Notations and Definitions

All the notations and definitions for sets, decision variables and parameters are summarized as follows.

1. Sets

FA Set of internal and external supply chain members.

Herein, FA^V: set of vendors (abbreviated as V ); FA^M: set of manufacturing centers (abbreviated as M ); FA^Ds: set of simple processing DCs (abbreviated as Ds; FA^Dd: set of deep processing DCs (abbreviated as Dd); FA^Dn: set of non-bonded DC (abbreviated as Dn); FA^B: set of brand companies (abbreviated as B); FA^I: set of all internal supply chain members (abbreviated as I); FA^DC: set of all DCs, including deep processing DCs, simple processing DCs and non-bonded DCs (abbreviated as DC); FA_C: set of chain members in international logistics zones (abbreviated as C); FA_T: set of chain members in tax areas (abbreviated as T).

G Set of types of goods. Herein, G^r : set of modular components (abbreviated as r); G^s: set of semi-products (abbreviated as s ); G^p : set of finished products (abbreviated as p).

N Set of countries.

SN Set of simple and deep process product lines. Herein,

SNrs: set of product lines involving transformation of modular components r into semi-products s (abbreviated as rs); SN^rp: set of product lines involving transformation of modular components r into finished products p (abbreviated as rp ); SN^sp: set of product lines involving transformation of semi-products s into finished products p (abbreviated as sp); SN^ss : set of product lines involving simple processing of semi-products s (abbreviated as ss ); SN^pp : set of product lines involving simple processing of finished products p

(abbreviated as pp); SN_I : set of product lines of inbound flow (abbreviated as I ); SN_O: set of product lines of outbound flow (abbreviated as O); SN^R: set of product lines of sum of corresponding modular components (abbreviated as R); SN^S: set of product lines of sum of corresponding semi-products (abbreviated as S); SN^P: set of product lines of sum of corresponding finished products (abbreviated as P).

Ω Set of transportation modes. Here, Ω(θ λ^x, ^y) is the set of

available transportation modes between a given chain member θ in the country x∈N and another given chain member λ in the country y∈N.

2. Decision variables

gotr_{θ αβ}x Binary decision variable indicates whether goods transformation occurs at a given chain member θ in country x∈N when transferring ancestor goods α into descendant goods β.

x f

gt_{θ α} Binary decision variable indicates whether goods α is in progress in a product line f at a given chain member θ in country x∈N.

oi_θx Operating income of a given chain member θ in country

x∈N for the period of analysis (dollar/unit of time).

ord_{θ α}x Number of nodes visited on the transfer path from the origin up to node θ^x for goods α (i.e., the visit number of the θ^xth node).

x ym

qu_{θ λ α} Binary decision variable representing whether goods α∈G

is shipped from a given chain member θ in country x∈N

to another given chain member λ in country y∈N, using transportation mode m∈Ω(θ λ^x, ^y).

3. Parameters

BR_λy Required finished products for a given chain member λ in country y∈N (units of p/unit of time).

BOM_αβ Units of ancestor goods α∈G required to make one unit of descendant goods B∈G (α-units/ β-unit).

BN A big number.

CSF Cycle stock factor (%).

COT_θx Corporate tax rate (%) of country x∈N of a supply chain member θ.

CPRICE_α International contract price of goods α∈G (dollar/unit of goods α )

DRT_{θ α}x Value added tax drawback rate (%) on the value of goods α∈G of country x∈N of supply chain member θ.

x y

DUTY_{θ λ α} Import duty rate (%) on the value of goods α∈G shipped from a given chain member θ in country x∈N to another given chain member λ in country y∈N.

E_θx Exchange rate of country x∈N of supply chain member θ (monetary units of the respective country/dollar).

FIX_θx Fixed cost associated with a given chain member θ in country x∈N (monetary units of country of member θ per unit of time).

x ym

FS_{θ λ} Frequency of goods shipments from a given chain member θ in country x∈N to another given chain member λ in country y∈N , using transportation mode m∈Ω(θ λ^x, ^y)

(units of time ).

H Holding cost ($/($. unit of time)).

IV_{θ α}x Inventory value of goods α∈G, given in monetary units of a given chain member θ in country x∈N per unit of goods α .

NODE Number of DC nodes.

y x

PROC_{λ θ α} Procurement cost (including total cost and taxes) of goods

α∈G shipped from a given chain member λ in country

y∈N to another given chain member θ in country x∈N

(monetary units of country of member λ/unit of goods α ).

PPA_θx Simple processing capacity of finished products in a given chain member θ in country x∈N (finished product units/unit of time).

PPC_θx Simple processing cost of finished products in a given chain member θ in country x∈N (monetary units of country of member θ/unit of finished product).

RSA_θx Capacity to transform goods associated with a given chain member θ in country x∈N for transferring modular components into semi-products (semi-product units/unit of time).

RSC_θx Cost of transforming goods associated with a given chain member θ in country x∈N for transferring modular components into semi-products (monetary units of country of member θ/ unit of semi-products).

RPA_θx Capacity to transform goods associated with a given chain member θ in country x∈N for transferring modular components into finished products (finished product units/unit of time).

RPC_θx Cost of transforming goods associated with a given chain

member θ in country x∈N for transferring modular components into finished products in country x∈N

(monetary units of country of member θ/ unit of finished products).

SPA_θx Capacity to transform goods associated with a given chain member θ in country x∈N for transferring semi-products into finished products (finished product units/unit of time).

SPC_θx Cost of transforming goods associated with a given chain member θ in country x∈N for transferring semi-products into finished products (monetary units of country of member θ/ unit of finished products).

SSA_θx Simple processing capacity of semi-product in a given chain member θ in country x∈N (semi-product units/unit of time).

SSC_θx Simple processing cost of semi-product in a given chain member θ in country x∈N (monetary units of country of member θ/unit of s).

SSF_{θ α}x Safety stock factor of goods α∈G at a given chain member θ in country x∈N.

TP_{θ α}x Transfer price of goods α∈G shipped from a given chain member θ in country x∈N (monetary units of country of member θ/unit of goods α ).

x ym

TRC_{θ λ} Transportation cost per weight unit of goods shipped from a given chain member θ in country x∈N to another given chain member λ in country y∈N , using transportation mode m∈Ω(θ λ^x, ^y) (monetary units of country of member θ/weight unit).

x ym

TT_{θ λ} Average transportation time from a given chain member θ in country x∈N to another given chain member λ in country y∈N , using transportation mode m∈Ω(θ λ^x, ^y)

(units of time).

VC_θx Capacity of a given chain member θ in country x∈N for supplying modular components (modular component units/unit of time).

VAT_{θ α}x Value added tax rate (%) on the value of goods α∈G of country x∈N of supply chain member θ.

W_α Weight of a unit of goods α∈G (weight units/unit of goods).