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

IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 4, OCTOBER 2008 573 Hybrid Nested Partitions and Mathematical Programming 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 difficult to solve, especially when different kinds of real constraints are considered. Conventionally, standard mathematical programming is a general approach for discrete optimization, but may suffer from the unacceptable long solution time in applications. On the other hand, some heuristics/metaheuristics methods are more powerful in finding approximate solutions efficiently, but mostly are problem and constraint dependent. In this paper, we develop a new Hybrid Nested Partitions and Mathematical Programming Approach, which creates compliance between Mathematical Programming and the heuristics/metaheuristics methods. Potentially applicable to many different types of problems, the hybrid approach can provide approximate solutions efficiently, and in the meantime can easily handle different kinds of constraints. The applications of the hybrid approach to the local pickup and delivery problem (LPDP) and the discrete facility location problem (DFLP) are presented in this paper. Note to Practitioners The Hybrid Nested Partitions and Mathematical Programming Approach provided in this paper is easy to implement and potentially applicable to many types of problems. For a given type of problem, the hybrid approach is very suitable for situations where many different kinds of real constraints are considered. Also, practitioners can design customized procedures to evaluate partial solutions and incorporate domain knowledge and local search into the nested partition framework to further improve the performance of the algorithm. Index Terms Discrete facility location problem, discrete optimization problem, mathematical programming, nested partitions, pickup and delivery problem. I. INTRODUCTION L ARGE-SCALE discrete optimization problems rise in many applications, and are difficult to solve, especially when different kinds of real constraints are considered which complicates 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. First published March 3, 2008; current version published October 1, 2008. This paper was recommended for publication by Associate Editor F. Chen and Editor N. Viswanadham upon evaluation of the reviewers comments. This work was supported in part by the Air Force Office of Scientific Research (AFOSR) under Contract FA9550-07-1-0390, in part by the National Science Foundation under Grant CMMI-0400294 and Grant CMMI-0646697, and in part by Schneider National, Inc. L. Pi is with the Department of Industrial and System Engineering, University of Wisconsin Madison, Madison, WI 53706 USA (e-mail: lpi@wisc.edu). Y. Pan is with CombineNet, Inc., Pittsburgh, PA 15222 USA (e-mail: ypan@combinenet.com). L. Shi is with the Department of Industrial and System Engineering, University of Wisconsin Madison, Madison, WI 53706 USA, and also with the Center for Intelligent and Networked Systems, Tsinghua University, Beijing, China (e-mail: leyuan@engr.wisc.edu). Digital Object Identifier 10.1109/TASE.2008.916761 lution approach, which tries to obtain the optimal solution of the problem and also mostly provides feasible solution(s) before the optimal solution is found. The most commonly used general exact algorithms are mathematical programming-based algorithms, such as branch-and-bound, branch-and-cut, column generation, etc. [44]. Also, there are some other specialized exact solution methods [2], [11], [13], [19], [25], most of which deal with certain types of problems with relatively simple problem structure. The exact solution approach generally cannot handle large-scale problems in many applications due to its unacceptable long solution time. The second type of approximate approach includes local search [22], nested partitions [40], approximate dynamic programming [36], which tries to generate approximate solutions efficiently. When applying to real problems, to achieve high performance, most approximate algorithms are highly problematic and constraint dependent [3], [6], [7], [16], [21], [23]. In this paper, we develop a new Hybrid Nested Partitions and Mathematical Programming (HNP-MP) Approach, which creates compliance between mathematical programming and the Nested Partitions methods. Potentially applicable to many different types of problems, the hybrid approach can provide approximate solutions efficiently, and in the meantime can easily handle different kinds of constraints. The applications of the hybrid approach to the local pickup and delivery problem (LPDP) and the discrete facility location problem (DFLP) are presented in this paper. The rest of this paper is organized as follows. In Section II, we presented the new HNP-MP approach. In Section III, we apply the HNP-MP approach to the LPDP. In Section IV, the HNP-MP is 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 discrete optimization problems. The NP method [40] is a partitioning and sampling-based strategy that focuses computation effort on the most promising region of the solution space, while maintaining a global perspective on the problem. In each iteration of the algorithm, the entire solution space is viewed as the union of a promising region and a surrounding region. The actual NP iteration comprises four steps, which we outline next. 1) Partitioning. This step partitions the current most promising region into several subregions and aggregates the remaining regions into the surrounding region. With an appropriate partitioning scheme, most of the good solutions would be clustered together in a few subregions after the partitioning. 2) Random Sampling. Samples are taken from the subregions and the surrounding 1545-5955/$25.00 2008 IEEE

574 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 4, OCTOBER 2008 region according to some sampling procedure. The procedure should guarantee a positive probability for each solution in a given region to be selected. As we would like to obtain high-quality samples, it is often beneficial to utilize problem structure in the sampling procedure (e.g., the weighted sampling method [45]). 3) Calculation of the Promise Index. For each region, we calculate the promise index to determine the most promising region. 4) Backtracking. The new most promising region is either a child of the current most promising region or the surrounding region. If more than one region is equally promising, ties are broken arbitrarily. When the new most promising region is the surrounding region, backtracking is performed. The algorithm can be devised to backtrack to either the root node or any other node along the path leading to the current promising region. Some previous successful applications of the NP method can be found in [31], [32], [39], [41], and [42]. For efficiency considerations, in this paper, we propose a variant of the NP method which is used in the hybrid approach. During each iteration of the NP algorithm, we first sample the promising region and surrounding region, and then calculate the promise index of both regions. If the surrounding region is more preferable, we backtrack. Otherwise, we partition the current promising region into subregions based on the sampling results, and choose a subregion to be the next promising region. The rest of the regions are aggregated to form the new surrounding region. For large sample space, this variant of NP is likely to be efficient, especially when the sampling procedure can deliver high-quality samples. A. Hybrid NP and MP Approach In the standard NP method, complete solutions/samples are generated in the sampling step. However, when dealing with integer linear programming (ILP)/mixed integer linear programming (MILP) problems, we find it more advantageous to only sample a number of partial solutions, where not all variables are fixed. Each partial solution represents a set of samples, and a problem associated with the partial solution is solved to select the best sample. Furthermore, domain knowledge-based partitioning and sampling are used in most previous applications of NP. In contrast, we develop some general NP-oriented techniques, which can be either applied alone or combined with domain knowledge. An outline of the HNP-MP approach is provided next. Hybrid NP and MP Approach for ILP/MILP Problems S0) Set the initial promising region as the overall solution space. Set the initial surrounding region as. Goto S1. S1) If stopping conditions hold, restart (go to S0) or stop; otherwise, go to S2. S2) Obtain the LP solution for the current promising region. Do the LP solution-based biased sampling over the promising region and surrounding region to generate partial solutions. Go to S3. S3) Evaluate these partial solutions by solving the embedded problems, and obtain samples. Calculate the promise index for both the promising region and the surrounding region. If the promising region is more promising, go to S4; otherwise, go to S5. S4) Perform the partitioning and get a new nested promising region. Go to S1. S5) Carry out backtracking. The resulting region is set to the next promising region. Go to S1. B. Sampling The first step to apply the HNP-MP method to ILP/MILP problems is to determine a proper form of partial solutions such that we can fully leverage the capability of ILP/MILP solvers or specialized algorithms to efficiently solve the small scale subproblems associated with the partial solutions. Biased sampling can be used to obtain partial solutions that contain high-quality samples. A large number of optimization problems belong to the following category: Here, s are binary variables, s can be either real variables or integer variables.,, and are some given parameters (or parameter matrix). Without loss of generality, we reasonably assume that no relation of or for certain can be deduced from Constraints (1). The existence of Constraint (2) provides a potential promising form of partial solutions by fixing some -variable(s) to zero. We develop a procedure, called the LP solution-based sampling, for biased sampling. It generates partial solutions based on the LP solution of the subproblem associated with the target sampling region, 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 for variable is positively correlated to the value of. 3) Based on the sampling weights, a partial solution can be generated: randomly select variables from all based on standard weighted sampling [45] with the weights calculated in step 2, fix the other -variables to zero, and the remaining problem is the subproblem associated with the partial solution. [Constraints (1) are not considered in this step.] Some revisions can be made to the above procedure, depending on the structure of the problem. For many problems, the LP lower bound is tight, the LP solution over the solution region would provide some useful information, and the above sampling procedure can potentially be very effective. This sampling procedure can be embedded into the hybrid approach easily. In each iteration of the hybrid approach, the (1) (2)

PI et al.: HYBRID NESTED PARTITIONS AND MATHEMATICAL PROGRAMMING APPROACH AND ITS APPLICATIONS 575 new promising region after the partitioning step can be represented by adding some constraints to the previous promising region, and the promising region after the backtracking step can be represented by dropping some constraints from previous promising region. So, in each iteration, we solve the LP relaxed problem on the promising region, and obtain the LP solution to calculate the sampling weights. Based on these weights, the weighted sampling can be performed on the overall solution space, and a partial solution would possibly consist of samples from both the promising region and the surrounding region. Also, we can choose to sample over the promising region and surrounding region separately, and in this case, for the surrounding region, we can generate partial solutions based on the LP solution on the entire solution space. Fig. 1. Calculating the promise index: branching and cutoff. C. Calculating the Promise Index To calculate the promise index, we first need to evaluate the partial solutions generated in the sampling step, and obtain a good sample within each partial solution. The top-ranking samples obtained in the partial solution evaluation step will be used to calculate the promise index and guide the partitioning/backtracking step. We consider using the general mathematical programming approach 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 of evaluating each partial solution. Apply the value of the current best sample as the feasible bound. Only the best sample will be used to determine the next promising region. Our limited computational experience indicates that on average at least 60% 90% of partial solutions will be dominated without going through the full evaluation process, and that nondominated ones can often be evaluated much faster. Set a computation time limit beforehand. Certain partial solutions may be very difficult to evaluate; therefore, we may be better off simply discarding them in order to save the total computation time of the hybrid approach. Choose a tolerance for the optimality gap. Generally speaking, to obtain an approximate solution for the problem associated with each partial solution can be a lot easier than to obtain an exact solution. Such an approximate solution is often adequate for our purposes. After the evaluation of the partial solutions, the promise index can be calculated in some standard manner. For example, for both the promising region and the surrounding region, we can define the promise index as the value of the best sample within that region [40]. Also, for some problems, evaluating a partial solution can be viewed as a branching process on the overall solution region, as shown in Fig. 1. Each time a partial solution is sampled and evaluated, the overall solution space is actually branched into two parts: the small part associated with this partial solution is evaluated and this branch will be cut off from the solution space; and the other part is not fully explored and constitute the later promising region and surrounding region. So, each time a partial solution is evaluated, a cut which cuts off the partial solution is Fig. 2. Partitioning and backtracking. added to the overall solution space to improve the efficiency of the algorithm. D. Effective Partitioning After comparing the promise index of the promising region with the index of the surrounding region, if the index of promising region is better than the index of surrounding region, we further partition the current promising region and generate a new promising region (shown in Fig. 2). An effective partitioning scheme will keep good solutions clustered in the next promising region. In this subsection, we provide some general techniques to guide the effective partitioning step. First, we keep the current best sample in the next promising region, which provide a set of available partitioning attributes. Certain constraints(s) can be constructed to satisfy each available attribute. Each available attribute can potentially be used to partition the current promising region into two subregions. Fig. 3 shows a situation where a single attribute is used for partitioning: the subregion with the partitioning attribute satisfied contains the current best sample, and will become the promising region of next iteration; the other one will be aggregated into the surrounding region. Also, we can use multiple attributes in the partitioning, for example, Fig. 4 shows a situation where both two attributes are used for partitioning: the subregion with both partitioning attributes satisfied contains the current best sample, and will become the promising region of next iteration; the other one will be aggregated into the surrounding region. Then, we can use LP solution-based partitioning to calculate the partitioning index, which is used to select the partitioning attribute(s) from all available ones. Each available partitioning attribute can be denoted as certain decision variable or the combination of some decision variables. We can calculate the portion of each attribute that is satisfied by the LP solution on the current promising region, and this value is defined as the partitioning index. In most cases, we suggest choosing the available attribute with the best partitioning index value when a single

576 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 4, OCTOBER 2008 TABLE I FLEXIBLE BACKTRACKING 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 available attributes with top partitioning index values when multiple attributes are used in the partitioning. E. Backtracking In each iteration, we backtrack (shown in Fig. 2) when the promise index of the surrounding region is better, i.e., the best sample 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 the current best solution. For example, as shown in Table I, at the beginning of the first iteration, the promising region is the entire solution space. Then, partitioning is performed, and the subregion with attribute A satisfied is selected as the next promising region. In the second iteration, partitioning is performed again, and the subregion with both attribute A and attribute B satisfied becomes the promising region for the third iteration. In the third iteration, the best sample with attribute B satisfied and attribute A unsatisfied appears in the surrounding region, and backtracking is in order. Then, we drop the constraint which let attribute A be satisfied, and obtain a backtracking area with attribute B satisfied as the promising region for the fourth iteration. This new promising region after backtracking is neither the root node region nor a region along the path leading to the previous promising region. F. Stopping Conditions For some situations, we may stop partitioning the current promising region (in this case, we say that one partitioning round is completed), and do a restarting if time permits. Basically, we can adopt two stopping conditions. Stopping Condition 1: If the promising region becomes sufficiently small, solve the problem on the promising region using standard MILP solvers directly, and stop. Standard MILP solvers can find a good or even optimal solution within a small promising region very quickly. It is usually not difficult to verify this stopping condition. When the attributes of the best sample are all used for partitioning, this stopping condition is activated. Stopping Condition 2: If the gap between the global lower bound (this can be the best solution so far) and LP upper bound of the current promising region becomes sufficiently small, we can stop partitioning the current promising region. In many applications, we generally have the maximum running time requirement. The HNP-MP approach should complete at least one partitioning round before reaching the time limit to obtain satisfactory performance. According to the overall time limit of the algorithm and the scale of the problems associated with the partial solutions, the number of partial solutions generated in each iteration and the number of attributes used if backtracking 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 logistics sector has increasingly intensified. The efforts of businesses to maintain viable profit margins are further complicated by rising personnel and fuel costs. For instance, the local pickup and delivery problem (LPDP), a variant of the vehicle routing problem (VRP), has drawn a great deal of interest lately. In this paper, we aim to provide a new general solution approach for solving this type of problem. The LPDP is concerned with the optimal movement of a set of loads in a local service area over a relatively short planning horizon. The basic operations involved in LPDP can be described as follows [43]: At the beginning of each work day, a fixed number of vehicles are positioned throughout the service area. A vehicle can serve only one load at a time. After the delivery of a load, it runs for another load immediately or becomes idle. Served loads generate revenues and unserved ones may be subcontracted to other carriers (for some nominal fee) or simply lost (without generating any revenue). Empty movements of vehicles incur costs. The optimization objective is to maximize the overall profit over a set planning horizon, e.g., from the decision epoch to the end of the day. To achieve this objective, a carrier must balance between serving as many loads as possible and minimizing empty movements. In this section, we consider both load-specific constraints [14], [46] and driver-specific constraints [24], [29], [34]. Particularly, as a demonstrating example, on the load side, we

PI et al.: HYBRID NESTED PARTITIONS AND MATHEMATICAL PROGRAMMING APPROACH AND ITS APPLICATIONS 577 consider the time-window constraints (or sometimes, pickup time-window constraints), one of the most important attributes of 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 either will be served within a given time window or will not be served at all. Also, on the driver side, we consider: 1) Homing driver constraints: as discussed in [34], the most important consideration in creating driver satisfaction in the planning process is to allow a driver to return home each day, should the driver prefer so. Creating a personalized, predetermined work schedule for the driver will clearly make the driver s life easier. 2) Driver qualifications and preference constraints: for some special loads, such as just-in-time loads, they can only be served by qualified drivers, and drivers may have preference over types of loads, which should be accommodated whenever possible. We provide a MILP formulation for LPDP. We use a continuous time index and continuous location space to realistically capture the properties of real problems. We first define the following 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 and then travel to the origin of node. The service time for the start and end nodes is assumed to be negligible. : earliest start time of each node. For means the initial available time of vehicle. : latest start time of each node. For means the latest time by which vehicle must reach 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 time window. : 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 and prefers 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 network: starting nodes of vehicles, nodes representing loads and end nodes of vehicles. Such a formulation generally provides a tight LP lower bound, and in this problem, the schedule of each vehicle is viewed as a different commodity in the network. The decision variables of this formulation are defined as: Variables: : 0-1 variables. if vehicle serves load and then go to serve load ; otherwise. : service start time of each node. Then, the formulation can be stated as follows. Objective: Subject to: (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) Here, (3) are the revenues of served loads, and (4) are the costs of empty movements. Constraints (5) (7) are standard multicommodity 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 between consecutive nodes. Constraints (13) and (14) are the pickup time-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 immediately after load ; hence,, set. According to our tests, standard mathematical programming method can solve small-scale LPDPs, however, for medium-scale and large-scale instances, more efficient method is needed. B. Applying HNP-MP to the LPDP Detailed description of HNP-MP when specialized for the LPDP is presented in this section. Our solution approach can also be applied to problems with other nonhomogeneous resource constraints or load constraints. 1) Sampling: For the LPDP, a feasible solution provides a schedule for each vehicle which satisfies the constraints of the problem, and where the load sequence each vehicle serves and

578 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 4, OCTOBER 2008 Fig. 6. LPDP: to assign load L1. Fig. 5. LPDP: a partial solution. the service start time of each load are fixed. We define partial solutions as feasible solutions to the problem in the form of assigning each load to a certain vehicle, without the fixation of load sequence for each vehicle and the service start time for each load. Here, if load is assigned to vehicle, we can represent the assignment by fixing some variables in the original problem: fix.given a partial solution and its associated subproblem, each load can be either served by its assigned vehicle or dropped, and each vehicle 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-negative number, such as 0.01 or 0.001. Based on our experience, this linear weight calculation procedure provides good partial solutions for the LPDP, although more complicated weight functions can be conceived. Then, after the normalization step, we can 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 and are assigned to vehicle, and the other loads are assigned to vehicle. Then, in the LP-based sampling step, as shown in 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 sampling procedure, load is assigned to vehicle with probability, and assigned to vehicle with probability. 2) Calculating the Promise Index: For the LPDP, to evaluate a partial solution, let each vehicle make as many profits as possible among its available loads, and obtain a sample where each vehicle has a fixed schedule of serving loads. For many LPDPs (including the LPDP in this paper), the problem associated with each partial solution consists of several subproblems separable by vehicles, and the evaluation of the partial solution can be even more efficient if we solve these subproblems separately. When applying the HNP-MP approach to LPDPs with different concerns and constraints, different forms of subprob- Fig. 7. Calculating the promise index: a sample. lems associated with partial solutions need to be solved, and all other 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 vehicle with its assigned loads ; the other is to fix the schedule of vehicle with its assigned loads. Fig. 7 shows the sample obtained after the partial solution evaluation step: vehicle will serve loads and go home, vehicle will serve loads and go home, and load is dropped without realizing any revenue. 3) Partitioning, Backtracking, and Stopping: For the LPDP, each available partitioning attribute fixes a load to be served by a certain vehicle, which leads to Constraints. Then, the value of can be used for the attribute. We can select one attribute (or several attributes) with the best (or top-ranking) LP solutionbased partitioning index value for partitioning. For example, in Fig. 5, suppose that we have the sample in Fig. 7 to be the current best sample, then in Table II, column Attributes of the best sample has six available attributes for partitioning corresponding to this sample. Column Index-LPsolution shows the LP solution-based partitioning index for each available partitioning attribute. Attribute has the greatest Index-LP-solution value, and thus can be chosen as the partitioning attribute. Backtracking and stopping are performed in the standard manner when needed. For the simple example, if in next iteration 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 original promising region to get the promising region for next iteration. C. Computational Results In this section, we report our computational experience with the proposed algorithms on randomly generated instances.

PI et al.: HYBRID NESTED PARTITIONS AND MATHEMATICAL PROGRAMMING APPROACH AND ITS APPLICATIONS 579 TABLE II INDEX FOR EFFECTIVE PARTITIONING TABLE IV LPDP: SCALE SETTINGS TABLE III LPDP: PARAMETER SETTINGS (Randomly generated instances capturing realistic properties are used in many recent research on solving LPDPs and PDPs [35], [43], [46], and our generation of testing data follows the style of these works.) 1) Testing Instances: The experiment settings are described as follows. Map and Locations: We generate 60 locations in a rectangle map of square miles. For each location pair, the distance between the two locations is the Euclidean distance on the map. Loads: Generate loads randomly on the origin destination location pairs. The handling time of each load is. The earliest starting time for each load is generated randomly on the time horizon from 7 am to 6 pm, and the length of the pickup timewindow is set to Uniform(0,4) hour. The net revenue of serving a load is set to, where is the rate of revenue per service time unit. Vehicles: For each vehicle, the initial and homing locations are randomly assigned among the locations. (These two locations are not necessarily the same, since our model and algorithm also intent to support some running horizon systems.) Each vehicle s working time is randomly set to from Uniform(7,9) am to Uniform(4,6) pm individually. The speed of each vehicle is 40 miles per hour. The cost rate of empty movements of vehicles is 10 per hour. Qualification/Preference: For each vehicle-load pair, the probability that vehicle is qualified and prefers to serve load is set to. Overall, we generated 42 testing instances, with six different groups of parameter settings (as shown in Table III) and seven different groups of scale settings (as shown in Table IV). All these settings are of common properties and scales in real applications. For example, as indicated in [43], the typical size of local sub fleet handled by a single load manager is around 20 as we used in our testing. 2) Algorithm Settings: We first test all instances through CPLEX 9 with default CPLEX parameter settings in our computer with Pentium 4 2.8 GHz CPU and 1 GB memory. For all instances, we set a time limit of 30 min. We implement our hybrid algorithm in AMPL, and limit the computation time to be within 30 min. (For the computation time, we only count the time for the LP solution calculating and partial solution evaluation, since with good implementation in highly efficient programming language such as C++, all other computation time can be neglected.) The settings of the hybrid algorithm are described as follows. The dual simplex method is used to solve the LP problem on the promising region in each iteration. To make a fair comparison, for the evaluation of each partial solution, we do not use specialized algorithms, but just call CPLEX to solve the problem associated with that partial solution: we apply the value of the current best solution as the feasible bound, set the MILP tolerance gap to be 0.01, and set the computation time limit to be 2 s. According to our experiences, most partial solutions can be evaluated or cut off within 1 s. (In applications, with better specialized algorithms, the evaluation of partial solutions can 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 certain vehicles if backtracking is not needed. Stop the algorithm when the stopping conditions are met, and no restarting is used. For many of the instances, the computation time is much smaller than the 30 min time limit. To show the generality of our approach, no domain knowledge is used in the tests. (In applications, we may combine special knowledge to further improve the performance of our approach.) For comparison purpose, we test two other approaches. One is local branching (LB) [15], which is a recently developed advantageous mathematical programming-based local search approach. For the LPDP, the initial solution is randomly generated, and CPLEX is used to search improved solution within the neighborhood of the initial point. The improved solution is again used to construct a neighborhood for further searching, until no improvement can be found. Multiple initial solutions are used within the time limit in the tests, and the best solution obtained is reported. The other one is a myopic approach, which is one most popular method used in applications [16]. This myopic approach is briefly 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 time are added, and assigning a load to a vehicle is only allowed

580 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 4, OCTOBER 2008 TABLE V LPDP: RESULTS ON EASY INSTANCES when the profit of the assignment is bigger than a predetermined parameter. Repeat the above process until no profits can be made. Then,, if load is assigned to vehicle, in the original MILP problem, let load only be available to vehicle by fixing some variables: fix. Resolve the MILP problem to obtain the final schedule. For each instance, run the myopic approach twice with set to 0 and 10, respectively, and the better result is selected. The reason to compare the CPLEX, HNP-MP, LB, and the myopic approach is that each of the four methods can be easily used as or developed into a general solver capable of solving LPDPs with different concerns and constraints, including some ill-defined problems with no existing specialized algorithm. [We have also tested Lagrangian relaxation (LR) [17], [26] approach on the problem. According to our observation, the LR convergence performance on the LPDP is not promising, and thus the detail computational results are not reported in this paper.] 3) Testing Results: For 15 of these instances, they can be solved by CPLEX to obtain an optimal gap less than 10%, probably because by nature these instances are easy. We also test our hybrid approach on these easy instances. Detail computational results about these 15 instances are shown in Table V. (Ins is the instance index, CPub is the infeasible bound from CPLEX, CPlb is bound from the best solution of CPLEX, CPgap is the gap between CPub and CPlb, NP-CPlb is the bound from the best solution of the hybrid algorithm, and NP-CPgap is the gap between CPub and NP-CPlb.) The performance of our results on these easy instances is also good, with an average optimal gap of 1.8%. For some of these instances, CPLEX results are better, which is reasonable, since standard MILP algorithms can mostly solve easy problems efficiently. For all other 27 instances, the optimal gap of CPLEX results is greater than 10%, which we consider as difficult instances. We tested our hybrid algorithm, LB, and the myopic approach on these instances. The computational results are shown in Table VI. (LBlb is the bound from the best solution of the LB approach, LBgap is the gap between CPub and LBlb, MYlb is the bound from the best solution of the myopic approach, and MYgap is the gap between CPub and MYlb.) For all these difficult 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 average optimality gap of 2.3%, ranging from 1.2% 4.0%, while average optimal gap of CPLEX, LB, and the myopic results is 19.3%, 22.5%, and 17.8% respectively. For a given problem, it is possible for some highly specialized algorithms to outperform our algorithm. However, our approach is general, and can be applied to many different problems. Also, for a given problem, our general approach can be combined with specialized techniques to achieve greater efficiency. 4) Extension: The computational results of our hybrid approach alone are already very promising, however, they may still be improved if combined with some efficient local search algorithms. One way to do this is just to use the NP results (the best solutions when the algorithm ends) as initial solutions, and do the local search. We select those 17 instances with HNP-CP gap greater than 2.0%, and combine the LB procedure into our hybrid approach. The results are summarized in Table VII (Here, NP-CP-LBub is bound from the best solution of combining HNP-MP and LB, and NP-CP-LBgap is the gap between NP-CP-LBub and CPlb.) While the average solution gap of the NP-CP is 2.8%, the average NP-CP-LB solution gap is reduced to 2.0%, and for 14 of these instances we get improved solutions. Overall, the HNP-MP approach is not only useful for those LPDPs with no efficient specialized algorithm, but also potentially useful for some LPDPs where efficient specialized local search algorithms exist. (Also, another deeper level combination can be considered: use top samples in each iteration of the NP approach as the initial solutions and explore the local search procedure; then, the results of the local search procedure are viewed as new samples and used to determine the next promising region. To compensate the time spent on the local search procedure, in each iteration of the HNP-MP approach, we should probably take less partial solutions, and/or fix more loads when partitioning is performed.) Also, to show the generality of our approach, we test another LPDP where the hard service time-window constraints are replaced by some load precedence constraints [14]. A load precedence constraint means certain loads can only be served some time after the completion of some other loads. As indicated in Section II-C, with these job precedence constraints, we do not have separable problems for each partial solution. The testing instance generation is similar to that in Section III-C1. The load precedence 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 computational results are reported in Table VIII, where it is shown that the 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 applications and has been the focus of lots of research effort during recent decades [1], [4], [18], [27], [28]. The basic elements of FLPs are described as below: there is a set of locations potentially usable for some facilities, and each facility located at certain place will lead to an operating cost; there is a set of tasks which will be served or routed through opened facilities with certain routing

PI et al.: HYBRID NESTED PARTITIONS AND MATHEMATICAL PROGRAMMING APPROACH AND ITS APPLICATIONS 581 TABLE VI LPDP: RESULTS ON DIFFICULT INSTANCES TABLE VII LPDP: NP-CP-LB RESULTS TABLE VIII LPDP: RESULTS ON INSTANCES WITH PRECEDENCE CONSTRAINTS cost; the overall objective is to minimize the total costs of the system, which balance the operating costs of facilities and the routing costs of tasks. Naturally, the DFLPs can be viewed as two-level decision problems: the first level is to decide the locations of facilities, and the second level it to decide the routes of tasks. Discrete facility location problems (DFLPs) are the type of FLPs with discrete location for facilities, and are commonly used in the real world. Large-scale DFLPs are generally difficult to 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 is a real DFLP from intermodal movement industry, and has been paid much research effort due to its significant economic impact. The reason for choosing the IHLP for demonstration is that IHLP is a DFLP with a set of typical real concerns such as the concave transportation cost function. In this paper, we do not use any specific technique dedicated to the IHLP, and most steps in our solution approach are easily applicable to other DFLPs. Our computational tests are also based on the IHLP. The formulations of the IHLP we consider in this paper is presented in the section. We first describe some notations as follows. Sets: : set of origin/destination terminal locations. : 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.

582 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 4, OCTOBER 2008 : transportation cost function of the flow between location to location. Due to the scale economies which is an crucial consideration in transportation industry,, we can assume to be a nondecreasing concave function of the amount of the flow from location to location. Furthermore, in the formulation, we assume that these functions are piecewise linear, and the values of these functions are given. (Refer to [30] and [37] on how to calculate the values of these cost functions.) : cost rate per unit amount if flow is not moved or moved by other more expensive methods such as pure truck movement. :, if movement from to are allowed;, otherwise. Then, we define the decision variables of this problem as follows. : the amount of flow moved through intermodal rail line. : the amount of flow from location to location. : 0-1 facility location variables. if hub is opened;, otherwise. : the amount of flow that is not moved through the intermodal operations. Then, the formulation of the problem is described as follows: Objective: (15) Subject to: (16) (17) (18) (19) (20) (21) (22) (23) (24) Here, in the objective function, (15) is the cost of hub operations, (16) is the cost of flows moved by the intermodal operations, and (17) is the cost of flows not moved by the intermodal operations. Constraint (18) is the requirement that all the flows should be covered. Constraints (19) and (20) require that each flow can only be routed via opened hub in a intermodal movement. Constraints (21) (23) are the relationships between variables s and variables s. Constraint (24) is the restrictions of Fig. 8. DFLP: a partial solution. the movements between terminals and hubs. In the formulation above, term (16) includes concave piecewise linear functions, leading to one major difficulty of solving the problem. There is standard procedures to linearized term (16), [10], however, the problem size will be increased due to the new variables and constraints introduced. As mentioned before, our solution approach is a general approach, and can be potentially applied to many other DFLPs. Particularly, for the IHLP, some of the other typical concerns and constraints 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, including the IHLP. For demonstration purpose, we use the example in Fig. 8 with seven hubs, and four flows ; 1) Sampling: First, for the IHLP, we define partial solutions as feasible solutions to the problem in the form of letting a set of hubs be closed, and no flow can go through these closed hubs. In a given problem, to achieve high performance, the number of hubs closed need to be controlled properly. The form of partial solutions described above is used in our tests. Alternatively, for the IHLP, we can also choose to take partial solutions in the form of fixing all the variables, and similarly the number of opened hubs and closed hubs in a partial solution need to be controlled properly. For other DFLPs, partial solutions can be generated in the form of fixing some or all of the facility location variables, by which each partial solution corresponds to an easier subproblem. For the example in Fig. 8, in a partial solution, hub, and are closed. Second, for the IHLP, denote the part of the LP solution for the current promising region as. Then, the basic rule for calculating the sampling weights is:, the probability that hub is not closed is positively correlated to the value of. In our computational tests, a linear function of is used to calculate 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 sampling weights as

PI et al.: HYBRID NESTED PARTITIONS AND MATHEMATICAL PROGRAMMING APPROACH AND ITS APPLICATIONS 583 TABLE IX DFLP: SCALE SETTINGS TABLE X DFLP: TRANSPORTATION COST FUNCTION SETTINGS Fig. 9. DFLP: a sample. Based on these weights, we can sample a set of hubs to be potentially open, and let all other hubs to be closed. Again, according to our experience, this linear weight calculation procedure provides good partial solutions for the IHLP. For other DFLPs, the LP solution of the facility location variables can be used in the LP-based sampling procedure. 2) Calculating the Promise Index: For the DFLP (and many other DFLPs), if only partial of variables (the facility location variables) are fixed (to 0) in a partial solution, each partial solution corresponds to a relatively small problem with the same structure of the original problem; if the partial solutions are in the form of fixing all the variables (the facility location variables), the subproblem associated with each partial solution is a pure routing problem. For both cases, a standard integer programming algorithm can be used to evaluate partial solutions efficiently. Fig. 9 shows a sample which is contained by the partial solution in Fig. 8. After the evaluation of the partial solutions, the promise index can be calculated through some standard ways. 3) Partitioning, Backtracking, and Stopping: For the IHLP and many other DFLPs, each available partitioning attribute let a certain hub/facility open, which leads to Constraint (for the IHLP). Then, for the IHLP, the value of is used as the LP solution-based partitioning index for the attribute that hub is open. We can select one attribute (or several attributes) with the best (or top) LP solution-based partitioning index value for partitioning. For other DFLPs, the LP solution of the facility location variables can be used to calculate the LP solution-based partitioning index. Backtracking and stopping are performed in the standard manner when needed. C. Computational Results In this section, we report our computational experience on applying HNP-MP to the IHLP. 1) Testing Instances: We randomly generated a set of instances with typical settings to test our solution approach. The experiment settings are described as follows. Map and Locations: We generate a rectangle map of 500 500 square miles. terminal locations and ramp locations are randomly generated over the map. For each location pair, the distance between the two locations is the Euclidean distance on the map. Flows: flows are randomly generated on the origin destination 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 amount of unmoved flows is. The transportation cost function (cost rate per mile) between a location pair is set to a four piece linear concave function: for each location pair with truck movement, the cost rate (the slop for each piece of the cost function) is randomly generated over the range (S1, S2), and the three nondifferentiable points are set to and ; for each location pair with truck movement, the cost rate is randomly generated over the range (S3, S4), and the three nondifferentiable points are set to and. Routing: When the distance between location and location is less than a predetermined parameter ; otherwise,. This constraint on the possible movement, adopted by many big transportation companies nowadays, actually reduces the problem size and thus the solution time of the problem. Here, is set to 200 miles. Overall, we generated 21 testing instances of common properties and scales in real applications, with seven different groups of scale settings (as shown in Table X) and three different groups of parameter settings (as shown in Table IX). 2) Algorithm Settings: We first test all instances through CPLEX 9.1 with default CPLEX parameter settings in our computer with P4 2.8G CPU and 1G memory. The reason we used CPLEX is that integer programming is one widely used approach to solve DFLPs in the literature. For all instances, we set a time limit of 2 h. Then, we test the HNP-MP approach on these instances. Some algorithm setups are described as follows. Dual simplex method is used to solve the LP problem on the promising region in each iteration. CPLEX is used to evaluate each partial solution: we apply the value of the current best solution as the feasible bound, set the MILP tolerance gap to be 0.005, and set the computation time limit to be 2 min.

584 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 4, OCTOBER 2008 TABLE XI DFLP: 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 backtracking 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 knowledge is used in the tests. (In applications, we may combine special knowledge to further improve the performance of our approach.) For comparison purpose, we also test the LR approach [17] which is widely used to solve DFLPs in the literature. The settings of the LR approach are briefly described as follows. Constraints (21) (23) are relaxed in the LR subproblem, and multipliers corresponding to these constraints are added to the objective function. Subgradient algorithm is used to update the multipliers in each iteration of the LR procedure. In each iteration, after obtaining the integer solution of the LR subproblem, fix the s variables, and solve the original problem to get the feasible integer solution. The best feasible solution is reported when the algorithm ends. 3) Testing Results: Detail computation results about these 21 instances are shown in Table XI. (Here, CPlb is the infeasible bound from CPLEX and CPub is bound from the best solution of CPLEX, LRub is the solution of the LR approach, LRgap is the gap between LRub and CPlb, NP-CPub is the solution of the HNP-MP approach, NP-CPgap is the gap between NP-CPub and CPlb.) The HNP-MP approach is superior to CPLEX and LR for each instances, mostly with a significant improvement ratio. V. CONCLUSION The Nested Partitions method is a general framework that can be combined with many local searches, metaheuristic algorithms, and domain knowledge. Previously, the NP method has been combined with genetic algorithms (GAs) [41], local search [42], and domain knowledge [31]. Numerical results show that the hybrid algorithm such as NP/GA or NP/TS performs much more efficiently than the GA or local search algorithm alone. In this paper, we exploit well-known exact algorithms such as MIP or MP and NP metaheuristic framework so that each complements the strengths of the other. The efficiency and novelty of our approach are demonstrated through two important, but difficult problems, i.e., LPDP and DFLP problems. We have also showed that our HNP-MP approach has the advantage of being easily adjusted to different kinds of constraints. In the standard HNP-MP approach, for evaluating each partial solution, we use the MILP solvers. For a specific problem, it is possible to apply or develop some efficient heuristic method to replace the MILP solvers for partial solution evaluation and improve the efficiency of the hybrid algorithm. We can also combine some cuts [44] to make a stronger LP formulation, which is very useful for some discrete optimization problems with loose LP relaxed problems. The LP solution corresponding to the stronger LP formulation is used in the sampling/partitioning step. However, by adding these cuts, we may increase the computation time for obtaining the LP solution, and we may also lose come useful information, especially when cuts are applied to the simplified LP problems. More computational tests need to be done to see the performance of combining cutting planes in the hybrid approach in the future. On the other hand, for many discrete optimization problems, such as LPDPs and DFLPs, it is relatively easy to evaluate partial solutions, and the bottleneck of the hybrid approach is the LP solution calculating step. For some larger scale instances, even obtaining a LP solution on the root node is not easy. We plan to further investigate using simplified LP solutions [38] in our algorithm to reduce the computation time of the LP solution calculating process. ACKNOWLEDGMENT The authors thank the Associate Editor and the three anonymous referees for helpful comments that improved this paper.

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[44] L. A. Wolsey, Integer Programming. : Wiley, 1998. [45] C. K. Wong and M. C. Easton, An efficient method for weighted sampling without replacement, SIAM J. Comput., vol. 9, no. 1, pp. 111 113, 1980. [46] H. Xu, Z.-L. Chen, S. Rajagopal, and S. Arunapuram, Solving a practical pickup and delivery problem, Transp. Sci., vol. 37, no. 3, pp. 347 364, 2001. Liang Pi (S 07) received the M.E. degree in industrial engineering from the University of Wisconsin Madison, Madison, in 2005, and the B.E. degree in computer science from the Special Class of the Gifted Young, University of Science and Technology of China, Hefei, in 2004. He is currently working towards the Ph.D. degree at the Department of Industrial and Systems Engineering, University of Wisconsin-Madison. His research interest include large-scale optimization 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 Applied Mathematics (SIAM), and the CFA Institute.

586 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 4, OCTOBER 2008 Yunpeng Pan (M 06) received the B.S. degree in computational mathematics from Nanjing University, Nanjing, China, in 1995, the M.S. degree in operations research from the University of Delaware, Newark, in 1998, and the M.S. degree in computer sciences and the Ph.D. degree in industrial engineering from the University of Wisconsin-Madison, Madison, in 2001 and 2003, respectively. He is an Algorithms and Formulations Architect with CombinetNet, Inc., Pittsburgh, PA. His research interests are concerned with developing industrial strength techniques and methods for solving difficult mixed-integer programming problems that arise in E-Commerce, Combinatorial Auctions, Procurement (Reverse) Auctions, and Healthcare Informatics. His work appears in Mathematical Programming, Operations Research Letters, the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, the European Journal of Operational Research, and the Journal of Systems Science and Systems Engineering. Dr. Pan is a member of the Institute for Operations Research and the Management Sciences (INFORMS), and the Mathematical Programming Society. Leyuan Shi (SM 06) received the B.S. degree in mathematics from Nanjing Normal University, Nanjing, China, in 1982, the M.S. degree in applied mathematics from Tsinghua University, Beijing, China, in 1985, and the M.S. degree in engineering and the Ph.D. degree in applied mathematics from Harvard University, Cambridge, MA, in 1990 and 1992, respectively. She is a Professor with the Department of Industrial and Systems Engineering, University of Wisconsin-Madison. She has been involved in undergraduate and graduate teaching, as well as research and professional service. Her research is devoted to the theory and applications of large-scale optimization algorithms, discrete-event simulation and modeling, and analysis of discrete dynamic systems. She has published many papers in these areas. Her work 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 Management Sciences (INFORMS). She is currently an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, INFORMS Journal on Computing, and the Journal of Discrete Event Dynamic Systems.