# Fill-rate service level constrained distribution network design

Pablo Escalona, Alejandro Angulo, Luce Brotcorne, Bernard Fortz, Paulina Tapia

► To cite this version:

Pablo Escalona, Alejandro Angulo, Luce Brotcorne, Bernard Fortz, Paulina Tapia. Fill-rate service level constrained distribution network design. International Transactions in Operational Research, 2023, 31 (1), pp.5-28. 10.1111/itor.13331. hal-04398370

HAL Id: hal-04398370

https://inria.hal.science/hal-04398370

Submitted on 5 Apr 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Fill-rate service level constrained Distribution Network design

Pablo Escalona<sup>a,*</sup>, Alejandro Angulo<sup>b</sup>, Luce Brotcorne<sup>d</sup>, Bernard Fortz<sup>c,d,e</sup> and Paulina Tapia<sup>a</sup>

<sup>a</sup>Department of Industrial Engineering, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile.

<sup>b</sup>Department of Electrical Engineering, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile.

<sup>c</sup>HEC Liège, Management School of the University of Liège, Rue Louvrex 14, 4000, Liège, Belgium.

<sup>d</sup>INOCS Team, INRIA Lille-Nord Europe, 40 Avenue Halley, 59650 Villeneuve d'Ascq, France.

<sup>e</sup>Département d'Informatique, Faculté des Sciences, Université libre de Bruxelles, Boulevard du Triomphe, 1050 Bruxelles, Belgium.

E-mail: pablo.escalona@usm.cl [Escalona]; alejandro.angulo@usm.cl [Angulo];

Luce.Brotcorne@inria.fr [Brotcorne]; bernard.fortz@uliege.be [Fortz]; paulina tapiae@sansano.usm.cl [Tapia]

## Abstract

The design of distribution networks that simultaneously consider location and inventory decisions seeks to balance costs and product availability. The most commonly observed measure of product availability in practical settings is the fill-rate service level. However, the optimal design of a distribution network that considers the fill-rate to control shortages of fast-moving consumer goods (FMCG) is considered intractable and has only been addressed by heuristic methods. This paper addresses the optimal design of a distribution network for FMCG able to provide high fill-rate service level under a continuous review (r, Q) policy. Considering the exact formulation for the provided fill-rate, we formulated a joint location-inventory model with fill-rate service level constraints as a convex MINLP for which a novel decomposition-based outer approximation algorithm is proposed. Numerical experiments have shown that our solution approach provides good-quality solutions that are on average 0.15% and, at worst, 2.2% from the optimal solution.

**Keywords:** Location; Inventory; Fill-rate; Network design; Conic integer programming; Outer approximation

## 1. Introduction

Fast-moving consumer goods (FMCG) are non-durable products that meet consumers' daily needs (Bala and Kumar (2011)). They include pre-packaged food and beverages, health and beauty items, household cleaning products, pet care items, and many others (Khan and Yu (2019)). For the manufacturer, FMCG

<footnote>*Corresponding author.</footnote>
# Fill-rate service level constrained Distribution Network design

Pablo Escalona<sup>a,*</sup>, Alejandro Angulo<sup>b</sup>, Luce Brotcorne<sup>d</sup>, Bernard Fortz<sup>c,d,e</sup> and Paulina Tapia<sup>a</sup>

<sup>a</sup>Department of Industrial Engineering, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile.

<sup>b</sup>Department of Electrical Engineering, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile.

<sup>c</sup>HEC Liège, Management School of the University of Liège, Rue Louvrex 14, 4000, Liège, Belgium.

<sup>d</sup>INOCS Team, INRIA Lille-Nord Europe, 40 Avenue Halley, 59650 Villeneuve d'Ascq, France.

<sup>e</sup>Département d'Informatique, Faculté des Sciences, Université libre de Bruxelles, Boulevard du Triomphe, 1050 Bruxelles, Belgium.

E-mail: pablo.escalona@usm.cl [Escalona]; alejandro.angulo@usm.cl [Angulo];

Luce.Brotcorne@inria.fr [Brotcorne]; bernard.fortz@uliege.be [Fortz]; paulina tapiae@sansano.usm.cl [Tapia]

## Abstract

The design of distribution networks that simultaneously consider location and inventory decisions seeks to balance costs and product availability. The most commonly observed measure of product availability in practical settings is the fill-rate service level. However, the optimal design of a distribution network that considers the fill-rate to control shortages of fast-moving consumer goods (FMCG) is considered intractable and has only been addressed by heuristic methods. This paper addresses the optimal design of a distribution network for FMCG able to provide high fill-rate service level under a continuous review (r, Q) policy. Considering the exact formulation for the provided fill-rate, we formulated a joint location-inventory model with fill-rate service level constraints as a convex MINLP for which a novel decomposition-based outer approximation algorithm is proposed. Numerical experiments have shown that our solution approach provides good-quality solutions that are on average 0.15% and, at worst, 2.2% from the optimal solution.

**Keywords:** Location; Inventory; Fill-rate; Network design; Conic integer programming; Outer approximation

## 1. Introduction

Fast-moving consumer goods (FMCG) are non-durable products that meet consumers' daily needs (Bala and Kumar (2011)). They include pre-packaged food and beverages, health and beauty items, household cleaning products, pet care items, and many others (Khan and Yu (2019)). For the manufacturer, FMCG

<footnote>*Corresponding author.</footnote>
are products with high volume demand, high inventory turnover, and low margins (Park (2015)). These characteristics of FMCGs are aligned with a low-cost strategy for distribution network design. However, the sales function of FMCG has been concentrated in large retail chains (Webster et al. (2006)) that require high service levels in terms of product availability at the supplier's cost. Consequently, the design of distribution networks that integrate location and inventory decisions simultaneously, defined as the *joint location-inventory problem* (JLIP) according to Daskin et al. (2002), is a relevant issue for FMCG suppliers to design distribution networks that balance cost and service level in terms of product availability.

The most commonly product availability measure observed in practical settings is the *fill-rate* service level (Babiloni and Guijarro, 2020; Teunter et al., 2017; Tan et al., 2017; Bijvank, 2014), defined as the fraction ($\beta$) of demand met directly from inventory on-hand in any period (Schneider (1981)). However, the most common used product availability measure in JLIP is the $\alpha_L$ service-level, defined as the probability ($\alpha_L$) that stockouts do not occur during a replenishment cycle (Schneider (1981)) where $L$ refers to the lead time. In practical settings, fill-rate controls the shortage size at an arbitrary time, and $\alpha_L$ service-level controls the frequency of replenishment cycle stockouts. Craig et al. (2016) found that retailers do not order more frequently in response to increased supplier service levels and infer that retail buyers place larger (and not more frequent) orders for products. Thus, the fill-rate service level is not only the most common measure of product availability in practical settings but also the most important in the supplier-retailer relationship. From the literature review, we observe that the fill-rate has been scarcely addressed in the design of distribution networks for FMCG, where it is usually more convenient and efficient to model demand over a period of time using a continuous distribution, e.g., normal or gamma distributions (Axsäter, 2015; Ramaekers et al., 2008; Peterson and Silver, 1979).

The objective of this paper is to determine the optimal configuration of a FMCG distribution network able to provide fill-rate service level. The optimal design of the distribution network should determine simultaneously the number and location of distribution centers (DCs), the allocation of customers to DCs, and the parameters of the inventory control system that ensure meeting the required fill-rate so that fixed installation costs, transportation costs, and inventory costs are minimized. We assume at each DC: (i) a continuous review ($r, Q$) policy with full-backorders, where $Q$ is a fixed order quantity and $r$ is the reorder point, (ii) normally distributed demand as an approximation for the FMCG demand, and (iii) fill-rate service level to control the shortage. We first model the JLIP with fill-rate service level constraints under a continuous review ($r, Q$) policy with full-backorders as a mixed integer nonlinear problem (MINLP), denoted as $\beta$-JLIP. Then, using conic integer programming, the $\beta$-JLIP model is reformulated as a convex MINLP with second-order cone constraints for which a novel decomposition-based outer approximation (OA) algorithm is proposed to solve it.

The main contributions are presented as: (1) We address for the first time the modeling and optimal solution of a FMCG distribution network design problem that takes into account providing and fulfilling high fill-rate service levels. Thus, we extend the design alternatives of FMCG distribution networks by considering the fill-rate service level to control shortages. (2) We show how to formulate the JLIP with fill-rate service level constraints as a convex MINLP. Using operations that preserve de convexity, we show that the inequality function of the fill-rate service level constraint is convex and, consequently, can be replaced by its outer-approximation. (3) We propose a novel decomposition-based OA algorithm to solve the JLIP with fill-rate service level constraints. Different from the OA approach described by Duran and Grossmann (1986), our decomposition approach uses a conic quadratic mixed-integer programming
are products with high volume demand, high inventory turnover, and low margins (Park (2015)). These characteristics of FMCGs are aligned with a low-cost strategy for distribution network design. However, the sales function of FMCG has been concentrated in large retail chains (Webster et al. (2006)) that require high service levels in terms of product availability at the supplier's cost. Consequently, the design of distribution networks that integrate location and inventory decisions simultaneously, defined as the *joint location-inventory problem* (JLIP) according to Daskin et al. (2002), is a relevant issue for FMCG suppliers to design distribution networks that balance cost and service level in terms of product availability.

The most commonly product availability measure observed in practical settings is the *fill-rate* service level (Babiloni and Guijarro, 2020; Teunter et al., 2017; Tan et al., 2017; Bijvank, 2014), defined as the fraction ($\beta$) of demand met directly from inventory on-hand in any period (Schneider (1981)). However, the most common used product availability measure in JLIP is the $\alpha_L$ service-level, defined as the probability ($\alpha_L$) that stockouts do not occur during a replenishment cycle (Schneider (1981)) where $L$ refers to the lead time. In practical settings, fill-rate controls the shortage size at an arbitrary time, and $\alpha_L$ service-level controls the frequency of replenishment cycle stockouts. Craig et al. (2016) found that retailers do not order more frequently in response to increased supplier service levels and infer that retail buyers place larger (and not more frequent) orders for products. Thus, the fill-rate service level is not only the most common measure of product availability in practical settings but also the most important in the supplier-retailer relationship. From the literature review, we observe that the fill-rate has been scarcely addressed in the design of distribution networks for FMCG, where it is usually more convenient and efficient to model demand over a period of time using a continuous distribution, e.g., normal or gamma distributions (Axsäter, 2015; Ramaekers et al., 2008; Peterson and Silver, 1979).

The objective of this paper is to determine the optimal configuration of a FMCG distribution network able to provide fill-rate service level. The optimal design of the distribution network should determine simultaneously the number and location of distribution centers (DCs), the allocation of customers to DCs, and the parameters of the inventory control system that ensure meeting the required fill-rate so that fixed installation costs, transportation costs, and inventory costs are minimized. We assume at each DC: (i) a continuous review ($r, Q$) policy with full-backorders, where $Q$ is a fixed order quantity and $r$ is the reorder point, (ii) normally distributed demand as an approximation for the FMCG demand, and (iii) fill-rate service level to control the shortage. We first model the JLIP with fill-rate service level constraints under a continuous review ($r, Q$) policy with full-backorders as a mixed integer nonlinear problem (MINLP), denoted as $\beta$-JLIP. Then, using conic integer programming, the $\beta$-JLIP model is reformulated as a convex MINLP with second-order cone constraints for which a novel decomposition-based outer approximation (OA) algorithm is proposed to solve it.

The main contributions are presented as: (1) We address for the first time the modeling and optimal solution of a FMCG distribution network design problem that takes into account providing and fulfilling high fill-rate service levels. Thus, we extend the design alternatives of FMCG distribution networks by considering the fill-rate service level to control shortages. (2) We show how to formulate the JLIP with fill-rate service level constraints as a convex MINLP. Using operations that preserve de convexity, we show that the inequality function of the fill-rate service level constraint is convex and, consequently, can be replaced by its outer-approximation. (3) We propose a novel decomposition-based OA algorithm to solve the JLIP with fill-rate service level constraints. Different from the OA approach described by Duran and Grossmann (1986), our decomposition approach uses a conic quadratic mixed-integer programming
(CQMIP) master problem and a convex nonlinear programming (NLP) primal problem (subproblem).

The paper is organized as follows. A review of related work is discussed in Section 2. Section 3 presents the JLIP model under a continuous review (r, Q) policy with full-backorders, fill-rate service levels, and normally distributed demand. In Section 4, we present the JLIP with fill-rate service level constraints as a convex MINLP with second-order cone constraints. In Section 5, we describe the solution approach based on OA scheme. Numerical results are reported in Section 6. Finally, conclusions and suggested future extensions are described in Section 7.

## 2. Related literature

To the best of our knowledge, JLIP with fill-rate service level has been studied mainly for the design of slow-moving items distribution networks. These items are products with low demand and/or low inventory turnover. They include high-value or high-priority service parts or spare parts. For convenience, the term "slow-moving item" is limited to items with 12 or fewer movements per year (Johnston et al. (2003)). A base stock inventory policy is typically chosen for slow-moving items (Wheatley et al. (2015)) and its demand is modeled over a period of time using an independent Poisson process (Sherbrooke (2006); muc (2005)), i.e., discrete distribution demand. In contrast, we study the design of distribution networks for fast-moving items using a continuous review (r, Q) policy with fill-rate and continuous distribution demand, which leads to a different optimization problem and solution approach.

Candas and Kutanoglu (2007) studied the design of a service parts distribution network using a multi-product JLIP under a continuous review base-stock policy with full-backorder and fill-rate. Customers follow a Poisson demand distribution to approximate slow-moving items. They developed a service level constraint that simultaneously ensures coverage and availability, denoted as a *time-based service coverage constraint*. The resulting model is a MINLP. They use piece-wise linear step functions to linearize the fill-rate, obtaining an integer linear problem (IP) approximating the original optimization model (neither relaxation nor constriction). A feasible solution is obtained using post processing. Jeet et al. (2009) studied the design of a service parts distribution network using a single-product JLIP under a continuous review base-stock policy with lost-sale and fill-rate. The model is formulated as a non-convex MINLP. They reformulate their model as a convex MINLP and solve it using an OA approach. Wheatley et al. (2015) addressed the same problem as Candas and Kutanoglu (2007). However, they present an exact methodology to solve the JLIP using Benders decomposition. All the above authors assume fill-rate defined as the fraction of demand met directly from inventory on-hand during the replenishment cycle (lead time). In contrast, the definition of fill-rate that we use in this paper is for any period according to Schneider (1981).

The JLIP under (r, Q) policy is a well-studied problem using $\alpha_L$ service-level. Daskin et al. (2002) studied the JLIP under EOQ quantity, and stochastic reorder point as approximations of the optimal parameters of a continuous review (r, Q) policy with $\alpha_L$ service-level. Customers follow normally distribution demand as an approximation of non-negative demand. The model is formulated as a nonlinear integer problem (INLP) because the EOQ quantity and stochastic reorder point are functions of the location-allocation variables. They solved the model by Lagrangian relaxation. Several solution approaches have been developed to address the formulation presented by Daskin et al. (2002), e.g., column generation (Shu et al., 2005; Shen et al., 2003), Lagrangean relaxation (You and Grossmann (2008)),
(CQMIP) master problem and a convex nonlinear programming (NLP) primal problem (subproblem).

The paper is organized as follows. A review of related work is discussed in Section 2. Section 3 presents the JLIP model under a continuous review (r, Q) policy with full-backorders, fill-rate service levels, and normally distributed demand. In Section 4, we present the JLIP with fill-rate service level constraints as a convex MINLP with second-order cone constraints. In Section 5, we describe the solution approach based on OA scheme. Numerical results are reported in Section 6. Finally, conclusions and suggested future extensions are described in Section 7.

## 2. Related literature

To the best of our knowledge, JLIP with fill-rate service level has been studied mainly for the design of slow-moving items distribution networks. These items are products with low demand and/or low inventory turnover. They include high-value or high-priority service parts or spare parts. For convenience, the term "slow-moving item" is limited to items with 12 or fewer movements per year (Johnston et al. (2003)). A base stock inventory policy is typically chosen for slow-moving items (Wheatley et al. (2015)) and its demand is modeled over a period of time using an independent Poisson process (Sherbrooke (2006); muc (2005)), i.e., discrete distribution demand. In contrast, we study the design of distribution networks for fast-moving items using a continuous review (r, Q) policy with fill-rate and continuous distribution demand, which leads to a different optimization problem and solution approach.

Candas and Kutanoglu (2007) studied the design of a service parts distribution network using a multi-product JLIP under a continuous review base-stock policy with full-backorder and fill-rate. Customers follow a Poisson demand distribution to approximate slow-moving items. They developed a service level constraint that simultaneously ensures coverage and availability, denoted as a *time-based service coverage constraint*. The resulting model is a MINLP. They use piece-wise linear step functions to linearize the fill-rate, obtaining an integer linear problem (IP) approximating the original optimization model (neither relaxation nor constriction). A feasible solution is obtained using post processing. Jeet et al. (2009) studied the design of a service parts distribution network using a single-product JLIP under a continuous review base-stock policy with lost-sale and fill-rate. The model is formulated as a non-convex MINLP. They reformulate their model as a convex MINLP and solve it using an OA approach. Wheatley et al. (2015) addressed the same problem as Candas and Kutanoglu (2007). However, they present an exact methodology to solve the JLIP using Benders decomposition. All the above authors assume fill-rate defined as the fraction of demand met directly from inventory on-hand during the replenishment cycle (lead time). In contrast, the definition of fill-rate that we use in this paper is for any period according to Schneider (1981).

The JLIP under (r, Q) policy is a well-studied problem using $\alpha_L$ service-level. Daskin et al. (2002) studied the JLIP under EOQ quantity, and stochastic reorder point as approximations of the optimal parameters of a continuous review (r, Q) policy with $\alpha_L$ service-level. Customers follow normally distribution demand as an approximation of non-negative demand. The model is formulated as a nonlinear integer problem (INLP) because the EOQ quantity and stochastic reorder point are functions of the location-allocation variables. They solved the model by Lagrangian relaxation. Several solution approaches have been developed to address the formulation presented by Daskin et al. (2002), e.g., column generation (Shu et al., 2005; Shen et al., 2003), Lagrangean relaxation (You and Grossmann (2008)),
and the CQMIP reformulation of Atamtürk et al. (2012) which can be solved directly using standard optimization solvers. Different generalizations have been made to the model of Daskin et al. (2002), e.g., the capacitated JLIP (Atamtürk et al., 2012; Ozsen et al., 2008; Miranda and Garrido, 2004), the multi-commodity JLIP (Atamtürk et al., 2012; Shen, 2005), the correlated demand JLIP (Atamtürk et al., 2012; Shahabi et al., 2014), the multi-echelon JLIP (Diabat et al. (2015)), the stochastic JLIP (Snyder et al. (2007)), the JLIP with differentiated service levels (Escalona et al., 2018, 2015), the stochastic capacity JLIP (Tapia-Ubeda et al., 2018; Miranda and Garrido, 2008, 2006), and the JLIP with supplier selection (Perez Loaiza et al. (2017)). All of the above authors approximate the optimal parameters of the continuous review (r, Q) policy by the EOQ quantity and stochastic reorder point, or approximate the inventory on-hand to the inventory level by considering the backorders negligible.

The JLIP with service level constraints (fill-rate or $\alpha_L$) considers the inventories in single-echelon settings. More precisely, inventory policy is considered only at DCs. This approach is denoted *non-coordinated* in the sense that DCs observe uncertain demand from retailers, but do not observe retailers’ inventory control policy. Recently, Darmawan et al. (2021) studied the coordinated JLIP where DCs and retailers operate under a continuous review (r, Q) policy with full-backorders and normally distributed demand. They use the *economic order quantity* (EOQ) approximating the replenishment lot size at DCs and retailers. The model is formulated as a MINLP with fill-rate service level constraints for the end-customers (retailers). Due to the complexity of the optimization problem, they used genetic algorithms to solve it without guaranteeing the quality-solution in terms of the optimality gap. Numerical experiments show that the inclusion of coordinated inventory control in DCs and retailers results in lower total cost. In contrast, we address a non-coordinated JLIP, we do not approximate the continuous review (r, Q) policy parameters, and we guarantee the quality-solution in terms of optimality gap.

On the other hand, in this paper we show how to formulate the JLIP with fill-rate service level constraints as a convex MINLP. Convex MINLP problems are an important subclass of MINLP problems, where their convex properties are exploited to derive efficient decomposition algorithms. Decomposition algorithms for convex MINLP are: Branch and Bound (B&B), extended cutting plane (ECP), extended supporting hyperplane (ESH), outer approximation (OA), and generalized Benders decomposition (GBD). In B&B methods (Leyffer (2001); Quesada and Grossmann (1992); Dakin (1965)) integer relaxed convex subproblems are solved in each node of a B&B tree. The decomposition methods ECP (Westerlund and Pettersson (1995)), ESH (Kronqvist et al. (2016)), and OA (Coey et al. (2020); Muts et al. (2020); De Mauri et al. (2020); Su et al. (2018); Duran and Grossmann (1986)), exploit the properties of convex MINLPs to derive linearizations of nonlinear constraints based on their gradients. These linearizations are equivalent to first-order Taylor expansions of nonlinear inequalities. The GBD (Floudas (1995); Geoffrion (1972)) exploits the nonlinear convex duality to derive linearizations of nonlinear constraints. Of the methods for solving convex MINLP, OA has been shown to be one of the most efficient algorithms (Bernal et al. (2022); Kronqvist et al. (2019); Su et al. (2015)).

In summary, the design of FMCG distribution networks able to provide fill-rate service level in terms of product availability under a continuous review (r, Q) policy with full-backorder has been scarcely studied. This paper contributes to extend the design alternatives of distribution networks for FMCG by considering the JLIP with fill-rate service level constraints and an efficient OA-based solution approach.
and the CQMIP reformulation of Atamtürk et al. (2012) which can be solved directly using standard optimization solvers. Different generalizations have been made to the model of Daskin et al. (2002), e.g., the capacitated JLIP (Atamtürk et al., 2012; Ozsen et al., 2008; Miranda and Garrido, 2004), the multi-commodity JLIP (Atamtürk et al., 2012; Shen, 2005), the correlated demand JLIP (Atamtürk et al., 2012; Shahabi et al., 2014), the multi-echelon JLIP (Diabat et al. (2015)), the stochastic JLIP (Snyder et al. (2007)), the JLIP with differentiated service levels (Escalona et al., 2018, 2015), the stochastic capacity JLIP (Tapia-Ubeda et al., 2018; Miranda and Garrido, 2008, 2006), and the JLIP with supplier selection (Perez Loaiza et al. (2017)). All of the above authors approximate the optimal parameters of the continuous review (r, Q) policy by the EOQ quantity and stochastic reorder point, or approximate the inventory on-hand to the inventory level by considering the backorders negligible.

The JLIP with service level constraints (fill-rate or $\alpha_L$) considers the inventories in single-echelon settings. More precisely, inventory policy is considered only at DCs. This approach is denoted *non-coordinated* in the sense that DCs observe uncertain demand from retailers, but do not observe retailers’ inventory control policy. Recently, Darmawan et al. (2021) studied the coordinated JLIP where DCs and retailers operate under a continuous review (r, Q) policy with full-backorders and normally distributed demand. They use the *economic order quantity* (EOQ) approximating the replenishment lot size at DCs and retailers. The model is formulated as a MINLP with fill-rate service level constraints for the end-customers (retailers). Due to the complexity of the optimization problem, they used genetic algorithms to solve it without guaranteeing the quality-solution in terms of the optimality gap. Numerical experiments show that the inclusion of coordinated inventory control in DCs and retailers results in lower total cost. In contrast, we address a non-coordinated JLIP, we do not approximate the continuous review (r, Q) policy parameters, and we guarantee the quality-solution in terms of optimality gap.

On the other hand, in this paper we show how to formulate the JLIP with fill-rate service level con-
straints as a convex MINLP. Convex MINLP problems are an important subclass of MINLP problems,
where their convex properties are exploited to derive efficient decomposition algorithms. Decompo-
sition algorithms for convex MINLP are: Branch and Bound (B&B), extended cutting plane (ECP),
extended supporting hyperplane (ESH), outer approximation (OA), and generalized Benders decompo-
sition (GBD). In B&B methods (Leyffer (2001); Quesada and Grossmann (1992); Dakin (1965)) integer
relaxed convex subproblems are solved in each node of a B&B tree. The decomposition methods ECP
(Westerlund and Pettersson (1995)), ESH (Kronqvist et al. (2016)), and OA (Coey et al. (2020); Muts
et al. (2020); De Mauri et al. (2020); Su et al. (2018); Duran and Grossmann (1986)), exploit the proper-
ties of convex MINLPs to derive linearizations of nonlinear constraints based on their gradients. These
linearizations are equivalent to first-order Taylor expansions of nonlinear inequalities. The GBD (Floudas
(1995); Geoffrion (1972)) exploits the nonlinear convex duality to derive linearizations of nonlinear con-
straints. Of the methods for solving convex MINLP, OA has been shown to be one of the most efficient
algorithms (Bernal et al. (2022); Kronqvist et al. (2019); Su et al. (2015)).

In summary, the design of FMCG distribution networks able to provide fill-rate service level in terms
of product availability under a continuous review (r, Q) policy with full-backorder has been scarcely
studied. This paper contributes to extend the design alternatives of distribution networks for FMCG by
considering the JLIP with fill-rate service level constraints and an efficient OA-based solution approach.
### 3. Problem description and formulation

The distribution network design problem we consider is as follows. A supplier ships a high demand volume product (FMCG) to a set of retailers (customers) through a set of DCs. Retailers require high fill-rate service levels in terms of product availability. Each retailer is supplied by a single DC (single source allocation) and the DCs are supplied by a single plant of unlimited capacity.

For the FMCG supplier, the distribution network design problem involves simultaneously determining: (i) Which DCs should be installed?, (ii) Which retailers should be supplied by each DC?, and (iii) What inventory policy parameters values should each DC use to satisfy the fill-rate service level required by the retailers?, such that the system-wide cost is minimized. The general structure of the concerned supply chain is shown in Figure 1.

<figure><img src="image_1.png" /><figcaption>Fig. 1. Supply chain network structure (three echelons)</figcaption></figure>

The plant location, site candidates to install the DCs, and retailers' location are known. Let $J$ ($j = 1, ..., |J|$) be the set of site candidates to install DCs. Each DC follows a continuous review ($r, Q$) inventory policy with full-backorders and deterministic lead time, i.e., at each DC $j$, an order quantity of constant size, $Q_j$, is placed to the plant whenever the inventory position (i.e., inventory on-hand plus out-standing orders minus backorders) falls below a fixed reorder point, $r_j$, which arrives at a fixed $L_j > 0$ time unit later. Product shortages at each DC are controlled using the fill-rate service level defined as the fraction ($\beta$) of demand met directly from on-hand inventory in any period according to Schneider (1981).

The costs associated with the distribution network design are the fixed cost of installation, transportation costs, and inventory costs. Backorder costs are not considered because we have assumed that inventory shortages are controlled by a service level constraint.

Let $I$ ($i = 1, ..., |I|$) be the set of retailers. We assume that the demand per time unit at each retailer is independent and normally distributed, as an approximation of FMCG demand, where $\mu_i$ and $\sigma_i^2$ are the mean and variance demand per time unit of retailer $i$, respectively. Let the location variable $X_j$ be equal to 1 if site $j$ is selected as a DC and 0 otherwise, and let the allocation variable $Y_{ij}$ be equal to 1 if retail $i$ is served by the DC located at $j$ and 0 otherwise. The demand at candidate DC $j$ is defined by
### 3. Problem description and formulation

The distribution network design problem we consider is as follows. A supplier ships a high demand volume product (FMCG) to a set of retailers (customers) through a set of DCs. Retailers require high fill-rate service levels in terms of product availability. Each retailer is supplied by a single DC (single source allocation) and the DCs are supplied by a single plant of unlimited capacity.

For the FMCG supplier, the distribution network design problem involves simultaneously determining: (i) Which DCs should be installed?, (ii) Which retailers should be supplied by each DC?, and (iii) What inventory policy parameters values should each DC use to satisfy the fill-rate service level required by the retailers?, such that the system-wide cost is minimized. The general structure of the concerned supply chain is shown in Figure 1.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Supply chain network structure (three echelons)</figcaption></figure>

The plant location, site candidates to install the DCs, and retailers' location are known. Let $J$ ($j = 1, ..., |J|$) be the set of site candidates to install DCs. Each DC follows a continuous review ($r, Q$) inventory policy with full-backorders and deterministic lead time, i.e., at each DC $j$, an order quantity of constant size, $Q_j$, is placed to the plant whenever the inventory position (i.e., inventory on-hand plus out-standing orders minus backorders) falls below a fixed reorder point, $r_j$, which arrives at a fixed $L_j > 0$ time unit later. Product shortages at each DC are controlled using the fill-rate service level defined as the fraction ($\beta$) of demand met directly from on-hand inventory in any period according to Schneider (1981).

The costs associated with the distribution network design are the fixed cost of installation, transportation costs, and inventory costs. Backorder costs are not considered because we have assumed that inventory shortages are controlled by a service level constraint.

Let $I$ ($i = 1, ..., |I|$) be the set of retailers. We assume that the demand per time unit at each retailer is independent and normally distributed, as an approximation of FMCG demand, where $\mu_i$ and $\sigma_i^2$ are the mean and variance demand per time unit of retailer $i$, respectively. Let the location variable $X_j$ be equal to 1 if site $j$ is selected as a DC and 0 otherwise, and let the allocation variable $Y_{ij}$ be equal to 1 if retail $i$ is served by the DC located at $j$ and 0 otherwise. The demand at candidate DC $j$ is defined by
the allocation variables. More precisely, under the assumption of independent and normally distributed demand, it is easy to show that $\mu_j = \sum_{i \in I} \mu_i Y_{ij} \ge 0$ and $\sigma_j^2 = \sum_{i \in I} \sigma_i^2 Y_{ij} \ge 0$, where $\mu_j$ and $\sigma_j^2$ are the mean and variance demand per time unit at DC j, respectively.

In what follows, the general cost function for each DC, the provided fill-rate, and the JLIP with fill-rate service level constraints are presented.

### 3.1. Cost function

The average cost per time unit at DC j considering fixed installation cost, linear transportation costs, and a continuous review (r, Q) policy is defined as:

$$AC_j(Q_j, r_j, X_j, \mathbf{Y}) = f_j X_j + a_j \sum_{i \in I} \mu_i Y_{ij} + \sum_{i \in I} d_{ij} \mu_i Y_{ij} + S_j \frac{\sum_{i \in I} \mu_i Y_{ij}}{Q_j} + h_j \left( \frac{Q_j}{2} + r_j - L_j \sum_{i \in I} \mu_i Y_{ij} \right), \quad (1)$$

where $f_j$ is the fixed cost per time unit of installing a DC at candidate site j, $a_j$ is the transportation cost per unit from the plant to DC j, $d_{ij}$ is the transportation cost per unit from DC j to retailer i, $S_j$ is the ordering cost at DC j, and $h_j$ is the holding cost per unit and time unit at DC j.

The first term in (1) is the fixed cost per time unit, the second and third terms are the supply and distribution costs per time unit, respectively, and the fourth term is the ordering cost per time unit. The fifth term is approximated the holding cost per time unit because similar to Daskin et al. (2002), we approximate the inventory on-hand to the inventory level.

The cost function (1) has a time-base unit because it considers investment and operational costs. To make the five terms in (1) comparable, the fixed cost of installing a DC at candidate site j (one-time investment) must be converted to fixed installation cost per time unit using an appropriate discount rate.

### 3.2. The fill-rate service level constraint

The shortage in inventory systems is an inevitable random phenomenon under uncertain demand. In this paper, we assume that shortage events at each DC are controlled using the fill-rate service level defined by Schneider (1981). Under normally distributed demand, the fill-rate is equivalent to the ready-rate service level defined as the probability ($\alpha$) of not being out of stock at an arbitrary time or equivalently as the fraction of time with a positive stock-on-hand (Escalona et al., 2021; Axsäter, 2015; Silver et al., 1998). Thus, the fill-rate under normally distributed demand explicitly controls the backorders size and stockouts frequency, both at an arbitrary time.

Let $\beta(r_j, Q_j, \mathbf{Y})$ be the provided fill-rate at DC j. Following Axsäter (2006), the provided fill-rate under a continuous review (r, Q) policy with full-backorders and normally distributed demand is defined
the allocation variables. More precisely, under the assumption of independent and normally distributed demand, it is easy to show that $\mu_j = \sum_{i \in I} \mu_i Y_{ij} \ge 0$ and $\sigma_j^2 = \sum_{i \in I} \sigma_i^2 Y_{ij} \ge 0$, where $\mu_j$ and $\sigma_j^2$ are the mean and variance demand per time unit at DC j, respectively.

In what follows, the general cost function for each DC, the provided fill-rate, and the JLIP with fill-rate service level constraints are presented.

### 3.1. Cost function

The average cost per time unit at DC j considering fixed installation cost, linear transportation costs, and a continuous review (r, Q) policy is defined as:

$$AC_j(Q_j, r_j, X_j, \mathbf{Y}) = f_j X_j + a_j \sum_{i \in I} \mu_i Y_{ij} + \sum_{i \in I} d_{ij} \mu_i Y_{ij} + S_j \frac{\sum_{i \in I} \mu_i Y_{ij}}{Q_j} + h_j \left( \frac{Q_j}{2} + r_j - L_j \sum_{i \in I} \mu_i Y_{ij} \right), \quad (1)$$

where $f_j$ is the fixed cost per time unit of installing a DC at candidate site j, $a_j$ is the transportation cost per unit from the plant to DC j, $d_{ij}$ is the transportation cost per unit from DC j to retailer i, $S_j$ is the ordering cost at DC j, and $h_j$ is the holding cost per unit and time unit at DC j.

The first term in (1) is the fixed cost per time unit, the second and third terms are the supply and distribution costs per time unit, respectively, and the fourth term is the ordering cost per time unit. The fifth term is approximated the holding cost per time unit because similar to Daskin et al. (2002), we approximate the inventory on-hand to the inventory level.

The cost function (1) has a time-base unit because it considers investment and operational costs. To make the five terms in (1) comparable, the fixed cost of installing a DC at candidate site j (one-time investment) must be converted to fixed installation cost per time unit using an appropriate discount rate.

### 3.2. The fill-rate service level constraint

The shortage in inventory systems is an inevitable random phenomenon under uncertain demand. In this paper, we assume that shortage events at each DC are controlled using the fill-rate service level defined by Schneider (1981). Under normally distributed demand, the fill-rate is equivalent to the ready-rate service level defined as the probability (α) of not being out of stock at an arbitrary time or equivalently as the fraction of time with a positive stock-on-hand (Escalona et al., 2021; Axsäter, 2015; Silver et al., 1998). Thus, the fill-rate under normally distributed demand explicitly controls the backorders size and stockouts frequency, both at an arbitrary time.

Let $\beta(r_j, Q_j, \mathbf{Y})$ be the provided fill-rate at DC j. Following Axsäter (2006), the provided fill-rate under a continuous review (r, Q) policy with full-backorders and normally distributed demand is defined
as:

$$ \beta(Q_j, r_j, \mathbf{Y}) = 1 - \frac{\sqrt{L_j \sum_{i \in I} \sigma_i^2 Y_{ij}}}{Q_j} \left[ G \left( \frac{r_j - L_j \sum_{i \in I} \mu_i Y_{ij}}{\sqrt{L_j \sum_{i \in I} \sigma_i^2 Y_{ij}}} \right) - G \left( \frac{r_j + Q_j - L_j \sum_{i \in I} \mu_i Y_{ij}}{\sqrt{L_j \sum_{i \in I} \sigma_i^2 Y_{ij}}} \right) \right], \quad (2) $$

where $G(x) = \int_{x}^{\infty} (v-x)\varphi(v) dv = \varphi(x) - x(1 - \Phi(x))$ with $\Phi(x)$ and $\varphi(x)$ the distribution and density functions of the standard normal distribution, respectively.

Let $\bar{\beta} \in (0, 1)$ be the preset fill-rate for the network distribution, i.e., it is the minimum fraction of demand that the decision-maker at DC $j$ requires to meet directly from on-hand inventory at an arbitrary time. Thus, the fill-rate service level constraint is defined as $\beta(Q_j, r_j, \mathbf{Y}) \ge \bar{\beta}$ for any DC $j \in J$.

### 3.3. The JILP under fill-rate service level constraints

We can formulate a location-inventory model under continuous review $(r, Q)$ policy with fill-rate service level constraints and normally distributed demand as a MINLP with nonlinear constraint and nonlinear objective function, denoted $\beta$-JLIP, as follows:

$$ \beta\text{-JLIP: } \min_{\mathbf{X},\mathbf{Y},r,\mathbf{Q}} \sum_{j \in J} \left\{ f_j X_j + \sum_{i \in I} \hat{d}_{ij} Y_{ij} + S_j \frac{\sum_{i \in I} \mu_i Y_{ij}}{Q_j} + h_j \left( \frac{Q_j}{2} + r_j - L_j \sum_{i \in I} \mu_i Y_{ij} \right) \right\} \quad (3) $$

$$ \text{s.t:} \quad \sum_{j \in J} Y_{ij} = 1 \quad \forall i \in I \quad (4) $$

$$ Y_{ij} \le X_j \quad \forall i \in I, j \in J \quad (5) $$

$$ \beta(Q_j, r_j, \mathbf{Y}) \ge \bar{\beta} \quad \forall j \in J \quad (6) $$

$$ r_j \ge L_j \sum_{i \in I} \mu_i Y_{ij} \quad \forall i \in I, j \in J \quad (7) $$

$$ Q_j \ge 0 \quad \forall j \in J \quad (8) $$

$$ X_j, Y_{ij} \in \{0, 1\} \quad \forall i \in I, j \in J, \quad (9) $$

where $\hat{d}_{ij} = (a_j + d_{ij})\mu_i$, and $\beta(Q_j, r_j, \mathbf{Y})$ is the provided fill-rate at candidate DC $j$ according to (2).

The objective function in (3) is the average total costs per time unit including fixed, supply and distribution, ordering, and holding costs. Constraints (4) establish that each retailer is served by only one DC. Constraints (5) ensure that the retailers are assigned to an installed DC. Constraint (6) ensures that the fill-rate provided by each DC is greater than or equal to the preset fill-rate defined for the distribution network. Constraint (7) ensures that the safety stock at each DC $j$ is non-negative because, by definition, the safety stock in a continuous review $(r, Q)$ policy is $r_j - \mathbb{E}(D_j(L_j))$, where $\mathbb{E}(D_j(L_j)) = L_j \sum_{i \in I} \mu_i Y_{ij}$ is the expected demand during lead time at DC $j$. Constraint (8) is a non-negativity constraint of order quantity decision variables. Constraint (9) is an integrality constraint of location-allocation decision variables, respectively. A glossary of terms is presented in Appendix A.
as:

$$ \beta(Q_j, r_j, \mathbf{Y}) = 1 - \frac{\sqrt{L_j \sum_{i \in I} \sigma_i^2 Y_{ij}}}{Q_j} \left[ G \left( \frac{r_j - L_j \sum_{i \in I} \mu_i Y_{ij}}{\sqrt{L_j \sum_{i \in I} \sigma_i^2 Y_{ij}}} \right) - G \left( \frac{r_j + Q_j - L_j \sum_{i \in I} \mu_i Y_{ij}}{\sqrt{L_j \sum_{i \in I} \sigma_i^2 Y_{ij}}} \right) \right], \quad (2) $$

where $G(x) = \int_{x}^{\infty} (v-x)\varphi(v) dv = \varphi(x) - x(1 - \Phi(x))$ with $\Phi(x)$ and $\varphi(x)$ the distribution and density functions of the standard normal distribution, respectively.

Let $\bar{\beta} \in (0, 1)$ be the preset fill-rate for the network distribution, i.e., it is the minimum fraction of demand that the decision-maker at DC $j$ requires to meet directly from on-hand inventory at an arbitrary time. Thus, the fill-rate service level constraint is defined as $\beta(Q_j, r_j, \mathbf{Y}) \ge \bar{\beta}$ for any DC $j \in J$.

### 3.3. The JILP under fill-rate service level constraints

We can formulate a location-inventory model under continuous review $(r, Q)$ policy with fill-rate service level constraints and normally distributed demand as a MINLP with nonlinear constraint and nonlinear objective function, denoted $\beta$-JLIP, as follows:

$$ \beta\text{-JLIP: } \min_{\mathbf{X},\mathbf{Y},r,\mathbf{Q}} \sum_{j \in J} \left\{ f_j X_j + \sum_{i \in I} \hat{d}_{ij} Y_{ij} + S_j \frac{\sum_{i \in I} \mu_i Y_{ij}}{Q_j} + h_j \left( \frac{Q_j}{2} + r_j - L_j \sum_{i \in I} \mu_i Y_{ij} \right) \right\} \quad (3) $$

$$ \text{s.t:} \quad \sum_{j \in J} Y_{ij} = 1 \quad \forall i \in I \quad (4) $$

$$ Y_{ij} \le X_j \quad \forall i \in I, j \in J \quad (5) $$

$$ \beta(Q_j, r_j, \mathbf{Y}) \ge \bar{\beta} \quad \forall j \in J \quad (6) $$

$$ r_j \ge L_j \sum_{i \in I} \mu_i Y_{ij} \quad \forall i \in I, j \in J \quad (7) $$

$$ Q_j \ge 0 \quad \forall j \in J \quad (8) $$

$$ X_j, Y_{ij} \in \{0, 1\} \quad \forall i \in I, j \in J, \quad (9) $$

where $\hat{d}_{ij} = (a_j + d_{ij})\mu_i$, and $\beta(Q_j, r_j, \mathbf{Y})$ is the provided fill-rate at candidate DC $j$ according to (2).

The objective function in (3) is the average total costs per time unit including fixed, supply and distribution, ordering, and holding costs. Constraints (4) establish that each retailer is served by only one DC. Constraints (5) ensure that the retailers are assigned to an installed DC. Constraint (6) ensures that the fill-rate provided by each DC is greater than or equal to the preset fill-rate defined for the distribution network. Constraint (7) ensures that the safety stock at each DC $j$ is non-negative because, by definition, the safety stock in a continuous review $(r, Q)$ policy is $r_j - \mathbb{E}(D_j(L_j))$, where $\mathbb{E}(D_j(L_j)) = L_j \sum_{i \in I} \mu_i Y_{ij}$ is the expected demand during lead time at DC $j$. Constraint (8) is a non-negativity constraint of order quantity decision variables. Constraint (9) is an integrality constraint of location-allocation decision variables, respectively. A glossary of terms is presented in Appendix A.
4. An equivalent convex MINLP with second-order cone constraints of β-JLIP

The β-JLIP model is a non-convex MINLP in its objective function and constraint (6). In this section, we show how to reformulate the β-JLIP model as a convex MINLP using conic integer programming.

Similar to Atamtürk et al. (2012), we introduce a set of two non-negative continuous variables, C1j and C2j, which exploits the fact that $Y_{ij} = Y_{ij}^2$. These variables are defined as follows:

$$C1_j^2 = \sum_{i \in I} \mu_i Y_{ij}^2 \quad \forall j \in J \qquad (10)$$

$$C2_j^2 = \sum_{i \in I} (\sigma_i Y_{ij})^2 \quad \forall j \in J \qquad (11)$$

$$C1_j \ge 0 \quad \forall j \in J \qquad (12)$$

$$C2_j \ge 0 \quad \forall j \in J. \qquad (13)$$

Furthermore, we define the safety stock at DC j as $v_j = r_j - L_j C1_j^2$ using (10). By implementing the change of variables in β-JLIP, we observe that the resulting objective function is increasing in C1j, and C2j is not present in the objective function. Consequently, constraints (10) and (11) can be further relaxed as the following inequalities constraints:

$$C1_j^2 \ge \sum_{i \in I} \mu_i Y_{ij}^2 \quad \forall j \in J \qquad (14)$$

$$C2_j^2 \ge \sum_{i \in I} (\sigma_i Y_{ij})^2 \quad \forall j \in J. \qquad (15)$$

Thus, the β-JLIP model can be reformulated as the following MINLP with second-order cone con-

$$\beta\text{-CQLIP: } \min_{X,Y,V,Q,C1,C2} \sum_{j \in J} \left\{ f_j X_j + \sum_{i \in I} \hat{d}_{ij} Y_{ij} + g_j(v_j, Q_j, C1_j) \right\} \qquad (16)$$

$$\text{s.t:} \quad b_j(v_j, Q_j, C2_j) \le 0 \quad \forall j \in J \qquad (17)$$

$$v_j \ge 0 \quad \forall j \in J \qquad (18)$$

$$(4), (5), (8), (9), (12), (13), (14), (15),$$

where

$$g_j(v_j, Q_j, C1_j) = S_j \frac{C1_j^2}{Q_j} + h_j \left( \frac{Q_j}{2} + v_j \right), \qquad (19)$$

and

$$b_j(v_j, Q_j, C2_j) = -\sqrt{L_j} C2_j (1 - \bar{\beta}) + L_j \frac{C2_j^2}{Q_j} \left[ G \left( \frac{v_j}{\sqrt{L_j C2_j}} \right) - G \left( \frac{v_j + Q_j}{\sqrt{L_j C2_j}} \right) \right]. \qquad (20)$$
4. An equivalent convex MINLP with second-order cone constraints of β-JLIP

The β-JLIP model is a non-convex MINLP in its objective function and constraint (6). In this section, we show how to reformulate the β-JLIP model as a convex MINLP using conic integer programming.

Similar to Atamtürk et al. (2012), we introduce a set of two non-negative continuous variables, C1j and C2j, which exploits the fact that $Y_{ij} = Y_{ij}^2$. These variables are defined as follows:

$$C1_j^2 = \sum_{i \in I} \mu_i Y_{ij}^2 \quad \forall j \in J \qquad (10)$$

$$C2_j^2 = \sum_{i \in I} (\sigma_i Y_{ij})^2 \quad \forall j \in J \qquad (11)$$

$$C1_j \ge 0 \quad \forall j \in J \qquad (12)$$

$$C2_j \ge 0 \quad \forall j \in J. \qquad (13)$$

Furthermore, we define the safety stock at DC j as $v_j = r_j - L_j C1_j^2$ using (10). By implementing the change of variables in β-JLIP, we observe that the resulting objective function is increasing in C1j, and C2j is not present in the objective function. Consequently, constraints (10) and (11) can be further relaxed as the following inequalities constraints:

$$C1_j^2 \ge \sum_{i \in I} \mu_i Y_{ij}^2 \quad \forall j \in J \qquad (14)$$

$$C2_j^2 \ge \sum_{i \in I} (\sigma_i Y_{ij})^2 \quad \forall j \in J. \qquad (15)$$

Thus, the β-JLIP model can be reformulated as the following MINLP with second-order cone con-

$$\beta\text{-CQLIP: } \min_{X,Y,V,Q,C1,C2} \sum_{j \in J} \left\{ f_j X_j + \sum_{i \in I} \hat{d}_{ij} Y_{ij} + g_j(v_j, Q_j, C1_j) \right\} \qquad (16)$$

$$\text{s.t:} \quad b_j(v_j, Q_j, C2_j) \le 0 \quad \forall j \in J \qquad (17)$$

$$v_j \ge 0 \quad \forall j \in J \qquad (18)$$

$$(4), (5), (8), (9), (12), (13), (14), (15),$$

where

$$g_j(v_j, Q_j, C1_j) = S_j \frac{C1_j^2}{Q_j} + h_j \left( \frac{Q_j}{2} + v_j \right), \qquad (19)$$

and

$$b_j(v_j, Q_j, C2_j) = -\sqrt{L_j} C2_j (1 - \bar{\beta}) + L_j \frac{C2_j^2}{Q_j} \left[ G \left( \frac{v_j}{\sqrt{L_j C2_j}} \right) - G \left( \frac{v_j + Q_j}{\sqrt{L_j C2_j}} \right) \right]. \qquad (20)$$
Constraints (14) and (15), together with constraints (12) and (13), define second-order cone con- straints. Thus, β-CQLIP is a MINLP with second-order cone constraints.

**Proposition 1.** The function $g_j$ is jointly convex in $v_j$, $Q_j$, and $C1_j$, and the function of the inequality constraint (17), $b_j$, is jointly convex in $v_j$, $Q_j$, and $C2_j$.

*Proof.* The proof is provided in Appendix B.

Thus, under Proposition 1, it can be stated that β-CQLIP is a convex MINLP with second-order cone constraints. Furthermore, β-CQLIP is a reformulation of β-JLIP. Consequently, $Z_{\beta-JLIP}^* = Z_{\beta-CQLIP}^*$, where $Z_{\beta-JLIP}^*$ and $Z_{\beta-CQLIP}^*$ are the optimal objective function of β-JLIP and β-CQLIP, respectively.

## 5. An outer-approximation solution approach

The main consequence of Proposition 1 is that we can write an equivalent β-CQLIP model replac- ing $g_j$ and $b_j$ by their outer approximations, i.e., the intersection of an infinite set of supporting functions for each one. These supporting functions correspond to linearizations of $g_j$ and $b_j$ at all $(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}, C2_j^{(k)}) \in \mathcal{G} = \{(v_j, Q_j, C1_j, C2_j) : v_j \ge 0, Q_j \ge 0, C1_j \ge 0, C2_j \ge 0\}$. We denoted fixed $(v_j, Q_j, C1_j, C2_j)$ variables as $(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}, C2_j^{(k)})$. Thus, the following conditions are satisfied:

$$ g_j(v_j, Q_j, C1_j) \ge g_j(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}) + \frac{\partial g_j^{(k)}}{\partial v_j} (v_j - v_j^{(k)}) + \frac{\partial g_j^{(k)}}{\partial Q_j} (Q_j - Q_j^{(k)}) + \frac{\partial g_j^{(k)}}{\partial C1_j} (C1_j - C1_j^{(k)}), \quad (21) $$

$$ b_j(v_j, Q_j, C2_j) \ge b_j(v_j^{(k)}, Q_j^{(k)}, C2_j^{(k)}) + \frac{\partial b_j^{(k)}}{\partial v_j} (v_j - v_j^{(k)}) + \frac{\partial b_j^{(k)}}{\partial Q_j} (Q_j - Q_j^{(k)}) + \frac{\partial b_j^{(k)}}{\partial C2_j} (C2_j - C2_j^{(k)}), \quad (22) $$

for any $(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}, C2_j^{(k)}) \in \mathcal{G}$ due to the convexity of $g_j$ and $b_j$, respectively.

Points $(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}, C2_j^{(k)})$ for any $j \in J$ result from fixing $(X, Y)$ variables in β-CQLIP to a 0-1 combination, which are denoted as $(X^{(k)}, Y^{(k)})$. The formulation, denoted as *primal problem*, result
Constraints (14) and (15), together with constraints (12) and (13), define second-order cone con- straints. Thus, β-CQLIP is a MINLP with second-order cone constraints.

**Proposition 1.** The function $g_j$ is jointly convex in $v_j$, $Q_j$, and $C1_j$, and the function of the inequality constraint (17), $b_j$, is jointly convex in $v_j$, $Q_j$, and $C2_j$.

*Proof.* The proof is provided in Appendix B.

<figure><img src="image_3.png" /></figure>

Thus, under Proposition 1, it can be stated that β-CQLIP is a convex MINLP with second-order cone constraints. Furthermore, β-CQLIP is a reformulation of β-JLIP. Consequently, $Z_{\beta-JLIP}^* = Z_{\beta-CQLIP}^*$, where $Z_{\beta-JLIP}^*$ and $Z_{\beta-CQLIP}^*$ are the optimal objective function of β-JLIP and β-CQLIP, respectively.

**5. An outer-approximation solution approach**

The main consequence of Proposition 1 is that we can write an equivalent β-CQLIP model replac- ing $g_j$ and $b_j$ by their outer approximations, i.e., the intersection of an infinite set of supporting functions for each one. These supporting functions correspond to linearizations of $g_j$ and $b_j$ at all $(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}, C2_j^{(k)}) \in \mathcal{G} = \{(v_j, Q_j, C1_j, C2_j) : v_j \ge 0, Q_j \ge 0, C1_j \ge 0, C2_j \ge 0\}$. We denoted fixed $(v_j, Q_j, C1_j, C2_j)$ variables as $(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}, C2_j^{(k)})$. Thus, the following conditions are satisfied:

$$g_j(v_j, Q_j, C1_j) \ge g_j(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}) + \frac{\partial g_j^{(k)}}{\partial v_j} (v_j - v_j^{(k)}) + \frac{\partial g_j^{(k)}}{\partial Q_j} (Q_j - Q_j^{(k)}) + \frac{\partial g_j^{(k)}}{\partial C1_j} (C1_j - C1_j^{(k)}), \quad (21)$$

$$b_j(v_j, Q_j, C2_j) \ge b_j(v_j^{(k)}, Q_j^{(k)}, C2_j^{(k)}) + \frac{\partial b_j^{(k)}}{\partial v_j} (v_j - v_j^{(k)}) + \frac{\partial b_j^{(k)}}{\partial Q_j} (Q_j - Q_j^{(k)}) + \frac{\partial b_j^{(k)}}{\partial C2_j} (C2_j - C2_j^{(k)}), \quad (22)$$

for any $(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}, C2_j^{(k)}) \in \mathcal{G}$ due to the convexity of $g_j$ and $b_j$, respectively.

Points $(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}, C2_j^{(k)})$ for any $j \in J$ result from fixing $(\mathbf{X}, \mathbf{Y})$ variables in β-CQLIP to a 0-1 combination, which are denoted as $(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$. The formulation, denoted as *primal problem*, result
in the following NLP:

$$P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)}): \min_{\mathbf{v},\mathbf{Q},\mathbf{C1},\mathbf{C2}} \sum_{j \in J} \left\{ f_j X_j^{(k)} + \sum_{i \in I} \hat{d}_{ij} Y_{ij}^{(k)} + g_j(v_j, Q_j, C1_j) \right\} \quad (23)$$

$$\text{s.t.:} \quad C1_j \ge \sqrt{\sum_{i \in I} \mu_i Y_{ij}^{(k)}} \quad \forall j \in J \quad (24)$$

$$C2_j \ge \sqrt{\sum_{i \in I} \sigma_i^2 Y_{ij}^{(k)}} \quad \forall j \in J \quad (25)$$

$$ (8), (12), (13), (17), (18). $$

The primal problem, $P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$, is a convex NLP because a convex function on a convex set is minimized. Furthermore, the primal problem is separable in $|J|$ subproblems, one for each DC $j \in J$.

Using (21), (22), and introducing a scalar $W$ in $\beta$-CQLIP, we obtain the following equivalent CQMIP, denoted as the master problem (MP),

$$\text{MP:} \quad \min_{\mathbf{X},\mathbf{Y},\mathbf{v},\mathbf{Q},\mathbf{C1},\mathbf{C2},\mathbf{W}} \sum_{j \in J} \left\{ f_j X_j + \sum_{i \in I} \hat{d}_{ij} Y_{ij} \right\} + W \quad (26)$$

$$\text{s.t.:} \quad W \ge \sum_{j \in J} \left\{ g_j(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}) + \frac{\partial g_j^{(k)}}{\partial v_j} (v_j - v_j^{(k)}) + \frac{\partial g_j^{(k)}}{\partial Q_j} (Q_j - Q_j^{(k)}) + \frac{\partial g_j^{(k)}}{\partial C1_j} (C1_j - C1_j^{(k)}) \right\} \quad \forall k \in \mathcal{K} \quad (27)$$

$$0 \ge b_j(v_j^{(k)}, Q_j^{(k)}, C2_j^{(k)}) + \frac{\partial b_j^{(k)}}{\partial v_j} (v_j - v_j^{(k)}) + \frac{\partial b_j^{(k)}}{\partial Q_j} (Q_j - Q_j^{(k)}) + \frac{\partial b_j^{(k)}}{\partial C2_j} (C2_j - C2_j^{(k)}) \quad \forall j \in J, k \in \mathcal{K} \quad (28)$$

$$ (4), (5), (8), (9), (12), (13), (18), (14), (15), $$

where $\mathcal{K} = \{k : (\mathbf{v}^{(k)}, \mathbf{Q}^{(k)}, \mathbf{C1}^{(k)}, \mathbf{C2}^{(k)})$ is a feasible solution to $P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})\}$.

The MP model consist of valid linear supports, and hence relaxation of the nonlinear function $g_j$ and $b_j$, for all points $(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}, C2_j^{(k)})$ that result from fixing $(\mathbf{X}, \mathbf{Y}) = (\mathbf{X}^{(k)}, \mathbf{Y}^{(k)}) \in \mathcal{F} = \{(\mathbf{X}, \mathbf{Y}) : (4), (5), (9)\}$. As a result, it represents a relaxation of the $\beta$-CQLIP model, and hence it is a lower bound on its solution, and it is identical to its solution if all supports are included. Clearly, it is not efficient to solve the MP model directly since we need to know all feasible $(\mathbf{v}^{(k)}, \mathbf{Q}^{(k)}, \mathbf{C1}^{(k)}, \mathbf{C2}^{(k)})$ points which implies that we have to solve all the primal problems $P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$, $(\mathbf{X}, \mathbf{Y}) \in \mathcal{F}$, i.e., exhaustive enumeration of 0-1 alternatives.

The natural approach to solving the master problem is relaxation, i.e., to consider an iterative approach
in the following NLP:

$$P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)}) : \min_{\mathbf{v}, \mathbf{Q}, \mathbf{C1}, \mathbf{C2}} \sum_{j \in J} \left\{ f_j X_j^{(k)} + \sum_{i \in I} \hat{d}_{ij} Y_{ij}^{(k)} + g_j(v_j, Q_j, C1_j) \right\} \quad (23)$$

$$\text{s.t.:} \quad C1_j \ge \sqrt{\sum_{i \in I} \mu_i Y_{ij}^{(k)}} \quad \forall j \in J \quad (24)$$

$$C2_j \ge \sqrt{\sum_{i \in I} \sigma_i^2 Y_{ij}^{(k)}} \quad \forall j \in J \quad (25)$$

(8), (12), (13), (17), (18).

The primal problem, $P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$, is a convex NLP because a convex function on a convex set is minimized. Furthermore, the primal problem is separable in $|J|$ subproblems, one for each DC $j \in J$.

Using (21), (22), and introducing a scalar $W$ in $\beta$-CQLIP, we obtain the following equivalent CQMIP, denoted as the master problem (MP),

$$\text{MP :} \quad \min_{\mathbf{X}, \mathbf{Y}, \mathbf{v}, \mathbf{Q}, \mathbf{C1}, \mathbf{C2}, \mathbf{W}} \sum_{j \in J} \left\{ f_j X_j + \sum_{i \in I} \hat{d}_{ij} Y_{ij} \right\} + W \quad (26)$$

$$\text{s.t.:} \quad W \ge \sum_{j \in J} \left\{ g_j(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}) + \frac{\partial g_j^{(k)}}{\partial v_j} (v_j - v_j^{(k)}) + \frac{\partial g_j^{(k)}}{\partial Q_j} (Q_j - Q_j^{(k)}) + \frac{\partial g_j^{(k)}}{\partial C1_j} (C1_j - C1_j^{(k)}) \right\} \quad \forall k \in \mathcal{K} \quad (27)$$

$$0 \ge b_j(v_j^{(k)}, Q_j^{(k)}, C2_j^{(k)}) + \frac{\partial b_j^{(k)}}{\partial v_j} (v_j - v_j^{(k)}) + \frac{\partial b_j^{(k)}}{\partial Q_j} (Q_j - Q_j^{(k)}) + \frac{\partial b_j^{(k)}}{\partial C2_j} (C2_j - C2_j^{(k)}) \quad \forall j \in J, k \in \mathcal{K} \quad (28)$$

(4), (5), (8), (9), (12), (13), (18), (14), (15),

where $\mathcal{K} = \{k : (\mathbf{v}^{(k)}, \mathbf{Q}^{(k)}, \mathbf{C1}^{(k)}, \mathbf{C2}^{(k)})$ is a feasible solution to $P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})\}$.

The MP model consist of valid linear supports, and hence relaxation of the nonlinear function $g_j$ and $b_j$, for all points $(v_j^{(k)}, Q_j^{(k)}, C1_j^{(k)}, C2_j^{(k)})$ that result from fixing $(\mathbf{X}, \mathbf{Y}) = (\mathbf{X}^{(k)}, \mathbf{Y}^{(k)}) \in \mathcal{F} = \{(\mathbf{X}, \mathbf{Y}) : (4), (5), (9)\}$. As a result, it represents a relaxation of the $\beta$-CQLIP model, and hence it is a lower bound on its solution, and it is identical to its solution if all supports are included. Clearly, it is not efficient to solve the MP model directly since we need to know all feasible $(\mathbf{v}^{(k)}, \mathbf{Q}^{(k)}, \mathbf{C1}^{(k)}, \mathbf{C2}^{(k)})$ points which implies that we have to solve all the primal problems $P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$, $(\mathbf{X}, \mathbf{Y}) \in \mathcal{F}$, i.e., exhaustive enumeration of 0-1 alternatives.

The natural approach to solving the master problem is relaxation, i.e., to consider an iterative approach
where at each iteration, the linear supports around all previous linearization points are considered. Thus, at each iteration, a new set of linear support constraints are added that improve the relaxation and hence the lower bound.

The first step of the iterative procedure (iteration counter $k = 0$) considers solving the well-known Uncapacitated Facility Location Problem (UFLP) to obtain a feasible distribution network configuration in terms of location-allocation variables. Let $(\mathbf{X}^{(0)}, \mathbf{Y}^{(0)})$ and $Z_{\text{UFLP}}^*$ be the optimal variables and objective function of the UFLP. Then, we set the lower bound $LB^{(0)} = Z_{\text{UFLP}}^*$, solve the primal problem $P(\mathbf{X}^{(0)}, \mathbf{Y}^{(0)})$, from which we obtain an optimal solution to $(\mathbf{v}^{(0)}, \mathbf{Q}^{(0)}, \mathbf{C1}^{(0)}, \mathbf{C2}^{(0)})$, set the best upper bound as $\inf UB = UB^{(0)} = Z_P^{(0)}$, where $Z_P^{(0)}$ is the optimal objective function of $P(\mathbf{X}^{(0)}, \mathbf{Y}^{(0)})$, and the iteration counter $k = k + 1$. The second step of the iterative procedure is solving the master problem with $\mathcal{K} = \{0, ..., k-1\}$, denoted as the relaxed master problem (RMP). Let $(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$ be the optimal solution of the relaxed master problem, and set $LB^{(k)} = Z_{\text{RMP}}^*$ be the new current lower bound, where $Z_{\text{RMP}}^*$ is the optimal objective function of the RMP model. Furthermore, $(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$ is the next point to be consider in the primal problem $P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$, from which we obtain an optimal solution to $(\mathbf{v}^{(k)}, \mathbf{Q}^{(k)}, \mathbf{C1}^{(k)}, \mathbf{C2}^{(k)})$ and the current upper bound, i.e., $UB^{(k)} = Z_P^{(k)}$, where $Z_P^{(k)}$ is the optimal objective function of $P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$. If the current upper bound is strictly lower than the best upper bound found so far, then $\inf UB = UB^{(k)}$. Thus, if $\frac{\inf UB - LB^{(k)}}{\inf UB} \le \epsilon$, then terminate, and $\inf UB$ is an $\epsilon$-optimal objective function to $\beta$-CQLIP, i.e., $\inf UB \le Z_{\beta\text{-CQLIP}}^* = Z_{\beta\text{-JLIP}}^*$. Otherwise, $k = k + 1$ and return to step 2.

The pseudo-code of the OA approach to obtain an $\epsilon$-optimal solution of $\beta$-JLIP, denoted CQ-OA Algorithm, is presented in Appendix C.

# 6. Numerical experiments and discussion

This section presents numerical results to evaluate the CQ-OA Algorithm performance to obtain an $\epsilon$-optimal solution of $\beta$-JLIP, and then some managerial insights are discussed. Numerical experiments were performed with two data sets representing U.S. cities provided by Daskin (2011): (1) a 49-node set, and (2) an 88-node set. Each node is a retailer and a candidate site for installing a DC, i.e., there are as many candidate DC locations as there are retailer locations for each set.

Model RMP is solved using CPLEX 20.1. Model $P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$ is solved using MINOS 5.4. The CQ-OA Algorithm stopping criterion is $\epsilon = 10^{-4}$ or 7200 seconds of CPU time. All tests were done on a PC with a 2.3 GHz Intel Core i9 CPU and 16 GB main memory.

## 6.1. Performance of CQ-OA Algorithm

The performance of the $\epsilon$-optimal solution for $\beta$-JLIP resulting from the CQ-OA Algorithm is evaluated by CPU times and the optimality gap of those instances that reach the time limit. In order to cover a wide range of data, we generated test problems with heterogeneous parameters. In particular, 200 random instances were generated for each node set.

Test problems (test set) used the following common criteria and parameters. The time base defined
where at each iteration, the linear supports around all previous linearization points are considered. Thus, at each iteration, a new set of linear support constraints are added that improve the relaxation and hence the lower bound.

The first step of the iterative procedure (iteration counter $k = 0$) considers solving the well-known Uncapacitated Facility Location Problem (UFLP) to obtain a feasible distribution network configuration in terms of location-allocation variables. Let $(\mathbf{X}^{(0)}, \mathbf{Y}^{(0)})$ and $Z_{\text{UFLP}}^*$ be the optimal variables and objective function of the UFLP. Then, we set the lower bound $LB^{(0)} = Z_{\text{UFLP}}^*$, solve the primal problem $P(\mathbf{X}^{(0)}, \mathbf{Y}^{(0)})$, from which we obtain an optimal solution to $(\mathbf{v}^{(0)}, \mathbf{Q}^{(0)}, \mathbf{C1}^{(0)}, \mathbf{C2}^{(0)})$, set the best upper bound as $\inf UB = UB^{(0)} = Z_P^{(0)}$, where $Z_P^{(0)}$ is the optimal objective function of $P(\mathbf{X}^{(0)}, \mathbf{Y}^{(0)})$, and the iteration counter $k = k + 1$. The second step of the iterative procedure is solving the master problem with $\mathcal{K} = \{0, ..., k-1\}$, denoted as the relaxed master problem (RMP). Let $(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$ be the optimal solution of the relaxed master problem, and set $LB^{(k)} = Z_{\text{RMP}}^*$ be the new current lower bound, where $Z_{\text{RMP}}^*$ is the optimal objective function of the RMP model. Furthermore, $(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$ is the next point to be consider in the primal problem $P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$, from which we obtain an optimal solution to $(\mathbf{v}^{(k)}, \mathbf{Q}^{(k)}, \mathbf{C1}^{(k)}, \mathbf{C2}^{(k)})$ and the current upper bound, i.e., $UB^{(k)} = Z_P^{(k)}$, where $Z_P^{(k)}$ is the optimal objective function of $P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$. If the current upper bound is strictly lower than the best upper bound found so far, then $\inf UB = UB^{(k)}$. Thus, if $\frac{\inf UB - LB^{(k)}}{\inf UB} \le \epsilon$, then terminate, and $\inf UB$ is an $\epsilon$-optimal objective function to $\beta$-CQLIP, i.e., $\inf UB \le Z_{\beta\text{-CQLIP}}^* = Z_{\beta\text{-JLIP}}^*$. Otherwise, $k = k + 1$ and return to step 2.

The pseudo-code of the OA approach to obtain an $\epsilon$-optimal solution of $\beta$-JLIP, denoted CQ-OA Algorithm, is presented in Appendix C.

# 6. Numerical experiments and discussion

This section presents numerical results to evaluate the CQ-OA Algorithm performance to obtain an $\epsilon$-optimal solution of $\beta$-JLIP, and then some managerial insights are discussed. Numerical experiments were performed with two data sets representing U.S. cities provided by Daskin (2011): (1) a 49-node set, and (2) an 88-node set. Each node is a retailer and a candidate site for installing a DC, i.e., there are as many candidate DC locations as there are retailer locations for each set.

Model RMP is solved using CPLEX 20.1. Model $P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$ is solved using MINOS 5.4. The CQ-OA Algorithm stopping criterion is $\epsilon = 10^{-4}$ or 7200 seconds of CPU time. All tests were done on a PC with a 2.3 GHz Intel Core i9 CPU and 16 GB main memory.

## 6.1. Performance of CQ-OA Algorithm

The performance of the $\epsilon$-optimal solution for $\beta$-JLIP resulting from the CQ-OA Algorithm is evaluated by CPU times and the optimality gap of those instances that reach the time limit. In order to cover a wide range of data, we generated test problems with heterogeneous parameters. In particular, 200 random instances were generated for each node set.

Test problems (test set) used the following common criteria and parameters. The time base defined
for the test set is the month. The demand per time unit at each retailer is normally distributed with mean $\mu_i = U[3000, 12000]$ for any $i \in I$, and coefficient of variation $CV_i = U[0.1, 0.5]$ for any $i \in I$. It should be noted that the normal distribution is a good approximation of non-negative demand when $CV \le 0.5$ (Peterson and Silver (1979)) in which case the probability of being less than zero is less than 0.0228. The preset service-level for the distribution network is $\bar{\beta} = U[0.75, 0.99]$. The fixed cost of installing a DC at candidate site $j$ (per time unit) is $f_j = U[4000, 8000]$ for any $j \in J$. The transportation cost per unit from the plant to candidate DC site $j$ is $a_j = a = U[0.3, 0.7]$ for any $j \in J$. The transportation cost per unit from DC $j$ to retailer $i$, $d_{ij}$, is equal to the distance between DC $j$ and retailer $i$ multiplied by a transportation rate $c_{ij} = c = U[0.001, 0.01]$ for any $i \in I$ and $j \in J$. The holding cost per unit and time unit at DC $j$ is $h_j = h = U[0.25, 1.25]$ for any $j \in J$. The ordering cost at DC $j$ is $S_j = S = U[100, 500]$ for any $j \in J$, and the lead time is $L_j = U[0.2, 0.6]$ for any $j \in J$.

For all instances with 49 and 88 nodes, the CPU times of the CQ-OA Algorithm were computed and are shown in Figure 2.

<figure><img src="image_2.png" /><figcaption>Fig. 2. CPU time using CQ-OA Algorithm</figcaption></figure>

As expected, the CPU time increases with the number of nodes. Figure 2 shows that for the set of 49-nodes, 95% of the instances are solved in less than 7200 s, while for the set of 88-nodes, 63% of the instances are solved in less than 7200 s. We compute the relative optimality gap between the lower bound given by $LB^{(k)}$ and the upper bound given by $\inf UB$, i.e., $Gap(\%) = 100 \times (\inf UB - LB^{(k)}) / \inf UB$. Figure 3 shows the relative optimality gap for all instances and node sets.
for the test set is the month. The demand per time unit at each retailer is normally distributed with mean $\mu_i = U[3000, 12000]$ for any $i \in I$, and coefficient of variation $CV_i = U[0.1, 0.5]$ for any $i \in I$. It should be noted that the normal distribution is a good approximation of non-negative demand when $CV \le 0.5$ (Peterson and Silver (1979)) in which case the probability of being less than zero is less than 0.0228. The preset service-level for the distribution network is $\bar{\beta} = U[0.75, 0.99]$. The fixed cost of installing a DC at candidate site $j$ (per time unit) is $f_j = U[4000, 8000]$ for any $j \in J$. The transportation cost per unit from the plant to candidate DC site $j$ is $a_j = a = U[0.3, 0.7]$ for any $j \in J$. The transportation cost per unit from DC $j$ to retailer $i$, $d_{ij}$, is equal to the distance between DC $j$ and retailer $i$ multiplied by a transportation rate $c_{ij} = c = U[0.001, 0.01]$ for any $i \in I$ and $j \in J$. The holding cost per unit and time unit at DC $j$ is $h_j = h = U[0.25, 1.25]$ for any $j \in J$. The ordering cost at DC $j$ is $S_j = S = U[100, 500]$ for any $j \in J$, and the lead time is $L_j = U[0.2, 0.6]$ for any $j \in J$.

For all instances with 49 and 88 nodes, the CPU times of the CQ-OA Algorithm were computed and are shown in Figure 2.

<figure><img src="image_4.png" /><figcaption>Fig. 2. CPU time using CQ-OA Algorithm</figcaption></figure>

As expected, the CPU time increases with the number of nodes. Figure 2 shows that for the set of 49-nodes, 95% of the instances are solved in less than 7200 s, while for the set of 88-nodes, 63% of the instances are solved in less than 7200 s. We compute the relative optimality gap between the lower bound given by $LB^{(k)}$ and the upper bound given by $\inf UB$, i.e., $Gap(\%) = 100 \times (\inf UB - LB^{(k)}) / \inf UB$. Figure 3 shows the relative optimality gap for all instances and node sets.
<figure><img src="image_3.png" /><figcaption>Fig. 3. Relative gap using CQ-OA Algorithm</figcaption></figure>

From Figure 3, we observe that the optimality gap is increasing with respect to the number of nodes. The average optimality gap is 0.01% and 0.3% for 49 and 88 nodes, respectively. The maximum optimality gap is 1.2% and 2.2% for 49 and 88 nodes, respectively, which occurs in instances exceeding the maximum CPU time. Although instances of maximum optimality gap occur when the CQ-OA Algorithm exceeds the time limit, the results are still good-quality solutions for the $\beta$-JLIP.

## 6.2. Illustrative example

We use an instance of the 49-node test set as an illustrative example to show the network configuration in terms of the location-allocation variables and inventory policy parameters. Table D1, in Appendix D, shows the demand per time unit and the coefficient of variation for retailers, and the fixed cost of installation and lead time for DCs. Furthermore, $\bar{\beta} = 0.975$, $c = 0.001$, $a = 0.5$, $h = 0.75$, and $S = 250$.

The illustrative example is solved using the CQ-OA algorithm in 1028 seconds and optimality gap strictly smaller than $\epsilon$. The network configuration is shown in Figure 4 where nine DCs are installed.
<figure><img src="image_5.png" /><figcaption>Fig. 3. Relative gap using CQ-OA Algorithm</figcaption></figure>

From Figure 3, we observe that the optimality gap is increasing with respect to the number of nodes. The average optimality gap is 0.01% and 0.3% for 49 and 88 nodes, respectively. The maximum optimality gap is 1.2% and 2.2% for 49 and 88 nodes, respectively, which occurs in instances exceeding the maximum CPU time. Although instances of maximum optimality gap occur when the CQ-OA Algorithm exceeds the time limit, the results are still good-quality solutions for the $\beta$-JLIP.

## 6.2. Illustrative example

We use an instance of the 49-node test set as an illustrative example to show the network configuration in terms of the location-allocation variables and inventory policy parameters. Table D1, in Appendix D, shows the demand per time unit and the coefficient of variation for retailers, and the fixed cost of installation and lead time for DCs. Furthermore, $\bar{\beta} = 0.975$, $c = 0.001$, $a = 0.5$, $h = 0.75$, and $S = 250$.

The illustrative example is solved using the CQ-OA algorithm in 1028 seconds and optimality gap strictly smaller than $\epsilon$. The network configuration is shown in Figure 4 where nine DCs are installed.
<figure><img src="image_4.png" /><figcaption>Fig. 4. Network configuration of the illustrative example</figcaption></figure>

The total cost of the distribution network is 399 523 $/month, where 13% is fixed installation cost, 71% is transportation cost, 3% is ordering cost, and 13% is holding cost. The inventory policy parameters for each installed DC are shown in Table 1.

<table><caption>Table 1
Inventory policy parameters of the illustrative example</caption><thead><tr><th></th><th>j = 8</th><th>j = 11</th><th>j = 19</th><th>j = 21</th><th>j = 27</th><th>j = 30</th><th>j = 37</th><th>j = 41</th><th>j = 44</th></tr></thead><tbody><tr><th>r<sub>j</sub></th><td>10598.5</td><td>14267.4</td><td>20618.7</td><td>21436.0</td><td>18103.1</td><td>26407.1</td><td>10452.6</td><td>36306.7</td><td>13254.0</td></tr><tr><th>Q<sub>j</sub></th><td>4580.9</td><td>7396.7</td><td>7951.7</td><td>6397.2</td><td>8168.9</td><td>8936.9</td><td>5038.0</td><td>8525.2</td><td>6092.3</td></tr></tbody></table>

### 6.3. Managerial insights

We derive management insights using a base case for each set of nodes. The fixed installation cost ($f_j$ for any $j \in J$) and lead time ($L_j$ for any $j \in J$) for the base case of 49 and 88 nodes are in Table D1 (Appendix D) and Table E1 (Appendix E), respectively. Furthermore, the base case for each set of nodes considers: $\bar{\beta} = 0.975$, $\mu_i = \mu = 7500$ for any $i \in I$, $CV_i = CV = 0.3$ for any $i \in I$, $c_{ij} = c = 0.001$ for any $i \in I$ and $j \in J$, $a_j = a = 0.5$ for any $j \in J$, $h_j = h = 0.75$ for any $j \in J$, and $S_j = S = 250$ for any $j \in J$.

Using the base case for each set of nodes, we observe how the variation of the preset fill-rate, the mean demand per time unit, the coefficient of variation, the transportation rate, and the holding cost per unit and time unit affect the number of installed DCs, costs, and performance of the CQ-OA Algorithm. We vary $\bar{\beta}$, $\mu$, $CV$, $c$, and $h$ between their lower and upper bounds defined in Section 6.1. Thus, we solved 44 problems, 22 for each node set. Table E2, in Appendix E, shows the base case for each node set, the
<figure><img src="image_6.png" /><figcaption>Fig. 4. Network configuration of the illustrative example</figcaption></figure>

The total cost of the distribution network is 399 523 $/month, where 13% is fixed installation cost, 71% is transportation cost, 3% is ordering cost, and 13% is holding cost. The inventory policy parameters for each installed DC are shown in Table 1.

<table><caption>Table 1
Inventory policy parameters of the illustrative example</caption><thead><tr><th></th><th>j = 8</th><th>j = 11</th><th>j = 19</th><th>j = 21</th><th>j = 27</th><th>j = 30</th><th>j = 37</th><th>j = 41</th><th>j = 44</th></tr></thead><tbody><tr><th>r<sub>j</sub></th><td>10598.5</td><td>14267.4</td><td>20618.7</td><td>21436.0</td><td>18103.1</td><td>26407.1</td><td>10452.6</td><td>36306.7</td><td>13254.0</td></tr><tr><th>Q<sub>j</sub></th><td>4580.9</td><td>7396.7</td><td>7951.7</td><td>6397.2</td><td>8168.9</td><td>8936.9</td><td>5038.0</td><td>8525.2</td><td>6092.3</td></tr></tbody></table>

### 6.3. Managerial insights

We derive management insights using a base case for each set of nodes. The fixed installation cost ($f_j$ for any $j \in J$) and lead time ($L_j$ for any $j \in J$) for the base case of 49 and 88 nodes are in Table D1 (Appendix D) and Table E1 (Appendix E), respectively. Furthermore, the base case for each set of nodes considers: $\bar{\beta} = 0.975$, $\mu_i = \mu = 7500$ for any $i \in I$, $CV_i = CV = 0.3$ for any $i \in I$, $c_{ij} = c = 0.001$ for any $i \in I$ and $j \in J$, $a_j = a = 0.5$ for any $j \in J$, $h_j = h = 0.75$ for any $j \in J$, and $S_j = S = 250$ for any $j \in J$.

Using the base case for each set of nodes, we observe how the variation of the preset fill-rate, the mean demand per time unit, the coefficient of variation, the transportation rate, and the holding cost per unit and time unit affect the number of installed DCs, costs, and performance of the CQ-OA Algorithm. We vary $\bar{\beta}$, $\mu$, $CV$, $c$, and $h$ between their lower and upper bounds defined in Section 6.1. Thus, we solved 44 problems, 22 for each node set. Table E2, in Appendix E, shows the base case for each node set, the
disturbed parameters, results, cost components (FC: fixed cost, TC: transportation cost, OC: ordering cost, HC: holding cost), and the CQ-OA Algorithm performance.

We have the following comments derived from the numerical experiments in Table E2.

• Regarding the number of DCs installed:

— The number of DCs is non-decreasing with the demand per time unit ($\mu$) and transportation rate (c). This is because higher demand in the network or higher transportation rate induce higher transportation cost (TC) and, similar to the UFLP, the JLIP with fill-rate constraints increases the number of DCs to reduce the transportation cost. Figure 5 shows the effect of demand per time unit and transport rate on the number of DCs installed.

<figure><img src="image_5.png" /><figcaption>Fig. 5. Effect of demand and transportation rate with $\bar{\beta} = 0.975$, $CV = 0.3$, $h = 0.75$</figcaption></figure>

— The number of DCs is non-increasing with the preset fill-rate ($\bar{\beta}$), the coefficient of variation ($CV$), and the holding cost per unit and time unit ($h$). This is because a higher preset fill-rate, demand variability, or holding cost per unit and time unit, increases the holding cost (HC). Thus, JLIP with fill-rate constraints decrease the number of DCs by taking advantage of the pooling effect described by Eppen (1979). Figure 6 shows the effect of coefficient of variation and holding cost per unit and time unit on the number of DCs installed.
disturbed parameters, results, cost components (FC: fixed cost, TC: transportation cost, OC: ordering cost, HC: holding cost), and the CQ-OA Algorithm performance.

We have the following comments derived from the numerical experiments in Table E2.

• Regarding the number of DCs installed:

— The number of DCs is non-decreasing with the demand per time unit ($\mu$) and transportation rate ($c$). This is because higher demand in the network or higher transportation rate induce higher transportation cost (TC) and, similar to the UFLP, the JLIP with fill-rate constraints increases the number of DCs to reduce the transportation cost. Figure 5 shows the effect of demand per time unit and transport rate on the number of DCs installed.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Effect of demand and transportation rate with $\bar{\beta} = 0.975$, $CV = 0.3$, $h = 0.75$</figcaption></figure>

— The number of DCs is non-increasing with the preset fill-rate ($\bar{\beta}$), the coefficient of variation ($CV$), and the holding cost per unit and time unit ($h$). This is because a higher preset fill-rate, demand variability, or holding cost per unit and time unit, increases the holding cost (HC). Thus, JLIP with fill-rate constraints decrease the number of DCs by taking advantage of the pooling effect described by Eppen (1979). Figure 6 shows the effect of coefficient of variation and holding cost per unit and time unit on the number of DCs installed.
<figure><img src="image_6.png" /><figcaption>Fig. 6. Effect of coefficient of variation and holding cost with $\bar{\beta} = 0.975$, $\mu = 7500$, $c = 0.001$</figcaption></figure>

* Regarding the cost components as percentages of the total cost:

- The fixed installation cost (FC) increases its share of the total cost (as a percentage) when the transportation rate (c) increases. The fixed installation cost decreases its share of the total cost when the preset fill-rate ($\bar{\beta}$), the coefficient of variation (CV), or the holding cost per unit and time unit (h), increases.

- The transportation cost (TC) decreases its share in the total cost when the demand per time unit ($\mu$), the coefficient of variation (CV), the transportation rate (c), or the holding cost per unit and time unit (h), increases.

- The ordering cost (OC) has a low share in the total cost in the tested instances. We only observe that the ordering cost increases its share in the total cost when the holding cost per unit and time unit (h), increase.

- The holding cost (HC) increases its share in the total cost when the $\bar{\beta}$, $\mu$, CV, c, or h, increases.

These results are consistent with the increase and decrease of the number of DCs. Furthermore, we observe that the transportation rate is the parameter that most affects the total cost and the number of installed DCs.

* Regarding the difficulty of solving the JLIP with fill-rate service level constraints:

- As expected, as the size of the distribution network grows, it becomes more difficult to solve them.

- From the 49-node network, where 70% of the instances in Table E2 were solved in less than the time limit, we observed that as the preset fill-rate, demand per time unit, coefficient of variation, or the holding cost per unit and time unit increases, the JLIP with fill-rate constraints becomes more difficult to solve.

It should be noted that we include in this section instances with $CV = 1$ to explore the effect of coefficients of variation larger than the limits defined in section 6.1. Although, under normal distribution the probability of negative demand increases for CVs strictly greater than 0.5 (Peterson and Silver (1979)), we observe that the results in terms of the number of installed DCs and cost components are consistent
<figure><img src="image_8.png" /><figcaption>Fig. 6. Effect of coefficient of variation and holding cost with $\bar{\beta} = 0.975$, $\mu = 7500$, $c = 0.001$</figcaption></figure>

• Regarding the cost components as percentages of the total cost:

- The fixed installation cost (FC) increases its share of the total cost (as a percentage) when the transportation rate (c) increases. The fixed installation cost decreases its share of the total cost when the preset fill-rate ($\bar{\beta}$), the coefficient of variation (CV), or the holding cost per unit and time unit (h), increases.

- The transportation cost (TC) decreases its share in the total cost when the demand per time unit ($\mu$), the coefficient of variation (CV), the transportation rate (c), or the holding cost per unit and time unit (h), increases.

- The ordering cost (OC) has a low share in the total cost in the tested instances. We only observe that the ordering cost increases its share in the total cost when the holding cost per unit and time unit (h), increase.

- The holding cost (HC) increases its share in the total cost when the $\bar{\beta}$, $\mu$, CV, c, or h, increases.

These results are consistent with the increase and decrease of the number of DCs. Furthermore, we observe that the transportation rate is the parameter that most affects the total cost and the number of installed DCs.

• Regarding the difficulty of solving the JLIP with fill-rate service level constraints:

- As expected, as the size of the distribution network grows, it becomes more difficult to solve them.

- From the 49-node network, where 70% of the instances in Table E2 were solved in less than the time limit, we observed that as the preset fill-rate, demand per time unit, coefficient of variation, or the holding cost per unit and time unit increases, the JLIP with fill-rate constraints becomes more difficult to solve.

It should be noted that we include in this section instances with $CV = 1$ to explore the effect of coefficients of variation larger than the limits defined in section 6.1. Although, under normal distribution the probability of negative demand increases for CVs strictly greater than 0.5 (Peterson and Silver (1979)), we observe that the results in terms of the number of installed DCs and cost components are consistent
with the results obtained for CVs less than or equal to 0.5.

# 7. Conclusions and future work

In this study, we considered a joint location-inventory model for fast-moving consumer goods in which distribution centers observe uncertain demand from a set of retailers requiring a high fill-rate service level in terms of product availability. At each DC the inventory control system operates under a continuous review (r, Q) policy with full-backorders and shortage events are controlled using the fill-rate service level.

We formulate the joint location-inventory model with fill-rate constraints as a non-convex MINLP. Using conic integer programming, we show how to reformulate it as a convex MINLP with second-order conic constraints. Consequently, an outer-approximation approach is proposed to solve the convex MINLP with second-order conic constraints using a CQMIP master problem and a convex NLP primal problem.

Numerical results show that the OA solution approach can provide good-quality solutions since for the test set problems, the maximum optimality gap is 2.2%, a very good solution, which provides us with the distribution network configuration, including the DCs location, the allocation of the demands, and the inventory policy parameters that ensure to provide the fill-rate required by the retailers.

Several issues may be addressed in future research. The first is that our OA solution approach can be applied to a wide variety of JLIP problems. For example, the capacitated JLIP, the multi-commodity JLIP, the correlated demand JLIP, the stochastic capacity JLIP, and the coordinated JLIP are all under a continuous review (r, Q) policy with fill-rate service level. The second issue is related to the scalability of the OA solution approach. Scalability can be addressed by improving the performance of the OA scheme through valid inequalities, including extended polymatroid, and extended cover cuts similar to Atamtürk et al. (2012). The third issue is to consider the design of FMCG distribution networks with other product availability measures. For example, the fraction (1 − γ) of demand being on backorder and the fraction (δ$_{L}$) of the lead time without stockouts. The fourth issue is to relax the assumption of normally distributed demand by considering, for example, the gamma distribution.

# Acknowledgments

Pablo Escalona is grateful for the support of the National Agency for Research and Development (ANID) Chile through grant FONDECYT 11200287. Alejandro Angulo is grateful for the support of the National Agency for Research and Development (ANID) Chile through the Basal Project FB0008 Advanced Center for Electrical and Electronic Engineering, AC3E.

# References

1. 2005. Background Concepts: An Introduction to the (s-1, s) Policy under Poisson and Compound Poisson Demand, Springer New York, New York, NY. pp. 37–67.
with the results obtained for CVs less than or equal to 0.5.

# 7. Conclusions and future work

In this study, we considered a joint location-inventory model for fast-moving consumer goods in which distribution centers observe uncertain demand from a set of retailers requiring a high fill-rate service level in terms of product availability. At each DC the inventory control system operates under a continuous review (r, Q) policy with full-backorders and shortage events are controlled using the fill-rate service level.

We formulate the joint location-inventory model with fill-rate constraints as a non-convex MINLP. Using conic integer programming, we show how to reformulate it as a convex MINLP with second-order conic constraints. Consequently, an outer-approximation approach is proposed to solve the convex MINLP with second-order conic constraints using a CQMIP master problem and a convex NLP primal problem.

Numerical results show that the OA solution approach can provide good-quality solutions since for the test set problems, the maximum optimality gap is 2.2%, a very good solution, which provides us with the distribution network configuration, including the DCs location, the allocation of the demands, and the inventory policy parameters that ensure to provide the fill-rate required by the retailers.

Several issues may be addressed in future research. The first is that our OA solution approach can be applied to a wide variety of JLIP problems. For example, the capacitated JLIP, the multi-commodity JLIP, the correlated demand JLIP, the stochastic capacity JLIP, and the coordinated JLIP are all under a continuous review (r, Q) policy with fill-rate service level. The second issue is related to the scalability of the OA solution approach. Scalability can be addressed by improving the performance of the OA scheme through valid inequalities, including extended polymatroid, and extended cover cuts similar to Atamtürk et al. (2012). The third issue is to consider the design of FMCG distribution networks with other product availability measures. For example, the fraction (1 − γ) of demand being on backorder and the fraction (δ$_{L}$) of the lead time without stockouts. The fourth issue is to relax the assumption of normally distributed demand by considering, for example, the gamma distribution.

# Acknowledgments

Pablo Escalona is grateful for the support of the National Agency for Research and Development (ANID) Chile through grant FONDECYT 11200287. Alejandro Angulo is grateful for the support of the National Agency for Research and Development (ANID) Chile through the Basal Project FB0008 Advanced Center for Electrical and Electronic Engineering, AC3E.

# References

1. 2005. Background Concepts: An Introduction to the (s-1, s) Policy under Poisson and Compound Poisson Demand, Springer New York, New York, NY. pp. 37–67.
Alptekinoglu, A., Banarjee, A., Paul, A., Jain, N., 2012. Inventory pooling to deliver differentiated service. *Manufacturing and service Operations Management* 15, 1–12.

Amiri-Aref, M., Klibi, W., Babai, M.Z., 2018. The multi-sourcing location inventory problem with stochastic demand. *European Journal of Operational Research* 266, 1, 72–87.

Arslan, H., Graves, S., Roemer, T., 2007. A single-product inventory model for multiple demand classes. *Management Science* 53, 1486–1500.

Atamtürk, A., Berenguer, G., Shen, Z.J., 2012. A conic integer programming approach to stochastic joint location-inventory problems. *Operations Research* 60, 2, 366–381.

Axsäter, S., 1996. Using the deterministic eqo formula in stochastic inventory control. *Management Science* 42, 6, 830–834.

Axsäter, S., 2006. A simple procedure for determining order quantities under a fill rate constraint and normally distributed lead-time demand. *European journal of operational research* 174, 1, 480–491.

Axsäter, S., 2013. When is it feasible to model low discrete demand by a normal distribution? *OR spectrum* 35, 1, 153–162.

Axsäter, S., 2015. *Inventory control*, Vol. 225. Springer.

Babiloni, E., Guijarro, E., 2020. Fill rate: from its definition to its calculation for the continuous (s, q) inventory system with discrete demands and lost sales. *Central European Journal of Operations Research* 28, 1, 35–43.

Bagajewicz, M., Manousiouthakis, V., 1991. On the generalized benders decomposition. *Computers & chemical engineering* 15, 10, 691–700.

Bala, M., Kumar, D., 2011. Supply chain performance attributes for the fast moving consumer goods industry. *Journal of transport and supply chain management* 5, 1, 23–38.

Benders, J., 1962. Partitioning procedures for solving mixed-variables programming problems. *Numerisch Mathematik* 4, 238–252.

Berman, O., Krass, D., Tajbakhsh, M., 2012. A coordinated location-inventory model. *European Journal of Operational Research* 217, 500–508.

Bernal, D.E., Peng, Z., Kronqvist, J., Grossmann, I.E., 2022. Alternative regularizations for outer-approximation algorithms for convex minlp. *Journal of Global Optimization* 84, 4, 807–842.

Bijvank, M., 2014. Periodic review inventory systems with a service level criterion. *Journal of the Operational Research Society* 65, 12, 1853–1863.

Boyd, S., Boyd, S.P., Vandenberghe, L., 2004. *Convex optimization*. Cambridge university press.

Bretthauer, K.M., Mahar, S., Venakataramanan, M., 2010. Inventory and distribution strategies for retail/e-tail organizations. *Computers & Industrial Engineering* 58, 1, 119–132.

Candas, M.F., Kutanoglu, E., 2007. Benefits of considering inventory in service parts logistics network design problems with time-based service constraints. *IIE transactions* 39, 2, 159–176.

Coey, C., Lubin, M., Vielma, J.P., 2020. Outer approximation with conic certificates for mixed-integer convex problems. *Mathematical Programming Computation* 12, 2, 249–293.

Craig, N., DeHoratius, N., Raman, A., 2016. The impact of supplier inventory service level on retailer demand. *Manufacturing & Service Operations Management* 18, 4, 461–474.

Dakin, R.J., 1965. A tree-search algorithm for mixed integer programming problems. *The computer journal* 8, 3, 250–255.

Darmawan, A., Wong, H., Thorstenson, A., 2021. Supply chain network design with coordinated inventory control. *Transportation Research Part E: Logistics and Transportation Review* 145, 102168.

Daskin, M., Coullard, C., Shen, Z., 2002. An inventory-location model: Formulation, solution algorithm and computational results. *Annals of Operations Research* 110, 83–106.

Daskin, M.S., 2011. *Network and discrete location: models, algorithms, and applications*. John Wiley & Sons.

De Mauri, M., Gillis, J., Swevers, J., Pipeleers, G., 2020. A proximal-point outer approximation algorithm. *Computational Optimization and Applications* 77, 755–777.

Dehghani, E., Pishvaee, M.S., Jabalameli, M.S., 2018. A hybrid markov process-mathematical programming approach for joint location-inventory problem under supply disruptions. *RAIRO-Operations Research* 52, 4-5, 1147–1173.

Deshpande, V., Cohen, M., Donohue, K., 2003. A threshold inventory rationing policy for service differentiated demand classes. *Management Science* 49, 683–703.

Diabat, A., Abdallah, T., Henschel, A., 2015. A closed-loop location-inventory problem with spare parts consideration. *Computers & Operations Research* 54, 245–256.

Duran, M.A., Grossmann, I.E., 1986. An outer-approximation algorithm for a class of mixed-integer nonlinear programs.
Alptekinoglu, A., Banarjee, A., Paul, A., Jain, N., 2012. Inventory pooling to deliver differentiated service. *Manufacturing and service Operations Management* 15, 1–12.

Amiri-Aref, M., Klibi, W., Babai, M.Z., 2018. The multi-sourcing location inventory problem with stochastic demand. *European Journal of Operational Research* 266, 1, 72–87.

Arslan, H., Graves, S., Roemer, T., 2007. A single-product inventory model for multiple demand classes. *Management Science* 53, 1486–1500.

Atamtürk, A., Berenguer, G., Shen, Z.J., 2012. A conic integer programming approach to stochastic joint location-inventory problems. *Operations Research* 60, 2, 366–381.

Axsäter, S., 1996. Using the deterministic eqo formula in stochastic inventory control. *Management Science* 42, 6, 830–834.

Axsäter, S., 2006. A simple procedure for determining order quantities under a fill rate constraint and normally distributed lead-time demand. *European journal of operational research* 174, 1, 480–491.

Axsäter, S., 2013. When is it feasible to model low discrete demand by a normal distribution? *OR spectrum* 35, 1, 153–162.

Axsäter, S., 2015. *Inventory control*, Vol. 225. Springer.

Babiloni, E., Guijarro, E., 2020. Fill rate: from its definition to its calculation for the continuous (s, q) inventory system with discrete demands and lost sales. *Central European Journal of Operations Research* 28, 1, 35–43.

Bagajewicz, M., Manousiouthakis, V., 1991. On the generalized benders decomposition. *Computers & chemical engineering* 15, 10, 691–700.

Bala, M., Kumar, D., 2011. Supply chain performance attributes for the fast moving consumer goods industry. *Journal of transport and supply chain management* 5, 1, 23–38.

Benders, J., 1962. Partitioning procedures for solving mixed-variables programming problems. *Numerisch Mathematik* 4, 238–252.

Berman, O., Krass, D., Tajbakhsh, M., 2012. A coordinated location-inventory model. *European Journal of Operational Research* 217, 500–508.

Bernal, D.E., Peng, Z., Kronqvist, J., Grossmann, I.E., 2022. Alternative regularizations for outer-approximation algorithms for convex minlp. *Journal of Global Optimization* 84, 4, 807–842.

Bijvank, M., 2014. Periodic review inventory systems with a service level criterion. *Journal of the Operational Research Society* 65, 12, 1853–1863.

Boyd, S., Boyd, S.P., Vandenberghe, L., 2004. *Convex optimization*. Cambridge university press.

Bretthauer, K.M., Mahar, S., Venakataramanan, M., 2010. Inventory and distribution strategies for retail/e-tail organizations. *Computers & Industrial Engineering* 58, 1, 119–132.

Candas, M.F., Kutanoglu, E., 2007. Benefits of considering inventory in service parts logistics network design problems with time-based service constraints. *IIE transactions* 39, 2, 159–176.

Coey, C., Lubin, M., Vielma, J.P., 2020. Outer approximation with conic certificates for mixed-integer convex problems. *Mathematical Programming Computation* 12, 2, 249–293.

Craig, N., DeHoratius, N., Raman, A., 2016. The impact of supplier inventory service level on retailer demand. *Manufacturing & Service Operations Management* 18, 4, 461–474.

Dakin, R.J., 1965. A tree-search algorithm for mixed integer programming problems. *The computer journal* 8, 3, 250–255.

Darmawan, A., Wong, H., Thorstenson, A., 2021. Supply chain network design with coordinated inventory control. *Transportation Research Part E: Logistics and Transportation Review* 145, 102168.

Daskin, M., Coullard, C., Shen, Z., 2002. An inventory-location model: Formulation, solution algorithm and computational results. *Annals of Operations Research* 110, 83–106.

Daskin, M.S., 2011. *Network and discrete location: models, algorithms, and applications*. John Wiley & Sons.

De Mauri, M., Gillis, J., Swevers, J., Pipeleers, G., 2020. A proximal-point outer approximation algorithm. *Computational Optimization and Applications* 77, 755–777.

Dehghani, E., Pishvaee, M.S., Jabalameli, M.S., 2018. A hybrid markov process-mathematical programming approach for joint location-inventory problem under supply disruptions. *RAIRO-Operations Research* 52, 4-5, 1147–1173.

Deshpande, V., Cohen, M., Donohue, K., 2003. A threshold inventory rationing policy for service differentiated demand classes. *Management Science* 49, 683–703.

Diabat, A., Abdallah, T., Henschel, A., 2015. A closed-loop location-inventory problem with spare parts consideration. *Computers & Operations Research* 54, 245–256.

Duran, M.A., Grossmann, I.E., 1986. An outer-approximation algorithm for a class of mixed-integer nonlinear programs.
Mathematical programming 36, 3, 307–339.

E.A. Silver, R.P. D.F. Pyke, 1998. *Inventory Management and Production Planning and Scheduling*. John Wiley, New Jersey, USA.

Eppen, G., 1979. Effect of centralization on expected cost in a multi-location newsboy problem. *Management Science* 25, 498–501.

Erlenkotter, D., 1978. A dual-based procedure for uncapacitated facility location. *Oper Res* 26, 992–1009.

Escalona, P., Araya, D., Simpson, E., Ramirez, M., Stegmaier, R., 2021. On the shortage control in a continuous review (*q*, *r*) inventory policy using $\alpha_l$ service-level. *RAIRO-Operations Research* 55, 5, 2785–2806.

Escalona, P., Marianov, V., Ordóñez, F., Stegmaier, R., 2018. On the effect of inventory policies on distribution network design with several demand classes. *Transportation Research Part E: Logistics and Transportation Review* 111, 229–240.

Escalona, P., Ordóñez, F., Iturrieta, E., 2017a. Convex backorders of a rationing inventory policy with two different demand classes. *RAIRO-Operations Research* 51, 2, 359–373.

Escalona, P., Ordóñez, F., Kauak, I., 2017b. Critical level rationing in inventory systems with continuously distributed demand. *OR Spectrum* 39, 1, 273–301.

Escalona, P., Ordóñez, F., Marianov, V., 2015. Joint location-inventory problem with differentiated service levels using critical level policy. *Transportation Research Part E: Logistics and Transportation Review* 83, 141–157.

Farahani, M., Shavandi, H., Rahmani, D., 2017. A location-inventory model considering a strategy to mitigate disruption risk in supply chain by substitutable products. *Computers & Industrial Engineering* 108, 213–224.

Farahani, R.Z., Rashidi Bajgan, H., Fahimnia, B., Kaviani, M., 2015. Location-inventory problem in supply chains: a modelling review. *International Journal of Production Research* 53, 12, 3769–3788.

Fletcher, R., Leyffer, S., 1994. Solving mixed integer nonlinear programs by outer approximation. *Mathematical programming* 66, 1, 327–349.

Floudas, C., Aggarwal, A., Ciric, A., 1989. Global optimum search for nonconvex nlp and minlp problems. *Computers & Chemical Engineering* 13, 10, 1117–1132.

Floudas, C.A., 1995. *Nonlinear and mixed-integer optimization: fundamentals and applications*. Oxford University Press.

Floudas, C.A., Visweswaran, V., 1990. A global optimization algorithm (gop) for certain classes of nonconvex nlps—i. theory. *Computers & chemical engineering* 14, 12, 1397–1417.

Geoffrion, A.M., 1972. Generalized benders decomposition. *Journal of optimization theory and applications* 10, 4, 237–260.

Javid, A.A., Azad, N., 2010. Incorporating location, routing and inventory decisions in supply chain network design. *Transportation Research Part E: Logistics and Transportation Review* 46, 5, 582–597.

Jeet, V., Kutanoglu, E., Partani, A., 2009. Logistics network design with inventory stocking for low-demand parts: Modeling and optimization. *IIE Transactions* 41, 5, 389–407.

Johnston, F., Boylan, J.E., Shale, E.A., 2003. An examination of the size of orders from customers, their characterisation and the implications for inventory control of slow moving items. *Journal of the Operational Research Society* 54, 8, 833–837.

Khan, S.A.R., Yu, Z., 2019. *Strategic supply chain management*. Springer.

Kleijn, M., Dekker, R., 1998. An overview of inventory systems wiyh several demand classes. *Econometric Institute Report* 9838/A, Erasmus University, Rotterdam, The Netherlands

Kronqvist, J., Bernal, D.E., Lundell, A., Grossmann, I.E., 2019. A review and comparison of solvers for convex minlp. *Optimization and Engineering* 20, 397–455.

Kronqvist, J., Lundell, A., Westerlund, T., 2016. The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming. *Journal of Global Optimization* 64, 249–272.

Leyffer, S., 2001. Integrating sqp and branch-and-bound for mixed integer nonlinear programming. *Computational optimization and applications* 18, 295–309.

Liu, K., Zhou, Y., Zhang, Z., 2010. Capacitated location model with online demand pooling in a multi-channel supply chain. *European Journal of Operational Research* 207, 1, 218–231.

Mahar, S., Bretthauer, K.M., Venkataramanan, M., 2009. The value of virtual pooling in dual sales channel supply chains. *European Journal of Operational Research* 192, 2, 561–575.

Mak, H.Y., Shen, Z.J.M., 2009. A two-echelon inventory-location problem with service considerations. *Naval Research Logistics (NRL)* 56, 8, 730–744.

Miranda, P., Garrido, R., 2004. Incorporating inventory control decision into a strategic distribution network design model with stochastic demand. *Transportation Research Part E* 40, 183–207.
Mathematical programming 36, 3, 307–339.

E.A. Silver, R.P. D.F. Pyke, 1998. *Inventory Management and Production Planning and Scheduling*. John Wiley, New Jersey, USA.

Eppen, G., 1979. Effect of centralization on expected cost in a multi-location newsboy problem. *Management Science* 25, 498–501.

Erlenkotter, D., 1978. A dual-based procedure for uncapacitated facility location. *Oper Res* 26, 992–1009.

Escalona, P., Araya, D., Simpson, E., Ramirez, M., Stegmaier, R., 2021. On the shortage control in a continuous review (*q*, *r*) inventory policy using $\alpha_l$ service-level. *RAIRO-Operations Research* 55, 5, 2785–2806.

Escalona, P., Marianov, V., Ordóñez, F., Stegmaier, R., 2018. On the effect of inventory policies on distribution network design with several demand classes. *Transportation Research Part E: Logistics and Transportation Review* 111, 229–240.

Escalona, P., Ordóñez, F., Iturrieta, E., 2017a. Convex backorders of a rationing inventory policy with two different demand classes. *RAIRO-Operations Research* 51, 2, 359–373.

Escalona, P., Ordóñez, F., Kauak, I., 2017b. Critical level rationing in inventory systems with continuously distributed demand. *OR Spectrum* 39, 1, 273–301.

Escalona, P., Ordóñez, F., Marianov, V., 2015. Joint location-inventory problem with differentiated service levels using critical level policy. *Transportation Research Part E: Logistics and Transportation Review* 83, 141–157.

Farahani, M., Shavandi, H., Rahmani, D., 2017. A location-inventory model considering a strategy to mitigate disruption risk in supply chain by substitutable products. *Computers & Industrial Engineering* 108, 213–224.

Farahani, R.Z., Rashidi Bajgan, H., Fahimnia, B., Kaviani, M., 2015. Location-inventory problem in supply chains: a modelling review. *International Journal of Production Research* 53, 12, 3769–3788.

Fletcher, R., Leyffer, S., 1994. Solving mixed integer nonlinear programs by outer approximation. *Mathematical programming* 66, 1, 327–349.

Floudas, C., Aggarwal, A., Ciric, A., 1989. Global optimum search for nonconvex nlp and minlp problems. *Computers & Chemical Engineering* 13, 10, 1117–1132.

Floudas, C.A., 1995. *Nonlinear and mixed-integer optimization: fundamentals and applications*. Oxford University Press.

Floudas, C.A., Visweswaran, V., 1990. A global optimization algorithm (gop) for certain classes of nonconvex nlps—i. theory. *Computers & chemical engineering* 14, 12, 1397–1417.

Geoffrion, A.M., 1972. Generalized benders decomposition. *Journal of optimization theory and applications* 10, 4, 237–260.

Javid, A.A., Azad, N., 2010. Incorporating location, routing and inventory decisions in supply chain network design. *Transportation Research Part E: Logistics and Transportation Review* 46, 5, 582–597.

Jeet, V., Kutanoglu, E., Partani, A., 2009. Logistics network design with inventory stocking for low-demand parts: Modeling and optimization. *IIE Transactions* 41, 5, 389–407.

Johnston, F., Boylan, J.E., Shale, E.A., 2003. An examination of the size of orders from customers, their characterisation and the implications for inventory control of slow moving items. *Journal of the Operational Research Society* 54, 8, 833–837.

Khan, S.A.R., Yu, Z., 2019. *Strategic supply chain management*. Springer.

Kleijn, M., Dekker, R., 1998. An overview of inventory systems wiyh several demand classes. *Econometric Institute Report* 9838/A, Erasmus University, Rotterdam, The Netherlands

Kronqvist, J., Bernal, D.E., Lundell, A., Grossmann, I.E., 2019. A review and comparison of solvers for convex minlp. *Optimization and Engineering* 20, 397–455.

Kronqvist, J., Lundell, A., Westerlund, T., 2016. The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming. *Journal of Global Optimization* 64, 249–272.

Leyffer, S., 2001. Integrating sqp and branch-and-bound for mixed integer nonlinear programming. *Computational optimization and applications* 18, 295–309.

Liu, K., Zhou, Y., Zhang, Z., 2010. Capacitated location model with online demand pooling in a multi-channel supply chain. *European Journal of Operational Research* 207, 1, 218–231.

Mahar, S., Bretthauer, K.M., Venkataramanan, M., 2009. The value of virtual pooling in dual sales channel supply chains. *European Journal of Operational Research* 192, 2, 561–575.

Mak, H.Y., Shen, Z.J.M., 2009. A two-echelon inventory-location problem with service considerations. *Naval Research Logistics (NRL)* 56, 8, 730–744.

Miranda, P., Garrido, R., 2004. Incorporating inventory control decision into a strategic distribution network design model with stochastic demand. *Transportation Research Part E* 40, 183–207.
Miranda, P., Garrido, R., 2009a. Inventory service-level optimization within distribution network design problem. *International Journal of Production Economics* 122, 276–285.

Miranda, P.A., Garrido, R.A., 2006. A simultaneous inventory control and facility location model with stochastic capacity constraints. *Networks and Spatial Economics* 6, 1, 39–53.

Miranda, P.A., Garrido, R.A., 2008. Valid inequalities for lagrangian relaxation in an inventory location problem with stochastic capacity. *Transportation Research Part E: Logistics and Transportation Review* 44, 1, 47–65.

Miranda, P.A., Garrido, R.A., 2009b. Inventory service-level optimization within distribution network design problem. *International Journal of production economics* 122, 1, 276–285.

Moon, I., Kang, S., 1998. Rationing policies for some inventory systems. *Journal of Operations Research Society* 49, 509–518.

Muts, P., Nowak, I., Hendrix, E.M., 2020. The decomposition-based outer approximation algorithm for convex mixed-integer nonlinear programming. *Journal of Global Optimization* 77, 75–96.

Möllering, K., Thonemann, U., 2010. An optimal constant level rationing policy under service level constraints. *OR Spectrum* 32, 319–341.

Nahmias, S., Demmy, C., 1981. Operating characteristics of an inventory system with rationing. *Management Science* 27, 1236–1245.

Nowak, I., 2005. *Relaxation and decomposition methods for mixed integer nonlinear programming*, Vol. 152. Springer Science & Business Media.

Ozsen, L., Coullard, C., Daskin, M., 2008. Capacitated warehouse location model with risk pooling. *Naval Research Logistics* 55, 295–312.

Park, C., 2015. Influencing factors for sustainable design implementation in the front-end of new product development process within the fast-moving-consumer-goods sector,

Perez Loaiza, R.E., Olivares-Benitez, E., Miranda Gonzalez, P.A., Guerrero Campanur, A., Martinez Flores, J.L., 2017. Supply chain network design with efficiency, location, and inventory policy using a multiobjective evolutionary algorithm. *International Transactions in Operational Research* 24, 1-2, 251–275.

Peterson, R., Silver, E.A., 1979. *Decision systems for inventory management and production planning*. Wiley New York.

P.Melchiors, Dekker, R., Kleijn, M., 2000. Inventory rationing in an (s,q) inventory model with lost sales and two demand classes. *Journal of the Operational Research Society* 51, 111–122.

Quesada, I., Grossmann, I.E., 1992. An lp/nlp based branch and bound algorithm for convex minlp optimization problems. *Computers & chemical engineering* 16, 10-11, 937–947.

Ramaekers, K., Janssens, G.K., et al., 2008. On the choice of a demand distribution for inventory management models. *European Journal of Industrial Engineering* 2, 4, 479–491.

Sadjadi, S.J., Makui, A., Dehghani, E., Pourmohammad, M., 2015. Applying queuing approach for a stochastic location-inventory problem with two different mean inventory considerations. *Applied Mathematical Modelling*

Sadjadi, S.J., Makui, A., Dehghani, E., Pourmohammad, M., 2016. Applying queuing approach for a stochastic location-inventory problem with two different mean inventory considerations. *Applied Mathematical Modelling* 40, 1, 578–596.

Sahinidis, N., Grossmann, I.E., 1991. Convergence properties of generalized benders decomposition. *Computers & Chemical Engineering* 15, 7, 481–491.

Schneider, H., 1981. Effect of service-levels on order-points or order-levels in inventory models. *The International Journal of Production Research* 19, 6, 615–631.

Schneider, H., Ringuest, J.L., 1990. Power approximation for computing (s, s) policies using service level. *Management Science* 36, 7, 822–834.

Seifbarghy, M., Behroo, T., 2021. A location-inventory problem considering restrictions on storing different perishable products with uncertain demands and lead times. *Journal of Industrial and Systems Engineering* 13, 3, 281–302.

Shahabi, M., Tafreshian, A., Unnikrishnan, A., Boyles, S.D., 2018. Joint production–inventory–location problem with multi-variate normal demand. *Transportation Research Part B: Methodological* 110, 60–78.

Shahabi, M., Unnikrishnan, A., Jafari-Shirazi, E., Boyles, S.D., 2014. A three level location-inventory problem with correlated demand. *Transportation Research Part B: Methodological* 69, 1–18.

Shen, Z., Coullard, C., Daskin, M., 2003. A joint location-inventory model. *Transportation Science* 37, 40–55.

Shen, Z.J., Daskin, Z.M., 2005. Trade-offs between customer service and cost in integrated supply chain design. *Manufacturing and Service Operations Management* 7, 188–207.

Shen, Z.J.M., 2005. A multi-commodity supply chain design problem. *IIE Transactions* 37, 753–762.
Miranda, P., Garrido, R., 2009a. Inventory service-level optimization within distribution network design problem. *International Journal of Production Economics* 122, 276–285.

Miranda, P.A., Garrido, R.A., 2006. A simultaneous inventory control and facility location model with stochastic capacity constraints. *Networks and Spatial Economics* 6, 1, 39–53.

Miranda, P.A., Garrido, R.A., 2008. Valid inequalities for lagrangian relaxation in an inventory location problem with stochastic capacity. *Transportation Research Part E: Logistics and Transportation Review* 44, 1, 47–65.

Miranda, P.A., Garrido, R.A., 2009b. Inventory service-level optimization within distribution network design problem. *International Journal of production economics* 122, 1, 276–285.

Moon, I., Kang, S., 1998. Rationing policies for some inventory systems. *Journal of Operations Research Society* 49, 509–518.

Muts, P., Nowak, I., Hendrix, E.M., 2020. The decomposition-based outer approximation algorithm for convex mixed-integer nonlinear programming. *Journal of Global Optimization* 77, 75–96.

Möllering, K., Thonemann, U., 2010. An optimal constant level rationing policy under service level constraints. *OR Spectrum* 32, 319–341.

Nahmias, S., Demmy, C., 1981. Operating characteristics of an inventory system with rationing. *Management Science* 27, 1236–1245.

Nowak, I., 2005. *Relaxation and decomposition methods for mixed integer nonlinear programming*, Vol. 152. Springer Science & Business Media.

Ozsen, L., Coullard, C., Daskin, M., 2008. Capacitated warehouse location model with risk pooling. *Naval Research Logistics* 55, 295–312.

Park, C., 2015. Influencing factors for sustainable design implementation in the front-end of new product development process within the fast-moving-consumer-goods sector,

Perez Loaiza, R.E., Olivares-Benitez, E., Miranda Gonzalez, P.A., Guerrero Campanur, A., Martinez Flores, J.L., 2017. Supply chain network design with efficiency, location, and inventory policy using a multiobjective evolutionary algorithm. *International Transactions in Operational Research* 24, 1-2, 251–275.

Peterson, R., Silver, E.A., 1979. *Decision systems for inventory management and production planning*. Wiley New York.

P.Melchiors, Dekker, R., Kleijn, M., 2000. Inventory rationing in an (s,q) inventory model with lost sales and two demand classes. *Journal of the Operational Research Society* 51, 111–122.

Quesada, I., Grossmann, I.E., 1992. An lp/nlp based branch and bound algorithm for convex minlp optimization problems. *Computers & chemical engineering* 16, 10-11, 937–947.

Ramaekers, K., Janssens, G.K., et al., 2008. On the choice of a demand distribution for inventory management models. *European Journal of Industrial Engineering* 2, 4, 479–491.

Sadjadi, S.J., Makui, A., Dehghani, E., Pourmohammad, M., 2015. Applying queuing approach for a stochastic location-inventory problem with two different mean inventory considerations. *Applied Mathematical Modelling*

Sadjadi, S.J., Makui, A., Dehghani, E., Pourmohammad, M., 2016. Applying queuing approach for a stochastic location-inventory problem with two different mean inventory considerations. *Applied Mathematical Modelling* 40, 1, 578–596.

Sahinidis, N., Grossmann, I.E., 1991. Convergence properties of generalized benders decomposition. *Computers & Chemical Engineering* 15, 7, 481–491.

Schneider, H., 1981. Effect of service-levels on order-points or order-levels in inventory models. *The International Journal of Production Research* 19, 6, 615–631.

Schneider, H., Ringuest, J.L., 1990. Power approximation for computing (s, s) policies using service level. *Management Science* 36, 7, 822–834.

Seifbarghy, M., Behroo, T., 2021. A location-inventory problem considering restrictions on storing different perishable products with uncertain demands and lead times. *Journal of Industrial and Systems Engineering* 13, 3, 281–302.

Shahabi, M., Tafreshian, A., Unnikrishnan, A., Boyles, S.D., 2018. Joint production–inventory–location problem with multi-variate normal demand. *Transportation Research Part B: Methodological* 110, 60–78.

Shahabi, M., Unnikrishnan, A., Jafari-Shirazi, E., Boyles, S.D., 2014. A three level location-inventory problem with correlated demand. *Transportation Research Part B: Methodological* 69, 1–18.

Shen, Z., Coullard, C., Daskin, M., 2003. A joint location-inventory model. *Transportation Science* 37, 40–55.

Shen, Z.J., Daskin, Z.M., 2005. Trade-offs between customer service and cost in integrated supply chain design. *Manufacturing and Service Operations Management* 7, 188–207.

Shen, Z.J.M., 2005. A multi-commodity supply chain design problem. *IIE Transactions* 37, 753–762.
Shen, Z.J.M., Qi, L., 2007. Incorporating inventory and routing costs in strategic location models. *European journal of operational research* 179, 2, 372–389.

Sherbrooke, C.C., 2006. *Optimal inventory modeling of systems: multi-echelon techniques*, Vol. 72. Springer Science & Business Media.

Shu, J., Teo, C.P., Shen, Z.J.M., 2005. Stochastic transportation-inventory network design problem. *Operations Research* 53, 1, 48–60.

Silver, E.A., Pyke, D.F., Peterson, R., et al., 1998. *Inventory management and production planning and scheduling*, Vol. 3. Wiley New York.

Snyder, L., Daskin, M., Teo, C.P., 2007. The stochastic location model with risk pooling. *European Journal of Operational Research* 179, 1221–1238.

Sourirajan, K., Ozsen, L., Uzsoy, R., 2007. A single-product network design model with lead time and safety stock considerations. *IIE Transactions* 39, 5, 411–424.

Su, L., Tang, L., Bernal, D.E., Grossmann, I.E., 2018. Improved quadratic cuts for convex mixed-integer nonlinear programs. *Computers & Chemical Engineering* 109, 77–95.

Su, L., Tang, L., Grossmann, I.E., 2015. Computational strategies for improved minlp algorithms. *Computers & Chemical Engineering* 75, 40–48.

Tan, Y., Paul, A.A., Deng, Q., Wei, L., 2017. Mitigating inventory overstocking: Optimal order-up-to level to achieve a target fill rate over a finite horizon. *Production and Operations Management* 26, 11, 1971–1988.

Tanonkou, G.A., Benyoucef, L., Xie, X., 2008. Design of stochastic distribution networks using lagrangian relaxation. *IEEE Transactions on automation science and engineering* 5, 4, 597–608.

Tapia-Ubeda, F.J., Miranda, P.A., Macchi, M., 2018. A generalized benders decomposition based algorithm for an inventory location problem with stochastic inventory capacity constraints. *European Journal of Operational Research* 267, 3, 806–817.

Teunter, R.H., Syntetos, A.A., Babai, M.Z., 2017. Stock keeping unit fill rate specification. *European Journal of Operational Research* 259, 3, 917–925.

Tsao, Y.C., Mangotra, D., Lu, J.C., Dong, M., 2012. A continuous approximation approach for the integrated facility-inventory allocation problem. *European journal of operational research* 222, 2, 216–228.

Wang, M., Wu, J., Kafa, N., Klibi, W., 2020. Carbon emission-compliance green location-inventory problem with demand and carbon price uncertainties. *Transportation Research Part E: Logistics and Transportation Review* 142, 102038.

Webster, M., Beach, R., Fouweather, I., 2006. E-business strategy development: an fmcg sector case study. *Supply Chain Management: An International Journal* 11, 4, 353–362.

Westerlund, T., Pettersson, F., 1995. An extended cutting plane method for solving convex minlp problems. *Computers & chemical engineering* 19, 131–136.

Wheatley, D., Gzara, F., Jewkes, E., 2015. Logic-based benders decomposition for an inventory-location problem with service constraints. *Omega* 55, 10–23.

Yao, Z., Lee, L.H., Jaruphongsa, W., Tan, V., Hui, C.F., 2010. Multi-source facility location–allocation and inventory problem. *European Journal of Operational Research* 207, 2, 750–762.

You, F., Grossmann, I., 2008. Mixed-integer nonlinear programming models and algorithms for large-scale supply chain design with stochastic inventory management. *Industrial & Engineering Chemistry Research* 47, 7802–7817.

Zhang, Z.H., Berenguer, G., Shen, Z.J., 2014. A capacitated facility location model with bidirectional flows. *Transportation Science* 49, 1, 114–129.

Zheng, Y.S., 1992. On properties of stochastic inventory systems. *Management science* 38, 1, 87–103.

Zipkin, P., 1986. Inventory service-level measures: convexity and approximation. *Management Science* 32, 8, 975–981.
Shen, Z.J.M., Qi, L., 2007. Incorporating inventory and routing costs in strategic location models. *European journal of operational research* 179, 2, 372–389.

Sherbrooke, C.C., 2006. *Optimal inventory modeling of systems: multi-echelon techniques*, Vol. 72. Springer Science & Business Media.

Shu, J., Teo, C.P., Shen, Z.J.M., 2005. Stochastic transportation-inventory network design problem. *Operations Research* 53, 1, 48–60.

Silver, E.A., Pyke, D.F., Peterson, R., et al., 1998. *Inventory management and production planning and scheduling*, Vol. 3. Wiley New York.

Snyder, L., Daskin, M., Teo, C.P., 2007. The stochastic location model with risk pooling. *European Journal of Operational Research* 179, 1221–1238.

Sourirajan, K., Ozsen, L., Uzsoy, R., 2007. A single-product network design model with lead time and safety stock considerations. *IIE Transactions* 39, 5, 411–424.

Su, L., Tang, L., Bernal, D.E., Grossmann, I.E., 2018. Improved quadratic cuts for convex mixed-integer nonlinear programs. *Computers & Chemical Engineering* 109, 77–95.

Su, L., Tang, L., Grossmann, I.E., 2015. Computational strategies for improved minlp algorithms. *Computers & Chemical Engineering* 75, 40–48.

Tan, Y., Paul, A.A., Deng, Q., Wei, L., 2017. Mitigating inventory overstocking: Optimal order-up-to level to achieve a target fill rate over a finite horizon. *Production and Operations Management* 26, 11, 1971–1988.

Tanonkou, G.A., Benyoucef, L., Xie, X., 2008. Design of stochastic distribution networks using lagrangian relaxation. *IEEE Transactions on automation science and engineering* 5, 4, 597–608.

Tapia-Ubeda, F.J., Miranda, P.A., Macchi, M., 2018. A generalized benders decomposition based algorithm for an inventory location problem with stochastic inventory capacity constraints. *European Journal of Operational Research* 267, 3, 806–817.

Teunter, R.H., Syntetos, A.A., Babai, M.Z., 2017. Stock keeping unit fill rate specification. *European Journal of Operational Research* 259, 3, 917–925.

Tsao, Y.C., Mangotra, D., Lu, J.C., Dong, M., 2012. A continuous approximation approach for the integrated facility-inventory allocation problem. *European journal of operational research* 222, 2, 216–228.

Wang, M., Wu, J., Kafa, N., Klibi, W., 2020. Carbon emission-compliance green location-inventory problem with demand and carbon price uncertainties. *Transportation Research Part E: Logistics and Transportation Review* 142, 102038.

Webster, M., Beach, R., Fouweather, I., 2006. E-business strategy development: an fmcg sector case study. *Supply Chain Management: An International Journal* 11, 4, 353–362.

Westerlund, T., Pettersson, F., 1995. An extended cutting plane method for solving convex minlp problems. *Computers & chemical engineering* 19, 131–136.

Wheatley, D., Gzara, F., Jewkes, E., 2015. Logic-based benders decomposition for an inventory-location problem with service constraints. *Omega* 55, 10–23.

Yao, Z., Lee, L.H., Jaruphongsa, W., Tan, V., Hui, C.F., 2010. Multi-source facility location–allocation and inventory problem. *European Journal of Operational Research* 207, 2, 750–762.

You, F., Grossmann, I., 2008. Mixed-integer nonlinear programming models and algorithms for large-scale supply chain design with stochastic inventory management. *Industrial & Engineering Chemistry Research* 47, 7802–7817.

Zhang, Z.H., Berenguer, G., Shen, Z.J., 2014. A capacitated facility location model with bidirectional flows. *Transportation Science* 49, 1, 114–129.

Zheng, Y.S., 1992. On properties of stochastic inventory systems. *Management science* 38, 1, 87–103.

Zipkin, P., 1986. Inventory service-level measures: convexity and approximation. *Management Science* 32, 8, 975–981.
# Appendix A. Glossary of terms.

<table><caption>Table A1
Sets, parameters, and variables</caption><thead><tr><th colspan="2">Sets</th></tr></thead><tbody><tr><td>I</td><td>Set of retailers, indexed by i = 1, ..., |I|</td></tr><tr><td>J</td><td>Set of possible DC locations, indexed by j = 1, ..., |J|</td></tr><tr><th colspan="2">Parameters</th></tr><tr><td>μ<sub>i</sub></td><td>Mean demand per time unit of retailer i</td></tr><tr><td>σ<sup>2</sup><sub>i</sub></td><td>Variance of demand per time unit of retailer i</td></tr><tr><td>f<sub>j</sub></td><td>Fixed cost per time unit to install a DC at candidate site j</td></tr><tr><td>d<sub>ij</sub></td><td>Transportation cost per unit from DC j to retailer i</td></tr><tr><td>c<sub>ij</sub></td><td>Transportation rate from DC j to retailer i</td></tr><tr><td>a<sub>j</sub></td><td>Transportation cost per unit from the plant to DC j</td></tr><tr><td>S<sub>j</sub></td><td>Ordering cost at DC j</td></tr><tr><td>h<sub>j</sub></td><td>Holding cost per unit and time unit at DC j</td></tr><tr><td>L<sub>j</sub></td><td>Fixed lead time in time unit from the supplier to DC j</td></tr><tr><td>β</td><td>Preset fill-rate service level</td></tr><tr><th colspan="2">Variables</th></tr><tr><td>X<sub>j</sub></td><td>1 if a DC is located at candidate site j, and 0 otherwise</td></tr><tr><td>Y<sub>ij</sub></td><td>1 if DC j serves retailer i, and 0 otherwise</td></tr><tr><td>Q<sub>j</sub></td><td>Fixed order quantity at candidate DC j</td></tr><tr><td>r<sub>j</sub></td><td>Reorder point at candidate DC j</td></tr></tbody></table>

# Appendix B. Convexity proof of $g_j$ and $b_j$.

*Proof.* We first show that $g_j$ is jointly convex in $v_j$, $Q_j$, and $C1_j$. The first term of $g_j$ is convex in $Q_j$ and $C1_j$ because it is a quadratic-over-linear function (Boyd et al. (2004), section 3.1.5, page 73). The second term of $g_j$ is a linear function in $Q_j$ and $v_j$, respectively. Thus, $g_j$ is a convex function because it is a sum of convex functions.

The proof of the convexity of $b_j$ is as follows. The first term in (20) is convex in $C2_j$ because it is a linear function. Thus, the convexity proof of $b_j$ is reduced to proving the convexity of the second term of (20). Consider $\hat{b}_j: \mathbb{R}^2 \to \mathbb{R}$ defined as $\hat{b}_j(v_j, Q_j) = \frac{1}{Q_j}(G(v_j) - G(v_j + Q_j))$, which is jointly convex in $v_j$ and $Q_j$ according to Zipkin (1986) and Axsäter (2015), and its perspective function $\bar{b}_j: \mathbb{R}^3 \to \mathbb{R}$ is defined as:

$$ \bar{b}_j(v_j, Q_j, C2_j) = \sqrt{L_j C 2_j} \hat{b}_j \left( \frac{v_j}{\sqrt{L_j C 2_j}}, \frac{Q_j}{\sqrt{L_j C 2_j}} \right) = L_j \frac{C 2_j^2}{Q_j} \left[ G \left( \frac{v_j}{\sqrt{L_j C 2_j}} \right) - G \left( \frac{v_j + Q_j}{\sqrt{L_j C 2_j}} \right) \right]. $$

As the perspective operation preserves convexity (Boyd et al. (2004), section 3.2.6, page 89), we conclude that $b_j$ is the sum of convex functions. □
# Appendix A. Glossary of terms.

<table><caption>Table A1
Sets, parameters, and variables</caption><thead><tr><th colspan="2">Sets</th></tr></thead><tbody><tr><td>I</td><td>Set of retailers, indexed by i = 1, ..., |I|</td></tr><tr><td>J</td><td>Set of possible DC locations, indexed by j = 1, ..., |J|</td></tr><tr><th colspan="2">Parameters</th></tr><tr><td>μ<sub>i</sub></td><td>Mean demand per time unit of retailer i</td></tr><tr><td>σ<sup>2</sup><sub>i</sub></td><td>Variance of demand per time unit of retailer i</td></tr><tr><td>f<sub>j</sub></td><td>Fixed cost per time unit to install a DC at candidate site j</td></tr><tr><td>d<sub>ij</sub></td><td>Transportation cost per unit from DC j to retailer i</td></tr><tr><td>c<sub>ij</sub></td><td>Transportation rate from DC j to retailer i</td></tr><tr><td>a<sub>j</sub></td><td>Transportation cost per unit from the plant to DC j</td></tr><tr><td>S<sub>j</sub></td><td>Ordering cost at DC j</td></tr><tr><td>h<sub>j</sub></td><td>Holding cost per unit and time unit at DC j</td></tr><tr><td>L<sub>j</sub></td><td>Fixed lead time in time unit from the supplier to DC j</td></tr><tr><td>β</td><td>Preset fill-rate service level</td></tr><tr><th colspan="2">Variables</th></tr><tr><td>X<sub>j</sub></td><td>1 if a DC is located at candidate site j, and 0 otherwise</td></tr><tr><td>Y<sub>ij</sub></td><td>1 if DC j serves retailer i, and 0 otherwise</td></tr><tr><td>Q<sub>j</sub></td><td>Fixed order quantity at candidate DC j</td></tr><tr><td>r<sub>j</sub></td><td>Reorder point at candidate DC j</td></tr></tbody></table>

# Appendix B. Convexity proof of $g_j$ and $b_j$.

*Proof.* We first show that $g_j$ is jointly convex in $v_j$, $Q_j$, and $C1_j$. The first term of $g_j$ is convex in $Q_j$ and $C1_j$ because it is a quadratic-over-linear function (Boyd et al. (2004), section 3.1.5, page 73). The second term of $g_j$ is a linear function in $Q_j$ and $v_j$, respectively. Thus, $g_j$ is a convex function because it is a sum of convex functions.

The proof of the convexity of $b_j$ is as follows. The first term in (20) is convex in $C2_j$ because it is a linear function. Thus, the convexity proof of $b_j$ is reduced to proving the convexity of the second term of (20). Consider $\hat{b}_j: \mathbb{R}^2 \to \mathbb{R}$ defined as $\hat{b}_j(v_j, Q_j) = \frac{1}{Q_j}(G(v_j) - G(v_j + Q_j))$, which is jointly convex in $v_j$ and $Q_j$ according to Zipkin (1986) and Axsäter (2015), and its perspective function $\bar{b}_j: \mathbb{R}^3 \to \mathbb{R}$ is defined as:

$$ \bar{b}_j(v_j, Q_j, C2_j) = \sqrt{L_j C 2_j} \hat{b}_j \left( \frac{v_j}{\sqrt{L_j C 2_j}}, \frac{Q_j}{\sqrt{L_j C 2_j}} \right) = L_j \frac{C 2_j^2}{Q_j} \left[ G \left( \frac{v_j}{\sqrt{L_j C 2_j}} \right) - G \left( \frac{v_j + Q_j}{\sqrt{L_j C 2_j}} \right) \right]. $$

As the perspective operation preserves convexity (Boyd et al. (2004), section 3.2.6, page 89), we conclude that $b_j$ is the sum of convex functions. □
Appendix C. CQ - Outer Approximation Algorithm for β-JILP.

<caption>**Algorithm 1 CQ-OA Algorithm for β-JILP**</caption>

1: $k = 0$
2: $(\mathbf{X}^{(0)}, \mathbf{Y}^{(0)}) \leftarrow \text{solve UFLP}$
3: Set $LB^{(0)} = Z_{\text{UFLP}}^{*}$
4: $(\mathbf{v}^{(0)}, \mathbf{Q}^{(0)}, \mathbf{C1}^{(0)}, \mathbf{C2}^{(0)})$, $Z_{P}^{(0)} \leftarrow \text{solve } P(\mathbf{X}^{(0)}, \mathbf{Y}^{(0)})$
5: Set $\inf_{UB} UB = UB^{(0)} = Z_{P}^{(0)}$
6: **while** $\frac{\inf_{UB} UB - LB^{(k)}}{\inf_{UB}} > \epsilon$ **do**
7: &nbsp;&nbsp;&nbsp;&nbsp;$k = k + 1$
8: &nbsp;&nbsp;&nbsp;&nbsp;$(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$, $LB^{(k)} = Z_{\text{RMP}}^{*}$ $\leftarrow$ solve RMP
9: &nbsp;&nbsp;&nbsp;&nbsp;$(\mathbf{v}^{(k)}, \mathbf{Q}^{(k)}, \mathbf{C1}^{(k)}, \mathbf{C2}^{(k)})$, $UB^{(k)} = Z_{P}^{(k)} \leftarrow \text{solve } P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$
10: &nbsp;&nbsp;&nbsp;&nbsp;**if** $\inf_{UB} UB > UB^{(k)}$ **then**
11: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$\inf_{UB} UB = UB^{(k)}$
12: &nbsp;&nbsp;&nbsp;&nbsp;**end if**
13: **end while**
14: **while** $j \le |J|$ **do**
15: &nbsp;&nbsp;&nbsp;&nbsp;**if** $X_j = 1$ **then**
16: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$r_j = v_j + L_j C 1_j^2$
17: &nbsp;&nbsp;&nbsp;&nbsp;**else**
18: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$r_j = 0$
19: &nbsp;&nbsp;&nbsp;&nbsp;**end if**
20: **end while**
21: Return $(\mathbf{X}, \mathbf{Y}, \mathbf{r}, \mathbf{Q})$, $\inf_{UB}$
Appendix C. CQ - Outer Approximation Algorithm for β-JILP.

<caption>**Algorithm 1 CQ-OA Algorithm for β-JILP**</caption>

1: $k = 0$
2: $(\mathbf{X}^{(0)}, \mathbf{Y}^{(0)}) \leftarrow \text{solve UFLP}$
3: Set $LB^{(0)} = Z_{\text{UFLP}}^{*}$
4: $(\mathbf{v}^{(0)}, \mathbf{Q}^{(0)}, \mathbf{C1}^{(0)}, \mathbf{C2}^{(0)})$, $Z_{P}^{(0)} \leftarrow \text{solve } P(\mathbf{X}^{(0)}, \mathbf{Y}^{(0)})$
5: Set $\inf_{UB} UB = UB^{(0)} = Z_{P}^{(0)}$
6: **while** $\frac{\inf_{UB} UB - LB^{(k)}}{\inf_{UB}} > \epsilon$ **do**
7: &nbsp;&nbsp;&nbsp;&nbsp;$k = k + 1$
8: &nbsp;&nbsp;&nbsp;&nbsp;$(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$, $LB^{(k)} = Z_{\text{RMP}}^{*}$ $\leftarrow$ solve RMP
9: &nbsp;&nbsp;&nbsp;&nbsp;$(\mathbf{v}^{(k)}, \mathbf{Q}^{(k)}, \mathbf{C1}^{(k)}, \mathbf{C2}^{(k)})$, $UB^{(k)} = Z_{P}^{(k)} \leftarrow \text{solve } P(\mathbf{X}^{(k)}, \mathbf{Y}^{(k)})$
10: &nbsp;&nbsp;&nbsp;&nbsp;**if** $\inf_{UB} UB > UB^{(k)}$ **then**
11: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$\inf_{UB} UB = UB^{(k)}$
12: &nbsp;&nbsp;&nbsp;&nbsp;**end if**
13: **end while**
14: **while** $j \le |J|$ **do**
15: &nbsp;&nbsp;&nbsp;&nbsp;**if** $X_j = 1$ **then**
16: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$r_j = v_j + L_j C 1_j^2$
17: &nbsp;&nbsp;&nbsp;&nbsp;**else**
18: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$r_j = 0$
19: &nbsp;&nbsp;&nbsp;&nbsp;**end if**
20: **end while**
21: Return $(\mathbf{X}, \mathbf{Y}, \mathbf{r}, \mathbf{Q})$, $\inf_{UB}$
# Appendix D. Illustrative example data.

<table><caption>Table D1
Illustrative example data</caption><thead><tr><th>Node</th><th>μ<sub>i</sub></th><th>CV<sub>i</sub></th><th>f<sub>j</sub></th><th>L<sub>j</sub></th><th>Node</th><th>μ<sub>i</sub></th><th>CV<sub>i</sub></th><th>f<sub>j</sub></th><th>L<sub>j</sub></th><th>Node</th><th>μ<sub>i</sub></th><th>CV<sub>i</sub></th><th>f<sub>j</sub></th><th>L<sub>j</sub></th></tr></thead><tbody><tr><td>1</td><td>8990</td><td>0.27</td><td>7276</td><td>0.27</td><td>18</td><td>3791</td><td>0.46</td><td>5145</td><td>0.53</td><td>35</td><td>5342</td><td>0.42</td><td>5159</td><td>0.6</td></tr><tr><td>2</td><td>9020</td><td>0.23</td><td>5455</td><td>0.29</td><td>19</td><td>3326</td><td>0.44</td><td>5244</td><td>0.3</td><td>36</td><td>8470</td><td>0.32</td><td>5435</td><td>0.43</td></tr><tr><td>3</td><td>3014</td><td>0.49</td><td>6145</td><td>0.41</td><td>20</td><td>6496</td><td>0.23</td><td>4611</td><td>0.31</td><td>37</td><td>7831</td><td>0.2</td><td>6209</td><td>0.36</td></tr><tr><td>4</td><td>6963</td><td>0.15</td><td>5866</td><td>0.2</td><td>21</td><td>11304</td><td>0.28</td><td>4577</td><td>0.57</td><td>38</td><td>8998</td><td>0.41</td><td>5605</td><td>0.58</td></tr><tr><td>5</td><td>7831</td><td>0.2</td><td>6209</td><td>0.36</td><td>22</td><td>8465</td><td>0.49</td><td>7028</td><td>0.4</td><td>39</td><td>5416</td><td>0.15</td><td>5326</td><td>0.51</td></tr><tr><td>6</td><td>9656</td><td>0.4</td><td>6413</td><td>0.39</td><td>23</td><td>5710</td><td>0.42</td><td>7277</td><td>0.52</td><td>40</td><td>5829</td><td>0.19</td><td>5011</td><td>0.33</td></tr><tr><td>7</td><td>4486</td><td>0.11</td><td>7639</td><td>0.4</td><td>24</td><td>8988</td><td>0.37</td><td>7411</td><td>0.25</td><td>41</td><td>11860</td><td>0.26</td><td>6797</td><td>0.57</td></tr><tr><td>8</td><td>10967</td><td>0.23</td><td>4987</td><td>0.45</td><td>25</td><td>8566</td><td>0.14</td><td>6213</td><td>0.43</td><td>42</td><td>6679</td><td>0.32</td><td>7199</td><td>0.36</td></tr><tr><td>9</td><td>4251</td><td>0.1</td><td>5741</td><td>0.2</td><td>26</td><td>3687</td><td>0.5</td><td>5790</td><td>0.52</td><td>43</td><td>6706</td><td>0.22</td><td>4894</td><td>0.22</td></tr><tr><td>10</td><td>8182</td><td>0.26</td><td>7397</td><td>0.42</td><td>27</td><td>10608</td><td>0.17</td><td>5431</td><td>0.24</td><td>44</td><td>11370</td><td>0.46</td><td>6123</td><td>0.32</td></tr><tr><td>11</td><td>10543</td><td>0.31</td><td>4073</td><td>0.22</td><td>28</td><td>4549</td><td>0.46</td><td>6024</td><td>0.44</td><td>45</td><td>3211</td><td>0.41</td><td>6040</td><td>0.34</td></tr><tr><td>12</td><td>10025</td><td>0.33</td><td>4289</td><td>0.47</td><td>29</td><td>9959</td><td>0.42</td><td>5940</td><td>0.35</td><td>46</td><td>7185</td><td>0.35</td><td>7048</td><td>0.53</td></tr><tr><td>13</td><td>7144</td><td>0.4</td><td>7754</td><td>0.56</td><td>30</td><td>11351</td><td>0.32</td><td>6750</td><td>0.33</td><td>47</td><td>11428</td><td>0.24</td><td>5409</td><td>0.43</td></tr><tr><td>14</td><td>3121</td><td>0.4</td><td>6174</td><td>0.38</td><td>31</td><td>7444</td><td>0.35</td><td>7443</td><td>0.45</td><td>48</td><td>9639</td><td>0.43</td><td>4953</td><td>0.39</td></tr><tr><td>15</td><td>3913</td><td>0.22</td><td>6819</td><td>0.27</td><td>32</td><td>9393</td><td>0.25</td><td>7737</td><td>0.47</td><td>49</td><td>7620</td><td>0.37</td><td>7875</td><td>0.54</td></tr><tr><td>16</td><td>8275</td><td>0.5</td><td>5737</td><td>0.3</td><td>33</td><td>6524</td><td>0.44</td><td>5872</td><td>0.36</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>17</td><td>8025</td><td>0.35</td><td>4385</td><td>0.39</td><td>34</td><td>7192</td><td>0.41</td><td>4407</td><td>0.23</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

# Appendix E. Test set for managerial insights.

<table><caption>Table E1
Base case 88-nodes</caption><thead><tr><th>Node</th><th>f<sub>j</sub></th><th>L<sub>j</sub></th><th>Node</th><th>f<sub>j</sub></th><th>L<sub>j</sub></th><th>Node</th><th>f<sub>j</sub></th><th>L<sub>j</sub></th><th>Node</th><th>f<sub>j</sub></th><th>L<sub>j</sub></th><th>Node</th><th>f<sub>j</sub></th><th>L<sub>j</sub></th></tr></thead><tbody><tr><td>1</td><td>5482</td><td>0.2</td><td>19</td><td>5409</td><td>0.43</td><td>37</td><td>6666</td><td>0.31</td><td>55</td><td>7443</td><td>0.45</td><td>73</td><td>4407</td><td>0.23</td></tr><tr><td>2</td><td>5203</td><td>0.45</td><td>20</td><td>7754</td><td>0.56</td><td>38</td><td>5397</td><td>0.28</td><td>56</td><td>6750</td><td>0.33</td><td>74</td><td>6209</td><td>0.36</td></tr><tr><td>3</td><td>6797</td><td>0.36</td><td>21</td><td>6907</td><td>0.34</td><td>39</td><td>7075</td><td>0.34</td><td>57</td><td>5435</td><td>0.43</td><td>75</td><td>7591</td><td>0.55</td></tr><tr><td>4</td><td>4635</td><td>0.21</td><td>22</td><td>5811</td><td>0.45</td><td>40</td><td>7058</td><td>0.42</td><td>58</td><td>5737</td><td>0.3</td><td>76</td><td>7875</td><td>0.54</td></tr><tr><td>5</td><td>7659</td><td>0.3</td><td>23</td><td>7055</td><td>0.43</td><td>41</td><td>7276</td><td>0.27</td><td>59</td><td>7028</td><td>0.4</td><td>77</td><td>7048</td><td>0.53</td></tr><tr><td>6</td><td>7826</td><td>0.45</td><td>24</td><td>5863</td><td>0.32</td><td>42</td><td>5542</td><td>0.31</td><td>60</td><td>5872</td><td>0.36</td><td>78</td><td>5326</td><td>0.51</td></tr><tr><td>7</td><td>4767</td><td>0.25</td><td>25</td><td>4385</td><td>0.39</td><td>43</td><td>6465</td><td>0.5</td><td>61</td><td>7199</td><td>0.36</td><td>79</td><td>5011</td><td>0.33</td></tr><tr><td>8</td><td>7618</td><td>0.54</td><td>26</td><td>5790</td><td>0.52</td><td>44</td><td>5605</td><td>0.38</td><td>62</td><td>5159</td><td>0.6</td><td>80</td><td>6819</td><td>0.27</td></tr><tr><td>9</td><td>7411</td><td>0.25</td><td>27</td><td>6145</td><td>0.41</td><td>45</td><td>6096</td><td>0.58</td><td>63</td><td>5431</td><td>0.24</td><td>81</td><td>5145</td><td>0.53</td></tr><tr><td>10</td><td>5966</td><td>0.45</td><td>28</td><td>6841</td><td>0.55</td><td>46</td><td>4711</td><td>0.31</td><td>64</td><td>4987</td><td>0.45</td><td>82</td><td>5244</td><td>0.3</td></tr><tr><td>11</td><td>6597</td><td>0.27</td><td>29</td><td>6024</td><td>0.44</td><td>47</td><td>4016</td><td>0.38</td><td>65</td><td>6797</td><td>0.57</td><td>83</td><td>6040</td><td>0.34</td></tr><tr><td>12</td><td>4718</td><td>0.26</td><td>30</td><td>6970</td><td>0.54</td><td>48</td><td>6173</td><td>0.49</td><td>66</td><td>5866</td><td>0.2</td><td>84</td><td>7277</td><td>0.52</td></tr><tr><td>13</td><td>6174</td><td>0.38</td><td>31</td><td>4336</td><td>0.43</td><td>49</td><td>5075</td><td>0.46</td><td>67</td><td>7737</td><td>0.47</td><td>85</td><td>4894</td><td>0.22</td></tr><tr><td>14</td><td>4378</td><td>0.5</td><td>32</td><td>6877</td><td>0.25</td><td>50</td><td>5122</td><td>0.37</td><td>68</td><td>5940</td><td>0.35</td><td>86</td><td>5605</td><td>0.58</td></tr><tr><td>15</td><td>4015</td><td>0.44</td><td>33</td><td>7896</td><td>0.4</td><td>51</td><td>4611</td><td>0.31</td><td>69</td><td>7639</td><td>0.4</td><td>87</td><td>6123</td><td>0.32</td></tr><tr><td>16</td><td>6413</td><td>0.39</td><td>34</td><td>5831</td><td>0.39</td><td>52</td><td>4577</td><td>0.57</td><td>70</td><td>5455</td><td>0.29</td><td>88</td><td>4953</td><td>0.39</td></tr><tr><td>17</td><td>6606</td><td>0.26</td><td>35</td><td>7669</td><td>0.42</td><td>53</td><td>7397</td><td>0.42</td><td>71</td><td>6213</td><td>0.43</td><td></td><td></td><td></td></tr><tr><td>18</td><td>7940</td><td>0.49</td><td>36</td><td>4073</td><td>0.22</td><td>54</td><td>4289</td><td>0.47</td><td>72</td><td>5741</td><td>0.2</td><td></td><td></td><td></td></tr></tbody></table>
# Appendix D. Illustrative example data.

<table><caption>Table D1
Illustrative example data</caption><thead><tr><th>Node</th><th>μ<sub>i</sub></th><th>CV<sub>i</sub></th><th>f<sub>j</sub></th><th>L<sub>j</sub></th><th>Node</th><th>μ<sub>i</sub></th><th>CV<sub>i</sub></th><th>f<sub>j</sub></th><th>L<sub>j</sub></th><th>Node</th><th>μ<sub>i</sub></th><th>CV<sub>i</sub></th><th>f<sub>j</sub></th><th>L<sub>j</sub></th></tr></thead><tbody><tr><td>1</td><td>8990</td><td>0.27</td><td>7276</td><td>0.27</td><td>18</td><td>3791</td><td>0.46</td><td>5145</td><td>0.53</td><td>35</td><td>5342</td><td>0.42</td><td>5159</td><td>0.6</td></tr><tr><td>2</td><td>9020</td><td>0.23</td><td>5455</td><td>0.29</td><td>19</td><td>3326</td><td>0.44</td><td>5244</td><td>0.3</td><td>36</td><td>8470</td><td>0.32</td><td>5435</td><td>0.43</td></tr><tr><td>3</td><td>3014</td><td>0.49</td><td>6145</td><td>0.41</td><td>20</td><td>6496</td><td>0.23</td><td>4611</td><td>0.31</td><td>37</td><td>7831</td><td>0.2</td><td>6209</td><td>0.36</td></tr><tr><td>4</td><td>6963</td><td>0.15</td><td>5866</td><td>0.2</td><td>21</td><td>11304</td><td>0.28</td><td>4577</td><td>0.57</td><td>38</td><td>8998</td><td>0.41</td><td>5605</td><td>0.58</td></tr><tr><td>5</td><td>7831</td><td>0.2</td><td>6209</td><td>0.36</td><td>22</td><td>8465</td><td>0.49</td><td>7028</td><td>0.4</td><td>39</td><td>5416</td><td>0.15</td><td>5326</td><td>0.51</td></tr><tr><td>6</td><td>9656</td><td>0.4</td><td>6413</td><td>0.39</td><td>23</td><td>5710</td><td>0.42</td><td>7277</td><td>0.52</td><td>40</td><td>5829</td><td>0.19</td><td>5011</td><td>0.33</td></tr><tr><td>7</td><td>4486</td><td>0.11</td><td>7639</td><td>0.4</td><td>24</td><td>8988</td><td>0.37</td><td>7411</td><td>0.25</td><td>41</td><td>11860</td><td>0.26</td><td>6797</td><td>0.57</td></tr><tr><td>8</td><td>10967</td><td>0.23</td><td>4987</td><td>0.45</td><td>25</td><td>8566</td><td>0.14</td><td>6213</td><td>0.43</td><td>42</td><td>6679</td><td>0.32</td><td>7199</td><td>0.36</td></tr><tr><td>9</td><td>4251</td><td>0.1</td><td>5741</td><td>0.2</td><td>26</td><td>3687</td><td>0.5</td><td>5790</td><td>0.52</td><td>43</td><td>6706</td><td>0.22</td><td>4894</td><td>0.22</td></tr><tr><td>10</td><td>8182</td><td>0.26</td><td>7397</td><td>0.42</td><td>27</td><td>10608</td><td>0.17</td><td>5431</td><td>0.24</td><td>44</td><td>11370</td><td>0.46</td><td>6123</td><td>0.32</td></tr><tr><td>11</td><td>10543</td><td>0.31</td><td>4073</td><td>0.22</td><td>28</td><td>4549</td><td>0.46</td><td>6024</td><td>0.44</td><td>45</td><td>3211</td><td>0.41</td><td>6040</td><td>0.34</td></tr><tr><td>12</td><td>10025</td><td>0.33</td><td>4289</td><td>0.47</td><td>29</td><td>9959</td><td>0.42</td><td>5940</td><td>0.35</td><td>46</td><td>7185</td><td>0.35</td><td>7048</td><td>0.53</td></tr><tr><td>13</td><td>7144</td><td>0.4</td><td>7754</td><td>0.56</td><td>30</td><td>11351</td><td>0.32</td><td>6750</td><td>0.33</td><td>47</td><td>11428</td><td>0.24</td><td>5409</td><td>0.43</td></tr><tr><td>14</td><td>3121</td><td>0.4</td><td>6174</td><td>0.38</td><td>31</td><td>7444</td><td>0.35</td><td>7443</td><td>0.45</td><td>48</td><td>9639</td><td>0.43</td><td>4953</td><td>0.39</td></tr><tr><td>15</td><td>3913</td><td>0.22</td><td>6819</td><td>0.27</td><td>32</td><td>9393</td><td>0.25</td><td>7737</td><td>0.47</td><td>49</td><td>7620</td><td>0.37</td><td>7875</td><td>0.54</td></tr><tr><td>16</td><td>8275</td><td>0.5</td><td>5737</td><td>0.3</td><td>33</td><td>6524</td><td>0.44</td><td>5872</td><td>0.36</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>17</td><td>8025</td><td>0.35</td><td>4385</td><td>0.39</td><td>34</td><td>7192</td><td>0.41</td><td>4407</td><td>0.23</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

# Appendix E. Test set for managerial insights.

<table><caption>Table E1
Base case 88-nodes</caption><thead><tr><th>Node</th><th>f<sub>j</sub></th><th>L<sub>j</sub></th><th>Node</th><th>f<sub>j</sub></th><th>L<sub>j</sub></th><th>Node</th><th>f<sub>j</sub></th><th>L<sub>j</sub></th><th>Node</th><th>f<sub>j</sub></th><th>L<sub>j</sub></th><th>Node</th><th>f<sub>j</sub></th><th>L<sub>j</sub></th></tr></thead><tbody><tr><td>1</td><td>5482</td><td>0.2</td><td>19</td><td>5409</td><td>0.43</td><td>37</td><td>6666</td><td>0.31</td><td>55</td><td>7443</td><td>0.45</td><td>73</td><td>4407</td><td>0.23</td></tr><tr><td>2</td><td>5203</td><td>0.45</td><td>20</td><td>7754</td><td>0.56</td><td>38</td><td>5397</td><td>0.28</td><td>56</td><td>6750</td><td>0.33</td><td>74</td><td>6209</td><td>0.36</td></tr><tr><td>3</td><td>6797</td><td>0.36</td><td>21</td><td>6907</td><td>0.34</td><td>39</td><td>7075</td><td>0.34</td><td>57</td><td>5435</td><td>0.43</td><td>75</td><td>7591</td><td>0.55</td></tr><tr><td>4</td><td>4635</td><td>0.21</td><td>22</td><td>5811</td><td>0.45</td><td>40</td><td>7058</td><td>0.42</td><td>58</td><td>5737</td><td>0.3</td><td>76</td><td>7875</td><td>0.54</td></tr><tr><td>5</td><td>7659</td><td>0.3</td><td>23</td><td>7055</td><td>0.43</td><td>41</td><td>7276</td><td>0.27</td><td>59</td><td>7028</td><td>0.4</td><td>77</td><td>7048</td><td>0.53</td></tr><tr><td>6</td><td>7826</td><td>0.45</td><td>24</td><td>5863</td><td>0.32</td><td>42</td><td>5542</td><td>0.31</td><td>60</td><td>5872</td><td>0.36</td><td>78</td><td>5326</td><td>0.51</td></tr><tr><td>7</td><td>4767</td><td>0.25</td><td>25</td><td>4385</td><td>0.39</td><td>43</td><td>6465</td><td>0.5</td><td>61</td><td>7199</td><td>0.36</td><td>79</td><td>5011</td><td>0.33</td></tr><tr><td>8</td><td>7618</td><td>0.54</td><td>26</td><td>5790</td><td>0.52</td><td>44</td><td>5605</td><td>0.38</td><td>62</td><td>5159</td><td>0.6</td><td>80</td><td>6819</td><td>0.27</td></tr><tr><td>9</td><td>7411</td><td>0.25</td><td>27</td><td>6145</td><td>0.41</td><td>45</td><td>6096</td><td>0.58</td><td>63</td><td>5431</td><td>0.24</td><td>81</td><td>5145</td><td>0.53</td></tr><tr><td>10</td><td>5966</td><td>0.45</td><td>28</td><td>6841</td><td>0.55</td><td>46</td><td>4711</td><td>0.31</td><td>64</td><td>4987</td><td>0.45</td><td>82</td><td>5244</td><td>0.3</td></tr><tr><td>11</td><td>6597</td><td>0.27</td><td>29</td><td>6024</td><td>0.44</td><td>47</td><td>4016</td><td>0.38</td><td>65</td><td>6797</td><td>0.57</td><td>83</td><td>6040</td><td>0.34</td></tr><tr><td>12</td><td>4718</td><td>0.26</td><td>30</td><td>6970</td><td>0.54</td><td>48</td><td>6173</td><td>0.49</td><td>66</td><td>5866</td><td>0.2</td><td>84</td><td>7277</td><td>0.52</td></tr><tr><td>13</td><td>6174</td><td>0.38</td><td>31</td><td>4336</td><td>0.43</td><td>49</td><td>5075</td><td>0.46</td><td>67</td><td>7737</td><td>0.47</td><td>85</td><td>4894</td><td>0.22</td></tr><tr><td>14</td><td>4378</td><td>0.5</td><td>32</td><td>6877</td><td>0.25</td><td>50</td><td>5122</td><td>0.37</td><td>68</td><td>5940</td><td>0.35</td><td>86</td><td>5605</td><td>0.58</td></tr><tr><td>15</td><td>4015</td><td>0.44</td><td>33</td><td>7896</td><td>0.4</td><td>51</td><td>4611</td><td>0.31</td><td>69</td><td>7639</td><td>0.4</td><td>87</td><td>6123</td><td>0.32</td></tr><tr><td>16</td><td>6413</td><td>0.39</td><td>34</td><td>5831</td><td>0.39</td><td>52</td><td>4577</td><td>0.57</td><td>70</td><td>5455</td><td>0.29</td><td>88</td><td>4953</td><td>0.39</td></tr><tr><td>17</td><td>6606</td><td>0.26</td><td>35</td><td>7669</td><td>0.42</td><td>53</td><td>7397</td><td>0.42</td><td>71</td><td>6213</td><td>0.43</td><td></td><td></td><td></td></tr><tr><td>18</td><td>7940</td><td>0.49</td><td>36</td><td>4073</td><td>0.22</td><td>54</td><td>4289</td><td>0.47</td><td>72</td><td>5741</td><td>0.2</td><td></td><td></td><td></td></tr></tbody></table>
<table><caption>Table E2
Test set for managerial insights</caption><thead><tr><th rowspan="2">Data set</th><th rowspan="2">Vary</th><th colspan="10">Parameters</th><th colspan="3">Cost Component</th><th colspan="3">Performance</th></tr><tr><th>β</th><th>μ</th><th>CV</th><th>c</th><th>h</th><th>Results</th><th>† DC</th><th>inf U/B</th><th>FC</th><th>TC</th><th>OC</th><th>HC</th><th>Iterations</th><th>Gap(%)</th><th>CPU Time (s)</th></tr></thead><tbody><tr><td rowspan="10">49-nodes</td><td rowspan="2">Case base</td><td>β</td><td>0.975</td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>10</td><td>398480</td><td>59855</td><td>279325</td><td>14372</td><td>44727</td><td>51</td><td>&lt; e</td><td>2069</td></tr><tr><td>β</td><td><strong>0.75</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>10</td><td>369431</td><td>60874</td><td>277575</td><td>18647</td><td>12335</td><td>46</td><td>&lt; e</td><td>367</td></tr><tr><td rowspan="2">μ</td><td>β</td><td><strong>0.99</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>10</td><td>407729</td><td>48125</td><td>294075</td><td>13649</td><td>65</td><td>0.21%</td><td>7200*</td></tr><tr><td>μ</td><td><strong>3000</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>4</td><td>200438</td><td>20575</td><td>146190</td><td>5049</td><td>28624</td><td>31</td><td>&lt; e</td><td>1128</td></tr><tr><td rowspan="2">CV</td><td>β</td><td><strong>12000</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>13</td><td>570031</td><td>71709</td><td>428160</td><td>19514</td><td>5058</td><td>67</td><td>&lt; e</td><td>3344</td></tr><tr><td>CV</td><td><strong>7500</strong></td><td>7500</td><td><strong>0.1</strong></td><td>0.001</td><td>0.75</td><td>10</td><td>375173</td><td>51293</td><td>287550</td><td>15326</td><td>21004</td><td>38</td><td>&lt; e</td><td>508</td></tr><tr><td rowspan="2">c</td><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>10</td><td>423614</td><td>54943</td><td>287550</td><td>12467</td><td>68653</td><td>23</td><td>0.50%</td><td>7200*</td></tr><tr><td>β</td><td><strong>1</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>6</td><td>474525</td><td>32810</td><td>329925</td><td>7982</td><td>103808</td><td>9</td><td>0.41%</td><td>7200*</td></tr><tr><td rowspan="2">c, h</td><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.0005</td><td>0.75</td><td>2</td><td>248083</td><td>9301</td><td>213600</td><td>6352</td><td>18831</td><td>24</td><td>&lt; e</td><td>875</td></tr><tr><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.0005</td><td>0.75</td><td>5</td><td>334770</td><td>25678</td><td>266250</td><td>9711</td><td>33131</td><td>37</td><td>0.59%</td><td>7200*</td></tr><tr><td rowspan="2">h</td><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td>0.75</td><td>33</td><td>583887</td><td>186046</td><td>281325</td><td>24599</td><td>91917</td><td>3</td><td>&lt; e</td><td>6</td></tr><tr><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.01</td><td>0.75</td><td>46</td><td>616008</td><td>275139</td><td>198675</td><td>29572</td><td>112622</td><td>2</td><td>&lt; e</td><td>2</td></tr><tr><td rowspan="2">μ, CV, c, h</td><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td><strong>0.25</strong></td><td>10</td><td>362047</td><td>51669</td><td>281700</td><td>8619</td><td>20060</td><td>28</td><td>&lt; e</td><td>169</td></tr><tr><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td><strong>1.25</strong></td><td>8</td><td>430436</td><td>43252</td><td>303750</td><td>15103</td><td>68332</td><td>22</td><td>1.09%</td><td>7200*</td></tr><tr><td rowspan="2">c</td><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td><strong>1.25</strong></td><td>2</td><td>260217</td><td>8967</td><td>217875</td><td>7794</td><td>25581</td><td>25</td><td>&lt; e</td><td>1729</td></tr><tr><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.0005</td><td><strong>1.25</strong></td><td>4</td><td>356688</td><td>25820</td><td>265650</td><td>12176</td><td>53042</td><td>25</td><td>1.11%</td><td>7200*</td></tr><tr><td rowspan="2">c, h</td><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td><strong>1.25</strong></td><td>31</td><td>642570</td><td>173257</td><td>299175</td><td>28845</td><td>141293</td><td>6</td><td>&lt; e</td><td>30</td></tr><tr><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td><strong>1.25</strong></td><td>45</td><td>689130</td><td>269708</td><td>206550</td><td>35343</td><td>177530</td><td>2</td><td>&lt; e</td><td>6</td></tr><tr><td rowspan="2">μ, CV, c, h</td><td>β</td><td><strong>12000</strong></td><td>7500</td><td>0.5</td><td>0.0001</td><td><strong>1.25</strong></td><td>2</td><td>407203</td><td>9301</td><td>341760</td><td>9188</td><td>46954</td><td>21</td><td>&lt; e</td><td>877</td></tr><tr><td>β</td><td><strong>12000</strong></td><td>7500</td><td>0.5</td><td>0.0005</td><td><strong>1.25</strong></td><td>6</td><td>551520</td><td>29109</td><td>428520</td><td>14507</td><td>79384</td><td>17</td><td>1.17%</td><td>7200*</td></tr><tr><td rowspan="2">c</td><td>β</td><td><strong>12000</strong></td><td>7500</td><td>0.5</td><td>0.005</td><td><strong>1.25</strong></td><td>39</td><td>899027</td><td>232517</td><td>371040</td><td>38255</td><td>257215</td><td>2</td><td>&lt; e</td><td>10</td></tr><tr><td>β</td><td><strong>12000</strong></td><td>7500</td><td>0.5</td><td>0.01</td><td><strong>1.25</strong></td><td>46</td><td>924143</td><td>275139</td><td>317880</td><td>42189</td><td>288935</td><td>2</td><td>&lt; e</td><td>5</td></tr><tr><td rowspan="10">88-nodes</td><td rowspan="2">Case base</td><td>β</td><td>0.975</td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>16</td><td>679622</td><td>83508</td><td>493500</td><td>22820</td><td>79794</td><td>20</td><td>0.97%</td><td>7200*</td></tr><tr><td>β</td><td><strong>0.75</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>18</td><td>629774</td><td>99573</td><td>466500</td><td>32360</td><td>31341</td><td>63</td><td>0.79%</td><td>7200*</td></tr><tr><td rowspan="2">μ</td><td>β</td><td><strong>0.99</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>15</td><td>694137</td><td>90518</td><td>483000</td><td>24556</td><td>96063</td><td>20</td><td>1.03%</td><td>7200*</td></tr><tr><td>μ</td><td><strong>3000</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>9</td><td>344449</td><td>49007</td><td>231600</td><td>9963</td><td>53879</td><td>20</td><td>0.68%</td><td>7200*</td></tr><tr><td rowspan="2">CV</td><td>β</td><td><strong>12000</strong></td><td>7500</td><td><strong>0.1</strong></td><td>0.001</td><td>0.75</td><td>18</td><td>972603</td><td>142855</td><td>682800</td><td>38734</td><td>108214</td><td>19</td><td>1.03%</td><td>7200*</td></tr><tr><td>CV</td><td><strong>7500</strong></td><td>7500</td><td><strong>0.5</strong></td><td>0.001</td><td>0.75</td><td>13</td><td>640797</td><td>100018</td><td>466500</td><td>29338</td><td>44941</td><td>30</td><td>1.15%</td><td>7200*</td></tr><tr><td rowspan="2">c</td><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>11</td><td>808305</td><td>60165</td><td>543000</td><td>13984</td><td>191157</td><td>11</td><td>0.63%</td><td>7200*</td></tr><tr><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.0001</td><td>0.75</td><td>3</td><td>441576</td><td>19822</td><td>377550</td><td>12129</td><td>32075</td><td>31</td><td>1.07%</td><td>7200*</td></tr><tr><td rowspan="2">h</td><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td><strong>0.25</strong></td><td>20</td><td>576598</td><td>58360</td><td>433500</td><td>19217</td><td>65521</td><td>19</td><td>0.88%</td><td>7200*</td></tr><tr><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td><strong>0.25</strong></td><td>13</td><td>729018</td><td>69666</td><td>465000</td><td>15706</td><td>35805</td><td>62</td><td>0.01%</td><td>7200*</td></tr><tr><td rowspan="2">c, h</td><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td><strong>1.25</strong></td><td>2</td><td>459383</td><td>15919</td><td>386400</td><td>12959</td><td>44105</td><td>14</td><td>&lt; e</td><td>211</td></tr><tr><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td><strong>1.25</strong></td><td>9</td><td>615272</td><td>46106</td><td>456000</td><td>21331</td><td>91835</td><td>32</td><td>1.01%</td><td>7200*</td></tr><tr><td rowspan="2">μ, CV, c, h</td><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td><strong>1.25</strong></td><td>44</td><td>1048073</td><td>253041</td><td>536250</td><td>46104</td><td>212678</td><td>21</td><td>1.08%</td><td>7200*</td></tr><tr><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.01</td><td><strong>1.25</strong></td><td>65</td><td>1173544</td><td>415796</td><td>410325</td><td>58375</td><td>289048</td><td>35</td><td>&lt; e</td><td>1133</td></tr><tr><td rowspan="2">c</td><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.3</td><td>0.0001</td><td><strong>1.25</strong></td><td>3</td><td>728138</td><td>16294</td><td>617040</td><td>14969</td><td>79835</td><td>30</td><td>1.06%</td><td>7200*</td></tr><tr><td>β</td><td><strong>7500</strong></td><td>7500</td><td>0.5</td><td>0.0001</td><td><strong>1.25</strong></td><td>10</td><td>947215</td><td>53390</td><td>706800</td><td>26077</td><td>160948</td><td>16</td><td>0.98%</td><td>7200*</td></tr><tr><td rowspan="2">c, h</td><td>β</td><td><strong>12000</strong></td><td>7500</td><td>0.5</td><td>0.0005</td><td><strong>1.25</strong></td><td>53</td><td>1493316</td><td>306716</td><td>738000</td><td>59473</td><td>389127</td><td>39</td><td>0.09%</td><td>7200*</td></tr><tr><td>β</td><td><strong>12000</strong></td><td>7500</td><td>0.5</td><td>0.005</td><td><strong>1.25</strong></td><td>74</td><td>1604897</td><td>441261</td><td>610560</td><td>70767</td><td>482309</td><td>7</td><td>&lt; e</td><td>135</td></tr></tbody></table>

a Exceeds the time limit.
<table><caption>Table E2
Test set for managerial insights</caption><thead><tr><th rowspan="2">Data set</th><th rowspan="2">Vary</th><th colspan="10">Parameters</th><th colspan="3">Cost Component</th><th colspan="3">Performance</th></tr><tr><th>β</th><th>μ</th><th>CV</th><th>c</th><th>h</th><th>Results</th><th>† DC</th><th>inf/UB</th><th>FC</th><th>TC</th><th>OC</th><th>HC</th><th>Iterations</th><th>Gap(%)</th><th>CPU Time (s)</th></tr></thead><tbody><tr><td rowspan="10">49-nodes</td><td rowspan="2">Case base</td><td>0.975</td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>10</td><td>398480</td><td>59855</td><td>279325</td><td>14372</td><td>44727</td><td>51</td><td>&lt; e</td><td>2069</td></tr><tr><td><strong>β</strong></td><td><strong>0.75</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>10</td><td>369431</td><td>60874</td><td>277575</td><td>18647</td><td>12335</td><td>46</td><td>&lt; e</td><td>367</td></tr><tr><td rowspan="2">μ</td><td><strong>0.99</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>10</td><td>407729</td><td>48125</td><td>294075</td><td>13649</td><td>51879</td><td>65</td><td>0.21%</td><td>7200*</td></tr><tr><td><strong>3000</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>4</td><td>200438</td><td>20575</td><td>146190</td><td>5049</td><td>28624</td><td>31</td><td>&lt; e</td><td>1128</td></tr><tr><td rowspan="2">CV</td><td><strong>12000</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>13</td><td>570031</td><td>71709</td><td>428160</td><td>19514</td><td>5058</td><td>67</td><td>&lt; e</td><td>3344</td></tr><tr><td><strong>0.1</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>10</td><td>375173</td><td>51293</td><td>287550</td><td>15326</td><td>21004</td><td>38</td><td>&lt; e</td><td>508</td></tr><tr><td rowspan="2">c</td><td><strong>0.5</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>10</td><td>423614</td><td>54943</td><td>287550</td><td>12467</td><td>68653</td><td>23</td><td>0.50%</td><td>7200*</td></tr><tr><td><strong>1</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>6</td><td>474525</td><td>32810</td><td>329925</td><td>7982</td><td>103808</td><td>9</td><td>0.41%</td><td>7200*</td></tr><tr><td rowspan="2">c, h</td><td><strong>0.5</strong></td><td>7500</td><td>0.3</td><td>0.0001</td><td>0.75</td><td>2</td><td>248083</td><td>9301</td><td>213600</td><td>6352</td><td>18831</td><td>24</td><td>&lt; e</td><td>875</td></tr><tr><td><strong>0.0005</strong></td><td>7500</td><td>0.3</td><td>0.0005</td><td>0.75</td><td>5</td><td>334770</td><td>25678</td><td>266250</td><td>9711</td><td>33131</td><td>37</td><td>0.59%</td><td>7200*</td></tr><tr><td rowspan="2">h</td><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td>0.75</td><td>33</td><td>583887</td><td>186046</td><td>281325</td><td>24599</td><td>91917</td><td>3</td><td>&lt; e</td><td>6</td></tr><tr><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.01</td><td>0.75</td><td>46</td><td>616008</td><td>275139</td><td>198675</td><td>29572</td><td>112622</td><td>2</td><td>&lt; e</td><td>2</td></tr><tr><td rowspan="2">μ, CV, c, h</td><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td><strong>0.25</strong></td><td>10</td><td>362047</td><td>51669</td><td>281700</td><td>8619</td><td>20060</td><td>28</td><td>&lt; e</td><td>169</td></tr><tr><td><strong>12000</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td><strong>1.25</strong></td><td>8</td><td>430436</td><td>43252</td><td>303750</td><td>15103</td><td>68332</td><td>22</td><td>1.09%</td><td>7200*</td></tr><tr><td rowspan="2">c</td><td><strong>0.5</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td><strong>1.25</strong></td><td>2</td><td>260217</td><td>8967</td><td>217875</td><td>7794</td><td>25581</td><td>25</td><td>&lt; e</td><td>1729</td></tr><tr><td><strong>0.0005</strong></td><td>7500</td><td>0.3</td><td>0.0005</td><td><strong>1.25</strong></td><td>4</td><td>356688</td><td>25820</td><td>265650</td><td>12176</td><td>53042</td><td>25</td><td>1.11%</td><td>7200*</td></tr><tr><td rowspan="2">μ, CV, c, h</td><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td><strong>1.25</strong></td><td>31</td><td>642570</td><td>173257</td><td>299175</td><td>28845</td><td>141293</td><td>6</td><td>&lt; e</td><td>30</td></tr><tr><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.01</td><td><strong>1.25</strong></td><td>45</td><td>689130</td><td>269708</td><td>206550</td><td>35343</td><td>177530</td><td>2</td><td>&lt; e</td><td>6</td></tr><tr><td rowspan="2">c, h</td><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.0005</td><td><strong>1.25</strong></td><td>2</td><td>407203</td><td>9301</td><td>341760</td><td>9188</td><td>46954</td><td>21</td><td>&lt; e</td><td>877</td></tr><tr><td><strong>12000</strong></td><td>7500</td><td>0.3</td><td>0.0005</td><td><strong>1.25</strong></td><td>6</td><td>551520</td><td>29109</td><td>428520</td><td>14507</td><td>79384</td><td>17</td><td>1.17%</td><td>7200*</td></tr><tr><td rowspan="2">μ, CV, c, h</td><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td><strong>1.25</strong></td><td>39</td><td>899027</td><td>232517</td><td>371040</td><td>38255</td><td>257215</td><td>2</td><td>&lt; e</td><td>10</td></tr><tr><td><strong>12000</strong></td><td>7500</td><td>0.3</td><td>0.01</td><td><strong>1.25</strong></td><td>46</td><td>924143</td><td>275139</td><td>317880</td><td>42189</td><td>288935</td><td>2</td><td>&lt; e</td><td>5</td></tr><tr><td rowspan="10">88-nodes</td><td rowspan="2">Case base</td><td>0.975</td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>16</td><td>679622</td><td>83508</td><td>493500</td><td>22820</td><td>79794</td><td>20</td><td>0.97%</td><td>7200*</td></tr><tr><td><strong>β</strong></td><td><strong>0.75</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>18</td><td>629774</td><td>99573</td><td>466500</td><td>32360</td><td>31341</td><td>63</td><td>0.79%</td><td>7200*</td></tr><tr><td rowspan="2">μ</td><td><strong>0.99</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>15</td><td>694137</td><td>90518</td><td>483000</td><td>24556</td><td>96063</td><td>20</td><td>1.03%</td><td>7200*</td></tr><tr><td><strong>3000</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>9</td><td>344449</td><td>49007</td><td>231600</td><td>9963</td><td>53879</td><td>20</td><td>0.68%</td><td>7200*</td></tr><tr><td rowspan="2">CV</td><td><strong>12000</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>23</td><td>972603</td><td>142855</td><td>682800</td><td>38734</td><td>108214</td><td>19</td><td>1.03%</td><td>7200*</td></tr><tr><td><strong>0.1</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>18</td><td>640797</td><td>100018</td><td>466500</td><td>29338</td><td>44941</td><td>30</td><td>1.15%</td><td>7200*</td></tr><tr><td rowspan="2">c</td><td><strong>0.5</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>13</td><td>715544</td><td>75666</td><td>513000</td><td>18821</td><td>108057</td><td>12</td><td>0.73%</td><td>7200*</td></tr><tr><td><strong>1</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td>0.75</td><td>11</td><td>808305</td><td>60165</td><td>543000</td><td>13984</td><td>191157</td><td>11</td><td>0.63%</td><td>7200*</td></tr><tr><td rowspan="2">c</td><td><strong>0.5</strong></td><td>7500</td><td>0.3</td><td>0.0001</td><td>0.75</td><td>3</td><td>441576</td><td>19822</td><td>377550</td><td>12129</td><td>32075</td><td>31</td><td>1.07%</td><td>7200*</td></tr><tr><td><strong>0.0005</strong></td><td>7500</td><td>0.3</td><td>0.0005</td><td>0.75</td><td>10</td><td>576598</td><td>58360</td><td>433500</td><td>19217</td><td>65521</td><td>19</td><td>0.88%</td><td>7200*</td></tr><tr><td rowspan="2">h</td><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td>0.75</td><td>49</td><td>955426</td><td>280407</td><td>486000</td><td>40201</td><td>148818</td><td>62</td><td>0.01%</td><td>7200*</td></tr><tr><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.01</td><td>0.75</td><td>71</td><td>1056536</td><td>421779</td><td>402225</td><td>48478</td><td>184054</td><td>14</td><td>&lt; e</td><td>211</td></tr><tr><td rowspan="2">c, h</td><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td><strong>0.25</strong></td><td>20</td><td>616177</td><td>99666</td><td>465000</td><td>15706</td><td>35805</td><td>97</td><td>0.17%</td><td>7200*</td></tr><tr><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td><strong>1.25</strong></td><td>13</td><td>729018</td><td>69277</td><td>517500</td><td>25250</td><td>116991</td><td>19</td><td>1.25%</td><td>7200*</td></tr><tr><td rowspan="2">μ, CV, c, h</td><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.0005</td><td><strong>1.25</strong></td><td>2</td><td>459383</td><td>15919</td><td>386400</td><td>12959</td><td>44105</td><td>32</td><td>1.01%</td><td>7200*</td></tr><tr><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td><strong>1.25</strong></td><td>9</td><td>615272</td><td>46106</td><td>456000</td><td>21331</td><td>91835</td><td>21</td><td>1.08%</td><td>7200*</td></tr><tr><td rowspan="2">c</td><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.0005</td><td><strong>1.25</strong></td><td>44</td><td>1048073</td><td>253041</td><td>536250</td><td>46104</td><td>212678</td><td>31</td><td>0.24%</td><td>7200*</td></tr><tr><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.01</td><td><strong>1.25</strong></td><td>65</td><td>1173544</td><td>415796</td><td>410325</td><td>58375</td><td>289048</td><td>35</td><td>&lt; e</td><td>1133</td></tr><tr><td rowspan="2">μ, CV, c, h</td><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.001</td><td><strong>1.25</strong></td><td>3</td><td>728138</td><td>16294</td><td>617040</td><td>14969</td><td>79835</td><td>30</td><td>1.06%</td><td>7200*</td></tr><tr><td><strong>12000</strong></td><td>7500</td><td>0.3</td><td>0.0005</td><td><strong>1.25</strong></td><td>10</td><td>947215</td><td>53390</td><td>706800</td><td>26077</td><td>160948</td><td>16</td><td>0.98%</td><td>7200*</td></tr><tr><td rowspan="2">μ, CV, c, h</td><td><strong>0.975</strong></td><td>7500</td><td>0.3</td><td>0.005</td><td><strong>1.25</strong></td><td>53</td><td>1493316</td><td>306716</td><td>738000</td><td>59473</td><td>389127</td><td>39</td><td>0.09%</td><td>7200*</td></tr><tr><td><strong>12000</strong></td><td>7500</td><td>0.3</td><td>0.01</td><td><strong>1.25</strong></td><td>74</td><td>1604897</td><td>441261</td><td>610560</td><td>70767</td><td>482309</td><td>7</td><td>&lt; e</td><td>135</td></tr></tbody></table>

a Exceeds the time limit.