Fermented mulberry leaf meal as fishmeal replacer in the formulation of feed for carp Labeo rohita and catfish Heteropneustes fossilis—optimization by mathematical programming

Abstract

Search for cost-effective, eco-friendly and sustainable plant resources as potential feedstuff to replace fishmeal in the formulation of feed for fish has been in the forefront of aquaculture researches since the last few years. In this study, experiments were conducted to evaluate if replacement of fishmeal by the fermented leaf meal of mulberry (Morus indica) was viable in the formulation of feed for carp fish Labeo rohita and catfish Heteropneustes fossilis. Four iso-proteinous, iso-lipidic and iso-energetic experimental feed were formulated by replacing 0, 25, 50 and 75% of fishmeal by the fermented mulberry leaf meal (FMLM), and both species were grown on these feeds for 8 weeks. Since the results revealed differences in response to fishmeal replacement level between parameters, we determined optimum fishmeal replacement level (OFRL) for each parameter from the polynomial curve equation. While maximum weight gain and specific growth rate and minimum feed conversion ratio was found at 30–32% OFRL for L. rohita and at 52–53% OFRL for H. fossilis, other parameters responded differently in both fish. Therefore, we applied a two-phase fuzzy goal programming technique using all parameters, which showed overall OFRL for L. rohita and H. fossilis as 30.95% and 52%, respectively. We also applied the concept of ‘decision tree’ to identify the key factor behind utilization of FMLM. It was concluded that activity of amylase and subsequent utilization of carbohydrate was the key factor in utilizing FMLM. Interestingly, H. fossilis was found more efficient in utilizing carbohydrate of FMLM than L. rohita.

Appendices

Appendix 1

Table 8 Cubic polynomial for the observed values

Appendix 2

In a two-phase method, we follow the approach proposed by Wu et al. (2015). Before going through detail description, the following definitions are required to present.

Definition 1:

A generalized multiple objective optimization problem is as follows:

$$ \left\{\begin{array}{c} opt\kern1.25em \left({f}_1(x),{f}_2(x),\kern0.5em \dots \dots ..,\kern0.75em {f}_k(x)\right)\\ {}s.t.\kern1.25em x\in X=\left\{x|\ {g}_i(x)\le 0,\kern0.75em j=1,\dots \dots, m.\right\}\end{array}\right. $$

where “opt” denotes minimization or maximization; x = (x₁, x₂,. .., x_n) are the decision variables; f_i(x), (i = 1,. .., k) are multiple objectives to be optimized and g_i(x) represents constraints.

Definition 2:

A decision plan x⁰ ∈ X represents a Pareto optimal solution if there replacement level not exist another y ∈ X, such that f_k(y) ≤ f_k(x⁰) for all k and f_s(y) < f_s(x⁰) for at least once.

Definition 3:

A decision plan x⁰ ∈ X represents a fuzzy-efficient solution if there replacement level not exist another y ∈ X is, such that μ_k(f_k(y)) ≥ μ_k(f_k(x⁰)) is for all k and μ_s(f_s(y)) ≥ μ_s(f_s(x⁰)) for at least once.

Guu and Wu (1999) first introduced two-phase fuzzy goal programming. The authors remove the upper and lower bounds and formula at linear membership functions featuring both the continuously increasing property of the maximization objective function (μ_t(f_t)) and the continuously decreasing property of the minimization objective function (μ_r(f_r)). For the maximization objective function:

$$ {\mu}_{\mathrm{t}}\left({\mathrm{f}}_{\mathrm{t}}\right)=\left\{\begin{array}{c}\frac{{\mathrm{f}}_{\mathrm{t}}-{\mathrm{f}}_{\mathrm{t}}^{\mathrm{min}}}{{\mathrm{f}}_{\mathrm{t}}^{\mathrm{max}}-{\mathrm{f}}_{\mathrm{t}}^{\mathrm{min}}}\kern1.25em if{\mathrm{f}}_{\mathrm{t}}^{\mathrm{min}}\le {\mathrm{f}}_{\mathrm{t}}\le {\mathrm{f}}_{\mathrm{t}}^{\mathrm{max}}\\ {}0\kern9.5em if\kern0.75em {\mathrm{f}}_{\mathrm{t}}\le {\mathrm{f}}_{\mathrm{t}}^{\mathrm{min}}\end{array}\right. $$

${\mu}_{\mathrm{r}}\left({\mathrm{f}}_{\mathrm{r}}\right)=\left\{\begin{array}{c}\frac{{\mathrm{f}}_{\mathrm{r}}^{\mathrm{max}}-{\mathrm{f}}_{\mathrm{r}}}{{\mathrm{f}}_{\mathrm{r}}^{\mathrm{max}}-{\mathrm{f}}_{\mathrm{r}}^{\mathrm{min}}}\kern1.25em if{\mathrm{f}}_{\mathrm{r}}^{\mathrm{min}}\le {\mathrm{f}}_{\mathrm{r}}\le {\mathrm{f}}_{\mathrm{r}}^{\mathrm{max}}\\ {}0\kern9.5em if\kern0.75em {\mathrm{f}}_{\mathrm{r}}\le {\mathrm{f}}_{\mathrm{r}}^{\mathrm{max}}\end{array}\right.$

where the possible range for the r-th objective $\left[{\mathrm{f}}_{\mathrm{r}}^{\mathrm{min}},{\mathrm{f}}_{\mathrm{r}}^{\mathrm{max}}\right]$ (i = 1, …,12) is constructed from the solution of the problem by incorporating only one objective function while ignoring the other objective function (Tables 5 and 6).

Under the two-phase approach, one needs to determine the optimal solution by solving the following optimization problem in Phase I:

$$ {\displaystyle \begin{array}{l}\kern3.75em \operatorname{Max}\kern0.75em {Z}_1\kern0.5em =\lambda \\ {}\mathrm{s}.\mathrm{t}.{\upmu}_{\mathrm{k}}\left({\mathrm{f}}_{\mathrm{k}}\right)\ge \lambda, \lambda \kern0.5em \ge 0,x\in \mathrm{X},\kern1em k=1,\dots 12\end{array}} $$

If the values of the membership function is limited by an upper bound, the objective functions may not attain the lowest or highest possible value because the fuzzy goals are set by the decision maker subjectively. The two-phase approach provides the flexibility to reach optimal goals by relaxing this constraint in Phase I. In Phase II, we need to determine the following optimization problem to obtain the optimal replacement level.

$$ {\displaystyle \begin{array}{l}\kern4.75em \operatorname{Max}\kern0.75em {\mathrm{Z}}_2\kern0.5em ={\sum}_{k=1}^{12}{\rho}_{\mathrm{k}}\ \\ {}\mathrm{s}.\mathrm{t}.{\upmu}_{\mathrm{k}}\left({\mathrm{f}}_{\mathrm{k}}\right)-{\rho}_k\ge {\lambda}^{\ast },\kern0.5em \lambda \ge 0,\kern0.75em {\rho}_{\mathrm{k}}\ge 0,\kern0.75em \mathrm{x}\in \mathrm{X},\kern1em k=1,\dots 12\end{array}} $$

where λ^* is the optimal value of λ obtained in Phase I. It is to be noted that if ρ_k = 0, then there are no solutions with better efficiency for the model under Phase I (Wu et al. 2015). If ρ_k > 0 for some k, the solution obtained from Phase II is more efficient compared to the solution obtained from Phase I, and the decision maker would be able to obtain information for achieving subjective goals.