the vanpool assignment problem: optimization models and solution algorithms

17
The Vanpool Assignment Problem: Optimization models and solution algorithms Levent Kaan a , Eli V. Olinick b,a Bell Helicopter, a Textron Company, 1700 N. State Hwy 360 Grand Prairie, TX 75050, USA b Department of EMIS, Bobby B. Lyle School of Engineering, Southern Methodist University, P.O. Box 750123, Dallas, TX 75275, USA article info Article history: Received 31 October 2012 Received in revised form 28 May 2013 Accepted 31 May 2013 Available online 10 June 2013 Keywords: Transportation Vanpooling Carpooling Park-and-ride Integer programming Heuristics abstract We present optimization models and solution algorithms for the Vanpool Assignment Problem. A vanpool is typically a group of 9-15 passengers who share their commute to a common target location (typically an office building or corporate campus). Commuters in a vanpool drive from their homes to a park-and- ride location where they board a van and ride together to the target location; at the end of the work day they ride together back to the park-and-ride location. The Minimum Cost Vanpool Assignment Model (MCVAM) developed in this study is motivated by a program offered by Gulfstream Aerospace, a large employer in the Dallas/Fort-Worth area, Dallas Area Rapid Transit (DART), and Enterprise Rent-A-Car. Our MCVAM imposes constraints on the capacity of each van and quality-of-service constraints on the cost and travel time involved in joining a vanpool. The goal of the MCVAM is to minimize the total cost of a one-way trip to the target location for all employees (including those employees who opt-out of the program and choose not to join a vanpool). To the best of our knowledge, this is the first mathematical programming model proposed for the standard (one-stop) Vanpool Assignment Problem. The MCVAM models the current practice in vanpooling of using one park-and-ride location per vanpool. We also pres- ent a Two-Stop MCVAM (TSMCVAM) that offers significant cost savings compared to the MCVAM with little or no increase in trip times for most passengers by allowing vanpools to stop at a second park- and-ride location. We present heuristics for the TSMCVAM which are shown in a computational study to find solutions with optimality gaps ranging from 5% to 10% in CPU times ranging from 1 to 15 min for problem instances with up to 600 employees and 120 potential park-and-ride locations. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Commuters in urban areas must contend with increasing traffic congestion and rising fuel costs. As noted by Newman and Kenwor- thy (1999), and Farhan and Murray (2008), private vehicle trans- portation based on current technologies is not likely to be sustainable in the United States (US) or abroad. When well planned and managed, public transit systems (e.g., trains and buses) and private ride-sharing arrangements such as carpools and vanpools can provide effective transportation alternatives to traditional commuting in single-occupant vehicles. In this study we present mathematical optimization models and solution algorithms to determine an efficient assignment of potential riders to vanpools subject to quality-of-service constraints on trip cost and travel time for individual users. 1.1. Vanpooling A vanpool is typically a group of 9–15 passengers who share their commute to a common target location (typically an office building or corporate campus). Commuters in a vanpool drive from their homes to a park-and-ride location where they board a van and ride together to the target location; at the end of the work day they ride together back to the park-and-ride location. Vanpooling in the US dates back at least as far the OPEC oil embargo of the mid 1970s. The Federal Energy Administration (FEA) was an initiator and supporter of many vanpooling projects around the country. The longest established employer owned vanpool program, however, was started in 1973 at the 3 M Company headquarters in St. Paul, Minnesota to avoid building an expensive parking garage (Owens & Sever, 1977). Since then vanpooling has been promoted nationwide by state and local governments, and non-governmental organizations interested in environmental conservation and en- ergy efficiency (Fishbein, 1982; Leherr, 1977; Mielke, 2006; Puget Sound Regional Council, 2005; Shuck & Welch, 1982). The benefits of vanpools can be offered at low cost by employers as part of a travel benefits package. In the US, both 0360-8352/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2013.05.020 Corresponding author. Tel.: +1 214 768 3092. E-mail addresses: [email protected] (L. Kaan), [email protected] (E.V. Olinick). Computers & Industrial Engineering 66 (2013) 24–40 Contents lists available at SciVerse ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie

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Page 1: The Vanpool Assignment Problem: Optimization models and solution algorithms

Computers & Industrial Engineering 66 (2013) 24–40

Contents lists available at SciVerse ScienceDirect

Computers & Industrial Engineering

journal homepage: www.elsevier .com/ locate/caie

The Vanpool Assignment Problem: Optimization models and solutionalgorithms

0360-8352/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.cie.2013.05.020

⇑ Corresponding author. Tel.: +1 214 768 3092.E-mail addresses: [email protected] (L. Kaan), [email protected]

(E.V. Olinick).

Levent Kaan a, Eli V. Olinick b,⇑a Bell Helicopter, a Textron Company, 1700 N. State Hwy 360 Grand Prairie, TX 75050, USAb Department of EMIS, Bobby B. Lyle School of Engineering, Southern Methodist University, P.O. Box 750123, Dallas, TX 75275, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 31 October 2012Received in revised form 28 May 2013Accepted 31 May 2013Available online 10 June 2013

Keywords:TransportationVanpoolingCarpoolingPark-and-rideInteger programmingHeuristics

We present optimization models and solution algorithms for the Vanpool Assignment Problem. A vanpoolis typically a group of 9-15 passengers who share their commute to a common target location (typicallyan office building or corporate campus). Commuters in a vanpool drive from their homes to a park-and-ride location where they board a van and ride together to the target location; at the end of the work daythey ride together back to the park-and-ride location. The Minimum Cost Vanpool Assignment Model(MCVAM) developed in this study is motivated by a program offered by Gulfstream Aerospace, a largeemployer in the Dallas/Fort-Worth area, Dallas Area Rapid Transit (DART), and Enterprise Rent-A-Car.Our MCVAM imposes constraints on the capacity of each van and quality-of-service constraints on thecost and travel time involved in joining a vanpool. The goal of the MCVAM is to minimize the total costof a one-way trip to the target location for all employees (including those employees who opt-out of theprogram and choose not to join a vanpool). To the best of our knowledge, this is the first mathematicalprogramming model proposed for the standard (one-stop) Vanpool Assignment Problem. The MCVAMmodels the current practice in vanpooling of using one park-and-ride location per vanpool. We also pres-ent a Two-Stop MCVAM (TSMCVAM) that offers significant cost savings compared to the MCVAM withlittle or no increase in trip times for most passengers by allowing vanpools to stop at a second park-and-ride location. We present heuristics for the TSMCVAM which are shown in a computational studyto find solutions with optimality gaps ranging from 5% to 10% in CPU times ranging from 1 to 15 minfor problem instances with up to 600 employees and 120 potential park-and-ride locations.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Commuters in urban areas must contend with increasing trafficcongestion and rising fuel costs. As noted by Newman and Kenwor-thy (1999), and Farhan and Murray (2008), private vehicle trans-portation based on current technologies is not likely to besustainable in the United States (US) or abroad. When well plannedand managed, public transit systems (e.g., trains and buses) andprivate ride-sharing arrangements such as carpools and vanpoolscan provide effective transportation alternatives to traditionalcommuting in single-occupant vehicles. In this study we presentmathematical optimization models and solution algorithms todetermine an efficient assignment of potential riders to vanpoolssubject to quality-of-service constraints on trip cost and traveltime for individual users.

1.1. Vanpooling

A vanpool is typically a group of 9–15 passengers who sharetheir commute to a common target location (typically an officebuilding or corporate campus). Commuters in a vanpool drive fromtheir homes to a park-and-ride location where they board a van andride together to the target location; at the end of the work day theyride together back to the park-and-ride location. Vanpooling in theUS dates back at least as far the OPEC oil embargo of the mid 1970s.The Federal Energy Administration (FEA) was an initiator andsupporter of many vanpooling projects around the country. Thelongest established employer owned vanpool program, however,was started in 1973 at the 3 M Company headquarters in St. Paul,Minnesota to avoid building an expensive parking garage (Owens& Sever, 1977). Since then vanpooling has been promotednationwide by state and local governments, and non-governmentalorganizations interested in environmental conservation and en-ergy efficiency (Fishbein, 1982; Leherr, 1977; Mielke, 2006; PugetSound Regional Council, 2005; Shuck & Welch, 1982).

The benefits of vanpools can be offered at low cost byemployers as part of a travel benefits package. In the US, both

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L. Kaan, E.V. Olinick / Computers & Industrial Engineering 66 (2013) 24–40 25

employers and employees can receive financial incentives, in theform of tax deductions and credits, for participating in vanpoolingprograms (IRS, 2009; Kaan, 2011).

In addition to these tangible benefits, vanpooling programs canengender stronger ties between employer and employee, andthereby improve employee retention and recruiting efforts (Kaan,2011).

The vanpool-optimization models developed in this study aremotivated by a program offered by Gulfstream Aerospace, a largeemployer in the Dallas/Fort-Worth area, Dallas Area Rapid Transit(DART), and Enterprise Rent-A-Car. We describe this program, andthe resulting optimization models, in more detail in Section 2. Inthe following section, however, we briefly review the literature re-lated to vanpooling with an emphasis on optimization/operationsresearch modeling.

1.2. Related literature

Despite the increasing public interest in vanpooling applica-tions, the vanpooling idea has received little attention from theoptimization/operations research community. For this reason ourliterature survey is organized around vanpooling and two relatedterms: carpooling, and ridesharing. In general terms, carpoolingis the shared use of a car by the driver and one or more passengersfor commuting. Carpooling formation involves varying degrees offormality and regularity. Carpoolers use pool members’ privatecars for their shared journeys. Carpools can either be arranged sothat one particular participant always drives and the passengerscontribute money for gas, tolls and other charges, or the partici-pants can alternate driving and no money is exchanged. Collec-tively, the carpool participants make up the rules and determinethe schedule. The vanpools that we consider tend to be more for-malized than carpools with vans and vanpool assignments pro-vided to the participants by their employer or a third-partyservice. From a modeling perspective, carpool problems differ fromthe Vanpool Assignment Problems we consider because they in-volve more complex routing sub-problems; a vehicle routing typeproblem must be solved for each carpool driver to determine theoptimal route for picking up the other carpool passengers and thendriving to the target location. Also, the cost structures for carpoolsand vanpools are different because vanpool riders incur a costbased in part on the monthly van leases.

Ride-sharing is a special implementation of a carpooling ser-vice. Typical characteristics for this type of carpooling are: arrange-ment of one-time trips instead of recurrent appointments forcommuters, and automatic and instant matching of rides througha network service (Kaan, 2011). In many areas in the United States,local transportation agencies offer a related type of service, typi-cally known as dial-a-ride, for people whose ‘‘access [to privatetransportation] is limited, or whose disability or health conditionprevents them from using the regular fixed-route bus service’’ (Se-nior Services of Snohomish County, 2006). In addition to involvingmore complex routing decisions, the dynamic nature of optimiza-tion problems involving ride-sharing arrangements and dial-a-rideservices set them apart from the vanpooling assignment problemswe consider. Agatz, Erera, Savelsbergh, and Wang (2011) and Cor-deau and Laporte (2007) describe optimization models and solu-tion algorithms for dial-a-ride problems, and the latter providesan extensive review of the literature in this area. Our VanpoolAssignment Problem is most closely related to combinatorial opti-mization problems posed in the carpooling literature by Varren-trapp, Stutzle, and Maniezzo (2002) and Baldacci, Maniezzo, andMingozzi (2004).

Varrentrapp et al. (2002) consider two different optimizationproblems involving carpools: the daily car pooling problem (DCPP)and the long-term car pooling problem (LCPP). In the DCPP, drivers

(servers) declare their availability for picking up and later bringingback colleagues (users) on any given day. Given constraints on tim-ing and vehicle capacity, the goal in the DCPP is to assign users toservers and to identify the routes to be driven by the servers in or-der to minimize a travel-cost function that includes a penalty termfor unassigned users (solo drivers). In the LCPP carpool participantsalternate between acting as users on some days and servers on oth-ers. The problem is to maximize pool sizes while minimizing thetotal distance traveled by all users when they act as servers.

Given a set of potential carpool participants, Baldacci et al.(2004) study the problem of assigning riders to carpools and deter-mining routes for each car in such a way that participation is max-imized and the sum of the route costs is minimized. They developexact and heuristic methods for solving this problem, and presentempirical results demonstrating the effectiveness of their algo-rithms on a series of test problems. They also provide a survey ofrelatively recent papers on a the dial-a-ride problem, which is aspecial case of the problem they study, and the more general vehi-cle routing problem as it pertains to carpooling.

Calvo, de Luigi, Haastrup, and Maniezzon (2004) describe a sys-tem for organizing and optimizing carpooling services that inte-grates the Web, SMS (texting), and a Geographic InformationSystem (GIS) with an optimization module. Authors such as Isaac(2004), Bisson (2005), and Hurzeler (2006) have applied for, andsome cases received, patents for similar systems and communica-tion tools to facilitate shared-ride programs. Bailey (1983) con-ducted a computer simulation study to estimate the market forvanpooling in the Baltimore region and Nassar (1986) applied sim-ulation techniques to develop a nonprofit vanpooling service pro-gram university faculty and staff.

One of the goals of vanpooling is to help vanpool passengers re-duce their current commuting costs, and several studies (e.g., Con-cas, Winters, & Wambalaba, 2005; Heaton, Abkowitz, Damm, &Jacobson, 1981; Maxwell & McIntyre, 1979; Maxwell & William-son, 1980; Morris, 1981; Ungemah, Rivers, & Anderson, 2006) focuson the economic relationship between pricing and demand forvanpooling. In order to encourage employees to join a vanpool—and prevent them from leaving when new riders join—it is impor-tant that the fare structure is perceived as fair. Building on thework Bird (1976) and Norde, Moretti, and Tijs (2004) use a gametheoretical approach to address this problem for carpools, whichare generally smaller than the vanpools we consider. For relatedstudies measuring the effects of incentives, policies, and gas priceson participation in other shared-ride programs such as carpools,see Stern (1976), Hekimian and Hershey (1981), Jacobs, Fairbanks,Poche, and Bailey (1982), Giuliano, Levine, and Teal (1990), Willsonand Shoup (1990), Vugt, Van Lange, Meertens, and Joireman(1996), Ferguson (1997)Mildner, Strathman, and Bianco (1997),Bauserman (1999), Huang, Yang, and Bell (2000) and Washbrook,Haider, and Jaccard (2006).

The rest of this paper is structured as follows. Section 2 de-scribes the Minimum Cost Vanpool Assignment Model (MCVAM)and the Two-Stop Minimum Cost Vanpool Assignment Model(TSMCVAM). The mathematical formulations of these models arealso given in Section 2. In Section 3, we describe three heuristicmethods to solve the TSMCVAM. Section 4 is organized to presentthe experimental setup, baseline computational results obtainedby applying an exact method to solve the MCVAM and the TSMC-VAM, and comparative results demonstrating the efficacy of ourheuristics for the TSMCVAM. We summarize our findings and out-line directions for further study of the Vanpool Assignment Prob-lem in Section 5. In the Appendix we show that MCVAM andTSMCVAM belong to class of inherently difficult NP-hard optimiza-tion problems (Garey & Johnson, 1979).

We claim three main contributions for this work: (1) introduc-tion of the MCVAM, (2) introduction of the TSMCVAM, (3)

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26 L. Kaan, E.V. Olinick / Computers & Industrial Engineering 66 (2013) 24–40

heuristics for the TSMCVAM. The literature focuses on carpoolingand related shared-vehicle transportation models. To the best ofour knowledge, the MCVAM is the first mathematical program-ming model proposed for the standard (one-stop) Vanpool Assign-ment Problem. The MCVAM models the current practice invanpooling of using one park-and-ride location per vanpool. How-ever, the TSMVCAM can generate significant cost savings comparedto the MCVAM with little or no increase in trip times for most pas-sengers by allowing vanpools to stop at a second park-and-ridelocation. The heuristics introduced in this paper can find high-quality solutions to the TSMCVAM with reasonable CPU times.

2. Vanpool Assignment Models

In this section we present two vanpool assignment models. Thefirst model is called the Minimum Cost Vanpool Assignment Modeland it minimizes the total system cost for all passengers using thetraditional one-stop approach. The second model, called the Two-Stop Minimum Cost Vanpool Assignment Model, minimizes the to-tal system cost for all passengers using a two-stop approach. In ourmodel, the vanpool driver is a commuter who has committed totaking passengers to and from work each day. The commuter van-pool lease company (DART) pays for each van’s insurance, includ-ing collision and liability, as part of the lease cost. In most cases,vanpool participants (drivers and riders) make a monthly commit-ment, and may leave the vanpool at any time after giving a month’snotice. Vanpool riders and drivers may set their own policy aboutpick-up and drop-off locations, but because fares are based in parton the fuel expenses for the van, our model assumes that all van-pools are organized around park-and-ride locations. For example,shopping centers will typically allow their parking lots to be usedas vanpool park-and-ride locations.

Many of the models in the ridesharing literature include timewindows as part of the input. That is, potential riders specify de-sired time ranges for being picked up at their home, or origin point,and dropped off at their destination. We assume that every poten-tial vanpool participant is willing to arrive at, and depart from, theworkplace (target location) at particular times (e.g., arrive at 9 amand leave at 5 pm). So rather than specifying time windows, thepotential vanpool participants in our model indicate a maximumallowable commuting time that includes the time spent drivingto the park-and-ride location and the time spent riding in thevan to the target location. A large employer offering vanpoolingbenefits would segment its employees into different groups basedon desired arrival and departure times, and solve an instance of themodel described below for each group. For example, a manufac-turer with 24-h operations might solve the model for three differ-ent 8-h work shifts.

1 Note that in some vanpool programs the driver of the vanpool rides free, in thoseses we need to subtract 1 from N to determine passenger p’s maximum share of the

anpooling cost.

2.1. The MCVAM

In this section an Integer Linear Program (ILP) called the Mini-mum Cost Vanpool Assignment Model (MCVAM) is presented forthe Vanpool Assignment Problem. In the MCVAM there are a num-ber of passengers, a number of potential park-and-ride locations,and a number of van types. Any passenger can be assigned to joina vanpool at any location, incurring some cost that may varydepending on the passenger-location assignment and van type. Itis required to assign each passenger to at most one location in sucha way that the total cost of the assignment is minimized. TheMCVAM determines the assignment of passengers and vans tovanpools, and a set of park-and-ride locations for a given probleminstance. Because of the nature of the contracts involved in van-pooling programs, we assume that the vanpools are fairly stable

and only change periodically. Thus, in practice the model wouldbe run once a month or once a quarter.

2.1.1. Sets and constants in the MCVAMLet P denote the set of passengers (employees considering the

vanpool program), L denote the set of park-and-ride locations,and V denote the set of van types or sizes (e.g., V = {small, large}).For each p 2 P and ‘ 2 L, (x,y) coordinates are known and can beused to calculate shortest travel distances to a park-and-ride loca-tion or the target location, (0,0). Distances for potential vanpoolpassengers to go to park-and-ride locations (dp‘), distances of po-tential park-and-ride locations to the target location (s‘), and thedistance from each passenger’s home to the target location (hp)are given in miles. The parameters Ck and vk k 2 V represent theone way trip rental cost and average mileage per gallon of a typek van, respectively. The average price per gallon for unleaded gasis denoted by g. Calculations of individual driving costs use thestandard mileage rate, r. For each passenger driving to a park-and-ride location and for each passenger opting out of vanpooling,we use this constant to calculate incurred costs. The standard mile-age rate is assumed to include all the costs of operating a vehicleincluding depreciation, gas, maintenance, etc. A passenger p willopt-out of vanpooling if his commuting time in the vanpool ex-ceeds his current commuting time by the time allowance, tp. If avan of type k is used in a vanpool, then the number of passengersin that vanpool must be at least a minimum value (mk) and nomore than the van capacity, uk. Note that we assume that thevan supplier has a large enough fleet to support the MCVAM solu-tion. That is, it has at least |P|/mk vans of type k. The MCVAM alsouses three computed parameters. Let F‘k, ‘ 2 L, k 2 V, denote the pertrip fuel cost for driving a van of type k from a park-and-ride loca-tion ‘ to the target location. Using the notation introduced above,

F‘k ¼gvk

� �s‘:

Passenger p will not join a vanpool at a park-and-ride location ‘ ifhis current commuting cost is less than the vanpooling cost. Thus,we only allow passenger p to be assigned to a vanpool at location‘ with a van of type k if

rhp P rdp‘ þCk þ F‘k

N; ð1Þ

where N is the number of passengers1 assigned to the vanpool atlocation ‘. Rearranging (1) we get

N PCk þ F‘k

rðhp � dp‘Þ

� �;

and define

Tp‘k ¼Ck þ F‘k

rðhp � dp‘Þ

� �;8p 2 P;8‘ 2 L;8k 2 V ; ð2Þ

where Tp‘k is the minimum number of passengers that must be as-signed to location ‘ for passenger p to join a vanpool at location ‘

with a van of type k. The cost of passenger p driving his vehicle fromhome to target location is cp = rhp.

2.1.2. Decision variables in the MCVAMThere are four types of decision variables in the MCVAM. The

binary variable xp‘k is equal to one if, and only if, passenger p joinsa vanpool at location ‘ using van type k. To reduce the problemsize, xp‘k is only defined for combinations of p and ‘ where dp‘ 6 hp.It is assumed that a passenger can drive 30 miles/h as an average to

cav

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L. Kaan, E.V. Olinick / Computers & Industrial Engineering 66 (2013) 24–40 27

his assigned park-and-ride location, and 60 miles/h on averagefrom the park-and-ride location to the target location in the van.He also can average 45 miles/h driving his own car to the targetlocation. Thus, we restrict the allowable (p,‘) combinations tothose where

2dp‘ þ s‘ �43

hp 6 tp; 8p 2 P;8j 2 L:

That is, we do not assign passengers to vanpools that violate thetime allowance. We denote the set of allowable (p,‘) combinationsas A, and the set of park-and-ride locations to which passenger pcould be assigned is denoted by Lp = {‘ in L|(p, ‘) 2 A}. The binaryvariable y‘k is equal to one if, and only if, there is a vanpool meetingat location ‘ using a van of type k. The decision of passenger p toopt-out of vanpooling, wp, is the third type of decision variable. Fi-nally, decision variable Z1 is the objective function value of theMCVAM.

2.1.3. Objective function and constraints of the MCVAMThe decision maker solving the MCVAM is the employer, or the

transit agency that manages the vanpool program on behalf of theemployer. Thus, the model seeks a system optimal solution mini-mizing the total cost of a one-way trip to the target location. Eachpassenger is either assigned to a vanpool at one particular park-and-ride location or not assigned at all. The MCVAM minimizesthe total system cost rather than each individual’s commuting cost.The model assumes that the employees in set P have applied to jointhe vanpool program and have agreed to participate in their as-signed vanpool provided that doing so neither increases their cur-rent commuting cost (i.e., the cost of driving their own vehicle tothe target location) nor increases their commuting time beyondtheir stated time allowance. To avoid solutions that achieve systemoptimality at the expense of one or more individual passengers, theMCVAM is constrained to ensure that passengers are not assignedto vanpools that would be more expensive than their current com-muting cost.

The objective of the MCVAM is as follows:

MinimizeZ1; ð3Þ

where

Z1 ¼X‘2L

Xk2V

ðCk þ F‘kÞy‘k þXðp;‘Þ2A

Xk2V

rdp‘xp‘k þXp2P

cpwp: ð4Þ

The objective given in (4) is the sum of three main costs as follows.The total per-trip cost due to van rental and van fuel cost isX‘2L

Xk2V

ðCk þ F‘kÞy‘k:

The total cost of vanpool passengers driving to the park-and-ridelocations isXðp;‘Þ2A

Xk2V

rdp‘xp‘k;

and the total cost of passengers who opt-out of vanpooling isXp2P

cpwp:

The following equations and inequalities are the constraintsthat bound the MCVAM. The set of constraints given in inequalitiesof type (5) enforces the minimum van capacity limits:Xfp2Pjðp;‘Þ2Ag

xp‘k P mky‘k 8‘ 2 L;8k 2 V : ð5Þ

If a van of type k is assigned to park-and-ride location ‘ then y‘kequals one and the right-hand side of (5) becomes the minimumnumber of passengers needed to form a vanpool using a van of type

k. On the other hand, (5) is automatically satisfied when y‘k equalszero. The set of constraints given in inequalities of type (6) enforcesthe maximum van capacity limits:Xfp2Pjðp;‘Þ2Ag

xp‘k 6 uk 8‘ 2 L;8k 2 V : ð6Þ

The set of constraints given in equalities of type (7) ensures thateach passenger is either assigned to a vanpool or opts-out:X‘2Lp

Xk2V

xp‘k þwp ¼ 1 8p 2 P: ð7Þ

The set of constraints given in inequalities of type (8) ensuresthat variables x and y are linked logically:

xp‘k 6 y‘k 8ðp; ‘Þ 2 A;8k 2 V : ð8Þ

The set of constraints given in inequalities of type (9) ensuresthat passengers are not assigned vanpools that would be moreexpensive than their current commuting costs:

Tp‘kxp‘k 6X

fi2Pjði;‘Þ2Agxi‘k 8ðp; ‘Þ 2 A;8k 2 V : ð9Þ

Constraint sets (10) and (11) define the domains for the vari-ables xp‘k and y‘k, respectively:

xp‘k 2 f0;1g 8ðp; ‘Þ 2 A; 8k 2 V ; ð10Þ

y‘k 2 f0;1g 8‘ 2 L;8k 2 V : ð11Þ

2.2. The TSMCVAM

Studying the MCVAM solutions for problem instances in ourcomputational study (see Section 4), we found that typically therewere multiple vanpools with excess capacity and that in many ofthese cases the straight-line path between the vanpool’s park-and-location, i, and the target passed closely by another park-and-ride location, j, that was close to the starting location of a pas-senger, p, who had opted out of vanpooling. In several of thesecases we discovered that the total time required for a van to gofrom park-and-ride location i to park-and-ride location j, and thento the target was within the time allowance of passenger p, andalso within the time allowances of all the passengers in the van-pool. This observation motivated us to develop an alternative van-pool assignment model called the Two-Stop Minimum Cost VanpoolAssignment Model (TSMCVAM), which, as demonstrated by ourcomputational study (see Section 4), can produce significant costsavings over the traditional one-stop approach modeled by theMCVAM.

The TSMCVAM extends the MCVAM to consider vanpools thatstop at up to two park-and-ride locations instead of just one. Thisallows starting vanpools at some locations where there may not beenough potential passengers (i.e., fewer than mk) to start a vanpoolin the MCVAM. In such cases, a second park-and-ride location willeventually satisfy the constraint for the minimum number of pas-sengers to operate a vanpool. It is still required to assign each pas-senger to at most one location in such a way that the total cost ofthe assignment is minimized. The TSMCVAM determines theassignment of passengers and vans to vanpools and a set of firstand second (if any) park-and-ride locations for a given problem in-stance. It should also be noted that the vanpool price per passengeris assumed to be based on the first park-and-ride location, so everypassenger will pay equal shares as in the MCVAM.

2.2.1. Sets and Constants in the TSMCVAMIn addition to the constants, and parameters defined in this sec-

tion, the TSMCVAM uses all the sets, constants, and parameters de-

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28 L. Kaan, E.V. Olinick / Computers & Industrial Engineering 66 (2013) 24–40

fined previously in the MCVAM. Distances between park-and-ridelocations (bij) represent the driving distance between park-and-ride locations i and j and are given in miles. The TSMCVAM usestwo new computed parameters.

For i, j 2 L and k 2 V, let F�

ijk denote the per trip fuel cost for driv-ing a van of type k from park-and-ride location i to park-and-ridelocation j, and then to the target location. Using the notation intro-duced above,

F�

ijk ¼gvk

� �ðsj þ bi;jÞ: ð12Þ

Passenger p will not join a two-stop vanpool starting at park-and-ride location i and then going to park-and-ride location j ifhis current commuting cost is less than the vanpooling cost. Thus,we only allow passenger p to be assigned to a two-stop vanpoolstarting at location i and then stopping at location j with a van oftype k if

rhp P r minðdpi;dpjÞ þCk þ F

�ijk

N; ð13Þ

where N is the total number of passengers assigned to the vanpoolat locations i and j. We assume that if passenger p joins a two-stopvanpool that he will drive to the closer park-and-ride location. Rear-ranging (12) we get

N PCk þ F

�ijk

rðhp �minðdpi;dpjÞÞ

& ’;

and define

T�

pijk ¼Ck þ F

�ijk

rðhp �minðdpi; dpjÞÞ

& ’8p 2 P; i; j 2 L; k 2 V ; ð14Þ

where T�

pijk is the minimum number of passengers that must be as-signed to locations i and j for passenger p to join a vanpool thatstops at locations i and j with a van of type k.

2.2.2. Decision variables in the TSMCVAMThere are three new types of decision variables in the TSMC-

VAM. The binary variable Xpijk is equal to one if, and only if, passen-ger p is assigned to a van of type k which goes from park-and-ridelocation i to park-and-ride location j and then to the target. An-other binary variable, Yijk, is equal to one if, and only if, a type kvan goes from park-and-ride location i to park-and-ride location jand then to the target. To reduce the problem size, Xpijk and Yijk

are only defined for combinations of i and j where si P sj (i.e. j isnot farther from the target than i); we denote the set of allowable(i, j) combinations as S. In addition to our previous assumptions forthe MCVAM, the TSMCVAM assumes that a van can be driven45 miles/h as an average from a park-and-ride location to anotherpark-and-ride location. Thus, we restrict the allowable (p, i, j) com-binations to those where (p, j) is in A or

2dp;i þ43

bi;j þ sj �43

hp 6 tp:

That is, we do not assign passengers to two-stop vanpools thatviolate the time allowance. We denote the set of allowable (p, i, j)combinations as B where (p, i, j) is in B only if (i, j) is in S. For a pairof park-and-ride locations (i, j) 2 S, let Pij = {p in P| (p, i, j) in B} beset of passengers who could potentially be assigned to a two-stopvanpool making stops at i and j. The set of pairs of park-and-ridelocations to which passenger p could be assigned in a two-stopvanpool is denoted by Sp = {(i, j) 2 S| (p, i, j) 2 B}. Finally, decisionvariable Z2 is the objective function value of the TSMCVAM.

2.2.3. Objective Function and Constraints in the TSMCVAMThe TSMCVAM minimizes the total system cost rather than each

individual’s commuting cost. The problem is to minimize the totalcost of a one-way trip to the target location. The objective is asfollows:

MinimizeZ2; ð15Þ

where

Z3 ¼ Z1 þXði;jÞ2S

Xk2V

ðCk þ F�

ijkÞYijk þ rXðp;i;jÞ2B

Xk2V

minðdpi;dpjÞXpijk: ð16Þ

The objective given in (15) is the sum of three main costs as fol-lows. The total per-trip cost of one-stop vanpools (as in theMCVAM), Z1, and, the total per-trip van rental and fuel costs fortwo-stop vanpools,Xði;jÞ2S

Xk2V

ðCk þ F�

ijkÞYijk;

and the total cost of vanpool passengers driving to the park-and-ride location

rXðp;i;iÞ2B

Xk2V

minðdpi;dpjÞXpijk:

In addition to the constraints (5)-(11), which were previouslyintroduced in the MCVAM, we also have new equations and inequal-ities defined below as the constraints that bound the TSMCVAM.

The set of constraints given in inequalities of type (16) enforcesthe minimum van capacity limits for two-stop vanpools:Xp2Pij

Xpijk P mkYijk 8ði; jÞ 2 S;8k 2 V : ð17Þ

If a van of type k is assigned to park-and-ride locations i and j thenYijk equals one and the right-hand side of (16) becomes the mini-mum number of passengers needed to form a vanpool using a vanof type k. On the other hand, (16) is automatically satisfied when Yijk

equals zero. The set of constraints given in inequalities of type (17)enforces the maximum van capacity limits for two-stop vanpools:Xp2Pij

Xpijk 6 uk 8ði; jÞ 2 S;8k 2 V : ð18Þ

The set of constraints given in equalities of type (18) ensuresthat each passenger is either assigned to a vanpool group oropts-out:Xði;jÞ2Sp

Xk2V

Xpijk þX‘2Lp

Xk2V

xp‘k þwp ¼ 1 8p 2 P: ð19Þ

The set of constraints given in inequalities of type (19) ensures thatvariables X and Y are linked logically:

Xpijk 6 Yijk 8ðp; i; jÞ 2 B;8k 2 V : ð20Þ

The set of constraints given in inequalities of type (20) ensures thatpassengers are not assigned to vanpools that would be more expen-sive than their current commuting costs:

T�

pijkXpijk 6Xq2Pij

Xpijk 8ðp; i; jÞ 2 B; 8k 2 V : ð21Þ

The sets of constraints given in inequalities of type (21), and (22)define the boundary conditions for the X and Y variables,respectively:

Xpijk 2 f0;1g 8ðp; i; jÞ 2 B; 8k 2 V ; ð22Þ

Yijk 2 f0;1g 8ði; jÞ 2 S; 8k 2 V : ð23Þ

Thus the TSMCVAM minimizes Z2 subject to (5), (6), (8)-(11), and(16)-(22).

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L. Kaan, E.V. Olinick / Computers & Industrial Engineering 66 (2013) 24–40 29

3. Heuristics

In our computational study (see Section 4), we found it chal-lenging to solve the TSMCVAM to provable optimality with astraight-forward application of the state-of-the-art ILP solverCPLEX. For example, it took over 12 h on average to solve med-ium-sized TSMCVAM problem instances whereas the correspond-ing MCVAM problems were solved in a matter of seconds.Therefore, we developed the three heuristic algorithms describedin this section for finding high quality solutions to the TSMCVAMwithin reasonable time frames.

3.1. The Restricted Allowance Heuristic

In Section 2.2.2., the set B was defined as the set of passengersthat can be assigned to two-stop vanpools without violating theirtime allowance assuming that the distance to the target from thesecond park-and-ride stop is not farther than the initial park-and-ride stop. In this section we present the Restricted AllowanceHeuristic (RAH) which further restricts the possible (p, i, j) combi-nations for the set B by eliminating the combinations that are un-likely to be used in an optimal solution. The RAH assumes that acombination is unlikely to be used in an optimal solution if the dis-tance between the initial park-and-ride location and the secondpark-and-ride location (bij) is longer than the distance from the ini-tial park-and-ride location to the target location (si). That is, RAHsets Xpijk equal to zero for all k where bij > si. As we show in the nextsection, the RAH can significantly reduce the total size of the set B.

3.2. The Relaxed Restricted Allowance Heuristic

The Relaxed Restricted Allowance Heuristic (RRAH) relaxes theTSMCVAM to a linear program (LP) by allowing the binary vari-ables to be continuous on the range [0, 1]. The solution to this LPis then used to gain information on what park-and-ride locationsappear to be most useful. This park-and-ride selection informationis then used to solve the ILP faster by removing those park-and-rides that the LP solution did not use (i.e., those where the corre-sponding y and Y variables are zero). The RRAH is depicted in Fig. 1.

3.3. The Greedy Cover Heuristic

A set of park-and-ride locations is said to cover passenger p if itcontains at least one pair of locations (i, j) such that (p, i, j) is in B.The Greedy Cover Heuristic (GCH) selects a minimal set of two-stop park-and-ride combinations from the set B that covers all pas-sengers that can be covered. GCH then solves the RAH with the se-lected two-stop park-and-ride combinations, S. Intuitively, weexpect GCH to generate competitive results in terms of cost sinceeach iteration picks the most attractive park-and-ride combinationto add the new set S, and then lets the TSMCVAM assign passengersto vanpools in cases where there are multiple combinations cover-ing a potential passenger. As we show in Section 4, GCH was notonly effective, but relatively fast, because typically only a smallnumber of park-and-ride locations are required to cover all thepassengers in our test problems. The GCH can be described moreformally as follows:

1. Initialize Setsa. W� P is the set of passengers who opt-out of two-stop van-

pooling because there is no suitable two-stop park-and-ridecombination (i, j) to assign them to. That is, passenger p is inset W if there is no (i, j) in S such that (p, i, j) is in B.

b. S0 # S is the set of two-stop combinations selected as inputto the optimization model.

c. U # PnW is the set of passengers that are uncovered by theset of selected two-stop combinations S. That is, passenger pis in set U if there is no (i, j) in S0 such that (p, i, j) is in B.

d. Ii,j is the set of passengers that are covered by the two-stopcombination (i, j). That is, passenger p is in set Ii,j if (p, i, j)is in B.

2. While |U| > 0 (i.e., there are uncovered passengers) Do thefollowing:a. Find the two-stop combination that covers the most uncov-

ered passengers. That is, find (i, j) in S/S0 that maximizesjIij \ Uj.

b. Add the two-stop combination found in step 2(a) to S0. Thatis, S0 = S{(i, j)}. Note: if multiple combinations tie in step 2a,then add all of them to S0.

c. Remove the passengers covered by the two-stop combina-tion found in step 2(a) from the set of uncovered passengers.That is, let U = U/Iij.

3. Apply the RAH to the TSMCVAM with the additional set ofconstraints:

Yijk ¼ 0 8ði; jÞ 2 S n S0;8k 2 V : ð24Þ

4. Results

In this section we present our experimental results from apply-ing the MCVAM and the TSMCVAM to three problem sets that eachcontain 10 problem instances. Data used in this study are derivedfrom the Gulfstream Vanpooling Survey (GVS), which was given toGulfstream employees in November 2008. A total of 17 people re-sponded to GVS. It consists of ten questions and was sent to a lim-ited number of people (Kaan, 2011).

AMPL (Fourer, Gay, & Kernighan, 2003) modeling language Ver-sion 20051214 (Linux 2.6.9-5.Elsmp) with a direct link to CPLEX(ILOG, 2003) 10.0.0. was used to solve our problem instances. Alltest runs were made on a Dell PE2950 Server Dual Quad Core [email protected] GHz processor with 32 GB (32,768 MB) of RAM. In all ofour solutions the CPLEX mipgap parameter was set at 0.01%, unlessotherwise noted.

4.1. Experimental setup

The problem instances described earlier all use a common set of120 potential park-and-locations and two van types: small andlarge. The ten problem instances in problem sets 1, 2 and 3 eachhave 200, 400, and 600 passengers, respectively. The followingparameter values are derived from the GVS. One way trip rentals(Ckk 2 V) are $6.14 for a small van, and $6.60 for a large van. Aver-age mileage per gallon (vkk 2 V) is 16 for a small van and 13 for alarge van. At the time of study, the average price per gallon for un-leaded gas (g) was $3.89, and the standard mileage rate (r) was$0.55/mile (IRS, 2008). A small van requires the number of passen-gers to be at least a minimum value m1 = 6 and no more than thecapacity of u1 = 9. A large van requires the number of passengersto be at least a minimum value m2 = 10 and no more than the largevan capacity of u1 = 15.

The distance from each passenger’s home to the target location(hp) and the time allowance for passengers (tp, p 2 P) are derivedfrom GVS data. Based on a statistical analysis of GVS data, the val-ues are well approximated by a Normal distribution with a mean of27.91 miles and standard deviation of 6.54 miles. Thus, the hp val-ues in our test problem instances are generated from a truncatedNormal distribution with mean 27.91 and standard deviation 6.54.That is, the hp value for a given passenger p is the maximum of zeroand a random sample from the Normal distribution with mean and

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Fig. 1. The relaxed restricted allowance heuristic.

30 L. Kaan, E.V. Olinick / Computers & Industrial Engineering 66 (2013) 24–40

standard deviation of 27.91 and 6.54, respectively. Likewise thetime allowances for the passengers (tp, p 2 P) are generated fromthe truncated Normal distribution with a mean of 11.52 min andstandard deviation of 4.03 min.

In each problem instance the 120 park-and-ride locations aredistributed evenly throughout the Dallas/Fort Worth metro areaas illustrated in Fig. 2. The red points in the figure are park-and-ride locations. Observe that there is one park-and-ride locationevery 10 miles in a 50 mile by 50 mile grid containing the targetlocation at its center. The (x, y) coordinates of the passenger loca-tions (blue points) are determined by choosing a random angle foreach passenger relative to the target location and using theappropriate formulas to convert polar to Cartesian coordinates.The angles are selected from the uniform distribution on the inter-val [0, 360]. The (x, y) coordinates were used to calculate shortest(Euclidean)2 distances for potential vanpool passengers to go to

2 Euclidean distances are an approximation. More exact data could be input to themodel by using driving distances calculated with a web-based trip planning tool.

park-and-ride locations (dp‘) from their homes, and distances of po-tential park-and-ride locations to the target location (s‘).

4.2. Baseline results

In the following two sections we summarize our computationalresults for the MCVAM and the TSMCVAM. The summary includesthe number of constraints, and binary variables in the models, andthe CPU time and total cost of the solutions found for each problemset. For the TSMCVAM results we also include optimality gaps andthe cost savings compared to the MCVAM.

4.2.1. MCVAM resultsThe MCVAM results are summarized in Tables 1–3 for problem

sets 1, 2, and 3, respectively. In problem set 1, the number of con-straints ranged from 3198 to 3676 and the number of binary vari-ables ranged from 1696 to 1936. The averaged solution time wasless than 1 s of CPU time. The objective function values (totalper-trip cost) for the solutions ranged from $942.80 to $986.74with an average cost of $967.04. In problem set 2, the number of

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Fig. 2. Vanpool Assignment problem set 1, Instance 1.

L. Kaan, E.V. Olinick / Computers & Industrial Engineering 66 (2013) 24–40 31

constraints ranged from 7194 to 7592 and the number of binaryvariables ranged from 3796 to 3992. The average solution timewas 2 s. The objective function values for the solutions ranged from$1424.45 to $1540.41 with an average cost of $1475.50. In problemset 3, the number of constraints ranged from 10,414 to 11,057 andthe number of binary variables ranged from 5502 to 5820. Theaverage solution time was 26 s. The objective function values forthe solutions ranged from $2158.40 to $2282.93 with an averagecost of $2240.35.

4.2.2. TSMCVAM resultsThe TSMCVAM results can be seen in Tables 4–6 for problem

sets 1, 2, and 3, respectively. In problem set 1, the number of con-straints ranged from 179,532 to 209,136, and the number of binaryvariables ranged from 151,386 to 174,608. The solution time aver-age was 38 min and 17 s of CPU time. The objective function valuesfor the solutions ranged from $786.17 to $825.22 with an averagecost of $805.32. The optimality gaps ranged from 0.20% to 0.92%with an average of 0.70%, and the cost savings compared to theMCVAM was 16.71% on average. In problem set 2, the number ofconstraints ranged from 405,878 to 432,698, and the number ofbinary variables ranged from 342,309 to 365,538. The solutiontime average was 36 h, 41 min and 15 s. The objective function val-ues for the solutions ranged from $1341.05 to $1449.38 with anaverage cost of $1389.83. The optimality gaps ranged from 3.41%to 4.99% with an average of 4.32%, and the cost savings comparedto the MCVAM was 5.80% on average. In problem set 3, the numberof constraints ranged from 587,717 to 633,022, and the number ofbinary variables ranged from 497,534 to 531,775. The solutiontimes for this problem set topped the 8 h limit for all problem in-stances. The objective function values for the solutions rangedfrom $4935.53 to $6521.44 with an average cost of $5488.40. Theoptimality gaps ranged from 61.15% to 70.49% with an average of

64.14%, and so the cost savings compared to the MCVAM is notreported.

In the following sections we use the cost of the best TSMCVAMsolution found by CPLEX within the 8 h time limit and the reportedoptimality gap to derive a lower bound on the cost of an optimalsolution for a given problem instance. For example, Table 6 showsthat CPLEX found a solution with a cost of $5073.85 for Problem In-stance 1 from Data Set 3. Since the optimality gap reported for thissolution was 62.59%, it follows that the cost of an optimal solutionmust be at least $1898.13 (37.41% of $5037.85). These lowerbounds are used to measure the quality of the solutions found byour heuristics.

4.3. Heuristic results

In this section we give computational results obtained by solv-ing the TSMCVAM with the RAH, the RRAH, and the GCH heuristicson the largest problem set (600 passengers and 120 park-and-ridelocations). Results for the other problem sets exhibit similar trendsand can be found in Kaan (2011).

4.3.1. The RAH ResultsThe RAH results for problem set 3 are reported in two separate

tables since we ran the heuristic for this problem set with andwithout changing CPLEX’s default mipgap parameter value. Theseresults can be found in Tables 7 and 8, respectively. The last threecolumns of each table give the following results: (1) total CPU timesavings using the RAH compared to using the full B set, (2) cost in-crease using RAH compared to using the full B set (a negative valueindicates a savings), and (3) total number of binary variables in themodel after applying the RAH.

With RAH, the number of binary variables (after AMPL’s pre-solve step) ranged from 84,949 to 95,709 with an average of90,557. The average number of binary variables (after pre-solve)

Page 9: The Vanpool Assignment Problem: Optimization models and solution algorithms

Table 1The MCVAM results summary for problem set 1 (200 passengers).

ProblemInstance

TotalCost

# Opt-Out

# Opt-in

Ave. Cost Opt-Out

Max. Cost Opt-out

Ave. CostOpt-In

Max. CostOpt-in

# of Vantype 1

# of Vantype 2

# ofLocations

Total CPU time(hh:mm:ss)

#ofConstraints

# of Binaryvariables

1 $984.80 8 192 $14.96 $24.26 $15.37 $22.49 21 3 24 0:00:01 3400 17972 $964.72 6 194 $10.90 $23.86 $15.92 $25.97 25 1 26 0:00:01 3676 19363 $986.74 9 191 $13.97 $25.32 $15.36 $25.40 26 1 27 0:00:01 3299 17464 $952.22 6 194 $13.43 $20.34 $15.35 $23.17 25 2 27 0:00:01 3663 19295 $983.79 9 191 $12.00 $21.15 $15.30 $24.95 23 2 25 0:00:01 3198 16966 $960.57 10 190 $10.69 $21.89 $15.24 $21.83 21 3 24 0:00:01 3437 18157 $942.80 6 194 $10.00 $18.42 $15.31 $24.03 27 0 27 0:00:01 3637 19178 $971.24 10 190 $14.67 $23.19 $14.85 $23.93 23 2 25 0:00:01 3543 18679 $947.34 5 195 $9.06 $11.48 $15.17 $25.64 23 3 26 0:00:01 3386 179110 $976.13 6 194 $15.09 $25.18 $15.56 $26.25 26 0 26 0:00:01 3577 1886Min $942.80 5 190 $9.06 $11.48 $14.85 $21.83 21 0 24 0:00:01 3198 1696Mean $967.04 8 193 $12.48 $21.51 $15.34 $24.37 24 2 26 0:00:01 3482 1838Max $986.74 10 195 $15.09 $25.32 $15.92 $26.25 27 3 27 0:00:01 3676 1936

Table 2The MCVAM results for problem set 2 (400 passengers).

ProblemInstance

TotalCost

# Opt-Out

# Opt-In

Ave. Cost Opt-Out

Max. Cost Opt-out

Ave. CostOpt-In

Max. CostOpt-in

# of Vantype 1

# of Vantype 2

# ofLocations

Total CPU time(hh:mm:ss)

#ofConstraints

# of Binaryvariables

1 $1430.68 5 395 $10.77 $14.74 $15.34 $25.75 10 21 31 0:00:01 7194 37962 $1540.41 12 388 $12.57 $17.65 $15.51 $25.30 12 21 33 0:00:02 7296 38433 $1480.02 8 392 $15.55 $24.87 $15.42 $28.12 9 21 30 0:00:02 7435 39164 $1536.74 10 390 $14.20 $18.17 $15.20 $28.33 11 20 31 0:00:02 7310 38515 $1486.87 7 393 $14.13 $20.79 $15.46 $27.88 13 19 32 0:00:03 7592 39926 $1450.43 6 394 $9.80 $16.06 $15.69 $27.80 11 21 32 0:00:02 7570 39837 $1470.17 6 394 $11.21 $16.74 $15.36 $24.36 12 20 32 0:00:02 7449 39198 $1442.00 6 394 $15.69 $17.39 $15.46 $25.46 13 20 33 0:00:01 7268 38319 $1424.45 7 393 $12.82 $23.37 $15.36 $26.17 10 22 32 0:00:01 7267 383010 $1493.20 6 394 $13.64 $20.29 $15.28 $24.37 16 18 34 0:00:01 7405 3899Min $1424.45 5 388 $9.80 $14.74 $15.20 $24.36 9 18 30 0:00:01 7194 3796Mean $1475.50 7 393 $13.04 $19.01 $15.41 $26.35 12 20 32 0:00:02 7379 3886Max $1540.41 12 395 $15.69 $24.87 $15.69 $28.33 16 22 34 0:00:03 7592 3992

32L.K

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Table 3The MCVAM results for problem set 3 (600 passengers).

ProblemInstance

TotalCost

# Opt-Out

# Opt-In

Ave. Cost Opt-Out

Max. Cost Opt-out

Ave. CostOpt-In

Max. CostOpt-in

# of Vantype 1

# of Vantype 2

# ofLocations

Total CPU time(hh:mm:ss)

#ofConstraints

# of Binaryvariables

1 $2261.95 14 586 $14.61 $24.82 $14.94 $25.88 22 27 49 0:00:17 10,414 55022 $2212.84 11 589 $14.72 $24.02 $15.24 $27.83 19 29 48 0:00:27 10,863 57253 $2282.93 15 585 $15.72 $22.42 $15.27 $26.62 24 26 50 0:00:25 11,057 58204 $2240.18 10 590 $16.10 $24.00 $15.17 $26.65 22 27 49 0:00:33 10,512 55505 $2271.08 15 585 $14.73 $23.57 $15.18 $24.83 20 29 49 0:00:22 10,684 56346 $2278.50 13 587 $16.15 $24.95 $15.39 $26.96 20 29 49 0:00:31 10,985 57857 $2248.79 11 589 $13.79 $23.47 $15.40 $26.05 26 25 51 0:00:22 10,832 57108 $2170.24 10 590 $12.82 $23.12 $15.30 $29.42 18 30 48 0:00:16 10,682 56359 $2278.63 11 589 $15.56 $24.42 $15.15 $27.40 23 27 50 0:00:40 10,931 575910 $2158.40 5 595 $12.39 $21.61 $15.39 $26.46 22 28 50 0:00:25 11,019 5806Min $2158.40 5 585 $12.39 $21.61 $14.94 $24.83 18 25 48 0:00:16 10,414 5502Mean $2240.35 12 589 $14.66 $23.64 $15.24 $26.81 22 28 49 0:00:26 10,798 5693Max $2282.93 15 595 $16.15 $24.95 $15.40 $29.42 26 30 51 0:00:40 11,057 5820

Table 4The TSMCVAM results for problem set 1 (200 passengers).

Probleminstance

Totalcost

# Opt-Out

# Opt-In

Ave. CostOpt-Out

Max. CostOpt-out

Ave. CostOpt-In

Max. CostOpt-in

# of Vantype 1

# of Vantype 2

Total CPU time(hh:mm:ss)

#ofConstraints

# of Binaryvariables

# of Twostops

Optimalitygaps

Cost savings (%)vs. MCVAM

1 $806.12 2 198 $11.65 $12.90 $15.39 $24.26 16 5 0:05:35 190,392 160,681 21 0.77% 18.14%2 $812.72 3 197 $7.82 $11.07 $15.89 $25.97 12 7 2:18:00 209,136 174,608 19 0.56% 15.76%3 $816.06 3 197 $12.24 $20.28 $15.35 $25.40 13 6 0:08:29 186,164 156,559 18 0.75% 17.30%4 $786.17 3 197 $13.69 $16.49 $15.32 $23.17 11 8 0:10:47 206,431 174,076 19 0.92% 17.44%5 $803.55 3 197 $17.80 $21.15 $15.11 $24.95 10 8 0:05:40 179,532 151,386 18 0.20% 18.32%6 $798.46 3 197 $10.26 $16.99 $15.09 $21.89 10 9 2:16:48 195,959 165,986 19 0.70% 16.88%7 $801.16 2 198 $16.22 $18.42 $15.14 $24.03 10 9 0:16:27 207,968 174,014 19 0.74% 15.02%8 $786.33 4 196 $10.11 $19.88 $14.94 $23.93 12 7 0:37:50 200,192 169,150 19 0.88% 19.04%9 $817.35 2 198 $9.72 $11.48 $15.07 $25.64 19 4 0:08:11 191,133 161,053 23 0.84% 13.72%10 $825.22 1 199 $18.01 $18.01 $15.54 $26.25 13 7 0:15:07 202,957 170,372 20 0.63% 15.46%Min $786.17 1 196 $7.82 $11.07 $14.94 $21.89 10 4 0:05:35 179,532 151,386 18 0.20% 13.72%Mean $805.32 3 197 $12.75 $16.67 $15.28 $24.55 13 7 0:38:17 196,986 165,789 20 0.70% 16.71%Max $825.22 4 199 $18.01 $21.15 $15.89 $26.25 19 9 2:18:00 209,136 174,608 23 0.92% 19.04%

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.Olinick

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&Industrial

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Table 5The TSMCVAM results for problem set 2 (400 passengers).

Probleminstance

Totalcost

# Opt-Out

#Opt-In

Ave. CostOpt-Out

Max. CostOpt-out

Ave. CostOpt-In

Max. CostOpt-in

# of vantype 1

# of vantype 2

Total CPU time(hh:mm:ss)

#ofConstraints

# of Binaryvariables

# of Twostops

n2(%)

Cost savings (%) vs.MCVAM

1 $1378.40 4 396 $11.21 $14.74 $15.32 $25.75 2 27 18:46:25 405878 342309 23 4.77 3.65%2 $1449.38 9 391 $12.87 $17.65 $15.48 $25.30 6 23 25:30:25 413021 346701 23 4.91 5.91%3 $1341.05 0 400 $0.00 $0.00 $15.42 $28.12 1 27 6:44:46 420471 353639 23 3.70 9.39%4 $1419.01 6 394 $14.02 $16.18 $15.19 $28.33 2 26 6:31:41 413619 349708 20 3.41 7.66%5 $1431.55 6 394 $15.02 $20.79 $15.44 $27.88 2 28 63:02:41 432698 365538 26 4.99 3.72%6 $1348.88 2 398 $7.33 $16.06 $15.64 $27.80 4 27 77:46:38 427077 357286 20 4.23 7.00%7 $1351.85 3 397 $7.17 $10.06 $15.36 $24.36 2 27 82:25:10 425273 359231 20 4.20 8.17%8 $1383.17 6 394 $14.98 $16.69 $15.47 $25.46 0 27 12:10:17 412594 345611 18 4.24 4.08%9 $1368.43 4 396 $13.01 $21.52 $15.34 $26.17 3 26 12:50:28 410753 347138 20 4.14 3.93%10 $1426.56 5 395 $15.30 $20.29 $15.26 $24.37 1 27 13:04:03 418570 353472 22 4.58 4.46%Min $1341.05 0 391 $0.00 $0.00 $15.19 $24.36 0 23 6:31:41 405878 342309 18 3.41 3.65%Mean $1389.83 5 396 $11.09 $15.40 $15.39 $26.35 2 26 36:41:15 417995 352063 21 4.32 5.80%Max $1449.38 9 400 $15.30 $21.52 $15.64 $28.33 6 28 82:25:10 432698 365538 26 4.99 9.39%

Note: The CPLEX settings mipgap parameter was set at 0.05 for these runs.

Table 6The TSMCVAM results for problem set 3 (600 passengers) after 8 h of CPU time.

Probleminstance

Totalcost

# Opt-Out

#Opt-In

Ave. CostOpt-Out

Max. CostOpt-out

Ave. CostOpt-In

Max. CostOpt-in

# of Vantype 1

# of Vantype 2

#ofConstraints

# of Binaryvariables

# of Twostops

# of Onestops

Optimalitygaps (%)

Cost savings (%) vs.MCVAM

1 $5073.85 276 324 $14.31 $25.88 $15.47 $23.52 13 14 587,717 497,534 16 11 62.59 N/A2 $5296.43 288 312 $14.70 $27.83 $15.71 $24.17 3 19 615,944 519,898 14 8 62.23 N/A3 $6016.94 330 270 $15.41 $26.62 $15.12 $24.89 7 14 633,022 531,775 9 12 67.06 N/A4 $5034.16 268 332 $14.36 $26.65 $15.85 $24.78 11 16 593,301 500,609 16 11 61.46 N/A5 $5104.16 274 326 $14.53 $22.57 $15.71 $24.83 5 19 610,305 515,536 17 7 61.15 N/A6 $6256.66 351 249 $15.17 $24.95 $15.73 $26.96 10 11 623,071 526,392 14 7 68.53 N/A7 $6521.44 367 233 $15.44 $26.05 $15.26 $25.35 11 9 616,998 519,147 11 9 70.49 N/A8 $4977.42 266 334 $14.09 $24.40 $16.19 $29.42 8 18 609,317 515,279 13 13 61.37 N/A9 $5667.38 318 282 $14.77 $24.42 $15.59 $27.40 12 12 623,379 526,492 11 13 65.29 N/A10 $4935.53 255 345 $14.64 $24.95 $15.90 $26.46 4 21 623,863 525,380 21 4 61.22 N/AMin $4935.53 255 233 $14.09 $22.57 $15.12 $23.52 3 9 587,717 497,534 9 4 61.15 N/AMean $5488.40 299 301 $14.74 $25.43 $15.65 $25.78 8 15 613,692 517,804 14 9 64.14 N/AMax $6521.44 367 345 $15.44 $27.83 $16.19 $29.42 13 21 633,022 531,775 21 13 70.49 N/A

Note: The CPLEX settings for these runs were mipgap = 0.05 and timelimit = 8 h.

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Table 7The TSMCVAM results for problem set 3 (600 passengers) by using the RAH (1).

Probleminstance

Totalcost

#Opt-Out

#Opt-In

Ave. CostOpt-Out

Max. CostOpt-Out

Ave. CostOpt-In

Max. CostOpt-In

Total CPU time(hh:mm:ss)

Optimalitygaps (%)

CPU time improvement vs.TSMCVAM with full B Set (%)

Cost increase vs. TSMCVAMwith full B Set (%)

Number ofbinary variables

1 $2068.67 7 593 $12.73 $24.82 $14.96 $25.88 2:27:17 8.25 69.32 �145.27 84,9492 $2158.53 12 588 $12.34 $21.73 $15.29 $27.83 1:12:55 7.33 84.81 �145.37 90,6203 $2200.13 16 584 $15.96 $24.66 $15.26 $26.62 1:34:58 9.93 80.22 �173.48 95,7094 $2118.64 11 589 $12.91 $17.71 $15.22 $26.65 1:23:49 8.43 82.54 �137.61 86,3945 $2156.81 13 587 $13.14 $20.35 $15.22 $24.83 2:16:29 8.06 71.57 �136.65 90,1196 $2170.60 14 586 $14.68 $24.95 $15.42 $26.96 1:43:08 9.29 78.51 �188.25 92,5027 $2145.08 10 590 $14.32 $23.47 $15.39 $26.05 2:57:01 10.27 63.12 �204.02 92,6698 $2118.84 9 591 $11.68 $21.88 $15.31 $29.42 2:20:04 9.25 70.82 �134.91 88,7459 $2131.54 12 588 $13.26 $21.72 $15.19 $27.40 1:38:29 7.72 79.48 �165.88 91,01510 $2058.27 6 594 $13.41 $21.61 $15.38 $26.46 1:28:14 7.01 81.62 �139.79 92,843Min $2058.27 6 584 $11.68 $17.71 $14.96 $24.83 1:12:55 7.01 63.12 �204.02 84,949Mean $2132.71 11 589 $13.44 $22.29 $15.26 $26.81 1:54:14 8.55 76.20 �157.12 90,557Max $2200.13 16 594 $15.96 $24.95 $15.42 $29.42 2:57:01 10.27 84.81 �134.91 95,709

Note: The CPLEX settings for these runs were mipgap = 5% and timelimit = 8 h.

Table 8The TSMCVAM results for problem set 3 (600 passengers) by using the RAH (2).

Probleminstance

Totalcost

# Opt-Out

#Opt-In

Ave. CostOpt-Out

Max. CostOpt-Out

Ave. CostOpt-In

Max. CostOpt-In

Optimalitygaps (%)

CPU time improvement vs. TSMCVAMwith full AA Set (%)

Cost increase vs. TSMCVAM withfull B Set (%)

Number of binaryvariables

1 $2053.07 7 593 $12.73 $24.82 $14.96 $25.88 7.55 0 �147.13 84,9492 $2119.34 11 589 $12.95 $21.73 $15.27 $27.83 5.61 0 �149.91 90,6203 $2155.90 15 585 $15.38 $22.13 $15.28 $26.62 8.08 0 �179.09 95,7094 $2073.80 8 592 $12.33 $16.38 $15.22 $26.65 6.45 0 �142.75 86,3945 $2099.27 13 587 $13.14 $20.35 $15.22 $24.83 5.54 0 �143.14 90,1196 $2143.96 13 587 $14.76 $24.95 $15.42 $26.96 8.16 0 �191.83 92,5027 $2116.02 11 589 $13.46 $23.47 $15.40 $26.05 9.04 0 �208.19 92,6698 $2059.83 9 591 $11.68 $21.88 $15.31 $29.42 6.65 0 �141.64 88,7459 $2112.07 11 589 $13.76 $21.72 $15.18 $27.40 6.87 0 �168.33 91,01510 $2005.04 5 595 $12.39 $21.61 $15.39 $26.46 4.54 0 �146.16 92,843Min $2005.04 5 585 $11.68 $16.38 $14.96 $24.83 4.54 0 �208.19 84,949Mean $2093.83 10 590 $13.26 $21.90 $15.27 $26.81 6.85 0 �161.82 90,557Max $2155.90 15 595 $15.38 $24.95 $15.42 $29.42 9.04 0 �141.64 95,709

Note: The CPLEX settings for these runs were mipgap = default and timelimit = 8 h.

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Table 9The TSMCVAM results for problem set 3 (600 passengers) by using the RRAH.

Probleminstance

Totalcost

# Opt-Out

#Opt-In

Ave. CostOpt-Out

Max. CostOpt-Out

Ave. CostOpt-In

Max. CostOpt-In

Total CPU time(hh:mm:ss)

Optimalitygaps (%)

CPU time improvementvs. RAH (%)

Cost increase vs.RAH (%)

Number of binaryvariables

1 $2073.12 7 593 $12.73 $24.82 $14.96 $25.88 1:28:19 8.45 40.04 0.21 15,6062 $2184.91 15 585 $12.86 $21.73 $15.29 $27.83 0:11:40 8.45 84.00 1.21 15,1823 $2198.87 15 585 $15.33 $22.13 $15.28 $26.62 1:13:35 9.87 22.52 �0.06 13,6524 $2144.42 10 590 $12.68 $19.40 $15.22 $26.65 0:43:29 9.53 48.12 1.20 16,0335 $2118.67 13 587 $13.14 $20.35 $15.22 $24.83 0:30:52 6.41 77.38 �1.80 12,6236 $2196.91 13 587 $14.76 $24.95 $15.42 $26.96 0:57:43 10.38 44.04 1.20 16,4627 $2141.89 11 589 $14.57 $23.47 $15.38 $26.05 0:26:44 10.14 84.90 �0.15 13,3128 $2147.98 11 589 $12.67 $21.88 $15.31 $29.42 0:36:40 10.48 73.82 1.36 17,0059 $2157.77 11 589 $14.92 $21.72 $15.16 $27.40 0:52:38 8.84 46.56 1.22 17,42710 $2039.30 5 595 $12.39 $21.61 $15.39 $26.46 0:54:52 6.14 37.82 �0.93 14,782Min $2039.30 5 585 $12.39 $19.40 $14.96 $24.83 0:11:40 6.14 22.52 �1.80 12,623Mean $2140.38 11 589 $13.61 $22.21 $15.26 $26.81 0:47:39 8.87 55.92 0.35 15,208Max $2198.87 15 595 $15.33 $24.95 $15.42 $29.42 1:28:19 10.48 84.90 1.36 17,427

Table 10The TSMCVAM results for problem set 3 (600 passengers) by using the GCH.

Probleminstance

Totalcost

# Opt-Out

#Opt-In

Ave. CostOpt-Out

Max. CostOpt-Out

Ave. CostOpt-In

Max. CostOpt-In

Total CPU time(hh:mm:ss)

Optimalitygaps (%)

CPU time improvementvs. RAH (%)

Cost increase vs.RAH (%)

Number of binaryvariables

1 $2071.90 7 593 $12.73 $24.82 $14.96 $25.88 0:02:37 8.39 99.40 0.16 22,1162 $2135.84 11 589 $12.95 $21.73 $15.27 $27.83 0:02:06 6.34 94.58 �1.06 25,2543 $2221.32 15 585 $15.38 $22.13 $15.28 $26.62 0:02:27 10.78 98.56 0.95 25,1014 $2107.75 7 593 $13.42 $16.38 $15.20 $26.65 0:01:15 7.96 99.05 �0.52 16,8335 $2177.86 14 586 $13.61 $20.35 $15.21 $24.83 0:00:51 8.95 99.40 0.97 16,4276 $2246.29 18 582 $14.88 $24.95 $15.42 $26.96 0:01:20 12.35 98.58 3.37 16,6557 $2174.22 13 587 $13.81 $23.47 $15.40 $26.05 0:02:20 11.47 99.16 1.34 23,7848 $2130.79 12 588 $12.45 $21.88 $15.31 $29.42 0:01:51 9.76 99.06 0.56 20,5429 $2204.08 15 585 $15.93 $23.12 $15.14 $27.40 0:01:21 10.76 98.46 3.29 19,67810 $2115.88 5 595 $12.39 $21.61 $15.39 $26.46 0:03:08 9.54 99.09 2.72 21,695Min $2071.90 5 582 $12.39 $16.38 $14.96 $24.83 0:00:51 6.34 94.58 �1.06 16,427Mean $2158.59 12 588 $13.76 $22.04 $15.26 $26.81 0:01:56 9.63 98.53 1.18 20,809Max $2246.29 18 595 $15.93 $24.95 $15.42 $29.42 0:03:08 12.35 99.40 3.37 25,254

Note: CPLEX settings for these runs were mipgap = 0.05 and timelimit = 8 h.

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Page 14: The Vanpool Assignment Problem: Optimization models and solution algorithms

Tabl

e11

The

TSM

CVA

Mre

sult

sfo

rpr

oble

mse

t3

(600

pass

enge

rs)

byus

ing

the

GCH

(2).

Prob

lem

inst

ance

Tota

lco

st#

Opt

-O

ut

# Opt

-In

Ave

.Cos

tO

pt-O

ut

Max

.Cos

tO

pt-O

ut

Ave

.Cos

tO

pt-I

nM

ax.C

ost

Opt

-In

Tota

lC

PUti

me

(hh

:mm

:ss)

Opt

imal

ity

gaps

(%)

CPU

tim

eim

prov

emen

tvs

.RA

H(%

)C

ost

incr

ease

vs.

RA

H(%

)N

um

ber

ofbi

nar

yva

riab

les

1$2

062.

067

593

$12.

73$2

4.82

$14.

96$2

5.88

0:07

:24

7.96

91.6

2�

0.54

22,1

162

$211

4.14

1158

9$1

2.95

$21.

73$1

5.27

$27.

830:

02:4

05.

3877

.14

�3.

3525

,254

3$2

188.

4915

585

$15.

38$2

2.13

$15.

28$2

6.62

0:13

:30

9.45

81.6

5�

0.47

25,1

014

$210

6.74

759

3$1

3.42

$16.

38$1

5.20

$26.

650:

02:2

87.

9294

.33

�1.

7916

,833

5$2

124.

6712

588

$13.

84$2

0.35

$15.

20$2

4.83

0:04

:21

6.67

85.9

10.

2816

,427

6$2

166.

8812

588

$14.

88$2

4.95

$15.

42$2

6.96

0:01

:00

9.13

98.2

7�

1.39

16,6

557

$211

5.78

1059

0$1

4.32

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47$1

5.39

$26.

050:

02:0

39.

0392

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2323

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8$2

089.

819

591

$12.

45$2

1.88

$15.

31$2

9.42

0:04

:20

7.99

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8�

2.78

20,5

429

$213

4.19

1059

0$1

4.67

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72$1

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400:

08:0

57.

8484

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1119

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10$2

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255

595

$12.

39$2

1.61

$15.

39$2

6.46

0:15

:09

7.10

72.3

91.

0221

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Min

$206

0.25

558

5$1

2.39

$16.

38$1

4.96

$24.

830:

01:0

05.

3872

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�3.

3516

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Mea

n$2

116.

3010

590

$13.

70$2

1.90

$15.

26$2

6.81

0:06

:06

7.84

86.6

5�

1.14

20,8

09M

ax$2

188.

4915

595

$15.

38$2

4.95

$15.

42$2

9.42

0:15

:09

9.45

98.2

71.

0225

,254

Not

e:C

PLEX

sett

ings

for

thes

eru

ns

wer

em

ipga

p=

defa

ult

and

tim

elim

it=

8h

.

L. Kaan, E.V. Olinick / Computers & Industrial Engineering 66 (2013) 24–40 37

using the full B set was 571,804. The solution times for problemset 3 using CPLEX with a preset 5% mipgap ranged from 1 h,12 min and 55 s to 2 h and 57 s with an average of 1 h, 54 minand 14 s of CPU time. This was a time savings of 63.12% to84.81%, with an average of 76.20% compared to the TSMCVAMresults with the full B set. To compute the improved optimalitygaps we used lower bounds derived by the solving TSMCVAMwith the full B set described in Section 4.2.2 and compared themto the upper bounds obtained by the RAH. Running the RAH foran average of 1 h 54 min and 14 s with a 5% mipgap resulted inoptimality gaps ranging from 7.01% to 10.27%, with an averageoptimality gap of 8.55%. On the other hand, running the RAHfor an average of 8 h with the default mipgap resulted in optimal-ity gaps ranging from 4.54% to 9.04%, with an average of 6.85%.As a result, adding approximately 6 more hours of CPU timehelped reduce the average optimality gap from 8.55% to 6.85%.The objective function values (total per-trip cost) using the 5%mipgap ranged from $2058.27 to $2200.13 with an average costof $2132.71. The objective function values (total per-trip cost)using the default mipgap ranged from $2005.04 to $2155.90 withan average cost of $2093.83. This resulted in cost increases rang-ing from 134.91% to 204.02%, with an average of 157.12% for 5%mipgap and 141.64% to 208.19%, with an average of 161.82% (anegative numbers indicates savings). These results indicate sig-nificant reductions in terms of CPU times and also significantsavings in terms of total cost. Statistical analysis and significancetests can be found in Kaan (2011).

4.3.2. The RRAH resultsThe RRAH results for problem set 3 are reported in Table 9.

With RRAH, the number of binary variables (after pre-solve) ran-ged from 12,623 to 17,427 with an average of 15,208. The aver-age number of binary variables (after pre-solve) using the full Bset was 571,804. The solution times ranged from 11 min and40 s to 1 h, 28 min and 19 s with an average of 47 min and 39 sof CPU time. This was a savings of 22.52% to 84.90%, with anaverage of 55.92% time savings compared to the RAH results(Section 4.3.1). Running the RRAH for an average of 47 min and39 s resulted in optimality gaps ranging from 6.14% to 10.48%,with an average optimality gap of 8.87%. The objective functionvalues ranged from $2039.30 to $2198.87 with an average costof $2140.38. This resulted in cost increases ranging from�1.80% to 1.36%, with an average of 0.35% (a negative numbersindicates savings) compared to the RHA results. These resultsrepresent statistically significant reductions in terms of CPU timeand statistically insignificant cost increases compared to RAH(Kaan, 2011).

4.3.3. The GCH resultsThe GCH results for problem set 3 are reported in two sepa-

rate tables since we ran the heuristic for this problem set withand without changing CPLEX’s default mipgap parameter value.These results can be found in Tables 10 and 11, respectively.With GCH, the number of binary variables (after pre-solve) ran-ged from 16,427 to 25,254 with an average of 20,809. The aver-age number of binary variables using the full B set was 571,804.The solution times with a preset 5% mipgap ranged from 51 s to3 min and 8 s with an average of 1 min and 56 s of CPU time. Thiswas a savings of 94.58% to 99.40%, with an average of 98.53%time savings compared to the RAH results (Section 4.3.1). Run-ning the GCH for an average of 1 min and 56 s with a 5% mipgapresulted in optimality gaps ranging from 6.34% to 12.35%, with anaverage optimality gap of 9.63%. On the other hand, running theRAH with the default mipgap resulted in optimality gaps rangingfrom 5.38% to 9.45%, with an average of 7.84%. The objectivefunction values using the 5% mipgap ranged from $2071.90 to

Page 15: The Vanpool Assignment Problem: Optimization models and solution algorithms

Table 12One-stop and two-stop vanpool models vs. heuristics performance metrics.

One-stop Two-stop Heuristics Number of passengers

MCVAM TSMCVAM RAH (1%) RAH (5%) RRAH GCH(1%) GCH(S%)

COST Min $942.80 $786.17 $783.92 N/A $802.54 $841.44 N/A 200 Problem setsMean $967.04 $805.32 $807.82 N/A $828.51 $867.80 N/AMax $986.74 $825.22 $826.22 N/A $839.94 $898.23 N/AMin $1424.45 $1341.05 N/A $1343.82 $1357.37 $1347.80 $1370.08 400Mean $1475.50 $1389.64 N/A $1393.77 $1417.51 $1391.54 $1420.71Max $1540.41 $1449.38 N/A $1452.73 $1478.98 $1443.50 $1497.80Min $2158.40 $4935.53 $2005.04 $2058.27 $2039.30 $2060.25 $2071.90 600Mean $2240.35 $5488.40 $2093.83 $2132.71 $2140.38 $2116.30 $2158.59Max $2282.93 $6521.44 $215590 $2200.13 $219887 $2188.49 $224629

TIME Min 0:00:01 0:05:35 0:00:23 N/A 0:00:06 0:00:05 N/A 200Mean 0:00:01 0:38:17 0:02:05 N/A 0:00:11 0:00:08 N/AMax 0:00:01 2:18:00 0:08:55 N/A 0:00:17 0:00:10 N/AMin 0:00:01 6:31:41 N/A 0:31:30 0:05:09 0:00:59 0:00:48 400Mean 0:00:02 12:41:15 N/A 0:56:31 0:09:39 0:07:41 0:01:26Max 0:00:03 10:25:10 N/A 2:42:09 0:18:09 0:27:51 0:03:57Min 0:00:16 8:00:00 8:00:00 1:12:55 0:11:40 0:01:00 0:00:51 600Mean 0:00:26 8:00:00 8:00:00 1:54:14 0:47:39 0:06:06 0:01:56Max 0:00:40 8:00:00 8:00:00 2:57:01 1:28:19 0:15:09 0:03:08

38 L. Kaan, E.V. Olinick / Computers & Industrial Engineering 66 (2013) 24–40

$2246.29 with an average cost of $2158.59. The objective functionvalues using the default mipgap ranged from $2060.25 to $2188.49with an average cost of $2116.30. This resulted in cost increasesranging from�1.06% to 3.37%, with an average of 1.18% for 5% mip-gap and �3.35% to 1.02%, with an average of �1.14% for defaultmipgap (a negative numbers indicates savings). As with RRAH,these results represent statistically significant reductions in termsof CPU time and statistically insignificant cost increases comparedto RAH (Kaan, 2011). The cost and the time metrics results for theMCVAM and the TSMCVAM in addition to all three heuristics canbe seen in Table 12. However, more detailed results for all problemsets can be seen in Kaan (2011).

5. Conclusion and directions for further study

We have developed two ILP models to solve the VanpoolAssignment Problem: the MCVAM and the TSMCVAM. The TSMC-VAM is an extension of the MCVAM, and both models minimize to-tal per-trip costs. The literature focuses on carpooling and relatedshared-vehicle transportation models, and to the best of ourknowledge, these are the first mathematical programming modelsproposed for the Vanpool Assignment Problem. As shown in Ta-ble 12, the TSMCVAM significantly improved vanpool savings perpassenger, even when solved with a 5% mipgap, and we considerthis model to be an important contribution of this work. TheMCVAM models the current practice in vanpooling of using onepark-and-ride location per vanpool. Based on our results, however,we recommend the TSMVCAM which can generate significant costsavings with little or no increase in trip times for most passengersby allowing vanpools to stop at a second park-and-ride location.

Solving the TSMCVAM with a straight-forward application ofthe state-of-the-art solver CPLEX proved to be challenging. There-fore, another contribution of this work is the development of heu-ristics to solve the TSMCVAM model for various size problem setsin significantly shorter times. These heuristics produced competi-tive results in terms of solution quality and solution time. On thelargest problems in our study (600 potential passengers and 120park-and-ride locations), the GCH was found to be the best of theseheuristics; it produced optimality gaps ranging from 5.38% to9.45% with an average optimality gap of 7.84% in CPU times rang-ing from 1 min to 15 min and 9 s with an average of 6 min and 6 s.

An efficient heuristic for the Interval Flow Model (Jones, 2009),which is a network flow model in which flow on a given arc must

either be zero or between positive lower and upper bounds, hasshown promise in solving problems with a similar structure tothe Vanpool Assignment Problem considered in this paper, andan implementation of this procedure for our models is the subjectof an on-going investigation. The models given here have one tar-get and multiple starting points; however, it is possible to use thesame van for multi-target vanpool programs. Our models could beextended to accommodate these types of vanpools. Another natu-ral extension of our models would be to allow multiple vans of thesame type to use the same one- or two-stop vanpooling routes.This can be achieved in our current framework by having multipleelements of the set L represent the same park-and-ride location.However, this approach has the drawback of increasing the num-ber of variables in the corresponding ILP. The TSMCVAM assumesthat the two-stop vanpool riders pay an equal share regardless ofwhether they join the vanpool at the first or second park-and-ridelocation. In practice, this might not be viewed as equitable, and soit may be advisable to adapt the model to improve fairness with re-gards to distributing the cost among riders in two-stop vanpools.

If there are only a few changes in the passenger set from1 month to the next, then it would be desirable to minimize thenumber of passengers whose vanpool assignments change. In ourexperience most commuters like to stick to a routine and we sus-pect that it would be difficult to sustain participation in the van-pool program if there is too much change from month to month.Thus, it could be quite valuable to extend our models to find re-optimized solutions that are as close as possible to the previousassignments as passengers come and in and out of the program.

If provably optimal TSMCVAM solutions are desired, it may bepossible to adapt exact algorithms that have been developed fora related class of problems known as hub location problems. Inhub location problems flows of commodities from multiple originsare aggregated at intermediate transshipment points called hubsand then routed to common destinations (possibly through otherhubs). Aggregation allows for flow on higher throughput routesand affords an economy scale in many transportation and telecom-munications applications (Contreras, Cordeau, & Laporte, 2011b).Hub location problems have been addressed in the literature withmethods based on Bender’s decomposition (Contreras, Cordeau, &Laporte, 2011a) and Lagrangian relaxation (Contreras et al.,2011b). However, there are differences between the vanpoolassignment and hub location problems that would have to be ad-dressed to apply these frameworks. For example, hub location typ-

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L. Kaan, E.V. Olinick / Computers & Industrial Engineering 66 (2013) 24–40 39

ically assumes that all flow must go through a hub (Contreras et al.,2011b), and this is not a requirement in the vanpool assignmentmodel (riders can opt-out of vanpooling and drive directly fromtheir home to the target location). Conversely hubs are generallynot required to have a minimum level of utilization whereas van-pools must have a minimum number of passengers depending onthe van type.

Finally, we note that in our computational studies the sets ofallowable assignments of passengers to vanpools based, or stop-ping at, particular park-and-ride locations (A and B) are based onsimplifying assumptions about average driving speeds (e.g., all pas-sengers can drive their own car with an average speed of 30 miles/h to their assigned park-and-ride location and 45 miles/h to thetarget location). In practice it would be desirable to obtain actualaverage driving speeds for each passenger to the target locationand to the candidate park-and-ride locations, as well as averagedriving times between the pairs of park-and-ride locations in S.Incorporating this data into our models would then be a straight-forward matter of adjusting the A and B sets accordingly. In caseswhere the driving times are highly variable around their averages,it could be useful to develop stochastic programming or chanceconstrained extensions of our vanpool assignment models that uti-lize more information about the distribution of the driving times toprovide more robust solutions.

Appendix Appendix:. Computational Complexity of MCVAM andTSMCVAM

In this section we establish the computational complexity of theVanpool Assignment Problems studied in this paper and show thatthey belong to class of inherently difficult NP-hard optimizationproblems (Garey & Johnson, 1979). We begin by outlining a poly-nomial reduction of the capacitated concentrator location problem(CCLP), which is known from the network design literature to beNP-hard (Pirkul (1987)), to the MCVAM. The goal of the CCLP isto determine a minimum cost design for a communications net-work connecting remote sites (end-user nodes) to a central sitevia devices called concentrators that allow end-user nodes to sharehigh capacity transmission lines (Narasimhan & Pirkul, 1992). Theinput to CCLP consists of a set I of end-user nodes, a set of potentialconcentrator locations, J, a fixed cost, f, for installing a concentratorat any location, a concentrator capacity, Q, which is the maximumnumber of end-user nodes that can be connected to any concentra-tor, and a connection cost, cij, for connecting end-user node i to aconcentrator installed at location j. Formulated as an ILP, the CCLPis stated as

min fXj2J

/j þXi2I

Xj2J

cijxij; ð25Þ

s:t:Xj2J

xij ¼ 1 8i 2 I; ð26Þ

Xi2I

xij 6 Q/j 8j 2 J; ð27Þ

xij 2 f0;1g 8i 2 I;8j 2 J; ð28Þ

/j 2 f0;1g 8j 2 J; ð29Þ

where /j = 1 if a concentrator is installed at location j, and xij = 1 ifend-user node i is connected to the concentrator installed atlocation j. Constraint set (25) ensures that each end-user node isconnected to exactly one concentrator, and the concentrator capac-ities are enforced by (26). Constraint sets (27) and (28) give the do-mains for the decision variables. The total cost of the network is

given by (24), which is the total fixed cost of installing the concen-trators plus the cost of connecting the end-user nodes to the se-lected concentrators.

For a given CCLP instance we may derive an equivalent MCVAMinstance with a single van size (i.e., |V| = 1) in which the passengerscorrespond to end-user nodes and park-and-ride locations corre-spond to concentrator locations. Note that since there is only onevan type in the derived MCVAM instance we may drop the sub-script k on parameters and variables having to do with van types.

To establish a one-to-one correspondence between passengersand end-user nodes, and between park-and-ride locations and po-tential concentrator locations we let P = I and L = J. Notice that thisalso establishes a one-to-one correspondence between the xpl andxij variables, and between the y‘ and /j variables. Next, we definethe parameters of the MCVAM instance as r = 1, dp‘ = cij, C = f, u = Q,m = 0, g = 0, s‘ = 0 for all ‘ in L, and hp = f + cmax + 1 for all p in P,where cmax is the largest cij value in the CCLP instance. Recall thatcp = rhp, which gives cp = f + cmax + 1 for every passenger p. Finally,we let the time allowances be sufficiently large so that any passen-ger may be assigned to any park-and-ride location. That is, we lettp = 2 cmax for every passenger p.

Given a feasible solution to the CCLP instance, let y‘ = /j for thecorresponding j, xp‘ = xij for the corresponding i–j pair, and wp = 0for every passenger p. Since the CCLP solution satisfies the concen-trator capacity constraint set (26), the corresponding MCVAM solu-tions satisfies the van-capacity constraint set (6) and the linkingconstraints (8). Furthermore, since the CCLP solution assigns eachend-user node to one of the selected concentrator locations, thecorresponding MCVAM solution satisfies (7) by assigning everypassenger to a vanpool. Observe that with the minimum numberof passengers per van set to m = 0, constraint set (5) is trivially sat-isfied by the MCVAM solution. Setting g = 0 implies that F‘ = 0 sothat with r = 1 and hp = f + cmax + 1, we get

Tp‘ ¼C

rðhp � dp‘Þ

� �¼ f

f þ cmax þ 1� cp‘

� �¼ 1; 8p 2 P;8‘ 2 L;

which means that (9) is also trivially satisfied. The objective func-tion of the derived MCVAM instance is

fX‘2L

y‘ þXp2P

X‘2L

cp‘xp‘ þ ðf þ cmax þ 1ÞXp2P

wp:

Thus, for every feasible solution to the given CCLP instancethere is a corresponding feasible solution to the derived MCVAMinstance with the same objective function value. By design, the costof any solution to the MCVAM in which at least one passenger optsout of vanpooling has a greater cost than every solution in whichall passengers join vanpools. Therefore, if wp is positive for any pas-senger p in the optimal MCVAM solution it means that the givenCCLP instance is infeasible. Therefore, solving the derived MCVAMinstance is equivalent to finding an optimal solution to the CCLP in-stance or determining that it is infeasible. Since CCLP is NP-hard, itfollows that MCVAM is also NP-hard.

Out empirical results indicate that the TSMCVAM is more diffi-cult to solve via integer programming than the MCVAM. From atheoretical perspective, we argue that TSMCVAM is NP-hard be-cause it contains MCVAM as a special case. Specifically, consideran instance of TSMCVAM in which the distances between thepark-and-ride locations (bij’s) are so large relative to the timeallowances that the set of allowable two-stop combinations (Sp)is empty for every passenger p. Such an instance only allows forone-stop vanpools and is effectively an instance of MCVAM.

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