A fuzzy genetic algorithm for driver scheduling

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Abstract

This paper presents a hybrid genetic algorithm (GA) for the bi-objective public transport driver scheduling problem. A greedy heuristic is used, which constructs a schedule by sequentially selecting shifts, from a very large set of pre-generated legal potential shifts, to cover the remaining work. Individual shifts and the schedule as a whole have to be evaluated in the process. Fuzzy set theory is applied on such evaluations. For individual shifts, their structural efficiency is assessed by fuzzified criteria identified from practical knowledge of the problem domain. A GA is used to derive a near-optimal weight distribution amongst the fuzzified criteria, so that a single-valued weighted evaluation can be computed for each shift. The corresponding schedule constructed utilising the weight distribution is evaluated by the GA’s fitness function, in which the two objectives of minimising the number of shifts and minimising the total cost are formulated as a fuzzy goal. Comparative results on real-world problems are presented.

Introduction

Bus and rail driver scheduling is a process of partitioning blocks of work, each of which is serviced by one vehicle, into a set of legal driver shifts. The basic objectives are to minimise the total number of shifts and the total shift costs. This problem has attracted much research interest since the 1960’s. Wren and Rousseau [33] gave an overview of the approaches, many of which have been reported in a series of international workshop conferences [4], [6], [32].

The driver scheduling problem can be formulated as a set covering integer linear programme (ILP). All the legal potential shifts are first constructed. Then, a least cost subset covering all the work is selected to form a solution schedule. A typical problem may have a solution schedule requiring over 100 shifts chosen from a potential set of about 50,000.

Set covering is one of the oldest and most studied NP-hard problems [3], [15], [18], [22]. Given a ground set U of m elements, and a weight for each set, the goal is to cover U with the smallest possible number of sets. In the case of driver scheduling, there is the additional objective of minimising the total weight.

Since the set covering problem is unlikely to be solved optimally in polynomial time, there has been a lot of work in exploring the possibility of obtaining efficiently near-optimal solutions. One of the polynomial time algorithms is the greedy algorithm: at each step choose the unused set which covers the largest number of remaining elements. Johnson [15] and Lovász [22] proved in the mid 70’s that the performance ratio of the greedy method is no worse than ln(m)+1, where m is the size of the ground set, and Chvátal [3] extended their results to the weighted case. Slavı́k [26] gave a better performance ratio in 1996, which is exactly ln(m)−lnln(m)+Θ(1), however, the simple greedy method is not suitable for large size set covering problems due to its poor approximation guarantee. In this paper, we develop a more comprehensive evaluation of the sets (potential shifts) when the next one is chosen to cover more of the remaining elements (work pieces).

The new polynomial time genetic algorithm with fuzzy evaluation (GAFE) for driver scheduling evaluates potential shifts based on fuzzified criteria. These criteria are represented by fuzzy membership functions about the structure and generally the goodness of a shift. The fuzzy membership functions are weighted and combined to yield an overall evaluation. While it might be possible to derive a general relatively robust set of weights by experiments, these weights will have to be fine-tuned for individual problems for best performance. A genetic algorithm (GA) is therefore proposed for determining the set of weights. With respect to the GA, the driver schedules constructed are only by-products. Nevertheless, the number of shifts and the total cost of the schedule are the factors used in the GA’s fitness function. Since the driver scheduling here is bi-objective, the minimisation of these two factors is treated as a fuzzy goal in the GA’s fitness function.

The GAFE algorithm belongs to the general class of hybrid GAs (memetic algorithms, genetic local search) [1], [23], [25], i.e. algorithms that hybridize genetic operations with local or constructive heuristics. Furthermore, there are some similarities between the idea introduced in this paper and the GRASP algorithm [8] and the adaptive multi-start (AMS) technique [2]: they all apply adaptive construction heuristics to obtain individual feasible solutions, and perform searches based on multiple solutions to improve the local optimum. However, the formations of multi-start are very different: GAFE is based on an evolutionary mechanism, while GRASP is purely randomized and AMS maintains a constant number of best solutions found so far.

This paper is organized in the following way. We shall first present the greedy algorithm framework in GAFE. The mathematical model of fuzzy comprehensive evaluation is then discussed. The GA follows a simple ‘standard’ scheme and will be briefly described. Comparative results using real-life problems, some of which are very large instances, are presented. Finally, conclusions are discussed.

Section snippets

Fuzzy comprehensive evaluation for driver scheduling

Definition 2.1

A relief opportunity (RO) is a time and place where a driver can leave the current vehicle, for reasons such as taking a meal-break, or transferring to another vehicle. The work between two consecutive ROs on the same vehicle is called a piece of work.

Definition 2.2

The work a single driver carried out in a day is called a shift, which is composed of several spells of work. A spell contains a number of consecutive pieces of work on the same vehicle.

Definition 2.3

A driver schedule is a solution that contains a set of shifts

Using GA to produce near-optimal weights

GAs [5], [11], [12] are search algorithms based on the mechanics of genetics and natural selection. They employ multiple concurrent search points called “chromosomes”, which are processed through three genetic operations of selection, crossover and mutation to generate new search points called “offspring” for the next iterations.

GAs are useful approaches to problems requiring an efficient search over a very large solution space, and have attracted much research interest over the last two

Computational results

The major concern about using heuristics is the quality of the obtained solution. Due to lack of knowledge about the true optimum, an effective method of assessing the quality of a heuristic solution is by comparing it to the best known solution. Therefore, this paper applies relative percentage deviation (RPD) over the best known schedule to measure the quality of a heuristic schedule.RPD=heuristicschedulebestknownschedulebestknownschedule×100%.

Table 1 shows the sizes and the best known

Conclusions

A new scheduling algorithm based on fuzzy set theory is presented in this paper. The new algorithm is novel because it is the first time that fuzzy set theory has been applied to the driver scheduling problem. An effective method is proposed to solve the problem about ranking the potential shifts in each iteration. Unlike the simple greedy algorithms, the new approach employs fuzzy comprehensive evaluation which depends on five fuzzified criteria about the structure of a shift including total

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