Author’s Accepted Manuscript

Benders decomposition for a strategic network design problem under NAFTA local content requirements

Katharina Mariel, Stefan Minner

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S0305-0483(16)30293-6 http://dx.doi.org/10.1016/j.omega.2016.06.002 OME1680

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Received date: 18 November 2014Revised date:7 May 2016Accepted date: 6 June 2016

Cite this article as: Katharina Mariel and Stefan Minner, Benders decompositio for a strategic network design problem under NAFTA local content requirements Omega, http://dx.doi.org/10.1016/j.omega.2016.06.002

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Benders decomposition for a strategic network design problem under NAFTA local content requirements

Katharina Mariela, Stefan Minnerb

aDepartment of Production Network Planning, Daimler Trucks, DE-70546 Stuttgart, Germany bTUM School of Management, Technische Universität München, Arcisstrasse 21, DE-80333 Munich, Germany

Abstract

Global production and sourcing strategies of multinational corporations are strongly inu-enced by the increasing number of Free Trade Agreements (FTAs). Based on the special local content requirements of the North American Free Trade Agreement (NAFTA) for automotive goods, the impact on strategic network design decisions is investigated. The presented model explicitly integrates the dierent NAFTA legal options to calculate the lo-cal content for automotive goods. Furthermore, it considers the possibility to underachieve the local content requirement and to pay penalty duties for NAFTA cross-border deliveries instead. Plant xed costs have a signicant impact on the local content fulllment and have to be allocated in accordance with the actual plant utilization and the dierent local content calculation options. Due to the resulting non-linearity of the mixed-integer pro-gram, a solution algorithm based on Benders decomposition is presented. In addition, we introduce multiple Benders cuts to improve the eciency and applicability to real-world planning problems. Compared to piecewise linearization approaches, the run-time can be improved signicantly. In a numerical study, the impact of local content requirements on the strategic network design is shown and the dierent NAFTA options to calculate the local content for automotive goods are compared with each other. Furthermore, compu-tational experiments are performed to evaluate the applicability and eciency of Benders decomposition.

Keywords: local content, network design, Benders decomposition, non-linear mixed-integer program

1. Introduction

Automotive original equipment manufacturers (OEM) are increasingly confronted with complex tari trade barriers, in particular local content (LC) requirements of Free Trade Agreements (FTA), like the North American Free Trade Agreement (NAFTA), the Mercado Comum do Sul (MERCOSUL) or the Association of Southeast Asian Nations (ASEAN). The number of listed FTA by the World Trade Organization (WTO) has skyrocketed in the last decades. By February 2014, 583 notications of FTAs had been received by the WTO. Of these, 377 were in force in 2014 [1]. To promote trade and to boost the economy

Preprint submitted to Omega June 11, 2016

in the free-trade area, duties between free-trade members are removed under the condition of certain LC requirements. Along with rules of origin (ROO), LC requirements determine the origin of goods manufactured out of raw materials and parts that criss-cross the globe. If products qualify as originating from the free trade area, the OEMs are allowed to ship these products between the participating countries without any duty payments. As the automotive industry traditionally is a highly protected one, duty rates are very high, e.g. 25 percent of the FOB-value (free on board) for trucks imported to the US. Given the expanding global production footprint as a result of rising international trading activities, 'tari engineering' gets more important within global strategic network design.

Inspired by real-world planning problems of a global OEM, this paper presents a strate-gic network design model that takes the special NAFTA LC rules for automotive goods into account. With the participating countries Canada, Mexico and the US, the NAFTA builds one of the largest free trade areas. In the territory of the NAFTA, the OEM runs several production plants, where dierent vehicle types are produced. As the main production is located in Mexico, but most of the vehicles are sold in the US, duties for cross-border deliveries represent a high nancial risk exposure. Hence, it is crucial to include the LC requirements in decisions on investment and product exibility at the production plants, such as production and sourcing strategies. Together with the dierent ways of calculating the NAFTA LC for automotive goods - the so-called election to average (EA) options - these decisions can be levers to inuence the LC.

The aim of this paper is to study how the global OEM can leverage its sourcing and production network using NAFTA LC requirements to lower overall material and produc-tion costs. Moreover, based on the work of Stephan [2], our main contribution is to provide an improved solution framework to assist decision makers in the design of their real-world supply chains. We rst deploy a reformulation of the presented non-linear strategic net-work design model under NAFTA LC requirements for the dierent EA-options introduced by Stephan [2]. Secondly, motivated by the resulting problem structure, we use Benders decomposition (BD) to decompose our problem into two sub-problems that can be solved with well-known solution algorithms. To facilitate the solution of real-world instances, even for multiple periods, we present two dierent versions of the BD-cut generation. Thirdly, we provide insights on the EA-choice and show the overall eciency of BD compared with linearization approaches using secants approximation in our numerical study.

The remainder is organized as follows. In Section 2, we provide an overview on research focusing on LC requirements in the context of supply chain design. The mathematical model and the underlying assumptions, including the basic NAFTA LC rules for automotive goods, are presented in Section 3. Section 4 describes the presented solution algorithm and the application of BD. Numerical results are discussed in Section 5. Finally, Section 6 summarizes our work and future research directions are suggested.

2. Literature review

In the context of strategic network design, many dierent global factors have been studied over the last decades. For a detailed literature review on strategic network design

2

problems, see Melo et al. [3]. Lu and Van Mieghem [4] and Wang et al. [5] provide managerial insights on the impact of trade barriers, including LC requirements, on global production and sourcing strategies by analyzing and comparing dierent strategies of a rm. Motivated by a multinational automotive OEM, Lu and Van Mieghem [4] analyze oshoring decisions from a network capacity investment perspective and show under which circumstances commonality is the preferred strategy. Wang et al. [5] investigate three often deployed sourcing strategies of multinational rms and point out how industry and country characteristics inuence the rm's strategy preference.

Arntzen et al. [6] developed a Global Supply Chain Model (GSCM) for the Digital Equipment Corporation. With the GSCM, dierent supply chain congurations can be analyzed subject to two dierent objectives: minimization of total costs and minimization of weighted cumulative production and distribution times. As global aspects, duties and duty drawbacks, such as LC requirements are modeled. Furthermore, oset requirements to ensure that a local value added in a nation accounts at least for some minimum fraction of the value sold there are included.

Munson and Rosenblatt [7] explicitly consider global sourcing decisions under LC re-quirements. They rst investigate a single-plant model, where the suppliers are selected such that LC requirements are met. Further, they extend the model to a general multiple plant mixed-integer program. To minimize total costs, the number of production sites and their location is optimized. Furthermore, they determine the optimal set of suppliers and optimal allocation of purchased quantities for a particular market. They investigate the nancial impact of LC rules on a corporation and the negative impact for a local industry if the required LC is set too high by the government.

The special rules created by the NAFTA for non-automotive products have been incor-porated by Wilhelm et al. [8]. They consider decisions on total revenues, costs, taxes and material ows through the whole supply chain and take transfer prices and LC require-ments into account. To prevent penalty duties charged for cross-border deliveries within the territory of the NAFTA, upper bounds on the value of non-originating materials are set.

Kouvelis et al. [9] study the design of global facility networks and propose a linear mixed-integer program that takes various governmental policies to stimulate production investments, like subsidies, trade taris or taxes, into account. The basic model maximizes the discounted after-tax prot for multiple periods. To get a better understanding of the global characteristics, they analyze dierent special cases. Hence, the impact of nancing and tax incentives such as regional trading blocks and LC requirements on the supply chain design is investigated. The calculation of the LC is based on deterministic transfer prices so that the proposed model formulation is linear. The dierence to previous models lies in the optional LC fulllment and the consideration of penalties for LC requirement failures. The results show that a reasonable increase in LC-requirements within a country or a free-trade area forces rms to move production sites to these regions and that the opposite eect may occur if the LC-requirement is very low. Therefore, the authors recommend an early integration of LC rules in the strategic network design.

Motivated by the Japan-Singapore Economic Partnership Agreement (JSEPA), Li et

3

al. [10] develop a material sourcing model. This extends Munson and Rosenblatt [7] by the following three aspects: First, they take into account an obligatory LC fulllment for each unit of a product. Second, based on the country of shipment, the related dif-ferent transportation costs are considered and third, solution methods based on dynamic programming and column generation techniques for realistic problem sizes are presented. The LC is calculated based on FOB (free on board) prices for each produced unit of a product. They assume a single-product manufacturer who requires multiple components sourced from local or global suppliers. The objective is to determine the optimal sourcing strategies in order to minimize total costs.

Guo et al. [11] propose a multi-stage production sourcing problem with the opportunity to exploit duty liberations together with LC requirements of FTAs. The model supports sourcing and plant location decisions for a single product. The LC is calculated as the cumulative value added in a country for all stages of production. This simplied assumption is valid for many FTAs, but not for automotive goods in the NAFTA. To solve large real-world applications, they present a solution algorithm that embeds a variant of very large-scale neighborhood local search into a simulated annealing framework.

A strategic network design model under consideration of the special NAFTA LC-rules for automotive goods is presented by Stephan [2]. Following a general introduction of LC requirements, the NAFTA LC-rules with dierent EA-options for automotive goods are explained. Afterwards, the LC-rules are integrated into a strategic network design model considering production, sourcing, exibility and investment decisions. If the LC require-ments are not fullled, the resulting duty payments are calculated simultaneously. For the calculation of the LC and the duty payments, the explicit allocation of the xed costs as determined by the plant utilization is required. To model the xed cost allocation, a non-linear constraint and a continuous decision variable representing the xed costs of a plant allocated to the production volume of a certain product shipped to a market are introduced. The resulting non-linear problem is linearized based on piecewise linear func-tions that interpolate the included non-linear constraint by the usage of SoS2-variables. To countervail and lower the high solution times of the linearization approach, Stephan [2] presents a solution approach based on secants and tangents approximation. In the provided four numerical examples, the key impact of the NAFTA LC regulations and the design of automotive production networks is illustrated. The examples are based on one data-set and a basic production network with a maximum of three production plants, three markets and three products for one period. Based on this work, our contribution is to provide an ecient solution algorithm, suciently fast to optimize larger production networks even with multiple-periods within reasonable computation time. Therefor, we introduce a refor-mulation of the presented problem and suggest BD for solving the problems. Furthermore, we present an extensive numerical study for various data-sets and production networks to study the impact of the dierent EA-options provided by the NAFTA.

In context of global automotive production network design, two recent publications by Mariel and Minner [12] and Haentsch and Huchzermeier [13] study the impact of duty drawbacks. Similarly as LC-requirements, also duty drawbacks may have a signicant impact on the global production footprint. In this case, duties are refunded, if the imported

4

materials are used for export manufacturing. Mariel and Minner [12] present a model that simultaneously considers strategic capacity adjustments and duties and duty drawbacks for multi-stage, multi-product production processes. Similarly, Haentsch and Huchzermeier[13] present a case study where the impact of possible future duty rate changes or FTA, like between the European Union and the United States, is investigated.

3. Mathematical formulation

A multinational automotive OEM operates several production plants j in the territory of the NAFTA, where various products p, e.g. heavy-duty or medium-duty trucks are produced and delivered to the dierent markets m. The plants and markets are located in dierent nations n. Jn denotes the set of plants per nation and Mj the set of export mar-kets per plant. The manufacturer sources the components c required for the production ofthe vehicles from multiple suppliers, e.g. in-house production plants or external suppliers. They are located either globally or locally within the NAFTA area. The multiple sourcing opportunities for the components of a vehicle are expressed by the dierent product con-gurations v (v 2 Vp) per product. To explain, for the production of a heavy-duty (HD) truck, two dierent engines may be used: i) an engine with a global supply base or ii) an engine with a local supply base. The result of which is that two product congurations must be considered. Given the strategic focus of the underlying problem, we introduce Ttime periods. Table 1 summarizes the used notation.

Sets

N set of NAFTA nations, n 2 NJ ; Jn set of plants j 2 J and plants per nation n, j 2 JnM; Mj set of markets m 2 M, and export markets of plant j, m 2 MjP set of products, p 2 PC set of components, c 2 CVp set of possible product congurations of product p, v 2 VpT set of periods, t 2 TParameters

Cjtfix xed costs of plant

j in period

t, ($)

Cinv

initial costs to allocate productp to plant j, ($)

jp

Cs

unit sourcing costs of componentc at plant j, ($/unit)

jc

Ca

unit production cost of productp at plant j, ($/unit)

jp

Cd

unit transportation costs for shipping productp from plant j to

jpm

market m, ($/unit)L

jc regional value content percentage of component c at plant jbased on Cs

jc, (%/unit)B

vc bill of material coecient, amount of component c required forproducing one unit of product conguration v, (units)

Dmpt demand for product p at market m in period t, (units)

Kj per period production capacity of plant j, (units)

Rt discount factor of period t

5

Tjmp duty rate for shipping the non-originating product p from plant

j to market m, (%/unit)required regional value content percentage, (%)

Pjtfix unit penalty costs for not allocated xed costs of plant

j in

period t, ($/unit)Pmpt

d unit penalty costs for unfullled demand of product p

at market

m in period t, ($/unit)M suciently large number

Table 1: Summary of sets and parameters

Based on the forecasted demand Dmpt and provided capacity Kj at the plants, a cost op-timal production and distribution program for the planning horizon should be developed. The production-distribution plan includes the component sourcing and the product distribution strat-egy. Moreover, the trade-o between the LC fulllment, along with duty-payments, and the minimization of the total costs is investigated. The product exibility provision at the plants may be used as a lever to inuence the LC fulllment. Note that usually the product-to-plantallocations are associated with high initial costs Cinv

jp to install the required production resourcesfix

at the plants. Furthermore, the plants are characterized by xed costs Cjt that may providea signicant contribution to the LC fulllment. To increase the readability of the mathematicalmodel, some cost aggregation parameters for the calculation of the LC and the total costs are

XC

jmvp =

BvcCjcs + Cjp

a + Cjpmd 8j 2 J ; m 2 M; p 2 P; v 2 Vp (1)

c2C

X

8j 2 J ; p 2 P; v 2 Vp (2)Cjvpor = BvcLjcCjc

s + Cjpa

c2C

X

Bvc(1 Ljc)Cjcs 8j 2 J ; p 2 P; v 2 Vp (3)Cjvp

nor =c2C

Cjmvp summarizes the total variable cost per produced unit of product p in product congura-tion v at plant j shipped to market m, including the variable material costs

P 2 BvcC

s

c C jc, variableproduction costs Ca Cd B

jp and transportation costs jpm. vc are the bill of material coecients, rep-

v. Given the multiple sourcing options, the LC Ljc of the components varies depending on thesourcing region. Generally, globally sourced components have a lower LC and lower sourcing costs, whereas local components are characterized by a high LC and higher sourcing costs. Cjvp

or

determines the originating value per produced unit of product p in product conguration v at plant j. In addition to the variable production costs, the gathered local value added by sourcedcomponents accounts for the originating value per produced unit. The variable non-originating value

C

jvpnor

represents the value added obtained outside the territory of the NAFTA per producedunit of product p in product conguration v at plant j.

3.1. LC requirements of the NAFTAThe calculation of the LC is based on the very detailed LC regulations for the automotive

industry provided by the NAFTA. To qualify as originating, automobiles have to fulll a certain

6

value added LC. According to the NAFTA legislation [14], the LC for automotive goods is dened as Regional Value Content (RVC) and has to be calculated based on the Net-Cost method (Article 403.1 NAFTA). In the following, the terms LC and RVC are used as synonyms. Equation (4) determines the RVC based on the total net costs (NC) for the entire production process (Article 402.3 NAFTA). Expenses for sales promotion (including marketing and after-sales services), roy-alties, shipping, packing costs, and non-allowable interest costs are not allowed to be considered in the NC. The value of non-originating materials (VNM) is the free-on-board transaction value of the components used for the production of a nal product not obtained in the territory of the NAFTA.

RV C =N C V N M

100%N C

Depending on the vehicle type, the required RVC ( ) varies. For light duty vehicles (e.g. passenger cars), the threshold is 62.5 %, whereas for heavy duty vehicles (e.g. heavy duty trucks) it is 60% (Article 403.5 NAFTA). As opposed for other FTAs where the LC is calculated based on one unit of the product, the NAFTA provides special cost aggregation regulations for the calculation of the RVC for automotive goods. The manufacturer may decide to determine the RVC for

"the same class of motor vehicles produced in the same plant in the territory of a Party;" (per plant)

"the same model line of motor vehicles in the same class of vehicles produced in the same plant in the territory of a Party;" (per plant and per product)

"the same model line of motor vehicles produced in the territory of a Party;" (per nation and per product)(Article 403.3 NAFTA).

In this context, a class of motor vehicles is dened based on the tari nomenclature (e.g. cars and trucks belong to dierent classes), whereas a model line refers to a group of motor vehicles having the same platform or model name. In the following, the terms model line and product are used as synonyms. Thus, a global OEM has the opportunity to choose between the above described three categories. Furthermore, the manufacturer may decide to average over a scal year for the total production volume or only for the production volume exported to another NAFTA-member (Article 403.4 NAFTA). Through the combination of the dierent categories, the manufacturer has the choice to select among the following six EA-options: per plant and total production (EA1), per plant and NAFTA export volume (EA2), per plant and per product and total production (EA3), per plant and per product and NAFTA export volume (EA4), per nation and per product and total production (EA5), per nation and per product and NAFTA export volumes (EA6).

3.2. Basic model for EA1

The strategic network design problem under NAFTA LC requirements is formulated as a non-linear mixed-integer program for multiple periods. Alternatively, t may also be interpretedas a scenario index for analyzing dierent parameter settings for a single-period. Hence, the impact of uncertain demand on the dierent EA-methods may be addressed. We rst present the mathematical model for EA-method 1, which is referred to as basic model. Moreover, the required changes for other EA-methods are provided in Section 3.3.

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From the strategic perspective, the OEM has to decide which products to allocate to the plants x

jp and which

plants to operate in period t,

u

jt. Hence, based on these strategic settings, the optimal production and distribution

program z

jmtv has to be determined. Moreover, the binary decision variable yjt determines if the required RVC is fullled or not. If the LC-requirement is not satised, duty payments d jmpt calculated based on the given duty rates Tjmp are determined.

For the exact LC calculation the xed costs of a plant have to be considered. Depending on the EA-option and the actual production program of the plant, the total xed costs have to be divided according to the considered EA-option. Furthermore, to determine the exact duty value, the proportional xed costs of the export volume have to be determined. Hence, we introduce

the continuous decision variable c

jt dened as the xed cost allocated to one produced unit. Through this procedure we can ensure the simultaneous exact calculation of the LC fulllment and possible resulting duty payments. This introduction of c jt and the resulting changes in the objective function and constraints represents the main dierence compared to the model proposed by Stephan [2].The considered decision variables are summarized in Table 2.

Decision variables

ujt 2 f0; 1g plant opening; if set to 1, plant j is open in period txjp 2 f0; 1g allocation of products to plants; if set to 1, product p can be

produced at plant j, else notyjt 2 f0; 1g RVC fulllment; if set to 1, plant j does not fulll the required

RVC in period tz

jmtv 0

production quantity of product conguration v in plant jshipped to market m in period t, (units)

djmpt

0total duty costs caused by the production of product p in plantj shipped to market m in period t, ($)

cjt 0 allocated xed costs per produced unit at plant j in period t,($/unit)

Table 2: Summary of decision variables

Following the above notations, the mathematical model can be formulated as follows:

min xjpCjpinv + Rt

X X X

2djmpt +

X

zjmtv

(cjt + Cjmvp)

3(5)

X X X 4 5

X X

j2J p2P t2T j2J p2P m2M v2Vp

8m 2 M; p 2 P; t 2 Ts:t:z

jmtv D

mpt (6)j2J v2Vp

X X X8j 2 J ; t 2 T

zjmtv

Kju

jt (7)m2M p2P v2Vp

X X zjmtv

Kjx

jp 8j 2 J ; p 2 P; t 2 T (8)v2Vp m2M

zjmtv = Cjtfixujt 8j 2 J ; t 2 T (9)

cjtX X X

m2M p2P v2Vp

8

X X X X X XCjvp

or + c

jt z

jmtv Cjvpnor + Cjvp

or + c

jt z

jmtv y

jtM

m2M p2P v2Vp m2M p2P v2Vp

X8j 2 J ; t 2 T (10)

Cjvpor + Cjvp

nor + cjtd

jmpt T

jmp zjmtv M (1 yjt)v2Vp

8j 2 J ; m 2 M; p 2 P; t 2 T (11)z

jmtv 0

8j 2 J ; m 2 M; p 2 P; v 2 Vp; t 2 T (12)c

jt; d

jmpt 0

8j 2 J ; m 2 M; p 2 P; t 2 T (13)

xjp; ujt; yjt 2 f0; 1g 8j 2 J ; p 2 P; t 2 T (14)

The objective function (5) minimizes the costs for product-to-plant allocations and the to-tal discounted operating costs. Total operating costs comprise the duty payments if the RVC-requirements are not fullled and the material, production and transportation costs. Constraints(6) ensure that demand is satised. Constraints (7) and (8) model the available production ca-pacities at the plants and the product to plant allocation xjp. If plant j is producing in period t, the binary decision variable ujt equals 1 and constraint (9) determines the xed cost allocation to the production volume.

In the basic model, the calculation of the RVC is based on the total production quantity for the same class of vehicles produced in the same plant (EA1). Constraints (10) determine the RVC of the plants according to EA1. If the calculated RVC is smaller than the required threshold( ), the binary decision variables

y

jt are set to 1 and duties have to be paid for all cross-border deliveries. Hence, the resulting duty payments are calculated according to constraints (11) as a fraction of the customs value, which is based on the total production costs of the nal product. Finally, constraints (12)-(14) dene non-negativity and binary conditions.

By reformulating and restructuring constraints (10), the suciently large number M can bereplaced to provide a proper formulation of the LC-fulllment constraints:

X X X X X X(1 ) Cjvp

or + cjt zjmtv ( yjt) Cjvpnorzjmtv 8j 2 J ; t 2 T (15)

m2M p2P v2Vp m2M p2P v2Vp

3.3. Election to average methods EA2-6In case of a dierent EA-method, the following redenitions of variables yjt, along with

mod-ications to constraints (15) and (11), are necessary. For EA2, the averaging per plant and NAFTA export volume, the summations in the RVC determination are restricted only to produc-tion volumes for export markets located in other NAFTA-member states. Hence, constraint (15) is modied as follows:

X X X X X X(1 ) Cjvp

or + cjt zjmtv ( yjt) Cjvpnorzjmtv8j 2 J ; t 2 T (16)

m2Mj p2P v2Vp m2Mj p2P v2Vp

With EA3 and EA4, the manufacturer decides to average per plant, product and total produc-tion (EA3) or NAFTA export volume (EA4), respectively. As the RVC fulllment also depends

9

on the product, the related decision variables on the RVC-fulllment ( yjt) require an additional index

p

(y

jpt). Constraints (15) are modied for EA3 (17) and EA4 (18), respectively, as follows:X X

Cjvpor

X X

(1 )+

c

jt zjmtv ( yjpt) Cjvpnorzjmtv8j 2 J ; p 2 P; t 2 T (17)

m2M v2Vp m2M v2Vp

X X X X(1 ) Cjvp

or + cjt zjmtv ( yjpt) Cjvpnorzjmtv8j 2 J ; p 2 P; t 2 T

m2Mj v2Vp m2Mj v2Vp

(18)

For every product p manufactured at plant j in period t, the LC-fulllment is determined. Inconstraints (17), the determination of the originating and non-originating costs are based on the total production volume of the dierent product congurations v of product p delivered to allmarkets. Conversely, the summations in constraints (18) are restricted to NAFTA export markets of plant j.

The duty constraints (11) for EA3 and EA4 change to:d

jmpt T

jmp Cjvpor + Cjvp

nor + cjt zjmtv M (1 yjpt) 8j 2 J ; m 2 M; p 2 P; t 2 T (19)v2Vp

XThe opportunity to average over the same product produced at dierent plants located in one member-state of the NAFTA is provided by options EA5 (total production) and EA6 (NAFTA export volume). Thus, for EA5 and EA6 the RVC fulllment yjt requires dierent indices n and

p (y

npt). Constraints (15) are modied for EA5 (20) and EA6 (21) as follows:X X X

Cjvpor

X X X

(1 ) + cjt zjmtv ( ynpt) Cjvpnorzjmtv

j2Jn m2M v2Vp j2Jn m2M v2Vp

8p 2 P; n 2 N ; t 2 T (20)X X X X X X

(1 ) Cjvpor + cjt zjmtv ( ynpt) Cjvp

norzjmtvj2Jn m2Mj v2Vp j2Jn m2Mj v2Vp

8p 2 P; n 2 N ; t 2 T (21)

Constraints (20) and (21) determine the LC-fulllment for product p produced in nation n in period t. Hence, the originating value and non-originating costs are averaged over all product congurations v of product p delivered to all markets m produced at all plants j located in nation n. For EA6, just production volumes exported to another NAFTA-nation are included in (21).

For both of these EA-methods, the duty calculation depends on whether the RVC is met for product p in nation n in period t or not.

XCjvp

ordjmpt

Tjmp + Cjvp

nor + cjt zjmtv M (1 ynpt)v2Vp

8n 2 N ; j 2 Jn; m 2 M; p 2 P; t 2 T (22)

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1. Initialization: k = 0; y0 = initial values of the y-variables (any feasible values for y) lbd = 1; ubd = 1; = convergence tolerance2. Solve Primal P (yk); obtaining an optimal solution xk

and optimal dual variables k; if Z(P (yk)) < ubd, then

update ubd3. If xk and yk are feasible solutions to the original problem (5)-(14), then update x

and y

4. Create M aster(k) by adding a new BD cut to the M aster(k 1)5. Solve the M aster(k); obtaining an optimal solution kb and y-variables yk; update lbd = kb

6. Evaluate the solutions: if abs(ubd lbd)=abs(lbd) < , then stop and x and y is the best found solution to the original problem (5)-(14); otherwise replace k by k + 1 and go to step 2.

Table 3: Algorithm outline

4. Solution by Benders decomposition

The basic idea of BD [15] is to separate and iterate a given problem into two more easily solvable subproblems, namely Primal problem and Master problem. To decompose the original problem, the decision variables are divided into complicating variables y and non-complicatingvariables x. By xing one type of these variables, the remaining optimization problem can besolved easily by using standard optimization solvers. Through iterating the Primal and Master problems in each iteration k, a Benders-cut is added to the Master. In case of minimization(maximization) the Master provides a lower (upper) and the Primal an upper (lower) bound to the optimal solution. The algorithm terminates if the dierence between upper and lower bound is less or equal to a small number. Table 3 presents a detailed algorithm outline.

Starting with the pioneering paper of Georion and Graves [16], BD has been successfully applied to various network design and facility location problems. The application of BD to xed charge network design problems is reviewed by Costa [17]. Dogan and Goetschalckx [18] and Cordeau et al. [19] investigate the integrated strategic supply chain design and the determination of a tactical production-distribution program. Dierent variants of the hub-and-spoke network design problems are solved by BD [Camargo et al. [20]-[22]]. For automotive applications, BD is used by Bihlmaier et al. [23] to solve the strategic and tactical production planning problem under uncertainty. Bidhandi et al. [24] and Uester et al. [25] investigate closed-loop supply chain network design problems. To solve the distribution network design problem, Uester and Argahari[25] and Tang et al. [26] present improved BD algorithms. For nonlinear problems, BD is used to solve a supply chain reconguration and supplier selection problem by Osman and Demirli [27] or a vendor selection problem under capacity constraints by Keskin et al. [28]. Furthermore, various improvement and acceleration techniques for BD have been developed.

As our problem includes bilinear terms in the objective function (5) and the constraints (9),

(11) and (15) due to the multiplication of the continuous decision variables z jmtv and cjt, we use BD as a heuristic solution approach. We separate the bilinear terms from the cost allocation by assigning one class to the Master problem and the other one to the Primal slave problem. As a result, we only have to solve linear problems. However, the generated Benders cuts are not valid inequalities for the problem, because the corresponding dual rays are only valid for the particular Master problem solution at the time of their creation. Notably, this implies that the procedure does not yield valid lower bounds on the optimal solution value. Sahinidis and Grossmann [29] and

11

Bagajewicz and Manousiouthakis [30] discuss dierent implementations of BD and demonstrate that a straightforward implementation of BD to non-convex problems, especially including bilinear terms, does not always identify the global optimum. Depending on the considered case, it may converge to a local optimum or, in the worst case, the algorithm may not converge. Alternatively, one might follow the proposal by Li et al. [31], however, again at the expense of solvability problems. They present a novel decomposition method based on BD to obtain -optimal solutions

for non-convex optimization problems. The BD algorithm is extended by an additional convex lower bounding problem in order to provide valid lower bounds to the original problem.

According to the basic idea of BD, we dene the binary decision variables for the product to plant allocation xjp, the plant opening ujt, and the RVC-fulllment yjt, respectively yjpt or ynpt for other EA-methods as complicating variables y. Corresponding to common decomposition

schemes the allocated unit xed costs

c

jt

are dened as continuous complicating variables. The remaining decision variables zjmtv and djmpt represent the non-complicating variables x.

4.1. Model improvements

To support the convergence of BD, we introduce some improvements instead of a straightfor-ward implementation. Saharidis and Ierapetritou [32] show that producing more optimality cuts than feasibility cuts leads to a faster convergence of BD and therefore improves the BD algorithm. In some cases, there is not enough network capacity provided to fulll the demand by xing the binary decision variables. Therefore, to avoid the generation of feasibility cuts, we introduce the continuous decision variable

dd

jmtv for unfullled demand and modify constraints (6) as follows:X X

(zjmtv + ddjmtv) Dmpt 8m 2 M; p 2 P; t 2 T (23)j2J v2Vp

Moreover, to obtain only non-negative dual variables, we change the equality constraints for the xed cost allocation (9) by introducing the continuous decision variable dm jt for not allocated xed costs to the following inequality constraints:

cjt zjmtv + dmjt Cjt

fixujt 8j 2 J ; t 2 T (24)

X X Xm2M p2P v2Vp

In order to ensure that total demand is satised and total xed costs are allocated, two penalty terms are introduced to the objective function:

X X X X X X 4 Xmin xjpCjp

inv + Rt2djmpt +

zjmtv

(cjt+

j2J p2P t2T j2J p2P m2M v2Vp

3Cjmvp) + dmjtPjtfix + dd

jmtvP

mptd (25)

X 5v2Vp

Thereby, the constant for penalizing unfullled demand Pmptd is determined as follows:

h i

d max

Cjmvp +

max Cfix D

8m 2 M; p 2 P; t 2 T

(26)

Pmpt

=j2J ;v2Vp j2J jt mpt

12

The constant for penalizing not allocated xed costs Pjtfix can be set equal to the plant xed costs

Cjtfix.

Given the minimization problem, both penalty terms will be zero in the optimal solution.As facility location problems are known as highly degenerated ones, we add the following

con-straints (27) to the Primal. According to Wentges [33], these constraints support BD convergence through the generation of "better" dual variables:

zjmtv Dmptxjp 8j 2 J ; m 2 M; p 2 P; v 2 Vp; t 2 T (27)

In the following, the Primal(k), Dual(k) and Master(k) are presented in detail for the basic model. For other EA-methods, the RVC related decision variables and constraints have to be modied as presented in Section 3.3.

4.2. PrimalThe binary decision variables uk , xk , and yk

jt jp jt provide the information whether or not a plant

the allocated xed cost per b b b cjt. Based on these decisions, an optimal

is open, the degree of product exibility, and the RVC-fulllment in iteration k. Furthermore,k

produced unit is set to

babove described BD algorithm and model

production program zjmtv and resulting duty payments djmpt are determined. Thus, following theimprovements, the presented strategic network design

model for EA1 reduces to the following linear sub-problem of iteration k:min

j2J p2P xjpkCjp

inv + t2TRt

j2J p2P m2M2

djmpt + v2Vp zjmtv cjtk+

X X b X X X X4 X b3

C jmv p) + dm jtP jtfix + ddjmtvPmptd (28)

X 5

X X

v2Vp

8m 2 M; p 2 P; t 2 Ts:t:(z

jmtv +

dd

jmtv)

Dmpt (29)

j2J v2Vp

X X X8j 2 J ; t 2 T (30)zjmtvKjujt

k

zjmtvKjxjpk

b 8j 2 J ; p 2 P; t 2 T (31)

m2M p2P v2Vp

bk X X fix k (32)

bv2Vp m2M

bX X X or k k nor

8j 2 J ; t 2 Tcjt

zjmtv

+ dm

jt C

jtu

jt

m2M p2P v2Vp

m2M p2P v2Vp n(1) C

jvp + cjty

jt Cjvp o

zjmtv 0 8j 2 J ; t 2 T (33)X X X or nor b k b k

djmpt

Tjmp v2Vp

Cjvp

+ C

jvp + cjt zjmtvM 1 yjt

X b b (34)

zjmtv

+ D

mptx

jpk 8j 2 J ; m 2 M; p 2 P; t 2 T

0 j ; m 2 M; p 2 P; v 2 Vp; t 2 T (35)

zjmtv; ddjmtv 0b

82 J

8j 2 J ; m 2 M; v 2 V; t 2 T (36)13

djmpt; dmjt 0 8j 2 J ; m 2 M; p 2 P; t 2 T (37)

4.3. DualGiven the dual variables

mpt, jt

, jpt

, jt

, jt

, jmpt

and jmtv, associated with constraints (29)-(35), the Dual(k)

of the Primal(k) can be written as follows:max j2J m2M p2P v2Vp t2T Dmpt mpt Kjujt

k jt Kjxjpk jpt + Cjt

fixujtk jt

X X X X Xb b b

M 1 yjtk

jmpt Dmptxjpk jmtv

1(38)

b b Aor nor jt k n jvp k jt jvp o

s:t: mpt jt jpt + ckjt + (1 ) Cor + cjt

k yk Cnor jt

T + C + c Rjmp

Cjvp jt jmpt jmtv t bcjt + Cjmvp bjvp b

bb

8j 2 J ; m 2 M; p 2 P; v 2 Vp; t 2 T (39)jmpt

Rt 8j 2 J ; m 2 M; p 2 P; t 2 T (40)

jt PjtfixRt 8j 2 J ; t 2 T (41)

mpt P

mptdR

t 8m 2 M; p 2 P; t 2 T (42)

mpt; jt

; jpt

; jt; jt

; jmpt

; jmtv

08j 2 J ; m 2 M; p 2 P; v 2 Vp; t 2 T (43)

4.4. MasterBased on the optimal solution of zk dk and the dual variables k , k , k k k

jmpt jmtv

jmtv,

b Master(k) of bour b b

be written as:

jmpt

b

mpt jt jpt, jt, jt,

b b

k and k associated with constraints (29)-(35), the Benders problem can

b bmin b 2

djmptk (cjt + Cjmvp)

3 (44)s:t: b xjpCjp

inv + Rt + zjmtvk

X X X X X X 4b X b 5j2J p2P

0t2T j2J p2P m2M v2Vp

mptk zjmtv

k Dmpt1

X X X b @X X

b Am2M p2P t2T j2J v2Vp

+ Kjujt1

X X

jtk

0

X X X

zjmtvk

b @ b A

j2J t2T m2M p2P v2Vp

+ Kjxjp1

jptk

0 zjmtvk

X X X b @ X X b Aj2J p2P t2T v2Vp m2M

Cjtfixujt

1

X X

jtk 0

cjt

X X X

zjmtvk

b @ b A

j2J t2T m2M p2P v2Vp

14

X Xjtk

0 (1 )Cjvpor + cjt ( yjt) Cjvp

nor zjmtvk 1

b @ X X X

b Aj2J t2T m2M p2P v2Vp

zjmtvk + M (1 yjt)

3jmpt

k 2djmpt

k Tjmp

X

Cjvpor+ Cjvp

nor + cjt

X X X X b 4b

k b 5j2J m2M p2P t2T v2Vp

8k 2 Kj2J m2M p2P v2Vp t2T jmtv zjmtv + Dmptxjp (45)

X X X X Xb bCfix

cjt

jt

8j 2 J ; t 2 T (46)u

jtminP rodjt

cjt

Cjtfix

8j 2 J ; t 2 T (47)u

jtmaxP rodjt

b; cjt 0 8j 2 J ; t 2 T (48)

xjp; yjt; ujt 2 f0; 1g 8j 2 J ; p 2 P; t 2 T (49)

As the Master(k) includes binary and continuous decision variables, it is a mixed-integer linear program. To accelerate the convergence, constraints (46) and (47) are introduced. These two constraints provide bounds on the allocated unit xed cost c jt if plant j is open in period t (u jt = 1), otherwise the bounds are zero, implying that plant j is closed in period t (u jt = 0) and cjt = 0). According to the total demand and network capacity, minimum minP rodjt and

maximum production maxP rod

jt

volumes can be determined for every plant. Hence, ecient upper and lower bounds to the allocated unit xed cost cjt can be provided. In order to satisfy total market demand, minP rodjt is the minimum production volume plant j has to produce in period t. For this case, the upper bound (46) of c jt is dened based on minP rodjt. To avoid zerodivision problems, it is assumed that

minP rod

jt is at least one unit and the upper bound equals the plant related

xed cost

Cjtfix. If the total demand is lower than the capacity of plant

j in

period t, maxP rodjt equals the total demand. Otherwise maxP rodjt

is set to the plant capacity

Kj. Consequently, the lower bound (47) is identical to the allocated unit xed cost per capacity unit.

To get feasible starting solutions of the complicating y-variables, the strategic network design

problem is solved without considering the LC-relevant constraints. Based on the resulting product exibility and production program, initial allocated unit xed costs c jt and the initial RVC-

fulllment y

jt can be determined. As a stopping criterion, we use maximum computation time.4.5. Improved Benders cuts

Especially for larger problem instances with multiple periods, the straight forward implemen-tation of BD with the generation of one cut per iteration leads to a high solution time. Therefore, we introduce a multi-cut version to accelerate the BD algorithm. In each iteration, we simulta-neously add multiple cuts to the Master problem by solving more than one subproblem in each iteration. As the main computation time in the underlying problem is spent to solve the Master, the solution space of the Master can be further restricted and therefore the overall run-time can be reduced signicantly through multiple cuts. To apply this improvement, the subproblems have to be separable into independent problems. Given our special problem structure and based on the works of Dogan and Goetschalckx [18], Uester et al. [34] and Tang et al. [26], we present a

15

disaggregation of the primal BD-cuts due to the considered multiple time-periods. The coupling constraints between dierent time periods are imposed by the opening of a plant. As these con-straints only occur in the Master problem, each period can be solved separately in the Primal problem. As a result, in every iteration k multiple-cuts are added to the Master(k) simultaneously.The number of added cuts in every iteration is equal to the number of considered time-periods T .

min b

s:t: b xjpCjpinv + Rt

2djmpt

k +z

jmtvk

(c

jt+

C

jmvp)

3

X X X X X 4b

X b 5j2J p2P

0j2J p2P m2M v2Vp

mptk zjmtv

k Dmpt 1

X X b @X X b Am2M p2P j2J v2Vp

1

X

jtk 0

X X X

zjmtvk +

K

ju

jt

b @ b Aj2J m2M p2P v2Vp

+ Kjxjp1

jptk

0 zjmtvk

X X b @ X X b Aj2J p2P v2Vp m2M

Cjtfixujt

1

X

jtk 0

cjt

X X X

zjmtvk

b @ b Aj2J

jtk 0 m2M p2P v2Vp

1

X

(1 ) Cjvpor + cjt ( yjt) Cjvp

nor zjmtvk

b @ X X X

b Aj2J m2M p2P v2Vp

3jmptk 2

djmptk T

jmp Cjvpor + Cjvp

nor + cjt zjmtvk + M (1 yjt)

X X X b 4b

k X b 5j2J m2M p2P v2Vp

8k 2 K; t 2 Tj2J m2M p2P v2Vp jmtvzjmtv + D

mptxjp

X X X X

bb(46) to (49)

5. Numerical results

(50)

(51)

In this section, we provide insights on the dierent EA-methods and evaluate the modeling and solution approach regarding computational performance and applicability to real world planning problems. First, we compare the EA-methods as presented in Sections 3.2 and 3.3 with each other and analyze the impact of cost and network utilization changes. In the second part, the computa-tional eciency of the presented BD approach is compared to a brute-force secant approximation approach using SoS2-variables and dierent BD-cut strategies are tested for various network sizes. For reasons of condentiality and the strategic focus of the underlying optimization problem, we introduce an altered data set that is based on the actual facts of a global OEM.

The Benders decomposition framework and the brute-force approach are implemented in

16

Instance Plants Markets Products Total product congurationsS1 2 2 2 4S2 2 2 2 6S3 2 2 2 8M1 3 2 3 6M2 3 2 3 9M3 3 2 3 12L1 4 2 4 8L2 4 2 4 12L3 4 2 4 16

Table 4: Characteristics of the 9 dierent problem instances concerning network size and structure

XPRESS-MP 7.3. All numerical tests were performed on a PC with an Intel Core i7 proces-sor 3.4 GHz with 8.00 GB RAM running a 64-Bit version of Windows 7.

5.1. Test design

We created 144 dierent test instances with three main dimensions: (i) network size and structure, (ii) network utilization and (iii) xed cost ratio. The xed cost ratio is dened as the xed cost portion of the total average network operating costs (including sourcing, production and transportation costs). In doing so, we determined the total average network operating costs based on the average variable costs per unit multiplied with the total demand. Table 4 summarizes the characteristics of the test problems regarding (i) network size and structure. Depending on the considered number of plants, markets and products, the cases are classied as small (S1-S3), medium (M1-M3) and large (L1-L3). In S-instances, one plant in Mexico and one in the US, in M-instances two plants in Mexico and one in the US and in L-instances three plants in Mexico and one in the US are considered. For every network-size, two, three or four numbers of product congurations per product are taken into account. In all instances, the two main markets Mexico and the US are considered. The relation of total demand and total network capacity determines(ii) the network utilization of an instance. We distinguish low network utilization (70%, denoted by the sux "u") and high network utilization (90%, denoted by the sux "U") instances. To test the impact of the xed cost on the optimal EA-choice, we investigate dierent xed cost ratios. The various combinations of the provided xed and variable cost parameters result in networks where xed costs account between 15% and 45% of the total average network operating cost. The nine instances concerning the network size and structure, together with four network utilization scenarios and four dierent cost scenarios regarding the xed cost ratio, yields a total of 144 test cases. As planning horizon, we rst consider one period, since the LC calculation is required for one scal year of the OEM and then we introduce multiple periods in Section 5.3.2. The used numbers are illustrative gures that take practical settings into account.

To capture changes to the production footprint of the automotive OEM, two dierent demand and capacity inputs for Mexico and the US are created as shown in Table 5. In each case, 40% of the total demand per market are demanded of product one. The remaining demand is equally allocated to the other products per instance. Given the fact that not all Mexican facilities are of identical size, for M-instances the allocation of the total capacity of Mexico is assumed to be 60% for plant one and 40% for plant two. For L-instances, the capacity assignment is (50%;

17

total demand total capacityMexico US Mexico US

u1 70 70 100 100u2 70 140 200 100U1 90 90 100 100U2 90 180 200 100

Table 5: Network utilization scenarios

variable sourcing cost variable production costMexico US Mexico US

min max min max - -v p1 8.00 12.00 10.00 15.00 1.88 3.75

p2 5.33 8.00 6.67 10.00 1.25 2.50p3 3.56 5.33 4.44 6.67 2.17 1.67p4 2.37 3.56 2.96 4.44 0.56 1.11

V p1 16.00 24.00 20.00 30.00 4.50 9.00p2 10.67 16.00 13.33 20.00 3.00 6.00p3 7.11 10.67 8.89 13.33 5.20 4.00p4 4.74 7.11 5.93 8.89 1.33 2.67

Table 6: Variable sourcing cost and production cost scenarios

30%; 20%). We consider two dierent scenarios of the xed costs (denoted by "f" and "F"). To determine the xed costs of a plant, we introduce xed cost per capacity unit that are multiplied with the available capacity. For f-instances, the unit xed cost per available unit of capacity are two monetary units for Mexico and three for the US and for F-instances two for Mexico and four for the US.

To focus our evaluation on the dierent sourcing and production strategies resulting from the EA-methods, the investments for the product to plant allocation are assumed to be 500 for all plants and products. The variable sourcing cost and RVC values per product conguration v ofproduct p and plant j are generated in two steps. Again, two variable sourcing cost scenarios(denoted by "v" and "V") are determined. In Table 6, minimum and maximum sourcing costs per product are presented. Depending on the number of product congurations per product and the provided range of the sourcing costs, they are equally allocated to the considered product congurations. In the second step, the RVC values are determined uniformly in the interval [0; 0.9] with respect to the number of product congurations per product. Thus, for M3V the variable sourcing cost of the four product congurations per product one for Mexico are (16; 18.67; 21.34;24) with (0; 0.3; 0.6; 0.9) RVC. Hence, low costs with low RVC values are linked to represent product congurations with mainly globally sourced components and high costs with high RVC values for local product congurations.

Table 6 contains the variable production cost per produced unit of product p. Again, a costadvantage up to 50% is assumed for Mexico. As the transportation costs are excluded from the RVC calculation, they are assumed to be 0.8 for domestic and 1.5 for foreign shipments. The required RVC is set to = 60%, which is the required threshold for heavy duty vehicles [14]. Forall cross-border deliveries not compliant to the required RVC, it is assumed that duty payments

18

100%

M1U2vf M1U2Vf

21.2%

26.8% 32.3% 26.7% 28.4% 26.8% 26.8% 30.8%

38.9%

30.8% 30.7%

46.8%

78.8%

73.2% 67.7% 73.3% 71.6% 73.2% 73.2% 69.2%

61.1%

69.2% 69.3%

53.2%

0%

EA1 EA2 EA3 EA4 EA5 EA6 EA1 EA2 EA3 EA4 EA5 EA6

global sourcing local sourcing

Figure 1: Mix of global and local components for M1U2vf and M1U2Vf

amount to Tjmp = 25%.

5.2. Comparison of election to average methods

We use M1 to provide insights into the various EA methods. This class includes two production sites (MX 1 and MX 2) in Mexico and one (US 1) in the US. Thus, the impact of EA5 and EA6, where the averaging for production networks within one country, here Mexico, is applicable and can be investigated. For M1, the components of the three dierent trucks can be either sourced globally (0% RVC) or locally (90% RVC).

Figure 1 shows the split between globally and locally sourced components used for the pro-duction of the vehicles for every EA-method and illustrates the impact of the xed costs on the sourcing strategies by varying the the variable costs. In instance M1U2vf the xed cost account for 35%, whereas in M1U2Vf for 18% on average. For EA-methods, based on the total produc-tion volume (EA1, EA3 and EA5) except for EA1, the fraction of locally sourced components is higher compared to options where the calculation is based on the export volume. To explain these eects, the RVC-requirement is not fullled for EA1 at MX1 and all exports to the US are subject to duties. The benets of global sourcing are higher compared to the duty payments for US exports. For EA-methods based on the export volume (EA2, EA4 and EA6), the RVC calcu-lation is separated for home and export markets. In this case, home markets are not subject to RVC-requirements and can be totally supplied with the most cost-ecient product congurations, in our numerical set-up global ones. As a consequence, the fraction of locally sourced components is decreasing. The remaining local components are used to obtain the required RVC in order to qualify for duty-free cross-border deliveries.

In case of the xed cost ratio being lower (M1U2Vf) than (M1U2vf), local sourcing becomes more protable. Hence, according to Figure 1 for all EA-methods of instance M1U2Vf, the portion of locally sourced components increases. The reason is that in order to qualify for duty-free cross-border deliveries, the missing RVC obtained through the higher xed cost can be balanced through a higher portion of local components in M1U2Vf. Thus, we can observe that it might be more protable to increase the portion of locally sourced components, instead of paying duties for unfullled RVC-requirements.

Based on the total variable operating costs including sourcing, production and transportation costs, the impact of network utilization changes on the dierent EA-methods is illustrated. For M1U2vf and M1u2vf, Figure 2 shows the relative operating cost changes for the dierent EA-methods compared to EA1. Due to the eect that home markets are mainly supplied with global

19

4.00%

1.64%

2.00%

0.00%

2.00%‐1.72%‐

‐4.00% ‐2.89%M1U2vf

3.13%‐ 3.10%‐

6.00%‐ 5.11%‐M1u2vf

8.00%‐7.74%‐ 7.73%‐ 7.74%‐

10.00%‐EA1 EA2 EA3 EA4 EA5 EA6

Figure 2: Relative variable operating cost changes compared to EA1 for low and high utilization networks

EA1 EA2 EA3 EA4 EA5 EA6u1 v f 3,573 3,548 3,536 3,504 3,543 3,512

F 3,599 3,528 3,577 3,551 3,617 3,587V f 5,040 4,912 5,040 4,904 5,042 4,926

F 5,128 4,996 5,120 4,990 5,135 4,995u2 v f 4,337 4,282 4,415 4,347 4,385 4,276

F 4,483 4,355 4,418 4,363 4,415 4,373V f 6,426 6,322 6,429 6,282 6,430 6,290

F 6,551 6,362 6,523 6,410 6,517 6,399U1 v f 4,489 4,419 4,469 4,419 4,473 4,418

F 4,570 4,512 4,555 4,514 4,560 4,513V f 6,280 6,138 6,247 6,139 6,269 6,138

F 6,369 6,232 6,333 6,232 6,356 6,236U2 v f 5,526 5,455 5,507 5,454 5,545 5,454

F 5,615 5,552 5,600 5,541 5,636 5,552V f 8,100 7,934 8,065 7,933 8,143 7,933

F 8,187 8,030 8,155 8,031 8,233 8,030

Table 7: Average total costs over the network size per EA-method

product congurations, the cost saving potentials obtained via global sourcing strategies are higher for EA-methods based on the export volume. If the network utilization decreases to 70% (M1u2vf), the overall operating cost savings gathered through a change in the RVC-calculation increases to more than 7%. Instance M1u2vf shows that the under-utilization allows to take a higher advantage of the production network regarding RVC-calculation. The highest total cost savings can be earned through the application of EA4 for M1U2vf. As opposed to this, Figure 2 shows that for M1u2vf, the preferred EA-method is EA2 or EA6.

The results in Table 7 present the average objective over the network size for all network utilization scenarios and combinations of cost structures. EA-methods based on the export volume are more favorable as the OEM can capitalize more on the benets of global sourcing. For low network utilization scenarios, EA4 would be the preferred method, whereas for a 90% utilization, EA6 is better because the OEM can leverage more on the production network within Mexico.

5.3. Performance study

We rst compare the BD single-cut approach to a standard linearization approach using se-cants and SoS2 variables. For the single-period problem, we investigate the solution quality and computation time provided by the BD heuristics compared to an approximative solution approach

20

in Section 5.3.1. Secondly, the eectiveness and applicability of the two dierent BD versions are tested for problems with multiple periods. The detailed comparison results are reported in Sec-tion 5.3.2. Finally, in Section 5.3.3 we present the applicability of BD multiple-cuts to larger production networks.

5.3.1. Comparison of BD single-cuts to secants approximationTo determine the results by using secants approximation, we used the model formulation

as presented by Stephan [2]. Depending on the number of considered interpolation points, the approximation becomes more accurate. Therefore, we tested the secants approximation with 10, 100 and 1000 interpolation points and compared the results to BD single-cuts (BD-st). But due to the fact that L-test instances are hard to solve to optimality within a limited computation time of 10,000 seconds by using the brute-force linearization approach, the comparison is only conducted for S- and M-test problems.

Tables 8 and 9 present the results regarding computation time and solution quality. To evaluate the computation time performance, the average run time in seconds ( (t)) and thestandard deviation ( (t)) of the run time are reported. Only instances solved within the providedmaximum run time of 10,000 seconds are considered. The number of unsolved instances (# max t) is shown separately. The solution quality is measured by the relative average gap (av. relative gap (%) presented in Table 9. Based on the best found solution obtained by the dierent methods for each instance, the relative gap to this solution is determined. From the results, it is apparent that the average computation time is increasing with the network size and that EA-methods 3 and 5 are most dicult to solve, as the problems are getting larger and the number of non-linear terms and binary decision variables is increasing. According to this, the main drivers of the computation time and the relative gap to the best found solution are the network size and structure, whereas the average network operating costs have no signicant impact on the performance. The computation time for low network utilization scenarios is smaller than for high utilization networks. For the secants approximation, this fact leads to a higher number of unsolved instances within the maximum run time for high utilization networks. In contrast, the relative gap to the best found solution is higher for low utilization networks and gets smaller with the number of used interpolation points, which leads to an increase of the computation time.

Compared to the secants approximation, BD single-cuts provides quite constant computation times for the dierent EA-methods. Only thirteen out of 864 instances did not converge with the provided starting solution based on the optimal production program without considering the LC-requirements. As opposed to this, fty instances could not be solved within the given maximum computation time by the secants approximation. The maximum run time was 1,803 seconds for L3U2VF-EA3 with BD single-cuts. The relative gap to the best found solution shows that the deviations are very low for all categories. Figure 3 illustrates the relationship between computation time and solution quality for low and high utilization S-instances.

Furthermore, in Tables 8 and 9, the performance of BD single-cuts and secants approxima-tion with 10 interpolation points for L-instances is reported. The results show that, even for L-instances, the computation time of BD is very low. To summarize, the performance study illus-trates the capability of BD to provide good solutions and computation times if compared to the linearization approach, especially for more complex EA-methods and realistic instances of large size.

21

EA1 EA2 EA3 EA4 EA5 EA6u S (t)( (t)) BD-st 2.6 (1.7) 2.0 (1.2) 3.9 (1.8) 2.3 (1.2) 3.0 (1.5) 1.9 (1.0)

Sec 10 0.2 (0.1) 0.2 (0.2) 0.6 (0.3) 0.3 (0.1) 0.6 (0.2) 0.4 (0.2)Sec 100 0.3 (0.1) 0.4 (0.2) 0.9 (0.3) 0.9 (0.2) 0.8 (0.3) 1.1 (0.3)Sec 1000 1.5 (0.4) 1.6 (0.4) 5.3 (2.1) 4.9 (1.2) 5.2 (1.8) 5.3 (1.8)

iter BD-st 36 29 39 31 39 31# max. t BD-st - - - - - -

Sec 10 - - - - - -Sec 100 - - - - - -Sec 1000 - - - - - -

M (t)( (t)) BD-st 14.6 (6.9) 10.2 (3.0) 20.9 (10.5) 14.2 (5.3) 14.7 (5.6) 10.9 (2.9)Sec 10 0.3 (0.1) 0.4 (0.2) 1.7 (0.7) 1.4 (0.5) 1.7 (0.5) 1.3 (0.3)Sec 100 0.7 (0.2) 0.7 (0.1) 9.7 (6.4) 7.0 (4.2) 15.1 (17.2) 8.7 (7.6)Sec 1000 5.0 (1.4) 4.7 (1.1) 76.6 (53.0) 294.8 (402.8) 59.7 (35.6) 85.2 (72.4)

iter BD-st 77 62 88 69 80 66# max. t BD-st - - - - - 1

Sec 10 - - - - - -Sec 100 - - - 2 2 4Sec 1000 - - - - 1 1

L (t)( (t)) BD-st 96.0 (54.2) 55.2 (36.9) 179.8 (207.3) 85.3 (61.4) 102.0 (72.9) 48.9 (18.9)Sec 10 0.6 (0.1) 0.6 (0.1) 8.5 (5.5) 9.8 (14.8) 8.2 (3.0) 5.5 (2.0)

iter BD-st 202 150 272 183 211 144# max. t BD-st 4 - 1 1 - -

Sec 10 - - - - - -U S (t)( (t)) BD-st 0.4 (0.1) 0.4 (0.1) 1.7 (0.7) 0.4 (0.0) 1.2 (0.6) 0.3 (0.0)

Sec 10 0.2 (0.1) 0.3 (0.2) 0.4 (0.2) 0.2 (0.1) 0.4 (0.1) 0.3 (0.1)Sec 100 0.3 (0.1) 0.3 (0.2) 0.8 (0.2) 1.9 (1.9) 0.72 (0.2) 2.10 (2.3)Sec 1000 1.6 (0.4) 1.1 (0.2) 6.9 (3.7) 88.3 (253.0) 7.16 (3.8) 7.14 (3.8)

iter BD-st 19 17 30 20 30 20# max. t BD-st - - - - - -

Sec 10 - - - - - -Sec 100 - - - - - -Sec 1000 - - - - - -

M (t)( (t)) BD-st 16.4 (3.6) 12.8 (4.1) 23.0 (7.9) 18.3 (6.6) 18.4 (6.7) 12.1 (3.1)Sec 10 0.4 (0.1) 0.4 (0.2) 1.0 (0.2) 1.0 (0.1) 0.8 (0.2) 0.9 (0.2)Sec 100 0.6 (0.1) 0.6 (0.1) 16.5 (28.3) 9.6 (10.7) 12.7 (12.5) 12.3 (13.0)Sec 1000 4.2 (0.6) 3.8 (0.7) 127.1 (115.6) 166.4 (121.1) 117.6 (232.1) 404.7 (1,136.9)

iter BD-st 96 81 115 95 103 83# max. t BD-st 1 - - - - -

Sec 10 - - - - - -Sec 100 - - 2 6 7 5Sec 1000 - - - 1 12 4

L (t)( (t)) BD-st 285.2 (95.0) 232.8 (92.5) 889.3 (524.8) 494.4 (224.9) 867.3 (484.9) 287.3 (100.3)Sec 10 0.8 (0.1) 0.6 (0.1) 10.5 (27.2) 4.6 (1.6) 10.2 (6.9) 5.1 (2.2)

iter BD-st 412 359 681 532 679 410# max. t BD-st - 1 1 1 1 1

Sec 10 - - - - - -

Table 8: Computation time comparison results secants approximation vs. BD single-cuts

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EA1 EA2 EA3 EA4 EA5 EA6av. relative gap (%) u S BD-st 0.05 0.09 0.05 0.08 0.05 0.07

sec10 0.00 0.00 1.88 2.26 1.88 2.24sec100 0.00 0.01 4.39 5.30 4.39 5.29sec1000 0.00 0.01 0.26 0.19 0.26 2.96

M BD-st 0.04 0.11 0.00 0.05 0.01 0.04sec10 0.88 0.61 8.89 9.39 9.44 9.33sec100 0.88 0.61 6.13 5.80 5.39 4.99sec1000 0.88 0.61 2.31 2.44 0.73 1.78

L BD-st 1.31 1.31 0.02 0.01 0.01 0.10sec10 0.46 0.54 5.32 5.59 5.48 5.49

U S BD-st 0.00 0.02 0.07 0.02 0.07 0.02sec10 0.01 0.00 0.27 0.29 0.27 0.29sec100 0.03 0.02 0.05 0.04 0.05 0.04sec1000 0.02 0.02 0.01 0.00 0.00 2.36

M BD-st 0.04 0.05 0.04 0.05 0.07 0.05sec10 0.00 0.00 0.00 0.00 0.00 0.00sec100 0.00 0.00 3.63 2.12 2.68 3.67sec1000 0.00 0.00 0.24 0.03 0.07 1.16

L BD-st 0.13 0.06 0.02 0.04 0.05 0.06sec10 0.00 0.02 0.07 0.08 0.14 0.02

Table 9: Relative gap to the best found solution for all methods

5,300

5,100

4,900

4,700

4,500

4,300

4,100

u u U

5,300 1,000.0

100.0

o b j .

10.0

(sect.av

1.0

).

4,100 0.1EA1 EA2 EA3 EA4EA1 EA5 EA2 EA6 EA3 EA1 EA4EA2 EA5EA3 EA46 EA5 EA6

4,100 0.1EA1 EA2 EA3 EA4 EA5 EA6

BD obj. Sec 10 obj. Sec 100 obj. Sec 1,000 obj.

BD time Sec 10 time Sec 100 time Sec 1,000 time

Figure 3: Comparison of BD single-cuts with secants approximation for S-instances

23

5.3.2. BD multiple-cuts for multiple periodsTo balance the usually high investments associated with the product-to-plant allocation and the

overall total costs including possible duty payments, multiple-periods are included in many real-world problems. By extending the demand to ve time-periods, we investigate the eective-ness of multiple-cuts compared to single-cuts and show that our presented solution approach is able to solve realistic problem instances. For every instance, the single-period demand presented in Table 5 is multiplied by the following parameters [0:8; 0:9; 1:0; 1:1; 1:0] to determine the demandfor periods one to ve. The computation time limit is set to four hours.

Table 10 summarizes the results of the single- (BD-st) and multiple-cuts (BD-ml) compared to the secants approximation approach with 10 interpolation points (sec10). Although a comparison is limited to the S-instances, the results clearly demonstrate a signicant reduction in computation time by the BD multiple-cuts algorithm; in fact it provides a good solution quality. Hardly any of the M-instances or L-instances could be solved within the given time limit of four hours with BD single-cuts or the secants approximation. By using multiple-cuts, all S-instances converge within the provided time-limit, whereas by using single-cuts (secants approximation) 47 (46) out of 144 u- instances and 50 (45) out of 144 U-instances cannot be solved within the provided time-limit. The multi-cut procedure outperforms the single-cut version and the secants approximation, even for those EA-methods where all instances can be solved with single cuts. The lowest average computation with single-cuts is 376 seconds for u-EA2 problems. As opposed to this, multiple-cuts only require one second on average. Again, EA3 and EA5 instances are the most dicult problem instances. These classes include the most instances not converging within the provided time-limit resulting in the highest average computation time for single-cuts, while the average computation time of multiple-cuts only runs up to nine seconds. Furthermore, the average number of required iterations by multiple-cuts is quite constant for the dierent EA-methods. By using the multi-cut version of the BD algorithm, it is possible to solve the M- and L-instances. However, the number of unsolved instances within the provided time-limit of four hours is increasing from M-to L-instances. Again, EA3 problems are most dicult.

To measure the solution quality, we generated valid lower bounds (LB) to the solution as follows: As a starting solution for the xed cost per unit, we assumed the xed cost per capacity unit. Additionally, we introduce a continuous decision variable for non-allocated xed costs per EA-category to the linear MIP, e.g. for EA2 per plant and market, in the xed cost allocation constraint (9) and the LC constraint (15). By using this approach, we ensure the exact calculation of the LC requirement per EA-category as the total sum of xed costs are considered even if the plant is not fully utilized. However, in the generated solution, possible duties might be underestimated due to the lower xed cost portion in the product value. With an increasing utilization of the production network, the generated lower bounds become stronger. It can be seen from Table 11 that the average gap to the lower bounds for the u-Instances and dierent methods is higher than for U-Instances. Looking at the details for u-Instances, it can be seen that BD-st provides better solutions than BD-ml, but with a signicant disadvantage in computation time. Compared with Sec 10, especially for the complex EA methods EA3 to EA6, the gap is lower with BD-ml. More importantly, as for the U-instances the average gap is very small, we can conclude that, using BD-ml, solutions reasonably close to the optimal one can be found.

Furthermore, to compare the eectiveness of the proposed BD multiple-cuts algorithm with a non-linear optimization solver, we used the S-instances for EA1 and EA3, representing the simplest and most dicult solvable EA-options. As non-linear optimization solver, we used the provided

24

EA1 EA2 EA3 EA4 EA5 EA6u S (t)( (t)) BD-st 1,744.9 321.3 - 3,139.0 14,304.4 1,799.7

(2,436.1) (667.9) - (2,416.0) (-) (1,545.5)BD-ml 3.5 2.5 8.8 4.3 9.4 4.1

(1.7) (1.2) (4.1) (1.9) (4.1) (1.7)sec 10 1.4 0.7 455.6 2,930.5 167.5 183.0

(1.2) (0.2) (842.4) (5,734.8) (172.2) (291.8)iter BD-st 1,048 420 - 1,801 3,243 1,262

BD-ml 23 20 33 25 34 24# max. t BD-st - - 24 - 23 -

BD-ml - - - - - -sec 10 - - 13 8 13 8

M (t)( (t)) BD-ml 303.9 79.7 2,478.4 160.0 477.8 89.6(208.1) (20.9) (4,314.4) (90.3) (357.6) ( 26.7)

iter BD-ml 118 75 156 113 134 83# max. t BD-ml 12 11 9 - - -

L (t)( (t)) BD-ml 4,130.7 1,793.5 6,585.0 3,750.2 4,683.6 4,627.6(2,810.2) (2,141.7) (1,388.8) (3,353.0) (1,784.7) (4,876.2)

iter BD-ml 313 289 448 362 346 346# max. t BD-ml 12 10 21 13 17 2

U S (t)( (t)) BD-st 1,933.7 278.9 - 1,522.4 - 1,599.8(1,412.5) (376.1) - (2,363.8) - (2,326.0)

BD-ml 1.3 1.0 6.4 1.8 7.3 1.8(0.8) (0.7) (3.6) (1.1) (4.0) (0.7)

sec 10 0.9 0.6 1,243.1 141.2 67.0 1,061.3(0.2) (0.1) (3,966.9) (263.8) (94.5) (3,415.4)

iter BD-st 1,419 403 - 1,101 - 1,002BD-ml 13 11 26 16 28 17

# max. t BD-st - 1 24 1 24 -BD-ml - - - - - -sec 10 - - 6 2 6 6

M (t)( (t)) BD-ml 90.0 22.8 4,314.9 126.4 823.0 38.8(42.1) (8.3) (3,728.3) (101.9) (969.8) (12.7)

iter BD-ml 73 46 154 88 114 60# max. t BD-ml 12 12 7 - - -

L (t)( (t)) BD-ml 10,509.9 3,314.0 12,612.0 867.5 5,064.5 1,226.2(2,701.7) (1,249.1) - (512.1) (3,334.8) (1,153.9)

iter BD-ml 239 235 214 195 217 175# max. t BD-ml 20 12 23 18 18 -

Table 10: Computation time comparison BD single- vs. multiple-cuts and secants approximation

EA 1 EA2 EA 3 EA 4 EA 5 EA 6u av. LB S 17,150 16,804 17,205 16,772 17,280 16.804

M 17,261 16,798 17,023 16,656 17,224 16,798L 16,544 16,153 16,533 16,186 16,719 16,281

av. gap (%) S BD-st 0.26 3.17 - 0.99 0.22 0.36BD-ml 2.01 2.36 2.55 2.53 2.05 2.21Sec 10 1.90 1.96 3.14 2.93 2.45 2.61

M BD-ml 0.64 1.17 1.34 1.63 0.36 0.85L BD-ml 1.01 1.71 1.42 1.76 1.04 2.44

U av. LB S 21,821 21,257 21,712 21,228 21,766 21,257M 21,311 20,655 21,065 20,575 21,203 20,655L 20,713 20,413 20,556 20,356 20,968 20,414

av. gap (%) S BD-st 0.04 5.32 - 0.26 - 0.12BD-ml 0.02 0.12 0.40 0.28 0.11 0.13Sec 10 0.06 0.10 0.57 0.87 0.33 0.45

M BD-ml 0.10 0.40 0.69 0.75 0.17 0.32L BD-ml 0.11 0.53 0.37 0.49 0.19 0.45

Table 11: Solution quality comparison BD single- vs. multiple-cuts and secants approximation

25

EA1 EA3BD-ml Xpress BD-ml Xpress

(t)( (t)) u 3.50 (1.73) 1.63 (0.58) 8.77 (4.09) 30.90 (23.40)U 1.29 (0.81) 2.29 (1.21) 6.42 (3.56) 43.23 (35.36)

av. LB u 17,283 17,205U 21,821 21,712

av. gap (%) u 2.01 0.02 2.55 0.50U 0.02 0.00 0.04 0.03

Table 12: Solution comparison BD multiple-cuts vs. Xpress-MP

optimization module for non-linear problems by XPRESS-MP. As the underlying optimization problem is non-convex, it cannot be ensured that the solution found by the used non-linear solver of XPRESS-MP is a global one. The results in Table 12 show that, even for S-instances, BD multiple-cuts outperforms the non-linear optimization solver in computation time. For EA1 high-utilization networks and all EA3 instances, the average required computation times by BD multiple-cuts are lower than with Xpress. However, the relative gaps obtained via BD multiple-cuts are, especially for low utilization networks, higher than compared with Xpress.

5.3.3. BD multiple-cuts for dierent network structuresTo test the algorithmic performance of BD multiple-cuts, we rst extended the network size and

then altered the number of considered plants and products. Following the test generation procedure as described in Section 5.1, we considered the dierent network utilizations and xed cost ratios to create 16 instances per network size with a maximum of six plants and six products. We only take two product congurations per product and one time period into account. For the calculation of the instances, we set = 0:1 and increased the maximum run time to eight hours.

In Tables 13 and 14, the results of the enlarged networks are presented based on the L-instances. Table 13 shows that, again, u-instances with a lower utilization of the production capacity can be solved easier than U-instances. Furthermore, EA-methods based on the total production of a plant are approved to be more complicated than EA-methods based only on the export volume. However, looking at the details for u-instances, it seems that this is not the case for production networks with six plants and products. For example, EA4 requires the highest computation time on average, even though, on average, less iterations are necessary than with EA3. The reason is that for EA3, two instances cannot be solved within the provided maximum run-time, but they can be nished for EA4 within the time limit. More importantly, for U-instances starting with ve products and plants, the average computation time increases signicantly. As a result, none of the EA3 instances for six products and plants can be solved within the provided maximum run-time. To evaluate the provided solutions, the average lower bound and the corresponding average gap per EA-method and network-size are presented in Table14. If the instances can be solved within the provided maximum run-time, the obtained results are reasonably close to the lower bound. The average gap is similar for all EA-methods and the dierent production networks. Thus, we can conclude that even for the large instances BD-ml can provide good solutions.

Finally, we analyzed the impact of uneven production networks for the considered number of plants and products on the dierent EA-methods. Table 15 summarizes the average total costs per EA-method and production networks obtained by the BD. Independent of the number of considered products and plants, again, EA-methods based on the export volume only are preferred. This means that for u-instances EA2 is most favorable, whereas for U-instances EA4

26

#plants #products EA1 EA2 EA3 EA4 EA5 EA6u (t)( (t)) 4 4 34.9 22.5 52.8 27.8 46.9 23.2

(18.7) (9.8) (34.3) (14.0) (29.8) (13.4)5 5 532.1 161.1 938.2 172.8 409.7 126.2

(406.4) (76.5) (1,101.6) (81.4) (321.5) (52.4)6 6 1,109.4 1,142.6 1,486.7 2,560.5 1,100.6 1,211.1

(979.1) (1,472.6) (840.8) (3,976.8) (752.1) (1,554.9)iter 4 4 118 74 153 91 137 81

5 5 509 288 575 273 456 2476 6 620 495 716 620 645 532

# max. t 4 4 - - - - - -5 5 - - - - - -6 6 2 - 2 - 2 -

U (t)( (t)) 4 4 77.5 52.8 262.9 109.8 196.4 58.2(24.5) (15.5) (138.9) (29.1) (64.5) (14.0)

5 5 3,693.1 5,227.7 11,828.0 2,337.9 7,076.2 1,951.7(464.3) (4,981.3) (6,761.3) (769.3) (4,447.8) (1,427.9)

6 6 14,414.4 10,287.4 - 4,440.0 20,038.6 6,770.1(1,962.8) (8,025.3) - (1,005.1) (0.0) (4,635.0)

iter 4 4 184 143 376 241 357 1605 5 1,033 992 1,608 1,040 1,355 7636 6 611 482 - 379 819 479

# max. t 4 4 - - - - - -5 5 3 4 4 36 6 6 4 8 6 7 4

Table 13: Computation time BD multiple-cuts for large networks

#plants #products EA1 EA2 EA3 EA4 EA5 EA6u av. LB 4 4 5,152 5,044 5,132 5,026 5,152 5,044

5 5 5,524 5,428 5,503 5,416 5,520 5,4256 6 6,100 6,007 6,086 5,992 6,117 6,007

av. gap (%) 4 4 0.50 0.73 0.77 0.91 0.34 0.555 5 0.55 0.55 0.90 0.71 0.61 0.676 6 0.40 0.56 0.63 0.78 0.46 0.55

U av. LB 4 4 6,674 6,612 6,650 6,606 6,723 6,6125 5 7,218 7,180 7,205 7,176 7,283 7,1796 6 8,126 8,043 8,113 8,039 8,146 8,041

av. gap (%) 4 4 0.50 0.45 0.54 0.53 0.38 0.535 5 0.42 0.38 0.49 0.31 0.29 0.316 6 0.33 0.50 - 0.37 0.48 0.53

Table 14: Solution quality BD multiple-cuts for large networks

27

#plants #products EA 1 EA 2 EA 3 EA 4 EA 5 EA 6u 3 4 5,058 4,980 5,060 4,976 5,067 4,971

5 5,415 5,331 5,413 5,341 5,410 5,3394 3 5,072 4,987 5,074 4,958 5,092 4,977

5 5,542 5,451 5,544 5,460 5,541 5,4615 3 5,374 5,301 5,340 5,305 5,357 5,293

4 5,339 5,266 5,337 5,269 5,366 5,269U 3 4 6,274 6,146 6,257 6,144 6,255 6,145

5 6,465 6,393 6,455 6,399 6,513 6,4054 3 6,452 6,371 6,428 6,371 6,474 6,368

5 6,959 6,897 6,941 6,891 7,013 6,9065 3 6,679 6,637 6,673 6,636 6,759 6,644

4 6,709 6,648 6,689 6,638 6,761 6,647

Table 15: Average total costs for dierent network structures

is the best solution. Hence, the selection of the EA-method is driven by the network utilization and the cost relation between products and plants, rather than the considered network size and structure.

6. Conclusion

In this paper, we present a strategic network design problem taking the special NAFTA LC-requirements for automotive goods, along with an optional LC-fulllment, into account. The resulting model is bilinear and we suggest the application of BD. Through the presented de-composition scheme, we create two sub-problems that can be solved iteratively with standard MIP-solvers and therefore can be easily integrated into standard optimization tools of the OEM. Furthermore, we present two dierent types of BD-cuts to increase the performance of BD in or-der to solve realistic problem M-instances, even for multiple periods. In the conducted numerical studies, we give insights into the LC-planning problem and show that the EA-methods based on the export volume are promising for total cost reduction as the benets of global sourcing can be better utilized. Moreover, we demonstrate that our solution framework is able to solve real-istic problem instances within reasonable computation time. Especially for EA-methods 3 to 6, which provide the highest cost saving potentials, BD outperforms linearization approaches based on SoS2-variables.

Acknowledgments

The authors would like to thank two anonymous referees for the helpful comments, remarks and suggestions to improve this article.

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