An application of extended NSGA-II in interval valued multi-objective scheduling problem of crews

Abstract

In public or private transport industry, crew scheduling of any public transport is an important problem. It is associated with the assignment of crews for their to and fro trips of public/private sector transportation at an optimum time/cost. This paper deals with a crew scheduling problem with day-to-day allotment of duties of a set of crew members for their to and fro trips such that the total standby times (or rest times or waiting times) and overall service times (along with standby times) will be minimized separately. Here the standby times and service times (along with standby times) are considered as uncertain due to uncertain arrival of transport vehicle. This uncertainty is represented by interval and the corresponding problem is formulated as an optimization problem with two interval valued objectives. To solve the said problem, the existing Non-dominated Sorting Genetic Algorithm-II (NSGA-II) developed by Deb et al. (IEEE Trans Evol Comput 6:182–197, 2002) is extended and ENSGA-II (Extended NSGA-II) has been developed with interval fitness (for 0–1 programming problem) and column exchange crossover and mutation. Then, the problem is formulated as optimization problem with multiple objectives in crisp environment considering centre-radius form of interval. To illustrate and validate the problem along with solution methodology, three numerical examples are solved and the results are compared for different approaches. Then to investigate the impact of different genetic algorithm parameters along with the stability of the algorithm, sensitivity analyses are done on the overall service time and total standby time.

Appendix A: Interval ranking

To solve multi-objective optimization problems in which the objective functions are interval valued, order relations between intervals becomes very important. In this case, interval ranking (order relation) plays an important role to compare the solutions. Till now, several researchers have defined interval ranking upon the following criteria:

• (a) Set properties

• (b) Fuzzy approach

• (c) Probabilistic approach

• (d) Value-based approach

• (e) Based on some particular indices/functions

Revising the limitations of the earlier proposed definitions, modified definitions of interval ranking/order relations were proposed by Bhunia and Samanta (2014) for minimization and maximization problems separately. The definition of interval ranking for minimization problem is as follows:

Definition

For minimization problems the order relation $\le^{\min }$ between two intervals $M = \left[ {m_{L} ,m_{U} } \right] = \left\langle {m_{c} ,m_{r} } \right\rangle$ and $N = \left[ {n_{L} ,n_{U} } \right] = \left\langle {n_{c} ,n_{r} } \right\rangle$ is as follows:

$$ M \le^{\min } N \Leftrightarrow \left\{ {\begin{array}{*{20}c} {m_{c} < n_{c} } \\ {m_{r} \le n_{r} } \\ \end{array} } \right.\begin{array}{*{20}c} {} \\ {} \\ \end{array} \begin{array}{*{20}c} {if} \\ {if} \\ \end{array} \begin{array}{*{20}c} {m_{c} \ne n_{c} } \\ {m_{c} = n_{c} } \\ \end{array} $$

And $M <^{\min } N \Leftrightarrow M \le^{\min } N$ and $M \ne N$.