Optimal distribution of financial incentives to foster off-hour deliveries in urban areas

Transportation Research Part A 46 (2012) 1205–1215

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Optimal distribution of financial incentives to foster off-hour deliveries in urban areas Michael A. Silas a,⇑, José Holguín-Veras b,1, Sergio Jara-Díaz c a

CENTRA Technology, 4121 Wilson Boulevard, Arlington, VA 22203, United States Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute, 110 8th Street, JEC 4030, Troy, NY 12180-3590, United States c Departamento de Ingeniería Civil, Universidad de Chile, Blanco Encalada 2120, 4to piso Of. 446 (Casilla 228-3), Santiago, Chile b

a r t i c l e

i n f o

Article history: Received 28 January 2011 Received in revised form 21 March 2012 Accepted 16 May 2012

Keywords: Transportation policy Off-hour deliveries Optimal incentives Transportation economics Karush–Kuhn–Tucker conditions Optimization

a b s t r a c t The main objective of this paper is to develop mathematical formulations to gain insight into the best way to distribute financial incentives to receivers of urban deliveries to maximize participation in off-hour deliveries. The paper considers two different types of incentive budgets: exogenous, and endogenous. The exogenous case represents the condition in which an external decision maker determines the incentive budget that is to be distributed among potential participants in off-hour deliveries. In the case of an endogenous incentive budget, the entity distributing the incentives must raise the necessary funds using revenue generation mechanisms such as tolls and fines. The optimal incentives are obtained from the Karush–Kuhn–Tucker conditions of a mathematical program that maximizes the number of truck trips shifted to the off-hours as a function of the incentives. The mathematical models developed in this paper provide guidelines about how to optimally distribute financial incentives to foster off-hour deliveries. Published by Elsevier Ltd.

1. Introduction Urban congestion is one of the factors that explain increases in the costs of living and conducting business, and a major source of concerns to the general public, the business sector, and decision makers. As a way to reduce the severity of the problem, a number of transportation demand management strategies have been proposed and implemented, primarily focusing on passenger transport. Examples include: congestion pricing, park and ride systems, among others. In the freight side, the number of applications and proposed concepts is considerably smaller. One of the most important ones involves offhour deliveries (OHDs) programs, whose main objective is to induce a shift of truck traffic from the regular business hours to the off-hours. There have been several attempts to foster OHD in urban areas. For historical reasons, it is important to mention that the first initiative on record was reported in Julius Caesar’s collection of laws called the ‘‘Lex Juliana Municipalis,’’ that mandated that good deliveries in Rome be done during the evening hours (Dessau, 1892), with the specific intent of reducing congestion. The scheme had to be abandoned after numerous Roman citizens complained about night noise (Roth and Roth, 2000). Quite tellingly, noise at night remains today an obstacle to OHD. Since Julius Caesar’s times, a number of studies have focused on OHD with somewhat contradictory results. The first one reported took place in London in 1968 and involved 100 companies changing their shipping and receiving operations to the off-hours. The study found that OHD were effective when they involved relatively small number of deliveries with large shipment sizes (Churchill, 1970). Later on, the Organization for Environmental Growth in 1979 conducted interviews of carriers, ⇑ Corresponding author. 1

E-mail addresses: [email protected] (M.A. Silas), [email protected] (J. Holguín-Veras), [email protected] (S. Jara-Díaz). Tel.: +1 518 276 6221; fax: +1 518 276 4833.

0965-8564/$ - see front matter Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.tra.2012.05.015

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third-party carriers, receivers and officials from public agencies about OHD (Organization for Environmental Growth, 1979). They found that the impacts of OHD were not clear, and that pilot testing was needed to reach definitive conclusions. Another study conducted in the late 1970s (Noel et al., 1980) found that: (1) transportation companies like OHD because of the cost and time savings derived from the higher productivity; (2) carriers are generally willing to make OHD if a sufficient number of their customers request it; and, (3) security issues are an obstacle to the implementation of OHD. A study on OHD in Los Angeles investigated its legal implications. The proposal considered, i.e., banning trucks from the Los Angeles metropolitan area during the peak hours, was strongly opposed by the receiving community because of the additional operational costs that they would incur (Nelson et al., 1991). The proposal was abandoned after it was clear that the business sector would be negatively impacted. A Port Authority of New York and New Jersey’s study on OHD (Vilain and Wolfrom, 2000), analyzed how peak-hour traffic could be reduced on the New York City area interstates. Using a small carrier survey, the study concluded that trucking firms: already attempt to avoid peak periods for travel; are very concerned with meeting customer demands, while not violating district curfews and union-agreed working hours; and, have major doubts about the usefulness of peak-hour tolls to reduce congestion (Vilain and Wolfrom, 2000). Yannis et al. (2006) used simulations and interview data to quantify the impacts of OHD in Athens. The study concluded that shifting truck traffic to off-hours would significantly reduce traffic congestion and improve environmental conditions (Yannis et al., 2006). A number of pilot tests have been conducted to understand how OHD would impact traffic and the environment in Barcelona, Dublin, London, and New York City (NYC). In Barcelona, OHD were piloted with twenty supermarkets. The original seven smaller daily deliveries were replaced with two larger night deliveries to these locations (NICHES, 2008). Likewise, in Dublin off-hour delivery operations were pilot tested with supermarkets, and combined with external consolidation centers and delivery curfews. The Barcelona and Dublin studies concluded that these operations lead to: (a) reductions in logistical delays, traffic congestion, emissions, and energy consumption, (b) increases in road safety, and (c) economic benefits from lower shipping costs and higher profit margins (NICHES, 2008). In London’s borough of Wadsworth, OHD were pilot tested using low noise equipment and larger trucks to deliver to the Sainsburys’ Garret Lane grocery store (London Noise Abatement Society, 2008). The pilot suggested that OHD operations saved the companies about seven hundred working hours per year, or about $25,000 in savings. Other findings include increased efficiency of workers, increased sales, and positive customer feedback about service and product availability (London Noise Abatement Society, 2008). The NYC pilot, conducted at the end of 2010, took a different approach than all previous pilots that preceded it. To start with, the pilot had access to theory and empirical evidence (Holguín-Veras et al., 2006; Holguín-Veras, 2008, 2011) that demonstrated that: (1) receivers are the key decision makers concerning delivery times; (2) carriers cannot unilaterally change delivery times without the consent of receivers; and that (3) receivers oppose OHD because of the extra costs that they would incur. Based on this insight, the NYC pilot offered receivers a financial incentive for participation. As a result, 33 companies (both receivers of goods and vendors) participated in a 1 month trial of OHD and, at the end, the participants declared to be extremely satisfied with experience (Brom et al., in press). The success of the pilot was widely reported in the popular press and influential trade publications, that deemed the project a sustainability concept the private sector should support (Journal of Commerce, 2009, 2010; Transport Topics, 2010; Wall Street Journal, 2010). In response to this outcome, the City of New York adopted OHD as part of the city’s sustainability strategy (City of New York, 2011); the New York City Department of Transportation and the United States Department of Transportation decided to fund a larger implementation of OHD; and the Environmental Protection Agency and the United States Department of Transportation decided to study the creation of a program to foster OHD throughout the United States. The success of the pilot hinged on the recognition that without addressing the negative impacts on the receivers via financial incentives OHD simply cannot be sustained by the private sector. The role played by the incentives brings to the forefront the key policy question about what is the optimal way to distribute an incentive budget. This paper attempts to gain insight on how to optimally distribute financial incentives to receivers to maximize participation in off-hour deliveries, given a budget constraint. In doing so, the paper builds on the previous research conducted by the authors. The formulations developed here consider a budget constraint, and a number of different penalties and incentives (e.g., time of day tolls, parking fines, per mile penalties to regular-hour carriers, and regular-hour delivery surcharges to receivers). This is achieved by using mathematical programming to derive the optimal financial incentives to be distributed to the various industry classes. This paper has four sections including this introduction. The second section discusses the economic rationale for the distribution of financial incentives to receivers. The third section defines the notation used and derives the optimal financial incentives given to receivers. The fourth section discusses numerical experiments and policy implications. The paper concludes with a summary of the key findings produced.

2. Rationale The need and rationale for the provision of incentive to the receivers of deliveries in urban areas—as an alternative to the more traditional approach of using cordon time of day tolls, time-distance pricing, and the like—was made evident by the analyses of the behavioral impacts of freight road pricing (Holguín-Veras et al., 2006), confirmed with the use of advance econometric models estimated with stated preference data (Holguín-Veras et al., 2007, 2008), established theoretically (Holguín-Veras, 2008, Holguín-Veras, 2011), and successfully validated by a pilot test in New York City (Brom et al., in press;

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Holguín-Veras et al., in press). For that reason, only a succinct summary of this research is presented here. This section provides the reader with a brief discussion of the key findings from the literature on freight road pricing and OHD. To start with, it is important to discuss the limitation of the alternative forms of freight road pricing that could be deemed suitable alternative to OHD policies. The use of freight road pricing is predicated under the assumption that, in response to large tolls in the congested periods, freight traffic (and implicitly the customers) would move to the off-hours. Interestingly, the handful of publications discussing the subject, though endorsing the concept, seemed to have perceived the existence of some unique factors that could hamper the effectiveness of freight road pricing (Jacobi, 1973; Hicks, 1977; Button, 1978; Button and Pearman, 1981). In this context, though it seems logical to expect that higher tolls could reduce freight traffic, this conjecture was contradicted by the data gathered by the first project that studied the behavioral impacts of freight road pricing (Holguín-Veras et al., 2006). The data clearly showed that: (1) the vast majority of carriers (89%) could not pass the toll costs to their customers, which eliminates the possibility that they could change behavior; and, (2) as a consequence of (1), the majority of the carriers (43%) were forced to enact productivity increases to help them cope with the toll increases without impacting their customers. Not surprisingly, in response to the question about why they could not change behavior, 75% indicated ‘‘customer requirements’’ as the reason. The research on OHD provided other important pieces of information (Holguín-Veras et al., 2007, 2008). As part of this effort, game theoretic analyses and advanced econometrics were used to study the interactions between carriers and receivers. These analyses proved that these interactions conform to the ‘‘Battle of the Sexes’’ game (Rasmusen, 2001), which is a game in which two players must reach a common decision though both want a different outcome (in this case, the carriers prefer OHD while the receivers prefer regular hour deliveries). This game is known to have two Nash equilibria where the outcome is determined by the player with the most clout. The obvious implication is that, since the majority of deliveries are made in the regular hours (95% in New York City), the receivers are the dominant player because otherwise the carriers would select OHD (Holguín-Veras et al., 2007). Moreover, these analyses led to the conclusion that providing incentives to the receivers to is the key for them to change their preferred outcome from regular to off-hours. This conjecture was tested and confirmed by discrete choice models estimated with stated preference data collected from a sample of carriers and receivers. The models indicated that indeed providing incentives to receivers not only could lead to a significant switch of truck traffic to the off-hours, but that doing so was more effective than freight road pricing. Silas and Holguín-Veras (2009) developed a Behavioral Micro-Simulation to test policies to foster OHD. This research confirmed that financial incentives to receivers and traveling rewards to carriers for off-hour travel were effective stimuli at increasing OHD. It was also found that time-distance pricing and parking fine enforcement were more effective in fostering participation in OHD than time of day cordon tolls (Silas and Holguín-Veras, 2009), for reasons related to the nature of the cost functions. All of this reflects the fact that, in order for OHD to take place, both carriers and receivers have to be made better off. This necessary condition was exploited by Holguín-Veras (2008) to develop insight into the effectiveness of alternative policies, and to identify potential industry segments naturally inclined to participate. A comprehensive economic explanation of the limitations of freight road pricing, the role of financial incentives, and the existence of a market failure that prevents the participants from reaching the social optimal outcome, i.e., OHD, is provided in Holguín-Veras (2011). These analyses indicate that the root of the market failure is that the savings accrued by the carriers are not large enough to compensate the receiver costs, which leads to a situation in which there is no compensation mechanism that carriers could use to entice receivers to do OHD. As a result, since the receivers cannot be compensated by the carriers, they refuse to do OHD and, in doing so, force the carriers to travel in the congested hours of the day (Holguín-Veras, 2011). Two additional publications (Brom et al., in press; Holguín-Veras et al., in press) discuss the results of the pilot test in NYC and the anticipated impacts of a full scale implementation. Brom et al. (in press) discusses the opinion surveys conducted with pilot test participants, which revealed a degree of satisfaction. Holguín-Veras et al. (in press) presents the results of the economic analyses conducted that revealed that shifting 20–40% of the truck traffic—which is the potential shift of OHD in NYC—could lead to economic savings in the range of $100–$200/million per year. The economic analyses conclude that OHD are indeed beneficial to NYC as they: reduce congestion and environmental pollution produced by freight traffic, increase the sustainability of urban freight operations, increase quality of life, and do so with the support of the private sector. These results, combined with the identification of the market failure, provide the rationale for the provision of financial incentives to the receivers of goods. However, in spite of the research conducted, the important question of how best to distribute an incentive budget has not been answered. This is the main focus of this paper.

3. Optimal allocation of incentives In the most general case concerning the allocation of an incentive budget, there are two important decisions to make. The first one is related to the total incentive budget to be distributed among receivers. The second is associated with the allocation of the incentive budget among the different industry segments. These decisions are interrelated in the case of the endogenous incentive. In this case, the budget determines the amounts to be offered to the participating businesses; while the financial mechanisms used to raise the incentive budget (e.g., tolls, parking fines) determine the total funds available. The decisions concerning the incentive budget have important welfare implications. On a conceptual fashion, it seems reasonable to expect decreasing marginal benefits and increasing marginal costs for OHD. The former is a consequence of

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the non-linear nature of congestion, and the latter reflects the fact that attracting businesses to the off-hours is bound to be increasingly difficult and expensive, as the businesses that remain in the regular hours are those that are not naturally inclined to do OHD. This leads to a situation in which the optimal level of participation (at marginal benefits equal marginal costs) is somewhere in between the status quo (minimal amount of OHD), and full participation (100% OHD). The analyses conducted for the New York City metropolitan area confirmed this conjecture as they found, using traffic simulations and economic analyses, that the optimal amount of OHD is between 14% and 21% of the total truck traffic (Holguín-Veras et al., 2010). Since computing the optimal amount of OHD—which implicitly determines the optimal incentive budget—is best done using traffic modeling techniques, the authors decided to focus on the subproblem of how to optimally allocate an incentive budget among multiple industry segments to maximize participation in OHD. The models developed here consider the case in which a decision maker seeks to maximize participation in OHD by distributing a given incentive budget to receivers in exchange for their commitment to OHD. Two different assumptions about the source of the budget are made. In the first case the source of the funds is exogenous, e.g., from general taxes. In the second case, the budget is endogenously generated by the same agency that distribute the incentives using revenue generation mechanisms that target either receivers (i.e., regular hour delivery penalty to receivers), or carriers (i.e., time of day toll surcharge, time-distance tolls, and parking fines). The mathematical models presented assume that there are different combinations of receivers and carriers, each one representing a duplet (or class) denoted by i. The class i is characterized in terms of their willingness to do OHD in exchange for a financial incentive wi, and a set of operational characteristics. In the base case conditions, there are a total of yBi receivers in each class i accepting regular-hour deliveries, that produce ti delivery tours/day per receiver, with ri delivery stops, and that travel a total of di miles. The paper considers different revenue generation mechanisms, though not all of them are necessarily active at a given time. In its most general form: each tour has a probability q of being assessed a parking fine at a stop in the amount of  dollars, receivers are charged p t dollars for each tour they generate during the regular-hours, and carriers are charged s dollars for each regular-hour tour they make, and f dollars per mile travelled during the regular-hours. A fraction / 6 1 of total revenues collected from penalties is used for incentives given to receivers, to account for administrative overhead or other uses of the funds. The objective of these mathematical models is to compute the average daily optimal incentives wi that maximizes the total number of off-hour tours, T, subject to a budget constraint that, as said, could be exogenous or endogenous. The budget constraint, or incentive budget, is the total amount of funds that could be allocated to the receivers on a daily basis. 3.1. Mathematical formulations This section shows the derivations of the optimal incentives, and the conceptual analyses of the key results. Two cases are considered: an exogenous budget, and an endogenous (self-sustaining) system in which the incentive budget is determined by the revenue generation capacity of the system itself. 3.1.1. Case 1: Exogenous incentive budget The objective here is to maximize the number of off-hour truck tours, given an external budget to distribute financial incentives to receivers. This is the case, for instance, if the incentive budget comes from general tax revenues. The budget P constraint considered requires that the total financial incentives given to off-hour receivers i wi yOi be less than or equal to the given budget constraint (B). The mathematical program is:

Maximize : T ¼

X t i yOi

X i Subject to : wi yOi 6 B

ð1Þ ð2Þ

i

The Lagrangian, L, is:

! X X O O L¼ t i yi  k wi yi  B i

ð3Þ

i

The complementary slackness condition is:

! X O k wi yi  B 6 0

ð4Þ

i

The partial derivative of L with respect to wi is:

  @yO @yO @L @ti @wi ¼ yOi þ t i i  k yOi þ wi i @wi @wi @wi @wi @wi

ð5Þ

M.A. Silas et al. / Transportation Research Part A 46 (2012) 1205–1215

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From the complementary slackness condition:

yOi

@yO @yO @t i þ ti i  kyOi  kwi i ¼ 0 @wi @wi @wi

ð6Þ

Solving for wi:

kwi

@yOi @yO @t i ¼ yOi þ t i i  kyOi @wi @wi @wi

ð7Þ

wi ¼

yOi @t i @wi ti @yOi @wi @wi þ  yOi k @wi @yOi k @wi @yOi @yOi

ð8Þ

wi ¼

yOi @t i t i @wi þ  yOi k @yOi k @yOi

ð9Þ

Defining the elasticity of ti with respect to the number of off-hour receivers yOi and the direct elasticity of yOi with respect to the financial incentive, wi:

gti ¼

@t i yOi @yOi t i

ð10Þ

@yOi wi @wi yOi

ð11Þ

gyOi ¼

Then, wi can be rewritten as:

wi ¼

yOi t t 1 wi g i þ i  yOi k ti yOi k gyO yOi

ð12Þ

i

From where, one can obtain:

wi ¼ t i

gyOi 1 þ gyO i

ð1 þ gti Þ k

! ð13Þ

Defining:

gyOi hi ¼ 1 þ gyO i

! ð14Þ

Eq. (13) becomes:

wi ¼ t i

ð1 þ gti Þ k

hi

ð15Þ

Eq. (15) indicates that the optimal financial incentive is proportional to the average number of truck tours produced, ti, the elasticity gti , and hi. This implies that receiver classes that generate higher amounts of truck tours that are elastic with respect to yOi , should receive a larger financial incentive, if they agree to OHD. As shown in Eq. (14), the term hi approaches one as the elasticity gyO increases, which constrains the incentive given to i off-hour receivers. In cases where the response is inelastic, gyO < 1, leads to 0 6 hi < 0:5; an unit response with gyO ¼ 1 proi i duces hi ¼ 0:5; while an elastic one (gyO > 1) yields 0 < hi 6 1. The elasticity gti measures the relative change in the number i O of tours transferred to the off-hours as function of the number of receivers in the off hours, yi . It is not clear if this parameter is positive or negative: it would be positive if yOi enhances the ability of the carriers to create off-hour tours, and negative if it does the opposite, e.g., if carriers are able to consolidate tours. This is an empirical issue that deserves to be addressed by future research. Lastly, 1=k represents the amount of money required to induce a switch of one tour to the off-hours. This term decreases as k increases and it tends to zero as k approaches infinity. 3.1.2. Case 2: Endogenous incentive budget This formulation represents a case in which the incentive budget is a function of the total revenue generated by instruments available to decision makers. In its more general version, this case considers: a time of day cordon toll surcharge, regular hour delivery penalties to receivers, parking fines, and time-distance pricing. The formulation is shown below, and requires that the total incentives given to receivers be less than or equal to the revenues from the penalties and toll surcharges. The left hand side of the constraint represents the total incentives to all classes, while the right hand side represents P the revenues. The term / i ðyBi  yOi Þeqri t i represents the expected revenues collected from parking fines, the term

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P / i ðyBi  yOi Þðri p t þ sÞt i are the revenues from regular-hour delivery penalties to receivers and carriers, and the term P / i ðyBi  yOi Þf  ti di is the revenue from time-distance pricing. The mathematical problem is:

Maximize : T ¼

X t i yOi

ð16Þ

i

X X wi yOi 6 / t i ðyBi  yOi Þ½eqri þ ðri pt þ sÞ þ f  di 

Subject to :

i

ð17Þ

i

As shown, the term in square brackets is the revenue raised from a regular-hour tour of industry segment i. Designating this term as:

ji ¼ ðeqri þ ðri pt þ sÞ þ f  di Þ

ð18Þ

The Lagrangian becomes:

L¼

( ) X X X t i yOi  k wi yOi  / t i ðyBi  yOi Þji i

i

ð19Þ

i

The partial derivative of L with respect to wi is:

   @yO @yO @yO @L @ti @t i ¼ t i i þ yOi  k wi i þ yOi  /ji ti i þ ðyBi  yOi Þ @wi @wi @wi @wi @wi @wi

ð20Þ

From the complementary condition, solving for wi:

kwi

  @yOi @yO @yO @t i @t i ¼ ti i þ yOi  kyOi þ k/ji ðyBi  yOi Þ  ti i @wi @wi @wi @wi @wi

wi ¼

  t i yOi @t i O @wi B O @t i  y þ / j ðy  y Þ  t þ i i i i i k k @yOi @yOi @yOi

ð21Þ

ð22Þ

Expressing wi in terms of gti and gyO : i

  ti ti wi @ti wi ¼ þ gti  þ /ji ðyBi  yOi Þ O  t i k k gyO @yi

ð23Þ

i

wi ¼

  O   O   y @t i t i yi @t i ti ti wi t þ /ji yBi i   t þ gti  i i k k gyO t i @yOi yOi ti @yOi

ð24Þ

  yB ti ti wi þ /ji t i gti Oi  gti  1 þ gti  k k gyO yi

ð25Þ

i

wi ¼

i   Thus, since 1 þ g1O ¼ h1 , wi becomes:



wi ¼ t i

wi ¼ t i

y i

ð1 þ gti Þ k



ð1 þ gti Þ k

i

   yB hi  /ji 1 þ gti 1  Oi yi

ð26Þ

  yB hi  /ji  /ji gti 1  Oi yi

ð27Þ

As shown, Eq. (27) is a general form of Eq. (15), as the latter is obtained for ji = 0. Eq. (27) has a number of interesting features. The first one is related to the role of ji. To isolate the effects of ji, it is best to disregard the role of gti . As shown, the net effect of ji is to lower the value of the incentive wi as industry segments with high values of ji would receive smaller incentives than other with low values of ji. Thus, industry segments with high values of ji would receive smaller incentives than those with low values. In essence, the various industry segments receive a different depending on whether or not they are ‘‘funders’’ or ‘‘recipients’’ of the incentives. The implication is that, in order to maximize participation in off-hour deliveries, it is advantageous to raise revenues from those industry segments well positioned to contribute, and then hand out these incentives to those segments inclined to switch to the off-hours. Conceptually, this policy makes sense. In a real life context, the industry segment—and the nature of the impact—depends on the specific revenue mechanisms used. In the case of a time of day toll surcharge (s), the industry segments making many trips to the tolled area would receive the smaller incentive. (Needless to say, this stands in contradiction with policies that provide frequent use discounts.) If parking fines are used as the primary funding mechanism, then the industry segments making long tours would be ones that would play the role of ‘‘funders’’ thus receiving the smaller incentives, with those carriers making short tours as the ‘‘recipients’’ of the largest incentives. Something similar would happen if penalties for off-hour deliveries are charged to receivers (with the obvious difference that while parking fines are paid by the carriers, these ones would be paid by the receivers); or if

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time-distance pricing is used (where the carriers would be charged as a function of the distance travelled in the regular hours). A second important aspect of Eq. (28) has to do with the role of gti which measures the scale effects and could be positive or negative. To gain insight into the role of gti , Eq. (28) has been rewritten as shown below:

wi ¼ t i

 ð1 þ gti Þ k

þ /ji gti

 yRi hi  / j i yOi

ð28Þ

where yRi ¼ yBi  yOi is the number of receivers in the regular hours. Furthermore, it can be seen that Eq. (28) is a more general form of Eq. (15). As shown in Eq. (29), a positive value of gti would lead to higher values of both zi and wi, and negative values would lead to the opposite. In this way, if the number of receivers in the off-hours increases the ability of the carriers to create off-hour tours increasing the incentives to the receiver is justified. Conversely, if the opposite happens (for instance, when the carriers are able to consolidate off-hour tours), reducing the incentive would be appropriate. Eq. (29) also shows that the role of gti is amplified by the ratio yRi =yOi . As a result, underrepresented classes with high values of yRi =yOi and positive gti would receive larger incentives than others. Defining:

zi ¼ /ji gti

yRi  /ji yOi

ð29Þ

Consequently, wi can be rewritten as:

wi ¼ t i

 ð1 þ gti Þ k

 þ zi hi

ð30Þ

Eq. (30) is a general solution, from where the solution for an exogenous budget could be obtained for zi = 0, and the endogenous case if zi – 0. It should also be noted that there is a linear relationship between zi and wi. To gain additional insight into the findings discussed, the authors conducted a number of numerical experiments. These are discussed in the next chapter. 4. Numerical experiments In order to understand the impacts predicted by the formulations developed and assess the reasonableness of solutions obtained, numerical experiments were conducted. These experiments have several assumptions. Since it is known that classes of receivers vary in characteristics and receptiveness to OHD, behavioral models were created with assumed parameters as shown in Fig. 1. As seen in the figure, the number of receivers accepting OHD is dependent upon the financial incentives given. It is also assumed that the average number of tours shifted to the off-hours for carriers (see Fig. 2) depends on the number of receivers willing to accept OHD. The experiments assume that information is available about the behavioral responses of the different classes of receivers in response to a financial incentive. The models assumed are shown in Fig. 1. As shown in the figure, the number of receivers accepting OHD depends on the financial incentive. It is also assumed that the average number of tours shifted to the offhours for carriers (shown in Fig. 2) depends on the number of receivers willing to accept OHD. 4.1. Exogenous incentive budget The numerical experiments discussed here are intended to help understand the impacts of distributing incentives for participating in OHD when no revenue generation mechanisms are used (i.e., zi = 0) and using an external budget. Assuming that

% of OH Receivers (yi)

100.00% 90.00% 80.00% 70.00% 60.00% 50.00%

Class 1

40.00%

Class 2

30.00%

Class 3

20.00%

Class 4

10.00%

Class 5

0.00% $0

$10,000 $20,000 $30,000 $40,000 $50,000 $60,000

Financial Incentives to Receivers (wi) Fig. 1. Behavioral models versus incentives for accepting OHD.

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% of Off-Hour Tours per day(ti)

100% 90% 80% 70% 60% 50%

Class 1

40%

Class 2

30%

Class 3

20%

Class 4

10%

Class 5

0% 0

100

200

300

400

500

Off-Hour Receivers (yi) Fig. 2. Off-hour tour generation models versus off-hour receivers.

tour characteristics are the same across all classes (i.e., di = 150 and ri = 5.5"i), the results are shown in Fig. 3. The results show there is a mildly non-linear relationship between the amount of incentives given to receivers and the incentive budget. However, user classes that do not switch a large number of tours to the off-hours receive smaller incentives than those classes that do. Fig. 4 shows the total number of tours shifted to the off-hours per day as a function of the incentive budget. As shown, the rapid increase at the beginning is followed by smaller increases in the number of tours transferred to the off-hours. The breakdown by classes is shown in Fig. 4. It should be also noted that the shapes exhibited in Figs. 4 and 5 are similar to the shapes of the behavioral models in Fig. 1. This shows that receivers’ willingness to accept OHD influences the number of tours shifted to the off-hours.

Financial Incentive (wi)

$40,000

Class 1

$35,000

Class 2

$30,000

Class 3

$25,000

Class 4 Class 5

$20,000 $15,000 $10,000 $5,000 $0 $0

$5,000,000

$10,000,000

$15,000,000

$20,000,000

Budget (B) Fig. 3. Financial incentives versus exogenous budget.

Total OH Tours per day (T)

2500 2000 1500 1000 500 0 $0

$5,000,000

$10,000,000

$15,000,000

$20,000,000

Budget (B) Fig. 4. Total off-hour tours per day as a function of the exogenous budget.

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Total OH Tours per day

1200

1213

Class 1 Class 2 Class 3 Class 4 Class 5

800

400

0 $0

$5,000,000

$10,000,000

$15,000,000

$20,000,000

Budget (B) Fig. 5. Total off-hour tours per day as a function of the exogenous budget by class.

4.2. Endogenous incentive budget This section considers the case in which the incentive budget is determined by the revenue instruments used (i.e., zi – 0). As in the previous section, the first set of experiments considers the role of willingness to participate in OHD while keeping tour characteristics constant for all industry classes. The first analysis focuses on how regular-hour traveling penalties and average distance travelled per tour influence the amount of incentives given to the different classes of receivers. Fig. 6 shows that results for Class 1. The results indicate that incentives given to the class are higher as the traveling penalties and the average distance travelled per tour increases. This is due to the larger revenue streams from the penalties for longer distances travelled and given as incentives, and not a direct effect of penalty.

Fig. 6. Off-hour incentives given versus the regular-hour traveling penalty assessed and average distance travelled per tour.

OH Tours per day

3000 2500 2000 1500 1000 500 0 $0

$10

$20

$30

$40

$50

$60

Expected Revenue Per Regular Hour Tour Fig. 7. Off-hour tours versus revenues generated per regular-hour tour.

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The results shown in Fig. 7 show the number tours shifted to the off-hours in terms of penalties collected per regular hour tour. This figure indicates that the number off-hour tours increases as the regular-hour tour revenues increases. The figure shows that, at first, the number of off-hour tours increases rapidly at the beginning and then tapers down. 5. Conclusions The main objective of this paper is to gain insight into the optimal way to distribute a given incentive budget among an arbitrary number of industry classes. Secondary objectives include understanding how the varying attributes of different market segments influence the optimal incentives. These objectives were achieved through the mathematical models developed. The mathematical models developed compute the optimal incentives to various industry classes in exchange for their commitment to off-hour deliveries. Two alternative funding structures are considered. The first case considered a budget that is exogenous. In the second case, the budget is a function of various revenue generation mechanisms (i.e., toll surcharges, traveling penalties, parking fines, and delivery penalties to receivers). The models provide insight into the optimal way to distribute financial incentives. In both the exogenous and endogenous budget, the financial incentives are a function of the amount of tours that each class transfers to the off-hours and its corresponding elasticity, i.e., ti ð1 þ gti Þ. In general, larger incentives will be given to classes that generate more tours. The optimal incentive also depends on the term hi which is the ratio of class elasticity of receiver participation with respect to the financial incentive provided. This term increases with the class elasticity gyO . The more elastic the receiver class, the higher hi, i and the optimal incentive. However, the imposition of the constraint that the incentive budget be generated by the system itself has a direct impact in the magnitude of the incentives distributed. The analytical derivations show that, while in the exogenous budget case the incentives only depend on ti ð1 þ gti Þ and hi, in the endogenous (self-sustaining) case the incentives also depend on the revenue generation of the class. In essence, the analysis shows that the larger the revenue generation of the class, the lower its optimal incentive. The net effect of this is to segregate the various industry classes into net ‘‘funders’’ and ‘‘recipients’’ of the incentives. The implication is that to maximize participation in off-hour deliveries, it is advantageous to raise revenues from those industry segments well positioned to contribute, and then hand out these incentives to those segments inclined to switch to the off-hours. Conceptually, this policy makes sense. In closing, this paper has gained insight into how best to incentivize receivers to participated in OHD programs. However, although progress has been made toward developing a good understanding of how to bring about OHD, there are still a number of open questions that must be answered before OHD becomes a standard tool in the arsenal of transportation demand management techniques. References Brom, M., Holguín-Veras, J., Hodge, S., 2011. Off-hour deliveries in manhattan: experiences of pilot test participants. Transportation Research Record 2238, 77–85. http://dx.doi.org/10.3141/2238-10. Button, K., Pearman, A.D., 1981. The Economics of Urban Freight. McMillan, London. Button, K.J., 1978. A note on the road pricing of commercial traffic. Transportation Planning and Technology 4 (3), 175–178. Churchill, J.D.C. 1970. Operation ‘‘MoonDrop’’: an experiment in out of hours goods delivery. 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