Economic and environmental considerations in a continuous review inventory control system with integrated transportation decisions

Transportation Research Part E 80 (2015) 142–165

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Economic and environmental considerations in a continuous review inventory control system with integrated transportation decisions Brian Schaefer, Dinçer Konur ⇑ Engineering Management and Systems Engineering, Missouri University of Science and Technology, United States

a r t i c l e

i n f o

Article history: Received 18 February 2015 Received in revised form 2 April 2015 Accepted 6 May 2015 Available online 14 June 2015 Keywords: Sustainability Inventory control Transportation Stochastic demand

a b s t r a c t Sustainability throughout supply chains is gaining more importance and re-planning inventory operations can help companies curb emissions. In this study, we present two bi-objective integrated continuous review inventory control and transportation models with less-than-truckload and truckload carriers. Solution methods to approximate the Pareto Frontiers are proposed. Numerical studies illustrate the effects of demand variance and lead time on expected costs and carbon emissions as well as the changes in expected costs and carbon emissions due to sustainability considerations. Sample examples illustrate the use of the methods to compare different carriers in terms of not only economic but also environmental considerations. Published by Elsevier Ltd.

1. Introduction Sustainability throughout supply chains is gaining more importance every day. Recent review papers on sustainable supply chain management document the necessity and importance of integrating sustainability with supply chain/operations management (see, e.g., Corbett and Kleindorfer, 2001a,b; Linton et al., 2007; Dekker et al., 2012). Mainly, companies are motivated to green their operations by regulatory requirements and increasing awareness of customers on climate change (Srivastava, 2007). In particular, policy makers implement environmental regulations, such as carbon taxing, carbon cap, and carbon trading as incentives for firms to increase sustainability of their operations. Aside from these regulations, companies today are able to get a competitive advantage by selling greener products (Bouchery et al., 2012). For instance, as a result of a survey among 582 European companies, Loebich et al. (2011) note that while the environmental regulations were the main reasons why companies green their operations in 2008, brand image improvement and executive board decisions were the top motivations for taking green actions in 2010. In another survey study, Kiron et al. (2012) highlight that two-thirds of 2874 managers/executives from 113 countries taking part in the survey see sustainability as a crucial factor in competition. Aside from manufacturing; inventory holding, freight transportation, logistics, and warehousing operations are the main supply chain activities that generate emissions in many industries. Particularly, inventory control is an important activity and appears in any type of organization (Tsou et al., 2010). Inventory control policy of a company derives the level of transportation, logistics, and warehousing activities; hence, it is the key determinant of the emissions generated. It is thus not surprising that there is an increasing number of studies focusing on sustainable inventory control models. In this paper, we analyze sustainability in an inventory control system with integrated transportation decisions under stochastic demand. ⇑ Corresponding author at: 206 EMSE, 600 W. 14th Street, Rolla, MO 65409, United States. E-mail address: [email protected] (D. Konur). http://dx.doi.org/10.1016/j.tre.2015.05.006 1366-5545/Published by Elsevier Ltd.

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From a practical point, many companies plan their inventory control and inbound transportation operations simultaneously as one affects the other. Inventory control policy, for instance, determines the frequency and amount of shipments. These, in turn, determine the transportation costs and emissions. Therefore, transportation carrier selection integrated with inventory control is an important decision that affects a company’s economic and environmental performance. For instance, there might be many carriers available for outsourcing inbound transportation, which is a common business practice (see, e.g., Gurler et al., 2014); and, each carrier might have different charges. Furthermore, since different carriers operate different transportation vehicles, they will also have distinct environmental performance. Therefore, a company, who wishes to plan its inventory control and transportation operations, should compare different carriers in terms of not only costs but also environmental performance. In this study, we provide tools that can be used to compare different carriers in terms of economic and environmental performance. Particularly, the most of the studies integrating sustainability with inventory control analyze the deterministic demand inventory control models. Nevertheless, deterministic demand assumption is very restrictive. There is a limited number of studies that focus on inventory control with environmental considerations in stochastic demand scenarios and the majority of these studies investigate single-period decisions. In this study, we investigate the continuous review inventory control model over a long planning horizon with sustainability under stochastic demand. Specifically, we analyze the ðQ ; RÞ policy, in which a retailer orders Q units whenever his/her inventory level is R. In the classic ðQ ; RÞ model, the retailer’s objective is to minimize expected costs due to inventory holding, order setups, and shortages. However, as noted by Dekker et al. (2012), profit maximization (or cost minimization) is not the only objective for companies. Many studies on sustainable supply chains, therefore, consider not only economic objectives such as cost minimization or profit maximization but also environmental objectives such as emission minimization (see, e.g., Li et al., 2008; Kim et al., 2009; Ramudhin et al., 2010; Wang et al., 2011; Bouchery et al., 2012; Chaabane et al., 2012). Similar to Bouchery et al. (2012), we formulate a sustainable continuous review inventory control model by considering two objectives: cost minimization and emission minimization. Multi-objective continuous review inventory control models have been analyzed in the literature for the classical ðQ ; RÞ settings (see, e.g., Agrell, 1995; Puerto and Fernandez, 1998; Tsou, 2008, 2009). To the best knowledge of the authors, the current study is the first that introduces an environmental objective into a continuous review inventory control model. Furthermore, we contribute to the sustainable inventory control models by analyses of stochastic demand inventory systems with integrated transportation decisions. Specifically, a major share of greenhouse gas (GHG) emissions comes from the transportation sector. According to ECOFYS (2010), around 15% of 2010 global GHG emissions resulted from transportation, including passenger and freight transportation. For instance, EEA (2013) data shows that 25% of France’s GHG emissions, 17% of Germany’s GHG emissions, and 20% of the U.K.’s GHG emissions in 2010 were generated from transportation sector. In the European Union (EU), approximately 25% of the total GHG emissions was due to transportation in 2010 (EU, 2013). In the U.S., transportation sector generated 27% of the total GHG emissions in 2011 (EPA, 2013). While the passenger transportation is the major player in transportation GHG emissions, freight transportation significantly contributes to transportation GHG emissions. Among freight transportation modes, the major contributor to GHG emissions is the road transportation. For instance, in 2010 in the EU, more than 70% of transportation emissions comes from road transportation including light-duty and heavy-duty vehicles (EU, 2013) and almost 19% and 22% of 2010 U.S. transportation emissions were generated by light-duty and medium/heavy-duty trucks, respectively (EPA, 2013). The statistics on the fact that freight trucks are the main contributors to GHG emissions of freight transportation are not surprising as freight trucks are the most common transportation mode. Furthermore, considering the anticipated increases in freight transportation in both European countries and the U.S. (see, e.g., Toptal and Bingol, 2011; FHWA, 2008), it is important to consider freight trucks in transportation integrated with inventory control decisions to accurately model sustainable inventory control decisions. To this end, we consider the proposed sustainable ðQ ; RÞ model with two different common types of road freight transportation: less-than-truckload (LTL) transportation and truckload (TL) transportation. In LTL transportation, the retailer is charged on the number of units (or volume or weight units) shipped. The settings of the ðQ ; RÞ model with LTL transportation are, therefore, parallel to the classical ðQ ; RÞ model. On the other hand, in TL transportation, the retailer is charged on the number of trucks used for transportation, which requires explicit transportation modeling. TL transportation has been integrated with inventory control decisions in the supply chain and logistics literature (see, e.g., Aucamp, 1982; Lee, 1986; Toptal et al., 2003; Toptal and Çetinkaya, 2006; Toptal, 2009; Toptal and Bingol, 2011; Konur and Toptal, 2012). Following the modeling approach in these studies, we model TL transportation explicitly by taking per truck costs and per truck capacities into account. Furthermore, as noted previously, trucks are the main contributors to freight transportation emissions; hence, to accurately model the carbon emissions in the case of TL transportation, we include TL transportation emissions explicitly in modeling the retailer’s emissions. We note that Hoen et al. (2014a) and Pan et al. (2012) define similar functions to model carbon emissions functions due to freight transportation. In order to minimize carbon emissions within a supply chain network, Pan et al. (2012) formulate a transportation problem with two modes of transportation (rail and trucks). Specifically, Pan et al. (2012) note that a piecewise discontinuous carbon emissions function is observed in TL and rail transportation. Recently, Konur (2014) and Konur and Schaefer (2014) model carbon emissions from trucks considering truck characteristics. This study presents two bi-objective ðQ ; RÞ models: one for LTL transportation and one for TL transportation. For each of these models, we focus on approximating a set of Pareto efficient ðQ ; RÞ policies, i.e., a Pareto Frontier, among which the retailer can select a policy regarding his/her sensitiveness to the environment and/or how much he/she is willing to pay

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to be more sustainable. In approximating the Pareto Frontier for the sustainable ðQ ; RÞ model with LTL transportation, we propose a method that adopts a normalized weighted approach, a common approach used for multi-objective optimization problems. This method is then utilized in approximating the Pareto Frontier for the sustainable ðQ ; RÞ model with TL transportation. Particularly, given the number of trucks to be used for inbound shipment, the normalized weighted approach can be used to generate a set of Pareto efficient ðQ ; RÞ policies considering the given transportation capacity. Then, we use a dominance relation between two sets of Pareto efficient solutions, each for different transportation capacities, to approximate the Pareto Frontier of the sustainable ðQ ; RÞ model with TL transportation. Our contribution in this study lies in including sustainability in stochastic inventory control models with explicit transportation decisions. We explicitly model the effects of freight trucks in cost as well as emission calculations. The methods presented in this study can be used by a retailer to compare different less-than-truckload carriers, less-than-truckload carriers to truckload carriers, and different truckload carriers in terms of not only economic but also environmental aspects. These methods suggest a set of efficient policies for the retailer, for each transportation mode available, so that the retailer can select a policy to adopt regarding his/her cost and green goals. Numerical studies are presented to analyze the effects of demand variance and lead time on the retailer’s costs and emissions. It has been demonstrated for deterministic demand inventory control models that carbon emissions can be significantly reduced with low cost increases (see, e.g., Chen et al., 2013). We further generalize these results for a continuous review inventory control model under stochastic demand both with LTL and TL transportation. Finally, we demonstrate how the retailer can utilize the tools presented in this study to adopt a policy and select a carrier considering his/her cost and environmental goals. The rest of the study is organized as follows. Section 2 reviews the studies in the intersection of sustainability and inventory control. Section 3 discusses the details of the settings and presents the bi-objective formulations of the sustainable ðQ ; RÞ models with LTL and TL transportation. In Section 4, a method to approximate the Pareto Frontier of each model is presented. The results of a set of numerical studies are given in Section 5 to analyze the effects of stochastic demand on costs and emissions, changes in costs and emissions due to sustainability considerations, and to illustrate the practical use of the methods discussed. Section 6 summarizes the results/findings and contributions of this study and proposes several future research directions. Appendix A includes the details of the notation, design of the numerical studies, the tables and illustrative example settings of Section 5. 2. Literature review In this section, we review the studies in the intersection of sustainability and inventory control. In general, the sustainability has been integrated into inventory control models using three approaches: associating direct costs with environmental damage, planning inventory control decisions under environmental regulations, and regarding environmental objectives in addition to the economic objectives. We review how these approaches are adopted in deterministic and stochastic demand models. Majority of the sustainable inventory control models assume deterministic demand and most of the studies with deterministic demand investigate the well-known Economic Order Quantity (EOQ) model. The EOQ model simply balances the inventory holding and order setup costs in case of deterministic demand and recommends an optimal order quantity for the retailer to minimize the total inventory related costs per unit time. The EOQ model has been integrated with sustainability by associating direct costs with the environmental damage such as waste disposal and emissions from transportation, inventory holding, warehousing activities, and packaging (see, e.g., Bonney and Jaber, 2011; Digiesi et al., 2012; Battini et al., 2014). Similar to Battini et al. (2014), we consider different transportation modes; however, our focus is on a stochastic inventory control model and we do not associate direct costs with emissions. The EOQ model and its extensions have also been analyzed with environmental regulations. The classical EOQ model subject to carbon trading (Hua et al., 2011), carbon cap and carbon taxing (Chen et al., 2013), carbon trading, carbon cap, carbon taxing, and carbon offsets (Arslan and Turkay, 2013) are some of the studies that formulate the classical EOQ model under environmental regulations. In a recent study, Toptal et al. (2014) extend the EOQ model by considering emissions reduction investment decisions along with order quantity decisions under carbon cap, carbon taxing, and carbon trading policies. Konur (2014) analyze the EOQ model under carbon cap regulation with the availability of heterogeneous truck types for inbound shipment and it is discussed that consideration of different truck types for shipment may not only reduce costs but also abate emissions. In Konur and Schaefer (2014), EOQ model with LTL and TL transportation, with single truck type, has been analyzed under carbon cap, taxing, trading, and offset policies. They discuss that a retailer’s mode selection depends on the regulation parameters. In this study, we also consider LTL and TL transportation. In particular, we include explicit truck costs, capacities, and emissions in case of TL transportation. As noted before, TL transportation costs have been previously modeled in integrated inventory control and transportation studies (see, e.g., Aucamp, 1982; Lee, 1986; Toptal et al., 2003; Toptal and Çetinkaya, 2006; Toptal, 2009; Toptal and Bingol, 2011; Konur and Toptal, 2012). In calculating the TL transportation emissions, we explicitly consider the freight trucks and their loads. In a recent review of green road freight transportation, Demir et al. (2014) note that emissions from a freight truck depend on the empty vehicle weight and the load carried by the vehicle. Therefore, we model TL transportation emissions with two components: fixed emissions due to the empty weight of a truck and variable emissions due to the load carried by the truck. Considering fixed and variable emissions from freight trucks results in a piecewise carbon emissions function when a TL carrier is used for inbound transportation. We note that Pan

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et al. (2012), Hoen et al. (2014a), Konur (2014), Konur and Schaefer (2014), Battini et al. (2014), and Palak et al. (2014) define similar emission functions considering the number of trucks and the total load carried by the trucks. Furthermore, Mallidis et al. (2014) and Hoen et al. (2014b) also note the fixed and variable emissions from transportation. Bouchery et al. (2012) define the sustainable EOQ model where the cost as well as a set of sustainability criteria are simultaneously minimized. They focus on determining Pareto efficient order quantity decisions for the proposed sustainable EOQ model. Furthermore, they discuss that sustainability criteria can be reduced with relatively low cost increases. In our study, similar to Bouchery et al. (2012), we formulate a multi-objective inventory control model with economic (cost minimization) and environmental (emissions minimization) objectives. Aside from the studies cited so far, we note that inventory control models under deterministic demand other than the EOQ models have also been analyzed with sustainability such as the two-echelon vendor–buyer coordination problem with deterministic demand under the settings of the EOQ model (see, e.g., Saadany et al., 2011; Wahab et al., 2011; Zavanella et al., 2013; Jaber et al., 2013; Chan et al., 2013) and economic lot sizing problem (see, e.g., Benjaafar et al., 2012; Absi et al., 2013; Palak et al., 2014; Helmrich et al., 2015). In practice, deterministic demand assumption is very restrictive. Compared to the deterministic inventory control models with sustainability considerations, there is a limited number of studies examining sustainability of inventory control systems under stochastic demand. Particularly, the single-period stochastic inventory control model, i.e., the Newsvendor problem, has been recently analyzed with environmental regulations. Song and Leng (2012) revisit the Newsvendor problem with carbon cap, trading, and taxing regulations. Similarly, Liu et al. (2013) study the Newsvendor problem with carbon trading. Using the settings of the Newsvendor problem, Hoen et al. (2014a) analyze transportation mode selection problem under environmental regulations. Rosic and Jammernegg (2013) focus on the single-item Newsvendor problem with dual sourcing (onshore and offshore suppliers) under carbon taxing and carbon trading regulations. Choi (2013a,b) study sourcing and supplier selection models with stochastic demand for single-order decisions in fashion and apparel industries under carbon taxing. Zhang and Xu (2013) extend the single-item Newsvendor model to the multi-item Newsvendor model and analyze the multi-item Newsvendor problem under carbon trading. Table 1 classify the aforementioned single-echelon inventory control models based on consideration of explicit transportation. Our study analyzes stochastic inventory control with explicit transportation considerations. Different than the cited stochastic inventory control studies with environmental considerations mentioned above, we do not consider single-period decisions in this study. The inventory control model of interest is considered for a long planning horizon. Specifically, we revisit the continuous review inventory control model with an environmental objective. To the best knowledge of the authors, the ðQ ; RÞ model has not been examined with sustainability considerations. The study by Arikan et al. (2014) simulates a ðQ ; RÞ model with different transportation modes and conducts a numerical analysis on how carbon emissions change depending on the demand variability for each transportation mode. In this study, we analytically model and investigate a bi-objective ðQ ; RÞ model with LTL and TL transportation. The next section explains the settings of the problem and gives the mathematical formulations of the models.

3. Sustainable ðQ ; RÞ model We consider a retailer’s continuous review inventory control policy for a single product. In particular, the demand per unit time for the product is a random variable with mean k and standard deviation #. We assume that the demand per unit time is normally distributed (the methods discussed in the rest of the paper are also valid for uniform and exponential demand distributions). The retailer adopts a ðQ ; RÞ policy such that an order of Q units is placed whenever R units are left in the inventory. That is, Q and R denote the order quantity and the re-order point, respectively. Let f ðDðsÞÞ an FðDðsÞÞ denote the probability density and cumulative distribution functions of the lead time demand, DðsÞ, respectively, given that the lead time is s time units. Furthermore, let lðsÞ and rðsÞ denote the expected lead time demand and the standard deviation of the pffiffiffi lead time demand. Note that one can show that lðsÞ ¼ ks and rðsÞ ¼ # s (see, e.g., Nahmias, 2009). We assume that the retailer will have a positive safety stock, which is common in practice; hence, we have R P lðsÞ. In this setting, the retailer is subject to procurement costs, inventory holding costs, order set-up costs, penalty costs associated with shortages, and transportation costs. We assume that shortages are backordered. Particularly, let c denote the unit procurement cost, h denote the inventory holding cost per unit per unit time, K denote the set-up cost per order, and p denote Table 1 Classification of single-echelon inventory control models with environmental considerations. Without explicit transportation Deterministic inventory control models Hua et al. (2011), Bonney and Jaber (2011), Bouchery et al. (2012), Digiesi et al. (2012), Chen et al. (2013), Arslan and Turkay (2013), Toptal et al. (2014) Stochastic inventory control models Song and Leng (2012), Liu et al. (2013) Rosic and Jammernegg (2013), Choi (2013a,b) Zhang and Xu (2013)

With explicit transportation Battini et al. (2014), Palak et al. (2014) Konur (2014), Konur and Schaefer (2014)

Hoen et al. (2014a) This study

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the penalty cost per unit shortage. It is well known that under a classical ðQ ; RÞ model, the retailer’s expected purchase cost,   inventory holding cost, order set-up cost, and penalty cost per unit time amount to ck; h R  l þ Q2 ; Kk , and pknðRÞ , respectively, Q Q where nðR; sÞ is the expected number of shortages backordered during the lead time of s time units (Nahmias, 2009). Throughout supply chains, carbon emissions are generated due to procurement from the energy used in purchasing or processing a product or material handling required, inventory holding from the energy used for heating/refrigeration or warehousing activities, and order placement from the energy used in transportation or order initiation (see, e.g., b and K b denote the emissions Benjaafar et al., 2012; Chen et al., 2013). Following the same line with the literature, we let b c ; h, generated from unit procurement, inventory holding per unit per unit time, and order set-up per order, respectively. Furthermore, we assume that additional carbon emissions are generated due to backordered shortages. The carbon emissions generated from backorders can be due to the fact that the retailer ships the items to the backordered customers (see, e.g., Anderson et al., 2012) or the customer may need to return to the retailer’s store to pick his/her backordered item and consumer’s travel generates emissions (see, e.g., Cachon, 2014). Let b p denote the carbon emissions generated due to unit backorder. Similar to the cost components, one can observe that the retailer’s expected carbon emissions per unit time from b sÞ Kk b R  l þ Q ; b procurement, inventory holding, order set-up, and shortages amount to b , and p knðR; , respectively, given c k; h Q 2 Q that the lead time is s time units. Up to this point, transportation costs and transportation emissions are not explicitly considered. Transportation costs constitute a major part of the total costs in retailing industries and, as noted in Section 1, freight transportation significantly contributes to the carbon emissions generated through a supply chain. In what follows, we model the retailer’s integrated inventory control and transportation decisions with economic and environmental considerations. As discussed in Section 1, the companies adjust their operations to curb carbon emissions. Therefore, similar to Li et al. (2008), Kim et al. (2009), Ramudhin et al. (2010), Wang et al. (2011), Chaabane et al. (2012), and Bouchery et al. (2012), we consider that the retailer wishes to minimize not only costs but also carbon emissions of the ðQ ; RÞ policy adopted. We refer to this problem as sustainable ðQ ; RÞ model. Furthermore, for inbound transportation, we consider that the retailer can use either a LTL-carrier or a TL-carrier, which are two most common modes of road transportation (see, e.g., Konur and Schaefer, 2014). The super-/sub-scripts 1 and 2 are associated with LTL and TL transportation, respectively. The next The notation used throughout the models and possible metrics for each notation are summarized in Appendix A.1. Additional notation will be defined as needed. 3.1. Sustainable ðQ ; RÞ model with LTL transportation In the case the retailer uses a LTL-carrier, we assume that there is a unit transportation cost. Let t denote the per unit transportation cost under LTL transportation (t might be calculated depending on the distance from the point of origin to the retailer and the LTL-carrier’s charge per unit distance per unit transported). Furthermore, let sltl denote the delivery lead time of the LTL-carrier. Then, unit transportation cost can be included within the unit procurement cost and the retailer’s expected cost per unit time under LTL transportation reads as

  Q Kk pknðR; sltl Þ þ C 1 ðQ; RÞ ¼ ðc þ tÞk þ h R  l þ þ : 2 Q Q

ð1Þ

When the retailer uses a LTL-carrier, we assume that transportation emissions are proportional to the quantity shipped. Let bt denote the per unit transportation emission under LTL transportation. Then, similar to the cost function, unit transportation emissions can be included within the unit procurement emissions and the retailer’s expected carbon emissions per unit time under LTL transportation reads as

  b Q Kk b p knðR; sltl Þ h Rlþ þ E1 ðQ ; RÞ ¼ ðb þ : c þ btÞk þ b 2 Q Q

ð2Þ

As noted previously, minimization of inventory and transportation related costs is not necessarily the only objective of a company (Bouchery et al., 2012) and an assumption that the company will only focus on emissions minimization is not realistic. In what follows, we, therefore, present a bi-objective optimization model in which the retailer aims to minimize both expected costs and expected carbon emissions due to his/her inventory control and transportation decisions with a LTL carrier. We refer to this model as the sustainable ðQ ; RÞ model with less-than-truckload transportation (S-(Q,R)-LTL), which is stated as follows:

  Q Kk pknðR; sltl Þ þ S-ðQ ; RÞ-LTL : min C 1 ðQ; RÞ ¼ ðc þ tÞk þ h R  l þ þ 2 Q Q   b Q Kk b p knðR; sltl Þ h Rlþ þ þ c þ btÞk þ b min E1 ðQ ; RÞ ¼ ðb 2 Q Q s:t: R P lðsltl Þ Q P 0:

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3.2. Sustainable ðQ ; RÞ model with TL transportation As noted in Section 1, freight trucks are commonly used for road transportation and the retailers may prefer TL-carriers in practice. In another case, if the retailer operates his/her own fleet of trucks such as Wal-mart, Pepsi Co., the retailer will be subject to explicit truck costs and capacities. Therefore, we also formulate the sustainable ðQ ; RÞ model by assuming that the retailer is subject to TL transportation costs in the inbound transportation. In doing so, we follow a similar approach to the inventory control studies with TL transportation in the literature (see, e.g., Aucamp, 1982; Lee, 1986; Toptal et al., 2003; Toptal and Çetinkaya, 2006; Toptal, 2009; Toptal and Bingol, 2011; Konur and Toptal, 2012; Konur and Schaefer, 2014; Konur, 2014) by explicitly considering per truck capacities and per truck costs. In the case the retailer uses a TL-carrier, he/she is charged based on the number of trucks used for inbound shipment. Particularly, let v denote the capacity of one truck (generally, the freight trucks have weight and volume capacity; hence, given a specific weight and volume capacity, v can be the minimum of the number of units that can be shipped with one truck considering the weight and volume capacities of the truck) and w the cost charged by a TL-carrier for one truck (w can be considered as the fixed cost for the driver and the insurance for the truck). In addition, we assume that there is a unit transportation cost for using the TL-carrier. Let e denote the per unit transportation cost under TL transportation (similar to t; e can be defined depending on the truck’s average cost per unit per unit distance). It should be noted that t > e in a practical case as t is the LTL carrier’s only charge while TL carrier also charges the retailer for the truck. Then, if     the retailer’s order quantity is Q , the retailer will use Qv trucks, that is, he/she will pay eQ þ Qv w for inbound shipment at each order. Furthermore, let stl denote the delivery lead time of the TL-carrier. Then, the retailer’s expected cost per unit time under TL transportation reads as

      K þ Qv w k pknðR; stl Þ Q þ C 2 ðQ; RÞ ¼ ðc þ eÞk þ h R  l þ þ : 2 Q Q

ð3Þ

b When the retailer uses a TL-carrier, he/she is responsible for the emissions generated from the trucks used. Recall that K is used to denote the amount of emissions generated with each inventory replenishment. Particularly, Hua et al. (2011) attrib to the transportation emissions generated for shipping an order. The key assumption in their modeling approach is bute K that a single truck can deliver any order amount. Nevertheless, in practice, the retailer may have to use multiple trucks for shipping his/her order. Therefore, in what follows, we explicitly model TL transportation emissions. Ligterink et al. (2012) note that truck characteristics such as fuel type, engine type, build year, and vehicle mass influence emission generation o f a truck. Particularly, sustainable supply chain and logistics studies that explicitly account for such truck characteristics mostly focus on vehicle routing models (see, e.g., Bektas and Laporte, 2011; Suzuki, 2011; Jabali et al., 2012; Erdogan and Miller-Hooks, 2012; Demir et al., 2012). As noted by Ligterink et al. (2012), a truck’s empty weight is effective in the amount b as the carbon emissions generated by an empty truck, i.e., the of carbon emissions generated by that truck, thus, we define w truck’s weight. Furthermore, depending on the aforementioned characteristics of the truck, each unit loaded into the truck will result in additional emission generation. Let b e denote the emissions generated by unit loaded to the truck. Then, the retailer’s expected carbon emissions per unit time under TL transportation reads as

 

  bþ Q w b k b K v Q p knðR; stl Þ h Rlþ þ E ðQ; RÞ ¼ ðb þ : cþb e Þk þ b 2 Q Q 2

ð4Þ

We note that similar emissions functions are defined in the literature (see, e.g., Pan et al., 2012; Hoen et al., 2014a; Battini et al., 2014; Konur, 2014; Konur and Schaefer, 2014). Similar to S-(Q,R)-LTL, we next formulate the sustainable ðQ ; RÞ model when the retailer uses a TL carrier. The sustainable ðQ ; RÞ model with truckload transportation (S-(Q,R)-TL) is stated as follows:

      K þ Qv w k pknðR; stl Þ Q þ S-ðQ ; RÞ-TL : min C 2 ðQ ; RÞ ¼ ðc þ eÞk þ h R  l þ þ 2 Q Q

Q    b b Kþ v w k b p knðR; stl Þ b RlþQ þ min E2 ðQ ; RÞ ¼ ðbc þ b þ e Þk þ h Q 2 Q s:t: R P lðstl Þ Q P 0:

4. Solution analysis Two common methods used for multi-objective optimization problems are Pareto Front generation/approximation and reduction to a single-objective formulation. Pareto Front (PF) consists of non-dominated solutions (Pareto-efficient solutions) and it provides a set of alternative solutions to the decision maker. The decision maker then can select a solution from the PF to adopt. On the other hand, reduction to a single-objective formulation (for instance, by assigning weights to the

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objective functions or focusing on finding a solution to minimize the deviations from the optimum solutions of each individual objective function) results in a single solution and pre-models the decision maker’s preferences. In solving S-(Q,R)-LTL and S-(Q,R)-TL, we focus on generating a set of Pareto-efficient solutions for each model. This enables the decision maker to compare different ðQ ; RÞ policies and adopt one regarding costs and carbon emissions. Weighted approaches are one of the most common methods used to solve multi-objective optimization problems by reducing the problems of interest to single-objective models (Marler and Arora, 2004). Nevertheless, these approaches can also be used to approximate the PF under certain convexity assumptions. The constrained approach, introduced by Lin (1976), is another method that can be used to approximate the PF of multi-objective optimization problems. Particularly, this approach reformulates the multi-objective optimization problem to a single objective problem such that one of the objective functions is used as the single objective function and the other objective functions are included in the constraints with bounds on their values. Compared to the weighted approach, this approach does not require convexity assumptions; however, solving the constrained subproblems can be challenging. In a set of preliminary numerical studies conducted, we observed that the weighted approach is computationally more efficient for the problem of interest in this study. Therefore, in this section, we first propose a normalized weighted approach for approximating the PF of S-(Q,R)-LTL. Then, utilizing the analysis of S-(Q,R)-LTL, we propose a method to approximate the PF of S-(Q,R)-TL. 4.1. Pareto Front approximation for S-(Q,R)-LTL Let the PF of S-(Q,R)-LTL be denoted by PF 1 . To approximate PF 1 , we focus on generating a set of Pareto-efficient ðQ ; RÞ solutions. For normally distributed demand, PF 1 is convex as both of the objective functions are convex for R P l (see, e.g., Brooks and Lu, 1969; Hariga, 2010) and the feasible region is convex (Ehrgott, 2005). In the case of convex PFs, weighted sum approaches can be used to generate the full PF (see, e.g., Das and Dennis, 1997; Marler and Arora, 2010). Specifically, we use the normalized weighted approach to approximate PF 1 . The normalized weighted approach for multi-objective optimization models associates weights to the normalized objective functions. We use the cost of the cost minimizing solution under LTL transportation and carbon emissions of the carbon emission minimizing solution under LTL transportation to normalize C 1 ðQ ; RÞ and E1 ðQ ; RÞ, respectively. Particularly, let ðQ C1 ; RC1 Þ and ðQ E1 ; RE1 Þ denote the cost minimizing and emission minimizing ðQ ; RÞ policies under LTL transportation. Then, for a given weight x such that x 2 ½0; 1, the solution of the following optimization model is defined to be in PF 1 :

S-ðQ ; RÞ-LTLðxÞ : min M1 ðQ ; RjxÞ ¼ x

C 1 ðQ ; RÞ C

1

ðQ C1 ; RC1 Þ

þ ð1  xÞ

E1 ðQ ; RÞ EðQ E1 ; RE1 Þ

s:t: R P lðsltl Þ Q P 0: C E x C E Let ðQ x 1 ; R1 Þ be the solution of S-(Q,R)-LTLðxÞ. Given ðQ 1 ; R1 Þ and ðQ 1 ; R1 Þ, one can derive that

  e p ðxÞknðR; sltl Þ e xÞ R  l þ Q þ K ðxÞk þ e c ðxÞk þ hð M1 ðQ ; RjxÞ ¼ e ; 2 Q Q

ð5Þ

e xÞ ¼ xh=C 1 ðQ C ; RC Þ þ ð1  xÞ h=E b 1 ðQ C ; RC Þ, K e ðxÞ ¼ xK=C 1 where e c ðxÞ ¼ xðc þ tÞ=C 1 ðQ C1 ; RC1 Þ þ ð1  xÞðb c þ btÞ=E1 ðQ C1 ; RC1 Þ, hð 1 1 1 1 C C 1 C C C 1 C C 1 C 1 b =E ðQ ; R Þ, and e p ðxÞ ¼ xp=C ðQ 1 ; R1 Þ þ ð1  xÞ b p =E ðQ 1 ; R1 Þ. Note that M ðQ ; RjxÞ has a very similar ðQ 1 ; R1 Þ þ ð1  xÞ K 1 1 functional form with Eqs. (1) and (2). An efficient method to heuristically find the minimizer of the expected cost per unit time of the classical ðQ ; RÞ model, i.e., C 1 ðQ ; RÞ, is stated by Hadley and Whitin (1963). We take advantage of this method in solving S-(Q,R)-LTLðxÞ as follows. Particularly, we first find a minimizer for M 1 ðQ ; RjxÞ using the method of Hadley and Whitin (1963). This method iteratively solves the following two equations, implied by the first order conditions, until a pre-determined precision is reached between two consecutive iterations:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u e ðxÞ þ e u2k K p ðxÞnðR; sltl Þ Q ¼t e hðxÞ 1  FðRÞ ¼

e xÞ Q hð : e p ðxÞk

ð6Þ

ð7Þ

Then, if the resulting solution is feasible for S-(Q,R)-LTLðxÞ, it is accepted as the solution of S-(Q,R)-LTLðxÞ. On the other hand, the resulting solution can be infeasible if R < lðsltl Þ (note that Eq. (6) implies Q P 0). In this case, we set R ¼ lðsltl Þ and solve for Q using Eq. (6). Notice that given R, the Q minimizing M1 ðQ ; RjxÞ will be given by Eq. (6) due to convexity of M1 ðQ ; RjxÞ with respect to Q . This routine to solve S-(Q,R)-LTLðxÞ is summarized as follows:

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Routine 1. Solving S-(Q,R)-LTLðxÞ 0: Let x; C 1 ðQ C1 ; RC1 Þ and E1 ðQ E1 ; RE1 Þ be given: x 1: Determine ðQ x 1 ; R1 Þ using the iterative method of Hadley and Whitin (1963) ltl 2: If Rx < l ð s Þ 1 x ltl 3: Set Rx 1 ¼ lðs Þ and determine Q 1 using Eq. (6) x x 4: Return ðQ 1 ; R1 Þ.

Routine 1 takes C 1 ðQ C1 ; RC1 Þ and E1 ðQ E1 ; RE1 Þ values as input data. To determine ðQ C1 ; RC1 Þ, we execute Routine 1 with

x ¼ 1; C 1 ðQ C1 ; RC1 Þ ¼ 1, and E1 ðQ E1 ; RE1 Þ > 0. The resulting ðQ x1 ; Rx1 Þ is taken as the ðQ C1 ; RC1 Þ. Similarly, ðQ E1 ; RE1 Þ can be estimated by executing Routine 1 with x ¼ 0; E1 ðQ E1 ; RE1 Þ ¼ 1, and C 1 ðQ C1 ; RC1 Þ > 0. We note that Routine 1 is an heuristic approach for solving S-(Q,R)-LTLðxÞ. S-(Q,R)-LTLðxÞ is a non-linear optimization problem with inequality constraints. Interior point methods are commonly used to solve non-linear optimization problems (see, e.g., Forsgren et al., 2002). However, in a set of numerical studies conducted to analyze the efficiency of Routine 1 compared to interior point method, we observed that Routine 1 finds the same solutions or very close solutions (sometimes better) with the interior point method solutions in less computational time. Appendix A.2 gives the details of the numerical studies comparing Routine 1 to interior point method. Therefore, we use Routine 1 to solve S-(Q,R)-LTLðxÞ. C x x E x C E Notice that when x ¼ 1; ðQ x 1 ; R1 Þ ¼ ðQ 1 ; R1 Þ and when x ¼ 0; ðQ 1 ; R1 Þ ¼ ðQ 1 ; R1 Þ. This implies that both of the cost min-

imizing and carbon emission minimizing ðQ ; RÞ policies under LTL transportation are in PF 1 . Through solving S-(Q,R)-LTLðxÞ with different x values, an approximation for the PF 1 can be achieved. The following algorithm determines ‘ þ 1 number of solutions within PF 1 including ðQ C1 ; RC1 Þ and ðQ E1 ; RE1 Þ. Algorithm 1. Approximating PF 1 0: Given problem parameters and ‘, set PF 1 ¼ £: 1: Execute Routine 1 with x ¼ 1; C 1 ðQ C1 ; RC1 Þ ¼ 1, and E1 ðQ E1 ; RE1 Þ > 0 C x C 2: Set ðQ x 1 ; R1 Þ ¼ ðQ 1 ; R1 Þ 3: Execute Routine 1 with x ¼ 0; E1 ðQ C1 ; RC1 Þ ¼ 1, and C 1 ðQ E1 ; RE1 Þ > 0 E x E 4: Set ðQ x 1 ; R1 Þ ¼ ðQ 1 ; R1 Þ 5: For j ¼ 1 : ‘ þ 1 6: Execute Routine 1 with x ¼ j1 ‘ x 7: Set PF 1 :¼ PF 1 [ fðQ x 1 ; R1 Þg 8: End 9: Return PF 1 .

4.2. Pareto Front approximation for S-(Q,R)-TL Let PF of S-(Q,R)-TL be denoted by PF 2 . To approximate PF 2 , we focus on generating a set of Pareto-efficient ðQ ; RÞ solutions. To do so, we first analyze the PF of S-(Q,R)-TL given that at most a specific number of trucks can be used for inbound shipment. Then, comparing the PFs with different number of trucks available, we generate a set of ðQ ; RÞ solutions for S-(Q,R)-TL.     Particularly, let the number of trucks that can be used for inbound transportation be m, i.e., Qv 6 m. Given Qv 6 m, S-(Q,R)-TL reduces to

  Q ðK þ mwÞk pknðR; stl Þ þ S-ðQ ; RÞ-TLðmÞ : min C 2 ðQ; RjmÞ ¼ ðc þ eÞk þ h R  l þ þ 2 Q Q   b b b p knðR; stl Þ b R  l þ Q þ ð K þ m wÞk þ min E2 ðQ ; RjmÞ ¼ ðb cþb e Þk þ h 2 Q Q s:t: Q 6 mv R P lðstl Þ Q P 0:

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Let PF 2 ðmÞ denote the PF of S-(Q,R)-TL(m). Similar to PF 1 , one can observe that PF 2 ðmÞ is convex as both C 2 ðQ ; RjmÞ and E2 ðQ ; RjmÞ are convex for R P lðstl Þ and the feasible region is convex. Therefore, the normalized weighted approach can be used to approximate PF 2 ðmÞ. Now, let ðQ C2 ðmÞ; RC2 ðmÞÞ and ðQ E2 ðmÞ; RE2 ðmÞÞ be the cost and emission minimizing solutions of S-(Q,R)-TL(m), respectively. Then, for a given weight h such that h 2 ½0; 1, the solution of the following optimization model is defined to be in PF 2 ðmÞ:

S-ðQ ; RÞ-TLðm; hÞ : min M 2 ðQ ; Rjm; hÞ ¼ h

C 2 ðQ ; RjmÞ C

2

ðQ C2 ðmÞ; RC2 ðmÞjmÞ

þ ð1  hÞ

E2 ðQ ; RjmÞ E ðQ E2 ðmÞ; RE2 ðmÞjmÞ 2

s:t: Q 6 mv R P lðstl Þ Q P 0: Let ðQ h2 ðmÞ; Rh2 ðmÞÞ be the solution of S-(Q,R)-TLðm; hÞ; thus, ðQ h2 ðmÞ; Rh2 ðmÞÞ 2 PF 2 ðmÞ given h 2 ½0; 1. It should be remarked S 2 that PF 2 # 1 m¼1 PF ðmÞ. Note that S-(Q,R)-TL(m) is very similar to S-(Q,R)-LTL: the only difference is the additional upper bound constraint on the order quantity due to the fixed transportation capacity. A lower bound constraint on Q , i.e., Q > ðm  1Þv , which would force not to pay for empty trucks, is not included in S-(Q,R)-TLðm; hÞ because any ðQ h2 ðmÞ; Rh2 ðmÞÞ is not in PF 2 if Q h2 ðmÞ 6 ðm  1Þv (since both expected costs and carbon emissions will be less with the same order quantity and re-order point with fewer trucks). Notice that, given ðQ C2 ðmÞ; RC2 ðmÞÞ and ðQ E2 ðmÞ; RE2 ðmÞÞ; M 2 ðQ ; Rjm; hÞ has a very similar functional form with M 1 ðQ ; RjxÞ. Therefore, in solving S-(Q,R)-TLðm; hÞ, we first apply the iterative method of Hadley and Whitin (1963) to find a minimizer of M2 ðQ ; Rjm; hÞ. Equivalent versions of Eqs. (6) and (7) for M 2 ðQ ; Rjm; hÞ can be derived to be:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u e ðm; hÞ þ e u2kð K p ðm; hÞnðR; stl ÞÞ Q ¼t e hðm; hÞ 1  FðRÞ ¼

e hÞ Q hðm; ; e p ðm; hÞk

ð8Þ

ð9Þ

e b  hÞ=E2 ðQ E ðmÞ; RE ðmÞjmÞ, K e ðm; hÞ ¼ ðK þ mwÞh=C 2 ðQ C ðmÞ; RC ðmÞjmÞ þ ð K bþ hÞ ¼ hh=C 2 ðQ C2 ðmÞ; RC2 ðmÞjmÞ þ hð1 where hðm; 2 2 2 2 E 2 C E 2 E C 2 E b  hÞ=E ðQ 2 ðmÞ; R2 ðmÞjmÞ, and e p ðhÞ ¼ ph=C ðQ 2 ðmÞ; R2 ðmÞjmÞ þ b p ð1  hÞ=E ðQ 2 ðmÞ; R2 ðmÞjmÞ. Then, if the resulting m wÞð1 solution is feasible for S-(Q,R)-TLðm; hÞ, it is accepted as the solution of S-(Q,R)-TLðm; hÞ. On the other hand, the resulting solution can be infeasible in three cases: (i) the order quantity is smaller or greater than the truck capacity available, i.e., Q 6 ðm  1Þv or Q > mv , (ii) the safety stock constraint is not satisfied, i.e., R < lðstl Þ, and (iii) both Q 6 ðm  1Þv or Q > mv and R < lðstl Þ. In cases (i) and (iii), we set Q ¼ ðm  1Þv þ  if Q 6 ðm  1Þv and Q ¼ mv if Q > mv , and solve for R using Eq. (9). Note that given Q , the R minimizing M 2 ðQ ; Rjm; hÞ will be given by Eq. (9) due to convexity of M 2 ðQ ; Rjm; hÞ with respect to R. Finally, if the updated R value through Eq. (9) does not satisfy the safety stock constraint, we let R ¼ lðstl Þ. In case (ii), we set R ¼ lðstl Þ and solve for Q using Eq. (8) for M 2 ðQ ; Rjm; hÞ. Note that given R, the Q minimizing M2 ðQ ; Rjm; hÞ will be given by Eq. (8) due to convexity of M 2 ðQ ; Rjm; hÞ with respect to Q . Finally, if the updated Q value through Eq. (8) is below the truck capacity available we set Q ¼ ðm  1Þv þ  and if it is over the truck capacity available we set Q ¼ mv . This routine to solve S-(Q,R)-TLðm; hÞ is summarized as follows: Routine 2. Solving S-(Q,R)-TLðm; hÞ 0: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10:

Let h; C 2 ðQ C2 ðmÞ; RC2 ðmÞjmÞ, and E2 ðQ E2 ðmÞ; RE2 ðmÞjmÞ be given: Determine ðQ h2 ðmÞ; Rh2 ðmÞÞ using the iterative method of Hadley and Whitin (1963) If Q h2 ðmÞ > mv (Q h2 ðmÞ 6 ðm  1Þv ) Set Q h2 ðmÞ ¼ mv (Q h2 ðmÞ ¼ ðm  1Þv þ ) and determine Rh2 ðmÞ using Eq. (9) If Rh2 ðmÞ < lðstl Þ Set Rh2 ðmÞ ¼ lðstl Þ h If R2 ðmÞ < lðstl Þ Set Rh2 ðmÞ ¼ l and determine Q h2 ðmÞ using Eq. (8) If Q h2 ðmÞ > mv (Q h2 ðmÞ 6 ðm  1Þv ) Set Q h2 ðmÞ ¼ mv (Q h2 ðmÞ ¼ ðm  1Þv þ ) Return ðQ h2 ðmÞ; Rh2 ðmÞÞ.

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Note that, similar to Routine 1, C 2 ðQ C2 ðmÞ; RC2 ðmÞjmÞ and E2 ðQ E2 ðmÞ; RE2 ðmÞjmÞ are input for Routine 2. To determine C ðQ 2 ðmÞ; RC2 ðmÞÞ, we can execute Routine 2 with h ¼ 1; C 2 ðQ C2 ðmÞ; RC2 ðmÞjmÞ ¼ 1, and E2 ðQ E2 ðmÞ; RE2 ðmÞjmÞ > 0. The resulting ðQ h2 ðmÞ; Rh2 ðmÞÞ is taken as the ðQ C2 ðmÞ; RC2 ðmÞÞ. Similarly, ðQ E2 ðmÞ; RE2 ðmÞÞ can be estimated by executing Routine 2 with h ¼ 0; E2 ðQ E2 ðmÞ; RE2 ðmÞjmÞ ¼ 1, and C 2 ðQ C2 ðmÞ; RC2 ðmÞjmÞ > 0. Similar to Routine 1, Routine 2 is also an heuristic approach for solving S-(Q,R)-TLðm; hÞ, which is a non-linear optimization problem with inequality constraints. Therefore, we compare Routine 2 to interior point method in Appendix A.3. As Routine 2 finds the same solutions or very close solutions (sometimes better) with the interior point method solutions in less computational time, we use Routine 2 in approximating PF 2 ðmÞ. PF 2 ðmÞ can be approximated via executing Routine 2 with different weight values such that h 2 ½0; 1. Specifically, to generate ‘ þ 1 solutions in PF 2 ðmÞ, similar to Algorithm 1, Routine 2 can be run with h values increasing from 0 to 1 in increments of 1=‘. Note that both ðQ C2 ðmÞ; RC2 ðmÞÞ and ðQ E2 ðmÞ; RE2 ðmÞÞ will be in PF 2 ðmÞ. The following algorithm determines ‘ þ 1 number of solutions within PF 2 ðmÞ including ðQ C2 ðmÞ; RC2 ðmÞÞ and ðQ E2 ðmÞ; RE2 ðmÞÞ. Algorithm 2. Approximating PF 2 ðmÞ 0: Given problem parameters, m, and ‘, set PF 2 ðmÞ ¼ £: 1: Execute Routine 2 with h ¼ 1; C 2 ðQ C2 ðmÞ; RC2 ðmÞjmÞ ¼ 1, and E2 ðQ E2 ðmÞ; RE2 ðmÞjmÞ > 0 2: Set ðQ h2 ðmÞ; Rh2 ðmÞÞ ¼ ðQ C2 ðmÞ; RC2 ðmÞÞ 3: Execute Routine 2 with h ¼ 0; C 2 ðQ C2 ðmÞ; RC2 ðmÞjmÞ > 0, and E2 ðQ E2 ðmÞ; RE2 ðmÞjmÞ ¼ 1 4: Set ðQ h2 ðmÞ; Rh2 ðmÞÞ ¼ ðQ E2 ðmÞ; RE2 ðmÞÞ 5: For j ¼ 1 : ‘ þ 1 6: Execute Routine 2 with h ¼ j1 ‘ 7: Set PF 2 ðmÞ :¼ PF 2 ðmÞ [ fðQ h2 ðmÞ; Rh2 ðmÞÞg 8: End 9: Return PF 2 ðmÞ.

Ideally, our purpose is to approximate PF of S-(Q,R)-TL, PF 2 . In generating PF 2 , starting from one truck, we evaluate whether adding more trucks can result in new Pareto-efficient ðQ ; RÞ solutions. Particularly, to determine whether addition of a new truck generates new Pareto-efficient solutions, we use the dominance relation between two sets of solutions as follows. Let PFðVÞ be the set of Pareto-efficient solutions within a given set of solutions V. Definition 1. Suppose that PFðV a Þ – PFðV b Þ. PFðV a Þ dominates PFðV b Þ if PFðV a Þ ¼ PFðV a

S

V b Þ.

The dominance relation between two PFs is represented as PFðV a Þ  PFðV b Þ as we are considering minimization of both objective functions. Given a set of solutions V, one can determine PFðVÞ using the following routine. Routine 3. Finding the Pareto-efficient solutions in a set V 0: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:

Let C i and Ei denote the cost and emissions of the ith member of V. Set i ¼ 1: While i 6 jVj  1 Set j ¼ i þ 1 While j 6 jVj If C i < C j and Ei < Ej Set V :¼ V  fðC j ; Ej Þg and j ¼ j  1 Else, if C i > C j and Ei > Ej Set V :¼ V  fðC i ; Ei Þg and j ¼ jVj and i ¼ i  1 Set j ¼ j þ 1 End Set i ¼ i þ 1 End Return PFðVÞ ¼ V.

In approximating PF 2 , if using any number of additional trucks less than or equal to L does not generate new Pareto-efficient solutions, we stop considering additional trucks. The following algorithm describes the steps of approximating PF 2 .

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Algorithm 3. Approximating PF 2

0: 1: 2: 3: 4: 5: 6: 7: 8:

Given problem parameters, L, and ‘, set S ¼ £; PF 2 ¼ £; counter ¼ 0, and m ¼ 1: While counter 6 L S Run Algorithm 2 and let S :¼ S PF 2 ðmÞ Run Routine 3 and get PFðSÞ If PF 2 ¼ PFðSÞ, set counter ¼ counter þ 1 Else set PF 2 ¼ PFðSÞ and counter ¼ 0 Set m ¼ m þ 1 End Return PF 2 .

Algorithm 3, at an intermediate iteration k with counter ¼ 0, generates a set of Pareto efficient ðQ ; RÞ solutions, which S

S

k kþL 2 2 require less than or equal to k trucks. If PF m¼1 PF ðmÞ  PF m¼k PF ðmÞ , it means that considering using 1; 2; . . . ; L additional trucks for inbound transportation does not generate new Pareto efficient ðQ ; RÞ solutions; hence, the algorithm terminates. In the next section, we present a set of numerical studies.

5. Numerical studies In this section, we conduct a set of numerical studies to provide insights on the two models discussed. Particularly, we focus on three sets of numerical analyses. In analyses (i), we investigate the effects of the demand variability and lead time duration on the expected costs and carbon emissions because it is possible to invest in improving demand variability and lead time duration. Specifically, we discuss that reducing demand variability and lead time duration can be green investment. In analyses (ii), we investigate how sustainable different policies are. In particular, we investigate the changes in costs and carbon emissions when the retailer adopts a sustainable policy instead of the traditional cost-minimizing policy. We discuss that the increase in costs is relatively less than the decrease in emissions when a sustainable policy is adopted. Finally, in analysis (iii), we illustrate how the tools provided in this study can be used compare of different carriers. We discuss that the retailer’s economic and environmental targets affect the retailer’s carrier selection. For instance, we discuss that a carrier with good economic performance can perform poorly in terms of environmental performance or vice versa. The routines and the algorithms discussed in Section 4 are coded in Matlab 2014 and all problem instances are solved using a personal computer with dual 3 GHz processer and 16 GB RAM. Throughout the numerical analysis, we assume that the demand per unit time is normally distributed with mean k and standard deviation #. For analyses (i)–(ii), the details of the design of the problem instance generation and the definitions of the parameter values are explained in Appendix A.4. The tables discussed in Sections 5.1 and 5.2 are described and presented in Appendix A.5 and A.6, respectively. 5.1. Effects of demand variability and lead time This set of numerical studies focuses on demonstrating the changes in the retailer’s expected costs and carbon emissions per unit time with both LTL and TL transportation as the lead time demand variability and the lead time duration change. In particular, for each transportation mode, we compare the changes observed in the cost-minimizing ðQ ; RÞ policy, the emission-minimizing ðQ ; RÞ policy, and the average of the ðQ ; RÞ policies in the PF. The average of the ðQ ; RÞ policies in a PF is calculated by taking the average of the expected costs and carbon emissions per unit time over all Pareto-efficient ðQ ; RÞ policies that lay on the approximated PF. Prior to the numerical investigation, it should be remarked that while it has been discussed in the literature that the expected costs per unit time (the expected carbon emissions per unit time) of the cost-minimizing policy (the emission-minimizing policy) will increase as the lead time duration or the lead time demand variance increases, it is possible that the expected carbon emissions per unit time (the expected costs per unit time) of the cost-minimizing policy (the emission-minimizing policy) might increase or decrease with an increase in the lead time duration and the lead time demand variance. For investigating the effects of the demand variability, we randomly generate 50 problem instances for each of the 10 different values of the lead time demand standard deviation, r, starting from 10 increasing up to 100 in increments of 10. Similarly, for investigating the effects of the lead time duration, we randomly generate 50 problem instances for each of the 10 different values of the lead time duration, s, starting from 0.1 increasing up to 1 in increments of 0.1 such that sltl ¼ s and stl ¼ s. Tables 4 and 5 summarize the average changes (average over 50 problem instances for each r and s value) in the expected costs and carbon emissions per unit time for the cost-minimizing ðQ ; RÞ policy, the emission-minimizing ðQ ; RÞ policy, and the average of the ðQ ; RÞ policies in the PF, for both LTL and TL transportation.

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Figs. 1 and 2, constructed from Table 4, illustrate the changes in the expected costs and carbon emissions per unit time as

r increases under LTL and TL transportation, respectively. We have the following observation from Figs. 1 and 2. O.1 As the standard deviation of the lead time demand increases, both the expected costs and carbon emissions per unit time under the cost-minimizing, emission-minimizing, and the average PF ðQ ; RÞ policies tend to increase with LTL and TL transportation. Results similar to observation O.1 are discussed through simulation of real-life cases by Daccarett-Garcia (2009) and Arikan et al. (2014). We note that these results are not surprising as the cost and emission functions have similar forms and it has been discussed in the literature that the expected costs increase with increasing demand variability. Nevertheless, observation O.1 has important implications about green technology investment. Particularly, in recent studies, inventory control and transportation models have been analyzed with green technology investment decisions (see, e.g., Bae et al., 2011; Swami and Shah, 2013; Toptal et al., 2014). Observation O.1 then suggests that an investment in demand variance reduction is actually a green investment. Furthermore, if the demand variance can be reduced with an investment such that the cost of the investment is compensated by the decrease in the expected costs, it is possible to reduce carbon emissions without additional costs. Figs. 3 and 4, constructed from Table 5, illustrate the changes in the expected costs and carbon emissions per unit time as s increases under LTL and TL transportation, respectively. We have the following observation from Figs. 3 and 4. O.2 As the standard deviation of the lead time demand increases, both the expected costs and carbon emissions per unit time under the cost-minimizing, emission-minimizing, and the average PF ðQ ; RÞ policies tend to increase with LTL and TL transportation. Similar to observation O.1, observation O.2 also suggests that if an investment for reducing lead time is compensated by the decrease in costs, it will provide additional benefits by reducing the expected carbon emissions. We note that, specially for TL transportation, the lead time can be changed by controlling the speed of trucks; however, the speed of the truck affects the emission generation rate (b e in our setting). The study by Jabali et al. (2012), for instance, analyzes a green vehicle routing problem, where the truck speed is a decision variable and the emission generation rate of a truck is a function of the speed (furthermore, this function is a convex function with decreasing and increasing sections). Depending on the range of the speed, an increase in speed (or decrease in lead time) can increase or decrease the emission generation rate. We believe that the current observations and the models presented in this study will be helpful in analyzing sustainable stochastic demand inventory systems with controllable lead time (where the delivery speed is a decision variable). We pose this problem as a future research direction in Section 6. 5.2. Sustainability analysis This set of numerical studies focuses on demonstrating the changes in the retailer’s expected costs and carbon emissions per unit time when the retailer adopts a ðQ ; RÞ policy from the set of policies in the PF. Recall that the cost-minimizing and the emission-minimizing policies are in the PF both with LTL and TL transportation. We refer to the any ðQ ; RÞ policy in PF 1

Expected Costs Per Unit Time

(a) σ vs. Expected Costs 9300 9200

Cost−Minimizing (Q,R) Emission−Minimizing (Q,R) Average−PF (Q,R)

9100 9000 8900 8800 8700 8600 10

20

30

40

50

60

70

80

90

Standard Deviation of Lead Time Demand (σ)

100

Expected Emissions Per Unit Time

and PF 2 other than the cost-minimizing policy, denoted by ðQ C ; RC Þ, as a sustainable ðQ ; RÞ policy, denoted by ðQ S ; RS Þ. In the following analysis, we specifically focus on the percent changes in the expected costs and carbon emissions per unit time due to preferring a sustainable policy instead of the cost-minimizing policy. First, we define the following three measures:

(b) σ vs. Expected Emissions 9200 9000

Cost−Minimizing (Q,R) Emission−Minimizing (Q,R) Average−PF (Q,R)

8800 8600 8400 8200 10

20

30

40

50

60

70

80

90

Standard Deviation of Lead Time Demand (σ)

Fig. 1. r vs. costs and emissions under different policies with LTL transportation.

100

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(a) σ vs. Expected Costs Expected Costs Per Unit Time

9800 Cost−Minimizing (Q,R) Emission−Minimizing (Q,R) Average−PF (Q,R)

9700 9600 9500 9400 9300 9200 9100 10

20

30

40

50

60

70

80

90

100

Expected Emissions Per Unit Time

154

(b) σ vs. Expected Emissions 9600 9400

Cost−Minimizing (Q,R) Emission−Minimizing (Q,R) Average−PF (Q,R)

9200 9000 8800 8600 8400 10

Standard Deviation of Lead Time Demand (σ)

20

30

40

50

60

70

80

90

100

Standard Deviation of Lead Time Demand (σ)

(a) τ vs. Expected Costs Expected Costs Per Unit Time

9600 Cost−Minimizing (Q,R) Emission−Minimizing (Q,R) Average−PF (Q,R)

9500 9400 9300 9200 9100 9000 8900 8800 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Expected Emissions Per Unit Time

Fig. 2. r vs. costs and carbon emissions under different policies with TL transportation.

(b) τ vs. Expected Emissions 10000 9750

Cost−Minimizing (Q,R) Emission−Minimizing (Q,R) Average−PF (Q,R)

9500 9250 9000 8750 8500 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Lead Time (τ ltl )

ltl

Lead Time (τ )

Expected Costs Per Unit Time

(a) τ vs. Expected Costs 10100 10000

Cost−Minimizing (Q,R) Emission−Minimizing (Q,R) Average−PF (Q,R)

9900 9800 9700 9600 9500 9400 9300 0.1

0.2

0.3

0.4

0.5

0.6

Lead Time (τ tl)

0.7

0.8

0.9

1

Expected Emissions Per Unit Time

Fig. 3. s vs. costs and emissions under different policies with LTL transportation.

(b) τ vs. Expected Emissions 10200 10000

Cost−Minimizing (Q,R) Emission−Minimizing (Q,R) Average−PF (Q,R)

9800 9600 9400 9200 9000 8800 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Lead Time ( τ tl)

Fig. 4. s vs. costs and carbon emissions under different policies with TL transportation.

DC i ¼ DEi ¼

C i ðQ S ; RS Þ  C i ðQ C ; RC Þ C i ðQ C ; RC Þ Ei ðQ S ; RS Þ  Ei ðQ C ; RC Þ

CoRi ¼

Ei ðQ C ; RC Þ

 100%;

ð10Þ

 100%;

ð11Þ

C i ðQ S ; RS Þ  C i ðQ C ; RC Þ Ei ðQ C ; RC Þ  Ei ðQ S ; RS Þ

;

ð12Þ

DC i ; DEi , and CoRi define the percent change in the expected costs per unit time, the percent change in the expected carbon emissions per unit time, and the cost of emission reduction, respectively, due to preferring a sustainable ðQ ; RÞ policy from PF i over the cost-minimizing ðQ ; RÞ policy such that i ¼ 1 defines LTL transportation and i ¼ 2 defines TL transportation. We note that, similar to Chen et al. (2013) and Toptal et al. (2014), the cost of emission reduction is defined as the increase in the

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expected costs per unit decrease in the expected carbon emissions. Recall that each problem instance solved has a set of solucC i ; D d i as the average of cE i , and CoR tions, i.e., a set of sustainable ðQ ; RÞ policies. For a single problem instance, we define D

DC i ; DEi , and CoRi values over all sustainable ðQ ; RÞ policies in the PF returned for the problem instance. cC i ; D d i values. After that, we calculate cE i , and CoR In Tables 6 and 7, for each problem instance solved, we first calculate D cC i ; D d i values over all the 50 problem instances solved for each r and s value, respectively. cE i , and CoR the averages of the D Note that a positive (negative) value for percent changes in Tables 6 and 7 indicate an increase (decrease). Fig. 5 illustrates the percent changes in the expected costs and carbon emissions per unit time with LTL and TL transportation as the standard deviation of the lead time demand and the lead time duration change. The following observations are based on the average results in Tables 6 and 7. O.3 When the retailer prefers a ðQ S ; RS Þ policy over ðQ C ; RC Þ, the percent increase in the expected costs per unit time is less than the percent decrease in the expected carbon emissions per unit time both with LTL and TL transportation. We note that similar results to observation O.3 are given in Chen et al. (2013). That is, the cost increase percentage is lower than the emission reduction percentage when the retailer adopts a sustainable strategy both with LTL and TL transportation. O.4 As the standard deviation of the lead time demand or the lead time duration increases, the percent increase in the expected costs per unit time, the percent decrease in the carbon emissions per unit time, and the average cost of emissions reduction all increase under LTL transportation. Observation O.4 concludes that the cost increase percentage and the emission reduction percentage due to adopting a sustainable strategy will both increase with an increase in the demand variance or the lead time duration when LTL carrier is used. That is, it will be more costly but also more environmentally friendly to adopt a sustainable strategy in scenarios with high demand variances or long lead time durations with LTL transportation. As both the cost increase and emission reduction incline with demand variance and lead time duration, one cannot say that unit emission reduction will be cheaper or more expensive or unit investment will abate more or less emissions in scenarios with high demand variance and long delivery lead times. However, observation O.2 further implies that the cost of emission reduction with LTL transportation increases with an increase in the demand variance or the lead time duration. That is, it will cost more to achieve unit emission reduction (or less emission reduction will be achieved with unit cost) with LTL transportation in scenarios with high demand variances or long lead time durations. Therefore, as a managerial insight, it can be noted that a retailer using LTL carrier is less likely to adopt a sustainable strategy in the settings with high demand variances or long lead time durations. O.5 As the standard deviation of the lead time demand or the lead time duration increases, the percent increase in the expected costs per unit time, the percent decrease in the carbon emissions per unit time, and the average cost of emissions reduction do not follow a strictly increasing or decreasing pattern under TL transportation. Observation O.5 concludes that the cost increase percentage and the emission reduction percentage due to adopting a sustainable strategy can increase or decrease with an increase in the demand variance or the lead time duration when TL carrier is used. Furthermore, unlike LTL transportation, the cost of emission reduction with TL transportation is not strictly increasing or decreasing as the demand variance or the lead time duration increases. These imply that the benefits of adopting a sustainable strategy with TL transportation is not positively or negatively affected by the demand variance and the lead time duration. Therefore, unlike LTL transportation, it is still possible that a retailer using TL transportation might or might not prefer adopting a sustainable strategy in the settings with high demand variances or long lead time durations. This

−1.6

1.2

−1.8

1.1

−2

1

−2.2

0.9

−2.4 1

ΔC

0.8 10

20

30

40

50

60

2

ΔC

70

1

2

ΔE

80

ΔE

90

−2.6 100

Standard Deviation of Lead Time Demand (σ)

−1.9

1.3

1

2

ΔE

1.25

ΔE

1

ΔC

2

ΔC

−2

1.2

−2.1

1.15

−2.2

1.1

−2.3

1.05

−2.4

1

−2.5

0.95

−2.6

0.9 0.1

−2.7 0.2

0.3

0.4

0.5

0.6

0.7

Lead Time (τ)

Fig. 5. Percent changes in costs and emissions as r and s change.

0.8

0.9

1

Percent Change in Expected Emissions (%)

1.3

Percent Change in Expected Costs (%)

 i and τ vs. ΔE i (b) τ vs. ΔC Percent Changes in Expected Emissions (%)

Percent Change in Expected Costs (%)

 i and σ vs. ΔE i (a) σ vs. ΔC

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basically follows from the fact that TL transportation costs and emissions are not strictly increasing with the retailer’s order quantity. When TL transportation is used, the benefits of adopting a sustainable policy heavily depends on the characteristics of the freight trucks used. 5.3. Transportation mode comparison In this set of numerical studies, we present examples on how the models and solution methods proposed in this paper can be used by a retailer for comparing two LTL carriers, a LTL carrier and a TL carrier, and two TL carriers, not only in terms of economic, but also environmental aspects. Prior to discussing examples, in the case of a single carrier option, the retailer can adopt a ðQ ; RÞ policy from the PF depending on his/her economic and environmental goals. Furthermore, in case of two carriers of any type, i.e., two LTL or two TL or one LTL and one TL carriers, the retailer can approximate a PF with each carrier and compare the PFs. If one of the PFs dominates the other PF, the carrier with the dominating PF would be preferred as it enables ðQ ; RÞ policies with lower expected costs and as well as lower expected carbon emissions per unit time. In the case there is no dominance relation between the two PFs, as in the cases for the following examples, the preference will depend on the retailer’s economic and environmental goals. The settings of the following three examples are given in Appendix A.7. For Examples 1–3, the points where two different PFs intersect (see, e.g., Fig. 6) is estimated by assuming a linear line between the two points of each PF, where these two points are the first points that locally dominate the points of the other PF. Example 1. Consider that Retailer R is planning to adopt a ðQ ; RÞ policy for a single product. Suppose that there are two LTL carriers available for his/her inbound shipment: LTL Carrier A and LTL Carrier B. LTL Carriers A and B have different per unit transportation costs, per unit emission generation rates, and delivery lead times (see Table 9). Fig. 6a shows Retailer R’s sets of Pareto-efficient ðQ ; RÞ policies when he/she contracts with LTL Carrier A and LTL Carrier B for his/her inbound transportation. The intersection point of the two PFs is when the expected costs per unit time amount to 8845 and the expected carbon emissions per unit time amount to 8885 per unit time.  If Retailer R does not have environmental considerations (i.e., he/she only wants to minimize the expected costs per unit time), he/she would prefer to contract with LTL Carrier B as the cost-minimizing policy with LTL Carrier B has lower expected costs per unit time compared to the cost-minimizing policy with LTL Carrier A.  If Retailer R does not have economic considerations (i.e., he/she only wants to minimize the expected carbon emissions per unit time), the retailer would prefer to contract with LTL Carrier A as the emission-minimizing policy with LTL Carrier A has lower expected carbon emissions per unit time compared to the emission-minimizing policy with LTL Carrier B.  If Retailer R does have both economic and environmental considerations, depending on the level of his/her economic and environmental goals, he/she can prefer LTL Carrier A or LTL Carrier B: – If Retailer R wants to reduce costs subject to environmental considerations, his/her preferences are defined as follows. If Retailer R targets his/her expected carbon emissions per unit time to be less than 8885, he/she would prefer LTL Carrier A because LTL Carrier A results in lower expected costs per unit time at any expected carbon emissions per unit time lower than 8885. If Retailer R, on the other hand, can have his/her expected carbon emissions per unit time greater than 8885, he/she would prefer LTL Carrier B because LTL Carrier B results in lower expected costs per unit time at any expected carbon emissions per unit time greater than 8885. – If Retailer R wants to reduce carbon emissions subject to economic considerations, his/her preferences are defined as follows. If Retailer R targets his/her expected costs per unit time to be less than 8845, he/she would prefer LTL Carrier B because LTL Carrier B results in lower expected carbon emissions per unit time at any expected costs per unit time lower than 8845. If Retailer R, on the other hand, can have his/her expected costs per unit time greater than 8845, he/she would prefer LTL Carrier A because LTL Carrier A results in lower expected carbon emissions per unit time at any expected costs per unit time greater than 8845. Example 2. Consider that Retailer R is planning to adopt a ðQ ; RÞ policy for a single product. Suppose that there are two carriers available for his/her inbound shipment: LTL Carrier A and TL Carrier A. Fig. 6b shows Retailer R’s sets of Pareto-efficient ðQ ; RÞ policies when he/she contracts with LTL Carrier A and TL Carrier A for his/her inbound transportation. The first intersection point of the two PFs (point 1 in Fig. 6b) is when the expected costs and the expected carbon emissions per unit time amount to 8836 and 8909, respectively. The second intersection point of the two PFs (point 2 in Fig. 6b) is when the expected costs and the expected carbon emissions per unit time amount to 8880 and 8849, respectively. Similar to Example 1, Retailer R’s preference of carrier will depend on his/her economic and environmental considerations. It can be observed from Fig. 6b that if Retailer R wants to reduce his/her expected costs per unit time, he/she would prefer TL Carrier A when his/her expected carbon emissions per unit time can be greater than 8909 or has to be less than 8849 as TL Carrier A results in lower expected costs per unit time for expected carbon emissions per unit time greater than 8909 or lower than 8849. On the other hand, when the expected carbon emissions per unit time can be greater than 8849 and has to be less than 8909, Retailer R would prefer LTL Carrier A as LTL Carrier A results in lower expected costs per unit time when the expected carbon emissions per unit time is between 8849 and 8909. It can be further observed from Fig. 6b

Expected Emissions Per Unit Time

B. Schaefer, D. Konur / Transportation Research Part E 80 (2015) 142–165

157

(a) LTL Carrier vs. LTL Carrier 8980

LTL Carrier A LTL Carrier B

8960 8940 8920 8900

8845, 8885 8880 8860 8840 8800

8820

8840

8860

8880

8900

8920

Expected Emissions Per Unit Time

Expected Costs Per Unit Time

(b) LTL Carrier vs. TL Carrier 9000

LTL Carrier A TL Carrier A 8950

1) 8836,8909 8900

2) 8880,8849 8850

8800 8800

8820

8840

8860

8880

8900

8920

Expected Emissions Per Unit Time

Expected Costs Per Unit Time

(c) TL Carrier vs. TL Carrier 9000

TL Carrier A TL Carrier B 8950

1) 8849, 8899

8900

2) 8873, 8858 3) 8890, 8841

8850 8800 8750 8800

8820

8840

8860

8880

8900

8920

8940

Expected Costs Per Unit Time Fig. 6. Comparison of different carriers.

that if Retailer R wants to reduce his/her expected carbon emissions per unit time, he/she would prefer TL Carrier A when his/her expected costs per unit time has to be lower than 8836 or can be greater than 8880 as TL Carrier A results in lower expected carbon emissions per unit time for expected costs per unit time lower than 8836 or greater than 8880. On the other hand, when the expected costs per unit time should be less than 8880 but can be greater than 8836, Retailer R would prefer LTL Carrier A as LTL Carrier A results in lower expected carbon emissions per unit time when the expected costs per unit time is between 8836 and 8880. Example 3. Consider that Retailer R is planning to adopt a ðQ ; RÞ policy for a single product. Suppose that there are two TL carriers available for his/her inbound shipment: TL Carrier A and TL Carrier B. TL Carriers A and B have different per unit transportation costs, per unit emission generation rates, truck charges and emissions, and delivery lead times (see Table 10). Fig. 6c shows Retailer R’s sets of Pareto-efficient ðQ ; RÞ policies when he/she contracts with TL Carrier A and TL Carrier B for his/her inbound transportation. The three intersection points of the two PFs are when the expected costs per unit time and expected carbon emissions per unit time amount to 8849 and 8899, 8873 and 8858, and 8890 and 8841, respectively, for points 1, 2, and 3. Similar to Examples 1 and 2, Retailer R’s preference of carrier will depend on his/her economic and environmental considerations. It can be observed from Fig. 6c that if Retailer R’s objective is to minimize the expected costs per unit time, he/she would prefer TL Carrier A when the expected carbon emissions can be greater 8899 or has to be between 8841

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Table 2 Routine 1 vs. interior point method.

x

Routine 1

IPM

Qx 1

Rx 1

x M 1 ðQ x 1 ; R1 jxÞ

cpu

Qx 1

Rx 1

x M1 ðQ x 1 ; R1 jxÞ

cpu

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

260.8 269.0 277.5 286.2 295.4 305.1 315.6 326.9 339.5 353.8 370.8

691.2 689.6 688.1 686.4 684.6 682.7 680.5 678.1 675.2 671.6 666.8

1.0000 1.0021 1.0037 1.0050 1.0059 1.0063 1.0062 1.0057 1.0046 1.0028 1.0000

0.0006 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0006 0.0006

260.9 269.1 277.5 286.2 295.5 305.2 315.6 327.0 339.5 353.8 370.8

691.2 689.6 688.1 686.4 684.6 682.7 680.5 678.1 675.2 671.6 666.8

1.0000 1.0021 1.0037 1.0050 1.0059 1.0063 1.0062 1.0057 1.0046 1.0028 1.0000

0.0719 0.0718 0.0721 0.0723 0.0728 0.0721 0.0727 0.0724 0.0732 0.0729 0.0723

Avg

309.2

681.3

1.0038

0.0005

309.2

681.3

1.0038

0.0724

and 8858. On the other hand, Retailer R would prefer TL Carrier B when the expected carbon emissions per unit time has to be less than 8841 or has to be between 8858 and 8899. It can be further observed from Fig. 6c that if Retailer R’s objective is to minimize the expected carbon emissions per unit time, he/she would prefer TL Carrier A when the expected costs per unit time has to less than 8849 or can be between 8873 and 8890. On the other hand, Retailer R would prefer TL Carrier B when the expected cots per unit time can be greater than 8890 or between 8849 and 8873. We note that while Examples 1–3 compare two carriers, similar analyses can be made when the retailer has more than two carrier options. Furthermore, we note that similar analyses can be used when the retailer is subject to carbon cap constraints. The targeted carbon emissions level per unit time can be considered as the carbon cap regulated by governmental agencies. Nevertheless, as noted by Benjaafar et al. (2012) and Chen et al. (2013), companies not only have carbon caps because of governmental regulations but also because of the green goals they set.

6. Conclusions and future research In this study, we integrate sustainability into continuous review inventory control systems by formulating a bi-objective ðQ ; RÞ model with expected costs and expected carbon emissions minimization. We analyze the sustainable ðQ ; RÞ model with two different transportation modes: less-than-truckload (LTL) and truckload (TL) transportation. For each case, a method is proposed to approximate the Pareto Frontier by determining a set of Pareto Efficient ðQ ; RÞ policies. Particularly, for the sustainable ðQ ; RÞ model with LTL transportation, we develop a normalized weighted approach based method for approximating the Pareto Frontier. For the sustainable ðQ ; RÞ model with TL transportation, utilizing the method developed for the LTL transportation, we develop a method that compares Pareto Frontiers with given transportation capacities and, then, generates a set of Pareto efficient policies with different numbers of trucks used. Defining a sustainable ðQ ; RÞ model also enables analyses of the effects of demand variance and lead time duration on the expected costs as well as expected carbon emissions. It is observed that both expected costs and carbon emissions tend to increase as demand variance and lead time duration increase with both LTL and TL transportation. The managerial insight of these observations is that an investment opportunity to reduce the demand variance or the lead time duration can be a free or a low-cost green action if the investment spending is fully or partially compensated by the reduction in the expected costs because the expected carbon emissions might also be reduced with the reduction of the demand variance of the lead time duration. Through a set of numerical analyses, we further discuss that adopting a sustainable ðQ ; RÞ policy instead of a cost minimizing ðQ ; RÞ policy for a continuous review inventory control system with LTL or TL transportation can reduce carbon emissions without significant cost increases. These observations generalize the results of Chen et al. (2013) for deterministic inventory control to stochastic continuous review inventory control with both LTL and TL transportation. Finally, we illustrate how the methods proposed in this study can be used by a retailer to select a carrier from a set of available carriers considering not only economic but also environmental goals. An immediate future research direction would be to analyze continuous review inventory control systems under carbon emission regulation policies. A ðQ ; RÞ model with carbon taxing, carbon trading, carbon cap, and carbon offset policies can be studied. We believe that the emissions function defined in this study will be utilized in such future research studies. Furthermore, analyses of integrated investment decisions on lead time/demand variance reduction and inventory control decisions is an interesting research area. Especially, as mentioned in Section 5, the sustainable ðQ ; RÞ model under TL transportation with lead time flexibility due to controllable truck speed is an important and practical future research direction. The sustainable ðQ ; RÞ model with TL transportation can also be studied by considering availability of different truck types

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for the inbound shipments. Finally, the sustainable continuous review inventory control model can be extended to multi-item and/or multi-echelon supply chain systems. Acknowledgements The authors thank the editor and three anonymous reviewers for their helpful comments and suggestions that have helped improve the manuscript. This study has been partially supported by the University of Missouri Inter-Disciplinary Inter-Campus research program and the U.S. Department of Transportationthrough the National University Transportation Center at Missouri University of Science and Technology under Project # 00043159. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsors. Appendix A A.1. Notation and possible metrics

Notation

Description

Metric

Demand parameters k: #: s: DðsÞ: f ðDðsÞÞ: FðDðsÞÞ: lðsÞ: rðsÞ:

expected demand rate standard deviation of demand rate lead time duration random variable defining lead time demand probability density function of DðsÞ cumulative distribution function of DðsÞ expected lead time demand standard deviation of lead time demand

units/year units year units

units units

Retailer’s cost parameters c: Per unit procurement cost K: Fixed order setup cost h: Inventory holding cost per unit per unit time p: Unit backorder cost

$/unit $/order $/unit/year $/unit

Retailer’s emission parameters b c: Emissions b: Emissions K b h: Emissions b p: Emissions

CO2 CO2 CO2 CO2

due due due due

to to to to

per unit procurement order placement inventory holding per unit per unit time per unit backorder

lbs/unit lbs/order lbs/unit/year lbs/unit

Retailer’s decision variables Q: Retailer’s order quantity per order R: Retailer’s re-order quantity

units units

LTL transportation parameters t: Transportation cost per unit by LTL carrier bt: Emissions due to per unit transportation with LTL carrier sltl : LTL carrier’s delivery lead time

$/unit CO2 lbs/unit year

TL transportation parameters e: Transportation cost per unit by TL carrier w: Transportation cost per truck by TL carrier v: Transportation capacity per truck by TL carrier b w: Emissions due to per empty truck transportation with TL carrier b e: Emissions due to per unit transportation with TL carrier stl : TL carrier’s delivery lead time

$/unit $/truck units/truck CO2 lbs/truck CO2 lbs/unit year

We note that emissions are given in terms of carbon emissions as other greenhouse gas emissions can be measured in terms of equivalent CO2 emissions (see, e.g., EPA, 2013).

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A.2. Comparison of Routine 1 to interior point method Recall that Routine 1 is proposed to solve S-(Q,R)-LTLðxÞ, which also can be solved by the interior point method (IPM). To compare Routine 1 to IPM, we consider 11 x values, starting from 0 increasing up to 1 in increments of 0.1 and for each x value, we solve 50 problem instances using the design explained in Appendix A.4. In each problem instance, ðQ C ; RC Þ and x 1 ðQ E ; RE Þ are determined using IPM. Table 2 demonstrates the average Q x 1 ; R1 ; M ðQ ; RjxÞ, and cpu time in seconds over all 50 problem instances solved with Routine 1 and IPM, respectively, for each x value. As it can be seen from Table 2, Routine 1 and IPM find the same or very close solutions on average and Routine 1 is more efficient in terms of computational times. Therefore, we use Routine 1 to solve S-(Q,R)-LTLðxÞ.

A.3. Comparison of Routine 2 to interior point method Recall that Routine 2 is proposed to solve S-(Q,R)-TLðm; hÞ and the IPM is another method available to solve S-(Q,R)-TLðm; hÞ. To compare Routine 2 to IPM, we consider problem instances for different m values increasing from 1 to 10 in increments of 1. For each m value, we generate 50 problem instances using the design explained in Appendix A.4 and each problem is then solved with 11 different h values starting from 0 increasing up to 1 in increments of 0.1. That is, for each m value, we solve 550 different problem instances with Routine 2 and IPM. In each problem instance with any m value, ðQ C2 ðmÞ; RC2 ðmÞÞ and ðQ E2 ðmÞ; RE2 ðmÞÞ are determined using IPM. Table 3 demonstrates the average Q h2 ðmÞ; Rh2 ðmÞ; M 2 ðQ ; Rjm; hÞ, and cpu time in seconds over all 550 problem instances solved with Routine 2 and IPM for each m. It can be observed from Table 3 that Routine 2 and IPM find the same or very close solutions on average and Routine 2 is more efficient in terms of computational times. Therefore, we use Routine 2 to solve S-(Q,R)-TLðm; hÞ. A.4. Design details for the numerical studies of Sections 5.1 and 5.2 In all of the problem instances solved, we assume that k ¼ 2000 units and # ¼ 100. Note that in analyses (i) and (ii), the standard deviation of the lead time demand and lead time duration will vary; thus, different demand characteristics will be captured. Furthermore, we assume that the retailer orders from a single supplier; hence, his/her procurement costs and procurement emissions are fixed per unit time. This further suggests that they are not effective in decision making. Therefore, we simply assume that c ¼ b c ¼ 1. c ), we define a lower and an upper bound for each In generating the retailer’s cost and emission parameters (except c and b parameter. The parameter value in a problem instance is then determined by randomly generating a value from a uniform distribution defined within the lower and upper bounds of the parameter. U½a; b denotes a uniform distribution with bounds a and b.  The uniform distribution for each cost parameter of the retailer is designed as follows: h  U½12; 8; K  U½50; 250, and p  U½2; 8. We note that similar values are assumed in many inventory control studies as well as in the numerical analysis of the studies focusing on inventory control models with carbon emission considerations (see, e.g., Benjaafar et al., 2012; Hua et al., 2011; Chen et al., 2013; Toptal et al., 2014; Konur, 2014; Konur and Schaefer, 2014).  The uniform distribution for each emission parameter of the retailer is designed as follows: Following the similar values in related literature (see, e.g., Benjaafar et al., 2012; Hua et al., 2011; Chen et al., 2013; Arikan et al., 2014; Konur, 2014; b  U½50; 150. In defining the range for b Konur and Schaefer, 2014), we let b h  U½5; 10 and K p , we assume that it is defined similar to the relation between h and p; therefore, we assume that b p  U½5; 10.

Table 3 Routine 2 vs. interior point method. m

Routine 2

IPM

Q h2 ðmÞ

Rh2 ðmÞ

M 2 ðQ h2 ðmÞ; Rh2 ðmÞÞ

cpu

Q h2 ðmÞ

Rh2 ðmÞ

M2 ðQ h2 ðmÞ; Rh2 ðmÞÞ

cpu

1 2 3 4 5 6 7 8 9 10

132.2 264.4 395.3 518.3 632.6 735.9 830.5 917.3 995.8 1067.2

657.9 638.9 626.1 616.3 608.4 601.9 596.4 592.0 588.7 585.9

1.4128 1.1938 1.1102 1.0659 1.0391 1.0214 1.0090 1.0001 0.9934 0.9882

0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005

132.2 264.4 395.3 518.3 632.6 735.9 830.4 916.9 994.6 1064.9

657.9 638.9 626.1 616.3 608.4 601.9 596.4 592.0 588.7 586.0

1.4128 1.1938 1.1102 1.0659 1.0391 1.0214 1.0090 1.0001 0.9934 0.9882

0.0469 0.0490 0.0498 0.0505 0.0522 0.0539 0.0549 0.0558 0.0571 0.0584

Avg

649.0

611.2

1.0834

0.0005

648.5

611.3

1.0834

0.0528

161

B. Schaefer, D. Konur / Transportation Research Part E 80 (2015) 142–165 Table 4 Expected costs and emissions with LTL and TL transportation as r changes.

r

LTL transportation 1

TL transportation 1

ðQ C ; RC Þ

1

ðQ E ; RE Þ

ðQ S ; RS Þ

ðQ C ; RC Þ

2

ðQ E ; RE Þ

2

ðQ S ; RS Þ

2

C1

E1

C1

E1

C1

E1

C2

E2

C2

E2

C2

E2

10 20 30 40 50 60 70 80 90 100

8684 8724 8764 8804 8844 8884 8923 8963 9002 9041

8453 8534 8616 8696 8777 8857 8937 9016 9095 9174

8896 8938 8979 9020 9061 9102 9142 9183 9224 9264

8226 8298 8369 8440 8511 8582 8652 8722 8792 8862

8758 8799 8841 8882 8923 8964 9004 9045 9085 9125

8299 8373 8446 8518 8591 8663 8736 8808 8879 8951

9168 9207 9245 9283 9321 9358 9396 9433 9471 9508

8802 8874 8958 9034 9116 9188 9265 9347 9424 9502

9387 9434 9480 9527 9573 9609 9655 9676 9722 9754

8533 8605 8677 8749 8821 8893 8964 9035 9106 9177

9262 9304 9346 9388 9429 9466 9507 9539 9579 9613

8654 8724 8794 8866 8936 9006 9076 9148 9220 9290

Avg

8863

8816

9081

8546

8943

8626

9339

9151

9582

8856

9443

8971

Table 5 Expected costs and emissions with LTL and TL transportation as s changes.

s

LTL transportation ðQ C ; RC Þ 1

1

TL transportation 1

1

ðQ E ; RE Þ 1

ðQ S ; RS Þ 1

ðQ C ; RC Þ 2

2

2

2

ðQ E ; RE Þ 2

ðQ S ; RS Þ 2

E1

C

E1

C

E1

C

E2

C

E2

C

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

8877 8972 9044 9105 9158 9205 9249 9290 9328 9363

8845 9038 9184 9307 9415 9512 9601 9683 9759 9830

9094 9191 9266 9329 9384 9434 9479 9522 9561 9599

8574 8746 8876 8986 9082 9169 9248 9321 9390 9455

8956 9054 9128 9190 9245 9294 9339 9381 9420 9457

8655 8830 8964 9077 9175 9264 9346 9421 9492 9558

9349 9439 9507 9565 9615 9660 9702 9740 9776 9809

9177 9364 9497 9622 9721 9813 9890 9969 10,038 10,125

9595 9705 9767 9821 9873 9919 9966 10,013 10,056 10,086

8881 9053 9186 9296 9392 9479 9559 9633 9702 9768

9455 9553 9617 9673 9722 9768 9811 9851 9888 9922

8996 9164 9295 9406 9504 9592 9670 9743 9811 9878

Avg

9159

9417

9386

9085

9247

9178

9616

9722

9880

9395

9726

9506

C

E2

Table 6 Percent changes and cost of reduction with LTL and TL transportation as r changes.

r

LTL transportation

TL transportation

cC 1 (%) D

cE 1 (%) D

d1 CoR

cC 2 (%) D

cE 2 (%) D

d2 CoR

10 20 30 40 50 60 70 80 90 100

0.86 0.87 0.88 0.88 0.89 0.90 0.91 0.92 0.93 0.94

1.75 1.83 1.91 1.99 2.06 2.12 2.19 2.25 2.30 2.36

0.473 0.457 0.446 0.439 0.433 0.428 0.423 0.420 0.416 0.413

1.10 1.15 1.18 1.22 1.25 1.24 1.27 1.22 1.25 1.21

1.68 1.70 1.85 1.88 1.99 1.99 2.06 2.13 2.16 2.22

0.916 1.227 0.899 0.974 1.064 0.768 0.822 0.888 1.112 0.660

Avg

0.90

2.07

0.435

1.21

1.97

0.933

In generating transportation parameters related to costs, we adopt values from integrated inventory control and truckload transportation studies. Specifically, we assume that w  U½150; 250 and v  U½50; 200 (similar values are defined in integrated inventory control and truckload transportation, see, e.g., Toptal et al., 2003; Toptal and Çetinkaya, 2006; Toptal, 2009; Konur and Toptal, 2012). Note that w defines the per truck cost for a single shipment from the supplier to the retailer store. We let v to be rounded to the nearest multiplier of 10 for practical purposes. To define t and e, we consider that the distance between supplier and the retailer store g ¼ 100 units. Specifically, let t 0 and e0 be the transportation charge per unit distance per unit shipped by LTL and TL carriers, respectively. Note that one should have t 0 > e0 , otherwise, the retailer would always prefer LTL carrier as TL carrier will have additional truck charges as well. We assume that

162

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Table 7 Percent changes and cost of reduction with LTL and TL transportation as s changes.

s

LTL transportation

TL transportation

cC 1 (%) D

cE 1 (%) D

d1 CoR

cC 2 (%) D

cE 2 (%) D

d2 CoR

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.90 0.92 0.93 0.95 0.96 0.97 0.98 0.99 1.00 1.00

2.09 2.23 2.33 2.41 2.47 2.53 2.58 2.62 2.65 2.68

0.430 0.420 0.415 0.410 0.407 0.404 0.402 0.400 0.398 0.397

1.21 1.30 1.26 1.25 1.21 1.21 1.23 1.24 1.25 1.25

1.99 2.13 2.13 2.25 2.23 2.24 2.22 2.24 2.24 2.44

0.727 0.925 1.318 1.587 0.556 0.577 0.593 0.581 0.591 0.668

Avg

0.96

2.46

0.408

1.24

2.21

0.812

e0  U½0:005; 0:015, and e ¼ e0 g (similar values are defined in Konur and Schaefer (2014), Palak et al. (2014)). Similarly, we assume that t 0  U½0:02; 0:03 and t ¼ t 0 g. Consider truck capacity and truck charge of the TL carrier, these ranges for e and t capture that cases where the total transportation cost per unit charged by the TL carrier (including the truck charges) can be lower or higher than the transportation cost per unit charged by the LTL carrier. In generating transportation parameters related to emissions, we focus on the following observations from the literature. Generally, emission characteristics for trucks are given for empty truck and full truck per mile or kilometer (km)(see, e.g., Pan b e and w b f denote the carbon emissions generated by unit distance by empty and full truck, et al., 2012; Reed et al., 2010). Let w b f  1:5 w b e for different respectively. It is observed from the values given by Pan et al. (2012) and Reed et al. (2010) that w b e varies between 1 and 2 CO2 kg per unit distance (similar numbers can also be deducted from a simulation truck types and w b e  U½1; 2; hence, we define study provided by Daccarett-Garcia (2009)). Therefore, we consider problem instances with w b f ¼ bw b e where b  U½1:2; 1:8, we consider that b b  U½100; 200 as g ¼ 100 distance units. To capture w e  U½0:5; 1:5. In w generating bt, we follow a discussion similar to the relation between t and e, and consider that bt  U½1:5; 3 to capture the cases where the total transportation emission per unit with the TL carrier (including the truck emissions) can be lower or higher than the transportation emission per unit with the LTL carrier. In all problem instances solved, we assume that s  U½0:1; 0:5; stl ¼ s and sltl ¼ s. Furthermore, we set ‘ ¼ 25 in Algorithms 1–3, and L ¼ 30 in Algorithm 3 as the termination criteria. A.5. Tables of Sections 5.1 For each r value, Table 4 summarizes the changes in averages over 50 problem instances solved with LTL transportation 1

for expected costs (C 1 ) and carbon emissions (E1 ) for the cost-minimizing ðQ ; RÞ policy (ðQ C ; RC Þ ), emission-minimizing S 1

E 1

ðQ ; RÞ policy (ðQ E ; R Þ ), and the average of the ðQ ; RÞ policies in PF 1 (ðQ S ; R Þ ) and the changes in averages over 50 problem instances solved with TL transportation in expected costs (C 2 ) and emissions (E2 ) for the cost-minimizing ðQ ; RÞ policy 2

2

2

(ðQ C ; RC Þ ), emission-minimizing policy ðQ ; RÞ policy (ðQ E ; RE Þ ), and the average of the ðQ ; RÞ policies in PF 2 (ðQ S ; RS Þ ). For each s value, Table 5 is constructed similar to Table 4. A.6. Tables of Sections 5.2 In Tables 6 and 7, for a single problem instance solved, we first calculate the DC 1 ; DE1 or DC 2 ; DE2 for any sustainable policy in the PF. Then, we are taking the average of these values. The average values of DC 1 ; DE1 ; DC 2 , and DE2 for the probcC 1 ; D cC 2 , and D cE 1 ; D cE 2 , respectively. Similarly, we calculate CoR1 and CoR2 for any sustainable lem instance are denoted by D Table 8 Retailer R’s cost and emission parameters. Cost parameters

Emission parameters

Parameter

Value

Parameter

Value

h K p

2 100 5

b h b K

6 150 7

b p

B. Schaefer, D. Konur / Transportation Research Part E 80 (2015) 142–165

163

Table 9 Cost and emission parameters for LTL carriers.

t bt

s

LTL Carrier A

LTL Carrier B

2.77 1.96 0.3

2.82 2.12 0.15

Table 10 Cost and emission parameters for TL Carriers A and B.

w

v e b w b e

s

TL Carrier A

TL Carrier B

250 120 0.7 150 0.75 0.25

200 120 1.15 175 0.565 0.20

d 1 and policy in the PF of the single problem instance. Then, we are determining the average of these values, denoted by CoR cC 1 ; D cC 2 , and D d 1 and CoR d 2 over 50 d 2 , respectively. For each r value, Table 6 summarizes the averages of D cE 1 ; D cE 2 ; CoR CoR problem instances solved for that r value. For each s value, Table 7 is constructed similar to Table 6. A.7. Examples of Section 5.3 In Examples 1–3, the same retailer has been considered to control inventory and transportation of a single product such that the demand per unit time for the product is normally distributed k ¼ 2000 units and # ¼ 200. It is assumed that c¼b c ¼ 1. The retailer has the cost and emission parameters given in Table 8. LTL Carriers A and B have the parameter values given in Table 9. TL Carriers A and B have the parameter values given in Table 10. References Absi, N., Dauzere-Peres, S., Kedad-Sidhoum, S., Penz, B., Rapine, C., 2013. Lot sizing with carbon emission constraints. Eur. J. Oper. Res. 227, 55–61. Agrell, P.J., 1995. A multicriteria framework for inventory control. Int. J. Prod. Econ. 41, 59–70. Anderson, D.R., Sweeney, D.J., Williams, T.A., Camm, J.D., Cochran, J.J., Fry, M.J., Ohlmann, J.W., 2012. Quantitative Methods for Business, 12th ed. SouthWestern Cengage Learning, Ohio, US. Arikan, E., Fichtinger, J., Ries, J.M., 2014. Impact of transportation lead-time variability on the economic and environmental performance of inventory systems. Int. J. Prod. Econ. 157, 279–288. Arslan, M.C., Turkay, M., 2013. EOQ revisited with sustainability considerations. Found. Comput. Decis. Sci. 38, 223–249. Aucamp, D., 1982. Nonlinear freight costs in the EOQ problem. Eur. J. Oper. Res. 9, 61–62. Bae, S.H., Sarkis, J., Yoo, C.S., 2011. Greening transportation fleets: insights from a two-stage game theoretic model. Transp. Res. Part E 47, 793–807. Battini, D., Persona, A., Sgarbossa, F., 2014. A sustainable EOQ model: theoretical formulation and applications. Int. J. Prod. Econ. 149, 145–153. Bektas, T., Laporte, G., 2011. The pollution-routing problem. Transp. Res. Part B 45, 1232–1250. Benjaafar, S., Li, Y., Daskin, M., 2012. Carbon footprint and the management of supply chains: insights from simple models. IEEE Trans. Autom. Sci. Eng. PP (99), 1. Bonney, M., Jaber, M.Y., 2011. Environmentally responsible inventory models: non-classical models for a non-classical era. Int. J. Prod. Econ. 133, 43–53. Bouchery, Y., Ghaffari, A., Jemai, Z., Dallery, Y., 2012. Including sustainability criteria into inventory models. Eur. J. Oper. Res. 222, 229–240. Brooks, R.S., Lu, J.Y., 1969. On the convexity of the backorder function for an EOQ policy. Manage. Sci. 15, 453–454. Cachon, G.P., 2014. Retail store density and the cost of greenhouse gas emissions. Manage. Sci. 60, 1907–1925. Chaabane, A., Ramudhin, A., Paquet, M., 2012. Design of sustainable supply chains under the emission trading scheme. Int. J. Prod. Econ. 135, 37–49. Chan, C.K., Lee, Y., Campbell, J., 2013. Environmental performance: impacts of vendor–buyer coordination. Int. J. Prod. Econ. 145, 683–695. Chen, X., Benjaafar, S., Elomri, A., 2013. The carbon constrained EOQ. Oper. Res. Lett. 41, 172–179. Choi, T.-M., 2013a. Local sourcing and fashion quick response system: the impacts of carbon footprint tax. Transp. Res. Part E: Logist. Transp. Rev. 55, 43–54. Choi, T.-M., 2013b. Optimal apparel supplier selection with forecast updates under carbon emission taxation scheme. Comput. Oper. Res. 40, 2646–2655. Corbett, C.J., Kleindorfer, P.R., 2001a. Environmental management and operations management: introduction to Part 1 (manufacturing and eco-logistics). Prod. Oper. Manage. 10, 107–111. Corbett, C.J., Kleindorfer, P.R., 2001b. Environmental management and operations management: introduction to Part 2 (integrating operations and environmental management systems). Prod. Oper. Manage. 10, 225–227. Daccarett-Garcia, J.Y., 2009. Modelling the Environmental Impact of Demand Variability Upon Supply Chains in the Beverage Industry. Master’s Thesis. Department of Industrial and Systems Engineering, Rochester Institute of Technology. Das, I., Dennis, J., 1997. A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct. Optimiz. 14, 63–69. Dekker, R., Bloemhof, J., Mallidis, I., 2012. Operations research for green logistics an overview of aspects, issues, contributions and challenges. Eur. J. Oper. Res. 219, 671–679. Demir, E., Bektas, T., Laporte, G., 2012. An adaptive large neighborhood search heuristic for the pollution-routing problem. Eur. J. Oper. Res. 223, 346–359.

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