Facility location based on adjusted present value

Rezaei, Hamidreza; Bostel, Nathalie; Hovelaque, Vincent; Péton, Olivier; Viviani, Jean-Laurent Article Facility location based on adjusted present value Operations Research Perspectives Provided in Cooperation with: Elsevier Suggested Citation: Rezaei, Hamidreza; Bostel, Nathalie; Hovelaque, Vincent; Péton, Olivier; Viviani, Jean-Laurent (2025) : Facility location based on adjusted present value, Operations Research Perspectives, ISSN 2214-7160, Elsevier, Amsterdam, Vol. 14, pp. 1-22, https://doi.org/10.1016/j.orp.2024.100319 This Version is available at: https://hdl.handle.net/10419/325799 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Contents lists available at ScienceDirect Operations Research Perspectives journal homepage: www.elsevier.com/locate/orp Facility location based on Adjusted Present Value Hamidreza Rezaei a,b, Nathalie Bostel b,c, Vincent Hovelaque d, Olivier Péton a,b,∗, Jean-Laurent Viviani d aIMT Atlantique, 4 rue Alfred Kastler, F-44307 Nantes Cedex, France bLaboratoire des Sciences du Numérique de Nantes (LS2N, UMR CNRS 6004), Nantes, France cNantes Université, Nantes, France dUniversité de Rennes, CNRS, CREM – UMR 6211, Rennes, France ARTICLE INFO Keywords: Facility location Supply chain Finance Adjusted present value ABSTRACT Supply chain network design aims to optimize strategic decisions such as facility location decisions. These decisions have a major impact on the supply chain, but also on the financial value of the company. However, financial considerations are often omitted from facility location mathematical models. This paper addresses the challenge of identifying a relevant financial indicator that can be practically implemented in facility location models across different industries. This paper makes several contributions: the Adjusted Present Value (APV) is identified as such a financial indicator; we propose a mathematical formulation that embeds the APV in a facility location model maximizing firm value; computational experiments demonstrate the tractability of the model. Finally, we compare the mathematical model with a sequential approach that first optimizes logistical decisions and then financial decisions. The proposed model improves the sequential approach up to 5.5%, increases the market coverage and anticipates facility location decisions. 1. Introduction Optimizing strategic decisions in supply chain network design involves key decisions such as locating facilities, determining their capacity, and deciding when to open them. It also involves determining the product flows in the logistics network under consideration. Supply chain network design has been the subject of extensive literature and many reviews (see, e.g. [1,2]). Most of this literature identifies the facility location problem and its extensions as the core of supply chain network design models. The problem is known to be NP-hard. The primary goals of facility location in the context of supply chain management are (i) to define the network itself, by locating facilities and defining the allocation of product flows to these facilities and (ii) to determine the optimal product flows in this logistics network. Classically, this network is optimized either by minimizing total logistics costs or by maximizing the profit generated by the distribution of goods. Ideally, the corresponding mathematical models should take into account the interactions between supply chain management and other company departments (e.g. marketing, human resources, finance) as well as external factors (e.g. competitors, consumer behavior, financial markets). ∗Correspondence to: IMT Atlantique, 4 rue Alfred Kastler, F-44307 Nantes, France. E-mail address: [email protected] (O. Péton). Despite their importance, it is striking that financial instruments rarely appear as full components of facility location models. However, most companies finance their strategic investments by resorting to debt. It therefore becomes clear that the strategic planning of logistical and financial decisions must go hand in hand. This would enhance cost management by enabling more effective capital allocation and debt financing, and reducing operational expenses. By analyzing financial implications, companies can strategically allocate resources to ensure that facility investments are both cost-effective and in line with their overall financial objectives. This assessment of location decisions further enhances resource allocation, leading to investments that are efficient and aligned with financial goals. Such integration improves investment strategies, secures advantageous financing terms, and facilitates better negotiation opportunities, resulting in lower borrowing costs and greater financial stability. In addition to internal financial benefits, well-informed location decisions can have other positive effects, like local economic development. Choosing sites that contribute to job creation and leveraging available economic incentives can stimulate local economies, creating additional value beyond the company’s immediate financial interests [3]. Although these factors contribute to the company’s reputation and long-term success, they are beyond the scope of this paper. https://doi.org/10.1016/j.orp.2024.100319 Received 21 June 2024; Received in revised form 2 October 2024; Accepted 19 November 2024 Operations Research Perspectives 14 (2025) 100319 Available online 20 December 2024 2214-7160/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). H. Rezaei et al. Fig. 1. Trade-off theory of capital structure. [4] mentioned the strong interaction between the financial factors and the strategic planning. In closed-loop supply chains, [5] noted few studies consider these as decision variables. Traditional supply chain management approaches consider the cash flows generated by operational decisions, but not those resulting from financial decisions. [6] observe that the majority of models only broadly examine the costs and revenues associated with the design of supply chains and ignore other financial elements that are related to its future performance. Therefore, a strong motivation of this paper is to jointly consider the impact of financial and operational decisions on the location strategy. Financial theory seeks to maximize the value of the company and states that the value of any asset is equal to the present value of the future cash flows discounted at an appropriate discount rate called the cost of capital. The discounted cash flows valuation models can be found in any finance text book for readers that would like to explore the model more in detail, for example [7,8]. In corporate finance, key decisions about sources of finance are based on both equity and debt. The theoretical and practical question is what should be the mix of equity and debt that maximizes the company’s value? To answer this question, the trade-off theory of capital structure [9] explores two opposite consequences of the debt: (i) the tax advantages associated with the fact that paying interest on debt reduces the corporate taxes and (ii) the distress disadvantages associated with the fact that, as the firm’s leverage increases, so does the probability of default and hence the Expected Bankruptcy Costs (EBC). The trade-off theory states that the advantages of debt (tax shield benefits) are balanced by its disadvantages (bankruptcy costs). According to this theory, there is an optimal debt level that maximizes a firm’s value (Fig. 1). As debt increases (horizontal axis), both the tax shield benefit and the bankruptcy cost of debt rise. The horizontal solid line represents the unlevered value of the firm, i.e., the value of the firm without taking into account the cash flow consequences of financial decisions. The tax shield, represented by the solid diagonal line, is linearly increasing, while the cost of bankruptcy is convex non-linear. The value of the firm (blue curve) is therefore a concave non-linear function. The peak of this curve indicates the optimal debt value. This paper explicitly introduces the benefits and disadvantages of debt in order to determine the optimal mix of debt and equity a company should settle to fund its facility location strategy. The starting point for most discussions on the impact of financing on firm valuation is the seminal work of Modigliani and Miller [10]. Since then, several methods have been proposed for incorporating the effects of debt into cash flow discounting valuation models. The Fig. 2. Calculation of the APV. two main approaches are either (i) to incorporate the consequences of financial decisions into the discounting rate: Free Cash Flow (FCF) discounted at the Weighted Average Cost of Capital (WACC), Capital Cash Flows (CCF) discounted at the pre-tax WACC, Economic Value Added (EVA) discounted at the WACC (also known as the Market Value Added (MVA)) [11]; or (ii) to include them in the Adjusted Present Value (APV). According to [12], all of the methods proposed in the literature lead to the same valuation of the company when properly applied. However, some methods are more appropriate than others in specific cases. This paper focuses on the APV [13], which represents the value of a leveraged firm (taking into account the cash flow consequences of financial decisions). It is a widely used measure of value creation in the context of discounted cash flow valuation models. The APV has several main advantages: (i) It offers detailed information on the factors contributing to the firm’s value [14], distinguishing the unlevered firm value from the value added by financial decisions. (ii) It allows a detailed analysis of the value derived from the choice of a particular financial structure by isolating the contribution of tax benefits to the corporate value creation [14]. (iii) As explained by [15], the change in leverage requires a periodic complex reassessment of the WACC. On the contrary, the APV works under both constant and variable debt ratios over the forecast period. Thus, maximizing APV satisfies our goal of incorporating financial considerations into a facility location model. More specifically, APV allows the advantages and disadvantages of debt to be incorporated into the company’s cash flows. If the company does not use debt, APV is reduced to the classic NPV model. The unlevered firm value, also known as the base case Net Present Value (NPV) [16], is the value generated by the operational decisions alone. This is why we call it Operationally Generated Value (OGV), as opposed to the Financially Generated Value (FGV). OGV is calculated by discounting future operational cash flows at the unlevered cost of equity (or debt-free cost of capital). FGV is the present value of the advantages and disadvantages of debt. As shown in Fig. 2, the APV is determined by two types of decisions. First, logistical decisions consist of selecting facilities from a set of candidate locations to deliver goods to a set of customers. This affects only the OGV. Second, financial decisions determine the debt level needed to finance logistics investments, staggered over time. This only affects the FGV. This led us to compare two approaches, which consist of solving the proposed MILP model sequentially (OGV then FGV) or all at once. These two approaches are described in detail in Section 6. The main contributions of this paper are: (i) to propose the Adjusted Present Value (APV) as a financial indicator used to optimize the future impact of strategic supply chain decisions on the future value of the firm (instead of the classical cost function), (ii) to propose a mixed integer linear programming (MILP) model that integrates facility location decisions and their financial consequences over a strategic horizon, (iii) to assess the tractability of this MILP by state-of-the-art solvers, and (iv) to evaluate the potential benefits of integrating financial considerations into a facility location model, through a comparison Operations Research Perspectives 14 (2025) 100319 2 H. Rezaei et al. with a sequential approach that optimizes logistical decisions first and the financial decisions second. The structure of the paper is as follows. Section 2positions this work in the related literature. Section 3describes the assumptions and the global structure of our problem. Section 4presents the mathematical formulation. Section 5describes the data generation principles used to create new instances. Section 6presents the results of our computational experiments. Section 7presents some managerial insights and concludes with future research directions. 2. State of the art In their review on facility location and supply chain management, [1] classify financial factors in three categories: (i) international factors, including taxes, duties, tariffs, exchange rates, transfer prices, and local content rules, (ii) financing and taxation incentives offered by governments and (iii) investment expenditures, usually limited by the total available budget. In the first category, [17] pleads for the consideration of different tax regimes and duties, exchange rates, transfer prices and differences in operating costs. [18] propose a facility location model in offshoring context, with both tactical and strategic decision levels. The financial decisions are the transfer pricing and two variables allocating logistics costs to various stakeholders of the supply chain. Budget constraints can be found in numerous facility location models. For example, [19] addressed a facility location problem considering budget constraints with the aim of minimizing the overall distance traveled. Similarly, [20] proposed a multi-period mathematical model minimizing total business costs. The available budget limitation is the incorporated financial factor. [21] addressed a multi-period stochastic facility location problem to maximize the benefits of the firm, with a budget constraint on investments and the possibility of setting a target for the return on investment. See also, e.g., [22–26] In many papers, the only financial consideration is to optimize NPV. Although NPV is an appropriate measure of the profitability of an investment, it does not capture the value contributed by financing decisions [16,27]. In the following paragraphs, we review publications according to their approach measuring financial values. Free cash flow (FCF) discounted at WACC. [27], propose an integrated strategic-tactical model whose objective function, called Corporate Value, is the difference between the FCF discounted at WACC and net debt. To cope with the difficulty of a variable WACC rate, they consider it as a fixed parameter over the planning horizon. This approach favorably compares with the traditional NPV-oriented model. [28] extend the work of [27] by introducing the uncertainty of demand, price, and interest rates to the model. The presented stochastic model is then compared with a deterministic model, indicating a significant performance improvement. Economic value added (EVA). EVA is an absolute key figure based on earnings that focuses on the performance of a single period [29], while all the following publications deal with a multi-period planning horizon. In these cases, the objective function is the sum of forecasted EVAs. [30] develop a mathematical model for designing a four-echelon supply chain under demand uncertainty, optimizing net created value, measured by EVA. They assume a constant WACC and compare their financial model with a non-financial one that ignores financial analysis. While the financial model creates more shareholder value, the non-financial model shows higher return on equity. The same authors propose a bi-objective MILP model that captures trade-off values between financial performance, measured with EVA, as the first objective and credit solvency, using Altman Z-score [31], as the second objective function [32] . Our APV model relies on a similar trade-off between the opposite effects of interest tax shield benefit and the present value of the debt. [33] address a three-echelon, multiperiod, multi-item closed-loop, strategic and tactical facility location problem. EVA is maximized while the WACC is defined as a constant parameter. [34] propose a multi-product, multi-period, four-echelon model addressing financial decisions like cash and risk management, capital structure, and revenue/cost management. They evaluate financial dimensions (corporate value, Change in Equity, and EVA) against profit maximization. [35] propose a fuzzy MILP model to design a global supply chain network that considers logistical and financial flows simultaneously. To assess the financial performance, EVA is maximized. [36] propose a MINLP model analyzing the impact of economic uncertainty on supply chain financial health. The model designs a multi-period closed-loop supply chain to maximize EVA under demand uncertainty. Besides EVA, the authors also use NPV and WACC as objective functions, with a fixed WACC rate. EVA is also used as the objective function in the multi-period model proposed by [37], which considers debt repayments and new capital entries as decision variables. Still, the WACC is assumed to be a fixed parameter. Their model leads to higher EVA in comparison to the model of [30]. [38] address a stochastic supply chain network design model which aims at maximizing EVA. Interest rates (short-term and longterm), expected return on stock market, and risk-free rate of interest are subject to uncertainty. Market value added (MVA). MVA is a tool to transform EVA to a multi-period basis. It measures the present value of future EVAs by discounting them at the WACC rate [29]. Unfortunately, the MVA loses the EVA’s property of being compatible with fluctuating debt ratios. [39] address a multi-period, multi-product sustainable supply chain network design problem maximizing the MVA of the firm. Yet, the authors consider the WACC rate as a fixed parameter. The reviewed literature provides a range of methodologies for integrating financial considerations into facility location models, each with its strengths and limitations. Free Cash Flow (FCF) and Economic Value Added (EVA) are commonly used but have constraints related to fixed financial parameters. Market Value Added (MVA) addresses multi-period analysis but struggles with fluctuating debt ratios. Notably, Adjusted Present Value (APV) is identified as a promising approach for handling both multi-period models and variable debt ratios. Table 1summarizes these approaches and highlights the potential for APV to offer a more adaptable and comprehensive financial valuation framework for facility location decisions. To our knowledge, Capital Cash Flow (CCF) and Adjusted Present Value (APV) have not been employed as valuation methods in facility location models and remain relatively unexplored in this context. This observation is corroborated by the recent review conducted by [6]. 3. Problem description and assumptions In this section, we describe the main settings and assumptions of the multi-period capacitated facility location problem that will be formally presented in Section 4. The notations are summarized in Appendix A. 3.1. General setting The main goal of this paper is to propose a mathematical model for the joint optimization of facility location and financing decisions. We want to ensure that: (i) the differences between the proposed model and traditional models based on cost minimization must be tangible and interpretable, and (ii) the logistical assumptions must be as ‘‘simple’’ as possible, in order to highlight the financial interpretations. The first feature motivated us to consider a company starting from a blank page. There is no operating facility at the beginning of the time horizon, nor current loans. Therefore, fulfilling all customer demands is not mandatory. This assumption supports a gradual expansion of Operations Research Perspectives 14 (2025) 100319 3 H. Rezaei et al. Table 1 Financial valuation approaches in facility locations models. Valuation approach Compatibility with References Multiple Fluctuating periods debt ratio Free Cash Flow (FCF) ✓✕[27] [28] [34] Economic Value Added (EVA) ✕✓[30] [32] [34] [33] [35] [36] [37] [38] Market Value Added (MVA) ✓✕[39] Capital Cash Flow (CCF) ✓✕– Adjusted Present Value (APV) ✓ ✓ this paper Fig. 3. Time scale of supply chain decisions and their financial impact. the logistics network, allowing the company to avoid the obligation of opening numerous candidate facilities in period 1 with no additional openings afterward. Based on the second feature, we consider a logistics network with two layers: a set of candidate production facilities and a set of customers to be served from the selected facilities. The model is based on a strategic time horizon (time periods are typically years). All data and parameters are assumed deterministic. The main goal of the model is to determine which candidate facilities to open, when to open them over the planning horizon, which customers will be served by the chosen facilities and how to finance these decisions with an appropriate mix of debt and equity. The objective function to be maximized is the company’s APV. 3.2. Time horizon Maximizing APV requires computing the present value of the future operational and financial cash flows over multiple periods. The time horizon = {1,…, 𝑇}represents the set of periods in which the location decisions are applicable, i.e., the company can borrow and invest money at any period 𝑡∈. Period 0 represents the initial state of the supply chain. The mathematical model presented in Section 4is assumed to be solved at period 0, and concerns decisions that apply in any period 𝑡∈. Even so, the logistical and financial decisions taken in this time horizon will have a much longer impact on the company’s cash flows. Assume that a new facility starts operating at some period 𝑡∈and that some money will be borrowed to finance this decision (possibly in addition to the use of internal and/or external equity). Given a payback period of 𝑁time periods and a facility lifetime 𝐿, the impact of this decision on the debt will be observed until period 𝑡+𝑁while the associated cash flow will be observed until period 𝑡+𝐿− 1. Time horizon needs to be extended. To do so, we define the time horizon extension ′= {𝑇+ 1,…, 𝑇+ max(𝐿, 𝑁)} as the set of periods during which the selected facilities are still operating and the financial impact of logistical decisions can be observed after the time horizon .Fig. 3illustrates the case where a new facility is selected at some period 𝑡∈and generates cash flow until period 𝑇+𝐿− 1. 3.3. Logistics and market features Each candidate facility 𝑗∈𝐽has a known capacity 𝐶𝑗. Each facility has a fixed opening cost 𝑂𝑗paid once if the facility 𝑗∈𝐽is selected. It also has a fixed yearly running cost 𝐹𝑗paid every year provided the facility is operating. The unitary processing cost 𝜇𝑗is paid for each unit of product manufactured in the facility. Since the model is initialized by a blank page, we consider that a new opened facility will not be closed during the horizon . The list of candidate facilities in facility location models is generally much larger than the total number of facilities actually selected. Thus, we consider an upper bound 𝐽𝑚𝑎𝑥 on the total number of selected facilities. The risk associated with the selection and all logistical operations is assumed to be the same in each facility. Our multi-period model addresses markets with time-variable and deterministic customers’ demands. We assume that reliable demand forecast is available for the whole time horizon . The demand of customer 𝑖∈in period 𝑡∈is denoted 𝐷𝑖𝑡. The selling price to each customer can depend on many factors, including the customer’s market, its demand level and its negotiation skills. For these reasons, we assume a selling price 𝑃𝑖for each customer 𝑖∈. We do not force the company to serve all customers at every period. Unsatisfied customers’ demands are simply lost and back orders are forbidden. Following the all-ornothing principle, a customer’s demand is either entirely fulfilled or not at all. We apply the idea of incremental service [40]: customers that are being served in a period must be served in all subsequent periods (the allocation of customers to facilities might change in different periods). 3.4. Financial features We assume all facilities operate in a homogeneous financial environment with a single tax rate 𝜂and no exchange rates. Consequently, the equity cost of capital 𝐾𝐸is location-independent and remains constant over time, as all investments are subject to the same business risk within the industry and market. This homogeneity means all candidate facilities and logistical operations share similar systematic risks. Additionally, we assume that investments related to location decisions carry the same risk as the company’s ‘‘business as usual.’’ For a new facility 𝑗∈opened in period 𝑡∈, the firm has to finance its fixed opening cost 𝑂𝑗by an appropriate mix of debt and equity. 3.4.1. Debt financing Debt financing amounts to borrowing money from a bank. There are |𝑇|discrete periods in which borrowing is possible. In light of this, we define |𝑇|separate loans, each of which is defined at a period 𝑡∈𝑇 Operations Research Perspectives 14 (2025) 100319 4 H. Rezaei et al. and may or may not be activated. We will refer to the loan that was triggered in period 𝑡as 𝑙𝑡throughout this study. We assume that the loans have the same reimbursement duration 𝑁, and a constant annuity repayment method (homogeneous markets). However, the cost of loan 𝑙𝑡, denoted by 𝐾 𝐷𝑡, depends on the accumulated debt ratio of period 𝑡, such that the higher the debt ratio, the higher the cost of loan. The increase in leverage leads to a higher risk of bankruptcy for the company, prompting debt holders to demand a higher risk premium. For each loan 𝑙𝑡, the annuity amount 𝐴𝑡associated with a loan amount 𝐵𝑡, a cost of loan 𝐾 𝐷𝑡and a number 𝑁of equally sized payments is given by: 𝐴𝑡=𝐵𝑡 𝐾 𝐷𝑡× (1 +𝐾 𝐷𝑡)𝑁 (1 +𝐾 𝐷𝑡)𝑁− 1. Assuming that the company will only use loans with the same maturity 𝑁,𝑁≥𝑇, and constant annuity repayment method, the amount borrowed in period 𝑡will be repaid from period 𝑡+ 1to period to 𝑡+𝑁. 3.4.2. Equity financing In equity financing, money is supported by shareholders in the form of internal financing (company’s cash holdings) represented by the variables 𝐼 𝐸𝑡and external equity represented by the variables 𝐸 𝐸𝑡. The company will first seek internal financing based on available cash and then external financing. The amount of cash available depends on the level of retained earnings and the amortization policy. The parameter 𝛿 defines the payout ratio, i.e., the fraction of earnings paid as dividends. The remaining amount will be retained by the company. It will be added to existing cash holdings. In an accounting approach, depreciation represents a yearly decrease in tangible assets’ value over their lifetime. Among different methods to depreciate the assets (see, e.g., [27]) we consider a straightline depreciation scheme, between the initial value 𝑂𝑗and the salvage value 𝑆 𝑉𝑗of the selected assets 𝑗∈. The amount of depreciation, as a non-cash accounting expense, enables the company to build up reserves that can be used for future investments. As mentioned above, the change from NPV to APV is made by adding two complementary elements: Tax Shield Benefits and Expected Bankruptcy Cost (see Fig. 2). EBC is the present value of direct (legal and accounting costs) and indirect bankruptcy costs, a continuum of costs that increase at an accelerating rate as exposure to bankruptcy increases, e.g., increased interest expenses, lost credit, lost sales, inefficient operations. Direct costs take the form of administrative expenses (trustee’s fees, legal fees, referee’s fees), and in the time lost by executives in liquidation [41]. [8] calculates EBC as the product of the probability of bankruptcy, denoted 𝑝, by the bankruptcy costs 𝛾×𝑂 𝐺 𝑉, where 0< 𝛾 <1is a known fixed parameter. We choose to express bankruptcy costs as a fraction of the firm value before bankruptcy as usually done both in theoretical [42] and empirical studies [43,44]. The various approaches only differ on the measures of the firm value (book versus market value, total versus equity value), in the model, we choose the market operational value of the company. A company will go bankrupt in period 𝑡∈if the market value of its assets falls below the value of debt at this period. The higher the debt threshold relative to the company’s assets, the more difficult it will be for the asset value to reach it. 4. Mathematical formulation This section details the mathematical model maximizing APV subject to logistical and financial constraints. Sections 4.1 and 4.2 enumerate the model constraints related to the Operationally Generated Value (OGV) and the Financially Generated Value (FGV), respectively. 4.1. OGV: Operationally generated value For each candidate facility 𝑗∈, the opening cost 𝑂𝑗is paid once if the facility is selected, and yearly operating fixed costs 𝐹𝑗paid at every period when the facility is operating. In addition, each operating facility has a processing cost 𝜇𝑗for each unit of product processed by this facility. We assume a lifetime 𝐿 > 𝑇for all candidate facilities which is consistent in a strategic point of view. The distance between a customer 𝑖∈and a facility 𝑗∈is denoted as 𝐷 𝑖𝑠𝑡𝑖𝑗 . We assume that the transportation cost between two locations is proportional to the distance traveled and the load carried, with a unit transportation cost 𝜔over the whole network. Finally, we consider a selling price 𝑃𝑖to customer 𝑖∈. Due to trade rules between geographical areas as well as various logistical constraints, some customers might not be delivered by some facilities. Thus, we introduce an accessibility binary parameter 𝑉𝑖𝑗 which takes the value 1 if the customer 𝑖∈is accessible from facility 𝑗∈. We consider two families of binary decision variables and one family of continuous decision variables. The variable 𝑦𝑗 𝑡, takes the value 1 if the facility 𝑗∈is operating in period 𝑡∈, and 0 otherwise. The variable 𝑥𝑖𝑡 takes value 1 if customer 𝑖∈is served in period 𝑡∈, and 0 otherwise. The variable 𝑞𝑖𝑗 𝑡denotes the quantity delivered by facility 𝑗∈to customer 𝑖∈in period 𝑡∈. 𝑦𝑗0= 0 ∀𝑗∈(1) 𝑦𝑗 ,𝑡−1 ≤𝑦𝑗 𝑡∀𝑗∈, 𝑡∈(2) ∑ 𝑗∈ 𝑦𝑗 ,𝑇 ≤𝐽𝑚𝑎𝑥.(3) 𝑥𝑖,𝑡−1 ≤𝑥𝑖𝑡 ∀𝑖∈, 𝑡∈(4) 𝑞𝑖𝑗 𝑡≤𝑉𝑖𝑗 𝐷𝑖𝑡 𝑦𝑗 𝑡∀𝑖∈, 𝑗∈, 𝑡∈(5) ∑ 𝑗∈ 𝑞𝑖𝑗 𝑡=𝐷𝑖𝑡 𝑥𝑖𝑡 ∀𝑖∈, 𝑡∈(6) ∑ 𝑖∈ 𝑞𝑖𝑗 𝑡≤𝐶𝑗𝑦𝑗 𝑡∀𝑗∈, 𝑡∈(7) 𝑒𝑗 𝑡=𝐹𝑗𝑦𝑗 𝑡+𝜇𝑗∑ 𝑖∈ 𝑞𝑖𝑗 𝑡+𝜔∑ 𝑖∈ (𝐷 𝑖𝑠𝑡𝑖𝑗 𝑞𝑖𝑗 𝑡) ∀𝑗∈, 𝑡∈(8) 𝑟𝑗 𝑡=∑ 𝑖∈ (𝑃𝑖𝑞𝑖𝑗 𝑡) ∀𝑗∈, 𝑡∈(9) Constraints (1) establish initial conditions: in period 0, none of the candidate facilities is selected. Constraints (2) state that selected facilities cannot be closed during the time horizon (neither during the complementary time horizon ′whereas 𝑦𝑗 ,𝑡 is frozen when 𝑡∈′). Constraints (3) define an upper bound 𝐽𝑚𝑎𝑥 on the number of selected facilities. Constraints (4) impose to serve in period 𝑡a customer who was served in period 𝑡− 1. Constraints (5) state that the variable 𝑞𝑖𝑗 𝑡is strictly positive only if facility 𝑗is opened and accessible (𝑦𝑗 𝑡= 1) and 𝑉𝑖𝑗 = 1. This quantity cannot exceed the demand 𝐷𝑖𝑡. Constraints (6) calculate the total quantity delivered to each customer, which is either 0 when 𝑥𝑖𝑡 = 0 or the total demand 𝐷𝑖𝑡 when 𝑥𝑖𝑡 = 1(no partial satisfaction of a particular customer’s demand). Some customers may be delivered from several different facilities, at a given period. The capacity constraints (7) enforce the total quantity shipped by a selected facility 𝑗∈to be at most equal to its capacity 𝐶𝑗. Constraints (8) calculate the total amount of logistics expenses 𝑒𝑗 𝑡 related to facility 𝑗∈in period 𝑡∈𝑇. This amount is the sum of the yearly fixed cost, the processing cost and the transportation cost. Constraints (9) calculate the total revenue 𝑟𝑗 𝑡generated by facility 𝑗∈in period 𝑡∈. Operations Research Perspectives 14 (2025) 100319 5 H. Rezaei et al. 4.1.1. Calculation of OGV We consider a tax rate 𝜂and the equity cost of capital of the unlevered company 𝐾𝐸. For each candidate facility 𝑗∈, the initial value and the salvage value and denoted 𝑂𝑗and 𝑆 𝑉𝑗, respectively. A facility 𝑗∈opened in period 𝜏∈has a lifetime 𝐿 > 𝑇. It generates variable cash flows between period 𝜏and 𝑇and a constant cash flow 𝐶 𝐹𝑗 𝑇after period 𝑇, as long as is it active. The calculation of OGV requires determining which facilities are operating at each period 𝑡∈∪′. We extend the definition of binary variables 𝑦𝑗 𝑡to ′as follows: for any 𝑗∈, 𝑡∈′,𝑦𝑗 𝑡is equal to 1 if facility 𝑗is operating in period 𝑡, and 0otherwise. If 𝑦𝑗 𝑇= 1 and the facility was opened in period 𝜏∈, then 𝑦𝑗 𝑡= 1for all 𝑇+ 1≤𝑡≤𝜏+𝐿− 1, and 0 otherwise. Then, for all 𝑡≥𝜏+𝐿,𝑦𝑡𝑗 = 0. If 𝑦𝑗 𝑇= 0, all variables 𝑦𝑗 𝑡are equal to 0. 𝐷 𝑒𝑝𝑗=𝑂𝑗−𝑆 𝑉𝑗 𝐿∀𝑗∈(10) 𝐸 𝐵 𝐼 𝑇𝑗 𝑡=𝑟𝑗 𝑡−𝑒𝑗 𝑡−𝐷 𝑒𝑝𝑗𝑦𝑗 ,𝑡−1 ∀𝑗∈, 𝑡∈(11) 𝐶 𝐹𝑗 𝑡= (1 −𝜂)𝐸 𝐵 𝐼 𝑇𝑗 𝑡+𝐷 𝑒𝑝𝑗𝑦𝑗 ,𝑡−1 ∀𝑗∈, 𝑡∈(12) 𝐹 𝐶 𝐹𝑗 𝑡= (1 −𝜂)𝐸 𝐵 𝐼 𝑇𝑗 𝑡− (𝑂𝑗(𝑦𝑗 𝑡−𝑦𝑗 ,𝑡−1) −𝐷 𝑒𝑝𝑗𝑦𝑗 ,𝑡−1) ∀𝑗∈, 𝑡∈(13) 𝑦𝑗 ,𝑡−1 ≥𝑦𝑗 𝑡∀𝑡∈′∖{𝑇+ 1} (14) ∑ 𝑡∈ 𝑦𝑗 𝑡+∑ 𝑡∈′ 𝑦𝑗 𝑡=𝐿 𝑦𝑗 𝑇∀𝑗∈(15) Constraints (10) calculate the straight-line depreciation 𝐷 𝑒𝑝𝑗of facility 𝑗∈as a linear function of its initial value (opening cost) 𝑂𝑗, salvage value 𝑆 𝑉𝑗, and lifetime 𝐿. Constraints (11) calculate the Earning Before Interest and Taxes 𝐸 𝐵 𝐼 𝑇𝑡associated with facility 𝑗∈in period 𝑡∈. It is the difference between the revenues and expenses of facility 𝑗∈(before interest and tax) in period 𝑡and the depreciation factor 𝐷 𝑒𝑝𝑗occurring if the facility 𝑗operates in period 𝑡− 1. Constraints (12) calculate the operating cash flow associated with facility 𝑗∈in period 𝑡∈. The first term is the accounting result after tax. The second term reintroduces the depreciation because it is not a cash outflow. In constraints (13), the free cash flows 𝐹 𝐶 𝐹𝑗 𝑡measures the ability of the firm to generate cash. It is the difference between operating and non-operating cash inflows and outflows associated with each facility. Constraints (14) and (15) model the extension of variables 𝑦𝑗 𝑡to the periods 𝑡∈′. The Operationally Generated Value (OGV) is defined by: 𝑂 𝐺 𝑉=∑ 𝑗∈(∑ 𝑡∈ 𝐹 𝐶 𝐹𝑗 𝑡 (1 +𝐾𝐸)𝑡+∑ 𝑡∈′ 𝐶 𝐹𝑗 𝑇𝑦𝑗 𝑡 (1 +𝐾𝐸)𝑡).(16) This expression is non-linear due to the product of variables 𝐶 𝐹𝑗 𝑇 and 𝑦𝑗 𝑡. The linearization process is explained in Appendix C.2. 4.2. FGV: Financially generated value If a facility 𝑗∈is opened in period 𝑡∈, the firm has to decide how to finance its initial value 𝑂𝑗. The company can mix debt financing, detailed in Section 4.2.1 and equity financing, detailed in Section 4.2.2. We consider three families of continuous decision variables: 𝑏𝑜𝑟𝑟𝑜𝑤𝑡represent the amount of money borrowed in period 𝑡∈;𝐼 𝐸𝑡and 𝐸 𝐸𝑡represent the internal equity the external equity in period 𝑡∈, respectively. Eq. (17) states that facility sunk costs are funded by the debt, internal funding or by external equity. ∑ 𝑗∈ 𝑂𝑗(𝑦𝑗 𝑡−𝑦𝑗 ,𝑡−1) =𝑏𝑜𝑟𝑟𝑜𝑤𝑡+𝐼 𝐸𝑡+𝐸 𝐸𝑡∀𝑡∈.(17) 4.2.1. Debt financing constraint For loan 𝑙 𝑡, we define 𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑡,𝑟𝑒𝑝𝑎𝑦 𝑡𝑡 and 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡 respectively as the total amount still alive, the annuity amount repaid, and the value of the interest associated with that loan in period 𝑡∈∪′. 𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑡 =⎧ ⎪ ⎨ ⎪ ⎩ 0𝑡 < 𝑡 𝑏𝑜𝑟𝑟𝑜𝑤 𝑡𝑡= 𝑡 𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡,𝑡−1 −𝑟𝑒𝑝𝑎𝑦 𝑡𝑡 +𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡 𝑡 > 𝑡 (18) 𝑟𝑒𝑝𝑎𝑦 𝑡𝑡 =⎧ ⎪ ⎨ ⎪ ⎩ 0𝑡≤ 𝑡 𝑏𝑜𝑟𝑟𝑜𝑤 𝑡 𝐾 𝐷 𝑡(1+𝐾 𝐷 𝑡)𝑁 (1+𝐾 𝐷 𝑡)𝑁−1  𝑡 < 𝑡≤ 𝑡+𝑁 0𝑡 > 𝑡+𝑁 (19) 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡 =𝐾 𝐷 𝑡𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡,𝑡−1 ∀𝑡∈∪′(20) Constraints (18) calculate the value of the loan balance associated with loan 𝑙 𝑡, in period 𝑡∈∪′. Its value is 0until its activation in period 𝑡= 𝑡. As it is activated (𝑡= 𝑡), the balance equals the amount borrowed (=𝑏𝑜𝑟𝑟𝑜𝑤 𝑡), then for 𝑡 > 𝑡, it is decreased by the debt amortization during the period, which is equal to 𝑟𝑒𝑝𝑎𝑦 𝑡𝑡 -𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡. The repayment formula is given by (19). After loan 𝑙 𝑡is activated in period  𝑡∈, its repayment starts at the beginning of the following period (𝑡= 𝑡+ 1) until it is fully repaid (𝑡= 𝑡+𝑁). Constraints (20) calculate 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡. Both constraints (19) and (20) are non-linear. Their linearization is explained in Appendices C.3, and C.4, respectively. 4.2.2. Equity financing constraint 𝑁 𝑂 𝑃 𝐴𝑇𝑡= (1 −𝜂)(∑ 𝑗∈ 𝐸 𝐵 𝐼 𝑇𝑗 𝑡−∑  𝑡∈ 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡) ∀𝑡∈(21) 𝑐 𝑎𝑠ℎ𝑡=𝑐 𝑎𝑠ℎ𝑡−1 + (1 −𝛿)𝑁 𝑂 𝑃 𝐴𝑇𝑡+𝐸 𝐸𝑡 +∑  𝑡∈ (𝑏𝑜𝑟𝑟𝑜𝑤 𝑡𝑡 −𝑟𝑒𝑝𝑎𝑦 𝑡𝑡 +𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡) −∑ 𝑗∈ (𝑂𝑗(𝑦𝑗 𝑡−𝑦𝑗 ,𝑡−1) −𝐷 𝑒𝑝𝑗𝑦𝑗 ,𝑡−1) ∀𝑡∈(22) 𝐼 𝐸𝑡≤max(0, 𝑐 𝑎𝑠ℎ𝑡−1) ∀𝑡∈(23) To ease the presentation calculation of internal equity variables 𝐼 𝐸𝑡, we introduce two intermediate variables: the Net Operating Profit After Taxes of period 𝑡∈, denoted 𝑁 𝑂 𝑃 𝐴𝑇𝑡, is obtained by removing the interest expenses from the EBIT (see constraints (11)) in period 𝑡∈, and then multiplying the result by the term (1 −𝜂), where 𝜂is the firm tax rate. The cash flows 𝑐 𝑎𝑠ℎ𝑡generated in period 𝑡and owned by shareholders (constraints (22)) are equal to the residual cash flows once all stakeholders, including lenders and State, have been remunerated. Finally, constraints (23) state that 𝐼 𝐸𝑡cannot exceed the cash available at the end of the preceding period, 𝑐 𝑎𝑠ℎ𝑡−1. The linearization of this equation is explained in Appendix C.5. 4.2.3. Calculation of FGV and APV The probability of default is denoted by 𝑝. The variable 𝑒𝑞 𝑢𝑖𝑡𝑦𝑡 represents the value of the equity in period 𝑡∈. The parameter 𝛽 >1 models the volatility of the company’s assets. The parameter 0< 𝜁 <1 is the maximum percentage of the total assets that can be used for debts related to new facilities. We recall that the parameter 0< 𝛾 <1, defined in Section 3.4 is used to calculate the bankruptcy cost. Since cash flows and tax shield benefits share the same systematic risk, they are discounted at the same rate 𝐾𝐸. 𝑒𝑞 𝑢𝑖𝑡𝑦𝑡=𝑒𝑞 𝑢𝑖𝑡𝑦𝑡−1 + (1 −𝛿)𝑁 𝑂 𝑃 𝐴𝑇𝑡+𝐸 𝐸𝑡∀𝑡∈(24) 𝑝=(∑ 𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑇 ∑ 𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑇 +𝑒𝑞 𝑢𝑖𝑡𝑦𝑇)𝛽 (25) ∑  𝑡∈ 𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑡 ≤𝜁(𝑐 𝑎𝑠ℎ𝑡+∑ 𝑗∈∑ 𝑡′≤𝑡 (𝑂𝑗(𝑦𝑗 𝑡′−𝑦𝑗 ,𝑡′−1) −𝐷 𝑒𝑝𝑗𝑦𝑗 ,𝑡′−1)) ∀𝑡∈.(26) Operations Research Perspectives 14 (2025) 100319 6 H. Rezaei et al. 𝐹 𝐺 𝑉= (1 −𝑝)𝜂∑ 𝑡∈∪′∑ 𝑡∈𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡 (1 +𝐾𝐸)𝑡−𝑝 𝛾 𝑂 𝐺 𝑉(27) In constraints (24), the value of equity in period 𝑡∈is defined as its value at time 𝑡− 1plus the retained earnings at time 𝑡plus the new cash provided by shareholders at time 𝑡. This value is used to calculate the probability of default in constraint (25). This probability is an increasing function of the debt ratio ∑ 𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑇 ∕(∑ 𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑇 + 𝑒𝑞 𝑢𝑖𝑡𝑦𝑇)in period 𝑇. The probability of default is influenced by the company’s capacity to generate enough cash flow to meet its debt obligations. The debt ratio quantifies the relative significance of the debt in relation to cash flow. Since we do not directly model cash flow risk, we approximate it indirectly using the coefficient 𝛽. The linearization of this constraint is explained in Appendix C.8. To avoid financial distress in earlier periods, we consider an upper bound on the level of debt, set by Constraints (26). The right-hand side represents the net value of the total assets in period 𝑡∈𝑇, multiplied by parameter 𝜁. Constraint (27) calculates the value of FGV. The first term corresponds to the tax shield benefit (TSB) and the second one to the expected bankruptcy cost (EBC) as illustrated in Fig. 2. As debt levels increase, so do interest payments and the resulting tax shield benefits. However, as shown in Eq. (25), a higher debt level also raises the probability of default and the corresponding bankruptcy costs. Finally, the objective function to be maximized is the APV, which is the sum of OGV and FGV, as written in constraint (28): 𝐴𝑃 𝑉= (1 −𝑝𝛾)𝑂 𝐺 𝑉+ (1 −𝑝)𝜂(∑ 𝑡∈∪′∑ 𝑡∈𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡 (1 +𝐾𝐸)𝑡).(28) The present value of tax shields is computed by discounting the annual interest amount ∑ 𝑡∈𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡 at the rate 𝐾𝐸, multiplied by the firm tax rate 𝜂. The company will receive the interest tax shields with a probability 1 −𝑝(i.e., if it is not bankrupted) and pay a bankruptcy cost 𝛾 𝑂 𝐺 𝑉 with probability 𝑝. From this equation, it is clear that the leverage has a mixed impact on APV. On the one hand, it has a negative impact through the bankruptcy risk (𝑝) and, on the other hand, a positive one through the tax shield benefit (last term of the APV function). 5. Data generation As explained in Section 2, we did not find a related benchmark instance that could be used to validate and experiment with this model. Therefore, new instances were generated following the generation rules used by different authors. As mentioned above, APV is particularly relevant for evaluating strategic decisions in large-scale activities. Thus, the generated instances mimic a supply chain with multiple markets, each market having its own costs, product price, etc. We generated instances with 60 to 270 customers. Following [45], the number ||of candidate facilities is defined as 10% of the number of customers. The maximum number of open facilities (𝐽𝑚𝑎𝑥) is defined as ⌈0.5||⌉. In all instances, ||= 5and |′|= 10 periods were considered. The next subsections detail the steps followed by the data generation. 5.1. Logistics data and parameters All locations were generated in a 1000 ×1000 grid. Both axes of the grid are decomposed into 5 intervals of size 200, defining 25 squared areas called regions. Markets. The grid is partitioned into two to five markets, that is a connected set of regions. Each market receives an economic index that is used as a proxy to indicate its economic situation and generate the costs and prices in accordance. The affected costs and prices are, specifically the processing cost (𝜇), the opening cost parameter (𝜑), and the selling price (𝑃) as shown in Table 2. The economic indices are generated with a uniform distribution between 50 and 150. The higher the economic index, the higher the market’s costs and prices. Customers and candidate facilities. To generate the location of customers and candidate facilities, we used two approaches: the coordinates of the customers and candidate facilities were either randomly generated with a uniform distribution over the entire grid or we used a clustered pattern. In the latter case, we located around 60% of customers (resp. candidate facilities) in a subset of 4 or 5 regions, then the remaining 40% were randomly located with a uniform distribution over the grid. Customers’ demand. We generated the customers’ demand according to two different demand profiles: First, following [46], the customers’ demands at each period are generated with a uniform distribution in the interval [100, 300]. Then, each customer’s demand still lies in the interval [100, 300], with the additional property that the total demand grows by a factor in the interval [1.05, 1.25] between two successive periods. Note that although the sum of customers’ demands grows over time, the individual demand of some customers may decrease between two successive periods due to the uneven distribution of growth." Capacity of the facilities. Each candidate location has a given capacity. We have generated three sizes of facilities, named small,medium and large, representing 80%, 100% and 130% of the ratio 𝐷∕𝐽𝑚𝑎𝑥 respectively, where 𝐷is the average demand per period. The capacity of each facility is randomly chosen such that about 1/3 of facilities are small, 1/3 are medium and 1/3 are large. Logistics costs. Several costs in the mathematical model are affected by variations in the cost of living across different market areas. To account for this, we introduced an intermediate parameter, referred to as the economic index, to generate these costs. For each market, the economic index is generated with a uniform distribution between values 50 and 150. The range [50,150] is decomposed into 5 intervals, and for each interval, the processing cost, the opening cost and the selling price are generated according to a uniform distribution, as detailed in Table 2. Note that as we generate the yearly fixed cost of facility as a percentage of the opening cost, this data is also directly affected by the economic index. •The Processing costs (𝜇𝑗) of small-sized facilities is defined with a uniform distribution on an interval depending on the economic index of the market where facility 𝑗lies (see line 2 of Table 2). The fictitious monetary unit used in the rest of this article is called relative money unit (𝑟𝑚𝑢). To model economies of scale, the processing costs at average-sized and large-sized facilities are obtained by multiplying these values by 0.98 and 0.96, respectively. These parameters have been adjusted in such a way that the relative part of processing costs roughly represents around 35% –45% of the total costs in each instance. •The value of the fixed opening cost (𝑂𝑗), represented in monetary value, is strongly related to the value of the real estate. To model economies of scale as the capacity grows, following [47], we assume that the fixed opening cost of a facility 𝑗is roughly proportional to the square root of its capacity. We set 𝑂𝑗= 𝜑𝑗√𝐶𝑗, where 𝐶𝑗represents the capacity of facility 𝑗and 𝜑𝑗 is the opening cost parameter at location 𝑗(mainly determined by the cost of the local real estate). The value of 𝜑𝑗is generated randomly with a uniform distribution between 625 and 750 (see line 3 of Table 2). These intervals have been set by successive adjustments in such a way that the relative part of the fixed opening costs roughly represent 25% –35% of the total logistics costs (see, e.g. [47,48] for similar approaches). •The Fixed yearly running cost (𝐹𝑗) is set at 5% of the fixed opening cost of each facility, 𝑂𝑗, per year. Thus, it represents around 10% –15% of the total costs. •The transportation costs are considered proportional to the Euclidean distance traveled. We assume they are similar in all markets. To have the transportation cost representing 10% to 20% of total costs [49], the unit transportation cost, 𝜔, is set at 0.002 𝑟𝑚𝑢 in all the instances. Operations Research Perspectives 14 (2025) 100319 7 H. Rezaei et al. Table 2 Generation of logistical data. Economic index [50, 70[ [70, 90[ [90, 110[ [110, 130[ [130, 150] Processing cost 𝜇(r mu∕unit) [1, 1.1] [1.1, 1.2] [1.2, 1.3] [1.3, 1.4] [1.4,1.5] Opening cost parameter 𝜑[625, 650] [650, 675] [675, 700] [700, 725] [725, 750] Selling price 𝑃(r mu∕unit) [3, 3.4] [3.4, 3.8] [3.8, 4.2] [4.2, 4.6] [4.6, 5] Table 3 Calculation of the cost of loan 𝐾 𝐷𝑡. Debt ratio interval Cost of loan (in %) (in %) (0,30] 3.2 (30,40] 3.5 (40,50] 4.0 (50,60] 5.0 (60,70] 6.6 (70,80] 9.6 Selling price. The price 𝑃𝑖proposed to customer 𝑖∈depends on the market in which 𝑖lies. Its value lies in the interval [3,5] (see line 4 of Table 2). Accessibility parameters. The accessibility parameter 𝑉𝑖𝑗 is set to value 1 if the distance between facility 𝑗∈and customer 𝑖∈is less than or equal to 750 (roughly half of the longest possible distance in the grid), and 0 otherwise. 5.2. Financial parameters The average cost of equity 𝐾𝐸is set at 9% [50]. In order to estimate the cost of loan 𝐾 𝐷𝑡in relation to the probability of bankruptcy, we employ an arbitrage equation used by a risk-neutral bank1. This approach allows us to quantify the expected rate of return on loans by taking into account their associated risks. Using the arbitrage equation, we modeled the cost of loan as a piecewise linear function varying between 3.2% and 9.6%. We considered six debt ratio intervals as defined in Table 3. In all instances, we consider a lifetime value 𝐿and a number of annuities 𝑁that are both equal to 10 years. The depreciation 𝐷 𝑒𝑝𝑗calculated with this lifetime value, and the salvage value 𝑆 𝑉𝑗 is assumed to be negligible [27]. The value of the bankruptcy cost parameter 𝛾is set to 0.5 and the upper bound of the debt ratio 𝜁is set to 0.8 in order to ensure an acceptable financial situation. The value of the bankruptcy probability parameter 𝛽in constraint (25) is set to 3. The corporate tax rate 𝜂is set at 30%. 5.3. Set of instances Following the principles described above, 32 instances were generated: 16 instances have random locations and 16 instances have clustered locations, 16 instances have random demands and 16 instances have growing demands. 1The arbitrage equation used by a risk-neutral bank can be expressed in the form of the following equation: 𝐹(1 +𝑟𝑓)𝑁= (1 +𝐾 𝐷)𝑁(𝑝𝑅 + (1 −𝑝)𝐹). Where 𝐹,𝑟𝑓,𝑁,𝐾 𝐷,𝑝, and 𝑅are loan facial value, risk-free rate, loan duration, cost of loan, probability of bankruptcy, and value given default or the amount that lender recovers if the company defaults on its debt, respectively. Note that, we set the risk-free rate to 3% in accordance with the cost of debt reported by the KPMG report [50]. According to this report, the cost of debt for industrial manufacturing companies with a debt ratio of 25%–30% is around 3%. Setting 𝑟𝑓to 3% lets us obtain almost the same cost of debt for the same amount of debt ratios. Instance names are formed by their size followed by one of the letters A, B, C, or D. Note that there are 4 different types of instances due to applying four patterns to generate the customers’ and facilities’ coordinates as well as the customers’ demands. For each size of instance, Table 4enumerates the number of binary and continuous variables associated to the logistical and financial part of the model, as well as the number of constraints. Note that the binary variables in the financial part of the model come from the linearization of nonlinear constraints. 6. Numerical experiments 6.1. Integrated and sequential approaches As represented in Fig. 2, APV can be decomposed into a logistical part (OGV) and a financial part (FGV). The whole mathematical model can be decomposed into two sub-problems presented in Sections 4.1 and 4.2, respectively. The logistical sub-problem determines which facilities should be opened, which customers should be served as well as the associated product flows. Once the logistical decisions have been fixed, the financial sub-problem optimizes the value of APV by maximizing FGV (see Fig. 4). The Sequential Approach mimics the decision process followed in classical facility location models. The logistical and financial subproblems are solved sequentially: First, the 𝑂 𝐺 𝑉formula defined by Eq. (16) is maximized, subject to constraints (1)—(15). Then, the 𝑂 𝐺 𝑉is considered constant and the 𝐹 𝐺 𝑉is optimized. This amounts to maximizing the APV formula defined in constraint (28), subject to constraints (17)—(26). The Integrated Approach considers the whole mathematical model (1)–(28) at once and maximizes APV by simultaneously determining the value of all logistical and financial variables. By nature, the objective function values of the optimal solutions in the integrated approach are higher than those in the sequential approach. The relative gap between the optimal solutions of the two approaches measures the benefit of introducing financial considerations into facility location models. This comparison underscores the tangible advantages of integrating financial decisions into classical facility location models, demonstrating the benefits of adopting an integrated financial perspective. In return, the integrated approach is likely to yield computational difficulties due to the size of the mathematical model. An objective of the numerical experiments is to explore the practical limits of such a model. 6.2. Assessment of the sequential and the integrated approaches All numerical experiments were run on an Intel(R) Xeon(R) Gold 6230 CPU @ 2.10 GHz using ten cores. The mathematical model was solved with the IBM ILOG CPLEX Optimization Studio 20.1.0 solver, with a time limit of 6 h (21 600 seconds). Table 5compares the computational time needed to solve the integrated and the sequential approaches, respectively. Columns 2 and 4 indicate the computational time (in seconds) to get an optimal solution. Columns 3 and 5 indicate the optimality gap when no optimal solution could be obtained after 3 h of computation. This table shows that both approaches are tractable for instances with up to 150 customers. As expected, the sequential approach is easier to solve than the integrated approach: 29 instances could be solved to optimality with the sequential approach within 6 h, only Operations Research Perspectives 14 (2025) 100319 8 H. Rezaei et al. For practical implementation of the integrated decision model, company management must consider the optimal mix of debt and equity financing for its strategic projects. The benefits from debt-related tax savings are relatively straightforward to calculate, as they depend on the easily observable corporate tax rates in the countries where the locations are established. In contrast, estimating bankruptcy costs is far more challenging. The probability of a company’s bankruptcy can be estimated using several factors: (i) the credit rating assigned to the company’s debt by rating agencies, (ii) the interest rate set by lenders (with higher rates indicating a greater risk of bankruptcy), (iii) econometric models based on the company’s financial ratios (scoring functions). Bankruptcy costs will be assessed by considering the costs of bankruptcy procedures in each country for direct costs and by comparing the bankruptcy costs of previously defaulted companies for indirect costs. Another managerial challenge is bringing the operational and financial teams together to collaboratively develop the implementation strategy. Since the goal of this paper was to highlight the interactions between logistical and financial decisions rather than describing a particular supply chain, we considered a simple two-echelon supply chain. This model can be extended in many directions. Considering a supply chain with already active facilities, customers and current loans only requires modifying the initial conditions. Relaxing the incremental customer satisfaction, allowing partial customer satisfaction or setting singlesourcing constraints would slightly modify the logistics constraints, but these rules do not affect the financial part of the model. Extending the model to more complex supply chains (e.g., with additional echelons, selection of raw material suppliers, resizing of facilities, selection of subcontractors, selection of transportation modes [51]) would require a set of more elaborated logistical constraints but, once again, these logistical rules do not affect the financial part of the model. However, introducing the possibility of closing facilities has direct financial consequences. A realistic assumption is that, once opened, facilities should be operating for a minimum number of periods (which is generally larger than 𝑇). Assuming that facilities already operating in period 0 can be closed before the end of their lifetime has consequences both on the value of OGV and FGV. Closing a facility and selling it modifies the cash flows. All associated loans must be the object of early reimbursement, with possible financial penalties. Finally, constraints (15), which assumed 𝑦𝑗 𝑇= 1for any operating facility, should be adapted to the case of closing facilities. Our model can easily apply to a realistic case where the taxes and cost of debt differ from one location to another. In that case, the difference between both approaches would probably be more significant. A further possible extension is the consideration of stochastic data or parameters. Several factors, such as demand, logistics costs, disruption probabilities, and financial parameters, are subject to uncertainty. A stochastic model incorporating multiple uncertain parameters would provide a more realistic representation. However, managing several stochastic parameters simultaneously remains a significant challenge in stochastic optimization. Robust optimization might be a more tractable approach, though it may yield very conservative results. A possible solution is to utilizing methods like Approximate Dynamic Programming and Stochastic Dynamic Optimization, which handle both uncertainty and time-dependent decision-making. Other possible extensions of the model would be to explore financial risk-aware facility location models [52], or to extend the model by integrating tactical variables such as the working capital. The costly linearization mechanism prevents MILP solvers to solve much larger instances than those presented in this paper. Table 5illustrates the model’s limited scalability. While the solutions obtained from the integrated approach are relatively close to those from the sequential approach, a significant gap between the lower and upper bounds emerges for instances with more than 210 customers. For even larger instances, such as those with 270 customers, the sequential approach also struggles to find optimal solutions. To tackle larger instances or more complex logistics networks not addressed in this study, alternative solution methods are required. Utilizing heuristic methods or a combination of exact and heuristic approaches could provide an effective strategy for solving these larger problems. CRediT authorship contribution statement Hamidreza Rezaei: Writing – review & editing, Writing – original draft, Validation, Software, Methodology, Investigation, Formal analysis, Conceptualization. Nathalie Bostel: Writing – review & editing, Writing – original draft, Validation, Supervision, Methodology, Investigation, Formal analysis, Conceptualization. Vincent Hovelaque: Writing – review & editing, Writing – original draft, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization. Olivier Péton: Writing – review & editing, Writing – original draft, Validation, Supervision, Methodology, Investigation, Formal analysis, Conceptualization. Jean-Laurent Viviani: Writing – review & editing, Writing – original draft, Validation, Methodology. Declaration of competing interest None Acknowledgments This work has been supported by ANR under the FILEAS FOG (ANR 17-CE10-0001-01) project. Appendix A. Notations Sets Set of customers (𝑖) Set of candidate facilities (𝑗) Set of time periods (𝑡) ′Extended set of time periods (𝑡) Decision variables 𝑦𝑗 𝑡= 1if 𝑗∈is operating in period 𝑡∈∪′, 0 otherwise {0,1} 𝑏𝑜𝑟𝑟𝑜𝑤𝑡Amount borrowed in period 𝑡∈[0,∞[ 𝐸 𝐸𝑡External equity in period 𝑡∈[0,∞[ 𝐼 𝐸𝑡Invested amount of internal equity in period 𝑡∈ [0,∞[ Intermediate logistical variables 𝑥𝑖𝑡 = 1if 𝑖∈is served in period 𝑡∈, 0 otherwise {0,1} 𝑒𝑗 𝑡Expenses occurred in period 𝑡∈𝑇for 𝑗∈[0,∞[ 𝑞𝑖𝑗 𝑡Quantity shipped from 𝑗∈to 𝑖∈in period 𝑡∈𝑇[0,∞[ 𝑟𝑗 𝑡Revenues obtained in period 𝑡∈𝑇related to 𝑗∈[0,∞[ Operations Research Perspectives 14 (2025) 100319 15 H. Rezaei et al. Intermediate financial variables 𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑡 Value of loan balance 𝑙 𝑡in period 𝑡∈[0,∞[ 𝑐 𝑎𝑠ℎ𝑡Cash level in period 𝑡∈] − ∞,∞[ 𝐶 𝐹𝑗 𝑡Cash flow generated by 𝑗∈𝐽in period 𝑡∈𝑇 ] − ∞,∞[ 𝐸 𝐵 𝐼 𝑇𝑗 𝑡Earnings before interest and tax by 𝑗∈ in period 𝑡∈ ] − ∞,∞[ 𝑒𝑞 𝑢𝑖𝑡𝑦𝑡Amount of equity in period 𝑡∈[0,∞[ 𝐹 𝐶 𝐹𝑗 𝑡Free cash flow associated with 𝑗∈in period 𝑡∈ ] − ∞,∞[ 𝐹 𝐺 𝑉Financially generated value of the firm ] − ∞,∞[ 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡 Interest of loan 𝑙 𝑡in period 𝑡∈[0,∞[ 𝐾 𝐷𝑡Cost of loan 𝑙𝑡[0,1] 𝑂 𝐺 𝑉Operationally generated value of the firm ] − ∞,∞[ 𝑁 𝑂 𝑃 𝐴𝑇𝑡Net operating profit after tax in period 𝑡∈ ] − ∞,∞[ 𝑝Probability of bankruptcy [0,1] 𝑟𝑒𝑝𝑎𝑦 𝑡𝑡 Repayment of loan 𝑙 𝑡in period 𝑡∈[0,∞[ Logistical parameters 𝐶𝑗Yearly production capacity of facility 𝑗∈ 𝐷𝑖𝑡 Demand of 𝑖∈in period 𝑡∈ 𝐷 𝑒𝑝𝑗Depreciation of 𝑗∈ 𝐹𝑗Fixed yearly running cost of 𝑗∈ 𝐽𝑚𝑎𝑥 Max number of facilities to be located 𝐿Facility lifetime 𝑂𝑗Opening cost of 𝑗∈ 𝑃𝑖Product selling price for 𝑖∈ 𝑆 𝑉𝑗Salvage Value of 𝑗∈ 𝑉𝑖𝑗 Equals 1 if 𝑗∈can serve 𝑖∈, 0 otherwise 𝜇𝑗Processing cost of 𝑗∈ 𝜔Unit transportation cost Financial parameters 𝐾𝐸Cost of equity rate 𝑁Number of annuities 𝛽Bankruptcy probability parameter 𝛾Bankruptcy cost parameter 𝛿Dividend payout ratio 𝜂Firm tax rate 𝜁Upper bound of debt ratio Appendix B. Detailed results: accumulated capacity and total investment Table B.12 reports the accumulated capacity installed at each period along with the total amount of investment in each solution (sum of the fixed cost of the selected facilities). For the sake of clarity, the capacity and investment values in Table B.12 represent thousands of units. Appendix C. Linearization procedures In the proposed mathematical model, several constraints are nonlinear due to the product of two decision variables. We first recall several well-known linearization techniques (Appendix C.1) and then explain how these techniques are applied to our model. C.1. Classical linearization procedures •Linearization 1: product of a real and a binary variable Let 𝑢and 𝑣be two real positive variables and 𝑏a binary variable. Given an upper bound 𝑈of variable 𝑢, the expression 𝑣=𝑏𝑢 can be linearized by: 𝑣≤𝑢 𝑣≥𝑢−𝑈(1 −𝐵) 𝑣≤𝑈 𝑏 𝑣≥0 •Linearization 2: product of two continuous variables There is no exact way to linearize a product of two continuous variables 𝑢and 𝑣. In our model, most continuous variables represent large monetary values. The consequence of rounding down these variables to the nearest integer value is then negligible. Assuming that variable 𝑢is rounded down, we use a power-of-two decomposition to represent 𝑢as a set of binary variables 𝑏𝑖: 𝑢=𝑏0+ 2𝑏1+ 4𝑏2+ 8𝑏3+⋯+ 2⌊𝑙 𝑜𝑔2𝑈⌋𝑏𝑙 𝑜𝑔2𝑈, where U is an upper bound of 𝑣. For example, the value 100 = 4 + 32 + 64 can be represented by the vector (0,0,1,0,0,1,1). The product 𝑢𝑣 can now be rewritten as follows: 𝑢𝑣 =𝑣 𝑖=⌊𝑙 𝑜𝑔2𝑈⌋ ∑ 𝑖=0 2𝑖𝑏𝑖. It is a weighted sum of the terms 𝑣𝑏𝑖, where 𝑣is a continuous variable and 𝑏𝑖is an integer variable. These terms can be linearized with Linearization 1. •Linearization 3: Piece-wise linearization Consider a general non-linear function 𝑓(𝑢)of a single variable 𝑢, where 𝑢∈ [𝑢0, 𝑢ℎ]. We consider intermediate values 𝑢1,…, 𝑢ℎ−1 and intervals of the form [𝑢𝑖, 𝑢𝑖+1]0≤𝑖≤ℎ−1. Let 𝜆𝑖be a vector of binary variables, where 𝜆𝑖= 1if and only if 𝑢∈ [𝑢𝑖, 𝑢𝑖+1]. We define a vector 𝜉of continuous variables such that 𝑢= ℎ−1 ∑ 𝑖=0 𝜉𝑖 𝜆𝑖𝑢𝑖≤𝜉𝑖≤𝜆𝑖𝑢𝑖+1 ℎ−1 ∑ 𝑖=0 𝜆𝑖= 1. Only one value in vector 𝜉is strictly positive and it corresponds to the value 𝑢. Hence, the function 𝑓(𝑢)is approximated by selecting the appropriate interval and considering the piece-wise linear approximation of 𝑓(𝑢)in this interval: 𝑓(𝑢) =∑ 𝑖 𝜆𝑖𝑓(𝑢𝑖) +∑ 𝑖((𝜉𝑖−𝜆𝑖𝑢𝑖)𝑓(𝑢𝑖+1) −𝑓(𝑢𝑖) 𝑢𝑖+1 −𝑢𝑖). Since only one value of vectors 𝜆and 𝜉is strictly positive, this expression reduces to 𝑓(𝑢) =𝜆𝑖𝑓(𝑢𝑖) + (𝜉𝑖−𝑢𝑖)𝑓(𝑢𝑖+1) −𝑓(𝑢𝑖) 𝑢𝑖+1 −𝑢𝑖 for some 0≤𝑖≤ℎ− 1. •Linearization 4: Logical constraints Logical constraints allow the expression of logical operators such as logical-or, logical-and, and conditional statements (if ... then ...) in the linear programming context. – Or-condition: Consider a general mathematical expression 𝑔(𝑥)and two parameters 𝑏1and 𝑏2. The expression (𝑔(𝑥)≤𝑏1) ∨ (𝑔(𝑥)≤𝑏2) Operations Research Perspectives 14 (2025) 100319 16 H. Rezaei et al. Table B.12 Accumulated capacity installed at each period and overall investment with the sequential and the integrated approaches. 1st part Instance Approach Total capacity installed (×103) Investment t=1 t=2 t=3 t=4 t=5 (×103r mu) 60-A Integrated 10.5 10.5 10.5 14 14 146.1 Sequential 10.5 10.5 10.5 10.5 10.5 104.7 60-B Integrated 10.5 10.5 10.5 16.5 16.5 162.4 Sequential 4.5 9 9 15 15 154.7 60-C Integrated 12 12 12 12 12 111.9 Sequential 6 6 6 12 12 111.9 60-D Integrated 6 9.5 9.5 15.5 15.5 153.3 Sequential 6 6 6 12 12 111.9 90-A Integrated 16.5 16.5 16.5 22.5 22.5 211.3 Sequential 12 12 12 18 18 165.7 90-B Integrated 12 12 16.5 22.5 22.5 211.3 Sequential 6 12 12 22.5 22.5 211.3 90-C Integrated 10.5 10.5 10.5 16.5 16.5 154.8 Sequential 10.5 10.5 10.5 16.5 16.5 154.8 90-D Integrated 6 10.5 10.5 16.5 16.5 154.8 Sequential 6 10.5 10.5 16.5 16.5 154.8 120-A Integrated 16.5 22.5 22.5 28.5 28.5 250.6 Sequential 16.5 16.5 22.5 28.5 28.5 250.6 120-B Integrated 12 16.5 22.5 28.5 28.5 250.6 Sequential 12 16.5 22.5 28.5 28.5 250.6 120-C Integrated 18 18 18 28.5 28.5 243.7 Sequential 12 12 12 22.5 22.5 191.8 120-D Integrated 12 18 22.5 28.5 28.5 243.7 Sequential 6 12 18 24 24 201.4 150-A Integrated 32 32 36.5 41 41 341.2 Sequential 27.5 27.5 32 36.5 36.5 296.6 150-B Integrated 21.5 27.5 32 41 41 341.2 Sequential 21.5 27.5 32 36.5 36.5 296.6 150-C Integrated 33 33 36.5 41 41 370.5 Sequential 28.5 28.5 33 37.5 37.5 331.2 150-D Integrated 22.5 28.5 33 41 41 370.5 Sequential 22.5 28.5 33 37.5 37.5 331.2 2nd part Instance Approach Total capacity installed (×103) Investment t=1 t=2 t=3 t=4 t=5 (×103r mu) 180-A Integrated 33 37.5 37.5 43.5 43.5 391.5 Sequential 33 33 33 39 39 346.9 180-B Integrated 28.5 33 39 45 45 400.7 Sequential 22.5 28.5 33 45 45 400.7 180-C Integrated 34.5 34.5 34.5 42.5 42.5 385.4 Sequential 34.5 34.5 34.5 39 39 346.6 180-D Integrated 30 33.5 41.5 46 46 424.2 Sequential 24 30 34.5 42.5 42.5 385.4 210-A Integrated 28.5 33 43.5 53 53 502.1 Sequential 28.5 28.5 37.5 49.5 49.5 463.1 210-B Integrated 28.5 33 39 55.5 55.5 514.2 Sequential 22.5 28.5 33 49.5 49.5 463.1 210-C Integrated 30 30 39.5 53.5 53.5 505 Sequential 30 30 39.5 49 49 460.7 210-D Integrated 24 30 36 53.5 53.5 505 Sequential 24 27.5 33.5 50 50 466 240-A Integrated 40.5 52.5 63 63 63 559.1 Sequential 57 57 57 63 63 559.1 240-B Integrated 39 43.5 49.5 59 59 539.4 Sequential 39 45 51 57 57 505.8 240-C Integrated 57 57 57 63 63 553.9 Sequential 57 57 57 63 63 553.9 (continued on next page) Operations Research Perspectives 14 (2025) 100319 17 H. Rezaei et al. Table B.12 (continued). 2nd part Instance Approach Total capacity installed (×103) Investment t=1 t=2 t=3 t=4 t=5 (×103r mu) 240-D Integrated 36 45 51 54.5 60.5 541.9 Sequential 40.5 45 51 57 57 503.2 270-A Integrated 15 34.5 40.5 57 57 521.4 Sequential 39 39 39 45 45 402.1 270-B Integrated 33 43.5 48 60 60 543 Sequential 28.5 33 39 51 51 453.8 270-C Integrated 45 45 54 58.5 58.5 535.9 Sequential 39 39 39 54 54 497.6 270-D Integrated 33 43.5 49.5 64.5 64.5 592.2 Sequential 34.5 40.5 45 64.5 64.5 592.2 can be linearized by: 𝑔(𝑥)≤𝑏1+𝑀×𝜗 𝑔(𝑥)≤𝑏2+𝑀× (1 −𝜗) where 𝑀is a large enough positive value and 𝜗a binary variable. – Conditional expression: Consider the following expression {𝑔(𝑥)≥0⇒𝜌= 1 𝑔(𝑥)<0⇒𝜌= 0, where 𝜌is a binary variable. This conditional expression is linearized by: {𝑔(𝑥) +𝜖≤𝜌×𝑀 −𝑔(𝑥)≤(1 −𝜌) ×𝑀 where 𝜖is a relatively small positive value. C.2. Linearization of the OGV second term The second term of the OGV formula (16) is non-linear due to the product of continuous variables 𝐶 𝐹𝑗 𝑇and binary variables 𝑦𝑗 𝑡. This expression can be linearized with Linearization 1 of Appendix C.1. Since the cash flow variables 𝐶 𝐹𝑗 𝑇measure the cash flows of facility 𝑗∈after period 𝑇, its maximal value corresponds to the case where all customers’ demands are served by facility 𝑗. Hence, the upper bound 𝑈can be set to ∑𝑖∈𝑃𝑖𝐷𝑖𝑇 . C.3. Linearization of constraints (19) According to the constraints (19), the repayment for loan 𝑙 𝑡can be a positive value only for periods  𝑡 < 𝑡≤ 𝑡+𝑁. This amount depends on the amount borrowed (𝑏𝑜𝑟𝑟𝑜𝑤 𝑡), and its associated interest rate, 𝐾 𝐷 𝑡. We linearize the product of 𝑏𝑜𝑟𝑟𝑜𝑤 𝑡and 𝐾 𝐷 𝑡. First, 𝐾 𝐷 𝑡is written as follows: 𝐾 𝐷 𝑡= 𝑛 ∑ 𝑚=1 𝑖𝑟𝑚×𝜃 𝑡𝑚 ∀ 𝑡∈, where 𝑛represents the number of debt ratio intervals and 𝑖𝑟𝑚is the interest rate associated with the interval 𝑚∈ [1, 𝑛].𝜃 𝑡𝑚 is a binary variable that takes the value of 1 if the debt ratio after activating loan 𝑙 𝑡belongs to the interval 𝑚, and 0 otherwise. Thus, if 𝜃 𝑡𝑚 = 1, the interest rate of loan 𝑙 𝑡is 𝑖𝑟𝑚. Now the repayment formula can be rewritten as follows: 𝑟𝑒𝑝𝑎𝑦 𝑡𝑡 =𝑏𝑜𝑟𝑟𝑜𝑤 𝑡×𝐾 𝐷 𝑡(1 +𝐾 𝐷 𝑡)𝑁 (1 +𝐾 𝐷 𝑡)𝑁− 1 =𝑏𝑜𝑟𝑟𝑜𝑤 𝑡× 𝑛 ∑ 𝑚=1 𝑖𝑟𝑚(1 +𝑖𝑟𝑚)𝑁 (1 +𝑖𝑟𝑚)𝑁− 1×𝜃 𝑡𝑚 = 𝑛 ∑ 𝑚=1 𝑖𝑟𝑚(1 +𝑖𝑟𝑚)𝑁 (1 +𝑖𝑟𝑚)𝑁− 1×𝜃 𝑡𝑚 ×𝑏𝑜𝑟𝑟𝑜𝑤 𝑡, where the term 𝜃 𝑡𝑚 ×𝑏𝑜𝑟𝑟𝑜𝑤 𝑡is a product of real and binary variables, that can be linearized according to the Linearization 1 of Appendix C.1, where the upper bound 𝑈can be set to the maximum investment size which equals ∑𝑗∈𝐽𝑂𝑗. C.4. Linearization of constraints (20) The linearization of product 𝐾 𝐷 𝑡×𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡,𝑡−1 follows the same principle as in Appendix C.3. We rewrite 𝐾 𝐷 𝑡=∑𝑛 𝑚=1 𝑖𝑟𝑚×𝜃 𝑡𝑚, and obtain the following equation: 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡 = 𝑛 ∑ 𝑚=1 𝑖𝑟𝑚×𝜃 𝑡𝑚 ×𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡,𝑡−1. Using again the procedure explained in Appendix C.1 concludes the linearization. For the product of binary (𝜃 𝑡𝑚) and real (𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡,𝑡−1) variables, the upper bound 𝑈is set to the maximum possible amount to borrow i.e., ∑𝑗∈𝐽𝑂𝑗. C.5. Linearization of constraints (23) In constraints (23), the expression max(0, 𝑐 𝑎𝑠ℎ𝑡−1)is non-linear due to the max operator. We introduce the continuous variables 𝐼 𝐸 𝑀 𝐴𝑋𝑡 and auxiliary binary variables 𝑤𝑡satisfying: 𝐼 𝐸 𝑀 𝐴𝑋𝑡≥𝑐 𝑎𝑠ℎ𝑡−1 ∀𝑡∈ 𝐼 𝐸 𝑀 𝐴𝑋𝑡≤𝑐 𝑎𝑠ℎ𝑡−1 +𝑈(1 −𝑤𝑡) ∀𝑡∈ 𝐼 𝐸 𝑀 𝐴𝑋𝑡≤𝑈 𝑤𝑡∀𝑡∈ 𝐼 𝐸 𝑀 𝐴𝑋𝑡≥0 ∀𝑡∈, where 𝑈=∑𝑖∈∑𝑗∈∑𝑡∈𝑃𝑖𝐷𝑖𝑡. C.6. Linearization of the tax shield benefit - objective function’s second term The tax shield benefit is calculated as: (1 −𝑝)𝜂∑ 𝑡∈∪′∑ 𝑡∈𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡 (1 +𝐾𝐸)𝑡 where both 𝑝and ∑ 𝑡∈𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑡 are continuous variables. We use Linearization 2 presented in Appendix C.1. 𝑈can be set to the maximum value of interest i.e., the interest paid if all candidate facilities are opened using debt financing. Ignoring the impact of the time value of money: 𝑈=𝑁×𝐾 𝐷 𝑡×∑𝑗∈𝐽𝑂𝑗, where 𝐾 𝐷 𝑡 is set to its maximum possible value. C.7. Linearization of the expected bankruptcy cost (EBC) - objective function’s third term EBC is estimated as 𝑝×𝛾×𝑂 𝐺 𝑉where 𝑝and 𝑂 𝐺 𝑉are continuous variables. We use Linearization 2 of Appendix C.1. The maximum value for OGV can be obtained if all customers are served at all periods without any cost. Thus: Operations Research Perspectives 14 (2025) 100319 18 H. Rezaei et al. Table D.13 Parameter values for sensitivity analysis. Parameter Baseline Alternative Alternative value 1 value 2 Tax rate (𝜂) 30% 20% 40% Debt ratio upper bound (𝜁) 0.8 0.7 0.9 Bankruptcy cost parameter (𝛾) 0.5 0.4 0.6 Facility fixed yearly running cost (𝐹) 5% 7.5% 10% Unit transport cost (𝜔) 0.002 r mu 0.001 r mu 0.003 r mu 𝑈= (1 −𝜂) ×(∑ 𝑖∈∑ 𝑗∈∑ 𝑡∈ 𝑃𝑖𝐷𝑖𝑡 +∑ 𝑖∈∑ 𝑗∈∑ 𝑡∈′ 𝑃𝑖𝐷𝑖𝑇 ). C.8. Linearization of the probability of bankruptcy 𝑝eq. (25) In Eq. (25), the probability of bankruptcy is given by 𝑝=(∑ 𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑇 ∑ 𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑇 +𝑒𝑞 𝑢𝑖𝑡𝑦𝑇)𝛽 . We first consider the debt ratio: 𝜅=∑ 𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑇 ∑ 𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑇 +𝑒𝑞 𝑢𝑖𝑡𝑦𝑇 . Then, we introduce the notation 𝑇 𝐴=∑ 𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑇 +𝑒𝑞 𝑢𝑖𝑡𝑦𝑇representing the total assets in period 𝑡. Using a power-of-two decomposition, 𝑇 𝐴can be represented by a set of binary variables 𝑏𝑖. The debt ratio 𝜅can then be rewritten as: 𝜅=∑ 𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑇 ∑𝑗=𝑈 𝑗=0 2𝑗𝑏𝑗 . Using a cross multiplication, we have 𝜅×∑𝑗=𝑈 𝑗=0 2𝑗𝑏𝑗=∑ 𝑡∈𝑏𝑎𝑙 𝑎𝑛𝑐 𝑒 𝑡𝑇 . This formulation does not contain any ratio, but still multiplies the continuous variable 𝜅with a set of binary variables 𝑏𝑗. Thus, Linearization 1 of Appendix C.1 can be used, with 𝑈= 1. Finally, we use a piece-wise linearization (Linearization 3 of Appendix C.1) to linearize the expression 𝑝(𝜅) =𝜅𝛽. C.9. Linearization of debt ratio intervals The debt ratio intervals impose different interest rates, i.e., 𝐾 𝐷 𝑡, for loan 𝑙 𝑡. 𝛼1𝑚≤𝜅 𝑡≤𝛼2𝑚⇒𝐾 𝐷 𝑡=𝑖𝑟𝑚, where 𝑚represents an interval number, 𝜅 𝑡denotes the debt ratio at  𝑡, and 𝛼1𝑚and 𝛼2𝑚are the debt ratio’s lower and upper bounds associated with interval 𝑚. Interval 𝑚is a conditional expression that can be linearized using Linearization 4 of Appendix C.1. To this end, the interest rate of loan 𝑙 𝑡, is replaced by the term ∑𝑛 𝑚=1 𝑖𝑟𝑚×𝜃 𝑡𝑚, where 𝜃 𝑡𝑚 takes the value of 1 if 𝜅 𝑡, the debt ratio after activating loan 𝑙 𝑡, stands in the interval 𝑚, and 0 otherwise. Appendix D. Sensitivity analysis We conduct a sensitivity analysis to assess the impact of various parameters on the objective function (APV) and fill rate within the integrated approach. Five key parameters are examined: Tax Rate (𝜂), Debt Ratio Upper Bound (𝜁), Bankruptcy Cost Parameter (𝛾), Facility Fixed Yearly Running Cost (𝐹), and Unit Transport Cost (𝜔). For each parameter, three values are tested: the baseline value used in prior numerical experiments and two alternative values as defined in Table D.13. In this table, the Facility Fixed Yearly Running Cost (𝐹) is calculated as a percentage of the facility opening cost (𝑂). The sensitivity of these parameters is tested on a representative subset of six instances. These selected instances — 60-A, 60-C, 120-B, 120-D, 210-A, 210-B — represent different sizes and patterns with an equal distribution across instance types. All instances were solved under the same settings as those specified in Section 6.2. The key observations regarding the influence of these parameters on APV and fill rate are outlined in the following subsections. Fig. D.7 visualizes the detailed results of this analysis. D.1. Tax rate APV:. The impact of the tax rate on APV is almost linear. A reduction in the tax rate will result in an increase in APV, due to the decrease in the tax burden on the firm’s net income, which will in turn lead to an increase in OGV. However, the advantage of the tax shield is reduced as the tax rate is lowered. While lower debt levels reduce bankruptcy costs, the overall impact on APV remains positive at lower tax rates. Conversely, higher tax rates increase the benefit from the tax shield, partially offsetting the decline in OGV. This is accompanied by higher debt and a greater risk of bankruptcy, which ultimately reduces APV at higher tax rates. Fill rate:. Lowering the tax rate generally reduces the fill rate for most instances, except for 210-A. Conversely, raising the tax rate increases the fill rate for half of the instances, while the others remain close to the baseline. When the tax rate decreases, the firm experiences a lower tax burden, which leads to an increase in OGV. Consequently, the expected bankruptcy cost – directly linked to OGV – tends to rise as a result of this increase, making the firm more cautious about increasing its debt ratio. This reduces the incentive to raise debt, while the diminished tax shield further lowers FGV. These combined effects reduce the motivation to open new facilities, aligning with earlier findings (obtained through comparison of the integrated and sequential approaches) that financial considerations make investment more appealing. Similar reasoning applies to the cases where the fill rate increases when the tax rate is raised. Thus, the observed decline in the fill rate under lower and higher tax rates can be explained by the interplay between rising bankruptcy costs, reduced debt, and reduced tax shield benefits. D.2. Debt ratio upper bound Considering value 0.9 as debt ratio upper bound amount to create a new debt ratio interval (80%,90%] in Table 3. We associate this interval with a 15.9% cost of loan. APV:. The results of the sensitivity analysis reveal a positive correlation between APV and the debt ratio upper bound. While increasing the firm’s debt ratio raises the probability of bankruptcy, it also increases the benefit of the tax shield. In all instances, the increase in the tax shield benefit outweighs the increase in expected bankruptcy costs, resulting in a higher APV when the firm is allowed to borrow more. Conversely, decreasing the upper bound to 0.7 reduces APV across all instances. The increase in APV when raising the upper bound to 0.9 is larger than the decrease when lowering it to 0.7. This asymmetry is due to the sharp rise in loan costs as the debt ratio increases. A higher debt ratio, like 0.9, raises loan interest rates and boosts the tax shield benefit. In contrast, lowering the upper bound to 0.7 reduces loan costs, but the decrease in the tax shield is less significant than the benefit gained at a 0.9 debt ratio. Fill rate:. Decreasing the debt ratio upper bound generally results in a decreased or unchanged fill rate, while increasing the upper bound typically leads to a higher fill rate. A higher tax shield makes opening new facilities more appealing, while a lower one has the opposite effect. The greater fill rate increase with a higher debt ratio upper bound is due to the larger tax shield benefit outweighing the reduction from a lower bound. Operations Research Perspectives 14 (2025) 100319 19 H. Rezaei et al. Fig. D.7. Impact of variation of tax rate 𝜂on APV (a) and on the fill rate (b) Fig. D.8. Impact of variation of the debt ratio upper bound 𝜁on APV (a) and on the fill rate (b) Table D.14 Calculation of the cost of loan 𝐾 𝐷𝑡for the alternative 𝛾levels. Debt ratio interval Cost of Loan (in %) 𝛾= 0.5𝛾= 0.4𝛾= 0.6 (baseline) (0,30] 3.2 3.2 3.2 (30,40] 3.5 3.4 3.5 (40,50] 4.0 3.9 4.1 (50,60] 5.0 4.9 5.1 (60,70] 6.6 6.5 6.8 (70,80] 9.6 9.4 9.8 D.3. Bankruptcy cost parameter Changing the bankruptcy cost parameter impacts the loan cost. A higher (or lower) bankruptcy cost parameter results in a lower (or higher) recovery value in the event of default, affecting how much the company can recover. As a result, the loan cost adjusts accordingly, and we updated it using the arbitrage equation, as shown in Table D.14. APV:. Reducing the bankruptcy cost parameter increases both OGV and the debt ratio, potentially resulting in the same or higher APV. In all instances, lower bankruptcy costs lead to higher APV. Conversely, increasing bankruptcy costs consistently reduces APV, confirming a negative correlation between the bankruptcy cost parameter and APV. Fill rate:. Decreasing the bankruptcy cost parameter may have a positive effect on the fill rate. It provides an opportunity to invest more, which in turn can lead to serving more customers. This is reflected in the results, where lowering the bankruptcy cost either increases the fill rate or keeps it unchanged. Conversely, increasing the bankruptcy cost reduces the attractiveness of further investments. As the cost increases, the impact of FGV diminishes, making it less profitable to invest further or serve additional customers. This results in a decreased or unchanged fill rate. The sharp reduction in fill rate for instance 60-C is primarily due to the limited number of candidate facilities in this instance. When the bankruptcy cost increases, the FGV effect decreases and there are no viable alternative facilities to replace the one(s) that are closed. This finding supports the earlier observation that the integration of financial dimensions can enhance the attractiveness of investment, ultimately leading to improved customer satisfaction. D.4. Facility fixed yearly running cost APV:. There is a negative linear relationship between APV and the facility’s fixed yearly running cost. This cost, generated as a percentage of the facility’s opening cost, directly impacts APV. The higher the percentage, the higher the yearly running cost, leading to a reduction in OGV and, consequently, a lower APV. Fill rate:. A similar trend can be observed between the increase in fixed yearly running costs and the fill rate. As this cost rises, the fill rate tends to decrease or remain constant across all instances. For most instances, increasing the fixed yearly running cost from 5% to 7.5% and then to 10% consistently reduces the fill rate, indicating that fewer facilities are being opened as the costs rise (see Figs. D.8–D.11). Operations Research Perspectives 14 (2025) 100319 20 H. Rezaei et al. Fig. D.9. Impact of variation of the bankruptcy cost parameter 𝛾on APV (a) and on the fill rate (b) Fig. D.10. Impact of variation of the facility fixed yearly cost 𝐹on APV (a) and on the fill rate (b) Fig. D.11. Impact of variation of the unit transport cost 𝜔on APV (a) and on the fill rate (b) D.5. Unit transport cost APV:. This parameter has a negative effect on APV across all instances. Decreasing the unit transport cost leads to a higher APV, while increasing it reduces APV. This is because increasing transport costs raises the firm’s total expenses, leading to a lower OGV, which directly lowers APV. Fill rate:. The relationship between unit transport cost and fill rate is more complex. 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