Modelling and Solving Fleet Assignment in a. Flexible Environment. Hani El Sakkout. IC-Parc, Imperial College, London SW7 2AZ.

Transcription

1 Modelling and Solving Fleet Assignment in a Flexible Environment Hani El Sakkout IC-Parc, Imperial College, London SW7 2AZ hhe@doc.ic.ac.uk Keywords Combinatorial Optimization, Constraint Satisfaction, Finite Domain Propagation, Constraint Repair, Fleet Assignment Abstract This paper reports on the modelling and solution of a large industrial combinatorial optimization problem using the ECL i PS e Constraint Logic Programming (CLP) platform. The problem is eet assignment: the assignment of a set of aircraft of dierent types to a predened shedule of ights. A single underlying model was developed and solved in ECL i PS e. Three optimization methods were applied in three stages of model development and solution. The three methods were used to meet dierent objectives. The rst was depth rst search branch and bound optimization in conjunction with Finite Domain propagation. This straightforward approach was used to develop and validate the model in the ECL i PS e platform. The second was mixed integer programming. The CLP model and solutions were exported to a mathematical solver and used to produce a globally optimal solution. Knowing the optimal solution is desirable, however obtaining good solutions close to the user's actual assignment is more useful in practice. The third method used a repair-based extension of the rst approach, and obtained high quality solutions close to the user's assignment. This multistage devolopment process resulted in the end user commissioning the development of an ECL i PS e -based application combining repair with Finite Domain propagation, with the purpose of improving their day to day eet assignment process. The exercise clearly reveals the exibility of the ECL i PS e platform in mapping a single model to dierent problem solving approaches, and illustrates how this exibility has important practical benets. 1

2 1 Introduction 1.1 Paper Outline This paper describes our experience in solving a large industrial eet assignment problem using the ECL i PS e Constraint Logic Programming platform. Three alternative solution strategies were used at dierent stages of the project to achieve dierent objectives. The British Airways (BA) shorthaul eet assignment requires the feasible allocation of aircraft of various types to a xed set of trips. Solutions to this problem must also satisfy a set of maintenance constraints and minimise a cost function. This large constraint satisfaction problem involves over 10,000 constraints and 8000 variables. The rst solution method involved depth rst search search in conjunction with a Finite Domain (FD) solver (Section 2). This approach was used to develop, verify and enhance a model for the problem. This model is formalised in Sections 1.3 and 1.4. In the second stage of the project, the model and solutions obtained in the rst stage were exported to a mixed integer solver. The globally optimal solution was obtained; Section 3 briey discusses this method. The global optimal is useful since it provides a measure against which other solutions are tested, but it is more useful in practice to have good quality solutions that minimise disruption to the user's current eet assignment. The third method used a repair hybrid based on the rst approach (Section 4). Knowledge of the problem structure obtained in the rst stage of the project was found to be invaluable. This method produced good quality solutions that were close to the user's current eet assignment. This multistage development process convinced the user to commission the development of a practical ECL i PS e -based repair hybrid based on the third method. The exercise demonstrates the exibility and practicality of the ECL i PS e platform in the modelling and solution of a large scale industrial problem. The implications of sharing the same model among a pure FD propagation implementation, mixed integer programming and a repairbased hybrid are discussed, and some conclusions are drawn in Section Background References The concept of arc consistency was introduced in [Mac77]; its combination with backtrack search was described in [HE80]; the notion of value propagation is due to [SS80]; the application of constraint methods to real arithmetic was surveyed in [Dav87]; and [Van89] describes in detail the integration of Finite Domain (FD) propagation methods into Logic Programming to produce FD Constraint Logic Programming (CLP) [Kow79, JL87]. ECL i PS e is a exible platform based originally on FD CLP and extended to facilitate incorporation of a number of other constraint solvers [ECR95].

3 1.3 The Fleet Assignment Problem Problem Description Fleet assignment involves the assignment of aircraft types to a xed set of ights. Each ight departs from a specied origin station at a specied time. The destination station is also xed, but the time of arrival is dependent on the aircraft type chosen. Naturally, the choice of aircraft type for a particular ight is limited to those available at the departure station at the time of departure. Requirements dictate that a certain percentage of each aircraft type must start and end the day at an airport capable of carrying out maintenance. The cost of each ight also depends on the type of aircraft chosen. The global cost function is dened as the sum of the ight costs. As explained below, detail of the user's original problem that may handled by a preprocessing stage has not been included in the model since it is not relevant to the search and optimization process The Problem Components A set of stations s 2 S The total number of aircraft overnighting at each station on(s) 2 N is given as part of the problem specication. The relation maint(s) holds i s is a maintenance station. A set of plane types p 2 P Each plane type is associated with a eet fs(p), and a required nightly maintenance percentage mpc(p). A set of ight legs l 2 L Each leg has an origin o(l) and a destination d(l), as well as a xed local departure time t dep (l) 2 T. A choice of assignments relation C : (L P ) This relation denes the space of possible assignment pairs (l 2 L; p 2 P ) of plane types to legs. Two functions are dened for each pair (l; p) 2 C : a cost c : LP! N, and the local time t ready : L P! T after which an aircraft of type p is ready for a subsequent departure from d(l) Feasibility and Optimality The task of solving the eet assignment problem is essentially to assign a particular plane type to each leg. Let each leg l 2 L be associated with a plane type variable pt l whose domain is dened as follows: dom(pt l ) = fp : (l; p) 2 Cg. Let each pair (s; p) be associated with an overnight variable on sp signifying the number of aircraft of plane type p overnighting at s, and the set of all such variables be ON.

4 A feasible solution is an assignment to all the plane type variables fpt l : l 2 Lg such that 9ON 8>< >: where delta(s; p; t) = (i) 8s : p2p on sp = on(s) (ii) P 8p : s2s on sp = fs(p) (iii) 8p : s2s^maint(s) on sp mpc(p) fs(p) (iv) 8s; p : jfl : o(l) = s ^ pt l = pgj = jfl : d(l) = s ^ pt l = pgj (v) 8s; p; t : on sp delta(s; p; t) l : pt l = p o(l) = s t dep (l) t? l : pt l = p d(l) = s t ready (l) t. An optimal solution is a feasible solution minimising the global cost function P l c(l; pt l ). 1.4 A Practical Model The problem formulation given above is not well suited to direct encoding of the constraints. Specically, constraints (iv) and (v) cannot be used for ecient constraint handling, as explained below. Constraint (iv) requires, for each station, the daily quantity of outgoing aircraft of each plane type to be equal to the daily quantity of incoming aircraft of the same type. Constraint (v) requires these quantities to be indexed to particular time points. To address this problem two sets of peripheral variables are dened: Boolean plane type variables, and Delta variables. This section introduces the more practical representation of the constraints that is common to all three methods presented here Boolean Plane Type Variables Since both constraints need a notion of plane type quantity, which is dependent on the assignment of plane type variables, it is necessary to introduce Boolean variables (in Binary representation). These are dened as follows: 8l; p : b lp = 1, pt l = p Constraints on quantities of aircraft of particular plane types are then dened in terms of these booleans directly Delta Variables In the more abstract specication above, constraint (v) as stated implies the computation of the dierence between departures and arrivals up to every time point in the day for each station and plane type pair (this value is designated delta(s; p; t) in the problem formulation), in order that the maximum may be identied. 1 An enhanced representation is more 1 It would be more precise to speak about ready times instead of arrival times. An aircraft must remain out of action for a minimum time in between an arrival and a subsequent departure.

5 practical and ecient; changes only occur to the delta(s; p; t) values after departure and arrival time-points from s. In addition, since the maximum is being computed, we need only observe the value of delta(s; p; t) at departure time-points ftj9l : t = t dep (l)^o(l) = sg. The value of delta(s; p; t) may also be constructed incrementally left-to-right on the time line, if the line is divided into adjacent intervals separated by ight departures. In other words the set of relevant time-points may be reduced to: MT (s) = ft start g [ ftj9l : t = t dep (l) ^ o(l) = sg [ ft end g where t start is an initial time point before which no arrivals or departures take place at s, and t end is a nal time point after which no arrivals or departures take place at s. Let I(s) be the set of time line intervals (t; t 0 ] such that t immediately precedes t 0 in an ordering of MT (s). We introduce delta variables of the form delta spt corresponding to the values for delta(s; p; t) for all t 2 MT (s)? ft start g. Constraints on these variables are dened for each station s, plane type p, and pair of adjacent intervals (t; t 0 ] and (t 0 ; t 00 ] in I(s): Constraint (v): delta spt 00 = P o(l)=s^t dep (l)=t 00 b lp? P d(l)=s^t 0 <t ready (l)t 00 b lp + delta spt 0 Constraint (iv): delta sptend = 0 2 The Pure FD Propagation Method 2.1 Motivation The constraint model given in Sections 1.3 and 1.4 is the product of an iterative process. Improvements to the user's original specication and data were brought about through the gradual removal of pre-processable computation, the correction of minor specication and data inconsistencies, and the subsequent incorporation of other reductions and enhancements as our understanding of the problem evolved. This model development and testing stage of the project made extensive use of the ECL i PS e \pure" Finite Domain propagation optimizer. The optimizer coupled depth rst branch and bound optimization with Finite Domain propagation. As the evolving model was tested on larger subsets of the data, the pure FD optimization process was made more ecient, primarily through the inclusion of redundant constraints, and the application of various search ordering strategies, as briey described below. 2.2 Enhancing Performance Redundant Constraints If chosen well, redundant constraints may produce signicant improvements in the behaviour of the Finite Domain solver. The earlier pruning they typically give rise to will reduce

6 the number of nodes searched as well as the number of backtracks. General Delta Constraints These redundant constraints are produced by performing total ow reasoning on all incoming and outgoing aircraft at a station. This is independent of plane type, unlike the ow reasoning of constraint (v). This allows us to relate the expected total number of aircraft on the ground at departure time-points with the total numbers of each plane type. By linking the new delta(s; t) variables with the existing delta(s; p; t) variables the two forms of reasoning interact. The General Delta Constraints are redundant in that, if ow logic is consistent for each plane type, it necessarily follows that the total ow logic is consistent. delta st 00 = X o(l)=s^t dep (l)=t 00^p2P b lp? X d(l)=s^t 0 <t ready (l)t 00^p2P b lp + delta st 0 delta st 00 = X p2p delta spt 00 Zero-Capacity Equality Constraints If a station is at Zero-Capacity (has no available aircraft) at two time-points in the day, then it is possible to deduce that the incoming aircraft are equal to the outgoing aircraft between the two time-points. For example, if a station has no available aircraft at, say, 10am, and a single arrival occurs at 10.30am followed by a single departure at 11am, and there are no other arrivals or departures this interval, the departure at 11am must be carried out by the aircraft arriving at 10.30am. The above example is the simplest case of this form of Zero-Capacity reasoning: 8t 1 ; t 2 2 MT (s) : t 1 < t 2 ^ delta st1 = delta st2 = 0 ) f pt l : o(l) = s ^ t 1 < t dep (l) t 2 g = f pt l : d(l) = s ^ t 1 < t ready (l) t 2 g This type of equality reasoning may be performed at pre-processing stage, or dynamically taking into account run-time assignments. In addition, it may be performed (as described above) independent of plane types, or it may take into the station's capacity with respect to each plane type. The implementation of the pure FD optimizer was enhanced by the simplest form of these constraints i.e. pre-processed Zero-Capacity equality constraints independent of plane type Search Ordering In the pure FD method, search ordering is accomplished by specifying the order in which variables and values are labelled. Search ordering is useful because it may direct the search towards more promising areas.

7 To solve the eet assignment problem, the labelling procedure must label (assign values to) the ight Plane Type variables. 2 There are almost 400 such variables in the BA shorthaul eet assignment, one for each ight. All combinations of the following labelling heuristics were attempted: Variable ordering heuristics Smallest Domain First Most Constrained Variable First Earliest Departure/Arrival First Divide By Station, and Order By Size, Maintenance Capability, etc. Value ordering heuristics Most Available Plane Type First Best Cost Plane Type First 2.3 Results The best combination involved the Maintenance Station First, Ascending Station Size, and Earliest Departure/Arrival First variable ordering heuristics, with no value ordering. This combination reached a rst feasible solution in 329 CPU seconds on a Sun SPARCstation- 20. However, the subsequent optimization process for the above combination was slow (the algorithm showed a 1% improvement at 1 CPU hour and a 3% improvement at 10 CPU hours). 3 The Mixed Integer Programming Method 3.1 Motivation Once the model had been developed and veried the task was to identify an optimization method that produced good quality solutions quickly. The nature of the model, involving a linear optimization function and a set of linear equalities and inequalities, indicated that a mathematical solver capable of solving mixed integer problems might be suitable. At this stage, several other alternative optimization strategies were also candidates for implementation. These were mostly repair algorithms (the method in Section 4 is one example). The solver has an advantage over repair methods in that it is complete; it can prove the global optimality of solutions. 2 Apart from the overnight variables, all other variables are dependent in the sense that they will be instantiated via constraint satisfaction after the complete assignment of the Plane Type variables.

8 Hence the mixed integer programming tool FortMP was chosen for the second method [EH95]. The FD model and a feasible solution were translated for its use. The process of exporting the model and solutions to the FortMP solver was straightforward. The FortMP solver used this model and solutions to quickly obtain the global optimum for this problem; the details of how this was accomplished are detailed in [HEW + 96]. 3.2 Results The FortMP solver obtained the optimum in two hours and 28 minutes. The optimal was 9% better than the user's actual assignment. [HEW + 96] details other results, as well as how the FD feasible solutions were used. The redundant constraints and variables of the pure FD model were incorporated without change. Interestingly, while they resulted in some performance degradation, the degradation was not major (around 10% of the CPU time). 4 The Repair Hybrid 4.1 Motivation A disadvantage of the mixed integer programming method described in Section 3 is that, while it produces the global optimum quickly, its solutions may not bear any relationship to the user's current eet assignment. While knowing the optimal is desirable, a good solution that is close to the current assignment is more useful in practice. Another weakness is that global solutions tend not to be robust. Their real use is as a measure against which other solutions are judged. Hence the need was for a repair algorithm that focussed on local improvements to the user's actual assignment. Furthermore, as shown in this section, the introduction of repair may be computationally benecial to FD propagation. Achieving this benet requires some knowledge of the structure of the problem Analyzing Pure FD Propagation Applied to the eet assignment problem, pure FD implementation is capable of producing a rst feasible solution in reasonable time. However, its performance especially in the optimization phase can be improved. A close examination of the algorithm's search revealed two reasons for the relatively slow performance. Structural Reasons The nature and conguration of the constraints of this problem reduce the eectiveness of constraint propagation during the early stages of search. When a minority of the plane type variables have been labelled, assigning a Plane Type variable representing some ight leg l 0 departing from station s 0 to plane type p 0 often results in the following propagation sequence:

9 { the minimum of a single delta spt variable is incremented to reect the increase in the minimum no. of planes of type p 0 on the ground at the time of departure { the minimum of a single on sp variable is raised by 1 to ensure that a sucient quantity of planes of type p 0 overnight at s 0 { the maxima of up to 60 on sp variables are decremented by 1, to reect the xed total of overnighting aircraft at s 0, and the xed eet size of p 0 { The maxima of several hundred delta spt variables are reduced to reect the decremented maxima of overnight variables Unfortunately, the reduced maxima for delta spt variables translate to useful reductions in the domains of other Plane Type variables in only a small minority cases. This propagation sequence reects the nature of the problem: making a single decision about the plane type of a particular ight may have global implications for the possible quantities of aircraft on the ground before other ight departures (through the eet size and maintenance constraints), but this inferred information does not usually translate into denite information about the possible plane types for other ights until most have already been xed. Once this occurs, however, Finite Domain propagation reductions on the remaining Plane Type variables increase dramatically. Scale A well known deciency of depth rst tree search is that a signicant backtrack overhead is incurred when an early combination of variable labels (close to the search tree root) are inconsistent with all possible later assignments and this inconsistency is not immediately detected. In fact, FD propagation was incorporated into chronological backtracking precisely with a view to addressing this problem; inconsistency is detected earlier using FD propagation. However, in problems where the structure of the problem does not allow FD propagation to detect inconsistencies at an early stage the weakness of depth rst search re-emerges, and becomes more apparent as the size of the problem is increased (eet assignment includes over 400 labelling variables and 7500 secondary variables) Improving Performance Various techniques, such as the introduction of redundant constraints (Section 2.2.1) may be used to hasten FD propagation's detection of inconsistency. Repair is also excellent for this purpose. In fact, knowledge of the structure of the problem may be used together with repair to eectively counter both weaknesses outlined above by (1) localizing FD propagation and (2) improving its eectiveness. 1. Localising Propagation Localising the scope of propagation is possible by labelling those variables in constraints that connect separable portions of the constraint graph [KK88]. In the BA

10 eet assignment the constraint graph may be decomposed into relatively independent subgraphs, corresponding to the stations of the problem, by labelling the overnight variables; these exist in all the constraints linking stations. The stations are still not completely independent though, since labelling or pruning a Plane Type variable has implications for both the departure and arrival stations. 2. Improving Propagation Ecacy The diminished eectiveness of propagation for some problems has been attributed by Minton et. al to lack of informedness, and used as an argument for repair methods [MJPL92]. Tong el al. also report this deciency for global constraints and suggests the use of constrained generators to satisfy them, while applying constraint propagation for the remainder of the problem [TBMV92]. According to Tong et al., a global constraint is one that ranges over a large number of variables without achieving much useful propagation. Constrained generators are procedures which repair an existing solution or modify it such that a global constraint is made redundant. While our constraints do not individually correspond to Tong's denition of a global constraint, the set of constraints on the overnight variables of the eet assignment problem act collectively as a global constraint. 4.2 Incorporating Repair The incorporation of repair for eet assignment depends on a division of the problem variables into repair variables and logical variables. Repair variables start the search with an initial assignment that is propagated onto the remaining logical variables of the problem, reducing their domains. In eect the repair variable assignment reduces search focus to logical sub-problems. Once the logical sub-problem has been searched exhaustively (or a time-out terminates search) logical variable assignments are undone. A reassignment of a subset of the repair variables creates a new logical sub-problem that is searched in a similar manner. In order not to search the same logical sub-problem more than once, those that have already been searched are recorded using nogood labels. Performing a repair labelling on the overnight variables addresses the functional and computational objectives outlined in the previous subsection. First, by starting the search with an overnight allocation equivalent to that of the user's current eet assignment, search nds good solutions close to that region achieving the funcionality required. The overnight variables of the problem are the repair variables that are reassigned to create logical subproblems. Perturbations are made to the user's solution by reassigning subsets of these overnight variables. Second, the problem is decomposed into relatively independent subgraphs corresponding to stations thus limiting the scope of propagation on labelling. This is because the constraints linking stations are on overnight variables, and these are assigned before logical sub-problems are searched. Third, the global constraint part of the problem is handled by eciently by a repair procedure that subsumes the global constraint. If a consistent solution of a constraint is initially given to the xed overnight, eet size and minimum maintenance constraints (Constraints (i),(ii) and (iii) of Section 1.3) operators

11 may be found to perturb this solution without introducing inconsistency; in this way global constraints are subsumed by the repair operator. Fourth, the eectiveness of propagation is enhanced by the tightening of the problem achieved through xing the overnight variables. The domain reductions on plane type and other variables resulting from overnight variable assignments greatly reduce the scope of the subsequent search. The highest level of the repair labelling implementation is outlined here: Nogoods = ;; OvernightLabels = Some Complete Label Set Satisfying Maintenance and Fleet Size Constraints; Repeat OvernightVariables = OvernightLabels; LabelPlaneTypeVariables(TimeOut, NewCost, BoolSuccess) if(boolsuccess) then AddConstraint(CostVariable NewCost) else Nogoods = Nogoods [ OvernightLabels; Repeat NewOvernightLabels = RandomSwapTwoPlaneLocations(OvernightLabels); Until NewOvernightLabels =2 Nogoods; OvernightLabels = NewOvernightLabels; Until False; LabelPlaneTypeVariables(TimeOut, NewCost, BoolSuccess) is a procedure that sets BoolSuccess to be True only when NewCost is a new cost that has been found through labelling the Plane Type variables in CPU time less than Timeout; otherwise the procedure returns before a xed time Timeout with BoolSuccess set to False. AddConstraint(Constraint) is a procedure that adds a constraint to the FD constraint store. Finally, RandomSwapTwo- PlaneLocations(OvernightLabels) is a repair procedure that selects a random pair of overnight variables from dierent stations, decrements one and increments the other thus eecting a swap between of two planes at dierent stations and of dierent plane type. Unlike many pure Logic Programming environments ECL i PS e makes the incorporation of repair easy through its support of destructive variable assignment, meta-variable structures, and through a high degree of propagation control for individual constraints. 4.3 Results Following the introduction of the repair component, the performance of FD propagation improved dramatically. A rst solution is reached in under 1 minute, with a 4% improvement over the rst feasible solution achieved in 10 minutes and a 6% improvement after 1 CPU hour. Heuristics such as Best-Cost-Value-First and Smallest-Domain-Variable-First achieved an increase in performance due to the tightening of the constraint network resulting from overnight variable labelling. The solutions produced at the start of the search

12 were close to the actual eet assignment, which meant they were of practical use. 5 Conclusion Generally speaking, a conict of interests exists in systems which generalise over several alternative solution techniques: exibility in switching between them versus specializing the problem description for ecient execution by any single one. By providing a separation and a high quality interface between problem descriptions and the techniques used for solving them, the ECL i PS e environment achieved exibility in this instance. It allowed alternative models to be quickly tested for correctness and eciency, and facilitated the application of several alternative algorithms to a chosen model. Moreover, the eet assignment problem showed that there is in fact much common ground among solution methods: Many of the solution method parameters were similar (decision variables, search ordering heuristics, etc.). For example, the overnight and (Boolean) plane type variables and emerged as critical variables that had to be given priority across all three methods, either through early labelling in the rst and third methods, or through explicit integer declarations in the second. Model specialization did not signicantly degrade performance of other methods. The FortMP solver was not signicantly disadvantaged by the introduction of redundant constraints, for example. Finally, the benets resulting from a exible platform outweigh any minor losses in eciency; we believe it is initially far more important to quickly and simply apply a range of possibly hybrid solution methods to a new problem than to focus on specializing any single one. The BA shorthaul eet assignment problem solved here is in fact an approximation of the \real problem" tailored in the past for easy representation and solution by a mathematical solver. This work has convinced the user to invest in the development of an ECL i PS e -based prototype to improve their more detailed day to day eet assignment process, using a repair FD propagation hybrid approach akin to the third method described in this paper. In conclusion: the process of rapidly constructing and testing tailored algorithms is supported in the current ECL i PS e environment. Future versions of ECL i PS e which further facilitate the tasks of solution exploration and integration are currently being developed at IC-Parc. References [Dav87] E. Davis. Constraint propagation with interval labels. Articial Intelligence, 32:281{331, 1987.

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