Supply-Chain Network Design

SCND · MULTI-ECHELON STRATEGIC MIP

Logistics · Network & Facility · Strategic

Supply-Chain Network Design (SCND) is the archetypal strategic logistics problem: given a set of suppliers, candidate plants, candidate distribution centres, and customers, decide which facilities to open, at what capacity, and how product should flow from tier to tier — all to minimise total fixed + variable cost subject to demand, capacity, and service-level constraints. A decade-long decision that locks in a carrier's or manufacturer's operating cost structure. The foundational paper is Geoffrion & Graves (1974); the best survey is Melo, Nickel & Saldanha-da-Gama (2009) "Facility location and supply chain management".

Problem scope

What SCND decides

SCND integrates facility location (which sites to open) with multi-commodity flow (how product moves) across a multi-echelon network — typically suppliers → plants → DCs → customer zones. Unlike the single-tier CFLP/UFLP, SCND must decide flows through intermediate tiers, respect capacities at each tier, and often handle multiple products with different demand patterns.

Extensions add complexity: multi-period (facility opening/closing over a planning horizon), stochastic demand (two-stage or robust), multi-objective (cost vs. service vs. CO₂), and closed-loop (reverse flows for returns and remanufacturing). The MIRHA / Geoffrion-Graves family of benchmarks remains the reference for deterministic SCND; more recent reviews cover stochastic and sustainable variants.

Formulation sketch

Multi-commodity 2-echelon MIP (Geoffrion-Graves 1974)

Notation

SymbolMeaning
$I$set of candidate plants / DCs (facilities to open)
$J$set of customers / demand zones
$K$set of products
$d_j^k$demand at customer $j$ for product $k$
$f_i$fixed opening cost at facility $i$
$u_i$capacity (in common unit) at facility $i$
$c_{ij}^k$unit flow cost of product $k$ from $i$ to $j$
$y_i \in \{0,1\}$1 if facility $i$ is opened
$x_{ij}^k \ge 0$flow of product $k$ from $i$ to $j$

Objective

Minimise fixed opening + variable flow cost
$$\min \; \sum_{i \in I} f_i \, y_i \;+\; \sum_{i \in I} \sum_{j \in J} \sum_{k \in K} c_{ij}^k \, x_{ij}^k$$

Constraints

(1) Demand must be met for every product at every customer
$$\sum_{i \in I} x_{ij}^k = d_j^k \quad \forall j \in J,\, k \in K$$
(2) Flow only from open facilities, capacity-constrained
$$\sum_{j \in J} \sum_{k \in K} x_{ij}^k \le u_i \, y_i \quad \forall i \in I$$
(3) Variable bounds
$$y_i \in \{0,1\}, \quad x_{ij}^k \ge 0$$

Extensions. Multi-echelon (plants → DCs → customers) adds intermediate nodes with flow conservation. Multi-period adds $t$ on every variable with open/close transitions. Stochastic SCND (two-stage SP or robust) hedges against demand-scenario uncertainty. Closed-loop SCND (Govindan, Fattahi & Keyvanshokooh 2017 review) adds reverse flows from customers back to remanufacturing facilities.

Solution methods. Commercial MIP (Gurobi, CPLEX) handles deterministic instances up to ~10,000 variables. Decomposition (Benders, Lagrangian) is standard for larger or stochastic cases. See Cordeau, Pasin & Solomon (2006) and Snyder (2006) for stochastic-SCND treatments.

Key references

DOIs & permanent URLs

Geoffrion, A. M., & Graves, G. W. (1974).
“Multicommodity distribution system design by Benders decomposition.”
Management Science, 20(5), 822–844. doi:10.1287/mnsc.20.5.822
Melo, M. T., Nickel, S., & Saldanha-da-Gama, F. (2009).
“Facility location and supply chain management — A review.”
European Journal of Operational Research, 196(2), 401–412. doi:10.1016/j.ejor.2008.05.007
Snyder, L. V. (2006).
“Facility location under uncertainty: A review.”
IIE Transactions, 38(7), 547–564. doi:10.1080/07408170500216480
Cordeau, J.-F., Pasin, F., & Solomon, M. M. (2006).
“An integrated model for logistics network design.”
Annals of Operations Research, 144, 59–82. doi:10.1007/s10479-006-0001-3
Govindan, K., Fattahi, M., & Keyvanshokooh, E. (2017).
“Supply chain network design under uncertainty: A comprehensive review and future research directions.”
European Journal of Operational Research, 263(1), 108–141. doi:10.1016/j.ejor.2017.04.009
Simchi-Levi, D., Kaminsky, P., & Simchi-Levi, E. (2008).
Designing and Managing the Supply Chain (3rd ed.).
McGraw-Hill.

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