Development and implementation
6.1 MOO test problems
The author used known MOO test problems to validate the implementation of the MOO CEM metaheuristic as well as the C-Interface inter-process communication procedure.
The MOO test problems were modelled as deterministic models in TPS. In total, five test problems were used from Coello Coello et al. (2007) and they are presented in Table 6.1. The results obtained by MOOSolver for the five problems are shown in Figures 6.1 and 6.2 where they were compared with results obtained by a MATLAB implementation of the MOO CEM together with the known true results of the test problems.
Table 6.1: Standard MOO test functions.
The comparisons in Figures6.1and 6.2show that the MOO CEM implementation in MOOSolver is valid. Additional (and more rigorous) tests for the metaheuristic can be found inBekker & Aldrich(2011). For the purpose of this study, the results obtained in this chapter are considered sufficient validation material as the algorithm used here (Algorithm9) is the same as the one in Bekker & Aldrich (2011). More importantly, the results in this section validate the C-Interface IPC procedure (Figure5.1) developed to integrate the suite with the discrete-event simulation software.
(a) MOP1 by MOOSolver. (b) MOP1 in MATLAB.
Figure 6.1: Comparison between MOO test problems results obtained by MOOSolver (a) and results obtained in MATLAB together with the known results (b).
(a) MOP2 by MOOSolver. (b) MOP2 in MATLAB.
(c) MOP3 by MOOSolver. (d) MOP3 in MATLAB.
(e) MOP4 by MOOSolver. (f) MOP4 in MATLAB.
(g) MOP6 by MOOSolver. (h) MOP6 in MATLAB.
Figure 6.2: Comparison between MOO test problems results obtained by MOOSolver
6.2 The buffer allocation problem
In her work,Yoon(2018) solved the BAP as a small-scale SO problem with preselected solutions using a MATLAB implementation of the MMY algorithm. She used the BAP to validate the procedure. The same BAP is solved here with MOOSolver and a similar validation approach is used. The results obtained here are compared to those inYoon (2018). Successful comparison would thus validate theMMY implementation in MOOSolver and, consequently, the COM-Interface IPC procedure as well.
The validation process inYoon(2018) was as follows: Ten scenarios were preselected to be compared using theMMY procedure. To ensure the validity of the results obtained by the procedure, 10 000 observations were run per scenario before using the procedure and the means obtained from these observations were considered as true means. The Pareto ranking algorithm (Algorithm1) was subsequently used to determine the true relaxed Pareto set, which would be used to validate the estimated relaxed Pareto set to be obtained by the MMY procedure. The same true relaxed Pareto set is used in this study to validate the MOOSolver implementation of the procedure.
6.2.1 Specifics of the BAP solved in Yoon (2018)
It was assumed for this variant of the BAP that a total of, say, nt buffer spaces was available to be arranged in the 4 niches considered for the BAPs in this study (i.e.
the number of machines in series for all the BAPs in this study is m = 5, see Figure 3.3). The problem had therefore a total number of ntm−2+m−2 feasible solutions. With nt selected as 6 in this case, the total number of potential solutions amounted to 93 = 84.
From the 84, 10 were selected and the true mean values to their respective objective functions were obtained in the manner described in the previous paragraph. Because the total number of storage spaces used in this case was known and kept constant, objective WP of Chapter3 was calculated differently here. Instead of being the sum of buffer spaces used in a solution like in Chapter3, here it is rather the average work-in-progress rate; still to be minimised. The selected IZ values for TRand WP were 0.2 and 0.12, respectively. Additional information about the problem is summarised in Table 6.2, and observations on the BAP model were made over a period of 100 days. The system was treated as a terminating system.
Table 6.2: Machines information for the BAP.
Machine 1 2 3 4 5
Processing times (µi) 60 min 55 min 50 min 46 min 43 min
ODFs (λi) 100 100 100 100 100
Repair times (βi) 120 min 120 min 120 min 120 min 120 min
6.2.2 Results and validation
The ten solutions preselected and their estimated true mean performances are shown in Tables6.3 and6.4, respectively.
Table 6.3: Selected solutions in the BAP.
System 1 2 3 4 5
(x1, x2, x3, x4) (1,1,1,3) (1,1,2,2) (1,2,1,2) (1,2,2,1) (2,1,1,2)
System 6 7 8 9 10
(x1, x2, x3, x4) (2,1,2,1) (2,2,1,1) (3,1,1,1) (1,3,1,1) (1,1,3,1)
Table 6.4: Estimated true means in the BAP.
System 1 2 3 4 5
(TR, WP) (16.42, 1.65) (16.66, 1.76) (16.97, 2.02) (17.14, 2.13) (17.04, 2.42)
System 6 7 8 9 10
(TR, WP) (17.28, 2.55) (17.48, 2.83) (17.23, 3.19) (17.18, 2.37) (16.73, 1.86) From Table 6.4, the true relaxed Pareto set can thus be obtained following the definition in Section 4.2.2.1. Figure 6.3 illustrates the true relaxed Pareto set for the BAP in this section. It follows from the figure that, except Systems 5 and 8, every other system in the set can be considered as relaxed Pareto-optimal.
With this information at hand, the MMY in MOOSolver could then be tested for validation. The result obtained by the suite is summarised in Table 6.5. The table shows that every system in the set preselected can be considered as relaxed Pareto-optimal except for Systems 5 and 8. The status column indicates, in effect, that the necessary number of observations was made for all considered systems. The reader can also observe that the highest number of observations made on a system during the
16.4 16.6 16.8 17 17.2 17.4 17.6
Figure 6.3: The true relaxed Pareto set for the BAP (Yoon,2018).
execution of the procedure was 143 (made on System 9), which is way less than 10 000. With the lowest number of observations made being 15, one can only imagine how inefficient using an LFC-based approach (see Section 3.1.3) would be in this case.
The result in Table6.5is in effect valid when compared to the true relaxed Pareto set.
This, therefore, validates the implementation of theMMY procedure in MOOSolver and shows the efficiency of the procedure. Consequently as a result, the COM-Interface IPC procedure (Figure5.2) is also valid.
Table 6.5: BAP result as obtained by MOOSolver.
Solution TR WP ST2
6.3 Chapter summary
In this chapter, the author validated the multi-objective optimisation suite developed in the previous chapters. To do this, standard deterministic MOO test problems were used as well as a variant of the buffer allocation problem in the literature with known solutions. The BAP is a stochastic problem.
In the next chapter, the MOO suite is used to solve well know problems from the literature as well as in practice and the solution approach proposed in Section4.1.1is tested.