# A Mathematical Programming Approach to Supply Chain Network Design considering Shareholder Value Creation

Authors: Abbas Biglar; Nima Hamta; Mona Ahmadi Rad
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

One of the main goals of supply chain management is to ensure proper flows of products and information through all nodes to supply them in the right place at the right time. To achieve this objective, it is very important to consider flows of products and finances among supply chain nodes. Traditionally, operational and financial processes have been optimized as separate problems. The developed model addresses the problem of designing a supply chain network and tries to integrate both areas of operations and financial aspects to maximize the value created and measured by the Shareholder Value Analysis (SVA). The results show that with appropriate financial decisions, creating more value for the company and its shareholders is achievable. The developed model with a new financial approach is able to improve the total created shareholder value as much as 0.7% larger than the SVA obtained without financial aspects and 0.93% larger than the value created by the basic model. The main reason for an increase in value creation is due to new operational and financial aspects, which mainly show the possibility of closing facilities and bank debt repayments. To validate and show the applicability of the proposed model, it was solved by GAMS-BARON solver with data provided from the literature. Sensitivity analyses on financial parameters were performed to evaluate the results.


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## Body

## 1. Introduction

In recent years, increasing costs and organizational concerns regarding the funding and allocation of financial resources have resulted in great attention being paid to financial flow and its effects on planning decisions through the supply chain networks. One of the main goals of supply chain management (SCM) is to maximize the profitability and competitiveness of a company since it provides an opportunity to enhance synergy [1]. The overall financial performance of a company can be affected by its strategic decisions and operational actions. Financial decisions in supply chain management can also affect future tactical and operational decisions [2]. Therefore, they should be simultaneously considered for optimizing the supply chain network [3]. Many researchers have mentioned the importance of financial decisions in supply chain management and suggested considering them when modeling a supply chain [4]. However, a limited number of studies have optimization models that merge supply chain planning with financial decisions such as investment, financing, and dividend decisions. Based on the literature, there are two different approaches in this field of research. In the first approach, financial aspects are considered endogenous variables and optimized with other variables. In the second approach, financial aspects are considered as known parameters and applied in objective functions and constraints.This study aims to enrich the literature on supply chain network design by using mathematical programming techniques and financial considerations to address the problem of designing a supply chain network. The objective function of the model is to maximize the company value, measured by Shareholder Value Analysis (SVA), which is one of the most prominent metrics being used in business today. In order to integrate financial aspects into supply chain network design, a mixed-integer nonlinear programming (MINLP) model has been developed that considers operational and financial decisions simultaneously for designing a deterministic multi-echelon, multiproduct, and multiperiod supply chain network. To show the model applicability, the data from a case study were employed and solved using BARON solver in GAMS software. The major contributions of this study can be summarized as follows:(i)
This study presents a mathematical model to solve a supply chain network design problem that considers tactical, strategic, and financial decisions at the same time.(ii)
Maximizing the creation of economic value for shareholders as measured by shareholder value analysis (SVA) as a new objective function instead of traditional approaches such as maximizing profits or minimizing costs. It has not been yet used in the general model in supply chain network design problems.(iii)
Providing the possibility of opening or closing facilities in order to deal with market fluctuations at any time period of the planning horizon.(iv)
The proposed model considers the amount of loan, bank repayment, and new capital from shareholders as decision variables; therefore, it provides an accounts payable policy for the company managers instead of considering that all payments should be paid in cash. This is a contribution to the literature because previous studies considered them as parameters.(v)
At the strategic level, the model specifies the number and location of each facility. At the tactical level, it determines the products quantities to be produced and stored to satisfy customers’ demand. Regarding financial decisions, the model specifies the amount of investment and their sources such as cash, bank debt, or shareholders’ capital as decision variables and it provides a repayment policy for managers.(vi)
Regarding the constraints, in addition to common operational constraints, a lower limit and/or upper limit values for performance ratios, efficiency ratios, liquidity ratios, and leverage ratios are taken into account in order to support the financial health of the corporation. To retain better financial performance, the proposed model provides a balance between new capital entries, loans, and repayments. With consideration of large cost of new capital entries, the model imposes an upper bound on it and to avoid an ever-increasing debt, it considers a lower bound for bank repayments. Besides these benefits, the proposed model also provides an accounts payable guideline for managers.(vii)
In contrast with basic models in previous studies which have too many assumptions, the presented model uses accounting principles with fewer assumptions makes it more realistic. For example, we use the net liabilities in the analysis of financial statements that balances bank loans and payments, determines the exact value of deprecation by knowing the lifetime of each asset in each time period, and apply real cash value instead of a predetermined proportion of profit.The main steps of this study can be outlined as follows:(i)
Addressing a supply chain network design problem that simultaneously considers operations, financial decisions, and considerations.(ii)
Developing a mixed-integer nonlinear programming (MINLP) model to model the problem.(iii)
Integrating new financial considerations in the developed model to ensure financial health and growth of the company.(iv)
Testing the applicability and efficiency of the proposed model with data as reported in the literature.(v)
Comparing the results obtained by the proposed model with the basic model through different criteria to show its applicability and advantages.The remaining sections of this paper are organized as follows: In Section2, the relevant literature is reviewed. Section 3 describes the problem and presents a mathematical model for designing a supply chain with financial considerations. Section 4 illustrates a numerical example and discusses the results. Finally, the conclusions and some suggestions for future studies are given in Section 5.


## 2. Literature Review

As mentioned in the previous section, the available published studies on supply chain network design that simultaneously take operations and financial dimensions into account are still rare. This section presents an overview of the selected studies that consider financial issues in the supply chain planning models.Longinidis and Georgiadis [5] introduced a (MINLP) SCN design model that integrates the sale and leaseback (SLB) technique model to find the optimal configuration of a SCN under uncertainty in product demand. Their model’s financial objectives are maximizing net operating profits after taxes (NOPAT) and unearned profit on SLB (UPSLB).Ramezani et al. [6] presented a financial approach to model a supply chain network design that considers financial and physical flows for long-term and mid-term decisions. They applied the change in a company’s equity as the objective function instead of traditional approaches such as maximizing profit or minimizing cost.Mussawi and Jaber [7] formulated a nonlinear program to find the optimal order amounts and the payment time of the supplier by using cash management integration. In their model, maximizing cash level and loan amount are financial decisions that need to be made to minimize inventory and financial costs.Badri et al. [8] proposed a stochastic MILP programming model for a value-based supply chain network design. In their model, to maximize the company’s value (EVA), decisions on financial flow and physical flow (raw materials and finished products) are integrated.Mohammadi et al. [9] developed a MILP model to consider financial and physical flows in mid-term and long-term decisions. The objective functions of their study are maximizing the economic value added (EVA), shareholders’ equity, and corporate value. Saberi et al. [10] considered a trade-off between funding and its effect on the environment in order to optimize NPV in a forward supply chain. Steinrücke and Albrecht [11] developed a mathematical model for maximizing payments to investors via the SCND with financial planning. Alavi and Jabbarzadeh [12] presented a stochastic robust optimization model in order to maximize expected supply chain profit under demand uncertainty. They also considered accounting for financial resources of trade credit and bank credit. In order to solve the model, they developed a solution method based on the Lagrangian relaxation method.Yousefi and Pishvaee [13] developed a MIP model considering the operational and financial aspects of a global supply chain. They also considered economic value added index to measure the financial performance of the global supply chain. Polo et al. [14] proposed a MINLP model in order to maximize EVA in the robust design of a closed-loop supply chain. Paz and Escobar [15] considered the problem of designing a global supply chain of consumer products by considering decisions regarding the location of facilities, transfer pricing, plant capacities, the flow of products, and transfer pricing through a supply chain. The objective function of the proposed mathematical model was to maximize the total profit after tax by considering the determination of global revenues in different facilities and their division over the chain. The problem was solved by using a mixed-integer linear programming model.Wang and Huang [16] proposed a general framework to design a flexible capital-constrained global supply chain (CCGSC), which coordinated both the material flow and cash flow. They also applied a scenario-based mix-integer linear programming model to maximize the quasi-shareholder value (QSC) of a CCGSC under uncertain demand and exchange rates.Kees et al. [17] developed a novel multiperiod approach that provides an alternative framework to determine managerial strategies, integrating financial aspects with logistic decisions in a public hospital supply chain. They also addressed the lack of certainty in data through fuzzy constraints and considered two conflicting objectives: the total cost and total product shortage. To deal with a multicriteria optimization, they applied fuzzy mixed-integer goal programming (FMIGP). Zhang and Wang [18] presented a model that simultaneously focused on multinational enterprises with a global supply chain network design using transfer pricing strategy to achieve the objective of after tax income maximization of the whole global supply chain. The effect of transfer price over the global supply chain was also studied.Brahm et al. [19] presented a new approach to address the problem of joint planning of physical and financial flows. In their research, supply chain contracts were combined and supply chain tactical planning was also considered within an uncertain condition; budgetary, environmental, and contractual constraints were also incorporated. They also developed and implemented a planning model on a rolling horizon basis in order to minimize the effect of disturbances due to existing uncertainties.Yazdi Moghaddam [20] presented a mathematical model that integrated strategic and tactical aspects of a supply chain as well as financial flows. His study compared the traditional approach (maximize profit) with a new approach (maximize the change in equity). The results showed that the new approach led to a change in equity.Goli et al. [21] addressed a closed-loop supply chain network design with uncertain parameters. They developed a mathematical model to incorporate the financial flow, constraints of debts, and employment under fuzzy uncertainty with three objective functions: maximize the cash flow, increase maximize the reliability of consumed raw materials, and maximize the total gobs created in a supply chain.Wang and Fei [22] developed a stochastic programming model for production decisions of manufacturing/remanufacturing. Their model integrated physical and financial operations based on scenario analysis, which took downward substitution between new and remanufactured products into account and selected financial performance indicators, i.e., economic value added, as the optimal objective function.Haghighatpanah et al. [23] proposed a scenario-based optimization model to deal with the SCND problem by considering sale and leaseback (SLB) transactions. The model is formulated based on accounting standards of sales to maximize the supply chain’s benefit after tax.Mohammadi et al. [24] presented a multiproduct, multistage, and multiobjective programming model to design a sustainable plastic closed-loop supply chain network.Escobar et al. [25] considered the design problem of a supply chain for mass-consumer products, taking financial criteria and demand scenarios into account. An established supply chain was adopted as the starting point. The central problem lies in determining the closure and consolidation of distribution centers. The problem was solved using a multiobjective mixed-integer linear programming model, considering two objective functions: the maximization of net present value (NPV) of the supply chain and the minimization of financial risk. Yousefi et al. [26] developed a MILP model which considers financial and physical flows and evaluates the financial performance of EVA and some financial ratios simultaneously. In order to handle the uncertainty of the exchange rate, quality, and quantity of returned products, fuzzy mathematical programming is applied. Tsao et al. [27] applied an approximation approach to examine the impacts of dynamic discounting regarding credit payment on a supply chain network design problem. Badakhshan and Ball [28] developed a MILP model and a simulation-based model to consider the financial and physical flows in a supply chain planning problem under economic uncertainty. They applied the economic value added (EVA) index to measure the financial performance of the supply chain. Goli and Kianfar [29] developed a biobjective mathematical model and Fuzzy ɛ-constraint method for a closed-loop mask supply chain design with the objectives of increasing the total profit and reducing the total environmental impact is presented. In their problem, there are some potential locations for collection, recycling, and disposal centers and the model should decide about the location of the established centers as well as the amount of produced masks and raw materials. Tirkolaee and Aydin [30] designed a bilevel DSS to configure supply chain and transportation networks and address the sustainable development of the problem by developing two MILP models. They applied a fuzzy weighted goal programming approach to deal with multi-objectiveness. Babaeinesami et al. [31] addressed a closed-loop supply chain (CLSC) network design considering suppliers, assembly centers, retailers, customers, collection centers, refurbishing centers, disassembly centers, and disposal centers to design a distribution network based on customers’ needs and simultaneously minimize the total cost and total CO2 emission. To tackle the complexity of the problem, a self-adaptive, nondominated sorting genetic algorithm II (NSGA-II) algorithm is designed, which is then evaluated against the ε-constraint method. Darvazeh et al. [32] proposed a hybrid methodology to expose the process of this problem which helps managers learn how they can determine the optimal number of suppliers. They addressed this gap by developing an integrated approach based on multicriteria decision-making (MCDM) comprising best-worst method (BWM), simple additive weighting (SAW), and a technique for order preference by similarity to ideal solution (TOPSIS), and simulation to determine the optimal number of suppliers. Table 1 presented an overview of studies which integrate financial aspects in supply change management.Table 1
Summarized literature review.
PaperPeriodFinished productParametersObjective functionFinancial futuresSingleMultipleSingleMultipleDeterministicStochasticProfit/costChange in equityEVASVAFinancial ratiosFinancing (loan, cash, …)TaxReceive-payment planningLonginidis and Georgiadis [5]✓✓✓✓✓✓Ramezani et al. [6]✓✓✓✓✓✓Badri et al. [8]✓✓✓✓✓Jin et al. [33]✓✓✓✓✓✓✓Borges et al. [34]✓✓✓✓✓✓✓✓Mohammadi et al. [9]✓✓✓✓✓✓✓✓✓Steinrücke and Albrecht [11]✓✓✓✓✓Alavi and Jabbarzadeh [12]✓✓✓✓✓Yousefi and Pishvaee [13]✓✓✓✓✓✓✓Polo et al. [14]✓✓✓✓✓Paz and Escobar [15]✓✓✓✓Zhang and Wang [18]✓✓✓✓✓Brahm et al. [19]✓✓✓✓✓Yazdi Moghaddam [20]✓✓✓✓✓Goli et al. [21]✓✓✓✓Wang and Fei [22]✓✓✓✓✓✓Haghighatpanah et al. [23]✓✓✓✓Mohammadi et al. [24]✓✓✓✓Escobar et al. [25]✓✓✓✓✓Yousefi et al. [26]✓✓✓✓✓✓✓Tsao et al. [35]✓✓✓✓Badakhshan and Ball [28]✓✓✓✓This study✓✓✓✓✓✓✓✓✓Based on the abovementioned works, this study suggests a mathematical model that simultaneously considers the physical and financial aspects of a supply chain planning problem. We develop a deterministic mixed-integer nonlinear programming (MINLP) model to specify the number and location of facilities and the links between them. The model also determines the quantities to be produced, stored, and transported in order to meet customers’ demands as well as maximize shareholder value analysis (SVA). In financial decisions, we consider the amount to invest, the source of the money needed (cash, bank loan, or new capital from shareholders), and repayments to the bank.


## 3. Problem Definition and Assumptions

In this study, a multi-echelon, multiperiod, and multiproduct supply chain was discussed. Its semantic structure is shown in Figure1. The supply chain consists of plants, warehouses, distribution centers and customer zones. The problem incorporates operational and financial decisions simultaneously; therefore, the mathematical formulation needs proper variables and parameters.Figure 1
The supply chain network considered in the proposed model.The objective function and financial constraints are calculated based on the studies by Brealey et al. [36] and Borges et al. [34]. The goals of the proposed model are to determine as follows:(i)
Strategic decisions about the facilities (plants, warehouses, and distribution centers) to be established (opening or closing) in given locations and the supply routes among them for each time period.(ii)
Tactical operation decisions regarding the quantity produced for each product at each factory, the materials flow between facilities and the levels of inventory that consist of maximum inventory at plants, products safety stock, and max and min inventory of products at warehouses and distribution centers.(iii)
Financial decisions for determining the amount of bank loans, new capital entries and total investments to establish the network and the quantity of repayments to the bank for each time period.These three kinds of decisions were made for maximizing the value of the company at the end of planning horizon that was measured by SVA as an indicator of the corporation’s profitability. As presented in the previous sections, supply chain strategic decisions and their operation impact corporate finances and, consequently, financial value created for shareholders. Shareholder value analysis is a method for valuing the entire equity in a company. It assumes that the value of a business is the net present value of its future cash flows, discounted at the appropriate cost of capital. Once the value of a business is calculated in this way, the next stage is to calculate the shareholder value using the following equation:(1)shareholder value=value of business–debt.This method was first developed by Alfred Rappaport in the 1980s. That shows how well the company utilizes its properties in order to create value. This method is one of the most accepted lines of thought on how the corporate performance relates to the shareholder value [37].Moreover, the assumptions of the proposed model can be summarized as follows:(i)
In each duration, the demand of each customer zone is clear.(ii)
To satisfy customers’ demands, the company can decide what kind of facilities to be involved at a particular time.(iii)
Products can be kept at the company as inventory or distributed among warehouses.(iv)
There is no any backorder.(v)
Transportation of products among different facilities has capacity limitations.(vi)
Cost and revenue are derived from the operation of a firm.(vii)
Fixed and variable expenses are related to transportation and production.(viii)
The establishment of facilities has fixed costs.(ix)
Financial considerations are defined regarding capital cost, financial ratios, tax and depreciation rates, and long-term borrowing.


### 3.1. Mathematical Formulation

The indices, parameters and decision variables applied in the mathematical model of this study are defined in Table2: (DC: distribution center, WH: warehouse, CZ: customer zone).Table 2
Notations.
IndicesEResources of production indexed byeIProducts indexed byiJLocations of plant, indexed byjKLocations of DC, indexed byKLLocations of CZ, indexed bylMLocations of WH, indexed bymTPlanning periods indexed bys and tParametersAjtPPlant market pricej during the time t, with j∈J and t∈TAmtWWarehouse market pricem during the time t, with m∈M and t∈TAktDDistribution center market priceK at time period t, with K∈K and t∈TCjtP+Cost for establishing a plant at locationj during the time period t, with j∈J and t∈TCmtW+Cost for establishing a WH at locationm during the time t, with m∈M and t∈TCktD+Cost for establishing a DC at locationK at time period t, with K∈K and t∈TCjtP−Cost for closing a plant at locationj during the time period t, with j∈J and t∈TCmtW−Cost for closing a WH at locationm during the time t, with m∈M and t∈TCktD−Cost for closing a DC at locationK during the time t, with K∈K and t∈TCijtFPFixed production cost for producti at plant j at time period t, with i∈I,j∈J, and t∈TCijtVPPUnit production cost for producti at plant j at time period t, with i∈I,j∈J, and t∈TCijmtFTPWFixed transportation cost of producti from plant j to WH m at time period t, with i∈I,j∈J,m∈M, and t∈TCijmtVTPWUnit transportation cost of producti from plant j to WH m at time period t, with i∈I,j∈J,m∈M, and t∈TCimktFTWDFixed transportation cost of producti from WH m to D.C K at time period t, with i∈I,m∈M,K∈K, and t∈TCimktVTWDUnit transportation cost of producti from W.H m to D.C K at time period t, with i∈I,m∈M,K∈K, and t∈TCikltFTDCFixed transportation cost of producti from DC K to CZ l at time period t, with i∈I,K∈K,l∈L, and t∈TCikltVTDCUnit transportation cost of producti from D.C K to CZ l at time period t, with i∈I,K∈K,l∈L, and t∈TCijtIPUnit inventory cost of producti at plant j at time period t, with i∈I,j∈J, and t∈TCimtIWUnit inventory cost of producti at WH m at time period t, with i∈I,m∈M, and t∈TCiktIDUnit inventory cost of producti at DC K at time period t, with i∈I,K∈K, and t∈TDkmaxMaximum capacity of DCK, with K∈KDkminMinimum capacity of DCK, with K∈KIijtmaxMaximum inventory level of producti being held at plant j at the end of time period t, with i∈I,j∈J, and t∈TOiltDemand of producti from customer zone l at time period t, with i∈I,l∈L, and t∈TPijmaxMaximum production capacity of producti at plant j with i∈I end j∈JPijminMinimum production capacity of producti at plant j with i∈I end j∈JPRiltUnit selling price of producti at CZ l at time period t, with i∈I,l∈L, and t∈TQimPWMaximum limit of products that can be transferred from plantj to WH m, with j∈J end m∈MQmkWDMaximum limit of products that can be transferred from WHm to D.C K, with m∈M end K∈KQklDCMaximum limit of products that can be transferred from DCK to C.Z l, with K∈K end l∈LRjeAvailable quantity of resourcee at plant j, with e∈E and j∈JWmmaxMaximum capacity of WHm, with m∈MWmminMinimum capacity of WHm, with m∈MSSiktDSafety stock of producti at DC K, during the time t with j∈J,K∈K, and t∈TSSimtWSafety stock of producti at WH m, during the time t with i∈I,m∈M, and t∈TCRtCash ratio lower bound during the timet, with t∈TCURtCurrent ratio lower bound during the timet, with t∈TCCRtCash coverage ratio lower bound during the timet, with t∈TATRtAssets turnover ratio lower bound during the timet, with t∈TCPtNew capital entries upper bound during the time periodt, with t∈TLTDRtLong-term debt ratio upper bound during the time periodt, with t∈TTDRtTotal debt ratio upper bound during the timet, with t∈TROEtReturn on equity ratio lower bound during the time periodt, with t∈TPMRtProfit margin ratio lower bound during the time periodt, with t∈TROAtReturn on assets ratio lower bound during the time periodt, with t∈TQRtLower bound for quick ratio during the time periodt, with t∈TACDPRstAccumulated depreciation rate of a facility opened at time periods and closed during the time t, with s and t∈TIRtLong-term interest rate during the time periodt, with t∈TTRtTax rate at the time periodt, with t∈TrtRate of capital cost during the timet, with t∈TDPRstDepreciation rate of a facility at the end of time periodt, with s and t∈TϱeijCoefficient relating resource utilization rate ofe to produce product i in plant j, with e∈E, i∈I, and j∈JγtCoefficient relating loans during the timet, with t∈TμtCoefficient relating payables outstanding at time periodt, with t∈TαtCoefficient relating revenues outstanding at time periodt, with t∈TDecisions and auxiliary variablesqijtPInventory level of producti being held at plant j at time period t, with i∈I,j∈J, and t∈TqimtWInventory level of producti being held at WH m at time period t, with i∈I.m∈M, and t∈TqiktDInventory level of producti being held at DC K at time period t, with i∈I.K∈K, and t∈TpijtProduct quantityi produced at plant j at time period t, with i∈I,j∈J, and t∈TxijmtPWProduct quantityi transferred from plant j to WH m in time period t, with i∈I,j∈J,m∈M, and t∈TximktWDProduct quantityi transferred from WH m to DC K in time period t, with i∈I,m∈M,K∈K, and t∈TxikltDCQuantity of producti transferred from DC K to C.Z l during time period t, with i∈I,K∈K,l∈L and t∈TyjtP+1,if a plant at locationjis opened at time periodt;0,otherwise.withj∈Jandt∈TyjtP−1,if a plant at locationjis closed at time periodt;0,otherwise.withj∈Jandt∈TymtW+1,if a W.H at location m is opened at time periodt;0,otherwise.with m∈Mandt∈TymtW−1,if a W.H at locationmis closed at time periodt;0,otherwise.with m∈Mandt∈TyktD+1,if a D.C at locationKis opened at time periodt;0,otherwise.withK∈Kandt∈TyktD−1,if a D.C at locationKis closed at time periodt;0,otherwise.withK∈K,andt∈Tuijt1,if productiis produced at plantjat time periodt;0,otherwise.withi∈I.j∈J.andt∈T.zjmtPW1,if  plantjsupplies W.H m at time periodt;0,otherwise.withj∈J.m∈M,andt∈T.zmktWD1,if W.H m supplies D.CKat time periodt;0,otherwise.with m∈M.K∈Kandt∈T.zkltDC1,if D.CKsupplies C.Zlat time periodt;0,otherwise.withK∈K.l∈L,andt∈TwjstP−1,if plantjwas opened at time periodsand closed at time periodt0,otherwise.withj∈J,andsandt∈TwjstP+1,if plantjwas opened at time periodsand is still open at time periodt0,otherwise.withKj∈J and s and t∈TwmstW−1,if W.H m was opened at time periodsand closed at time periodt;0,otherwise. with m∈M, and s and t∈TwmstW+1,if  W.H m was opened at time periodsand is still open at time periodt;0,otherwise. with m∈M, and s and t∈TwkstD+1,if  D.CKwas opened at time periodsand is still open at time periodt;0,otherwise. with K∈K, and s and t∈TwkstD−1,if D.CKwas opened at time periodsand closed at time periodt;0,otherwise. with K∈K, and s and t∈TncptNew capital entries from shareholders during the time periodt, with t∈TrptRepaid amount to the bank during the time periodt, with t∈TCAtCurrent assets during the time periodt, with t∈TbtBank debts during the time periodt, with t∈TDPVtDepreciation value at time periodt, with t∈TCStCost of sales at time periodt, with t∈TCtCash during the time periodt, with t∈TFAItInvestment of fixed assets during the time periodt, with t∈TFADtDivestment of fixed assets during the time periodt, with t∈TIPtInterest paid(interest expense) during the time periodt, with t∈TICtCost of holding inventory during the time periodt, with t∈TLTDtLong-term debt during the time periodt, with t∈TIVtInventory value at time periodt, with t∈TNOItNonoperating income during the time periodt, with t∈TPCtProduction cost during the time periodt, with t∈TNFAtNet fixed assets during the time periodt, with t∈TREVtRevenues from sales during the time periodt, with t∈TTCtTransportation cost during the time periodt, with t∈T


### 3.2. Objective Function

As presented in the previous sections, strategic and operational decisions in supply chain management impact company’s financial performance and, consequently, the financial value created for shareholders. Shareholder value is the value delivered to the equity owners of a corporation. It is created when earnings exceed the total costs of invested capital [38, 39]. In accordance with it, in this work, the shareholder value analysis (SVA) as an objective function has been applied in order to maximize shareholder value created with the supply chain network configuration.SVA calculates the shareholder value (or equity value) by deducting the long-term liabilities value at the end of the project lifetime (LTDT) from the firm value for the time period under analysis. (2) shows the objective function.(2)maxSVA=DFCF−LTDT.Now, we explainDFCF, LTDT, and other components involved to calculate them.As given by (3), the discounted free cash flow DFCF is obtained by adding the summation of the discounted free cash flows (FCFFt) to the terminal value of a firm VT over the planning period.(3)DFCF=∑t∈TFCFFt1+rtt+VT1+rTT.Note thatT shows the number of time periods of the planning horizon. rT is a parameter to show the discount rate and cost of capital and represents the time value of money and investment risk. VT shows the final value of the firm, that is, the value of total future cash flows, beyond the planning horizon. In this study, VT is calculated by the growing perpetuity model, which presumes that free cash flows grow at a fixed rate g constantly. (4) shows how the terminal value of the firm is calculated.(4)VT=FCFFT+1rT−g,∀t∈T.Because we estimateFCFFT+1 based on an adjustment to FCFF from the last period of the planning horizon, making it grow at the fixed rate g (see (5)), therefore modification in the FCFF is needed since we have assumed stability beyond the planning horizon. This means that nonoperating income is considered zero and new investments will be offset by depreciation.(5)FCFFT+1=REVT−CST−DPVT1−TRT−mWCT1−g.


#### 3.2.1. Free Cash Flow to the Firm (FCFF)

The free cash flow to the firm represents the quantity of cash flow from operations after accounting for depreciation expenses, taxes, working capital, and investments. It is calculated by (6), which deducts the net fixed asset investment FAIt−DPVt and the changes in working capital ΔWCt from the operating income after taxes. In this equation, (REVt) is the revenue, the nonoperating income (NOI), the cost of sales (CSt), and depreciation (DPVt).Note that operating earnings are a taxable revenue; it means that in order to get net income, taxes must be subtracted from incomes. The tax rateTRt is according to current tax laws.As shown in (6), depreciation is considered a cost because it decreases taxable income, and it is not related to a real payment (cash outflow). This means that in order to calculate the FCFFt, depreciation has to be added again.(6)FCFFt=REVt+NOIt−CSt−DPVt1−TRt−FAIt−DPVt−ΔWCt,∀t∈T.Next, the free cash flow components will be explained in more detail.


#### 3.2.2. Revenues

The revenuesREVt coming from selling products/providing services are calculated as follows:(7)REVt=∑i∈I.l∈LPRiltOilt,∀t∈T.


#### 3.2.3. Nonoperating Income

The nonoperating incomeNOIt is the portion of a firm’s income that is derived from activities not related to its core business operations including gains/losses from property or property sales. Therefore, in a period that physical assets are not sold, the nonoperating income will be zero. In this model, we have assumed that if there is a decision to close a facility, it will be sold. As shown in (8), the NOIt consists of three income components derived from the sale of plants, warehouses, or distribution centers. The profit or loss from selling a plant is the difference between the cash inflow resulting from alienation and calculated by the market price of the plant for the period (AjtP) minus the cost of closing it (CjtP−) and the plant net value.(8)NOIt=∑j∈JAjtP−CjtP−yjtP−−∑S=1tCjsP+1−ACDPRstwjstP−+∑m∈MAmtw−Cmtw−ymtw−−∑S=1tCmsw+1−ACDPRstwmstw−+∑k∈KAktD−CktD−yktD−−∑S=1tCksD+1−ACDPRstwkstD−,∀t∈T.


#### 3.2.4. Cost of Sales

As expressed in (9), the cost of sales CSt represents all the expenditures that are needed for producing and delivering products to customers. It consists of four parts: costs of production PCt, costs of transportation TCt, costs of inventory holding ICt, and changes in inventory value IVt−IVt−1.(9)CSt=PCt+TCt+ICt−IVt−IVt−1,∀t∈T.Production costs have a fixed and variable part as follows:(10)PCt=∑i∈I∑j∈JCijtVPPpijt+CijtFPPuijt,∀t∈T.In (10), CijtVPP and CijtFPP represent the variable and fixed cost of production, respectively, at plant j, in time period t. Also, pijt is the quantity of product i produced in plant j at time period t and uijt is a binary value which has the value 1 if the product i is produced in plant j at the time period t and zero, otherwise.According to (11), transportation costs include three fixed and variable costs incurred while transporting products among facilities [34].(11)TCt=∑i∈I∑j∈J∑m∈MCijmtVTPWxijmtPW+CijmtFTPWzjmtPW,+∑i∈I∑m∈M∑k∈KCimktVTWDximktWD+CimktFTWDzmktWD,+∑i∈I∑k∈K∑lkLCikltVTDCxikltDC+CikltVTDCziktDC∀t∈T.The total inventory holding cost has three parts related to the average quantity held at each facility during the time period [34].(12)ICt=∑i∈I∑j∈JCijtIPqijtP+qijt−1P2+∑i∈I∑m∈MCimtIWqimtW+qimt−1W2+∑i∈I∑j∈JCijtIPqijtP+qijt−1P2+∑i∈I∑k∈KCiktIDqiktD+qikt−1D2∀t∈T.Based on accounting principles, the value of inventory is calculated by historical cost. In this case, (14) shows the production price for each product at each time period.(13)IVt=∑i∈I∑j∈J∑m∈M∑k∈KCijtVPPqijtP+qimtW+qiktD∀t∈T.


#### 3.2.5. Depreciation

The value of fixed assets such as plants, warehouses, and distribution centers should be modified for devaluation [34]. Based on this accounting rule, the total depreciation value at the time period t (DPVt) is calculated by the summation of the depreciated value of plants, warehouses, and distribution centers that are operating during the time period t [36]. In this model, we assume that fixed assets existing before the planning horizon have been completely depreciated.(14)DPVt=∑j∈J∑s=1tDPRstCjsP+WjstP++∑m∈M∑s=1tDPRstCmsW+WmstW++∑k∈K∑s=1tDPRstCksD+WkstW+∀t∈T.In (14), WjstP+, WmstW+, and WkstW+ are binary variables set to 1 if a facility opened at the time period s is still open at the time period t.


#### 3.2.6. Fixed Assets Investment

Fixed assets are long-term tangible properties that a firm owns and utilizes in its operations to generate income. In our model, (FAIt) represents fixed assets investment at the time period t which is the needed finance to establish facilities (plants, warehouses, and distribution centers) in the time period t.(15)FAIt=∑j∈JCjtP+yjtP++∑m∈MCmtW+ymtW++∑k∈KCktD+yktD+∀t∈T.


#### 3.2.7. Changes in Working Capital

The changes in working capitalΔWCt are obtained by the difference between the working capital in two successive periods. Working capital is calculated by adding receivable accounts to the value of inventory and deducting payable accounts. It is assumed that the accounts receivable and the accounts payable are a portion of the revenues and of the operational costs, respectively, at the end of time period t. Therefore, ΔWCt can be obtained as follows:(16)ΔWCt=αtREVt−αt−1REVt−1+IVt−IVt−1−μtPCt+TCt+ICt−μt−1PCt−1+TCt−1+ICt−1,∀t∈T.Note thatαt and μt represent the amount of revenues and payments (in percentage), respectively, which are outstanding in the current time period and defined by the company policy on payables and receivables.


#### 3.2.8. Long-Term Liabilities Calculation

Long-term liabilities are represented by long-term debtLTDt which is incurred to finance fixed assets investments, and calculated by (17). This is a function of the previous period debt value and current period loans Bt and bank repayments (RPt).(17)LTDt=LTDt−1+Bt−RPt∀t∈T.


### 3.3. The Model Constraints

The model constraints can be categorized into two groups that should be satisfied as financial constraints and operational constraints.


#### 3.3.1. Financial Constraints

Financial ratios are one of the beneficial parts of financial statements which prepare standard tools to evaluate the overall financial condition of a company’s performance, efficiency, liquidity, and leverage. The financial constraints enforce financial ratios in order to support the financial health of the corporation. This study used the ratio categories defined by Brealey et al. [36] and Borges et al. [34] and sets upper/lower limits value for them.(1) Performance Ratios. Performance ratios measure the financial performance of the company. In this study we considered two common measures, that is, return on equity (ROE) and return on assets (ROA). (20) and (21) present the least values of ROEt and ROAt that have to be satisfied in each time duration.(i)
Return on equity (ROE)Return on equity illustrates the marginal investment income of shareholders and is calculated by dividing the net income by shareholders’ equity. The net incomeNIt is what the business has left over after all expenses. Also, (EBITt) is named earnings before interests and taxes. They are calculated in the following equations:(18)EBITt=REVt+NOIt−CSt−DPVt∀t∈T,(19)NIt=EBITt−IRt∗LTDt1−TRt∀t∈T,Et=Et−1+EBITt−IRt∗LTDt1−TRt+NCPt,t∈T.According to the previous descriptions, theROE equation can be written as follows:(20)EBITt−IRt∗LTDt1−TRtEt≥ROEt∀t∈T.(ii)
Return on assets (ROA)The return on assets (ROA) is a measure of financial performance and represents the percentage of how profitable a company’s assets are for generating revenue. It is calculated by (21). Note that in this equation, NOPAT, (NFAt), and (CAt) are the net operating profit after taxes, net fixed assets, and the current assets, respectively.(21)EBITt1−TRt+CAt≥ROAt∀t∈T.(22) shows how the current net fixed assets (NFAt) are calculated from those of the previous period, which are increased/decreased in an amount equal to the value of the investment (FAIt)/divestment (FADt) in fixed assets of depreciation in time period t as follows:(22)NFAt=NFAt−1+FAIt−FADt−DPVt∀t∈T.Investment expresses the ownership of fixed assets, while divestment represents sales fixed assets. In this study, we have assumed that before the planning horizon, existing assets were completely depreciated, also (FADt) shows the net value (accounting value of the asset after depreciation) of the assets bought during the planning horizon and until time period t:(23)FADt=∑s=1t∑j∈JCjsP+1−ACDPRstWjstP−+∑m∈MCmsW+1−ACDPRstWmstW−+∑k∈KCksD+1−ACDPRstWkstD−∀t∈T.DPVt and FAIt refer to (14) and (15). Current assets are any assets that can reasonably be expected to be sold, consumed, or exhausted through the normal operations of a business. In this study, current assets (CAt) consist of cash and banks (Ct); accounts receivable, here represented as a percent of the revenues (αtREVt), and inventory value (IVt).(24)CAt=Ct+αtREVt+IVt∀t∈T.Eq. (25) shows the cash function at each duration (Ct). The cash at time period t is the available cash in the previous period, cash inflows, and cash outflows [34]. Cash inflows come from different sources:(i)
Customer and receivables (αt−1REVt−1) and product sales 1−αtREVt),(ii)
Fixed assets sales,(iii)
New capital entries (NCPt),(iv)
Loans of the period to finance investments (Bt).Also, cash outflows come from different sources:(i)
Repayments of debt to the bank (RPt),(ii)
Costs of interest are calculated by multiplying an interest rate by the debt of the period (IRtLTDt,(iii)
Accounts payableμt−1PCt−1+TCt−1+ICt−1 and payments to suppliers (1−μtPCt+TCt+ICt),(iv)
Payment of income taxes of the previous period,(v)
The amount invested in new assets.(25)Ct=Ct−1+αt−1REVt−1+1−αtREVt+∑j∈JAjtP−CjtP−yjtP−+∑m∈MAmtW−CmtW−ymtW−+∑k∈KAktD−CktD−yjtD−+NCPt+Bt−RPt−IRtLTDt−μt−1PCt−1+TCt−1+ICt−1−1−μtPCt+TCt+ICt−TRt−1EBITt−1−IRt−1LTDt−1−FAIt∀t∈T.Note thatREVt is defined in (7) and income taxes are due only if there is a taxable income.(2) Efficiency Ratios. Efficiency ratios measure how well the company utilizes its different assets. These ratios allow the company to evaluate its efficiency. In this study, we considered profit margin (PMR) and asset turnover (ATR) as efficiency ratios.(i)
Profit margin (PMR)Profit margin measures the proportion of sales that finds its way into profits. It is defined as the ratio of net income to sales and must attain a minimum value at each time duration (PMRt). Its ratios are given by(26)EBITt−IRtLTDt1−TRtREVt≥PMRt∀t∈T.(ii)
Asset turnover (ATR)Asset turnover displays the incomes generated per monetary unit of total assets, measuring how hard the firm’s assets are working. It is given by the ratio of sales revenue to total assets in time periodt. (27) shows asset turnover ratios.(27)REVtNFAt+CAt≥ATRt,∀t∈T.(3) Liquidity Ratios. Liquidity ratios determine how quickly assets can be converted into cash. The liquidity ratios analysis helps the company to evaluate its ability to keep more liquid assets.(i)
Current ratio (CUR)Current ratio is the ratio of current assets to its current liabilities and must attain a minimum value (CURt). (28) shows current ratio constraint as follows:(28)CAtSTDt≥CURt,∀t∈T.As in our model, short-term loans are negligible; thus, short-term debt (STDt) is due to accounts payable and taxes as follows:(29)STDt=μtPCt+TCt+ICt+EBITt−IRtLTDtTRt∀t∈T.(ii)
Quick ratio (QR)Quick ratio is the ratio of current assets (except inventory) to its current liabilities which must satisfy a threshold value (QRt) as follows:(30)Ct+αtREVtSTDt≥QRt,∀t∈T.(iii)
Cash ratio (CR)The cash ratio is the ratio of its current liabilities which must satisfy a threshold value (CRt) as follows:(31)CtSTDt≥CRt,∀t∈T.(4) Leverage Ratios. Leverage ratios assess the firm’s ability to meet the financial obligations.(i)
Long-term debt to equity ratio (LTDR)It provides an indication on how much debt a company is using to finance its assets. This ratio must be below a given limit.(32)LTDtEt≥LTDRt,∀t∈T.(ii)
Total debt ratio (TDR)The total debt ratio provides an indication on the total amount of debt relative to assets. It is obtained by dividing total debt by total assets and must be lower a given limit.(33)STDt+LTDtNFAt+CAt≥LTDt∀t,∈T.(iii)
Cash coverage ratio (CCR)The cash coverage ratio measures the firm’s capacity to meet interest payments in cash, thus it must satisfy a given lower limit.(34)EBITt+DPRtIRtLTDt≥CCRt,∀t∈T.(5) Other Financial Constraints. (35) shows that new capital entries are limited to the quantity that company partners desire to invest in the company.(35)NCPt≤CPt∀t∈Tsss.Commonly, banks constrain the repayment (RPt) to be at least the interest costs to barricade a growing debt.(36)RPt≥IRtLTDt∀t∈T.Furthermore, because repayments (RPt) are part of the debt, in each period they must satisfy the constraint.(37)RPt≥LTDt∀t∈T.For each time period, the company may limit the amount borrowed to the percentage of the value of investments as follows:(38)Bt≤γtFAIt,∀t∈T.


#### 3.3.2. Operational Constraints

(1) At the Plant Level. (39) and (40) show that production constraints enforce the production quantities in each time period, each plant, and for each product to be in a specified range.(39)pijt≤Pijmax∑s=0twjstP+∀i∈I,j∈J,andt∈T,(40)pijt≤Pijmin∑s=0twjstP+∀i∈I,j∈J,andt∈T.Production quantities are also collectively limited by the available quantity of each time period, each resource, and each plant constraint (41). Note that the availability of the resources is fixed over time.(41)∑i∈Itρijepijt≤Rje,∀j∈J,e∈E.andt∈T.Because production has a fixed cost, in equation (42), a binary variable uijt is used to show the existence of production that assumes the value 1 whenever some non-zero quantity is produced.(42)pijt≤Muijt,∀i∈I,j∈J,andt∈T.Plants might send all or part of the products to the warehouses that have been established. This is stated in the following equations:(43)∑i∈I∑m∈MxijmtPW≤M∑s=0twjstp+,∀j∈J,andt∈T,(44)∑i∈I∑j∈JxijmtPW≤M∑s=0twmstW+,∀m∈M,andt∈T.The total production quantity sent by each plant to each warehouse must satisfy the transport capacity, which is shown by (45) (Note that M is a sufficiently large number).(45)∑i∈IxijmtPW≤QjmPWZjmtPW,∀j∈J,m∈M,andt∈T.Eq. (46) is for inventory balance at each plant and each product in each time period. The available inventory is calculated by the available inventory in the previous period, plus the produced quantity in the current period minus the quantity sent to warehouses.(46)qijtP=qijt−1P+pijt−∑m∈MxijmtPW,∀i∈I,j∈J,andt∈T.Eq. (47) shows that at each plant and in each time period, inventory for each product is limited.(47)qijtP≤Iijtmax,∀i∈I,j∈J,andt∈T.Finally, the proper auxiliary variables associated with the closing/remaining open status of the facilities should be set to confirm the accuracy of the opening and closing decisions in the model. During the whole planning period, if a plant was not initially open, it can only be opened at most once.(48)∑t∈TyjtP+≤1,∀j∈J.Throughout the planning period, a plant can be closed at most once if it was opened before the following equation:(49)∑t∈TyjtP−≤1,∀j∈J.(50)yjtP−≤∑s=0t−1yjsP+∀j∈Jandt∈T.It is impossible for a plant to be opened and closed in the same time period.(51)yjtP++yjtP−≤1,∀j∈Jandt∈T.Eq. (52) illustrates that if a plant was opened in the time period s and then closed in the time period t, therefore all decision variables: opening (yjsP+), closing (yjtP−), and closing status (wjstP−) should be set to 1.(52)yjsP++yjtP−≤wjstP−+1∀j∈J,s=0.…T−1.andt=s+1.…T,If only a closing decision was made, the closing status variable would be set to 1.(53)wjstP−≤yjtP−∀j∈J,S=0.…T−1,andt=s+1.…T.Also, the opening status variablewjstP+ would be set to 1 if an opening decision was made.(54)wjstP+≤yjsP+∀j∈J,s∈T,andt=s.…T.If a plant was opened in the time periods and is yet open in the time period t, in any of the periods in the internal s + 1 and t, a closing decision would be impossible [34].(55)wjstP+−yjsP++∑v=s+1tyjvP−≤0∀j∈J,s=0.…T−1.andt=s+1.…T.(2) At the Warehouse Level. (56) and (57) show that the stored quantities in each warehouse for each product and time period to be within a prespecified range.(56)∑i∈IqimtW≤Wmmax∑s=0tWmstW+,∀m∈Mandt∈T,(57)∑iqimtW≥Wmmin∑s=0tWmstW+,∀m∈Mandt∈T.Active warehouses might send all or part of their products to distribution centers in operation as stated in the following equations:(58)∑i∈I∑k∈KximktWD≤M∑s=0tWmstD+,∀m∈Mandt∈T.(59)∑i∈I∑m∈MximktWD≤M∑s=0tWkstD+,∀k∈Kandt∈T.Eq. (60) shows that the total quantity sent by warehouses to distribution centers in each time period, if any, must satisfy the transport capacity.(60)∑i∈IximktWD≤QmkWDZmktWD,∀m∈M.k∈Kandt∈T.Eq. (61) is for inventory balance at warehouses and shows that for each warehouse and each product in each time period, the available inventory is calculated by the available inventory in the previous period plus the quantity received from the plants in the current period minus the quantity sent to distribution centers.(61)qimtW=qimt−1W+∑j∈JxijmtPW−∑k∈KximktWD,∀i∈I,m∈M.k∈Kandt∈T.Moreover, for each product, safety stock is defined in each time period at each warehouse.(62)qimtW≥SSimtw∑s=0tWmstW+∀i∈I,m∈M,k∈Kandt∈T.Now the proper auxiliary variables associated with the closing/remaining open status of the facilities should be set to confirm the accuracy of the opening and closing decisions in the model. Equations (64) to (67) show that during the whole planning period, firstly, if a warehouse was not initially open, it could only be opened at most once. Secondly, it also could be closed at most once if it was opened before. Finally, a warehouse cannot be opened and closed in the same time period.(63)∑t∈TymtW+≤1,∀m∈M,(64)∑t∈TymtW−≤1,∀m∈M,(65)ymtW−≤∑s=0t−1ymtW+,∀m∈Mandt∈T,(66)ymtW++ymtW−≤1,∀m∈Mandt∈T.Eq. (67) illustrates that if a plant was opened in the time period s then closed in the time period t, therefore all decision variables: opening ymsW+, closing ymtW−, and closing status wmstW− should be set to 1.(67)ymsW++ymtW−≤wmstW−+1∀m∈M,s=0.…T−1,andt=s+1.…T.If only a closing decision was made, a closing status variable would be set to 1:(68)WmstW−≤ymtW−∀m∈M,s=0.…T−1.andt=s+1.…T.The opening status variable (WmstW+) would be set to 1 if an opening decision was made.(69)WmstW+≤ymsW+,∀m∈M,s∈T,andt=s+1.…T.If a warehouse was opened in the time periods and is yet to open in the time period t, in any of the periods in the internal s + 1 and t, a closing decision is impossible [34].(70)WmstW+−ymsW++∑v=s+1tymvW−≤0∀m∈M,s=0.…T−1,andt=s+1.…T.(3) At the Distribution Center Level. (71) and (72) show that the stored quantities in each distribution center for each product and time period must be within a prespecified range.(71)∑i∈IqiktD≤Dkmax∑s=0tWkstD+,∀k∈Kandt∈T.(72)∑i∈IqiktD≥Dkmin∑s=0tWkstD+∀k∈Kandt∈T.Active distribution centers might send all or part of their products to customer zones as stated by(73)∑i∈I∑l∈LxikltDC≤M∑s=0tWkstD+,∀k∈Kandt∈T.Eq. (74) shows that the total quantity sent by distribution centers to customer zones in each time period, if any, must satisfy the transport capacity.(74)∑i∈IxikltDC≤QklDCZkltDC∀k∈K.l∈L,andt∈T.Note that customer zones do not hold inventory, so the total product received by each customer zone from the distribution centers has to be the same as the market demand.(75)∑k∈KxikltDC=Oilt∀i∈I,l∈L,andt∈T.Eq. (76) is for inventory balance at distribution centers. It shows that for each distribution center and each product in each time period, the available inventory is calculated by the inventory available in the previous period, plus the quantity received from the warehouses minus the quantity sent to the customer zones.(76)qiktD=qikt−1D+∑m∈MximktWD−∑k∈KxikltDC,∀i∈I,m∈M,andt∈T.Also, at each warehouse, safety stock is defined for each product and time period.(77)qiktD≥SSiktD∀i∈I,m∈M,k∈K,andt∈T.Now the proper auxiliary variables associated with the closing/remaining open status of the facilities should be set to confirm the accuracy of the opening and closing decisions in the model. Equations (79) to (82) shows that during the whole planning period, firstly, if a distribution center was not initially open, it could only opened at most once. Secondly, it could also be closed at most once if it was opened before. Finally, a distribution center cannot be opened and closed in the same time period.(78)∑t∈TyktD+≤1,∀k∈K,∑t∈TyktD−≤1,∀k∈K,yktD−≤∑s=0t−1yksD+∀k∈K,andt∈T,yktD++yktD−≤1∀k∈K,andt∈T.Eq. (79) illustrates that if a plant was opened in the time period s then closed in the time period t, therefore, all decision variables: opening yksD+, closing yktD−, and closing status wkstD− should be set to 1.(79)yksD++yktD−≤wkstD−+1∀k∈K.s=0.…T−1,andt=s+1.….T.If only a closing decision was made, a closing status variable would be set to 1.(80)wkstD−≤yktD−,∀k∈K,s=0.…T−1,andt=s+1.….T.Also, an opening status variable (wkstD+) would be set to 1 if an opening decision was made.(81)wkstD+≤yksD+∀k∈K,s=1.…T,andt=s.….T.If a distribution center was opened in the time periods and is yet open in the time period t, in any of the periods in the internal s + 1 and t, a closing decision would be impossible [34].(82)wkstD+≤yksD++∑v=s+1tykvD−≤0,∀k∈K,s=0.…T−1,andt=s+1.….T.


## 3.1. Mathematical Formulation

The indices, parameters and decision variables applied in the mathematical model of this study are defined in Table2: (DC: distribution center, WH: warehouse, CZ: customer zone).Table 2
Notations.
IndicesEResources of production indexed byeIProducts indexed byiJLocations of plant, indexed byjKLocations of DC, indexed byKLLocations of CZ, indexed bylMLocations of WH, indexed bymTPlanning periods indexed bys and tParametersAjtPPlant market pricej during the time t, with j∈J and t∈TAmtWWarehouse market pricem during the time t, with m∈M and t∈TAktDDistribution center market priceK at time period t, with K∈K and t∈TCjtP+Cost for establishing a plant at locationj during the time period t, with j∈J and t∈TCmtW+Cost for establishing a WH at locationm during the time t, with m∈M and t∈TCktD+Cost for establishing a DC at locationK at time period t, with K∈K and t∈TCjtP−Cost for closing a plant at locationj during the time period t, with j∈J and t∈TCmtW−Cost for closing a WH at locationm during the time t, with m∈M and t∈TCktD−Cost for closing a DC at locationK during the time t, with K∈K and t∈TCijtFPFixed production cost for producti at plant j at time period t, with i∈I,j∈J, and t∈TCijtVPPUnit production cost for producti at plant j at time period t, with i∈I,j∈J, and t∈TCijmtFTPWFixed transportation cost of producti from plant j to WH m at time period t, with i∈I,j∈J,m∈M, and t∈TCijmtVTPWUnit transportation cost of producti from plant j to WH m at time period t, with i∈I,j∈J,m∈M, and t∈TCimktFTWDFixed transportation cost of producti from WH m to D.C K at time period t, with i∈I,m∈M,K∈K, and t∈TCimktVTWDUnit transportation cost of producti from W.H m to D.C K at time period t, with i∈I,m∈M,K∈K, and t∈TCikltFTDCFixed transportation cost of producti from DC K to CZ l at time period t, with i∈I,K∈K,l∈L, and t∈TCikltVTDCUnit transportation cost of producti from D.C K to CZ l at time period t, with i∈I,K∈K,l∈L, and t∈TCijtIPUnit inventory cost of producti at plant j at time period t, with i∈I,j∈J, and t∈TCimtIWUnit inventory cost of producti at WH m at time period t, with i∈I,m∈M, and t∈TCiktIDUnit inventory cost of producti at DC K at time period t, with i∈I,K∈K, and t∈TDkmaxMaximum capacity of DCK, with K∈KDkminMinimum capacity of DCK, with K∈KIijtmaxMaximum inventory level of producti being held at plant j at the end of time period t, with i∈I,j∈J, and t∈TOiltDemand of producti from customer zone l at time period t, with i∈I,l∈L, and t∈TPijmaxMaximum production capacity of producti at plant j with i∈I end j∈JPijminMinimum production capacity of producti at plant j with i∈I end j∈JPRiltUnit selling price of producti at CZ l at time period t, with i∈I,l∈L, and t∈TQimPWMaximum limit of products that can be transferred from plantj to WH m, with j∈J end m∈MQmkWDMaximum limit of products that can be transferred from WHm to D.C K, with m∈M end K∈KQklDCMaximum limit of products that can be transferred from DCK to C.Z l, with K∈K end l∈LRjeAvailable quantity of resourcee at plant j, with e∈E and j∈JWmmaxMaximum capacity of WHm, with m∈MWmminMinimum capacity of WHm, with m∈MSSiktDSafety stock of producti at DC K, during the time t with j∈J,K∈K, and t∈TSSimtWSafety stock of producti at WH m, during the time t with i∈I,m∈M, and t∈TCRtCash ratio lower bound during the timet, with t∈TCURtCurrent ratio lower bound during the timet, with t∈TCCRtCash coverage ratio lower bound during the timet, with t∈TATRtAssets turnover ratio lower bound during the timet, with t∈TCPtNew capital entries upper bound during the time periodt, with t∈TLTDRtLong-term debt ratio upper bound during the time periodt, with t∈TTDRtTotal debt ratio upper bound during the timet, with t∈TROEtReturn on equity ratio lower bound during the time periodt, with t∈TPMRtProfit margin ratio lower bound during the time periodt, with t∈TROAtReturn on assets ratio lower bound during the time periodt, with t∈TQRtLower bound for quick ratio during the time periodt, with t∈TACDPRstAccumulated depreciation rate of a facility opened at time periods and closed during the time t, with s and t∈TIRtLong-term interest rate during the time periodt, with t∈TTRtTax rate at the time periodt, with t∈TrtRate of capital cost during the timet, with t∈TDPRstDepreciation rate of a facility at the end of time periodt, with s and t∈TϱeijCoefficient relating resource utilization rate ofe to produce product i in plant j, with e∈E, i∈I, and j∈JγtCoefficient relating loans during the timet, with t∈TμtCoefficient relating payables outstanding at time periodt, with t∈TαtCoefficient relating revenues outstanding at time periodt, with t∈TDecisions and auxiliary variablesqijtPInventory level of producti being held at plant j at time period t, with i∈I,j∈J, and t∈TqimtWInventory level of producti being held at WH m at time period t, with i∈I.m∈M, and t∈TqiktDInventory level of producti being held at DC K at time period t, with i∈I.K∈K, and t∈TpijtProduct quantityi produced at plant j at time period t, with i∈I,j∈J, and t∈TxijmtPWProduct quantityi transferred from plant j to WH m in time period t, with i∈I,j∈J,m∈M, and t∈TximktWDProduct quantityi transferred from WH m to DC K in time period t, with i∈I,m∈M,K∈K, and t∈TxikltDCQuantity of producti transferred from DC K to C.Z l during time period t, with i∈I,K∈K,l∈L and t∈TyjtP+1,if a plant at locationjis opened at time periodt;0,otherwise.withj∈Jandt∈TyjtP−1,if a plant at locationjis closed at time periodt;0,otherwise.withj∈Jandt∈TymtW+1,if a W.H at location m is opened at time periodt;0,otherwise.with m∈Mandt∈TymtW−1,if a W.H at locationmis closed at time periodt;0,otherwise.with m∈Mandt∈TyktD+1,if a D.C at locationKis opened at time periodt;0,otherwise.withK∈Kandt∈TyktD−1,if a D.C at locationKis closed at time periodt;0,otherwise.withK∈K,andt∈Tuijt1,if productiis produced at plantjat time periodt;0,otherwise.withi∈I.j∈J.andt∈T.zjmtPW1,if  plantjsupplies W.H m at time periodt;0,otherwise.withj∈J.m∈M,andt∈T.zmktWD1,if W.H m supplies D.CKat time periodt;0,otherwise.with m∈M.K∈Kandt∈T.zkltDC1,if D.CKsupplies C.Zlat time periodt;0,otherwise.withK∈K.l∈L,andt∈TwjstP−1,if plantjwas opened at time periodsand closed at time periodt0,otherwise.withj∈J,andsandt∈TwjstP+1,if plantjwas opened at time periodsand is still open at time periodt0,otherwise.withKj∈J and s and t∈TwmstW−1,if W.H m was opened at time periodsand closed at time periodt;0,otherwise. with m∈M, and s and t∈TwmstW+1,if  W.H m was opened at time periodsand is still open at time periodt;0,otherwise. with m∈M, and s and t∈TwkstD+1,if  D.CKwas opened at time periodsand is still open at time periodt;0,otherwise. with K∈K, and s and t∈TwkstD−1,if D.CKwas opened at time periodsand closed at time periodt;0,otherwise. with K∈K, and s and t∈TncptNew capital entries from shareholders during the time periodt, with t∈TrptRepaid amount to the bank during the time periodt, with t∈TCAtCurrent assets during the time periodt, with t∈TbtBank debts during the time periodt, with t∈TDPVtDepreciation value at time periodt, with t∈TCStCost of sales at time periodt, with t∈TCtCash during the time periodt, with t∈TFAItInvestment of fixed assets during the time periodt, with t∈TFADtDivestment of fixed assets during the time periodt, with t∈TIPtInterest paid(interest expense) during the time periodt, with t∈TICtCost of holding inventory during the time periodt, with t∈TLTDtLong-term debt during the time periodt, with t∈TIVtInventory value at time periodt, with t∈TNOItNonoperating income during the time periodt, with t∈TPCtProduction cost during the time periodt, with t∈TNFAtNet fixed assets during the time periodt, with t∈TREVtRevenues from sales during the time periodt, with t∈TTCtTransportation cost during the time periodt, with t∈T


## 3.2. Objective Function

As presented in the previous sections, strategic and operational decisions in supply chain management impact company’s financial performance and, consequently, the financial value created for shareholders. Shareholder value is the value delivered to the equity owners of a corporation. It is created when earnings exceed the total costs of invested capital [38, 39]. In accordance with it, in this work, the shareholder value analysis (SVA) as an objective function has been applied in order to maximize shareholder value created with the supply chain network configuration.SVA calculates the shareholder value (or equity value) by deducting the long-term liabilities value at the end of the project lifetime (LTDT) from the firm value for the time period under analysis. (2) shows the objective function.(2)maxSVA=DFCF−LTDT.Now, we explainDFCF, LTDT, and other components involved to calculate them.As given by (3), the discounted free cash flow DFCF is obtained by adding the summation of the discounted free cash flows (FCFFt) to the terminal value of a firm VT over the planning period.(3)DFCF=∑t∈TFCFFt1+rtt+VT1+rTT.Note thatT shows the number of time periods of the planning horizon. rT is a parameter to show the discount rate and cost of capital and represents the time value of money and investment risk. VT shows the final value of the firm, that is, the value of total future cash flows, beyond the planning horizon. In this study, VT is calculated by the growing perpetuity model, which presumes that free cash flows grow at a fixed rate g constantly. (4) shows how the terminal value of the firm is calculated.(4)VT=FCFFT+1rT−g,∀t∈T.Because we estimateFCFFT+1 based on an adjustment to FCFF from the last period of the planning horizon, making it grow at the fixed rate g (see (5)), therefore modification in the FCFF is needed since we have assumed stability beyond the planning horizon. This means that nonoperating income is considered zero and new investments will be offset by depreciation.(5)FCFFT+1=REVT−CST−DPVT1−TRT−mWCT1−g.


### 3.2.1. Free Cash Flow to the Firm (FCFF)

The free cash flow to the firm represents the quantity of cash flow from operations after accounting for depreciation expenses, taxes, working capital, and investments. It is calculated by (6), which deducts the net fixed asset investment FAIt−DPVt and the changes in working capital ΔWCt from the operating income after taxes. In this equation, (REVt) is the revenue, the nonoperating income (NOI), the cost of sales (CSt), and depreciation (DPVt).Note that operating earnings are a taxable revenue; it means that in order to get net income, taxes must be subtracted from incomes. The tax rateTRt is according to current tax laws.As shown in (6), depreciation is considered a cost because it decreases taxable income, and it is not related to a real payment (cash outflow). This means that in order to calculate the FCFFt, depreciation has to be added again.(6)FCFFt=REVt+NOIt−CSt−DPVt1−TRt−FAIt−DPVt−ΔWCt,∀t∈T.Next, the free cash flow components will be explained in more detail.


### 3.2.2. Revenues

The revenuesREVt coming from selling products/providing services are calculated as follows:(7)REVt=∑i∈I.l∈LPRiltOilt,∀t∈T.


### 3.2.3. Nonoperating Income

The nonoperating incomeNOIt is the portion of a firm’s income that is derived from activities not related to its core business operations including gains/losses from property or property sales. Therefore, in a period that physical assets are not sold, the nonoperating income will be zero. In this model, we have assumed that if there is a decision to close a facility, it will be sold. As shown in (8), the NOIt consists of three income components derived from the sale of plants, warehouses, or distribution centers. The profit or loss from selling a plant is the difference between the cash inflow resulting from alienation and calculated by the market price of the plant for the period (AjtP) minus the cost of closing it (CjtP−) and the plant net value.(8)NOIt=∑j∈JAjtP−CjtP−yjtP−−∑S=1tCjsP+1−ACDPRstwjstP−+∑m∈MAmtw−Cmtw−ymtw−−∑S=1tCmsw+1−ACDPRstwmstw−+∑k∈KAktD−CktD−yktD−−∑S=1tCksD+1−ACDPRstwkstD−,∀t∈T.


### 3.2.4. Cost of Sales

As expressed in (9), the cost of sales CSt represents all the expenditures that are needed for producing and delivering products to customers. It consists of four parts: costs of production PCt, costs of transportation TCt, costs of inventory holding ICt, and changes in inventory value IVt−IVt−1.(9)CSt=PCt+TCt+ICt−IVt−IVt−1,∀t∈T.Production costs have a fixed and variable part as follows:(10)PCt=∑i∈I∑j∈JCijtVPPpijt+CijtFPPuijt,∀t∈T.In (10), CijtVPP and CijtFPP represent the variable and fixed cost of production, respectively, at plant j, in time period t. Also, pijt is the quantity of product i produced in plant j at time period t and uijt is a binary value which has the value 1 if the product i is produced in plant j at the time period t and zero, otherwise.According to (11), transportation costs include three fixed and variable costs incurred while transporting products among facilities [34].(11)TCt=∑i∈I∑j∈J∑m∈MCijmtVTPWxijmtPW+CijmtFTPWzjmtPW,+∑i∈I∑m∈M∑k∈KCimktVTWDximktWD+CimktFTWDzmktWD,+∑i∈I∑k∈K∑lkLCikltVTDCxikltDC+CikltVTDCziktDC∀t∈T.The total inventory holding cost has three parts related to the average quantity held at each facility during the time period [34].(12)ICt=∑i∈I∑j∈JCijtIPqijtP+qijt−1P2+∑i∈I∑m∈MCimtIWqimtW+qimt−1W2+∑i∈I∑j∈JCijtIPqijtP+qijt−1P2+∑i∈I∑k∈KCiktIDqiktD+qikt−1D2∀t∈T.Based on accounting principles, the value of inventory is calculated by historical cost. In this case, (14) shows the production price for each product at each time period.(13)IVt=∑i∈I∑j∈J∑m∈M∑k∈KCijtVPPqijtP+qimtW+qiktD∀t∈T.


### 3.2.5. Depreciation

The value of fixed assets such as plants, warehouses, and distribution centers should be modified for devaluation [34]. Based on this accounting rule, the total depreciation value at the time period t (DPVt) is calculated by the summation of the depreciated value of plants, warehouses, and distribution centers that are operating during the time period t [36]. In this model, we assume that fixed assets existing before the planning horizon have been completely depreciated.(14)DPVt=∑j∈J∑s=1tDPRstCjsP+WjstP++∑m∈M∑s=1tDPRstCmsW+WmstW++∑k∈K∑s=1tDPRstCksD+WkstW+∀t∈T.In (14), WjstP+, WmstW+, and WkstW+ are binary variables set to 1 if a facility opened at the time period s is still open at the time period t.


### 3.2.6. Fixed Assets Investment

Fixed assets are long-term tangible properties that a firm owns and utilizes in its operations to generate income. In our model, (FAIt) represents fixed assets investment at the time period t which is the needed finance to establish facilities (plants, warehouses, and distribution centers) in the time period t.(15)FAIt=∑j∈JCjtP+yjtP++∑m∈MCmtW+ymtW++∑k∈KCktD+yktD+∀t∈T.


### 3.2.7. Changes in Working Capital

The changes in working capitalΔWCt are obtained by the difference between the working capital in two successive periods. Working capital is calculated by adding receivable accounts to the value of inventory and deducting payable accounts. It is assumed that the accounts receivable and the accounts payable are a portion of the revenues and of the operational costs, respectively, at the end of time period t. Therefore, ΔWCt can be obtained as follows:(16)ΔWCt=αtREVt−αt−1REVt−1+IVt−IVt−1−μtPCt+TCt+ICt−μt−1PCt−1+TCt−1+ICt−1,∀t∈T.Note thatαt and μt represent the amount of revenues and payments (in percentage), respectively, which are outstanding in the current time period and defined by the company policy on payables and receivables.


### 3.2.8. Long-Term Liabilities Calculation

Long-term liabilities are represented by long-term debtLTDt which is incurred to finance fixed assets investments, and calculated by (17). This is a function of the previous period debt value and current period loans Bt and bank repayments (RPt).(17)LTDt=LTDt−1+Bt−RPt∀t∈T.


## 3.2.1. Free Cash Flow to the Firm (FCFF)

The free cash flow to the firm represents the quantity of cash flow from operations after accounting for depreciation expenses, taxes, working capital, and investments. It is calculated by (6), which deducts the net fixed asset investment FAIt−DPVt and the changes in working capital ΔWCt from the operating income after taxes. In this equation, (REVt) is the revenue, the nonoperating income (NOI), the cost of sales (CSt), and depreciation (DPVt).Note that operating earnings are a taxable revenue; it means that in order to get net income, taxes must be subtracted from incomes. The tax rateTRt is according to current tax laws.As shown in (6), depreciation is considered a cost because it decreases taxable income, and it is not related to a real payment (cash outflow). This means that in order to calculate the FCFFt, depreciation has to be added again.(6)FCFFt=REVt+NOIt−CSt−DPVt1−TRt−FAIt−DPVt−ΔWCt,∀t∈T.Next, the free cash flow components will be explained in more detail.


## 3.2.2. Revenues

The revenuesREVt coming from selling products/providing services are calculated as follows:(7)REVt=∑i∈I.l∈LPRiltOilt,∀t∈T.


## 3.2.3. Nonoperating Income

The nonoperating incomeNOIt is the portion of a firm’s income that is derived from activities not related to its core business operations including gains/losses from property or property sales. Therefore, in a period that physical assets are not sold, the nonoperating income will be zero. In this model, we have assumed that if there is a decision to close a facility, it will be sold. As shown in (8), the NOIt consists of three income components derived from the sale of plants, warehouses, or distribution centers. The profit or loss from selling a plant is the difference between the cash inflow resulting from alienation and calculated by the market price of the plant for the period (AjtP) minus the cost of closing it (CjtP−) and the plant net value.(8)NOIt=∑j∈JAjtP−CjtP−yjtP−−∑S=1tCjsP+1−ACDPRstwjstP−+∑m∈MAmtw−Cmtw−ymtw−−∑S=1tCmsw+1−ACDPRstwmstw−+∑k∈KAktD−CktD−yktD−−∑S=1tCksD+1−ACDPRstwkstD−,∀t∈T.


## 3.2.4. Cost of Sales

As expressed in (9), the cost of sales CSt represents all the expenditures that are needed for producing and delivering products to customers. It consists of four parts: costs of production PCt, costs of transportation TCt, costs of inventory holding ICt, and changes in inventory value IVt−IVt−1.(9)CSt=PCt+TCt+ICt−IVt−IVt−1,∀t∈T.Production costs have a fixed and variable part as follows:(10)PCt=∑i∈I∑j∈JCijtVPPpijt+CijtFPPuijt,∀t∈T.In (10), CijtVPP and CijtFPP represent the variable and fixed cost of production, respectively, at plant j, in time period t. Also, pijt is the quantity of product i produced in plant j at time period t and uijt is a binary value which has the value 1 if the product i is produced in plant j at the time period t and zero, otherwise.According to (11), transportation costs include three fixed and variable costs incurred while transporting products among facilities [34].(11)TCt=∑i∈I∑j∈J∑m∈MCijmtVTPWxijmtPW+CijmtFTPWzjmtPW,+∑i∈I∑m∈M∑k∈KCimktVTWDximktWD+CimktFTWDzmktWD,+∑i∈I∑k∈K∑lkLCikltVTDCxikltDC+CikltVTDCziktDC∀t∈T.The total inventory holding cost has three parts related to the average quantity held at each facility during the time period [34].(12)ICt=∑i∈I∑j∈JCijtIPqijtP+qijt−1P2+∑i∈I∑m∈MCimtIWqimtW+qimt−1W2+∑i∈I∑j∈JCijtIPqijtP+qijt−1P2+∑i∈I∑k∈KCiktIDqiktD+qikt−1D2∀t∈T.Based on accounting principles, the value of inventory is calculated by historical cost. In this case, (14) shows the production price for each product at each time period.(13)IVt=∑i∈I∑j∈J∑m∈M∑k∈KCijtVPPqijtP+qimtW+qiktD∀t∈T.


## 3.2.5. Depreciation

The value of fixed assets such as plants, warehouses, and distribution centers should be modified for devaluation [34]. Based on this accounting rule, the total depreciation value at the time period t (DPVt) is calculated by the summation of the depreciated value of plants, warehouses, and distribution centers that are operating during the time period t [36]. In this model, we assume that fixed assets existing before the planning horizon have been completely depreciated.(14)DPVt=∑j∈J∑s=1tDPRstCjsP+WjstP++∑m∈M∑s=1tDPRstCmsW+WmstW++∑k∈K∑s=1tDPRstCksD+WkstW+∀t∈T.In (14), WjstP+, WmstW+, and WkstW+ are binary variables set to 1 if a facility opened at the time period s is still open at the time period t.


## 3.2.6. Fixed Assets Investment

Fixed assets are long-term tangible properties that a firm owns and utilizes in its operations to generate income. In our model, (FAIt) represents fixed assets investment at the time period t which is the needed finance to establish facilities (plants, warehouses, and distribution centers) in the time period t.(15)FAIt=∑j∈JCjtP+yjtP++∑m∈MCmtW+ymtW++∑k∈KCktD+yktD+∀t∈T.


## 3.2.7. Changes in Working Capital

The changes in working capitalΔWCt are obtained by the difference between the working capital in two successive periods. Working capital is calculated by adding receivable accounts to the value of inventory and deducting payable accounts. It is assumed that the accounts receivable and the accounts payable are a portion of the revenues and of the operational costs, respectively, at the end of time period t. Therefore, ΔWCt can be obtained as follows:(16)ΔWCt=αtREVt−αt−1REVt−1+IVt−IVt−1−μtPCt+TCt+ICt−μt−1PCt−1+TCt−1+ICt−1,∀t∈T.Note thatαt and μt represent the amount of revenues and payments (in percentage), respectively, which are outstanding in the current time period and defined by the company policy on payables and receivables.


## 3.2.8. Long-Term Liabilities Calculation

Long-term liabilities are represented by long-term debtLTDt which is incurred to finance fixed assets investments, and calculated by (17). This is a function of the previous period debt value and current period loans Bt and bank repayments (RPt).(17)LTDt=LTDt−1+Bt−RPt∀t∈T.


## 3.3. The Model Constraints

The model constraints can be categorized into two groups that should be satisfied as financial constraints and operational constraints.


### 3.3.1. Financial Constraints

Financial ratios are one of the beneficial parts of financial statements which prepare standard tools to evaluate the overall financial condition of a company’s performance, efficiency, liquidity, and leverage. The financial constraints enforce financial ratios in order to support the financial health of the corporation. This study used the ratio categories defined by Brealey et al. [36] and Borges et al. [34] and sets upper/lower limits value for them.(1) Performance Ratios. Performance ratios measure the financial performance of the company. In this study we considered two common measures, that is, return on equity (ROE) and return on assets (ROA). (20) and (21) present the least values of ROEt and ROAt that have to be satisfied in each time duration.(i)
Return on equity (ROE)Return on equity illustrates the marginal investment income of shareholders and is calculated by dividing the net income by shareholders’ equity. The net incomeNIt is what the business has left over after all expenses. Also, (EBITt) is named earnings before interests and taxes. They are calculated in the following equations:(18)EBITt=REVt+NOIt−CSt−DPVt∀t∈T,(19)NIt=EBITt−IRt∗LTDt1−TRt∀t∈T,Et=Et−1+EBITt−IRt∗LTDt1−TRt+NCPt,t∈T.According to the previous descriptions, theROE equation can be written as follows:(20)EBITt−IRt∗LTDt1−TRtEt≥ROEt∀t∈T.(ii)
Return on assets (ROA)The return on assets (ROA) is a measure of financial performance and represents the percentage of how profitable a company’s assets are for generating revenue. It is calculated by (21). Note that in this equation, NOPAT, (NFAt), and (CAt) are the net operating profit after taxes, net fixed assets, and the current assets, respectively.(21)EBITt1−TRt+CAt≥ROAt∀t∈T.(22) shows how the current net fixed assets (NFAt) are calculated from those of the previous period, which are increased/decreased in an amount equal to the value of the investment (FAIt)/divestment (FADt) in fixed assets of depreciation in time period t as follows:(22)NFAt=NFAt−1+FAIt−FADt−DPVt∀t∈T.Investment expresses the ownership of fixed assets, while divestment represents sales fixed assets. In this study, we have assumed that before the planning horizon, existing assets were completely depreciated, also (FADt) shows the net value (accounting value of the asset after depreciation) of the assets bought during the planning horizon and until time period t:(23)FADt=∑s=1t∑j∈JCjsP+1−ACDPRstWjstP−+∑m∈MCmsW+1−ACDPRstWmstW−+∑k∈KCksD+1−ACDPRstWkstD−∀t∈T.DPVt and FAIt refer to (14) and (15). Current assets are any assets that can reasonably be expected to be sold, consumed, or exhausted through the normal operations of a business. In this study, current assets (CAt) consist of cash and banks (Ct); accounts receivable, here represented as a percent of the revenues (αtREVt), and inventory value (IVt).(24)CAt=Ct+αtREVt+IVt∀t∈T.Eq. (25) shows the cash function at each duration (Ct). The cash at time period t is the available cash in the previous period, cash inflows, and cash outflows [34]. Cash inflows come from different sources:(i)
Customer and receivables (αt−1REVt−1) and product sales 1−αtREVt),(ii)
Fixed assets sales,(iii)
New capital entries (NCPt),(iv)
Loans of the period to finance investments (Bt).Also, cash outflows come from different sources:(i)
Repayments of debt to the bank (RPt),(ii)
Costs of interest are calculated by multiplying an interest rate by the debt of the period (IRtLTDt,(iii)
Accounts payableμt−1PCt−1+TCt−1+ICt−1 and payments to suppliers (1−μtPCt+TCt+ICt),(iv)
Payment of income taxes of the previous period,(v)
The amount invested in new assets.(25)Ct=Ct−1+αt−1REVt−1+1−αtREVt+∑j∈JAjtP−CjtP−yjtP−+∑m∈MAmtW−CmtW−ymtW−+∑k∈KAktD−CktD−yjtD−+NCPt+Bt−RPt−IRtLTDt−μt−1PCt−1+TCt−1+ICt−1−1−μtPCt+TCt+ICt−TRt−1EBITt−1−IRt−1LTDt−1−FAIt∀t∈T.Note thatREVt is defined in (7) and income taxes are due only if there is a taxable income.(2) Efficiency Ratios. Efficiency ratios measure how well the company utilizes its different assets. These ratios allow the company to evaluate its efficiency. In this study, we considered profit margin (PMR) and asset turnover (ATR) as efficiency ratios.(i)
Profit margin (PMR)Profit margin measures the proportion of sales that finds its way into profits. It is defined as the ratio of net income to sales and must attain a minimum value at each time duration (PMRt). Its ratios are given by(26)EBITt−IRtLTDt1−TRtREVt≥PMRt∀t∈T.(ii)
Asset turnover (ATR)Asset turnover displays the incomes generated per monetary unit of total assets, measuring how hard the firm’s assets are working. It is given by the ratio of sales revenue to total assets in time periodt. (27) shows asset turnover ratios.(27)REVtNFAt+CAt≥ATRt,∀t∈T.(3) Liquidity Ratios. Liquidity ratios determine how quickly assets can be converted into cash. The liquidity ratios analysis helps the company to evaluate its ability to keep more liquid assets.(i)
Current ratio (CUR)Current ratio is the ratio of current assets to its current liabilities and must attain a minimum value (CURt). (28) shows current ratio constraint as follows:(28)CAtSTDt≥CURt,∀t∈T.As in our model, short-term loans are negligible; thus, short-term debt (STDt) is due to accounts payable and taxes as follows:(29)STDt=μtPCt+TCt+ICt+EBITt−IRtLTDtTRt∀t∈T.(ii)
Quick ratio (QR)Quick ratio is the ratio of current assets (except inventory) to its current liabilities which must satisfy a threshold value (QRt) as follows:(30)Ct+αtREVtSTDt≥QRt,∀t∈T.(iii)
Cash ratio (CR)The cash ratio is the ratio of its current liabilities which must satisfy a threshold value (CRt) as follows:(31)CtSTDt≥CRt,∀t∈T.(4) Leverage Ratios. Leverage ratios assess the firm’s ability to meet the financial obligations.(i)
Long-term debt to equity ratio (LTDR)It provides an indication on how much debt a company is using to finance its assets. This ratio must be below a given limit.(32)LTDtEt≥LTDRt,∀t∈T.(ii)
Total debt ratio (TDR)The total debt ratio provides an indication on the total amount of debt relative to assets. It is obtained by dividing total debt by total assets and must be lower a given limit.(33)STDt+LTDtNFAt+CAt≥LTDt∀t,∈T.(iii)
Cash coverage ratio (CCR)The cash coverage ratio measures the firm’s capacity to meet interest payments in cash, thus it must satisfy a given lower limit.(34)EBITt+DPRtIRtLTDt≥CCRt,∀t∈T.(5) Other Financial Constraints. (35) shows that new capital entries are limited to the quantity that company partners desire to invest in the company.(35)NCPt≤CPt∀t∈Tsss.Commonly, banks constrain the repayment (RPt) to be at least the interest costs to barricade a growing debt.(36)RPt≥IRtLTDt∀t∈T.Furthermore, because repayments (RPt) are part of the debt, in each period they must satisfy the constraint.(37)RPt≥LTDt∀t∈T.For each time period, the company may limit the amount borrowed to the percentage of the value of investments as follows:(38)Bt≤γtFAIt,∀t∈T.


### 3.3.2. Operational Constraints

(1) At the Plant Level. (39) and (40) show that production constraints enforce the production quantities in each time period, each plant, and for each product to be in a specified range.(39)pijt≤Pijmax∑s=0twjstP+∀i∈I,j∈J,andt∈T,(40)pijt≤Pijmin∑s=0twjstP+∀i∈I,j∈J,andt∈T.Production quantities are also collectively limited by the available quantity of each time period, each resource, and each plant constraint (41). Note that the availability of the resources is fixed over time.(41)∑i∈Itρijepijt≤Rje,∀j∈J,e∈E.andt∈T.Because production has a fixed cost, in equation (42), a binary variable uijt is used to show the existence of production that assumes the value 1 whenever some non-zero quantity is produced.(42)pijt≤Muijt,∀i∈I,j∈J,andt∈T.Plants might send all or part of the products to the warehouses that have been established. This is stated in the following equations:(43)∑i∈I∑m∈MxijmtPW≤M∑s=0twjstp+,∀j∈J,andt∈T,(44)∑i∈I∑j∈JxijmtPW≤M∑s=0twmstW+,∀m∈M,andt∈T.The total production quantity sent by each plant to each warehouse must satisfy the transport capacity, which is shown by (45) (Note that M is a sufficiently large number).(45)∑i∈IxijmtPW≤QjmPWZjmtPW,∀j∈J,m∈M,andt∈T.Eq. (46) is for inventory balance at each plant and each product in each time period. The available inventory is calculated by the available inventory in the previous period, plus the produced quantity in the current period minus the quantity sent to warehouses.(46)qijtP=qijt−1P+pijt−∑m∈MxijmtPW,∀i∈I,j∈J,andt∈T.Eq. (47) shows that at each plant and in each time period, inventory for each product is limited.(47)qijtP≤Iijtmax,∀i∈I,j∈J,andt∈T.Finally, the proper auxiliary variables associated with the closing/remaining open status of the facilities should be set to confirm the accuracy of the opening and closing decisions in the model. During the whole planning period, if a plant was not initially open, it can only be opened at most once.(48)∑t∈TyjtP+≤1,∀j∈J.Throughout the planning period, a plant can be closed at most once if it was opened before the following equation:(49)∑t∈TyjtP−≤1,∀j∈J.(50)yjtP−≤∑s=0t−1yjsP+∀j∈Jandt∈T.It is impossible for a plant to be opened and closed in the same time period.(51)yjtP++yjtP−≤1,∀j∈Jandt∈T.Eq. (52) illustrates that if a plant was opened in the time period s and then closed in the time period t, therefore all decision variables: opening (yjsP+), closing (yjtP−), and closing status (wjstP−) should be set to 1.(52)yjsP++yjtP−≤wjstP−+1∀j∈J,s=0.…T−1.andt=s+1.…T,If only a closing decision was made, the closing status variable would be set to 1.(53)wjstP−≤yjtP−∀j∈J,S=0.…T−1,andt=s+1.…T.Also, the opening status variablewjstP+ would be set to 1 if an opening decision was made.(54)wjstP+≤yjsP+∀j∈J,s∈T,andt=s.…T.If a plant was opened in the time periods and is yet open in the time period t, in any of the periods in the internal s + 1 and t, a closing decision would be impossible [34].(55)wjstP+−yjsP++∑v=s+1tyjvP−≤0∀j∈J,s=0.…T−1.andt=s+1.…T.(2) At the Warehouse Level. (56) and (57) show that the stored quantities in each warehouse for each product and time period to be within a prespecified range.(56)∑i∈IqimtW≤Wmmax∑s=0tWmstW+,∀m∈Mandt∈T,(57)∑iqimtW≥Wmmin∑s=0tWmstW+,∀m∈Mandt∈T.Active warehouses might send all or part of their products to distribution centers in operation as stated in the following equations:(58)∑i∈I∑k∈KximktWD≤M∑s=0tWmstD+,∀m∈Mandt∈T.(59)∑i∈I∑m∈MximktWD≤M∑s=0tWkstD+,∀k∈Kandt∈T.Eq. (60) shows that the total quantity sent by warehouses to distribution centers in each time period, if any, must satisfy the transport capacity.(60)∑i∈IximktWD≤QmkWDZmktWD,∀m∈M.k∈Kandt∈T.Eq. (61) is for inventory balance at warehouses and shows that for each warehouse and each product in each time period, the available inventory is calculated by the available inventory in the previous period plus the quantity received from the plants in the current period minus the quantity sent to distribution centers.(61)qimtW=qimt−1W+∑j∈JxijmtPW−∑k∈KximktWD,∀i∈I,m∈M.k∈Kandt∈T.Moreover, for each product, safety stock is defined in each time period at each warehouse.(62)qimtW≥SSimtw∑s=0tWmstW+∀i∈I,m∈M,k∈Kandt∈T.Now the proper auxiliary variables associated with the closing/remaining open status of the facilities should be set to confirm the accuracy of the opening and closing decisions in the model. Equations (64) to (67) show that during the whole planning period, firstly, if a warehouse was not initially open, it could only be opened at most once. Secondly, it also could be closed at most once if it was opened before. Finally, a warehouse cannot be opened and closed in the same time period.(63)∑t∈TymtW+≤1,∀m∈M,(64)∑t∈TymtW−≤1,∀m∈M,(65)ymtW−≤∑s=0t−1ymtW+,∀m∈Mandt∈T,(66)ymtW++ymtW−≤1,∀m∈Mandt∈T.Eq. (67) illustrates that if a plant was opened in the time period s then closed in the time period t, therefore all decision variables: opening ymsW+, closing ymtW−, and closing status wmstW− should be set to 1.(67)ymsW++ymtW−≤wmstW−+1∀m∈M,s=0.…T−1,andt=s+1.…T.If only a closing decision was made, a closing status variable would be set to 1:(68)WmstW−≤ymtW−∀m∈M,s=0.…T−1.andt=s+1.…T.The opening status variable (WmstW+) would be set to 1 if an opening decision was made.(69)WmstW+≤ymsW+,∀m∈M,s∈T,andt=s+1.…T.If a warehouse was opened in the time periods and is yet to open in the time period t, in any of the periods in the internal s + 1 and t, a closing decision is impossible [34].(70)WmstW+−ymsW++∑v=s+1tymvW−≤0∀m∈M,s=0.…T−1,andt=s+1.…T.(3) At the Distribution Center Level. (71) and (72) show that the stored quantities in each distribution center for each product and time period must be within a prespecified range.(71)∑i∈IqiktD≤Dkmax∑s=0tWkstD+,∀k∈Kandt∈T.(72)∑i∈IqiktD≥Dkmin∑s=0tWkstD+∀k∈Kandt∈T.Active distribution centers might send all or part of their products to customer zones as stated by(73)∑i∈I∑l∈LxikltDC≤M∑s=0tWkstD+,∀k∈Kandt∈T.Eq. (74) shows that the total quantity sent by distribution centers to customer zones in each time period, if any, must satisfy the transport capacity.(74)∑i∈IxikltDC≤QklDCZkltDC∀k∈K.l∈L,andt∈T.Note that customer zones do not hold inventory, so the total product received by each customer zone from the distribution centers has to be the same as the market demand.(75)∑k∈KxikltDC=Oilt∀i∈I,l∈L,andt∈T.Eq. (76) is for inventory balance at distribution centers. It shows that for each distribution center and each product in each time period, the available inventory is calculated by the inventory available in the previous period, plus the quantity received from the warehouses minus the quantity sent to the customer zones.(76)qiktD=qikt−1D+∑m∈MximktWD−∑k∈KxikltDC,∀i∈I,m∈M,andt∈T.Also, at each warehouse, safety stock is defined for each product and time period.(77)qiktD≥SSiktD∀i∈I,m∈M,k∈K,andt∈T.Now the proper auxiliary variables associated with the closing/remaining open status of the facilities should be set to confirm the accuracy of the opening and closing decisions in the model. Equations (79) to (82) shows that during the whole planning period, firstly, if a distribution center was not initially open, it could only opened at most once. Secondly, it could also be closed at most once if it was opened before. Finally, a distribution center cannot be opened and closed in the same time period.(78)∑t∈TyktD+≤1,∀k∈K,∑t∈TyktD−≤1,∀k∈K,yktD−≤∑s=0t−1yksD+∀k∈K,andt∈T,yktD++yktD−≤1∀k∈K,andt∈T.Eq. (79) illustrates that if a plant was opened in the time period s then closed in the time period t, therefore, all decision variables: opening yksD+, closing yktD−, and closing status wkstD− should be set to 1.(79)yksD++yktD−≤wkstD−+1∀k∈K.s=0.…T−1,andt=s+1.….T.If only a closing decision was made, a closing status variable would be set to 1.(80)wkstD−≤yktD−,∀k∈K,s=0.…T−1,andt=s+1.….T.Also, an opening status variable (wkstD+) would be set to 1 if an opening decision was made.(81)wkstD+≤yksD+∀k∈K,s=1.…T,andt=s.….T.If a distribution center was opened in the time periods and is yet open in the time period t, in any of the periods in the internal s + 1 and t, a closing decision would be impossible [34].(82)wkstD+≤yksD++∑v=s+1tykvD−≤0,∀k∈K,s=0.…T−1,andt=s+1.….T.


## 3.3.1. Financial Constraints

Financial ratios are one of the beneficial parts of financial statements which prepare standard tools to evaluate the overall financial condition of a company’s performance, efficiency, liquidity, and leverage. The financial constraints enforce financial ratios in order to support the financial health of the corporation. This study used the ratio categories defined by Brealey et al. [36] and Borges et al. [34] and sets upper/lower limits value for them.(1) Performance Ratios. Performance ratios measure the financial performance of the company. In this study we considered two common measures, that is, return on equity (ROE) and return on assets (ROA). (20) and (21) present the least values of ROEt and ROAt that have to be satisfied in each time duration.(i)
Return on equity (ROE)Return on equity illustrates the marginal investment income of shareholders and is calculated by dividing the net income by shareholders’ equity. The net incomeNIt is what the business has left over after all expenses. Also, (EBITt) is named earnings before interests and taxes. They are calculated in the following equations:(18)EBITt=REVt+NOIt−CSt−DPVt∀t∈T,(19)NIt=EBITt−IRt∗LTDt1−TRt∀t∈T,Et=Et−1+EBITt−IRt∗LTDt1−TRt+NCPt,t∈T.According to the previous descriptions, theROE equation can be written as follows:(20)EBITt−IRt∗LTDt1−TRtEt≥ROEt∀t∈T.(ii)
Return on assets (ROA)The return on assets (ROA) is a measure of financial performance and represents the percentage of how profitable a company’s assets are for generating revenue. It is calculated by (21). Note that in this equation, NOPAT, (NFAt), and (CAt) are the net operating profit after taxes, net fixed assets, and the current assets, respectively.(21)EBITt1−TRt+CAt≥ROAt∀t∈T.(22) shows how the current net fixed assets (NFAt) are calculated from those of the previous period, which are increased/decreased in an amount equal to the value of the investment (FAIt)/divestment (FADt) in fixed assets of depreciation in time period t as follows:(22)NFAt=NFAt−1+FAIt−FADt−DPVt∀t∈T.Investment expresses the ownership of fixed assets, while divestment represents sales fixed assets. In this study, we have assumed that before the planning horizon, existing assets were completely depreciated, also (FADt) shows the net value (accounting value of the asset after depreciation) of the assets bought during the planning horizon and until time period t:(23)FADt=∑s=1t∑j∈JCjsP+1−ACDPRstWjstP−+∑m∈MCmsW+1−ACDPRstWmstW−+∑k∈KCksD+1−ACDPRstWkstD−∀t∈T.DPVt and FAIt refer to (14) and (15). Current assets are any assets that can reasonably be expected to be sold, consumed, or exhausted through the normal operations of a business. In this study, current assets (CAt) consist of cash and banks (Ct); accounts receivable, here represented as a percent of the revenues (αtREVt), and inventory value (IVt).(24)CAt=Ct+αtREVt+IVt∀t∈T.Eq. (25) shows the cash function at each duration (Ct). The cash at time period t is the available cash in the previous period, cash inflows, and cash outflows [34]. Cash inflows come from different sources:(i)
Customer and receivables (αt−1REVt−1) and product sales 1−αtREVt),(ii)
Fixed assets sales,(iii)
New capital entries (NCPt),(iv)
Loans of the period to finance investments (Bt).Also, cash outflows come from different sources:(i)
Repayments of debt to the bank (RPt),(ii)
Costs of interest are calculated by multiplying an interest rate by the debt of the period (IRtLTDt,(iii)
Accounts payableμt−1PCt−1+TCt−1+ICt−1 and payments to suppliers (1−μtPCt+TCt+ICt),(iv)
Payment of income taxes of the previous period,(v)
The amount invested in new assets.(25)Ct=Ct−1+αt−1REVt−1+1−αtREVt+∑j∈JAjtP−CjtP−yjtP−+∑m∈MAmtW−CmtW−ymtW−+∑k∈KAktD−CktD−yjtD−+NCPt+Bt−RPt−IRtLTDt−μt−1PCt−1+TCt−1+ICt−1−1−μtPCt+TCt+ICt−TRt−1EBITt−1−IRt−1LTDt−1−FAIt∀t∈T.Note thatREVt is defined in (7) and income taxes are due only if there is a taxable income.(2) Efficiency Ratios. Efficiency ratios measure how well the company utilizes its different assets. These ratios allow the company to evaluate its efficiency. In this study, we considered profit margin (PMR) and asset turnover (ATR) as efficiency ratios.(i)
Profit margin (PMR)Profit margin measures the proportion of sales that finds its way into profits. It is defined as the ratio of net income to sales and must attain a minimum value at each time duration (PMRt). Its ratios are given by(26)EBITt−IRtLTDt1−TRtREVt≥PMRt∀t∈T.(ii)
Asset turnover (ATR)Asset turnover displays the incomes generated per monetary unit of total assets, measuring how hard the firm’s assets are working. It is given by the ratio of sales revenue to total assets in time periodt. (27) shows asset turnover ratios.(27)REVtNFAt+CAt≥ATRt,∀t∈T.(3) Liquidity Ratios. Liquidity ratios determine how quickly assets can be converted into cash. The liquidity ratios analysis helps the company to evaluate its ability to keep more liquid assets.(i)
Current ratio (CUR)Current ratio is the ratio of current assets to its current liabilities and must attain a minimum value (CURt). (28) shows current ratio constraint as follows:(28)CAtSTDt≥CURt,∀t∈T.As in our model, short-term loans are negligible; thus, short-term debt (STDt) is due to accounts payable and taxes as follows:(29)STDt=μtPCt+TCt+ICt+EBITt−IRtLTDtTRt∀t∈T.(ii)
Quick ratio (QR)Quick ratio is the ratio of current assets (except inventory) to its current liabilities which must satisfy a threshold value (QRt) as follows:(30)Ct+αtREVtSTDt≥QRt,∀t∈T.(iii)
Cash ratio (CR)The cash ratio is the ratio of its current liabilities which must satisfy a threshold value (CRt) as follows:(31)CtSTDt≥CRt,∀t∈T.(4) Leverage Ratios. Leverage ratios assess the firm’s ability to meet the financial obligations.(i)
Long-term debt to equity ratio (LTDR)It provides an indication on how much debt a company is using to finance its assets. This ratio must be below a given limit.(32)LTDtEt≥LTDRt,∀t∈T.(ii)
Total debt ratio (TDR)The total debt ratio provides an indication on the total amount of debt relative to assets. It is obtained by dividing total debt by total assets and must be lower a given limit.(33)STDt+LTDtNFAt+CAt≥LTDt∀t,∈T.(iii)
Cash coverage ratio (CCR)The cash coverage ratio measures the firm’s capacity to meet interest payments in cash, thus it must satisfy a given lower limit.(34)EBITt+DPRtIRtLTDt≥CCRt,∀t∈T.(5) Other Financial Constraints. (35) shows that new capital entries are limited to the quantity that company partners desire to invest in the company.(35)NCPt≤CPt∀t∈Tsss.Commonly, banks constrain the repayment (RPt) to be at least the interest costs to barricade a growing debt.(36)RPt≥IRtLTDt∀t∈T.Furthermore, because repayments (RPt) are part of the debt, in each period they must satisfy the constraint.(37)RPt≥LTDt∀t∈T.For each time period, the company may limit the amount borrowed to the percentage of the value of investments as follows:(38)Bt≤γtFAIt,∀t∈T.


## 3.3.2. Operational Constraints

(1) At the Plant Level. (39) and (40) show that production constraints enforce the production quantities in each time period, each plant, and for each product to be in a specified range.(39)pijt≤Pijmax∑s=0twjstP+∀i∈I,j∈J,andt∈T,(40)pijt≤Pijmin∑s=0twjstP+∀i∈I,j∈J,andt∈T.Production quantities are also collectively limited by the available quantity of each time period, each resource, and each plant constraint (41). Note that the availability of the resources is fixed over time.(41)∑i∈Itρijepijt≤Rje,∀j∈J,e∈E.andt∈T.Because production has a fixed cost, in equation (42), a binary variable uijt is used to show the existence of production that assumes the value 1 whenever some non-zero quantity is produced.(42)pijt≤Muijt,∀i∈I,j∈J,andt∈T.Plants might send all or part of the products to the warehouses that have been established. This is stated in the following equations:(43)∑i∈I∑m∈MxijmtPW≤M∑s=0twjstp+,∀j∈J,andt∈T,(44)∑i∈I∑j∈JxijmtPW≤M∑s=0twmstW+,∀m∈M,andt∈T.The total production quantity sent by each plant to each warehouse must satisfy the transport capacity, which is shown by (45) (Note that M is a sufficiently large number).(45)∑i∈IxijmtPW≤QjmPWZjmtPW,∀j∈J,m∈M,andt∈T.Eq. (46) is for inventory balance at each plant and each product in each time period. The available inventory is calculated by the available inventory in the previous period, plus the produced quantity in the current period minus the quantity sent to warehouses.(46)qijtP=qijt−1P+pijt−∑m∈MxijmtPW,∀i∈I,j∈J,andt∈T.Eq. (47) shows that at each plant and in each time period, inventory for each product is limited.(47)qijtP≤Iijtmax,∀i∈I,j∈J,andt∈T.Finally, the proper auxiliary variables associated with the closing/remaining open status of the facilities should be set to confirm the accuracy of the opening and closing decisions in the model. During the whole planning period, if a plant was not initially open, it can only be opened at most once.(48)∑t∈TyjtP+≤1,∀j∈J.Throughout the planning period, a plant can be closed at most once if it was opened before the following equation:(49)∑t∈TyjtP−≤1,∀j∈J.(50)yjtP−≤∑s=0t−1yjsP+∀j∈Jandt∈T.It is impossible for a plant to be opened and closed in the same time period.(51)yjtP++yjtP−≤1,∀j∈Jandt∈T.Eq. (52) illustrates that if a plant was opened in the time period s and then closed in the time period t, therefore all decision variables: opening (yjsP+), closing (yjtP−), and closing status (wjstP−) should be set to 1.(52)yjsP++yjtP−≤wjstP−+1∀j∈J,s=0.…T−1.andt=s+1.…T,If only a closing decision was made, the closing status variable would be set to 1.(53)wjstP−≤yjtP−∀j∈J,S=0.…T−1,andt=s+1.…T.Also, the opening status variablewjstP+ would be set to 1 if an opening decision was made.(54)wjstP+≤yjsP+∀j∈J,s∈T,andt=s.…T.If a plant was opened in the time periods and is yet open in the time period t, in any of the periods in the internal s + 1 and t, a closing decision would be impossible [34].(55)wjstP+−yjsP++∑v=s+1tyjvP−≤0∀j∈J,s=0.…T−1.andt=s+1.…T.(2) At the Warehouse Level. (56) and (57) show that the stored quantities in each warehouse for each product and time period to be within a prespecified range.(56)∑i∈IqimtW≤Wmmax∑s=0tWmstW+,∀m∈Mandt∈T,(57)∑iqimtW≥Wmmin∑s=0tWmstW+,∀m∈Mandt∈T.Active warehouses might send all or part of their products to distribution centers in operation as stated in the following equations:(58)∑i∈I∑k∈KximktWD≤M∑s=0tWmstD+,∀m∈Mandt∈T.(59)∑i∈I∑m∈MximktWD≤M∑s=0tWkstD+,∀k∈Kandt∈T.Eq. (60) shows that the total quantity sent by warehouses to distribution centers in each time period, if any, must satisfy the transport capacity.(60)∑i∈IximktWD≤QmkWDZmktWD,∀m∈M.k∈Kandt∈T.Eq. (61) is for inventory balance at warehouses and shows that for each warehouse and each product in each time period, the available inventory is calculated by the available inventory in the previous period plus the quantity received from the plants in the current period minus the quantity sent to distribution centers.(61)qimtW=qimt−1W+∑j∈JxijmtPW−∑k∈KximktWD,∀i∈I,m∈M.k∈Kandt∈T.Moreover, for each product, safety stock is defined in each time period at each warehouse.(62)qimtW≥SSimtw∑s=0tWmstW+∀i∈I,m∈M,k∈Kandt∈T.Now the proper auxiliary variables associated with the closing/remaining open status of the facilities should be set to confirm the accuracy of the opening and closing decisions in the model. Equations (64) to (67) show that during the whole planning period, firstly, if a warehouse was not initially open, it could only be opened at most once. Secondly, it also could be closed at most once if it was opened before. Finally, a warehouse cannot be opened and closed in the same time period.(63)∑t∈TymtW+≤1,∀m∈M,(64)∑t∈TymtW−≤1,∀m∈M,(65)ymtW−≤∑s=0t−1ymtW+,∀m∈Mandt∈T,(66)ymtW++ymtW−≤1,∀m∈Mandt∈T.Eq. (67) illustrates that if a plant was opened in the time period s then closed in the time period t, therefore all decision variables: opening ymsW+, closing ymtW−, and closing status wmstW− should be set to 1.(67)ymsW++ymtW−≤wmstW−+1∀m∈M,s=0.…T−1,andt=s+1.…T.If only a closing decision was made, a closing status variable would be set to 1:(68)WmstW−≤ymtW−∀m∈M,s=0.…T−1.andt=s+1.…T.The opening status variable (WmstW+) would be set to 1 if an opening decision was made.(69)WmstW+≤ymsW+,∀m∈M,s∈T,andt=s+1.…T.If a warehouse was opened in the time periods and is yet to open in the time period t, in any of the periods in the internal s + 1 and t, a closing decision is impossible [34].(70)WmstW+−ymsW++∑v=s+1tymvW−≤0∀m∈M,s=0.…T−1,andt=s+1.…T.(3) At the Distribution Center Level. (71) and (72) show that the stored quantities in each distribution center for each product and time period must be within a prespecified range.(71)∑i∈IqiktD≤Dkmax∑s=0tWkstD+,∀k∈Kandt∈T.(72)∑i∈IqiktD≥Dkmin∑s=0tWkstD+∀k∈Kandt∈T.Active distribution centers might send all or part of their products to customer zones as stated by(73)∑i∈I∑l∈LxikltDC≤M∑s=0tWkstD+,∀k∈Kandt∈T.Eq. (74) shows that the total quantity sent by distribution centers to customer zones in each time period, if any, must satisfy the transport capacity.(74)∑i∈IxikltDC≤QklDCZkltDC∀k∈K.l∈L,andt∈T.Note that customer zones do not hold inventory, so the total product received by each customer zone from the distribution centers has to be the same as the market demand.(75)∑k∈KxikltDC=Oilt∀i∈I,l∈L,andt∈T.Eq. (76) is for inventory balance at distribution centers. It shows that for each distribution center and each product in each time period, the available inventory is calculated by the inventory available in the previous period, plus the quantity received from the warehouses minus the quantity sent to the customer zones.(76)qiktD=qikt−1D+∑m∈MximktWD−∑k∈KxikltDC,∀i∈I,m∈M,andt∈T.Also, at each warehouse, safety stock is defined for each product and time period.(77)qiktD≥SSiktD∀i∈I,m∈M,k∈K,andt∈T.Now the proper auxiliary variables associated with the closing/remaining open status of the facilities should be set to confirm the accuracy of the opening and closing decisions in the model. Equations (79) to (82) shows that during the whole planning period, firstly, if a distribution center was not initially open, it could only opened at most once. Secondly, it could also be closed at most once if it was opened before. Finally, a distribution center cannot be opened and closed in the same time period.(78)∑t∈TyktD+≤1,∀k∈K,∑t∈TyktD−≤1,∀k∈K,yktD−≤∑s=0t−1yksD+∀k∈K,andt∈T,yktD++yktD−≤1∀k∈K,andt∈T.Eq. (79) illustrates that if a plant was opened in the time period s then closed in the time period t, therefore, all decision variables: opening yksD+, closing yktD−, and closing status wkstD− should be set to 1.(79)yksD++yktD−≤wkstD−+1∀k∈K.s=0.…T−1,andt=s+1.….T.If only a closing decision was made, a closing status variable would be set to 1.(80)wkstD−≤yktD−,∀k∈K,s=0.…T−1,andt=s+1.….T.Also, an opening status variable (wkstD+) would be set to 1 if an opening decision was made.(81)wkstD+≤yksD+∀k∈K,s=1.…T,andt=s.….T.If a distribution center was opened in the time periods and is yet open in the time period t, in any of the periods in the internal s + 1 and t, a closing decision would be impossible [34].(82)wkstD+≤yksD++∑v=s+1tykvD−≤0,∀k∈K,s=0.…T−1,andt=s+1.….T.


## 4. Case Study Implementation and Evaluation

### 4.1. Input Parameters of the Model

In order to evaluate the applicability and efficiency of the developed model presented in the previous section, we applied the data of a real company which is located in the UK as shown in Figure2 and studied by Longinidis and Georgiadis [5]. Note that, because of some data incongruity and missing data, their case study could not be directly applied and we have considered the following assumptions regarding the missing information:(i)
This company has three plants in three different locations and four possible locations for warehouses and six potential locations for distribution centers.(ii)
Each plant is able to produce six of seven products within its limitations of production capacity. Each plant also holds about two times the average annual demand as initial inventories.(iii)
In each time duration, each warehouse and also distribution centers have an upper and lower bound handling capacity and need safety stock.(iv)
Initial inventories are considered about two times the average annual demand.(v)
Safety stock for each product at each facility is equal to the total quantity transferred from the facility during a period of 15 days.(vi)
Product flows among plants, warehouses, distribution centers, and customer zones have upper bounds.(vii)
Prices and demands of products in each customer zone are known.(viii)
The company has a 4-year planning horizon.(ix)
Before the planning horizon, balance sheet data are integrated into the optimization process.(x)
All tangible assets have been deprecated. Short-term liabilities (accounts payables and taxes of previous profits) should be paid in one year.(xi)
The real value of cash has been calculated instead of considering it as a percent of net income.Figure 2
The case study supply chain network.


### 4.2. Comparison between Basic Model and Developed Models

Now, in order to display the improvements in the proposed model, we compared the results of the basic model presented by Longinidis and Georgiadis [5] with our developed models which have a new objective function, accurate calculations, and additional financial considerations. All the problems were solved by Branch and Reduce Optimization Navigator (BARON) solver in GAMS software on a personal computer with core i5 CPU 2.50 GHz and 8 GB of RAM on Windows 8.


#### 4.2.1. Basic Model

The basic model was considered with the same decision-making assumptions and objective function presented by Longinidis and Georgiadis [5]. Its objective is to maximize the company’s net created value which is measured by Economic Value Added (EVA) index. The model was solved and the total value created amounts to 85,855,590 (monetary units). The optimal results of the basic model will be used to compare them with results obtained from other developed models. In this way, it is possible to show the advantages of the proposed approach clearly.


#### 4.2.2. The First Developed Model with New Objective Function

According to what is explained in Section2, SVA is one of the most accepted methods to measure the value of a company. SVA determines the financial value of a company by looking at the returns it provides for its stockholders and is based on the view that the objective of company managers is to maximize the wealth of company stockholders. SVA calculates the shareholder value by deducting the value of long-term liabilities at the end of planning horizon from the value of the firm for the time period [34]. In this study, the final value of the company is obtained by the discounted free cash flow (DFCF) method with a fixed growth rate (0.5%).Now, in the first stage of developing the model, shareholder value analysis (SVA) is applied as an objective function in the basic model. The model was solved and the total value created amounts is 86,855,590 monetary units. The optimal network structure is shown in Figure3. The total production quantities for the whole planning horizon is only 1407 units: plant 1 and plant 3 produce 809 and 598, respectively; plant 2 does not produce at all. Therefore, reducing inventory was clearly shown and had these results: (i) decreasing production quantities to reduce the product quantities in stock. (ii) The large flows lead to establishing a new distribution center to meet customers’ demands. In order to reduce the need for working capital, SVA tends to reduce the inventory. Therefore, the produced quantity by SVA model is smaller than the EVA model. This feature of SVA model also makes a large number of flows between warehouses and distribution centers and between distribution centers and customer zones. The total quantities transported from plants to warehouses for both models are compared in Table 3.Figure 3
Network structure and produced products for the developed model.Table 3
Total products transported from plants to warehouses.
Developed modelBasic modelW1W2W3W4W1W2W3W4Plant 179017471Plant 262101498Plant 335023201According to Table3, by SVA model, warehouse 1 receives more products supplying distribution centers 1 and 6. Similarly, warehouse 2 receives more quantity; therefore, it supplies distribution centers 1, 2, 5, and 6. But by EVA model, warehouse 2 just supplied distribution center 2.As shown in Tables4 and 5, by applying the model with SVA as the objective function, inventory was stored in five distribution centers (all distribution centers except 4); therefore, total flows between distribution centers and customer zones are much larger than total flows transported when EVA was the objective function.Table 4
Total products transported from warehouses to distribution centers.
Developed modelBasic modelDC1DC2DC3DC4DC5DC6DC1DC2DC3DC4DC5DC6W1529825437471W2105230350833211498W316132983201W4Table 5
Total products transported from distribution centers to customer zones (SVA base model).
Developed modelBasic modelCZ1CZ2CZ3CZ4CZ5CZ6CZ7CZ8CZ1CZ2CZ3CZ4CZ5CZ6CZ7CZ8DC113491141672123904144313492018124114131458DC215167281498DC3149834662081614981559DC4DC5508DC6531Note that since distribution center 6 has the lowest inventory cost among others, it received most of the inventory transferred from warehouses to distribution centers. It receives 5864 units but it only supplies customer zone 6 with 531 units and 5333 units are kept as inventory. Also, the model with SVA as the objective function tends to reduce the inventory quantities to decrease the need for working capital. Only 878 units stay at the plants as inventory.


#### 4.2.3. The Second Developed Model with New Financial Aspects

Now, in the second phase of model development, we add new financial aspects to the previous version of the model to make it similar to real conditions. These new features include the possibility of closing and opening facilities at any time period of the planning horizon, repayments obligation to the bank, adding the possibility of new capital entries from shareholders, and adoption of an accounts payable policy. To better understand the effect of these aspects, we explained them separately.First, to test the possibility of closing and opening facilities at any time period, we considered two times of the establishment price of each facility as selling prices. The value created for shareholders is 87,397,697 monetary units, which is 0.88% larger than the value created by the basic model which is the gains resulting from selling the plants. Then the new model with the obligation of bank repayments created 89,407,636 monetary units, which is 3.02% larger than the value created by the model with SVA as objective function. The network structure remains the same. By repaying to the bank every year, long-term debt is reduced and a lower amount is deducted from the free cash flow that was generated over the planning horizon, creating more value for shareholders.Next, in order to consider an account payable policy, it is assumed that 60% of payments to suppliers are made in cash and 40% are made in credit. In this situation, the value created for shareholders is 88,549,322 monetary units, that is, 0.96% smaller. Because more amount of money (working capital) is needed to support operating expenses and pay suppliers, the free cash flow decreases and the value created is 858,314 monetary units lower.Finally, we add the possibility of raising new capital from shareholders and also set per year limit of 60,000 monetary units for the new capital entries. This limit shows the maximum that shareholders are willing to invest in the company to receive dividends in the future. The new developed model was solved optimally and the value for shareholders increased to 92,460,308 monetary units, which is 3.18% larger than the value without these financial considerations created, and 6.3% larger than the value created by the basic model. Figures4 to 6 display the network structure during the planning horizon. As it can be seen, the flows between facilities and the quantities transported change during the time.Figure 4
Network structure for the complete model in year 1 and for the developed model with new financial aspects.Figure 5
Network structure in year 2 and for the developed model with new financial aspects.Figure 6
Network structure in year 3 and for the developed model with new financial aspects.According to Figures7 to 6, plants only produce during the first two years and their total quantity is 1394 units. The total quantity produced by the SVA model is much lower than the quantity production when EVA was the objective function. Therefore, the need for working capital and payments to suppliers is smaller. These changes lead to an increase in the value created for shareholders. Also, by using EVA as the objective function, the value of the company improves by creating higher inventories (which are a part of current assets) [37].Figure 7
Network structure in year 4 and for the developed model with new financial aspects.Plant 2 closes at the start of the second year with a final inventory of 3341 units, reducing its initial inventory by 76%. Plant 1 and plant 3 are closed at the beginning of the third year, with the final inventory of 1971 and 881 units. This means an inventory reduction of 245% and 285%, respectively. Note that products 2, 4, and 7 at plant 1 which were not sold within the planning horizon are considered as the final inventory. Also, products like 3 and 6 at plant 1 that were produced in the years 1 and 2, have no final inventory. As explained before, in accordance with the evolution of the number of flows among facilities, the product quantities transported from plants to warehouses increase from year 1 to year 2. Table6 presented the operating costs (production, transportation, and inventory holding costs) that resulted from the decisions described above. As we can see, the largest portion of the operating costs is transportation costs (50.58%), then inventory holding costs (40.27%), and production costs (9.15%). There are production costs in the first and second years. Also, due to high inventory at the beginning of the planning horizon, there is no production in the years 3 and 4. In these two years, from plants to warehouses and from warehouses to distribution centers, there are no transportation costs because plants are closed and the warehouses are not operating. As shown in Table 6, inventory costs decrease over time. The inventory costs at plants in years 3 and 4 refer to products that were already in inventory at the beginning of the planning horizon and the ones customers did not request. It is important to note that although the final inventory at the distribution centers is equal to zero, there is an inventory cost since inventory is calculated based on its average during a year.Table 6
Production, transportation, and inventory costs for year for the developed model with new financial aspects.
Year 1Year 2Year 3Year 4TotalProduction costs101390,1020091,115Transportation costs162,717209,85660,41771,303504,293Inventory costs141,402109,54289,50260,991401,437According to financial decisions made by the final model, managers are provided with an accounts payable policy in Table7. It shows that the company has enough cash (based on the initial balance sheet) and does not need bank loans. Therefore, all capital entries are captured from shareholders. As we can see, production costs by the developed model are low, since high levels of inventory and money are available for investment. Therefore, the company is in a good condition for repayments to the bank, decreasing debt and maximizing the value of the corporate which is measured by SVA.Table 7
Financial decisions for each year for the developed model with new financial aspects.
Financial decisionsYear 1Year 2Year 3Year 4TotalLoans00000New capital entries60,00060,00060, 00060,000240,000Investment300,000000300,000Repayments540,000270,000135,00067,5001,012,500


### 4.3. Financial Sensitivity Analysis

In this section, we test the performance of proposed models in several cases by changing some financial parameters. These parameters are important because they are suggestive of the economic environment and in many cases are accepted conditions that the company has no impact on them. The cost of capital rate at time periodt (rt) is an important parameter. Also, one of the important financial parameters affecting the company’s wealth is the tax rate (Trt). Moreover, we selected the depreciation (DPRst) rate as a financial parameter for the sensitivity test.Table8 shows the effects on the developed model by changing these parameters from −15% to +15%. The results show that the developed model with new financial aspects was robust against changes in these financial parameters.Table 8
Sensitivity analysis of value created according to changes in financial parameters.
ParameterChange (%)−15−10−5−2+2+5+10+15Cost of capital rate at time periodt (rt)105,947,496101,350,94096,869,75294,204,96490,717,78088,114,17283,796,38479,838,788Tax rate (Trt)99,756,84097,326,66494,896,18493,435,23691,484,46890,020,78487,580,19685,139,760Depreciation rateDPRst93,832,79293,377,62892,919,88092,644,30492,275,78091,998,60891,534,32491,070,724


### 4.4. Results and Discussions

In the previous section, the optimal results of a basic model were used to compare them with the results obtained from other developed models to show the advantages of the developed models. We carried out two phases of development in order to improve the basic model: (i) applying a new objective function, which maximizes the value of the company measured by the SVA method, (ii) adding new financial aspects to the previous version of the model to make it more realistic.In the first step, we applied SVA as a new objective function instead of EVA. The model with the new objective was solved and the total value created for shareholder’s amounts to 86,635,307 monetary units.In the second step, the new financial aspects were integrated into the previous version of the model. The total value created by the complete version of the model was 92,460,308 monetary units, which is 0.7% larger than the SVA obtained without financial aspects and 0.93% larger than the value created in the basic model. The main reasons for an increase in value creation for shareholders are due to new operational and financial aspects, which mainly show the possibility of closing facilities and bank debt repayments. Bank repayments which reduce debt and new capital enables the company to choose better operational options. The value created by each model is reported in Table9.Table 9
Values obtained by each model.
ModelValue created (monetary units)The basic model85, 855, 590The first developed model with new objective function86, 635, 307The second developed model with new financial aspects92, 460, 308The main reasons for an increase in the value created are due to both operational and financial aspects such as the possibility of closing facilities and bank repayments.In terms of the type of objective function in this study instead of EVA, which is based on conventional accounting principles, SVA is applied as an objective function, that is, one of the most accepted methods of measuring how corporate performance relates to shareholder value. As mentioned before, the SVA for a company is calculated by adding the present value of cash flows to their terminal value, which represents the value of the company discounted at the proper cost of capital. The EVA for measuring a company's financial performance deducts its cost of capital from its net operating profit after taxes. As explained in the previous sections, since EVA is based on accounting principles, making unreasonable decisions is possible. For example, increasing current assets by higher inventories in order to make more EVA.


### 4.5. Managerial Insight

As a result of decreasing profit margins and the competitive landscape, supply chain managers are forced to design and optimize the operation of their supply chain networks by considering operational and financial performance indexes at the same time [40, 41]. Therefore, they need comprehensive decision support models that track and measure the financial impact of their production and distribution decision by integrating various financial performances. Moreover, this integration makes a “common language” between supply chain managers and financial managers and improves cooperation between them [42, 43]. This study suggests a mathematical programming decision model that considers the physical and financial aspects of a supply chain planning problem simultaneously. A deterministic mixed-integer nonlinear programming (MINLP) model has been developed to specify the number and location of facilities and the links between them. The model also determines the quantities to be produced, stored, and transported in order to meet customers' demands. According to financial decisions made by the model, managers are provided with an accounts payable policy since we consider the amount to invest, the source of the money needed (cash, bank loan, or new capital from shareholders), and repayments. It enables supply chain managers to take holistic decisions without underestimating the basic objective of a profit company which is the creation of value for shareholders measured by the SVA index. This objective indicates a satisfactory financial status in order to guarantee new funds from shareholders and financial institutions.


## 4.1. Input Parameters of the Model

In order to evaluate the applicability and efficiency of the developed model presented in the previous section, we applied the data of a real company which is located in the UK as shown in Figure2 and studied by Longinidis and Georgiadis [5]. Note that, because of some data incongruity and missing data, their case study could not be directly applied and we have considered the following assumptions regarding the missing information:(i)
This company has three plants in three different locations and four possible locations for warehouses and six potential locations for distribution centers.(ii)
Each plant is able to produce six of seven products within its limitations of production capacity. Each plant also holds about two times the average annual demand as initial inventories.(iii)
In each time duration, each warehouse and also distribution centers have an upper and lower bound handling capacity and need safety stock.(iv)
Initial inventories are considered about two times the average annual demand.(v)
Safety stock for each product at each facility is equal to the total quantity transferred from the facility during a period of 15 days.(vi)
Product flows among plants, warehouses, distribution centers, and customer zones have upper bounds.(vii)
Prices and demands of products in each customer zone are known.(viii)
The company has a 4-year planning horizon.(ix)
Before the planning horizon, balance sheet data are integrated into the optimization process.(x)
All tangible assets have been deprecated. Short-term liabilities (accounts payables and taxes of previous profits) should be paid in one year.(xi)
The real value of cash has been calculated instead of considering it as a percent of net income.Figure 2
The case study supply chain network.


## 4.2. Comparison between Basic Model and Developed Models

Now, in order to display the improvements in the proposed model, we compared the results of the basic model presented by Longinidis and Georgiadis [5] with our developed models which have a new objective function, accurate calculations, and additional financial considerations. All the problems were solved by Branch and Reduce Optimization Navigator (BARON) solver in GAMS software on a personal computer with core i5 CPU 2.50 GHz and 8 GB of RAM on Windows 8.


### 4.2.1. Basic Model

The basic model was considered with the same decision-making assumptions and objective function presented by Longinidis and Georgiadis [5]. Its objective is to maximize the company’s net created value which is measured by Economic Value Added (EVA) index. The model was solved and the total value created amounts to 85,855,590 (monetary units). The optimal results of the basic model will be used to compare them with results obtained from other developed models. In this way, it is possible to show the advantages of the proposed approach clearly.


### 4.2.2. The First Developed Model with New Objective Function

According to what is explained in Section2, SVA is one of the most accepted methods to measure the value of a company. SVA determines the financial value of a company by looking at the returns it provides for its stockholders and is based on the view that the objective of company managers is to maximize the wealth of company stockholders. SVA calculates the shareholder value by deducting the value of long-term liabilities at the end of planning horizon from the value of the firm for the time period [34]. In this study, the final value of the company is obtained by the discounted free cash flow (DFCF) method with a fixed growth rate (0.5%).Now, in the first stage of developing the model, shareholder value analysis (SVA) is applied as an objective function in the basic model. The model was solved and the total value created amounts is 86,855,590 monetary units. The optimal network structure is shown in Figure3. The total production quantities for the whole planning horizon is only 1407 units: plant 1 and plant 3 produce 809 and 598, respectively; plant 2 does not produce at all. Therefore, reducing inventory was clearly shown and had these results: (i) decreasing production quantities to reduce the product quantities in stock. (ii) The large flows lead to establishing a new distribution center to meet customers’ demands. In order to reduce the need for working capital, SVA tends to reduce the inventory. Therefore, the produced quantity by SVA model is smaller than the EVA model. This feature of SVA model also makes a large number of flows between warehouses and distribution centers and between distribution centers and customer zones. The total quantities transported from plants to warehouses for both models are compared in Table 3.Figure 3
Network structure and produced products for the developed model.Table 3
Total products transported from plants to warehouses.
Developed modelBasic modelW1W2W3W4W1W2W3W4Plant 179017471Plant 262101498Plant 335023201According to Table3, by SVA model, warehouse 1 receives more products supplying distribution centers 1 and 6. Similarly, warehouse 2 receives more quantity; therefore, it supplies distribution centers 1, 2, 5, and 6. But by EVA model, warehouse 2 just supplied distribution center 2.As shown in Tables4 and 5, by applying the model with SVA as the objective function, inventory was stored in five distribution centers (all distribution centers except 4); therefore, total flows between distribution centers and customer zones are much larger than total flows transported when EVA was the objective function.Table 4
Total products transported from warehouses to distribution centers.
Developed modelBasic modelDC1DC2DC3DC4DC5DC6DC1DC2DC3DC4DC5DC6W1529825437471W2105230350833211498W316132983201W4Table 5
Total products transported from distribution centers to customer zones (SVA base model).
Developed modelBasic modelCZ1CZ2CZ3CZ4CZ5CZ6CZ7CZ8CZ1CZ2CZ3CZ4CZ5CZ6CZ7CZ8DC113491141672123904144313492018124114131458DC215167281498DC3149834662081614981559DC4DC5508DC6531Note that since distribution center 6 has the lowest inventory cost among others, it received most of the inventory transferred from warehouses to distribution centers. It receives 5864 units but it only supplies customer zone 6 with 531 units and 5333 units are kept as inventory. Also, the model with SVA as the objective function tends to reduce the inventory quantities to decrease the need for working capital. Only 878 units stay at the plants as inventory.


### 4.2.3. The Second Developed Model with New Financial Aspects

Now, in the second phase of model development, we add new financial aspects to the previous version of the model to make it similar to real conditions. These new features include the possibility of closing and opening facilities at any time period of the planning horizon, repayments obligation to the bank, adding the possibility of new capital entries from shareholders, and adoption of an accounts payable policy. To better understand the effect of these aspects, we explained them separately.First, to test the possibility of closing and opening facilities at any time period, we considered two times of the establishment price of each facility as selling prices. The value created for shareholders is 87,397,697 monetary units, which is 0.88% larger than the value created by the basic model which is the gains resulting from selling the plants. Then the new model with the obligation of bank repayments created 89,407,636 monetary units, which is 3.02% larger than the value created by the model with SVA as objective function. The network structure remains the same. By repaying to the bank every year, long-term debt is reduced and a lower amount is deducted from the free cash flow that was generated over the planning horizon, creating more value for shareholders.Next, in order to consider an account payable policy, it is assumed that 60% of payments to suppliers are made in cash and 40% are made in credit. In this situation, the value created for shareholders is 88,549,322 monetary units, that is, 0.96% smaller. Because more amount of money (working capital) is needed to support operating expenses and pay suppliers, the free cash flow decreases and the value created is 858,314 monetary units lower.Finally, we add the possibility of raising new capital from shareholders and also set per year limit of 60,000 monetary units for the new capital entries. This limit shows the maximum that shareholders are willing to invest in the company to receive dividends in the future. The new developed model was solved optimally and the value for shareholders increased to 92,460,308 monetary units, which is 3.18% larger than the value without these financial considerations created, and 6.3% larger than the value created by the basic model. Figures4 to 6 display the network structure during the planning horizon. As it can be seen, the flows between facilities and the quantities transported change during the time.Figure 4
Network structure for the complete model in year 1 and for the developed model with new financial aspects.Figure 5
Network structure in year 2 and for the developed model with new financial aspects.Figure 6
Network structure in year 3 and for the developed model with new financial aspects.According to Figures7 to 6, plants only produce during the first two years and their total quantity is 1394 units. The total quantity produced by the SVA model is much lower than the quantity production when EVA was the objective function. Therefore, the need for working capital and payments to suppliers is smaller. These changes lead to an increase in the value created for shareholders. Also, by using EVA as the objective function, the value of the company improves by creating higher inventories (which are a part of current assets) [37].Figure 7
Network structure in year 4 and for the developed model with new financial aspects.Plant 2 closes at the start of the second year with a final inventory of 3341 units, reducing its initial inventory by 76%. Plant 1 and plant 3 are closed at the beginning of the third year, with the final inventory of 1971 and 881 units. This means an inventory reduction of 245% and 285%, respectively. Note that products 2, 4, and 7 at plant 1 which were not sold within the planning horizon are considered as the final inventory. Also, products like 3 and 6 at plant 1 that were produced in the years 1 and 2, have no final inventory. As explained before, in accordance with the evolution of the number of flows among facilities, the product quantities transported from plants to warehouses increase from year 1 to year 2. Table6 presented the operating costs (production, transportation, and inventory holding costs) that resulted from the decisions described above. As we can see, the largest portion of the operating costs is transportation costs (50.58%), then inventory holding costs (40.27%), and production costs (9.15%). There are production costs in the first and second years. Also, due to high inventory at the beginning of the planning horizon, there is no production in the years 3 and 4. In these two years, from plants to warehouses and from warehouses to distribution centers, there are no transportation costs because plants are closed and the warehouses are not operating. As shown in Table 6, inventory costs decrease over time. The inventory costs at plants in years 3 and 4 refer to products that were already in inventory at the beginning of the planning horizon and the ones customers did not request. It is important to note that although the final inventory at the distribution centers is equal to zero, there is an inventory cost since inventory is calculated based on its average during a year.Table 6
Production, transportation, and inventory costs for year for the developed model with new financial aspects.
Year 1Year 2Year 3Year 4TotalProduction costs101390,1020091,115Transportation costs162,717209,85660,41771,303504,293Inventory costs141,402109,54289,50260,991401,437According to financial decisions made by the final model, managers are provided with an accounts payable policy in Table7. It shows that the company has enough cash (based on the initial balance sheet) and does not need bank loans. Therefore, all capital entries are captured from shareholders. As we can see, production costs by the developed model are low, since high levels of inventory and money are available for investment. Therefore, the company is in a good condition for repayments to the bank, decreasing debt and maximizing the value of the corporate which is measured by SVA.Table 7
Financial decisions for each year for the developed model with new financial aspects.
Financial decisionsYear 1Year 2Year 3Year 4TotalLoans00000New capital entries60,00060,00060, 00060,000240,000Investment300,000000300,000Repayments540,000270,000135,00067,5001,012,500


## 4.2.1. Basic Model

The basic model was considered with the same decision-making assumptions and objective function presented by Longinidis and Georgiadis [5]. Its objective is to maximize the company’s net created value which is measured by Economic Value Added (EVA) index. The model was solved and the total value created amounts to 85,855,590 (monetary units). The optimal results of the basic model will be used to compare them with results obtained from other developed models. In this way, it is possible to show the advantages of the proposed approach clearly.


## 4.2.2. The First Developed Model with New Objective Function

According to what is explained in Section2, SVA is one of the most accepted methods to measure the value of a company. SVA determines the financial value of a company by looking at the returns it provides for its stockholders and is based on the view that the objective of company managers is to maximize the wealth of company stockholders. SVA calculates the shareholder value by deducting the value of long-term liabilities at the end of planning horizon from the value of the firm for the time period [34]. In this study, the final value of the company is obtained by the discounted free cash flow (DFCF) method with a fixed growth rate (0.5%).Now, in the first stage of developing the model, shareholder value analysis (SVA) is applied as an objective function in the basic model. The model was solved and the total value created amounts is 86,855,590 monetary units. The optimal network structure is shown in Figure3. The total production quantities for the whole planning horizon is only 1407 units: plant 1 and plant 3 produce 809 and 598, respectively; plant 2 does not produce at all. Therefore, reducing inventory was clearly shown and had these results: (i) decreasing production quantities to reduce the product quantities in stock. (ii) The large flows lead to establishing a new distribution center to meet customers’ demands. In order to reduce the need for working capital, SVA tends to reduce the inventory. Therefore, the produced quantity by SVA model is smaller than the EVA model. This feature of SVA model also makes a large number of flows between warehouses and distribution centers and between distribution centers and customer zones. The total quantities transported from plants to warehouses for both models are compared in Table 3.Figure 3
Network structure and produced products for the developed model.Table 3
Total products transported from plants to warehouses.
Developed modelBasic modelW1W2W3W4W1W2W3W4Plant 179017471Plant 262101498Plant 335023201According to Table3, by SVA model, warehouse 1 receives more products supplying distribution centers 1 and 6. Similarly, warehouse 2 receives more quantity; therefore, it supplies distribution centers 1, 2, 5, and 6. But by EVA model, warehouse 2 just supplied distribution center 2.As shown in Tables4 and 5, by applying the model with SVA as the objective function, inventory was stored in five distribution centers (all distribution centers except 4); therefore, total flows between distribution centers and customer zones are much larger than total flows transported when EVA was the objective function.Table 4
Total products transported from warehouses to distribution centers.
Developed modelBasic modelDC1DC2DC3DC4DC5DC6DC1DC2DC3DC4DC5DC6W1529825437471W2105230350833211498W316132983201W4Table 5
Total products transported from distribution centers to customer zones (SVA base model).
Developed modelBasic modelCZ1CZ2CZ3CZ4CZ5CZ6CZ7CZ8CZ1CZ2CZ3CZ4CZ5CZ6CZ7CZ8DC113491141672123904144313492018124114131458DC215167281498DC3149834662081614981559DC4DC5508DC6531Note that since distribution center 6 has the lowest inventory cost among others, it received most of the inventory transferred from warehouses to distribution centers. It receives 5864 units but it only supplies customer zone 6 with 531 units and 5333 units are kept as inventory. Also, the model with SVA as the objective function tends to reduce the inventory quantities to decrease the need for working capital. Only 878 units stay at the plants as inventory.


## 4.2.3. The Second Developed Model with New Financial Aspects

Now, in the second phase of model development, we add new financial aspects to the previous version of the model to make it similar to real conditions. These new features include the possibility of closing and opening facilities at any time period of the planning horizon, repayments obligation to the bank, adding the possibility of new capital entries from shareholders, and adoption of an accounts payable policy. To better understand the effect of these aspects, we explained them separately.First, to test the possibility of closing and opening facilities at any time period, we considered two times of the establishment price of each facility as selling prices. The value created for shareholders is 87,397,697 monetary units, which is 0.88% larger than the value created by the basic model which is the gains resulting from selling the plants. Then the new model with the obligation of bank repayments created 89,407,636 monetary units, which is 3.02% larger than the value created by the model with SVA as objective function. The network structure remains the same. By repaying to the bank every year, long-term debt is reduced and a lower amount is deducted from the free cash flow that was generated over the planning horizon, creating more value for shareholders.Next, in order to consider an account payable policy, it is assumed that 60% of payments to suppliers are made in cash and 40% are made in credit. In this situation, the value created for shareholders is 88,549,322 monetary units, that is, 0.96% smaller. Because more amount of money (working capital) is needed to support operating expenses and pay suppliers, the free cash flow decreases and the value created is 858,314 monetary units lower.Finally, we add the possibility of raising new capital from shareholders and also set per year limit of 60,000 monetary units for the new capital entries. This limit shows the maximum that shareholders are willing to invest in the company to receive dividends in the future. The new developed model was solved optimally and the value for shareholders increased to 92,460,308 monetary units, which is 3.18% larger than the value without these financial considerations created, and 6.3% larger than the value created by the basic model. Figures4 to 6 display the network structure during the planning horizon. As it can be seen, the flows between facilities and the quantities transported change during the time.Figure 4
Network structure for the complete model in year 1 and for the developed model with new financial aspects.Figure 5
Network structure in year 2 and for the developed model with new financial aspects.Figure 6
Network structure in year 3 and for the developed model with new financial aspects.According to Figures7 to 6, plants only produce during the first two years and their total quantity is 1394 units. The total quantity produced by the SVA model is much lower than the quantity production when EVA was the objective function. Therefore, the need for working capital and payments to suppliers is smaller. These changes lead to an increase in the value created for shareholders. Also, by using EVA as the objective function, the value of the company improves by creating higher inventories (which are a part of current assets) [37].Figure 7
Network structure in year 4 and for the developed model with new financial aspects.Plant 2 closes at the start of the second year with a final inventory of 3341 units, reducing its initial inventory by 76%. Plant 1 and plant 3 are closed at the beginning of the third year, with the final inventory of 1971 and 881 units. This means an inventory reduction of 245% and 285%, respectively. Note that products 2, 4, and 7 at plant 1 which were not sold within the planning horizon are considered as the final inventory. Also, products like 3 and 6 at plant 1 that were produced in the years 1 and 2, have no final inventory. As explained before, in accordance with the evolution of the number of flows among facilities, the product quantities transported from plants to warehouses increase from year 1 to year 2. Table6 presented the operating costs (production, transportation, and inventory holding costs) that resulted from the decisions described above. As we can see, the largest portion of the operating costs is transportation costs (50.58%), then inventory holding costs (40.27%), and production costs (9.15%). There are production costs in the first and second years. Also, due to high inventory at the beginning of the planning horizon, there is no production in the years 3 and 4. In these two years, from plants to warehouses and from warehouses to distribution centers, there are no transportation costs because plants are closed and the warehouses are not operating. As shown in Table 6, inventory costs decrease over time. The inventory costs at plants in years 3 and 4 refer to products that were already in inventory at the beginning of the planning horizon and the ones customers did not request. It is important to note that although the final inventory at the distribution centers is equal to zero, there is an inventory cost since inventory is calculated based on its average during a year.Table 6
Production, transportation, and inventory costs for year for the developed model with new financial aspects.
Year 1Year 2Year 3Year 4TotalProduction costs101390,1020091,115Transportation costs162,717209,85660,41771,303504,293Inventory costs141,402109,54289,50260,991401,437According to financial decisions made by the final model, managers are provided with an accounts payable policy in Table7. It shows that the company has enough cash (based on the initial balance sheet) and does not need bank loans. Therefore, all capital entries are captured from shareholders. As we can see, production costs by the developed model are low, since high levels of inventory and money are available for investment. Therefore, the company is in a good condition for repayments to the bank, decreasing debt and maximizing the value of the corporate which is measured by SVA.Table 7
Financial decisions for each year for the developed model with new financial aspects.
Financial decisionsYear 1Year 2Year 3Year 4TotalLoans00000New capital entries60,00060,00060, 00060,000240,000Investment300,000000300,000Repayments540,000270,000135,00067,5001,012,500


## 4.3. Financial Sensitivity Analysis

In this section, we test the performance of proposed models in several cases by changing some financial parameters. These parameters are important because they are suggestive of the economic environment and in many cases are accepted conditions that the company has no impact on them. The cost of capital rate at time periodt (rt) is an important parameter. Also, one of the important financial parameters affecting the company’s wealth is the tax rate (Trt). Moreover, we selected the depreciation (DPRst) rate as a financial parameter for the sensitivity test.Table8 shows the effects on the developed model by changing these parameters from −15% to +15%. The results show that the developed model with new financial aspects was robust against changes in these financial parameters.Table 8
Sensitivity analysis of value created according to changes in financial parameters.
ParameterChange (%)−15−10−5−2+2+5+10+15Cost of capital rate at time periodt (rt)105,947,496101,350,94096,869,75294,204,96490,717,78088,114,17283,796,38479,838,788Tax rate (Trt)99,756,84097,326,66494,896,18493,435,23691,484,46890,020,78487,580,19685,139,760Depreciation rateDPRst93,832,79293,377,62892,919,88092,644,30492,275,78091,998,60891,534,32491,070,724


## 4.4. Results and Discussions

In the previous section, the optimal results of a basic model were used to compare them with the results obtained from other developed models to show the advantages of the developed models. We carried out two phases of development in order to improve the basic model: (i) applying a new objective function, which maximizes the value of the company measured by the SVA method, (ii) adding new financial aspects to the previous version of the model to make it more realistic.In the first step, we applied SVA as a new objective function instead of EVA. The model with the new objective was solved and the total value created for shareholder’s amounts to 86,635,307 monetary units.In the second step, the new financial aspects were integrated into the previous version of the model. The total value created by the complete version of the model was 92,460,308 monetary units, which is 0.7% larger than the SVA obtained without financial aspects and 0.93% larger than the value created in the basic model. The main reasons for an increase in value creation for shareholders are due to new operational and financial aspects, which mainly show the possibility of closing facilities and bank debt repayments. Bank repayments which reduce debt and new capital enables the company to choose better operational options. The value created by each model is reported in Table9.Table 9
Values obtained by each model.
ModelValue created (monetary units)The basic model85, 855, 590The first developed model with new objective function86, 635, 307The second developed model with new financial aspects92, 460, 308The main reasons for an increase in the value created are due to both operational and financial aspects such as the possibility of closing facilities and bank repayments.In terms of the type of objective function in this study instead of EVA, which is based on conventional accounting principles, SVA is applied as an objective function, that is, one of the most accepted methods of measuring how corporate performance relates to shareholder value. As mentioned before, the SVA for a company is calculated by adding the present value of cash flows to their terminal value, which represents the value of the company discounted at the proper cost of capital. The EVA for measuring a company's financial performance deducts its cost of capital from its net operating profit after taxes. As explained in the previous sections, since EVA is based on accounting principles, making unreasonable decisions is possible. For example, increasing current assets by higher inventories in order to make more EVA.


## 4.5. Managerial Insight

As a result of decreasing profit margins and the competitive landscape, supply chain managers are forced to design and optimize the operation of their supply chain networks by considering operational and financial performance indexes at the same time [40, 41]. Therefore, they need comprehensive decision support models that track and measure the financial impact of their production and distribution decision by integrating various financial performances. Moreover, this integration makes a “common language” between supply chain managers and financial managers and improves cooperation between them [42, 43]. This study suggests a mathematical programming decision model that considers the physical and financial aspects of a supply chain planning problem simultaneously. A deterministic mixed-integer nonlinear programming (MINLP) model has been developed to specify the number and location of facilities and the links between them. The model also determines the quantities to be produced, stored, and transported in order to meet customers' demands. According to financial decisions made by the model, managers are provided with an accounts payable policy since we consider the amount to invest, the source of the money needed (cash, bank loan, or new capital from shareholders), and repayments. It enables supply chain managers to take holistic decisions without underestimating the basic objective of a profit company which is the creation of value for shareholders measured by the SVA index. This objective indicates a satisfactory financial status in order to guarantee new funds from shareholders and financial institutions.


## 5. Conclusions and Future Research

The main purpose of a supply chain is to satisfy demand, improve responsiveness and profitability, and build a good network to facilitate the financial success of a company. Many of the previous studies emphasize that strategic decisions such as supply chain decisions have a significant impact on shareholder value creation. Investment decisions also should be considered as critical inputs to financial planning. Since these kinds of decisions for supply chain networks plays a key role in financial health of companies, therefore, financial considerations should also be regarded when modeling supply chains.However, studies on supply chain models integrating financial aspects are limited. In these studies, financial aspects have been considered as known parameters or endogenous variables in constraints and objective functions.Based on the abovementioned concerns, this study suggests a mathematical model that considers the physical and financial aspects of a supply chain planning problem, simultaneously. A deterministic mixed-integer nonlinear programming (MINLP) model was developed to specify the number and location of facilities and the links between them. The model also determines the quantities to be produced, stored, and transported in order to meet customers’ demands as well as maximize the shareholder value measured by SVA method. In financial decisions, the amount of investment, the source of the money needed (cash, bank loan, or new capital from shareholders), and repayments to the bank were considered. To demonstrate the applicability and efficiency of the proposed model, data of Longinidis and Georgiadis [5] were used. The results show that with appropriate financial decisions, creating more value for the company and its shareholders is achievable. The model could be used as an effective strategic decision tool by supply chain managers, supporting their decisions with figures and indexes convenient for financial managers. The major contributions of this study can be summarized as follow:(i)
This study presents a mathematical model to solve a supply chain network design problem that considers tactical, strategic, and financial decisions at the same time.(ii)
Maximizing the creation of economic value for shareholders measured by shareholder value analysis (SVA) as a new objective function instead of traditional approaches such as maximizing profits or minimizing costs. It has not been still used in the general model in supply chain network design problems.(iii)
The proposed model considers the amount of loan, bank repayment, and new capital from shareholders as decision variables; therefore, it provides an accounts payable policy for the company managers instead of considering that all payments should be paid in cash. Previous studies of the literature consider them as parameters.(iv)
At the strategic level, the model specifies the number and location of each facility. At the tactical level, it determines the products quantities to be produced and stored to satisfy customers demand. Regarding financial decisions, the model specifies the amount of investment and their sources such as cash, bank debt, or shareholders’ capital as decision variables and it provides a repayment policy for managers.(v)
Regarding the constraints, in addition to common operational constraints, lower limit and/or upper limit values for performance ratios, efficiency ratios, liquidity ratios, and leverage ratios were considered in order to support the financial health of the corporation. To retain a better financial performance, the proposed model provides a balance among new capital entries, loans, and repayment. With consideration of large cost of new capital entries, the model imposes upper bound on it and avoid an ever-increasing debt. It considers lower bound for bank repayments. Besides, these benefits of our model provides managers with an accounts payable guideline.(vi)
Providing the possibility of opening or closing facilities in order to deal with market fluctuations at any time period of the planning horizon.(vii)
In contrast with basic models in previous studies which have too many assumptions, the presented model uses accounting principles with less assumptions that made it more realistic. For example, we use the net liabilities in the analysis of financial statements that balances bank loans and payments, determines the exact value of deprecation by knowing the lifetime of each asset in each time period, and applies real cash value instead of predetermined proportion of profit.This work can be extended in the following directions: in order to make the model similar to real conditions, future studies can consider uncertainty in some parameters such as product prices and demand. Using financial ratios as objective functions in our model, we can look for ways to increase and improve the firm soundness and its optimal results through experiments. The green supply chain with a closed-loop structure can be the other research trend for the model considering environmental, social, technological, and economic facets; such facets can be included in the supply chain network design. The problem would get more complicated with such developments. Therefore, following other solutions, such as metaheuristics, can be considered as other suggestions for future studies.



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*Source: 2072301-2022-10-06.xml*