carbon constrained integrated inventory control and truckload transportation with heterogeneous...
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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
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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
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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
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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
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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
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(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
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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