A simultaneous sequencing and allocation problem for military pilot training: Integer programming approaches

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Highlights

  • This paper presents a number of formulations for the scheduling of pilot training.

  • Two MILP models are polynomial in size, one has a quadrative objective function.

  • The third MILP model which integrates a modified Knuth’s Dancing Link Algorithm.

  • We present a column generation formulation as a heuristic approach.

  • We compare numerical results for all models and approaches mentioned above.

Abstract

In this paper, we study a unique, rich combinatorial optimization problem that arose from helicopter aircrew training for the Royal Australian Navy. Each pilot trainee (student) has to complete a syllabus. A syllabus is a sequence of courses (commonly known as subjects), and each course is associated with a pass rate. A pre-requisite structure exists amongst some courses. Each course has a number of repeated sessions, each spanning the same amount of time, but occupying a different set of (possibly overlapping) time slots. A feasible schedule is a sequence of course sessions such that each course in the syllabus is covered by exactly one session, and that all pre-requisite requirements are observed. The optimization problem is to simultaneously assemble course sessions to form feasible schedules, allocate students to these schedules with an objective to minimize the total time-span in completing the syllabus, while ensuring that the class size limits for each course session is not exceeded. The problem is different from the school or university time tabling family of problems due to the assembly component required. This paper is to serve as a pilot study: we derive a number of mixed-integer linear programming models and investigate their performance using test instances provided for us by our industry partner. For each of these models, we propose a number of solution strategies as topics for future research papers. From our numerical testing, it appears that the Column Generation-based approach is computationally the most promising method.

Introduction

In this paper, we study a complex simultaneous assembly and allocation problem that arose from the scheduling of classes for the Royal Australian Navy pilot trainees. Each successful pilot trainee student has to complete a syllabus. A syllabus is a sequence of courses (i.e., subjects). Some of the courses are pre-requisites of others. Each course has a number of repeated sessions with the same time span but occupying a different set of (possibly overlapping) time slots. Each course sessions has a class size limit. A student will be required to do no more than one session for each course. There is a pass rate associated with each course. The pass rates are obtained from historical data and can be represented as a distribution. To graduate, a student must pass all courses, and there is no opportunity to retake a course if he or she fails it.

A feasible schedule is a sequence of course sessions such that each course in the syllabus is covered by exactly one session, and that all pre-requisite requirements and class size limits are satisfied. The optimisation problem is to simultaneously assemble course sessions to form feasible schedules, and to allocate students to these schedules with an objective to minimize the total time-span in completing the syllabus over all students. Students enter the program at different times, so the start time of a schedule allocated to a student must be no earlier than the time the student enters the program. In what follows, we refer to this problem as the Simultaneous Sequencing and Allocation Problem (SSAP). There is a significant difference between the SSAP and the school and university timetabling family of problems due to the absence of the assembly component in the latter, and similarly, a significant difference between the SSAP and the Generalized Vehicle Routing Problem with Time-windows (GVRPTW) due to the existence of pass rate in the SSAP.

A Toy Example. Table 1 presents an example with 6 courses and 13 course sessions in total, where the start and end days (times) are listed in the last two columns. The notation i,j represent the jth session of Course i.

In this example, 1,1 2,1 3,2 4,2 5,1 7,2 is a time and pre-requisite feasible schedule, and so is 1,1 3,1 2,2 4,3 5,2 7,2. So for each feasible schedule, the decision making problem is not only to determine the sequence of appearance of the courses in each schedule, but also for each course, which session is to be taken. The SSAP is to simultaneously generate feasible schedules and allocate them to students so that each student is allocated a schedule with all class size constraints satisfied, and that the total span for all students is minimized. There are a number of different ways the total span is determined, which we discuss in detail in Section 2.

We now describe the SSAP mathematically. Consider a directed graph D=(V,A), where V represent all the course sessions, with V partitioned to k clusters, (i.e., V={V1,,Vk}, for Vi={vi,1,,vi,|Vi|},i=1,,k). In the context of an SSAP, each cluster Vi represents a course, and the vertices vi,j in the cluster represent a session of the course. Each vertex has a capacity of Ni,j and a time-window [σi,j,i,j] associated with it, for σi,j and i,j the start time and end time respectively. The time-window for all vertices in the same cluster have the same span though the actual time-windows are different and may overlap. Precedence relations exist between some clusters. A feasible path, i.e., a sequence of visits to each cluster exactly once, to exactly one vertex in each cluster without inducing any time-clashes) represents a feasible schedule. This is in some ways similar to a Generalized Vehicle Routing Problem with Time-windows (GVRPTW), however there are substantial differences.

If we describe the SSAP using the GVRPTW terminology, we are considering a set of “vehicles”, each entering the journey at a different time, and each allocated a feasible path. It is possible that the same path be allocated to more than one vehicle. A vertex vi,j may be visited by different vehicles, but the capacity of the vertex Mi,j must not be exceeded. There may be a wait between the time a vehicle enters the journey and the start time of the first visit in the path. Not all vehicles finish the entire path. Some may exit at a particular vertex, and the exit rate for each cluster may be different though the same exit rate is filtered through all vertices in the same cluster. The exit rate of a vertex may vary from time to time but it follows a distribution. The optimization problem is to find the allocation such that the sum of span for travelling time of all vehicles is minimized.

Complex scheduling and timetabling problems exist in a large number of domains, e.g., nurse rostering in healthcare, (see, e.g., (Bellanti, Carello, Croce, & Tadei, 2004)), train timetabling in transportation, (see, e.g., Barrena, Canca, Coelho, & Laporte, 2014), school and university timetabling, (see, e.g., Babaei et al., 2015, Burke et al., 1997, Burke and Petrovic, 2002, Lewis, 2008, MirHassani and Habibi, 2013, Schaerf, 1999), scheduling hospital trainees (see, e.g., Beliën & Demeulemeester, 2006), military aircraft fleet management, (see, e.g., Looker, Mak-Hau, & Marlow, 2017), to name but a few. From the 1980s, one can easily find a number of studies that are based on graphs and networks. E.g., certain timetabling problems were reduced to graph colouring problems, (see, e.g., de Werra, 1985). The Korea Army Training Center has used an edge-colouring based heuristic to generate “good” schedules in a reasonably short amount of computation time (see, e.g., Lee, Kim, & Lee, 2009). Integer programming models were developed since the late 1980s, however, they were computationally expensive at the time, prompting further research for solution methodologies for real-world timetabling problems, see, e.g., (Burke et al., 2005, Carter, 1989, Costa, 1994). Heuristics approaches has been used in timetabling as well as scheduling family of problems, e.g., job shop scheduling (Battiti and Tecchiolli, 1994, Dell’Amico and Trubian, 1993, Glover and Laguna, 1997, Hertz and Widmer, 1996, Nowicki and Smutnicki, 1996). Meta-heuristic algorithms, such as Tabu Search, Simulated Annealing, and evolutionary algorithms, their hybrids and hyper-heuristics are recent successes in scheduling problems in various domains (Pillay, 2016, Teoh et al., 2015). In vehicle routing family problems, recent research focused on complex “rich” optimization problems, see, e.g., (Lam & Hentenryck, Jul 2016) wherein a branch-and-price-and-check algorithm is proposed. However, we are not aware of any active research in the cases when vehicles are allowed to “disappear” midway through their allocated routes.

In the education domain, the majority of work in the literature are for examination timetabling and course timetabling. The latter is further catergorized into post-enrolment and curriculum-based timetabling (Bettinelli et al., 2015, Müller and Rudová, 2016, Rudová et al., 2011). Examination timetabling problems concern the scheduling of non-recurring events (examinations). On the other hand, course timetabling is focused on scheduling lectures, tutorials, etc that recurs on a weekly basis. In post-enrolment course timetabling, classes are scheduled after students have enrolled whereas in curriculum-based course timetabling, the schedule of classes are defined before enrolment. Research in the area of exam timetabling can be found in, e.g., (Burke & Petrovic, 2002), wherein a graph-based hyper-heuristic method is proposed and a thorough review of past meta-heuristics methods is provided; and (Merlot, Boland, Hughes, & Stuckey, 2003) wherein a hybrid constraint programming and simulated annealing approach is presented. University course tabling, on the other hand, can be found in, e.g., (Boland et al., 2008, Sabar et al., 2012). The latter implemented a honey-bee optimisation algorithm.

Applications of Operations Research (OR) techniques in military training have been used to explore policy effects, bottlenecks and resource utilisation (Davenport et al., 2007, Nguyen et al., 2016, Novak et al., 2015, Séguin and Hunter, 2013). A review of system dynamics simulations in workforce planning can be found in Wang (2005). Yang and Ignizio (1987) addressed a class of military training-related scheduling problem where heuristic algorithms were proposed for a peacetime scheduling of training activities of battalions where decisions must be made regarding the kind of training tasks to be performed, by which battalion, at what time, and the objective is to minimize the makespan. The heuristic algorithm proposed is a two-phase one, with a greedy heuristic employed to generate a solution in Phase 1, an exchange heuristic is used to improve the solution in Phase 2. In military and other aircrew training applications, heuristic methods can be found in, e.g., Lee et al. (2009), whilst hybrid heuristic-integer programming were employed to schedule examination timetables for cadets in, (see, e.g., Wang, Bussieck, Guignard, Meeraus, & O’Brien, 2010). In scheduling army battalion training exercises, a two-phase heuristic is proposed in Yang and Ignizio (1987), wherein the first phase is to find a feasible solution, and then an exchange heuristic is applied to improve the cost in the second phase, starting from this feasible solution obtained. (Scott, 2005) applies integer programming to produce multi-year schedules at the Defence Language Institute. In Qi, Bard, and Yu (2004), a Branch and bound along with a family of heuristics is applied to the scheduling of the retraining of Continental Airlines pilots. Raffensperger and Schrage (2008), on the other hand, provides a measure of readiness for training in a tank battalion and used a hybrid dynamic programming and column generation approach. A resource scheduling problem for the US Army’s basic combat training program is presented in McGinnis and Fernandez-Gaucherand (1994). The decision problem was formulated as a Dynamic Programming problem, however, to solve problems of real-world sizes. A 3-phase heuristic approach was proposed, which Phase 1 deriving an initial feasible training resource schedule, and if necessary, in Phase 2, a change in the level of resources is made, and in Phase 3, an improvement heuristic is applied. In Brown, Carlyle, Dell, and Brau (2013), an intratheatre military airlift problem was presented. The problem has some strong similarity to the problem under study in this paper. The methodology presented in the paper is very interesting–an ILP model is solved, with variables representing paths that can be described on a directed graph, and the ILP is to allocate passengers to these path variables. The variables are generated by an interesting stack iterative algorithm. In other military related recent ILP-based work, Squires and Hoffman (2015) solves a military maintenance planning and scheduling problem, wherein the ILPs are solved using Benders decomposition.

A resource allocation problem for fighter squadron training has been studied in Vestli, Lundsbakken, Fagerholt, and Hvattum (2015), and a column generation approach is proposed. The nature of the optimization problem solved therein, however, is fundamentally different. There is no schedule assembly required. The decision problem is to allocate classes into time slots, (whereas the SSAP has course-sessions in given time slots). Most constraints discussed in the paper are Knapsack-type constraints. The column generation subproblem has each column as a feasible fighter duty or a simulator duty, and a duty describes all the training to be performed by a single pilot over the planning horizon. Heuristic pricing was experimented.

In any case, the scheduling and timetabling problems reviewed above are very different in problem definition when compared to the SSAP which arise from a military navy pilot training scheduling problem where the sequence of the curriculum for each student can be different and is also a part of the optimization problem. For aircrew training a significant issue is the length of the overall training process. Student morale can be affected if students spend a significant amount of time waiting between courses. A long waiting time will also induce a waste in financial terms, as students in Australian Defence are paid salaries throughout these waiting periods. Accordingly, in this work we adopt the simple view that the cost of a schedule for an individual student is the total time (makespan) either from the start date of the first course (or the day a student enters the program) to the end date of the last course. In Yu, Pachon, Thengvall, Darryal, and Al (2004), a commercial airline pilot planning and training optimization problem is studied where two MILP models were presented. The first MIP was to solve the Phase 1 pilot-transitioning problem, where the output of the decision problem consists of classes to be scheduled, the pilots in each class, and their associated training curricula, and the available training resources. Unlike the SSAP, there is no schedule assembly required. The second MILP was to solve the Phase 2 training-class-scheduling problem which is an assignment problem–assigning training resources to classes, with three sets of simple knapsack and assignment constraints, and a set of flow balance constraints. No pass rate is considered. A rolling-horizon approach was adopted. For example, if there were 10 classes to schedule, the first three classes together. Then, the schedule of the first class is fixed, and classes two through four are scheduled together afterwards. This process is repeated until all 10 classes are scheduled. Commercial solver was used in obtaining the solution. The economic benefit of the optimization was significant. Our problem is much more complex in the way that the curricula for the pilot students are to be determined simultaneously, though we are not required to allocate resources.

To the best of our knowledge, there are very few existing work closely related to the pilot training scheduling problem under study in this paper. The only existing work (see Nguyen et al., 2018) is a complete enumeration of feasible schedules using Knuth’s Dancing Link by formulating the problem as an Exact Cover Problem, followed by assigning students to these schedules using a MILP model.

This paper serves as a pilot study and the first of a series of Mathematical Programming methods for the SSAP. We develop a number of mixed-integer linear programming (MILP) models, analyse their performance, and propose exact and heuristic methodologies based on these MILP models. Given that the pass rates for the courses are stochastic in nature, the solution methodologies we develop in this paper are for solving instantaneous problem instances with pass rates generated by simulation. The idea is, by executing simulation and optimisation iteratively for a large number of times, we are able to obtain important insights as to how the total span changes with the variations in pass rates, and can therefore perform a thorough capacity analysis for the Royal Australian Navy. Hence, it is crucial to develop a scalable and fast solution methodology, and the insights we obtain from this paper will be used to guide research directions for future papers in this series.

An outline of the rest of the paper is as follows. In Section 2, we present the first two MILP models, both are polynomial in size and are based on directed graphs with vertices representing courses and arcs representing possible connections between two courses. The first ILP model uses binary variables to assemble schedules and each student is presented by a schedule. The second MILP model uses general integer variables to represent the number of students assigned to each schedule, hence the objective function is quadratic as both the span and the number of students allocated to a schedule are decision variables. In Section 3, we present the third integer programming model–a very simple model that uses a binary variable for each schedule. However, these schedules must be generated in advance. In Nguyen et al. (2018), an approach based on a modified Knuth’s Dancing Link Algorithm (RDLX) for Exact Covering is used to generate feasible schedules. In this paper, we provide a number of new insights on the RDLX. When solving larger problems, however, RDLX fails to generate all feasible schedules quickly enough, so we attempted a column generation approach in Section 3. It appears that for most test instances, the Linear Programming-relaxation provides naturally integer solutions, and we will explain why this is the case. For cases where the LP-relaxation did not produce an integer solution, we use column generation as a heuristic approach. We present computational comparisons for the various integer programming models throughout the paper for the convenience of the readers. We conclude our findings in Section 4, and discuss future research directions in Section 5.

Section snippets

Two polynomial-size MILPs that simultaneously assemble and allocation schedules

In this section we introduce two polynomial-size MILPs. First we introduce the notation used throughout the paper (see Table 2).

Notice that some course sessions can be eliminated in pre-processing if i,jσi,j,jTCi,iτ(Ci) or σi,ji,j,jTi,iρ(Ci). The complexity for a single iteration of elimination routine given above is at worst O(nC2), however, further elimination may be performed in an iterative manner. In any case, these cases are rare in our dataset so we did not

An alternative MILP formulation

In this section, we present a third formulation that is exponential in size with each decision variable representing a feasible schedule. In Nguyen et al. (2018), we formulated the feasible-schedule generation subproblem as a Generalised Exact Cover Problem (GECP)–instead of finding just one exact cover for all columns (column of an Exact Cover table, not a decision variable in this MILP), we are required to find all feasible exact covers for a particular subset of columns. One can of course

Conclusions

In this paper, we studied a simultaneous schedule assembly and allocation problem for the Royal Australian Navy pilot training. The problem is fundamentally different from the school or university time tabling family of problems due to the assembly component required and is also fundamentally different from the Generalized Vehicle Routing-Family problems due to the complexities induced by the pass rates and that the capacity constraints are on vertices (rather than on vehicles). The

Future research directions

We now discuss future research directions. First of all, with the column generation approach, instead of solving the pricing subproblem given in Model 4 exactly, we can derive and implement a rapid heuristic pricing algorithm, and only invoke the exact pricer when the heuristic pricer cannot find a new column.

Secondly, to implement a branch-and-price approach for exact solutions for larger scale problems, a specialised branching rule is required to be developed so as to avoid re-generating

CRediT authorship contribution statement

Vicky Mak-Hau: Conceptualization, Formal analysis, Investigation, Methodology, Supervision, Validation, Writing - original draft, Writing - review & editing, Brendan Hill: Resources, Software, Conceptualization, Data curation, Investigation. David Kirszenblat: Resources, Software, Conceptualization, Data curation, Investigation. Bill Moran: Resources, Software, Conceptualization, Data curation, Investigation. Vivian Nguyen: Resources, Software, Conceptualization, Data curation, Investigation.

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