Taking advantage of solving the resource constrained multi-project scheduling problems using multi-modal genetic algorithms

Abstract

In this paper, for the first time, multi-modal genetic algorithms (MMGAs) are proposed to optimize the resource constrained multi-project scheduling problem (RCMPSP). In problems where the landscape has both multiple local and global optima, such as the RCMPSP, a MMGAs approach can provide managers with an advantage in decision-making because they can choose between alternative solutions equally good. Alternative optima are achieved because the diversification techniques of MMGAs introduce diversity in population, decreasing the possibility of the optimization process getting caught in a unique local or global optimum. To compare the performance of a MMGAs approach with other alternative approaches, commonly accepted by researchers to solve the RCMPSP such as classical genetic algorithms and dispatching heuristics based on priority rules, we analyse two time-based objective functions (makespan and average percent delay) and three coding systems [random keys (RK), activity list (AL), and a new proposal called priority rule (PR)]. We have found that MMGAs significantly improve the efficacy (the algorithm’s capability to find the best optimum) and the multi-solution-based efficacy (the algorithm’s capability to find multiple optima) of the other two approaches. For makespan the PR is the best code in terms of the efficacy and multi-solution-based efficacy, and the RK is the best code for the average percent delay.

Appendices

Appendix 1: OX and OBM operators

In the adapted OX operator, two random integers (between 0 and the string size $-$1) are generated. These numbers indicate the string positions to create the parents substrings. The parent substrings are directly inherited by their children. To better understand, we focus on the example of Fig. 4 where the random numbers are 5 and 7. The father 1’s substring [2 1 3] is inherited by the child-1 and the father-2’s substring [3 2 1] is inherited by the child-2.

Fig. 4

OX crossover operator adapted to permutation coding with repetition

The other children string items are reached as follows: For child-1, a new string is created inheriting the father-2’s string items beginning from the next position of the second random number, going back to the beginning of the father-2’s sequence, and finishing at the second random number position: (2 3 3 1 1 3 / 2 2 1 1 3 2 1). The inherited activities from the father 2 (that is, the second activity of project 2, the third activity of project 1, and the second activity of project 3) are removed from the previous string (2 3 X 1 1 3 X 2 X 1 3 2 1) and the following string is obtained (2 3 1 1 3 2 1 3 2 1). The child-1’s string is filled beginning from the next position of the second random number: (2, 3, 1, 1, 3, 2), going back to the beginning of the child-1’s sequence, and finishing at the first random number position: (1, 3, 2, 1). The other child-2 string items can be found in a similar way.

In the OBM operator, two random integers (between 0 and the string size minus 1) are generated. These numbers indicate the string positions whose items exchange their values (see Fig. 5).

Fig. 5

OBM mutation operator

Appendix 2

See Table 15.

Table 15 Priority rules for the RCMPSP and for PR code