# The Method of Production Scheduling with Uncertainties Using the Ants Colony Optimisation
## 5 Predictive-Reactive Scheduling Case Study  
This section introduces various ACO parameter data sets to verify the performance of ACO and MIDOS I or MIDOS II for predictive scheduling in various job shop sizes. Job shop scheduling problems are investigated to fine-tune the parameters where 9 jobs have to be performed on 8 machines (9 \(\times\) 8) and 11 jobs have to be performed on 10 machines (11 \(\times\) 10). The first machine is the most heavily loaded. The failure-free time of the bottleneck _MTTF_ equals 66. The repair time of the bottleneck _MTTR_ equals 6. The increased probability of the bottleneck failure occurs in time horizon [\(a\), \(b+MTTR\)] where: \(a=60\) and \(b=72\). The objective is to find an approach which is able to generate stable and robust schedules in the event of the bottleneck failure. The objective is to achieve a robust and stable schedule for the problem, \(C_{max}(k)\)\(\rightarrow\) min (15). Computer simulation of the Ant Colony Optimisation is run for the parameter of the relative importance between the pheromone trace and the reciprocal of the distance \(\beta=1\); pheromone evaporation factor \(\alpha=\rho=\{0.2,0.4,0.6,0.8\}\); number of ants, \(K=\{10,\) 15, 20, 25\}; number of iterations, \(E=\{10,\) 20, 30, 40\(\}\) and parameter \(q_{0}\) which decides about exploration or exploatation selection by an ant, \(q_{0}=\{0.3,0.4,0.5,0.6\}\). The ACO is run 10 times for each set of input parameters \(\{\rho,K,E,q_{0}\}\). First, the influence of the number of iterations, \(E=\{10,\) 20, 30, 40\(\}\) over the quality of basic schedules generation for single criterion problem is investigated and for unchanging pheromone evaporation factor \(\alpha=\rho=\{0.6\}\) and number of ants \(K=\{10\}\). The parameter \(q_{0}\) is equal to 0.5 to get an equal chance of choosing to explore and exploit by ants. By observing the first and third quartiles of \(C_{max}\) and the best schedules achieved the following conclusion can be drawn that a larger number of iteration is, the higher chances of achieving a better solution are in scheduling problem (11 \(\times\) 10) (Figure 3b). By observing the first and third quartiles of \(C_{max}\) and the best schedules achieved for the scheduling problem (9 \(\times\) 8), the opposite phenomenon can be observed. The smaller the number of iterations is, the greater the chances of achieving a better solution are (Figure 3a). Next, the influence of the number of ants, \(K=\{10,\) 15, 20, 25\(\}\) over the quality of basic schedules generation for single criterion problem is investigated for unchanging pheromone evaporation factor \(\alpha=\rho=\{0.6\}\), number of iterations \(E=\{20\}\), parameter \(q_{0}\) is equal to 0.5. Observing the results of the achieved value of the makespan criterion for the basic schedules (Figure 4), the following conclusion can be drawn that a larger the ant population is, the greater the chances of achieving a better solution are. This phenomenon is noticed for both sizes of scheduling problems (9 \(\times\) 8) and (11 \(\times\) 10). Then, the simulations are continued for the number of ants, \(K=\{15\}\), iteration size, \(E=\{20\}\), parameter \(q_{0}=0.5\) and changing values of pheromone evaporation factor \(\alpha=\rho=\{0.2,0.4,0.6,0.8\}\). Observing average values of \(C_{max}\) and the best schedules achieved (Figure 5) the following conclusion can be drawn that the higher values of pheromone evaporation factors \(\alpha=\rho\) are, the higer chances of achieving a better solution are. Algthough the average quality of population does not increase with the parameter values, better solutions are achieved for scheduling problem (9 \(\times\) 8). The best schedule is achieved for \(C_{max}\) equals 152 for scheduling problem (9 \(\times\) 8), for the pheromone evapora  
Figure 3: The \(C_{max}\) of the basic schedules achieved by the ACO for different iteration numbers. tion factor \(\alpha=\rho=0.8\) (Figure 5a).