Multi-objective parameter estimation on cultivation of yeast Kluyveromyces marxianus var. lactis MC5

Figures & data

Table 1. Results after application of the PROMETHEE II method.

Rank	Model	Form	ϕ	ϕ^+	ϕ^-
1	Han-Levenspiel – Monod	ADD	0.6154	0.8042	0.1888
2	Tessier – Moser	ADD	0.3636	0.6783	0.3147
3	Tessier – Monod	ADD	0.2378	0.6084	0.3706
4	Tessier – Monod	MUL	0.1469	0.5734	0.4266
5	Mink – Haldane	MUL	0.1049	0.5455	0.4406
6	Moser – Monod	MUL	0.0559	0.5245	0.4685
7	Tessier – Moser	MUL	0.0350	0.5175	0.4825
8	Han-Levenspiel – Monod	MUL	−0.0769	0.4615	0.5385
9	Monod – Monod	MUL	−0.1329	0.4336	0.5664
10	Haldane – Haldane	MUL	−0.1399	0.4266	0.5664
11	Moser – Mink	MUL	−0.1469	0.4126	0.5594
12	Monod – Tessier	MUL	−0.1888	0.4056	0.5944
13	Mink – Mink	MUL	−0.3566	0.3147	0.6713
14	Tessier – Mink	MUL	−0.5175	0.2308	0.7483

ADD: additive form; MUL: multiplicative form of specific growth rate; ϕ⁺: positive outranking flow; ϕ⁻: negative outranking flow, outranking for PROMETHEE II method: $\phi ={\phi }^{+}-{\phi }^{-}$.

Table 2. Various weights for 4^th batch process.

Weight					Objective functions, Qk(θ)×10^-3
Case	1^st	2^nd	3^rd	4^th	1^st	2^nd	3^rd	4^th
1	1	1	0	0	5.941	5.736	8.462	9.496
2	0	0	1	1	5.735	3.914	7.537	8.314
3	1	1	1	1	5.941	5.736	8.462	9.496
4	1	1	1	1	3.685	5.698	3.830	3.292
5	1	0	0	0	5.941	5.736	8.462	9.496
6	0	1	0	0	5.869	3.929	7.475	8.171

Table 3. Optimal estimated parameters of the fourth case.

Parameter	Search bound	Optimal values
${\mu }_{1\mathrm{\text{max}}}$	[0.1, 1.0]	0.367
KS	[5.0, 150.0]	143.080
Sm	[45, 150]	126.730
n	[0.1, 3.0]	1.090
m	[0.1, 3.0]	2.125
${\mu }_{2\mathrm{\text{max}}}$	[0.1, 1.0]	0.633
KC	[0.1, 15.0] × 10^−3	0.715 × 10^−3
YX/S	[0.01, 2.5]	0.056
YX/C	[1.0, 25.0]	3.419
kla	[80.0, 180.0]	145.769

Figure 1. Concentration of biomass, lactose and oxygen for the 1st batch experiment. Xe_j, Se_j and Ce_j – the experimental data for the four batch experiments, j = 1, …, 4; X_k,j – biomass concentration results obtained by using the model parameters for the first (k = 1) and the second case (k = 2), j = 1, …, 4. The indices ‘_k,j’ for the concentration of lactose (S_k,j) and oxygen (C_k,j) are the same as those for X_k,j; Xm_j, Sm_j and Cm_j – simulated results obtained by the optimal solutions (case 4 in Table 2) of the model.

Figure 2. Concentration of biomass, lactose and oxygen for the 2nd batch experiment.

Figure 3. Concentration of biomass, lactose and oxygen for the 3rd batch experiment.

Figure 4. Concentration of biomass, lactose and oxygen for the 4th batch experiment.

Figure 5. Concentration of biomass, lactose and oxygen for the tested results of the 5th batch experiment. Xe_j, Se_j and Ce_j – experimental data of biomass, lactose and oxygen concentration for 5th and 6th test batch processes; Xm_j, Sm_j and Cm_j (j = 5, 6) – simulated data, respectively.

Figure 6. Concentration of biomass, lactose and oxygen for the tested results of the 6^th batch experiment.

Data availability statement

All data that support the findings reported in this study are available from the corresponding authors [MP] and are shown in the Supplemental Appendix, which is an integral part of the article.