Author's Accepted Manuscript

Carbon Constrained Integrated InventoryControl and Truckload Transportation withHeterogeneous Freight Trucks

Dinçer Konur

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

To appear in: Int. J. Production Economics

Received date: 15 July 2013Accepted date: 8 March 2014

Cite this article as: Dinçer Konur, Carbon Constrained Integrated InventoryControl and Truckload Transportation with Heterogeneous Freight Trucks, Int.J. Production Economics, http://dx.doi.org/10.1016/j.ijpe.2014.03.009

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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 environmen-tal considerations. Particularly, explicit transportation modeling is included with inventory controldecisions to capture per truck costs and per truck capacities. Furthermore, a carbon cap constrainton the total emissions is formulated by considering emission characteristics of various trucks thatcan 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 studiesillustrate the efficiency of the proposed method. Furthermore, numerical examples are presented toshow that both costs and emissions can be reduced by considering heterogeneous trucks for inboundtransportation.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 (multi-

objective 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.

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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 Cetinkaya, 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 less-

than-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 H(Q) = pλ+ AλQ + hQ

2

s.t. Q ≥ 0.

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

2Aλh , 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

Cetinkaya, 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 H(Q,x) = pλ+ AλQ + hQ

2 + λQ

n∑i=1

xiRi

s.t. Q ≤n∑

i=1

xiPi

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

(∑n

i=1 xiRi 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

A as the emissions generated per order (CO2 lbs/order), h as the emissions due to holding one unit

in inventory per unit time (CO2 lbs/item/year), and p as the emissions per unit due to procurement

(CO2 lbs/unit). We note that A is defined as the emissions generated by an empty truck in Hua

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).

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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 =∑n

i=1 qi and the total carbon emissions generated

per unit time amount to

E(Q,x) = pλ+Aλ

Q+

hQ

2+

λ

Q

(n∑

i=1

xie0i +

n∑i=1

eiqi

), (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+ AλQ + hQ

2 + λQ

n∑i=1

xiRi

s.t. E(Q,x) = pλ+AλQ +

hQ2 + λ

Q

(n∑

i=1

xie0i +

n∑i=1

eiqi

)≤ C

Q =n∑

i=1

qi,

qi ≤ xiPi ∀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 P-

TL-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

emissions, especially, in warehousing activities. Ulku (2012), for instance, notes that emissions can be

reduced with considering different packaging technologies and sizes. Under LTL transportation for the

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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 < q∗i ≤ x∗iPi, ∀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 q∗i = x∗iPi ∀i, i �= j and (x∗j − 1)Pj <

q∗j ≤ x∗jPj 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∗ = argmini=1,2,...,n{ei}. Then, there exists an optimum solution (Q∗,x∗) such that

q∗i = x∗iPi ∀i, i �= j∗ and (x∗j∗ − 1)Pj∗ < q∗j∗ ≤ 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 q∗∗i = x∗∗i Pi ∀i, i �= j∗ and (x∗∗j∗ − 1)Pj∗ < q∗∗j∗ ≤ x∗∗j∗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∗ =

argmini=1,2,...,n{ei}. It then follows that q∗i = x∗iPi ∀i, i �= j∗ and q∗j∗ = (x∗j∗ − 1)Pj∗ + v∗j∗ where

v∗j∗ 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 v∗j∗ to

find Q∗, and let it be denoted by Q∗(x∗).

Now, let us define the following functions:

H(vj∗ |x = x∗) = pλ+

λ

(A+

n∑i=1

x∗iRi

)(

n∑i=1

x∗iPi − Pj∗ + vj∗

) +h

2

(n∑

i=1

x∗iPi − Pj∗ + vj∗

)(2)

E(vj∗ |x = x∗) = pλ+

λ

(A+

n∑i=1

x∗i (e0i + eiPi)− ej∗Pj∗ + ej∗vj∗

)(

n∑i=1

x∗iPi − Pj∗ + vj∗

) +h

2

(n∑

i=1

x∗iPi − 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 v∗j∗ given x∗ = x∗, and, therefore, solves Q∗(x∗):

(P-TL-Cap(j∗)) min H(vj∗|x = x∗)s.t. E(vj∗ |x = x∗) ≤ C

ε ≤ vj∗ ≤ Pj∗,

where ε 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 v∗j∗. 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

w1 =Φ−

√Φ2 − 2hΨ

h, (4)

w2 =Φ+

√Φ2 − 2hΨ

h, (5)

where Φ = C − h (∑n

i=1 x∗iPi − Pj∗) − λ(p + ej∗) and Ψ = λA + λp (

∑ni=1 x

∗iPi − Pj∗) +

h (∑n

i=1 x∗iPi − Pj∗)

2 /2 + λ(∑n

i=1 x∗i (e

0i + eiPi)− ej∗Pj∗

) − C (∑n

i=1 x∗iPi − Pj∗). This implies the

following reformulation of P-TL-Cap(j∗):

(P-TL-Cap(j∗)) min H(vj∗ |x = x∗)s.t. max{0, w1}+ ε ≤ 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

w0 =

√√√√√√2λ

(A+

n∑i=1

x∗iRi

)h

+ Pj∗ −n∑

i=1

x∗iPi. (6)

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 v∗j∗ be the optimum solution of P-TL-Cap(j∗).

Then

v∗j∗ =

⎧⎪⎪⎨⎪⎪⎩max{0, w1}+ ε if w0 < max{0, w1}+ ε,

w0 if max{0, w1}+ ε ≤ w0 ≤ min{Pj∗ , w2},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) = x0iRi ∀i, i �= j∗ and

qj∗(x0) = (x0j∗ − 1)Rj∗ + vj∗(x

0), where vj∗(x0) is the solution of P-TL-Cap(j∗) under x0. If P-TL-

Cap(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

Step 1.2. Set x[−i] = x and if x[−i]i > 0, set x

[−i]i = x

[−i]i − 1

Step 1.3. Solve P-TL-Cap(j∗) under x[−i]

Step 1.4. If it is infeasible, set H(Q∗(x[−i]),x[−i]) = ∞Step 1.5. End

Step 1.6. x[−1] = argmin{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

Step 2.2. Set x[+i] = x and let x[+i]i = x

[+i]i + 1

Step 2.3. Solve P-TL-Cap(j∗) under x[+i]

Step 2.4. If it is infeasible, set H(Q∗(x[+i]),x[+i]) = ∞Step 2.5. End

Step 2.6. x[+1] = argmin{H(Q∗(x[+i]),x[+i])}

Step 3: Let x = argmin{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},h = {$0.75/unit/year, $1.5/unit/year}, A = {100lbs of CO2/order, 200lbs CO2/order}, and h =

{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 A h A h Data Set A h A h

1 150 0.75 100 0.5 9 300 0.75 100 0.52 150 0.75 100 1 10 300 0.75 100 13 150 0.75 200 0.5 11 300 0.75 200 0.54 150 0.75 200 1 12 300 0.75 200 15 150 1.5 100 0.5 13 300 1.5 100 0.56 150 1.5 100 1 14 300 1.5 100 17 150 1.5 200 0.5 15 300 1.5 200 0.58 150 1.5 200 1 16 300 1.5 200 1

For each data set, four different problem sizes are considered: n = 5, n = 10,

n = 15, and n = 20. 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 ∼ U [250 units/truck, 500 units/truck], per empty truck emissions

e0i ∼ 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,∑n

i=1 xi is the total number of trucks used per order, TT 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

BARON FTSHn Q

∑ni=1 xi TT H(Q,x) E(Q,x) cpu Q

∑ni=1 xi TT H(Q,x) E(Q,x) cpu

5 4612.2 10.47 1.30 67904 107295 0.220 4559.7 10.34 1.29 67880 107287 0.02810 4602.7 10.10 1.33 67460 107218 0.222 4588.8 10.06 1.33 67466 107197 0.07315 4589.8 10.27 1.52 67468 107364 0.268 4596.3 10.29 1.54 67503 107355 0.14920 4599.4 9.81 1.54 66612 107074 0.283 4606.1 9.84 1.54 66631 107047 0.171avg 4601.0 10.16 1.42 67361 107238 0.248 4587.7 10.13 1.43 67370 107222 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 (∑n

i=1 xi), and number of different trucks types used

(TT ) 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-TL-

Cap. 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 BARON FTSHSet Q

∑ni=1 xi TT H(Q,x) E(Q,x) cpu Q

∑ni=1 xi TT H(Q,x) E(Q,x) cpu

1 4586.9 10.02 1.70 66357 106129 0.257 4590.6 10.04 1.77 66337 106122 0.0942 3876.8 8.67 1.33 66109 107184 0.227 3915.9 8.75 1.34 66110 107170 0.0813 5174.4 11.34 1.31 65311 106975 0.237 5099.9 11.18 1.30 65320 106960 0.1024 4504.9 10.03 1.58 65625 108312 0.286 4504.9 10.03 1.58 65625 108312 0.1085 3762.9 8.21 1.45 67503 106260 0.244 3658.1 7.98 1.40 67507 106242 0.0956 3167.0 7.13 1.59 67726 107121 0.270 3207.1 7.23 1.63 67916 107111 0.0757 3807.1 8.37 1.16 66853 107436 0.262 3803.6 8.37 1.09 66871 107405 0.1178 3752.8 8.31 1.34 66822 108365 0.232 3658.8 8.06 1.33 66778 108288 0.0949 5580.3 12.13 1.46 67495 106276 0.236 5535.7 12.06 1.52 67461 106275 0.14810 5331.4 11.83 1.14 67346 107679 0.225 5381.5 11.96 1.16 67370 107666 0.10211 6394.7 14.15 1.53 67044 106949 0.248 6405.9 14.18 1.53 67044 106932 0.12312 5664.6 12.54 1.31 66703 108428 0.228 5701.6 12.62 1.34 66701 108415 0.12913 4534.5 9.97 1.65 69695 106176 0.229 4549.9 10.00 1.66 69692 106169 0.10714 3904.5 8.70 1.34 69414 107244 0.237 3935.0 8.77 1.36 69428 107232 0.10115 5086.5 11.22 1.35 68919 107027 0.267 4968.0 10.96 1.30 68902 107006 0.11016 4487.4 9.98 1.54 68855 108242 0.284 4487.4 9.98 1.54 68855 108242 0.105avg 4601.0 10.16 1.42 67361 107238 0.248 4587.7 10.13 1.43 67370 107222 0.106

4.2 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

Table 4: Effects of carbon cap C with BARON and FTSH

Average BARON FTSHCap Q

∑ni=1 xi TT H(Q,x) E(Q,x) cpu Q

∑ni=1 xi TT H(Q,x) E(Q,x) cpu

107081 4551.3 10.10 1.56 67756 107039 0.261 4586.0 10.19 1.55 67770 107035 0.074107124 4574.7 10.14 1.56 67627 107090 0.241 4549.6 10.10 1.54 67610 107076 0.076107168 4586.7 10.17 1.55 67537 107135 0.250 4583.2 10.16 1.53 67542 107125 0.087107211 4560.1 10.08 1.53 67428 107173 0.252 4543.9 10.04 1.51 67449 107163 0.091107255 4582.8 10.12 1.49 67371 107214 0.245 4549.9 10.05 1.48 67390 107202 0.100107298 4602.9 10.15 1.47 67304 107255 0.264 4557.9 10.05 1.46 67315 107252 0.109107342 4619.9 10.19 1.43 67246 107297 0.250 4609.0 10.16 1.41 67257 107289 0.119107385 4651.2 10.25 1.37 67174 107335 0.247 4615.6 10.16 1.36 67169 107328 0.126107429 4658.0 10.25 1.27 67137 107372 0.254 4655.3 10.24 1.31 67118 107360 0.133107472 4622.6 10.16 1.00 67031 107466 0.218 4626.8 10.18 1.12 67077 107386 0.142avg 4601.0 10.16 1.42 67361 107238 0.248 4587.7 10.13 1.43 67370 107222 0.106

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

Figure 1: Carbon Cap vs. Costs and Emissions

(a) n = 5

2 4 6 8 106.75

6.765

6.78

6.795

6.81

6.825

6.84x 10

4

Carbon Cap (C)

Tot

al C

osts

2 4 6 8 101.0715

1.072

1.0725

1.073

1.0735

1.074

1.0745x 10

5

Tot

al E

mis

sion

s

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

(b) n = 10

2 4 6 8 106.72

6.73

6.74

6.75

6.76

6.77

6.78x 10

4

Carbon Cap (C)

Tot

al C

osts

2 4 6 8 101.07

1.071

1.072

1.073

1.074

1.075

1.076x 10

5

Tot

al E

mis

sion

s

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

(c) n = 15

2 4 6 8 106.7

6.72

6.74

6.76

6.78

6.8

6.82x 10

4

Carbon Cap (C)

Tot

al C

osts

2 4 6 8 101.07

1.0715

1.073

1.0745

1.076

1.0775

1.079x 10

5

Tot

al E

mis

sion

s

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

(d) n = 20

2 4 6 8 106.6

6.625

6.65

6.675

6.7

6.725

6.75x 10

4

Carbon Cap (C)

Tot

al C

osts

2 4 6 8 101.068

1.069

1.07

1.071

1.072

1.073

1.074x 10

5

Tot

al E

mis

sion

s

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

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

(a) n = 5

1 2 3 4 5 6 7 8 9 1010

10.2

10.4

10.6

10.8

11

Carbon Cap (C)

Num

ber

of T

ruck

s

1 2 3 4 5 6 7 8 9 101

1.1

1.2

1.3

1.4

1.5

Num

ber

of T

ruck

Typ

es

FTSH Truck NumberBARON Truck NumberFTSH Truck TypesBARON Truck Types

(b) n = 10

1 2 3 4 5 6 7 8 9 109.9

10

10.1

10.2

10.3

10.4

Carbon Cap (C)

Num

ber

of T

ruck

s

1 2 3 4 5 6 7 8 9 101

1.1

1.2

1.3

1.4

1.5

Num

ber

of T

ruck

Typ

es

FTSH Truck TypesBARON Truck TypesFTSH Truck NumberBARON Truck Number

(c) n = 15

1 2 3 4 5 6 7 8 9 1010.1

10.2

10.3

10.4

10.5

Carbon Cap (C)

Num

ber

of T

ruck

s

1 2 3 4 5 6 7 8 9 101

1.2

1.4

1.6

1.8

Num

ber

of T

ruck

Typ

es

FTSH Truck TypesBARON Truck TypesFTSH Truck NumberBARON Truck Number

(d) n = 20

1 2 3 4 5 6 7 8 9 109.7

9.8

9.9

10

10.1

Carbon Cap (C)

Num

ber

of T

ruck

s

1 2 3 4 5 6 7 8 9 101

1.2

1.4

1.6

1.8

Num

ber

of T

ruck

Typ

es

FTSH Truck TypesBARON Truck TypesFTSH Truck NumberBARON Truck Number

Example 1 Consider the following problem parameters: λ = 50, 000 units/year, p = $1/unit, A =

$150/order, h = $1.5/unit/year, p = 1lbs CO2/unit, A = 200lbs CO2/order, h = 1lbs CO2/unit/year.

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

S-TT-S M-TT-STruck Type i Ri e0i xi qi H(Q,x) E(Q,x) xi qi H(Q,x) E(Q,x)

1 120 40 6 3000 0 02 100 50 0 0 66750 108830 5 2500 66000 1087503 130 30 0 0 3 1500

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 =

1lbs CO2/unit, A = 100lbs CO2/order, h = 0.5lbs CO2/unit/year. Suppose that the retailer can

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

S-TT-S M-TT-STruck Type Ri Pi ei xi qi H(Q,x) E(Q,x) xi qi H(Q,x) E(Q,x)

1 110 500 0.770 13 6500 1 5002 100 500 0.804 0 0 65745 94894 9 4500 65537 948603 120 400 0.567 0 0 2 800

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-managed-

delivery 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.

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6 Appendix

6.1 Notation and Possible Metrics

Notation Description Metric

λ: Demand rate units/yearp: Per unit procurement cost $/unit

A: Fixed order setup cost $/order

h: Inventory holding cost per unit per unit time $/unit/year

C: Maximum emissions allowed per unit time CO2 lbs/year

p: Emissions due to per unit procurement CO2 lbs/unit

A: Emissions due to order placement CO2 lbs/order

h: Emissions due to inventory holding per unit per unit time CO2 lbs/unit/year

i: Index used for different truck types i = {1, 2, 3, . . . , n}Pi: Per truck capacity for trucks of type i units/truck

Ri: Per truck cost for trucks of type i $/truck

ei: Emissions due to shipping one unit with type i trucks CO2 lbs/unit

e0i : Emissions due to empty-weight of type i truck CO2 lbs/truck

qi: Total order quantity shipped by type i trucks per order units/order

Q: Retailer’s total order quantity per order Q =∑n

i=1 qi units/order

Q: n-vector of qi values Q = [q1, q2, . . . , qn]t

xi: Number of type i trucks used per order trucks

x: n-vector of xi values x = [x1, x2, . . . , xn]t

H(Q,x): Retailer’s total costs per unit time $/year

E(Q,x): Retailer’s total emissions per unit time CO2 lbs/year

6.2 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 q∗j ≤ (x∗j − 1)Pj for some j, j ∈ {1, 2, . . . , n}. Now, consider (Q∗∗,x∗∗) such that q∗∗i = q∗i∀i, i ∈ {1, 2, . . . , n} and x∗∗i = x∗i ∀i, i �= j, i ∈ {1, 2, . . . , n} and x∗∗j = x∗j − 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 q∗i = x∗iPi ∀i, i �= j, i �= k, (x∗j − 1)Pj < q∗j ≤ x∗jPj , and

(x∗k − 1)Pk < q∗k ≤ x∗kPk. Without loss of generality, let ek < ej. Now, consider (Q∗∗,x∗∗) such that

q∗∗i = q∗i ∀i, i �= j, i �= k and x∗∗i = x∗i ∀i, i ∈ {1, 2, . . . , n} and let q∗∗k = x∗∗k Pk and q∗∗j = q∗j − (x∗kPk− q∗k).

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 q∗i = x∗iPi ∀i, i �= j and (x∗j − 1)Pj < q∗j ≤ x∗jPj only for truck type j. �

6.4 Proof of Lemma 1

We first prove convexity of H(vj∗ |x = x∗). One can show that

∂2H(vj∗ |x = x∗)∂v2j∗

=

2λ

(A+

n∑i=1

x∗iRi

)(

n∑i=1

x∗iPi − Pj∗ + vj∗

)3 .

It follows from the above equality that∂2H(vj∗ |x=x∗)

∂v2j∗

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

x∗). Similarly, one can show that

∂2E(vj∗ |x = x∗)∂v2j∗

=

2λ

(A+

n∑i=1

x∗i (e0i + eiPi)− ej∗Pj∗ + ej∗vj∗

)(

n∑i=1

x∗iPi − Pj∗ + vj∗

)3 − 2λej∗(n∑

i=1

x∗iPi − Pj∗ + vj∗

)2 .

To establish a contradiction, let us assume that∂2E(vj∗ |x=x∗)

∂v2j∗

< 0 or vj∗ ≥ 0. It then follows that(A+

n∑i=1

x∗i (e0i + eiPi)− ej∗Pj∗ + ej∗vj∗

)< ej∗

(n∑

i=1

x∗iPi − Pj∗ + vj∗

).

The above inequality implies that A+∑n

i=1 x∗i e

0i +∑n

i=1 x∗i (ei − ej∗)Pi < 0. Nevertheless, by definition,

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

i=1 x∗i e

0i+∑n

i=1 x∗i (ei−ej∗)Pi > 0, which is a contradiction.

Therefore,∂2E(vj∗ |x=x∗)

∂v2j∗

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

6.5 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}+ ε, H(vj∗ |x = x∗) is increasing over max{0, w1}+ ε ≤

27

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

max{0, w1}+ ε ≤ vj∗ ≤ min{Pj∗ , w2} and v∗j∗ = min{Pj∗ , w2}; if max{0, w1}+ ε ≤ w0 ≤ min{Pj∗ , w2},H(vj∗ |x = x∗) is minimized at w0 and v∗j∗ = 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

(P-TL-Capi) : min Hi(Q) = pλ+ AλQ + hQ

2 +⌈QPi

⌉RiλQ

s.t. Ei(Q) = (p+ ei)λ+AλQ +

⌈QPi

⌉e0iλQ ≤ C

Q ≥ 0.

First, we note that Hi(Q) is minimized by ordering Qc such that

Qc = argmin{Hi(min{Qik+1, (k + 1)Pi}),Hi(kPi)}, (7)

where Qik+1 =

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

√2Aλh ≤ (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

Qe = argmin{Ei(min{Qin+1, (n+ 1)Pi}), Ei(nPi)}, (8)

where Qin+1 =

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

√2 Aλh

≤ (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

H�i (Q) = pλ+ Aλ

Q + hQ2 + �Riλ

Q and E�i (Q) = (p+ei)λ+

AλQ +

�e0iλQ , respectively. Note that these functions

are convex and E�i (Q) ≤ C for q�1i ≤ Q ≤ q�2i such that

q�1i =C − eλ−

√(C − (p+ ei)λ)2 − 2hλ(A+ �e0i )

h, (9)

q�2i =C − eλ+

√(C − (p + ei)λ)2 − 2hλ(A+ �e0i )

h. (10)

Now let us define S�i = {((�− 1)Pi, �P ]∩ [q�1i , q�2i ]}, that is, S�

i defines the set of feasible order quantities

that can be carried with � trucks of type i. For notational simplicity, let S�i = [q�1i , q�1i ]. Then, H�

i (Q)

over Q ∈ S�i is minimized by Q

∗(�)i , where

28

Q∗(�)i =

⎧⎪⎪⎪⎨⎪⎪⎪⎩q�1i if Q

i� < q�1i ,

Qeoqt1 if q�1i ≤ Q

i� ≤ q�2i ,

q�1i if q�1i < Qi�,

(11)

where Qi� =

√2(A+ �e0i )λ/h. Equation (11) follows from the convexity of H�

i (Q) over S�i .

A solution method for P-TL-Capi is already implied by Equation (11): one can find H�i (Q

∗(�)i )

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

(9) and (10) that q�1i ≤ q(�+1)1i ≤ q

(�+2)2i ≤ q�2i , which indicates that any Q such that Q ≥ q

(1)1i is not

feasible for P-TL-Capi. Therefore, one can define M =

⌈q(1)1iPi

⌉. Then, the solution of P-TL-Capi, Q∗

i is

given by

Q∗i = argmin{H1

i (Q∗(1)i ),H2

i (Q∗(2)i ), . . . ,HM

i (Q∗(M)i )}. (12)

29