --> B{"Choosing a supply source or a stocking network?"}
  B -->|"Source"| C["Landed cost: price, duty, freight, pipeline and safety stock"]
  B -->|"Stocking network"| D{"Candidate sites known?"}
  D -->|"No"| E["Centre of gravity and real-estate screen to find candidates"]
  E --> F
  D -->|"Yes"| F{"Fewer than about 10 candidates?"}
  F -->|"Yes"| G["Enumerate all combinations in a spreadsheet"]
  F -->|"No"| H["Mixed-integer model in Solver or an optimisation tool"]
  G --> I["Add safety stock by pooled demand per site"]
  H --> I
  I --> J{"Service promise binding?"}
  J -->|"Yes"| K["Remove lanes beyond the promise and re-solve: the difference is the price of the promise"]
  J -->|"No"| L["Sensitivity: freight, demand scenarios, fixed costs"]
  K --> L
  C --> L
  L --> M["Recommendation with the cost of each option and its risks"]
```

## Worked example A (fully solved): Northfield distribution network

<!-- Guidance: Rebuild the distance and cost tables first, then the assignment rule, then the 15 configurations. The numbers come from a re-runnable script; if yours differ, check the 1.2 road factor and that demand is in thousands. -->

**Situation (*Example*, fictional).** Northfield Outdoor Co. serves five regions from leased distribution centres and
wants to redesign its network. Imports land at a port at map position (100, 200). Coordinates are in km on a map grid;
road distance is 1.2 times straight-line distance.

| Region | Position (km) | Annual demand (thousand units) | Weekly demand SD (units) |
|---|---|---|---|
| R1 North | (100, 800) | 300 | 2,700 |
| R2 West | (150, 300) | 450 | 3,600 |
| R3 Central | (500, 500) | 600 | 4,200 |
| R4 East | (850, 600) | 350 | 3,000 |
| R5 South | (600, 100) | 300 | 2,700 |

| Candidate site | Position (km) | Fixed cost (USD m a year) | Road km from port | Inbound USD per unit |
|---|---|---|---|---|
| S1 | (200, 650) | 0.70 | 553.2 | 0.664 |
| S2 | (450, 450) | 0.95 | 516.1 | 0.619 |
| S3 | (800, 550) | 0.65 | 939.1 | 1.127 |
| S4 | (500, 150) | 0.60 | 483.7 | 0.580 |

Cost rules: outbound USD 0.40 per unit plus USD 0.006 per unit per road km (`=0.40+0.006*road_km`); inbound full
trucks USD 0.0012 per unit per road km. Safety stock at each site covers a 97% cycle service level ($z$ = 1.8808,
`=NORM.S.INV(0.97)`) over a 2-week replenishment lead time, on the pooled SD of the regions it serves, valued at
USD 40 a unit with a 25% holding rate. Regions are independent. The service promise is next-day delivery, which
means a site within 450 road km.

**A1. Road distance (km) and outbound cost per unit (USD).**

| Site | R1 North | R2 West | R3 Central | R4 East | R5 South |
|---|---|---|---|---|---|
| S1 km / USD | 216.3 / 1.698 | 424.3 / 2.946 | 402.5 / 2.815 | 782.3 / 5.094 | 816.1 / 5.297 |
| S2 km / USD | 594.0 / 3.964 | 402.5 / 2.815 | 84.9 / 0.909 | 512.6 / 3.476 | 456.9 / 3.142 |
| S3 km / USD | 892.0 / 5.752 | 835.7 / 5.414 | 365.0 / 2.590 | 84.9 / 0.909 | 590.9 / 3.946 |
| S4 km / USD | 915.9 / 5.895 | 456.9 / 3.142 | 420.0 / 2.920 | 684.1 / 4.505 | 134.2 / 1.205 |

**A2. All 15 networks.** Each region is served by the open site with the lowest inbound plus outbound cost per unit.
Costs in USD m a year; components are rounded, so a total can differ from their sum by 0.001.

| Open sites *(Example)* | Fixed | Outbound | Inbound | Safety stock holding | Total | Demand within next-day reach |
|---|---|---|---|---|---|---|
| S2 + S3 | 1.600 | 4.262 | 1.416 | 0.259 | 7.537 | 70.0% |
| S2 | 0.950 | 5.160 | 1.239 | 0.196 | 7.545 | 52.5% |
| S2 + S3 + S4 | 2.200 | 3.681 | 1.405 | 0.315 | 7.601 | 85.0% |
| S2 + S4 | 1.550 | 4.579 | 1.227 | 0.254 | 7.610 | 67.5% |
| S1 + S2 + S3 | 2.300 | 3.582 | 1.430 | 0.315 | 7.627 | 85.0% |
| S1 + S2 | 1.650 | 4.481 | 1.252 | 0.254 | 7.637 | 67.5% |
| S1 + S2 + S3 + S4 | 2.900 | 3.001 | 1.418 | 0.371 | 7.690 | 100.0% |
| S1 + S2 + S4 | 2.250 | 3.900 | 1.240 | 0.311 | 7.701 | 82.5% |
| S1 + S3 + S4 | 1.950 | 4.204 | 1.465 | 0.315 | 7.934 | 100.0% |
| S1 + S3 | 1.350 | 5.026 | 1.629 | 0.271 | 8.276 | 85.0% |
| S1 + S4 | 1.300 | 5.462 | 1.273 | 0.271 | 8.307 | 82.5% |
| S3 + S4 | 1.250 | 5.614 | 1.352 | 0.259 | 8.475 | 62.5% |
| S4 | 0.600 | 6.872 | 1.161 | 0.196 | 8.829 | 45.0% |
| S1 | 0.700 | 6.896 | 1.328 | 0.196 | 9.119 | 67.5% |
| S3 | 0.650 | 7.218 | 2.254 | 0.196 | 10.317 | 47.5% |

**A3. Reading the result.**

- The cheapest network is **S2 + S3** at USD 7.537m: S2 (central) serves R1, R2, R3 and R5, S3 serves R4 East. A
  mixed-integer model without the inventory line (`scipy.optimize.milp` or Solver) opens the same two sites at USD
  7.278m; adding back the USD 0.259m of safety stock holding gives the same total.
- One central site (S2) costs only USD 0.008m more. The cost curve is flat: the best network with one, two, three and
  four sites costs 7.545, 7.537, 7.601 and 7.690.
- Safety stock rises with the number of sites: 19,578 units with one site (pooled SD 7,360.7 a week,
  `=1.8808*SQRT(2700^2+3600^2+4200^2+3000^2+2700^2)*SQRT(2)`) and 37,056 units with four.
- **The price of the promise.** Only two networks reach every region next day. The cheaper is all four sites at USD
  7.690m: next-day delivery everywhere costs USD 0.153m a year more than the cheapest network, 2.0% of network cost.
  If next-day service for R1 North and R5 South (30% of demand) adds more than that in margin, open all four.

```vega-lite
{
  "$schema": "https://vega.github.io/schema/vega-lite/v5.json",
  "description": "Northfield network: cost components of the best network for each number of sites, USD m a year (fictional, computed)",
  "width": 360,
  "height": 220,
  "data": {"values": [
    {"sites": "1 (S2)", "component": "Fixed", "usd_m": 0.95}, {"sites": "1 (S2)", "component": "Outbound", "usd_m": 5.16},
    {"sites": "1 (S2)", "component": "Inbound", "usd_m": 1.239}, {"sites": "1 (S2)", "component": "Safety stock", "usd_m": 0.196},
    {"sites": "2 (S2+S3)", "component": "Fixed", "usd_m": 1.6}, {"sites": "2 (S2+S3)", "component": "Outbound", "usd_m": 4.262},
    {"sites": "2 (S2+S3)", "component": "Inbound", "usd_m": 1.416}, {"sites": "2 (S2+S3)", "component": "Safety stock", "usd_m": 0.259},
    {"sites": "3 (S2+S3+S4)", "component": "Fixed", "usd_m": 2.2}, {"sites": "3 (S2+S3+S4)", "component": "Outbound", "usd_m": 3.681},
    {"sites": "3 (S2+S3+S4)", "component": "Inbound", "usd_m": 1.405}, {"sites": "3 (S2+S3+S4)", "component": "Safety stock", "usd_m": 0.315},
    {"sites": "4 (all)", "component": "Fixed", "usd_m": 2.9}, {"sites": "4 (all)", "component": "Outbound", "usd_m": 3.001},
    {"sites": "4 (all)", "component": "Inbound", "usd_m": 1.418}, {"sites": "4 (all)", "component": "Safety stock", "usd_m": 0.371}
  ]},
  "mark": "bar",
  "encoding": {
    "x": {"field": "sites", "type": "nominal", "title": "Number of sites (best network)", "sort": null},
    "y": {"field": "usd_m", "type": "quantitative", "title": "Annual cost (USD m)", "stack": "zero"},
    "color": {"field": "component", "type": "nominal", "title": "Cost"}
  }
}
```

**A4. Centre of gravity as a screen.** Weights are annual demand in thousands.

| Quantity | Result | Excel |
|---|---|---|
| $\bar{x}$ | 437.5 km | `=SUMPRODUCT(w,x)/SUM(w)` |
| $\bar{y}$ | 457.5 km | `=SUMPRODUCT(w,y)/SUM(w)` |
| Nearest candidate | S2 at (450, 450) | `=SQRT((450-437.5)^2+(450-457.5)^2)` = 14.6 km away |
| Weighted straight-line distance from the centre of gravity | 607.7 million unit-km | `=SUMPRODUCT(w,SQRT((x-437.5)^2+(y-457.5)^2))/1000` |
| Point that minimises weighted distance (Weiszfeld) | (500, 500), the R3 Central demand point: 582.5 million unit-km | Iterate, or check the vertex condition |

The centre of gravity pointed at S2, which is the backbone of the best networks. But the distance-minimising point is
R3 itself: R3 carries a weight of 600 against a combined pull of 325.18 from the other regions, so the optimum sits on
it. Neither point accounts for inbound freight from the port, fixed costs or the service promise; that is what the
location model adds.

## Worked example B (half solved): Corvane Robotics spare-parts depots

<!-- Guidance: Fill each [ ] with a number and an Excel formula, then check the answer key. Before computing, predict whether correlation between regions makes pooling more or less valuable. -->

**Situation (*Example*, fictional).** Corvane Robotics keeps a critical drive-module kit for its warehouse robots in four
regional depots. Each depot holds 500 kits of safety stock for the same service level, and the four regions have equal,
independent demand. A kit is valued at USD 2,500 and the holding rate is 20% a year. Operations proposes one central
depot with express courier delivery.

1. Square-root law: safety stock with one depot = 2,000 x $\sqrt{1/4}$ = [ ] kits (`=2000*SQRT(1/4)`).
2. With two depots: [ ] kits (`=2000*SQRT(2/4)`).
3. Annual holding cost of safety stock: four depots USD [ ]; one depot USD [ ]; two depots USD [ ]
   (`=kits*2500*0.2`).
4. Now suppose demand in the four regions is correlated with $\rho$ = 0.3 (the same customers' robots run the same
   seasonal peaks). Pooled safety stock with one depot = $500\sqrt{4 + 4 \times 3 \times 0.3}$ = [ ] kits
   (`=500*SQRT(4+12*0.3)`); saving versus four depots = [ ]%.
5. A robot down at a customer's warehouse costs the customer about USD 9,000 an hour (*Example*). Courier delivery from
   one central depot takes 6 hours longer on average than from a regional depot. Which costs, risks and contract terms
   (service-level agreement penalties) would you add before recommending consolidation?

## Exercise C (unsolved): near-shore or far-shore for a backpack line

<!-- Guidance: Solve on paper first, then confirm with Excel. Write the cost lines you will include before you compute anything. -->

Northfield sources a backpack line, 52,000 units a year (*Example*, fictional). Two offers:

| Item | Far-shore supplier | Near-shore supplier |
|---|---|---|
| Price per unit (FOB) | USD 62.00 | USD 70.00 |
| Duty rate *(Example; check the tariff schedule)* | 12% | 0% (trade agreement) |
| Freight per unit | USD 3.10 | USD 1.40 |
| Transit time | 6 weeks | 1 week |
| Replenishment lead time (order to receipt) | 10 weeks | 3 weeks |
| Weekly demand SD | 250 units | 250 units |

Holding cost rate 20% a year; pipeline stock is valued at the purchase price and safety stock at the landed unit cost
(price plus duty plus freight); cycle service level 95%.

1. Compute the landed unit cost (price plus duty plus freight) for each option.
2. Compute pipeline inventory (units and annual holding cost) for each, using Little's Law with weekly demand.
3. Compute safety stock (units and annual holding cost) for each.
4. Compute the total annual cost and the cost per unit. Which option is cheaper, and by how much a year?
5. Name two factors outside this calculation (one strategic, one risk) that could change the decision, and say which
   Incoterms rule you would want in each contract and why.

## Connections to other cards

<!-- Guidance: Interleave: after this card, solve one problem from a neighbouring card and decide which model fits before you compute. -->

- **mba/mba-inventory-card** (core): safety stock, service levels and Little's Law, used here for pooling and
  pipeline stock.
- **mba/mba-decision-analytics-guide** (core): linear programming and Solver, the engine of the location model.
- **mba/mba-supply-chain-guide**: where network design sits in the elective, with Fisher's (1997) efficient versus
  responsive logic.
- **mba/mba-bullwhip-effect-card**: lead times set by the network drive order amplification.
- **mba/mba-sourcing-procurement-card**: supplier selection and contracts after the landed-cost screen.
- **mba/mba-supply-chain-resilience-card**: single-site and single-source networks look cheapest until a disruption.
- **mba/mba-npv-irr-card** and **finance/capex-request**: a facility decision is an investment decision.
- **mba/mba-relevant-costs-card**: only costs that change with the network belong in the comparison (sunk lease costs
  do not).

## Common mistakes and misconceptions

<!-- Guidance: Read this list before an exam. Copy any mistake you make into mba/mba-error-log with the correct reasoning. -->

- **Treating the centre of gravity as the answer.** It minimises squared distance, ignores costs and real estate, and
  can point to a lake. Use it to find candidates.
- **Forgetting inventory.** A model with only fixed and transport costs opens too many sites; each extra site adds
  safety stock.
- **Applying the square-root law blindly.** It assumes equal, independent markets and the same service level; with
  correlated demand the saving shrinks (Worked example B).
- **Mixing time units.** Annual demand with weekly standard deviations, or fixed costs per month with transport per
  year.
- **Ignoring the inbound leg.** A site near customers but far from the port can lose on inbound freight (S3 in Worked
  example A).
- **Comparing suppliers on price.** Landed cost includes duty, freight, pipeline and safety stock, and the far-shore
  price advantage can disappear (Exercise C).
- **Over-trusting the optimum.** When options are within 1 to 2% of each other, the ranking can flip with a freight
  rate; present the flat region and the robust choice.
- **Forgetting the service promise.** The cheapest network may not reach customers on time; price the promise
  explicitly.
- **Hard-coding today's duties and rates.** Tariffs and freight change; date every rate and re-run before a decision.
- **Including sunk costs.** A lease already signed and not cancellable does not belong in the comparison.

## Self-test

<!-- Guidance: Answer from memory, in writing, before scrolling to the answer key. Return after two days and after a week. Mark misses in mba/mba-error-log. -->

1. Name the four cost lines that change with the number of distribution centres and the direction each moves as you add
   sites.
2. Why is the total cost curve often flat across the number of sites, and what does that imply for the recommendation?
3. A company consolidates nine equal, independent stocking points with 1,000 units of safety stock each into one.
   What safety stock does the square-root law predict?
4. A supplier ships 2,000 units a week with a 5-week transit time. How many units are in the pipeline?
5. Two customers sit on a line at km 0 (weight 100) and km 400 (weight 300). Where is the centre of gravity, and where
   is the point that minimises total weighted distance?
6. A unit costs USD 50 at the factory, duty is 8% and freight USD 2. What is the landed unit cost before inventory?
7. In the facility location model, what does the constraint $x_{ij} \le y_j$ do?
8. How do you compute the cost of a delivery promise with the model?
9. Why does correlation between regions reduce the value of pooling?
10. What did Hakimi (1964) show that made location a combinatorial choice?
11. Which of the following belongs in a network comparison: (a) a non-cancellable lease on an existing site, (b) the
    severance cost of closing a site, (c) the safety stock change?
12. Interleaving: which card first? (a) Orders from a distributor swing more than store sales. (b) How many depots for
    spare parts? (c) How much of a seasonal jacket to buy before the season?

## My notes and next steps

<!-- Guidance: This section is yours. Rate your confidence honestly and plan the next review. -->

| Item | My entry |
|---|---|
| Confidence with network design (1 = lost, 5 = could teach it) | [ ] |
| A network I know that looks over- or under-built, and why | [ ] |
| Questions for my study group or professor | [ ] |
| Mistakes logged in mba/mba-error-log | [Date and short description] |
| Next review date (spacing: +2 days, +1 week, +1 month) | YYYY-MM-DD |
| Next card to study | [mba/mba-sourcing-procurement-card / mba/mba-supply-chain-resilience-card] |

## Sources and further reading

<!-- Guidance: Cite the primary work for each model. Textbooks are named for further reading only; nothing on this card is copied from them. -->

- Weber, A. (1909). *Über den Standort der Industrien*. Mohr. English translation: *Theory of the Location of
  Industries* (1929), University of Chicago Press.
- Weiszfeld, E. (1937). Sur le point pour lequel la somme des distances de n points donnés est minimum. *Tôhoku
  Mathematical Journal*, 43. English translation by F. Plastria (2009), *Annals of Operations Research*,
  167(1), 7-41.
- Hakimi, S. L. (1964). Optimum locations of switching centers and the absolute centers and medians of a graph.
  *Operations Research*, 12(3), 450-459.
- Geoffrion, A. M., and Graves, G. W. (1974). Multicommodity distribution system design by Benders decomposition.
  *Management Science*, 20(5), 822-844.
- Maister, D. H. (1976). Centralisation of inventories and the "square root law". *International Journal of Physical
  Distribution*, 6(3), 124-134.
- Eppen, G. D. (1979). Effects of centralization on expected costs in a multi-location newsboy problem. *Management
  Science*, 25(5), 498-501.
- Fisher, M. L. (1997). What is the right supply chain for your product? *Harvard Business Review*, March-April 1997.
- International Chamber of Commerce. Incoterms 2020 (in force from 1 January 2020),
  https://iccwbo.org/business-solutions/incoterms-rules/ (as of October 2026; check for a newer edition).
- Textbooks (further reading, by name only): Chopra and Meindl, *Supply Chain Management: Strategy, Planning, and
  Operation*; Simchi-Levi, Kaminsky and Simchi-Levi, *Designing and Managing the Supply Chain*; Watson, Lewis, Cacioppi
  and Jayaraman, *Supply Chain Network Design*.
- Open resource: MIT OpenCourseWare, ESD.273J Logistics and Supply Chain Management (Fall 2009),
  https://ocw.mit.edu/courses/esd-273j-logistics-and-supply-chain-management-fall-2009/.

## Answer key

<!-- Guidance: Check your answers only after writing your own. For each miss, write down why your first answer was wrong. -->

**Worked example B.** (1) 1,000 kits. (2) 1,414.21 kits. (3) Four depots: 2,000 x 2,500 x 0.2 = USD 1,000,000; one
depot USD 500,000; two depots USD 707,106.78. (4) $500\sqrt{7.6}$ = 1,378.40 kits; saving 621.60 kits, 31.08% (against
50% with independent demand). (5) Add the customer downtime cost (6 extra hours x USD 9,000 per incident, times the
expected number of urgent incidents), courier cost per shipment, service-level agreement penalties and credits, the
risk of a single depot outage (fire, strike, IT), and customer contract terms on response time. A two-depot network or
a forward-stocking scheme for the largest customers is a common compromise.

**Exercise C.** (1) Far: 62 x 1.12 + 3.10 = USD 72.54 (`=62*(1+0.12)+3.10`); near: 70 + 1.40 = USD 71.40. (2) Weekly
demand 1,000. Far pipeline 6,000 units, holding 6,000 x 62 x 0.20 = USD 74,400; near 1,000 units, USD 14,000.
(3) Far safety stock `=NORM.S.INV(0.95)*250*SQRT(10)` = 1,300.37 units, holding 1,300.37 x 72.54 x 0.20 = USD
18,865.78; near `=NORM.S.INV(0.95)*250*SQRT(3)` = 712.24 units, holding USD 10,170.82. (4) Far total 72.54 x 52,000 +
74,400 + 18,865.78 = USD 3,865,345.78 (USD 74.33 a unit); near USD 3,736,970.82 (USD 71.86 a unit). Near-shore is
cheaper by USD 128,374.96 a year even though its price is USD 8 higher. (5) Strategic: speed to market and the option to
reorder in season (worth more for innovative products, Fisher 1997). Risk: concentration and disruption risk of the
far-shore route, and duty changes. Incoterms: for example FCA or FOB when you control main carriage and want to manage
freight, or DAP or DDP when you want the supplier to carry delivery risk to your site; which rule fits depends on who
can manage carriage and customs better. Check the rule text in Incoterms 2020.

**Self-test.**

1. Fixed facility cost rises; outbound transport falls; inbound transport usually rises a little; safety stock rises.
2. The rising and falling cost lines offset each other near the optimum. Several networks are near-equal, so choose on
   service, risk and flexibility, and show the sensitivity.
3. 9,000 x $\sqrt{1/9}$ = 3,000 units.
4. 2,000 x 5 = 10,000 units.
5. Centre of gravity: (100 x 0 + 300 x 400) / 400 = km 300. The weighted-distance minimiser is km 400, at the heavier
   customer: moving from 400 towards 0 adds 300 units of weighted distance per km and saves only 100.
6. 50 x 1.08 + 2 = USD 56.00.
7. It prevents assigning a region to a site that is not open.
8. Solve without the promise, then remove every lane that breaks it and solve again; the difference in total cost is
   the price of the promise.
9. Pooling works because high and low demands cancel out; when regions move together, they cancel less.
10. That, on a network, a location minimising total weighted distance exists among the nodes (later extended to
    several facilities), so the search is over a finite set of candidates.
11. (b) and (c). The non-cancellable lease is sunk and stays the same in every option; severance and inventory change.
12. (a) mba/mba-bullwhip-effect-card. (b) This card, risk pooling. (c) The newsvendor model in mba/mba-inventory-card.

## Revision history

<!-- Guidance: Add a row each time you change demand data, rate cards or candidate sites. -->

| Version | Date | Author | Change summary |
|---|---|---|---|
| 0.1 | YYYY-MM-DD | [Your name] | First draft from the template |
| 0.2 | YYYY-MM-DD | [Your name] | *Example:* Replaced Northfield data with own regions and rate cards, service promise priced |

## Standard and licence note

Based on the published location and supply chain literature, chiefly Weber (1909), Hakimi (1964), Geoffrion and Graves
(1974), Maister (1976) and Eppen (1979), with the ICC Incoterms rules named as a dated fact only; all text, data and
examples are original and fictional, and textbooks are named as further reading only. This template is not affiliated
with or endorsed by any business school, publisher, standards body or course. Check your course materials for the
notation your professor uses and the official sources for current duty rates and trade terms. Template licence: MIT.