Scheduling technicians for planned maintenance of geographically distributed equipment

Transportation Research Part E 43 (2007) 591–609 www.elsevier.com/locate/tre

Scheduling technicians for planned maintenance of geographically distributed equipment Hao Tang a, Elise Miller-Hooks

b,*

, Robert Tomastik

c

a

Operations Research Group, FedEx Express Corporation, Memphis, TN 38125, United States Department of Civil and Environmental Engineering, University of Maryland, 1173 Glenn L. Martin Hall, College Park, MD 20742, United States United Technologies Research Center, United Technologies Corporation, East Hartford, CT 06108, United States b

c

Received 23 August 2005; received in revised form 25 February 2006; accepted 28 March 2006

Abstract A real-world planned maintenance scheduling problem that exists at several business units within United Technologies Corporation (UTC) is addressed in this paper. The scheduling problem is formulated as a multiple tour maximum collection problem with time-dependent rewards and an adaptive memory tabu search heuristic is developed to solve it. The effectiveness of the proposed solution approach is examined using real-world problem instances supplied by UTC. Relevant upper bounds are derived for the application. Results of numerical experiments indicate that the proposed tabu search heuristic is able to obtain near optimal solutions for large-size (i.e., actual) problem instances in reasonable computation time.  2006 Elsevier Ltd. All rights reserved. Keywords: Multiple tour maximum collection problem; Time dependent; Selective traveling salesman problem; Tabu search; Maintenance scheduling

1. Introduction This paper addresses a planned-maintenance scheduling problem that exists at Otis, Carrier, Chubb and other business units within United Technologies Corporation (UTC). UTC manufactures, installs and services building equipment, including, for example: heating, ventilation and air conditioning (HVAC) systems; distributed power equipment; escalators; elevators; moving sidewalks; security system equipment; and fire detection and suppression equipment. Such pieces of equipment require regular maintenance and are located in buildings that are geographically dispersed. A technician should complete each maintenance procedure as close as possible to pre-specified recommended intervals of time that depend on the type of procedure, the last time that procedure was completed, the type of equipment that is involved, the equipment’s condition, and the *

Corresponding author. Tel.: +1 301 405 2046; fax: +1 301 405 2585. E-mail address: [email protected] (E. Miller-Hooks).

1366-5545/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2006.03.004

592

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

equipment’s expected future usage. Each procedure that is to be performed in the future on a particular piece of equipment is referred to as a task. The scheduling problem is to determine, for each technician and each day in the scheduling horizon (typically, 1 or 2 weeks), the tasks to perform and the sequence in which to perform them. This selection of tasks to perform and the choice of sequence are determined, as best as possible, such that tasks are performed close to their recommended maintenance dates while simultaneously considering the key complicating constraint on workday duration limitations. The tasks to consider in the schedule include all tasks due several weeks prior to or beyond the schedule horizon such that opportunities for minimizing visits to the same pieces of equipment are exploited. Each technician handles multiple types of equipment in an assigned geographic territory. Since each procedure requires only one technician, we can consider each scheduling problem of each technician independently. There is no limit on the number of tasks that each service technician can complete in a day, but the total travel time between equipment locations plus total service time must not exceed the duration of a workday (typically, 8 h). Each task requires between 10 and 960 min to complete. Tasks with service times that exceed the workday duration are handled separately. Determination of an optimal or nearly optimal set of schedules by service technicians or managers is extremely difficult, as such determination requires the simultaneous consideration of a very large number of tasks and pieces of equipment, the travel time matrix, and target maintenance dates for each procedure. Thus, an optimization technique that can automate this process could provide significant improvement over the existing manual scheduling process. In this paper, the planned maintenance scheduling problem is modeled as a Multiple Tour Maximum Collection Problem with Time-Dependent rewards (MTMCPTD). The Maximum Tour Maximum Collection Problem (MTMCP) with time-invariant rewards has been addressed previously in the literature. The authors are not aware of any work in the literature that addresses the MTMCP with time-dependent rewards. The objective of the MTMCPTD is to determine a set of tours, each corresponding to a technician’s schedule on a particular day, such that the total reward collected during the scheduling horizon is maximized. Tours are simultaneously scheduled for multiple days. The associated reward for completing a task on a given day is a function of the day to which it is assigned, i.e., the rewards are time-dependent. Greater reward is assigned to tasks that are more past their due date, in order that these tasks will be selected. The reward is based on the ‘‘urgency’’ of the task. Note that the rewards are not actual monetary rewards or profit received for completing service; rather, they are values that are used internally to the algorithm that are set to try to force tasks that are most urgent to be scheduled earliest. By employing time-dependent rewards, the proposed framework can explicitly model details of the realworld scheduling problem. In addition to its utility in modeling task urgency, one can model the relative desirability associated with performing a specific task on a given day. The desirability depends on the level of inconvenience that will be incurred in taking the piece of equipment out of service. The cost in terms of convenience and, thus, customer satisfaction can change dramatically from one day to the next. For instance, taking an elevator out of service to complete maintenance is more costly on a day when a major convention is being held in the building as compared with the cost on the day prior to the convention. The proposed framework with time-dependent rewards can also support condition-based maintenance, where the equipment condition is detected or estimated. Probability failure curves for each piece of equipment are known and can be used to estimate the urgency associated with completing a maintenance task on a given day. The scheduling horizon may be either short-term (on the order of days or weeks) or long-term (on the order of months). In the particular application studied here, like in many related applications, procedures need not be repeated on a given piece of equipment in a short-term horizon, because the suggested interval before repeating a procedure on a given piece of equipment might range between several months and several years. On the contrary, if the horizon is long-term, such an assumption might not be reasonable. As the MTMCPTD is shown to be NP-hard, a tabu search heuristic is proposed in this paper for the shortterm planned maintenance scheduling problem. Tight upper bounds are derived for the scheduling application. Results of numerical experiments using problem instances supplied by UTC indicate that the heuristic is able to obtain near-optimal solutions in reasonable computing time. The MTMCPTD and its application to this short-term scheduling problem are described next. The long-term scheduling problem is considered in Section 5.

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

593

The MTMCPTD is defined on a directed graph G = (V, A), where vertex set V = {1, 2, . . . , n} represents a central depot (vertex 1) and a set of maintenance tasks (Vn{1}) and arc set A = {(i, j) j i, j 2 V} represents the links between all pairs of vertices. Here, n is the number of vertices. Each vertex i (i 2 Vn{1}) is associated with an on-site service time si and a set of rewards rik, k 2 H, where H = {1, 2, . . . , m} is the planning horizon and rik denotes the reward (or profit or urgency) for completing task i on day k. Here, m is the number of days in the planning horizon. Each arc (i, j) ((i, j) 2 A) is associated with a non-negative travel time tij. The MTMCPTD seeks m tours that start and end at the depot, visiting a subset of vertices in Vn{1}, such that the total collected reward is maximized and the duration of any tour (total service time plus total travel time) does not exceed a pre-specified limit T. In this work, time-windows for service calls and the impact on the suggested schedule of emergency calls (i.e., calls that must be serviced within only hours of their appearance) are not considered. An illustrative example of a 2-day (m = 2) and one-technician planned-maintenance scheduling problem is given in Fig. 1. Let vertices 2–9 represent maintenance tasks that need be scheduled. The location of each vertex corresponds with the associated equipment location. Suppose that the maximum working duration for each day is 8 h (T = 8 h). Without loss of generality, breaks are ignored. Service time si and reward ri for vertex i and travel times between tasks are shown in the figure. There are two values for each task reward; each value corresponding to the reward received from serving the equipment on the first or second day. For example, ‘‘r3,1 = 3 and r3,2 = 2’’ indicate that the reward received from servicing task 3 on day 1 is 3 and on day 2 is 2. Travel times are assumed to be symmetric. Rewards are greater for completing tasks with greater urgency or higher priority. The problem is to select a set of tasks to complete on each day within the tour duration limit T over the 2-day planning horizon so that the overall reward is maximized. Note that there may be some savings incurred by completing a less urgent task and, thus, it is not necessarily best to only consider the tasks with the highest rewards (i.e., with greatest urgency or priority). A solution employing two tours with 23-unit total reward is also depicted in the figure. It appears that this is the optimal solution for the example problem. Note that three tasks (vertices 2–4) have been omitted from the schedule. To include them would require exclusion of other tasks due to tour duration restrictions and would result in a lower total reward. The optimal solution schedules for this dynamic problem differs from the optimal solution to the static variant of this problem. For example, if one assumes that the reward received

4

r4,1 = 4, r4,2 = 3 s4 = 0.9

2nd day schedule

3

Total duration = 6.3 Total reward = 12

r3,1 = 3, r3,2 = 2 s3 = 2.1

r5,1 = 3, r5,2 = 6 s5 = 1.6 5

r6,1 = 4, r6,2 = 6 s6 = 2.2

6

1 2

r2,1 = 5, r2,2 = 7 s2 = 2

7

1st day schedule Total duration = 8 Total reward = 11

r9,1 = 5, r9,2 = 4 s9 = 0.7

r8,1 = 3, r8,2 = 3 s8 = 2 8

9

r7,1 = 3, r7,2 = 4 s7 = 2

1 2 3 4 5 6 7 8 9

1 0

2 1 0

3 1 0.8 0

4 1 1.7 0.6 0

5 0.5 1.4 1 0.8 0

6 1.1 2 1.8 1.2 0.9 0

7 0.4 1.2 1.1 1.1 0.5 0.9 0

Travel Time Matrix (hours)

Fig. 1. An illustrative example.

8 0.8 1.4 1.5 1.7 1.1 1.1 0.7 0

9 1.5 1.1 1.7 1.9 1.4 1.5 0.9 0.7 0

594

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

from servicing a task on the second day is the same as on the first day, the optimal solutions would be identical on the first day, but the second day’s schedule will be (1, 3, 4, 5, 1) instead of (1, 5, 6, 1) as indicated in Fig. 1. Despite that there are many related scheduling applications where rewards should be taken as a function of time, it appears that there has been no published work where this factor has been considered. Erkut and Zhang (1996) addressed a related, albeit different problem variant: the Selective Traveling Salesperson Problem (STSP) with time-dependent rewards. In this problem formulation, a tour can be any length, but rewards are only received from vertices visited prior to some duration limit T. It is further assumed that all vertices must be visited. They formulated this problem as a mixed integer program and solved it by a penalty-based greedy heuristic and an exact implicit enumeration algorithm. The largest problem solved to optimality had 20 vertices. The static version of the MTMCPTD, the Multiple Tour Maximum Collection Problem (MTMCP), and its variant, the Team Orienteering Problem (TOP), have received some attention in the literature. In the TOP, the tours are permitted to have non-identical starting and terminus vertices. Butt and Cavalier (1994) proposed a greedy construction procedure for the MTMCP. Chao et al. (1996a) developed a 5-step metaheuristic to solve the TOP, whose structure was based on a deterministic variant of simulated annealing (Dueck and Scheuer, 1990). Tang and Miller-Hooks (2005) designed a tabu search heuristic for the TOP that outperformed other existing heuristic techniques based on experiments conducted on a set of benchmark problems. The only exact algorithm that addresses the TOP is based on column generation (Butt and Ryan, 1999). This algorithm is capable of solving some small- to moderate-size problems with up to 100 vertices, provided the number of vertices in each tour remains relatively small. The MTMCP is the multi-tour version of the STSP. A number of works have addressed the STSP or the related Orienteering Problem (OP, the single-tour version of the TOP), several of which propose exact algorithms (Laporte and Martello, 1990; Ramesh et al., 1992; Fischetti et al., 1998; Gendreau et al., 1998b). Since the STSP (like the MTMCP) is NP-hard (Golden et al., 1987), most research has focused on providing heuristic approaches, including construction heuristics (Tsiligirides, 1984; Golden et al., 1987, 1988; Gendreau et al., 1998b), heuristics with both construction and improvement phases (Ramesh and Brown, 1991), and metaheuristics (Chao et al., 1996b; Gendreau et al., 1998a). On first glance, it might appear that the long-term maintenance scheduling problem can be modeled as an Inventory Routing Problem (IRP). A review of the literature related to the IRP can be found in (Savelsbergh, 2003). In the IRP, vendors determine the timing and size of deliveries to their customers. The delivery time and size of delivery is similar to scheduling revisit time and service type for maintenance tasks in the long-term planning problem. However, there are significant differences between these problem classes. For example, the objective of the IRP is to minimize long-term delivery costs, while the objective of the maintenance scheduling problem is to maximize the total reward collected by servicing the tasks. In addition, the problem elements of the IRP are static, while in the maintenance scheduling problem task rewards change with passing time. Thus, techniques developed for the IRP would require significant changes for use in addressing the maintenance scheduling problem. In the remainder of this paper, an integer programming formulation is given for the MTMCPTD (Section 2) and a tabu search heuristic for its solution is presented (Section 3). Upper bounds for the MTMCPTD are also derived (Section 4). Results and analyses of numerous computational experiments conducted on the data from a real-world application are provided (Section 5). Additional experiments were conducted to assess an approach to the long-term scheduling problem, where tasks may be repeated during the planning horizon. In the proposed approach, the short-term formulation is repeatedly solved in a rolling horizon framework. Conclusions follow (Section 6). 2. Mathematical formulation The MTMCPTD described in Section 1 can be formulated as a three-index integer program. This formulation (denoted as P) is given for n P 3. It is assumed that the total reward for servicing customers (or equipment) is maximized over a planning horizon of m days. The following additional notation is used: yik = 1, if task i is serviced on day k; 0, otherwise.

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

595

xijk = 1, if task j is serviced immediately after task i on day k; 0, otherwise. VC = Vn{1}. U = a subset of tasks in V, i.e., U  V. ðPÞ

max

XX

ð1Þ

rik y ik

i2V C k2H

s:t:

X

x1ik ¼

i2V C

X

xi1k ¼ 1

X

xijk ¼

j2V nfig

X

xjik ¼ y ik

X

ð2Þ

ði 2 V C ; k 2 H Þ

ð3Þ

j2V nfig

XX ðtij þ si Þxijk 6 T i2V

ðk 2 H Þ

i2V C

ðk 2 H Þ

ð4Þ

j2V

y ik 6 1

ði 2 V C Þ

ð5Þ

k2H

X X

xijk 6 jU j  1 ðU  V C ; n  1 P jU j P 2; k 2 H Þ

ð6Þ

i2U j2U nfig

xijk 2 f0; 1g y ik 2 f0; 1g

ði 2 V ; j 2 V ; i 6¼ j; k 2 H Þ C

ði 2 V ; k 2 H Þ

ð7Þ ð8Þ

In formulation (P), the objective (1) is to maximize the total reward received from servicing the selected tasks over m days. Constraints (2) ensure that each tour starts and ends at the depot. Constraints (3) guarantee the connectivity of the service tour for each day, while the duration limit on each tour is respected by constraints (4). Constraints (5) ensure that all maintenance tasks can be scheduled at most once during the planning horizon. Constraints (6) prevent sub-tours that do not include the depot. Integrality requirements are given by constraints (7) and (8). It is assumed that the maintenance tasks will be completed at most once during a planning horizon of m days, consistent with the short-term problem as discussed in Section 1. Since the MTMCP is NP-hard (Butt and Cavalier, 1994) and is a special case (where all rewards are static) of the MTMCPTD formulated in (1)–(8), it follows that the MTMCPTD is also NP-hard. 3. Tabu search heuristic for the MTMCPTD Since the MTMCPTD is NP-hard and most real-world service scheduling problems are large, it is unlikely that optimal solutions can be obtained within reasonable computing time. In this section, a tabu search heuristic embedded in an Adaptive Memory Procedure (AMP) is proposed for solving the MTMCPTD. The main operations within the AMP are included in three steps: partial solution generation and storage, solution construction by combining partial solutions, and partial solution update. A partial solution is defined herein as a single tour, i.e., one of the required m tours. At the start of the AMP, a certain amount of storage space (adaptive memory) is allotted. A set of partial solutions is generated by heuristic techniques and is stored in the adaptive memory. A linked list structure can be employed to store the single tours, as shown in Fig. 2. In each linked list, there are four components for each tour: a unit for recording the time-dependent tour reward, a unit for recording the date on which the tour is scheduled in the original solution, a pointer to the actual address of the stored tour, and the elements of the stored tour. Note that if a tour is the kth tour (1 6 k 6 m) of the original solution, its scheduled date recorded in a linked list is k. Documenting the scheduled date of a tour is necessary, because the reward is time-dependent. Such a structure allows for convenient update of stored partial solutions. For instance, stored single tours can be sorted according to their objective values. Based on the sorted tour objective values, existing tours can be readily deleted and new tours can be inserted into the adaptive memory.

596

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

Reward of Tour 1

Scheduled Date

Reward of Tour 2

Scheduled Date

Reward of Tour N

Scheduled Date

1 → 5 →7 → 1

1 → 6 → 2 → 15 → 18 → 1 1→ 4 → 3 → 8 → 1

Fig. 2. Data structure for storing feasible solutions in the adaptive memory.

Solution construction is performed by combining select partial solutions in the adaptive memory, where selection preference is probabilistically biased to those partial solutions with preferred objective values. The complete solution constructed through this process is improved by the tabu search procedure and the solutions maintained in the adaptive memory are updated using the components (i.e., individual tours) of the improved solution. For example, the lowest-reward tours in the adaptive memory can be replaced by individual tours of the improved solution provided that these individual tours have better total rewards. As discussed in Golden et al. (1997), the AMP works in a similar way to genetic algorithms, with the exception that offspring (in AMP, the complete solutions constructed by combining stored partial solutions) can be generated from more than two parents. The idea of using such an AMP was introduced by Rochat and Taillard (1995). Several published studies show that the AMP is very effective in providing high quality solutions, especially when implemented in conjunction with tabu search (e.g., Rochat and Taillard, 1995; Gendreau et al., 1999). The main procedures of the proposed technique are summarized in Fig. 3, where Steps 1, 2 and 4 implement the AMP operations and Step 3 is the embedded tabu search procedure. Text given in bold in the figure is discussed in greater detail in the following subsections. 3.1. Generating vehicle tours in the AMP A randomized cheapest insertion procedure, where the first non-depot vertex included in each tour is randomly chosen, is employed in generating the set of initial solutions for the AMP. At each step, a vertex is

Fig. 3. Overview of the tabu search heuristic.

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

597

selected for inclusion in one of the tours. In which tour the vertex will be included depends on the reward obtained from its inclusion. Between which pair of vertices it will be inserted within the tour depends on a ratio of the duration to reward incurred from its inclusion in the particular location. Additional details of this initial solution generation procedure are provided in Fig. 4.  Note that one pass of the procedure in Fig. 4 will generate m feasible tours; thus, executing this procedure W times will create at least w tours. m 3.2. Constructing a solution within the AMP Step 2 of the proposed tabu search heuristic constructs a feasible solution for the MTMCPTD with m tours, as follows. First, a candidate list, CL, is created to include all individual tours in the adaptive memory. Each tour is assigned with a selection probability, which equals its objective value relative to the combined objective values of all tours. Second, the MTMCPTD solution, denoted as S, is constructed by combining no more than m tours in the adaptive memory. More specifically, if a tour is selected as one of the tours of solution S, remove from CL all tours having at least one vertex in common with that tour. If the recorded scheduled date of this tour is k, allocate the tour to the kth day (tour) of solution S if the kth day has not yet been scheduled. If the kth day of solution S has already been scheduled, allocate the selected tour to an available day of solution S so that the total time-dependent reward of the selected tour is maximized. Repeat this process until CL is empty. Third, check whether the current solution S contains m tours. If the number of tours in S is equal to m, go to Step 3; otherwise, construct (from the prior partial solution) new tours (until the number of tours in S equals m) using the vertices that are not yet included in S (similar to Step 3 of Fig. 4) and go to Step 3. 3.3. Generating neighborhood solutions In Step 3B of the tabu search procedure, neighborhood solutions are generated and improvements to these solutions are considered. The tabu search procedure starts from a feasible solution (denoted as S) constructed in the AMP. In one tabu iteration, neighbors of this solution will be generated and the best non-tabu solution from the generated neighborhood solutions will be selected to replace solution S. Similar to most tabu search implementations, a tabu solution will be selected if it is better than the best solution obtained in all previous iterations. Any valid neighborhood solution (or neighbor) of solution S has a different total reward than solution S. Even if the neighboring solution contains the same set of vertices as solution S, its total reward may differ from that of S, because some of the vertices may be assigned to different tours and, thus, may lead to different values of reward. To generate neighborhood solutions of solution S, vertices from select tours of S are randomly removed and vertices chosen from set S (i.e., the set of vertices not included in S) are randomly inserted in the select tours of S, as follows. At each iteration, two tours of S are chosen at random. For each select tour, some randomly chosen vertices are removed from the tour (the number of vertices to be removed can be chosen randomly, as in the experiments conducted as part of this work). For each such tour, vertices from set S are

Step 1. Create m initial tours τk = {1, 1} and set their duration such that D(τk) = 0 (k = 1, 2, …, m). Let Ω = {j | t1j + tj1 + sj ≤ T, ∀ j ∈V \{1}}. Let H = {1, 2, …, m}. Step 2. For each k ∈H, randomly select and remove a vertex j from set Ω, and insert it in τk.

(

{

Step 3. Select j ∈ Ω and two vertices p and q on τk where k = t | max r jt t ∈H

}) such that D(τ ) + k

tpj + tjq – tpq + sj ≤ T and the evaluation function (tpj + tjq – tpq + sj)/rjk is minimal. If no j exists such that D(τk) + tpj + tjq – tpq + sj ≤ T, then H = H \{k}. Otherwise, insert j between the selected p and q, remove j from Ω and update the duration of τk. Step 4. If H = ∅, stop; otherwise, go to Step 3. Fig. 4. Initial solution generation for the MTMCPTD.

598

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

randomly inserted until the total duration of the resulting tour is no less than T(1 + e), where e P 0 is a pre-set parameter. Each vertex is inserted in the location that leads to the minimal increase in tour duration. In this process, infeasible solutions (i.e., solutions for which the tour limitation constraint is not enforced) are permitted when e > 0. Permitting intermediate infeasible solutions may aid in moving the search process out of a local optimum. First introduced by Gendreau et al. (1994), empirical results in several different contexts (and also in the numerical experiments described in Section 5) have shown that this strategy is very effective in providing high quality solutions. Infeasible solutions are guided towards feasibility through neighborhood generation and improvement. A quasi-reward is used in the tabu search heuristic. Here, a quasi-reward of a solution is defined as its true total reward less an adaptively adjusted penalty cost that is incurred if at least one tour contained in this solution has duration greater than T, as follows: Rquasi ð e SÞ ¼

m X k¼1

Rðsk Þ  g

m X

maxðDðsk Þ  T ; 0Þ

ð9Þ

k¼1

A neighborhood solution, denoted as e S, consists of m individual tours sk (k = 1, 2, . . . , m). R(Æ) represents the total reward of a tour and parameter g will be adaptively adjusted (see Section 3.6). Parameter g in this equation helps to guide the search by affecting when the search will alternate between feasible and infeasible solution spaces. For example, if penalty parameter g is set to a high value, infeasible solutions with at least one tour exceeding time limit T will be given a low quasi-reward by Eq. (9) and, thus, are less likely to be chosen as the candidate to enter a subsequent iteration in the tabu search (see Step 3C of Fig. 3). Conversely, if parameter g is set to a low value (i.e., infeasible solutions are under-penalized), infeasible solutions are more likely to be selected as candidates. This is because infeasible solutions can have higher quasi-rewards than feasible solutions. Therefore, to guide the search process so that it alternates between feasible and infeasible solution neighborhoods, we set parameter g lower if some feasible solutions have been chosen as candidates in the last several iterations to allow more infeasible candidate solutions and we set parameter g higher if no feasible solutions have been selected in the last several iterations. 3.4. Improving neighborhood solutions Three tour improvement procedures are employed in the tabu search heuristic for the MTMCPTD, including vertex relocation, vertex addition, and a random vertex insertion (RVI) procedure. Similar techniques are described by Tang and Miller-Hooks (2005) for a related time-invariant problem. They are repeated here for completeness with adaptations for the time-varying problem and are given for a current solution S and two select tours from solution S, s1 and s2. 3.4.1. Vertex relocation for the select tours Two vertex relocation schemes are considered for the two select tours. The first scheme reduces the total tour duration by relocating vertices in the tour of longer duration to the tour of shorter duration, as follows. Assume D(s1) > D(s2). Scan s1 starting from the first vertex to examine if any vertex can be removed from s1 and inserted in s2 such that the updated total duration of s1 and s2 is reduced. If such a vertex is found and the corresponding relocation increases the total rewards (note that rewards are time-dependent, hence relocating vertices between select tours may result in different total rewards), then perform this relocation operation immediately. Otherwise, examine the next vertex. This procedure stops when all vertices on s1 have been examined. The second scheme improves the total duration–reward ratio via vertex exchange between the two tours. To illustrate, let sk[i] denote the ith vertex (other than the depot) visited in tour sk (k = 1, 2) and define a duration– reward ratio  for each vertex pair (s1[i], s2[j]) according to expression cðs1 ½i; s2 ½jÞ :¼ ðts1 ½i1;s1 ½i þ ts1 ½i;s1 ½iþ1 þ ts2 ½j1;s2 ½j þ ts2 ½j;s2 ½jþ1 Þ=ðrs1 ½is1 þ rs2 ½js2 Þ Vertex exchange will be conducted between tours s1 and s2, as follows. Starting from the first vertex in s1, examine each vertex pair (s1[1], b), where b is a vertex in s2. If the duration–reward ratio decreases when vertex s1[1] is exchanged with vertex b in s2, make this exchange and repeat this evaluation starting from the second vertex in s1. This procedure stops when all vertex pairs in s1 and s2 have been examined.

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

599

3.4.2. Addition of vertices to select tours At chosen iterations, a greedy procedure is employed to examine if additional vertices can be inserted in the select tours. This addition of vertices will result in improved total reward collected by the tours. The greedy procedure follows directly from Steps 3 and 4 in Fig. 4, in which set X is replaced by S and set H only consists of the indices of the select tours. 3.4.3. The random vertex insertion (RVI) procedure The RVI procedure uses a randomized greedy method that builds on the idea proposed by Hart and Shogan (1987). The procedure seeks to improve each tour’s duration by random vertex removal and reinsertion within each tour. No vertex exchange between tours is considered. Details of the RVI can be found in Tang and Miller-Hooks (2005). 3.5. Selecting non-tabu neighborhood solutions A solution that was recently identified as S and used to generate neighborhood solutions is prohibited from being considered to replace the incumbent S in an effort to guide the search process out of a local optimum. Specifically, a tabu list is employed to prevent a move made at iteration qT from being repeated until at least iteration (qT + h), where h is the tabu duration that may be set deterministically or chosen randomly from a pre-specified interval. In this study, whenever a vertex is inserted in a solution tour at iteration qT, it may not be removed from that tour until iteration (qT + h), where h is randomly chosen on interval [hmin, hmax]. At each iteration, the non-tabu neighborhood solution with the maximum total reward will be selected to override the current solution S. However, similar to most tabu search procedures in the literature, in the proposed heuristic a move will not be accepted if it is forbidden in the tabu list unless its tabu status is overridden by the aspiration criterion. This can occur when the move results in a better solution than the best found thus far. 3.6. Adjusting heuristic parameters In Step 3C, heuristic parameters will be adjusted based on solution quality and the neighborhood size used in generating neighborhood solutions in Step 3B. Parameter g employed in expression (9) is adaptively adjusted, similar to Gendreau et al. (1994). More specifically, if at least one of the best non-tabu neighborhood solutions chosen in the last d iterations are feasible, then set g = g/2; if they are all infeasible, set g = 2g. By adaptively adjusting parameter g, the tabu search heuristic is able to alternate between neighborhood solution generation stages where more feasible or infeasible neighbors will be explored. Another important parameter in the tabu search heuristic is the number of neighbor solutions that are randomly produced (i.e., the neighborhood size) in Step 3B, denoted by b. In the implementation employed herein, a method of switching between small and large neighborhood sizes is proposed. That is, during the stage where a large number of neighborhood solutions are generated, once an improved solution (over the current best feasible solution) is found, the algorithm immediately returns to the stage where a small number of neighborhood solutions are generated. The algorithm returns to the small neighborhood stage, because an improvement in the best available solution may indicate that the search process has successfully moved out of the current local optimum. Intuitively, one would expect that if the search process stays in the large neighborhood stage (i.e., does not return to small neighborhood stage), the result obtained should be much better than alternating between the two stages. However, empirical results conducted in a related study (Tang and Miller-Hooks, 2005) show that alternating between small and large neighborhood stages during the course of the tabu search enables the search to evolve in an efficient way without leading to solutions that degrade in quality. This alternation in neighborhood exploration stages is similar in nature to variable depth search (e.g., Lin and Kernighan, 1973) and variable neighborhood search (Mladenovic´ and Hansen, 1997). Step 3 of the tabu search heuristic presented in Fig. 3 will be repeated or terminated based on heuristic parameters and the quality of the incumbent solution, as follows. Let a denote a pre-set value of maximum

600

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

non-improvement iterations in Step 3. If the current neighborhood size is small and qT > a/2, set the neighborhood size parameter to the large size and go to Step 3B; if the current neighborhood size parameter is small and qT 6 a/2, go to Step 3B. If a new improved feasible solution is found, set qT = 0 and go to Step 3A; if the neighborhood size is large and 0 < qT 6 a, go to Step 3B. For all other cases, go to Step 4 (i.e., Step 3 terminates). 3.7. Updating solution tours within AMP After each pass of Step 3 in the proposed tabu search heuristic (see Fig. 3), the objective of incumbent solution S may be improved and, therefore, tours included in solution S may be modified during the process. These tours are used to update the solution tours maintained in the adaptive memory, as follows. First, individual solution tours of the improved solution S are inserted in the adaptive memory based on their objective values. Second, the worst tours (i.e., tours with smallest objective values) in the adaptive memory are removed so that the number of single solution tours included remains constant. This updating mechanism ensures that better solutions tours generated during the tabu search process may be selected in future iterations while limiting the size of computer memory required by the heuristic. 4. Upper bounds Since the MTMCPTD is NP-hard and the problem instances that will need to be addressed in the actual application are large, it is difficult to obtain exact solutions with reasonable computational effort. Thus, in this section, several methods for computing upper bounds on the optimal solution to the MTMCPTD are provided. The quality of the solutions obtained by the proposed tabu search heuristic can be compared with valid upper bounds for each problem instance. Data received from UTC indicate that service times at each piece of equipment are, in general, greater than the travel times between them. Further, it is reasonable to allow the rewards obtained from completing the tasks to remain constant within a given day, but to vary from one day to the next. If UTC were to offer priority service for all or some of the customers (e.g., those willing to pay for higher level service), it may be necessary to allow the rewards to vary by time of day, consistent with, for example, allowing the customer to choose a preferred time period during the day for service completion. Given these characteristics, a tight upper bound for this service scheduling application may be obtained by modeling the problem as a type of Multiple Knapsack Problem (MKSP) (see, for example, Martello and Toth (1990), for an introduction to the MKSP). In this transformation, each workday corresponds with a knapsack and each maintenance task can be viewed as an item that can be placed into a knapsack, i.e., assigning each task to a particular day. Each knapsack or day has a volume or weight restriction, T, akin to the tour duration restriction in the planned maintenance scheduling problem. The objective is to maximize the total reward of all items included in the knapsacks, i.e., of all scheduled tasks. While the reward received from incorporating items in a knapsack is typically a constant across knapsacks, in this problem, the reward must depend on the knapsack or day to which the item or task is assigned. Thus, this problem is referred to as the Assignment-Dependent Multiple Knapsack Problem (ADMKSP). The ADMKSP is formulated as an integer program, as follows. XX max rik zik ð10Þ i2V C k2H

s:t:

X

si zik 6 T

ðk 2 H Þ

ð11Þ

i2V C

X

zik 6 1

ði 2 V C Þ

ð12Þ

ði 2 V C ; k 2 H Þ

ð13Þ

k2H

zik 2 f0; 1g

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

601

Here, zij = 1 if task i is assigned to day (knapsack) k, and 0 otherwise; si is the transformed total time (service and travel times) for service provided at task i. The transformed total time si provides estimated lower bounds of the time required for servicing task i in any day’s schedule, as addressed in the following propositions. The objective (10) of the ADMKSP is to maximize the total reward received from assigning tasks to daily schedules. Constraints (11) ensure that the total service time of all tasks assigned to day k does not exceed the duration limit T, akin to not exceeding a weight or volume restriction in the knapsack problem. Constraints (12) ensure that each task i is assigned at most once. Integrality requirements are respected by (13). It is known that the MKSP is NP-hard (Martello and Toth, 1990). Since the MKSP is a special case of the ADMKSP when all rewards are static, the ADMKSP is at least as hard as the MKSP. Proposition 1. The ADMKSP solution provides a valid upper bound on the solution of the MTMCPTD for si ¼ si ; i 2 V . Proof. If si ¼ si ; i 2 V , the effect of travel times is simply ignored. Suppose the capacity of each knapsack in the ADMKSP is limited to T. If each knapsack of the ADMKSP corresponds with each day in the MTMCPTD, then the set of vertices included in each feasible tour for any given day for the MTMCPTD will be a feasible set of vertices to include in a knapsack. This is because the duration of each tour in the MTMCPTD, also with a duration limitation of T, is computed from both travel time between vertices and service times at these vertices, while the capacity consumed by each vertex included in each knapsack only accounts for service times (travel times are assumed to contribute zero in the capacity computations). By this argument, any feasible solution to the MTMCPTD is also feasible to the ADMKSP. Or, equivalently, feasible solutions to the MTMCPTD are a subset of all feasible solutions to the ADMKSP. Hence, the optimal solution value for the ADMKSP is at least as large as that for the MTMCPTD. It follows that the ADMKSP solution provides an upper bound for the MTMCPTD. h In Proposition 1, travel times between select vertices are simply ignored. Better upper bounds may be obtained by under-estimating, but not ignoring, the effect of travel times, as presented in the following propositions. Proposition 2. The ADMKSP solution provides a valid upper bound on the solution of the MTMCPTD for 2t1i si ¼ si ð1 þ T 2t Þ; i 2 V , if travel times satisfy the triangular inequality (i.e., tij + tjl P til, " i, j, l 2 V). 1i Proof. Similar to the logic for proving Proposition 1, to show that the ADMKSP solution is an upper bound 2t1i for the MTMCPTD when si ¼ si ð1 þ T 2t Þ; i 2 V , is equivalent to showing that vertices visited by any tour in 1i a feasible MTMCPTD solution satisfies inequality (11). Without loss of generality, suppose that a feasible MTMCPTD tour visits a subset of vertices in V, denoted as X (note that depot 1 2X  V). Since the total duration of this tour is limited by T (by MTMCPTD definition), we have X si þ LTSPðXÞ 6 T ð14Þ i2X

where LTSP(X) denotes the optimal TSP solution value (in terms of total travel time) over set X. In addition, one can show (through the triangular inequality) that ð15Þ

LTSPðXÞ P 2tMAX where tMAX = maxi2Xn{1}{t1i}. By (15) and (14), the following inequalities follow: ðT  2tMAX Þ  LTSPðXÞ P

X

! si þ LTSPðXÞ  2tMAX

i2X

P

X i2X

ðby ð14ÞÞ P

i2X

! si

 LTSPðXÞ

X

 ð2tMAX Þ ¼ 2

X i2X

ðtMAX  si Þ P 2

X ðt1i  si Þ i2X

! si

 LTSPðXÞ

ðby ð15ÞÞ

602

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

Thus, we have P 2 i2X ðt1i  si Þ 6 LTSPðXÞ T  2tMAX P X 2 i2X ðt1i  si Þ X ) si þ 6 si þ LTSPðXÞ T  2tMAX i2X i2X X X t1i  si X ) si þ 2 6 si þ LTSPðXÞ 6 T T  2t1i i2X i2X i2X  X  2t1i si 1 þ ) 6T T  2t1i i2X X si 6 T )

ðby ð14ÞÞ

i2X

This shows that vertices visited by any feasible MTMCPTD tour satisfy inequality (11). Thus, the correctness of Proposition 2 follows. h Another estimate for si , taking advantage of the information on the minimal travel time arc emanating from vertex i, can be described as follows. Proposition 3. The ADMKSP solution provides a valid upper bound on the solution of the MTMCPTD for si ¼ si þ tMINðiÞ ; i 2 V , where tMIN(i) = min{j2Vn{i}tij}. Proof. Variable tMIN(i) represents the minimum travel time from vertex i to any other vertex. Thus, si ¼ si þ tMINðiÞ is the lower bound on the total time for visiting vertex i and any of its induced arcs. For any vertex set X (X  V) that can be visited in a feasible MTMCPTD tour, it follows that X X X si ¼ ðsi þ tMINðiÞ Þ 6 si þ LTSPðXÞ 6 T i2X

i2X

i2X

The vertices of any tour that is included in a feasible MTMCPTD solution will also satisfy inequality (11). h 5. Computational experiments In this section, the performance of the proposed tabu search heuristic is examined on an actual maintenance scheduling problem provided by UTC. The computational experiments seek to determine how the tabu search parameters should be set to obtain good solutions with relatively small computational effort, how good the quality of the solutions generated by the proposed tabu search heuristic is, if the solution quality suffers when the parameters are not tuned for the problem instance, what the effect on solution quality is of pre-maturely terminating the heuristic as compared with solutions obtained at the natural termination, and what the effect of employing varying length planning horizons is, including repeated tasks, on the resulting tours. The tabu search procedure was implemented in C++ and run on a DEC Alpha XP1000 professional workstation with 1 Gb ram and 2 Gb swap, 533 MHz Risc processor, running Digital UNIX 5.0 operating system, using Digital’s C++ compiler. 5.1. Problem instances Actual data sets for two select technicians assigned to service building equipment for UTC were provided for this study. Ideally, the first technician would complete 2344 maintenance tasks and the second technician would complete 2315 maintenance tasks in approximately 90 buildings in the near future. Historical service records for select regions were studied to estimate the service times for each task. The average travel time between buildings is 20.3 min and approximately 75% of all tasks have a service time less than or equal to 60 min. In addition to solving the problem directly, two additional smaller-scale problems were constructed for each technician by clustering maintenance tasks located within close geographic proximity to one another

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

603

and with similar due dates (i.e., recommended maintenance dates based on recommended service intervals). Thus, for each technician, three problem instances were used in the experiments, two of which were generated by clustering: instances of size 1276, 1754 and 2344 for the first technician (Tech I) and instances of size 1120, 1689 and 2315 for the second technician (Tech II). Maintenance task rewards were generated with a piecewise linear function based on the recommended interval and the elapsed time since the last scheduled service. The slope of each portion of the function over a specific time period was steeper for tasks associated with procedures that must be repeated often as compared with tasks involving procedures that should be repeated over a longer time period. The function forced the rewards to be higher for tasks that are approaching their due dates and lower for tasks that need not be repeated in the near-term. In addition, time-dependent rewards were set in such a way that overdue tasks will have higher rewards earlier in time to improve the chances that these tasks will be scheduled early in the planning horizon. A maximum reward value was specified to prevent the setting of excessively high values for long overdue tasks. 5.2. Reducing computational complexity The instances encountered in this maintenance service scheduling problem are significantly larger than most VRP-related problems attempted in the literature (rarely with more than 150 vertices). To address a problem of this magnitude, it is necessary to reduce the computational effort required by the tabu search heuristic. The RVI procedure and vertex addition described in Section 3.4 are applied only to the best j solutions selected in Step 3B of the tabu search to achieve this reduction. Preliminary numerical experiments were conducted on a set of benchmark problems for the static MTMCP to evaluate the impact of reducing the number of solutions in which such improvements are attempted. Results of these experiments indicate that this implementation can lead to significant computational savings with only a slight degradation in solution quality. This concept of selective implementation of the improvement procedures has potential use in other metaheuristics, in which a significant portion of computing effort is devoted to the improvement of candidate solutions. 5.3. Experimental results The experimental results on this maintenance scheduling problem are reported in Tables 1–4 presented in Sections 5.3.1–5.3.4. Some column headings used within these tables are defined as follows: n the number of tasks; m the number of days in the planning horizon; TABU results for the tabu search procedure; Table 1 Parameter sets and solution summary Parameter

k

a

w

b

Best_M1

Best_M2

Best_M3

Best_M4

Best_M5

Total

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

3 3 3 3 7 7 7 10 10 10 3 3 7 7 10 10

25 50 100 25 25 50 100 10 20 40 25 50 25 50 10 20

1500 1500 1500 3000 1500 1500 1500 3000 3000 3000 1500 1500 1500 1500 3000 3000

n n n n n n n n n n 2n 2n 2n 2n 2n 2n

4

4

2

1

1

12

2

2

3

3

2

12

1 1

1

2 1

1

1 1

1

1

1

604

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

Table 2 Results for problem instances of the first technician n

m

TABU REWARD

BOUND

TABU/BOUND (%)

TIME

1276

5 10

7667.47 14 064.30

170.6 250.4

8035.79 14 987.12

95.4 93.8

1754

5 10

9247.26 16 188.80

370.8 747.1

9547.61 17 710.11

96.9 91.4

2344

5 10

11 007.90 17 308.20

802.0 2062.9

11 649.73 18 255.52

94.5 94.8

Avg

94.5

BOUND

TABU/BOUND (%)

Table 3 Results for problem instances of the second technician n

m

TABU REWARD

TIME

1120

5 10

7045.83 13 105.90

141.4 223.2

7818.82 14 451.05

90.1 90.7

1689

5 10

8220.26 13 903.30

443.3 565.8

9046.48 15 181.41

90.9 91.6

2315

5 10

8914.19 14 529.20

731.7 1002.3

9350.96 15 301.50

95.3 95.0

Avg

92.3

Table 4 Results for the rolling horizon framework Week

One-week schedule Reward

Two-week schedule c.p.u.

1 2 3 4 5 6 7 8 9 10 11 12

11 007.90 7601.01 6744.78 5608.13 5563.22 4914.44 4526.80 4176.18 4426.99 4978.55 5053.97 5141.31

800.7 662.2 704.1 1051.5 727.7 1264.5 1313.1 1103.3 932.0 1625.7 1003.2 961.9

Total

69 743.28

12 149.9

Three-week schedule

Reward

c.p.u.

Reward

17 176.50

1589.1

19 597.50

2292.0

14 431.40

1414.0 18 383.60

2080.4

13 227.10

1351.0

11 538.20

1659.9

17 444.40

2281.0

11 796.40

1275.2 18 512.70

3935.8

12 901.20

837.5

81 070.80

8126.7

73 938.20

10 589.2

REWARD the total reward of the schedule obtained by the tabu search procedure; TIME c.p.u. time in seconds; BOUND the ADMKSP upper bound for the corresponding MTMCPTD; Avg average results; TABU/BOUND the ratio of the tabu search solution reward divided by the upper bound; AMPTIME the c.p.u. time for the first call to the AMP.

c.p.u.

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

605

5.3.1. Setting heuristic parameters The tabu search heuristic employs a set of parameters whose values need be tuned before the heuristic is run. These parameters include the number of adaptive memory iterations (k), the number of partial solutions stored in adaptive memory (w), penalty factor (g), neighborhood size (b), retrospect length (d), tour duration violation coefficient (e), the number of select candidate solutions (j), and tabu tenure interval [hmin, hmax]. Among these parameters, the setting of parameters k, a, w, and b affect solution quality of the heuristic more significantly than that of the other parameters. Therefore, the parameter settings for k, a, w, and b were systematically tested. Recall that the tabu search heuristic alternates between two neighborhood sizes (i.e., the small and large neighborhood sizes). Only one neighborhood size parameter, b, is employed, which denotes the small neighborhood size. The large neighborhood size is fixed to be 2 · b. Other parameters that less significantly affect the heuristic performance are set based on a number of trial runs. Their values are (d, e, j, hmin, hmax) = (6, 0, 1, 5, 10) for the small neighborhood size, and (d, e, j, hmin, hmax) = (6, 0.02, 1, 6, 12) for the large neighborhood size. For all the experiments conducted in this and successive subsections, it is assumed that duration limit T for each workday is 420 min (7 h), corresponding to an eight-hour workday with a one-hour lunch-break. To assess the impact of employing different parameter settings on solution quality and computation time, a number of different measures were proposed: Reward c:p:u: Reward=Bound 2¼ c:p:u:=minimumðc:p:u:Þ Reward  Bound  0:90 3¼ c:p:u:=minimumðc:p:u:Þ Reward  Bound  0:92 4¼ c:p:u:=minimumðc:p:u:Þ Reward  Bound  0:94 5¼ c:p:u:=minimumðc:p:u:Þ

Measure 1 ¼ Measure Measure Measure Measure

Here, c.p.u. denotes the time required to solve a problem instance with the given set of parameters and minimum (c.p.u.) denotes the minimum c.p.u. time for solving all problem instances for a given set of parameters. These measures are used to evaluate different degrees of trade-off between solution quality and computation time. The higher the value of each measure, the better the performance. Sixteen combinations of the selected parameters on three problem instances (e.g., 1276-task, 1754-task and 2344-task cases for Tech I) for two planning horizons (5- and 10-day schedules) each were tested. The results of these experiments are given in Table 1. The number under each Best_* column indicates the number of times the parameter set resulted in the highest value for measure * of all parameter sets. A blank indicates that the parameter set never led to the best value of the given measure for any problem instance or planning horizon. From Table 1, parameter sets 1 and 4 clearly perform best in terms of trade-offs between solution quality and computation time. Based on a comparison of the solutions and the number of times a parameter set resulted in the best value for each measure, parameter set 4 was chosen. Thus, all remaining experiments employed this set of parameters. 5.3.2. Solution quality for the tabu search In this subsection, solution quality of the tabu search heuristic is examined through experiments on the technician data sets. Results for the problem instances for Tech I are reported in Table 2, where the ‘‘BOUND’’ for each instance was obtained by solving the ADMKSP using the mixed integer solver of CPLEX (version 6.0). Recall that there are three variants of the ADMKSP upper bounds proposed in Section 4. Since in the provided data sets travel time between the depot (which denotes the home of each technician) and the

606

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

location of each first scheduled maintenance task is set to zero (i.e., the paid hours start when each technician arrives at the first scheduled stop), these three upper bounds are equivalent. Hence, only one value is provided for each problem instance under column ‘‘BOUND.’’ As the ADMKSP is NP-hard, upper bounds reported in Table 2 (and also for Table 3) are best upper bounds for the ADMKSP found by the mixed integer solver within approximately 100 c.p.u. seconds. Although only upper bounds were obtained, according to the CPLEX output, these bounds are very close to the optimal solution values for the corresponding ADMKSP problems (generally less than 1% above the optimal). It is important to note that if a solution obtained via the tabu search procedure is within X% of the upper bound, as reported in Table 2 (and Table 3), it is actually better than X% if as compared to the true optimal solution of the maintenance scheduling problem. This is because the optimal solution is likely to have a reward that is lower than the upper bound. In Table 2, two planning horizons were applied to each problem instance, i.e., a planning horizon of a week (five workdays) and a planning horizon of two weeks (10 workdays). Results reported in this table show that the solutions obtained by the proposed tabu search are at least 94.5% optimal on average. In addition, computational times are very reasonable for such large-size problem instances. Additionally, the average result of the largest problem instance (i.e., the original data set with 2344 tasks) is slightly better than those of the instances with reduced size. This may be attributed to the following. First, the ADMKSP upper bounds of the original data set are slightly tighter as compared with smaller-size instances, because inter-task travel times have been underestimated due to clustering. Second, the tabu search heuristic provided high-quality solutions that do not degrade with increasing problem size. Results in Table 2 also indicate that the total reward decreases with the reduction in problem size. However, by reducing problem size, the computation time significantly decreases. The 1754-task case provides a nice compromise, as the total reward (for the 5- and 10-day schedules) reported by the tabu search procedure for the 1754-task problem is approximately 90% of that of the 2344-task case, while the total c.p.u. time is reduced by approximately 60%. Note that the average reward received for each day of the 5-day schedule is higher than that of the 10-day schedule. This is expected, because the majority of tasks included in the 5-day schedule will have relatively high rewards. The 10-day schedule will include, perhaps, the same set of tasks as the 5-day schedule and will fill its remaining days by choosing from a set of tasks with lower average rewards. As described earlier, the heuristic parameters for the proposed tabu search were tuned based on a small number of experiments conducted on the problem instances derived from Tech I. This tabu search procedure (without retuning the parameters) was also used to solve the problem instances for Tech II. Results in Table 3 indicate that the average solution is above 92.3% of optimality. This average performance is acceptable for most applications, particularly given the large problem size and limited required computational effort. 5.3.3. Effects of limiting computation time The tabu search heuristic can be stopped prior to its natural termination (i.e., prior to meeting the pre-set stopping criteria) and the best known solution can be output. As with many heuristic procedures, a solution that is very close to optimal may be found very early in the process and a great deal of computation time may be used in searching for solutions that are only slightly better than this solution in succeeding iterations. In this subsection, the effect on solution quality of limiting the tabu search computation time is examined. Specifically, the tabu search procedure is halted after a number of different c.p.u. time limits and solution values are reported. These values were then compared with the best solution found at the natural termination of the heuristic. This analysis is of particular interest in real-world practice as maintenance scheduling needs to be conducted for many technicians and, as such, it is unrealistic to assume that a natural termination of the tabu search for scheduling each of the technicians can be achieved in reasonable time. The problem instance with n = 1754 for Tech I is used for this set of experiments. The results are presented in Fig. 5, where solutions of a 5-day and a 10-day planning horizon are compared when the tabu search procedure is forced to terminate at specific times after an initial solution has been constructed in the AMP. It was noted that the AMP required approximately 40 c.p.u. seconds for each case. In this figure, solution ratios are percentage values of solutions output at pre-specified times divided by solutions obtained after a natural termination of the tabu search. These results indicate that for the 5-day

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

607

100% 400 secs

Solution Ratio

98% 96% 94% 92%

50 secs 1 sec

100 secs

200 secs

5 secs

5-day planning 10-day planning

90%

Fig. 5. Effects of limiting computation time.

planning horizon, the solution obtained in only one c.p.u. second after the AMP is first run is equivalent to the best solution found at the procedure’s natural termination, i.e., found after approximately 330 c.p.u. seconds after the first run of the AMP. Similarly, the best solution found for the 10-day planning horizon was achieved after approximately 400 c.p.u. seconds after the AMP, a little more than half of the total 747 c.p.u. seconds required for natural termination. Moreover, very good solutions were found with significantly less computation time. For example, the resulting solution for the 10-day planning horizon was within 96% of the best solution achieved after natural termination of the tabu search heuristic only five c.p.u. seconds after the AMP. Based on limited testing, these results indicate that very good solutions can often be attained in a small portion of the computation time required for natural termination. 5.3.4. Schedules based on a rolling horizon framework The proposed experiments described thus far address the short-term scheduling problem, assuming that each maintenance task will be assigned only once during the scheduling time horizon. However, if the time horizon is sufficiently large, there may be tasks with revisit interval durations short enough to warrant more than one service visit. This consideration of repeating maintenance procedures may be substantial in the longterm scheduling problem. Thus, it is of particular interest to consider scheduling of maintenance tasks over a relatively long time horizon, where the time-dependent reward for each task when scheduled the second or higher time is conditioned on the last time the task was scheduled. Upon choosing which day the task will first be assigned, the reward for this task can be updated and the task can be re-entered in the pool of tasks to consider for future scheduling. To solve this long-term scheduling problem, a rolling horizon approach is devised in which the short-term problem is solved repeatedly given appropriate updates to the task rewards. The approach begins by discretizing the (long-term) time horizon into a set of equi-duration sub-intervals of time (e.g., T1, T2, . . .). A series of smaller problems is then solved in increasing order of time. Here, the duration of each discrete sub-interval of time is referred to as the roll period. Once the problem is solved for time horizon T1, the rewards for all tasks assigned in T1 are reset and these tasks are then reconsidered in T2. T2 is then solved and the same resetting of task rewards could be completed for T3, etc. When time horizon Tk is considered, all assignments in Tk1 are set permanently. This process is justified as follows. Since the expected revisit interval for each task is determined only by its equipment type (assuming that the same type of maintenance service is provided for the same task on each visit) and its time-dependent reward is reset after each visit, the maintenance service needed for each task can be associated with a stationary renewal process. Given this interpretation, an optimal strategy for providing maintenance to each task would be to schedule the next visit at a constant interval after any given maintenance visit, assuming such an ideal schedule is feasible. This implies that an equi-duration periodic schedule could be appropriate given time-dependent task rewards that reflect the level of urgency. A planning horizon of 12 weeks is considered in the experiments in which three roll periods are examined: one, two, and three weeks. Once the tasks have been scheduled within a roll period, the time-dependent rewards for successive roll periods within the planning horizon are recomputed based on the currently scheduled date. The problem instance with n = 2344 for Tech I is used for this set of experiments. In Table 4 the total reward received within each roll period and the required computation time are given. The total reward for the entire 12-week horizon is reported in the last row of the table. For example, at the end

608

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

of the third week, the total reward for the three-week schedule is 19 597.5. The reward for the second threeweek schedule, for weeks four through six, is 18 383.6. The total reward achieved with the three-week roll period over the 12-week horizon is 73 938.2. From Table 4, one can see that the best overall reward achieved within the 12-week planning horizon employed a two-week roll period. Further, this result was achieved in the lowest total computation time of all three roll periods tested. The fact that the two-week schedule generates higher rewards than the one- and three-week schedules indicates that there are trade-offs between employing longer and shorter roll periods. It was expected that as the number of weeks in the roll period increases, the total reward would decrease, because repeating tasks are not available for scheduling as often as they might be in shorter roll periods. For example, a task that may need to be performed multiple times during a roll period can only be considered for scheduling once during that period. Thus, scheduling over a short roll period (e.g., one week) can reduce the adverse effect of the no-repeating task assumption. On the other hand, if a short roll period is maintained, fewer combinations of task assignments can be considered simultaneously than would be considered in a longer roll period. The schedules of shorter roll periods are myopic relative to those of longer roll periods. It is observed that a decrease of total reward occurred in moving from a roll period of two weeks to a roll period of three weeks, but the same did not occur in moving from a roll period of one week to a roll period of two weeks. These results indicate that the proposed rolling horizon technique for the long-term planning problem is promising. Other factors, including heuristic solution quality and time-dependent reward resetting, may also affect the results of the rolling horizon experiments and might be further investigated in implementing such an approach. 6. Conclusions In this paper, a real-world planned maintenance scheduling problem that exists at UTC is considered. The primary contributions of this work are a formulation, the MTMCPTD, to model this problem and a tabu search-based heuristic embedded in an adaptive memory procedure (AMP) devised for its solution. Numerical experiments conducted on the test data sets for the real-world application show that the heuristic is able to provide near optimal solutions for actual large-size problems within reasonable computation time. In this tabu search heuristic, the AMP and the mechanism for updating the set of solutions maintained within the AMP allow a relatively large pool of good and diversified solutions to be stored and utilized during the search process. By alternating between small and large sizes of neighborhood stages during the tabu search, the heuristic evolves efficiently without loss of solution quality. Both random and greedy procedures are employed in neighborhood solution exploration. These features are largely responsible for the efficiency and effectiveness of this tabu search procedure in the computational experiments. Several upper bounds for the MTMCPTD based on a variant of the multiple knapsack problem are proposed. These upper bounds appear to be tight for the data sets tested in the experiments. A rolling horizon approach for allowing tasks to be scheduled more than once in a long-term planning horizon is also examined in this paper. In this approach, the planning horizon is divided into planning periods and the MTMCPTD is solved within each period. Task rewards are updated at the end of each period. In addition to providing solutions for the service scheduling problem over a long-term planning horizon, experiments with varying length roll periods can also provide insight into the preferred roll period duration. Due to the complexity of this real-world application, several issues identified in the numerical experiments may require further investigation. For example, the multiple knapsack problem based upper bounds may not be tight for MTMCPTD instances where travel times are more significant than in UTC’s operations. In addition, it was observed that heuristic solution quality and time-dependent reward resetting may affect the results of the rolling horizon technique for the long-term scheduling problem, and the heuristic may cherry pick highreward tasks when a short roll period is employed (e.g., one week). These topics are the subject of future research by the authors. Rescheduling in the presence of emergency calls is a subject of ongoing research by the authors.

H. Tang et al. / Transportation Research Part E 43 (2007) 591–609

609

Acknowledgements The authors are grateful to Dr. Arthur Hsu from the United Technologies Research Center, who provided valuable comments to an earlier version of the paper and suggested a similar upper bound to that of Proposition 2. This work was supported by United Technologies Corporation and NSF grant CMS 0350211. This support is gratefully acknowledged but implies no endorsement of the findings. References Butt, S., Cavalier, T., 1994. A heuristic for the multiple tour maximum collection problem. Computers and Operations Research 21, 101– 111. Butt, S., Ryan, D., 1999. An optimal solution procedure for the multiple tour maximum collection problem using column generation. Computer and Operations Research 26, 427–441. Chao, I., Golden, B., Wasil, E., 1996a. The team orienteering problem. European Journal of Operational Research 88, 464–474. Chao, I., Golden, B., Wasil, E., 1996b. A fast and effective heuristic for the orienteering problem. European Journal of Operational Research 88, 475–489. Dueck, G., Scheuer, T., 1990. Threshold accepting: a general purpose optimization algorithm appearing superior to simulated annealing. Journal of Computational Physics 90, 161–175. Erkut, E., Zhang, J., 1996. On the maximum collection problem with time-dependent rewards. Naval Research Logistics 43, 749–763. Fischetti, M., Salazar, J., Toth, P., 1998. Solving the orienteering problem through branch-and-cut. INFORMS Journal on Computing 10, 33–148. Gendreau, M., Hertz, A., Laporte, G., 1994. A tabu search heuristic for the vehicle routing problem. Management Science 40, 1276–1290. Gendreau, M., Laporte, G., Semet, F., 1998a. A tabu search heuristic for the undirected selective traveling salesman problem. European Journal of Operational Research 106, 539–545. Gendreau, M., Laporte, G., Semet, F., 1998b. A branch-and-cut algorithm for the undirected selective traveling salesman problem. Networks 32, 263–273. ´ ., 1999. A tabu search heuristic for the heterogeneous fleet vehicle routing Gendreau, M., Laporte, G., Musaraganyi, C., Taillard, E problem. Computers and Operations Research 26, 1153–1173. Golden, B., Levy, L., Vohra, R., 1987. The orienteering problem. Naval Research Logistics 34, 307–318. Golden, B., Wang, Q., Liu, L., 1988. A multifaceted heuristic for the orienteering problem. Naval Research Logistics 35, 359–366. Golden, B., Laporte, G., Taillard, E., 1997. An adaptive memory heuristic for a class of vehicle routing problems with minmax objective. Computers and Operations Research 24, 445–452. Hart, J., Shogan, A., 1987. Semi-greedy heuristics: an empirical study. Operations Research Letters 6, 107–114. Laporte, G., Martello, S., 1990. The selective traveling salesman problem. Discrete Applied Mathematics 26, 193–207. Lin, S., Kernighan, B., 1973. An effective heuristic algorithm for the traveling salesman problem. Operations Research 21, 498–516. Martello, S., Toth, P., 1990. Knapsack Problems: Algorithms and Computer Implementations. John Wiley and Sons, Chichester. Mladenovic´, N., Hansen, P., 1997. Variable neighborhood search. Computers and Operations Research 24, 1097–1110. Ramesh, R., Brown, K., 1991. An efficient four-phase heuristic for the generalized orienteering problem. Computers and Operations Research 18, 151–165. Ramesh, R., Yoon, Y., Karwan, M., 1992. An optimal algorithm for the orienteering tour problem. ORSA Journal on Computing 4, 155– 165. Rochat, Y., Taillard, E´., 1995. Probabilistic diversification and intensification in local search for vehicle routing. Journal of Heuristics 1, 147–167. Savelsbergh, M., 2003. The inventory routing problem. Working Paper, Georgia Institute of Technology. Tang, H., Miller-Hooks, E., 2005. A tabu search heuristic for the team orienteering problem. Computers and Operations Research 32, 1379–1407. Tsiligirides, T., 1984. Heuristic methods applied to orienteering. Journal of the Operational Research Society 35, 797–809.