Carbon constrained integrated inventory control and truckload transportation with heterogeneous freight trucks

Author's Accepted Manuscript

Carbon Constrained Integrated Inventory Control and Truckload Transportation with Heterogeneous Freight Trucks Dinçer Konur

www.elsevier.com/locate/ijpe

PII: DOI: Reference:

S0925-5273(14)00089-9 http://dx.doi.org/10.1016/j.ijpe.2014.03.009 PROECO5719

To appear in:

Int. J. Production Economics

Received date: 15 July 2013 Accepted date: 8 March 2014 Cite this article as: Dinçer Konur, Carbon Constrained Integrated Inventory Control and Truckload Transportation with Heterogeneous Freight Trucks, Int. J. Production Economics, http://dx.doi.org/10.1016/j.ijpe.2014.03.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Carbon Constrained Integrated Inventory Control and Truckload Transportation with Heterogeneous Freight Trucks

March 15, 2014 Abstract This paper analyzes an integrated inventory control and transportation problem with environmental considerations. Particularly, explicit transportation modeling is included with inventory control decisions to capture per truck costs and per truck capacities. Furthermore, a carbon cap constraint on the total emissions is formulated by considering emission characteristics of various trucks that can be used for inbound transportation. Due to complexity of the resulting optimization problem, a heuristic search method is proposed based on the properties of the problem. Numerical studies illustrate the efficiency of the proposed method. Furthermore, numerical examples are presented to show that both costs and emissions can be reduced by considering heterogeneous trucks for inbound transportation. Keywords: Inventory Control, Carbon Emissions, Truckload Transportation

1

Introduction and Literature Review

Environmental awareness throughout supply chains is growing due to the regulatory policies legislated by governments (such as Kyoto Protocol, UNFCCC, 1997), voluntary organizations established to curb emissions (such as Regional Greenhouse Gas Initiative and the Western Climate Initiative) and the concerns of environmentally sensitive customers (see, e.g., Liu et al., 2012, Zavanella et al., 2013). As a result, supply chain agents review their carbon footprints, and they replan their operations or invest in carbon emissions abatement projects to fulfill their environmental responsibilities (Bouchery et al., 2011).

Supply chain operations such as inventory holding, freight transportation, logistics,

and warehousing activities are the main contributors to emissions generated in many manufacturing, retailing, transportation, health, and service industries. In particular, transportation is one of the major contributors to greenhouse gas (GHG) emissions: approximately 13% of global GHG emissions in 2004 was due to transportation sector (Rogner et al., 2007). Contribution of transportation to 2010 GHG emissions of Europe Union was almost 20%: 25% of France GHG emissions, 17% of Germany GHG emissions, and 20% of the U.K. GHG emissions were due to transportation in 2010 (EEA, 2013). Furthermore, emissions from road transportation 1

constitutes the majority of transportation emissions. Leonardi and Baumgartner (2004), for instance, note that 6% of total emissions and 29% of transportation emissions in Germany in 2001 were due to road freight transportation. In the U.S., transportation sector generated almost 27% of the national GHG emissions in 2010 (EPA, 2013). Furthermore, while passenger cars are the biggest GHG emitters in the transportation sector, freight trucks generate the majority of the U.S. GHG emissions due to freight transportation. In particular, light-duty trucks (18.9% of 2010 U.S. GHG emissions generated by transportation sector) and medium- and heavy-duty trucks (21.9% of 2010 U.S. GHG emissions generated by transportation sector) are the largest GHG emissions generators from freight transportation in the U.S. in 2010 (EPA, 2013). A 50% increase in freight transportation from 2000 to 2020 is estimated for European countries (see, e.g. Toptal and Bingol, 2011). In the U.S., over 68% of freight is transported with trucks and a dramatic increase in the U.S. freight truck traffic is expected by 2040 (FHWA, 2008). Given that the freight trucks is the most common mode for freight transportation, the above statistics are not surprising. It is, therefore, crucial to explicitly consider transportation in replanning supply chain operations for achieving environmental goals. We refer the reader to review papers by Corbett and Kleindorfer (2001a,b), Kleindorfer et al. (2005), Linton et al. (2007), Srivastava (2007), Sbihi and Eglese (2010), Sarkis et al. (2011), and Dekker et al. (2012) for the class of supply chain and operations management and logistics problems studied with environmental considerations. This study analyzes an integrated inventory control and inbound transportation problem with carbon emissions constraint. Particularly, this paper focuses on the economic order quantity (EOQ) model with truckload transportation and carbon emissions constraint. Inventory control models have been recently studied with carbon emissions regulation policies. Most of these studies focus on the variants of the EOQ model as the EOQ model is a commonly used inventory control policy in practice in case of deterministic demand. Specifically, Chen et al. (2013) models the EOQ model with carbon emissions constraint, i.e., the carbon cap policy. They provide solutions for the carbon-constrained EOQ model and discuss the conditions under which carbon emissions reduction is relatively more than the increases in costs due to carbon cap. Hua et al. (2011a) also studies the EOQ model. They formulate and solve the EOQ model in the presence of an emissions trading system such as the European Union Emissions Trading System and the New Zealand Emissions Trading Scheme. That is, their focus is on the EOQ model under the carbon cap and trade policy, which allows trading carbon emissions at a specified carbon trading price. This study is then extended to include pricing decisions within the EOQ model by Hua et al. (2011b). Arslan and Turkay (2013) revisit the EOQ model with carbon cap, carbon cap and trade, carbon taxing (where emissions are taxed), and carbon offsetting (where carbon abatement projects can be used to 2

curb emissions in case emissions exceed the carbon cap) policies. In a recent study, Toptal et al. (2014) jointly analyze inventory control and carbon emissions reduction investment decisions in an EOQ model under carbon cap, cap-and-trade, and tax policies. Similar to the above studies, we analyze the EOQ model but further include joint transportation decisions. Specifically, we study this model under a carbon cap policy. In a carbon cap policy, the emissions of a company is restricted by a mandatory emissions limit, which is referred to as the carbon cap (see, e.g., Chen et al., 2013). While the carbon cap can be imposed by governmental agencies, the companies can also define their carbon cap in the view of their green goals (Benjaafar et al., 2012, Toptal et al., 2014). For instance, a survey conducted among 582 European companies by Loebich et al. (2011) documents that company management decisions are the main motivation for greening operations in 2011 while the main motivation for greening operations in 2008 was environmental regulations. Therefore, we analyze the model of interest with carbon cap policy. Furthermore, it should be noted that, instead of studying EOQ model with carbon emissions regulation policies, Bouchery et al. (2011, 2012) analyze multi-objective EOQ model with cost and environmental impacts minimization (multiobjective optimization models with cost and environmental objectives have also been studied for different supply chain management problems). They focus on generating Pareto efficient inventory decisions. A very commonly used method to generate Pareto efficient solutions is the constrained method, which is introduced by Lin (1976). The constrained method is guaranteed to generate Pareto efficient solutions independent of the problem properties (such as convexity requirements). The EOQ model with joint transportation decisions under carbon cap policy is the subproblem required by the constrained method if one wishes to solve bi-objective EOQ model with joint transportation decisions, where both costs and carbon emissions are minimized. Therefore, the analysis presented in this study can be utilized in multi-objective models for similar settings. It should be noted that inventory control systems other than the classical EOQ model have also been analyzed with environmental considerations. Letmathe and Balakrishnan (2005) study a product mix problem with carbon cap, carbon trading, and carbon taxing policies. Benjaafar et al. (2012) and Absi et al. (2013) focus on lot-sizing problems with carbon emissions regulations. Song and Leng (2012) analyze the single period stochastic demand model (i.e., the newsvendor model) and Hoen et al. (2012) study transportation mode selection problem in the setting of newsvendor model with carbon emissions regulations. Jiang and Klabjan (2012) characterize the optimal emissions reduction investment and capacity planning in case of stochastic demand. Liu et al. (2012) and Zavanella et al. (2013) investigate two echelon supply chains with environmentally sensitive customers and Jaber et al. (2013) model the vendor-buyer coordination problem with carbon trading and emissions reduction investment. 3

As noted previously, freight transportation, especially, freight trucks are major contributors to carbon emissions.

Nevertheless, the studies focusing on the EOQ models with carbon emissions

considerations fail to model not only explicit transportation costs but also explicit transportation emissions. Particularly, these studies assume less-than-truckload (LTL) transportation, that is, a single truck is considered to have sufficiently large capacity to carry any shipment. On the other hand, truckload (TL) transportation is common in practice and supply chain agents should consider TL transportation costs and emissions in controlling their inventory and transportation operations. The studies that account for basic truck characteristics such as truck capacity and truck emissions in the context of environmentally sensitive logistics operations focus on vehicle routing problems (see, e.g., Bektas and Laporte, 2011, Suzuki, 2011, Jabali et al., 2012, Erdogan and Miller-Hooks, 2012, and Demir et al., 2012). In the supply chain literature, TL transportation costs are modeled in various inventory control models (see, e.g., Aucamp, 1982, Lee, 1986, Toptal et al., 2003, Toptal and C ¸ etinkaya, 2006, Toptal, 2009, Toptal and Bingol, 2011, Konur and Toptal, 2012). These studies account for the per truck capacities and per truck costs in the context of inventory control. In this study, similar to these studies, we model transportation costs by explicitly accounting for per truck capacities and per truck costs. Furthermore, transportation emissions are formulated considering truck capacities and truck characteristics. In particular, a retailer who operates under the basic EOQ model settings is considered. Additional to the retailer’s inventory holding and order setup costs, the retailer is subject to inbound transportation costs, which are determined by the numbers of specific trucks used for inbound transportation. It is also assumed that a fixed amount of carbon emissions is generated by each empty truck and emissions due to transportation increase with the loads of the trucks. We note that Hoen et al. (2012) and Pan et al. (2010) similarly define transportation emissions from freight trucks. In this study, we contribute to environmental inventory control studies by analyzing the deterministic inventory control models with carbon emissions considerations and explicit modeling of TL transportation costs as well as emissions. Furthermore, we consider availability of different truck types for inbound transportation. In practice, it can be the case that a retailer outsources its inbound transportation from a TL carrier, who offers a set of trucks with different characteristics. Moreover, it can be the case that the retailer has different TL carriers available in the market and each TL carrier offers trucks with different per truck costs and per truck capacities. In these cases, the retailer has to consider different truck types in modeling his/her transportation costs. Additionally, trucks types with distinct truck characteristics will have varying emissions generations (see, e.g., Demir et al., 2011). For instance, fuel type used, type of engine, year of built, vehicle mass, and driving characteristics (such as drag force, resistance) are all 4

effective on the emissions generated by a freight truck (Ligterink et al., 2012). McKinnon (2005) and Mallidis et al. (2010), for instance, list sets of British and EURO truck types, respectively, and their emissions characteristics. Reed et al. (2010) also note that different truck types should be considered in calculating transportation emissions. In the analysis of a beverage industry in the U.S., Daccarett-Garcia (2009) notes that fleet management (truck configurations used) has significant effect on not only costs but also carbon emissions. According to the results of survey conducted by Leonardi and Baumgartner (2004) among 200 German companies, selecting the optimum vehicle categories is crucial for reducing fuel consumption (and; thus, emissions) from logistic activities. In a recent study, Bae et al. (2011) analyze competitive firms’ investment decisions for greening their transportation fleets. Based on these studies, it is an important decision to choose truck configurations (fleet management) for managing emissions as well as costs due to transportation. The current study, therefore, models emissions due to freight transportation considering different truck types. To the best of our knowledge, this study is first in explicitly considering different truck characteristics (cost and emission characteristics) in an integrated inventory control and transportation problem with carbon emissions constraint.

We contribute to the current body of literature on carbon sensitive

inventory models by modeling the EOQ model with TL transportation costs and emissions in the presence of heterogeneous trucks. The complexity of the problem is stated and an efficient heuristic solution method is developed. Furthermore, it is illustrated that considering different trucks in the inventory control and inbound transportation planning can reduce not only total costs but also emissions. The rest of the paper is organized as follows. Section 2 models the integrated inventory control and inbound transportation problem with carbon cap. Particularly, the retailer’s cost and emissions functions are formulated with heterogeneous freight trucks. In Section 3, the properties of the problem of interest are discussed and a heuristic solution method is proposed. Results of a set of numerical studies are documented in Section 4 to illustrate the efficiency of the proposed heuristic method and analyze the effects of carbon cap. Furthermore, sample examples are solved to show the benefits of explicitly modeling different truck types. Section 5 summarizes the contributions and the findings of the paper and discusses possible future research directions.

2

Problem Formulation

In this study, a retailer’s deterministic inventory control with truckload transportation is considered. In particular, the retailer assumes the basic economic order quantity (EOQ) model. In this setting, the

5

demand rate, λ per unit time (items per year), is deterministic and constant over time, the delivery lead time is fixed, and a long planning horizon is considered. Under the basic EOQ model, the retailer determines his/her order quantity, Q (units/order), that minimizes total costs per unit time. The retailer’s total costs include purchase costs, setup costs, and inventory holding costs. Specifically, let p denote the per unit purchase cost ($/item), A denote the setup cost per order ($/order), and h denote the inventory carrying cost per unit per unit time ($/item/year). Most of the EOQ studies assume lessthan-truckload (LTL) transportation for the shipment of the order, that is, a per unit transportation cost is assumed. Therefore, the transportation costs can be included within the purchase costs. It is well known that the retailer’s problem of determining the optimum order quantity to minimize total costs per unit time reads (P-LTL)

min s.t.

H(Q) = pλ + Q ≥ 0.

Aλ Q

+

hQ 2

The optimum solution of the P-LTL is achieved when the retailer orders Qeoq =



2Aλ h ,

which is also

known as the economic order quantity. Note that p does not affect Qeoq , hence, LTL transportation costs are not effective on the retailer’s inventory control policy. In practice, however, trucks are commonly used for shipping freight. Therefore, we extend the classical EOQ model by assuming that the retailer is subject to truckload (TL) transportation costs in his/her inbound transportation. Furthermore, as discussed in Section 1, the retailer should also determine his/her truck fleet configuration in some practical cases. To capture truck fleet management decisions, we assume that the retailer can use n different truck types. Let truck types be indexed by i, i ∈ {1, 2, . . . , n} and let xi denote the number of trucks of type i used by the retailer. Similar to studies on TL modeling in inventory control, (see, e.g., Aucamp, 1982, Lee, 1986, Toptal et al., 2003, Toptal and C ¸ etinkaya, 2006, Toptal, 2009, Toptal and Bingol, 2011, Konur and Toptal, 2012), it is assumed that each truck of type i has a capacity of Pi (units/truck) and cost of Ri ($/truck). Pi can be determined using the weight and/or volume limit of truck type i and the unit volume/weight of the product being carried. Ri is the fixed cost per truck of type i charged by the TL carriers. Ri depends on the distance of the delivery and, generally, does not consider the amount delivered. Therefore, we ignore per unit transportation costs in case of TL transportation. We note that more generalized transportation cost structures can be considered such as freight discounts, non-linear cost functions, different transportation modes. The models and methods presented next can be extended and modified for such cases. We pose such cases as future research directions. Then the retailer’s integrated inventory control and TL transportation problem reads

6

(P-TL)

min s.t.

H(Q, x) = pλ + Q≤

n 

Aλ Q

+

hQ 2

+

λ Q

n 

xi Ri

i=1

xi Pi

i=1

xi ∈ {0, 1, 2, . . .} ∀i, i = 1, 2, . . . , n, where x = [x1 , x2 , . . . , xn ]t . The first constraint of P-TL enforces the order quantity to be less than or equal to the total capacities of the trucks used and the second set of constraints are the integer definitions of xi values. Note that the last term of H(Q, x) is the TL transportation cost per unit time  ( ni=1 xi Ri is the total transportation cost per order shipment). Remark 1 P-TL is an NP-complete problem. Remark 1 follows from the fact that the special case of P-TL, when Q is fixed, is the integer knapsack problem, which is known to be NP-complete (see, e.g., Papadimitriou, 1981). In what follows, we formulate the retailer’s problem under carbon emissions constraint. Considering the recent discussions on carbon emissions, we assume that the retailer is subject to an upper limit on his/her carbon emissions per unit time, which is known as the carbon cap∗ . The carbon cap, C (CO2 lbs/year), can be imposed by government agencies as carbon regulatory policies or can be set by the company itself to achieve the company’s green goals (see, e.g., Chen et al., 2013). For instance, carbon cap policy was considered by the Congressional Budget Office (CBO), Congress of the United States as an option to reduce CO2 emissions (CBO, 2008). As aforementioned, inventory holding and transportation are the main contributors to carbon emissions of a retailer. In particular, similar to Hua et al. (2011a) and Chen et al. (2013), we define  as the emissions generated per order (CO2 lbs/order),  h as the emissions due to holding one unit A in inventory per unit time (CO2 lbs/item/year), and p as the emissions per unit due to procurement  is defined as the emissions generated by an empty truck in Hua (CO2 lbs/unit). We note that A et al. (2011a) and they consider that each unit loaded to the truck generates unit carbon emissions. The underlying assumption of Hua et al. (2011a) is LTL transportation, i.e., a single-truck is assumed to have sufficient capacity to ship any order size. However, as noted previously, TL transportation is common in practice. Furthermore, each truck type has different weights and fuel efficiency, hence; the carbon levels emitted vary for different truck types. To capture different truck characteristics for modeling the carbon emissions, we represent emissions generated by transportation considering each truck type and the load it carries. Let e0i and ei denote the emissions generated by an empty truck of type i (CO2 lbs/truck) and emissions generated by carrying one unit with truck type i (CO2 lbs/unit), respectively. ∗

Other greenhouse gas emissions can be calculated in terms of CO2 (see, e.g., EPA, 2013).

7

In this setting, the carbon emissions generated by shipping Q units not only depends on the number of trucks of each truck type used but also how much each truck type carries. In particular, let qi denote the  quantity that is carried by type i trucks. Then, Q = ni=1 qi and the total carbon emissions generated per unit time amount to   hQ λ Aλ + + E(Q, x) = pλ + Q 2 Q



n 

xi e0i +

n 

i=1

 ei qi ,

(1)

i=1

where Q = [q1 , q2 , . . . , qn ]t . The first, second, and third terms of Equation (1) represent the emissions per unit time due to procurement, order setup, and inventory holding, respectively. In the last term of Equation (1), the first component represents the emissions generated by empty trucks and the second component accounts for the emissions generated by the loads of the trucks. Then the retailer’s integrated inventory control and TL transportation problem with carbon cap can be formulated as follows: (P-TL-Cap)

min

H(Q, x) = pλ +

Aλ Q

s.t.

E(Q, x) = pλ +

 Aλ Q

Q=

n 

+

hQ 2

+

 hQ 2

+

λ Q

n 

xi Ri i=1   n n   λ +Q xi e0i + ei qi ≤ C i=1

i=1

qi ,

i=1

qi ≤ xi Pi ∀i, i = 1, 2, . . . , n, xi ∈ {0, 1, 2, . . .} ∀i, i = 1, 2, . . . , n. The first constraint of P-TL-Cap is the carbon cap constraint. The second constraint defines the total order quantity. The third set of constraints ensures that load of trucks of type i does not exceed the capacities of type i trucks used. The last set of constraints gives the integer definitions of xi values. Remark 2 P-TL-Cap is an NP-complete problem. Remark 2 is a direct implication of Remark 1 considering the fact that P-TL is a special case of PTL-Cap when C → ∞. In particular, P-TL-Cap is a mixed-integer-nonlinear programming problem (MINLP). Prior to analysis of P-TL-Cap, it is worthwhile to mention that P-TL-Cap also applies to the following practical scenario. Consider the retailer defined above without the TL transportation. The retailer can use different packages for his/her outbound shipment. For instance, de Kok (2000) studies a capacity allocation problem in case of different package sizes. Let each package type have a specific size and specific cost as well as varying emissions characteristics. We note that packaging also contributes to ¨ u (2012), for instance, notes that emissions can be emissions, especially, in warehousing activities. Ulk¨ reduced with considering different packaging technologies and sizes. Under LTL transportation for the 8

outbound shipment, the integrated inventory control and packaged shipment decisions can be formulated similar to P-TL-Cap in case of carbon cap. Appendix 6.1 summarizes the notation used throughout the paper and the metrics assumed for each parameter. Additional notation will be defined as needed. In the next section, we analyze the properties of P-TL-Cap and propose a heuristic method utilizing these properties.

3

Solution Analysis

In this section, we first analyze the retailer’s order quantity decisions, Q, given the retailer’s truck choices, x. Then, the results are used in a search heuristic to find truck choices. Let (Q∗ , x∗ ) denote an optimum solution of P-TL-Cap. Theorem 1 If x∗i > 0 then (x∗i − 1)Pi < qi∗ ≤ x∗i Pi , ∀i, i ∈ {1, 2, . . . , n}. Proof: All proofs are presented in Appendix. Theorem 1 implies that there will be no empty truck of any truck type i. This result is intuitive as the retailer will neither pay to use an empty truck nor generate carbon emissions with an empty truck. Theorem 2 There exists an optimum solution (Q∗ , x∗ ) such that qi∗ = x∗i Pi ∀i, i = j and (x∗j − 1)Pj < qj∗ ≤ x∗j Pj for a single truck type j. Theorem 2 indicates that there exists an optimum solution such that at most one truck, among all trucks, might have less than its full truckload. That is, all of the trucks will be loaded to their full capacities except one truck, which might carry less than its capacity. The following corollary is a direct result of Theorem 2. Corollary 1 Let j ∗ = arg mini=1,2,...,n {ei }. Then, there exists an optimum solution (Q∗ , x∗ ) such that qi∗ = x∗i Pi ∀i, i = j ∗ and (x∗j ∗ − 1)Pj ∗ < qj∗∗ ≤ x∗j ∗ Pj ∗ . In particular, Corollary 1 follows from the fact that given (Q∗ , x∗ ) is optimum, one can construct another feasible solution, say (Q∗∗ , x∗∗ ), such that Q∗ = Q∗∗ and x∗ = x∗∗ (thus, H(Q∗ , x∗ ) = H(Q∗∗ , x∗∗ )) ∗∗ ∗ ∗∗ ∗∗ ∗∗ ∗∗ and qi∗∗ = x∗∗ i Pi ∀i, i = j and (xj ∗ − 1)Pj ∗ < qj ∗ ≤ xj ∗ Pj ∗ (where, one can show that E(Q , x ) ≤

E(Q∗ , x∗ ) ≤ C, i.e., (Q∗∗ , x∗∗ ) is feasible, see proof of Theorem 2). Intuitively, Corollary 1 suggests a retailer how to allocate his/her total order to the selected trucks. Specifically, given a total order quantity and a set of selected trucks, the retailer can significantly reduce his/her carbon emissions by loading trucks taking their emissions characteristics into account. Considering Theorems 1 and 2, all of the selected trucks will carry loads. Therefore, the retailer can 9

reduce emissions by first filling the trucks that are environmentally more friendly. Then, a truck with the highest emission rate per load will be loaded until it is not costly beneficial to increase the load of that truck (see Section 3.1). Thus, Corollary 1 directs the retailer to an optimum solution which also minimizes the carbon emissions per unit time for the given truck choices. In what follows, we, therefore, focus on determining an optimum solution satisfying Corollary 1.

3.1

Truck Load Decisions ∗

Let any optimum solution in the form of Corollary 1 be denoted by (Q , x∗ ).

Recall that j ∗ =

arg mini=1,2,...,n {ei }. It then follows that qi∗ = x∗i Pi ∀i, i = j ∗ and qj∗∗ = (x∗j ∗ − 1)Pj ∗ + vj∗∗ where vj∗∗ denotes the quantity shipped by the one of the trucks of type j ∗ , which is the only truck with ∗

possibly less than full truckload under solution (Q , x∗ ). Then, given x∗ , one needs to determine vj∗∗ to ∗

find Q , and let it be denoted by Q∗ (x∗ ). Now, let us define the following functions:  λ A+ H(vj ∗ |x = x∗ ) = pλ + 

n 

n  i=1

x∗i Pi

 x∗i Ri

− Pj ∗ + vj ∗

h + 2

 n 

 x∗i Pi − Pj ∗ + vj ∗

(2)

i=1

i=1

 + λ A ∗

E(vj ∗ |x = x ) = pλ +

n  i=1

 x∗i (e0i + ei Pi ) − ej ∗ Pj ∗ + ej ∗ vj ∗



n  i=1

 x∗i Pi − Pj ∗ + vj ∗

 h + 2



n  i=1

 x∗i Pi

− Pj ∗ + vj ∗

. (3)

Equation (2), given the retailer’s truck choices are defined by x∗ , describes the retailer’s costs per unit time as a function of the load of the single type j ∗ truck with possible less than truckload, i.e., vj ∗ . Similarly, Equation (3) is the retailer’s carbon emissions per unit time as a function of vj ∗ given x = x∗ . The following problem then determines vj∗∗ given x∗ = x∗ , and, therefore, solves Q∗ (x∗ ): (P-TL-Cap(j ∗ ))

min s.t.

H(vj ∗ |x = x∗ ) E(vj ∗ |x = x∗ ) ≤ C  ≤ vj ∗ ≤ Pj ∗ ,

where  is a very small positive number. The first constraint of P-TL-Cap(j ∗ ) is the carbon cap constraint and the second constraint guarantees that the load of the truck with less than truckload is not exceeding the truck capacity. The following lemma characterizes H(vj ∗ |x = x∗ ) and E(vj ∗ |x = x∗ ). Lemma 1 H(vj ∗ |x = x∗ ) and E(vj ∗ |x = x∗ ) are convex with respect to vj ∗ over vj ∗ ≥ 0.

10

Lemma 1 implies that P-TL-Cap(j ∗ ) is a convex optimization problem; hence, KKT conditions are necessary and sufficient for finding the optimum vj ∗ value, denoted by vj∗∗ . However, E(vj ∗ |x = x∗ ) is a quadratic convex function and one can show that E(vj ∗ |x = x∗ ) ≤ C for vj ∗ ∈ [w1 , w2 ] such that  Φ − Φ2 − 2 hΨ , (4) w1 =  h  Φ + Φ2 − 2 hΨ , (5) w2 =  h    + λ p + ej ∗ ) and Ψ = λA p ( ni=1 x∗i Pi − Pj ∗ ) + where Φ = C −  h ( ni=1 x∗i Pi − Pj ∗ ) − λ( n

n  ∗ 0 ∗  This implies the h ( ni=1 x∗i Pi − Pj ∗ )2 /2 + λ i=1 xi (ei + ei Pi ) − ej ∗ Pj ∗ − C ( i=1 xi Pi − Pj ∗ ). following reformulation of P-TL-Cap(j ∗ ): (P-TL-Cap(j ∗ ))

min s.t.

H(vj ∗ |x = x∗ ) max{0, w1 } +  ≤ vj ∗ ≤ min{Pj ∗ , w2 },

Note that if w1 > Pj ∗ or w2 ≤ 0, P-TL-Cap(j ∗ ) is infeasible. It further follows from Equation (2) and Lemma 1 that H(vj ∗ |x = x∗ ) is minimized at    n  2λ A + x∗i Ri n 

i=1 + Pj ∗ − x∗i Pi . w0 = h

(6)

i=1

The following theorem is a direct result of Lemma 1 and Equations (4)-(6). Theorem 3 Suppose that P-TL-Cap(j ∗ ) is feasible and let vj∗∗ be the optimum solution of P-TL-Cap(j ∗ ). Then vj∗∗

⎧ ⎪ max{0, w1 } +  ⎪ ⎨ = w0 ⎪ ⎪ ⎩ min{P ∗ , w } j

2

if w0 < max{0, w1 } + , if max{0, w1 } +  ≤ w0 ≤ min{Pj ∗ , w2 }, if min{Pj ∗ , w2 } < w0 .

Corollary 1 and Theorem 3 define necessary conditions for an optimum solution in the form of Corollary 1. While these conditions are not sufficient for optimality, they can be used in checking whether a given truck choices vector x is not optimal. In particular, given x, one can contradict the assumption that x is optimum if Q∗ (x) determined via Corollary 1 and Theorem 3 is not feasible. In what follows, we utilize this approach in a search algorithm in finding retailer’s truck choices.

3.2

Truck Choices

Suppose that x0 is assumed to be optimum. Then, Q∗ (x0 ) is defined as qi (x0 ) = x0i Ri ∀i, i = j ∗ and qj ∗ (x0 ) = (x0j ∗ − 1)Rj ∗ + vj ∗ (x0 ), where vj ∗ (x0 ) is the solution of P-TL-Cap(j ∗ ) under x0 . If P-TLCap(j ∗ ) under x0 is infeasible, i.e., when w1 and w2 does not have real values or when w2 is not positive, 11

this implies that x0 is not optimum; thus, another truck choices vector should be considered. On the other hand, if Q∗ (x0 ) is feasible, we check whether including or excluding a truck lowers the retailer’s costs. In doing so, each option to add a truck is evaluated and each option to exclude a truck is evaluated. This approach is referred to as full truckload search heuristic since for any given truck choices vector x, we first fill all of the trucks to their full capacities and then find the load of an additional truck.

Algorithm 1 Full Truckload Search Heuristic (FTSH) for truck choices: Step 0: Let x be a given truck choices vector. Set x∗ = x. Step 1: Find the best truck choices with one less truckload, x[−1] Step 1.1. For i = 1 : n [−i]

[−i]

Step 1.2.

Set x[−i] = x and if xi

Step 1.3.

Solve P-TL-Cap(j ∗ ) under x[−i]

Step 1.4.

> 0, set xi

[−i]

= xi

−1

If it is infeasible, set H(Q∗ (x[−i] ), x[−i] ) = ∞

Step 1.5. End Step 1.6. x[−1] = arg min{H(Q∗ (x[−i] ), x[−i] )} Step 2: Find the best truck choices with one more truckload, x[+1] Step 2.1. For i = 1 : n [+i]

[+i]

Step 2.2.

Set x[+i] = x and let xi

Step 2.3.

Solve P-TL-Cap(j ∗ ) under x[+i]

Step 2.4.

= xi

+1

If it is infeasible, set H(Q∗ (x[+i] ), x[+i] ) = ∞

Step 2.5. End Step 2.6. x[+1] = arg min{H(Q∗ (x[+i] ), x[+i] )} Step 3: Let x = arg min{H(Q∗ (x[−1] ), x[−1] ), H(Q∗ (x), x), H(Q∗ (x[+1] ), x[+1] )}. If x = x∗ , stop and return x∗ . Else, let x = x and go to Step 1. There are two points to be highlighted about FTSH: FTSH might end up with infeasible solutions unless the starting solution is feasible, and FTSH will find the local minimum for truck choices. To overcome the infeasibility and to improve the search for better truck choices, we run FTSH with n different feasible starting solutions. Feasible starting solutions are achieved as follows. For each truck type, we find the optimum solution to P-TL-Cap assuming that only this specific truck type is available for inbound transportation. The method to optimally solve P-TL-Cap with single truck type is explained 12

in the Appendix 6.6. It should be pointed out that FTSH will find a local minimum solution and each iteration of FTSH runs in O(n). In the next section, we compare FTSH to BARON, a solver for MINLPs. The main advantage of FTSH is that it seeks the optimal solution with less carbon emissions. As is observed in the numerical studies, FTSH finds solutions very close to BARON in terms of costs in less computational time, but the main advantage of FTSH is that it utilizes Corollary 1 and finds solutions with less carbon emissions. Next section further analyzes the effects of carbon cap on the retailer’s costs and emissions as well as benefits of managing truck fleet instead of using a single truck type for inbound transportation.

4

Numerical Analysis

This section documents the results of a set of numerical studies conducted. In particular, we focus on three sets of analyses: (i) efficiency analysis of FTSH, (ii) effects of carbon cap, and (iii) benefits of having heterogeneous trucks available for inbound shipments. In analyses (i) and (ii), 16 different data sets are considered, each of which corresponds to a combination of A = {$150/order, $300/order},  = {100lbs of CO2 /order, 200lbs CO2 /order}, and  h = h = {$0.75/unit/year, $1.5/unit/year}, A {0.5lbs CO2 /unit/year, 1lbs CO2 /unit/year}. The specifications of the data sets are summarized in Table 1. For all of the data sets, it is assumed that λ = 50, 000 units/year, p = $1/unit, and p = 1lbs CO2 /unit. Table 1: Data Set Specifications Data Set 1 2 3 4 5 6 7 8

A 150 150 150 150 150 150 150 150

h 0.75 0.75 0.75 0.75 1.5 1.5 1.5 1.5

 A 100 100 200 200 100 100 200 200

 h 0.5 1 0.5 1 0.5 1 0.5 1

Data Set 9 10 11 12 13 14 15 16

A 300 300 300 300 300 300 300 300

h 0.75 0.75 0.75 0.75 1.5 1.5 1.5 1.5

For each data set, four different problem sizes are considered: n

=

15, and n

=

20.

 h 0.5 1 0.5 1 0.5 1 0.5 1

 A 100 100 200 200 100 100 200 200 n

=

5, n

=

10,

For a given data set and a given problem size, 10 problem

instances are generated by randomly selecting per truck costs Ri ∼ U [$100/truck, $150/truck], per truck capacities Pi e0i

∼

U [250 units/truck, 500 units/truck], per empty truck emissions

∼ U [25lbs CO2 /truck, 50lbs CO2 /truck], and emissions from unit load for trucks ei

∼

U [0.5lbs CO2 /unit, 1lbs CO2 /unit], where U [a, b] denotes a uniform distribution with lower and upper 13

bounds equal to a and b, respectively. Any generated problem instance is solved for 10 different values of carbon cap. Particularly, for the given problem instance, maximum value of carbon cap is determined by solving P-TL-Cap without the carbon cap constraint (i.e., maximization of H(Q, x)) and calculating the emissions under the corresponding solution; and minimum value of carbon cap is determined by minimum value of E(Q, x) (BARON is used for determining minimum and maximum values for carbon cap). Then, 10 values of carbon cap between maximum and minimum values are generated in equal increments.

4.1

Analysis of the Full Truckload Search Heuristic

To analyze the efficiency of FTSH, we compare it to BARON, which is a popular solver for MINLPs. Particularly, FTSH is coded in Matlab and P-TL-Cap is coded in GAMS and BARON is used as the solver option with default settings. Table 2 summarizes the average results of the problems solved for each problem size n (for any problem size, 1600 problems are solved with BARON and FTSH). In Table  2, Q is the total order quantity per order, ni=1 xi is the total number of trucks used per order, T T is the number of different truck types used per order, H(Q, x) is the retailer’s costs per unit time ($/year), E(Q, x) is the retailer’s emissions per unit time (CO2 lbs/unit), and cpu is the computational time in seconds. We have the following observations from Table 2. Table 2: Comparison of BARON and FTSH for varying n values n 5 10 15 20 avg

Q 4612.2 4602.7 4589.8 4599.4 4601.0

n

i=1 xi 10.47 10.10 10.27 9.81 10.16

BARON TT H(Q, x) 1.30 67904 1.33 67460 1.52 67468 1.54 66612 1.42 67361

E(Q, x) 107295 107218 107364 107074 107238

cpu 0.220 0.222 0.268 0.283 0.248

Q 4559.7 4588.8 4596.3 4606.1 4587.7

n

i=1 xi 10.34 10.06 10.29 9.84 10.13

FTSH TT H(Q, x) 1.29 67880 1.33 67466 1.54 67503 1.54 66631 1.43 67370

E(Q, x) 107287 107197 107355 107047 107222

cpu 0.028 0.073 0.149 0.171 0.106

• In terms of cpu time, FTSH is more efficient than BARON. Particularly, FTSH is almost 2.5 times faster than BARON on average. While the cpu time of BARON is relatively affected less with problem size (n) compared to FTSH, FTSH is still computationally more efficient compared to BARON for all problem sizes. • In terms of solution quality (H(Q, x)), BARON is slightly better than FTSH; however, there are problem instances where FTSH was able to find better quality solutions compared to BARON. • In terms of emissions of the solutions generated (E(Q, x)), FTSH finds solutions with less emissions compared to BARON. This can be explained as FTSH seeks to find the best solution with minimum emissions (as explained in Corollary 1). 14

 • Total order quantity (Q), number of trucks ( ni=1 xi ), and number of different trucks types used (T T ) do not have significant differences depending on the solution methodology. Based on these observations, it can be concluded that FTSH is an efficient solution method for P-TLCap. It might also be prefered over BARON due the fact that it, on average, generated solutions with less emissions while not increasing costs significantly. Finally, to show that efficiency of FTSH does not depend on problem characteristics, we compare BARON to FTSH for different data sets in Table 3. Table 3 documents the average statistics of problems solved for each data set (400 problems are solved for each data set). As can be observed from Table 3, the differences in computational times and the deviations from costs and emissions do not follow a specific pattern for different data sets. Table 3: Comparison of BARON and FTSH for different data sets Data Set 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 avg

4.2

Q 4586.9 3876.8 5174.4 4504.9 3762.9 3167.0 3807.1 3752.8 5580.3 5331.4 6394.7 5664.6 4534.5 3904.5 5086.5 4487.4 4601.0

n

i=1 xi 10.02 8.67 11.34 10.03 8.21 7.13 8.37 8.31 12.13 11.83 14.15 12.54 9.97 8.70 11.22 9.98 10.16

BARON TT H(Q, x) 1.70 66357 1.33 66109 1.31 65311 1.58 65625 1.45 67503 1.59 67726 1.16 66853 1.34 66822 1.46 67495 1.14 67346 1.53 67044 1.31 66703 1.65 69695 1.34 69414 1.35 68919 1.54 68855 1.42 67361

E(Q, x) 106129 107184 106975 108312 106260 107121 107436 108365 106276 107679 106949 108428 106176 107244 107027 108242 107238

cpu 0.257 0.227 0.237 0.286 0.244 0.270 0.262 0.232 0.236 0.225 0.248 0.228 0.229 0.237 0.267 0.284 0.248

Q 4590.6 3915.9 5099.9 4504.9 3658.1 3207.1 3803.6 3658.8 5535.7 5381.5 6405.9 5701.6 4549.9 3935.0 4968.0 4487.4 4587.7

n

i=1 xi 10.04 8.75 11.18 10.03 7.98 7.23 8.37 8.06 12.06 11.96 14.18 12.62 10.00 8.77 10.96 9.98 10.13

FTSH TT H(Q, x) 1.77 66337 1.34 66110 1.30 65320 1.58 65625 1.40 67507 1.63 67916 1.09 66871 1.33 66778 1.52 67461 1.16 67370 1.53 67044 1.34 66701 1.66 69692 1.36 69428 1.30 68902 1.54 68855 1.43 67370

E(Q, x) 106122 107170 106960 108312 106242 107111 107405 108288 106275 107666 106932 108415 106169 107232 107006 108242 107222

Analysis of Carbon Cap

To analyze the effects of carbon cap, we next document the problem statistics achieved with both BARON and FTSH for increasing values of carbon cap. Table 4 summarizes the average problem statistics for each point of carbon cap considered for each problem instances. In particular, as mentioned previously, each problem instance is solved with 10 different carbon cap values, increasing from minimum carbon cap to maximum carbon cap values in equal increments. In Table 4, the average results for each step of carbon cap is presented (for each carbon cap step, 640 problems are solved). The following observations are based on Table 4 and hold with both BARON and FTSH solutions. • As carbon cap increases, while the retailer’s cost per unit time decreases, the retailer’s emission 15

cpu 0.094 0.081 0.102 0.108 0.095 0.075 0.117 0.094 0.148 0.102 0.123 0.129 0.107 0.101 0.110 0.105 0.106

Table 4: Effects of carbon cap C with BARON and FTSH Average Cap 107081 107124 107168 107211 107255 107298 107342 107385 107429 107472 avg

Q 4551.3 4574.7 4586.7 4560.1 4582.8 4602.9 4619.9 4651.2 4658.0 4622.6 4601.0

n

i=1 xi 10.10 10.14 10.17 10.08 10.12 10.15 10.19 10.25 10.25 10.16 10.16

BARON TT H(Q, x) 1.56 67756 1.56 67627 1.55 67537 1.53 67428 1.49 67371 1.47 67304 1.43 67246 1.37 67174 1.27 67137 1.00 67031 1.42 67361

E(Q, x) 107039 107090 107135 107173 107214 107255 107297 107335 107372 107466 107238

cpu 0.261 0.241 0.250 0.252 0.245 0.264 0.250 0.247 0.254 0.218 0.248

Q 4586.0 4549.6 4583.2 4543.9 4549.9 4557.9 4609.0 4615.6 4655.3 4626.8 4587.7

n

i=1 xi 10.19 10.10 10.16 10.04 10.05 10.05 10.16 10.16 10.24 10.18 10.13

FTSH TT H(Q, x) 1.55 67770 1.54 67610 1.53 67542 1.51 67449 1.48 67390 1.46 67315 1.41 67257 1.36 67169 1.31 67118 1.12 67077 1.43 67370

E(Q, x) 107035 107076 107125 107163 107202 107252 107289 107328 107360 107386 107222

per unit time increases. This result is expected as the retailer’s feasible set of order quantities and truck choices increase as carbon cap increases and the retailer generates more emissions to reduce costs when carbon cap is larger. • While the number of trucks used for shipment does not follow an increasing or decreasing pattern with increasing carbon cap, the number of different truck types used decreases as carbon cap increases. That is, as carbon cap constraint get more restricting, the retailer prefers to use distinct trucks to better manage his/her emissions while avoiding high cost increases. To illustrate the changes in costs and emissions as carbon cap increases, Figure 1 depicts these changes under both BARON and FTSH solutions for different problem sizes (for each problem size and carbon cap step combination, 160 problems are solved). As can be seen in Figures 1a-1d, the retailer’s costs increase and emissions decrease as carbon cap increases. One may also observe that while the costs under BARON are less than the costs under FTSH, the emissions under BARON are more than FTSH.

Similarly, to illustrate the changes in number of trucks and number of truck types used as carbon cap increases, Figure 2 shows the changes under both BARON and FTSH solutions for different problem sizes (for each problem size and carbon cap step combination, 160 problems are solved). As can be seen in Figures 2a-2d, the number of trucks used by the retailer does not follow a specific pattern. On the other hand, the number of truck types used for shipment tend to decrease as carbon cap increases.

4.3

Analysis of Single Truck Type vs. Multiple Truck Types

In this section, through sample scenarios, we show the possible benefits of using multiple trucks in inbound shipment. In particular, in absence of the formulation approach of this study, a retailer 16

cpu 0.074 0.076 0.087 0.091 0.100 0.109 0.119 0.126 0.133 0.142 0.106

Figure 1: Carbon Cap vs. Costs and Emissions (a) n = 5

(b) n = 10

BARON H(Q,x) FTSH H(Q,x) BARON E(Q,x) FTSH E(Q,x)

6.81

6.78

1.074

6.77

1.0735

6.795

1.073

6.78

1.0725

6.765 6.75

2

4

6 Carbon Cap (C)

8

5

x 10

x 10 1.076 BARON E(Q,x) FTSH E(Q,x) BARON H(Q,x) FTSH H(Q,x)

6.76

1.073

6.74

1.072

1.072

6.73

1.071

1.0715 10

6.72

2

4

x 10

BARON E(Q,x) FTSH E(Q,x) BARON H(Q,x) FTSH H(Q,x)

1.073

6.72

1.0715 6 Carbon Cap (C)

8

Total Costs

6.74

4

1.07 10

5

x 10

x 10 1.074 BARON E(Q,x) FTSH E(Q,x) BARON H(Q,x) FTSH H(Q,x)

6.725 Total Emissions

Total Costs

1.076 1.0745

2

6.75

1.0775

6.76

6.7

8

4

x 10 1.079

6.78

6 Carbon Cap (C)

(d) n = 20 5

6.8

1.074

6.75

(c) n = 15 4

6.82

1.075

1.07 10

6.7

1.073 1.072

6.675

1.071

6.65

1.07

6.625

1.069

6.6

2

4

6 Carbon Cap (C)

8

Total Emissions

Total Costs

6.825

x 10 1.0745

Total Emissions

x 10

4

Total Costs

6.84

5

Total Emissions

4

1.068 10

can naively solve his/her integrated inventory control and inbound transportation problem with consideration of single truck type, that is, the retailer prefers to use a single truck type for his/her inbound transportation. In such a case, the retailer can solve P-TL-Cap with each single truck type and adopt the solution which gives the minimum costs among all solutions, each of which considers a different truck type. In Appendix 6.6, an exact solution method to solve P-TL-Cap with single truck type is presented. We refer to this approach as single truck-type shipment (S-TT-S). On the other hand, as mentioned in previous sections, the retailer can utilize different trucks for inbound transportation; hence, he/she would solve P-TL-Cap. We refer to this approach as multiple truck-types shipment (M-TT-S). It should be noted that the retailer’s total costs per unit time under M-TT-S will always be less than or equal to the retailer’s total costs per unit time under S-TT-S. On the other hand, the retailer’s emissions per unit time can increase or decrease due to M-TT-S. In what follows, we discuss two examples where M-TT-S not only reduces costs but also emissions† . †

Both of the examples are solved with BARON and FTSH, the solution with the minimum costs is assumed to be adopted by the retailer.

17

Figure 2: Carbon Cap vs. Number of Trucks and Truck Types (b) n = 10

Number of Trucks

10.8

1.5 1.4

10.6

1.3

10.4

1.2

10.2

1.1

2

3

4

5 6 7 Carbon Cap (C)

8

10.3

1 10

9

FTSH Truck Types BARON Truck Types FTSH Truck Number BARON Truck Number

1.3

10.1

1.2

10

1.1

9.9 1

2

3

4

1.6

10.3

1.4

10.2

1.2

Number of Trucks

10.4

4

5 6 7 Carbon Cap (C)

8

9

10.1

Number of Trucks

1.8 Number of Truck Types

FTSH Truck Types BARON Truck Types FTSH Truck Number BARON Truck Number

3

5 6 7 Carbon Cap (C)

8

1 10

9

(d) n = 20

10.5

2

1.4

10.2

(c) n = 15

10.1 1

1.5

1 10

FTSH Truck Types BARON Truck Types FTSH Truck Number BARON Truck Number

1.8

10

1.6

9.9

1.4

9.8

1.2

9.7 1

2

3

4

5 6 7 Carbon Cap (C)

8

9

Number of Truck Types

10 1

10.4

Number of Trucks

FTSH Truck Number BARON Truck Number FTSH Truck Types BARON Truck Types

Number of Truck Types

11

Number of Truck Types

(a) n = 5

1 10

Example 1 Consider the following problem parameters: λ = 50, 000 units/year, p = $1/unit, A =  = 200lbs CO2 /order,  h = 1lbs CO2 /unit/year. $150/order, h = $1.5/unit/year, p = 1lbs CO2 /unit, A Suppose that the retailer can use 3 different truck types such that each truck has capacity of 500 units, i.e., Pi = 500units/truck ∀i = 1, 2, 3 and each truck type’s emission from carrying a unit is 1, i.e., ei = 1lbs CO2 /unit ∀i = 1, 2, 3. Table 5 presents the solution of the retailer’s integrated inventory control and inbound transportation problem with carbon cap constraint, where C = 108850lbs CO2 /year. Table 5: Solution of Example 1 Truck Type i 1 2 3

Ri 120 100 130

e0i

40 50 30

xi 6 0 0

qi 3000 0 0

S-TT-S H(Q, x)

E(Q, x)

66750

108830

xi 0 5 3

qi 0 2500 1500

M-TT-S H(Q, x)

E(Q, x)

66000

108750

As can be seen in Table 5, when the retailer decides to use single truck type for inbound transportation, he/she will order 3000 units and use 6 trucks of type 1 for inbound transportation. 18

The resulting annual costs and emissions are $66,750 and 108,830 lbs of CO2 , respectively. On the other hand, if the retailer decides to use different truck types for inbound transportation, he/she will order 4000 units in total and use 5 trucks of type 2 to ship 2,500 units and 3 trucks of type 3 to ship 1,500 units. The resulting annual costs and emissions are $66,000 and 108,750 lbs of CO2 , respectively. That is, both costs and emissions are reduced by considering using different truck types for the inbound transportation. Example 2 λ = 50, 000 units/year, p = $1/unit, A = $300/order, h = $0.75/unit/year, p =  = 100lbs CO2 /order,  h = 0.5lbs CO2 /unit/year. Suppose that the retailer can 1lbs CO2 /unit, A use 3 different truck types such that empty truck’s emissions generation is 40 for each truck type, i.e., e0i = 40lbs CO2 /truck ∀i = 1, 2, 3. Table 6 presents the solution of the retailer’s integrated inventory control and inbound transportation problem with carbon cap constraint, where C = 94900lbs CO2 /year. Table 6: Solution of Example 2 Truck Type 1 2 3

Ri 110 100 120

Pi 500 500 400

ei 0.770 0.804 0.567

xi 13 0 0

qi 6500 0 0

S-TT-S H(Q, x)

E(Q, x)

65745

94894

xi 1 9 2

qi 500 4500 800

M-TT-S H(Q, x)

E(Q, x)

65537

94860

As can be seen in Table 6, when the retailer decides to use single truck type for inbound transportation, he/she will order 6500 units and use 13 trucks of type 1 for inbound transportation. The resulting annual costs and emissions are $65,745 and 94,894 lbs of CO2 , respectively. On the other hand, if the retailer decides to use different truck types for inbound transportation, he/she will order 5800 units in total and use 1 truck of type 1 to ship 500 units, 9 trucks of type 2 to ship 4,500 units, and 2 trucks of type 3 to ship 800 units. The resulting costs and emissions per unit time are $65,537 and 94,860 lbs of CO2 , respectively. Similar to Example 1, both costs and emissions are reduced by considering using different truck types for the inbound transportation.

5

Conclusions and Future Research

This paper studies an integrated inventory control and truckload transportation with carbon emissions considerations.

An EOQ model is formulated with explicit transportation costs.

A carbon cap

constraint is considered to limit the emissions from inventory holding, order placement, and truckload transportation.

We consider different truck types in modeling truckload transportation costs and

emissions. Specifically, to capture the fact that different truck types have distinct characteristics, each 19

truck type is considered to have distinct per truck capacities and per truck costs as well as distinct emissions generations. In particular, each truck type is assumed to generate a specific amount of emissions due to its empty vehicle weight and the rates of emissions generated due to the loads of the trucks depend on the truck type. The resulting integrated inventory control and truckload transportation problem with heterogeneous truck types subject to carbon cap constraint is a mixed-integer-nonlinear programming model. The complexity of this problem is shown to be NP-hard; thus, heuristic methods are provided to solve this problem. Particularly, an heuristic local search algorithm is proposed based on the properties of the problem. For the special case with single truck type, an exact solution method is provided in the Appendix. The solution of the problem with single truck type has been utilized in the starting process of the local search heuristic. Through comparing the heuristic method to a commercial solver (BARON) over a set of numerical studies, it is observed that the heuristic method is efficient in terms of solution time and it finds good quality solutions. It should also be noted that the solutions found by the heuristic method result in less carbon emissions. Another set of numerical studies is conducted to analyze the effects of carbon cap on the retailer’s integrated inventory control and transportation decisions. As expected, as carbon cap increases, the retailer’s cost decrease while the emissions increase. It is observed that, as the carbon cap gets tighter, the retailer tends to increase the number of different truck types used for inbound transportation. On the other hand, the number of trucks used for shipment does not follow an increasing or decreasing pattern as carbon cap increases. Furthermore, through two sample scenarios, it is discussed that considering heterogeneous trucks for inbound transportation not only decreases costs but also reduces the emissions. This study contributes to the body of literature on environmentally sensitive inventory models by integrating practical aspects of truckload transportation with heterogeneous trucks. There is a growing awareness on emissions throughout supply chains and freight transportation, especially, is the major contributor to emissions throughout supply chains. It is also known that freight trucks are the most common transportation mode. It is, therefore, important to model freight truck costs as well as freight truck emissions in inventory control models as inventory control determines the freight amounts shipped throughout the supply chains. We believe that the modeling approach of this study and the analyses of the formulated model along with the insights gained will pioneer modeling integrated inventory control and transportation problems with environmental considerations. One of the possible future research directions is to study inventory control models with further generalized transportation models. For instance, different transportation modes, freight discounts, and nonlinear transportation costs can be analyzed in environmentally sensitive integrated inventory 20

control and transportation models. One can also analyze multi-item or multi-echelon inventory systems with truckload transportation. For instance, it is important to analyze the effects of vendor-manageddelivery on carbon emissions. Another research direction is to study stochastic inventory control with environmental considerations and integrated transportation decisions. As discussed in Section 1, there are limited studies focusing on environmentally sensitive inventory models with stochastic demand and these studies focus on the single-period inventory decisions (see, e.g., Song and Leng, 2012, Hoen et al., 2012). It is an important research area to study continuous or periodic inventory review systems (such as (Q, R) or (s, S) inventory models) with environmental considerations integrated with transportation. The modeling approach and findings of this study can be used in these aforementioned studies. Finally, the tools provided throughout this study can be used in policy development for truck weight limits. For instance, McKinnon (2005) study the effects of increasing freight vehicle loads in U.K. on environment. Through statistical analyses, it is noted that increasing freight loads has environmental benefits. The current study can be used in finding analytical results on the effects of truck characteristics on emissions.

References Absi, N., Dauzere-Peres, S., Kedad-Sidhoum, S., Penz, B., Rapine, C., 2013. Lot sizing with carbon emission constraints. European Journal of Operational Research 227, 55–61. Arslan, M. C., Turkay, M., January 2013. EOQ revisited with sustainability considerations, Working paper, downloaded from http://home.ku.edu.tr/ mturkay/pub/EOQ Sustainability.pdf. Aucamp, D., 1982. Nonlinear freight costs in the EOQ problem. European Journal of Operational Research 9, 61–62. Bae, S. H., Sarkis, J., Yoo, C. S., 2011. Greening transportation fleets: Insights from a two-stage game theoretic model. Transportation Research Part E 47, 793–807. Bektas, T., Laporte, G., 2011. The pollution-routing problem. Transportation Research 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 Transactions on Automation Science and Engineering PP (99), 1. Bouchery, Y., Ghaffari, A., Jemai, Z., Dallery, Y., 2011. Adjust or invest: What is the best option to green a supply chain? Tech. rep., Laboratoire Genie Industriel, Ecole Centrale Paris. 21

Bouchery, Y., Ghaffari, A., Jemai, Z., Dallery, Y., 2012. Including sustainability criteria into inventory models. European Journal of Operational Research 222, 229–240. CBO, 2008. Policy options for reducing CO2 emissions. Tech. rep., Congressional Budget Office, Congress of the United States. Chen, X., Benjaafar, S., Elomri, A., 2013. The carbon constrained EOQ. Operations Research Letters 41, 172–179. Corbett, C. J., Kleindorfer, P. R., 2001a. Environmental management and operations management: introduction to part 1 (manufacturing and eco-logistics). Production and Operations Management 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). Production and Operations Management 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. de Kok, T. G., 2000. Capacity allocation and outsourcing in a process industry. International Journal of Production Economics 68, 229–239. Dekker, R., Bloemhof, J., Mallidis, I., 2012. Operations research for green logistics an overview of aspects, issues, contributions and challenges. European Journal of Operational Research 219, 671– 679. Demir, E., Bektas, T., Laporte, G., 2011. A comparative analysis of several vehicle emission models for road freight transportation. Transportation Research Part D 16, 347–357. Demir, E., Bektas, T., Laporte, G., 2012. An adaptive large neighborhood search heuristic for the pollution-routing problem. European Journal of Operational Research 223, 346–359. EEA,

2013.

Greenhouse

gas

-

data

viewer

2010.

European

Environment

Agency,

http://www.eea.europa.eu/data-and-maps/data/data-viewers/greenhouse-gases-viewer. EPA, 2013. Inventory of U.S. greenhouse gas emissions and sinks:

1990-2010. Tech. rep., U.S.

Environmental Protection Agency, 1200 Pennsylvania Ave., N.W. Washington, DC 20460 U.S.A. 22

Erdogan, S., Miller-Hooks, E., 2012. A green vehicle routing problem. Transportation Research Part E 48, 100–114. FHWA, 2008. Freight facts and figures 2008. Tech. rep., U.S. Department of Transportation, Federal Highway Administration. Hoen, K. M. R., Tan, T., Fransoo, J. C., van Houtum, G. J., 2012. Effect of carbon emission regulations on transport mode selection under stochastic demand. Flexible Services and Manufacturing Journal http://dx.doi.org/10.1007/s10696-012-9151-6, 1–26. Hua, G., Cheng, T., Wang, S., 2011a. Managing carbon footprints in inventory management. International Journal of Production Economics 132 (2), 178–185. Hua, G., Qiao, H., Li, J., March 2011b. Optimal order lot sizing and pricing with carbon trade, available at SSRN: http://ssrn.com/abstract=1796507 or http://dx.doi.org/10.2139/ssrn.1796507. Jabali, O., Woensel, T. V., de Kok, A., 2012. Analysis of travel times and CO2 emissions in timedependent vehicle routing. Production and Operations Management 21, 1060–1074. Jaber, M. Y., Glock, C. H., Saadany, A. M. E., 2013. Supply chain coordination with emissions reduction incentives. International Journal of Production Research 51, 68–82. Jiang, der

Y., green

Klabjan, house

D., gas

2012. emissions

Optimal

emissions

regulations,

reduction

retrieved

from

investment (March

un2013):

http://www.klabjan.dynresmanagement.com/articles/CarbonRegulations.pdf. Kleindorfer, P. R., Singhal, K., Wassenhove, L. N. V., 2005. Sustainable operations management. Production and Operations Management 14, 482–492. Konur, D., Toptal, A., 2012. Analysis and applications of replenishment problems under stepwise transportation costs and generalized wholesale prices. International Journal of Production Economics 140 (1), 521–529. Lee, C.-Y., 1986. The economic order quantity for freight discount costs. IIE Transactions 18 (3), 318– 320. Leonardi, J., Baumgartner, M., 2004. CO2 efficiency in road freight transportation: Status quo, measures and potential. Transportation Research Part D 9, 451–464.

23

Letmathe, P., Balakrishnan, N., 2005. Environmental considerations on the optimal product mix. European Journal of Operational Research 167, 398–412. Ligterink, N. E., Tavasszy, L. A., de Lange, R., 2012. A velocity and payload dependent emission model for heavy-duty road freight transportation. Transportation Research Part D 17, 487–491. Lin, J., 1976. Multiple-objective problems: Pareto-optimal solutions by method of proper equality constraints. IEEE Transactions on Automatic Control 21 (5), 641–650. Linton, J. D., Klassen, R., Jayaraman, V., 2007. Sustainable supply chains: An introduction. Journal of Operations Management 25, 1075–1082. Liu, Z. L., Anderson, T. D., Cruz, J. M., 2012. Consumer environmental awareness and competition in two-stage supply chains. European Journal of Operational Research 218, 602–613. Loebich, M., Donval, Y., Houot, X., 2011. Green supply chain:

from awareness to ac-

tion. White paper, BearingPoint Management and Technology Consultants, downloaded from: http://www.bearingpoint.com/en-uk/download/TAP-SC EN.pdf. Mallidis, I., Dekker, R., Vlachos, D., 2010. Greening supply chains: impact on cost and design. Tech. Rep. EI2010-39b, Report Econometric Institute. McKinnon, A. C., 2005. The economic and environmental benefits of increasing maximum truck weight: the british experience. Transportation Research Part D 10, 77–95. Pan, S., Ballot, E., Fontane, F., 2010. The reduction of greenhouse gas emissions from freight transport by pooling supply chains. International Journal of Production Economics In Press. Papadimitriou, C. H., 1981. On the complexity of integer programming. Journal of the Association for Computing Machinery 28, 765–768. Reed, B. D., Smas, M. J., Rzepka, R. A., Guiffrida, A. L., 2010. Introducing green transportation costs in supply chain modeling. In: Proceedings of the First Annual Kent State International Symposium on Green Supply Chains. pp. 189–197. Rogner, H.-H., Zhou, D., Crabbe, R. B. P., Edenhofer, O., Hare, B., Kuijpers, L., Yamaguch, M., 2007. Contribution of Working Group III to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, 2007. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA. 24

Sarkis, J., Zhu, Q., hung Lai, K., 2011. An organizational theoretic review of green supply chain management literature. International Journal of Production Economics 130, 1–15. Sbihi, A., Eglese, R. W., 2010. Combinatorial optimization and green logistics. Annals of Operations Research 175, 159–175. Song, J., Leng, M., 2012. Analysis of the single-period problem under carbon emissions policies. In: Choi, T.-M. (Ed.), Handbook of Newsvendor Problems. Vol. 176 of International Series in Operations Research and Management Science. Springer New York. Srivastava, S. K., 2007. Green supply-chain management:

a state-of-the-art literature review.

International Journal of Management Reviews 9, 53–80. Suzuki, Y., 2011. A new truck-routing approach for reducing fuel consumption and pollutants emission. Transportation Research Part D 16, 73–77. Toptal, A., 2009. Replenishment decisions under an all-units discount schedule and stepwise freight costs. European Journal of Operational Research 198, 504–510. Toptal, A., Bingol, S. O., 2011. Transportation pricing of a truckload carrier. European Journal of Operational Research 214, 559–567. Toptal, A., C ¸ etinkaya, S., 2006. Contractual agreements for coordination and vendor-managed delivery under explicit transportation considerations. Naval Research Logistics 53 (5), 397–417. Toptal, A., C ¸ etinkaya, S., Lee, C.-Y., 2003. The buyer-vendor coordination problem: Modeling inbound and outbound cargo capacity and costs. IIE Transactions 35 (11), 987 – 1002. Toptal, A., Ozlu, H., Konur, D., 2014. Joint decisions on inventory replenishment and emission reduction investment under different emission regulations. International Journal of Production Research 52 (1), 243–269. ¨ u, A., 2012. Dare to care: Shipment consolidation reduces not only costs, but also environmental Ulk¨ damage. International Journal of Production Economics 139 (2), 438–446. UNFCCC, 1997. Kyoto Protocol. United Nations Framework Convetion on Climate Change, accessed December 2012 http://unfccc.int/resource/docs/convkp/kpeng.pdf.

25

Zavanella,

L. E.,

Zanoni,

S.,

Mazzoldi,

L.,

Jaber,

M. Y.,

2013. A joint economic lot

size model with price and environmental sensitive demand, retrieved from (March 2013): http://www.medifas.net/IGLS/Papers2012/Paper146.pdf.

6 6.1

Appendix Notation and Possible Metrics Notation λ: p: A: h: C: p:  A:

 h: i: Pi : Ri : ei : e0i : qi : Q: Q: xi : x: H(Q, x): E(Q, x):

6.2

Description Demand rate Per unit procurement cost Fixed order setup cost Inventory holding cost per unit per unit time Maximum emissions allowed per unit time Emissions due to per unit procurement

Metric units/year $/unit $/order $/unit/year CO2 lbs/year CO2 lbs/unit

Emissions due to order placement

CO2 lbs/order

Emissions due to inventory holding per unit per unit time Index used for different truck types Per truck capacity for trucks of type i Per truck cost for trucks of type i Emissions due to shipping one unit with type i trucks Emissions due to empty-weight of type i truck Total order quantity shipped by type i trucks per order Retailer’s total order quantity per order n-vector of qi values Number of type i trucks used per order n-vector of xi values Retailer’s total costs per unit time Retailer’s total emissions per unit time

CO2 lbs/unit/year i = {1, 2, 3, . . . , n} units/truck $/truck CO2 lbs/unit CO2 lbs/truck units/order  Q = ni=1 qi units/order Q = [q1 , q2 , . . . , qn ]t trucks x = [x1 , x2 , . . . , xn ]t $/year CO2 lbs/year

Proof of Theorem 1

Suppose that there exists an empty truck in the optimum solution (Q∗ , x∗ ) of P-TL-Cap such that x∗j > 0 and qj∗ ≤ (x∗j − 1)Pj for some j, j ∈ {1, 2, . . . , n}. Now, consider (Q∗∗ , x∗∗ ) such that qi∗∗ = qi∗ ∗ ∗∗ ∗ ∀i, i ∈ {1, 2, . . . , n} and x∗∗ i = xi ∀i, i = j, i ∈ {1, 2, . . . , n} and xj = xj − 1. It then follows that

(Q∗∗ , x∗∗ ) is a feasible solution of P-TL-Cap. Furthermore, one can easily discuss that H(Q∗ , x∗ ) > H(Q∗∗ , x∗∗ ). Thus, (Q∗ , x∗ ) is not optimum, which is a contradiction.

26



6.3

Proof of Theorem 2

Suppose that (Q∗ , x∗ ) is optimum such that qi∗ = x∗i Pi ∀i, i = j, i = k, (x∗j − 1)Pj < qj∗ ≤ x∗j Pj , and (x∗k − 1)Pk < qk∗ ≤ x∗k Pk . Without loss of generality, let ek < ej . Now, consider (Q∗∗ , x∗∗ ) such that ∗ ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ qi∗∗ = qi∗ ∀i, i = j, i = k and x∗∗ i = xi ∀i, i ∈ {1, 2, . . . , n} and let qk = xk Pk and qj = qj − (xk Pk − qk ).

Note that Q∗ = Q∗∗ and x∗ = x∗∗ . It then follows that H(Q∗ , x∗ ) = H(Q∗∗ , x∗∗ ). Furthermore, as ek < ej , it follows from Equation (1) that E(Q∗ , x∗ ) ≥ E(Q∗∗ , x∗∗ ). Since, (Q∗ , x∗ ) is optimum and feasible, E(Q∗ , x∗ ) ≤ C; thus, we have E(Q∗∗ , x∗∗ ) ≤ C as well. That is, (Q∗∗ , x∗∗ ) is feasible and have the same objective function value with (Q∗ , x∗ ). Therefore, (Q∗∗ , x∗∗ ) is also an optimum solution of P-TL-Cap such that qi∗ = x∗i Pi ∀i, i = j and (x∗j − 1)Pj < qj∗ ≤ x∗j Pj only for truck type j.

6.4



Proof of Lemma 1

We first prove convexity of H(vj ∗ |x = x∗ ). One can show that   n  ∗ 2λ A + xi Ri ∂ 2 H(vj ∗ |x = x∗ ) i=1 = 3 . n ∂vj2∗  x∗i Pi − Pj ∗ + vj ∗ i=1

It follows from the above equality that

∂ 2 H(vj ∗ |x=x∗ ) ∂vj2∗

≥ 0 for vj ∗ ≥ 0, which proves convexity of H(vj ∗ |x =

x∗ ). Similarly, one can show that   n  + 2λ A x∗i (e0i + ei Pi ) − ej ∗ Pj ∗ + ej ∗ vj ∗ 2 ∗ 2λej ∗ ∂ E(vj ∗ |x = x ) i=1 = −  n 3 2 . 2 n ∂vj ∗   ∗ ∗ xi Pi − Pj ∗ + vj ∗ xi Pi − Pj ∗ + vj ∗ i=1

i=1

To establish a contradiction, let us assume that  + A

n  i=1

x∗i (e0i + ei Pi ) − ej ∗ Pj ∗ + ej ∗ vj ∗ n

∗ 0 i=1 xi ei +

 ei −ej ∗ ≥ 0 ∀i, i ∈ {1, 2, . . . , n}. That is, A+ Therefore,

6.5

< 0 or vj ∗ ≥ 0. It then follows that



+ The above inequality implies that A ∂ 2 E(vj ∗ |x=x∗ ) ∂vj2∗

∂ 2 E(vj ∗ |x=x∗ ) ∂vj2∗

< ej ∗

∗ i=1 xi (ei

∗ 0 i=1 xi ei +

n  i=1

n

n



n

 x∗i Pi − Pj ∗ + vj ∗

.

− ej ∗ )Pi < 0. Nevertheless, by definition,

∗ i=1 xi (ei −ej ∗ )Pi

≥ 0, which proves convexity of E(vj ∗ |x = x∗ ).

> 0, which is a contradiction. 

Proof of Theorem 3

First note that w0 is derived by solving

∂H(vj ∗ |x=x∗ ) ∂vj ∗

= 0. The result then follows from convexity of

H(vj ∗ |x = x∗ ). In particular, if w0 < max{0, w1 } + , H(vj ∗ |x = x∗ ) is increasing over max{0, w1 } +  ≤ 27

vj ∗ ≤ min{Pj ∗ , w2 } and vj∗∗ = max{0, w1 } + ; if min{Pj ∗ , w2 } < w0 , H(vj ∗ |x = x∗ ) is decreasing over max{0, w1 } +  ≤ vj ∗ ≤ min{Pj ∗ , w2 } and vj∗∗ = min{Pj ∗ , w2 }; if max{0, w1 } +  ≤ w0 ≤ min{Pj ∗ , w2 }, H(vj ∗ |x = x∗ ) is minimized at w0 and vj∗∗ = w0 .

6.6



Solution of P-TL-Cap with Single Truck Type

Suppose that the retailer can only choose a single truck type for inbound transportation and let this truck type be i. Then, P-TL-Cap with truck type i only, P-TL-Capi , reads   hQ Q Ri λ + + (P-TL-Capi ) : min Hi (Q) = pλ + Aλ Q 2 Pi  Q0  Q ei λ s.t. Ei (Q) = ( p + ei )λ + Aλ Q + Pi Q ≤C Q ≥ 0.

First, we note that Hi (Q) is minimized by ordering Qc such that i , (k + 1)Pi }), Hi (kPi )}, Qc = arg min{Hi (min{Q k+1 i where Q k+1 =



2(A + (k + 1)Ri )λ/h and k is the unique integer such that kPi <

(7) 

2Aλ h

≤ (k + 1)Pi .

The proof of Equation (7) follows from the piecewise convex structure of Hi (Q). We refer the reader to Toptal et al. (2003) for detailed discussion on derivation of Equation (7). Following a similar method, Ei (Q) is minimized by ordering Qe such that i

Qe = arg min{Ei (min{Qn+1 , (n + 1)Pi }), Ei (nPi )}, where

i Qn+1

 =

2(A + (n + 1)e0i )λ/ h and n is the unique integer such that nPi <

(8) 

 2Aλ  h

≤ (n + 1)Pi .

Note that, for feasibility of P-TL-Capi , it should be the case that Ei (Qe ) ≤ C; hence, we assume that Ei (Qe ) ≤ C. In the following discussion, we utilize the piecewise convex structures of Hi (Q) and Ei (Q) functions to find the solution of P-TL-Capi . In particular, for ( − 1)P < Q ≤ P such that  ∈ {1, 2, . . .}, Hi (Q) and Ei (Q) are defined by 

hQ Ri λ  p + ei )λ + Aλ Hi (Q) = pλ + Aλ Q + 2 + Q and Ei (Q) = ( Q +

e0i λ Q ,

respectively. Note that these functions

are convex and Ei (Q) ≤ C for qi1 ≤ Q ≤ qi2 such that   + e0 ) C − eλ − (C − ( p + ei )λ)2 − 2 hλ(A i , qi1 =  h   + e0 ) C − eλ + (C − ( p + ei )λ)2 − 2 hλ(A i . qi2 =  h

(9) (10)

Now let us define Si = {(( − 1)Pi , P ] ∩ [qi1 , qi2 ]}, that is, Si defines the set of feasible order quantities qi1 , qi1 ]. Then, Hi (Q) that can be carried with  trucks of type i. For notational simplicity, let Si = [ ∗()

over Q ∈ Si is minimized by Qi

, where 28

∗()

Qi

i

where Q =



=

⎧ ⎪ q1 ⎪ ⎪ ⎨ i

Qeoq t1 ⎪ ⎪ ⎪ ⎩ q1 i

i

if Q < qi1 , i

if qi1 ≤ Q ≤ qi2 ,

(11)

i

if qi1 < Q ,

 + e0 )λ/ 2(A h. Equation (11) follows from the convexity of Hi (Q) over Si . i ∗()

A solution method for P-TL-Capi is already implied by Equation (11): one can find Hi (Qi

)

values for  = {1, 2, . . . , M } and compare them, where M is the maximum number of trucks of type i that can be used for inbound transportation. M can be calculated as follows. It follows from Equations (+1)1

(9) and (10) that qi1 ≤ qi

(+2)2

(1)1

≤ qi2 , which indicates is not  (1)1  that any Q such that Q ≥ qi q feasible for P-TL-Capi . Therefore, one can define M = iPi . Then, the solution of P-TL-Capi , Q∗i is ≤ qi

given by ∗(1)

Q∗i = arg min{Hi1 (Qi

∗(2)

), Hi2 (Qi

29

∗(M )

), . . . , HiM (Qi

)}.

(12)