IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND · PDF fileMathematical Programming (HNP-MP)...

IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 4, OCTOBER 2008 573

Hybrid Nested Partitions and MathematicalProgramming Approach and Its Applications

Liang Pi, Student Member, IEEE, Yunpeng Pan, Member, IEEE, and Leyuan Shi, Senior Member, IEEE

Abstract—Large-scale discrete optimization problems are diffi-cult to solve, especially when different kinds of real constraints areconsidered. Conventionally, standard mathematical programmingis a general approach for discrete optimization, but may sufferfrom the unacceptable long solution time in applications. On theother hand, some heuristics/metaheuristics methods are morepowerful in finding approximate solutions efficiently, but mostlyare problem and constraint dependent. In this paper, we developa new Hybrid Nested Partitions and Mathematical ProgrammingApproach, which creates compliance between Mathematical Pro-gramming and the heuristics/metaheuristics methods. Potentiallyapplicable to many different types of problems, the hybrid ap-proach can provide approximate solutions efficiently, and in themeantime can easily handle different kinds of constraints. Theapplications of the hybrid approach to the local pickup and de-livery problem (LPDP) and the discrete facility location problem(DFLP) are presented in this paper.

Note to Practitioners—The Hybrid Nested Partitions and Math-ematical Programming Approach provided in this paper is easy toimplement and potentially applicable to many types of problems.For a given type of problem, the hybrid approach is very suitablefor situations where many different kinds of real constraints areconsidered. Also, practitioners can design customized proceduresto evaluate partial solutions and incorporate domain knowledgeand local search into the nested partition framework to further im-prove the performance of the algorithm.

Index Terms—Discrete facility location problem, discrete opti-mization problem, mathematical programming, nested partitions,pickup and delivery problem.

I. INTRODUCTION

L ARGE-SCALE discrete optimization problems rise inmany applications, and are difficult to solve, especially

when different kinds of real constraints are considered whichcomplicates the problem structure.

Conventionally, to solve discrete optimization problems,there are two types of approaches. The first is the exact so-

Manuscript received March 12, 2007; revised September 5, 2007. Firstpublished March 3, 2008; current version published October 1, 2008. Thispaper was recommended for publication by Associate Editor F. Chen andEditor N. Viswanadham upon evaluation of the reviewers’ comments. Thiswork was supported in part by the Air Force Office of Scientific Research(AFOSR) under Contract FA9550-07-1-0390, in part by the National ScienceFoundation under Grant CMMI-0400294 and Grant CMMI-0646697, and inpart by Schneider National, Inc.

L. Pi is with the Department of Industrial and System Engineering, Universityof Wisconsin–Madison, Madison, WI 53706 USA (e-mail: [email protected]).

Y. Pan is with CombineNet, Inc., Pittsburgh, PA 15222 USA (e-mail:[email protected]).

L. Shi is with the Department of Industrial and System Engineering, Uni-versity of Wisconsin–Madison, Madison, WI 53706 USA, and also with theCenter for Intelligent and Networked Systems, Tsinghua University, Beijing,China (e-mail: [email protected]).

Digital Object Identifier 10.1109/TASE.2008.916761

lution approach, which tries to obtain the optimal solutionof the problem and also mostly provides feasible solution(s)before the optimal solution is found. The most commonly usedgeneral exact algorithms are mathematical programming-basedalgorithms, such as branch-and-bound, branch-and-cut, columngeneration, etc. [44]. Also, there are some other specializedexact solution methods [2], [11], [13], [19], [25], most of whichdeal with certain types of problems with relatively simpleproblem structure. The exact solution approach generallycannot handle large-scale problems in many applications due toits unacceptable long solution time. The second type of approx-imate approach includes local search [22], nested partitions[40], approximate dynamic programming [36], which tries togenerate approximate solutions efficiently. When applying toreal problems, to achieve high performance, most approximatealgorithms are highly problematic and constraint dependent[3], [6], [7], [16], [21], [23].

In this paper, we develop a new Hybrid Nested Partitions andMathematical Programming (HNP-MP) Approach, which cre-ates compliance between mathematical programming and theNested Partitions methods. Potentially applicable to many dif-ferent types of problems, the hybrid approach can provide ap-proximate solutions efficiently, and in the meantime can easilyhandle different kinds of constraints. The applications of the hy-brid approach to the local pickup and delivery problem (LPDP)and the discrete facility location problem (DFLP) are presentedin this paper.

The rest of this paper is organized as follows. In Section II, wepresented the new HNP-MP approach. In Section III, we applythe HNP-MP approach to the LPDP. In Section IV, the HNP-MPis applied to the DFLP, and we conclude in Section V.

II. HYBRID NESTED PARTITIONS AND MATHEMATICAL

PROGRAMMING (HNP-MP) APPROACH

We develop a new hybrid NP and MP approach for discreteoptimization problems. The NP method [40] is a partitioningand sampling-based strategy that focuses computation effort onthe most promising region of the solution space, while main-taining a global perspective on the problem. In each iteration ofthe algorithm, the entire solution space is viewed as the unionof a promising region and a surrounding region. The actual NPiteration comprises four steps, which we outline next.

1) Partitioning.This step partitions the current most promising regioninto several subregions and aggregates the remaining re-gions into the surrounding region. With an appropriatepartitioning scheme, most of the good solutions would beclustered together in a few subregions after the partitioning.

2) Random Sampling.Samples are taken from the subregions and the surrounding

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574 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 4, OCTOBER 2008

region according to some sampling procedure. The proce-dure should guarantee a positive probability for each so-lution in a given region to be selected. As we would liketo obtain high-quality samples, it is often beneficial to uti-lize problem structure in the sampling procedure (e.g., theweighted sampling method [45]).

3) Calculation of the Promise Index.For each region, we calculate the promise index to deter-mine the most promising region.

4) Backtracking.The new most promising region is either a child of the cur-rent most promising region or the surrounding region. Ifmore than one region is equally promising, ties are brokenarbitrarily. When the new most promising region is the sur-rounding region, backtracking is performed. The algorithmcan be devised to backtrack to either the root node or anyother node along the path leading to the current promisingregion.

Some previous successful applications of the NP method canbe found in [31], [32], [39], [41], and [42]. For efficiency con-siderations, in this paper, we propose a variant of the NP methodwhich is used in the hybrid approach. During each iteration ofthe NP algorithm, we first sample the promising region and sur-rounding region, and then calculate the promise index of bothregions. If the surrounding region is more preferable, we back-track. Otherwise, we partition the current promising region intosubregions based on the sampling results, and choose a subre-gion to be the next promising region. The rest of the regionsare aggregated to form the new surrounding region. For largesample space, this variant of NP is likely to be efficient, es-pecially when the sampling procedure can deliver high-qualitysamples.

A. Hybrid NP and MP Approach

In the standard NP method, complete solutions/samples aregenerated in the sampling step. However, when dealing with in-teger linear programming (ILP)/mixed integer linear program-ming (MILP) problems, we find it more advantageous to onlysample a number of partial solutions, where not all variables arefixed. Each partial solution represents a set of samples, and aproblem associated with the partial solution is solved to selectthe best sample. Furthermore, domain knowledge-based parti-tioning and sampling are used in most previous applicationsof NP. In contrast, we develop some general NP-oriented tech-niques, which can be either applied alone or combined with do-main knowledge. An outline of the HNP-MP approach is pro-vided next.

Hybrid NP and MP Approach for ILP/MILP Problems

S0) Set the initial promising region as the overall solutionspace. Set the initial surrounding region as . Go toS1.

S1) If stopping conditions hold, restart (go to S0) or stop;otherwise, go to S2.

S2) Obtain the LP solution for the current promisingregion. Do the LP solution-based biased samplingover the promising region and surrounding region togenerate partial solutions. Go to S3.

S3) Evaluate these partial solutions by solving theembedded problems, and obtain samples. Calculatethe promise index for both the promising region andthe surrounding region. If the promising region ismore promising, go to S4; otherwise, go to S5.

S4) Perform the partitioning and get a new nestedpromising region. Go to S1.

S5) Carry out backtracking. The resulting region is set tothe next promising region. Go to S1.

B. Sampling

The first step to apply the HNP-MP method to ILP/MILPproblems is to determine a proper form of partial solutions suchthat we can fully leverage the capability of ILP/MILP solvers orspecialized algorithms to efficiently solve the small scale sub-problems associated with the partial solutions.

Biased sampling can be used to obtain partial solutions thatcontain high-quality samples. A large number of optimizationproblems belong to the following category:

(1)

(2)

Here, ’s are binary variables, ’s can be either real variablesor integer variables. , , and are some given pa-rameters (or parameter matrix). Without loss of generality, wereasonably assume that no relation of or for cer-tain can be deduced from Constraints (1). Theexistence of Constraint (2) provides a potential promising formof partial solutions by fixing some -variable(s) to zero. We de-velop a procedure, called the LP solution-based sampling, forbiased sampling. It generates partial solutions based on the LPsolution of the subproblem associated with the target samplingregion, according to the follows steps.

1) Obtain the LP solution. Denote it by .2) Calculate the sampling weights of variable , based on the

value of . , the sampling weight forvariable is positively correlated to the value of .

3) Based on the sampling weights, a partial solution can begenerated: randomly select variables from all basedon standard weighted sampling [45] with the weights cal-culated in step 2, fix the other -variables to zero,and the remaining problem is the subproblem associatedwith the partial solution. [Constraints (1) are not consid-ered in this step.]

Some revisions can be made to the above procedure, de-pending on the structure of the problem. For many problems,the LP lower bound is tight, the LP solution over the solu-tion region would provide some useful information, and theabove sampling procedure can potentially be very effective.This sampling procedure can be embedded into the hybridapproach easily. In each iteration of the hybrid approach, the

PI et al.: HYBRID NESTED PARTITIONS AND MATHEMATICAL PROGRAMMING APPROACH AND ITS APPLICATIONS 575

new promising region after the partitioning step can be repre-sented by adding some constraints to the previous promisingregion, and the promising region after the backtracking stepcan be represented by dropping some constraints from previouspromising region. So, in each iteration, we solve the LP relaxedproblem on the promising region, and obtain the LP solutionto calculate the sampling weights. Based on these weights, theweighted sampling can be performed on the overall solutionspace, and a partial solution would possibly consist of samplesfrom both the promising region and the surrounding region.Also, we can choose to sample over the promising regionand surrounding region separately, and in this case, for thesurrounding region, we can generate partial solutions based onthe LP solution on the entire solution space.

C. Calculating the Promise Index

To calculate the promise index, we first need to evaluate thepartial solutions generated in the sampling step, and obtain agood sample within each partial solution. The top-ranking sam-ples obtained in the partial solution evaluation step will be usedto calculate the promise index and guide the partitioning/back-tracking step.

We consider using the general mathematical programmingapproach to solve the problem associated with partial solutions,by which we can deal with different kinds of constraints easily.Several techniques can be applied to improve the efficiency ofevaluating each partial solution.

• Apply the value of the current best sample as the feasiblebound. Only the best sample will be used to determine thenext promising region. Our limited computational experi-ence indicates that on average at least 60%–90% of partialsolutions will be dominated without going through the fullevaluation process, and that nondominated ones can oftenbe evaluated much faster.

• Set a computation time limit beforehand. Certain partialsolutions may be very difficult to evaluate; therefore, wemay be better off simply discarding them in order to savethe total computation time of the hybrid approach.

• Choose a tolerance for the optimality gap. Generallyspeaking, to obtain an approximate solution for theproblem associated with each partial solution can be a loteasier than to obtain an exact solution. Such an approxi-mate solution is often adequate for our purposes.

After the evaluation of the partial solutions, the promise indexcan be calculated in some standard manner. For example, forboth the promising region and the surrounding region, we candefine the promise index as the value of the best sample withinthat region [40].

Also, for some problems, evaluating a partial solution can beviewed as a branching process on the overall solution region,as shown in Fig. 1. Each time a partial solution is sampled andevaluated, the overall solution space is actually branched intotwo parts: the small part associated with this partial solution isevaluated and this branch will be cut off from the solution space;and the other part is not fully explored and constitute the laterpromising region and surrounding region. So, each time a partialsolution is evaluated, a cut which cuts off the partial solution is

Fig. 1. Calculating the promise index: branching and cutoff.

Fig. 2. Partitioning and backtracking.

added to the overall solution space to improve the efficiency ofthe algorithm.

D. Effective Partitioning

After comparing the promise index of the promising re-gion with the index of the surrounding region, if the indexof promising region is better than the index of surroundingregion, we further partition the current promising region andgenerate a new promising region (shown in Fig. 2). An effectivepartitioning scheme will keep good solutions clustered in thenext promising region. In this subsection, we provide somegeneral techniques to guide the effective partitioning step.

First, we keep the current best sample in the next promisingregion, which provide a set of available partitioning attributes.Certain constraints(s) can be constructed to satisfy each avail-able attribute. Each available attribute can potentially be usedto partition the current promising region into two subregions.Fig. 3 shows a situation where a single attribute is used for par-titioning: the subregion with the partitioning attribute satisfiedcontains the current best sample, and will become the promisingregion of next iteration; the other one will be aggregated into thesurrounding region. Also, we can use multiple attributes in thepartitioning, for example, Fig. 4 shows a situation where bothtwo attributes are used for partitioning: the subregion with bothpartitioning attributes satisfied contains the current best sample,and will become the promising region of next iteration; the otherone will be aggregated into the surrounding region.

Then, we can use LP solution-based partitioning to calculatethe partitioning index, which is used to select the partitioningattribute(s) from all available ones. Each available partitioningattribute can be denoted as certain decision variable or the com-bination of some decision variables. We can calculate the por-tion of each attribute that is satisfied by the LP solution on thecurrent promising region, and this value is defined as the parti-tioning index. In most cases, we suggest choosing the availableattribute with the best partitioning index value when a single


Fig. 3. Effective partitioning: partitioning with one attribute.

Fig. 4. Effective partitioning: partitioning with multiple attributes.

attribute is used in the partitioning, or choosing the availableattributes with top partitioning index values when multiple at-tributes are used in the partitioning.

E. Backtracking

In each iteration, we backtrack (shown in Fig. 2) when thepromise index of the surrounding region is better, i.e., the bestsample so far appears in the surrounding region. In most cases,the backtracking area can be determined with some flexibility,provided that it contains the original promising region and thecurrent best solution. For example, as shown in Table I, at thebeginning of the first iteration, the promising region is the entiresolution space. Then, partitioning is performed, and the subre-gion with attribute A satisfied is selected as the next promisingregion. In the second iteration, partitioning is performed again,and the subregion with both attribute A and attribute B satis-fied becomes the promising region for the third iteration. In thethird iteration, the best sample with attribute B satisfied andattribute A unsatisfied appears in the surrounding region, andbacktracking is in order. Then, we drop the constraint whichlet attribute A be satisfied, and obtain a backtracking area withattribute B satisfied as the promising region for the fourth iter-ation. This new promising region after backtracking is neitherthe root node region nor a region along the path leading to theprevious promising region.

F. Stopping Conditions

For some situations, we may stop partitioning the currentpromising region (in this case, we say that one partitioninground is completed), and do a restarting if time permits. Basi-cally, we can adopt two stopping conditions.

Stopping Condition 1: If the promising region becomessufficiently small, solve the problem on the promising region

TABLE IFLEXIBLE BACKTRACKING

using standard MILP solvers directly, and stop. Standard MILPsolvers can find a good or even optimal solution within a smallpromising region very quickly. It is usually not difficult toverify this stopping condition. When the attributes of the bestsample are all used for partitioning, this stopping condition isactivated.

Stopping Condition 2: If the gap between the global lowerbound (this can be the best solution so far) and LP upper boundof the current promising region becomes sufficiently small, wecan stop partitioning the current promising region.

In many applications, we generally have the maximum run-ning time requirement. The HNP-MP approach should completeat least one partitioning round before reaching the time limit toobtain satisfactory performance. According to the overall timelimit of the algorithm and the scale of the problems associatedwith the partial solutions, the number of partial solutions gener-ated in each iteration and the number of attributes used if back-tracking is not performed need to be managed properly.

III. HNP-MP FOR THE LOCAL PICKUP AND

DELIVERY PROBLEM (LPDP)

A. Problem Description

In recent years, competition in the transportation and logisticssector has increasingly intensified. The efforts of businesses tomaintain viable profit margins are further complicated by risingpersonnel and fuel costs. For instance, the local pickup and de-livery problem (LPDP), a variant of the vehicle routing problem(VRP), has drawn a great deal of interest lately. In this paper, weaim to provide a new general solution approach for solving thistype of problem.

The LPDP is concerned with the optimal movement of a setof loads in a local service area over a relatively short plan-ning horizon. The basic operations involved in LPDP can bedescribed as follows [43]: At the beginning of each work day, afixed number of vehicles are positioned throughout the servicearea. A vehicle can serve only one load at a time. After the de-livery of a load, it runs for another load immediately or becomesidle. Served loads generate revenues and unserved ones may besubcontracted to other carriers (for some nominal fee) or simplylost (without generating any revenue). Empty movements of ve-hicles incur costs. The optimization objective is to maximize theoverall profit over a set planning horizon, e.g., from the decisionepoch to the end of the day. To achieve this objective, a carriermust balance between serving as many loads as possible andminimizing empty movements.

In this section, we consider both load-specific constraints[14], [46] and driver-specific constraints [24], [29], [34]. Par-ticularly, as a demonstrating example, on the load side, we


consider the time-window constraints (or sometimes, pickuptime-window constraints), one of the most important attributesof loads that have been considered in various formulations[5], [6], [12], [20], [43]. As often considered in applications,service time-window constraints enforce that each load eitherwill be served within a given time window or will not be servedat all. Also, on the driver side, we consider: 1) Homing driverconstraints: as discussed in [34], the most important considera-tion in creating driver satisfaction in the planning process is toallow a driver to return home each day, should the driver preferso. Creating a personalized, predetermined work schedule forthe driver will clearly make the driver’s life easier. 2) Driverqualifications and preference constraints: for some specialloads, such as just-in-time loads, they can only be served byqualified drivers, and drivers may have preference over types ofloads, which should be accommodated whenever possible.

We provide a MILP formulation for LPDP. We use a contin-uous time index and continuous location space to realisticallycapture the properties of real problems. We first define the fol-lowing notation.Sets:

• : set of vehicles or drivers.• : set of nodes representing loads.• : set of start nodes. For

is the start node of vehicle .• : set of end nodes.

For is the end node of vehicle .Parameters:

• : time needed to serve node andthen travel to the origin of node . The service time for thestart and end nodes is assumed to be negligible.

• : earliest start time of each node. Formeans the initial available time of vehicle .

• : latest start time of each node. Formeans the latest time by which vehicle mustreach home. We assume,

, to make the problem feasible. Without loss of generality,this parameter can be revised to account for delivery time,and is denoted as the load pickup timewindow.

• : net revenue from served loads.• : cost of traveling from the

destination of node to the origin of node .• : driver qualifications/preference index.

, if vehicle is qualified andprefers to serve load , otherwise. Assume

.Now, we give a formulation based on the multicommodity

network flow [33]. Three kinds of nodes are included in the net-work: starting nodes of vehicles, nodes representing loads andend nodes of vehicles. Such a formulation generally provides atight LP lower bound, and in this problem, the schedule of eachvehicle is viewed as a different commodity in the network. Thedecision variables of this formulation are defined as:Variables:

• :0-1 variables. if vehicle serves load and thengo to serve load ; otherwise.

• : service start time of each node.

Then, the formulation can be stated as follows.Objective:

(3)

(4)

Subject to:

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

Here, (3) are the revenues of served loads, and (4) are thecosts of empty movements. Constraints (5)–(7) are standardmulticommodity network flow constraints. Constraints (8)require each load to be served no more than once. Constraints(9) are the driver qualifications and preference constraints.Constraints (10)–(12) are the temporal relations betweenconsecutive nodes. Constraints (13) and (14) are the pickuptime-window constraints.

In this formulation, the number of variables can be large.Hence, we apply some rules to delete some unnecessary ones.For example, from (10), we can deduce the relation that

if , load cannot be served immedi-ately after load ; hence, , set . According to ourtests, standard mathematical programming method can solvesmall-scale LPDPs, however, for medium-scale and large-scaleinstances, more efficient method is needed.

B. Applying HNP-MP to the LPDP

Detailed description of HNP-MP when specialized for theLPDP is presented in this section. Our solution approach canalso be applied to problems with other nonhomogeneous re-source constraints or load constraints.

1) Sampling: For the LPDP, a feasible solution provides aschedule for each vehicle which satisfies the constraints of theproblem, and where the load sequence each vehicle serves and


Fig. 5. LPDP: a partial solution.

the service start time of each load are fixed. We define partialsolutions as feasible solutions to the problem in the form ofassigning each load to a certain vehicle, without the fixation ofload sequence for each vehicle and the service start time for eachload. Here, if load is assigned to vehicle , we can representthe assignment by fixing some variables in the original problem:

fix . Givena partial solution and its associated subproblem, each load canbe either served by its assigned vehicle or dropped, and eachvehicle has a set of available loads.

Then, let the LP solution for the current promising region be, and , denote .

Then, the basic rule for calculating the sampling weight is:, the probability that load is assigned to a certain vehicle is

positively correlated to the value of .In our tests, a linear function of is used to calculate the

sampling weights. if , define ;otherwise, let , ( is a very small non-negativenumber, such as 0.01 or 0.001. Based on our experience, thislinear weight calculation procedure provides good partial so-lutions for the LPDP, although more complicated weight func-tions can be conceived. Then, after the normalization step, wecan obtain the sampling weights as: (assign load to vehicle

) .To explain this step, we take a simple example in Fig. 5.

There the partial solution could be: loads andare assigned to vehicle , and the other loads are assignedto vehicle . Then, in the LP-based sampling step, as shownin Fig. 6, to assign load to either vehicle or vehicle

, we first obtain the LP solution to determine and, and the sampling weights are calculated (

is used). Then, for each partial solution generated in the sam-pling procedure, load is assigned to vehicle with prob-ability , and as-signed to vehicle with probability

.2) Calculating the Promise Index: For the LPDP, to eval-

uate a partial solution, let each vehicle make as many profits aspossible among its available loads, and obtain a sample whereeach vehicle has a fixed schedule of serving loads. For manyLPDPs (including the LPDP in this paper), the problem associ-ated with each partial solution consists of several subproblemsseparable by vehicles, and the evaluation of the partial solutioncan be even more efficient if we solve these subproblems sep-arately. When applying the HNP-MP approach to LPDPs withdifferent concerns and constraints, different forms of subprob-

Fig. 6. LPDP: to assign load L1.

Fig. 7. Calculating the promise index: a sample.

lems associated with partial solutions need to be solved, and allother steps of the HNP-MP approach can be directly applied.

For example, in the evaluation of the partial solution in Fig. 5,there are two subproblems: one is to fix the schedule of ve-hicle with its assigned loads ; the other is to fixthe schedule of vehicle with its assigned loads .Fig. 7 shows the sample obtained after the partial solution eval-uation step: vehicle will serve loads and go home,vehicle will serve loads and go home, and loadis dropped without realizing any revenue.

3) Partitioning, Backtracking, and Stopping: For the LPDP,each available partitioning attribute fixes a load to be servedby a certain vehicle , which leads to Constraints

. Then, the value of can beused for the attribute . We can select one attribute (orseveral attributes) with the best (or top-ranking) LP solution-based partitioning index value for partitioning.

For example, in Fig. 5, suppose that we have the sample inFig. 7 to be the current best sample, then in Table II, column“Attributes of the best sample” has six available attributes forpartitioning corresponding to this sample. Column “Index-LP-solution” shows the LP solution-based partitioning index foreach available partitioning attribute. Attribute hasthe greatest Index-LP-solution value, and thus can be chosen asthe partitioning attribute.

Backtracking and stopping are performed in the standardmanner when needed. For the simple example, if in next iter-ation we get the best sample in surrounding region with load

not served by vehicle K1, backtracking will be performed:the constraints that fix to are relaxed from the originalpromising region to get the promising region for next iteration.

C. Computational Results

In this section, we report our computational experience withthe proposed algorithms on randomly generated instances.


TABLE IIINDEX FOR EFFECTIVE PARTITIONING

TABLE IIILPDP: PARAMETER SETTINGS

(Randomly generated instances capturing realistic propertiesare used in many recent research on solving LPDPs and PDPs[35], [43], [46], and our generation of testing data follows thestyle of these works.)

1) Testing Instances: The experiment settings are describedas follows.

Map and Locations: We generate 60 locations in a rectanglemap of square miles. For each location pair, the distancebetween the two locations is the Euclidean distance on the map.

Loads: Generate loads randomly on the origin–destinationlocation pairs. The handling time of each load is . The earlieststarting time for each load is generated randomly on the timehorizon from 7 am to 6 pm, and the length of the pickup time-window is set to Uniform(0,4) hour. The net revenue of servinga load is set to ,where is the rate of revenue per service time unit.

Vehicles: For each vehicle, the initial and homing loca-tions are randomly assigned among the locations. (These twolocations are not necessarily the same, since our model andalgorithm also intent to support some running horizon sys-tems.) Each vehicle’s working time is randomly set to fromUniform(7,9) am to Uniform(4,6) pm individually. The speedof each vehicle is 40 miles per hour. The cost rate of emptymovements of vehicles is 10 per hour.

Qualification/Preference: For each vehicle-load pair ,the probability that vehicle is qualified and prefers to serveload is set to .

Overall, we generated 42 testing instances, with six differentgroups of parameter settings (as shown in Table III) and sevendifferent groups of scale settings (as shown in Table IV). Allthese settings are of common properties and scales in real ap-plications. For example, as indicated in [43], the typical size oflocal sub fleet handled by a single load manager is around 20 aswe used in our testing.

2) Algorithm Settings: We first test all instances throughCPLEX 9 with default CPLEX parameter settings in our com-puter with Pentium 4 2.8 GHz CPU and 1 GB memory. For allinstances, we set a time limit of 30 min.

TABLE IVLPDP: SCALE SETTINGS

We implement our hybrid algorithm in AMPL, and limit thecomputation time to be within 30 min. (For the computationtime, we only count the time for the LP solution calculating andpartial solution evaluation, since with good implementation inhighly efficient programming language such as C++, all othercomputation time can be neglected.) The settings of the hybridalgorithm are described as follows.

• The dual simplex method is used to solve the LP problemon the promising region in each iteration.

• To make a fair comparison, for the evaluation of each par-tial solution, we do not use specialized algorithms, but justcall CPLEX to solve the problem associated with that par-tial solution: we apply the value of the current best solu-tion as the feasible bound, set the MILP tolerance gap tobe 0.01, and set the computation time limit to be 2 s. Ac-cording to our experiences, most partial solutions can beevaluated or cut off within 1 s. (In applications, with betterspecialized algorithms, the evaluation of partial solutionscan be even more efficient.)

• LP solution-based partitioning is used.• Depending on the scale of the problem, in each iteration,

take 20–100 partial solutions, and fix 2–10 loads to certainvehicles if backtracking is not needed.

• Stop the algorithm when the stopping conditions are met,and no restarting is used. For many of the instances, thecomputation time is much smaller than the 30 min timelimit.

• To show the generality of our approach, no domain knowl-edge is used in the tests. (In applications, we may combinespecial knowledge to further improve the performance ofour approach.)

For comparison purpose, we test two other approaches. Oneis local branching (LB) [15], which is a recently developed ad-vantageous mathematical programming-based local search ap-proach. For the LPDP, the initial solution is randomly gener-ated, and CPLEX is used to search improved solution withinthe neighborhood of the initial point. The improved solution isagain used to construct a neighborhood for further searching,until no improvement can be found. Multiple initial solutionsare used within the time limit in the tests, and the best solutionobtained is reported.

The other one is a myopic approach, which is one most pop-ular method used in applications [16]. This myopic approach isbriefly described as follows.

• In each iteration, assign at most one load to each vehicle,and maximize the profits (revenue - empty movement cost)for this stage. Then, let the vehicle serve its assigned load,and update the location and available time of the vehicle.Constraints that guarantee that each can get home on timeare added, and assigning a load to a vehicle is only allowed


TABLE VLPDP: RESULTS ON EASY INSTANCES

when the profit of the assignment is bigger than a prede-termined parameter . Repeat the above process until noprofits can be made.

• Then, , if load is assigned to vehicle ,in the original MILP problem, let load only be availableto vehicle by fixing some variables:

fix . Resolve the MILPproblem to obtain the final schedule.

• For each instance, run the myopic approach twice withset to 0 and 10, respectively, and the better result is selected.

The reason to compare the CPLEX, HNP-MP, LB, and themyopic approach is that each of the four methods can be easilyused as or developed into a general solver capable of solvingLPDPs with different concerns and constraints, including someill-defined problems with no existing specialized algorithm. [Wehave also tested Lagrangian relaxation (LR) [17], [26] approachon the problem. According to our observation, the LR conver-gence performance on the LPDP is not promising, and thus thedetail computational results are not reported in this paper.]

3) Testing Results: For 15 of these instances, they can besolved by CPLEX to obtain an optimal gap less than 10%, prob-ably because by nature these instances are easy. We also test ourhybrid approach on these easy instances. Detail computationalresults about these 15 instances are shown in Table V. (Ins isthe instance index, CPub is the infeasible bound from CPLEX,CPlb is bound from the best solution of CPLEX, CPgap is thegap between CPub and CPlb, NP-CPlb is the bound from thebest solution of the hybrid algorithm, and NP-CPgap is the gapbetween CPub and NP-CPlb.) The performance of our resultson these easy instances is also good, with an average optimalgap of 1.8%. For some of these instances, CPLEX results arebetter, which is reasonable, since standard MILP algorithms canmostly solve easy problems efficiently.

For all other 27 instances, the optimal gap of CPLEX re-sults is greater than 10%, which we consider as difficult in-stances. We tested our hybrid algorithm, LB, and the myopic ap-proach on these instances. The computational results are shownin Table VI. (LBlb is the bound from the best solution of the LBapproach, LBgap is the gap between CPub and LBlb, MYlb isthe bound from the best solution of the myopic approach, andMYgap is the gap between CPub and MYlb.) For all these dif-ficult instances, our hybrid approach outperforms CPLEX, LB,and the myopic approach by a significant margin. The solution

quality of the hybrid algorithm is very promising, with an av-erage optimality gap of 2.3%, ranging from 1.2%–4.0%, whileaverage optimal gap of CPLEX, LB, and the myopic results is19.3%, 22.5%, and 17.8% respectively.

For a given problem, it is possible for some highly specializedalgorithms to outperform our algorithm. However, our approachis general, and can be applied to many different problems. Also,for a given problem, our general approach can be combined withspecialized techniques to achieve greater efficiency.

4) Extension: The computational results of our hybrid ap-proach alone are already very promising, however, they maystill be improved if combined with some efficient local searchalgorithms. One way to do this is just to use the NP results(the best solutions when the algorithm ends) as initial solu-tions, and do the local search. We select those 17 instances withHNP-CP gap greater than 2.0%, and combine the LB proce-dure into our hybrid approach. The results are summarized inTable VII (Here, NP-CP-LBub is bound from the best solution ofcombining HNP-MP and LB, and NP-CP-LBgap is the gap be-tween NP-CP-LBub and CPlb.) While the average solution gapof the NP-CP is 2.8%, the average NP-CP-LB solution gap isreduced to 2.0%, and for 14 of these instances we get improvedsolutions. Overall, the HNP-MP approach is not only useful forthose LPDPs with no efficient specialized algorithm, but alsopotentially useful for some LPDPs where efficient specializedlocal search algorithms exist. (Also, another deeper level com-bination can be considered: use top samples in each iteration ofthe NP approach as the initial solutions and explore the localsearch procedure; then, the results of the local search proce-dure are viewed as new samples and used to determine the nextpromising region. To compensate the time spent on the localsearch procedure, in each iteration of the HNP-MP approach,we should probably take less partial solutions, and/or fix moreloads when partitioning is performed.)

Also, to show the generality of our approach, we test anotherLPDP where the hard service time-window constraints are re-placed by some load precedence constraints [14]. A load prece-dence constraint means certain loads can only be served sometime after the completion of some other loads. As indicated inSection II-C, with these job precedence constraints, we do nothave separable problems for each partial solution. The testinginstance generation is similar to that in Section III-C1. The loadprecedence constraints are randomly generated among job pairs,and the number of these constraints are set to 20 (A) and 40 (B),respectively. For parameter settings, group in Table III is used,and the scale settings are presented in Table IV. The computa-tional results are reported in Table VIII, where it is shown thatthe HNP-MP approach is superior to CPLEX.

IV. HNP-MP FOR DISCRETE FACILITY LOCATION PROBLEM

(DFLP)

A. Problem Description

Facility location problems (FLPs) rise in many applicationsand has been the focus of lots of research effort during recentdecades [1], [4], [18], [27], [28]. The basic elements of FLPsare described as below: there is a set of locations potentially us-able for some facilities, and each facility located at certain placewill lead to an operating cost; there is a set of tasks which will beserved or routed through opened facilities with certain routing


TABLE VILPDP: RESULTS ON DIFFICULT INSTANCES

TABLE VIILPDP: NP-CP-LB RESULTS

cost; the overall objective is to minimize the total costs of thesystem, which balance the operating costs of facilities and therouting costs of tasks. Naturally, the DFLPs can be viewed astwo-level decision problems: the first level is to decide the loca-tions of facilities, and the second level it to decide the routes oftasks. Discrete facility location problems (DFLPs) are the typeof FLPs with discrete location for facilities, and are commonlyused in the real world. Large-scale DFLPs are generally difficultto solve, especially when lots of real constraints are considered.

The intermodal hub location problem (IHLP) [8], [30], [37]is selected to demonstrate our solution approach. The IHLP isa real DFLP from intermodal movement industry, and has beenpaid much research effort due to its significant economic im-pact. The reason for choosing the IHLP for demonstration isthat IHLP is a DFLP with a set of typical real concerns such as

TABLE VIIILPDP: RESULTS ON INSTANCES WITH PRECEDENCE CONSTRAINTS

the concave transportation cost function. In this paper, we do notuse any specific technique dedicated to the IHLP, and most stepsin our solution approach are easily applicable to other DFLPs.Our computational tests are also based on the IHLP. The formu-lations of the IHLP we consider in this paper is presented in thesection. We first describe some notations as follows.Sets:

• : set of origin/destination terminal loca-tions.

• : set of intermodal hub locations.• : set of demanding flows, i.e., movement

demand from certain origin to certain destination.Parameters:

• : origin terminal of flow .• : destination terminal of flow .• : amount of flow .• : operating cost of hub , if is opened.


• : transportationcost function of the flow between location to location .Due to the scale economies which is an crucial considera-tion in transportation industry,

, we can assume to be a nondecreasing con-cave function of the amount of the flow from locationto location . Furthermore, in the formulation, we assumethat these functions are piecewise linear, and the values ofthese functions are given. (Refer to [30] and [37] on howto calculate the values of these cost functions.)

• : cost rate per unit amount if flow is notmoved or moved by other more expensive methods such aspure truck movement.

• : , if movement fromto are allowed; , otherwise.

Then, we define the decision variables of this problem asfollows.

• : the amount of flowmoved through intermodal rail line .

• : the amountof flow from location to location .

• : 0-1 facility location variables. if hubis opened; , otherwise.

• : the amount of flow that is not movedthrough the intermodal operations.

Then, the formulation of the problem is described as follows:Objective:

(15)

(16)

(17)

Subject to:(18)

(19)

(20)

(21)

(22)

(23)

(24)

Here, in the objective function, (15) is the cost of hub opera-tions, (16) is the cost of flows moved by the intermodal opera-tions, and (17) is the cost of flows not moved by the intermodaloperations. Constraint (18) is the requirement that all the flowsshould be covered. Constraints (19) and (20) require that eachflow can only be routed via opened hub in a intermodal move-ment. Constraints (21)–(23) are the relationships between vari-ables ’s and variables ’s. Constraint (24) is the restrictions of

Fig. 8. DFLP: a partial solution.

the movements between terminals and hubs. In the formulationabove, term (16) includes concave piecewise linear functions,leading to one major difficulty of solving the problem. There isstandard procedures to linearized term (16), [10], however, theproblem size will be increased due to the new variables and con-straints introduced.

As mentioned before, our solution approach is a general ap-proach, and can be potentially applied to many other DFLPs.Particularly, for the IHLP, some of the other typical concerns andconstraints from applications include hub capacity constraints,establishing certain number of hubs [8], [9].

B. Applying HNP-MP to DFLP

The HNP-MP approach can be applied to many DFLPs, in-cluding the IHLP. For demonstration purpose, we use the ex-ample in Fig. 8 with seven hubs , and four flows ;

1) Sampling: First, for the IHLP, we define partial solutionsas feasible solutions to the problem in the form of letting a set ofhubs be closed, and no flow can go through these closed hubs.In a given problem, to achieve high performance, the numberof hubs closed need to be controlled properly. The form of par-tial solutions described above is used in our tests. Alternatively,for the IHLP, we can also choose to take partial solutions in theform of fixing all the variables, and similarly the number ofopened hubs and closed hubs in a partial solution need to becontrolled properly. For other DFLPs, partial solutions can begenerated in the form of fixing some or all of the facility loca-tion variables, by which each partial solution corresponds to aneasier subproblem. For the example in Fig. 8, in a partial solu-tion, hub , and are closed.

Second, for the IHLP, denote the part of the LP solutionfor the current promising region as . Then, the basic rule forcalculating the sampling weights is: , the probabilitythat hub is not closed is positively correlated to the value of

. In our computational tests, a linear function of is used tocalculate the sampling weights. , define( is a very small non-negative number, such as 0.02 or 0.01).Then, after the normalization step, we can obtain the samplingweights as


Fig. 9. DFLP: a sample.

Based on these weights, we can sample a set of hubs to bepotentially open, and let all other hubs to be closed. Again, ac-cording to our experience, this linear weight calculation pro-cedure provides good partial solutions for the IHLP. For otherDFLPs, the LP solution of the facility location variables can beused in the LP-based sampling procedure.

2) Calculating the Promise Index: For the DFLP (and manyother DFLPs), if only partial of variables (the facility locationvariables) are fixed (to 0) in a partial solution, each partial so-lution corresponds to a relatively small problem with the samestructure of the original problem; if the partial solutions are inthe form of fixing all the variables (the facility location vari-ables), the subproblem associated with each partial solution isa pure routing problem. For both cases, a standard integer pro-gramming algorithm can be used to evaluate partial solutionsefficiently. Fig. 9 shows a sample which is contained by thepartial solution in Fig. 8. After the evaluation of the partial so-lutions, the promise index can be calculated through some stan-dard ways.

3) Partitioning, Backtracking, and Stopping: For the IHLPand many other DFLPs, each available partitioning attribute leta certain hub/facility open, which leads to Constraint(for the IHLP). Then, for the IHLP, the value of is used asthe LP solution-based partitioning index for the attribute thathub is open. We can select one attribute (or several attributes)with the best (or top) LP solution-based partitioning index valuefor partitioning. For other DFLPs, the LP solution of the facilitylocation variables can be used to calculate the LP solution-basedpartitioning index.

Backtracking and stopping are performed in the standardmanner when needed.

C. Computational Results

In this section, we report our computational experience onapplying HNP-MP to the IHLP.

1) Testing Instances: We randomly generated a set of in-stances with typical settings to test our solution approach. Theexperiment settings are described as follows.

Map and Locations: We generate a rectangle map of500 500 square miles. terminal locations and ramp lo-cations are randomly generated over the map. For each location

TABLE IXDFLP: SCALE SETTINGS

TABLE XDFLP: TRANSPORTATION COST FUNCTION SETTINGS

pair, the distance between the two locations is the Euclideandistance on the map.

Flows: flows are randomly generated on the origin–desti-nation location pairs with distance larger than 300 miles.

is randomly generated by Uniform(10,50).Cost Functions: The cost of open a certain hub

is set to . The cost rate per unit amountof unmoved flows is . Thetransportation cost function (cost rate per mile) between a loca-tion pair is set to a four piece linear concave function: for eachlocation pair with truck movement, the cost rate (the slop foreach piece of the cost function) is randomly generated over therange (S1, S2), and the three nondifferentiable points are set to

and ; for each locationpair with truck movement, the cost rate is randomly generatedover the range (S3, S4), and the three nondifferentiable pointsare set to and .

Routing: When the distance between location and locationis less than a predetermined parameter ; oth-

erwise, . This constraint on the possible movement,adopted by many big transportation companies nowadays, actu-ally reduces the problem size and thus the solution time of theproblem. Here, is set to 200 miles.

Overall, we generated 21 testing instances of common prop-erties and scales in real applications, with seven different groupsof scale settings (as shown in Table X) and three different groupsof parameter settings (as shown in Table IX).

2) Algorithm Settings: We first test all instances throughCPLEX 9.1 with default CPLEX parameter settings in our com-puter with P4 2.8G CPU and 1G memory. The reason we usedCPLEX is that integer programming is one widely used ap-proach to solve DFLPs in the literature. For all instances, weset a time limit of 2 h. Then, we test the HNP-MP approach onthese instances. Some algorithm setups are described as follows.

• Dual simplex method is used to solve the LP problem onthe promising region in each iteration.

• CPLEX is used to evaluate each partial solution: we applythe value of the current best solution as the feasible bound,set the MILP tolerance gap to be 0.005, and set the com-putation time limit to be 2 min.


TABLE XIDFLP: CPLEX RESULTS AND LR RESULTS VERSUS HNP-MP RESULTS

• Depending on the scale of the problem, in each iteration,take 8–20 partial solutions, and fix 2–3 hubs open if back-tracking is not needed.

• Stop the algorithm when the stopping conditions are met,and no restart is used.

• To show the generality of our approach, no domain knowl-edge is used in the tests. (In applications, we may combinespecial knowledge to further improve the performance ofour approach.)

For comparison purpose, we also test the LR approach [17]which is widely used to solve DFLPs in the literature. The set-tings of the LR approach are briefly described as follows.

• Constraints (21)–(23) are relaxed in the LR subproblem,and multipliers corresponding to these constraints areadded to the objective function.

• Subgradient algorithm is used to update the multipliers ineach iteration of the LR procedure.

• In each iteration, after obtaining the integer solution of theLR subproblem, fix the ’s variables, and solve the orig-inal problem to get the feasible integer solution. The bestfeasible solution is reported when the algorithm ends.

3) Testing Results: Detail computation results about these 21instances are shown in Table XI. (Here, CPlb is the infeasiblebound from CPLEX and CPub is bound from the best solutionof CPLEX, LRub is the solution of the LR approach, LRgapis the gap between LRub and CPlb, NP-CPub is the solution ofthe HNP-MP approach, NP-CPgap is the gap between NP-CPuband CPlb.) The HNP-MP approach is superior to CPLEX andLR for each instances, mostly with a significant improvementratio.

V. CONCLUSION

The Nested Partitions method is a general framework thatcan be combined with many local searches, metaheuristic algo-rithms, and domain knowledge. Previously, the NP method hasbeen combined with genetic algorithms (GAs) [41], local search

[42], and domain knowledge [31]. Numerical results show thatthe hybrid algorithm such as NP/GA or NP/TS performs muchmore efficiently than the GA or local search algorithm alone.

In this paper, we exploit well-known exact algorithms suchas MIP or MP and NP metaheuristic framework so that eachcomplements the strengths of the other. The efficiency and nov-elty of our approach are demonstrated through two important,but difficult problems, i.e., LPDP and DFLP problems. We havealso showed that our HNP-MP approach has the advantage ofbeing easily adjusted to different kinds of constraints.

In the standard HNP-MP approach, for evaluating each partialsolution, we use the MILP solvers. For a specific problem, itis possible to apply or develop some efficient heuristic methodto replace the MILP solvers for partial solution evaluation andimprove the efficiency of the hybrid algorithm.

We can also combine some cuts [44] to make a stronger LPformulation, which is very useful for some discrete optimiza-tion problems with loose LP relaxed problems. The LP solutioncorresponding to the stronger LP formulation is used in the sam-pling/partitioning step. However, by adding these cuts, we mayincrease the computation time for obtaining the LP solution, andwe may also lose come useful information, especially when cutsare applied to the simplified LP problems. More computationaltests need to be done to see the performance of combining cut-ting planes in the hybrid approach in the future. On the otherhand, for many discrete optimization problems, such as LPDPsand DFLPs, it is relatively easy to evaluate partial solutions, andthe bottleneck of the hybrid approach is the LP solution cal-culating step. For some larger scale instances, even obtaininga LP solution on the root node is not easy. We plan to furtherinvestigate using simplified LP solutions [38] in our algorithmto reduce the computation time of the LP solution calculatingprocess.

ACKNOWLEDGMENT

The authors thank the Associate Editor and the three anony-mous referees for helpful comments that improved this paper.


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Liang Pi (S’07) received the M.E. degree inindustrial engineering from the University ofWisconsin–Madison, Madison, in 2005, and theB.E. degree in computer science from the SpecialClass of the Gifted Young, University of Science andTechnology of China, Hefei, in 2004. He is currentlyworking towards the Ph.D. degree at the Departmentof Industrial and Systems Engineering, University ofWisconsin-Madison.

His research interest include large-scale opti-mization techniques, machine learning, logistics and

supply chain management, financial engineering, etc.Mr. Pi is a student member of the Institute for Operations Research and the

Management Sciences (INFORMS), the Society for Industrial and AppliedMathematics (SIAM), and the CFA Institute.


Yunpeng Pan (M’06) received the B.S. degree incomputational mathematics from Nanjing Univer-sity, Nanjing, China, in 1995, the M.S. degree inoperations research from the University of Delaware,Newark, in 1998, and the M.S. degree in computersciences and the Ph.D. degree in industrial engi-neering from the University of Wisconsin-Madison,Madison, in 2001 and 2003, respectively.

He is an Algorithms and Formulations Archi-tect with CombinetNet, Inc., Pittsburgh, PA. Hisresearch interests are concerned with developing

industrial strength techniques and methods for solving difficult mixed-integerprogramming problems that arise in E-Commerce, Combinatorial Auctions,Procurement (Reverse) Auctions, and Healthcare Informatics. His work ap-pears in Mathematical Programming, Operations Research Letters, the IEEETRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, the EuropeanJournal of Operational Research, and the Journal of Systems Science andSystems Engineering.

Dr. Pan is a member of the Institute for Operations Research and the Man-agement Sciences (INFORMS), and the Mathematical Programming Society.

Leyuan Shi (SM’06) received the B.S. degreein mathematics from Nanjing Normal University,Nanjing, China, in 1982, the M.S. degree in appliedmathematics from Tsinghua University, Beijing,China, in 1985, and the M.S. degree in engineeringand the Ph.D. degree in applied mathematics fromHarvard University, Cambridge, MA, in 1990 and1992, respectively.

She is a Professor with the Department of In-dustrial and Systems Engineering, University ofWisconsin-Madison. She has been involved in

undergraduate and graduate teaching, as well as research and professionalservice. Her research is devoted to the theory and applications of large-scaleoptimization algorithms, discrete-event simulation and modeling, and analysisof discrete dynamic systems. She has published many papers in these areas. Herwork has appeared in Discrete Event Dynamic Systems, Operations Research,Management Science, the IEEE TRANSACTIONS, and the IIE TRANSACTIONS.

Dr. Shi is a member of the Institute for Operations Research and the Manage-ment Sciences (INFORMS). She is currently an Associate Editor of the IEEETRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, INFORMSJournal on Computing, and the Journal of Discrete Event Dynamic Systems.

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